LMFDB - Embedded newform 177.4.a.b.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [177,4,Mod(1,177)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(177, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("177.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	177.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$10.4433380710$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 41x^{5} - 7x^{4} + 484x^{3} + 63x^{2} - 1736x - 44 $ x^7 - 41x^5 - 7x^4 + 484x^3 + 63x^2 - 1736*x - 44
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.0253269$ of defining polynomial
Character	$\chi$	$=$	177.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0253269 q^{2} -3.00000 q^{3} -7.99936 q^{4} +8.85515 q^{5} -0.0759806 q^{6} +13.8749 q^{7} -0.405214 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+0.0253269 q^{2} -3.00000 q^{3} -7.99936 q^{4} +8.85515 q^{5} -0.0759806 q^{6} +13.8749 q^{7} -0.405214 q^{8} +9.00000 q^{9} +0.224273 q^{10} -33.5949 q^{11} +23.9981 q^{12} -15.9269 q^{13} +0.351408 q^{14} -26.5655 q^{15} +63.9846 q^{16} -14.1822 q^{17} +0.227942 q^{18} -111.821 q^{19} -70.8356 q^{20} -41.6247 q^{21} -0.850854 q^{22} -188.829 q^{23} +1.21564 q^{24} -46.5862 q^{25} -0.403379 q^{26} -27.0000 q^{27} -110.990 q^{28} -56.6729 q^{29} -0.672820 q^{30} -18.4624 q^{31} +4.86224 q^{32} +100.785 q^{33} -0.359190 q^{34} +122.864 q^{35} -71.9942 q^{36} +53.8076 q^{37} -2.83209 q^{38} +47.7807 q^{39} -3.58823 q^{40} -68.8886 q^{41} -1.05422 q^{42} -344.210 q^{43} +268.738 q^{44} +79.6964 q^{45} -4.78246 q^{46} +291.289 q^{47} -191.954 q^{48} -150.487 q^{49} -1.17988 q^{50} +42.5466 q^{51} +127.405 q^{52} +636.830 q^{53} -0.683826 q^{54} -297.488 q^{55} -5.62230 q^{56} +335.464 q^{57} -1.43535 q^{58} +59.0000 q^{59} +212.507 q^{60} -489.425 q^{61} -0.467594 q^{62} +124.874 q^{63} -511.754 q^{64} -141.035 q^{65} +2.55256 q^{66} +91.2469 q^{67} +113.448 q^{68} +566.488 q^{69} +3.11177 q^{70} +179.941 q^{71} -3.64692 q^{72} -414.334 q^{73} +1.36278 q^{74} +139.759 q^{75} +894.500 q^{76} -466.126 q^{77} +1.21014 q^{78} -889.652 q^{79} +566.594 q^{80} +81.0000 q^{81} -1.74473 q^{82} +812.868 q^{83} +332.971 q^{84} -125.585 q^{85} -8.71778 q^{86} +170.019 q^{87} +13.6131 q^{88} -623.403 q^{89} +2.01846 q^{90} -220.984 q^{91} +1510.51 q^{92} +55.3871 q^{93} +7.37743 q^{94} -990.196 q^{95} -14.5867 q^{96} +1229.07 q^{97} -3.81137 q^{98} -302.354 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 21 q^{3} + 26 q^{4} - 2 q^{5} - 59 q^{7} - 21 q^{8} + 63 q^{9}+O(q^{10}) $ 7 * q - 21 * q^3 + 26 * q^4 - 2 * q^5 - 59 * q^7 - 21 * q^8 + 63 * q^9	\( 7 q - 21 q^{3} + 26 q^{4} - 2 q^{5} - 59 q^{7} - 21 q^{8} + 63 q^{9} - 71 q^{10} - 5 q^{11} - 78 q^{12} - 67 q^{13} - 65 q^{14} + 6 q^{15} - 94 q^{16} - 23 q^{17} - 176 q^{19} - 207 q^{20} + 177 q^{21} - 704 q^{22} - 218 q^{23} + 63 q^{24} - 183 q^{25} + 58 q^{26} - 189 q^{27} - 938 q^{28} + 168 q^{29} + 213 q^{30} - 604 q^{31} - 448 q^{32} + 15 q^{33} - 610 q^{34} - 336 q^{35} + 234 q^{36} - 505 q^{37} - 453 q^{38} + 201 q^{39} - 1080 q^{40} - 265 q^{41} + 195 q^{42} - 493 q^{43} + 504 q^{44} - 18 q^{45} + 381 q^{46} - 244 q^{47} + 282 q^{48} + 770 q^{49} + 1639 q^{50} + 69 q^{51} + 160 q^{52} + 686 q^{53} - 116 q^{55} + 2190 q^{56} + 528 q^{57} + 1584 q^{58} + 413 q^{59} + 621 q^{60} - 838 q^{61} + 286 q^{62} - 531 q^{63} + 205 q^{64} + 490 q^{65} + 2112 q^{66} - 1504 q^{67} + 3047 q^{68} + 654 q^{69} + 1530 q^{70} - 1267 q^{71} - 189 q^{72} - 666 q^{73} + 528 q^{74} + 549 q^{75} - 64 q^{76} + 1109 q^{77} - 174 q^{78} - 2741 q^{79} + 1213 q^{80} + 567 q^{81} + 953 q^{82} - 2025 q^{83} + 2814 q^{84} - 1274 q^{85} + 4394 q^{86} - 504 q^{87} - 1639 q^{88} + 616 q^{89} - 639 q^{90} - 2415 q^{91} + 218 q^{92} + 1812 q^{93} + 900 q^{94} + 2554 q^{95} + 1344 q^{96} - 1298 q^{97} - 172 q^{98} - 45 q^{99}+O(q^{100}) \) 7 * q - 21 * q^3 + 26 * q^4 - 2 * q^5 - 59 * q^7 - 21 * q^8 + 63 * q^9 - 71 * q^10 - 5 * q^11 - 78 * q^12 - 67 * q^13 - 65 * q^14 + 6 * q^15 - 94 * q^16 - 23 * q^17 - 176 * q^19 - 207 * q^20 + 177 * q^21 - 704 * q^22 - 218 * q^23 + 63 * q^24 - 183 * q^25 + 58 * q^26 - 189 * q^27 - 938 * q^28 + 168 * q^29 + 213 * q^30 - 604 * q^31 - 448 * q^32 + 15 * q^33 - 610 * q^34 - 336 * q^35 + 234 * q^36 - 505 * q^37 - 453 * q^38 + 201 * q^39 - 1080 * q^40 - 265 * q^41 + 195 * q^42 - 493 * q^43 + 504 * q^44 - 18 * q^45 + 381 * q^46 - 244 * q^47 + 282 * q^48 + 770 * q^49 + 1639 * q^50 + 69 * q^51 + 160 * q^52 + 686 * q^53 - 116 * q^55 + 2190 * q^56 + 528 * q^57 + 1584 * q^58 + 413 * q^59 + 621 * q^60 - 838 * q^61 + 286 * q^62 - 531 * q^63 + 205 * q^64 + 490 * q^65 + 2112 * q^66 - 1504 * q^67 + 3047 * q^68 + 654 * q^69 + 1530 * q^70 - 1267 * q^71 - 189 * q^72 - 666 * q^73 + 528 * q^74 + 549 * q^75 - 64 * q^76 + 1109 * q^77 - 174 * q^78 - 2741 * q^79 + 1213 * q^80 + 567 * q^81 + 953 * q^82 - 2025 * q^83 + 2814 * q^84 - 1274 * q^85 + 4394 * q^86 - 504 * q^87 - 1639 * q^88 + 616 * q^89 - 639 * q^90 - 2415 * q^91 + 218 * q^92 + 1812 * q^93 + 900 * q^94 + 2554 * q^95 + 1344 * q^96 - 1298 * q^97 - 172 * q^98 - 45 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.0253269	0.00895440	0.00447720	−	0.999990i	$-0.498575\pi$
$2$	0.0253269	0.00895440	0.00447720	+	0.999990i	$0.498575\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−7.99936	−0.999920
$5$	8.85515	0.792029	0.396015	−	0.918244i	$-0.370393\pi$
$5$	8.85515	0.792029	0.396015	+	0.918244i	$0.370393\pi$
$6$	−0.0759806	−0.00516983
$7$	13.8749	0.749174	0.374587	−	0.927192i	$-0.377784\pi$
$7$	13.8749	0.749174	0.374587	+	0.927192i	$0.377784\pi$
$8$	−0.405214	−0.0179081
$9$	9.00000	0.333333
$10$	0.224273	0.00709215
$11$	−33.5949	−0.920841	−0.460420	−	0.887701i	$-0.652301\pi$
$11$	−33.5949	−0.920841	−0.460420	+	0.887701i	$0.652301\pi$
$12$	23.9981	0.577304
$13$	−15.9269	−0.339794	−0.169897	−	0.985462i	$-0.554344\pi$
$13$	−15.9269	−0.339794	−0.169897	+	0.985462i	$0.554344\pi$
$14$	0.351408	0.00670841
$15$	−26.5655	−0.457278
$16$	63.9846	0.999759
$17$	−14.1822	−0.202334	−0.101167	−	0.994869i	$-0.532258\pi$
$17$	−14.1822	−0.202334	−0.101167	+	0.994869i	$0.532258\pi$
$18$	0.227942	0.00298480
$19$	−111.821	−1.35019	−0.675095	−	0.737731i	$-0.735897\pi$
$19$	−111.821	−1.35019	−0.675095	+	0.737731i	$0.735897\pi$
$20$	−70.8356	−0.791966
$21$	−41.6247	−0.432536
$22$	−0.850854	−0.00824558
$23$	−188.829	−1.71190	−0.855949	−	0.517060i	$-0.827026\pi$
$23$	−188.829	−1.71190	−0.855949	+	0.517060i	$0.827026\pi$
$24$	1.21564	0.0103392
$25$	−46.5862	−0.372690
$26$	−0.403379	−0.00304266
$27$	−27.0000	−0.192450
$28$	−110.990	−0.749114
$29$	−56.6729	−0.362893	−0.181446	−	0.983401i	$-0.558078\pi$
$29$	−56.6729	−0.362893	−0.181446	+	0.983401i	$0.558078\pi$
$30$	−0.672820	−0.00409465
$31$	−18.4624	−0.106966	−0.0534829	−	0.998569i	$-0.517032\pi$
$31$	−18.4624	−0.106966	−0.0534829	+	0.998569i	$0.517032\pi$
$32$	4.86224	0.0268603
$33$	100.785	0.531648
$34$	−0.359190	−0.00181178
$35$	122.864	0.593368
$36$	−71.9942	−0.333307
$37$	53.8076	0.239079	0.119539	−	0.992829i	$-0.461858\pi$
$37$	53.8076	0.239079	0.119539	+	0.992829i	$0.461858\pi$
$38$	−2.83209	−0.0120901
$39$	47.7807	0.196180
$40$	−3.58823	−0.0141837
$41$	−68.8886	−0.262405	−0.131202	−	0.991356i	$-0.541884\pi$
$41$	−68.8886	−0.262405	−0.131202	+	0.991356i	$0.541884\pi$
$42$	−1.05422	−0.00387310
$43$	−344.210	−1.22073	−0.610367	−	0.792118i	$-0.708978\pi$
$43$	−344.210	−1.22073	−0.610367	+	0.792118i	$0.708978\pi$
$44$	268.738	0.920767
$45$	79.6964	0.264010
$46$	−4.78246	−0.0153290
$47$	291.289	0.904017	0.452009	−	0.892014i	$-0.350708\pi$
$47$	291.289	0.904017	0.452009	+	0.892014i	$0.350708\pi$
$48$	−191.954	−0.577211
$49$	−150.487	−0.438738
$50$	−1.17988	−0.00333721
$51$	42.5466	0.116818
$52$	127.405	0.339767
$53$	636.830	1.65048	0.825238	−	0.564785i	$-0.191041\pi$
$53$	636.830	1.65048	0.825238	+	0.564785i	$0.191041\pi$
$54$	−0.683826	−0.00172328
$55$	−297.488	−0.729333
$56$	−5.62230	−0.0134163
$57$	335.464	0.779532
$58$	−1.43535	−0.00324949
$59$	59.0000	0.130189
$59$	59.0000	0.130189
$60$	212.507	0.457242
$61$	−489.425	−1.02729	−0.513643	−	0.858004i	$-0.671704\pi$
$61$	−489.425	−1.02729	−0.513643	+	0.858004i	$0.671704\pi$
$62$	−0.467594	−0.000957814 0
$63$	124.874	0.249725
$64$	−511.754	−0.999519
$65$	−141.035	−0.269127
$66$	2.55256	0.00476059
$67$	91.2469	0.166382	0.0831909	−	0.996534i	$-0.473489\pi$
$67$	91.2469	0.166382	0.0831909	+	0.996534i	$0.473489\pi$
$68$	113.448	0.202318
$69$	566.488	0.988365
$70$	3.11177	0.00531325
$71$	179.941	0.300776	0.150388	−	0.988627i	$-0.451948\pi$
$71$	179.941	0.300776	0.150388	+	0.988627i	$0.451948\pi$
$72$	−3.64692	−0.00596936
$73$	−414.334	−0.664304	−0.332152	−	0.943226i	$-0.607775\pi$
$73$	−414.334	−0.664304	−0.332152	+	0.943226i	$0.607775\pi$
$74$	1.36278	0.00214081
$75$	139.759	0.215173
$76$	894.500	1.35008
$77$	−466.126	−0.689870
$78$	1.21014	0.00175668
$79$	−889.652	−1.26701	−0.633504	−	0.773739i	$-0.718384\pi$
$79$	−889.652	−1.26701	−0.633504	+	0.773739i	$0.718384\pi$
$80$	566.594	0.791839
$81$	81.0000	0.111111
$82$	−1.74473	−0.00234968
$83$	812.868	1.07499	0.537493	−	0.843268i	$-0.319371\pi$
$83$	812.868	1.07499	0.537493	+	0.843268i	$0.319371\pi$
$84$	332.971	0.432501
$85$	−125.585	−0.160255
$86$	−8.71778	−0.0109310
$87$	170.019	0.209516
$88$	13.6131	0.0164905
$89$	−623.403	−0.742479	−0.371240	−	0.928537i	$-0.621067\pi$
$89$	−623.403	−0.742479	−0.371240	+	0.928537i	$0.621067\pi$
$90$	2.01846	0.00236405
$91$	−220.984	−0.254565
$92$	1510.51	1.71176
$93$	55.3871	0.0617567
$94$	7.37743	0.00809493
$95$	−990.196	−1.06939
$96$	−14.5867	−0.0155078
$97$	1229.07	1.28653	0.643266	−	0.765643i	$-0.277579\pi$
$97$	1229.07	1.28653	0.643266	+	0.765643i	$0.277579\pi$
$98$	−3.81137	−0.00392864
$99$	−302.354	−0.306947
$100$	372.660	0.372660
$101$	792.722	0.780978	0.390489	−	0.920608i	$-0.372306\pi$
$101$	792.722	0.780978	0.390489	+	0.920608i	$0.372306\pi$
$102$	1.07757	0.00104603
$103$	56.2822	0.0538413	0.0269206	−	0.999638i	$-0.491430\pi$
$103$	56.2822	0.0538413	0.0269206	+	0.999638i	$0.491430\pi$
$104$	6.45380	0.00608507
$105$	−368.593	−0.342581
$106$	16.1289	0.0147790
$107$	−342.964	−0.309865	−0.154933	−	0.987925i	$-0.549516\pi$
$107$	−342.964	−0.309865	−0.154933	+	0.987925i	$0.549516\pi$
$108$	215.983	0.192435
$109$	1581.04	1.38932	0.694662	−	0.719336i	$-0.255554\pi$
$109$	1581.04	1.38932	0.694662	+	0.719336i	$0.255554\pi$
$110$	−7.53444	−0.00653074
$111$	−161.423	−0.138032
$112$	887.780	0.748994
$113$	−142.494	−0.118626	−0.0593128	−	0.998239i	$-0.518891\pi$
$113$	−142.494	−0.118626	−0.0593128	+	0.998239i	$0.518891\pi$
$114$	8.49626	0.00698024
$115$	−1672.11	−1.35587
$116$	453.346	0.362863
$117$	−143.342	−0.113265
$118$	1.49429	0.00116576
$119$	−196.776	−0.151584
$120$	10.7647	0.00818898
$121$	−202.382	−0.152053
$122$	−12.3956	−0.00919873
$123$	206.666	0.151500
$124$	147.687	0.106957
$125$	−1519.42	−1.08721
$126$	3.16267	0.00223614
$127$	−1616.84	−1.12970	−0.564848	−	0.825195i	$-0.691065\pi$
$127$	−1616.84	−1.12970	−0.564848	+	0.825195i	$0.691065\pi$
$128$	−51.8590	−0.0358104
$129$	1032.63	0.704792
$130$	−3.57198	−0.00240987
$131$	−617.730	−0.411995	−0.205998	−	0.978553i	$-0.566044\pi$
$131$	−617.730	−0.411995	−0.205998	+	0.978553i	$0.566044\pi$
$132$	−806.213	−0.531605
$133$	−1551.51	−1.01153
$134$	2.31100	0.00148985
$135$	−239.089	−0.152426
$136$	5.74682	0.00362342
$137$	149.739	0.0933798	0.0466899	−	0.998909i	$-0.485133\pi$
$137$	149.739	0.0933798	0.0466899	+	0.998909i	$0.485133\pi$
$138$	14.3474	0.00885022
$139$	−1014.64	−0.619139	−0.309569	−	0.950877i	$-0.600185\pi$
$139$	−1014.64	−0.619139	−0.309569	+	0.950877i	$0.600185\pi$
$140$	−982.836	−0.593320
$141$	−873.866	−0.521935
$142$	4.55734	0.00269327
$143$	535.063	0.312897
$144$	575.861	0.333253
$145$	−501.847	−0.287422
$146$	−10.4938	−0.00594844
$147$	451.462	0.253306
$148$	−430.426	−0.239060
$149$	1456.96	0.801066	0.400533	−	0.916282i	$-0.368825\pi$
$149$	1456.96	0.801066	0.400533	+	0.916282i	$0.368825\pi$
$150$	3.53965	0.00192674
$151$	620.816	0.334578	0.167289	−	0.985908i	$-0.446499\pi$
$151$	620.816	0.334578	0.167289	+	0.985908i	$0.446499\pi$
$152$	45.3116	0.0241793
$153$	−127.640	−0.0674448
$154$	−11.8055	−0.00617737
$155$	−163.487	−0.0847200
$156$	−382.215	−0.196165
$157$	2085.11	1.05993	0.529967	−	0.848018i	$-0.322204\pi$
$157$	2085.11	1.05993	0.529967	+	0.848018i	$0.322204\pi$
$158$	−22.5321	−0.0113453
$159$	−1910.49	−0.952903
$160$	43.0559	0.0212742
$161$	−2619.99	−1.28251
$162$	2.05148	0.000994934 0
$163$	−3928.28	−1.88765	−0.943823	−	0.330451i	$-0.892799\pi$
$163$	−3928.28	−1.88765	−0.943823	+	0.330451i	$0.892799\pi$
$164$	551.065	0.262384
$165$	892.464	0.421080
$166$	20.5874	0.00962586
$167$	−1209.74	−0.560552	−0.280276	−	0.959919i	$-0.590426\pi$
$167$	−1209.74	−0.560552	−0.280276	+	0.959919i	$0.590426\pi$
$168$	16.8669	0.00774589
$169$	−1943.33	−0.884540
$170$	−3.18069	−0.00143499
$171$	−1006.39	−0.450063
$172$	2753.46	1.22064
$173$	920.080	0.404349	0.202174	−	0.979350i	$-0.435199\pi$
$173$	920.080	0.404349	0.202174	+	0.979350i	$0.435199\pi$
$174$	4.30604	0.00187609
$175$	−646.379	−0.279210
$176$	−2149.56	−0.920619
$177$	−177.000	−0.0751646
$178$	−15.7889	−0.00664846
$179$	4056.88	1.69400	0.846999	−	0.531594i	$-0.178407\pi$
$179$	4056.88	1.69400	0.846999	+	0.531594i	$0.178407\pi$
$180$	−637.520	−0.263989
$181$	2038.25	0.837026	0.418513	−	0.908211i	$-0.362551\pi$
$181$	2038.25	0.837026	0.418513	+	0.908211i	$0.362551\pi$
$182$	−5.59684	−0.00227948
$183$	1468.27	0.593104
$184$	76.5163	0.0306568
$185$	476.475	0.189357
$186$	1.40278	0.000552994 0
$187$	476.449	0.186318
$188$	−2330.12	−0.903945
$189$	−374.622	−0.144179
$190$	−25.0786	−0.00957574
$191$	−1362.37	−0.516115	−0.258057	−	0.966130i	$-0.583082\pi$
$191$	−1362.37	−0.516115	−0.258057	+	0.966130i	$0.583082\pi$
$192$	1535.26	0.577073
$193$	−2837.51	−1.05828	−0.529142	−	0.848533i	$-0.677486\pi$
$193$	−2837.51	−1.05828	−0.529142	+	0.848533i	$0.677486\pi$
$194$	31.1286	0.0115201
$195$	423.106	0.155381
$196$	1203.80	0.438703
$197$	4236.08	1.53202	0.766011	−	0.642828i	$-0.222239\pi$
$197$	4236.08	1.53202	0.766011	+	0.642828i	$0.222239\pi$
$198$	−7.65769	−0.00274853
$199$	2678.88	0.954274	0.477137	−	0.878829i	$-0.341675\pi$
$199$	2678.88	0.954274	0.477137	+	0.878829i	$0.341675\pi$
$200$	18.8774	0.00667416
$201$	−273.741	−0.0960606
$202$	20.0772	0.00699319
$203$	−786.330	−0.271870
$204$	−340.345	−0.116808
$205$	−610.020	−0.207832
$206$	1.42545	0.000482116 0
$207$	−1699.46	−0.570633
$208$	−1019.08	−0.339713
$209$	3756.63	1.24331
$210$	−9.33531	−0.00306761
$211$	−4807.45	−1.56852	−0.784262	−	0.620430i	$-0.786958\pi$
$211$	−4807.45	−1.56852	−0.784262	+	0.620430i	$0.786958\pi$
$212$	−5094.23	−1.65034
$213$	−539.823	−0.173653
$214$	−8.68621	−0.00277466
$215$	−3048.04	−0.966858
$216$	10.9408	0.00344641
$217$	−256.163	−0.0801360
$218$	40.0428	0.0124406
$219$	1243.00	0.383536
$220$	2379.71	0.729274
$221$	225.878	0.0687521
$222$	−4.08833	−0.00123600
$223$	3465.13	1.04055	0.520274	−	0.853999i	$-0.325830\pi$
$223$	3465.13	1.04055	0.520274	+	0.853999i	$0.325830\pi$
$224$	67.4631	0.0201231
$225$	−419.276	−0.124230
$226$	−3.60892	−0.00106222
$227$	−759.521	−0.222076	−0.111038	−	0.993816i	$-0.535417\pi$
$227$	−759.521	−0.222076	−0.111038	+	0.993816i	$0.535417\pi$
$228$	−2683.50	−0.779470
$229$	6541.52	1.88767	0.943834	−	0.330419i	$-0.107190\pi$
$229$	6541.52	1.88767	0.943834	+	0.330419i	$0.107190\pi$
$230$	−42.3494	−0.0121410
$231$	1398.38	0.398297
$232$	22.9646	0.00649871
$233$	−6389.44	−1.79651	−0.898254	−	0.439477i	$-0.855164\pi$
$233$	−6389.44	−1.79651	−0.898254	+	0.439477i	$0.855164\pi$
$234$	−3.63041	−0.00101422
$235$	2579.41	0.716008
$236$	−471.962	−0.130178
$237$	2668.96	0.731508
$238$	−4.98373	−0.00135734
$239$	1601.37	0.433406	0.216703	−	0.976238i	$-0.430470\pi$
$239$	1601.37	0.433406	0.216703	+	0.976238i	$0.430470\pi$
$240$	−1699.78	−0.457168
$241$	5739.23	1.53401	0.767004	−	0.641642i	$-0.221746\pi$
$241$	5739.23	1.53401	0.767004	+	0.641642i	$0.221746\pi$
$242$	−5.12570	−0.00136154
$243$	−243.000	−0.0641500
$244$	3915.08	1.02720
$245$	−1332.59	−0.347493
$246$	5.23420	0.00135659
$247$	1780.97	0.458787
$248$	7.48120	0.00191555
$249$	−2438.60	−0.620643
$250$	−38.4822	−0.00973532
$251$	−2418.34	−0.608145	−0.304072	−	0.952649i	$-0.598347\pi$
$251$	−2418.34	−0.608145	−0.304072	+	0.952649i	$0.598347\pi$
$252$	−998.913	−0.249705
$253$	6343.71	1.57639
$254$	−40.9495	−0.0101158
$255$	376.756	0.0925231
$256$	4092.72	0.999198
$257$	2849.27	0.691566	0.345783	−	0.938314i	$-0.387613\pi$
$257$	2849.27	0.691566	0.345783	+	0.938314i	$0.387613\pi$
$258$	26.1533	0.00631099
$259$	746.575	0.179112
$260$	1128.19	0.269106
$261$	−510.056	−0.120964
$262$	−15.6452	−0.00368917
$263$	6916.57	1.62165	0.810825	−	0.585289i	$-0.199019\pi$
$263$	6916.57	1.62165	0.810825	+	0.585289i	$0.199019\pi$
$264$	−40.8394	−0.00952079
$265$	5639.22	1.30723
$266$	−39.2949	−0.00905762
$267$	1870.21	0.428670
$268$	−729.916	−0.166368
$269$	5951.84	1.34903	0.674517	−	0.738260i	$-0.264352\pi$
$269$	5951.84	1.34903	0.674517	+	0.738260i	$0.264352\pi$
$270$	−6.05538	−0.00136488
$271$	−2810.97	−0.630089	−0.315044	−	0.949077i	$-0.602019\pi$
$271$	−2810.97	−0.630089	−0.315044	+	0.949077i	$0.602019\pi$
$272$	−907.441	−0.202286
$273$	662.953	0.146973
$274$	3.79241	0.000836161 0
$275$	1565.06	0.343188
$276$	−4531.54	−0.988285
$277$	−2077.94	−0.450727	−0.225363	−	0.974275i	$-0.572357\pi$
$277$	−2077.94	−0.450727	−0.225363	+	0.974275i	$0.572357\pi$
$278$	−25.6976	−0.00554402
$279$	−166.161	−0.0356553
$280$	−49.7863	−0.0106261
$281$	3252.61	0.690513	0.345257	−	0.938508i	$-0.387792\pi$
$281$	3252.61	0.690513	0.345257	+	0.938508i	$0.387792\pi$
$282$	−22.1323	−0.00467361
$283$	−2305.82	−0.484335	−0.242167	−	0.970234i	$-0.577858\pi$
$283$	−2305.82	−0.484335	−0.242167	+	0.970234i	$0.577858\pi$
$284$	−1439.41	−0.300751
$285$	2970.59	0.617412
$286$	13.5515	0.00280180
$287$	−955.823	−0.196587
$288$	43.7602	0.00895345
$289$	−4711.87	−0.959061
$290$	−12.7102	−0.00257369
$291$	−3687.22	−0.742779
$292$	3314.41	0.664250
$293$	−5556.39	−1.10788	−0.553939	−	0.832558i	$-0.686876\pi$
$293$	−5556.39	−1.10788	−0.553939	+	0.832558i	$0.686876\pi$
$294$	11.4341	0.00226820
$295$	522.454	0.103113
$296$	−21.8036	−0.00428144
$297$	907.063	0.177216
$298$	36.9003	0.00717307
$299$	3007.47	0.581693
$300$	−1117.98	−0.215155
$301$	−4775.89	−0.914543
$302$	15.7233	0.00299595
$303$	−2378.17	−0.450898
$304$	−7154.85	−1.34986
$305$	−4333.93	−0.813640
$306$	−3.23271	−0.000603928 0
$307$	−4370.47	−0.812494	−0.406247	−	0.913763i	$-0.633163\pi$
$307$	−4370.47	−0.812494	−0.406247	+	0.913763i	$0.633163\pi$
$308$	3728.71	0.689815
$309$	−168.847	−0.0310853
$310$	−4.14062	−0.000758617 0
$311$	−6394.69	−1.16595	−0.582974	−	0.812491i	$-0.698111\pi$
$311$	−6394.69	−1.16595	−0.582974	+	0.812491i	$0.698111\pi$
$312$	−19.3614	−0.00351322
$313$	1266.98	0.228798	0.114399	−	0.993435i	$-0.463506\pi$
$313$	1266.98	0.228798	0.114399	+	0.993435i	$0.463506\pi$
$314$	52.8093	0.00949108
$315$	1105.78	0.197789
$316$	7116.64	1.26691
$317$	−236.906	−0.0419746	−0.0209873	−	0.999780i	$-0.506681\pi$
$317$	−236.906	−0.0419746	−0.0209873	+	0.999780i	$0.506681\pi$
$318$	−48.3867	−0.00853268
$319$	1903.92	0.334166
$320$	−4531.66	−0.791648
$321$	1028.89	0.178901
$322$	−66.3561	−0.0114841
$323$	1585.87	0.273190
$324$	−647.948	−0.111102
$325$	741.974	0.126638
$326$	−99.4910	−0.0169027
$327$	−4743.12	−0.802126
$328$	27.9146	0.00469917
$329$	4041.60	0.677266
$330$	22.6033	0.00377052
$331$	−549.458	−0.0912415	−0.0456208	−	0.998959i	$-0.514527\pi$
$331$	−549.458	−0.0912415	−0.0456208	+	0.998959i	$0.514527\pi$
$332$	−6502.42	−1.07490
$333$	484.268	0.0796929
$334$	−30.6388	−0.00501941
$335$	808.005	0.131779
$336$	−2663.34	−0.432432
$337$	−4347.11	−0.702677	−0.351339	−	0.936248i	$-0.614273\pi$
$337$	−4347.11	−0.702677	−0.351339	+	0.936248i	$0.614273\pi$
$338$	−49.2186	−0.00792052
$339$	427.481	0.0684885
$340$	1004.60	0.160242
$341$	620.241	0.0984984
$342$	−25.4888	−0.00403005
$343$	−6847.08	−1.07787
$344$	139.479	0.0218610
$345$	5016.34	0.782814
$346$	23.3027	0.00362070
$347$	4628.06	0.715987	0.357993	−	0.933724i	$-0.383461\pi$
$347$	4628.06	0.715987	0.357993	+	0.933724i	$0.383461\pi$
$348$	−1360.04	−0.209499
$349$	3589.52	0.550552	0.275276	−	0.961365i	$-0.411231\pi$
$349$	3589.52	0.550552	0.275276	+	0.961365i	$0.411231\pi$
$350$	−16.3708	−0.00250015
$351$	430.026	0.0653935
$352$	−163.347	−0.0247341
$353$	−6895.16	−1.03964	−0.519819	−	0.854276i	$-0.674001\pi$
$353$	−6895.16	−1.03964	−0.519819	+	0.854276i	$0.674001\pi$
$354$	−4.48286	−0.000673054 0
$355$	1593.41	0.238223
$356$	4986.83	0.742420
$357$	590.329	0.0875169
$358$	102.748	0.0151687
$359$	−4916.16	−0.722744	−0.361372	−	0.932422i	$-0.617692\pi$
$359$	−4916.16	−0.722744	−0.361372	+	0.932422i	$0.617692\pi$
$360$	−32.2941	−0.00472791
$361$	5645.03	0.823011
$362$	51.6224	0.00749507
$363$	607.146	0.0877876
$364$	1767.73	0.254545
$365$	−3669.00	−0.526148
$366$	37.1868	0.00531089
$367$	3909.42	0.556050	0.278025	−	0.960574i	$-0.410320\pi$
$367$	3909.42	0.556050	0.278025	+	0.960574i	$0.410320\pi$
$368$	−12082.2	−1.71149
$369$	−619.998	−0.0874683
$370$	12.0676	0.00169558
$371$	8835.95	1.23649
$372$	−443.061	−0.0617518
$373$	−12322.0	−1.71048	−0.855238	−	0.518236i	$-0.826589\pi$
$373$	−12322.0	−1.71048	−0.855238	+	0.518236i	$0.826589\pi$
$374$	12.0670	0.00166836
$375$	4558.27	0.627701
$376$	−118.034	−0.0161892
$377$	902.623	0.123309
$378$	−9.48801	−0.00129103
$379$	−2231.61	−0.302453	−0.151227	−	0.988499i	$-0.548322\pi$
$379$	−2231.61	−0.302453	−0.151227	+	0.988499i	$0.548322\pi$
$380$	7920.93	1.06930
$381$	4850.52	0.652230
$382$	−34.5047	−0.00462150
$383$	6278.86	0.837689	0.418845	−	0.908058i	$-0.362435\pi$
$383$	6278.86	0.837689	0.418845	+	0.908058i	$0.362435\pi$
$384$	155.577	0.0206752
$385$	−4127.62	−0.546397
$386$	−71.8653	−0.00947630
$387$	−3097.89	−0.406912
$388$	−9831.80	−1.28643
$389$	5321.38	0.693585	0.346792	−	0.937942i	$-0.387271\pi$
$389$	5321.38	0.693585	0.346792	+	0.937942i	$0.387271\pi$
$390$	10.7159	0.00139134
$391$	2678.01	0.346376
$392$	60.9795	0.00785696
$393$	1853.19	0.237865
$394$	107.287	0.0137183
$395$	−7878.01	−1.00351
$396$	2418.64	0.306922
$397$	−14422.3	−1.82326	−0.911630	−	0.411011i	$-0.865176\pi$
$397$	−14422.3	−1.82326	−0.911630	+	0.411011i	$0.865176\pi$
$398$	67.8476	0.00854495
$399$	4654.53	0.584005
$400$	−2980.80	−0.372600
$401$	5306.39	0.660820	0.330410	−	0.943838i	$-0.392813\pi$
$401$	5306.39	0.660820	0.330410	+	0.943838i	$0.392813\pi$
$402$	−6.93299	−0.000860165 0
$403$	294.048	0.0363464
$404$	−6341.27	−0.780915
$405$	717.268	0.0880032
$406$	−19.9153	−0.00243443
$407$	−1807.66	−0.220153
$408$	−17.2404	−0.00209198
$409$	−8289.85	−1.00222	−0.501108	−	0.865385i	$-0.667074\pi$
$409$	−8289.85	−1.00222	−0.501108	+	0.865385i	$0.667074\pi$
$410$	−15.4499	−0.00186101
$411$	−449.216	−0.0539129
$412$	−450.222	−0.0538370
$413$	818.619	0.0975342
$414$	−43.0421	−0.00510967
$415$	7198.07	0.851420
$416$	−77.4404	−0.00912699
$417$	3043.91	0.357460
$418$	95.1437	0.0111331
$419$	−14094.4	−1.64333	−0.821666	−	0.569969i	$-0.806955\pi$
$419$	−14094.4	−1.64333	−0.821666	+	0.569969i	$0.806955\pi$
$420$	2948.51	0.342554
$421$	4731.09	0.547694	0.273847	−	0.961773i	$-0.411704\pi$
$421$	4731.09	0.547694	0.273847	+	0.961773i	$0.411704\pi$
$422$	−121.758	−0.0140452
$423$	2621.60	0.301339
$424$	−258.052	−0.0295569
$425$	660.695	0.0754080
$426$	−13.6720	−0.00155496
$427$	−6790.72	−0.769616
$428$	2743.49	0.309841
$429$	−1605.19	−0.180651
$430$	−77.1973	−0.00865763
$431$	−5813.25	−0.649685	−0.324843	−	0.945768i	$-0.605311\pi$
$431$	−5813.25	−0.649685	−0.324843	+	0.945768i	$0.605311\pi$
$432$	−1727.58	−0.192404
$433$	−6041.84	−0.670559	−0.335280	−	0.942119i	$-0.608831\pi$
$433$	−6041.84	−0.670559	−0.335280	+	0.942119i	$0.608831\pi$
$434$	−6.48782	−0.000717570 0
$435$	1505.54	0.165943
$436$	−12647.3	−1.38921
$437$	21115.2	2.31139
$438$	31.4814	0.00343433
$439$	−900.540	−0.0979053	−0.0489526	−	0.998801i	$-0.515588\pi$
$439$	−900.540	−0.0979053	−0.0489526	+	0.998801i	$0.515588\pi$
$440$	120.546	0.0130610
$441$	−1354.38	−0.146246
$442$	5.72079	0.000615634 0
$443$	−883.492	−0.0947538	−0.0473769	−	0.998877i	$-0.515086\pi$
$443$	−883.492	−0.0947538	−0.0473769	+	0.998877i	$0.515086\pi$
$444$	1291.28	0.138021
$445$	−5520.33	−0.588065
$446$	87.7609	0.00931748
$447$	−4370.88	−0.462496
$448$	−7100.53	−0.748814
$449$	2759.86	0.290079	0.145040	−	0.989426i	$-0.453669\pi$
$449$	2759.86	0.290079	0.145040	+	0.989426i	$0.453669\pi$
$450$	−10.6190	−0.00111240
$451$	2314.31	0.241633
$452$	1139.86	0.118616
$453$	−1862.45	−0.193169
$454$	−19.2363	−0.00198856
$455$	−1956.85	−0.201623
$456$	−135.935	−0.0139599
$457$	953.929	0.0976432	0.0488216	−	0.998808i	$-0.484453\pi$
$457$	953.929	0.0976432	0.0488216	+	0.998808i	$0.484453\pi$
$458$	165.676	0.0169029
$459$	382.919	0.0389393
$460$	13375.8	1.35576
$461$	−615.985	−0.0622327	−0.0311163	−	0.999516i	$-0.509906\pi$
$461$	−615.985	−0.0622327	−0.0311163	+	0.999516i	$0.509906\pi$
$462$	35.4165	0.00356651
$463$	571.209	0.0573355	0.0286677	−	0.999589i	$-0.490874\pi$
$463$	571.209	0.0573355	0.0286677	+	0.999589i	$0.490874\pi$
$464$	−3626.19	−0.362805
$465$	490.461	0.0489131
$466$	−161.825	−0.0160867
$467$	−15066.7	−1.49294	−0.746470	−	0.665419i	$-0.768253\pi$
$467$	−15066.7	−1.49294	−0.746470	+	0.665419i	$0.768253\pi$
$468$	1146.65	0.113256
$469$	1266.04	0.124649
$470$	65.3283	0.00641142
$471$	−6255.32	−0.611953
$472$	−23.9076	−0.00233143
$473$	11563.7	1.12410
$474$	67.5963	0.00655021
$475$	5209.34	0.503202
$476$	1574.08	0.151572
$477$	5731.47	0.550159
$478$	40.5577	0.00388089
$479$	10104.3	0.963840	0.481920	−	0.876215i	$-0.339940\pi$
$479$	10104.3	0.963840	0.481920	+	0.876215i	$0.339940\pi$
$480$	−129.168	−0.0122826
$481$	−856.988	−0.0812376
$482$	145.357	0.0137361
$483$	7859.97	0.740457
$484$	1618.93	0.152040
$485$	10883.6	1.01897
$486$	−6.15443	−0.000574425 0
$487$	−13243.9	−1.23232	−0.616160	−	0.787621i	$-0.711313\pi$
$487$	−13243.9	−1.23232	−0.616160	+	0.787621i	$0.711313\pi$
$488$	198.322	0.0183967
$489$	11784.8	1.08983
$490$	−33.7503	−0.00311160
$491$	−1801.88	−0.165616	−0.0828081	−	0.996566i	$-0.526389\pi$
$491$	−1801.88	−0.165616	−0.0828081	+	0.996566i	$0.526389\pi$
$492$	−1653.19	−0.151487
$493$	803.745	0.0734257
$494$	45.1064	0.00410816
$495$	−2677.39	−0.243111
$496$	−1181.31	−0.106940
$497$	2496.66	0.225333
$498$	−61.7622	−0.00555749
$499$	18836.7	1.68987	0.844937	−	0.534867i	$-0.179638\pi$
$499$	18836.7	1.68987	0.844937	+	0.534867i	$0.179638\pi$
$500$	12154.4	1.08712
$501$	3629.21	0.323635
$502$	−61.2490	−0.00544557
$503$	11244.9	0.996786	0.498393	−	0.866951i	$-0.333924\pi$
$503$	11244.9	0.996786	0.498393	+	0.866951i	$0.333924\pi$
$504$	−50.6007	−0.00447209
$505$	7019.67	0.618557
$506$	160.666	0.0141156
$507$	5830.00	0.510689
$508$	12933.7	1.12961
$509$	5157.16	0.449091	0.224545	−	0.974464i	$-0.427910\pi$
$509$	5157.16	0.449091	0.224545	+	0.974464i	$0.427910\pi$
$510$	9.54206	0.000828489 0
$511$	−5748.85	−0.497679
$512$	518.528	0.0447577
$513$	3019.18	0.259844
$514$	72.1631	0.00619256
$515$	498.388	0.0426439
$516$	−8260.39	−0.704735
$517$	−9785.81	−0.832456
$518$	18.9084	0.00160384
$519$	−2760.24	−0.233451
$520$	57.1494	0.00481955
$521$	−20667.2	−1.73790	−0.868952	−	0.494896i	$-0.835206\pi$
$521$	−20667.2	−1.73790	−0.868952	+	0.494896i	$0.835206\pi$
$522$	−12.9181	−0.00108316
$523$	−22161.5	−1.85288	−0.926439	−	0.376445i	$-0.877146\pi$
$523$	−22161.5	−1.85288	−0.926439	+	0.376445i	$0.877146\pi$
$524$	4941.45	0.411962
$525$	1939.14	0.161202
$526$	175.175	0.0145209
$527$	261.837	0.0216429
$528$	6448.67	0.531520
$529$	23489.5	1.93059
$530$	142.824	0.0117054
$531$	531.000	0.0433963
$532$	12411.1	1.01145
$533$	1097.18	0.0891637
$534$	47.3666	0.00383849
$535$	−3037.00	−0.245422
$536$	−36.9745	−0.00297958
$537$	−12170.7	−0.978031
$538$	150.741	0.0120798
$539$	5055.60	0.404008
$540$	1912.56	0.152414
$541$	3396.28	0.269903	0.134952	−	0.990852i	$-0.456912\pi$
$541$	3396.28	0.269903	0.134952	+	0.990852i	$0.456912\pi$
$542$	−71.1930	−0.00564207
$543$	−6114.74	−0.483257
$544$	−68.9572	−0.00543477
$545$	14000.4	1.10038
$546$	16.7905	0.00131606
$547$	4790.05	0.374420	0.187210	−	0.982320i	$-0.440055\pi$
$547$	4790.05	0.374420	0.187210	+	0.982320i	$0.440055\pi$
$548$	−1197.81	−0.0933724
$549$	−4404.82	−0.342429
$550$	39.6381	0.00307304
$551$	6337.24	0.489974
$552$	−229.549	−0.0176997
$553$	−12343.8	−0.949210
$554$	−52.6277	−0.00403599
$555$	−1429.42	−0.109326
$556$	8116.44	0.619089
$557$	17592.3	1.33825	0.669127	−	0.743148i	$-0.266668\pi$
$557$	17592.3	1.33825	0.669127	+	0.743148i	$0.266668\pi$
$558$	−4.20835	−0.000319271 0
$559$	5482.21	0.414799
$560$	7861.43	0.593225
$561$	−1429.35	−0.107571
$562$	82.3784	0.00618313
$563$	−22917.8	−1.71558	−0.857789	−	0.514002i	$-0.828162\pi$
$563$	−22917.8	−1.71558	−0.857789	+	0.514002i	$0.828162\pi$
$564$	6990.36	0.521893
$565$	−1261.80	−0.0939549
$566$	−58.3992	−0.00433693
$567$	1123.87	0.0832416
$568$	−72.9146	−0.00538631
$569$	3650.54	0.268961	0.134480	−	0.990916i	$-0.457063\pi$
$569$	3650.54	0.268961	0.134480	+	0.990916i	$0.457063\pi$
$570$	75.2357	0.00552856
$571$	5473.59	0.401160	0.200580	−	0.979677i	$-0.435717\pi$
$571$	5473.59	0.401160	0.200580	+	0.979677i	$0.435717\pi$
$572$	−4280.16	−0.312871
$573$	4087.12	0.297979
$574$	−24.2080	−0.00176032
$575$	8796.85	0.638007
$576$	−4605.78	−0.333173
$577$	11228.4	0.810129	0.405065	−	0.914288i	$-0.367249\pi$
$577$	11228.4	0.810129	0.405065	+	0.914288i	$0.367249\pi$
$578$	−119.337	−0.00858782
$579$	8512.54	0.611000
$580$	4014.45	0.287398
$581$	11278.5	0.805352
$582$	−93.3858	−0.00665115
$583$	−21394.2	−1.51983
$584$	167.894	0.0118964
$585$	−1269.32	−0.0897090
$586$	−140.726	−0.00992038
$587$	−21794.4	−1.53246	−0.766228	−	0.642569i	$-0.777869\pi$
$587$	−21794.4	−1.53246	−0.766228	+	0.642569i	$0.777869\pi$
$588$	−3611.40	−0.253285
$589$	2064.49	0.144424
$590$	13.2321	0.000923319 0
$591$	−12708.2	−0.884513
$592$	3442.86	0.239021
$593$	10553.5	0.730824	0.365412	−	0.930846i	$-0.380928\pi$
$593$	10553.5	0.730824	0.365412	+	0.930846i	$0.380928\pi$
$594$	22.9731	0.00158686
$595$	−1742.49	−0.120059
$596$	−11654.8	−0.801002
$597$	−8036.63	−0.550950
$598$	76.1698	0.00520872
$599$	−12542.5	−0.855544	−0.427772	−	0.903887i	$-0.640701\pi$
$599$	−12542.5	−0.855544	−0.427772	+	0.903887i	$0.640701\pi$
$600$	−56.6321	−0.00385333
$601$	−19080.4	−1.29502	−0.647508	−	0.762059i	$-0.724189\pi$
$601$	−19080.4	−1.29502	−0.647508	+	0.762059i	$0.724189\pi$
$602$	−120.958	−0.00818919
$603$	821.222	0.0554606
$604$	−4966.13	−0.334551
$605$	−1792.12	−0.120430
$606$	−60.2315	−0.00403752
$607$	3062.83	0.204805	0.102402	−	0.994743i	$-0.467347\pi$
$607$	3062.83	0.204805	0.102402	+	0.994743i	$0.467347\pi$
$608$	−543.703	−0.0362665
$609$	2358.99	0.156964
$610$	−109.765	−0.00728566
$611$	−4639.32	−0.307180
$612$	1021.04	0.0674394
$613$	−757.895	−0.0499366	−0.0249683	−	0.999688i	$-0.507948\pi$
$613$	−757.895	−0.0499366	−0.0249683	+	0.999688i	$0.507948\pi$
$614$	−110.690	−0.00727540
$615$	1830.06	0.119992
$616$	188.881	0.0123543
$617$	−29721.4	−1.93928	−0.969641	−	0.244531i	$-0.921366\pi$
$617$	−29721.4	−1.93928	−0.969641	+	0.244531i	$0.921366\pi$
$618$	−4.27636	−0.000278350 0
$619$	−16540.8	−1.07404	−0.537021	−	0.843569i	$-0.680450\pi$
$619$	−16540.8	−1.07404	−0.537021	+	0.843569i	$0.680450\pi$
$620$	1307.79	0.0847132
$621$	5098.39	0.329455
$622$	−161.958	−0.0104404
$623$	−8649.66	−0.556246
$624$	3057.23	0.196133
$625$	−7631.44	−0.488412
$626$	32.0886	0.00204875
$627$	−11269.9	−0.717825
$628$	−16679.5	−1.05985
$629$	−763.109	−0.0483739
$630$	28.0059	0.00177108
$631$	3442.24	0.217169	0.108584	−	0.994087i	$-0.465368\pi$
$631$	3442.24	0.217169	0.108584	+	0.994087i	$0.465368\pi$
$632$	360.499	0.0226897
$633$	14422.3	0.905587
$634$	−6.00009	−0.000375858 0
$635$	−14317.4	−0.894752
$636$	15282.7	0.952827
$637$	2396.80	0.149081
$638$	48.2203	0.00299226
$639$	1619.47	0.100259
$640$	−459.220	−0.0283629
$641$	−1578.29	−0.0972524	−0.0486262	−	0.998817i	$-0.515484\pi$
$641$	−1578.29	−0.0972524	−0.0486262	+	0.998817i	$0.515484\pi$
$642$	26.0586	0.00160195
$643$	−23885.3	−1.46492	−0.732460	−	0.680810i	$-0.761628\pi$
$643$	−23885.3	−1.46492	−0.732460	+	0.680810i	$0.761628\pi$
$644$	20958.2	1.28241
$645$	9144.11	0.558216
$646$	40.1652	0.00244625
$647$	−28377.5	−1.72432	−0.862161	−	0.506634i	$-0.830889\pi$
$647$	−28377.5	−1.72432	−0.862161	+	0.506634i	$0.830889\pi$
$648$	−32.8223	−0.00198979
$649$	−1982.10	−0.119883
$650$	18.7919	0.00113397
$651$	768.490	0.0462665
$652$	31423.7	1.88749
$653$	−3579.86	−0.214534	−0.107267	−	0.994230i	$-0.534210\pi$
$653$	−3579.86	−0.214534	−0.107267	+	0.994230i	$0.534210\pi$
$654$	−120.129	−0.00718256
$655$	−5470.10	−0.326312
$656$	−4407.81	−0.262342
$657$	−3729.01	−0.221435
$658$	102.361	0.00606451
$659$	27873.3	1.64763	0.823816	−	0.566857i	$-0.191841\pi$
$659$	27873.3	1.64763	0.823816	+	0.566857i	$0.191841\pi$
$660$	−7139.14	−0.421047
$661$	−10031.8	−0.590307	−0.295154	−	0.955450i	$-0.595371\pi$
$661$	−10031.8	−0.590307	−0.295154	+	0.955450i	$0.595371\pi$
$662$	−13.9161	−0.000817013 0
$663$	−677.635	−0.0396941
$664$	−329.385	−0.0192509
$665$	−13738.9	−0.801159
$666$	12.2650	0.000713602 0
$667$	10701.5	0.621235
$668$	9677.11	0.560507
$669$	−10395.4	−0.600761
$670$	20.4642	0.00118000
$671$	16442.2	0.945966
$672$	−202.389	−0.0116181
$673$	−1426.11	−0.0816828	−0.0408414	−	0.999166i	$-0.513004\pi$
$673$	−1426.11	−0.0816828	−0.0408414	+	0.999166i	$0.513004\pi$
$674$	−110.099	−0.00629206
$675$	1257.83	0.0717242
$676$	15545.4	0.884469
$677$	17441.8	0.990165	0.495082	−	0.868846i	$-0.335138\pi$
$677$	17441.8	0.990165	0.495082	+	0.868846i	$0.335138\pi$
$678$	10.8268	0.000613273 0
$679$	17053.3	0.963836
$680$	50.8889	0.00286986
$681$	2278.56	0.128216
$682$	15.7088	0.000881994 0
$683$	−6216.52	−0.348270	−0.174135	−	0.984722i	$-0.555713\pi$
$683$	−6216.52	−0.348270	−0.174135	+	0.984722i	$0.555713\pi$
$684$	8050.50	0.450027
$685$	1325.96	0.0739596
$686$	−173.415	−0.00965164
$687$	−19624.6	−1.08985
$688$	−22024.2	−1.22044
$689$	−10142.7	−0.560823
$690$	127.048	0.00700963
$691$	21688.1	1.19400	0.597001	−	0.802240i	$-0.296359\pi$
$691$	21688.1	1.19400	0.597001	+	0.802240i	$0.296359\pi$
$692$	−7360.05	−0.404317
$693$	−4195.13	−0.229957
$694$	117.214	0.00641123
$695$	−8984.76	−0.490376
$696$	−68.8939	−0.00375203
$697$	976.991	0.0530935
$698$	90.9113	0.00492986
$699$	19168.3	1.03721
$700$	5170.62	0.279187
$701$	−29389.5	−1.58349	−0.791744	−	0.610853i	$-0.790827\pi$
$701$	−29389.5	−1.58349	−0.791744	+	0.610853i	$0.790827\pi$
$702$	10.8912	0.000585560 0
$703$	−6016.84	−0.322802
$704$	17192.3	0.920398
$705$	−7738.22	−0.413387
$706$	−174.633	−0.00930934
$707$	10998.9	0.585088
$708$	1415.89	0.0751586
$709$	−8186.60	−0.433645	−0.216823	−	0.976211i	$-0.569569\pi$
$709$	−8186.60	−0.433645	−0.216823	+	0.976211i	$0.569569\pi$
$710$	40.3560	0.00213314
$711$	−8006.87	−0.422336
$712$	252.612	0.0132964
$713$	3486.24	0.183114
$714$	14.9512	0.000783661 0
$715$	4738.07	0.247823
$716$	−32452.5	−1.69386
$717$	−4804.11	−0.250227
$718$	−124.511	−0.00647174
$719$	−28735.1	−1.49046	−0.745229	−	0.666808i	$-0.767660\pi$
$719$	−28735.1	−1.49046	−0.745229	+	0.666808i	$0.767660\pi$
$720$	5099.34	0.263946
$721$	780.910	0.0403365
$722$	142.971	0.00736957
$723$	−17217.7	−0.885660
$724$	−16304.7	−0.836959
$725$	2640.17	0.135246
$726$	15.3771	0.000786086 0
$727$	−12141.5	−0.619397	−0.309698	−	0.950835i	$-0.600228\pi$
$727$	−12141.5	−0.619397	−0.309698	+	0.950835i	$0.600228\pi$
$728$	89.5458	0.00455878
$729$	729.000	0.0370370
$730$	−92.9242	−0.00471134
$731$	4881.66	0.246997
$732$	−11745.3	−0.593056
$733$	20396.0	1.02775	0.513877	−	0.857864i	$-0.328209\pi$
$733$	20396.0	1.02775	0.513877	+	0.857864i	$0.328209\pi$
$734$	99.0135	0.00497909
$735$	3997.76	0.200625
$736$	−918.134	−0.0459822
$737$	−3065.43	−0.153211
$738$	−15.7026	−0.000783226 0
$739$	15016.7	0.747492	0.373746	−	0.927531i	$-0.378073\pi$
$739$	15016.7	0.747492	0.373746	+	0.927531i	$0.378073\pi$
$740$	−3811.49	−0.189342
$741$	−5342.91	−0.264881
$742$	223.787	0.0110721
$743$	8964.84	0.442649	0.221324	−	0.975200i	$-0.428962\pi$
$743$	8964.84	0.442649	0.221324	+	0.975200i	$0.428962\pi$
$744$	−22.4436	−0.00110594
$745$	12901.6	0.634468
$746$	−312.077	−0.0153163
$747$	7315.81	0.358329
$748$	−3811.29	−0.186303
$749$	−4758.59	−0.232143
$750$	115.447	0.00562069
$751$	6506.21	0.316132	0.158066	−	0.987429i	$-0.449474\pi$
$751$	6506.21	0.316132	0.158066	+	0.987429i	$0.449474\pi$
$752$	18638.0	0.903800
$753$	7255.03	0.351113
$754$	22.8606	0.00110416
$755$	5497.42	0.264996
$756$	2996.74	0.144167
$757$	−1562.84	−0.0750364	−0.0375182	−	0.999296i	$-0.511945\pi$
$757$	−1562.84	−0.0750364	−0.0375182	+	0.999296i	$0.511945\pi$
$758$	−56.5196	−0.00270829
$759$	−19031.1	−0.910126
$760$	401.241	0.0191507
$761$	−2971.60	−0.141551	−0.0707757	−	0.997492i	$-0.522547\pi$
$761$	−2971.60	−0.141551	−0.0707757	+	0.997492i	$0.522547\pi$
$762$	122.849	0.00584033
$763$	21936.8	1.04085
$764$	10898.1	0.516073
$765$	−1130.27	−0.0534182
$766$	159.024	0.00750100
$767$	−939.687	−0.0442375
$768$	−12278.1	−0.576887
$769$	18566.0	0.870619	0.435310	−	0.900281i	$-0.356639\pi$
$769$	18566.0	0.870619	0.435310	+	0.900281i	$0.356639\pi$
$770$	−104.540	−0.00489266
$771$	−8547.81	−0.399276
$772$	22698.3	1.05820
$773$	−4922.01	−0.229020	−0.114510	−	0.993422i	$-0.536530\pi$
$773$	−4922.01	−0.229020	−0.114510	+	0.993422i	$0.536530\pi$
$774$	−78.4600	−0.00364365
$775$	860.092	0.0398650
$776$	−498.038	−0.0230393
$777$	−2239.72	−0.103410
$778$	134.774	0.00621064
$779$	7703.23	0.354296
$780$	−3384.57	−0.155368
$781$	−6045.10	−0.276966
$782$	67.8257	0.00310159
$783$	1530.17	0.0698387
$784$	−9628.86	−0.438633
$785$	18463.9	0.839499
$786$	46.9355	0.00212994
$787$	22392.8	1.01425	0.507126	−	0.861872i	$-0.330708\pi$
$787$	22392.8	1.01425	0.507126	+	0.861872i	$0.330708\pi$
$788$	−33885.9	−1.53190
$789$	−20749.7	−0.936260
$790$	−199.525	−0.00898581
$791$	−1977.09	−0.0888712
$792$	122.518	0.00549683
$793$	7795.02	0.349066
$794$	−365.272	−0.0163262
$795$	−16917.7	−0.754727
$796$	−21429.3	−0.954197
$797$	−11623.6	−0.516597	−0.258298	−	0.966065i	$-0.583162\pi$
$797$	−11623.6	−0.516597	−0.258298	+	0.966065i	$0.583162\pi$
$798$	117.885	0.00522942
$799$	−4131.11	−0.182914
$800$	−226.513	−0.0100106
$801$	−5610.63	−0.247493
$802$	134.394	0.00591724
$803$	13919.5	0.611718
$804$	2189.75	0.0960529
$805$	−23200.4	−1.01578
$806$	7.44732	0.000325460 0
$807$	−17855.5	−0.778865
$808$	−321.222	−0.0139858
$809$	−11274.6	−0.489978	−0.244989	−	0.969526i	$-0.578784\pi$
$809$	−11274.6	−0.489978	−0.244989	+	0.969526i	$0.578784\pi$
$810$	18.1661	0.000788016 0
$811$	20634.8	0.893449	0.446724	−	0.894672i	$-0.352590\pi$
$811$	20634.8	0.893449	0.446724	+	0.894672i	$0.352590\pi$
$812$	6290.14	0.271848
$813$	8432.90	0.363782
$814$	−45.7824	−0.00197134
$815$	−34785.5	−1.49507
$816$	2722.32	0.116790
$817$	38490.1	1.64822
$818$	−209.956	−0.00897425
$819$	−1988.86	−0.0848551
$820$	4879.77	0.207816
$821$	42062.6	1.78806	0.894029	−	0.448009i	$-0.147867\pi$
$821$	42062.6	1.78806	0.894029	+	0.448009i	$0.147867\pi$
$822$	−11.3772	−0.000482758 0
$823$	32974.8	1.39663	0.698317	−	0.715789i	$-0.253933\pi$
$823$	32974.8	1.39663	0.698317	+	0.715789i	$0.253933\pi$
$824$	−22.8063	−0.000964194 0
$825$	−4695.18	−0.198140
$826$	20.7331	0.000873360 0
$827$	9364.19	0.393742	0.196871	−	0.980429i	$-0.436922\pi$
$827$	9364.19	0.393742	0.196871	+	0.980429i	$0.436922\pi$
$828$	13594.6	0.570587
$829$	−18611.4	−0.779734	−0.389867	−	0.920871i	$-0.627479\pi$
$829$	−18611.4	−0.779734	−0.389867	+	0.920871i	$0.627479\pi$
$830$	182.305	0.00762396
$831$	6233.82	0.260227
$832$	8150.65	0.339631
$833$	2134.24	0.0887718
$834$	77.0927	0.00320084
$835$	−10712.4	−0.443973
$836$	−30050.6	−1.24321
$837$	498.484	0.0205856
$838$	−356.967	−0.0147151
$839$	30534.1	1.25644	0.628221	−	0.778035i	$-0.283784\pi$
$839$	30534.1	1.25644	0.628221	+	0.778035i	$0.283784\pi$
$840$	149.359	0.00613497
$841$	−21177.2	−0.868309
$842$	119.824	0.00490427
$843$	−9757.82	−0.398668
$844$	38456.5	1.56840
$845$	−17208.5	−0.700581
$846$	66.3969	0.00269831
$847$	−2808.03	−0.113914
$848$	40747.3	1.65008
$849$	6917.46	0.279631
$850$	16.7333	0.000675233 0
$851$	−10160.5	−0.409278
$852$	4318.24	0.173639
$853$	1881.16	0.0755098	0.0377549	−	0.999287i	$-0.487979\pi$
$853$	1881.16	0.0755098	0.0377549	+	0.999287i	$0.487979\pi$
$854$	−171.988	−0.00689145
$855$	−8911.76	−0.356463
$856$	138.974	0.00554910
$857$	−24782.3	−0.987803	−0.493902	−	0.869518i	$-0.664430\pi$
$857$	−24782.3	−0.987803	−0.493902	+	0.869518i	$0.664430\pi$
$858$	−40.6544	−0.00161762
$859$	34731.4	1.37953	0.689767	−	0.724032i	$-0.257713\pi$
$859$	34731.4	1.37953	0.689767	+	0.724032i	$0.257713\pi$
$860$	24382.3	0.966780
$861$	2867.47	0.113500
$862$	−147.231	−0.00581754
$863$	−15140.1	−0.597190	−0.298595	−	0.954380i	$-0.596518\pi$
$863$	−15140.1	−0.597190	−0.298595	+	0.954380i	$0.596518\pi$
$864$	−131.280	−0.00516927
$865$	8147.45	0.320256
$866$	−153.021	−0.00600446
$867$	14135.6	0.553714
$868$	2049.14	0.0801295
$869$	29887.8	1.16671
$870$	38.1306	0.00148592
$871$	−1453.28	−0.0565356
$872$	−640.660	−0.0248801
$873$	11061.7	0.428844
$874$	534.781	0.0206971
$875$	−21081.8	−0.814510
$876$	−9943.23	−0.383505
$877$	50253.4	1.93493	0.967467	−	0.252996i	$-0.0814158\pi$
$877$	50253.4	1.93493	0.967467	+	0.252996i	$0.0814158\pi$
$878$	−22.8079	−0.000876683 0
$879$	16669.2	0.639633
$880$	−19034.7	−0.729157
$881$	−6565.72	−0.251084	−0.125542	−	0.992088i	$-0.540067\pi$
$881$	−6565.72	−0.251084	−0.125542	+	0.992088i	$0.540067\pi$
$882$	−34.3023	−0.00130955
$883$	35396.1	1.34901	0.674504	−	0.738271i	$-0.264357\pi$
$883$	35396.1	1.34901	0.674504	+	0.738271i	$0.264357\pi$
$884$	−1806.88	−0.0687466
$885$	−1567.36	−0.0595326
$886$	−22.3761	−0.000848464 0
$887$	−40472.0	−1.53204	−0.766019	−	0.642818i	$-0.777765\pi$
$887$	−40472.0	−1.53204	−0.766019	+	0.642818i	$0.777765\pi$
$888$	65.4107	0.00247189
$889$	−22433.5	−0.846339
$890$	−139.813	−0.00526577
$891$	−2721.19	−0.102316
$892$	−27718.8	−1.04046
$893$	−32572.3	−1.22059
$894$	−110.701	−0.00414137
$895$	35924.3	1.34170
$896$	−719.539	−0.0268282
$897$	−9022.40	−0.335841
$898$	69.8985	0.00259749
$899$	1046.31	0.0388171
$900$	3353.94	0.124220
$901$	−9031.63	−0.333948
$902$	58.6142	0.00216368
$903$	14327.7	0.528012
$904$	57.7404	0.00212436
$905$	18049.0	0.662949
$906$	−47.1700	−0.00172971
$907$	19742.8	0.722765	0.361382	−	0.932418i	$-0.382305\pi$
$907$	19742.8	0.722765	0.361382	+	0.932418i	$0.382305\pi$
$908$	6075.68	0.222058
$909$	7134.50	0.260326
$910$	−49.5609	−0.00180541
$911$	54399.1	1.97840	0.989201	−	0.146568i	$-0.0468228\pi$
$911$	54399.1	1.97840	0.989201	+	0.146568i	$0.0468228\pi$
$912$	21464.5	0.779345
$913$	−27308.2	−0.989891
$914$	24.1600	0.000874336 0
$915$	13001.8	0.469755
$916$	−52328.0	−1.88752
$917$	−8570.95	−0.308656
$918$	9.69814	0.000348678 0
$919$	14039.6	0.503943	0.251972	−	0.967735i	$-0.418921\pi$
$919$	14039.6	0.503943	0.251972	+	0.967735i	$0.418921\pi$
$920$	677.563	0.0242811
$921$	13111.4	0.469094
$922$	−15.6010	−0.000557256 0
$923$	−2865.90	−0.102202
$924$	−11186.1	−0.398265
$925$	−2506.69	−0.0891022
$926$	14.4669	0.000513405 0
$927$	506.540	0.0179471
$928$	−275.557	−0.00974742
$929$	49759.1	1.75731	0.878656	−	0.477456i	$-0.158441\pi$
$929$	49759.1	1.75731	0.878656	+	0.477456i	$0.158441\pi$
$930$	12.4219	0.000437988 0
$931$	16827.7	0.592380
$932$	51111.4	1.79636
$933$	19184.1	0.673160
$934$	−381.592	−0.0133684
$935$	4219.03	0.147569
$936$	58.0842	0.00202836
$937$	−24113.8	−0.840728	−0.420364	−	0.907355i	$-0.638098\pi$
$937$	−24113.8	−0.840728	−0.420364	+	0.907355i	$0.638098\pi$
$938$	32.0649	0.00111616
$939$	−3800.93	−0.132097
$940$	−20633.6	−0.715950
$941$	−7779.87	−0.269518	−0.134759	−	0.990878i	$-0.543026\pi$
$941$	−7779.87	−0.269518	−0.134759	+	0.990878i	$0.543026\pi$
$942$	−158.428	−0.00547968
$943$	13008.2	0.449210
$944$	3775.09	0.130158
$945$	−3317.34	−0.114194
$946$	292.873	0.0100657
$947$	42730.7	1.46627	0.733137	−	0.680081i	$-0.238055\pi$
$947$	42730.7	1.46627	0.733137	+	0.680081i	$0.238055\pi$
$948$	−21349.9	−0.731449
$949$	6599.06	0.225727
$950$	131.936	0.00450587
$951$	710.718	0.0242341
$952$	79.7365	0.00271457
$953$	−15153.9	−0.515094	−0.257547	−	0.966266i	$-0.582914\pi$
$953$	−15153.9	−0.515094	−0.257547	+	0.966266i	$0.582914\pi$
$954$	145.160	0.00492634
$955$	−12064.0	−0.408778
$956$	−12809.9	−0.433371
$957$	−5711.76	−0.192931
$958$	255.912	0.00863061
$959$	2077.61	0.0699578
$960$	13595.0	0.457058
$961$	−29450.1	−0.988558
$962$	−21.7048	−0.000727435 0
$963$	−3086.68	−0.103288
$964$	−45910.1	−1.53389
$965$	−25126.6	−0.838191
$966$	199.068	0.00663035
$967$	22252.0	0.739996	0.369998	−	0.929033i	$-0.379358\pi$
$967$	22252.0	0.739996	0.369998	+	0.929033i	$0.379358\pi$
$968$	82.0080	0.00272297
$969$	−4757.62	−0.157726
$970$	275.649	0.00912427
$971$	−35050.0	−1.15840	−0.579201	−	0.815184i	$-0.696636\pi$
$971$	−35050.0	−1.15840	−0.579201	+	0.815184i	$0.696636\pi$
$972$	1943.84	0.0641449
$973$	−14078.0	−0.463843
$974$	−335.428	−0.0110347
$975$	−2225.92	−0.0731145
$976$	−31315.7	−1.02704
$977$	−28934.1	−0.947476	−0.473738	−	0.880666i	$-0.657096\pi$
$977$	−28934.1	−0.947476	−0.473738	+	0.880666i	$0.657096\pi$
$978$	298.473	0.00975880
$979$	20943.2	0.683705
$980$	10659.8	0.347466
$981$	14229.4	0.463108
$982$	−45.6359	−0.00148299
$983$	−18336.9	−0.594972	−0.297486	−	0.954726i	$-0.596148\pi$
$983$	−18336.9	−0.594972	−0.297486	+	0.954726i	$0.596148\pi$
$984$	−83.7439	−0.00271307
$985$	37511.1	1.21341
$986$	20.3563	0.000657483 0
$987$	−12124.8	−0.391020
$988$	−14246.6	−0.458750
$989$	64997.1	2.08977
$990$	−67.8100	−0.00217691
$991$	−13126.3	−0.420758	−0.210379	−	0.977620i	$-0.567470\pi$
$991$	−13126.3	−0.420758	−0.210379	+	0.977620i	$0.567470\pi$
$992$	−89.7684	−0.00287314
$993$	1648.37	0.0526783
$994$	63.2327	0.00201772
$995$	23721.9	0.755813
$996$	19507.3	0.620594
$997$	−18514.9	−0.588137	−0.294069	−	0.955784i	$-0.595009\pi$
$997$	−18514.9	−0.588137	−0.294069	+	0.955784i	$0.595009\pi$
$998$	477.075	0.0151318
$999$	−1452.81	−0.0460107

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.a.b.1.4	✓	7
3.2	odd	2		531.4.a.c.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.b.1.4	✓	7	1.1	even	1	trivial
531.4.a.c.1.4		7	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0253269 q^{2} -3.00000 q^{3} -7.99936 q^{4} +8.85515 q^{5} -0.0759806 q^{6} +13.8749 q^{7} -0.405214 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+0.0253269 q^{2} -3.00000 q^{3} -7.99936 q^{4} +8.85515 q^{5} -0.0759806 q^{6} +13.8749 q^{7} -0.405214 q^{8} +9.00000 q^{9} +0.224273 q^{10} -33.5949 q^{11} +23.9981 q^{12} -15.9269 q^{13} +0.351408 q^{14} -26.5655 q^{15} +63.9846 q^{16} -14.1822 q^{17} +0.227942 q^{18} -111.821 q^{19} -70.8356 q^{20} -41.6247 q^{21} -0.850854 q^{22} -188.829 q^{23} +1.21564 q^{24} -46.5862 q^{25} -0.403379 q^{26} -27.0000 q^{27} -110.990 q^{28} -56.6729 q^{29} -0.672820 q^{30} -18.4624 q^{31} +4.86224 q^{32} +100.785 q^{33} -0.359190 q^{34} +122.864 q^{35} -71.9942 q^{36} +53.8076 q^{37} -2.83209 q^{38} +47.7807 q^{39} -3.58823 q^{40} -68.8886 q^{41} -1.05422 q^{42} -344.210 q^{43} +268.738 q^{44} +79.6964 q^{45} -4.78246 q^{46} +291.289 q^{47} -191.954 q^{48} -150.487 q^{49} -1.17988 q^{50} +42.5466 q^{51} +127.405 q^{52} +636.830 q^{53} -0.683826 q^{54} -297.488 q^{55} -5.62230 q^{56} +335.464 q^{57} -1.43535 q^{58} +59.0000 q^{59} +212.507 q^{60} -489.425 q^{61} -0.467594 q^{62} +124.874 q^{63} -511.754 q^{64} -141.035 q^{65} +2.55256 q^{66} +91.2469 q^{67} +113.448 q^{68} +566.488 q^{69} +3.11177 q^{70} +179.941 q^{71} -3.64692 q^{72} -414.334 q^{73} +1.36278 q^{74} +139.759 q^{75} +894.500 q^{76} -466.126 q^{77} +1.21014 q^{78} -889.652 q^{79} +566.594 q^{80} +81.0000 q^{81} -1.74473 q^{82} +812.868 q^{83} +332.971 q^{84} -125.585 q^{85} -8.71778 q^{86} +170.019 q^{87} +13.6131 q^{88} -623.403 q^{89} +2.01846 q^{90} -220.984 q^{91} +1510.51 q^{92} +55.3871 q^{93} +7.37743 q^{94} -990.196 q^{95} -14.5867 q^{96} +1229.07 q^{97} -3.81137 q^{98} -302.354 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 21 q^{3} + 26 q^{4} - 2 q^{5} - 59 q^{7} - 21 q^{8} + 63 q^{9}+O(q^{10}) \) 7 * q - 21 * q^3 + 26 * q^4 - 2 * q^5 - 59 * q^7 - 21 * q^8 + 63 * q^9	\( 7 q - 21 q^{3} + 26 q^{4} - 2 q^{5} - 59 q^{7} - 21 q^{8} + 63 q^{9} - 71 q^{10} - 5 q^{11} - 78 q^{12} - 67 q^{13} - 65 q^{14} + 6 q^{15} - 94 q^{16} - 23 q^{17} - 176 q^{19} - 207 q^{20} + 177 q^{21} - 704 q^{22} - 218 q^{23} + 63 q^{24} - 183 q^{25} + 58 q^{26} - 189 q^{27} - 938 q^{28} + 168 q^{29} + 213 q^{30} - 604 q^{31} - 448 q^{32} + 15 q^{33} - 610 q^{34} - 336 q^{35} + 234 q^{36} - 505 q^{37} - 453 q^{38} + 201 q^{39} - 1080 q^{40} - 265 q^{41} + 195 q^{42} - 493 q^{43} + 504 q^{44} - 18 q^{45} + 381 q^{46} - 244 q^{47} + 282 q^{48} + 770 q^{49} + 1639 q^{50} + 69 q^{51} + 160 q^{52} + 686 q^{53} - 116 q^{55} + 2190 q^{56} + 528 q^{57} + 1584 q^{58} + 413 q^{59} + 621 q^{60} - 838 q^{61} + 286 q^{62} - 531 q^{63} + 205 q^{64} + 490 q^{65} + 2112 q^{66} - 1504 q^{67} + 3047 q^{68} + 654 q^{69} + 1530 q^{70} - 1267 q^{71} - 189 q^{72} - 666 q^{73} + 528 q^{74} + 549 q^{75} - 64 q^{76} + 1109 q^{77} - 174 q^{78} - 2741 q^{79} + 1213 q^{80} + 567 q^{81} + 953 q^{82} - 2025 q^{83} + 2814 q^{84} - 1274 q^{85} + 4394 q^{86} - 504 q^{87} - 1639 q^{88} + 616 q^{89} - 639 q^{90} - 2415 q^{91} + 218 q^{92} + 1812 q^{93} + 900 q^{94} + 2554 q^{95} + 1344 q^{96} - 1298 q^{97} - 172 q^{98} - 45 q^{99}+O(q^{100}) \) 7 * q - 21 * q^3 + 26 * q^4 - 2 * q^5 - 59 * q^7 - 21 * q^8 + 63 * q^9 - 71 * q^10 - 5 * q^11 - 78 * q^12 - 67 * q^13 - 65 * q^14 + 6 * q^15 - 94 * q^16 - 23 * q^17 - 176 * q^19 - 207 * q^20 + 177 * q^21 - 704 * q^22 - 218 * q^23 + 63 * q^24 - 183 * q^25 + 58 * q^26 - 189 * q^27 - 938 * q^28 + 168 * q^29 + 213 * q^30 - 604 * q^31 - 448 * q^32 + 15 * q^33 - 610 * q^34 - 336 * q^35 + 234 * q^36 - 505 * q^37 - 453 * q^38 + 201 * q^39 - 1080 * q^40 - 265 * q^41 + 195 * q^42 - 493 * q^43 + 504 * q^44 - 18 * q^45 + 381 * q^46 - 244 * q^47 + 282 * q^48 + 770 * q^49 + 1639 * q^50 + 69 * q^51 + 160 * q^52 + 686 * q^53 - 116 * q^55 + 2190 * q^56 + 528 * q^57 + 1584 * q^58 + 413 * q^59 + 621 * q^60 - 838 * q^61 + 286 * q^62 - 531 * q^63 + 205 * q^64 + 490 * q^65 + 2112 * q^66 - 1504 * q^67 + 3047 * q^68 + 654 * q^69 + 1530 * q^70 - 1267 * q^71 - 189 * q^72 - 666 * q^73 + 528 * q^74 + 549 * q^75 - 64 * q^76 + 1109 * q^77 - 174 * q^78 - 2741 * q^79 + 1213 * q^80 + 567 * q^81 + 953 * q^82 - 2025 * q^83 + 2814 * q^84 - 1274 * q^85 + 4394 * q^86 - 504 * q^87 - 1639 * q^88 + 616 * q^89 - 639 * q^90 - 2415 * q^91 + 218 * q^92 + 1812 * q^93 + 900 * q^94 + 2554 * q^95 + 1344 * q^96 - 1298 * q^97 - 172 * q^98 - 45 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more