LMFDB - Embedded newform 177.4.a.b.1.2

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			1.2
Root			$3.06012$ 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)

$f(q)$	$=$	$q-3.06012 q^{2} -3.00000 q^{3} +1.36432 q^{4} +0.675250 q^{5} +9.18035 q^{6} -3.38755 q^{7} +20.3060 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-3.06012 q^{2} -3.00000 q^{3} +1.36432 q^{4} +0.675250 q^{5} +9.18035 q^{6} -3.38755 q^{7} +20.3060 q^{8} +9.00000 q^{9} -2.06635 q^{10} -8.79097 q^{11} -4.09296 q^{12} +59.4118 q^{13} +10.3663 q^{14} -2.02575 q^{15} -73.0532 q^{16} +31.2380 q^{17} -27.5411 q^{18} -77.5189 q^{19} +0.921258 q^{20} +10.1626 q^{21} +26.9014 q^{22} +73.1225 q^{23} -60.9179 q^{24} -124.544 q^{25} -181.807 q^{26} -27.0000 q^{27} -4.62171 q^{28} -61.5678 q^{29} +6.19904 q^{30} -278.253 q^{31} +61.1037 q^{32} +26.3729 q^{33} -95.5920 q^{34} -2.28744 q^{35} +12.2789 q^{36} -326.365 q^{37} +237.217 q^{38} -178.235 q^{39} +13.7116 q^{40} +453.551 q^{41} -31.0989 q^{42} -174.901 q^{43} -11.9937 q^{44} +6.07725 q^{45} -223.763 q^{46} -480.883 q^{47} +219.160 q^{48} -331.525 q^{49} +381.119 q^{50} -93.7140 q^{51} +81.0568 q^{52} -185.568 q^{53} +82.6232 q^{54} -5.93610 q^{55} -68.7874 q^{56} +232.557 q^{57} +188.405 q^{58} +59.0000 q^{59} -2.76378 q^{60} +807.823 q^{61} +851.486 q^{62} -30.4879 q^{63} +397.441 q^{64} +40.1178 q^{65} -80.7042 q^{66} -129.081 q^{67} +42.6187 q^{68} -219.367 q^{69} +6.99985 q^{70} -272.032 q^{71} +182.754 q^{72} -610.054 q^{73} +998.716 q^{74} +373.632 q^{75} -105.761 q^{76} +29.7798 q^{77} +545.422 q^{78} +886.175 q^{79} -49.3292 q^{80} +81.0000 q^{81} -1387.92 q^{82} +51.4871 q^{83} +13.8651 q^{84} +21.0935 q^{85} +535.217 q^{86} +184.703 q^{87} -178.509 q^{88} +182.350 q^{89} -18.5971 q^{90} -201.261 q^{91} +99.7625 q^{92} +834.758 q^{93} +1471.56 q^{94} -52.3447 q^{95} -183.311 q^{96} -1353.44 q^{97} +1014.50 q^{98} -79.1187 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−3.06012	−1.08192	−0.540958	−	0.841050i	$-0.681938\pi$
$2$	−3.06012	−1.08192	−0.540958	+	0.841050i	$0.681938\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	1.36432	0.170540
$5$	0.675250	0.0603962	0.0301981	−	0.999544i	$-0.490386\pi$
$5$	0.675250	0.0603962	0.0301981	+	0.999544i	$0.490386\pi$
$6$	9.18035	0.624644
$7$	−3.38755	−0.182910	−0.0914552	−	0.995809i	$-0.529152\pi$
$7$	−3.38755	−0.182910	−0.0914552	+	0.995809i	$0.529152\pi$
$8$	20.3060	0.897405
$9$	9.00000	0.333333
$10$	−2.06635	−0.0653436
$11$	−8.79097	−0.240962	−0.120481	−	0.992716i	$-0.538444\pi$
$11$	−8.79097	−0.240962	−0.120481	+	0.992716i	$0.538444\pi$
$12$	−4.09296	−0.0984614
$13$	59.4118	1.26753	0.633764	−	0.773526i	$-0.281509\pi$
$13$	59.4118	1.26753	0.633764	+	0.773526i	$0.281509\pi$
$14$	10.3663	0.197894
$15$	−2.02575	−0.0348698
$16$	−73.0532	−1.14146
$17$	31.2380	0.445666	0.222833	−	0.974857i	$-0.428469\pi$
$17$	31.2380	0.445666	0.222833	+	0.974857i	$0.428469\pi$
$18$	−27.5411	−0.360638
$19$	−77.5189	−0.936003	−0.468002	−	0.883728i	$-0.655026\pi$
$19$	−77.5189	−0.936003	−0.468002	+	0.883728i	$0.655026\pi$
$20$	0.921258	0.0103000
$21$	10.1626	0.105603
$22$	26.9014	0.260700
$23$	73.1225	0.662917	0.331458	−	0.943470i	$-0.392459\pi$
$23$	73.1225	0.662917	0.331458	+	0.943470i	$0.392459\pi$
$24$	−60.9179	−0.518117
$25$	−124.544	−0.996352
$26$	−181.807	−1.37136
$27$	−27.0000	−0.192450
$28$	−4.62171	−0.0311936
$29$	−61.5678	−0.394236	−0.197118	−	0.980380i	$-0.563158\pi$
$29$	−61.5678	−0.394236	−0.197118	+	0.980380i	$0.563158\pi$
$30$	6.19904	0.0377261
$31$	−278.253	−1.61212	−0.806059	−	0.591835i	$-0.798404\pi$
$31$	−278.253	−1.61212	−0.806059	+	0.591835i	$0.798404\pi$
$32$	61.1037	0.337554
$33$	26.3729	0.139119
$34$	−95.5920	−0.482173
$35$	−2.28744	−0.0110471
$36$	12.2789	0.0568467
$37$	−326.365	−1.45011	−0.725055	−	0.688691i	$-0.758186\pi$
$37$	−326.365	−1.45011	−0.725055	+	0.688691i	$0.758186\pi$
$38$	237.217	1.01268
$39$	−178.235	−0.731808
$40$	13.7116	0.0541999
$41$	453.551	1.72763	0.863815	−	0.503809i	$-0.168069\pi$
$41$	453.551	1.72763	0.863815	+	0.503809i	$0.168069\pi$
$42$	−31.0989	−0.114254
$43$	−174.901	−0.620282	−0.310141	−	0.950691i	$-0.600376\pi$
$43$	−174.901	−0.620282	−0.310141	+	0.950691i	$0.600376\pi$
$44$	−11.9937	−0.0410936
$45$	6.07725	0.0201321
$46$	−223.763	−0.717220
$47$	−480.883	−1.49243	−0.746213	−	0.665707i	$-0.768130\pi$
$47$	−480.883	−1.49243	−0.746213	+	0.665707i	$0.768130\pi$
$48$	219.160	0.659020
$49$	−331.525	−0.966544
$50$	381.119	1.07797
$51$	−93.7140	−0.257306
$52$	81.0568	0.216165
$53$	−185.568	−0.480938	−0.240469	−	0.970657i	$-0.577301\pi$
$53$	−185.568	−0.480938	−0.240469	+	0.970657i	$0.577301\pi$
$54$	82.6232	0.208215
$55$	−5.93610	−0.0145532
$56$	−68.7874	−0.164145
$57$	232.557	0.540402
$58$	188.405	0.426530
$59$	59.0000	0.130189
$59$	59.0000	0.130189
$60$	−2.76378	−0.00594670
$61$	807.823	1.69559	0.847796	−	0.530322i	$-0.177929\pi$
$61$	807.823	1.69559	0.847796	+	0.530322i	$0.177929\pi$
$62$	851.486	1.74417
$63$	−30.4879	−0.0609702
$64$	397.441	0.776252
$65$	40.1178	0.0765539
$66$	−80.7042	−0.150515
$67$	−129.081	−0.235369	−0.117685	−	0.993051i	$-0.537547\pi$
$67$	−129.081	−0.235369	−0.117685	+	0.993051i	$0.537547\pi$
$68$	42.6187	0.0760040
$69$	−219.367	−0.382735
$70$	6.99985	0.0119520
$71$	−272.032	−0.454708	−0.227354	−	0.973812i	$-0.573007\pi$
$71$	−272.032	−0.454708	−0.227354	+	0.973812i	$0.573007\pi$
$72$	182.754	0.299135
$73$	−610.054	−0.978102	−0.489051	−	0.872255i	$-0.662657\pi$
$73$	−610.054	−0.978102	−0.489051	+	0.872255i	$0.662657\pi$
$74$	998.716	1.56890
$75$	373.632	0.575244
$76$	−105.761	−0.159626
$77$	29.7798	0.0440744
$78$	545.422	0.791754
$79$	886.175	1.26206	0.631028	−	0.775760i	$-0.282633\pi$
$79$	886.175	1.26206	0.631028	+	0.775760i	$0.282633\pi$
$80$	−49.3292	−0.0689396
$81$	81.0000	0.111111
$82$	−1387.92	−1.86915
$83$	51.4871	0.0680896	0.0340448	−	0.999420i	$-0.489161\pi$
$83$	51.4871	0.0680896	0.0340448	+	0.999420i	$0.489161\pi$
$84$	13.8651	0.0180096
$85$	21.0935	0.0269166
$86$	535.217	0.671092
$87$	184.703	0.227612
$88$	−178.509	−0.216240
$89$	182.350	0.217180	0.108590	−	0.994087i	$-0.465366\pi$
$89$	182.350	0.217180	0.108590	+	0.994087i	$0.465366\pi$
$90$	−18.5971	−0.0217812
$91$	−201.261	−0.231844
$92$	99.7625	0.113054
$93$	834.758	0.930757
$94$	1471.56	1.61468
$95$	−52.3447	−0.0565311
$96$	−183.311	−0.194887
$97$	−1353.44	−1.41671	−0.708354	−	0.705857i	$-0.750562\pi$
$97$	−1353.44	−1.41671	−0.708354	+	0.705857i	$0.750562\pi$
$98$	1014.50	1.04572
$99$	−79.1187	−0.0803205
$100$	−169.918	−0.169918
$101$	−406.406	−0.400385	−0.200192	−	0.979757i	$-0.564157\pi$
$101$	−406.406	−0.400385	−0.200192	+	0.979757i	$0.564157\pi$
$102$	286.776	0.278383
$103$	−304.606	−0.291395	−0.145697	−	0.989329i	$-0.546543\pi$
$103$	−304.606	−0.291395	−0.145697	+	0.989329i	$0.546543\pi$
$104$	1206.41	1.13749
$105$	6.86233	0.00637805
$106$	567.859	0.520334
$107$	−1515.70	−1.36942	−0.684711	−	0.728815i	$-0.740071\pi$
$107$	−1515.70	−1.36942	−0.684711	+	0.728815i	$0.740071\pi$
$108$	−36.8367	−0.0328205
$109$	−1.57624	−0.00138511	−0.000692553	−	1.00000i	$-0.500220\pi$
$109$	−1.57624	−0.00138511	−0.000692553		1.00000i	$0.500220\pi$
$110$	18.1652	0.0157453
$111$	979.095	0.837222
$112$	247.471	0.208784
$113$	−577.813	−0.481027	−0.240514	−	0.970646i	$-0.577316\pi$
$113$	−577.813	−0.481027	−0.240514	+	0.970646i	$0.577316\pi$
$114$	−711.651	−0.584669
$115$	49.3760	0.0400377
$116$	−83.9983	−0.0672331
$117$	534.706	0.422510
$118$	−180.547	−0.140853
$119$	−105.820	−0.0815171
$120$	−41.1348	−0.0312923
$121$	−1253.72	−0.941938
$122$	−2472.03	−1.83449
$123$	−1360.65	−0.997448
$124$	−379.626	−0.274931
$125$	−168.505	−0.120572
$126$	93.2967	0.0659645
$127$	291.667	0.203789	0.101895	−	0.994795i	$-0.467510\pi$
$127$	291.667	0.203789	0.101895	+	0.994795i	$0.467510\pi$
$128$	−1705.05	−1.17739
$129$	524.702	0.358120
$130$	−122.765	−0.0828249
$131$	−1160.30	−0.773863	−0.386932	−	0.922108i	$-0.626465\pi$
$131$	−1160.30	−0.773863	−0.386932	+	0.922108i	$0.626465\pi$
$132$	35.9811	0.0237254
$133$	262.599	0.171205
$134$	395.003	0.254649
$135$	−18.2318	−0.0116233
$136$	634.318	0.399943
$137$	571.419	0.356348	0.178174	−	0.983999i	$-0.442981\pi$
$137$	571.419	0.356348	0.178174	+	0.983999i	$0.442981\pi$
$138$	671.290	0.414087
$139$	−576.322	−0.351677	−0.175838	−	0.984419i	$-0.556264\pi$
$139$	−576.322	−0.351677	−0.175838	+	0.984419i	$0.556264\pi$
$140$	−3.12081	−0.00188397
$141$	1442.65	0.861653
$142$	832.450	0.491956
$143$	−522.287	−0.305426
$144$	−657.479	−0.380485
$145$	−41.5737	−0.0238104
$146$	1866.84	1.05822
$147$	994.574	0.558034
$148$	−445.267	−0.247302
$149$	−1845.62	−1.01476	−0.507380	−	0.861723i	$-0.669386\pi$
$149$	−1845.62	−1.01476	−0.507380	+	0.861723i	$0.669386\pi$
$150$	−1143.36	−0.622365
$151$	−1957.64	−1.05503	−0.527517	−	0.849545i	$-0.676877\pi$
$151$	−1957.64	−1.05503	−0.527517	+	0.849545i	$0.676877\pi$
$152$	−1574.10	−0.839974
$153$	281.142	0.148555
$154$	−91.1298	−0.0476847
$155$	−187.890	−0.0973658
$156$	−243.170	−0.124803
$157$	791.623	0.402410	0.201205	−	0.979549i	$-0.435514\pi$
$157$	791.623	0.402410	0.201205	+	0.979549i	$0.435514\pi$
$158$	−2711.80	−1.36544
$159$	556.703	0.277669
$160$	41.2603	0.0203870
$161$	−247.706	−0.121254
$162$	−247.870	−0.120213
$163$	−3364.00	−1.61650	−0.808248	−	0.588843i	$-0.799584\pi$
$163$	−3364.00	−1.61650	−0.808248	+	0.588843i	$0.799584\pi$
$164$	618.790	0.294630
$165$	17.8083	0.00840227
$166$	−157.556	−0.0736672
$167$	514.992	0.238630	0.119315	−	0.992856i	$-0.461930\pi$
$167$	514.992	0.238630	0.119315	+	0.992856i	$0.461930\pi$
$168$	206.362	0.0947690
$169$	1332.76	0.606629
$170$	−64.5485	−0.0291214
$171$	−697.670	−0.312001
$172$	−238.621	−0.105783
$173$	2625.47	1.15382	0.576910	−	0.816808i	$-0.304258\pi$
$173$	2625.47	1.15382	0.576910	+	0.816808i	$0.304258\pi$
$174$	−565.214	−0.246257
$175$	421.899	0.182243
$176$	642.208	0.275047
$177$	−177.000	−0.0751646
$178$	−558.012	−0.234971
$179$	−165.711	−0.0691947	−0.0345973	−	0.999401i	$-0.511015\pi$
$179$	−165.711	−0.0691947	−0.0345973	+	0.999401i	$0.511015\pi$
$180$	8.29133	0.00343333
$181$	1680.47	0.690100	0.345050	−	0.938584i	$-0.387862\pi$
$181$	1680.47	0.690100	0.345050	+	0.938584i	$0.387862\pi$
$182$	615.881	0.250836
$183$	−2423.47	−0.978951
$184$	1484.82	0.594905
$185$	−220.378	−0.0875812
$186$	−2554.46	−1.00700
$187$	−274.612	−0.107388
$188$	−656.079	−0.254519
$189$	91.4638	0.0352011
$190$	160.181	0.0611618
$191$	−825.467	−0.312716	−0.156358	−	0.987700i	$-0.549975\pi$
$191$	−825.467	−0.312716	−0.156358	+	0.987700i	$0.549975\pi$
$192$	−1192.32	−0.448169
$193$	3110.12	1.15995	0.579977	−	0.814633i	$-0.303061\pi$
$193$	3110.12	1.15995	0.579977	+	0.814633i	$0.303061\pi$
$194$	4141.68	1.53276
$195$	−120.354	−0.0441984
$196$	−452.306	−0.164835
$197$	−1252.31	−0.452909	−0.226455	−	0.974022i	$-0.572713\pi$
$197$	−1252.31	−0.452909	−0.226455	+	0.974022i	$0.572713\pi$
$198$	242.113	0.0869000
$199$	854.395	0.304354	0.152177	−	0.988353i	$-0.451372\pi$
$199$	854.395	0.304354	0.152177	+	0.988353i	$0.451372\pi$
$200$	−2528.99	−0.894132
$201$	387.242	0.135890
$202$	1243.65	0.433182
$203$	208.564	0.0721099
$204$	−127.856	−0.0438809
$205$	306.261	0.104342
$206$	932.129	0.315265
$207$	658.102	0.220972
$208$	−4340.22	−1.44683
$209$	681.466	0.225541
$210$	−20.9995	−0.00690051
$211$	5555.73	1.81266	0.906332	−	0.422566i	$-0.138870\pi$
$211$	5555.73	1.81266	0.906332	+	0.422566i	$0.138870\pi$
$212$	−253.174	−0.0820192
$213$	816.096	0.262526
$214$	4638.22	1.48160
$215$	−118.102	−0.0374627
$216$	−548.261	−0.172706
$217$	942.595	0.294873
$218$	4.82349	0.00149857
$219$	1830.16	0.564707
$220$	−8.09875	−0.00248190
$221$	1855.91	0.564895
$222$	−2996.15	−0.905803
$223$	−2120.53	−0.636776	−0.318388	−	0.947961i	$-0.603141\pi$
$223$	−2120.53	−0.636776	−0.318388	+	0.947961i	$0.603141\pi$
$224$	−206.992	−0.0617421
$225$	−1120.90	−0.332117
$226$	1768.18	0.520431
$227$	3373.06	0.986247	0.493123	−	0.869959i	$-0.335855\pi$
$227$	3373.06	0.986247	0.493123	+	0.869959i	$0.335855\pi$
$228$	317.282	0.0921602
$229$	1545.12	0.445872	0.222936	−	0.974833i	$-0.428436\pi$
$229$	1545.12	0.445872	0.222936	+	0.974833i	$0.428436\pi$
$230$	−151.096	−0.0433173
$231$	−89.3395	−0.0254464
$232$	−1250.19	−0.353790
$233$	4860.03	1.36649	0.683243	−	0.730191i	$-0.260569\pi$
$233$	4860.03	1.36649	0.683243	+	0.730191i	$0.260569\pi$
$234$	−1636.26	−0.457120
$235$	−324.717	−0.0901369
$236$	80.4950	0.0222024
$237$	−2658.52	−0.728648
$238$	323.823	0.0881945
$239$	6957.12	1.88292	0.941462	−	0.337119i	$-0.109452\pi$
$239$	6957.12	1.88292	0.941462	+	0.337119i	$0.109452\pi$
$240$	147.988	0.0398023
$241$	−463.881	−0.123988	−0.0619942	−	0.998077i	$-0.519746\pi$
$241$	−463.881	−0.123988	−0.0619942	+	0.998077i	$0.519746\pi$
$242$	3836.53	1.01910
$243$	−243.000	−0.0641500
$244$	1102.13	0.289167
$245$	−223.862	−0.0583756
$246$	4163.76	1.07915
$247$	−4605.54	−1.18641
$248$	−5650.19	−1.44672
$249$	−154.461	−0.0393116
$250$	515.644	0.130449
$251$	−7550.40	−1.89871	−0.949356	−	0.314202i	$-0.898263\pi$
$251$	−7550.40	−1.89871	−0.949356	+	0.314202i	$0.898263\pi$
$252$	−41.5954	−0.0103979
$253$	−642.817	−0.159737
$254$	−892.534	−0.220483
$255$	−63.2804	−0.0155403
$256$	2038.11	0.497586
$257$	69.0755	0.0167658	0.00838290	−	0.999965i	$-0.497332\pi$
$257$	69.0755	0.0167658	0.00838290	+	0.999965i	$0.497332\pi$
$258$	−1605.65	−0.387455
$259$	1105.58	0.265240
$260$	54.7336	0.0130555
$261$	−554.110	−0.131412
$262$	3550.66	0.837254
$263$	−6387.24	−1.49754	−0.748772	−	0.662828i	$-0.769356\pi$
$263$	−6387.24	−1.49754	−0.748772	+	0.662828i	$0.769356\pi$
$264$	535.527	0.124846
$265$	−125.305	−0.0290468
$266$	−803.585	−0.185229
$267$	−547.049	−0.125389
$268$	−176.108	−0.0401399
$269$	−1359.76	−0.308201	−0.154100	−	0.988055i	$-0.549248\pi$
$269$	−1359.76	−0.308201	−0.154100	+	0.988055i	$0.549248\pi$
$270$	55.7913	0.0125754
$271$	−3184.33	−0.713780	−0.356890	−	0.934146i	$-0.616163\pi$
$271$	−3184.33	−0.713780	−0.356890	+	0.934146i	$0.616163\pi$
$272$	−2282.04	−0.508709
$273$	603.782	0.133855
$274$	−1748.61	−0.385538
$275$	1094.86	0.240083
$276$	−299.288	−0.0652717
$277$	2086.69	0.452626	0.226313	−	0.974055i	$-0.427333\pi$
$277$	2086.69	0.452626	0.226313	+	0.974055i	$0.427333\pi$
$278$	1763.61	0.380484
$279$	−2504.27	−0.537373
$280$	−46.4487	−0.00991372
$281$	6730.70	1.42890	0.714448	−	0.699688i	$-0.246678\pi$
$281$	6730.70	1.42890	0.714448	+	0.699688i	$0.246678\pi$
$282$	−4414.68	−0.932235
$283$	6977.08	1.46553	0.732764	−	0.680483i	$-0.238230\pi$
$283$	6977.08	1.46553	0.732764	+	0.680483i	$0.238230\pi$
$284$	−371.139	−0.0775460
$285$	157.034	0.0326382
$286$	1598.26	0.330445
$287$	−1536.43	−0.316002
$288$	549.934	0.112518
$289$	−3937.19	−0.801382
$290$	127.220	0.0257608
$291$	4060.31	0.817937
$292$	−832.310	−0.166806
$293$	1173.58	0.233999	0.116999	−	0.993132i	$-0.462672\pi$
$293$	1173.58	0.233999	0.116999	+	0.993132i	$0.462672\pi$
$294$	−3043.51	−0.603746
$295$	39.8398	0.00786292
$296$	−6627.16	−1.30134
$297$	237.356	0.0463731
$298$	5647.82	1.09788
$299$	4344.34	0.840266
$300$	509.754	0.0981023
$301$	592.485	0.113456
$302$	5990.60	1.14146
$303$	1219.22	0.231162
$304$	5663.01	1.06841
$305$	545.483	0.102407
$306$	−860.328	−0.160724
$307$	−6437.77	−1.19682	−0.598409	−	0.801191i	$-0.704200\pi$
$307$	−6437.77	−1.19682	−0.598409	+	0.801191i	$0.704200\pi$
$308$	40.6293	0.00751646
$309$	913.817	0.168237
$310$	574.966	0.105342
$311$	−3665.91	−0.668407	−0.334204	−	0.942501i	$-0.608467\pi$
$311$	−3665.91	−0.668407	−0.334204	+	0.942501i	$0.608467\pi$
$312$	−3619.24	−0.656728
$313$	−1795.66	−0.324270	−0.162135	−	0.986769i	$-0.551838\pi$
$313$	−1795.66	−0.324270	−0.162135	+	0.986769i	$0.551838\pi$
$314$	−2422.46	−0.435373
$315$	−20.5870	−0.00368237
$316$	1209.03	0.215231
$317$	793.810	0.140646	0.0703230	−	0.997524i	$-0.477597\pi$
$317$	793.810	0.140646	0.0703230	+	0.997524i	$0.477597\pi$
$318$	−1703.58	−0.300415
$319$	541.240	0.0949958
$320$	268.372	0.0468827
$321$	4547.09	0.790636
$322$	758.009	0.131187
$323$	−2421.54	−0.417145
$324$	110.510	0.0189489
$325$	−7399.39	−1.26291
$326$	10294.2	1.74891
$327$	4.72873	0.000799692 0
$328$	9209.80	1.55038
$329$	1629.02	0.272980
$330$	−54.4955	−0.00909055
$331$	3322.57	0.551737	0.275868	−	0.961195i	$-0.411035\pi$
$331$	3322.57	0.551737	0.275868	+	0.961195i	$0.411035\pi$
$332$	70.2449	0.0116120
$333$	−2937.29	−0.483370
$334$	−1575.94	−0.258178
$335$	−87.1619	−0.0142154
$336$	−742.414	−0.120542
$337$	2092.35	0.338213	0.169106	−	0.985598i	$-0.445912\pi$
$337$	2092.35	0.338213	0.169106	+	0.985598i	$0.445912\pi$
$338$	−4078.42	−0.656322
$339$	1733.44	0.277721
$340$	28.7783	0.00459036
$341$	2446.11	0.388458
$342$	2134.95	0.337559
$343$	2284.99	0.359701
$344$	−3551.53	−0.556644
$345$	−148.128	−0.0231158
$346$	−8034.25	−1.24834
$347$	−9919.25	−1.53456	−0.767281	−	0.641311i	$-0.778391\pi$
$347$	−9919.25	−1.53456	−0.767281	+	0.641311i	$0.778391\pi$
$348$	251.995	0.0388171
$349$	1359.74	0.208554	0.104277	−	0.994548i	$-0.466747\pi$
$349$	1359.74	0.208554	0.104277	+	0.994548i	$0.466747\pi$
$350$	−1291.06	−0.197172
$351$	−1604.12	−0.243936
$352$	−537.161	−0.0813374
$353$	−1123.35	−0.169376	−0.0846880	−	0.996408i	$-0.526989\pi$
$353$	−1123.35	−0.169376	−0.0846880	+	0.996408i	$0.526989\pi$
$354$	541.641	0.0813217
$355$	−183.690	−0.0274626
$356$	248.784	0.0370380
$357$	317.461	0.0470639
$358$	507.096	0.0748628
$359$	4438.64	0.652541	0.326271	−	0.945276i	$-0.394208\pi$
$359$	4438.64	0.652541	0.326271	+	0.945276i	$0.394208\pi$
$360$	123.404	0.0180666
$361$	−849.815	−0.123898
$362$	−5142.43	−0.746630
$363$	3761.16	0.543828
$364$	−274.584	−0.0395388
$365$	−411.939	−0.0590736
$366$	7416.10	1.05914
$367$	−12343.3	−1.75563	−0.877817	−	0.478996i	$-0.841001\pi$
$367$	−12343.3	−1.75563	−0.877817	+	0.478996i	$0.841001\pi$
$368$	−5341.83	−0.756690
$369$	4081.96	0.575877
$370$	674.383	0.0947554
$371$	628.620	0.0879685
$372$	1138.88	0.158731
$373$	8458.96	1.17423	0.587116	−	0.809503i	$-0.300263\pi$
$373$	8458.96	1.17423	0.587116	+	0.809503i	$0.300263\pi$
$374$	840.346	0.116185
$375$	505.514	0.0696123
$376$	−9764.79	−1.33931
$377$	−3657.85	−0.499706
$378$	−279.890	−0.0380846
$379$	−7901.32	−1.07088	−0.535440	−	0.844573i	$-0.679854\pi$
$379$	−7901.32	−1.07088	−0.535440	+	0.844573i	$0.679854\pi$
$380$	−71.4150	−0.00964082
$381$	−875.000	−0.117658
$382$	2526.03	0.338332
$383$	1775.56	0.236885	0.118443	−	0.992961i	$-0.462210\pi$
$383$	1775.56	0.236885	0.118443	+	0.992961i	$0.462210\pi$
$384$	5115.14	0.679768
$385$	20.1088	0.00266193
$386$	−9517.33	−1.25497
$387$	−1574.11	−0.206761
$388$	−1846.52	−0.241606
$389$	12845.2	1.67423	0.837115	−	0.547027i	$-0.184241\pi$
$389$	12845.2	1.67423	0.837115	+	0.547027i	$0.184241\pi$
$390$	368.296	0.0478190
$391$	2284.20	0.295440
$392$	−6731.92	−0.867381
$393$	3480.91	0.446790
$394$	3832.20	0.490009
$395$	598.390	0.0762234
$396$	−107.943	−0.0136979
$397$	3824.54	0.483497	0.241749	−	0.970339i	$-0.422279\pi$
$397$	3824.54	0.483497	0.241749	+	0.970339i	$0.422279\pi$
$398$	−2614.55	−0.329285
$399$	−787.798	−0.0988452
$400$	9098.34	1.13729
$401$	−10107.1	−1.25867	−0.629334	−	0.777135i	$-0.716672\pi$
$401$	−10107.1	−1.25867	−0.629334	+	0.777135i	$0.716672\pi$
$402$	−1185.01	−0.147022
$403$	−16531.5	−2.04341
$404$	−554.468	−0.0682817
$405$	54.6953	0.00671069
$406$	−638.230	−0.0780168
$407$	2869.07	0.349421
$408$	−1902.95	−0.230907
$409$	−863.888	−0.104441	−0.0522207	−	0.998636i	$-0.516630\pi$
$409$	−863.888	−0.104441	−0.0522207	+	0.998636i	$0.516630\pi$
$410$	−937.194	−0.112890
$411$	−1714.26	−0.205737
$412$	−415.580	−0.0496945
$413$	−199.865	−0.0238129
$414$	−2013.87	−0.239073
$415$	34.7666	0.00411235
$416$	3630.28	0.427859
$417$	1728.97	0.203041
$418$	−2085.37	−0.244016
$419$	10373.1	1.20945	0.604725	−	0.796434i	$-0.293283\pi$
$419$	10373.1	1.20945	0.604725	+	0.796434i	$0.293283\pi$
$420$	9.36243	0.00108771
$421$	4058.69	0.469853	0.234927	−	0.972013i	$-0.424515\pi$
$421$	4058.69	0.469853	0.234927	+	0.972013i	$0.424515\pi$
$422$	−17001.2	−1.96115
$423$	−4327.95	−0.497475
$424$	−3768.13	−0.431596
$425$	−3890.51	−0.444041
$426$	−2497.35	−0.284031
$427$	−2736.54	−0.310142
$428$	−2067.90	−0.233541
$429$	1566.86	0.176338
$430$	361.405	0.0405314
$431$	−13813.1	−1.54375	−0.771873	−	0.635777i	$-0.780680\pi$
$431$	−13813.1	−1.54375	−0.771873	+	0.635777i	$0.780680\pi$
$432$	1972.44	0.219673
$433$	−14065.7	−1.56110	−0.780551	−	0.625093i	$-0.785061\pi$
$433$	−14065.7	−1.56110	−0.780551	+	0.625093i	$0.785061\pi$
$434$	−2884.45	−0.319028
$435$	124.721	0.0137469
$436$	−2.15050	−0.000236216 0
$437$	−5668.37	−0.620492
$438$	−5600.51	−0.610965
$439$	15470.0	1.68187	0.840937	−	0.541133i	$-0.182004\pi$
$439$	15470.0	1.68187	0.840937	+	0.541133i	$0.182004\pi$
$440$	−120.538	−0.0130601
$441$	−2983.72	−0.322181
$442$	−5679.29	−0.611168
$443$	805.886	0.0864307	0.0432153	−	0.999066i	$-0.486240\pi$
$443$	805.886	0.0864307	0.0432153	+	0.999066i	$0.486240\pi$
$444$	1335.80	0.142780
$445$	123.132	0.0131169
$446$	6489.06	0.688937
$447$	5536.86	0.585871
$448$	−1346.35	−0.141985
$449$	15047.6	1.58160	0.790800	−	0.612074i	$-0.209665\pi$
$449$	15047.6	1.58160	0.790800	+	0.612074i	$0.209665\pi$
$450$	3430.07	0.359323
$451$	−3987.16	−0.416292
$452$	−788.323	−0.0820345
$453$	5872.91	0.609124
$454$	−10322.0	−1.06704
$455$	−135.901	−0.0140025
$456$	4722.29	0.484959
$457$	2132.43	0.218273	0.109137	−	0.994027i	$-0.465191\pi$
$457$	2132.43	0.218273	0.109137	+	0.994027i	$0.465191\pi$
$458$	−4728.26	−0.482395
$459$	−843.426	−0.0857685
$460$	67.3647	0.00682803
$461$	−7140.37	−0.721389	−0.360695	−	0.932684i	$-0.617460\pi$
$461$	−7140.37	−0.721389	−0.360695	+	0.932684i	$0.617460\pi$
$462$	273.390	0.0275308
$463$	10919.2	1.09602	0.548011	−	0.836471i	$-0.315385\pi$
$463$	10919.2	1.09602	0.548011	+	0.836471i	$0.315385\pi$
$464$	4497.72	0.450003
$465$	563.671	0.0562142
$466$	−14872.3	−1.47842
$467$	13946.8	1.38197	0.690987	−	0.722867i	$-0.257176\pi$
$467$	13946.8	1.38197	0.690987	+	0.722867i	$0.257176\pi$
$468$	729.511	0.0720549
$469$	437.268	0.0430515
$470$	993.671	0.0975205
$471$	−2374.87	−0.232332
$472$	1198.05	0.116832
$473$	1537.55	0.149464
$474$	8135.40	0.788336
$475$	9654.52	0.932589
$476$	−144.373	−0.0139019
$477$	−1670.11	−0.160313
$478$	−21289.6	−2.03716
$479$	18107.2	1.72723	0.863613	−	0.504155i	$-0.168196\pi$
$479$	18107.2	1.72723	0.863613	+	0.504155i	$0.168196\pi$
$480$	−123.781	−0.0117704
$481$	−19389.9	−1.83806
$482$	1419.53	0.134145
$483$	743.118	0.0700063
$484$	−1710.48	−0.160638
$485$	−913.908	−0.0855638
$486$	743.609	0.0694049
$487$	−15930.5	−1.48230	−0.741150	−	0.671340i	$-0.765719\pi$
$487$	−15930.5	−1.48230	−0.741150	+	0.671340i	$0.765719\pi$
$488$	16403.6	1.52163
$489$	10092.0	0.933284
$490$	685.044	0.0631574
$491$	2619.27	0.240745	0.120373	−	0.992729i	$-0.461591\pi$
$491$	2619.27	0.240745	0.120373	+	0.992729i	$0.461591\pi$
$492$	−1856.37	−0.170105
$493$	−1923.25	−0.175698
$494$	14093.5	1.28360
$495$	−53.4249	−0.00485105
$496$	20327.2	1.84016
$497$	921.522	0.0831709
$498$	472.669	0.0425318
$499$	−9350.73	−0.838870	−0.419435	−	0.907785i	$-0.637772\pi$
$499$	−9350.73	−0.838870	−0.419435	+	0.907785i	$0.637772\pi$
$500$	−229.895	−0.0205624
$501$	−1544.98	−0.137773
$502$	23105.1	2.05425
$503$	−7300.43	−0.647137	−0.323568	−	0.946205i	$-0.604883\pi$
$503$	−7300.43	−0.647137	−0.323568	+	0.946205i	$0.604883\pi$
$504$	−619.087	−0.0547149
$505$	−274.426	−0.0241817
$506$	1967.10	0.172822
$507$	−3998.29	−0.350238
$508$	397.927	0.0347542
$509$	16241.5	1.41432	0.707162	−	0.707052i	$-0.249975\pi$
$509$	16241.5	1.41432	0.707162	+	0.707052i	$0.249975\pi$
$510$	193.645	0.0168133
$511$	2066.59	0.178905
$512$	7403.50	0.639046
$513$	2093.01	0.180134
$514$	−211.379	−0.0181392
$515$	−205.685	−0.0175991
$516$	715.862	0.0610738
$517$	4227.43	0.359617
$518$	−3383.20	−0.286968
$519$	−7876.41	−0.666158
$520$	814.631	0.0686999
$521$	21182.1	1.78120	0.890598	−	0.454792i	$-0.150286\pi$
$521$	21182.1	1.78120	0.890598	+	0.454792i	$0.150286\pi$
$522$	1695.64	0.142177
$523$	19761.6	1.65222	0.826112	−	0.563505i	$-0.190548\pi$
$523$	19761.6	1.65222	0.826112	+	0.563505i	$0.190548\pi$
$524$	−1583.03	−0.131975
$525$	−1265.70	−0.105218
$526$	19545.7	1.62022
$527$	−8692.06	−0.718467
$528$	−1926.63	−0.158798
$529$	−6820.11	−0.560541
$530$	383.447	0.0314262
$531$	531.000	0.0433963
$532$	358.270	0.0291973
$533$	26946.3	2.18982
$534$	1674.04	0.135660
$535$	−1023.48	−0.0827079
$536$	−2621.11	−0.211221
$537$	497.134	0.0399496
$538$	4161.02	0.333447
$539$	2914.42	0.232900
$540$	−24.8740	−0.00198223
$541$	1805.49	0.143482	0.0717412	−	0.997423i	$-0.477144\pi$
$541$	1805.49	0.143482	0.0717412	+	0.997423i	$0.477144\pi$
$542$	9744.44	0.772250
$543$	−5041.40	−0.398429
$544$	1908.76	0.150436
$545$	−1.06436	−8.36552e−5 0
$546$	−1847.64	−0.144820
$547$	−2017.11	−0.157670	−0.0788349	−	0.996888i	$-0.525120\pi$
$547$	−2017.11	−0.157670	−0.0788349	+	0.996888i	$0.525120\pi$
$548$	779.599	0.0607716
$549$	7270.41	0.565198
$550$	−3350.41	−0.259749
$551$	4772.67	0.369006
$552$	−4454.46	−0.343468
$553$	−3001.96	−0.230843
$554$	−6385.53	−0.489703
$555$	661.134	0.0505650
$556$	−786.289	−0.0599750
$557$	22836.9	1.73722	0.868611	−	0.495495i	$-0.165013\pi$
$557$	22836.9	1.73722	0.868611	+	0.495495i	$0.165013\pi$
$558$	7663.37	0.581392
$559$	−10391.2	−0.786225
$560$	167.105	0.0126098
$561$	823.837	0.0620007
$562$	−20596.7	−1.54594
$563$	−11521.8	−0.862494	−0.431247	−	0.902234i	$-0.641926\pi$
$563$	−11521.8	−0.862494	−0.431247	+	0.902234i	$0.641926\pi$
$564$	1968.24	0.146946
$565$	−390.168	−0.0290522
$566$	−21350.7	−1.58558
$567$	−274.392	−0.0203234
$568$	−5523.87	−0.408057
$569$	−431.235	−0.0317721	−0.0158860	−	0.999874i	$-0.505057\pi$
$569$	−431.235	−0.0317721	−0.0158860	+	0.999874i	$0.505057\pi$
$570$	−480.543	−0.0353118
$571$	7544.48	0.552937	0.276468	−	0.961023i	$-0.410836\pi$
$571$	7544.48	0.552937	0.276468	+	0.961023i	$0.410836\pi$
$572$	−712.568	−0.0520874
$573$	2476.40	0.180547
$574$	4701.65	0.341887
$575$	−9106.97	−0.660499
$576$	3576.97	0.258751
$577$	−12507.5	−0.902416	−0.451208	−	0.892419i	$-0.649007\pi$
$577$	−12507.5	−0.902416	−0.451208	+	0.892419i	$0.649007\pi$
$578$	12048.3	0.867027
$579$	−9330.36	−0.669700
$580$	−56.7198	−0.00406063
$581$	−174.415	−0.0124543
$582$	−12425.0	−0.884938
$583$	1631.32	0.115887
$584$	−12387.7	−0.877753
$585$	361.061	0.0255180
$586$	−3591.31	−0.253167
$587$	12553.1	0.882659	0.441330	−	0.897345i	$-0.354507\pi$
$587$	12553.1	0.882659	0.441330	+	0.897345i	$0.354507\pi$
$588$	1356.92	0.0951673
$589$	21569.9	1.50895
$590$	−121.914	−0.00850701
$591$	3756.92	0.261487
$592$	23842.0	1.65524
$593$	−4198.56	−0.290749	−0.145374	−	0.989377i	$-0.546439\pi$
$593$	−4198.56	−0.290749	−0.145374	+	0.989377i	$0.546439\pi$
$594$	−726.338	−0.0501717
$595$	−71.4552	−0.00492332
$596$	−2518.02	−0.173057
$597$	−2563.18	−0.175719
$598$	−13294.2	−0.909097
$599$	−4991.05	−0.340449	−0.170224	−	0.985405i	$-0.554449\pi$
$599$	−4991.05	−0.340449	−0.170224	+	0.985405i	$0.554449\pi$
$600$	7586.96	0.516227
$601$	−8600.34	−0.583719	−0.291859	−	0.956461i	$-0.594274\pi$
$601$	−8600.34	−0.583719	−0.291859	+	0.956461i	$0.594274\pi$
$602$	−1813.07	−0.122750
$603$	−1161.73	−0.0784564
$604$	−2670.84	−0.179926
$605$	−846.574	−0.0568895
$606$	−3730.95	−0.250098
$607$	−17833.1	−1.19246	−0.596230	−	0.802814i	$-0.703335\pi$
$607$	−17833.1	−1.19246	−0.596230	+	0.802814i	$0.703335\pi$
$608$	−4736.70	−0.315951
$609$	−625.692	−0.0416327
$610$	−1669.24	−0.110796
$611$	−28570.2	−1.89169
$612$	383.568	0.0253347
$613$	−22215.0	−1.46371	−0.731856	−	0.681459i	$-0.761346\pi$
$613$	−22215.0	−1.46371	−0.731856	+	0.681459i	$0.761346\pi$
$614$	19700.3	1.29485
$615$	−918.782	−0.0602421
$616$	604.708	0.0395526
$617$	18251.1	1.19086	0.595432	−	0.803406i	$-0.296981\pi$
$617$	18251.1	1.19086	0.595432	+	0.803406i	$0.296981\pi$
$618$	−2796.39	−0.182018
$619$	9872.40	0.641042	0.320521	−	0.947241i	$-0.396142\pi$
$619$	9872.40	0.641042	0.320521	+	0.947241i	$0.396142\pi$
$620$	−256.343	−0.0166048
$621$	−1974.31	−0.127578
$622$	11218.1	0.723160
$623$	−617.719	−0.0397245
$624$	13020.7	0.835327
$625$	15454.2	0.989070
$626$	5494.92	0.350833
$627$	−2044.40	−0.130216
$628$	1080.03	0.0686271
$629$	−10195.0	−0.646265
$630$	62.9986	0.00398401
$631$	−9843.54	−0.621023	−0.310511	−	0.950570i	$-0.600500\pi$
$631$	−9843.54	−0.621023	−0.310511	+	0.950570i	$0.600500\pi$
$632$	17994.6	1.13258
$633$	−16667.2	−1.04654
$634$	−2429.15	−0.152167
$635$	196.948	0.0123081
$636$	759.522	0.0473538
$637$	−19696.5	−1.22512
$638$	−1656.26	−0.102777
$639$	−2448.29	−0.151569
$640$	−1151.33	−0.0711100
$641$	−16239.8	−1.00068	−0.500338	−	0.865830i	$-0.666791\pi$
$641$	−16239.8	−1.00068	−0.500338	+	0.865830i	$0.666791\pi$
$642$	−13914.6	−0.855401
$643$	−4102.04	−0.251584	−0.125792	−	0.992057i	$-0.540147\pi$
$643$	−4102.04	−0.251584	−0.125792	+	0.992057i	$0.540147\pi$
$644$	−337.951	−0.0206788
$645$	354.305	0.0216291
$646$	7410.19	0.451316
$647$	−29339.0	−1.78274	−0.891371	−	0.453274i	$-0.850256\pi$
$647$	−29339.0	−1.78274	−0.891371	+	0.453274i	$0.850256\pi$
$648$	1644.78	0.0997117
$649$	−518.667	−0.0313705
$650$	22643.0	1.36636
$651$	−2827.78	−0.170245
$652$	−4589.58	−0.275677
$653$	−22467.1	−1.34641	−0.673206	−	0.739455i	$-0.735083\pi$
$653$	−22467.1	−1.34641	−0.673206	+	0.739455i	$0.735083\pi$
$654$	−14.4705	−0.000865198 0
$655$	−783.494	−0.0467384
$656$	−33133.4	−1.97201
$657$	−5490.49	−0.326034
$658$	−4984.98	−0.295342
$659$	14150.3	0.836447	0.418223	−	0.908344i	$-0.362653\pi$
$659$	14150.3	0.836447	0.418223	+	0.908344i	$0.362653\pi$
$660$	24.2963	0.00143293
$661$	−10577.7	−0.622428	−0.311214	−	0.950340i	$-0.600736\pi$
$661$	−10577.7	−0.622428	−0.311214	+	0.950340i	$0.600736\pi$
$662$	−10167.4	−0.596932
$663$	−5567.72	−0.326142
$664$	1045.49	0.0611040
$665$	177.320	0.0103401
$666$	8988.44	0.522965
$667$	−4501.99	−0.261346
$668$	702.615	0.0406961
$669$	6361.58	0.367643
$670$	266.726	0.0153799
$671$	−7101.55	−0.408573
$672$	620.976	0.0356468
$673$	−9785.95	−0.560506	−0.280253	−	0.959926i	$-0.590418\pi$
$673$	−9785.95	−0.560506	−0.280253	+	0.959926i	$0.590418\pi$
$674$	−6402.85	−0.365918
$675$	3362.69	0.191748
$676$	1818.32	0.103455
$677$	−25327.9	−1.43786	−0.718929	−	0.695084i	$-0.755367\pi$
$677$	−25327.9	−1.43786	−0.718929	+	0.695084i	$0.755367\pi$
$678$	−5304.53	−0.300471
$679$	4584.83	0.259131
$680$	428.323	0.0241551
$681$	−10119.2	−0.569410
$682$	−7485.39	−0.420279
$683$	11813.3	0.661819	0.330909	−	0.943662i	$-0.392645\pi$
$683$	11813.3	0.661819	0.330909	+	0.943662i	$0.392645\pi$
$684$	−951.847	−0.0532087
$685$	385.851	0.0215220
$686$	−6992.32	−0.389166
$687$	−4635.37	−0.257424
$688$	12777.1	0.708024
$689$	−11024.9	−0.609602
$690$	453.289	0.0250093
$691$	10486.4	0.577309	0.288655	−	0.957433i	$-0.406792\pi$
$691$	10486.4	0.577309	0.288655	+	0.957433i	$0.406792\pi$
$692$	3581.99	0.196773
$693$	268.019	0.0146915
$694$	30354.1	1.66027
$695$	−389.162	−0.0212399
$696$	3750.58	0.204261
$697$	14168.0	0.769946
$698$	−4160.97	−0.225637
$699$	−14580.1	−0.788942
$700$	575.606	0.0310798
$701$	10841.7	0.584144	0.292072	−	0.956396i	$-0.405655\pi$
$701$	10841.7	0.584144	0.292072	+	0.956396i	$0.405655\pi$
$702$	4908.79	0.263918
$703$	25299.5	1.35731
$704$	−3493.89	−0.187047
$705$	974.150	0.0520406
$706$	3437.57	0.183250
$707$	1376.72	0.0732346
$708$	−241.485	−0.0128186
$709$	−15139.3	−0.801930	−0.400965	−	0.916093i	$-0.631325\pi$
$709$	−15139.3	−0.801930	−0.400965	+	0.916093i	$0.631325\pi$
$710$	562.112	0.0297123
$711$	7975.57	0.420685
$712$	3702.79	0.194899
$713$	−20346.5	−1.06870
$714$	−971.468	−0.0509191
$715$	−352.675	−0.0184466
$716$	−226.084	−0.0118005
$717$	−20871.4	−1.08711
$718$	−13582.8	−0.705994
$719$	−26507.9	−1.37494	−0.687468	−	0.726215i	$-0.741278\pi$
$719$	−26507.9	−1.37494	−0.687468	+	0.726215i	$0.741278\pi$
$720$	−443.963	−0.0229799
$721$	1031.87	0.0532992
$722$	2600.53	0.134047
$723$	1391.64	0.0715847
$724$	2292.70	0.117690
$725$	7667.90	0.392798
$726$	−11509.6	−0.588376
$727$	−4196.55	−0.214087	−0.107044	−	0.994254i	$-0.534138\pi$
$727$	−4196.55	−0.214087	−0.107044	+	0.994254i	$0.534138\pi$
$728$	−4086.79	−0.208058
$729$	729.000	0.0370370
$730$	1260.58	0.0639127
$731$	−5463.55	−0.276439
$732$	−3306.39	−0.166950
$733$	19609.3	0.988110	0.494055	−	0.869431i	$-0.335514\pi$
$733$	19609.3	0.988110	0.494055	+	0.869431i	$0.335514\pi$
$734$	37772.1	1.89945
$735$	671.586	0.0337032
$736$	4468.05	0.223770
$737$	1134.75	0.0567149
$738$	−12491.3	−0.623050
$739$	2865.95	0.142660	0.0713299	−	0.997453i	$-0.477276\pi$
$739$	2865.95	0.142660	0.0713299	+	0.997453i	$0.477276\pi$
$740$	−300.667	−0.0149361
$741$	13816.6	0.684975
$742$	−1923.65	−0.0951745
$743$	6771.12	0.334332	0.167166	−	0.985929i	$-0.446538\pi$
$743$	6771.12	0.334332	0.167166	+	0.985929i	$0.446538\pi$
$744$	16950.6	0.835266
$745$	−1246.26	−0.0612876
$746$	−25885.4	−1.27042
$747$	463.383	0.0226965
$748$	−374.659	−0.0183140
$749$	5134.50	0.250482
$750$	−1546.93	−0.0753146
$751$	−15205.4	−0.738819	−0.369409	−	0.929267i	$-0.620440\pi$
$751$	−15205.4	−0.738819	−0.369409	+	0.929267i	$0.620440\pi$
$752$	35130.1	1.70354
$753$	22651.2	1.09622
$754$	11193.5	0.540639
$755$	−1321.89	−0.0637201
$756$	124.786	0.00600321
$757$	11748.5	0.564077	0.282038	−	0.959403i	$-0.408989\pi$
$757$	11748.5	0.564077	0.282038	+	0.959403i	$0.408989\pi$
$758$	24179.0	1.15860
$759$	1928.45	0.0922244
$760$	−1062.91	−0.0507313
$761$	−33071.4	−1.57535	−0.787673	−	0.616094i	$-0.788714\pi$
$761$	−33071.4	−1.57535	−0.787673	+	0.616094i	$0.788714\pi$
$762$	2677.60	0.127296
$763$	5.33960	0.000253351 0
$764$	−1126.20	−0.0533306
$765$	189.841	0.00897219
$766$	−5433.43	−0.256290
$767$	3505.30	0.165018
$768$	−6114.34	−0.287282
$769$	32399.7	1.51933	0.759665	−	0.650315i	$-0.225363\pi$
$769$	32399.7	1.51933	0.759665	+	0.650315i	$0.225363\pi$
$770$	−61.5354	−0.00287998
$771$	−207.227	−0.00967974
$772$	4243.20	0.197819
$773$	−35967.3	−1.67355	−0.836774	−	0.547548i	$-0.815561\pi$
$773$	−35967.3	−1.67355	−0.836774	+	0.547548i	$0.815561\pi$
$774$	4816.95	0.223697
$775$	34654.7	1.60624
$776$	−27482.8	−1.27136
$777$	−3316.73	−0.153137
$778$	−39307.7	−1.81137
$779$	−35158.8	−1.61707
$780$	−164.201	−0.00753761
$781$	2391.43	0.109567
$782$	−6989.92	−0.319641
$783$	1662.33	0.0758708
$784$	24218.9	1.10327
$785$	534.543	0.0243040
$786$	−10652.0	−0.483389
$787$	−2130.77	−0.0965106	−0.0482553	−	0.998835i	$-0.515366\pi$
$787$	−2130.77	−0.0965106	−0.0482553	+	0.998835i	$0.515366\pi$
$788$	−1708.55	−0.0772392
$789$	19161.7	0.864607
$790$	−1831.14	−0.0824673
$791$	1957.37	0.0879849
$792$	−1606.58	−0.0720800
$793$	47994.3	2.14921
$794$	−11703.6	−0.523103
$795$	375.914	0.0167702
$796$	1165.67	0.0519046
$797$	25863.7	1.14949	0.574743	−	0.818334i	$-0.305102\pi$
$797$	25863.7	1.14949	0.574743	+	0.818334i	$0.305102\pi$
$798$	2410.75	0.106942
$799$	−15021.8	−0.665124
$800$	−7610.11	−0.336322
$801$	1641.15	0.0723934
$802$	30929.0	1.36177
$803$	5362.97	0.235685
$804$	528.323	0.0231748
$805$	−167.263	−0.00732331
$806$	50588.3	2.21079
$807$	4079.28	0.177940
$808$	−8252.46	−0.359307
$809$	26720.2	1.16123	0.580614	−	0.814179i	$-0.302813\pi$
$809$	26720.2	1.16123	0.580614	+	0.814179i	$0.302813\pi$
$810$	−167.374	−0.00726040
$811$	23248.5	1.00662	0.503308	−	0.864107i	$-0.332116\pi$
$811$	23248.5	1.00662	0.503308	+	0.864107i	$0.332116\pi$
$812$	284.548	0.0122976
$813$	9553.00	0.412101
$814$	−8779.68	−0.378044
$815$	−2271.54	−0.0976302
$816$	6846.11	0.293703
$817$	13558.1	0.580586
$818$	2643.60	0.112997
$819$	−1811.34	−0.0772814
$820$	417.838	0.0177946
$821$	7799.17	0.331538	0.165769	−	0.986165i	$-0.446989\pi$
$821$	7799.17	0.331538	0.165769	+	0.986165i	$0.446989\pi$
$822$	5245.83	0.222590
$823$	2499.10	0.105848	0.0529242	−	0.998599i	$-0.483146\pi$
$823$	2499.10	0.105848	0.0529242	+	0.998599i	$0.483146\pi$
$824$	−6185.31	−0.261499
$825$	−3284.59	−0.138612
$826$	611.612	0.0257636
$827$	−39871.3	−1.67649	−0.838246	−	0.545292i	$-0.816419\pi$
$827$	−39871.3	−1.67649	−0.838246	+	0.545292i	$0.816419\pi$
$828$	897.863	0.0376846
$829$	8721.76	0.365403	0.182702	−	0.983168i	$-0.441516\pi$
$829$	8721.76	0.365403	0.182702	+	0.983168i	$0.441516\pi$
$830$	−106.390	−0.00444922
$831$	−6260.08	−0.261324
$832$	23612.7	0.983922
$833$	−10356.2	−0.430756
$834$	−5290.84	−0.219673
$835$	347.749	0.0144124
$836$	929.739	0.0384638
$837$	7512.82	0.310252
$838$	−31742.9	−1.30852
$839$	−11850.6	−0.487637	−0.243818	−	0.969821i	$-0.578400\pi$
$839$	−11850.6	−0.487637	−0.243818	+	0.969821i	$0.578400\pi$
$840$	139.346	0.00572369
$841$	−20598.4	−0.844578
$842$	−12420.1	−0.508341
$843$	−20192.1	−0.824974
$844$	7579.80	0.309132
$845$	899.950	0.0366381
$846$	13244.0	0.538226
$847$	4247.04	0.172290
$848$	13556.3	0.548969
$849$	−20931.2	−0.846123
$850$	11905.4	0.480414
$851$	−23864.6	−0.961303
$852$	1113.42	0.0447712
$853$	−25857.7	−1.03793	−0.518963	−	0.854797i	$-0.673682\pi$
$853$	−25857.7	−1.03793	−0.518963	+	0.854797i	$0.673682\pi$
$854$	8374.14	0.335547
$855$	−471.102	−0.0188437
$856$	−30777.7	−1.22893
$857$	−13217.7	−0.526847	−0.263424	−	0.964680i	$-0.584852\pi$
$857$	−13217.7	−0.526847	−0.263424	+	0.964680i	$0.584852\pi$
$858$	−4794.78	−0.190782
$859$	45527.8	1.80837	0.904184	−	0.427142i	$-0.140480\pi$
$859$	45527.8	1.80837	0.904184	+	0.427142i	$0.140480\pi$
$860$	−161.129	−0.00638889
$861$	4609.28	0.182444
$862$	42269.8	1.67020
$863$	5044.12	0.198961	0.0994807	−	0.995039i	$-0.468282\pi$
$863$	5044.12	0.198961	0.0994807	+	0.995039i	$0.468282\pi$
$864$	−1649.80	−0.0649622
$865$	1772.85	0.0696864
$866$	43042.9	1.68898
$867$	11811.6	0.462678
$868$	1286.00	0.0502877
$869$	−7790.33	−0.304107
$870$	−381.661	−0.0148730
$871$	−7668.93	−0.298337
$872$	−32.0071	−0.00124300
$873$	−12180.9	−0.472236
$874$	17345.9	0.671320
$875$	570.818	0.0220539
$876$	2496.93	0.0963053
$877$	−16427.0	−0.632498	−0.316249	−	0.948676i	$-0.602424\pi$
$877$	−16427.0	−0.632498	−0.316249	+	0.948676i	$0.602424\pi$
$878$	−47340.0	−1.81965
$879$	−3520.75	−0.135099
$880$	433.651	0.0166118
$881$	49555.6	1.89508	0.947542	−	0.319631i	$-0.103559\pi$
$881$	49555.6	1.89508	0.947542	+	0.319631i	$0.103559\pi$
$882$	9130.54	0.348573
$883$	−10096.6	−0.384798	−0.192399	−	0.981317i	$-0.561627\pi$
$883$	−10096.6	−0.384798	−0.192399	+	0.981317i	$0.561627\pi$
$884$	2532.05	0.0963373
$885$	−119.519	−0.00453966
$886$	−2466.11	−0.0935106
$887$	17531.6	0.663647	0.331824	−	0.943341i	$-0.392336\pi$
$887$	17531.6	0.663647	0.331824	+	0.943341i	$0.392336\pi$
$888$	19881.5	0.751327
$889$	−988.035	−0.0372752
$890$	−376.798	−0.0141913
$891$	−712.068	−0.0267735
$892$	−2893.08	−0.108596
$893$	37277.6	1.39692
$894$	−16943.4	−0.633863
$895$	−111.897	−0.00417910
$896$	5775.93	0.215357
$897$	−13033.0	−0.485128
$898$	−46047.3	−1.71116
$899$	17131.4	0.635555
$900$	−1529.26	−0.0566394
$901$	−5796.77	−0.214338
$902$	12201.2	0.450393
$903$	−1777.45	−0.0655039
$904$	−11733.0	−0.431676
$905$	1134.74	0.0416794
$906$	−17971.8	−0.659021
$907$	14205.5	0.520052	0.260026	−	0.965602i	$-0.416269\pi$
$907$	14205.5	0.520052	0.260026	+	0.965602i	$0.416269\pi$
$908$	4601.94	0.168195
$909$	−3657.65	−0.133462
$910$	415.874	0.0151495
$911$	−6382.81	−0.232132	−0.116066	−	0.993242i	$-0.537028\pi$
$911$	−6382.81	−0.232132	−0.116066	+	0.993242i	$0.537028\pi$
$912$	−16989.0	−0.616845
$913$	−452.621	−0.0164070
$914$	−6525.48	−0.236153
$915$	−1636.45	−0.0591249
$916$	2108.04	0.0760390
$917$	3930.58	0.141548
$918$	2580.98	0.0927943
$919$	−4515.26	−0.162073	−0.0810363	−	0.996711i	$-0.525823\pi$
$919$	−4515.26	−0.162073	−0.0810363	+	0.996711i	$0.525823\pi$
$920$	1002.63	0.0359300
$921$	19313.3	0.690983
$922$	21850.4	0.780482
$923$	−16161.9	−0.576356
$924$	−121.888	−0.00433963
$925$	40646.8	1.44482
$926$	−33414.0	−1.18580
$927$	−2741.45	−0.0971316
$928$	−3762.02	−0.133076
$929$	−33860.3	−1.19583	−0.597913	−	0.801561i	$-0.704003\pi$
$929$	−33860.3	−1.19583	−0.597913	+	0.801561i	$0.704003\pi$
$930$	−1724.90	−0.0608190
$931$	25699.4	0.904688
$932$	6630.65	0.233041
$933$	10997.7	0.385905
$934$	−42678.9	−1.49518
$935$	−185.432	−0.00648586
$936$	10857.7	0.379162
$937$	−22085.0	−0.769996	−0.384998	−	0.922917i	$-0.625798\pi$
$937$	−22085.0	−0.769996	−0.384998	+	0.922917i	$0.625798\pi$
$938$	−1338.09	−0.0465780
$939$	5386.97	0.187217
$940$	−443.018	−0.0153720
$941$	−14767.9	−0.511603	−0.255801	−	0.966729i	$-0.582339\pi$
$941$	−14767.9	−0.511603	−0.255801	+	0.966729i	$0.582339\pi$
$942$	7267.38	0.251363
$943$	33164.8	1.14527
$944$	−4310.14	−0.148605
$945$	61.7610	0.00212602
$946$	−4705.07	−0.161707
$947$	39640.1	1.36022	0.680111	−	0.733109i	$-0.261932\pi$
$947$	39640.1	1.36022	0.680111	+	0.733109i	$0.261932\pi$
$948$	−3627.08	−0.124264
$949$	−36244.4	−1.23977
$950$	−29544.0	−1.00898
$951$	−2381.43	−0.0812020
$952$	−2148.78	−0.0731538
$953$	−10159.5	−0.345328	−0.172664	−	0.984981i	$-0.555237\pi$
$953$	−10159.5	−0.345328	−0.172664	+	0.984981i	$0.555237\pi$
$954$	5110.73	0.173445
$955$	−557.397	−0.0188869
$956$	9491.75	0.321114
$957$	−1623.72	−0.0548458
$958$	−55410.3	−1.86871
$959$	−1935.71	−0.0651797
$960$	−805.116	−0.0270677
$961$	47633.6	1.59892
$962$	59335.5	1.98862
$963$	−13641.3	−0.456474
$964$	−632.882	−0.0211450
$965$	2100.11	0.0700569
$966$	−2274.03	−0.0757408
$967$	2430.67	0.0808325	0.0404163	−	0.999183i	$-0.487132\pi$
$967$	2430.67	0.0808325	0.0404163	+	0.999183i	$0.487132\pi$
$968$	−25458.0	−0.845300
$969$	7264.61	0.240839
$970$	2796.67	0.0925728
$971$	7053.09	0.233105	0.116552	−	0.993185i	$-0.462816\pi$
$971$	7053.09	0.233105	0.116552	+	0.993185i	$0.462816\pi$
$972$	−331.530	−0.0109402
$973$	1952.32	0.0643253
$974$	48749.2	1.60372
$975$	22198.2	0.729139
$976$	−59014.1	−1.93545
$977$	36193.6	1.18519	0.592597	−	0.805499i	$-0.298103\pi$
$977$	36193.6	1.18519	0.592597	+	0.805499i	$0.298103\pi$
$978$	−30882.7	−1.00973
$979$	−1603.03	−0.0523321
$980$	−305.420	−0.00995538
$981$	−14.1862	−0.000461702 0
$982$	−8015.27	−0.260466
$983$	−12172.1	−0.394944	−0.197472	−	0.980309i	$-0.563273\pi$
$983$	−12172.1	−0.394944	−0.197472	+	0.980309i	$0.563273\pi$
$984$	−27629.4	−0.895115
$985$	−845.620	−0.0273540
$986$	5885.39	0.190090
$987$	−4887.05	−0.157605
$988$	−6283.44	−0.202331
$989$	−12789.2	−0.411195
$990$	163.487	0.00524843
$991$	37357.4	1.19747	0.598737	−	0.800946i	$-0.295670\pi$
$991$	37357.4	1.19747	0.598737	+	0.800946i	$0.295670\pi$
$992$	−17002.3	−0.544176
$993$	−9967.70	−0.318545
$994$	−2819.97	−0.0899838
$995$	576.930	0.0183818
$996$	−210.735	−0.00670420
$997$	−43827.1	−1.39220	−0.696098	−	0.717947i	$-0.745082\pi$
$997$	−43827.1	−1.39220	−0.696098	+	0.717947i	$0.745082\pi$
$998$	28614.3	0.907586
$999$	8811.86	0.279074

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.a.b.1.2	✓	7
3.2	odd	2		531.4.a.c.1.6		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.b.1.2	✓	7	1.1	even	1	trivial
531.4.a.c.1.6		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)

\(f(q)\)	\(=\)	\(q-3.06012 q^{2} -3.00000 q^{3} +1.36432 q^{4} +0.675250 q^{5} +9.18035 q^{6} -3.38755 q^{7} +20.3060 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-3.06012 q^{2} -3.00000 q^{3} +1.36432 q^{4} +0.675250 q^{5} +9.18035 q^{6} -3.38755 q^{7} +20.3060 q^{8} +9.00000 q^{9} -2.06635 q^{10} -8.79097 q^{11} -4.09296 q^{12} +59.4118 q^{13} +10.3663 q^{14} -2.02575 q^{15} -73.0532 q^{16} +31.2380 q^{17} -27.5411 q^{18} -77.5189 q^{19} +0.921258 q^{20} +10.1626 q^{21} +26.9014 q^{22} +73.1225 q^{23} -60.9179 q^{24} -124.544 q^{25} -181.807 q^{26} -27.0000 q^{27} -4.62171 q^{28} -61.5678 q^{29} +6.19904 q^{30} -278.253 q^{31} +61.1037 q^{32} +26.3729 q^{33} -95.5920 q^{34} -2.28744 q^{35} +12.2789 q^{36} -326.365 q^{37} +237.217 q^{38} -178.235 q^{39} +13.7116 q^{40} +453.551 q^{41} -31.0989 q^{42} -174.901 q^{43} -11.9937 q^{44} +6.07725 q^{45} -223.763 q^{46} -480.883 q^{47} +219.160 q^{48} -331.525 q^{49} +381.119 q^{50} -93.7140 q^{51} +81.0568 q^{52} -185.568 q^{53} +82.6232 q^{54} -5.93610 q^{55} -68.7874 q^{56} +232.557 q^{57} +188.405 q^{58} +59.0000 q^{59} -2.76378 q^{60} +807.823 q^{61} +851.486 q^{62} -30.4879 q^{63} +397.441 q^{64} +40.1178 q^{65} -80.7042 q^{66} -129.081 q^{67} +42.6187 q^{68} -219.367 q^{69} +6.99985 q^{70} -272.032 q^{71} +182.754 q^{72} -610.054 q^{73} +998.716 q^{74} +373.632 q^{75} -105.761 q^{76} +29.7798 q^{77} +545.422 q^{78} +886.175 q^{79} -49.3292 q^{80} +81.0000 q^{81} -1387.92 q^{82} +51.4871 q^{83} +13.8651 q^{84} +21.0935 q^{85} +535.217 q^{86} +184.703 q^{87} -178.509 q^{88} +182.350 q^{89} -18.5971 q^{90} -201.261 q^{91} +99.7625 q^{92} +834.758 q^{93} +1471.56 q^{94} -52.3447 q^{95} -183.311 q^{96} -1353.44 q^{97} +1014.50 q^{98} -79.1187 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 21 q^{3} + 26 q^{4} - 2 q^{5} - 59 q^{7} - 21 q^{8} + 63 q^{9}+O(q^{10}) \) 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