LMFDB - Embedded newform 169.4.a.k.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,4,Mod(1,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("169.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	169.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:	$9.97132279097$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 46x^{7} + 145x^{6} + 680x^{5} - 1501x^{4} - 3203x^{3} + 4784x^{2} + 3584x - 4096 $ x^9 - 4x^8 - 46x^7 + 145x^6 + 680x^5 - 1501x^4 - 3203x^3 + 4784x^2 + 3584x - 4096
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 13^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.39012$ of defining polynomial
Character	$\chi$	$=$	169.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.390115 q^{2} +3.60967 q^{3} -7.84781 q^{4} +7.52136 q^{5} +1.40819 q^{6} -19.5446 q^{7} -6.18247 q^{8} -13.9703 q^{9} +O(q^{10})$	\(q+0.390115 q^{2} +3.60967 q^{3} -7.84781 q^{4} +7.52136 q^{5} +1.40819 q^{6} -19.5446 q^{7} -6.18247 q^{8} -13.9703 q^{9} +2.93420 q^{10} -45.8243 q^{11} -28.3280 q^{12} -7.62463 q^{14} +27.1496 q^{15} +60.3706 q^{16} +86.5200 q^{17} -5.45003 q^{18} -148.737 q^{19} -59.0262 q^{20} -70.5494 q^{21} -17.8767 q^{22} -91.5351 q^{23} -22.3167 q^{24} -68.4292 q^{25} -147.889 q^{27} +153.382 q^{28} +258.901 q^{29} +10.5915 q^{30} -31.2317 q^{31} +73.0113 q^{32} -165.410 q^{33} +33.7528 q^{34} -147.002 q^{35} +109.636 q^{36} -148.436 q^{37} -58.0245 q^{38} -46.5006 q^{40} -95.9135 q^{41} -27.5224 q^{42} -80.9930 q^{43} +359.620 q^{44} -105.076 q^{45} -35.7093 q^{46} -94.3777 q^{47} +217.918 q^{48} +38.9898 q^{49} -26.6953 q^{50} +312.308 q^{51} +493.555 q^{53} -57.6938 q^{54} -344.661 q^{55} +120.834 q^{56} -536.891 q^{57} +101.001 q^{58} +575.686 q^{59} -213.065 q^{60} -40.2865 q^{61} -12.1840 q^{62} +273.043 q^{63} -454.482 q^{64} -64.5291 q^{66} -601.141 q^{67} -678.992 q^{68} -330.411 q^{69} -57.3476 q^{70} +518.800 q^{71} +86.3710 q^{72} +1055.21 q^{73} -57.9072 q^{74} -247.007 q^{75} +1167.26 q^{76} +895.615 q^{77} -320.840 q^{79} +454.069 q^{80} -156.633 q^{81} -37.4173 q^{82} +32.4841 q^{83} +553.658 q^{84} +650.748 q^{85} -31.5966 q^{86} +934.547 q^{87} +283.307 q^{88} -450.795 q^{89} -40.9916 q^{90} +718.350 q^{92} -112.736 q^{93} -36.8182 q^{94} -1118.70 q^{95} +263.546 q^{96} -231.743 q^{97} +15.2105 q^{98} +640.179 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 5 q^{2} + q^{3} + 37 q^{4} - 30 q^{5} - 48 q^{6} - 38 q^{7} - 60 q^{8} + 66 q^{9}+O(q^{10}) $ 9 * q - 5 * q^2 + q^3 + 37 * q^4 - 30 * q^5 - 48 * q^6 - 38 * q^7 - 60 * q^8 + 66 * q^9	\( 9 q - 5 q^{2} + q^{3} + 37 q^{4} - 30 q^{5} - 48 q^{6} - 38 q^{7} - 60 q^{8} + 66 q^{9} - 147 q^{10} - 181 q^{11} + 39 q^{12} - 147 q^{14} - 218 q^{15} + 269 q^{16} - 55 q^{17} - 79 q^{18} - 161 q^{19} - 370 q^{20} - 188 q^{21} + 340 q^{22} - 204 q^{23} - 798 q^{24} + 307 q^{25} - 668 q^{27} - 344 q^{28} + 280 q^{29} + 521 q^{30} - 706 q^{31} - 680 q^{32} - 500 q^{33} - 216 q^{34} + 20 q^{35} - 909 q^{36} - 298 q^{37} - 739 q^{38} + 13 q^{40} - 1201 q^{41} - 4 q^{42} - 533 q^{43} - 355 q^{44} + 90 q^{45} + 840 q^{46} - 956 q^{47} - 132 q^{48} + 403 q^{49} + 1156 q^{50} + 470 q^{51} - 278 q^{53} + 2555 q^{54} - 250 q^{55} + 250 q^{56} + 810 q^{57} + 2877 q^{58} - 1377 q^{59} + 3157 q^{60} - 136 q^{61} + 2035 q^{62} + 944 q^{63} + 284 q^{64} + 3279 q^{66} + 931 q^{67} - 1536 q^{68} - 2050 q^{69} + 4854 q^{70} - 2046 q^{71} + 4342 q^{72} + 45 q^{73} - 1990 q^{74} + 2393 q^{75} + 3608 q^{76} - 718 q^{77} + 412 q^{79} + 787 q^{80} - 835 q^{81} + 2757 q^{82} - 3709 q^{83} + 1539 q^{84} + 2106 q^{85} - 125 q^{86} - 786 q^{87} - 636 q^{88} - 1663 q^{89} - 1280 q^{90} + 4010 q^{92} + 1186 q^{93} - 2531 q^{94} - 1614 q^{95} + 3084 q^{96} + 1087 q^{97} + 282 q^{98} - 1357 q^{99}+O(q^{100}) \) 9 * q - 5 * q^2 + q^3 + 37 * q^4 - 30 * q^5 - 48 * q^6 - 38 * q^7 - 60 * q^8 + 66 * q^9 - 147 * q^10 - 181 * q^11 + 39 * q^12 - 147 * q^14 - 218 * q^15 + 269 * q^16 - 55 * q^17 - 79 * q^18 - 161 * q^19 - 370 * q^20 - 188 * q^21 + 340 * q^22 - 204 * q^23 - 798 * q^24 + 307 * q^25 - 668 * q^27 - 344 * q^28 + 280 * q^29 + 521 * q^30 - 706 * q^31 - 680 * q^32 - 500 * q^33 - 216 * q^34 + 20 * q^35 - 909 * q^36 - 298 * q^37 - 739 * q^38 + 13 * q^40 - 1201 * q^41 - 4 * q^42 - 533 * q^43 - 355 * q^44 + 90 * q^45 + 840 * q^46 - 956 * q^47 - 132 * q^48 + 403 * q^49 + 1156 * q^50 + 470 * q^51 - 278 * q^53 + 2555 * q^54 - 250 * q^55 + 250 * q^56 + 810 * q^57 + 2877 * q^58 - 1377 * q^59 + 3157 * q^60 - 136 * q^61 + 2035 * q^62 + 944 * q^63 + 284 * q^64 + 3279 * q^66 + 931 * q^67 - 1536 * q^68 - 2050 * q^69 + 4854 * q^70 - 2046 * q^71 + 4342 * q^72 + 45 * q^73 - 1990 * q^74 + 2393 * q^75 + 3608 * q^76 - 718 * q^77 + 412 * q^79 + 787 * q^80 - 835 * q^81 + 2757 * q^82 - 3709 * q^83 + 1539 * q^84 + 2106 * q^85 - 125 * q^86 - 786 * q^87 - 636 * q^88 - 1663 * q^89 - 1280 * q^90 + 4010 * q^92 + 1186 * q^93 - 2531 * q^94 - 1614 * q^95 + 3084 * q^96 + 1087 * q^97 + 282 * q^98 - 1357 * 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 $

$2$	0.390115	0.137927	0.0689633	−	0.997619i	$-0.478031\pi$
$2$	0.390115	0.137927	0.0689633	+	0.997619i	$0.478031\pi$
$3$	3.60967	0.694681	0.347340	−	0.937739i	$-0.387085\pi$
$3$	3.60967	0.694681	0.347340	+	0.937739i	$0.387085\pi$
$4$	−7.84781	−0.980976
$5$	7.52136	0.672731	0.336365	−	0.941732i	$-0.390802\pi$
$5$	7.52136	0.672731	0.336365	+	0.941732i	$0.390802\pi$
$6$	1.40819	0.0958150
$7$	−19.5446	−1.05531	−0.527654	−	0.849460i	$-0.676928\pi$
$7$	−19.5446	−1.05531	−0.527654	+	0.849460i	$0.676928\pi$
$8$	−6.18247	−0.273229
$9$	−13.9703	−0.517418
$10$	2.93420	0.0927875
$11$	−45.8243	−1.25605	−0.628024	−	0.778194i	$-0.716136\pi$
$11$	−45.8243	−1.25605	−0.628024	+	0.778194i	$0.716136\pi$
$12$	−28.3280	−0.681465
$13$	0	0
$13$	0	0
$14$	−7.62463	−0.145555
$15$	27.1496	0.467333
$16$	60.3706	0.943291
$17$	86.5200	1.23436	0.617182	−	0.786821i	$-0.288274\pi$
$17$	86.5200	1.23436	0.617182	+	0.786821i	$0.288274\pi$
$18$	−5.45003	−0.0713658
$19$	−148.737	−1.79593	−0.897963	−	0.440072i	$-0.854953\pi$
$19$	−148.737	−1.79593	−0.897963	+	0.440072i	$0.854953\pi$
$20$	−59.0262	−0.659933
$21$	−70.5494	−0.733102
$22$	−17.8767	−0.173242
$23$	−91.5351	−0.829843	−0.414922	−	0.909857i	$-0.636191\pi$
$23$	−91.5351	−0.829843	−0.414922	+	0.909857i	$0.636191\pi$
$24$	−22.3167	−0.189807
$25$	−68.4292	−0.547433
$26$	0	0
$27$	−147.889	−1.05412
$28$	153.382	1.03523
$29$	258.901	1.65782	0.828909	−	0.559383i	$-0.188962\pi$
$29$	258.901	1.65782	0.828909	+	0.559383i	$0.188962\pi$
$30$	10.5915	0.0644577
$31$	−31.2317	−0.180948	−0.0904739	−	0.995899i	$-0.528838\pi$
$31$	−31.2317	−0.180948	−0.0904739	+	0.995899i	$0.528838\pi$
$32$	73.0113	0.403334
$33$	−165.410	−0.872553
$34$	33.7528	0.170252
$35$	−147.002	−0.709938
$36$	109.636	0.507575
$37$	−148.436	−0.659533	−0.329767	−	0.944062i	$-0.606970\pi$
$37$	−148.436	−0.659533	−0.329767	+	0.944062i	$0.606970\pi$
$38$	−58.0245	−0.247706
$39$	0	0
$40$	−46.5006	−0.183810
$41$	−95.9135	−0.365346	−0.182673	−	0.983174i	$-0.558475\pi$
$41$	−95.9135	−0.365346	−0.182673	+	0.983174i	$0.558475\pi$
$42$	−27.5224	−0.101114
$43$	−80.9930	−0.287240	−0.143620	−	0.989633i	$-0.545874\pi$
$43$	−80.9930	−0.287240	−0.143620	+	0.989633i	$0.545874\pi$
$44$	359.620	1.23215
$45$	−105.076	−0.348083
$46$	−35.7093	−0.114457
$47$	−94.3777	−0.292902	−0.146451	−	0.989218i	$-0.546785\pi$
$47$	−94.3777	−0.292902	−0.146451	+	0.989218i	$0.546785\pi$
$48$	217.918	0.655286
$49$	38.9898	0.113673
$50$	−26.6953	−0.0755056
$51$	312.308	0.857489
$52$	0	0
$53$	493.555	1.27915	0.639575	−	0.768729i	$-0.279110\pi$
$53$	493.555	1.27915	0.639575	+	0.768729i	$0.279110\pi$
$54$	−57.6938	−0.145391
$55$	−344.661	−0.844983
$56$	120.834	0.288341
$57$	−536.891	−1.24759
$58$	101.001	0.228657
$59$	575.686	1.27030	0.635152	−	0.772387i	$-0.280938\pi$
$59$	575.686	1.27030	0.635152	+	0.772387i	$0.280938\pi$
$60$	−213.065	−0.458443
$61$	−40.2865	−0.0845599	−0.0422800	−	0.999106i	$-0.513462\pi$
$61$	−40.2865	−0.0845599	−0.0422800	+	0.999106i	$0.513462\pi$
$62$	−12.1840	−0.0249575
$63$	273.043	0.546035
$64$	−454.482	−0.887660
$65$	0	0
$66$	−64.5291	−0.120348
$67$	−601.141	−1.09613	−0.548067	−	0.836434i	$-0.684636\pi$
$67$	−601.141	−1.09613	−0.548067	+	0.836434i	$0.684636\pi$
$68$	−678.992	−1.21088
$69$	−330.411	−0.576476
$70$	−57.3476	−0.0979193
$71$	518.800	0.867187	0.433593	−	0.901109i	$-0.357245\pi$
$71$	518.800	0.867187	0.433593	+	0.901109i	$0.357245\pi$
$72$	86.3710	0.141374
$73$	1055.21	1.69182	0.845908	−	0.533328i	$-0.179059\pi$
$73$	1055.21	1.69182	0.845908	+	0.533328i	$0.179059\pi$
$74$	−57.9072	−0.0909672
$75$	−247.007	−0.380291
$76$	1167.26	1.76176
$77$	895.615	1.32552
$78$	0	0
$79$	−320.840	−0.456928	−0.228464	−	0.973552i	$-0.573370\pi$
$79$	−320.840	−0.456928	−0.228464	+	0.973552i	$0.573370\pi$
$80$	454.069	0.634581
$81$	−156.633	−0.214860
$82$	−37.4173	−0.0503909
$83$	32.4841	0.0429590	0.0214795	−	0.999769i	$-0.493162\pi$
$83$	32.4841	0.0429590	0.0214795	+	0.999769i	$0.493162\pi$
$84$	553.658	0.719155
$85$	650.748	0.830394
$86$	−31.5966	−0.0396180
$87$	934.547	1.15165
$88$	283.307	0.343189
$89$	−450.795	−0.536901	−0.268450	−	0.963293i	$-0.586512\pi$
$89$	−450.795	−0.536901	−0.268450	+	0.963293i	$0.586512\pi$
$90$	−40.9916	−0.0480099
$91$	0	0
$92$	718.350	0.814057
$93$	−112.736	−0.125701
$94$	−36.8182	−0.0403990
$95$	−1118.70	−1.20817
$96$	263.546	0.280189
$97$	−231.743	−0.242576	−0.121288	−	0.992617i	$-0.538702\pi$
$97$	−231.743	−0.242576	−0.121288	+	0.992617i	$0.538702\pi$
$98$	15.2105	0.0156785
$99$	640.179	0.649903
$100$	537.019	0.537019
$101$	570.125	0.561679	0.280840	−	0.959755i	$-0.409387\pi$
$101$	570.125	0.561679	0.280840	+	0.959755i	$0.409387\pi$
$102$	121.836	0.118270
$103$	969.551	0.927502	0.463751	−	0.885965i	$-0.346503\pi$
$103$	969.551	0.927502	0.463751	+	0.885965i	$0.346503\pi$
$104$	0	0
$105$	−530.627	−0.493180
$106$	192.543	0.176429
$107$	−343.156	−0.310039	−0.155019	−	0.987911i	$-0.549544\pi$
$107$	−343.156	−0.310039	−0.155019	+	0.987911i	$0.549544\pi$
$108$	1160.61	1.03407
$109$	83.1640	0.0730795	0.0365398	−	0.999332i	$-0.488366\pi$
$109$	83.1640	0.0730795	0.0365398	+	0.999332i	$0.488366\pi$
$110$	−134.457	−0.116546
$111$	−535.805	−0.458165
$112$	−1179.92	−0.995461
$113$	−2116.18	−1.76171	−0.880856	−	0.473384i	$-0.843032\pi$
$113$	−2116.18	−1.76171	−0.880856	+	0.473384i	$0.843032\pi$
$114$	−209.449	−0.172077
$115$	−688.469	−0.558261
$116$	−2031.81	−1.62628
$117$	0	0
$118$	224.584	0.175209
$119$	−1691.00	−1.30263
$120$	−167.852	−0.127689
$121$	768.863	0.577658
$122$	−15.7164	−0.0116631
$123$	−346.216	−0.253799
$124$	245.101	0.177506
$125$	−1454.85	−1.04101
$126$	106.518	0.0753128
$127$	−1176.69	−0.822161	−0.411081	−	0.911599i	$-0.634848\pi$
$127$	−1176.69	−0.822161	−0.411081	+	0.911599i	$0.634848\pi$
$128$	−761.391	−0.525766
$129$	−292.358	−0.199540
$130$	0	0
$131$	775.336	0.517110	0.258555	−	0.965996i	$-0.416754\pi$
$131$	775.336	0.517110	0.258555	+	0.965996i	$0.416754\pi$
$132$	1298.11	0.855954
$133$	2907.00	1.89525
$134$	−234.514	−0.151186
$135$	−1112.33	−0.709140
$136$	−534.907	−0.337264
$137$	−2542.62	−1.58563	−0.792814	−	0.609464i	$-0.791385\pi$
$137$	−2542.62	−1.58563	−0.792814	+	0.609464i	$0.791385\pi$
$138$	−128.899	−0.0795114
$139$	−286.317	−0.174713	−0.0873564	−	0.996177i	$-0.527842\pi$
$139$	−286.317	−0.174713	−0.0873564	+	0.996177i	$0.527842\pi$
$140$	1153.64	0.696432
$141$	−340.672	−0.203473
$142$	202.392	0.119608
$143$	0	0
$144$	−843.395	−0.488076
$145$	1947.29	1.11527
$146$	411.652	0.233346
$147$	140.740	0.0789664
$148$	1164.90	0.646987
$149$	−2354.13	−1.29435	−0.647173	−	0.762343i	$-0.724049\pi$
$149$	−2354.13	−1.29435	−0.647173	+	0.762343i	$0.724049\pi$
$150$	−96.3610	−0.0524523
$151$	165.158	0.0890089	0.0445045	−	0.999009i	$-0.485829\pi$
$151$	165.158	0.0890089	0.0445045	+	0.999009i	$0.485829\pi$
$152$	919.562	0.490699
$153$	−1208.71	−0.638682
$154$	349.393	0.182824
$155$	−234.905	−0.121729
$156$	0	0
$157$	−3095.72	−1.57367	−0.786833	−	0.617166i	$-0.788281\pi$
$157$	−3095.72	−1.57367	−0.786833	+	0.617166i	$0.788281\pi$
$158$	−125.165	−0.0630226
$159$	1781.57	0.888601
$160$	549.144	0.271335
$161$	1789.01	0.875740
$162$	−61.1048	−0.0296349
$163$	299.310	0.143827	0.0719134	−	0.997411i	$-0.477089\pi$
$163$	299.310	0.143827	0.0719134	+	0.997411i	$0.477089\pi$
$164$	752.711	0.358395
$165$	−1244.11	−0.586993
$166$	12.6726	0.00592519
$167$	−3005.17	−1.39250	−0.696249	−	0.717801i	$-0.745149\pi$
$167$	−3005.17	−1.39250	−0.696249	+	0.717801i	$0.745149\pi$
$168$	436.170	0.200305
$169$	0	0
$170$	253.867	0.114533
$171$	2077.90	0.929245
$172$	635.618	0.281776
$173$	455.853	0.200334	0.100167	−	0.994971i	$-0.468062\pi$
$173$	455.853	0.200334	0.100167	+	0.994971i	$0.468062\pi$
$174$	364.581	0.158844
$175$	1337.42	0.577710
$176$	−2766.44	−1.18482
$177$	2078.03	0.882455
$178$	−175.862	−0.0740529
$179$	1364.19	0.569635	0.284818	−	0.958582i	$-0.408067\pi$
$179$	1364.19	0.569635	0.284818	+	0.958582i	$0.408067\pi$
$180$	824.614	0.341461
$181$	−2026.11	−0.832041	−0.416021	−	0.909355i	$-0.636576\pi$
$181$	−2026.11	−0.832041	−0.416021	+	0.909355i	$0.636576\pi$
$182$	0	0
$183$	−145.421	−0.0587422
$184$	565.914	0.226738
$185$	−1116.44	−0.443688
$186$	−43.9801	−0.0173375
$187$	−3964.71	−1.55042
$188$	740.658	0.287330
$189$	2890.43	1.11242
$190$	−436.423	−0.166639
$191$	2161.06	0.818686	0.409343	−	0.912380i	$-0.365758\pi$
$191$	2161.06	0.818686	0.409343	+	0.912380i	$0.365758\pi$
$192$	−1640.53	−0.616641
$193$	1207.88	0.450491	0.225246	−	0.974302i	$-0.427682\pi$
$193$	1207.88	0.450491	0.225246	+	0.974302i	$0.427682\pi$
$194$	−90.4063	−0.0334577
$195$	0	0
$196$	−305.985	−0.111510
$197$	−4926.76	−1.78181	−0.890906	−	0.454187i	$-0.849930\pi$
$197$	−4926.76	−1.78181	−0.890906	+	0.454187i	$0.849930\pi$
$198$	249.743	0.0896389
$199$	1009.45	0.359589	0.179795	−	0.983704i	$-0.442457\pi$
$199$	1009.45	0.359589	0.179795	+	0.983704i	$0.442457\pi$
$200$	423.061	0.149575
$201$	−2169.92	−0.761464
$202$	222.415	0.0774705
$203$	−5060.11	−1.74951
$204$	−2450.94	−0.841176
$205$	−721.400	−0.245779
$206$	378.237	0.127927
$207$	1278.77	0.429376
$208$	0	0
$209$	6815.76	2.25577
$210$	−207.006	−0.0680226
$211$	−4911.32	−1.60241	−0.801206	−	0.598388i	$-0.795808\pi$
$211$	−4911.32	−1.60241	−0.801206	+	0.598388i	$0.795808\pi$
$212$	−3873.32	−1.25482
$213$	1872.70	0.602418
$214$	−133.871	−0.0427626
$215$	−609.178	−0.193235
$216$	914.321	0.288017
$217$	610.410	0.190956
$218$	32.4435	0.0100796
$219$	3808.95	1.17527
$220$	2704.83	0.828908
$221$	0	0
$222$	−209.026	−0.0631932
$223$	1364.58	0.409771	0.204885	−	0.978786i	$-0.434318\pi$
$223$	1364.58	0.409771	0.204885	+	0.978786i	$0.434318\pi$
$224$	−1426.97	−0.425641
$225$	955.976	0.283252
$226$	−825.554	−0.242987
$227$	4169.88	1.21923	0.609614	−	0.792699i	$-0.291325\pi$
$227$	4169.88	1.21923	0.609614	+	0.792699i	$0.291325\pi$
$228$	4213.42	1.22386
$229$	3506.89	1.01197	0.505987	−	0.862541i	$-0.331128\pi$
$229$	3506.89	1.01197	0.505987	+	0.862541i	$0.331128\pi$
$230$	−268.582	−0.0769991
$231$	3232.87	0.920811
$232$	−1600.65	−0.452965
$233$	−570.253	−0.160337	−0.0801684	−	0.996781i	$-0.525546\pi$
$233$	−570.253	−0.160337	−0.0801684	+	0.996781i	$0.525546\pi$
$234$	0	0
$235$	−709.848	−0.197044
$236$	−4517.87	−1.24614
$237$	−1158.13	−0.317419
$238$	−659.683	−0.179668
$239$	231.056	0.0625347	0.0312674	−	0.999511i	$-0.490046\pi$
$239$	231.056	0.0625347	0.0312674	+	0.999511i	$0.490046\pi$
$240$	1639.04	0.440831
$241$	3088.50	0.825508	0.412754	−	0.910842i	$-0.364567\pi$
$241$	3088.50	0.825508	0.412754	+	0.910842i	$0.364567\pi$
$242$	299.945	0.0796744
$243$	3427.62	0.904863
$244$	316.161	0.0829513
$245$	293.256	0.0764713
$246$	−135.064	−0.0350056
$247$	0	0
$248$	193.089	0.0494403
$249$	117.257	0.0298428
$250$	−567.559	−0.143582
$251$	−2445.64	−0.615009	−0.307504	−	0.951547i	$-0.599494\pi$
$251$	−2445.64	−0.615009	−0.307504	+	0.951547i	$0.599494\pi$
$252$	−2142.79	−0.535648
$253$	4194.53	1.04232
$254$	−459.045	−0.113398
$255$	2348.98	0.576859
$256$	3338.83	0.815143
$257$	3273.99	0.794654	0.397327	−	0.917677i	$-0.369938\pi$
$257$	3273.99	0.794654	0.397327	+	0.917677i	$0.369938\pi$
$258$	−114.053	−0.0275219
$259$	2901.12	0.696010
$260$	0	0
$261$	−3616.93	−0.857786
$262$	302.471	0.0713233
$263$	−4631.27	−1.08584	−0.542921	−	0.839784i	$-0.682682\pi$
$263$	−4631.27	−1.08584	−0.542921	+	0.839784i	$0.682682\pi$
$264$	1022.65	0.238407
$265$	3712.20	0.860524
$266$	1134.06	0.261406
$267$	−1627.22	−0.372975
$268$	4717.64	1.07528
$269$	−2838.51	−0.643373	−0.321686	−	0.946846i	$-0.604250\pi$
$269$	−2838.51	−0.643373	−0.321686	+	0.946846i	$0.604250\pi$
$270$	−433.936	−0.0978093
$271$	7052.31	1.58080	0.790401	−	0.612589i	$-0.209872\pi$
$271$	7052.31	1.58080	0.790401	+	0.612589i	$0.209872\pi$
$272$	5223.26	1.16436
$273$	0	0
$274$	−991.917	−0.218700
$275$	3135.72	0.687603
$276$	2593.01	0.565510
$277$	−1938.31	−0.420439	−0.210220	−	0.977654i	$-0.567418\pi$
$277$	−1938.31	−0.420439	−0.210220	+	0.977654i	$0.567418\pi$
$278$	−111.697	−0.0240975
$279$	436.317	0.0936258
$280$	908.834	0.193976
$281$	−3290.74	−0.698609	−0.349305	−	0.937009i	$-0.613582\pi$
$281$	−3290.74	−0.698609	−0.349305	+	0.937009i	$0.613582\pi$
$282$	−132.901	−0.0280644
$283$	−7918.08	−1.66318	−0.831592	−	0.555388i	$-0.812570\pi$
$283$	−7918.08	−1.66318	−0.831592	+	0.555388i	$0.812570\pi$
$284$	−4071.45	−0.850690
$285$	−4038.15	−0.839296
$286$	0	0
$287$	1874.59	0.385552
$288$	−1019.99	−0.208693
$289$	2572.71	0.523653
$290$	759.667	0.153825
$291$	−836.514	−0.168513
$292$	−8281.06	−1.65963
$293$	5675.43	1.13161	0.565806	−	0.824539i	$-0.308565\pi$
$293$	5675.43	1.13161	0.565806	+	0.824539i	$0.308565\pi$
$294$	54.9049	0.0108916
$295$	4329.94	0.854572
$296$	917.702	0.180204
$297$	6776.91	1.32403
$298$	−918.381	−0.178525
$299$	0	0
$300$	1938.46	0.373057
$301$	1582.97	0.303126
$302$	64.4306	0.0122767
$303$	2057.96	0.390188
$304$	−8979.33	−1.69408
$305$	−303.009	−0.0568861
$306$	−471.536	−0.0880913
$307$	−4338.86	−0.806618	−0.403309	−	0.915064i	$-0.632140\pi$
$307$	−4338.86	−0.806618	−0.403309	+	0.915064i	$0.632140\pi$
$308$	−7028.62	−1.30030
$309$	3499.76	0.644318
$310$	−91.6401	−0.0167897
$311$	−5234.75	−0.954454	−0.477227	−	0.878780i	$-0.658358\pi$
$311$	−5234.75	−0.954454	−0.477227	+	0.878780i	$0.658358\pi$
$312$	0	0
$313$	2167.86	0.391484	0.195742	−	0.980655i	$-0.437288\pi$
$313$	2167.86	0.391484	0.195742	+	0.980655i	$0.437288\pi$
$314$	−1207.69	−0.217050
$315$	2053.66	0.367335
$316$	2517.89	0.448236
$317$	4863.71	0.861744	0.430872	−	0.902413i	$-0.358206\pi$
$317$	4863.71	0.861744	0.430872	+	0.902413i	$0.358206\pi$
$318$	695.017	0.122562
$319$	−11864.0	−2.08230
$320$	−3418.32	−0.597156
$321$	−1238.68	−0.215378
$322$	697.922	0.120788
$323$	−12868.7	−2.21682
$324$	1229.22	0.210772
$325$	0	0
$326$	116.765	0.0198375
$327$	300.194	0.0507669
$328$	592.983	0.0998231
$329$	1844.57	0.309102
$330$	−485.347	−0.0809620
$331$	−2685.26	−0.445908	−0.222954	−	0.974829i	$-0.571570\pi$
$331$	−2685.26	−0.445908	−0.222954	+	0.974829i	$0.571570\pi$
$332$	−254.929	−0.0421417
$333$	2073.70	0.341255
$334$	−1172.36	−0.192062
$335$	−4521.39	−0.737404
$336$	−4259.11	−0.691528
$337$	−6518.36	−1.05364	−0.526821	−	0.849976i	$-0.676616\pi$
$337$	−6518.36	−1.05364	−0.526821	+	0.849976i	$0.676616\pi$
$338$	0	0
$339$	−7638.71	−1.22383
$340$	−5106.95	−0.814597
$341$	1431.17	0.227279
$342$	810.620	0.128168
$343$	5941.75	0.935347
$344$	500.737	0.0784824
$345$	−2485.14	−0.387813
$346$	177.835	0.0276314
$347$	75.7963	0.0117261	0.00586305	−	0.999983i	$-0.498134\pi$
$347$	75.7963	0.0117261	0.00586305	+	0.999983i	$0.498134\pi$
$348$	−7334.15	−1.12975
$349$	−3682.09	−0.564750	−0.282375	−	0.959304i	$-0.591122\pi$
$349$	−3682.09	−0.564750	−0.282375	+	0.959304i	$0.591122\pi$
$350$	521.747	0.0796816
$351$	0	0
$352$	−3345.69	−0.506607
$353$	10031.5	1.51254	0.756268	−	0.654261i	$-0.227020\pi$
$353$	10031.5	1.51254	0.756268	+	0.654261i	$0.227020\pi$
$354$	810.673	0.121714
$355$	3902.08	0.583383
$356$	3537.75	0.526687
$357$	−6103.93	−0.904914
$358$	532.193	0.0785678
$359$	−6869.76	−1.00995	−0.504975	−	0.863134i	$-0.668498\pi$
$359$	−6869.76	−1.00995	−0.504975	+	0.863134i	$0.668498\pi$
$360$	649.627	0.0951066
$361$	15263.7	2.22535
$362$	−790.416	−0.114761
$363$	2775.34	0.401288
$364$	0	0
$365$	7936.59	1.13814
$366$	−56.7309	−0.00810211
$367$	8883.29	1.26350	0.631749	−	0.775173i	$-0.282337\pi$
$367$	8883.29	1.26350	0.631749	+	0.775173i	$0.282337\pi$
$368$	−5526.03	−0.782784
$369$	1339.94	0.189037
$370$	−435.541	−0.0611964
$371$	−9646.31	−1.34990
$372$	884.732	0.123310
$373$	−5454.50	−0.757166	−0.378583	−	0.925567i	$-0.623589\pi$
$373$	−5454.50	−0.757166	−0.378583	+	0.925567i	$0.623589\pi$
$374$	−1546.70	−0.213844
$375$	−5251.53	−0.723167
$376$	583.487	0.0800294
$377$	0	0
$378$	1127.60	0.153433
$379$	6642.15	0.900222	0.450111	−	0.892973i	$-0.351384\pi$
$379$	6642.15	0.900222	0.450111	+	0.892973i	$0.351384\pi$
$380$	8779.37	1.18519
$381$	−4247.46	−0.571140
$382$	843.064	0.112919
$383$	1262.33	0.168413	0.0842066	−	0.996448i	$-0.473164\pi$
$383$	1262.33	0.168413	0.0842066	+	0.996448i	$0.473164\pi$
$384$	−2748.37	−0.365240
$385$	6736.24	0.891716
$386$	471.211	0.0621347
$387$	1131.50	0.148623
$388$	1818.67	0.237962
$389$	−2793.42	−0.364093	−0.182046	−	0.983290i	$-0.558272\pi$
$389$	−2793.42	−0.364093	−0.182046	+	0.983290i	$0.558272\pi$
$390$	0	0
$391$	−7919.62	−1.02433
$392$	−241.053	−0.0310588
$393$	2798.71	0.359227
$394$	−1922.00	−0.245759
$395$	−2413.15	−0.307390
$396$	−5024.00	−0.637539
$397$	6848.91	0.865836	0.432918	−	0.901433i	$-0.357484\pi$
$397$	6848.91	0.865836	0.432918	+	0.901433i	$0.357484\pi$
$398$	393.803	0.0495969
$399$	10493.3	1.31660
$400$	−4131.11	−0.516389
$401$	11024.5	1.37291	0.686456	−	0.727171i	$-0.259165\pi$
$401$	11024.5	1.37291	0.686456	+	0.727171i	$0.259165\pi$
$402$	−846.518	−0.105026
$403$	0	0
$404$	−4474.24	−0.550994
$405$	−1178.09	−0.144543
$406$	−1974.03	−0.241304
$407$	6801.97	0.828406
$408$	−1930.84	−0.234291
$409$	3530.56	0.426833	0.213417	−	0.976961i	$-0.431541\pi$
$409$	3530.56	0.426833	0.213417	+	0.976961i	$0.431541\pi$
$410$	−281.429	−0.0338995
$411$	−9178.03	−1.10151
$412$	−7608.86	−0.909858
$413$	−11251.5	−1.34056
$414$	498.869	0.0592224
$415$	244.325	0.0288998
$416$	0	0
$417$	−1033.51	−0.121370
$418$	2658.93	0.311131
$419$	−4179.22	−0.487275	−0.243637	−	0.969866i	$-0.578341\pi$
$419$	−4179.22	−0.487275	−0.243637	+	0.969866i	$0.578341\pi$
$420$	4164.26	0.483798
$421$	−6209.31	−0.718820	−0.359410	−	0.933180i	$-0.617022\pi$
$421$	−6209.31	−0.718820	−0.359410	+	0.933180i	$0.617022\pi$
$422$	−1915.98	−0.221015
$423$	1318.48	0.151553
$424$	−3051.39	−0.349501
$425$	−5920.49	−0.675732
$426$	730.568	0.0830895
$427$	787.382	0.0892367
$428$	2693.03	0.304141
$429$	0	0
$430$	−237.650	−0.0266523
$431$	−11880.2	−1.32773	−0.663864	−	0.747854i	$-0.731084\pi$
$431$	−11880.2	−1.32773	−0.663864	+	0.747854i	$0.731084\pi$
$432$	−8928.16	−0.994343
$433$	8725.71	0.968432	0.484216	−	0.874949i	$-0.339105\pi$
$433$	8725.71	0.968432	0.484216	+	0.874949i	$0.339105\pi$
$434$	238.130	0.0263378
$435$	7029.06	0.774754
$436$	−652.655	−0.0716893
$437$	13614.7	1.49034
$438$	1485.93	0.162101
$439$	−1200.37	−0.130503	−0.0652514	−	0.997869i	$-0.520785\pi$
$439$	−1200.37	−0.130503	−0.0652514	+	0.997869i	$0.520785\pi$
$440$	2130.86	0.230874
$441$	−544.699	−0.0588165
$442$	0	0
$443$	2258.86	0.242261	0.121130	−	0.992637i	$-0.461348\pi$
$443$	2258.86	0.242261	0.121130	+	0.992637i	$0.461348\pi$
$444$	4204.90	0.449449
$445$	−3390.59	−0.361190
$446$	532.343	0.0565183
$447$	−8497.62	−0.899158
$448$	8882.65	0.936754
$449$	14662.4	1.54112	0.770561	−	0.637367i	$-0.219976\pi$
$449$	14662.4	1.54112	0.770561	+	0.637367i	$0.219976\pi$
$450$	372.941	0.0390680
$451$	4395.16	0.458892
$452$	16607.4	1.72820
$453$	596.165	0.0618328
$454$	1626.73	0.168164
$455$	0	0
$456$	3319.31	0.340879
$457$	−9334.94	−0.955514	−0.477757	−	0.878492i	$-0.658550\pi$
$457$	−9334.94	−0.955514	−0.477757	+	0.878492i	$0.658550\pi$
$458$	1368.09	0.139578
$459$	−12795.4	−1.30117
$460$	5402.97	0.547641
$461$	−10736.9	−1.08474	−0.542370	−	0.840140i	$-0.682473\pi$
$461$	−10736.9	−1.08474	−0.542370	+	0.840140i	$0.682473\pi$
$462$	1261.19	0.127004
$463$	−10650.0	−1.06900	−0.534501	−	0.845168i	$-0.679501\pi$
$463$	−10650.0	−1.06900	−0.534501	+	0.845168i	$0.679501\pi$
$464$	15630.0	1.56380
$465$	−847.929	−0.0845630
$466$	−222.464	−0.0221147
$467$	−2638.11	−0.261407	−0.130703	−	0.991422i	$-0.541724\pi$
$467$	−2638.11	−0.261407	−0.130703	+	0.991422i	$0.541724\pi$
$468$	0	0
$469$	11749.0	1.15676
$470$	−276.923	−0.0271776
$471$	−11174.5	−1.09320
$472$	−3559.16	−0.347084
$473$	3711.45	0.360787
$474$	−451.803	−0.0437806
$475$	10177.9	0.983149
$476$	13270.6	1.27785
$477$	−6895.11	−0.661856
$478$	90.1386	0.00862520
$479$	3225.31	0.307658	0.153829	−	0.988097i	$-0.450840\pi$
$479$	3225.31	0.307658	0.153829	+	0.988097i	$0.450840\pi$
$480$	1982.23	0.188491
$481$	0	0
$482$	1204.87	0.113860
$483$	6457.75	0.608360
$484$	−6033.89	−0.566669
$485$	−1743.02	−0.163189
$486$	1337.17	0.124805
$487$	1496.39	0.139236	0.0696181	−	0.997574i	$-0.477822\pi$
$487$	1496.39	0.139236	0.0696181	+	0.997574i	$0.477822\pi$
$488$	249.070	0.0231043
$489$	1080.41	0.0999137
$490$	114.404	0.0105474
$491$	−518.408	−0.0476485	−0.0238243	−	0.999716i	$-0.507584\pi$
$491$	−518.408	−0.0476485	−0.0238243	+	0.999716i	$0.507584\pi$
$492$	2717.04	0.248970
$493$	22400.1	2.04635
$494$	0	0
$495$	4815.01	0.437210
$496$	−1885.48	−0.170686
$497$	−10139.7	−0.915148
$498$	45.7437	0.00411611
$499$	2405.01	0.215757	0.107879	−	0.994164i	$-0.465594\pi$
$499$	2405.01	0.215757	0.107879	+	0.994164i	$0.465594\pi$
$500$	11417.4	1.02120
$501$	−10847.7	−0.967341
$502$	−954.080	−0.0848260
$503$	2413.76	0.213965	0.106983	−	0.994261i	$-0.465881\pi$
$503$	2413.76	0.213965	0.106983	+	0.994261i	$0.465881\pi$
$504$	−1688.08	−0.149193
$505$	4288.12	0.377859
$506$	1636.35	0.143764
$507$	0	0
$508$	9234.45	0.806520
$509$	−15678.3	−1.36528	−0.682641	−	0.730754i	$-0.739169\pi$
$509$	−15678.3	−1.36528	−0.682641	+	0.730754i	$0.739169\pi$
$510$	916.374	0.0795642
$511$	−20623.6	−1.78539
$512$	7393.65	0.638196
$513$	21996.6	1.89312
$514$	1277.24	0.109604
$515$	7292.34	0.623959
$516$	2294.37	0.195744
$517$	4324.79	0.367899
$518$	1131.77	0.0959983
$519$	1645.48	0.139168
$520$	0	0
$521$	−11691.5	−0.983135	−0.491568	−	0.870839i	$-0.663576\pi$
$521$	−11691.5	−0.983135	−0.491568	+	0.870839i	$0.663576\pi$
$522$	−1411.02	−0.118311
$523$	−3878.95	−0.324311	−0.162156	−	0.986765i	$-0.551845\pi$
$523$	−3878.95	−0.324311	−0.162156	+	0.986765i	$0.551845\pi$
$524$	−6084.69	−0.507273
$525$	4827.63	0.401324
$526$	−1806.73	−0.149767
$527$	−2702.17	−0.223355
$528$	−9985.92	−0.823071
$529$	−3788.32	−0.311360
$530$	1448.19	0.118689
$531$	−8042.50	−0.657278
$532$	−22813.6	−1.85920
$533$	0	0
$534$	−634.804	−0.0514431
$535$	−2581.00	−0.208573
$536$	3716.54	0.299496
$537$	4924.29	0.395715
$538$	−1107.35	−0.0887382
$539$	−1786.68	−0.142779
$540$	8729.34	0.695650
$541$	16353.0	1.29958	0.649788	−	0.760115i	$-0.274858\pi$
$541$	16353.0	1.29958	0.649788	+	0.760115i	$0.274858\pi$
$542$	2751.22	0.218035
$543$	−7313.58	−0.578003
$544$	6316.93	0.497861
$545$	625.506	0.0491628
$546$	0	0
$547$	2748.67	0.214853	0.107426	−	0.994213i	$-0.465739\pi$
$547$	2748.67	0.214853	0.107426	+	0.994213i	$0.465739\pi$
$548$	19954.0	1.55546
$549$	562.814	0.0437529
$550$	1223.29	0.0948387
$551$	−38508.1	−2.97732
$552$	2042.76	0.157510
$553$	6270.68	0.482200
$554$	−756.163	−0.0579897
$555$	−4029.98	−0.308222
$556$	2246.96	0.171389
$557$	16765.6	1.27537	0.637686	−	0.770297i	$-0.279892\pi$
$557$	16765.6	1.27537	0.637686	+	0.770297i	$0.279892\pi$
$558$	170.214	0.0129135
$559$	0	0
$560$	−8874.58	−0.669677
$561$	−14311.3	−1.07705
$562$	−1283.77	−0.0963568
$563$	18492.4	1.38430	0.692151	−	0.721752i	$-0.256663\pi$
$563$	18492.4	1.38430	0.692151	+	0.721752i	$0.256663\pi$
$564$	2673.53	0.199603
$565$	−15916.5	−1.18516
$566$	−3088.96	−0.229397
$567$	3061.32	0.226743
$568$	−3207.47	−0.236941
$569$	1563.28	0.115178	0.0575888	−	0.998340i	$-0.481659\pi$
$569$	1563.28	0.115178	0.0575888	+	0.998340i	$0.481659\pi$
$570$	−1575.34	−0.115761
$571$	9165.98	0.671776	0.335888	−	0.941902i	$-0.390964\pi$
$571$	9165.98	0.671776	0.335888	+	0.941902i	$0.390964\pi$
$572$	0	0
$573$	7800.72	0.568726
$574$	731.305	0.0531778
$575$	6263.67	0.454284
$576$	6349.25	0.459292
$577$	18762.5	1.35372	0.676858	−	0.736114i	$-0.263341\pi$
$577$	18762.5	1.35372	0.676858	+	0.736114i	$0.263341\pi$
$578$	1003.65	0.0722257
$579$	4360.03	0.312948
$580$	−15281.9	−1.09405
$581$	−634.888	−0.0453349
$582$	−326.337	−0.0232424
$583$	−22616.8	−1.60667
$584$	−6523.79	−0.462254
$585$	0	0
$586$	2214.07	0.156079
$587$	12646.3	0.889212	0.444606	−	0.895726i	$-0.353344\pi$
$587$	12646.3	0.889212	0.444606	+	0.895726i	$0.353344\pi$
$588$	−1104.50	−0.0774642
$589$	4645.31	0.324969
$590$	1689.18	0.117868
$591$	−17784.0	−1.23779
$592$	−8961.17	−0.622132
$593$	−9662.74	−0.669142	−0.334571	−	0.942371i	$-0.608591\pi$
$593$	−9662.74	−0.669142	−0.334571	+	0.942371i	$0.608591\pi$
$594$	2643.78	0.182619
$595$	−12718.6	−0.876321
$596$	18474.7	1.26972
$597$	3643.79	0.249800
$598$	0	0
$599$	−26968.7	−1.83959	−0.919794	−	0.392402i	$-0.871644\pi$
$599$	−26968.7	−1.83959	−0.919794	+	0.392402i	$0.871644\pi$
$600$	1527.11	0.103907
$601$	11280.1	0.765600	0.382800	−	0.923831i	$-0.374960\pi$
$601$	11280.1	0.765600	0.382800	+	0.923831i	$0.374960\pi$
$602$	617.542	0.0418092
$603$	8398.11	0.567160
$604$	−1296.13	−0.0873156
$605$	5782.89	0.388608
$606$	802.843	0.0538173
$607$	12052.6	0.805931	0.402965	−	0.915215i	$-0.367979\pi$
$607$	12052.6	0.805931	0.402965	+	0.915215i	$0.367979\pi$
$608$	−10859.5	−0.724358
$609$	−18265.3	−1.21535
$610$	−118.208	−0.00784610
$611$	0	0
$612$	9485.73	0.626532
$613$	11259.7	0.741883	0.370941	−	0.928656i	$-0.379035\pi$
$613$	11259.7	0.741883	0.370941	+	0.928656i	$0.379035\pi$
$614$	−1692.65	−0.111254
$615$	−2604.01	−0.170738
$616$	−5537.12	−0.362170
$617$	−24058.2	−1.56976	−0.784882	−	0.619645i	$-0.787277\pi$
$617$	−24058.2	−1.56976	−0.784882	+	0.619645i	$0.787277\pi$
$618$	1365.31	0.0888686
$619$	−2793.41	−0.181384	−0.0906919	−	0.995879i	$-0.528908\pi$
$619$	−2793.41	−0.181384	−0.0906919	+	0.995879i	$0.528908\pi$
$620$	1843.49	0.119413
$621$	13537.1	0.874756
$622$	−2042.15	−0.131645
$623$	8810.59	0.566595
$624$	0	0
$625$	−2388.81	−0.152884
$626$	845.714	0.0539960
$627$	24602.6	1.56704
$628$	24294.6	1.54373
$629$	−12842.7	−0.814104
$630$	801.163	0.0506652
$631$	−24850.0	−1.56777	−0.783883	−	0.620908i	$-0.786764\pi$
$631$	−24850.0	−1.56777	−0.783883	+	0.620908i	$0.786764\pi$
$632$	1983.59	0.124846
$633$	−17728.2	−1.11317
$634$	1897.41	0.118857
$635$	−8850.32	−0.553093
$636$	−13981.4	−0.871696
$637$	0	0
$638$	−4628.31	−0.287205
$639$	−7247.79	−0.448698
$640$	−5726.69	−0.353699
$641$	−7794.71	−0.480300	−0.240150	−	0.970736i	$-0.577197\pi$
$641$	−7794.71	−0.480300	−0.240150	+	0.970736i	$0.577197\pi$
$642$	−483.228	−0.0297064
$643$	27709.9	1.69949	0.849744	−	0.527195i	$-0.176756\pi$
$643$	27709.9	1.69949	0.849744	+	0.527195i	$0.176756\pi$
$644$	−14039.8	−0.859080
$645$	−2198.93	−0.134237
$646$	−5020.28	−0.305759
$647$	11150.9	0.677571	0.338786	−	0.940864i	$-0.389984\pi$
$647$	11150.9	0.677571	0.338786	+	0.940864i	$0.389984\pi$
$648$	968.378	0.0587060
$649$	−26380.4	−1.59556
$650$	0	0
$651$	2203.38	0.132653
$652$	−2348.93	−0.141091
$653$	29137.5	1.74615	0.873077	−	0.487583i	$-0.162121\pi$
$653$	29137.5	1.74615	0.873077	+	0.487583i	$0.162121\pi$
$654$	117.110	0.00700211
$655$	5831.58	0.347876
$656$	−5790.36	−0.344627
$657$	−14741.6	−0.875377
$658$	719.595	0.0426333
$659$	−5300.12	−0.313298	−0.156649	−	0.987654i	$-0.550069\pi$
$659$	−5300.12	−0.313298	−0.156649	+	0.987654i	$0.550069\pi$
$660$	9763.54	0.575826
$661$	−27412.5	−1.61305	−0.806523	−	0.591203i	$-0.798653\pi$
$661$	−27412.5	−1.61305	−0.806523	+	0.591203i	$0.798653\pi$
$662$	−1047.56	−0.0615025
$663$	0	0
$664$	−200.832	−0.0117377
$665$	21864.6	1.27499
$666$	808.981	0.0470681
$667$	−23698.6	−1.37573
$668$	23584.0	1.36601
$669$	4925.67	0.284660
$670$	−1763.87	−0.101708
$671$	1846.10	0.106211
$672$	−5150.90	−0.295685
$673$	−21282.1	−1.21896	−0.609482	−	0.792800i	$-0.708623\pi$
$673$	−21282.1	−1.21896	−0.609482	+	0.792800i	$0.708623\pi$
$674$	−2542.91	−0.145325
$675$	10119.9	0.577061
$676$	0	0
$677$	−13544.2	−0.768904	−0.384452	−	0.923145i	$-0.625610\pi$
$677$	−13544.2	−0.768904	−0.384452	+	0.923145i	$0.625610\pi$
$678$	−2979.98	−0.168798
$679$	4529.31	0.255992
$680$	−4023.23	−0.226888
$681$	15051.9	0.846974
$682$	558.322	0.0313479
$683$	10350.1	0.579849	0.289924	−	0.957050i	$-0.406370\pi$
$683$	10350.1	0.579849	0.289924	+	0.957050i	$0.406370\pi$
$684$	−16307.0	−0.911567
$685$	−19124.0	−1.06670
$686$	2317.97	0.129009
$687$	12658.7	0.702999
$688$	−4889.60	−0.270951
$689$	0	0
$690$	−969.492	−0.0534898
$691$	−25714.8	−1.41569	−0.707843	−	0.706370i	$-0.750331\pi$
$691$	−25714.8	−1.41569	−0.707843	+	0.706370i	$0.750331\pi$
$692$	−3577.45	−0.196523
$693$	−12512.0	−0.685847
$694$	29.5693	0.00161734
$695$	−2153.49	−0.117535
$696$	−5777.81	−0.314666
$697$	−8298.43	−0.450969
$698$	−1436.44	−0.0778940
$699$	−2058.42	−0.111383
$700$	−10495.8	−0.566720
$701$	7431.30	0.400394	0.200197	−	0.979756i	$-0.435842\pi$
$701$	7431.30	0.400394	0.200197	+	0.979756i	$0.435842\pi$
$702$	0	0
$703$	22077.9	1.18447
$704$	20826.3	1.11494
$705$	−2562.32	−0.136883
$706$	3913.46	0.208619
$707$	−11142.8	−0.592744
$708$	−16308.0	−0.865668
$709$	−19986.8	−1.05870	−0.529351	−	0.848403i	$-0.677565\pi$
$709$	−19986.8	−1.05870	−0.529351	+	0.848403i	$0.677565\pi$
$710$	1522.26	0.0804641
$711$	4482.23	0.236423
$712$	2787.03	0.146697
$713$	2858.80	0.150158
$714$	−2381.24	−0.124812
$715$	0	0
$716$	−10705.9	−0.558798
$717$	834.037	0.0434417
$718$	−2680.00	−0.139299
$719$	36101.9	1.87256	0.936281	−	0.351251i	$-0.114244\pi$
$719$	36101.9	1.87256	0.936281	+	0.351251i	$0.114244\pi$
$720$	−6343.48	−0.328344
$721$	−18949.5	−0.978800
$722$	5954.59	0.306935
$723$	11148.4	0.573465
$724$	15900.5	0.816213
$725$	−17716.4	−0.907545
$726$	1082.70	0.0553483
$727$	−1751.90	−0.0893735	−0.0446868	−	0.999001i	$-0.514229\pi$
$727$	−1751.90	−0.0893735	−0.0446868	+	0.999001i	$0.514229\pi$
$728$	0	0
$729$	16601.6	0.843451
$730$	3096.18	0.156979
$731$	−7007.52	−0.354559
$732$	1141.24	0.0576247
$733$	−20031.3	−1.00938	−0.504688	−	0.863302i	$-0.668393\pi$
$733$	−20031.3	−1.00938	−0.504688	+	0.863302i	$0.668393\pi$
$734$	3465.51	0.174270
$735$	1058.56	0.0531231
$736$	−6683.10	−0.334704
$737$	27546.8	1.37680
$738$	522.731	0.0260732
$739$	18632.9	0.927502	0.463751	−	0.885966i	$-0.346503\pi$
$739$	18632.9	0.927502	0.463751	+	0.885966i	$0.346503\pi$
$740$	8761.62	0.435248
$741$	0	0
$742$	−3763.17	−0.186187
$743$	4907.69	0.242323	0.121161	−	0.992633i	$-0.461338\pi$
$743$	4907.69	0.242323	0.121161	+	0.992633i	$0.461338\pi$
$744$	696.988	0.0343452
$745$	−17706.2	−0.870747
$746$	−2127.88	−0.104433
$747$	−453.813	−0.0222278
$748$	31114.3	1.52093
$749$	6706.84	0.327186
$750$	−2048.70	−0.0997440
$751$	31156.9	1.51389	0.756945	−	0.653479i	$-0.226691\pi$
$751$	31156.9	1.51389	0.756945	+	0.653479i	$0.226691\pi$
$752$	−5697.64	−0.276292
$753$	−8827.93	−0.427235
$754$	0	0
$755$	1242.21	0.0598790
$756$	−22683.5	−1.09126
$757$	−11047.0	−0.530396	−0.265198	−	0.964194i	$-0.585437\pi$
$757$	−11047.0	−0.530396	−0.265198	+	0.964194i	$0.585437\pi$
$758$	2591.20	0.124165
$759$	15140.9	0.724082
$760$	6916.35	0.330109
$761$	27606.1	1.31501	0.657503	−	0.753452i	$-0.271613\pi$
$761$	27606.1	1.31501	0.657503	+	0.753452i	$0.271613\pi$
$762$	−1657.00	−0.0787753
$763$	−1625.40	−0.0771213
$764$	−16959.6	−0.803112
$765$	−9091.14	−0.429661
$766$	492.456	0.0232287
$767$	0	0
$768$	12052.1	0.566264
$769$	−3694.72	−0.173257	−0.0866287	−	0.996241i	$-0.527609\pi$
$769$	−3694.72	−0.173257	−0.0866287	+	0.996241i	$0.527609\pi$
$770$	2627.91	0.122991
$771$	11818.0	0.552031
$772$	−9479.18	−0.441921
$773$	19808.0	0.921662	0.460831	−	0.887488i	$-0.347551\pi$
$773$	19808.0	0.921662	0.460831	+	0.887488i	$0.347551\pi$
$774$	441.414	0.0204991
$775$	2137.16	0.0990569
$776$	1432.74	0.0662789
$777$	10472.1	0.483505
$778$	−1089.76	−0.0502181
$779$	14265.9	0.656133
$780$	0	0
$781$	−23773.6	−1.08923
$782$	−3089.56	−0.141282
$783$	−38288.7	−1.74754
$784$	2353.84	0.107227
$785$	−23284.0	−1.05865
$786$	1091.82	0.0495469
$787$	−4116.79	−0.186464	−0.0932322	−	0.995644i	$-0.529720\pi$
$787$	−4116.79	−0.186464	−0.0932322	+	0.995644i	$0.529720\pi$
$788$	38664.3	1.74792
$789$	−16717.4	−0.754314
$790$	−941.408	−0.0423972
$791$	41359.8	1.85915
$792$	−3957.89	−0.177572
$793$	0	0
$794$	2671.86	0.119422
$795$	13399.8	0.597789
$796$	−7922.00	−0.352748
$797$	25359.3	1.12707	0.563533	−	0.826094i	$-0.309442\pi$
$797$	25359.3	1.12707	0.563533	+	0.826094i	$0.309442\pi$
$798$	4093.59	0.181594
$799$	−8165.55	−0.361548
$800$	−4996.10	−0.220799
$801$	6297.74	0.277802
$802$	4300.83	0.189361
$803$	−48354.1	−2.12500
$804$	17029.1	0.746978
$805$	13455.8	0.589137
$806$	0	0
$807$	−10246.1	−0.446939
$808$	−3524.78	−0.153467
$809$	−5558.73	−0.241576	−0.120788	−	0.992678i	$-0.538542\pi$
$809$	−5558.73	−0.241576	−0.120788	+	0.992678i	$0.538542\pi$
$810$	−459.591	−0.0199363
$811$	15021.4	0.650399	0.325199	−	0.945646i	$-0.394569\pi$
$811$	15021.4	0.650399	0.325199	+	0.945646i	$0.394569\pi$
$812$	39710.8	1.71623
$813$	25456.5	1.09815
$814$	2653.55	0.114259
$815$	2251.22	0.0967567
$816$	18854.2	0.808861
$817$	12046.6	0.515862
$818$	1377.32	0.0588717
$819$	0	0
$820$	5661.41	0.241104
$821$	−15901.6	−0.675966	−0.337983	−	0.941152i	$-0.609745\pi$
$821$	−15901.6	−0.675966	−0.337983	+	0.941152i	$0.609745\pi$
$822$	−3580.49	−0.151927
$823$	−40279.3	−1.70601	−0.853006	−	0.521901i	$-0.825223\pi$
$823$	−40279.3	−1.70601	−0.853006	+	0.521901i	$0.825223\pi$
$824$	−5994.23	−0.253421
$825$	11318.9	0.477664
$826$	−4389.39	−0.184899
$827$	−5251.09	−0.220796	−0.110398	−	0.993887i	$-0.535213\pi$
$827$	−5251.09	−0.220796	−0.110398	+	0.993887i	$0.535213\pi$
$828$	−10035.6	−0.421208
$829$	33964.4	1.42296	0.711479	−	0.702708i	$-0.248026\pi$
$829$	33964.4	1.42296	0.711479	+	0.702708i	$0.248026\pi$
$830$	95.3148	0.00398606
$831$	−6996.65	−0.292071
$832$	0	0
$833$	3373.40	0.140314
$834$	−403.188	−0.0167401
$835$	−22603.0	−0.936776
$836$	−53488.8	−2.21286
$837$	4618.83	0.190741
$838$	−1630.38	−0.0672081
$839$	15485.0	0.637188	0.318594	−	0.947891i	$-0.396789\pi$
$839$	15485.0	0.637188	0.318594	+	0.947891i	$0.396789\pi$
$840$	3280.59	0.134751
$841$	42640.8	1.74836
$842$	−2422.35	−0.0991444
$843$	−11878.5	−0.485311
$844$	38543.1	1.57193
$845$	0	0
$846$	514.361	0.0209032
$847$	−15027.1	−0.609606
$848$	29796.2	1.20661
$849$	−28581.6	−1.15538
$850$	−2309.67	−0.0932014
$851$	13587.1	0.547309
$852$	−14696.6	−0.590958
$853$	−20057.8	−0.805118	−0.402559	−	0.915394i	$-0.631879\pi$
$853$	−20057.8	−0.805118	−0.402559	+	0.915394i	$0.631879\pi$
$854$	307.170	0.0123081
$855$	15628.6	0.625132
$856$	2121.55	0.0847117
$857$	−8066.23	−0.321514	−0.160757	−	0.986994i	$-0.551394\pi$
$857$	−8066.23	−0.321514	−0.160757	+	0.986994i	$0.551394\pi$
$858$	0	0
$859$	39719.0	1.57764	0.788821	−	0.614623i	$-0.210692\pi$
$859$	39719.0	1.57764	0.788821	+	0.614623i	$0.210692\pi$
$860$	4780.71	0.189559
$861$	6766.64	0.267835
$862$	−4634.66	−0.183129
$863$	24473.8	0.965351	0.482676	−	0.875799i	$-0.339665\pi$
$863$	24473.8	0.965351	0.482676	+	0.875799i	$0.339665\pi$
$864$	−10797.6	−0.425163
$865$	3428.63	0.134771
$866$	3404.03	0.133572
$867$	9286.62	0.363772
$868$	−4790.38	−0.187323
$869$	14702.3	0.573924
$870$	2742.15	0.106859
$871$	0	0
$872$	−514.159	−0.0199675
$873$	3237.51	0.125513
$874$	5311.28	0.205557
$875$	28434.4	1.09858
$876$	−29891.9	−1.15291
$877$	25326.6	0.975162	0.487581	−	0.873078i	$-0.337879\pi$
$877$	25326.6	0.975162	0.487581	+	0.873078i	$0.337879\pi$
$878$	−468.284	−0.0179998
$879$	20486.4	0.786109
$880$	−20807.4	−0.797064
$881$	1327.44	0.0507634	0.0253817	−	0.999678i	$-0.491920\pi$
$881$	1327.44	0.0507634	0.0253817	+	0.999678i	$0.491920\pi$
$882$	−212.496	−0.00811235
$883$	−2112.05	−0.0804941	−0.0402470	−	0.999190i	$-0.512814\pi$
$883$	−2112.05	−0.0804941	−0.0402470	+	0.999190i	$0.512814\pi$
$884$	0	0
$885$	15629.6	0.593655
$886$	881.214	0.0334142
$887$	−40935.4	−1.54958	−0.774790	−	0.632219i	$-0.782144\pi$
$887$	−40935.4	−1.54958	−0.774790	+	0.632219i	$0.782144\pi$
$888$	3312.60	0.125184
$889$	22997.9	0.867632
$890$	−1322.72	−0.0498177
$891$	7177.58	0.269874
$892$	−10708.9	−0.401975
$893$	14037.4	0.526030
$894$	−3315.05	−0.124018
$895$	10260.6	0.383211
$896$	14881.0	0.554845
$897$	0	0
$898$	5720.04	0.212562
$899$	−8085.93	−0.299979
$900$	−7502.32	−0.277864
$901$	42702.3	1.57894
$902$	1714.62	0.0632934
$903$	5714.01	0.210576
$904$	13083.2	0.481351
$905$	−15239.1	−0.559740
$906$	232.573	0.00852838
$907$	153.968	0.00563663	0.00281831	−	0.999996i	$-0.499103\pi$
$907$	153.968	0.00563663	0.00281831	+	0.999996i	$0.499103\pi$
$908$	−32724.4	−1.19603
$909$	−7964.82	−0.290623
$910$	0	0
$911$	−733.607	−0.0266800	−0.0133400	−	0.999911i	$-0.504246\pi$
$911$	−733.607	−0.0266800	−0.0133400	+	0.999911i	$0.504246\pi$
$912$	−32412.4	−1.17684
$913$	−1488.56	−0.0539586
$914$	−3641.70	−0.131791
$915$	−1093.76	−0.0395177
$916$	−27521.4	−0.992722
$917$	−15153.6	−0.545710
$918$	−4991.67	−0.179466
$919$	−51106.4	−1.83443	−0.917216	−	0.398390i	$-0.869569\pi$
$919$	−51106.4	−1.83443	−0.917216	+	0.398390i	$0.869569\pi$
$920$	4256.44	0.152533
$921$	−15661.8	−0.560342
$922$	−4188.61	−0.149615
$923$	0	0
$924$	−25371.0	−0.903294
$925$	10157.4	0.361051
$926$	−4154.73	−0.147444
$927$	−13544.9	−0.479907
$928$	18902.7	0.668655
$929$	−30645.2	−1.08228	−0.541139	−	0.840933i	$-0.682007\pi$
$929$	−30645.2	−1.08228	−0.541139	+	0.840933i	$0.682007\pi$
$930$	−330.790	−0.0116635
$931$	−5799.22	−0.204148
$932$	4475.23	0.157287
$933$	−18895.7	−0.663041
$934$	−1029.17	−0.0360550
$935$	−29820.0	−1.04302
$936$	0	0
$937$	−24422.2	−0.851482	−0.425741	−	0.904845i	$-0.639987\pi$
$937$	−24422.2	−0.851482	−0.425741	+	0.904845i	$0.639987\pi$
$938$	4583.48	0.159548
$939$	7825.24	0.271956
$940$	5570.75	0.193296
$941$	−16475.9	−0.570774	−0.285387	−	0.958412i	$-0.592122\pi$
$941$	−16475.9	−0.570774	−0.285387	+	0.958412i	$0.592122\pi$
$942$	−4359.35	−0.150781
$943$	8779.46	0.303180
$944$	34754.5	1.19827
$945$	21740.0	0.748361
$946$	1447.89	0.0497622
$947$	−10122.1	−0.347332	−0.173666	−	0.984805i	$-0.555561\pi$
$947$	−10122.1	−0.347332	−0.173666	+	0.984805i	$0.555561\pi$
$948$	9088.76	0.311381
$949$	0	0
$950$	3970.57	0.135602
$951$	17556.4	0.598637
$952$	10454.5	0.355917
$953$	38097.2	1.29495	0.647475	−	0.762086i	$-0.275825\pi$
$953$	38097.2	1.29495	0.647475	+	0.762086i	$0.275825\pi$
$954$	−2689.89	−0.0912875
$955$	16254.1	0.550756
$956$	−1813.29	−0.0613451
$957$	−42824.9	−1.44653
$958$	1258.24	0.0424342
$959$	49694.5	1.67332
$960$	−12339.0	−0.414833
$961$	−28815.6	−0.967258
$962$	0	0
$963$	4794.00	0.160420
$964$	−24237.9	−0.809804
$965$	9084.87	0.303059
$966$	2519.27	0.0839090
$967$	−44515.5	−1.48037	−0.740187	−	0.672401i	$-0.765263\pi$
$967$	−44515.5	−1.48037	−0.740187	+	0.672401i	$0.765263\pi$
$968$	−4753.47	−0.157833
$969$	−46451.8	−1.53999
$970$	−679.978	−0.0225080
$971$	−25444.3	−0.840934	−0.420467	−	0.907308i	$-0.638134\pi$
$971$	−25444.3	−0.840934	−0.420467	+	0.907308i	$0.638134\pi$
$972$	−26899.3	−0.887649
$973$	5595.94	0.184376
$974$	583.765	0.0192044
$975$	0	0
$976$	−2432.12	−0.0797646
$977$	39467.9	1.29242	0.646208	−	0.763161i	$-0.276354\pi$
$977$	39467.9	1.29242	0.646208	+	0.763161i	$0.276354\pi$
$978$	421.484	0.0137808
$979$	20657.4	0.674374
$980$	−2301.42	−0.0750165
$981$	−1161.83	−0.0378127
$982$	−202.239	−0.00657200
$983$	−14970.4	−0.485740	−0.242870	−	0.970059i	$-0.578089\pi$
$983$	−14970.4	−0.485740	−0.242870	+	0.970059i	$0.578089\pi$
$984$	2140.47	0.0693452
$985$	−37055.9	−1.19868
$986$	8738.63	0.282246
$987$	6658.28	0.214727
$988$	0	0
$989$	7413.71	0.238364
$990$	1878.41	0.0603028
$991$	58422.4	1.87270	0.936352	−	0.351063i	$-0.114180\pi$
$991$	58422.4	1.87270	0.936352	+	0.351063i	$0.114180\pi$
$992$	−2280.27	−0.0729825
$993$	−9692.91	−0.309763
$994$	−3955.66	−0.126223
$995$	7592.46	0.241907
$996$	−920.210	−0.0292751
$997$	20131.6	0.639493	0.319746	−	0.947503i	$-0.396402\pi$
$997$	20131.6	0.639493	0.319746	+	0.947503i	$0.396402\pi$
$998$	938.230	0.0297587
$999$	21952.1	0.695228

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.a.k.1.6	✓	9
3.2	odd	2		1521.4.a.bh.1.4		9
13.2	odd	12		169.4.e.h.147.11		36
13.3	even	3		169.4.c.l.22.4		18
13.4	even	6		169.4.c.k.146.6		18
13.5	odd	4		169.4.b.g.168.8		18
13.6	odd	12		169.4.e.h.23.8		36
13.7	odd	12		169.4.e.h.23.11		36
13.8	odd	4		169.4.b.g.168.11		18
13.9	even	3		169.4.c.l.146.4		18
13.10	even	6		169.4.c.k.22.6		18
13.11	odd	12		169.4.e.h.147.8		36
13.12	even	2		169.4.a.l.1.4	yes	9
39.38	odd	2		1521.4.a.bg.1.6		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.4.a.k.1.6	✓	9	1.1	even	1	trivial
169.4.a.l.1.4	yes	9	13.12	even	2
169.4.b.g.168.8		18	13.5	odd	4
169.4.b.g.168.11		18	13.8	odd	4
169.4.c.k.22.6		18	13.10	even	6
169.4.c.k.146.6		18	13.4	even	6
169.4.c.l.22.4		18	13.3	even	3
169.4.c.l.146.4		18	13.9	even	3
169.4.e.h.23.8		36	13.6	odd	12
169.4.e.h.23.11		36	13.7	odd	12
169.4.e.h.147.8		36	13.11	odd	12
169.4.e.h.147.11		36	13.2	odd	12
1521.4.a.bg.1.6		9	39.38	odd	2
1521.4.a.bh.1.4		9	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:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	169.a (trivial)

\(f(q)\)	\(=\)	\(q+0.390115 q^{2} +3.60967 q^{3} -7.84781 q^{4} +7.52136 q^{5} +1.40819 q^{6} -19.5446 q^{7} -6.18247 q^{8} -13.9703 q^{9} +O(q^{10})\)	\(q+0.390115 q^{2} +3.60967 q^{3} -7.84781 q^{4} +7.52136 q^{5} +1.40819 q^{6} -19.5446 q^{7} -6.18247 q^{8} -13.9703 q^{9} +2.93420 q^{10} -45.8243 q^{11} -28.3280 q^{12} -7.62463 q^{14} +27.1496 q^{15} +60.3706 q^{16} +86.5200 q^{17} -5.45003 q^{18} -148.737 q^{19} -59.0262 q^{20} -70.5494 q^{21} -17.8767 q^{22} -91.5351 q^{23} -22.3167 q^{24} -68.4292 q^{25} -147.889 q^{27} +153.382 q^{28} +258.901 q^{29} +10.5915 q^{30} -31.2317 q^{31} +73.0113 q^{32} -165.410 q^{33} +33.7528 q^{34} -147.002 q^{35} +109.636 q^{36} -148.436 q^{37} -58.0245 q^{38} -46.5006 q^{40} -95.9135 q^{41} -27.5224 q^{42} -80.9930 q^{43} +359.620 q^{44} -105.076 q^{45} -35.7093 q^{46} -94.3777 q^{47} +217.918 q^{48} +38.9898 q^{49} -26.6953 q^{50} +312.308 q^{51} +493.555 q^{53} -57.6938 q^{54} -344.661 q^{55} +120.834 q^{56} -536.891 q^{57} +101.001 q^{58} +575.686 q^{59} -213.065 q^{60} -40.2865 q^{61} -12.1840 q^{62} +273.043 q^{63} -454.482 q^{64} -64.5291 q^{66} -601.141 q^{67} -678.992 q^{68} -330.411 q^{69} -57.3476 q^{70} +518.800 q^{71} +86.3710 q^{72} +1055.21 q^{73} -57.9072 q^{74} -247.007 q^{75} +1167.26 q^{76} +895.615 q^{77} -320.840 q^{79} +454.069 q^{80} -156.633 q^{81} -37.4173 q^{82} +32.4841 q^{83} +553.658 q^{84} +650.748 q^{85} -31.5966 q^{86} +934.547 q^{87} +283.307 q^{88} -450.795 q^{89} -40.9916 q^{90} +718.350 q^{92} -112.736 q^{93} -36.8182 q^{94} -1118.70 q^{95} +263.546 q^{96} -231.743 q^{97} +15.2105 q^{98} +640.179 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 5 q^{2} + q^{3} + 37 q^{4} - 30 q^{5} - 48 q^{6} - 38 q^{7} - 60 q^{8} + 66 q^{9}+O(q^{10}) \) 9 * q - 5 * q^2 + q^3 + 37 * q^4 - 30 * q^5 - 48 * q^6 - 38 * q^7 - 60 * q^8 + 66 * q^9	\( 9 q - 5 q^{2} + q^{3} + 37 q^{4} - 30 q^{5} - 48 q^{6} - 38 q^{7} - 60 q^{8} + 66 q^{9} - 147 q^{10} - 181 q^{11} + 39 q^{12} - 147 q^{14} - 218 q^{15} + 269 q^{16} - 55 q^{17} - 79 q^{18} - 161 q^{19} - 370 q^{20} - 188 q^{21} + 340 q^{22} - 204 q^{23} - 798 q^{24} + 307 q^{25} - 668 q^{27} - 344 q^{28} + 280 q^{29} + 521 q^{30} - 706 q^{31} - 680 q^{32} - 500 q^{33} - 216 q^{34} + 20 q^{35} - 909 q^{36} - 298 q^{37} - 739 q^{38} + 13 q^{40} - 1201 q^{41} - 4 q^{42} - 533 q^{43} - 355 q^{44} + 90 q^{45} + 840 q^{46} - 956 q^{47} - 132 q^{48} + 403 q^{49} + 1156 q^{50} + 470 q^{51} - 278 q^{53} + 2555 q^{54} - 250 q^{55} + 250 q^{56} + 810 q^{57} + 2877 q^{58} - 1377 q^{59} + 3157 q^{60} - 136 q^{61} + 2035 q^{62} + 944 q^{63} + 284 q^{64} + 3279 q^{66} + 931 q^{67} - 1536 q^{68} - 2050 q^{69} + 4854 q^{70} - 2046 q^{71} + 4342 q^{72} + 45 q^{73} - 1990 q^{74} + 2393 q^{75} + 3608 q^{76} - 718 q^{77} + 412 q^{79} + 787 q^{80} - 835 q^{81} + 2757 q^{82} - 3709 q^{83} + 1539 q^{84} + 2106 q^{85} - 125 q^{86} - 786 q^{87} - 636 q^{88} - 1663 q^{89} - 1280 q^{90} + 4010 q^{92} + 1186 q^{93} - 2531 q^{94} - 1614 q^{95} + 3084 q^{96} + 1087 q^{97} + 282 q^{98} - 1357 q^{99}+O(q^{100}) \) 9 * q - 5 * q^2 + q^3 + 37 * q^4 - 30 * q^5 - 48 * q^6 - 38 * q^7 - 60 * q^8 + 66 * q^9 - 147 * q^10 - 181 * q^11 + 39 * q^12 - 147 * q^14 - 218 * q^15 + 269 * q^16 - 55 * q^17 - 79 * q^18 - 161 * q^19 - 370 * q^20 - 188 * q^21 + 340 * q^22 - 204 * q^23 - 798 * q^24 + 307 * q^25 - 668 * q^27 - 344 * q^28 + 280 * q^29 + 521 * q^30 - 706 * q^31 - 680 * q^32 - 500 * q^33 - 216 * q^34 + 20 * q^35 - 909 * q^36 - 298 * q^37 - 739 * q^38 + 13 * q^40 - 1201 * q^41 - 4 * q^42 - 533 * q^43 - 355 * q^44 + 90 * q^45 + 840 * q^46 - 956 * q^47 - 132 * q^48 + 403 * q^49 + 1156 * q^50 + 470 * q^51 - 278 * q^53 + 2555 * q^54 - 250 * q^55 + 250 * q^56 + 810 * q^57 + 2877 * q^58 - 1377 * q^59 + 3157 * q^60 - 136 * q^61 + 2035 * q^62 + 944 * q^63 + 284 * q^64 + 3279 * q^66 + 931 * q^67 - 1536 * q^68 - 2050 * q^69 + 4854 * q^70 - 2046 * q^71 + 4342 * q^72 + 45 * q^73 - 1990 * q^74 + 2393 * q^75 + 3608 * q^76 - 718 * q^77 + 412 * q^79 + 787 * q^80 - 835 * q^81 + 2757 * q^82 - 3709 * q^83 + 1539 * q^84 + 2106 * q^85 - 125 * q^86 - 786 * q^87 - 636 * q^88 - 1663 * q^89 - 1280 * q^90 + 4010 * q^92 + 1186 * q^93 - 2531 * q^94 - 1614 * q^95 + 3084 * q^96 + 1087 * q^97 + 282 * q^98 - 1357 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more