LMFDB - Embedded newform 289.4.b.f.288.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [289,4,Mod(288,289)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(289, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

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

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

chi := DirichletCharacter("289.288");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 289 = 17^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	289.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$17.0515519917$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			288.2
Character	$\chi$	$=$	289.288
Dual form			289.4.b.f.288.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.44354 q^{2} +0.537336i q^{3} +11.7450 q^{4} +17.2592i q^{5} -2.38767i q^{6} +6.40763i q^{7} -16.6411 q^{8} +26.7113 q^{9} +O(q^{10})$	\(q-4.44354 q^{2} +0.537336i q^{3} +11.7450 q^{4} +17.2592i q^{5} -2.38767i q^{6} +6.40763i q^{7} -16.6411 q^{8} +26.7113 q^{9} -76.6918i q^{10} -55.3227i q^{11} +6.31102i q^{12} +58.6637 q^{13} -28.4725i q^{14} -9.27398 q^{15} -20.0149 q^{16} -118.692 q^{18} +91.1866 q^{19} +202.709i q^{20} -3.44305 q^{21} +245.829i q^{22} -120.879i q^{23} -8.94185i q^{24} -172.879 q^{25} -260.674 q^{26} +28.8610i q^{27} +75.2576i q^{28} -215.755i q^{29} +41.2093 q^{30} -17.5166i q^{31} +222.065 q^{32} +29.7269 q^{33} -110.590 q^{35} +313.724 q^{36} -8.40485i q^{37} -405.191 q^{38} +31.5221i q^{39} -287.211i q^{40} -99.9530i q^{41} +15.2993 q^{42} +81.5297 q^{43} -649.766i q^{44} +461.015i q^{45} +537.130i q^{46} +195.351 q^{47} -10.7547i q^{48} +301.942 q^{49} +768.195 q^{50} +689.006 q^{52} -260.322 q^{53} -128.245i q^{54} +954.825 q^{55} -106.630i q^{56} +48.9979i q^{57} +958.715i q^{58} +536.401 q^{59} -108.923 q^{60} +265.689i q^{61} +77.8357i q^{62} +171.156i q^{63} -826.636 q^{64} +1012.49i q^{65} -132.093 q^{66} +514.794 q^{67} +64.9526 q^{69} +491.412 q^{70} -704.023i q^{71} -444.504 q^{72} +184.948i q^{73} +37.3473i q^{74} -92.8943i q^{75} +1070.99 q^{76} +354.487 q^{77} -140.070i q^{78} +34.8358i q^{79} -345.440i q^{80} +705.696 q^{81} +444.145i q^{82} +647.682 q^{83} -40.4386 q^{84} -362.280 q^{86} +115.933 q^{87} +920.630i q^{88} -1060.24 q^{89} -2048.53i q^{90} +375.895i q^{91} -1419.72i q^{92} +9.41231 q^{93} -868.051 q^{94} +1573.81i q^{95} +119.324i q^{96} -256.409i q^{97} -1341.69 q^{98} -1477.74i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 96 q^{4} + 102 q^{8} - 216 q^{9}+O(q^{10}) $ 24 * q + 96 * q^4 + 102 * q^8 - 216 * q^9	$ 24 q + 96 q^{4} + 102 q^{8} - 216 q^{9} - 144 q^{13} + 276 q^{15} + 384 q^{16} + 378 q^{18} + 132 q^{19} + 492 q^{21} - 888 q^{25} - 1056 q^{26} - 3780 q^{30} + 2706 q^{32} + 1932 q^{33} - 132 q^{35} + 1326 q^{36} - 120 q^{38} - 1608 q^{42} - 972 q^{43} - 1776 q^{47} + 1140 q^{49} + 870 q^{50} + 450 q^{52} - 2052 q^{53} + 1944 q^{55} + 1584 q^{59} - 2916 q^{60} - 3714 q^{64} + 1188 q^{66} + 1248 q^{67} - 3012 q^{69} + 3300 q^{70} + 2910 q^{72} - 1350 q^{76} + 2016 q^{77} + 5040 q^{81} - 1344 q^{83} - 1554 q^{84} + 5556 q^{86} - 1452 q^{87} - 1812 q^{89} - 264 q^{93} - 1470 q^{94} + 3822 q^{98}+O(q^{100}) $ 24 * q + 96 * q^4 + 102 * q^8 - 216 * q^9 - 144 * q^13 + 276 * q^15 + 384 * q^16 + 378 * q^18 + 132 * q^19 + 492 * q^21 - 888 * q^25 - 1056 * q^26 - 3780 * q^30 + 2706 * q^32 + 1932 * q^33 - 132 * q^35 + 1326 * q^36 - 120 * q^38 - 1608 * q^42 - 972 * q^43 - 1776 * q^47 + 1140 * q^49 + 870 * q^50 + 450 * q^52 - 2052 * q^53 + 1944 * q^55 + 1584 * q^59 - 2916 * q^60 - 3714 * q^64 + 1188 * q^66 + 1248 * q^67 - 3012 * q^69 + 3300 * q^70 + 2910 * q^72 - 1350 * q^76 + 2016 * q^77 + 5040 * q^81 - 1344 * q^83 - 1554 * q^84 + 5556 * q^86 - 1452 * q^87 - 1812 * q^89 - 264 * q^93 - 1470 * q^94 + 3822 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/289\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$-1$

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 $

$2$	−4.44354	−1.57103	−0.785514	−	0.618844i	$-0.787601\pi$
$2$	−4.44354	−1.57103	−0.785514	+	0.618844i	$0.787601\pi$
$3$	0.537336i	0.103410i	0.998662	+	0.0517052i	$0.0164656\pi$
$3$	0.537336i	0.103410i	−0.998662	+	0.0517052i	$0.983534\pi$
$4$	11.7450	1.46813
$5$	17.2592i	1.54371i	0.635800	+	0.771854i	$0.280670\pi$
$5$	17.2592i	1.54371i	−0.635800	+	0.771854i	$0.719330\pi$
$6$	− 2.38767i	− 0.162461i
$7$	6.40763i	0.345979i	0.984924	+	0.172990i	$0.0553427\pi$
$7$	6.40763i	0.345979i	−0.984924	+	0.172990i	$0.944657\pi$
$8$	−16.6411	−0.735439
$9$	26.7113	0.989306
$10$	− 76.6918i	− 2.42521i
$11$	− 55.3227i	− 1.51640i	−0.652020	−	0.758201i	$-0.726078\pi$
$11$	− 55.3227i	− 1.51640i	0.652020	−	0.758201i	$-0.273922\pi$
$12$	6.31102i	0.151819i
$13$	58.6637	1.25157	0.625784	−	0.779996i	$-0.284779\pi$
$13$	58.6637	1.25157	0.625784	+	0.779996i	$0.284779\pi$
$14$	− 28.4725i	− 0.543543i
$15$	−9.27398	−0.159635
$16$	−20.0149	−0.312732
$17$	0	0
$17$	0	0
$18$	−118.692	−1.55423
$19$	91.1866	1.10103	0.550517	−	0.834824i	$-0.314431\pi$
$19$	91.1866	1.10103	0.550517	+	0.834824i	$0.314431\pi$
$20$	202.709i	2.26636i
$21$	−3.44305	−0.0357779
$22$	245.829i	2.38231i
$23$	− 120.879i	− 1.09587i	−0.836521	−	0.547935i	$-0.815414\pi$
$23$	− 120.879i	− 1.09587i	0.836521	−	0.547935i	$-0.184586\pi$
$24$	− 8.94185i	− 0.0760520i
$25$	−172.879	−1.38303
$26$	−260.674	−1.96625
$27$	28.8610i	0.205715i
$28$	75.2576i	0.507941i
$29$	− 215.755i	− 1.38154i	−0.723074	−	0.690771i	$-0.757271\pi$
$29$	− 215.755i	− 1.38154i	0.723074	−	0.690771i	$-0.242729\pi$
$30$	41.2093	0.250792
$31$	− 17.5166i	− 0.101486i	−0.998712	−	0.0507432i	$-0.983841\pi$
$31$	− 17.5166i	− 0.101486i	0.998712	−	0.0507432i	$-0.0161590\pi$
$32$	222.065	1.22675
$33$	29.7269	0.156812
$34$	0	0
$35$	−110.590	−0.534091
$36$	313.724	1.45243
$37$	− 8.40485i	− 0.0373446i	−0.999826	−	0.0186723i	$-0.994056\pi$
$37$	− 8.40485i	− 0.0373446i	0.999826	−	0.0186723i	$-0.00594392\pi$
$38$	−405.191	−1.72975
$39$	31.5221i	0.129425i
$40$	− 287.211i	− 1.13530i
$41$	− 99.9530i	− 0.380733i	−0.981713	−	0.190366i	$-0.939032\pi$
$41$	− 99.9530i	− 0.380733i	0.981713	−	0.190366i	$-0.0609676\pi$
$42$	15.2993	0.0562080
$43$	81.5297	0.289143	0.144572	−	0.989494i	$-0.453820\pi$
$43$	81.5297	0.289143	0.144572	+	0.989494i	$0.453820\pi$
$44$	− 649.766i	− 2.22627i
$45$	461.015i	1.52720i
$46$	537.130i	1.72164i
$47$	195.351	0.606275	0.303137	−	0.952947i	$-0.401966\pi$
$47$	195.351	0.606275	0.303137	+	0.952947i	$0.401966\pi$
$48$	− 10.7547i	− 0.0323398i
$49$	301.942	0.880298
$50$	768.195	2.17278
$51$	0	0
$52$	689.006	1.83746
$53$	−260.322	−0.674678	−0.337339	−	0.941383i	$-0.609527\pi$
$53$	−260.322	−0.674678	−0.337339	+	0.941383i	$0.609527\pi$
$54$	− 128.245i	− 0.323184i
$55$	954.825	2.34088
$56$	− 106.630i	− 0.254447i
$57$	48.9979i	0.113858i
$58$	958.715i	2.17044i
$59$	536.401	1.18362	0.591809	−	0.806078i	$-0.298414\pi$
$59$	536.401	1.18362	0.591809	+	0.806078i	$0.298414\pi$
$60$	−108.923	−0.234365
$61$	265.689i	0.557671i	0.960339	+	0.278836i	$0.0899485\pi$
$61$	265.689i	0.557671i	−0.960339	+	0.278836i	$0.910052\pi$
$62$	77.8357i	0.159438i
$63$	171.156i	0.342280i
$64$	−826.636	−1.61452
$65$	1012.49i	1.93206i
$66$	−132.093	−0.246356
$67$	514.794	0.938687	0.469344	−	0.883016i	$-0.344491\pi$
$67$	514.794	0.938687	0.469344	+	0.883016i	$0.344491\pi$
$68$	0	0
$69$	64.9526	0.113324
$70$	491.412	0.839071
$71$	− 704.023i	− 1.17679i	−0.808573	−	0.588396i	$-0.799760\pi$
$71$	− 704.023i	− 1.17679i	0.808573	−	0.588396i	$-0.200240\pi$
$72$	−444.504	−0.727574
$73$	184.948i	0.296528i	0.988948	+	0.148264i	$0.0473685\pi$
$73$	184.948i	0.296528i	−0.988948	+	0.148264i	$0.952632\pi$
$74$	37.3473i	0.0586693i
$75$	− 92.8943i	− 0.143020i
$76$	1070.99	1.61646
$77$	354.487	0.524644
$78$	− 140.070i	− 0.203330i
$79$	34.8358i	0.0496118i	0.999692	+	0.0248059i	$0.00789678\pi$
$79$	34.8358i	0.0496118i	−0.999692	+	0.0248059i	$0.992103\pi$
$80$	− 345.440i	− 0.482767i
$81$	705.696	0.968033
$82$	444.145i	0.598141i
$83$	647.682	0.856534	0.428267	−	0.903652i	$-0.359124\pi$
$83$	647.682	0.856534	0.428267	+	0.903652i	$0.359124\pi$
$84$	−40.4386	−0.0525264
$85$	0	0
$86$	−362.280	−0.454252
$87$	115.933	0.142866
$88$	920.630i	1.11522i
$89$	−1060.24	−1.26275	−0.631377	−	0.775476i	$-0.717510\pi$
$89$	−1060.24	−1.26275	−0.631377	+	0.775476i	$0.717510\pi$
$90$	− 2048.53i	− 2.39927i
$91$	375.895i	0.433017i
$92$	− 1419.72i	− 1.60887i
$93$	9.41231	0.0104947
$94$	−868.051	−0.952474
$95$	1573.81i	1.69967i
$96$	119.324i	0.126859i
$97$	− 256.409i	− 0.268395i	−0.990955	−	0.134198i	$-0.957154\pi$
$97$	− 256.409i	− 0.268395i	0.990955	−	0.134198i	$-0.0428457\pi$
$98$	−1341.69	−1.38297
$99$	− 1477.74i	− 1.50019i
$100$	−2030.47	−2.03047
$101$	−1465.07	−1.44337	−0.721685	−	0.692222i	$-0.756632\pi$
$101$	−1465.07	−1.44337	−0.721685	+	0.692222i	$0.756632\pi$
$102$	0	0
$103$	−25.6805	−0.0245668	−0.0122834	−	0.999925i	$-0.503910\pi$
$103$	−25.6805	−0.0245668	−0.0122834	+	0.999925i	$0.503910\pi$
$104$	−976.227	−0.920452
$105$	− 59.4242i	− 0.0552306i
$106$	1156.75	1.05994
$107$	2030.70i	1.83472i	0.398060	+	0.917359i	$0.369683\pi$
$107$	2030.70i	1.83472i	−0.398060	+	0.917359i	$0.630317\pi$
$108$	338.973i	0.302015i
$109$	716.479i	0.629598i	0.949158	+	0.314799i	$0.101937\pi$
$109$	716.479i	0.629598i	−0.949158	+	0.314799i	$0.898063\pi$
$110$	−4242.80	−3.67759
$111$	4.51623	0.00386182
$112$	− 128.248i	− 0.108199i
$113$	− 16.9632i	− 0.0141218i	−0.999975	−	0.00706090i	$-0.997752\pi$
$113$	− 16.9632i	− 0.0141218i	0.999975	−	0.00706090i	$-0.00224757\pi$
$114$	− 217.724i	− 0.178875i
$115$	2086.27	1.69170
$116$	− 2534.05i	− 2.02828i
$117$	1566.98	1.23818
$118$	−2383.52	−1.85950
$119$	0	0
$120$	154.329	0.117402
$121$	−1729.60	−1.29948
$122$	− 1180.60i	− 0.876117i
$123$	53.7084	0.0393717
$124$	− 205.733i	− 0.148995i
$125$	− 826.356i	− 0.591292i
$126$	− 760.537i	− 0.537730i
$127$	1138.85	0.795724	0.397862	−	0.917445i	$-0.369752\pi$
$127$	1138.85	0.795724	0.397862	+	0.917445i	$0.369752\pi$
$128$	1896.67	1.30971
$129$	43.8089i	0.0299004i
$130$	− 4499.02i	− 3.03531i
$131$	− 618.036i	− 0.412199i	−0.978531	−	0.206100i	$-0.933923\pi$
$131$	− 618.036i	− 0.412199i	0.978531	−	0.206100i	$-0.0660771\pi$
$132$	349.143	0.230219
$133$	584.290i	0.380935i
$134$	−2287.50	−1.47470
$135$	−498.117	−0.317564
$136$	0	0
$137$	−2255.17	−1.40637	−0.703183	−	0.711009i	$-0.748238\pi$
$137$	−2255.17	−1.40637	−0.703183	+	0.711009i	$0.748238\pi$
$138$	−288.619	−0.178036
$139$	1014.29i	0.618930i	0.950911	+	0.309465i	$0.100150\pi$
$139$	1014.29i	0.618930i	−0.950911	+	0.309465i	$0.899850\pi$
$140$	−1298.88	−0.784113
$141$	104.969i	0.0626951i
$142$	3128.35i	1.84877i
$143$	− 3245.44i	− 1.89788i
$144$	−534.622	−0.309388
$145$	3723.76	2.13270
$146$	− 821.822i	− 0.465853i
$147$	162.245i	0.0910320i
$148$	− 98.7150i	− 0.0548265i
$149$	−345.080	−0.189732	−0.0948661	−	0.995490i	$-0.530242\pi$
$149$	−345.080	−0.189732	−0.0948661	+	0.995490i	$0.530242\pi$
$150$	412.779i	0.224688i
$151$	−2555.94	−1.37748	−0.688741	−	0.725007i	$-0.741836\pi$
$151$	−2555.94	−1.37748	−0.688741	+	0.725007i	$0.741836\pi$
$152$	−1517.44	−0.809743
$153$	0	0
$154$	−1575.18	−0.824230
$155$	302.322	0.156665
$156$	370.228i	0.190012i
$157$	3651.08	1.85598	0.927988	−	0.372610i	$-0.121537\pi$
$157$	3651.08	1.85598	0.927988	+	0.372610i	$0.121537\pi$
$158$	− 154.794i	− 0.0779415i
$159$	− 139.880i	− 0.0697688i
$160$	3832.66i	1.89374i
$161$	774.547	0.379148
$162$	−3135.79	−1.52081
$163$	− 211.756i	− 0.101755i	−0.998705	−	0.0508773i	$-0.983798\pi$
$163$	− 211.756i	− 0.101755i	0.998705	−	0.0508773i	$-0.0162018\pi$
$164$	− 1173.95i	− 0.558963i
$165$	513.062i	0.242072i
$166$	−2878.00	−1.34564
$167$	977.319i	0.452858i	0.974028	+	0.226429i	$0.0727051\pi$
$167$	977.319i	0.452858i	−0.974028	+	0.226429i	$0.927295\pi$
$168$	57.2960	0.0263124
$169$	1244.43	0.566424
$170$	0	0
$171$	2435.71	1.08926
$172$	957.567	0.424499
$173$	3607.58i	1.58543i	0.609594	+	0.792714i	$0.291333\pi$
$173$	3607.58i	1.58543i	−0.609594	+	0.792714i	$0.708667\pi$
$174$	−515.152	−0.224446
$175$	− 1107.75i	− 0.478501i
$176$	1107.28i	0.474228i
$177$	288.228i	0.122398i
$178$	4711.21	1.98382
$179$	−3180.10	−1.32789	−0.663943	−	0.747783i	$-0.731118\pi$
$179$	−3180.10	−1.32789	−0.663943	+	0.747783i	$0.731118\pi$
$180$	5414.62i	2.24212i
$181$	1094.34i	0.449401i	0.974428	+	0.224701i	$0.0721404\pi$
$181$	1094.34i	0.449401i	−0.974428	+	0.224701i	$0.927860\pi$
$182$	− 1670.30i	− 0.680281i
$183$	−142.764	−0.0576690
$184$	2011.56i	0.805945i
$185$	145.061	0.0576491
$186$	−41.8239	−0.0164875
$187$	0	0
$188$	2294.40	0.890088
$189$	−184.931	−0.0711731
$190$	− 6993.26i	− 2.67023i
$191$	−2323.98	−0.880406	−0.440203	−	0.897898i	$-0.645094\pi$
$191$	−2323.98	−0.880406	−0.440203	+	0.897898i	$0.645094\pi$
$192$	− 444.182i	− 0.166959i
$193$	− 2521.70i	− 0.940498i	−0.882534	−	0.470249i	$-0.844164\pi$
$193$	− 2521.70i	− 0.940498i	0.882534	−	0.470249i	$-0.155836\pi$
$194$	1139.36i	0.421656i
$195$	−544.046	−0.199795
$196$	3546.31	1.29239
$197$	2196.87i	0.794521i	0.917706	+	0.397260i	$0.130039\pi$
$197$	2196.87i	0.794521i	−0.917706	+	0.397260i	$0.869961\pi$
$198$	6566.39i	2.35683i
$199$	3596.88i	1.28129i	0.767839	+	0.640643i	$0.221332\pi$
$199$	3596.88i	1.28129i	−0.767839	+	0.640643i	$0.778668\pi$
$200$	2876.90	1.01714
$201$	276.617i	0.0970700i
$202$	6510.11	2.26757
$203$	1382.48	0.477985
$204$	0	0
$205$	1725.11	0.587740
$206$	114.112	0.0385951
$207$	− 3228.83i	− 1.08415i
$208$	−1174.15	−0.391406
$209$	− 5044.69i	− 1.66961i
$210$	264.054i	0.0867687i
$211$	− 3677.29i	− 1.19979i	−0.800080	−	0.599893i	$-0.795210\pi$
$211$	− 3677.29i	− 1.19979i	0.800080	−	0.599893i	$-0.204790\pi$
$212$	−3057.48	−0.990513
$213$	378.297	0.121692
$214$	− 9023.47i	− 2.88239i
$215$	1407.14i	0.446353i
$216$	− 480.278i	− 0.151291i
$217$	112.240	0.0351122
$218$	− 3183.70i	− 0.989116i
$219$	−99.3792	−0.0306640
$220$	11214.4	3.43671
$221$	0	0
$222$	−20.0680	−0.00606702
$223$	−972.699	−0.292093	−0.146047	−	0.989278i	$-0.546655\pi$
$223$	−972.699	−0.292093	−0.146047	+	0.989278i	$0.546655\pi$
$224$	1422.91i	0.424430i
$225$	−4617.82	−1.36824
$226$	75.3766i	0.0221857i
$227$	− 4696.68i	− 1.37326i	−0.727008	−	0.686629i	$-0.759090\pi$
$227$	− 4696.68i	− 1.37326i	0.727008	−	0.686629i	$-0.240910\pi$
$228$	575.480i	0.167158i
$229$	2105.80	0.607664	0.303832	−	0.952726i	$-0.401734\pi$
$229$	2105.80	0.607664	0.303832	+	0.952726i	$0.401734\pi$
$230$	−9270.42	−2.65771
$231$	190.479i	0.0542537i
$232$	3590.40i	1.01604i
$233$	641.400i	0.180341i	0.995926	+	0.0901706i	$0.0287412\pi$
$233$	641.400i	0.180341i	−0.995926	+	0.0901706i	$0.971259\pi$
$234$	−6962.94	−1.94522
$235$	3371.60i	0.935911i
$236$	6300.03	1.73770
$237$	−18.7185	−0.00513038
$238$	0	0
$239$	4322.85	1.16997	0.584984	−	0.811045i	$-0.301101\pi$
$239$	4322.85	1.16997	0.584984	+	0.811045i	$0.301101\pi$
$240$	185.617	0.0499231
$241$	2462.38i	0.658156i	0.944303	+	0.329078i	$0.106738\pi$
$241$	2462.38i	0.658156i	−0.944303	+	0.329078i	$0.893262\pi$
$242$	7685.56	2.04151
$243$	1158.44i	0.305820i
$244$	3120.52i	0.818732i
$245$	5211.28i	1.35892i
$246$	−238.655	−0.0618540
$247$	5349.35	1.37802
$248$	291.495i	0.0746370i
$249$	348.023i	0.0885746i
$250$	3671.94i	0.928936i
$251$	2631.96	0.661865	0.330932	−	0.943654i	$-0.392637\pi$
$251$	2631.96	0.661865	0.330932	+	0.943654i	$0.392637\pi$
$252$	2010.23i	0.502509i
$253$	−6687.35	−1.66178
$254$	−5060.54	−1.25010
$255$	0	0
$256$	−1814.81	−0.443068
$257$	3375.79	0.819361	0.409680	−	0.912229i	$-0.365640\pi$
$257$	3375.79	0.819361	0.409680	+	0.912229i	$0.365640\pi$
$258$	− 194.666i	− 0.0469744i
$259$	53.8552	0.0129204
$260$	11891.7i	2.83650i
$261$	− 5763.09i	− 1.36677i
$262$	2746.27i	0.647576i
$263$	1603.60	0.375977	0.187989	−	0.982171i	$-0.439803\pi$
$263$	1603.60	0.375977	0.187989	+	0.982171i	$0.439803\pi$
$264$	−494.688	−0.115325
$265$	− 4492.94i	− 1.04151i
$266$	− 2596.31i	− 0.598459i
$267$	− 569.705i	− 0.130582i
$268$	6046.26	1.37811
$269$	− 6543.59i	− 1.48316i	−0.670864	−	0.741580i	$-0.734077\pi$
$269$	− 6543.59i	− 1.48316i	0.670864	−	0.741580i	$-0.265923\pi$
$270$	2213.40	0.498901
$271$	−356.005	−0.0797998	−0.0398999	−	0.999204i	$-0.512704\pi$
$271$	−356.005	−0.0797998	−0.0398999	+	0.999204i	$0.512704\pi$
$272$	0	0
$273$	−201.982	−0.0447784
$274$	10020.9	2.20944
$275$	9564.15i	2.09724i
$276$	762.869	0.166374
$277$	3175.91i	0.688887i	0.938807	+	0.344444i	$0.111932\pi$
$277$	3175.91i	0.688887i	−0.938807	+	0.344444i	$0.888068\pi$
$278$	− 4507.05i	− 0.972356i
$279$	− 467.891i	− 0.100401i
$280$	1840.34	0.392791
$281$	−1667.30	−0.353960	−0.176980	−	0.984214i	$-0.556633\pi$
$281$	−1667.30	−0.353960	−0.176980	+	0.984214i	$0.556633\pi$
$282$	− 466.435i	− 0.0984957i
$283$	− 7800.28i	− 1.63844i	−0.573479	−	0.819220i	$-0.694407\pi$
$283$	− 7800.28i	− 1.63844i	0.573479	−	0.819220i	$-0.305593\pi$
$284$	− 8268.76i	− 1.72768i
$285$	−845.663	−0.175764
$286$	14421.2i	2.98162i
$287$	640.461	0.131726
$288$	5931.65	1.21363
$289$	0	0
$290$	−16546.6	−3.35052
$291$	137.778	0.0277549
$292$	2172.21i	0.435340i
$293$	−2403.16	−0.479161	−0.239580	−	0.970877i	$-0.577010\pi$
$293$	−2403.16	−0.479161	−0.239580	+	0.970877i	$0.577010\pi$
$294$	− 720.939i	− 0.143014i
$295$	9257.84i	1.82716i
$296$	139.866i	0.0274646i
$297$	1596.67	0.311947
$298$	1533.38	0.298074
$299$	− 7091.21i	− 1.37156i
$300$	− 1091.04i	− 0.209971i
$301$	522.412i	0.100038i
$302$	11357.4	2.16406
$303$	− 787.238i	− 0.149259i
$304$	−1825.09	−0.344329
$305$	−4585.57	−0.860881
$306$	0	0
$307$	−4027.80	−0.748791	−0.374395	−	0.927269i	$-0.622150\pi$
$307$	−4027.80	−0.748791	−0.374395	+	0.927269i	$0.622150\pi$
$308$	4163.46	0.770244
$309$	− 13.7991i	− 0.00254046i
$310$	−1343.38	−0.246125
$311$	3944.44i	0.719192i	0.933108	+	0.359596i	$0.117085\pi$
$311$	3944.44i	0.719192i	−0.933108	+	0.359596i	$0.882915\pi$
$312$	− 524.562i	− 0.0951843i
$313$	− 6540.41i	− 1.18111i	−0.806999	−	0.590553i	$-0.798910\pi$
$313$	− 6540.41i	− 1.18111i	0.806999	−	0.590553i	$-0.201090\pi$
$314$	−16223.7	−2.91579
$315$	−2954.01	−0.528380
$316$	409.147i	0.0728364i
$317$	3711.41i	0.657582i	0.944403	+	0.328791i	$0.106641\pi$
$317$	3711.41i	0.657582i	−0.944403	+	0.328791i	$0.893359\pi$
$318$	621.563i	0.109609i
$319$	−11936.2	−2.09497
$320$	− 14267.1i	− 2.49235i
$321$	−1091.17	−0.189729
$322$	−3441.73	−0.595652
$323$	0	0
$324$	8288.41	1.42119
$325$	−10141.7	−1.73096
$326$	940.945i	0.159859i
$327$	−384.990	−0.0651070
$328$	1663.33i	0.280005i
$329$	1251.74i	0.209759i
$330$	− 2279.81i	− 0.380301i
$331$	−3510.10	−0.582877	−0.291439	−	0.956590i	$-0.594134\pi$
$331$	−3510.10	−0.582877	−0.291439	+	0.956590i	$0.594134\pi$
$332$	7607.03	1.25750
$333$	− 224.504i	− 0.0369452i
$334$	− 4342.75i	− 0.711451i
$335$	8884.92i	1.44906i
$336$	68.9121	0.0111889
$337$	− 5718.46i	− 0.924346i	−0.886790	−	0.462173i	$-0.847070\pi$
$337$	− 5718.46i	− 0.924346i	0.886790	−	0.462173i	$-0.152930\pi$
$338$	−5529.68	−0.889867
$339$	9.11494	0.00146034
$340$	0	0
$341$	−969.067	−0.153894
$342$	−10823.2	−1.71126
$343$	4132.55i	0.650544i
$344$	−1356.74	−0.212647
$345$	1121.03i	0.174940i
$346$	− 16030.4i	− 2.49075i
$347$	− 677.787i	− 0.104857i	−0.998625	−	0.0524287i	$-0.983304\pi$
$347$	− 677.787i	− 0.104857i	0.998625	−	0.0524287i	$-0.0166962\pi$
$348$	1361.63	0.209745
$349$	−12161.8	−1.86534	−0.932670	−	0.360731i	$-0.882527\pi$
$349$	−12161.8	−1.86534	−0.932670	+	0.360731i	$0.882527\pi$
$350$	4922.31i	0.751738i
$351$	1693.09i	0.257466i
$352$	− 12285.3i	− 1.86025i
$353$	5681.58	0.856657	0.428328	−	0.903623i	$-0.359103\pi$
$353$	5681.58	0.856657	0.428328	+	0.903623i	$0.359103\pi$
$354$	− 1280.75i	− 0.192291i
$355$	12150.9	1.81662
$356$	−12452.5	−1.85388
$357$	0	0
$358$	14130.9	2.08614
$359$	5883.91	0.865017	0.432508	−	0.901630i	$-0.357629\pi$
$359$	5883.91	0.865017	0.432508	+	0.901630i	$0.357629\pi$
$360$	− 7671.78i	− 1.12316i
$361$	1456.00	0.212276
$362$	− 4862.74i	− 0.706022i
$363$	− 929.379i	− 0.134380i
$364$	4414.89i	0.635723i
$365$	−3192.05	−0.457752
$366$	634.377	0.0905996
$367$	11526.1i	1.63939i	0.572801	+	0.819695i	$0.305857\pi$
$367$	11526.1i	1.63939i	−0.572801	+	0.819695i	$0.694143\pi$
$368$	2419.37i	0.342714i
$369$	− 2669.87i	− 0.376661i
$370$	−644.583	−0.0905683
$371$	− 1668.05i	− 0.233425i
$372$	110.548	0.0154076
$373$	3533.27	0.490471	0.245236	−	0.969464i	$-0.421135\pi$
$373$	3533.27	0.490471	0.245236	+	0.969464i	$0.421135\pi$
$374$	0	0
$375$	444.031	0.0611458
$376$	−3250.86	−0.445878
$377$	− 12657.0i	− 1.72909i
$378$	821.746	0.111815
$379$	− 13568.8i	− 1.83901i	−0.393082	−	0.919503i	$-0.628591\pi$
$379$	− 13568.8i	− 1.83901i	0.393082	−	0.919503i	$-0.371409\pi$
$380$	18484.4i	2.49534i
$381$	611.947i	0.0822861i
$382$	10326.7	1.38314
$383$	12948.4	1.72750	0.863748	−	0.503923i	$-0.168111\pi$
$383$	12948.4	1.72750	0.863748	+	0.503923i	$0.168111\pi$
$384$	1019.15i	0.135438i
$385$	6118.16i	0.809897i
$386$	11205.3i	1.47755i
$387$	2177.76	0.286051
$388$	− 3011.52i	− 0.394038i
$389$	−8901.44	−1.16021	−0.580104	−	0.814542i	$-0.696988\pi$
$389$	−8901.44	−1.16021	−0.580104	+	0.814542i	$0.696988\pi$
$390$	2417.49	0.313883
$391$	0	0
$392$	−5024.64	−0.647405
$393$	332.093	0.0426257
$394$	− 9761.88i	− 1.24821i
$395$	−601.237	−0.0765862
$396$	− 17356.1i	− 2.20246i
$397$	7180.09i	0.907703i	0.891077	+	0.453852i	$0.149950\pi$
$397$	7180.09i	0.907703i	−0.891077	+	0.453852i	$0.850050\pi$
$398$	− 15982.9i	− 2.01293i
$399$	−313.960	−0.0393926
$400$	3460.15	0.432519
$401$	− 5087.50i	− 0.633560i	−0.948499	−	0.316780i	$-0.897398\pi$
$401$	− 5087.50i	− 0.633560i	0.948499	−	0.316780i	$-0.102602\pi$
$402$	− 1229.16i	− 0.152500i
$403$	− 1027.59i	− 0.127017i
$404$	−17207.3	−2.11905
$405$	12179.7i	1.49436i
$406$	−6143.09	−0.750927
$407$	−464.979	−0.0566294
$408$	0	0
$409$	−5287.62	−0.639256	−0.319628	−	0.947543i	$-0.603558\pi$
$409$	−5287.62	−0.639256	−0.319628	+	0.947543i	$0.603558\pi$
$410$	−7665.57	−0.923355
$411$	− 1211.78i	− 0.145433i
$412$	−301.618	−0.0360671
$413$	3437.06i	0.409507i
$414$	14347.4i	1.70323i
$415$	11178.5i	1.32224i
$416$	13027.2	1.53536
$417$	−545.017	−0.0640038
$418$	22416.3i	2.62300i
$419$	− 15292.9i	− 1.78307i	−0.452956	−	0.891533i	$-0.649630\pi$
$419$	− 15292.9i	− 1.78307i	0.452956	−	0.891533i	$-0.350370\pi$
$420$	− 697.938i	− 0.0810854i
$421$	−1484.20	−0.171818	−0.0859089	−	0.996303i	$-0.527379\pi$
$421$	−1484.20	−0.171818	−0.0859089	+	0.996303i	$0.527379\pi$
$422$	16340.2i	1.88490i
$423$	5218.08	0.599792
$424$	4332.04	0.496185
$425$	0	0
$426$	−1680.98	−0.191182
$427$	−1702.43	−0.192943
$428$	23850.6i	2.69360i
$429$	1743.89	0.196261
$430$	− 6252.66i	− 0.701232i
$431$	− 2361.94i	− 0.263969i	−0.991252	−	0.131984i	$-0.957865\pi$
$431$	− 2361.94i	− 0.263969i	0.991252	−	0.131984i	$-0.0421348\pi$
$432$	− 577.649i	− 0.0643337i
$433$	12384.3	1.37449	0.687244	−	0.726426i	$-0.258820\pi$
$433$	12384.3	1.37449	0.687244	+	0.726426i	$0.258820\pi$
$434$	−498.742	−0.0551622
$435$	2000.91i	0.220543i
$436$	8415.05i	0.924330i
$437$	− 11022.5i	− 1.20659i
$438$	441.595	0.0481740
$439$	7612.65i	0.827636i	0.910360	+	0.413818i	$0.135805\pi$
$439$	7612.65i	0.827636i	−0.910360	+	0.413818i	$0.864195\pi$
$440$	−15889.3	−1.72158
$441$	8065.26	0.870885
$442$	0	0
$443$	11492.3	1.23254	0.616272	−	0.787533i	$-0.288642\pi$
$443$	11492.3	1.23254	0.616272	+	0.787533i	$0.288642\pi$
$444$	53.0432	0.00566963
$445$	− 18298.9i	− 1.94932i
$446$	4322.22	0.458886
$447$	− 185.424i	− 0.0196203i
$448$	− 5296.78i	− 0.558592i
$449$	13654.4i	1.43517i	0.696472	+	0.717584i	$0.254752\pi$
$449$	13654.4i	1.43517i	−0.696472	+	0.717584i	$0.745248\pi$
$450$	20519.5	2.14955
$451$	−5529.67	−0.577344
$452$	− 199.233i	− 0.0207326i
$453$	− 1373.40i	− 0.142446i
$454$	20869.9i	2.15743i
$455$	−6487.64	−0.668451
$456$	− 815.377i	− 0.0837358i
$457$	−12274.9	−1.25645	−0.628223	−	0.778034i	$-0.716217\pi$
$457$	−12274.9	−1.25645	−0.628223	+	0.778034i	$0.716217\pi$
$458$	−9357.20	−0.954657
$459$	0	0
$460$	24503.3	2.48363
$461$	932.142	0.0941739	0.0470869	−	0.998891i	$-0.485006\pi$
$461$	932.142	0.0941739	0.0470869	+	0.998891i	$0.485006\pi$
$462$	− 846.400i	− 0.0852340i
$463$	5569.01	0.558993	0.279496	−	0.960147i	$-0.409832\pi$
$463$	5569.01	0.558993	0.279496	+	0.960147i	$0.409832\pi$
$464$	4318.31i	0.432053i
$465$	162.449i	0.0162008i
$466$	− 2850.08i	− 0.283321i
$467$	−14662.2	−1.45286	−0.726428	−	0.687243i	$-0.758821\pi$
$467$	−14662.2	−1.45286	−0.726428	+	0.687243i	$0.758821\pi$
$468$	18404.2	1.81781
$469$	3298.61i	0.324766i
$470$	− 14981.8i	− 1.47034i
$471$	1961.86i	0.191927i
$472$	−8926.29	−0.870478
$473$	− 4510.45i	− 0.438458i
$474$	83.1765	0.00805996
$475$	−15764.3	−1.52277
$476$	0	0
$477$	−6953.53	−0.667464
$478$	−19208.8	−1.83805
$479$	− 12651.6i	− 1.20682i	−0.797432	−	0.603409i	$-0.793809\pi$
$479$	− 12651.6i	− 1.20682i	0.797432	−	0.603409i	$-0.206191\pi$
$480$	−2059.43	−0.195833
$481$	− 493.060i	− 0.0467393i
$482$	− 10941.7i	− 1.03398i
$483$	416.192i	0.0392079i
$484$	−20314.2	−1.90780
$485$	4425.40	0.414324
$486$	− 5147.58i	− 0.480451i
$487$	13319.7i	1.23937i	0.784852	+	0.619683i	$0.212739\pi$
$487$	13319.7i	1.23937i	−0.784852	+	0.619683i	$0.787261\pi$
$488$	− 4421.34i	− 0.410133i
$489$	113.784	0.0105225
$490$	− 23156.5i	− 2.13491i
$491$	19380.9	1.78136	0.890679	−	0.454633i	$-0.150229\pi$
$491$	19380.9	1.78136	0.890679	+	0.454633i	$0.150229\pi$
$492$	630.805	0.0578026
$493$	0	0
$494$	−23770.0	−2.16491
$495$	25504.6	2.31585
$496$	350.593i	0.0317381i
$497$	4511.12	0.407146
$498$	− 1546.45i	− 0.139153i
$499$	20101.1i	1.80330i	0.432466	+	0.901650i	$0.357644\pi$
$499$	20101.1i	1.80330i	−0.432466	+	0.901650i	$0.642356\pi$
$500$	− 9705.56i	− 0.868091i
$501$	−525.149	−0.0468302
$502$	−11695.2	−1.03981
$503$	− 6166.84i	− 0.546652i	−0.961921	−	0.273326i	$-0.911876\pi$
$503$	− 6166.84i	− 0.546652i	0.961921	−	0.273326i	$-0.0881238\pi$
$504$	− 2848.22i	− 0.251726i
$505$	− 25286.0i	− 2.22814i
$506$	29715.5	2.61070
$507$	668.679i	0.0585741i
$508$	13375.8	1.16822
$509$	14496.0	1.26233	0.631165	−	0.775649i	$-0.282577\pi$
$509$	14496.0	1.26233	0.631165	+	0.775649i	$0.282577\pi$
$510$	0	0
$511$	−1185.08	−0.102592
$512$	−7109.16	−0.613639
$513$	2631.74i	0.226499i
$514$	−15000.4	−1.28724
$515$	− 443.225i	− 0.0379239i
$516$	514.535i	0.0438976i
$517$	− 10807.4i	− 0.919357i
$518$	−239.307	−0.0202984
$519$	−1938.48	−0.163950
$520$	− 16848.9i	− 1.42091i
$521$	− 2642.20i	− 0.222182i	−0.993810	−	0.111091i	$-0.964565\pi$
$521$	− 2642.20i	− 0.222182i	0.993810	−	0.111091i	$-0.0354346\pi$
$522$	25608.5i	2.14723i
$523$	12574.5	1.05133	0.525665	−	0.850692i	$-0.323817\pi$
$523$	12574.5	1.05133	0.525665	+	0.850692i	$0.323817\pi$
$524$	− 7258.84i	− 0.605160i
$525$	595.232	0.0494820
$526$	−7125.63	−0.590670
$527$	0	0
$528$	−594.980	−0.0490401
$529$	−2444.72	−0.200930
$530$	19964.5i	1.63623i
$531$	14328.0	1.17096
$532$	6862.49i	0.559261i
$533$	− 5863.61i	− 0.476513i
$534$	2531.50i	0.205148i
$535$	−35048.2	−2.83227
$536$	−8566.72	−0.690347
$537$	− 1708.78i	− 0.137317i
$538$	29076.7i	2.33008i
$539$	− 16704.3i	− 1.33489i
$540$	−5850.39	−0.466224
$541$	− 18289.3i	− 1.45345i	−0.686927	−	0.726726i	$-0.741041\pi$
$541$	− 18289.3i	− 1.45345i	0.686927	−	0.726726i	$-0.258959\pi$
$542$	1581.92	0.125368
$543$	−588.028	−0.0464728
$544$	0	0
$545$	−12365.8	−0.971916
$546$	897.515	0.0703481
$547$	9989.86i	0.780869i	0.920631	+	0.390435i	$0.127675\pi$
$547$	9989.86i	0.780869i	−0.920631	+	0.390435i	$0.872325\pi$
$548$	−26487.0	−2.06472
$549$	7096.88i	0.551708i
$550$	− 42498.6i	− 3.29482i
$551$	− 19674.0i	− 1.52112i
$552$	−1080.88	−0.0833431
$553$	−223.215	−0.0171647
$554$	− 14112.3i	− 1.08226i
$555$	77.9464i	0.00596152i
$556$	11912.9i	0.908668i
$557$	−4181.95	−0.318124	−0.159062	−	0.987269i	$-0.550847\pi$
$557$	−4181.95	−0.318124	−0.159062	+	0.987269i	$0.550847\pi$
$558$	2079.09i	0.157733i
$559$	4782.84	0.361883
$560$	2213.45	0.167027
$561$	0	0
$562$	7408.70	0.556080
$563$	−10429.1	−0.780697	−0.390348	−	0.920667i	$-0.627645\pi$
$563$	−10429.1	−0.780697	−0.390348	+	0.920667i	$0.627645\pi$
$564$	1232.87i	0.0920443i
$565$	292.771	0.0217999
$566$	34660.8i	2.57403i
$567$	4521.84i	0.334920i
$568$	11715.7i	0.865458i
$569$	−8440.98	−0.621906	−0.310953	−	0.950425i	$-0.600648\pi$
$569$	−8440.98	−0.621906	−0.310953	+	0.950425i	$0.600648\pi$
$570$	3757.73	0.276130
$571$	27092.6i	1.98562i	0.119694	+	0.992811i	$0.461809\pi$
$571$	27092.6i	1.98562i	−0.119694	+	0.992811i	$0.538191\pi$
$572$	− 38117.7i	− 2.78633i
$573$	− 1248.76i	− 0.0910431i
$574$	−2845.91	−0.206944
$575$	20897.5i	1.51562i
$576$	−22080.5	−1.59726
$577$	−10864.3	−0.783857	−0.391929	−	0.919996i	$-0.628192\pi$
$577$	−10864.3	−0.783857	−0.391929	+	0.919996i	$0.628192\pi$
$578$	0	0
$579$	1355.00	0.0972572
$580$	43735.5	3.13107
$581$	4150.11i	0.296343i
$582$	−612.220	−0.0436037
$583$	14401.7i	1.02308i
$584$	− 3077.73i	− 0.218078i
$585$	27044.8i	1.91140i
$586$	10678.5	0.752774
$587$	−25173.8	−1.77008	−0.885038	−	0.465518i	$-0.845868\pi$
$587$	−25173.8	−1.77008	−0.885038	+	0.465518i	$0.845868\pi$
$588$	1905.56i	0.133646i
$589$	− 1597.28i	− 0.111740i
$590$	− 41137.5i	− 2.87052i
$591$	−1180.46	−0.0821617
$592$	168.222i	0.0116788i
$593$	−777.712	−0.0538563	−0.0269282	−	0.999637i	$-0.508573\pi$
$593$	−777.712	−0.0538563	−0.0269282	+	0.999637i	$0.508573\pi$
$594$	−7094.86	−0.490077
$595$	0	0
$596$	−4052.97	−0.278551
$597$	−1932.73	−0.132498
$598$	31510.0i	2.15475i
$599$	6690.40	0.456365	0.228182	−	0.973618i	$-0.426722\pi$
$599$	6690.40	0.456365	0.228182	+	0.973618i	$0.426722\pi$
$600$	1545.86i	0.105182i
$601$	− 1574.80i	− 0.106884i	−0.998571	−	0.0534421i	$-0.982981\pi$
$601$	− 1574.80i	− 0.106884i	0.998571	−	0.0534421i	$-0.0170193\pi$
$602$	− 2321.36i	− 0.157162i
$603$	13750.8	0.928649
$604$	−30019.6	−2.02232
$605$	− 29851.6i	− 2.00601i
$606$	3498.12i	0.234491i
$607$	24567.4i	1.64276i	0.570378	+	0.821382i	$0.306797\pi$
$607$	24567.4i	1.64276i	−0.570378	+	0.821382i	$0.693203\pi$
$608$	20249.4	1.35069
$609$	742.856i	0.0494286i
$610$	20376.1	1.35247
$611$	11460.0	0.758795
$612$	0	0
$613$	−27992.8	−1.84440	−0.922201	−	0.386710i	$-0.873611\pi$
$613$	−27992.8	−1.84440	−0.922201	+	0.386710i	$0.873611\pi$
$614$	17897.7	1.17637
$615$	926.962i	0.0607784i
$616$	−5899.05	−0.385843
$617$	3787.41i	0.247124i	0.992337	+	0.123562i	$0.0394317\pi$
$617$	3787.41i	0.247124i	−0.992337	+	0.123562i	$0.960568\pi$
$618$	61.3167i	0.00399113i
$619$	4066.72i	0.264063i	0.991246	+	0.132032i	$0.0421500\pi$
$619$	4066.72i	0.264063i	−0.991246	+	0.132032i	$0.957850\pi$
$620$	3550.78	0.230004
$621$	3488.69	0.225437
$622$	− 17527.3i	− 1.12987i
$623$	− 6793.62i	− 0.436887i
$624$	− 630.911i	− 0.0404754i
$625$	−7347.68	−0.470251
$626$	29062.6i	1.85555i
$627$	2710.70	0.172655
$628$	42882.0	2.72481
$629$	0	0
$630$	13126.2	0.830099
$631$	11686.8	0.737309	0.368655	−	0.929566i	$-0.379818\pi$
$631$	11686.8	0.737309	0.368655	+	0.929566i	$0.379818\pi$
$632$	− 579.705i	− 0.0364864i
$633$	1975.94	0.124070
$634$	− 16491.8i	− 1.03308i
$635$	19655.7i	1.22836i
$636$	− 1642.90i	− 0.102429i
$637$	17713.1	1.10175
$638$	53038.8	3.29126
$639$	− 18805.4i	− 1.16421i
$640$	32734.9i	2.02181i
$641$	23652.6i	1.45744i	0.684810	+	0.728721i	$0.259885\pi$
$641$	23652.6i	1.45744i	−0.684810	+	0.728721i	$0.740115\pi$
$642$	4848.64	0.298069
$643$	− 2712.61i	− 0.166368i	−0.996534	−	0.0831841i	$-0.973491\pi$
$643$	− 2712.61i	− 0.166368i	0.996534	−	0.0831841i	$-0.0265089\pi$
$644$	9097.06	0.556637
$645$	−756.105	−0.0461575
$646$	0	0
$647$	3672.97	0.223183	0.111592	−	0.993754i	$-0.464405\pi$
$647$	3672.97	0.223183	0.111592	+	0.993754i	$0.464405\pi$
$648$	−11743.5	−0.711929
$649$	− 29675.2i	− 1.79484i
$650$	45065.2	2.71939
$651$	60.3106i	0.00363097i
$652$	− 2487.08i	− 0.149389i
$653$	26090.8i	1.56357i	0.623548	+	0.781785i	$0.285691\pi$
$653$	26090.8i	1.56357i	−0.623548	+	0.781785i	$0.714309\pi$
$654$	1710.72	0.102285
$655$	10666.8	0.636315
$656$	2000.54i	0.119067i
$657$	4940.19i	0.293357i
$658$	− 5562.14i	− 0.329536i
$659$	−14303.7	−0.845515	−0.422758	−	0.906243i	$-0.638938\pi$
$659$	−14303.7	−0.845515	−0.422758	+	0.906243i	$0.638938\pi$
$660$	6025.92i	0.355392i
$661$	9780.49	0.575517	0.287759	−	0.957703i	$-0.407090\pi$
$661$	9780.49	0.575517	0.287759	+	0.957703i	$0.407090\pi$
$662$	15597.2	0.915716
$663$	0	0
$664$	−10778.1	−0.629928
$665$	−10084.4	−0.588052
$666$	997.593i	0.0580419i
$667$	−26080.2	−1.51399
$668$	11478.6i	0.664852i
$669$	− 522.667i	− 0.0302055i
$670$	− 39480.4i	− 2.27651i
$671$	14698.6	0.845654
$672$	−764.582	−0.0438905
$673$	21380.1i	1.22458i	0.790634	+	0.612289i	$0.209751\pi$
$673$	21380.1i	1.22458i	−0.790634	+	0.612289i	$0.790249\pi$
$674$	25410.2i	1.45217i
$675$	− 4989.47i	− 0.284511i
$676$	14615.9	0.831581
$677$	− 7244.51i	− 0.411269i	−0.978629	−	0.205635i	$-0.934074\pi$
$677$	− 7244.51i	− 0.411269i	0.978629	−	0.205635i	$-0.0659258\pi$
$678$	−40.5026	−0.00229424
$679$	1642.97	0.0928593
$680$	0	0
$681$	2523.70	0.142009
$682$	4306.08	0.241772
$683$	− 4140.24i	− 0.231950i	−0.993252	−	0.115975i	$-0.963001\pi$
$683$	− 4140.24i	− 0.231950i	0.993252	−	0.115975i	$-0.0369992\pi$
$684$	28607.4	1.59917
$685$	− 38922.4i	− 2.17102i
$686$	− 18363.1i	− 1.02202i
$687$	1131.52i	0.0628388i
$688$	−1631.81	−0.0904244
$689$	−15271.4	−0.844406
$690$	− 4981.33i	− 0.274835i
$691$	− 30750.3i	− 1.69290i	−0.532466	−	0.846451i	$-0.678735\pi$
$691$	− 30750.3i	− 1.69290i	0.532466	−	0.846451i	$-0.321265\pi$
$692$	42371.0i	2.32761i
$693$	9468.81	0.519034
$694$	3011.77i	0.164734i
$695$	−17505.9	−0.955448
$696$	−1929.25	−0.105069
$697$	0	0
$698$	54041.2	2.93050
$699$	−344.647	−0.0186492
$700$	− 13010.5i	− 0.702500i
$701$	−20466.3	−1.10271	−0.551356	−	0.834270i	$-0.685890\pi$
$701$	−20466.3	−1.10271	−0.551356	+	0.834270i	$0.685890\pi$
$702$	− 7523.32i	− 0.404487i
$703$	− 766.410i	− 0.0411176i
$704$	45731.8i	2.44827i
$705$	−1811.68	−0.0967830
$706$	−25246.3	−1.34583
$707$	− 9387.65i	− 0.499376i
$708$	3385.24i	0.179696i
$709$	− 24814.8i	− 1.31444i	−0.753697	−	0.657222i	$-0.771731\pi$
$709$	− 24814.8i	− 1.31444i	0.753697	−	0.657222i	$-0.228269\pi$
$710$	−53992.8	−2.85396
$711$	930.509i	0.0490813i
$712$	17643.5	0.928678
$713$	−2117.39	−0.111216
$714$	0	0
$715$	56013.6	2.92978
$716$	−37350.3	−1.94950
$717$	2322.83i	0.120987i
$718$	−26145.4	−1.35897
$719$	24507.8i	1.27119i	0.772021	+	0.635597i	$0.219246\pi$
$719$	24507.8i	1.27119i	−0.772021	+	0.635597i	$0.780754\pi$
$720$	− 9227.14i	− 0.477604i
$721$	− 164.551i	− 0.00849960i
$722$	−6469.79	−0.333491
$723$	−1323.12	−0.0680601
$724$	12853.0i	0.659778i
$725$	37299.6i	1.91072i
$726$	4129.73i	0.211114i
$727$	−7468.67	−0.381015	−0.190507	−	0.981686i	$-0.561013\pi$
$727$	−7468.67	−0.381015	−0.190507	+	0.981686i	$0.561013\pi$
$728$	− 6255.30i	− 0.318457i
$729$	18431.3	0.936408
$730$	14184.0	0.719140
$731$	0	0
$732$	−1676.77	−0.0846654
$733$	−2801.95	−0.141190	−0.0705952	−	0.997505i	$-0.522490\pi$
$733$	−2801.95	−0.141190	−0.0705952	+	0.997505i	$0.522490\pi$
$734$	− 51216.5i	− 2.57552i
$735$	−2800.21	−0.140527
$736$	− 26843.0i	− 1.34436i
$737$	− 28479.8i	− 1.42343i
$738$	11863.7i	0.591745i
$739$	−5867.20	−0.292055	−0.146027	−	0.989281i	$-0.546649\pi$
$739$	−5867.20	−0.292055	−0.146027	+	0.989281i	$0.546649\pi$
$740$	1703.74	0.0846361
$741$	2874.40i	0.142502i
$742$	7412.02i	0.366717i
$743$	23585.7i	1.16457i	0.812984	+	0.582286i	$0.197841\pi$
$743$	23585.7i	1.16457i	−0.812984	+	0.582286i	$0.802159\pi$
$744$	−156.631	−0.00771824
$745$	− 5955.80i	− 0.292891i
$746$	−15700.2	−0.770543
$747$	17300.4	0.847375
$748$	0	0
$749$	−13011.9	−0.634775
$750$	−1973.07	−0.0960616
$751$	− 15142.8i	− 0.735775i	−0.929870	−	0.367887i	$-0.880081\pi$
$751$	− 15142.8i	− 0.735775i	0.929870	−	0.367887i	$-0.119919\pi$
$752$	−3909.93	−0.189602
$753$	1414.25i	0.0684437i
$754$	56241.8i	2.71645i
$755$	− 44113.5i	− 2.12643i
$756$	−2172.01	−0.104491
$757$	13149.7	0.631353	0.315676	−	0.948867i	$-0.397769\pi$
$757$	13149.7	0.631353	0.315676	+	0.948867i	$0.397769\pi$
$758$	60293.5i	2.88913i
$759$	− 3593.36i	− 0.171845i
$760$	− 26189.8i	− 1.25001i
$761$	−12831.3	−0.611213	−0.305607	−	0.952158i	$-0.598859\pi$
$761$	−12831.3	−0.611213	−0.305607	+	0.952158i	$0.598859\pi$
$762$	− 2719.21i	− 0.129274i
$763$	−4590.93	−0.217828
$764$	−27295.2	−1.29255
$765$	0	0
$766$	−57536.6	−2.71394
$767$	31467.3	1.48138
$768$	− 975.162i	− 0.0458179i
$769$	−11365.1	−0.532949	−0.266474	−	0.963842i	$-0.585859\pi$
$769$	−11365.1	−0.532949	−0.266474	+	0.963842i	$0.585859\pi$
$770$	− 27186.3i	− 1.27237i
$771$	1813.93i	0.0847304i
$772$	− 29617.4i	− 1.38077i
$773$	16551.8	0.770150	0.385075	−	0.922885i	$-0.374176\pi$
$773$	16551.8	0.770150	0.385075	+	0.922885i	$0.374176\pi$
$774$	−9676.96	−0.449394
$775$	3028.26i	0.140359i
$776$	4266.91i	0.197388i
$777$	28.9383i	0.00133611i
$778$	39553.9	1.82272
$779$	− 9114.37i	− 0.419199i
$780$	−6389.83	−0.293324
$781$	−38948.5	−1.78449
$782$	0	0
$783$	6226.91	0.284204
$784$	−6043.33	−0.275298
$785$	63014.7i	2.86509i
$786$	−1475.67	−0.0669661
$787$	− 32051.7i	− 1.45174i	−0.687832	−	0.725870i	$-0.741438\pi$
$787$	− 32051.7i	− 1.45174i	0.687832	−	0.725870i	$-0.258562\pi$
$788$	25802.3i	1.16646i
$789$	861.670i	0.0388799i
$790$	2671.62	0.120319
$791$	108.694	0.00488585
$792$	24591.2i	1.10330i
$793$	15586.3i	0.697964i
$794$	− 31905.0i	− 1.42603i
$795$	2414.22	0.107703
$796$	42245.4i	1.88109i
$797$	−27679.0	−1.23016	−0.615082	−	0.788463i	$-0.710877\pi$
$797$	−27679.0	−1.23016	−0.615082	+	0.788463i	$0.710877\pi$
$798$	1395.09	0.0618869
$799$	0	0
$800$	−38390.5	−1.69664
$801$	−28320.3	−1.24925
$802$	22606.5i	0.995340i
$803$	10231.8	0.449655
$804$	3248.87i	0.142511i
$805$	13368.0i	0.585294i
$806$	4566.13i	0.199547i
$807$	3516.11	0.153374
$808$	24380.4	1.06151
$809$	− 7438.60i	− 0.323272i	−0.986850	−	0.161636i	$-0.948323\pi$
$809$	− 7438.60i	− 0.323272i	0.986850	−	0.161636i	$-0.0516771\pi$
$810$	− 54121.1i	− 2.34768i
$811$	10261.1i	0.444288i	0.975014	+	0.222144i	$0.0713055\pi$
$811$	10261.1i	0.444288i	−0.975014	+	0.222144i	$0.928694\pi$
$812$	16237.2	0.701742
$813$	− 191.294i	− 0.00825213i
$814$	2066.15	0.0889663
$815$	3654.73	0.157079
$816$	0	0
$817$	7434.42	0.318357
$818$	23495.7	1.00429
$819$	10040.6i	0.428386i
$820$	20261.4	0.862876
$821$	− 21739.5i	− 0.924134i	−0.886845	−	0.462067i	$-0.847108\pi$
$821$	− 21739.5i	− 0.924134i	0.886845	−	0.462067i	$-0.152892\pi$
$822$	5384.60i	0.228479i
$823$	− 23728.9i	− 1.00503i	−0.864569	−	0.502514i	$-0.832408\pi$
$823$	− 23728.9i	− 1.00503i	0.864569	−	0.502514i	$-0.167592\pi$
$824$	427.352	0.0180674
$825$	−5139.16	−0.216876
$826$	− 15272.7i	− 0.643347i
$827$	− 21101.5i	− 0.887270i	−0.896207	−	0.443635i	$-0.853689\pi$
$827$	− 21101.5i	− 0.887270i	0.896207	−	0.443635i	$-0.146311\pi$
$828$	− 37922.6i	− 1.59167i
$829$	−1621.77	−0.0679448	−0.0339724	−	0.999423i	$-0.510816\pi$
$829$	−1621.77	−0.0679448	−0.0339724	+	0.999423i	$0.510816\pi$
$830$	− 49671.9i	− 2.07727i
$831$	−1706.53	−0.0712381
$832$	−48493.6	−2.02069
$833$	0	0
$834$	2421.80	0.100552
$835$	−16867.7	−0.699080
$836$	− 59250.0i	− 2.45120i
$837$	505.547	0.0208773
$838$	67954.3i	2.80124i
$839$	1287.05i	0.0529607i	0.999649	+	0.0264803i	$0.00842994\pi$
$839$	1287.05i	0.0529607i	−0.999649	+	0.0264803i	$0.991570\pi$
$840$	988.883i	0.0406187i
$841$	−22161.3	−0.908658
$842$	6595.08	0.269930
$843$	− 895.900i	− 0.0366031i
$844$	− 43189.8i	− 1.76144i
$845$	21477.9i	0.874392i
$846$	−23186.7	−0.942289
$847$	− 11082.7i	− 0.449592i
$848$	5210.30	0.210994
$849$	4191.37	0.169432
$850$	0	0
$851$	−1015.97	−0.0409248
$852$	4443.10	0.178660
$853$	18173.4i	0.729479i	0.931110	+	0.364740i	$0.118842\pi$
$853$	18173.4i	0.729479i	−0.931110	+	0.364740i	$0.881158\pi$
$854$	7564.83	0.303118
$855$	42038.4i	1.68150i
$856$	− 33793.0i	− 1.34932i
$857$	39590.2i	1.57803i	0.614371	+	0.789017i	$0.289410\pi$
$857$	39590.2i	1.57803i	−0.614371	+	0.789017i	$0.710590\pi$
$858$	−7749.04	−0.308331
$859$	17747.9	0.704948	0.352474	−	0.935822i	$-0.385341\pi$
$859$	17747.9	0.704948	0.352474	+	0.935822i	$0.385341\pi$
$860$	16526.8i	0.655302i
$861$	344.143i	0.0136218i
$862$	10495.3i	0.414702i
$863$	−8155.37	−0.321683	−0.160841	−	0.986980i	$-0.551421\pi$
$863$	−8155.37	−0.321683	−0.160841	+	0.986980i	$0.551421\pi$
$864$	6409.03i	0.252361i
$865$	−62263.8	−2.44744
$866$	−55030.3	−2.15936
$867$	0	0
$868$	1318.26	0.0515491
$869$	1927.21	0.0752315
$870$	− 8891.11i	− 0.346479i
$871$	30199.7	1.17483
$872$	− 11923.0i	− 0.463031i
$873$	− 6849.00i	− 0.265525i
$874$	48979.1i	1.89559i
$875$	5294.98	0.204575
$876$	−1167.21	−0.0450187
$877$	− 37979.1i	− 1.46233i	−0.682202	−	0.731164i	$-0.738977\pi$
$877$	− 37979.1i	− 1.46233i	0.682202	−	0.731164i	$-0.261023\pi$
$878$	− 33827.1i	− 1.30024i
$879$	− 1291.30i	− 0.0495502i
$880$	−19110.7	−0.732069
$881$	18289.7i	0.699427i	0.936857	+	0.349714i	$0.113721\pi$
$881$	18289.7i	0.699427i	−0.936857	+	0.349714i	$0.886279\pi$
$882$	−35838.3	−1.36818
$883$	18414.9	0.701825	0.350912	−	0.936408i	$-0.385871\pi$
$883$	18414.9	0.701825	0.350912	+	0.936408i	$0.385871\pi$
$884$	0	0
$885$	−4974.57	−0.188947
$886$	−51066.6	−1.93636
$887$	50689.5i	1.91881i	0.282028	+	0.959406i	$0.408993\pi$
$887$	50689.5i	1.91881i	−0.282028	+	0.959406i	$0.591007\pi$
$888$	−75.1549	−0.00284013
$889$	7297.35i	0.275304i
$890$	81311.6i	3.06244i
$891$	− 39041.0i	− 1.46793i
$892$	−11424.4	−0.428830
$893$	17813.4	0.667529
$894$	823.939i	0.0308240i
$895$	− 54885.8i	− 2.04987i
$896$	12153.1i	0.453133i
$897$	3810.36	0.141833
$898$	− 60673.8i	− 2.25469i
$899$	−3779.30	−0.140208
$900$	−54236.4	−2.00875
$901$	0	0
$902$	24571.3	0.907023
$903$	−280.711	−0.0103449
$904$	282.286i	0.0103857i
$905$	−18887.4	−0.693744
$906$	6102.76i	0.223786i
$907$	− 44568.0i	− 1.63160i	−0.578338	−	0.815798i	$-0.696298\pi$
$907$	− 44568.0i	− 1.63160i	0.578338	−	0.815798i	$-0.303702\pi$
$908$	− 55162.5i	− 2.01612i
$909$	−39134.0	−1.42794
$910$	28828.1	1.05016
$911$	7809.03i	0.284001i	0.989867	+	0.142000i	$0.0453535\pi$
$911$	7809.03i	0.284001i	−0.989867	+	0.142000i	$0.954647\pi$
$912$	− 980.685i	− 0.0356072i
$913$	− 35831.5i	− 1.29885i
$914$	54543.9	1.97391
$915$	− 2463.99i	− 0.0890241i
$916$	24732.6	0.892128
$917$	3960.15	0.142612
$918$	0	0
$919$	−36710.8	−1.31771	−0.658856	−	0.752269i	$-0.728959\pi$
$919$	−36710.8	−1.31771	−0.658856	+	0.752269i	$0.728959\pi$
$920$	−34717.8	−1.24414
$921$	− 2164.28i	− 0.0774328i
$922$	−4142.00	−0.147950
$923$	− 41300.6i	− 1.47284i
$924$	2237.18i	0.0796512i
$925$	1453.02i	0.0516488i
$926$	−24746.1	−0.878193
$927$	−685.960	−0.0243041
$928$	− 47911.7i	− 1.69481i
$929$	15068.9i	0.532180i	0.963948	+	0.266090i	$0.0857319\pi$
$929$	15068.9i	0.532180i	−0.963948	+	0.266090i	$0.914268\pi$
$930$	− 721.847i	− 0.0254519i
$931$	27533.1	0.969238
$932$	7533.25i	0.264764i
$933$	−2119.49	−0.0743719
$934$	65151.8	2.28248
$935$	0	0
$936$	−26076.3	−0.910609
$937$	−38416.7	−1.33940	−0.669700	−	0.742632i	$-0.733577\pi$
$937$	−38416.7	−1.33940	−0.669700	+	0.742632i	$0.733577\pi$
$938$	− 14657.5i	− 0.510217i
$939$	3514.40	0.122139
$940$	39599.5i	1.37404i
$941$	− 34621.7i	− 1.19940i	−0.800224	−	0.599701i	$-0.795286\pi$
$941$	− 34621.7i	− 1.19940i	0.800224	−	0.599701i	$-0.204714\pi$
$942$	− 8717.59i	− 0.301523i
$943$	−12082.2	−0.417233
$944$	−10736.0	−0.370155
$945$	− 3191.75i	− 0.109871i
$946$	20042.3i	0.688829i
$947$	20331.6i	0.697665i	0.937185	+	0.348832i	$0.113422\pi$
$947$	20331.6i	0.697665i	−0.937185	+	0.348832i	$0.886578\pi$
$948$	−219.849	−0.00753204
$949$	10849.7i	0.371124i
$950$	70049.1	2.39231
$951$	−1994.28	−0.0680009
$952$	0	0
$953$	17934.5	0.609607	0.304803	−	0.952415i	$-0.401409\pi$
$953$	17934.5	0.609607	0.304803	+	0.952415i	$0.401409\pi$
$954$	30898.2	1.04860
$955$	− 40110.0i	− 1.35909i
$956$	50771.9	1.71766
$957$	− 6413.73i	− 0.216642i
$958$	56217.8i	1.89594i
$959$	− 14450.3i	− 0.486573i
$960$	7666.21	0.257735
$961$	29484.2	0.989701
$962$	2190.93i	0.0734287i
$963$	54242.5i	1.81510i
$964$	28920.6i	0.966256i
$965$	43522.5	1.45185
$966$	− 1849.37i	− 0.0615966i
$967$	−8758.07	−0.291252	−0.145626	−	0.989340i	$-0.546520\pi$
$967$	−8758.07	−0.291252	−0.145626	+	0.989340i	$0.546520\pi$
$968$	28782.5	0.955686
$969$	0	0
$970$	−19664.4	−0.650914
$971$	52844.2	1.74650	0.873250	−	0.487273i	$-0.162008\pi$
$971$	52844.2	1.74650	0.873250	+	0.487273i	$0.162008\pi$
$972$	13605.9i	0.448982i
$973$	−6499.22	−0.214137
$974$	− 59186.4i	− 1.94708i
$975$	− 5449.52i	− 0.178999i
$976$	− 5317.72i	− 0.174402i
$977$	45377.7	1.48594	0.742970	−	0.669325i	$-0.233417\pi$
$977$	45377.7	1.48594	0.742970	+	0.669325i	$0.233417\pi$
$978$	−505.604	−0.0165311
$979$	58655.3i	1.91484i
$980$	61206.5i	1.99507i
$981$	19138.1i	0.622866i
$982$	−86119.6	−2.79856
$983$	− 20092.8i	− 0.651944i	−0.945379	−	0.325972i	$-0.894309\pi$
$983$	− 20092.8i	− 0.651944i	0.945379	−	0.325972i	$-0.105691\pi$
$984$	−893.765	−0.0289555
$985$	−37916.2	−1.22651
$986$	0	0
$987$	−672.604	−0.0216912
$988$	62828.1	2.02311
$989$	− 9855.23i	− 0.316863i
$990$	−113331.	−3.63826
$991$	8915.20i	0.285773i	0.989739	+	0.142886i	$0.0456383\pi$
$991$	8915.20i	0.285773i	−0.989739	+	0.142886i	$0.954362\pi$
$992$	− 3889.83i	− 0.124498i
$993$	− 1886.10i	− 0.0602756i
$994$	−20045.3	−0.639637
$995$	−62079.1	−1.97793
$996$	4087.53i	0.130039i
$997$	− 12587.1i	− 0.399837i	−0.979812	−	0.199919i	$-0.935932\pi$
$997$	− 12587.1i	− 0.399837i	0.979812	−	0.199919i	$-0.0640678\pi$
$998$	− 89319.8i	− 2.83303i
$999$	242.572	0.00768234

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	289.4.b.f.288.2		24
17.4	even	4		289.4.a.i.1.12	yes	12
17.13	even	4		289.4.a.h.1.12	✓	12
17.16	even	2	inner	289.4.b.f.288.1		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
289.4.a.h.1.12	✓	12	17.13	even	4
289.4.a.i.1.12	yes	12	17.4	even	4
289.4.b.f.288.1		24	17.16	even	2	inner
289.4.b.f.288.2		24	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 289 = 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	289.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.44354 q^{2} +0.537336i q^{3} +11.7450 q^{4} +17.2592i q^{5} -2.38767i q^{6} +6.40763i q^{7} -16.6411 q^{8} +26.7113 q^{9} +O(q^{10})\)	\(q-4.44354 q^{2} +0.537336i q^{3} +11.7450 q^{4} +17.2592i q^{5} -2.38767i q^{6} +6.40763i q^{7} -16.6411 q^{8} +26.7113 q^{9} -76.6918i q^{10} -55.3227i q^{11} +6.31102i q^{12} +58.6637 q^{13} -28.4725i q^{14} -9.27398 q^{15} -20.0149 q^{16} -118.692 q^{18} +91.1866 q^{19} +202.709i q^{20} -3.44305 q^{21} +245.829i q^{22} -120.879i q^{23} -8.94185i q^{24} -172.879 q^{25} -260.674 q^{26} +28.8610i q^{27} +75.2576i q^{28} -215.755i q^{29} +41.2093 q^{30} -17.5166i q^{31} +222.065 q^{32} +29.7269 q^{33} -110.590 q^{35} +313.724 q^{36} -8.40485i q^{37} -405.191 q^{38} +31.5221i q^{39} -287.211i q^{40} -99.9530i q^{41} +15.2993 q^{42} +81.5297 q^{43} -649.766i q^{44} +461.015i q^{45} +537.130i q^{46} +195.351 q^{47} -10.7547i q^{48} +301.942 q^{49} +768.195 q^{50} +689.006 q^{52} -260.322 q^{53} -128.245i q^{54} +954.825 q^{55} -106.630i q^{56} +48.9979i q^{57} +958.715i q^{58} +536.401 q^{59} -108.923 q^{60} +265.689i q^{61} +77.8357i q^{62} +171.156i q^{63} -826.636 q^{64} +1012.49i q^{65} -132.093 q^{66} +514.794 q^{67} +64.9526 q^{69} +491.412 q^{70} -704.023i q^{71} -444.504 q^{72} +184.948i q^{73} +37.3473i q^{74} -92.8943i q^{75} +1070.99 q^{76} +354.487 q^{77} -140.070i q^{78} +34.8358i q^{79} -345.440i q^{80} +705.696 q^{81} +444.145i q^{82} +647.682 q^{83} -40.4386 q^{84} -362.280 q^{86} +115.933 q^{87} +920.630i q^{88} -1060.24 q^{89} -2048.53i q^{90} +375.895i q^{91} -1419.72i q^{92} +9.41231 q^{93} -868.051 q^{94} +1573.81i q^{95} +119.324i q^{96} -256.409i q^{97} -1341.69 q^{98} -1477.74i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 96 q^{4} + 102 q^{8} - 216 q^{9}+O(q^{10}) \) 24 * q + 96 * q^4 + 102 * q^8 - 216 * q^9	\( 24 q + 96 q^{4} + 102 q^{8} - 216 q^{9} - 144 q^{13} + 276 q^{15} + 384 q^{16} + 378 q^{18} + 132 q^{19} + 492 q^{21} - 888 q^{25} - 1056 q^{26} - 3780 q^{30} + 2706 q^{32} + 1932 q^{33} - 132 q^{35} + 1326 q^{36} - 120 q^{38} - 1608 q^{42} - 972 q^{43} - 1776 q^{47} + 1140 q^{49} + 870 q^{50} + 450 q^{52} - 2052 q^{53} + 1944 q^{55} + 1584 q^{59} - 2916 q^{60} - 3714 q^{64} + 1188 q^{66} + 1248 q^{67} - 3012 q^{69} + 3300 q^{70} + 2910 q^{72} - 1350 q^{76} + 2016 q^{77} + 5040 q^{81} - 1344 q^{83} - 1554 q^{84} + 5556 q^{86} - 1452 q^{87} - 1812 q^{89} - 264 q^{93} - 1470 q^{94} + 3822 q^{98}+O(q^{100}) \) 24 * q + 96 * q^4 + 102 * q^8 - 216 * q^9 - 144 * q^13 + 276 * q^15 + 384 * q^16 + 378 * q^18 + 132 * q^19 + 492 * q^21 - 888 * q^25 - 1056 * q^26 - 3780 * q^30 + 2706 * q^32 + 1932 * q^33 - 132 * q^35 + 1326 * q^36 - 120 * q^38 - 1608 * q^42 - 972 * q^43 - 1776 * q^47 + 1140 * q^49 + 870 * q^50 + 450 * q^52 - 2052 * q^53 + 1944 * q^55 + 1584 * q^59 - 2916 * q^60 - 3714 * q^64 + 1188 * q^66 + 1248 * q^67 - 3012 * q^69 + 3300 * q^70 + 2910 * q^72 - 1350 * q^76 + 2016 * q^77 + 5040 * q^81 - 1344 * q^83 - 1554 * q^84 + 5556 * q^86 - 1452 * q^87 - 1812 * q^89 - 264 * q^93 - 1470 * q^94 + 3822 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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