LMFDB - Embedded newform 5239.2.a.r.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5239,2,Mod(1,5239)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5239, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("5239.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5239 = 13^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5239.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:	$41.8336256189$
Analytic rank:	$0$
Dimension:	$34$
Twist minimal:	no (minimal twist has level 403)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	5239.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.32498 q^{2} -1.54906 q^{3} -0.244422 q^{4} -1.40687 q^{5} +2.05248 q^{6} +0.535818 q^{7} +2.97382 q^{8} -0.600418 q^{9} +O(q^{10})$	\(q-1.32498 q^{2} -1.54906 q^{3} -0.244422 q^{4} -1.40687 q^{5} +2.05248 q^{6} +0.535818 q^{7} +2.97382 q^{8} -0.600418 q^{9} +1.86407 q^{10} -3.98764 q^{11} +0.378624 q^{12} -0.709950 q^{14} +2.17932 q^{15} -3.45141 q^{16} +0.287725 q^{17} +0.795543 q^{18} +0.134018 q^{19} +0.343869 q^{20} -0.830014 q^{21} +5.28355 q^{22} -2.52396 q^{23} -4.60662 q^{24} -3.02072 q^{25} +5.57726 q^{27} -0.130966 q^{28} +2.53898 q^{29} -2.88756 q^{30} -1.00000 q^{31} -1.37458 q^{32} +6.17709 q^{33} -0.381231 q^{34} -0.753825 q^{35} +0.146755 q^{36} +1.58013 q^{37} -0.177571 q^{38} -4.18377 q^{40} -9.71919 q^{41} +1.09975 q^{42} -0.405850 q^{43} +0.974667 q^{44} +0.844708 q^{45} +3.34420 q^{46} +0.0603089 q^{47} +5.34644 q^{48} -6.71290 q^{49} +4.00241 q^{50} -0.445703 q^{51} -0.245150 q^{53} -7.38977 q^{54} +5.61008 q^{55} +1.59343 q^{56} -0.207602 q^{57} -3.36410 q^{58} +8.55531 q^{59} -0.532674 q^{60} -13.7204 q^{61} +1.32498 q^{62} -0.321715 q^{63} +8.72412 q^{64} -8.18453 q^{66} -8.87098 q^{67} -0.0703264 q^{68} +3.90976 q^{69} +0.998805 q^{70} +3.42636 q^{71} -1.78553 q^{72} +10.0870 q^{73} -2.09364 q^{74} +4.67928 q^{75} -0.0327569 q^{76} -2.13665 q^{77} -11.5503 q^{79} +4.85568 q^{80} -6.83824 q^{81} +12.8778 q^{82} +3.36143 q^{83} +0.202874 q^{84} -0.404791 q^{85} +0.537744 q^{86} -3.93302 q^{87} -11.8585 q^{88} -14.4699 q^{89} -1.11922 q^{90} +0.616910 q^{92} +1.54906 q^{93} -0.0799082 q^{94} -0.188545 q^{95} +2.12930 q^{96} -1.23442 q^{97} +8.89447 q^{98} +2.39425 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 34 q + 8 q^{2} + 32 q^{4} + 16 q^{5} + 12 q^{6} + 8 q^{7} + 24 q^{8} + 34 q^{9}+O(q^{10}) $ 34 * q + 8 * q^2 + 32 * q^4 + 16 * q^5 + 12 * q^6 + 8 * q^7 + 24 * q^8 + 34 * q^9	\( 34 q + 8 q^{2} + 32 q^{4} + 16 q^{5} + 12 q^{6} + 8 q^{7} + 24 q^{8} + 34 q^{9} + 8 q^{10} + 26 q^{11} + 8 q^{12} - 4 q^{14} + 16 q^{15} + 36 q^{16} - 6 q^{17} + 64 q^{18} + 4 q^{19} + 40 q^{20} + 32 q^{21} + 20 q^{22} - 8 q^{23} - 16 q^{24} + 36 q^{25} - 6 q^{27} + 24 q^{28} + 32 q^{30} - 34 q^{31} + 36 q^{32} + 40 q^{33} + 16 q^{34} - 30 q^{35} + 40 q^{36} + 2 q^{37} + 18 q^{38} + 4 q^{40} + 80 q^{41} + 16 q^{42} + 12 q^{43} + 108 q^{44} + 12 q^{45} + 48 q^{46} + 24 q^{47} + 46 q^{48} + 22 q^{49} - 44 q^{50} - 28 q^{51} + 10 q^{53} + 48 q^{54} + 6 q^{55} - 2 q^{56} + 66 q^{57} - 44 q^{58} + 64 q^{59} + 48 q^{60} - 6 q^{61} - 8 q^{62} - 52 q^{63} - 12 q^{64} + 4 q^{66} + 16 q^{67} - 58 q^{68} - 28 q^{69} + 72 q^{70} + 52 q^{71} + 152 q^{72} + 42 q^{73} - 8 q^{74} - 4 q^{75} - 48 q^{76} + 10 q^{77} + 8 q^{79} + 48 q^{80} + 58 q^{81} - 42 q^{82} + 44 q^{83} - 8 q^{84} + 96 q^{85} + 16 q^{86} + 20 q^{87} + 64 q^{88} + 74 q^{89} - 26 q^{90} + 24 q^{92} - 8 q^{94} - 32 q^{95} + 50 q^{96} + 40 q^{97} + 72 q^{98} + 74 q^{99}+O(q^{100}) \) 34 * q + 8 * q^2 + 32 * q^4 + 16 * q^5 + 12 * q^6 + 8 * q^7 + 24 * q^8 + 34 * q^9 + 8 * q^10 + 26 * q^11 + 8 * q^12 - 4 * q^14 + 16 * q^15 + 36 * q^16 - 6 * q^17 + 64 * q^18 + 4 * q^19 + 40 * q^20 + 32 * q^21 + 20 * q^22 - 8 * q^23 - 16 * q^24 + 36 * q^25 - 6 * q^27 + 24 * q^28 + 32 * q^30 - 34 * q^31 + 36 * q^32 + 40 * q^33 + 16 * q^34 - 30 * q^35 + 40 * q^36 + 2 * q^37 + 18 * q^38 + 4 * q^40 + 80 * q^41 + 16 * q^42 + 12 * q^43 + 108 * q^44 + 12 * q^45 + 48 * q^46 + 24 * q^47 + 46 * q^48 + 22 * q^49 - 44 * q^50 - 28 * q^51 + 10 * q^53 + 48 * q^54 + 6 * q^55 - 2 * q^56 + 66 * q^57 - 44 * q^58 + 64 * q^59 + 48 * q^60 - 6 * q^61 - 8 * q^62 - 52 * q^63 - 12 * q^64 + 4 * q^66 + 16 * q^67 - 58 * q^68 - 28 * q^69 + 72 * q^70 + 52 * q^71 + 152 * q^72 + 42 * q^73 - 8 * q^74 - 4 * q^75 - 48 * q^76 + 10 * q^77 + 8 * q^79 + 48 * q^80 + 58 * q^81 - 42 * q^82 + 44 * q^83 - 8 * q^84 + 96 * q^85 + 16 * q^86 + 20 * q^87 + 64 * q^88 + 74 * q^89 - 26 * q^90 + 24 * q^92 - 8 * q^94 - 32 * q^95 + 50 * q^96 + 40 * q^97 + 72 * q^98 + 74 * 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 $

$2$	−1.32498	−0.936904	−0.468452	−	0.883489i	$-0.655188\pi$
$2$	−1.32498	−0.936904	−0.468452	+	0.883489i	$0.655188\pi$
$3$	−1.54906	−0.894349	−0.447175	−	0.894447i	$-0.647570\pi$
$3$	−1.54906	−0.894349	−0.447175	+	0.894447i	$0.647570\pi$
$4$	−0.244422	−0.122211
$5$	−1.40687	−0.629170	−0.314585	−	0.949229i	$-0.601865\pi$
$5$	−1.40687	−0.629170	−0.314585	+	0.949229i	$0.601865\pi$
$6$	2.05248	0.837919
$7$	0.535818	0.202520	0.101260	−	0.994860i	$-0.467713\pi$
$7$	0.535818	0.202520	0.101260	+	0.994860i	$0.467713\pi$
$8$	2.97382	1.05140
$9$	−0.600418	−0.200139
$10$	1.86407	0.589472
$11$	−3.98764	−1.20232	−0.601159	−	0.799129i	$-0.705294\pi$
$11$	−3.98764	−1.20232	−0.601159	+	0.799129i	$0.705294\pi$
$12$	0.378624	0.109299
$13$	0	0
$13$	0	0
$14$	−0.709950	−0.189742
$15$	2.17932	0.562698
$16$	−3.45141	−0.862854
$17$	0.287725	0.0697836	0.0348918	−	0.999391i	$-0.488891\pi$
$17$	0.287725	0.0697836	0.0348918	+	0.999391i	$0.488891\pi$
$18$	0.795543	0.187511
$19$	0.134018	0.0307458	0.0153729	−	0.999882i	$-0.495106\pi$
$19$	0.134018	0.0307458	0.0153729	+	0.999882i	$0.495106\pi$
$20$	0.343869	0.0768915
$21$	−0.830014	−0.181124
$22$	5.28355	1.12646
$23$	−2.52396	−0.526281	−0.263141	−	0.964757i	$-0.584758\pi$
$23$	−2.52396	−0.526281	−0.263141	+	0.964757i	$0.584758\pi$
$24$	−4.60662	−0.940322
$25$	−3.02072	−0.604145
$26$	0	0
$27$	5.57726	1.07334
$28$	−0.130966	−0.0247502
$29$	2.53898	0.471476	0.235738	−	0.971817i	$-0.424249\pi$
$29$	2.53898	0.471476	0.235738	+	0.971817i	$0.424249\pi$
$30$	−2.88756	−0.527194
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−1.37458	−0.242993
$33$	6.17709	1.07529
$34$	−0.381231	−0.0653806
$35$	−0.753825	−0.127420
$36$	0.146755	0.0244592
$37$	1.58013	0.259772	0.129886	−	0.991529i	$-0.458539\pi$
$37$	1.58013	0.259772	0.129886	+	0.991529i	$0.458539\pi$
$38$	−0.177571	−0.0288059
$39$	0	0
$40$	−4.18377	−0.661512
$41$	−9.71919	−1.51788	−0.758941	−	0.651159i	$-0.774283\pi$
$41$	−9.71919	−1.51788	−0.758941	+	0.651159i	$0.774283\pi$
$42$	1.09975	0.169696
$43$	−0.405850	−0.0618916	−0.0309458	−	0.999521i	$-0.509852\pi$
$43$	−0.405850	−0.0618916	−0.0309458	+	0.999521i	$0.509852\pi$
$44$	0.974667	0.146937
$45$	0.844708	0.125922
$46$	3.34420	0.493075
$47$	0.0603089	0.00879696	0.00439848	−	0.999990i	$-0.498600\pi$
$47$	0.0603089	0.00879696	0.00439848	+	0.999990i	$0.498600\pi$
$48$	5.34644	0.771692
$49$	−6.71290	−0.958986
$50$	4.00241	0.566026
$51$	−0.445703	−0.0624109
$52$	0	0
$53$	−0.245150	−0.0336739	−0.0168369	−	0.999858i	$-0.505360\pi$
$53$	−0.245150	−0.0336739	−0.0168369	+	0.999858i	$0.505360\pi$
$54$	−7.38977	−1.00562
$55$	5.61008	0.756463
$56$	1.59343	0.212931
$57$	−0.207602	−0.0274975
$58$	−3.36410	−0.441728
$59$	8.55531	1.11381	0.556903	−	0.830577i	$-0.311989\pi$
$59$	8.55531	1.11381	0.556903	+	0.830577i	$0.311989\pi$
$60$	−0.532674	−0.0687679
$61$	−13.7204	−1.75671	−0.878356	−	0.478008i	$-0.841359\pi$
$61$	−13.7204	−1.75671	−0.878356	+	0.478008i	$0.841359\pi$
$62$	1.32498	0.168273
$63$	−0.321715	−0.0405323
$64$	8.72412	1.09051
$65$	0	0
$66$	−8.18453	−1.00745
$67$	−8.87098	−1.08376	−0.541881	−	0.840455i	$-0.682288\pi$
$67$	−8.87098	−1.08376	−0.541881	+	0.840455i	$0.682288\pi$
$68$	−0.0703264	−0.00852832
$69$	3.90976	0.470679
$70$	0.998805	0.119380
$71$	3.42636	0.406634	0.203317	−	0.979113i	$-0.434828\pi$
$71$	3.42636	0.406634	0.203317	+	0.979113i	$0.434828\pi$
$72$	−1.78553	−0.210427
$73$	10.0870	1.18059	0.590297	−	0.807186i	$-0.299010\pi$
$73$	10.0870	1.18059	0.590297	+	0.807186i	$0.299010\pi$
$74$	−2.09364	−0.243381
$75$	4.67928	0.540316
$76$	−0.0327569	−0.00375748
$77$	−2.13665	−0.243494
$78$	0	0
$79$	−11.5503	−1.29951	−0.649755	−	0.760143i	$-0.725129\pi$
$79$	−11.5503	−1.29951	−0.649755	+	0.760143i	$0.725129\pi$
$80$	4.85568	0.542882
$81$	−6.83824	−0.759805
$82$	12.8778	1.42211
$83$	3.36143	0.368965	0.184483	−	0.982836i	$-0.440939\pi$
$83$	3.36143	0.368965	0.184483	+	0.982836i	$0.440939\pi$
$84$	0.202874	0.0221353
$85$	−0.404791	−0.0439058
$86$	0.537744	0.0579865
$87$	−3.93302	−0.421664
$88$	−11.8585	−1.26412
$89$	−14.4699	−1.53381	−0.766903	−	0.641763i	$-0.778203\pi$
$89$	−14.4699	−1.53381	−0.766903	+	0.641763i	$0.778203\pi$
$90$	−1.11922	−0.117977
$91$	0	0
$92$	0.616910	0.0643173
$93$	1.54906	0.160630
$94$	−0.0799082	−0.00824190
$95$	−0.188545	−0.0193444
$96$	2.12930	0.217321
$97$	−1.23442	−0.125337	−0.0626683	−	0.998034i	$-0.519961\pi$
$97$	−1.23442	−0.125337	−0.0626683	+	0.998034i	$0.519961\pi$
$98$	8.89447	0.898477
$99$	2.39425	0.240631
$100$	0.738331	0.0738331
$101$	3.24608	0.322997	0.161499	−	0.986873i	$-0.448367\pi$
$101$	3.24608	0.322997	0.161499	+	0.986873i	$0.448367\pi$
$102$	0.590549	0.0584731
$103$	−4.44909	−0.438381	−0.219191	−	0.975682i	$-0.570342\pi$
$103$	−4.44909	−0.438381	−0.219191	+	0.975682i	$0.570342\pi$
$104$	0	0
$105$	1.16772	0.113958
$106$	0.324819	0.0315492
$107$	−5.51676	−0.533325	−0.266663	−	0.963790i	$-0.585921\pi$
$107$	−5.51676	−0.533325	−0.266663	+	0.963790i	$0.585921\pi$
$108$	−1.36320	−0.131174
$109$	−9.63137	−0.922518	−0.461259	−	0.887266i	$-0.652602\pi$
$109$	−9.63137	−0.922518	−0.461259	+	0.887266i	$0.652602\pi$
$110$	−7.43326	−0.708733
$111$	−2.44771	−0.232327
$112$	−1.84933	−0.174745
$113$	3.97665	0.374092	0.187046	−	0.982351i	$-0.440109\pi$
$113$	3.97665	0.374092	0.187046	+	0.982351i	$0.440109\pi$
$114$	0.275068	0.0257625
$115$	3.55087	0.331121
$116$	−0.620582	−0.0576196
$117$	0	0
$118$	−11.3356	−1.04353
$119$	0.154168	0.0141326
$120$	6.48090	0.591623
$121$	4.90128	0.445571
$122$	18.1792	1.64587
$123$	15.0556	1.35752
$124$	0.244422	0.0219497
$125$	11.2841	1.00928
$126$	0.426266	0.0379748
$127$	−14.1662	−1.25705	−0.628523	−	0.777791i	$-0.716340\pi$
$127$	−14.1662	−1.25705	−0.628523	+	0.777791i	$0.716340\pi$
$128$	−8.81015	−0.778714
$129$	0.628686	0.0553527
$130$	0	0
$131$	−11.2863	−0.986091	−0.493046	−	0.870003i	$-0.664116\pi$
$131$	−11.2863	−0.986091	−0.493046	+	0.870003i	$0.664116\pi$
$132$	−1.50982	−0.131413
$133$	0.0718092	0.00622665
$134$	11.7539	1.01538
$135$	−7.84646	−0.675316
$136$	0.855643	0.0733708
$137$	6.75116	0.576790	0.288395	−	0.957511i	$-0.406878\pi$
$137$	6.75116	0.576790	0.288395	+	0.957511i	$0.406878\pi$
$138$	−5.18036	−0.440981
$139$	−13.5185	−1.14662	−0.573311	−	0.819338i	$-0.694341\pi$
$139$	−13.5185	−1.14662	−0.573311	+	0.819338i	$0.694341\pi$
$140$	0.184251	0.0155721
$141$	−0.0934220	−0.00786755
$142$	−4.53986	−0.380977
$143$	0	0
$144$	2.07229	0.172691
$145$	−3.57200	−0.296639
$146$	−13.3651	−1.10610
$147$	10.3987	0.857668
$148$	−0.386218	−0.0317469
$149$	−5.24084	−0.429346	−0.214673	−	0.976686i	$-0.568869\pi$
$149$	−5.24084	−0.429346	−0.214673	+	0.976686i	$0.568869\pi$
$150$	−6.19996	−0.506225
$151$	18.4654	1.50269	0.751347	−	0.659907i	$-0.229404\pi$
$151$	18.4654	1.50269	0.751347	+	0.659907i	$0.229404\pi$
$152$	0.398545	0.0323263
$153$	−0.172755	−0.0139664
$154$	2.83102	0.228130
$155$	1.40687	0.113002
$156$	0	0
$157$	−17.9544	−1.43292	−0.716460	−	0.697629i	$-0.754239\pi$
$157$	−17.9544	−1.43292	−0.716460	+	0.697629i	$0.754239\pi$
$158$	15.3040	1.21752
$159$	0.379751	0.0301162
$160$	1.93385	0.152884
$161$	−1.35238	−0.106583
$162$	9.06055	0.711864
$163$	−15.3702	−1.20389	−0.601943	−	0.798539i	$-0.705607\pi$
$163$	−15.3702	−1.20389	−0.601943	+	0.798539i	$0.705607\pi$
$164$	2.37558	0.185502
$165$	−8.69034	−0.676542
$166$	−4.45384	−0.345685
$167$	−12.5179	−0.968663	−0.484332	−	0.874885i	$-0.660937\pi$
$167$	−12.5179	−0.968663	−0.484332	+	0.874885i	$0.660937\pi$
$168$	−2.46831	−0.190434
$169$	0	0
$170$	0.536341	0.0411355
$171$	−0.0804667	−0.00615345
$172$	0.0991987	0.00756383
$173$	8.33885	0.633991	0.316996	−	0.948427i	$-0.397326\pi$
$173$	8.33885	0.633991	0.316996	+	0.948427i	$0.397326\pi$
$174$	5.21119	0.395059
$175$	−1.61856	−0.122352
$176$	13.7630	1.03743
$177$	−13.2527	−0.996132
$178$	19.1724	1.43703
$179$	−12.0947	−0.904001	−0.452001	−	0.892018i	$-0.649289\pi$
$179$	−12.0947	−0.904001	−0.452001	+	0.892018i	$0.649289\pi$
$180$	−0.206465	−0.0153890
$181$	−20.0263	−1.48854	−0.744272	−	0.667876i	$-0.767203\pi$
$181$	−20.0263	−1.48854	−0.744272	+	0.667876i	$0.767203\pi$
$182$	0	0
$183$	21.2536	1.57111
$184$	−7.50579	−0.553334
$185$	−2.22303	−0.163441
$186$	−2.05248	−0.150495
$187$	−1.14734	−0.0839022
$188$	−0.0147408	−0.00107508
$189$	2.98840	0.217374
$190$	0.249819	0.0181238
$191$	−5.06448	−0.366453	−0.183227	−	0.983071i	$-0.558654\pi$
$191$	−5.06448	−0.366453	−0.183227	+	0.983071i	$0.558654\pi$
$192$	−13.5142	−0.975301
$193$	15.3566	1.10539	0.552697	−	0.833382i	$-0.313599\pi$
$193$	15.3566	1.10539	0.552697	+	0.833382i	$0.313599\pi$
$194$	1.63559	0.117428
$195$	0	0
$196$	1.64078	0.117199
$197$	−5.04262	−0.359272	−0.179636	−	0.983733i	$-0.557492\pi$
$197$	−5.04262	−0.359272	−0.179636	+	0.983733i	$0.557492\pi$
$198$	−3.17234	−0.225448
$199$	−0.136258	−0.00965907	−0.00482954	−	0.999988i	$-0.501537\pi$
$199$	−0.136258	−0.00965907	−0.00482954	+	0.999988i	$0.501537\pi$
$200$	−8.98309	−0.635200
$201$	13.7417	0.969262
$202$	−4.30100	−0.302618
$203$	1.36043	0.0954835
$204$	0.108940	0.00762730
$205$	13.6736	0.955006
$206$	5.89496	0.410721
$207$	1.51543	0.105330
$208$	0	0
$209$	−0.534415	−0.0369663
$210$	−1.54721	−0.106767
$211$	−0.181250	−0.0124778	−0.00623888	−	0.999981i	$-0.501986\pi$
$211$	−0.181250	−0.0124778	−0.00623888	+	0.999981i	$0.501986\pi$
$212$	0.0599199	0.00411532
$213$	−5.30763	−0.363673
$214$	7.30961	0.499675
$215$	0.570978	0.0389403
$216$	16.5858	1.12852
$217$	−0.535818	−0.0363737
$218$	12.7614	0.864311
$219$	−15.6254	−1.05586
$220$	−1.37123	−0.0924481
$221$	0	0
$222$	3.24318	0.217668
$223$	18.8128	1.25980	0.629898	−	0.776678i	$-0.283097\pi$
$223$	18.8128	1.25980	0.629898	+	0.776678i	$0.283097\pi$
$224$	−0.736523	−0.0492110
$225$	1.81370	0.120913
$226$	−5.26899	−0.350488
$227$	17.1957	1.14132	0.570658	−	0.821188i	$-0.306688\pi$
$227$	17.1957	1.14132	0.570658	+	0.821188i	$0.306688\pi$
$228$	0.0507424	0.00336050
$229$	−11.3595	−0.750658	−0.375329	−	0.926892i	$-0.622470\pi$
$229$	−11.3595	−0.750658	−0.375329	+	0.926892i	$0.622470\pi$
$230$	−4.70484	−0.310228
$231$	3.30980	0.217769
$232$	7.55046	0.495712
$233$	−16.8113	−1.10135	−0.550674	−	0.834721i	$-0.685629\pi$
$233$	−16.8113	−1.10135	−0.550674	+	0.834721i	$0.685629\pi$
$234$	0	0
$235$	−0.0848466	−0.00553478
$236$	−2.09111	−0.136119
$237$	17.8921	1.16222
$238$	−0.204270	−0.0132409
$239$	−10.4405	−0.675338	−0.337669	−	0.941265i	$-0.609638\pi$
$239$	−10.4405	−0.675338	−0.337669	+	0.941265i	$0.609638\pi$
$240$	−7.52174	−0.485526
$241$	25.5433	1.64539	0.822696	−	0.568482i	$-0.192469\pi$
$241$	25.5433	1.64539	0.822696	+	0.568482i	$0.192469\pi$
$242$	−6.49410	−0.417457
$243$	−6.13893	−0.393813
$244$	3.35356	0.214689
$245$	9.44416	0.603365
$246$	−19.9484	−1.27186
$247$	0	0
$248$	−2.97382	−0.188838
$249$	−5.20706	−0.329984
$250$	−14.9512	−0.945599
$251$	−21.7182	−1.37084	−0.685421	−	0.728147i	$-0.740382\pi$
$251$	−21.7182	−1.37084	−0.685421	+	0.728147i	$0.740382\pi$
$252$	0.0786342	0.00495349
$253$	10.0646	0.632758
$254$	18.7699	1.17773
$255$	0.627045	0.0392671
$256$	−5.77494	−0.360934
$257$	17.5337	1.09372	0.546862	−	0.837223i	$-0.315822\pi$
$257$	17.5337	1.09372	0.546862	+	0.837223i	$0.315822\pi$
$258$	−0.832998	−0.0518602
$259$	0.846662	0.0526090
$260$	0	0
$261$	−1.52445	−0.0943609
$262$	14.9542	0.923873
$263$	10.0344	0.618747	0.309373	−	0.950941i	$-0.399881\pi$
$263$	10.0344	0.618747	0.309373	+	0.950941i	$0.399881\pi$
$264$	18.3695	1.13057
$265$	0.344893	0.0211866
$266$	−0.0951459	−0.00583377
$267$	22.4147	1.37176
$268$	2.16826	0.132448
$269$	−6.39447	−0.389878	−0.194939	−	0.980815i	$-0.562451\pi$
$269$	−6.39447	−0.389878	−0.194939	+	0.980815i	$0.562451\pi$
$270$	10.3964	0.632706
$271$	−15.4161	−0.936463	−0.468232	−	0.883606i	$-0.655109\pi$
$271$	−15.4161	−0.936463	−0.468232	+	0.883606i	$0.655109\pi$
$272$	−0.993059	−0.0602130
$273$	0	0
$274$	−8.94516	−0.540397
$275$	12.0456	0.726375
$276$	−0.955630	−0.0575222
$277$	18.9399	1.13799	0.568993	−	0.822342i	$-0.307333\pi$
$277$	18.9399	1.13799	0.568993	+	0.822342i	$0.307333\pi$
$278$	17.9117	1.07427
$279$	0.600418	0.0359461
$280$	−2.24174	−0.133970
$281$	5.30476	0.316455	0.158228	−	0.987403i	$-0.449422\pi$
$281$	5.30476	0.316455	0.158228	+	0.987403i	$0.449422\pi$
$282$	0.123782	0.00737114
$283$	10.8844	0.647009	0.323505	−	0.946227i	$-0.395139\pi$
$283$	10.8844	0.647009	0.323505	+	0.946227i	$0.395139\pi$
$284$	−0.837477	−0.0496951
$285$	0.292068	0.0173006
$286$	0	0
$287$	−5.20772	−0.307402
$288$	0.825320	0.0486325
$289$	−16.9172	−0.995130
$290$	4.73284	0.277922
$291$	1.91219	0.112095
$292$	−2.46549	−0.144282
$293$	24.1661	1.41180	0.705900	−	0.708311i	$-0.250543\pi$
$293$	24.1661	1.41180	0.705900	+	0.708311i	$0.250543\pi$
$294$	−13.7781	−0.803553
$295$	−12.0362	−0.700774
$296$	4.69902	0.273125
$297$	−22.2401	−1.29050
$298$	6.94402	0.402256
$299$	0	0
$300$	−1.14372	−0.0660326
$301$	−0.217462	−0.0125343
$302$	−24.4664	−1.40788
$303$	−5.02837	−0.288873
$304$	−0.462551	−0.0265291
$305$	19.3027	1.10527
$306$	0.228898	0.0130852
$307$	7.61124	0.434396	0.217198	−	0.976128i	$-0.430308\pi$
$307$	7.61124	0.434396	0.217198	+	0.976128i	$0.430308\pi$
$308$	0.522244	0.0297576
$309$	6.89189	0.392066
$310$	−1.86407	−0.105872
$311$	30.7134	1.74160	0.870798	−	0.491641i	$-0.163603\pi$
$311$	30.7134	1.74160	0.870798	+	0.491641i	$0.163603\pi$
$312$	0	0
$313$	−12.0572	−0.681511	−0.340755	−	0.940152i	$-0.610683\pi$
$313$	−12.0572	−0.681511	−0.340755	+	0.940152i	$0.610683\pi$
$314$	23.7893	1.34251
$315$	0.452610	0.0255017
$316$	2.82315	0.158814
$317$	19.7478	1.10914	0.554572	−	0.832136i	$-0.312882\pi$
$317$	19.7478	1.10914	0.554572	+	0.832136i	$0.312882\pi$
$318$	−0.503163	−0.0282160
$319$	−10.1245	−0.566865
$320$	−12.2737	−0.686119
$321$	8.54578	0.476979
$322$	1.79188	0.0998577
$323$	0.0385603	0.00214555
$324$	1.67142	0.0928565
$325$	0	0
$326$	20.3652	1.12793
$327$	14.9196	0.825053
$328$	−28.9031	−1.59591
$329$	0.0323146	0.00178156
$330$	11.5146	0.633855
$331$	9.98775	0.548976	0.274488	−	0.961590i	$-0.411492\pi$
$331$	9.98775	0.548976	0.274488	+	0.961590i	$0.411492\pi$
$332$	−0.821608	−0.0450916
$333$	−0.948738	−0.0519905
$334$	16.5860	0.907544
$335$	12.4803	0.681871
$336$	2.86472	0.156283
$337$	19.0018	1.03509	0.517547	−	0.855655i	$-0.326845\pi$
$337$	19.0018	1.03509	0.517547	+	0.855655i	$0.326845\pi$
$338$	0	0
$339$	−6.16006	−0.334569
$340$	0.0989399	0.00536577
$341$	3.98764	0.215943
$342$	0.106617	0.00576519
$343$	−7.34762	−0.396734
$344$	−1.20693	−0.0650731
$345$	−5.50051	−0.296137
$346$	−11.0488	−0.593989
$347$	9.77071	0.524519	0.262260	−	0.964997i	$-0.415532\pi$
$347$	9.77071	0.524519	0.262260	+	0.964997i	$0.415532\pi$
$348$	0.961317	0.0515320
$349$	34.9720	1.87201	0.936005	−	0.351986i	$-0.114493\pi$
$349$	34.9720	1.87201	0.936005	+	0.351986i	$0.114493\pi$
$350$	2.14456	0.114632
$351$	0	0
$352$	5.48132	0.292155
$353$	26.5869	1.41508	0.707538	−	0.706675i	$-0.249806\pi$
$353$	26.5869	1.41508	0.707538	+	0.706675i	$0.249806\pi$
$354$	17.5596	0.933280
$355$	−4.82043	−0.255842
$356$	3.53676	0.187448
$357$	−0.238816	−0.0126395
$358$	16.0253	0.846962
$359$	1.80327	0.0951732	0.0475866	−	0.998867i	$-0.484847\pi$
$359$	1.80327	0.0951732	0.0475866	+	0.998867i	$0.484847\pi$
$360$	2.51201	0.132395
$361$	−18.9820	−0.999055
$362$	26.5345	1.39462
$363$	−7.59236	−0.398496
$364$	0	0
$365$	−14.1911	−0.742795
$366$	−28.1607	−1.47198
$367$	17.3015	0.903133	0.451567	−	0.892237i	$-0.350865\pi$
$367$	17.3015	0.903133	0.451567	+	0.892237i	$0.350865\pi$
$368$	8.71122	0.454104
$369$	5.83557	0.303788
$370$	2.94548	0.153128
$371$	−0.131356	−0.00681964
$372$	−0.378624	−0.0196307
$373$	−23.0579	−1.19389	−0.596947	−	0.802281i	$-0.703620\pi$
$373$	−23.0579	−1.19389	−0.596947	+	0.802281i	$0.703620\pi$
$374$	1.52021	0.0786083
$375$	−17.4797	−0.902649
$376$	0.179348	0.00924915
$377$	0	0
$378$	−3.95957	−0.203658
$379$	−23.4719	−1.20567	−0.602836	−	0.797865i	$-0.705963\pi$
$379$	−23.4719	−1.20567	−0.602836	+	0.797865i	$0.705963\pi$
$380$	0.0460846	0.00236409
$381$	21.9442	1.12424
$382$	6.71035	0.343331
$383$	−30.8138	−1.57451	−0.787256	−	0.616627i	$-0.788499\pi$
$383$	−30.8138	−1.57451	−0.787256	+	0.616627i	$0.788499\pi$
$384$	13.6474	0.696443
$385$	3.00598	0.153199
$386$	−20.3473	−1.03565
$387$	0.243680	0.0123869
$388$	0.301720	0.0153175
$389$	26.2280	1.32981	0.664906	−	0.746927i	$-0.268472\pi$
$389$	26.2280	1.32981	0.664906	+	0.746927i	$0.268472\pi$
$390$	0	0
$391$	−0.726206	−0.0367258
$392$	−19.9629	−1.00828
$393$	17.4832	0.881910
$394$	6.68138	0.336603
$395$	16.2497	0.817614
$396$	−0.585207	−0.0294078
$397$	14.8452	0.745057	0.372529	−	0.928021i	$-0.378491\pi$
$397$	14.8452	0.745057	0.372529	+	0.928021i	$0.378491\pi$
$398$	0.180539	0.00904962
$399$	−0.111237	−0.00556880
$400$	10.4258	0.521288
$401$	3.79093	0.189310	0.0946549	−	0.995510i	$-0.469825\pi$
$401$	3.79093	0.189310	0.0946549	+	0.995510i	$0.469825\pi$
$402$	−18.2075	−0.908106
$403$	0	0
$404$	−0.793414	−0.0394738
$405$	9.62050	0.478047
$406$	−1.80255	−0.0894588
$407$	−6.30099	−0.312328
$408$	−1.32544	−0.0656191
$409$	6.63051	0.327857	0.163929	−	0.986472i	$-0.447583\pi$
$409$	6.63051	0.327857	0.163929	+	0.986472i	$0.447583\pi$
$410$	−18.1173	−0.894749
$411$	−10.4579	−0.515852
$412$	1.08745	0.0535750
$413$	4.58409	0.225568
$414$	−2.00792	−0.0986837
$415$	−4.72909	−0.232142
$416$	0	0
$417$	20.9409	1.02548
$418$	0.708091	0.0346338
$419$	19.4864	0.951973	0.475987	−	0.879453i	$-0.342091\pi$
$419$	19.4864	0.951973	0.475987	+	0.879453i	$0.342091\pi$
$420$	−0.285416	−0.0139269
$421$	23.0190	1.12188	0.560939	−	0.827857i	$-0.310440\pi$
$421$	23.0190	1.12188	0.560939	+	0.827857i	$0.310440\pi$
$422$	0.240153	0.0116905
$423$	−0.0362105	−0.00176062
$424$	−0.729030	−0.0354048
$425$	−0.869139	−0.0421594
$426$	7.03251	0.340726
$427$	−7.35161	−0.355770
$428$	1.34842	0.0651782
$429$	0	0
$430$	−0.756535	−0.0364834
$431$	6.75661	0.325454	0.162727	−	0.986671i	$-0.447971\pi$
$431$	6.75661	0.325454	0.162727	+	0.986671i	$0.447971\pi$
$432$	−19.2494	−0.926138
$433$	3.24915	0.156144	0.0780720	−	0.996948i	$-0.475124\pi$
$433$	3.24915	0.156144	0.0780720	+	0.996948i	$0.475124\pi$
$434$	0.709950	0.0340787
$435$	5.53324	0.265299
$436$	2.35412	0.112742
$437$	−0.338255	−0.0161809
$438$	20.7033	0.989243
$439$	25.0536	1.19574	0.597872	−	0.801591i	$-0.296013\pi$
$439$	25.0536	1.19574	0.597872	+	0.801591i	$0.296013\pi$
$440$	16.6834	0.795348
$441$	4.03054	0.191931
$442$	0	0
$443$	15.3744	0.730462	0.365231	−	0.930917i	$-0.380990\pi$
$443$	15.3744	0.730462	0.365231	+	0.930917i	$0.380990\pi$
$444$	0.598275	0.0283929
$445$	20.3572	0.965025
$446$	−24.9266	−1.18031
$447$	8.11837	0.383986
$448$	4.67454	0.220851
$449$	−4.92299	−0.232330	−0.116165	−	0.993230i	$-0.537060\pi$
$449$	−4.92299	−0.232330	−0.116165	+	0.993230i	$0.537060\pi$
$450$	−2.40312	−0.113284
$451$	38.7566	1.82498
$452$	−0.971980	−0.0457181
$453$	−28.6040	−1.34393
$454$	−22.7840	−1.06930
$455$	0	0
$456$	−0.617369	−0.0289110
$457$	−1.48307	−0.0693752	−0.0346876	−	0.999398i	$-0.511044\pi$
$457$	−1.48307	−0.0693752	−0.0346876	+	0.999398i	$0.511044\pi$
$458$	15.0511	0.703294
$459$	1.60472	0.0749018
$460$	−0.867911	−0.0404666
$461$	−18.9732	−0.883671	−0.441835	−	0.897096i	$-0.645673\pi$
$461$	−18.9732	−0.883671	−0.441835	+	0.897096i	$0.645673\pi$
$462$	−4.38542	−0.204028
$463$	−17.4966	−0.813138	−0.406569	−	0.913620i	$-0.633275\pi$
$463$	−17.4966	−0.813138	−0.406569	+	0.913620i	$0.633275\pi$
$464$	−8.76306	−0.406815
$465$	−2.17932	−0.101064
$466$	22.2747	1.03186
$467$	25.7440	1.19129	0.595645	−	0.803248i	$-0.296896\pi$
$467$	25.7440	1.19129	0.595645	+	0.803248i	$0.296896\pi$
$468$	0	0
$469$	−4.75323	−0.219484
$470$	0.112420	0.00518556
$471$	27.8125	1.28153
$472$	25.4419	1.17106
$473$	1.61838	0.0744134
$474$	−23.7067	−1.08889
$475$	−0.404831	−0.0185749
$476$	−0.0376821	−0.00172716
$477$	0.147192	0.00673947
$478$	13.8334	0.632727
$479$	12.3270	0.563234	0.281617	−	0.959527i	$-0.409129\pi$
$479$	12.3270	0.563234	0.281617	+	0.959527i	$0.409129\pi$
$480$	−2.99564	−0.136732
$481$	0	0
$482$	−33.8445	−1.54157
$483$	2.09492	0.0953221
$484$	−1.19798	−0.0544536
$485$	1.73667	0.0788580
$486$	8.13398	0.368965
$487$	−40.7098	−1.84474	−0.922369	−	0.386309i	$-0.873750\pi$
$487$	−40.7098	−1.84474	−0.922369	+	0.386309i	$0.873750\pi$
$488$	−40.8019	−1.84701
$489$	23.8093	1.07669
$490$	−12.5133	−0.565295
$491$	33.7082	1.52123	0.760614	−	0.649204i	$-0.224898\pi$
$491$	33.7082	1.52123	0.760614	+	0.649204i	$0.224898\pi$
$492$	−3.67992	−0.165903
$493$	0.730528	0.0329013
$494$	0	0
$495$	−3.36839	−0.151398
$496$	3.45141	0.154973
$497$	1.83590	0.0823515
$498$	6.89926	0.309163
$499$	−11.8836	−0.531984	−0.265992	−	0.963975i	$-0.585699\pi$
$499$	−11.8836	−0.531984	−0.265992	+	0.963975i	$0.585699\pi$
$500$	−2.75808	−0.123345
$501$	19.3909	0.866323
$502$	28.7763	1.28435
$503$	3.65132	0.162804	0.0814021	−	0.996681i	$-0.474060\pi$
$503$	3.65132	0.162804	0.0814021	+	0.996681i	$0.474060\pi$
$504$	−0.956722	−0.0426158
$505$	−4.56681	−0.203220
$506$	−13.3355	−0.592833
$507$	0	0
$508$	3.46253	0.153625
$509$	31.9251	1.41506	0.707528	−	0.706686i	$-0.249810\pi$
$509$	31.9251	1.41506	0.707528	+	0.706686i	$0.249810\pi$
$510$	−0.830824	−0.0367895
$511$	5.40480	0.239094
$512$	25.2720	1.11687
$513$	0.747452	0.0330008
$514$	−23.2319	−1.02471
$515$	6.25927	0.275817
$516$	−0.153665	−0.00676471
$517$	−0.240490	−0.0105767
$518$	−1.12181	−0.0492896
$519$	−12.9174	−0.567010
$520$	0	0
$521$	26.0587	1.14165	0.570827	−	0.821070i	$-0.306623\pi$
$521$	26.0587	1.14165	0.570827	+	0.821070i	$0.306623\pi$
$522$	2.01987	0.0884071
$523$	0.266182	0.0116393	0.00581966	−	0.999983i	$-0.498148\pi$
$523$	0.266182	0.0116393	0.00581966	+	0.999983i	$0.498148\pi$
$524$	2.75863	0.120511
$525$	2.50724	0.109425
$526$	−13.2954	−0.579706
$527$	−0.287725	−0.0125335
$528$	−21.3197	−0.927820
$529$	−16.6296	−0.723028
$530$	−0.456977	−0.0198498
$531$	−5.13676	−0.222916
$532$	−0.0175517	−0.000760965 0
$533$	0	0
$534$	−29.6991	−1.28521
$535$	7.76135	0.335552
$536$	−26.3807	−1.13947
$537$	18.7354	0.808493
$538$	8.47256	0.365278
$539$	26.7686	1.15301
$540$	1.91785	0.0825310
$541$	43.2673	1.86021	0.930103	−	0.367298i	$-0.119717\pi$
$541$	43.2673	1.86021	0.930103	+	0.367298i	$0.119717\pi$
$542$	20.4261	0.877376
$543$	31.0219	1.33128
$544$	−0.395500	−0.0169569
$545$	13.5501	0.580421
$546$	0	0
$547$	17.2262	0.736538	0.368269	−	0.929719i	$-0.379951\pi$
$547$	17.2262	0.736538	0.368269	+	0.929719i	$0.379951\pi$
$548$	−1.65013	−0.0704901
$549$	8.23795	0.351587
$550$	−15.9602	−0.680543
$551$	0.340268	0.0144959
$552$	11.6269	0.494874
$553$	−6.18886	−0.263177
$554$	−25.0950	−1.06618
$555$	3.44361	0.146173
$556$	3.30421	0.140130
$557$	−31.7448	−1.34507	−0.672535	−	0.740065i	$-0.734795\pi$
$557$	−31.7448	−1.34507	−0.672535	+	0.740065i	$0.734795\pi$
$558$	−0.795543	−0.0336780
$559$	0	0
$560$	2.60176	0.109945
$561$	1.77730	0.0750378
$562$	−7.02871	−0.296488
$563$	−44.7339	−1.88531	−0.942654	−	0.333770i	$-0.891679\pi$
$563$	−44.7339	−1.88531	−0.942654	+	0.333770i	$0.891679\pi$
$564$	0.0228344	0.000961501 0
$565$	−5.59462	−0.235367
$566$	−14.4216	−0.606186
$567$	−3.66406	−0.153876
$568$	10.1894	0.427536
$569$	24.4279	1.02407	0.512036	−	0.858964i	$-0.328892\pi$
$569$	24.4279	1.02407	0.512036	+	0.858964i	$0.328892\pi$
$570$	−0.386985	−0.0162090
$571$	−29.1484	−1.21982	−0.609911	−	0.792470i	$-0.708795\pi$
$571$	−29.1484	−1.21982	−0.609911	+	0.792470i	$0.708795\pi$
$572$	0	0
$573$	7.84518	0.327737
$574$	6.90013	0.288006
$575$	7.62418	0.317950
$576$	−5.23812	−0.218255
$577$	−23.5979	−0.982394	−0.491197	−	0.871049i	$-0.663440\pi$
$577$	−23.5979	−0.982394	−0.491197	+	0.871049i	$0.663440\pi$
$578$	22.4150	0.932341
$579$	−23.7883	−0.988609
$580$	0.873076	0.0362525
$581$	1.80112	0.0747230
$582$	−2.53362	−0.105022
$583$	0.977568	0.0404867
$584$	29.9969	1.24128
$585$	0	0
$586$	−32.0197	−1.32272
$587$	−36.3504	−1.50034	−0.750172	−	0.661243i	$-0.770029\pi$
$587$	−36.3504	−1.50034	−0.750172	+	0.661243i	$0.770029\pi$
$588$	−2.54166	−0.104816
$589$	−0.134018	−0.00552211
$590$	15.9477	0.656558
$591$	7.81131	0.321314
$592$	−5.45368	−0.224145
$593$	−18.8775	−0.775206	−0.387603	−	0.921826i	$-0.626697\pi$
$593$	−18.8775	−0.775206	−0.387603	+	0.921826i	$0.626697\pi$
$594$	29.4677	1.20908
$595$	−0.216895	−0.00889181
$596$	1.28098	0.0524708
$597$	0.211072	0.00863859
$598$	0	0
$599$	10.7418	0.438897	0.219448	−	0.975624i	$-0.429574\pi$
$599$	10.7418	0.438897	0.219448	+	0.975624i	$0.429574\pi$
$600$	13.9153	0.568091
$601$	−31.9355	−1.30267	−0.651337	−	0.758788i	$-0.725792\pi$
$601$	−31.9355	−1.30267	−0.651337	+	0.758788i	$0.725792\pi$
$602$	0.288133	0.0117434
$603$	5.32629	0.216903
$604$	−4.51335	−0.183646
$605$	−6.89545	−0.280340
$606$	6.66251	0.270646
$607$	−19.5300	−0.792700	−0.396350	−	0.918100i	$-0.629723\pi$
$607$	−19.5300	−0.792700	−0.396350	+	0.918100i	$0.629723\pi$
$608$	−0.184218	−0.00747102
$609$	−2.10739	−0.0853956
$610$	−25.5758	−1.03553
$611$	0	0
$612$	0.0422252	0.00170685
$613$	20.0808	0.811056	0.405528	−	0.914083i	$-0.367088\pi$
$613$	20.0808	0.811056	0.405528	+	0.914083i	$0.367088\pi$
$614$	−10.0848	−0.406987
$615$	−21.1812	−0.854109
$616$	−6.35401	−0.256010
$617$	11.3472	0.456822	0.228411	−	0.973565i	$-0.426647\pi$
$617$	11.3472	0.456822	0.228411	+	0.973565i	$0.426647\pi$
$618$	−9.13164	−0.367328
$619$	21.5688	0.866922	0.433461	−	0.901172i	$-0.357292\pi$
$619$	21.5688	0.866922	0.433461	+	0.901172i	$0.357292\pi$
$620$	−0.343869	−0.0138101
$621$	−14.0768	−0.564881
$622$	−40.6947	−1.63171
$623$	−7.75323	−0.310627
$624$	0	0
$625$	−0.771606	−0.0308643
$626$	15.9755	0.638510
$627$	0.827840	0.0330608
$628$	4.38845	0.175118
$629$	0.454643	0.0181278
$630$	−0.599700	−0.0238926
$631$	9.35343	0.372354	0.186177	−	0.982516i	$-0.440390\pi$
$631$	9.35343	0.372354	0.186177	+	0.982516i	$0.440390\pi$
$632$	−34.3485	−1.36631
$633$	0.280767	0.0111595
$634$	−26.1654	−1.03916
$635$	19.9299	0.790896
$636$	−0.0928195	−0.00368053
$637$	0	0
$638$	13.4148	0.531098
$639$	−2.05725	−0.0813834
$640$	12.3947	0.489944
$641$	−8.52094	−0.336557	−0.168278	−	0.985739i	$-0.553821\pi$
$641$	−8.52094	−0.336557	−0.168278	+	0.985739i	$0.553821\pi$
$642$	−11.3230	−0.446884
$643$	−26.7882	−1.05642	−0.528211	−	0.849113i	$-0.677137\pi$
$643$	−26.7882	−1.05642	−0.528211	+	0.849113i	$0.677137\pi$
$644$	0.330552	0.0130256
$645$	−0.884478	−0.0348263
$646$	−0.0510918	−0.00201018
$647$	7.09378	0.278885	0.139443	−	0.990230i	$-0.455469\pi$
$647$	7.09378	0.278885	0.139443	+	0.990230i	$0.455469\pi$
$648$	−20.3357	−0.798862
$649$	−34.1155	−1.33915
$650$	0	0
$651$	0.830014	0.0325308
$652$	3.75681	0.147128
$653$	−32.6615	−1.27814	−0.639072	−	0.769147i	$-0.720682\pi$
$653$	−32.6615	−1.27814	−0.639072	+	0.769147i	$0.720682\pi$
$654$	−19.7681	−0.772996
$655$	15.8784	0.620419
$656$	33.5449	1.30971
$657$	−6.05642	−0.236283
$658$	−0.0428163	−0.00166915
$659$	−26.9292	−1.04901	−0.524507	−	0.851406i	$-0.675750\pi$
$659$	−26.9292	−1.04901	−0.524507	+	0.851406i	$0.675750\pi$
$660$	2.12411	0.0826809
$661$	26.8745	1.04530	0.522649	−	0.852548i	$-0.324944\pi$
$661$	26.8745	1.04530	0.522649	+	0.852548i	$0.324944\pi$
$662$	−13.2336	−0.514338
$663$	0	0
$664$	9.99630	0.387932
$665$	−0.101026	−0.00391762
$666$	1.25706	0.0487101
$667$	−6.40827	−0.248129
$668$	3.05965	0.118381
$669$	−29.1421	−1.12670
$670$	−16.5362	−0.638848
$671$	54.7118	2.11213
$672$	1.14092	0.0440118
$673$	−21.1625	−0.815755	−0.407878	−	0.913037i	$-0.633731\pi$
$673$	−21.1625	−0.815755	−0.407878	+	0.913037i	$0.633731\pi$
$674$	−25.1771	−0.969784
$675$	−16.8474	−0.648455
$676$	0	0
$677$	23.2073	0.891930	0.445965	−	0.895051i	$-0.352861\pi$
$677$	23.2073	0.891930	0.445965	+	0.895051i	$0.352861\pi$
$678$	8.16197	0.313459
$679$	−0.661426	−0.0253832
$680$	−1.20378	−0.0461627
$681$	−26.6371	−1.02074
$682$	−5.28355	−0.202318
$683$	−27.8546	−1.06583	−0.532913	−	0.846170i	$-0.678903\pi$
$683$	−27.8546	−1.06583	−0.532913	+	0.846170i	$0.678903\pi$
$684$	0.0196678	0.000752019 0
$685$	−9.49798	−0.362899
$686$	9.73547	0.371702
$687$	17.5965	0.671350
$688$	1.40076	0.0534034
$689$	0	0
$690$	7.28808	0.277452
$691$	−6.91338	−0.262997	−0.131499	−	0.991316i	$-0.541979\pi$
$691$	−6.91338	−0.262997	−0.131499	+	0.991316i	$0.541979\pi$
$692$	−2.03820	−0.0774807
$693$	1.28288	0.0487327
$694$	−12.9460	−0.491424
$695$	19.0187	0.721421
$696$	−11.6961	−0.443340
$697$	−2.79646	−0.105923
$698$	−46.3373	−1.75389
$699$	26.0417	0.984989
$700$	0.395611	0.0149527
$701$	−14.6629	−0.553810	−0.276905	−	0.960897i	$-0.589309\pi$
$701$	−14.6629	−0.553810	−0.276905	+	0.960897i	$0.589309\pi$
$702$	0	0
$703$	0.211766	0.00798689
$704$	−34.7886	−1.31115
$705$	0.131432	0.00495003
$706$	−35.2271	−1.32579
$707$	1.73931	0.0654135
$708$	3.23924	0.121738
$709$	−5.97946	−0.224563	−0.112282	−	0.993676i	$-0.535816\pi$
$709$	−5.97946	−0.224563	−0.112282	+	0.993676i	$0.535816\pi$
$710$	6.38698	0.239699
$711$	6.93501	0.260083
$712$	−43.0309	−1.61265
$713$	2.52396	0.0945229
$714$	0.316427	0.0118420
$715$	0	0
$716$	2.95621	0.110479
$717$	16.1729	0.603988
$718$	−2.38931	−0.0891681
$719$	13.1205	0.489312	0.244656	−	0.969610i	$-0.421325\pi$
$719$	13.1205	0.489312	0.244656	+	0.969610i	$0.421325\pi$
$720$	−2.91544	−0.108652
$721$	−2.38390	−0.0887811
$722$	25.1509	0.936018
$723$	−39.5681	−1.47155
$724$	4.89487	0.181916
$725$	−7.66955	−0.284840
$726$	10.0597	0.373352
$727$	20.4585	0.758765	0.379383	−	0.925240i	$-0.376136\pi$
$727$	20.4585	0.758765	0.379383	+	0.925240i	$0.376136\pi$
$728$	0	0
$729$	30.0243	1.11201
$730$	18.8029	0.695928
$731$	−0.116773	−0.00431902
$732$	−5.19485	−0.192007
$733$	−14.8462	−0.548355	−0.274178	−	0.961679i	$-0.588406\pi$
$733$	−14.8462	−0.548355	−0.274178	+	0.961679i	$0.588406\pi$
$734$	−22.9242	−0.846149
$735$	−14.6296	−0.539619
$736$	3.46937	0.127883
$737$	35.3743	1.30303
$738$	−7.73203	−0.284620
$739$	28.7965	1.05930	0.529649	−	0.848217i	$-0.322324\pi$
$739$	28.7965	1.05930	0.529649	+	0.848217i	$0.322324\pi$
$740$	0.543358	0.0199742
$741$	0	0
$742$	0.174044	0.00638935
$743$	15.8717	0.582276	0.291138	−	0.956681i	$-0.405966\pi$
$743$	15.8717	0.582276	0.291138	+	0.956681i	$0.405966\pi$
$744$	4.60662	0.168887
$745$	7.37317	0.270132
$746$	30.5513	1.11856
$747$	−2.01827	−0.0738445
$748$	0.280436	0.0102538
$749$	−2.95598	−0.108009
$750$	23.1603	0.845695
$751$	−0.0303943	−0.00110910	−0.000554552	−	1.00000i	$-0.500177\pi$
$751$	−0.0303943	−0.00110910	−0.000554552		1.00000i	$0.500177\pi$
$752$	−0.208151	−0.00759048
$753$	33.6428	1.22601
$754$	0	0
$755$	−25.9784	−0.945451
$756$	−0.730429	−0.0265655
$757$	19.5744	0.711444	0.355722	−	0.934592i	$-0.384235\pi$
$757$	19.5744	0.711444	0.355722	+	0.934592i	$0.384235\pi$
$758$	31.0999	1.12960
$759$	−15.5907	−0.565907
$760$	−0.560700	−0.0203387
$761$	−17.1652	−0.622237	−0.311119	−	0.950371i	$-0.600704\pi$
$761$	−17.1652	−0.622237	−0.311119	+	0.950371i	$0.600704\pi$
$762$	−29.0757	−1.05330
$763$	−5.16066	−0.186829
$764$	1.23787	0.0447846
$765$	0.243044	0.00878727
$766$	40.8277	1.47517
$767$	0	0
$768$	8.94573	0.322801
$769$	3.66144	0.132035	0.0660175	−	0.997818i	$-0.478971\pi$
$769$	3.66144	0.132035	0.0660175	+	0.997818i	$0.478971\pi$
$770$	−3.98287	−0.143533
$771$	−27.1608	−0.978171
$772$	−3.75350	−0.135091
$773$	14.2056	0.510941	0.255471	−	0.966817i	$-0.417770\pi$
$773$	14.2056	0.510941	0.255471	+	0.966817i	$0.417770\pi$
$774$	−0.322871	−0.0116054
$775$	3.02072	0.108508
$776$	−3.67095	−0.131779
$777$	−1.31153	−0.0470508
$778$	−34.7516	−1.24591
$779$	−1.30254	−0.0466685
$780$	0	0
$781$	−13.6631	−0.488903
$782$	0.962210	0.0344086
$783$	14.1605	0.506056
$784$	23.1690	0.827464
$785$	25.2595	0.901550
$786$	−23.1649	−0.826265
$787$	33.0064	1.17655	0.588276	−	0.808660i	$-0.299807\pi$
$787$	33.0064	1.17655	0.588276	+	0.808660i	$0.299807\pi$
$788$	1.23253	0.0439069
$789$	−15.5439	−0.553376
$790$	−21.5306	−0.766025
$791$	2.13076	0.0757611
$792$	7.12007	0.253001
$793$	0	0
$794$	−19.6696	−0.698047
$795$	−0.534259	−0.0189482
$796$	0.0333044	0.00118044
$797$	38.2497	1.35488	0.677438	−	0.735580i	$-0.263090\pi$
$797$	38.2497	1.35488	0.677438	+	0.735580i	$0.263090\pi$
$798$	0.147387	0.00521743
$799$	0.0173524	0.000613883 0
$800$	4.15222	0.146803
$801$	8.68799	0.306975
$802$	−5.02291	−0.177365
$803$	−40.2234	−1.41945
$804$	−3.35876	−0.118454
$805$	1.90262	0.0670586
$806$	0	0
$807$	9.90541	0.348687
$808$	9.65327	0.339601
$809$	16.0489	0.564248	0.282124	−	0.959378i	$-0.408961\pi$
$809$	16.0489	0.564248	0.282124	+	0.959378i	$0.408961\pi$
$810$	−12.7470	−0.447884
$811$	−46.8490	−1.64509	−0.822545	−	0.568700i	$-0.807446\pi$
$811$	−46.8490	−1.64509	−0.822545	+	0.568700i	$0.807446\pi$
$812$	−0.332519	−0.0116691
$813$	23.8805	0.837525
$814$	8.34870	0.292622
$815$	21.6238	0.757449
$816$	1.53831	0.0538515
$817$	−0.0543912	−0.00190291
$818$	−8.78530	−0.307171
$819$	0	0
$820$	−3.34213	−0.116712
$821$	−16.2131	−0.565840	−0.282920	−	0.959144i	$-0.591303\pi$
$821$	−16.2131	−0.565840	−0.282920	+	0.959144i	$0.591303\pi$
$822$	13.8566	0.483304
$823$	0.581902	0.0202838	0.0101419	−	0.999949i	$-0.496772\pi$
$823$	0.581902	0.0202838	0.0101419	+	0.999949i	$0.496772\pi$
$824$	−13.2308	−0.460916
$825$	−18.6593	−0.649633
$826$	−6.07384	−0.211336
$827$	5.91297	0.205614	0.102807	−	0.994701i	$-0.467218\pi$
$827$	5.91297	0.205614	0.102807	+	0.994701i	$0.467218\pi$
$828$	−0.370404	−0.0128724
$829$	−43.0082	−1.49374	−0.746868	−	0.664972i	$-0.768443\pi$
$829$	−43.0082	−1.49374	−0.746868	+	0.664972i	$0.768443\pi$
$830$	6.26596	0.217495
$831$	−29.3390	−1.01776
$832$	0	0
$833$	−1.93147	−0.0669215
$834$	−27.7463	−0.960777
$835$	17.6110	0.609454
$836$	0.130623	0.00451768
$837$	−5.57726	−0.192778
$838$	−25.8191	−0.891907
$839$	−11.9162	−0.411393	−0.205696	−	0.978616i	$-0.565946\pi$
$839$	−11.9162	−0.411393	−0.205696	+	0.978616i	$0.565946\pi$
$840$	3.47259	0.119816
$841$	−22.5536	−0.777710
$842$	−30.4998	−1.05109
$843$	−8.21738	−0.283022
$844$	0.0443015	0.00152492
$845$	0	0
$846$	0.0479783	0.00164953
$847$	2.62619	0.0902370
$848$	0.846113	0.0290556
$849$	−16.8605	−0.578652
$850$	1.15159	0.0394993
$851$	−3.98818	−0.136713
$852$	1.29730	0.0444448
$853$	−20.2202	−0.692327	−0.346164	−	0.938174i	$-0.612516\pi$
$853$	−20.2202	−0.692327	−0.346164	+	0.938174i	$0.612516\pi$
$854$	9.74076	0.333322
$855$	0.113206	0.00387156
$856$	−16.4058	−0.560740
$857$	−20.8507	−0.712248	−0.356124	−	0.934439i	$-0.615902\pi$
$857$	−20.8507	−0.712248	−0.356124	+	0.934439i	$0.615902\pi$
$858$	0	0
$859$	17.2363	0.588094	0.294047	−	0.955791i	$-0.404998\pi$
$859$	17.2363	0.588094	0.294047	+	0.955791i	$0.404998\pi$
$860$	−0.139559	−0.00475894
$861$	8.06706	0.274925
$862$	−8.95239	−0.304920
$863$	22.3118	0.759504	0.379752	−	0.925088i	$-0.376009\pi$
$863$	22.3118	0.759504	0.379752	+	0.925088i	$0.376009\pi$
$864$	−7.66637	−0.260815
$865$	−11.7317	−0.398888
$866$	−4.30506	−0.146292
$867$	26.2058	0.889994
$868$	0.130966	0.00444527
$869$	46.0585	1.56243
$870$	−7.33145	−0.248559
$871$	0	0
$872$	−28.6420	−0.969939
$873$	0.741169	0.0250848
$874$	0.448182	0.0151600
$875$	6.04622	0.204400
$876$	3.81918	0.129038
$877$	−9.16052	−0.309329	−0.154664	−	0.987967i	$-0.549430\pi$
$877$	−9.16052	−0.309329	−0.154664	+	0.987967i	$0.549430\pi$
$878$	−33.1956	−1.12030
$879$	−37.4348	−1.26264
$880$	−19.3627	−0.652717
$881$	56.0116	1.88708	0.943540	−	0.331260i	$-0.107474\pi$
$881$	56.0116	1.88708	0.943540	+	0.331260i	$0.107474\pi$
$882$	−5.34040	−0.179821
$883$	−17.4540	−0.587375	−0.293687	−	0.955902i	$-0.594882\pi$
$883$	−17.4540	−0.587375	−0.293687	+	0.955902i	$0.594882\pi$
$884$	0	0
$885$	18.6448	0.626737
$886$	−20.3709	−0.684373
$887$	47.5313	1.59595	0.797973	−	0.602694i	$-0.205906\pi$
$887$	47.5313	1.59595	0.797973	+	0.602694i	$0.205906\pi$
$888$	−7.27906	−0.244269
$889$	−7.59050	−0.254577
$890$	−26.9730	−0.904136
$891$	27.2685	0.913528
$892$	−4.59825	−0.153961
$893$	0.00808247	0.000270470 0
$894$	−10.7567	−0.359758
$895$	17.0157	0.568771
$896$	−4.72064	−0.157705
$897$	0	0
$898$	6.52287	0.217671
$899$	−2.53898	−0.0846796
$900$	−0.443307	−0.0147769
$901$	−0.0705357	−0.00234989
$902$	−51.3518	−1.70983
$903$	0.336861	0.0112100
$904$	11.8258	0.393321
$905$	28.1744	0.936548
$906$	37.8998	1.25914
$907$	−22.4006	−0.743801	−0.371900	−	0.928273i	$-0.621294\pi$
$907$	−22.4006	−0.743801	−0.371900	+	0.928273i	$0.621294\pi$
$908$	−4.20300	−0.139481
$909$	−1.94901	−0.0646445
$910$	0	0
$911$	59.3261	1.96556	0.982781	−	0.184775i	$-0.0591557\pi$
$911$	59.3261	1.96556	0.982781	+	0.184775i	$0.0591557\pi$
$912$	0.716519	0.0237263
$913$	−13.4042	−0.443614
$914$	1.96504	0.0649979
$915$	−29.9010	−0.988498
$916$	2.77651	0.0917386
$917$	−6.04742	−0.199703
$918$	−2.12622	−0.0701758
$919$	47.8640	1.57889	0.789444	−	0.613822i	$-0.210369\pi$
$919$	47.8640	1.57889	0.789444	+	0.613822i	$0.210369\pi$
$920$	10.5597	0.348141
$921$	−11.7902	−0.388502
$922$	25.1392	0.827915
$923$	0	0
$924$	−0.808987	−0.0266137
$925$	−4.77314	−0.156940
$926$	23.1827	0.761832
$927$	2.67131	0.0877373
$928$	−3.49002	−0.114565
$929$	26.5604	0.871418	0.435709	−	0.900088i	$-0.356498\pi$
$929$	26.5604	0.871418	0.435709	+	0.900088i	$0.356498\pi$
$930$	2.88756	0.0946868
$931$	−0.899649	−0.0294848
$932$	4.10906	0.134597
$933$	−47.5768	−1.55759
$934$	−34.1103	−1.11612
$935$	1.61416	0.0527887
$936$	0	0
$937$	−7.11322	−0.232379	−0.116189	−	0.993227i	$-0.537068\pi$
$937$	−7.11322	−0.232379	−0.116189	+	0.993227i	$0.537068\pi$
$938$	6.29795	0.205635
$939$	18.6772	0.609509
$940$	0.0207384	0.000676411 0
$941$	−54.1823	−1.76629	−0.883146	−	0.469098i	$-0.844579\pi$
$941$	−54.1823	−1.76629	−0.883146	+	0.469098i	$0.844579\pi$
$942$	−36.8510	−1.20067
$943$	24.5308	0.798833
$944$	−29.5279	−0.961052
$945$	−4.20428	−0.136765
$946$	−2.14433	−0.0697182
$947$	44.1004	1.43307	0.716535	−	0.697552i	$-0.245727\pi$
$947$	44.1004	1.43307	0.716535	+	0.697552i	$0.245727\pi$
$948$	−4.37322	−0.142036
$949$	0	0
$950$	0.536394	0.0174029
$951$	−30.5904	−0.991963
$952$	0.458469	0.0148591
$953$	6.82747	0.221163	0.110582	−	0.993867i	$-0.464729\pi$
$953$	6.82747	0.221163	0.110582	+	0.993867i	$0.464729\pi$
$954$	−0.195027	−0.00631423
$955$	7.12506	0.230561
$956$	2.55188	0.0825337
$957$	15.6835	0.506975
$958$	−16.3330	−0.527696
$959$	3.61739	0.116812
$960$	19.0126	0.613630
$961$	1.00000	0.0322581
$962$	0	0
$963$	3.31236	0.106739
$964$	−6.24335	−0.201085
$965$	−21.6047	−0.695481
$966$	−2.77573	−0.0893076
$967$	−26.8389	−0.863080	−0.431540	−	0.902094i	$-0.642030\pi$
$967$	−26.8389	−0.863080	−0.431540	+	0.902094i	$0.642030\pi$
$968$	14.5755	0.468475
$969$	−0.0597322	−0.00191887
$970$	−2.30105	−0.0738824
$971$	−54.6509	−1.75383	−0.876916	−	0.480644i	$-0.840403\pi$
$971$	−54.6509	−1.75383	−0.876916	+	0.480644i	$0.840403\pi$
$972$	1.50049	0.0481282
$973$	−7.24344	−0.232214
$974$	53.9398	1.72834
$975$	0	0
$976$	47.3546	1.51578
$977$	36.9930	1.18351	0.591756	−	0.806117i	$-0.298435\pi$
$977$	36.9930	1.18351	0.591756	+	0.806117i	$0.298435\pi$
$978$	−31.5469	−1.00876
$979$	57.7007	1.84412
$980$	−2.30836	−0.0737378
$981$	5.78285	0.184632
$982$	−44.6627	−1.42525
$983$	−45.7906	−1.46049	−0.730246	−	0.683184i	$-0.760595\pi$
$983$	−45.7906	−1.46049	−0.730246	+	0.683184i	$0.760595\pi$
$984$	44.7726	1.42730
$985$	7.09430	0.226043
$986$	−0.967937	−0.0308254
$987$	−0.0500572	−0.00159334
$988$	0	0
$989$	1.02435	0.0325724
$990$	4.46306	0.141845
$991$	12.6196	0.400876	0.200438	−	0.979706i	$-0.435763\pi$
$991$	12.6196	0.400876	0.200438	+	0.979706i	$0.435763\pi$
$992$	1.37458	0.0436428
$993$	−15.4716	−0.490976
$994$	−2.43254	−0.0771555
$995$	0.191697	0.00607720
$996$	1.27272	0.0403277
$997$	29.7544	0.942332	0.471166	−	0.882045i	$-0.343833\pi$
$997$	29.7544	0.942332	0.471166	+	0.882045i	$0.343833\pi$
$998$	15.7456	0.498418
$999$	8.81279	0.278824

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5239.2.a.r.1.9		34
13.6	odd	12		403.2.r.a.218.25	✓	68
13.11	odd	12		403.2.r.a.342.25	yes	68
13.12	even	2		5239.2.a.q.1.26		34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.r.a.218.25	✓	68	13.6	odd	12
403.2.r.a.342.25	yes	68	13.11	odd	12
5239.2.a.q.1.26		34	13.12	even	2
5239.2.a.r.1.9		34	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:	\( N \)	\(=\)	\( 5239 = 13^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5239.a (trivial)

\(f(q)\)	\(=\)	\(q-1.32498 q^{2} -1.54906 q^{3} -0.244422 q^{4} -1.40687 q^{5} +2.05248 q^{6} +0.535818 q^{7} +2.97382 q^{8} -0.600418 q^{9} +O(q^{10})\)	\(q-1.32498 q^{2} -1.54906 q^{3} -0.244422 q^{4} -1.40687 q^{5} +2.05248 q^{6} +0.535818 q^{7} +2.97382 q^{8} -0.600418 q^{9} +1.86407 q^{10} -3.98764 q^{11} +0.378624 q^{12} -0.709950 q^{14} +2.17932 q^{15} -3.45141 q^{16} +0.287725 q^{17} +0.795543 q^{18} +0.134018 q^{19} +0.343869 q^{20} -0.830014 q^{21} +5.28355 q^{22} -2.52396 q^{23} -4.60662 q^{24} -3.02072 q^{25} +5.57726 q^{27} -0.130966 q^{28} +2.53898 q^{29} -2.88756 q^{30} -1.00000 q^{31} -1.37458 q^{32} +6.17709 q^{33} -0.381231 q^{34} -0.753825 q^{35} +0.146755 q^{36} +1.58013 q^{37} -0.177571 q^{38} -4.18377 q^{40} -9.71919 q^{41} +1.09975 q^{42} -0.405850 q^{43} +0.974667 q^{44} +0.844708 q^{45} +3.34420 q^{46} +0.0603089 q^{47} +5.34644 q^{48} -6.71290 q^{49} +4.00241 q^{50} -0.445703 q^{51} -0.245150 q^{53} -7.38977 q^{54} +5.61008 q^{55} +1.59343 q^{56} -0.207602 q^{57} -3.36410 q^{58} +8.55531 q^{59} -0.532674 q^{60} -13.7204 q^{61} +1.32498 q^{62} -0.321715 q^{63} +8.72412 q^{64} -8.18453 q^{66} -8.87098 q^{67} -0.0703264 q^{68} +3.90976 q^{69} +0.998805 q^{70} +3.42636 q^{71} -1.78553 q^{72} +10.0870 q^{73} -2.09364 q^{74} +4.67928 q^{75} -0.0327569 q^{76} -2.13665 q^{77} -11.5503 q^{79} +4.85568 q^{80} -6.83824 q^{81} +12.8778 q^{82} +3.36143 q^{83} +0.202874 q^{84} -0.404791 q^{85} +0.537744 q^{86} -3.93302 q^{87} -11.8585 q^{88} -14.4699 q^{89} -1.11922 q^{90} +0.616910 q^{92} +1.54906 q^{93} -0.0799082 q^{94} -0.188545 q^{95} +2.12930 q^{96} -1.23442 q^{97} +8.89447 q^{98} +2.39425 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q + 8 q^{2} + 32 q^{4} + 16 q^{5} + 12 q^{6} + 8 q^{7} + 24 q^{8} + 34 q^{9}+O(q^{10}) \) 34 * q + 8 * q^2 + 32 * q^4 + 16 * q^5 + 12 * q^6 + 8 * q^7 + 24 * q^8 + 34 * q^9	\( 34 q + 8 q^{2} + 32 q^{4} + 16 q^{5} + 12 q^{6} + 8 q^{7} + 24 q^{8} + 34 q^{9} + 8 q^{10} + 26 q^{11} + 8 q^{12} - 4 q^{14} + 16 q^{15} + 36 q^{16} - 6 q^{17} + 64 q^{18} + 4 q^{19} + 40 q^{20} + 32 q^{21} + 20 q^{22} - 8 q^{23} - 16 q^{24} + 36 q^{25} - 6 q^{27} + 24 q^{28} + 32 q^{30} - 34 q^{31} + 36 q^{32} + 40 q^{33} + 16 q^{34} - 30 q^{35} + 40 q^{36} + 2 q^{37} + 18 q^{38} + 4 q^{40} + 80 q^{41} + 16 q^{42} + 12 q^{43} + 108 q^{44} + 12 q^{45} + 48 q^{46} + 24 q^{47} + 46 q^{48} + 22 q^{49} - 44 q^{50} - 28 q^{51} + 10 q^{53} + 48 q^{54} + 6 q^{55} - 2 q^{56} + 66 q^{57} - 44 q^{58} + 64 q^{59} + 48 q^{60} - 6 q^{61} - 8 q^{62} - 52 q^{63} - 12 q^{64} + 4 q^{66} + 16 q^{67} - 58 q^{68} - 28 q^{69} + 72 q^{70} + 52 q^{71} + 152 q^{72} + 42 q^{73} - 8 q^{74} - 4 q^{75} - 48 q^{76} + 10 q^{77} + 8 q^{79} + 48 q^{80} + 58 q^{81} - 42 q^{82} + 44 q^{83} - 8 q^{84} + 96 q^{85} + 16 q^{86} + 20 q^{87} + 64 q^{88} + 74 q^{89} - 26 q^{90} + 24 q^{92} - 8 q^{94} - 32 q^{95} + 50 q^{96} + 40 q^{97} + 72 q^{98} + 74 q^{99}+O(q^{100}) \) 34 * q + 8 * q^2 + 32 * q^4 + 16 * q^5 + 12 * q^6 + 8 * q^7 + 24 * q^8 + 34 * q^9 + 8 * q^10 + 26 * q^11 + 8 * q^12 - 4 * q^14 + 16 * q^15 + 36 * q^16 - 6 * q^17 + 64 * q^18 + 4 * q^19 + 40 * q^20 + 32 * q^21 + 20 * q^22 - 8 * q^23 - 16 * q^24 + 36 * q^25 - 6 * q^27 + 24 * q^28 + 32 * q^30 - 34 * q^31 + 36 * q^32 + 40 * q^33 + 16 * q^34 - 30 * q^35 + 40 * q^36 + 2 * q^37 + 18 * q^38 + 4 * q^40 + 80 * q^41 + 16 * q^42 + 12 * q^43 + 108 * q^44 + 12 * q^45 + 48 * q^46 + 24 * q^47 + 46 * q^48 + 22 * q^49 - 44 * q^50 - 28 * q^51 + 10 * q^53 + 48 * q^54 + 6 * q^55 - 2 * q^56 + 66 * q^57 - 44 * q^58 + 64 * q^59 + 48 * q^60 - 6 * q^61 - 8 * q^62 - 52 * q^63 - 12 * q^64 + 4 * q^66 + 16 * q^67 - 58 * q^68 - 28 * q^69 + 72 * q^70 + 52 * q^71 + 152 * q^72 + 42 * q^73 - 8 * q^74 - 4 * q^75 - 48 * q^76 + 10 * q^77 + 8 * q^79 + 48 * q^80 + 58 * q^81 - 42 * q^82 + 44 * q^83 - 8 * q^84 + 96 * q^85 + 16 * q^86 + 20 * q^87 + 64 * q^88 + 74 * q^89 - 26 * q^90 + 24 * q^92 - 8 * q^94 - 32 * q^95 + 50 * q^96 + 40 * q^97 + 72 * q^98 + 74 * q^99

Introduction

L-functions

Modular forms

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