LMFDB - Embedded newform 6034.2.a.r.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6034,2,Mod(1,6034)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6034, 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("6034.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6034 = 2 \cdot 7 \cdot 431 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6034.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:	$48.1817325796$
Analytic rank:	$0$
Dimension:	$31$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	6034.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +0.747767 q^{3} +1.00000 q^{4} -1.63599 q^{5} +0.747767 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.44084 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.747767 q^{3} +1.00000 q^{4} -1.63599 q^{5} +0.747767 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.44084 q^{9} -1.63599 q^{10} -4.29976 q^{11} +0.747767 q^{12} +6.70451 q^{13} -1.00000 q^{14} -1.22334 q^{15} +1.00000 q^{16} -1.42662 q^{17} -2.44084 q^{18} +3.01277 q^{19} -1.63599 q^{20} -0.747767 q^{21} -4.29976 q^{22} +0.971058 q^{23} +0.747767 q^{24} -2.32355 q^{25} +6.70451 q^{26} -4.06848 q^{27} -1.00000 q^{28} +9.17402 q^{29} -1.22334 q^{30} -8.60874 q^{31} +1.00000 q^{32} -3.21522 q^{33} -1.42662 q^{34} +1.63599 q^{35} -2.44084 q^{36} -0.352370 q^{37} +3.01277 q^{38} +5.01342 q^{39} -1.63599 q^{40} -0.0784354 q^{41} -0.747767 q^{42} +7.08354 q^{43} -4.29976 q^{44} +3.99319 q^{45} +0.971058 q^{46} +8.20358 q^{47} +0.747767 q^{48} +1.00000 q^{49} -2.32355 q^{50} -1.06678 q^{51} +6.70451 q^{52} +11.5449 q^{53} -4.06848 q^{54} +7.03435 q^{55} -1.00000 q^{56} +2.25285 q^{57} +9.17402 q^{58} -1.43005 q^{59} -1.22334 q^{60} -3.40442 q^{61} -8.60874 q^{62} +2.44084 q^{63} +1.00000 q^{64} -10.9685 q^{65} -3.21522 q^{66} -5.22623 q^{67} -1.42662 q^{68} +0.726125 q^{69} +1.63599 q^{70} +9.88223 q^{71} -2.44084 q^{72} +2.30766 q^{73} -0.352370 q^{74} -1.73748 q^{75} +3.01277 q^{76} +4.29976 q^{77} +5.01342 q^{78} +14.0440 q^{79} -1.63599 q^{80} +4.28025 q^{81} -0.0784354 q^{82} -14.1578 q^{83} -0.747767 q^{84} +2.33393 q^{85} +7.08354 q^{86} +6.86003 q^{87} -4.29976 q^{88} +11.2815 q^{89} +3.99319 q^{90} -6.70451 q^{91} +0.971058 q^{92} -6.43733 q^{93} +8.20358 q^{94} -4.92885 q^{95} +0.747767 q^{96} +9.41142 q^{97} +1.00000 q^{98} +10.4951 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q + 31 q^{2} + 7 q^{3} + 31 q^{4} + 15 q^{5} + 7 q^{6} - 31 q^{7} + 31 q^{8} + 42 q^{9}+O(q^{10}) $ 31 * q + 31 * q^2 + 7 * q^3 + 31 * q^4 + 15 * q^5 + 7 * q^6 - 31 * q^7 + 31 * q^8 + 42 * q^9	\( 31 q + 31 q^{2} + 7 q^{3} + 31 q^{4} + 15 q^{5} + 7 q^{6} - 31 q^{7} + 31 q^{8} + 42 q^{9} + 15 q^{10} + 12 q^{11} + 7 q^{12} + 26 q^{13} - 31 q^{14} + 6 q^{15} + 31 q^{16} + 33 q^{17} + 42 q^{18} + 34 q^{19} + 15 q^{20} - 7 q^{21} + 12 q^{22} - 14 q^{23} + 7 q^{24} + 58 q^{25} + 26 q^{26} + 28 q^{27} - 31 q^{28} + 11 q^{29} + 6 q^{30} + 19 q^{31} + 31 q^{32} + 43 q^{33} + 33 q^{34} - 15 q^{35} + 42 q^{36} + 2 q^{37} + 34 q^{38} - 16 q^{39} + 15 q^{40} + 53 q^{41} - 7 q^{42} + 22 q^{43} + 12 q^{44} + 43 q^{45} - 14 q^{46} + 27 q^{47} + 7 q^{48} + 31 q^{49} + 58 q^{50} + 17 q^{51} + 26 q^{52} + 11 q^{53} + 28 q^{54} + 19 q^{55} - 31 q^{56} + 45 q^{57} + 11 q^{58} + 54 q^{59} + 6 q^{60} + 41 q^{61} + 19 q^{62} - 42 q^{63} + 31 q^{64} + 30 q^{65} + 43 q^{66} + 13 q^{67} + 33 q^{68} + 17 q^{69} - 15 q^{70} + 43 q^{71} + 42 q^{72} + 42 q^{73} + 2 q^{74} + 62 q^{75} + 34 q^{76} - 12 q^{77} - 16 q^{78} - 12 q^{79} + 15 q^{80} + 63 q^{81} + 53 q^{82} + 35 q^{83} - 7 q^{84} + 16 q^{85} + 22 q^{86} - 4 q^{87} + 12 q^{88} + 115 q^{89} + 43 q^{90} - 26 q^{91} - 14 q^{92} + q^{93} + 27 q^{94} - 13 q^{95} + 7 q^{96} + 32 q^{97} + 31 q^{98} + 34 q^{99}+O(q^{100}) \) 31 * q + 31 * q^2 + 7 * q^3 + 31 * q^4 + 15 * q^5 + 7 * q^6 - 31 * q^7 + 31 * q^8 + 42 * q^9 + 15 * q^10 + 12 * q^11 + 7 * q^12 + 26 * q^13 - 31 * q^14 + 6 * q^15 + 31 * q^16 + 33 * q^17 + 42 * q^18 + 34 * q^19 + 15 * q^20 - 7 * q^21 + 12 * q^22 - 14 * q^23 + 7 * q^24 + 58 * q^25 + 26 * q^26 + 28 * q^27 - 31 * q^28 + 11 * q^29 + 6 * q^30 + 19 * q^31 + 31 * q^32 + 43 * q^33 + 33 * q^34 - 15 * q^35 + 42 * q^36 + 2 * q^37 + 34 * q^38 - 16 * q^39 + 15 * q^40 + 53 * q^41 - 7 * q^42 + 22 * q^43 + 12 * q^44 + 43 * q^45 - 14 * q^46 + 27 * q^47 + 7 * q^48 + 31 * q^49 + 58 * q^50 + 17 * q^51 + 26 * q^52 + 11 * q^53 + 28 * q^54 + 19 * q^55 - 31 * q^56 + 45 * q^57 + 11 * q^58 + 54 * q^59 + 6 * q^60 + 41 * q^61 + 19 * q^62 - 42 * q^63 + 31 * q^64 + 30 * q^65 + 43 * q^66 + 13 * q^67 + 33 * q^68 + 17 * q^69 - 15 * q^70 + 43 * q^71 + 42 * q^72 + 42 * q^73 + 2 * q^74 + 62 * q^75 + 34 * q^76 - 12 * q^77 - 16 * q^78 - 12 * q^79 + 15 * q^80 + 63 * q^81 + 53 * q^82 + 35 * q^83 - 7 * q^84 + 16 * q^85 + 22 * q^86 - 4 * q^87 + 12 * q^88 + 115 * q^89 + 43 * q^90 - 26 * q^91 - 14 * q^92 + q^93 + 27 * q^94 - 13 * q^95 + 7 * q^96 + 32 * q^97 + 31 * q^98 + 34 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0.747767	0.431724	0.215862	−	0.976424i	$-0.430744\pi$
$3$	0.747767	0.431724	0.215862	+	0.976424i	$0.430744\pi$
$4$	1.00000	0.500000
$5$	−1.63599	−0.731635	−0.365817	−	0.930687i	$-0.619211\pi$
$5$	−1.63599	−0.731635	−0.365817	+	0.930687i	$0.619211\pi$
$6$	0.747767	0.305275
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	−2.44084	−0.813615
$10$	−1.63599	−0.517344
$11$	−4.29976	−1.29643	−0.648214	−	0.761459i	$-0.724484\pi$
$11$	−4.29976	−1.29643	−0.648214	+	0.761459i	$0.724484\pi$
$12$	0.747767	0.215862
$13$	6.70451	1.85950	0.929749	−	0.368194i	$-0.120024\pi$
$13$	6.70451	1.85950	0.929749	+	0.368194i	$0.120024\pi$
$14$	−1.00000	−0.267261
$15$	−1.22334	−0.315864
$16$	1.00000	0.250000
$17$	−1.42662	−0.346006	−0.173003	−	0.984921i	$-0.555347\pi$
$17$	−1.42662	−0.346006	−0.173003	+	0.984921i	$0.555347\pi$
$18$	−2.44084	−0.575313
$19$	3.01277	0.691177	0.345588	−	0.938386i	$-0.387679\pi$
$19$	3.01277	0.691177	0.345588	+	0.938386i	$0.387679\pi$
$20$	−1.63599	−0.365817
$21$	−0.747767	−0.163176
$22$	−4.29976	−0.916712
$23$	0.971058	0.202480	0.101240	−	0.994862i	$-0.467719\pi$
$23$	0.971058	0.202480	0.101240	+	0.994862i	$0.467719\pi$
$24$	0.747767	0.152637
$25$	−2.32355	−0.464711
$26$	6.70451	1.31486
$27$	−4.06848	−0.782980
$28$	−1.00000	−0.188982
$29$	9.17402	1.70357	0.851786	−	0.523890i	$-0.175520\pi$
$29$	9.17402	1.70357	0.851786	+	0.523890i	$0.175520\pi$
$30$	−1.22334	−0.223350
$31$	−8.60874	−1.54618	−0.773088	−	0.634299i	$-0.781289\pi$
$31$	−8.60874	−1.54618	−0.773088	+	0.634299i	$0.781289\pi$
$32$	1.00000	0.176777
$33$	−3.21522	−0.559698
$34$	−1.42662	−0.244663
$35$	1.63599	0.276532
$36$	−2.44084	−0.406807
$37$	−0.352370	−0.0579292	−0.0289646	−	0.999580i	$-0.509221\pi$
$37$	−0.352370	−0.0579292	−0.0289646	+	0.999580i	$0.509221\pi$
$38$	3.01277	0.488736
$39$	5.01342	0.802789
$40$	−1.63599	−0.258672
$41$	−0.0784354	−0.0122495	−0.00612477	−	0.999981i	$-0.501950\pi$
$41$	−0.0784354	−0.0122495	−0.00612477	+	0.999981i	$0.501950\pi$
$42$	−0.747767	−0.115383
$43$	7.08354	1.08023	0.540115	−	0.841591i	$-0.318381\pi$
$43$	7.08354	1.08023	0.540115	+	0.841591i	$0.318381\pi$
$44$	−4.29976	−0.648214
$45$	3.99319	0.595269
$46$	0.971058	0.143175
$47$	8.20358	1.19662	0.598308	−	0.801267i	$-0.295840\pi$
$47$	8.20358	1.19662	0.598308	+	0.801267i	$0.295840\pi$
$48$	0.747767	0.107931
$49$	1.00000	0.142857
$50$	−2.32355	−0.328600
$51$	−1.06678	−0.149379
$52$	6.70451	0.929749
$53$	11.5449	1.58582	0.792908	−	0.609342i	$-0.208566\pi$
$53$	11.5449	1.58582	0.792908	+	0.609342i	$0.208566\pi$
$54$	−4.06848	−0.553651
$55$	7.03435	0.948511
$56$	−1.00000	−0.133631
$57$	2.25285	0.298397
$58$	9.17402	1.20461
$59$	−1.43005	−0.186177	−0.0930884	−	0.995658i	$-0.529674\pi$
$59$	−1.43005	−0.186177	−0.0930884	+	0.995658i	$0.529674\pi$
$60$	−1.22334	−0.157932
$61$	−3.40442	−0.435892	−0.217946	−	0.975961i	$-0.569936\pi$
$61$	−3.40442	−0.435892	−0.217946	+	0.975961i	$0.569936\pi$
$62$	−8.60874	−1.09331
$63$	2.44084	0.307517
$64$	1.00000	0.125000
$65$	−10.9685	−1.36047
$66$	−3.21522	−0.395766
$67$	−5.22623	−0.638485	−0.319243	−	0.947673i	$-0.603429\pi$
$67$	−5.22623	−0.638485	−0.319243	+	0.947673i	$0.603429\pi$
$68$	−1.42662	−0.173003
$69$	0.726125	0.0874152
$70$	1.63599	0.195538
$71$	9.88223	1.17280	0.586402	−	0.810020i	$-0.300544\pi$
$71$	9.88223	1.17280	0.586402	+	0.810020i	$0.300544\pi$
$72$	−2.44084	−0.287656
$73$	2.30766	0.270091	0.135045	−	0.990839i	$-0.456882\pi$
$73$	2.30766	0.270091	0.135045	+	0.990839i	$0.456882\pi$
$74$	−0.352370	−0.0409621
$75$	−1.73748	−0.200626
$76$	3.01277	0.345588
$77$	4.29976	0.490003
$78$	5.01342	0.567658
$79$	14.0440	1.58007	0.790036	−	0.613060i	$-0.210062\pi$
$79$	14.0440	1.58007	0.790036	+	0.613060i	$0.210062\pi$
$80$	−1.63599	−0.182909
$81$	4.28025	0.475584
$82$	−0.0784354	−0.00866174
$83$	−14.1578	−1.55402	−0.777012	−	0.629486i	$-0.783265\pi$
$83$	−14.1578	−1.55402	−0.777012	+	0.629486i	$0.783265\pi$
$84$	−0.747767	−0.0815881
$85$	2.33393	0.253150
$86$	7.08354	0.763838
$87$	6.86003	0.735472
$88$	−4.29976	−0.458356
$89$	11.2815	1.19584	0.597918	−	0.801558i	$-0.295995\pi$
$89$	11.2815	1.19584	0.597918	+	0.801558i	$0.295995\pi$
$90$	3.99319	0.420919
$91$	−6.70451	−0.702824
$92$	0.971058	0.101240
$93$	−6.43733	−0.667520
$94$	8.20358	0.846135
$95$	−4.92885	−0.505689
$96$	0.747767	0.0763187
$97$	9.41142	0.955585	0.477793	−	0.878473i	$-0.341437\pi$
$97$	9.41142	0.955585	0.477793	+	0.878473i	$0.341437\pi$
$98$	1.00000	0.101015
$99$	10.4951	1.05479
$100$	−2.32355	−0.232355
$101$	5.95861	0.592903	0.296452	−	0.955048i	$-0.404197\pi$
$101$	5.95861	0.592903	0.296452	+	0.955048i	$0.404197\pi$
$102$	−1.06678	−0.105627
$103$	3.50190	0.345052	0.172526	−	0.985005i	$-0.444807\pi$
$103$	3.50190	0.345052	0.172526	+	0.985005i	$0.444807\pi$
$104$	6.70451	0.657432
$105$	1.22334	0.119385
$106$	11.5449	1.12134
$107$	4.73600	0.457846	0.228923	−	0.973444i	$-0.426480\pi$
$107$	4.73600	0.457846	0.228923	+	0.973444i	$0.426480\pi$
$108$	−4.06848	−0.391490
$109$	−7.09960	−0.680019	−0.340009	−	0.940422i	$-0.610430\pi$
$109$	−7.09960	−0.680019	−0.340009	+	0.940422i	$0.610430\pi$
$110$	7.03435	0.670699
$111$	−0.263490	−0.0250094
$112$	−1.00000	−0.0944911
$113$	−2.68036	−0.252147	−0.126074	−	0.992021i	$-0.540238\pi$
$113$	−2.68036	−0.252147	−0.126074	+	0.992021i	$0.540238\pi$
$114$	2.25285	0.210999
$115$	−1.58864	−0.148141
$116$	9.17402	0.851786
$117$	−16.3647	−1.51291
$118$	−1.43005	−0.131647
$119$	1.42662	0.130778
$120$	−1.22334	−0.111675
$121$	7.48796	0.680723
$122$	−3.40442	−0.308222
$123$	−0.0586514	−0.00528842
$124$	−8.60874	−0.773088
$125$	11.9812	1.07163
$126$	2.44084	0.217448
$127$	−17.7440	−1.57453	−0.787263	−	0.616618i	$-0.788502\pi$
$127$	−17.7440	−1.57453	−0.787263	+	0.616618i	$0.788502\pi$
$128$	1.00000	0.0883883
$129$	5.29684	0.466361
$130$	−10.9685	−0.962000
$131$	−6.99691	−0.611323	−0.305661	−	0.952140i	$-0.598878\pi$
$131$	−6.99691	−0.611323	−0.305661	+	0.952140i	$0.598878\pi$
$132$	−3.21522	−0.279849
$133$	−3.01277	−0.261240
$134$	−5.22623	−0.451477
$135$	6.65598	0.572856
$136$	−1.42662	−0.122332
$137$	2.99018	0.255468	0.127734	−	0.991808i	$-0.459230\pi$
$137$	2.99018	0.255468	0.127734	+	0.991808i	$0.459230\pi$
$138$	0.726125	0.0618119
$139$	18.5734	1.57538	0.787689	−	0.616073i	$-0.211278\pi$
$139$	18.5734	1.57538	0.787689	+	0.616073i	$0.211278\pi$
$140$	1.63599	0.138266
$141$	6.13437	0.516607
$142$	9.88223	0.829298
$143$	−28.8278	−2.41070
$144$	−2.44084	−0.203404
$145$	−15.0086	−1.24639
$146$	2.30766	0.190983
$147$	0.747767	0.0616748
$148$	−0.352370	−0.0289646
$149$	11.4868	0.941031	0.470516	−	0.882392i	$-0.344068\pi$
$149$	11.4868	0.941031	0.470516	+	0.882392i	$0.344068\pi$
$150$	−1.73748	−0.141864
$151$	−1.48700	−0.121010	−0.0605050	−	0.998168i	$-0.519271\pi$
$151$	−1.48700	−0.121010	−0.0605050	+	0.998168i	$0.519271\pi$
$152$	3.01277	0.244368
$153$	3.48216	0.281516
$154$	4.29976	0.346485
$155$	14.0838	1.13124
$156$	5.01342	0.401394
$157$	0.859916	0.0686287	0.0343144	−	0.999411i	$-0.489075\pi$
$157$	0.859916	0.0686287	0.0343144	+	0.999411i	$0.489075\pi$
$158$	14.0440	1.11728
$159$	8.63290	0.684634
$160$	−1.63599	−0.129336
$161$	−0.971058	−0.0765301
$162$	4.28025	0.336289
$163$	−21.9323	−1.71787	−0.858937	−	0.512081i	$-0.828875\pi$
$163$	−21.9323	−1.71787	−0.858937	+	0.512081i	$0.828875\pi$
$164$	−0.0784354	−0.00612477
$165$	5.26005	0.409495
$166$	−14.1578	−1.09886
$167$	16.5814	1.28311	0.641553	−	0.767078i	$-0.278290\pi$
$167$	16.5814	1.28311	0.641553	+	0.767078i	$0.278290\pi$
$168$	−0.747767	−0.0576915
$169$	31.9505	2.45773
$170$	2.33393	0.179004
$171$	−7.35370	−0.562352
$172$	7.08354	0.540115
$173$	5.68618	0.432312	0.216156	−	0.976359i	$-0.430648\pi$
$173$	5.68618	0.432312	0.216156	+	0.976359i	$0.430648\pi$
$174$	6.86003	0.520057
$175$	2.32355	0.175644
$176$	−4.29976	−0.324107
$177$	−1.06935	−0.0803769
$178$	11.2815	0.845583
$179$	2.93803	0.219599	0.109799	−	0.993954i	$-0.464979\pi$
$179$	2.93803	0.219599	0.109799	+	0.993954i	$0.464979\pi$
$180$	3.99319	0.297634
$181$	26.3697	1.96004	0.980021	−	0.198896i	$-0.0637355\pi$
$181$	26.3697	1.96004	0.980021	+	0.198896i	$0.0637355\pi$
$182$	−6.70451	−0.496972
$183$	−2.54572	−0.188185
$184$	0.971058	0.0715873
$185$	0.576472	0.0423830
$186$	−6.43733	−0.472008
$187$	6.13413	0.448572
$188$	8.20358	0.598308
$189$	4.06848	0.295939
$190$	−4.92885	−0.357576
$191$	13.1186	0.949227	0.474613	−	0.880194i	$-0.342588\pi$
$191$	13.1186	0.949227	0.474613	+	0.880194i	$0.342588\pi$
$192$	0.747767	0.0539654
$193$	19.6332	1.41323	0.706614	−	0.707599i	$-0.250222\pi$
$193$	19.6332	1.41323	0.706614	+	0.707599i	$0.250222\pi$
$194$	9.41142	0.675701
$195$	−8.20187	−0.587348
$196$	1.00000	0.0714286
$197$	14.0412	1.00039	0.500197	−	0.865912i	$-0.333261\pi$
$197$	14.0412	1.00039	0.500197	+	0.865912i	$0.333261\pi$
$198$	10.4951	0.745851
$199$	−13.8196	−0.979644	−0.489822	−	0.871822i	$-0.662938\pi$
$199$	−13.8196	−0.979644	−0.489822	+	0.871822i	$0.662938\pi$
$200$	−2.32355	−0.164300
$201$	−3.90800	−0.275649
$202$	5.95861	0.419246
$203$	−9.17402	−0.643890
$204$	−1.06678	−0.0746895
$205$	0.128319	0.00896220
$206$	3.50190	0.243989
$207$	−2.37020	−0.164740
$208$	6.70451	0.464874
$209$	−12.9542	−0.896061
$210$	1.22334	0.0844182
$211$	−26.6372	−1.83378	−0.916889	−	0.399142i	$-0.869308\pi$
$211$	−26.6372	−1.83378	−0.916889	+	0.399142i	$0.869308\pi$
$212$	11.5449	0.792908
$213$	7.38960	0.506327
$214$	4.73600	0.323746
$215$	−11.5886	−0.790334
$216$	−4.06848	−0.276825
$217$	8.60874	0.584399
$218$	−7.09960	−0.480846
$219$	1.72559	0.116605
$220$	7.03435	0.474256
$221$	−9.56480	−0.643398
$222$	−0.263490	−0.0176843
$223$	25.4960	1.70734	0.853670	−	0.520814i	$-0.174372\pi$
$223$	25.4960	1.70734	0.853670	+	0.520814i	$0.174372\pi$
$224$	−1.00000	−0.0668153
$225$	5.67143	0.378095
$226$	−2.68036	−0.178295
$227$	16.1673	1.07306	0.536530	−	0.843881i	$-0.319735\pi$
$227$	16.1673	1.07306	0.536530	+	0.843881i	$0.319735\pi$
$228$	2.25285	0.149199
$229$	14.5978	0.964652	0.482326	−	0.875992i	$-0.339792\pi$
$229$	14.5978	0.964652	0.482326	+	0.875992i	$0.339792\pi$
$230$	−1.58864	−0.104752
$231$	3.21522	0.211546
$232$	9.17402	0.602304
$233$	−18.4000	−1.20543	−0.602713	−	0.797958i	$-0.705914\pi$
$233$	−18.4000	−1.20543	−0.602713	+	0.797958i	$0.705914\pi$
$234$	−16.3647	−1.06979
$235$	−13.4209	−0.875485
$236$	−1.43005	−0.0930884
$237$	10.5016	0.682154
$238$	1.42662	0.0924741
$239$	7.47535	0.483540	0.241770	−	0.970334i	$-0.422272\pi$
$239$	7.47535	0.483540	0.241770	+	0.970334i	$0.422272\pi$
$240$	−1.22334	−0.0789660
$241$	−29.5944	−1.90634	−0.953170	−	0.302435i	$-0.902200\pi$
$241$	−29.5944	−1.90634	−0.953170	+	0.302435i	$0.902200\pi$
$242$	7.48796	0.481344
$243$	15.4061	0.988301
$244$	−3.40442	−0.217946
$245$	−1.63599	−0.104519
$246$	−0.0586514	−0.00373948
$247$	20.1992	1.28524
$248$	−8.60874	−0.546656
$249$	−10.5868	−0.670908
$250$	11.9812	0.757759
$251$	−25.4400	−1.60576	−0.802879	−	0.596141i	$-0.796700\pi$
$251$	−25.4400	−1.60576	−0.802879	+	0.596141i	$0.796700\pi$
$252$	2.44084	0.153759
$253$	−4.17532	−0.262500
$254$	−17.7440	−1.11336
$255$	1.74524	0.109291
$256$	1.00000	0.0625000
$257$	6.10514	0.380828	0.190414	−	0.981704i	$-0.439017\pi$
$257$	6.10514	0.380828	0.190414	+	0.981704i	$0.439017\pi$
$258$	5.29684	0.329767
$259$	0.352370	0.0218952
$260$	−10.9685	−0.680237
$261$	−22.3923	−1.38605
$262$	−6.99691	−0.432271
$263$	28.5441	1.76010	0.880051	−	0.474879i	$-0.157508\pi$
$263$	28.5441	1.76010	0.880051	+	0.474879i	$0.157508\pi$
$264$	−3.21522	−0.197883
$265$	−18.8873	−1.16024
$266$	−3.01277	−0.184725
$267$	8.43592	0.516270
$268$	−5.22623	−0.319243
$269$	−9.77129	−0.595766	−0.297883	−	0.954602i	$-0.596281\pi$
$269$	−9.77129	−0.595766	−0.297883	+	0.954602i	$0.596281\pi$
$270$	6.65598	0.405070
$271$	−23.0418	−1.39969	−0.699844	−	0.714295i	$-0.746747\pi$
$271$	−23.0418	−1.39969	−0.699844	+	0.714295i	$0.746747\pi$
$272$	−1.42662	−0.0865016
$273$	−5.01342	−0.303426
$274$	2.99018	0.180643
$275$	9.99072	0.602463
$276$	0.726125	0.0437076
$277$	−6.07868	−0.365232	−0.182616	−	0.983184i	$-0.558457\pi$
$277$	−6.07868	−0.365232	−0.182616	+	0.983184i	$0.558457\pi$
$278$	18.5734	1.11396
$279$	21.0126	1.25799
$280$	1.63599	0.0977688
$281$	3.59496	0.214458	0.107229	−	0.994234i	$-0.465802\pi$
$281$	3.59496	0.214458	0.107229	+	0.994234i	$0.465802\pi$
$282$	6.13437	0.365296
$283$	15.9846	0.950188	0.475094	−	0.879935i	$-0.342414\pi$
$283$	15.9846	0.950188	0.475094	+	0.879935i	$0.342414\pi$
$284$	9.88223	0.586402
$285$	−3.68563	−0.218318
$286$	−28.8278	−1.70462
$287$	0.0784354	0.00462989
$288$	−2.44084	−0.143828
$289$	−14.9648	−0.880280
$290$	−15.0086	−0.881333
$291$	7.03755	0.412549
$292$	2.30766	0.135045
$293$	16.5823	0.968746	0.484373	−	0.874862i	$-0.339048\pi$
$293$	16.5823	0.968746	0.484373	+	0.874862i	$0.339048\pi$
$294$	0.747767	0.0436107
$295$	2.33954	0.136213
$296$	−0.352370	−0.0204811
$297$	17.4935	1.01508
$298$	11.4868	0.665410
$299$	6.51047	0.376510
$300$	−1.73748	−0.100313
$301$	−7.08354	−0.408289
$302$	−1.48700	−0.0855670
$303$	4.45565	0.255970
$304$	3.01277	0.172794
$305$	5.56958	0.318913
$306$	3.48216	0.199062
$307$	−8.25517	−0.471147	−0.235574	−	0.971856i	$-0.575697\pi$
$307$	−8.25517	−0.471147	−0.235574	+	0.971856i	$0.575697\pi$
$308$	4.29976	0.245002
$309$	2.61860	0.148967
$310$	14.0838	0.799905
$311$	−16.2859	−0.923490	−0.461745	−	0.887013i	$-0.652777\pi$
$311$	−16.2859	−0.923490	−0.461745	+	0.887013i	$0.652777\pi$
$312$	5.01342	0.283829
$313$	−31.4794	−1.77932	−0.889661	−	0.456622i	$-0.849059\pi$
$313$	−31.4794	−1.77932	−0.889661	+	0.456622i	$0.849059\pi$
$314$	0.859916	0.0485278
$315$	−3.99319	−0.224990
$316$	14.0440	0.790036
$317$	0.123332	0.00692699	0.00346349	−	0.999994i	$-0.498898\pi$
$317$	0.123332	0.00692699	0.00346349	+	0.999994i	$0.498898\pi$
$318$	8.63290	0.484109
$319$	−39.4461	−2.20856
$320$	−1.63599	−0.0914544
$321$	3.54142	0.197663
$322$	−0.971058	−0.0541149
$323$	−4.29808	−0.239152
$324$	4.28025	0.237792
$325$	−15.5783	−0.864128
$326$	−21.9323	−1.21472
$327$	−5.30885	−0.293580
$328$	−0.0784354	−0.00433087
$329$	−8.20358	−0.452278
$330$	5.26005	0.289556
$331$	−6.37897	−0.350620	−0.175310	−	0.984513i	$-0.556093\pi$
$331$	−6.37897	−0.350620	−0.175310	+	0.984513i	$0.556093\pi$
$332$	−14.1578	−0.777012
$333$	0.860080	0.0471321
$334$	16.5814	0.907293
$335$	8.55003	0.467138
$336$	−0.747767	−0.0407940
$337$	−29.0346	−1.58161	−0.790806	−	0.612066i	$-0.790339\pi$
$337$	−29.0346	−1.58161	−0.790806	+	0.612066i	$0.790339\pi$
$338$	31.9505	1.73788
$339$	−2.00428	−0.108858
$340$	2.33393	0.126575
$341$	37.0155	2.00450
$342$	−7.35370	−0.397643
$343$	−1.00000	−0.0539949
$344$	7.08354	0.381919
$345$	−1.18793	−0.0639560
$346$	5.68618	0.305691
$347$	−1.76658	−0.0948352	−0.0474176	−	0.998875i	$-0.515099\pi$
$347$	−1.76658	−0.0948352	−0.0474176	+	0.998875i	$0.515099\pi$
$348$	6.86003	0.367736
$349$	−14.3536	−0.768329	−0.384164	−	0.923265i	$-0.625510\pi$
$349$	−14.3536	−0.768329	−0.384164	+	0.923265i	$0.625510\pi$
$350$	2.32355	0.124199
$351$	−27.2772	−1.45595
$352$	−4.29976	−0.229178
$353$	24.5673	1.30758	0.653792	−	0.756674i	$-0.273177\pi$
$353$	24.5673	1.30758	0.653792	+	0.756674i	$0.273177\pi$
$354$	−1.06935	−0.0568351
$355$	−16.1672	−0.858064
$356$	11.2815	0.597918
$357$	1.06678	0.0564600
$358$	2.93803	0.155280
$359$	30.4314	1.60611	0.803054	−	0.595906i	$-0.203207\pi$
$359$	30.4314	1.60611	0.803054	+	0.595906i	$0.203207\pi$
$360$	3.99319	0.210459
$361$	−9.92321	−0.522274
$362$	26.3697	1.38596
$363$	5.59925	0.293884
$364$	−6.70451	−0.351412
$365$	−3.77529	−0.197608
$366$	−2.54572	−0.133067
$367$	−0.231440	−0.0120811	−0.00604053	−	0.999982i	$-0.501923\pi$
$367$	−0.231440	−0.0120811	−0.00604053	+	0.999982i	$0.501923\pi$
$368$	0.971058	0.0506199
$369$	0.191449	0.00996641
$370$	0.576472	0.0299693
$371$	−11.5449	−0.599382
$372$	−6.43733	−0.333760
$373$	−2.57991	−0.133583	−0.0667914	−	0.997767i	$-0.521276\pi$
$373$	−2.57991	−0.133583	−0.0667914	+	0.997767i	$0.521276\pi$
$374$	6.13413	0.317188
$375$	8.95916	0.462649
$376$	8.20358	0.423067
$377$	61.5073	3.16779
$378$	4.06848	0.209260
$379$	−32.3446	−1.66143	−0.830715	−	0.556698i	$-0.812068\pi$
$379$	−32.3446	−1.66143	−0.830715	+	0.556698i	$0.812068\pi$
$380$	−4.92885	−0.252845
$381$	−13.2684	−0.679760
$382$	13.1186	0.671205
$383$	31.0748	1.58785	0.793924	−	0.608018i	$-0.208035\pi$
$383$	31.0748	1.58785	0.793924	+	0.608018i	$0.208035\pi$
$384$	0.747767	0.0381593
$385$	−7.03435	−0.358504
$386$	19.6332	0.999303
$387$	−17.2898	−0.878891
$388$	9.41142	0.477793
$389$	−16.2158	−0.822175	−0.411087	−	0.911596i	$-0.634851\pi$
$389$	−16.2158	−0.822175	−0.411087	+	0.911596i	$0.634851\pi$
$390$	−8.20187	−0.415318
$391$	−1.38533	−0.0700592
$392$	1.00000	0.0505076
$393$	−5.23206	−0.263923
$394$	14.0412	0.707385
$395$	−22.9758	−1.15604
$396$	10.4951	0.527396
$397$	−12.1890	−0.611748	−0.305874	−	0.952072i	$-0.598949\pi$
$397$	−12.1890	−0.611748	−0.305874	+	0.952072i	$0.598949\pi$
$398$	−13.8196	−0.692713
$399$	−2.25285	−0.112784
$400$	−2.32355	−0.116178
$401$	21.3361	1.06547	0.532737	−	0.846281i	$-0.321163\pi$
$401$	21.3361	1.06547	0.532737	+	0.846281i	$0.321163\pi$
$402$	−3.90800	−0.194913
$403$	−57.7174	−2.87511
$404$	5.95861	0.296452
$405$	−7.00243	−0.347954
$406$	−9.17402	−0.455299
$407$	1.51511	0.0751010
$408$	−1.06678	−0.0528135
$409$	5.87159	0.290331	0.145166	−	0.989407i	$-0.453628\pi$
$409$	5.87159	0.290331	0.145166	+	0.989407i	$0.453628\pi$
$410$	0.128319	0.00633723
$411$	2.23596	0.110292
$412$	3.50190	0.172526
$413$	1.43005	0.0703682
$414$	−2.37020	−0.116489
$415$	23.1620	1.13698
$416$	6.70451	0.328716
$417$	13.8886	0.680128
$418$	−12.9542	−0.633610
$419$	−34.3354	−1.67739	−0.838696	−	0.544600i	$-0.816682\pi$
$419$	−34.3354	−1.67739	−0.838696	+	0.544600i	$0.816682\pi$
$420$	1.22334	0.0596927
$421$	−23.6248	−1.15140	−0.575700	−	0.817661i	$-0.695270\pi$
$421$	−23.6248	−1.15140	−0.575700	+	0.817661i	$0.695270\pi$
$422$	−26.6372	−1.29668
$423$	−20.0237	−0.973584
$424$	11.5449	0.560670
$425$	3.31483	0.160793
$426$	7.38960	0.358027
$427$	3.40442	0.164752
$428$	4.73600	0.228923
$429$	−21.5565	−1.04076
$430$	−11.5886	−0.558851
$431$	−1.00000	−0.0481683
$431$	−1.00000	−0.0481683
$432$	−4.06848	−0.195745
$433$	15.5321	0.746427	0.373213	−	0.927746i	$-0.378256\pi$
$433$	15.5321	0.746427	0.373213	+	0.927746i	$0.378256\pi$
$434$	8.60874	0.413233
$435$	−11.2229	−0.538097
$436$	−7.09960	−0.340009
$437$	2.92557	0.139949
$438$	1.72559	0.0824519
$439$	38.9562	1.85928	0.929638	−	0.368473i	$-0.120119\pi$
$439$	38.9562	1.85928	0.929638	+	0.368473i	$0.120119\pi$
$440$	7.03435	0.335349
$441$	−2.44084	−0.116231
$442$	−9.56480	−0.454951
$443$	−18.2551	−0.867328	−0.433664	−	0.901075i	$-0.642779\pi$
$443$	−18.2551	−0.867328	−0.433664	+	0.901075i	$0.642779\pi$
$444$	−0.263490	−0.0125047
$445$	−18.4563	−0.874915
$446$	25.4960	1.20727
$447$	8.58941	0.406265
$448$	−1.00000	−0.0472456
$449$	−15.1071	−0.712948	−0.356474	−	0.934305i	$-0.616021\pi$
$449$	−15.1071	−0.712948	−0.356474	+	0.934305i	$0.616021\pi$
$450$	5.67143	0.267354
$451$	0.337253	0.0158806
$452$	−2.68036	−0.126074
$453$	−1.11193	−0.0522428
$454$	16.1673	0.758768
$455$	10.9685	0.514211
$456$	2.25285	0.105499
$457$	9.03293	0.422542	0.211271	−	0.977427i	$-0.432240\pi$
$457$	9.03293	0.422542	0.211271	+	0.977427i	$0.432240\pi$
$458$	14.5978	0.682112
$459$	5.80418	0.270916
$460$	−1.58864	−0.0740705
$461$	−21.2711	−0.990696	−0.495348	−	0.868695i	$-0.664959\pi$
$461$	−21.2711	−0.990696	−0.495348	+	0.868695i	$0.664959\pi$
$462$	3.21522	0.149586
$463$	−5.05460	−0.234907	−0.117453	−	0.993078i	$-0.537473\pi$
$463$	−5.05460	−0.234907	−0.117453	+	0.993078i	$0.537473\pi$
$464$	9.17402	0.425893
$465$	10.5314	0.488381
$466$	−18.4000	−0.852365
$467$	21.6082	0.999910	0.499955	−	0.866051i	$-0.333350\pi$
$467$	21.6082	0.999910	0.499955	+	0.866051i	$0.333350\pi$
$468$	−16.3647	−0.756457
$469$	5.22623	0.241325
$470$	−13.4209	−0.619062
$471$	0.643017	0.0296286
$472$	−1.43005	−0.0658235
$473$	−30.4576	−1.40044
$474$	10.5016	0.482356
$475$	−7.00033	−0.321197
$476$	1.42662	0.0653890
$477$	−28.1793	−1.29024
$478$	7.47535	0.341915
$479$	−3.16401	−0.144567	−0.0722837	−	0.997384i	$-0.523029\pi$
$479$	−3.16401	−0.144567	−0.0722837	+	0.997384i	$0.523029\pi$
$480$	−1.22334	−0.0558374
$481$	−2.36247	−0.107719
$482$	−29.5944	−1.34799
$483$	−0.726125	−0.0330398
$484$	7.48796	0.340362
$485$	−15.3969	−0.699139
$486$	15.4061	0.698834
$487$	−25.1661	−1.14038	−0.570192	−	0.821511i	$-0.693131\pi$
$487$	−25.1661	−1.14038	−0.570192	+	0.821511i	$0.693131\pi$
$488$	−3.40442	−0.154111
$489$	−16.4003	−0.741647
$490$	−1.63599	−0.0739063
$491$	−37.9585	−1.71304	−0.856522	−	0.516110i	$-0.827380\pi$
$491$	−37.9585	−1.71304	−0.856522	+	0.516110i	$0.827380\pi$
$492$	−0.0586514	−0.00264421
$493$	−13.0878	−0.589447
$494$	20.1992	0.908803
$495$	−17.1697	−0.771723
$496$	−8.60874	−0.386544
$497$	−9.88223	−0.443278
$498$	−10.5868	−0.474404
$499$	40.8353	1.82804	0.914019	−	0.405671i	$-0.132962\pi$
$499$	40.8353	1.82804	0.914019	+	0.405671i	$0.132962\pi$
$500$	11.9812	0.535817
$501$	12.3990	0.553947
$502$	−25.4400	−1.13544
$503$	44.4759	1.98308	0.991542	−	0.129786i	$-0.0414290\pi$
$503$	44.4759	1.98308	0.991542	+	0.129786i	$0.0414290\pi$
$504$	2.44084	0.108724
$505$	−9.74819	−0.433789
$506$	−4.17532	−0.185615
$507$	23.8915	1.06106
$508$	−17.7440	−0.787263
$509$	9.06432	0.401769	0.200885	−	0.979615i	$-0.435618\pi$
$509$	9.06432	0.401769	0.200885	+	0.979615i	$0.435618\pi$
$510$	1.74524	0.0772803
$511$	−2.30766	−0.102085
$512$	1.00000	0.0441942
$513$	−12.2574	−0.541178
$514$	6.10514	0.269286
$515$	−5.72905	−0.252452
$516$	5.29684	0.233180
$517$	−35.2734	−1.55132
$518$	0.352370	0.0154822
$519$	4.25194	0.186639
$520$	−10.9685	−0.481000
$521$	0.192659	0.00844056	0.00422028	−	0.999991i	$-0.498657\pi$
$521$	0.192659	0.00844056	0.00422028	+	0.999991i	$0.498657\pi$
$522$	−22.3923	−0.980086
$523$	−9.41293	−0.411599	−0.205799	−	0.978594i	$-0.565979\pi$
$523$	−9.41293	−0.411599	−0.205799	+	0.978594i	$0.565979\pi$
$524$	−6.99691	−0.305661
$525$	1.73748	0.0758297
$526$	28.5441	1.24458
$527$	12.2814	0.534986
$528$	−3.21522	−0.139925
$529$	−22.0570	−0.959002
$530$	−18.8873	−0.820412
$531$	3.49053	0.151476
$532$	−3.01277	−0.130620
$533$	−0.525871	−0.0227780
$534$	8.43592	0.365058
$535$	−7.74802	−0.334976
$536$	−5.22623	−0.225739
$537$	2.19696	0.0948060
$538$	−9.77129	−0.421270
$539$	−4.29976	−0.185204
$540$	6.65598	0.286428
$541$	16.8102	0.722728	0.361364	−	0.932425i	$-0.382311\pi$
$541$	16.8102	0.722728	0.361364	+	0.932425i	$0.382311\pi$
$542$	−23.0418	−0.989729
$543$	19.7184	0.846196
$544$	−1.42662	−0.0611658
$545$	11.6148	0.497525
$546$	−5.01342	−0.214554
$547$	−3.37853	−0.144455	−0.0722277	−	0.997388i	$-0.523011\pi$
$547$	−3.37853	−0.144455	−0.0722277	+	0.997388i	$0.523011\pi$
$548$	2.99018	0.127734
$549$	8.30966	0.354648
$550$	9.99072	0.426006
$551$	27.6392	1.17747
$552$	0.726125	0.0309059
$553$	−14.0440	−0.597211
$554$	−6.07868	−0.258258
$555$	0.431066	0.0182978
$556$	18.5734	0.787689
$557$	23.6075	1.00028	0.500141	−	0.865944i	$-0.333281\pi$
$557$	23.6075	1.00028	0.500141	+	0.865944i	$0.333281\pi$
$558$	21.0126	0.889534
$559$	47.4917	2.00869
$560$	1.63599	0.0691330
$561$	4.58690	0.193659
$562$	3.59496	0.151644
$563$	−9.64514	−0.406494	−0.203247	−	0.979127i	$-0.565149\pi$
$563$	−9.64514	−0.406494	−0.203247	+	0.979127i	$0.565149\pi$
$564$	6.13437	0.258303
$565$	4.38503	0.184480
$566$	15.9846	0.671884
$567$	−4.28025	−0.179754
$568$	9.88223	0.414649
$569$	−37.7920	−1.58432	−0.792161	−	0.610312i	$-0.791044\pi$
$569$	−37.7920	−1.58432	−0.792161	+	0.610312i	$0.791044\pi$
$570$	−3.68563	−0.154374
$571$	14.6572	0.613383	0.306692	−	0.951809i	$-0.400778\pi$
$571$	14.6572	0.613383	0.306692	+	0.951809i	$0.400778\pi$
$572$	−28.8278	−1.20535
$573$	9.80964	0.409804
$574$	0.0784354	0.00327383
$575$	−2.25630	−0.0940944
$576$	−2.44084	−0.101702
$577$	27.2329	1.13372	0.566860	−	0.823814i	$-0.308158\pi$
$577$	27.2329	1.13372	0.566860	+	0.823814i	$0.308158\pi$
$578$	−14.9648	−0.622452
$579$	14.6811	0.610124
$580$	−15.0086	−0.623196
$581$	14.1578	0.587366
$582$	7.03755	0.291716
$583$	−49.6404	−2.05589
$584$	2.30766	0.0954915
$585$	26.7724	1.10690
$586$	16.5823	0.685007
$587$	−10.8067	−0.446039	−0.223019	−	0.974814i	$-0.571591\pi$
$587$	−10.8067	−0.446039	−0.223019	+	0.974814i	$0.571591\pi$
$588$	0.747767	0.0308374
$589$	−25.9362	−1.06868
$590$	2.33954	0.0963175
$591$	10.4995	0.431894
$592$	−0.352370	−0.0144823
$593$	23.8462	0.979248	0.489624	−	0.871934i	$-0.337134\pi$
$593$	23.8462	0.979248	0.489624	+	0.871934i	$0.337134\pi$
$594$	17.4935	0.717768
$595$	−2.33393	−0.0956818
$596$	11.4868	0.470516
$597$	−10.3338	−0.422936
$598$	6.51047	0.266233
$599$	20.3864	0.832966	0.416483	−	0.909144i	$-0.363263\pi$
$599$	20.3864	0.832966	0.416483	+	0.909144i	$0.363263\pi$
$600$	−1.73748	−0.0709322
$601$	−8.12767	−0.331535	−0.165767	−	0.986165i	$-0.553010\pi$
$601$	−8.12767	−0.331535	−0.165767	+	0.986165i	$0.553010\pi$
$602$	−7.08354	−0.288704
$603$	12.7564	0.519481
$604$	−1.48700	−0.0605050
$605$	−12.2502	−0.498041
$606$	4.45565	0.180998
$607$	18.6539	0.757140	0.378570	−	0.925573i	$-0.376416\pi$
$607$	18.6539	0.757140	0.378570	+	0.925573i	$0.376416\pi$
$608$	3.01277	0.122184
$609$	−6.86003	−0.277982
$610$	5.56958	0.225506
$611$	55.0010	2.22510
$612$	3.48216	0.140758
$613$	32.7182	1.32148	0.660739	−	0.750616i	$-0.270243\pi$
$613$	32.7182	1.32148	0.660739	+	0.750616i	$0.270243\pi$
$614$	−8.25517	−0.333152
$615$	0.0959528	0.00386919
$616$	4.29976	0.173242
$617$	22.2871	0.897245	0.448622	−	0.893721i	$-0.351915\pi$
$617$	22.2871	0.897245	0.448622	+	0.893721i	$0.351915\pi$
$618$	2.61860	0.105336
$619$	4.23366	0.170165	0.0850826	−	0.996374i	$-0.472885\pi$
$619$	4.23366	0.170165	0.0850826	+	0.996374i	$0.472885\pi$
$620$	14.0838	0.565618
$621$	−3.95073	−0.158537
$622$	−16.2859	−0.653006
$623$	−11.2815	−0.451983
$624$	5.01342	0.200697
$625$	−7.98334	−0.319334
$626$	−31.4794	−1.25817
$627$	−9.68672	−0.386850
$628$	0.859916	0.0343144
$629$	0.502698	0.0200439
$630$	−3.99319	−0.159092
$631$	−8.12261	−0.323356	−0.161678	−	0.986844i	$-0.551691\pi$
$631$	−8.12261	−0.323356	−0.161678	+	0.986844i	$0.551691\pi$
$632$	14.0440	0.558640
$633$	−19.9184	−0.791685
$634$	0.123332	0.00489812
$635$	29.0289	1.15198
$636$	8.63290	0.342317
$637$	6.70451	0.265643
$638$	−39.4461	−1.56169
$639$	−24.1210	−0.954211
$640$	−1.63599	−0.0646680
$641$	4.72659	0.186689	0.0933446	−	0.995634i	$-0.470244\pi$
$641$	4.72659	0.186689	0.0933446	+	0.995634i	$0.470244\pi$
$642$	3.54142	0.139769
$643$	0.374898	0.0147845	0.00739226	−	0.999973i	$-0.497647\pi$
$643$	0.374898	0.0147845	0.00739226	+	0.999973i	$0.497647\pi$
$644$	−0.971058	−0.0382650
$645$	−8.66555	−0.341206
$646$	−4.29808	−0.169106
$647$	−11.2900	−0.443854	−0.221927	−	0.975063i	$-0.571235\pi$
$647$	−11.2900	−0.443854	−0.221927	+	0.975063i	$0.571235\pi$
$648$	4.28025	0.168144
$649$	6.14888	0.241365
$650$	−15.5783	−0.611031
$651$	6.43733	0.252299
$652$	−21.9323	−0.858937
$653$	−38.3918	−1.50239	−0.751193	−	0.660083i	$-0.770521\pi$
$653$	−38.3918	−1.50239	−0.751193	+	0.660083i	$0.770521\pi$
$654$	−5.30885	−0.207592
$655$	11.4468	0.447265
$656$	−0.0784354	−0.00306239
$657$	−5.63263	−0.219750
$658$	−8.20358	−0.319809
$659$	−6.11616	−0.238252	−0.119126	−	0.992879i	$-0.538009\pi$
$659$	−6.11616	−0.238252	−0.119126	+	0.992879i	$0.538009\pi$
$660$	5.26005	0.204747
$661$	−32.9523	−1.28170	−0.640848	−	0.767668i	$-0.721417\pi$
$661$	−32.9523	−1.28170	−0.640848	+	0.767668i	$0.721417\pi$
$662$	−6.37897	−0.247926
$663$	−7.15224	−0.277770
$664$	−14.1578	−0.549430
$665$	4.92885	0.191133
$666$	0.860080	0.0333274
$667$	8.90850	0.344938
$668$	16.5814	0.641553
$669$	19.0651	0.737099
$670$	8.55003	0.330317
$671$	14.6382	0.565102
$672$	−0.747767	−0.0288457
$673$	12.2700	0.472974	0.236487	−	0.971635i	$-0.424004\pi$
$673$	12.2700	0.472974	0.236487	+	0.971635i	$0.424004\pi$
$674$	−29.0346	−1.11837
$675$	9.45334	0.363859
$676$	31.9505	1.22887
$677$	35.8480	1.37775	0.688876	−	0.724879i	$-0.258105\pi$
$677$	35.8480	1.37775	0.688876	+	0.724879i	$0.258105\pi$
$678$	−2.00428	−0.0769741
$679$	−9.41142	−0.361177
$680$	2.33393	0.0895021
$681$	12.0894	0.463265
$682$	37.0155	1.41740
$683$	−49.2171	−1.88324	−0.941620	−	0.336677i	$-0.890697\pi$
$683$	−49.2171	−1.88324	−0.941620	+	0.336677i	$0.890697\pi$
$684$	−7.35370	−0.281176
$685$	−4.89188	−0.186909
$686$	−1.00000	−0.0381802
$687$	10.9158	0.416463
$688$	7.08354	0.270058
$689$	77.4030	2.94882
$690$	−1.18793	−0.0452237
$691$	9.77899	0.372010	0.186005	−	0.982549i	$-0.440446\pi$
$691$	9.77899	0.372010	0.186005	+	0.982549i	$0.440446\pi$
$692$	5.68618	0.216156
$693$	−10.4951	−0.398674
$694$	−1.76658	−0.0670586
$695$	−30.3859	−1.15260
$696$	6.86003	0.260029
$697$	0.111897	0.00423842
$698$	−14.3536	−0.543290
$699$	−13.7589	−0.520411
$700$	2.32355	0.0878220
$701$	−39.3046	−1.48451	−0.742257	−	0.670115i	$-0.766245\pi$
$701$	−39.3046	−1.48451	−0.742257	+	0.670115i	$0.766245\pi$
$702$	−27.2772	−1.02951
$703$	−1.06161	−0.0400393
$704$	−4.29976	−0.162053
$705$	−10.0357	−0.377968
$706$	24.5673	0.924602
$707$	−5.95861	−0.224096
$708$	−1.06935	−0.0401885
$709$	−12.1452	−0.456123	−0.228062	−	0.973647i	$-0.573239\pi$
$709$	−12.1452	−0.456123	−0.228062	+	0.973647i	$0.573239\pi$
$710$	−16.1672	−0.606743
$711$	−34.2792	−1.28557
$712$	11.2815	0.422792
$713$	−8.35958	−0.313069
$714$	1.06678	0.0399232
$715$	47.1619	1.76375
$716$	2.93803	0.109799
$717$	5.58982	0.208756
$718$	30.4314	1.13569
$719$	−32.6273	−1.21679	−0.608396	−	0.793634i	$-0.708187\pi$
$719$	−32.6273	−1.21679	−0.608396	+	0.793634i	$0.708187\pi$
$720$	3.99319	0.148817
$721$	−3.50190	−0.130417
$722$	−9.92321	−0.369304
$723$	−22.1297	−0.823012
$724$	26.3697	0.980021
$725$	−21.3163	−0.791668
$726$	5.59925	0.207808
$727$	−45.9710	−1.70497	−0.852484	−	0.522753i	$-0.824905\pi$
$727$	−45.9710	−1.70497	−0.852484	+	0.522753i	$0.824905\pi$
$728$	−6.70451	−0.248486
$729$	−1.32060	−0.0489110
$730$	−3.77529	−0.139730
$731$	−10.1055	−0.373766
$732$	−2.54572	−0.0940923
$733$	47.7399	1.76331	0.881656	−	0.471892i	$-0.156429\pi$
$733$	47.7399	1.76331	0.881656	+	0.471892i	$0.156429\pi$
$734$	−0.231440	−0.00854260
$735$	−1.22334	−0.0451234
$736$	0.971058	0.0357937
$737$	22.4715	0.827750
$738$	0.191449	0.00704732
$739$	−46.4225	−1.70768	−0.853840	−	0.520535i	$-0.825732\pi$
$739$	−46.4225	−1.70768	−0.853840	+	0.520535i	$0.825732\pi$
$740$	0.576472	0.0211915
$741$	15.1043	0.554869
$742$	−11.5449	−0.423827
$743$	36.1937	1.32782	0.663909	−	0.747814i	$-0.268896\pi$
$743$	36.1937	1.32782	0.663909	+	0.747814i	$0.268896\pi$
$744$	−6.43733	−0.236004
$745$	−18.7922	−0.688491
$746$	−2.57991	−0.0944573
$747$	34.5570	1.26438
$748$	6.13413	0.224286
$749$	−4.73600	−0.173050
$750$	8.95916	0.327142
$751$	−38.1574	−1.39238	−0.696191	−	0.717857i	$-0.745123\pi$
$751$	−38.1574	−1.39238	−0.696191	+	0.717857i	$0.745123\pi$
$752$	8.20358	0.299154
$753$	−19.0232	−0.693244
$754$	61.5073	2.23996
$755$	2.43270	0.0885351
$756$	4.06848	0.147969
$757$	18.7595	0.681827	0.340913	−	0.940095i	$-0.389264\pi$
$757$	18.7595	0.681827	0.340913	+	0.940095i	$0.389264\pi$
$758$	−32.3446	−1.17481
$759$	−3.12216	−0.113327
$760$	−4.92885	−0.178788
$761$	7.25779	0.263095	0.131547	−	0.991310i	$-0.458005\pi$
$761$	7.25779	0.263095	0.131547	+	0.991310i	$0.458005\pi$
$762$	−13.2684	−0.480663
$763$	7.09960	0.257023
$764$	13.1186	0.474613
$765$	−5.69676	−0.205967
$766$	31.0748	1.12278
$767$	−9.58780	−0.346195
$768$	0.747767	0.0269827
$769$	15.6889	0.565755	0.282877	−	0.959156i	$-0.408711\pi$
$769$	15.6889	0.565755	0.282877	+	0.959156i	$0.408711\pi$
$770$	−7.03435	−0.253500
$771$	4.56522	0.164412
$772$	19.6332	0.706614
$773$	−27.6668	−0.995107	−0.497553	−	0.867433i	$-0.665768\pi$
$773$	−27.6668	−0.995107	−0.497553	+	0.867433i	$0.665768\pi$
$774$	−17.2898	−0.621470
$775$	20.0029	0.718524
$776$	9.41142	0.337850
$777$	0.263490	0.00945267
$778$	−16.2158	−0.581365
$779$	−0.236308	−0.00846660
$780$	−8.20187	−0.293674
$781$	−42.4912	−1.52046
$782$	−1.38533	−0.0495393
$783$	−37.3243	−1.33386
$784$	1.00000	0.0357143
$785$	−1.40681	−0.0502112
$786$	−5.23206	−0.186621
$787$	26.7951	0.955144	0.477572	−	0.878593i	$-0.341517\pi$
$787$	26.7951	0.955144	0.477572	+	0.878593i	$0.341517\pi$
$788$	14.0412	0.500197
$789$	21.3443	0.759878
$790$	−22.9758	−0.817441
$791$	2.68036	0.0953026
$792$	10.4951	0.372925
$793$	−22.8250	−0.810539
$794$	−12.1890	−0.432571
$795$	−14.1233	−0.500902
$796$	−13.8196	−0.489822
$797$	11.6180	0.411530	0.205765	−	0.978601i	$-0.434032\pi$
$797$	11.6180	0.411530	0.205765	+	0.978601i	$0.434032\pi$
$798$	−2.25285	−0.0797500
$799$	−11.7034	−0.414036
$800$	−2.32355	−0.0821500
$801$	−27.5363	−0.972949
$802$	21.3361	0.753405
$803$	−9.92237	−0.350153
$804$	−3.90800	−0.137825
$805$	1.58864	0.0559921
$806$	−57.7174	−2.03301
$807$	−7.30665	−0.257206
$808$	5.95861	0.209623
$809$	23.4156	0.823247	0.411623	−	0.911354i	$-0.364962\pi$
$809$	23.4156	0.823247	0.411623	+	0.911354i	$0.364962\pi$
$810$	−7.00243	−0.246040
$811$	54.7256	1.92168	0.960838	−	0.277112i	$-0.0893772\pi$
$811$	54.7256	1.92168	0.960838	+	0.277112i	$0.0893772\pi$
$812$	−9.17402	−0.321945
$813$	−17.2299	−0.604279
$814$	1.51511	0.0531044
$815$	35.8810	1.25686
$816$	−1.06678	−0.0373448
$817$	21.3411	0.746630
$818$	5.87159	0.205295
$819$	16.3647	0.571828
$820$	0.128319	0.00448110
$821$	16.7944	0.586130	0.293065	−	0.956093i	$-0.405325\pi$
$821$	16.7944	0.586130	0.293065	+	0.956093i	$0.405325\pi$
$822$	2.23596	0.0779879
$823$	−7.80178	−0.271953	−0.135977	−	0.990712i	$-0.543417\pi$
$823$	−7.80178	−0.271953	−0.135977	+	0.990712i	$0.543417\pi$
$824$	3.50190	0.121994
$825$	7.47073	0.260098
$826$	1.43005	0.0497579
$827$	11.2293	0.390481	0.195240	−	0.980755i	$-0.437451\pi$
$827$	11.2293	0.390481	0.195240	+	0.980755i	$0.437451\pi$
$828$	−2.37020	−0.0823702
$829$	43.6573	1.51628	0.758140	−	0.652092i	$-0.226108\pi$
$829$	43.6573	1.51628	0.758140	+	0.652092i	$0.226108\pi$
$830$	23.1620	0.803964
$831$	−4.54544	−0.157679
$832$	6.70451	0.232437
$833$	−1.42662	−0.0494295
$834$	13.8886	0.480923
$835$	−27.1269	−0.938766
$836$	−12.9542	−0.448030
$837$	35.0245	1.21062
$838$	−34.3354	−1.18610
$839$	29.1956	1.00795	0.503973	−	0.863720i	$-0.331871\pi$
$839$	29.1956	1.00795	0.503973	+	0.863720i	$0.331871\pi$
$840$	1.22334	0.0422091
$841$	55.1626	1.90216
$842$	−23.6248	−0.814163
$843$	2.68820	0.0925864
$844$	−26.6372	−0.916889
$845$	−52.2706	−1.79816
$846$	−20.0237	−0.688428
$847$	−7.48796	−0.257289
$848$	11.5449	0.396454
$849$	11.9528	0.410218
$850$	3.31483	0.113698
$851$	−0.342171	−0.0117295
$852$	7.38960	0.253164
$853$	−17.3807	−0.595105	−0.297553	−	0.954705i	$-0.596170\pi$
$853$	−17.3807	−0.595105	−0.297553	+	0.954705i	$0.596170\pi$
$854$	3.40442	0.116497
$855$	12.0305	0.411436
$856$	4.73600	0.161873
$857$	−42.6979	−1.45853	−0.729266	−	0.684231i	$-0.760138\pi$
$857$	−42.6979	−1.45853	−0.729266	+	0.684231i	$0.760138\pi$
$858$	−21.5565	−0.735927
$859$	27.8769	0.951148	0.475574	−	0.879676i	$-0.342240\pi$
$859$	27.8769	0.951148	0.475574	+	0.879676i	$0.342240\pi$
$860$	−11.5886	−0.395167
$861$	0.0586514	0.00199883
$862$	−1.00000	−0.0340601
$863$	−5.37955	−0.183122	−0.0915611	−	0.995799i	$-0.529186\pi$
$863$	−5.37955	−0.183122	−0.0915611	+	0.995799i	$0.529186\pi$
$864$	−4.06848	−0.138413
$865$	−9.30250	−0.316295
$866$	15.5321	0.527804
$867$	−11.1902	−0.380037
$868$	8.60874	0.292200
$869$	−60.3858	−2.04845
$870$	−11.2229	−0.380492
$871$	−35.0393	−1.18726
$872$	−7.09960	−0.240423
$873$	−22.9718	−0.777478
$874$	2.92557	0.0989590
$875$	−11.9812	−0.405039
$876$	1.72559	0.0583023
$877$	−33.2757	−1.12364	−0.561820	−	0.827259i	$-0.689899\pi$
$877$	−33.2757	−1.12364	−0.561820	+	0.827259i	$0.689899\pi$
$878$	38.9562	1.31471
$879$	12.3997	0.418231
$880$	7.03435	0.237128
$881$	24.3402	0.820042	0.410021	−	0.912076i	$-0.365521\pi$
$881$	24.3402	0.820042	0.410021	+	0.912076i	$0.365521\pi$
$882$	−2.44084	−0.0821875
$883$	3.85509	0.129734	0.0648670	−	0.997894i	$-0.479338\pi$
$883$	3.85509	0.129734	0.0648670	+	0.997894i	$0.479338\pi$
$884$	−9.56480	−0.321699
$885$	1.74943	0.0588066
$886$	−18.2551	−0.613293
$887$	−11.6607	−0.391527	−0.195764	−	0.980651i	$-0.562719\pi$
$887$	−11.6607	−0.391527	−0.195764	+	0.980651i	$0.562719\pi$
$888$	−0.263490	−0.00884216
$889$	17.7440	0.595115
$890$	−18.4563	−0.618658
$891$	−18.4041	−0.616560
$892$	25.4960	0.853670
$893$	24.7155	0.827073
$894$	8.58941	0.287273
$895$	−4.80658	−0.160666
$896$	−1.00000	−0.0334077
$897$	4.86832	0.162548
$898$	−15.1071	−0.504130
$899$	−78.9767	−2.63402
$900$	5.67143	0.189048
$901$	−16.4702	−0.548702
$902$	0.337253	0.0112293
$903$	−5.29684	−0.176268
$904$	−2.68036	−0.0891474
$905$	−43.1404	−1.43403
$906$	−1.11193	−0.0369413
$907$	30.7784	1.02198	0.510990	−	0.859587i	$-0.329279\pi$
$907$	30.7784	1.02198	0.510990	+	0.859587i	$0.329279\pi$
$908$	16.1673	0.536530
$909$	−14.5440	−0.482395
$910$	10.9685	0.363602
$911$	−50.0982	−1.65983	−0.829914	−	0.557892i	$-0.811610\pi$
$911$	−50.0982	−1.65983	−0.829914	+	0.557892i	$0.811610\pi$
$912$	2.25285	0.0745993
$913$	60.8753	2.01468
$914$	9.03293	0.298783
$915$	4.16475	0.137682
$916$	14.5978	0.482326
$917$	6.99691	0.231058
$918$	5.80418	0.191567
$919$	−30.0523	−0.991334	−0.495667	−	0.868513i	$-0.665076\pi$
$919$	−30.0523	−0.991334	−0.495667	+	0.868513i	$0.665076\pi$
$920$	−1.58864	−0.0523758
$921$	−6.17294	−0.203405
$922$	−21.2711	−0.700528
$923$	66.2555	2.18083
$924$	3.21522	0.105773
$925$	0.818750	0.0269203
$926$	−5.05460	−0.166104
$927$	−8.54758	−0.280739
$928$	9.17402	0.301152
$929$	−52.7929	−1.73208	−0.866040	−	0.499974i	$-0.833343\pi$
$929$	−52.7929	−1.73208	−0.866040	+	0.499974i	$0.833343\pi$
$930$	10.5314	0.345338
$931$	3.01277	0.0987396
$932$	−18.4000	−0.602713
$933$	−12.1781	−0.398692
$934$	21.6082	0.707043
$935$	−10.0353	−0.328191
$936$	−16.3647	−0.534896
$937$	16.7898	0.548499	0.274250	−	0.961659i	$-0.411571\pi$
$937$	16.7898	0.548499	0.274250	+	0.961659i	$0.411571\pi$
$938$	5.22623	0.170642
$939$	−23.5393	−0.768175
$940$	−13.4209	−0.437743
$941$	11.6227	0.378888	0.189444	−	0.981892i	$-0.439331\pi$
$941$	11.6227	0.378888	0.189444	+	0.981892i	$0.439331\pi$
$942$	0.643017	0.0209506
$943$	−0.0761653	−0.00248028
$944$	−1.43005	−0.0465442
$945$	−6.65598	−0.216519
$946$	−30.4576	−0.990261
$947$	−23.3109	−0.757502	−0.378751	−	0.925499i	$-0.623646\pi$
$947$	−23.3109	−0.757502	−0.378751	+	0.925499i	$0.623646\pi$
$948$	10.5016	0.341077
$949$	15.4717	0.502233
$950$	−7.00033	−0.227121
$951$	0.0922233	0.00299054
$952$	1.42662	0.0462370
$953$	−23.2525	−0.753221	−0.376611	−	0.926372i	$-0.622911\pi$
$953$	−23.2525	−0.753221	−0.376611	+	0.926372i	$0.622911\pi$
$954$	−28.1793	−0.912339
$955$	−21.4618	−0.694487
$956$	7.47535	0.241770
$957$	−29.4965	−0.953486
$958$	−3.16401	−0.102225
$959$	−2.99018	−0.0965578
$960$	−1.22334	−0.0394830
$961$	43.1104	1.39066
$962$	−2.36247	−0.0761690
$963$	−11.5598	−0.372511
$964$	−29.5944	−0.953170
$965$	−32.1196	−1.03397
$966$	−0.726125	−0.0233627
$967$	−53.1699	−1.70983	−0.854914	−	0.518769i	$-0.826390\pi$
$967$	−53.1699	−1.70983	−0.854914	+	0.518769i	$0.826390\pi$
$968$	7.48796	0.240672
$969$	−3.21396	−0.103247
$970$	−15.3969	−0.494366
$971$	2.55552	0.0820105	0.0410053	−	0.999159i	$-0.486944\pi$
$971$	2.55552	0.0820105	0.0410053	+	0.999159i	$0.486944\pi$
$972$	15.4061	0.494150
$973$	−18.5734	−0.595437
$974$	−25.1661	−0.806373
$975$	−11.6489	−0.373064
$976$	−3.40442	−0.108973
$977$	−25.1041	−0.803152	−0.401576	−	0.915826i	$-0.631538\pi$
$977$	−25.1041	−0.803152	−0.401576	+	0.915826i	$0.631538\pi$
$978$	−16.4003	−0.524423
$979$	−48.5077	−1.55031
$980$	−1.63599	−0.0522596
$981$	17.3290	0.553273
$982$	−37.9585	−1.21131
$983$	36.7066	1.17076	0.585380	−	0.810759i	$-0.300945\pi$
$983$	36.7066	1.17076	0.585380	+	0.810759i	$0.300945\pi$
$984$	−0.0586514	−0.00186974
$985$	−22.9712	−0.731923
$986$	−13.0878	−0.416802
$987$	−6.13437	−0.195259
$988$	20.1992	0.642621
$989$	6.87853	0.218725
$990$	−17.1697	−0.545690
$991$	−13.3662	−0.424590	−0.212295	−	0.977206i	$-0.568094\pi$
$991$	−13.3662	−0.424590	−0.212295	+	0.977206i	$0.568094\pi$
$992$	−8.60874	−0.273328
$993$	−4.76999	−0.151371
$994$	−9.88223	−0.313445
$995$	22.6086	0.716742
$996$	−10.5868	−0.335454
$997$	−39.6061	−1.25434	−0.627169	−	0.778883i	$-0.715786\pi$
$997$	−39.6061	−1.25434	−0.627169	+	0.778883i	$0.715786\pi$
$998$	40.8353	1.29262
$999$	1.43361	0.0453574

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6034.2.a.r.1.19	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6034.2.a.r.1.19	✓	31	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:	\( N \)	\(=\)	\( 6034 = 2 \cdot 7 \cdot 431 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6034.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.747767 q^{3} +1.00000 q^{4} -1.63599 q^{5} +0.747767 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.44084 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.747767 q^{3} +1.00000 q^{4} -1.63599 q^{5} +0.747767 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.44084 q^{9} -1.63599 q^{10} -4.29976 q^{11} +0.747767 q^{12} +6.70451 q^{13} -1.00000 q^{14} -1.22334 q^{15} +1.00000 q^{16} -1.42662 q^{17} -2.44084 q^{18} +3.01277 q^{19} -1.63599 q^{20} -0.747767 q^{21} -4.29976 q^{22} +0.971058 q^{23} +0.747767 q^{24} -2.32355 q^{25} +6.70451 q^{26} -4.06848 q^{27} -1.00000 q^{28} +9.17402 q^{29} -1.22334 q^{30} -8.60874 q^{31} +1.00000 q^{32} -3.21522 q^{33} -1.42662 q^{34} +1.63599 q^{35} -2.44084 q^{36} -0.352370 q^{37} +3.01277 q^{38} +5.01342 q^{39} -1.63599 q^{40} -0.0784354 q^{41} -0.747767 q^{42} +7.08354 q^{43} -4.29976 q^{44} +3.99319 q^{45} +0.971058 q^{46} +8.20358 q^{47} +0.747767 q^{48} +1.00000 q^{49} -2.32355 q^{50} -1.06678 q^{51} +6.70451 q^{52} +11.5449 q^{53} -4.06848 q^{54} +7.03435 q^{55} -1.00000 q^{56} +2.25285 q^{57} +9.17402 q^{58} -1.43005 q^{59} -1.22334 q^{60} -3.40442 q^{61} -8.60874 q^{62} +2.44084 q^{63} +1.00000 q^{64} -10.9685 q^{65} -3.21522 q^{66} -5.22623 q^{67} -1.42662 q^{68} +0.726125 q^{69} +1.63599 q^{70} +9.88223 q^{71} -2.44084 q^{72} +2.30766 q^{73} -0.352370 q^{74} -1.73748 q^{75} +3.01277 q^{76} +4.29976 q^{77} +5.01342 q^{78} +14.0440 q^{79} -1.63599 q^{80} +4.28025 q^{81} -0.0784354 q^{82} -14.1578 q^{83} -0.747767 q^{84} +2.33393 q^{85} +7.08354 q^{86} +6.86003 q^{87} -4.29976 q^{88} +11.2815 q^{89} +3.99319 q^{90} -6.70451 q^{91} +0.971058 q^{92} -6.43733 q^{93} +8.20358 q^{94} -4.92885 q^{95} +0.747767 q^{96} +9.41142 q^{97} +1.00000 q^{98} +10.4951 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q + 31 q^{2} + 7 q^{3} + 31 q^{4} + 15 q^{5} + 7 q^{6} - 31 q^{7} + 31 q^{8} + 42 q^{9}+O(q^{10}) \) 31 * q + 31 * q^2 + 7 * q^3 + 31 * q^4 + 15 * q^5 + 7 * q^6 - 31 * q^7 + 31 * q^8 + 42 * q^9	\( 31 q + 31 q^{2} + 7 q^{3} + 31 q^{4} + 15 q^{5} + 7 q^{6} - 31 q^{7} + 31 q^{8} + 42 q^{9} + 15 q^{10} + 12 q^{11} + 7 q^{12} + 26 q^{13} - 31 q^{14} + 6 q^{15} + 31 q^{16} + 33 q^{17} + 42 q^{18} + 34 q^{19} + 15 q^{20} - 7 q^{21} + 12 q^{22} - 14 q^{23} + 7 q^{24} + 58 q^{25} + 26 q^{26} + 28 q^{27} - 31 q^{28} + 11 q^{29} + 6 q^{30} + 19 q^{31} + 31 q^{32} + 43 q^{33} + 33 q^{34} - 15 q^{35} + 42 q^{36} + 2 q^{37} + 34 q^{38} - 16 q^{39} + 15 q^{40} + 53 q^{41} - 7 q^{42} + 22 q^{43} + 12 q^{44} + 43 q^{45} - 14 q^{46} + 27 q^{47} + 7 q^{48} + 31 q^{49} + 58 q^{50} + 17 q^{51} + 26 q^{52} + 11 q^{53} + 28 q^{54} + 19 q^{55} - 31 q^{56} + 45 q^{57} + 11 q^{58} + 54 q^{59} + 6 q^{60} + 41 q^{61} + 19 q^{62} - 42 q^{63} + 31 q^{64} + 30 q^{65} + 43 q^{66} + 13 q^{67} + 33 q^{68} + 17 q^{69} - 15 q^{70} + 43 q^{71} + 42 q^{72} + 42 q^{73} + 2 q^{74} + 62 q^{75} + 34 q^{76} - 12 q^{77} - 16 q^{78} - 12 q^{79} + 15 q^{80} + 63 q^{81} + 53 q^{82} + 35 q^{83} - 7 q^{84} + 16 q^{85} + 22 q^{86} - 4 q^{87} + 12 q^{88} + 115 q^{89} + 43 q^{90} - 26 q^{91} - 14 q^{92} + q^{93} + 27 q^{94} - 13 q^{95} + 7 q^{96} + 32 q^{97} + 31 q^{98} + 34 q^{99}+O(q^{100}) \) 31 * q + 31 * q^2 + 7 * q^3 + 31 * q^4 + 15 * q^5 + 7 * q^6 - 31 * q^7 + 31 * q^8 + 42 * q^9 + 15 * q^10 + 12 * q^11 + 7 * q^12 + 26 * q^13 - 31 * q^14 + 6 * q^15 + 31 * q^16 + 33 * q^17 + 42 * q^18 + 34 * q^19 + 15 * q^20 - 7 * q^21 + 12 * q^22 - 14 * q^23 + 7 * q^24 + 58 * q^25 + 26 * q^26 + 28 * q^27 - 31 * q^28 + 11 * q^29 + 6 * q^30 + 19 * q^31 + 31 * q^32 + 43 * q^33 + 33 * q^34 - 15 * q^35 + 42 * q^36 + 2 * q^37 + 34 * q^38 - 16 * q^39 + 15 * q^40 + 53 * q^41 - 7 * q^42 + 22 * q^43 + 12 * q^44 + 43 * q^45 - 14 * q^46 + 27 * q^47 + 7 * q^48 + 31 * q^49 + 58 * q^50 + 17 * q^51 + 26 * q^52 + 11 * q^53 + 28 * q^54 + 19 * q^55 - 31 * q^56 + 45 * q^57 + 11 * q^58 + 54 * q^59 + 6 * q^60 + 41 * q^61 + 19 * q^62 - 42 * q^63 + 31 * q^64 + 30 * q^65 + 43 * q^66 + 13 * q^67 + 33 * q^68 + 17 * q^69 - 15 * q^70 + 43 * q^71 + 42 * q^72 + 42 * q^73 + 2 * q^74 + 62 * q^75 + 34 * q^76 - 12 * q^77 - 16 * q^78 - 12 * q^79 + 15 * q^80 + 63 * q^81 + 53 * q^82 + 35 * q^83 - 7 * q^84 + 16 * q^85 + 22 * q^86 - 4 * q^87 + 12 * q^88 + 115 * q^89 + 43 * q^90 - 26 * q^91 - 14 * q^92 + q^93 + 27 * q^94 - 13 * q^95 + 7 * q^96 + 32 * q^97 + 31 * q^98 + 34 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more