LMFDB - Embedded newform 6034.2.a.o.1.15

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:	$1$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
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)

$f(q)$	$=$	$q-1.00000 q^{2} +0.0257691 q^{3} +1.00000 q^{4} +3.06772 q^{5} -0.0257691 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.99934 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +0.0257691 q^{3} +1.00000 q^{4} +3.06772 q^{5} -0.0257691 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.99934 q^{9} -3.06772 q^{10} -2.07556 q^{11} +0.0257691 q^{12} +1.32404 q^{13} +1.00000 q^{14} +0.0790523 q^{15} +1.00000 q^{16} -1.76795 q^{17} +2.99934 q^{18} +6.17760 q^{19} +3.06772 q^{20} -0.0257691 q^{21} +2.07556 q^{22} -8.31468 q^{23} -0.0257691 q^{24} +4.41089 q^{25} -1.32404 q^{26} -0.154597 q^{27} -1.00000 q^{28} +3.32569 q^{29} -0.0790523 q^{30} -5.80859 q^{31} -1.00000 q^{32} -0.0534853 q^{33} +1.76795 q^{34} -3.06772 q^{35} -2.99934 q^{36} +8.74256 q^{37} -6.17760 q^{38} +0.0341193 q^{39} -3.06772 q^{40} -2.56551 q^{41} +0.0257691 q^{42} -2.36764 q^{43} -2.07556 q^{44} -9.20111 q^{45} +8.31468 q^{46} -5.87757 q^{47} +0.0257691 q^{48} +1.00000 q^{49} -4.41089 q^{50} -0.0455585 q^{51} +1.32404 q^{52} +1.59613 q^{53} +0.154597 q^{54} -6.36723 q^{55} +1.00000 q^{56} +0.159191 q^{57} -3.32569 q^{58} +11.9804 q^{59} +0.0790523 q^{60} +5.45935 q^{61} +5.80859 q^{62} +2.99934 q^{63} +1.00000 q^{64} +4.06178 q^{65} +0.0534853 q^{66} +2.72798 q^{67} -1.76795 q^{68} -0.214262 q^{69} +3.06772 q^{70} -13.8607 q^{71} +2.99934 q^{72} +13.4987 q^{73} -8.74256 q^{74} +0.113665 q^{75} +6.17760 q^{76} +2.07556 q^{77} -0.0341193 q^{78} -8.49905 q^{79} +3.06772 q^{80} +8.99402 q^{81} +2.56551 q^{82} -12.8421 q^{83} -0.0257691 q^{84} -5.42357 q^{85} +2.36764 q^{86} +0.0856999 q^{87} +2.07556 q^{88} -12.8646 q^{89} +9.20111 q^{90} -1.32404 q^{91} -8.31468 q^{92} -0.149682 q^{93} +5.87757 q^{94} +18.9511 q^{95} -0.0257691 q^{96} +9.65527 q^{97} -1.00000 q^{98} +6.22530 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 25 q^{2} - 4 q^{3} + 25 q^{4} + 4 q^{6} - 25 q^{7} - 25 q^{8} + 25 q^{9}+O(q^{10}) $ 25 * q - 25 * q^2 - 4 * q^3 + 25 * q^4 + 4 * q^6 - 25 * q^7 - 25 * q^8 + 25 * q^9	\( 25 q - 25 q^{2} - 4 q^{3} + 25 q^{4} + 4 q^{6} - 25 q^{7} - 25 q^{8} + 25 q^{9} - 13 q^{11} - 4 q^{12} + 17 q^{13} + 25 q^{14} - 18 q^{15} + 25 q^{16} - 4 q^{17} - 25 q^{18} - 9 q^{19} + 4 q^{21} + 13 q^{22} - 14 q^{23} + 4 q^{24} + 23 q^{25} - 17 q^{26} - 7 q^{27} - 25 q^{28} - 4 q^{29} + 18 q^{30} - 15 q^{31} - 25 q^{32} - 15 q^{33} + 4 q^{34} + 25 q^{36} + 13 q^{37} + 9 q^{38} - 31 q^{39} - 31 q^{41} - 4 q^{42} + 29 q^{43} - 13 q^{44} + 10 q^{45} + 14 q^{46} - 31 q^{47} - 4 q^{48} + 25 q^{49} - 23 q^{50} - 9 q^{51} + 17 q^{52} + 23 q^{53} + 7 q^{54} - 48 q^{55} + 25 q^{56} + 32 q^{57} + 4 q^{58} - 50 q^{59} - 18 q^{60} - 2 q^{61} + 15 q^{62} - 25 q^{63} + 25 q^{64} - 4 q^{65} + 15 q^{66} - 8 q^{67} - 4 q^{68} - 57 q^{69} - 61 q^{71} - 25 q^{72} + 31 q^{73} - 13 q^{74} - 21 q^{75} - 9 q^{76} + 13 q^{77} + 31 q^{78} - 10 q^{79} + 61 q^{81} + 31 q^{82} - 47 q^{83} + 4 q^{84} + 2 q^{85} - 29 q^{86} + 17 q^{87} + 13 q^{88} - 44 q^{89} - 10 q^{90} - 17 q^{91} - 14 q^{92} - 13 q^{93} + 31 q^{94} - 7 q^{95} + 4 q^{96} + 10 q^{97} - 25 q^{98} - 47 q^{99}+O(q^{100}) \) 25 * q - 25 * q^2 - 4 * q^3 + 25 * q^4 + 4 * q^6 - 25 * q^7 - 25 * q^8 + 25 * q^9 - 13 * q^11 - 4 * q^12 + 17 * q^13 + 25 * q^14 - 18 * q^15 + 25 * q^16 - 4 * q^17 - 25 * q^18 - 9 * q^19 + 4 * q^21 + 13 * q^22 - 14 * q^23 + 4 * q^24 + 23 * q^25 - 17 * q^26 - 7 * q^27 - 25 * q^28 - 4 * q^29 + 18 * q^30 - 15 * q^31 - 25 * q^32 - 15 * q^33 + 4 * q^34 + 25 * q^36 + 13 * q^37 + 9 * q^38 - 31 * q^39 - 31 * q^41 - 4 * q^42 + 29 * q^43 - 13 * q^44 + 10 * q^45 + 14 * q^46 - 31 * q^47 - 4 * q^48 + 25 * q^49 - 23 * q^50 - 9 * q^51 + 17 * q^52 + 23 * q^53 + 7 * q^54 - 48 * q^55 + 25 * q^56 + 32 * q^57 + 4 * q^58 - 50 * q^59 - 18 * q^60 - 2 * q^61 + 15 * q^62 - 25 * q^63 + 25 * q^64 - 4 * q^65 + 15 * q^66 - 8 * q^67 - 4 * q^68 - 57 * q^69 - 61 * q^71 - 25 * q^72 + 31 * q^73 - 13 * q^74 - 21 * q^75 - 9 * q^76 + 13 * q^77 + 31 * q^78 - 10 * q^79 + 61 * q^81 + 31 * q^82 - 47 * q^83 + 4 * q^84 + 2 * q^85 - 29 * q^86 + 17 * q^87 + 13 * q^88 - 44 * q^89 - 10 * q^90 - 17 * q^91 - 14 * q^92 - 13 * q^93 + 31 * q^94 - 7 * q^95 + 4 * q^96 + 10 * q^97 - 25 * q^98 - 47 * 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
$3$	0.0257691	0.0148778	0.00743889	−	0.999972i	$-0.497632\pi$
$3$	0.0257691	0.0148778	0.00743889	+	0.999972i	$0.497632\pi$
$4$	1.00000	0.500000
$5$	3.06772	1.37192	0.685962	−	0.727637i	$-0.259381\pi$
$5$	3.06772	1.37192	0.685962	+	0.727637i	$0.259381\pi$
$6$	−0.0257691	−0.0105202
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	−2.99934	−0.999779
$10$	−3.06772	−0.970097
$11$	−2.07556	−0.625805	−0.312903	−	0.949785i	$-0.601301\pi$
$11$	−2.07556	−0.625805	−0.312903	+	0.949785i	$0.601301\pi$
$12$	0.0257691	0.00743889
$13$	1.32404	0.367222	0.183611	−	0.982999i	$-0.441221\pi$
$13$	1.32404	0.367222	0.183611	+	0.982999i	$0.441221\pi$
$14$	1.00000	0.267261
$15$	0.0790523	0.0204112
$16$	1.00000	0.250000
$17$	−1.76795	−0.428791	−0.214396	−	0.976747i	$-0.568778\pi$
$17$	−1.76795	−0.428791	−0.214396	+	0.976747i	$0.568778\pi$
$18$	2.99934	0.706950
$19$	6.17760	1.41724	0.708619	−	0.705591i	$-0.249318\pi$
$19$	6.17760	1.41724	0.708619	+	0.705591i	$0.249318\pi$
$20$	3.06772	0.685962
$21$	−0.0257691	−0.00562328
$22$	2.07556	0.442511
$23$	−8.31468	−1.73373	−0.866865	−	0.498543i	$-0.833869\pi$
$23$	−8.31468	−1.73373	−0.866865	+	0.498543i	$0.833869\pi$
$24$	−0.0257691	−0.00526009
$25$	4.41089	0.882178
$26$	−1.32404	−0.259665
$27$	−0.154597	−0.0297523
$28$	−1.00000	−0.188982
$29$	3.32569	0.617564	0.308782	−	0.951133i	$-0.400079\pi$
$29$	3.32569	0.617564	0.308782	+	0.951133i	$0.400079\pi$
$30$	−0.0790523	−0.0144329
$31$	−5.80859	−1.04325	−0.521627	−	0.853174i	$-0.674675\pi$
$31$	−5.80859	−1.04325	−0.521627	+	0.853174i	$0.674675\pi$
$32$	−1.00000	−0.176777
$33$	−0.0534853	−0.00931060
$34$	1.76795	0.303201
$35$	−3.06772	−0.518539
$36$	−2.99934	−0.499889
$37$	8.74256	1.43727	0.718634	−	0.695389i	$-0.244768\pi$
$37$	8.74256	1.43727	0.718634	+	0.695389i	$0.244768\pi$
$38$	−6.17760	−1.00214
$39$	0.0341193	0.00546346
$40$	−3.06772	−0.485049
$41$	−2.56551	−0.400666	−0.200333	−	0.979728i	$-0.564202\pi$
$41$	−2.56551	−0.400666	−0.200333	+	0.979728i	$0.564202\pi$
$42$	0.0257691	0.00397626
$43$	−2.36764	−0.361061	−0.180531	−	0.983569i	$-0.557782\pi$
$43$	−2.36764	−0.361061	−0.180531	+	0.983569i	$0.557782\pi$
$44$	−2.07556	−0.312903
$45$	−9.20111	−1.37162
$46$	8.31468	1.22593
$47$	−5.87757	−0.857331	−0.428666	−	0.903463i	$-0.641016\pi$
$47$	−5.87757	−0.857331	−0.428666	+	0.903463i	$0.641016\pi$
$48$	0.0257691	0.00371945
$49$	1.00000	0.142857
$50$	−4.41089	−0.623794
$51$	−0.0455585	−0.00637946
$52$	1.32404	0.183611
$53$	1.59613	0.219245	0.109623	−	0.993973i	$-0.465036\pi$
$53$	1.59613	0.219245	0.109623	+	0.993973i	$0.465036\pi$
$54$	0.154597	0.0210380
$55$	−6.36723	−0.858557
$56$	1.00000	0.133631
$57$	0.159191	0.0210854
$58$	−3.32569	−0.436684
$59$	11.9804	1.55972	0.779859	−	0.625956i	$-0.215291\pi$
$59$	11.9804	1.55972	0.779859	+	0.625956i	$0.215291\pi$
$60$	0.0790523	0.0102056
$61$	5.45935	0.698998	0.349499	−	0.936937i	$-0.386352\pi$
$61$	5.45935	0.698998	0.349499	+	0.936937i	$0.386352\pi$
$62$	5.80859	0.737691
$63$	2.99934	0.377881
$64$	1.00000	0.125000
$65$	4.06178	0.503801
$66$	0.0534853	0.00658359
$67$	2.72798	0.333275	0.166638	−	0.986018i	$-0.446709\pi$
$67$	2.72798	0.333275	0.166638	+	0.986018i	$0.446709\pi$
$68$	−1.76795	−0.214396
$69$	−0.214262	−0.0257941
$70$	3.06772	0.366662
$71$	−13.8607	−1.64496	−0.822479	−	0.568796i	$-0.807409\pi$
$71$	−13.8607	−1.64496	−0.822479	+	0.568796i	$0.807409\pi$
$72$	2.99934	0.353475
$73$	13.4987	1.57991	0.789953	−	0.613168i	$-0.210105\pi$
$73$	13.4987	1.57991	0.789953	+	0.613168i	$0.210105\pi$
$74$	−8.74256	−1.01630
$75$	0.113665	0.0131249
$76$	6.17760	0.708619
$77$	2.07556	0.236532
$78$	−0.0341193	−0.00386325
$79$	−8.49905	−0.956217	−0.478109	−	0.878301i	$-0.658678\pi$
$79$	−8.49905	−0.956217	−0.478109	+	0.878301i	$0.658678\pi$
$80$	3.06772	0.342981
$81$	8.99402	0.999336
$82$	2.56551	0.283313
$83$	−12.8421	−1.40961	−0.704803	−	0.709403i	$-0.748965\pi$
$83$	−12.8421	−1.40961	−0.704803	+	0.709403i	$0.748965\pi$
$84$	−0.0257691	−0.00281164
$85$	−5.42357	−0.588269
$86$	2.36764	0.255309
$87$	0.0856999	0.00918799
$88$	2.07556	0.221255
$89$	−12.8646	−1.36365	−0.681825	−	0.731515i	$-0.738813\pi$
$89$	−12.8646	−1.36365	−0.681825	+	0.731515i	$0.738813\pi$
$90$	9.20111	0.969883
$91$	−1.32404	−0.138797
$92$	−8.31468	−0.866865
$93$	−0.149682	−0.0155213
$94$	5.87757	0.606225
$95$	18.9511	1.94435
$96$	−0.0257691	−0.00263005
$97$	9.65527	0.980345	0.490172	−	0.871626i	$-0.336934\pi$
$97$	9.65527	0.980345	0.490172	+	0.871626i	$0.336934\pi$
$98$	−1.00000	−0.101015
$99$	6.22530	0.625667
$100$	4.41089	0.441089
$101$	−15.2235	−1.51480	−0.757399	−	0.652952i	$-0.773530\pi$
$101$	−15.2235	−1.51480	−0.757399	+	0.652952i	$0.773530\pi$
$102$	0.0455585	0.00451096
$103$	4.61972	0.455194	0.227597	−	0.973755i	$-0.426913\pi$
$103$	4.61972	0.455194	0.227597	+	0.973755i	$0.426913\pi$
$104$	−1.32404	−0.129833
$105$	−0.0790523	−0.00771471
$106$	−1.59613	−0.155030
$107$	−12.2657	−1.18577	−0.592887	−	0.805286i	$-0.702012\pi$
$107$	−12.2657	−1.18577	−0.592887	+	0.805286i	$0.702012\pi$
$108$	−0.154597	−0.0148761
$109$	−10.5765	−1.01305	−0.506523	−	0.862226i	$-0.669070\pi$
$109$	−10.5765	−1.01305	−0.506523	+	0.862226i	$0.669070\pi$
$110$	6.36723	0.607092
$111$	0.225288	0.0213834
$112$	−1.00000	−0.0944911
$113$	−16.3691	−1.53987	−0.769936	−	0.638121i	$-0.779712\pi$
$113$	−16.3691	−1.53987	−0.769936	+	0.638121i	$0.779712\pi$
$114$	−0.159191	−0.0149096
$115$	−25.5071	−2.37855
$116$	3.32569	0.308782
$117$	−3.97124	−0.367141
$118$	−11.9804	−1.10289
$119$	1.76795	0.162068
$120$	−0.0790523	−0.00721645
$121$	−6.69205	−0.608368
$122$	−5.45935	−0.494266
$123$	−0.0661109	−0.00596102
$124$	−5.80859	−0.521627
$125$	−1.80723	−0.161644
$126$	−2.99934	−0.267202
$127$	−19.4858	−1.72909	−0.864543	−	0.502559i	$-0.832392\pi$
$127$	−19.4858	−1.72909	−0.864543	+	0.502559i	$0.832392\pi$
$128$	−1.00000	−0.0883883
$129$	−0.0610119	−0.00537179
$130$	−4.06178	−0.356241
$131$	3.89655	0.340443	0.170222	−	0.985406i	$-0.445552\pi$
$131$	3.89655	0.340443	0.170222	+	0.985406i	$0.445552\pi$
$132$	−0.0534853	−0.00465530
$133$	−6.17760	−0.535666
$134$	−2.72798	−0.235661
$135$	−0.474261	−0.0408179
$136$	1.76795	0.151601
$137$	20.4494	1.74711	0.873557	−	0.486723i	$-0.161808\pi$
$137$	20.4494	1.74711	0.873557	+	0.486723i	$0.161808\pi$
$138$	0.214262	0.0182392
$139$	9.68142	0.821167	0.410584	−	0.911823i	$-0.365325\pi$
$139$	9.68142	0.821167	0.410584	+	0.911823i	$0.365325\pi$
$140$	−3.06772	−0.259269
$141$	−0.151460	−0.0127552
$142$	13.8607	1.16316
$143$	−2.74812	−0.229810
$144$	−2.99934	−0.249945
$145$	10.2023	0.847252
$146$	−13.4987	−1.11716
$147$	0.0257691	0.00212540
$148$	8.74256	0.718634
$149$	−11.2756	−0.923736	−0.461868	−	0.886949i	$-0.652821\pi$
$149$	−11.2756	−0.923736	−0.461868	+	0.886949i	$0.652821\pi$
$150$	−0.113665	−0.00928067
$151$	−15.3788	−1.25151	−0.625755	−	0.780020i	$-0.715209\pi$
$151$	−15.3788	−1.25151	−0.625755	+	0.780020i	$0.715209\pi$
$152$	−6.17760	−0.501070
$153$	5.30268	0.428696
$154$	−2.07556	−0.167253
$155$	−17.8191	−1.43127
$156$	0.0341193	0.00273173
$157$	−19.7157	−1.57349	−0.786744	−	0.617280i	$-0.788235\pi$
$157$	−19.7157	−1.57349	−0.786744	+	0.617280i	$0.788235\pi$
$158$	8.49905	0.676148
$159$	0.0411308	0.00326188
$160$	−3.06772	−0.242524
$161$	8.31468	0.655288
$162$	−8.99402	−0.706637
$163$	−0.882952	−0.0691582	−0.0345791	−	0.999402i	$-0.511009\pi$
$163$	−0.882952	−0.0691582	−0.0345791	+	0.999402i	$0.511009\pi$
$164$	−2.56551	−0.200333
$165$	−0.164078	−0.0127734
$166$	12.8421	0.996742
$167$	14.8703	1.15070	0.575348	−	0.817909i	$-0.304867\pi$
$167$	14.8703	1.15070	0.575348	+	0.817909i	$0.304867\pi$
$168$	0.0257691	0.00198813
$169$	−11.2469	−0.865148
$170$	5.42357	0.415969
$171$	−18.5287	−1.41693
$172$	−2.36764	−0.180531
$173$	−0.660547	−0.0502205	−0.0251102	−	0.999685i	$-0.507994\pi$
$173$	−0.660547	−0.0502205	−0.0251102	+	0.999685i	$0.507994\pi$
$174$	−0.0856999	−0.00649689
$175$	−4.41089	−0.333432
$176$	−2.07556	−0.156451
$177$	0.308724	0.0232051
$178$	12.8646	0.964246
$179$	4.62726	0.345857	0.172929	−	0.984934i	$-0.444677\pi$
$179$	4.62726	0.345857	0.172929	+	0.984934i	$0.444677\pi$
$180$	−9.20111	−0.685811
$181$	−11.9879	−0.891053	−0.445527	−	0.895269i	$-0.646984\pi$
$181$	−11.9879	−0.891053	−0.445527	+	0.895269i	$0.646984\pi$
$182$	1.32404	0.0981443
$183$	0.140682	0.0103995
$184$	8.31468	0.612966
$185$	26.8197	1.97182
$186$	0.149682	0.0109752
$187$	3.66949	0.268340
$188$	−5.87757	−0.428666
$189$	0.154597	0.0112453
$190$	−18.9511	−1.37486
$191$	−9.00395	−0.651503	−0.325751	−	0.945455i	$-0.605617\pi$
$191$	−9.00395	−0.651503	−0.325751	+	0.945455i	$0.605617\pi$
$192$	0.0257691	0.00185972
$193$	1.11141	0.0800010	0.0400005	−	0.999200i	$-0.487264\pi$
$193$	1.11141	0.0800010	0.0400005	+	0.999200i	$0.487264\pi$
$194$	−9.65527	−0.693208
$195$	0.104668	0.00749545
$196$	1.00000	0.0714286
$197$	2.09526	0.149281	0.0746405	−	0.997211i	$-0.476219\pi$
$197$	2.09526	0.149281	0.0746405	+	0.997211i	$0.476219\pi$
$198$	−6.22530	−0.442413
$199$	−12.0129	−0.851569	−0.425784	−	0.904825i	$-0.640002\pi$
$199$	−12.0129	−0.851569	−0.425784	+	0.904825i	$0.640002\pi$
$200$	−4.41089	−0.311897
$201$	0.0702975	0.00495840
$202$	15.2235	1.07112
$203$	−3.32569	−0.233417
$204$	−0.0455585	−0.00318973
$205$	−7.87026	−0.549683
$206$	−4.61972	−0.321871
$207$	24.9385	1.73335
$208$	1.32404	0.0918056
$209$	−12.8220	−0.886915
$210$	0.0790523	0.00545512
$211$	−21.9753	−1.51284	−0.756420	−	0.654086i	$-0.773054\pi$
$211$	−21.9753	−1.51284	−0.756420	+	0.654086i	$0.773054\pi$
$212$	1.59613	0.109623
$213$	−0.357176	−0.0244733
$214$	12.2657	0.838469
$215$	−7.26324	−0.495349
$216$	0.154597	0.0105190
$217$	5.80859	0.394313
$218$	10.5765	0.716332
$219$	0.347850	0.0235055
$220$	−6.36723	−0.429279
$221$	−2.34084	−0.157462
$222$	−0.225288	−0.0151203
$223$	16.4257	1.09995	0.549973	−	0.835182i	$-0.314638\pi$
$223$	16.4257	1.09995	0.549973	+	0.835182i	$0.314638\pi$
$224$	1.00000	0.0668153
$225$	−13.2297	−0.881982
$226$	16.3691	1.08885
$227$	25.5805	1.69784	0.848918	−	0.528524i	$-0.177254\pi$
$227$	25.5805	1.69784	0.848918	+	0.528524i	$0.177254\pi$
$228$	0.159191	0.0105427
$229$	0.928990	0.0613894	0.0306947	−	0.999529i	$-0.490228\pi$
$229$	0.928990	0.0613894	0.0306947	+	0.999529i	$0.490228\pi$
$230$	25.5071	1.68189
$231$	0.0534853	0.00351907
$232$	−3.32569	−0.218342
$233$	−24.9451	−1.63421	−0.817104	−	0.576491i	$-0.804422\pi$
$233$	−24.9451	−1.63421	−0.817104	+	0.576491i	$0.804422\pi$
$234$	3.97124	0.259608
$235$	−18.0307	−1.17619
$236$	11.9804	0.779859
$237$	−0.219013	−0.0142264
$238$	−1.76795	−0.114599
$239$	1.74058	0.112589	0.0562944	−	0.998414i	$-0.482071\pi$
$239$	1.74058	0.112589	0.0562944	+	0.998414i	$0.482071\pi$
$240$	0.0790523	0.00510280
$241$	16.8490	1.08534	0.542671	−	0.839945i	$-0.317413\pi$
$241$	16.8490	1.08534	0.542671	+	0.839945i	$0.317413\pi$
$242$	6.69205	0.430181
$243$	0.695560	0.0446202
$244$	5.45935	0.349499
$245$	3.06772	0.195989
$246$	0.0661109	0.00421508
$247$	8.17938	0.520442
$248$	5.80859	0.368846
$249$	−0.330930	−0.0209718
$250$	1.80723	0.114299
$251$	22.7725	1.43738	0.718692	−	0.695328i	$-0.244741\pi$
$251$	22.7725	1.43738	0.718692	+	0.695328i	$0.244741\pi$
$252$	2.99934	0.188940
$253$	17.2576	1.08498
$254$	19.4858	1.22265
$255$	−0.139761	−0.00875214
$256$	1.00000	0.0625000
$257$	5.30935	0.331188	0.165594	−	0.986194i	$-0.447046\pi$
$257$	5.30935	0.331188	0.165594	+	0.986194i	$0.447046\pi$
$258$	0.0610119	0.00379843
$259$	−8.74256	−0.543236
$260$	4.06178	0.251901
$261$	−9.97485	−0.617428
$262$	−3.89655	−0.240730
$263$	−24.2024	−1.49238	−0.746192	−	0.665731i	$-0.768120\pi$
$263$	−24.2024	−1.49238	−0.746192	+	0.665731i	$0.768120\pi$
$264$	0.0534853	0.00329179
$265$	4.89647	0.300788
$266$	6.17760	0.378773
$267$	−0.331510	−0.0202881
$268$	2.72798	0.166638
$269$	12.9205	0.787777	0.393889	−	0.919158i	$-0.371130\pi$
$269$	12.9205	0.787777	0.393889	+	0.919158i	$0.371130\pi$
$270$	0.474261	0.0288626
$271$	−4.94649	−0.300478	−0.150239	−	0.988650i	$-0.548004\pi$
$271$	−4.94649	−0.300478	−0.150239	+	0.988650i	$0.548004\pi$
$272$	−1.76795	−0.107198
$273$	−0.0341193	−0.00206499
$274$	−20.4494	−1.23540
$275$	−9.15506	−0.552071
$276$	−0.214262	−0.0128970
$277$	−13.7834	−0.828162	−0.414081	−	0.910240i	$-0.635897\pi$
$277$	−13.7834	−0.828162	−0.414081	+	0.910240i	$0.635897\pi$
$278$	−9.68142	−0.580653
$279$	17.4219	1.04302
$280$	3.06772	0.183331
$281$	−4.49712	−0.268276	−0.134138	−	0.990963i	$-0.542826\pi$
$281$	−4.49712	−0.268276	−0.134138	+	0.990963i	$0.542826\pi$
$282$	0.151460	0.00901928
$283$	−0.00538604	−0.000320167 0	−0.000160083	−	1.00000i	$-0.500051\pi$
$283$	−0.00538604	−0.000320167 0	−0.000160083		1.00000i	$0.500051\pi$
$284$	−13.8607	−0.822479
$285$	0.488353	0.0289276
$286$	2.74812	0.162500
$287$	2.56551	0.151437
$288$	2.99934	0.176738
$289$	−13.8743	−0.816138
$290$	−10.2023	−0.599098
$291$	0.248808	0.0145854
$292$	13.4987	0.789953
$293$	−27.0514	−1.58036	−0.790180	−	0.612874i	$-0.790013\pi$
$293$	−27.0514	−1.58036	−0.790180	+	0.612874i	$0.790013\pi$
$294$	−0.0257691	−0.00150288
$295$	36.7525	2.13982
$296$	−8.74256	−0.508151
$297$	0.320876	0.0186191
$298$	11.2756	0.653180
$299$	−11.0090	−0.636664
$300$	0.113665	0.00656243
$301$	2.36764	0.136468
$302$	15.3788	0.884951
$303$	−0.392297	−0.0225369
$304$	6.17760	0.354310
$305$	16.7477	0.958973
$306$	−5.30268	−0.303134
$307$	22.3575	1.27601	0.638004	−	0.770033i	$-0.279760\pi$
$307$	22.3575	1.27601	0.638004	+	0.770033i	$0.279760\pi$
$308$	2.07556	0.118266
$309$	0.119046	0.00677229
$310$	17.8191	1.01206
$311$	9.03450	0.512299	0.256150	−	0.966637i	$-0.417546\pi$
$311$	9.03450	0.512299	0.256150	+	0.966637i	$0.417546\pi$
$312$	−0.0341193	−0.00193162
$313$	18.1511	1.02596	0.512982	−	0.858400i	$-0.328541\pi$
$313$	18.1511	1.02596	0.512982	+	0.858400i	$0.328541\pi$
$314$	19.7157	1.11262
$315$	9.20111	0.518424
$316$	−8.49905	−0.478109
$317$	−24.1174	−1.35457	−0.677285	−	0.735720i	$-0.736844\pi$
$317$	−24.1174	−1.35457	−0.677285	+	0.735720i	$0.736844\pi$
$318$	−0.0411308	−0.00230650
$319$	−6.90266	−0.386475
$320$	3.06772	0.171491
$321$	−0.316077	−0.0176417
$322$	−8.31468	−0.463359
$323$	−10.9217	−0.607699
$324$	8.99402	0.499668
$325$	5.84019	0.323955
$326$	0.882952	0.0489022
$327$	−0.272547	−0.0150719
$328$	2.56551	0.141657
$329$	5.87757	0.324041
$330$	0.164078	0.00903218
$331$	−26.7049	−1.46784	−0.733918	−	0.679238i	$-0.762310\pi$
$331$	−26.7049	−1.46784	−0.733918	+	0.679238i	$0.762310\pi$
$332$	−12.8421	−0.704803
$333$	−26.2219	−1.43695
$334$	−14.8703	−0.813665
$335$	8.36866	0.457229
$336$	−0.0257691	−0.00140582
$337$	3.18113	0.173287	0.0866436	−	0.996239i	$-0.472386\pi$
$337$	3.18113	0.173287	0.0866436	+	0.996239i	$0.472386\pi$
$338$	11.2469	0.611752
$339$	−0.421816	−0.0229099
$340$	−5.42357	−0.294135
$341$	12.0561	0.652873
$342$	18.5287	1.00192
$343$	−1.00000	−0.0539949
$344$	2.36764	0.127654
$345$	−0.657294	−0.0353875
$346$	0.660547	0.0355112
$347$	−33.8395	−1.81660	−0.908299	−	0.418322i	$-0.862618\pi$
$347$	−33.8395	−1.81660	−0.908299	+	0.418322i	$0.862618\pi$
$348$	0.0856999	0.00459400
$349$	−2.05253	−0.109869	−0.0549347	−	0.998490i	$-0.517495\pi$
$349$	−2.05253	−0.109869	−0.0549347	+	0.998490i	$0.517495\pi$
$350$	4.41089	0.235772
$351$	−0.204693	−0.0109257
$352$	2.07556	0.110628
$353$	−14.4006	−0.766464	−0.383232	−	0.923652i	$-0.625189\pi$
$353$	−14.4006	−0.766464	−0.383232	+	0.923652i	$0.625189\pi$
$354$	−0.308724	−0.0164085
$355$	−42.5206	−2.25676
$356$	−12.8646	−0.681825
$357$	0.0455585	0.00241121
$358$	−4.62726	−0.244558
$359$	31.9487	1.68619	0.843093	−	0.537768i	$-0.180732\pi$
$359$	31.9487	1.68619	0.843093	+	0.537768i	$0.180732\pi$
$360$	9.20111	0.484941
$361$	19.1628	1.00857
$362$	11.9879	0.630070
$363$	−0.172448	−0.00905117
$364$	−1.32404	−0.0693985
$365$	41.4102	2.16751
$366$	−0.140682	−0.00735359
$367$	33.0481	1.72510	0.862550	−	0.505973i	$-0.168866\pi$
$367$	33.0481	1.72510	0.862550	+	0.505973i	$0.168866\pi$
$368$	−8.31468	−0.433433
$369$	7.69483	0.400577
$370$	−26.8197	−1.39429
$371$	−1.59613	−0.0828669
$372$	−0.149682	−0.00776065
$373$	27.9336	1.44634	0.723172	−	0.690667i	$-0.242683\pi$
$373$	27.9336	1.44634	0.723172	+	0.690667i	$0.242683\pi$
$374$	−3.66949	−0.189745
$375$	−0.0465707	−0.00240490
$376$	5.87757	0.303112
$377$	4.40334	0.226783
$378$	−0.154597	−0.00795163
$379$	5.85783	0.300897	0.150448	−	0.988618i	$-0.451928\pi$
$379$	5.85783	0.300897	0.150448	+	0.988618i	$0.451928\pi$
$380$	18.9511	0.972173
$381$	−0.502131	−0.0257250
$382$	9.00395	0.460682
$383$	31.6095	1.61517	0.807585	−	0.589752i	$-0.200774\pi$
$383$	31.6095	1.61517	0.807585	+	0.589752i	$0.200774\pi$
$384$	−0.0257691	−0.00131502
$385$	6.36723	0.324504
$386$	−1.11141	−0.0565692
$387$	7.10134	0.360981
$388$	9.65527	0.490172
$389$	30.2214	1.53228	0.766142	−	0.642671i	$-0.222174\pi$
$389$	30.2214	1.53228	0.766142	+	0.642671i	$0.222174\pi$
$390$	−0.104668	−0.00530008
$391$	14.6999	0.743408
$392$	−1.00000	−0.0505076
$393$	0.100411	0.00506505
$394$	−2.09526	−0.105558
$395$	−26.0727	−1.31186
$396$	6.22530	0.312833
$397$	6.27078	0.314722	0.157361	−	0.987541i	$-0.449701\pi$
$397$	6.27078	0.314722	0.157361	+	0.987541i	$0.449701\pi$
$398$	12.0129	0.602150
$399$	−0.159191	−0.00796952
$400$	4.41089	0.220544
$401$	36.2314	1.80931	0.904655	−	0.426145i	$-0.140129\pi$
$401$	36.2314	1.80931	0.904655	+	0.426145i	$0.140129\pi$
$402$	−0.0702975	−0.00350612
$403$	−7.69080	−0.383106
$404$	−15.2235	−0.757399
$405$	27.5911	1.37101
$406$	3.32569	0.165051
$407$	−18.1457	−0.899450
$408$	0.0455585	0.00225548
$409$	4.85125	0.239879	0.119940	−	0.992781i	$-0.461730\pi$
$409$	4.85125	0.239879	0.119940	+	0.992781i	$0.461730\pi$
$410$	7.87026	0.388685
$411$	0.526963	0.0259932
$412$	4.61972	0.227597
$413$	−11.9804	−0.589518
$414$	−24.9385	−1.22566
$415$	−39.3960	−1.93387
$416$	−1.32404	−0.0649163
$417$	0.249481	0.0122172
$418$	12.8220	0.627144
$419$	−7.53934	−0.368321	−0.184160	−	0.982896i	$-0.558957\pi$
$419$	−7.53934	−0.368321	−0.184160	+	0.982896i	$0.558957\pi$
$420$	−0.0790523	−0.00385736
$421$	−10.8775	−0.530139	−0.265070	−	0.964229i	$-0.585395\pi$
$421$	−10.8775	−0.530139	−0.265070	+	0.964229i	$0.585395\pi$
$422$	21.9753	1.06974
$423$	17.6288	0.857142
$424$	−1.59613	−0.0775149
$425$	−7.79823	−0.378270
$426$	0.357176	0.0173053
$427$	−5.45935	−0.264196
$428$	−12.2657	−0.592887
$429$	−0.0708166	−0.00341906
$430$	7.26324	0.350265
$431$	−1.00000	−0.0481683
$431$	−1.00000	−0.0481683
$432$	−0.154597	−0.00743807
$433$	−14.5828	−0.700803	−0.350401	−	0.936600i	$-0.613955\pi$
$433$	−14.5828	−0.700803	−0.350401	+	0.936600i	$0.613955\pi$
$434$	−5.80859	−0.278821
$435$	0.262903	0.0126052
$436$	−10.5765	−0.506523
$437$	−51.3648	−2.45711
$438$	−0.347850	−0.0166209
$439$	−28.2708	−1.34929	−0.674646	−	0.738141i	$-0.735704\pi$
$439$	−28.2708	−1.34929	−0.674646	+	0.738141i	$0.735704\pi$
$440$	6.36723	0.303546
$441$	−2.99934	−0.142826
$442$	2.34084	0.111342
$443$	−38.2783	−1.81866	−0.909328	−	0.416080i	$-0.863404\pi$
$443$	−38.2783	−1.81866	−0.909328	+	0.416080i	$0.863404\pi$
$444$	0.225288	0.0106917
$445$	−39.4651	−1.87083
$446$	−16.4257	−0.777779
$447$	−0.290563	−0.0137431
$448$	−1.00000	−0.0472456
$449$	−3.38324	−0.159665	−0.0798326	−	0.996808i	$-0.525439\pi$
$449$	−3.38324	−0.159665	−0.0798326	+	0.996808i	$0.525439\pi$
$450$	13.2297	0.623656
$451$	5.32487	0.250739
$452$	−16.3691	−0.769936
$453$	−0.396298	−0.0186197
$454$	−25.5805	−1.20055
$455$	−4.06178	−0.190419
$456$	−0.159191	−0.00745481
$457$	−8.03439	−0.375833	−0.187916	−	0.982185i	$-0.560173\pi$
$457$	−8.03439	−0.375833	−0.187916	+	0.982185i	$0.560173\pi$
$458$	−0.928990	−0.0434089
$459$	0.273321	0.0127575
$460$	−25.5071	−1.18927
$461$	−25.3806	−1.18209	−0.591047	−	0.806637i	$-0.701285\pi$
$461$	−25.3806	−1.18209	−0.591047	+	0.806637i	$0.701285\pi$
$462$	−0.0534853	−0.00248836
$463$	−17.8053	−0.827483	−0.413741	−	0.910394i	$-0.635778\pi$
$463$	−17.8053	−0.827483	−0.413741	+	0.910394i	$0.635778\pi$
$464$	3.32569	0.154391
$465$	−0.459182	−0.0212941
$466$	24.9451	1.15556
$467$	2.03176	0.0940189	0.0470094	−	0.998894i	$-0.485031\pi$
$467$	2.03176	0.0940189	0.0470094	+	0.998894i	$0.485031\pi$
$468$	−3.97124	−0.183570
$469$	−2.72798	−0.125966
$470$	18.0307	0.831695
$471$	−0.508057	−0.0234100
$472$	−11.9804	−0.551443
$473$	4.91418	0.225954
$474$	0.219013	0.0100596
$475$	27.2487	1.25026
$476$	1.76795	0.0810339
$477$	−4.78733	−0.219197
$478$	−1.74058	−0.0796123
$479$	−15.3851	−0.702965	−0.351482	−	0.936194i	$-0.614322\pi$
$479$	−15.3851	−0.702965	−0.351482	+	0.936194i	$0.614322\pi$
$480$	−0.0790523	−0.00360823
$481$	11.5755	0.527797
$482$	−16.8490	−0.767453
$483$	0.214262	0.00974924
$484$	−6.69205	−0.304184
$485$	29.6196	1.34496
$486$	−0.695560	−0.0315512
$487$	1.63665	0.0741635	0.0370817	−	0.999312i	$-0.488194\pi$
$487$	1.63665	0.0741635	0.0370817	+	0.999312i	$0.488194\pi$
$488$	−5.45935	−0.247133
$489$	−0.0227529	−0.00102892
$490$	−3.06772	−0.138585
$491$	−17.6215	−0.795248	−0.397624	−	0.917548i	$-0.630165\pi$
$491$	−17.6215	−0.795248	−0.397624	+	0.917548i	$0.630165\pi$
$492$	−0.0661109	−0.00298051
$493$	−5.87965	−0.264806
$494$	−8.17938	−0.368008
$495$	19.0975	0.858367
$496$	−5.80859	−0.260813
$497$	13.8607	0.621735
$498$	0.330930	0.0148293
$499$	−21.4771	−0.961447	−0.480724	−	0.876872i	$-0.659626\pi$
$499$	−21.4771	−0.961447	−0.480724	+	0.876872i	$0.659626\pi$
$500$	−1.80723	−0.0808218
$501$	0.383193	0.0171198
$502$	−22.7725	−1.01638
$503$	−33.1414	−1.47770	−0.738851	−	0.673869i	$-0.764631\pi$
$503$	−33.1414	−1.47770	−0.738851	+	0.673869i	$0.764631\pi$
$504$	−2.99934	−0.133601
$505$	−46.7015	−2.07819
$506$	−17.2576	−0.767195
$507$	−0.289823	−0.0128715
$508$	−19.4858	−0.864543
$509$	−5.49911	−0.243744	−0.121872	−	0.992546i	$-0.538890\pi$
$509$	−5.49911	−0.243744	−0.121872	+	0.992546i	$0.538890\pi$
$510$	0.139761	0.00618870
$511$	−13.4987	−0.597148
$512$	−1.00000	−0.0441942
$513$	−0.955041	−0.0421661
$514$	−5.30935	−0.234185
$515$	14.1720	0.624492
$516$	−0.0610119	−0.00268590
$517$	12.1992	0.536522
$518$	8.74256	0.384126
$519$	−0.0170217	−0.000747169 0
$520$	−4.06178	−0.178121
$521$	9.04760	0.396383	0.198191	−	0.980163i	$-0.436493\pi$
$521$	9.04760	0.396383	0.198191	+	0.980163i	$0.436493\pi$
$522$	9.97485	0.436587
$523$	5.76450	0.252064	0.126032	−	0.992026i	$-0.459776\pi$
$523$	5.76450	0.252064	0.126032	+	0.992026i	$0.459776\pi$
$524$	3.89655	0.170222
$525$	−0.113665	−0.00496073
$526$	24.2024	1.05528
$527$	10.2693	0.447338
$528$	−0.0534853	−0.00232765
$529$	46.1339	2.00582
$530$	−4.89647	−0.212689
$531$	−35.9333	−1.55937
$532$	−6.17760	−0.267833
$533$	−3.39684	−0.147133
$534$	0.331510	0.0143459
$535$	−37.6278	−1.62679
$536$	−2.72798	−0.117831
$537$	0.119240	0.00514559
$538$	−12.9205	−0.557042
$539$	−2.07556	−0.0894007
$540$	−0.474261	−0.0204089
$541$	12.3616	0.531466	0.265733	−	0.964047i	$-0.414386\pi$
$541$	12.3616	0.531466	0.265733	+	0.964047i	$0.414386\pi$
$542$	4.94649	0.212470
$543$	−0.308917	−0.0132569
$544$	1.76795	0.0758003
$545$	−32.4458	−1.38982
$546$	0.0341193	0.00146017
$547$	41.4608	1.77274	0.886368	−	0.462982i	$-0.153220\pi$
$547$	41.4608	1.77274	0.886368	+	0.462982i	$0.153220\pi$
$548$	20.4494	0.873557
$549$	−16.3744	−0.698843
$550$	9.15506	0.390373
$551$	20.5448	0.875236
$552$	0.214262	0.00911958
$553$	8.49905	0.361416
$554$	13.7834	0.585599
$555$	0.691119	0.0293364
$556$	9.68142	0.410584
$557$	−7.83443	−0.331955	−0.165978	−	0.986130i	$-0.553078\pi$
$557$	−7.83443	−0.331955	−0.165978	+	0.986130i	$0.553078\pi$
$558$	−17.4219	−0.737528
$559$	−3.13484	−0.132590
$560$	−3.06772	−0.129635
$561$	0.0945594	0.00399230
$562$	4.49712	0.189699
$563$	2.68344	0.113093	0.0565467	−	0.998400i	$-0.481991\pi$
$563$	2.68344	0.113093	0.0565467	+	0.998400i	$0.481991\pi$
$564$	−0.151460	−0.00637760
$565$	−50.2156	−2.11259
$566$	0.00538604	0.000226392 0
$567$	−8.99402	−0.377714
$568$	13.8607	0.581580
$569$	−5.79560	−0.242964	−0.121482	−	0.992594i	$-0.538765\pi$
$569$	−5.79560	−0.242964	−0.121482	+	0.992594i	$0.538765\pi$
$570$	−0.488353	−0.0204549
$571$	−23.8267	−0.997118	−0.498559	−	0.866856i	$-0.666137\pi$
$571$	−23.8267	−0.997118	−0.498559	+	0.866856i	$0.666137\pi$
$572$	−2.74812	−0.114905
$573$	−0.232024	−0.00969292
$574$	−2.56551	−0.107082
$575$	−36.6751	−1.52946
$576$	−2.99934	−0.124972
$577$	−37.6193	−1.56611	−0.783055	−	0.621952i	$-0.786340\pi$
$577$	−37.6193	−1.56611	−0.783055	+	0.621952i	$0.786340\pi$
$578$	13.8743	0.577097
$579$	0.0286400	0.00119024
$580$	10.2023	0.423626
$581$	12.8421	0.532781
$582$	−0.248808	−0.0103134
$583$	−3.31286	−0.137205
$584$	−13.4987	−0.558581
$585$	−12.1826	−0.503690
$586$	27.0514	1.11748
$587$	17.9369	0.740336	0.370168	−	0.928965i	$-0.379300\pi$
$587$	17.9369	0.740336	0.370168	+	0.928965i	$0.379300\pi$
$588$	0.0257691	0.00106270
$589$	−35.8831	−1.47854
$590$	−36.7525	−1.51308
$591$	0.0539929	0.00222097
$592$	8.74256	0.359317
$593$	42.8433	1.75936	0.879682	−	0.475562i	$-0.157755\pi$
$593$	42.8433	1.75936	0.879682	+	0.475562i	$0.157755\pi$
$594$	−0.320876	−0.0131657
$595$	5.42357	0.222345
$596$	−11.2756	−0.461868
$597$	−0.309560	−0.0126695
$598$	11.0090	0.450190
$599$	1.85871	0.0759447	0.0379723	−	0.999279i	$-0.487910\pi$
$599$	1.85871	0.0759447	0.0379723	+	0.999279i	$0.487910\pi$
$600$	−0.113665	−0.00464034
$601$	36.3631	1.48328	0.741640	−	0.670798i	$-0.234048\pi$
$601$	36.3631	1.48328	0.741640	+	0.670798i	$0.234048\pi$
$602$	−2.36764	−0.0964977
$603$	−8.18212	−0.333202
$604$	−15.3788	−0.625755
$605$	−20.5293	−0.834635
$606$	0.392297	0.0159360
$607$	−21.3873	−0.868082	−0.434041	−	0.900893i	$-0.642913\pi$
$607$	−21.3873	−0.868082	−0.434041	+	0.900893i	$0.642913\pi$
$608$	−6.17760	−0.250535
$609$	−0.0856999	−0.00347273
$610$	−16.7477	−0.678096
$611$	−7.78213	−0.314831
$612$	5.30268	0.214348
$613$	15.1772	0.613002	0.306501	−	0.951870i	$-0.400842\pi$
$613$	15.1772	0.613002	0.306501	+	0.951870i	$0.400842\pi$
$614$	−22.3575	−0.902274
$615$	−0.202809	−0.00817807
$616$	−2.07556	−0.0836267
$617$	31.6220	1.27305	0.636526	−	0.771255i	$-0.280371\pi$
$617$	31.6220	1.27305	0.636526	+	0.771255i	$0.280371\pi$
$618$	−0.119046	−0.00478873
$619$	−15.3520	−0.617047	−0.308524	−	0.951217i	$-0.599835\pi$
$619$	−15.3520	−0.617047	−0.308524	+	0.951217i	$0.599835\pi$
$620$	−17.8191	−0.715633
$621$	1.28543	0.0515824
$622$	−9.03450	−0.362250
$623$	12.8646	0.515411
$624$	0.0341193	0.00136586
$625$	−27.5985	−1.10394
$626$	−18.1511	−0.725466
$627$	−0.330411	−0.0131953
$628$	−19.7157	−0.786744
$629$	−15.4564	−0.616288
$630$	−9.20111	−0.366581
$631$	−39.0785	−1.55569	−0.777845	−	0.628456i	$-0.783687\pi$
$631$	−39.0785	−1.55569	−0.777845	+	0.628456i	$0.783687\pi$
$632$	8.49905	0.338074
$633$	−0.566283	−0.0225077
$634$	24.1174	0.957826
$635$	−59.7769	−2.37218
$636$	0.0411308	0.00163094
$637$	1.32404	0.0524603
$638$	6.90266	0.273279
$639$	41.5728	1.64459
$640$	−3.06772	−0.121262
$641$	29.1655	1.15197	0.575984	−	0.817461i	$-0.304619\pi$
$641$	29.1655	1.15197	0.575984	+	0.817461i	$0.304619\pi$
$642$	0.316077	0.0124746
$643$	−35.9915	−1.41936	−0.709682	−	0.704522i	$-0.751162\pi$
$643$	−35.9915	−1.41936	−0.709682	+	0.704522i	$0.751162\pi$
$644$	8.31468	0.327644
$645$	−0.187167	−0.00736970
$646$	10.9217	0.429708
$647$	−10.5237	−0.413729	−0.206864	−	0.978370i	$-0.566326\pi$
$647$	−10.5237	−0.413729	−0.206864	+	0.978370i	$0.566326\pi$
$648$	−8.99402	−0.353319
$649$	−24.8661	−0.976079
$650$	−5.84019	−0.229071
$651$	0.149682	0.00586650
$652$	−0.882952	−0.0345791
$653$	19.1836	0.750713	0.375356	−	0.926881i	$-0.377520\pi$
$653$	19.1836	0.750713	0.375356	+	0.926881i	$0.377520\pi$
$654$	0.272547	0.0106574
$655$	11.9535	0.467063
$656$	−2.56551	−0.100166
$657$	−40.4872	−1.57956
$658$	−5.87757	−0.229131
$659$	−15.9875	−0.622783	−0.311391	−	0.950282i	$-0.600795\pi$
$659$	−15.9875	−0.622783	−0.311391	+	0.950282i	$0.600795\pi$
$660$	−0.164078	−0.00638672
$661$	−8.46917	−0.329412	−0.164706	−	0.986343i	$-0.552668\pi$
$661$	−8.46917	−0.329412	−0.164706	+	0.986343i	$0.552668\pi$
$662$	26.7049	1.03792
$663$	−0.0603212	−0.00234268
$664$	12.8421	0.498371
$665$	−18.9511	−0.734893
$666$	26.2219	1.01608
$667$	−27.6520	−1.07069
$668$	14.8703	0.575348
$669$	0.423275	0.0163648
$670$	−8.36866	−0.323310
$671$	−11.3312	−0.437437
$672$	0.0257691	0.000994064 0
$673$	13.0805	0.504217	0.252108	−	0.967699i	$-0.418876\pi$
$673$	13.0805	0.504217	0.252108	+	0.967699i	$0.418876\pi$
$674$	−3.18113	−0.122533
$675$	−0.681912	−0.0262468
$676$	−11.2469	−0.432574
$677$	−46.5146	−1.78770	−0.893850	−	0.448365i	$-0.852006\pi$
$677$	−46.5146	−1.78770	−0.893850	+	0.448365i	$0.852006\pi$
$678$	0.421816	0.0161997
$679$	−9.65527	−0.370535
$680$	5.42357	0.207985
$681$	0.659186	0.0252601
$682$	−12.0561	−0.461651
$683$	−2.27285	−0.0869683	−0.0434842	−	0.999054i	$-0.513846\pi$
$683$	−2.27285	−0.0869683	−0.0434842	+	0.999054i	$0.513846\pi$
$684$	−18.5287	−0.708463
$685$	62.7331	2.39691
$686$	1.00000	0.0381802
$687$	0.0239392	0.000913339 0
$688$	−2.36764	−0.0902653
$689$	2.11334	0.0805117
$690$	0.657294	0.0250228
$691$	21.2705	0.809167	0.404584	−	0.914501i	$-0.367416\pi$
$691$	21.2705	0.809167	0.404584	+	0.914501i	$0.367416\pi$
$692$	−0.660547	−0.0251102
$693$	−6.22530	−0.236480
$694$	33.8395	1.28453
$695$	29.6999	1.12658
$696$	−0.0856999	−0.00324845
$697$	4.53570	0.171802
$698$	2.05253	0.0776895
$699$	−0.642812	−0.0243134
$700$	−4.41089	−0.166716
$701$	16.1394	0.609578	0.304789	−	0.952420i	$-0.401414\pi$
$701$	16.1394	0.609578	0.304789	+	0.952420i	$0.401414\pi$
$702$	0.204693	0.00772564
$703$	54.0080	2.03695
$704$	−2.07556	−0.0782256
$705$	−0.464635	−0.0174992
$706$	14.4006	0.541972
$707$	15.2235	0.572540
$708$	0.308724	0.0116026
$709$	−32.0623	−1.20413	−0.602063	−	0.798448i	$-0.705654\pi$
$709$	−32.0623	−1.20413	−0.602063	+	0.798448i	$0.705654\pi$
$710$	42.5206	1.59577
$711$	25.4915	0.956006
$712$	12.8646	0.482123
$713$	48.2965	1.80872
$714$	−0.0455585	−0.00170498
$715$	−8.43046	−0.315281
$716$	4.62726	0.172929
$717$	0.0448532	0.00167507
$718$	−31.9487	−1.19231
$719$	−21.0170	−0.783804	−0.391902	−	0.920007i	$-0.628183\pi$
$719$	−21.0170	−0.783804	−0.391902	+	0.920007i	$0.628183\pi$
$720$	−9.20111	−0.342905
$721$	−4.61972	−0.172047
$722$	−19.1628	−0.713164
$723$	0.434184	0.0161475
$724$	−11.9879	−0.445527
$725$	14.6692	0.544801
$726$	0.172448	0.00640015
$727$	−4.86718	−0.180514	−0.0902569	−	0.995919i	$-0.528769\pi$
$727$	−4.86718	−0.180514	−0.0902569	+	0.995919i	$0.528769\pi$
$728$	1.32404	0.0490721
$729$	−26.9641	−0.998672
$730$	−41.4102	−1.53266
$731$	4.18587	0.154820
$732$	0.140682	0.00519977
$733$	46.8557	1.73066	0.865328	−	0.501207i	$-0.167110\pi$
$733$	46.8557	1.73066	0.865328	+	0.501207i	$0.167110\pi$
$734$	−33.0481	−1.21983
$735$	0.0790523	0.00291589
$736$	8.31468	0.306483
$737$	−5.66208	−0.208565
$738$	−7.69483	−0.283251
$739$	15.7410	0.579041	0.289520	−	0.957172i	$-0.406504\pi$
$739$	15.7410	0.579041	0.289520	+	0.957172i	$0.406504\pi$
$740$	26.8197	0.985912
$741$	0.210775	0.00774302
$742$	1.59613	0.0585957
$743$	28.6952	1.05272	0.526362	−	0.850260i	$-0.323556\pi$
$743$	28.6952	1.05272	0.526362	+	0.850260i	$0.323556\pi$
$744$	0.149682	0.00548761
$745$	−34.5905	−1.26730
$746$	−27.9336	−1.02272
$747$	38.5179	1.40929
$748$	3.66949	0.134170
$749$	12.2657	0.448181
$750$	0.0465707	0.00170052
$751$	−26.6315	−0.971796	−0.485898	−	0.874015i	$-0.661507\pi$
$751$	−26.6315	−0.971796	−0.485898	+	0.874015i	$0.661507\pi$
$752$	−5.87757	−0.214333
$753$	0.586825	0.0213851
$754$	−4.40334	−0.160360
$755$	−47.1778	−1.71698
$756$	0.154597	0.00562265
$757$	−12.2940	−0.446834	−0.223417	−	0.974723i	$-0.571721\pi$
$757$	−12.2940	−0.446834	−0.223417	+	0.974723i	$0.571721\pi$
$758$	−5.85783	−0.212766
$759$	0.444713	0.0161421
$760$	−18.9511	−0.687430
$761$	−20.8409	−0.755480	−0.377740	−	0.925912i	$-0.623299\pi$
$761$	−20.8409	−0.755480	−0.377740	+	0.925912i	$0.623299\pi$
$762$	0.502131	0.0181903
$763$	10.5765	0.382896
$764$	−9.00395	−0.325751
$765$	16.2671	0.588139
$766$	−31.6095	−1.14210
$767$	15.8625	0.572763
$768$	0.0257691	0.000929862 0
$769$	20.4775	0.738436	0.369218	−	0.929343i	$-0.379626\pi$
$769$	20.4775	0.738436	0.369218	+	0.929343i	$0.379626\pi$
$770$	−6.36723	−0.229459
$771$	0.136817	0.00492735
$772$	1.11141	0.0400005
$773$	11.7857	0.423901	0.211950	−	0.977280i	$-0.432018\pi$
$773$	11.7857	0.423901	0.211950	+	0.977280i	$0.432018\pi$
$774$	−7.10134	−0.255252
$775$	−25.6210	−0.920335
$776$	−9.65527	−0.346604
$777$	−0.225288	−0.00808215
$778$	−30.2214	−1.08349
$779$	−15.8487	−0.567839
$780$	0.104668	0.00374772
$781$	28.7686	1.02942
$782$	−14.6999	−0.525669
$783$	−0.514142	−0.0183740
$784$	1.00000	0.0357143
$785$	−60.4823	−2.15871
$786$	−0.100411	−0.00358153
$787$	9.46024	0.337221	0.168611	−	0.985683i	$-0.446072\pi$
$787$	9.46024	0.337221	0.168611	+	0.985683i	$0.446072\pi$
$788$	2.09526	0.0746405
$789$	−0.623674	−0.0222034
$790$	26.0727	0.927624
$791$	16.3691	0.582017
$792$	−6.22530	−0.221207
$793$	7.22839	0.256688
$794$	−6.27078	−0.222542
$795$	0.126178	0.00447506
$796$	−12.0129	−0.425784
$797$	50.7447	1.79747	0.898734	−	0.438493i	$-0.144488\pi$
$797$	50.7447	1.79747	0.898734	+	0.438493i	$0.144488\pi$
$798$	0.159191	0.00563531
$799$	10.3913	0.367616
$800$	−4.41089	−0.155948
$801$	38.5854	1.36335
$802$	−36.2314	−1.27938
$803$	−28.0174	−0.988713
$804$	0.0702975	0.00247920
$805$	25.5071	0.899006
$806$	7.69080	0.270897
$807$	0.332950	0.0117204
$808$	15.2235	0.535562
$809$	24.3120	0.854764	0.427382	−	0.904071i	$-0.359436\pi$
$809$	24.3120	0.854764	0.427382	+	0.904071i	$0.359436\pi$
$810$	−27.5911	−0.969453
$811$	−1.74848	−0.0613975	−0.0306987	−	0.999529i	$-0.509773\pi$
$811$	−1.74848	−0.0613975	−0.0306987	+	0.999529i	$0.509773\pi$
$812$	−3.32569	−0.116709
$813$	−0.127467	−0.00447045
$814$	18.1457	0.636007
$815$	−2.70865	−0.0948798
$816$	−0.0455585	−0.00159487
$817$	−14.6263	−0.511710
$818$	−4.85125	−0.169620
$819$	3.97124	0.138766
$820$	−7.87026	−0.274841
$821$	29.3926	1.02581	0.512904	−	0.858446i	$-0.328570\pi$
$821$	29.3926	1.02581	0.512904	+	0.858446i	$0.328570\pi$
$822$	−0.526963	−0.0183800
$823$	−21.7776	−0.759121	−0.379560	−	0.925167i	$-0.623925\pi$
$823$	−21.7776	−0.759121	−0.379560	+	0.925167i	$0.623925\pi$
$824$	−4.61972	−0.160935
$825$	−0.235918	−0.00821360
$826$	11.9804	0.416852
$827$	37.0781	1.28933	0.644666	−	0.764465i	$-0.276997\pi$
$827$	37.0781	1.28933	0.644666	+	0.764465i	$0.276997\pi$
$828$	24.9385	0.866673
$829$	35.1626	1.22125	0.610624	−	0.791921i	$-0.290919\pi$
$829$	35.1626	1.22125	0.610624	+	0.791921i	$0.290919\pi$
$830$	39.3960	1.36746
$831$	−0.355185	−0.0123212
$832$	1.32404	0.0459028
$833$	−1.76795	−0.0612559
$834$	−0.249481	−0.00863883
$835$	45.6178	1.57867
$836$	−12.8220	−0.443458
$837$	0.897993	0.0310392
$838$	7.53934	0.260442
$839$	−18.8524	−0.650858	−0.325429	−	0.945566i	$-0.605509\pi$
$839$	−18.8524	−0.650858	−0.325429	+	0.945566i	$0.605509\pi$
$840$	0.0790523	0.00272756
$841$	−17.9398	−0.618614
$842$	10.8775	0.374865
$843$	−0.115887	−0.00399135
$844$	−21.9753	−0.756420
$845$	−34.5024	−1.18692
$846$	−17.6288	−0.606091
$847$	6.69205	0.229942
$848$	1.59613	0.0548113
$849$	−0.000138793 0	−4.76337e−6 0
$850$	7.79823	0.267477
$851$	−72.6916	−2.49184
$852$	−0.357176	−0.0122367
$853$	40.2119	1.37683	0.688414	−	0.725318i	$-0.258307\pi$
$853$	40.2119	1.37683	0.688414	+	0.725318i	$0.258307\pi$
$854$	5.45935	0.186815
$855$	−56.8408	−1.94391
$856$	12.2657	0.419235
$857$	3.79586	0.129664	0.0648320	−	0.997896i	$-0.479349\pi$
$857$	3.79586	0.129664	0.0648320	+	0.997896i	$0.479349\pi$
$858$	0.0708166	0.00241764
$859$	18.6015	0.634676	0.317338	−	0.948312i	$-0.397211\pi$
$859$	18.6015	0.634676	0.317338	+	0.948312i	$0.397211\pi$
$860$	−7.26324	−0.247675
$861$	0.0661109	0.00225305
$862$	1.00000	0.0340601
$863$	−29.7990	−1.01437	−0.507185	−	0.861837i	$-0.669314\pi$
$863$	−29.7990	−1.01437	−0.507185	+	0.861837i	$0.669314\pi$
$864$	0.154597	0.00525951
$865$	−2.02637	−0.0688987
$866$	14.5828	0.495542
$867$	−0.357529	−0.0121423
$868$	5.80859	0.197156
$869$	17.6403	0.598406
$870$	−0.262903	−0.00891325
$871$	3.61195	0.122386
$872$	10.5765	0.358166
$873$	−28.9594	−0.980128
$874$	51.3648	1.73744
$875$	1.80723	0.0610955
$876$	0.347850	0.0117528
$877$	15.9240	0.537714	0.268857	−	0.963180i	$-0.413354\pi$
$877$	15.9240	0.537714	0.268857	+	0.963180i	$0.413354\pi$
$878$	28.2708	0.954094
$879$	−0.697090	−0.0235123
$880$	−6.36723	−0.214639
$881$	−40.0978	−1.35093	−0.675465	−	0.737392i	$-0.736057\pi$
$881$	−40.0978	−1.35093	−0.675465	+	0.737392i	$0.736057\pi$
$882$	2.99934	0.100993
$883$	25.9146	0.872095	0.436047	−	0.899924i	$-0.356378\pi$
$883$	25.9146	0.872095	0.436047	+	0.899924i	$0.356378\pi$
$884$	−2.34084	−0.0787308
$885$	0.947079	0.0318357
$886$	38.2783	1.28598
$887$	−8.26231	−0.277421	−0.138711	−	0.990333i	$-0.544296\pi$
$887$	−8.26231	−0.277421	−0.138711	+	0.990333i	$0.544296\pi$
$888$	−0.225288	−0.00756016
$889$	19.4858	0.653533
$890$	39.4651	1.32287
$891$	−18.6676	−0.625390
$892$	16.4257	0.549973
$893$	−36.3093	−1.21504
$894$	0.290563	0.00971787
$895$	14.1951	0.474490
$896$	1.00000	0.0334077
$897$	−0.283691	−0.00947216
$898$	3.38324	0.112900
$899$	−19.3175	−0.644276
$900$	−13.2297	−0.440991
$901$	−2.82188	−0.0940104
$902$	−5.32487	−0.177299
$903$	0.0610119	0.00203035
$904$	16.3691	0.544427
$905$	−36.7755	−1.22246
$906$	0.396298	0.0131661
$907$	−2.18793	−0.0726490	−0.0363245	−	0.999340i	$-0.511565\pi$
$907$	−2.18793	−0.0726490	−0.0363245	+	0.999340i	$0.511565\pi$
$908$	25.5805	0.848918
$909$	45.6605	1.51446
$910$	4.06178	0.134647
$911$	−41.0084	−1.35867	−0.679335	−	0.733828i	$-0.737732\pi$
$911$	−41.0084	−1.35867	−0.679335	+	0.733828i	$0.737732\pi$
$912$	0.159191	0.00527135
$913$	26.6546	0.882139
$914$	8.03439	0.265754
$915$	0.431574	0.0142674
$916$	0.928990	0.0306947
$917$	−3.89655	−0.128676
$918$	−0.273321	−0.00902093
$919$	−15.4199	−0.508655	−0.254327	−	0.967118i	$-0.581854\pi$
$919$	−15.4199	−0.508655	−0.254327	+	0.967118i	$0.581854\pi$
$920$	25.5071	0.840944
$921$	0.576131	0.0189842
$922$	25.3806	0.835867
$923$	−18.3520	−0.604065
$924$	0.0534853	0.00175954
$925$	38.5624	1.26793
$926$	17.8053	0.585119
$927$	−13.8561	−0.455094
$928$	−3.32569	−0.109171
$929$	−23.7505	−0.779228	−0.389614	−	0.920978i	$-0.627392\pi$
$929$	−23.7505	−0.779228	−0.389614	+	0.920978i	$0.627392\pi$
$930$	0.459182	0.0150572
$931$	6.17760	0.202463
$932$	−24.9451	−0.817104
$933$	0.232811	0.00762188
$934$	−2.03176	−0.0664814
$935$	11.2570	0.368142
$936$	3.97124	0.129804
$937$	21.6493	0.707250	0.353625	−	0.935387i	$-0.384949\pi$
$937$	21.6493	0.707250	0.353625	+	0.935387i	$0.384949\pi$
$938$	2.72798	0.0890716
$939$	0.467739	0.0152641
$940$	−18.0307	−0.588097
$941$	55.5084	1.80952	0.904761	−	0.425920i	$-0.140049\pi$
$941$	55.5084	1.80952	0.904761	+	0.425920i	$0.140049\pi$
$942$	0.508057	0.0165534
$943$	21.3314	0.694646
$944$	11.9804	0.389929
$945$	0.474261	0.0154277
$946$	−4.91418	−0.159774
$947$	−41.8830	−1.36101	−0.680507	−	0.732741i	$-0.738240\pi$
$947$	−41.8830	−1.36101	−0.680507	+	0.732741i	$0.738240\pi$
$948$	−0.219013	−0.00711320
$949$	17.8728	0.580176
$950$	−27.2487	−0.884065
$951$	−0.621485	−0.0201530
$952$	−1.76795	−0.0572996
$953$	38.0320	1.23198	0.615989	−	0.787755i	$-0.288756\pi$
$953$	38.0320	1.23198	0.615989	+	0.787755i	$0.288756\pi$
$954$	4.78733	0.154995
$955$	−27.6216	−0.893813
$956$	1.74058	0.0562944
$957$	−0.177875	−0.00574989
$958$	15.3851	0.497071
$959$	−20.4494	−0.660347
$960$	0.0790523	0.00255140
$961$	2.73970	0.0883775
$962$	−11.5755	−0.373209
$963$	36.7891	1.18551
$964$	16.8490	0.542671
$965$	3.40949	0.109755
$966$	−0.214262	−0.00689376
$967$	−53.9053	−1.73348	−0.866738	−	0.498763i	$-0.833788\pi$
$967$	−53.9053	−1.73348	−0.866738	+	0.498763i	$0.833788\pi$
$968$	6.69205	0.215091
$969$	−0.281442	−0.00904122
$970$	−29.6196	−0.951030
$971$	−52.4044	−1.68174	−0.840869	−	0.541238i	$-0.817956\pi$
$971$	−52.4044	−1.68174	−0.840869	+	0.541238i	$0.817956\pi$
$972$	0.695560	0.0223101
$973$	−9.68142	−0.310372
$974$	−1.63665	−0.0524415
$975$	0.150496	0.00481974
$976$	5.45935	0.174750
$977$	−14.0479	−0.449432	−0.224716	−	0.974424i	$-0.572145\pi$
$977$	−14.0479	−0.449432	−0.224716	+	0.974424i	$0.572145\pi$
$978$	0.0227529	0.000727557 0
$979$	26.7014	0.853379
$980$	3.06772	0.0979946
$981$	31.7225	1.01282
$982$	17.6215	0.562325
$983$	49.8345	1.58947	0.794737	−	0.606953i	$-0.207609\pi$
$983$	49.8345	1.58947	0.794737	+	0.606953i	$0.207609\pi$
$984$	0.0661109	0.00210754
$985$	6.42766	0.204802
$986$	5.87965	0.187246
$987$	0.151460	0.00482101
$988$	8.17938	0.260221
$989$	19.6861	0.625983
$990$	−19.0975	−0.606957
$991$	23.2262	0.737805	0.368903	−	0.929468i	$-0.379734\pi$
$991$	23.2262	0.737805	0.368903	+	0.929468i	$0.379734\pi$
$992$	5.80859	0.184423
$993$	−0.688162	−0.0218382
$994$	−13.8607	−0.439633
$995$	−36.8520	−1.16829
$996$	−0.330930	−0.0104859
$997$	−21.2557	−0.673174	−0.336587	−	0.941652i	$-0.609273\pi$
$997$	−21.2557	−0.673174	−0.336587	+	0.941652i	$0.609273\pi$
$998$	21.4771	0.679846
$999$	−1.35158	−0.0427620

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6034.2.a.o.1.15	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6034.2.a.o.1.15	✓	25	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +0.0257691 q^{3} +1.00000 q^{4} +3.06772 q^{5} -0.0257691 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.99934 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +0.0257691 q^{3} +1.00000 q^{4} +3.06772 q^{5} -0.0257691 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.99934 q^{9} -3.06772 q^{10} -2.07556 q^{11} +0.0257691 q^{12} +1.32404 q^{13} +1.00000 q^{14} +0.0790523 q^{15} +1.00000 q^{16} -1.76795 q^{17} +2.99934 q^{18} +6.17760 q^{19} +3.06772 q^{20} -0.0257691 q^{21} +2.07556 q^{22} -8.31468 q^{23} -0.0257691 q^{24} +4.41089 q^{25} -1.32404 q^{26} -0.154597 q^{27} -1.00000 q^{28} +3.32569 q^{29} -0.0790523 q^{30} -5.80859 q^{31} -1.00000 q^{32} -0.0534853 q^{33} +1.76795 q^{34} -3.06772 q^{35} -2.99934 q^{36} +8.74256 q^{37} -6.17760 q^{38} +0.0341193 q^{39} -3.06772 q^{40} -2.56551 q^{41} +0.0257691 q^{42} -2.36764 q^{43} -2.07556 q^{44} -9.20111 q^{45} +8.31468 q^{46} -5.87757 q^{47} +0.0257691 q^{48} +1.00000 q^{49} -4.41089 q^{50} -0.0455585 q^{51} +1.32404 q^{52} +1.59613 q^{53} +0.154597 q^{54} -6.36723 q^{55} +1.00000 q^{56} +0.159191 q^{57} -3.32569 q^{58} +11.9804 q^{59} +0.0790523 q^{60} +5.45935 q^{61} +5.80859 q^{62} +2.99934 q^{63} +1.00000 q^{64} +4.06178 q^{65} +0.0534853 q^{66} +2.72798 q^{67} -1.76795 q^{68} -0.214262 q^{69} +3.06772 q^{70} -13.8607 q^{71} +2.99934 q^{72} +13.4987 q^{73} -8.74256 q^{74} +0.113665 q^{75} +6.17760 q^{76} +2.07556 q^{77} -0.0341193 q^{78} -8.49905 q^{79} +3.06772 q^{80} +8.99402 q^{81} +2.56551 q^{82} -12.8421 q^{83} -0.0257691 q^{84} -5.42357 q^{85} +2.36764 q^{86} +0.0856999 q^{87} +2.07556 q^{88} -12.8646 q^{89} +9.20111 q^{90} -1.32404 q^{91} -8.31468 q^{92} -0.149682 q^{93} +5.87757 q^{94} +18.9511 q^{95} -0.0257691 q^{96} +9.65527 q^{97} -1.00000 q^{98} +6.22530 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 25 q^{2} - 4 q^{3} + 25 q^{4} + 4 q^{6} - 25 q^{7} - 25 q^{8} + 25 q^{9}+O(q^{10}) \) 25 * q - 25 * q^2 - 4 * q^3 + 25 * q^4 + 4 * q^6 - 25 * q^7 - 25 * q^8 + 25 * q^9	\( 25 q - 25 q^{2} - 4 q^{3} + 25 q^{4} + 4 q^{6} - 25 q^{7} - 25 q^{8} + 25 q^{9} - 13 q^{11} - 4 q^{12} + 17 q^{13} + 25 q^{14} - 18 q^{15} + 25 q^{16} - 4 q^{17} - 25 q^{18} - 9 q^{19} + 4 q^{21} + 13 q^{22} - 14 q^{23} + 4 q^{24} + 23 q^{25} - 17 q^{26} - 7 q^{27} - 25 q^{28} - 4 q^{29} + 18 q^{30} - 15 q^{31} - 25 q^{32} - 15 q^{33} + 4 q^{34} + 25 q^{36} + 13 q^{37} + 9 q^{38} - 31 q^{39} - 31 q^{41} - 4 q^{42} + 29 q^{43} - 13 q^{44} + 10 q^{45} + 14 q^{46} - 31 q^{47} - 4 q^{48} + 25 q^{49} - 23 q^{50} - 9 q^{51} + 17 q^{52} + 23 q^{53} + 7 q^{54} - 48 q^{55} + 25 q^{56} + 32 q^{57} + 4 q^{58} - 50 q^{59} - 18 q^{60} - 2 q^{61} + 15 q^{62} - 25 q^{63} + 25 q^{64} - 4 q^{65} + 15 q^{66} - 8 q^{67} - 4 q^{68} - 57 q^{69} - 61 q^{71} - 25 q^{72} + 31 q^{73} - 13 q^{74} - 21 q^{75} - 9 q^{76} + 13 q^{77} + 31 q^{78} - 10 q^{79} + 61 q^{81} + 31 q^{82} - 47 q^{83} + 4 q^{84} + 2 q^{85} - 29 q^{86} + 17 q^{87} + 13 q^{88} - 44 q^{89} - 10 q^{90} - 17 q^{91} - 14 q^{92} - 13 q^{93} + 31 q^{94} - 7 q^{95} + 4 q^{96} + 10 q^{97} - 25 q^{98} - 47 q^{99}+O(q^{100}) \) 25 * q - 25 * q^2 - 4 * q^3 + 25 * q^4 + 4 * q^6 - 25 * q^7 - 25 * q^8 + 25 * q^9 - 13 * q^11 - 4 * q^12 + 17 * q^13 + 25 * q^14 - 18 * q^15 + 25 * q^16 - 4 * q^17 - 25 * q^18 - 9 * q^19 + 4 * q^21 + 13 * q^22 - 14 * q^23 + 4 * q^24 + 23 * q^25 - 17 * q^26 - 7 * q^27 - 25 * q^28 - 4 * q^29 + 18 * q^30 - 15 * q^31 - 25 * q^32 - 15 * q^33 + 4 * q^34 + 25 * q^36 + 13 * q^37 + 9 * q^38 - 31 * q^39 - 31 * q^41 - 4 * q^42 + 29 * q^43 - 13 * q^44 + 10 * q^45 + 14 * q^46 - 31 * q^47 - 4 * q^48 + 25 * q^49 - 23 * q^50 - 9 * q^51 + 17 * q^52 + 23 * q^53 + 7 * q^54 - 48 * q^55 + 25 * q^56 + 32 * q^57 + 4 * q^58 - 50 * q^59 - 18 * q^60 - 2 * q^61 + 15 * q^62 - 25 * q^63 + 25 * q^64 - 4 * q^65 + 15 * q^66 - 8 * q^67 - 4 * q^68 - 57 * q^69 - 61 * q^71 - 25 * q^72 + 31 * q^73 - 13 * q^74 - 21 * q^75 - 9 * q^76 + 13 * q^77 + 31 * q^78 - 10 * q^79 + 61 * q^81 + 31 * q^82 - 47 * q^83 + 4 * q^84 + 2 * q^85 - 29 * q^86 + 17 * q^87 + 13 * q^88 - 44 * q^89 - 10 * q^90 - 17 * q^91 - 14 * q^92 - 13 * q^93 + 31 * q^94 - 7 * q^95 + 4 * q^96 + 10 * q^97 - 25 * q^98 - 47 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more