LMFDB - Embedded newform 4002.2.a.bg.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4002,2,Mod(1,4002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4002, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4002 = 2 \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4002.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:	$31.9561308889$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 13x^{5} + 16x^{4} + 19x^{3} - 8x^{2} - 10x - 2 $ x^7 - x^6 - 13x^5 + 16x^4 + 19x^3 - 8x^2 - 10*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.571135$ of defining polynomial
Character	$\chi$	$=$	4002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.26164 q^{5} +1.00000 q^{6} -5.18548 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.26164 q^{5} +1.00000 q^{6} -5.18548 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.26164 q^{10} +1.67849 q^{11} -1.00000 q^{12} +3.65861 q^{13} +5.18548 q^{14} -2.26164 q^{15} +1.00000 q^{16} -3.94013 q^{17} -1.00000 q^{18} -2.02647 q^{19} +2.26164 q^{20} +5.18548 q^{21} -1.67849 q^{22} +1.00000 q^{23} +1.00000 q^{24} +0.115013 q^{25} -3.65861 q^{26} -1.00000 q^{27} -5.18548 q^{28} +1.00000 q^{29} +2.26164 q^{30} +10.1123 q^{31} -1.00000 q^{32} -1.67849 q^{33} +3.94013 q^{34} -11.7277 q^{35} +1.00000 q^{36} -0.341829 q^{37} +2.02647 q^{38} -3.65861 q^{39} -2.26164 q^{40} -6.99527 q^{41} -5.18548 q^{42} -8.58420 q^{43} +1.67849 q^{44} +2.26164 q^{45} -1.00000 q^{46} -8.56649 q^{47} -1.00000 q^{48} +19.8892 q^{49} -0.115013 q^{50} +3.94013 q^{51} +3.65861 q^{52} +10.6734 q^{53} +1.00000 q^{54} +3.79614 q^{55} +5.18548 q^{56} +2.02647 q^{57} -1.00000 q^{58} -0.919230 q^{59} -2.26164 q^{60} +10.0494 q^{61} -10.1123 q^{62} -5.18548 q^{63} +1.00000 q^{64} +8.27446 q^{65} +1.67849 q^{66} +8.06080 q^{67} -3.94013 q^{68} -1.00000 q^{69} +11.7277 q^{70} -5.56591 q^{71} -1.00000 q^{72} -5.95173 q^{73} +0.341829 q^{74} -0.115013 q^{75} -2.02647 q^{76} -8.70376 q^{77} +3.65861 q^{78} -12.7138 q^{79} +2.26164 q^{80} +1.00000 q^{81} +6.99527 q^{82} -3.91365 q^{83} +5.18548 q^{84} -8.91115 q^{85} +8.58420 q^{86} -1.00000 q^{87} -1.67849 q^{88} +6.19047 q^{89} -2.26164 q^{90} -18.9716 q^{91} +1.00000 q^{92} -10.1123 q^{93} +8.56649 q^{94} -4.58315 q^{95} +1.00000 q^{96} -11.7633 q^{97} -19.8892 q^{98} +1.67849 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{2} - 7 q^{3} + 7 q^{4} + 2 q^{5} + 7 q^{6} - 5 q^{7} - 7 q^{8} + 7 q^{9}+O(q^{10}) $ 7 * q - 7 * q^2 - 7 * q^3 + 7 * q^4 + 2 * q^5 + 7 * q^6 - 5 * q^7 - 7 * q^8 + 7 * q^9	\( 7 q - 7 q^{2} - 7 q^{3} + 7 q^{4} + 2 q^{5} + 7 q^{6} - 5 q^{7} - 7 q^{8} + 7 q^{9} - 2 q^{10} - q^{11} - 7 q^{12} + q^{13} + 5 q^{14} - 2 q^{15} + 7 q^{16} - q^{17} - 7 q^{18} - 9 q^{19} + 2 q^{20} + 5 q^{21} + q^{22} + 7 q^{23} + 7 q^{24} + q^{25} - q^{26} - 7 q^{27} - 5 q^{28} + 7 q^{29} + 2 q^{30} - 19 q^{31} - 7 q^{32} + q^{33} + q^{34} + 11 q^{35} + 7 q^{36} - 18 q^{37} + 9 q^{38} - q^{39} - 2 q^{40} + 4 q^{41} - 5 q^{42} - 9 q^{43} - q^{44} + 2 q^{45} - 7 q^{46} - 11 q^{47} - 7 q^{48} + 12 q^{49} - q^{50} + q^{51} + q^{52} + 8 q^{53} + 7 q^{54} - 11 q^{55} + 5 q^{56} + 9 q^{57} - 7 q^{58} + 12 q^{59} - 2 q^{60} - 5 q^{61} + 19 q^{62} - 5 q^{63} + 7 q^{64} + 23 q^{65} - q^{66} - 13 q^{67} - q^{68} - 7 q^{69} - 11 q^{70} - q^{71} - 7 q^{72} - 26 q^{73} + 18 q^{74} - q^{75} - 9 q^{76} + 6 q^{77} + q^{78} - 38 q^{79} + 2 q^{80} + 7 q^{81} - 4 q^{82} - 6 q^{83} + 5 q^{84} - 25 q^{85} + 9 q^{86} - 7 q^{87} + q^{88} + 20 q^{89} - 2 q^{90} + 2 q^{91} + 7 q^{92} + 19 q^{93} + 11 q^{94} - 31 q^{95} + 7 q^{96} - 8 q^{97} - 12 q^{98} - q^{99}+O(q^{100}) \) 7 * q - 7 * q^2 - 7 * q^3 + 7 * q^4 + 2 * q^5 + 7 * q^6 - 5 * q^7 - 7 * q^8 + 7 * q^9 - 2 * q^10 - q^11 - 7 * q^12 + q^13 + 5 * q^14 - 2 * q^15 + 7 * q^16 - q^17 - 7 * q^18 - 9 * q^19 + 2 * q^20 + 5 * q^21 + q^22 + 7 * q^23 + 7 * q^24 + q^25 - q^26 - 7 * q^27 - 5 * q^28 + 7 * q^29 + 2 * q^30 - 19 * q^31 - 7 * q^32 + q^33 + q^34 + 11 * q^35 + 7 * q^36 - 18 * q^37 + 9 * q^38 - q^39 - 2 * q^40 + 4 * q^41 - 5 * q^42 - 9 * q^43 - q^44 + 2 * q^45 - 7 * q^46 - 11 * q^47 - 7 * q^48 + 12 * q^49 - q^50 + q^51 + q^52 + 8 * q^53 + 7 * q^54 - 11 * q^55 + 5 * q^56 + 9 * q^57 - 7 * q^58 + 12 * q^59 - 2 * q^60 - 5 * q^61 + 19 * q^62 - 5 * q^63 + 7 * q^64 + 23 * q^65 - q^66 - 13 * q^67 - q^68 - 7 * q^69 - 11 * q^70 - q^71 - 7 * q^72 - 26 * q^73 + 18 * q^74 - q^75 - 9 * q^76 + 6 * q^77 + q^78 - 38 * q^79 + 2 * q^80 + 7 * q^81 - 4 * q^82 - 6 * q^83 + 5 * q^84 - 25 * q^85 + 9 * q^86 - 7 * q^87 + q^88 + 20 * q^89 - 2 * q^90 + 2 * q^91 + 7 * q^92 + 19 * q^93 + 11 * q^94 - 31 * q^95 + 7 * q^96 - 8 * q^97 - 12 * q^98 - 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	2.26164	1.01144	0.505718	−	0.862699i	$-0.331228\pi$
$5$	2.26164	1.01144	0.505718	+	0.862699i	$0.331228\pi$
$6$	1.00000	0.408248
$7$	−5.18548	−1.95993	−0.979963	−	0.199179i	$-0.936172\pi$
$7$	−5.18548	−1.95993	−0.979963	+	0.199179i	$0.936172\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−2.26164	−0.715193
$11$	1.67849	0.506083	0.253042	−	0.967455i	$-0.418569\pi$
$11$	1.67849	0.506083	0.253042	+	0.967455i	$0.418569\pi$
$12$	−1.00000	−0.288675
$13$	3.65861	1.01472	0.507358	−	0.861735i	$-0.330622\pi$
$13$	3.65861	1.01472	0.507358	+	0.861735i	$0.330622\pi$
$14$	5.18548	1.38588
$15$	−2.26164	−0.583953
$16$	1.00000	0.250000
$17$	−3.94013	−0.955621	−0.477811	−	0.878463i	$-0.658570\pi$
$17$	−3.94013	−0.955621	−0.477811	+	0.878463i	$0.658570\pi$
$18$	−1.00000	−0.235702
$19$	−2.02647	−0.464905	−0.232452	−	0.972608i	$-0.574675\pi$
$19$	−2.02647	−0.464905	−0.232452	+	0.972608i	$0.574675\pi$
$20$	2.26164	0.505718
$21$	5.18548	1.13156
$22$	−1.67849	−0.357855
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	1.00000	0.204124
$25$	0.115013	0.0230027
$26$	−3.65861	−0.717512
$27$	−1.00000	−0.192450
$28$	−5.18548	−0.979963
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	2.26164	0.412917
$31$	10.1123	1.81622	0.908109	−	0.418733i	$-0.137526\pi$
$31$	10.1123	1.81622	0.908109	+	0.418733i	$0.137526\pi$
$32$	−1.00000	−0.176777
$33$	−1.67849	−0.292187
$34$	3.94013	0.675726
$35$	−11.7277	−1.98234
$36$	1.00000	0.166667
$37$	−0.341829	−0.0561963	−0.0280982	−	0.999605i	$-0.508945\pi$
$37$	−0.341829	−0.0561963	−0.0280982	+	0.999605i	$0.508945\pi$
$38$	2.02647	0.328737
$39$	−3.65861	−0.585846
$40$	−2.26164	−0.357597
$41$	−6.99527	−1.09248	−0.546239	−	0.837630i	$-0.683941\pi$
$41$	−6.99527	−1.09248	−0.546239	+	0.837630i	$0.683941\pi$
$42$	−5.18548	−0.800136
$43$	−8.58420	−1.30908	−0.654539	−	0.756028i	$-0.727137\pi$
$43$	−8.58420	−1.30908	−0.654539	+	0.756028i	$0.727137\pi$
$44$	1.67849	0.253042
$45$	2.26164	0.337145
$46$	−1.00000	−0.147442
$47$	−8.56649	−1.24955	−0.624775	−	0.780805i	$-0.714809\pi$
$47$	−8.56649	−1.24955	−0.624775	+	0.780805i	$0.714809\pi$
$48$	−1.00000	−0.144338
$49$	19.8892	2.84131
$50$	−0.115013	−0.0162653
$51$	3.94013	0.551728
$52$	3.65861	0.507358
$53$	10.6734	1.46610	0.733050	−	0.680175i	$-0.238096\pi$
$53$	10.6734	1.46610	0.733050	+	0.680175i	$0.238096\pi$
$54$	1.00000	0.136083
$55$	3.79614	0.511871
$56$	5.18548	0.692939
$57$	2.02647	0.268413
$58$	−1.00000	−0.131306
$59$	−0.919230	−0.119674	−0.0598368	−	0.998208i	$-0.519058\pi$
$59$	−0.919230	−0.119674	−0.0598368	+	0.998208i	$0.519058\pi$
$60$	−2.26164	−0.291976
$61$	10.0494	1.28670	0.643350	−	0.765573i	$-0.277544\pi$
$61$	10.0494	1.28670	0.643350	+	0.765573i	$0.277544\pi$
$62$	−10.1123	−1.28426
$63$	−5.18548	−0.653309
$64$	1.00000	0.125000
$65$	8.27446	1.02632
$66$	1.67849	0.206608
$67$	8.06080	0.984783	0.492391	−	0.870374i	$-0.336123\pi$
$67$	8.06080	0.984783	0.492391	+	0.870374i	$0.336123\pi$
$68$	−3.94013	−0.477811
$69$	−1.00000	−0.120386
$70$	11.7277	1.40173
$71$	−5.56591	−0.660552	−0.330276	−	0.943884i	$-0.607142\pi$
$71$	−5.56591	−0.660552	−0.330276	+	0.943884i	$0.607142\pi$
$72$	−1.00000	−0.117851
$73$	−5.95173	−0.696597	−0.348299	−	0.937384i	$-0.613241\pi$
$73$	−5.95173	−0.696597	−0.348299	+	0.937384i	$0.613241\pi$
$74$	0.341829	0.0397368
$75$	−0.115013	−0.0132806
$76$	−2.02647	−0.232452
$77$	−8.70376	−0.991886
$78$	3.65861	0.414256
$79$	−12.7138	−1.43041	−0.715204	−	0.698915i	$-0.753666\pi$
$79$	−12.7138	−1.43041	−0.715204	+	0.698915i	$0.753666\pi$
$80$	2.26164	0.252859
$81$	1.00000	0.111111
$82$	6.99527	0.772498
$83$	−3.91365	−0.429579	−0.214790	−	0.976660i	$-0.568907\pi$
$83$	−3.91365	−0.429579	−0.214790	+	0.976660i	$0.568907\pi$
$84$	5.18548	0.565782
$85$	−8.91115	−0.966550
$86$	8.58420	0.925658
$87$	−1.00000	−0.107211
$88$	−1.67849	−0.178927
$89$	6.19047	0.656189	0.328094	−	0.944645i	$-0.393594\pi$
$89$	6.19047	0.656189	0.328094	+	0.944645i	$0.393594\pi$
$90$	−2.26164	−0.238398
$91$	−18.9716	−1.98877
$92$	1.00000	0.104257
$93$	−10.1123	−1.04859
$94$	8.56649	0.883566
$95$	−4.58315	−0.470221
$96$	1.00000	0.102062
$97$	−11.7633	−1.19438	−0.597192	−	0.802098i	$-0.703717\pi$
$97$	−11.7633	−1.19438	−0.597192	+	0.802098i	$0.703717\pi$
$98$	−19.8892	−2.00911
$99$	1.67849	0.168694
$100$	0.115013	0.0115013
$101$	3.59695	0.357910	0.178955	−	0.983857i	$-0.442728\pi$
$101$	3.59695	0.357910	0.178955	+	0.983857i	$0.442728\pi$
$102$	−3.94013	−0.390131
$103$	8.17997	0.805997	0.402998	−	0.915201i	$-0.367968\pi$
$103$	8.17997	0.805997	0.402998	+	0.915201i	$0.367968\pi$
$104$	−3.65861	−0.358756
$105$	11.7277	1.14450
$106$	−10.6734	−1.03669
$107$	−3.70502	−0.358178	−0.179089	−	0.983833i	$-0.557315\pi$
$107$	−3.70502	−0.358178	−0.179089	+	0.983833i	$0.557315\pi$
$108$	−1.00000	−0.0962250
$109$	−16.5982	−1.58982	−0.794911	−	0.606726i	$-0.792482\pi$
$109$	−16.5982	−1.58982	−0.794911	+	0.606726i	$0.792482\pi$
$110$	−3.79614	−0.361947
$111$	0.341829	0.0324450
$112$	−5.18548	−0.489982
$113$	−3.48554	−0.327892	−0.163946	−	0.986469i	$-0.552422\pi$
$113$	−3.48554	−0.327892	−0.163946	+	0.986469i	$0.552422\pi$
$114$	−2.02647	−0.189797
$115$	2.26164	0.210899
$116$	1.00000	0.0928477
$117$	3.65861	0.338239
$118$	0.919230	0.0846220
$119$	20.4314	1.87295
$120$	2.26164	0.206458
$121$	−8.18268	−0.743880
$122$	−10.0494	−0.909834
$123$	6.99527	0.630742
$124$	10.1123	0.908109
$125$	−11.0481	−0.988170
$126$	5.18548	0.461959
$127$	−8.80505	−0.781322	−0.390661	−	0.920535i	$-0.627753\pi$
$127$	−8.80505	−0.781322	−0.390661	+	0.920535i	$0.627753\pi$
$128$	−1.00000	−0.0883883
$129$	8.58420	0.755796
$130$	−8.27446	−0.725718
$131$	12.1320	1.05998	0.529989	−	0.848004i	$-0.322196\pi$
$131$	12.1320	1.05998	0.529989	+	0.848004i	$0.322196\pi$
$132$	−1.67849	−0.146094
$133$	10.5082	0.911179
$134$	−8.06080	−0.696347
$135$	−2.26164	−0.194651
$136$	3.94013	0.337863
$137$	9.66441	0.825686	0.412843	−	0.910802i	$-0.364536\pi$
$137$	9.66441	0.825686	0.412843	+	0.910802i	$0.364536\pi$
$138$	1.00000	0.0851257
$139$	−13.4990	−1.14497	−0.572484	−	0.819916i	$-0.694020\pi$
$139$	−13.4990	−1.14497	−0.572484	+	0.819916i	$0.694020\pi$
$140$	−11.7277	−0.991170
$141$	8.56649	0.721428
$142$	5.56591	0.467080
$143$	6.14093	0.513531
$144$	1.00000	0.0833333
$145$	2.26164	0.187819
$146$	5.95173	0.492569
$147$	−19.8892	−1.64043
$148$	−0.341829	−0.0280982
$149$	14.8484	1.21643	0.608216	−	0.793772i	$-0.291885\pi$
$149$	14.8484	1.21643	0.608216	+	0.793772i	$0.291885\pi$
$150$	0.115013	0.00939080
$151$	−2.65666	−0.216196	−0.108098	−	0.994140i	$-0.534476\pi$
$151$	−2.65666	−0.216196	−0.108098	+	0.994140i	$0.534476\pi$
$152$	2.02647	0.164369
$153$	−3.94013	−0.318540
$154$	8.70376	0.701369
$155$	22.8703	1.83699
$156$	−3.65861	−0.292923
$157$	−18.7191	−1.49395	−0.746974	−	0.664854i	$-0.768494\pi$
$157$	−18.7191	−1.49395	−0.746974	+	0.664854i	$0.768494\pi$
$158$	12.7138	1.01145
$159$	−10.6734	−0.846453
$160$	−2.26164	−0.178798
$161$	−5.18548	−0.408673
$162$	−1.00000	−0.0785674
$163$	−2.31941	−0.181670	−0.0908351	−	0.995866i	$-0.528954\pi$
$163$	−2.31941	−0.181670	−0.0908351	+	0.995866i	$0.528954\pi$
$164$	−6.99527	−0.546239
$165$	−3.79614	−0.295529
$166$	3.91365	0.303759
$167$	−0.957962	−0.0741294	−0.0370647	−	0.999313i	$-0.511801\pi$
$167$	−0.957962	−0.0741294	−0.0370647	+	0.999313i	$0.511801\pi$
$168$	−5.18548	−0.400068
$169$	0.385424	0.0296480
$170$	8.91115	0.683454
$171$	−2.02647	−0.154968
$172$	−8.58420	−0.654539
$173$	−11.5729	−0.879874	−0.439937	−	0.898029i	$-0.644999\pi$
$173$	−11.5729	−0.879874	−0.439937	+	0.898029i	$0.644999\pi$
$174$	1.00000	0.0758098
$175$	−0.596399	−0.0450835
$176$	1.67849	0.126521
$177$	0.919230	0.0690936
$178$	−6.19047	−0.463995
$179$	−10.3390	−0.772776	−0.386388	−	0.922336i	$-0.626277\pi$
$179$	−10.3390	−0.772776	−0.386388	+	0.922336i	$0.626277\pi$
$180$	2.26164	0.168573
$181$	24.7065	1.83642	0.918209	−	0.396097i	$-0.129636\pi$
$181$	24.7065	1.83642	0.918209	+	0.396097i	$0.129636\pi$
$182$	18.9716	1.40627
$183$	−10.0494	−0.742876
$184$	−1.00000	−0.0737210
$185$	−0.773094	−0.0568390
$186$	10.1123	0.741468
$187$	−6.61346	−0.483624
$188$	−8.56649	−0.624775
$189$	5.18548	0.377188
$190$	4.58315	0.332497
$191$	12.1868	0.881808	0.440904	−	0.897554i	$-0.354658\pi$
$191$	12.1868	0.881808	0.440904	+	0.897554i	$0.354658\pi$
$192$	−1.00000	−0.0721688
$193$	7.91470	0.569713	0.284856	−	0.958570i	$-0.408054\pi$
$193$	7.91470	0.569713	0.284856	+	0.958570i	$0.408054\pi$
$194$	11.7633	0.844557
$195$	−8.27446	−0.592546
$196$	19.8892	1.42066
$197$	1.08793	0.0775118	0.0387559	−	0.999249i	$-0.487661\pi$
$197$	1.08793	0.0775118	0.0387559	+	0.999249i	$0.487661\pi$
$198$	−1.67849	−0.119285
$199$	−16.4282	−1.16456	−0.582281	−	0.812988i	$-0.697840\pi$
$199$	−16.4282	−1.16456	−0.582281	+	0.812988i	$0.697840\pi$
$200$	−0.115013	−0.00813267
$201$	−8.06080	−0.568565
$202$	−3.59695	−0.253081
$203$	−5.18548	−0.363949
$204$	3.94013	0.275864
$205$	−15.8208	−1.10497
$206$	−8.17997	−0.569926
$207$	1.00000	0.0695048
$208$	3.65861	0.253679
$209$	−3.40141	−0.235281
$210$	−11.7277	−0.809287
$211$	−18.3924	−1.26618	−0.633092	−	0.774076i	$-0.718215\pi$
$211$	−18.3924	−1.26618	−0.633092	+	0.774076i	$0.718215\pi$
$212$	10.6734	0.733050
$213$	5.56591	0.381370
$214$	3.70502	0.253270
$215$	−19.4144	−1.32405
$216$	1.00000	0.0680414
$217$	−52.4370	−3.55965
$218$	16.5982	1.12417
$219$	5.95173	0.402181
$220$	3.79614	0.255935
$221$	−14.4154	−0.969684
$222$	−0.341829	−0.0229421
$223$	−7.91788	−0.530220	−0.265110	−	0.964218i	$-0.585408\pi$
$223$	−7.91788	−0.530220	−0.265110	+	0.964218i	$0.585408\pi$
$224$	5.18548	0.346469
$225$	0.115013	0.00766756
$226$	3.48554	0.231855
$227$	−25.9727	−1.72387	−0.861935	−	0.507018i	$-0.830748\pi$
$227$	−25.9727	−1.72387	−0.861935	+	0.507018i	$0.830748\pi$
$228$	2.02647	0.134206
$229$	23.9829	1.58483	0.792417	−	0.609980i	$-0.208822\pi$
$229$	23.9829	1.58483	0.792417	+	0.609980i	$0.208822\pi$
$230$	−2.26164	−0.149128
$231$	8.70376	0.572665
$232$	−1.00000	−0.0656532
$233$	−2.23951	−0.146715	−0.0733577	−	0.997306i	$-0.523371\pi$
$233$	−2.23951	−0.146715	−0.0733577	+	0.997306i	$0.523371\pi$
$234$	−3.65861	−0.239171
$235$	−19.3743	−1.26384
$236$	−0.919230	−0.0598368
$237$	12.7138	0.825847
$238$	−20.4314	−1.32437
$239$	15.6532	1.01252	0.506260	−	0.862381i	$-0.331028\pi$
$239$	15.6532	1.01252	0.506260	+	0.862381i	$0.331028\pi$
$240$	−2.26164	−0.145988
$241$	3.74945	0.241524	0.120762	−	0.992682i	$-0.461466\pi$
$241$	3.74945	0.241524	0.120762	+	0.992682i	$0.461466\pi$
$242$	8.18268	0.526002
$243$	−1.00000	−0.0641500
$244$	10.0494	0.643350
$245$	44.9821	2.87380
$246$	−6.99527	−0.446002
$247$	−7.41407	−0.471746
$248$	−10.1123	−0.642130
$249$	3.91365	0.248018
$250$	11.0481	0.698742
$251$	−30.4348	−1.92103	−0.960515	−	0.278229i	$-0.910253\pi$
$251$	−30.4348	−1.92103	−0.960515	+	0.278229i	$0.910253\pi$
$252$	−5.18548	−0.326654
$253$	1.67849	0.105526
$254$	8.80505	0.552478
$255$	8.91115	0.558038
$256$	1.00000	0.0625000
$257$	1.93369	0.120620	0.0603101	−	0.998180i	$-0.480791\pi$
$257$	1.93369	0.120620	0.0603101	+	0.998180i	$0.480791\pi$
$258$	−8.58420	−0.534429
$259$	1.77255	0.110141
$260$	8.27446	0.513160
$261$	1.00000	0.0618984
$262$	−12.1320	−0.749518
$263$	27.5921	1.70140	0.850700	−	0.525651i	$-0.176178\pi$
$263$	27.5921	1.70140	0.850700	+	0.525651i	$0.176178\pi$
$264$	1.67849	0.103304
$265$	24.1393	1.48287
$266$	−10.5082	−0.644301
$267$	−6.19047	−0.378851
$268$	8.06080	0.492391
$269$	0.980263	0.0597677	0.0298838	−	0.999553i	$-0.490486\pi$
$269$	0.980263	0.0597677	0.0298838	+	0.999553i	$0.490486\pi$
$270$	2.26164	0.137639
$271$	−23.1859	−1.40844	−0.704221	−	0.709981i	$-0.748704\pi$
$271$	−23.1859	−1.40844	−0.704221	+	0.709981i	$0.748704\pi$
$272$	−3.94013	−0.238905
$273$	18.9716	1.14822
$274$	−9.66441	−0.583848
$275$	0.193049	0.0116413
$276$	−1.00000	−0.0601929
$277$	3.29919	0.198229	0.0991146	−	0.995076i	$-0.468399\pi$
$277$	3.29919	0.198229	0.0991146	+	0.995076i	$0.468399\pi$
$278$	13.4990	0.809615
$279$	10.1123	0.605406
$280$	11.7277	0.700863
$281$	14.9206	0.890090	0.445045	−	0.895508i	$-0.353188\pi$
$281$	14.9206	0.890090	0.445045	+	0.895508i	$0.353188\pi$
$282$	−8.56649	−0.510127
$283$	−7.35703	−0.437330	−0.218665	−	0.975800i	$-0.570170\pi$
$283$	−7.35703	−0.437330	−0.218665	+	0.975800i	$0.570170\pi$
$284$	−5.56591	−0.330276
$285$	4.58315	0.271482
$286$	−6.14093	−0.363121
$287$	36.2738	2.14118
$288$	−1.00000	−0.0589256
$289$	−1.47539	−0.0867879
$290$	−2.26164	−0.132808
$291$	11.7633	0.689578
$292$	−5.95173	−0.348299
$293$	−15.0931	−0.881746	−0.440873	−	0.897569i	$-0.645331\pi$
$293$	−15.0931	−0.881746	−0.440873	+	0.897569i	$0.645331\pi$
$294$	19.8892	1.15996
$295$	−2.07897	−0.121042
$296$	0.341829	0.0198684
$297$	−1.67849	−0.0973958
$298$	−14.8484	−0.860147
$299$	3.65861	0.211583
$300$	−0.115013	−0.00664030
$301$	44.5132	2.56570
$302$	2.65666	0.152874
$303$	−3.59695	−0.206639
$304$	−2.02647	−0.116226
$305$	22.7282	1.30141
$306$	3.94013	0.225242
$307$	−26.6685	−1.52205	−0.761027	−	0.648721i	$-0.775304\pi$
$307$	−26.6685	−1.52205	−0.761027	+	0.648721i	$0.775304\pi$
$308$	−8.70376	−0.495943
$309$	−8.17997	−0.465342
$310$	−22.8703	−1.29895
$311$	−23.3847	−1.32602	−0.663012	−	0.748609i	$-0.730722\pi$
$311$	−23.3847	−1.32602	−0.663012	+	0.748609i	$0.730722\pi$
$312$	3.65861	0.207128
$313$	3.76959	0.213070	0.106535	−	0.994309i	$-0.466024\pi$
$313$	3.76959	0.213070	0.106535	+	0.994309i	$0.466024\pi$
$314$	18.7191	1.05638
$315$	−11.7277	−0.660780
$316$	−12.7138	−0.715204
$317$	−29.0727	−1.63288	−0.816442	−	0.577427i	$-0.804057\pi$
$317$	−29.0727	−1.63288	−0.816442	+	0.577427i	$0.804057\pi$
$318$	10.6734	0.598533
$319$	1.67849	0.0939773
$320$	2.26164	0.126429
$321$	3.70502	0.206794
$322$	5.18548	0.288975
$323$	7.98456	0.444273
$324$	1.00000	0.0555556
$325$	0.420789	0.0233412
$326$	2.31941	0.128460
$327$	16.5982	0.917884
$328$	6.99527	0.386249
$329$	44.4213	2.44903
$330$	3.79614	0.208970
$331$	−33.3842	−1.83496	−0.917482	−	0.397777i	$-0.869782\pi$
$331$	−33.3842	−1.83496	−0.917482	+	0.397777i	$0.869782\pi$
$332$	−3.91365	−0.214790
$333$	−0.341829	−0.0187321
$334$	0.957962	0.0524174
$335$	18.2306	0.996045
$336$	5.18548	0.282891
$337$	18.7050	1.01893	0.509463	−	0.860492i	$-0.329844\pi$
$337$	18.7050	1.01893	0.509463	+	0.860492i	$0.329844\pi$
$338$	−0.385424	−0.0209643
$339$	3.48554	0.189308
$340$	−8.91115	−0.483275
$341$	16.9733	0.919158
$342$	2.02647	0.109579
$343$	−66.8365	−3.60883
$344$	8.58420	0.462829
$345$	−2.26164	−0.121763
$346$	11.5729	0.622165
$347$	−30.8979	−1.65869	−0.829343	−	0.558740i	$-0.811285\pi$
$347$	−30.8979	−1.65869	−0.829343	+	0.558740i	$0.811285\pi$
$348$	−1.00000	−0.0536056
$349$	−5.57364	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$349$	−5.57364	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$350$	0.596399	0.0318789
$351$	−3.65861	−0.195282
$352$	−1.67849	−0.0894637
$353$	−17.1035	−0.910329	−0.455164	−	0.890407i	$-0.650419\pi$
$353$	−17.1035	−0.910329	−0.455164	+	0.890407i	$0.650419\pi$
$354$	−0.919230	−0.0488565
$355$	−12.5881	−0.668106
$356$	6.19047	0.328094
$357$	−20.4314	−1.08135
$358$	10.3390	0.546435
$359$	−2.26293	−0.119433	−0.0597164	−	0.998215i	$-0.519020\pi$
$359$	−2.26293	−0.119433	−0.0597164	+	0.998215i	$0.519020\pi$
$360$	−2.26164	−0.119199
$361$	−14.8934	−0.783864
$362$	−24.7065	−1.29854
$363$	8.18268	0.429479
$364$	−18.9716	−0.994384
$365$	−13.4607	−0.704564
$366$	10.0494	0.525293
$367$	13.2727	0.692831	0.346416	−	0.938081i	$-0.387399\pi$
$367$	13.2727	0.692831	0.346416	+	0.938081i	$0.387399\pi$
$368$	1.00000	0.0521286
$369$	−6.99527	−0.364159
$370$	0.773094	0.0401912
$371$	−55.3465	−2.87345
$372$	−10.1123	−0.524297
$373$	0.0831012	0.00430282	0.00215141	−	0.999998i	$-0.499315\pi$
$373$	0.0831012	0.00430282	0.00215141	+	0.999998i	$0.499315\pi$
$374$	6.61346	0.341974
$375$	11.0481	0.570520
$376$	8.56649	0.441783
$377$	3.65861	0.188428
$378$	−5.18548	−0.266712
$379$	−7.42482	−0.381387	−0.190694	−	0.981650i	$-0.561074\pi$
$379$	−7.42482	−0.381387	−0.190694	+	0.981650i	$0.561074\pi$
$380$	−4.58315	−0.235111
$381$	8.80505	0.451096
$382$	−12.1868	−0.623533
$383$	20.7513	1.06034	0.530171	−	0.847891i	$-0.322128\pi$
$383$	20.7513	1.06034	0.530171	+	0.847891i	$0.322128\pi$
$384$	1.00000	0.0510310
$385$	−19.6848	−1.00323
$386$	−7.91470	−0.402848
$387$	−8.58420	−0.436359
$388$	−11.7633	−0.597192
$389$	−36.3610	−1.84358	−0.921788	−	0.387695i	$-0.873271\pi$
$389$	−36.3610	−1.84358	−0.921788	+	0.387695i	$0.873271\pi$
$390$	8.27446	0.418993
$391$	−3.94013	−0.199261
$392$	−19.8892	−1.00455
$393$	−12.1320	−0.611979
$394$	−1.08793	−0.0548091
$395$	−28.7539	−1.44677
$396$	1.67849	0.0843472
$397$	2.50661	0.125803	0.0629017	−	0.998020i	$-0.479965\pi$
$397$	2.50661	0.125803	0.0629017	+	0.998020i	$0.479965\pi$
$398$	16.4282	0.823469
$399$	−10.5082	−0.526069
$400$	0.115013	0.00575067
$401$	−26.5916	−1.32792	−0.663961	−	0.747767i	$-0.731126\pi$
$401$	−26.5916	−1.32792	−0.663961	+	0.747767i	$0.731126\pi$
$402$	8.06080	0.402036
$403$	36.9969	1.84295
$404$	3.59695	0.178955
$405$	2.26164	0.112382
$406$	5.18548	0.257351
$407$	−0.573756	−0.0284400
$408$	−3.94013	−0.195065
$409$	−26.1269	−1.29189	−0.645945	−	0.763384i	$-0.723537\pi$
$409$	−26.1269	−1.29189	−0.645945	+	0.763384i	$0.723537\pi$
$410$	15.8208	0.781332
$411$	−9.66441	−0.476710
$412$	8.17997	0.402998
$413$	4.76665	0.234551
$414$	−1.00000	−0.0491473
$415$	−8.85128	−0.434492
$416$	−3.65861	−0.179378
$417$	13.4990	0.661048
$418$	3.40141	0.166368
$419$	34.8966	1.70481	0.852405	−	0.522883i	$-0.175143\pi$
$419$	34.8966	1.70481	0.852405	+	0.522883i	$0.175143\pi$
$420$	11.7277	0.572252
$421$	−19.2371	−0.937560	−0.468780	−	0.883315i	$-0.655306\pi$
$421$	−19.2371	−0.937560	−0.468780	+	0.883315i	$0.655306\pi$
$422$	18.3924	0.895328
$423$	−8.56649	−0.416517
$424$	−10.6734	−0.518345
$425$	−0.453167	−0.0219818
$426$	−5.56591	−0.269669
$427$	−52.1112	−2.52184
$428$	−3.70502	−0.179089
$429$	−6.14093	−0.296487
$430$	19.4144	0.936244
$431$	−4.25421	−0.204918	−0.102459	−	0.994737i	$-0.532671\pi$
$431$	−4.25421	−0.204918	−0.102459	+	0.994737i	$0.532671\pi$
$432$	−1.00000	−0.0481125
$433$	1.85387	0.0890912	0.0445456	−	0.999007i	$-0.485816\pi$
$433$	1.85387	0.0890912	0.0445456	+	0.999007i	$0.485816\pi$
$434$	52.4370	2.51706
$435$	−2.26164	−0.108437
$436$	−16.5982	−0.794911
$437$	−2.02647	−0.0969393
$438$	−5.95173	−0.284385
$439$	−13.0700	−0.623797	−0.311898	−	0.950115i	$-0.600965\pi$
$439$	−13.0700	−0.623797	−0.311898	+	0.950115i	$0.600965\pi$
$440$	−3.79614	−0.180974
$441$	19.8892	0.947103
$442$	14.4154	0.685670
$443$	16.2061	0.769977	0.384988	−	0.922921i	$-0.374205\pi$
$443$	16.2061	0.769977	0.384988	+	0.922921i	$0.374205\pi$
$444$	0.341829	0.0162225
$445$	14.0006	0.663693
$446$	7.91788	0.374922
$447$	−14.8484	−0.702307
$448$	−5.18548	−0.244991
$449$	−0.298743	−0.0140985	−0.00704927	−	0.999975i	$-0.502244\pi$
$449$	−0.298743	−0.0140985	−0.00704927	+	0.999975i	$0.502244\pi$
$450$	−0.115013	−0.00542178
$451$	−11.7415	−0.552884
$452$	−3.48554	−0.163946
$453$	2.65666	0.124821
$454$	25.9727	1.21896
$455$	−42.9070	−2.01151
$456$	−2.02647	−0.0948983
$457$	−6.82339	−0.319185	−0.159592	−	0.987183i	$-0.551018\pi$
$457$	−6.82339	−0.319185	−0.159592	+	0.987183i	$0.551018\pi$
$458$	−23.9829	−1.12065
$459$	3.94013	0.183909
$460$	2.26164	0.105449
$461$	2.68303	0.124961	0.0624805	−	0.998046i	$-0.480099\pi$
$461$	2.68303	0.124961	0.0624805	+	0.998046i	$0.480099\pi$
$462$	−8.70376	−0.404936
$463$	−8.11062	−0.376932	−0.188466	−	0.982080i	$-0.560352\pi$
$463$	−8.11062	−0.376932	−0.188466	+	0.982080i	$0.560352\pi$
$464$	1.00000	0.0464238
$465$	−22.8703	−1.06059
$466$	2.23951	0.103744
$467$	−27.5000	−1.27255	−0.636275	−	0.771462i	$-0.719526\pi$
$467$	−27.5000	−1.27255	−0.636275	+	0.771462i	$0.719526\pi$
$468$	3.65861	0.169119
$469$	−41.7991	−1.93010
$470$	19.3743	0.893670
$471$	18.7191	0.862531
$472$	0.919230	0.0423110
$473$	−14.4085	−0.662502
$474$	−12.7138	−0.583962
$475$	−0.233071	−0.0106941
$476$	20.4314	0.936474
$477$	10.6734	0.488700
$478$	−15.6532	−0.715960
$479$	24.4909	1.11902	0.559508	−	0.828825i	$-0.310990\pi$
$479$	24.4909	1.11902	0.559508	+	0.828825i	$0.310990\pi$
$480$	2.26164	0.103229
$481$	−1.25062	−0.0570233
$482$	−3.74945	−0.170783
$483$	5.18548	0.235947
$484$	−8.18268	−0.371940
$485$	−26.6044	−1.20804
$486$	1.00000	0.0453609
$487$	−19.6246	−0.889276	−0.444638	−	0.895710i	$-0.646668\pi$
$487$	−19.6246	−0.889276	−0.444638	+	0.895710i	$0.646668\pi$
$488$	−10.0494	−0.454917
$489$	2.31941	0.104887
$490$	−44.9821	−2.03209
$491$	23.4593	1.05870	0.529351	−	0.848403i	$-0.322436\pi$
$491$	23.4593	1.05870	0.529351	+	0.848403i	$0.322436\pi$
$492$	6.99527	0.315371
$493$	−3.94013	−0.177454
$494$	7.41407	0.333575
$495$	3.79614	0.170624
$496$	10.1123	0.454055
$497$	28.8619	1.29463
$498$	−3.91365	−0.175375
$499$	−17.4292	−0.780237	−0.390118	−	0.920765i	$-0.627566\pi$
$499$	−17.4292	−0.780237	−0.390118	+	0.920765i	$0.627566\pi$
$500$	−11.0481	−0.494085
$501$	0.957962	0.0427986
$502$	30.4348	1.35837
$503$	42.5737	1.89827	0.949133	−	0.314877i	$-0.101963\pi$
$503$	42.5737	1.89827	0.949133	+	0.314877i	$0.101963\pi$
$504$	5.18548	0.230980
$505$	8.13500	0.362003
$506$	−1.67849	−0.0746179
$507$	−0.385424	−0.0171173
$508$	−8.80505	−0.390661
$509$	12.3239	0.546248	0.273124	−	0.961979i	$-0.411943\pi$
$509$	12.3239	0.546248	0.273124	+	0.961979i	$0.411943\pi$
$510$	−8.91115	−0.394592
$511$	30.8626	1.36528
$512$	−1.00000	−0.0441942
$513$	2.02647	0.0894710
$514$	−1.93369	−0.0852913
$515$	18.5001	0.815214
$516$	8.58420	0.377898
$517$	−14.3787	−0.632377
$518$	−1.77255	−0.0778812
$519$	11.5729	0.507995
$520$	−8.27446	−0.362859
$521$	24.1714	1.05897	0.529483	−	0.848320i	$-0.322386\pi$
$521$	24.1714	1.05897	0.529483	+	0.848320i	$0.322386\pi$
$522$	−1.00000	−0.0437688
$523$	−8.50983	−0.372109	−0.186054	−	0.982539i	$-0.559570\pi$
$523$	−8.50983	−0.372109	−0.186054	+	0.982539i	$0.559570\pi$
$524$	12.1320	0.529989
$525$	0.596399	0.0260290
$526$	−27.5921	−1.20307
$527$	−39.8437	−1.73562
$528$	−1.67849	−0.0730468
$529$	1.00000	0.0434783
$530$	−24.1393	−1.04854
$531$	−0.919230	−0.0398912
$532$	10.5082	0.455590
$533$	−25.5930	−1.10855
$534$	6.19047	0.267888
$535$	−8.37942	−0.362274
$536$	−8.06080	−0.348173
$537$	10.3390	0.446162
$538$	−0.980263	−0.0422621
$539$	33.3837	1.43794
$540$	−2.26164	−0.0973255
$541$	−13.7964	−0.593152	−0.296576	−	0.955009i	$-0.595845\pi$
$541$	−13.7964	−0.593152	−0.296576	+	0.955009i	$0.595845\pi$
$542$	23.1859	0.995919
$543$	−24.7065	−1.06026
$544$	3.94013	0.168932
$545$	−37.5392	−1.60800
$546$	−18.9716	−0.811911
$547$	−27.5569	−1.17825	−0.589123	−	0.808043i	$-0.700527\pi$
$547$	−27.5569	−1.17825	−0.589123	+	0.808043i	$0.700527\pi$
$548$	9.66441	0.412843
$549$	10.0494	0.428900
$550$	−0.193049	−0.00823162
$551$	−2.02647	−0.0863307
$552$	1.00000	0.0425628
$553$	65.9269	2.80350
$554$	−3.29919	−0.140169
$555$	0.773094	0.0328160
$556$	−13.4990	−0.572484
$557$	−23.8652	−1.01120	−0.505601	−	0.862768i	$-0.668729\pi$
$557$	−23.8652	−1.01120	−0.505601	+	0.862768i	$0.668729\pi$
$558$	−10.1123	−0.428087
$559$	−31.4062	−1.32834
$560$	−11.7277	−0.495585
$561$	6.61346	0.279220
$562$	−14.9206	−0.629388
$563$	33.8372	1.42607	0.713034	−	0.701129i	$-0.247320\pi$
$563$	33.8372	1.42607	0.713034	+	0.701129i	$0.247320\pi$
$564$	8.56649	0.360714
$565$	−7.88303	−0.331642
$566$	7.35703	0.309239
$567$	−5.18548	−0.217770
$568$	5.56591	0.233540
$569$	−19.3483	−0.811122	−0.405561	−	0.914068i	$-0.632924\pi$
$569$	−19.3483	−0.811122	−0.405561	+	0.914068i	$0.632924\pi$
$570$	−4.58315	−0.191967
$571$	−11.3806	−0.476264	−0.238132	−	0.971233i	$-0.576535\pi$
$571$	−11.3806	−0.476264	−0.238132	+	0.971233i	$0.576535\pi$
$572$	6.14093	0.256765
$573$	−12.1868	−0.509112
$574$	−36.2738	−1.51404
$575$	0.115013	0.00479639
$576$	1.00000	0.0416667
$577$	−20.5766	−0.856614	−0.428307	−	0.903633i	$-0.640890\pi$
$577$	−20.5766	−0.856614	−0.428307	+	0.903633i	$0.640890\pi$
$578$	1.47539	0.0613683
$579$	−7.91470	−0.328924
$580$	2.26164	0.0939095
$581$	20.2942	0.841944
$582$	−11.7633	−0.487605
$583$	17.9151	0.741969
$584$	5.95173	0.246284
$585$	8.27446	0.342107
$586$	15.0931	0.623489
$587$	29.9829	1.23753	0.618764	−	0.785577i	$-0.287634\pi$
$587$	29.9829	1.23753	0.618764	+	0.785577i	$0.287634\pi$
$588$	−19.8892	−0.820216
$589$	−20.4923	−0.844369
$590$	2.07897	0.0855897
$591$	−1.08793	−0.0447515
$592$	−0.341829	−0.0140491
$593$	7.75453	0.318440	0.159220	−	0.987243i	$-0.449102\pi$
$593$	7.75453	0.318440	0.159220	+	0.987243i	$0.449102\pi$
$594$	1.67849	0.0688692
$595$	46.2086	1.89437
$596$	14.8484	0.608216
$597$	16.4282	0.672360
$598$	−3.65861	−0.149612
$599$	−4.01034	−0.163858	−0.0819291	−	0.996638i	$-0.526108\pi$
$599$	−4.01034	−0.163858	−0.0819291	+	0.996638i	$0.526108\pi$
$600$	0.115013	0.00469540
$601$	−29.1456	−1.18888	−0.594438	−	0.804142i	$-0.702625\pi$
$601$	−29.1456	−1.18888	−0.594438	+	0.804142i	$0.702625\pi$
$602$	−44.5132	−1.81422
$603$	8.06080	0.328261
$604$	−2.65666	−0.108098
$605$	−18.5063	−0.752387
$606$	3.59695	0.146116
$607$	−6.54187	−0.265526	−0.132763	−	0.991148i	$-0.542385\pi$
$607$	−6.54187	−0.265526	−0.132763	+	0.991148i	$0.542385\pi$
$608$	2.02647	0.0821843
$609$	5.18548	0.210126
$610$	−22.7282	−0.920239
$611$	−31.3414	−1.26794
$612$	−3.94013	−0.159270
$613$	9.34783	0.377555	0.188778	−	0.982020i	$-0.439547\pi$
$613$	9.34783	0.377555	0.188778	+	0.982020i	$0.439547\pi$
$614$	26.6685	1.07625
$615$	15.8208	0.637955
$616$	8.70376	0.350685
$617$	−13.0888	−0.526936	−0.263468	−	0.964668i	$-0.584866\pi$
$617$	−13.0888	−0.526936	−0.263468	+	0.964668i	$0.584866\pi$
$618$	8.17997	0.329047
$619$	−30.4964	−1.22576	−0.612878	−	0.790178i	$-0.709988\pi$
$619$	−30.4964	−1.22576	−0.612878	+	0.790178i	$0.709988\pi$
$620$	22.8703	0.918494
$621$	−1.00000	−0.0401286
$622$	23.3847	0.937640
$623$	−32.1005	−1.28608
$624$	−3.65861	−0.146462
$625$	−25.5618	−1.02247
$626$	−3.76959	−0.150663
$627$	3.40141	0.135839
$628$	−18.7191	−0.746974
$629$	1.34685	0.0537024
$630$	11.7277	0.467242
$631$	42.8187	1.70458	0.852292	−	0.523067i	$-0.175212\pi$
$631$	42.8187	1.70458	0.852292	+	0.523067i	$0.175212\pi$
$632$	12.7138	0.505726
$633$	18.3924	0.731032
$634$	29.0727	1.15462
$635$	−19.9138	−0.790257
$636$	−10.6734	−0.423227
$637$	72.7667	2.88312
$638$	−1.67849	−0.0664520
$639$	−5.56591	−0.220184
$640$	−2.26164	−0.0893992
$641$	16.5300	0.652897	0.326448	−	0.945215i	$-0.394148\pi$
$641$	16.5300	0.652897	0.326448	+	0.945215i	$0.394148\pi$
$642$	−3.70502	−0.146225
$643$	31.6166	1.24684	0.623419	−	0.781888i	$-0.285743\pi$
$643$	31.6166	1.24684	0.623419	+	0.781888i	$0.285743\pi$
$644$	−5.18548	−0.204336
$645$	19.4144	0.764440
$646$	−7.98456	−0.314148
$647$	−7.66830	−0.301472	−0.150736	−	0.988574i	$-0.548164\pi$
$647$	−7.66830	−0.301472	−0.150736	+	0.988574i	$0.548164\pi$
$648$	−1.00000	−0.0392837
$649$	−1.54292	−0.0605648
$650$	−0.420789	−0.0165047
$651$	52.4370	2.05517
$652$	−2.31941	−0.0908351
$653$	−13.2880	−0.519999	−0.260000	−	0.965609i	$-0.583722\pi$
$653$	−13.2880	−0.519999	−0.260000	+	0.965609i	$0.583722\pi$
$654$	−16.5982	−0.649042
$655$	27.4382	1.07210
$656$	−6.99527	−0.273119
$657$	−5.95173	−0.232199
$658$	−44.4213	−1.73172
$659$	3.17353	0.123623	0.0618116	−	0.998088i	$-0.480312\pi$
$659$	3.17353	0.123623	0.0618116	+	0.998088i	$0.480312\pi$
$660$	−3.79614	−0.147764
$661$	5.63282	0.219091	0.109546	−	0.993982i	$-0.465060\pi$
$661$	5.63282	0.219091	0.109546	+	0.993982i	$0.465060\pi$
$662$	33.3842	1.29752
$663$	14.4154	0.559847
$664$	3.91365	0.151879
$665$	23.7658	0.921599
$666$	0.341829	0.0132456
$667$	1.00000	0.0387202
$668$	−0.957962	−0.0370647
$669$	7.91788	0.306123
$670$	−18.2306	−0.704310
$671$	16.8679	0.651177
$672$	−5.18548	−0.200034
$673$	14.8683	0.573130	0.286565	−	0.958061i	$-0.407487\pi$
$673$	14.8683	0.573130	0.286565	+	0.958061i	$0.407487\pi$
$674$	−18.7050	−0.720490
$675$	−0.115013	−0.00442687
$676$	0.385424	0.0148240
$677$	−4.96188	−0.190700	−0.0953502	−	0.995444i	$-0.530397\pi$
$677$	−4.96188	−0.190700	−0.0953502	+	0.995444i	$0.530397\pi$
$678$	−3.48554	−0.133861
$679$	60.9984	2.34090
$680$	8.91115	0.341727
$681$	25.9727	0.995277
$682$	−16.9733	−0.649943
$683$	36.6077	1.40075	0.700377	−	0.713773i	$-0.253015\pi$
$683$	36.6077	1.40075	0.700377	+	0.713773i	$0.253015\pi$
$684$	−2.02647	−0.0774841
$685$	21.8574	0.835128
$686$	66.8365	2.55183
$687$	−23.9829	−0.915004
$688$	−8.58420	−0.327269
$689$	39.0497	1.48767
$690$	2.26164	0.0860991
$691$	29.9333	1.13871	0.569357	−	0.822090i	$-0.307192\pi$
$691$	29.9333	1.13871	0.569357	+	0.822090i	$0.307192\pi$
$692$	−11.5729	−0.439937
$693$	−8.70376	−0.330629
$694$	30.8979	1.17287
$695$	−30.5298	−1.15806
$696$	1.00000	0.0379049
$697$	27.5623	1.04399
$698$	5.57364	0.210966
$699$	2.23951	0.0847062
$700$	−0.596399	−0.0225418
$701$	33.7050	1.27302	0.636510	−	0.771269i	$-0.280377\pi$
$701$	33.7050	1.27302	0.636510	+	0.771269i	$0.280377\pi$
$702$	3.65861	0.138085
$703$	0.692707	0.0261259
$704$	1.67849	0.0632604
$705$	19.3743	0.729679
$706$	17.1035	0.643700
$707$	−18.6519	−0.701477
$708$	0.919230	0.0345468
$709$	1.39042	0.0522183	0.0261092	−	0.999659i	$-0.491688\pi$
$709$	1.39042	0.0522183	0.0261092	+	0.999659i	$0.491688\pi$
$710$	12.5881	0.472422
$711$	−12.7138	−0.476803
$712$	−6.19047	−0.231998
$713$	10.1123	0.378708
$714$	20.4314	0.764627
$715$	13.8886	0.519403
$716$	−10.3390	−0.386388
$717$	−15.6532	−0.584579
$718$	2.26293	0.0844517
$719$	6.93817	0.258750	0.129375	−	0.991596i	$-0.458703\pi$
$719$	6.93817	0.258750	0.129375	+	0.991596i	$0.458703\pi$
$720$	2.26164	0.0842863
$721$	−42.4171	−1.57969
$722$	14.8934	0.554275
$723$	−3.74945	−0.139444
$724$	24.7065	0.918209
$725$	0.115013	0.00427149
$726$	−8.18268	−0.303688
$727$	50.1096	1.85846	0.929231	−	0.369498i	$-0.120470\pi$
$727$	50.1096	1.85846	0.929231	+	0.369498i	$0.120470\pi$
$728$	18.9716	0.703136
$729$	1.00000	0.0370370
$730$	13.4607	0.498202
$731$	33.8228	1.25098
$732$	−10.0494	−0.371438
$733$	−22.7042	−0.838600	−0.419300	−	0.907848i	$-0.637724\pi$
$733$	−22.7042	−0.838600	−0.419300	+	0.907848i	$0.637724\pi$
$734$	−13.2727	−0.489906
$735$	−44.9821	−1.65919
$736$	−1.00000	−0.0368605
$737$	13.5299	0.498382
$738$	6.99527	0.257499
$739$	16.9804	0.624634	0.312317	−	0.949978i	$-0.398895\pi$
$739$	16.9804	0.624634	0.312317	+	0.949978i	$0.398895\pi$
$740$	−0.773094	−0.0284195
$741$	7.41407	0.272363
$742$	55.3465	2.03183
$743$	21.3220	0.782229	0.391115	−	0.920342i	$-0.372090\pi$
$743$	21.3220	0.782229	0.391115	+	0.920342i	$0.372090\pi$
$744$	10.1123	0.370734
$745$	33.5818	1.23034
$746$	−0.0831012	−0.00304255
$747$	−3.91365	−0.143193
$748$	−6.61346	−0.241812
$749$	19.2123	0.702002
$750$	−11.0481	−0.403419
$751$	−30.6682	−1.11910	−0.559549	−	0.828798i	$-0.689025\pi$
$751$	−30.6682	−1.11910	−0.559549	+	0.828798i	$0.689025\pi$
$752$	−8.56649	−0.312388
$753$	30.4348	1.10911
$754$	−3.65861	−0.133239
$755$	−6.00842	−0.218669
$756$	5.18548	0.188594
$757$	39.2460	1.42642	0.713210	−	0.700950i	$-0.247241\pi$
$757$	39.2460	1.42642	0.713210	+	0.700950i	$0.247241\pi$
$758$	7.42482	0.269682
$759$	−1.67849	−0.0609253
$760$	4.58315	0.166248
$761$	22.8777	0.829317	0.414659	−	0.909977i	$-0.363901\pi$
$761$	22.8777	0.829317	0.414659	+	0.909977i	$0.363901\pi$
$762$	−8.80505	−0.318973
$763$	86.0697	3.11593
$764$	12.1868	0.440904
$765$	−8.91115	−0.322183
$766$	−20.7513	−0.749775
$767$	−3.36311	−0.121435
$768$	−1.00000	−0.0360844
$769$	48.8512	1.76162	0.880810	−	0.473469i	$-0.156998\pi$
$769$	48.8512	1.76162	0.880810	+	0.473469i	$0.156998\pi$
$770$	19.6848	0.709390
$771$	−1.93369	−0.0696401
$772$	7.91470	0.284856
$773$	−49.5308	−1.78150	−0.890750	−	0.454493i	$-0.849820\pi$
$773$	−49.5308	−1.78150	−0.890750	+	0.454493i	$0.849820\pi$
$774$	8.58420	0.308553
$775$	1.16305	0.0417779
$776$	11.7633	0.422278
$777$	−1.77255	−0.0635898
$778$	36.3610	1.30360
$779$	14.1757	0.507898
$780$	−8.27446	−0.296273
$781$	−9.34231	−0.334294
$782$	3.94013	0.140899
$783$	−1.00000	−0.0357371
$784$	19.8892	0.710328
$785$	−42.3359	−1.51103
$786$	12.1320	0.432735
$787$	−11.2970	−0.402693	−0.201347	−	0.979520i	$-0.564532\pi$
$787$	−11.2970	−0.402693	−0.201347	+	0.979520i	$0.564532\pi$
$788$	1.08793	0.0387559
$789$	−27.5921	−0.982304
$790$	28.7539	1.02302
$791$	18.0742	0.642644
$792$	−1.67849	−0.0596425
$793$	36.7670	1.30563
$794$	−2.50661	−0.0889564
$795$	−24.1393	−0.856133
$796$	−16.4282	−0.582281
$797$	−52.0174	−1.84255	−0.921276	−	0.388910i	$-0.872852\pi$
$797$	−52.0174	−1.84255	−0.921276	+	0.388910i	$0.872852\pi$
$798$	10.5082	0.371987
$799$	33.7531	1.19410
$800$	−0.115013	−0.00406634
$801$	6.19047	0.218730
$802$	26.5916	0.938982
$803$	−9.98991	−0.352536
$804$	−8.06080	−0.284282
$805$	−11.7277	−0.413346
$806$	−36.9969	−1.30316
$807$	−0.980263	−0.0345069
$808$	−3.59695	−0.126540
$809$	4.37247	0.153728	0.0768639	−	0.997042i	$-0.475509\pi$
$809$	4.37247	0.153728	0.0768639	+	0.997042i	$0.475509\pi$
$810$	−2.26164	−0.0794659
$811$	−11.3391	−0.398169	−0.199085	−	0.979982i	$-0.563797\pi$
$811$	−11.3391	−0.398169	−0.199085	+	0.979982i	$0.563797\pi$
$812$	−5.18548	−0.181975
$813$	23.1859	0.813164
$814$	0.573756	0.0201101
$815$	−5.24567	−0.183748
$816$	3.94013	0.137932
$817$	17.3956	0.608597
$818$	26.1269	0.913505
$819$	−18.9716	−0.662923
$820$	−15.8208	−0.552485
$821$	23.2357	0.810931	0.405465	−	0.914110i	$-0.367109\pi$
$821$	23.2357	0.810931	0.405465	+	0.914110i	$0.367109\pi$
$822$	9.66441	0.337085
$823$	−44.9685	−1.56750	−0.783752	−	0.621073i	$-0.786697\pi$
$823$	−44.9685	−1.56750	−0.783752	+	0.621073i	$0.786697\pi$
$824$	−8.17997	−0.284963
$825$	−0.193049	−0.00672109
$826$	−4.76665	−0.165853
$827$	−17.3178	−0.602199	−0.301099	−	0.953593i	$-0.597354\pi$
$827$	−17.3178	−0.602199	−0.301099	+	0.953593i	$0.597354\pi$
$828$	1.00000	0.0347524
$829$	37.0366	1.28633	0.643167	−	0.765726i	$-0.277620\pi$
$829$	37.0366	1.28633	0.643167	+	0.765726i	$0.277620\pi$
$830$	8.85128	0.307232
$831$	−3.29919	−0.114448
$832$	3.65861	0.126839
$833$	−78.3659	−2.71522
$834$	−13.4990	−0.467431
$835$	−2.16657	−0.0749771
$836$	−3.40141	−0.117640
$837$	−10.1123	−0.349531
$838$	−34.8966	−1.20548
$839$	38.9850	1.34591	0.672956	−	0.739682i	$-0.265024\pi$
$839$	38.9850	1.34591	0.672956	+	0.739682i	$0.265024\pi$
$840$	−11.7277	−0.404643
$841$	1.00000	0.0344828
$842$	19.2371	0.662955
$843$	−14.9206	−0.513893
$844$	−18.3924	−0.633092
$845$	0.871690	0.0299870
$846$	8.56649	0.294522
$847$	42.4311	1.45795
$848$	10.6734	0.366525
$849$	7.35703	0.252493
$850$	0.453167	0.0155435
$851$	−0.341829	−0.0117177
$852$	5.56591	0.190685
$853$	−20.5551	−0.703793	−0.351896	−	0.936039i	$-0.614463\pi$
$853$	−20.5551	−0.703793	−0.351896	+	0.936039i	$0.614463\pi$
$854$	52.1112	1.78321
$855$	−4.58315	−0.156740
$856$	3.70502	0.126635
$857$	−38.6247	−1.31940	−0.659698	−	0.751531i	$-0.729316\pi$
$857$	−38.6247	−1.31940	−0.659698	+	0.751531i	$0.729316\pi$
$858$	6.14093	0.209648
$859$	−21.7884	−0.743412	−0.371706	−	0.928351i	$-0.621227\pi$
$859$	−21.7884	−0.743412	−0.371706	+	0.928351i	$0.621227\pi$
$860$	−19.4144	−0.662024
$861$	−36.2738	−1.23621
$862$	4.25421	0.144899
$863$	47.7394	1.62507	0.812534	−	0.582913i	$-0.198087\pi$
$863$	47.7394	1.62507	0.812534	+	0.582913i	$0.198087\pi$
$864$	1.00000	0.0340207
$865$	−26.1738	−0.889936
$866$	−1.85387	−0.0629970
$867$	1.47539	0.0501070
$868$	−52.4370	−1.77983
$869$	−21.3399	−0.723906
$870$	2.26164	0.0766768
$871$	29.4913	0.999275
$872$	16.5982	0.562087
$873$	−11.7633	−0.398128
$874$	2.02647	0.0685465
$875$	57.2896	1.93674
$876$	5.95173	0.201090
$877$	56.3532	1.90291	0.951456	−	0.307784i	$-0.0995874\pi$
$877$	56.3532	1.90291	0.951456	+	0.307784i	$0.0995874\pi$
$878$	13.0700	0.441091
$879$	15.0931	0.509076
$880$	3.79614	0.127968
$881$	37.3886	1.25966	0.629828	−	0.776735i	$-0.283126\pi$
$881$	37.3886	1.25966	0.629828	+	0.776735i	$0.283126\pi$
$882$	−19.8892	−0.669703
$883$	−40.4172	−1.36015	−0.680074	−	0.733144i	$-0.738052\pi$
$883$	−40.4172	−1.36015	−0.680074	+	0.733144i	$0.738052\pi$
$884$	−14.4154	−0.484842
$885$	2.07897	0.0698837
$886$	−16.2061	−0.544456
$887$	5.45256	0.183079	0.0915395	−	0.995801i	$-0.470821\pi$
$887$	5.45256	0.183079	0.0915395	+	0.995801i	$0.470821\pi$
$888$	−0.341829	−0.0114710
$889$	45.6584	1.53133
$890$	−14.0006	−0.469302
$891$	1.67849	0.0562315
$892$	−7.91788	−0.265110
$893$	17.3598	0.580922
$894$	14.8484	0.496606
$895$	−23.3832	−0.781613
$896$	5.18548	0.173235
$897$	−3.65861	−0.122157
$898$	0.298743	0.00996918
$899$	10.1123	0.337263
$900$	0.115013	0.00383378
$901$	−42.0544	−1.40104
$902$	11.7415	0.390948
$903$	−44.5132	−1.48131
$904$	3.48554	0.115927
$905$	55.8771	1.85742
$906$	−2.65666	−0.0882618
$907$	32.8885	1.09205	0.546023	−	0.837770i	$-0.316141\pi$
$907$	32.8885	1.09205	0.546023	+	0.837770i	$0.316141\pi$
$908$	−25.9727	−0.861935
$909$	3.59695	0.119303
$910$	42.9070	1.42235
$911$	−30.2278	−1.00149	−0.500747	−	0.865594i	$-0.666941\pi$
$911$	−30.2278	−1.00149	−0.500747	+	0.865594i	$0.666941\pi$
$912$	2.02647	0.0671032
$913$	−6.56902	−0.217403
$914$	6.82339	0.225698
$915$	−22.7282	−0.751372
$916$	23.9829	0.792417
$917$	−62.9103	−2.07748
$918$	−3.94013	−0.130044
$919$	3.63191	0.119806	0.0599028	−	0.998204i	$-0.480921\pi$
$919$	3.63191	0.119806	0.0599028	+	0.998204i	$0.480921\pi$
$920$	−2.26164	−0.0745640
$921$	26.6685	0.878758
$922$	−2.68303	−0.0883608
$923$	−20.3635	−0.670272
$924$	8.70376	0.286333
$925$	−0.0393149	−0.00129267
$926$	8.11062	0.266531
$927$	8.17997	0.268666
$928$	−1.00000	−0.0328266
$929$	−50.9359	−1.67115	−0.835576	−	0.549375i	$-0.814866\pi$
$929$	−50.9359	−1.67115	−0.835576	+	0.549375i	$0.814866\pi$
$930$	22.8703	0.749948
$931$	−40.3049	−1.32094
$932$	−2.23951	−0.0733577
$933$	23.3847	0.765580
$934$	27.5000	0.899829
$935$	−14.9573	−0.489155
$936$	−3.65861	−0.119585
$937$	36.4955	1.19226	0.596128	−	0.802889i	$-0.296705\pi$
$937$	36.4955	1.19226	0.596128	+	0.802889i	$0.296705\pi$
$938$	41.7991	1.36479
$939$	−3.76959	−0.123016
$940$	−19.3743	−0.631920
$941$	−35.5367	−1.15846	−0.579232	−	0.815163i	$-0.696647\pi$
$941$	−35.5367	−1.15846	−0.579232	+	0.815163i	$0.696647\pi$
$942$	−18.7191	−0.609901
$943$	−6.99527	−0.227797
$944$	−0.919230	−0.0299184
$945$	11.7277	0.381501
$946$	14.4085	0.468460
$947$	18.3895	0.597579	0.298789	−	0.954319i	$-0.403417\pi$
$947$	18.3895	0.597579	0.298789	+	0.954319i	$0.403417\pi$
$948$	12.7138	0.412923
$949$	−21.7751	−0.706848
$950$	0.233071	0.00756184
$951$	29.0727	0.942746
$952$	−20.4314	−0.662187
$953$	29.3478	0.950667	0.475333	−	0.879806i	$-0.342327\pi$
$953$	29.3478	0.950667	0.475333	+	0.879806i	$0.342327\pi$
$954$	−10.6734	−0.345563
$955$	27.5622	0.891892
$956$	15.6532	0.506260
$957$	−1.67849	−0.0542578
$958$	−24.4909	−0.791263
$959$	−50.1146	−1.61828
$960$	−2.26164	−0.0729941
$961$	71.2582	2.29865
$962$	1.25062	0.0403216
$963$	−3.70502	−0.119393
$964$	3.74945	0.120762
$965$	17.9002	0.576228
$966$	−5.18548	−0.166840
$967$	−23.3078	−0.749528	−0.374764	−	0.927120i	$-0.622276\pi$
$967$	−23.3078	−0.749528	−0.374764	+	0.927120i	$0.622276\pi$
$968$	8.18268	0.263001
$969$	−7.98456	−0.256501
$970$	26.6044	0.854215
$971$	41.2512	1.32382	0.661908	−	0.749585i	$-0.269747\pi$
$971$	41.2512	1.32382	0.661908	+	0.749585i	$0.269747\pi$
$972$	−1.00000	−0.0320750
$973$	69.9986	2.24405
$974$	19.6246	0.628813
$975$	−0.420789	−0.0134760
$976$	10.0494	0.321675
$977$	29.1039	0.931116	0.465558	−	0.885017i	$-0.345854\pi$
$977$	29.1039	0.931116	0.465558	+	0.885017i	$0.345854\pi$
$978$	−2.31941	−0.0741665
$979$	10.3906	0.332086
$980$	44.9821	1.43690
$981$	−16.5982	−0.529941
$982$	−23.4593	−0.748615
$983$	15.1901	0.484489	0.242244	−	0.970215i	$-0.422116\pi$
$983$	15.1901	0.484489	0.242244	+	0.970215i	$0.422116\pi$
$984$	−6.99527	−0.223001
$985$	2.46051	0.0783982
$986$	3.94013	0.125479
$987$	−44.4213	−1.41395
$988$	−7.41407	−0.235873
$989$	−8.58420	−0.272962
$990$	−3.79614	−0.120649
$991$	46.4337	1.47501	0.737507	−	0.675339i	$-0.236003\pi$
$991$	46.4337	1.47501	0.737507	+	0.675339i	$0.236003\pi$
$992$	−10.1123	−0.321065
$993$	33.3842	1.05942
$994$	−28.8619	−0.915443
$995$	−37.1546	−1.17788
$996$	3.91365	0.124009
$997$	−7.74371	−0.245246	−0.122623	−	0.992453i	$-0.539131\pi$
$997$	−7.74371	−0.245246	−0.122623	+	0.992453i	$0.539131\pi$
$998$	17.4292	0.551711
$999$	0.341829	0.0108150

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.bg.1.6	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.bg.1.6	✓	7	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:	\( N \)	\(=\)	\( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4002.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.26164 q^{5} +1.00000 q^{6} -5.18548 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.26164 q^{5} +1.00000 q^{6} -5.18548 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.26164 q^{10} +1.67849 q^{11} -1.00000 q^{12} +3.65861 q^{13} +5.18548 q^{14} -2.26164 q^{15} +1.00000 q^{16} -3.94013 q^{17} -1.00000 q^{18} -2.02647 q^{19} +2.26164 q^{20} +5.18548 q^{21} -1.67849 q^{22} +1.00000 q^{23} +1.00000 q^{24} +0.115013 q^{25} -3.65861 q^{26} -1.00000 q^{27} -5.18548 q^{28} +1.00000 q^{29} +2.26164 q^{30} +10.1123 q^{31} -1.00000 q^{32} -1.67849 q^{33} +3.94013 q^{34} -11.7277 q^{35} +1.00000 q^{36} -0.341829 q^{37} +2.02647 q^{38} -3.65861 q^{39} -2.26164 q^{40} -6.99527 q^{41} -5.18548 q^{42} -8.58420 q^{43} +1.67849 q^{44} +2.26164 q^{45} -1.00000 q^{46} -8.56649 q^{47} -1.00000 q^{48} +19.8892 q^{49} -0.115013 q^{50} +3.94013 q^{51} +3.65861 q^{52} +10.6734 q^{53} +1.00000 q^{54} +3.79614 q^{55} +5.18548 q^{56} +2.02647 q^{57} -1.00000 q^{58} -0.919230 q^{59} -2.26164 q^{60} +10.0494 q^{61} -10.1123 q^{62} -5.18548 q^{63} +1.00000 q^{64} +8.27446 q^{65} +1.67849 q^{66} +8.06080 q^{67} -3.94013 q^{68} -1.00000 q^{69} +11.7277 q^{70} -5.56591 q^{71} -1.00000 q^{72} -5.95173 q^{73} +0.341829 q^{74} -0.115013 q^{75} -2.02647 q^{76} -8.70376 q^{77} +3.65861 q^{78} -12.7138 q^{79} +2.26164 q^{80} +1.00000 q^{81} +6.99527 q^{82} -3.91365 q^{83} +5.18548 q^{84} -8.91115 q^{85} +8.58420 q^{86} -1.00000 q^{87} -1.67849 q^{88} +6.19047 q^{89} -2.26164 q^{90} -18.9716 q^{91} +1.00000 q^{92} -10.1123 q^{93} +8.56649 q^{94} -4.58315 q^{95} +1.00000 q^{96} -11.7633 q^{97} -19.8892 q^{98} +1.67849 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{2} - 7 q^{3} + 7 q^{4} + 2 q^{5} + 7 q^{6} - 5 q^{7} - 7 q^{8} + 7 q^{9}+O(q^{10}) \) 7 * q - 7 * q^2 - 7 * q^3 + 7 * q^4 + 2 * q^5 + 7 * q^6 - 5 * q^7 - 7 * q^8 + 7 * q^9	\( 7 q - 7 q^{2} - 7 q^{3} + 7 q^{4} + 2 q^{5} + 7 q^{6} - 5 q^{7} - 7 q^{8} + 7 q^{9} - 2 q^{10} - q^{11} - 7 q^{12} + q^{13} + 5 q^{14} - 2 q^{15} + 7 q^{16} - q^{17} - 7 q^{18} - 9 q^{19} + 2 q^{20} + 5 q^{21} + q^{22} + 7 q^{23} + 7 q^{24} + q^{25} - q^{26} - 7 q^{27} - 5 q^{28} + 7 q^{29} + 2 q^{30} - 19 q^{31} - 7 q^{32} + q^{33} + q^{34} + 11 q^{35} + 7 q^{36} - 18 q^{37} + 9 q^{38} - q^{39} - 2 q^{40} + 4 q^{41} - 5 q^{42} - 9 q^{43} - q^{44} + 2 q^{45} - 7 q^{46} - 11 q^{47} - 7 q^{48} + 12 q^{49} - q^{50} + q^{51} + q^{52} + 8 q^{53} + 7 q^{54} - 11 q^{55} + 5 q^{56} + 9 q^{57} - 7 q^{58} + 12 q^{59} - 2 q^{60} - 5 q^{61} + 19 q^{62} - 5 q^{63} + 7 q^{64} + 23 q^{65} - q^{66} - 13 q^{67} - q^{68} - 7 q^{69} - 11 q^{70} - q^{71} - 7 q^{72} - 26 q^{73} + 18 q^{74} - q^{75} - 9 q^{76} + 6 q^{77} + q^{78} - 38 q^{79} + 2 q^{80} + 7 q^{81} - 4 q^{82} - 6 q^{83} + 5 q^{84} - 25 q^{85} + 9 q^{86} - 7 q^{87} + q^{88} + 20 q^{89} - 2 q^{90} + 2 q^{91} + 7 q^{92} + 19 q^{93} + 11 q^{94} - 31 q^{95} + 7 q^{96} - 8 q^{97} - 12 q^{98} - q^{99}+O(q^{100}) \) 7 * q - 7 * q^2 - 7 * q^3 + 7 * q^4 + 2 * q^5 + 7 * q^6 - 5 * q^7 - 7 * q^8 + 7 * q^9 - 2 * q^10 - q^11 - 7 * q^12 + q^13 + 5 * q^14 - 2 * q^15 + 7 * q^16 - q^17 - 7 * q^18 - 9 * q^19 + 2 * q^20 + 5 * q^21 + q^22 + 7 * q^23 + 7 * q^24 + q^25 - q^26 - 7 * q^27 - 5 * q^28 + 7 * q^29 + 2 * q^30 - 19 * q^31 - 7 * q^32 + q^33 + q^34 + 11 * q^35 + 7 * q^36 - 18 * q^37 + 9 * q^38 - q^39 - 2 * q^40 + 4 * q^41 - 5 * q^42 - 9 * q^43 - q^44 + 2 * q^45 - 7 * q^46 - 11 * q^47 - 7 * q^48 + 12 * q^49 - q^50 + q^51 + q^52 + 8 * q^53 + 7 * q^54 - 11 * q^55 + 5 * q^56 + 9 * q^57 - 7 * q^58 + 12 * q^59 - 2 * q^60 - 5 * q^61 + 19 * q^62 - 5 * q^63 + 7 * q^64 + 23 * q^65 - q^66 - 13 * q^67 - q^68 - 7 * q^69 - 11 * q^70 - q^71 - 7 * q^72 - 26 * q^73 + 18 * q^74 - q^75 - 9 * q^76 + 6 * q^77 + q^78 - 38 * q^79 + 2 * q^80 + 7 * q^81 - 4 * q^82 - 6 * q^83 + 5 * q^84 - 25 * q^85 + 9 * q^86 - 7 * q^87 + q^88 + 20 * q^89 - 2 * q^90 + 2 * q^91 + 7 * q^92 + 19 * q^93 + 11 * q^94 - 31 * q^95 + 7 * q^96 - 8 * q^97 - 12 * q^98 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more