LMFDB - Embedded newform 6026.2.a.j.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6026,2,Mod(1,6026)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6026, 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("6026.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6026 = 2 \cdot 23 \cdot 131 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6026.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:	$48.1178522580$
Analytic rank:	$0$
Dimension:	$33$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	6026.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -0.987482 q^{3} +1.00000 q^{4} -4.26130 q^{5} +0.987482 q^{6} +0.584751 q^{7} -1.00000 q^{8} -2.02488 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.987482 q^{3} +1.00000 q^{4} -4.26130 q^{5} +0.987482 q^{6} +0.584751 q^{7} -1.00000 q^{8} -2.02488 q^{9} +4.26130 q^{10} +6.24613 q^{11} -0.987482 q^{12} -6.01959 q^{13} -0.584751 q^{14} +4.20795 q^{15} +1.00000 q^{16} -4.53777 q^{17} +2.02488 q^{18} -6.84209 q^{19} -4.26130 q^{20} -0.577431 q^{21} -6.24613 q^{22} +1.00000 q^{23} +0.987482 q^{24} +13.1587 q^{25} +6.01959 q^{26} +4.96198 q^{27} +0.584751 q^{28} +4.25524 q^{29} -4.20795 q^{30} -3.36375 q^{31} -1.00000 q^{32} -6.16794 q^{33} +4.53777 q^{34} -2.49180 q^{35} -2.02488 q^{36} -9.15630 q^{37} +6.84209 q^{38} +5.94423 q^{39} +4.26130 q^{40} -10.9436 q^{41} +0.577431 q^{42} +8.62265 q^{43} +6.24613 q^{44} +8.62862 q^{45} -1.00000 q^{46} -6.91155 q^{47} -0.987482 q^{48} -6.65807 q^{49} -13.1587 q^{50} +4.48096 q^{51} -6.01959 q^{52} -7.39363 q^{53} -4.96198 q^{54} -26.6166 q^{55} -0.584751 q^{56} +6.75644 q^{57} -4.25524 q^{58} +2.42383 q^{59} +4.20795 q^{60} -6.41763 q^{61} +3.36375 q^{62} -1.18405 q^{63} +1.00000 q^{64} +25.6513 q^{65} +6.16794 q^{66} -7.04733 q^{67} -4.53777 q^{68} -0.987482 q^{69} +2.49180 q^{70} -12.4714 q^{71} +2.02488 q^{72} +7.39978 q^{73} +9.15630 q^{74} -12.9939 q^{75} -6.84209 q^{76} +3.65243 q^{77} -5.94423 q^{78} -6.20525 q^{79} -4.26130 q^{80} +1.17478 q^{81} +10.9436 q^{82} -7.62028 q^{83} -0.577431 q^{84} +19.3368 q^{85} -8.62265 q^{86} -4.20197 q^{87} -6.24613 q^{88} +5.91911 q^{89} -8.62862 q^{90} -3.51996 q^{91} +1.00000 q^{92} +3.32164 q^{93} +6.91155 q^{94} +29.1562 q^{95} +0.987482 q^{96} +16.3234 q^{97} +6.65807 q^{98} -12.6477 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q - 33 q^{2} + 3 q^{3} + 33 q^{4} - 4 q^{5} - 3 q^{6} + 11 q^{7} - 33 q^{8} + 44 q^{9}+O(q^{10}) $ 33 * q - 33 * q^2 + 3 * q^3 + 33 * q^4 - 4 * q^5 - 3 * q^6 + 11 * q^7 - 33 * q^8 + 44 * q^9	\( 33 q - 33 q^{2} + 3 q^{3} + 33 q^{4} - 4 q^{5} - 3 q^{6} + 11 q^{7} - 33 q^{8} + 44 q^{9} + 4 q^{10} + 5 q^{11} + 3 q^{12} + 15 q^{13} - 11 q^{14} + 16 q^{15} + 33 q^{16} + 2 q^{17} - 44 q^{18} + 32 q^{19} - 4 q^{20} + 8 q^{21} - 5 q^{22} + 33 q^{23} - 3 q^{24} + 49 q^{25} - 15 q^{26} + 15 q^{27} + 11 q^{28} + 20 q^{29} - 16 q^{30} + 25 q^{31} - 33 q^{32} - 6 q^{33} - 2 q^{34} + 15 q^{35} + 44 q^{36} + 6 q^{37} - 32 q^{38} + 25 q^{39} + 4 q^{40} + 2 q^{41} - 8 q^{42} + 31 q^{43} + 5 q^{44} + 2 q^{45} - 33 q^{46} + 4 q^{47} + 3 q^{48} + 72 q^{49} - 49 q^{50} + 26 q^{51} + 15 q^{52} - 65 q^{53} - 15 q^{54} - 4 q^{55} - 11 q^{56} + 12 q^{57} - 20 q^{58} + 8 q^{59} + 16 q^{60} + 23 q^{61} - 25 q^{62} - 14 q^{63} + 33 q^{64} + 5 q^{65} + 6 q^{66} + 31 q^{67} + 2 q^{68} + 3 q^{69} - 15 q^{70} + 20 q^{71} - 44 q^{72} + 22 q^{73} - 6 q^{74} - 32 q^{75} + 32 q^{76} + 2 q^{77} - 25 q^{78} + 53 q^{79} - 4 q^{80} + 17 q^{81} - 2 q^{82} + 45 q^{83} + 8 q^{84} + 60 q^{85} - 31 q^{86} + 11 q^{87} - 5 q^{88} - 54 q^{89} - 2 q^{90} + 38 q^{91} + 33 q^{92} + 63 q^{93} - 4 q^{94} + 44 q^{95} - 3 q^{96} - 72 q^{98} + 43 q^{99}+O(q^{100}) \) 33 * q - 33 * q^2 + 3 * q^3 + 33 * q^4 - 4 * q^5 - 3 * q^6 + 11 * q^7 - 33 * q^8 + 44 * q^9 + 4 * q^10 + 5 * q^11 + 3 * q^12 + 15 * q^13 - 11 * q^14 + 16 * q^15 + 33 * q^16 + 2 * q^17 - 44 * q^18 + 32 * q^19 - 4 * q^20 + 8 * q^21 - 5 * q^22 + 33 * q^23 - 3 * q^24 + 49 * q^25 - 15 * q^26 + 15 * q^27 + 11 * q^28 + 20 * q^29 - 16 * q^30 + 25 * q^31 - 33 * q^32 - 6 * q^33 - 2 * q^34 + 15 * q^35 + 44 * q^36 + 6 * q^37 - 32 * q^38 + 25 * q^39 + 4 * q^40 + 2 * q^41 - 8 * q^42 + 31 * q^43 + 5 * q^44 + 2 * q^45 - 33 * q^46 + 4 * q^47 + 3 * q^48 + 72 * q^49 - 49 * q^50 + 26 * q^51 + 15 * q^52 - 65 * q^53 - 15 * q^54 - 4 * q^55 - 11 * q^56 + 12 * q^57 - 20 * q^58 + 8 * q^59 + 16 * q^60 + 23 * q^61 - 25 * q^62 - 14 * q^63 + 33 * q^64 + 5 * q^65 + 6 * q^66 + 31 * q^67 + 2 * q^68 + 3 * q^69 - 15 * q^70 + 20 * q^71 - 44 * q^72 + 22 * q^73 - 6 * q^74 - 32 * q^75 + 32 * q^76 + 2 * q^77 - 25 * q^78 + 53 * q^79 - 4 * q^80 + 17 * q^81 - 2 * q^82 + 45 * q^83 + 8 * q^84 + 60 * q^85 - 31 * q^86 + 11 * q^87 - 5 * q^88 - 54 * q^89 - 2 * q^90 + 38 * q^91 + 33 * q^92 + 63 * q^93 - 4 * q^94 + 44 * q^95 - 3 * q^96 - 72 * q^98 + 43 * 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
$3$	−0.987482	−0.570123	−0.285061	−	0.958509i	$-0.592014\pi$
$3$	−0.987482	−0.570123	−0.285061	+	0.958509i	$0.592014\pi$
$4$	1.00000	0.500000
$5$	−4.26130	−1.90571	−0.952856	−	0.303424i	$-0.901870\pi$
$5$	−4.26130	−1.90571	−0.952856	+	0.303424i	$0.901870\pi$
$6$	0.987482	0.403138
$7$	0.584751	0.221015	0.110508	−	0.993875i	$-0.464752\pi$
$7$	0.584751	0.221015	0.110508	+	0.993875i	$0.464752\pi$
$8$	−1.00000	−0.353553
$9$	−2.02488	−0.674960
$10$	4.26130	1.34754
$11$	6.24613	1.88328	0.941640	−	0.336621i	$-0.109284\pi$
$11$	6.24613	1.88328	0.941640	+	0.336621i	$0.109284\pi$
$12$	−0.987482	−0.285061
$13$	−6.01959	−1.66953	−0.834766	−	0.550604i	$-0.814397\pi$
$13$	−6.01959	−1.66953	−0.834766	+	0.550604i	$0.814397\pi$
$14$	−0.584751	−0.156281
$15$	4.20795	1.08649
$16$	1.00000	0.250000
$17$	−4.53777	−1.10057	−0.550285	−	0.834977i	$-0.685481\pi$
$17$	−4.53777	−1.10057	−0.550285	+	0.834977i	$0.685481\pi$
$18$	2.02488	0.477269
$19$	−6.84209	−1.56968	−0.784841	−	0.619697i	$-0.787256\pi$
$19$	−6.84209	−1.56968	−0.784841	+	0.619697i	$0.787256\pi$
$20$	−4.26130	−0.952856
$21$	−0.577431	−0.126006
$22$	−6.24613	−1.33168
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0.987482	0.201569
$25$	13.1587	2.63173
$26$	6.01959	1.18054
$27$	4.96198	0.954933
$28$	0.584751	0.110508
$29$	4.25524	0.790178	0.395089	−	0.918643i	$-0.370714\pi$
$29$	4.25524	0.790178	0.395089	+	0.918643i	$0.370714\pi$
$30$	−4.20795	−0.768264
$31$	−3.36375	−0.604148	−0.302074	−	0.953284i	$-0.597679\pi$
$31$	−3.36375	−0.604148	−0.302074	+	0.953284i	$0.597679\pi$
$32$	−1.00000	−0.176777
$33$	−6.16794	−1.07370
$34$	4.53777	0.778221
$35$	−2.49180	−0.421191
$36$	−2.02488	−0.337480
$37$	−9.15630	−1.50529	−0.752644	−	0.658428i	$-0.771222\pi$
$37$	−9.15630	−1.50529	−0.752644	+	0.658428i	$0.771222\pi$
$38$	6.84209	1.10993
$39$	5.94423	0.951839
$40$	4.26130	0.673771
$41$	−10.9436	−1.70910	−0.854548	−	0.519372i	$-0.826166\pi$
$41$	−10.9436	−1.70910	−0.854548	+	0.519372i	$0.826166\pi$
$42$	0.577431	0.0890996
$43$	8.62265	1.31494	0.657471	−	0.753480i	$-0.271626\pi$
$43$	8.62265	1.31494	0.657471	+	0.753480i	$0.271626\pi$
$44$	6.24613	0.941640
$45$	8.62862	1.28628
$46$	−1.00000	−0.147442
$47$	−6.91155	−1.00815	−0.504077	−	0.863659i	$-0.668167\pi$
$47$	−6.91155	−1.00815	−0.504077	+	0.863659i	$0.668167\pi$
$48$	−0.987482	−0.142531
$49$	−6.65807	−0.951152
$50$	−13.1587	−1.86092
$51$	4.48096	0.627460
$52$	−6.01959	−0.834766
$53$	−7.39363	−1.01559	−0.507797	−	0.861477i	$-0.669540\pi$
$53$	−7.39363	−1.01559	−0.507797	+	0.861477i	$0.669540\pi$
$54$	−4.96198	−0.675239
$55$	−26.6166	−3.58899
$56$	−0.584751	−0.0781407
$57$	6.75644	0.894912
$58$	−4.25524	−0.558740
$59$	2.42383	0.315555	0.157778	−	0.987475i	$-0.449567\pi$
$59$	2.42383	0.315555	0.157778	+	0.987475i	$0.449567\pi$
$60$	4.20795	0.543245
$61$	−6.41763	−0.821693	−0.410846	−	0.911705i	$-0.634767\pi$
$61$	−6.41763	−0.821693	−0.410846	+	0.911705i	$0.634767\pi$
$62$	3.36375	0.427197
$63$	−1.18405	−0.149176
$64$	1.00000	0.125000
$65$	25.6513	3.18165
$66$	6.16794	0.759221
$67$	−7.04733	−0.860969	−0.430484	−	0.902598i	$-0.641657\pi$
$67$	−7.04733	−0.860969	−0.430484	+	0.902598i	$0.641657\pi$
$68$	−4.53777	−0.550285
$69$	−0.987482	−0.118879
$70$	2.49180	0.297827
$71$	−12.4714	−1.48009	−0.740044	−	0.672558i	$-0.765195\pi$
$71$	−12.4714	−1.48009	−0.740044	+	0.672558i	$0.765195\pi$
$72$	2.02488	0.238634
$73$	7.39978	0.866078	0.433039	−	0.901375i	$-0.357441\pi$
$73$	7.39978	0.866078	0.433039	+	0.901375i	$0.357441\pi$
$74$	9.15630	1.06440
$75$	−12.9939	−1.50041
$76$	−6.84209	−0.784841
$77$	3.65243	0.416234
$78$	−5.94423	−0.673051
$79$	−6.20525	−0.698145	−0.349073	−	0.937096i	$-0.613503\pi$
$79$	−6.20525	−0.698145	−0.349073	+	0.937096i	$0.613503\pi$
$80$	−4.26130	−0.476428
$81$	1.17478	0.130531
$82$	10.9436	1.20851
$83$	−7.62028	−0.836434	−0.418217	−	0.908347i	$-0.637345\pi$
$83$	−7.62028	−0.836434	−0.418217	+	0.908347i	$0.637345\pi$
$84$	−0.577431	−0.0630029
$85$	19.3368	2.09737
$86$	−8.62265	−0.929804
$87$	−4.20197	−0.450498
$88$	−6.24613	−0.665840
$89$	5.91911	0.627425	0.313712	−	0.949518i	$-0.398427\pi$
$89$	5.91911	0.627425	0.313712	+	0.949518i	$0.398427\pi$
$90$	−8.62862	−0.909536
$91$	−3.51996	−0.368992
$92$	1.00000	0.104257
$93$	3.32164	0.344439
$94$	6.91155	0.712872
$95$	29.1562	2.99136
$96$	0.987482	0.100784
$97$	16.3234	1.65739	0.828694	−	0.559702i	$-0.189085\pi$
$97$	16.3234	1.65739	0.828694	+	0.559702i	$0.189085\pi$
$98$	6.65807	0.672566
$99$	−12.6477	−1.27114
$100$	13.1587	1.31587
$101$	−15.6103	−1.55329	−0.776643	−	0.629940i	$-0.783079\pi$
$101$	−15.6103	−1.55329	−0.776643	+	0.629940i	$0.783079\pi$
$102$	−4.48096	−0.443681
$103$	−3.37237	−0.332290	−0.166145	−	0.986101i	$-0.553132\pi$
$103$	−3.37237	−0.332290	−0.166145	+	0.986101i	$0.553132\pi$
$104$	6.01959	0.590269
$105$	2.46061	0.240131
$106$	7.39363	0.718133
$107$	−6.16401	−0.595898	−0.297949	−	0.954582i	$-0.596303\pi$
$107$	−6.16401	−0.595898	−0.297949	+	0.954582i	$0.596303\pi$
$108$	4.96198	0.477466
$109$	8.95249	0.857493	0.428747	−	0.903425i	$-0.358955\pi$
$109$	8.95249	0.857493	0.428747	+	0.903425i	$0.358955\pi$
$110$	26.6166	2.53780
$111$	9.04168	0.858199
$112$	0.584751	0.0552538
$113$	−8.75315	−0.823427	−0.411713	−	0.911313i	$-0.635069\pi$
$113$	−8.75315	−0.823427	−0.411713	+	0.911313i	$0.635069\pi$
$114$	−6.75644	−0.632798
$115$	−4.26130	−0.397368
$116$	4.25524	0.395089
$117$	12.1889	1.12687
$118$	−2.42383	−0.223131
$119$	−2.65347	−0.243243
$120$	−4.20795	−0.384132
$121$	28.0142	2.54674
$122$	6.41763	0.581025
$123$	10.8066	0.974395
$124$	−3.36375	−0.302074
$125$	−34.7666	−3.10961
$126$	1.18405	0.105484
$127$	−17.0305	−1.51121	−0.755607	−	0.655026i	$-0.772658\pi$
$127$	−17.0305	−1.51121	−0.755607	+	0.655026i	$0.772658\pi$
$128$	−1.00000	−0.0883883
$129$	−8.51471	−0.749678
$130$	−25.6513	−2.24976
$131$	−1.00000	−0.0873704
$131$	−1.00000	−0.0873704
$132$	−6.16794	−0.536850
$133$	−4.00092	−0.346924
$134$	7.04733	0.608797
$135$	−21.1445	−1.81983
$136$	4.53777	0.389110
$137$	−20.1253	−1.71942	−0.859709	−	0.510784i	$-0.829355\pi$
$137$	−20.1253	−1.71942	−0.859709	+	0.510784i	$0.829355\pi$
$138$	0.987482	0.0840600
$139$	−7.41635	−0.629047	−0.314523	−	0.949250i	$-0.601845\pi$
$139$	−7.41635	−0.629047	−0.314523	+	0.949250i	$0.601845\pi$
$140$	−2.49180	−0.210596
$141$	6.82503	0.574771
$142$	12.4714	1.04658
$143$	−37.5991	−3.14420
$144$	−2.02488	−0.168740
$145$	−18.1328	−1.50585
$146$	−7.39978	−0.612410
$147$	6.57472	0.542274
$148$	−9.15630	−0.752644
$149$	−0.519839	−0.0425869	−0.0212935	−	0.999773i	$-0.506778\pi$
$149$	−0.519839	−0.0425869	−0.0212935	+	0.999773i	$0.506778\pi$
$150$	12.9939	1.06095
$151$	−3.75626	−0.305680	−0.152840	−	0.988251i	$-0.548842\pi$
$151$	−3.75626	−0.305680	−0.152840	+	0.988251i	$0.548842\pi$
$152$	6.84209	0.554967
$153$	9.18844	0.742841
$154$	−3.65243	−0.294322
$155$	14.3340	1.15133
$156$	5.94423	0.475919
$157$	−7.24367	−0.578108	−0.289054	−	0.957313i	$-0.593341\pi$
$157$	−7.24367	−0.578108	−0.289054	+	0.957313i	$0.593341\pi$
$158$	6.20525	0.493663
$159$	7.30107	0.579013
$160$	4.26130	0.336885
$161$	0.584751	0.0460849
$162$	−1.17478	−0.0922994
$163$	6.24569	0.489200	0.244600	−	0.969624i	$-0.421343\pi$
$163$	6.24569	0.489200	0.244600	+	0.969624i	$0.421343\pi$
$164$	−10.9436	−0.854548
$165$	26.2835	2.04616
$166$	7.62028	0.591448
$167$	−7.38204	−0.571239	−0.285620	−	0.958343i	$-0.592199\pi$
$167$	−7.38204	−0.571239	−0.285620	+	0.958343i	$0.592199\pi$
$168$	0.577431	0.0445498
$169$	23.2354	1.78734
$170$	−19.3368	−1.48306
$171$	13.8544	1.05947
$172$	8.62265	0.657471
$173$	1.08865	0.0827686	0.0413843	−	0.999143i	$-0.486823\pi$
$173$	1.08865	0.0827686	0.0413843	+	0.999143i	$0.486823\pi$
$174$	4.20197	0.318550
$175$	7.69455	0.581653
$176$	6.24613	0.470820
$177$	−2.39348	−0.179905
$178$	−5.91911	−0.443656
$179$	−2.54066	−0.189898	−0.0949489	−	0.995482i	$-0.530269\pi$
$179$	−2.54066	−0.189898	−0.0949489	+	0.995482i	$0.530269\pi$
$180$	8.62862	0.643139
$181$	8.02694	0.596638	0.298319	−	0.954466i	$-0.403574\pi$
$181$	8.02694	0.596638	0.298319	+	0.954466i	$0.403574\pi$
$182$	3.51996	0.260917
$183$	6.33729	0.468466
$184$	−1.00000	−0.0737210
$185$	39.0178	2.86864
$186$	−3.32164	−0.243555
$187$	−28.3435	−2.07268
$188$	−6.91155	−0.504077
$189$	2.90152	0.211055
$190$	−29.1562	−2.11521
$191$	−12.7863	−0.925185	−0.462592	−	0.886571i	$-0.653081\pi$
$191$	−12.7863	−0.925185	−0.462592	+	0.886571i	$0.653081\pi$
$192$	−0.987482	−0.0712653
$193$	6.22728	0.448250	0.224125	−	0.974560i	$-0.428048\pi$
$193$	6.22728	0.448250	0.224125	+	0.974560i	$0.428048\pi$
$194$	−16.3234	−1.17195
$195$	−25.3301	−1.81393
$196$	−6.65807	−0.475576
$197$	−11.5953	−0.826132	−0.413066	−	0.910701i	$-0.635542\pi$
$197$	−11.5953	−0.826132	−0.413066	+	0.910701i	$0.635542\pi$
$198$	12.6477	0.898831
$199$	−6.39672	−0.453452	−0.226726	−	0.973959i	$-0.572802\pi$
$199$	−6.39672	−0.453452	−0.226726	+	0.973959i	$0.572802\pi$
$200$	−13.1587	−0.930459
$201$	6.95911	0.490858
$202$	15.6103	1.09834
$203$	2.48825	0.174641
$204$	4.48096	0.313730
$205$	46.6338	3.25704
$206$	3.37237	0.234964
$207$	−2.02488	−0.140739
$208$	−6.01959	−0.417383
$209$	−42.7366	−2.95615
$210$	−2.46061	−0.169798
$211$	12.1955	0.839576	0.419788	−	0.907622i	$-0.362104\pi$
$211$	12.1955	0.839576	0.419788	+	0.907622i	$0.362104\pi$
$212$	−7.39363	−0.507797
$213$	12.3153	0.843832
$214$	6.16401	0.421363
$215$	−36.7437	−2.50590
$216$	−4.96198	−0.337620
$217$	−1.96696	−0.133526
$218$	−8.95249	−0.606339
$219$	−7.30714	−0.493771
$220$	−26.6166	−1.79449
$221$	27.3155	1.83744
$222$	−9.04168	−0.606838
$223$	−9.76432	−0.653867	−0.326934	−	0.945047i	$-0.606015\pi$
$223$	−9.76432	−0.653867	−0.326934	+	0.945047i	$0.606015\pi$
$224$	−0.584751	−0.0390703
$225$	−26.6447	−1.77632
$226$	8.75315	0.582251
$227$	25.9857	1.72473	0.862365	−	0.506286i	$-0.168982\pi$
$227$	25.9857	1.72473	0.862365	+	0.506286i	$0.168982\pi$
$228$	6.75644	0.447456
$229$	1.31639	0.0869894	0.0434947	−	0.999054i	$-0.486151\pi$
$229$	1.31639	0.0869894	0.0434947	+	0.999054i	$0.486151\pi$
$230$	4.26130	0.280982
$231$	−3.60671	−0.237304
$232$	−4.25524	−0.279370
$233$	−13.1517	−0.861596	−0.430798	−	0.902448i	$-0.641768\pi$
$233$	−13.1517	−0.861596	−0.430798	+	0.902448i	$0.641768\pi$
$234$	−12.1889	−0.796816
$235$	29.4522	1.92125
$236$	2.42383	0.157778
$237$	6.12757	0.398028
$238$	2.65347	0.171999
$239$	7.06742	0.457154	0.228577	−	0.973526i	$-0.426593\pi$
$239$	7.06742	0.457154	0.228577	+	0.973526i	$0.426593\pi$
$240$	4.20795	0.271622
$241$	6.40805	0.412779	0.206389	−	0.978470i	$-0.433829\pi$
$241$	6.40805	0.412779	0.206389	+	0.978470i	$0.433829\pi$
$242$	−28.0142	−1.80082
$243$	−16.0460	−1.02935
$244$	−6.41763	−0.410846
$245$	28.3720	1.81262
$246$	−10.8066	−0.689001
$247$	41.1865	2.62064
$248$	3.36375	0.213599
$249$	7.52489	0.476870
$250$	34.7666	2.19883
$251$	20.3180	1.28246	0.641229	−	0.767349i	$-0.278425\pi$
$251$	20.3180	1.28246	0.641229	+	0.767349i	$0.278425\pi$
$252$	−1.18405	−0.0745882
$253$	6.24613	0.392691
$254$	17.0305	1.06859
$255$	−19.0947	−1.19576
$256$	1.00000	0.0625000
$257$	−7.31453	−0.456268	−0.228134	−	0.973630i	$-0.573262\pi$
$257$	−7.31453	−0.456268	−0.228134	+	0.973630i	$0.573262\pi$
$258$	8.51471	0.530103
$259$	−5.35416	−0.332691
$260$	25.6513	1.59082
$261$	−8.61634	−0.533338
$262$	1.00000	0.0617802
$263$	−10.7237	−0.661251	−0.330626	−	0.943762i	$-0.607260\pi$
$263$	−10.7237	−0.661251	−0.330626	+	0.943762i	$0.607260\pi$
$264$	6.16794	0.379611
$265$	31.5065	1.93543
$266$	4.00092	0.245312
$267$	−5.84501	−0.357709
$268$	−7.04733	−0.430484
$269$	26.0087	1.58578	0.792888	−	0.609368i	$-0.208577\pi$
$269$	26.0087	1.58578	0.792888	+	0.609368i	$0.208577\pi$
$270$	21.1445	1.28681
$271$	25.7787	1.56594	0.782972	−	0.622056i	$-0.213703\pi$
$271$	25.7787	1.56594	0.782972	+	0.622056i	$0.213703\pi$
$272$	−4.53777	−0.275143
$273$	3.47590	0.210371
$274$	20.1253	1.21581
$275$	82.1908	4.95629
$276$	−0.987482	−0.0594394
$277$	−1.93257	−0.116117	−0.0580586	−	0.998313i	$-0.518491\pi$
$277$	−1.93257	−0.116117	−0.0580586	+	0.998313i	$0.518491\pi$
$278$	7.41635	0.444803
$279$	6.81120	0.407776
$280$	2.49180	0.148914
$281$	−30.6259	−1.82699	−0.913494	−	0.406852i	$-0.866626\pi$
$281$	−30.6259	−1.82699	−0.913494	+	0.406852i	$0.866626\pi$
$282$	−6.82503	−0.406425
$283$	−17.1596	−1.02003	−0.510016	−	0.860165i	$-0.670361\pi$
$283$	−17.1596	−1.02003	−0.510016	+	0.860165i	$0.670361\pi$
$284$	−12.4714	−0.740044
$285$	−28.7912	−1.70544
$286$	37.5991	2.22328
$287$	−6.39926	−0.377736
$288$	2.02488	0.119317
$289$	3.59134	0.211255
$290$	18.1328	1.06480
$291$	−16.1190	−0.944914
$292$	7.39978	0.433039
$293$	2.26558	0.132357	0.0661784	−	0.997808i	$-0.478919\pi$
$293$	2.26558	0.132357	0.0661784	+	0.997808i	$0.478919\pi$
$294$	−6.57472	−0.383445
$295$	−10.3286	−0.601357
$296$	9.15630	0.532199
$297$	30.9932	1.79841
$298$	0.519839	0.0301135
$299$	−6.01959	−0.348122
$300$	−12.9939	−0.750206
$301$	5.04211	0.290622
$302$	3.75626	0.216149
$303$	15.4149	0.885564
$304$	−6.84209	−0.392421
$305$	27.3474	1.56591
$306$	−9.18844	−0.525268
$307$	−4.03773	−0.230446	−0.115223	−	0.993340i	$-0.536758\pi$
$307$	−4.03773	−0.230446	−0.115223	+	0.993340i	$0.536758\pi$
$308$	3.65243	0.208117
$309$	3.33015	0.189446
$310$	−14.3340	−0.814114
$311$	−1.76503	−0.100086	−0.0500429	−	0.998747i	$-0.515936\pi$
$311$	−1.76503	−0.100086	−0.0500429	+	0.998747i	$0.515936\pi$
$312$	−5.94423	−0.336526
$313$	10.2480	0.579253	0.289626	−	0.957140i	$-0.406469\pi$
$313$	10.2480	0.579253	0.289626	+	0.957140i	$0.406469\pi$
$314$	7.24367	0.408784
$315$	5.04560	0.284287
$316$	−6.20525	−0.349073
$317$	−3.56685	−0.200334	−0.100167	−	0.994971i	$-0.531938\pi$
$317$	−3.56685	−0.200334	−0.100167	+	0.994971i	$0.531938\pi$
$318$	−7.30107	−0.409424
$319$	26.5788	1.48813
$320$	−4.26130	−0.238214
$321$	6.08685	0.339735
$322$	−0.584751	−0.0325869
$323$	31.0478	1.72755
$324$	1.17478	0.0652655
$325$	−79.2098	−4.39377
$326$	−6.24569	−0.345917
$327$	−8.84042	−0.488876
$328$	10.9436	0.604257
$329$	−4.04154	−0.222817
$330$	−26.2835	−1.44686
$331$	18.3010	1.00592	0.502958	−	0.864311i	$-0.332245\pi$
$331$	18.3010	1.00592	0.502958	+	0.864311i	$0.332245\pi$
$332$	−7.62028	−0.418217
$333$	18.5404	1.01601
$334$	7.38204	0.403927
$335$	30.0308	1.64076
$336$	−0.577431	−0.0315014
$337$	15.8723	0.864618	0.432309	−	0.901725i	$-0.357699\pi$
$337$	15.8723	0.864618	0.432309	+	0.901725i	$0.357699\pi$
$338$	−23.2354	−1.26384
$339$	8.64357	0.469454
$340$	19.3368	1.04868
$341$	−21.0105	−1.13778
$342$	−13.8544	−0.749160
$343$	−7.98657	−0.431234
$344$	−8.62265	−0.464902
$345$	4.20795	0.226549
$346$	−1.08865	−0.0585263
$347$	−14.9535	−0.802746	−0.401373	−	0.915915i	$-0.631467\pi$
$347$	−14.9535	−0.802746	−0.401373	+	0.915915i	$0.631467\pi$
$348$	−4.20197	−0.225249
$349$	1.73558	0.0929036	0.0464518	−	0.998921i	$-0.485209\pi$
$349$	1.73558	0.0929036	0.0464518	+	0.998921i	$0.485209\pi$
$350$	−7.69455	−0.411291
$351$	−29.8690	−1.59429
$352$	−6.24613	−0.332920
$353$	18.1288	0.964897	0.482449	−	0.875924i	$-0.339747\pi$
$353$	18.1288	0.964897	0.482449	+	0.875924i	$0.339747\pi$
$354$	2.39348	0.127212
$355$	53.1446	2.82062
$356$	5.91911	0.313712
$357$	2.62025	0.138678
$358$	2.54066	0.134278
$359$	−1.72285	−0.0909284	−0.0454642	−	0.998966i	$-0.514477\pi$
$359$	−1.72285	−0.0909284	−0.0454642	+	0.998966i	$0.514477\pi$
$360$	−8.62862	−0.454768
$361$	27.8142	1.46390
$362$	−8.02694	−0.421887
$363$	−27.6635	−1.45196
$364$	−3.51996	−0.184496
$365$	−31.5327	−1.65049
$366$	−6.33729	−0.331255
$367$	−6.81137	−0.355551	−0.177775	−	0.984071i	$-0.556890\pi$
$367$	−6.81137	−0.355551	−0.177775	+	0.984071i	$0.556890\pi$
$368$	1.00000	0.0521286
$369$	22.1594	1.15357
$370$	−39.0178	−2.02844
$371$	−4.32343	−0.224462
$372$	3.32164	0.172219
$373$	20.5503	1.06405	0.532026	−	0.846728i	$-0.321431\pi$
$373$	20.5503	1.06405	0.532026	+	0.846728i	$0.321431\pi$
$374$	28.3435	1.46561
$375$	34.3313	1.77286
$376$	6.91155	0.356436
$377$	−25.6148	−1.31923
$378$	−2.90152	−0.149238
$379$	29.2302	1.50146	0.750728	−	0.660611i	$-0.229703\pi$
$379$	29.2302	1.50146	0.750728	+	0.660611i	$0.229703\pi$
$380$	29.1562	1.49568
$381$	16.8173	0.861577
$382$	12.7863	0.654205
$383$	22.1859	1.13365	0.566824	−	0.823839i	$-0.308172\pi$
$383$	22.1859	1.13365	0.566824	+	0.823839i	$0.308172\pi$
$384$	0.987482	0.0503922
$385$	−15.5641	−0.793221
$386$	−6.22728	−0.316960
$387$	−17.4598	−0.887533
$388$	16.3234	0.828694
$389$	17.6403	0.894399	0.447200	−	0.894434i	$-0.352421\pi$
$389$	17.6403	0.894399	0.447200	+	0.894434i	$0.352421\pi$
$390$	25.3301	1.28264
$391$	−4.53777	−0.229485
$392$	6.65807	0.336283
$393$	0.987482	0.0498119
$394$	11.5953	0.584164
$395$	26.4424	1.33046
$396$	−12.6477	−0.635569
$397$	3.94087	0.197787	0.0988933	−	0.995098i	$-0.468470\pi$
$397$	3.94087	0.197787	0.0988933	+	0.995098i	$0.468470\pi$
$398$	6.39672	0.320639
$399$	3.95083	0.197789
$400$	13.1587	0.657934
$401$	−3.11942	−0.155776	−0.0778881	−	0.996962i	$-0.524818\pi$
$401$	−3.11942	−0.155776	−0.0778881	+	0.996962i	$0.524818\pi$
$402$	−6.95911	−0.347089
$403$	20.2484	1.00864
$404$	−15.6103	−0.776643
$405$	−5.00609	−0.248754
$406$	−2.48825	−0.123490
$407$	−57.1915	−2.83488
$408$	−4.48096	−0.221841
$409$	13.0090	0.643254	0.321627	−	0.946866i	$-0.395770\pi$
$409$	13.0090	0.643254	0.321627	+	0.946866i	$0.395770\pi$
$410$	−46.6338	−2.30308
$411$	19.8733	0.980280
$412$	−3.37237	−0.166145
$413$	1.41733	0.0697425
$414$	2.02488	0.0995174
$415$	32.4723	1.59400
$416$	6.01959	0.295134
$417$	7.32351	0.358634
$418$	42.7366	2.09032
$419$	2.22916	0.108902	0.0544508	−	0.998516i	$-0.482659\pi$
$419$	2.22916	0.108902	0.0544508	+	0.998516i	$0.482659\pi$
$420$	2.46061	0.120065
$421$	−27.8708	−1.35834	−0.679171	−	0.733980i	$-0.737660\pi$
$421$	−27.8708	−1.35834	−0.679171	+	0.733980i	$0.737660\pi$
$422$	−12.1955	−0.593670
$423$	13.9951	0.680463
$424$	7.39363	0.359066
$425$	−59.7110	−2.89641
$426$	−12.3153	−0.596679
$427$	−3.75272	−0.181607
$428$	−6.16401	−0.297949
$429$	37.1285	1.79258
$430$	36.7437	1.77194
$431$	29.4130	1.41677	0.708387	−	0.705824i	$-0.249423\pi$
$431$	29.4130	1.41677	0.708387	+	0.705824i	$0.249423\pi$
$432$	4.96198	0.238733
$433$	−2.19823	−0.105640	−0.0528200	−	0.998604i	$-0.516821\pi$
$433$	−2.19823	−0.105640	−0.0528200	+	0.998604i	$0.516821\pi$
$434$	1.96696	0.0944171
$435$	17.9058	0.858520
$436$	8.95249	0.428747
$437$	−6.84209	−0.327301
$438$	7.30714	0.349149
$439$	19.0174	0.907653	0.453826	−	0.891090i	$-0.350059\pi$
$439$	19.0174	0.907653	0.453826	+	0.891090i	$0.350059\pi$
$440$	26.6166	1.26890
$441$	13.4818	0.641990
$442$	−27.3155	−1.29926
$443$	0.172624	0.00820160	0.00410080	−	0.999992i	$-0.498695\pi$
$443$	0.172624	0.00820160	0.00410080	+	0.999992i	$0.498695\pi$
$444$	9.04168	0.429099
$445$	−25.2231	−1.19569
$446$	9.76432	0.462354
$447$	0.513332	0.0242798
$448$	0.584751	0.0276269
$449$	10.7223	0.506019	0.253009	−	0.967464i	$-0.418580\pi$
$449$	10.7223	0.506019	0.253009	+	0.967464i	$0.418580\pi$
$450$	26.6447	1.25604
$451$	−68.3549	−3.21871
$452$	−8.75315	−0.411713
$453$	3.70924	0.174275
$454$	−25.9857	−1.21957
$455$	14.9996	0.703192
$456$	−6.75644	−0.316399
$457$	−28.2535	−1.32164	−0.660821	−	0.750543i	$-0.729792\pi$
$457$	−28.2535	−1.32164	−0.660821	+	0.750543i	$0.729792\pi$
$458$	−1.31639	−0.0615108
$459$	−22.5163	−1.05097
$460$	−4.26130	−0.198684
$461$	3.44997	0.160681	0.0803406	−	0.996767i	$-0.474399\pi$
$461$	3.44997	0.160681	0.0803406	+	0.996767i	$0.474399\pi$
$462$	3.60671	0.167799
$463$	−11.0166	−0.511984	−0.255992	−	0.966679i	$-0.582402\pi$
$463$	−11.0166	−0.511984	−0.255992	+	0.966679i	$0.582402\pi$
$464$	4.25524	0.197544
$465$	−14.1545	−0.656400
$466$	13.1517	0.609240
$467$	−37.2921	−1.72567	−0.862836	−	0.505483i	$-0.831314\pi$
$467$	−37.2921	−1.72567	−0.862836	+	0.505483i	$0.831314\pi$
$468$	12.1889	0.563434
$469$	−4.12094	−0.190287
$470$	−29.4522	−1.35853
$471$	7.15299	0.329592
$472$	−2.42383	−0.111566
$473$	53.8582	2.47640
$474$	−6.12757	−0.281449
$475$	−90.0328	−4.13099
$476$	−2.65347	−0.121621
$477$	14.9712	0.685485
$478$	−7.06742	−0.323256
$479$	25.5502	1.16742	0.583709	−	0.811963i	$-0.301601\pi$
$479$	25.5502	1.16742	0.583709	+	0.811963i	$0.301601\pi$
$480$	−4.20795	−0.192066
$481$	55.1172	2.51313
$482$	−6.40805	−0.291879
$483$	−0.577431	−0.0262740
$484$	28.0142	1.27337
$485$	−69.5588	−3.15850
$486$	16.0460	0.727861
$487$	−20.5437	−0.930925	−0.465463	−	0.885067i	$-0.654112\pi$
$487$	−20.5437	−0.930925	−0.465463	+	0.885067i	$0.654112\pi$
$488$	6.41763	0.290512
$489$	−6.16750	−0.278904
$490$	−28.3720	−1.28172
$491$	8.31143	0.375090	0.187545	−	0.982256i	$-0.439947\pi$
$491$	8.31143	0.375090	0.187545	+	0.982256i	$0.439947\pi$
$492$	10.8066	0.487197
$493$	−19.3093	−0.869646
$494$	−41.1865	−1.85307
$495$	53.8955	2.42242
$496$	−3.36375	−0.151037
$497$	−7.29269	−0.327122
$498$	−7.52489	−0.337198
$499$	14.2634	0.638518	0.319259	−	0.947667i	$-0.396566\pi$
$499$	14.2634	0.638518	0.319259	+	0.947667i	$0.396566\pi$
$500$	−34.7666	−1.55481
$501$	7.28963	0.325677
$502$	−20.3180	−0.906835
$503$	−16.8394	−0.750833	−0.375417	−	0.926856i	$-0.622500\pi$
$503$	−16.8394	−0.750833	−0.375417	+	0.926856i	$0.622500\pi$
$504$	1.18405	0.0527418
$505$	66.5203	2.96012
$506$	−6.24613	−0.277675
$507$	−22.9445	−1.01900
$508$	−17.0305	−0.755607
$509$	35.9561	1.59373	0.796864	−	0.604159i	$-0.206491\pi$
$509$	35.9561	1.59373	0.796864	+	0.604159i	$0.206491\pi$
$510$	19.0947	0.845529
$511$	4.32703	0.191416
$512$	−1.00000	−0.0441942
$513$	−33.9503	−1.49894
$514$	7.31453	0.322630
$515$	14.3707	0.633248
$516$	−8.51471	−0.374839
$517$	−43.1705	−1.89864
$518$	5.35416	0.235248
$519$	−1.07502	−0.0471883
$520$	−25.6513	−1.12488
$521$	−1.25284	−0.0548879	−0.0274440	−	0.999623i	$-0.508737\pi$
$521$	−1.25284	−0.0548879	−0.0274440	+	0.999623i	$0.508737\pi$
$522$	8.61634	0.377127
$523$	15.7380	0.688174	0.344087	−	0.938938i	$-0.388189\pi$
$523$	15.7380	0.688174	0.344087	+	0.938938i	$0.388189\pi$
$524$	−1.00000	−0.0436852
$525$	−7.59823	−0.331614
$526$	10.7237	0.467575
$527$	15.2639	0.664907
$528$	−6.16794	−0.268425
$529$	1.00000	0.0434783
$530$	−31.5065	−1.36855
$531$	−4.90796	−0.212987
$532$	−4.00092	−0.173462
$533$	65.8757	2.85339
$534$	5.84501	0.252939
$535$	26.2667	1.13561
$536$	7.04733	0.304398
$537$	2.50886	0.108265
$538$	−26.0087	−1.12131
$539$	−41.5872	−1.79129
$540$	−21.1445	−0.909913
$541$	43.6084	1.87487	0.937436	−	0.348159i	$-0.113193\pi$
$541$	43.6084	1.87487	0.937436	+	0.348159i	$0.113193\pi$
$542$	−25.7787	−1.10729
$543$	−7.92645	−0.340157
$544$	4.53777	0.194555
$545$	−38.1492	−1.63413
$546$	−3.47590	−0.148755
$547$	−37.3565	−1.59725	−0.798625	−	0.601829i	$-0.794439\pi$
$547$	−37.3565	−1.59725	−0.798625	+	0.601829i	$0.794439\pi$
$548$	−20.1253	−0.859709
$549$	12.9949	0.554610
$550$	−82.1908	−3.50463
$551$	−29.1147	−1.24033
$552$	0.987482	0.0420300
$553$	−3.62853	−0.154301
$554$	1.93257	0.0821072
$555$	−38.5293	−1.63548
$556$	−7.41635	−0.314523
$557$	6.45737	0.273607	0.136804	−	0.990598i	$-0.456317\pi$
$557$	6.45737	0.273607	0.136804	+	0.990598i	$0.456317\pi$
$558$	−6.81120	−0.288341
$559$	−51.9048	−2.19534
$560$	−2.49180	−0.105298
$561$	27.9887	1.18168
$562$	30.6259	1.29188
$563$	−27.4708	−1.15776	−0.578879	−	0.815414i	$-0.696510\pi$
$563$	−27.4708	−1.15776	−0.578879	+	0.815414i	$0.696510\pi$
$564$	6.82503	0.287386
$565$	37.2998	1.56921
$566$	17.1596	0.721272
$567$	0.686954	0.0288493
$568$	12.4714	0.523290
$569$	−0.564241	−0.0236542	−0.0118271	−	0.999930i	$-0.503765\pi$
$569$	−0.564241	−0.0236542	−0.0118271	+	0.999930i	$0.503765\pi$
$570$	28.7912	1.20593
$571$	−1.56866	−0.0656462	−0.0328231	−	0.999461i	$-0.510450\pi$
$571$	−1.56866	−0.0656462	−0.0328231	+	0.999461i	$0.510450\pi$
$572$	−37.5991	−1.57210
$573$	12.6262	0.527469
$574$	6.39926	0.267100
$575$	13.1587	0.548755
$576$	−2.02488	−0.0843700
$577$	−27.8834	−1.16080	−0.580400	−	0.814331i	$-0.697104\pi$
$577$	−27.8834	−1.16080	−0.580400	+	0.814331i	$0.697104\pi$
$578$	−3.59134	−0.149380
$579$	−6.14933	−0.255557
$580$	−18.1328	−0.752925
$581$	−4.45597	−0.184865
$582$	16.1190	0.668155
$583$	−46.1816	−1.91265
$584$	−7.39978	−0.306205
$585$	−51.9407	−2.14748
$586$	−2.26558	−0.0935903
$587$	−13.4297	−0.554303	−0.277151	−	0.960826i	$-0.589390\pi$
$587$	−13.4297	−0.554303	−0.277151	+	0.960826i	$0.589390\pi$
$588$	6.57472	0.271137
$589$	23.0151	0.948320
$590$	10.3286	0.425224
$591$	11.4502	0.470997
$592$	−9.15630	−0.376322
$593$	15.4112	0.632863	0.316431	−	0.948615i	$-0.397515\pi$
$593$	15.4112	0.632863	0.316431	+	0.948615i	$0.397515\pi$
$594$	−30.9932	−1.27167
$595$	11.3072	0.463550
$596$	−0.519839	−0.0212935
$597$	6.31665	0.258523
$598$	6.01959	0.246159
$599$	−5.02862	−0.205464	−0.102732	−	0.994709i	$-0.532758\pi$
$599$	−5.02862	−0.205464	−0.102732	+	0.994709i	$0.532758\pi$
$600$	12.9939	0.530476
$601$	−42.0008	−1.71325	−0.856624	−	0.515942i	$-0.827442\pi$
$601$	−42.0008	−1.71325	−0.856624	+	0.515942i	$0.827442\pi$
$602$	−5.04211	−0.205501
$603$	14.2700	0.581120
$604$	−3.75626	−0.152840
$605$	−119.377	−4.85336
$606$	−15.4149	−0.626189
$607$	23.7173	0.962653	0.481327	−	0.876541i	$-0.340155\pi$
$607$	23.7173	0.962653	0.481327	+	0.876541i	$0.340155\pi$
$608$	6.84209	0.277483
$609$	−2.45711	−0.0995670
$610$	−27.3474	−1.10727
$611$	41.6047	1.68315
$612$	9.18844	0.371421
$613$	39.2301	1.58449	0.792244	−	0.610205i	$-0.208913\pi$
$613$	39.2301	1.58449	0.792244	+	0.610205i	$0.208913\pi$
$614$	4.03773	0.162950
$615$	−46.0500	−1.85691
$616$	−3.65243	−0.147161
$617$	−27.8306	−1.12042	−0.560209	−	0.828352i	$-0.689279\pi$
$617$	−27.8306	−1.12042	−0.560209	+	0.828352i	$0.689279\pi$
$618$	−3.33015	−0.133958
$619$	4.95584	0.199192	0.0995960	−	0.995028i	$-0.468245\pi$
$619$	4.95584	0.199192	0.0995960	+	0.995028i	$0.468245\pi$
$620$	14.3340	0.575666
$621$	4.96198	0.199117
$622$	1.76503	0.0707714
$623$	3.46121	0.138670
$624$	5.94423	0.237960
$625$	82.3573	3.29429
$626$	−10.2480	−0.409593
$627$	42.2016	1.68537
$628$	−7.24367	−0.289054
$629$	41.5492	1.65667
$630$	−5.04560	−0.201021
$631$	−5.97630	−0.237913	−0.118956	−	0.992899i	$-0.537955\pi$
$631$	−5.97630	−0.237913	−0.118956	+	0.992899i	$0.537955\pi$
$632$	6.20525	0.246832
$633$	−12.0429	−0.478661
$634$	3.56685	0.141658
$635$	72.5721	2.87994
$636$	7.30107	0.289506
$637$	40.0788	1.58798
$638$	−26.5788	−1.05226
$639$	25.2532	0.999001
$640$	4.26130	0.168443
$641$	39.3797	1.55540	0.777701	−	0.628634i	$-0.216386\pi$
$641$	39.3797	1.55540	0.777701	+	0.628634i	$0.216386\pi$
$642$	−6.08685	−0.240229
$643$	42.4376	1.67358	0.836788	−	0.547527i	$-0.184431\pi$
$643$	42.4376	1.67358	0.836788	+	0.547527i	$0.184431\pi$
$644$	0.584751	0.0230424
$645$	36.2837	1.42867
$646$	−31.0478	−1.22156
$647$	−15.4548	−0.607591	−0.303795	−	0.952737i	$-0.598254\pi$
$647$	−15.4548	−0.607591	−0.303795	+	0.952737i	$0.598254\pi$
$648$	−1.17478	−0.0461497
$649$	15.1395	0.594279
$650$	79.2098	3.10686
$651$	1.94234	0.0761261
$652$	6.24569	0.244600
$653$	14.0215	0.548705	0.274352	−	0.961629i	$-0.411537\pi$
$653$	14.0215	0.548705	0.274352	+	0.961629i	$0.411537\pi$
$654$	8.84042	0.345688
$655$	4.26130	0.166503
$656$	−10.9436	−0.427274
$657$	−14.9837	−0.584568
$658$	4.04154	0.157556
$659$	−39.3251	−1.53189	−0.765944	−	0.642907i	$-0.777728\pi$
$659$	−39.3251	−1.53189	−0.765944	+	0.642907i	$0.777728\pi$
$660$	26.2835	1.02308
$661$	−17.1900	−0.668615	−0.334308	−	0.942464i	$-0.608502\pi$
$661$	−17.1900	−0.668615	−0.334308	+	0.942464i	$0.608502\pi$
$662$	−18.3010	−0.711290
$663$	−26.9735	−1.04757
$664$	7.62028	0.295724
$665$	17.0491	0.661136
$666$	−18.5404	−0.718427
$667$	4.25524	0.164763
$668$	−7.38204	−0.285620
$669$	9.64209	0.372785
$670$	−30.0308	−1.16019
$671$	−40.0854	−1.54748
$672$	0.577431	0.0222749
$673$	−38.3261	−1.47736	−0.738682	−	0.674054i	$-0.764551\pi$
$673$	−38.3261	−1.47736	−0.738682	+	0.674054i	$0.764551\pi$
$674$	−15.8723	−0.611377
$675$	65.2930	2.51313
$676$	23.2354	0.893669
$677$	22.5496	0.866651	0.433325	−	0.901238i	$-0.357340\pi$
$677$	22.5496	0.866651	0.433325	+	0.901238i	$0.357340\pi$
$678$	−8.64357	−0.331954
$679$	9.54511	0.366308
$680$	−19.3368	−0.741532
$681$	−25.6604	−0.983308
$682$	21.0105	0.804532
$683$	−25.3375	−0.969512	−0.484756	−	0.874650i	$-0.661092\pi$
$683$	−25.3375	−0.969512	−0.484756	+	0.874650i	$0.661092\pi$
$684$	13.8544	0.529736
$685$	85.7598	3.27671
$686$	7.98657	0.304929
$687$	−1.29991	−0.0495946
$688$	8.62265	0.328735
$689$	44.5066	1.69557
$690$	−4.20795	−0.160194
$691$	21.5161	0.818513	0.409256	−	0.912419i	$-0.365788\pi$
$691$	21.5161	0.818513	0.409256	+	0.912419i	$0.365788\pi$
$692$	1.08865	0.0413843
$693$	−7.39574	−0.280941
$694$	14.9535	0.567627
$695$	31.6033	1.19878
$696$	4.20197	0.159275
$697$	49.6593	1.88098
$698$	−1.73558	−0.0656928
$699$	12.9871	0.491215
$700$	7.69455	0.290827
$701$	−10.2852	−0.388467	−0.194234	−	0.980955i	$-0.562222\pi$
$701$	−10.2852	−0.388467	−0.194234	+	0.980955i	$0.562222\pi$
$702$	29.8690	1.12733
$703$	62.6482	2.36282
$704$	6.24613	0.235410
$705$	−29.0835	−1.09535
$706$	−18.1288	−0.682286
$707$	−9.12817	−0.343300
$708$	−2.39348	−0.0899526
$709$	39.0511	1.46659	0.733297	−	0.679909i	$-0.237981\pi$
$709$	39.0511	1.46659	0.733297	+	0.679909i	$0.237981\pi$
$710$	−53.1446	−1.99448
$711$	12.5649	0.471220
$712$	−5.91911	−0.221828
$713$	−3.36375	−0.125974
$714$	−2.62025	−0.0980603
$715$	160.221	5.99193
$716$	−2.54066	−0.0949489
$717$	−6.97895	−0.260634
$718$	1.72285	0.0642961
$719$	−28.1476	−1.04973	−0.524864	−	0.851186i	$-0.675884\pi$
$719$	−28.1476	−1.04973	−0.524864	+	0.851186i	$0.675884\pi$
$720$	8.62862	0.321570
$721$	−1.97200	−0.0734410
$722$	−27.8142	−1.03514
$723$	−6.32783	−0.235335
$724$	8.02694	0.298319
$725$	55.9933	2.07954
$726$	27.6635	1.02669
$727$	48.3709	1.79398	0.896990	−	0.442052i	$-0.145749\pi$
$727$	48.3709	1.79398	0.896990	+	0.442052i	$0.145749\pi$
$728$	3.51996	0.130458
$729$	12.3208	0.456326
$730$	31.5327	1.16708
$731$	−39.1276	−1.44719
$732$	6.33729	0.234233
$733$	36.2722	1.33974	0.669872	−	0.742476i	$-0.266349\pi$
$733$	36.2722	1.33974	0.669872	+	0.742476i	$0.266349\pi$
$734$	6.81137	0.251412
$735$	−28.0168	−1.03342
$736$	−1.00000	−0.0368605
$737$	−44.0186	−1.62145
$738$	−22.1594	−0.815698
$739$	19.1053	0.702799	0.351399	−	0.936226i	$-0.385706\pi$
$739$	19.1053	0.702799	0.351399	+	0.936226i	$0.385706\pi$
$740$	39.0178	1.43432
$741$	−40.6709	−1.49408
$742$	4.32343	0.158718
$743$	42.7308	1.56764	0.783820	−	0.620988i	$-0.213268\pi$
$743$	42.7308	1.56764	0.783820	+	0.620988i	$0.213268\pi$
$744$	−3.32164	−0.121777
$745$	2.21519	0.0811583
$746$	−20.5503	−0.752399
$747$	15.4302	0.564560
$748$	−28.3435	−1.03634
$749$	−3.60442	−0.131702
$750$	−34.3313	−1.25360
$751$	34.8122	1.27032	0.635158	−	0.772382i	$-0.280935\pi$
$751$	34.8122	1.27032	0.635158	+	0.772382i	$0.280935\pi$
$752$	−6.91155	−0.252038
$753$	−20.0636	−0.731159
$754$	25.6148	0.932835
$755$	16.0066	0.582539
$756$	2.90152	0.105527
$757$	−20.7555	−0.754371	−0.377186	−	0.926138i	$-0.623108\pi$
$757$	−20.7555	−0.754371	−0.377186	+	0.926138i	$0.623108\pi$
$758$	−29.2302	−1.06169
$759$	−6.16794	−0.223882
$760$	−29.1562	−1.05761
$761$	10.2267	0.370717	0.185358	−	0.982671i	$-0.440655\pi$
$761$	10.2267	0.370717	0.185358	+	0.982671i	$0.440655\pi$
$762$	−16.8173	−0.609227
$763$	5.23498	0.189519
$764$	−12.7863	−0.462592
$765$	−39.1547	−1.41564
$766$	−22.1859	−0.801610
$767$	−14.5904	−0.526830
$768$	−0.987482	−0.0356327
$769$	−30.6382	−1.10484	−0.552422	−	0.833565i	$-0.686296\pi$
$769$	−30.6382	−1.10484	−0.552422	+	0.833565i	$0.686296\pi$
$770$	15.5641	0.560892
$771$	7.22297	0.260129
$772$	6.22728	0.224125
$773$	−19.7197	−0.709270	−0.354635	−	0.935005i	$-0.615395\pi$
$773$	−19.7197	−0.709270	−0.354635	+	0.935005i	$0.615395\pi$
$774$	17.4598	0.627581
$775$	−44.2625	−1.58996
$776$	−16.3234	−0.585975
$777$	5.28713	0.189675
$778$	−17.6403	−0.632436
$779$	74.8767	2.68274
$780$	−25.3301	−0.906965
$781$	−77.8983	−2.78742
$782$	4.53777	0.162270
$783$	21.1144	0.754567
$784$	−6.65807	−0.237788
$785$	30.8674	1.10171
$786$	−0.987482	−0.0352223
$787$	22.3378	0.796255	0.398127	−	0.917330i	$-0.369660\pi$
$787$	22.3378	0.796255	0.398127	+	0.917330i	$0.369660\pi$
$788$	−11.5953	−0.413066
$789$	10.5895	0.376994
$790$	−26.4424	−0.940779
$791$	−5.11841	−0.181990
$792$	12.6477	0.449415
$793$	38.6315	1.37184
$794$	−3.94087	−0.139856
$795$	−31.1121	−1.10343
$796$	−6.39672	−0.226726
$797$	−33.5106	−1.18701	−0.593503	−	0.804832i	$-0.702256\pi$
$797$	−33.5106	−1.18701	−0.593503	+	0.804832i	$0.702256\pi$
$798$	−3.95083	−0.139858
$799$	31.3630	1.10954
$800$	−13.1587	−0.465229
$801$	−11.9855	−0.423487
$802$	3.11942	0.110150
$803$	46.2200	1.63107
$804$	6.95911	0.245429
$805$	−2.49180	−0.0878244
$806$	−20.2484	−0.713220
$807$	−25.6831	−0.904087
$808$	15.6103	0.549170
$809$	31.7335	1.11569	0.557845	−	0.829945i	$-0.311628\pi$
$809$	31.7335	1.11569	0.557845	+	0.829945i	$0.311628\pi$
$810$	5.00609	0.175896
$811$	−5.23351	−0.183773	−0.0918867	−	0.995769i	$-0.529290\pi$
$811$	−5.23351	−0.183773	−0.0918867	+	0.995769i	$0.529290\pi$
$812$	2.48825	0.0873206
$813$	−25.4560	−0.892781
$814$	57.1915	2.00456
$815$	−26.6147	−0.932274
$816$	4.48096	0.156865
$817$	−58.9969	−2.06404
$818$	−13.0090	−0.454849
$819$	7.12750	0.249055
$820$	46.6338	1.62852
$821$	20.7086	0.722736	0.361368	−	0.932423i	$-0.382310\pi$
$821$	20.7086	0.722736	0.361368	+	0.932423i	$0.382310\pi$
$822$	−19.8733	−0.693162
$823$	−40.0967	−1.39768	−0.698842	−	0.715276i	$-0.746301\pi$
$823$	−40.0967	−1.39768	−0.698842	+	0.715276i	$0.746301\pi$
$824$	3.37237	0.117482
$825$	−81.1619	−2.82570
$826$	−1.41733	−0.0493154
$827$	−53.5561	−1.86233	−0.931163	−	0.364603i	$-0.881205\pi$
$827$	−53.5561	−1.86233	−0.931163	+	0.364603i	$0.881205\pi$
$828$	−2.02488	−0.0703694
$829$	−48.9300	−1.69941	−0.849704	−	0.527259i	$-0.823220\pi$
$829$	−48.9300	−1.69941	−0.849704	+	0.527259i	$0.823220\pi$
$830$	−32.4723	−1.12713
$831$	1.90838	0.0662010
$832$	−6.01959	−0.208692
$833$	30.2128	1.04681
$834$	−7.32351	−0.253592
$835$	31.4571	1.08862
$836$	−42.7366	−1.47808
$837$	−16.6909	−0.576921
$838$	−2.22916	−0.0770051
$839$	−30.8239	−1.06416	−0.532080	−	0.846694i	$-0.678590\pi$
$839$	−30.8239	−1.06416	−0.532080	+	0.846694i	$0.678590\pi$
$840$	−2.46061	−0.0848990
$841$	−10.8930	−0.375619
$842$	27.8708	0.960492
$843$	30.2425	1.04161
$844$	12.1955	0.419788
$845$	−99.0130	−3.40615
$846$	−13.9951	−0.481160
$847$	16.3813	0.562869
$848$	−7.39363	−0.253898
$849$	16.9448	0.581544
$850$	59.7110	2.04807
$851$	−9.15630	−0.313874
$852$	12.3153	0.421916
$853$	48.4053	1.65737	0.828683	−	0.559718i	$-0.189090\pi$
$853$	48.4053	1.65737	0.828683	+	0.559718i	$0.189090\pi$
$854$	3.75272	0.128415
$855$	−59.0378	−2.01905
$856$	6.16401	0.210682
$857$	−29.0852	−0.993532	−0.496766	−	0.867885i	$-0.665479\pi$
$857$	−29.0852	−0.993532	−0.496766	+	0.867885i	$0.665479\pi$
$858$	−37.1285	−1.26754
$859$	−29.1587	−0.994883	−0.497442	−	0.867497i	$-0.665727\pi$
$859$	−29.1587	−0.994883	−0.497442	+	0.867497i	$0.665727\pi$
$860$	−36.7437	−1.25295
$861$	6.31915	0.215356
$862$	−29.4130	−1.00181
$863$	−21.6476	−0.736894	−0.368447	−	0.929649i	$-0.620110\pi$
$863$	−21.6476	−0.736894	−0.368447	+	0.929649i	$0.620110\pi$
$864$	−4.96198	−0.168810
$865$	−4.63907	−0.157733
$866$	2.19823	0.0746988
$867$	−3.54638	−0.120441
$868$	−1.96696	−0.0667629
$869$	−38.7588	−1.31480
$870$	−17.9058	−0.607065
$871$	42.4220	1.43742
$872$	−8.95249	−0.303170
$873$	−33.0529	−1.11867
$874$	6.84209	0.231437
$875$	−20.3298	−0.687272
$876$	−7.30714	−0.246885
$877$	30.9335	1.04455	0.522275	−	0.852777i	$-0.325084\pi$
$877$	30.9335	1.04455	0.522275	+	0.852777i	$0.325084\pi$
$878$	−19.0174	−0.641807
$879$	−2.23722	−0.0754596
$880$	−26.6166	−0.897247
$881$	−25.5011	−0.859152	−0.429576	−	0.903031i	$-0.641337\pi$
$881$	−25.5011	−0.859152	−0.429576	+	0.903031i	$0.641337\pi$
$882$	−13.4818	−0.453955
$883$	0.279284	0.00939865	0.00469932	−	0.999989i	$-0.498504\pi$
$883$	0.279284	0.00939865	0.00469932	+	0.999989i	$0.498504\pi$
$884$	27.3155	0.918719
$885$	10.1993	0.342847
$886$	−0.172624	−0.00579941
$887$	−45.8190	−1.53845	−0.769226	−	0.638977i	$-0.779358\pi$
$887$	−45.8190	−1.53845	−0.769226	+	0.638977i	$0.779358\pi$
$888$	−9.04168	−0.303419
$889$	−9.95861	−0.334001
$890$	25.2231	0.845481
$891$	7.33783	0.245827
$892$	−9.76432	−0.326934
$893$	47.2895	1.58248
$894$	−0.513332	−0.0171684
$895$	10.8265	0.361890
$896$	−0.584751	−0.0195352
$897$	5.94423	0.198472
$898$	−10.7223	−0.357809
$899$	−14.3136	−0.477384
$900$	−26.6447	−0.888158
$901$	33.5506	1.11773
$902$	68.3549	2.27597
$903$	−4.97899	−0.165690
$904$	8.75315	0.291125
$905$	−34.2052	−1.13702
$906$	−3.70924	−0.123231
$907$	−1.67737	−0.0556963	−0.0278481	−	0.999612i	$-0.508865\pi$
$907$	−1.67737	−0.0556963	−0.0278481	+	0.999612i	$0.508865\pi$
$908$	25.9857	0.862365
$909$	31.6091	1.04841
$910$	−14.9996	−0.497232
$911$	−46.8072	−1.55079	−0.775396	−	0.631475i	$-0.782450\pi$
$911$	−46.8072	−1.55079	−0.775396	+	0.631475i	$0.782450\pi$
$912$	6.75644	0.223728
$913$	−47.5973	−1.57524
$914$	28.2535	0.934542
$915$	−27.0051	−0.892761
$916$	1.31639	0.0434947
$917$	−0.584751	−0.0193102
$918$	22.5163	0.743149
$919$	−43.9419	−1.44951	−0.724756	−	0.689006i	$-0.758047\pi$
$919$	−43.9419	−1.44951	−0.724756	+	0.689006i	$0.758047\pi$
$920$	4.26130	0.140491
$921$	3.98719	0.131382
$922$	−3.44997	−0.113619
$923$	75.0729	2.47106
$924$	−3.60671	−0.118652
$925$	−120.485	−3.96152
$926$	11.0166	0.362027
$927$	6.82865	0.224282
$928$	−4.25524	−0.139685
$929$	16.5555	0.543167	0.271583	−	0.962415i	$-0.412453\pi$
$929$	16.5555	0.543167	0.271583	+	0.962415i	$0.412453\pi$
$930$	14.1545	0.464145
$931$	45.5551	1.49301
$932$	−13.1517	−0.430798
$933$	1.74294	0.0570612
$934$	37.2921	1.22023
$935$	120.780	3.94993
$936$	−12.1889	−0.398408
$937$	2.11899	0.0692245	0.0346122	−	0.999401i	$-0.488980\pi$
$937$	2.11899	0.0692245	0.0346122	+	0.999401i	$0.488980\pi$
$938$	4.12094	0.134553
$939$	−10.1197	−0.330245
$940$	29.4522	0.960625
$941$	−46.0949	−1.50265	−0.751325	−	0.659932i	$-0.770585\pi$
$941$	−46.0949	−1.50265	−0.751325	+	0.659932i	$0.770585\pi$
$942$	−7.15299	−0.233057
$943$	−10.9436	−0.356371
$944$	2.42383	0.0788888
$945$	−12.3643	−0.402209
$946$	−53.8582	−1.75108
$947$	−40.9427	−1.33046	−0.665230	−	0.746639i	$-0.731666\pi$
$947$	−40.9427	−1.33046	−0.665230	+	0.746639i	$0.731666\pi$
$948$	6.12757	0.199014
$949$	−44.5436	−1.44595
$950$	90.0328	2.92105
$951$	3.52220	0.114215
$952$	2.65347	0.0859993
$953$	−39.2327	−1.27087	−0.635436	−	0.772153i	$-0.719180\pi$
$953$	−39.2327	−1.27087	−0.635436	+	0.772153i	$0.719180\pi$
$954$	−14.9712	−0.484711
$955$	54.4863	1.76314
$956$	7.06742	0.228577
$957$	−26.2461	−0.848414
$958$	−25.5502	−0.825490
$959$	−11.7683	−0.380018
$960$	4.20795	0.135811
$961$	−19.6852	−0.635005
$962$	−55.1172	−1.77705
$963$	12.4814	0.402207
$964$	6.40805	0.206389
$965$	−26.5363	−0.854234
$966$	0.577431	0.0185785
$967$	23.7667	0.764285	0.382143	−	0.924103i	$-0.375186\pi$
$967$	23.7667	0.764285	0.382143	+	0.924103i	$0.375186\pi$
$968$	−28.0142	−0.900410
$969$	−30.6591	−0.984913
$970$	69.5588	2.23340
$971$	23.8475	0.765303	0.382652	−	0.923893i	$-0.375011\pi$
$971$	23.8475	0.765303	0.382652	+	0.923893i	$0.375011\pi$
$972$	−16.0460	−0.514676
$973$	−4.33672	−0.139029
$974$	20.5437	0.658264
$975$	78.2182	2.50499
$976$	−6.41763	−0.205423
$977$	34.8455	1.11481	0.557403	−	0.830242i	$-0.311798\pi$
$977$	34.8455	1.11481	0.557403	+	0.830242i	$0.311798\pi$
$978$	6.16750	0.197215
$979$	36.9716	1.18162
$980$	28.3720	0.906311
$981$	−18.1277	−0.578774
$982$	−8.31143	−0.265229
$983$	23.1947	0.739797	0.369899	−	0.929072i	$-0.379392\pi$
$983$	23.1947	0.739797	0.369899	+	0.929072i	$0.379392\pi$
$984$	−10.8066	−0.344500
$985$	49.4111	1.57437
$986$	19.3093	0.614933
$987$	3.99095	0.127033
$988$	41.1865	1.31032
$989$	8.62265	0.274184
$990$	−53.8955	−1.71291
$991$	30.9724	0.983872	0.491936	−	0.870631i	$-0.336289\pi$
$991$	30.9724	0.983872	0.491936	+	0.870631i	$0.336289\pi$
$992$	3.36375	0.106799
$993$	−18.0719	−0.573495
$994$	7.29269	0.231310
$995$	27.2583	0.864148
$996$	7.52489	0.238435
$997$	−5.66795	−0.179506	−0.0897529	−	0.995964i	$-0.528608\pi$
$997$	−5.66795	−0.179506	−0.0897529	+	0.995964i	$0.528608\pi$
$998$	−14.2634	−0.451501
$999$	−45.4334	−1.43745

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6026.2.a.j.1.12	✓	33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6026.2.a.j.1.12	✓	33	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:	\( N \)	\(=\)	\( 6026 = 2 \cdot 23 \cdot 131 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6026.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.987482 q^{3} +1.00000 q^{4} -4.26130 q^{5} +0.987482 q^{6} +0.584751 q^{7} -1.00000 q^{8} -2.02488 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.987482 q^{3} +1.00000 q^{4} -4.26130 q^{5} +0.987482 q^{6} +0.584751 q^{7} -1.00000 q^{8} -2.02488 q^{9} +4.26130 q^{10} +6.24613 q^{11} -0.987482 q^{12} -6.01959 q^{13} -0.584751 q^{14} +4.20795 q^{15} +1.00000 q^{16} -4.53777 q^{17} +2.02488 q^{18} -6.84209 q^{19} -4.26130 q^{20} -0.577431 q^{21} -6.24613 q^{22} +1.00000 q^{23} +0.987482 q^{24} +13.1587 q^{25} +6.01959 q^{26} +4.96198 q^{27} +0.584751 q^{28} +4.25524 q^{29} -4.20795 q^{30} -3.36375 q^{31} -1.00000 q^{32} -6.16794 q^{33} +4.53777 q^{34} -2.49180 q^{35} -2.02488 q^{36} -9.15630 q^{37} +6.84209 q^{38} +5.94423 q^{39} +4.26130 q^{40} -10.9436 q^{41} +0.577431 q^{42} +8.62265 q^{43} +6.24613 q^{44} +8.62862 q^{45} -1.00000 q^{46} -6.91155 q^{47} -0.987482 q^{48} -6.65807 q^{49} -13.1587 q^{50} +4.48096 q^{51} -6.01959 q^{52} -7.39363 q^{53} -4.96198 q^{54} -26.6166 q^{55} -0.584751 q^{56} +6.75644 q^{57} -4.25524 q^{58} +2.42383 q^{59} +4.20795 q^{60} -6.41763 q^{61} +3.36375 q^{62} -1.18405 q^{63} +1.00000 q^{64} +25.6513 q^{65} +6.16794 q^{66} -7.04733 q^{67} -4.53777 q^{68} -0.987482 q^{69} +2.49180 q^{70} -12.4714 q^{71} +2.02488 q^{72} +7.39978 q^{73} +9.15630 q^{74} -12.9939 q^{75} -6.84209 q^{76} +3.65243 q^{77} -5.94423 q^{78} -6.20525 q^{79} -4.26130 q^{80} +1.17478 q^{81} +10.9436 q^{82} -7.62028 q^{83} -0.577431 q^{84} +19.3368 q^{85} -8.62265 q^{86} -4.20197 q^{87} -6.24613 q^{88} +5.91911 q^{89} -8.62862 q^{90} -3.51996 q^{91} +1.00000 q^{92} +3.32164 q^{93} +6.91155 q^{94} +29.1562 q^{95} +0.987482 q^{96} +16.3234 q^{97} +6.65807 q^{98} -12.6477 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q - 33 q^{2} + 3 q^{3} + 33 q^{4} - 4 q^{5} - 3 q^{6} + 11 q^{7} - 33 q^{8} + 44 q^{9}+O(q^{10}) \) 33 * q - 33 * q^2 + 3 * q^3 + 33 * q^4 - 4 * q^5 - 3 * q^6 + 11 * q^7 - 33 * q^8 + 44 * q^9	\( 33 q - 33 q^{2} + 3 q^{3} + 33 q^{4} - 4 q^{5} - 3 q^{6} + 11 q^{7} - 33 q^{8} + 44 q^{9} + 4 q^{10} + 5 q^{11} + 3 q^{12} + 15 q^{13} - 11 q^{14} + 16 q^{15} + 33 q^{16} + 2 q^{17} - 44 q^{18} + 32 q^{19} - 4 q^{20} + 8 q^{21} - 5 q^{22} + 33 q^{23} - 3 q^{24} + 49 q^{25} - 15 q^{26} + 15 q^{27} + 11 q^{28} + 20 q^{29} - 16 q^{30} + 25 q^{31} - 33 q^{32} - 6 q^{33} - 2 q^{34} + 15 q^{35} + 44 q^{36} + 6 q^{37} - 32 q^{38} + 25 q^{39} + 4 q^{40} + 2 q^{41} - 8 q^{42} + 31 q^{43} + 5 q^{44} + 2 q^{45} - 33 q^{46} + 4 q^{47} + 3 q^{48} + 72 q^{49} - 49 q^{50} + 26 q^{51} + 15 q^{52} - 65 q^{53} - 15 q^{54} - 4 q^{55} - 11 q^{56} + 12 q^{57} - 20 q^{58} + 8 q^{59} + 16 q^{60} + 23 q^{61} - 25 q^{62} - 14 q^{63} + 33 q^{64} + 5 q^{65} + 6 q^{66} + 31 q^{67} + 2 q^{68} + 3 q^{69} - 15 q^{70} + 20 q^{71} - 44 q^{72} + 22 q^{73} - 6 q^{74} - 32 q^{75} + 32 q^{76} + 2 q^{77} - 25 q^{78} + 53 q^{79} - 4 q^{80} + 17 q^{81} - 2 q^{82} + 45 q^{83} + 8 q^{84} + 60 q^{85} - 31 q^{86} + 11 q^{87} - 5 q^{88} - 54 q^{89} - 2 q^{90} + 38 q^{91} + 33 q^{92} + 63 q^{93} - 4 q^{94} + 44 q^{95} - 3 q^{96} - 72 q^{98} + 43 q^{99}+O(q^{100}) \) 33 * q - 33 * q^2 + 3 * q^3 + 33 * q^4 - 4 * q^5 - 3 * q^6 + 11 * q^7 - 33 * q^8 + 44 * q^9 + 4 * q^10 + 5 * q^11 + 3 * q^12 + 15 * q^13 - 11 * q^14 + 16 * q^15 + 33 * q^16 + 2 * q^17 - 44 * q^18 + 32 * q^19 - 4 * q^20 + 8 * q^21 - 5 * q^22 + 33 * q^23 - 3 * q^24 + 49 * q^25 - 15 * q^26 + 15 * q^27 + 11 * q^28 + 20 * q^29 - 16 * q^30 + 25 * q^31 - 33 * q^32 - 6 * q^33 - 2 * q^34 + 15 * q^35 + 44 * q^36 + 6 * q^37 - 32 * q^38 + 25 * q^39 + 4 * q^40 + 2 * q^41 - 8 * q^42 + 31 * q^43 + 5 * q^44 + 2 * q^45 - 33 * q^46 + 4 * q^47 + 3 * q^48 + 72 * q^49 - 49 * q^50 + 26 * q^51 + 15 * q^52 - 65 * q^53 - 15 * q^54 - 4 * q^55 - 11 * q^56 + 12 * q^57 - 20 * q^58 + 8 * q^59 + 16 * q^60 + 23 * q^61 - 25 * q^62 - 14 * q^63 + 33 * q^64 + 5 * q^65 + 6 * q^66 + 31 * q^67 + 2 * q^68 + 3 * q^69 - 15 * q^70 + 20 * q^71 - 44 * q^72 + 22 * q^73 - 6 * q^74 - 32 * q^75 + 32 * q^76 + 2 * q^77 - 25 * q^78 + 53 * q^79 - 4 * q^80 + 17 * q^81 - 2 * q^82 + 45 * q^83 + 8 * q^84 + 60 * q^85 - 31 * q^86 + 11 * q^87 - 5 * q^88 - 54 * q^89 - 2 * q^90 + 38 * q^91 + 33 * q^92 + 63 * q^93 - 4 * q^94 + 44 * q^95 - 3 * q^96 - 72 * q^98 + 43 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more