LMFDB - Embedded newform 845.2.a.l.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [845,2,Mod(1,845)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(845, 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("845.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 845 = 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	845.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:	$6.74735897080$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.4752.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 3x^{2} + 4x + 1 $ x^4 - 2x^3 - 3x^2 + 4*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.49551$ of defining polynomial
Character	$\chi$	$=$	845.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.49551 q^{2} +0.0947876 q^{3} +0.236543 q^{4} +1.00000 q^{5} +0.141756 q^{6} -4.82684 q^{7} -2.63726 q^{8} -2.99102 q^{9} +O(q^{10})$	\(q+1.49551 q^{2} +0.0947876 q^{3} +0.236543 q^{4} +1.00000 q^{5} +0.141756 q^{6} -4.82684 q^{7} -2.63726 q^{8} -2.99102 q^{9} +1.49551 q^{10} +1.06939 q^{11} +0.0224214 q^{12} -7.21857 q^{14} +0.0947876 q^{15} -4.41713 q^{16} +3.55889 q^{17} -4.47309 q^{18} -5.73205 q^{19} +0.236543 q^{20} -0.457524 q^{21} +1.59928 q^{22} -7.08580 q^{23} -0.249980 q^{24} +1.00000 q^{25} -0.567874 q^{27} -1.14176 q^{28} +1.47309 q^{29} +0.141756 q^{30} +1.46410 q^{31} -1.33133 q^{32} +0.101365 q^{33} +5.32235 q^{34} -4.82684 q^{35} -0.707504 q^{36} +0.0253983 q^{37} -8.57233 q^{38} -2.63726 q^{40} -0.267949 q^{41} -0.684231 q^{42} +3.55889 q^{43} +0.252957 q^{44} -2.99102 q^{45} -10.5969 q^{46} -6.51793 q^{47} -0.418689 q^{48} +16.2984 q^{49} +1.49551 q^{50} +0.337339 q^{51} +0.991015 q^{53} -0.849260 q^{54} +1.06939 q^{55} +12.7296 q^{56} -0.543327 q^{57} +2.20301 q^{58} -8.72307 q^{59} +0.0224214 q^{60} +6.33734 q^{61} +2.18958 q^{62} +14.4371 q^{63} +6.84325 q^{64} +0.151592 q^{66} -5.17316 q^{67} +0.841831 q^{68} -0.671646 q^{69} -7.21857 q^{70} +7.76488 q^{71} +7.88809 q^{72} -10.1088 q^{73} +0.0379833 q^{74} +0.0947876 q^{75} -1.35588 q^{76} -5.16177 q^{77} +8.78347 q^{79} -4.41713 q^{80} +8.91922 q^{81} -0.400720 q^{82} +0.725474 q^{83} -0.108224 q^{84} +3.55889 q^{85} +5.32235 q^{86} +0.139630 q^{87} -2.82026 q^{88} -13.5065 q^{89} -4.47309 q^{90} -1.67610 q^{92} +0.138779 q^{93} -9.74761 q^{94} -5.73205 q^{95} -0.126194 q^{96} +3.43870 q^{97} +24.3743 q^{98} -3.19856 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 4 q^{5} + 4 q^{6} - 10 q^{7} - 6 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 4 * q^5 + 4 * q^6 - 10 * q^7 - 6 * q^8 + 4 * q^9	\( 4 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 4 q^{5} + 4 q^{6} - 10 q^{7} - 6 q^{8} + 4 q^{9} - 2 q^{10} - 10 q^{12} + 2 q^{14} - 2 q^{15} + 2 q^{16} - 2 q^{17} - 20 q^{18} - 16 q^{19} + 2 q^{20} - 4 q^{21} + 12 q^{22} - 10 q^{23} + 24 q^{24} + 4 q^{25} - 2 q^{27} - 8 q^{28} + 8 q^{29} + 4 q^{30} - 8 q^{31} - 4 q^{32} - 18 q^{33} + 4 q^{34} - 10 q^{35} + 20 q^{36} + 2 q^{37} + 8 q^{38} - 6 q^{40} - 8 q^{41} - 4 q^{42} - 2 q^{43} - 12 q^{44} + 4 q^{45} - 16 q^{46} - 8 q^{47} - 28 q^{48} + 12 q^{49} - 2 q^{50} + 4 q^{51} - 12 q^{53} + 16 q^{54} + 12 q^{56} + 14 q^{57} - 22 q^{58} - 12 q^{59} - 10 q^{60} + 28 q^{61} + 4 q^{62} - 4 q^{63} + 4 q^{64} + 6 q^{66} - 30 q^{67} + 14 q^{68} - 16 q^{69} + 2 q^{70} - 4 q^{71} - 12 q^{72} + 8 q^{73} - 10 q^{74} - 2 q^{75} - 20 q^{76} + 18 q^{77} - 8 q^{79} + 2 q^{80} - 8 q^{81} + 4 q^{82} + 12 q^{83} + 28 q^{84} - 2 q^{85} + 4 q^{86} - 22 q^{87} - 18 q^{88} + 12 q^{89} - 20 q^{90} + 22 q^{92} - 8 q^{93} - 32 q^{94} - 16 q^{95} - 4 q^{96} - 2 q^{97} + 24 q^{98} - 24 q^{99}+O(q^{100}) \) 4 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 4 * q^5 + 4 * q^6 - 10 * q^7 - 6 * q^8 + 4 * q^9 - 2 * q^10 - 10 * q^12 + 2 * q^14 - 2 * q^15 + 2 * q^16 - 2 * q^17 - 20 * q^18 - 16 * q^19 + 2 * q^20 - 4 * q^21 + 12 * q^22 - 10 * q^23 + 24 * q^24 + 4 * q^25 - 2 * q^27 - 8 * q^28 + 8 * q^29 + 4 * q^30 - 8 * q^31 - 4 * q^32 - 18 * q^33 + 4 * q^34 - 10 * q^35 + 20 * q^36 + 2 * q^37 + 8 * q^38 - 6 * q^40 - 8 * q^41 - 4 * q^42 - 2 * q^43 - 12 * q^44 + 4 * q^45 - 16 * q^46 - 8 * q^47 - 28 * q^48 + 12 * q^49 - 2 * q^50 + 4 * q^51 - 12 * q^53 + 16 * q^54 + 12 * q^56 + 14 * q^57 - 22 * q^58 - 12 * q^59 - 10 * q^60 + 28 * q^61 + 4 * q^62 - 4 * q^63 + 4 * q^64 + 6 * q^66 - 30 * q^67 + 14 * q^68 - 16 * q^69 + 2 * q^70 - 4 * q^71 - 12 * q^72 + 8 * q^73 - 10 * q^74 - 2 * q^75 - 20 * q^76 + 18 * q^77 - 8 * q^79 + 2 * q^80 - 8 * q^81 + 4 * q^82 + 12 * q^83 + 28 * q^84 - 2 * q^85 + 4 * q^86 - 22 * q^87 - 18 * q^88 + 12 * q^89 - 20 * q^90 + 22 * q^92 - 8 * q^93 - 32 * q^94 - 16 * q^95 - 4 * q^96 - 2 * q^97 + 24 * q^98 - 24 * 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 $

$2$	1.49551	1.05748	0.528742	−	0.848783i	$-0.322664\pi$
$2$	1.49551	1.05748	0.528742	+	0.848783i	$0.322664\pi$
$3$	0.0947876	0.0547256	0.0273628	−	0.999626i	$-0.491289\pi$
$3$	0.0947876	0.0547256	0.0273628	+	0.999626i	$0.491289\pi$
$4$	0.236543	0.118272
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0.141756	0.0578715
$7$	−4.82684	−1.82437	−0.912187	−	0.409775i	$-0.865607\pi$
$7$	−4.82684	−1.82437	−0.912187	+	0.409775i	$0.865607\pi$
$8$	−2.63726	−0.932413
$9$	−2.99102	−0.997005
$10$	1.49551	0.472921
$11$	1.06939	0.322433	0.161217	−	0.986919i	$-0.448458\pi$
$11$	1.06939	0.322433	0.161217	+	0.986919i	$0.448458\pi$
$12$	0.0224214	0.00647249
$13$	0	0
$13$	0	0
$14$	−7.21857	−1.92924
$15$	0.0947876	0.0244740
$16$	−4.41713	−1.10428
$17$	3.55889	0.863157	0.431579	−	0.902075i	$-0.357957\pi$
$17$	3.55889	0.863157	0.431579	+	0.902075i	$0.357957\pi$
$18$	−4.47309	−1.05432
$19$	−5.73205	−1.31502	−0.657511	−	0.753445i	$-0.728391\pi$
$19$	−5.73205	−1.31502	−0.657511	+	0.753445i	$0.728391\pi$
$20$	0.236543	0.0528927
$21$	−0.457524	−0.0998400
$22$	1.59928	0.340968
$23$	−7.08580	−1.47749	−0.738746	−	0.673984i	$-0.764582\pi$
$23$	−7.08580	−1.47749	−0.738746	+	0.673984i	$0.764582\pi$
$24$	−0.249980	−0.0510269
$25$	1.00000	0.200000
$26$	0	0
$27$	−0.567874	−0.109287
$28$	−1.14176	−0.215772
$29$	1.47309	0.273545	0.136773	−	0.990602i	$-0.456327\pi$
$29$	1.47309	0.273545	0.136773	+	0.990602i	$0.456327\pi$
$30$	0.141756	0.0258809
$31$	1.46410	0.262960	0.131480	−	0.991319i	$-0.458027\pi$
$31$	1.46410	0.262960	0.131480	+	0.991319i	$0.458027\pi$
$32$	−1.33133	−0.235348
$33$	0.101365	0.0176454
$34$	5.32235	0.912775
$35$	−4.82684	−0.815885
$36$	−0.707504	−0.117917
$37$	0.0253983	0.00417545	0.00208772	−	0.999998i	$-0.499335\pi$
$37$	0.0253983	0.00417545	0.00208772	+	0.999998i	$0.499335\pi$
$38$	−8.57233	−1.39061
$39$	0	0
$40$	−2.63726	−0.416988
$41$	−0.267949	−0.0418466	−0.0209233	−	0.999781i	$-0.506661\pi$
$41$	−0.267949	−0.0418466	−0.0209233	+	0.999781i	$0.506661\pi$
$42$	−0.684231	−0.105579
$43$	3.55889	0.542726	0.271363	−	0.962477i	$-0.412526\pi$
$43$	3.55889	0.542726	0.271363	+	0.962477i	$0.412526\pi$
$44$	0.252957	0.0381347
$45$	−2.99102	−0.445874
$46$	−10.5969	−1.56242
$47$	−6.51793	−0.950738	−0.475369	−	0.879787i	$-0.657685\pi$
$47$	−6.51793	−0.950738	−0.475369	+	0.879787i	$0.657685\pi$
$48$	−0.418689	−0.0604326
$49$	16.2984	2.32834
$50$	1.49551	0.211497
$51$	0.337339	0.0472368
$52$	0	0
$53$	0.991015	0.136126	0.0680632	−	0.997681i	$-0.478318\pi$
$53$	0.991015	0.136126	0.0680632	+	0.997681i	$0.478318\pi$
$54$	−0.849260	−0.115570
$55$	1.06939	0.144196
$56$	12.7296	1.70107
$57$	−0.543327	−0.0719655
$58$	2.20301	0.289270
$59$	−8.72307	−1.13565	−0.567823	−	0.823151i	$-0.692214\pi$
$59$	−8.72307	−1.13565	−0.567823	+	0.823151i	$0.692214\pi$
$60$	0.0224214	0.00289458
$61$	6.33734	0.811413	0.405707	−	0.914003i	$-0.367026\pi$
$61$	6.33734	0.811413	0.405707	+	0.914003i	$0.367026\pi$
$62$	2.18958	0.278076
$63$	14.4371	1.81891
$64$	6.84325	0.855406
$65$	0	0
$66$	0.151592	0.0186597
$67$	−5.17316	−0.632002	−0.316001	−	0.948759i	$-0.602340\pi$
$67$	−5.17316	−0.632002	−0.316001	+	0.948759i	$0.602340\pi$
$68$	0.841831	0.102087
$69$	−0.671646	−0.0808567
$70$	−7.21857	−0.862785
$71$	7.76488	0.921521	0.460761	−	0.887524i	$-0.347577\pi$
$71$	7.76488	0.921521	0.460761	+	0.887524i	$0.347577\pi$
$72$	7.88809	0.929621
$73$	−10.1088	−1.18314	−0.591572	−	0.806252i	$-0.701493\pi$
$73$	−10.1088	−1.18314	−0.591572	+	0.806252i	$0.701493\pi$
$74$	0.0379833	0.00441547
$75$	0.0947876	0.0109451
$76$	−1.35588	−0.155530
$77$	−5.16177	−0.588238
$78$	0	0
$79$	8.78347	0.988218	0.494109	−	0.869400i	$-0.335494\pi$
$79$	8.78347	0.988218	0.494109	+	0.869400i	$0.335494\pi$
$80$	−4.41713	−0.493851
$81$	8.91922	0.991024
$82$	−0.400720	−0.0442521
$83$	0.725474	0.0796311	0.0398155	−	0.999207i	$-0.487323\pi$
$83$	0.725474	0.0796311	0.0398155	+	0.999207i	$0.487323\pi$
$84$	−0.108224	−0.0118082
$85$	3.55889	0.386016
$86$	5.32235	0.573923
$87$	0.139630	0.0149699
$88$	−2.82026	−0.300641
$89$	−13.5065	−1.43169	−0.715845	−	0.698259i	$-0.753958\pi$
$89$	−13.5065	−1.43169	−0.715845	+	0.698259i	$0.753958\pi$
$90$	−4.47309	−0.471505
$91$	0	0
$92$	−1.67610	−0.174745
$93$	0.138779	0.0143907
$94$	−9.74761	−1.00539
$95$	−5.73205	−0.588096
$96$	−0.126194	−0.0128796
$97$	3.43870	0.349147	0.174574	−	0.984644i	$-0.444145\pi$
$97$	3.43870	0.349147	0.174574	+	0.984644i	$0.444145\pi$
$98$	24.3743	2.46218
$99$	−3.19856	−0.321467
$100$	0.236543	0.0236543
$101$	2.85527	0.284110	0.142055	−	0.989859i	$-0.454629\pi$
$101$	2.85527	0.284110	0.142055	+	0.989859i	$0.454629\pi$
$102$	0.504492	0.0499522
$103$	−5.54488	−0.546354	−0.273177	−	0.961964i	$-0.588074\pi$
$103$	−5.54488	−0.546354	−0.273177	+	0.961964i	$0.588074\pi$
$104$	0	0
$105$	−0.457524	−0.0446498
$106$	1.48207	0.143951
$107$	−4.44111	−0.429338	−0.214669	−	0.976687i	$-0.568867\pi$
$107$	−4.44111	−0.429338	−0.214669	+	0.976687i	$0.568867\pi$
$108$	−0.134327	−0.0129256
$109$	−13.7804	−1.31993	−0.659963	−	0.751298i	$-0.729428\pi$
$109$	−13.7804	−1.31993	−0.659963	+	0.751298i	$0.729428\pi$
$110$	1.59928	0.152485
$111$	0.00240744	0.000228504 0
$112$	21.3208	2.01463
$113$	−8.04399	−0.756715	−0.378358	−	0.925660i	$-0.623511\pi$
$113$	−8.04399	−0.756715	−0.378358	+	0.925660i	$0.623511\pi$
$114$	−0.812550	−0.0761023
$115$	−7.08580	−0.660755
$116$	0.348448	0.0323526
$117$	0	0
$118$	−13.0454	−1.20093
$119$	−17.1782	−1.57472
$120$	−0.249980	−0.0228199
$121$	−9.85641	−0.896037
$122$	9.47754	0.858056
$123$	−0.0253983	−0.00229008
$124$	0.346323	0.0311007
$125$	1.00000	0.0894427
$126$	21.5909	1.92347
$127$	0.706653	0.0627053	0.0313526	−	0.999508i	$-0.490019\pi$
$127$	0.706653	0.0627053	0.0313526	+	0.999508i	$0.490019\pi$
$128$	12.8968	1.13993
$129$	0.337339	0.0297010
$130$	0	0
$131$	6.26554	0.547423	0.273711	−	0.961812i	$-0.411749\pi$
$131$	6.26554	0.547423	0.273711	+	0.961812i	$0.411749\pi$
$132$	0.0239772	0.00208694
$133$	27.6677	2.39909
$134$	−7.73650	−0.668332
$135$	−0.567874	−0.0488748
$136$	−9.38573	−0.804820
$137$	−16.3058	−1.39310	−0.696549	−	0.717509i	$-0.745282\pi$
$137$	−16.3058	−1.39310	−0.696549	+	0.717509i	$0.745282\pi$
$138$	−1.00445	−0.0855046
$139$	−6.82528	−0.578913	−0.289456	−	0.957191i	$-0.593475\pi$
$139$	−6.82528	−0.578913	−0.289456	+	0.957191i	$0.593475\pi$
$140$	−1.14176	−0.0964960
$141$	−0.617819	−0.0520297
$142$	11.6124	0.974494
$143$	0	0
$144$	13.2117	1.10098
$145$	1.47309	0.122333
$146$	−15.1178	−1.25116
$147$	1.54488	0.127420
$148$	0.00600778	0.000493837 0
$149$	8.43955	0.691395	0.345698	−	0.938346i	$-0.387642\pi$
$149$	8.43955	0.691395	0.345698	+	0.938346i	$0.387642\pi$
$150$	0.141756	0.0115743
$151$	1.37017	0.111503	0.0557513	−	0.998445i	$-0.482245\pi$
$151$	1.37017	0.111503	0.0557513	+	0.998445i	$0.482245\pi$
$152$	15.1169	1.22614
$153$	−10.6447	−0.860572
$154$	−7.71947	−0.622052
$155$	1.46410	0.117599
$156$	0	0
$157$	11.9700	0.955311	0.477656	−	0.878547i	$-0.341487\pi$
$157$	11.9700	0.955311	0.477656	+	0.878547i	$0.341487\pi$
$158$	13.1357	1.04502
$159$	0.0939360	0.00744961
$160$	−1.33133	−0.105251
$161$	34.2020	2.69550
$162$	13.3388	1.04799
$163$	22.5713	1.76792	0.883962	−	0.467559i	$-0.154866\pi$
$163$	22.5713	1.76792	0.883962	+	0.467559i	$0.154866\pi$
$164$	−0.0633815	−0.00494927
$165$	0.101365	0.00789124
$166$	1.08495	0.0842085
$167$	−8.19700	−0.634303	−0.317152	−	0.948375i	$-0.602726\pi$
$167$	−8.19700	−0.634303	−0.317152	+	0.948375i	$0.602726\pi$
$168$	1.20661	0.0930922
$169$	0	0
$170$	5.32235	0.408205
$171$	17.1447	1.31108
$172$	0.841831	0.0641890
$173$	−9.16772	−0.697009	−0.348505	−	0.937307i	$-0.613310\pi$
$173$	−9.16772	−0.697009	−0.348505	+	0.937307i	$0.613310\pi$
$174$	0.208818	0.0158305
$175$	−4.82684	−0.364875
$176$	−4.72364	−0.356057
$177$	−0.826838	−0.0621490
$178$	−20.1991	−1.51399
$179$	−10.0370	−0.750200	−0.375100	−	0.926984i	$-0.622392\pi$
$179$	−10.0370	−0.750200	−0.375100	+	0.926984i	$0.622392\pi$
$180$	−0.707504	−0.0527342
$181$	17.0238	1.26537	0.632686	−	0.774408i	$-0.281952\pi$
$181$	17.0238	1.26537	0.632686	+	0.774408i	$0.281952\pi$
$182$	0	0
$183$	0.600701	0.0444051
$184$	18.6871	1.37763
$185$	0.0253983	0.00186732
$186$	0.207545	0.0152179
$187$	3.80584	0.278310
$188$	−1.54177	−0.112445
$189$	2.74104	0.199381
$190$	−8.57233	−0.621902
$191$	−3.87741	−0.280559	−0.140280	−	0.990112i	$-0.544800\pi$
$191$	−3.87741	−0.280559	−0.140280	+	0.990112i	$0.544800\pi$
$192$	0.648655	0.0468127
$193$	−1.25394	−0.0902608	−0.0451304	−	0.998981i	$-0.514370\pi$
$193$	−1.25394	−0.0902608	−0.0451304	+	0.998981i	$0.514370\pi$
$194$	5.14261	0.369218
$195$	0	0
$196$	3.85527	0.275376
$197$	15.2820	1.08879	0.544397	−	0.838828i	$-0.316758\pi$
$197$	15.2820	1.08879	0.544397	+	0.838828i	$0.316758\pi$
$198$	−4.78347	−0.339946
$199$	−13.2296	−0.937822	−0.468911	−	0.883246i	$-0.655353\pi$
$199$	−13.2296	−0.937822	−0.468911	+	0.883246i	$0.655353\pi$
$200$	−2.63726	−0.186483
$201$	−0.490352	−0.0345867
$202$	4.27007	0.300441
$203$	−7.11035	−0.499049
$204$	0.0797951	0.00558678
$205$	−0.267949	−0.0187144
$206$	−8.29242	−0.577760
$207$	21.1937	1.47307
$208$	0	0
$209$	−6.12979	−0.424007
$210$	−0.684231	−0.0472164
$211$	−4.81042	−0.331163	−0.165582	−	0.986196i	$-0.552950\pi$
$211$	−4.81042	−0.331163	−0.165582	+	0.986196i	$0.552950\pi$
$212$	0.234418	0.0160999
$213$	0.736014	0.0504309
$214$	−6.64172	−0.454018
$215$	3.55889	0.242714
$216$	1.49763	0.101901
$217$	−7.06698	−0.479738
$218$	−20.6088	−1.39580
$219$	−0.958188	−0.0647484
$220$	0.252957	0.0170543
$221$	0	0
$222$	0.00360034	0.000241639 0
$223$	−14.7132	−0.985271	−0.492635	−	0.870236i	$-0.663966\pi$
$223$	−14.7132	−0.985271	−0.492635	+	0.870236i	$0.663966\pi$
$224$	6.42612	0.429363
$225$	−2.99102	−0.199401
$226$	−12.0299	−0.800214
$227$	−14.9028	−0.989134	−0.494567	−	0.869140i	$-0.664673\pi$
$227$	−14.9028	−0.989134	−0.494567	+	0.869140i	$0.664673\pi$
$228$	−0.128520	−0.00851147
$229$	19.3074	1.27587	0.637933	−	0.770092i	$-0.279790\pi$
$229$	19.3074	1.27587	0.637933	+	0.770092i	$0.279790\pi$
$230$	−10.5969	−0.698737
$231$	−0.489272	−0.0321917
$232$	−3.88492	−0.255057
$233$	−21.1937	−1.38845	−0.694224	−	0.719759i	$-0.744252\pi$
$233$	−21.1937	−1.38845	−0.694224	+	0.719759i	$0.744252\pi$
$234$	0	0
$235$	−6.51793	−0.425183
$236$	−2.06338	−0.134315
$237$	0.832564	0.0540808
$238$	−25.6901	−1.66524
$239$	14.8971	0.963612	0.481806	−	0.876278i	$-0.339981\pi$
$239$	14.8971	0.963612	0.481806	+	0.876278i	$0.339981\pi$
$240$	−0.418689	−0.0270263
$241$	−9.39168	−0.604971	−0.302486	−	0.953154i	$-0.597816\pi$
$241$	−9.39168	−0.604971	−0.302486	+	0.953154i	$0.597816\pi$
$242$	−14.7403	−0.947544
$243$	2.54905	0.163522
$244$	1.49905	0.0959671
$245$	16.2984	1.04126
$246$	−0.0379833	−0.00242173
$247$	0	0
$248$	−3.86122	−0.245188
$249$	0.0687659	0.00435786
$250$	1.49551	0.0945842
$251$	11.3163	0.714281	0.357140	−	0.934051i	$-0.383752\pi$
$251$	11.3163	0.714281	0.357140	+	0.934051i	$0.383752\pi$
$252$	3.41501	0.215125
$253$	−7.57748	−0.476392
$254$	1.05680	0.0663098
$255$	0.337339	0.0211250
$256$	5.60076	0.350047
$257$	−26.5319	−1.65502	−0.827508	−	0.561453i	$-0.810242\pi$
$257$	−26.5319	−1.65502	−0.827508	+	0.561453i	$0.810242\pi$
$258$	0.504492	0.0314083
$259$	−0.122593	−0.00761758
$260$	0	0
$261$	−4.40602	−0.272726
$262$	9.37017	0.578891
$263$	−14.1408	−0.871957	−0.435979	−	0.899957i	$-0.643598\pi$
$263$	−14.1408	−0.871957	−0.435979	+	0.899957i	$0.643598\pi$
$264$	−0.267326	−0.0164528
$265$	0.991015	0.0608776
$266$	41.3772	2.53700
$267$	−1.28025	−0.0783502
$268$	−1.22368	−0.0747479
$269$	24.7745	1.51053	0.755264	−	0.655421i	$-0.227509\pi$
$269$	24.7745	1.51053	0.755264	+	0.655421i	$0.227509\pi$
$270$	−0.849260	−0.0516843
$271$	−18.7171	−1.13698	−0.568492	−	0.822689i	$-0.692473\pi$
$271$	−18.7171	−1.13698	−0.568492	+	0.822689i	$0.692473\pi$
$272$	−15.7201	−0.953170
$273$	0	0
$274$	−24.3854	−1.47318
$275$	1.06939	0.0644866
$276$	−0.158873	−0.00956305
$277$	−22.6647	−1.36179	−0.680893	−	0.732382i	$-0.738408\pi$
$277$	−22.6647	−1.36179	−0.680893	+	0.732382i	$0.738408\pi$
$278$	−10.2073	−0.612191
$279$	−4.37915	−0.262173
$280$	12.7296	0.760742
$281$	27.8384	1.66070	0.830351	−	0.557241i	$-0.188140\pi$
$281$	27.8384	1.66070	0.830351	+	0.557241i	$0.188140\pi$
$282$	−0.923953	−0.0550206
$283$	7.92007	0.470799	0.235400	−	0.971899i	$-0.424360\pi$
$283$	7.92007	0.470799	0.235400	+	0.971899i	$0.424360\pi$
$284$	1.83673	0.108990
$285$	−0.543327	−0.0321839
$286$	0	0
$287$	1.29335	0.0763439
$288$	3.98203	0.234643
$289$	−4.33431	−0.254959
$290$	2.20301	0.129365
$291$	0.325946	0.0191073
$292$	−2.39117	−0.139932
$293$	0.272971	0.0159471	0.00797356	−	0.999968i	$-0.497462\pi$
$293$	0.272971	0.0159471	0.00797356	+	0.999968i	$0.497462\pi$
$294$	2.31038	0.134744
$295$	−8.72307	−0.507877
$296$	−0.0669819	−0.00389324
$297$	−0.607278	−0.0352379
$298$	12.6214	0.731139
$299$	0	0
$300$	0.0224214	0.00129450
$301$	−17.1782	−0.990134
$302$	2.04909	0.117912
$303$	0.270644	0.0155481
$304$	25.3192	1.45216
$305$	6.33734	0.362875
$306$	−15.9192	−0.910041
$307$	−6.85224	−0.391078	−0.195539	−	0.980696i	$-0.562646\pi$
$307$	−6.85224	−0.391078	−0.195539	+	0.980696i	$0.562646\pi$
$308$	−1.22098	−0.0695719
$309$	−0.525586	−0.0298995
$310$	2.18958	0.124360
$311$	10.6447	0.603605	0.301803	−	0.953370i	$-0.402412\pi$
$311$	10.6447	0.603605	0.301803	+	0.953370i	$0.402412\pi$
$312$	0	0
$313$	17.8236	1.00745	0.503724	−	0.863865i	$-0.331963\pi$
$313$	17.8236	1.00745	0.503724	+	0.863865i	$0.331963\pi$
$314$	17.9012	1.01023
$315$	14.4371	0.813441
$316$	2.07767	0.116878
$317$	−8.17161	−0.458963	−0.229482	−	0.973313i	$-0.573703\pi$
$317$	−8.17161	−0.458963	−0.229482	+	0.973313i	$0.573703\pi$
$318$	0.140482	0.00787784
$319$	1.57530	0.0882000
$320$	6.84325	0.382549
$321$	−0.420962	−0.0234958
$322$	51.1494	2.85044
$323$	−20.3997	−1.13507
$324$	2.10978	0.117210
$325$	0	0
$326$	33.7556	1.86955
$327$	−1.30621	−0.0722338
$328$	0.706653	0.0390184
$329$	31.4610	1.73450
$330$	0.151592	0.00834486
$331$	−24.9395	−1.37080	−0.685400	−	0.728167i	$-0.740373\pi$
$331$	−24.9395	−1.37080	−0.685400	+	0.728167i	$0.740373\pi$
$332$	0.171606	0.00941809
$333$	−0.0759666	−0.00416294
$334$	−12.2587	−0.670765
$335$	−5.17316	−0.282640
$336$	2.02095	0.110252
$337$	−19.6057	−1.06799	−0.533996	−	0.845487i	$-0.679310\pi$
$337$	−19.6057	−1.06799	−0.533996	+	0.845487i	$0.679310\pi$
$338$	0	0
$339$	−0.762471	−0.0414117
$340$	0.841831	0.0456547
$341$	1.56569	0.0847871
$342$	25.6400	1.38645
$343$	−44.8817	−2.42339
$344$	−9.38573	−0.506045
$345$	−0.671646	−0.0361602
$346$	−13.7104	−0.737076
$347$	17.0810	0.916955	0.458478	−	0.888706i	$-0.348395\pi$
$347$	17.0810	0.916955	0.458478	+	0.888706i	$0.348395\pi$
$348$	0.0330286	0.00177052
$349$	28.3719	1.51871	0.759357	−	0.650674i	$-0.225514\pi$
$349$	28.3719	1.51871	0.759357	+	0.650674i	$0.225514\pi$
$350$	−7.21857	−0.385849
$351$	0	0
$352$	−1.42371	−0.0758840
$353$	−21.2520	−1.13113	−0.565564	−	0.824704i	$-0.691342\pi$
$353$	−21.2520	−1.13113	−0.565564	+	0.824704i	$0.691342\pi$
$354$	−1.23654	−0.0657215
$355$	7.76488	0.412117
$356$	−3.19488	−0.169328
$357$	−1.62828	−0.0861776
$358$	−15.0104	−0.793325
$359$	−32.6519	−1.72330	−0.861650	−	0.507502i	$-0.830569\pi$
$359$	−32.6519	−1.72330	−0.861650	+	0.507502i	$0.830569\pi$
$360$	7.88809	0.415739
$361$	13.8564	0.729285
$362$	25.4593	1.33811
$363$	−0.934265	−0.0490362
$364$	0	0
$365$	−10.1088	−0.529118
$366$	0.898353	0.0469577
$367$	−5.91837	−0.308936	−0.154468	−	0.987998i	$-0.549366\pi$
$367$	−5.91837	−0.308936	−0.154468	+	0.987998i	$0.549366\pi$
$368$	31.2989	1.63157
$369$	0.801440	0.0417213
$370$	0.0379833	0.00197466
$371$	−4.78347	−0.248345
$372$	0.0328271	0.00170201
$373$	−13.3185	−0.689607	−0.344803	−	0.938675i	$-0.612054\pi$
$373$	−13.3185	−0.689607	−0.344803	+	0.938675i	$0.612054\pi$
$374$	5.69166	0.294309
$375$	0.0947876	0.00489481
$376$	17.1895	0.886480
$377$	0	0
$378$	4.09924	0.210842
$379$	−25.4186	−1.30566	−0.652832	−	0.757503i	$-0.726419\pi$
$379$	−25.4186	−1.30566	−0.652832	+	0.757503i	$0.726419\pi$
$380$	−1.35588	−0.0695550
$381$	0.0669819	0.00343159
$382$	−5.79869	−0.296687
$383$	10.8268	0.553226	0.276613	−	0.960981i	$-0.410788\pi$
$383$	10.8268	0.553226	0.276613	+	0.960981i	$0.410788\pi$
$384$	1.22246	0.0623832
$385$	−5.16177	−0.263068
$386$	−1.87528	−0.0954493
$387$	−10.6447	−0.541100
$388$	0.813402	0.0412942
$389$	−23.0370	−1.16802	−0.584011	−	0.811746i	$-0.698518\pi$
$389$	−23.0370	−1.16802	−0.584011	+	0.811746i	$0.698518\pi$
$390$	0	0
$391$	−25.2176	−1.27531
$392$	−42.9831	−2.17097
$393$	0.593896	0.0299581
$394$	22.8543	1.15138
$395$	8.78347	0.441944
$396$	−0.756597	−0.0380205
$397$	−21.0864	−1.05830	−0.529149	−	0.848529i	$-0.677489\pi$
$397$	−21.0864	−1.05830	−0.529149	+	0.848529i	$0.677489\pi$
$398$	−19.7850	−0.991731
$399$	2.62255	0.131292
$400$	−4.41713	−0.220857
$401$	19.7769	0.987611	0.493805	−	0.869572i	$-0.335605\pi$
$401$	19.7769	0.987611	0.493805	+	0.869572i	$0.335605\pi$
$402$	−0.733324	−0.0365749
$403$	0	0
$404$	0.675394	0.0336021
$405$	8.91922	0.443200
$406$	−10.6336	−0.527736
$407$	0.0271606	0.00134630
$408$	−0.889650	−0.0440443
$409$	31.8809	1.57641	0.788204	−	0.615414i	$-0.211011\pi$
$409$	31.8809	1.57641	0.788204	+	0.615414i	$0.211011\pi$
$410$	−0.400720	−0.0197902
$411$	−1.54559	−0.0762382
$412$	−1.31160	−0.0646181
$413$	42.1048	2.07184
$414$	31.6954	1.55774
$415$	0.725474	0.0356121
$416$	0	0
$417$	−0.646952	−0.0316814
$418$	−9.16715	−0.448380
$419$	30.7296	1.50124	0.750621	−	0.660733i	$-0.229755\pi$
$419$	30.7296	1.50124	0.750621	+	0.660733i	$0.229755\pi$
$420$	−0.108224	−0.00528080
$421$	−17.9820	−0.876391	−0.438195	−	0.898880i	$-0.644382\pi$
$421$	−17.9820	−0.876391	−0.438195	+	0.898880i	$0.644382\pi$
$422$	−7.19403	−0.350200
$423$	19.4952	0.947890
$424$	−2.61357	−0.126926
$425$	3.55889	0.172631
$426$	1.10071	0.0533298
$427$	−30.5893	−1.48032
$428$	−1.05051	−0.0507785
$429$	0	0
$430$	5.32235	0.256666
$431$	−4.89949	−0.236000	−0.118000	−	0.993014i	$-0.537648\pi$
$431$	−4.89949	−0.236000	−0.118000	+	0.993014i	$0.537648\pi$
$432$	2.50837	0.120684
$433$	−19.2394	−0.924588	−0.462294	−	0.886727i	$-0.652974\pi$
$433$	−19.2394	−0.924588	−0.462294	+	0.886727i	$0.652974\pi$
$434$	−10.5687	−0.507315
$435$	0.139630	0.00669476
$436$	−3.25967	−0.156110
$437$	40.6162	1.94294
$438$	−1.43298	−0.0684703
$439$	−8.55974	−0.408534	−0.204267	−	0.978915i	$-0.565481\pi$
$439$	−8.55974	−0.408534	−0.204267	+	0.978915i	$0.565481\pi$
$440$	−2.82026	−0.134451
$441$	−48.7487	−2.32137
$442$	0	0
$443$	−37.9652	−1.80378	−0.901891	−	0.431965i	$-0.857821\pi$
$443$	−37.9652	−1.80378	−0.901891	+	0.431965i	$0.857821\pi$
$444$	0.000569463 0	2.70255e−5 0
$445$	−13.5065	−0.640271
$446$	−22.0037	−1.04191
$447$	0.799965	0.0378370
$448$	−33.0313	−1.56058
$449$	26.7062	1.26035	0.630173	−	0.776455i	$-0.282984\pi$
$449$	26.7062	1.26035	0.630173	+	0.776455i	$0.282984\pi$
$450$	−4.47309	−0.210863
$451$	−0.286542	−0.0134927
$452$	−1.90275	−0.0894979
$453$	0.129875	0.00610205
$454$	−22.2873	−1.04599
$455$	0	0
$456$	1.43290	0.0671016
$457$	4.27127	0.199801	0.0999007	−	0.994997i	$-0.468147\pi$
$457$	4.27127	0.199801	0.0999007	+	0.994997i	$0.468147\pi$
$458$	28.8743	1.34921
$459$	−2.02100	−0.0943322
$460$	−1.67610	−0.0781485
$461$	−20.6423	−0.961407	−0.480704	−	0.876883i	$-0.659619\pi$
$461$	−20.6423	−0.961407	−0.480704	+	0.876883i	$0.659619\pi$
$462$	−0.731710	−0.0340422
$463$	32.1040	1.49200	0.745999	−	0.665947i	$-0.231972\pi$
$463$	32.1040	1.49200	0.745999	+	0.665947i	$0.231972\pi$
$464$	−6.50682	−0.302071
$465$	0.138779	0.00643571
$466$	−31.6954	−1.46826
$467$	23.3774	1.08178	0.540888	−	0.841095i	$-0.318088\pi$
$467$	23.3774	1.08178	0.540888	+	0.841095i	$0.318088\pi$
$468$	0	0
$469$	24.9700	1.15301
$470$	−9.74761	−0.449624
$471$	1.13461	0.0522800
$472$	23.0050	1.05889
$473$	3.80584	0.174993
$474$	1.24511	0.0571896
$475$	−5.73205	−0.263005
$476$	−4.06338	−0.186245
$477$	−2.96414	−0.135719
$478$	22.2787	1.01900
$479$	5.17534	0.236467	0.118234	−	0.992986i	$-0.462277\pi$
$479$	5.17534	0.236467	0.118234	+	0.992986i	$0.462277\pi$
$480$	−0.126194	−0.00575992
$481$	0	0
$482$	−14.0453	−0.639747
$483$	3.24193	0.147513
$484$	−2.33147	−0.105976
$485$	3.43870	0.156143
$486$	3.81213	0.172922
$487$	30.7729	1.39445	0.697227	−	0.716850i	$-0.254417\pi$
$487$	30.7729	1.39445	0.697227	+	0.716850i	$0.254417\pi$
$488$	−16.7132	−0.756572
$489$	2.13948	0.0967508
$490$	24.3743	1.10112
$491$	35.7983	1.61556	0.807778	−	0.589487i	$-0.200670\pi$
$491$	35.7983	1.61556	0.807778	+	0.589487i	$0.200670\pi$
$492$	−0.00600778	−0.000270852 0
$493$	5.24255	0.236113
$494$	0	0
$495$	−3.19856	−0.143765
$496$	−6.46713	−0.290383
$497$	−37.4798	−1.68120
$498$	0.102840	0.00460837
$499$	−28.8971	−1.29361	−0.646805	−	0.762655i	$-0.723895\pi$
$499$	−28.8971	−1.29361	−0.646805	+	0.762655i	$0.723895\pi$
$500$	0.236543	0.0105785
$501$	−0.776974	−0.0347126
$502$	16.9237	0.755340
$503$	−7.86321	−0.350603	−0.175302	−	0.984515i	$-0.556090\pi$
$503$	−7.86321	−0.350603	−0.175302	+	0.984515i	$0.556090\pi$
$504$	−38.0746	−1.69598
$505$	2.85527	0.127058
$506$	−11.3322	−0.503777
$507$	0	0
$508$	0.167154	0.00741625
$509$	28.0113	1.24158	0.620790	−	0.783977i	$-0.286812\pi$
$509$	28.0113	1.24158	0.620790	+	0.783977i	$0.286812\pi$
$510$	0.504492	0.0223393
$511$	48.7935	2.15850
$512$	−17.4176	−0.769757
$513$	3.25508	0.143715
$514$	−39.6787	−1.75015
$515$	−5.54488	−0.244337
$516$	0.0797951	0.00351278
$517$	−6.97020	−0.306549
$518$	−0.183339	−0.00805546
$519$	−0.868986	−0.0381443
$520$	0	0
$521$	−37.5609	−1.64557	−0.822786	−	0.568351i	$-0.807581\pi$
$521$	−37.5609	−1.64557	−0.822786	+	0.568351i	$0.807581\pi$
$522$	−6.58924	−0.288403
$523$	−45.3106	−1.98129	−0.990647	−	0.136450i	$-0.956431\pi$
$523$	−45.3106	−1.98129	−0.990647	+	0.136450i	$0.956431\pi$
$524$	1.48207	0.0647446
$525$	−0.457524	−0.0199680
$526$	−21.1476	−0.922080
$527$	5.21058	0.226976
$528$	−0.447742	−0.0194855
$529$	27.2086	1.18298
$530$	1.48207	0.0643770
$531$	26.0908	1.13225
$532$	6.54460	0.283744
$533$	0	0
$534$	−1.91463	−0.0828540
$535$	−4.44111	−0.192006
$536$	13.6430	0.589287
$537$	−0.951383	−0.0410552
$538$	37.0504	1.59736
$539$	17.4293	0.750733
$540$	−0.134327	−0.00578050
$541$	−19.7445	−0.848882	−0.424441	−	0.905456i	$-0.639529\pi$
$541$	−19.7445	−0.848882	−0.424441	+	0.905456i	$0.639529\pi$
$542$	−27.9916	−1.20234
$543$	1.61365	0.0692483
$544$	−4.73806	−0.203143
$545$	−13.7804	−0.590289
$546$	0	0
$547$	−11.8312	−0.505867	−0.252934	−	0.967484i	$-0.581395\pi$
$547$	−11.8312	−0.505867	−0.252934	+	0.967484i	$0.581395\pi$
$548$	−3.85702	−0.164764
$549$	−18.9551	−0.808983
$550$	1.59928	0.0681935
$551$	−8.44381	−0.359718
$552$	1.77131	0.0753919
$553$	−42.3964	−1.80288
$554$	−33.8952	−1.44007
$555$	0.00240744	0.000102190 0
$556$	−1.61447	−0.0684689
$557$	4.04621	0.171443	0.0857217	−	0.996319i	$-0.472680\pi$
$557$	4.04621	0.171443	0.0857217	+	0.996319i	$0.472680\pi$
$558$	−6.54905	−0.277244
$559$	0	0
$560$	21.3208	0.900968
$561$	0.360746	0.0152307
$562$	41.6326	1.75617
$563$	3.89926	0.164334	0.0821671	−	0.996619i	$-0.473816\pi$
$563$	3.89926	0.164334	0.0821671	+	0.996619i	$0.473816\pi$
$564$	−0.146141	−0.00615364
$565$	−8.04399	−0.338413
$566$	11.8445	0.497863
$567$	−43.0516	−1.80800
$568$	−20.4780	−0.859239
$569$	−17.3356	−0.726744	−0.363372	−	0.931644i	$-0.618375\pi$
$569$	−17.3356	−0.726744	−0.363372	+	0.931644i	$0.618375\pi$
$570$	−0.812550	−0.0340340
$571$	29.5118	1.23503	0.617515	−	0.786559i	$-0.288140\pi$
$571$	29.5118	1.23503	0.617515	+	0.786559i	$0.288140\pi$
$572$	0	0
$573$	−0.367530	−0.0153538
$574$	1.93421	0.0807324
$575$	−7.08580	−0.295498
$576$	−20.4683	−0.852845
$577$	28.3684	1.18099	0.590496	−	0.807041i	$-0.298932\pi$
$577$	28.3684	1.18099	0.590496	+	0.807041i	$0.298932\pi$
$578$	−6.48199	−0.269615
$579$	−0.118858	−0.00493958
$580$	0.348448	0.0144685
$581$	−3.50174	−0.145277
$582$	0.487455	0.0202057
$583$	1.05978	0.0438917
$584$	26.6595	1.10318
$585$	0	0
$586$	0.408230	0.0168638
$587$	34.3877	1.41933	0.709667	−	0.704538i	$-0.248846\pi$
$587$	34.3877	1.41933	0.709667	+	0.704538i	$0.248846\pi$
$588$	0.365432	0.0150701
$589$	−8.39230	−0.345799
$590$	−13.0454	−0.537071
$591$	1.44854	0.0595850
$592$	−0.112187	−0.00461088
$593$	5.47612	0.224877	0.112439	−	0.993659i	$-0.464134\pi$
$593$	5.47612	0.224877	0.112439	+	0.993659i	$0.464134\pi$
$594$	−0.908189	−0.0372635
$595$	−17.1782	−0.704237
$596$	1.99632	0.0817724
$597$	−1.25400	−0.0513229
$598$	0	0
$599$	38.6039	1.57731	0.788657	−	0.614833i	$-0.210777\pi$
$599$	38.6039	1.57731	0.788657	+	0.614833i	$0.210777\pi$
$600$	−0.249980	−0.0102054
$601$	6.57896	0.268361	0.134181	−	0.990957i	$-0.457160\pi$
$601$	6.57896	0.268361	0.134181	+	0.990957i	$0.457160\pi$
$602$	−25.6901	−1.04705
$603$	15.4730	0.630109
$604$	0.324103	0.0131876
$605$	−9.85641	−0.400720
$606$	0.404750	0.0164418
$607$	−16.7664	−0.680525	−0.340263	−	0.940330i	$-0.610516\pi$
$607$	−16.7664	−0.680525	−0.340263	+	0.940330i	$0.610516\pi$
$608$	7.63126	0.309488
$609$	−0.673973	−0.0273108
$610$	9.47754	0.383734
$611$	0	0
$612$	−2.51793	−0.101781
$613$	28.7789	1.16237	0.581184	−	0.813772i	$-0.302590\pi$
$613$	28.7789	1.16237	0.581184	+	0.813772i	$0.302590\pi$
$614$	−10.2476	−0.413558
$615$	−0.0253983	−0.00102416
$616$	13.6129	0.548481
$617$	−37.3291	−1.50281	−0.751406	−	0.659840i	$-0.770624\pi$
$617$	−37.3291	−1.50281	−0.751406	+	0.659840i	$0.770624\pi$
$618$	−0.786018	−0.0316183
$619$	12.7535	0.512606	0.256303	−	0.966597i	$-0.417496\pi$
$619$	12.7535	0.512606	0.256303	+	0.966597i	$0.417496\pi$
$620$	0.346323	0.0139087
$621$	4.02384	0.161471
$622$	15.9192	0.638303
$623$	65.1939	2.61194
$624$	0	0
$625$	1.00000	0.0400000
$626$	26.6553	1.06536
$627$	−0.581028	−0.0232040
$628$	2.83143	0.112986
$629$	0.0903896	0.00360407
$630$	21.5909	0.860201
$631$	28.7242	1.14349	0.571746	−	0.820430i	$-0.306266\pi$
$631$	28.7242	1.14349	0.571746	+	0.820430i	$0.306266\pi$
$632$	−23.1643	−0.921427
$633$	−0.455969	−0.0181231
$634$	−12.2207	−0.485346
$635$	0.706653	0.0280427
$636$	0.0222199	0.000881077 0
$637$	0	0
$638$	2.35588	0.0932701
$639$	−23.2249	−0.918762
$640$	12.8968	0.509791
$641$	−22.3970	−0.884630	−0.442315	−	0.896860i	$-0.645843\pi$
$641$	−22.3970	−0.884630	−0.442315	+	0.896860i	$0.645843\pi$
$642$	−0.629552	−0.0248464
$643$	−14.1642	−0.558581	−0.279290	−	0.960207i	$-0.590099\pi$
$643$	−14.1642	−0.558581	−0.279290	+	0.960207i	$0.590099\pi$
$644$	8.09025	0.318801
$645$	0.337339	0.0132827
$646$	−30.5080	−1.20032
$647$	−23.6097	−0.928193	−0.464096	−	0.885785i	$-0.653621\pi$
$647$	−23.6097	−0.928193	−0.464096	+	0.885785i	$0.653621\pi$
$648$	−23.5223	−0.924044
$649$	−9.32835	−0.366170
$650$	0	0
$651$	−0.669862	−0.0262540
$652$	5.33910	0.209095
$653$	33.2765	1.30221	0.651105	−	0.758987i	$-0.274306\pi$
$653$	33.2765	1.30221	0.651105	+	0.758987i	$0.274306\pi$
$654$	−1.95345	−0.0763861
$655$	6.26554	0.244815
$656$	1.18357	0.0462105
$657$	30.2356	1.17960
$658$	47.0502	1.83421
$659$	23.0908	0.899491	0.449745	−	0.893157i	$-0.351515\pi$
$659$	23.0908	0.899491	0.449745	+	0.893157i	$0.351515\pi$
$660$	0.0239772	0.000933310 0
$661$	−13.4365	−0.522620	−0.261310	−	0.965255i	$-0.584155\pi$
$661$	−13.4365	−0.522620	−0.261310	+	0.965255i	$0.584155\pi$
$662$	−37.2972	−1.44960
$663$	0	0
$664$	−1.91326	−0.0742491
$665$	27.6677	1.07291
$666$	−0.113609	−0.00440224
$667$	−10.4380	−0.404161
$668$	−1.93895	−0.0750200
$669$	−1.39463	−0.0539196
$670$	−7.73650	−0.298887
$671$	6.77708	0.261626
$672$	0.609116	0.0234972
$673$	−1.94524	−0.0749835	−0.0374918	−	0.999297i	$-0.511937\pi$
$673$	−1.94524	−0.0749835	−0.0374918	+	0.999297i	$0.511937\pi$
$674$	−29.3205	−1.12938
$675$	−0.567874	−0.0218575
$676$	0	0
$677$	24.8683	0.955768	0.477884	−	0.878423i	$-0.341404\pi$
$677$	24.8683	0.955768	0.477884	+	0.878423i	$0.341404\pi$
$678$	−1.14028	−0.0437922
$679$	−16.5981	−0.636975
$680$	−9.38573	−0.359926
$681$	−1.41260	−0.0541310
$682$	2.34151	0.0896610
$683$	−14.6221	−0.559500	−0.279750	−	0.960073i	$-0.590252\pi$
$683$	−14.6221	−0.559500	−0.279750	+	0.960073i	$0.590252\pi$
$684$	4.05545	0.155064
$685$	−16.3058	−0.623013
$686$	−67.1210	−2.56269
$687$	1.83010	0.0698226
$688$	−15.7201	−0.599323
$689$	0	0
$690$	−1.00445	−0.0382388
$691$	−3.52451	−0.134079	−0.0670393	−	0.997750i	$-0.521355\pi$
$691$	−3.52451	−0.134079	−0.0670393	+	0.997750i	$0.521355\pi$
$692$	−2.16856	−0.0824364
$693$	15.4389	0.586477
$694$	25.5447	0.969665
$695$	−6.82528	−0.258898
$696$	−0.368242	−0.0139582
$697$	−0.953601	−0.0361202
$698$	42.4304	1.60602
$699$	−2.00890	−0.0759837
$700$	−1.14176	−0.0431543
$701$	−1.53457	−0.0579599	−0.0289800	−	0.999580i	$-0.509226\pi$
$701$	−1.53457	−0.0579599	−0.0289800	+	0.999580i	$0.509226\pi$
$702$	0	0
$703$	−0.145584	−0.00549081
$704$	7.31810	0.275811
$705$	−0.617819	−0.0232684
$706$	−31.7825	−1.19615
$707$	−13.7819	−0.518322
$708$	−0.195583	−0.00735046
$709$	14.0052	0.525978	0.262989	−	0.964799i	$-0.415292\pi$
$709$	14.0052	0.525978	0.262989	+	0.964799i	$0.415292\pi$
$710$	11.6124	0.435807
$711$	−26.2715	−0.985258
$712$	35.6203	1.33493
$713$	−10.3743	−0.388522
$714$	−2.43510	−0.0911314
$715$	0	0
$716$	−2.37418	−0.0887274
$717$	1.41206	0.0527343
$718$	−48.8312	−1.82236
$719$	22.4761	0.838218	0.419109	−	0.907936i	$-0.362343\pi$
$719$	22.4761	0.838218	0.419109	+	0.907936i	$0.362343\pi$
$720$	13.2117	0.492372
$721$	26.7643	0.996753
$722$	20.7224	0.771206
$723$	−0.890215	−0.0331074
$724$	4.02687	0.149658
$725$	1.47309	0.0547091
$726$	−1.39720	−0.0518550
$727$	−10.3421	−0.383566	−0.191783	−	0.981437i	$-0.561427\pi$
$727$	−10.3421	−0.383566	−0.191783	+	0.981437i	$0.561427\pi$
$728$	0	0
$729$	−26.5160	−0.982075
$730$	−15.1178	−0.559534
$731$	12.6657	0.468458
$732$	0.142092	0.00525186
$733$	27.3533	1.01032	0.505159	−	0.863026i	$-0.331434\pi$
$733$	27.3533	1.01032	0.505159	+	0.863026i	$0.331434\pi$
$734$	−8.85096	−0.326695
$735$	1.54488	0.0569839
$736$	9.43355	0.347725
$737$	−5.53212	−0.203778
$738$	1.19856	0.0441196
$739$	−13.4517	−0.494827	−0.247413	−	0.968910i	$-0.579581\pi$
$739$	−13.4517	−0.494827	−0.247413	+	0.968910i	$0.579581\pi$
$740$	0.00600778	0.000220851 0
$741$	0	0
$742$	−7.15372	−0.262621
$743$	16.3926	0.601388	0.300694	−	0.953721i	$-0.402782\pi$
$743$	16.3926	0.601388	0.300694	+	0.953721i	$0.402782\pi$
$744$	−0.365996	−0.0134181
$745$	8.43955	0.309201
$746$	−19.9179	−0.729248
$747$	−2.16990	−0.0793926
$748$	0.900245	0.0329162
$749$	21.4365	0.783274
$750$	0.141756	0.00517618
$751$	−27.6655	−1.00953	−0.504764	−	0.863257i	$-0.668421\pi$
$751$	−27.6655	−1.00953	−0.504764	+	0.863257i	$0.668421\pi$
$752$	28.7906	1.04988
$753$	1.07265	0.0390895
$754$	0	0
$755$	1.37017	0.0498654
$756$	0.648373	0.0235811
$757$	22.9978	0.835870	0.417935	−	0.908477i	$-0.362754\pi$
$757$	22.9978	0.835870	0.417935	+	0.908477i	$0.362754\pi$
$758$	−38.0136	−1.38072
$759$	−0.718251	−0.0260709
$760$	15.1169	0.548349
$761$	7.66442	0.277835	0.138918	−	0.990304i	$-0.455638\pi$
$761$	7.66442	0.277835	0.138918	+	0.990304i	$0.455638\pi$
$762$	0.100172	0.00362885
$763$	66.5160	2.40804
$764$	−0.917174	−0.0331822
$765$	−10.6447	−0.384860
$766$	16.1916	0.585027
$767$	0	0
$768$	0.530882	0.0191566
$769$	7.23095	0.260755	0.130377	−	0.991464i	$-0.458381\pi$
$769$	7.23095	0.260755	0.130377	+	0.991464i	$0.458381\pi$
$770$	−7.71947	−0.278190
$771$	−2.51490	−0.0905718
$772$	−0.296612	−0.0106753
$773$	33.5995	1.20849	0.604246	−	0.796798i	$-0.293475\pi$
$773$	33.5995	1.20849	0.604246	+	0.796798i	$0.293475\pi$
$774$	−15.9192	−0.572204
$775$	1.46410	0.0525921
$776$	−9.06877	−0.325550
$777$	−0.0116203	−0.000416877 0
$778$	−34.4520	−1.23516
$779$	1.53590	0.0550293
$780$	0	0
$781$	8.30368	0.297129
$782$	−37.7131	−1.34862
$783$	−0.836527	−0.0298950
$784$	−71.9921	−2.57115
$785$	11.9700	0.427228
$786$	0.888175	0.0316802
$787$	2.97168	0.105929	0.0529645	−	0.998596i	$-0.483133\pi$
$787$	2.97168	0.105929	0.0529645	+	0.998596i	$0.483133\pi$
$788$	3.61484	0.128773
$789$	−1.34037	−0.0477184
$790$	13.1357	0.467349
$791$	38.8270	1.38053
$792$	8.43544	0.299740
$793$	0	0
$794$	−31.5349	−1.11913
$795$	0.0939360	0.00333156
$796$	−3.12937	−0.110918
$797$	−22.5751	−0.799650	−0.399825	−	0.916591i	$-0.630929\pi$
$797$	−22.5751	−0.799650	−0.399825	+	0.916591i	$0.630929\pi$
$798$	3.92205	0.138839
$799$	−23.1966	−0.820636
$800$	−1.33133	−0.0470696
$801$	40.3983	1.42740
$802$	29.5765	1.04438
$803$	−10.8102	−0.381485
$804$	−0.115989	−0.00409063
$805$	34.2020	1.20546
$806$	0	0
$807$	2.34831	0.0826646
$808$	−7.53009	−0.264908
$809$	13.6584	0.480204	0.240102	−	0.970748i	$-0.422819\pi$
$809$	13.6584	0.480204	0.240102	+	0.970748i	$0.422819\pi$
$810$	13.3388	0.468676
$811$	−14.1147	−0.495636	−0.247818	−	0.968807i	$-0.579713\pi$
$811$	−14.1147	−0.495636	−0.247818	+	0.968807i	$0.579713\pi$
$812$	−1.68190	−0.0590233
$813$	−1.77415	−0.0622222
$814$	0.0406189	0.00142369
$815$	22.5713	0.790640
$816$	−1.49007	−0.0521629
$817$	−20.3997	−0.713696
$818$	47.6781	1.66703
$819$	0	0
$820$	−0.0633815	−0.00221338
$821$	−1.58562	−0.0553383	−0.0276692	−	0.999617i	$-0.508808\pi$
$821$	−1.58562	−0.0553383	−0.0276692	+	0.999617i	$0.508808\pi$
$822$	−2.31144	−0.0806206
$823$	18.5648	0.647127	0.323563	−	0.946206i	$-0.395119\pi$
$823$	18.5648	0.647127	0.323563	+	0.946206i	$0.395119\pi$
$824$	14.6233	0.509427
$825$	0.101365	0.00352907
$826$	62.9681	2.19094
$827$	−9.01023	−0.313316	−0.156658	−	0.987653i	$-0.550072\pi$
$827$	−9.01023	−0.313316	−0.156658	+	0.987653i	$0.550072\pi$
$828$	5.01324	0.174222
$829$	47.1177	1.63647	0.818233	−	0.574887i	$-0.194954\pi$
$829$	47.1177	1.63647	0.818233	+	0.574887i	$0.194954\pi$
$830$	1.08495	0.0376592
$831$	−2.14833	−0.0745247
$832$	0	0
$833$	58.0041	2.00972
$834$	−0.967522	−0.0335025
$835$	−8.19700	−0.283669
$836$	−1.44996	−0.0501479
$837$	−0.831425	−0.0287383
$838$	45.9564	1.58754
$839$	53.4766	1.84622	0.923108	−	0.384541i	$-0.125640\pi$
$839$	53.4766	1.84622	0.923108	+	0.384541i	$0.125640\pi$
$840$	1.20661	0.0416321
$841$	−26.8300	−0.925173
$842$	−26.8923	−0.926769
$843$	2.63874	0.0908830
$844$	−1.13787	−0.0391672
$845$	0	0
$846$	29.1553	1.00238
$847$	47.5753	1.63471
$848$	−4.37745	−0.150322
$849$	0.750724	0.0257648
$850$	5.32235	0.182555
$851$	−0.179967	−0.00616919
$852$	0.174099	0.00596454
$853$	−27.7756	−0.951019	−0.475510	−	0.879711i	$-0.657736\pi$
$853$	−27.7756	−0.951019	−0.475510	+	0.879711i	$0.657736\pi$
$854$	−45.7465	−1.56541
$855$	17.1447	0.586335
$856$	11.7124	0.400321
$857$	53.6917	1.83407	0.917037	−	0.398801i	$-0.130574\pi$
$857$	53.6917	1.83407	0.917037	+	0.398801i	$0.130574\pi$
$858$	0	0
$859$	2.08958	0.0712955	0.0356477	−	0.999364i	$-0.488651\pi$
$859$	2.08958	0.0712955	0.0356477	+	0.999364i	$0.488651\pi$
$860$	0.841831	0.0287062
$861$	0.122593	0.00417797
$862$	−7.32722	−0.249566
$863$	−1.75413	−0.0597113	−0.0298557	−	0.999554i	$-0.509505\pi$
$863$	−1.75413	−0.0597113	−0.0298557	+	0.999554i	$0.509505\pi$
$864$	0.756028	0.0257206
$865$	−9.16772	−0.311712
$866$	−28.7727	−0.977737
$867$	−0.410839	−0.0139528
$868$	−1.67165	−0.0567394
$869$	9.39295	0.318634
$870$	0.208818	0.00707960
$871$	0	0
$872$	36.3426	1.23072
$873$	−10.2852	−0.348102
$874$	60.7418	2.05462
$875$	−4.82684	−0.163177
$876$	−0.226653	−0.00765789
$877$	21.5672	0.728272	0.364136	−	0.931346i	$-0.381364\pi$
$877$	21.5672	0.728272	0.364136	+	0.931346i	$0.381364\pi$
$878$	−12.8012	−0.432018
$879$	0.0258742	0.000872716 0
$880$	−4.72364	−0.159234
$881$	25.0263	0.843158	0.421579	−	0.906792i	$-0.361476\pi$
$881$	25.0263	0.843158	0.421579	+	0.906792i	$0.361476\pi$
$882$	−72.9040	−2.45481
$883$	−48.7832	−1.64169	−0.820843	−	0.571154i	$-0.806496\pi$
$883$	−48.7832	−1.64169	−0.820843	+	0.571154i	$0.806496\pi$
$884$	0	0
$885$	−0.826838	−0.0277939
$886$	−56.7772	−1.90747
$887$	−33.7933	−1.13467	−0.567334	−	0.823488i	$-0.692025\pi$
$887$	−33.7933	−1.13467	−0.567334	+	0.823488i	$0.692025\pi$
$888$	−0.00634905	−0.000213060 0
$889$	−3.41090	−0.114398
$890$	−20.1991	−0.677076
$891$	9.53812	0.319539
$892$	−3.48031	−0.116530
$893$	37.3611	1.25024
$894$	1.19635	0.0400121
$895$	−10.0370	−0.335500
$896$	−62.2508	−2.07965
$897$	0	0
$898$	39.9394	1.33279
$899$	2.15675	0.0719316
$900$	−0.707504	−0.0235835
$901$	3.52691	0.117499
$902$	−0.428526	−0.0142683
$903$	−1.62828	−0.0541857
$904$	21.2141	0.705571
$905$	17.0238	0.565892
$906$	0.194229	0.00645281
$907$	−34.6270	−1.14977	−0.574885	−	0.818234i	$-0.694953\pi$
$907$	−34.6270	−1.14977	−0.574885	+	0.818234i	$0.694953\pi$
$908$	−3.52516	−0.116986
$909$	−8.54015	−0.283259
$910$	0	0
$911$	31.1865	1.03326	0.516628	−	0.856210i	$-0.327187\pi$
$911$	31.1865	1.03326	0.516628	+	0.856210i	$0.327187\pi$
$912$	2.39995	0.0794703
$913$	0.775814	0.0256757
$914$	6.38771	0.211287
$915$	0.600701	0.0198586
$916$	4.56702	0.150899
$917$	−30.2428	−0.998704
$918$	−3.02242	−0.0997548
$919$	−51.9220	−1.71275	−0.856374	−	0.516356i	$-0.827288\pi$
$919$	−51.9220	−1.71275	−0.856374	+	0.516356i	$0.827288\pi$
$920$	18.6871	0.616096
$921$	−0.649507	−0.0214020
$922$	−30.8707	−1.01667
$923$	0	0
$924$	−0.115734	−0.00380736
$925$	0.0253983	0.000835090 0
$926$	48.0117	1.57776
$927$	16.5848	0.544717
$928$	−1.96117	−0.0643784
$929$	20.4915	0.672304	0.336152	−	0.941808i	$-0.390874\pi$
$929$	20.4915	0.672304	0.336152	+	0.941808i	$0.390874\pi$
$930$	0.207545	0.00680565
$931$	−93.4231	−3.06182
$932$	−5.01324	−0.164214
$933$	1.00898	0.0330327
$934$	34.9610	1.14396
$935$	3.80584	0.124464
$936$	0	0
$937$	−39.6806	−1.29631	−0.648154	−	0.761510i	$-0.724459\pi$
$937$	−39.6806	−1.29631	−0.648154	+	0.761510i	$0.724459\pi$
$938$	37.3428	1.21929
$939$	1.68945	0.0551333
$940$	−1.54177	−0.0502870
$941$	−19.6189	−0.639557	−0.319779	−	0.947492i	$-0.603609\pi$
$941$	−19.6189	−0.639557	−0.319779	+	0.947492i	$0.603609\pi$
$942$	1.69682	0.0552853
$943$	1.89864	0.0618281
$944$	38.5309	1.25408
$945$	2.74104	0.0891659
$946$	5.69166	0.185052
$947$	57.2124	1.85915	0.929576	−	0.368631i	$-0.120173\pi$
$947$	57.2124	1.85915	0.929576	+	0.368631i	$0.120173\pi$
$948$	0.196937	0.00639623
$949$	0	0
$950$	−8.57233	−0.278123
$951$	−0.774567	−0.0251170
$952$	45.3034	1.46829
$953$	27.4770	0.890066	0.445033	−	0.895514i	$-0.353192\pi$
$953$	27.4770	0.890066	0.445033	+	0.895514i	$0.353192\pi$
$954$	−4.43290	−0.143520
$955$	−3.87741	−0.125470
$956$	3.52380	0.113968
$957$	0.149319	0.00482680
$958$	7.73976	0.250060
$959$	78.7055	2.54153
$960$	0.648655	0.0209353
$961$	−28.8564	−0.930852
$962$	0	0
$963$	13.2834	0.428053
$964$	−2.22154	−0.0715509
$965$	−1.25394	−0.0403659
$966$	4.84833	0.155992
$967$	−10.3643	−0.333293	−0.166647	−	0.986017i	$-0.553294\pi$
$967$	−10.3643	−0.333293	−0.166647	+	0.986017i	$0.553294\pi$
$968$	25.9939	0.835477
$969$	−1.93364	−0.0621175
$970$	5.14261	0.165119
$971$	41.7515	1.33987	0.669935	−	0.742420i	$-0.266322\pi$
$971$	41.7515	1.33987	0.669935	+	0.742420i	$0.266322\pi$
$972$	0.602961	0.0193400
$973$	32.9445	1.05615
$974$	46.0212	1.47461
$975$	0	0
$976$	−27.9929	−0.896030
$977$	−13.7938	−0.441303	−0.220652	−	0.975353i	$-0.570818\pi$
$977$	−13.7938	−0.441303	−0.220652	+	0.975353i	$0.570818\pi$
$978$	3.19961	0.102312
$979$	−14.4437	−0.461624
$980$	3.85527	0.123152
$981$	41.2175	1.31597
$982$	53.5367	1.70842
$983$	37.9997	1.21200	0.606002	−	0.795463i	$-0.292773\pi$
$983$	37.9997	1.21200	0.606002	+	0.795463i	$0.292773\pi$
$984$	0.0669819	0.00213530
$985$	15.2820	0.486924
$986$	7.84028	0.249685
$987$	2.98211	0.0949217
$988$	0	0
$989$	−25.2176	−0.801873
$990$	−4.78347	−0.152029
$991$	−52.5530	−1.66940	−0.834700	−	0.550705i	$-0.814359\pi$
$991$	−52.5530	−1.66940	−0.834700	+	0.550705i	$0.814359\pi$
$992$	−1.94920	−0.0618873
$993$	−2.36396	−0.0750179
$994$	−56.0513	−1.77784
$995$	−13.2296	−0.419407
$996$	0.0162661	0.000515411 0
$997$	18.5899	0.588749	0.294375	−	0.955690i	$-0.404889\pi$
$997$	18.5899	0.588749	0.294375	+	0.955690i	$0.404889\pi$
$998$	−43.2158	−1.36797
$999$	−0.0144230	−0.000456324 0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.a.l.1.4		4
3.2	odd	2		7605.2.a.cj.1.1		4
5.4	even	2		4225.2.a.bl.1.1		4
13.2	odd	12		65.2.m.a.56.4	yes	8
13.3	even	3		845.2.e.n.191.1		8
13.4	even	6		845.2.e.m.146.4		8
13.5	odd	4		845.2.c.g.506.2		8
13.6	odd	12		845.2.m.g.361.1		8
13.7	odd	12		65.2.m.a.36.4	✓	8
13.8	odd	4		845.2.c.g.506.7		8
13.9	even	3		845.2.e.n.146.1		8
13.10	even	6		845.2.e.m.191.4		8
13.11	odd	12		845.2.m.g.316.1		8
13.12	even	2		845.2.a.m.1.1		4
39.2	even	12		585.2.bu.c.316.1		8
39.20	even	12		585.2.bu.c.361.1		8
39.38	odd	2		7605.2.a.cf.1.4		4
52.7	even	12		1040.2.da.b.881.3		8
52.15	even	12		1040.2.da.b.641.3		8
65.2	even	12		325.2.m.c.199.1		8
65.7	even	12		325.2.m.b.49.4		8
65.28	even	12		325.2.m.b.199.4		8
65.33	even	12		325.2.m.c.49.1		8
65.54	odd	12		325.2.n.d.251.1		8
65.59	odd	12		325.2.n.d.101.1		8
65.64	even	2		4225.2.a.bi.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.m.a.36.4	✓	8	13.7	odd	12
65.2.m.a.56.4	yes	8	13.2	odd	12
325.2.m.b.49.4		8	65.7	even	12
325.2.m.b.199.4		8	65.28	even	12
325.2.m.c.49.1		8	65.33	even	12
325.2.m.c.199.1		8	65.2	even	12
325.2.n.d.101.1		8	65.59	odd	12
325.2.n.d.251.1		8	65.54	odd	12
585.2.bu.c.316.1		8	39.2	even	12
585.2.bu.c.361.1		8	39.20	even	12
845.2.a.l.1.4		4	1.1	even	1	trivial
845.2.a.m.1.1		4	13.12	even	2
845.2.c.g.506.2		8	13.5	odd	4
845.2.c.g.506.7		8	13.8	odd	4
845.2.e.m.146.4		8	13.4	even	6
845.2.e.m.191.4		8	13.10	even	6
845.2.e.n.146.1		8	13.9	even	3
845.2.e.n.191.1		8	13.3	even	3
845.2.m.g.316.1		8	13.11	odd	12
845.2.m.g.361.1		8	13.6	odd	12
1040.2.da.b.641.3		8	52.15	even	12
1040.2.da.b.881.3		8	52.7	even	12
4225.2.a.bi.1.4		4	65.64	even	2
4225.2.a.bl.1.1		4	5.4	even	2
7605.2.a.cf.1.4		4	39.38	odd	2
7605.2.a.cj.1.1		4	3.2	odd	2

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:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.a (trivial)

\(f(q)\)	\(=\)	\(q+1.49551 q^{2} +0.0947876 q^{3} +0.236543 q^{4} +1.00000 q^{5} +0.141756 q^{6} -4.82684 q^{7} -2.63726 q^{8} -2.99102 q^{9} +O(q^{10})\)	\(q+1.49551 q^{2} +0.0947876 q^{3} +0.236543 q^{4} +1.00000 q^{5} +0.141756 q^{6} -4.82684 q^{7} -2.63726 q^{8} -2.99102 q^{9} +1.49551 q^{10} +1.06939 q^{11} +0.0224214 q^{12} -7.21857 q^{14} +0.0947876 q^{15} -4.41713 q^{16} +3.55889 q^{17} -4.47309 q^{18} -5.73205 q^{19} +0.236543 q^{20} -0.457524 q^{21} +1.59928 q^{22} -7.08580 q^{23} -0.249980 q^{24} +1.00000 q^{25} -0.567874 q^{27} -1.14176 q^{28} +1.47309 q^{29} +0.141756 q^{30} +1.46410 q^{31} -1.33133 q^{32} +0.101365 q^{33} +5.32235 q^{34} -4.82684 q^{35} -0.707504 q^{36} +0.0253983 q^{37} -8.57233 q^{38} -2.63726 q^{40} -0.267949 q^{41} -0.684231 q^{42} +3.55889 q^{43} +0.252957 q^{44} -2.99102 q^{45} -10.5969 q^{46} -6.51793 q^{47} -0.418689 q^{48} +16.2984 q^{49} +1.49551 q^{50} +0.337339 q^{51} +0.991015 q^{53} -0.849260 q^{54} +1.06939 q^{55} +12.7296 q^{56} -0.543327 q^{57} +2.20301 q^{58} -8.72307 q^{59} +0.0224214 q^{60} +6.33734 q^{61} +2.18958 q^{62} +14.4371 q^{63} +6.84325 q^{64} +0.151592 q^{66} -5.17316 q^{67} +0.841831 q^{68} -0.671646 q^{69} -7.21857 q^{70} +7.76488 q^{71} +7.88809 q^{72} -10.1088 q^{73} +0.0379833 q^{74} +0.0947876 q^{75} -1.35588 q^{76} -5.16177 q^{77} +8.78347 q^{79} -4.41713 q^{80} +8.91922 q^{81} -0.400720 q^{82} +0.725474 q^{83} -0.108224 q^{84} +3.55889 q^{85} +5.32235 q^{86} +0.139630 q^{87} -2.82026 q^{88} -13.5065 q^{89} -4.47309 q^{90} -1.67610 q^{92} +0.138779 q^{93} -9.74761 q^{94} -5.73205 q^{95} -0.126194 q^{96} +3.43870 q^{97} +24.3743 q^{98} -3.19856 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 4 q^{5} + 4 q^{6} - 10 q^{7} - 6 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 4 * q^5 + 4 * q^6 - 10 * q^7 - 6 * q^8 + 4 * q^9	\( 4 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 4 q^{5} + 4 q^{6} - 10 q^{7} - 6 q^{8} + 4 q^{9} - 2 q^{10} - 10 q^{12} + 2 q^{14} - 2 q^{15} + 2 q^{16} - 2 q^{17} - 20 q^{18} - 16 q^{19} + 2 q^{20} - 4 q^{21} + 12 q^{22} - 10 q^{23} + 24 q^{24} + 4 q^{25} - 2 q^{27} - 8 q^{28} + 8 q^{29} + 4 q^{30} - 8 q^{31} - 4 q^{32} - 18 q^{33} + 4 q^{34} - 10 q^{35} + 20 q^{36} + 2 q^{37} + 8 q^{38} - 6 q^{40} - 8 q^{41} - 4 q^{42} - 2 q^{43} - 12 q^{44} + 4 q^{45} - 16 q^{46} - 8 q^{47} - 28 q^{48} + 12 q^{49} - 2 q^{50} + 4 q^{51} - 12 q^{53} + 16 q^{54} + 12 q^{56} + 14 q^{57} - 22 q^{58} - 12 q^{59} - 10 q^{60} + 28 q^{61} + 4 q^{62} - 4 q^{63} + 4 q^{64} + 6 q^{66} - 30 q^{67} + 14 q^{68} - 16 q^{69} + 2 q^{70} - 4 q^{71} - 12 q^{72} + 8 q^{73} - 10 q^{74} - 2 q^{75} - 20 q^{76} + 18 q^{77} - 8 q^{79} + 2 q^{80} - 8 q^{81} + 4 q^{82} + 12 q^{83} + 28 q^{84} - 2 q^{85} + 4 q^{86} - 22 q^{87} - 18 q^{88} + 12 q^{89} - 20 q^{90} + 22 q^{92} - 8 q^{93} - 32 q^{94} - 16 q^{95} - 4 q^{96} - 2 q^{97} + 24 q^{98} - 24 q^{99}+O(q^{100}) \) 4 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 4 * q^5 + 4 * q^6 - 10 * q^7 - 6 * q^8 + 4 * q^9 - 2 * q^10 - 10 * q^12 + 2 * q^14 - 2 * q^15 + 2 * q^16 - 2 * q^17 - 20 * q^18 - 16 * q^19 + 2 * q^20 - 4 * q^21 + 12 * q^22 - 10 * q^23 + 24 * q^24 + 4 * q^25 - 2 * q^27 - 8 * q^28 + 8 * q^29 + 4 * q^30 - 8 * q^31 - 4 * q^32 - 18 * q^33 + 4 * q^34 - 10 * q^35 + 20 * q^36 + 2 * q^37 + 8 * q^38 - 6 * q^40 - 8 * q^41 - 4 * q^42 - 2 * q^43 - 12 * q^44 + 4 * q^45 - 16 * q^46 - 8 * q^47 - 28 * q^48 + 12 * q^49 - 2 * q^50 + 4 * q^51 - 12 * q^53 + 16 * q^54 + 12 * q^56 + 14 * q^57 - 22 * q^58 - 12 * q^59 - 10 * q^60 + 28 * q^61 + 4 * q^62 - 4 * q^63 + 4 * q^64 + 6 * q^66 - 30 * q^67 + 14 * q^68 - 16 * q^69 + 2 * q^70 - 4 * q^71 - 12 * q^72 + 8 * q^73 - 10 * q^74 - 2 * q^75 - 20 * q^76 + 18 * q^77 - 8 * q^79 + 2 * q^80 - 8 * q^81 + 4 * q^82 + 12 * q^83 + 28 * q^84 - 2 * q^85 + 4 * q^86 - 22 * q^87 - 18 * q^88 + 12 * q^89 - 20 * q^90 + 22 * q^92 - 8 * q^93 - 32 * q^94 - 16 * q^95 - 4 * q^96 - 2 * q^97 + 24 * q^98 - 24 * q^99

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