LMFDB - Embedded newform 4650.2.a.ci.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4650,2,Mod(1,4650)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4650.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4650, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4650.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-3,-3,3,0,3,-6,-3,3,0,8,-3,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$37.1304369399$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1708.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 8x - 2 $ x^3 - x^2 - 8*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 930)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.47090$ of defining polynomial
Character	$\chi$	$=$	4650.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -0.470896 q^{11} -1.00000 q^{12} -6.47090 q^{13} +2.00000 q^{14} +1.00000 q^{16} -7.04712 q^{17} -1.00000 q^{18} -7.04712 q^{19} +2.00000 q^{21} +0.470896 q^{22} -6.94179 q^{23} +1.00000 q^{24} +6.47090 q^{26} -1.00000 q^{27} -2.00000 q^{28} -6.94179 q^{29} -1.00000 q^{31} -1.00000 q^{32} +0.470896 q^{33} +7.04712 q^{34} +1.00000 q^{36} +1.78935 q^{37} +7.04712 q^{38} +6.47090 q^{39} +2.00000 q^{41} -2.00000 q^{42} +0.210649 q^{43} -0.470896 q^{44} +6.94179 q^{46} +7.04712 q^{47} -1.00000 q^{48} -3.00000 q^{49} +7.04712 q^{51} -6.47090 q^{52} +3.15244 q^{53} +1.00000 q^{54} +2.00000 q^{56} +7.04712 q^{57} +6.94179 q^{58} -7.15244 q^{59} +11.2578 q^{61} +1.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -0.470896 q^{66} +8.26025 q^{67} -7.04712 q^{68} +6.94179 q^{69} +0.260246 q^{71} -1.00000 q^{72} -11.8836 q^{73} -1.78935 q^{74} -7.04712 q^{76} +0.941791 q^{77} -6.47090 q^{78} -1.89468 q^{79} +1.00000 q^{81} -2.00000 q^{82} +11.7783 q^{83} +2.00000 q^{84} -0.210649 q^{86} +6.94179 q^{87} +0.470896 q^{88} +12.0942 q^{89} +12.9418 q^{91} -6.94179 q^{92} +1.00000 q^{93} -7.04712 q^{94} +1.00000 q^{96} +3.52910 q^{97} +3.00000 q^{98} -0.470896 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{6} - 6 q^{7} - 3 q^{8} + 3 q^{9} + 8 q^{11} - 3 q^{12} - 10 q^{13} + 6 q^{14} + 3 q^{16} - 2 q^{17} - 3 q^{18} - 2 q^{19} + 6 q^{21} - 8 q^{22} - 2 q^{23} + 3 q^{24}+ \cdots + 8 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^6 - 6 * q^7 - 3 * q^8 + 3 * q^9 + 8 * q^11 - 3 * q^12 - 10 * q^13 + 6 * q^14 + 3 * q^16 - 2 * q^17 - 3 * q^18 - 2 * q^19 + 6 * q^21 - 8 * q^22 - 2 * q^23 + 3 * q^24 + 10 * q^26 - 3 * q^27 - 6 * q^28 - 2 * q^29 - 3 * q^31 - 3 * q^32 - 8 * q^33 + 2 * q^34 + 3 * q^36 + 6 * q^37 + 2 * q^38 + 10 * q^39 + 6 * q^41 - 6 * q^42 + 8 * q^44 + 2 * q^46 + 2 * q^47 - 3 * q^48 - 9 * q^49 + 2 * q^51 - 10 * q^52 - 10 * q^53 + 3 * q^54 + 6 * q^56 + 2 * q^57 + 2 * q^58 - 2 * q^59 + 14 * q^61 + 3 * q^62 - 6 * q^63 + 3 * q^64 + 8 * q^66 + 16 * q^67 - 2 * q^68 + 2 * q^69 - 8 * q^71 - 3 * q^72 + 2 * q^73 - 6 * q^74 - 2 * q^76 - 16 * q^77 - 10 * q^78 - 6 * q^79 + 3 * q^81 - 6 * q^82 - 2 * q^83 + 6 * q^84 + 2 * q^87 - 8 * q^88 - 2 * q^89 + 20 * q^91 - 2 * q^92 + 3 * q^93 - 2 * q^94 + 3 * q^96 + 20 * q^97 + 9 * q^98 + 8 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.470896	−0.141980	−0.0709902	−	0.997477i	$-0.522616\pi$
$11$	−0.470896	−0.141980	−0.0709902	+	0.997477i	$0.522616\pi$
$12$	−1.00000	−0.288675
$13$	−6.47090	−1.79470	−0.897352	−	0.441316i	$-0.854512\pi$
$13$	−6.47090	−1.79470	−0.897352	+	0.441316i	$0.854512\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	−7.04712	−1.70918	−0.854588	−	0.519306i	$-0.826190\pi$
$17$	−7.04712	−1.70918	−0.854588	+	0.519306i	$0.826190\pi$
$18$	−1.00000	−0.235702
$19$	−7.04712	−1.61672	−0.808360	−	0.588689i	$-0.799644\pi$
$19$	−7.04712	−1.61672	−0.808360	+	0.588689i	$0.799644\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0.470896	0.100395
$23$	−6.94179	−1.44746	−0.723732	−	0.690081i	$-0.757575\pi$
$23$	−6.94179	−1.44746	−0.723732	+	0.690081i	$0.757575\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	6.47090	1.26905
$27$	−1.00000	−0.192450
$28$	−2.00000	−0.377964
$29$	−6.94179	−1.28906	−0.644529	−	0.764580i	$-0.722947\pi$
$29$	−6.94179	−1.28906	−0.644529	+	0.764580i	$0.722947\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−1.00000	−0.176777
$33$	0.470896	0.0819724
$34$	7.04712	1.20857
$35$	0	0
$36$	1.00000	0.166667
$37$	1.78935	0.294167	0.147084	−	0.989124i	$-0.453011\pi$
$37$	1.78935	0.294167	0.147084	+	0.989124i	$0.453011\pi$
$38$	7.04712	1.14319
$39$	6.47090	1.03617
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	−2.00000	−0.308607
$43$	0.210649	0.0321237	0.0160619	−	0.999871i	$-0.494887\pi$
$43$	0.210649	0.0321237	0.0160619	+	0.999871i	$0.494887\pi$
$44$	−0.470896	−0.0709902
$45$	0	0
$46$	6.94179	1.02351
$47$	7.04712	1.02793	0.513964	−	0.857812i	$-0.328177\pi$
$47$	7.04712	1.02793	0.513964	+	0.857812i	$0.328177\pi$
$48$	−1.00000	−0.144338
$49$	−3.00000	−0.428571
$50$	0	0
$51$	7.04712	0.986794
$52$	−6.47090	−0.897352
$53$	3.15244	0.433021	0.216510	−	0.976280i	$-0.430532\pi$
$53$	3.15244	0.433021	0.216510	+	0.976280i	$0.430532\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	2.00000	0.267261
$57$	7.04712	0.933413
$58$	6.94179	0.911502
$59$	−7.15244	−0.931168	−0.465584	−	0.885004i	$-0.654156\pi$
$59$	−7.15244	−0.931168	−0.465584	+	0.885004i	$0.654156\pi$
$60$	0	0
$61$	11.2578	1.44141	0.720705	−	0.693242i	$-0.243818\pi$
$61$	11.2578	1.44141	0.720705	+	0.693242i	$0.243818\pi$
$62$	1.00000	0.127000
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	0	0
$66$	−0.470896	−0.0579632
$67$	8.26025	1.00915	0.504575	−	0.863368i	$-0.331649\pi$
$67$	8.26025	1.00915	0.504575	+	0.863368i	$0.331649\pi$
$68$	−7.04712	−0.854588
$69$	6.94179	0.835693
$70$	0	0
$71$	0.260246	0.0308855	0.0154428	−	0.999881i	$-0.495084\pi$
$71$	0.260246	0.0308855	0.0154428	+	0.999881i	$0.495084\pi$
$72$	−1.00000	−0.117851
$73$	−11.8836	−1.39087	−0.695434	−	0.718590i	$-0.744788\pi$
$73$	−11.8836	−1.39087	−0.695434	+	0.718590i	$0.744788\pi$
$74$	−1.78935	−0.208008
$75$	0	0
$76$	−7.04712	−0.808360
$77$	0.941791	0.107327
$78$	−6.47090	−0.732685
$79$	−1.89468	−0.213168	−0.106584	−	0.994304i	$-0.533991\pi$
$79$	−1.89468	−0.213168	−0.106584	+	0.994304i	$0.533991\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	11.7783	1.29283	0.646416	−	0.762985i	$-0.276267\pi$
$83$	11.7783	1.29283	0.646416	+	0.762985i	$0.276267\pi$
$84$	2.00000	0.218218
$85$	0	0
$86$	−0.210649	−0.0227149
$87$	6.94179	0.744238
$88$	0.470896	0.0501976
$89$	12.0942	1.28199	0.640993	−	0.767547i	$-0.278523\pi$
$89$	12.0942	1.28199	0.640993	+	0.767547i	$0.278523\pi$
$90$	0	0
$91$	12.9418	1.35667
$92$	−6.94179	−0.723732
$93$	1.00000	0.103695
$94$	−7.04712	−0.726854
$95$	0	0
$96$	1.00000	0.102062
$97$	3.52910	0.358326	0.179163	−	0.983819i	$-0.442661\pi$
$97$	3.52910	0.358326	0.179163	+	0.983819i	$0.442661\pi$
$98$	3.00000	0.303046
$99$	−0.470896	−0.0473268
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	−7.04712	−0.697768
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	6.47090	0.634524
$105$	0	0
$106$	−3.15244	−0.306192
$107$	−16.7311	−1.61746	−0.808730	−	0.588180i	$-0.799845\pi$
$107$	−16.7311	−1.61746	−0.808730	+	0.588180i	$0.799845\pi$
$108$	−1.00000	−0.0962250
$109$	−11.1524	−1.06821	−0.534105	−	0.845418i	$-0.679351\pi$
$109$	−11.1524	−1.06821	−0.534105	+	0.845418i	$0.679351\pi$
$110$	0	0
$111$	−1.78935	−0.169838
$112$	−2.00000	−0.188982
$113$	−16.3049	−1.53383	−0.766917	−	0.641746i	$-0.778210\pi$
$113$	−16.3049	−1.53383	−0.766917	+	0.641746i	$0.778210\pi$
$114$	−7.04712	−0.660023
$115$	0	0
$116$	−6.94179	−0.644529
$117$	−6.47090	−0.598235
$118$	7.15244	0.658436
$119$	14.0942	1.29202
$120$	0	0
$121$	−10.7783	−0.979842
$122$	−11.2578	−1.01923
$123$	−2.00000	−0.180334
$124$	−1.00000	−0.0898027
$125$	0	0
$126$	2.00000	0.178174
$127$	−0.941791	−0.0835704	−0.0417852	−	0.999127i	$-0.513305\pi$
$127$	−0.941791	−0.0835704	−0.0417852	+	0.999127i	$0.513305\pi$
$128$	−1.00000	−0.0883883
$129$	−0.210649	−0.0185466
$130$	0	0
$131$	−3.15244	−0.275430	−0.137715	−	0.990472i	$-0.543976\pi$
$131$	−3.15244	−0.275430	−0.137715	+	0.990472i	$0.543976\pi$
$132$	0.470896	0.0409862
$133$	14.0942	1.22212
$134$	−8.26025	−0.713577
$135$	0	0
$136$	7.04712	0.604285
$137$	−9.67293	−0.826414	−0.413207	−	0.910637i	$-0.635591\pi$
$137$	−9.67293	−0.826414	−0.413207	+	0.910637i	$0.635591\pi$
$138$	−6.94179	−0.590924
$139$	5.05821	0.429032	0.214516	−	0.976721i	$-0.431183\pi$
$139$	5.05821	0.429032	0.214516	+	0.976721i	$0.431183\pi$
$140$	0	0
$141$	−7.04712	−0.593474
$142$	−0.260246	−0.0218394
$143$	3.04712	0.254813
$144$	1.00000	0.0833333
$145$	0	0
$146$	11.8836	0.983492
$147$	3.00000	0.247436
$148$	1.78935	0.147084
$149$	13.4127	1.09881	0.549405	−	0.835556i	$-0.314854\pi$
$149$	13.4127	1.09881	0.549405	+	0.835556i	$0.314854\pi$
$150$	0	0
$151$	19.5676	1.59239	0.796195	−	0.605041i	$-0.206843\pi$
$151$	19.5676	1.59239	0.796195	+	0.605041i	$0.206843\pi$
$152$	7.04712	0.571597
$153$	−7.04712	−0.569726
$154$	−0.941791	−0.0758917
$155$	0	0
$156$	6.47090	0.518086
$157$	21.2467	1.69567	0.847835	−	0.530261i	$-0.177906\pi$
$157$	21.2467	1.69567	0.847835	+	0.530261i	$0.177906\pi$
$158$	1.89468	0.150732
$159$	−3.15244	−0.250005
$160$	0	0
$161$	13.8836	1.09418
$162$	−1.00000	−0.0785674
$163$	4.26025	0.333688	0.166844	−	0.985983i	$-0.446642\pi$
$163$	4.26025	0.333688	0.166844	+	0.985983i	$0.446642\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	−11.7783	−0.914170
$167$	−2.00000	−0.154765	−0.0773823	−	0.997001i	$-0.524656\pi$
$167$	−2.00000	−0.154765	−0.0773823	+	0.997001i	$0.524656\pi$
$168$	−2.00000	−0.154303
$169$	28.8725	2.22096
$170$	0	0
$171$	−7.04712	−0.538906
$172$	0.210649	0.0160619
$173$	−17.9889	−1.36767	−0.683836	−	0.729636i	$-0.739689\pi$
$173$	−17.9889	−1.36767	−0.683836	+	0.729636i	$0.739689\pi$
$174$	−6.94179	−0.526256
$175$	0	0
$176$	−0.470896	−0.0354951
$177$	7.15244	0.537610
$178$	−12.0942	−0.906501
$179$	−12.0496	−0.900629	−0.450315	−	0.892870i	$-0.648688\pi$
$179$	−12.0496	−0.900629	−0.450315	+	0.892870i	$0.648688\pi$
$180$	0	0
$181$	−13.6729	−1.01630	−0.508151	−	0.861268i	$-0.669671\pi$
$181$	−13.6729	−1.01630	−0.508151	+	0.861268i	$0.669671\pi$
$182$	−12.9418	−0.959309
$183$	−11.2578	−0.832198
$184$	6.94179	0.511756
$185$	0	0
$186$	−1.00000	−0.0733236
$187$	3.31846	0.242669
$188$	7.04712	0.513964
$189$	2.00000	0.145479
$190$	0	0
$191$	−14.8254	−1.07273	−0.536363	−	0.843987i	$-0.680202\pi$
$191$	−14.8254	−1.07273	−0.536363	+	0.843987i	$0.680202\pi$
$192$	−1.00000	−0.0721688
$193$	8.04960	0.579423	0.289711	−	0.957114i	$-0.406441\pi$
$193$	8.04960	0.579423	0.289711	+	0.957114i	$0.406441\pi$
$194$	−3.52910	−0.253375
$195$	0	0
$196$	−3.00000	−0.214286
$197$	−16.3049	−1.16167	−0.580837	−	0.814020i	$-0.697275\pi$
$197$	−16.3049	−1.16167	−0.580837	+	0.814020i	$0.697275\pi$
$198$	0.470896	0.0334651
$199$	−3.98891	−0.282766	−0.141383	−	0.989955i	$-0.545155\pi$
$199$	−3.98891	−0.282766	−0.141383	+	0.989955i	$0.545155\pi$
$200$	0	0
$201$	−8.26025	−0.582633
$202$	0	0
$203$	13.8836	0.974436
$204$	7.04712	0.493397
$205$	0	0
$206$	−6.00000	−0.418040
$207$	−6.94179	−0.482488
$208$	−6.47090	−0.448676
$209$	3.31846	0.229542
$210$	0	0
$211$	2.11642	0.145700	0.0728501	−	0.997343i	$-0.476791\pi$
$211$	2.11642	0.145700	0.0728501	+	0.997343i	$0.476791\pi$
$212$	3.15244	0.216510
$213$	−0.260246	−0.0178318
$214$	16.7311	1.14372
$215$	0	0
$216$	1.00000	0.0680414
$217$	2.00000	0.135769
$218$	11.1524	0.755339
$219$	11.8836	0.803018
$220$	0	0
$221$	45.6011	3.06747
$222$	1.78935	0.120093
$223$	2.58731	0.173259	0.0866297	−	0.996241i	$-0.472390\pi$
$223$	2.58731	0.173259	0.0866297	+	0.996241i	$0.472390\pi$
$224$	2.00000	0.133631
$225$	0	0
$226$	16.3049	1.08458
$227$	−4.94179	−0.327998	−0.163999	−	0.986460i	$-0.552439\pi$
$227$	−4.94179	−0.327998	−0.163999	+	0.986460i	$0.552439\pi$
$228$	7.04712	0.466707
$229$	−11.6791	−0.771774	−0.385887	−	0.922546i	$-0.626105\pi$
$229$	−11.6791	−0.771774	−0.385887	+	0.922546i	$0.626105\pi$
$230$	0	0
$231$	−0.941791	−0.0619653
$232$	6.94179	0.455751
$233$	−13.7894	−0.903370	−0.451685	−	0.892177i	$-0.649177\pi$
$233$	−13.7894	−0.903370	−0.451685	+	0.892177i	$0.649177\pi$
$234$	6.47090	0.423016
$235$	0	0
$236$	−7.15244	−0.465584
$237$	1.89468	0.123072
$238$	−14.0942	−0.913593
$239$	−12.7311	−0.823509	−0.411755	−	0.911295i	$-0.635084\pi$
$239$	−12.7311	−0.823509	−0.411755	+	0.911295i	$0.635084\pi$
$240$	0	0
$241$	−3.15244	−0.203067	−0.101533	−	0.994832i	$-0.532375\pi$
$241$	−3.15244	−0.203067	−0.101533	+	0.994832i	$0.532375\pi$
$242$	10.7783	0.692853
$243$	−1.00000	−0.0641500
$244$	11.2578	0.720705
$245$	0	0
$246$	2.00000	0.127515
$247$	45.6011	2.90153
$248$	1.00000	0.0635001
$249$	−11.7783	−0.746417
$250$	0	0
$251$	3.05821	0.193032	0.0965162	−	0.995331i	$-0.469230\pi$
$251$	3.05821	0.193032	0.0965162	+	0.995331i	$0.469230\pi$
$252$	−2.00000	−0.125988
$253$	3.26886	0.205511
$254$	0.941791	0.0590932
$255$	0	0
$256$	1.00000	0.0625000
$257$	16.0942	1.00393	0.501965	−	0.864888i	$-0.332611\pi$
$257$	16.0942	1.00393	0.501965	+	0.864888i	$0.332611\pi$
$258$	0.210649	0.0131145
$259$	−3.57870	−0.222370
$260$	0	0
$261$	−6.94179	−0.429686
$262$	3.15244	0.194758
$263$	0.0942314	0.00581056	0.00290528	−	0.999996i	$-0.499075\pi$
$263$	0.0942314	0.00581056	0.00290528	+	0.999996i	$0.499075\pi$
$264$	−0.470896	−0.0289816
$265$	0	0
$266$	−14.0942	−0.864173
$267$	−12.0942	−0.740155
$268$	8.26025	0.504575
$269$	−23.8836	−1.45621	−0.728104	−	0.685467i	$-0.759598\pi$
$269$	−23.8836	−1.45621	−0.728104	+	0.685467i	$0.759598\pi$
$270$	0	0
$271$	24.4102	1.48281	0.741407	−	0.671055i	$-0.234159\pi$
$271$	24.4102	1.48281	0.741407	+	0.671055i	$0.234159\pi$
$272$	−7.04712	−0.427294
$273$	−12.9418	−0.783273
$274$	9.67293	0.584363
$275$	0	0
$276$	6.94179	0.417847
$277$	−7.41269	−0.445385	−0.222693	−	0.974889i	$-0.571485\pi$
$277$	−7.41269	−0.445385	−0.222693	+	0.974889i	$0.571485\pi$
$278$	−5.05821	−0.303371
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−11.8836	−0.708915	−0.354458	−	0.935072i	$-0.615334\pi$
$281$	−11.8836	−0.708915	−0.354458	+	0.935072i	$0.615334\pi$
$282$	7.04712	0.419650
$283$	22.9864	1.36640	0.683201	−	0.730231i	$-0.260587\pi$
$283$	22.9864	1.36640	0.683201	+	0.730231i	$0.260587\pi$
$284$	0.260246	0.0154428
$285$	0	0
$286$	−3.04712	−0.180180
$287$	−4.00000	−0.236113
$288$	−1.00000	−0.0589256
$289$	32.6618	1.92128
$290$	0	0
$291$	−3.52910	−0.206880
$292$	−11.8836	−0.695434
$293$	−14.4213	−0.842501	−0.421251	−	0.906944i	$-0.638409\pi$
$293$	−14.4213	−0.842501	−0.421251	+	0.906944i	$0.638409\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	−1.78935	−0.104004
$297$	0.470896	0.0273241
$298$	−13.4127	−0.776976
$299$	44.9196	2.59777
$300$	0	0
$301$	−0.421299	−0.0242832
$302$	−19.5676	−1.12599
$303$	0	0
$304$	−7.04712	−0.404180
$305$	0	0
$306$	7.04712	0.402857
$307$	24.9418	1.42350	0.711752	−	0.702431i	$-0.247902\pi$
$307$	24.9418	1.42350	0.711752	+	0.702431i	$0.247902\pi$
$308$	0.941791	0.0536635
$309$	−6.00000	−0.341328
$310$	0	0
$311$	−24.4487	−1.38636	−0.693180	−	0.720765i	$-0.743791\pi$
$311$	−24.4487	−1.38636	−0.693180	+	0.720765i	$0.743791\pi$
$312$	−6.47090	−0.366342
$313$	−14.1885	−0.801979	−0.400990	−	0.916083i	$-0.631334\pi$
$313$	−14.1885	−0.801979	−0.400990	+	0.916083i	$0.631334\pi$
$314$	−21.2467	−1.19902
$315$	0	0
$316$	−1.89468	−0.106584
$317$	21.9889	1.23502	0.617510	−	0.786563i	$-0.288141\pi$
$317$	21.9889	1.23502	0.617510	+	0.786563i	$0.288141\pi$
$318$	3.15244	0.176780
$319$	3.26886	0.183021
$320$	0	0
$321$	16.7311	0.933841
$322$	−13.8836	−0.773702
$323$	49.6618	2.76326
$324$	1.00000	0.0555556
$325$	0	0
$326$	−4.26025	−0.235953
$327$	11.1524	0.616731
$328$	−2.00000	−0.110432
$329$	−14.0942	−0.777040
$330$	0	0
$331$	−15.3631	−0.844432	−0.422216	−	0.906495i	$-0.638748\pi$
$331$	−15.3631	−0.844432	−0.422216	+	0.906495i	$0.638748\pi$
$332$	11.7783	0.646416
$333$	1.78935	0.0980558
$334$	2.00000	0.109435
$335$	0	0
$336$	2.00000	0.109109
$337$	−13.9778	−0.761420	−0.380710	−	0.924694i	$-0.624320\pi$
$337$	−13.9778	−0.761420	−0.380710	+	0.924694i	$0.624320\pi$
$338$	−28.8725	−1.57046
$339$	16.3049	0.885560
$340$	0	0
$341$	0.470896	0.0255004
$342$	7.04712	0.381064
$343$	20.0000	1.07990
$344$	−0.210649	−0.0113574
$345$	0	0
$346$	17.9889	0.967090
$347$	−17.6618	−0.948137	−0.474069	−	0.880488i	$-0.657215\pi$
$347$	−17.6618	−0.948137	−0.474069	+	0.880488i	$0.657215\pi$
$348$	6.94179	0.372119
$349$	−7.67293	−0.410723	−0.205361	−	0.978686i	$-0.565837\pi$
$349$	−7.67293	−0.410723	−0.205361	+	0.978686i	$0.565837\pi$
$350$	0	0
$351$	6.47090	0.345391
$352$	0.470896	0.0250988
$353$	−11.0471	−0.587979	−0.293989	−	0.955809i	$-0.594983\pi$
$353$	−11.0471	−0.587979	−0.293989	+	0.955809i	$0.594983\pi$
$354$	−7.15244	−0.380148
$355$	0	0
$356$	12.0942	0.640993
$357$	−14.0942	−0.745946
$358$	12.0496	0.636841
$359$	−35.9282	−1.89622	−0.948109	−	0.317944i	$-0.897007\pi$
$359$	−35.9282	−1.89622	−0.948109	+	0.317944i	$0.897007\pi$
$360$	0	0
$361$	30.6618	1.61378
$362$	13.6729	0.718633
$363$	10.7783	0.565712
$364$	12.9418	0.678334
$365$	0	0
$366$	11.2578	0.588453
$367$	−19.2963	−1.00726	−0.503629	−	0.863920i	$-0.668002\pi$
$367$	−19.2963	−1.00726	−0.503629	+	0.863920i	$0.668002\pi$
$368$	−6.94179	−0.361866
$369$	2.00000	0.104116
$370$	0	0
$371$	−6.30488	−0.327333
$372$	1.00000	0.0518476
$373$	24.0942	1.24755	0.623776	−	0.781603i	$-0.285598\pi$
$373$	24.0942	1.24755	0.623776	+	0.781603i	$0.285598\pi$
$374$	−3.31846	−0.171593
$375$	0	0
$376$	−7.04712	−0.363427
$377$	44.9196	2.31348
$378$	−2.00000	−0.102869
$379$	25.1413	1.29142	0.645712	−	0.763581i	$-0.276561\pi$
$379$	25.1413	1.29142	0.645712	+	0.763581i	$0.276561\pi$
$380$	0	0
$381$	0.941791	0.0482494
$382$	14.8254	0.758532
$383$	−22.9196	−1.17114	−0.585569	−	0.810623i	$-0.699129\pi$
$383$	−22.9196	−1.17114	−0.585569	+	0.810623i	$0.699129\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−8.04960	−0.409714
$387$	0.210649	0.0107079
$388$	3.52910	0.179163
$389$	−21.0360	−1.06657	−0.533284	−	0.845936i	$-0.679042\pi$
$389$	−21.0360	−1.06657	−0.533284	+	0.845936i	$0.679042\pi$
$390$	0	0
$391$	48.9196	2.47397
$392$	3.00000	0.151523
$393$	3.15244	0.159020
$394$	16.3049	0.821428
$395$	0	0
$396$	−0.470896	−0.0236634
$397$	38.4983	1.93217	0.966087	−	0.258216i	$-0.0831345\pi$
$397$	38.4983	1.93217	0.966087	+	0.258216i	$0.0831345\pi$
$398$	3.98891	0.199946
$399$	−14.0942	−0.705594
$400$	0	0
$401$	−12.3545	−0.616953	−0.308477	−	0.951232i	$-0.599819\pi$
$401$	−12.3545	−0.616953	−0.308477	+	0.951232i	$0.599819\pi$
$402$	8.26025	0.411984
$403$	6.47090	0.322338
$404$	0	0
$405$	0	0
$406$	−13.8836	−0.689031
$407$	−0.842597	−0.0417660
$408$	−7.04712	−0.348884
$409$	−32.6147	−1.61269	−0.806347	−	0.591443i	$-0.798559\pi$
$409$	−32.6147	−1.61269	−0.806347	+	0.591443i	$0.798559\pi$
$410$	0	0
$411$	9.67293	0.477131
$412$	6.00000	0.295599
$413$	14.3049	0.703897
$414$	6.94179	0.341170
$415$	0	0
$416$	6.47090	0.317262
$417$	−5.05821	−0.247702
$418$	−3.31846	−0.162311
$419$	10.9418	0.534541	0.267271	−	0.963621i	$-0.413878\pi$
$419$	10.9418	0.534541	0.267271	+	0.963621i	$0.413878\pi$
$420$	0	0
$421$	−22.4213	−1.09275	−0.546374	−	0.837542i	$-0.683992\pi$
$421$	−22.4213	−1.09275	−0.546374	+	0.837542i	$0.683992\pi$
$422$	−2.11642	−0.103026
$423$	7.04712	0.342642
$424$	−3.15244	−0.153096
$425$	0	0
$426$	0.260246	0.0126090
$427$	−22.5155	−1.08960
$428$	−16.7311	−0.808730
$429$	−3.04712	−0.147116
$430$	0	0
$431$	33.1303	1.59583	0.797914	−	0.602771i	$-0.205937\pi$
$431$	33.1303	1.59583	0.797914	+	0.602771i	$0.205937\pi$
$432$	−1.00000	−0.0481125
$433$	9.78935	0.470446	0.235223	−	0.971941i	$-0.424418\pi$
$433$	9.78935	0.470446	0.235223	+	0.971941i	$0.424418\pi$
$434$	−2.00000	−0.0960031
$435$	0	0
$436$	−11.1524	−0.534105
$437$	48.9196	2.34014
$438$	−11.8836	−0.567820
$439$	−22.6147	−1.07934	−0.539671	−	0.841876i	$-0.681451\pi$
$439$	−22.6147	−1.07934	−0.539671	+	0.841876i	$0.681451\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	−45.6011	−2.16903
$443$	−0.210649	−0.0100083	−0.00500413	−	0.999987i	$-0.501593\pi$
$443$	−0.210649	−0.0100083	−0.00500413	+	0.999987i	$0.501593\pi$
$444$	−1.78935	−0.0849188
$445$	0	0
$446$	−2.58731	−0.122513
$447$	−13.4127	−0.634398
$448$	−2.00000	−0.0944911
$449$	−27.4127	−1.29368	−0.646842	−	0.762624i	$-0.723911\pi$
$449$	−27.4127	−1.29368	−0.646842	+	0.762624i	$0.723911\pi$
$450$	0	0
$451$	−0.941791	−0.0443472
$452$	−16.3049	−0.766917
$453$	−19.5676	−0.919366
$454$	4.94179	0.231930
$455$	0	0
$456$	−7.04712	−0.330011
$457$	−26.9196	−1.25925	−0.629623	−	0.776901i	$-0.716791\pi$
$457$	−26.9196	−1.25925	−0.629623	+	0.776901i	$0.716791\pi$
$458$	11.6791	0.545727
$459$	7.04712	0.328931
$460$	0	0
$461$	29.0360	1.35234	0.676171	−	0.736745i	$-0.263638\pi$
$461$	29.0360	1.35234	0.676171	+	0.736745i	$0.263638\pi$
$462$	0.941791	0.0438161
$463$	16.8922	0.785047	0.392523	−	0.919742i	$-0.371602\pi$
$463$	16.8922	0.785047	0.392523	+	0.919742i	$0.371602\pi$
$464$	−6.94179	−0.322265
$465$	0	0
$466$	13.7894	0.638779
$467$	−7.76716	−0.359421	−0.179711	−	0.983719i	$-0.557516\pi$
$467$	−7.76716	−0.359421	−0.179711	+	0.983719i	$0.557516\pi$
$468$	−6.47090	−0.299117
$469$	−16.5205	−0.762845
$470$	0	0
$471$	−21.2467	−0.978995
$472$	7.15244	0.329218
$473$	−0.0991938	−0.00456094
$474$	−1.89468	−0.0870253
$475$	0	0
$476$	14.0942	0.646008
$477$	3.15244	0.144340
$478$	12.7311	0.582309
$479$	32.4487	1.48262	0.741310	−	0.671163i	$-0.234205\pi$
$479$	32.4487	1.48262	0.741310	+	0.671163i	$0.234205\pi$
$480$	0	0
$481$	−11.5787	−0.527943
$482$	3.15244	0.143590
$483$	−13.8836	−0.631725
$484$	−10.7783	−0.489921
$485$	0	0
$486$	1.00000	0.0453609
$487$	−35.4847	−1.60797	−0.803983	−	0.594652i	$-0.797290\pi$
$487$	−35.4847	−1.60797	−0.803983	+	0.594652i	$0.797290\pi$
$488$	−11.2578	−0.509615
$489$	−4.26025	−0.192655
$490$	0	0
$491$	25.9828	1.17259	0.586293	−	0.810099i	$-0.300587\pi$
$491$	25.9828	1.17259	0.586293	+	0.810099i	$0.300587\pi$
$492$	−2.00000	−0.0901670
$493$	48.9196	2.20323
$494$	−45.6011	−2.05169
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	−0.520492	−0.0233473
$498$	11.7783	0.527796
$499$	7.90577	0.353911	0.176955	−	0.984219i	$-0.443375\pi$
$499$	7.90577	0.353911	0.176955	+	0.984219i	$0.443375\pi$
$500$	0	0
$501$	2.00000	0.0893534
$502$	−3.05821	−0.136495
$503$	−9.14135	−0.407593	−0.203796	−	0.979013i	$-0.565328\pi$
$503$	−9.14135	−0.407593	−0.203796	+	0.979013i	$0.565328\pi$
$504$	2.00000	0.0890871
$505$	0	0
$506$	−3.26886	−0.145318
$507$	−28.8725	−1.28227
$508$	−0.941791	−0.0417852
$509$	28.5155	1.26393	0.631964	−	0.774997i	$-0.282249\pi$
$509$	28.5155	1.26393	0.631964	+	0.774997i	$0.282249\pi$
$510$	0	0
$511$	23.7672	1.05140
$512$	−1.00000	−0.0441942
$513$	7.04712	0.311138
$514$	−16.0942	−0.709886
$515$	0	0
$516$	−0.210649	−0.00927332
$517$	−3.31846	−0.145945
$518$	3.57870	0.157239
$519$	17.9889	0.789625
$520$	0	0
$521$	23.3631	1.02356	0.511778	−	0.859118i	$-0.328987\pi$
$521$	23.3631	1.02356	0.511778	+	0.859118i	$0.328987\pi$
$522$	6.94179	0.303834
$523$	42.0942	1.84065	0.920326	−	0.391152i	$-0.127923\pi$
$523$	42.0942	1.84065	0.920326	+	0.391152i	$0.127923\pi$
$524$	−3.15244	−0.137715
$525$	0	0
$526$	−0.0942314	−0.00410868
$527$	7.04712	0.306977
$528$	0.470896	0.0204931
$529$	25.1885	1.09515
$530$	0	0
$531$	−7.15244	−0.310389
$532$	14.0942	0.611062
$533$	−12.9418	−0.560571
$534$	12.0942	0.523369
$535$	0	0
$536$	−8.26025	−0.356788
$537$	12.0496	0.519978
$538$	23.8836	1.02969
$539$	1.41269	0.0608487
$540$	0	0
$541$	−44.5925	−1.91718	−0.958591	−	0.284785i	$-0.908078\pi$
$541$	−44.5925	−1.91718	−0.958591	+	0.284785i	$0.908078\pi$
$542$	−24.4102	−1.04851
$543$	13.6729	0.586762
$544$	7.04712	0.302143
$545$	0	0
$546$	12.9418	0.553858
$547$	−4.52049	−0.193282	−0.0966411	−	0.995319i	$-0.530810\pi$
$547$	−4.52049	−0.193282	−0.0966411	+	0.995319i	$0.530810\pi$
$548$	−9.67293	−0.413207
$549$	11.2578	0.480470
$550$	0	0
$551$	48.9196	2.08405
$552$	−6.94179	−0.295462
$553$	3.78935	0.161140
$554$	7.41269	0.314935
$555$	0	0
$556$	5.05821	0.214516
$557$	−20.5155	−0.869271	−0.434635	−	0.900606i	$-0.643123\pi$
$557$	−20.5155	−0.869271	−0.434635	+	0.900606i	$0.643123\pi$
$558$	1.00000	0.0423334
$559$	−1.36309	−0.0576525
$560$	0	0
$561$	−3.31846	−0.140105
$562$	11.8836	0.501279
$563$	34.6147	1.45884	0.729418	−	0.684068i	$-0.239791\pi$
$563$	34.6147	1.45884	0.729418	+	0.684068i	$0.239791\pi$
$564$	−7.04712	−0.296737
$565$	0	0
$566$	−22.9864	−0.966192
$567$	−2.00000	−0.0839921
$568$	−0.260246	−0.0109197
$569$	7.25163	0.304004	0.152002	−	0.988380i	$-0.451428\pi$
$569$	7.25163	0.304004	0.152002	+	0.988380i	$0.451428\pi$
$570$	0	0
$571$	−0.0942314	−0.00394346	−0.00197173	−	0.999998i	$-0.500628\pi$
$571$	−0.0942314	−0.00394346	−0.00197173	+	0.999998i	$0.500628\pi$
$572$	3.04712	0.127406
$573$	14.8254	0.619339
$574$	4.00000	0.166957
$575$	0	0
$576$	1.00000	0.0416667
$577$	9.41269	0.391855	0.195928	−	0.980618i	$-0.437228\pi$
$577$	9.41269	0.391855	0.195928	+	0.980618i	$0.437228\pi$
$578$	−32.6618	−1.35855
$579$	−8.04960	−0.334530
$580$	0	0
$581$	−23.5565	−0.977289
$582$	3.52910	0.146286
$583$	−1.48447	−0.0614805
$584$	11.8836	0.491746
$585$	0	0
$586$	14.4213	0.595738
$587$	42.2938	1.74565	0.872826	−	0.488032i	$-0.162285\pi$
$587$	42.2938	1.74565	0.872826	+	0.488032i	$0.162285\pi$
$588$	3.00000	0.123718
$589$	7.04712	0.290371
$590$	0	0
$591$	16.3049	0.670693
$592$	1.78935	0.0735419
$593$	−5.88854	−0.241814	−0.120907	−	0.992664i	$-0.538580\pi$
$593$	−5.88854	−0.241814	−0.120907	+	0.992664i	$0.538580\pi$
$594$	−0.470896	−0.0193211
$595$	0	0
$596$	13.4127	0.549405
$597$	3.98891	0.163255
$598$	−44.9196	−1.83690
$599$	−0.780739	−0.0319001	−0.0159501	−	0.999873i	$-0.505077\pi$
$599$	−0.780739	−0.0319001	−0.0159501	+	0.999873i	$0.505077\pi$
$600$	0	0
$601$	−25.5787	−1.04338	−0.521688	−	0.853136i	$-0.674698\pi$
$601$	−25.5787	−1.04338	−0.521688	+	0.853136i	$0.674698\pi$
$602$	0.421299	0.0171708
$603$	8.26025	0.336383
$604$	19.5676	0.796195
$605$	0	0
$606$	0	0
$607$	7.46228	0.302885	0.151442	−	0.988466i	$-0.451608\pi$
$607$	7.46228	0.302885	0.151442	+	0.988466i	$0.451608\pi$
$608$	7.04712	0.285798
$609$	−13.8836	−0.562591
$610$	0	0
$611$	−45.6011	−1.84483
$612$	−7.04712	−0.284863
$613$	−4.68651	−0.189286	−0.0946431	−	0.995511i	$-0.530171\pi$
$613$	−4.68651	−0.189286	−0.0946431	+	0.995511i	$0.530171\pi$
$614$	−24.9418	−1.00657
$615$	0	0
$616$	−0.941791	−0.0379458
$617$	7.69512	0.309794	0.154897	−	0.987931i	$-0.450495\pi$
$617$	7.69512	0.309794	0.154897	+	0.987931i	$0.450495\pi$
$618$	6.00000	0.241355
$619$	33.7894	1.35811	0.679054	−	0.734088i	$-0.262390\pi$
$619$	33.7894	1.35811	0.679054	+	0.734088i	$0.262390\pi$
$620$	0	0
$621$	6.94179	0.278564
$622$	24.4487	0.980304
$623$	−24.1885	−0.969090
$624$	6.47090	0.259043
$625$	0	0
$626$	14.1885	0.567085
$627$	−3.31846	−0.132526
$628$	21.2467	0.847835
$629$	−12.6098	−0.502784
$630$	0	0
$631$	−34.3049	−1.36566	−0.682828	−	0.730579i	$-0.739250\pi$
$631$	−34.3049	−1.36566	−0.682828	+	0.730579i	$0.739250\pi$
$632$	1.89468	0.0753661
$633$	−2.11642	−0.0841201
$634$	−21.9889	−0.873291
$635$	0	0
$636$	−3.15244	−0.125002
$637$	19.4127	0.769159
$638$	−3.26886	−0.129415
$639$	0.260246	0.0102952
$640$	0	0
$641$	3.83399	0.151433	0.0757167	−	0.997129i	$-0.475876\pi$
$641$	3.83399	0.151433	0.0757167	+	0.997129i	$0.475876\pi$
$642$	−16.7311	−0.660325
$643$	12.9640	0.511249	0.255625	−	0.966776i	$-0.417719\pi$
$643$	12.9640	0.511249	0.255625	+	0.966776i	$0.417719\pi$
$644$	13.8836	0.547090
$645$	0	0
$646$	−49.6618	−1.95392
$647$	29.3459	1.15371	0.576853	−	0.816848i	$-0.304281\pi$
$647$	29.3459	1.15371	0.576853	+	0.816848i	$0.304281\pi$
$648$	−1.00000	−0.0392837
$649$	3.36805	0.132208
$650$	0	0
$651$	−2.00000	−0.0783862
$652$	4.26025	0.166844
$653$	−4.52662	−0.177140	−0.0885702	−	0.996070i	$-0.528230\pi$
$653$	−4.52662	−0.177140	−0.0885702	+	0.996070i	$0.528230\pi$
$654$	−11.1524	−0.436095
$655$	0	0
$656$	2.00000	0.0780869
$657$	−11.8836	−0.463623
$658$	14.0942	0.549450
$659$	−15.8836	−0.618737	−0.309368	−	0.950942i	$-0.600118\pi$
$659$	−15.8836	−0.618737	−0.309368	+	0.950942i	$0.600118\pi$
$660$	0	0
$661$	12.8254	0.498849	0.249425	−	0.968394i	$-0.419759\pi$
$661$	12.8254	0.498849	0.249425	+	0.968394i	$0.419759\pi$
$662$	15.3631	0.597103
$663$	−45.6011	−1.77100
$664$	−11.7783	−0.457085
$665$	0	0
$666$	−1.78935	−0.0693359
$667$	48.1885	1.86586
$668$	−2.00000	−0.0773823
$669$	−2.58731	−0.100031
$670$	0	0
$671$	−5.30123	−0.204652
$672$	−2.00000	−0.0771517
$673$	27.9828	1.07866	0.539328	−	0.842096i	$-0.318678\pi$
$673$	27.9828	1.07866	0.539328	+	0.842096i	$0.318678\pi$
$674$	13.9778	0.538405
$675$	0	0
$676$	28.8725	1.11048
$677$	−27.6729	−1.06356	−0.531779	−	0.846883i	$-0.678476\pi$
$677$	−27.6729	−1.06356	−0.531779	+	0.846883i	$0.678476\pi$
$678$	−16.3049	−0.626185
$679$	−7.05821	−0.270869
$680$	0	0
$681$	4.94179	0.189370
$682$	−0.470896	−0.0180315
$683$	−8.18846	−0.313323	−0.156661	−	0.987652i	$-0.550073\pi$
$683$	−8.18846	−0.313323	−0.156661	+	0.987652i	$0.550073\pi$
$684$	−7.04712	−0.269453
$685$	0	0
$686$	−20.0000	−0.763604
$687$	11.6791	0.445584
$688$	0.210649	0.00803093
$689$	−20.3991	−0.777144
$690$	0	0
$691$	22.4152	0.852713	0.426357	−	0.904555i	$-0.359797\pi$
$691$	22.4152	0.852713	0.426357	+	0.904555i	$0.359797\pi$
$692$	−17.9889	−0.683836
$693$	0.941791	0.0357757
$694$	17.6618	0.670434
$695$	0	0
$696$	−6.94179	−0.263128
$697$	−14.0942	−0.533857
$698$	7.67293	0.290425
$699$	13.7894	0.521561
$700$	0	0
$701$	−20.4709	−0.773175	−0.386588	−	0.922253i	$-0.626346\pi$
$701$	−20.4709	−0.773175	−0.386588	+	0.922253i	$0.626346\pi$
$702$	−6.47090	−0.244228
$703$	−12.6098	−0.475586
$704$	−0.470896	−0.0177475
$705$	0	0
$706$	11.0471	0.415764
$707$	0	0
$708$	7.15244	0.268805
$709$	−48.7200	−1.82972	−0.914860	−	0.403771i	$-0.867699\pi$
$709$	−48.7200	−1.82972	−0.914860	+	0.403771i	$0.867699\pi$
$710$	0	0
$711$	−1.89468	−0.0710559
$712$	−12.0942	−0.453250
$713$	6.94179	0.259972
$714$	14.0942	0.527463
$715$	0	0
$716$	−12.0496	−0.450315
$717$	12.7311	0.475453
$718$	35.9282	1.34083
$719$	50.8032	1.89464	0.947320	−	0.320290i	$-0.103780\pi$
$719$	50.8032	1.89464	0.947320	+	0.320290i	$0.103780\pi$
$720$	0	0
$721$	−12.0000	−0.446903
$722$	−30.6618	−1.14112
$723$	3.15244	0.117241
$724$	−13.6729	−0.508151
$725$	0	0
$726$	−10.7783	−0.400019
$727$	−33.1352	−1.22892	−0.614459	−	0.788949i	$-0.710625\pi$
$727$	−33.1352	−1.22892	−0.614459	+	0.788949i	$0.710625\pi$
$728$	−12.9418	−0.479655
$729$	1.00000	0.0370370
$730$	0	0
$731$	−1.48447	−0.0549051
$732$	−11.2578	−0.416099
$733$	−2.96398	−0.109477	−0.0547385	−	0.998501i	$-0.517433\pi$
$733$	−2.96398	−0.109477	−0.0547385	+	0.998501i	$0.517433\pi$
$734$	19.2963	0.712238
$735$	0	0
$736$	6.94179	0.255878
$737$	−3.88971	−0.143279
$738$	−2.00000	−0.0736210
$739$	−6.52049	−0.239860	−0.119930	−	0.992782i	$-0.538267\pi$
$739$	−6.52049	−0.239860	−0.119930	+	0.992782i	$0.538267\pi$
$740$	0	0
$741$	−45.6011	−1.67520
$742$	6.30488	0.231459
$743$	20.0942	0.737186	0.368593	−	0.929591i	$-0.379840\pi$
$743$	20.0942	0.737186	0.368593	+	0.929591i	$0.379840\pi$
$744$	−1.00000	−0.0366618
$745$	0	0
$746$	−24.0942	−0.882152
$747$	11.7783	0.430944
$748$	3.31846	0.121335
$749$	33.4623	1.22269
$750$	0	0
$751$	−46.0942	−1.68200	−0.841001	−	0.541033i	$-0.818033\pi$
$751$	−46.0942	−1.68200	−0.841001	+	0.541033i	$0.818033\pi$
$752$	7.04712	0.256982
$753$	−3.05821	−0.111447
$754$	−44.9196	−1.63588
$755$	0	0
$756$	2.00000	0.0727393
$757$	28.2827	1.02795	0.513976	−	0.857805i	$-0.328172\pi$
$757$	28.2827	1.02795	0.513976	+	0.857805i	$0.328172\pi$
$758$	−25.1413	−0.913175
$759$	−3.26886	−0.118652
$760$	0	0
$761$	−41.0634	−1.48855	−0.744274	−	0.667874i	$-0.767204\pi$
$761$	−41.0634	−1.48855	−0.744274	+	0.667874i	$0.767204\pi$
$762$	−0.941791	−0.0341175
$763$	22.3049	0.807491
$764$	−14.8254	−0.536363
$765$	0	0
$766$	22.9196	0.828119
$767$	46.2827	1.67117
$768$	−1.00000	−0.0360844
$769$	−32.4933	−1.17174	−0.585870	−	0.810405i	$-0.699247\pi$
$769$	−32.4933	−1.17174	−0.585870	+	0.810405i	$0.699247\pi$
$770$	0	0
$771$	−16.0942	−0.579620
$772$	8.04960	0.289711
$773$	−50.3991	−1.81273	−0.906365	−	0.422495i	$-0.861154\pi$
$773$	−50.3991	−1.81273	−0.906365	+	0.422495i	$0.861154\pi$
$774$	−0.210649	−0.00757163
$775$	0	0
$776$	−3.52910	−0.126687
$777$	3.57870	0.128385
$778$	21.0360	0.754178
$779$	−14.0942	−0.504978
$780$	0	0
$781$	−0.122549	−0.00438514
$782$	−48.9196	−1.74936
$783$	6.94179	0.248079
$784$	−3.00000	−0.107143
$785$	0	0
$786$	−3.15244	−0.112444
$787$	26.6147	0.948712	0.474356	−	0.880333i	$-0.342681\pi$
$787$	26.6147	0.948712	0.474356	+	0.880333i	$0.342681\pi$
$788$	−16.3049	−0.580837
$789$	−0.0942314	−0.00335473
$790$	0	0
$791$	32.6098	1.15947
$792$	0.470896	0.0167325
$793$	−72.8478	−2.58690
$794$	−38.4983	−1.36625
$795$	0	0
$796$	−3.98891	−0.141383
$797$	−2.52049	−0.0892804	−0.0446402	−	0.999003i	$-0.514214\pi$
$797$	−2.52049	−0.0892804	−0.0446402	+	0.999003i	$0.514214\pi$
$798$	14.0942	0.498930
$799$	−49.6618	−1.75691
$800$	0	0
$801$	12.0942	0.427329
$802$	12.3545	0.436252
$803$	5.59593	0.197476
$804$	−8.26025	−0.291316
$805$	0	0
$806$	−6.47090	−0.227928
$807$	23.8836	0.840742
$808$	0	0
$809$	36.1216	1.26997	0.634985	−	0.772525i	$-0.281006\pi$
$809$	36.1216	1.26997	0.634985	+	0.772525i	$0.281006\pi$
$810$	0	0
$811$	−7.57870	−0.266124	−0.133062	−	0.991108i	$-0.542481\pi$
$811$	−7.57870	−0.266124	−0.133062	+	0.991108i	$0.542481\pi$
$812$	13.8836	0.487218
$813$	−24.4102	−0.856103
$814$	0.842597	0.0295330
$815$	0	0
$816$	7.04712	0.246698
$817$	−1.48447	−0.0519350
$818$	32.6147	1.14035
$819$	12.9418	0.452223
$820$	0	0
$821$	−32.3049	−1.12745	−0.563724	−	0.825963i	$-0.690632\pi$
$821$	−32.3049	−1.12745	−0.563724	+	0.825963i	$0.690632\pi$
$822$	−9.67293	−0.337382
$823$	12.8922	0.449394	0.224697	−	0.974429i	$-0.427861\pi$
$823$	12.8922	0.449394	0.224697	+	0.974429i	$0.427861\pi$
$824$	−6.00000	−0.209020
$825$	0	0
$826$	−14.3049	−0.497730
$827$	−13.4734	−0.468515	−0.234258	−	0.972175i	$-0.575266\pi$
$827$	−13.4734	−0.468515	−0.234258	+	0.972175i	$0.575266\pi$
$828$	−6.94179	−0.241244
$829$	50.2827	1.74639	0.873195	−	0.487371i	$-0.162044\pi$
$829$	50.2827	1.74639	0.873195	+	0.487371i	$0.162044\pi$
$830$	0	0
$831$	7.41269	0.257143
$832$	−6.47090	−0.224338
$833$	21.1413	0.732504
$834$	5.05821	0.175151
$835$	0	0
$836$	3.31846	0.114771
$837$	1.00000	0.0345651
$838$	−10.9418	−0.377978
$839$	−6.20569	−0.214244	−0.107122	−	0.994246i	$-0.534164\pi$
$839$	−6.20569	−0.214244	−0.107122	+	0.994246i	$0.534164\pi$
$840$	0	0
$841$	19.1885	0.661671
$842$	22.4213	0.772689
$843$	11.8836	0.409292
$844$	2.11642	0.0728501
$845$	0	0
$846$	−7.04712	−0.242285
$847$	21.5565	0.740691
$848$	3.15244	0.108255
$849$	−22.9864	−0.788892
$850$	0	0
$851$	−12.4213	−0.425797
$852$	−0.260246	−0.00891589
$853$	−45.2245	−1.54846	−0.774228	−	0.632906i	$-0.781862\pi$
$853$	−45.2245	−1.54846	−0.774228	+	0.632906i	$0.781862\pi$
$854$	22.5155	0.770466
$855$	0	0
$856$	16.7311	0.571859
$857$	4.51553	0.154248	0.0771238	−	0.997022i	$-0.475426\pi$
$857$	4.51553	0.154248	0.0771238	+	0.997022i	$0.475426\pi$
$858$	3.04712	0.104027
$859$	−11.6951	−0.399032	−0.199516	−	0.979895i	$-0.563937\pi$
$859$	−11.6951	−0.399032	−0.199516	+	0.979895i	$0.563937\pi$
$860$	0	0
$861$	4.00000	0.136320
$862$	−33.1303	−1.12842
$863$	−3.38528	−0.115236	−0.0576181	−	0.998339i	$-0.518351\pi$
$863$	−3.38528	−0.115236	−0.0576181	+	0.998339i	$0.518351\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−9.78935	−0.332656
$867$	−32.6618	−1.10925
$868$	2.00000	0.0678844
$869$	0.892194	0.0302656
$870$	0	0
$871$	−53.4512	−1.81112
$872$	11.1524	0.377669
$873$	3.52910	0.119442
$874$	−48.9196	−1.65473
$875$	0	0
$876$	11.8836	0.401509
$877$	−37.0582	−1.25137	−0.625683	−	0.780077i	$-0.715180\pi$
$877$	−37.0582	−1.25137	−0.625683	+	0.780077i	$0.715180\pi$
$878$	22.6147	0.763210
$879$	14.4213	0.486418
$880$	0	0
$881$	−35.5020	−1.19609	−0.598046	−	0.801462i	$-0.704056\pi$
$881$	−35.5020	−1.19609	−0.598046	+	0.801462i	$0.704056\pi$
$882$	3.00000	0.101015
$883$	−27.0138	−0.909088	−0.454544	−	0.890724i	$-0.650198\pi$
$883$	−27.0138	−0.909088	−0.454544	+	0.890724i	$0.650198\pi$
$884$	45.6011	1.53373
$885$	0	0
$886$	0.210649	0.00707690
$887$	−36.6098	−1.22924	−0.614618	−	0.788825i	$-0.710690\pi$
$887$	−36.6098	−1.22924	−0.614618	+	0.788825i	$0.710690\pi$
$888$	1.78935	0.0600467
$889$	1.88358	0.0631733
$890$	0	0
$891$	−0.470896	−0.0157756
$892$	2.58731	0.0866297
$893$	−49.6618	−1.66187
$894$	13.4127	0.448587
$895$	0	0
$896$	2.00000	0.0668153
$897$	−44.9196	−1.49982
$898$	27.4127	0.914773
$899$	6.94179	0.231522
$900$	0	0
$901$	−22.2156	−0.740109
$902$	0.941791	0.0313582
$903$	0.421299	0.0140199
$904$	16.3049	0.542292
$905$	0	0
$906$	19.5676	0.650090
$907$	−57.8118	−1.91961	−0.959805	−	0.280669i	$-0.909444\pi$
$907$	−57.8118	−1.91961	−0.959805	+	0.280669i	$0.909444\pi$
$908$	−4.94179	−0.163999
$909$	0	0
$910$	0	0
$911$	−49.8836	−1.65272	−0.826358	−	0.563145i	$-0.809591\pi$
$911$	−49.8836	−1.65272	−0.826358	+	0.563145i	$0.809591\pi$
$912$	7.04712	0.233353
$913$	−5.54633	−0.183557
$914$	26.9196	0.890421
$915$	0	0
$916$	−11.6791	−0.385887
$917$	6.30488	0.208206
$918$	−7.04712	−0.232589
$919$	50.3819	1.66195	0.830973	−	0.556313i	$-0.187785\pi$
$919$	50.3819	1.66195	0.830973	+	0.556313i	$0.187785\pi$
$920$	0	0
$921$	−24.9418	−0.821860
$922$	−29.0360	−0.956250
$923$	−1.68403	−0.0554304
$924$	−0.941791	−0.0309827
$925$	0	0
$926$	−16.8922	−0.555112
$927$	6.00000	0.197066
$928$	6.94179	0.227875
$929$	8.93683	0.293208	0.146604	−	0.989195i	$-0.453166\pi$
$929$	8.93683	0.293208	0.146604	+	0.989195i	$0.453166\pi$
$930$	0	0
$931$	21.1413	0.692880
$932$	−13.7894	−0.451685
$933$	24.4487	0.800415
$934$	7.76716	0.254149
$935$	0	0
$936$	6.47090	0.211508
$937$	3.42991	0.112050	0.0560251	−	0.998429i	$-0.482157\pi$
$937$	3.42991	0.112050	0.0560251	+	0.998429i	$0.482157\pi$
$938$	16.5205	0.539413
$939$	14.1885	0.463023
$940$	0	0
$941$	45.2467	1.47500	0.737500	−	0.675347i	$-0.236006\pi$
$941$	45.2467	1.47500	0.737500	+	0.675347i	$0.236006\pi$
$942$	21.2467	0.692254
$943$	−13.8836	−0.452112
$944$	−7.15244	−0.232792
$945$	0	0
$946$	0.0991938	0.00322507
$947$	8.41021	0.273295	0.136647	−	0.990620i	$-0.456367\pi$
$947$	8.41021	0.273295	0.136647	+	0.990620i	$0.456367\pi$
$948$	1.89468	0.0615362
$949$	76.8974	2.49620
$950$	0	0
$951$	−21.9889	−0.713039
$952$	−14.0942	−0.456797
$953$	−9.37418	−0.303660	−0.151830	−	0.988407i	$-0.548517\pi$
$953$	−9.37418	−0.303660	−0.151830	+	0.988407i	$0.548517\pi$
$954$	−3.15244	−0.102064
$955$	0	0
$956$	−12.7311	−0.411755
$957$	−3.26886	−0.105667
$958$	−32.4487	−1.04837
$959$	19.3459	0.624711
$960$	0	0
$961$	1.00000	0.0322581
$962$	11.5787	0.373312
$963$	−16.7311	−0.539154
$964$	−3.15244	−0.101533
$965$	0	0
$966$	13.8836	0.446697
$967$	27.8168	0.894527	0.447263	−	0.894402i	$-0.352399\pi$
$967$	27.8168	0.894527	0.447263	+	0.894402i	$0.352399\pi$
$968$	10.7783	0.346426
$969$	−49.6618	−1.59537
$970$	0	0
$971$	45.6557	1.46516	0.732581	−	0.680680i	$-0.238316\pi$
$971$	45.6557	1.46516	0.732581	+	0.680680i	$0.238316\pi$
$972$	−1.00000	−0.0320750
$973$	−10.1164	−0.324317
$974$	35.4847	1.13700
$975$	0	0
$976$	11.2578	0.360352
$977$	19.0410	0.609175	0.304588	−	0.952484i	$-0.401481\pi$
$977$	19.0410	0.609175	0.304588	+	0.952484i	$0.401481\pi$
$978$	4.26025	0.136228
$979$	−5.69512	−0.182017
$980$	0	0
$981$	−11.1524	−0.356070
$982$	−25.9828	−0.829144
$983$	−17.4795	−0.557510	−0.278755	−	0.960362i	$-0.589922\pi$
$983$	−17.4795	−0.557510	−0.278755	+	0.960362i	$0.589922\pi$
$984$	2.00000	0.0637577
$985$	0	0
$986$	−48.9196	−1.55792
$987$	14.0942	0.448624
$988$	45.6011	1.45077
$989$	−1.46228	−0.0464979
$990$	0	0
$991$	−3.34587	−0.106285	−0.0531425	−	0.998587i	$-0.516924\pi$
$991$	−3.34587	−0.106285	−0.0531425	+	0.998587i	$0.516924\pi$
$992$	1.00000	0.0317500
$993$	15.3631	0.487533
$994$	0.520492	0.0165090
$995$	0	0
$996$	−11.7783	−0.373208
$997$	−41.6458	−1.31894	−0.659468	−	0.751733i	$-0.729218\pi$
$997$	−41.6458	−1.31894	−0.659468	+	0.751733i	$0.729218\pi$
$998$	−7.90577	−0.250253
$999$	−1.78935	−0.0566126

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4650.2.a.ci.1.1		3
5.2	odd	4		930.2.d.i.559.1	✓	6
5.3	odd	4		930.2.d.i.559.4	yes	6
5.4	even	2		4650.2.a.cp.1.1		3
15.2	even	4		2790.2.d.j.559.6		6
15.8	even	4		2790.2.d.j.559.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.d.i.559.1	✓	6	5.2	odd	4
930.2.d.i.559.4	yes	6	5.3	odd	4
2790.2.d.j.559.3		6	15.8	even	4
2790.2.d.j.559.6		6	15.2	even	4
4650.2.a.ci.1.1		3	1.1	even	1	trivial
4650.2.a.cp.1.1		3	5.4	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4650.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -0.470896 q^{11} -1.00000 q^{12} -6.47090 q^{13} +2.00000 q^{14} +1.00000 q^{16} -7.04712 q^{17} -1.00000 q^{18} -7.04712 q^{19} +2.00000 q^{21} +0.470896 q^{22} -6.94179 q^{23} +1.00000 q^{24} +6.47090 q^{26} -1.00000 q^{27} -2.00000 q^{28} -6.94179 q^{29} -1.00000 q^{31} -1.00000 q^{32} +0.470896 q^{33} +7.04712 q^{34} +1.00000 q^{36} +1.78935 q^{37} +7.04712 q^{38} +6.47090 q^{39} +2.00000 q^{41} -2.00000 q^{42} +0.210649 q^{43} -0.470896 q^{44} +6.94179 q^{46} +7.04712 q^{47} -1.00000 q^{48} -3.00000 q^{49} +7.04712 q^{51} -6.47090 q^{52} +3.15244 q^{53} +1.00000 q^{54} +2.00000 q^{56} +7.04712 q^{57} +6.94179 q^{58} -7.15244 q^{59} +11.2578 q^{61} +1.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -0.470896 q^{66} +8.26025 q^{67} -7.04712 q^{68} +6.94179 q^{69} +0.260246 q^{71} -1.00000 q^{72} -11.8836 q^{73} -1.78935 q^{74} -7.04712 q^{76} +0.941791 q^{77} -6.47090 q^{78} -1.89468 q^{79} +1.00000 q^{81} -2.00000 q^{82} +11.7783 q^{83} +2.00000 q^{84} -0.210649 q^{86} +6.94179 q^{87} +0.470896 q^{88} +12.0942 q^{89} +12.9418 q^{91} -6.94179 q^{92} +1.00000 q^{93} -7.04712 q^{94} +1.00000 q^{96} +3.52910 q^{97} +3.00000 q^{98} -0.470896 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{6} - 6 q^{7} - 3 q^{8} + 3 q^{9} + 8 q^{11} - 3 q^{12} - 10 q^{13} + 6 q^{14} + 3 q^{16} - 2 q^{17} - 3 q^{18} - 2 q^{19} + 6 q^{21} - 8 q^{22} - 2 q^{23} + 3 q^{24}+ \cdots + 8 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^6 - 6 * q^7 - 3 * q^8 + 3 * q^9 + 8 * q^11 - 3 * q^12 - 10 * q^13 + 6 * q^14 + 3 * q^16 - 2 * q^17 - 3 * q^18 - 2 * q^19 + 6 * q^21 - 8 * q^22 - 2 * q^23 + 3 * q^24 + 10 * q^26 - 3 * q^27 - 6 * q^28 - 2 * q^29 - 3 * q^31 - 3 * q^32 - 8 * q^33 + 2 * q^34 + 3 * q^36 + 6 * q^37 + 2 * q^38 + 10 * q^39 + 6 * q^41 - 6 * q^42 + 8 * q^44 + 2 * q^46 + 2 * q^47 - 3 * q^48 - 9 * q^49 + 2 * q^51 - 10 * q^52 - 10 * q^53 + 3 * q^54 + 6 * q^56 + 2 * q^57 + 2 * q^58 - 2 * q^59 + 14 * q^61 + 3 * q^62 - 6 * q^63 + 3 * q^64 + 8 * q^66 + 16 * q^67 - 2 * q^68 + 2 * q^69 - 8 * q^71 - 3 * q^72 + 2 * q^73 - 6 * q^74 - 2 * q^76 - 16 * q^77 - 10 * q^78 - 6 * q^79 + 3 * q^81 - 6 * q^82 - 2 * q^83 + 6 * q^84 + 2 * q^87 - 8 * q^88 - 2 * q^89 + 20 * q^91 - 2 * q^92 + 3 * q^93 - 2 * q^94 + 3 * q^96 + 20 * q^97 + 9 * q^98 + 8 * q^99

Introduction

L-functions

Modular forms

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