LMFDB - Embedded newform 4650.2.a.ce.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4650,2,Mod(1,4650)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$37.1304369399$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			1.2
Root			$-1.56155$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.56155 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.56155 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.56155 q^{11} +1.00000 q^{12} -2.00000 q^{13} -1.56155 q^{14} +1.00000 q^{16} -5.12311 q^{17} -1.00000 q^{18} +4.68466 q^{19} +1.56155 q^{21} +1.56155 q^{22} -5.56155 q^{23} -1.00000 q^{24} +2.00000 q^{26} +1.00000 q^{27} +1.56155 q^{28} -1.12311 q^{29} +1.00000 q^{31} -1.00000 q^{32} -1.56155 q^{33} +5.12311 q^{34} +1.00000 q^{36} -5.12311 q^{37} -4.68466 q^{38} -2.00000 q^{39} -1.12311 q^{41} -1.56155 q^{42} +7.80776 q^{43} -1.56155 q^{44} +5.56155 q^{46} +3.12311 q^{47} +1.00000 q^{48} -4.56155 q^{49} -5.12311 q^{51} -2.00000 q^{52} -11.5616 q^{53} -1.00000 q^{54} -1.56155 q^{56} +4.68466 q^{57} +1.12311 q^{58} -4.87689 q^{59} +6.00000 q^{61} -1.00000 q^{62} +1.56155 q^{63} +1.00000 q^{64} +1.56155 q^{66} -9.36932 q^{67} -5.12311 q^{68} -5.56155 q^{69} +4.68466 q^{71} -1.00000 q^{72} -9.80776 q^{73} +5.12311 q^{74} +4.68466 q^{76} -2.43845 q^{77} +2.00000 q^{78} -16.6847 q^{79} +1.00000 q^{81} +1.12311 q^{82} +2.24621 q^{83} +1.56155 q^{84} -7.80776 q^{86} -1.12311 q^{87} +1.56155 q^{88} -1.31534 q^{89} -3.12311 q^{91} -5.56155 q^{92} +1.00000 q^{93} -3.12311 q^{94} -1.00000 q^{96} +6.00000 q^{97} +4.56155 q^{98} -1.56155 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - q^7 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9} + q^{11} + 2 q^{12} - 4 q^{13} + q^{14} + 2 q^{16} - 2 q^{17} - 2 q^{18} - 3 q^{19} - q^{21} - q^{22} - 7 q^{23} - 2 q^{24} + 4 q^{26} + 2 q^{27} - q^{28} + 6 q^{29} + 2 q^{31} - 2 q^{32} + q^{33} + 2 q^{34} + 2 q^{36} - 2 q^{37} + 3 q^{38} - 4 q^{39} + 6 q^{41} + q^{42} - 5 q^{43} + q^{44} + 7 q^{46} - 2 q^{47} + 2 q^{48} - 5 q^{49} - 2 q^{51} - 4 q^{52} - 19 q^{53} - 2 q^{54} + q^{56} - 3 q^{57} - 6 q^{58} - 18 q^{59} + 12 q^{61} - 2 q^{62} - q^{63} + 2 q^{64} - q^{66} + 6 q^{67} - 2 q^{68} - 7 q^{69} - 3 q^{71} - 2 q^{72} + q^{73} + 2 q^{74} - 3 q^{76} - 9 q^{77} + 4 q^{78} - 21 q^{79} + 2 q^{81} - 6 q^{82} - 12 q^{83} - q^{84} + 5 q^{86} + 6 q^{87} - q^{88} - 15 q^{89} + 2 q^{91} - 7 q^{92} + 2 q^{93} + 2 q^{94} - 2 q^{96} + 12 q^{97} + 5 q^{98} + q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - q^7 - 2 * q^8 + 2 * q^9 + q^11 + 2 * q^12 - 4 * q^13 + q^14 + 2 * q^16 - 2 * q^17 - 2 * q^18 - 3 * q^19 - q^21 - q^22 - 7 * q^23 - 2 * q^24 + 4 * q^26 + 2 * q^27 - q^28 + 6 * q^29 + 2 * q^31 - 2 * q^32 + q^33 + 2 * q^34 + 2 * q^36 - 2 * q^37 + 3 * q^38 - 4 * q^39 + 6 * q^41 + q^42 - 5 * q^43 + q^44 + 7 * q^46 - 2 * q^47 + 2 * q^48 - 5 * q^49 - 2 * q^51 - 4 * q^52 - 19 * q^53 - 2 * q^54 + q^56 - 3 * q^57 - 6 * q^58 - 18 * q^59 + 12 * q^61 - 2 * q^62 - q^63 + 2 * q^64 - q^66 + 6 * q^67 - 2 * q^68 - 7 * q^69 - 3 * q^71 - 2 * q^72 + q^73 + 2 * q^74 - 3 * q^76 - 9 * q^77 + 4 * q^78 - 21 * q^79 + 2 * q^81 - 6 * q^82 - 12 * q^83 - q^84 + 5 * q^86 + 6 * q^87 - q^88 - 15 * q^89 + 2 * q^91 - 7 * q^92 + 2 * q^93 + 2 * q^94 - 2 * q^96 + 12 * q^97 + 5 * q^98 + q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	1.56155	0.590211	0.295106	−	0.955465i	$-0.404645\pi$
$7$	1.56155	0.590211	0.295106	+	0.955465i	$0.404645\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.56155	−0.470826	−0.235413	−	0.971895i	$-0.575644\pi$
$11$	−1.56155	−0.470826	−0.235413	+	0.971895i	$0.575644\pi$
$12$	1.00000	0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−1.56155	−0.417343
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.12311	−1.24254	−0.621268	−	0.783598i	$-0.713382\pi$
$17$	−5.12311	−1.24254	−0.621268	+	0.783598i	$0.713382\pi$
$18$	−1.00000	−0.235702
$19$	4.68466	1.07473	0.537367	−	0.843348i	$-0.319419\pi$
$19$	4.68466	1.07473	0.537367	+	0.843348i	$0.319419\pi$
$20$	0	0
$21$	1.56155	0.340759
$22$	1.56155	0.332924
$23$	−5.56155	−1.15966	−0.579832	−	0.814736i	$-0.696882\pi$
$23$	−5.56155	−1.15966	−0.579832	+	0.814736i	$0.696882\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	2.00000	0.392232
$27$	1.00000	0.192450
$28$	1.56155	0.295106
$29$	−1.12311	−0.208555	−0.104278	−	0.994548i	$-0.533253\pi$
$29$	−1.12311	−0.208555	−0.104278	+	0.994548i	$0.533253\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−1.00000	−0.176777
$33$	−1.56155	−0.271831
$34$	5.12311	0.878605
$35$	0	0
$36$	1.00000	0.166667
$37$	−5.12311	−0.842233	−0.421117	−	0.907006i	$-0.638362\pi$
$37$	−5.12311	−0.842233	−0.421117	+	0.907006i	$0.638362\pi$
$38$	−4.68466	−0.759952
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−1.12311	−0.175400	−0.0876998	−	0.996147i	$-0.527952\pi$
$41$	−1.12311	−0.175400	−0.0876998	+	0.996147i	$0.527952\pi$
$42$	−1.56155	−0.240953
$43$	7.80776	1.19067	0.595336	−	0.803477i	$-0.297019\pi$
$43$	7.80776	1.19067	0.595336	+	0.803477i	$0.297019\pi$
$44$	−1.56155	−0.235413
$45$	0	0
$46$	5.56155	0.820006
$47$	3.12311	0.455552	0.227776	−	0.973714i	$-0.426855\pi$
$47$	3.12311	0.455552	0.227776	+	0.973714i	$0.426855\pi$
$48$	1.00000	0.144338
$49$	−4.56155	−0.651650
$50$	0	0
$51$	−5.12311	−0.717378
$52$	−2.00000	−0.277350
$53$	−11.5616	−1.58810	−0.794051	−	0.607852i	$-0.792032\pi$
$53$	−11.5616	−1.58810	−0.794051	+	0.607852i	$0.792032\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−1.56155	−0.208671
$57$	4.68466	0.620498
$58$	1.12311	0.147471
$59$	−4.87689	−0.634918	−0.317459	−	0.948272i	$-0.602830\pi$
$59$	−4.87689	−0.634918	−0.317459	+	0.948272i	$0.602830\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	−1.00000	−0.127000
$63$	1.56155	0.196737
$64$	1.00000	0.125000
$65$	0	0
$66$	1.56155	0.192214
$67$	−9.36932	−1.14464	−0.572322	−	0.820029i	$-0.693957\pi$
$67$	−9.36932	−1.14464	−0.572322	+	0.820029i	$0.693957\pi$
$68$	−5.12311	−0.621268
$69$	−5.56155	−0.669532
$70$	0	0
$71$	4.68466	0.555967	0.277983	−	0.960586i	$-0.410334\pi$
$71$	4.68466	0.555967	0.277983	+	0.960586i	$0.410334\pi$
$72$	−1.00000	−0.117851
$73$	−9.80776	−1.14791	−0.573956	−	0.818886i	$-0.694592\pi$
$73$	−9.80776	−1.14791	−0.573956	+	0.818886i	$0.694592\pi$
$74$	5.12311	0.595549
$75$	0	0
$76$	4.68466	0.537367
$77$	−2.43845	−0.277887
$78$	2.00000	0.226455
$79$	−16.6847	−1.87717	−0.938585	−	0.345047i	$-0.887863\pi$
$79$	−16.6847	−1.87717	−0.938585	+	0.345047i	$0.887863\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	1.12311	0.124026
$83$	2.24621	0.246554	0.123277	−	0.992372i	$-0.460660\pi$
$83$	2.24621	0.246554	0.123277	+	0.992372i	$0.460660\pi$
$84$	1.56155	0.170379
$85$	0	0
$86$	−7.80776	−0.841933
$87$	−1.12311	−0.120410
$88$	1.56155	0.166462
$89$	−1.31534	−0.139426	−0.0697130	−	0.997567i	$-0.522208\pi$
$89$	−1.31534	−0.139426	−0.0697130	+	0.997567i	$0.522208\pi$
$90$	0	0
$91$	−3.12311	−0.327390
$92$	−5.56155	−0.579832
$93$	1.00000	0.103695
$94$	−3.12311	−0.322124
$95$	0	0
$96$	−1.00000	−0.102062
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	4.56155	0.460786
$99$	−1.56155	−0.156942
$100$	0	0
$101$	3.56155	0.354388	0.177194	−	0.984176i	$-0.443298\pi$
$101$	3.56155	0.354388	0.177194	+	0.984176i	$0.443298\pi$
$102$	5.12311	0.507263
$103$	18.2462	1.79785	0.898926	−	0.438100i	$-0.144348\pi$
$103$	18.2462	1.79785	0.898926	+	0.438100i	$0.144348\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	11.5616	1.12296
$107$	−1.56155	−0.150961	−0.0754805	−	0.997147i	$-0.524049\pi$
$107$	−1.56155	−0.150961	−0.0754805	+	0.997147i	$0.524049\pi$
$108$	1.00000	0.0962250
$109$	−19.3693	−1.85524	−0.927622	−	0.373520i	$-0.878151\pi$
$109$	−19.3693	−1.85524	−0.927622	+	0.373520i	$0.878151\pi$
$110$	0	0
$111$	−5.12311	−0.486264
$112$	1.56155	0.147553
$113$	−10.6847	−1.00513	−0.502564	−	0.864540i	$-0.667610\pi$
$113$	−10.6847	−1.00513	−0.502564	+	0.864540i	$0.667610\pi$
$114$	−4.68466	−0.438758
$115$	0	0
$116$	−1.12311	−0.104278
$117$	−2.00000	−0.184900
$118$	4.87689	0.448955
$119$	−8.00000	−0.733359
$120$	0	0
$121$	−8.56155	−0.778323
$122$	−6.00000	−0.543214
$123$	−1.12311	−0.101267
$124$	1.00000	0.0898027
$125$	0	0
$126$	−1.56155	−0.139114
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−1.00000	−0.0883883
$129$	7.80776	0.687435
$130$	0	0
$131$	9.36932	0.818601	0.409301	−	0.912400i	$-0.365773\pi$
$131$	9.36932	0.818601	0.409301	+	0.912400i	$0.365773\pi$
$132$	−1.56155	−0.135916
$133$	7.31534	0.634321
$134$	9.36932	0.809386
$135$	0	0
$136$	5.12311	0.439303
$137$	−3.75379	−0.320708	−0.160354	−	0.987060i	$-0.551264\pi$
$137$	−3.75379	−0.320708	−0.160354	+	0.987060i	$0.551264\pi$
$138$	5.56155	0.473431
$139$	−8.87689	−0.752928	−0.376464	−	0.926431i	$-0.622860\pi$
$139$	−8.87689	−0.752928	−0.376464	+	0.926431i	$0.622860\pi$
$140$	0	0
$141$	3.12311	0.263013
$142$	−4.68466	−0.393128
$143$	3.12311	0.261167
$144$	1.00000	0.0833333
$145$	0	0
$146$	9.80776	0.811696
$147$	−4.56155	−0.376231
$148$	−5.12311	−0.421117
$149$	−20.0540	−1.64289	−0.821443	−	0.570291i	$-0.806830\pi$
$149$	−20.0540	−1.64289	−0.821443	+	0.570291i	$0.806830\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	−4.68466	−0.379976
$153$	−5.12311	−0.414179
$154$	2.43845	0.196496
$155$	0	0
$156$	−2.00000	−0.160128
$157$	20.0540	1.60048	0.800241	−	0.599679i	$-0.204705\pi$
$157$	20.0540	1.60048	0.800241	+	0.599679i	$0.204705\pi$
$158$	16.6847	1.32736
$159$	−11.5616	−0.916891
$160$	0	0
$161$	−8.68466	−0.684447
$162$	−1.00000	−0.0785674
$163$	−9.36932	−0.733862	−0.366931	−	0.930248i	$-0.619591\pi$
$163$	−9.36932	−0.733862	−0.366931	+	0.930248i	$0.619591\pi$
$164$	−1.12311	−0.0876998
$165$	0	0
$166$	−2.24621	−0.174340
$167$	2.43845	0.188693	0.0943464	−	0.995539i	$-0.469924\pi$
$167$	2.43845	0.188693	0.0943464	+	0.995539i	$0.469924\pi$
$168$	−1.56155	−0.120476
$169$	−9.00000	−0.692308
$170$	0	0
$171$	4.68466	0.358245
$172$	7.80776	0.595336
$173$	8.24621	0.626948	0.313474	−	0.949597i	$-0.398507\pi$
$173$	8.24621	0.626948	0.313474	+	0.949597i	$0.398507\pi$
$174$	1.12311	0.0851424
$175$	0	0
$176$	−1.56155	−0.117706
$177$	−4.87689	−0.366570
$178$	1.31534	0.0985890
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	24.0540	1.78792	0.893959	−	0.448149i	$-0.147917\pi$
$181$	24.0540	1.78792	0.893959	+	0.448149i	$0.147917\pi$
$182$	3.12311	0.231500
$183$	6.00000	0.443533
$184$	5.56155	0.410003
$185$	0	0
$186$	−1.00000	−0.0733236
$187$	8.00000	0.585018
$188$	3.12311	0.227776
$189$	1.56155	0.113586
$190$	0	0
$191$	2.24621	0.162530	0.0812651	−	0.996693i	$-0.474104\pi$
$191$	2.24621	0.162530	0.0812651	+	0.996693i	$0.474104\pi$
$192$	1.00000	0.0721688
$193$	18.4924	1.33111	0.665557	−	0.746347i	$-0.268194\pi$
$193$	18.4924	1.33111	0.665557	+	0.746347i	$0.268194\pi$
$194$	−6.00000	−0.430775
$195$	0	0
$196$	−4.56155	−0.325825
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	1.56155	0.110975
$199$	3.80776	0.269925	0.134963	−	0.990851i	$-0.456909\pi$
$199$	3.80776	0.269925	0.134963	+	0.990851i	$0.456909\pi$
$200$	0	0
$201$	−9.36932	−0.660861
$202$	−3.56155	−0.250590
$203$	−1.75379	−0.123092
$204$	−5.12311	−0.358689
$205$	0	0
$206$	−18.2462	−1.27127
$207$	−5.56155	−0.386555
$208$	−2.00000	−0.138675
$209$	−7.31534	−0.506013
$210$	0	0
$211$	−11.3153	−0.778980	−0.389490	−	0.921031i	$-0.627349\pi$
$211$	−11.3153	−0.778980	−0.389490	+	0.921031i	$0.627349\pi$
$212$	−11.5616	−0.794051
$213$	4.68466	0.320988
$214$	1.56155	0.106746
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	1.56155	0.106005
$218$	19.3693	1.31186
$219$	−9.80776	−0.662747
$220$	0	0
$221$	10.2462	0.689235
$222$	5.12311	0.343840
$223$	12.8769	0.862301	0.431150	−	0.902280i	$-0.358108\pi$
$223$	12.8769	0.862301	0.431150	+	0.902280i	$0.358108\pi$
$224$	−1.56155	−0.104336
$225$	0	0
$226$	10.6847	0.710733
$227$	4.68466	0.310932	0.155466	−	0.987841i	$-0.450312\pi$
$227$	4.68466	0.310932	0.155466	+	0.987841i	$0.450312\pi$
$228$	4.68466	0.310249
$229$	−12.4384	−0.821956	−0.410978	−	0.911645i	$-0.634813\pi$
$229$	−12.4384	−0.821956	−0.410978	+	0.911645i	$0.634813\pi$
$230$	0	0
$231$	−2.43845	−0.160438
$232$	1.12311	0.0737355
$233$	−20.0540	−1.31378	−0.656890	−	0.753987i	$-0.728128\pi$
$233$	−20.0540	−1.31378	−0.656890	+	0.753987i	$0.728128\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	−4.87689	−0.317459
$237$	−16.6847	−1.08379
$238$	8.00000	0.518563
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	−4.24621	−0.273523	−0.136761	−	0.990604i	$-0.543669\pi$
$241$	−4.24621	−0.273523	−0.136761	+	0.990604i	$0.543669\pi$
$242$	8.56155	0.550357
$243$	1.00000	0.0641500
$244$	6.00000	0.384111
$245$	0	0
$246$	1.12311	0.0716066
$247$	−9.36932	−0.596155
$248$	−1.00000	−0.0635001
$249$	2.24621	0.142348
$250$	0	0
$251$	−16.4924	−1.04099	−0.520496	−	0.853864i	$-0.674253\pi$
$251$	−16.4924	−1.04099	−0.520496	+	0.853864i	$0.674253\pi$
$252$	1.56155	0.0983686
$253$	8.68466	0.546000
$254$	0	0
$255$	0	0
$256$	1.00000	0.0625000
$257$	−10.6847	−0.666491	−0.333245	−	0.942840i	$-0.608144\pi$
$257$	−10.6847	−0.666491	−0.333245	+	0.942840i	$0.608144\pi$
$258$	−7.80776	−0.486090
$259$	−8.00000	−0.497096
$260$	0	0
$261$	−1.12311	−0.0695185
$262$	−9.36932	−0.578838
$263$	−12.4924	−0.770316	−0.385158	−	0.922851i	$-0.625853\pi$
$263$	−12.4924	−0.770316	−0.385158	+	0.922851i	$0.625853\pi$
$264$	1.56155	0.0961069
$265$	0	0
$266$	−7.31534	−0.448532
$267$	−1.31534	−0.0804976
$268$	−9.36932	−0.572322
$269$	16.2462	0.990549	0.495274	−	0.868737i	$-0.335067\pi$
$269$	16.2462	0.990549	0.495274	+	0.868737i	$0.335067\pi$
$270$	0	0
$271$	−10.0540	−0.610736	−0.305368	−	0.952234i	$-0.598779\pi$
$271$	−10.0540	−0.610736	−0.305368	+	0.952234i	$0.598779\pi$
$272$	−5.12311	−0.310634
$273$	−3.12311	−0.189019
$274$	3.75379	0.226775
$275$	0	0
$276$	−5.56155	−0.334766
$277$	−19.3693	−1.16379	−0.581895	−	0.813264i	$-0.697688\pi$
$277$	−19.3693	−1.16379	−0.581895	+	0.813264i	$0.697688\pi$
$278$	8.87689	0.532401
$279$	1.00000	0.0598684
$280$	0	0
$281$	13.1231	0.782859	0.391429	−	0.920208i	$-0.371981\pi$
$281$	13.1231	0.782859	0.391429	+	0.920208i	$0.371981\pi$
$282$	−3.12311	−0.185978
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	4.68466	0.277983
$285$	0	0
$286$	−3.12311	−0.184673
$287$	−1.75379	−0.103523
$288$	−1.00000	−0.0589256
$289$	9.24621	0.543895
$290$	0	0
$291$	6.00000	0.351726
$292$	−9.80776	−0.573956
$293$	6.49242	0.379291	0.189646	−	0.981853i	$-0.439266\pi$
$293$	6.49242	0.379291	0.189646	+	0.981853i	$0.439266\pi$
$294$	4.56155	0.266035
$295$	0	0
$296$	5.12311	0.297774
$297$	−1.56155	−0.0906105
$298$	20.0540	1.16170
$299$	11.1231	0.643266
$300$	0	0
$301$	12.1922	0.702749
$302$	0	0
$303$	3.56155	0.204606
$304$	4.68466	0.268684
$305$	0	0
$306$	5.12311	0.292868
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	−2.43845	−0.138943
$309$	18.2462	1.03799
$310$	0	0
$311$	−18.2462	−1.03465	−0.517324	−	0.855790i	$-0.673072\pi$
$311$	−18.2462	−1.03465	−0.517324	+	0.855790i	$0.673072\pi$
$312$	2.00000	0.113228
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	−20.0540	−1.13171
$315$	0	0
$316$	−16.6847	−0.938585
$317$	−25.1231	−1.41105	−0.705527	−	0.708683i	$-0.749290\pi$
$317$	−25.1231	−1.41105	−0.705527	+	0.708683i	$0.749290\pi$
$318$	11.5616	0.648340
$319$	1.75379	0.0981933
$320$	0	0
$321$	−1.56155	−0.0871574
$322$	8.68466	0.483977
$323$	−24.0000	−1.33540
$324$	1.00000	0.0555556
$325$	0	0
$326$	9.36932	0.518918
$327$	−19.3693	−1.07113
$328$	1.12311	0.0620131
$329$	4.87689	0.268872
$330$	0	0
$331$	10.2462	0.563183	0.281591	−	0.959534i	$-0.409138\pi$
$331$	10.2462	0.563183	0.281591	+	0.959534i	$0.409138\pi$
$332$	2.24621	0.123277
$333$	−5.12311	−0.280744
$334$	−2.43845	−0.133426
$335$	0	0
$336$	1.56155	0.0851897
$337$	10.0000	0.544735	0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000	0.544735	0.272367	+	0.962193i	$0.412193\pi$
$338$	9.00000	0.489535
$339$	−10.6847	−0.580311
$340$	0	0
$341$	−1.56155	−0.0845628
$342$	−4.68466	−0.253317
$343$	−18.0540	−0.974823
$344$	−7.80776	−0.420966
$345$	0	0
$346$	−8.24621	−0.443319
$347$	2.24621	0.120583	0.0602915	−	0.998181i	$-0.480797\pi$
$347$	2.24621	0.120583	0.0602915	+	0.998181i	$0.480797\pi$
$348$	−1.12311	−0.0602048
$349$	−19.3693	−1.03682	−0.518408	−	0.855133i	$-0.673475\pi$
$349$	−19.3693	−1.03682	−0.518408	+	0.855133i	$0.673475\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	1.56155	0.0832310
$353$	−16.2462	−0.864699	−0.432349	−	0.901706i	$-0.642315\pi$
$353$	−16.2462	−0.864699	−0.432349	+	0.901706i	$0.642315\pi$
$354$	4.87689	0.259204
$355$	0	0
$356$	−1.31534	−0.0697130
$357$	−8.00000	−0.423405
$358$	12.0000	0.634220
$359$	−19.3153	−1.01942	−0.509712	−	0.860345i	$-0.670248\pi$
$359$	−19.3153	−1.01942	−0.509712	+	0.860345i	$0.670248\pi$
$360$	0	0
$361$	2.94602	0.155054
$362$	−24.0540	−1.26425
$363$	−8.56155	−0.449365
$364$	−3.12311	−0.163695
$365$	0	0
$366$	−6.00000	−0.313625
$367$	9.36932	0.489074	0.244537	−	0.969640i	$-0.421364\pi$
$367$	9.36932	0.489074	0.244537	+	0.969640i	$0.421364\pi$
$368$	−5.56155	−0.289916
$369$	−1.12311	−0.0584665
$370$	0	0
$371$	−18.0540	−0.937316
$372$	1.00000	0.0518476
$373$	−6.68466	−0.346118	−0.173059	−	0.984911i	$-0.555365\pi$
$373$	−6.68466	−0.346118	−0.173059	+	0.984911i	$0.555365\pi$
$374$	−8.00000	−0.413670
$375$	0	0
$376$	−3.12311	−0.161062
$377$	2.24621	0.115686
$378$	−1.56155	−0.0803176
$379$	30.0540	1.54377	0.771885	−	0.635763i	$-0.219314\pi$
$379$	30.0540	1.54377	0.771885	+	0.635763i	$0.219314\pi$
$380$	0	0
$381$	0	0
$382$	−2.24621	−0.114926
$383$	−6.24621	−0.319166	−0.159583	−	0.987184i	$-0.551015\pi$
$383$	−6.24621	−0.319166	−0.159583	+	0.987184i	$0.551015\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−18.4924	−0.941240
$387$	7.80776	0.396891
$388$	6.00000	0.304604
$389$	24.2462	1.22933	0.614666	−	0.788788i	$-0.289291\pi$
$389$	24.2462	1.22933	0.614666	+	0.788788i	$0.289291\pi$
$390$	0	0
$391$	28.4924	1.44092
$392$	4.56155	0.230393
$393$	9.36932	0.472620
$394$	6.00000	0.302276
$395$	0	0
$396$	−1.56155	−0.0784710
$397$	28.0540	1.40799	0.703994	−	0.710206i	$-0.251398\pi$
$397$	28.0540	1.40799	0.703994	+	0.710206i	$0.251398\pi$
$398$	−3.80776	−0.190866
$399$	7.31534	0.366225
$400$	0	0
$401$	−1.31534	−0.0656850	−0.0328425	−	0.999461i	$-0.510456\pi$
$401$	−1.31534	−0.0656850	−0.0328425	+	0.999461i	$0.510456\pi$
$402$	9.36932	0.467299
$403$	−2.00000	−0.0996271
$404$	3.56155	0.177194
$405$	0	0
$406$	1.75379	0.0870391
$407$	8.00000	0.396545
$408$	5.12311	0.253632
$409$	−12.6307	−0.624547	−0.312274	−	0.949992i	$-0.601091\pi$
$409$	−12.6307	−0.624547	−0.312274	+	0.949992i	$0.601091\pi$
$410$	0	0
$411$	−3.75379	−0.185161
$412$	18.2462	0.898926
$413$	−7.61553	−0.374736
$414$	5.56155	0.273335
$415$	0	0
$416$	2.00000	0.0980581
$417$	−8.87689	−0.434703
$418$	7.31534	0.357805
$419$	22.2462	1.08680	0.543399	−	0.839474i	$-0.317137\pi$
$419$	22.2462	1.08680	0.543399	+	0.839474i	$0.317137\pi$
$420$	0	0
$421$	18.4924	0.901266	0.450633	−	0.892709i	$-0.351198\pi$
$421$	18.4924	0.901266	0.450633	+	0.892709i	$0.351198\pi$
$422$	11.3153	0.550822
$423$	3.12311	0.151851
$424$	11.5616	0.561479
$425$	0	0
$426$	−4.68466	−0.226972
$427$	9.36932	0.453413
$428$	−1.56155	−0.0754805
$429$	3.12311	0.150785
$430$	0	0
$431$	−13.7538	−0.662497	−0.331248	−	0.943544i	$-0.607470\pi$
$431$	−13.7538	−0.662497	−0.331248	+	0.943544i	$0.607470\pi$
$432$	1.00000	0.0481125
$433$	18.3002	0.879451	0.439725	−	0.898132i	$-0.355076\pi$
$433$	18.3002	0.879451	0.439725	+	0.898132i	$0.355076\pi$
$434$	−1.56155	−0.0749569
$435$	0	0
$436$	−19.3693	−0.927622
$437$	−26.0540	−1.24633
$438$	9.80776	0.468633
$439$	−31.6155	−1.50893	−0.754463	−	0.656342i	$-0.772103\pi$
$439$	−31.6155	−1.50893	−0.754463	+	0.656342i	$0.772103\pi$
$440$	0	0
$441$	−4.56155	−0.217217
$442$	−10.2462	−0.487363
$443$	14.0540	0.667725	0.333862	−	0.942622i	$-0.391648\pi$
$443$	14.0540	0.667725	0.333862	+	0.942622i	$0.391648\pi$
$444$	−5.12311	−0.243132
$445$	0	0
$446$	−12.8769	−0.609739
$447$	−20.0540	−0.948520
$448$	1.56155	0.0737764
$449$	−22.4924	−1.06148	−0.530742	−	0.847534i	$-0.678087\pi$
$449$	−22.4924	−1.06148	−0.530742	+	0.847534i	$0.678087\pi$
$450$	0	0
$451$	1.75379	0.0825827
$452$	−10.6847	−0.502564
$453$	0	0
$454$	−4.68466	−0.219862
$455$	0	0
$456$	−4.68466	−0.219379
$457$	−24.7386	−1.15722	−0.578612	−	0.815603i	$-0.696406\pi$
$457$	−24.7386	−1.15722	−0.578612	+	0.815603i	$0.696406\pi$
$458$	12.4384	0.581210
$459$	−5.12311	−0.239126
$460$	0	0
$461$	−15.3693	−0.715820	−0.357910	−	0.933756i	$-0.616511\pi$
$461$	−15.3693	−0.715820	−0.357910	+	0.933756i	$0.616511\pi$
$462$	2.43845	0.113447
$463$	18.7386	0.870858	0.435429	−	0.900223i	$-0.356597\pi$
$463$	18.7386	0.870858	0.435429	+	0.900223i	$0.356597\pi$
$464$	−1.12311	−0.0521389
$465$	0	0
$466$	20.0540	0.928982
$467$	26.2462	1.21453	0.607265	−	0.794499i	$-0.292267\pi$
$467$	26.2462	1.21453	0.607265	+	0.794499i	$0.292267\pi$
$468$	−2.00000	−0.0924500
$469$	−14.6307	−0.675582
$470$	0	0
$471$	20.0540	0.924038
$472$	4.87689	0.224477
$473$	−12.1922	−0.560600
$474$	16.6847	0.766352
$475$	0	0
$476$	−8.00000	−0.366679
$477$	−11.5616	−0.529367
$478$	−24.0000	−1.09773
$479$	6.43845	0.294180	0.147090	−	0.989123i	$-0.453009\pi$
$479$	6.43845	0.294180	0.147090	+	0.989123i	$0.453009\pi$
$480$	0	0
$481$	10.2462	0.467187
$482$	4.24621	0.193410
$483$	−8.68466	−0.395166
$484$	−8.56155	−0.389161
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−24.0000	−1.08754	−0.543772	−	0.839233i	$-0.683004\pi$
$487$	−24.0000	−1.08754	−0.543772	+	0.839233i	$0.683004\pi$
$488$	−6.00000	−0.271607
$489$	−9.36932	−0.423695
$490$	0	0
$491$	2.93087	0.132268	0.0661341	−	0.997811i	$-0.478933\pi$
$491$	2.93087	0.132268	0.0661341	+	0.997811i	$0.478933\pi$
$492$	−1.12311	−0.0506335
$493$	5.75379	0.259138
$494$	9.36932	0.421545
$495$	0	0
$496$	1.00000	0.0449013
$497$	7.31534	0.328138
$498$	−2.24621	−0.100655
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	2.43845	0.108942
$502$	16.4924	0.736093
$503$	−9.75379	−0.434900	−0.217450	−	0.976071i	$-0.569774\pi$
$503$	−9.75379	−0.434900	−0.217450	+	0.976071i	$0.569774\pi$
$504$	−1.56155	−0.0695571
$505$	0	0
$506$	−8.68466	−0.386080
$507$	−9.00000	−0.399704
$508$	0	0
$509$	−21.6155	−0.958091	−0.479046	−	0.877790i	$-0.659017\pi$
$509$	−21.6155	−0.958091	−0.479046	+	0.877790i	$0.659017\pi$
$510$	0	0
$511$	−15.3153	−0.677511
$512$	−1.00000	−0.0441942
$513$	4.68466	0.206833
$514$	10.6847	0.471280
$515$	0	0
$516$	7.80776	0.343718
$517$	−4.87689	−0.214486
$518$	8.00000	0.351500
$519$	8.24621	0.361968
$520$	0	0
$521$	−15.7538	−0.690186	−0.345093	−	0.938568i	$-0.612153\pi$
$521$	−15.7538	−0.690186	−0.345093	+	0.938568i	$0.612153\pi$
$522$	1.12311	0.0491570
$523$	−39.4233	−1.72386	−0.861930	−	0.507027i	$-0.830744\pi$
$523$	−39.4233	−1.72386	−0.861930	+	0.507027i	$0.830744\pi$
$524$	9.36932	0.409301
$525$	0	0
$526$	12.4924	0.544696
$527$	−5.12311	−0.223166
$528$	−1.56155	−0.0679579
$529$	7.93087	0.344820
$530$	0	0
$531$	−4.87689	−0.211639
$532$	7.31534	0.317160
$533$	2.24621	0.0972942
$534$	1.31534	0.0569204
$535$	0	0
$536$	9.36932	0.404693
$537$	−12.0000	−0.517838
$538$	−16.2462	−0.700424
$539$	7.12311	0.306814
$540$	0	0
$541$	36.2462	1.55835	0.779173	−	0.626809i	$-0.215639\pi$
$541$	36.2462	1.55835	0.779173	+	0.626809i	$0.215639\pi$
$542$	10.0540	0.431855
$543$	24.0540	1.03225
$544$	5.12311	0.219651
$545$	0	0
$546$	3.12311	0.133657
$547$	−35.1231	−1.50176	−0.750878	−	0.660441i	$-0.770369\pi$
$547$	−35.1231	−1.50176	−0.750878	+	0.660441i	$0.770369\pi$
$548$	−3.75379	−0.160354
$549$	6.00000	0.256074
$550$	0	0
$551$	−5.26137	−0.224142
$552$	5.56155	0.236715
$553$	−26.0540	−1.10793
$554$	19.3693	0.822923
$555$	0	0
$556$	−8.87689	−0.376464
$557$	6.19224	0.262373	0.131187	−	0.991358i	$-0.458121\pi$
$557$	6.19224	0.262373	0.131187	+	0.991358i	$0.458121\pi$
$558$	−1.00000	−0.0423334
$559$	−15.6155	−0.660466
$560$	0	0
$561$	8.00000	0.337760
$562$	−13.1231	−0.553565
$563$	16.4924	0.695073	0.347536	−	0.937667i	$-0.387018\pi$
$563$	16.4924	0.695073	0.347536	+	0.937667i	$0.387018\pi$
$564$	3.12311	0.131506
$565$	0	0
$566$	0	0
$567$	1.56155	0.0655791
$568$	−4.68466	−0.196564
$569$	10.1922	0.427281	0.213640	−	0.976912i	$-0.431468\pi$
$569$	10.1922	0.427281	0.213640	+	0.976912i	$0.431468\pi$
$570$	0	0
$571$	32.8769	1.37586	0.687928	−	0.725779i	$-0.258521\pi$
$571$	32.8769	1.37586	0.687928	+	0.725779i	$0.258521\pi$
$572$	3.12311	0.130584
$573$	2.24621	0.0938368
$574$	1.75379	0.0732017
$575$	0	0
$576$	1.00000	0.0416667
$577$	−6.87689	−0.286289	−0.143144	−	0.989702i	$-0.545721\pi$
$577$	−6.87689	−0.286289	−0.143144	+	0.989702i	$0.545721\pi$
$578$	−9.24621	−0.384592
$579$	18.4924	0.768519
$580$	0	0
$581$	3.50758	0.145519
$582$	−6.00000	−0.248708
$583$	18.0540	0.747719
$584$	9.80776	0.405848
$585$	0	0
$586$	−6.49242	−0.268200
$587$	−32.4924	−1.34111	−0.670553	−	0.741862i	$-0.733943\pi$
$587$	−32.4924	−1.34111	−0.670553	+	0.741862i	$0.733943\pi$
$588$	−4.56155	−0.188115
$589$	4.68466	0.193028
$590$	0	0
$591$	−6.00000	−0.246807
$592$	−5.12311	−0.210558
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	1.56155	0.0640713
$595$	0	0
$596$	−20.0540	−0.821443
$597$	3.80776	0.155841
$598$	−11.1231	−0.454858
$599$	−23.4233	−0.957050	−0.478525	−	0.878074i	$-0.658828\pi$
$599$	−23.4233	−0.957050	−0.478525	+	0.878074i	$0.658828\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	−12.1922	−0.496918
$603$	−9.36932	−0.381548
$604$	0	0
$605$	0	0
$606$	−3.56155	−0.144678
$607$	30.0540	1.21985	0.609927	−	0.792458i	$-0.291199\pi$
$607$	30.0540	1.21985	0.609927	+	0.792458i	$0.291199\pi$
$608$	−4.68466	−0.189988
$609$	−1.75379	−0.0710671
$610$	0	0
$611$	−6.24621	−0.252695
$612$	−5.12311	−0.207089
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	0	0
$615$	0	0
$616$	2.43845	0.0982478
$617$	19.5616	0.787518	0.393759	−	0.919214i	$-0.371174\pi$
$617$	19.5616	0.787518	0.393759	+	0.919214i	$0.371174\pi$
$618$	−18.2462	−0.733970
$619$	−13.3693	−0.537358	−0.268679	−	0.963230i	$-0.586587\pi$
$619$	−13.3693	−0.537358	−0.268679	+	0.963230i	$0.586587\pi$
$620$	0	0
$621$	−5.56155	−0.223177
$622$	18.2462	0.731606
$623$	−2.05398	−0.0822908
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	−10.0000	−0.399680
$627$	−7.31534	−0.292147
$628$	20.0540	0.800241
$629$	26.2462	1.04650
$630$	0	0
$631$	−24.3002	−0.967375	−0.483688	−	0.875241i	$-0.660703\pi$
$631$	−24.3002	−0.967375	−0.483688	+	0.875241i	$0.660703\pi$
$632$	16.6847	0.663680
$633$	−11.3153	−0.449744
$634$	25.1231	0.997766
$635$	0	0
$636$	−11.5616	−0.458445
$637$	9.12311	0.361471
$638$	−1.75379	−0.0694332
$639$	4.68466	0.185322
$640$	0	0
$641$	−38.4924	−1.52036	−0.760180	−	0.649713i	$-0.774889\pi$
$641$	−38.4924	−1.52036	−0.760180	+	0.649713i	$0.774889\pi$
$642$	1.56155	0.0616296
$643$	−30.0540	−1.18521	−0.592607	−	0.805492i	$-0.701901\pi$
$643$	−30.0540	−1.18521	−0.592607	+	0.805492i	$0.701901\pi$
$644$	−8.68466	−0.342223
$645$	0	0
$646$	24.0000	0.944267
$647$	17.0691	0.671057	0.335528	−	0.942030i	$-0.391085\pi$
$647$	17.0691	0.671057	0.335528	+	0.942030i	$0.391085\pi$
$648$	−1.00000	−0.0392837
$649$	7.61553	0.298936
$650$	0	0
$651$	1.56155	0.0612021
$652$	−9.36932	−0.366931
$653$	−23.7538	−0.929558	−0.464779	−	0.885427i	$-0.653866\pi$
$653$	−23.7538	−0.929558	−0.464779	+	0.885427i	$0.653866\pi$
$654$	19.3693	0.757400
$655$	0	0
$656$	−1.12311	−0.0438499
$657$	−9.80776	−0.382637
$658$	−4.87689	−0.190121
$659$	18.7386	0.729954	0.364977	−	0.931017i	$-0.381077\pi$
$659$	18.7386	0.729954	0.364977	+	0.931017i	$0.381077\pi$
$660$	0	0
$661$	−6.49242	−0.252526	−0.126263	−	0.991997i	$-0.540298\pi$
$661$	−6.49242	−0.252526	−0.126263	+	0.991997i	$0.540298\pi$
$662$	−10.2462	−0.398230
$663$	10.2462	0.397930
$664$	−2.24621	−0.0871699
$665$	0	0
$666$	5.12311	0.198516
$667$	6.24621	0.241854
$668$	2.43845	0.0943464
$669$	12.8769	0.497849
$670$	0	0
$671$	−9.36932	−0.361698
$672$	−1.56155	−0.0602382
$673$	−28.2462	−1.08881	−0.544406	−	0.838822i	$-0.683245\pi$
$673$	−28.2462	−1.08881	−0.544406	+	0.838822i	$0.683245\pi$
$674$	−10.0000	−0.385186
$675$	0	0
$676$	−9.00000	−0.346154
$677$	29.4233	1.13083	0.565414	−	0.824807i	$-0.308716\pi$
$677$	29.4233	1.13083	0.565414	+	0.824807i	$0.308716\pi$
$678$	10.6847	0.410342
$679$	9.36932	0.359561
$680$	0	0
$681$	4.68466	0.179517
$682$	1.56155	0.0597949
$683$	−19.3153	−0.739081	−0.369541	−	0.929215i	$-0.620485\pi$
$683$	−19.3153	−0.739081	−0.369541	+	0.929215i	$0.620485\pi$
$684$	4.68466	0.179122
$685$	0	0
$686$	18.0540	0.689304
$687$	−12.4384	−0.474556
$688$	7.80776	0.297668
$689$	23.1231	0.880920
$690$	0	0
$691$	−22.0540	−0.838973	−0.419486	−	0.907762i	$-0.637790\pi$
$691$	−22.0540	−0.838973	−0.419486	+	0.907762i	$0.637790\pi$
$692$	8.24621	0.313474
$693$	−2.43845	−0.0926289
$694$	−2.24621	−0.0852650
$695$	0	0
$696$	1.12311	0.0425712
$697$	5.75379	0.217940
$698$	19.3693	0.733139
$699$	−20.0540	−0.758511
$700$	0	0
$701$	25.8078	0.974746	0.487373	−	0.873194i	$-0.337955\pi$
$701$	25.8078	0.974746	0.487373	+	0.873194i	$0.337955\pi$
$702$	2.00000	0.0754851
$703$	−24.0000	−0.905177
$704$	−1.56155	−0.0588532
$705$	0	0
$706$	16.2462	0.611434
$707$	5.56155	0.209164
$708$	−4.87689	−0.183285
$709$	−20.0540	−0.753143	−0.376571	−	0.926388i	$-0.622897\pi$
$709$	−20.0540	−0.753143	−0.376571	+	0.926388i	$0.622897\pi$
$710$	0	0
$711$	−16.6847	−0.625724
$712$	1.31534	0.0492945
$713$	−5.56155	−0.208282
$714$	8.00000	0.299392
$715$	0	0
$716$	−12.0000	−0.448461
$717$	24.0000	0.896296
$718$	19.3153	0.720842
$719$	−43.1231	−1.60822	−0.804110	−	0.594480i	$-0.797358\pi$
$719$	−43.1231	−1.60822	−0.804110	+	0.594480i	$0.797358\pi$
$720$	0	0
$721$	28.4924	1.06111
$722$	−2.94602	−0.109640
$723$	−4.24621	−0.157918
$724$	24.0540	0.893959
$725$	0	0
$726$	8.56155	0.317749
$727$	14.0540	0.521233	0.260617	−	0.965442i	$-0.416074\pi$
$727$	14.0540	0.521233	0.260617	+	0.965442i	$0.416074\pi$
$728$	3.12311	0.115750
$729$	1.00000	0.0370370
$730$	0	0
$731$	−40.0000	−1.47945
$732$	6.00000	0.221766
$733$	−26.4924	−0.978520	−0.489260	−	0.872138i	$-0.662733\pi$
$733$	−26.4924	−0.978520	−0.489260	+	0.872138i	$0.662733\pi$
$734$	−9.36932	−0.345828
$735$	0	0
$736$	5.56155	0.205002
$737$	14.6307	0.538928
$738$	1.12311	0.0413421
$739$	−12.9848	−0.477655	−0.238828	−	0.971062i	$-0.576763\pi$
$739$	−12.9848	−0.477655	−0.238828	+	0.971062i	$0.576763\pi$
$740$	0	0
$741$	−9.36932	−0.344190
$742$	18.0540	0.662782
$743$	−11.4233	−0.419080	−0.209540	−	0.977800i	$-0.567197\pi$
$743$	−11.4233	−0.419080	−0.209540	+	0.977800i	$0.567197\pi$
$744$	−1.00000	−0.0366618
$745$	0	0
$746$	6.68466	0.244743
$747$	2.24621	0.0821846
$748$	8.00000	0.292509
$749$	−2.43845	−0.0890989
$750$	0	0
$751$	36.8769	1.34566	0.672828	−	0.739798i	$-0.265079\pi$
$751$	36.8769	1.34566	0.672828	+	0.739798i	$0.265079\pi$
$752$	3.12311	0.113888
$753$	−16.4924	−0.601017
$754$	−2.24621	−0.0818022
$755$	0	0
$756$	1.56155	0.0567931
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	−30.0540	−1.09161
$759$	8.68466	0.315233
$760$	0	0
$761$	41.4233	1.50159	0.750797	−	0.660533i	$-0.229670\pi$
$761$	41.4233	1.50159	0.750797	+	0.660533i	$0.229670\pi$
$762$	0	0
$763$	−30.2462	−1.09499
$764$	2.24621	0.0812651
$765$	0	0
$766$	6.24621	0.225685
$767$	9.75379	0.352189
$768$	1.00000	0.0360844
$769$	−16.4384	−0.592786	−0.296393	−	0.955066i	$-0.595784\pi$
$769$	−16.4384	−0.592786	−0.296393	+	0.955066i	$0.595784\pi$
$770$	0	0
$771$	−10.6847	−0.384799
$772$	18.4924	0.665557
$773$	15.5616	0.559710	0.279855	−	0.960042i	$-0.409714\pi$
$773$	15.5616	0.559710	0.279855	+	0.960042i	$0.409714\pi$
$774$	−7.80776	−0.280644
$775$	0	0
$776$	−6.00000	−0.215387
$777$	−8.00000	−0.286998
$778$	−24.2462	−0.869269
$779$	−5.26137	−0.188508
$780$	0	0
$781$	−7.31534	−0.261764
$782$	−28.4924	−1.01889
$783$	−1.12311	−0.0401365
$784$	−4.56155	−0.162913
$785$	0	0
$786$	−9.36932	−0.334192
$787$	−10.9309	−0.389643	−0.194822	−	0.980839i	$-0.562413\pi$
$787$	−10.9309	−0.389643	−0.194822	+	0.980839i	$0.562413\pi$
$788$	−6.00000	−0.213741
$789$	−12.4924	−0.444742
$790$	0	0
$791$	−16.6847	−0.593238
$792$	1.56155	0.0554874
$793$	−12.0000	−0.426132
$794$	−28.0540	−0.995598
$795$	0	0
$796$	3.80776	0.134963
$797$	−38.9848	−1.38091	−0.690457	−	0.723373i	$-0.742591\pi$
$797$	−38.9848	−1.38091	−0.690457	+	0.723373i	$0.742591\pi$
$798$	−7.31534	−0.258960
$799$	−16.0000	−0.566039
$800$	0	0
$801$	−1.31534	−0.0464753
$802$	1.31534	0.0464463
$803$	15.3153	0.540467
$804$	−9.36932	−0.330430
$805$	0	0
$806$	2.00000	0.0704470
$807$	16.2462	0.571894
$808$	−3.56155	−0.125295
$809$	−25.3153	−0.890040	−0.445020	−	0.895521i	$-0.646803\pi$
$809$	−25.3153	−0.890040	−0.445020	+	0.895521i	$0.646803\pi$
$810$	0	0
$811$	53.6695	1.88459	0.942296	−	0.334782i	$-0.108663\pi$
$811$	53.6695	1.88459	0.942296	+	0.334782i	$0.108663\pi$
$812$	−1.75379	−0.0615459
$813$	−10.0540	−0.352608
$814$	−8.00000	−0.280400
$815$	0	0
$816$	−5.12311	−0.179345
$817$	36.5767	1.27966
$818$	12.6307	0.441621
$819$	−3.12311	−0.109130
$820$	0	0
$821$	−21.6155	−0.754387	−0.377194	−	0.926134i	$-0.623111\pi$
$821$	−21.6155	−0.754387	−0.377194	+	0.926134i	$0.623111\pi$
$822$	3.75379	0.130928
$823$	−15.6155	−0.544323	−0.272162	−	0.962252i	$-0.587739\pi$
$823$	−15.6155	−0.544323	−0.272162	+	0.962252i	$0.587739\pi$
$824$	−18.2462	−0.635637
$825$	0	0
$826$	7.61553	0.264978
$827$	−18.2462	−0.634483	−0.317241	−	0.948345i	$-0.602757\pi$
$827$	−18.2462	−0.634483	−0.317241	+	0.948345i	$0.602757\pi$
$828$	−5.56155	−0.193277
$829$	−37.4233	−1.29976	−0.649882	−	0.760035i	$-0.725182\pi$
$829$	−37.4233	−1.29976	−0.649882	+	0.760035i	$0.725182\pi$
$830$	0	0
$831$	−19.3693	−0.671914
$832$	−2.00000	−0.0693375
$833$	23.3693	0.809699
$834$	8.87689	0.307382
$835$	0	0
$836$	−7.31534	−0.253006
$837$	1.00000	0.0345651
$838$	−22.2462	−0.768483
$839$	34.9309	1.20595	0.602974	−	0.797761i	$-0.293982\pi$
$839$	34.9309	1.20595	0.602974	+	0.797761i	$0.293982\pi$
$840$	0	0
$841$	−27.7386	−0.956505
$842$	−18.4924	−0.637291
$843$	13.1231	0.451984
$844$	−11.3153	−0.389490
$845$	0	0
$846$	−3.12311	−0.107375
$847$	−13.3693	−0.459375
$848$	−11.5616	−0.397025
$849$	0	0
$850$	0	0
$851$	28.4924	0.976708
$852$	4.68466	0.160494
$853$	38.7926	1.32823	0.664117	−	0.747629i	$-0.268808\pi$
$853$	38.7926	1.32823	0.664117	+	0.747629i	$0.268808\pi$
$854$	−9.36932	−0.320611
$855$	0	0
$856$	1.56155	0.0533728
$857$	53.2311	1.81834	0.909169	−	0.416427i	$-0.136718\pi$
$857$	53.2311	1.81834	0.909169	+	0.416427i	$0.136718\pi$
$858$	−3.12311	−0.106621
$859$	−22.7386	−0.775832	−0.387916	−	0.921695i	$-0.626805\pi$
$859$	−22.7386	−0.775832	−0.387916	+	0.921695i	$0.626805\pi$
$860$	0	0
$861$	−1.75379	−0.0597690
$862$	13.7538	0.468456
$863$	30.5464	1.03981	0.519906	−	0.854224i	$-0.325967\pi$
$863$	30.5464	1.03981	0.519906	+	0.854224i	$0.325967\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−18.3002	−0.621866
$867$	9.24621	0.314018
$868$	1.56155	0.0530026
$869$	26.0540	0.883821
$870$	0	0
$871$	18.7386	0.634934
$872$	19.3693	0.655928
$873$	6.00000	0.203069
$874$	26.0540	0.881289
$875$	0	0
$876$	−9.80776	−0.331374
$877$	50.0000	1.68838	0.844190	−	0.536044i	$-0.180082\pi$
$877$	50.0000	1.68838	0.844190	+	0.536044i	$0.180082\pi$
$878$	31.6155	1.06697
$879$	6.49242	0.218984
$880$	0	0
$881$	42.4924	1.43161	0.715803	−	0.698302i	$-0.246061\pi$
$881$	42.4924	1.43161	0.715803	+	0.698302i	$0.246061\pi$
$882$	4.56155	0.153595
$883$	31.4233	1.05748	0.528739	−	0.848784i	$-0.322665\pi$
$883$	31.4233	1.05748	0.528739	+	0.848784i	$0.322665\pi$
$884$	10.2462	0.344617
$885$	0	0
$886$	−14.0540	−0.472153
$887$	57.3693	1.92627	0.963137	−	0.269013i	$-0.0866974\pi$
$887$	57.3693	1.92627	0.963137	+	0.269013i	$0.0866974\pi$
$888$	5.12311	0.171920
$889$	0	0
$890$	0	0
$891$	−1.56155	−0.0523140
$892$	12.8769	0.431150
$893$	14.6307	0.489597
$894$	20.0540	0.670705
$895$	0	0
$896$	−1.56155	−0.0521678
$897$	11.1231	0.371390
$898$	22.4924	0.750582
$899$	−1.12311	−0.0374577
$900$	0	0
$901$	59.2311	1.97327
$902$	−1.75379	−0.0583948
$903$	12.1922	0.405732
$904$	10.6847	0.355366
$905$	0	0
$906$	0	0
$907$	−24.0000	−0.796907	−0.398453	−	0.917189i	$-0.630453\pi$
$907$	−24.0000	−0.796907	−0.398453	+	0.917189i	$0.630453\pi$
$908$	4.68466	0.155466
$909$	3.56155	0.118129
$910$	0	0
$911$	−44.4924	−1.47410	−0.737050	−	0.675838i	$-0.763782\pi$
$911$	−44.4924	−1.47410	−0.737050	+	0.675838i	$0.763782\pi$
$912$	4.68466	0.155125
$913$	−3.50758	−0.116084
$914$	24.7386	0.818281
$915$	0	0
$916$	−12.4384	−0.410978
$917$	14.6307	0.483148
$918$	5.12311	0.169088
$919$	39.6155	1.30680	0.653398	−	0.757015i	$-0.273343\pi$
$919$	39.6155	1.30680	0.653398	+	0.757015i	$0.273343\pi$
$920$	0	0
$921$	0	0
$922$	15.3693	0.506161
$923$	−9.36932	−0.308395
$924$	−2.43845	−0.0802190
$925$	0	0
$926$	−18.7386	−0.615790
$927$	18.2462	0.599284
$928$	1.12311	0.0368677
$929$	11.5616	0.379322	0.189661	−	0.981850i	$-0.439261\pi$
$929$	11.5616	0.379322	0.189661	+	0.981850i	$0.439261\pi$
$930$	0	0
$931$	−21.3693	−0.700351
$932$	−20.0540	−0.656890
$933$	−18.2462	−0.597354
$934$	−26.2462	−0.858802
$935$	0	0
$936$	2.00000	0.0653720
$937$	−16.6307	−0.543301	−0.271650	−	0.962396i	$-0.587569\pi$
$937$	−16.6307	−0.543301	−0.271650	+	0.962396i	$0.587569\pi$
$938$	14.6307	0.477709
$939$	10.0000	0.326338
$940$	0	0
$941$	27.3693	0.892214	0.446107	−	0.894980i	$-0.352810\pi$
$941$	27.3693	0.892214	0.446107	+	0.894980i	$0.352810\pi$
$942$	−20.0540	−0.653394
$943$	6.24621	0.203405
$944$	−4.87689	−0.158729
$945$	0	0
$946$	12.1922	0.396404
$947$	0.876894	0.0284952	0.0142476	−	0.999898i	$-0.495465\pi$
$947$	0.876894	0.0284952	0.0142476	+	0.999898i	$0.495465\pi$
$948$	−16.6847	−0.541893
$949$	19.6155	0.636747
$950$	0	0
$951$	−25.1231	−0.814673
$952$	8.00000	0.259281
$953$	58.1080	1.88230	0.941151	−	0.337988i	$-0.109746\pi$
$953$	58.1080	1.88230	0.941151	+	0.337988i	$0.109746\pi$
$954$	11.5616	0.374319
$955$	0	0
$956$	24.0000	0.776215
$957$	1.75379	0.0566919
$958$	−6.43845	−0.208017
$959$	−5.86174	−0.189285
$960$	0	0
$961$	1.00000	0.0322581
$962$	−10.2462	−0.330351
$963$	−1.56155	−0.0503203
$964$	−4.24621	−0.136761
$965$	0	0
$966$	8.68466	0.279424
$967$	25.7538	0.828186	0.414093	−	0.910235i	$-0.364099\pi$
$967$	25.7538	0.828186	0.414093	+	0.910235i	$0.364099\pi$
$968$	8.56155	0.275179
$969$	−24.0000	−0.770991
$970$	0	0
$971$	−28.8769	−0.926704	−0.463352	−	0.886174i	$-0.653353\pi$
$971$	−28.8769	−0.926704	−0.463352	+	0.886174i	$0.653353\pi$
$972$	1.00000	0.0320750
$973$	−13.8617	−0.444387
$974$	24.0000	0.769010
$975$	0	0
$976$	6.00000	0.192055
$977$	22.9848	0.735350	0.367675	−	0.929954i	$-0.380154\pi$
$977$	22.9848	0.735350	0.367675	+	0.929954i	$0.380154\pi$
$978$	9.36932	0.299598
$979$	2.05398	0.0656453
$980$	0	0
$981$	−19.3693	−0.618415
$982$	−2.93087	−0.0935278
$983$	24.9848	0.796893	0.398446	−	0.917192i	$-0.369549\pi$
$983$	24.9848	0.796893	0.398446	+	0.917192i	$0.369549\pi$
$984$	1.12311	0.0358033
$985$	0	0
$986$	−5.75379	−0.183238
$987$	4.87689	0.155233
$988$	−9.36932	−0.298078
$989$	−43.4233	−1.38078
$990$	0	0
$991$	23.3153	0.740636	0.370318	−	0.928905i	$-0.379249\pi$
$991$	23.3153	0.740636	0.370318	+	0.928905i	$0.379249\pi$
$992$	−1.00000	−0.0317500
$993$	10.2462	0.325154
$994$	−7.31534	−0.232029
$995$	0	0
$996$	2.24621	0.0711739
$997$	62.4924	1.97915	0.989577	−	0.144002i	$-0.0459971\pi$
$997$	62.4924	1.97915	0.989577	+	0.144002i	$0.0459971\pi$
$998$	−20.0000	−0.633089
$999$	−5.12311	−0.162088

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4650.2.a.ce.1.2		2
5.2	odd	4		4650.2.d.bd.3349.2		4
5.3	odd	4		4650.2.d.bd.3349.3		4
5.4	even	2		930.2.a.p.1.1	✓	2
15.14	odd	2		2790.2.a.be.1.1		2
20.19	odd	2		7440.2.a.bl.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.p.1.1	✓	2	5.4	even	2
2790.2.a.be.1.1		2	15.14	odd	2
4650.2.a.ce.1.2		2	1.1	even	1	trivial
4650.2.d.bd.3349.2		4	5.2	odd	4
4650.2.d.bd.3349.3		4	5.3	odd	4
7440.2.a.bl.1.2		2	20.19	odd	2

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.56155 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.56155 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.56155 q^{11} +1.00000 q^{12} -2.00000 q^{13} -1.56155 q^{14} +1.00000 q^{16} -5.12311 q^{17} -1.00000 q^{18} +4.68466 q^{19} +1.56155 q^{21} +1.56155 q^{22} -5.56155 q^{23} -1.00000 q^{24} +2.00000 q^{26} +1.00000 q^{27} +1.56155 q^{28} -1.12311 q^{29} +1.00000 q^{31} -1.00000 q^{32} -1.56155 q^{33} +5.12311 q^{34} +1.00000 q^{36} -5.12311 q^{37} -4.68466 q^{38} -2.00000 q^{39} -1.12311 q^{41} -1.56155 q^{42} +7.80776 q^{43} -1.56155 q^{44} +5.56155 q^{46} +3.12311 q^{47} +1.00000 q^{48} -4.56155 q^{49} -5.12311 q^{51} -2.00000 q^{52} -11.5616 q^{53} -1.00000 q^{54} -1.56155 q^{56} +4.68466 q^{57} +1.12311 q^{58} -4.87689 q^{59} +6.00000 q^{61} -1.00000 q^{62} +1.56155 q^{63} +1.00000 q^{64} +1.56155 q^{66} -9.36932 q^{67} -5.12311 q^{68} -5.56155 q^{69} +4.68466 q^{71} -1.00000 q^{72} -9.80776 q^{73} +5.12311 q^{74} +4.68466 q^{76} -2.43845 q^{77} +2.00000 q^{78} -16.6847 q^{79} +1.00000 q^{81} +1.12311 q^{82} +2.24621 q^{83} +1.56155 q^{84} -7.80776 q^{86} -1.12311 q^{87} +1.56155 q^{88} -1.31534 q^{89} -3.12311 q^{91} -5.56155 q^{92} +1.00000 q^{93} -3.12311 q^{94} -1.00000 q^{96} +6.00000 q^{97} +4.56155 q^{98} -1.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9} + q^{11} + 2 q^{12} - 4 q^{13} + q^{14} + 2 q^{16} - 2 q^{17} - 2 q^{18} - 3 q^{19} - q^{21} - q^{22} - 7 q^{23} - 2 q^{24} + 4 q^{26} + 2 q^{27} - q^{28} + 6 q^{29} + 2 q^{31} - 2 q^{32} + q^{33} + 2 q^{34} + 2 q^{36} - 2 q^{37} + 3 q^{38} - 4 q^{39} + 6 q^{41} + q^{42} - 5 q^{43} + q^{44} + 7 q^{46} - 2 q^{47} + 2 q^{48} - 5 q^{49} - 2 q^{51} - 4 q^{52} - 19 q^{53} - 2 q^{54} + q^{56} - 3 q^{57} - 6 q^{58} - 18 q^{59} + 12 q^{61} - 2 q^{62} - q^{63} + 2 q^{64} - q^{66} + 6 q^{67} - 2 q^{68} - 7 q^{69} - 3 q^{71} - 2 q^{72} + q^{73} + 2 q^{74} - 3 q^{76} - 9 q^{77} + 4 q^{78} - 21 q^{79} + 2 q^{81} - 6 q^{82} - 12 q^{83} - q^{84} + 5 q^{86} + 6 q^{87} - q^{88} - 15 q^{89} + 2 q^{91} - 7 q^{92} + 2 q^{93} + 2 q^{94} - 2 q^{96} + 12 q^{97} + 5 q^{98} + q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - q^7 - 2 * q^8 + 2 * q^9 + q^11 + 2 * q^12 - 4 * q^13 + q^14 + 2 * q^16 - 2 * q^17 - 2 * q^18 - 3 * q^19 - q^21 - q^22 - 7 * q^23 - 2 * q^24 + 4 * q^26 + 2 * q^27 - q^28 + 6 * q^29 + 2 * q^31 - 2 * q^32 + q^33 + 2 * q^34 + 2 * q^36 - 2 * q^37 + 3 * q^38 - 4 * q^39 + 6 * q^41 + q^42 - 5 * q^43 + q^44 + 7 * q^46 - 2 * q^47 + 2 * q^48 - 5 * q^49 - 2 * q^51 - 4 * q^52 - 19 * q^53 - 2 * q^54 + q^56 - 3 * q^57 - 6 * q^58 - 18 * q^59 + 12 * q^61 - 2 * q^62 - q^63 + 2 * q^64 - q^66 + 6 * q^67 - 2 * q^68 - 7 * q^69 - 3 * q^71 - 2 * q^72 + q^73 + 2 * q^74 - 3 * q^76 - 9 * q^77 + 4 * q^78 - 21 * q^79 + 2 * q^81 - 6 * q^82 - 12 * q^83 - q^84 + 5 * q^86 + 6 * q^87 - q^88 - 15 * q^89 + 2 * q^91 - 7 * q^92 + 2 * q^93 + 2 * q^94 - 2 * q^96 + 12 * q^97 + 5 * q^98 + q^99

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