LMFDB - Embedded newform 6325.2.a.m.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6325,2,Mod(1,6325)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6325.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 6325 = 5^{2} \cdot 11 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6325.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,-3,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$50.5053792785$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.8639957.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 4x^{4} + 10x^{3} + 6x^{2} - 7x - 2 $ x^6 - 3x^5 - 4x^4 + 10x^3 + 6x^2 - 7*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 253)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.255514$ of defining polynomial
Character	$\chi$	$=$	6325.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.25551 q^{2} -0.646163 q^{3} -0.423685 q^{4} +0.811267 q^{6} -0.798084 q^{7} +3.04297 q^{8} -2.58247 q^{9} +1.00000 q^{11} +0.273769 q^{12} +3.19289 q^{13} +1.00201 q^{14} -2.97312 q^{16} -0.503752 q^{17} +3.24233 q^{18} +3.98359 q^{19} +0.515693 q^{21} -1.25551 q^{22} +1.00000 q^{23} -1.96625 q^{24} -4.00871 q^{26} +3.60719 q^{27} +0.338136 q^{28} -6.16897 q^{29} -3.28168 q^{31} -2.35314 q^{32} -0.646163 q^{33} +0.632468 q^{34} +1.09415 q^{36} -5.17919 q^{37} -5.00145 q^{38} -2.06312 q^{39} -11.8209 q^{41} -0.647459 q^{42} +7.08974 q^{43} -0.423685 q^{44} -1.25551 q^{46} -1.22773 q^{47} +1.92112 q^{48} -6.36306 q^{49} +0.325506 q^{51} -1.35278 q^{52} +3.00201 q^{53} -4.52888 q^{54} -2.42855 q^{56} -2.57405 q^{57} +7.74523 q^{58} +9.03304 q^{59} +2.63648 q^{61} +4.12020 q^{62} +2.06103 q^{63} +8.90065 q^{64} +0.811267 q^{66} -8.99026 q^{67} +0.213432 q^{68} -0.646163 q^{69} +9.80465 q^{71} -7.85839 q^{72} -1.31433 q^{73} +6.50255 q^{74} -1.68779 q^{76} -0.798084 q^{77} +2.59028 q^{78} +15.6898 q^{79} +5.41659 q^{81} +14.8413 q^{82} +2.92233 q^{83} -0.218491 q^{84} -8.90127 q^{86} +3.98616 q^{87} +3.04297 q^{88} +6.20642 q^{89} -2.54819 q^{91} -0.423685 q^{92} +2.12050 q^{93} +1.54144 q^{94} +1.52051 q^{96} +15.3359 q^{97} +7.98891 q^{98} -2.58247 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{2} - 7 q^{3} + 5 q^{4} + 3 q^{6} + q^{7} - 3 q^{8} + 9 q^{9} + 6 q^{11} - 11 q^{12} + 3 q^{13} - 8 q^{14} - q^{16} - 5 q^{17} + 11 q^{18} + q^{19} - 15 q^{21} - 3 q^{22} + 6 q^{23} - 3 q^{24}+ \cdots + 9 q^{99}+O(q^{100}) $ 6 * q - 3 * q^2 - 7 * q^3 + 5 * q^4 + 3 * q^6 + q^7 - 3 * q^8 + 9 * q^9 + 6 * q^11 - 11 * q^12 + 3 * q^13 - 8 * q^14 - q^16 - 5 * q^17 + 11 * q^18 + q^19 - 15 * q^21 - 3 * q^22 + 6 * q^23 - 3 * q^24 + 15 * q^26 - 16 * q^27 + 6 * q^29 - 8 * q^31 + 6 * q^32 - 7 * q^33 - 6 * q^34 - 8 * q^36 - 2 * q^37 - 9 * q^38 + q^39 - 2 * q^41 + 31 * q^42 - 10 * q^43 + 5 * q^44 - 3 * q^46 - 14 * q^47 - 6 * q^48 - 5 * q^49 - 24 * q^51 + 11 * q^52 + 4 * q^53 - 13 * q^54 - 13 * q^56 + 19 * q^57 + 6 * q^58 + 39 * q^59 - 22 * q^61 + 35 * q^63 - 33 * q^64 + 3 * q^66 - 8 * q^67 - 17 * q^68 - 7 * q^69 + 18 * q^71 - 6 * q^72 + 9 * q^73 + 15 * q^74 - 25 * q^76 + q^77 - 20 * q^78 - 15 * q^79 + 18 * q^81 + 8 * q^82 - 43 * q^83 - 4 * q^84 - 55 * q^86 + 28 * q^87 - 3 * q^88 + 17 * q^89 - 9 * q^91 + 5 * q^92 - 27 * q^93 + 63 * q^94 + 26 * q^96 - 19 * q^97 - 34 * q^98 + 9 * 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 $

$2$	−1.25551	−0.887782	−0.443891	−	0.896081i	$-0.646402\pi$
$2$	−1.25551	−0.887782	−0.443891	+	0.896081i	$0.646402\pi$
$3$	−0.646163	−0.373062	−0.186531	−	0.982449i	$-0.559725\pi$
$3$	−0.646163	−0.373062	−0.186531	+	0.982449i	$0.559725\pi$
$4$	−0.423685	−0.211842
$5$	0	0
$5$	0	0
$6$	0.811267	0.331198
$7$	−0.798084	−0.301647	−0.150824	−	0.988561i	$-0.548193\pi$
$7$	−0.798084	−0.301647	−0.150824	+	0.988561i	$0.548193\pi$
$8$	3.04297	1.07585
$9$	−2.58247	−0.860824
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0.273769	0.0790304
$13$	3.19289	0.885547	0.442774	−	0.896633i	$-0.353995\pi$
$13$	3.19289	0.885547	0.442774	+	0.896633i	$0.353995\pi$
$14$	1.00201	0.267797
$15$	0	0
$16$	−2.97312	−0.743281
$17$	−0.503752	−0.122178	−0.0610889	−	0.998132i	$-0.519457\pi$
$17$	−0.503752	−0.122178	−0.0610889	+	0.998132i	$0.519457\pi$
$18$	3.24233	0.764225
$19$	3.98359	0.913898	0.456949	−	0.889493i	$-0.348942\pi$
$19$	3.98359	0.913898	0.456949	+	0.889493i	$0.348942\pi$
$20$	0	0
$21$	0.515693	0.112533
$22$	−1.25551	−0.267676
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−1.96625	−0.401360
$25$	0	0
$26$	−4.00871	−0.786173
$27$	3.60719	0.694204
$28$	0.338136	0.0639017
$29$	−6.16897	−1.14555	−0.572774	−	0.819713i	$-0.694133\pi$
$29$	−6.16897	−1.14555	−0.572774	+	0.819713i	$0.694133\pi$
$30$	0	0
$31$	−3.28168	−0.589407	−0.294704	−	0.955589i	$-0.595221\pi$
$31$	−3.28168	−0.589407	−0.294704	+	0.955589i	$0.595221\pi$
$32$	−2.35314	−0.415981
$33$	−0.646163	−0.112483
$34$	0.632468	0.108467
$35$	0	0
$36$	1.09415	0.182359
$37$	−5.17919	−0.851454	−0.425727	−	0.904852i	$-0.639982\pi$
$37$	−5.17919	−0.851454	−0.425727	+	0.904852i	$0.639982\pi$
$38$	−5.00145	−0.811343
$39$	−2.06312	−0.330364
$40$	0	0
$41$	−11.8209	−1.84611	−0.923056	−	0.384666i	$-0.874317\pi$
$41$	−11.8209	−1.84611	−0.923056	+	0.384666i	$0.874317\pi$
$42$	−0.647459	−0.0999051
$43$	7.08974	1.08118	0.540588	−	0.841288i	$-0.318202\pi$
$43$	7.08974	1.08118	0.540588	+	0.841288i	$0.318202\pi$
$44$	−0.423685	−0.0638728
$45$	0	0
$46$	−1.25551	−0.185115
$47$	−1.22773	−0.179083	−0.0895416	−	0.995983i	$-0.528540\pi$
$47$	−1.22773	−0.179083	−0.0895416	+	0.995983i	$0.528540\pi$
$48$	1.92112	0.277290
$49$	−6.36306	−0.909009
$50$	0	0
$51$	0.325506	0.0455800
$52$	−1.35278	−0.187596
$53$	3.00201	0.412357	0.206179	−	0.978514i	$-0.433897\pi$
$53$	3.00201	0.412357	0.206179	+	0.978514i	$0.433897\pi$
$54$	−4.52888	−0.616302
$55$	0	0
$56$	−2.42855	−0.324528
$57$	−2.57405	−0.340941
$58$	7.74523	1.01700
$59$	9.03304	1.17600	0.588001	−	0.808861i	$-0.299915\pi$
$59$	9.03304	1.17600	0.588001	+	0.808861i	$0.299915\pi$
$60$	0	0
$61$	2.63648	0.337567	0.168783	−	0.985653i	$-0.446016\pi$
$61$	2.63648	0.337567	0.168783	+	0.985653i	$0.446016\pi$
$62$	4.12020	0.523266
$63$	2.06103	0.259666
$64$	8.90065	1.11258
$65$	0	0
$66$	0.811267	0.0998600
$67$	−8.99026	−1.09834	−0.549168	−	0.835712i	$-0.685055\pi$
$67$	−8.99026	−1.09834	−0.549168	+	0.835712i	$0.685055\pi$
$68$	0.213432	0.0258824
$69$	−0.646163	−0.0777889
$70$	0	0
$71$	9.80465	1.16360	0.581799	−	0.813333i	$-0.302349\pi$
$71$	9.80465	1.16360	0.581799	+	0.813333i	$0.302349\pi$
$72$	−7.85839	−0.926120
$73$	−1.31433	−0.153830	−0.0769151	−	0.997038i	$-0.524507\pi$
$73$	−1.31433	−0.153830	−0.0769151	+	0.997038i	$0.524507\pi$
$74$	6.50255	0.755906
$75$	0	0
$76$	−1.68779	−0.193602
$77$	−0.798084	−0.0909501
$78$	2.59028	0.293292
$79$	15.6898	1.76524	0.882618	−	0.470091i	$-0.155779\pi$
$79$	15.6898	1.76524	0.882618	+	0.470091i	$0.155779\pi$
$80$	0	0
$81$	5.41659	0.601843
$82$	14.8413	1.63895
$83$	2.92233	0.320767	0.160384	−	0.987055i	$-0.448727\pi$
$83$	2.92233	0.320767	0.160384	+	0.987055i	$0.448727\pi$
$84$	−0.218491	−0.0238393
$85$	0	0
$86$	−8.90127	−0.959849
$87$	3.98616	0.427361
$88$	3.04297	0.324382
$89$	6.20642	0.657879	0.328940	−	0.944351i	$-0.393309\pi$
$89$	6.20642	0.657879	0.328940	+	0.944351i	$0.393309\pi$
$90$	0	0
$91$	−2.54819	−0.267123
$92$	−0.423685	−0.0441722
$93$	2.12050	0.219886
$94$	1.54144	0.158987
$95$	0	0
$96$	1.52051	0.155187
$97$	15.3359	1.55712	0.778561	−	0.627569i	$-0.215950\pi$
$97$	15.3359	1.55712	0.778561	+	0.627569i	$0.215950\pi$
$98$	7.98891	0.807002
$99$	−2.58247	−0.259548
$100$	0	0
$101$	−5.69653	−0.566826	−0.283413	−	0.958998i	$-0.591467\pi$
$101$	−5.69653	−0.566826	−0.283413	+	0.958998i	$0.591467\pi$
$102$	−0.408677	−0.0404651
$103$	−0.992886	−0.0978319	−0.0489160	−	0.998803i	$-0.515577\pi$
$103$	−0.992886	−0.0978319	−0.0489160	+	0.998803i	$0.515577\pi$
$104$	9.71585	0.952718
$105$	0	0
$106$	−3.76906	−0.366084
$107$	5.98921	0.578999	0.289500	−	0.957178i	$-0.406511\pi$
$107$	5.98921	0.578999	0.289500	+	0.957178i	$0.406511\pi$
$108$	−1.52831	−0.147062
$109$	−15.3585	−1.47108	−0.735541	−	0.677480i	$-0.763072\pi$
$109$	−15.3585	−1.47108	−0.735541	+	0.677480i	$0.763072\pi$
$110$	0	0
$111$	3.34660	0.317645
$112$	2.37280	0.224209
$113$	−15.7470	−1.48135	−0.740675	−	0.671863i	$-0.765494\pi$
$113$	−15.7470	−1.48135	−0.740675	+	0.671863i	$0.765494\pi$
$114$	3.23175	0.302681
$115$	0	0
$116$	2.61370	0.242676
$117$	−8.24554	−0.762301
$118$	−11.3411	−1.04403
$119$	0.402037	0.0368546
$120$	0	0
$121$	1.00000	0.0909091
$122$	−3.31014	−0.299686
$123$	7.63822	0.688715
$124$	1.39040	0.124861
$125$	0	0
$126$	−2.58765	−0.230526
$127$	−9.06863	−0.804710	−0.402355	−	0.915484i	$-0.631808\pi$
$127$	−9.06863	−0.804710	−0.402355	+	0.915484i	$0.631808\pi$
$128$	−6.46860	−0.571749
$129$	−4.58113	−0.403346
$130$	0	0
$131$	10.9241	0.954442	0.477221	−	0.878783i	$-0.341644\pi$
$131$	10.9241	0.954442	0.477221	+	0.878783i	$0.341644\pi$
$132$	0.273769	0.0238286
$133$	−3.17924	−0.275675
$134$	11.2874	0.975083
$135$	0	0
$136$	−1.53290	−0.131445
$137$	−22.4269	−1.91606	−0.958029	−	0.286672i	$-0.907451\pi$
$137$	−22.4269	−1.91606	−0.958029	+	0.286672i	$0.907451\pi$
$138$	0.811267	0.0690596
$139$	−0.760327	−0.0644901	−0.0322451	−	0.999480i	$-0.510266\pi$
$139$	−0.760327	−0.0644901	−0.0322451	+	0.999480i	$0.510266\pi$
$140$	0	0
$141$	0.793315	0.0668092
$142$	−12.3099	−1.03302
$143$	3.19289	0.267002
$144$	7.67801	0.639834
$145$	0	0
$146$	1.65015	0.136568
$147$	4.11158	0.339117
$148$	2.19434	0.180374
$149$	5.59496	0.458357	0.229179	−	0.973384i	$-0.426396\pi$
$149$	5.59496	0.458357	0.229179	+	0.973384i	$0.426396\pi$
$150$	0	0
$151$	7.13755	0.580845	0.290423	−	0.956898i	$-0.406204\pi$
$151$	7.13755	0.580845	0.290423	+	0.956898i	$0.406204\pi$
$152$	12.1219	0.983219
$153$	1.30093	0.105174
$154$	1.00201	0.0807439
$155$	0	0
$156$	0.874114	0.0699851
$157$	18.8473	1.50418	0.752090	−	0.659061i	$-0.229046\pi$
$157$	18.8473	1.50418	0.752090	+	0.659061i	$0.229046\pi$
$158$	−19.6987	−1.56715
$159$	−1.93979	−0.153835
$160$	0	0
$161$	−0.798084	−0.0628978
$162$	−6.80060	−0.534306
$163$	19.5160	1.52861	0.764305	−	0.644854i	$-0.223082\pi$
$163$	19.5160	1.52861	0.764305	+	0.644854i	$0.223082\pi$
$164$	5.00833	0.391084
$165$	0	0
$166$	−3.66903	−0.284772
$167$	−20.5303	−1.58868	−0.794342	−	0.607471i	$-0.792184\pi$
$167$	−20.5303	−1.58868	−0.794342	+	0.607471i	$0.792184\pi$
$168$	1.56924	0.121069
$169$	−2.80548	−0.215806
$170$	0	0
$171$	−10.2875	−0.786706
$172$	−3.00381	−0.229039
$173$	−4.67283	−0.355268	−0.177634	−	0.984097i	$-0.556844\pi$
$173$	−4.67283	−0.355268	−0.177634	+	0.984097i	$0.556844\pi$
$174$	−5.00468	−0.379404
$175$	0	0
$176$	−2.97312	−0.224108
$177$	−5.83681	−0.438722
$178$	−7.79225	−0.584054
$179$	−4.68491	−0.350166	−0.175083	−	0.984554i	$-0.556019\pi$
$179$	−4.68491	−0.350166	−0.175083	+	0.984554i	$0.556019\pi$
$180$	0	0
$181$	10.2400	0.761133	0.380566	−	0.924754i	$-0.375729\pi$
$181$	10.2400	0.761133	0.380566	+	0.924754i	$0.375729\pi$
$182$	3.19929	0.237147
$183$	−1.70360	−0.125933
$184$	3.04297	0.224331
$185$	0	0
$186$	−2.66232	−0.195211
$187$	−0.503752	−0.0368380
$188$	0.520171	0.0379374
$189$	−2.87884	−0.209405
$190$	0	0
$191$	−6.39892	−0.463009	−0.231505	−	0.972834i	$-0.574365\pi$
$191$	−6.39892	−0.463009	−0.231505	+	0.972834i	$0.574365\pi$
$192$	−5.75127	−0.415062
$193$	−2.17363	−0.156462	−0.0782308	−	0.996935i	$-0.524927\pi$
$193$	−2.17363	−0.156462	−0.0782308	+	0.996935i	$0.524927\pi$
$194$	−19.2544	−1.38238
$195$	0	0
$196$	2.69593	0.192566
$197$	9.15789	0.652473	0.326236	−	0.945288i	$-0.394219\pi$
$197$	9.15789	0.652473	0.326236	+	0.945288i	$0.394219\pi$
$198$	3.24233	0.230422
$199$	7.01847	0.497526	0.248763	−	0.968564i	$-0.419976\pi$
$199$	7.01847	0.497526	0.248763	+	0.968564i	$0.419976\pi$
$200$	0	0
$201$	5.80917	0.409748
$202$	7.15208	0.503218
$203$	4.92336	0.345552
$204$	−0.137912	−0.00965576
$205$	0	0
$206$	1.24658	0.0868535
$207$	−2.58247	−0.179494
$208$	−9.49284	−0.658210
$209$	3.98359	0.275551
$210$	0	0
$211$	−14.4490	−0.994707	−0.497354	−	0.867548i	$-0.665695\pi$
$211$	−14.4490	−0.994707	−0.497354	+	0.867548i	$0.665695\pi$
$212$	−1.27190	−0.0873547
$213$	−6.33540	−0.434095
$214$	−7.51954	−0.514025
$215$	0	0
$216$	10.9766	0.746861
$217$	2.61906	0.177793
$218$	19.2829	1.30600
$219$	0.849268	0.0573882
$220$	0	0
$221$	−1.60842	−0.108194
$222$	−4.20170	−0.282000
$223$	−24.2905	−1.62661	−0.813305	−	0.581838i	$-0.802334\pi$
$223$	−24.2905	−1.62661	−0.813305	+	0.581838i	$0.802334\pi$
$224$	1.87801	0.125480
$225$	0	0
$226$	19.7705	1.31512
$227$	−21.5364	−1.42942	−0.714709	−	0.699422i	$-0.753441\pi$
$227$	−21.5364	−1.42942	−0.714709	+	0.699422i	$0.753441\pi$
$228$	1.09058	0.0722257
$229$	6.71062	0.443450	0.221725	−	0.975109i	$-0.428831\pi$
$229$	6.71062	0.443450	0.221725	+	0.975109i	$0.428831\pi$
$230$	0	0
$231$	0.515693	0.0339301
$232$	−18.7720	−1.23244
$233$	15.0861	0.988326	0.494163	−	0.869369i	$-0.335475\pi$
$233$	15.0861	0.988326	0.494163	+	0.869369i	$0.335475\pi$
$234$	10.3524	0.676757
$235$	0	0
$236$	−3.82716	−0.249127
$237$	−10.1381	−0.658543
$238$	−0.504763	−0.0327189
$239$	−30.6490	−1.98252	−0.991260	−	0.131919i	$-0.957886\pi$
$239$	−30.6490	−1.98252	−0.991260	+	0.131919i	$0.957886\pi$
$240$	0	0
$241$	−13.3710	−0.861301	−0.430651	−	0.902519i	$-0.641716\pi$
$241$	−13.3710	−0.861301	−0.430651	+	0.902519i	$0.641716\pi$
$242$	−1.25551	−0.0807075
$243$	−14.3216	−0.918729
$244$	−1.11704	−0.0715109
$245$	0	0
$246$	−9.58989	−0.611429
$247$	12.7191	0.809300
$248$	−9.98606	−0.634115
$249$	−1.88830	−0.119666
$250$	0	0
$251$	25.4838	1.60852	0.804262	−	0.594276i	$-0.202561\pi$
$251$	25.4838	1.60852	0.804262	+	0.594276i	$0.202561\pi$
$252$	−0.873227	−0.0550081
$253$	1.00000	0.0628695
$254$	11.3858	0.714408
$255$	0	0
$256$	−9.67988	−0.604992
$257$	14.1016	0.879634	0.439817	−	0.898087i	$-0.355043\pi$
$257$	14.1016	0.879634	0.439817	+	0.898087i	$0.355043\pi$
$258$	5.75167	0.358083
$259$	4.13343	0.256839
$260$	0	0
$261$	15.9312	0.986117
$262$	−13.7153	−0.847337
$263$	10.5011	0.647527	0.323764	−	0.946138i	$-0.395052\pi$
$263$	10.5011	0.647527	0.323764	+	0.946138i	$0.395052\pi$
$264$	−1.96625	−0.121015
$265$	0	0
$266$	3.99158	0.244739
$267$	−4.01036	−0.245430
$268$	3.80903	0.232674
$269$	20.5718	1.25429	0.627143	−	0.778904i	$-0.284224\pi$
$269$	20.5718	1.25429	0.627143	+	0.778904i	$0.284224\pi$
$270$	0	0
$271$	−28.7365	−1.74562	−0.872810	−	0.488060i	$-0.837705\pi$
$271$	−28.7365	−1.74562	−0.872810	+	0.488060i	$0.837705\pi$
$272$	1.49772	0.0908124
$273$	1.64655	0.0996536
$274$	28.1573	1.70104
$275$	0	0
$276$	0.273769	0.0164790
$277$	13.7543	0.826416	0.413208	−	0.910637i	$-0.364408\pi$
$277$	13.7543	0.826416	0.413208	+	0.910637i	$0.364408\pi$
$278$	0.954601	0.0572532
$279$	8.47486	0.507376
$280$	0	0
$281$	7.54904	0.450338	0.225169	−	0.974320i	$-0.427707\pi$
$281$	7.54904	0.450338	0.225169	+	0.974320i	$0.427707\pi$
$282$	−0.996019	−0.0593120
$283$	27.0020	1.60510	0.802551	−	0.596584i	$-0.203476\pi$
$283$	27.0020	1.60510	0.802551	+	0.596584i	$0.203476\pi$
$284$	−4.15408	−0.246499
$285$	0	0
$286$	−4.00871	−0.237040
$287$	9.43406	0.556875
$288$	6.07693	0.358086
$289$	−16.7462	−0.985073
$290$	0	0
$291$	−9.90947	−0.580903
$292$	0.556859	0.0325877
$293$	10.4872	0.612666	0.306333	−	0.951924i	$-0.400898\pi$
$293$	10.4872	0.612666	0.306333	+	0.951924i	$0.400898\pi$
$294$	−5.16214	−0.301062
$295$	0	0
$296$	−15.7601	−0.916038
$297$	3.60719	0.209310
$298$	−7.02455	−0.406921
$299$	3.19289	0.184649
$300$	0	0
$301$	−5.65821	−0.326134
$302$	−8.96129	−0.515664
$303$	3.68089	0.211462
$304$	−11.8437	−0.679283
$305$	0	0
$306$	−1.63333	−0.0933714
$307$	−19.2068	−1.09619	−0.548096	−	0.836415i	$-0.684647\pi$
$307$	−19.2068	−1.09619	−0.548096	+	0.836415i	$0.684647\pi$
$308$	0.338136	0.0192671
$309$	0.641566	0.0364974
$310$	0	0
$311$	−16.6171	−0.942270	−0.471135	−	0.882061i	$-0.656155\pi$
$311$	−16.6171	−0.942270	−0.471135	+	0.882061i	$0.656155\pi$
$312$	−6.27803	−0.355423
$313$	−20.8071	−1.17609	−0.588044	−	0.808829i	$-0.700102\pi$
$313$	−20.8071	−1.17609	−0.588044	+	0.808829i	$0.700102\pi$
$314$	−23.6631	−1.33538
$315$	0	0
$316$	−6.64751	−0.373952
$317$	0.842442	0.0473163	0.0236581	−	0.999720i	$-0.492469\pi$
$317$	0.842442	0.0473163	0.0236581	+	0.999720i	$0.492469\pi$
$318$	2.43543	0.136572
$319$	−6.16897	−0.345396
$320$	0	0
$321$	−3.87001	−0.216003
$322$	1.00201	0.0558396
$323$	−2.00674	−0.111658
$324$	−2.29492	−0.127496
$325$	0	0
$326$	−24.5026	−1.35707
$327$	9.92412	0.548805
$328$	−35.9706	−1.98614
$329$	0.979834	0.0540200
$330$	0	0
$331$	−10.2249	−0.562014	−0.281007	−	0.959706i	$-0.590668\pi$
$331$	−10.2249	−0.562014	−0.281007	+	0.959706i	$0.590668\pi$
$332$	−1.23815	−0.0679521
$333$	13.3751	0.732952
$334$	25.7761	1.41041
$335$	0	0
$336$	−1.53322	−0.0836438
$337$	−29.3644	−1.59958	−0.799791	−	0.600279i	$-0.795056\pi$
$337$	−29.3644	−1.59958	−0.799791	+	0.600279i	$0.795056\pi$
$338$	3.52232	0.191589
$339$	10.1751	0.552636
$340$	0	0
$341$	−3.28168	−0.177713
$342$	12.9161	0.698424
$343$	10.6648	0.575848
$344$	21.5739	1.16318
$345$	0	0
$346$	5.86680	0.315401
$347$	−24.9778	−1.34088	−0.670438	−	0.741965i	$-0.733894\pi$
$347$	−24.9778	−1.34088	−0.670438	+	0.741965i	$0.733894\pi$
$348$	−1.68887	−0.0905332
$349$	−30.7570	−1.64638	−0.823191	−	0.567764i	$-0.807808\pi$
$349$	−30.7570	−1.64638	−0.823191	+	0.567764i	$0.807808\pi$
$350$	0	0
$351$	11.5173	0.614750
$352$	−2.35314	−0.125423
$353$	18.1605	0.966588	0.483294	−	0.875458i	$-0.339440\pi$
$353$	18.1605	0.966588	0.483294	+	0.875458i	$0.339440\pi$
$354$	7.32820	0.389490
$355$	0	0
$356$	−2.62956	−0.139367
$357$	−0.259781	−0.0137491
$358$	5.88197	0.310872
$359$	−0.950635	−0.0501726	−0.0250863	−	0.999685i	$-0.507986\pi$
$359$	−0.950635	−0.0501726	−0.0250863	+	0.999685i	$0.507986\pi$
$360$	0	0
$361$	−3.13102	−0.164790
$362$	−12.8565	−0.675720
$363$	−0.646163	−0.0339148
$364$	1.07963	0.0565879
$365$	0	0
$366$	2.13889	0.111801
$367$	28.6929	1.49776	0.748878	−	0.662708i	$-0.230593\pi$
$367$	28.6929	1.49776	0.748878	+	0.662708i	$0.230593\pi$
$368$	−2.97312	−0.154985
$369$	30.5271	1.58918
$370$	0	0
$371$	−2.39585	−0.124387
$372$	−0.898424	−0.0465811
$373$	−8.58485	−0.444507	−0.222253	−	0.974989i	$-0.571341\pi$
$373$	−8.58485	−0.444507	−0.222253	+	0.974989i	$0.571341\pi$
$374$	0.632468	0.0327041
$375$	0	0
$376$	−3.73595	−0.192667
$377$	−19.6968	−1.01444
$378$	3.61442	0.185906
$379$	10.0573	0.516610	0.258305	−	0.966063i	$-0.416836\pi$
$379$	10.0573	0.516610	0.258305	+	0.966063i	$0.416836\pi$
$380$	0	0
$381$	5.85981	0.300207
$382$	8.03393	0.411052
$383$	19.3152	0.986963	0.493481	−	0.869756i	$-0.335724\pi$
$383$	19.3152	0.986963	0.493481	+	0.869756i	$0.335724\pi$
$384$	4.17977	0.213298
$385$	0	0
$386$	2.72903	0.138904
$387$	−18.3091	−0.930702
$388$	−6.49757	−0.329864
$389$	15.0719	0.764178	0.382089	−	0.924126i	$-0.375205\pi$
$389$	15.0719	0.764178	0.382089	+	0.924126i	$0.375205\pi$
$390$	0	0
$391$	−0.503752	−0.0254758
$392$	−19.3626	−0.977959
$393$	−7.05874	−0.356066
$394$	−11.4979	−0.579254
$395$	0	0
$396$	1.09415	0.0549833
$397$	−12.9854	−0.651720	−0.325860	−	0.945418i	$-0.605654\pi$
$397$	−12.9854	−0.651720	−0.325860	+	0.945418i	$0.605654\pi$
$398$	−8.81179	−0.441695
$399$	2.05431	0.102844
$400$	0	0
$401$	−12.3840	−0.618429	−0.309214	−	0.950992i	$-0.600066\pi$
$401$	−12.3840	−0.618429	−0.309214	+	0.950992i	$0.600066\pi$
$402$	−7.29350	−0.363767
$403$	−10.4780	−0.521948
$404$	2.41353	0.120078
$405$	0	0
$406$	−6.18134	−0.306775
$407$	−5.17919	−0.256723
$408$	0.990505	0.0490373
$409$	−0.984934	−0.0487019	−0.0243509	−	0.999703i	$-0.507752\pi$
$409$	−0.984934	−0.0487019	−0.0243509	+	0.999703i	$0.507752\pi$
$410$	0	0
$411$	14.4914	0.714809
$412$	0.420670	0.0207249
$413$	−7.20912	−0.354738
$414$	3.24233	0.159352
$415$	0	0
$416$	−7.51332	−0.368371
$417$	0.491295	0.0240588
$418$	−5.00145	−0.244629
$419$	−12.0822	−0.590255	−0.295127	−	0.955458i	$-0.595362\pi$
$419$	−12.0822	−0.590255	−0.295127	+	0.955458i	$0.595362\pi$
$420$	0	0
$421$	−37.5161	−1.82842	−0.914210	−	0.405240i	$-0.867188\pi$
$421$	−37.5161	−1.82842	−0.914210	+	0.405240i	$0.867188\pi$
$422$	18.1409	0.883083
$423$	3.17059	0.154159
$424$	9.13501	0.443635
$425$	0	0
$426$	7.95419	0.385382
$427$	−2.10413	−0.101826
$428$	−2.53754	−0.122657
$429$	−2.06312	−0.0996086
$430$	0	0
$431$	−8.43869	−0.406478	−0.203239	−	0.979129i	$-0.565147\pi$
$431$	−8.43869	−0.406478	−0.203239	+	0.979129i	$0.565147\pi$
$432$	−10.7246	−0.515988
$433$	−8.74254	−0.420140	−0.210070	−	0.977686i	$-0.567369\pi$
$433$	−8.74254	−0.420140	−0.210070	+	0.977686i	$0.567369\pi$
$434$	−3.28826	−0.157842
$435$	0	0
$436$	6.50718	0.311637
$437$	3.98359	0.190561
$438$	−1.06627	−0.0509483
$439$	2.75687	0.131578	0.0657891	−	0.997834i	$-0.479044\pi$
$439$	2.75687	0.131578	0.0657891	+	0.997834i	$0.479044\pi$
$440$	0	0
$441$	16.4324	0.782497
$442$	2.01940	0.0960529
$443$	−0.793474	−0.0376991	−0.0188496	−	0.999822i	$-0.506000\pi$
$443$	−0.793474	−0.0376991	−0.0188496	+	0.999822i	$0.506000\pi$
$444$	−1.41790	−0.0672907
$445$	0	0
$446$	30.4970	1.44408
$447$	−3.61526	−0.170996
$448$	−7.10347	−0.335607
$449$	−38.9043	−1.83601	−0.918003	−	0.396574i	$-0.870199\pi$
$449$	−38.9043	−1.83601	−0.918003	+	0.396574i	$0.870199\pi$
$450$	0	0
$451$	−11.8209	−0.556624
$452$	6.67175	0.313813
$453$	−4.61202	−0.216692
$454$	27.0392	1.26901
$455$	0	0
$456$	−7.83275	−0.366802
$457$	−4.02954	−0.188494	−0.0942470	−	0.995549i	$-0.530044\pi$
$457$	−4.02954	−0.188494	−0.0942470	+	0.995549i	$0.530044\pi$
$458$	−8.42528	−0.393688
$459$	−1.81713	−0.0848163
$460$	0	0
$461$	−8.05034	−0.374942	−0.187471	−	0.982270i	$-0.560029\pi$
$461$	−8.05034	−0.374942	−0.187471	+	0.982270i	$0.560029\pi$
$462$	−0.647459	−0.0301225
$463$	3.83757	0.178347	0.0891735	−	0.996016i	$-0.471577\pi$
$463$	3.83757	0.178347	0.0891735	+	0.996016i	$0.471577\pi$
$464$	18.3411	0.851464
$465$	0	0
$466$	−18.9409	−0.877418
$467$	−32.1096	−1.48585	−0.742927	−	0.669372i	$-0.766563\pi$
$467$	−32.1096	−1.48585	−0.742927	+	0.669372i	$0.766563\pi$
$468$	3.49351	0.161487
$469$	7.17498	0.331310
$470$	0	0
$471$	−12.1784	−0.561153
$472$	27.4873	1.26520
$473$	7.08974	0.325987
$474$	12.7286	0.584643
$475$	0	0
$476$	−0.170337	−0.00780737
$477$	−7.75260	−0.354967
$478$	38.4803	1.76005
$479$	−41.4512	−1.89395	−0.946976	−	0.321305i	$-0.895879\pi$
$479$	−41.4512	−1.89395	−0.946976	+	0.321305i	$0.895879\pi$
$480$	0	0
$481$	−16.5366	−0.754002
$482$	16.7875	0.764648
$483$	0.515693	0.0234648
$484$	−0.423685	−0.0192584
$485$	0	0
$486$	17.9809	0.815631
$487$	−21.7836	−0.987109	−0.493554	−	0.869715i	$-0.664303\pi$
$487$	−21.7836	−0.987109	−0.493554	+	0.869715i	$0.664303\pi$
$488$	8.02273	0.363172
$489$	−12.6105	−0.570267
$490$	0	0
$491$	16.5105	0.745110	0.372555	−	0.928010i	$-0.378482\pi$
$491$	16.5105	0.745110	0.372555	+	0.928010i	$0.378482\pi$
$492$	−3.23619	−0.145899
$493$	3.10763	0.139961
$494$	−15.9691	−0.718482
$495$	0	0
$496$	9.75684	0.438095
$497$	−7.82494	−0.350996
$498$	2.37079	0.106238
$499$	−5.88608	−0.263497	−0.131748	−	0.991283i	$-0.542059\pi$
$499$	−5.88608	−0.263497	−0.131748	+	0.991283i	$0.542059\pi$
$500$	0	0
$501$	13.2659	0.592678
$502$	−31.9953	−1.42802
$503$	−25.4624	−1.13531	−0.567656	−	0.823266i	$-0.692150\pi$
$503$	−25.4624	−1.13531	−0.567656	+	0.823266i	$0.692150\pi$
$504$	6.27166	0.279362
$505$	0	0
$506$	−1.25551	−0.0558144
$507$	1.81280	0.0805093
$508$	3.84224	0.170472
$509$	−37.9786	−1.68337	−0.841687	−	0.539966i	$-0.818437\pi$
$509$	−37.9786	−1.68337	−0.841687	+	0.539966i	$0.818437\pi$
$510$	0	0
$511$	1.04894	0.0464025
$512$	25.0904	1.10885
$513$	14.3696	0.634431
$514$	−17.7048	−0.780924
$515$	0	0
$516$	1.94095	0.0854457
$517$	−1.22773	−0.0539956
$518$	−5.18958	−0.228017
$519$	3.01941	0.132537
$520$	0	0
$521$	30.5870	1.34004	0.670020	−	0.742343i	$-0.266286\pi$
$521$	30.5870	1.34004	0.670020	+	0.742343i	$0.266286\pi$
$522$	−20.0018	−0.875457
$523$	2.66147	0.116378	0.0581890	−	0.998306i	$-0.481467\pi$
$523$	2.66147	0.116378	0.0581890	+	0.998306i	$0.481467\pi$
$524$	−4.62837	−0.202191
$525$	0	0
$526$	−13.1843	−0.574863
$527$	1.65315	0.0720125
$528$	1.92112	0.0836061
$529$	1.00000	0.0434783
$530$	0	0
$531$	−23.3276	−1.01233
$532$	1.34699	0.0583996
$533$	−37.7427	−1.63482
$534$	5.03506	0.217888
$535$	0	0
$536$	−27.3571	−1.18165
$537$	3.02721	0.130634
$538$	−25.8282	−1.11353
$539$	−6.36306	−0.274076
$540$	0	0
$541$	41.1086	1.76740	0.883698	−	0.468058i	$-0.155046\pi$
$541$	41.1086	1.76740	0.883698	+	0.468058i	$0.155046\pi$
$542$	36.0791	1.54973
$543$	−6.61670	−0.283950
$544$	1.18540	0.0508236
$545$	0	0
$546$	−2.06726	−0.0884707
$547$	7.10820	0.303925	0.151962	−	0.988386i	$-0.451441\pi$
$547$	7.10820	0.303925	0.151962	+	0.988386i	$0.451441\pi$
$548$	9.50192	0.405902
$549$	−6.80864	−0.290586
$550$	0	0
$551$	−24.5746	−1.04691
$552$	−1.96625	−0.0836894
$553$	−12.5217	−0.532479
$554$	−17.2687	−0.733678
$555$	0	0
$556$	0.322139	0.0136617
$557$	17.5181	0.742265	0.371133	−	0.928580i	$-0.378970\pi$
$557$	17.5181	0.742265	0.371133	+	0.928580i	$0.378970\pi$
$558$	−10.6403	−0.450440
$559$	22.6367	0.957432
$560$	0	0
$561$	0.325506	0.0137429
$562$	−9.47793	−0.399802
$563$	−32.5836	−1.37324	−0.686618	−	0.727019i	$-0.740905\pi$
$563$	−32.5836	−1.37324	−0.686618	+	0.727019i	$0.740905\pi$
$564$	−0.336115	−0.0141530
$565$	0	0
$566$	−33.9014	−1.42498
$567$	−4.32289	−0.181544
$568$	29.8353	1.25186
$569$	42.5643	1.78439	0.892194	−	0.451653i	$-0.149165\pi$
$569$	42.5643	1.78439	0.892194	+	0.451653i	$0.149165\pi$
$570$	0	0
$571$	−3.70786	−0.155169	−0.0775845	−	0.996986i	$-0.524721\pi$
$571$	−3.70786	−0.155169	−0.0775845	+	0.996986i	$0.524721\pi$
$572$	−1.35278	−0.0565624
$573$	4.13474	0.172731
$574$	−11.8446	−0.494384
$575$	0	0
$576$	−22.9857	−0.957737
$577$	6.75679	0.281289	0.140644	−	0.990060i	$-0.455083\pi$
$577$	6.75679	0.281289	0.140644	+	0.990060i	$0.455083\pi$
$578$	21.0251	0.874530
$579$	1.40452	0.0583700
$580$	0	0
$581$	−2.33227	−0.0967587
$582$	12.4415	0.515716
$583$	3.00201	0.124330
$584$	−3.99945	−0.165498
$585$	0	0
$586$	−13.1668	−0.543915
$587$	−3.71910	−0.153504	−0.0767519	−	0.997050i	$-0.524455\pi$
$587$	−3.71910	−0.153504	−0.0767519	+	0.997050i	$0.524455\pi$
$588$	−1.74201	−0.0718393
$589$	−13.0729	−0.538658
$590$	0	0
$591$	−5.91749	−0.243413
$592$	15.3984	0.632869
$593$	−22.6333	−0.929439	−0.464719	−	0.885458i	$-0.653845\pi$
$593$	−22.6333	−0.929439	−0.464719	+	0.885458i	$0.653845\pi$
$594$	−4.52888	−0.185822
$595$	0	0
$596$	−2.37050	−0.0970994
$597$	−4.53508	−0.185608
$598$	−4.00871	−0.163928
$599$	−30.4468	−1.24402	−0.622010	−	0.783009i	$-0.713684\pi$
$599$	−30.4468	−1.24402	−0.622010	+	0.783009i	$0.713684\pi$
$600$	0	0
$601$	11.3551	0.463186	0.231593	−	0.972813i	$-0.425606\pi$
$601$	11.3551	0.463186	0.231593	+	0.972813i	$0.425606\pi$
$602$	7.10396	0.289536
$603$	23.2171	0.945474
$604$	−3.02407	−0.123048
$605$	0	0
$606$	−4.62141	−0.187732
$607$	−9.14323	−0.371112	−0.185556	−	0.982634i	$-0.559409\pi$
$607$	−9.14323	−0.371112	−0.185556	+	0.982634i	$0.559409\pi$
$608$	−9.37396	−0.380164
$609$	−3.18129	−0.128912
$610$	0	0
$611$	−3.92001	−0.158587
$612$	−0.551182	−0.0222802
$613$	3.40904	0.137690	0.0688449	−	0.997627i	$-0.478069\pi$
$613$	3.40904	0.137690	0.0688449	+	0.997627i	$0.478069\pi$
$614$	24.1145	0.973180
$615$	0	0
$616$	−2.42855	−0.0978489
$617$	−1.68640	−0.0678919	−0.0339460	−	0.999424i	$-0.510807\pi$
$617$	−1.68640	−0.0678919	−0.0339460	+	0.999424i	$0.510807\pi$
$618$	−0.805495	−0.0324018
$619$	−7.71276	−0.310002	−0.155001	−	0.987914i	$-0.549538\pi$
$619$	−7.71276	−0.310002	−0.155001	+	0.987914i	$0.549538\pi$
$620$	0	0
$621$	3.60719	0.144751
$622$	20.8630	0.836530
$623$	−4.95324	−0.198448
$624$	6.13392	0.245553
$625$	0	0
$626$	26.1237	1.04411
$627$	−2.57405	−0.102798
$628$	−7.98532	−0.318649
$629$	2.60903	0.104029
$630$	0	0
$631$	−39.9860	−1.59182	−0.795908	−	0.605417i	$-0.793006\pi$
$631$	−39.9860	−1.59182	−0.795908	+	0.605417i	$0.793006\pi$
$632$	47.7435	1.89913
$633$	9.33638	0.371088
$634$	−1.05770	−0.0420065
$635$	0	0
$636$	0.821857	0.0325888
$637$	−20.3165	−0.804970
$638$	7.74523	0.306637
$639$	−25.3202	−1.00165
$640$	0	0
$641$	−8.13445	−0.321291	−0.160646	−	0.987012i	$-0.551358\pi$
$641$	−8.13445	−0.321291	−0.160646	+	0.987012i	$0.551358\pi$
$642$	4.85885	0.191764
$643$	29.6220	1.16818	0.584088	−	0.811690i	$-0.301452\pi$
$643$	29.6220	1.16818	0.584088	+	0.811690i	$0.301452\pi$
$644$	0.338136	0.0133244
$645$	0	0
$646$	2.51949	0.0991281
$647$	−18.3325	−0.720727	−0.360363	−	0.932812i	$-0.617347\pi$
$647$	−18.3325	−0.720727	−0.360363	+	0.932812i	$0.617347\pi$
$648$	16.4825	0.647494
$649$	9.03304	0.354578
$650$	0	0
$651$	−1.69234	−0.0663280
$652$	−8.26863	−0.323824
$653$	4.41518	0.172779	0.0863896	−	0.996261i	$-0.472467\pi$
$653$	4.41518	0.172779	0.0863896	+	0.996261i	$0.472467\pi$
$654$	−12.4599	−0.487220
$655$	0	0
$656$	35.1449	1.37218
$657$	3.39421	0.132421
$658$	−1.23020	−0.0479580
$659$	−10.5431	−0.410703	−0.205351	−	0.978688i	$-0.565834\pi$
$659$	−10.5431	−0.410703	−0.205351	+	0.978688i	$0.565834\pi$
$660$	0	0
$661$	−10.2501	−0.398681	−0.199341	−	0.979930i	$-0.563880\pi$
$661$	−10.2501	−0.398681	−0.199341	+	0.979930i	$0.563880\pi$
$662$	12.8376	0.498946
$663$	1.03930	0.0403632
$664$	8.89256	0.345098
$665$	0	0
$666$	−16.7927	−0.650702
$667$	−6.16897	−0.238863
$668$	8.69838	0.336550
$669$	15.6956	0.606827
$670$	0	0
$671$	2.63648	0.101780
$672$	−1.21350	−0.0468117
$673$	−29.0456	−1.11962	−0.559812	−	0.828620i	$-0.689127\pi$
$673$	−29.0456	−1.11962	−0.559812	+	0.828620i	$0.689127\pi$
$674$	36.8674	1.42008
$675$	0	0
$676$	1.18864	0.0457169
$677$	−21.8323	−0.839084	−0.419542	−	0.907736i	$-0.637809\pi$
$677$	−21.8323	−0.839084	−0.419542	+	0.907736i	$0.637809\pi$
$678$	−12.7750	−0.490621
$679$	−12.2393	−0.469702
$680$	0	0
$681$	13.9160	0.533262
$682$	4.12020	0.157771
$683$	−48.4518	−1.85396	−0.926979	−	0.375114i	$-0.877604\pi$
$683$	−48.4518	−1.85396	−0.926979	+	0.375114i	$0.877604\pi$
$684$	4.35866	0.166658
$685$	0	0
$686$	−13.3899	−0.511227
$687$	−4.33616	−0.165435
$688$	−21.0787	−0.803617
$689$	9.58506	0.365162
$690$	0	0
$691$	23.0694	0.877601	0.438800	−	0.898585i	$-0.355404\pi$
$691$	23.0694	0.877601	0.438800	+	0.898585i	$0.355404\pi$
$692$	1.97980	0.0752609
$693$	2.06103	0.0782921
$694$	31.3599	1.19041
$695$	0	0
$696$	12.1298	0.459778
$697$	5.95480	0.225554
$698$	38.6158	1.46163
$699$	−9.74811	−0.368707
$700$	0	0
$701$	35.6219	1.34542	0.672711	−	0.739905i	$-0.265130\pi$
$701$	35.6219	1.34542	0.672711	+	0.739905i	$0.265130\pi$
$702$	−14.4602	−0.545764
$703$	−20.6318	−0.778142
$704$	8.90065	0.335456
$705$	0	0
$706$	−22.8008	−0.858120
$707$	4.54631	0.170982
$708$	2.47297	0.0929398
$709$	21.1768	0.795312	0.397656	−	0.917534i	$-0.369824\pi$
$709$	21.1768	0.795312	0.397656	+	0.917534i	$0.369824\pi$
$710$	0	0
$711$	−40.5184	−1.51956
$712$	18.8859	0.707781
$713$	−3.28168	−0.122900
$714$	0.326159	0.0122062
$715$	0	0
$716$	1.98492	0.0741801
$717$	19.8043	0.739604
$718$	1.19354	0.0445424
$719$	−47.0596	−1.75503	−0.877513	−	0.479553i	$-0.840799\pi$
$719$	−47.0596	−1.75503	−0.877513	+	0.479553i	$0.840799\pi$
$720$	0	0
$721$	0.792406	0.0295108
$722$	3.93104	0.146298
$723$	8.63984	0.321319
$724$	−4.33853	−0.161240
$725$	0	0
$726$	0.811267	0.0301089
$727$	34.5022	1.27961	0.639807	−	0.768535i	$-0.279014\pi$
$727$	34.5022	1.27961	0.639807	+	0.768535i	$0.279014\pi$
$728$	−7.75407	−0.287385
$729$	−6.99570	−0.259100
$730$	0	0
$731$	−3.57147	−0.132096
$732$	0.721787	0.0266780
$733$	19.9498	0.736863	0.368431	−	0.929655i	$-0.379895\pi$
$733$	19.9498	0.736863	0.368431	+	0.929655i	$0.379895\pi$
$734$	−36.0243	−1.32968
$735$	0	0
$736$	−2.35314	−0.0867380
$737$	−8.99026	−0.331160
$738$	−38.3272	−1.41084
$739$	9.42242	0.346609	0.173305	−	0.984868i	$-0.444555\pi$
$739$	9.42242	0.346609	0.173305	+	0.984868i	$0.444555\pi$
$740$	0	0
$741$	−8.21864	−0.301919
$742$	3.00803	0.110428
$743$	−7.25579	−0.266189	−0.133095	−	0.991103i	$-0.542491\pi$
$743$	−7.25579	−0.266189	−0.133095	+	0.991103i	$0.542491\pi$
$744$	6.45262	0.236565
$745$	0	0
$746$	10.7784	0.394625
$747$	−7.54684	−0.276124
$748$	0.213432	0.00780385
$749$	−4.77990	−0.174654
$750$	0	0
$751$	−18.9612	−0.691904	−0.345952	−	0.938252i	$-0.612444\pi$
$751$	−18.9612	−0.691904	−0.345952	+	0.938252i	$0.612444\pi$
$752$	3.65020	0.133109
$753$	−16.4667	−0.600079
$754$	24.7296	0.900600
$755$	0	0
$756$	1.21972	0.0443608
$757$	8.51568	0.309508	0.154754	−	0.987953i	$-0.450542\pi$
$757$	8.51568	0.309508	0.154754	+	0.987953i	$0.450542\pi$
$758$	−12.6271	−0.458637
$759$	−0.646163	−0.0234542
$760$	0	0
$761$	−3.63058	−0.131609	−0.0658043	−	0.997833i	$-0.520961\pi$
$761$	−3.63058	−0.131609	−0.0658043	+	0.997833i	$0.520961\pi$
$762$	−7.35707	−0.266519
$763$	12.2574	0.443748
$764$	2.71112	0.0980850
$765$	0	0
$766$	−24.2506	−0.876208
$767$	28.8414	1.04140
$768$	6.25478	0.225700
$769$	−11.1413	−0.401764	−0.200882	−	0.979615i	$-0.564381\pi$
$769$	−11.1413	−0.401764	−0.200882	+	0.979615i	$0.564381\pi$
$770$	0	0
$771$	−9.11194	−0.328158
$772$	0.920935	0.0331452
$773$	−40.1436	−1.44387	−0.721933	−	0.691963i	$-0.756746\pi$
$773$	−40.1436	−1.44387	−0.721933	+	0.691963i	$0.756746\pi$
$774$	22.9873	0.826261
$775$	0	0
$776$	46.6666	1.67523
$777$	−2.67087	−0.0958169
$778$	−18.9230	−0.678424
$779$	−47.0895	−1.68716
$780$	0	0
$781$	9.80465	0.350838
$782$	0.632468	0.0226170
$783$	−22.2526	−0.795244
$784$	18.9182	0.675649
$785$	0	0
$786$	8.86235	0.316109
$787$	32.1916	1.14751	0.573754	−	0.819028i	$-0.305487\pi$
$787$	32.1916	1.14751	0.573754	+	0.819028i	$0.305487\pi$
$788$	−3.88006	−0.138221
$789$	−6.78544	−0.241568
$790$	0	0
$791$	12.5674	0.446846
$792$	−7.85839	−0.279236
$793$	8.41798	0.298931
$794$	16.3034	0.578585
$795$	0	0
$796$	−2.97362	−0.105397
$797$	10.4731	0.370977	0.185489	−	0.982646i	$-0.440613\pi$
$797$	10.4731	0.370977	0.185489	+	0.982646i	$0.440613\pi$
$798$	−2.57921	−0.0913031
$799$	0.618473	0.0218800
$800$	0	0
$801$	−16.0279	−0.566318
$802$	15.5483	0.549030
$803$	−1.31433	−0.0463815
$804$	−2.46126	−0.0868018
$805$	0	0
$806$	13.1553	0.463376
$807$	−13.2928	−0.467927
$808$	−17.3344	−0.609821
$809$	−2.51048	−0.0882637	−0.0441318	−	0.999026i	$-0.514052\pi$
$809$	−2.51048	−0.0882637	−0.0441318	+	0.999026i	$0.514052\pi$
$810$	0	0
$811$	−28.1415	−0.988183	−0.494091	−	0.869410i	$-0.664499\pi$
$811$	−28.1415	−0.988183	−0.494091	+	0.869410i	$0.664499\pi$
$812$	−2.08595	−0.0732025
$813$	18.5685	0.651225
$814$	6.50255	0.227914
$815$	0	0
$816$	−0.967769	−0.0338787
$817$	28.2426	0.988084
$818$	1.23660	0.0432367
$819$	6.58064	0.229946
$820$	0	0
$821$	9.55356	0.333421	0.166711	−	0.986006i	$-0.446685\pi$
$821$	9.55356	0.333421	0.166711	+	0.986006i	$0.446685\pi$
$822$	−18.1942	−0.634595
$823$	38.7711	1.35147	0.675737	−	0.737143i	$-0.263826\pi$
$823$	38.7711	1.35147	0.675737	+	0.737143i	$0.263826\pi$
$824$	−3.02132	−0.105253
$825$	0	0
$826$	9.05116	0.314930
$827$	−28.9099	−1.00530	−0.502648	−	0.864491i	$-0.667641\pi$
$827$	−28.9099	−1.00530	−0.502648	+	0.864491i	$0.667641\pi$
$828$	1.09415	0.0380245
$829$	−13.0709	−0.453972	−0.226986	−	0.973898i	$-0.572887\pi$
$829$	−13.0709	−0.453972	−0.226986	+	0.973898i	$0.572887\pi$
$830$	0	0
$831$	−8.88752	−0.308305
$832$	28.4187	0.985243
$833$	3.20541	0.111061
$834$	−0.616828	−0.0213590
$835$	0	0
$836$	−1.68779	−0.0583733
$837$	−11.8376	−0.409169
$838$	15.1694	0.524018
$839$	−9.03560	−0.311944	−0.155972	−	0.987762i	$-0.549851\pi$
$839$	−9.03560	−0.311944	−0.155972	+	0.987762i	$0.549851\pi$
$840$	0	0
$841$	9.05619	0.312282
$842$	47.1019	1.62324
$843$	−4.87791	−0.168004
$844$	6.12180	0.210721
$845$	0	0
$846$	−3.98072	−0.136860
$847$	−0.798084	−0.0274225
$848$	−8.92533	−0.306497
$849$	−17.4477	−0.598803
$850$	0	0
$851$	−5.17919	−0.177540
$852$	2.68421	0.0919596
$853$	13.4429	0.460275	0.230137	−	0.973158i	$-0.426082\pi$
$853$	13.4429	0.460275	0.230137	+	0.973158i	$0.426082\pi$
$854$	2.64177	0.0903995
$855$	0	0
$856$	18.2250	0.622918
$857$	−40.8692	−1.39607	−0.698033	−	0.716065i	$-0.745941\pi$
$857$	−40.8692	−1.39607	−0.698033	+	0.716065i	$0.745941\pi$
$858$	2.59028	0.0884308
$859$	33.6168	1.14699	0.573495	−	0.819209i	$-0.305587\pi$
$859$	33.6168	1.14699	0.573495	+	0.819209i	$0.305587\pi$
$860$	0	0
$861$	−6.09594	−0.207749
$862$	10.5949	0.360864
$863$	−6.28825	−0.214055	−0.107027	−	0.994256i	$-0.534133\pi$
$863$	−6.28825	−0.214055	−0.107027	+	0.994256i	$0.534133\pi$
$864$	−8.48823	−0.288775
$865$	0	0
$866$	10.9764	0.372993
$867$	10.8208	0.367494
$868$	−1.10965	−0.0376641
$869$	15.6898	0.532239
$870$	0	0
$871$	−28.7049	−0.972627
$872$	−46.7356	−1.58267
$873$	−39.6045	−1.34041
$874$	−5.00145	−0.169177
$875$	0	0
$876$	−0.359822	−0.0121573
$877$	46.4269	1.56773	0.783863	−	0.620934i	$-0.213246\pi$
$877$	46.4269	1.56773	0.783863	+	0.620934i	$0.213246\pi$
$878$	−3.46129	−0.116813
$879$	−6.77642	−0.228563
$880$	0	0
$881$	−6.31123	−0.212631	−0.106315	−	0.994332i	$-0.533905\pi$
$881$	−6.31123	−0.212631	−0.106315	+	0.994332i	$0.533905\pi$
$882$	−20.6312	−0.694687
$883$	−33.7642	−1.13625	−0.568127	−	0.822941i	$-0.692332\pi$
$883$	−33.7642	−1.13625	−0.568127	+	0.822941i	$0.692332\pi$
$884$	0.681464	0.0229201
$885$	0	0
$886$	0.996218	0.0334686
$887$	−14.5976	−0.490139	−0.245069	−	0.969506i	$-0.578811\pi$
$887$	−14.5976	−0.490139	−0.245069	+	0.969506i	$0.578811\pi$
$888$	10.1836	0.341739
$889$	7.23753	0.242739
$890$	0	0
$891$	5.41659	0.181463
$892$	10.2915	0.344585
$893$	−4.89078	−0.163664
$894$	4.53901	0.151807
$895$	0	0
$896$	5.16249	0.172467
$897$	−2.06312	−0.0688857
$898$	48.8448	1.62997
$899$	20.2446	0.675195
$900$	0	0
$901$	−1.51227	−0.0503809
$902$	14.8413	0.494161
$903$	3.65613	0.121668
$904$	−47.9176	−1.59371
$905$	0	0
$906$	5.79046	0.192375
$907$	48.0826	1.59656	0.798278	−	0.602290i	$-0.205745\pi$
$907$	48.0826	1.59656	0.798278	+	0.602290i	$0.205745\pi$
$908$	9.12462	0.302811
$909$	14.7111	0.487938
$910$	0	0
$911$	−28.6152	−0.948065	−0.474033	−	0.880507i	$-0.657202\pi$
$911$	−28.6152	−0.948065	−0.474033	+	0.880507i	$0.657202\pi$
$912$	7.65296	0.253415
$913$	2.92233	0.0967150
$914$	5.05915	0.167342
$915$	0	0
$916$	−2.84319	−0.0939416
$917$	−8.71834	−0.287905
$918$	2.28143	0.0752984
$919$	8.89355	0.293371	0.146686	−	0.989183i	$-0.453139\pi$
$919$	8.89355	0.293371	0.146686	+	0.989183i	$0.453139\pi$
$920$	0	0
$921$	12.4108	0.408948
$922$	10.1073	0.332866
$923$	31.3051	1.03042
$924$	−0.218491	−0.00718782
$925$	0	0
$926$	−4.81812	−0.158333
$927$	2.56410	0.0842161
$928$	14.5165	0.476526
$929$	−18.1893	−0.596771	−0.298385	−	0.954445i	$-0.596448\pi$
$929$	−18.1893	−0.596771	−0.298385	+	0.954445i	$0.596448\pi$
$930$	0	0
$931$	−25.3478	−0.830741
$932$	−6.39176	−0.209369
$933$	10.7374	0.351525
$934$	40.3140	1.31912
$935$	0	0
$936$	−25.0909	−0.820123
$937$	−33.3478	−1.08943	−0.544713	−	0.838623i	$-0.683361\pi$
$937$	−33.3478	−1.08943	−0.544713	+	0.838623i	$0.683361\pi$
$938$	−9.00829	−0.294131
$939$	13.4448	0.438755
$940$	0	0
$941$	29.2596	0.953836	0.476918	−	0.878948i	$-0.341754\pi$
$941$	29.2596	0.953836	0.476918	+	0.878948i	$0.341754\pi$
$942$	15.2902	0.498182
$943$	−11.8209	−0.384941
$944$	−26.8563	−0.874099
$945$	0	0
$946$	−8.90127	−0.289405
$947$	−34.5660	−1.12324	−0.561621	−	0.827394i	$-0.689822\pi$
$947$	−34.5660	−1.12324	−0.561621	+	0.827394i	$0.689822\pi$
$948$	4.29537	0.139507
$949$	−4.19649	−0.136224
$950$	0	0
$951$	−0.544355	−0.0176519
$952$	1.22339	0.0396502
$953$	−28.4736	−0.922350	−0.461175	−	0.887309i	$-0.652572\pi$
$953$	−28.4736	−0.922350	−0.461175	+	0.887309i	$0.652572\pi$
$954$	9.73350	0.315134
$955$	0	0
$956$	12.9855	0.419982
$957$	3.98616	0.128854
$958$	52.0425	1.68142
$959$	17.8985	0.577974
$960$	0	0
$961$	−20.2306	−0.652599
$962$	20.7619	0.669390
$963$	−15.4670	−0.498417
$964$	5.66508	0.182460
$965$	0	0
$966$	−0.647459	−0.0208317
$967$	10.9313	0.351526	0.175763	−	0.984433i	$-0.443761\pi$
$967$	10.9313	0.351526	0.175763	+	0.984433i	$0.443761\pi$
$968$	3.04297	0.0978048
$969$	1.29668	0.0416554
$970$	0	0
$971$	−7.59300	−0.243671	−0.121835	−	0.992550i	$-0.538878\pi$
$971$	−7.59300	−0.243671	−0.121835	+	0.992550i	$0.538878\pi$
$972$	6.06782	0.194626
$973$	0.606805	0.0194533
$974$	27.3496	0.876338
$975$	0	0
$976$	−7.83858	−0.250907
$977$	−23.0302	−0.736802	−0.368401	−	0.929667i	$-0.620095\pi$
$977$	−23.0302	−0.736802	−0.368401	+	0.929667i	$0.620095\pi$
$978$	15.8327	0.506273
$979$	6.20642	0.198358
$980$	0	0
$981$	39.6630	1.26634
$982$	−20.7292	−0.661496
$983$	−21.6864	−0.691688	−0.345844	−	0.938292i	$-0.612407\pi$
$983$	−21.6864	−0.691688	−0.345844	+	0.938292i	$0.612407\pi$
$984$	23.2429	0.740956
$985$	0	0
$986$	−3.90168	−0.124255
$987$	−0.633133	−0.0201528
$988$	−5.38890	−0.171444
$989$	7.08974	0.225441
$990$	0	0
$991$	32.0034	1.01662	0.508310	−	0.861174i	$-0.330270\pi$
$991$	32.0034	1.01662	0.508310	+	0.861174i	$0.330270\pi$
$992$	7.72227	0.245182
$993$	6.60698	0.209666
$994$	9.82432	0.311608
$995$	0	0
$996$	0.800044	0.0253504
$997$	44.7240	1.41642	0.708211	−	0.706001i	$-0.249502\pi$
$997$	44.7240	1.41642	0.708211	+	0.706001i	$0.249502\pi$
$998$	7.39005	0.233928
$999$	−18.6823	−0.591082

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6325.2.a.m.1.3		6
5.4	even	2		253.2.a.d.1.4	✓	6
15.14	odd	2		2277.2.a.m.1.3		6
20.19	odd	2		4048.2.a.bc.1.4		6
55.54	odd	2		2783.2.a.h.1.3		6
115.114	odd	2		5819.2.a.e.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
253.2.a.d.1.4	✓	6	5.4	even	2
2277.2.a.m.1.3		6	15.14	odd	2
2783.2.a.h.1.3		6	55.54	odd	2
4048.2.a.bc.1.4		6	20.19	odd	2
5819.2.a.e.1.4		6	115.114	odd	2
6325.2.a.m.1.3		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6325 = 5^{2} \cdot 11 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6325.a (trivial)

\(f(q)\)	\(=\)	\(q-1.25551 q^{2} -0.646163 q^{3} -0.423685 q^{4} +0.811267 q^{6} -0.798084 q^{7} +3.04297 q^{8} -2.58247 q^{9} +1.00000 q^{11} +0.273769 q^{12} +3.19289 q^{13} +1.00201 q^{14} -2.97312 q^{16} -0.503752 q^{17} +3.24233 q^{18} +3.98359 q^{19} +0.515693 q^{21} -1.25551 q^{22} +1.00000 q^{23} -1.96625 q^{24} -4.00871 q^{26} +3.60719 q^{27} +0.338136 q^{28} -6.16897 q^{29} -3.28168 q^{31} -2.35314 q^{32} -0.646163 q^{33} +0.632468 q^{34} +1.09415 q^{36} -5.17919 q^{37} -5.00145 q^{38} -2.06312 q^{39} -11.8209 q^{41} -0.647459 q^{42} +7.08974 q^{43} -0.423685 q^{44} -1.25551 q^{46} -1.22773 q^{47} +1.92112 q^{48} -6.36306 q^{49} +0.325506 q^{51} -1.35278 q^{52} +3.00201 q^{53} -4.52888 q^{54} -2.42855 q^{56} -2.57405 q^{57} +7.74523 q^{58} +9.03304 q^{59} +2.63648 q^{61} +4.12020 q^{62} +2.06103 q^{63} +8.90065 q^{64} +0.811267 q^{66} -8.99026 q^{67} +0.213432 q^{68} -0.646163 q^{69} +9.80465 q^{71} -7.85839 q^{72} -1.31433 q^{73} +6.50255 q^{74} -1.68779 q^{76} -0.798084 q^{77} +2.59028 q^{78} +15.6898 q^{79} +5.41659 q^{81} +14.8413 q^{82} +2.92233 q^{83} -0.218491 q^{84} -8.90127 q^{86} +3.98616 q^{87} +3.04297 q^{88} +6.20642 q^{89} -2.54819 q^{91} -0.423685 q^{92} +2.12050 q^{93} +1.54144 q^{94} +1.52051 q^{96} +15.3359 q^{97} +7.98891 q^{98} -2.58247 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{2} - 7 q^{3} + 5 q^{4} + 3 q^{6} + q^{7} - 3 q^{8} + 9 q^{9} + 6 q^{11} - 11 q^{12} + 3 q^{13} - 8 q^{14} - q^{16} - 5 q^{17} + 11 q^{18} + q^{19} - 15 q^{21} - 3 q^{22} + 6 q^{23} - 3 q^{24}+ \cdots + 9 q^{99}+O(q^{100}) \) 6 * q - 3 * q^2 - 7 * q^3 + 5 * q^4 + 3 * q^6 + q^7 - 3 * q^8 + 9 * q^9 + 6 * q^11 - 11 * q^12 + 3 * q^13 - 8 * q^14 - q^16 - 5 * q^17 + 11 * q^18 + q^19 - 15 * q^21 - 3 * q^22 + 6 * q^23 - 3 * q^24 + 15 * q^26 - 16 * q^27 + 6 * q^29 - 8 * q^31 + 6 * q^32 - 7 * q^33 - 6 * q^34 - 8 * q^36 - 2 * q^37 - 9 * q^38 + q^39 - 2 * q^41 + 31 * q^42 - 10 * q^43 + 5 * q^44 - 3 * q^46 - 14 * q^47 - 6 * q^48 - 5 * q^49 - 24 * q^51 + 11 * q^52 + 4 * q^53 - 13 * q^54 - 13 * q^56 + 19 * q^57 + 6 * q^58 + 39 * q^59 - 22 * q^61 + 35 * q^63 - 33 * q^64 + 3 * q^66 - 8 * q^67 - 17 * q^68 - 7 * q^69 + 18 * q^71 - 6 * q^72 + 9 * q^73 + 15 * q^74 - 25 * q^76 + q^77 - 20 * q^78 - 15 * q^79 + 18 * q^81 + 8 * q^82 - 43 * q^83 - 4 * q^84 - 55 * q^86 + 28 * q^87 - 3 * q^88 + 17 * q^89 - 9 * q^91 + 5 * q^92 - 27 * q^93 + 63 * q^94 + 26 * q^96 - 19 * q^97 - 34 * q^98 + 9 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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