LMFDB - Embedded newform 4275.2.a.bd.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4275,2,Mod(1,4275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4275, 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("4275.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4275 = 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4275.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:	$34.1360468641$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 285)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	4275.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.311108 q^{2} -1.90321 q^{4} +2.68889 q^{7} +1.21432 q^{8} +0.214320 q^{11} -3.11753 q^{13} -0.836535 q^{14} +3.42864 q^{16} -0.474572 q^{17} -1.00000 q^{19} -0.0666765 q^{22} -2.47457 q^{23} +0.969888 q^{26} -5.11753 q^{28} +2.83654 q^{29} -4.90321 q^{31} -3.49532 q^{32} +0.147643 q^{34} +1.93332 q^{37} +0.311108 q^{38} -5.45875 q^{41} +10.7906 q^{43} -0.407896 q^{44} +0.769859 q^{46} -9.52543 q^{47} +0.230141 q^{49} +5.93332 q^{52} -13.7605 q^{53} +3.26517 q^{56} -0.882468 q^{58} +6.56199 q^{59} +7.76049 q^{61} +1.52543 q^{62} -5.76986 q^{64} -6.85728 q^{67} +0.903212 q^{68} +2.00000 q^{71} +16.6637 q^{73} -0.601472 q^{74} +1.90321 q^{76} +0.576283 q^{77} +1.37778 q^{79} +1.69826 q^{82} -1.65878 q^{83} -3.35704 q^{86} +0.260253 q^{88} -16.5827 q^{89} -8.38271 q^{91} +4.70964 q^{92} +2.96343 q^{94} +9.05731 q^{97} -0.0715987 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + q^{4} + 8 q^{7} - 3 q^{8} - 6 q^{11} + 4 q^{13} + 4 q^{14} - 3 q^{16} - 8 q^{17} - 3 q^{19} - 14 q^{23} - 4 q^{26} - 2 q^{28} + 2 q^{29} - 8 q^{31} + 3 q^{32} - 6 q^{34} + 6 q^{37} + q^{38}+ \cdots + 33 q^{98}+O(q^{100}) $ 3 * q - q^2 + q^4 + 8 * q^7 - 3 * q^8 - 6 * q^11 + 4 * q^13 + 4 * q^14 - 3 * q^16 - 8 * q^17 - 3 * q^19 - 14 * q^23 - 4 * q^26 - 2 * q^28 + 2 * q^29 - 8 * q^31 + 3 * q^32 - 6 * q^34 + 6 * q^37 + q^38 - 10 * q^41 + 6 * q^43 - 8 * q^44 - 4 * q^46 - 22 * q^47 + 7 * q^49 + 18 * q^52 - 8 * q^53 - 10 * q^56 - 16 * q^58 + 6 * q^59 - 10 * q^61 - 2 * q^62 - 11 * q^64 + 6 * q^67 - 4 * q^68 + 6 * q^71 + 10 * q^73 - 22 * q^74 - q^76 - 18 * q^77 + 4 * q^79 + 32 * q^82 + 2 * q^83 - 30 * q^86 + 14 * q^88 - 16 * q^89 + 8 * q^91 - 6 * q^92 + 16 * q^94 + 33 * q^98

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 $

$2$	−0.311108	−0.219986	−0.109993	−	0.993932i	$-0.535083\pi$
$2$	−0.311108	−0.219986	−0.109993	+	0.993932i	$0.535083\pi$
$3$	0	0
$3$	0	0
$4$	−1.90321	−0.951606
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.68889	1.01631	0.508153	−	0.861267i	$-0.330328\pi$
$7$	2.68889	1.01631	0.508153	+	0.861267i	$0.330328\pi$
$8$	1.21432	0.429327
$9$	0	0
$10$	0	0
$11$	0.214320	0.0646198	0.0323099	−	0.999478i	$-0.489714\pi$
$11$	0.214320	0.0646198	0.0323099	+	0.999478i	$0.489714\pi$
$12$	0	0
$13$	−3.11753	−0.864648	−0.432324	−	0.901718i	$-0.642306\pi$
$13$	−3.11753	−0.864648	−0.432324	+	0.901718i	$0.642306\pi$
$14$	−0.836535	−0.223573
$15$	0	0
$16$	3.42864	0.857160
$17$	−0.474572	−0.115101	−0.0575504	−	0.998343i	$-0.518329\pi$
$17$	−0.474572	−0.115101	−0.0575504	+	0.998343i	$0.518329\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	−0.0666765	−0.0142155
$23$	−2.47457	−0.515984	−0.257992	−	0.966147i	$-0.583061\pi$
$23$	−2.47457	−0.515984	−0.257992	+	0.966147i	$0.583061\pi$
$24$	0	0
$25$	0	0
$26$	0.969888	0.190211
$27$	0	0
$28$	−5.11753	−0.967123
$29$	2.83654	0.526731	0.263366	−	0.964696i	$-0.415167\pi$
$29$	2.83654	0.526731	0.263366	+	0.964696i	$0.415167\pi$
$30$	0	0
$31$	−4.90321	−0.880643	−0.440321	−	0.897840i	$-0.645136\pi$
$31$	−4.90321	−0.880643	−0.440321	+	0.897840i	$0.645136\pi$
$32$	−3.49532	−0.617890
$33$	0	0
$34$	0.147643	0.0253206
$35$	0	0
$36$	0	0
$37$	1.93332	0.317836	0.158918	−	0.987292i	$-0.449199\pi$
$37$	1.93332	0.317836	0.158918	+	0.987292i	$0.449199\pi$
$38$	0.311108	0.0504684
$39$	0	0
$40$	0	0
$41$	−5.45875	−0.852514	−0.426257	−	0.904602i	$-0.640168\pi$
$41$	−5.45875	−0.852514	−0.426257	+	0.904602i	$0.640168\pi$
$42$	0	0
$43$	10.7906	1.64555	0.822776	−	0.568366i	$-0.192424\pi$
$43$	10.7906	1.64555	0.822776	+	0.568366i	$0.192424\pi$
$44$	−0.407896	−0.0614926
$45$	0	0
$46$	0.769859	0.113509
$47$	−9.52543	−1.38943	−0.694713	−	0.719287i	$-0.744469\pi$
$47$	−9.52543	−1.38943	−0.694713	+	0.719287i	$0.744469\pi$
$48$	0	0
$49$	0.230141	0.0328773
$50$	0	0
$51$	0	0
$52$	5.93332	0.822804
$53$	−13.7605	−1.89015	−0.945074	−	0.326855i	$-0.894011\pi$
$53$	−13.7605	−1.89015	−0.945074	+	0.326855i	$0.894011\pi$
$54$	0	0
$55$	0	0
$56$	3.26517	0.436327
$57$	0	0
$58$	−0.882468	−0.115874
$59$	6.56199	0.854299	0.427149	−	0.904181i	$-0.359518\pi$
$59$	6.56199	0.854299	0.427149	+	0.904181i	$0.359518\pi$
$60$	0	0
$61$	7.76049	0.993629	0.496815	−	0.867857i	$-0.334503\pi$
$61$	7.76049	0.993629	0.496815	+	0.867857i	$0.334503\pi$
$62$	1.52543	0.193729
$63$	0	0
$64$	−5.76986	−0.721232
$65$	0	0
$66$	0	0
$67$	−6.85728	−0.837750	−0.418875	−	0.908044i	$-0.637575\pi$
$67$	−6.85728	−0.837750	−0.418875	+	0.908044i	$0.637575\pi$
$68$	0.903212	0.109531
$69$	0	0
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	0	0
$73$	16.6637	1.95034	0.975169	−	0.221460i	$-0.0710822\pi$
$73$	16.6637	1.95034	0.975169	+	0.221460i	$0.0710822\pi$
$74$	−0.601472	−0.0699197
$75$	0	0
$76$	1.90321	0.218313
$77$	0.576283	0.0656735
$78$	0	0
$79$	1.37778	0.155013	0.0775064	−	0.996992i	$-0.475304\pi$
$79$	1.37778	0.155013	0.0775064	+	0.996992i	$0.475304\pi$
$80$	0	0
$81$	0	0
$82$	1.69826	0.187541
$83$	−1.65878	−0.182075	−0.0910374	−	0.995847i	$-0.529018\pi$
$83$	−1.65878	−0.182075	−0.0910374	+	0.995847i	$0.529018\pi$
$84$	0	0
$85$	0	0
$86$	−3.35704	−0.361999
$87$	0	0
$88$	0.260253	0.0277430
$89$	−16.5827	−1.75777	−0.878883	−	0.477037i	$-0.841711\pi$
$89$	−16.5827	−1.75777	−0.878883	+	0.477037i	$0.841711\pi$
$90$	0	0
$91$	−8.38271	−0.878746
$92$	4.70964	0.491013
$93$	0	0
$94$	2.96343	0.305655
$95$	0	0
$96$	0	0
$97$	9.05731	0.919630	0.459815	−	0.888015i	$-0.347916\pi$
$97$	9.05731	0.919630	0.459815	+	0.888015i	$0.347916\pi$
$98$	−0.0715987	−0.00723256
$99$	0	0
$100$	0	0
$101$	−9.80642	−0.975776	−0.487888	−	0.872906i	$-0.662232\pi$
$101$	−9.80642	−0.975776	−0.487888	+	0.872906i	$0.662232\pi$
$102$	0	0
$103$	−0.561993	−0.0553748	−0.0276874	−	0.999617i	$-0.508814\pi$
$103$	−0.561993	−0.0553748	−0.0276874	+	0.999617i	$0.508814\pi$
$104$	−3.78568	−0.371216
$105$	0	0
$106$	4.28100	0.415807
$107$	−11.0923	−1.07234	−0.536169	−	0.844111i	$-0.680129\pi$
$107$	−11.0923	−1.07234	−0.536169	+	0.844111i	$0.680129\pi$
$108$	0	0
$109$	−8.19358	−0.784802	−0.392401	−	0.919794i	$-0.628355\pi$
$109$	−8.19358	−0.784802	−0.392401	+	0.919794i	$0.628355\pi$
$110$	0	0
$111$	0	0
$112$	9.21924	0.871136
$113$	14.1891	1.33480	0.667401	−	0.744699i	$-0.267407\pi$
$113$	14.1891	1.33480	0.667401	+	0.744699i	$0.267407\pi$
$114$	0	0
$115$	0	0
$116$	−5.39853	−0.501241
$117$	0	0
$118$	−2.04149	−0.187934
$119$	−1.27607	−0.116978
$120$	0	0
$121$	−10.9541	−0.995824
$122$	−2.41435	−0.218585
$123$	0	0
$124$	9.33185	0.838025
$125$	0	0
$126$	0	0
$127$	16.2766	1.44431	0.722155	−	0.691731i	$-0.243152\pi$
$127$	16.2766	1.44431	0.722155	+	0.691731i	$0.243152\pi$
$128$	8.78568	0.776552
$129$	0	0
$130$	0	0
$131$	−13.8272	−1.20809	−0.604043	−	0.796952i	$-0.706444\pi$
$131$	−13.8272	−1.20809	−0.604043	+	0.796952i	$0.706444\pi$
$132$	0	0
$133$	−2.68889	−0.233157
$134$	2.13335	0.184294
$135$	0	0
$136$	−0.576283	−0.0494158
$137$	11.1240	0.950386	0.475193	−	0.879882i	$-0.342378\pi$
$137$	11.1240	0.950386	0.475193	+	0.879882i	$0.342378\pi$
$138$	0	0
$139$	9.80642	0.831770	0.415885	−	0.909417i	$-0.363472\pi$
$139$	9.80642	0.831770	0.415885	+	0.909417i	$0.363472\pi$
$140$	0	0
$141$	0	0
$142$	−0.622216	−0.0522152
$143$	−0.668149	−0.0558734
$144$	0	0
$145$	0	0
$146$	−5.18421	−0.429048
$147$	0	0
$148$	−3.67952	−0.302455
$149$	−2.56199	−0.209887	−0.104943	−	0.994478i	$-0.533466\pi$
$149$	−2.56199	−0.209887	−0.104943	+	0.994478i	$0.533466\pi$
$150$	0	0
$151$	−12.7699	−1.03920	−0.519598	−	0.854411i	$-0.673918\pi$
$151$	−12.7699	−1.03920	−0.519598	+	0.854411i	$0.673918\pi$
$152$	−1.21432	−0.0984943
$153$	0	0
$154$	−0.179286	−0.0144473
$155$	0	0
$156$	0	0
$157$	−10.4286	−0.832296	−0.416148	−	0.909297i	$-0.636620\pi$
$157$	−10.4286	−0.832296	−0.416148	+	0.909297i	$0.636620\pi$
$158$	−0.428639	−0.0341007
$159$	0	0
$160$	0	0
$161$	−6.65386	−0.524398
$162$	0	0
$163$	−3.67952	−0.288203	−0.144101	−	0.989563i	$-0.546029\pi$
$163$	−3.67952	−0.288203	−0.144101	+	0.989563i	$0.546029\pi$
$164$	10.3892	0.811257
$165$	0	0
$166$	0.516060	0.0400540
$167$	−16.7096	−1.29303	−0.646515	−	0.762901i	$-0.723774\pi$
$167$	−16.7096	−1.29303	−0.646515	+	0.762901i	$0.723774\pi$
$168$	0	0
$169$	−3.28100	−0.252384
$170$	0	0
$171$	0	0
$172$	−20.5368	−1.56592
$173$	5.33185	0.405373	0.202687	−	0.979244i	$-0.435033\pi$
$173$	5.33185	0.405373	0.202687	+	0.979244i	$0.435033\pi$
$174$	0	0
$175$	0	0
$176$	0.734825	0.0553895
$177$	0	0
$178$	5.15902	0.386685
$179$	−4.66370	−0.348582	−0.174291	−	0.984694i	$-0.555763\pi$
$179$	−4.66370	−0.348582	−0.174291	+	0.984694i	$0.555763\pi$
$180$	0	0
$181$	−25.2257	−1.87501	−0.937506	−	0.347970i	$-0.886871\pi$
$181$	−25.2257	−1.87501	−0.937506	+	0.347970i	$0.886871\pi$
$182$	2.60793	0.193312
$183$	0	0
$184$	−3.00492	−0.221526
$185$	0	0
$186$	0	0
$187$	−0.101710	−0.00743779
$188$	18.1289	1.32219
$189$	0	0
$190$	0	0
$191$	−18.4909	−1.33795	−0.668976	−	0.743284i	$-0.733267\pi$
$191$	−18.4909	−1.33795	−0.668976	+	0.743284i	$0.733267\pi$
$192$	0	0
$193$	6.16839	0.444010	0.222005	−	0.975046i	$-0.428740\pi$
$193$	6.16839	0.444010	0.222005	+	0.975046i	$0.428740\pi$
$194$	−2.81780	−0.202306
$195$	0	0
$196$	−0.438007	−0.0312862
$197$	18.1891	1.29592	0.647961	−	0.761674i	$-0.275622\pi$
$197$	18.1891	1.29592	0.647961	+	0.761674i	$0.275622\pi$
$198$	0	0
$199$	14.3970	1.02058	0.510288	−	0.860004i	$-0.329539\pi$
$199$	14.3970	1.02058	0.510288	+	0.860004i	$0.329539\pi$
$200$	0	0
$201$	0	0
$202$	3.05086	0.214657
$203$	7.62714	0.535320
$204$	0	0
$205$	0	0
$206$	0.174840	0.0121817
$207$	0	0
$208$	−10.6889	−0.741141
$209$	−0.214320	−0.0148248
$210$	0	0
$211$	−23.0923	−1.58974	−0.794871	−	0.606778i	$-0.792462\pi$
$211$	−23.0923	−1.58974	−0.794871	+	0.606778i	$0.792462\pi$
$212$	26.1891	1.79868
$213$	0	0
$214$	3.45091	0.235900
$215$	0	0
$216$	0	0
$217$	−13.1842	−0.895002
$218$	2.54909	0.172646
$219$	0	0
$220$	0	0
$221$	1.47949	0.0995216
$222$	0	0
$223$	−10.6637	−0.714094	−0.357047	−	0.934086i	$-0.616216\pi$
$223$	−10.6637	−0.714094	−0.357047	+	0.934086i	$0.616216\pi$
$224$	−9.39853	−0.627966
$225$	0	0
$226$	−4.41435	−0.293638
$227$	−9.09679	−0.603775	−0.301888	−	0.953344i	$-0.597617\pi$
$227$	−9.09679	−0.603775	−0.301888	+	0.953344i	$0.597617\pi$
$228$	0	0
$229$	−15.0366	−0.993644	−0.496822	−	0.867852i	$-0.665500\pi$
$229$	−15.0366	−0.993644	−0.496822	+	0.867852i	$0.665500\pi$
$230$	0	0
$231$	0	0
$232$	3.44446	0.226140
$233$	−11.2859	−0.739365	−0.369683	−	0.929158i	$-0.620534\pi$
$233$	−11.2859	−0.739365	−0.369683	+	0.929158i	$0.620534\pi$
$234$	0	0
$235$	0	0
$236$	−12.4889	−0.812956
$237$	0	0
$238$	0.396997	0.0257335
$239$	1.26517	0.0818374	0.0409187	−	0.999162i	$-0.486972\pi$
$239$	1.26517	0.0818374	0.0409187	+	0.999162i	$0.486972\pi$
$240$	0	0
$241$	2.26671	0.146011	0.0730057	−	0.997332i	$-0.476741\pi$
$241$	2.26671	0.146011	0.0730057	+	0.997332i	$0.476741\pi$
$242$	3.40790	0.219068
$243$	0	0
$244$	−14.7699	−0.945543
$245$	0	0
$246$	0	0
$247$	3.11753	0.198364
$248$	−5.95407	−0.378084
$249$	0	0
$250$	0	0
$251$	16.8780	1.06533	0.532666	−	0.846326i	$-0.321190\pi$
$251$	16.8780	1.06533	0.532666	+	0.846326i	$0.321190\pi$
$252$	0	0
$253$	−0.530350	−0.0333428
$254$	−5.06376	−0.317729
$255$	0	0
$256$	8.80642	0.550401
$257$	23.2716	1.45164	0.725822	−	0.687882i	$-0.241460\pi$
$257$	23.2716	1.45164	0.725822	+	0.687882i	$0.241460\pi$
$258$	0	0
$259$	5.19850	0.323019
$260$	0	0
$261$	0	0
$262$	4.30174	0.265762
$263$	−26.3827	−1.62683	−0.813414	−	0.581686i	$-0.802393\pi$
$263$	−26.3827	−1.62683	−0.813414	+	0.581686i	$0.802393\pi$
$264$	0	0
$265$	0	0
$266$	0.836535	0.0512913
$267$	0	0
$268$	13.0509	0.797208
$269$	−2.89676	−0.176618	−0.0883092	−	0.996093i	$-0.528146\pi$
$269$	−2.89676	−0.176618	−0.0883092	+	0.996093i	$0.528146\pi$
$270$	0	0
$271$	−8.47013	−0.514524	−0.257262	−	0.966342i	$-0.582820\pi$
$271$	−8.47013	−0.514524	−0.257262	+	0.966342i	$0.582820\pi$
$272$	−1.62714	−0.0986597
$273$	0	0
$274$	−3.46076	−0.209072
$275$	0	0
$276$	0	0
$277$	−10.1017	−0.606953	−0.303476	−	0.952839i	$-0.598147\pi$
$277$	−10.1017	−0.606953	−0.303476	+	0.952839i	$0.598147\pi$
$278$	−3.05086	−0.182978
$279$	0	0
$280$	0	0
$281$	−25.6938	−1.53276	−0.766382	−	0.642385i	$-0.777945\pi$
$281$	−25.6938	−1.53276	−0.766382	+	0.642385i	$0.777945\pi$
$282$	0	0
$283$	1.05731	0.0628505	0.0314252	−	0.999506i	$-0.489995\pi$
$283$	1.05731	0.0628505	0.0314252	+	0.999506i	$0.489995\pi$
$284$	−3.80642	−0.225870
$285$	0	0
$286$	0.207866	0.0122914
$287$	−14.6780	−0.866415
$288$	0	0
$289$	−16.7748	−0.986752
$290$	0	0
$291$	0	0
$292$	−31.7146	−1.85595
$293$	−26.2306	−1.53241	−0.766205	−	0.642597i	$-0.777857\pi$
$293$	−26.2306	−1.53241	−0.766205	+	0.642597i	$0.777857\pi$
$294$	0	0
$295$	0	0
$296$	2.34767	0.136456
$297$	0	0
$298$	0.797056	0.0461722
$299$	7.71456	0.446144
$300$	0	0
$301$	29.0148	1.67238
$302$	3.97280	0.228609
$303$	0	0
$304$	−3.42864	−0.196646
$305$	0	0
$306$	0	0
$307$	−16.3368	−0.932389	−0.466194	−	0.884682i	$-0.654375\pi$
$307$	−16.3368	−0.932389	−0.466194	+	0.884682i	$0.654375\pi$
$308$	−1.09679	−0.0624953
$309$	0	0
$310$	0	0
$311$	−16.5827	−0.940321	−0.470160	−	0.882581i	$-0.655804\pi$
$311$	−16.5827	−0.940321	−0.470160	+	0.882581i	$0.655804\pi$
$312$	0	0
$313$	−5.18421	−0.293029	−0.146514	−	0.989209i	$-0.546805\pi$
$313$	−5.18421	−0.293029	−0.146514	+	0.989209i	$0.546805\pi$
$314$	3.24443	0.183094
$315$	0	0
$316$	−2.62222	−0.147511
$317$	−15.6084	−0.876655	−0.438328	−	0.898815i	$-0.644429\pi$
$317$	−15.6084	−0.876655	−0.438328	+	0.898815i	$0.644429\pi$
$318$	0	0
$319$	0.607926	0.0340373
$320$	0	0
$321$	0	0
$322$	2.07007	0.115360
$323$	0.474572	0.0264059
$324$	0	0
$325$	0	0
$326$	1.14473	0.0634007
$327$	0	0
$328$	−6.62867	−0.366007
$329$	−25.6128	−1.41208
$330$	0	0
$331$	16.2908	0.895426	0.447713	−	0.894177i	$-0.352239\pi$
$331$	16.2908	0.895426	0.447713	+	0.894177i	$0.352239\pi$
$332$	3.15701	0.173263
$333$	0	0
$334$	5.19850	0.284449
$335$	0	0
$336$	0	0
$337$	−27.0859	−1.47546	−0.737731	−	0.675095i	$-0.764103\pi$
$337$	−27.0859	−1.47546	−0.737731	+	0.675095i	$0.764103\pi$
$338$	1.02074	0.0555211
$339$	0	0
$340$	0	0
$341$	−1.05086	−0.0569070
$342$	0	0
$343$	−18.2034	−0.982892
$344$	13.1032	0.706479
$345$	0	0
$346$	−1.65878	−0.0891766
$347$	2.24935	0.120752	0.0603758	−	0.998176i	$-0.480770\pi$
$347$	2.24935	0.120752	0.0603758	+	0.998176i	$0.480770\pi$
$348$	0	0
$349$	−2.81579	−0.150726	−0.0753629	−	0.997156i	$-0.524012\pi$
$349$	−2.81579	−0.150726	−0.0753629	+	0.997156i	$0.524012\pi$
$350$	0	0
$351$	0	0
$352$	−0.749115	−0.0399280
$353$	27.8479	1.48219	0.741097	−	0.671398i	$-0.234306\pi$
$353$	27.8479	1.48219	0.741097	+	0.671398i	$0.234306\pi$
$354$	0	0
$355$	0	0
$356$	31.5605	1.67270
$357$	0	0
$358$	1.45091	0.0766832
$359$	−32.7130	−1.72653	−0.863264	−	0.504753i	$-0.831584\pi$
$359$	−32.7130	−1.72653	−0.863264	+	0.504753i	$0.831584\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	7.84791	0.412477
$363$	0	0
$364$	15.9541	0.836220
$365$	0	0
$366$	0	0
$367$	18.7906	0.980861	0.490431	−	0.871480i	$-0.336839\pi$
$367$	18.7906	0.980861	0.490431	+	0.871480i	$0.336839\pi$
$368$	−8.48442	−0.442281
$369$	0	0
$370$	0	0
$371$	−37.0005	−1.92097
$372$	0	0
$373$	−5.33969	−0.276479	−0.138239	−	0.990399i	$-0.544144\pi$
$373$	−5.33969	−0.276479	−0.138239	+	0.990399i	$0.544144\pi$
$374$	0.0316429	0.00163621
$375$	0	0
$376$	−11.5669	−0.596518
$377$	−8.84299	−0.455437
$378$	0	0
$379$	−18.0143	−0.925332	−0.462666	−	0.886533i	$-0.653107\pi$
$379$	−18.0143	−0.925332	−0.462666	+	0.886533i	$0.653107\pi$
$380$	0	0
$381$	0	0
$382$	5.75265	0.294331
$383$	0.295286	0.0150884	0.00754421	−	0.999972i	$-0.497599\pi$
$383$	0.295286	0.0150884	0.00754421	+	0.999972i	$0.497599\pi$
$384$	0	0
$385$	0	0
$386$	−1.91903	−0.0976762
$387$	0	0
$388$	−17.2380	−0.875126
$389$	22.7052	1.15120	0.575599	−	0.817732i	$-0.304769\pi$
$389$	22.7052	1.15120	0.575599	+	0.817732i	$0.304769\pi$
$390$	0	0
$391$	1.17436	0.0593901
$392$	0.279465	0.0141151
$393$	0	0
$394$	−5.65878	−0.285085
$395$	0	0
$396$	0	0
$397$	0.682439	0.0342506	0.0171253	−	0.999853i	$-0.494549\pi$
$397$	0.682439	0.0342506	0.0171253	+	0.999853i	$0.494549\pi$
$398$	−4.47902	−0.224513
$399$	0	0
$400$	0	0
$401$	3.23659	0.161628	0.0808139	−	0.996729i	$-0.474248\pi$
$401$	3.23659	0.161628	0.0808139	+	0.996729i	$0.474248\pi$
$402$	0	0
$403$	15.2859	0.761446
$404$	18.6637	0.928554
$405$	0	0
$406$	−2.37286	−0.117763
$407$	0.414349	0.0205385
$408$	0	0
$409$	12.5303	0.619586	0.309793	−	0.950804i	$-0.399740\pi$
$409$	12.5303	0.619586	0.309793	+	0.950804i	$0.399740\pi$
$410$	0	0
$411$	0	0
$412$	1.06959	0.0526950
$413$	17.6445	0.868229
$414$	0	0
$415$	0	0
$416$	10.8968	0.534258
$417$	0	0
$418$	0.0666765	0.00326126
$419$	−28.0810	−1.37184	−0.685922	−	0.727675i	$-0.740601\pi$
$419$	−28.0810	−1.37184	−0.685922	+	0.727675i	$0.740601\pi$
$420$	0	0
$421$	10.3368	0.503784	0.251892	−	0.967755i	$-0.418947\pi$
$421$	10.3368	0.503784	0.251892	+	0.967755i	$0.418947\pi$
$422$	7.18421	0.349722
$423$	0	0
$424$	−16.7096	−0.811492
$425$	0	0
$426$	0	0
$427$	20.8671	1.00983
$428$	21.1111	1.02044
$429$	0	0
$430$	0	0
$431$	0.815792	0.0392953	0.0196477	−	0.999807i	$-0.493746\pi$
$431$	0.815792	0.0392953	0.0196477	+	0.999807i	$0.493746\pi$
$432$	0	0
$433$	31.5877	1.51801	0.759003	−	0.651087i	$-0.225687\pi$
$433$	31.5877	1.51801	0.759003	+	0.651087i	$0.225687\pi$
$434$	4.10171	0.196888
$435$	0	0
$436$	15.5941	0.746823
$437$	2.47457	0.118375
$438$	0	0
$439$	17.8350	0.851218	0.425609	−	0.904907i	$-0.360060\pi$
$439$	17.8350	0.851218	0.425609	+	0.904907i	$0.360060\pi$
$440$	0	0
$441$	0	0
$442$	−0.460282	−0.0218934
$443$	−24.4429	−1.16132	−0.580659	−	0.814147i	$-0.697205\pi$
$443$	−24.4429	−1.16132	−0.580659	+	0.814147i	$0.697205\pi$
$444$	0	0
$445$	0	0
$446$	3.31756	0.157091
$447$	0	0
$448$	−15.5145	−0.732993
$449$	−27.2652	−1.28672	−0.643362	−	0.765562i	$-0.722461\pi$
$449$	−27.2652	−1.28672	−0.643362	+	0.765562i	$0.722461\pi$
$450$	0	0
$451$	−1.16992	−0.0550893
$452$	−27.0049	−1.27020
$453$	0	0
$454$	2.83008	0.132822
$455$	0	0
$456$	0	0
$457$	−5.27607	−0.246804	−0.123402	−	0.992357i	$-0.539381\pi$
$457$	−5.27607	−0.246804	−0.123402	+	0.992357i	$0.539381\pi$
$458$	4.67799	0.218588
$459$	0	0
$460$	0	0
$461$	0.876015	0.0408001	0.0204000	−	0.999792i	$-0.493506\pi$
$461$	0.876015	0.0408001	0.0204000	+	0.999792i	$0.493506\pi$
$462$	0	0
$463$	−22.8321	−1.06110	−0.530549	−	0.847655i	$-0.678014\pi$
$463$	−22.8321	−1.06110	−0.530549	+	0.847655i	$0.678014\pi$
$464$	9.72546	0.451493
$465$	0	0
$466$	3.51114	0.162650
$467$	36.1388	1.67230	0.836151	−	0.548499i	$-0.184801\pi$
$467$	36.1388	1.67230	0.836151	+	0.548499i	$0.184801\pi$
$468$	0	0
$469$	−18.4385	−0.851410
$470$	0	0
$471$	0	0
$472$	7.96836	0.366773
$473$	2.31264	0.106335
$474$	0	0
$475$	0	0
$476$	2.42864	0.111317
$477$	0	0
$478$	−0.393606	−0.0180031
$479$	7.59210	0.346892	0.173446	−	0.984843i	$-0.444510\pi$
$479$	7.59210	0.346892	0.173446	+	0.984843i	$0.444510\pi$
$480$	0	0
$481$	−6.02720	−0.274817
$482$	−0.705190	−0.0321205
$483$	0	0
$484$	20.8479	0.947632
$485$	0	0
$486$	0	0
$487$	25.0638	1.13575	0.567874	−	0.823116i	$-0.307766\pi$
$487$	25.0638	1.13575	0.567874	+	0.823116i	$0.307766\pi$
$488$	9.42372	0.426592
$489$	0	0
$490$	0	0
$491$	31.7768	1.43407	0.717033	−	0.697039i	$-0.245500\pi$
$491$	31.7768	1.43407	0.717033	+	0.697039i	$0.245500\pi$
$492$	0	0
$493$	−1.34614	−0.0606272
$494$	−0.969888	−0.0436373
$495$	0	0
$496$	−16.8113	−0.754852
$497$	5.37778	0.241227
$498$	0	0
$499$	16.7654	0.750523	0.375261	−	0.926919i	$-0.377553\pi$
$499$	16.7654	0.750523	0.375261	+	0.926919i	$0.377553\pi$
$500$	0	0
$501$	0	0
$502$	−5.25088	−0.234358
$503$	26.5575	1.18414	0.592071	−	0.805886i	$-0.298310\pi$
$503$	26.5575	1.18414	0.592071	+	0.805886i	$0.298310\pi$
$504$	0	0
$505$	0	0
$506$	0.164996	0.00733496
$507$	0	0
$508$	−30.9777	−1.37441
$509$	19.1733	0.849842	0.424921	−	0.905230i	$-0.360302\pi$
$509$	19.1733	0.849842	0.424921	+	0.905230i	$0.360302\pi$
$510$	0	0
$511$	44.8069	1.98214
$512$	−20.3111	−0.897633
$513$	0	0
$514$	−7.23999	−0.319342
$515$	0	0
$516$	0	0
$517$	−2.04149	−0.0897845
$518$	−1.61729	−0.0710598
$519$	0	0
$520$	0	0
$521$	33.3481	1.46101	0.730504	−	0.682908i	$-0.239285\pi$
$521$	33.3481	1.46101	0.730504	+	0.682908i	$0.239285\pi$
$522$	0	0
$523$	12.5303	0.547914	0.273957	−	0.961742i	$-0.411667\pi$
$523$	12.5303	0.547914	0.273957	+	0.961742i	$0.411667\pi$
$524$	26.3160	1.14962
$525$	0	0
$526$	8.20787	0.357880
$527$	2.32693	0.101363
$528$	0	0
$529$	−16.8765	−0.733760
$530$	0	0
$531$	0	0
$532$	5.11753	0.221873
$533$	17.0178	0.737124
$534$	0	0
$535$	0	0
$536$	−8.32693	−0.359669
$537$	0	0
$538$	0.901204	0.0388537
$539$	0.0493238	0.00212453
$540$	0	0
$541$	−28.3684	−1.21965	−0.609827	−	0.792535i	$-0.708761\pi$
$541$	−28.3684	−1.21965	−0.609827	+	0.792535i	$0.708761\pi$
$542$	2.63512	0.113188
$543$	0	0
$544$	1.65878	0.0711196
$545$	0	0
$546$	0	0
$547$	−25.8765	−1.10640	−0.553199	−	0.833049i	$-0.686593\pi$
$547$	−25.8765	−1.10640	−0.553199	+	0.833049i	$0.686593\pi$
$548$	−21.1713	−0.904393
$549$	0	0
$550$	0	0
$551$	−2.83654	−0.120840
$552$	0	0
$553$	3.70471	0.157540
$554$	3.14272	0.133521
$555$	0	0
$556$	−18.6637	−0.791517
$557$	33.0509	1.40041	0.700205	−	0.713942i	$-0.253092\pi$
$557$	33.0509	1.40041	0.700205	+	0.713942i	$0.253092\pi$
$558$	0	0
$559$	−33.6400	−1.42282
$560$	0	0
$561$	0	0
$562$	7.99355	0.337187
$563$	10.9032	0.459516	0.229758	−	0.973248i	$-0.426207\pi$
$563$	10.9032	0.459516	0.229758	+	0.973248i	$0.426207\pi$
$564$	0	0
$565$	0	0
$566$	−0.328937	−0.0138263
$567$	0	0
$568$	2.42864	0.101903
$569$	−18.6113	−0.780227	−0.390113	−	0.920767i	$-0.627564\pi$
$569$	−18.6113	−0.780227	−0.390113	+	0.920767i	$0.627564\pi$
$570$	0	0
$571$	34.6222	1.44889	0.724447	−	0.689330i	$-0.242095\pi$
$571$	34.6222	1.44889	0.724447	+	0.689330i	$0.242095\pi$
$572$	1.27163	0.0531695
$573$	0	0
$574$	4.56644	0.190599
$575$	0	0
$576$	0	0
$577$	29.3590	1.22223	0.611117	−	0.791541i	$-0.290721\pi$
$577$	29.3590	1.22223	0.611117	+	0.791541i	$0.290721\pi$
$578$	5.21877	0.217072
$579$	0	0
$580$	0	0
$581$	−4.46028	−0.185044
$582$	0	0
$583$	−2.94914	−0.122141
$584$	20.2351	0.837333
$585$	0	0
$586$	8.16055	0.337109
$587$	12.5892	0.519611	0.259806	−	0.965661i	$-0.416341\pi$
$587$	12.5892	0.519611	0.259806	+	0.965661i	$0.416341\pi$
$588$	0	0
$589$	4.90321	0.202033
$590$	0	0
$591$	0	0
$592$	6.62867	0.272437
$593$	−41.5210	−1.70506	−0.852531	−	0.522676i	$-0.824934\pi$
$593$	−41.5210	−1.70506	−0.852531	+	0.522676i	$0.824934\pi$
$594$	0	0
$595$	0	0
$596$	4.87601	0.199729
$597$	0	0
$598$	−2.40006	−0.0981457
$599$	−23.9398	−0.978153	−0.489076	−	0.872241i	$-0.662666\pi$
$599$	−23.9398	−0.978153	−0.489076	+	0.872241i	$0.662666\pi$
$600$	0	0
$601$	43.2070	1.76245	0.881224	−	0.472698i	$-0.156720\pi$
$601$	43.2070	1.76245	0.881224	+	0.472698i	$0.156720\pi$
$602$	−9.02672	−0.367902
$603$	0	0
$604$	24.3037	0.988905
$605$	0	0
$606$	0	0
$607$	−28.0513	−1.13857	−0.569284	−	0.822141i	$-0.692780\pi$
$607$	−28.0513	−1.13857	−0.569284	+	0.822141i	$0.692780\pi$
$608$	3.49532	0.141754
$609$	0	0
$610$	0	0
$611$	29.6958	1.20136
$612$	0	0
$613$	27.5526	1.11284	0.556420	−	0.830901i	$-0.312175\pi$
$613$	27.5526	1.11284	0.556420	+	0.830901i	$0.312175\pi$
$614$	5.08250	0.205113
$615$	0	0
$616$	0.699791	0.0281954
$617$	−8.94470	−0.360100	−0.180050	−	0.983657i	$-0.557626\pi$
$617$	−8.94470	−0.360100	−0.180050	+	0.983657i	$0.557626\pi$
$618$	0	0
$619$	17.2573	0.693631	0.346815	−	0.937933i	$-0.387263\pi$
$619$	17.2573	0.693631	0.346815	+	0.937933i	$0.387263\pi$
$620$	0	0
$621$	0	0
$622$	5.15902	0.206858
$623$	−44.5892	−1.78643
$624$	0	0
$625$	0	0
$626$	1.61285	0.0644624
$627$	0	0
$628$	19.8479	0.792018
$629$	−0.917502	−0.0365832
$630$	0	0
$631$	−16.4572	−0.655152	−0.327576	−	0.944825i	$-0.606232\pi$
$631$	−16.4572	−0.655152	−0.327576	+	0.944825i	$0.606232\pi$
$632$	1.67307	0.0665512
$633$	0	0
$634$	4.85590	0.192852
$635$	0	0
$636$	0	0
$637$	−0.717472	−0.0284273
$638$	−0.189130	−0.00748774
$639$	0	0
$640$	0	0
$641$	16.3289	0.644954	0.322477	−	0.946577i	$-0.395484\pi$
$641$	16.3289	0.644954	0.322477	+	0.946577i	$0.395484\pi$
$642$	0	0
$643$	39.8731	1.57244	0.786221	−	0.617946i	$-0.212035\pi$
$643$	39.8731	1.57244	0.786221	+	0.617946i	$0.212035\pi$
$644$	12.6637	0.499020
$645$	0	0
$646$	−0.147643	−0.00580894
$647$	−31.5067	−1.23866	−0.619328	−	0.785132i	$-0.712595\pi$
$647$	−31.5067	−1.23866	−0.619328	+	0.785132i	$0.712595\pi$
$648$	0	0
$649$	1.40636	0.0552046
$650$	0	0
$651$	0	0
$652$	7.00291	0.274255
$653$	45.2400	1.77038	0.885189	−	0.465232i	$-0.154029\pi$
$653$	45.2400	1.77038	0.885189	+	0.465232i	$0.154029\pi$
$654$	0	0
$655$	0	0
$656$	−18.7161	−0.730741
$657$	0	0
$658$	7.96836	0.310639
$659$	47.5812	1.85350	0.926750	−	0.375678i	$-0.122590\pi$
$659$	47.5812	1.85350	0.926750	+	0.375678i	$0.122590\pi$
$660$	0	0
$661$	7.84791	0.305248	0.152624	−	0.988284i	$-0.451228\pi$
$661$	7.84791	0.305248	0.152624	+	0.988284i	$0.451228\pi$
$662$	−5.06821	−0.196981
$663$	0	0
$664$	−2.01429	−0.0781696
$665$	0	0
$666$	0	0
$667$	−7.01921	−0.271785
$668$	31.8020	1.23046
$669$	0	0
$670$	0	0
$671$	1.66323	0.0642081
$672$	0	0
$673$	−41.5462	−1.60149	−0.800744	−	0.599007i	$-0.795562\pi$
$673$	−41.5462	−1.60149	−0.800744	+	0.599007i	$0.795562\pi$
$674$	8.42663	0.324582
$675$	0	0
$676$	6.24443	0.240170
$677$	4.05884	0.155994	0.0779969	−	0.996954i	$-0.475148\pi$
$677$	4.05884	0.155994	0.0779969	+	0.996954i	$0.475148\pi$
$678$	0	0
$679$	24.3541	0.934626
$680$	0	0
$681$	0	0
$682$	0.326929	0.0125188
$683$	−30.3180	−1.16009	−0.580044	−	0.814585i	$-0.696965\pi$
$683$	−30.3180	−1.16009	−0.580044	+	0.814585i	$0.696965\pi$
$684$	0	0
$685$	0	0
$686$	5.66323	0.216223
$687$	0	0
$688$	36.9971	1.41050
$689$	42.8988	1.63431
$690$	0	0
$691$	−25.1941	−0.958427	−0.479213	−	0.877698i	$-0.659078\pi$
$691$	−25.1941	−0.958427	−0.479213	+	0.877698i	$0.659078\pi$
$692$	−10.1476	−0.385756
$693$	0	0
$694$	−0.699791	−0.0265637
$695$	0	0
$696$	0	0
$697$	2.59057	0.0981249
$698$	0.876015	0.0331577
$699$	0	0
$700$	0	0
$701$	3.98126	0.150370	0.0751851	−	0.997170i	$-0.476045\pi$
$701$	3.98126	0.150370	0.0751851	+	0.997170i	$0.476045\pi$
$702$	0	0
$703$	−1.93332	−0.0729167
$704$	−1.23659	−0.0466059
$705$	0	0
$706$	−8.66370	−0.326063
$707$	−26.3684	−0.991686
$708$	0	0
$709$	−47.8149	−1.79573	−0.897863	−	0.440275i	$-0.854881\pi$
$709$	−47.8149	−1.79573	−0.897863	+	0.440275i	$0.854881\pi$
$710$	0	0
$711$	0	0
$712$	−20.1367	−0.754656
$713$	12.1334	0.454398
$714$	0	0
$715$	0	0
$716$	8.87601	0.331712
$717$	0	0
$718$	10.1773	0.379813
$719$	−5.13182	−0.191385	−0.0956923	−	0.995411i	$-0.530506\pi$
$719$	−5.13182	−0.191385	−0.0956923	+	0.995411i	$0.530506\pi$
$720$	0	0
$721$	−1.51114	−0.0562777
$722$	−0.311108	−0.0115782
$723$	0	0
$724$	48.0098	1.78427
$725$	0	0
$726$	0	0
$727$	39.7081	1.47269	0.736346	−	0.676605i	$-0.236550\pi$
$727$	39.7081	1.47269	0.736346	+	0.676605i	$0.236550\pi$
$728$	−10.1793	−0.377269
$729$	0	0
$730$	0	0
$731$	−5.12092	−0.189404
$732$	0	0
$733$	−11.3363	−0.418716	−0.209358	−	0.977839i	$-0.567137\pi$
$733$	−11.3363	−0.418716	−0.209358	+	0.977839i	$0.567137\pi$
$734$	−5.84590	−0.215776
$735$	0	0
$736$	8.64941	0.318822
$737$	−1.46965	−0.0541353
$738$	0	0
$739$	−39.9496	−1.46957	−0.734785	−	0.678300i	$-0.762717\pi$
$739$	−39.9496	−1.46957	−0.734785	+	0.678300i	$0.762717\pi$
$740$	0	0
$741$	0	0
$742$	11.5111	0.422587
$743$	−44.3783	−1.62808	−0.814040	−	0.580808i	$-0.802737\pi$
$743$	−44.3783	−1.62808	−0.814040	+	0.580808i	$0.802737\pi$
$744$	0	0
$745$	0	0
$746$	1.66122	0.0608215
$747$	0	0
$748$	0.193576	0.00707784
$749$	−29.8261	−1.08982
$750$	0	0
$751$	−31.9037	−1.16418	−0.582091	−	0.813124i	$-0.697765\pi$
$751$	−31.9037	−1.16418	−0.582091	+	0.813124i	$0.697765\pi$
$752$	−32.6593	−1.19096
$753$	0	0
$754$	2.75112	0.100190
$755$	0	0
$756$	0	0
$757$	24.2449	0.881196	0.440598	−	0.897704i	$-0.354766\pi$
$757$	24.2449	0.881196	0.440598	+	0.897704i	$0.354766\pi$
$758$	5.60439	0.203560
$759$	0	0
$760$	0	0
$761$	−0.308193	−0.0111720	−0.00558600	−	0.999984i	$-0.501778\pi$
$761$	−0.308193	−0.0111720	−0.00558600	+	0.999984i	$0.501778\pi$
$762$	0	0
$763$	−22.0316	−0.797599
$764$	35.1920	1.27320
$765$	0	0
$766$	−0.0918659	−0.00331925
$767$	−20.4572	−0.738667
$768$	0	0
$769$	22.0830	0.796332	0.398166	−	0.917313i	$-0.369647\pi$
$769$	22.0830	0.796332	0.398166	+	0.917313i	$0.369647\pi$
$770$	0	0
$771$	0	0
$772$	−11.7397	−0.422523
$773$	−11.0968	−0.399124	−0.199562	−	0.979885i	$-0.563952\pi$
$773$	−11.0968	−0.399124	−0.199562	+	0.979885i	$0.563952\pi$
$774$	0	0
$775$	0	0
$776$	10.9985	0.394822
$777$	0	0
$778$	−7.06376	−0.253248
$779$	5.45875	0.195580
$780$	0	0
$781$	0.428639	0.0153379
$782$	−0.365354	−0.0130650
$783$	0	0
$784$	0.789071	0.0281811
$785$	0	0
$786$	0	0
$787$	22.4286	0.799495	0.399747	−	0.916625i	$-0.369098\pi$
$787$	22.4286	0.799495	0.399747	+	0.916625i	$0.369098\pi$
$788$	−34.6178	−1.23321
$789$	0	0
$790$	0	0
$791$	38.1530	1.35657
$792$	0	0
$793$	−24.1936	−0.859139
$794$	−0.212312	−0.00753467
$795$	0	0
$796$	−27.4005	−0.971186
$797$	8.36980	0.296474	0.148237	−	0.988952i	$-0.452640\pi$
$797$	8.36980	0.296474	0.148237	+	0.988952i	$0.452640\pi$
$798$	0	0
$799$	4.52051	0.159924
$800$	0	0
$801$	0	0
$802$	−1.00693	−0.0355559
$803$	3.57136	0.126031
$804$	0	0
$805$	0	0
$806$	−4.75557	−0.167508
$807$	0	0
$808$	−11.9081	−0.418927
$809$	−2.94914	−0.103686	−0.0518432	−	0.998655i	$-0.516510\pi$
$809$	−2.94914	−0.103686	−0.0518432	+	0.998655i	$0.516510\pi$
$810$	0	0
$811$	−22.1017	−0.776096	−0.388048	−	0.921639i	$-0.626851\pi$
$811$	−22.1017	−0.776096	−0.388048	+	0.921639i	$0.626851\pi$
$812$	−14.5161	−0.509414
$813$	0	0
$814$	−0.128907	−0.00451820
$815$	0	0
$816$	0	0
$817$	−10.7906	−0.377515
$818$	−3.89829	−0.136301
$819$	0	0
$820$	0	0
$821$	13.6316	0.475746	0.237873	−	0.971296i	$-0.423550\pi$
$821$	13.6316	0.475746	0.237873	+	0.971296i	$0.423550\pi$
$822$	0	0
$823$	49.2509	1.71678	0.858389	−	0.512999i	$-0.171466\pi$
$823$	49.2509	1.71678	0.858389	+	0.512999i	$0.171466\pi$
$824$	−0.682439	−0.0237739
$825$	0	0
$826$	−5.48934	−0.190999
$827$	−31.3733	−1.09096	−0.545479	−	0.838125i	$-0.683652\pi$
$827$	−31.3733	−1.09096	−0.545479	+	0.838125i	$0.683652\pi$
$828$	0	0
$829$	−40.9719	−1.42301	−0.711506	−	0.702680i	$-0.751987\pi$
$829$	−40.9719	−1.42301	−0.711506	+	0.702680i	$0.751987\pi$
$830$	0	0
$831$	0	0
$832$	17.9877	0.623612
$833$	−0.109219	−0.00378420
$834$	0	0
$835$	0	0
$836$	0.407896	0.0141074
$837$	0	0
$838$	8.73621	0.301787
$839$	3.12399	0.107852	0.0539260	−	0.998545i	$-0.482826\pi$
$839$	3.12399	0.107852	0.0539260	+	0.998545i	$0.482826\pi$
$840$	0	0
$841$	−20.9541	−0.722554
$842$	−3.21585	−0.110826
$843$	0	0
$844$	43.9496	1.51281
$845$	0	0
$846$	0	0
$847$	−29.4543	−1.01206
$848$	−47.1798	−1.62016
$849$	0	0
$850$	0	0
$851$	−4.78415	−0.163999
$852$	0	0
$853$	−14.9175	−0.510766	−0.255383	−	0.966840i	$-0.582201\pi$
$853$	−14.9175	−0.510766	−0.255383	+	0.966840i	$0.582201\pi$
$854$	−6.49193	−0.222149
$855$	0	0
$856$	−13.4697	−0.460383
$857$	12.0459	0.411481	0.205741	−	0.978607i	$-0.434040\pi$
$857$	12.0459	0.411481	0.205741	+	0.978607i	$0.434040\pi$
$858$	0	0
$859$	49.7560	1.69765	0.848827	−	0.528670i	$-0.177309\pi$
$859$	49.7560	1.69765	0.848827	+	0.528670i	$0.177309\pi$
$860$	0	0
$861$	0	0
$862$	−0.253799	−0.00864444
$863$	57.0103	1.94065	0.970327	−	0.241797i	$-0.0777367\pi$
$863$	57.0103	1.94065	0.970327	+	0.241797i	$0.0777367\pi$
$864$	0	0
$865$	0	0
$866$	−9.82717	−0.333941
$867$	0	0
$868$	25.0923	0.851690
$869$	0.295286	0.0100169
$870$	0	0
$871$	21.3778	0.724359
$872$	−9.94962	−0.336937
$873$	0	0
$874$	−0.769859	−0.0260409
$875$	0	0
$876$	0	0
$877$	32.1051	1.08411	0.542056	−	0.840342i	$-0.317646\pi$
$877$	32.1051	1.08411	0.542056	+	0.840342i	$0.317646\pi$
$878$	−5.54861	−0.187256
$879$	0	0
$880$	0	0
$881$	−35.0736	−1.18166	−0.590830	−	0.806796i	$-0.701200\pi$
$881$	−35.0736	−1.18166	−0.590830	+	0.806796i	$0.701200\pi$
$882$	0	0
$883$	−27.3842	−0.921553	−0.460776	−	0.887516i	$-0.652429\pi$
$883$	−27.3842	−0.921553	−0.460776	+	0.887516i	$0.652429\pi$
$884$	−2.81579	−0.0947053
$885$	0	0
$886$	7.60439	0.255474
$887$	17.3274	0.581797	0.290899	−	0.956754i	$-0.406046\pi$
$887$	17.3274	0.581797	0.290899	+	0.956754i	$0.406046\pi$
$888$	0	0
$889$	43.7659	1.46786
$890$	0	0
$891$	0	0
$892$	20.2953	0.679536
$893$	9.52543	0.318756
$894$	0	0
$895$	0	0
$896$	23.6237	0.789214
$897$	0	0
$898$	8.48241	0.283062
$899$	−13.9081	−0.463862
$900$	0	0
$901$	6.53035	0.217557
$902$	0.363971	0.0121189
$903$	0	0
$904$	17.2301	0.573066
$905$	0	0
$906$	0	0
$907$	−6.52051	−0.216510	−0.108255	−	0.994123i	$-0.534526\pi$
$907$	−6.52051	−0.216510	−0.108255	+	0.994123i	$0.534526\pi$
$908$	17.3131	0.574556
$909$	0	0
$910$	0	0
$911$	54.7467	1.81384	0.906919	−	0.421305i	$-0.138428\pi$
$911$	54.7467	1.81384	0.906919	+	0.421305i	$0.138428\pi$
$912$	0	0
$913$	−0.355509	−0.0117656
$914$	1.64143	0.0542936
$915$	0	0
$916$	28.6178	0.945558
$917$	−37.1798	−1.22778
$918$	0	0
$919$	−17.1655	−0.566237	−0.283118	−	0.959085i	$-0.591369\pi$
$919$	−17.1655	−0.566237	−0.283118	+	0.959085i	$0.591369\pi$
$920$	0	0
$921$	0	0
$922$	−0.272535	−0.00897546
$923$	−6.23506	−0.205230
$924$	0	0
$925$	0	0
$926$	7.10324	0.233427
$927$	0	0
$928$	−9.91459	−0.325462
$929$	−28.8287	−0.945839	−0.472919	−	0.881106i	$-0.656800\pi$
$929$	−28.8287	−0.945839	−0.472919	+	0.881106i	$0.656800\pi$
$930$	0	0
$931$	−0.230141	−0.00754257
$932$	21.4795	0.703584
$933$	0	0
$934$	−11.2430	−0.367884
$935$	0	0
$936$	0	0
$937$	23.6829	0.773687	0.386844	−	0.922145i	$-0.373565\pi$
$937$	23.6829	0.773687	0.386844	+	0.922145i	$0.373565\pi$
$938$	5.73636	0.187299
$939$	0	0
$940$	0	0
$941$	−7.92888	−0.258474	−0.129237	−	0.991614i	$-0.541253\pi$
$941$	−7.92888	−0.258474	−0.129237	+	0.991614i	$0.541253\pi$
$942$	0	0
$943$	13.5081	0.439883
$944$	22.4987	0.732271
$945$	0	0
$946$	−0.719480	−0.0233923
$947$	−42.8113	−1.39118	−0.695591	−	0.718438i	$-0.744857\pi$
$947$	−42.8113	−1.39118	−0.695591	+	0.718438i	$0.744857\pi$
$948$	0	0
$949$	−51.9496	−1.68636
$950$	0	0
$951$	0	0
$952$	−1.54956	−0.0502216
$953$	41.5353	1.34546	0.672730	−	0.739888i	$-0.265122\pi$
$953$	41.5353	1.34546	0.672730	+	0.739888i	$0.265122\pi$
$954$	0	0
$955$	0	0
$956$	−2.40790	−0.0778769
$957$	0	0
$958$	−2.36196	−0.0763116
$959$	29.9112	0.965883
$960$	0	0
$961$	−6.95851	−0.224468
$962$	1.87511	0.0604559
$963$	0	0
$964$	−4.31402	−0.138945
$965$	0	0
$966$	0	0
$967$	2.64740	0.0851348	0.0425674	−	0.999094i	$-0.486446\pi$
$967$	2.64740	0.0851348	0.0425674	+	0.999094i	$0.486446\pi$
$968$	−13.3017	−0.427534
$969$	0	0
$970$	0	0
$971$	−39.0736	−1.25393	−0.626966	−	0.779047i	$-0.715703\pi$
$971$	−39.0736	−1.25393	−0.626966	+	0.779047i	$0.715703\pi$
$972$	0	0
$973$	26.3684	0.845333
$974$	−7.79753	−0.249849
$975$	0	0
$976$	26.6079	0.851699
$977$	12.1160	0.387625	0.193813	−	0.981039i	$-0.437915\pi$
$977$	12.1160	0.387625	0.193813	+	0.981039i	$0.437915\pi$
$978$	0	0
$979$	−3.55401	−0.113587
$980$	0	0
$981$	0	0
$982$	−9.88601	−0.315475
$983$	30.1003	0.960051	0.480026	−	0.877254i	$-0.340627\pi$
$983$	30.1003	0.960051	0.480026	+	0.877254i	$0.340627\pi$
$984$	0	0
$985$	0	0
$986$	0.418795	0.0133372
$987$	0	0
$988$	−5.93332	−0.188764
$989$	−26.7021	−0.849078
$990$	0	0
$991$	39.0420	1.24021	0.620104	−	0.784519i	$-0.287090\pi$
$991$	39.0420	1.24021	0.620104	+	0.784519i	$0.287090\pi$
$992$	17.1383	0.544141
$993$	0	0
$994$	−1.67307	−0.0530666
$995$	0	0
$996$	0	0
$997$	−51.3403	−1.62596	−0.812982	−	0.582289i	$-0.802157\pi$
$997$	−51.3403	−1.62596	−0.812982	+	0.582289i	$0.802157\pi$
$998$	−5.21585	−0.165105
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4275.2.a.bd.1.2		3
3.2	odd	2		1425.2.a.x.1.2		3
5.2	odd	4		855.2.c.e.514.3		6
5.3	odd	4		855.2.c.e.514.4		6
5.4	even	2		4275.2.a.bi.1.2		3
15.2	even	4		285.2.c.a.229.4	yes	6
15.8	even	4		285.2.c.a.229.3	✓	6
15.14	odd	2		1425.2.a.s.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.c.a.229.3	✓	6	15.8	even	4
285.2.c.a.229.4	yes	6	15.2	even	4
855.2.c.e.514.3		6	5.2	odd	4
855.2.c.e.514.4		6	5.3	odd	4
1425.2.a.s.1.2		3	15.14	odd	2
1425.2.a.x.1.2		3	3.2	odd	2
4275.2.a.bd.1.2		3	1.1	even	1	trivial
4275.2.a.bi.1.2		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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4275.a (trivial)

\(f(q)\)	\(=\)	\(q-0.311108 q^{2} -1.90321 q^{4} +2.68889 q^{7} +1.21432 q^{8} +0.214320 q^{11} -3.11753 q^{13} -0.836535 q^{14} +3.42864 q^{16} -0.474572 q^{17} -1.00000 q^{19} -0.0666765 q^{22} -2.47457 q^{23} +0.969888 q^{26} -5.11753 q^{28} +2.83654 q^{29} -4.90321 q^{31} -3.49532 q^{32} +0.147643 q^{34} +1.93332 q^{37} +0.311108 q^{38} -5.45875 q^{41} +10.7906 q^{43} -0.407896 q^{44} +0.769859 q^{46} -9.52543 q^{47} +0.230141 q^{49} +5.93332 q^{52} -13.7605 q^{53} +3.26517 q^{56} -0.882468 q^{58} +6.56199 q^{59} +7.76049 q^{61} +1.52543 q^{62} -5.76986 q^{64} -6.85728 q^{67} +0.903212 q^{68} +2.00000 q^{71} +16.6637 q^{73} -0.601472 q^{74} +1.90321 q^{76} +0.576283 q^{77} +1.37778 q^{79} +1.69826 q^{82} -1.65878 q^{83} -3.35704 q^{86} +0.260253 q^{88} -16.5827 q^{89} -8.38271 q^{91} +4.70964 q^{92} +2.96343 q^{94} +9.05731 q^{97} -0.0715987 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + q^{4} + 8 q^{7} - 3 q^{8} - 6 q^{11} + 4 q^{13} + 4 q^{14} - 3 q^{16} - 8 q^{17} - 3 q^{19} - 14 q^{23} - 4 q^{26} - 2 q^{28} + 2 q^{29} - 8 q^{31} + 3 q^{32} - 6 q^{34} + 6 q^{37} + q^{38}+ \cdots + 33 q^{98}+O(q^{100}) \) 3 * q - q^2 + q^4 + 8 * q^7 - 3 * q^8 - 6 * q^11 + 4 * q^13 + 4 * q^14 - 3 * q^16 - 8 * q^17 - 3 * q^19 - 14 * q^23 - 4 * q^26 - 2 * q^28 + 2 * q^29 - 8 * q^31 + 3 * q^32 - 6 * q^34 + 6 * q^37 + q^38 - 10 * q^41 + 6 * q^43 - 8 * q^44 - 4 * q^46 - 22 * q^47 + 7 * q^49 + 18 * q^52 - 8 * q^53 - 10 * q^56 - 16 * q^58 + 6 * q^59 - 10 * q^61 - 2 * q^62 - 11 * q^64 + 6 * q^67 - 4 * q^68 + 6 * q^71 + 10 * q^73 - 22 * q^74 - q^76 - 18 * q^77 + 4 * q^79 + 32 * q^82 + 2 * q^83 - 30 * q^86 + 14 * q^88 - 16 * q^89 + 8 * q^91 - 6 * q^92 + 16 * q^94 + 33 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more