LMFDB - Embedded newform 4275.2.a.bi.1.3

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			1.3
Root			$2.17009$ 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)

$f(q)$	$=$	$q+2.17009 q^{2} +2.70928 q^{4} -0.829914 q^{7} +1.53919 q^{8} -2.53919 q^{11} -4.24846 q^{13} -1.80098 q^{14} -2.07838 q^{16} +1.36910 q^{17} -1.00000 q^{19} -5.51026 q^{22} +3.36910 q^{23} -9.21953 q^{26} -2.24846 q^{28} +3.80098 q^{29} -0.290725 q^{31} -7.58864 q^{32} +2.97107 q^{34} -7.51026 q^{37} -2.17009 q^{38} -10.1412 q^{41} -5.35350 q^{43} -6.87936 q^{44} +7.31124 q^{46} +8.63090 q^{47} -6.31124 q^{49} -11.5103 q^{52} -1.86603 q^{53} -1.27739 q^{56} +8.24846 q^{58} -10.0989 q^{59} -7.86603 q^{61} -0.630898 q^{62} -12.3112 q^{64} -4.15676 q^{67} +3.70928 q^{68} +2.00000 q^{71} +3.57531 q^{73} -16.2979 q^{74} -2.70928 q^{76} +2.10731 q^{77} -2.34017 q^{79} -22.0072 q^{82} -10.3896 q^{83} -11.6176 q^{86} -3.90829 q^{88} +12.0566 q^{89} +3.52586 q^{91} +9.12783 q^{92} +18.7298 q^{94} +18.6875 q^{97} -13.6959 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.17009	1.53448	0.767241	−	0.641358i	$-0.221629\pi$
$2$	2.17009	1.53448	0.767241	+	0.641358i	$0.221629\pi$
$3$	0	0
$3$	0	0
$4$	2.70928	1.35464
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.829914	−0.313678	−0.156839	−	0.987624i	$-0.550130\pi$
$7$	−0.829914	−0.313678	−0.156839	+	0.987624i	$0.550130\pi$
$8$	1.53919	0.544185
$9$	0	0
$10$	0	0
$11$	−2.53919	−0.765594	−0.382797	−	0.923832i	$-0.625039\pi$
$11$	−2.53919	−0.765594	−0.382797	+	0.923832i	$0.625039\pi$
$12$	0	0
$13$	−4.24846	−1.17831	−0.589156	−	0.808019i	$-0.700540\pi$
$13$	−4.24846	−1.17831	−0.589156	+	0.808019i	$0.700540\pi$
$14$	−1.80098	−0.481333
$15$	0	0
$16$	−2.07838	−0.519594
$17$	1.36910	0.332056	0.166028	−	0.986121i	$-0.446906\pi$
$17$	1.36910	0.332056	0.166028	+	0.986121i	$0.446906\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	−5.51026	−1.17479
$23$	3.36910	0.702506	0.351253	−	0.936281i	$-0.385756\pi$
$23$	3.36910	0.702506	0.351253	+	0.936281i	$0.385756\pi$
$24$	0	0
$25$	0	0
$26$	−9.21953	−1.80810
$27$	0	0
$28$	−2.24846	−0.424920
$29$	3.80098	0.705825	0.352913	−	0.935656i	$-0.385191\pi$
$29$	3.80098	0.705825	0.352913	+	0.935656i	$0.385191\pi$
$30$	0	0
$31$	−0.290725	−0.0522157	−0.0261078	−	0.999659i	$-0.508311\pi$
$31$	−0.290725	−0.0522157	−0.0261078	+	0.999659i	$0.508311\pi$
$32$	−7.58864	−1.34149
$33$	0	0
$34$	2.97107	0.509534
$35$	0	0
$36$	0	0
$37$	−7.51026	−1.23468	−0.617340	−	0.786697i	$-0.711790\pi$
$37$	−7.51026	−1.23468	−0.617340	+	0.786697i	$0.711790\pi$
$38$	−2.17009	−0.352035
$39$	0	0
$40$	0	0
$41$	−10.1412	−1.58378	−0.791891	−	0.610662i	$-0.790903\pi$
$41$	−10.1412	−1.58378	−0.791891	+	0.610662i	$0.790903\pi$
$42$	0	0
$43$	−5.35350	−0.816402	−0.408201	−	0.912892i	$-0.633844\pi$
$43$	−5.35350	−0.816402	−0.408201	+	0.912892i	$0.633844\pi$
$44$	−6.87936	−1.03710
$45$	0	0
$46$	7.31124	1.07798
$47$	8.63090	1.25895	0.629473	−	0.777022i	$-0.283271\pi$
$47$	8.63090	1.25895	0.629473	+	0.777022i	$0.283271\pi$
$48$	0	0
$49$	−6.31124	−0.901606
$50$	0	0
$51$	0	0
$52$	−11.5103	−1.59619
$53$	−1.86603	−0.256319	−0.128160	−	0.991754i	$-0.540907\pi$
$53$	−1.86603	−0.256319	−0.128160	+	0.991754i	$0.540907\pi$
$54$	0	0
$55$	0	0
$56$	−1.27739	−0.170699
$57$	0	0
$58$	8.24846	1.08308
$59$	−10.0989	−1.31476	−0.657382	−	0.753557i	$-0.728336\pi$
$59$	−10.0989	−1.31476	−0.657382	+	0.753557i	$0.728336\pi$
$60$	0	0
$61$	−7.86603	−1.00714	−0.503571	−	0.863954i	$-0.667981\pi$
$61$	−7.86603	−1.00714	−0.503571	+	0.863954i	$0.667981\pi$
$62$	−0.630898	−0.0801241
$63$	0	0
$64$	−12.3112	−1.53891
$65$	0	0
$66$	0	0
$67$	−4.15676	−0.507829	−0.253914	−	0.967227i	$-0.581718\pi$
$67$	−4.15676	−0.507829	−0.253914	+	0.967227i	$0.581718\pi$
$68$	3.70928	0.449816
$69$	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
$73$	3.57531	0.418458	0.209229	−	0.977867i	$-0.432905\pi$
$73$	3.57531	0.418458	0.209229	+	0.977867i	$0.432905\pi$
$74$	−16.2979	−1.89459
$75$	0	0
$76$	−2.70928	−0.310775
$77$	2.10731	0.240150
$78$	0	0
$79$	−2.34017	−0.263290	−0.131645	−	0.991297i	$-0.542026\pi$
$79$	−2.34017	−0.263290	−0.131645	+	0.991297i	$0.542026\pi$
$80$	0	0
$81$	0	0
$82$	−22.0072	−2.43029
$83$	−10.3896	−1.14041	−0.570205	−	0.821503i	$-0.693136\pi$
$83$	−10.3896	−1.14041	−0.570205	+	0.821503i	$0.693136\pi$
$84$	0	0
$85$	0	0
$86$	−11.6176	−1.25275
$87$	0	0
$88$	−3.90829	−0.416625
$89$	12.0566	1.27800	0.639000	−	0.769206i	$-0.279348\pi$
$89$	12.0566	1.27800	0.639000	+	0.769206i	$0.279348\pi$
$90$	0	0
$91$	3.52586	0.369610
$92$	9.12783	0.951642
$93$	0	0
$94$	18.7298	1.93183
$95$	0	0
$96$	0	0
$97$	18.6875	1.89743	0.948716	−	0.316130i	$-0.102384\pi$
$97$	18.6875	1.89743	0.948716	+	0.316130i	$0.102384\pi$
$98$	−13.6959	−1.38350
$99$	0	0
$100$	0	0
$101$	−0.581449	−0.0578564	−0.0289282	−	0.999581i	$-0.509209\pi$
$101$	−0.581449	−0.0578564	−0.0289282	+	0.999581i	$0.509209\pi$
$102$	0	0
$103$	−16.0989	−1.58627	−0.793136	−	0.609045i	$-0.791553\pi$
$103$	−16.0989	−1.58627	−0.793136	+	0.609045i	$0.791553\pi$
$104$	−6.53919	−0.641220
$105$	0	0
$106$	−4.04945	−0.393317
$107$	−14.6537	−1.41663	−0.708313	−	0.705899i	$-0.750543\pi$
$107$	−14.6537	−1.41663	−0.708313	+	0.705899i	$0.750543\pi$
$108$	0	0
$109$	−17.4186	−1.66839	−0.834197	−	0.551466i	$-0.814069\pi$
$109$	−17.4186	−1.66839	−0.834197	+	0.551466i	$0.814069\pi$
$110$	0	0
$111$	0	0
$112$	1.72487	0.162985
$113$	6.94441	0.653275	0.326638	−	0.945150i	$-0.394084\pi$
$113$	6.94441	0.653275	0.326638	+	0.945150i	$0.394084\pi$
$114$	0	0
$115$	0	0
$116$	10.2979	0.956137
$117$	0	0
$118$	−21.9155	−2.01748
$119$	−1.13624	−0.104159
$120$	0	0
$121$	−4.55252	−0.413865
$122$	−17.0700	−1.54544
$123$	0	0
$124$	−0.787653	−0.0707333
$125$	0	0
$126$	0	0
$127$	22.4124	1.98878	0.994390	−	0.105778i	$-0.0337332\pi$
$127$	22.4124	1.98878	0.994390	+	0.105778i	$0.0337332\pi$
$128$	−11.5392	−1.01993
$129$	0	0
$130$	0	0
$131$	7.37629	0.644469	0.322235	−	0.946660i	$-0.395566\pi$
$131$	7.37629	0.644469	0.322235	+	0.946660i	$0.395566\pi$
$132$	0	0
$133$	0.829914	0.0719626
$134$	−9.02052	−0.779254
$135$	0	0
$136$	2.10731	0.180700
$137$	22.1978	1.89649	0.948243	−	0.317546i	$-0.102859\pi$
$137$	22.1978	1.89649	0.948243	+	0.317546i	$0.102859\pi$
$138$	0	0
$139$	0.581449	0.0493179	0.0246589	−	0.999696i	$-0.492150\pi$
$139$	0.581449	0.0493179	0.0246589	+	0.999696i	$0.492150\pi$
$140$	0	0
$141$	0	0
$142$	4.34017	0.364219
$143$	10.7877	0.902109
$144$	0	0
$145$	0	0
$146$	7.75872	0.642117
$147$	0	0
$148$	−20.3474	−1.67254
$149$	14.0989	1.15503	0.577513	−	0.816381i	$-0.304023\pi$
$149$	14.0989	1.15503	0.577513	+	0.816381i	$0.304023\pi$
$150$	0	0
$151$	−19.3112	−1.57153	−0.785763	−	0.618527i	$-0.787730\pi$
$151$	−19.3112	−1.57153	−0.785763	+	0.618527i	$0.787730\pi$
$152$	−1.53919	−0.124845
$153$	0	0
$154$	4.57304	0.368506
$155$	0	0
$156$	0	0
$157$	4.92162	0.392788	0.196394	−	0.980525i	$-0.437077\pi$
$157$	4.92162	0.392788	0.196394	+	0.980525i	$0.437077\pi$
$158$	−5.07838	−0.404014
$159$	0	0
$160$	0	0
$161$	−2.79606	−0.220361
$162$	0	0
$163$	−20.3474	−1.59373	−0.796864	−	0.604159i	$-0.793509\pi$
$163$	−20.3474	−1.59373	−0.796864	+	0.604159i	$0.793509\pi$
$164$	−27.4752	−2.14545
$165$	0	0
$166$	−22.5464	−1.74994
$167$	2.87217	0.222255	0.111128	−	0.993806i	$-0.464554\pi$
$167$	2.87217	0.222255	0.111128	+	0.993806i	$0.464554\pi$
$168$	0	0
$169$	5.04945	0.388419
$170$	0	0
$171$	0	0
$172$	−14.5041	−1.10593
$173$	4.78765	0.363999	0.181999	−	0.983299i	$-0.441743\pi$
$173$	4.78765	0.363999	0.181999	+	0.983299i	$0.441743\pi$
$174$	0	0
$175$	0	0
$176$	5.27739	0.397799
$177$	0	0
$178$	26.1639	1.96107
$179$	15.5753	1.16415	0.582077	−	0.813134i	$-0.302240\pi$
$179$	15.5753	1.16415	0.582077	+	0.813134i	$0.302240\pi$
$180$	0	0
$181$	11.6742	0.867737	0.433868	−	0.900976i	$-0.357148\pi$
$181$	11.6742	0.867737	0.433868	+	0.900976i	$0.357148\pi$
$182$	7.65142	0.567161
$183$	0	0
$184$	5.18568	0.382294
$185$	0	0
$186$	0	0
$187$	−3.47641	−0.254220
$188$	23.3835	1.70542
$189$	0	0
$190$	0	0
$191$	22.9516	1.66072	0.830360	−	0.557228i	$-0.188135\pi$
$191$	22.9516	1.66072	0.830360	+	0.557228i	$0.188135\pi$
$192$	0	0
$193$	2.98667	0.214985	0.107493	−	0.994206i	$-0.465718\pi$
$193$	2.98667	0.214985	0.107493	+	0.994206i	$0.465718\pi$
$194$	40.5536	2.91158
$195$	0	0
$196$	−17.0989	−1.22135
$197$	2.94441	0.209780	0.104890	−	0.994484i	$-0.466551\pi$
$197$	2.94441	0.209780	0.104890	+	0.994484i	$0.466551\pi$
$198$	0	0
$199$	16.4657	1.16722	0.583612	−	0.812032i	$-0.301639\pi$
$199$	16.4657	1.16722	0.583612	+	0.812032i	$0.301639\pi$
$200$	0	0
$201$	0	0
$202$	−1.26180	−0.0887796
$203$	−3.15449	−0.221402
$204$	0	0
$205$	0	0
$206$	−34.9360	−2.43411
$207$	0	0
$208$	8.82991	0.612244
$209$	2.53919	0.175639
$210$	0	0
$211$	2.65368	0.182687	0.0913436	−	0.995819i	$-0.470884\pi$
$211$	2.65368	0.182687	0.0913436	+	0.995819i	$0.470884\pi$
$212$	−5.05559	−0.347219
$213$	0	0
$214$	−31.7998	−2.17379
$215$	0	0
$216$	0	0
$217$	0.241276	0.0163789
$218$	−37.7998	−2.56012
$219$	0	0
$220$	0	0
$221$	−5.81658	−0.391266
$222$	0	0
$223$	−9.57531	−0.641210	−0.320605	−	0.947213i	$-0.603886\pi$
$223$	−9.57531	−0.641210	−0.320605	+	0.947213i	$0.603886\pi$
$224$	6.29791	0.420797
$225$	0	0
$226$	15.0700	1.00244
$227$	13.7093	0.909917	0.454958	−	0.890513i	$-0.349654\pi$
$227$	13.7093	0.909917	0.454958	+	0.890513i	$0.349654\pi$
$228$	0	0
$229$	0.729794	0.0482262	0.0241131	−	0.999709i	$-0.492324\pi$
$229$	0.729794	0.0482262	0.0241131	+	0.999709i	$0.492324\pi$
$230$	0	0
$231$	0	0
$232$	5.85043	0.384100
$233$	−5.23513	−0.342965	−0.171482	−	0.985187i	$-0.554856\pi$
$233$	−5.23513	−0.342965	−0.171482	+	0.985187i	$0.554856\pi$
$234$	0	0
$235$	0	0
$236$	−27.3607	−1.78103
$237$	0	0
$238$	−2.46573	−0.159830
$239$	−3.27739	−0.211997	−0.105998	−	0.994366i	$-0.533804\pi$
$239$	−3.27739	−0.211997	−0.105998	+	0.994366i	$0.533804\pi$
$240$	0	0
$241$	−20.0410	−1.29096	−0.645478	−	0.763779i	$-0.723342\pi$
$241$	−20.0410	−1.29096	−0.645478	+	0.763779i	$0.723342\pi$
$242$	−9.87936	−0.635069
$243$	0	0
$244$	−21.3112	−1.36431
$245$	0	0
$246$	0	0
$247$	4.24846	0.270323
$248$	−0.447480	−0.0284150
$249$	0	0
$250$	0	0
$251$	−6.11450	−0.385944	−0.192972	−	0.981204i	$-0.561813\pi$
$251$	−6.11450	−0.385944	−0.192972	+	0.981204i	$0.561813\pi$
$252$	0	0
$253$	−8.55479	−0.537835
$254$	48.6369	3.05175
$255$	0	0
$256$	−0.418551	−0.0261594
$257$	7.22672	0.450791	0.225395	−	0.974267i	$-0.427633\pi$
$257$	7.22672	0.450791	0.225395	+	0.974267i	$0.427633\pi$
$258$	0	0
$259$	6.23287	0.387291
$260$	0	0
$261$	0	0
$262$	16.0072	0.988927
$263$	14.4741	0.892514	0.446257	−	0.894905i	$-0.352757\pi$
$263$	14.4741	0.892514	0.446257	+	0.894905i	$0.352757\pi$
$264$	0	0
$265$	0	0
$266$	1.80098	0.110425
$267$	0	0
$268$	−11.2618	−0.687924
$269$	−24.2401	−1.47794	−0.738971	−	0.673737i	$-0.764688\pi$
$269$	−24.2401	−1.47794	−0.738971	+	0.673737i	$0.764688\pi$
$270$	0	0
$271$	20.9939	1.27529	0.637643	−	0.770332i	$-0.279910\pi$
$271$	20.9939	1.27529	0.637643	+	0.770332i	$0.279910\pi$
$272$	−2.84551	−0.172535
$273$	0	0
$274$	48.1711	2.91012
$275$	0	0
$276$	0	0
$277$	6.52359	0.391965	0.195982	−	0.980607i	$-0.437210\pi$
$277$	6.52359	0.391965	0.195982	+	0.980607i	$0.437210\pi$
$278$	1.26180	0.0756774
$279$	0	0
$280$	0	0
$281$	−15.6442	−0.933256	−0.466628	−	0.884454i	$-0.654531\pi$
$281$	−15.6442	−0.933256	−0.466628	+	0.884454i	$0.654531\pi$
$282$	0	0
$283$	26.6875	1.58641	0.793205	−	0.608955i	$-0.208411\pi$
$283$	26.6875	1.58641	0.793205	+	0.608955i	$0.208411\pi$
$284$	5.41855	0.321532
$285$	0	0
$286$	23.4101	1.38427
$287$	8.41628	0.496797
$288$	0	0
$289$	−15.1256	−0.889739
$290$	0	0
$291$	0	0
$292$	9.68649	0.566859
$293$	−18.8599	−1.10181	−0.550903	−	0.834569i	$-0.685717\pi$
$293$	−18.8599	−1.10181	−0.550903	+	0.834569i	$0.685717\pi$
$294$	0	0
$295$	0	0
$296$	−11.5597	−0.671894
$297$	0	0
$298$	30.5958	1.77237
$299$	−14.3135	−0.827772
$300$	0	0
$301$	4.44295	0.256087
$302$	−41.9071	−2.41148
$303$	0	0
$304$	2.07838	0.119203
$305$	0	0
$306$	0	0
$307$	−1.97334	−0.112624	−0.0563122	−	0.998413i	$-0.517934\pi$
$307$	−1.97334	−0.112624	−0.0563122	+	0.998413i	$0.517934\pi$
$308$	5.70928	0.325316
$309$	0	0
$310$	0	0
$311$	12.0566	0.683669	0.341835	−	0.939760i	$-0.388952\pi$
$311$	12.0566	0.683669	0.341835	+	0.939760i	$0.388952\pi$
$312$	0	0
$313$	−7.75872	−0.438549	−0.219274	−	0.975663i	$-0.570369\pi$
$313$	−7.75872	−0.438549	−0.219274	+	0.975663i	$0.570369\pi$
$314$	10.6803	0.602727
$315$	0	0
$316$	−6.34017	−0.356663
$317$	−33.2001	−1.86470	−0.932351	−	0.361555i	$-0.882246\pi$
$317$	−33.2001	−1.86470	−0.932351	+	0.361555i	$0.882246\pi$
$318$	0	0
$319$	−9.65142	−0.540376
$320$	0	0
$321$	0	0
$322$	−6.06770	−0.338140
$323$	−1.36910	−0.0761789
$324$	0	0
$325$	0	0
$326$	−44.1555	−2.44555
$327$	0	0
$328$	−15.6092	−0.861871
$329$	−7.16290	−0.394903
$330$	0	0
$331$	−8.42082	−0.462850	−0.231425	−	0.972853i	$-0.574339\pi$
$331$	−8.42082	−0.462850	−0.231425	+	0.972853i	$0.574339\pi$
$332$	−28.1483	−1.54484
$333$	0	0
$334$	6.23287	0.341047
$335$	0	0
$336$	0	0
$337$	27.2956	1.48689	0.743444	−	0.668798i	$-0.233191\pi$
$337$	27.2956	1.48689	0.743444	+	0.668798i	$0.233191\pi$
$338$	10.9577	0.596022
$339$	0	0
$340$	0	0
$341$	0.738205	0.0399760
$342$	0	0
$343$	11.0472	0.596492
$344$	−8.24005	−0.444274
$345$	0	0
$346$	10.3896	0.558550
$347$	−1.49466	−0.0802376	−0.0401188	−	0.999195i	$-0.512774\pi$
$347$	−1.49466	−0.0802376	−0.0401188	+	0.999195i	$0.512774\pi$
$348$	0	0
$349$	−15.7587	−0.843545	−0.421773	−	0.906702i	$-0.638592\pi$
$349$	−15.7587	−0.843545	−0.421773	+	0.906702i	$0.638592\pi$
$350$	0	0
$351$	0	0
$352$	19.2690	1.02704
$353$	5.33403	0.283902	0.141951	−	0.989874i	$-0.454662\pi$
$353$	5.33403	0.283902	0.141951	+	0.989874i	$0.454662\pi$
$354$	0	0
$355$	0	0
$356$	32.6647	1.73123
$357$	0	0
$358$	33.7998	1.78637
$359$	−28.4501	−1.50154	−0.750770	−	0.660563i	$-0.770317\pi$
$359$	−28.4501	−1.50154	−0.750770	+	0.660563i	$0.770317\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	25.3340	1.33153
$363$	0	0
$364$	9.55252	0.500688
$365$	0	0
$366$	0	0
$367$	−13.3535	−0.697047	−0.348524	−	0.937300i	$-0.613317\pi$
$367$	−13.3535	−0.697047	−0.348524	+	0.937300i	$0.613317\pi$
$368$	−7.00227	−0.365018
$369$	0	0
$370$	0	0
$371$	1.54864	0.0804016
$372$	0	0
$373$	35.1533	1.82017	0.910084	−	0.414425i	$-0.136017\pi$
$373$	35.1533	1.82017	0.910084	+	0.414425i	$0.136017\pi$
$374$	−7.54411	−0.390097
$375$	0	0
$376$	13.2846	0.685100
$377$	−16.1483	−0.831682
$378$	0	0
$379$	−31.9916	−1.64330	−0.821649	−	0.569994i	$-0.806945\pi$
$379$	−31.9916	−1.64330	−0.821649	+	0.569994i	$0.806945\pi$
$380$	0	0
$381$	0	0
$382$	49.8069	2.54835
$383$	−5.94214	−0.303629	−0.151815	−	0.988409i	$-0.548512\pi$
$383$	−5.94214	−0.303629	−0.151815	+	0.988409i	$0.548512\pi$
$384$	0	0
$385$	0	0
$386$	6.48133	0.329891
$387$	0	0
$388$	50.6297	2.57033
$389$	−21.4908	−1.08963	−0.544813	−	0.838558i	$-0.683399\pi$
$389$	−21.4908	−1.08963	−0.544813	+	0.838558i	$0.683399\pi$
$390$	0	0
$391$	4.61265	0.233272
$392$	−9.71420	−0.490641
$393$	0	0
$394$	6.38962	0.321904
$395$	0	0
$396$	0	0
$397$	−24.7792	−1.24363	−0.621817	−	0.783162i	$-0.713605\pi$
$397$	−24.7792	−1.24363	−0.621817	+	0.783162i	$0.713605\pi$
$398$	35.7321	1.79109
$399$	0	0
$400$	0	0
$401$	−29.2606	−1.46120	−0.730602	−	0.682804i	$-0.760760\pi$
$401$	−29.2606	−1.46120	−0.730602	+	0.682804i	$0.760760\pi$
$402$	0	0
$403$	1.23513	0.0615264
$404$	−1.57531	−0.0783744
$405$	0	0
$406$	−6.84551	−0.339737
$407$	19.0700	0.945263
$408$	0	0
$409$	3.44521	0.170355	0.0851774	−	0.996366i	$-0.472854\pi$
$409$	3.44521	0.170355	0.0851774	+	0.996366i	$0.472854\pi$
$410$	0	0
$411$	0	0
$412$	−43.6163	−2.14882
$413$	8.38121	0.412412
$414$	0	0
$415$	0	0
$416$	32.2401	1.58070
$417$	0	0
$418$	5.51026	0.269516
$419$	−36.4813	−1.78223	−0.891115	−	0.453778i	$-0.850076\pi$
$419$	−36.4813	−1.78223	−0.891115	+	0.453778i	$0.850076\pi$
$420$	0	0
$421$	−7.97334	−0.388597	−0.194298	−	0.980942i	$-0.562243\pi$
$421$	−7.97334	−0.388597	−0.194298	+	0.980942i	$0.562243\pi$
$422$	5.75872	0.280330
$423$	0	0
$424$	−2.87217	−0.139485
$425$	0	0
$426$	0	0
$427$	6.52813	0.315918
$428$	−39.7009	−1.91901
$429$	0	0
$430$	0	0
$431$	13.7587	0.662734	0.331367	−	0.943502i	$-0.392490\pi$
$431$	13.7587	0.662734	0.331367	+	0.943502i	$0.392490\pi$
$432$	0	0
$433$	5.24232	0.251930	0.125965	−	0.992035i	$-0.459797\pi$
$433$	5.24232	0.251930	0.125965	+	0.992035i	$0.459797\pi$
$434$	0.523590	0.0251331
$435$	0	0
$436$	−47.1917	−2.26007
$437$	−3.36910	−0.161166
$438$	0	0
$439$	36.5646	1.74513	0.872567	−	0.488494i	$-0.162454\pi$
$439$	36.5646	1.74513	0.872567	+	0.488494i	$0.162454\pi$
$440$	0	0
$441$	0	0
$442$	−12.6225	−0.600390
$443$	32.9132	1.56375	0.781877	−	0.623433i	$-0.214263\pi$
$443$	32.9132	1.56375	0.781877	+	0.623433i	$0.214263\pi$
$444$	0	0
$445$	0	0
$446$	−20.7792	−0.983926
$447$	0	0
$448$	10.2173	0.482721
$449$	−22.7226	−1.07235	−0.536173	−	0.844108i	$-0.680131\pi$
$449$	−22.7226	−1.07235	−0.536173	+	0.844108i	$0.680131\pi$
$450$	0	0
$451$	25.7503	1.21253
$452$	18.8143	0.884951
$453$	0	0
$454$	29.7503	1.39625
$455$	0	0
$456$	0	0
$457$	5.13624	0.240263	0.120132	−	0.992758i	$-0.461668\pi$
$457$	5.13624	0.240263	0.120132	+	0.992758i	$0.461668\pi$
$458$	1.58372	0.0740022
$459$	0	0
$460$	0	0
$461$	34.1978	1.59275	0.796375	−	0.604803i	$-0.206748\pi$
$461$	34.1978	1.59275	0.796375	+	0.604803i	$0.206748\pi$
$462$	0	0
$463$	−6.56198	−0.304961	−0.152480	−	0.988306i	$-0.548726\pi$
$463$	−6.56198	−0.304961	−0.152480	+	0.988306i	$0.548726\pi$
$464$	−7.89988	−0.366743
$465$	0	0
$466$	−11.3607	−0.526274
$467$	21.7548	1.00669	0.503347	−	0.864084i	$-0.332102\pi$
$467$	21.7548	1.00669	0.503347	+	0.864084i	$0.332102\pi$
$468$	0	0
$469$	3.44975	0.159295
$470$	0	0
$471$	0	0
$472$	−15.5441	−0.715476
$473$	13.5936	0.625032
$474$	0	0
$475$	0	0
$476$	−3.07838	−0.141097
$477$	0	0
$478$	−7.11223	−0.325306
$479$	1.12064	0.0512033	0.0256016	−	0.999672i	$-0.491850\pi$
$479$	1.12064	0.0512033	0.0256016	+	0.999672i	$0.491850\pi$
$480$	0	0
$481$	31.9071	1.45484
$482$	−43.4908	−1.98095
$483$	0	0
$484$	−12.3340	−0.560638
$485$	0	0
$486$	0	0
$487$	28.6369	1.29766	0.648830	−	0.760933i	$-0.275259\pi$
$487$	28.6369	1.29766	0.648830	+	0.760933i	$0.275259\pi$
$488$	−12.1073	−0.548072
$489$	0	0
$490$	0	0
$491$	−26.1867	−1.18179	−0.590895	−	0.806748i	$-0.701225\pi$
$491$	−26.1867	−1.18179	−0.590895	+	0.806748i	$0.701225\pi$
$492$	0	0
$493$	5.20394	0.234374
$494$	9.21953	0.414806
$495$	0	0
$496$	0.604236	0.0271310
$497$	−1.65983	−0.0744534
$498$	0	0
$499$	−7.05172	−0.315678	−0.157839	−	0.987465i	$-0.550453\pi$
$499$	−7.05172	−0.315678	−0.157839	+	0.987465i	$0.550453\pi$
$500$	0	0
$501$	0	0
$502$	−13.2690	−0.592224
$503$	20.4619	0.912349	0.456175	−	0.889890i	$-0.349219\pi$
$503$	20.4619	0.912349	0.456175	+	0.889890i	$0.349219\pi$
$504$	0	0
$505$	0	0
$506$	−18.5646	−0.825298
$507$	0	0
$508$	60.7214	2.69408
$509$	1.82765	0.0810090	0.0405045	−	0.999179i	$-0.487103\pi$
$509$	1.82765	0.0810090	0.0405045	+	0.999179i	$0.487103\pi$
$510$	0	0
$511$	−2.96719	−0.131261
$512$	22.1701	0.979789
$513$	0	0
$514$	15.6826	0.691730
$515$	0	0
$516$	0	0
$517$	−21.9155	−0.963842
$518$	13.5259	0.594292
$519$	0	0
$520$	0	0
$521$	−19.1084	−0.837152	−0.418576	−	0.908182i	$-0.637471\pi$
$521$	−19.1084	−0.837152	−0.418576	+	0.908182i	$0.637471\pi$
$522$	0	0
$523$	−3.44521	−0.150649	−0.0753243	−	0.997159i	$-0.523999\pi$
$523$	−3.44521	−0.150649	−0.0753243	+	0.997159i	$0.523999\pi$
$524$	19.9844	0.873023
$525$	0	0
$526$	31.4101	1.36955
$527$	−0.398032	−0.0173385
$528$	0	0
$529$	−11.6491	−0.506485
$530$	0	0
$531$	0	0
$532$	2.24846	0.0974833
$533$	43.0843	1.86619
$534$	0	0
$535$	0	0
$536$	−6.39803	−0.276353
$537$	0	0
$538$	−52.6030	−2.26788
$539$	16.0254	0.690265
$540$	0	0
$541$	−2.48255	−0.106733	−0.0533666	−	0.998575i	$-0.516995\pi$
$541$	−2.48255	−0.106733	−0.0533666	+	0.998575i	$0.516995\pi$
$542$	45.5585	1.95690
$543$	0	0
$544$	−10.3896	−0.445451
$545$	0	0
$546$	0	0
$547$	20.6491	0.882894	0.441447	−	0.897287i	$-0.354465\pi$
$547$	20.6491	0.882894	0.441447	+	0.897287i	$0.354465\pi$
$548$	60.1399	2.56905
$549$	0	0
$550$	0	0
$551$	−3.80098	−0.161927
$552$	0	0
$553$	1.94214	0.0825882
$554$	14.1568	0.601463
$555$	0	0
$556$	1.57531	0.0668079
$557$	−31.2618	−1.32460	−0.662302	−	0.749237i	$-0.730421\pi$
$557$	−31.2618	−1.32460	−0.662302	+	0.749237i	$0.730421\pi$
$558$	0	0
$559$	22.7442	0.961976
$560$	0	0
$561$	0	0
$562$	−33.9493	−1.43207
$563$	−6.29072	−0.265122	−0.132561	−	0.991175i	$-0.542320\pi$
$563$	−6.29072	−0.265122	−0.132561	+	0.991175i	$0.542320\pi$
$564$	0	0
$565$	0	0
$566$	57.9143	2.43432
$567$	0	0
$568$	3.07838	0.129166
$569$	−17.9265	−0.751520	−0.375760	−	0.926717i	$-0.622618\pi$
$569$	−17.9265	−0.751520	−0.375760	+	0.926717i	$0.622618\pi$
$570$	0	0
$571$	38.3402	1.60449	0.802243	−	0.596997i	$-0.203640\pi$
$571$	38.3402	1.60449	0.802243	+	0.596997i	$0.203640\pi$
$572$	29.2267	1.22203
$573$	0	0
$574$	18.2641	0.762327
$575$	0	0
$576$	0	0
$577$	18.6947	0.778271	0.389136	−	0.921180i	$-0.372774\pi$
$577$	18.6947	0.778271	0.389136	+	0.921180i	$0.372774\pi$
$578$	−32.8238	−1.36529
$579$	0	0
$580$	0	0
$581$	8.62249	0.357721
$582$	0	0
$583$	4.73820	0.196236
$584$	5.50307	0.227719
$585$	0	0
$586$	−40.9276	−1.69070
$587$	42.0060	1.73377	0.866886	−	0.498507i	$-0.166118\pi$
$587$	42.0060	1.73377	0.866886	+	0.498507i	$0.166118\pi$
$588$	0	0
$589$	0.290725	0.0119791
$590$	0	0
$591$	0	0
$592$	15.6092	0.641532
$593$	10.2679	0.421654	0.210827	−	0.977523i	$-0.432384\pi$
$593$	10.2679	0.421654	0.210827	+	0.977523i	$0.432384\pi$
$594$	0	0
$595$	0	0
$596$	38.1978	1.56464
$597$	0	0
$598$	−31.0616	−1.27020
$599$	−3.56093	−0.145496	−0.0727478	−	0.997350i	$-0.523177\pi$
$599$	−3.56093	−0.145496	−0.0727478	+	0.997350i	$0.523177\pi$
$600$	0	0
$601$	−38.0288	−1.55123	−0.775613	−	0.631209i	$-0.782559\pi$
$601$	−38.0288	−1.55123	−0.775613	+	0.631209i	$0.782559\pi$
$602$	9.64158	0.392961
$603$	0	0
$604$	−52.3195	−2.12885
$605$	0	0
$606$	0	0
$607$	−12.2868	−0.498708	−0.249354	−	0.968412i	$-0.580218\pi$
$607$	−12.2868	−0.498708	−0.249354	+	0.968412i	$0.580218\pi$
$608$	7.58864	0.307760
$609$	0	0
$610$	0	0
$611$	−36.6681	−1.48343
$612$	0	0
$613$	11.2762	0.455440	0.227720	−	0.973727i	$-0.426873\pi$
$613$	11.2762	0.455440	0.227720	+	0.973727i	$0.426873\pi$
$614$	−4.28231	−0.172820
$615$	0	0
$616$	3.24354	0.130686
$617$	−19.6248	−0.790063	−0.395031	−	0.918668i	$-0.629266\pi$
$617$	−19.6248	−0.790063	−0.395031	+	0.918668i	$0.629266\pi$
$618$	0	0
$619$	−27.2183	−1.09400	−0.546998	−	0.837134i	$-0.684230\pi$
$619$	−27.2183	−1.09400	−0.546998	+	0.837134i	$0.684230\pi$
$620$	0	0
$621$	0	0
$622$	26.1639	1.04908
$623$	−10.0060	−0.400881
$624$	0	0
$625$	0	0
$626$	−16.8371	−0.672946
$627$	0	0
$628$	13.3340	0.532086
$629$	−10.2823	−0.409983
$630$	0	0
$631$	−38.9048	−1.54878	−0.774388	−	0.632711i	$-0.781942\pi$
$631$	−38.9048	−1.54878	−0.774388	+	0.632711i	$0.781942\pi$
$632$	−3.60197	−0.143279
$633$	0	0
$634$	−72.0470	−2.86135
$635$	0	0
$636$	0	0
$637$	26.8131	1.06237
$638$	−20.9444	−0.829197
$639$	0	0
$640$	0	0
$641$	−41.9143	−1.65551	−0.827757	−	0.561087i	$-0.810383\pi$
$641$	−41.9143	−1.65551	−0.827757	+	0.561087i	$0.810383\pi$
$642$	0	0
$643$	−25.0712	−0.988711	−0.494356	−	0.869260i	$-0.664596\pi$
$643$	−25.0712	−0.988711	−0.494356	+	0.869260i	$0.664596\pi$
$644$	−7.57531	−0.298509
$645$	0	0
$646$	−2.97107	−0.116895
$647$	−13.7237	−0.539532	−0.269766	−	0.962926i	$-0.586946\pi$
$647$	−13.7237	−0.539532	−0.269766	+	0.962926i	$0.586946\pi$
$648$	0	0
$649$	25.6430	1.00658
$650$	0	0
$651$	0	0
$652$	−55.1266	−2.15892
$653$	−22.3174	−0.873347	−0.436673	−	0.899620i	$-0.643843\pi$
$653$	−22.3174	−0.873347	−0.436673	+	0.899620i	$0.643843\pi$
$654$	0	0
$655$	0	0
$656$	21.0772	0.822925
$657$	0	0
$658$	−15.5441	−0.605972
$659$	36.7070	1.42990	0.714951	−	0.699175i	$-0.246449\pi$
$659$	36.7070	1.42990	0.714951	+	0.699175i	$0.246449\pi$
$660$	0	0
$661$	−25.3340	−0.985380	−0.492690	−	0.870205i	$-0.663986\pi$
$661$	−25.3340	−0.985380	−0.492690	+	0.870205i	$0.663986\pi$
$662$	−18.2739	−0.710235
$663$	0	0
$664$	−15.9916	−0.620594
$665$	0	0
$666$	0	0
$667$	12.8059	0.495847
$668$	7.78151	0.301076
$669$	0	0
$670$	0	0
$671$	19.9733	0.771062
$672$	0	0
$673$	28.6732	1.10527	0.552635	−	0.833424i	$-0.313623\pi$
$673$	28.6732	1.10527	0.552635	+	0.833424i	$0.313623\pi$
$674$	59.2339	2.28160
$675$	0	0
$676$	13.6803	0.526167
$677$	41.4512	1.59310	0.796549	−	0.604574i	$-0.206657\pi$
$677$	41.4512	1.59310	0.796549	+	0.604574i	$0.206657\pi$
$678$	0	0
$679$	−15.5090	−0.595182
$680$	0	0
$681$	0	0
$682$	1.60197	0.0613425
$683$	−32.3279	−1.23699	−0.618496	−	0.785788i	$-0.712258\pi$
$683$	−32.3279	−1.23699	−0.618496	+	0.785788i	$0.712258\pi$
$684$	0	0
$685$	0	0
$686$	23.9733	0.915306
$687$	0	0
$688$	11.1266	0.424198
$689$	7.92777	0.302024
$690$	0	0
$691$	4.13009	0.157116	0.0785581	−	0.996910i	$-0.474968\pi$
$691$	4.13009	0.157116	0.0785581	+	0.996910i	$0.474968\pi$
$692$	12.9711	0.493086
$693$	0	0
$694$	−3.24354	−0.123123
$695$	0	0
$696$	0	0
$697$	−13.8843	−0.525905
$698$	−34.1978	−1.29441
$699$	0	0
$700$	0	0
$701$	−40.3545	−1.52417	−0.762085	−	0.647477i	$-0.775824\pi$
$701$	−40.3545	−1.52417	−0.762085	+	0.647477i	$0.775824\pi$
$702$	0	0
$703$	7.51026	0.283255
$704$	31.2606	1.17818
$705$	0	0
$706$	11.5753	0.435642
$707$	0.482553	0.0181483
$708$	0	0
$709$	43.6802	1.64044	0.820222	−	0.572046i	$-0.193850\pi$
$709$	43.6802	1.64044	0.820222	+	0.572046i	$0.193850\pi$
$710$	0	0
$711$	0	0
$712$	18.5574	0.695469
$713$	−0.979481	−0.0366819
$714$	0	0
$715$	0	0
$716$	42.1978	1.57701
$717$	0	0
$718$	−61.7392	−2.30409
$719$	−11.7431	−0.437945	−0.218972	−	0.975731i	$-0.570270\pi$
$719$	−11.7431	−0.437945	−0.218972	+	0.975731i	$0.570270\pi$
$720$	0	0
$721$	13.3607	0.497578
$722$	2.17009	0.0807623
$723$	0	0
$724$	31.6286	1.17547
$725$	0	0
$726$	0	0
$727$	−43.6358	−1.61836	−0.809181	−	0.587559i	$-0.800089\pi$
$727$	−43.6358	−1.61836	−0.809181	+	0.587559i	$0.800089\pi$
$728$	5.42696	0.201137
$729$	0	0
$730$	0	0
$731$	−7.32950	−0.271091
$732$	0	0
$733$	31.5753	1.16626	0.583130	−	0.812379i	$-0.301828\pi$
$733$	31.5753	1.16626	0.583130	+	0.812379i	$0.301828\pi$
$734$	−28.9783	−1.06961
$735$	0	0
$736$	−25.5669	−0.942408
$737$	10.5548	0.388791
$738$	0	0
$739$	−3.18956	−0.117330	−0.0586649	−	0.998278i	$-0.518684\pi$
$739$	−3.18956	−0.117330	−0.0586649	+	0.998278i	$0.518684\pi$
$740$	0	0
$741$	0	0
$742$	3.36069	0.123375
$743$	2.11118	0.0774518	0.0387259	−	0.999250i	$-0.487670\pi$
$743$	2.11118	0.0774518	0.0387259	+	0.999250i	$0.487670\pi$
$744$	0	0
$745$	0	0
$746$	76.2856	2.79302
$747$	0	0
$748$	−9.41855	−0.344376
$749$	12.1613	0.444364
$750$	0	0
$751$	11.2579	0.410807	0.205404	−	0.978677i	$-0.434149\pi$
$751$	11.2579	0.410807	0.205404	+	0.978677i	$0.434149\pi$
$752$	−17.9383	−0.654141
$753$	0	0
$754$	−35.0433	−1.27620
$755$	0	0
$756$	0	0
$757$	6.86830	0.249633	0.124816	−	0.992180i	$-0.460166\pi$
$757$	6.86830	0.249633	0.124816	+	0.992180i	$0.460166\pi$
$758$	−69.4245	−2.52161
$759$	0	0
$760$	0	0
$761$	45.9565	1.66592	0.832961	−	0.553331i	$-0.186644\pi$
$761$	45.9565	1.66592	0.832961	+	0.553331i	$0.186644\pi$
$762$	0	0
$763$	14.4559	0.523338
$764$	62.1822	2.24967
$765$	0	0
$766$	−12.8950	−0.465914
$767$	42.9048	1.54920
$768$	0	0
$769$	−25.8310	−0.931488	−0.465744	−	0.884919i	$-0.654213\pi$
$769$	−25.8310	−0.931488	−0.465744	+	0.884919i	$0.654213\pi$
$770$	0	0
$771$	0	0
$772$	8.09171	0.291227
$773$	15.7093	0.565023	0.282512	−	0.959264i	$-0.408832\pi$
$773$	15.7093	0.565023	0.282512	+	0.959264i	$0.408832\pi$
$774$	0	0
$775$	0	0
$776$	28.7636	1.03255
$777$	0	0
$778$	−46.6369	−1.67201
$779$	10.1412	0.363345
$780$	0	0
$781$	−5.07838	−0.181719
$782$	10.0098	0.357951
$783$	0	0
$784$	13.1171	0.468470
$785$	0	0
$786$	0	0
$787$	−16.9216	−0.603191	−0.301595	−	0.953436i	$-0.597519\pi$
$787$	−16.9216	−0.603191	−0.301595	+	0.953436i	$0.597519\pi$
$788$	7.97721	0.284176
$789$	0	0
$790$	0	0
$791$	−5.76326	−0.204918
$792$	0	0
$793$	33.4186	1.18673
$794$	−53.7731	−1.90834
$795$	0	0
$796$	44.6102	1.58117
$797$	−48.3728	−1.71345	−0.856726	−	0.515771i	$-0.827505\pi$
$797$	−48.3728	−1.71345	−0.856726	+	0.515771i	$0.827505\pi$
$798$	0	0
$799$	11.8166	0.418041
$800$	0	0
$801$	0	0
$802$	−63.4980	−2.24219
$803$	−9.07838	−0.320369
$804$	0	0
$805$	0	0
$806$	2.68035	0.0944112
$807$	0	0
$808$	−0.894960	−0.0314846
$809$	−4.73820	−0.166586	−0.0832932	−	0.996525i	$-0.526544\pi$
$809$	−4.73820	−0.166586	−0.0832932	+	0.996525i	$0.526544\pi$
$810$	0	0
$811$	−18.5236	−0.650451	−0.325226	−	0.945636i	$-0.605440\pi$
$811$	−18.5236	−0.650451	−0.325226	+	0.945636i	$0.605440\pi$
$812$	−8.54638	−0.299919
$813$	0	0
$814$	41.3835	1.45049
$815$	0	0
$816$	0	0
$817$	5.35350	0.187295
$818$	7.47641	0.261407
$819$	0	0
$820$	0	0
$821$	39.5174	1.37917	0.689584	−	0.724206i	$-0.257793\pi$
$821$	39.5174	1.37917	0.689584	+	0.724206i	$0.257793\pi$
$822$	0	0
$823$	−30.7310	−1.07122	−0.535608	−	0.844467i	$-0.679918\pi$
$823$	−30.7310	−1.07122	−0.535608	+	0.844467i	$0.679918\pi$
$824$	−24.7792	−0.863226
$825$	0	0
$826$	18.1880	0.632840
$827$	−2.70313	−0.0939971	−0.0469986	−	0.998895i	$-0.514966\pi$
$827$	−2.70313	−0.0939971	−0.0469986	+	0.998895i	$0.514966\pi$
$828$	0	0
$829$	25.5318	0.886757	0.443378	−	0.896335i	$-0.353780\pi$
$829$	25.5318	0.886757	0.443378	+	0.896335i	$0.353780\pi$
$830$	0	0
$831$	0	0
$832$	52.3039	1.81331
$833$	−8.64074	−0.299384
$834$	0	0
$835$	0	0
$836$	6.87936	0.237928
$837$	0	0
$838$	−79.1676	−2.73480
$839$	−30.1978	−1.04254	−0.521272	−	0.853391i	$-0.674542\pi$
$839$	−30.1978	−1.04254	−0.521272	+	0.853391i	$0.674542\pi$
$840$	0	0
$841$	−14.5525	−0.501811
$842$	−17.3028	−0.596295
$843$	0	0
$844$	7.18956	0.247475
$845$	0	0
$846$	0	0
$847$	3.77820	0.129820
$848$	3.87832	0.133182
$849$	0	0
$850$	0	0
$851$	−25.3028	−0.867370
$852$	0	0
$853$	24.2823	0.831411	0.415705	−	0.909499i	$-0.363535\pi$
$853$	24.2823	0.831411	0.415705	+	0.909499i	$0.363535\pi$
$854$	14.1666	0.484771
$855$	0	0
$856$	−22.5548	−0.770907
$857$	−18.4475	−0.630154	−0.315077	−	0.949066i	$-0.602030\pi$
$857$	−18.4475	−0.630154	−0.315077	+	0.949066i	$0.602030\pi$
$858$	0	0
$859$	3.77101	0.128665	0.0643326	−	0.997929i	$-0.479508\pi$
$859$	3.77101	0.128665	0.0643326	+	0.997929i	$0.479508\pi$
$860$	0	0
$861$	0	0
$862$	29.8576	1.01695
$863$	−2.07999	−0.0708035	−0.0354018	−	0.999373i	$-0.511271\pi$
$863$	−2.07999	−0.0708035	−0.0354018	+	0.999373i	$0.511271\pi$
$864$	0	0
$865$	0	0
$866$	11.3763	0.386582
$867$	0	0
$868$	0.653684	0.0221875
$869$	5.94214	0.201573
$870$	0	0
$871$	17.6598	0.598380
$872$	−26.8104	−0.907916
$873$	0	0
$874$	−7.31124	−0.247307
$875$	0	0
$876$	0	0
$877$	−38.1015	−1.28660	−0.643299	−	0.765615i	$-0.722435\pi$
$877$	−38.1015	−1.28660	−0.643299	+	0.765615i	$0.722435\pi$
$878$	79.3484	2.67788
$879$	0	0
$880$	0	0
$881$	35.0082	1.17946	0.589729	−	0.807601i	$-0.299235\pi$
$881$	35.0082	1.17946	0.589729	+	0.807601i	$0.299235\pi$
$882$	0	0
$883$	−2.28950	−0.0770479	−0.0385239	−	0.999258i	$-0.512266\pi$
$883$	−2.28950	−0.0770479	−0.0385239	+	0.999258i	$0.512266\pi$
$884$	−15.7587	−0.530023
$885$	0	0
$886$	71.4245	2.39955
$887$	23.1506	0.777321	0.388661	−	0.921381i	$-0.372938\pi$
$887$	23.1506	0.777321	0.388661	+	0.921381i	$0.372938\pi$
$888$	0	0
$889$	−18.6004	−0.623836
$890$	0	0
$891$	0	0
$892$	−25.9421	−0.868607
$893$	−8.63090	−0.288822
$894$	0	0
$895$	0	0
$896$	9.57653	0.319929
$897$	0	0
$898$	−49.3100	−1.64550
$899$	−1.10504	−0.0368551
$900$	0	0
$901$	−2.55479	−0.0851123
$902$	55.8804	1.86061
$903$	0	0
$904$	10.6888	0.355503
$905$	0	0
$906$	0	0
$907$	13.8166	0.458772	0.229386	−	0.973336i	$-0.426328\pi$
$907$	13.8166	0.458772	0.229386	+	0.973336i	$0.426328\pi$
$908$	37.1422	1.23261
$909$	0	0
$910$	0	0
$911$	−13.4063	−0.444169	−0.222085	−	0.975027i	$-0.571286\pi$
$911$	−13.4063	−0.444169	−0.222085	+	0.975027i	$0.571286\pi$
$912$	0	0
$913$	26.3812	0.873091
$914$	11.1461	0.368679
$915$	0	0
$916$	1.97721	0.0653290
$917$	−6.12168	−0.202156
$918$	0	0
$919$	40.1133	1.32321	0.661607	−	0.749850i	$-0.269875\pi$
$919$	40.1133	1.32321	0.661607	+	0.749850i	$0.269875\pi$
$920$	0	0
$921$	0	0
$922$	74.2122	2.44405
$923$	−8.49693	−0.279680
$924$	0	0
$925$	0	0
$926$	−14.2401	−0.467957
$927$	0	0
$928$	−28.8443	−0.946860
$929$	10.1399	0.332680	0.166340	−	0.986068i	$-0.446805\pi$
$929$	10.1399	0.332680	0.166340	+	0.986068i	$0.446805\pi$
$930$	0	0
$931$	6.31124	0.206843
$932$	−14.1834	−0.464593
$933$	0	0
$934$	47.2099	1.54476
$935$	0	0
$936$	0	0
$937$	−9.23060	−0.301551	−0.150775	−	0.988568i	$-0.548177\pi$
$937$	−9.23060	−0.301551	−0.150775	+	0.988568i	$0.548177\pi$
$938$	7.48625	0.244435
$939$	0	0
$940$	0	0
$941$	16.8527	0.549382	0.274691	−	0.961533i	$-0.411424\pi$
$941$	16.8527	0.549382	0.274691	+	0.961533i	$0.411424\pi$
$942$	0	0
$943$	−34.1666	−1.11262
$944$	20.9893	0.683144
$945$	0	0
$946$	29.4992	0.959102
$947$	25.3958	0.825251	0.412626	−	0.910901i	$-0.364612\pi$
$947$	25.3958	0.825251	0.412626	+	0.910901i	$0.364612\pi$
$948$	0	0
$949$	−15.1896	−0.493074
$950$	0	0
$951$	0	0
$952$	−1.74888	−0.0566816
$953$	−24.2595	−0.785843	−0.392922	−	0.919572i	$-0.628536\pi$
$953$	−24.2595	−0.785843	−0.392922	+	0.919572i	$0.628536\pi$
$954$	0	0
$955$	0	0
$956$	−8.87936	−0.287179
$957$	0	0
$958$	2.43188	0.0785705
$959$	−18.4222	−0.594885
$960$	0	0
$961$	−30.9155	−0.997274
$962$	69.2411	2.23242
$963$	0	0
$964$	−54.2967	−1.74878
$965$	0	0
$966$	0	0
$967$	−24.7454	−0.795758	−0.397879	−	0.917438i	$-0.630254\pi$
$967$	−24.7454	−0.795758	−0.397879	+	0.917438i	$0.630254\pi$
$968$	−7.00719	−0.225220
$969$	0	0
$970$	0	0
$971$	31.0082	0.995102	0.497551	−	0.867435i	$-0.334233\pi$
$971$	31.0082	0.995102	0.497551	+	0.867435i	$0.334233\pi$
$972$	0	0
$973$	−0.482553	−0.0154699
$974$	62.1445	1.99124
$975$	0	0
$976$	16.3486	0.523305
$977$	−22.5152	−0.720324	−0.360162	−	0.932890i	$-0.617279\pi$
$977$	−22.5152	−0.720324	−0.360162	+	0.932890i	$0.617279\pi$
$978$	0	0
$979$	−30.6141	−0.978430
$980$	0	0
$981$	0	0
$982$	−56.8275	−1.81344
$983$	39.3667	1.25560	0.627801	−	0.778374i	$-0.283955\pi$
$983$	39.3667	1.25560	0.627801	+	0.778374i	$0.283955\pi$
$984$	0	0
$985$	0	0
$986$	11.2930	0.359642
$987$	0	0
$988$	11.5103	0.366190
$989$	−18.0365	−0.573527
$990$	0	0
$991$	−23.4641	−0.745362	−0.372681	−	0.927959i	$-0.621562\pi$
$991$	−23.4641	−0.745362	−0.372681	+	0.927959i	$0.621562\pi$
$992$	2.20620	0.0700470
$993$	0	0
$994$	−3.60197	−0.114247
$995$	0	0
$996$	0	0
$997$	−41.0493	−1.30004	−0.650022	−	0.759916i	$-0.725240\pi$
$997$	−41.0493	−1.30004	−0.650022	+	0.759916i	$0.725240\pi$
$998$	−15.3028	−0.484403
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4275.2.a.bi.1.3		3
3.2	odd	2		1425.2.a.s.1.1		3
5.2	odd	4		855.2.c.e.514.6		6
5.3	odd	4		855.2.c.e.514.1		6
5.4	even	2		4275.2.a.bd.1.1		3
15.2	even	4		285.2.c.a.229.1	✓	6
15.8	even	4		285.2.c.a.229.6	yes	6
15.14	odd	2		1425.2.a.x.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.c.a.229.1	✓	6	15.2	even	4
285.2.c.a.229.6	yes	6	15.8	even	4
855.2.c.e.514.1		6	5.3	odd	4
855.2.c.e.514.6		6	5.2	odd	4
1425.2.a.s.1.1		3	3.2	odd	2
1425.2.a.x.1.3		3	15.14	odd	2
4275.2.a.bd.1.1		3	5.4	even	2
4275.2.a.bi.1.3		3	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+2.17009 q^{2} +2.70928 q^{4} -0.829914 q^{7} +1.53919 q^{8} -2.53919 q^{11} -4.24846 q^{13} -1.80098 q^{14} -2.07838 q^{16} +1.36910 q^{17} -1.00000 q^{19} -5.51026 q^{22} +3.36910 q^{23} -9.21953 q^{26} -2.24846 q^{28} +3.80098 q^{29} -0.290725 q^{31} -7.58864 q^{32} +2.97107 q^{34} -7.51026 q^{37} -2.17009 q^{38} -10.1412 q^{41} -5.35350 q^{43} -6.87936 q^{44} +7.31124 q^{46} +8.63090 q^{47} -6.31124 q^{49} -11.5103 q^{52} -1.86603 q^{53} -1.27739 q^{56} +8.24846 q^{58} -10.0989 q^{59} -7.86603 q^{61} -0.630898 q^{62} -12.3112 q^{64} -4.15676 q^{67} +3.70928 q^{68} +2.00000 q^{71} +3.57531 q^{73} -16.2979 q^{74} -2.70928 q^{76} +2.10731 q^{77} -2.34017 q^{79} -22.0072 q^{82} -10.3896 q^{83} -11.6176 q^{86} -3.90829 q^{88} +12.0566 q^{89} +3.52586 q^{91} +9.12783 q^{92} +18.7298 q^{94} +18.6875 q^{97} -13.6959 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + q^{4} - 8 q^{7} + 3 q^{8} - 6 q^{11} - 4 q^{13} + 4 q^{14} - 3 q^{16} + 8 q^{17} - 3 q^{19} + 14 q^{23} - 4 q^{26} + 2 q^{28} + 2 q^{29} - 8 q^{31} - 3 q^{32} - 6 q^{34} - 6 q^{37} - q^{38}+ \cdots - 33 q^{98}+O(q^{100}) \) 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

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