LMFDB - Embedded newform 4275.2.a.t.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:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
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			$1.73205$ 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)

$f(q)$	$=$	$q+1.73205 q^{2} +1.00000 q^{4} -0.732051 q^{7} -1.73205 q^{8} -1.26795 q^{11} +2.73205 q^{13} -1.26795 q^{14} -5.00000 q^{16} +1.00000 q^{19} -2.19615 q^{22} -3.46410 q^{23} +4.73205 q^{26} -0.732051 q^{28} +2.19615 q^{29} -4.92820 q^{31} -5.19615 q^{32} -4.19615 q^{37} +1.73205 q^{38} -4.73205 q^{41} +6.19615 q^{43} -1.26795 q^{44} -6.00000 q^{46} +3.46410 q^{47} -6.46410 q^{49} +2.73205 q^{52} -2.53590 q^{53} +1.26795 q^{56} +3.80385 q^{58} -9.46410 q^{59} -13.4641 q^{61} -8.53590 q^{62} +1.00000 q^{64} -8.00000 q^{67} -16.3923 q^{71} +3.07180 q^{73} -7.26795 q^{74} +1.00000 q^{76} +0.928203 q^{77} +2.92820 q^{79} -8.19615 q^{82} +0.928203 q^{83} +10.7321 q^{86} +2.19615 q^{88} -7.26795 q^{89} -2.00000 q^{91} -3.46410 q^{92} +6.00000 q^{94} -4.19615 q^{97} -11.1962 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} + 2 q^{7} - 6 q^{11} + 2 q^{13} - 6 q^{14} - 10 q^{16} + 2 q^{19} + 6 q^{22} + 6 q^{26} + 2 q^{28} - 6 q^{29} + 4 q^{31} + 2 q^{37} - 6 q^{41} + 2 q^{43} - 6 q^{44} - 12 q^{46} - 6 q^{49}+ \cdots - 12 q^{98}+O(q^{100}) $ 2 * q + 2 * q^4 + 2 * q^7 - 6 * q^11 + 2 * q^13 - 6 * q^14 - 10 * q^16 + 2 * q^19 + 6 * q^22 + 6 * q^26 + 2 * q^28 - 6 * q^29 + 4 * q^31 + 2 * q^37 - 6 * q^41 + 2 * q^43 - 6 * q^44 - 12 * q^46 - 6 * q^49 + 2 * q^52 - 12 * q^53 + 6 * q^56 + 18 * q^58 - 12 * q^59 - 20 * q^61 - 24 * q^62 + 2 * q^64 - 16 * q^67 - 12 * q^71 + 20 * q^73 - 18 * q^74 + 2 * q^76 - 12 * q^77 - 8 * q^79 - 6 * q^82 - 12 * q^83 + 18 * q^86 - 6 * q^88 - 18 * q^89 - 4 * q^91 + 12 * q^94 + 2 * q^97 - 12 * 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 $

$2$	1.73205	1.22474	0.612372	−	0.790569i	$-0.290215\pi$
$2$	1.73205	1.22474	0.612372	+	0.790569i	$0.290215\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.732051	−0.276689	−0.138345	−	0.990384i	$-0.544178\pi$
$7$	−0.732051	−0.276689	−0.138345	+	0.990384i	$0.544178\pi$
$8$	−1.73205	−0.612372
$9$	0	0
$10$	0	0
$11$	−1.26795	−0.382301	−0.191151	−	0.981561i	$-0.561222\pi$
$11$	−1.26795	−0.382301	−0.191151	+	0.981561i	$0.561222\pi$
$12$	0	0
$13$	2.73205	0.757735	0.378867	−	0.925451i	$-0.376314\pi$
$13$	2.73205	0.757735	0.378867	+	0.925451i	$0.376314\pi$
$14$	−1.26795	−0.338874
$15$	0	0
$16$	−5.00000	−1.25000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	−2.19615	−0.468221
$23$	−3.46410	−0.722315	−0.361158	−	0.932505i	$-0.617618\pi$
$23$	−3.46410	−0.722315	−0.361158	+	0.932505i	$0.617618\pi$
$24$	0	0
$25$	0	0
$26$	4.73205	0.928032
$27$	0	0
$28$	−0.732051	−0.138345
$29$	2.19615	0.407815	0.203908	−	0.978990i	$-0.434636\pi$
$29$	2.19615	0.407815	0.203908	+	0.978990i	$0.434636\pi$
$30$	0	0
$31$	−4.92820	−0.885131	−0.442566	−	0.896736i	$-0.645932\pi$
$31$	−4.92820	−0.885131	−0.442566	+	0.896736i	$0.645932\pi$
$32$	−5.19615	−0.918559
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.19615	−0.689843	−0.344922	−	0.938631i	$-0.612095\pi$
$37$	−4.19615	−0.689843	−0.344922	+	0.938631i	$0.612095\pi$
$38$	1.73205	0.280976
$39$	0	0
$40$	0	0
$41$	−4.73205	−0.739022	−0.369511	−	0.929226i	$-0.620475\pi$
$41$	−4.73205	−0.739022	−0.369511	+	0.929226i	$0.620475\pi$
$42$	0	0
$43$	6.19615	0.944904	0.472452	−	0.881356i	$-0.343369\pi$
$43$	6.19615	0.944904	0.472452	+	0.881356i	$0.343369\pi$
$44$	−1.26795	−0.191151
$45$	0	0
$46$	−6.00000	−0.884652
$47$	3.46410	0.505291	0.252646	−	0.967559i	$-0.418699\pi$
$47$	3.46410	0.505291	0.252646	+	0.967559i	$0.418699\pi$
$48$	0	0
$49$	−6.46410	−0.923443
$50$	0	0
$51$	0	0
$52$	2.73205	0.378867
$53$	−2.53590	−0.348332	−0.174166	−	0.984716i	$-0.555723\pi$
$53$	−2.53590	−0.348332	−0.174166	+	0.984716i	$0.555723\pi$
$54$	0	0
$55$	0	0
$56$	1.26795	0.169437
$57$	0	0
$58$	3.80385	0.499470
$59$	−9.46410	−1.23212	−0.616061	−	0.787699i	$-0.711272\pi$
$59$	−9.46410	−1.23212	−0.616061	+	0.787699i	$0.711272\pi$
$60$	0	0
$61$	−13.4641	−1.72390	−0.861951	−	0.506992i	$-0.830757\pi$
$61$	−13.4641	−1.72390	−0.861951	+	0.506992i	$0.830757\pi$
$62$	−8.53590	−1.08406
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−16.3923	−1.94541	−0.972704	−	0.232048i	$-0.925457\pi$
$71$	−16.3923	−1.94541	−0.972704	+	0.232048i	$0.925457\pi$
$72$	0	0
$73$	3.07180	0.359527	0.179763	−	0.983710i	$-0.442467\pi$
$73$	3.07180	0.359527	0.179763	+	0.983710i	$0.442467\pi$
$74$	−7.26795	−0.844882
$75$	0	0
$76$	1.00000	0.114708
$77$	0.928203	0.105779
$78$	0	0
$79$	2.92820	0.329449	0.164724	−	0.986340i	$-0.447327\pi$
$79$	2.92820	0.329449	0.164724	+	0.986340i	$0.447327\pi$
$80$	0	0
$81$	0	0
$82$	−8.19615	−0.905114
$83$	0.928203	0.101884	0.0509418	−	0.998702i	$-0.483778\pi$
$83$	0.928203	0.101884	0.0509418	+	0.998702i	$0.483778\pi$
$84$	0	0
$85$	0	0
$86$	10.7321	1.15727
$87$	0	0
$88$	2.19615	0.234111
$89$	−7.26795	−0.770401	−0.385201	−	0.922833i	$-0.625868\pi$
$89$	−7.26795	−0.770401	−0.385201	+	0.922833i	$0.625868\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	−3.46410	−0.361158
$93$	0	0
$94$	6.00000	0.618853
$95$	0	0
$96$	0	0
$97$	−4.19615	−0.426055	−0.213027	−	0.977046i	$-0.568332\pi$
$97$	−4.19615	−0.426055	−0.213027	+	0.977046i	$0.568332\pi$
$98$	−11.1962	−1.13098
$99$	0	0
$100$	0	0
$101$	−10.3923	−1.03407	−0.517036	−	0.855963i	$-0.672965\pi$
$101$	−10.3923	−1.03407	−0.517036	+	0.855963i	$0.672965\pi$
$102$	0	0
$103$	17.8564	1.75944	0.879722	−	0.475488i	$-0.157729\pi$
$103$	17.8564	1.75944	0.879722	+	0.475488i	$0.157729\pi$
$104$	−4.73205	−0.464016
$105$	0	0
$106$	−4.39230	−0.426618
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	6.39230	0.612272	0.306136	−	0.951988i	$-0.400964\pi$
$109$	6.39230	0.612272	0.306136	+	0.951988i	$0.400964\pi$
$110$	0	0
$111$	0	0
$112$	3.66025	0.345861
$113$	5.07180	0.477115	0.238557	−	0.971128i	$-0.423326\pi$
$113$	5.07180	0.477115	0.238557	+	0.971128i	$0.423326\pi$
$114$	0	0
$115$	0	0
$116$	2.19615	0.203908
$117$	0	0
$118$	−16.3923	−1.50903
$119$	0	0
$120$	0	0
$121$	−9.39230	−0.853846
$122$	−23.3205	−2.11134
$123$	0	0
$124$	−4.92820	−0.442566
$125$	0	0
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	12.1244	1.07165
$129$	0	0
$130$	0	0
$131$	−15.1244	−1.32142	−0.660711	−	0.750641i	$-0.729745\pi$
$131$	−15.1244	−1.32142	−0.660711	+	0.750641i	$0.729745\pi$
$132$	0	0
$133$	−0.732051	−0.0634769
$134$	−13.8564	−1.19701
$135$	0	0
$136$	0	0
$137$	−7.85641	−0.671218	−0.335609	−	0.942001i	$-0.608942\pi$
$137$	−7.85641	−0.671218	−0.335609	+	0.942001i	$0.608942\pi$
$138$	0	0
$139$	12.3923	1.05110	0.525551	−	0.850762i	$-0.323859\pi$
$139$	12.3923	1.05110	0.525551	+	0.850762i	$0.323859\pi$
$140$	0	0
$141$	0	0
$142$	−28.3923	−2.38263
$143$	−3.46410	−0.289683
$144$	0	0
$145$	0	0
$146$	5.32051	0.440328
$147$	0	0
$148$	−4.19615	−0.344922
$149$	−7.85641	−0.643622	−0.321811	−	0.946804i	$-0.604292\pi$
$149$	−7.85641	−0.643622	−0.321811	+	0.946804i	$0.604292\pi$
$150$	0	0
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	−1.73205	−0.140488
$153$	0	0
$154$	1.60770	0.129552
$155$	0	0
$156$	0	0
$157$	14.3923	1.14863	0.574315	−	0.818634i	$-0.305268\pi$
$157$	14.3923	1.14863	0.574315	+	0.818634i	$0.305268\pi$
$158$	5.07180	0.403490
$159$	0	0
$160$	0	0
$161$	2.53590	0.199857
$162$	0	0
$163$	−12.7321	−0.997251	−0.498626	−	0.866817i	$-0.666162\pi$
$163$	−12.7321	−0.997251	−0.498626	+	0.866817i	$0.666162\pi$
$164$	−4.73205	−0.369511
$165$	0	0
$166$	1.60770	0.124781
$167$	−3.46410	−0.268060	−0.134030	−	0.990977i	$-0.542792\pi$
$167$	−3.46410	−0.268060	−0.134030	+	0.990977i	$0.542792\pi$
$168$	0	0
$169$	−5.53590	−0.425838
$170$	0	0
$171$	0	0
$172$	6.19615	0.472452
$173$	−6.92820	−0.526742	−0.263371	−	0.964695i	$-0.584834\pi$
$173$	−6.92820	−0.526742	−0.263371	+	0.964695i	$0.584834\pi$
$174$	0	0
$175$	0	0
$176$	6.33975	0.477876
$177$	0	0
$178$	−12.5885	−0.943545
$179$	11.3205	0.846135	0.423067	−	0.906098i	$-0.360953\pi$
$179$	11.3205	0.846135	0.423067	+	0.906098i	$0.360953\pi$
$180$	0	0
$181$	18.3923	1.36709	0.683545	−	0.729909i	$-0.260437\pi$
$181$	18.3923	1.36709	0.683545	+	0.729909i	$0.260437\pi$
$182$	−3.46410	−0.256776
$183$	0	0
$184$	6.00000	0.442326
$185$	0	0
$186$	0	0
$187$	0	0
$188$	3.46410	0.252646
$189$	0	0
$190$	0	0
$191$	−17.6603	−1.27785	−0.638926	−	0.769269i	$-0.720621\pi$
$191$	−17.6603	−1.27785	−0.638926	+	0.769269i	$0.720621\pi$
$192$	0	0
$193$	7.12436	0.512822	0.256411	−	0.966568i	$-0.417460\pi$
$193$	7.12436	0.512822	0.256411	+	0.966568i	$0.417460\pi$
$194$	−7.26795	−0.521808
$195$	0	0
$196$	−6.46410	−0.461722
$197$	−24.0000	−1.70993	−0.854965	−	0.518686i	$-0.826421\pi$
$197$	−24.0000	−1.70993	−0.854965	+	0.518686i	$0.826421\pi$
$198$	0	0
$199$	19.3205	1.36959	0.684797	−	0.728734i	$-0.259891\pi$
$199$	19.3205	1.36959	0.684797	+	0.728734i	$0.259891\pi$
$200$	0	0
$201$	0	0
$202$	−18.0000	−1.26648
$203$	−1.60770	−0.112838
$204$	0	0
$205$	0	0
$206$	30.9282	2.15487
$207$	0	0
$208$	−13.6603	−0.947168
$209$	−1.26795	−0.0877059
$210$	0	0
$211$	14.9282	1.02770	0.513850	−	0.857880i	$-0.328219\pi$
$211$	14.9282	1.02770	0.513850	+	0.857880i	$0.328219\pi$
$212$	−2.53590	−0.174166
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3.60770	0.244906
$218$	11.0718	0.749877
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−9.85641	−0.660034	−0.330017	−	0.943975i	$-0.607054\pi$
$223$	−9.85641	−0.660034	−0.330017	+	0.943975i	$0.607054\pi$
$224$	3.80385	0.254155
$225$	0	0
$226$	8.78461	0.584344
$227$	−10.3923	−0.689761	−0.344881	−	0.938647i	$-0.612081\pi$
$227$	−10.3923	−0.689761	−0.344881	+	0.938647i	$0.612081\pi$
$228$	0	0
$229$	−25.4641	−1.68272	−0.841358	−	0.540479i	$-0.818243\pi$
$229$	−25.4641	−1.68272	−0.841358	+	0.540479i	$0.818243\pi$
$230$	0	0
$231$	0	0
$232$	−3.80385	−0.249735
$233$	19.8564	1.30084	0.650418	−	0.759576i	$-0.274594\pi$
$233$	19.8564	1.30084	0.650418	+	0.759576i	$0.274594\pi$
$234$	0	0
$235$	0	0
$236$	−9.46410	−0.616061
$237$	0	0
$238$	0	0
$239$	20.1962	1.30638	0.653190	−	0.757194i	$-0.273430\pi$
$239$	20.1962	1.30638	0.653190	+	0.757194i	$0.273430\pi$
$240$	0	0
$241$	−16.9282	−1.09044	−0.545221	−	0.838293i	$-0.683554\pi$
$241$	−16.9282	−1.09044	−0.545221	+	0.838293i	$0.683554\pi$
$242$	−16.2679	−1.04574
$243$	0	0
$244$	−13.4641	−0.861951
$245$	0	0
$246$	0	0
$247$	2.73205	0.173836
$248$	8.53590	0.542030
$249$	0	0
$250$	0	0
$251$	10.0526	0.634512	0.317256	−	0.948340i	$-0.397239\pi$
$251$	10.0526	0.634512	0.317256	+	0.948340i	$0.397239\pi$
$252$	0	0
$253$	4.39230	0.276142
$254$	6.92820	0.434714
$255$	0	0
$256$	19.0000	1.18750
$257$	−24.0000	−1.49708	−0.748539	−	0.663090i	$-0.769245\pi$
$257$	−24.0000	−1.49708	−0.748539	+	0.663090i	$0.769245\pi$
$258$	0	0
$259$	3.07180	0.190872
$260$	0	0
$261$	0	0
$262$	−26.1962	−1.61840
$263$	6.00000	0.369976	0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000	0.369976	0.184988	+	0.982741i	$0.440775\pi$
$264$	0	0
$265$	0	0
$266$	−1.26795	−0.0777430
$267$	0	0
$268$	−8.00000	−0.488678
$269$	30.5885	1.86501	0.932506	−	0.361156i	$-0.117618\pi$
$269$	30.5885	1.86501	0.932506	+	0.361156i	$0.117618\pi$
$270$	0	0
$271$	−20.3923	−1.23874	−0.619372	−	0.785098i	$-0.712613\pi$
$271$	−20.3923	−1.23874	−0.619372	+	0.785098i	$0.712613\pi$
$272$	0	0
$273$	0	0
$274$	−13.6077	−0.822071
$275$	0	0
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	21.4641	1.28733
$279$	0	0
$280$	0	0
$281$	−4.73205	−0.282290	−0.141145	−	0.989989i	$-0.545078\pi$
$281$	−4.73205	−0.282290	−0.141145	+	0.989989i	$0.545078\pi$
$282$	0	0
$283$	26.9808	1.60384	0.801920	−	0.597432i	$-0.203812\pi$
$283$	26.9808	1.60384	0.801920	+	0.597432i	$0.203812\pi$
$284$	−16.3923	−0.972704
$285$	0	0
$286$	−6.00000	−0.354787
$287$	3.46410	0.204479
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	3.07180	0.179763
$293$	−27.7128	−1.61900	−0.809500	−	0.587120i	$-0.800262\pi$
$293$	−27.7128	−1.61900	−0.809500	+	0.587120i	$0.800262\pi$
$294$	0	0
$295$	0	0
$296$	7.26795	0.422441
$297$	0	0
$298$	−13.6077	−0.788273
$299$	−9.46410	−0.547323
$300$	0	0
$301$	−4.53590	−0.261445
$302$	24.2487	1.39536
$303$	0	0
$304$	−5.00000	−0.286770
$305$	0	0
$306$	0	0
$307$	11.6077	0.662486	0.331243	−	0.943545i	$-0.392532\pi$
$307$	11.6077	0.662486	0.331243	+	0.943545i	$0.392532\pi$
$308$	0.928203	0.0528893
$309$	0	0
$310$	0	0
$311$	26.4449	1.49955	0.749775	−	0.661692i	$-0.230162\pi$
$311$	26.4449	1.49955	0.749775	+	0.661692i	$0.230162\pi$
$312$	0	0
$313$	14.3923	0.813501	0.406751	−	0.913539i	$-0.366662\pi$
$313$	14.3923	0.813501	0.406751	+	0.913539i	$0.366662\pi$
$314$	24.9282	1.40678
$315$	0	0
$316$	2.92820	0.164724
$317$	23.3205	1.30981	0.654905	−	0.755711i	$-0.272709\pi$
$317$	23.3205	1.30981	0.654905	+	0.755711i	$0.272709\pi$
$318$	0	0
$319$	−2.78461	−0.155908
$320$	0	0
$321$	0	0
$322$	4.39230	0.244774
$323$	0	0
$324$	0	0
$325$	0	0
$326$	−22.0526	−1.22138
$327$	0	0
$328$	8.19615	0.452557
$329$	−2.53590	−0.139809
$330$	0	0
$331$	29.7128	1.63316	0.816582	−	0.577230i	$-0.195866\pi$
$331$	29.7128	1.63316	0.816582	+	0.577230i	$0.195866\pi$
$332$	0.928203	0.0509418
$333$	0	0
$334$	−6.00000	−0.328305
$335$	0	0
$336$	0	0
$337$	19.1244	1.04177	0.520885	−	0.853627i	$-0.325602\pi$
$337$	19.1244	1.04177	0.520885	+	0.853627i	$0.325602\pi$
$338$	−9.58846	−0.521543
$339$	0	0
$340$	0	0
$341$	6.24871	0.338387
$342$	0	0
$343$	9.85641	0.532196
$344$	−10.7321	−0.578633
$345$	0	0
$346$	−12.0000	−0.645124
$347$	12.9282	0.694022	0.347011	−	0.937861i	$-0.387197\pi$
$347$	12.9282	0.694022	0.347011	+	0.937861i	$0.387197\pi$
$348$	0	0
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	0	0
$351$	0	0
$352$	6.58846	0.351166
$353$	26.7846	1.42560	0.712800	−	0.701367i	$-0.247427\pi$
$353$	26.7846	1.42560	0.712800	+	0.701367i	$0.247427\pi$
$354$	0	0
$355$	0	0
$356$	−7.26795	−0.385201
$357$	0	0
$358$	19.6077	1.03630
$359$	−17.6603	−0.932073	−0.466036	−	0.884766i	$-0.654318\pi$
$359$	−17.6603	−0.932073	−0.466036	+	0.884766i	$0.654318\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	31.8564	1.67434
$363$	0	0
$364$	−2.00000	−0.104828
$365$	0	0
$366$	0	0
$367$	−5.80385	−0.302958	−0.151479	−	0.988460i	$-0.548404\pi$
$367$	−5.80385	−0.302958	−0.151479	+	0.988460i	$0.548404\pi$
$368$	17.3205	0.902894
$369$	0	0
$370$	0	0
$371$	1.85641	0.0963798
$372$	0	0
$373$	−4.19615	−0.217269	−0.108634	−	0.994082i	$-0.534648\pi$
$373$	−4.19615	−0.217269	−0.108634	+	0.994082i	$0.534648\pi$
$374$	0	0
$375$	0	0
$376$	−6.00000	−0.309426
$377$	6.00000	0.309016
$378$	0	0
$379$	7.07180	0.363254	0.181627	−	0.983368i	$-0.441864\pi$
$379$	7.07180	0.363254	0.181627	+	0.983368i	$0.441864\pi$
$380$	0	0
$381$	0	0
$382$	−30.5885	−1.56504
$383$	−30.9282	−1.58036	−0.790179	−	0.612877i	$-0.790012\pi$
$383$	−30.9282	−1.58036	−0.790179	+	0.612877i	$0.790012\pi$
$384$	0	0
$385$	0	0
$386$	12.3397	0.628077
$387$	0	0
$388$	−4.19615	−0.213027
$389$	19.8564	1.00676	0.503380	−	0.864065i	$-0.332090\pi$
$389$	19.8564	1.00676	0.503380	+	0.864065i	$0.332090\pi$
$390$	0	0
$391$	0	0
$392$	11.1962	0.565491
$393$	0	0
$394$	−41.5692	−2.09423
$395$	0	0
$396$	0	0
$397$	4.92820	0.247339	0.123670	−	0.992323i	$-0.460534\pi$
$397$	4.92820	0.247339	0.123670	+	0.992323i	$0.460534\pi$
$398$	33.4641	1.67740
$399$	0	0
$400$	0	0
$401$	−4.05256	−0.202375	−0.101188	−	0.994867i	$-0.532264\pi$
$401$	−4.05256	−0.202375	−0.101188	+	0.994867i	$0.532264\pi$
$402$	0	0
$403$	−13.4641	−0.670695
$404$	−10.3923	−0.517036
$405$	0	0
$406$	−2.78461	−0.138198
$407$	5.32051	0.263728
$408$	0	0
$409$	−5.60770	−0.277283	−0.138641	−	0.990343i	$-0.544274\pi$
$409$	−5.60770	−0.277283	−0.138641	+	0.990343i	$0.544274\pi$
$410$	0	0
$411$	0	0
$412$	17.8564	0.879722
$413$	6.92820	0.340915
$414$	0	0
$415$	0	0
$416$	−14.1962	−0.696024
$417$	0	0
$418$	−2.19615	−0.107417
$419$	−10.0526	−0.491100	−0.245550	−	0.969384i	$-0.578968\pi$
$419$	−10.0526	−0.491100	−0.245550	+	0.969384i	$0.578968\pi$
$420$	0	0
$421$	22.7846	1.11045	0.555227	−	0.831699i	$-0.312631\pi$
$421$	22.7846	1.11045	0.555227	+	0.831699i	$0.312631\pi$
$422$	25.8564	1.25867
$423$	0	0
$424$	4.39230	0.213309
$425$	0	0
$426$	0	0
$427$	9.85641	0.476985
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−23.3205	−1.12331	−0.561655	−	0.827372i	$-0.689835\pi$
$431$	−23.3205	−1.12331	−0.561655	+	0.827372i	$0.689835\pi$
$432$	0	0
$433$	−20.5885	−0.989418	−0.494709	−	0.869059i	$-0.664725\pi$
$433$	−20.5885	−0.989418	−0.494709	+	0.869059i	$0.664725\pi$
$434$	6.24871	0.299948
$435$	0	0
$436$	6.39230	0.306136
$437$	−3.46410	−0.165710
$438$	0	0
$439$	13.0718	0.623883	0.311941	−	0.950101i	$-0.399021\pi$
$439$	13.0718	0.623883	0.311941	+	0.950101i	$0.399021\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	29.3205	1.39306	0.696530	−	0.717528i	$-0.254726\pi$
$443$	29.3205	1.39306	0.696530	+	0.717528i	$0.254726\pi$
$444$	0	0
$445$	0	0
$446$	−17.0718	−0.808373
$447$	0	0
$448$	−0.732051	−0.0345861
$449$	−11.6603	−0.550281	−0.275141	−	0.961404i	$-0.588724\pi$
$449$	−11.6603	−0.550281	−0.275141	+	0.961404i	$0.588724\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	5.07180	0.238557
$453$	0	0
$454$	−18.0000	−0.844782
$455$	0	0
$456$	0	0
$457$	−11.4641	−0.536268	−0.268134	−	0.963382i	$-0.586407\pi$
$457$	−11.4641	−0.536268	−0.268134	+	0.963382i	$0.586407\pi$
$458$	−44.1051	−2.06090
$459$	0	0
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	−9.51666	−0.442277	−0.221138	−	0.975242i	$-0.570977\pi$
$463$	−9.51666	−0.442277	−0.221138	+	0.975242i	$0.570977\pi$
$464$	−10.9808	−0.509769
$465$	0	0
$466$	34.3923	1.59319
$467$	27.4641	1.27089	0.635444	−	0.772147i	$-0.280817\pi$
$467$	27.4641	1.27089	0.635444	+	0.772147i	$0.280817\pi$
$468$	0	0
$469$	5.85641	0.270424
$470$	0	0
$471$	0	0
$472$	16.3923	0.754517
$473$	−7.85641	−0.361238
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	34.9808	1.59998
$479$	19.5167	0.891739	0.445869	−	0.895098i	$-0.352895\pi$
$479$	19.5167	0.891739	0.445869	+	0.895098i	$0.352895\pi$
$480$	0	0
$481$	−11.4641	−0.522718
$482$	−29.3205	−1.33551
$483$	0	0
$484$	−9.39230	−0.426923
$485$	0	0
$486$	0	0
$487$	11.6077	0.525995	0.262997	−	0.964797i	$-0.415289\pi$
$487$	11.6077	0.525995	0.262997	+	0.964797i	$0.415289\pi$
$488$	23.3205	1.05567
$489$	0	0
$490$	0	0
$491$	22.0526	0.995218	0.497609	−	0.867401i	$-0.334211\pi$
$491$	22.0526	0.995218	0.497609	+	0.867401i	$0.334211\pi$
$492$	0	0
$493$	0	0
$494$	4.73205	0.212905
$495$	0	0
$496$	24.6410	1.10641
$497$	12.0000	0.538274
$498$	0	0
$499$	17.4641	0.781801	0.390900	−	0.920433i	$-0.372164\pi$
$499$	17.4641	0.781801	0.390900	+	0.920433i	$0.372164\pi$
$500$	0	0
$501$	0	0
$502$	17.4115	0.777115
$503$	−36.9282	−1.64655	−0.823274	−	0.567645i	$-0.807855\pi$
$503$	−36.9282	−1.64655	−0.823274	+	0.567645i	$0.807855\pi$
$504$	0	0
$505$	0	0
$506$	7.60770	0.338203
$507$	0	0
$508$	4.00000	0.177471
$509$	−28.0526	−1.24341	−0.621704	−	0.783252i	$-0.713559\pi$
$509$	−28.0526	−1.24341	−0.621704	+	0.783252i	$0.713559\pi$
$510$	0	0
$511$	−2.24871	−0.0994771
$512$	8.66025	0.382733
$513$	0	0
$514$	−41.5692	−1.83354
$515$	0	0
$516$	0	0
$517$	−4.39230	−0.193173
$518$	5.32051	0.233770
$519$	0	0
$520$	0	0
$521$	−40.7321	−1.78450	−0.892252	−	0.451538i	$-0.850875\pi$
$521$	−40.7321	−1.78450	−0.892252	+	0.451538i	$0.850875\pi$
$522$	0	0
$523$	−43.3205	−1.89427	−0.947137	−	0.320830i	$-0.896038\pi$
$523$	−43.3205	−1.89427	−0.947137	+	0.320830i	$0.896038\pi$
$524$	−15.1244	−0.660711
$525$	0	0
$526$	10.3923	0.453126
$527$	0	0
$528$	0	0
$529$	−11.0000	−0.478261
$530$	0	0
$531$	0	0
$532$	−0.732051	−0.0317384
$533$	−12.9282	−0.559983
$534$	0	0
$535$	0	0
$536$	13.8564	0.598506
$537$	0	0
$538$	52.9808	2.28416
$539$	8.19615	0.353033
$540$	0	0
$541$	−13.7128	−0.589560	−0.294780	−	0.955565i	$-0.595246\pi$
$541$	−13.7128	−0.589560	−0.294780	+	0.955565i	$0.595246\pi$
$542$	−35.3205	−1.51715
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−8.67949	−0.371108	−0.185554	−	0.982634i	$-0.559408\pi$
$547$	−8.67949	−0.371108	−0.185554	+	0.982634i	$0.559408\pi$
$548$	−7.85641	−0.335609
$549$	0	0
$550$	0	0
$551$	2.19615	0.0935592
$552$	0	0
$553$	−2.14359	−0.0911549
$554$	−3.46410	−0.147176
$555$	0	0
$556$	12.3923	0.525551
$557$	−12.9282	−0.547786	−0.273893	−	0.961760i	$-0.588311\pi$
$557$	−12.9282	−0.547786	−0.273893	+	0.961760i	$0.588311\pi$
$558$	0	0
$559$	16.9282	0.715987
$560$	0	0
$561$	0	0
$562$	−8.19615	−0.345734
$563$	−20.5359	−0.865485	−0.432742	−	0.901518i	$-0.642454\pi$
$563$	−20.5359	−0.865485	−0.432742	+	0.901518i	$0.642454\pi$
$564$	0	0
$565$	0	0
$566$	46.7321	1.96429
$567$	0	0
$568$	28.3923	1.19131
$569$	16.0526	0.672958	0.336479	−	0.941691i	$-0.390764\pi$
$569$	16.0526	0.672958	0.336479	+	0.941691i	$0.390764\pi$
$570$	0	0
$571$	14.2487	0.596290	0.298145	−	0.954521i	$-0.403632\pi$
$571$	14.2487	0.596290	0.298145	+	0.954521i	$0.403632\pi$
$572$	−3.46410	−0.144841
$573$	0	0
$574$	6.00000	0.250435
$575$	0	0
$576$	0	0
$577$	47.1769	1.96400	0.982000	−	0.188879i	$-0.0604855\pi$
$577$	47.1769	1.96400	0.982000	+	0.188879i	$0.0604855\pi$
$578$	−29.4449	−1.22474
$579$	0	0
$580$	0	0
$581$	−0.679492	−0.0281901
$582$	0	0
$583$	3.21539	0.133168
$584$	−5.32051	−0.220164
$585$	0	0
$586$	−48.0000	−1.98286
$587$	3.46410	0.142979	0.0714894	−	0.997441i	$-0.477225\pi$
$587$	3.46410	0.142979	0.0714894	+	0.997441i	$0.477225\pi$
$588$	0	0
$589$	−4.92820	−0.203063
$590$	0	0
$591$	0	0
$592$	20.9808	0.862304
$593$	2.78461	0.114350	0.0571751	−	0.998364i	$-0.481791\pi$
$593$	2.78461	0.114350	0.0571751	+	0.998364i	$0.481791\pi$
$594$	0	0
$595$	0	0
$596$	−7.85641	−0.321811
$597$	0	0
$598$	−16.3923	−0.670331
$599$	−13.8564	−0.566157	−0.283079	−	0.959097i	$-0.591356\pi$
$599$	−13.8564	−0.566157	−0.283079	+	0.959097i	$0.591356\pi$
$600$	0	0
$601$	15.1769	0.619079	0.309540	−	0.950887i	$-0.399825\pi$
$601$	15.1769	0.619079	0.309540	+	0.950887i	$0.399825\pi$
$602$	−7.85641	−0.320203
$603$	0	0
$604$	14.0000	0.569652
$605$	0	0
$606$	0	0
$607$	32.3923	1.31476	0.657382	−	0.753558i	$-0.271664\pi$
$607$	32.3923	1.31476	0.657382	+	0.753558i	$0.271664\pi$
$608$	−5.19615	−0.210732
$609$	0	0
$610$	0	0
$611$	9.46410	0.382877
$612$	0	0
$613$	−21.6077	−0.872727	−0.436363	−	0.899771i	$-0.643734\pi$
$613$	−21.6077	−0.872727	−0.436363	+	0.899771i	$0.643734\pi$
$614$	20.1051	0.811377
$615$	0	0
$616$	−1.60770	−0.0647759
$617$	−27.7128	−1.11568	−0.557838	−	0.829950i	$-0.688369\pi$
$617$	−27.7128	−1.11568	−0.557838	+	0.829950i	$0.688369\pi$
$618$	0	0
$619$	19.3205	0.776557	0.388278	−	0.921542i	$-0.373070\pi$
$619$	19.3205	0.776557	0.388278	+	0.921542i	$0.373070\pi$
$620$	0	0
$621$	0	0
$622$	45.8038	1.83657
$623$	5.32051	0.213162
$624$	0	0
$625$	0	0
$626$	24.9282	0.996331
$627$	0	0
$628$	14.3923	0.574315
$629$	0	0
$630$	0	0
$631$	−21.0718	−0.838855	−0.419427	−	0.907789i	$-0.637769\pi$
$631$	−21.0718	−0.838855	−0.419427	+	0.907789i	$0.637769\pi$
$632$	−5.07180	−0.201745
$633$	0	0
$634$	40.3923	1.60418
$635$	0	0
$636$	0	0
$637$	−17.6603	−0.699725
$638$	−4.82309	−0.190948
$639$	0	0
$640$	0	0
$641$	−17.4115	−0.687715	−0.343857	−	0.939022i	$-0.611734\pi$
$641$	−17.4115	−0.687715	−0.343857	+	0.939022i	$0.611734\pi$
$642$	0	0
$643$	1.80385	0.0711368	0.0355684	−	0.999367i	$-0.488676\pi$
$643$	1.80385	0.0711368	0.0355684	+	0.999367i	$0.488676\pi$
$644$	2.53590	0.0999284
$645$	0	0
$646$	0	0
$647$	−31.8564	−1.25240	−0.626202	−	0.779661i	$-0.715392\pi$
$647$	−31.8564	−1.25240	−0.626202	+	0.779661i	$0.715392\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	0	0
$652$	−12.7321	−0.498626
$653$	30.9282	1.21031	0.605157	−	0.796106i	$-0.293110\pi$
$653$	30.9282	1.21031	0.605157	+	0.796106i	$0.293110\pi$
$654$	0	0
$655$	0	0
$656$	23.6603	0.923778
$657$	0	0
$658$	−4.39230	−0.171230
$659$	−18.9282	−0.737338	−0.368669	−	0.929561i	$-0.620186\pi$
$659$	−18.9282	−0.737338	−0.368669	+	0.929561i	$0.620186\pi$
$660$	0	0
$661$	−23.1769	−0.901477	−0.450739	−	0.892656i	$-0.648839\pi$
$661$	−23.1769	−0.901477	−0.450739	+	0.892656i	$0.648839\pi$
$662$	51.4641	2.00021
$663$	0	0
$664$	−1.60770	−0.0623907
$665$	0	0
$666$	0	0
$667$	−7.60770	−0.294571
$668$	−3.46410	−0.134030
$669$	0	0
$670$	0	0
$671$	17.0718	0.659049
$672$	0	0
$673$	7.12436	0.274624	0.137312	−	0.990528i	$-0.456154\pi$
$673$	7.12436	0.274624	0.137312	+	0.990528i	$0.456154\pi$
$674$	33.1244	1.27590
$675$	0	0
$676$	−5.53590	−0.212919
$677$	35.3205	1.35748	0.678739	−	0.734380i	$-0.262527\pi$
$677$	35.3205	1.35748	0.678739	+	0.734380i	$0.262527\pi$
$678$	0	0
$679$	3.07180	0.117885
$680$	0	0
$681$	0	0
$682$	10.8231	0.414437
$683$	−18.9282	−0.724268	−0.362134	−	0.932126i	$-0.617952\pi$
$683$	−18.9282	−0.724268	−0.362134	+	0.932126i	$0.617952\pi$
$684$	0	0
$685$	0	0
$686$	17.0718	0.651804
$687$	0	0
$688$	−30.9808	−1.18113
$689$	−6.92820	−0.263944
$690$	0	0
$691$	−8.39230	−0.319258	−0.159629	−	0.987177i	$-0.551030\pi$
$691$	−8.39230	−0.319258	−0.159629	+	0.987177i	$0.551030\pi$
$692$	−6.92820	−0.263371
$693$	0	0
$694$	22.3923	0.850000
$695$	0	0
$696$	0	0
$697$	0	0
$698$	−38.1051	−1.44230
$699$	0	0
$700$	0	0
$701$	−21.7128	−0.820082	−0.410041	−	0.912067i	$-0.634486\pi$
$701$	−21.7128	−0.820082	−0.410041	+	0.912067i	$0.634486\pi$
$702$	0	0
$703$	−4.19615	−0.158261
$704$	−1.26795	−0.0477876
$705$	0	0
$706$	46.3923	1.74600
$707$	7.60770	0.286117
$708$	0	0
$709$	33.1769	1.24599	0.622993	−	0.782228i	$-0.285917\pi$
$709$	33.1769	1.24599	0.622993	+	0.782228i	$0.285917\pi$
$710$	0	0
$711$	0	0
$712$	12.5885	0.471772
$713$	17.0718	0.639344
$714$	0	0
$715$	0	0
$716$	11.3205	0.423067
$717$	0	0
$718$	−30.5885	−1.14155
$719$	−5.66025	−0.211092	−0.105546	−	0.994414i	$-0.533659\pi$
$719$	−5.66025	−0.211092	−0.105546	+	0.994414i	$0.533659\pi$
$720$	0	0
$721$	−13.0718	−0.486819
$722$	1.73205	0.0644603
$723$	0	0
$724$	18.3923	0.683545
$725$	0	0
$726$	0	0
$727$	−8.33975	−0.309304	−0.154652	−	0.987969i	$-0.549426\pi$
$727$	−8.33975	−0.309304	−0.154652	+	0.987969i	$0.549426\pi$
$728$	3.46410	0.128388
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−22.7846	−0.841569	−0.420784	−	0.907161i	$-0.638245\pi$
$733$	−22.7846	−0.841569	−0.420784	+	0.907161i	$0.638245\pi$
$734$	−10.0526	−0.371047
$735$	0	0
$736$	18.0000	0.663489
$737$	10.1436	0.373644
$738$	0	0
$739$	33.8564	1.24543	0.622714	−	0.782450i	$-0.286030\pi$
$739$	33.8564	1.24543	0.622714	+	0.782450i	$0.286030\pi$
$740$	0	0
$741$	0	0
$742$	3.21539	0.118041
$743$	−44.7846	−1.64299	−0.821494	−	0.570217i	$-0.806859\pi$
$743$	−44.7846	−1.64299	−0.821494	+	0.570217i	$0.806859\pi$
$744$	0	0
$745$	0	0
$746$	−7.26795	−0.266099
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	26.0000	0.948753	0.474377	−	0.880322i	$-0.342673\pi$
$751$	26.0000	0.948753	0.474377	+	0.880322i	$0.342673\pi$
$752$	−17.3205	−0.631614
$753$	0	0
$754$	10.3923	0.378465
$755$	0	0
$756$	0	0
$757$	16.2487	0.590569	0.295285	−	0.955409i	$-0.404585\pi$
$757$	16.2487	0.590569	0.295285	+	0.955409i	$0.404585\pi$
$758$	12.2487	0.444893
$759$	0	0
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	0	0
$763$	−4.67949	−0.169409
$764$	−17.6603	−0.638926
$765$	0	0
$766$	−53.5692	−1.93553
$767$	−25.8564	−0.933621
$768$	0	0
$769$	48.6410	1.75404	0.877020	−	0.480454i	$-0.159528\pi$
$769$	48.6410	1.75404	0.877020	+	0.480454i	$0.159528\pi$
$770$	0	0
$771$	0	0
$772$	7.12436	0.256411
$773$	37.1769	1.33716	0.668580	−	0.743640i	$-0.266902\pi$
$773$	37.1769	1.33716	0.668580	+	0.743640i	$0.266902\pi$
$774$	0	0
$775$	0	0
$776$	7.26795	0.260904
$777$	0	0
$778$	34.3923	1.23302
$779$	−4.73205	−0.169543
$780$	0	0
$781$	20.7846	0.743732
$782$	0	0
$783$	0	0
$784$	32.3205	1.15430
$785$	0	0
$786$	0	0
$787$	−43.3205	−1.54421	−0.772105	−	0.635495i	$-0.780796\pi$
$787$	−43.3205	−1.54421	−0.772105	+	0.635495i	$0.780796\pi$
$788$	−24.0000	−0.854965
$789$	0	0
$790$	0	0
$791$	−3.71281	−0.132012
$792$	0	0
$793$	−36.7846	−1.30626
$794$	8.53590	0.302928
$795$	0	0
$796$	19.3205	0.684797
$797$	3.21539	0.113895	0.0569475	−	0.998377i	$-0.481863\pi$
$797$	3.21539	0.113895	0.0569475	+	0.998377i	$0.481863\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	−7.01924	−0.247858
$803$	−3.89488	−0.137447
$804$	0	0
$805$	0	0
$806$	−23.3205	−0.821430
$807$	0	0
$808$	18.0000	0.633238
$809$	26.7846	0.941697	0.470848	−	0.882214i	$-0.343948\pi$
$809$	26.7846	0.941697	0.470848	+	0.882214i	$0.343948\pi$
$810$	0	0
$811$	−45.5692	−1.60015	−0.800076	−	0.599899i	$-0.795207\pi$
$811$	−45.5692	−1.60015	−0.800076	+	0.599899i	$0.795207\pi$
$812$	−1.60770	−0.0564190
$813$	0	0
$814$	9.21539	0.322999
$815$	0	0
$816$	0	0
$817$	6.19615	0.216776
$818$	−9.71281	−0.339601
$819$	0	0
$820$	0	0
$821$	39.4641	1.37731	0.688653	−	0.725091i	$-0.258202\pi$
$821$	39.4641	1.37731	0.688653	+	0.725091i	$0.258202\pi$
$822$	0	0
$823$	38.9808	1.35878	0.679392	−	0.733776i	$-0.262244\pi$
$823$	38.9808	1.35878	0.679392	+	0.733776i	$0.262244\pi$
$824$	−30.9282	−1.07744
$825$	0	0
$826$	12.0000	0.417533
$827$	29.3205	1.01957	0.509787	−	0.860301i	$-0.329724\pi$
$827$	29.3205	1.01957	0.509787	+	0.860301i	$0.329724\pi$
$828$	0	0
$829$	−42.1051	−1.46237	−0.731186	−	0.682179i	$-0.761033\pi$
$829$	−42.1051	−1.46237	−0.731186	+	0.682179i	$0.761033\pi$
$830$	0	0
$831$	0	0
$832$	2.73205	0.0947168
$833$	0	0
$834$	0	0
$835$	0	0
$836$	−1.26795	−0.0438529
$837$	0	0
$838$	−17.4115	−0.601472
$839$	40.3923	1.39450	0.697249	−	0.716829i	$-0.254407\pi$
$839$	40.3923	1.39450	0.697249	+	0.716829i	$0.254407\pi$
$840$	0	0
$841$	−24.1769	−0.833687
$842$	39.4641	1.36002
$843$	0	0
$844$	14.9282	0.513850
$845$	0	0
$846$	0	0
$847$	6.87564	0.236250
$848$	12.6795	0.435416
$849$	0	0
$850$	0	0
$851$	14.5359	0.498284
$852$	0	0
$853$	35.1769	1.20443	0.602217	−	0.798332i	$-0.294284\pi$
$853$	35.1769	1.20443	0.602217	+	0.798332i	$0.294284\pi$
$854$	17.0718	0.584185
$855$	0	0
$856$	0	0
$857$	6.24871	0.213452	0.106726	−	0.994288i	$-0.465963\pi$
$857$	6.24871	0.213452	0.106726	+	0.994288i	$0.465963\pi$
$858$	0	0
$859$	32.0000	1.09183	0.545913	−	0.837842i	$-0.316183\pi$
$859$	32.0000	1.09183	0.545913	+	0.837842i	$0.316183\pi$
$860$	0	0
$861$	0	0
$862$	−40.3923	−1.37577
$863$	−12.0000	−0.408485	−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000	−0.408485	−0.204242	+	0.978920i	$0.565473\pi$
$864$	0	0
$865$	0	0
$866$	−35.6603	−1.21178
$867$	0	0
$868$	3.60770	0.122453
$869$	−3.71281	−0.125949
$870$	0	0
$871$	−21.8564	−0.740576
$872$	−11.0718	−0.374938
$873$	0	0
$874$	−6.00000	−0.202953
$875$	0	0
$876$	0	0
$877$	−28.8756	−0.975061	−0.487531	−	0.873106i	$-0.662102\pi$
$877$	−28.8756	−0.975061	−0.487531	+	0.873106i	$0.662102\pi$
$878$	22.6410	0.764097
$879$	0	0
$880$	0	0
$881$	−15.4641	−0.520999	−0.260499	−	0.965474i	$-0.583887\pi$
$881$	−15.4641	−0.520999	−0.260499	+	0.965474i	$0.583887\pi$
$882$	0	0
$883$	14.9808	0.504143	0.252071	−	0.967709i	$-0.418888\pi$
$883$	14.9808	0.504143	0.252071	+	0.967709i	$0.418888\pi$
$884$	0	0
$885$	0	0
$886$	50.7846	1.70614
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	0	0
$889$	−2.92820	−0.0982088
$890$	0	0
$891$	0	0
$892$	−9.85641	−0.330017
$893$	3.46410	0.115922
$894$	0	0
$895$	0	0
$896$	−8.87564	−0.296514
$897$	0	0
$898$	−20.1962	−0.673954
$899$	−10.8231	−0.360970
$900$	0	0
$901$	0	0
$902$	10.3923	0.346026
$903$	0	0
$904$	−8.78461	−0.292172
$905$	0	0
$906$	0	0
$907$	11.6077	0.385427	0.192714	−	0.981255i	$-0.438271\pi$
$907$	11.6077	0.385427	0.192714	+	0.981255i	$0.438271\pi$
$908$	−10.3923	−0.344881
$909$	0	0
$910$	0	0
$911$	41.0718	1.36077	0.680385	−	0.732855i	$-0.261813\pi$
$911$	41.0718	1.36077	0.680385	+	0.732855i	$0.261813\pi$
$912$	0	0
$913$	−1.17691	−0.0389502
$914$	−19.8564	−0.656792
$915$	0	0
$916$	−25.4641	−0.841358
$917$	11.0718	0.365623
$918$	0	0
$919$	−59.4256	−1.96027	−0.980135	−	0.198330i	$-0.936448\pi$
$919$	−59.4256	−1.96027	−0.980135	+	0.198330i	$0.936448\pi$
$920$	0	0
$921$	0	0
$922$	−10.3923	−0.342252
$923$	−44.7846	−1.47410
$924$	0	0
$925$	0	0
$926$	−16.4833	−0.541676
$927$	0	0
$928$	−11.4115	−0.374602
$929$	22.3923	0.734668	0.367334	−	0.930089i	$-0.380271\pi$
$929$	22.3923	0.734668	0.367334	+	0.930089i	$0.380271\pi$
$930$	0	0
$931$	−6.46410	−0.211852
$932$	19.8564	0.650418
$933$	0	0
$934$	47.5692	1.55651
$935$	0	0
$936$	0	0
$937$	−32.2487	−1.05352	−0.526760	−	0.850014i	$-0.676593\pi$
$937$	−32.2487	−1.05352	−0.526760	+	0.850014i	$0.676593\pi$
$938$	10.1436	0.331200
$939$	0	0
$940$	0	0
$941$	30.5885	0.997155	0.498578	−	0.866845i	$-0.333856\pi$
$941$	30.5885	0.997155	0.498578	+	0.866845i	$0.333856\pi$
$942$	0	0
$943$	16.3923	0.533807
$944$	47.3205	1.54015
$945$	0	0
$946$	−13.6077	−0.442424
$947$	55.8564	1.81509	0.907545	−	0.419956i	$-0.137954\pi$
$947$	55.8564	1.81509	0.907545	+	0.419956i	$0.137954\pi$
$948$	0	0
$949$	8.39230	0.272426
$950$	0	0
$951$	0	0
$952$	0	0
$953$	10.1436	0.328583	0.164292	−	0.986412i	$-0.447466\pi$
$953$	10.1436	0.328583	0.164292	+	0.986412i	$0.447466\pi$
$954$	0	0
$955$	0	0
$956$	20.1962	0.653190
$957$	0	0
$958$	33.8038	1.09215
$959$	5.75129	0.185719
$960$	0	0
$961$	−6.71281	−0.216542
$962$	−19.8564	−0.640196
$963$	0	0
$964$	−16.9282	−0.545221
$965$	0	0
$966$	0	0
$967$	−29.1244	−0.936576	−0.468288	−	0.883576i	$-0.655129\pi$
$967$	−29.1244	−0.936576	−0.468288	+	0.883576i	$0.655129\pi$
$968$	16.2679	0.522872
$969$	0	0
$970$	0	0
$971$	−27.7128	−0.889346	−0.444673	−	0.895693i	$-0.646680\pi$
$971$	−27.7128	−0.889346	−0.444673	+	0.895693i	$0.646680\pi$
$972$	0	0
$973$	−9.07180	−0.290828
$974$	20.1051	0.644210
$975$	0	0
$976$	67.3205	2.15488
$977$	51.0333	1.63270	0.816350	−	0.577557i	$-0.195994\pi$
$977$	51.0333	1.63270	0.816350	+	0.577557i	$0.195994\pi$
$978$	0	0
$979$	9.21539	0.294525
$980$	0	0
$981$	0	0
$982$	38.1962	1.21889
$983$	−6.67949	−0.213043	−0.106521	−	0.994310i	$-0.533971\pi$
$983$	−6.67949	−0.213043	−0.106521	+	0.994310i	$0.533971\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	2.73205	0.0869181
$989$	−21.4641	−0.682519
$990$	0	0
$991$	26.9282	0.855403	0.427701	−	0.903920i	$-0.359324\pi$
$991$	26.9282	0.855403	0.427701	+	0.903920i	$0.359324\pi$
$992$	25.6077	0.813045
$993$	0	0
$994$	20.7846	0.659248
$995$	0	0
$996$	0	0
$997$	38.3923	1.21590	0.607948	−	0.793977i	$-0.291993\pi$
$997$	38.3923	1.21590	0.607948	+	0.793977i	$0.291993\pi$
$998$	30.2487	0.957506
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4275.2.a.t.1.2		2
3.2	odd	2		1425.2.a.o.1.1		2
5.4	even	2		855.2.a.f.1.1		2
15.2	even	4		1425.2.c.k.799.2		4
15.8	even	4		1425.2.c.k.799.3		4
15.14	odd	2		285.2.a.e.1.2	✓	2
60.59	even	2		4560.2.a.bh.1.1		2
285.284	even	2		5415.2.a.r.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.a.e.1.2	✓	2	15.14	odd	2
855.2.a.f.1.1		2	5.4	even	2
1425.2.a.o.1.1		2	3.2	odd	2
1425.2.c.k.799.2		4	15.2	even	4
1425.2.c.k.799.3		4	15.8	even	4
4275.2.a.t.1.2		2	1.1	even	1	trivial
4560.2.a.bh.1.1		2	60.59	even	2
5415.2.a.r.1.1		2	285.284	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)

\(f(q)\)	\(=\)	\(q+1.73205 q^{2} +1.00000 q^{4} -0.732051 q^{7} -1.73205 q^{8} -1.26795 q^{11} +2.73205 q^{13} -1.26795 q^{14} -5.00000 q^{16} +1.00000 q^{19} -2.19615 q^{22} -3.46410 q^{23} +4.73205 q^{26} -0.732051 q^{28} +2.19615 q^{29} -4.92820 q^{31} -5.19615 q^{32} -4.19615 q^{37} +1.73205 q^{38} -4.73205 q^{41} +6.19615 q^{43} -1.26795 q^{44} -6.00000 q^{46} +3.46410 q^{47} -6.46410 q^{49} +2.73205 q^{52} -2.53590 q^{53} +1.26795 q^{56} +3.80385 q^{58} -9.46410 q^{59} -13.4641 q^{61} -8.53590 q^{62} +1.00000 q^{64} -8.00000 q^{67} -16.3923 q^{71} +3.07180 q^{73} -7.26795 q^{74} +1.00000 q^{76} +0.928203 q^{77} +2.92820 q^{79} -8.19615 q^{82} +0.928203 q^{83} +10.7321 q^{86} +2.19615 q^{88} -7.26795 q^{89} -2.00000 q^{91} -3.46410 q^{92} +6.00000 q^{94} -4.19615 q^{97} -11.1962 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 2 q^{7} - 6 q^{11} + 2 q^{13} - 6 q^{14} - 10 q^{16} + 2 q^{19} + 6 q^{22} + 6 q^{26} + 2 q^{28} - 6 q^{29} + 4 q^{31} + 2 q^{37} - 6 q^{41} + 2 q^{43} - 6 q^{44} - 12 q^{46} - 6 q^{49}+ \cdots - 12 q^{98}+O(q^{100}) \) 2 * q + 2 * q^4 + 2 * q^7 - 6 * q^11 + 2 * q^13 - 6 * q^14 - 10 * q^16 + 2 * q^19 + 6 * q^22 + 6 * q^26 + 2 * q^28 - 6 * q^29 + 4 * q^31 + 2 * q^37 - 6 * q^41 + 2 * q^43 - 6 * q^44 - 12 * q^46 - 6 * q^49 + 2 * q^52 - 12 * q^53 + 6 * q^56 + 18 * q^58 - 12 * q^59 - 20 * q^61 - 24 * q^62 + 2 * q^64 - 16 * q^67 - 12 * q^71 + 20 * q^73 - 18 * q^74 + 2 * q^76 - 12 * q^77 - 8 * q^79 - 6 * q^82 - 12 * q^83 + 18 * q^86 - 6 * q^88 - 18 * q^89 - 4 * q^91 + 12 * q^94 + 2 * q^97 - 12 * q^98

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