LMFDB - Embedded newform 2106.2.a.s.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2106,2,Mod(1,2106)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2106.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2106, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2106 = 2 \cdot 3^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2106.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-4,0,4,-2,0,0,-4,0,2,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$16.8164946657$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{7})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 1 $ x^4 - 5*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.456850$ of defining polynomial
Character	$\chi$	$=$	2106.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} -0.397613 q^{5} +2.64575 q^{7} -1.00000 q^{8} +0.397613 q^{10} -4.98019 q^{11} +1.00000 q^{13} -2.64575 q^{14} +1.00000 q^{16} +2.37780 q^{17} -0.784036 q^{19} -0.397613 q^{20} +4.98019 q^{22} -6.89389 q^{23} -4.84190 q^{25} -1.00000 q^{26} +2.64575 q^{28} +2.62594 q^{29} -2.11847 q^{31} -1.00000 q^{32} -2.37780 q^{34} -1.05199 q^{35} +0.708497 q^{37} +0.784036 q^{38} +0.397613 q^{40} +5.80759 q^{41} -6.66888 q^{43} -4.98019 q^{44} +6.89389 q^{46} -10.2515 q^{47} +4.84190 q^{50} +1.00000 q^{52} -2.78404 q^{53} +1.98019 q^{55} -2.64575 q^{56} -2.62594 q^{58} -9.53059 q^{59} -4.86171 q^{61} +2.11847 q^{62} +1.00000 q^{64} -0.397613 q^{65} +14.6293 q^{67} +2.37780 q^{68} +1.05199 q^{70} +15.3382 q^{71} +8.62925 q^{73} -0.708497 q^{74} -0.784036 q^{76} -13.1763 q^{77} -13.7036 q^{79} -0.397613 q^{80} -5.80759 q^{82} -5.77542 q^{83} -0.945446 q^{85} +6.66888 q^{86} +4.98019 q^{88} -4.51609 q^{89} +2.64575 q^{91} -6.89389 q^{92} +10.2515 q^{94} +0.311743 q^{95} -14.4910 q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} - 2 q^{5} - 4 q^{8} + 2 q^{10} - 2 q^{11} + 4 q^{13} + 4 q^{16} - 8 q^{17} - 6 q^{19} - 2 q^{20} + 2 q^{22} - 6 q^{23} + 12 q^{25} - 4 q^{26} - 18 q^{29} - 4 q^{31} - 4 q^{32} + 8 q^{34}+ \cdots + 6 q^{97}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 - 2 * q^5 - 4 * q^8 + 2 * q^10 - 2 * q^11 + 4 * q^13 + 4 * q^16 - 8 * q^17 - 6 * q^19 - 2 * q^20 + 2 * q^22 - 6 * q^23 + 12 * q^25 - 4 * q^26 - 18 * q^29 - 4 * q^31 - 4 * q^32 + 8 * q^34 - 14 * q^35 + 24 * q^37 + 6 * q^38 + 2 * q^40 - 2 * q^41 - 12 * q^43 - 2 * q^44 + 6 * q^46 - 8 * q^47 - 12 * q^50 + 4 * q^52 - 14 * q^53 - 10 * q^55 + 18 * q^58 - 10 * q^59 - 6 * q^61 + 4 * q^62 + 4 * q^64 - 2 * q^65 + 8 * q^67 - 8 * q^68 + 14 * q^70 + 8 * q^71 - 16 * q^73 - 24 * q^74 - 6 * q^76 - 14 * q^77 - 10 * q^79 - 2 * q^80 + 2 * q^82 - 6 * q^83 - 16 * q^85 + 12 * q^86 + 2 * q^88 - 14 * q^89 - 6 * q^92 + 8 * q^94 - 26 * q^95 + 6 * q^97

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−0.397613	−0.177818	−0.0889090	−	0.996040i	$-0.528338\pi$
$5$	−0.397613	−0.177818	−0.0889090	+	0.996040i	$0.528338\pi$
$6$	0	0
$7$	2.64575	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$7$	2.64575	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0.397613	0.125736
$11$	−4.98019	−1.50158	−0.750792	−	0.660539i	$-0.770328\pi$
$11$	−4.98019	−1.50158	−0.750792	+	0.660539i	$0.770328\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−2.64575	−0.707107
$15$	0	0
$16$	1.00000	0.250000
$17$	2.37780	0.576702	0.288351	−	0.957525i	$-0.406893\pi$
$17$	2.37780	0.576702	0.288351	+	0.957525i	$0.406893\pi$
$18$	0	0
$19$	−0.784036	−0.179870	−0.0899352	−	0.995948i	$-0.528666\pi$
$19$	−0.784036	−0.179870	−0.0899352	+	0.995948i	$0.528666\pi$
$20$	−0.397613	−0.0889090
$21$	0	0
$22$	4.98019	1.06178
$23$	−6.89389	−1.43748	−0.718738	−	0.695281i	$-0.755280\pi$
$23$	−6.89389	−1.43748	−0.718738	+	0.695281i	$0.755280\pi$
$24$	0	0
$25$	−4.84190	−0.968381
$26$	−1.00000	−0.196116
$27$	0	0
$28$	2.64575	0.500000
$29$	2.62594	0.487625	0.243812	−	0.969822i	$-0.421602\pi$
$29$	2.62594	0.487625	0.243812	+	0.969822i	$0.421602\pi$
$30$	0	0
$31$	−2.11847	−0.380489	−0.190245	−	0.981737i	$-0.560928\pi$
$31$	−2.11847	−0.380489	−0.190245	+	0.981737i	$0.560928\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−2.37780	−0.407790
$35$	−1.05199	−0.177818
$36$	0	0
$37$	0.708497	0.116476	0.0582381	−	0.998303i	$-0.481452\pi$
$37$	0.708497	0.116476	0.0582381	+	0.998303i	$0.481452\pi$
$38$	0.784036	0.127188
$39$	0	0
$40$	0.397613	0.0628682
$41$	5.80759	0.906993	0.453497	−	0.891258i	$-0.350176\pi$
$41$	5.80759	0.906993	0.453497	+	0.891258i	$0.350176\pi$
$42$	0	0
$43$	−6.66888	−1.01699	−0.508497	−	0.861064i	$-0.669799\pi$
$43$	−6.66888	−1.01699	−0.508497	+	0.861064i	$0.669799\pi$
$44$	−4.98019	−0.750792
$45$	0	0
$46$	6.89389	1.01645
$47$	−10.2515	−1.49533	−0.747664	−	0.664077i	$-0.768825\pi$
$47$	−10.2515	−1.49533	−0.747664	+	0.664077i	$0.768825\pi$
$48$	0	0
$49$	0	0
$50$	4.84190	0.684749
$51$	0	0
$52$	1.00000	0.138675
$53$	−2.78404	−0.382417	−0.191208	−	0.981549i	$-0.561241\pi$
$53$	−2.78404	−0.382417	−0.191208	+	0.981549i	$0.561241\pi$
$54$	0	0
$55$	1.98019	0.267009
$56$	−2.64575	−0.353553
$57$	0	0
$58$	−2.62594	−0.344803
$59$	−9.53059	−1.24078	−0.620389	−	0.784295i	$-0.713025\pi$
$59$	−9.53059	−1.24078	−0.620389	+	0.784295i	$0.713025\pi$
$60$	0	0
$61$	−4.86171	−0.622479	−0.311239	−	0.950332i	$-0.600744\pi$
$61$	−4.86171	−0.622479	−0.311239	+	0.950332i	$0.600744\pi$
$62$	2.11847	0.269046
$63$	0	0
$64$	1.00000	0.125000
$65$	−0.397613	−0.0493178
$66$	0	0
$67$	14.6293	1.78725	0.893624	−	0.448817i	$-0.148154\pi$
$67$	14.6293	1.78725	0.893624	+	0.448817i	$0.148154\pi$
$68$	2.37780	0.288351
$69$	0	0
$70$	1.05199	0.125736
$71$	15.3382	1.82031	0.910154	−	0.414271i	$-0.135963\pi$
$71$	15.3382	1.82031	0.910154	+	0.414271i	$0.135963\pi$
$72$	0	0
$73$	8.62925	1.00998	0.504989	−	0.863126i	$-0.331497\pi$
$73$	8.62925	1.00998	0.504989	+	0.863126i	$0.331497\pi$
$74$	−0.708497	−0.0823611
$75$	0	0
$76$	−0.784036	−0.0899352
$77$	−13.1763	−1.50158
$78$	0	0
$79$	−13.7036	−1.54178	−0.770889	−	0.636970i	$-0.780188\pi$
$79$	−13.7036	−1.54178	−0.770889	+	0.636970i	$0.780188\pi$
$80$	−0.397613	−0.0444545
$81$	0	0
$82$	−5.80759	−0.641341
$83$	−5.77542	−0.633934	−0.316967	−	0.948437i	$-0.602664\pi$
$83$	−5.77542	−0.633934	−0.316967	+	0.948437i	$0.602664\pi$
$84$	0	0
$85$	−0.945446	−0.102548
$86$	6.66888	0.719123
$87$	0	0
$88$	4.98019	0.530890
$89$	−4.51609	−0.478704	−0.239352	−	0.970933i	$-0.576935\pi$
$89$	−4.51609	−0.478704	−0.239352	+	0.970933i	$0.576935\pi$
$90$	0	0
$91$	2.64575	0.277350
$92$	−6.89389	−0.718738
$93$	0	0
$94$	10.2515	1.05736
$95$	0.311743	0.0319842
$96$	0	0
$97$	−14.4910	−1.47133	−0.735667	−	0.677343i	$-0.763131\pi$
$97$	−14.4910	−1.47133	−0.735667	+	0.677343i	$0.763131\pi$
$98$	0	0
$99$	0	0
$100$	−4.84190	−0.484190
$101$	5.29482	0.526854	0.263427	−	0.964679i	$-0.415147\pi$
$101$	5.29482	0.526854	0.263427	+	0.964679i	$0.415147\pi$
$102$	0	0
$103$	7.33775	0.723010	0.361505	−	0.932370i	$-0.382263\pi$
$103$	7.33775	0.723010	0.361505	+	0.932370i	$0.382263\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	2.78404	0.270410
$107$	−1.84721	−0.178577	−0.0892884	−	0.996006i	$-0.528459\pi$
$107$	−1.84721	−0.178577	−0.0892884	+	0.996006i	$0.528459\pi$
$108$	0	0
$109$	6.64906	0.636865	0.318432	−	0.947946i	$-0.396844\pi$
$109$	6.64906	0.636865	0.318432	+	0.947946i	$0.396844\pi$
$110$	−1.98019	−0.188804
$111$	0	0
$112$	2.64575	0.250000
$113$	−13.9373	−1.31111	−0.655553	−	0.755149i	$-0.727565\pi$
$113$	−13.9373	−1.31111	−0.655553	+	0.755149i	$0.727565\pi$
$114$	0	0
$115$	2.74110	0.255609
$116$	2.62594	0.243812
$117$	0	0
$118$	9.53059	0.877362
$119$	6.29107	0.576702
$120$	0	0
$121$	13.8023	1.25475
$122$	4.86171	0.440159
$123$	0	0
$124$	−2.11847	−0.190245
$125$	3.91327	0.350014
$126$	0	0
$127$	−7.10080	−0.630094	−0.315047	−	0.949076i	$-0.602020\pi$
$127$	−7.10080	−0.630094	−0.315047	+	0.949076i	$0.602020\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0.397613	0.0348730
$131$	−4.25933	−0.372139	−0.186070	−	0.982537i	$-0.559575\pi$
$131$	−4.25933	−0.372139	−0.186070	+	0.982537i	$0.559575\pi$
$132$	0	0
$133$	−2.07437	−0.179870
$134$	−14.6293	−1.26377
$135$	0	0
$136$	−2.37780	−0.203895
$137$	−2.45003	−0.209320	−0.104660	−	0.994508i	$-0.533375\pi$
$137$	−2.45003	−0.209320	−0.104660	+	0.994508i	$0.533375\pi$
$138$	0	0
$139$	1.23909	0.105098	0.0525490	−	0.998618i	$-0.483265\pi$
$139$	1.23909	0.105098	0.0525490	+	0.998618i	$0.483265\pi$
$140$	−1.05199	−0.0889090
$141$	0	0
$142$	−15.3382	−1.28715
$143$	−4.98019	−0.416464
$144$	0	0
$145$	−1.04411	−0.0867085
$146$	−8.62925	−0.714162
$147$	0	0
$148$	0.708497	0.0582381
$149$	−13.8221	−1.13235	−0.566175	−	0.824285i	$-0.691577\pi$
$149$	−13.8221	−1.13235	−0.566175	+	0.824285i	$0.691577\pi$
$150$	0	0
$151$	4.72460	0.384483	0.192241	−	0.981348i	$-0.438424\pi$
$151$	4.72460	0.384483	0.192241	+	0.981348i	$0.438424\pi$
$152$	0.784036	0.0635938
$153$	0	0
$154$	13.1763	1.06178
$155$	0.842333	0.0676578
$156$	0	0
$157$	−21.6817	−1.73039	−0.865193	−	0.501439i	$-0.832804\pi$
$157$	−21.6817	−1.73039	−0.865193	+	0.501439i	$0.832804\pi$
$158$	13.7036	1.09020
$159$	0	0
$160$	0.397613	0.0314341
$161$	−18.2395	−1.43748
$162$	0	0
$163$	−22.6293	−1.77246	−0.886230	−	0.463246i	$-0.846685\pi$
$163$	−22.6293	−1.77246	−0.886230	+	0.463246i	$0.846685\pi$
$164$	5.80759	0.453497
$165$	0	0
$166$	5.77542	0.448259
$167$	1.41742	0.109684	0.0548418	−	0.998495i	$-0.482535\pi$
$167$	1.41742	0.109684	0.0548418	+	0.998495i	$0.482535\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0.945446	0.0725124
$171$	0	0
$172$	−6.66888	−0.508497
$173$	15.1908	1.15494	0.577469	−	0.816412i	$-0.304040\pi$
$173$	15.1908	1.15494	0.577469	+	0.816412i	$0.304040\pi$
$174$	0	0
$175$	−12.8105	−0.968381
$176$	−4.98019	−0.375396
$177$	0	0
$178$	4.51609	0.338495
$179$	−6.92820	−0.517838	−0.258919	−	0.965899i	$-0.583366\pi$
$179$	−6.92820	−0.517838	−0.258919	+	0.965899i	$0.583366\pi$
$180$	0	0
$181$	−23.0590	−1.71397	−0.856983	−	0.515346i	$-0.827664\pi$
$181$	−23.0590	−1.71397	−0.856983	+	0.515346i	$0.827664\pi$
$182$	−2.64575	−0.196116
$183$	0	0
$184$	6.89389	0.508224
$185$	−0.281708	−0.0207116
$186$	0	0
$187$	−11.8419	−0.865966
$188$	−10.2515	−0.747664
$189$	0	0
$190$	−0.311743	−0.0226162
$191$	−7.01407	−0.507521	−0.253760	−	0.967267i	$-0.581667\pi$
$191$	−7.01407	−0.507521	−0.253760	+	0.967267i	$0.581667\pi$
$192$	0	0
$193$	20.0128	1.44055	0.720276	−	0.693687i	$-0.244015\pi$
$193$	20.0128	1.44055	0.720276	+	0.693687i	$0.244015\pi$
$194$	14.4910	1.04039
$195$	0	0
$196$	0	0
$197$	−2.30558	−0.164265	−0.0821327	−	0.996621i	$-0.526173\pi$
$197$	−2.30558	−0.164265	−0.0821327	+	0.996621i	$0.526173\pi$
$198$	0	0
$199$	−2.04625	−0.145055	−0.0725273	−	0.997366i	$-0.523106\pi$
$199$	−2.04625	−0.145055	−0.0725273	+	0.997366i	$0.523106\pi$
$200$	4.84190	0.342374
$201$	0	0
$202$	−5.29482	−0.372542
$203$	6.94758	0.487625
$204$	0	0
$205$	−2.30917	−0.161280
$206$	−7.33775	−0.511245
$207$	0	0
$208$	1.00000	0.0693375
$209$	3.90465	0.270090
$210$	0	0
$211$	−9.90134	−0.681636	−0.340818	−	0.940129i	$-0.610704\pi$
$211$	−9.90134	−0.681636	−0.340818	+	0.940129i	$0.610704\pi$
$212$	−2.78404	−0.191208
$213$	0	0
$214$	1.84721	0.126273
$215$	2.65163	0.180840
$216$	0	0
$217$	−5.60496	−0.380489
$218$	−6.64906	−0.450331
$219$	0	0
$220$	1.98019	0.133504
$221$	2.37780	0.159948
$222$	0	0
$223$	−16.6850	−1.11731	−0.558654	−	0.829400i	$-0.688682\pi$
$223$	−16.6850	−1.11731	−0.558654	+	0.829400i	$0.688682\pi$
$224$	−2.64575	−0.176777
$225$	0	0
$226$	13.9373	0.927092
$227$	9.97976	0.662380	0.331190	−	0.943564i	$-0.392550\pi$
$227$	9.97976	0.662380	0.331190	+	0.943564i	$0.392550\pi$
$228$	0	0
$229$	−20.3064	−1.34189	−0.670943	−	0.741508i	$-0.734111\pi$
$229$	−20.3064	−1.34189	−0.670943	+	0.741508i	$0.734111\pi$
$230$	−2.74110	−0.180743
$231$	0	0
$232$	−2.62594	−0.172401
$233$	3.81878	0.250177	0.125088	−	0.992146i	$-0.460079\pi$
$233$	3.81878	0.250177	0.125088	+	0.992146i	$0.460079\pi$
$234$	0	0
$235$	4.07611	0.265896
$236$	−9.53059	−0.620389
$237$	0	0
$238$	−6.29107	−0.407790
$239$	−12.3955	−0.801797	−0.400898	−	0.916123i	$-0.631302\pi$
$239$	−12.3955	−0.801797	−0.400898	+	0.916123i	$0.631302\pi$
$240$	0	0
$241$	12.9229	0.832437	0.416218	−	0.909265i	$-0.363355\pi$
$241$	12.9229	0.832437	0.416218	+	0.909265i	$0.363355\pi$
$242$	−13.8023	−0.887244
$243$	0	0
$244$	−4.86171	−0.311239
$245$	0	0
$246$	0	0
$247$	−0.784036	−0.0498870
$248$	2.11847	0.134523
$249$	0	0
$250$	−3.91327	−0.247497
$251$	14.3906	0.908326	0.454163	−	0.890919i	$-0.349938\pi$
$251$	14.3906	0.908326	0.454163	+	0.890919i	$0.349938\pi$
$252$	0	0
$253$	34.3329	2.15849
$254$	7.10080	0.445544
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.4798	1.15274	0.576368	−	0.817190i	$-0.304469\pi$
$257$	18.4798	1.15274	0.576368	+	0.817190i	$0.304469\pi$
$258$	0	0
$259$	1.87451	0.116476
$260$	−0.397613	−0.0246589
$261$	0	0
$262$	4.25933	0.263142
$263$	−0.846783	−0.0522148	−0.0261074	−	0.999659i	$-0.508311\pi$
$263$	−0.846783	−0.0522148	−0.0261074	+	0.999659i	$0.508311\pi$
$264$	0	0
$265$	1.10697	0.0680006
$266$	2.07437	0.127188
$267$	0	0
$268$	14.6293	0.893624
$269$	−4.55828	−0.277923	−0.138962	−	0.990298i	$-0.544376\pi$
$269$	−4.55828	−0.277923	−0.138962	+	0.990298i	$0.544376\pi$
$270$	0	0
$271$	10.5665	0.641870	0.320935	−	0.947101i	$-0.396003\pi$
$271$	10.5665	0.641870	0.320935	+	0.947101i	$0.396003\pi$
$272$	2.37780	0.144175
$273$	0	0
$274$	2.45003	0.148012
$275$	24.1136	1.45410
$276$	0	0
$277$	10.6574	0.640339	0.320170	−	0.947360i	$-0.396260\pi$
$277$	10.6574	0.640339	0.320170	+	0.947360i	$0.396260\pi$
$278$	−1.23909	−0.0743155
$279$	0	0
$280$	1.05199	0.0628682
$281$	17.4650	1.04187	0.520936	−	0.853596i	$-0.325583\pi$
$281$	17.4650	1.04187	0.520936	+	0.853596i	$0.325583\pi$
$282$	0	0
$283$	−24.2668	−1.44251	−0.721256	−	0.692668i	$-0.756435\pi$
$283$	−24.2668	−1.44251	−0.721256	+	0.692668i	$0.756435\pi$
$284$	15.3382	0.910154
$285$	0	0
$286$	4.98019	0.294485
$287$	15.3654	0.906993
$288$	0	0
$289$	−11.3461	−0.667415
$290$	1.04411	0.0613122
$291$	0	0
$292$	8.62925	0.504989
$293$	−26.4386	−1.54456	−0.772278	−	0.635284i	$-0.780883\pi$
$293$	−26.4386	−1.54456	−0.772278	+	0.635284i	$0.780883\pi$
$294$	0	0
$295$	3.78949	0.220633
$296$	−0.708497	−0.0411806
$297$	0	0
$298$	13.8221	0.800692
$299$	−6.89389	−0.398684
$300$	0	0
$301$	−17.6442	−1.01699
$302$	−4.72460	−0.271870
$303$	0	0
$304$	−0.784036	−0.0449676
$305$	1.93308	0.110688
$306$	0	0
$307$	1.56807	0.0894947	0.0447473	−	0.998998i	$-0.485752\pi$
$307$	1.56807	0.0894947	0.0447473	+	0.998998i	$0.485752\pi$
$308$	−13.1763	−0.750792
$309$	0	0
$310$	−0.842333	−0.0478413
$311$	3.46410	0.196431	0.0982156	−	0.995165i	$-0.468687\pi$
$311$	3.46410	0.196431	0.0982156	+	0.995165i	$0.468687\pi$
$312$	0	0
$313$	−11.9823	−0.677281	−0.338641	−	0.940916i	$-0.609967\pi$
$313$	−11.9823	−0.677281	−0.338641	+	0.940916i	$0.609967\pi$
$314$	21.6817	1.22357
$315$	0	0
$316$	−13.7036	−0.770889
$317$	−24.1458	−1.35616	−0.678081	−	0.734987i	$-0.737188\pi$
$317$	−24.1458	−1.35616	−0.678081	+	0.734987i	$0.737188\pi$
$318$	0	0
$319$	−13.0777	−0.732209
$320$	−0.397613	−0.0222273
$321$	0	0
$322$	18.2395	1.01645
$323$	−1.86428	−0.103732
$324$	0	0
$325$	−4.84190	−0.268580
$326$	22.6293	1.25332
$327$	0	0
$328$	−5.80759	−0.320671
$329$	−27.1228	−1.49533
$330$	0	0
$331$	2.62925	0.144517	0.0722584	−	0.997386i	$-0.476979\pi$
$331$	2.62925	0.144517	0.0722584	+	0.997386i	$0.476979\pi$
$332$	−5.77542	−0.316967
$333$	0	0
$334$	−1.41742	−0.0775580
$335$	−5.81678	−0.317805
$336$	0	0
$337$	26.0392	1.41845	0.709224	−	0.704984i	$-0.249046\pi$
$337$	26.0392	1.41845	0.709224	+	0.704984i	$0.249046\pi$
$338$	−1.00000	−0.0543928
$339$	0	0
$340$	−0.945446	−0.0512740
$341$	10.5504	0.571336
$342$	0	0
$343$	−18.5203	−1.00000
$344$	6.66888	0.359562
$345$	0	0
$346$	−15.1908	−0.816665
$347$	14.1775	0.761089	0.380544	−	0.924763i	$-0.375737\pi$
$347$	14.1775	0.761089	0.380544	+	0.924763i	$0.375737\pi$
$348$	0	0
$349$	14.4319	0.772523	0.386262	−	0.922389i	$-0.373766\pi$
$349$	14.4319	0.772523	0.386262	+	0.922389i	$0.373766\pi$
$350$	12.8105	0.684749
$351$	0	0
$352$	4.98019	0.265445
$353$	33.0414	1.75861	0.879307	−	0.476255i	$-0.158006\pi$
$353$	33.0414	1.75861	0.879307	+	0.476255i	$0.158006\pi$
$354$	0	0
$355$	−6.09866	−0.323683
$356$	−4.51609	−0.239352
$357$	0	0
$358$	6.92820	0.366167
$359$	16.6661	0.879605	0.439802	−	0.898095i	$-0.355048\pi$
$359$	16.6661	0.879605	0.439802	+	0.898095i	$0.355048\pi$
$360$	0	0
$361$	−18.3853	−0.967647
$362$	23.0590	1.21196
$363$	0	0
$364$	2.64575	0.138675
$365$	−3.43111	−0.179592
$366$	0	0
$367$	−28.7143	−1.49887	−0.749436	−	0.662076i	$-0.769675\pi$
$367$	−28.7143	−1.49887	−0.749436	+	0.662076i	$0.769675\pi$
$368$	−6.89389	−0.359369
$369$	0	0
$370$	0.281708	0.0146453
$371$	−7.36587	−0.382417
$372$	0	0
$373$	0.476036	0.0246482	0.0123241	−	0.999924i	$-0.496077\pi$
$373$	0.476036	0.0246482	0.0123241	+	0.999924i	$0.496077\pi$
$374$	11.8419	0.612330
$375$	0	0
$376$	10.2515	0.528678
$377$	2.62594	0.135243
$378$	0	0
$379$	16.5537	0.850307	0.425154	−	0.905121i	$-0.360220\pi$
$379$	16.5537	0.850307	0.425154	+	0.905121i	$0.360220\pi$
$380$	0.311743	0.0159921
$381$	0	0
$382$	7.01407	0.358871
$383$	36.8745	1.88420	0.942100	−	0.335333i	$-0.108849\pi$
$383$	36.8745	1.88420	0.942100	+	0.335333i	$0.108849\pi$
$384$	0	0
$385$	5.23909	0.267009
$386$	−20.0128	−1.01862
$387$	0	0
$388$	−14.4910	−0.735667
$389$	10.8494	0.550084	0.275042	−	0.961432i	$-0.411308\pi$
$389$	10.8494	0.550084	0.275042	+	0.961432i	$0.411308\pi$
$390$	0	0
$391$	−16.3923	−0.828994
$392$	0	0
$393$	0	0
$394$	2.30558	0.116153
$395$	5.44874	0.274156
$396$	0	0
$397$	−32.5080	−1.63153	−0.815766	−	0.578382i	$-0.803684\pi$
$397$	−32.5080	−1.63153	−0.815766	+	0.578382i	$0.803684\pi$
$398$	2.04625	0.102569
$399$	0	0
$400$	−4.84190	−0.242095
$401$	4.63182	0.231302	0.115651	−	0.993290i	$-0.463105\pi$
$401$	4.63182	0.231302	0.115651	+	0.993290i	$0.463105\pi$
$402$	0	0
$403$	−2.11847	−0.105529
$404$	5.29482	0.263427
$405$	0	0
$406$	−6.94758	−0.344803
$407$	−3.52845	−0.174899
$408$	0	0
$409$	4.46727	0.220892	0.110446	−	0.993882i	$-0.464772\pi$
$409$	4.46727	0.220892	0.110446	+	0.993882i	$0.464772\pi$
$410$	2.30917	0.114042
$411$	0	0
$412$	7.33775	0.361505
$413$	−25.2156	−1.24078
$414$	0	0
$415$	2.29638	0.112725
$416$	−1.00000	−0.0490290
$417$	0	0
$418$	−3.90465	−0.190983
$419$	−37.3650	−1.82540	−0.912701	−	0.408628i	$-0.866007\pi$
$419$	−37.3650	−1.82540	−0.912701	+	0.408628i	$0.866007\pi$
$420$	0	0
$421$	23.9009	1.16486	0.582430	−	0.812881i	$-0.302102\pi$
$421$	23.9009	1.16486	0.582430	+	0.812881i	$0.302102\pi$
$422$	9.90134	0.481990
$423$	0	0
$424$	2.78404	0.135205
$425$	−11.5131	−0.558467
$426$	0	0
$427$	−12.8629	−0.622479
$428$	−1.84721	−0.0892884
$429$	0	0
$430$	−2.65163	−0.127873
$431$	−1.11060	−0.0534956	−0.0267478	−	0.999642i	$-0.508515\pi$
$431$	−1.11060	−0.0534956	−0.0267478	+	0.999642i	$0.508515\pi$
$432$	0	0
$433$	−3.59665	−0.172844	−0.0864220	−	0.996259i	$-0.527543\pi$
$433$	−3.59665	−0.172844	−0.0864220	+	0.996259i	$0.527543\pi$
$434$	5.60496	0.269046
$435$	0	0
$436$	6.64906	0.318432
$437$	5.40506	0.258559
$438$	0	0
$439$	40.9819	1.95596	0.977981	−	0.208696i	$-0.0669220\pi$
$439$	40.9819	1.95596	0.977981	+	0.208696i	$0.0669220\pi$
$440$	−1.98019	−0.0944018
$441$	0	0
$442$	−2.37780	−0.113101
$443$	14.1040	0.670100	0.335050	−	0.942200i	$-0.391247\pi$
$443$	14.1040	0.670100	0.335050	+	0.942200i	$0.391247\pi$
$444$	0	0
$445$	1.79566	0.0851223
$446$	16.6850	0.790057
$447$	0	0
$448$	2.64575	0.125000
$449$	10.6913	0.504552	0.252276	−	0.967655i	$-0.418821\pi$
$449$	10.6913	0.504552	0.252276	+	0.967655i	$0.418821\pi$
$450$	0	0
$451$	−28.9229	−1.36193
$452$	−13.9373	−0.655553
$453$	0	0
$454$	−9.97976	−0.468373
$455$	−1.05199	−0.0493178
$456$	0	0
$457$	35.9732	1.68275	0.841377	−	0.540449i	$-0.181745\pi$
$457$	35.9732	1.68275	0.841377	+	0.540449i	$0.181745\pi$
$458$	20.3064	0.948857
$459$	0	0
$460$	2.74110	0.127805
$461$	15.6486	0.728830	0.364415	−	0.931237i	$-0.381269\pi$
$461$	15.6486	0.728830	0.364415	+	0.931237i	$0.381269\pi$
$462$	0	0
$463$	7.11965	0.330878	0.165439	−	0.986220i	$-0.447096\pi$
$463$	7.11965	0.330878	0.165439	+	0.986220i	$0.447096\pi$
$464$	2.62594	0.121906
$465$	0	0
$466$	−3.81878	−0.176902
$467$	−19.2321	−0.889954	−0.444977	−	0.895542i	$-0.646788\pi$
$467$	−19.2321	−0.889954	−0.444977	+	0.895542i	$0.646788\pi$
$468$	0	0
$469$	38.7054	1.78725
$470$	−4.07611	−0.188017
$471$	0	0
$472$	9.53059	0.438681
$473$	33.2123	1.52710
$474$	0	0
$475$	3.79623	0.174183
$476$	6.29107	0.288351
$477$	0	0
$478$	12.3955	0.566956
$479$	−25.0761	−1.14576	−0.572878	−	0.819640i	$-0.694173\pi$
$479$	−25.0761	−1.14576	−0.572878	+	0.819640i	$0.694173\pi$
$480$	0	0
$481$	0.708497	0.0323047
$482$	−12.9229	−0.588622
$483$	0	0
$484$	13.8023	0.627376
$485$	5.76180	0.261630
$486$	0	0
$487$	23.4304	1.06173	0.530866	−	0.847456i	$-0.321867\pi$
$487$	23.4304	1.06173	0.530866	+	0.847456i	$0.321867\pi$
$488$	4.86171	0.220079
$489$	0	0
$490$	0	0
$491$	38.3354	1.73005	0.865027	−	0.501725i	$-0.167301\pi$
$491$	38.3354	1.73005	0.865027	+	0.501725i	$0.167301\pi$
$492$	0	0
$493$	6.24397	0.281214
$494$	0.784036	0.0352755
$495$	0	0
$496$	−2.11847	−0.0951223
$497$	40.5810	1.82031
$498$	0	0
$499$	−3.09749	−0.138663	−0.0693313	−	0.997594i	$-0.522087\pi$
$499$	−3.09749	−0.138663	−0.0693313	+	0.997594i	$0.522087\pi$
$500$	3.91327	0.175007
$501$	0	0
$502$	−14.3906	−0.642284
$503$	15.4125	0.687211	0.343606	−	0.939114i	$-0.388352\pi$
$503$	15.4125	0.687211	0.343606	+	0.939114i	$0.388352\pi$
$504$	0	0
$505$	−2.10529	−0.0936841
$506$	−34.3329	−1.52628
$507$	0	0
$508$	−7.10080	−0.315047
$509$	−32.5971	−1.44484	−0.722421	−	0.691454i	$-0.756971\pi$
$509$	−32.5971	−1.44484	−0.722421	+	0.691454i	$0.756971\pi$
$510$	0	0
$511$	22.8309	1.00998
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−18.4798	−0.815108
$515$	−2.91759	−0.128564
$516$	0	0
$517$	51.0542	2.24536
$518$	−1.87451	−0.0823611
$519$	0	0
$520$	0.397613	0.0174365
$521$	39.5026	1.73064	0.865320	−	0.501220i	$-0.167115\pi$
$521$	39.5026	1.73064	0.865320	+	0.501220i	$0.167115\pi$
$522$	0	0
$523$	−22.3589	−0.977684	−0.488842	−	0.872372i	$-0.662581\pi$
$523$	−22.3589	−0.977684	−0.488842	+	0.872372i	$0.662581\pi$
$524$	−4.25933	−0.186070
$525$	0	0
$526$	0.846783	0.0369215
$527$	−5.03731	−0.219429
$528$	0	0
$529$	24.5257	1.06634
$530$	−1.10697	−0.0480837
$531$	0	0
$532$	−2.07437	−0.0899352
$533$	5.80759	0.251555
$534$	0	0
$535$	0.734476	0.0317542
$536$	−14.6293	−0.631887
$537$	0	0
$538$	4.55828	0.196521
$539$	0	0
$540$	0	0
$541$	−19.5549	−0.840730	−0.420365	−	0.907355i	$-0.638098\pi$
$541$	−19.5549	−0.840730	−0.420365	+	0.907355i	$0.638098\pi$
$542$	−10.5665	−0.453870
$543$	0	0
$544$	−2.37780	−0.101947
$545$	−2.64376	−0.113246
$546$	0	0
$547$	−25.3840	−1.08534	−0.542671	−	0.839946i	$-0.682587\pi$
$547$	−25.3840	−1.08534	−0.542671	+	0.839946i	$0.682587\pi$
$548$	−2.45003	−0.104660
$549$	0	0
$550$	−24.1136	−1.02821
$551$	−2.05883	−0.0877092
$552$	0	0
$553$	−36.2564	−1.54178
$554$	−10.6574	−0.452788
$555$	0	0
$556$	1.23909	0.0525490
$557$	35.8125	1.51742	0.758712	−	0.651426i	$-0.225829\pi$
$557$	35.8125	1.51742	0.758712	+	0.651426i	$0.225829\pi$
$558$	0	0
$559$	−6.66888	−0.282063
$560$	−1.05199	−0.0444545
$561$	0	0
$562$	−17.4650	−0.736715
$563$	−5.62751	−0.237171	−0.118586	−	0.992944i	$-0.537836\pi$
$563$	−5.62751	−0.237171	−0.118586	+	0.992944i	$0.537836\pi$
$564$	0	0
$565$	5.54164	0.233138
$566$	24.2668	1.02001
$567$	0	0
$568$	−15.3382	−0.643576
$569$	−4.86659	−0.204018	−0.102009	−	0.994783i	$-0.532527\pi$
$569$	−4.86659	−0.204018	−0.102009	+	0.994783i	$0.532527\pi$
$570$	0	0
$571$	−28.2101	−1.18056	−0.590278	−	0.807200i	$-0.700982\pi$
$571$	−28.2101	−1.18056	−0.590278	+	0.807200i	$0.700982\pi$
$572$	−4.98019	−0.208232
$573$	0	0
$574$	−15.3654	−0.641341
$575$	33.3795	1.39202
$576$	0	0
$577$	37.1660	1.54724	0.773621	−	0.633649i	$-0.218444\pi$
$577$	37.1660	1.54724	0.773621	+	0.633649i	$0.218444\pi$
$578$	11.3461	0.471934
$579$	0	0
$580$	−1.04411	−0.0433542
$581$	−15.2803	−0.633934
$582$	0	0
$583$	13.8650	0.574231
$584$	−8.62925	−0.357081
$585$	0	0
$586$	26.4386	1.09217
$587$	28.4315	1.17350	0.586748	−	0.809770i	$-0.300408\pi$
$587$	28.4315	1.17350	0.586748	+	0.809770i	$0.300408\pi$
$588$	0	0
$589$	1.66096	0.0684387
$590$	−3.78949	−0.156011
$591$	0	0
$592$	0.708497	0.0291191
$593$	25.0305	1.02788	0.513939	−	0.857827i	$-0.328186\pi$
$593$	25.0305	1.02788	0.513939	+	0.857827i	$0.328186\pi$
$594$	0	0
$595$	−2.50141	−0.102548
$596$	−13.8221	−0.566175
$597$	0	0
$598$	6.89389	0.281912
$599$	35.2862	1.44175	0.720877	−	0.693063i	$-0.243739\pi$
$599$	35.2862	1.44175	0.720877	+	0.693063i	$0.243739\pi$
$600$	0	0
$601$	25.7710	1.05122	0.525610	−	0.850726i	$-0.323837\pi$
$601$	25.7710	1.05122	0.525610	+	0.850726i	$0.323837\pi$
$602$	17.6442	0.719123
$603$	0	0
$604$	4.72460	0.192241
$605$	−5.48797	−0.223118
$606$	0	0
$607$	45.5187	1.84755	0.923773	−	0.382940i	$-0.125088\pi$
$607$	45.5187	1.84755	0.923773	+	0.382940i	$0.125088\pi$
$608$	0.784036	0.0317969
$609$	0	0
$610$	−1.93308	−0.0782682
$611$	−10.2515	−0.414729
$612$	0	0
$613$	46.1224	1.86286	0.931432	−	0.363915i	$-0.118560\pi$
$613$	46.1224	1.86286	0.931432	+	0.363915i	$0.118560\pi$
$614$	−1.56807	−0.0632823
$615$	0	0
$616$	13.1763	0.530890
$617$	43.1991	1.73913	0.869565	−	0.493819i	$-0.164400\pi$
$617$	43.1991	1.73913	0.869565	+	0.493819i	$0.164400\pi$
$618$	0	0
$619$	−21.7660	−0.874848	−0.437424	−	0.899255i	$-0.644109\pi$
$619$	−21.7660	−0.874848	−0.437424	+	0.899255i	$0.644109\pi$
$620$	0.842333	0.0338289
$621$	0	0
$622$	−3.46410	−0.138898
$623$	−11.9484	−0.478704
$624$	0	0
$625$	22.6536	0.906142
$626$	11.9823	0.478910
$627$	0	0
$628$	−21.6817	−0.865193
$629$	1.68467	0.0671721
$630$	0	0
$631$	−33.8160	−1.34620	−0.673098	−	0.739554i	$-0.735037\pi$
$631$	−33.8160	−1.34620	−0.673098	+	0.739554i	$0.735037\pi$
$632$	13.7036	0.545101
$633$	0	0
$634$	24.1458	0.958951
$635$	2.82337	0.112042
$636$	0	0
$637$	0	0
$638$	13.0777	0.517750
$639$	0	0
$640$	0.397613	0.0157170
$641$	−49.5273	−1.95621	−0.978106	−	0.208109i	$-0.933269\pi$
$641$	−49.5273	−1.95621	−0.978106	+	0.208109i	$0.933269\pi$
$642$	0	0
$643$	−20.2771	−0.799652	−0.399826	−	0.916591i	$-0.630930\pi$
$643$	−20.2771	−0.799652	−0.399826	+	0.916591i	$0.630930\pi$
$644$	−18.2395	−0.718738
$645$	0	0
$646$	1.86428	0.0733493
$647$	−8.72043	−0.342836	−0.171418	−	0.985198i	$-0.554835\pi$
$647$	−8.72043	−0.342836	−0.171418	+	0.985198i	$0.554835\pi$
$648$	0	0
$649$	47.4641	1.86313
$650$	4.84190	0.189915
$651$	0	0
$652$	−22.6293	−0.886230
$653$	−0.281708	−0.0110241	−0.00551204	−	0.999985i	$-0.501755\pi$
$653$	−0.281708	−0.0110241	−0.00551204	+	0.999985i	$0.501755\pi$
$654$	0	0
$655$	1.69357	0.0661731
$656$	5.80759	0.226748
$657$	0	0
$658$	27.1228	1.05736
$659$	−16.9216	−0.659171	−0.329586	−	0.944126i	$-0.606909\pi$
$659$	−16.9216	−0.659171	−0.329586	+	0.944126i	$0.606909\pi$
$660$	0	0
$661$	−0.668875	−0.0260162	−0.0130081	−	0.999915i	$-0.504141\pi$
$661$	−0.668875	−0.0260162	−0.0130081	+	0.999915i	$0.504141\pi$
$662$	−2.62925	−0.102189
$663$	0	0
$664$	5.77542	0.224130
$665$	0.824795	0.0319842
$666$	0	0
$667$	−18.1029	−0.700949
$668$	1.41742	0.0548418
$669$	0	0
$670$	5.81678	0.224722
$671$	24.2123	0.934704
$672$	0	0
$673$	46.8718	1.80677	0.903387	−	0.428826i	$-0.141073\pi$
$673$	46.8718	1.80677	0.903387	+	0.428826i	$0.141073\pi$
$674$	−26.0392	−1.00299
$675$	0	0
$676$	1.00000	0.0384615
$677$	42.6413	1.63884	0.819419	−	0.573195i	$-0.194296\pi$
$677$	42.6413	1.63884	0.819419	+	0.573195i	$0.194296\pi$
$678$	0	0
$679$	−38.3395	−1.47133
$680$	0.945446	0.0362562
$681$	0	0
$682$	−10.5504	−0.403996
$683$	−15.7709	−0.603458	−0.301729	−	0.953394i	$-0.597564\pi$
$683$	−15.7709	−0.603458	−0.301729	+	0.953394i	$0.597564\pi$
$684$	0	0
$685$	0.974164	0.0372209
$686$	18.5203	0.707107
$687$	0	0
$688$	−6.66888	−0.254248
$689$	−2.78404	−0.106063
$690$	0	0
$691$	−27.8425	−1.05918	−0.529589	−	0.848254i	$-0.677654\pi$
$691$	−27.8425	−1.05918	−0.529589	+	0.848254i	$0.677654\pi$
$692$	15.1908	0.577469
$693$	0	0
$694$	−14.1775	−0.538171
$695$	−0.492678	−0.0186883
$696$	0	0
$697$	13.8093	0.523065
$698$	−14.4319	−0.546256
$699$	0	0
$700$	−12.8105	−0.484190
$701$	−22.3494	−0.844124	−0.422062	−	0.906567i	$-0.638694\pi$
$701$	−22.3494	−0.844124	−0.422062	+	0.906567i	$0.638694\pi$
$702$	0	0
$703$	−0.555488	−0.0209506
$704$	−4.98019	−0.187698
$705$	0	0
$706$	−33.0414	−1.24353
$707$	14.0088	0.526854
$708$	0	0
$709$	2.94972	0.110779	0.0553896	−	0.998465i	$-0.482360\pi$
$709$	2.94972	0.110779	0.0553896	+	0.998465i	$0.482360\pi$
$710$	6.09866	0.228879
$711$	0	0
$712$	4.51609	0.169248
$713$	14.6045	0.546944
$714$	0	0
$715$	1.98019	0.0740549
$716$	−6.92820	−0.258919
$717$	0	0
$718$	−16.6661	−0.621975
$719$	−25.0053	−0.932542	−0.466271	−	0.884642i	$-0.654403\pi$
$719$	−25.0053	−0.932542	−0.466271	+	0.884642i	$0.654403\pi$
$720$	0	0
$721$	19.4139	0.723010
$722$	18.3853	0.684230
$723$	0	0
$724$	−23.0590	−0.856983
$725$	−12.7145	−0.472207
$726$	0	0
$727$	23.0149	0.853577	0.426788	−	0.904352i	$-0.359645\pi$
$727$	23.0149	0.853577	0.426788	+	0.904352i	$0.359645\pi$
$728$	−2.64575	−0.0980581
$729$	0	0
$730$	3.43111	0.126991
$731$	−15.8573	−0.586502
$732$	0	0
$733$	8.25676	0.304970	0.152485	−	0.988306i	$-0.451272\pi$
$733$	8.25676	0.304970	0.152485	+	0.988306i	$0.451272\pi$
$734$	28.7143	1.05986
$735$	0	0
$736$	6.89389	0.254112
$737$	−72.8564	−2.68370
$738$	0	0
$739$	20.4749	0.753180	0.376590	−	0.926380i	$-0.377097\pi$
$739$	20.4749	0.753180	0.376590	+	0.926380i	$0.377097\pi$
$740$	−0.281708	−0.0103558
$741$	0	0
$742$	7.36587	0.270410
$743$	7.93522	0.291115	0.145558	−	0.989350i	$-0.453502\pi$
$743$	7.93522	0.291115	0.145558	+	0.989350i	$0.453502\pi$
$744$	0	0
$745$	5.49585	0.201352
$746$	−0.476036	−0.0174289
$747$	0	0
$748$	−11.8419	−0.432983
$749$	−4.88726	−0.178577
$750$	0	0
$751$	7.40061	0.270052	0.135026	−	0.990842i	$-0.456888\pi$
$751$	7.40061	0.270052	0.135026	+	0.990842i	$0.456888\pi$
$752$	−10.2515	−0.373832
$753$	0	0
$754$	−2.62594	−0.0956311
$755$	−1.87856	−0.0683680
$756$	0	0
$757$	−7.88601	−0.286622	−0.143311	−	0.989678i	$-0.545775\pi$
$757$	−7.88601	−0.286622	−0.143311	+	0.989678i	$0.545775\pi$
$758$	−16.5537	−0.601258
$759$	0	0
$760$	−0.311743	−0.0113081
$761$	−28.5896	−1.03637	−0.518187	−	0.855268i	$-0.673393\pi$
$761$	−28.5896	−1.03637	−0.518187	+	0.855268i	$0.673393\pi$
$762$	0	0
$763$	17.5918	0.636865
$764$	−7.01407	−0.253760
$765$	0	0
$766$	−36.8745	−1.33233
$767$	−9.53059	−0.344130
$768$	0	0
$769$	−18.8961	−0.681410	−0.340705	−	0.940170i	$-0.610666\pi$
$769$	−18.8961	−0.681410	−0.340705	+	0.940170i	$0.610666\pi$
$770$	−5.23909	−0.188804
$771$	0	0
$772$	20.0128	0.720276
$773$	10.7728	0.387472	0.193736	−	0.981054i	$-0.437939\pi$
$773$	10.7728	0.387472	0.193736	+	0.981054i	$0.437939\pi$
$774$	0	0
$775$	10.2574	0.368458
$776$	14.4910	0.520195
$777$	0	0
$778$	−10.8494	−0.388968
$779$	−4.55336	−0.163141
$780$	0	0
$781$	−76.3870	−2.73334
$782$	16.3923	0.586188
$783$	0	0
$784$	0	0
$785$	8.62092	0.307694
$786$	0	0
$787$	40.4741	1.44275	0.721373	−	0.692547i	$-0.243512\pi$
$787$	40.4741	1.44275	0.721373	+	0.692547i	$0.243512\pi$
$788$	−2.30558	−0.0821327
$789$	0	0
$790$	−5.44874	−0.193857
$791$	−36.8745	−1.31111
$792$	0	0
$793$	−4.86171	−0.172645
$794$	32.5080	1.15367
$795$	0	0
$796$	−2.04625	−0.0725273
$797$	−36.8601	−1.30565	−0.652827	−	0.757507i	$-0.726417\pi$
$797$	−36.8601	−1.30565	−0.652827	+	0.757507i	$0.726417\pi$
$798$	0	0
$799$	−24.3759	−0.862358
$800$	4.84190	0.171187
$801$	0	0
$802$	−4.63182	−0.163555
$803$	−42.9753	−1.51657
$804$	0	0
$805$	7.25227	0.255609
$806$	2.11847	0.0746201
$807$	0	0
$808$	−5.29482	−0.186271
$809$	−37.4159	−1.31547	−0.657736	−	0.753249i	$-0.728486\pi$
$809$	−37.4159	−1.31547	−0.657736	+	0.753249i	$0.728486\pi$
$810$	0	0
$811$	−51.0901	−1.79402	−0.897008	−	0.442015i	$-0.854264\pi$
$811$	−51.0901	−1.79402	−0.897008	+	0.442015i	$0.854264\pi$
$812$	6.94758	0.243812
$813$	0	0
$814$	3.52845	0.123672
$815$	8.99769	0.315175
$816$	0	0
$817$	5.22864	0.182927
$818$	−4.46727	−0.156194
$819$	0	0
$820$	−2.30917	−0.0806399
$821$	−4.47390	−0.156140	−0.0780700	−	0.996948i	$-0.524876\pi$
$821$	−4.47390	−0.156140	−0.0780700	+	0.996948i	$0.524876\pi$
$822$	0	0
$823$	14.8894	0.519013	0.259507	−	0.965741i	$-0.416440\pi$
$823$	14.8894	0.519013	0.259507	+	0.965741i	$0.416440\pi$
$824$	−7.33775	−0.255623
$825$	0	0
$826$	25.2156	0.877362
$827$	11.1361	0.387242	0.193621	−	0.981076i	$-0.437977\pi$
$827$	11.1361	0.387242	0.193621	+	0.981076i	$0.437977\pi$
$828$	0	0
$829$	12.8943	0.447838	0.223919	−	0.974608i	$-0.428115\pi$
$829$	12.8943	0.447838	0.223919	+	0.974608i	$0.428115\pi$
$830$	−2.29638	−0.0797086
$831$	0	0
$832$	1.00000	0.0346688
$833$	0	0
$834$	0	0
$835$	−0.563587	−0.0195037
$836$	3.90465	0.135045
$837$	0	0
$838$	37.3650	1.29075
$839$	−31.0863	−1.07322	−0.536609	−	0.843831i	$-0.680295\pi$
$839$	−31.0863	−1.07322	−0.536609	+	0.843831i	$0.680295\pi$
$840$	0	0
$841$	−22.1044	−0.762222
$842$	−23.9009	−0.823681
$843$	0	0
$844$	−9.90134	−0.340818
$845$	−0.397613	−0.0136783
$846$	0	0
$847$	36.5174	1.25475
$848$	−2.78404	−0.0956042
$849$	0	0
$850$	11.5131	0.394896
$851$	−4.88430	−0.167432
$852$	0	0
$853$	−26.7349	−0.915387	−0.457693	−	0.889110i	$-0.651324\pi$
$853$	−26.7349	−0.915387	−0.457693	+	0.889110i	$0.651324\pi$
$854$	12.8629	0.440159
$855$	0	0
$856$	1.84721	0.0631364
$857$	36.1679	1.23547	0.617735	−	0.786386i	$-0.288050\pi$
$857$	36.1679	1.23547	0.617735	+	0.786386i	$0.288050\pi$
$858$	0	0
$859$	−26.0104	−0.887465	−0.443732	−	0.896159i	$-0.646346\pi$
$859$	−26.0104	−0.887465	−0.443732	+	0.896159i	$0.646346\pi$
$860$	2.65163	0.0904199
$861$	0	0
$862$	1.11060	0.0378271
$863$	28.5281	0.971106	0.485553	−	0.874207i	$-0.338618\pi$
$863$	28.5281	0.971106	0.485553	+	0.874207i	$0.338618\pi$
$864$	0	0
$865$	−6.04008	−0.205369
$866$	3.59665	0.122219
$867$	0	0
$868$	−5.60496	−0.190245
$869$	68.2466	2.31511
$870$	0	0
$871$	14.6293	0.495693
$872$	−6.64906	−0.225166
$873$	0	0
$874$	−5.40506	−0.182829
$875$	10.3535	0.350014
$876$	0	0
$877$	16.2932	0.550184	0.275092	−	0.961418i	$-0.411292\pi$
$877$	16.2932	0.550184	0.275092	+	0.961418i	$0.411292\pi$
$878$	−40.9819	−1.38307
$879$	0	0
$880$	1.98019	0.0667522
$881$	−25.8643	−0.871390	−0.435695	−	0.900094i	$-0.643497\pi$
$881$	−25.8643	−0.871390	−0.435695	+	0.900094i	$0.643497\pi$
$882$	0	0
$883$	33.0410	1.11192	0.555959	−	0.831210i	$-0.312351\pi$
$883$	33.0410	1.11192	0.555959	+	0.831210i	$0.312351\pi$
$884$	2.37780	0.0799741
$885$	0	0
$886$	−14.1040	−0.473832
$887$	−54.3144	−1.82370	−0.911849	−	0.410525i	$-0.865345\pi$
$887$	−54.3144	−1.82370	−0.911849	+	0.410525i	$0.865345\pi$
$888$	0	0
$889$	−18.7870	−0.630094
$890$	−1.79566	−0.0601905
$891$	0	0
$892$	−16.6850	−0.558654
$893$	8.03751	0.268965
$894$	0	0
$895$	2.75475	0.0920810
$896$	−2.64575	−0.0883883
$897$	0	0
$898$	−10.6913	−0.356772
$899$	−5.56299	−0.185536
$900$	0	0
$901$	−6.61989	−0.220540
$902$	28.9229	0.963027
$903$	0	0
$904$	13.9373	0.463546
$905$	9.16858	0.304774
$906$	0	0
$907$	1.56807	0.0520670	0.0260335	−	0.999661i	$-0.491712\pi$
$907$	1.56807	0.0520670	0.0260335	+	0.999661i	$0.491712\pi$
$908$	9.97976	0.331190
$909$	0	0
$910$	1.05199	0.0348730
$911$	−43.0753	−1.42715	−0.713574	−	0.700580i	$-0.752925\pi$
$911$	−43.0753	−1.42715	−0.713574	+	0.700580i	$0.752925\pi$
$912$	0	0
$913$	28.7627	0.951905
$914$	−35.9732	−1.18989
$915$	0	0
$916$	−20.3064	−0.670943
$917$	−11.2691	−0.372139
$918$	0	0
$919$	−58.5403	−1.93106	−0.965532	−	0.260283i	$-0.916184\pi$
$919$	−58.5403	−1.93106	−0.965532	+	0.260283i	$0.916184\pi$
$920$	−2.74110	−0.0903714
$921$	0	0
$922$	−15.6486	−0.515360
$923$	15.3382	0.504862
$924$	0	0
$925$	−3.43048	−0.112793
$926$	−7.11965	−0.233966
$927$	0	0
$928$	−2.62594	−0.0862007
$929$	−13.1255	−0.430635	−0.215317	−	0.976544i	$-0.569079\pi$
$929$	−13.1255	−0.430635	−0.215317	+	0.976544i	$0.569079\pi$
$930$	0	0
$931$	0	0
$932$	3.81878	0.125088
$933$	0	0
$934$	19.2321	0.629293
$935$	4.70850	0.153984
$936$	0	0
$937$	−6.22330	−0.203306	−0.101653	−	0.994820i	$-0.532413\pi$
$937$	−6.22330	−0.203306	−0.101653	+	0.994820i	$0.532413\pi$
$938$	−38.7054	−1.26377
$939$	0	0
$940$	4.07611	0.132948
$941$	25.8234	0.841819	0.420910	−	0.907103i	$-0.361711\pi$
$941$	25.8234	0.841819	0.420910	+	0.907103i	$0.361711\pi$
$942$	0	0
$943$	−40.0369	−1.30378
$944$	−9.53059	−0.310194
$945$	0	0
$946$	−33.2123	−1.07982
$947$	−8.03962	−0.261253	−0.130626	−	0.991432i	$-0.541699\pi$
$947$	−8.03962	−0.261253	−0.130626	+	0.991432i	$0.541699\pi$
$948$	0	0
$949$	8.62925	0.280117
$950$	−3.79623	−0.123166
$951$	0	0
$952$	−6.29107	−0.203895
$953$	−25.8564	−0.837571	−0.418786	−	0.908085i	$-0.637544\pi$
$953$	−25.8564	−0.837571	−0.418786	+	0.908085i	$0.637544\pi$
$954$	0	0
$955$	2.78889	0.0902463
$956$	−12.3955	−0.400898
$957$	0	0
$958$	25.0761	0.810172
$959$	−6.48217	−0.209320
$960$	0	0
$961$	−26.5121	−0.855228
$962$	−0.708497	−0.0228429
$963$	0	0
$964$	12.9229	0.416218
$965$	−7.95735	−0.256156
$966$	0	0
$967$	35.4304	1.13936	0.569682	−	0.821865i	$-0.307066\pi$
$967$	35.4304	1.13936	0.569682	+	0.821865i	$0.307066\pi$
$968$	−13.8023	−0.443622
$969$	0	0
$970$	−5.76180	−0.185000
$971$	−27.2073	−0.873125	−0.436563	−	0.899674i	$-0.643804\pi$
$971$	−27.2073	−0.873125	−0.436563	+	0.899674i	$0.643804\pi$
$972$	0	0
$973$	3.27832	0.105098
$974$	−23.4304	−0.750757
$975$	0	0
$976$	−4.86171	−0.155620
$977$	27.1057	0.867189	0.433594	−	0.901108i	$-0.357245\pi$
$977$	27.1057	0.867189	0.433594	+	0.901108i	$0.357245\pi$
$978$	0	0
$979$	22.4910	0.718814
$980$	0	0
$981$	0	0
$982$	−38.3354	−1.22333
$983$	−20.7309	−0.661213	−0.330607	−	0.943769i	$-0.607253\pi$
$983$	−20.7309	−0.661213	−0.330607	+	0.943769i	$0.607253\pi$
$984$	0	0
$985$	0.916727	0.0292094
$986$	−6.24397	−0.198848
$987$	0	0
$988$	−0.784036	−0.0249435
$989$	45.9745	1.46190
$990$	0	0
$991$	22.0916	0.701764	0.350882	−	0.936420i	$-0.385882\pi$
$991$	22.0916	0.701764	0.350882	+	0.936420i	$0.385882\pi$
$992$	2.11847	0.0672616
$993$	0	0
$994$	−40.5810	−1.28715
$995$	0.813615	0.0257933
$996$	0	0
$997$	28.1396	0.891189	0.445595	−	0.895235i	$-0.352992\pi$
$997$	28.1396	0.891189	0.445595	+	0.895235i	$0.352992\pi$
$998$	3.09749	0.0980493
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2106.2.a.s.1.3	✓	4
3.2	odd	2		2106.2.a.u.1.2	yes	4
9.2	odd	6		2106.2.e.bi.1405.3		8
9.4	even	3		2106.2.e.bj.703.2		8
9.5	odd	6		2106.2.e.bi.703.3		8
9.7	even	3		2106.2.e.bj.1405.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2106.2.a.s.1.3	✓	4	1.1	even	1	trivial
2106.2.a.u.1.2	yes	4	3.2	odd	2
2106.2.e.bi.703.3		8	9.5	odd	6
2106.2.e.bi.1405.3		8	9.2	odd	6
2106.2.e.bj.703.2		8	9.4	even	3
2106.2.e.bj.1405.2		8	9.7	even	3

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2106 = 2 \cdot 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2106.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -0.397613 q^{5} +2.64575 q^{7} -1.00000 q^{8} +0.397613 q^{10} -4.98019 q^{11} +1.00000 q^{13} -2.64575 q^{14} +1.00000 q^{16} +2.37780 q^{17} -0.784036 q^{19} -0.397613 q^{20} +4.98019 q^{22} -6.89389 q^{23} -4.84190 q^{25} -1.00000 q^{26} +2.64575 q^{28} +2.62594 q^{29} -2.11847 q^{31} -1.00000 q^{32} -2.37780 q^{34} -1.05199 q^{35} +0.708497 q^{37} +0.784036 q^{38} +0.397613 q^{40} +5.80759 q^{41} -6.66888 q^{43} -4.98019 q^{44} +6.89389 q^{46} -10.2515 q^{47} +4.84190 q^{50} +1.00000 q^{52} -2.78404 q^{53} +1.98019 q^{55} -2.64575 q^{56} -2.62594 q^{58} -9.53059 q^{59} -4.86171 q^{61} +2.11847 q^{62} +1.00000 q^{64} -0.397613 q^{65} +14.6293 q^{67} +2.37780 q^{68} +1.05199 q^{70} +15.3382 q^{71} +8.62925 q^{73} -0.708497 q^{74} -0.784036 q^{76} -13.1763 q^{77} -13.7036 q^{79} -0.397613 q^{80} -5.80759 q^{82} -5.77542 q^{83} -0.945446 q^{85} +6.66888 q^{86} +4.98019 q^{88} -4.51609 q^{89} +2.64575 q^{91} -6.89389 q^{92} +10.2515 q^{94} +0.311743 q^{95} -14.4910 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} - 2 q^{5} - 4 q^{8} + 2 q^{10} - 2 q^{11} + 4 q^{13} + 4 q^{16} - 8 q^{17} - 6 q^{19} - 2 q^{20} + 2 q^{22} - 6 q^{23} + 12 q^{25} - 4 q^{26} - 18 q^{29} - 4 q^{31} - 4 q^{32} + 8 q^{34}+ \cdots + 6 q^{97}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 - 2 * q^5 - 4 * q^8 + 2 * q^10 - 2 * q^11 + 4 * q^13 + 4 * q^16 - 8 * q^17 - 6 * q^19 - 2 * q^20 + 2 * q^22 - 6 * q^23 + 12 * q^25 - 4 * q^26 - 18 * q^29 - 4 * q^31 - 4 * q^32 + 8 * q^34 - 14 * q^35 + 24 * q^37 + 6 * q^38 + 2 * q^40 - 2 * q^41 - 12 * q^43 - 2 * q^44 + 6 * q^46 - 8 * q^47 - 12 * q^50 + 4 * q^52 - 14 * q^53 - 10 * q^55 + 18 * q^58 - 10 * q^59 - 6 * q^61 + 4 * q^62 + 4 * q^64 - 2 * q^65 + 8 * q^67 - 8 * q^68 + 14 * q^70 + 8 * q^71 - 16 * q^73 - 24 * q^74 - 6 * q^76 - 14 * q^77 - 10 * q^79 - 2 * q^80 + 2 * q^82 - 6 * q^83 - 16 * q^85 + 12 * q^86 + 2 * q^88 - 14 * q^89 - 6 * q^92 + 8 * q^94 - 26 * q^95 + 6 * q^97

Introduction

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