LMFDB - Embedded newform 7400.2.a.o.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7400,2,Mod(1,7400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7400.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 7400 = 2^{3} \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7400.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,-1,0,0,0,2,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$59.0892974957$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.998068.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 10x^{2} + 3x - 2 $ x^5 - 2x^4 - 6x^3 + 10x^2 + 3x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1480)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.75031$ of defining polynomial
Character	$\chi$	$=$	7400.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.14266 q^{3} +3.63076 q^{7} -1.69433 q^{9} -1.42153 q^{11} +5.50061 q^{13} +3.01251 q^{17} +2.20924 q^{19} +4.14872 q^{21} -6.84305 q^{23} -5.36402 q^{27} +6.35495 q^{29} +7.54039 q^{31} -1.62432 q^{33} +1.00000 q^{37} +6.28531 q^{39} -8.54989 q^{41} -4.05751 q^{43} +10.4163 q^{47} +6.18244 q^{49} +3.44226 q^{51} +8.13744 q^{53} +2.52440 q^{57} +1.23303 q^{59} +2.91870 q^{61} -6.15173 q^{63} -7.81578 q^{67} -7.81926 q^{69} -0.806278 q^{71} -2.80628 q^{73} -5.16122 q^{77} +0.754853 q^{79} -1.04623 q^{81} -7.41508 q^{83} +7.26153 q^{87} +14.8681 q^{89} +19.9714 q^{91} +8.61609 q^{93} +5.94249 q^{97} +2.40854 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{3} + 2 q^{7} + 6 q^{9} + 8 q^{11} + 4 q^{13} + q^{17} + 10 q^{19} + 5 q^{21} - 4 q^{23} - q^{27} + 11 q^{29} - 3 q^{31} + 3 q^{33} + 5 q^{37} + 18 q^{39} + 16 q^{41} - 31 q^{43} - 5 q^{47} + 7 q^{49}+ \cdots + 37 q^{99}+O(q^{100}) $ 5 * q - q^3 + 2 * q^7 + 6 * q^9 + 8 * q^11 + 4 * q^13 + q^17 + 10 * q^19 + 5 * q^21 - 4 * q^23 - q^27 + 11 * q^29 - 3 * q^31 + 3 * q^33 + 5 * q^37 + 18 * q^39 + 16 * q^41 - 31 * q^43 - 5 * q^47 + 7 * q^49 + 34 * q^51 - 8 * q^53 + 8 * q^57 + 24 * q^59 - 15 * q^61 - 19 * q^63 - 12 * q^67 + 10 * q^69 + 5 * q^71 - 5 * q^73 + 4 * q^77 + 4 * q^79 + 17 * q^81 - 27 * q^83 + 4 * q^87 + 16 * q^89 + 26 * q^91 - 14 * q^93 + 19 * q^97 + 37 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.14266	0.659714	0.329857	−	0.944031i	$-0.393000\pi$
$3$	1.14266	0.659714	0.329857	+	0.944031i	$0.393000\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.63076	1.37230	0.686150	−	0.727460i	$-0.259299\pi$
$7$	3.63076	1.37230	0.686150	+	0.727460i	$0.259299\pi$
$8$	0	0
$9$	−1.69433	−0.564778
$10$	0	0
$11$	−1.42153	−0.428606	−0.214303	−	0.976767i	$-0.568748\pi$
$11$	−1.42153	−0.428606	−0.214303	+	0.976767i	$0.568748\pi$
$12$	0	0
$13$	5.50061	1.52560	0.762798	−	0.646637i	$-0.223825\pi$
$13$	5.50061	1.52560	0.762798	+	0.646637i	$0.223825\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.01251	0.730640	0.365320	−	0.930882i	$-0.380960\pi$
$17$	3.01251	0.730640	0.365320	+	0.930882i	$0.380960\pi$
$18$	0	0
$19$	2.20924	0.506834	0.253417	−	0.967357i	$-0.418446\pi$
$19$	2.20924	0.506834	0.253417	+	0.967357i	$0.418446\pi$
$20$	0	0
$21$	4.14872	0.905324
$22$	0	0
$23$	−6.84305	−1.42688	−0.713438	−	0.700719i	$-0.752863\pi$
$23$	−6.84305	−1.42688	−0.713438	+	0.700719i	$0.752863\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.36402	−1.03231
$28$	0	0
$29$	6.35495	1.18008	0.590042	−	0.807373i	$-0.299111\pi$
$29$	6.35495	1.18008	0.590042	+	0.807373i	$0.299111\pi$
$30$	0	0
$31$	7.54039	1.35429	0.677147	−	0.735847i	$-0.263216\pi$
$31$	7.54039	1.35429	0.677147	+	0.735847i	$0.263216\pi$
$32$	0	0
$33$	−1.62432	−0.282757
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	6.28531	1.00646
$40$	0	0
$41$	−8.54989	−1.33527	−0.667634	−	0.744489i	$-0.732693\pi$
$41$	−8.54989	−1.33527	−0.667634	+	0.744489i	$0.732693\pi$
$42$	0	0
$43$	−4.05751	−0.618765	−0.309382	−	0.950938i	$-0.600122\pi$
$43$	−4.05751	−0.618765	−0.309382	+	0.950938i	$0.600122\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.4163	1.51937	0.759687	−	0.650289i	$-0.225352\pi$
$47$	10.4163	1.51937	0.759687	+	0.650289i	$0.225352\pi$
$48$	0	0
$49$	6.18244	0.883206
$50$	0	0
$51$	3.44226	0.482013
$52$	0	0
$53$	8.13744	1.11776	0.558881	−	0.829248i	$-0.311231\pi$
$53$	8.13744	1.11776	0.558881	+	0.829248i	$0.311231\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.52440	0.334365
$58$	0	0
$59$	1.23303	0.160526	0.0802631	−	0.996774i	$-0.474424\pi$
$59$	1.23303	0.160526	0.0802631	+	0.996774i	$0.474424\pi$
$60$	0	0
$61$	2.91870	0.373701	0.186851	−	0.982388i	$-0.440172\pi$
$61$	2.91870	0.373701	0.186851	+	0.982388i	$0.440172\pi$
$62$	0	0
$63$	−6.15173	−0.775045
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−7.81578	−0.954849	−0.477424	−	0.878673i	$-0.658430\pi$
$67$	−7.81578	−0.954849	−0.477424	+	0.878673i	$0.658430\pi$
$68$	0	0
$69$	−7.81926	−0.941329
$70$	0	0
$71$	−0.806278	−0.0956876	−0.0478438	−	0.998855i	$-0.515235\pi$
$71$	−0.806278	−0.0956876	−0.0478438	+	0.998855i	$0.515235\pi$
$72$	0	0
$73$	−2.80628	−0.328450	−0.164225	−	0.986423i	$-0.552512\pi$
$73$	−2.80628	−0.328450	−0.164225	+	0.986423i	$0.552512\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−5.16122	−0.588176
$78$	0	0
$79$	0.754853	0.0849276	0.0424638	−	0.999098i	$-0.486479\pi$
$79$	0.754853	0.0849276	0.0424638	+	0.999098i	$0.486479\pi$
$80$	0	0
$81$	−1.04623	−0.116248
$82$	0	0
$83$	−7.41508	−0.813911	−0.406955	−	0.913448i	$-0.633410\pi$
$83$	−7.41508	−0.813911	−0.406955	+	0.913448i	$0.633410\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	7.26153	0.778517
$88$	0	0
$89$	14.8681	1.57601	0.788006	−	0.615668i	$-0.211114\pi$
$89$	14.8681	1.57601	0.788006	+	0.615668i	$0.211114\pi$
$90$	0	0
$91$	19.9714	2.09357
$92$	0	0
$93$	8.61609	0.893447
$94$	0	0
$95$	0	0
$96$	0	0
$97$	5.94249	0.603368	0.301684	−	0.953408i	$-0.402451\pi$
$97$	5.94249	0.603368	0.301684	+	0.953408i	$0.402451\pi$
$98$	0	0
$99$	2.40854	0.242067
$100$	0	0
$101$	−0.694334	−0.0690888	−0.0345444	−	0.999403i	$-0.510998\pi$
$101$	−0.694334	−0.0690888	−0.0345444	+	0.999403i	$0.510998\pi$
$102$	0	0
$103$	11.9714	1.17958	0.589789	−	0.807557i	$-0.299211\pi$
$103$	11.9714	1.17958	0.589789	+	0.807557i	$0.299211\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.31259	0.126893	0.0634465	−	0.997985i	$-0.479791\pi$
$107$	1.31259	0.126893	0.0634465	+	0.997985i	$0.479791\pi$
$108$	0	0
$109$	5.05873	0.484539	0.242269	−	0.970209i	$-0.422108\pi$
$109$	5.05873	0.484539	0.242269	+	0.970209i	$0.422108\pi$
$110$	0	0
$111$	1.14266	0.108456
$112$	0	0
$113$	−12.4012	−1.16660	−0.583302	−	0.812255i	$-0.698240\pi$
$113$	−12.4012	−1.16660	−0.583302	+	0.812255i	$0.698240\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−9.31988	−0.861623
$118$	0	0
$119$	10.9377	1.00266
$120$	0	0
$121$	−8.97926	−0.816297
$122$	0	0
$123$	−9.76960	−0.880895
$124$	0	0
$125$	0	0
$126$	0	0
$127$	6.59704	0.585393	0.292696	−	0.956205i	$-0.405447\pi$
$127$	6.59704	0.585393	0.292696	+	0.956205i	$0.405447\pi$
$128$	0	0
$129$	−4.63634	−0.408207
$130$	0	0
$131$	3.43973	0.300531	0.150265	−	0.988646i	$-0.451987\pi$
$131$	3.43973	0.300531	0.150265	+	0.988646i	$0.451987\pi$
$132$	0	0
$133$	8.02122	0.695528
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.45561	−0.551540	−0.275770	−	0.961224i	$-0.588933\pi$
$137$	−6.45561	−0.551540	−0.275770	+	0.961224i	$0.588933\pi$
$138$	0	0
$139$	−2.78554	−0.236267	−0.118133	−	0.992998i	$-0.537691\pi$
$139$	−2.78554	−0.236267	−0.118133	+	0.992998i	$0.537691\pi$
$140$	0	0
$141$	11.9023	1.00235
$142$	0	0
$143$	−7.81926	−0.653880
$144$	0	0
$145$	0	0
$146$	0	0
$147$	7.06441	0.582663
$148$	0	0
$149$	−5.55552	−0.455126	−0.227563	−	0.973763i	$-0.573076\pi$
$149$	−5.55552	−0.455126	−0.227563	+	0.973763i	$0.573076\pi$
$150$	0	0
$151$	−4.41847	−0.359570	−0.179785	−	0.983706i	$-0.557540\pi$
$151$	−4.41847	−0.359570	−0.179785	+	0.983706i	$0.557540\pi$
$152$	0	0
$153$	−5.10419	−0.412650
$154$	0	0
$155$	0	0
$156$	0	0
$157$	16.9555	1.35319	0.676597	−	0.736353i	$-0.263454\pi$
$157$	16.9555	1.35319	0.676597	+	0.736353i	$0.263454\pi$
$158$	0	0
$159$	9.29830	0.737403
$160$	0	0
$161$	−24.8455	−1.95810
$162$	0	0
$163$	−17.0709	−1.33709	−0.668546	−	0.743671i	$-0.733083\pi$
$163$	−17.0709	−1.33709	−0.668546	+	0.743671i	$0.733083\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−0.418474	−0.0323825	−0.0161912	−	0.999869i	$-0.505154\pi$
$167$	−0.418474	−0.0323825	−0.0161912	+	0.999869i	$0.505154\pi$
$168$	0	0
$169$	17.2567	1.32744
$170$	0	0
$171$	−3.74319	−0.286249
$172$	0	0
$173$	−14.9805	−1.13895	−0.569473	−	0.822010i	$-0.692853\pi$
$173$	−14.9805	−1.13895	−0.569473	+	0.822010i	$0.692853\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.40893	0.105901
$178$	0	0
$179$	18.0656	1.35029	0.675144	−	0.737686i	$-0.264081\pi$
$179$	18.0656	1.35029	0.675144	+	0.737686i	$0.264081\pi$
$180$	0	0
$181$	−16.9918	−1.26299	−0.631494	−	0.775381i	$-0.717558\pi$
$181$	−16.9918	−1.26299	−0.631494	+	0.775381i	$0.717558\pi$
$182$	0	0
$183$	3.33507	0.246536
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4.28236	−0.313157
$188$	0	0
$189$	−19.4755	−1.41663
$190$	0	0
$191$	14.4621	1.04644	0.523219	−	0.852198i	$-0.324731\pi$
$191$	14.4621	1.04644	0.523219	+	0.852198i	$0.324731\pi$
$192$	0	0
$193$	11.9524	0.860354	0.430177	−	0.902745i	$-0.358451\pi$
$193$	11.9524	0.860354	0.430177	+	0.902745i	$0.358451\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.44313	−0.174066	−0.0870328	−	0.996205i	$-0.527738\pi$
$197$	−2.44313	−0.174066	−0.0870328	+	0.996205i	$0.527738\pi$
$198$	0	0
$199$	16.4720	1.16767	0.583834	−	0.811873i	$-0.301552\pi$
$199$	16.4720	1.16767	0.583834	+	0.811873i	$0.301552\pi$
$200$	0	0
$201$	−8.93075	−0.629927
$202$	0	0
$203$	23.0733	1.61943
$204$	0	0
$205$	0	0
$206$	0	0
$207$	11.5944	0.805868
$208$	0	0
$209$	−3.14049	−0.217232
$210$	0	0
$211$	15.4526	1.06380	0.531899	−	0.846808i	$-0.321479\pi$
$211$	15.4526	1.06380	0.531899	+	0.846808i	$0.321479\pi$
$212$	0	0
$213$	−0.921299	−0.0631264
$214$	0	0
$215$	0	0
$216$	0	0
$217$	27.3774	1.85850
$218$	0	0
$219$	−3.20661	−0.216683
$220$	0	0
$221$	16.5706	1.11466
$222$	0	0
$223$	17.3298	1.16049	0.580243	−	0.814443i	$-0.302957\pi$
$223$	17.3298	1.16049	0.580243	+	0.814443i	$0.302957\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−17.8421	−1.18422	−0.592111	−	0.805857i	$-0.701705\pi$
$227$	−17.8421	−1.18422	−0.592111	+	0.805857i	$0.701705\pi$
$228$	0	0
$229$	10.9455	0.723302	0.361651	−	0.932314i	$-0.382213\pi$
$229$	10.9455	0.723302	0.361651	+	0.932314i	$0.382213\pi$
$230$	0	0
$231$	−5.89751	−0.388028
$232$	0	0
$233$	−20.9787	−1.37436	−0.687179	−	0.726488i	$-0.741151\pi$
$233$	−20.9787	−1.37436	−0.687179	+	0.726488i	$0.741151\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0.862539	0.0560279
$238$	0	0
$239$	13.2680	0.858234	0.429117	−	0.903249i	$-0.358825\pi$
$239$	13.2680	0.858234	0.429117	+	0.903249i	$0.358825\pi$
$240$	0	0
$241$	16.3960	1.05616	0.528080	−	0.849195i	$-0.322912\pi$
$241$	16.3960	1.05616	0.528080	+	0.849195i	$0.322912\pi$
$242$	0	0
$243$	14.8966	0.955615
$244$	0	0
$245$	0	0
$246$	0	0
$247$	12.1522	0.773223
$248$	0	0
$249$	−8.47290	−0.536948
$250$	0	0
$251$	−12.7989	−0.807857	−0.403929	−	0.914791i	$-0.632356\pi$
$251$	−12.7989	−0.807857	−0.403929	+	0.914791i	$0.632356\pi$
$252$	0	0
$253$	9.72758	0.611568
$254$	0	0
$255$	0	0
$256$	0	0
$257$	23.5992	1.47208	0.736038	−	0.676940i	$-0.236694\pi$
$257$	23.5992	1.47208	0.736038	+	0.676940i	$0.236694\pi$
$258$	0	0
$259$	3.63076	0.225605
$260$	0	0
$261$	−10.7674	−0.666486
$262$	0	0
$263$	11.1300	0.686307	0.343153	−	0.939279i	$-0.388505\pi$
$263$	11.1300	0.686307	0.343153	+	0.939279i	$0.388505\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	16.9891	1.03972
$268$	0	0
$269$	25.4780	1.55342	0.776712	−	0.629856i	$-0.216886\pi$
$269$	25.4780	1.55342	0.776712	+	0.629856i	$0.216886\pi$
$270$	0	0
$271$	−11.0630	−0.672030	−0.336015	−	0.941857i	$-0.609079\pi$
$271$	−11.0630	−0.672030	−0.336015	+	0.941857i	$0.609079\pi$
$272$	0	0
$273$	22.8205	1.38116
$274$	0	0
$275$	0	0
$276$	0	0
$277$	11.0631	0.664720	0.332360	−	0.943153i	$-0.392155\pi$
$277$	11.0631	0.664720	0.332360	+	0.943153i	$0.392155\pi$
$278$	0	0
$279$	−12.7759	−0.764876
$280$	0	0
$281$	−23.3661	−1.39390	−0.696952	−	0.717117i	$-0.745461\pi$
$281$	−23.3661	−1.39390	−0.696952	+	0.717117i	$0.745461\pi$
$282$	0	0
$283$	−18.9295	−1.12524	−0.562621	−	0.826715i	$-0.690207\pi$
$283$	−18.9295	−1.12524	−0.562621	+	0.826715i	$0.690207\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−31.0426	−1.83239
$288$	0	0
$289$	−7.92481	−0.466165
$290$	0	0
$291$	6.79023	0.398050
$292$	0	0
$293$	−17.1019	−0.999103	−0.499551	−	0.866284i	$-0.666502\pi$
$293$	−17.1019	−0.999103	−0.499551	+	0.866284i	$0.666502\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	7.62509	0.442452
$298$	0	0
$299$	−37.6410	−2.17683
$300$	0	0
$301$	−14.7319	−0.849130
$302$	0	0
$303$	−0.793386	−0.0455788
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−11.6537	−0.665110	−0.332555	−	0.943084i	$-0.607911\pi$
$307$	−11.6537	−0.665110	−0.332555	+	0.943084i	$0.607911\pi$
$308$	0	0
$309$	13.6792	0.778184
$310$	0	0
$311$	−29.8993	−1.69543	−0.847716	−	0.530451i	$-0.822023\pi$
$311$	−29.8993	−1.69543	−0.847716	+	0.530451i	$0.822023\pi$
$312$	0	0
$313$	31.1655	1.76158	0.880789	−	0.473509i	$-0.157013\pi$
$313$	31.1655	1.76158	0.880789	+	0.473509i	$0.157013\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4.64628	−0.260961	−0.130480	−	0.991451i	$-0.541652\pi$
$317$	−4.64628	−0.260961	−0.130480	+	0.991451i	$0.541652\pi$
$318$	0	0
$319$	−9.03372	−0.505791
$320$	0	0
$321$	1.49984	0.0837130
$322$	0	0
$323$	6.65534	0.370313
$324$	0	0
$325$	0	0
$326$	0	0
$327$	5.78040	0.319657
$328$	0	0
$329$	37.8191	2.08504
$330$	0	0
$331$	11.6576	0.640760	0.320380	−	0.947289i	$-0.396189\pi$
$331$	11.6576	0.640760	0.320380	+	0.947289i	$0.396189\pi$
$332$	0	0
$333$	−1.69433	−0.0928489
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−6.83052	−0.372082	−0.186041	−	0.982542i	$-0.559566\pi$
$337$	−6.83052	−0.372082	−0.186041	+	0.982542i	$0.559566\pi$
$338$	0	0
$339$	−14.1703	−0.769625
$340$	0	0
$341$	−10.7189	−0.580459
$342$	0	0
$343$	−2.96837	−0.160277
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.61256	−0.0865665	−0.0432833	−	0.999063i	$-0.513782\pi$
$347$	−1.61256	−0.0865665	−0.0432833	+	0.999063i	$0.513782\pi$
$348$	0	0
$349$	7.20928	0.385904	0.192952	−	0.981208i	$-0.438194\pi$
$349$	7.20928	0.385904	0.192952	+	0.981208i	$0.438194\pi$
$350$	0	0
$351$	−29.5054	−1.57488
$352$	0	0
$353$	−30.6980	−1.63389	−0.816946	−	0.576715i	$-0.804334\pi$
$353$	−30.6980	−1.63389	−0.816946	+	0.576715i	$0.804334\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	12.4980	0.661466
$358$	0	0
$359$	−17.9589	−0.947834	−0.473917	−	0.880570i	$-0.657160\pi$
$359$	−17.9589	−0.947834	−0.473917	+	0.880570i	$0.657160\pi$
$360$	0	0
$361$	−14.1193	−0.743120
$362$	0	0
$363$	−10.2602	−0.538522
$364$	0	0
$365$	0	0
$366$	0	0
$367$	35.9363	1.87586	0.937930	−	0.346825i	$-0.112740\pi$
$367$	35.9363	1.87586	0.937930	+	0.346825i	$0.112740\pi$
$368$	0	0
$369$	14.4864	0.754131
$370$	0	0
$371$	29.5451	1.53391
$372$	0	0
$373$	19.7825	1.02430	0.512149	−	0.858896i	$-0.328849\pi$
$373$	19.7825	1.02430	0.512149	+	0.858896i	$0.328849\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	34.9561	1.80033
$378$	0	0
$379$	10.7458	0.551975	0.275988	−	0.961161i	$-0.410995\pi$
$379$	10.7458	0.551975	0.275988	+	0.961161i	$0.410995\pi$
$380$	0	0
$381$	7.53816	0.386191
$382$	0	0
$383$	−22.5473	−1.15211	−0.576056	−	0.817410i	$-0.695409\pi$
$383$	−22.5473	−1.15211	−0.576056	+	0.817410i	$0.695409\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.87478	0.349465
$388$	0	0
$389$	32.8486	1.66549	0.832744	−	0.553658i	$-0.186769\pi$
$389$	32.8486	1.66549	0.832744	+	0.553658i	$0.186769\pi$
$390$	0	0
$391$	−20.6147	−1.04253
$392$	0	0
$393$	3.93044	0.198264
$394$	0	0
$395$	0	0
$396$	0	0
$397$	9.15912	0.459683	0.229842	−	0.973228i	$-0.426179\pi$
$397$	9.15912	0.459683	0.229842	+	0.973228i	$0.426179\pi$
$398$	0	0
$399$	9.16550	0.458849
$400$	0	0
$401$	13.7457	0.686428	0.343214	−	0.939257i	$-0.388484\pi$
$401$	13.7457	0.686428	0.343214	+	0.939257i	$0.388484\pi$
$402$	0	0
$403$	41.4768	2.06611
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.42153	−0.0704624
$408$	0	0
$409$	26.8879	1.32952	0.664761	−	0.747056i	$-0.268533\pi$
$409$	26.8879	1.32952	0.664761	+	0.747056i	$0.268533\pi$
$410$	0	0
$411$	−7.37655	−0.363858
$412$	0	0
$413$	4.47682	0.220290
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−3.18292	−0.155868
$418$	0	0
$419$	3.93220	0.192100	0.0960502	−	0.995376i	$-0.469379\pi$
$419$	3.93220	0.192100	0.0960502	+	0.995376i	$0.469379\pi$
$420$	0	0
$421$	11.7293	0.571649	0.285824	−	0.958282i	$-0.407733\pi$
$421$	11.7293	0.571649	0.285824	+	0.958282i	$0.407733\pi$
$422$	0	0
$423$	−17.6487	−0.858109
$424$	0	0
$425$	0	0
$426$	0	0
$427$	10.5971	0.512830
$428$	0	0
$429$	−8.93474	−0.431373
$430$	0	0
$431$	15.7910	0.760627	0.380313	−	0.924858i	$-0.375816\pi$
$431$	15.7910	0.760627	0.380313	+	0.924858i	$0.375816\pi$
$432$	0	0
$433$	−24.2648	−1.16609	−0.583047	−	0.812439i	$-0.698140\pi$
$433$	−24.2648	−1.16609	−0.583047	+	0.812439i	$0.698140\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−15.1179	−0.723188
$438$	0	0
$439$	−35.0198	−1.67140	−0.835702	−	0.549184i	$-0.814939\pi$
$439$	−35.0198	−1.67140	−0.835702	+	0.549184i	$0.814939\pi$
$440$	0	0
$441$	−10.4751	−0.498815
$442$	0	0
$443$	−39.3708	−1.87056	−0.935282	−	0.353904i	$-0.884854\pi$
$443$	−39.3708	−1.87056	−0.935282	+	0.353904i	$0.884854\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−6.34806	−0.300253
$448$	0	0
$449$	−17.3911	−0.820738	−0.410369	−	0.911920i	$-0.634600\pi$
$449$	−17.3911	−0.820738	−0.410369	+	0.911920i	$0.634600\pi$
$450$	0	0
$451$	12.1539	0.572305
$452$	0	0
$453$	−5.04880	−0.237213
$454$	0	0
$455$	0	0
$456$	0	0
$457$	40.5649	1.89754	0.948772	−	0.315960i	$-0.102327\pi$
$457$	40.5649	1.89754	0.948772	+	0.315960i	$0.102327\pi$
$458$	0	0
$459$	−16.1591	−0.754244
$460$	0	0
$461$	33.0855	1.54094	0.770472	−	0.637474i	$-0.220021\pi$
$461$	33.0855	1.54094	0.770472	+	0.637474i	$0.220021\pi$
$462$	0	0
$463$	−26.4617	−1.22978	−0.614890	−	0.788613i	$-0.710800\pi$
$463$	−26.4617	−1.22978	−0.614890	+	0.788613i	$0.710800\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−18.2121	−0.842756	−0.421378	−	0.906885i	$-0.638453\pi$
$467$	−18.2121	−0.842756	−0.421378	+	0.906885i	$0.638453\pi$
$468$	0	0
$469$	−28.3772	−1.31034
$470$	0	0
$471$	19.3743	0.892721
$472$	0	0
$473$	5.76786	0.265206
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−13.7875	−0.631288
$478$	0	0
$479$	−30.3881	−1.38847	−0.694235	−	0.719749i	$-0.744257\pi$
$479$	−30.3881	−1.38847	−0.694235	+	0.719749i	$0.744257\pi$
$480$	0	0
$481$	5.50061	0.250806
$482$	0	0
$483$	−28.3899	−1.29178
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−35.7014	−1.61779	−0.808893	−	0.587955i	$-0.799933\pi$
$487$	−35.7014	−1.61779	−0.808893	+	0.587955i	$0.799933\pi$
$488$	0	0
$489$	−19.5061	−0.882098
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	19.1443	0.862217
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.92740	−0.131312
$498$	0	0
$499$	19.8175	0.887151	0.443576	−	0.896237i	$-0.353710\pi$
$499$	19.8175	0.887151	0.443576	+	0.896237i	$0.353710\pi$
$500$	0	0
$501$	−0.478172	−0.0213632
$502$	0	0
$503$	18.8136	0.838858	0.419429	−	0.907788i	$-0.362230\pi$
$503$	18.8136	0.838858	0.419429	+	0.907788i	$0.362230\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	19.7185	0.875731
$508$	0	0
$509$	21.1937	0.939395	0.469698	−	0.882827i	$-0.344363\pi$
$509$	21.1937	0.939395	0.469698	+	0.882827i	$0.344363\pi$
$510$	0	0
$511$	−10.1889	−0.450732
$512$	0	0
$513$	−11.8504	−0.523207
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−14.8070	−0.651213
$518$	0	0
$519$	−17.1176	−0.751378
$520$	0	0
$521$	24.2087	1.06060	0.530301	−	0.847810i	$-0.322079\pi$
$521$	24.2087	1.06060	0.530301	+	0.847810i	$0.322079\pi$
$522$	0	0
$523$	−36.9848	−1.61723	−0.808615	−	0.588338i	$-0.799783\pi$
$523$	−36.9848	−1.61723	−0.808615	+	0.588338i	$0.799783\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	22.7155	0.989502
$528$	0	0
$529$	23.8274	1.03597
$530$	0	0
$531$	−2.08916	−0.0906617
$532$	0	0
$533$	−47.0296	−2.03708
$534$	0	0
$535$	0	0
$536$	0	0
$537$	20.6428	0.890804
$538$	0	0
$539$	−8.78850	−0.378547
$540$	0	0
$541$	17.1110	0.735661	0.367831	−	0.929893i	$-0.380101\pi$
$541$	17.1110	0.735661	0.367831	+	0.929893i	$0.380101\pi$
$542$	0	0
$543$	−19.4158	−0.833210
$544$	0	0
$545$	0	0
$546$	0	0
$547$	33.7522	1.44314	0.721569	−	0.692342i	$-0.243421\pi$
$547$	33.7522	1.44314	0.721569	+	0.692342i	$0.243421\pi$
$548$	0	0
$549$	−4.94525	−0.211058
$550$	0	0
$551$	14.0396	0.598106
$552$	0	0
$553$	2.74069	0.116546
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.77792	−0.117704	−0.0588520	−	0.998267i	$-0.518744\pi$
$557$	−2.77792	−0.117704	−0.0588520	+	0.998267i	$0.518744\pi$
$558$	0	0
$559$	−22.3188	−0.943984
$560$	0	0
$561$	−4.89327	−0.206594
$562$	0	0
$563$	5.17299	0.218015	0.109008	−	0.994041i	$-0.465233\pi$
$563$	5.17299	0.218015	0.109008	+	0.994041i	$0.465233\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−3.79861	−0.159527
$568$	0	0
$569$	−45.7741	−1.91895	−0.959476	−	0.281791i	$-0.909071\pi$
$569$	−45.7741	−1.91895	−0.959476	+	0.281791i	$0.909071\pi$
$570$	0	0
$571$	−12.2360	−0.512061	−0.256030	−	0.966669i	$-0.582415\pi$
$571$	−12.2360	−0.512061	−0.256030	+	0.966669i	$0.582415\pi$
$572$	0	0
$573$	16.5252	0.690349
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−43.7417	−1.82099	−0.910494	−	0.413521i	$-0.864299\pi$
$577$	−43.7417	−1.82099	−0.910494	+	0.413521i	$0.864299\pi$
$578$	0	0
$579$	13.6575	0.567587
$580$	0	0
$581$	−26.9224	−1.11693
$582$	0	0
$583$	−11.5676	−0.479080
$584$	0	0
$585$	0	0
$586$	0	0
$587$	32.3324	1.33450	0.667250	−	0.744834i	$-0.267471\pi$
$587$	32.3324	1.33450	0.667250	+	0.744834i	$0.267471\pi$
$588$	0	0
$589$	16.6585	0.686402
$590$	0	0
$591$	−2.79166	−0.114833
$592$	0	0
$593$	−17.4912	−0.718276	−0.359138	−	0.933285i	$-0.616929\pi$
$593$	−17.4912	−0.718276	−0.359138	+	0.933285i	$0.616929\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	18.8218	0.770326
$598$	0	0
$599$	−16.1914	−0.661562	−0.330781	−	0.943708i	$-0.607312\pi$
$599$	−16.1914	−0.661562	−0.330781	+	0.943708i	$0.607312\pi$
$600$	0	0
$601$	−1.66627	−0.0679685	−0.0339843	−	0.999422i	$-0.510820\pi$
$601$	−1.66627	−0.0679685	−0.0339843	+	0.999422i	$0.510820\pi$
$602$	0	0
$603$	13.2425	0.539278
$604$	0	0
$605$	0	0
$606$	0	0
$607$	36.6648	1.48818	0.744088	−	0.668081i	$-0.232884\pi$
$607$	36.6648	1.48818	0.744088	+	0.668081i	$0.232884\pi$
$608$	0	0
$609$	26.3649	1.06836
$610$	0	0
$611$	57.2961	2.31795
$612$	0	0
$613$	17.4611	0.705248	0.352624	−	0.935765i	$-0.385290\pi$
$613$	17.4611	0.705248	0.352624	+	0.935765i	$0.385290\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	15.4269	0.621065	0.310532	−	0.950563i	$-0.399493\pi$
$617$	15.4269	0.621065	0.310532	+	0.950563i	$0.399493\pi$
$618$	0	0
$619$	−6.66491	−0.267886	−0.133943	−	0.990989i	$-0.542764\pi$
$619$	−6.66491	−0.267886	−0.133943	+	0.990989i	$0.542764\pi$
$620$	0	0
$621$	36.7062	1.47297
$622$	0	0
$623$	53.9824	2.16276
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−3.58850	−0.143311
$628$	0	0
$629$	3.01251	0.120116
$630$	0	0
$631$	37.7713	1.50365	0.751827	−	0.659361i	$-0.229173\pi$
$631$	37.7713	1.50365	0.751827	+	0.659361i	$0.229173\pi$
$632$	0	0
$633$	17.6570	0.701802
$634$	0	0
$635$	0	0
$636$	0	0
$637$	34.0072	1.34741
$638$	0	0
$639$	1.36610	0.0540422
$640$	0	0
$641$	19.4063	0.766504	0.383252	−	0.923644i	$-0.374804\pi$
$641$	19.4063	0.766504	0.383252	+	0.923644i	$0.374804\pi$
$642$	0	0
$643$	−39.3776	−1.55290	−0.776451	−	0.630178i	$-0.782982\pi$
$643$	−39.3776	−1.55290	−0.776451	+	0.630178i	$0.782982\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−25.0461	−0.984664	−0.492332	−	0.870408i	$-0.663855\pi$
$647$	−25.0461	−0.984664	−0.492332	+	0.870408i	$0.663855\pi$
$648$	0	0
$649$	−1.75278	−0.0688026
$650$	0	0
$651$	31.2830	1.22608
$652$	0	0
$653$	−11.8911	−0.465334	−0.232667	−	0.972556i	$-0.574745\pi$
$653$	−11.8911	−0.465334	−0.232667	+	0.972556i	$0.574745\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	4.75477	0.185501
$658$	0	0
$659$	−26.4031	−1.02852	−0.514260	−	0.857634i	$-0.671933\pi$
$659$	−26.4031	−1.02852	−0.514260	+	0.857634i	$0.671933\pi$
$660$	0	0
$661$	−26.2909	−1.02260	−0.511299	−	0.859403i	$-0.670835\pi$
$661$	−26.2909	−1.02260	−0.511299	+	0.859403i	$0.670835\pi$
$662$	0	0
$663$	18.9346	0.735357
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−43.4872	−1.68383
$668$	0	0
$669$	19.8020	0.765589
$670$	0	0
$671$	−4.14901	−0.160171
$672$	0	0
$673$	−1.12801	−0.0434815	−0.0217407	−	0.999764i	$-0.506921\pi$
$673$	−1.12801	−0.0434815	−0.0217407	+	0.999764i	$0.506921\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	34.9004	1.34133	0.670666	−	0.741760i	$-0.266008\pi$
$677$	34.9004	1.34133	0.670666	+	0.741760i	$0.266008\pi$
$678$	0	0
$679$	21.5758	0.828002
$680$	0	0
$681$	−20.3874	−0.781247
$682$	0	0
$683$	−27.7382	−1.06137	−0.530687	−	0.847568i	$-0.678066\pi$
$683$	−27.7382	−1.06137	−0.530687	+	0.847568i	$0.678066\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	12.5070	0.477172
$688$	0	0
$689$	44.7609	1.70525
$690$	0	0
$691$	−30.7533	−1.16991	−0.584955	−	0.811065i	$-0.698888\pi$
$691$	−30.7533	−1.16991	−0.584955	+	0.811065i	$0.698888\pi$
$692$	0	0
$693$	8.74484	0.332189
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−25.7566	−0.975601
$698$	0	0
$699$	−23.9714	−0.906682
$700$	0	0
$701$	1.46049	0.0551619	0.0275809	−	0.999620i	$-0.491220\pi$
$701$	1.46049	0.0551619	0.0275809	+	0.999620i	$0.491220\pi$
$702$	0	0
$703$	2.20924	0.0833229
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.52096	−0.0948106
$708$	0	0
$709$	23.4773	0.881708	0.440854	−	0.897579i	$-0.354676\pi$
$709$	23.4773	0.881708	0.440854	+	0.897579i	$0.354676\pi$
$710$	0	0
$711$	−1.27897	−0.0479653
$712$	0	0
$713$	−51.5993	−1.93241
$714$	0	0
$715$	0	0
$716$	0	0
$717$	15.1607	0.566188
$718$	0	0
$719$	25.7124	0.958912	0.479456	−	0.877566i	$-0.340834\pi$
$719$	25.7124	0.958912	0.479456	+	0.877566i	$0.340834\pi$
$720$	0	0
$721$	43.4654	1.61874
$722$	0	0
$723$	18.7350	0.696762
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−37.2155	−1.38025	−0.690124	−	0.723691i	$-0.742444\pi$
$727$	−37.2155	−1.38025	−0.690124	+	0.723691i	$0.742444\pi$
$728$	0	0
$729$	20.1604	0.746680
$730$	0	0
$731$	−12.2233	−0.452094
$732$	0	0
$733$	45.9003	1.69537	0.847683	−	0.530503i	$-0.177997\pi$
$733$	45.9003	1.69537	0.847683	+	0.530503i	$0.177997\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11.1103	0.409254
$738$	0	0
$739$	26.7807	0.985142	0.492571	−	0.870272i	$-0.336057\pi$
$739$	26.7807	0.985142	0.492571	+	0.870272i	$0.336057\pi$
$740$	0	0
$741$	13.8857	0.510106
$742$	0	0
$743$	−3.78292	−0.138782	−0.0693909	−	0.997590i	$-0.522106\pi$
$743$	−3.78292	−0.138782	−0.0693909	+	0.997590i	$0.522106\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12.5636	0.459679
$748$	0	0
$749$	4.76571	0.174135
$750$	0	0
$751$	−28.5727	−1.04263	−0.521317	−	0.853363i	$-0.674559\pi$
$751$	−28.5727	−1.04263	−0.521317	+	0.853363i	$0.674559\pi$
$752$	0	0
$753$	−14.6247	−0.532954
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−30.6449	−1.11381	−0.556904	−	0.830577i	$-0.688011\pi$
$757$	−30.6449	−1.11381	−0.556904	+	0.830577i	$0.688011\pi$
$758$	0	0
$759$	11.1153	0.403459
$760$	0	0
$761$	20.9555	0.759635	0.379818	−	0.925061i	$-0.375987\pi$
$761$	20.9555	0.759635	0.379818	+	0.925061i	$0.375987\pi$
$762$	0	0
$763$	18.3671	0.664932
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6.78240	0.244898
$768$	0	0
$769$	18.3605	0.662095	0.331047	−	0.943614i	$-0.392598\pi$
$769$	18.3605	0.662095	0.331047	+	0.943614i	$0.392598\pi$
$770$	0	0
$771$	26.9658	0.971148
$772$	0	0
$773$	48.4623	1.74307	0.871534	−	0.490334i	$-0.163125\pi$
$773$	48.4623	1.74307	0.871534	+	0.490334i	$0.163125\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	4.14872	0.148834
$778$	0	0
$779$	−18.8887	−0.676759
$780$	0	0
$781$	1.14614	0.0410123
$782$	0	0
$783$	−34.0880	−1.21821
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−1.54910	−0.0552194	−0.0276097	−	0.999619i	$-0.508790\pi$
$787$	−1.54910	−0.0552194	−0.0276097	+	0.999619i	$0.508790\pi$
$788$	0	0
$789$	12.7178	0.452766
$790$	0	0
$791$	−45.0257	−1.60093
$792$	0	0
$793$	16.0546	0.570117
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−49.0430	−1.73719	−0.868597	−	0.495520i	$-0.834978\pi$
$797$	−49.0430	−1.73719	−0.868597	+	0.495520i	$0.834978\pi$
$798$	0	0
$799$	31.3792	1.11012
$800$	0	0
$801$	−25.1915	−0.890097
$802$	0	0
$803$	3.98920	0.140776
$804$	0	0
$805$	0	0
$806$	0	0
$807$	29.1127	1.02481
$808$	0	0
$809$	−12.7273	−0.447468	−0.223734	−	0.974650i	$-0.571825\pi$
$809$	−12.7273	−0.447468	−0.223734	+	0.974650i	$0.571825\pi$
$810$	0	0
$811$	−30.7930	−1.08129	−0.540645	−	0.841251i	$-0.681820\pi$
$811$	−30.7930	−1.08129	−0.540645	+	0.841251i	$0.681820\pi$
$812$	0	0
$813$	−12.6412	−0.443347
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−8.96400	−0.313611
$818$	0	0
$819$	−33.8383	−1.18240
$820$	0	0
$821$	28.0073	0.977463	0.488731	−	0.872434i	$-0.337460\pi$
$821$	28.0073	0.977463	0.488731	+	0.872434i	$0.337460\pi$
$822$	0	0
$823$	−1.24000	−0.0432237	−0.0216119	−	0.999766i	$-0.506880\pi$
$823$	−1.24000	−0.0432237	−0.0216119	+	0.999766i	$0.506880\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	14.1170	0.490895	0.245447	−	0.969410i	$-0.421065\pi$
$827$	14.1170	0.490895	0.245447	+	0.969410i	$0.421065\pi$
$828$	0	0
$829$	−14.1516	−0.491507	−0.245753	−	0.969332i	$-0.579035\pi$
$829$	−14.1516	−0.491507	−0.245753	+	0.969332i	$0.579035\pi$
$830$	0	0
$831$	12.6414	0.438525
$832$	0	0
$833$	18.6246	0.645306
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−40.4468	−1.39805
$838$	0	0
$839$	1.43182	0.0494319	0.0247159	−	0.999695i	$-0.492132\pi$
$839$	1.43182	0.0494319	0.0247159	+	0.999695i	$0.492132\pi$
$840$	0	0
$841$	11.3853	0.392598
$842$	0	0
$843$	−26.6994	−0.919578
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−32.6016	−1.12020
$848$	0	0
$849$	−21.6299	−0.742337
$850$	0	0
$851$	−6.84305	−0.234577
$852$	0	0
$853$	39.6167	1.35645	0.678226	−	0.734854i	$-0.262749\pi$
$853$	39.6167	1.35645	0.678226	+	0.734854i	$0.262749\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−38.5710	−1.31756	−0.658780	−	0.752336i	$-0.728927\pi$
$857$	−38.5710	−1.31756	−0.658780	+	0.752336i	$0.728927\pi$
$858$	0	0
$859$	−12.9737	−0.442656	−0.221328	−	0.975199i	$-0.571039\pi$
$859$	−12.9737	−0.442656	−0.221328	+	0.975199i	$0.571039\pi$
$860$	0	0
$861$	−35.4711	−1.20885
$862$	0	0
$863$	−20.3870	−0.693982	−0.346991	−	0.937868i	$-0.612797\pi$
$863$	−20.3870	−0.693982	−0.346991	+	0.937868i	$0.612797\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−9.05534	−0.307535
$868$	0	0
$869$	−1.07304	−0.0364005
$870$	0	0
$871$	−42.9916	−1.45671
$872$	0	0
$873$	−10.0686	−0.340769
$874$	0	0
$875$	0	0
$876$	0	0
$877$	15.4158	0.520554	0.260277	−	0.965534i	$-0.416186\pi$
$877$	15.4158	0.520554	0.260277	+	0.965534i	$0.416186\pi$
$878$	0	0
$879$	−19.5416	−0.659122
$880$	0	0
$881$	19.1361	0.644711	0.322355	−	0.946619i	$-0.395525\pi$
$881$	19.1361	0.644711	0.322355	+	0.946619i	$0.395525\pi$
$882$	0	0
$883$	−3.29325	−0.110827	−0.0554134	−	0.998463i	$-0.517648\pi$
$883$	−3.29325	−0.110827	−0.0554134	+	0.998463i	$0.517648\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−53.8424	−1.80785	−0.903925	−	0.427691i	$-0.859327\pi$
$887$	−53.8424	−1.80785	−0.903925	+	0.427691i	$0.859327\pi$
$888$	0	0
$889$	23.9523	0.803334
$890$	0	0
$891$	1.48724	0.0498245
$892$	0	0
$893$	23.0121	0.770070
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−43.0107	−1.43609
$898$	0	0
$899$	47.9188	1.59818
$900$	0	0
$901$	24.5141	0.816682
$902$	0	0
$903$	−16.8335	−0.560183
$904$	0	0
$905$	0	0
$906$	0	0
$907$	39.2250	1.30244	0.651222	−	0.758888i	$-0.274257\pi$
$907$	39.2250	1.30244	0.651222	+	0.758888i	$0.274257\pi$
$908$	0	0
$909$	1.17643	0.0390199
$910$	0	0
$911$	−17.0376	−0.564479	−0.282240	−	0.959344i	$-0.591077\pi$
$911$	−17.0376	−0.564479	−0.282240	+	0.959344i	$0.591077\pi$
$912$	0	0
$913$	10.5407	0.348847
$914$	0	0
$915$	0	0
$916$	0	0
$917$	12.4889	0.412418
$918$	0	0
$919$	−53.6082	−1.76837	−0.884185	−	0.467136i	$-0.845286\pi$
$919$	−53.6082	−1.76837	−0.884185	+	0.467136i	$0.845286\pi$
$920$	0	0
$921$	−13.3162	−0.438782
$922$	0	0
$923$	−4.43502	−0.145981
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−20.2836	−0.666200
$928$	0	0
$929$	−18.7137	−0.613977	−0.306989	−	0.951713i	$-0.599321\pi$
$929$	−18.7137	−0.613977	−0.306989	+	0.951713i	$0.599321\pi$
$930$	0	0
$931$	13.6585	0.447638
$932$	0	0
$933$	−34.1646	−1.11850
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−44.9946	−1.46991	−0.734955	−	0.678116i	$-0.762797\pi$
$937$	−44.9946	−1.46991	−0.734955	+	0.678116i	$0.762797\pi$
$938$	0	0
$939$	35.6115	1.16214
$940$	0	0
$941$	−40.2103	−1.31082	−0.655408	−	0.755275i	$-0.727503\pi$
$941$	−40.2103	−1.31082	−0.655408	+	0.755275i	$0.727503\pi$
$942$	0	0
$943$	58.5074	1.90526
$944$	0	0
$945$	0	0
$946$	0	0
$947$	30.8063	1.00107	0.500535	−	0.865717i	$-0.333137\pi$
$947$	30.8063	1.00107	0.500535	+	0.865717i	$0.333137\pi$
$948$	0	0
$949$	−15.4362	−0.501082
$950$	0	0
$951$	−5.30910	−0.172159
$952$	0	0
$953$	−28.5659	−0.925341	−0.462671	−	0.886530i	$-0.653109\pi$
$953$	−28.5659	−0.925341	−0.462671	+	0.886530i	$0.653109\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−10.3224	−0.333677
$958$	0	0
$959$	−23.4388	−0.756878
$960$	0	0
$961$	25.8576	0.834115
$962$	0	0
$963$	−2.22397	−0.0716664
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−43.4628	−1.39767	−0.698835	−	0.715282i	$-0.746298\pi$
$967$	−43.4628	−1.39767	−0.698835	+	0.715282i	$0.746298\pi$
$968$	0	0
$969$	7.60477	0.244301
$970$	0	0
$971$	−32.6988	−1.04935	−0.524676	−	0.851302i	$-0.675814\pi$
$971$	−32.6988	−1.04935	−0.524676	+	0.851302i	$0.675814\pi$
$972$	0	0
$973$	−10.1136	−0.324228
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−58.9072	−1.88461	−0.942304	−	0.334758i	$-0.891345\pi$
$977$	−58.9072	−1.88461	−0.942304	+	0.334758i	$0.891345\pi$
$978$	0	0
$979$	−21.1353	−0.675488
$980$	0	0
$981$	−8.57119	−0.273657
$982$	0	0
$983$	−27.4016	−0.873975	−0.436987	−	0.899468i	$-0.643955\pi$
$983$	−27.4016	−0.873975	−0.436987	+	0.899468i	$0.643955\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	43.2143	1.37553
$988$	0	0
$989$	27.7658	0.882900
$990$	0	0
$991$	−6.34631	−0.201597	−0.100799	−	0.994907i	$-0.532140\pi$
$991$	−6.34631	−0.201597	−0.100799	+	0.994907i	$0.532140\pi$
$992$	0	0
$993$	13.3206	0.422718
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−61.3970	−1.94446	−0.972231	−	0.234024i	$-0.924811\pi$
$997$	−61.3970	−1.94446	−0.972231	+	0.234024i	$0.924811\pi$
$998$	0	0
$999$	−5.36402	−0.169710

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7400.2.a.o.1.4		5
5.4	even	2		1480.2.a.j.1.2	✓	5
20.19	odd	2		2960.2.a.x.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1480.2.a.j.1.2	✓	5	5.4	even	2
2960.2.a.x.1.4		5	20.19	odd	2
7400.2.a.o.1.4		5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7400 = 2^{3} \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7400.a (trivial)

\(f(q)\)	\(=\)	\(q+1.14266 q^{3} +3.63076 q^{7} -1.69433 q^{9} -1.42153 q^{11} +5.50061 q^{13} +3.01251 q^{17} +2.20924 q^{19} +4.14872 q^{21} -6.84305 q^{23} -5.36402 q^{27} +6.35495 q^{29} +7.54039 q^{31} -1.62432 q^{33} +1.00000 q^{37} +6.28531 q^{39} -8.54989 q^{41} -4.05751 q^{43} +10.4163 q^{47} +6.18244 q^{49} +3.44226 q^{51} +8.13744 q^{53} +2.52440 q^{57} +1.23303 q^{59} +2.91870 q^{61} -6.15173 q^{63} -7.81578 q^{67} -7.81926 q^{69} -0.806278 q^{71} -2.80628 q^{73} -5.16122 q^{77} +0.754853 q^{79} -1.04623 q^{81} -7.41508 q^{83} +7.26153 q^{87} +14.8681 q^{89} +19.9714 q^{91} +8.61609 q^{93} +5.94249 q^{97} +2.40854 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{3} + 2 q^{7} + 6 q^{9} + 8 q^{11} + 4 q^{13} + q^{17} + 10 q^{19} + 5 q^{21} - 4 q^{23} - q^{27} + 11 q^{29} - 3 q^{31} + 3 q^{33} + 5 q^{37} + 18 q^{39} + 16 q^{41} - 31 q^{43} - 5 q^{47} + 7 q^{49}+ \cdots + 37 q^{99}+O(q^{100}) \) 5 * q - q^3 + 2 * q^7 + 6 * q^9 + 8 * q^11 + 4 * q^13 + q^17 + 10 * q^19 + 5 * q^21 - 4 * q^23 - q^27 + 11 * q^29 - 3 * q^31 + 3 * q^33 + 5 * q^37 + 18 * q^39 + 16 * q^41 - 31 * q^43 - 5 * q^47 + 7 * q^49 + 34 * q^51 - 8 * q^53 + 8 * q^57 + 24 * q^59 - 15 * q^61 - 19 * q^63 - 12 * q^67 + 10 * q^69 + 5 * q^71 - 5 * q^73 + 4 * q^77 + 4 * q^79 + 17 * q^81 - 27 * q^83 + 4 * q^87 + 16 * q^89 + 26 * q^91 - 14 * q^93 + 19 * q^97 + 37 * q^99

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