LMFDB - Embedded newform 5796.2.a.v.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5796,2,Mod(1,5796)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5796, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5796.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5796 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5796.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:	$46.2812930115$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 18x^{4} + 28x^{3} + 82x^{2} - 82x - 1 $ x^6 - 2x^5 - 18x^4 + 28x^3 + 82x^2 - 82*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.0120505$ of defining polynomial
Character	$\chi$	$=$	5796.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.01205 q^{5} -1.00000 q^{7} +O(q^{10})$	$q+1.01205 q^{5} -1.00000 q^{7} +1.02940 q^{11} +6.58951 q^{13} +5.10831 q^{17} -1.43975 q^{19} -1.00000 q^{23} -3.97575 q^{25} -0.199765 q^{29} +2.26938 q^{31} -1.01205 q^{35} +5.54276 q^{37} -4.81744 q^{41} +1.21182 q^{43} +10.7356 q^{47} +1.00000 q^{49} +12.4984 q^{53} +1.04181 q^{55} -12.2459 q^{59} -8.23396 q^{61} +6.66891 q^{65} +6.76517 q^{67} -0.801467 q^{71} -3.54841 q^{73} -1.02940 q^{77} -5.55336 q^{79} -9.32563 q^{83} +5.16987 q^{85} +12.9999 q^{89} -6.58951 q^{91} -1.45710 q^{95} -0.175664 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{5} - 6 q^{7}+O(q^{10}) $ 6 * q + 4 * q^5 - 6 * q^7	$ 6 q + 4 q^{5} - 6 q^{7} + q^{11} + 4 q^{17} - 3 q^{19} - 6 q^{23} + 12 q^{25} + 4 q^{31} - 4 q^{35} + 2 q^{37} + 13 q^{41} + 4 q^{43} - 7 q^{47} + 6 q^{49} + 19 q^{53} + 8 q^{55} + 25 q^{59} + 7 q^{61} + 28 q^{65} + 4 q^{67} - 2 q^{71} + 2 q^{73} - q^{77} + 6 q^{83} - 14 q^{85} + 38 q^{89} - 4 q^{97}+O(q^{100}) $ 6 * q + 4 * q^5 - 6 * q^7 + q^11 + 4 * q^17 - 3 * q^19 - 6 * q^23 + 12 * q^25 + 4 * q^31 - 4 * q^35 + 2 * q^37 + 13 * q^41 + 4 * q^43 - 7 * q^47 + 6 * q^49 + 19 * q^53 + 8 * q^55 + 25 * q^59 + 7 * q^61 + 28 * q^65 + 4 * q^67 - 2 * q^71 + 2 * q^73 - q^77 + 6 * q^83 - 14 * q^85 + 38 * q^89 - 4 * q^97

Display 100 coefficients 10 coefficients

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.01205	0.452603	0.226301	−	0.974057i	$-0.427337\pi$
$5$	1.01205	0.452603	0.226301	+	0.974057i	$0.427337\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.02940	0.310376	0.155188	−	0.987885i	$-0.450402\pi$
$11$	1.02940	0.310376	0.155188	+	0.987885i	$0.450402\pi$
$12$	0	0
$13$	6.58951	1.82760	0.913800	−	0.406164i	$-0.133134\pi$
$13$	6.58951	1.82760	0.913800	+	0.406164i	$0.133134\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.10831	1.23895	0.619473	−	0.785018i	$-0.287346\pi$
$17$	5.10831	1.23895	0.619473	+	0.785018i	$0.287346\pi$
$18$	0	0
$19$	−1.43975	−0.330301	−0.165150	−	0.986268i	$-0.552811\pi$
$19$	−1.43975	−0.330301	−0.165150	+	0.986268i	$0.552811\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−3.97575	−0.795151
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.199765	−0.0370954	−0.0185477	−	0.999828i	$-0.505904\pi$
$29$	−0.199765	−0.0370954	−0.0185477	+	0.999828i	$0.505904\pi$
$30$	0	0
$31$	2.26938	0.407593	0.203797	−	0.979013i	$-0.434672\pi$
$31$	2.26938	0.407593	0.203797	+	0.979013i	$0.434672\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.01205	−0.171068
$36$	0	0
$37$	5.54276	0.911224	0.455612	−	0.890179i	$-0.349421\pi$
$37$	5.54276	0.911224	0.455612	+	0.890179i	$0.349421\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.81744	−0.752358	−0.376179	−	0.926547i	$-0.622762\pi$
$41$	−4.81744	−0.752358	−0.376179	+	0.926547i	$0.622762\pi$
$42$	0	0
$43$	1.21182	0.184800	0.0924001	−	0.995722i	$-0.470546\pi$
$43$	1.21182	0.184800	0.0924001	+	0.995722i	$0.470546\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.7356	1.56595	0.782976	−	0.622052i	$-0.213701\pi$
$47$	10.7356	1.56595	0.782976	+	0.622052i	$0.213701\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	12.4984	1.71679	0.858394	−	0.512991i	$-0.171463\pi$
$53$	12.4984	1.71679	0.858394	+	0.512991i	$0.171463\pi$
$54$	0	0
$55$	1.04181	0.140477
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−12.2459	−1.59428	−0.797138	−	0.603797i	$-0.793654\pi$
$59$	−12.2459	−1.59428	−0.797138	+	0.603797i	$0.793654\pi$
$60$	0	0
$61$	−8.23396	−1.05425	−0.527126	−	0.849787i	$-0.676730\pi$
$61$	−8.23396	−1.05425	−0.527126	+	0.849787i	$0.676730\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.66891	0.827177
$66$	0	0
$67$	6.76517	0.826497	0.413249	−	0.910618i	$-0.364394\pi$
$67$	6.76517	0.826497	0.413249	+	0.910618i	$0.364394\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−0.801467	−0.0951167	−0.0475583	−	0.998868i	$-0.515144\pi$
$71$	−0.801467	−0.0951167	−0.0475583	+	0.998868i	$0.515144\pi$
$72$	0	0
$73$	−3.54841	−0.415310	−0.207655	−	0.978202i	$-0.566583\pi$
$73$	−3.54841	−0.415310	−0.207655	+	0.978202i	$0.566583\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.02940	−0.117311
$78$	0	0
$79$	−5.55336	−0.624801	−0.312401	−	0.949950i	$-0.601133\pi$
$79$	−5.55336	−0.624801	−0.312401	+	0.949950i	$0.601133\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−9.32563	−1.02362	−0.511811	−	0.859098i	$-0.671025\pi$
$83$	−9.32563	−1.02362	−0.511811	+	0.859098i	$0.671025\pi$
$84$	0	0
$85$	5.16987	0.560751
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12.9999	1.37798	0.688991	−	0.724770i	$-0.258054\pi$
$89$	12.9999	1.37798	0.688991	+	0.724770i	$0.258054\pi$
$90$	0	0
$91$	−6.58951	−0.690768
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.45710	−0.149495
$96$	0	0
$97$	−0.175664	−0.0178360	−0.00891799	−	0.999960i	$-0.502839\pi$
$97$	−0.175664	−0.0178360	−0.00891799	+	0.999960i	$0.502839\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	5.78854	0.575981	0.287991	−	0.957633i	$-0.407013\pi$
$101$	5.78854	0.575981	0.287991	+	0.957633i	$0.407013\pi$
$102$	0	0
$103$	−4.78412	−0.471393	−0.235697	−	0.971827i	$-0.575737\pi$
$103$	−4.78412	−0.471393	−0.235697	+	0.971827i	$0.575737\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.74761	0.265622	0.132811	−	0.991141i	$-0.457600\pi$
$107$	2.74761	0.265622	0.132811	+	0.991141i	$0.457600\pi$
$108$	0	0
$109$	5.58587	0.535029	0.267515	−	0.963554i	$-0.413798\pi$
$109$	5.58587	0.535029	0.267515	+	0.963554i	$0.413798\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	8.29602	0.780424	0.390212	−	0.920725i	$-0.372402\pi$
$113$	8.29602	0.780424	0.390212	+	0.920725i	$0.372402\pi$
$114$	0	0
$115$	−1.01205	−0.0943742
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.10831	−0.468278
$120$	0	0
$121$	−9.94034	−0.903667
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−9.08392	−0.812490
$126$	0	0
$127$	−0.856982	−0.0760449	−0.0380224	−	0.999277i	$-0.512106\pi$
$127$	−0.856982	−0.0760449	−0.0380224	+	0.999277i	$0.512106\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−15.2167	−1.32949	−0.664744	−	0.747071i	$-0.731459\pi$
$131$	−15.2167	−1.32949	−0.664744	+	0.747071i	$0.731459\pi$
$132$	0	0
$133$	1.43975	0.124842
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0.894978	0.0764631	0.0382316	−	0.999269i	$-0.487828\pi$
$137$	0.894978	0.0764631	0.0382316	+	0.999269i	$0.487828\pi$
$138$	0	0
$139$	−19.1993	−1.62847	−0.814233	−	0.580538i	$-0.802842\pi$
$139$	−19.1993	−1.62847	−0.814233	+	0.580538i	$0.802842\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	6.78324	0.567243
$144$	0	0
$145$	−0.202172	−0.0167895
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.21588	0.591148	0.295574	−	0.955320i	$-0.404489\pi$
$149$	7.21588	0.591148	0.295574	+	0.955320i	$0.404489\pi$
$150$	0	0
$151$	0.419287	0.0341211	0.0170605	−	0.999854i	$-0.494569\pi$
$151$	0.419287	0.0341211	0.0170605	+	0.999854i	$0.494569\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.29673	0.184478
$156$	0	0
$157$	22.2660	1.77702	0.888509	−	0.458860i	$-0.151742\pi$
$157$	22.2660	1.77702	0.888509	+	0.458860i	$0.151742\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	16.1472	1.26475	0.632373	−	0.774664i	$-0.282081\pi$
$163$	16.1472	1.26475	0.632373	+	0.774664i	$0.282081\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.49600	0.657440	0.328720	−	0.944427i	$-0.393383\pi$
$167$	8.49600	0.657440	0.328720	+	0.944427i	$0.393383\pi$
$168$	0	0
$169$	30.4216	2.34012
$170$	0	0
$171$	0	0
$172$	0	0
$173$	25.6119	1.94723	0.973617	−	0.228188i	$-0.0732802\pi$
$173$	25.6119	1.94723	0.973617	+	0.228188i	$0.0732802\pi$
$174$	0	0
$175$	3.97575	0.300539
$176$	0	0
$177$	0	0
$178$	0	0
$179$	14.9180	1.11503	0.557513	−	0.830168i	$-0.311756\pi$
$179$	14.9180	1.11503	0.557513	+	0.830168i	$0.311756\pi$
$180$	0	0
$181$	−17.0896	−1.27026	−0.635129	−	0.772406i	$-0.719053\pi$
$181$	−17.0896	−1.27026	−0.635129	+	0.772406i	$0.719053\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.60955	0.412422
$186$	0	0
$187$	5.25849	0.384539
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.9007	1.51232	0.756161	−	0.654385i	$-0.227072\pi$
$191$	20.9007	1.51232	0.756161	+	0.654385i	$0.227072\pi$
$192$	0	0
$193$	−3.60481	−0.259480	−0.129740	−	0.991548i	$-0.541414\pi$
$193$	−3.60481	−0.259480	−0.129740	+	0.991548i	$0.541414\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−9.23294	−0.657820	−0.328910	−	0.944361i	$-0.606681\pi$
$197$	−9.23294	−0.657820	−0.328910	+	0.944361i	$0.606681\pi$
$198$	0	0
$199$	0.134213	0.00951413	0.00475706	−	0.999989i	$-0.498486\pi$
$199$	0.134213	0.00951413	0.00475706	+	0.999989i	$0.498486\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.199765	0.0140208
$204$	0	0
$205$	−4.87549	−0.340519
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.48208	−0.102517
$210$	0	0
$211$	−15.4926	−1.06655	−0.533277	−	0.845940i	$-0.679040\pi$
$211$	−15.4926	−1.06655	−0.533277	+	0.845940i	$0.679040\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.22642	0.0836411
$216$	0	0
$217$	−2.26938	−0.154056
$218$	0	0
$219$	0	0
$220$	0	0
$221$	33.6612	2.26430
$222$	0	0
$223$	7.65795	0.512814	0.256407	−	0.966569i	$-0.417461\pi$
$223$	7.65795	0.512814	0.256407	+	0.966569i	$0.417461\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.97735	0.330358	0.165179	−	0.986264i	$-0.447180\pi$
$227$	4.97735	0.330358	0.165179	+	0.986264i	$0.447180\pi$
$228$	0	0
$229$	−19.9999	−1.32163	−0.660816	−	0.750548i	$-0.729790\pi$
$229$	−19.9999	−1.32163	−0.660816	+	0.750548i	$0.729790\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	22.5185	1.47524	0.737618	−	0.675218i	$-0.235950\pi$
$233$	22.5185	1.47524	0.737618	+	0.675218i	$0.235950\pi$
$234$	0	0
$235$	10.8650	0.708754
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−10.0393	−0.649385	−0.324693	−	0.945820i	$-0.605261\pi$
$239$	−10.0393	−0.649385	−0.324693	+	0.945820i	$0.605261\pi$
$240$	0	0
$241$	−9.42915	−0.607385	−0.303692	−	0.952770i	$-0.598220\pi$
$241$	−9.42915	−0.607385	−0.303692	+	0.952770i	$0.598220\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.01205	0.0646575
$246$	0	0
$247$	−9.48723	−0.603658
$248$	0	0
$249$	0	0
$250$	0	0
$251$	26.2269	1.65543	0.827714	−	0.561150i	$-0.189641\pi$
$251$	26.2269	1.65543	0.827714	+	0.561150i	$0.189641\pi$
$252$	0	0
$253$	−1.02940	−0.0647178
$254$	0	0
$255$	0	0
$256$	0	0
$257$	13.4573	0.839444	0.419722	−	0.907653i	$-0.362128\pi$
$257$	13.4573	0.839444	0.419722	+	0.907653i	$0.362128\pi$
$258$	0	0
$259$	−5.54276	−0.344410
$260$	0	0
$261$	0	0
$262$	0	0
$263$	20.6077	1.27072	0.635361	−	0.772215i	$-0.280851\pi$
$263$	20.6077	1.27072	0.635361	+	0.772215i	$0.280851\pi$
$264$	0	0
$265$	12.6490	0.777023
$266$	0	0
$267$	0	0
$268$	0	0
$269$	23.1151	1.40935	0.704677	−	0.709528i	$-0.251092\pi$
$269$	23.1151	1.40935	0.704677	+	0.709528i	$0.251092\pi$
$270$	0	0
$271$	16.6551	1.01173	0.505863	−	0.862614i	$-0.331174\pi$
$271$	16.6551	1.01173	0.505863	+	0.862614i	$0.331174\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.09264	−0.246796
$276$	0	0
$277$	−21.3377	−1.28206	−0.641029	−	0.767516i	$-0.721492\pi$
$277$	−21.3377	−1.28206	−0.641029	+	0.767516i	$0.721492\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−6.67248	−0.398047	−0.199023	−	0.979995i	$-0.563777\pi$
$281$	−6.67248	−0.398047	−0.199023	+	0.979995i	$0.563777\pi$
$282$	0	0
$283$	−18.6496	−1.10860	−0.554302	−	0.832315i	$-0.687015\pi$
$283$	−18.6496	−1.10860	−0.554302	+	0.832315i	$0.687015\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.81744	0.284364
$288$	0	0
$289$	9.09481	0.534989
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−8.64025	−0.504769	−0.252384	−	0.967627i	$-0.581215\pi$
$293$	−8.64025	−0.504769	−0.252384	+	0.967627i	$0.581215\pi$
$294$	0	0
$295$	−12.3934	−0.721574
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−6.58951	−0.381081
$300$	0	0
$301$	−1.21182	−0.0698479
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−8.33319	−0.477157
$306$	0	0
$307$	21.1826	1.20895	0.604477	−	0.796623i	$-0.293382\pi$
$307$	21.1826	1.20895	0.604477	+	0.796623i	$0.293382\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2.70340	0.153296	0.0766480	−	0.997058i	$-0.475578\pi$
$311$	2.70340	0.153296	0.0766480	+	0.997058i	$0.475578\pi$
$312$	0	0
$313$	3.32288	0.187821	0.0939103	−	0.995581i	$-0.470063\pi$
$313$	3.32288	0.187821	0.0939103	+	0.995581i	$0.470063\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−11.0703	−0.621772	−0.310886	−	0.950447i	$-0.600626\pi$
$317$	−11.0703	−0.621772	−0.310886	+	0.950447i	$0.600626\pi$
$318$	0	0
$319$	−0.205638	−0.0115135
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−7.35468	−0.409225
$324$	0	0
$325$	−26.1983	−1.45322
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−10.7356	−0.591874
$330$	0	0
$331$	13.7214	0.754197	0.377099	−	0.926173i	$-0.376922\pi$
$331$	13.7214	0.754197	0.377099	+	0.926173i	$0.376922\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	6.84669	0.374075
$336$	0	0
$337$	−14.1498	−0.770790	−0.385395	−	0.922752i	$-0.625935\pi$
$337$	−14.1498	−0.770790	−0.385395	+	0.922752i	$0.625935\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.33610	0.126507
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−27.4376	−1.47293	−0.736463	−	0.676478i	$-0.763506\pi$
$347$	−27.4376	−1.47293	−0.736463	+	0.676478i	$0.763506\pi$
$348$	0	0
$349$	24.3786	1.30496	0.652479	−	0.757807i	$-0.273729\pi$
$349$	24.3786	1.30496	0.652479	+	0.757807i	$0.273729\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	33.1213	1.76287	0.881434	−	0.472306i	$-0.156578\pi$
$353$	33.1213	1.76287	0.881434	+	0.472306i	$0.156578\pi$
$354$	0	0
$355$	−0.811126	−0.0430501
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7.38253	0.389635	0.194818	−	0.980839i	$-0.437588\pi$
$359$	7.38253	0.389635	0.194818	+	0.980839i	$0.437588\pi$
$360$	0	0
$361$	−16.9271	−0.890901
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3.59117	−0.187970
$366$	0	0
$367$	−2.79712	−0.146009	−0.0730043	−	0.997332i	$-0.523259\pi$
$367$	−2.79712	−0.146009	−0.0730043	+	0.997332i	$0.523259\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−12.4984	−0.648885
$372$	0	0
$373$	11.0979	0.574628	0.287314	−	0.957836i	$-0.407238\pi$
$373$	11.0979	0.574628	0.287314	+	0.957836i	$0.407238\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.31635	−0.0677956
$378$	0	0
$379$	8.05851	0.413938	0.206969	−	0.978348i	$-0.433640\pi$
$379$	8.05851	0.413938	0.206969	+	0.978348i	$0.433640\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	15.3362	0.783645	0.391822	−	0.920041i	$-0.371845\pi$
$383$	15.3362	0.783645	0.391822	+	0.920041i	$0.371845\pi$
$384$	0	0
$385$	−1.04181	−0.0530953
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−19.1968	−0.973316	−0.486658	−	0.873593i	$-0.661784\pi$
$389$	−19.1968	−0.973316	−0.486658	+	0.873593i	$0.661784\pi$
$390$	0	0
$391$	−5.10831	−0.258338
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−5.62028	−0.282787
$396$	0	0
$397$	−2.96169	−0.148643	−0.0743215	−	0.997234i	$-0.523679\pi$
$397$	−2.96169	−0.148643	−0.0743215	+	0.997234i	$0.523679\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.46263	−0.0730403	−0.0365201	−	0.999333i	$-0.511627\pi$
$401$	−1.46263	−0.0730403	−0.0365201	+	0.999333i	$0.511627\pi$
$402$	0	0
$403$	14.9541	0.744918
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5.70571	0.282822
$408$	0	0
$409$	9.61710	0.475535	0.237768	−	0.971322i	$-0.423584\pi$
$409$	9.61710	0.475535	0.237768	+	0.971322i	$0.423584\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	12.2459	0.602580
$414$	0	0
$415$	−9.43801	−0.463294
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−8.76001	−0.427954	−0.213977	−	0.976839i	$-0.568642\pi$
$419$	−8.76001	−0.427954	−0.213977	+	0.976839i	$0.568642\pi$
$420$	0	0
$421$	−34.5428	−1.68351	−0.841757	−	0.539856i	$-0.818479\pi$
$421$	−34.5428	−1.68351	−0.841757	+	0.539856i	$0.818479\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−20.3094	−0.985149
$426$	0	0
$427$	8.23396	0.398469
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−27.3733	−1.31853	−0.659264	−	0.751912i	$-0.729132\pi$
$431$	−27.3733	−1.31853	−0.659264	+	0.751912i	$0.729132\pi$
$432$	0	0
$433$	9.37038	0.450312	0.225156	−	0.974323i	$-0.427711\pi$
$433$	9.37038	0.450312	0.225156	+	0.974323i	$0.427711\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.43975	0.0688725
$438$	0	0
$439$	−3.49992	−0.167042	−0.0835210	−	0.996506i	$-0.526617\pi$
$439$	−3.49992	−0.167042	−0.0835210	+	0.996506i	$0.526617\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3.73489	−0.177450	−0.0887250	−	0.996056i	$-0.528279\pi$
$443$	−3.73489	−0.177450	−0.0887250	+	0.996056i	$0.528279\pi$
$444$	0	0
$445$	13.1565	0.623678
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−10.5667	−0.498674	−0.249337	−	0.968417i	$-0.580213\pi$
$449$	−10.5667	−0.498674	−0.249337	+	0.968417i	$0.580213\pi$
$450$	0	0
$451$	−4.95907	−0.233514
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−6.66891	−0.312643
$456$	0	0
$457$	5.39511	0.252373	0.126186	−	0.992007i	$-0.459726\pi$
$457$	5.39511	0.252373	0.126186	+	0.992007i	$0.459726\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	19.9610	0.929677	0.464838	−	0.885396i	$-0.346112\pi$
$461$	19.9610	0.929677	0.464838	+	0.885396i	$0.346112\pi$
$462$	0	0
$463$	−19.0813	−0.886785	−0.443392	−	0.896328i	$-0.646225\pi$
$463$	−19.0813	−0.886785	−0.443392	+	0.896328i	$0.646225\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.33552	−0.0618006	−0.0309003	−	0.999522i	$-0.509837\pi$
$467$	−1.33552	−0.0618006	−0.0309003	+	0.999522i	$0.509837\pi$
$468$	0	0
$469$	−6.76517	−0.312387
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.24744	0.0573575
$474$	0	0
$475$	5.72408	0.262639
$476$	0	0
$477$	0	0
$478$	0	0
$479$	8.86659	0.405125	0.202562	−	0.979269i	$-0.435073\pi$
$479$	8.86659	0.405125	0.202562	+	0.979269i	$0.435073\pi$
$480$	0	0
$481$	36.5240	1.66535
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−0.177781	−0.00807261
$486$	0	0
$487$	−35.3937	−1.60384	−0.801922	−	0.597429i	$-0.796189\pi$
$487$	−35.3937	−1.60384	−0.801922	+	0.597429i	$0.796189\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	30.2780	1.36642	0.683212	−	0.730220i	$-0.260582\pi$
$491$	30.2780	1.36642	0.683212	+	0.730220i	$0.260582\pi$
$492$	0	0
$493$	−1.02046	−0.0459593
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.801467	0.0359507
$498$	0	0
$499$	−1.32868	−0.0594798	−0.0297399	−	0.999558i	$-0.509468\pi$
$499$	−1.32868	−0.0594798	−0.0297399	+	0.999558i	$0.509468\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	11.0600	0.493139	0.246570	−	0.969125i	$-0.420697\pi$
$503$	11.0600	0.493139	0.246570	+	0.969125i	$0.420697\pi$
$504$	0	0
$505$	5.85829	0.260691
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−14.6115	−0.647644	−0.323822	−	0.946118i	$-0.604968\pi$
$509$	−14.6115	−0.647644	−0.323822	+	0.946118i	$0.604968\pi$
$510$	0	0
$511$	3.54841	0.156972
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−4.84177	−0.213354
$516$	0	0
$517$	11.0513	0.486034
$518$	0	0
$519$	0	0
$520$	0	0
$521$	26.5629	1.16374	0.581870	−	0.813282i	$-0.302321\pi$
$521$	26.5629	1.16374	0.581870	+	0.813282i	$0.302321\pi$
$522$	0	0
$523$	−33.1185	−1.44817	−0.724085	−	0.689711i	$-0.757738\pi$
$523$	−33.1185	−1.44817	−0.724085	+	0.689711i	$0.757738\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	11.5927	0.504986
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−31.7445	−1.37501
$534$	0	0
$535$	2.78072	0.120221
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.02940	0.0443394
$540$	0	0
$541$	23.1641	0.995901	0.497951	−	0.867205i	$-0.334086\pi$
$541$	23.1641	0.995901	0.497951	+	0.867205i	$0.334086\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5.65318	0.242156
$546$	0	0
$547$	36.2142	1.54841	0.774203	−	0.632937i	$-0.218151\pi$
$547$	36.2142	1.54841	0.774203	+	0.632937i	$0.218151\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0.287611	0.0122527
$552$	0	0
$553$	5.55336	0.236153
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−6.05880	−0.256720	−0.128360	−	0.991728i	$-0.540971\pi$
$557$	−6.05880	−0.256720	−0.128360	+	0.991728i	$0.540971\pi$
$558$	0	0
$559$	7.98527	0.337741
$560$	0	0
$561$	0	0
$562$	0	0
$563$	30.1090	1.26894	0.634472	−	0.772946i	$-0.281218\pi$
$563$	30.1090	1.26894	0.634472	+	0.772946i	$0.281218\pi$
$564$	0	0
$565$	8.39599	0.353222
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−40.1792	−1.68440	−0.842199	−	0.539167i	$-0.818739\pi$
$569$	−40.1792	−1.68440	−0.842199	+	0.539167i	$0.818739\pi$
$570$	0	0
$571$	23.4706	0.982213	0.491107	−	0.871099i	$-0.336593\pi$
$571$	23.4706	0.982213	0.491107	+	0.871099i	$0.336593\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.97575	0.165800
$576$	0	0
$577$	30.3074	1.26171	0.630856	−	0.775900i	$-0.282704\pi$
$577$	30.3074	1.26171	0.630856	+	0.775900i	$0.282704\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	9.32563	0.386892
$582$	0	0
$583$	12.8659	0.532849
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−23.8780	−0.985552	−0.492776	−	0.870156i	$-0.664018\pi$
$587$	−23.8780	−0.985552	−0.492776	+	0.870156i	$0.664018\pi$
$588$	0	0
$589$	−3.26734	−0.134628
$590$	0	0
$591$	0	0
$592$	0	0
$593$	36.4385	1.49635	0.748174	−	0.663502i	$-0.230931\pi$
$593$	36.4385	1.49635	0.748174	+	0.663502i	$0.230931\pi$
$594$	0	0
$595$	−5.16987	−0.211944
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−30.6055	−1.25051	−0.625253	−	0.780422i	$-0.715004\pi$
$599$	−30.6055	−1.25051	−0.625253	+	0.780422i	$0.715004\pi$
$600$	0	0
$601$	4.56735	0.186306	0.0931531	−	0.995652i	$-0.470305\pi$
$601$	4.56735	0.186306	0.0931531	+	0.995652i	$0.470305\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−10.0601	−0.409002
$606$	0	0
$607$	10.9639	0.445012	0.222506	−	0.974931i	$-0.428576\pi$
$607$	10.9639	0.445012	0.222506	+	0.974931i	$0.428576\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	70.7425	2.86193
$612$	0	0
$613$	−11.7327	−0.473878	−0.236939	−	0.971525i	$-0.576144\pi$
$613$	−11.7327	−0.473878	−0.236939	+	0.971525i	$0.576144\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	34.4005	1.38491	0.692456	−	0.721460i	$-0.256529\pi$
$617$	34.4005	1.38491	0.692456	+	0.721460i	$0.256529\pi$
$618$	0	0
$619$	−14.8111	−0.595307	−0.297654	−	0.954674i	$-0.596204\pi$
$619$	−14.8111	−0.595307	−0.297654	+	0.954674i	$0.596204\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−12.9999	−0.520828
$624$	0	0
$625$	10.6854	0.427415
$626$	0	0
$627$	0	0
$628$	0	0
$629$	28.3141	1.12896
$630$	0	0
$631$	22.3832	0.891062	0.445531	−	0.895267i	$-0.353015\pi$
$631$	22.3832	0.891062	0.445531	+	0.895267i	$0.353015\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−0.867309	−0.0344181
$636$	0	0
$637$	6.58951	0.261086
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−34.6974	−1.37046	−0.685232	−	0.728325i	$-0.740299\pi$
$641$	−34.6974	−1.37046	−0.685232	+	0.728325i	$0.740299\pi$
$642$	0	0
$643$	−31.8954	−1.25783	−0.628916	−	0.777473i	$-0.716501\pi$
$643$	−31.8954	−1.25783	−0.628916	+	0.777473i	$0.716501\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−16.3671	−0.643457	−0.321729	−	0.946832i	$-0.604264\pi$
$647$	−16.3671	−0.643457	−0.321729	+	0.946832i	$0.604264\pi$
$648$	0	0
$649$	−12.6059	−0.494825
$650$	0	0
$651$	0	0
$652$	0	0
$653$	24.6734	0.965546	0.482773	−	0.875746i	$-0.339630\pi$
$653$	24.6734	0.965546	0.482773	+	0.875746i	$0.339630\pi$
$654$	0	0
$655$	−15.4000	−0.601730
$656$	0	0
$657$	0	0
$658$	0	0
$659$	4.60360	0.179331	0.0896654	−	0.995972i	$-0.471420\pi$
$659$	4.60360	0.179331	0.0896654	+	0.995972i	$0.471420\pi$
$660$	0	0
$661$	−23.4279	−0.911238	−0.455619	−	0.890175i	$-0.650582\pi$
$661$	−23.4279	−0.911238	−0.455619	+	0.890175i	$0.650582\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.45710	0.0565038
$666$	0	0
$667$	0.199765	0.00773493
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−8.47605	−0.327214
$672$	0	0
$673$	−5.90664	−0.227684	−0.113842	−	0.993499i	$-0.536316\pi$
$673$	−5.90664	−0.227684	−0.113842	+	0.993499i	$0.536316\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0.784705	0.0301587	0.0150793	−	0.999886i	$-0.495200\pi$
$677$	0.784705	0.0301587	0.0150793	+	0.999886i	$0.495200\pi$
$678$	0	0
$679$	0.175664	0.00674137
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−35.5147	−1.35893	−0.679465	−	0.733707i	$-0.737788\pi$
$683$	−35.5147	−1.35893	−0.679465	+	0.733707i	$0.737788\pi$
$684$	0	0
$685$	0.905763	0.0346074
$686$	0	0
$687$	0	0
$688$	0	0
$689$	82.3583	3.13760
$690$	0	0
$691$	−21.9610	−0.835437	−0.417719	−	0.908576i	$-0.637170\pi$
$691$	−21.9610	−0.835437	−0.417719	+	0.908576i	$0.637170\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−19.4307	−0.737048
$696$	0	0
$697$	−24.6090	−0.932131
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−27.6018	−1.04251	−0.521253	−	0.853402i	$-0.674535\pi$
$701$	−27.6018	−1.04251	−0.521253	+	0.853402i	$0.674535\pi$
$702$	0	0
$703$	−7.98017	−0.300978
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5.78854	−0.217700
$708$	0	0
$709$	52.7085	1.97951	0.989755	−	0.142779i	$-0.0456039\pi$
$709$	52.7085	1.97951	0.989755	+	0.142779i	$0.0456039\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2.26938	−0.0849891
$714$	0	0
$715$	6.86498	0.256736
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−31.7044	−1.18237	−0.591187	−	0.806535i	$-0.701340\pi$
$719$	−31.7044	−1.18237	−0.591187	+	0.806535i	$0.701340\pi$
$720$	0	0
$721$	4.78412	0.178170
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0.794217	0.0294965
$726$	0	0
$727$	−10.8339	−0.401808	−0.200904	−	0.979611i	$-0.564388\pi$
$727$	−10.8339	−0.401808	−0.200904	+	0.979611i	$0.564388\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	6.19033	0.228958
$732$	0	0
$733$	1.87413	0.0692225	0.0346113	−	0.999401i	$-0.488981\pi$
$733$	1.87413	0.0692225	0.0346113	+	0.999401i	$0.488981\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.96407	0.256525
$738$	0	0
$739$	−3.54368	−0.130357	−0.0651783	−	0.997874i	$-0.520762\pi$
$739$	−3.54368	−0.130357	−0.0651783	+	0.997874i	$0.520762\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	21.8271	0.800759	0.400379	−	0.916350i	$-0.368878\pi$
$743$	21.8271	0.800759	0.400379	+	0.916350i	$0.368878\pi$
$744$	0	0
$745$	7.30284	0.267555
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2.74761	−0.100396
$750$	0	0
$751$	−13.1396	−0.479471	−0.239735	−	0.970838i	$-0.577061\pi$
$751$	−13.1396	−0.479471	−0.239735	+	0.970838i	$0.577061\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0.424340	0.0154433
$756$	0	0
$757$	−27.5762	−1.00227	−0.501136	−	0.865368i	$-0.667085\pi$
$757$	−27.5762	−1.00227	−0.501136	+	0.865368i	$0.667085\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	12.6187	0.457429	0.228714	−	0.973494i	$-0.426548\pi$
$761$	12.6187	0.457429	0.228714	+	0.973494i	$0.426548\pi$
$762$	0	0
$763$	−5.58587	−0.202222
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−80.6942	−2.91370
$768$	0	0
$769$	−35.3043	−1.27311	−0.636553	−	0.771233i	$-0.719640\pi$
$769$	−35.3043	−1.27311	−0.636553	+	0.771233i	$0.719640\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	40.8736	1.47012	0.735060	−	0.678002i	$-0.237154\pi$
$773$	40.8736	1.47012	0.735060	+	0.678002i	$0.237154\pi$
$774$	0	0
$775$	−9.02251	−0.324098
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.93590	0.248504
$780$	0	0
$781$	−0.825031	−0.0295219
$782$	0	0
$783$	0	0
$784$	0	0
$785$	22.5343	0.804283
$786$	0	0
$787$	−40.2778	−1.43575	−0.717874	−	0.696173i	$-0.754885\pi$
$787$	−40.2778	−1.43575	−0.717874	+	0.696173i	$0.754885\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−8.29602	−0.294973
$792$	0	0
$793$	−54.2578	−1.92675
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−27.7638	−0.983443	−0.491722	−	0.870752i	$-0.663632\pi$
$797$	−27.7638	−0.983443	−0.491722	+	0.870752i	$0.663632\pi$
$798$	0	0
$799$	54.8409	1.94013
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−3.65273	−0.128902
$804$	0	0
$805$	1.01205	0.0356701
$806$	0	0
$807$	0	0
$808$	0	0
$809$	14.5064	0.510019	0.255010	−	0.966938i	$-0.417921\pi$
$809$	14.5064	0.510019	0.255010	+	0.966938i	$0.417921\pi$
$810$	0	0
$811$	−24.0518	−0.844572	−0.422286	−	0.906463i	$-0.638772\pi$
$811$	−24.0518	−0.844572	−0.422286	+	0.906463i	$0.638772\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	16.3418	0.572428
$816$	0	0
$817$	−1.74471	−0.0610397
$818$	0	0
$819$	0	0
$820$	0	0
$821$	16.7846	0.585785	0.292893	−	0.956145i	$-0.405382\pi$
$821$	16.7846	0.585785	0.292893	+	0.956145i	$0.405382\pi$
$822$	0	0
$823$	48.1167	1.67724	0.838622	−	0.544714i	$-0.183362\pi$
$823$	48.1167	1.67724	0.838622	+	0.544714i	$0.183362\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−50.5084	−1.75635	−0.878175	−	0.478340i	$-0.841239\pi$
$827$	−50.5084	−1.75635	−0.878175	+	0.478340i	$0.841239\pi$
$828$	0	0
$829$	24.1859	0.840011	0.420006	−	0.907522i	$-0.362028\pi$
$829$	24.1859	0.840011	0.420006	+	0.907522i	$0.362028\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	5.10831	0.176992
$834$	0	0
$835$	8.59838	0.297559
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−34.6347	−1.19572	−0.597862	−	0.801599i	$-0.703983\pi$
$839$	−34.6347	−1.19572	−0.597862	+	0.801599i	$0.703983\pi$
$840$	0	0
$841$	−28.9601	−0.998624
$842$	0	0
$843$	0	0
$844$	0	0
$845$	30.7882	1.05915
$846$	0	0
$847$	9.94034	0.341554
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−5.54276	−0.190003
$852$	0	0
$853$	11.1720	0.382523	0.191261	−	0.981539i	$-0.438742\pi$
$853$	11.1720	0.382523	0.191261	+	0.981539i	$0.438742\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	14.4753	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$857$	14.4753	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$858$	0	0
$859$	49.6507	1.69406	0.847030	−	0.531546i	$-0.178389\pi$
$859$	49.6507	1.69406	0.847030	+	0.531546i	$0.178389\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−38.1298	−1.29795	−0.648977	−	0.760808i	$-0.724803\pi$
$863$	−38.1298	−1.29795	−0.648977	+	0.760808i	$0.724803\pi$
$864$	0	0
$865$	25.9205	0.881323
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−5.71662	−0.193923
$870$	0	0
$871$	44.5791	1.51051
$872$	0	0
$873$	0	0
$874$	0	0
$875$	9.08392	0.307092
$876$	0	0
$877$	−7.69627	−0.259884	−0.129942	−	0.991522i	$-0.541479\pi$
$877$	−7.69627	−0.259884	−0.129942	+	0.991522i	$0.541479\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	23.0677	0.777170	0.388585	−	0.921413i	$-0.372964\pi$
$881$	23.0677	0.777170	0.388585	+	0.921413i	$0.372964\pi$
$882$	0	0
$883$	15.5167	0.522178	0.261089	−	0.965315i	$-0.415918\pi$
$883$	15.5167	0.522178	0.261089	+	0.965315i	$0.415918\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−56.1028	−1.88375	−0.941874	−	0.335967i	$-0.890937\pi$
$887$	−56.1028	−1.88375	−0.941874	+	0.335967i	$0.890937\pi$
$888$	0	0
$889$	0.856982	0.0287423
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−15.4566	−0.517235
$894$	0	0
$895$	15.0978	0.504664
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−0.453343	−0.0151199
$900$	0	0
$901$	63.8457	2.12701
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−17.2955	−0.574922
$906$	0	0
$907$	−23.4803	−0.779652	−0.389826	−	0.920889i	$-0.627465\pi$
$907$	−23.4803	−0.779652	−0.389826	+	0.920889i	$0.627465\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−36.7096	−1.21624	−0.608122	−	0.793843i	$-0.708077\pi$
$911$	−36.7096	−1.21624	−0.608122	+	0.793843i	$0.708077\pi$
$912$	0	0
$913$	−9.59981	−0.317707
$914$	0	0
$915$	0	0
$916$	0	0
$917$	15.2167	0.502499
$918$	0	0
$919$	−7.03056	−0.231917	−0.115958	−	0.993254i	$-0.536994\pi$
$919$	−7.03056	−0.231917	−0.115958	+	0.993254i	$0.536994\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−5.28127	−0.173835
$924$	0	0
$925$	−22.0366	−0.724560
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−42.7266	−1.40181	−0.700907	−	0.713253i	$-0.747221\pi$
$929$	−42.7266	−1.40181	−0.700907	+	0.713253i	$0.747221\pi$
$930$	0	0
$931$	−1.43975	−0.0471858
$932$	0	0
$933$	0	0
$934$	0	0
$935$	5.32186	0.174043
$936$	0	0
$937$	−16.5542	−0.540801	−0.270400	−	0.962748i	$-0.587156\pi$
$937$	−16.5542	−0.540801	−0.270400	+	0.962748i	$0.587156\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−28.8055	−0.939031	−0.469516	−	0.882924i	$-0.655571\pi$
$941$	−28.8055	−0.939031	−0.469516	+	0.882924i	$0.655571\pi$
$942$	0	0
$943$	4.81744	0.156877
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7.17697	0.233220	0.116610	−	0.993178i	$-0.462797\pi$
$947$	7.17697	0.233220	0.116610	+	0.993178i	$0.462797\pi$
$948$	0	0
$949$	−23.3823	−0.759021
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−29.5417	−0.956951	−0.478476	−	0.878101i	$-0.658810\pi$
$953$	−29.5417	−0.956951	−0.478476	+	0.878101i	$0.658810\pi$
$954$	0	0
$955$	21.1526	0.684482
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−0.894978	−0.0289003
$960$	0	0
$961$	−25.8499	−0.833868
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−3.64825	−0.117441
$966$	0	0
$967$	−10.9596	−0.352438	−0.176219	−	0.984351i	$-0.556387\pi$
$967$	−10.9596	−0.352438	−0.176219	+	0.984351i	$0.556387\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	28.3852	0.910924	0.455462	−	0.890255i	$-0.349474\pi$
$971$	28.3852	0.910924	0.455462	+	0.890255i	$0.349474\pi$
$972$	0	0
$973$	19.1993	0.615502
$974$	0	0
$975$	0	0
$976$	0	0
$977$	32.4347	1.03768	0.518839	−	0.854872i	$-0.326364\pi$
$977$	32.4347	1.03768	0.518839	+	0.854872i	$0.326364\pi$
$978$	0	0
$979$	13.3821	0.427692
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−16.0707	−0.512576	−0.256288	−	0.966600i	$-0.582500\pi$
$983$	−16.0707	−0.512576	−0.256288	+	0.966600i	$0.582500\pi$
$984$	0	0
$985$	−9.34420	−0.297731
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.21182	−0.0385335
$990$	0	0
$991$	35.1154	1.11548	0.557739	−	0.830016i	$-0.311669\pi$
$991$	35.1154	1.11548	0.557739	+	0.830016i	$0.311669\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0.135831	0.00430612
$996$	0	0
$997$	−7.75167	−0.245498	−0.122749	−	0.992438i	$-0.539171\pi$
$997$	−7.75167	−0.245498	−0.122749	+	0.992438i	$0.539171\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5796.2.a.v.1.4	yes	6
3.2	odd	2		5796.2.a.u.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5796.2.a.u.1.3	✓	6	3.2	odd	2
5796.2.a.v.1.4	yes	6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5796 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5796.a (trivial)

\(f(q)\)	\(=\)	\(q+1.01205 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+1.01205 q^{5} -1.00000 q^{7} +1.02940 q^{11} +6.58951 q^{13} +5.10831 q^{17} -1.43975 q^{19} -1.00000 q^{23} -3.97575 q^{25} -0.199765 q^{29} +2.26938 q^{31} -1.01205 q^{35} +5.54276 q^{37} -4.81744 q^{41} +1.21182 q^{43} +10.7356 q^{47} +1.00000 q^{49} +12.4984 q^{53} +1.04181 q^{55} -12.2459 q^{59} -8.23396 q^{61} +6.66891 q^{65} +6.76517 q^{67} -0.801467 q^{71} -3.54841 q^{73} -1.02940 q^{77} -5.55336 q^{79} -9.32563 q^{83} +5.16987 q^{85} +12.9999 q^{89} -6.58951 q^{91} -1.45710 q^{95} -0.175664 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{5} - 6 q^{7}+O(q^{10}) \) 6 * q + 4 * q^5 - 6 * q^7	\( 6 q + 4 q^{5} - 6 q^{7} + q^{11} + 4 q^{17} - 3 q^{19} - 6 q^{23} + 12 q^{25} + 4 q^{31} - 4 q^{35} + 2 q^{37} + 13 q^{41} + 4 q^{43} - 7 q^{47} + 6 q^{49} + 19 q^{53} + 8 q^{55} + 25 q^{59} + 7 q^{61} + 28 q^{65} + 4 q^{67} - 2 q^{71} + 2 q^{73} - q^{77} + 6 q^{83} - 14 q^{85} + 38 q^{89} - 4 q^{97}+O(q^{100}) \) 6 * q + 4 * q^5 - 6 * q^7 + q^11 + 4 * q^17 - 3 * q^19 - 6 * q^23 + 12 * q^25 + 4 * q^31 - 4 * q^35 + 2 * q^37 + 13 * q^41 + 4 * q^43 - 7 * q^47 + 6 * q^49 + 19 * q^53 + 8 * q^55 + 25 * q^59 + 7 * q^61 + 28 * q^65 + 4 * q^67 - 2 * q^71 + 2 * q^73 - q^77 + 6 * q^83 - 14 * q^85 + 38 * q^89 - 4 * q^97

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