LMFDB - Embedded newform 7600.2.a.cd.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7600,2,Mod(1,7600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7600, 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("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7600 = 2^{4} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7600.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:	$60.6863055362$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 190)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.12489$ of defining polynomial
Character	$\chi$	$=$	7600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.12489 q^{3} +4.12489 q^{7} +1.51514 q^{9} +O(q^{10})$	$q-2.12489 q^{3} +4.12489 q^{7} +1.51514 q^{9} +2.64002 q^{11} -2.51514 q^{13} +0.515138 q^{17} -1.00000 q^{19} -8.76491 q^{21} +3.09461 q^{23} +3.15516 q^{27} -7.79518 q^{29} -3.67030 q^{31} -5.60975 q^{33} -10.2498 q^{37} +5.34438 q^{39} +8.88979 q^{41} +8.64002 q^{43} -4.96972 q^{47} +10.0147 q^{49} -1.09461 q^{51} -5.48486 q^{53} +2.12489 q^{57} -3.15516 q^{59} -12.6400 q^{61} +6.24977 q^{63} -7.40493 q^{67} -6.57569 q^{69} -11.1396 q^{71} +2.70436 q^{73} +10.8898 q^{77} -16.7493 q^{79} -11.2498 q^{81} +3.28005 q^{83} +16.5639 q^{87} -7.60975 q^{89} -10.3747 q^{91} +7.79897 q^{93} -3.93945 q^{97} +4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} + 4 q^{7} + 5 q^{9}+O(q^{10}) $ 3 * q + 2 * q^3 + 4 * q^7 + 5 * q^9	$ 3 q + 2 q^{3} + 4 q^{7} + 5 q^{9} - 8 q^{13} + 2 q^{17} - 3 q^{19} - 10 q^{21} + 2 q^{27} - 8 q^{29} - 4 q^{31} - 8 q^{33} - 14 q^{37} - 10 q^{39} + 2 q^{41} + 18 q^{43} - 14 q^{47} - 3 q^{49} + 6 q^{51} - 16 q^{53} - 2 q^{57} - 2 q^{59} - 30 q^{61} + 2 q^{63} + 2 q^{67} - 22 q^{69} + 8 q^{71} - 10 q^{73} + 8 q^{77} - 17 q^{81} - 6 q^{83} + 6 q^{87} - 14 q^{89} - 6 q^{91} - 4 q^{93} - 10 q^{97} + 12 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 + 4 * q^7 + 5 * q^9 - 8 * q^13 + 2 * q^17 - 3 * q^19 - 10 * q^21 + 2 * q^27 - 8 * q^29 - 4 * q^31 - 8 * q^33 - 14 * q^37 - 10 * q^39 + 2 * q^41 + 18 * q^43 - 14 * q^47 - 3 * q^49 + 6 * q^51 - 16 * q^53 - 2 * q^57 - 2 * q^59 - 30 * q^61 + 2 * q^63 + 2 * q^67 - 22 * q^69 + 8 * q^71 - 10 * q^73 + 8 * q^77 - 17 * q^81 - 6 * q^83 + 6 * q^87 - 14 * q^89 - 6 * q^91 - 4 * q^93 - 10 * q^97 + 12 * q^99

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
$3$	−2.12489	−1.22680	−0.613402	−	0.789771i	$-0.710199\pi$
$3$	−2.12489	−1.22680	−0.613402	+	0.789771i	$0.710199\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.12489	1.55906	0.779530	−	0.626365i	$-0.215458\pi$
$7$	4.12489	1.55906	0.779530	+	0.626365i	$0.215458\pi$
$8$	0	0
$9$	1.51514	0.505046
$10$	0	0
$11$	2.64002	0.795997	0.397999	−	0.917386i	$-0.369705\pi$
$11$	2.64002	0.795997	0.397999	+	0.917386i	$0.369705\pi$
$12$	0	0
$13$	−2.51514	−0.697574	−0.348787	−	0.937202i	$-0.613406\pi$
$13$	−2.51514	−0.697574	−0.348787	+	0.937202i	$0.613406\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.515138	0.124939	0.0624697	−	0.998047i	$-0.480102\pi$
$17$	0.515138	0.124939	0.0624697	+	0.998047i	$0.480102\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−8.76491	−1.91266
$22$	0	0
$23$	3.09461	0.645271	0.322635	−	0.946523i	$-0.395431\pi$
$23$	3.09461	0.645271	0.322635	+	0.946523i	$0.395431\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3.15516	0.607211
$28$	0	0
$29$	−7.79518	−1.44753	−0.723765	−	0.690047i	$-0.757590\pi$
$29$	−7.79518	−1.44753	−0.723765	+	0.690047i	$0.757590\pi$
$30$	0	0
$31$	−3.67030	−0.659205	−0.329603	−	0.944120i	$-0.606915\pi$
$31$	−3.67030	−0.659205	−0.329603	+	0.944120i	$0.606915\pi$
$32$	0	0
$33$	−5.60975	−0.976532
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−10.2498	−1.68505	−0.842526	−	0.538656i	$-0.818932\pi$
$37$	−10.2498	−1.68505	−0.842526	+	0.538656i	$0.818932\pi$
$38$	0	0
$39$	5.34438	0.855786
$40$	0	0
$41$	8.88979	1.38835	0.694176	−	0.719805i	$-0.255769\pi$
$41$	8.88979	1.38835	0.694176	+	0.719805i	$0.255769\pi$
$42$	0	0
$43$	8.64002	1.31759	0.658796	−	0.752322i	$-0.271066\pi$
$43$	8.64002	1.31759	0.658796	+	0.752322i	$0.271066\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.96972	−0.724909	−0.362454	−	0.932002i	$-0.618061\pi$
$47$	−4.96972	−0.724909	−0.362454	+	0.932002i	$0.618061\pi$
$48$	0	0
$49$	10.0147	1.43067
$50$	0	0
$51$	−1.09461	−0.153276
$52$	0	0
$53$	−5.48486	−0.753404	−0.376702	−	0.926335i	$-0.622942\pi$
$53$	−5.48486	−0.753404	−0.376702	+	0.926335i	$0.622942\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.12489	0.281448
$58$	0	0
$59$	−3.15516	−0.410767	−0.205384	−	0.978682i	$-0.565844\pi$
$59$	−3.15516	−0.410767	−0.205384	+	0.978682i	$0.565844\pi$
$60$	0	0
$61$	−12.6400	−1.61839	−0.809195	−	0.587541i	$-0.800096\pi$
$61$	−12.6400	−1.61839	−0.809195	+	0.587541i	$0.800096\pi$
$62$	0	0
$63$	6.24977	0.787397
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−7.40493	−0.904656	−0.452328	−	0.891852i	$-0.649406\pi$
$67$	−7.40493	−0.904656	−0.452328	+	0.891852i	$0.649406\pi$
$68$	0	0
$69$	−6.57569	−0.791620
$70$	0	0
$71$	−11.1396	−1.32202	−0.661012	−	0.750376i	$-0.729873\pi$
$71$	−11.1396	−1.32202	−0.661012	+	0.750376i	$0.729873\pi$
$72$	0	0
$73$	2.70436	0.316521	0.158261	−	0.987397i	$-0.449411\pi$
$73$	2.70436	0.316521	0.158261	+	0.987397i	$0.449411\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	10.8898	1.24101
$78$	0	0
$79$	−16.7493	−1.88444	−0.942222	−	0.334988i	$-0.891268\pi$
$79$	−16.7493	−1.88444	−0.942222	+	0.334988i	$0.891268\pi$
$80$	0	0
$81$	−11.2498	−1.24997
$82$	0	0
$83$	3.28005	0.360032	0.180016	−	0.983664i	$-0.442385\pi$
$83$	3.28005	0.360032	0.180016	+	0.983664i	$0.442385\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	16.5639	1.77583
$88$	0	0
$89$	−7.60975	−0.806632	−0.403316	−	0.915061i	$-0.632142\pi$
$89$	−7.60975	−0.806632	−0.403316	+	0.915061i	$0.632142\pi$
$90$	0	0
$91$	−10.3747	−1.08756
$92$	0	0
$93$	7.79897	0.808715
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−3.93945	−0.399990	−0.199995	−	0.979797i	$-0.564093\pi$
$97$	−3.93945	−0.399990	−0.199995	+	0.979797i	$0.564093\pi$
$98$	0	0
$99$	4.00000	0.402015
$100$	0	0
$101$	−13.7990	−1.37305	−0.686524	−	0.727107i	$-0.740864\pi$
$101$	−13.7990	−1.37305	−0.686524	+	0.727107i	$0.740864\pi$
$102$	0	0
$103$	4.57947	0.451229	0.225614	−	0.974217i	$-0.427561\pi$
$103$	4.57947	0.451229	0.225614	+	0.974217i	$0.427561\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.3747	−1.00296	−0.501478	−	0.865170i	$-0.667210\pi$
$107$	−10.3747	−1.00296	−0.501478	+	0.865170i	$0.667210\pi$
$108$	0	0
$109$	3.01468	0.288754	0.144377	−	0.989523i	$-0.453882\pi$
$109$	3.01468	0.288754	0.144377	+	0.989523i	$0.453882\pi$
$110$	0	0
$111$	21.7796	2.06723
$112$	0	0
$113$	19.2001	1.80620	0.903098	−	0.429435i	$-0.141287\pi$
$113$	19.2001	1.80620	0.903098	+	0.429435i	$0.141287\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−3.81078	−0.352307
$118$	0	0
$119$	2.12489	0.194788
$120$	0	0
$121$	−4.03028	−0.366389
$122$	0	0
$123$	−18.8898	−1.70324
$124$	0	0
$125$	0	0
$126$	0	0
$127$	14.3103	1.26984	0.634918	−	0.772580i	$-0.281034\pi$
$127$	14.3103	1.26984	0.634918	+	0.772580i	$0.281034\pi$
$128$	0	0
$129$	−18.3591	−1.61643
$130$	0	0
$131$	6.56009	0.573158	0.286579	−	0.958057i	$-0.407482\pi$
$131$	6.56009	0.573158	0.286579	+	0.958057i	$0.407482\pi$
$132$	0	0
$133$	−4.12489	−0.357673
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.45459	0.551452	0.275726	−	0.961236i	$-0.411082\pi$
$137$	6.45459	0.551452	0.275726	+	0.961236i	$0.411082\pi$
$138$	0	0
$139$	23.0596	1.95589	0.977946	−	0.208856i	$-0.0669740\pi$
$139$	23.0596	1.95589	0.977946	+	0.208856i	$0.0669740\pi$
$140$	0	0
$141$	10.5601	0.889320
$142$	0	0
$143$	−6.64002	−0.555267
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−21.2800	−1.75515
$148$	0	0
$149$	16.0294	1.31318	0.656588	−	0.754249i	$-0.271999\pi$
$149$	16.0294	1.31318	0.656588	+	0.754249i	$0.271999\pi$
$150$	0	0
$151$	14.3103	1.16456	0.582279	−	0.812989i	$-0.302161\pi$
$151$	14.3103	1.16456	0.582279	+	0.812989i	$0.302161\pi$
$152$	0	0
$153$	0.780505	0.0631001
$154$	0	0
$155$	0	0
$156$	0	0
$157$	18.0294	1.43890	0.719450	−	0.694544i	$-0.244394\pi$
$157$	18.0294	1.43890	0.719450	+	0.694544i	$0.244394\pi$
$158$	0	0
$159$	11.6547	0.924278
$160$	0	0
$161$	12.7649	1.00602
$162$	0	0
$163$	2.70058	0.211525	0.105763	−	0.994391i	$-0.466272\pi$
$163$	2.70058	0.211525	0.105763	+	0.994391i	$0.466272\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.95035	0.692599	0.346299	−	0.938124i	$-0.387438\pi$
$167$	8.95035	0.692599	0.346299	+	0.938124i	$0.387438\pi$
$168$	0	0
$169$	−6.67408	−0.513391
$170$	0	0
$171$	−1.51514	−0.115866
$172$	0	0
$173$	−0.310323	−0.0235934	−0.0117967	−	0.999930i	$-0.503755\pi$
$173$	−0.310323	−0.0235934	−0.0117967	+	0.999930i	$0.503755\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.70436	0.503930
$178$	0	0
$179$	1.52982	0.114344	0.0571720	−	0.998364i	$-0.481792\pi$
$179$	1.52982	0.114344	0.0571720	+	0.998364i	$0.481792\pi$
$180$	0	0
$181$	−14.7493	−1.09631	−0.548154	−	0.836377i	$-0.684669\pi$
$181$	−14.7493	−1.09631	−0.548154	+	0.836377i	$0.684669\pi$
$182$	0	0
$183$	26.8586	1.98544
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.35998	0.0994513
$188$	0	0
$189$	13.0147	0.946679
$190$	0	0
$191$	−18.1249	−1.31147	−0.655735	−	0.754991i	$-0.727641\pi$
$191$	−18.1249	−1.31147	−0.655735	+	0.754991i	$0.727641\pi$
$192$	0	0
$193$	4.56009	0.328243	0.164121	−	0.986440i	$-0.447521\pi$
$193$	4.56009	0.328243	0.164121	+	0.986440i	$0.447521\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.14048	0.152503	0.0762515	−	0.997089i	$-0.475705\pi$
$197$	2.14048	0.152503	0.0762515	+	0.997089i	$0.475705\pi$
$198$	0	0
$199$	−16.5639	−1.17418	−0.587091	−	0.809521i	$-0.699727\pi$
$199$	−16.5639	−1.17418	−0.587091	+	0.809521i	$0.699727\pi$
$200$	0	0
$201$	15.7346	1.10984
$202$	0	0
$203$	−32.1542	−2.25679
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.68876	0.325891
$208$	0	0
$209$	−2.64002	−0.182614
$210$	0	0
$211$	15.2838	1.05218	0.526091	−	0.850428i	$-0.323657\pi$
$211$	15.2838	1.05218	0.526091	+	0.850428i	$0.323657\pi$
$212$	0	0
$213$	23.6703	1.62186
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−15.1396	−1.02774
$218$	0	0
$219$	−5.74645	−0.388309
$220$	0	0
$221$	−1.29564	−0.0871544
$222$	0	0
$223$	−3.75023	−0.251134	−0.125567	−	0.992085i	$-0.540075\pi$
$223$	−3.75023	−0.251134	−0.125567	+	0.992085i	$0.540075\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−24.1542	−1.60317	−0.801587	−	0.597878i	$-0.796011\pi$
$227$	−24.1542	−1.60317	−0.801587	+	0.597878i	$0.796011\pi$
$228$	0	0
$229$	−13.0596	−0.863005	−0.431503	−	0.902112i	$-0.642016\pi$
$229$	−13.0596	−0.863005	−0.431503	+	0.902112i	$0.642016\pi$
$230$	0	0
$231$	−23.1396	−1.52247
$232$	0	0
$233$	6.43899	0.421832	0.210916	−	0.977504i	$-0.432355\pi$
$233$	6.43899	0.421832	0.210916	+	0.977504i	$0.432355\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	35.5904	2.31184
$238$	0	0
$239$	−22.8742	−1.47961	−0.739804	−	0.672822i	$-0.765082\pi$
$239$	−22.8742	−1.47961	−0.739804	+	0.672822i	$0.765082\pi$
$240$	0	0
$241$	4.96972	0.320128	0.160064	−	0.987107i	$-0.448830\pi$
$241$	4.96972	0.320128	0.160064	+	0.987107i	$0.448830\pi$
$242$	0	0
$243$	14.4390	0.926262
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.51514	0.160034
$248$	0	0
$249$	−6.96972	−0.441688
$250$	0	0
$251$	−15.9201	−1.00487	−0.502433	−	0.864616i	$-0.667562\pi$
$251$	−15.9201	−1.00487	−0.502433	+	0.864616i	$0.667562\pi$
$252$	0	0
$253$	8.16984	0.513634
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.04965	0.0654756	0.0327378	−	0.999464i	$-0.489577\pi$
$257$	1.04965	0.0654756	0.0327378	+	0.999464i	$0.489577\pi$
$258$	0	0
$259$	−42.2791	−2.62710
$260$	0	0
$261$	−11.8108	−0.731069
$262$	0	0
$263$	−11.9394	−0.736218	−0.368109	−	0.929783i	$-0.619995\pi$
$263$	−11.9394	−0.736218	−0.368109	+	0.929783i	$0.619995\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	16.1698	0.989578
$268$	0	0
$269$	−15.9394	−0.971845	−0.485923	−	0.874002i	$-0.661516\pi$
$269$	−15.9394	−0.971845	−0.485923	+	0.874002i	$0.661516\pi$
$270$	0	0
$271$	−1.46548	−0.0890218	−0.0445109	−	0.999009i	$-0.514173\pi$
$271$	−1.46548	−0.0890218	−0.0445109	+	0.999009i	$0.514173\pi$
$272$	0	0
$273$	22.0450	1.33422
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−32.4995	−1.95271	−0.976354	−	0.216177i	$-0.930641\pi$
$277$	−32.4995	−1.95271	−0.976354	+	0.216177i	$0.930641\pi$
$278$	0	0
$279$	−5.56101	−0.332929
$280$	0	0
$281$	1.04965	0.0626171	0.0313085	−	0.999510i	$-0.490033\pi$
$281$	1.04965	0.0626171	0.0313085	+	0.999510i	$0.490033\pi$
$282$	0	0
$283$	−9.34060	−0.555241	−0.277620	−	0.960691i	$-0.589546\pi$
$283$	−9.34060	−0.555241	−0.277620	+	0.960691i	$0.589546\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	36.6694	2.16453
$288$	0	0
$289$	−16.7346	−0.984390
$290$	0	0
$291$	8.37088	0.490709
$292$	0	0
$293$	−25.5748	−1.49409	−0.747047	−	0.664771i	$-0.768529\pi$
$293$	−25.5748	−1.49409	−0.747047	+	0.664771i	$0.768529\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	8.32970	0.483338
$298$	0	0
$299$	−7.78337	−0.450124
$300$	0	0
$301$	35.6391	2.05420
$302$	0	0
$303$	29.3212	1.68446
$304$	0	0
$305$	0	0
$306$	0	0
$307$	9.43991	0.538764	0.269382	−	0.963033i	$-0.413181\pi$
$307$	9.43991	0.538764	0.269382	+	0.963033i	$0.413181\pi$
$308$	0	0
$309$	−9.73085	−0.553569
$310$	0	0
$311$	17.4655	0.990377	0.495188	−	0.868786i	$-0.335099\pi$
$311$	17.4655	0.990377	0.495188	+	0.868786i	$0.335099\pi$
$312$	0	0
$313$	10.4546	0.590928	0.295464	−	0.955354i	$-0.404526\pi$
$313$	10.4546	0.590928	0.295464	+	0.955354i	$0.404526\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4.33348	−0.243393	−0.121696	−	0.992567i	$-0.538833\pi$
$317$	−4.33348	−0.243393	−0.121696	+	0.992567i	$0.538833\pi$
$318$	0	0
$319$	−20.5795	−1.15223
$320$	0	0
$321$	22.0450	1.23043
$322$	0	0
$323$	−0.515138	−0.0286630
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−6.40585	−0.354244
$328$	0	0
$329$	−20.4995	−1.13018
$330$	0	0
$331$	4.56387	0.250853	0.125427	−	0.992103i	$-0.459970\pi$
$331$	4.56387	0.250853	0.125427	+	0.992103i	$0.459970\pi$
$332$	0	0
$333$	−15.5298	−0.851029
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−13.0109	−0.708749	−0.354374	−	0.935104i	$-0.615306\pi$
$337$	−13.0109	−0.708749	−0.354374	+	0.935104i	$0.615306\pi$
$338$	0	0
$339$	−40.7980	−2.21585
$340$	0	0
$341$	−9.68968	−0.524725
$342$	0	0
$343$	12.4352	0.671438
$344$	0	0
$345$	0	0
$346$	0	0
$347$	13.2195	0.709660	0.354830	−	0.934931i	$-0.384539\pi$
$347$	13.2195	0.709660	0.354830	+	0.934931i	$0.384539\pi$
$348$	0	0
$349$	−20.4390	−1.09407	−0.547037	−	0.837108i	$-0.684244\pi$
$349$	−20.4390	−1.09407	−0.547037	+	0.837108i	$0.684244\pi$
$350$	0	0
$351$	−7.93567	−0.423575
$352$	0	0
$353$	−10.7044	−0.569735	−0.284868	−	0.958567i	$-0.591950\pi$
$353$	−10.7044	−0.569735	−0.284868	+	0.958567i	$0.591950\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−4.51514	−0.238966
$358$	0	0
$359$	4.31410	0.227690	0.113845	−	0.993499i	$-0.463683\pi$
$359$	4.31410	0.227690	0.113845	+	0.993499i	$0.463683\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	8.56387	0.449487
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−12.8099	−0.668669	−0.334335	−	0.942454i	$-0.608512\pi$
$367$	−12.8099	−0.668669	−0.334335	+	0.942454i	$0.608512\pi$
$368$	0	0
$369$	13.4693	0.701182
$370$	0	0
$371$	−22.6244	−1.17460
$372$	0	0
$373$	−0.704357	−0.0364702	−0.0182351	−	0.999834i	$-0.505805\pi$
$373$	−0.704357	−0.0364702	−0.0182351	+	0.999834i	$0.505805\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	19.6060	1.00976
$378$	0	0
$379$	−6.12489	−0.314614	−0.157307	−	0.987550i	$-0.550281\pi$
$379$	−6.12489	−0.314614	−0.157307	+	0.987550i	$0.550281\pi$
$380$	0	0
$381$	−30.4078	−1.55784
$382$	0	0
$383$	18.6400	0.952461	0.476230	−	0.879321i	$-0.342003\pi$
$383$	18.6400	0.952461	0.476230	+	0.879321i	$0.342003\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	13.0908	0.665444
$388$	0	0
$389$	13.9201	0.705776	0.352888	−	0.935666i	$-0.385200\pi$
$389$	13.9201	0.705776	0.352888	+	0.935666i	$0.385200\pi$
$390$	0	0
$391$	1.59415	0.0806197
$392$	0	0
$393$	−13.9394	−0.703152
$394$	0	0
$395$	0	0
$396$	0	0
$397$	28.3784	1.42427	0.712136	−	0.702041i	$-0.247728\pi$
$397$	28.3784	1.42427	0.712136	+	0.702041i	$0.247728\pi$
$398$	0	0
$399$	8.76491	0.438794
$400$	0	0
$401$	5.54920	0.277114	0.138557	−	0.990354i	$-0.455754\pi$
$401$	5.54920	0.277114	0.138557	+	0.990354i	$0.455754\pi$
$402$	0	0
$403$	9.23131	0.459844
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−27.0596	−1.34130
$408$	0	0
$409$	−5.01090	−0.247773	−0.123886	−	0.992296i	$-0.539536\pi$
$409$	−5.01090	−0.247773	−0.123886	+	0.992296i	$0.539536\pi$
$410$	0	0
$411$	−13.7153	−0.676524
$412$	0	0
$413$	−13.0147	−0.640411
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−48.9991	−2.39950
$418$	0	0
$419$	15.8889	0.776222	0.388111	−	0.921613i	$-0.373128\pi$
$419$	15.8889	0.776222	0.388111	+	0.921613i	$0.373128\pi$
$420$	0	0
$421$	−2.38647	−0.116310	−0.0581548	−	0.998308i	$-0.518522\pi$
$421$	−2.38647	−0.116310	−0.0581548	+	0.998308i	$0.518522\pi$
$422$	0	0
$423$	−7.52982	−0.366112
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−52.1386	−2.52317
$428$	0	0
$429$	14.1093	0.681203
$430$	0	0
$431$	2.35906	0.113632	0.0568160	−	0.998385i	$-0.481905\pi$
$431$	2.35906	0.113632	0.0568160	+	0.998385i	$0.481905\pi$
$432$	0	0
$433$	0.640023	0.0307576	0.0153788	−	0.999882i	$-0.495105\pi$
$433$	0.640023	0.0307576	0.0153788	+	0.999882i	$0.495105\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.09461	−0.148035
$438$	0	0
$439$	25.4499	1.21466	0.607328	−	0.794451i	$-0.292241\pi$
$439$	25.4499	1.21466	0.607328	+	0.794451i	$0.292241\pi$
$440$	0	0
$441$	15.1736	0.722553
$442$	0	0
$443$	−35.6685	−1.69466	−0.847330	−	0.531067i	$-0.821791\pi$
$443$	−35.6685	−1.69466	−0.847330	+	0.531067i	$0.821791\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−34.0606	−1.61101
$448$	0	0
$449$	−11.4399	−0.539883	−0.269941	−	0.962877i	$-0.587004\pi$
$449$	−11.4399	−0.539883	−0.269941	+	0.962877i	$0.587004\pi$
$450$	0	0
$451$	23.4693	1.10512
$452$	0	0
$453$	−30.4078	−1.42868
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−19.4849	−0.911463	−0.455732	−	0.890117i	$-0.650622\pi$
$457$	−19.4849	−0.911463	−0.455732	+	0.890117i	$0.650622\pi$
$458$	0	0
$459$	1.62534	0.0758645
$460$	0	0
$461$	−29.3893	−1.36880	−0.684399	−	0.729108i	$-0.739935\pi$
$461$	−29.3893	−1.36880	−0.684399	+	0.729108i	$0.739935\pi$
$462$	0	0
$463$	12.1892	0.566481	0.283241	−	0.959049i	$-0.408591\pi$
$463$	12.1892	0.566481	0.283241	+	0.959049i	$0.408591\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−17.8889	−0.827799	−0.413899	−	0.910323i	$-0.635833\pi$
$467$	−17.8889	−0.827799	−0.413899	+	0.910323i	$0.635833\pi$
$468$	0	0
$469$	−30.5445	−1.41041
$470$	0	0
$471$	−38.3103	−1.76525
$472$	0	0
$473$	22.8099	1.04880
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−8.31032	−0.380504
$478$	0	0
$479$	1.15894	0.0529534	0.0264767	−	0.999649i	$-0.491571\pi$
$479$	1.15894	0.0529534	0.0264767	+	0.999649i	$0.491571\pi$
$480$	0	0
$481$	25.7796	1.17545
$482$	0	0
$483$	−27.1240	−1.23418
$484$	0	0
$485$	0	0
$486$	0	0
$487$	30.6888	1.39064	0.695320	−	0.718700i	$-0.255263\pi$
$487$	30.6888	1.39064	0.695320	+	0.718700i	$0.255263\pi$
$488$	0	0
$489$	−5.73841	−0.259500
$490$	0	0
$491$	−3.67030	−0.165638	−0.0828192	−	0.996565i	$-0.526392\pi$
$491$	−3.67030	−0.165638	−0.0828192	+	0.996565i	$0.526392\pi$
$492$	0	0
$493$	−4.01560	−0.180853
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−45.9494	−2.06111
$498$	0	0
$499$	−31.3893	−1.40518	−0.702590	−	0.711595i	$-0.747973\pi$
$499$	−31.3893	−1.40518	−0.702590	+	0.711595i	$0.747973\pi$
$500$	0	0
$501$	−19.0185	−0.849682
$502$	0	0
$503$	−18.1542	−0.809458	−0.404729	−	0.914437i	$-0.632634\pi$
$503$	−18.1542	−0.809458	−0.404729	+	0.914437i	$0.632634\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	14.1817	0.629829
$508$	0	0
$509$	11.5298	0.511050	0.255525	−	0.966802i	$-0.417752\pi$
$509$	11.5298	0.511050	0.255525	+	0.966802i	$0.417752\pi$
$510$	0	0
$511$	11.1552	0.493475
$512$	0	0
$513$	−3.15516	−0.139304
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−13.1202	−0.577025
$518$	0	0
$519$	0.659401	0.0289445
$520$	0	0
$521$	−42.5895	−1.86588	−0.932939	−	0.360035i	$-0.882765\pi$
$521$	−42.5895	−1.86588	−0.932939	+	0.360035i	$0.882765\pi$
$522$	0	0
$523$	1.21571	0.0531594	0.0265797	−	0.999647i	$-0.491538\pi$
$523$	1.21571	0.0531594	0.0265797	+	0.999647i	$0.491538\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.89071	−0.0823607
$528$	0	0
$529$	−13.4234	−0.583626
$530$	0	0
$531$	−4.78051	−0.207456
$532$	0	0
$533$	−22.3591	−0.968478
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−3.25069	−0.140278
$538$	0	0
$539$	26.4390	1.13881
$540$	0	0
$541$	−1.92007	−0.0825503	−0.0412751	−	0.999148i	$-0.513142\pi$
$541$	−1.92007	−0.0825503	−0.0412751	+	0.999148i	$0.513142\pi$
$542$	0	0
$543$	31.3406	1.34495
$544$	0	0
$545$	0	0
$546$	0	0
$547$	10.9697	0.469032	0.234516	−	0.972112i	$-0.424650\pi$
$547$	10.9697	0.469032	0.234516	+	0.972112i	$0.424650\pi$
$548$	0	0
$549$	−19.1514	−0.817361
$550$	0	0
$551$	7.79518	0.332086
$552$	0	0
$553$	−69.0890	−2.93796
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−23.5005	−0.995746	−0.497873	−	0.867250i	$-0.665886\pi$
$557$	−23.5005	−0.995746	−0.497873	+	0.867250i	$0.665886\pi$
$558$	0	0
$559$	−21.7309	−0.919117
$560$	0	0
$561$	−2.88979	−0.122007
$562$	0	0
$563$	−7.87890	−0.332056	−0.166028	−	0.986121i	$-0.553094\pi$
$563$	−7.87890	−0.332056	−0.166028	+	0.986121i	$0.553094\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−46.4040	−1.94879
$568$	0	0
$569$	−1.09083	−0.0457299	−0.0228650	−	0.999739i	$-0.507279\pi$
$569$	−1.09083	−0.0457299	−0.0228650	+	0.999739i	$0.507279\pi$
$570$	0	0
$571$	−21.4886	−0.899272	−0.449636	−	0.893212i	$-0.648446\pi$
$571$	−21.4886	−0.899272	−0.449636	+	0.893212i	$0.648446\pi$
$572$	0	0
$573$	38.5133	1.60892
$574$	0	0
$575$	0	0
$576$	0	0
$577$	16.1055	0.670481	0.335241	−	0.942133i	$-0.391182\pi$
$577$	16.1055	0.670481	0.335241	+	0.942133i	$0.391182\pi$
$578$	0	0
$579$	−9.68968	−0.402689
$580$	0	0
$581$	13.5298	0.561311
$582$	0	0
$583$	−14.4802	−0.599707
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0.480164	0.0198185	0.00990925	−	0.999951i	$-0.496846\pi$
$587$	0.480164	0.0198185	0.00990925	+	0.999951i	$0.496846\pi$
$588$	0	0
$589$	3.67030	0.151232
$590$	0	0
$591$	−4.54828	−0.187091
$592$	0	0
$593$	−40.4683	−1.66184	−0.830918	−	0.556395i	$-0.812184\pi$
$593$	−40.4683	−1.66184	−0.830918	+	0.556395i	$0.812184\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	35.1963	1.44049
$598$	0	0
$599$	36.1698	1.47786	0.738930	−	0.673782i	$-0.235331\pi$
$599$	36.1698	1.47786	0.738930	+	0.673782i	$0.235331\pi$
$600$	0	0
$601$	24.3903	0.994899	0.497450	−	0.867493i	$-0.334270\pi$
$601$	24.3903	0.994899	0.497450	+	0.867493i	$0.334270\pi$
$602$	0	0
$603$	−11.2195	−0.456893
$604$	0	0
$605$	0	0
$606$	0	0
$607$	21.6897	0.880357	0.440178	−	0.897910i	$-0.354915\pi$
$607$	21.6897	0.880357	0.440178	+	0.897910i	$0.354915\pi$
$608$	0	0
$609$	68.3241	2.76863
$610$	0	0
$611$	12.4995	0.505677
$612$	0	0
$613$	26.2304	1.05944	0.529718	−	0.848174i	$-0.322298\pi$
$613$	26.2304	1.05944	0.529718	+	0.848174i	$0.322298\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	22.7200	0.914671	0.457335	−	0.889294i	$-0.348804\pi$
$617$	22.7200	0.914671	0.457335	+	0.889294i	$0.348804\pi$
$618$	0	0
$619$	32.9192	1.32313	0.661566	−	0.749887i	$-0.269892\pi$
$619$	32.9192	1.32313	0.661566	+	0.749887i	$0.269892\pi$
$620$	0	0
$621$	9.76399	0.391816
$622$	0	0
$623$	−31.3893	−1.25759
$624$	0	0
$625$	0	0
$626$	0	0
$627$	5.60975	0.224032
$628$	0	0
$629$	−5.28005	−0.210529
$630$	0	0
$631$	−17.2876	−0.688209	−0.344104	−	0.938931i	$-0.611817\pi$
$631$	−17.2876	−0.688209	−0.344104	+	0.938931i	$0.611817\pi$
$632$	0	0
$633$	−32.4764	−1.29082
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−25.1883	−0.997997
$638$	0	0
$639$	−16.8780	−0.667683
$640$	0	0
$641$	−9.29942	−0.367305	−0.183653	−	0.982991i	$-0.558792\pi$
$641$	−9.29942	−0.367305	−0.183653	+	0.982991i	$0.558792\pi$
$642$	0	0
$643$	−3.40115	−0.134128	−0.0670642	−	0.997749i	$-0.521363\pi$
$643$	−3.40115	−0.134128	−0.0670642	+	0.997749i	$0.521363\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	14.9348	0.587146	0.293573	−	0.955937i	$-0.405156\pi$
$647$	14.9348	0.587146	0.293573	+	0.955937i	$0.405156\pi$
$648$	0	0
$649$	−8.32970	−0.326969
$650$	0	0
$651$	32.1698	1.26084
$652$	0	0
$653$	−21.1202	−0.826497	−0.413248	−	0.910618i	$-0.635606\pi$
$653$	−21.1202	−0.826497	−0.413248	+	0.910618i	$0.635606\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	4.09747	0.159858
$658$	0	0
$659$	−27.6547	−1.07727	−0.538637	−	0.842538i	$-0.681061\pi$
$659$	−27.6547	−1.07727	−0.538637	+	0.842538i	$0.681061\pi$
$660$	0	0
$661$	10.2342	0.398063	0.199032	−	0.979993i	$-0.436220\pi$
$661$	10.2342	0.398063	0.199032	+	0.979993i	$0.436220\pi$
$662$	0	0
$663$	2.75309	0.106921
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−24.1231	−0.934048
$668$	0	0
$669$	7.96881	0.308092
$670$	0	0
$671$	−33.3700	−1.28823
$672$	0	0
$673$	13.4693	0.519202	0.259601	−	0.965716i	$-0.416409\pi$
$673$	13.4693	0.519202	0.259601	+	0.965716i	$0.416409\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−15.1433	−0.582006	−0.291003	−	0.956722i	$-0.593989\pi$
$677$	−15.1433	−0.582006	−0.291003	+	0.956722i	$0.593989\pi$
$678$	0	0
$679$	−16.2498	−0.623609
$680$	0	0
$681$	51.3250	1.96678
$682$	0	0
$683$	15.8789	0.607589	0.303795	−	0.952738i	$-0.401746\pi$
$683$	15.8789	0.607589	0.303795	+	0.952738i	$0.401746\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	27.7502	1.05874
$688$	0	0
$689$	13.7952	0.525555
$690$	0	0
$691$	−27.3094	−1.03890	−0.519449	−	0.854501i	$-0.673863\pi$
$691$	−27.3094	−1.03890	−0.519449	+	0.854501i	$0.673863\pi$
$692$	0	0
$693$	16.4995	0.626766
$694$	0	0
$695$	0	0
$696$	0	0
$697$	4.57947	0.173460
$698$	0	0
$699$	−13.6821	−0.517505
$700$	0	0
$701$	−20.9697	−0.792016	−0.396008	−	0.918247i	$-0.629605\pi$
$701$	−20.9697	−0.792016	−0.396008	+	0.918247i	$0.629605\pi$
$702$	0	0
$703$	10.2498	0.386577
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−56.9192	−2.14067
$708$	0	0
$709$	37.5592	1.41056	0.705282	−	0.708927i	$-0.250820\pi$
$709$	37.5592	1.41056	0.705282	+	0.708927i	$0.250820\pi$
$710$	0	0
$711$	−25.3775	−0.951731
$712$	0	0
$713$	−11.3581	−0.425366
$714$	0	0
$715$	0	0
$716$	0	0
$717$	48.6050	1.81519
$718$	0	0
$719$	−3.94323	−0.147058	−0.0735288	−	0.997293i	$-0.523426\pi$
$719$	−3.94323	−0.147058	−0.0735288	+	0.997293i	$0.523426\pi$
$720$	0	0
$721$	18.8898	0.703493
$722$	0	0
$723$	−10.5601	−0.392734
$724$	0	0
$725$	0	0
$726$	0	0
$727$	33.2139	1.23183	0.615917	−	0.787811i	$-0.288786\pi$
$727$	33.2139	1.23183	0.615917	+	0.787811i	$0.288786\pi$
$728$	0	0
$729$	3.06811	0.113634
$730$	0	0
$731$	4.45080	0.164619
$732$	0	0
$733$	8.62065	0.318411	0.159205	−	0.987245i	$-0.449107\pi$
$733$	8.62065	0.318411	0.159205	+	0.987245i	$0.449107\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−19.5492	−0.720104
$738$	0	0
$739$	45.1689	1.66157	0.830783	−	0.556597i	$-0.187893\pi$
$739$	45.1689	1.66157	0.830783	+	0.556597i	$0.187893\pi$
$740$	0	0
$741$	−5.34438	−0.196331
$742$	0	0
$743$	−24.7905	−0.909475	−0.454737	−	0.890626i	$-0.650267\pi$
$743$	−24.7905	−0.909475	−0.454737	+	0.890626i	$0.650267\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	4.96972	0.181833
$748$	0	0
$749$	−42.7943	−1.56367
$750$	0	0
$751$	6.76869	0.246993	0.123497	−	0.992345i	$-0.460589\pi$
$751$	6.76869	0.246993	0.123497	+	0.992345i	$0.460589\pi$
$752$	0	0
$753$	33.8283	1.23277
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−45.2101	−1.64319	−0.821595	−	0.570072i	$-0.806915\pi$
$757$	−45.2101	−1.64319	−0.821595	+	0.570072i	$0.806915\pi$
$758$	0	0
$759$	−17.3600	−0.630127
$760$	0	0
$761$	19.8851	0.720834	0.360417	−	0.932791i	$-0.382634\pi$
$761$	19.8851	0.720834	0.360417	+	0.932791i	$0.382634\pi$
$762$	0	0
$763$	12.4352	0.450185
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7.93567	0.286540
$768$	0	0
$769$	8.07615	0.291233	0.145617	−	0.989341i	$-0.453483\pi$
$769$	8.07615	0.291233	0.145617	+	0.989341i	$0.453483\pi$
$770$	0	0
$771$	−2.23039	−0.0803257
$772$	0	0
$773$	−45.9456	−1.65255	−0.826275	−	0.563267i	$-0.809544\pi$
$773$	−45.9456	−1.65255	−0.826275	+	0.563267i	$0.809544\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	89.8383	3.22293
$778$	0	0
$779$	−8.88979	−0.318510
$780$	0	0
$781$	−29.4087	−1.05233
$782$	0	0
$783$	−24.5951	−0.878956
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−17.8439	−0.636067	−0.318034	−	0.948079i	$-0.603022\pi$
$787$	−17.8439	−0.636067	−0.318034	+	0.948079i	$0.603022\pi$
$788$	0	0
$789$	25.3700	0.903194
$790$	0	0
$791$	79.1983	2.81597
$792$	0	0
$793$	31.7914	1.12895
$794$	0	0
$795$	0	0
$796$	0	0
$797$	37.3326	1.32239	0.661194	−	0.750215i	$-0.270050\pi$
$797$	37.3326	1.32239	0.661194	+	0.750215i	$0.270050\pi$
$798$	0	0
$799$	−2.56009	−0.0905696
$800$	0	0
$801$	−11.5298	−0.407386
$802$	0	0
$803$	7.13957	0.251950
$804$	0	0
$805$	0	0
$806$	0	0
$807$	33.8695	1.19226
$808$	0	0
$809$	−38.6950	−1.36044	−0.680221	−	0.733007i	$-0.738116\pi$
$809$	−38.6950	−1.36044	−0.680221	+	0.733007i	$0.738116\pi$
$810$	0	0
$811$	−13.3444	−0.468585	−0.234292	−	0.972166i	$-0.575277\pi$
$811$	−13.3444	−0.468585	−0.234292	+	0.972166i	$0.575277\pi$
$812$	0	0
$813$	3.11399	0.109212
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−8.64002	−0.302276
$818$	0	0
$819$	−15.7190	−0.549268
$820$	0	0
$821$	−37.3482	−1.30346	−0.651730	−	0.758451i	$-0.725956\pi$
$821$	−37.3482	−1.30346	−0.651730	+	0.758451i	$0.725956\pi$
$822$	0	0
$823$	−51.5630	−1.79737	−0.898686	−	0.438593i	$-0.855477\pi$
$823$	−51.5630	−1.79737	−0.898686	+	0.438593i	$0.855477\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−48.5639	−1.68873	−0.844366	−	0.535767i	$-0.820022\pi$
$827$	−48.5639	−1.68873	−0.844366	+	0.535767i	$0.820022\pi$
$828$	0	0
$829$	0.325919	0.0113196	0.00565982	−	0.999984i	$-0.498198\pi$
$829$	0.325919	0.0113196	0.00565982	+	0.999984i	$0.498198\pi$
$830$	0	0
$831$	69.0578	2.39559
$832$	0	0
$833$	5.15894	0.178747
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−11.5804	−0.400277
$838$	0	0
$839$	17.1202	0.591055	0.295527	−	0.955334i	$-0.404505\pi$
$839$	17.1202	0.591055	0.295527	+	0.955334i	$0.404505\pi$
$840$	0	0
$841$	31.7649	1.09534
$842$	0	0
$843$	−2.23039	−0.0768188
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−16.6244	−0.571222
$848$	0	0
$849$	19.8477	0.681171
$850$	0	0
$851$	−31.7190	−1.08731
$852$	0	0
$853$	24.9092	0.852874	0.426437	−	0.904517i	$-0.359769\pi$
$853$	24.9092	0.852874	0.426437	+	0.904517i	$0.359769\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	11.6509	0.397988	0.198994	−	0.980001i	$-0.436233\pi$
$857$	11.6509	0.397988	0.198994	+	0.980001i	$0.436233\pi$
$858$	0	0
$859$	5.35998	0.182880	0.0914400	−	0.995811i	$-0.470853\pi$
$859$	5.35998	0.182880	0.0914400	+	0.995811i	$0.470853\pi$
$860$	0	0
$861$	−77.9182	−2.65545
$862$	0	0
$863$	−41.0790	−1.39835	−0.699173	−	0.714953i	$-0.746448\pi$
$863$	−41.0790	−1.39835	−0.699173	+	0.714953i	$0.746448\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	35.5592	1.20765
$868$	0	0
$869$	−44.2186	−1.50001
$870$	0	0
$871$	18.6244	0.631065
$872$	0	0
$873$	−5.96881	−0.202014
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−45.9532	−1.55173	−0.775865	−	0.630899i	$-0.782686\pi$
$877$	−45.9532	−1.55173	−0.775865	+	0.630899i	$0.782686\pi$
$878$	0	0
$879$	54.3435	1.83296
$880$	0	0
$881$	−6.90917	−0.232776	−0.116388	−	0.993204i	$-0.537132\pi$
$881$	−6.90917	−0.232776	−0.116388	+	0.993204i	$0.537132\pi$
$882$	0	0
$883$	48.5213	1.63287	0.816437	−	0.577435i	$-0.195946\pi$
$883$	48.5213	1.63287	0.816437	+	0.577435i	$0.195946\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	23.7990	0.799091	0.399546	−	0.916713i	$-0.369168\pi$
$887$	23.7990	0.799091	0.399546	+	0.916713i	$0.369168\pi$
$888$	0	0
$889$	59.0284	1.97975
$890$	0	0
$891$	−29.6997	−0.994976
$892$	0	0
$893$	4.96972	0.166305
$894$	0	0
$895$	0	0
$896$	0	0
$897$	16.5388	0.552213
$898$	0	0
$899$	28.6107	0.954219
$900$	0	0
$901$	−2.82546	−0.0941298
$902$	0	0
$903$	−75.7290	−2.52010
$904$	0	0
$905$	0	0
$906$	0	0
$907$	17.9726	0.596770	0.298385	−	0.954446i	$-0.403552\pi$
$907$	17.9726	0.596770	0.298385	+	0.954446i	$0.403552\pi$
$908$	0	0
$909$	−20.9073	−0.693453
$910$	0	0
$911$	−19.5592	−0.648024	−0.324012	−	0.946053i	$-0.605032\pi$
$911$	−19.5592	−0.648024	−0.324012	+	0.946053i	$0.605032\pi$
$912$	0	0
$913$	8.65940	0.286584
$914$	0	0
$915$	0	0
$916$	0	0
$917$	27.0596	0.893588
$918$	0	0
$919$	−35.4948	−1.17087	−0.585433	−	0.810720i	$-0.699076\pi$
$919$	−35.4948	−1.17087	−0.585433	+	0.810720i	$0.699076\pi$
$920$	0	0
$921$	−20.0587	−0.660957
$922$	0	0
$923$	28.0175	0.922209
$924$	0	0
$925$	0	0
$926$	0	0
$927$	6.93853	0.227891
$928$	0	0
$929$	17.9532	0.589026	0.294513	−	0.955648i	$-0.404843\pi$
$929$	17.9532	0.589026	0.294513	+	0.955648i	$0.404843\pi$
$930$	0	0
$931$	−10.0147	−0.328218
$932$	0	0
$933$	−37.1122	−1.21500
$934$	0	0
$935$	0	0
$936$	0	0
$937$	5.66652	0.185117	0.0925585	−	0.995707i	$-0.470495\pi$
$937$	5.66652	0.185117	0.0925585	+	0.995707i	$0.470495\pi$
$938$	0	0
$939$	−22.2148	−0.724953
$940$	0	0
$941$	−24.7044	−0.805339	−0.402670	−	0.915345i	$-0.631918\pi$
$941$	−24.7044	−0.805339	−0.402670	+	0.915345i	$0.631918\pi$
$942$	0	0
$943$	27.5104	0.895863
$944$	0	0
$945$	0	0
$946$	0	0
$947$	13.5904	0.441628	0.220814	−	0.975316i	$-0.429129\pi$
$947$	13.5904	0.441628	0.220814	+	0.975316i	$0.429129\pi$
$948$	0	0
$949$	−6.80183	−0.220797
$950$	0	0
$951$	9.20815	0.298595
$952$	0	0
$953$	−24.7375	−0.801326	−0.400663	−	0.916225i	$-0.631220\pi$
$953$	−24.7375	−0.801326	−0.400663	+	0.916225i	$0.631220\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	43.7290	1.41356
$958$	0	0
$959$	26.6244	0.859748
$960$	0	0
$961$	−17.5289	−0.565448
$962$	0	0
$963$	−15.7190	−0.506539
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−35.6897	−1.14770	−0.573851	−	0.818960i	$-0.694551\pi$
$967$	−35.6897	−1.14770	−0.573851	+	0.818960i	$0.694551\pi$
$968$	0	0
$969$	1.09461	0.0351639
$970$	0	0
$971$	−16.4995	−0.529495	−0.264748	−	0.964318i	$-0.585289\pi$
$971$	−16.4995	−0.529495	−0.264748	+	0.964318i	$0.585289\pi$
$972$	0	0
$973$	95.1184	3.04935
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−49.8501	−1.59485	−0.797423	−	0.603420i	$-0.793804\pi$
$977$	−49.8501	−1.59485	−0.797423	+	0.603420i	$0.793804\pi$
$978$	0	0
$979$	−20.0899	−0.642076
$980$	0	0
$981$	4.56766	0.145834
$982$	0	0
$983$	−5.19014	−0.165540	−0.0827698	−	0.996569i	$-0.526377\pi$
$983$	−5.19014	−0.165540	−0.0827698	+	0.996569i	$0.526377\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	43.5592	1.38650
$988$	0	0
$989$	26.7375	0.850203
$990$	0	0
$991$	32.4272	1.03008	0.515042	−	0.857165i	$-0.327776\pi$
$991$	32.4272	1.03008	0.515042	+	0.857165i	$0.327776\pi$
$992$	0	0
$993$	−9.69771	−0.307748
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−10.1992	−0.323012	−0.161506	−	0.986872i	$-0.551635\pi$
$997$	−10.1992	−0.323012	−0.161506	+	0.986872i	$0.551635\pi$
$998$	0	0
$999$	−32.3397	−1.02318

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.cd.1.1		3
4.3	odd	2		950.2.a.i.1.3		3
5.2	odd	4		1520.2.d.j.609.5		6
5.3	odd	4		1520.2.d.j.609.2		6
5.4	even	2		7600.2.a.bi.1.3		3
12.11	even	2		8550.2.a.cl.1.1		3
20.3	even	4		190.2.b.b.39.6	yes	6
20.7	even	4		190.2.b.b.39.1	✓	6
20.19	odd	2		950.2.a.n.1.1		3
60.23	odd	4		1710.2.d.d.1369.1		6
60.47	odd	4		1710.2.d.d.1369.4		6
60.59	even	2		8550.2.a.ck.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
190.2.b.b.39.1	✓	6	20.7	even	4
190.2.b.b.39.6	yes	6	20.3	even	4
950.2.a.i.1.3		3	4.3	odd	2
950.2.a.n.1.1		3	20.19	odd	2
1520.2.d.j.609.2		6	5.3	odd	4
1520.2.d.j.609.5		6	5.2	odd	4
1710.2.d.d.1369.1		6	60.23	odd	4
1710.2.d.d.1369.4		6	60.47	odd	4
7600.2.a.bi.1.3		3	5.4	even	2
7600.2.a.cd.1.1		3	1.1	even	1	trivial
8550.2.a.ck.1.3		3	60.59	even	2
8550.2.a.cl.1.1		3	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7600.a (trivial)

\(f(q)\)	\(=\)	\(q-2.12489 q^{3} +4.12489 q^{7} +1.51514 q^{9} +O(q^{10})\)	\(q-2.12489 q^{3} +4.12489 q^{7} +1.51514 q^{9} +2.64002 q^{11} -2.51514 q^{13} +0.515138 q^{17} -1.00000 q^{19} -8.76491 q^{21} +3.09461 q^{23} +3.15516 q^{27} -7.79518 q^{29} -3.67030 q^{31} -5.60975 q^{33} -10.2498 q^{37} +5.34438 q^{39} +8.88979 q^{41} +8.64002 q^{43} -4.96972 q^{47} +10.0147 q^{49} -1.09461 q^{51} -5.48486 q^{53} +2.12489 q^{57} -3.15516 q^{59} -12.6400 q^{61} +6.24977 q^{63} -7.40493 q^{67} -6.57569 q^{69} -11.1396 q^{71} +2.70436 q^{73} +10.8898 q^{77} -16.7493 q^{79} -11.2498 q^{81} +3.28005 q^{83} +16.5639 q^{87} -7.60975 q^{89} -10.3747 q^{91} +7.79897 q^{93} -3.93945 q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} + 4 q^{7} + 5 q^{9}+O(q^{10}) \) 3 * q + 2 * q^3 + 4 * q^7 + 5 * q^9	\( 3 q + 2 q^{3} + 4 q^{7} + 5 q^{9} - 8 q^{13} + 2 q^{17} - 3 q^{19} - 10 q^{21} + 2 q^{27} - 8 q^{29} - 4 q^{31} - 8 q^{33} - 14 q^{37} - 10 q^{39} + 2 q^{41} + 18 q^{43} - 14 q^{47} - 3 q^{49} + 6 q^{51} - 16 q^{53} - 2 q^{57} - 2 q^{59} - 30 q^{61} + 2 q^{63} + 2 q^{67} - 22 q^{69} + 8 q^{71} - 10 q^{73} + 8 q^{77} - 17 q^{81} - 6 q^{83} + 6 q^{87} - 14 q^{89} - 6 q^{91} - 4 q^{93} - 10 q^{97} + 12 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 + 4 * q^7 + 5 * q^9 - 8 * q^13 + 2 * q^17 - 3 * q^19 - 10 * q^21 + 2 * q^27 - 8 * q^29 - 4 * q^31 - 8 * q^33 - 14 * q^37 - 10 * q^39 + 2 * q^41 + 18 * q^43 - 14 * q^47 - 3 * q^49 + 6 * q^51 - 16 * q^53 - 2 * q^57 - 2 * q^59 - 30 * q^61 + 2 * q^63 + 2 * q^67 - 22 * q^69 + 8 * q^71 - 10 * q^73 + 8 * q^77 - 17 * q^81 - 6 * q^83 + 6 * q^87 - 14 * q^89 - 6 * q^91 - 4 * q^93 - 10 * q^97 + 12 * q^99

Introduction

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