LMFDB - Embedded newform 7600.2.a.bq.1.3

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:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 760)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.86081$ 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)

$f(q)$	$=$	$q+2.32340 q^{3} -1.39821 q^{7} +2.39821 q^{9} +O(q^{10})$	$q+2.32340 q^{3} -1.39821 q^{7} +2.39821 q^{9} -4.32340 q^{13} -0.601793 q^{17} -1.00000 q^{19} -3.24860 q^{21} +6.04502 q^{23} -1.39821 q^{27} +4.60179 q^{29} -2.79641 q^{31} -1.07480 q^{37} -10.0450 q^{39} -5.44322 q^{41} -8.64681 q^{43} -1.85039 q^{47} -5.04502 q^{49} -1.39821 q^{51} +3.11982 q^{53} -2.32340 q^{57} +6.69182 q^{59} -2.64681 q^{61} -3.35319 q^{63} -14.4134 q^{67} +14.0450 q^{69} +5.59283 q^{71} -12.6918 q^{73} +4.64681 q^{79} -10.4432 q^{81} -1.20359 q^{83} +10.6918 q^{87} +9.44322 q^{89} +6.04502 q^{91} -6.49720 q^{93} -4.51803 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} - q^{7} + 4 q^{9}+O(q^{10}) $ 3 * q - q^3 - q^7 + 4 * q^9	$ 3 q - q^{3} - q^{7} + 4 q^{9} - 5 q^{13} - 5 q^{17} - 3 q^{19} + 3 q^{21} - q^{23} - q^{27} + 17 q^{29} - 2 q^{31} - 8 q^{37} - 11 q^{39} + 6 q^{41} - 10 q^{43} + 4 q^{47} + 4 q^{49} - q^{51} - 5 q^{53} + q^{57} - 15 q^{59} + 8 q^{61} - 26 q^{63} + 3 q^{67} + 23 q^{69} + 4 q^{71} - 3 q^{73} - 2 q^{79} - 9 q^{81} - 10 q^{83} - 3 q^{87} + 6 q^{89} - q^{91} + 6 q^{93} + 4 q^{97}+O(q^{100}) $ 3 * q - q^3 - q^7 + 4 * q^9 - 5 * q^13 - 5 * q^17 - 3 * q^19 + 3 * q^21 - q^23 - q^27 + 17 * q^29 - 2 * q^31 - 8 * q^37 - 11 * q^39 + 6 * q^41 - 10 * q^43 + 4 * q^47 + 4 * q^49 - q^51 - 5 * q^53 + q^57 - 15 * q^59 + 8 * q^61 - 26 * q^63 + 3 * q^67 + 23 * q^69 + 4 * q^71 - 3 * q^73 - 2 * q^79 - 9 * q^81 - 10 * q^83 - 3 * q^87 + 6 * q^89 - q^91 + 6 * q^93 + 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
$3$	2.32340	1.34142	0.670709	−	0.741721i	$-0.265990\pi$
$3$	2.32340	1.34142	0.670709	+	0.741721i	$0.265990\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.39821	−0.528473	−0.264236	−	0.964458i	$-0.585120\pi$
$7$	−1.39821	−0.528473	−0.264236	+	0.964458i	$0.585120\pi$
$8$	0	0
$9$	2.39821	0.799402
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−4.32340	−1.19910	−0.599548	−	0.800339i	$-0.704653\pi$
$13$	−4.32340	−1.19910	−0.599548	+	0.800339i	$0.704653\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.601793	−0.145956	−0.0729781	−	0.997334i	$-0.523250\pi$
$17$	−0.601793	−0.145956	−0.0729781	+	0.997334i	$0.523250\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−3.24860	−0.708903
$22$	0	0
$23$	6.04502	1.26047	0.630236	−	0.776403i	$-0.282958\pi$
$23$	6.04502	1.26047	0.630236	+	0.776403i	$0.282958\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.39821	−0.269085
$28$	0	0
$29$	4.60179	0.854531	0.427266	−	0.904126i	$-0.359477\pi$
$29$	4.60179	0.854531	0.427266	+	0.904126i	$0.359477\pi$
$30$	0	0
$31$	−2.79641	−0.502251	−0.251125	−	0.967955i	$-0.580801\pi$
$31$	−2.79641	−0.502251	−0.251125	+	0.967955i	$0.580801\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.07480	−0.176697	−0.0883483	−	0.996090i	$-0.528159\pi$
$37$	−1.07480	−0.176697	−0.0883483	+	0.996090i	$0.528159\pi$
$38$	0	0
$39$	−10.0450	−1.60849
$40$	0	0
$41$	−5.44322	−0.850089	−0.425044	−	0.905173i	$-0.639741\pi$
$41$	−5.44322	−0.850089	−0.425044	+	0.905173i	$0.639741\pi$
$42$	0	0
$43$	−8.64681	−1.31863	−0.659313	−	0.751869i	$-0.729153\pi$
$43$	−8.64681	−1.31863	−0.659313	+	0.751869i	$0.729153\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.85039	−0.269908	−0.134954	−	0.990852i	$-0.543089\pi$
$47$	−1.85039	−0.269908	−0.134954	+	0.990852i	$0.543089\pi$
$48$	0	0
$49$	−5.04502	−0.720717
$50$	0	0
$51$	−1.39821	−0.195788
$52$	0	0
$53$	3.11982	0.428540	0.214270	−	0.976774i	$-0.431263\pi$
$53$	3.11982	0.428540	0.214270	+	0.976774i	$0.431263\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.32340	−0.307742
$58$	0	0
$59$	6.69182	0.871201	0.435601	−	0.900140i	$-0.356536\pi$
$59$	6.69182	0.871201	0.435601	+	0.900140i	$0.356536\pi$
$60$	0	0
$61$	−2.64681	−0.338889	−0.169445	−	0.985540i	$-0.554197\pi$
$61$	−2.64681	−0.338889	−0.169445	+	0.985540i	$0.554197\pi$
$62$	0	0
$63$	−3.35319	−0.422462
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−14.4134	−1.76088	−0.880441	−	0.474156i	$-0.842753\pi$
$67$	−14.4134	−1.76088	−0.880441	+	0.474156i	$0.842753\pi$
$68$	0	0
$69$	14.0450	1.69082
$70$	0	0
$71$	5.59283	0.663747	0.331873	−	0.943324i	$-0.392319\pi$
$71$	5.59283	0.663747	0.331873	+	0.943324i	$0.392319\pi$
$72$	0	0
$73$	−12.6918	−1.48547	−0.742733	−	0.669588i	$-0.766471\pi$
$73$	−12.6918	−1.48547	−0.742733	+	0.669588i	$0.766471\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	4.64681	0.522807	0.261403	−	0.965230i	$-0.415815\pi$
$79$	4.64681	0.522807	0.261403	+	0.965230i	$0.415815\pi$
$80$	0	0
$81$	−10.4432	−1.16036
$82$	0	0
$83$	−1.20359	−0.132111	−0.0660553	−	0.997816i	$-0.521041\pi$
$83$	−1.20359	−0.132111	−0.0660553	+	0.997816i	$0.521041\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	10.6918	1.14628
$88$	0	0
$89$	9.44322	1.00098	0.500490	−	0.865742i	$-0.333153\pi$
$89$	9.44322	1.00098	0.500490	+	0.865742i	$0.333153\pi$
$90$	0	0
$91$	6.04502	0.633690
$92$	0	0
$93$	−6.49720	−0.673728
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−4.51803	−0.458736	−0.229368	−	0.973340i	$-0.573666\pi$
$97$	−4.51803	−0.458736	−0.229368	+	0.973340i	$0.573666\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−5.05398	−0.502890	−0.251445	−	0.967872i	$-0.580906\pi$
$101$	−5.05398	−0.502890	−0.251445	+	0.967872i	$0.580906\pi$
$102$	0	0
$103$	−11.0748	−1.09123	−0.545616	−	0.838035i	$-0.683704\pi$
$103$	−11.0748	−1.09123	−0.545616	+	0.838035i	$0.683704\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.17380	−0.403496	−0.201748	−	0.979437i	$-0.564662\pi$
$107$	−4.17380	−0.403496	−0.201748	+	0.979437i	$0.564662\pi$
$108$	0	0
$109$	14.8414	1.42155	0.710776	−	0.703419i	$-0.248344\pi$
$109$	14.8414	1.42155	0.710776	+	0.703419i	$0.248344\pi$
$110$	0	0
$111$	−2.49720	−0.237024
$112$	0	0
$113$	−1.72161	−0.161956	−0.0809778	−	0.996716i	$-0.525804\pi$
$113$	−1.72161	−0.161956	−0.0809778	+	0.996716i	$0.525804\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−10.3684	−0.958561
$118$	0	0
$119$	0.841431	0.0771338
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	−12.6468	−1.14032
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−2.12878	−0.188899	−0.0944494	−	0.995530i	$-0.530109\pi$
$127$	−2.12878	−0.188899	−0.0944494	+	0.995530i	$0.530109\pi$
$128$	0	0
$129$	−20.0900	−1.76883
$130$	0	0
$131$	−1.59283	−0.139166	−0.0695831	−	0.997576i	$-0.522167\pi$
$131$	−1.59283	−0.139166	−0.0695831	+	0.997576i	$0.522167\pi$
$132$	0	0
$133$	1.39821	0.121240
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−9.89541	−0.845422	−0.422711	−	0.906265i	$-0.638921\pi$
$137$	−9.89541	−0.845422	−0.422711	+	0.906265i	$0.638921\pi$
$138$	0	0
$139$	−3.70079	−0.313897	−0.156948	−	0.987607i	$-0.550166\pi$
$139$	−3.70079	−0.313897	−0.156948	+	0.987607i	$0.550166\pi$
$140$	0	0
$141$	−4.29921	−0.362059
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−11.7216	−0.966782
$148$	0	0
$149$	18.3476	1.50309	0.751547	−	0.659680i	$-0.229308\pi$
$149$	18.3476	1.50309	0.751547	+	0.659680i	$0.229308\pi$
$150$	0	0
$151$	−13.9404	−1.13446	−0.567228	−	0.823561i	$-0.691984\pi$
$151$	−13.9404	−1.13446	−0.567228	+	0.823561i	$0.691984\pi$
$152$	0	0
$153$	−1.44322	−0.116678
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0.149606	0.0119399	0.00596994	−	0.999982i	$-0.498100\pi$
$157$	0.149606	0.0119399	0.00596994	+	0.999982i	$0.498100\pi$
$158$	0	0
$159$	7.24860	0.574851
$160$	0	0
$161$	−8.45219	−0.666126
$162$	0	0
$163$	−24.6468	−1.93049	−0.965244	−	0.261352i	$-0.915832\pi$
$163$	−24.6468	−1.93049	−0.965244	+	0.261352i	$0.915832\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−18.8656	−1.45987	−0.729933	−	0.683519i	$-0.760449\pi$
$167$	−18.8656	−1.45987	−0.729933	+	0.683519i	$0.760449\pi$
$168$	0	0
$169$	5.69182	0.437833
$170$	0	0
$171$	−2.39821	−0.183396
$172$	0	0
$173$	6.36842	0.484182	0.242091	−	0.970254i	$-0.422167\pi$
$173$	6.36842	0.484182	0.242091	+	0.970254i	$0.422167\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	15.5478	1.16865
$178$	0	0
$179$	−11.4432	−0.855307	−0.427653	−	0.903943i	$-0.640660\pi$
$179$	−11.4432	−0.855307	−0.427653	+	0.903943i	$0.640660\pi$
$180$	0	0
$181$	−3.29362	−0.244813	−0.122406	−	0.992480i	$-0.539061\pi$
$181$	−3.29362	−0.244813	−0.122406	+	0.992480i	$0.539061\pi$
$182$	0	0
$183$	−6.14961	−0.454592
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	1.95498	0.142204
$190$	0	0
$191$	−20.5422	−1.48638	−0.743191	−	0.669079i	$-0.766689\pi$
$191$	−20.5422	−1.48638	−0.743191	+	0.669079i	$0.766689\pi$
$192$	0	0
$193$	19.6620	1.41530	0.707652	−	0.706561i	$-0.249754\pi$
$193$	19.6620	1.41530	0.707652	+	0.706561i	$0.249754\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	9.74580	0.690862	0.345431	−	0.938444i	$-0.387733\pi$
$199$	9.74580	0.690862	0.345431	+	0.938444i	$0.387733\pi$
$200$	0	0
$201$	−33.4882	−2.36208
$202$	0	0
$203$	−6.43426	−0.451597
$204$	0	0
$205$	0	0
$206$	0	0
$207$	14.4972	1.00763
$208$	0	0
$209$	0	0
$210$	0	0
$211$	9.09899	0.626401	0.313200	−	0.949687i	$-0.398599\pi$
$211$	9.09899	0.626401	0.313200	+	0.949687i	$0.398599\pi$
$212$	0	0
$213$	12.9944	0.890362
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3.90997	0.265426
$218$	0	0
$219$	−29.4882	−1.99263
$220$	0	0
$221$	2.60179	0.175016
$222$	0	0
$223$	−13.8712	−0.928885	−0.464443	−	0.885603i	$-0.653745\pi$
$223$	−13.8712	−0.928885	−0.464443	+	0.885603i	$0.653745\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.3234	0.685188	0.342594	−	0.939483i	$-0.388694\pi$
$227$	10.3234	0.685188	0.342594	+	0.939483i	$0.388694\pi$
$228$	0	0
$229$	−14.7368	−0.973838	−0.486919	−	0.873447i	$-0.661879\pi$
$229$	−14.7368	−0.973838	−0.486919	+	0.873447i	$0.661879\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−13.1857	−0.863821	−0.431911	−	0.901916i	$-0.642160\pi$
$233$	−13.1857	−0.863821	−0.431911	+	0.901916i	$0.642160\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	10.7964	0.701303
$238$	0	0
$239$	0.452186	0.0292495	0.0146247	−	0.999893i	$-0.495345\pi$
$239$	0.452186	0.0292495	0.0146247	+	0.999893i	$0.495345\pi$
$240$	0	0
$241$	13.5333	0.871754	0.435877	−	0.900006i	$-0.356438\pi$
$241$	13.5333	0.871754	0.435877	+	0.900006i	$0.356438\pi$
$242$	0	0
$243$	−20.0692	−1.28744
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4.32340	0.275092
$248$	0	0
$249$	−2.79641	−0.177216
$250$	0	0
$251$	−21.2936	−1.34404	−0.672021	−	0.740532i	$-0.734573\pi$
$251$	−21.2936	−1.34404	−0.672021	+	0.740532i	$0.734573\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	28.3989	1.77147	0.885737	−	0.464188i	$-0.153654\pi$
$257$	28.3989	1.77147	0.885737	+	0.464188i	$0.153654\pi$
$258$	0	0
$259$	1.50280	0.0933793
$260$	0	0
$261$	11.0361	0.683115
$262$	0	0
$263$	11.1857	0.689737	0.344869	−	0.938651i	$-0.387923\pi$
$263$	11.1857	0.689737	0.344869	+	0.938651i	$0.387923\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	21.9404	1.34273
$268$	0	0
$269$	19.5928	1.19460	0.597298	−	0.802019i	$-0.296241\pi$
$269$	19.5928	1.19460	0.597298	+	0.802019i	$0.296241\pi$
$270$	0	0
$271$	−5.95498	−0.361740	−0.180870	−	0.983507i	$-0.557891\pi$
$271$	−5.95498	−0.361740	−0.180870	+	0.983507i	$0.557891\pi$
$272$	0	0
$273$	14.0450	0.850043
$274$	0	0
$275$	0	0
$276$	0	0
$277$	23.9404	1.43844	0.719220	−	0.694782i	$-0.244499\pi$
$277$	23.9404	1.43844	0.719220	+	0.694782i	$0.244499\pi$
$278$	0	0
$279$	−6.70638	−0.401501
$280$	0	0
$281$	10.7368	0.640506	0.320253	−	0.947332i	$-0.396232\pi$
$281$	10.7368	0.640506	0.320253	+	0.947332i	$0.396232\pi$
$282$	0	0
$283$	−16.0900	−0.956453	−0.478227	−	0.878237i	$-0.658720\pi$
$283$	−16.0900	−0.956453	−0.478227	+	0.878237i	$0.658720\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.61076	0.449249
$288$	0	0
$289$	−16.6378	−0.978697
$290$	0	0
$291$	−10.4972	−0.615357
$292$	0	0
$293$	−14.5630	−0.850782	−0.425391	−	0.905010i	$-0.639863\pi$
$293$	−14.5630	−0.850782	−0.425391	+	0.905010i	$0.639863\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−26.1350	−1.51143
$300$	0	0
$301$	12.0900	0.696858
$302$	0	0
$303$	−11.7424	−0.674585
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.77559	0.158411	0.0792057	−	0.996858i	$-0.474762\pi$
$307$	2.77559	0.158411	0.0792057	+	0.996858i	$0.474762\pi$
$308$	0	0
$309$	−25.7312	−1.46380
$310$	0	0
$311$	7.63785	0.433102	0.216551	−	0.976271i	$-0.430519\pi$
$311$	7.63785	0.433102	0.216551	+	0.976271i	$0.430519\pi$
$312$	0	0
$313$	32.5243	1.83838	0.919191	−	0.393812i	$-0.128844\pi$
$313$	32.5243	1.83838	0.919191	+	0.393812i	$0.128844\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10.8206	−0.607746	−0.303873	−	0.952713i	$-0.598280\pi$
$317$	−10.8206	−0.607746	−0.303873	+	0.952713i	$0.598280\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−9.69742	−0.541257
$322$	0	0
$323$	0.601793	0.0334846
$324$	0	0
$325$	0	0
$326$	0	0
$327$	34.4826	1.90689
$328$	0	0
$329$	2.58723	0.142639
$330$	0	0
$331$	−14.6918	−0.807536	−0.403768	−	0.914861i	$-0.632300\pi$
$331$	−14.6918	−0.807536	−0.403768	+	0.914861i	$0.632300\pi$
$332$	0	0
$333$	−2.57760	−0.141252
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.07480	0.0585483	0.0292741	−	0.999571i	$-0.490680\pi$
$337$	1.07480	0.0585483	0.0292741	+	0.999571i	$0.490680\pi$
$338$	0	0
$339$	−4.00000	−0.217250
$340$	0	0
$341$	0	0
$342$	0	0
$343$	16.8414	0.909352
$344$	0	0
$345$	0	0
$346$	0	0
$347$	32.0900	1.72268	0.861342	−	0.508026i	$-0.169625\pi$
$347$	32.0900	1.72268	0.861342	+	0.508026i	$0.169625\pi$
$348$	0	0
$349$	31.4737	1.68475	0.842374	−	0.538894i	$-0.181158\pi$
$349$	31.4737	1.68475	0.842374	+	0.538894i	$0.181158\pi$
$350$	0	0
$351$	6.04502	0.322659
$352$	0	0
$353$	11.4882	0.611457	0.305729	−	0.952119i	$-0.401100\pi$
$353$	11.4882	0.611457	0.305729	+	0.952119i	$0.401100\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.95498	0.103469
$358$	0	0
$359$	−26.0450	−1.37460	−0.687302	−	0.726372i	$-0.741205\pi$
$359$	−26.0450	−1.37460	−0.687302	+	0.726372i	$0.741205\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−25.5574	−1.34142
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−21.5512	−1.12496	−0.562481	−	0.826810i	$-0.690153\pi$
$367$	−21.5512	−1.12496	−0.562481	+	0.826810i	$0.690153\pi$
$368$	0	0
$369$	−13.0540	−0.679563
$370$	0	0
$371$	−4.36215	−0.226472
$372$	0	0
$373$	−24.9702	−1.29291	−0.646454	−	0.762953i	$-0.723749\pi$
$373$	−24.9702	−1.29291	−0.646454	+	0.762953i	$0.723749\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−19.8954	−1.02467
$378$	0	0
$379$	−28.6323	−1.47074	−0.735370	−	0.677666i	$-0.762992\pi$
$379$	−28.6323	−1.47074	−0.735370	+	0.677666i	$0.762992\pi$
$380$	0	0
$381$	−4.94602	−0.253392
$382$	0	0
$383$	17.9612	0.917777	0.458889	−	0.888494i	$-0.348248\pi$
$383$	17.9612	0.917777	0.458889	+	0.888494i	$0.348248\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−20.7368	−1.05411
$388$	0	0
$389$	−6.09003	−0.308777	−0.154388	−	0.988010i	$-0.549341\pi$
$389$	−6.09003	−0.308777	−0.154388	+	0.988010i	$0.549341\pi$
$390$	0	0
$391$	−3.63785	−0.183974
$392$	0	0
$393$	−3.70079	−0.186680
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−21.8325	−1.09574	−0.547870	−	0.836563i	$-0.684561\pi$
$397$	−21.8325	−1.09574	−0.547870	+	0.836563i	$0.684561\pi$
$398$	0	0
$399$	3.24860	0.162633
$400$	0	0
$401$	−12.7064	−0.634526	−0.317263	−	0.948338i	$-0.602764\pi$
$401$	−12.7064	−0.634526	−0.317263	+	0.948338i	$0.602764\pi$
$402$	0	0
$403$	12.0900	0.602247
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	16.8864	0.834981	0.417491	−	0.908681i	$-0.362910\pi$
$409$	16.8864	0.834981	0.417491	+	0.908681i	$0.362910\pi$
$410$	0	0
$411$	−22.9910	−1.13406
$412$	0	0
$413$	−9.35656	−0.460406
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−8.59843	−0.421067
$418$	0	0
$419$	−27.7908	−1.35767	−0.678835	−	0.734291i	$-0.737515\pi$
$419$	−27.7908	−1.35767	−0.678835	+	0.734291i	$0.737515\pi$
$420$	0	0
$421$	15.0090	0.731492	0.365746	−	0.930715i	$-0.380814\pi$
$421$	15.0090	0.731492	0.365746	+	0.930715i	$0.380814\pi$
$422$	0	0
$423$	−4.43763	−0.215765
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3.70079	0.179094
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0.736841	0.0354924	0.0177462	−	0.999843i	$-0.494351\pi$
$431$	0.736841	0.0354924	0.0177462	+	0.999843i	$0.494351\pi$
$432$	0	0
$433$	−31.4945	−1.51353	−0.756765	−	0.653687i	$-0.773221\pi$
$433$	−31.4945	−1.51353	−0.756765	+	0.653687i	$0.773221\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.04502	−0.289172
$438$	0	0
$439$	23.2340	1.10890	0.554450	−	0.832217i	$-0.312929\pi$
$439$	23.2340	1.10890	0.554450	+	0.832217i	$0.312929\pi$
$440$	0	0
$441$	−12.0990	−0.576143
$442$	0	0
$443$	16.4793	0.782954	0.391477	−	0.920188i	$-0.371964\pi$
$443$	16.4793	0.782954	0.391477	+	0.920188i	$0.371964\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	42.6289	2.01628
$448$	0	0
$449$	−2.29921	−0.108507	−0.0542533	−	0.998527i	$-0.517278\pi$
$449$	−2.29921	−0.108507	−0.0542533	+	0.998527i	$0.517278\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−32.3892	−1.52178
$454$	0	0
$455$	0	0
$456$	0	0
$457$	20.0450	0.937666	0.468833	−	0.883287i	$-0.344675\pi$
$457$	20.0450	0.937666	0.468833	+	0.883287i	$0.344675\pi$
$458$	0	0
$459$	0.841431	0.0392746
$460$	0	0
$461$	32.5872	1.51774	0.758869	−	0.651243i	$-0.225752\pi$
$461$	32.5872	1.51774	0.758869	+	0.651243i	$0.225752\pi$
$462$	0	0
$463$	2.58723	0.120239	0.0601195	−	0.998191i	$-0.480852\pi$
$463$	2.58723	0.120239	0.0601195	+	0.998191i	$0.480852\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−0.299213	−0.0138459	−0.00692295	−	0.999976i	$-0.502204\pi$
$467$	−0.299213	−0.0138459	−0.00692295	+	0.999976i	$0.502204\pi$
$468$	0	0
$469$	20.1530	0.930578
$470$	0	0
$471$	0.347596	0.0160164
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	7.48197	0.342576
$478$	0	0
$479$	35.0665	1.60223	0.801115	−	0.598511i	$-0.204241\pi$
$479$	35.0665	1.60223	0.801115	+	0.598511i	$0.204241\pi$
$480$	0	0
$481$	4.64681	0.211876
$482$	0	0
$483$	−19.6378	−0.893553
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−41.7521	−1.89197	−0.945983	−	0.324215i	$-0.894900\pi$
$487$	−41.7521	−1.89197	−0.945983	+	0.324215i	$0.894900\pi$
$488$	0	0
$489$	−57.2645	−2.58959
$490$	0	0
$491$	30.9765	1.39795	0.698974	−	0.715147i	$-0.253640\pi$
$491$	30.9765	1.39795	0.698974	+	0.715147i	$0.253640\pi$
$492$	0	0
$493$	−2.76932	−0.124724
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−7.81994	−0.350772
$498$	0	0
$499$	3.90997	0.175034	0.0875171	−	0.996163i	$-0.472107\pi$
$499$	3.90997	0.175034	0.0875171	+	0.996163i	$0.472107\pi$
$500$	0	0
$501$	−43.8325	−1.95829
$502$	0	0
$503$	16.4522	0.733567	0.366783	−	0.930306i	$-0.380459\pi$
$503$	16.4522	0.733567	0.366783	+	0.930306i	$0.380459\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	13.2244	0.587317
$508$	0	0
$509$	−25.4432	−1.12775	−0.563876	−	0.825860i	$-0.690690\pi$
$509$	−25.4432	−1.12775	−0.563876	+	0.825860i	$0.690690\pi$
$510$	0	0
$511$	17.7458	0.785028
$512$	0	0
$513$	1.39821	0.0617324
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	14.7964	0.649491
$520$	0	0
$521$	−27.3836	−1.19970	−0.599850	−	0.800113i	$-0.704773\pi$
$521$	−27.3836	−1.19970	−0.599850	+	0.800113i	$0.704773\pi$
$522$	0	0
$523$	41.5574	1.81718	0.908590	−	0.417689i	$-0.137160\pi$
$523$	41.5574	1.81718	0.908590	+	0.417689i	$0.137160\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.68286	0.0733066
$528$	0	0
$529$	13.5422	0.588792
$530$	0	0
$531$	16.0484	0.696441
$532$	0	0
$533$	23.5333	1.01934
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−26.5872	−1.14732
$538$	0	0
$539$	0	0
$540$	0	0
$541$	14.6468	0.629715	0.314858	−	0.949139i	$-0.398043\pi$
$541$	14.6468	0.629715	0.314858	+	0.949139i	$0.398043\pi$
$542$	0	0
$543$	−7.65240	−0.328396
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−24.9252	−1.06572	−0.532862	−	0.846202i	$-0.678884\pi$
$547$	−24.9252	−1.06572	−0.532862	+	0.846202i	$0.678884\pi$
$548$	0	0
$549$	−6.34760	−0.270909
$550$	0	0
$551$	−4.60179	−0.196043
$552$	0	0
$553$	−6.49720	−0.276289
$554$	0	0
$555$	0	0
$556$	0	0
$557$	44.9348	1.90395	0.951975	−	0.306176i	$-0.0990496\pi$
$557$	44.9348	1.90395	0.951975	+	0.306176i	$0.0990496\pi$
$558$	0	0
$559$	37.3836	1.58116
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−10.2605	−0.432427	−0.216213	−	0.976346i	$-0.569371\pi$
$563$	−10.2605	−0.432427	−0.216213	+	0.976346i	$0.569371\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	14.6018	0.613218
$568$	0	0
$569$	24.6773	1.03452	0.517262	−	0.855827i	$-0.326951\pi$
$569$	24.6773	1.03452	0.517262	+	0.855827i	$0.326951\pi$
$570$	0	0
$571$	10.9765	0.459351	0.229676	−	0.973267i	$-0.426233\pi$
$571$	10.9765	0.459351	0.229676	+	0.973267i	$0.426233\pi$
$572$	0	0
$573$	−47.7279	−1.99386
$574$	0	0
$575$	0	0
$576$	0	0
$577$	45.1295	1.87876	0.939382	−	0.342873i	$-0.111400\pi$
$577$	45.1295	1.87876	0.939382	+	0.342873i	$0.111400\pi$
$578$	0	0
$579$	45.6829	1.89851
$580$	0	0
$581$	1.68286	0.0698169
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.90997	0.326479	0.163240	−	0.986586i	$-0.447806\pi$
$587$	7.90997	0.326479	0.163240	+	0.986586i	$0.447806\pi$
$588$	0	0
$589$	2.79641	0.115224
$590$	0	0
$591$	4.64681	0.191144
$592$	0	0
$593$	7.29362	0.299513	0.149756	−	0.988723i	$-0.452151\pi$
$593$	7.29362	0.299513	0.149756	+	0.988723i	$0.452151\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	22.6434	0.926734
$598$	0	0
$599$	−20.6468	−0.843606	−0.421803	−	0.906688i	$-0.638603\pi$
$599$	−20.6468	−0.843606	−0.421803	+	0.906688i	$0.638603\pi$
$600$	0	0
$601$	45.1440	1.84146	0.920731	−	0.390197i	$-0.127593\pi$
$601$	45.1440	1.84146	0.920731	+	0.390197i	$0.127593\pi$
$602$	0	0
$603$	−34.5664	−1.40765
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−20.7577	−0.842528	−0.421264	−	0.906938i	$-0.638413\pi$
$607$	−20.7577	−0.842528	−0.421264	+	0.906938i	$0.638413\pi$
$608$	0	0
$609$	−14.9494	−0.605780
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	−30.3476	−1.22573	−0.612864	−	0.790188i	$-0.709983\pi$
$613$	−30.3476	−1.22573	−0.612864	+	0.790188i	$0.709983\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−47.2161	−1.90085	−0.950425	−	0.310955i	$-0.899351\pi$
$617$	−47.2161	−1.90085	−0.950425	+	0.310955i	$0.899351\pi$
$618$	0	0
$619$	22.7064	0.912647	0.456323	−	0.889814i	$-0.349166\pi$
$619$	22.7064	0.912647	0.456323	+	0.889814i	$0.349166\pi$
$620$	0	0
$621$	−8.45219	−0.339175
$622$	0	0
$623$	−13.2036	−0.528990
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0.646809	0.0257899
$630$	0	0
$631$	−32.4793	−1.29298	−0.646490	−	0.762923i	$-0.723764\pi$
$631$	−32.4793	−1.29298	−0.646490	+	0.762923i	$0.723764\pi$
$632$	0	0
$633$	21.1406	0.840265
$634$	0	0
$635$	0	0
$636$	0	0
$637$	21.8116	0.864209
$638$	0	0
$639$	13.4128	0.530601
$640$	0	0
$641$	10.9460	0.432342	0.216171	−	0.976356i	$-0.430643\pi$
$641$	10.9460	0.432342	0.216171	+	0.976356i	$0.430643\pi$
$642$	0	0
$643$	4.38924	0.173095	0.0865475	−	0.996248i	$-0.472417\pi$
$643$	4.38924	0.173095	0.0865475	+	0.996248i	$0.472417\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	16.4522	0.646802	0.323401	−	0.946262i	$-0.395174\pi$
$647$	16.4522	0.646802	0.323401	+	0.946262i	$0.395174\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	9.08444	0.356047
$652$	0	0
$653$	−1.53326	−0.0600009	−0.0300005	−	0.999550i	$-0.509551\pi$
$653$	−1.53326	−0.0600009	−0.0300005	+	0.999550i	$0.509551\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−30.4376	−1.18748
$658$	0	0
$659$	−22.1767	−0.863881	−0.431941	−	0.901902i	$-0.642171\pi$
$659$	−22.1767	−0.863881	−0.431941	+	0.901902i	$0.642171\pi$
$660$	0	0
$661$	−26.0755	−1.01422	−0.507109	−	0.861882i	$-0.669286\pi$
$661$	−26.0755	−1.01422	−0.507109	+	0.861882i	$0.669286\pi$
$662$	0	0
$663$	6.04502	0.234769
$664$	0	0
$665$	0	0
$666$	0	0
$667$	27.8179	1.07711
$668$	0	0
$669$	−32.2284	−1.24602
$670$	0	0
$671$	0	0
$672$	0	0
$673$	30.4168	1.17248	0.586241	−	0.810137i	$-0.300607\pi$
$673$	30.4168	1.17248	0.586241	+	0.810137i	$0.300607\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	44.2043	1.69891	0.849454	−	0.527663i	$-0.176932\pi$
$677$	44.2043	1.69891	0.849454	+	0.527663i	$0.176932\pi$
$678$	0	0
$679$	6.31714	0.242430
$680$	0	0
$681$	23.9854	0.919124
$682$	0	0
$683$	42.3989	1.62235	0.811174	−	0.584805i	$-0.198829\pi$
$683$	42.3989	1.62235	0.811174	+	0.584805i	$0.198829\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−34.2396	−1.30632
$688$	0	0
$689$	−13.4882	−0.513861
$690$	0	0
$691$	−7.79082	−0.296377	−0.148188	−	0.988959i	$-0.547344\pi$
$691$	−7.79082	−0.296377	−0.148188	+	0.988959i	$0.547344\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	3.27569	0.124076
$698$	0	0
$699$	−30.6356	−1.15875
$700$	0	0
$701$	−8.05957	−0.304406	−0.152203	−	0.988349i	$-0.548637\pi$
$701$	−8.05957	−0.304406	−0.152203	+	0.988349i	$0.548637\pi$
$702$	0	0
$703$	1.07480	0.0405370
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7.06651	0.265763
$708$	0	0
$709$	−35.2936	−1.32548	−0.662740	−	0.748850i	$-0.730606\pi$
$709$	−35.2936	−1.32548	−0.662740	+	0.748850i	$0.730606\pi$
$710$	0	0
$711$	11.1440	0.417933
$712$	0	0
$713$	−16.9044	−0.633074
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.05061	0.0392358
$718$	0	0
$719$	48.9315	1.82484	0.912418	−	0.409260i	$-0.134213\pi$
$719$	48.9315	1.82484	0.912418	+	0.409260i	$0.134213\pi$
$720$	0	0
$721$	15.4849	0.576687
$722$	0	0
$723$	31.4432	1.16939
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−3.29025	−0.122029	−0.0610143	−	0.998137i	$-0.519434\pi$
$727$	−3.29025	−0.122029	−0.0610143	+	0.998137i	$0.519434\pi$
$728$	0	0
$729$	−15.2992	−0.566638
$730$	0	0
$731$	5.20359	0.192462
$732$	0	0
$733$	−22.3892	−0.826966	−0.413483	−	0.910512i	$-0.635688\pi$
$733$	−22.3892	−0.826966	−0.413483	+	0.910512i	$0.635688\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−4.90437	−0.180410	−0.0902051	−	0.995923i	$-0.528752\pi$
$739$	−4.90437	−0.180410	−0.0902051	+	0.995923i	$0.528752\pi$
$740$	0	0
$741$	10.0450	0.369013
$742$	0	0
$743$	28.1592	1.03306	0.516531	−	0.856268i	$-0.327223\pi$
$743$	28.1592	1.03306	0.516531	+	0.856268i	$0.327223\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−2.88645	−0.105610
$748$	0	0
$749$	5.83584	0.213237
$750$	0	0
$751$	−42.9348	−1.56671	−0.783357	−	0.621572i	$-0.786494\pi$
$751$	−42.9348	−1.56671	−0.783357	+	0.621572i	$0.786494\pi$
$752$	0	0
$753$	−49.4737	−1.80292
$754$	0	0
$755$	0	0
$756$	0	0
$757$	25.0665	0.911058	0.455529	−	0.890221i	$-0.349450\pi$
$757$	25.0665	0.911058	0.455529	+	0.890221i	$0.349450\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−30.8898	−1.11975	−0.559877	−	0.828575i	$-0.689152\pi$
$761$	−30.8898	−1.11975	−0.559877	+	0.828575i	$0.689152\pi$
$762$	0	0
$763$	−20.7514	−0.751251
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−28.9315	−1.04465
$768$	0	0
$769$	7.78745	0.280823	0.140411	−	0.990093i	$-0.455157\pi$
$769$	7.78745	0.280823	0.140411	+	0.990093i	$0.455157\pi$
$770$	0	0
$771$	65.9821	2.37629
$772$	0	0
$773$	−20.7126	−0.744982	−0.372491	−	0.928036i	$-0.621496\pi$
$773$	−20.7126	−0.744982	−0.372491	+	0.928036i	$0.621496\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	3.49161	0.125261
$778$	0	0
$779$	5.44322	0.195024
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−6.43426	−0.229942
$784$	0	0
$785$	0	0
$786$	0	0
$787$	10.5035	0.374408	0.187204	−	0.982321i	$-0.440057\pi$
$787$	10.5035	0.374408	0.187204	+	0.982321i	$0.440057\pi$
$788$	0	0
$789$	25.9888	0.925226
$790$	0	0
$791$	2.40717	0.0855891
$792$	0	0
$793$	11.4432	0.406361
$794$	0	0
$795$	0	0
$796$	0	0
$797$	37.8871	1.34203	0.671015	−	0.741444i	$-0.265858\pi$
$797$	37.8871	1.34203	0.671015	+	0.741444i	$0.265858\pi$
$798$	0	0
$799$	1.11355	0.0393947
$800$	0	0
$801$	22.6468	0.800186
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	45.5221	1.60245
$808$	0	0
$809$	−33.9438	−1.19340	−0.596700	−	0.802464i	$-0.703522\pi$
$809$	−33.9438	−1.19340	−0.596700	+	0.802464i	$0.703522\pi$
$810$	0	0
$811$	−4.66137	−0.163683	−0.0818414	−	0.996645i	$-0.526080\pi$
$811$	−4.66137	−0.163683	−0.0818414	+	0.996645i	$0.526080\pi$
$812$	0	0
$813$	−13.8358	−0.485244
$814$	0	0
$815$	0	0
$816$	0	0
$817$	8.64681	0.302514
$818$	0	0
$819$	14.4972	0.506573
$820$	0	0
$821$	−26.5693	−0.927275	−0.463638	−	0.886025i	$-0.653456\pi$
$821$	−26.5693	−0.927275	−0.463638	+	0.886025i	$0.653456\pi$
$822$	0	0
$823$	24.7998	0.864466	0.432233	−	0.901762i	$-0.357726\pi$
$823$	24.7998	0.864466	0.432233	+	0.901762i	$0.357726\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.11422	−0.0735188	−0.0367594	−	0.999324i	$-0.511704\pi$
$827$	−2.11422	−0.0735188	−0.0367594	+	0.999324i	$0.511704\pi$
$828$	0	0
$829$	45.7279	1.58819	0.794097	−	0.607790i	$-0.207944\pi$
$829$	45.7279	1.58819	0.794097	+	0.607790i	$0.207944\pi$
$830$	0	0
$831$	55.6233	1.92955
$832$	0	0
$833$	3.03605	0.105193
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.90997	0.135148
$838$	0	0
$839$	−15.4432	−0.533159	−0.266580	−	0.963813i	$-0.585894\pi$
$839$	−15.4432	−0.533159	−0.266580	+	0.963813i	$0.585894\pi$
$840$	0	0
$841$	−7.82351	−0.269776
$842$	0	0
$843$	24.9460	0.859187
$844$	0	0
$845$	0	0
$846$	0	0
$847$	15.3803	0.528473
$848$	0	0
$849$	−37.3836	−1.28300
$850$	0	0
$851$	−6.49720	−0.222721
$852$	0	0
$853$	−38.1801	−1.30726	−0.653630	−	0.756814i	$-0.726755\pi$
$853$	−38.1801	−1.30726	−0.653630	+	0.756814i	$0.726755\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−52.2672	−1.78541	−0.892707	−	0.450638i	$-0.851196\pi$
$857$	−52.2672	−1.78541	−0.892707	+	0.450638i	$0.851196\pi$
$858$	0	0
$859$	39.8809	1.36072	0.680359	−	0.732879i	$-0.261824\pi$
$859$	39.8809	1.36072	0.680359	+	0.732879i	$0.261824\pi$
$860$	0	0
$861$	17.6829	0.602630
$862$	0	0
$863$	22.9557	0.781420	0.390710	−	0.920514i	$-0.372230\pi$
$863$	22.9557	0.781420	0.390710	+	0.920514i	$0.372230\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−38.6564	−1.31284
$868$	0	0
$869$	0	0
$870$	0	0
$871$	62.3151	2.11147
$872$	0	0
$873$	−10.8352	−0.366715
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.73057	0.0922050	0.0461025	−	0.998937i	$-0.485320\pi$
$877$	2.73057	0.0922050	0.0461025	+	0.998937i	$0.485320\pi$
$878$	0	0
$879$	−33.8358	−1.14125
$880$	0	0
$881$	9.14401	0.308070	0.154035	−	0.988065i	$-0.450773\pi$
$881$	9.14401	0.308070	0.154035	+	0.988065i	$0.450773\pi$
$882$	0	0
$883$	−11.0540	−0.371996	−0.185998	−	0.982550i	$-0.559552\pi$
$883$	−11.0540	−0.371996	−0.185998	+	0.982550i	$0.559552\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	37.1469	1.24727	0.623636	−	0.781715i	$-0.285655\pi$
$887$	37.1469	1.24727	0.623636	+	0.781715i	$0.285655\pi$
$888$	0	0
$889$	2.97648	0.0998279
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.85039	0.0619211
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−60.7223	−2.02746
$898$	0	0
$899$	−12.8685	−0.429189
$900$	0	0
$901$	−1.87748	−0.0625481
$902$	0	0
$903$	28.0900	0.934778
$904$	0	0
$905$	0	0
$906$	0	0
$907$	4.00627	0.133026	0.0665129	−	0.997786i	$-0.478813\pi$
$907$	4.00627	0.133026	0.0665129	+	0.997786i	$0.478813\pi$
$908$	0	0
$909$	−12.1205	−0.402011
$910$	0	0
$911$	−21.2161	−0.702921	−0.351461	−	0.936203i	$-0.614315\pi$
$911$	−21.2161	−0.702921	−0.351461	+	0.936203i	$0.614315\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.22711	0.0735455
$918$	0	0
$919$	23.5478	0.776771	0.388385	−	0.921497i	$-0.373033\pi$
$919$	23.5478	0.776771	0.388385	+	0.921497i	$0.373033\pi$
$920$	0	0
$921$	6.44882	0.212496
$922$	0	0
$923$	−24.1801	−0.795896
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−26.5597	−0.872334
$928$	0	0
$929$	−42.7223	−1.40167	−0.700836	−	0.713322i	$-0.747190\pi$
$929$	−42.7223	−1.40167	−0.700836	+	0.713322i	$0.747190\pi$
$930$	0	0
$931$	5.04502	0.165344
$932$	0	0
$933$	17.7458	0.580972
$934$	0	0
$935$	0	0
$936$	0	0
$937$	31.1890	1.01890	0.509451	−	0.860500i	$-0.329849\pi$
$937$	31.1890	1.01890	0.509451	+	0.860500i	$0.329849\pi$
$938$	0	0
$939$	75.5671	2.46604
$940$	0	0
$941$	20.2251	0.659319	0.329659	−	0.944100i	$-0.393066\pi$
$941$	20.2251	0.659319	0.329659	+	0.944100i	$0.393066\pi$
$942$	0	0
$943$	−32.9044	−1.07151
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−20.1801	−0.655764	−0.327882	−	0.944719i	$-0.606335\pi$
$947$	−20.1801	−0.655764	−0.327882	+	0.944719i	$0.606335\pi$
$948$	0	0
$949$	54.8719	1.78122
$950$	0	0
$951$	−25.1406	−0.815241
$952$	0	0
$953$	−20.6856	−0.670071	−0.335035	−	0.942206i	$-0.608748\pi$
$953$	−20.6856	−0.670071	−0.335035	+	0.942206i	$0.608748\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	13.8358	0.446782
$960$	0	0
$961$	−23.1801	−0.747744
$962$	0	0
$963$	−10.0096	−0.322556
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−53.9530	−1.73501	−0.867505	−	0.497428i	$-0.834278\pi$
$967$	−53.9530	−1.73501	−0.867505	+	0.497428i	$0.834278\pi$
$968$	0	0
$969$	1.39821	0.0449169
$970$	0	0
$971$	9.41277	0.302070	0.151035	−	0.988528i	$-0.451739\pi$
$971$	9.41277	0.302070	0.151035	+	0.988528i	$0.451739\pi$
$972$	0	0
$973$	5.17447	0.165886
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.21881	0.134972	0.0674859	−	0.997720i	$-0.478502\pi$
$977$	4.21881	0.134972	0.0674859	+	0.997720i	$0.478502\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	35.5928	1.13639
$982$	0	0
$983$	−10.3088	−0.328801	−0.164401	−	0.986394i	$-0.552569\pi$
$983$	−10.3088	−0.328801	−0.164401	+	0.986394i	$0.552569\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	6.01119	0.191338
$988$	0	0
$989$	−52.2701	−1.66209
$990$	0	0
$991$	−54.6356	−1.73556	−0.867779	−	0.496951i	$-0.834453\pi$
$991$	−54.6356	−1.73556	−0.867779	+	0.496951i	$0.834453\pi$
$992$	0	0
$993$	−34.1350	−1.08324
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−22.9169	−0.725786	−0.362893	−	0.931831i	$-0.618211\pi$
$997$	−22.9169	−0.725786	−0.362893	+	0.931831i	$0.618211\pi$
$998$	0	0
$999$	1.50280	0.0475464

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.bq.1.3		3
4.3	odd	2		3800.2.a.x.1.1		3
5.4	even	2		1520.2.a.s.1.1		3
20.3	even	4		3800.2.d.l.3649.2		6
20.7	even	4		3800.2.d.l.3649.5		6
20.19	odd	2		760.2.a.j.1.3	✓	3
40.19	odd	2		6080.2.a.bv.1.1		3
40.29	even	2		6080.2.a.bq.1.3		3
60.59	even	2		6840.2.a.bg.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.a.j.1.3	✓	3	20.19	odd	2
1520.2.a.s.1.1		3	5.4	even	2
3800.2.a.x.1.1		3	4.3	odd	2
3800.2.d.l.3649.2		6	20.3	even	4
3800.2.d.l.3649.5		6	20.7	even	4
6080.2.a.bq.1.3		3	40.29	even	2
6080.2.a.bv.1.1		3	40.19	odd	2
6840.2.a.bg.1.2		3	60.59	even	2
7600.2.a.bq.1.3		3	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:	\( N \)	\(=\)	\( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7600.a (trivial)

\(f(q)\)	\(=\)	\(q+2.32340 q^{3} -1.39821 q^{7} +2.39821 q^{9} +O(q^{10})\)	\(q+2.32340 q^{3} -1.39821 q^{7} +2.39821 q^{9} -4.32340 q^{13} -0.601793 q^{17} -1.00000 q^{19} -3.24860 q^{21} +6.04502 q^{23} -1.39821 q^{27} +4.60179 q^{29} -2.79641 q^{31} -1.07480 q^{37} -10.0450 q^{39} -5.44322 q^{41} -8.64681 q^{43} -1.85039 q^{47} -5.04502 q^{49} -1.39821 q^{51} +3.11982 q^{53} -2.32340 q^{57} +6.69182 q^{59} -2.64681 q^{61} -3.35319 q^{63} -14.4134 q^{67} +14.0450 q^{69} +5.59283 q^{71} -12.6918 q^{73} +4.64681 q^{79} -10.4432 q^{81} -1.20359 q^{83} +10.6918 q^{87} +9.44322 q^{89} +6.04502 q^{91} -6.49720 q^{93} -4.51803 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} - q^{7} + 4 q^{9}+O(q^{10}) \) 3 * q - q^3 - q^7 + 4 * q^9	\( 3 q - q^{3} - q^{7} + 4 q^{9} - 5 q^{13} - 5 q^{17} - 3 q^{19} + 3 q^{21} - q^{23} - q^{27} + 17 q^{29} - 2 q^{31} - 8 q^{37} - 11 q^{39} + 6 q^{41} - 10 q^{43} + 4 q^{47} + 4 q^{49} - q^{51} - 5 q^{53} + q^{57} - 15 q^{59} + 8 q^{61} - 26 q^{63} + 3 q^{67} + 23 q^{69} + 4 q^{71} - 3 q^{73} - 2 q^{79} - 9 q^{81} - 10 q^{83} - 3 q^{87} + 6 q^{89} - q^{91} + 6 q^{93} + 4 q^{97}+O(q^{100}) \) 3 * q - q^3 - q^7 + 4 * q^9 - 5 * q^13 - 5 * q^17 - 3 * q^19 + 3 * q^21 - q^23 - q^27 + 17 * q^29 - 2 * q^31 - 8 * q^37 - 11 * q^39 + 6 * q^41 - 10 * q^43 + 4 * q^47 + 4 * q^49 - q^51 - 5 * q^53 + q^57 - 15 * q^59 + 8 * q^61 - 26 * q^63 + 3 * q^67 + 23 * q^69 + 4 * q^71 - 3 * q^73 - 2 * q^79 - 9 * q^81 - 10 * q^83 - 3 * q^87 + 6 * q^89 - q^91 + 6 * q^93 + 4 * q^97

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