LMFDB - Embedded newform 7600.2.a.by.1.2

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:	$0$
Dimension:	$3$
Coefficient field:	3.3.257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 3 $ x^3 - x^2 - 4*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1900)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.713538$ 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)

$f(q)$	$=$	$q+0.286462 q^{3} +0.286462 q^{7} -2.91794 q^{9} +O(q^{10})$	$q+0.286462 q^{3} +0.286462 q^{7} -2.91794 q^{9} -4.26819 q^{11} -3.20440 q^{13} -0.286462 q^{17} +1.00000 q^{19} +0.0820605 q^{21} +0.936212 q^{23} -1.69527 q^{27} +2.26819 q^{29} +4.18613 q^{31} -1.22267 q^{33} -8.67699 q^{37} -0.917939 q^{39} +1.08206 q^{41} +4.42708 q^{43} -0.759053 q^{47} -6.91794 q^{49} -0.0820605 q^{51} -4.42708 q^{53} +0.286462 q^{57} +4.70050 q^{59} +10.1861 q^{61} -0.835879 q^{63} +9.82284 q^{67} +0.268189 q^{69} -4.83588 q^{71} -10.4726 q^{73} -1.22267 q^{77} -5.10407 q^{79} +8.26819 q^{81} +15.7355 q^{83} +0.649750 q^{87} -9.75382 q^{89} -0.917939 q^{91} +1.19917 q^{93} +9.61320 q^{97} +12.4543 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} + 2 q^{7} + q^{9}+O(q^{10}) $ 3 * q + 2 * q^3 + 2 * q^7 + q^9	$ 3 q + 2 q^{3} + 2 q^{7} + q^{9} + q^{11} - q^{13} - 2 q^{17} + 3 q^{19} + 10 q^{21} + 8 q^{23} + 11 q^{27} - 7 q^{29} - 11 q^{31} - 10 q^{33} + 5 q^{37} + 7 q^{39} + 13 q^{41} + 11 q^{43} + 19 q^{47} - 11 q^{49} - 10 q^{51} - 11 q^{53} + 2 q^{57} + 6 q^{59} + 7 q^{61} + 17 q^{63} + 3 q^{67} - 13 q^{69} + 5 q^{71} - 9 q^{73} - 10 q^{77} + 18 q^{79} + 11 q^{81} + 3 q^{83} + 6 q^{87} + 7 q^{91} - 13 q^{93} + 3 q^{97}+O(q^{100}) $ 3 * q + 2 * q^3 + 2 * q^7 + q^9 + q^11 - q^13 - 2 * q^17 + 3 * q^19 + 10 * q^21 + 8 * q^23 + 11 * q^27 - 7 * q^29 - 11 * q^31 - 10 * q^33 + 5 * q^37 + 7 * q^39 + 13 * q^41 + 11 * q^43 + 19 * q^47 - 11 * q^49 - 10 * q^51 - 11 * q^53 + 2 * q^57 + 6 * q^59 + 7 * q^61 + 17 * q^63 + 3 * q^67 - 13 * q^69 + 5 * q^71 - 9 * q^73 - 10 * q^77 + 18 * q^79 + 11 * q^81 + 3 * q^83 + 6 * q^87 + 7 * q^91 - 13 * q^93 + 3 * 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
$3$	0.286462	0.165389	0.0826945	−	0.996575i	$-0.473647\pi$
$3$	0.286462	0.165389	0.0826945	+	0.996575i	$0.473647\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.286462	0.108272	0.0541362	−	0.998534i	$-0.482759\pi$
$7$	0.286462	0.108272	0.0541362	+	0.998534i	$0.482759\pi$
$8$	0	0
$9$	−2.91794	−0.972646
$10$	0	0
$11$	−4.26819	−1.28691	−0.643454	−	0.765485i	$-0.722499\pi$
$11$	−4.26819	−1.28691	−0.643454	+	0.765485i	$0.722499\pi$
$12$	0	0
$13$	−3.20440	−0.888741	−0.444371	−	0.895843i	$-0.646573\pi$
$13$	−3.20440	−0.888741	−0.444371	+	0.895843i	$0.646573\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.286462	−0.0694773	−0.0347386	−	0.999396i	$-0.511060\pi$
$17$	−0.286462	−0.0694773	−0.0347386	+	0.999396i	$0.511060\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0.0820605	0.0179071
$22$	0	0
$23$	0.936212	0.195214	0.0976069	−	0.995225i	$-0.468881\pi$
$23$	0.936212	0.195214	0.0976069	+	0.995225i	$0.468881\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.69527	−0.326254
$28$	0	0
$29$	2.26819	0.421192	0.210596	−	0.977573i	$-0.432460\pi$
$29$	2.26819	0.421192	0.210596	+	0.977573i	$0.432460\pi$
$30$	0	0
$31$	4.18613	0.751851	0.375925	−	0.926650i	$-0.377325\pi$
$31$	4.18613	0.751851	0.375925	+	0.926650i	$0.377325\pi$
$32$	0	0
$33$	−1.22267	−0.212840
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.67699	−1.42649	−0.713244	−	0.700915i	$-0.752775\pi$
$37$	−8.67699	−1.42649	−0.713244	+	0.700915i	$0.752775\pi$
$38$	0	0
$39$	−0.917939	−0.146988
$40$	0	0
$41$	1.08206	0.168989	0.0844947	−	0.996424i	$-0.473072\pi$
$41$	1.08206	0.168989	0.0844947	+	0.996424i	$0.473072\pi$
$42$	0	0
$43$	4.42708	0.675123	0.337561	−	0.941304i	$-0.390398\pi$
$43$	4.42708	0.675123	0.337561	+	0.941304i	$0.390398\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.759053	−0.110719	−0.0553596	−	0.998466i	$-0.517631\pi$
$47$	−0.759053	−0.110719	−0.0553596	+	0.998466i	$0.517631\pi$
$48$	0	0
$49$	−6.91794	−0.988277
$50$	0	0
$51$	−0.0820605	−0.0114908
$52$	0	0
$53$	−4.42708	−0.608106	−0.304053	−	0.952655i	$-0.598340\pi$
$53$	−4.42708	−0.608106	−0.304053	+	0.952655i	$0.598340\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.286462	0.0379428
$58$	0	0
$59$	4.70050	0.611953	0.305976	−	0.952039i	$-0.401017\pi$
$59$	4.70050	0.611953	0.305976	+	0.952039i	$0.401017\pi$
$60$	0	0
$61$	10.1861	1.30420	0.652100	−	0.758133i	$-0.273888\pi$
$61$	10.1861	1.30420	0.652100	+	0.758133i	$0.273888\pi$
$62$	0	0
$63$	−0.835879	−0.105311
$64$	0	0
$65$	0	0
$66$	0	0
$67$	9.82284	1.20005	0.600025	−	0.799981i	$-0.295157\pi$
$67$	9.82284	1.20005	0.600025	+	0.799981i	$0.295157\pi$
$68$	0	0
$69$	0.268189	0.0322862
$70$	0	0
$71$	−4.83588	−0.573913	−0.286957	−	0.957944i	$-0.592644\pi$
$71$	−4.83588	−0.573913	−0.286957	+	0.957944i	$0.592644\pi$
$72$	0	0
$73$	−10.4726	−1.22572	−0.612862	−	0.790190i	$-0.709982\pi$
$73$	−10.4726	−1.22572	−0.612862	+	0.790190i	$0.709982\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.22267	−0.139337
$78$	0	0
$79$	−5.10407	−0.574253	−0.287126	−	0.957893i	$-0.592700\pi$
$79$	−5.10407	−0.574253	−0.287126	+	0.957893i	$0.592700\pi$
$80$	0	0
$81$	8.26819	0.918688
$82$	0	0
$83$	15.7355	1.72720	0.863600	−	0.504177i	$-0.168204\pi$
$83$	15.7355	1.72720	0.863600	+	0.504177i	$0.168204\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.649750	0.0696605
$88$	0	0
$89$	−9.75382	−1.03390	−0.516951	−	0.856015i	$-0.672933\pi$
$89$	−9.75382	−1.03390	−0.516951	+	0.856015i	$0.672933\pi$
$90$	0	0
$91$	−0.917939	−0.0962262
$92$	0	0
$93$	1.19917	0.124348
$94$	0	0
$95$	0	0
$96$	0	0
$97$	9.61320	0.976073	0.488037	−	0.872823i	$-0.337713\pi$
$97$	9.61320	0.976073	0.488037	+	0.872823i	$0.337713\pi$
$98$	0	0
$99$	12.4543	1.25171
$100$	0	0
$101$	0.731811	0.0728179	0.0364089	−	0.999337i	$-0.488408\pi$
$101$	0.731811	0.0728179	0.0364089	+	0.999337i	$0.488408\pi$
$102$	0	0
$103$	−3.10407	−0.305853	−0.152926	−	0.988238i	$-0.548870\pi$
$103$	−3.10407	−0.305853	−0.152926	+	0.988238i	$0.548870\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.1731	1.37016	0.685082	−	0.728466i	$-0.259766\pi$
$107$	14.1731	1.37016	0.685082	+	0.728466i	$0.259766\pi$
$108$	0	0
$109$	−9.61844	−0.921279	−0.460640	−	0.887587i	$-0.652380\pi$
$109$	−9.61844	−0.921279	−0.460640	+	0.887587i	$0.652380\pi$
$110$	0	0
$111$	−2.48563	−0.235925
$112$	0	0
$113$	−8.77733	−0.825701	−0.412851	−	0.910799i	$-0.635467\pi$
$113$	−8.77733	−0.825701	−0.412851	+	0.910799i	$0.635467\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	9.35025	0.864431
$118$	0	0
$119$	−0.0820605	−0.00752248
$120$	0	0
$121$	7.21744	0.656131
$122$	0	0
$123$	0.309969	0.0279490
$124$	0	0
$125$	0	0
$126$	0	0
$127$	6.77209	0.600926	0.300463	−	0.953793i	$-0.402859\pi$
$127$	6.77209	0.600926	0.300463	+	0.953793i	$0.402859\pi$
$128$	0	0
$129$	1.26819	0.111658
$130$	0	0
$131$	2.91794	0.254942	0.127471	−	0.991842i	$-0.459314\pi$
$131$	2.91794	0.254942	0.127471	+	0.991842i	$0.459314\pi$
$132$	0	0
$133$	0.286462	0.0248394
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.4360	1.23335	0.616677	−	0.787216i	$-0.288478\pi$
$137$	14.4360	1.23335	0.616677	+	0.787216i	$0.288478\pi$
$138$	0	0
$139$	2.29950	0.195041	0.0975205	−	0.995234i	$-0.468909\pi$
$139$	2.29950	0.195041	0.0975205	+	0.995234i	$0.468909\pi$
$140$	0	0
$141$	−0.217440	−0.0183117
$142$	0	0
$143$	13.6770	1.14373
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.98173	−0.163450
$148$	0	0
$149$	6.75382	0.553294	0.276647	−	0.960972i	$-0.410777\pi$
$149$	6.75382	0.553294	0.276647	+	0.960972i	$0.410777\pi$
$150$	0	0
$151$	11.8866	0.967320	0.483660	−	0.875256i	$-0.339307\pi$
$151$	11.8866	0.967320	0.483660	+	0.875256i	$0.339307\pi$
$152$	0	0
$153$	0.835879	0.0675768
$154$	0	0
$155$	0	0
$156$	0	0
$157$	14.1731	1.13114	0.565568	−	0.824702i	$-0.308657\pi$
$157$	14.1731	1.13114	0.565568	+	0.824702i	$0.308657\pi$
$158$	0	0
$159$	−1.26819	−0.100574
$160$	0	0
$161$	0.268189	0.0211363
$162$	0	0
$163$	7.89443	0.618340	0.309170	−	0.951007i	$-0.399949\pi$
$163$	7.89443	0.618340	0.309170	+	0.951007i	$0.399949\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	20.4726	1.58422	0.792108	−	0.610381i	$-0.208983\pi$
$167$	20.4726	1.58422	0.792108	+	0.610381i	$0.208983\pi$
$168$	0	0
$169$	−2.73181	−0.210139
$170$	0	0
$171$	−2.91794	−0.223140
$172$	0	0
$173$	24.0492	1.82843	0.914215	−	0.405229i	$-0.132808\pi$
$173$	24.0492	1.82843	0.914215	+	0.405229i	$0.132808\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.34651	0.101210
$178$	0	0
$179$	−18.6718	−1.39559	−0.697796	−	0.716296i	$-0.745836\pi$
$179$	−18.6718	−1.39559	−0.697796	+	0.716296i	$0.745836\pi$
$180$	0	0
$181$	−3.53638	−0.262857	−0.131428	−	0.991326i	$-0.541956\pi$
$181$	−3.53638	−0.262857	−0.131428	+	0.991326i	$0.541956\pi$
$182$	0	0
$183$	2.91794	0.215700
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.22267	0.0894108
$188$	0	0
$189$	−0.485629	−0.0353243
$190$	0	0
$191$	14.2682	1.03241	0.516205	−	0.856465i	$-0.327344\pi$
$191$	14.2682	1.03241	0.516205	+	0.856465i	$0.327344\pi$
$192$	0	0
$193$	8.18089	0.588874	0.294437	−	0.955671i	$-0.404868\pi$
$193$	8.18089	0.588874	0.294437	+	0.955671i	$0.404868\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−12.6404	−0.900595	−0.450297	−	0.892879i	$-0.648682\pi$
$197$	−12.6404	−0.900595	−0.450297	+	0.892879i	$0.648682\pi$
$198$	0	0
$199$	4.10407	0.290930	0.145465	−	0.989363i	$-0.453532\pi$
$199$	4.10407	0.290930	0.145465	+	0.989363i	$0.453532\pi$
$200$	0	0
$201$	2.81387	0.198475
$202$	0	0
$203$	0.649750	0.0456035
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−2.73181	−0.189874
$208$	0	0
$209$	−4.26819	−0.295237
$210$	0	0
$211$	−1.53638	−0.105769	−0.0528843	−	0.998601i	$-0.516841\pi$
$211$	−1.53638	−0.105769	−0.0528843	+	0.998601i	$0.516841\pi$
$212$	0	0
$213$	−1.38530	−0.0949189
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.19917	0.0814048
$218$	0	0
$219$	−3.00000	−0.202721
$220$	0	0
$221$	0.917939	0.0617473
$222$	0	0
$223$	1.40880	0.0943404	0.0471702	−	0.998887i	$-0.484980\pi$
$223$	1.40880	0.0943404	0.0471702	+	0.998887i	$0.484980\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	22.9179	1.52112	0.760559	−	0.649269i	$-0.224925\pi$
$227$	22.9179	1.52112	0.760559	+	0.649269i	$0.224925\pi$
$228$	0	0
$229$	3.78513	0.250128	0.125064	−	0.992149i	$-0.460086\pi$
$229$	3.78513	0.250128	0.125064	+	0.992149i	$0.460086\pi$
$230$	0	0
$231$	−0.350250	−0.0230447
$232$	0	0
$233$	−2.26819	−0.148594	−0.0742970	−	0.997236i	$-0.523671\pi$
$233$	−2.26819	−0.148594	−0.0742970	+	0.997236i	$0.523671\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−1.46212	−0.0949750
$238$	0	0
$239$	2.18613	0.141409	0.0707045	−	0.997497i	$-0.477475\pi$
$239$	2.18613	0.141409	0.0707045	+	0.997497i	$0.477475\pi$
$240$	0	0
$241$	19.9907	1.28771	0.643857	−	0.765146i	$-0.277333\pi$
$241$	19.9907	1.28771	0.643857	+	0.765146i	$0.277333\pi$
$242$	0	0
$243$	7.45432	0.478195
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−3.20440	−0.203891
$248$	0	0
$249$	4.50764	0.285660
$250$	0	0
$251$	−1.35025	−0.0852270	−0.0426135	−	0.999092i	$-0.513568\pi$
$251$	−1.35025	−0.0852270	−0.0426135	+	0.999092i	$0.513568\pi$
$252$	0	0
$253$	−3.99593	−0.251222
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.9582	0.745933	0.372967	−	0.927845i	$-0.378341\pi$
$257$	11.9582	0.745933	0.372967	+	0.927845i	$0.378341\pi$
$258$	0	0
$259$	−2.48563	−0.154449
$260$	0	0
$261$	−6.61844	−0.409671
$262$	0	0
$263$	−8.31370	−0.512645	−0.256322	−	0.966591i	$-0.582511\pi$
$263$	−8.31370	−0.512645	−0.256322	+	0.966591i	$0.582511\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−2.79410	−0.170996
$268$	0	0
$269$	−5.02201	−0.306197	−0.153099	−	0.988211i	$-0.548925\pi$
$269$	−5.02201	−0.306197	−0.153099	+	0.988211i	$0.548925\pi$
$270$	0	0
$271$	−6.96869	−0.423318	−0.211659	−	0.977344i	$-0.567887\pi$
$271$	−6.96869	−0.423318	−0.211659	+	0.977344i	$0.567887\pi$
$272$	0	0
$273$	−0.262955	−0.0159148
$274$	0	0
$275$	0	0
$276$	0	0
$277$	12.7083	0.763568	0.381784	−	0.924252i	$-0.375310\pi$
$277$	12.7083	0.763568	0.381784	+	0.924252i	$0.375310\pi$
$278$	0	0
$279$	−12.2149	−0.731285
$280$	0	0
$281$	−16.5364	−0.986478	−0.493239	−	0.869894i	$-0.664187\pi$
$281$	−16.5364	−0.986478	−0.493239	+	0.869894i	$0.664187\pi$
$282$	0	0
$283$	−26.1548	−1.55474	−0.777371	−	0.629042i	$-0.783447\pi$
$283$	−26.1548	−1.55474	−0.777371	+	0.629042i	$0.783447\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.309969	0.0182969
$288$	0	0
$289$	−16.9179	−0.995173
$290$	0	0
$291$	2.75382	0.161432
$292$	0	0
$293$	3.95449	0.231023	0.115512	−	0.993306i	$-0.463149\pi$
$293$	3.95449	0.231023	0.115512	+	0.993306i	$0.463149\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	7.23571	0.419859
$298$	0	0
$299$	−3.00000	−0.173494
$300$	0	0
$301$	1.26819	0.0730972
$302$	0	0
$303$	0.209636	0.0120433
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−20.7355	−1.18344	−0.591720	−	0.806144i	$-0.701551\pi$
$307$	−20.7355	−1.18344	−0.591720	+	0.806144i	$0.701551\pi$
$308$	0	0
$309$	−0.889198	−0.0505847
$310$	0	0
$311$	1.97799	0.112162	0.0560808	−	0.998426i	$-0.482140\pi$
$311$	1.97799	0.112162	0.0560808	+	0.998426i	$0.482140\pi$
$312$	0	0
$313$	8.67699	0.490453	0.245226	−	0.969466i	$-0.421138\pi$
$313$	8.67699	0.490453	0.245226	+	0.969466i	$0.421138\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.34502	0.131709	0.0658546	−	0.997829i	$-0.479023\pi$
$317$	2.34502	0.131709	0.0658546	+	0.997829i	$0.479023\pi$
$318$	0	0
$319$	−9.68106	−0.542035
$320$	0	0
$321$	4.06005	0.226610
$322$	0	0
$323$	−0.286462	−0.0159392
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−2.75532	−0.152369
$328$	0	0
$329$	−0.217440	−0.0119878
$330$	0	0
$331$	22.8579	1.25638	0.628192	−	0.778059i	$-0.283795\pi$
$331$	22.8579	1.25638	0.628192	+	0.778059i	$0.283795\pi$
$332$	0	0
$333$	25.3189	1.38747
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0.386795	0.0210701	0.0105350	−	0.999945i	$-0.496647\pi$
$337$	0.386795	0.0210701	0.0105350	+	0.999945i	$0.496647\pi$
$338$	0	0
$339$	−2.51437	−0.136562
$340$	0	0
$341$	−17.8672	−0.967563
$342$	0	0
$343$	−3.98696	−0.215276
$344$	0	0
$345$	0	0
$346$	0	0
$347$	14.1406	0.759108	0.379554	−	0.925170i	$-0.376077\pi$
$347$	14.1406	0.759108	0.379554	+	0.925170i	$0.376077\pi$
$348$	0	0
$349$	−2.81387	−0.150623	−0.0753115	−	0.997160i	$-0.523995\pi$
$349$	−2.81387	−0.150623	−0.0753115	+	0.997160i	$0.523995\pi$
$350$	0	0
$351$	5.43231	0.289955
$352$	0	0
$353$	13.3137	0.708617	0.354308	−	0.935129i	$-0.384716\pi$
$353$	13.3137	0.708617	0.354308	+	0.935129i	$0.384716\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−0.0235072	−0.00124413
$358$	0	0
$359$	−14.1861	−0.748715	−0.374358	−	0.927284i	$-0.622137\pi$
$359$	−14.1861	−0.748715	−0.374358	+	0.927284i	$0.622137\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	2.06752	0.108517
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−28.0272	−1.46301	−0.731505	−	0.681836i	$-0.761182\pi$
$367$	−28.0272	−1.46301	−0.731505	+	0.681836i	$0.761182\pi$
$368$	0	0
$369$	−3.15739	−0.164367
$370$	0	0
$371$	−1.26819	−0.0658411
$372$	0	0
$373$	24.3502	1.26081	0.630404	−	0.776267i	$-0.282889\pi$
$373$	24.3502	1.26081	0.630404	+	0.776267i	$0.282889\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−7.26819	−0.374331
$378$	0	0
$379$	−28.8866	−1.48381	−0.741903	−	0.670507i	$-0.766077\pi$
$379$	−28.8866	−1.48381	−0.741903	+	0.670507i	$0.766077\pi$
$380$	0	0
$381$	1.93995	0.0993865
$382$	0	0
$383$	−10.3398	−0.528338	−0.264169	−	0.964476i	$-0.585098\pi$
$383$	−10.3398	−0.528338	−0.264169	+	0.964476i	$0.585098\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−12.9179	−0.656656
$388$	0	0
$389$	33.9086	1.71924	0.859618	−	0.510937i	$-0.170702\pi$
$389$	33.9086	1.71924	0.859618	+	0.510937i	$0.170702\pi$
$390$	0	0
$391$	−0.268189	−0.0135629
$392$	0	0
$393$	0.835879	0.0421645
$394$	0	0
$395$	0	0
$396$	0	0
$397$	33.8866	1.70072	0.850361	−	0.526200i	$-0.176384\pi$
$397$	33.8866	1.70072	0.850361	+	0.526200i	$0.176384\pi$
$398$	0	0
$399$	0.0820605	0.00410816
$400$	0	0
$401$	−10.5051	−0.524598	−0.262299	−	0.964987i	$-0.584481\pi$
$401$	−10.5051	−0.524598	−0.262299	+	0.964987i	$0.584481\pi$
$402$	0	0
$403$	−13.4140	−0.668201
$404$	0	0
$405$	0	0
$406$	0	0
$407$	37.0350	1.83576
$408$	0	0
$409$	−6.78256	−0.335376	−0.167688	−	0.985840i	$-0.553630\pi$
$409$	−6.78256	−0.335376	−0.167688	+	0.985840i	$0.553630\pi$
$410$	0	0
$411$	4.13538	0.203983
$412$	0	0
$413$	1.34651	0.0662577
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0.658720	0.0322576
$418$	0	0
$419$	−18.6498	−0.911100	−0.455550	−	0.890210i	$-0.650557\pi$
$419$	−18.6498	−0.911100	−0.455550	+	0.890210i	$0.650557\pi$
$420$	0	0
$421$	9.64975	0.470300	0.235150	−	0.971959i	$-0.424442\pi$
$421$	9.64975	0.470300	0.235150	+	0.971959i	$0.424442\pi$
$422$	0	0
$423$	2.21487	0.107691
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2.91794	0.141209
$428$	0	0
$429$	3.91794	0.189160
$430$	0	0
$431$	33.7758	1.62692	0.813462	−	0.581618i	$-0.197580\pi$
$431$	33.7758	1.62692	0.813462	+	0.581618i	$0.197580\pi$
$432$	0	0
$433$	−1.08986	−0.0523755	−0.0261878	−	0.999657i	$-0.508337\pi$
$433$	−1.08986	−0.0523755	−0.0261878	+	0.999657i	$0.508337\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.936212	0.0447851
$438$	0	0
$439$	12.9687	0.618962	0.309481	−	0.950906i	$-0.399845\pi$
$439$	12.9687	0.618962	0.309481	+	0.950906i	$0.399845\pi$
$440$	0	0
$441$	20.1861	0.961244
$442$	0	0
$443$	−31.9437	−1.51769	−0.758845	−	0.651271i	$-0.774236\pi$
$443$	−31.9437	−1.51769	−0.758845	+	0.651271i	$0.774236\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1.93471	0.0915088
$448$	0	0
$449$	−5.05075	−0.238360	−0.119180	−	0.992873i	$-0.538026\pi$
$449$	−5.05075	−0.238360	−0.119180	+	0.992873i	$0.538026\pi$
$450$	0	0
$451$	−4.61844	−0.217474
$452$	0	0
$453$	3.40507	0.159984
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−35.0948	−1.64166	−0.820832	−	0.571170i	$-0.806490\pi$
$457$	−35.0948	−1.64166	−0.820832	+	0.571170i	$0.806490\pi$
$458$	0	0
$459$	0.485629	0.0226672
$460$	0	0
$461$	−27.8866	−1.29881	−0.649405	−	0.760443i	$-0.724982\pi$
$461$	−27.8866	−1.29881	−0.649405	+	0.760443i	$0.724982\pi$
$462$	0	0
$463$	−28.0037	−1.30144	−0.650722	−	0.759316i	$-0.725534\pi$
$463$	−28.0037	−1.30144	−0.650722	+	0.759316i	$0.725534\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17.4543	0.807690	0.403845	−	0.914828i	$-0.367674\pi$
$467$	17.4543	0.807690	0.403845	+	0.914828i	$0.367674\pi$
$468$	0	0
$469$	2.81387	0.129933
$470$	0	0
$471$	4.06005	0.187077
$472$	0	0
$473$	−18.8956	−0.868821
$474$	0	0
$475$	0	0
$476$	0	0
$477$	12.9179	0.591472
$478$	0	0
$479$	−26.0440	−1.18998	−0.594991	−	0.803733i	$-0.702844\pi$
$479$	−26.0440	−1.18998	−0.594991	+	0.803733i	$0.702844\pi$
$480$	0	0
$481$	27.8046	1.26778
$482$	0	0
$483$	0.0768261	0.00349571
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−21.9907	−0.996494	−0.498247	−	0.867035i	$-0.666023\pi$
$487$	−21.9907	−0.996494	−0.498247	+	0.867035i	$0.666023\pi$
$488$	0	0
$489$	2.26146	0.102267
$490$	0	0
$491$	33.9907	1.53398	0.766989	−	0.641660i	$-0.221754\pi$
$491$	33.9907	1.53398	0.766989	+	0.641660i	$0.221754\pi$
$492$	0	0
$493$	−0.649750	−0.0292633
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.38530	−0.0621390
$498$	0	0
$499$	−6.83331	−0.305901	−0.152950	−	0.988234i	$-0.548877\pi$
$499$	−6.83331	−0.305901	−0.152950	+	0.988234i	$0.548877\pi$
$500$	0	0
$501$	5.86462	0.262012
$502$	0	0
$503$	34.4685	1.53688	0.768438	−	0.639925i	$-0.221034\pi$
$503$	34.4685	1.53688	0.768438	+	0.639925i	$0.221034\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−0.782560	−0.0347547
$508$	0	0
$509$	18.9179	0.838523	0.419261	−	0.907866i	$-0.362289\pi$
$509$	18.9179	0.838523	0.419261	+	0.907866i	$0.362289\pi$
$510$	0	0
$511$	−3.00000	−0.132712
$512$	0	0
$513$	−1.69527	−0.0748478
$514$	0	0
$515$	0	0
$516$	0	0
$517$	3.23978	0.142485
$518$	0	0
$519$	6.88920	0.302402
$520$	0	0
$521$	−1.62774	−0.0713127	−0.0356563	−	0.999364i	$-0.511352\pi$
$521$	−1.62774	−0.0713127	−0.0356563	+	0.999364i	$0.511352\pi$
$522$	0	0
$523$	−1.16669	−0.0510158	−0.0255079	−	0.999675i	$-0.508120\pi$
$523$	−1.16669	−0.0510158	−0.0255079	+	0.999675i	$0.508120\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.19917	−0.0522365
$528$	0	0
$529$	−22.1235	−0.961892
$530$	0	0
$531$	−13.7158	−0.595214
$532$	0	0
$533$	−3.46736	−0.150188
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−5.34875	−0.230816
$538$	0	0
$539$	29.5271	1.27182
$540$	0	0
$541$	−23.9399	−1.02926	−0.514629	−	0.857413i	$-0.672070\pi$
$541$	−23.9399	−1.02926	−0.514629	+	0.857413i	$0.672070\pi$
$542$	0	0
$543$	−1.01304	−0.0434736
$544$	0	0
$545$	0	0
$546$	0	0
$547$	14.8139	0.633395	0.316698	−	0.948527i	$-0.397426\pi$
$547$	14.8139	0.633395	0.316698	+	0.948527i	$0.397426\pi$
$548$	0	0
$549$	−29.7225	−1.26853
$550$	0	0
$551$	2.26819	0.0966281
$552$	0	0
$553$	−1.46212	−0.0621757
$554$	0	0
$555$	0	0
$556$	0	0
$557$	39.3592	1.66770	0.833852	−	0.551988i	$-0.186131\pi$
$557$	39.3592	1.66770	0.833852	+	0.551988i	$0.186131\pi$
$558$	0	0
$559$	−14.1861	−0.600009
$560$	0	0
$561$	0.350250	0.0147876
$562$	0	0
$563$	9.55722	0.402789	0.201394	−	0.979510i	$-0.435453\pi$
$563$	9.55722	0.402789	0.201394	+	0.979510i	$0.435453\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.36852	0.0994686
$568$	0	0
$569$	21.2995	0.892922	0.446461	−	0.894803i	$-0.352684\pi$
$569$	21.2995	0.892922	0.446461	+	0.894803i	$0.352684\pi$
$570$	0	0
$571$	17.4543	0.730440	0.365220	−	0.930921i	$-0.380994\pi$
$571$	17.4543	0.730440	0.365220	+	0.930921i	$0.380994\pi$
$572$	0	0
$573$	4.08729	0.170749
$574$	0	0
$575$	0	0
$576$	0	0
$577$	37.2861	1.55224	0.776121	−	0.630584i	$-0.217185\pi$
$577$	37.2861	1.55224	0.776121	+	0.630584i	$0.217185\pi$
$578$	0	0
$579$	2.34352	0.0973932
$580$	0	0
$581$	4.50764	0.187008
$582$	0	0
$583$	18.8956	0.782576
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−7.73181	−0.319126	−0.159563	−	0.987188i	$-0.551009\pi$
$587$	−7.73181	−0.319126	−0.159563	+	0.987188i	$0.551009\pi$
$588$	0	0
$589$	4.18613	0.172486
$590$	0	0
$591$	−3.62101	−0.148948
$592$	0	0
$593$	6.74858	0.277131	0.138566	−	0.990353i	$-0.455751\pi$
$593$	6.74858	0.277131	0.138566	+	0.990353i	$0.455751\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.17566	0.0481166
$598$	0	0
$599$	16.9687	0.693322	0.346661	−	0.937991i	$-0.387315\pi$
$599$	16.9687	0.693322	0.346661	+	0.937991i	$0.387315\pi$
$600$	0	0
$601$	31.7251	1.29409	0.647046	−	0.762451i	$-0.276004\pi$
$601$	31.7251	1.29409	0.647046	+	0.762451i	$0.276004\pi$
$602$	0	0
$603$	−28.6625	−1.16723
$604$	0	0
$605$	0	0
$606$	0	0
$607$	3.84518	0.156071	0.0780356	−	0.996951i	$-0.475135\pi$
$607$	3.84518	0.156071	0.0780356	+	0.996951i	$0.475135\pi$
$608$	0	0
$609$	0.186129	0.00754232
$610$	0	0
$611$	2.43231	0.0984007
$612$	0	0
$613$	32.1208	1.29735	0.648674	−	0.761066i	$-0.275324\pi$
$613$	32.1208	1.29735	0.648674	+	0.761066i	$0.275324\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−35.2630	−1.41963	−0.709817	−	0.704387i	$-0.751222\pi$
$617$	−35.2630	−1.41963	−0.709817	+	0.704387i	$0.751222\pi$
$618$	0	0
$619$	−9.29020	−0.373405	−0.186702	−	0.982417i	$-0.559780\pi$
$619$	−9.29020	−0.373405	−0.186702	+	0.982417i	$0.559780\pi$
$620$	0	0
$621$	−1.58713	−0.0636893
$622$	0	0
$623$	−2.79410	−0.111943
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−1.22267	−0.0488289
$628$	0	0
$629$	2.48563	0.0991085
$630$	0	0
$631$	26.5271	1.05603	0.528013	−	0.849236i	$-0.322937\pi$
$631$	26.5271	1.05603	0.528013	+	0.849236i	$0.322937\pi$
$632$	0	0
$633$	−0.440114	−0.0174930
$634$	0	0
$635$	0	0
$636$	0	0
$637$	22.1679	0.878322
$638$	0	0
$639$	14.1108	0.558215
$640$	0	0
$641$	−7.86462	−0.310634	−0.155317	−	0.987865i	$-0.549640\pi$
$641$	−7.86462	−0.310634	−0.155317	+	0.987865i	$0.549640\pi$
$642$	0	0
$643$	−1.67176	−0.0659277	−0.0329638	−	0.999457i	$-0.510495\pi$
$643$	−1.67176	−0.0659277	−0.0329638	+	0.999457i	$0.510495\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	38.4909	1.51323	0.756616	−	0.653859i	$-0.226851\pi$
$647$	38.4909	1.51323	0.756616	+	0.653859i	$0.226851\pi$
$648$	0	0
$649$	−20.0626	−0.787527
$650$	0	0
$651$	0.343516	0.0134634
$652$	0	0
$653$	25.9985	1.01740	0.508700	−	0.860944i	$-0.330126\pi$
$653$	25.9985	1.01740	0.508700	+	0.860944i	$0.330126\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	30.5584	1.19220
$658$	0	0
$659$	−46.9594	−1.82928	−0.914639	−	0.404272i	$-0.867525\pi$
$659$	−46.9594	−1.82928	−0.914639	+	0.404272i	$0.867525\pi$
$660$	0	0
$661$	−35.7225	−1.38944	−0.694722	−	0.719278i	$-0.744473\pi$
$661$	−35.7225	−1.38944	−0.694722	+	0.719278i	$0.744473\pi$
$662$	0	0
$663$	0.262955	0.0102123
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.12351	0.0822225
$668$	0	0
$669$	0.403569	0.0156029
$670$	0	0
$671$	−43.4763	−1.67838
$672$	0	0
$673$	−27.3450	−1.05407	−0.527036	−	0.849843i	$-0.676697\pi$
$673$	−27.3450	−1.05407	−0.527036	+	0.849843i	$0.676697\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	39.9944	1.53711	0.768555	−	0.639783i	$-0.220976\pi$
$677$	39.9944	1.53711	0.768555	+	0.639783i	$0.220976\pi$
$678$	0	0
$679$	2.75382	0.105682
$680$	0	0
$681$	6.56512	0.251576
$682$	0	0
$683$	21.0130	0.804042	0.402021	−	0.915631i	$-0.368308\pi$
$683$	21.0130	0.804042	0.402021	+	0.915631i	$0.368308\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.08430	0.0413685
$688$	0	0
$689$	14.1861	0.540448
$690$	0	0
$691$	10.7445	0.408741	0.204370	−	0.978894i	$-0.434485\pi$
$691$	10.7445	0.408741	0.204370	+	0.978894i	$0.434485\pi$
$692$	0	0
$693$	3.56769	0.135525
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−0.309969	−0.0117409
$698$	0	0
$699$	−0.649750	−0.0245758
$700$	0	0
$701$	42.3029	1.59776	0.798879	−	0.601491i	$-0.205427\pi$
$701$	42.3029	1.59776	0.798879	+	0.601491i	$0.205427\pi$
$702$	0	0
$703$	−8.67699	−0.327259
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.209636	0.00788417
$708$	0	0
$709$	−40.5584	−1.52320	−0.761601	−	0.648046i	$-0.775586\pi$
$709$	−40.5584	−1.52320	−0.761601	+	0.648046i	$0.775586\pi$
$710$	0	0
$711$	14.8934	0.558545
$712$	0	0
$713$	3.91911	0.146772
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0.626243	0.0233875
$718$	0	0
$719$	−16.8866	−0.629765	−0.314882	−	0.949131i	$-0.601965\pi$
$719$	−16.8866	−0.629765	−0.314882	+	0.949131i	$0.601965\pi$
$720$	0	0
$721$	−0.889198	−0.0331155
$722$	0	0
$723$	5.72658	0.212974
$724$	0	0
$725$	0	0
$726$	0	0
$727$	27.4398	1.01769	0.508843	−	0.860860i	$-0.330074\pi$
$727$	27.4398	1.01769	0.508843	+	0.860860i	$0.330074\pi$
$728$	0	0
$729$	−22.6692	−0.839600
$730$	0	0
$731$	−1.26819	−0.0469057
$732$	0	0
$733$	14.0078	0.517390	0.258695	−	0.965959i	$-0.416708\pi$
$733$	14.0078	0.517390	0.258695	+	0.965959i	$0.416708\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−41.9257	−1.54435
$738$	0	0
$739$	19.7851	0.727808	0.363904	−	0.931437i	$-0.381444\pi$
$739$	19.7851	0.727808	0.363904	+	0.931437i	$0.381444\pi$
$740$	0	0
$741$	−0.917939	−0.0337213
$742$	0	0
$743$	41.8344	1.53475	0.767377	−	0.641196i	$-0.221561\pi$
$743$	41.8344	1.53475	0.767377	+	0.641196i	$0.221561\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−45.9154	−1.67996
$748$	0	0
$749$	4.06005	0.148351
$750$	0	0
$751$	−48.4983	−1.76973	−0.884865	−	0.465848i	$-0.845749\pi$
$751$	−48.4983	−1.76973	−0.884865	+	0.465848i	$0.845749\pi$
$752$	0	0
$753$	−0.386795	−0.0140956
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−32.2537	−1.17228	−0.586139	−	0.810210i	$-0.699353\pi$
$757$	−32.2537	−1.17228	−0.586139	+	0.810210i	$0.699353\pi$
$758$	0	0
$759$	−1.14468	−0.0415493
$760$	0	0
$761$	−25.6184	−0.928668	−0.464334	−	0.885660i	$-0.653706\pi$
$761$	−25.6184	−0.928668	−0.464334	+	0.885660i	$0.653706\pi$
$762$	0	0
$763$	−2.75532	−0.0997492
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−15.0623	−0.543868
$768$	0	0
$769$	42.3096	1.52572	0.762862	−	0.646561i	$-0.223793\pi$
$769$	42.3096	1.52572	0.762862	+	0.646561i	$0.223793\pi$
$770$	0	0
$771$	3.42558	0.123369
$772$	0	0
$773$	0.889198	0.0319822	0.0159911	−	0.999872i	$-0.494910\pi$
$773$	0.889198	0.0319822	0.0159911	+	0.999872i	$0.494910\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−0.712038	−0.0255442
$778$	0	0
$779$	1.08206	0.0387688
$780$	0	0
$781$	20.6404	0.738573
$782$	0	0
$783$	−3.84518	−0.137416
$784$	0	0
$785$	0	0
$786$	0	0
$787$	50.6650	1.80601	0.903007	−	0.429627i	$-0.141355\pi$
$787$	50.6650	1.80601	0.903007	+	0.429627i	$0.141355\pi$
$788$	0	0
$789$	−2.38156	−0.0847858
$790$	0	0
$791$	−2.51437	−0.0894007
$792$	0	0
$793$	−32.6404	−1.15910
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−33.4909	−1.18631	−0.593154	−	0.805089i	$-0.702117\pi$
$797$	−33.4909	−1.18631	−0.593154	+	0.805089i	$0.702117\pi$
$798$	0	0
$799$	0.217440	0.00769247
$800$	0	0
$801$	28.4611	1.00562
$802$	0	0
$803$	44.6990	1.57739
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1.43861	−0.0506416
$808$	0	0
$809$	−6.35025	−0.223263	−0.111631	−	0.993750i	$-0.535608\pi$
$809$	−6.35025	−0.223263	−0.111631	+	0.993750i	$0.535608\pi$
$810$	0	0
$811$	1.08463	0.0380865	0.0190433	−	0.999819i	$-0.493938\pi$
$811$	1.08463	0.0380865	0.0190433	+	0.999819i	$0.493938\pi$
$812$	0	0
$813$	−1.99627	−0.0700121
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4.42708	0.154884
$818$	0	0
$819$	2.67849	0.0935941
$820$	0	0
$821$	−12.3630	−0.431470	−0.215735	−	0.976452i	$-0.569215\pi$
$821$	−12.3630	−0.431470	−0.215735	+	0.976452i	$0.569215\pi$
$822$	0	0
$823$	27.0765	0.943827	0.471914	−	0.881645i	$-0.343563\pi$
$823$	27.0765	0.943827	0.471914	+	0.881645i	$0.343563\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−17.0765	−0.593808	−0.296904	−	0.954907i	$-0.595954\pi$
$827$	−17.0765	−0.593808	−0.296904	+	0.954907i	$0.595954\pi$
$828$	0	0
$829$	−28.5804	−0.992638	−0.496319	−	0.868140i	$-0.665315\pi$
$829$	−28.5804	−0.992638	−0.496319	+	0.868140i	$0.665315\pi$
$830$	0	0
$831$	3.64045	0.126286
$832$	0	0
$833$	1.98173	0.0686628
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−7.09660	−0.245294
$838$	0	0
$839$	55.9019	1.92995	0.964974	−	0.262346i	$-0.0844960\pi$
$839$	55.9019	1.92995	0.964974	+	0.262346i	$0.0844960\pi$
$840$	0	0
$841$	−23.8553	−0.822597
$842$	0	0
$843$	−4.73705	−0.163153
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.06752	0.0710409
$848$	0	0
$849$	−7.49236	−0.257137
$850$	0	0
$851$	−8.12351	−0.278470
$852$	0	0
$853$	4.93214	0.168873	0.0844367	−	0.996429i	$-0.473091\pi$
$853$	4.93214	0.168873	0.0844367	+	0.996429i	$0.473091\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−39.9411	−1.36436	−0.682181	−	0.731183i	$-0.738968\pi$
$857$	−39.9411	−1.36436	−0.682181	+	0.731183i	$0.738968\pi$
$858$	0	0
$859$	−1.99070	−0.0679217	−0.0339608	−	0.999423i	$-0.510812\pi$
$859$	−1.99070	−0.0679217	−0.0339608	+	0.999423i	$0.510812\pi$
$860$	0	0
$861$	0.0887944	0.00302611
$862$	0	0
$863$	31.9817	1.08867	0.544335	−	0.838868i	$-0.316782\pi$
$863$	31.9817	1.08867	0.544335	+	0.838868i	$0.316782\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−4.84635	−0.164591
$868$	0	0
$869$	21.7851	0.739010
$870$	0	0
$871$	−31.4763	−1.06653
$872$	0	0
$873$	−28.0507	−0.949374
$874$	0	0
$875$	0	0
$876$	0	0
$877$	22.2917	0.752737	0.376369	−	0.926470i	$-0.377173\pi$
$877$	22.2917	0.752737	0.376369	+	0.926470i	$0.377173\pi$
$878$	0	0
$879$	1.13281	0.0382087
$880$	0	0
$881$	35.6912	1.20247	0.601233	−	0.799073i	$-0.294676\pi$
$881$	35.6912	1.20247	0.601233	+	0.799073i	$0.294676\pi$
$882$	0	0
$883$	17.9948	0.605572	0.302786	−	0.953059i	$-0.402083\pi$
$883$	17.9948	0.605572	0.302786	+	0.953059i	$0.402083\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	24.3566	0.817813	0.408907	−	0.912576i	$-0.365910\pi$
$887$	24.3566	0.817813	0.408907	+	0.912576i	$0.365910\pi$
$888$	0	0
$889$	1.93995	0.0650637
$890$	0	0
$891$	−35.2902	−1.18227
$892$	0	0
$893$	−0.759053	−0.0254007
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−0.859386	−0.0286941
$898$	0	0
$899$	9.49493	0.316674
$900$	0	0
$901$	1.26819	0.0422495
$902$	0	0
$903$	0.363288	0.0120895
$904$	0	0
$905$	0	0
$906$	0	0
$907$	5.73821	0.190534	0.0952671	−	0.995452i	$-0.469629\pi$
$907$	5.73821	0.190534	0.0952671	+	0.995452i	$0.469629\pi$
$908$	0	0
$909$	−2.13538	−0.0708261
$910$	0	0
$911$	46.8291	1.55152	0.775759	−	0.631029i	$-0.217367\pi$
$911$	46.8291	1.55152	0.775759	+	0.631029i	$0.217367\pi$
$912$	0	0
$913$	−67.1623	−2.22275
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0.835879	0.0276032
$918$	0	0
$919$	−2.32151	−0.0765795	−0.0382897	−	0.999267i	$-0.512191\pi$
$919$	−2.32151	−0.0765795	−0.0382897	+	0.999267i	$0.512191\pi$
$920$	0	0
$921$	−5.93995	−0.195728
$922$	0	0
$923$	15.4961	0.510060
$924$	0	0
$925$	0	0
$926$	0	0
$927$	9.05748	0.297487
$928$	0	0
$929$	−25.5610	−0.838628	−0.419314	−	0.907841i	$-0.637729\pi$
$929$	−25.5610	−0.838628	−0.419314	+	0.907841i	$0.637729\pi$
$930$	0	0
$931$	−6.91794	−0.226726
$932$	0	0
$933$	0.566620	0.0185503
$934$	0	0
$935$	0	0
$936$	0	0
$937$	25.7501	0.841219	0.420609	−	0.907242i	$-0.361816\pi$
$937$	25.7501	0.841219	0.420609	+	0.907242i	$0.361816\pi$
$938$	0	0
$939$	2.48563	0.0811154
$940$	0	0
$941$	−29.1261	−0.949483	−0.474741	−	0.880125i	$-0.657458\pi$
$941$	−29.1261	−0.949483	−0.474741	+	0.880125i	$0.657458\pi$
$942$	0	0
$943$	1.01304	0.0329891
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−6.21711	−0.202029	−0.101014	−	0.994885i	$-0.532209\pi$
$947$	−6.21711	−0.202029	−0.101014	+	0.994885i	$0.532209\pi$
$948$	0	0
$949$	33.5584	1.08935
$950$	0	0
$951$	0.671758	0.0217832
$952$	0	0
$953$	−47.8266	−1.54925	−0.774627	−	0.632418i	$-0.782063\pi$
$953$	−47.8266	−1.54925	−0.774627	+	0.632418i	$0.782063\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−2.77326	−0.0896467
$958$	0	0
$959$	4.13538	0.133538
$960$	0	0
$961$	−13.4763	−0.434720
$962$	0	0
$963$	−41.3562	−1.33269
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−37.2301	−1.19724	−0.598620	−	0.801033i	$-0.704284\pi$
$967$	−37.2301	−1.19724	−0.598620	+	0.801033i	$0.704284\pi$
$968$	0	0
$969$	−0.0820605	−0.00263616
$970$	0	0
$971$	−37.8359	−1.21421	−0.607106	−	0.794621i	$-0.707669\pi$
$971$	−37.8359	−1.21421	−0.607106	+	0.794621i	$0.707669\pi$
$972$	0	0
$973$	0.658720	0.0211176
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−55.8684	−1.78739	−0.893694	−	0.448678i	$-0.851895\pi$
$977$	−55.8684	−1.78739	−0.893694	+	0.448678i	$0.851895\pi$
$978$	0	0
$979$	41.6311	1.33054
$980$	0	0
$981$	28.0660	0.896079
$982$	0	0
$983$	−6.93738	−0.221268	−0.110634	−	0.993861i	$-0.535288\pi$
$983$	−6.93738	−0.221268	−0.110634	+	0.993861i	$0.535288\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−0.0622883	−0.00198266
$988$	0	0
$989$	4.14468	0.131793
$990$	0	0
$991$	−36.5897	−1.16231	−0.581155	−	0.813793i	$-0.697399\pi$
$991$	−36.5897	−1.16231	−0.581155	+	0.813793i	$0.697399\pi$
$992$	0	0
$993$	6.54792	0.207792
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−33.7538	−1.06899	−0.534497	−	0.845170i	$-0.679499\pi$
$997$	−33.7538	−1.06899	−0.534497	+	0.845170i	$0.679499\pi$
$998$	0	0
$999$	14.7098	0.465398

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.by.1.2		3
4.3	odd	2		1900.2.a.f.1.2	✓	3
5.4	even	2		7600.2.a.bj.1.2		3
20.3	even	4		1900.2.c.g.1749.3		6
20.7	even	4		1900.2.c.g.1749.4		6
20.19	odd	2		1900.2.a.h.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1900.2.a.f.1.2	✓	3	4.3	odd	2
1900.2.a.h.1.2	yes	3	20.19	odd	2
1900.2.c.g.1749.3		6	20.3	even	4
1900.2.c.g.1749.4		6	20.7	even	4
7600.2.a.bj.1.2		3	5.4	even	2
7600.2.a.by.1.2		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)

\(f(q)\)	\(=\)	\(q+0.286462 q^{3} +0.286462 q^{7} -2.91794 q^{9} +O(q^{10})\)	\(q+0.286462 q^{3} +0.286462 q^{7} -2.91794 q^{9} -4.26819 q^{11} -3.20440 q^{13} -0.286462 q^{17} +1.00000 q^{19} +0.0820605 q^{21} +0.936212 q^{23} -1.69527 q^{27} +2.26819 q^{29} +4.18613 q^{31} -1.22267 q^{33} -8.67699 q^{37} -0.917939 q^{39} +1.08206 q^{41} +4.42708 q^{43} -0.759053 q^{47} -6.91794 q^{49} -0.0820605 q^{51} -4.42708 q^{53} +0.286462 q^{57} +4.70050 q^{59} +10.1861 q^{61} -0.835879 q^{63} +9.82284 q^{67} +0.268189 q^{69} -4.83588 q^{71} -10.4726 q^{73} -1.22267 q^{77} -5.10407 q^{79} +8.26819 q^{81} +15.7355 q^{83} +0.649750 q^{87} -9.75382 q^{89} -0.917939 q^{91} +1.19917 q^{93} +9.61320 q^{97} +12.4543 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} + 2 q^{7} + q^{9}+O(q^{10}) \) 3 * q + 2 * q^3 + 2 * q^7 + q^9	\( 3 q + 2 q^{3} + 2 q^{7} + q^{9} + q^{11} - q^{13} - 2 q^{17} + 3 q^{19} + 10 q^{21} + 8 q^{23} + 11 q^{27} - 7 q^{29} - 11 q^{31} - 10 q^{33} + 5 q^{37} + 7 q^{39} + 13 q^{41} + 11 q^{43} + 19 q^{47} - 11 q^{49} - 10 q^{51} - 11 q^{53} + 2 q^{57} + 6 q^{59} + 7 q^{61} + 17 q^{63} + 3 q^{67} - 13 q^{69} + 5 q^{71} - 9 q^{73} - 10 q^{77} + 18 q^{79} + 11 q^{81} + 3 q^{83} + 6 q^{87} + 7 q^{91} - 13 q^{93} + 3 q^{97}+O(q^{100}) \) 3 * q + 2 * q^3 + 2 * q^7 + q^9 + q^11 - q^13 - 2 * q^17 + 3 * q^19 + 10 * q^21 + 8 * q^23 + 11 * q^27 - 7 * q^29 - 11 * q^31 - 10 * q^33 + 5 * q^37 + 7 * q^39 + 13 * q^41 + 11 * q^43 + 19 * q^47 - 11 * q^49 - 10 * q^51 - 11 * q^53 + 2 * q^57 + 6 * q^59 + 7 * q^61 + 17 * q^63 + 3 * q^67 - 13 * q^69 + 5 * q^71 - 9 * q^73 - 10 * q^77 + 18 * q^79 + 11 * q^81 + 3 * q^83 + 6 * q^87 + 7 * q^91 - 13 * q^93 + 3 * q^97

Introduction

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