LMFDB - Embedded newform 7600.2.a.bg.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:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 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			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	$q+2.41421 q^{3} +1.58579 q^{7} +2.82843 q^{9} +O(q^{10})$	$q+2.41421 q^{3} +1.58579 q^{7} +2.82843 q^{9} -1.41421 q^{11} -0.171573 q^{13} -1.00000 q^{17} +1.00000 q^{19} +3.82843 q^{21} +9.24264 q^{23} -0.414214 q^{27} -5.82843 q^{29} +2.24264 q^{31} -3.41421 q^{33} -8.48528 q^{37} -0.414214 q^{39} +4.24264 q^{41} +10.2426 q^{43} -4.48528 q^{49} -2.41421 q^{51} +11.4853 q^{53} +2.41421 q^{57} +12.8995 q^{59} +5.75736 q^{61} +4.48528 q^{63} +13.2426 q^{67} +22.3137 q^{69} +10.5858 q^{71} +5.48528 q^{73} -2.24264 q^{77} -10.4853 q^{79} -9.48528 q^{81} -2.48528 q^{83} -14.0711 q^{87} -7.07107 q^{89} -0.272078 q^{91} +5.41421 q^{93} +11.6569 q^{97} -4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 6 q^{7}+O(q^{10}) $ 2 * q + 2 * q^3 + 6 * q^7	$ 2 q + 2 q^{3} + 6 q^{7} - 6 q^{13} - 2 q^{17} + 2 q^{19} + 2 q^{21} + 10 q^{23} + 2 q^{27} - 6 q^{29} - 4 q^{31} - 4 q^{33} + 2 q^{39} + 12 q^{43} + 8 q^{49} - 2 q^{51} + 6 q^{53} + 2 q^{57} + 6 q^{59} + 20 q^{61} - 8 q^{63} + 18 q^{67} + 22 q^{69} + 24 q^{71} - 6 q^{73} + 4 q^{77} - 4 q^{79} - 2 q^{81} + 12 q^{83} - 14 q^{87} - 26 q^{91} + 8 q^{93} + 12 q^{97} - 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 6 * q^7 - 6 * q^13 - 2 * q^17 + 2 * q^19 + 2 * q^21 + 10 * q^23 + 2 * q^27 - 6 * q^29 - 4 * q^31 - 4 * q^33 + 2 * q^39 + 12 * q^43 + 8 * q^49 - 2 * q^51 + 6 * q^53 + 2 * q^57 + 6 * q^59 + 20 * q^61 - 8 * q^63 + 18 * q^67 + 22 * q^69 + 24 * q^71 - 6 * q^73 + 4 * q^77 - 4 * q^79 - 2 * q^81 + 12 * q^83 - 14 * q^87 - 26 * q^91 + 8 * q^93 + 12 * q^97 - 8 * 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
$3$	2.41421	1.39385	0.696923	−	0.717146i	$-0.254552\pi$
$3$	2.41421	1.39385	0.696923	+	0.717146i	$0.254552\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.58579	0.599371	0.299685	−	0.954038i	$-0.403118\pi$
$7$	1.58579	0.599371	0.299685	+	0.954038i	$0.403118\pi$
$8$	0	0
$9$	2.82843	0.942809
$10$	0	0
$11$	−1.41421	−0.426401	−0.213201	−	0.977008i	$-0.568389\pi$
$11$	−1.41421	−0.426401	−0.213201	+	0.977008i	$0.568389\pi$
$12$	0	0
$13$	−0.171573	−0.0475858	−0.0237929	−	0.999717i	$-0.507574\pi$
$13$	−0.171573	−0.0475858	−0.0237929	+	0.999717i	$0.507574\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.00000	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	3.82843	0.835431
$22$	0	0
$23$	9.24264	1.92722	0.963612	−	0.267305i	$-0.0861332\pi$
$23$	9.24264	1.92722	0.963612	+	0.267305i	$0.0861332\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−0.414214	−0.0797154
$28$	0	0
$29$	−5.82843	−1.08231	−0.541156	−	0.840922i	$-0.682013\pi$
$29$	−5.82843	−1.08231	−0.541156	+	0.840922i	$0.682013\pi$
$30$	0	0
$31$	2.24264	0.402790	0.201395	−	0.979510i	$-0.435452\pi$
$31$	2.24264	0.402790	0.201395	+	0.979510i	$0.435452\pi$
$32$	0	0
$33$	−3.41421	−0.594338
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.48528	−1.39497	−0.697486	−	0.716599i	$-0.745698\pi$
$37$	−8.48528	−1.39497	−0.697486	+	0.716599i	$0.745698\pi$
$38$	0	0
$39$	−0.414214	−0.0663273
$40$	0	0
$41$	4.24264	0.662589	0.331295	−	0.943527i	$-0.392515\pi$
$41$	4.24264	0.662589	0.331295	+	0.943527i	$0.392515\pi$
$42$	0	0
$43$	10.2426	1.56199	0.780994	−	0.624538i	$-0.214713\pi$
$43$	10.2426	1.56199	0.780994	+	0.624538i	$0.214713\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−4.48528	−0.640754
$50$	0	0
$51$	−2.41421	−0.338058
$52$	0	0
$53$	11.4853	1.57762	0.788812	−	0.614634i	$-0.210696\pi$
$53$	11.4853	1.57762	0.788812	+	0.614634i	$0.210696\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.41421	0.319770
$58$	0	0
$59$	12.8995	1.67937	0.839686	−	0.543073i	$-0.182739\pi$
$59$	12.8995	1.67937	0.839686	+	0.543073i	$0.182739\pi$
$60$	0	0
$61$	5.75736	0.737154	0.368577	−	0.929597i	$-0.379845\pi$
$61$	5.75736	0.737154	0.368577	+	0.929597i	$0.379845\pi$
$62$	0	0
$63$	4.48528	0.565092
$64$	0	0
$65$	0	0
$66$	0	0
$67$	13.2426	1.61785	0.808923	−	0.587915i	$-0.200051\pi$
$67$	13.2426	1.61785	0.808923	+	0.587915i	$0.200051\pi$
$68$	0	0
$69$	22.3137	2.68625
$70$	0	0
$71$	10.5858	1.25630	0.628151	−	0.778092i	$-0.283812\pi$
$71$	10.5858	1.25630	0.628151	+	0.778092i	$0.283812\pi$
$72$	0	0
$73$	5.48528	0.642004	0.321002	−	0.947079i	$-0.395980\pi$
$73$	5.48528	0.642004	0.321002	+	0.947079i	$0.395980\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.24264	−0.255573
$78$	0	0
$79$	−10.4853	−1.17969	−0.589843	−	0.807518i	$-0.700810\pi$
$79$	−10.4853	−1.17969	−0.589843	+	0.807518i	$0.700810\pi$
$80$	0	0
$81$	−9.48528	−1.05392
$82$	0	0
$83$	−2.48528	−0.272795	−0.136398	−	0.990654i	$-0.543552\pi$
$83$	−2.48528	−0.272795	−0.136398	+	0.990654i	$0.543552\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−14.0711	−1.50858
$88$	0	0
$89$	−7.07107	−0.749532	−0.374766	−	0.927119i	$-0.622277\pi$
$89$	−7.07107	−0.749532	−0.374766	+	0.927119i	$0.622277\pi$
$90$	0	0
$91$	−0.272078	−0.0285215
$92$	0	0
$93$	5.41421	0.561428
$94$	0	0
$95$	0	0
$96$	0	0
$97$	11.6569	1.18357	0.591787	−	0.806094i	$-0.298423\pi$
$97$	11.6569	1.18357	0.591787	+	0.806094i	$0.298423\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	0	0
$101$	−1.07107	−0.106575	−0.0532876	−	0.998579i	$-0.516970\pi$
$101$	−1.07107	−0.106575	−0.0532876	+	0.998579i	$0.516970\pi$
$102$	0	0
$103$	−4.24264	−0.418040	−0.209020	−	0.977911i	$-0.567027\pi$
$103$	−4.24264	−0.418040	−0.209020	+	0.977911i	$0.567027\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.72792	−0.553739	−0.276870	−	0.960908i	$-0.589297\pi$
$107$	−5.72792	−0.553739	−0.276870	+	0.960908i	$0.589297\pi$
$108$	0	0
$109$	15.9706	1.52970	0.764851	−	0.644207i	$-0.222812\pi$
$109$	15.9706	1.52970	0.764851	+	0.644207i	$0.222812\pi$
$110$	0	0
$111$	−20.4853	−1.94438
$112$	0	0
$113$	1.75736	0.165318	0.0826592	−	0.996578i	$-0.473659\pi$
$113$	1.75736	0.165318	0.0826592	+	0.996578i	$0.473659\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−0.485281	−0.0448643
$118$	0	0
$119$	−1.58579	−0.145369
$120$	0	0
$121$	−9.00000	−0.818182
$122$	0	0
$123$	10.2426	0.923548
$124$	0	0
$125$	0	0
$126$	0	0
$127$	14.4853	1.28536	0.642680	−	0.766134i	$-0.277822\pi$
$127$	14.4853	1.28536	0.642680	+	0.766134i	$0.277822\pi$
$128$	0	0
$129$	24.7279	2.17717
$130$	0	0
$131$	−16.9706	−1.48272	−0.741362	−	0.671105i	$-0.765820\pi$
$131$	−16.9706	−1.48272	−0.741362	+	0.671105i	$0.765820\pi$
$132$	0	0
$133$	1.58579	0.137505
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−13.0000	−1.11066	−0.555332	−	0.831628i	$-0.687409\pi$
$137$	−13.0000	−1.11066	−0.555332	+	0.831628i	$0.687409\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0.242641	0.0202906
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−10.8284	−0.893114
$148$	0	0
$149$	6.34315	0.519651	0.259825	−	0.965656i	$-0.416335\pi$
$149$	6.34315	0.519651	0.259825	+	0.965656i	$0.416335\pi$
$150$	0	0
$151$	−6.48528	−0.527765	−0.263882	−	0.964555i	$-0.585003\pi$
$151$	−6.48528	−0.527765	−0.263882	+	0.964555i	$0.585003\pi$
$152$	0	0
$153$	−2.82843	−0.228665
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−0.343146	−0.0273860	−0.0136930	−	0.999906i	$-0.504359\pi$
$157$	−0.343146	−0.0273860	−0.0136930	+	0.999906i	$0.504359\pi$
$158$	0	0
$159$	27.7279	2.19897
$160$	0	0
$161$	14.6569	1.15512
$162$	0	0
$163$	−1.75736	−0.137647	−0.0688235	−	0.997629i	$-0.521925\pi$
$163$	−1.75736	−0.137647	−0.0688235	+	0.997629i	$0.521925\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−9.75736	−0.755047	−0.377524	−	0.926000i	$-0.623224\pi$
$167$	−9.75736	−0.755047	−0.377524	+	0.926000i	$0.623224\pi$
$168$	0	0
$169$	−12.9706	−0.997736
$170$	0	0
$171$	2.82843	0.216295
$172$	0	0
$173$	16.4853	1.25335	0.626676	−	0.779280i	$-0.284415\pi$
$173$	16.4853	1.25335	0.626676	+	0.779280i	$0.284415\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	31.1421	2.34079
$178$	0	0
$179$	0.343146	0.0256479	0.0128240	−	0.999918i	$-0.495918\pi$
$179$	0.343146	0.0256479	0.0128240	+	0.999918i	$0.495918\pi$
$180$	0	0
$181$	8.48528	0.630706	0.315353	−	0.948974i	$-0.397877\pi$
$181$	8.48528	0.630706	0.315353	+	0.948974i	$0.397877\pi$
$182$	0	0
$183$	13.8995	1.02748
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.41421	0.103418
$188$	0	0
$189$	−0.656854	−0.0477791
$190$	0	0
$191$	18.5563	1.34269	0.671345	−	0.741145i	$-0.265717\pi$
$191$	18.5563	1.34269	0.671345	+	0.741145i	$0.265717\pi$
$192$	0	0
$193$	11.6569	0.839079	0.419539	−	0.907737i	$-0.362192\pi$
$193$	11.6569	0.839079	0.419539	+	0.907737i	$0.362192\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.2426	1.44223	0.721114	−	0.692816i	$-0.243630\pi$
$197$	20.2426	1.44223	0.721114	+	0.692816i	$0.243630\pi$
$198$	0	0
$199$	−0.757359	−0.0536878	−0.0268439	−	0.999640i	$-0.508546\pi$
$199$	−0.757359	−0.0536878	−0.0268439	+	0.999640i	$0.508546\pi$
$200$	0	0
$201$	31.9706	2.25503
$202$	0	0
$203$	−9.24264	−0.648706
$204$	0	0
$205$	0	0
$206$	0	0
$207$	26.1421	1.81700
$208$	0	0
$209$	−1.41421	−0.0978232
$210$	0	0
$211$	−5.72792	−0.394326	−0.197163	−	0.980371i	$-0.563173\pi$
$211$	−5.72792	−0.394326	−0.197163	+	0.980371i	$0.563173\pi$
$212$	0	0
$213$	25.5563	1.75109
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3.55635	0.241421
$218$	0	0
$219$	13.2426	0.894855
$220$	0	0
$221$	0.171573	0.0115412
$222$	0	0
$223$	−20.8284	−1.39477	−0.697387	−	0.716694i	$-0.745654\pi$
$223$	−20.8284	−1.39477	−0.697387	+	0.716694i	$0.745654\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−25.2426	−1.67541	−0.837706	−	0.546121i	$-0.816104\pi$
$227$	−25.2426	−1.67541	−0.837706	+	0.546121i	$0.816104\pi$
$228$	0	0
$229$	−18.9706	−1.25361	−0.626805	−	0.779176i	$-0.715638\pi$
$229$	−18.9706	−1.25361	−0.626805	+	0.779176i	$0.715638\pi$
$230$	0	0
$231$	−5.41421	−0.356229
$232$	0	0
$233$	−8.97056	−0.587681	−0.293841	−	0.955854i	$-0.594933\pi$
$233$	−8.97056	−0.587681	−0.293841	+	0.955854i	$0.594933\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−25.3137	−1.64430
$238$	0	0
$239$	−12.8995	−0.834399	−0.417199	−	0.908815i	$-0.636988\pi$
$239$	−12.8995	−0.834399	−0.417199	+	0.908815i	$0.636988\pi$
$240$	0	0
$241$	−24.9706	−1.60850	−0.804248	−	0.594294i	$-0.797431\pi$
$241$	−24.9706	−1.60850	−0.804248	+	0.594294i	$0.797431\pi$
$242$	0	0
$243$	−21.6569	−1.38929
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−0.171573	−0.0109169
$248$	0	0
$249$	−6.00000	−0.380235
$250$	0	0
$251$	27.5563	1.73934	0.869671	−	0.493632i	$-0.164331\pi$
$251$	27.5563	1.73934	0.869671	+	0.493632i	$0.164331\pi$
$252$	0	0
$253$	−13.0711	−0.821771
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.72792	0.294920	0.147460	−	0.989068i	$-0.452890\pi$
$257$	4.72792	0.294920	0.147460	+	0.989068i	$0.452890\pi$
$258$	0	0
$259$	−13.4558	−0.836105
$260$	0	0
$261$	−16.4853	−1.02041
$262$	0	0
$263$	6.97056	0.429823	0.214912	−	0.976633i	$-0.431054\pi$
$263$	6.97056	0.429823	0.214912	+	0.976633i	$0.431054\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−17.0711	−1.04473
$268$	0	0
$269$	28.6274	1.74544	0.872722	−	0.488217i	$-0.162353\pi$
$269$	28.6274	1.74544	0.872722	+	0.488217i	$0.162353\pi$
$270$	0	0
$271$	18.7574	1.13943	0.569714	−	0.821843i	$-0.307054\pi$
$271$	18.7574	1.13943	0.569714	+	0.821843i	$0.307054\pi$
$272$	0	0
$273$	−0.656854	−0.0397546
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	0	0
$279$	6.34315	0.379754
$280$	0	0
$281$	−4.24264	−0.253095	−0.126547	−	0.991961i	$-0.540390\pi$
$281$	−4.24264	−0.253095	−0.126547	+	0.991961i	$0.540390\pi$
$282$	0	0
$283$	3.85786	0.229326	0.114663	−	0.993404i	$-0.463421\pi$
$283$	3.85786	0.229326	0.114663	+	0.993404i	$0.463421\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.72792	0.397137
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	28.1421	1.64972
$292$	0	0
$293$	−11.4853	−0.670977	−0.335489	−	0.942044i	$-0.608901\pi$
$293$	−11.4853	−0.670977	−0.335489	+	0.942044i	$0.608901\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0.585786	0.0339908
$298$	0	0
$299$	−1.58579	−0.0917084
$300$	0	0
$301$	16.2426	0.936210
$302$	0	0
$303$	−2.58579	−0.148550
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−6.34315	−0.362022	−0.181011	−	0.983481i	$-0.557937\pi$
$307$	−6.34315	−0.362022	−0.181011	+	0.983481i	$0.557937\pi$
$308$	0	0
$309$	−10.2426	−0.582683
$310$	0	0
$311$	13.2426	0.750921	0.375461	−	0.926838i	$-0.377485\pi$
$311$	13.2426	0.750921	0.375461	+	0.926838i	$0.377485\pi$
$312$	0	0
$313$	−25.9706	−1.46794	−0.733971	−	0.679180i	$-0.762335\pi$
$313$	−25.9706	−1.46794	−0.733971	+	0.679180i	$0.762335\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−7.48528	−0.420415	−0.210208	−	0.977657i	$-0.567414\pi$
$317$	−7.48528	−0.420415	−0.210208	+	0.977657i	$0.567414\pi$
$318$	0	0
$319$	8.24264	0.461499
$320$	0	0
$321$	−13.8284	−0.771828
$322$	0	0
$323$	−1.00000	−0.0556415
$324$	0	0
$325$	0	0
$326$	0	0
$327$	38.5563	2.13217
$328$	0	0
$329$	0	0
$330$	0	0
$331$	10.7574	0.591278	0.295639	−	0.955300i	$-0.404467\pi$
$331$	10.7574	0.591278	0.295639	+	0.955300i	$0.404467\pi$
$332$	0	0
$333$	−24.0000	−1.31519
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−33.8995	−1.84662	−0.923312	−	0.384052i	$-0.874528\pi$
$337$	−33.8995	−1.84662	−0.923312	+	0.384052i	$0.874528\pi$
$338$	0	0
$339$	4.24264	0.230429
$340$	0	0
$341$	−3.17157	−0.171750
$342$	0	0
$343$	−18.2132	−0.983421
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−22.4853	−1.20707	−0.603537	−	0.797335i	$-0.706242\pi$
$347$	−22.4853	−1.20707	−0.603537	+	0.797335i	$0.706242\pi$
$348$	0	0
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	0	0
$351$	0.0710678	0.00379332
$352$	0	0
$353$	19.4853	1.03710	0.518548	−	0.855048i	$-0.326473\pi$
$353$	19.4853	1.03710	0.518548	+	0.855048i	$0.326473\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−3.82843	−0.202622
$358$	0	0
$359$	31.2426	1.64892	0.824462	−	0.565918i	$-0.191478\pi$
$359$	31.2426	1.64892	0.824462	+	0.565918i	$0.191478\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−21.7279	−1.14042
$364$	0	0
$365$	0	0
$366$	0	0
$367$	25.4558	1.32878	0.664392	−	0.747384i	$-0.268691\pi$
$367$	25.4558	1.32878	0.664392	+	0.747384i	$0.268691\pi$
$368$	0	0
$369$	12.0000	0.624695
$370$	0	0
$371$	18.2132	0.945582
$372$	0	0
$373$	−9.00000	−0.466002	−0.233001	−	0.972476i	$-0.574855\pi$
$373$	−9.00000	−0.466002	−0.233001	+	0.972476i	$0.574855\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.00000	0.0515026
$378$	0	0
$379$	−2.75736	−0.141636	−0.0708180	−	0.997489i	$-0.522561\pi$
$379$	−2.75736	−0.141636	−0.0708180	+	0.997489i	$0.522561\pi$
$380$	0	0
$381$	34.9706	1.79160
$382$	0	0
$383$	3.75736	0.191992	0.0959960	−	0.995382i	$-0.469396\pi$
$383$	3.75736	0.191992	0.0959960	+	0.995382i	$0.469396\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	28.9706	1.47266
$388$	0	0
$389$	37.0711	1.87958	0.939789	−	0.341756i	$-0.111021\pi$
$389$	37.0711	1.87958	0.939789	+	0.341756i	$0.111021\pi$
$390$	0	0
$391$	−9.24264	−0.467420
$392$	0	0
$393$	−40.9706	−2.06669
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−24.0000	−1.20453	−0.602263	−	0.798298i	$-0.705734\pi$
$397$	−24.0000	−1.20453	−0.602263	+	0.798298i	$0.705734\pi$
$398$	0	0
$399$	3.82843	0.191661
$400$	0	0
$401$	−22.5858	−1.12788	−0.563940	−	0.825816i	$-0.690715\pi$
$401$	−22.5858	−1.12788	−0.563940	+	0.825816i	$0.690715\pi$
$402$	0	0
$403$	−0.384776	−0.0191671
$404$	0	0
$405$	0	0
$406$	0	0
$407$	12.0000	0.594818
$408$	0	0
$409$	−17.2132	−0.851138	−0.425569	−	0.904926i	$-0.639926\pi$
$409$	−17.2132	−0.851138	−0.425569	+	0.904926i	$0.639926\pi$
$410$	0	0
$411$	−31.3848	−1.54810
$412$	0	0
$413$	20.4558	1.00657
$414$	0	0
$415$	0	0
$416$	0	0
$417$	28.9706	1.41869
$418$	0	0
$419$	−16.5858	−0.810269	−0.405134	−	0.914257i	$-0.632775\pi$
$419$	−16.5858	−0.810269	−0.405134	+	0.914257i	$0.632775\pi$
$420$	0	0
$421$	2.51472	0.122560	0.0612799	−	0.998121i	$-0.480482\pi$
$421$	2.51472	0.122560	0.0612799	+	0.998121i	$0.480482\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	9.12994	0.441829
$428$	0	0
$429$	0.585786	0.0282820
$430$	0	0
$431$	−30.3848	−1.46358	−0.731792	−	0.681528i	$-0.761316\pi$
$431$	−30.3848	−1.46358	−0.731792	+	0.681528i	$0.761316\pi$
$432$	0	0
$433$	−21.5563	−1.03593	−0.517966	−	0.855401i	$-0.673311\pi$
$433$	−21.5563	−1.03593	−0.517966	+	0.855401i	$0.673311\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	9.24264	0.442135
$438$	0	0
$439$	14.2426	0.679764	0.339882	−	0.940468i	$-0.389613\pi$
$439$	14.2426	0.679764	0.339882	+	0.940468i	$0.389613\pi$
$440$	0	0
$441$	−12.6863	−0.604109
$442$	0	0
$443$	−4.24264	−0.201574	−0.100787	−	0.994908i	$-0.532136\pi$
$443$	−4.24264	−0.201574	−0.100787	+	0.994908i	$0.532136\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	15.3137	0.724314
$448$	0	0
$449$	−5.31371	−0.250769	−0.125385	−	0.992108i	$-0.540017\pi$
$449$	−5.31371	−0.250769	−0.125385	+	0.992108i	$0.540017\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	0	0
$453$	−15.6569	−0.735623
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.00000	0.140334	0.0701670	−	0.997535i	$-0.477647\pi$
$457$	3.00000	0.140334	0.0701670	+	0.997535i	$0.477647\pi$
$458$	0	0
$459$	0.414214	0.0193338
$460$	0	0
$461$	27.5563	1.28343	0.641714	−	0.766944i	$-0.278224\pi$
$461$	27.5563	1.28343	0.641714	+	0.766944i	$0.278224\pi$
$462$	0	0
$463$	−14.1421	−0.657241	−0.328620	−	0.944462i	$-0.606584\pi$
$463$	−14.1421	−0.657241	−0.328620	+	0.944462i	$0.606584\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−24.7279	−1.14427	−0.572136	−	0.820159i	$-0.693885\pi$
$467$	−24.7279	−1.14427	−0.572136	+	0.820159i	$0.693885\pi$
$468$	0	0
$469$	21.0000	0.969690
$470$	0	0
$471$	−0.828427	−0.0381719
$472$	0	0
$473$	−14.4853	−0.666034
$474$	0	0
$475$	0	0
$476$	0	0
$477$	32.4853	1.48740
$478$	0	0
$479$	−31.1127	−1.42158	−0.710788	−	0.703407i	$-0.751661\pi$
$479$	−31.1127	−1.42158	−0.710788	+	0.703407i	$0.751661\pi$
$480$	0	0
$481$	1.45584	0.0663808
$482$	0	0
$483$	35.3848	1.61006
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−1.79899	−0.0815200	−0.0407600	−	0.999169i	$-0.512978\pi$
$487$	−1.79899	−0.0815200	−0.0407600	+	0.999169i	$0.512978\pi$
$488$	0	0
$489$	−4.24264	−0.191859
$490$	0	0
$491$	2.44365	0.110280	0.0551402	−	0.998479i	$-0.482439\pi$
$491$	2.44365	0.110280	0.0551402	+	0.998479i	$0.482439\pi$
$492$	0	0
$493$	5.82843	0.262499
$494$	0	0
$495$	0	0
$496$	0	0
$497$	16.7868	0.752991
$498$	0	0
$499$	34.2426	1.53291	0.766456	−	0.642297i	$-0.222019\pi$
$499$	34.2426	1.53291	0.766456	+	0.642297i	$0.222019\pi$
$500$	0	0
$501$	−23.5563	−1.05242
$502$	0	0
$503$	39.7279	1.77138	0.885690	−	0.464277i	$-0.153686\pi$
$503$	39.7279	1.77138	0.885690	+	0.464277i	$0.153686\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−31.3137	−1.39069
$508$	0	0
$509$	−4.97056	−0.220316	−0.110158	−	0.993914i	$-0.535136\pi$
$509$	−4.97056	−0.220316	−0.110158	+	0.993914i	$0.535136\pi$
$510$	0	0
$511$	8.69848	0.384798
$512$	0	0
$513$	−0.414214	−0.0182880
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	39.7990	1.74698
$520$	0	0
$521$	0.686292	0.0300670	0.0150335	−	0.999887i	$-0.495215\pi$
$521$	0.686292	0.0300670	0.0150335	+	0.999887i	$0.495215\pi$
$522$	0	0
$523$	−27.7279	−1.21246	−0.606229	−	0.795290i	$-0.707318\pi$
$523$	−27.7279	−1.21246	−0.606229	+	0.795290i	$0.707318\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.24264	−0.0976910
$528$	0	0
$529$	62.4264	2.71419
$530$	0	0
$531$	36.4853	1.58333
$532$	0	0
$533$	−0.727922	−0.0315298
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0.828427	0.0357493
$538$	0	0
$539$	6.34315	0.273219
$540$	0	0
$541$	18.2426	0.784312	0.392156	−	0.919899i	$-0.371729\pi$
$541$	18.2426	0.784312	0.392156	+	0.919899i	$0.371729\pi$
$542$	0	0
$543$	20.4853	0.879108
$544$	0	0
$545$	0	0
$546$	0	0
$547$	5.31371	0.227198	0.113599	−	0.993527i	$-0.463762\pi$
$547$	5.31371	0.227198	0.113599	+	0.993527i	$0.463762\pi$
$548$	0	0
$549$	16.2843	0.694996
$550$	0	0
$551$	−5.82843	−0.248299
$552$	0	0
$553$	−16.6274	−0.707070
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−16.0000	−0.677942	−0.338971	−	0.940797i	$-0.610079\pi$
$557$	−16.0000	−0.677942	−0.338971	+	0.940797i	$0.610079\pi$
$558$	0	0
$559$	−1.75736	−0.0743284
$560$	0	0
$561$	3.41421	0.144148
$562$	0	0
$563$	−28.9706	−1.22096	−0.610482	−	0.792030i	$-0.709024\pi$
$563$	−28.9706	−1.22096	−0.610482	+	0.792030i	$0.709024\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−15.0416	−0.631689
$568$	0	0
$569$	28.2843	1.18574	0.592869	−	0.805299i	$-0.297995\pi$
$569$	28.2843	1.18574	0.592869	+	0.805299i	$0.297995\pi$
$570$	0	0
$571$	−6.24264	−0.261246	−0.130623	−	0.991432i	$-0.541698\pi$
$571$	−6.24264	−0.261246	−0.130623	+	0.991432i	$0.541698\pi$
$572$	0	0
$573$	44.7990	1.87150
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−2.31371	−0.0963209	−0.0481605	−	0.998840i	$-0.515336\pi$
$577$	−2.31371	−0.0963209	−0.0481605	+	0.998840i	$0.515336\pi$
$578$	0	0
$579$	28.1421	1.16955
$580$	0	0
$581$	−3.94113	−0.163505
$582$	0	0
$583$	−16.2426	−0.672701
$584$	0	0
$585$	0	0
$586$	0	0
$587$	21.7574	0.898022	0.449011	−	0.893526i	$-0.351776\pi$
$587$	21.7574	0.898022	0.449011	+	0.893526i	$0.351776\pi$
$588$	0	0
$589$	2.24264	0.0924064
$590$	0	0
$591$	48.8701	2.01025
$592$	0	0
$593$	−22.0000	−0.903432	−0.451716	−	0.892162i	$-0.649188\pi$
$593$	−22.0000	−0.903432	−0.451716	+	0.892162i	$0.649188\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.82843	−0.0748325
$598$	0	0
$599$	−27.2132	−1.11190	−0.555951	−	0.831215i	$-0.687646\pi$
$599$	−27.2132	−1.11190	−0.555951	+	0.831215i	$0.687646\pi$
$600$	0	0
$601$	−11.7574	−0.479593	−0.239796	−	0.970823i	$-0.577081\pi$
$601$	−11.7574	−0.479593	−0.239796	+	0.970823i	$0.577081\pi$
$602$	0	0
$603$	37.4558	1.52532
$604$	0	0
$605$	0	0
$606$	0	0
$607$	8.82843	0.358335	0.179167	−	0.983819i	$-0.442660\pi$
$607$	8.82843	0.358335	0.179167	+	0.983819i	$0.442660\pi$
$608$	0	0
$609$	−22.3137	−0.904197
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−47.6985	−1.92652	−0.963262	−	0.268564i	$-0.913451\pi$
$613$	−47.6985	−1.92652	−0.963262	+	0.268564i	$0.913451\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−12.4853	−0.502639	−0.251319	−	0.967904i	$-0.580864\pi$
$617$	−12.4853	−0.502639	−0.251319	+	0.967904i	$0.580864\pi$
$618$	0	0
$619$	−16.2426	−0.652847	−0.326423	−	0.945224i	$-0.605844\pi$
$619$	−16.2426	−0.652847	−0.326423	+	0.945224i	$0.605844\pi$
$620$	0	0
$621$	−3.82843	−0.153629
$622$	0	0
$623$	−11.2132	−0.449248
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−3.41421	−0.136351
$628$	0	0
$629$	8.48528	0.338330
$630$	0	0
$631$	5.02944	0.200219	0.100109	−	0.994976i	$-0.468081\pi$
$631$	5.02944	0.200219	0.100109	+	0.994976i	$0.468081\pi$
$632$	0	0
$633$	−13.8284	−0.549631
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0.769553	0.0304908
$638$	0	0
$639$	29.9411	1.18445
$640$	0	0
$641$	30.0416	1.18657	0.593287	−	0.804991i	$-0.297830\pi$
$641$	30.0416	1.18657	0.593287	+	0.804991i	$0.297830\pi$
$642$	0	0
$643$	2.48528	0.0980099	0.0490050	−	0.998799i	$-0.484395\pi$
$643$	2.48528	0.0980099	0.0490050	+	0.998799i	$0.484395\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−27.2426	−1.07102	−0.535509	−	0.844529i	$-0.679880\pi$
$647$	−27.2426	−1.07102	−0.535509	+	0.844529i	$0.679880\pi$
$648$	0	0
$649$	−18.2426	−0.716086
$650$	0	0
$651$	8.58579	0.336504
$652$	0	0
$653$	−4.97056	−0.194513	−0.0972566	−	0.995259i	$-0.531007\pi$
$653$	−4.97056	−0.194513	−0.0972566	+	0.995259i	$0.531007\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	15.5147	0.605287
$658$	0	0
$659$	0.899495	0.0350393	0.0175197	−	0.999847i	$-0.494423\pi$
$659$	0.899495	0.0350393	0.0175197	+	0.999847i	$0.494423\pi$
$660$	0	0
$661$	32.4558	1.26239	0.631193	−	0.775626i	$-0.282566\pi$
$661$	32.4558	1.26239	0.631193	+	0.775626i	$0.282566\pi$
$662$	0	0
$663$	0.414214	0.0160867
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−53.8701	−2.08586
$668$	0	0
$669$	−50.2843	−1.94410
$670$	0	0
$671$	−8.14214	−0.314324
$672$	0	0
$673$	12.0000	0.462566	0.231283	−	0.972887i	$-0.425708\pi$
$673$	12.0000	0.462566	0.231283	+	0.972887i	$0.425708\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−26.9411	−1.03543	−0.517716	−	0.855553i	$-0.673218\pi$
$677$	−26.9411	−1.03543	−0.517716	+	0.855553i	$0.673218\pi$
$678$	0	0
$679$	18.4853	0.709400
$680$	0	0
$681$	−60.9411	−2.33527
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−45.7990	−1.74734
$688$	0	0
$689$	−1.97056	−0.0750725
$690$	0	0
$691$	−5.51472	−0.209790	−0.104895	−	0.994483i	$-0.533451\pi$
$691$	−5.51472	−0.209790	−0.104895	+	0.994483i	$0.533451\pi$
$692$	0	0
$693$	−6.34315	−0.240956
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−4.24264	−0.160701
$698$	0	0
$699$	−21.6569	−0.819137
$700$	0	0
$701$	−16.9706	−0.640969	−0.320485	−	0.947254i	$-0.603846\pi$
$701$	−16.9706	−0.640969	−0.320485	+	0.947254i	$0.603846\pi$
$702$	0	0
$703$	−8.48528	−0.320028
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.69848	−0.0638781
$708$	0	0
$709$	−34.0000	−1.27690	−0.638448	−	0.769665i	$-0.720423\pi$
$709$	−34.0000	−1.27690	−0.638448	+	0.769665i	$0.720423\pi$
$710$	0	0
$711$	−29.6569	−1.11222
$712$	0	0
$713$	20.7279	0.776267
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−31.1421	−1.16302
$718$	0	0
$719$	24.8995	0.928594	0.464297	−	0.885679i	$-0.346307\pi$
$719$	24.8995	0.928594	0.464297	+	0.885679i	$0.346307\pi$
$720$	0	0
$721$	−6.72792	−0.250561
$722$	0	0
$723$	−60.2843	−2.24200
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−15.7279	−0.583316	−0.291658	−	0.956523i	$-0.594207\pi$
$727$	−15.7279	−0.583316	−0.291658	+	0.956523i	$0.594207\pi$
$728$	0	0
$729$	−23.8284	−0.882534
$730$	0	0
$731$	−10.2426	−0.378838
$732$	0	0
$733$	−12.0000	−0.443230	−0.221615	−	0.975134i	$-0.571133\pi$
$733$	−12.0000	−0.443230	−0.221615	+	0.975134i	$0.571133\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−18.7279	−0.689852
$738$	0	0
$739$	−26.7279	−0.983203	−0.491601	−	0.870820i	$-0.663588\pi$
$739$	−26.7279	−0.983203	−0.491601	+	0.870820i	$0.663588\pi$
$740$	0	0
$741$	−0.414214	−0.0152165
$742$	0	0
$743$	−18.7279	−0.687061	−0.343530	−	0.939142i	$-0.611623\pi$
$743$	−18.7279	−0.687061	−0.343530	+	0.939142i	$0.611623\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−7.02944	−0.257194
$748$	0	0
$749$	−9.08326	−0.331895
$750$	0	0
$751$	1.27208	0.0464188	0.0232094	−	0.999731i	$-0.492612\pi$
$751$	1.27208	0.0464188	0.0232094	+	0.999731i	$0.492612\pi$
$752$	0	0
$753$	66.5269	2.42438
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−23.6569	−0.859823	−0.429911	−	0.902871i	$-0.641455\pi$
$757$	−23.6569	−0.859823	−0.429911	+	0.902871i	$0.641455\pi$
$758$	0	0
$759$	−31.5563	−1.14542
$760$	0	0
$761$	10.0294	0.363567	0.181783	−	0.983339i	$-0.441813\pi$
$761$	10.0294	0.363567	0.181783	+	0.983339i	$0.441813\pi$
$762$	0	0
$763$	25.3259	0.916859
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.21320	−0.0799141
$768$	0	0
$769$	14.4558	0.521291	0.260646	−	0.965435i	$-0.416065\pi$
$769$	14.4558	0.521291	0.260646	+	0.965435i	$0.416065\pi$
$770$	0	0
$771$	11.4142	0.411073
$772$	0	0
$773$	19.9706	0.718291	0.359146	−	0.933282i	$-0.383068\pi$
$773$	19.9706	0.718291	0.359146	+	0.933282i	$0.383068\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−32.4853	−1.16540
$778$	0	0
$779$	4.24264	0.152008
$780$	0	0
$781$	−14.9706	−0.535689
$782$	0	0
$783$	2.41421	0.0862770
$784$	0	0
$785$	0	0
$786$	0	0
$787$	35.1838	1.25417	0.627083	−	0.778953i	$-0.284249\pi$
$787$	35.1838	1.25417	0.627083	+	0.778953i	$0.284249\pi$
$788$	0	0
$789$	16.8284	0.599108
$790$	0	0
$791$	2.78680	0.0990871
$792$	0	0
$793$	−0.987807	−0.0350780
$794$	0	0
$795$	0	0
$796$	0	0
$797$	30.5147	1.08089	0.540443	−	0.841380i	$-0.318257\pi$
$797$	30.5147	1.08089	0.540443	+	0.841380i	$0.318257\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−20.0000	−0.706665
$802$	0	0
$803$	−7.75736	−0.273751
$804$	0	0
$805$	0	0
$806$	0	0
$807$	69.1127	2.43288
$808$	0	0
$809$	10.7990	0.379672	0.189836	−	0.981816i	$-0.439204\pi$
$809$	10.7990	0.379672	0.189836	+	0.981816i	$0.439204\pi$
$810$	0	0
$811$	6.69848	0.235216	0.117608	−	0.993060i	$-0.462477\pi$
$811$	6.69848	0.235216	0.117608	+	0.993060i	$0.462477\pi$
$812$	0	0
$813$	45.2843	1.58819
$814$	0	0
$815$	0	0
$816$	0	0
$817$	10.2426	0.358345
$818$	0	0
$819$	−0.769553	−0.0268903
$820$	0	0
$821$	17.6569	0.616228	0.308114	−	0.951349i	$-0.400302\pi$
$821$	17.6569	0.616228	0.308114	+	0.951349i	$0.400302\pi$
$822$	0	0
$823$	9.72792	0.339094	0.169547	−	0.985522i	$-0.445770\pi$
$823$	9.72792	0.339094	0.169547	+	0.985522i	$0.445770\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	38.6985	1.34568	0.672839	−	0.739789i	$-0.265075\pi$
$827$	38.6985	1.34568	0.672839	+	0.739789i	$0.265075\pi$
$828$	0	0
$829$	−40.4558	−1.40509	−0.702545	−	0.711640i	$-0.747953\pi$
$829$	−40.4558	−1.40509	−0.702545	+	0.711640i	$0.747953\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	4.48528	0.155406
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−0.928932	−0.0321086
$838$	0	0
$839$	22.6274	0.781185	0.390593	−	0.920564i	$-0.372270\pi$
$839$	22.6274	0.781185	0.390593	+	0.920564i	$0.372270\pi$
$840$	0	0
$841$	4.97056	0.171399
$842$	0	0
$843$	−10.2426	−0.352775
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−14.2721	−0.490394
$848$	0	0
$849$	9.31371	0.319646
$850$	0	0
$851$	−78.4264	−2.68842
$852$	0	0
$853$	−39.9411	−1.36756	−0.683779	−	0.729689i	$-0.739665\pi$
$853$	−39.9411	−1.36756	−0.683779	+	0.729689i	$0.739665\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	20.7279	0.707228	0.353614	−	0.935392i	$-0.384953\pi$
$859$	20.7279	0.707228	0.353614	+	0.935392i	$0.384953\pi$
$860$	0	0
$861$	16.2426	0.553548
$862$	0	0
$863$	−24.7279	−0.841748	−0.420874	−	0.907119i	$-0.638277\pi$
$863$	−24.7279	−0.841748	−0.420874	+	0.907119i	$0.638277\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−38.6274	−1.31186
$868$	0	0
$869$	14.8284	0.503020
$870$	0	0
$871$	−2.27208	−0.0769864
$872$	0	0
$873$	32.9706	1.11588
$874$	0	0
$875$	0	0
$876$	0	0
$877$	47.1421	1.59188	0.795938	−	0.605378i	$-0.206978\pi$
$877$	47.1421	1.59188	0.795938	+	0.605378i	$0.206978\pi$
$878$	0	0
$879$	−27.7279	−0.935240
$880$	0	0
$881$	39.5980	1.33409	0.667045	−	0.745018i	$-0.267559\pi$
$881$	39.5980	1.33409	0.667045	+	0.745018i	$0.267559\pi$
$882$	0	0
$883$	−2.48528	−0.0836364	−0.0418182	−	0.999125i	$-0.513315\pi$
$883$	−2.48528	−0.0836364	−0.0418182	+	0.999125i	$0.513315\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.78680	−0.0935715	−0.0467857	−	0.998905i	$-0.514898\pi$
$887$	−2.78680	−0.0935715	−0.0467857	+	0.998905i	$0.514898\pi$
$888$	0	0
$889$	22.9706	0.770408
$890$	0	0
$891$	13.4142	0.449393
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−3.82843	−0.127827
$898$	0	0
$899$	−13.0711	−0.435945
$900$	0	0
$901$	−11.4853	−0.382630
$902$	0	0
$903$	39.2132	1.30493
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−20.6985	−0.687282	−0.343641	−	0.939101i	$-0.611660\pi$
$907$	−20.6985	−0.687282	−0.343641	+	0.939101i	$0.611660\pi$
$908$	0	0
$909$	−3.02944	−0.100480
$910$	0	0
$911$	16.2843	0.539522	0.269761	−	0.962927i	$-0.413055\pi$
$911$	16.2843	0.539522	0.269761	+	0.962927i	$0.413055\pi$
$912$	0	0
$913$	3.51472	0.116320
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−26.9117	−0.888702
$918$	0	0
$919$	−36.2132	−1.19456	−0.597282	−	0.802032i	$-0.703753\pi$
$919$	−36.2132	−1.19456	−0.597282	+	0.802032i	$0.703753\pi$
$920$	0	0
$921$	−15.3137	−0.504604
$922$	0	0
$923$	−1.81623	−0.0597821
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−12.0000	−0.394132
$928$	0	0
$929$	−17.8284	−0.584932	−0.292466	−	0.956276i	$-0.594476\pi$
$929$	−17.8284	−0.584932	−0.292466	+	0.956276i	$0.594476\pi$
$930$	0	0
$931$	−4.48528	−0.146999
$932$	0	0
$933$	31.9706	1.04667
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−27.0000	−0.882052	−0.441026	−	0.897494i	$-0.645385\pi$
$937$	−27.0000	−0.882052	−0.441026	+	0.897494i	$0.645385\pi$
$938$	0	0
$939$	−62.6985	−2.04609
$940$	0	0
$941$	24.8579	0.810343	0.405172	−	0.914241i	$-0.367212\pi$
$941$	24.8579	0.810343	0.405172	+	0.914241i	$0.367212\pi$
$942$	0	0
$943$	39.2132	1.27696
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	0	0
$949$	−0.941125	−0.0305502
$950$	0	0
$951$	−18.0711	−0.585995
$952$	0	0
$953$	6.72792	0.217939	0.108969	−	0.994045i	$-0.465245\pi$
$953$	6.72792	0.217939	0.108969	+	0.994045i	$0.465245\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	19.8995	0.643259
$958$	0	0
$959$	−20.6152	−0.665700
$960$	0	0
$961$	−25.9706	−0.837760
$962$	0	0
$963$	−16.2010	−0.522070
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−19.1127	−0.614623	−0.307311	−	0.951609i	$-0.599429\pi$
$967$	−19.1127	−0.614623	−0.307311	+	0.951609i	$0.599429\pi$
$968$	0	0
$969$	−2.41421	−0.0775557
$970$	0	0
$971$	−30.3431	−0.973758	−0.486879	−	0.873469i	$-0.661865\pi$
$971$	−30.3431	−0.973758	−0.486879	+	0.873469i	$0.661865\pi$
$972$	0	0
$973$	19.0294	0.610056
$974$	0	0
$975$	0	0
$976$	0	0
$977$	8.24264	0.263705	0.131853	−	0.991269i	$-0.457907\pi$
$977$	8.24264	0.263705	0.131853	+	0.991269i	$0.457907\pi$
$978$	0	0
$979$	10.0000	0.319601
$980$	0	0
$981$	45.1716	1.44222
$982$	0	0
$983$	−0.544156	−0.0173559	−0.00867794	−	0.999962i	$-0.502762\pi$
$983$	−0.544156	−0.0173559	−0.00867794	+	0.999962i	$0.502762\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	94.6690	3.01030
$990$	0	0
$991$	−22.2426	−0.706561	−0.353280	−	0.935517i	$-0.614934\pi$
$991$	−22.2426	−0.706561	−0.353280	+	0.935517i	$0.614934\pi$
$992$	0	0
$993$	25.9706	0.824151
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−23.2721	−0.737034	−0.368517	−	0.929621i	$-0.620134\pi$
$997$	−23.2721	−0.737034	−0.368517	+	0.929621i	$0.620134\pi$
$998$	0	0
$999$	3.51472	0.111201

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.bg.1.2		2
4.3	odd	2		950.2.a.g.1.1		2
5.2	odd	4		1520.2.d.e.609.1		4
5.3	odd	4		1520.2.d.e.609.4		4
5.4	even	2		7600.2.a.v.1.1		2
12.11	even	2		8550.2.a.bn.1.2		2
20.3	even	4		190.2.b.a.39.1	✓	4
20.7	even	4		190.2.b.a.39.4	yes	4
20.19	odd	2		950.2.a.f.1.2		2
60.23	odd	4		1710.2.d.c.1369.4		4
60.47	odd	4		1710.2.d.c.1369.2		4
60.59	even	2		8550.2.a.cb.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
190.2.b.a.39.1	✓	4	20.3	even	4
190.2.b.a.39.4	yes	4	20.7	even	4
950.2.a.f.1.2		2	20.19	odd	2
950.2.a.g.1.1		2	4.3	odd	2
1520.2.d.e.609.1		4	5.2	odd	4
1520.2.d.e.609.4		4	5.3	odd	4
1710.2.d.c.1369.2		4	60.47	odd	4
1710.2.d.c.1369.4		4	60.23	odd	4
7600.2.a.v.1.1		2	5.4	even	2
7600.2.a.bg.1.2		2	1.1	even	1	trivial
8550.2.a.bn.1.2		2	12.11	even	2
8550.2.a.cb.1.1		2	60.59	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)

\(f(q)\)	\(=\)	\(q+2.41421 q^{3} +1.58579 q^{7} +2.82843 q^{9} +O(q^{10})\)	\(q+2.41421 q^{3} +1.58579 q^{7} +2.82843 q^{9} -1.41421 q^{11} -0.171573 q^{13} -1.00000 q^{17} +1.00000 q^{19} +3.82843 q^{21} +9.24264 q^{23} -0.414214 q^{27} -5.82843 q^{29} +2.24264 q^{31} -3.41421 q^{33} -8.48528 q^{37} -0.414214 q^{39} +4.24264 q^{41} +10.2426 q^{43} -4.48528 q^{49} -2.41421 q^{51} +11.4853 q^{53} +2.41421 q^{57} +12.8995 q^{59} +5.75736 q^{61} +4.48528 q^{63} +13.2426 q^{67} +22.3137 q^{69} +10.5858 q^{71} +5.48528 q^{73} -2.24264 q^{77} -10.4853 q^{79} -9.48528 q^{81} -2.48528 q^{83} -14.0711 q^{87} -7.07107 q^{89} -0.272078 q^{91} +5.41421 q^{93} +11.6569 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 6 q^{7}+O(q^{10}) \) 2 * q + 2 * q^3 + 6 * q^7	\( 2 q + 2 q^{3} + 6 q^{7} - 6 q^{13} - 2 q^{17} + 2 q^{19} + 2 q^{21} + 10 q^{23} + 2 q^{27} - 6 q^{29} - 4 q^{31} - 4 q^{33} + 2 q^{39} + 12 q^{43} + 8 q^{49} - 2 q^{51} + 6 q^{53} + 2 q^{57} + 6 q^{59} + 20 q^{61} - 8 q^{63} + 18 q^{67} + 22 q^{69} + 24 q^{71} - 6 q^{73} + 4 q^{77} - 4 q^{79} - 2 q^{81} + 12 q^{83} - 14 q^{87} - 26 q^{91} + 8 q^{93} + 12 q^{97} - 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 6 * q^7 - 6 * q^13 - 2 * q^17 + 2 * q^19 + 2 * q^21 + 10 * q^23 + 2 * q^27 - 6 * q^29 - 4 * q^31 - 4 * q^33 + 2 * q^39 + 12 * q^43 + 8 * q^49 - 2 * q^51 + 6 * q^53 + 2 * q^57 + 6 * q^59 + 20 * q^61 - 8 * q^63 + 18 * q^67 + 22 * q^69 + 24 * q^71 - 6 * q^73 + 4 * q^77 - 4 * q^79 - 2 * q^81 + 12 * q^83 - 14 * q^87 - 26 * q^91 + 8 * q^93 + 12 * q^97 - 8 * q^99

Introduction

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