LMFDB - Embedded newform 7600.2.a.cd.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:	$1$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 190)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.363328$ 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)

$f(q)$	$=$	$q+1.36333 q^{3} +0.636672 q^{7} -1.14134 q^{9} +O(q^{10})$	$q+1.36333 q^{3} +0.636672 q^{7} -1.14134 q^{9} -3.50466 q^{11} +0.141336 q^{13} -2.14134 q^{17} -1.00000 q^{19} +0.867993 q^{21} +4.91934 q^{23} -5.64600 q^{27} +7.15066 q^{29} +7.78734 q^{31} -4.77801 q^{33} -3.27334 q^{37} +0.192688 q^{39} -4.23132 q^{41} +2.49534 q^{43} -10.2827 q^{47} -6.59465 q^{49} -2.91934 q^{51} -8.14134 q^{53} -1.36333 q^{57} +5.64600 q^{59} -6.49534 q^{61} -0.726656 q^{63} +8.37266 q^{67} +6.70668 q^{69} +8.95798 q^{71} +3.69735 q^{73} -2.23132 q^{77} +4.17997 q^{79} -4.27334 q^{81} -9.00933 q^{83} +9.74870 q^{87} -6.77801 q^{89} +0.0899847 q^{91} +10.6167 q^{93} -14.5653 q^{97} +4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} + 4 q^{7} + 5 q^{9}+O(q^{10}) $ 3 * q + 2 * q^3 + 4 * q^7 + 5 * q^9	$ 3 q + 2 q^{3} + 4 q^{7} + 5 q^{9} - 8 q^{13} + 2 q^{17} - 3 q^{19} - 10 q^{21} + 2 q^{27} - 8 q^{29} - 4 q^{31} - 8 q^{33} - 14 q^{37} - 10 q^{39} + 2 q^{41} + 18 q^{43} - 14 q^{47} - 3 q^{49} + 6 q^{51} - 16 q^{53} - 2 q^{57} - 2 q^{59} - 30 q^{61} + 2 q^{63} + 2 q^{67} - 22 q^{69} + 8 q^{71} - 10 q^{73} + 8 q^{77} - 17 q^{81} - 6 q^{83} + 6 q^{87} - 14 q^{89} - 6 q^{91} - 4 q^{93} - 10 q^{97} + 12 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 + 4 * q^7 + 5 * q^9 - 8 * q^13 + 2 * q^17 - 3 * q^19 - 10 * q^21 + 2 * q^27 - 8 * q^29 - 4 * q^31 - 8 * q^33 - 14 * q^37 - 10 * q^39 + 2 * q^41 + 18 * q^43 - 14 * q^47 - 3 * q^49 + 6 * q^51 - 16 * q^53 - 2 * q^57 - 2 * q^59 - 30 * q^61 + 2 * q^63 + 2 * q^67 - 22 * q^69 + 8 * q^71 - 10 * q^73 + 8 * q^77 - 17 * q^81 - 6 * q^83 + 6 * q^87 - 14 * q^89 - 6 * q^91 - 4 * q^93 - 10 * q^97 + 12 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.36333	0.787118	0.393559	−	0.919299i	$-0.371244\pi$
$3$	1.36333	0.787118	0.393559	+	0.919299i	$0.371244\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.636672	0.240639	0.120320	−	0.992735i	$-0.461608\pi$
$7$	0.636672	0.240639	0.120320	+	0.992735i	$0.461608\pi$
$8$	0	0
$9$	−1.14134	−0.380445
$10$	0	0
$11$	−3.50466	−1.05670	−0.528348	−	0.849028i	$-0.677188\pi$
$11$	−3.50466	−1.05670	−0.528348	+	0.849028i	$0.677188\pi$
$12$	0	0
$13$	0.141336	0.0391996	0.0195998	−	0.999808i	$-0.493761\pi$
$13$	0.141336	0.0391996	0.0195998	+	0.999808i	$0.493761\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.14134	−0.519350	−0.259675	−	0.965696i	$-0.583615\pi$
$17$	−2.14134	−0.519350	−0.259675	+	0.965696i	$0.583615\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0.867993	0.189412
$22$	0	0
$23$	4.91934	1.02575	0.512877	−	0.858462i	$-0.328580\pi$
$23$	4.91934	1.02575	0.512877	+	0.858462i	$0.328580\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.64600	−1.08657
$28$	0	0
$29$	7.15066	1.32785	0.663923	−	0.747801i	$-0.268890\pi$
$29$	7.15066	1.32785	0.663923	+	0.747801i	$0.268890\pi$
$30$	0	0
$31$	7.78734	1.39865	0.699323	−	0.714805i	$-0.253485\pi$
$31$	7.78734	1.39865	0.699323	+	0.714805i	$0.253485\pi$
$32$	0	0
$33$	−4.77801	−0.831744
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.27334	−0.538134	−0.269067	−	0.963121i	$-0.586715\pi$
$37$	−3.27334	−0.538134	−0.269067	+	0.963121i	$0.586715\pi$
$38$	0	0
$39$	0.192688	0.0308547
$40$	0	0
$41$	−4.23132	−0.660821	−0.330411	−	0.943837i	$-0.607187\pi$
$41$	−4.23132	−0.660821	−0.330411	+	0.943837i	$0.607187\pi$
$42$	0	0
$43$	2.49534	0.380535	0.190268	−	0.981732i	$-0.439064\pi$
$43$	2.49534	0.380535	0.190268	+	0.981732i	$0.439064\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.2827	−1.49988	−0.749941	−	0.661505i	$-0.769918\pi$
$47$	−10.2827	−1.49988	−0.749941	+	0.661505i	$0.769918\pi$
$48$	0	0
$49$	−6.59465	−0.942093
$50$	0	0
$51$	−2.91934	−0.408790
$52$	0	0
$53$	−8.14134	−1.11830	−0.559149	−	0.829067i	$-0.688872\pi$
$53$	−8.14134	−1.11830	−0.559149	+	0.829067i	$0.688872\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.36333	−0.180577
$58$	0	0
$59$	5.64600	0.735047	0.367523	−	0.930014i	$-0.380206\pi$
$59$	5.64600	0.735047	0.367523	+	0.930014i	$0.380206\pi$
$60$	0	0
$61$	−6.49534	−0.831643	−0.415821	−	0.909446i	$-0.636506\pi$
$61$	−6.49534	−0.831643	−0.415821	+	0.909446i	$0.636506\pi$
$62$	0	0
$63$	−0.726656	−0.0915501
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.37266	1.02288	0.511441	−	0.859318i	$-0.329112\pi$
$67$	8.37266	1.02288	0.511441	+	0.859318i	$0.329112\pi$
$68$	0	0
$69$	6.70668	0.807389
$70$	0	0
$71$	8.95798	1.06312	0.531558	−	0.847022i	$-0.321607\pi$
$71$	8.95798	1.06312	0.531558	+	0.847022i	$0.321607\pi$
$72$	0	0
$73$	3.69735	0.432742	0.216371	−	0.976311i	$-0.430578\pi$
$73$	3.69735	0.432742	0.216371	+	0.976311i	$0.430578\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.23132	−0.254283
$78$	0	0
$79$	4.17997	0.470283	0.235142	−	0.971961i	$-0.424445\pi$
$79$	4.17997	0.470283	0.235142	+	0.971961i	$0.424445\pi$
$80$	0	0
$81$	−4.27334	−0.474816
$82$	0	0
$83$	−9.00933	−0.988902	−0.494451	−	0.869205i	$-0.664631\pi$
$83$	−9.00933	−0.988902	−0.494451	+	0.869205i	$0.664631\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	9.74870	1.04517
$88$	0	0
$89$	−6.77801	−0.718467	−0.359234	−	0.933248i	$-0.616962\pi$
$89$	−6.77801	−0.718467	−0.359234	+	0.933248i	$0.616962\pi$
$90$	0	0
$91$	0.0899847	0.00943296
$92$	0	0
$93$	10.6167	1.10090
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−14.5653	−1.47889	−0.739443	−	0.673219i	$-0.764911\pi$
$97$	−14.5653	−1.47889	−0.739443	+	0.673219i	$0.764911\pi$
$98$	0	0
$99$	4.00000	0.402015
$100$	0	0
$101$	−16.6167	−1.65342	−0.826712	−	0.562626i	$-0.809791\pi$
$101$	−16.6167	−1.65342	−0.826712	+	0.562626i	$0.809791\pi$
$102$	0	0
$103$	9.06068	0.892775	0.446388	−	0.894840i	$-0.352710\pi$
$103$	9.06068	0.892775	0.446388	+	0.894840i	$0.352710\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.0899847	0.00869915	0.00434958	−	0.999991i	$-0.498615\pi$
$107$	0.0899847	0.00869915	0.00434958	+	0.999991i	$0.498615\pi$
$108$	0	0
$109$	−13.5946	−1.30213	−0.651066	−	0.759021i	$-0.725678\pi$
$109$	−13.5946	−1.30213	−0.651066	+	0.759021i	$0.725678\pi$
$110$	0	0
$111$	−4.46264	−0.423575
$112$	0	0
$113$	−11.5233	−1.08402	−0.542011	−	0.840371i	$-0.682337\pi$
$113$	−11.5233	−1.08402	−0.542011	+	0.840371i	$0.682337\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−0.161312	−0.0149133
$118$	0	0
$119$	−1.36333	−0.124976
$120$	0	0
$121$	1.28267	0.116607
$122$	0	0
$123$	−5.76868	−0.520144
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−3.29200	−0.292118	−0.146059	−	0.989276i	$-0.546659\pi$
$127$	−3.29200	−0.292118	−0.146059	+	0.989276i	$0.546659\pi$
$128$	0	0
$129$	3.40196	0.299526
$130$	0	0
$131$	−18.0187	−1.57430	−0.787149	−	0.616763i	$-0.788444\pi$
$131$	−18.0187	−1.57430	−0.787149	+	0.616763i	$0.788444\pi$
$132$	0	0
$133$	−0.636672	−0.0552064
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.4240	1.23233	0.616163	−	0.787619i	$-0.288686\pi$
$137$	14.4240	1.23233	0.616163	+	0.787619i	$0.288686\pi$
$138$	0	0
$139$	−15.4720	−1.31232	−0.656158	−	0.754624i	$-0.727819\pi$
$139$	−15.4720	−1.31232	−0.656158	+	0.754624i	$0.727819\pi$
$140$	0	0
$141$	−14.0187	−1.18058
$142$	0	0
$143$	−0.495336	−0.0414220
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−8.99067	−0.741538
$148$	0	0
$149$	−17.1893	−1.40820	−0.704101	−	0.710100i	$-0.748650\pi$
$149$	−17.1893	−1.40820	−0.704101	+	0.710100i	$0.748650\pi$
$150$	0	0
$151$	−3.29200	−0.267899	−0.133950	−	0.990988i	$-0.542766\pi$
$151$	−3.29200	−0.267899	−0.133950	+	0.990988i	$0.542766\pi$
$152$	0	0
$153$	2.44398	0.197584
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−15.1893	−1.21224	−0.606119	−	0.795374i	$-0.707274\pi$
$157$	−15.1893	−1.21224	−0.606119	+	0.795374i	$0.707274\pi$
$158$	0	0
$159$	−11.0993	−0.880233
$160$	0	0
$161$	3.13201	0.246837
$162$	0	0
$163$	−14.0700	−1.10205	−0.551024	−	0.834489i	$-0.685763\pi$
$163$	−14.0700	−1.10205	−0.551024	+	0.834489i	$0.685763\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−14.7967	−1.14500	−0.572500	−	0.819905i	$-0.694026\pi$
$167$	−14.7967	−1.14500	−0.572500	+	0.819905i	$0.694026\pi$
$168$	0	0
$169$	−12.9800	−0.998463
$170$	0	0
$171$	1.14134	0.0872802
$172$	0	0
$173$	17.2920	1.31469	0.657343	−	0.753591i	$-0.271680\pi$
$173$	17.2920	1.31469	0.657343	+	0.753591i	$0.271680\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	7.69735	0.578568
$178$	0	0
$179$	−17.7360	−1.32565	−0.662825	−	0.748774i	$-0.730643\pi$
$179$	−17.7360	−1.32565	−0.662825	+	0.748774i	$0.730643\pi$
$180$	0	0
$181$	6.17997	0.459354	0.229677	−	0.973267i	$-0.426233\pi$
$181$	6.17997	0.459354	0.229677	+	0.973267i	$0.426233\pi$
$182$	0	0
$183$	−8.85527	−0.654601
$184$	0	0
$185$	0	0
$186$	0	0
$187$	7.50466	0.548795
$188$	0	0
$189$	−3.59465	−0.261472
$190$	0	0
$191$	−14.6367	−1.05907	−0.529536	−	0.848287i	$-0.677634\pi$
$191$	−14.6367	−1.05907	−0.529536	+	0.848287i	$0.677634\pi$
$192$	0	0
$193$	−20.0187	−1.44097	−0.720487	−	0.693468i	$-0.756082\pi$
$193$	−20.0187	−1.44097	−0.720487	+	0.693468i	$0.756082\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.94865	0.708812	0.354406	−	0.935092i	$-0.384683\pi$
$197$	9.94865	0.708812	0.354406	+	0.935092i	$0.384683\pi$
$198$	0	0
$199$	−9.74870	−0.691067	−0.345534	−	0.938406i	$-0.612302\pi$
$199$	−9.74870	−0.691067	−0.345534	+	0.938406i	$0.612302\pi$
$200$	0	0
$201$	11.4147	0.805129
$202$	0	0
$203$	4.55263	0.319532
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−5.61462	−0.390243
$208$	0	0
$209$	3.50466	0.242423
$210$	0	0
$211$	20.7580	1.42904	0.714521	−	0.699614i	$-0.246645\pi$
$211$	20.7580	1.42904	0.714521	+	0.699614i	$0.246645\pi$
$212$	0	0
$213$	12.2127	0.836798
$214$	0	0
$215$	0	0
$216$	0	0
$217$	4.95798	0.336569
$218$	0	0
$219$	5.04070	0.340619
$220$	0	0
$221$	−0.302648	−0.0203583
$222$	0	0
$223$	−10.7267	−0.718310	−0.359155	−	0.933278i	$-0.616935\pi$
$223$	−10.7267	−0.718310	−0.359155	+	0.933278i	$0.616935\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.5526	0.833147	0.416574	−	0.909102i	$-0.363231\pi$
$227$	12.5526	0.833147	0.416574	+	0.909102i	$0.363231\pi$
$228$	0	0
$229$	25.4720	1.68324	0.841618	−	0.540074i	$-0.181604\pi$
$229$	25.4720	1.68324	0.841618	+	0.540074i	$0.181604\pi$
$230$	0	0
$231$	−3.04202	−0.200150
$232$	0	0
$233$	3.11203	0.203876	0.101938	−	0.994791i	$-0.467496\pi$
$233$	3.11203	0.203876	0.101938	+	0.994791i	$0.467496\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	5.69867	0.370168
$238$	0	0
$239$	1.54330	0.0998276	0.0499138	−	0.998754i	$-0.484105\pi$
$239$	1.54330	0.0998276	0.0499138	+	0.998754i	$0.484105\pi$
$240$	0	0
$241$	10.2827	0.662365	0.331183	−	0.943567i	$-0.392552\pi$
$241$	10.2827	0.662365	0.331183	+	0.943567i	$0.392552\pi$
$242$	0	0
$243$	11.1120	0.712837
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−0.141336	−0.00899300
$248$	0	0
$249$	−12.2827	−0.778383
$250$	0	0
$251$	2.51399	0.158682	0.0793409	−	0.996848i	$-0.474718\pi$
$251$	2.51399	0.158682	0.0793409	+	0.996848i	$0.474718\pi$
$252$	0	0
$253$	−17.2406	−1.08391
$254$	0	0
$255$	0	0
$256$	0	0
$257$	24.7967	1.54677	0.773387	−	0.633934i	$-0.218561\pi$
$257$	24.7967	1.54677	0.773387	+	0.633934i	$0.218561\pi$
$258$	0	0
$259$	−2.08405	−0.129496
$260$	0	0
$261$	−8.16131	−0.505173
$262$	0	0
$263$	−22.5653	−1.39144	−0.695719	−	0.718314i	$-0.744914\pi$
$263$	−22.5653	−1.39144	−0.695719	+	0.718314i	$0.744914\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−9.24065	−0.565519
$268$	0	0
$269$	−26.5653	−1.61972	−0.809859	−	0.586625i	$-0.800456\pi$
$269$	−26.5653	−1.61972	−0.809859	+	0.586625i	$0.800456\pi$
$270$	0	0
$271$	24.9380	1.51488	0.757438	−	0.652907i	$-0.226451\pi$
$271$	24.9380	1.51488	0.757438	+	0.652907i	$0.226451\pi$
$272$	0	0
$273$	0.122679	0.00742485
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−18.5467	−1.11436	−0.557181	−	0.830391i	$-0.688117\pi$
$277$	−18.5467	−1.11436	−0.557181	+	0.830391i	$0.688117\pi$
$278$	0	0
$279$	−8.88797	−0.532109
$280$	0	0
$281$	24.7967	1.47925	0.739623	−	0.673022i	$-0.235004\pi$
$281$	24.7967	1.47925	0.739623	+	0.673022i	$0.235004\pi$
$282$	0	0
$283$	13.5747	0.806931	0.403465	−	0.914995i	$-0.367806\pi$
$283$	13.5747	0.806931	0.403465	+	0.914995i	$0.367806\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.69396	−0.159020
$288$	0	0
$289$	−12.4147	−0.730275
$290$	0	0
$291$	−19.8573	−1.16406
$292$	0	0
$293$	15.6133	0.912139	0.456070	−	0.889944i	$-0.349257\pi$
$293$	15.6133	0.912139	0.456070	+	0.889944i	$0.349257\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	19.7873	1.14818
$298$	0	0
$299$	0.695281	0.0402091
$300$	0	0
$301$	1.58871	0.0915717
$302$	0	0
$303$	−22.6540	−1.30144
$304$	0	0
$305$	0	0
$306$	0	0
$307$	34.0187	1.94155	0.970774	−	0.239997i	$-0.0771464\pi$
$307$	34.0187	1.94155	0.970774	+	0.239997i	$0.0771464\pi$
$308$	0	0
$309$	12.3527	0.702719
$310$	0	0
$311$	−8.93800	−0.506828	−0.253414	−	0.967358i	$-0.581553\pi$
$311$	−8.93800	−0.506828	−0.253414	+	0.967358i	$0.581553\pi$
$312$	0	0
$313$	18.4240	1.04139	0.520693	−	0.853744i	$-0.325674\pi$
$313$	18.4240	1.04139	0.520693	+	0.853744i	$0.325674\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−33.5547	−1.88462	−0.942310	−	0.334742i	$-0.891351\pi$
$317$	−33.5547	−1.88462	−0.942310	+	0.334742i	$0.891351\pi$
$318$	0	0
$319$	−25.0607	−1.40313
$320$	0	0
$321$	0.122679	0.00684726
$322$	0	0
$323$	2.14134	0.119147
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−18.5340	−1.02493
$328$	0	0
$329$	−6.54669	−0.360931
$330$	0	0
$331$	−2.25130	−0.123742	−0.0618712	−	0.998084i	$-0.519707\pi$
$331$	−2.25130	−0.123742	−0.0618712	+	0.998084i	$0.519707\pi$
$332$	0	0
$333$	3.73599	0.204731
$334$	0	0
$335$	0	0
$336$	0	0
$337$	21.3620	1.16366	0.581831	−	0.813309i	$-0.302336\pi$
$337$	21.3620	1.16366	0.581831	+	0.813309i	$0.302336\pi$
$338$	0	0
$339$	−15.7101	−0.853254
$340$	0	0
$341$	−27.2920	−1.47794
$342$	0	0
$343$	−8.65533	−0.467344
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11.5560	0.620359	0.310180	−	0.950678i	$-0.399611\pi$
$347$	11.5560	0.620359	0.310180	+	0.950678i	$0.399611\pi$
$348$	0	0
$349$	−17.1120	−0.915986	−0.457993	−	0.888956i	$-0.651432\pi$
$349$	−17.1120	−0.915986	−0.457993	+	0.888956i	$0.651432\pi$
$350$	0	0
$351$	−0.797984	−0.0425932
$352$	0	0
$353$	−11.6974	−0.622587	−0.311294	−	0.950314i	$-0.600762\pi$
$353$	−11.6974	−0.622587	−0.311294	+	0.950314i	$0.600762\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1.85866	−0.0983709
$358$	0	0
$359$	4.47536	0.236200	0.118100	−	0.993002i	$-0.462320\pi$
$359$	4.47536	0.236200	0.118100	+	0.993002i	$0.462320\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	1.74870	0.0917831
$364$	0	0
$365$	0	0
$366$	0	0
$367$	18.7453	0.978497	0.489249	−	0.872144i	$-0.337271\pi$
$367$	18.7453	0.978497	0.489249	+	0.872144i	$0.337271\pi$
$368$	0	0
$369$	4.82936	0.251406
$370$	0	0
$371$	−5.18336	−0.269107
$372$	0	0
$373$	−1.69735	−0.0878855	−0.0439428	−	0.999034i	$-0.513992\pi$
$373$	−1.69735	−0.0878855	−0.0439428	+	0.999034i	$0.513992\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.01065	0.0520510
$378$	0	0
$379$	−2.63667	−0.135437	−0.0677184	−	0.997704i	$-0.521572\pi$
$379$	−2.63667	−0.135437	−0.0677184	+	0.997704i	$0.521572\pi$
$380$	0	0
$381$	−4.48808	−0.229931
$382$	0	0
$383$	12.4953	0.638482	0.319241	−	0.947674i	$-0.396572\pi$
$383$	12.4953	0.638482	0.319241	+	0.947674i	$0.396572\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.84802	−0.144773
$388$	0	0
$389$	−4.51399	−0.228869	−0.114434	−	0.993431i	$-0.536506\pi$
$389$	−4.51399	−0.228869	−0.114434	+	0.993431i	$0.536506\pi$
$390$	0	0
$391$	−10.5340	−0.532726
$392$	0	0
$393$	−24.5653	−1.23916
$394$	0	0
$395$	0	0
$396$	0	0
$397$	35.6774	1.79060	0.895298	−	0.445468i	$-0.146963\pi$
$397$	35.6774	1.79060	0.895298	+	0.445468i	$0.146963\pi$
$398$	0	0
$399$	−0.867993	−0.0434540
$400$	0	0
$401$	15.3434	0.766210	0.383105	−	0.923705i	$-0.374855\pi$
$401$	15.3434	0.766210	0.383105	+	0.923705i	$0.374855\pi$
$402$	0	0
$403$	1.10063	0.0548264
$404$	0	0
$405$	0	0
$406$	0	0
$407$	11.4720	0.568644
$408$	0	0
$409$	29.3620	1.45186	0.725929	−	0.687770i	$-0.241410\pi$
$409$	29.3620	1.45186	0.725929	+	0.687770i	$0.241410\pi$
$410$	0	0
$411$	19.6647	0.969986
$412$	0	0
$413$	3.59465	0.176881
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−21.0934	−1.03295
$418$	0	0
$419$	−25.1379	−1.22807	−0.614035	−	0.789279i	$-0.710454\pi$
$419$	−25.1379	−1.22807	−0.614035	+	0.789279i	$0.710454\pi$
$420$	0	0
$421$	14.5454	0.708898	0.354449	−	0.935075i	$-0.384668\pi$
$421$	14.5454	0.708898	0.354449	+	0.935075i	$0.384668\pi$
$422$	0	0
$423$	11.7360	0.570623
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−4.13540	−0.200126
$428$	0	0
$429$	−0.675305	−0.0326040
$430$	0	0
$431$	−19.4020	−0.934560	−0.467280	−	0.884109i	$-0.654766\pi$
$431$	−19.4020	−0.934560	−0.467280	+	0.884109i	$0.654766\pi$
$432$	0	0
$433$	−5.50466	−0.264537	−0.132269	−	0.991214i	$-0.542226\pi$
$433$	−5.50466	−0.264537	−0.132269	+	0.991214i	$0.542226\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.91934	−0.235324
$438$	0	0
$439$	−12.2500	−0.584660	−0.292330	−	0.956318i	$-0.594430\pi$
$439$	−12.2500	−0.584660	−0.292330	+	0.956318i	$0.594430\pi$
$440$	0	0
$441$	7.52671	0.358415
$442$	0	0
$443$	31.6006	1.50139	0.750695	−	0.660649i	$-0.229719\pi$
$443$	31.6006	1.50139	0.750695	+	0.660649i	$0.229719\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−23.4347	−1.10842
$448$	0	0
$449$	−36.0187	−1.69983	−0.849913	−	0.526923i	$-0.823345\pi$
$449$	−36.0187	−1.69983	−0.849913	+	0.526923i	$0.823345\pi$
$450$	0	0
$451$	14.8294	0.698287
$452$	0	0
$453$	−4.48808	−0.210868
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−22.1413	−1.03573	−0.517864	−	0.855463i	$-0.673273\pi$
$457$	−22.1413	−1.03573	−0.517864	+	0.855463i	$0.673273\pi$
$458$	0	0
$459$	12.0900	0.564312
$460$	0	0
$461$	−2.31537	−0.107837	−0.0539187	−	0.998545i	$-0.517171\pi$
$461$	−2.31537	−0.107837	−0.0539187	+	0.998545i	$0.517171\pi$
$462$	0	0
$463$	15.8387	0.736086	0.368043	−	0.929809i	$-0.380028\pi$
$463$	15.8387	0.736086	0.368043	+	0.929809i	$0.380028\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	23.1379	1.07070	0.535348	−	0.844631i	$-0.320180\pi$
$467$	23.1379	1.07070	0.535348	+	0.844631i	$0.320180\pi$
$468$	0	0
$469$	5.33063	0.246146
$470$	0	0
$471$	−20.7080	−0.954174
$472$	0	0
$473$	−8.74531	−0.402110
$474$	0	0
$475$	0	0
$476$	0	0
$477$	9.29200	0.425451
$478$	0	0
$479$	10.1214	0.462457	0.231228	−	0.972900i	$-0.425726\pi$
$479$	10.1214	0.462457	0.231228	+	0.972900i	$0.425726\pi$
$480$	0	0
$481$	−0.462642	−0.0210946
$482$	0	0
$483$	4.26995	0.194290
$484$	0	0
$485$	0	0
$486$	0	0
$487$	20.3854	0.923750	0.461875	−	0.886945i	$-0.347177\pi$
$487$	20.3854	0.923750	0.461875	+	0.886945i	$0.347177\pi$
$488$	0	0
$489$	−19.1820	−0.867442
$490$	0	0
$491$	7.78734	0.351438	0.175719	−	0.984440i	$-0.443775\pi$
$491$	7.78734	0.351438	0.175719	+	0.984440i	$0.443775\pi$
$492$	0	0
$493$	−15.3120	−0.689617
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.70329	0.255828
$498$	0	0
$499$	−4.31537	−0.193182	−0.0965912	−	0.995324i	$-0.530794\pi$
$499$	−4.31537	−0.193182	−0.0965912	+	0.995324i	$0.530794\pi$
$500$	0	0
$501$	−20.1727	−0.901250
$502$	0	0
$503$	18.5526	0.827221	0.413610	−	0.910454i	$-0.364268\pi$
$503$	18.5526	0.827221	0.413610	+	0.910454i	$0.364268\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−17.6960	−0.785908
$508$	0	0
$509$	−7.73599	−0.342892	−0.171446	−	0.985194i	$-0.554844\pi$
$509$	−7.73599	−0.342892	−0.171446	+	0.985194i	$0.554844\pi$
$510$	0	0
$511$	2.35400	0.104135
$512$	0	0
$513$	5.64600	0.249277
$514$	0	0
$515$	0	0
$516$	0	0
$517$	36.0373	1.58492
$518$	0	0
$519$	23.5747	1.03481
$520$	0	0
$521$	15.2080	0.666273	0.333136	−	0.942879i	$-0.391893\pi$
$521$	15.2080	0.666273	0.333136	+	0.942879i	$0.391893\pi$
$522$	0	0
$523$	−18.2113	−0.796327	−0.398163	−	0.917315i	$-0.630352\pi$
$523$	−18.2113	−0.796327	−0.398163	+	0.917315i	$0.630352\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−16.6753	−0.726388
$528$	0	0
$529$	1.19995	0.0521715
$530$	0	0
$531$	−6.44398	−0.279645
$532$	0	0
$533$	−0.598038	−0.0259039
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−24.1800	−1.04344
$538$	0	0
$539$	23.1120	0.995506
$540$	0	0
$541$	16.5140	0.709992	0.354996	−	0.934868i	$-0.384482\pi$
$541$	16.5140	0.709992	0.354996	+	0.934868i	$0.384482\pi$
$542$	0	0
$543$	8.42533	0.361565
$544$	0	0
$545$	0	0
$546$	0	0
$547$	16.2827	0.696197	0.348098	−	0.937458i	$-0.386828\pi$
$547$	16.2827	0.696197	0.348098	+	0.937458i	$0.386828\pi$
$548$	0	0
$549$	7.41336	0.316395
$550$	0	0
$551$	−7.15066	−0.304629
$552$	0	0
$553$	2.66127	0.113169
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−37.4533	−1.58695	−0.793474	−	0.608604i	$-0.791730\pi$
$557$	−37.4533	−1.58695	−0.793474	+	0.608604i	$0.791730\pi$
$558$	0	0
$559$	0.352681	0.0149168
$560$	0	0
$561$	10.2313	0.431967
$562$	0	0
$563$	−29.1307	−1.22771	−0.613856	−	0.789418i	$-0.710382\pi$
$563$	−29.1307	−1.22771	−0.613856	+	0.789418i	$0.710382\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2.72072	−0.114259
$568$	0	0
$569$	14.8480	0.622461	0.311231	−	0.950334i	$-0.399259\pi$
$569$	14.8480	0.622461	0.311231	+	0.950334i	$0.399259\pi$
$570$	0	0
$571$	−41.9087	−1.75382	−0.876912	−	0.480651i	$-0.840401\pi$
$571$	−41.9087	−1.75382	−0.876912	+	0.480651i	$0.840401\pi$
$572$	0	0
$573$	−19.9546	−0.833615
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−16.4427	−0.684517	−0.342259	−	0.939606i	$-0.611192\pi$
$577$	−16.4427	−0.684517	−0.342259	+	0.939606i	$0.611192\pi$
$578$	0	0
$579$	−27.2920	−1.13422
$580$	0	0
$581$	−5.73599	−0.237969
$582$	0	0
$583$	28.5327	1.18170
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−42.5327	−1.75551	−0.877755	−	0.479109i	$-0.840960\pi$
$587$	−42.5327	−1.75551	−0.877755	+	0.479109i	$0.840960\pi$
$588$	0	0
$589$	−7.78734	−0.320872
$590$	0	0
$591$	13.5633	0.557919
$592$	0	0
$593$	−3.92273	−0.161087	−0.0805437	−	0.996751i	$-0.525666\pi$
$593$	−3.92273	−0.161087	−0.0805437	+	0.996751i	$0.525666\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−13.2907	−0.543951
$598$	0	0
$599$	10.7594	0.439615	0.219808	−	0.975543i	$-0.429457\pi$
$599$	10.7594	0.439615	0.219808	+	0.975543i	$0.429457\pi$
$600$	0	0
$601$	25.2220	1.02883	0.514413	−	0.857542i	$-0.328010\pi$
$601$	25.2220	1.02883	0.514413	+	0.857542i	$0.328010\pi$
$602$	0	0
$603$	−9.55602	−0.389151
$604$	0	0
$605$	0	0
$606$	0	0
$607$	39.2920	1.59481	0.797407	−	0.603442i	$-0.206205\pi$
$607$	39.2920	1.59481	0.797407	+	0.603442i	$0.206205\pi$
$608$	0	0
$609$	6.20672	0.251509
$610$	0	0
$611$	−1.45331	−0.0587947
$612$	0	0
$613$	−9.80599	−0.396060	−0.198030	−	0.980196i	$-0.563454\pi$
$613$	−9.80599	−0.396060	−0.198030	+	0.980196i	$0.563454\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	35.0093	1.40942	0.704711	−	0.709494i	$-0.251077\pi$
$617$	35.0093	1.40942	0.704711	+	0.709494i	$0.251077\pi$
$618$	0	0
$619$	−13.4206	−0.539420	−0.269710	−	0.962942i	$-0.586928\pi$
$619$	−13.4206	−0.539420	−0.269710	+	0.962942i	$0.586928\pi$
$620$	0	0
$621$	−27.7746	−1.11456
$622$	0	0
$623$	−4.31537	−0.172891
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4.77801	0.190815
$628$	0	0
$629$	7.00933	0.279480
$630$	0	0
$631$	−40.5254	−1.61329	−0.806645	−	0.591036i	$-0.798719\pi$
$631$	−40.5254	−1.61329	−0.806645	+	0.591036i	$0.798719\pi$
$632$	0	0
$633$	28.3000	1.12482
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−0.932062	−0.0369296
$638$	0	0
$639$	−10.2241	−0.404458
$640$	0	0
$641$	−26.0700	−1.02970	−0.514852	−	0.857279i	$-0.672153\pi$
$641$	−26.0700	−1.02970	−0.514852	+	0.857279i	$0.672153\pi$
$642$	0	0
$643$	30.1400	1.18861	0.594303	−	0.804241i	$-0.297428\pi$
$643$	30.1400	1.18861	0.594303	+	0.804241i	$0.297428\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−20.1086	−0.790552	−0.395276	−	0.918562i	$-0.629351\pi$
$647$	−20.1086	−0.790552	−0.395276	+	0.918562i	$0.629351\pi$
$648$	0	0
$649$	−19.7873	−0.776721
$650$	0	0
$651$	6.75935	0.264920
$652$	0	0
$653$	28.0373	1.09718	0.548592	−	0.836090i	$-0.315164\pi$
$653$	28.0373	1.09718	0.548592	+	0.836090i	$0.315164\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−4.21992	−0.164635
$658$	0	0
$659$	−4.90069	−0.190904	−0.0954518	−	0.995434i	$-0.530430\pi$
$659$	−4.90069	−0.190904	−0.0954518	+	0.995434i	$0.530430\pi$
$660$	0	0
$661$	−8.03863	−0.312667	−0.156333	−	0.987704i	$-0.549967\pi$
$661$	−8.03863	−0.312667	−0.156333	+	0.987704i	$0.549967\pi$
$662$	0	0
$663$	−0.412609	−0.0160244
$664$	0	0
$665$	0	0
$666$	0	0
$667$	35.1766	1.36204
$668$	0	0
$669$	−14.6240	−0.565395
$670$	0	0
$671$	22.7640	0.878793
$672$	0	0
$673$	4.82936	0.186158	0.0930791	−	0.995659i	$-0.470329\pi$
$673$	4.82936	0.186158	0.0930791	+	0.995659i	$0.470329\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−12.8094	−0.492305	−0.246152	−	0.969231i	$-0.579166\pi$
$677$	−12.8094	−0.492305	−0.246152	+	0.969231i	$0.579166\pi$
$678$	0	0
$679$	−9.27334	−0.355878
$680$	0	0
$681$	17.1133	0.655785
$682$	0	0
$683$	37.1307	1.42077	0.710383	−	0.703815i	$-0.248522\pi$
$683$	37.1307	1.42077	0.710383	+	0.703815i	$0.248522\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	34.7267	1.32490
$688$	0	0
$689$	−1.15066	−0.0438368
$690$	0	0
$691$	18.1986	0.692308	0.346154	−	0.938178i	$-0.387487\pi$
$691$	18.1986	0.692308	0.346154	+	0.938178i	$0.387487\pi$
$692$	0	0
$693$	2.54669	0.0967406
$694$	0	0
$695$	0	0
$696$	0	0
$697$	9.06068	0.343198
$698$	0	0
$699$	4.24272	0.160474
$700$	0	0
$701$	−26.2827	−0.992683	−0.496341	−	0.868127i	$-0.665324\pi$
$701$	−26.2827	−0.992683	−0.496341	+	0.868127i	$0.665324\pi$
$702$	0	0
$703$	3.27334	0.123456
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−10.5794	−0.397879
$708$	0	0
$709$	−14.9253	−0.560531	−0.280265	−	0.959923i	$-0.590422\pi$
$709$	−14.9253	−0.560531	−0.280265	+	0.959923i	$0.590422\pi$
$710$	0	0
$711$	−4.77075	−0.178917
$712$	0	0
$713$	38.3086	1.43467
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.10402	0.0785761
$718$	0	0
$719$	−32.3327	−1.20581	−0.602903	−	0.797814i	$-0.705989\pi$
$719$	−32.3327	−1.20581	−0.602903	+	0.797814i	$0.705989\pi$
$720$	0	0
$721$	5.76868	0.214837
$722$	0	0
$723$	14.0187	0.521359
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−42.0246	−1.55861	−0.779303	−	0.626647i	$-0.784427\pi$
$727$	−42.0246	−1.55861	−0.779303	+	0.626647i	$0.784427\pi$
$728$	0	0
$729$	27.9694	1.03590
$730$	0	0
$731$	−5.34335	−0.197631
$732$	0	0
$733$	−26.5840	−0.981903	−0.490951	−	0.871187i	$-0.663351\pi$
$733$	−26.5840	−0.981903	−0.490951	+	0.871187i	$0.663351\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−29.3434	−1.08088
$738$	0	0
$739$	−8.14728	−0.299702	−0.149851	−	0.988709i	$-0.547879\pi$
$739$	−8.14728	−0.299702	−0.149851	+	0.988709i	$0.547879\pi$
$740$	0	0
$741$	−0.192688	−0.00707855
$742$	0	0
$743$	35.8247	1.31428	0.657139	−	0.753769i	$-0.271766\pi$
$743$	35.8247	1.31428	0.657139	+	0.753769i	$0.271766\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	10.2827	0.376223
$748$	0	0
$749$	0.0572907	0.00209336
$750$	0	0
$751$	14.8994	0.543686	0.271843	−	0.962342i	$-0.412367\pi$
$751$	14.8994	0.543686	0.271843	+	0.962342i	$0.412367\pi$
$752$	0	0
$753$	3.42740	0.124901
$754$	0	0
$755$	0	0
$756$	0	0
$757$	47.7920	1.73703	0.868514	−	0.495664i	$-0.165075\pi$
$757$	47.7920	1.73703	0.868514	+	0.495664i	$0.165075\pi$
$758$	0	0
$759$	−23.5047	−0.853165
$760$	0	0
$761$	−38.9053	−1.41032	−0.705158	−	0.709050i	$-0.749124\pi$
$761$	−38.9053	−1.41032	−0.705158	+	0.709050i	$0.749124\pi$
$762$	0	0
$763$	−8.65533	−0.313344
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0.797984	0.0288135
$768$	0	0
$769$	8.74663	0.315412	0.157706	−	0.987486i	$-0.449590\pi$
$769$	8.74663	0.315412	0.157706	+	0.987486i	$0.449590\pi$
$770$	0	0
$771$	33.8060	1.21749
$772$	0	0
$773$	23.4707	0.844181	0.422090	−	0.906554i	$-0.361297\pi$
$773$	23.4707	0.844181	0.422090	+	0.906554i	$0.361297\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−2.84124	−0.101929
$778$	0	0
$779$	4.23132	0.151603
$780$	0	0
$781$	−31.3947	−1.12339
$782$	0	0
$783$	−40.3727	−1.44280
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.26063	0.0449364	0.0224682	−	0.999748i	$-0.492848\pi$
$787$	1.26063	0.0449364	0.0224682	+	0.999748i	$0.492848\pi$
$788$	0	0
$789$	−30.7640	−1.09523
$790$	0	0
$791$	−7.33657	−0.260859
$792$	0	0
$793$	−0.918026	−0.0326000
$794$	0	0
$795$	0	0
$796$	0	0
$797$	38.6481	1.36898	0.684492	−	0.729020i	$-0.260024\pi$
$797$	38.6481	1.36898	0.684492	+	0.729020i	$0.260024\pi$
$798$	0	0
$799$	22.0187	0.778964
$800$	0	0
$801$	7.73599	0.273338
$802$	0	0
$803$	−12.9580	−0.457277
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−36.2173	−1.27491
$808$	0	0
$809$	51.6506	1.81594	0.907970	−	0.419036i	$-0.137632\pi$
$809$	51.6506	1.81594	0.907970	+	0.419036i	$0.137632\pi$
$810$	0	0
$811$	−8.19269	−0.287684	−0.143842	−	0.989601i	$-0.545946\pi$
$811$	−8.19269	−0.287684	−0.143842	+	0.989601i	$0.545946\pi$
$812$	0	0
$813$	33.9987	1.19239
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−2.49534	−0.0873007
$818$	0	0
$819$	−0.102703	−0.00358873
$820$	0	0
$821$	−49.9600	−1.74362	−0.871809	−	0.489846i	$-0.837053\pi$
$821$	−49.9600	−1.74362	−0.871809	+	0.489846i	$0.837053\pi$
$822$	0	0
$823$	−16.8421	−0.587078	−0.293539	−	0.955947i	$-0.594833\pi$
$823$	−16.8421	−0.587078	−0.293539	+	0.955947i	$0.594833\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−41.7487	−1.45174	−0.725872	−	0.687829i	$-0.758564\pi$
$827$	−41.7487	−1.45174	−0.725872	+	0.687829i	$0.758564\pi$
$828$	0	0
$829$	−5.98002	−0.207695	−0.103847	−	0.994593i	$-0.533115\pi$
$829$	−5.98002	−0.207695	−0.103847	+	0.994593i	$0.533115\pi$
$830$	0	0
$831$	−25.2852	−0.877135
$832$	0	0
$833$	14.1214	0.489276
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−43.9673	−1.51973
$838$	0	0
$839$	−32.0373	−1.10605	−0.553025	−	0.833164i	$-0.686527\pi$
$839$	−32.0373	−1.10605	−0.553025	+	0.833164i	$0.686527\pi$
$840$	0	0
$841$	22.1320	0.763173
$842$	0	0
$843$	33.8060	1.16434
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0.816641	0.0280601
$848$	0	0
$849$	18.5067	0.635150
$850$	0	0
$851$	−16.1027	−0.551994
$852$	0	0
$853$	40.8480	1.39861	0.699305	−	0.714824i	$-0.253493\pi$
$853$	40.8480	1.39861	0.699305	+	0.714824i	$0.253493\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−28.8667	−0.986067	−0.493033	−	0.870010i	$-0.664112\pi$
$857$	−28.8667	−0.986067	−0.493033	+	0.870010i	$0.664112\pi$
$858$	0	0
$859$	11.5047	0.392534	0.196267	−	0.980550i	$-0.437118\pi$
$859$	11.5047	0.392534	0.196267	+	0.980550i	$0.437118\pi$
$860$	0	0
$861$	−3.67276	−0.125167
$862$	0	0
$863$	−31.6074	−1.07593	−0.537964	−	0.842968i	$-0.680806\pi$
$863$	−31.6074	−1.07593	−0.537964	+	0.842968i	$0.680806\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−16.9253	−0.574813
$868$	0	0
$869$	−14.6494	−0.496947
$870$	0	0
$871$	1.18336	0.0400966
$872$	0	0
$873$	16.6240	0.562636
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−12.0641	−0.407375	−0.203687	−	0.979036i	$-0.565293\pi$
$877$	−12.0641	−0.407375	−0.203687	+	0.979036i	$0.565293\pi$
$878$	0	0
$879$	21.2861	0.717961
$880$	0	0
$881$	−22.8480	−0.769769	−0.384885	−	0.922965i	$-0.625759\pi$
$881$	−22.8480	−0.769769	−0.384885	+	0.922965i	$0.625759\pi$
$882$	0	0
$883$	−34.1773	−1.15016	−0.575079	−	0.818098i	$-0.695029\pi$
$883$	−34.1773	−1.15016	−0.575079	+	0.818098i	$0.695029\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	26.6167	0.893701	0.446851	−	0.894609i	$-0.352546\pi$
$887$	26.6167	0.893701	0.446851	+	0.894609i	$0.352546\pi$
$888$	0	0
$889$	−2.09592	−0.0702950
$890$	0	0
$891$	14.9766	0.501736
$892$	0	0
$893$	10.2827	0.344097
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0.947896	0.0316493
$898$	0	0
$899$	55.6846	1.85719
$900$	0	0
$901$	17.4333	0.580789
$902$	0	0
$903$	2.16593	0.0720777
$904$	0	0
$905$	0	0
$906$	0	0
$907$	13.1434	0.436420	0.218210	−	0.975902i	$-0.429978\pi$
$907$	13.1434	0.436420	0.218210	+	0.975902i	$0.429978\pi$
$908$	0	0
$909$	18.9652	0.629037
$910$	0	0
$911$	32.9253	1.09086	0.545432	−	0.838155i	$-0.316366\pi$
$911$	32.9253	1.09086	0.545432	+	0.838155i	$0.316366\pi$
$912$	0	0
$913$	31.5747	1.04497
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−11.4720	−0.378838
$918$	0	0
$919$	24.1273	0.795886	0.397943	−	0.917410i	$-0.369724\pi$
$919$	24.1273	0.795886	0.397943	+	0.917410i	$0.369724\pi$
$920$	0	0
$921$	46.3786	1.52823
$922$	0	0
$923$	1.26609	0.0416737
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−10.3413	−0.339652
$928$	0	0
$929$	−15.9359	−0.522841	−0.261420	−	0.965225i	$-0.584191\pi$
$929$	−15.9359	−0.522841	−0.261420	+	0.965225i	$0.584191\pi$
$930$	0	0
$931$	6.59465	0.216131
$932$	0	0
$933$	−12.1854	−0.398933
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−23.5547	−0.769498	−0.384749	−	0.923021i	$-0.625712\pi$
$937$	−23.5547	−0.769498	−0.384749	+	0.923021i	$0.625712\pi$
$938$	0	0
$939$	25.1180	0.819694
$940$	0	0
$941$	−25.6974	−0.837710	−0.418855	−	0.908053i	$-0.637568\pi$
$941$	−25.6974	−0.837710	−0.418855	+	0.908053i	$0.637568\pi$
$942$	0	0
$943$	−20.8153	−0.677840
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−16.3013	−0.529722	−0.264861	−	0.964287i	$-0.585326\pi$
$947$	−16.3013	−0.529722	−0.264861	+	0.964287i	$0.585326\pi$
$948$	0	0
$949$	0.522569	0.0169633
$950$	0	0
$951$	−45.7461	−1.48342
$952$	0	0
$953$	−10.2754	−0.332853	−0.166427	−	0.986054i	$-0.553223\pi$
$953$	−10.2754	−0.332853	−0.166427	+	0.986054i	$0.553223\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−34.1659	−1.10443
$958$	0	0
$959$	9.18336	0.296546
$960$	0	0
$961$	29.6426	0.956213
$962$	0	0
$963$	−0.102703	−0.00330955
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−53.2920	−1.71376	−0.856878	−	0.515520i	$-0.827599\pi$
$967$	−53.2920	−1.71376	−0.856878	+	0.515520i	$0.827599\pi$
$968$	0	0
$969$	2.91934	0.0937828
$970$	0	0
$971$	−2.54669	−0.0817271	−0.0408635	−	0.999165i	$-0.513011\pi$
$971$	−2.54669	−0.0817271	−0.0408635	+	0.999165i	$0.513011\pi$
$972$	0	0
$973$	−9.85057	−0.315795
$974$	0	0
$975$	0	0
$976$	0	0
$977$	49.2966	1.57714	0.788569	−	0.614946i	$-0.210822\pi$
$977$	49.2966	1.57714	0.788569	+	0.614946i	$0.210822\pi$
$978$	0	0
$979$	23.7546	0.759202
$980$	0	0
$981$	15.5161	0.495390
$982$	0	0
$983$	−36.7453	−1.17199	−0.585997	−	0.810313i	$-0.699297\pi$
$983$	−36.7453	−1.17199	−0.585997	+	0.810313i	$0.699297\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−8.92528	−0.284095
$988$	0	0
$989$	12.2754	0.390335
$990$	0	0
$991$	35.5674	1.12984	0.564918	−	0.825147i	$-0.308908\pi$
$991$	35.5674	1.12984	0.564918	+	0.825147i	$0.308908\pi$
$992$	0	0
$993$	−3.06926	−0.0973999
$994$	0	0
$995$	0	0
$996$	0	0
$997$	48.4299	1.53379	0.766896	−	0.641771i	$-0.221800\pi$
$997$	48.4299	1.53379	0.766896	+	0.641771i	$0.221800\pi$
$998$	0	0
$999$	18.4813	0.584722

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.cd.1.2		3
4.3	odd	2		950.2.a.i.1.2		3
5.2	odd	4		1520.2.d.j.609.3		6
5.3	odd	4		1520.2.d.j.609.4		6
5.4	even	2		7600.2.a.bi.1.2		3
12.11	even	2		8550.2.a.cl.1.2		3
20.3	even	4		190.2.b.b.39.5	yes	6
20.7	even	4		190.2.b.b.39.2	✓	6
20.19	odd	2		950.2.a.n.1.2		3
60.23	odd	4		1710.2.d.d.1369.2		6
60.47	odd	4		1710.2.d.d.1369.5		6
60.59	even	2		8550.2.a.ck.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
190.2.b.b.39.2	✓	6	20.7	even	4
190.2.b.b.39.5	yes	6	20.3	even	4
950.2.a.i.1.2		3	4.3	odd	2
950.2.a.n.1.2		3	20.19	odd	2
1520.2.d.j.609.3		6	5.2	odd	4
1520.2.d.j.609.4		6	5.3	odd	4
1710.2.d.d.1369.2		6	60.23	odd	4
1710.2.d.d.1369.5		6	60.47	odd	4
7600.2.a.bi.1.2		3	5.4	even	2
7600.2.a.cd.1.2		3	1.1	even	1	trivial
8550.2.a.ck.1.2		3	60.59	even	2
8550.2.a.cl.1.2		3	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.36333 q^{3} +0.636672 q^{7} -1.14134 q^{9} +O(q^{10})\)	\(q+1.36333 q^{3} +0.636672 q^{7} -1.14134 q^{9} -3.50466 q^{11} +0.141336 q^{13} -2.14134 q^{17} -1.00000 q^{19} +0.867993 q^{21} +4.91934 q^{23} -5.64600 q^{27} +7.15066 q^{29} +7.78734 q^{31} -4.77801 q^{33} -3.27334 q^{37} +0.192688 q^{39} -4.23132 q^{41} +2.49534 q^{43} -10.2827 q^{47} -6.59465 q^{49} -2.91934 q^{51} -8.14134 q^{53} -1.36333 q^{57} +5.64600 q^{59} -6.49534 q^{61} -0.726656 q^{63} +8.37266 q^{67} +6.70668 q^{69} +8.95798 q^{71} +3.69735 q^{73} -2.23132 q^{77} +4.17997 q^{79} -4.27334 q^{81} -9.00933 q^{83} +9.74870 q^{87} -6.77801 q^{89} +0.0899847 q^{91} +10.6167 q^{93} -14.5653 q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} + 4 q^{7} + 5 q^{9}+O(q^{10}) \) 3 * q + 2 * q^3 + 4 * q^7 + 5 * q^9	\( 3 q + 2 q^{3} + 4 q^{7} + 5 q^{9} - 8 q^{13} + 2 q^{17} - 3 q^{19} - 10 q^{21} + 2 q^{27} - 8 q^{29} - 4 q^{31} - 8 q^{33} - 14 q^{37} - 10 q^{39} + 2 q^{41} + 18 q^{43} - 14 q^{47} - 3 q^{49} + 6 q^{51} - 16 q^{53} - 2 q^{57} - 2 q^{59} - 30 q^{61} + 2 q^{63} + 2 q^{67} - 22 q^{69} + 8 q^{71} - 10 q^{73} + 8 q^{77} - 17 q^{81} - 6 q^{83} + 6 q^{87} - 14 q^{89} - 6 q^{91} - 4 q^{93} - 10 q^{97} + 12 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 + 4 * q^7 + 5 * q^9 - 8 * q^13 + 2 * q^17 - 3 * q^19 - 10 * q^21 + 2 * q^27 - 8 * q^29 - 4 * q^31 - 8 * q^33 - 14 * q^37 - 10 * q^39 + 2 * q^41 + 18 * q^43 - 14 * q^47 - 3 * q^49 + 6 * q^51 - 16 * q^53 - 2 * q^57 - 2 * q^59 - 30 * q^61 + 2 * q^63 + 2 * q^67 - 22 * q^69 + 8 * q^71 - 10 * q^73 + 8 * q^77 - 17 * q^81 - 6 * q^83 + 6 * q^87 - 14 * q^89 - 6 * q^91 - 4 * q^93 - 10 * q^97 + 12 * q^99

Introduction

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