LMFDB - Embedded newform 304.10.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,10,Mod(1,304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(304, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 10, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("304.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 304 = 2^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	304.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:	$156.570894194$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 34433x^{2} - 2723303x - 48270488 $ x^4 - x^3 - 34433x^2 - 2723303x - 48270488
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$219.264$ of defining polynomial
Character	$\chi$	$=$	304.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+66.6053 q^{3} -2418.56 q^{5} +1533.95 q^{7} -15246.7 q^{9} +O(q^{10})$	$q+66.6053 q^{3} -2418.56 q^{5} +1533.95 q^{7} -15246.7 q^{9} +5815.50 q^{11} -135546. q^{13} -161089. q^{15} -448133. q^{17} -130321. q^{19} +102169. q^{21} -2.07551e6 q^{23} +3.89629e6 q^{25} -2.32650e6 q^{27} -643630. q^{29} +194155. q^{31} +387343. q^{33} -3.70994e6 q^{35} +9.46515e6 q^{37} -9.02808e6 q^{39} -2.34337e7 q^{41} -3.80295e7 q^{43} +3.68751e7 q^{45} +2.28118e7 q^{47} -3.80006e7 q^{49} -2.98480e7 q^{51} -6.65359e7 q^{53} -1.40651e7 q^{55} -8.68007e6 q^{57} +9.14212e7 q^{59} -4.02765e7 q^{61} -2.33877e7 q^{63} +3.27826e8 q^{65} -2.42739e8 q^{67} -1.38240e8 q^{69} +1.83180e8 q^{71} -1.39336e8 q^{73} +2.59514e8 q^{75} +8.92069e6 q^{77} -2.51527e8 q^{79} +1.45144e8 q^{81} +1.30417e8 q^{83} +1.08383e9 q^{85} -4.28692e7 q^{87} +7.10883e8 q^{89} -2.07921e8 q^{91} +1.29318e7 q^{93} +3.15189e8 q^{95} -1.31644e9 q^{97} -8.86675e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 84 q^{3} - 1395 q^{5} - 12307 q^{7} + 16538 q^{9}+O(q^{10}) $ 4 * q - 84 * q^3 - 1395 * q^5 - 12307 * q^7 + 16538 * q^9	$ 4 q - 84 q^{3} - 1395 q^{5} - 12307 q^{7} + 16538 q^{9} + 104249 q^{11} + 120486 q^{13} + 591090 q^{15} - 412139 q^{17} - 521284 q^{19} + 2437006 q^{21} - 3010300 q^{23} + 9760585 q^{25} - 12387978 q^{27} + 6153240 q^{29} - 12774024 q^{31} - 3258022 q^{33} - 9823425 q^{35} + 20506048 q^{37} - 69881444 q^{39} + 11620300 q^{41} - 7698327 q^{43} - 124015815 q^{45} + 31581083 q^{47} + 18970383 q^{49} + 8594812 q^{51} + 72549422 q^{53} - 21332505 q^{55} + 10946964 q^{57} + 149234120 q^{59} + 129004373 q^{61} - 102967551 q^{63} + 124691700 q^{65} - 132595266 q^{67} - 45529972 q^{69} + 47138482 q^{71} - 39332795 q^{73} - 824627010 q^{75} - 165933719 q^{77} + 307010840 q^{79} + 1305551744 q^{81} + 746568232 q^{83} - 105005985 q^{85} + 82148208 q^{87} + 286943482 q^{89} - 3155781114 q^{91} + 1151901596 q^{93} + 181797795 q^{95} + 793519958 q^{97} + 1681833809 q^{99}+O(q^{100}) $ 4 * q - 84 * q^3 - 1395 * q^5 - 12307 * q^7 + 16538 * q^9 + 104249 * q^11 + 120486 * q^13 + 591090 * q^15 - 412139 * q^17 - 521284 * q^19 + 2437006 * q^21 - 3010300 * q^23 + 9760585 * q^25 - 12387978 * q^27 + 6153240 * q^29 - 12774024 * q^31 - 3258022 * q^33 - 9823425 * q^35 + 20506048 * q^37 - 69881444 * q^39 + 11620300 * q^41 - 7698327 * q^43 - 124015815 * q^45 + 31581083 * q^47 + 18970383 * q^49 + 8594812 * q^51 + 72549422 * q^53 - 21332505 * q^55 + 10946964 * q^57 + 149234120 * q^59 + 129004373 * q^61 - 102967551 * q^63 + 124691700 * q^65 - 132595266 * q^67 - 45529972 * q^69 + 47138482 * q^71 - 39332795 * q^73 - 824627010 * q^75 - 165933719 * q^77 + 307010840 * q^79 + 1305551744 * q^81 + 746568232 * q^83 - 105005985 * q^85 + 82148208 * q^87 + 286943482 * q^89 - 3155781114 * q^91 + 1151901596 * q^93 + 181797795 * q^95 + 793519958 * q^97 + 1681833809 * 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
$3$	66.6053	0.474748	0.237374	−	0.971418i	$-0.423713\pi$
$3$	66.6053	0.474748	0.237374	+	0.971418i	$0.423713\pi$
$4$	0	0
$5$	−2418.56	−1.73058	−0.865289	−	0.501273i	$-0.832865\pi$
$5$	−2418.56	−1.73058	−0.865289	+	0.501273i	$0.832865\pi$
$6$	0	0
$7$	1533.95	0.241474	0.120737	−	0.992685i	$-0.461474\pi$
$7$	1533.95	0.241474	0.120737	+	0.992685i	$0.461474\pi$
$8$	0	0
$9$	−15246.7	−0.774615
$10$	0	0
$11$	5815.50	0.119762	0.0598812	−	0.998206i	$-0.480928\pi$
$11$	5815.50	0.119762	0.0598812	+	0.998206i	$0.480928\pi$
$12$	0	0
$13$	−135546.	−1.31626	−0.658130	−	0.752904i	$-0.728652\pi$
$13$	−135546.	−1.31626	−0.658130	+	0.752904i	$0.728652\pi$
$14$	0	0
$15$	−161089.	−0.821588
$16$	0	0
$17$	−448133.	−1.30133	−0.650663	−	0.759366i	$-0.725509\pi$
$17$	−448133.	−1.30133	−0.650663	+	0.759366i	$0.725509\pi$
$18$	0	0
$19$	−130321.	−0.229416
$19$	−130321.	−0.229416
$20$	0	0
$21$	102169.	0.114639
$22$	0	0
$23$	−2.07551e6	−1.54650	−0.773249	−	0.634102i	$-0.781370\pi$
$23$	−2.07551e6	−1.54650	−0.773249	+	0.634102i	$0.781370\pi$
$24$	0	0
$25$	3.89629e6	1.99490
$26$	0	0
$27$	−2.32650e6	−0.842494
$28$	0	0
$29$	−643630.	−0.168984	−0.0844920	−	0.996424i	$-0.526927\pi$
$29$	−643630.	−0.168984	−0.0844920	+	0.996424i	$0.526927\pi$
$30$	0	0
$31$	194155.	0.0377591	0.0188796	−	0.999822i	$-0.493990\pi$
$31$	194155.	0.0377591	0.0188796	+	0.999822i	$0.493990\pi$
$32$	0	0
$33$	387343.	0.0568569
$34$	0	0
$35$	−3.70994e6	−0.417889
$36$	0	0
$37$	9.46515e6	0.830271	0.415135	−	0.909760i	$-0.363734\pi$
$37$	9.46515e6	0.830271	0.415135	+	0.909760i	$0.363734\pi$
$38$	0	0
$39$	−9.02808e6	−0.624892
$40$	0	0
$41$	−2.34337e7	−1.29513	−0.647565	−	0.762010i	$-0.724213\pi$
$41$	−2.34337e7	−1.29513	−0.647565	+	0.762010i	$0.724213\pi$
$42$	0	0
$43$	−3.80295e7	−1.69634	−0.848170	−	0.529724i	$-0.822296\pi$
$43$	−3.80295e7	−1.69634	−0.848170	+	0.529724i	$0.822296\pi$
$44$	0	0
$45$	3.68751e7	1.34053
$46$	0	0
$47$	2.28118e7	0.681897	0.340948	−	0.940082i	$-0.389252\pi$
$47$	2.28118e7	0.681897	0.340948	+	0.940082i	$0.389252\pi$
$48$	0	0
$49$	−3.80006e7	−0.941690
$50$	0	0
$51$	−2.98480e7	−0.617802
$52$	0	0
$53$	−6.65359e7	−1.15828	−0.579141	−	0.815227i	$-0.696612\pi$
$53$	−6.65359e7	−1.15828	−0.579141	+	0.815227i	$0.696612\pi$
$54$	0	0
$55$	−1.40651e7	−0.207258
$56$	0	0
$57$	−8.68007e6	−0.108915
$58$	0	0
$59$	9.14212e7	0.982230	0.491115	−	0.871095i	$-0.336590\pi$
$59$	9.14212e7	0.982230	0.491115	+	0.871095i	$0.336590\pi$
$60$	0	0
$61$	−4.02765e7	−0.372450	−0.186225	−	0.982507i	$-0.559625\pi$
$61$	−4.02765e7	−0.372450	−0.186225	+	0.982507i	$0.559625\pi$
$62$	0	0
$63$	−2.33877e7	−0.187049
$64$	0	0
$65$	3.27826e8	2.27789
$66$	0	0
$67$	−2.42739e8	−1.47164	−0.735822	−	0.677175i	$-0.763204\pi$
$67$	−2.42739e8	−1.47164	−0.735822	+	0.677175i	$0.763204\pi$
$68$	0	0
$69$	−1.38240e8	−0.734197
$70$	0	0
$71$	1.83180e8	0.855491	0.427745	−	0.903899i	$-0.359308\pi$
$71$	1.83180e8	0.855491	0.427745	+	0.903899i	$0.359308\pi$
$72$	0	0
$73$	−1.39336e8	−0.574264	−0.287132	−	0.957891i	$-0.592702\pi$
$73$	−1.39336e8	−0.574264	−0.287132	+	0.957891i	$0.592702\pi$
$74$	0	0
$75$	2.59514e8	0.947075
$76$	0	0
$77$	8.92069e6	0.0289195
$78$	0	0
$79$	−2.51527e8	−0.726546	−0.363273	−	0.931683i	$-0.618341\pi$
$79$	−2.51527e8	−0.726546	−0.363273	+	0.931683i	$0.618341\pi$
$80$	0	0
$81$	1.45144e8	0.374642
$82$	0	0
$83$	1.30417e8	0.301636	0.150818	−	0.988562i	$-0.451809\pi$
$83$	1.30417e8	0.301636	0.150818	+	0.988562i	$0.451809\pi$
$84$	0	0
$85$	1.08383e9	2.25205
$86$	0	0
$87$	−4.28692e7	−0.0802247
$88$	0	0
$89$	7.10883e8	1.20100	0.600500	−	0.799625i	$-0.294968\pi$
$89$	7.10883e8	1.20100	0.600500	+	0.799625i	$0.294968\pi$
$90$	0	0
$91$	−2.07921e8	−0.317842
$92$	0	0
$93$	1.29318e7	0.0179261
$94$	0	0
$95$	3.15189e8	0.397022
$96$	0	0
$97$	−1.31644e9	−1.50983	−0.754916	−	0.655822i	$-0.772322\pi$
$97$	−1.31644e9	−1.50983	−0.754916	+	0.655822i	$0.772322\pi$
$98$	0	0
$99$	−8.86675e7	−0.0927696
$100$	0	0
$101$	1.38054e9	1.32009	0.660044	−	0.751227i	$-0.270538\pi$
$101$	1.38054e9	1.32009	0.660044	+	0.751227i	$0.270538\pi$
$102$	0	0
$103$	9.35066e8	0.818606	0.409303	−	0.912399i	$-0.365772\pi$
$103$	9.35066e8	0.818606	0.409303	+	0.912399i	$0.365772\pi$
$104$	0	0
$105$	−2.47102e8	−0.198392
$106$	0	0
$107$	4.27772e8	0.315490	0.157745	−	0.987480i	$-0.449578\pi$
$107$	4.27772e8	0.315490	0.157745	+	0.987480i	$0.449578\pi$
$108$	0	0
$109$	−1.13801e8	−0.0772197	−0.0386098	−	0.999254i	$-0.512293\pi$
$109$	−1.13801e8	−0.0772197	−0.0386098	+	0.999254i	$0.512293\pi$
$110$	0	0
$111$	6.30429e8	0.394169
$112$	0	0
$113$	1.37096e9	0.790990	0.395495	−	0.918468i	$-0.370573\pi$
$113$	1.37096e9	0.790990	0.395495	+	0.918468i	$0.370573\pi$
$114$	0	0
$115$	5.01974e9	2.67634
$116$	0	0
$117$	2.06663e9	1.01959
$118$	0	0
$119$	−6.87413e8	−0.314236
$120$	0	0
$121$	−2.32413e9	−0.985657
$122$	0	0
$123$	−1.56081e9	−0.614860
$124$	0	0
$125$	−4.69966e9	−1.72176
$126$	0	0
$127$	−1.64706e9	−0.561813	−0.280906	−	0.959735i	$-0.590635\pi$
$127$	−1.64706e9	−0.561813	−0.280906	+	0.959735i	$0.590635\pi$
$128$	0	0
$129$	−2.53297e9	−0.805334
$130$	0	0
$131$	3.65669e9	1.08485	0.542423	−	0.840106i	$-0.317507\pi$
$131$	3.65669e9	1.08485	0.542423	+	0.840106i	$0.317507\pi$
$132$	0	0
$133$	−1.99906e8	−0.0553979
$134$	0	0
$135$	5.62678e9	1.45800
$136$	0	0
$137$	−3.24163e9	−0.786179	−0.393089	−	0.919500i	$-0.628594\pi$
$137$	−3.24163e9	−0.786179	−0.393089	+	0.919500i	$0.628594\pi$
$138$	0	0
$139$	−8.69038e9	−1.97457	−0.987285	−	0.158963i	$-0.949185\pi$
$139$	−8.69038e9	−1.97457	−0.987285	+	0.158963i	$0.949185\pi$
$140$	0	0
$141$	1.51938e9	0.323729
$142$	0	0
$143$	−7.88269e8	−0.157638
$144$	0	0
$145$	1.55666e9	0.292440
$146$	0	0
$147$	−2.53104e9	−0.447065
$148$	0	0
$149$	−7.11665e9	−1.18287	−0.591435	−	0.806353i	$-0.701438\pi$
$149$	−7.11665e9	−1.18287	−0.591435	+	0.806353i	$0.701438\pi$
$150$	0	0
$151$	−1.87879e9	−0.294091	−0.147046	−	0.989130i	$-0.546976\pi$
$151$	−1.87879e9	−0.294091	−0.147046	+	0.989130i	$0.546976\pi$
$152$	0	0
$153$	6.83256e9	1.00803
$154$	0	0
$155$	−4.69576e8	−0.0653451
$156$	0	0
$157$	4.95963e9	0.651480	0.325740	−	0.945459i	$-0.394387\pi$
$157$	4.95963e9	0.651480	0.325740	+	0.945459i	$0.394387\pi$
$158$	0	0
$159$	−4.43164e9	−0.549892
$160$	0	0
$161$	−3.18373e9	−0.373439
$162$	0	0
$163$	−2.87539e9	−0.319046	−0.159523	−	0.987194i	$-0.550996\pi$
$163$	−2.87539e9	−0.319046	−0.159523	+	0.987194i	$0.550996\pi$
$164$	0	0
$165$	−9.36812e8	−0.0983953
$166$	0	0
$167$	−1.47841e10	−1.47085	−0.735427	−	0.677604i	$-0.763018\pi$
$167$	−1.47841e10	−1.47085	−0.735427	+	0.677604i	$0.763018\pi$
$168$	0	0
$169$	7.76823e9	0.732541
$170$	0	0
$171$	1.98697e9	0.177709
$172$	0	0
$173$	1.45582e10	1.23566	0.617832	−	0.786310i	$-0.288011\pi$
$173$	1.45582e10	1.23566	0.617832	+	0.786310i	$0.288011\pi$
$174$	0	0
$175$	5.97672e9	0.481716
$176$	0	0
$177$	6.08914e9	0.466311
$178$	0	0
$179$	1.82403e10	1.32799	0.663994	−	0.747738i	$-0.268860\pi$
$179$	1.82403e10	1.32799	0.663994	+	0.747738i	$0.268860\pi$
$180$	0	0
$181$	2.81167e10	1.94720	0.973602	−	0.228252i	$-0.0733011\pi$
$181$	2.81167e10	1.94720	0.973602	+	0.228252i	$0.0733011\pi$
$182$	0	0
$183$	−2.68263e9	−0.176820
$184$	0	0
$185$	−2.28920e10	−1.43685
$186$	0	0
$187$	−2.60612e9	−0.155850
$188$	0	0
$189$	−3.56874e9	−0.203440
$190$	0	0
$191$	−2.82317e8	−0.0153492	−0.00767462	−	0.999971i	$-0.502443\pi$
$191$	−2.82317e8	−0.0153492	−0.00767462	+	0.999971i	$0.502443\pi$
$192$	0	0
$193$	1.43117e10	0.742477	0.371239	−	0.928538i	$-0.378933\pi$
$193$	1.43117e10	0.742477	0.371239	+	0.928538i	$0.378933\pi$
$194$	0	0
$195$	2.18349e10	1.08142
$196$	0	0
$197$	2.02116e10	0.956100	0.478050	−	0.878333i	$-0.341344\pi$
$197$	2.02116e10	0.956100	0.478050	+	0.878333i	$0.341344\pi$
$198$	0	0
$199$	3.95924e10	1.78967	0.894835	−	0.446398i	$-0.147293\pi$
$199$	3.95924e10	1.78967	0.894835	+	0.446398i	$0.147293\pi$
$200$	0	0
$201$	−1.61677e10	−0.698660
$202$	0	0
$203$	−9.87296e8	−0.0408052
$204$	0	0
$205$	5.66757e10	2.24132
$206$	0	0
$207$	3.16447e10	1.19794
$208$	0	0
$209$	−7.57882e8	−0.0274754
$210$	0	0
$211$	1.31017e10	0.455047	0.227523	−	0.973773i	$-0.426937\pi$
$211$	1.31017e10	0.455047	0.227523	+	0.973773i	$0.426937\pi$
$212$	0	0
$213$	1.22008e10	0.406142
$214$	0	0
$215$	9.19766e10	2.93565
$216$	0	0
$217$	2.97825e8	0.00911784
$218$	0	0
$219$	−9.28054e9	−0.272631
$220$	0	0
$221$	6.07426e10	1.71288
$222$	0	0
$223$	−6.50346e10	−1.76105	−0.880526	−	0.473997i	$-0.842811\pi$
$223$	−6.50346e10	−1.76105	−0.880526	+	0.473997i	$0.842811\pi$
$224$	0	0
$225$	−5.94058e10	−1.54528
$226$	0	0
$227$	−1.02990e10	−0.257442	−0.128721	−	0.991681i	$-0.541087\pi$
$227$	−1.02990e10	−0.257442	−0.128721	+	0.991681i	$0.541087\pi$
$228$	0	0
$229$	1.71789e10	0.412797	0.206398	−	0.978468i	$-0.433826\pi$
$229$	1.71789e10	0.412797	0.206398	+	0.978468i	$0.433826\pi$
$230$	0	0
$231$	5.94165e8	0.0137294
$232$	0	0
$233$	−6.52465e10	−1.45029	−0.725146	−	0.688595i	$-0.758228\pi$
$233$	−6.52465e10	−1.45029	−0.725146	+	0.688595i	$0.758228\pi$
$234$	0	0
$235$	−5.51716e10	−1.18008
$236$	0	0
$237$	−1.67530e10	−0.344926
$238$	0	0
$239$	1.43024e10	0.283543	0.141771	−	0.989899i	$-0.454720\pi$
$239$	1.43024e10	0.283543	0.141771	+	0.989899i	$0.454720\pi$
$240$	0	0
$241$	−4.79560e10	−0.915727	−0.457864	−	0.889022i	$-0.651385\pi$
$241$	−4.79560e10	−0.915727	−0.457864	+	0.889022i	$0.651385\pi$
$242$	0	0
$243$	5.54600e10	1.02035
$244$	0	0
$245$	9.19066e10	1.62967
$246$	0	0
$247$	1.76645e10	0.301971
$248$	0	0
$249$	8.68647e9	0.143201
$250$	0	0
$251$	1.25023e10	0.198819	0.0994096	−	0.995047i	$-0.468305\pi$
$251$	1.25023e10	0.198819	0.0994096	+	0.995047i	$0.468305\pi$
$252$	0	0
$253$	−1.20701e10	−0.185212
$254$	0	0
$255$	7.21891e10	1.06915
$256$	0	0
$257$	1.83653e10	0.262602	0.131301	−	0.991343i	$-0.458085\pi$
$257$	1.83653e10	0.262602	0.131301	+	0.991343i	$0.458085\pi$
$258$	0	0
$259$	1.45191e10	0.200489
$260$	0	0
$261$	9.81326e9	0.130897
$262$	0	0
$263$	−8.39157e10	−1.08154	−0.540770	−	0.841171i	$-0.681867\pi$
$263$	−8.39157e10	−1.08154	−0.540770	+	0.841171i	$0.681867\pi$
$264$	0	0
$265$	1.60921e11	2.00450
$266$	0	0
$267$	4.73485e10	0.570172
$268$	0	0
$269$	9.33233e9	0.108669	0.0543344	−	0.998523i	$-0.482696\pi$
$269$	9.33233e9	0.108669	0.0543344	+	0.998523i	$0.482696\pi$
$270$	0	0
$271$	1.57854e11	1.77785	0.888924	−	0.458055i	$-0.151454\pi$
$271$	1.57854e11	1.77785	0.888924	+	0.458055i	$0.151454\pi$
$272$	0	0
$273$	−1.38486e10	−0.150895
$274$	0	0
$275$	2.26589e10	0.238914
$276$	0	0
$277$	−1.12981e11	−1.15305	−0.576525	−	0.817080i	$-0.695592\pi$
$277$	−1.12981e11	−1.15305	−0.576525	+	0.817080i	$0.695592\pi$
$278$	0	0
$279$	−2.96024e9	−0.0292488
$280$	0	0
$281$	7.83548e10	0.749699	0.374849	−	0.927086i	$-0.377694\pi$
$281$	7.83548e10	0.749699	0.374849	+	0.927086i	$0.377694\pi$
$282$	0	0
$283$	−8.46004e10	−0.784032	−0.392016	−	0.919958i	$-0.628222\pi$
$283$	−8.46004e10	−0.784032	−0.392016	+	0.919958i	$0.628222\pi$
$284$	0	0
$285$	2.09932e10	0.188485
$286$	0	0
$287$	−3.59461e10	−0.312740
$288$	0	0
$289$	8.22349e10	0.693451
$290$	0	0
$291$	−8.76819e10	−0.716789
$292$	0	0
$293$	−1.06621e11	−0.845157	−0.422579	−	0.906326i	$-0.638875\pi$
$293$	−1.06621e11	−0.845157	−0.422579	+	0.906326i	$0.638875\pi$
$294$	0	0
$295$	−2.21107e11	−1.69983
$296$	0	0
$297$	−1.35298e10	−0.100899
$298$	0	0
$299$	2.81327e11	2.03559
$300$	0	0
$301$	−5.83354e10	−0.409622
$302$	0	0
$303$	9.19513e10	0.626709
$304$	0	0
$305$	9.74110e10	0.644553
$306$	0	0
$307$	−1.65292e11	−1.06201	−0.531006	−	0.847368i	$-0.678186\pi$
$307$	−1.65292e11	−1.06201	−0.531006	+	0.847368i	$0.678186\pi$
$308$	0	0
$309$	6.22803e10	0.388631
$310$	0	0
$311$	1.08969e11	0.660515	0.330257	−	0.943891i	$-0.392864\pi$
$311$	1.08969e11	0.660515	0.330257	+	0.943891i	$0.392864\pi$
$312$	0	0
$313$	9.48622e10	0.558655	0.279328	−	0.960196i	$-0.409888\pi$
$313$	9.48622e10	0.558655	0.279328	+	0.960196i	$0.409888\pi$
$314$	0	0
$315$	5.65645e10	0.323703
$316$	0	0
$317$	2.75000e11	1.52956	0.764780	−	0.644291i	$-0.222848\pi$
$317$	2.75000e11	1.52956	0.764780	+	0.644291i	$0.222848\pi$
$318$	0	0
$319$	−3.74303e9	−0.0202379
$320$	0	0
$321$	2.84919e10	0.149778
$322$	0	0
$323$	5.84011e10	0.298545
$324$	0	0
$325$	−5.28127e11	−2.62581
$326$	0	0
$327$	−7.57977e9	−0.0366599
$328$	0	0
$329$	3.49921e10	0.164660
$330$	0	0
$331$	1.28057e11	0.586378	0.293189	−	0.956055i	$-0.405284\pi$
$331$	1.28057e11	0.586378	0.293189	+	0.956055i	$0.405284\pi$
$332$	0	0
$333$	−1.44313e11	−0.643140
$334$	0	0
$335$	5.87078e11	2.54680
$336$	0	0
$337$	−3.03698e10	−0.128265	−0.0641324	−	0.997941i	$-0.520428\pi$
$337$	−3.03698e10	−0.128265	−0.0641324	+	0.997941i	$0.520428\pi$
$338$	0	0
$339$	9.13130e10	0.375521
$340$	0	0
$341$	1.12911e9	0.00452212
$342$	0	0
$343$	−1.20191e11	−0.468867
$344$	0	0
$345$	3.34341e11	1.27059
$346$	0	0
$347$	−4.03811e11	−1.49519	−0.747594	−	0.664156i	$-0.768791\pi$
$347$	−4.03811e11	−1.49519	−0.747594	+	0.664156i	$0.768791\pi$
$348$	0	0
$349$	6.29420e10	0.227105	0.113552	−	0.993532i	$-0.463777\pi$
$349$	6.29420e10	0.227105	0.113552	+	0.993532i	$0.463777\pi$
$350$	0	0
$351$	3.15349e11	1.10894
$352$	0	0
$353$	9.36033e10	0.320852	0.160426	−	0.987048i	$-0.448713\pi$
$353$	9.36033e10	0.320852	0.160426	+	0.987048i	$0.448713\pi$
$354$	0	0
$355$	−4.43031e11	−1.48049
$356$	0	0
$357$	−4.57853e10	−0.149183
$358$	0	0
$359$	−3.18605e11	−1.01234	−0.506171	−	0.862433i	$-0.668940\pi$
$359$	−3.18605e11	−1.01234	−0.506171	+	0.862433i	$0.668940\pi$
$360$	0	0
$361$	1.69836e10	0.0526316
$362$	0	0
$363$	−1.54799e11	−0.467939
$364$	0	0
$365$	3.36993e11	0.993810
$366$	0	0
$367$	−9.81213e10	−0.282336	−0.141168	−	0.989986i	$-0.545086\pi$
$367$	−9.81213e10	−0.282336	−0.141168	+	0.989986i	$0.545086\pi$
$368$	0	0
$369$	3.57287e11	1.00323
$370$	0	0
$371$	−1.02063e11	−0.279695
$372$	0	0
$373$	2.24989e11	0.601827	0.300913	−	0.953652i	$-0.402709\pi$
$373$	2.24989e11	0.601827	0.300913	+	0.953652i	$0.402709\pi$
$374$	0	0
$375$	−3.13022e11	−0.817400
$376$	0	0
$377$	8.72415e10	0.222427
$378$	0	0
$379$	−6.24000e11	−1.55349	−0.776744	−	0.629817i	$-0.783130\pi$
$379$	−6.24000e11	−1.55349	−0.776744	+	0.629817i	$0.783130\pi$
$380$	0	0
$381$	−1.09703e11	−0.266719
$382$	0	0
$383$	−5.36899e11	−1.27497	−0.637483	−	0.770465i	$-0.720024\pi$
$383$	−5.36899e11	−1.27497	−0.637483	+	0.770465i	$0.720024\pi$
$384$	0	0
$385$	−2.15752e10	−0.0500474
$386$	0	0
$387$	5.79826e11	1.31401
$388$	0	0
$389$	−1.30155e11	−0.288195	−0.144097	−	0.989564i	$-0.546028\pi$
$389$	−1.30155e11	−0.288195	−0.144097	+	0.989564i	$0.546028\pi$
$390$	0	0
$391$	9.30103e11	2.01250
$392$	0	0
$393$	2.43555e11	0.515028
$394$	0	0
$395$	6.08333e11	1.25735
$396$	0	0
$397$	−7.36706e11	−1.48846	−0.744230	−	0.667924i	$-0.767183\pi$
$397$	−7.36706e11	−1.48846	−0.744230	+	0.667924i	$0.767183\pi$
$398$	0	0
$399$	−1.33148e10	−0.0263000
$400$	0	0
$401$	−5.90469e11	−1.14037	−0.570187	−	0.821515i	$-0.693129\pi$
$401$	−5.90469e11	−1.14037	−0.570187	+	0.821515i	$0.693129\pi$
$402$	0	0
$403$	−2.63170e10	−0.0497008
$404$	0	0
$405$	−3.51039e11	−0.648348
$406$	0	0
$407$	5.50446e10	0.0994352
$408$	0	0
$409$	−4.27579e11	−0.755547	−0.377773	−	0.925898i	$-0.623310\pi$
$409$	−4.27579e11	−0.755547	−0.377773	+	0.925898i	$0.623310\pi$
$410$	0	0
$411$	−2.15910e11	−0.373237
$412$	0	0
$413$	1.40236e11	0.237183
$414$	0	0
$415$	−3.15421e11	−0.522005
$416$	0	0
$417$	−5.78825e11	−0.937422
$418$	0	0
$419$	−9.75902e11	−1.54683	−0.773416	−	0.633899i	$-0.781454\pi$
$419$	−9.75902e11	−1.54683	−0.773416	+	0.633899i	$0.781454\pi$
$420$	0	0
$421$	−6.82920e10	−0.105950	−0.0529750	−	0.998596i	$-0.516870\pi$
$421$	−6.82920e10	−0.105950	−0.0529750	+	0.998596i	$0.516870\pi$
$422$	0	0
$423$	−3.47805e11	−0.528207
$424$	0	0
$425$	−1.74606e12	−2.59602
$426$	0	0
$427$	−6.17821e10	−0.0899368
$428$	0	0
$429$	−5.25028e10	−0.0748385
$430$	0	0
$431$	2.97113e11	0.414738	0.207369	−	0.978263i	$-0.433510\pi$
$431$	2.97113e11	0.414738	0.207369	+	0.978263i	$0.433510\pi$
$432$	0	0
$433$	−4.51168e11	−0.616798	−0.308399	−	0.951257i	$-0.599793\pi$
$433$	−4.51168e11	−0.616798	−0.308399	+	0.951257i	$0.599793\pi$
$434$	0	0
$435$	1.03682e11	0.138835
$436$	0	0
$437$	2.70482e11	0.354791
$438$	0	0
$439$	−5.55482e11	−0.713805	−0.356903	−	0.934142i	$-0.616167\pi$
$439$	−5.55482e11	−0.713805	−0.356903	+	0.934142i	$0.616167\pi$
$440$	0	0
$441$	5.79385e11	0.729447
$442$	0	0
$443$	1.61691e12	1.99466	0.997329	−	0.0730386i	$-0.0232696\pi$
$443$	1.61691e12	1.99466	0.997329	+	0.0730386i	$0.0232696\pi$
$444$	0	0
$445$	−1.71931e12	−2.07842
$446$	0	0
$447$	−4.74006e11	−0.561565
$448$	0	0
$449$	4.52893e11	0.525880	0.262940	−	0.964812i	$-0.415308\pi$
$449$	4.52893e11	0.525880	0.262940	+	0.964812i	$0.415308\pi$
$450$	0	0
$451$	−1.36279e11	−0.155108
$452$	0	0
$453$	−1.25137e11	−0.139619
$454$	0	0
$455$	5.02868e11	0.550051
$456$	0	0
$457$	1.10310e12	1.18302	0.591511	−	0.806297i	$-0.298532\pi$
$457$	1.10310e12	1.18302	0.591511	+	0.806297i	$0.298532\pi$
$458$	0	0
$459$	1.04258e12	1.09636
$460$	0	0
$461$	2.22092e11	0.229023	0.114511	−	0.993422i	$-0.463470\pi$
$461$	2.22092e11	0.229023	0.114511	+	0.993422i	$0.463470\pi$
$462$	0	0
$463$	−9.97229e11	−1.00851	−0.504255	−	0.863555i	$-0.668233\pi$
$463$	−9.97229e11	−1.00851	−0.504255	+	0.863555i	$0.668233\pi$
$464$	0	0
$465$	−3.12762e10	−0.0310225
$466$	0	0
$467$	−8.73582e11	−0.849919	−0.424959	−	0.905212i	$-0.639712\pi$
$467$	−8.73582e11	−0.849919	−0.424959	+	0.905212i	$0.639712\pi$
$468$	0	0
$469$	−3.72349e11	−0.355363
$470$	0	0
$471$	3.30338e11	0.309289
$472$	0	0
$473$	−2.21161e11	−0.203158
$474$	0	0
$475$	−5.07769e11	−0.457662
$476$	0	0
$477$	1.01446e12	0.897223
$478$	0	0
$479$	−1.60658e12	−1.39442	−0.697209	−	0.716868i	$-0.745575\pi$
$479$	−1.60658e12	−1.39442	−0.697209	+	0.716868i	$0.745575\pi$
$480$	0	0
$481$	−1.28296e12	−1.09285
$482$	0	0
$483$	−2.12053e11	−0.177289
$484$	0	0
$485$	3.18389e12	2.61288
$486$	0	0
$487$	9.25381e11	0.745487	0.372744	−	0.927934i	$-0.378417\pi$
$487$	9.25381e11	0.745487	0.372744	+	0.927934i	$0.378417\pi$
$488$	0	0
$489$	−1.91516e11	−0.151466
$490$	0	0
$491$	9.28628e11	0.721066	0.360533	−	0.932746i	$-0.382595\pi$
$491$	9.28628e11	0.721066	0.360533	+	0.932746i	$0.382595\pi$
$492$	0	0
$493$	2.88432e11	0.219903
$494$	0	0
$495$	2.14447e11	0.160545
$496$	0	0
$497$	2.80989e11	0.206579
$498$	0	0
$499$	−1.19609e12	−0.863601	−0.431800	−	0.901969i	$-0.642122\pi$
$499$	−1.19609e12	−0.863601	−0.431800	+	0.901969i	$0.642122\pi$
$500$	0	0
$501$	−9.84696e11	−0.698285
$502$	0	0
$503$	−1.02039e12	−0.710741	−0.355370	−	0.934726i	$-0.615645\pi$
$503$	−1.02039e12	−0.710741	−0.355370	+	0.934726i	$0.615645\pi$
$504$	0	0
$505$	−3.33892e12	−2.28452
$506$	0	0
$507$	5.17405e11	0.347772
$508$	0	0
$509$	−5.49549e11	−0.362891	−0.181446	−	0.983401i	$-0.558078\pi$
$509$	−5.49549e11	−0.362891	−0.181446	+	0.983401i	$0.558078\pi$
$510$	0	0
$511$	−2.13735e11	−0.138670
$512$	0	0
$513$	3.03192e11	0.193281
$514$	0	0
$515$	−2.26151e12	−1.41666
$516$	0	0
$517$	1.32662e11	0.0816656
$518$	0	0
$519$	9.69654e11	0.586629
$520$	0	0
$521$	1.28517e12	0.764172	0.382086	−	0.924127i	$-0.375206\pi$
$521$	1.28517e12	0.764172	0.382086	+	0.924127i	$0.375206\pi$
$522$	0	0
$523$	−1.65568e12	−0.967654	−0.483827	−	0.875164i	$-0.660754\pi$
$523$	−1.65568e12	−0.967654	−0.483827	+	0.875164i	$0.660754\pi$
$524$	0	0
$525$	3.98081e11	0.228694
$526$	0	0
$527$	−8.70074e10	−0.0491370
$528$	0	0
$529$	2.50659e12	1.39166
$530$	0	0
$531$	−1.39388e12	−0.760849
$532$	0	0
$533$	3.17634e12	1.70473
$534$	0	0
$535$	−1.03459e12	−0.545980
$536$	0	0
$537$	1.21490e12	0.630460
$538$	0	0
$539$	−2.20993e11	−0.112779
$540$	0	0
$541$	−1.91314e12	−0.960192	−0.480096	−	0.877216i	$-0.659398\pi$
$541$	−1.91314e12	−0.960192	−0.480096	+	0.877216i	$0.659398\pi$
$542$	0	0
$543$	1.87272e12	0.924431
$544$	0	0
$545$	2.75235e11	0.133635
$546$	0	0
$547$	8.87680e11	0.423949	0.211974	−	0.977275i	$-0.432011\pi$
$547$	8.87680e11	0.423949	0.211974	+	0.977275i	$0.432011\pi$
$548$	0	0
$549$	6.14085e11	0.288505
$550$	0	0
$551$	8.38785e10	0.0387676
$552$	0	0
$553$	−3.85830e11	−0.175442
$554$	0	0
$555$	−1.52473e12	−0.682141
$556$	0	0
$557$	−1.28351e12	−0.565001	−0.282501	−	0.959267i	$-0.591164\pi$
$557$	−1.28351e12	−0.565001	−0.282501	+	0.959267i	$0.591164\pi$
$558$	0	0
$559$	5.15475e12	2.23283
$560$	0	0
$561$	−1.73581e11	−0.0739894
$562$	0	0
$563$	−4.00016e12	−1.67799	−0.838995	−	0.544139i	$-0.816856\pi$
$563$	−4.00016e12	−1.67799	−0.838995	+	0.544139i	$0.816856\pi$
$564$	0	0
$565$	−3.31574e12	−1.36887
$566$	0	0
$567$	2.22644e11	0.0904662
$568$	0	0
$569$	−1.23909e12	−0.495563	−0.247782	−	0.968816i	$-0.579702\pi$
$569$	−1.23909e12	−0.495563	−0.247782	+	0.968816i	$0.579702\pi$
$570$	0	0
$571$	5.04024e12	1.98421	0.992107	−	0.125395i	$-0.0400200\pi$
$571$	5.04024e12	1.98421	0.992107	+	0.125395i	$0.0400200\pi$
$572$	0	0
$573$	−1.88038e10	−0.00728702
$574$	0	0
$575$	−8.08679e12	−3.08511
$576$	0	0
$577$	−4.83898e12	−1.81745	−0.908725	−	0.417395i	$-0.862943\pi$
$577$	−4.83898e12	−1.81745	−0.908725	+	0.417395i	$0.862943\pi$
$578$	0	0
$579$	9.53234e11	0.352489
$580$	0	0
$581$	2.00053e11	0.0728372
$582$	0	0
$583$	−3.86940e11	−0.138719
$584$	0	0
$585$	−4.99827e12	−1.76449
$586$	0	0
$587$	1.95960e12	0.681233	0.340617	−	0.940202i	$-0.389364\pi$
$587$	1.95960e12	0.681233	0.340617	+	0.940202i	$0.389364\pi$
$588$	0	0
$589$	−2.53025e10	−0.00866254
$590$	0	0
$591$	1.34620e12	0.453906
$592$	0	0
$593$	−4.89414e12	−1.62529	−0.812643	−	0.582762i	$-0.801972\pi$
$593$	−4.89414e12	−1.62529	−0.812643	+	0.582762i	$0.801972\pi$
$594$	0	0
$595$	1.66255e12	0.543810
$596$	0	0
$597$	2.63706e12	0.849642
$598$	0	0
$599$	−4.50082e12	−1.42847	−0.714235	−	0.699906i	$-0.753225\pi$
$599$	−4.50082e12	−1.42847	−0.714235	+	0.699906i	$0.753225\pi$
$600$	0	0
$601$	−5.43075e12	−1.69795	−0.848974	−	0.528435i	$-0.822779\pi$
$601$	−5.43075e12	−1.69795	−0.848974	+	0.528435i	$0.822779\pi$
$602$	0	0
$603$	3.70098e12	1.13996
$604$	0	0
$605$	5.62104e12	1.70576
$606$	0	0
$607$	−5.89921e12	−1.76378	−0.881891	−	0.471454i	$-0.843729\pi$
$607$	−5.89921e12	−1.76378	−0.881891	+	0.471454i	$0.843729\pi$
$608$	0	0
$609$	−6.57591e10	−0.0193722
$610$	0	0
$611$	−3.09205e12	−0.897554
$612$	0	0
$613$	−2.44465e11	−0.0699268	−0.0349634	−	0.999389i	$-0.511131\pi$
$613$	−2.44465e11	−0.0699268	−0.0349634	+	0.999389i	$0.511131\pi$
$614$	0	0
$615$	3.77490e12	1.06406
$616$	0	0
$617$	−6.79288e11	−0.188699	−0.0943497	−	0.995539i	$-0.530077\pi$
$617$	−6.79288e11	−0.188699	−0.0943497	+	0.995539i	$0.530077\pi$
$618$	0	0
$619$	−1.31874e12	−0.361037	−0.180519	−	0.983572i	$-0.557778\pi$
$619$	−1.31874e12	−0.361037	−0.180519	+	0.983572i	$0.557778\pi$
$620$	0	0
$621$	4.82868e12	1.30292
$622$	0	0
$623$	1.09046e12	0.290010
$624$	0	0
$625$	3.75646e12	0.984732
$626$	0	0
$627$	−5.04790e10	−0.0130439
$628$	0	0
$629$	−4.24164e12	−1.08045
$630$	0	0
$631$	6.44908e12	1.61944	0.809721	−	0.586815i	$-0.199618\pi$
$631$	6.44908e12	1.61944	0.809721	+	0.586815i	$0.199618\pi$
$632$	0	0
$633$	8.72641e11	0.216032
$634$	0	0
$635$	3.98350e12	0.972261
$636$	0	0
$637$	5.15083e12	1.23951
$638$	0	0
$639$	−2.79290e12	−0.662676
$640$	0	0
$641$	−5.38181e12	−1.25912	−0.629561	−	0.776951i	$-0.716765\pi$
$641$	−5.38181e12	−1.25912	−0.629561	+	0.776951i	$0.716765\pi$
$642$	0	0
$643$	7.16838e12	1.65376	0.826878	−	0.562381i	$-0.190115\pi$
$643$	7.16838e12	1.65376	0.826878	+	0.562381i	$0.190115\pi$
$644$	0	0
$645$	6.12613e12	1.39369
$646$	0	0
$647$	1.24135e12	0.278501	0.139250	−	0.990257i	$-0.455531\pi$
$647$	1.24135e12	0.278501	0.139250	+	0.990257i	$0.455531\pi$
$648$	0	0
$649$	5.31661e11	0.117634
$650$	0	0
$651$	1.98367e10	0.00432867
$652$	0	0
$653$	6.46167e12	1.39071	0.695353	−	0.718669i	$-0.255248\pi$
$653$	6.46167e12	1.39071	0.695353	+	0.718669i	$0.255248\pi$
$654$	0	0
$655$	−8.84392e12	−1.87741
$656$	0	0
$657$	2.12443e12	0.444834
$658$	0	0
$659$	−3.48331e12	−0.719462	−0.359731	−	0.933056i	$-0.617131\pi$
$659$	−3.48331e12	−0.719462	−0.359731	+	0.933056i	$0.617131\pi$
$660$	0	0
$661$	−2.69783e12	−0.549678	−0.274839	−	0.961490i	$-0.588625\pi$
$661$	−2.69783e12	−0.549678	−0.274839	+	0.961490i	$0.588625\pi$
$662$	0	0
$663$	4.04578e12	0.813188
$664$	0	0
$665$	4.83484e11	0.0958704
$666$	0	0
$667$	1.33586e12	0.261333
$668$	0	0
$669$	−4.33165e12	−0.836056
$670$	0	0
$671$	−2.34228e11	−0.0446054
$672$	0	0
$673$	−7.15485e12	−1.34441	−0.672207	−	0.740364i	$-0.734653\pi$
$673$	−7.15485e12	−1.34441	−0.672207	+	0.740364i	$0.734653\pi$
$674$	0	0
$675$	−9.06474e12	−1.68069
$676$	0	0
$677$	3.99069e12	0.730127	0.365064	−	0.930983i	$-0.381047\pi$
$677$	3.99069e12	0.730127	0.365064	+	0.930983i	$0.381047\pi$
$678$	0	0
$679$	−2.01935e12	−0.364585
$680$	0	0
$681$	−6.85969e11	−0.122220
$682$	0	0
$683$	7.70306e12	1.35447	0.677236	−	0.735766i	$-0.263177\pi$
$683$	7.70306e12	1.35447	0.677236	+	0.735766i	$0.263177\pi$
$684$	0	0
$685$	7.84008e12	1.36054
$686$	0	0
$687$	1.14421e12	0.195974
$688$	0	0
$689$	9.01868e12	1.52460
$690$	0	0
$691$	6.94842e12	1.15940	0.579702	−	0.814828i	$-0.303169\pi$
$691$	6.94842e12	1.15940	0.579702	+	0.814828i	$0.303169\pi$
$692$	0	0
$693$	−1.36011e11	−0.0224014
$694$	0	0
$695$	2.10182e13	3.41715
$696$	0	0
$697$	1.05014e13	1.68539
$698$	0	0
$699$	−4.34576e12	−0.688523
$700$	0	0
$701$	8.89057e12	1.39059	0.695294	−	0.718725i	$-0.255274\pi$
$701$	8.89057e12	1.39059	0.695294	+	0.718725i	$0.255274\pi$
$702$	0	0
$703$	−1.23351e12	−0.190477
$704$	0	0
$705$	−3.67472e12	−0.560239
$706$	0	0
$707$	2.11768e12	0.318767
$708$	0	0
$709$	5.07939e12	0.754924	0.377462	−	0.926025i	$-0.376797\pi$
$709$	5.07939e12	0.754924	0.377462	+	0.926025i	$0.376797\pi$
$710$	0	0
$711$	3.83497e12	0.562793
$712$	0	0
$713$	−4.02972e11	−0.0583944
$714$	0	0
$715$	1.90647e12	0.272806
$716$	0	0
$717$	9.52615e11	0.134611
$718$	0	0
$719$	2.62697e11	0.0366585	0.0183293	−	0.999832i	$-0.494165\pi$
$719$	2.62697e11	0.0366585	0.0183293	+	0.999832i	$0.494165\pi$
$720$	0	0
$721$	1.43434e12	0.197672
$722$	0	0
$723$	−3.19412e12	−0.434739
$724$	0	0
$725$	−2.50777e12	−0.337106
$726$	0	0
$727$	5.69375e12	0.755951	0.377976	−	0.925816i	$-0.376620\pi$
$727$	5.69375e12	0.755951	0.377976	+	0.925816i	$0.376620\pi$
$728$	0	0
$729$	8.37055e11	0.109769
$730$	0	0
$731$	1.70423e13	2.20749
$732$	0	0
$733$	−1.28415e13	−1.64304	−0.821521	−	0.570178i	$-0.806874\pi$
$733$	−1.28415e13	−1.64304	−0.821521	+	0.570178i	$0.806874\pi$
$734$	0	0
$735$	6.12147e12	0.773682
$736$	0	0
$737$	−1.41165e12	−0.176248
$738$	0	0
$739$	−2.66001e12	−0.328083	−0.164042	−	0.986453i	$-0.552453\pi$
$739$	−2.66001e12	−0.328083	−0.164042	+	0.986453i	$0.552453\pi$
$740$	0	0
$741$	1.17655e12	0.143360
$742$	0	0
$743$	−1.02254e13	−1.23092	−0.615462	−	0.788166i	$-0.711031\pi$
$743$	−1.02254e13	−1.23092	−0.615462	+	0.788166i	$0.711031\pi$
$744$	0	0
$745$	1.72120e13	2.04705
$746$	0	0
$747$	−1.98844e12	−0.233652
$748$	0	0
$749$	6.56181e11	0.0761826
$750$	0	0
$751$	−7.08317e12	−0.812546	−0.406273	−	0.913752i	$-0.633172\pi$
$751$	−7.08317e12	−0.812546	−0.406273	+	0.913752i	$0.633172\pi$
$752$	0	0
$753$	8.32720e11	0.0943890
$754$	0	0
$755$	4.54396e12	0.508948
$756$	0	0
$757$	1.55185e13	1.71758	0.858792	−	0.512325i	$-0.171216\pi$
$757$	1.55185e13	1.71758	0.858792	+	0.512325i	$0.171216\pi$
$758$	0	0
$759$	−8.03934e11	−0.0879291
$760$	0	0
$761$	3.36663e12	0.363885	0.181943	−	0.983309i	$-0.441761\pi$
$761$	3.36663e12	0.363885	0.181943	+	0.983309i	$0.441761\pi$
$762$	0	0
$763$	−1.74565e11	−0.0186465
$764$	0	0
$765$	−1.65249e13	−1.74447
$766$	0	0
$767$	−1.23918e13	−1.29287
$768$	0	0
$769$	−9.31790e12	−0.960836	−0.480418	−	0.877040i	$-0.659515\pi$
$769$	−9.31790e12	−0.960836	−0.480418	+	0.877040i	$0.659515\pi$
$770$	0	0
$771$	1.22322e12	0.124670
$772$	0	0
$773$	−7.33840e12	−0.739254	−0.369627	−	0.929180i	$-0.620514\pi$
$773$	−7.33840e12	−0.739254	−0.369627	+	0.929180i	$0.620514\pi$
$774$	0	0
$775$	7.56487e11	0.0753258
$776$	0	0
$777$	9.67046e11	0.0951815
$778$	0	0
$779$	3.05390e12	0.297123
$780$	0	0
$781$	1.06528e12	0.102456
$782$	0	0
$783$	1.49741e12	0.142368
$784$	0	0
$785$	−1.19951e13	−1.12744
$786$	0	0
$787$	−1.81540e13	−1.68689	−0.843443	−	0.537219i	$-0.819475\pi$
$787$	−1.81540e13	−1.68689	−0.843443	+	0.537219i	$0.819475\pi$
$788$	0	0
$789$	−5.58923e12	−0.513459
$790$	0	0
$791$	2.10298e12	0.191003
$792$	0	0
$793$	5.45932e12	0.490241
$794$	0	0
$795$	1.07182e13	0.951632
$796$	0	0
$797$	5.44638e12	0.478129	0.239065	−	0.971004i	$-0.423159\pi$
$797$	5.44638e12	0.478129	0.239065	+	0.971004i	$0.423159\pi$
$798$	0	0
$799$	−1.02227e13	−0.887371
$800$	0	0
$801$	−1.08386e13	−0.930312
$802$	0	0
$803$	−8.10312e11	−0.0687753
$804$	0	0
$805$	7.70002e12	0.646265
$806$	0	0
$807$	6.21582e11	0.0515903
$808$	0	0
$809$	2.44611e11	0.0200774	0.0100387	−	0.999950i	$-0.496805\pi$
$809$	2.44611e11	0.0200774	0.0100387	+	0.999950i	$0.496805\pi$
$810$	0	0
$811$	−1.86901e13	−1.51711	−0.758555	−	0.651609i	$-0.774094\pi$
$811$	−1.86901e13	−1.51711	−0.758555	+	0.651609i	$0.774094\pi$
$812$	0	0
$813$	1.05139e13	0.844030
$814$	0	0
$815$	6.95430e12	0.552134
$816$	0	0
$817$	4.95605e12	0.389167
$818$	0	0
$819$	3.17011e12	0.246205
$820$	0	0
$821$	2.98096e12	0.228987	0.114494	−	0.993424i	$-0.463475\pi$
$821$	2.98096e12	0.228987	0.114494	+	0.993424i	$0.463475\pi$
$822$	0	0
$823$	1.01707e13	0.772774	0.386387	−	0.922337i	$-0.373723\pi$
$823$	1.01707e13	0.772774	0.386387	+	0.922337i	$0.373723\pi$
$824$	0	0
$825$	1.50920e12	0.113424
$826$	0	0
$827$	−1.99682e13	−1.48445	−0.742223	−	0.670153i	$-0.766229\pi$
$827$	−1.99682e13	−1.48445	−0.742223	+	0.670153i	$0.766229\pi$
$828$	0	0
$829$	2.35288e13	1.73023	0.865115	−	0.501574i	$-0.167245\pi$
$829$	2.35288e13	1.73023	0.865115	+	0.501574i	$0.167245\pi$
$830$	0	0
$831$	−7.52516e12	−0.547408
$832$	0	0
$833$	1.70293e13	1.22545
$834$	0	0
$835$	3.57561e13	2.54543
$836$	0	0
$837$	−4.51704e11	−0.0318119
$838$	0	0
$839$	−9.16607e12	−0.638637	−0.319319	−	0.947647i	$-0.603454\pi$
$839$	−9.16607e12	−0.638637	−0.319319	+	0.947647i	$0.603454\pi$
$840$	0	0
$841$	−1.40929e13	−0.971444
$842$	0	0
$843$	5.21884e12	0.355918
$844$	0	0
$845$	−1.87879e13	−1.26772
$846$	0	0
$847$	−3.56509e12	−0.238010
$848$	0	0
$849$	−5.63484e12	−0.372217
$850$	0	0
$851$	−1.96450e13	−1.28401
$852$	0	0
$853$	9.94411e12	0.643125	0.321562	−	0.946888i	$-0.395792\pi$
$853$	9.94411e12	0.643125	0.321562	+	0.946888i	$0.395792\pi$
$854$	0	0
$855$	−4.80560e12	−0.307539
$856$	0	0
$857$	−1.90800e13	−1.20827	−0.604137	−	0.796880i	$-0.706482\pi$
$857$	−1.90800e13	−1.20827	−0.604137	+	0.796880i	$0.706482\pi$
$858$	0	0
$859$	−3.10124e12	−0.194342	−0.0971710	−	0.995268i	$-0.530979\pi$
$859$	−3.10124e12	−0.194342	−0.0971710	+	0.995268i	$0.530979\pi$
$860$	0	0
$861$	−2.39420e12	−0.148473
$862$	0	0
$863$	2.40338e13	1.47494	0.737470	−	0.675380i	$-0.236020\pi$
$863$	2.40338e13	1.47494	0.737470	+	0.675380i	$0.236020\pi$
$864$	0	0
$865$	−3.52099e13	−2.13842
$866$	0	0
$867$	5.47728e12	0.329214
$868$	0	0
$869$	−1.46276e12	−0.0870129
$870$	0	0
$871$	3.29023e13	1.93707
$872$	0	0
$873$	2.00714e13	1.16954
$874$	0	0
$875$	−7.20905e12	−0.415759
$876$	0	0
$877$	−5.20893e12	−0.297338	−0.148669	−	0.988887i	$-0.547499\pi$
$877$	−5.20893e12	−0.297338	−0.148669	+	0.988887i	$0.547499\pi$
$878$	0	0
$879$	−7.10151e12	−0.401237
$880$	0	0
$881$	−7.36387e12	−0.411826	−0.205913	−	0.978570i	$-0.566016\pi$
$881$	−7.36387e12	−0.411826	−0.205913	+	0.978570i	$0.566016\pi$
$882$	0	0
$883$	−2.98715e13	−1.65362	−0.826808	−	0.562485i	$-0.809845\pi$
$883$	−2.98715e13	−1.65362	−0.826808	+	0.562485i	$0.809845\pi$
$884$	0	0
$885$	−1.47269e13	−0.806988
$886$	0	0
$887$	1.43208e13	0.776802	0.388401	−	0.921490i	$-0.373027\pi$
$887$	1.43208e13	0.776802	0.388401	+	0.921490i	$0.373027\pi$
$888$	0	0
$889$	−2.52650e12	−0.135663
$890$	0	0
$891$	8.44086e11	0.0448680
$892$	0	0
$893$	−2.97285e12	−0.156438
$894$	0	0
$895$	−4.41153e13	−2.29819
$896$	0	0
$897$	1.87379e13	0.966394
$898$	0	0
$899$	−1.24964e11	−0.00638069
$900$	0	0
$901$	2.98169e13	1.50730
$902$	0	0
$903$	−3.88544e12	−0.194467
$904$	0	0
$905$	−6.80020e13	−3.36979
$906$	0	0
$907$	3.98348e13	1.95448	0.977238	−	0.212146i	$-0.0680453\pi$
$907$	3.98348e13	1.95448	0.977238	+	0.212146i	$0.0680453\pi$
$908$	0	0
$909$	−2.10487e13	−1.02256
$910$	0	0
$911$	8.77892e12	0.422288	0.211144	−	0.977455i	$-0.432281\pi$
$911$	8.77892e12	0.422288	0.211144	+	0.977455i	$0.432281\pi$
$912$	0	0
$913$	7.58442e11	0.0361247
$914$	0	0
$915$	6.48809e12	0.306000
$916$	0	0
$917$	5.60918e12	0.261962
$918$	0	0
$919$	−2.89837e12	−0.134040	−0.0670200	−	0.997752i	$-0.521349\pi$
$919$	−2.89837e12	−0.134040	−0.0670200	+	0.997752i	$0.521349\pi$
$920$	0	0
$921$	−1.10093e13	−0.504188
$922$	0	0
$923$	−2.48293e13	−1.12605
$924$	0	0
$925$	3.68790e13	1.65631
$926$	0	0
$927$	−1.42567e13	−0.634104
$928$	0	0
$929$	1.09141e13	0.480749	0.240375	−	0.970680i	$-0.422730\pi$
$929$	1.09141e13	0.480749	0.240375	+	0.970680i	$0.422730\pi$
$930$	0	0
$931$	4.95228e12	0.216039
$932$	0	0
$933$	7.25793e12	0.313578
$934$	0	0
$935$	6.30304e12	0.269711
$936$	0	0
$937$	3.31892e13	1.40659	0.703296	−	0.710897i	$-0.251711\pi$
$937$	3.31892e13	1.40659	0.703296	+	0.710897i	$0.251711\pi$
$938$	0	0
$939$	6.31832e12	0.265220
$940$	0	0
$941$	−4.06875e13	−1.69164	−0.845819	−	0.533471i	$-0.820888\pi$
$941$	−4.06875e13	−1.69164	−0.845819	+	0.533471i	$0.820888\pi$
$942$	0	0
$943$	4.86368e13	2.00292
$944$	0	0
$945$	8.63120e12	0.352069
$946$	0	0
$947$	1.90875e13	0.771211	0.385606	−	0.922664i	$-0.373993\pi$
$947$	1.90875e13	0.771211	0.385606	+	0.922664i	$0.373993\pi$
$948$	0	0
$949$	1.88865e13	0.755882
$950$	0	0
$951$	1.83165e13	0.726155
$952$	0	0
$953$	−2.79671e12	−0.109832	−0.0549161	−	0.998491i	$-0.517489\pi$
$953$	−2.79671e12	−0.109832	−0.0549161	+	0.998491i	$0.517489\pi$
$954$	0	0
$955$	6.82800e11	0.0265631
$956$	0	0
$957$	−2.49306e11	−0.00960790
$958$	0	0
$959$	−4.97250e12	−0.189842
$960$	0	0
$961$	−2.64019e13	−0.998574
$962$	0	0
$963$	−6.52213e12	−0.244383
$964$	0	0
$965$	−3.46136e13	−1.28492
$966$	0	0
$967$	1.90809e10	0.000701745 0	0.000350873	−	1.00000i	$-0.499888\pi$
$967$	1.90809e10	0.000701745 0	0.000350873		1.00000i	$0.499888\pi$
$968$	0	0
$969$	3.88982e12	0.141733
$970$	0	0
$971$	4.13526e12	0.149285	0.0746426	−	0.997210i	$-0.476218\pi$
$971$	4.13526e12	0.149285	0.0746426	+	0.997210i	$0.476218\pi$
$972$	0	0
$973$	−1.33306e13	−0.476807
$974$	0	0
$975$	−3.51761e13	−1.24660
$976$	0	0
$977$	3.74336e13	1.31443	0.657213	−	0.753705i	$-0.271735\pi$
$977$	3.74336e13	1.31443	0.657213	+	0.753705i	$0.271735\pi$
$978$	0	0
$979$	4.13414e12	0.143835
$980$	0	0
$981$	1.73510e12	0.0598155
$982$	0	0
$983$	−1.37412e13	−0.469390	−0.234695	−	0.972069i	$-0.575409\pi$
$983$	−1.37412e13	−0.469390	−0.234695	+	0.972069i	$0.575409\pi$
$984$	0	0
$985$	−4.88830e13	−1.65461
$986$	0	0
$987$	2.33066e12	0.0781721
$988$	0	0
$989$	7.89307e13	2.62339
$990$	0	0
$991$	2.92261e13	0.962584	0.481292	−	0.876560i	$-0.340168\pi$
$991$	2.92261e13	0.962584	0.481292	+	0.876560i	$0.340168\pi$
$992$	0	0
$993$	8.52927e12	0.278381
$994$	0	0
$995$	−9.57564e13	−3.09716
$996$	0	0
$997$	1.27081e13	0.407335	0.203668	−	0.979040i	$-0.434714\pi$
$997$	1.27081e13	0.407335	0.203668	+	0.979040i	$0.434714\pi$
$998$	0	0
$999$	−2.20207e13	−0.699499

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.10.a.e.1.3		4
4.3	odd	2		38.10.a.d.1.2	✓	4
12.11	even	2		342.10.a.l.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.10.a.d.1.2	✓	4	4.3	odd	2
304.10.a.e.1.3		4	1.1	even	1	trivial
342.10.a.l.1.4		4	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:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	304.a (trivial)

\(f(q)\)	\(=\)	\(q+66.6053 q^{3} -2418.56 q^{5} +1533.95 q^{7} -15246.7 q^{9} +O(q^{10})\)	\(q+66.6053 q^{3} -2418.56 q^{5} +1533.95 q^{7} -15246.7 q^{9} +5815.50 q^{11} -135546. q^{13} -161089. q^{15} -448133. q^{17} -130321. q^{19} +102169. q^{21} -2.07551e6 q^{23} +3.89629e6 q^{25} -2.32650e6 q^{27} -643630. q^{29} +194155. q^{31} +387343. q^{33} -3.70994e6 q^{35} +9.46515e6 q^{37} -9.02808e6 q^{39} -2.34337e7 q^{41} -3.80295e7 q^{43} +3.68751e7 q^{45} +2.28118e7 q^{47} -3.80006e7 q^{49} -2.98480e7 q^{51} -6.65359e7 q^{53} -1.40651e7 q^{55} -8.68007e6 q^{57} +9.14212e7 q^{59} -4.02765e7 q^{61} -2.33877e7 q^{63} +3.27826e8 q^{65} -2.42739e8 q^{67} -1.38240e8 q^{69} +1.83180e8 q^{71} -1.39336e8 q^{73} +2.59514e8 q^{75} +8.92069e6 q^{77} -2.51527e8 q^{79} +1.45144e8 q^{81} +1.30417e8 q^{83} +1.08383e9 q^{85} -4.28692e7 q^{87} +7.10883e8 q^{89} -2.07921e8 q^{91} +1.29318e7 q^{93} +3.15189e8 q^{95} -1.31644e9 q^{97} -8.86675e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 84 q^{3} - 1395 q^{5} - 12307 q^{7} + 16538 q^{9}+O(q^{10}) \) 4 * q - 84 * q^3 - 1395 * q^5 - 12307 * q^7 + 16538 * q^9	\( 4 q - 84 q^{3} - 1395 q^{5} - 12307 q^{7} + 16538 q^{9} + 104249 q^{11} + 120486 q^{13} + 591090 q^{15} - 412139 q^{17} - 521284 q^{19} + 2437006 q^{21} - 3010300 q^{23} + 9760585 q^{25} - 12387978 q^{27} + 6153240 q^{29} - 12774024 q^{31} - 3258022 q^{33} - 9823425 q^{35} + 20506048 q^{37} - 69881444 q^{39} + 11620300 q^{41} - 7698327 q^{43} - 124015815 q^{45} + 31581083 q^{47} + 18970383 q^{49} + 8594812 q^{51} + 72549422 q^{53} - 21332505 q^{55} + 10946964 q^{57} + 149234120 q^{59} + 129004373 q^{61} - 102967551 q^{63} + 124691700 q^{65} - 132595266 q^{67} - 45529972 q^{69} + 47138482 q^{71} - 39332795 q^{73} - 824627010 q^{75} - 165933719 q^{77} + 307010840 q^{79} + 1305551744 q^{81} + 746568232 q^{83} - 105005985 q^{85} + 82148208 q^{87} + 286943482 q^{89} - 3155781114 q^{91} + 1151901596 q^{93} + 181797795 q^{95} + 793519958 q^{97} + 1681833809 q^{99}+O(q^{100}) \) 4 * q - 84 * q^3 - 1395 * q^5 - 12307 * q^7 + 16538 * q^9 + 104249 * q^11 + 120486 * q^13 + 591090 * q^15 - 412139 * q^17 - 521284 * q^19 + 2437006 * q^21 - 3010300 * q^23 + 9760585 * q^25 - 12387978 * q^27 + 6153240 * q^29 - 12774024 * q^31 - 3258022 * q^33 - 9823425 * q^35 + 20506048 * q^37 - 69881444 * q^39 + 11620300 * q^41 - 7698327 * q^43 - 124015815 * q^45 + 31581083 * q^47 + 18970383 * q^49 + 8594812 * q^51 + 72549422 * q^53 - 21332505 * q^55 + 10946964 * q^57 + 149234120 * q^59 + 129004373 * q^61 - 102967551 * q^63 + 124691700 * q^65 - 132595266 * q^67 - 45529972 * q^69 + 47138482 * q^71 - 39332795 * q^73 - 824627010 * q^75 - 165933719 * q^77 + 307010840 * q^79 + 1305551744 * q^81 + 746568232 * q^83 - 105005985 * q^85 + 82148208 * q^87 + 286943482 * q^89 - 3155781114 * q^91 + 1151901596 * q^93 + 181797795 * q^95 + 793519958 * q^97 + 1681833809 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more