LMFDB - Embedded newform 304.10.a.i.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [304,10,Mod(1,304)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

Level:	$ N $	$=$	$ 304 = 2^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	304.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,-7,0,3894] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$156.570894194$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 3356 x^{6} - 1330 x^{5} + 3186388 x^{4} - 1801192 x^{3} - 758043152 x^{2} + \cdots - 16080668672 $ x^8 - x^7 - 3356x^6 - 1330x^5 + 3186388x^4 - 1801192x^3 - 758043152x^2 + 6645763712x - 16080668672
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{16}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 19)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$34.1433$ 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)

$f(q)$	$=$	$q-6.24328 q^{3} -1240.15 q^{5} +8874.26 q^{7} -19644.0 q^{9} -14509.7 q^{11} +171819. q^{13} +7742.60 q^{15} +250400. q^{17} -130321. q^{19} -55404.5 q^{21} -370879. q^{23} -415154. q^{25} +245530. q^{27} +5.30618e6 q^{29} -3.66089e6 q^{31} +90587.9 q^{33} -1.10054e7 q^{35} -1.93497e7 q^{37} -1.07271e6 q^{39} +2.31432e7 q^{41} +1.20515e7 q^{43} +2.43615e7 q^{45} -6.98659e6 q^{47} +3.83990e7 q^{49} -1.56332e6 q^{51} -4.90402e7 q^{53} +1.79941e7 q^{55} +813630. q^{57} -6.35336e7 q^{59} -7.02244e6 q^{61} -1.74326e8 q^{63} -2.13081e8 q^{65} +1.38553e8 q^{67} +2.31550e6 q^{69} +1.11040e8 q^{71} +1.22351e8 q^{73} +2.59192e6 q^{75} -1.28763e8 q^{77} -6.37779e8 q^{79} +3.85120e8 q^{81} -7.84665e8 q^{83} -3.10534e8 q^{85} -3.31279e7 q^{87} +6.81105e8 q^{89} +1.52476e9 q^{91} +2.28560e7 q^{93} +1.61618e8 q^{95} +8.49995e8 q^{97} +2.85028e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 7 q^{3} + 3894 q^{5} + 7133 q^{7} + 102715 q^{9} - 172818 q^{11} + 109291 q^{13} - 457332 q^{15} + 583575 q^{17} - 1042568 q^{19} + 372767 q^{21} + 2292405 q^{23} + 205328 q^{25} + 15261551 q^{27}+ \cdots + 1682553420 q^{99}+O(q^{100}) $ 8 * q - 7 * q^3 + 3894 * q^5 + 7133 * q^7 + 102715 * q^9 - 172818 * q^11 + 109291 * q^13 - 457332 * q^15 + 583575 * q^17 - 1042568 * q^19 + 372767 * q^21 + 2292405 * q^23 + 205328 * q^25 + 15261551 * q^27 + 4546335 * q^29 + 8277416 * q^31 + 13959234 * q^33 - 15633768 * q^35 + 24485536 * q^37 - 16641725 * q^39 + 35051946 * q^41 - 36624268 * q^43 + 138146370 * q^45 - 9865524 * q^47 + 111765087 * q^49 - 109712553 * q^51 + 13963683 * q^53 - 261540660 * q^55 + 912247 * q^57 - 21215463 * q^59 + 180552112 * q^61 - 303758726 * q^63 - 21864624 * q^65 - 1767355 * q^67 + 504738939 * q^69 - 273631578 * q^71 - 263711543 * q^73 + 340714091 * q^75 + 441243702 * q^77 - 744472174 * q^79 + 1867545184 * q^81 - 832139910 * q^83 + 1013793492 * q^85 + 231595575 * q^87 - 291457734 * q^89 + 1351077415 * q^91 + 537395144 * q^93 - 507469974 * q^95 - 3483837140 * q^97 + 1682553420 * q^99

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
$3$	−6.24328	−0.0445007	−0.0222504	−	0.999752i	$-0.507083\pi$
$3$	−6.24328	−0.0445007	−0.0222504	+	0.999752i	$0.507083\pi$
$4$	0	0
$5$	−1240.15	−0.887379	−0.443689	−	0.896181i	$-0.646331\pi$
$5$	−1240.15	−0.887379	−0.443689	+	0.896181i	$0.646331\pi$
$6$	0	0
$7$	8874.26	1.39698	0.698492	−	0.715618i	$-0.253855\pi$
$7$	8874.26	1.39698	0.698492	+	0.715618i	$0.253855\pi$
$8$	0	0
$9$	−19644.0	−0.998020
$10$	0	0
$11$	−14509.7	−0.298807	−0.149403	−	0.988776i	$-0.547735\pi$
$11$	−14509.7	−0.298807	−0.149403	+	0.988776i	$0.547735\pi$
$12$	0	0
$13$	171819.	1.66850	0.834248	−	0.551390i	$-0.185902\pi$
$13$	171819.	1.66850	0.834248	+	0.551390i	$0.185902\pi$
$14$	0	0
$15$	7742.60	0.0394890
$16$	0	0
$17$	250400.	0.727134	0.363567	−	0.931568i	$-0.381559\pi$
$17$	250400.	0.727134	0.363567	+	0.931568i	$0.381559\pi$
$18$	0	0
$19$	−130321.	−0.229416
$19$	−130321.	−0.229416
$20$	0	0
$21$	−55404.5	−0.0621668
$22$	0	0
$23$	−370879.	−0.276349	−0.138174	−	0.990408i	$-0.544123\pi$
$23$	−370879.	−0.276349	−0.138174	+	0.990408i	$0.544123\pi$
$24$	0	0
$25$	−415154.	−0.212559
$26$	0	0
$27$	245530.	0.0889133
$28$	0	0
$29$	5.30618e6	1.39313	0.696564	−	0.717495i	$-0.254711\pi$
$29$	5.30618e6	1.39313	0.696564	+	0.717495i	$0.254711\pi$
$30$	0	0
$31$	−3.66089e6	−0.711966	−0.355983	−	0.934492i	$-0.615854\pi$
$31$	−3.66089e6	−0.711966	−0.355983	+	0.934492i	$0.615854\pi$
$32$	0	0
$33$	90587.9	0.0132971
$34$	0	0
$35$	−1.10054e7	−1.23965
$36$	0	0
$37$	−1.93497e7	−1.69733	−0.848666	−	0.528929i	$-0.822594\pi$
$37$	−1.93497e7	−1.69733	−0.848666	+	0.528929i	$0.822594\pi$
$38$	0	0
$39$	−1.07271e6	−0.0742493
$40$	0	0
$41$	2.31432e7	1.27907	0.639537	−	0.768760i	$-0.279126\pi$
$41$	2.31432e7	1.27907	0.639537	+	0.768760i	$0.279126\pi$
$42$	0	0
$43$	1.20515e7	0.537565	0.268783	−	0.963201i	$-0.413379\pi$
$43$	1.20515e7	0.537565	0.268783	+	0.963201i	$0.413379\pi$
$44$	0	0
$45$	2.43615e7	0.885622
$46$	0	0
$47$	−6.98659e6	−0.208846	−0.104423	−	0.994533i	$-0.533299\pi$
$47$	−6.98659e6	−0.208846	−0.104423	+	0.994533i	$0.533299\pi$
$48$	0	0
$49$	3.83990e7	0.951562
$50$	0	0
$51$	−1.56332e6	−0.0323580
$52$	0	0
$53$	−4.90402e7	−0.853711	−0.426855	−	0.904320i	$-0.640379\pi$
$53$	−4.90402e7	−0.853711	−0.426855	+	0.904320i	$0.640379\pi$
$54$	0	0
$55$	1.79941e7	0.265155
$56$	0	0
$57$	813630.	0.0102092
$58$	0	0
$59$	−6.35336e7	−0.682605	−0.341303	−	0.939954i	$-0.610868\pi$
$59$	−6.35336e7	−0.682605	−0.341303	+	0.939954i	$0.610868\pi$
$60$	0	0
$61$	−7.02244e6	−0.0649387	−0.0324694	−	0.999473i	$-0.510337\pi$
$61$	−7.02244e6	−0.0649387	−0.0324694	+	0.999473i	$0.510337\pi$
$62$	0	0
$63$	−1.74326e8	−1.39422
$64$	0	0
$65$	−2.13081e8	−1.48059
$66$	0	0
$67$	1.38553e8	0.840000	0.420000	−	0.907524i	$-0.362030\pi$
$67$	1.38553e8	0.840000	0.420000	+	0.907524i	$0.362030\pi$
$68$	0	0
$69$	2.31550e6	0.0122977
$70$	0	0
$71$	1.11040e8	0.518579	0.259290	−	0.965800i	$-0.416512\pi$
$71$	1.11040e8	0.518579	0.259290	+	0.965800i	$0.416512\pi$
$72$	0	0
$73$	1.22351e8	0.504259	0.252130	−	0.967693i	$-0.418869\pi$
$73$	1.22351e8	0.504259	0.252130	+	0.967693i	$0.418869\pi$
$74$	0	0
$75$	2.59192e6	0.00945902
$76$	0	0
$77$	−1.28763e8	−0.417428
$78$	0	0
$79$	−6.37779e8	−1.84225	−0.921124	−	0.389269i	$-0.872728\pi$
$79$	−6.37779e8	−1.84225	−0.921124	+	0.389269i	$0.872728\pi$
$80$	0	0
$81$	3.85120e8	0.994063
$82$	0	0
$83$	−7.84665e8	−1.81482	−0.907409	−	0.420250i	$-0.861943\pi$
$83$	−7.84665e8	−1.81482	−0.907409	+	0.420250i	$0.861943\pi$
$84$	0	0
$85$	−3.10534e8	−0.645243
$86$	0	0
$87$	−3.31279e7	−0.0619952
$88$	0	0
$89$	6.81105e8	1.15069	0.575346	−	0.817910i	$-0.304868\pi$
$89$	6.81105e8	1.15069	0.575346	+	0.817910i	$0.304868\pi$
$90$	0	0
$91$	1.52476e9	2.33086
$92$	0	0
$93$	2.28560e7	0.0316830
$94$	0	0
$95$	1.61618e8	0.203579
$96$	0	0
$97$	8.49995e8	0.974863	0.487432	−	0.873161i	$-0.337934\pi$
$97$	8.49995e8	0.974863	0.487432	+	0.873161i	$0.337934\pi$
$98$	0	0
$99$	2.85028e8	0.298215
$100$	0	0
$101$	4.49876e8	0.430177	0.215088	−	0.976595i	$-0.430996\pi$
$101$	4.49876e8	0.430177	0.215088	+	0.976595i	$0.430996\pi$
$102$	0	0
$103$	7.66689e8	0.671199	0.335600	−	0.942005i	$-0.391061\pi$
$103$	7.66689e8	0.671199	0.335600	+	0.942005i	$0.391061\pi$
$104$	0	0
$105$	6.87099e7	0.0551655
$106$	0	0
$107$	1.97149e9	1.45401	0.727006	−	0.686631i	$-0.240911\pi$
$107$	1.97149e9	1.45401	0.727006	+	0.686631i	$0.240911\pi$
$108$	0	0
$109$	−3.72061e8	−0.252462	−0.126231	−	0.992001i	$-0.540288\pi$
$109$	−3.72061e8	−0.252462	−0.126231	+	0.992001i	$0.540288\pi$
$110$	0	0
$111$	1.20806e8	0.0755325
$112$	0	0
$113$	2.17965e9	1.25758	0.628788	−	0.777576i	$-0.283551\pi$
$113$	2.17965e9	1.25758	0.628788	+	0.777576i	$0.283551\pi$
$114$	0	0
$115$	4.59946e8	0.245226
$116$	0	0
$117$	−3.37521e9	−1.66519
$118$	0	0
$119$	2.22212e9	1.01579
$120$	0	0
$121$	−2.14742e9	−0.910715
$122$	0	0
$123$	−1.44489e8	−0.0569197
$124$	0	0
$125$	2.93702e9	1.07600
$126$	0	0
$127$	−3.25291e9	−1.10957	−0.554785	−	0.831994i	$-0.687200\pi$
$127$	−3.25291e9	−1.10957	−0.554785	+	0.831994i	$0.687200\pi$
$128$	0	0
$129$	−7.52406e7	−0.0239221
$130$	0	0
$131$	−2.11327e8	−0.0626953	−0.0313476	−	0.999509i	$-0.509980\pi$
$131$	−2.11327e8	−0.0626953	−0.0313476	+	0.999509i	$0.509980\pi$
$132$	0	0
$133$	−1.15650e9	−0.320490
$134$	0	0
$135$	−3.04493e8	−0.0788998
$136$	0	0
$137$	−6.91279e9	−1.67653	−0.838264	−	0.545265i	$-0.816429\pi$
$137$	−6.91279e9	−1.67653	−0.838264	+	0.545265i	$0.816429\pi$
$138$	0	0
$139$	3.99615e9	0.907978	0.453989	−	0.891007i	$-0.350000\pi$
$139$	3.99615e9	0.907978	0.453989	+	0.891007i	$0.350000\pi$
$140$	0	0
$141$	4.36193e7	0.00929378
$142$	0	0
$143$	−2.49303e9	−0.498558
$144$	0	0
$145$	−6.58045e9	−1.23623
$146$	0	0
$147$	−2.39735e8	−0.0423452
$148$	0	0
$149$	9.81038e9	1.63060	0.815301	−	0.579038i	$-0.196572\pi$
$149$	9.81038e9	1.63060	0.815301	+	0.579038i	$0.196572\pi$
$150$	0	0
$151$	−1.13142e10	−1.77103	−0.885517	−	0.464607i	$-0.846196\pi$
$151$	−1.13142e10	−1.77103	−0.885517	+	0.464607i	$0.846196\pi$
$152$	0	0
$153$	−4.91887e9	−0.725694
$154$	0	0
$155$	4.54005e9	0.631784
$156$	0	0
$157$	9.30128e9	1.22178	0.610892	−	0.791714i	$-0.290811\pi$
$157$	9.30128e9	1.22178	0.610892	+	0.791714i	$0.290811\pi$
$158$	0	0
$159$	3.06172e8	0.0379908
$160$	0	0
$161$	−3.29128e9	−0.386054
$162$	0	0
$163$	3.82971e9	0.424934	0.212467	−	0.977168i	$-0.431850\pi$
$163$	3.82971e9	0.424934	0.212467	+	0.977168i	$0.431850\pi$
$164$	0	0
$165$	−1.12342e8	−0.0117996
$166$	0	0
$167$	7.83909e9	0.779905	0.389952	−	0.920835i	$-0.372491\pi$
$167$	7.83909e9	0.779905	0.389952	+	0.920835i	$0.372491\pi$
$168$	0	0
$169$	1.89171e10	1.78388
$170$	0	0
$171$	2.56003e9	0.228961
$172$	0	0
$173$	9.97767e9	0.846880	0.423440	−	0.905924i	$-0.360823\pi$
$173$	9.97767e9	0.846880	0.423440	+	0.905924i	$0.360823\pi$
$174$	0	0
$175$	−3.68419e9	−0.296941
$176$	0	0
$177$	3.96658e8	0.0303764
$178$	0	0
$179$	−6.49148e9	−0.472612	−0.236306	−	0.971679i	$-0.575937\pi$
$179$	−6.49148e9	−0.472612	−0.236306	+	0.971679i	$0.575937\pi$
$180$	0	0
$181$	1.36985e10	0.948679	0.474339	−	0.880342i	$-0.342687\pi$
$181$	1.36985e10	0.948679	0.474339	+	0.880342i	$0.342687\pi$
$182$	0	0
$183$	4.38430e7	0.00288982
$184$	0	0
$185$	2.39966e10	1.50618
$186$	0	0
$187$	−3.63322e9	−0.217272
$188$	0	0
$189$	2.17889e9	0.124210
$190$	0	0
$191$	2.21243e10	1.20287	0.601435	−	0.798921i	$-0.294596\pi$
$191$	2.21243e10	1.20287	0.601435	+	0.798921i	$0.294596\pi$
$192$	0	0
$193$	−5.70496e9	−0.295968	−0.147984	−	0.988990i	$-0.547278\pi$
$193$	−5.70496e9	−0.295968	−0.147984	+	0.988990i	$0.547278\pi$
$194$	0	0
$195$	1.33032e9	0.0658872
$196$	0	0
$197$	3.08490e10	1.45929	0.729647	−	0.683824i	$-0.239684\pi$
$197$	3.08490e10	1.45929	0.729647	+	0.683824i	$0.239684\pi$
$198$	0	0
$199$	1.72813e10	0.781156	0.390578	−	0.920570i	$-0.372275\pi$
$199$	1.72813e10	0.781156	0.390578	+	0.920570i	$0.372275\pi$
$200$	0	0
$201$	−8.65025e8	−0.0373806
$202$	0	0
$203$	4.70884e10	1.94618
$204$	0	0
$205$	−2.87010e10	−1.13502
$206$	0	0
$207$	7.28556e9	0.275801
$208$	0	0
$209$	1.89091e9	0.0685509
$210$	0	0
$211$	−8.20745e9	−0.285060	−0.142530	−	0.989790i	$-0.545524\pi$
$211$	−8.20745e9	−0.285060	−0.142530	+	0.989790i	$0.545524\pi$
$212$	0	0
$213$	−6.93251e8	−0.0230772
$214$	0	0
$215$	−1.49456e10	−0.477024
$216$	0	0
$217$	−3.24877e10	−0.994605
$218$	0	0
$219$	−7.63870e8	−0.0224399
$220$	0	0
$221$	4.30234e10	1.21322
$222$	0	0
$223$	3.07148e10	0.831718	0.415859	−	0.909429i	$-0.363481\pi$
$223$	3.07148e10	0.831718	0.415859	+	0.909429i	$0.363481\pi$
$224$	0	0
$225$	8.15529e9	0.212138
$226$	0	0
$227$	2.43129e10	0.607744	0.303872	−	0.952713i	$-0.401721\pi$
$227$	2.43129e10	0.607744	0.303872	+	0.952713i	$0.401721\pi$
$228$	0	0
$229$	5.56525e10	1.33729	0.668644	−	0.743582i	$-0.266875\pi$
$229$	5.56525e10	1.33729	0.668644	+	0.743582i	$0.266875\pi$
$230$	0	0
$231$	8.03901e8	0.0185758
$232$	0	0
$233$	−3.47756e10	−0.772988	−0.386494	−	0.922292i	$-0.626314\pi$
$233$	−3.47756e10	−0.772988	−0.386494	+	0.922292i	$0.626314\pi$
$234$	0	0
$235$	8.66442e9	0.185325
$236$	0	0
$237$	3.98183e9	0.0819814
$238$	0	0
$239$	4.50629e10	0.893364	0.446682	−	0.894693i	$-0.352606\pi$
$239$	4.50629e10	0.893364	0.446682	+	0.894693i	$0.352606\pi$
$240$	0	0
$241$	3.19551e10	0.610189	0.305094	−	0.952322i	$-0.401312\pi$
$241$	3.19551e10	0.610189	0.305094	+	0.952322i	$0.401312\pi$
$242$	0	0
$243$	−7.23717e9	−0.133150
$244$	0	0
$245$	−4.76204e10	−0.844396
$246$	0	0
$247$	−2.23916e10	−0.382779
$248$	0	0
$249$	4.89888e9	0.0807607
$250$	0	0
$251$	4.83205e10	0.768422	0.384211	−	0.923245i	$-0.374474\pi$
$251$	4.83205e10	0.768422	0.384211	+	0.923245i	$0.374474\pi$
$252$	0	0
$253$	5.38133e9	0.0825748
$254$	0	0
$255$	1.93875e9	0.0287138
$256$	0	0
$257$	1.61379e10	0.230754	0.115377	−	0.993322i	$-0.463192\pi$
$257$	1.61379e10	0.230754	0.115377	+	0.993322i	$0.463192\pi$
$258$	0	0
$259$	−1.71715e11	−2.37115
$260$	0	0
$261$	−1.04235e11	−1.39037
$262$	0	0
$263$	1.13746e11	1.46601	0.733003	−	0.680225i	$-0.238118\pi$
$263$	1.13746e11	1.46601	0.733003	+	0.680225i	$0.238118\pi$
$264$	0	0
$265$	6.08172e10	0.757565
$266$	0	0
$267$	−4.25233e9	−0.0512066
$268$	0	0
$269$	2.26757e10	0.264043	0.132022	−	0.991247i	$-0.457853\pi$
$269$	2.26757e10	0.264043	0.132022	+	0.991247i	$0.457853\pi$
$270$	0	0
$271$	−1.44599e11	−1.62856	−0.814278	−	0.580475i	$-0.802867\pi$
$271$	−1.44599e11	−1.62856	−0.814278	+	0.580475i	$0.802867\pi$
$272$	0	0
$273$	−9.51953e9	−0.103725
$274$	0	0
$275$	6.02374e9	0.0635140
$276$	0	0
$277$	6.06156e10	0.618622	0.309311	−	0.950961i	$-0.399902\pi$
$277$	6.06156e10	0.618622	0.309311	+	0.950961i	$0.399902\pi$
$278$	0	0
$279$	7.19146e10	0.710556
$280$	0	0
$281$	5.17469e10	0.495115	0.247557	−	0.968873i	$-0.420372\pi$
$281$	5.17469e10	0.495115	0.247557	+	0.968873i	$0.420372\pi$
$282$	0	0
$283$	1.36866e11	1.26840	0.634198	−	0.773170i	$-0.281330\pi$
$283$	1.36866e11	1.26840	0.634198	+	0.773170i	$0.281330\pi$
$284$	0	0
$285$	−1.00902e9	−0.00905940
$286$	0	0
$287$	2.05379e11	1.78684
$288$	0	0
$289$	−5.58877e10	−0.471276
$290$	0	0
$291$	−5.30676e9	−0.0433821
$292$	0	0
$293$	1.54294e11	1.22305	0.611525	−	0.791225i	$-0.290556\pi$
$293$	1.54294e11	1.22305	0.611525	+	0.791225i	$0.290556\pi$
$294$	0	0
$295$	7.87912e10	0.605729
$296$	0	0
$297$	−3.56255e9	−0.0265679
$298$	0	0
$299$	−6.37240e10	−0.461087
$300$	0	0
$301$	1.06948e11	0.750970
$302$	0	0
$303$	−2.80870e9	−0.0191432
$304$	0	0
$305$	8.70887e9	0.0576252
$306$	0	0
$307$	7.66818e10	0.492685	0.246343	−	0.969183i	$-0.420771\pi$
$307$	7.66818e10	0.492685	0.246343	+	0.969183i	$0.420771\pi$
$308$	0	0
$309$	−4.78665e9	−0.0298689
$310$	0	0
$311$	1.01619e11	0.615960	0.307980	−	0.951393i	$-0.400347\pi$
$311$	1.01619e11	0.615960	0.307980	+	0.951393i	$0.400347\pi$
$312$	0	0
$313$	1.80159e11	1.06098	0.530490	−	0.847691i	$-0.322008\pi$
$313$	1.80159e11	1.06098	0.530490	+	0.847691i	$0.322008\pi$
$314$	0	0
$315$	2.16191e11	1.23720
$316$	0	0
$317$	1.09242e10	0.0607610	0.0303805	−	0.999538i	$-0.490328\pi$
$317$	1.09242e10	0.0607610	0.0303805	+	0.999538i	$0.490328\pi$
$318$	0	0
$319$	−7.69908e10	−0.416276
$320$	0	0
$321$	−1.23086e10	−0.0647046
$322$	0	0
$323$	−3.26324e10	−0.166816
$324$	0	0
$325$	−7.13312e10	−0.354654
$326$	0	0
$327$	2.32288e9	0.0112347
$328$	0	0
$329$	−6.20009e10	−0.291754
$330$	0	0
$331$	−4.98916e10	−0.228455	−0.114228	−	0.993455i	$-0.536439\pi$
$331$	−4.98916e10	−0.228455	−0.114228	+	0.993455i	$0.536439\pi$
$332$	0	0
$333$	3.80106e11	1.69397
$334$	0	0
$335$	−1.71826e11	−0.745398
$336$	0	0
$337$	−4.09090e10	−0.172776	−0.0863882	−	0.996262i	$-0.527533\pi$
$337$	−4.09090e10	−0.172776	−0.0863882	+	0.996262i	$0.527533\pi$
$338$	0	0
$339$	−1.36082e10	−0.0559631
$340$	0	0
$341$	5.31183e10	0.212740
$342$	0	0
$343$	−1.73461e10	−0.0676672
$344$	0	0
$345$	−2.87157e9	−0.0109127
$346$	0	0
$347$	−2.01278e11	−0.745269	−0.372635	−	0.927978i	$-0.621545\pi$
$347$	−2.01278e11	−0.745269	−0.372635	+	0.927978i	$0.621545\pi$
$348$	0	0
$349$	−4.71409e11	−1.70092	−0.850460	−	0.526040i	$-0.823676\pi$
$349$	−4.71409e11	−1.70092	−0.850460	+	0.526040i	$0.823676\pi$
$350$	0	0
$351$	4.21866e10	0.148352
$352$	0	0
$353$	2.31623e11	0.793956	0.396978	−	0.917828i	$-0.370059\pi$
$353$	2.31623e11	0.793956	0.396978	+	0.917828i	$0.370059\pi$
$354$	0	0
$355$	−1.37706e11	−0.460176
$356$	0	0
$357$	−1.38733e10	−0.0452036
$358$	0	0
$359$	−7.53299e10	−0.239355	−0.119677	−	0.992813i	$-0.538186\pi$
$359$	−7.53299e10	−0.239355	−0.119677	+	0.992813i	$0.538186\pi$
$360$	0	0
$361$	1.69836e10	0.0526316
$362$	0	0
$363$	1.34069e10	0.0405275
$364$	0	0
$365$	−1.51733e11	−0.447469
$366$	0	0
$367$	−2.54619e11	−0.732646	−0.366323	−	0.930488i	$-0.619383\pi$
$367$	−2.54619e11	−0.732646	−0.366323	+	0.930488i	$0.619383\pi$
$368$	0	0
$369$	−4.54625e11	−1.27654
$370$	0	0
$371$	−4.35196e11	−1.19262
$372$	0	0
$373$	−6.37265e11	−1.70463	−0.852316	−	0.523027i	$-0.824803\pi$
$373$	−6.37265e11	−1.70463	−0.852316	+	0.523027i	$0.824803\pi$
$374$	0	0
$375$	−1.83366e10	−0.0478827
$376$	0	0
$377$	9.11700e11	2.32443
$378$	0	0
$379$	6.88255e11	1.71346	0.856728	−	0.515769i	$-0.172494\pi$
$379$	6.88255e11	1.71346	0.856728	+	0.515769i	$0.172494\pi$
$380$	0	0
$381$	2.03088e10	0.0493767
$382$	0	0
$383$	−4.52345e11	−1.07418	−0.537088	−	0.843526i	$-0.680476\pi$
$383$	−4.52345e11	−1.07418	−0.537088	+	0.843526i	$0.680476\pi$
$384$	0	0
$385$	1.59685e11	0.370417
$386$	0	0
$387$	−2.36739e11	−0.536501
$388$	0	0
$389$	−4.61174e10	−0.102115	−0.0510577	−	0.998696i	$-0.516259\pi$
$389$	−4.61174e10	−0.102115	−0.0510577	+	0.998696i	$0.516259\pi$
$390$	0	0
$391$	−9.28682e10	−0.200942
$392$	0	0
$393$	1.31938e9	0.00278999
$394$	0	0
$395$	7.90941e11	1.63477
$396$	0	0
$397$	7.38758e11	1.49260	0.746302	−	0.665607i	$-0.231827\pi$
$397$	7.38758e11	1.49260	0.746302	+	0.665607i	$0.231827\pi$
$398$	0	0
$399$	7.22037e9	0.0142620
$400$	0	0
$401$	7.45051e10	0.143892	0.0719460	−	0.997409i	$-0.477079\pi$
$401$	7.45051e10	0.143892	0.0719460	+	0.997409i	$0.477079\pi$
$402$	0	0
$403$	−6.29009e11	−1.18791
$404$	0	0
$405$	−4.77607e11	−0.882110
$406$	0	0
$407$	2.80758e11	0.507174
$408$	0	0
$409$	−7.43819e10	−0.131435	−0.0657177	−	0.997838i	$-0.520934\pi$
$409$	−7.43819e10	−0.131435	−0.0657177	+	0.997838i	$0.520934\pi$
$410$	0	0
$411$	4.31585e10	0.0746067
$412$	0	0
$413$	−5.63814e11	−0.953588
$414$	0	0
$415$	9.73102e11	1.61043
$416$	0	0
$417$	−2.49491e10	−0.0404057
$418$	0	0
$419$	2.32606e11	0.368688	0.184344	−	0.982862i	$-0.440984\pi$
$419$	2.32606e11	0.368688	0.184344	+	0.982862i	$0.440984\pi$
$420$	0	0
$421$	−5.41341e11	−0.839849	−0.419924	−	0.907559i	$-0.637943\pi$
$421$	−5.41341e11	−0.839849	−0.419924	+	0.907559i	$0.637943\pi$
$422$	0	0
$423$	1.37245e11	0.208432
$424$	0	0
$425$	−1.03955e11	−0.154559
$426$	0	0
$427$	−6.23190e10	−0.0907183
$428$	0	0
$429$	1.55647e10	0.0221862
$430$	0	0
$431$	−5.20995e11	−0.727253	−0.363626	−	0.931545i	$-0.618462\pi$
$431$	−5.20995e11	−0.727253	−0.363626	+	0.931545i	$0.618462\pi$
$432$	0	0
$433$	−1.26796e12	−1.73345	−0.866725	−	0.498786i	$-0.833779\pi$
$433$	−1.26796e12	−1.73345	−0.866725	+	0.498786i	$0.833779\pi$
$434$	0	0
$435$	4.10836e10	0.0550132
$436$	0	0
$437$	4.83334e10	0.0633987
$438$	0	0
$439$	−1.45353e12	−1.86782	−0.933910	−	0.357508i	$-0.883627\pi$
$439$	−1.45353e12	−1.86782	−0.933910	+	0.357508i	$0.883627\pi$
$440$	0	0
$441$	−7.54310e11	−0.949678
$442$	0	0
$443$	3.99595e11	0.492950	0.246475	−	0.969149i	$-0.420728\pi$
$443$	3.99595e11	0.492950	0.246475	+	0.969149i	$0.420728\pi$
$444$	0	0
$445$	−8.44672e11	−1.02110
$446$	0	0
$447$	−6.12490e10	−0.0725629
$448$	0	0
$449$	−2.62303e11	−0.304576	−0.152288	−	0.988336i	$-0.548664\pi$
$449$	−2.62303e11	−0.304576	−0.152288	+	0.988336i	$0.548664\pi$
$450$	0	0
$451$	−3.35800e11	−0.382196
$452$	0	0
$453$	7.06376e10	0.0788123
$454$	0	0
$455$	−1.89094e12	−2.06836
$456$	0	0
$457$	5.46879e11	0.586500	0.293250	−	0.956036i	$-0.405263\pi$
$457$	5.46879e11	0.586500	0.293250	+	0.956036i	$0.405263\pi$
$458$	0	0
$459$	6.14806e10	0.0646519
$460$	0	0
$461$	−5.18813e11	−0.535003	−0.267502	−	0.963557i	$-0.586198\pi$
$461$	−5.18813e11	−0.535003	−0.267502	+	0.963557i	$0.586198\pi$
$462$	0	0
$463$	4.45590e11	0.450630	0.225315	−	0.974286i	$-0.427659\pi$
$463$	4.45590e11	0.450630	0.225315	+	0.974286i	$0.427659\pi$
$464$	0	0
$465$	−2.83448e10	−0.0281148
$466$	0	0
$467$	1.84238e12	1.79247	0.896237	−	0.443575i	$-0.146290\pi$
$467$	1.84238e12	1.79247	0.896237	+	0.443575i	$0.146290\pi$
$468$	0	0
$469$	1.22956e12	1.17347
$470$	0	0
$471$	−5.80705e10	−0.0543702
$472$	0	0
$473$	−1.74862e11	−0.160628
$474$	0	0
$475$	5.41033e10	0.0487643
$476$	0	0
$477$	9.63347e11	0.852020
$478$	0	0
$479$	−1.55017e12	−1.34546	−0.672728	−	0.739890i	$-0.734877\pi$
$479$	−1.55017e12	−1.34546	−0.672728	+	0.739890i	$0.734877\pi$
$480$	0	0
$481$	−3.32464e12	−2.83199
$482$	0	0
$483$	2.05484e10	0.0171797
$484$	0	0
$485$	−1.05412e12	−0.865073
$486$	0	0
$487$	3.95615e11	0.318708	0.159354	−	0.987222i	$-0.449059\pi$
$487$	3.95615e11	0.318708	0.159354	+	0.987222i	$0.449059\pi$
$488$	0	0
$489$	−2.39099e10	−0.0189099
$490$	0	0
$491$	−1.94721e12	−1.51198	−0.755991	−	0.654582i	$-0.772844\pi$
$491$	−1.94721e12	−1.51198	−0.755991	+	0.654582i	$0.772844\pi$
$492$	0	0
$493$	1.32867e12	1.01299
$494$	0	0
$495$	−3.53477e11	−0.264630
$496$	0	0
$497$	9.85394e11	0.724447
$498$	0	0
$499$	−2.58799e12	−1.86858	−0.934288	−	0.356520i	$-0.883963\pi$
$499$	−2.58799e12	−1.86858	−0.934288	+	0.356520i	$0.883963\pi$
$500$	0	0
$501$	−4.89416e10	−0.0347063
$502$	0	0
$503$	2.66146e12	1.85381	0.926903	−	0.375302i	$-0.122461\pi$
$503$	2.66146e12	1.85381	0.926903	+	0.375302i	$0.122461\pi$
$504$	0	0
$505$	−5.57914e11	−0.381730
$506$	0	0
$507$	−1.18105e11	−0.0793839
$508$	0	0
$509$	2.00453e12	1.32368	0.661840	−	0.749645i	$-0.269776\pi$
$509$	2.00453e12	1.32368	0.661840	+	0.749645i	$0.269776\pi$
$510$	0	0
$511$	1.08577e12	0.704442
$512$	0	0
$513$	−3.19977e10	−0.0203981
$514$	0	0
$515$	−9.50809e11	−0.595608
$516$	0	0
$517$	1.01373e11	0.0624044
$518$	0	0
$519$	−6.22934e10	−0.0376868
$520$	0	0
$521$	1.29928e12	0.772562	0.386281	−	0.922381i	$-0.373759\pi$
$521$	1.29928e12	0.772562	0.386281	+	0.922381i	$0.373759\pi$
$522$	0	0
$523$	−2.43324e12	−1.42209	−0.711046	−	0.703145i	$-0.751778\pi$
$523$	−2.43324e12	−1.42209	−0.711046	+	0.703145i	$0.751778\pi$
$524$	0	0
$525$	2.30014e10	0.0132141
$526$	0	0
$527$	−9.16688e11	−0.517695
$528$	0	0
$529$	−1.66360e12	−0.923631
$530$	0	0
$531$	1.24806e12	0.681253
$532$	0	0
$533$	3.97643e12	2.13413
$534$	0	0
$535$	−2.44494e12	−1.29026
$536$	0	0
$537$	4.05281e10	0.0210316
$538$	0	0
$539$	−5.57156e11	−0.284333
$540$	0	0
$541$	1.00558e12	0.504697	0.252349	−	0.967636i	$-0.418797\pi$
$541$	1.00558e12	0.504697	0.252349	+	0.967636i	$0.418797\pi$
$542$	0	0
$543$	−8.55235e10	−0.0422169
$544$	0	0
$545$	4.61412e11	0.224029
$546$	0	0
$547$	7.27895e11	0.347637	0.173818	−	0.984778i	$-0.444389\pi$
$547$	7.27895e11	0.347637	0.173818	+	0.984778i	$0.444389\pi$
$548$	0	0
$549$	1.37949e11	0.0648101
$550$	0	0
$551$	−6.91506e11	−0.319605
$552$	0	0
$553$	−5.65982e12	−2.57359
$554$	0	0
$555$	−1.49817e11	−0.0670260
$556$	0	0
$557$	1.62693e11	0.0716177	0.0358088	−	0.999359i	$-0.488599\pi$
$557$	1.62693e11	0.0716177	0.0358088	+	0.999359i	$0.488599\pi$
$558$	0	0
$559$	2.07066e12	0.896926
$560$	0	0
$561$	2.26832e10	0.00966878
$562$	0	0
$563$	−6.20629e10	−0.0260342	−0.0130171	−	0.999915i	$-0.504144\pi$
$563$	−6.20629e10	−0.0260342	−0.0130171	+	0.999915i	$0.504144\pi$
$564$	0	0
$565$	−2.70310e12	−1.11595
$566$	0	0
$567$	3.41766e12	1.38869
$568$	0	0
$569$	4.55359e11	0.182116	0.0910582	−	0.995846i	$-0.470975\pi$
$569$	4.55359e11	0.182116	0.0910582	+	0.995846i	$0.470975\pi$
$570$	0	0
$571$	4.44357e12	1.74932	0.874661	−	0.484735i	$-0.161084\pi$
$571$	4.44357e12	1.74932	0.874661	+	0.484735i	$0.161084\pi$
$572$	0	0
$573$	−1.38128e11	−0.0535286
$574$	0	0
$575$	1.53972e11	0.0587404
$576$	0	0
$577$	−3.55560e12	−1.33543	−0.667716	−	0.744416i	$-0.732728\pi$
$577$	−3.55560e12	−1.33543	−0.667716	+	0.744416i	$0.732728\pi$
$578$	0	0
$579$	3.56176e10	0.0131708
$580$	0	0
$581$	−6.96332e12	−2.53527
$582$	0	0
$583$	7.11557e11	0.255094
$584$	0	0
$585$	4.18576e12	1.47766
$586$	0	0
$587$	−4.95575e12	−1.72281	−0.861405	−	0.507918i	$-0.830415\pi$
$587$	−4.95575e12	−1.72281	−0.861405	+	0.507918i	$0.830415\pi$
$588$	0	0
$589$	4.77091e11	0.163336
$590$	0	0
$591$	−1.92599e11	−0.0649397
$592$	0	0
$593$	3.26211e12	1.08331	0.541655	−	0.840601i	$-0.317798\pi$
$593$	3.26211e12	1.08331	0.541655	+	0.840601i	$0.317798\pi$
$594$	0	0
$595$	−2.75576e12	−0.901394
$596$	0	0
$597$	−1.07892e11	−0.0347620
$598$	0	0
$599$	−3.16477e12	−1.00443	−0.502217	−	0.864742i	$-0.667482\pi$
$599$	−3.16477e12	−1.00443	−0.502217	+	0.864742i	$0.667482\pi$
$600$	0	0
$601$	3.23436e11	0.101124	0.0505619	−	0.998721i	$-0.483899\pi$
$601$	3.23436e11	0.101124	0.0505619	+	0.998721i	$0.483899\pi$
$602$	0	0
$603$	−2.72174e12	−0.838336
$604$	0	0
$605$	2.66312e12	0.808149
$606$	0	0
$607$	2.87199e12	0.858686	0.429343	−	0.903141i	$-0.358745\pi$
$607$	2.87199e12	0.858686	0.429343	+	0.903141i	$0.358745\pi$
$608$	0	0
$609$	−2.93986e11	−0.0866062
$610$	0	0
$611$	−1.20043e12	−0.348458
$612$	0	0
$613$	3.75184e12	1.07318	0.536590	−	0.843843i	$-0.319712\pi$
$613$	3.75184e12	1.07318	0.536590	+	0.843843i	$0.319712\pi$
$614$	0	0
$615$	1.79188e11	0.0505093
$616$	0	0
$617$	−1.23662e12	−0.343520	−0.171760	−	0.985139i	$-0.554945\pi$
$617$	−1.23662e12	−0.343520	−0.171760	+	0.985139i	$0.554945\pi$
$618$	0	0
$619$	5.95594e12	1.63058	0.815290	−	0.579053i	$-0.196577\pi$
$619$	5.95594e12	1.63058	0.815290	+	0.579053i	$0.196577\pi$
$620$	0	0
$621$	−9.10618e10	−0.0245711
$622$	0	0
$623$	6.04430e12	1.60750
$624$	0	0
$625$	−2.83150e12	−0.742260
$626$	0	0
$627$	−1.18055e10	−0.00305057
$628$	0	0
$629$	−4.84517e12	−1.23419
$630$	0	0
$631$	−5.85099e11	−0.146926	−0.0734628	−	0.997298i	$-0.523405\pi$
$631$	−5.85099e11	−0.146926	−0.0734628	+	0.997298i	$0.523405\pi$
$632$	0	0
$633$	5.12414e10	0.0126854
$634$	0	0
$635$	4.03409e12	0.984609
$636$	0	0
$637$	6.59766e12	1.58768
$638$	0	0
$639$	−2.18126e12	−0.517552
$640$	0	0
$641$	−9.42630e11	−0.220536	−0.110268	−	0.993902i	$-0.535171\pi$
$641$	−9.42630e11	−0.220536	−0.110268	+	0.993902i	$0.535171\pi$
$642$	0	0
$643$	4.77642e12	1.10193	0.550963	−	0.834529i	$-0.314260\pi$
$643$	4.77642e12	1.10193	0.550963	+	0.834529i	$0.314260\pi$
$644$	0	0
$645$	9.33096e10	0.0212279
$646$	0	0
$647$	−1.98087e12	−0.444414	−0.222207	−	0.975000i	$-0.571326\pi$
$647$	−1.98087e12	−0.444414	−0.222207	+	0.975000i	$0.571326\pi$
$648$	0	0
$649$	9.21851e11	0.203967
$650$	0	0
$651$	2.02830e11	0.0442606
$652$	0	0
$653$	−7.31543e11	−0.157446	−0.0787228	−	0.996897i	$-0.525084\pi$
$653$	−7.31543e11	−0.157446	−0.0787228	+	0.996897i	$0.525084\pi$
$654$	0	0
$655$	2.62078e11	0.0556345
$656$	0	0
$657$	−2.40346e12	−0.503261
$658$	0	0
$659$	−6.80012e12	−1.40453	−0.702267	−	0.711914i	$-0.747829\pi$
$659$	−6.80012e12	−1.40453	−0.702267	+	0.711914i	$0.747829\pi$
$660$	0	0
$661$	1.11163e12	0.226493	0.113247	−	0.993567i	$-0.463875\pi$
$661$	1.11163e12	0.226493	0.113247	+	0.993567i	$0.463875\pi$
$662$	0	0
$663$	−2.68607e11	−0.0539892
$664$	0	0
$665$	1.43424e12	0.284396
$666$	0	0
$667$	−1.96795e12	−0.384989
$668$	0	0
$669$	−1.91761e11	−0.0370120
$670$	0	0
$671$	1.01893e11	0.0194041
$672$	0	0
$673$	2.47185e12	0.464466	0.232233	−	0.972660i	$-0.425397\pi$
$673$	2.47185e12	0.464466	0.232233	+	0.972660i	$0.425397\pi$
$674$	0	0
$675$	−1.01933e11	−0.0188993
$676$	0	0
$677$	4.80539e12	0.879183	0.439591	−	0.898198i	$-0.355123\pi$
$677$	4.80539e12	0.879183	0.439591	+	0.898198i	$0.355123\pi$
$678$	0	0
$679$	7.54308e12	1.36187
$680$	0	0
$681$	−1.51792e11	−0.0270450
$682$	0	0
$683$	2.32385e12	0.408615	0.204307	−	0.978907i	$-0.434506\pi$
$683$	2.32385e12	0.408615	0.204307	+	0.978907i	$0.434506\pi$
$684$	0	0
$685$	8.57289e12	1.48771
$686$	0	0
$687$	−3.47454e11	−0.0595103
$688$	0	0
$689$	−8.42602e12	−1.42441
$690$	0	0
$691$	−8.11211e10	−0.0135358	−0.00676788	−	0.999977i	$-0.502154\pi$
$691$	−8.11211e10	−0.0135358	−0.00676788	+	0.999977i	$0.502154\pi$
$692$	0	0
$693$	2.52941e12	0.416601
$694$	0	0
$695$	−4.95583e12	−0.805721
$696$	0	0
$697$	5.79505e12	0.930058
$698$	0	0
$699$	2.17114e11	0.0343985
$700$	0	0
$701$	−3.44255e12	−0.538454	−0.269227	−	0.963077i	$-0.586768\pi$
$701$	−3.44255e12	−0.538454	−0.269227	+	0.963077i	$0.586768\pi$
$702$	0	0
$703$	2.52168e12	0.389395
$704$	0	0
$705$	−5.40944e10	−0.00824710
$706$	0	0
$707$	3.99232e12	0.600950
$708$	0	0
$709$	4.26474e12	0.633848	0.316924	−	0.948451i	$-0.397350\pi$
$709$	4.26474e12	0.633848	0.316924	+	0.948451i	$0.397350\pi$
$710$	0	0
$711$	1.25285e13	1.83860
$712$	0	0
$713$	1.35775e12	0.196751
$714$	0	0
$715$	3.09173e12	0.442409
$716$	0	0
$717$	−2.81340e11	−0.0397553
$718$	0	0
$719$	−6.75064e12	−0.942031	−0.471015	−	0.882125i	$-0.656112\pi$
$719$	−6.75064e12	−0.942031	−0.471015	+	0.882125i	$0.656112\pi$
$720$	0	0
$721$	6.80380e12	0.937654
$722$	0	0
$723$	−1.99505e11	−0.0271538
$724$	0	0
$725$	−2.20288e12	−0.296122
$726$	0	0
$727$	−1.03379e12	−0.137254	−0.0686272	−	0.997642i	$-0.521862\pi$
$727$	−1.03379e12	−0.137254	−0.0686272	+	0.997642i	$0.521862\pi$
$728$	0	0
$729$	−7.53514e12	−0.988138
$730$	0	0
$731$	3.01768e12	0.390882
$732$	0	0
$733$	6.86393e12	0.878223	0.439112	−	0.898432i	$-0.355293\pi$
$733$	6.86393e12	0.878223	0.439112	+	0.898432i	$0.355293\pi$
$734$	0	0
$735$	2.97308e11	0.0375762
$736$	0	0
$737$	−2.01036e12	−0.250997
$738$	0	0
$739$	−4.93061e12	−0.608136	−0.304068	−	0.952650i	$-0.598345\pi$
$739$	−4.93061e12	−0.608136	−0.304068	+	0.952650i	$0.598345\pi$
$740$	0	0
$741$	1.39797e11	0.0170340
$742$	0	0
$743$	−1.44876e12	−0.174400	−0.0871999	−	0.996191i	$-0.527792\pi$
$743$	−1.44876e12	−0.174400	−0.0871999	+	0.996191i	$0.527792\pi$
$744$	0	0
$745$	−1.21663e13	−1.44696
$746$	0	0
$747$	1.54140e13	1.81122
$748$	0	0
$749$	1.74955e13	2.03123
$750$	0	0
$751$	−1.10310e13	−1.26542	−0.632709	−	0.774390i	$-0.718057\pi$
$751$	−1.10310e13	−1.26542	−0.632709	+	0.774390i	$0.718057\pi$
$752$	0	0
$753$	−3.01678e11	−0.0341953
$754$	0	0
$755$	1.40313e13	1.57158
$756$	0	0
$757$	6.01909e12	0.666192	0.333096	−	0.942893i	$-0.391907\pi$
$757$	6.01909e12	0.666192	0.333096	+	0.942893i	$0.391907\pi$
$758$	0	0
$759$	−3.35972e10	−0.00367464
$760$	0	0
$761$	1.31729e13	1.42381	0.711904	−	0.702277i	$-0.247833\pi$
$761$	1.31729e13	1.42381	0.711904	+	0.702277i	$0.247833\pi$
$762$	0	0
$763$	−3.30177e12	−0.352685
$764$	0	0
$765$	6.10013e12	0.643965
$766$	0	0
$767$	−1.09163e13	−1.13892
$768$	0	0
$769$	1.18311e12	0.121999	0.0609993	−	0.998138i	$-0.480571\pi$
$769$	1.18311e12	0.121999	0.0609993	+	0.998138i	$0.480571\pi$
$770$	0	0
$771$	−1.00754e11	−0.0102687
$772$	0	0
$773$	7.44943e12	0.750439	0.375220	−	0.926936i	$-0.377567\pi$
$773$	7.44943e12	0.750439	0.375220	+	0.926936i	$0.377567\pi$
$774$	0	0
$775$	1.51983e12	0.151335
$776$	0	0
$777$	1.07206e12	0.105518
$778$	0	0
$779$	−3.01604e12	−0.293440
$780$	0	0
$781$	−1.61115e12	−0.154955
$782$	0	0
$783$	1.30282e12	0.123868
$784$	0	0
$785$	−1.15350e13	−1.08418
$786$	0	0
$787$	1.78315e13	1.65692	0.828460	−	0.560048i	$-0.189217\pi$
$787$	1.78315e13	1.65692	0.828460	+	0.560048i	$0.189217\pi$
$788$	0	0
$789$	−7.10149e11	−0.0652383
$790$	0	0
$791$	1.93428e13	1.75681
$792$	0	0
$793$	−1.20659e12	−0.108350
$794$	0	0
$795$	−3.79699e11	−0.0337122
$796$	0	0
$797$	1.15025e13	1.00979	0.504894	−	0.863181i	$-0.331532\pi$
$797$	1.15025e13	1.00979	0.504894	+	0.863181i	$0.331532\pi$
$798$	0	0
$799$	−1.74944e12	−0.151859
$800$	0	0
$801$	−1.33796e13	−1.14841
$802$	0	0
$803$	−1.77527e12	−0.150676
$804$	0	0
$805$	4.08168e12	0.342577
$806$	0	0
$807$	−1.41571e11	−0.0117501
$808$	0	0
$809$	−1.47465e13	−1.21038	−0.605188	−	0.796083i	$-0.706902\pi$
$809$	−1.47465e13	−1.21038	−0.605188	+	0.796083i	$0.706902\pi$
$810$	0	0
$811$	1.72663e13	1.40154	0.700770	−	0.713387i	$-0.252840\pi$
$811$	1.72663e13	1.40154	0.700770	+	0.713387i	$0.252840\pi$
$812$	0	0
$813$	9.02771e11	0.0724719
$814$	0	0
$815$	−4.74941e12	−0.377077
$816$	0	0
$817$	−1.57056e12	−0.123326
$818$	0	0
$819$	−2.99525e13	−2.32624
$820$	0	0
$821$	3.86659e11	0.0297019	0.0148510	−	0.999890i	$-0.495273\pi$
$821$	3.86659e11	0.0297019	0.0148510	+	0.999890i	$0.495273\pi$
$822$	0	0
$823$	−1.82542e13	−1.38696	−0.693478	−	0.720478i	$-0.743922\pi$
$823$	−1.82542e13	−1.38696	−0.693478	+	0.720478i	$0.743922\pi$
$824$	0	0
$825$	−3.76079e10	−0.00282642
$826$	0	0
$827$	−1.39909e13	−1.04009	−0.520044	−	0.854140i	$-0.674084\pi$
$827$	−1.39909e13	−1.04009	−0.520044	+	0.854140i	$0.674084\pi$
$828$	0	0
$829$	−4.77942e12	−0.351463	−0.175731	−	0.984438i	$-0.556229\pi$
$829$	−4.77942e12	−0.351463	−0.175731	+	0.984438i	$0.556229\pi$
$830$	0	0
$831$	−3.78440e11	−0.0275291
$832$	0	0
$833$	9.61510e12	0.691913
$834$	0	0
$835$	−9.72164e12	−0.692071
$836$	0	0
$837$	−8.98857e11	−0.0633033
$838$	0	0
$839$	1.56344e13	1.08931	0.544657	−	0.838659i	$-0.316660\pi$
$839$	1.56344e13	1.08931	0.544657	+	0.838659i	$0.316660\pi$
$840$	0	0
$841$	1.36484e13	0.940804
$842$	0	0
$843$	−3.23070e11	−0.0220330
$844$	0	0
$845$	−2.34601e13	−1.58298
$846$	0	0
$847$	−1.90567e13	−1.27225
$848$	0	0
$849$	−8.54490e11	−0.0564446
$850$	0	0
$851$	7.17641e12	0.469056
$852$	0	0
$853$	1.16938e13	0.756286	0.378143	−	0.925747i	$-0.376563\pi$
$853$	1.16938e13	0.756286	0.378143	+	0.925747i	$0.376563\pi$
$854$	0	0
$855$	−3.17482e12	−0.203176
$856$	0	0
$857$	7.96244e12	0.504234	0.252117	−	0.967697i	$-0.418873\pi$
$857$	7.96244e12	0.504234	0.252117	+	0.967697i	$0.418873\pi$
$858$	0	0
$859$	−8.97508e12	−0.562431	−0.281215	−	0.959645i	$-0.590738\pi$
$859$	−8.97508e12	−0.562431	−0.281215	+	0.959645i	$0.590738\pi$
$860$	0	0
$861$	−1.28224e12	−0.0795159
$862$	0	0
$863$	−2.21958e13	−1.36214	−0.681071	−	0.732217i	$-0.738486\pi$
$863$	−2.21958e13	−1.36214	−0.681071	+	0.732217i	$0.738486\pi$
$864$	0	0
$865$	−1.23738e13	−0.751503
$866$	0	0
$867$	3.48922e11	0.0209721
$868$	0	0
$869$	9.25395e12	0.550476
$870$	0	0
$871$	2.38060e13	1.40154
$872$	0	0
$873$	−1.66973e13	−0.972933
$874$	0	0
$875$	2.60639e13	1.50315
$876$	0	0
$877$	−1.46163e13	−0.834334	−0.417167	−	0.908830i	$-0.636977\pi$
$877$	−1.46163e13	−0.834334	−0.417167	+	0.908830i	$0.636977\pi$
$878$	0	0
$879$	−9.63299e11	−0.0544266
$880$	0	0
$881$	1.76227e13	0.985556	0.492778	−	0.870155i	$-0.335982\pi$
$881$	1.76227e13	0.985556	0.492778	+	0.870155i	$0.335982\pi$
$882$	0	0
$883$	7.69996e12	0.426251	0.213126	−	0.977025i	$-0.431636\pi$
$883$	7.69996e12	0.426251	0.213126	+	0.977025i	$0.431636\pi$
$884$	0	0
$885$	−4.91915e11	−0.0269554
$886$	0	0
$887$	2.64731e13	1.43598	0.717989	−	0.696054i	$-0.245063\pi$
$887$	2.64731e13	1.43598	0.717989	+	0.696054i	$0.245063\pi$
$888$	0	0
$889$	−2.88672e13	−1.55005
$890$	0	0
$891$	−5.58797e12	−0.297033
$892$	0	0
$893$	9.10500e11	0.0479124
$894$	0	0
$895$	8.05040e12	0.419386
$896$	0	0
$897$	3.97847e11	0.0205187
$898$	0	0
$899$	−1.94253e13	−0.991859
$900$	0	0
$901$	−1.22797e13	−0.620762
$902$	0	0
$903$	−6.67705e11	−0.0334187
$904$	0	0
$905$	−1.69882e13	−0.841838
$906$	0	0
$907$	−8.94209e12	−0.438739	−0.219370	−	0.975642i	$-0.570400\pi$
$907$	−8.94209e12	−0.438739	−0.219370	+	0.975642i	$0.570400\pi$
$908$	0	0
$909$	−8.83738e12	−0.429325
$910$	0	0
$911$	3.74724e13	1.80251	0.901257	−	0.433285i	$-0.142646\pi$
$911$	3.74724e13	1.80251	0.901257	+	0.433285i	$0.142646\pi$
$912$	0	0
$913$	1.13852e13	0.542279
$914$	0	0
$915$	−5.43719e10	−0.00256436
$916$	0	0
$917$	−1.87537e12	−0.0875843
$918$	0	0
$919$	1.16571e13	0.539103	0.269551	−	0.962986i	$-0.413125\pi$
$919$	1.16571e13	0.539103	0.269551	+	0.962986i	$0.413125\pi$
$920$	0	0
$921$	−4.78746e11	−0.0219248
$922$	0	0
$923$	1.90787e13	0.865247
$924$	0	0
$925$	8.03312e12	0.360783
$926$	0	0
$927$	−1.50608e13	−0.669870
$928$	0	0
$929$	−3.03689e12	−0.133770	−0.0668850	−	0.997761i	$-0.521306\pi$
$929$	−3.03689e12	−0.133770	−0.0668850	+	0.997761i	$0.521306\pi$
$930$	0	0
$931$	−5.00419e12	−0.218303
$932$	0	0
$933$	−6.34435e11	−0.0274107
$934$	0	0
$935$	4.50574e12	0.192803
$936$	0	0
$937$	2.54301e13	1.07775	0.538877	−	0.842385i	$-0.318849\pi$
$937$	2.54301e13	1.07775	0.538877	+	0.842385i	$0.318849\pi$
$938$	0	0
$939$	−1.12479e12	−0.0472144
$940$	0	0
$941$	−1.16765e13	−0.485469	−0.242734	−	0.970093i	$-0.578044\pi$
$941$	−1.16765e13	−0.485469	−0.242734	+	0.970093i	$0.578044\pi$
$942$	0	0
$943$	−8.58333e12	−0.353470
$944$	0	0
$945$	−2.70215e12	−0.110222
$946$	0	0
$947$	2.43549e13	0.984038	0.492019	−	0.870584i	$-0.336259\pi$
$947$	2.43549e13	0.984038	0.492019	+	0.870584i	$0.336259\pi$
$948$	0	0
$949$	2.10222e13	0.841355
$950$	0	0
$951$	−6.82031e10	−0.00270391
$952$	0	0
$953$	−3.15922e13	−1.24069	−0.620344	−	0.784330i	$-0.713007\pi$
$953$	−3.15922e13	−1.24069	−0.620344	+	0.784330i	$0.713007\pi$
$954$	0	0
$955$	−2.74374e13	−1.06740
$956$	0	0
$957$	4.80675e11	0.0185246
$958$	0	0
$959$	−6.13459e13	−2.34208
$960$	0	0
$961$	−1.30375e13	−0.493104
$962$	0	0
$963$	−3.87280e13	−1.45113
$964$	0	0
$965$	7.07500e12	0.262636
$966$	0	0
$967$	8.21729e12	0.302210	0.151105	−	0.988518i	$-0.451717\pi$
$967$	8.21729e12	0.302210	0.151105	+	0.988518i	$0.451717\pi$
$968$	0	0
$969$	2.03733e11	0.00742343
$970$	0	0
$971$	1.56122e13	0.563610	0.281805	−	0.959472i	$-0.409067\pi$
$971$	1.56122e13	0.563610	0.281805	+	0.959472i	$0.409067\pi$
$972$	0	0
$973$	3.54629e13	1.26843
$974$	0	0
$975$	4.45341e11	0.0157823
$976$	0	0
$977$	2.49825e13	0.877224	0.438612	−	0.898676i	$-0.355470\pi$
$977$	2.49825e13	0.877224	0.438612	+	0.898676i	$0.355470\pi$
$978$	0	0
$979$	−9.88260e12	−0.343834
$980$	0	0
$981$	7.30878e12	0.251962
$982$	0	0
$983$	3.47691e13	1.18769	0.593845	−	0.804580i	$-0.297609\pi$
$983$	3.47691e13	1.18769	0.593845	+	0.804580i	$0.297609\pi$
$984$	0	0
$985$	−3.82574e13	−1.29495
$986$	0	0
$987$	3.87089e11	0.0129832
$988$	0	0
$989$	−4.46963e12	−0.148555
$990$	0	0
$991$	1.33317e13	0.439090	0.219545	−	0.975602i	$-0.429543\pi$
$991$	1.33317e13	0.439090	0.219545	+	0.975602i	$0.429543\pi$
$992$	0	0
$993$	3.11487e11	0.0101664
$994$	0	0
$995$	−2.14314e13	−0.693182
$996$	0	0
$997$	2.81190e13	0.901305	0.450653	−	0.892699i	$-0.351191\pi$
$997$	2.81190e13	0.901305	0.450653	+	0.892699i	$0.351191\pi$
$998$	0	0
$999$	−4.75093e12	−0.150915

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.10.a.i.1.5		8
4.3	odd	2		19.10.a.b.1.2	✓	8
12.11	even	2		171.10.a.f.1.7		8
76.75	even	2		361.10.a.c.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.10.a.b.1.2	✓	8	4.3	odd	2
171.10.a.f.1.7		8	12.11	even	2
304.10.a.i.1.5		8	1.1	even	1	trivial
361.10.a.c.1.7		8	76.75	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	304.a (trivial)

\(f(q)\)	\(=\)	\(q-6.24328 q^{3} -1240.15 q^{5} +8874.26 q^{7} -19644.0 q^{9} -14509.7 q^{11} +171819. q^{13} +7742.60 q^{15} +250400. q^{17} -130321. q^{19} -55404.5 q^{21} -370879. q^{23} -415154. q^{25} +245530. q^{27} +5.30618e6 q^{29} -3.66089e6 q^{31} +90587.9 q^{33} -1.10054e7 q^{35} -1.93497e7 q^{37} -1.07271e6 q^{39} +2.31432e7 q^{41} +1.20515e7 q^{43} +2.43615e7 q^{45} -6.98659e6 q^{47} +3.83990e7 q^{49} -1.56332e6 q^{51} -4.90402e7 q^{53} +1.79941e7 q^{55} +813630. q^{57} -6.35336e7 q^{59} -7.02244e6 q^{61} -1.74326e8 q^{63} -2.13081e8 q^{65} +1.38553e8 q^{67} +2.31550e6 q^{69} +1.11040e8 q^{71} +1.22351e8 q^{73} +2.59192e6 q^{75} -1.28763e8 q^{77} -6.37779e8 q^{79} +3.85120e8 q^{81} -7.84665e8 q^{83} -3.10534e8 q^{85} -3.31279e7 q^{87} +6.81105e8 q^{89} +1.52476e9 q^{91} +2.28560e7 q^{93} +1.61618e8 q^{95} +8.49995e8 q^{97} +2.85028e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 7 q^{3} + 3894 q^{5} + 7133 q^{7} + 102715 q^{9} - 172818 q^{11} + 109291 q^{13} - 457332 q^{15} + 583575 q^{17} - 1042568 q^{19} + 372767 q^{21} + 2292405 q^{23} + 205328 q^{25} + 15261551 q^{27}+ \cdots + 1682553420 q^{99}+O(q^{100}) \) 8 * q - 7 * q^3 + 3894 * q^5 + 7133 * q^7 + 102715 * q^9 - 172818 * q^11 + 109291 * q^13 - 457332 * q^15 + 583575 * q^17 - 1042568 * q^19 + 372767 * q^21 + 2292405 * q^23 + 205328 * q^25 + 15261551 * q^27 + 4546335 * q^29 + 8277416 * q^31 + 13959234 * q^33 - 15633768 * q^35 + 24485536 * q^37 - 16641725 * q^39 + 35051946 * q^41 - 36624268 * q^43 + 138146370 * q^45 - 9865524 * q^47 + 111765087 * q^49 - 109712553 * q^51 + 13963683 * q^53 - 261540660 * q^55 + 912247 * q^57 - 21215463 * q^59 + 180552112 * q^61 - 303758726 * q^63 - 21864624 * q^65 - 1767355 * q^67 + 504738939 * q^69 - 273631578 * q^71 - 263711543 * q^73 + 340714091 * q^75 + 441243702 * q^77 - 744472174 * q^79 + 1867545184 * q^81 - 832139910 * q^83 + 1013793492 * q^85 + 231595575 * q^87 - 291457734 * q^89 + 1351077415 * q^91 + 537395144 * q^93 - 507469974 * q^95 - 3483837140 * q^97 + 1682553420 * q^99

Introduction

L-functions

Modular forms

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