LMFDB - Embedded newform 304.8.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,8,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, 8, names="a")

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

chi := DirichletCharacter("304.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 304 = 2^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 8 $
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:	$94.9650477472$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{633}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 158 $ x^2 - x - 158
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-12.0797$ 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)

$f(q)$	$=$	$q+72.2392 q^{3} +467.472 q^{5} +1471.23 q^{7} +3031.51 q^{9} +O(q^{10})$	$q+72.2392 q^{3} +467.472 q^{5} +1471.23 q^{7} +3031.51 q^{9} +3547.04 q^{11} -12361.8 q^{13} +33769.8 q^{15} -13486.3 q^{17} +6859.00 q^{19} +106281. q^{21} +94084.9 q^{23} +140405. q^{25} +61006.6 q^{27} +33541.0 q^{29} -212096. q^{31} +256236. q^{33} +687760. q^{35} +33656.5 q^{37} -893007. q^{39} -480531. q^{41} +561340. q^{43} +1.41715e6 q^{45} -478445. q^{47} +1.34098e6 q^{49} -974237. q^{51} -93127.0 q^{53} +1.65814e6 q^{55} +495489. q^{57} -955034. q^{59} -1.71632e6 q^{61} +4.46005e6 q^{63} -5.77880e6 q^{65} -476255. q^{67} +6.79662e6 q^{69} -1.95518e6 q^{71} +1.21632e6 q^{73} +1.01428e7 q^{75} +5.21852e6 q^{77} -3.94544e6 q^{79} -2.22284e6 q^{81} +3.54252e6 q^{83} -6.30445e6 q^{85} +2.42298e6 q^{87} -7.50202e6 q^{89} -1.81871e7 q^{91} -1.53216e7 q^{93} +3.20639e6 q^{95} +1.27974e7 q^{97} +1.07529e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 69 q^{3} + 155 q^{5} + 2238 q^{7} + 855 q^{9}+O(q^{10}) $ 2 * q + 69 * q^3 + 155 * q^5 + 2238 * q^7 + 855 * q^9	$ 2 q + 69 q^{3} + 155 q^{5} + 2238 q^{7} + 855 q^{9} + 3295 q^{11} - 13427 q^{13} + 34782 q^{15} - 32256 q^{17} + 13718 q^{19} + 103797 q^{21} + 82525 q^{23} + 159919 q^{25} + 75141 q^{27} - 12749 q^{29} - 258944 q^{31} + 257052 q^{33} + 448167 q^{35} - 149260 q^{37} - 889557 q^{39} + 339130 q^{41} + 83869 q^{43} + 2097243 q^{45} - 1471025 q^{47} + 1105372 q^{49} - 913437 q^{51} - 945643 q^{53} + 1736899 q^{55} + 473271 q^{57} + 969009 q^{59} - 1506755 q^{61} + 2791179 q^{63} - 5445956 q^{65} + 1848219 q^{67} + 6834063 q^{69} + 3417184 q^{71} - 2499822 q^{73} + 10079553 q^{75} + 5025267 q^{77} - 2636926 q^{79} + 2491398 q^{81} + 10059354 q^{83} - 439425 q^{85} + 2572923 q^{87} - 3506160 q^{89} - 19003851 q^{91} - 15169884 q^{93} + 1063145 q^{95} + 5893526 q^{97} + 11301453 q^{99}+O(q^{100}) $ 2 * q + 69 * q^3 + 155 * q^5 + 2238 * q^7 + 855 * q^9 + 3295 * q^11 - 13427 * q^13 + 34782 * q^15 - 32256 * q^17 + 13718 * q^19 + 103797 * q^21 + 82525 * q^23 + 159919 * q^25 + 75141 * q^27 - 12749 * q^29 - 258944 * q^31 + 257052 * q^33 + 448167 * q^35 - 149260 * q^37 - 889557 * q^39 + 339130 * q^41 + 83869 * q^43 + 2097243 * q^45 - 1471025 * q^47 + 1105372 * q^49 - 913437 * q^51 - 945643 * q^53 + 1736899 * q^55 + 473271 * q^57 + 969009 * q^59 - 1506755 * q^61 + 2791179 * q^63 - 5445956 * q^65 + 1848219 * q^67 + 6834063 * q^69 + 3417184 * q^71 - 2499822 * q^73 + 10079553 * q^75 + 5025267 * q^77 - 2636926 * q^79 + 2491398 * q^81 + 10059354 * q^83 - 439425 * q^85 + 2572923 * q^87 - 3506160 * q^89 - 19003851 * q^91 - 15169884 * q^93 + 1063145 * q^95 + 5893526 * q^97 + 11301453 * 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
$3$	72.2392	1.54472	0.772358	−	0.635187i	$-0.219077\pi$
$3$	72.2392	1.54472	0.772358	+	0.635187i	$0.219077\pi$
$4$	0	0
$5$	467.472	1.67248	0.836240	−	0.548364i	$-0.184749\pi$
$5$	467.472	1.67248	0.836240	+	0.548364i	$0.184749\pi$
$6$	0	0
$7$	1471.23	1.62121	0.810603	−	0.585596i	$-0.199139\pi$
$7$	1471.23	1.62121	0.810603	+	0.585596i	$0.199139\pi$
$8$	0	0
$9$	3031.51	1.38615
$10$	0	0
$11$	3547.04	0.803511	0.401756	−	0.915747i	$-0.368400\pi$
$11$	3547.04	0.803511	0.401756	+	0.915747i	$0.368400\pi$
$12$	0	0
$13$	−12361.8	−1.56056	−0.780279	−	0.625431i	$-0.784923\pi$
$13$	−12361.8	−1.56056	−0.780279	+	0.625431i	$0.784923\pi$
$14$	0	0
$15$	33769.8	2.58351
$16$	0	0
$17$	−13486.3	−0.665764	−0.332882	−	0.942969i	$-0.608021\pi$
$17$	−13486.3	−0.665764	−0.332882	+	0.942969i	$0.608021\pi$
$18$	0	0
$19$	6859.00	0.229416
$19$	6859.00	0.229416
$20$	0	0
$21$	106281.	2.50430
$22$	0	0
$23$	94084.9	1.61240	0.806199	−	0.591644i	$-0.201521\pi$
$23$	94084.9	1.61240	0.806199	+	0.591644i	$0.201521\pi$
$24$	0	0
$25$	140405.	1.79719
$26$	0	0
$27$	61006.6	0.596490
$28$	0	0
$29$	33541.0	0.255378	0.127689	−	0.991814i	$-0.459244\pi$
$29$	33541.0	0.255378	0.127689	+	0.991814i	$0.459244\pi$
$30$	0	0
$31$	−212096.	−1.27869	−0.639346	−	0.768919i	$-0.720795\pi$
$31$	−212096.	−1.27869	−0.639346	+	0.768919i	$0.720795\pi$
$32$	0	0
$33$	256236.	1.24120
$34$	0	0
$35$	687760.	2.71143
$36$	0	0
$37$	33656.5	0.109235	0.0546176	−	0.998507i	$-0.482606\pi$
$37$	33656.5	0.109235	0.0546176	+	0.998507i	$0.482606\pi$
$38$	0	0
$39$	−893007.	−2.41062
$40$	0	0
$41$	−480531.	−1.08887	−0.544437	−	0.838801i	$-0.683257\pi$
$41$	−480531.	−1.08887	−0.544437	+	0.838801i	$0.683257\pi$
$42$	0	0
$43$	561340.	1.07668	0.538339	−	0.842728i	$-0.319052\pi$
$43$	561340.	1.07668	0.538339	+	0.842728i	$0.319052\pi$
$44$	0	0
$45$	1.41715e6	2.31830
$46$	0	0
$47$	−478445.	−0.672187	−0.336093	−	0.941829i	$-0.609106\pi$
$47$	−478445.	−0.672187	−0.336093	+	0.941829i	$0.609106\pi$
$48$	0	0
$49$	1.34098e6	1.62831
$50$	0	0
$51$	−974237.	−1.02842
$52$	0	0
$53$	−93127.0	−0.0859232	−0.0429616	−	0.999077i	$-0.513679\pi$
$53$	−93127.0	−0.0859232	−0.0429616	+	0.999077i	$0.513679\pi$
$54$	0	0
$55$	1.65814e6	1.34386
$56$	0	0
$57$	495489.	0.354382
$58$	0	0
$59$	−955034.	−0.605392	−0.302696	−	0.953087i	$-0.597887\pi$
$59$	−955034.	−0.605392	−0.302696	+	0.953087i	$0.597887\pi$
$60$	0	0
$61$	−1.71632e6	−0.968152	−0.484076	−	0.875026i	$-0.660844\pi$
$61$	−1.71632e6	−0.968152	−0.484076	+	0.875026i	$0.660844\pi$
$62$	0	0
$63$	4.46005e6	2.24723
$64$	0	0
$65$	−5.77880e6	−2.61000
$66$	0	0
$67$	−476255.	−0.193454	−0.0967270	−	0.995311i	$-0.530837\pi$
$67$	−476255.	−0.193454	−0.0967270	+	0.995311i	$0.530837\pi$
$68$	0	0
$69$	6.79662e6	2.49070
$70$	0	0
$71$	−1.95518e6	−0.648311	−0.324155	−	0.946004i	$-0.605080\pi$
$71$	−1.95518e6	−0.648311	−0.324155	+	0.946004i	$0.605080\pi$
$72$	0	0
$73$	1.21632e6	0.365948	0.182974	−	0.983118i	$-0.441428\pi$
$73$	1.21632e6	0.365948	0.182974	+	0.983118i	$0.441428\pi$
$74$	0	0
$75$	1.01428e7	2.77614
$76$	0	0
$77$	5.21852e6	1.30266
$78$	0	0
$79$	−3.94544e6	−0.900328	−0.450164	−	0.892946i	$-0.648634\pi$
$79$	−3.94544e6	−0.900328	−0.450164	+	0.892946i	$0.648634\pi$
$80$	0	0
$81$	−2.22284e6	−0.464740
$82$	0	0
$83$	3.54252e6	0.680048	0.340024	−	0.940417i	$-0.389565\pi$
$83$	3.54252e6	0.680048	0.340024	+	0.940417i	$0.389565\pi$
$84$	0	0
$85$	−6.30445e6	−1.11348
$86$	0	0
$87$	2.42298e6	0.394487
$88$	0	0
$89$	−7.50202e6	−1.12801	−0.564006	−	0.825771i	$-0.690740\pi$
$89$	−7.50202e6	−1.12801	−0.564006	+	0.825771i	$0.690740\pi$
$90$	0	0
$91$	−1.81871e7	−2.52999
$92$	0	0
$93$	−1.53216e7	−1.97522
$94$	0	0
$95$	3.20639e6	0.383693
$96$	0	0
$97$	1.27974e7	1.42371	0.711855	−	0.702326i	$-0.247855\pi$
$97$	1.27974e7	1.42371	0.711855	+	0.702326i	$0.247855\pi$
$98$	0	0
$99$	1.07529e7	1.11379
$100$	0	0
$101$	332855.	0.0321463	0.0160731	−	0.999871i	$-0.494884\pi$
$101$	332855.	0.0321463	0.0160731	+	0.999871i	$0.494884\pi$
$102$	0	0
$103$	−4.52517e6	−0.408042	−0.204021	−	0.978967i	$-0.565401\pi$
$103$	−4.52517e6	−0.408042	−0.204021	+	0.978967i	$0.565401\pi$
$104$	0	0
$105$	4.96833e7	4.18840
$106$	0	0
$107$	−7.33885e6	−0.579141	−0.289571	−	0.957157i	$-0.593513\pi$
$107$	−7.33885e6	−0.579141	−0.289571	+	0.957157i	$0.593513\pi$
$108$	0	0
$109$	−1.03920e7	−0.768612	−0.384306	−	0.923206i	$-0.625559\pi$
$109$	−1.03920e7	−0.768612	−0.384306	+	0.923206i	$0.625559\pi$
$110$	0	0
$111$	2.43132e6	0.168737
$112$	0	0
$113$	5.33971e6	0.348131	0.174066	−	0.984734i	$-0.444309\pi$
$113$	5.33971e6	0.348131	0.174066	+	0.984734i	$0.444309\pi$
$114$	0	0
$115$	4.39820e7	2.69670
$116$	0	0
$117$	−3.74749e7	−2.16317
$118$	0	0
$119$	−1.98414e7	−1.07934
$120$	0	0
$121$	−6.90567e6	−0.354370
$122$	0	0
$123$	−3.47132e7	−1.68200
$124$	0	0
$125$	2.91142e7	1.33328
$126$	0	0
$127$	3.43331e7	1.48730	0.743652	−	0.668566i	$-0.233092\pi$
$127$	3.43331e7	1.48730	0.743652	+	0.668566i	$0.233092\pi$
$128$	0	0
$129$	4.05507e7	1.66316
$130$	0	0
$131$	−250925.	−0.00975202	−0.00487601	−	0.999988i	$-0.501552\pi$
$131$	−250925.	−0.00975202	−0.00487601	+	0.999988i	$0.501552\pi$
$132$	0	0
$133$	1.00912e7	0.371930
$134$	0	0
$135$	2.85189e7	0.997618
$136$	0	0
$137$	−2.54613e7	−0.845978	−0.422989	−	0.906135i	$-0.639019\pi$
$137$	−2.54613e7	−0.845978	−0.422989	+	0.906135i	$0.639019\pi$
$138$	0	0
$139$	−4.33237e7	−1.36828	−0.684138	−	0.729352i	$-0.739822\pi$
$139$	−4.33237e7	−1.36828	−0.684138	+	0.729352i	$0.739822\pi$
$140$	0	0
$141$	−3.45625e7	−1.03834
$142$	0	0
$143$	−4.38478e7	−1.25393
$144$	0	0
$145$	1.56795e7	0.427114
$146$	0	0
$147$	9.68716e7	2.51528
$148$	0	0
$149$	−3.98620e7	−0.987204	−0.493602	−	0.869688i	$-0.664320\pi$
$149$	−3.98620e7	−0.987204	−0.493602	+	0.869688i	$0.664320\pi$
$150$	0	0
$151$	2.13810e6	0.0505369	0.0252684	−	0.999681i	$-0.491956\pi$
$151$	2.13810e6	0.0505369	0.0252684	+	0.999681i	$0.491956\pi$
$152$	0	0
$153$	−4.08837e7	−0.922847
$154$	0	0
$155$	−9.91489e7	−2.13859
$156$	0	0
$157$	−3.04702e7	−0.628387	−0.314193	−	0.949359i	$-0.601734\pi$
$157$	−3.04702e7	−0.628387	−0.314193	+	0.949359i	$0.601734\pi$
$158$	0	0
$159$	−6.72743e6	−0.132727
$160$	0	0
$161$	1.38421e8	2.61403
$162$	0	0
$163$	9.48952e7	1.71628	0.858139	−	0.513417i	$-0.171621\pi$
$163$	9.48952e7	1.71628	0.858139	+	0.513417i	$0.171621\pi$
$164$	0	0
$165$	1.19783e8	2.07588
$166$	0	0
$167$	1.10332e8	1.83313	0.916563	−	0.399890i	$-0.130952\pi$
$167$	1.10332e8	1.83313	0.916563	+	0.399890i	$0.130952\pi$
$168$	0	0
$169$	9.00657e7	1.43534
$170$	0	0
$171$	2.07931e7	0.318004
$172$	0	0
$173$	1.06112e8	1.55813	0.779066	−	0.626942i	$-0.215694\pi$
$173$	1.06112e8	1.55813	0.779066	+	0.626942i	$0.215694\pi$
$174$	0	0
$175$	2.06569e8	2.91361
$176$	0	0
$177$	−6.89909e7	−0.935159
$178$	0	0
$179$	−4.88351e7	−0.636424	−0.318212	−	0.948020i	$-0.603082\pi$
$179$	−4.88351e7	−0.636424	−0.318212	+	0.948020i	$0.603082\pi$
$180$	0	0
$181$	5.52074e6	0.0692026	0.0346013	−	0.999401i	$-0.488984\pi$
$181$	5.52074e6	0.0692026	0.0346013	+	0.999401i	$0.488984\pi$
$182$	0	0
$183$	−1.23986e8	−1.49552
$184$	0	0
$185$	1.57335e7	0.182693
$186$	0	0
$187$	−4.78363e7	−0.534948
$188$	0	0
$189$	8.97549e7	0.967034
$190$	0	0
$191$	1.15562e8	1.20005	0.600023	−	0.799983i	$-0.295158\pi$
$191$	1.15562e8	1.20005	0.600023	+	0.799983i	$0.295158\pi$
$192$	0	0
$193$	4.01954e6	0.0402463	0.0201232	−	0.999798i	$-0.493594\pi$
$193$	4.01954e6	0.0402463	0.0201232	+	0.999798i	$0.493594\pi$
$194$	0	0
$195$	−4.17456e8	−4.03171
$196$	0	0
$197$	8.92289e7	0.831522	0.415761	−	0.909474i	$-0.363515\pi$
$197$	8.92289e7	0.831522	0.415761	+	0.909474i	$0.363515\pi$
$198$	0	0
$199$	1.73811e8	1.56348	0.781740	−	0.623604i	$-0.214332\pi$
$199$	1.73811e8	1.56348	0.781740	+	0.623604i	$0.214332\pi$
$200$	0	0
$201$	−3.44043e7	−0.298832
$202$	0	0
$203$	4.93467e7	0.414020
$204$	0	0
$205$	−2.24635e8	−1.82112
$206$	0	0
$207$	2.85219e8	2.23502
$208$	0	0
$209$	2.43292e7	0.184338
$210$	0	0
$211$	1.48188e8	1.08598	0.542992	−	0.839738i	$-0.317291\pi$
$211$	1.48188e8	1.08598	0.542992	+	0.839738i	$0.317291\pi$
$212$	0	0
$213$	−1.41241e8	−1.00146
$214$	0	0
$215$	2.62411e8	1.80072
$216$	0	0
$217$	−3.12042e8	−2.07302
$218$	0	0
$219$	8.78662e7	0.565285
$220$	0	0
$221$	1.66714e8	1.03896
$222$	0	0
$223$	−1.38594e6	−0.00836907	−0.00418454	−	0.999991i	$-0.501332\pi$
$223$	−1.38594e6	−0.00836907	−0.00418454	+	0.999991i	$0.501332\pi$
$224$	0	0
$225$	4.25639e8	2.49117
$226$	0	0
$227$	−1.35147e7	−0.0766858	−0.0383429	−	0.999265i	$-0.512208\pi$
$227$	−1.35147e7	−0.0766858	−0.0383429	+	0.999265i	$0.512208\pi$
$228$	0	0
$229$	3.09882e7	0.170519	0.0852594	−	0.996359i	$-0.472828\pi$
$229$	3.09882e7	0.170519	0.0852594	+	0.996359i	$0.472828\pi$
$230$	0	0
$231$	3.76982e8	2.01224
$232$	0	0
$233$	−2.04142e8	−1.05727	−0.528635	−	0.848849i	$-0.677296\pi$
$233$	−2.04142e8	−1.05727	−0.528635	+	0.848849i	$0.677296\pi$
$234$	0	0
$235$	−2.23660e8	−1.12422
$236$	0	0
$237$	−2.85016e8	−1.39075
$238$	0	0
$239$	2.25595e8	1.06890	0.534449	−	0.845201i	$-0.320519\pi$
$239$	2.25595e8	1.06890	0.534449	+	0.845201i	$0.320519\pi$
$240$	0	0
$241$	4.16156e7	0.191512	0.0957562	−	0.995405i	$-0.469473\pi$
$241$	4.16156e7	0.191512	0.0957562	+	0.995405i	$0.469473\pi$
$242$	0	0
$243$	−2.93998e8	−1.31438
$244$	0	0
$245$	6.26872e8	2.72331
$246$	0	0
$247$	−8.47896e7	−0.358017
$248$	0	0
$249$	2.55909e8	1.05048
$250$	0	0
$251$	−4.55579e8	−1.81847	−0.909233	−	0.416287i	$-0.863331\pi$
$251$	−4.55579e8	−1.81847	−0.909233	+	0.416287i	$0.863331\pi$
$252$	0	0
$253$	3.33723e8	1.29558
$254$	0	0
$255$	−4.55428e8	−1.72000
$256$	0	0
$257$	−2.46264e8	−0.904972	−0.452486	−	0.891772i	$-0.649463\pi$
$257$	−2.46264e8	−0.904972	−0.452486	+	0.891772i	$0.649463\pi$
$258$	0	0
$259$	4.95165e7	0.177093
$260$	0	0
$261$	1.01680e8	0.353992
$262$	0	0
$263$	−4.04810e8	−1.37217	−0.686083	−	0.727523i	$-0.740671\pi$
$263$	−4.04810e8	−1.37217	−0.686083	+	0.727523i	$0.740671\pi$
$264$	0	0
$265$	−4.35343e7	−0.143705
$266$	0	0
$267$	−5.41940e8	−1.74246
$268$	0	0
$269$	−2.44584e8	−0.766118	−0.383059	−	0.923724i	$-0.625129\pi$
$269$	−2.44584e8	−0.766118	−0.383059	+	0.923724i	$0.625129\pi$
$270$	0	0
$271$	−7.88689e7	−0.240721	−0.120360	−	0.992730i	$-0.538405\pi$
$271$	−7.88689e7	−0.240721	−0.120360	+	0.992730i	$0.538405\pi$
$272$	0	0
$273$	−1.31382e9	−3.90811
$274$	0	0
$275$	4.98023e8	1.44406
$276$	0	0
$277$	−2.30329e8	−0.651133	−0.325567	−	0.945519i	$-0.605555\pi$
$277$	−2.30329e8	−0.651133	−0.325567	+	0.945519i	$0.605555\pi$
$278$	0	0
$279$	−6.42970e8	−1.77246
$280$	0	0
$281$	−5.44589e8	−1.46419	−0.732094	−	0.681203i	$-0.761457\pi$
$281$	−5.44589e8	−1.46419	−0.732094	+	0.681203i	$0.761457\pi$
$282$	0	0
$283$	3.69887e7	0.0970101	0.0485051	−	0.998823i	$-0.484554\pi$
$283$	3.69887e7	0.0970101	0.0485051	+	0.998823i	$0.484554\pi$
$284$	0	0
$285$	2.31627e8	0.592697
$286$	0	0
$287$	−7.06973e8	−1.76529
$288$	0	0
$289$	−2.28460e8	−0.556759
$290$	0	0
$291$	9.24477e8	2.19923
$292$	0	0
$293$	1.98050e8	0.459979	0.229990	−	0.973193i	$-0.426131\pi$
$293$	1.98050e8	0.459979	0.229990	+	0.973193i	$0.426131\pi$
$294$	0	0
$295$	−4.46452e8	−1.01251
$296$	0	0
$297$	2.16393e8	0.479287
$298$	0	0
$299$	−1.16306e9	−2.51624
$300$	0	0
$301$	8.25861e8	1.74552
$302$	0	0
$303$	2.40452e7	0.0496569
$304$	0	0
$305$	−8.02331e8	−1.61921
$306$	0	0
$307$	−3.28496e8	−0.647957	−0.323979	−	0.946064i	$-0.605021\pi$
$307$	−3.28496e8	−0.647957	−0.323979	+	0.946064i	$0.605021\pi$
$308$	0	0
$309$	−3.26895e8	−0.630309
$310$	0	0
$311$	5.87246e8	1.10703	0.553514	−	0.832840i	$-0.313287\pi$
$311$	5.87246e8	1.10703	0.553514	+	0.832840i	$0.313287\pi$
$312$	0	0
$313$	6.29976e8	1.16123	0.580616	−	0.814178i	$-0.302812\pi$
$313$	6.29976e8	1.16123	0.580616	+	0.814178i	$0.302812\pi$
$314$	0	0
$315$	2.08495e9	3.75845
$316$	0	0
$317$	8.00885e8	1.41209	0.706046	−	0.708166i	$-0.250477\pi$
$317$	8.00885e8	1.41209	0.706046	+	0.708166i	$0.250477\pi$
$318$	0	0
$319$	1.18971e8	0.205199
$320$	0	0
$321$	−5.30153e8	−0.894609
$322$	0	0
$323$	−9.25022e7	−0.152737
$324$	0	0
$325$	−1.73566e9	−2.80462
$326$	0	0
$327$	−7.50711e8	−1.18729
$328$	0	0
$329$	−7.03905e8	−1.08975
$330$	0	0
$331$	4.54262e8	0.688507	0.344253	−	0.938877i	$-0.388132\pi$
$331$	4.54262e8	0.688507	0.344253	+	0.938877i	$0.388132\pi$
$332$	0	0
$333$	1.02030e8	0.151416
$334$	0	0
$335$	−2.22636e8	−0.323548
$336$	0	0
$337$	−9.81183e8	−1.39651	−0.698257	−	0.715847i	$-0.746041\pi$
$337$	−9.81183e8	−1.39651	−0.698257	+	0.715847i	$0.746041\pi$
$338$	0	0
$339$	3.85737e8	0.537764
$340$	0	0
$341$	−7.52313e8	−1.02744
$342$	0	0
$343$	7.61275e8	1.01862
$344$	0	0
$345$	3.17723e9	4.16564
$346$	0	0
$347$	−2.69918e8	−0.346800	−0.173400	−	0.984852i	$-0.555475\pi$
$347$	−2.69918e8	−0.346800	−0.173400	+	0.984852i	$0.555475\pi$
$348$	0	0
$349$	1.09409e9	1.37773	0.688863	−	0.724892i	$-0.258110\pi$
$349$	1.09409e9	1.37773	0.688863	+	0.724892i	$0.258110\pi$
$350$	0	0
$351$	−7.54151e8	−0.930858
$352$	0	0
$353$	5.02386e8	0.607891	0.303946	−	0.952689i	$-0.401696\pi$
$353$	5.02386e8	0.607891	0.303946	+	0.952689i	$0.401696\pi$
$354$	0	0
$355$	−9.13994e8	−1.08429
$356$	0	0
$357$	−1.43333e9	−1.66727
$358$	0	0
$359$	4.89464e8	0.558329	0.279165	−	0.960243i	$-0.409942\pi$
$359$	4.89464e8	0.558329	0.279165	+	0.960243i	$0.409942\pi$
$360$	0	0
$361$	4.70459e7	0.0526316
$362$	0	0
$363$	−4.98860e8	−0.547401
$364$	0	0
$365$	5.68597e8	0.612040
$366$	0	0
$367$	5.54974e8	0.586059	0.293029	−	0.956103i	$-0.405337\pi$
$367$	5.54974e8	0.586059	0.293029	+	0.956103i	$0.405337\pi$
$368$	0	0
$369$	−1.45673e9	−1.50934
$370$	0	0
$371$	−1.37012e8	−0.139299
$372$	0	0
$373$	−1.29835e9	−1.29542	−0.647708	−	0.761889i	$-0.724272\pi$
$373$	−1.29835e9	−1.29542	−0.647708	+	0.761889i	$0.724272\pi$
$374$	0	0
$375$	2.10319e9	2.05954
$376$	0	0
$377$	−4.14628e8	−0.398532
$378$	0	0
$379$	−7.49227e8	−0.706929	−0.353465	−	0.935448i	$-0.614997\pi$
$379$	−7.49227e8	−0.706929	−0.353465	+	0.935448i	$0.614997\pi$
$380$	0	0
$381$	2.48020e9	2.29746
$382$	0	0
$383$	−2.12814e8	−0.193555	−0.0967774	−	0.995306i	$-0.530853\pi$
$383$	−2.12814e8	−0.193555	−0.0967774	+	0.995306i	$0.530853\pi$
$384$	0	0
$385$	2.43951e9	2.17867
$386$	0	0
$387$	1.70171e9	1.49244
$388$	0	0
$389$	3.13367e8	0.269917	0.134958	−	0.990851i	$-0.456910\pi$
$389$	3.13367e8	0.269917	0.134958	+	0.990851i	$0.456910\pi$
$390$	0	0
$391$	−1.26885e9	−1.07348
$392$	0	0
$393$	−1.81266e7	−0.0150641
$394$	0	0
$395$	−1.84438e9	−1.50578
$396$	0	0
$397$	1.91013e9	1.53213	0.766066	−	0.642762i	$-0.222211\pi$
$397$	1.91013e9	1.53213	0.766066	+	0.642762i	$0.222211\pi$
$398$	0	0
$399$	7.28980e8	0.574527
$400$	0	0
$401$	−1.27797e9	−0.989726	−0.494863	−	0.868971i	$-0.664782\pi$
$401$	−1.27797e9	−0.989726	−0.494863	+	0.868971i	$0.664782\pi$
$402$	0	0
$403$	2.62189e9	1.99547
$404$	0	0
$405$	−1.03912e9	−0.777269
$406$	0	0
$407$	1.19381e8	0.0877716
$408$	0	0
$409$	2.92650e8	0.211503	0.105752	−	0.994393i	$-0.466275\pi$
$409$	2.92650e8	0.211503	0.105752	+	0.994393i	$0.466275\pi$
$410$	0	0
$411$	−1.83931e9	−1.30680
$412$	0	0
$413$	−1.40508e9	−0.981465
$414$	0	0
$415$	1.65603e9	1.13737
$416$	0	0
$417$	−3.12967e9	−2.11360
$418$	0	0
$419$	−3.81573e8	−0.253413	−0.126706	−	0.991940i	$-0.540441\pi$
$419$	−3.81573e8	−0.253413	−0.126706	+	0.991940i	$0.540441\pi$
$420$	0	0
$421$	2.64639e9	1.72849	0.864244	−	0.503073i	$-0.167797\pi$
$421$	2.64639e9	1.72849	0.864244	+	0.503073i	$0.167797\pi$
$422$	0	0
$423$	−1.45041e9	−0.931751
$424$	0	0
$425$	−1.89354e9	−1.19650
$426$	0	0
$427$	−2.52511e9	−1.56957
$428$	0	0
$429$	−3.16753e9	−1.93696
$430$	0	0
$431$	−1.69877e9	−1.02203	−0.511015	−	0.859572i	$-0.670730\pi$
$431$	−1.69877e9	−1.02203	−0.511015	+	0.859572i	$0.670730\pi$
$432$	0	0
$433$	2.37930e9	1.40845	0.704226	−	0.709975i	$-0.251294\pi$
$433$	2.37930e9	1.40845	0.704226	+	0.709975i	$0.251294\pi$
$434$	0	0
$435$	1.13267e9	0.659770
$436$	0	0
$437$	6.45328e8	0.369909
$438$	0	0
$439$	−2.18333e9	−1.23167	−0.615835	−	0.787875i	$-0.711181\pi$
$439$	−2.18333e9	−1.23167	−0.615835	+	0.787875i	$0.711181\pi$
$440$	0	0
$441$	4.06520e9	2.25708
$442$	0	0
$443$	2.41144e9	1.31784	0.658922	−	0.752212i	$-0.271013\pi$
$443$	2.41144e9	1.31784	0.658922	+	0.752212i	$0.271013\pi$
$444$	0	0
$445$	−3.50699e9	−1.88658
$446$	0	0
$447$	−2.87960e9	−1.52495
$448$	0	0
$449$	−6.02088e8	−0.313905	−0.156952	−	0.987606i	$-0.550167\pi$
$449$	−6.02088e8	−0.313905	−0.156952	+	0.987606i	$0.550167\pi$
$450$	0	0
$451$	−1.70446e9	−0.874923
$452$	0	0
$453$	1.54455e8	0.0780652
$454$	0	0
$455$	−8.50196e9	−4.23135
$456$	0	0
$457$	−3.28085e9	−1.60798	−0.803989	−	0.594644i	$-0.797293\pi$
$457$	−3.28085e9	−1.60798	−0.803989	+	0.594644i	$0.797293\pi$
$458$	0	0
$459$	−8.22750e8	−0.397122
$460$	0	0
$461$	−3.50631e9	−1.66685	−0.833426	−	0.552632i	$-0.813624\pi$
$461$	−3.50631e9	−1.66685	−0.833426	+	0.552632i	$0.813624\pi$
$462$	0	0
$463$	3.85170e9	1.80351	0.901757	−	0.432244i	$-0.142278\pi$
$463$	3.85170e9	1.80351	0.901757	+	0.432244i	$0.142278\pi$
$464$	0	0
$465$	−7.16244e9	−3.30351
$466$	0	0
$467$	−4.73318e8	−0.215052	−0.107526	−	0.994202i	$-0.534293\pi$
$467$	−4.73318e8	−0.215052	−0.107526	+	0.994202i	$0.534293\pi$
$468$	0	0
$469$	−7.00682e8	−0.313629
$470$	0	0
$471$	−2.20115e9	−0.970680
$472$	0	0
$473$	1.99109e9	0.865123
$474$	0	0
$475$	9.63039e8	0.412303
$476$	0	0
$477$	−2.82315e8	−0.119102
$478$	0	0
$479$	−2.43257e9	−1.01132	−0.505662	−	0.862732i	$-0.668752\pi$
$479$	−2.43257e9	−1.01132	−0.505662	+	0.862732i	$0.668752\pi$
$480$	0	0
$481$	−4.16055e8	−0.170468
$482$	0	0
$483$	9.99941e9	4.03793
$484$	0	0
$485$	5.98244e9	2.38113
$486$	0	0
$487$	3.37943e9	1.32584	0.662921	−	0.748689i	$-0.269317\pi$
$487$	3.37943e9	1.32584	0.662921	+	0.748689i	$0.269317\pi$
$488$	0	0
$489$	6.85516e9	2.65116
$490$	0	0
$491$	1.62343e9	0.618938	0.309469	−	0.950910i	$-0.399849\pi$
$491$	1.62343e9	0.618938	0.309469	+	0.950910i	$0.399849\pi$
$492$	0	0
$493$	−4.52343e8	−0.170021
$494$	0	0
$495$	5.02667e9	1.86278
$496$	0	0
$497$	−2.87653e9	−1.05105
$498$	0	0
$499$	−3.88153e9	−1.39846	−0.699231	−	0.714896i	$-0.746474\pi$
$499$	−3.88153e9	−1.39846	−0.699231	+	0.714896i	$0.746474\pi$
$500$	0	0
$501$	7.97027e9	2.83166
$502$	0	0
$503$	−2.38583e9	−0.835893	−0.417947	−	0.908472i	$-0.637250\pi$
$503$	−2.38583e9	−0.835893	−0.417947	+	0.908472i	$0.637250\pi$
$504$	0	0
$505$	1.55600e8	0.0537639
$506$	0	0
$507$	6.50628e9	2.21720
$508$	0	0
$509$	−3.37295e9	−1.13370	−0.566849	−	0.823822i	$-0.691838\pi$
$509$	−3.37295e9	−1.13370	−0.566849	+	0.823822i	$0.691838\pi$
$510$	0	0
$511$	1.78949e9	0.593277
$512$	0	0
$513$	4.18444e8	0.136844
$514$	0	0
$515$	−2.11539e9	−0.682441
$516$	0	0
$517$	−1.69707e9	−0.540109
$518$	0	0
$519$	7.66547e9	2.40687
$520$	0	0
$521$	5.32384e9	1.64927	0.824637	−	0.565662i	$-0.191379\pi$
$521$	5.32384e9	1.64927	0.824637	+	0.565662i	$0.191379\pi$
$522$	0	0
$523$	−4.07644e9	−1.24602	−0.623010	−	0.782214i	$-0.714090\pi$
$523$	−4.07644e9	−1.24602	−0.623010	+	0.782214i	$0.714090\pi$
$524$	0	0
$525$	1.49224e10	4.50070
$526$	0	0
$527$	2.86038e9	0.851307
$528$	0	0
$529$	5.44713e9	1.59983
$530$	0	0
$531$	−2.89519e9	−0.839163
$532$	0	0
$533$	5.94023e9	1.69925
$534$	0	0
$535$	−3.43071e9	−0.968602
$536$	0	0
$537$	−3.52781e9	−0.983095
$538$	0	0
$539$	4.75652e9	1.30837
$540$	0	0
$541$	−3.63310e9	−0.986477	−0.493239	−	0.869894i	$-0.664187\pi$
$541$	−3.63310e9	−0.986477	−0.493239	+	0.869894i	$0.664187\pi$
$542$	0	0
$543$	3.98814e8	0.106898
$544$	0	0
$545$	−4.85798e9	−1.28549
$546$	0	0
$547$	5.74113e9	1.49983	0.749914	−	0.661535i	$-0.230095\pi$
$547$	5.74113e9	1.49983	0.749914	+	0.661535i	$0.230095\pi$
$548$	0	0
$549$	−5.20303e9	−1.34200
$550$	0	0
$551$	2.30058e8	0.0585877
$552$	0	0
$553$	−5.80466e9	−1.45962
$554$	0	0
$555$	1.13657e9	0.282210
$556$	0	0
$557$	9.95936e8	0.244196	0.122098	−	0.992518i	$-0.461038\pi$
$557$	9.95936e8	0.244196	0.122098	+	0.992518i	$0.461038\pi$
$558$	0	0
$559$	−6.93917e9	−1.68022
$560$	0	0
$561$	−3.45566e9	−0.826344
$562$	0	0
$563$	4.05283e9	0.957147	0.478573	−	0.878048i	$-0.341154\pi$
$563$	4.05283e9	0.957147	0.478573	+	0.878048i	$0.341154\pi$
$564$	0	0
$565$	2.49617e9	0.582242
$566$	0	0
$567$	−3.27031e9	−0.753440
$568$	0	0
$569$	3.35141e9	0.762666	0.381333	−	0.924438i	$-0.375465\pi$
$569$	3.35141e9	0.762666	0.381333	+	0.924438i	$0.375465\pi$
$570$	0	0
$571$	−4.22454e9	−0.949627	−0.474813	−	0.880087i	$-0.657484\pi$
$571$	−4.22454e9	−0.949627	−0.474813	+	0.880087i	$0.657484\pi$
$572$	0	0
$573$	8.34810e9	1.85373
$574$	0	0
$575$	1.32100e10	2.89778
$576$	0	0
$577$	−6.43062e9	−1.39360	−0.696799	−	0.717266i	$-0.745393\pi$
$577$	−6.43062e9	−1.39360	−0.696799	+	0.717266i	$0.745393\pi$
$578$	0	0
$579$	2.90369e8	0.0621691
$580$	0	0
$581$	5.21188e9	1.10250
$582$	0	0
$583$	−3.30325e8	−0.0690402
$584$	0	0
$585$	−1.75185e10	−3.61785
$586$	0	0
$587$	2.90754e9	0.593326	0.296663	−	0.954982i	$-0.404126\pi$
$587$	2.90754e9	0.593326	0.296663	+	0.954982i	$0.404126\pi$
$588$	0	0
$589$	−1.45476e9	−0.293352
$590$	0	0
$591$	6.44582e9	1.28447
$592$	0	0
$593$	−5.13346e8	−0.101093	−0.0505463	−	0.998722i	$-0.516096\pi$
$593$	−5.13346e8	−0.101093	−0.0505463	+	0.998722i	$0.516096\pi$
$594$	0	0
$595$	−9.27531e9	−1.80517
$596$	0	0
$597$	1.25560e10	2.41513
$598$	0	0
$599$	3.44859e9	0.655613	0.327807	−	0.944745i	$-0.393691\pi$
$599$	3.44859e9	0.655613	0.327807	+	0.944745i	$0.393691\pi$
$600$	0	0
$601$	−5.96985e9	−1.12177	−0.560883	−	0.827895i	$-0.689538\pi$
$601$	−5.96985e9	−1.12177	−0.560883	+	0.827895i	$0.689538\pi$
$602$	0	0
$603$	−1.44377e9	−0.268156
$604$	0	0
$605$	−3.22821e9	−0.592676
$606$	0	0
$607$	1.00863e10	1.83050	0.915249	−	0.402888i	$-0.131993\pi$
$607$	1.00863e10	1.83050	0.915249	+	0.402888i	$0.131993\pi$
$608$	0	0
$609$	3.56477e9	0.639544
$610$	0	0
$611$	5.91445e9	1.04899
$612$	0	0
$613$	1.24543e9	0.218378	0.109189	−	0.994021i	$-0.465175\pi$
$613$	1.24543e9	0.218378	0.109189	+	0.994021i	$0.465175\pi$
$614$	0	0
$615$	−1.62275e10	−2.81311
$616$	0	0
$617$	−5.96038e8	−0.102159	−0.0510794	−	0.998695i	$-0.516266\pi$
$617$	−5.96038e8	−0.102159	−0.0510794	+	0.998695i	$0.516266\pi$
$618$	0	0
$619$	3.90038e9	0.660981	0.330491	−	0.943809i	$-0.392786\pi$
$619$	3.90038e9	0.660981	0.330491	+	0.943809i	$0.392786\pi$
$620$	0	0
$621$	5.73979e9	0.961780
$622$	0	0
$623$	−1.10372e10	−1.82874
$624$	0	0
$625$	2.64094e9	0.432692
$626$	0	0
$627$	1.75752e9	0.284750
$628$	0	0
$629$	−4.53899e8	−0.0727248
$630$	0	0
$631$	−4.64916e9	−0.736668	−0.368334	−	0.929694i	$-0.620072\pi$
$631$	−4.64916e9	−0.736668	−0.368334	+	0.929694i	$0.620072\pi$
$632$	0	0
$633$	1.07050e10	1.67754
$634$	0	0
$635$	1.60498e10	2.48749
$636$	0	0
$637$	−1.65770e10	−2.54107
$638$	0	0
$639$	−5.92715e9	−0.898655
$640$	0	0
$641$	9.04607e9	1.35662	0.678308	−	0.734778i	$-0.262714\pi$
$641$	9.04607e9	1.35662	0.678308	+	0.734778i	$0.262714\pi$
$642$	0	0
$643$	−1.11196e10	−1.64949	−0.824746	−	0.565503i	$-0.808682\pi$
$643$	−1.11196e10	−1.64949	−0.824746	+	0.565503i	$0.808682\pi$
$644$	0	0
$645$	1.89563e10	2.78161
$646$	0	0
$647$	3.91564e9	0.568379	0.284189	−	0.958768i	$-0.408276\pi$
$647$	3.91564e9	0.568379	0.284189	+	0.958768i	$0.408276\pi$
$648$	0	0
$649$	−3.38754e9	−0.486439
$650$	0	0
$651$	−2.25417e10	−3.20223
$652$	0	0
$653$	−1.07233e10	−1.50707	−0.753536	−	0.657407i	$-0.771653\pi$
$653$	−1.07233e10	−1.50707	−0.753536	+	0.657407i	$0.771653\pi$
$654$	0	0
$655$	−1.17300e8	−0.0163100
$656$	0	0
$657$	3.68729e9	0.507258
$658$	0	0
$659$	−9.55747e8	−0.130090	−0.0650450	−	0.997882i	$-0.520719\pi$
$659$	−9.55747e8	−0.130090	−0.0650450	+	0.997882i	$0.520719\pi$
$660$	0	0
$661$	−1.17758e10	−1.58594	−0.792971	−	0.609260i	$-0.791467\pi$
$661$	−1.17758e10	−1.58594	−0.792971	+	0.609260i	$0.791467\pi$
$662$	0	0
$663$	1.20433e10	1.60490
$664$	0	0
$665$	4.71735e9	0.622046
$666$	0	0
$667$	3.15570e9	0.411771
$668$	0	0
$669$	−1.00119e8	−0.0129278
$670$	0	0
$671$	−6.08786e9	−0.777921
$672$	0	0
$673$	5.31058e9	0.671567	0.335783	−	0.941939i	$-0.390999\pi$
$673$	5.31058e9	0.671567	0.335783	+	0.941939i	$0.390999\pi$
$674$	0	0
$675$	8.56564e9	1.07200
$676$	0	0
$677$	1.06518e10	1.31936	0.659680	−	0.751547i	$-0.270692\pi$
$677$	1.06518e10	1.31936	0.659680	+	0.751547i	$0.270692\pi$
$678$	0	0
$679$	1.88280e10	2.30813
$680$	0	0
$681$	−9.76290e8	−0.118458
$682$	0	0
$683$	3.14122e8	0.0377247	0.0188624	−	0.999822i	$-0.493996\pi$
$683$	3.14122e8	0.0377247	0.0188624	+	0.999822i	$0.493996\pi$
$684$	0	0
$685$	−1.19025e10	−1.41488
$686$	0	0
$687$	2.23856e9	0.263403
$688$	0	0
$689$	1.15122e9	0.134088
$690$	0	0
$691$	1.44774e10	1.66923	0.834615	−	0.550833i	$-0.185690\pi$
$691$	1.44774e10	1.66923	0.834615	+	0.550833i	$0.185690\pi$
$692$	0	0
$693$	1.58200e10	1.80568
$694$	0	0
$695$	−2.02526e10	−2.28841
$696$	0	0
$697$	6.48056e9	0.724933
$698$	0	0
$699$	−1.47470e10	−1.63318
$700$	0	0
$701$	1.22852e10	1.34701	0.673503	−	0.739185i	$-0.264789\pi$
$701$	1.22852e10	1.34701	0.673503	+	0.739185i	$0.264789\pi$
$702$	0	0
$703$	2.30850e8	0.0250603
$704$	0	0
$705$	−1.61570e10	−1.73660
$706$	0	0
$707$	4.89707e8	0.0521157
$708$	0	0
$709$	−3.48632e9	−0.367371	−0.183686	−	0.982985i	$-0.558803\pi$
$709$	−3.48632e9	−0.367371	−0.183686	+	0.982985i	$0.558803\pi$
$710$	0	0
$711$	−1.19606e10	−1.24799
$712$	0	0
$713$	−1.99550e10	−2.06176
$714$	0	0
$715$	−2.04976e10	−2.09717
$716$	0	0
$717$	1.62968e10	1.65115
$718$	0	0
$719$	−6.30005e9	−0.632110	−0.316055	−	0.948741i	$-0.602358\pi$
$719$	−6.30005e9	−0.632110	−0.316055	+	0.948741i	$0.602358\pi$
$720$	0	0
$721$	−6.65758e9	−0.661520
$722$	0	0
$723$	3.00628e9	0.295832
$724$	0	0
$725$	4.70933e9	0.458962
$726$	0	0
$727$	4.41318e9	0.425973	0.212986	−	0.977055i	$-0.431681\pi$
$727$	4.41318e9	0.425973	0.212986	+	0.977055i	$0.431681\pi$
$728$	0	0
$729$	−1.63768e10	−1.56561
$730$	0	0
$731$	−7.57037e9	−0.716814
$732$	0	0
$733$	1.15847e10	1.08648	0.543240	−	0.839577i	$-0.317197\pi$
$733$	1.15847e10	1.08648	0.543240	+	0.839577i	$0.317197\pi$
$734$	0	0
$735$	4.52848e10	4.20675
$736$	0	0
$737$	−1.68930e9	−0.155443
$738$	0	0
$739$	−1.60174e10	−1.45995	−0.729974	−	0.683475i	$-0.760468\pi$
$739$	−1.60174e10	−1.45995	−0.729974	+	0.683475i	$0.760468\pi$
$740$	0	0
$741$	−6.12514e9	−0.553034
$742$	0	0
$743$	−6.54024e9	−0.584969	−0.292484	−	0.956270i	$-0.594482\pi$
$743$	−6.54024e9	−0.584969	−0.292484	+	0.956270i	$0.594482\pi$
$744$	0	0
$745$	−1.86344e10	−1.65108
$746$	0	0
$747$	1.07392e10	0.942648
$748$	0	0
$749$	−1.07972e10	−0.938908
$750$	0	0
$751$	−1.04818e10	−0.903015	−0.451508	−	0.892267i	$-0.649114\pi$
$751$	−1.04818e10	−0.903015	−0.451508	+	0.892267i	$0.649114\pi$
$752$	0	0
$753$	−3.29107e10	−2.80901
$754$	0	0
$755$	9.99502e8	0.0845219
$756$	0	0
$757$	4.11773e9	0.345003	0.172501	−	0.985009i	$-0.444815\pi$
$757$	4.11773e9	0.345003	0.172501	+	0.985009i	$0.444815\pi$
$758$	0	0
$759$	2.41079e10	2.00130
$760$	0	0
$761$	−1.90960e10	−1.57071	−0.785357	−	0.619043i	$-0.787521\pi$
$761$	−1.90960e10	−1.57071	−0.785357	+	0.619043i	$0.787521\pi$
$762$	0	0
$763$	−1.52891e10	−1.24608
$764$	0	0
$765$	−1.91120e10	−1.54344
$766$	0	0
$767$	1.18059e10	0.944750
$768$	0	0
$769$	−1.20473e10	−0.955315	−0.477658	−	0.878546i	$-0.658514\pi$
$769$	−1.20473e10	−0.955315	−0.477658	+	0.878546i	$0.658514\pi$
$770$	0	0
$771$	−1.77899e10	−1.39792
$772$	0	0
$773$	4.62608e9	0.360234	0.180117	−	0.983645i	$-0.442352\pi$
$773$	4.62608e9	0.360234	0.180117	+	0.983645i	$0.442352\pi$
$774$	0	0
$775$	−2.97793e10	−2.29805
$776$	0	0
$777$	3.57703e9	0.273558
$778$	0	0
$779$	−3.29596e9	−0.249805
$780$	0	0
$781$	−6.93512e9	−0.520925
$782$	0	0
$783$	2.04622e9	0.152330
$784$	0	0
$785$	−1.42440e10	−1.05096
$786$	0	0
$787$	−5.71149e9	−0.417675	−0.208837	−	0.977950i	$-0.566968\pi$
$787$	−5.71149e9	−0.417675	−0.208837	+	0.977950i	$0.566968\pi$
$788$	0	0
$789$	−2.92432e10	−2.11961
$790$	0	0
$791$	7.85596e9	0.564393
$792$	0	0
$793$	2.12168e10	1.51086
$794$	0	0
$795$	−3.14488e9	−0.221983
$796$	0	0
$797$	−1.15032e10	−0.804848	−0.402424	−	0.915453i	$-0.631832\pi$
$797$	−1.15032e10	−0.804848	−0.402424	+	0.915453i	$0.631832\pi$
$798$	0	0
$799$	6.45244e9	0.447517
$800$	0	0
$801$	−2.27424e10	−1.56359
$802$	0	0
$803$	4.31435e9	0.294043
$804$	0	0
$805$	6.47078e10	4.37191
$806$	0	0
$807$	−1.76686e10	−1.18344
$808$	0	0
$809$	1.91918e10	1.27437	0.637184	−	0.770712i	$-0.280099\pi$
$809$	1.91918e10	1.27437	0.637184	+	0.770712i	$0.280099\pi$
$810$	0	0
$811$	5.04377e9	0.332034	0.166017	−	0.986123i	$-0.446909\pi$
$811$	5.04377e9	0.332034	0.166017	+	0.986123i	$0.446909\pi$
$812$	0	0
$813$	−5.69743e9	−0.371845
$814$	0	0
$815$	4.43609e10	2.87044
$816$	0	0
$817$	3.85023e9	0.247007
$818$	0	0
$819$	−5.51343e10	−3.50694
$820$	0	0
$821$	2.85707e10	1.80185	0.900927	−	0.433972i	$-0.142888\pi$
$821$	2.85707e10	1.80185	0.900927	+	0.433972i	$0.142888\pi$
$822$	0	0
$823$	−1.82564e10	−1.14161	−0.570803	−	0.821087i	$-0.693368\pi$
$823$	−1.82564e10	−1.14161	−0.570803	+	0.821087i	$0.693368\pi$
$824$	0	0
$825$	3.59768e10	2.23066
$826$	0	0
$827$	1.92073e9	0.118086	0.0590428	−	0.998255i	$-0.481195\pi$
$827$	1.92073e9	0.118086	0.0590428	+	0.998255i	$0.481195\pi$
$828$	0	0
$829$	4.75211e9	0.289698	0.144849	−	0.989454i	$-0.453730\pi$
$829$	4.75211e9	0.289698	0.144849	+	0.989454i	$0.453730\pi$
$830$	0	0
$831$	−1.66388e10	−1.00582
$832$	0	0
$833$	−1.80848e10	−1.08407
$834$	0	0
$835$	5.15770e10	3.06587
$836$	0	0
$837$	−1.29392e10	−0.762728
$838$	0	0
$839$	1.17511e10	0.686927	0.343463	−	0.939166i	$-0.388400\pi$
$839$	1.17511e10	0.686927	0.343463	+	0.939166i	$0.388400\pi$
$840$	0	0
$841$	−1.61249e10	−0.934782
$842$	0	0
$843$	−3.93407e10	−2.26176
$844$	0	0
$845$	4.21032e10	2.40058
$846$	0	0
$847$	−1.01598e10	−0.574507
$848$	0	0
$849$	2.67204e9	0.149853
$850$	0	0
$851$	3.16656e9	0.176130
$852$	0	0
$853$	−1.98045e10	−1.09255	−0.546276	−	0.837605i	$-0.683955\pi$
$853$	−1.98045e10	−1.09255	−0.546276	+	0.837605i	$0.683955\pi$
$854$	0	0
$855$	9.72020e9	0.531856
$856$	0	0
$857$	1.83237e10	0.994445	0.497223	−	0.867623i	$-0.334353\pi$
$857$	1.83237e10	0.994445	0.497223	+	0.867623i	$0.334353\pi$
$858$	0	0
$859$	−2.65234e10	−1.42775	−0.713876	−	0.700272i	$-0.753062\pi$
$859$	−2.65234e10	−1.42775	−0.713876	+	0.700272i	$0.753062\pi$
$860$	0	0
$861$	−5.10712e10	−2.72687
$862$	0	0
$863$	1.77522e10	0.940188	0.470094	−	0.882616i	$-0.344220\pi$
$863$	1.77522e10	0.940188	0.470094	+	0.882616i	$0.344220\pi$
$864$	0	0
$865$	4.96045e10	2.60594
$866$	0	0
$867$	−1.65038e10	−0.860034
$868$	0	0
$869$	−1.39946e10	−0.723423
$870$	0	0
$871$	5.88737e9	0.301897
$872$	0	0
$873$	3.87955e10	1.97348
$874$	0	0
$875$	4.28338e10	2.16152
$876$	0	0
$877$	1.22608e9	0.0613792	0.0306896	−	0.999529i	$-0.490230\pi$
$877$	1.22608e9	0.0613792	0.0306896	+	0.999529i	$0.490230\pi$
$878$	0	0
$879$	1.43070e10	0.710537
$880$	0	0
$881$	6.34486e9	0.312612	0.156306	−	0.987709i	$-0.450041\pi$
$881$	6.34486e9	0.312612	0.156306	+	0.987709i	$0.450041\pi$
$882$	0	0
$883$	2.24216e10	1.09598	0.547992	−	0.836484i	$-0.315392\pi$
$883$	2.24216e10	1.09598	0.547992	+	0.836484i	$0.315392\pi$
$884$	0	0
$885$	−3.22513e10	−1.56403
$886$	0	0
$887$	−3.38047e10	−1.62647	−0.813233	−	0.581938i	$-0.802295\pi$
$887$	−3.38047e10	−1.62647	−0.813233	+	0.581938i	$0.802295\pi$
$888$	0	0
$889$	5.05120e10	2.41123
$890$	0	0
$891$	−7.88450e9	−0.373424
$892$	0	0
$893$	−3.28166e9	−0.154210
$894$	0	0
$895$	−2.28291e10	−1.06441
$896$	0	0
$897$	−8.40185e10	−3.88688
$898$	0	0
$899$	−7.11391e9	−0.326550
$900$	0	0
$901$	1.25593e9	0.0572045
$902$	0	0
$903$	5.96596e10	2.69633
$904$	0	0
$905$	2.58079e9	0.115740
$906$	0	0
$907$	1.56762e10	0.697616	0.348808	−	0.937194i	$-0.386587\pi$
$907$	1.56762e10	0.697616	0.348808	+	0.937194i	$0.386587\pi$
$908$	0	0
$909$	1.00905e9	0.0445595
$910$	0	0
$911$	−2.35920e10	−1.03383	−0.516916	−	0.856036i	$-0.672920\pi$
$911$	−2.35920e10	−1.03383	−0.516916	+	0.856036i	$0.672920\pi$
$912$	0	0
$913$	1.25655e10	0.546426
$914$	0	0
$915$	−5.79598e10	−2.50123
$916$	0	0
$917$	−3.69169e8	−0.0158100
$918$	0	0
$919$	3.39353e10	1.44227	0.721137	−	0.692793i	$-0.243620\pi$
$919$	3.39353e10	1.44227	0.721137	+	0.692793i	$0.243620\pi$
$920$	0	0
$921$	−2.37303e10	−1.00091
$922$	0	0
$923$	2.41696e10	1.01173
$924$	0	0
$925$	4.72554e9	0.196316
$926$	0	0
$927$	−1.37181e10	−0.565607
$928$	0	0
$929$	7.75319e9	0.317267	0.158634	−	0.987338i	$-0.449291\pi$
$929$	7.75319e9	0.317267	0.158634	+	0.987338i	$0.449291\pi$
$930$	0	0
$931$	9.19780e9	0.373560
$932$	0	0
$933$	4.24222e10	1.71004
$934$	0	0
$935$	−2.23621e10	−0.894690
$936$	0	0
$937$	−3.05613e10	−1.21362	−0.606811	−	0.794846i	$-0.707552\pi$
$937$	−3.05613e10	−1.21362	−0.606811	+	0.794846i	$0.707552\pi$
$938$	0	0
$939$	4.55090e10	1.79377
$940$	0	0
$941$	−4.59802e10	−1.79890	−0.899451	−	0.437022i	$-0.856033\pi$
$941$	−4.59802e10	−1.79890	−0.899451	+	0.437022i	$0.856033\pi$
$942$	0	0
$943$	−4.52107e10	−1.75570
$944$	0	0
$945$	4.19579e10	1.61734
$946$	0	0
$947$	3.07383e10	1.17613	0.588065	−	0.808813i	$-0.299890\pi$
$947$	3.07383e10	1.17613	0.588065	+	0.808813i	$0.299890\pi$
$948$	0	0
$949$	−1.50359e10	−0.571083
$950$	0	0
$951$	5.78553e10	2.18128
$952$	0	0
$953$	1.04326e10	0.390451	0.195226	−	0.980758i	$-0.437456\pi$
$953$	1.04326e10	0.390451	0.195226	+	0.980758i	$0.437456\pi$
$954$	0	0
$955$	5.40220e10	2.00705
$956$	0	0
$957$	8.59441e9	0.316974
$958$	0	0
$959$	−3.74595e10	−1.37150
$960$	0	0
$961$	1.74720e10	0.635054
$962$	0	0
$963$	−2.22478e10	−0.802776
$964$	0	0
$965$	1.87902e9	0.0673111
$966$	0	0
$967$	−1.29860e10	−0.461829	−0.230915	−	0.972974i	$-0.574172\pi$
$967$	−1.29860e10	−0.461829	−0.230915	+	0.972974i	$0.574172\pi$
$968$	0	0
$969$	−6.68229e9	−0.235935
$970$	0	0
$971$	5.17386e10	1.81362	0.906812	−	0.421536i	$-0.138509\pi$
$971$	5.17386e10	1.81362	0.906812	+	0.421536i	$0.138509\pi$
$972$	0	0
$973$	−6.37393e10	−2.21826
$974$	0	0
$975$	−1.25383e11	−4.33234
$976$	0	0
$977$	3.29063e10	1.12888	0.564440	−	0.825474i	$-0.309092\pi$
$977$	3.29063e10	1.12888	0.564440	+	0.825474i	$0.309092\pi$
$978$	0	0
$979$	−2.66100e10	−0.906370
$980$	0	0
$981$	−3.15035e10	−1.06541
$982$	0	0
$983$	2.58802e9	0.0869021	0.0434510	−	0.999056i	$-0.486165\pi$
$983$	2.58802e9	0.0869021	0.0434510	+	0.999056i	$0.486165\pi$
$984$	0	0
$985$	4.17120e10	1.39070
$986$	0	0
$987$	−5.08495e10	−1.68336
$988$	0	0
$989$	5.28136e10	1.73603
$990$	0	0
$991$	4.62321e10	1.50899	0.754494	−	0.656306i	$-0.227882\pi$
$991$	4.62321e10	1.50899	0.754494	+	0.656306i	$0.227882\pi$
$992$	0	0
$993$	3.28155e10	1.06355
$994$	0	0
$995$	8.12520e10	2.61489
$996$	0	0
$997$	2.54695e10	0.813930	0.406965	−	0.913444i	$-0.366587\pi$
$997$	2.54695e10	0.813930	0.406965	+	0.913444i	$0.366587\pi$
$998$	0	0
$999$	2.05326e9	0.0651577

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.8.a.d.1.2		2
4.3	odd	2		38.8.a.d.1.1	✓	2
12.11	even	2		342.8.a.g.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.a.d.1.1	✓	2	4.3	odd	2
304.8.a.d.1.2		2	1.1	even	1	trivial
342.8.a.g.1.1		2	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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	304.a (trivial)

\(f(q)\)	\(=\)	\(q+72.2392 q^{3} +467.472 q^{5} +1471.23 q^{7} +3031.51 q^{9} +O(q^{10})\)	\(q+72.2392 q^{3} +467.472 q^{5} +1471.23 q^{7} +3031.51 q^{9} +3547.04 q^{11} -12361.8 q^{13} +33769.8 q^{15} -13486.3 q^{17} +6859.00 q^{19} +106281. q^{21} +94084.9 q^{23} +140405. q^{25} +61006.6 q^{27} +33541.0 q^{29} -212096. q^{31} +256236. q^{33} +687760. q^{35} +33656.5 q^{37} -893007. q^{39} -480531. q^{41} +561340. q^{43} +1.41715e6 q^{45} -478445. q^{47} +1.34098e6 q^{49} -974237. q^{51} -93127.0 q^{53} +1.65814e6 q^{55} +495489. q^{57} -955034. q^{59} -1.71632e6 q^{61} +4.46005e6 q^{63} -5.77880e6 q^{65} -476255. q^{67} +6.79662e6 q^{69} -1.95518e6 q^{71} +1.21632e6 q^{73} +1.01428e7 q^{75} +5.21852e6 q^{77} -3.94544e6 q^{79} -2.22284e6 q^{81} +3.54252e6 q^{83} -6.30445e6 q^{85} +2.42298e6 q^{87} -7.50202e6 q^{89} -1.81871e7 q^{91} -1.53216e7 q^{93} +3.20639e6 q^{95} +1.27974e7 q^{97} +1.07529e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 69 q^{3} + 155 q^{5} + 2238 q^{7} + 855 q^{9}+O(q^{10}) \) 2 * q + 69 * q^3 + 155 * q^5 + 2238 * q^7 + 855 * q^9	\( 2 q + 69 q^{3} + 155 q^{5} + 2238 q^{7} + 855 q^{9} + 3295 q^{11} - 13427 q^{13} + 34782 q^{15} - 32256 q^{17} + 13718 q^{19} + 103797 q^{21} + 82525 q^{23} + 159919 q^{25} + 75141 q^{27} - 12749 q^{29} - 258944 q^{31} + 257052 q^{33} + 448167 q^{35} - 149260 q^{37} - 889557 q^{39} + 339130 q^{41} + 83869 q^{43} + 2097243 q^{45} - 1471025 q^{47} + 1105372 q^{49} - 913437 q^{51} - 945643 q^{53} + 1736899 q^{55} + 473271 q^{57} + 969009 q^{59} - 1506755 q^{61} + 2791179 q^{63} - 5445956 q^{65} + 1848219 q^{67} + 6834063 q^{69} + 3417184 q^{71} - 2499822 q^{73} + 10079553 q^{75} + 5025267 q^{77} - 2636926 q^{79} + 2491398 q^{81} + 10059354 q^{83} - 439425 q^{85} + 2572923 q^{87} - 3506160 q^{89} - 19003851 q^{91} - 15169884 q^{93} + 1063145 q^{95} + 5893526 q^{97} + 11301453 q^{99}+O(q^{100}) \) 2 * q + 69 * q^3 + 155 * q^5 + 2238 * q^7 + 855 * q^9 + 3295 * q^11 - 13427 * q^13 + 34782 * q^15 - 32256 * q^17 + 13718 * q^19 + 103797 * q^21 + 82525 * q^23 + 159919 * q^25 + 75141 * q^27 - 12749 * q^29 - 258944 * q^31 + 257052 * q^33 + 448167 * q^35 - 149260 * q^37 - 889557 * q^39 + 339130 * q^41 + 83869 * q^43 + 2097243 * q^45 - 1471025 * q^47 + 1105372 * q^49 - 913437 * q^51 - 945643 * q^53 + 1736899 * q^55 + 473271 * q^57 + 969009 * q^59 - 1506755 * q^61 + 2791179 * q^63 - 5445956 * q^65 + 1848219 * q^67 + 6834063 * q^69 + 3417184 * q^71 - 2499822 * q^73 + 10079553 * q^75 + 5025267 * q^77 - 2636926 * q^79 + 2491398 * q^81 + 10059354 * q^83 - 439425 * q^85 + 2572923 * q^87 - 3506160 * q^89 - 19003851 * q^91 - 15169884 * q^93 + 1063145 * q^95 + 5893526 * q^97 + 11301453 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more