LMFDB - Embedded newform 304.10.a.e.1.2

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			1.2
Root			$-26.2676$ 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)

$f(q)$	$=$	$q-25.2570 q^{3} +2126.71 q^{5} -11469.0 q^{7} -19045.1 q^{9} +O(q^{10})$	$q-25.2570 q^{3} +2126.71 q^{5} -11469.0 q^{7} -19045.1 q^{9} +10430.4 q^{11} +144659. q^{13} -53714.4 q^{15} -654214. q^{17} -130321. q^{19} +289672. q^{21} -1.63517e6 q^{23} +2.56979e6 q^{25} +978155. q^{27} +3.60690e6 q^{29} -3.62937e6 q^{31} -263440. q^{33} -2.43912e7 q^{35} -1.06553e7 q^{37} -3.65364e6 q^{39} +1.31851e7 q^{41} +2.73590e6 q^{43} -4.05035e7 q^{45} +5.26213e7 q^{47} +9.11835e7 q^{49} +1.65235e7 q^{51} +5.87078e7 q^{53} +2.21825e7 q^{55} +3.29152e6 q^{57} +7.23340e7 q^{59} -7.46069e7 q^{61} +2.18427e8 q^{63} +3.07647e8 q^{65} -1.12380e8 q^{67} +4.12994e7 q^{69} -3.11980e8 q^{71} +3.47317e6 q^{73} -6.49052e7 q^{75} -1.19626e8 q^{77} -1.96071e8 q^{79} +3.50159e8 q^{81} +3.30282e8 q^{83} -1.39133e9 q^{85} -9.10995e7 q^{87} -6.40446e8 q^{89} -1.65908e9 q^{91} +9.16671e7 q^{93} -2.77156e8 q^{95} +1.56258e9 q^{97} -1.98648e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−25.2570	−0.180026	−0.0900132	−	0.995941i	$-0.528691\pi$
$3$	−25.2570	−0.180026	−0.0900132	+	0.995941i	$0.528691\pi$
$4$	0	0
$5$	2126.71	1.52175	0.760877	−	0.648897i	$-0.224769\pi$
$5$	2126.71	1.52175	0.760877	+	0.648897i	$0.224769\pi$
$6$	0	0
$7$	−11469.0	−1.80544	−0.902720	−	0.430229i	$-0.858433\pi$
$7$	−11469.0	−1.80544	−0.902720	+	0.430229i	$0.858433\pi$
$8$	0	0
$9$	−19045.1	−0.967591
$10$	0	0
$11$	10430.4	0.214800	0.107400	−	0.994216i	$-0.465748\pi$
$11$	10430.4	0.214800	0.107400	+	0.994216i	$0.465748\pi$
$12$	0	0
$13$	144659.	1.40475	0.702375	−	0.711807i	$-0.252123\pi$
$13$	144659.	1.40475	0.702375	+	0.711807i	$0.252123\pi$
$14$	0	0
$15$	−53714.4	−0.273956
$16$	0	0
$17$	−654214.	−1.89976	−0.949882	−	0.312608i	$-0.898797\pi$
$17$	−654214.	−1.89976	−0.949882	+	0.312608i	$0.898797\pi$
$18$	0	0
$19$	−130321.	−0.229416
$19$	−130321.	−0.229416
$20$	0	0
$21$	289672.	0.325027
$22$	0	0
$23$	−1.63517e6	−1.21839	−0.609196	−	0.793020i	$-0.708508\pi$
$23$	−1.63517e6	−1.21839	−0.609196	+	0.793020i	$0.708508\pi$
$24$	0	0
$25$	2.56979e6	1.31573
$26$	0	0
$27$	978155.	0.354218
$28$	0	0
$29$	3.60690e6	0.946985	0.473493	−	0.880798i	$-0.342993\pi$
$29$	3.60690e6	0.946985	0.473493	+	0.880798i	$0.342993\pi$
$30$	0	0
$31$	−3.62937e6	−0.705836	−0.352918	−	0.935654i	$-0.614811\pi$
$31$	−3.62937e6	−0.705836	−0.352918	+	0.935654i	$0.614811\pi$
$32$	0	0
$33$	−263440.	−0.0386696
$34$	0	0
$35$	−2.43912e7	−2.74743
$36$	0	0
$37$	−1.06553e7	−0.934668	−0.467334	−	0.884081i	$-0.654785\pi$
$37$	−1.06553e7	−0.934668	−0.467334	+	0.884081i	$0.654785\pi$
$38$	0	0
$39$	−3.65364e6	−0.252892
$40$	0	0
$41$	1.31851e7	0.728710	0.364355	−	0.931260i	$-0.381289\pi$
$41$	1.31851e7	0.728710	0.364355	+	0.931260i	$0.381289\pi$
$42$	0	0
$43$	2.73590e6	0.122037	0.0610187	−	0.998137i	$-0.480565\pi$
$43$	2.73590e6	0.122037	0.0610187	+	0.998137i	$0.480565\pi$
$44$	0	0
$45$	−4.05035e7	−1.47243
$46$	0	0
$47$	5.26213e7	1.57297	0.786486	−	0.617608i	$-0.211898\pi$
$47$	5.26213e7	1.57297	0.786486	+	0.617608i	$0.211898\pi$
$48$	0	0
$49$	9.11835e7	2.25961
$50$	0	0
$51$	1.65235e7	0.342008
$52$	0	0
$53$	5.87078e7	1.02201	0.511004	−	0.859578i	$-0.329274\pi$
$53$	5.87078e7	1.02201	0.511004	+	0.859578i	$0.329274\pi$
$54$	0	0
$55$	2.21825e7	0.326872
$56$	0	0
$57$	3.29152e6	0.0413009
$58$	0	0
$59$	7.23340e7	0.777157	0.388578	−	0.921416i	$-0.372966\pi$
$59$	7.23340e7	0.777157	0.388578	+	0.921416i	$0.372966\pi$
$60$	0	0
$61$	−7.46069e7	−0.689914	−0.344957	−	0.938619i	$-0.612106\pi$
$61$	−7.46069e7	−0.689914	−0.344957	+	0.938619i	$0.612106\pi$
$62$	0	0
$63$	2.18427e8	1.74693
$64$	0	0
$65$	3.07647e8	2.13768
$66$	0	0
$67$	−1.12380e8	−0.681322	−0.340661	−	0.940186i	$-0.610651\pi$
$67$	−1.12380e8	−0.681322	−0.340661	+	0.940186i	$0.610651\pi$
$68$	0	0
$69$	4.12994e7	0.219343
$70$	0	0
$71$	−3.11980e8	−1.45701	−0.728507	−	0.685038i	$-0.759786\pi$
$71$	−3.11980e8	−1.45701	−0.728507	+	0.685038i	$0.759786\pi$
$72$	0	0
$73$	3.47317e6	0.0143144	0.00715720	−	0.999974i	$-0.497722\pi$
$73$	3.47317e6	0.0143144	0.00715720	+	0.999974i	$0.497722\pi$
$74$	0	0
$75$	−6.49052e7	−0.236867
$76$	0	0
$77$	−1.19626e8	−0.387808
$78$	0	0
$79$	−1.96071e8	−0.566359	−0.283179	−	0.959067i	$-0.591389\pi$
$79$	−1.96071e8	−0.566359	−0.283179	+	0.959067i	$0.591389\pi$
$80$	0	0
$81$	3.50159e8	0.903822
$82$	0	0
$83$	3.30282e8	0.763895	0.381948	−	0.924184i	$-0.375253\pi$
$83$	3.30282e8	0.763895	0.381948	+	0.924184i	$0.375253\pi$
$84$	0	0
$85$	−1.39133e9	−2.89097
$86$	0	0
$87$	−9.10995e7	−0.170482
$88$	0	0
$89$	−6.40446e8	−1.08200	−0.541000	−	0.841023i	$-0.681954\pi$
$89$	−6.40446e8	−1.08200	−0.541000	+	0.841023i	$0.681954\pi$
$90$	0	0
$91$	−1.65908e9	−2.53619
$92$	0	0
$93$	9.16671e7	0.127069
$94$	0	0
$95$	−2.77156e8	−0.349114
$96$	0	0
$97$	1.56258e9	1.79213	0.896064	−	0.443925i	$-0.146414\pi$
$97$	1.56258e9	1.79213	0.896064	+	0.443925i	$0.146414\pi$
$98$	0	0
$99$	−1.98648e8	−0.207838
$100$	0	0
$101$	2.55498e8	0.244310	0.122155	−	0.992511i	$-0.461019\pi$
$101$	2.55498e8	0.244310	0.122155	+	0.992511i	$0.461019\pi$
$102$	0	0
$103$	9.05731e8	0.792925	0.396462	−	0.918051i	$-0.370238\pi$
$103$	9.05731e8	0.792925	0.396462	+	0.918051i	$0.370238\pi$
$104$	0	0
$105$	6.16049e8	0.494610
$106$	0	0
$107$	3.97489e8	0.293155	0.146578	−	0.989199i	$-0.453174\pi$
$107$	3.97489e8	0.293155	0.146578	+	0.989199i	$0.453174\pi$
$108$	0	0
$109$	5.91761e8	0.401538	0.200769	−	0.979639i	$-0.435656\pi$
$109$	5.91761e8	0.401538	0.200769	+	0.979639i	$0.435656\pi$
$110$	0	0
$111$	2.69120e8	0.168265
$112$	0	0
$113$	−2.57125e9	−1.48351	−0.741757	−	0.670668i	$-0.766007\pi$
$113$	−2.57125e9	−1.48351	−0.741757	+	0.670668i	$0.766007\pi$
$114$	0	0
$115$	−3.47753e9	−1.85409
$116$	0	0
$117$	−2.75503e9	−1.35922
$118$	0	0
$119$	7.50316e9	3.42991
$120$	0	0
$121$	−2.24915e9	−0.953861
$122$	0	0
$123$	−3.33015e8	−0.131187
$124$	0	0
$125$	1.31147e9	0.480468
$126$	0	0
$127$	−1.47739e9	−0.503938	−0.251969	−	0.967735i	$-0.581078\pi$
$127$	−1.47739e9	−0.503938	−0.251969	+	0.967735i	$0.581078\pi$
$128$	0	0
$129$	−6.91007e7	−0.0219699
$130$	0	0
$131$	5.81972e9	1.72656	0.863279	−	0.504726i	$-0.168407\pi$
$131$	5.81972e9	1.72656	0.863279	+	0.504726i	$0.168407\pi$
$132$	0	0
$133$	1.49465e9	0.414196
$134$	0	0
$135$	2.08026e9	0.539033
$136$	0	0
$137$	−2.81502e8	−0.0682715	−0.0341358	−	0.999417i	$-0.510868\pi$
$137$	−2.81502e8	−0.0682715	−0.0341358	+	0.999417i	$0.510868\pi$
$138$	0	0
$139$	3.71990e9	0.845210	0.422605	−	0.906314i	$-0.361116\pi$
$139$	3.71990e9	0.845210	0.422605	+	0.906314i	$0.361116\pi$
$140$	0	0
$141$	−1.32905e9	−0.283176
$142$	0	0
$143$	1.50884e9	0.301740
$144$	0	0
$145$	7.67085e9	1.44108
$146$	0	0
$147$	−2.30302e9	−0.406790
$148$	0	0
$149$	2.98264e9	0.495750	0.247875	−	0.968792i	$-0.420268\pi$
$149$	2.98264e9	0.495750	0.247875	+	0.968792i	$0.420268\pi$
$150$	0	0
$151$	1.07704e10	1.68592	0.842960	−	0.537977i	$-0.180811\pi$
$151$	1.07704e10	1.68592	0.842960	+	0.537977i	$0.180811\pi$
$152$	0	0
$153$	1.24596e10	1.83819
$154$	0	0
$155$	−7.71864e9	−1.07411
$156$	0	0
$157$	4.65738e8	0.0611777	0.0305889	−	0.999532i	$-0.490262\pi$
$157$	4.65738e8	0.0611777	0.0305889	+	0.999532i	$0.490262\pi$
$158$	0	0
$159$	−1.48278e9	−0.183988
$160$	0	0
$161$	1.87537e10	2.19973
$162$	0	0
$163$	3.04519e9	0.337885	0.168943	−	0.985626i	$-0.445965\pi$
$163$	3.04519e9	0.337885	0.168943	+	0.985626i	$0.445965\pi$
$164$	0	0
$165$	−5.60262e8	−0.0588455
$166$	0	0
$167$	5.43187e9	0.540413	0.270206	−	0.962802i	$-0.412908\pi$
$167$	5.43187e9	0.540413	0.270206	+	0.962802i	$0.412908\pi$
$168$	0	0
$169$	1.03216e10	0.973322
$170$	0	0
$171$	2.48197e9	0.221980
$172$	0	0
$173$	−5.67736e8	−0.0481880	−0.0240940	−	0.999710i	$-0.507670\pi$
$173$	−5.67736e8	−0.0481880	−0.0240940	+	0.999710i	$0.507670\pi$
$174$	0	0
$175$	−2.94728e10	−2.37548
$176$	0	0
$177$	−1.82694e9	−0.139909
$178$	0	0
$179$	1.14816e9	0.0835920	0.0417960	−	0.999126i	$-0.486692\pi$
$179$	1.14816e9	0.0835920	0.0417960	+	0.999126i	$0.486692\pi$
$180$	0	0
$181$	−1.40584e10	−0.973606	−0.486803	−	0.873512i	$-0.661837\pi$
$181$	−1.40584e10	−0.973606	−0.486803	+	0.873512i	$0.661837\pi$
$182$	0	0
$183$	1.88435e9	0.124203
$184$	0	0
$185$	−2.26608e10	−1.42233
$186$	0	0
$187$	−6.82370e9	−0.408068
$188$	0	0
$189$	−1.12184e10	−0.639519
$190$	0	0
$191$	1.97985e10	1.07642	0.538209	−	0.842811i	$-0.319101\pi$
$191$	1.97985e10	1.07642	0.538209	+	0.842811i	$0.319101\pi$
$192$	0	0
$193$	2.34501e10	1.21657	0.608284	−	0.793719i	$-0.291858\pi$
$193$	2.34501e10	1.21657	0.608284	+	0.793719i	$0.291858\pi$
$194$	0	0
$195$	−7.77025e9	−0.384839
$196$	0	0
$197$	6.29979e8	0.0298008	0.0149004	−	0.999889i	$-0.495257\pi$
$197$	6.29979e8	0.0298008	0.0149004	+	0.999889i	$0.495257\pi$
$198$	0	0
$199$	6.30124e9	0.284831	0.142416	−	0.989807i	$-0.454513\pi$
$199$	6.30124e9	0.284831	0.142416	+	0.989807i	$0.454513\pi$
$200$	0	0
$201$	2.83838e9	0.122656
$202$	0	0
$203$	−4.13674e10	−1.70973
$204$	0	0
$205$	2.80409e10	1.10892
$206$	0	0
$207$	3.11419e10	1.17890
$208$	0	0
$209$	−1.35930e9	−0.0492784
$210$	0	0
$211$	4.85098e10	1.68484	0.842419	−	0.538823i	$-0.181131\pi$
$211$	4.85098e10	1.68484	0.842419	+	0.538823i	$0.181131\pi$
$212$	0	0
$213$	7.87967e9	0.262301
$214$	0	0
$215$	5.81849e9	0.185711
$216$	0	0
$217$	4.16252e10	1.27435
$218$	0	0
$219$	−8.77219e7	−0.00257697
$220$	0	0
$221$	−9.46377e10	−2.66869
$222$	0	0
$223$	−2.29208e10	−0.620666	−0.310333	−	0.950628i	$-0.600440\pi$
$223$	−2.29208e10	−0.620666	−0.310333	+	0.950628i	$0.600440\pi$
$224$	0	0
$225$	−4.89419e10	−1.27309
$226$	0	0
$227$	2.40728e10	0.601742	0.300871	−	0.953665i	$-0.402723\pi$
$227$	2.40728e10	0.601742	0.300871	+	0.953665i	$0.402723\pi$
$228$	0	0
$229$	1.31840e10	0.316801	0.158400	−	0.987375i	$-0.449366\pi$
$229$	1.31840e10	0.316801	0.158400	+	0.987375i	$0.449366\pi$
$230$	0	0
$231$	3.02139e9	0.0698156
$232$	0	0
$233$	1.64719e10	0.366135	0.183068	−	0.983100i	$-0.441397\pi$
$233$	1.64719e10	0.366135	0.183068	+	0.983100i	$0.441397\pi$
$234$	0	0
$235$	1.11910e11	2.39367
$236$	0	0
$237$	4.95217e9	0.101959
$238$	0	0
$239$	−6.84337e9	−0.135669	−0.0678343	−	0.997697i	$-0.521609\pi$
$239$	−6.84337e9	−0.135669	−0.0678343	+	0.997697i	$0.521609\pi$
$240$	0	0
$241$	6.99690e10	1.33607	0.668034	−	0.744130i	$-0.267136\pi$
$241$	6.99690e10	1.33607	0.668034	+	0.744130i	$0.267136\pi$
$242$	0	0
$243$	−2.80970e10	−0.516930
$244$	0	0
$245$	1.93921e11	3.43857
$246$	0	0
$247$	−1.88520e10	−0.322272
$248$	0	0
$249$	−8.34194e9	−0.137521
$250$	0	0
$251$	6.10283e10	0.970509	0.485254	−	0.874373i	$-0.338727\pi$
$251$	6.10283e10	0.970509	0.485254	+	0.874373i	$0.338727\pi$
$252$	0	0
$253$	−1.70554e10	−0.261710
$254$	0	0
$255$	3.51407e10	0.520451
$256$	0	0
$257$	−5.46978e10	−0.782115	−0.391058	−	0.920366i	$-0.627891\pi$
$257$	−5.46978e10	−0.782115	−0.391058	+	0.920366i	$0.627891\pi$
$258$	0	0
$259$	1.22205e11	1.68749
$260$	0	0
$261$	−6.86937e10	−0.916294
$262$	0	0
$263$	−6.53058e9	−0.0841688	−0.0420844	−	0.999114i	$-0.513400\pi$
$263$	−6.53058e9	−0.0841688	−0.0420844	+	0.999114i	$0.513400\pi$
$264$	0	0
$265$	1.24855e11	1.55524
$266$	0	0
$267$	1.61757e10	0.194788
$268$	0	0
$269$	1.22877e11	1.43082	0.715409	−	0.698706i	$-0.246241\pi$
$269$	1.22877e11	1.43082	0.715409	+	0.698706i	$0.246241\pi$
$270$	0	0
$271$	−3.90560e10	−0.439872	−0.219936	−	0.975514i	$-0.570585\pi$
$271$	−3.90560e10	−0.439872	−0.219936	+	0.975514i	$0.570585\pi$
$272$	0	0
$273$	4.19035e10	0.456581
$274$	0	0
$275$	2.68039e10	0.282619
$276$	0	0
$277$	−2.55807e10	−0.261068	−0.130534	−	0.991444i	$-0.541669\pi$
$277$	−2.55807e10	−0.261068	−0.130534	+	0.991444i	$0.541669\pi$
$278$	0	0
$279$	6.91217e10	0.682961
$280$	0	0
$281$	−1.33338e10	−0.127578	−0.0637888	−	0.997963i	$-0.520318\pi$
$281$	−1.33338e10	−0.127578	−0.0637888	+	0.997963i	$0.520318\pi$
$282$	0	0
$283$	1.04221e11	0.965862	0.482931	−	0.875658i	$-0.339572\pi$
$283$	1.04221e11	0.965862	0.482931	+	0.875658i	$0.339572\pi$
$284$	0	0
$285$	7.00012e9	0.0628497
$286$	0	0
$287$	−1.51219e11	−1.31564
$288$	0	0
$289$	3.09408e11	2.60910
$290$	0	0
$291$	−3.94660e10	−0.322630
$292$	0	0
$293$	−2.48589e11	−1.97050	−0.985251	−	0.171114i	$-0.945263\pi$
$293$	−2.48589e11	−1.97050	−0.985251	+	0.171114i	$0.945263\pi$
$294$	0	0
$295$	1.53834e11	1.18264
$296$	0	0
$297$	1.02025e10	0.0760859
$298$	0	0
$299$	−2.36541e11	−1.71154
$300$	0	0
$301$	−3.13780e10	−0.220331
$302$	0	0
$303$	−6.45312e9	−0.0439823
$304$	0	0
$305$	−1.58668e11	−1.04988
$306$	0	0
$307$	−3.35661e10	−0.215664	−0.107832	−	0.994169i	$-0.534391\pi$
$307$	−3.35661e10	−0.215664	−0.107832	+	0.994169i	$0.534391\pi$
$308$	0	0
$309$	−2.28761e10	−0.142747
$310$	0	0
$311$	−6.13779e10	−0.372041	−0.186020	−	0.982546i	$-0.559559\pi$
$311$	−6.13779e10	−0.372041	−0.186020	+	0.982546i	$0.559559\pi$
$312$	0	0
$313$	−1.45987e11	−0.859738	−0.429869	−	0.902891i	$-0.641440\pi$
$313$	−1.45987e11	−0.859738	−0.429869	+	0.902891i	$0.641440\pi$
$314$	0	0
$315$	4.64533e11	2.65839
$316$	0	0
$317$	2.34017e10	0.130161	0.0650806	−	0.997880i	$-0.479270\pi$
$317$	2.34017e10	0.130161	0.0650806	+	0.997880i	$0.479270\pi$
$318$	0	0
$319$	3.76214e10	0.203412
$320$	0	0
$321$	−1.00394e10	−0.0527757
$322$	0	0
$323$	8.52578e10	0.435836
$324$	0	0
$325$	3.71742e11	1.84828
$326$	0	0
$327$	−1.49461e10	−0.0722875
$328$	0	0
$329$	−6.03511e11	−2.83991
$330$	0	0
$331$	−1.52176e11	−0.696818	−0.348409	−	0.937343i	$-0.613278\pi$
$331$	−1.52176e11	−0.696818	−0.348409	+	0.937343i	$0.613278\pi$
$332$	0	0
$333$	2.02931e11	0.904376
$334$	0	0
$335$	−2.39000e11	−1.03680
$336$	0	0
$337$	1.05261e11	0.444565	0.222282	−	0.974982i	$-0.428649\pi$
$337$	1.05261e11	0.444565	0.222282	+	0.974982i	$0.428649\pi$
$338$	0	0
$339$	6.49421e10	0.267072
$340$	0	0
$341$	−3.78558e10	−0.151613
$342$	0	0
$343$	−5.82967e11	−2.27416
$344$	0	0
$345$	8.78321e10	0.333785
$346$	0	0
$347$	2.81412e11	1.04198	0.520992	−	0.853562i	$-0.325562\pi$
$347$	2.81412e11	1.04198	0.520992	+	0.853562i	$0.325562\pi$
$348$	0	0
$349$	3.55958e11	1.28435	0.642176	−	0.766557i	$-0.278032\pi$
$349$	3.55958e11	1.28435	0.642176	+	0.766557i	$0.278032\pi$
$350$	0	0
$351$	1.41498e11	0.497588
$352$	0	0
$353$	1.35601e11	0.464813	0.232406	−	0.972619i	$-0.425340\pi$
$353$	1.35601e11	0.464813	0.232406	+	0.972619i	$0.425340\pi$
$354$	0	0
$355$	−6.63492e11	−2.21722
$356$	0	0
$357$	−1.89507e11	−0.617474
$358$	0	0
$359$	2.12047e11	0.673764	0.336882	−	0.941547i	$-0.390628\pi$
$359$	2.12047e11	0.673764	0.336882	+	0.941547i	$0.390628\pi$
$360$	0	0
$361$	1.69836e10	0.0526316
$362$	0	0
$363$	5.68069e10	0.171720
$364$	0	0
$365$	7.38644e9	0.0217830
$366$	0	0
$367$	4.48163e11	1.28955	0.644775	−	0.764372i	$-0.276951\pi$
$367$	4.48163e11	1.28955	0.644775	+	0.764372i	$0.276951\pi$
$368$	0	0
$369$	−2.51111e11	−0.705093
$370$	0	0
$371$	−6.73318e11	−1.84517
$372$	0	0
$373$	1.86916e11	0.499985	0.249993	−	0.968248i	$-0.419572\pi$
$373$	1.86916e11	0.499985	0.249993	+	0.968248i	$0.419572\pi$
$374$	0	0
$375$	−3.31239e10	−0.0864968
$376$	0	0
$377$	5.21769e11	1.33028
$378$	0	0
$379$	3.54895e11	0.883535	0.441768	−	0.897130i	$-0.354352\pi$
$379$	3.54895e11	0.883535	0.441768	+	0.897130i	$0.354352\pi$
$380$	0	0
$381$	3.73143e10	0.0907221
$382$	0	0
$383$	−6.87668e10	−0.163299	−0.0816497	−	0.996661i	$-0.526019\pi$
$383$	−6.87668e10	−0.163299	−0.0816497	+	0.996661i	$0.526019\pi$
$384$	0	0
$385$	−2.54410e11	−0.590147
$386$	0	0
$387$	−5.21055e10	−0.118082
$388$	0	0
$389$	7.25930e10	0.160739	0.0803696	−	0.996765i	$-0.474390\pi$
$389$	7.25930e10	0.160739	0.0803696	+	0.996765i	$0.474390\pi$
$390$	0	0
$391$	1.06975e12	2.31466
$392$	0	0
$393$	−1.46989e11	−0.310826
$394$	0	0
$395$	−4.16987e11	−0.861858
$396$	0	0
$397$	2.87651e11	0.581178	0.290589	−	0.956848i	$-0.406149\pi$
$397$	2.87651e11	0.581178	0.290589	+	0.956848i	$0.406149\pi$
$398$	0	0
$399$	−3.77503e10	−0.0745662
$400$	0	0
$401$	−4.88308e11	−0.943071	−0.471535	−	0.881847i	$-0.656300\pi$
$401$	−4.88308e11	−0.943071	−0.471535	+	0.881847i	$0.656300\pi$
$402$	0	0
$403$	−5.25020e11	−0.991524
$404$	0	0
$405$	7.44689e11	1.37539
$406$	0	0
$407$	−1.11139e11	−0.200766
$408$	0	0
$409$	5.83827e11	1.03164	0.515821	−	0.856696i	$-0.327487\pi$
$409$	5.83827e11	1.03164	0.515821	+	0.856696i	$0.327487\pi$
$410$	0	0
$411$	7.10990e9	0.0122907
$412$	0	0
$413$	−8.29596e11	−1.40311
$414$	0	0
$415$	7.02416e11	1.16246
$416$	0	0
$417$	−9.39535e10	−0.152160
$418$	0	0
$419$	2.09427e11	0.331947	0.165974	−	0.986130i	$-0.446923\pi$
$419$	2.09427e11	0.331947	0.165974	+	0.986130i	$0.446923\pi$
$420$	0	0
$421$	−4.21346e11	−0.653687	−0.326844	−	0.945078i	$-0.605985\pi$
$421$	−4.21346e11	−0.653687	−0.326844	+	0.945078i	$0.605985\pi$
$422$	0	0
$423$	−1.00218e12	−1.52199
$424$	0	0
$425$	−1.68119e12	−2.49958
$426$	0	0
$427$	8.55664e11	1.24560
$428$	0	0
$429$	−3.81089e10	−0.0543211
$430$	0	0
$431$	3.62152e11	0.505525	0.252763	−	0.967528i	$-0.418661\pi$
$431$	3.62152e11	0.505525	0.252763	+	0.967528i	$0.418661\pi$
$432$	0	0
$433$	−8.27434e10	−0.113120	−0.0565598	−	0.998399i	$-0.518013\pi$
$433$	−8.27434e10	−0.113120	−0.0565598	+	0.998399i	$0.518013\pi$
$434$	0	0
$435$	−1.93743e11	−0.259432
$436$	0	0
$437$	2.13097e11	0.279518
$438$	0	0
$439$	6.69663e11	0.860530	0.430265	−	0.902703i	$-0.358420\pi$
$439$	6.69663e11	0.860530	0.430265	+	0.902703i	$0.358420\pi$
$440$	0	0
$441$	−1.73660e12	−2.18638
$442$	0	0
$443$	−5.66791e11	−0.699208	−0.349604	−	0.936898i	$-0.613684\pi$
$443$	−5.66791e11	−0.699208	−0.349604	+	0.936898i	$0.613684\pi$
$444$	0	0
$445$	−1.36205e12	−1.64654
$446$	0	0
$447$	−7.53325e10	−0.0892480
$448$	0	0
$449$	3.81051e11	0.442461	0.221230	−	0.975222i	$-0.428993\pi$
$449$	3.81051e11	0.442461	0.221230	+	0.975222i	$0.428993\pi$
$450$	0	0
$451$	1.37525e11	0.156527
$452$	0	0
$453$	−2.72029e11	−0.303510
$454$	0	0
$455$	−3.52840e12	−3.85946
$456$	0	0
$457$	−1.09800e12	−1.17755	−0.588775	−	0.808297i	$-0.700390\pi$
$457$	−1.09800e12	−1.17755	−0.588775	+	0.808297i	$0.700390\pi$
$458$	0	0
$459$	−6.39923e11	−0.672931
$460$	0	0
$461$	6.53157e11	0.673540	0.336770	−	0.941587i	$-0.390665\pi$
$461$	6.53157e11	0.673540	0.336770	+	0.941587i	$0.390665\pi$
$462$	0	0
$463$	9.03119e11	0.913336	0.456668	−	0.889637i	$-0.349043\pi$
$463$	9.03119e11	0.913336	0.456668	+	0.889637i	$0.349043\pi$
$464$	0	0
$465$	1.94950e11	0.193368
$466$	0	0
$467$	1.07789e12	1.04869	0.524345	−	0.851506i	$-0.324310\pi$
$467$	1.07789e12	1.04869	0.524345	+	0.851506i	$0.324310\pi$
$468$	0	0
$469$	1.28888e12	1.23009
$470$	0	0
$471$	−1.17631e10	−0.0110136
$472$	0	0
$473$	2.85365e10	0.0262136
$474$	0	0
$475$	−3.34898e11	−0.301850
$476$	0	0
$477$	−1.11810e12	−0.988886
$478$	0	0
$479$	1.13640e12	0.986329	0.493165	−	0.869936i	$-0.335840\pi$
$479$	1.13640e12	0.986329	0.493165	+	0.869936i	$0.335840\pi$
$480$	0	0
$481$	−1.54138e12	−1.31297
$482$	0	0
$483$	−4.73661e11	−0.396010
$484$	0	0
$485$	3.32316e12	2.72718
$486$	0	0
$487$	−2.14926e11	−0.173144	−0.0865721	−	0.996246i	$-0.527591\pi$
$487$	−2.14926e11	−0.173144	−0.0865721	+	0.996246i	$0.527591\pi$
$488$	0	0
$489$	−7.69122e10	−0.0608283
$490$	0	0
$491$	−1.98317e12	−1.53991	−0.769953	−	0.638100i	$-0.779720\pi$
$491$	−1.98317e12	−1.53991	−0.769953	+	0.638100i	$0.779720\pi$
$492$	0	0
$493$	−2.35969e12	−1.79905
$494$	0	0
$495$	−4.22467e11	−0.316278
$496$	0	0
$497$	3.57809e12	2.63055
$498$	0	0
$499$	−1.66297e12	−1.20070	−0.600348	−	0.799739i	$-0.704971\pi$
$499$	−1.66297e12	−1.20070	−0.600348	+	0.799739i	$0.704971\pi$
$500$	0	0
$501$	−1.37193e11	−0.0972885
$502$	0	0
$503$	−2.73936e12	−1.90806	−0.954031	−	0.299707i	$-0.903111\pi$
$503$	−2.73936e12	−1.90806	−0.954031	+	0.299707i	$0.903111\pi$
$504$	0	0
$505$	5.43372e11	0.371780
$506$	0	0
$507$	−2.60693e11	−0.175224
$508$	0	0
$509$	−1.41595e12	−0.935016	−0.467508	−	0.883989i	$-0.654848\pi$
$509$	−1.41595e12	−0.935016	−0.467508	+	0.883989i	$0.654848\pi$
$510$	0	0
$511$	−3.98337e10	−0.0258438
$512$	0	0
$513$	−1.27474e11	−0.0812632
$514$	0	0
$515$	1.92623e12	1.20664
$516$	0	0
$517$	5.48860e11	0.337874
$518$	0	0
$519$	1.43393e10	0.00867510
$520$	0	0
$521$	−2.76092e12	−1.64166	−0.820832	−	0.571170i	$-0.806490\pi$
$521$	−2.76092e12	−1.64166	−0.820832	+	0.571170i	$0.806490\pi$
$522$	0	0
$523$	−5.34558e11	−0.312419	−0.156209	−	0.987724i	$-0.549927\pi$
$523$	−5.34558e11	−0.312419	−0.156209	+	0.987724i	$0.549927\pi$
$524$	0	0
$525$	7.44395e11	0.427648
$526$	0	0
$527$	2.37439e12	1.34092
$528$	0	0
$529$	8.72619e11	0.484478
$530$	0	0
$531$	−1.37761e12	−0.751969
$532$	0	0
$533$	1.90733e12	1.02366
$534$	0	0
$535$	8.45346e11	0.446110
$536$	0	0
$537$	−2.89991e10	−0.0150488
$538$	0	0
$539$	9.51079e11	0.485364
$540$	0	0
$541$	−3.40287e12	−1.70788	−0.853941	−	0.520369i	$-0.825794\pi$
$541$	−3.40287e12	−1.70788	−0.853941	+	0.520369i	$0.825794\pi$
$542$	0	0
$543$	3.55074e11	0.175275
$544$	0	0
$545$	1.25851e12	0.611042
$546$	0	0
$547$	1.07691e12	0.514321	0.257161	−	0.966369i	$-0.417213\pi$
$547$	1.07691e12	0.514321	0.257161	+	0.966369i	$0.417213\pi$
$548$	0	0
$549$	1.42090e12	0.667554
$550$	0	0
$551$	−4.70055e11	−0.217253
$552$	0	0
$553$	2.24873e12	1.02253
$554$	0	0
$555$	5.72342e11	0.256058
$556$	0	0
$557$	3.44597e12	1.51692	0.758460	−	0.651720i	$-0.225952\pi$
$557$	3.44597e12	1.51692	0.758460	+	0.651720i	$0.225952\pi$
$558$	0	0
$559$	3.95772e11	0.171432
$560$	0	0
$561$	1.72346e11	0.0734631
$562$	0	0
$563$	−3.12894e12	−1.31253	−0.656266	−	0.754530i	$-0.727865\pi$
$563$	−3.12894e12	−1.31253	−0.656266	+	0.754530i	$0.727865\pi$
$564$	0	0
$565$	−5.46832e12	−2.25754
$566$	0	0
$567$	−4.01596e12	−1.63180
$568$	0	0
$569$	−3.86048e12	−1.54396	−0.771980	−	0.635647i	$-0.780734\pi$
$569$	−3.86048e12	−1.54396	−0.771980	+	0.635647i	$0.780734\pi$
$570$	0	0
$571$	−4.04984e12	−1.59432	−0.797160	−	0.603768i	$-0.793665\pi$
$571$	−4.04984e12	−1.59432	−0.797160	+	0.603768i	$0.793665\pi$
$572$	0	0
$573$	−5.00050e11	−0.193784
$574$	0	0
$575$	−4.20204e12	−1.60308
$576$	0	0
$577$	1.81317e11	0.0681001	0.0340500	−	0.999420i	$-0.489159\pi$
$577$	1.81317e11	0.0681001	0.0340500	+	0.999420i	$0.489159\pi$
$578$	0	0
$579$	−5.92279e11	−0.219014
$580$	0	0
$581$	−3.78799e12	−1.37917
$582$	0	0
$583$	6.12345e11	0.219527
$584$	0	0
$585$	−5.85917e12	−2.06840
$586$	0	0
$587$	−4.63066e11	−0.160980	−0.0804900	−	0.996755i	$-0.525649\pi$
$587$	−4.63066e11	−0.160980	−0.0804900	+	0.996755i	$0.525649\pi$
$588$	0	0
$589$	4.72984e11	0.161930
$590$	0	0
$591$	−1.59114e10	−0.00536493
$592$	0	0
$593$	3.90685e12	1.29742	0.648711	−	0.761035i	$-0.275308\pi$
$593$	3.90685e12	1.29742	0.648711	+	0.761035i	$0.275308\pi$
$594$	0	0
$595$	1.59571e13	5.21948
$596$	0	0
$597$	−1.59150e11	−0.0512771
$598$	0	0
$599$	4.47991e11	0.142183	0.0710916	−	0.997470i	$-0.477352\pi$
$599$	4.47991e11	0.142183	0.0710916	+	0.997470i	$0.477352\pi$
$600$	0	0
$601$	2.01818e12	0.630994	0.315497	−	0.948927i	$-0.397829\pi$
$601$	2.01818e12	0.630994	0.315497	+	0.948927i	$0.397829\pi$
$602$	0	0
$603$	2.14029e12	0.659241
$604$	0	0
$605$	−4.78331e12	−1.45154
$606$	0	0
$607$	−6.23030e12	−1.86277	−0.931387	−	0.364032i	$-0.881400\pi$
$607$	−6.23030e12	−1.86277	−0.931387	+	0.364032i	$0.881400\pi$
$608$	0	0
$609$	1.04482e12	0.307796
$610$	0	0
$611$	7.61212e12	2.20963
$612$	0	0
$613$	3.59461e11	0.102820	0.0514102	−	0.998678i	$-0.483628\pi$
$613$	3.59461e11	0.102820	0.0514102	+	0.998678i	$0.483628\pi$
$614$	0	0
$615$	−7.08228e11	−0.199634
$616$	0	0
$617$	4.00051e12	1.11130	0.555650	−	0.831416i	$-0.312469\pi$
$617$	4.00051e12	1.11130	0.555650	+	0.831416i	$0.312469\pi$
$618$	0	0
$619$	3.72631e12	1.02017	0.510084	−	0.860125i	$-0.329614\pi$
$619$	3.72631e12	1.02017	0.510084	+	0.860125i	$0.329614\pi$
$620$	0	0
$621$	−1.59945e12	−0.431576
$622$	0	0
$623$	7.34525e12	1.95349
$624$	0	0
$625$	−2.22999e12	−0.584580
$626$	0	0
$627$	3.43318e10	0.00887141
$628$	0	0
$629$	6.97084e12	1.77565
$630$	0	0
$631$	−5.38677e11	−0.135268	−0.0676342	−	0.997710i	$-0.521545\pi$
$631$	−5.38677e11	−0.135268	−0.0676342	+	0.997710i	$0.521545\pi$
$632$	0	0
$633$	−1.22521e12	−0.303315
$634$	0	0
$635$	−3.14198e12	−0.766869
$636$	0	0
$637$	1.31905e13	3.17419
$638$	0	0
$639$	5.94168e12	1.40979
$640$	0	0
$641$	5.87686e12	1.37494	0.687471	−	0.726212i	$-0.258721\pi$
$641$	5.87686e12	1.37494	0.687471	+	0.726212i	$0.258721\pi$
$642$	0	0
$643$	−3.03822e12	−0.700923	−0.350462	−	0.936577i	$-0.613975\pi$
$643$	−3.03822e12	−0.700923	−0.350462	+	0.936577i	$0.613975\pi$
$644$	0	0
$645$	−1.46957e11	−0.0334328
$646$	0	0
$647$	−1.92485e12	−0.431844	−0.215922	−	0.976411i	$-0.569276\pi$
$647$	−1.92485e12	−0.431844	−0.215922	+	0.976411i	$0.569276\pi$
$648$	0	0
$649$	7.54472e11	0.166933
$650$	0	0
$651$	−1.05133e12	−0.229416
$652$	0	0
$653$	−4.80531e12	−1.03422	−0.517109	−	0.855920i	$-0.672992\pi$
$653$	−4.80531e12	−1.03422	−0.517109	+	0.855920i	$0.672992\pi$
$654$	0	0
$655$	1.23769e13	2.62740
$656$	0	0
$657$	−6.61468e10	−0.0138505
$658$	0	0
$659$	−7.75087e12	−1.60091	−0.800454	−	0.599394i	$-0.795408\pi$
$659$	−7.75087e12	−1.60091	−0.800454	+	0.599394i	$0.795408\pi$
$660$	0	0
$661$	4.58458e12	0.934100	0.467050	−	0.884231i	$-0.345317\pi$
$661$	4.58458e12	0.934100	0.467050	+	0.884231i	$0.345317\pi$
$662$	0	0
$663$	2.39026e12	0.480435
$664$	0	0
$665$	3.17869e12	0.630305
$666$	0	0
$667$	−5.89789e12	−1.15380
$668$	0	0
$669$	5.78910e11	0.111736
$670$	0	0
$671$	−7.78179e11	−0.148193
$672$	0	0
$673$	−7.38830e12	−1.38828	−0.694139	−	0.719841i	$-0.744215\pi$
$673$	−7.38830e12	−1.38828	−0.694139	+	0.719841i	$0.744215\pi$
$674$	0	0
$675$	2.51365e12	0.466056
$676$	0	0
$677$	−3.48101e12	−0.636878	−0.318439	−	0.947943i	$-0.603159\pi$
$677$	−3.48101e12	−0.636878	−0.318439	+	0.947943i	$0.603159\pi$
$678$	0	0
$679$	−1.79212e13	−3.23558
$680$	0	0
$681$	−6.08006e11	−0.108329
$682$	0	0
$683$	−6.60095e12	−1.16068	−0.580341	−	0.814373i	$-0.697081\pi$
$683$	−6.60095e12	−1.16068	−0.580341	+	0.814373i	$0.697081\pi$
$684$	0	0
$685$	−5.98675e11	−0.103892
$686$	0	0
$687$	−3.32987e11	−0.0570325
$688$	0	0
$689$	8.49259e12	1.43567
$690$	0	0
$691$	8.94184e12	1.49202	0.746012	−	0.665933i	$-0.231966\pi$
$691$	8.94184e12	1.49202	0.746012	+	0.665933i	$0.231966\pi$
$692$	0	0
$693$	2.27828e12	0.375239
$694$	0	0
$695$	7.91117e12	1.28620
$696$	0	0
$697$	−8.62586e12	−1.38438
$698$	0	0
$699$	−4.16030e11	−0.0659140
$700$	0	0
$701$	−9.13468e11	−0.142877	−0.0714385	−	0.997445i	$-0.522759\pi$
$701$	−9.13468e11	−0.142877	−0.0714385	+	0.997445i	$0.522759\pi$
$702$	0	0
$703$	1.38861e12	0.214428
$704$	0	0
$705$	−2.82652e12	−0.430924
$706$	0	0
$707$	−2.93030e12	−0.441088
$708$	0	0
$709$	2.22286e12	0.330372	0.165186	−	0.986262i	$-0.447178\pi$
$709$	2.22286e12	0.330372	0.165186	+	0.986262i	$0.447178\pi$
$710$	0	0
$711$	3.73419e12	0.548003
$712$	0	0
$713$	5.93463e12	0.859985
$714$	0	0
$715$	3.20888e12	0.459173
$716$	0	0
$717$	1.72843e11	0.0244239
$718$	0	0
$719$	1.38107e12	0.192724	0.0963622	−	0.995346i	$-0.469279\pi$
$719$	1.38107e12	0.192724	0.0963622	+	0.995346i	$0.469279\pi$
$720$	0	0
$721$	−1.03878e13	−1.43158
$722$	0	0
$723$	−1.76721e12	−0.240528
$724$	0	0
$725$	9.26898e12	1.24598
$726$	0	0
$727$	1.01054e13	1.34168	0.670838	−	0.741604i	$-0.265935\pi$
$727$	1.01054e13	1.34168	0.670838	+	0.741604i	$0.265935\pi$
$728$	0	0
$729$	−6.18254e12	−0.810761
$730$	0	0
$731$	−1.78987e12	−0.231842
$732$	0	0
$733$	−2.46166e12	−0.314964	−0.157482	−	0.987522i	$-0.550338\pi$
$733$	−2.46166e12	−0.314964	−0.157482	+	0.987522i	$0.550338\pi$
$734$	0	0
$735$	−4.89787e12	−0.619034
$736$	0	0
$737$	−1.17217e12	−0.146348
$738$	0	0
$739$	−8.80833e12	−1.08641	−0.543205	−	0.839600i	$-0.682789\pi$
$739$	−8.80833e12	−1.08641	−0.543205	+	0.839600i	$0.682789\pi$
$740$	0	0
$741$	4.76146e11	0.0580174
$742$	0	0
$743$	2.35539e11	0.0283539	0.0141770	−	0.999900i	$-0.495487\pi$
$743$	2.35539e11	0.0283539	0.0141770	+	0.999900i	$0.495487\pi$
$744$	0	0
$745$	6.34322e12	0.754409
$746$	0	0
$747$	−6.29025e12	−0.739138
$748$	0	0
$749$	−4.55879e12	−0.529275
$750$	0	0
$751$	6.15738e12	0.706344	0.353172	−	0.935558i	$-0.385103\pi$
$751$	6.15738e12	0.706344	0.353172	+	0.935558i	$0.385103\pi$
$752$	0	0
$753$	−1.54139e12	−0.174717
$754$	0	0
$755$	2.29056e13	2.56555
$756$	0	0
$757$	1.74954e12	0.193639	0.0968195	−	0.995302i	$-0.469133\pi$
$757$	1.74954e12	0.193639	0.0968195	+	0.995302i	$0.469133\pi$
$758$	0	0
$759$	4.30769e11	0.0471147
$760$	0	0
$761$	−1.52097e13	−1.64395	−0.821976	−	0.569522i	$-0.807128\pi$
$761$	−1.52097e13	−1.64395	−0.821976	+	0.569522i	$0.807128\pi$
$762$	0	0
$763$	−6.78688e12	−0.724953
$764$	0	0
$765$	2.64979e13	2.79728
$766$	0	0
$767$	1.04637e13	1.09171
$768$	0	0
$769$	−9.63035e12	−0.993055	−0.496528	−	0.868021i	$-0.665392\pi$
$769$	−9.63035e12	−0.993055	−0.496528	+	0.868021i	$0.665392\pi$
$770$	0	0
$771$	1.38150e12	0.140801
$772$	0	0
$773$	1.49129e13	1.50230	0.751148	−	0.660134i	$-0.229501\pi$
$773$	1.49129e13	1.50230	0.751148	+	0.660134i	$0.229501\pi$
$774$	0	0
$775$	−9.32673e12	−0.928692
$776$	0	0
$777$	−3.08653e12	−0.303792
$778$	0	0
$779$	−1.71829e12	−0.167178
$780$	0	0
$781$	−3.25407e12	−0.312966
$782$	0	0
$783$	3.52811e12	0.335439
$784$	0	0
$785$	9.90492e11	0.0930974
$786$	0	0
$787$	−1.51917e13	−1.41163	−0.705816	−	0.708396i	$-0.749419\pi$
$787$	−1.51917e13	−1.41163	−0.705816	+	0.708396i	$0.749419\pi$
$788$	0	0
$789$	1.64943e11	0.0151526
$790$	0	0
$791$	2.94896e13	2.67840
$792$	0	0
$793$	−1.07925e13	−0.969157
$794$	0	0
$795$	−3.15346e12	−0.279985
$796$	0	0
$797$	1.37336e13	1.20565	0.602827	−	0.797872i	$-0.294041\pi$
$797$	1.37336e13	1.20565	0.602827	+	0.797872i	$0.294041\pi$
$798$	0	0
$799$	−3.44256e13	−2.98828
$800$	0	0
$801$	1.21973e13	1.04693
$802$	0	0
$803$	3.62265e10	0.00307473
$804$	0	0
$805$	3.98837e13	3.34745
$806$	0	0
$807$	−3.10349e12	−0.257585
$808$	0	0
$809$	−1.54605e12	−0.126898	−0.0634491	−	0.997985i	$-0.520210\pi$
$809$	−1.54605e12	−0.126898	−0.0634491	+	0.997985i	$0.520210\pi$
$810$	0	0
$811$	−1.50745e13	−1.22363	−0.611814	−	0.791002i	$-0.709560\pi$
$811$	−1.50745e13	−1.22363	−0.611814	+	0.791002i	$0.709560\pi$
$812$	0	0
$813$	9.86438e11	0.0791885
$814$	0	0
$815$	6.47624e12	0.514178
$816$	0	0
$817$	−3.56546e11	−0.0279973
$818$	0	0
$819$	3.15974e13	2.45399
$820$	0	0
$821$	7.55550e12	0.580389	0.290194	−	0.956968i	$-0.406280\pi$
$821$	7.55550e12	0.580389	0.290194	+	0.956968i	$0.406280\pi$
$822$	0	0
$823$	−3.21334e12	−0.244150	−0.122075	−	0.992521i	$-0.538955\pi$
$823$	−3.21334e12	−0.244150	−0.122075	+	0.992521i	$0.538955\pi$
$824$	0	0
$825$	−6.76986e11	−0.0508788
$826$	0	0
$827$	−8.79853e12	−0.654087	−0.327043	−	0.945009i	$-0.606052\pi$
$827$	−8.79853e12	−0.654087	−0.327043	+	0.945009i	$0.606052\pi$
$828$	0	0
$829$	2.39558e13	1.76163	0.880816	−	0.473459i	$-0.156995\pi$
$829$	2.39558e13	1.76163	0.880816	+	0.473459i	$0.156995\pi$
$830$	0	0
$831$	6.46092e11	0.0469991
$832$	0	0
$833$	−5.96535e13	−4.29273
$834$	0	0
$835$	1.15520e13	0.822375
$836$	0	0
$837$	−3.55009e12	−0.250020
$838$	0	0
$839$	2.38372e13	1.66083	0.830417	−	0.557143i	$-0.188103\pi$
$839$	2.38372e13	1.66083	0.830417	+	0.557143i	$0.188103\pi$
$840$	0	0
$841$	−1.49741e12	−0.103219
$842$	0	0
$843$	3.36771e11	0.0229673
$844$	0	0
$845$	2.19511e13	1.48116
$846$	0	0
$847$	2.57955e13	1.72214
$848$	0	0
$849$	−2.63230e12	−0.173881
$850$	0	0
$851$	1.74232e13	1.13879
$852$	0	0
$853$	1.15908e13	0.749621	0.374811	−	0.927101i	$-0.377708\pi$
$853$	1.15908e13	0.749621	0.374811	+	0.927101i	$0.377708\pi$
$854$	0	0
$855$	5.27845e12	0.337800
$856$	0	0
$857$	2.40055e13	1.52019	0.760093	−	0.649815i	$-0.225154\pi$
$857$	2.40055e13	1.52019	0.760093	+	0.649815i	$0.225154\pi$
$858$	0	0
$859$	1.28254e13	0.803712	0.401856	−	0.915703i	$-0.368365\pi$
$859$	1.28254e13	0.803712	0.401856	+	0.915703i	$0.368365\pi$
$860$	0	0
$861$	3.81934e12	0.236850
$862$	0	0
$863$	−1.10966e13	−0.680994	−0.340497	−	0.940246i	$-0.610595\pi$
$863$	−1.10966e13	−0.680994	−0.340497	+	0.940246i	$0.610595\pi$
$864$	0	0
$865$	−1.20741e12	−0.0733302
$866$	0	0
$867$	−7.81472e12	−0.469707
$868$	0	0
$869$	−2.04510e12	−0.121654
$870$	0	0
$871$	−1.62567e13	−0.957087
$872$	0	0
$873$	−2.97594e13	−1.73405
$874$	0	0
$875$	−1.50412e13	−0.867455
$876$	0	0
$877$	2.34043e12	0.133597	0.0667986	−	0.997766i	$-0.478721\pi$
$877$	2.34043e12	0.133597	0.0667986	+	0.997766i	$0.478721\pi$
$878$	0	0
$879$	6.27860e12	0.354742
$880$	0	0
$881$	−1.38455e12	−0.0774315	−0.0387158	−	0.999250i	$-0.512327\pi$
$881$	−1.38455e12	−0.0774315	−0.0387158	+	0.999250i	$0.512327\pi$
$882$	0	0
$883$	6.13771e12	0.339769	0.169884	−	0.985464i	$-0.445661\pi$
$883$	6.13771e12	0.339769	0.169884	+	0.985464i	$0.445661\pi$
$884$	0	0
$885$	−3.88538e12	−0.212906
$886$	0	0
$887$	−8.64012e12	−0.468666	−0.234333	−	0.972156i	$-0.575291\pi$
$887$	−8.64012e12	−0.468666	−0.234333	+	0.972156i	$0.575291\pi$
$888$	0	0
$889$	1.69441e13	0.909830
$890$	0	0
$891$	3.65229e12	0.194141
$892$	0	0
$893$	−6.85766e12	−0.360864
$894$	0	0
$895$	2.44181e12	0.127206
$896$	0	0
$897$	5.97431e12	0.308121
$898$	0	0
$899$	−1.30908e13	−0.668417
$900$	0	0
$901$	−3.84075e13	−1.94157
$902$	0	0
$903$	7.92513e11	0.0396654
$904$	0	0
$905$	−2.98983e13	−1.48159
$906$	0	0
$907$	3.36715e13	1.65207	0.826037	−	0.563616i	$-0.190590\pi$
$907$	3.36715e13	1.65207	0.826037	+	0.563616i	$0.190590\pi$
$908$	0	0
$909$	−4.86599e12	−0.236392
$910$	0	0
$911$	−2.03568e13	−0.979211	−0.489606	−	0.871944i	$-0.662859\pi$
$911$	−2.03568e13	−0.979211	−0.489606	+	0.871944i	$0.662859\pi$
$912$	0	0
$913$	3.44497e12	0.164084
$914$	0	0
$915$	4.00747e12	0.189006
$916$	0	0
$917$	−6.67462e13	−3.11720
$918$	0	0
$919$	−2.44953e13	−1.13282	−0.566412	−	0.824122i	$-0.691669\pi$
$919$	−2.44953e13	−1.13282	−0.566412	+	0.824122i	$0.691669\pi$
$920$	0	0
$921$	8.47779e11	0.0388252
$922$	0	0
$923$	−4.51306e13	−2.04674
$924$	0	0
$925$	−2.73819e13	−1.22977
$926$	0	0
$927$	−1.72497e13	−0.767226
$928$	0	0
$929$	2.79588e13	1.23154	0.615768	−	0.787928i	$-0.288846\pi$
$929$	2.79588e13	1.23154	0.615768	+	0.787928i	$0.288846\pi$
$930$	0	0
$931$	−1.18831e13	−0.518391
$932$	0	0
$933$	1.55022e12	0.0669771
$934$	0	0
$935$	−1.45121e13	−0.620979
$936$	0	0
$937$	−8.07789e12	−0.342349	−0.171175	−	0.985241i	$-0.554756\pi$
$937$	−8.07789e12	−0.342349	−0.171175	+	0.985241i	$0.554756\pi$
$938$	0	0
$939$	3.68720e12	0.154775
$940$	0	0
$941$	1.80335e12	0.0749768	0.0374884	−	0.999297i	$-0.488064\pi$
$941$	1.80335e12	0.0749768	0.0374884	+	0.999297i	$0.488064\pi$
$942$	0	0
$943$	−2.15598e13	−0.887855
$944$	0	0
$945$	−2.38584e13	−0.973191
$946$	0	0
$947$	−1.33152e13	−0.537989	−0.268995	−	0.963142i	$-0.586691\pi$
$947$	−1.33152e13	−0.537989	−0.268995	+	0.963142i	$0.586691\pi$
$948$	0	0
$949$	5.02424e11	0.0201082
$950$	0	0
$951$	−5.91057e11	−0.0234324
$952$	0	0
$953$	−3.50858e13	−1.37789	−0.688943	−	0.724815i	$-0.741925\pi$
$953$	−3.50858e13	−1.37789	−0.688943	+	0.724815i	$0.741925\pi$
$954$	0	0
$955$	4.21057e13	1.63804
$956$	0	0
$957$	−9.50203e11	−0.0366195
$958$	0	0
$959$	3.22854e12	0.123260
$960$	0	0
$961$	−1.32673e13	−0.501795
$962$	0	0
$963$	−7.57021e12	−0.283654
$964$	0	0
$965$	4.98716e13	1.85132
$966$	0	0
$967$	−3.33371e13	−1.22605	−0.613026	−	0.790063i	$-0.710048\pi$
$967$	−3.33371e13	−1.22605	−0.613026	+	0.790063i	$0.710048\pi$
$968$	0	0
$969$	−2.15336e12	−0.0784619
$970$	0	0
$971$	3.31130e13	1.19540	0.597698	−	0.801722i	$-0.296082\pi$
$971$	3.31130e13	1.19540	0.597698	+	0.801722i	$0.296082\pi$
$972$	0	0
$973$	−4.26634e13	−1.52598
$974$	0	0
$975$	−9.38909e12	−0.332738
$976$	0	0
$977$	−1.28147e12	−0.0449969	−0.0224984	−	0.999747i	$-0.507162\pi$
$977$	−1.28147e12	−0.0449969	−0.0224984	+	0.999747i	$0.507162\pi$
$978$	0	0
$979$	−6.68009e12	−0.232413
$980$	0	0
$981$	−1.12701e13	−0.388525
$982$	0	0
$983$	−3.93077e13	−1.34272	−0.671362	−	0.741130i	$-0.734290\pi$
$983$	−3.93077e13	−1.34272	−0.671362	+	0.741130i	$0.734290\pi$
$984$	0	0
$985$	1.33979e12	0.0453495
$986$	0	0
$987$	1.52429e13	0.511258
$988$	0	0
$989$	−4.47366e12	−0.148689
$990$	0	0
$991$	3.13171e13	1.03145	0.515727	−	0.856753i	$-0.327522\pi$
$991$	3.13171e13	1.03145	0.515727	+	0.856753i	$0.327522\pi$
$992$	0	0
$993$	3.84350e12	0.125446
$994$	0	0
$995$	1.34010e13	0.433443
$996$	0	0
$997$	−8.04007e12	−0.257710	−0.128855	−	0.991663i	$-0.541130\pi$
$997$	−8.04007e12	−0.257710	−0.128855	+	0.991663i	$0.541130\pi$
$998$	0	0
$999$	−1.04225e13	−0.331076

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.10.a.d.1.3	✓	4	4.3	odd	2
304.10.a.e.1.2		4	1.1	even	1	trivial
342.10.a.l.1.1		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)

\(f(q)\)	\(=\)	\(q-25.2570 q^{3} +2126.71 q^{5} -11469.0 q^{7} -19045.1 q^{9} +O(q^{10})\)	\(q-25.2570 q^{3} +2126.71 q^{5} -11469.0 q^{7} -19045.1 q^{9} +10430.4 q^{11} +144659. q^{13} -53714.4 q^{15} -654214. q^{17} -130321. q^{19} +289672. q^{21} -1.63517e6 q^{23} +2.56979e6 q^{25} +978155. q^{27} +3.60690e6 q^{29} -3.62937e6 q^{31} -263440. q^{33} -2.43912e7 q^{35} -1.06553e7 q^{37} -3.65364e6 q^{39} +1.31851e7 q^{41} +2.73590e6 q^{43} -4.05035e7 q^{45} +5.26213e7 q^{47} +9.11835e7 q^{49} +1.65235e7 q^{51} +5.87078e7 q^{53} +2.21825e7 q^{55} +3.29152e6 q^{57} +7.23340e7 q^{59} -7.46069e7 q^{61} +2.18427e8 q^{63} +3.07647e8 q^{65} -1.12380e8 q^{67} +4.12994e7 q^{69} -3.11980e8 q^{71} +3.47317e6 q^{73} -6.49052e7 q^{75} -1.19626e8 q^{77} -1.96071e8 q^{79} +3.50159e8 q^{81} +3.30282e8 q^{83} -1.39133e9 q^{85} -9.10995e7 q^{87} -6.40446e8 q^{89} -1.65908e9 q^{91} +9.16671e7 q^{93} -2.77156e8 q^{95} +1.56258e9 q^{97} -1.98648e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 84 q^{3} - 1395 q^{5} - 12307 q^{7} + 16538 q^{9}+O(q^{10}) \) 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

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