LMFDB - Embedded newform 12.20.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [12,20,Mod(1,12)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(12, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 20, names="a")

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

chi := DirichletCharacter("12.1");

S:= CuspForms(chi, 20);

N := Newforms(S);

Level:	$ N $	$=$	$ 12 = 2^{2} \cdot 3 $
Weight:	$ k $	$=$	$ 20 $
Character orbit:	$[\chi]$	$=$	12.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:	$27.4580035868$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{193153}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 48288 $ x^2 - x - 48288
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{3}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$220.246$ of defining polynomial
Character	$\chi$	$=$	12.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-19683.0 q^{3} -3.48482e6 q^{5} +1.50425e8 q^{7} +3.87420e8 q^{9} +O(q^{10})$	$q-19683.0 q^{3} -3.48482e6 q^{5} +1.50425e8 q^{7} +3.87420e8 q^{9} -3.64092e9 q^{11} +3.93799e10 q^{13} +6.85917e10 q^{15} -7.49645e11 q^{17} +7.05586e11 q^{19} -2.96081e12 q^{21} +3.56777e12 q^{23} -6.92952e12 q^{25} -7.62560e12 q^{27} -1.14555e14 q^{29} -2.11767e14 q^{31} +7.16642e13 q^{33} -5.24203e14 q^{35} -1.03089e15 q^{37} -7.75114e14 q^{39} +3.07261e14 q^{41} -6.78071e13 q^{43} -1.35009e15 q^{45} -9.55161e15 q^{47} +1.12287e16 q^{49} +1.47553e16 q^{51} +4.64691e16 q^{53} +1.26879e16 q^{55} -1.38881e16 q^{57} +2.59683e16 q^{59} -2.33924e13 q^{61} +5.82776e16 q^{63} -1.37232e17 q^{65} +1.12682e17 q^{67} -7.02243e16 q^{69} -6.05925e17 q^{71} -7.22679e16 q^{73} +1.36394e17 q^{75} -5.47684e17 q^{77} -1.52111e18 q^{79} +1.50095e17 q^{81} +1.13367e17 q^{83} +2.61238e18 q^{85} +2.25479e18 q^{87} -3.99865e18 q^{89} +5.92370e18 q^{91} +4.16822e18 q^{93} -2.45884e18 q^{95} +1.29512e19 q^{97} -1.41057e18 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 39366 q^{3} + 624780 q^{5} + 4667104 q^{7} + 774840978 q^{9}+O(q^{10}) $ 2 * q - 39366 * q^3 + 624780 * q^5 + 4667104 * q^7 + 774840978 * q^9	$ 2 q - 39366 q^{3} + 624780 q^{5} + 4667104 q^{7} + 774840978 q^{9} + 9015783192 q^{11} + 25948151500 q^{13} - 12297544740 q^{15} - 330949868796 q^{17} - 1555799112920 q^{19} - 91862608032 q^{21} - 2955794188464 q^{23} - 9114209254450 q^{25} - 15251194969974 q^{27} - 171112962219588 q^{29} - 439686204047792 q^{31} - 177457660568136 q^{33} - 11\!\cdots\!40 q^{35}+ \cdots + 34\!\cdots\!88 q^{99}+O(q^{100}) $ 2 * q - 39366 * q^3 + 624780 * q^5 + 4667104 * q^7 + 774840978 * q^9 + 9015783192 * q^11 + 25948151500 * q^13 - 12297544740 * q^15 - 330949868796 * q^17 - 1555799112920 * q^19 - 91862608032 * q^21 - 2955794188464 * q^23 - 9114209254450 * q^25 - 15251194969974 * q^27 - 171112962219588 * q^29 - 439686204047792 * q^31 - 177457660568136 * q^33 - 1123207990107840 * q^35 - 912841755316868 * q^37 - 510737465974500 * q^39 - 2039868884446380 * q^41 - 5607123893986664 * q^43 + 242052573117420 * q^45 - 720086631286464 * q^47 + 21075072481456722 * q^49 + 6514086267511668 * q^51 + 56983015881991692 * q^53 + 64701905169678480 * q^55 + 30622793939604360 * q^57 + 17959203991017432 * q^59 + 29244471161814604 * q^61 + 1808131713893856 * q^63 - 192430646530745400 * q^65 - 37110594569246744 * q^67 + 58178897011536912 * q^69 - 604502474341905360 * q^71 - 659663813215165772 * q^73 + 179394980755339350 * q^75 - 2392494322775596416 * q^77 - 352041739978389776 * q^79 + 300189270593998242 * q^81 - 2492903589286330584 * q^83 + 4333046841673556760 * q^85 + 3368016435368150604 * q^87 + 2523728600070118740 * q^89 + 7881478354364513600 * q^91 + 8654343554272689936 * q^93 - 11752227437069091600 * q^95 + 17246738574828858628 * q^97 + 3492899132962620888 * 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
$3$	−19683.0	−0.577350
$3$	−19683.0	−0.577350
$4$	0	0
$5$	−3.48482e6	−0.797931	−0.398965	−	0.916966i	$-0.630631\pi$
$5$	−3.48482e6	−0.797931	−0.398965	+	0.916966i	$0.630631\pi$
$6$	0	0
$7$	1.50425e8	1.40892	0.704462	−	0.709741i	$-0.251188\pi$
$7$	1.50425e8	1.40892	0.704462	+	0.709741i	$0.251188\pi$
$8$	0	0
$9$	3.87420e8	0.333333
$10$	0	0
$11$	−3.64092e9	−0.465565	−0.232783	−	0.972529i	$-0.574783\pi$
$11$	−3.64092e9	−0.465565	−0.232783	+	0.972529i	$0.574783\pi$
$12$	0	0
$13$	3.93799e10	1.02994	0.514971	−	0.857208i	$-0.327803\pi$
$13$	3.93799e10	1.02994	0.514971	+	0.857208i	$0.327803\pi$
$14$	0	0
$15$	6.85917e10	0.460685
$16$	0	0
$17$	−7.49645e11	−1.53317	−0.766586	−	0.642141i	$-0.778046\pi$
$17$	−7.49645e11	−1.53317	−0.766586	+	0.642141i	$0.778046\pi$
$18$	0	0
$19$	7.05586e11	0.501639	0.250819	−	0.968034i	$-0.419300\pi$
$19$	7.05586e11	0.501639	0.250819	+	0.968034i	$0.419300\pi$
$20$	0	0
$21$	−2.96081e12	−0.813443
$22$	0	0
$23$	3.56777e12	0.413030	0.206515	−	0.978443i	$-0.433788\pi$
$23$	3.56777e12	0.413030	0.206515	+	0.978443i	$0.433788\pi$
$24$	0	0
$25$	−6.92952e12	−0.363307
$26$	0	0
$27$	−7.62560e12	−0.192450
$28$	0	0
$29$	−1.14555e14	−1.46634	−0.733170	−	0.680045i	$-0.761960\pi$
$29$	−1.14555e14	−1.46634	−0.733170	+	0.680045i	$0.761960\pi$
$30$	0	0
$31$	−2.11767e14	−1.43854	−0.719272	−	0.694729i	$-0.755524\pi$
$31$	−2.11767e14	−1.43854	−0.719272	+	0.694729i	$0.755524\pi$
$32$	0	0
$33$	7.16642e13	0.268794
$34$	0	0
$35$	−5.24203e14	−1.12422
$36$	0	0
$37$	−1.03089e15	−1.30405	−0.652027	−	0.758196i	$-0.726081\pi$
$37$	−1.03089e15	−1.30405	−0.652027	+	0.758196i	$0.726081\pi$
$38$	0	0
$39$	−7.75114e14	−0.594637
$40$	0	0
$41$	3.07261e14	0.146575	0.0732877	−	0.997311i	$-0.476651\pi$
$41$	3.07261e14	0.146575	0.0732877	+	0.997311i	$0.476651\pi$
$42$	0	0
$43$	−6.78071e13	−0.0205743	−0.0102871	−	0.999947i	$-0.503275\pi$
$43$	−6.78071e13	−0.0205743	−0.0102871	+	0.999947i	$0.503275\pi$
$44$	0	0
$45$	−1.35009e15	−0.265977
$46$	0	0
$47$	−9.55161e15	−1.24494	−0.622468	−	0.782645i	$-0.713870\pi$
$47$	−9.55161e15	−1.24494	−0.622468	+	0.782645i	$0.713870\pi$
$48$	0	0
$49$	1.12287e16	0.985069
$50$	0	0
$51$	1.47553e16	0.885177
$52$	0	0
$53$	4.64691e16	1.93439	0.967193	−	0.254041i	$-0.0817598\pi$
$53$	4.64691e16	1.93439	0.967193	+	0.254041i	$0.0817598\pi$
$54$	0	0
$55$	1.26879e16	0.371489
$56$	0	0
$57$	−1.38881e16	−0.289621
$58$	0	0
$59$	2.59683e16	0.390256	0.195128	−	0.980778i	$-0.437488\pi$
$59$	2.59683e16	0.390256	0.195128	+	0.980778i	$0.437488\pi$
$60$	0	0
$61$	−2.33924e13	−0.000256119 0	−0.000128059	−	1.00000i	$-0.500041\pi$
$61$	−2.33924e13	−0.000256119 0	−0.000128059		1.00000i	$0.500041\pi$
$62$	0	0
$63$	5.82776e16	0.469642
$64$	0	0
$65$	−1.37232e17	−0.821822
$66$	0	0
$67$	1.12682e17	0.505991	0.252995	−	0.967468i	$-0.418584\pi$
$67$	1.12682e17	0.505991	0.252995	+	0.967468i	$0.418584\pi$
$68$	0	0
$69$	−7.02243e16	−0.238463
$70$	0	0
$71$	−6.05925e17	−1.56843	−0.784214	−	0.620491i	$-0.786933\pi$
$71$	−6.05925e17	−1.56843	−0.784214	+	0.620491i	$0.786933\pi$
$72$	0	0
$73$	−7.22679e16	−0.143674	−0.0718370	−	0.997416i	$-0.522886\pi$
$73$	−7.22679e16	−0.143674	−0.0718370	+	0.997416i	$0.522886\pi$
$74$	0	0
$75$	1.36394e17	0.209755
$76$	0	0
$77$	−5.47684e17	−0.655946
$78$	0	0
$79$	−1.52111e18	−1.42792	−0.713960	−	0.700186i	$-0.753100\pi$
$79$	−1.52111e18	−1.42792	−0.713960	+	0.700186i	$0.753100\pi$
$80$	0	0
$81$	1.50095e17	0.111111
$82$	0	0
$83$	1.13367e17	0.0665646	0.0332823	−	0.999446i	$-0.489404\pi$
$83$	1.13367e17	0.0665646	0.0332823	+	0.999446i	$0.489404\pi$
$84$	0	0
$85$	2.61238e18	1.22337
$86$	0	0
$87$	2.25479e18	0.846592
$88$	0	0
$89$	−3.99865e18	−1.20979	−0.604893	−	0.796307i	$-0.706784\pi$
$89$	−3.99865e18	−1.20979	−0.604893	+	0.796307i	$0.706784\pi$
$90$	0	0
$91$	5.92370e18	1.45111
$92$	0	0
$93$	4.16822e18	0.830543
$94$	0	0
$95$	−2.45884e18	−0.400273
$96$	0	0
$97$	1.29512e19	1.72974	0.864868	−	0.502000i	$-0.167402\pi$
$97$	1.29512e19	1.72974	0.864868	+	0.502000i	$0.167402\pi$
$98$	0	0
$99$	−1.41057e18	−0.155188
$100$	0	0
$101$	−4.76443e18	−0.433469	−0.216734	−	0.976231i	$-0.569541\pi$
$101$	−4.76443e18	−0.433469	−0.216734	+	0.976231i	$0.569541\pi$
$102$	0	0
$103$	−1.75089e19	−1.32222	−0.661112	−	0.750287i	$-0.729915\pi$
$103$	−1.75089e19	−1.32222	−0.661112	+	0.750287i	$0.729915\pi$
$104$	0	0
$105$	1.03179e19	0.649071
$106$	0	0
$107$	−2.85016e19	−1.49873	−0.749365	−	0.662157i	$-0.769641\pi$
$107$	−2.85016e19	−1.49873	−0.749365	+	0.662157i	$0.769641\pi$
$108$	0	0
$109$	2.44094e19	1.07648	0.538239	−	0.842792i	$-0.319090\pi$
$109$	2.44094e19	1.07648	0.538239	+	0.842792i	$0.319090\pi$
$110$	0	0
$111$	2.02910e19	0.752896
$112$	0	0
$113$	−4.58487e19	−1.43576	−0.717879	−	0.696168i	$-0.754887\pi$
$113$	−4.58487e19	−1.43576	−0.717879	+	0.696168i	$0.754887\pi$
$114$	0	0
$115$	−1.24330e19	−0.329570
$116$	0	0
$117$	1.52566e19	0.343314
$118$	0	0
$119$	−1.12765e20	−2.16012
$120$	0	0
$121$	−4.79028e19	−0.783249
$122$	0	0
$123$	−6.04783e18	−0.0846254
$124$	0	0
$125$	9.06158e19	1.08782
$126$	0	0
$127$	1.82608e20	1.88532	0.942661	−	0.333752i	$-0.108315\pi$
$127$	1.82608e20	1.88532	0.942661	+	0.333752i	$0.108315\pi$
$128$	0	0
$129$	1.33465e18	0.0118786
$130$	0	0
$131$	−7.54821e19	−0.580453	−0.290226	−	0.956958i	$-0.593731\pi$
$131$	−7.54821e19	−0.580453	−0.290226	+	0.956958i	$0.593731\pi$
$132$	0	0
$133$	1.06138e20	0.706771
$134$	0	0
$135$	2.65738e19	0.153562
$136$	0	0
$137$	3.36532e20	1.69115	0.845573	−	0.533859i	$-0.179259\pi$
$137$	3.36532e20	1.69115	0.845573	+	0.533859i	$0.179259\pi$
$138$	0	0
$139$	−2.33633e20	−1.02304	−0.511520	−	0.859271i	$-0.670917\pi$
$139$	−2.33633e20	−1.02304	−0.511520	+	0.859271i	$0.670917\pi$
$140$	0	0
$141$	1.88004e20	0.718764
$142$	0	0
$143$	−1.43379e20	−0.479505
$144$	0	0
$145$	3.99205e20	1.17004
$146$	0	0
$147$	−2.21014e20	−0.568730
$148$	0	0
$149$	5.70775e19	0.129180	0.0645900	−	0.997912i	$-0.479426\pi$
$149$	5.70775e19	0.129180	0.0645900	+	0.997912i	$0.479426\pi$
$150$	0	0
$151$	−9.39406e20	−1.87315	−0.936574	−	0.350470i	$-0.886022\pi$
$151$	−9.39406e20	−1.87315	−0.936574	+	0.350470i	$0.886022\pi$
$152$	0	0
$153$	−2.90428e20	−0.511057
$154$	0	0
$155$	7.37971e20	1.14786
$156$	0	0
$157$	−2.91445e20	−0.401338	−0.200669	−	0.979659i	$-0.564312\pi$
$157$	−2.91445e20	−0.401338	−0.200669	+	0.979659i	$0.564312\pi$
$158$	0	0
$159$	−9.14651e20	−1.11682
$160$	0	0
$161$	5.36680e20	0.581928
$162$	0	0
$163$	−6.64161e20	−0.640459	−0.320229	−	0.947340i	$-0.603760\pi$
$163$	−6.64161e20	−0.640459	−0.320229	+	0.947340i	$0.603760\pi$
$164$	0	0
$165$	−2.49737e20	−0.214479
$166$	0	0
$167$	−9.84447e20	−0.754025	−0.377013	−	0.926208i	$-0.623049\pi$
$167$	−9.84447e20	−0.754025	−0.377013	+	0.926208i	$0.623049\pi$
$168$	0	0
$169$	8.88535e19	0.0607786
$170$	0	0
$171$	2.73359e20	0.167213
$172$	0	0
$173$	1.25238e21	0.685960	0.342980	−	0.939343i	$-0.388564\pi$
$173$	1.25238e21	0.685960	0.342980	+	0.939343i	$0.388564\pi$
$174$	0	0
$175$	−1.04237e21	−0.511872
$176$	0	0
$177$	−5.11134e20	−0.225314
$178$	0	0
$179$	4.38911e21	1.73889	0.869446	−	0.494029i	$-0.164476\pi$
$179$	4.38911e21	1.73889	0.869446	+	0.494029i	$0.164476\pi$
$180$	0	0
$181$	−6.83828e19	−0.0243781	−0.0121891	−	0.999926i	$-0.503880\pi$
$181$	−6.83828e19	−0.0243781	−0.0121891	+	0.999926i	$0.503880\pi$
$182$	0	0
$183$	4.60432e17	0.000147870 0
$184$	0	0
$185$	3.59246e21	1.04054
$186$	0	0
$187$	2.72940e21	0.713792
$188$	0	0
$189$	−1.14708e21	−0.271148
$190$	0	0
$191$	3.35651e21	0.717912	0.358956	−	0.933354i	$-0.383133\pi$
$191$	3.35651e21	0.717912	0.358956	+	0.933354i	$0.383133\pi$
$192$	0	0
$193$	1.69184e21	0.327767	0.163883	−	0.986480i	$-0.447598\pi$
$193$	1.69184e21	0.327767	0.163883	+	0.986480i	$0.447598\pi$
$194$	0	0
$195$	2.70113e21	0.474479
$196$	0	0
$197$	−9.30440e21	−1.48340	−0.741702	−	0.670730i	$-0.765981\pi$
$197$	−9.30440e21	−1.48340	−0.741702	+	0.670730i	$0.765981\pi$
$198$	0	0
$199$	−7.58115e20	−0.109807	−0.0549036	−	0.998492i	$-0.517485\pi$
$199$	−7.58115e20	−0.109807	−0.0549036	+	0.998492i	$0.517485\pi$
$200$	0	0
$201$	−2.21791e21	−0.292134
$202$	0	0
$203$	−1.72320e22	−2.06596
$204$	0	0
$205$	−1.07075e21	−0.116957
$206$	0	0
$207$	1.38223e21	0.137677
$208$	0	0
$209$	−2.56898e21	−0.233545
$210$	0	0
$211$	1.12497e22	0.934240	0.467120	−	0.884194i	$-0.345292\pi$
$211$	1.12497e22	0.934240	0.467120	+	0.884194i	$0.345292\pi$
$212$	0	0
$213$	1.19264e22	0.905532
$214$	0	0
$215$	2.36295e20	0.0164168
$216$	0	0
$217$	−3.18550e22	−2.02680
$218$	0	0
$219$	1.42245e21	0.0829502
$220$	0	0
$221$	−2.95209e22	−1.57908
$222$	0	0
$223$	1.20414e22	0.591263	0.295631	−	0.955302i	$-0.404470\pi$
$223$	1.20414e22	0.591263	0.295631	+	0.955302i	$0.404470\pi$
$224$	0	0
$225$	−2.68464e21	−0.121102
$226$	0	0
$227$	1.48331e22	0.615155	0.307578	−	0.951523i	$-0.400482\pi$
$227$	1.48331e22	0.615155	0.307578	+	0.951523i	$0.400482\pi$
$228$	0	0
$229$	3.25997e22	1.24387	0.621937	−	0.783068i	$-0.286346\pi$
$229$	3.25997e22	1.24387	0.621937	+	0.783068i	$0.286346\pi$
$230$	0	0
$231$	1.07801e22	0.378711
$232$	0	0
$233$	−2.09798e22	−0.679077	−0.339539	−	0.940592i	$-0.610271\pi$
$233$	−2.09798e22	−0.679077	−0.339539	+	0.940592i	$0.610271\pi$
$234$	0	0
$235$	3.32856e22	0.993373
$236$	0	0
$237$	2.99401e22	0.824410
$238$	0	0
$239$	4.90405e21	0.124674	0.0623368	−	0.998055i	$-0.480145\pi$
$239$	4.90405e21	0.124674	0.0623368	+	0.998055i	$0.480145\pi$
$240$	0	0
$241$	2.14629e22	0.504111	0.252055	−	0.967713i	$-0.418894\pi$
$241$	2.14629e22	0.504111	0.252055	+	0.967713i	$0.418894\pi$
$242$	0	0
$243$	−2.95431e21	−0.0641500
$244$	0	0
$245$	−3.91300e22	−0.786016
$246$	0	0
$247$	2.77859e22	0.516658
$248$	0	0
$249$	−2.23139e21	−0.0384311
$250$	0	0
$251$	8.08984e22	1.29134	0.645668	−	0.763618i	$-0.276579\pi$
$251$	8.08984e22	1.29134	0.645668	+	0.763618i	$0.276579\pi$
$252$	0	0
$253$	−1.29899e22	−0.192292
$254$	0	0
$255$	−5.14194e22	−0.706310
$256$	0	0
$257$	8.82149e22	1.12506	0.562532	−	0.826775i	$-0.309827\pi$
$257$	8.82149e22	1.12506	0.562532	+	0.826775i	$0.309827\pi$
$258$	0	0
$259$	−1.55071e23	−1.83731
$260$	0	0
$261$	−4.43811e22	−0.488780
$262$	0	0
$263$	−7.52166e22	−0.770432	−0.385216	−	0.922826i	$-0.625873\pi$
$263$	−7.52166e22	−0.770432	−0.385216	+	0.922826i	$0.625873\pi$
$264$	0	0
$265$	−1.61936e23	−1.54351
$266$	0	0
$267$	7.87055e22	0.698470
$268$	0	0
$269$	−5.72019e22	−0.472894	−0.236447	−	0.971644i	$-0.575983\pi$
$269$	−5.72019e22	−0.472894	−0.236447	+	0.971644i	$0.575983\pi$
$270$	0	0
$271$	3.15185e22	0.242861	0.121430	−	0.992600i	$-0.461252\pi$
$271$	3.15185e22	0.242861	0.121430	+	0.992600i	$0.461252\pi$
$272$	0	0
$273$	−1.16596e23	−0.837798
$274$	0	0
$275$	2.52298e22	0.169143
$276$	0	0
$277$	1.70323e23	1.06590	0.532950	−	0.846147i	$-0.321083\pi$
$277$	1.70323e23	1.06590	0.532950	+	0.846147i	$0.321083\pi$
$278$	0	0
$279$	−8.20430e22	−0.479514
$280$	0	0
$281$	2.56268e23	1.39954	0.699769	−	0.714369i	$-0.253286\pi$
$281$	2.56268e23	1.39954	0.699769	+	0.714369i	$0.253286\pi$
$282$	0	0
$283$	−3.22436e23	−1.64616	−0.823082	−	0.567923i	$-0.807747\pi$
$283$	−3.22436e23	−1.64616	−0.823082	+	0.567923i	$0.807747\pi$
$284$	0	0
$285$	4.83974e22	0.231098
$286$	0	0
$287$	4.62197e22	0.206514
$288$	0	0
$289$	3.22895e23	1.35062
$290$	0	0
$291$	−2.54919e23	−0.998663
$292$	0	0
$293$	3.97763e23	1.46010	0.730049	−	0.683395i	$-0.239497\pi$
$293$	3.97763e23	1.46010	0.730049	+	0.683395i	$0.239497\pi$
$294$	0	0
$295$	−9.04948e22	−0.311397
$296$	0	0
$297$	2.77642e22	0.0895980
$298$	0	0
$299$	1.40498e23	0.425397
$300$	0	0
$301$	−1.01999e22	−0.0289876
$302$	0	0
$303$	9.37783e22	0.250263
$304$	0	0
$305$	8.15182e19	0.000204365 0
$306$	0	0
$307$	−5.03215e22	−0.118560	−0.0592801	−	0.998241i	$-0.518881\pi$
$307$	−5.03215e22	−0.118560	−0.0592801	+	0.998241i	$0.518881\pi$
$308$	0	0
$309$	3.44627e23	0.763386
$310$	0	0
$311$	5.36483e23	1.11772	0.558859	−	0.829263i	$-0.311239\pi$
$311$	5.36483e23	1.11772	0.558859	+	0.829263i	$0.311239\pi$
$312$	0	0
$313$	1.19264e23	0.233796	0.116898	−	0.993144i	$-0.462705\pi$
$313$	1.19264e23	0.233796	0.116898	+	0.993144i	$0.462705\pi$
$314$	0	0
$315$	−2.03087e23	−0.374741
$316$	0	0
$317$	−1.12855e24	−1.96092	−0.980458	−	0.196726i	$-0.936969\pi$
$317$	−1.12855e24	−1.96092	−0.980458	+	0.196726i	$0.936969\pi$
$318$	0	0
$319$	4.17087e23	0.682677
$320$	0	0
$321$	5.60997e23	0.865292
$322$	0	0
$323$	−5.28939e23	−0.769098
$324$	0	0
$325$	−2.72884e23	−0.374184
$326$	0	0
$327$	−4.80450e23	−0.621505
$328$	0	0
$329$	−1.43680e24	−1.75402
$330$	0	0
$331$	−8.28009e23	−0.954266	−0.477133	−	0.878831i	$-0.658324\pi$
$331$	−8.28009e23	−0.954266	−0.477133	+	0.878831i	$0.658324\pi$
$332$	0	0
$333$	−3.99387e23	−0.434685
$334$	0	0
$335$	−3.92675e23	−0.403745
$336$	0	0
$337$	5.97855e23	0.580914	0.290457	−	0.956888i	$-0.406193\pi$
$337$	5.97855e23	0.580914	0.290457	+	0.956888i	$0.406193\pi$
$338$	0	0
$339$	9.02439e23	0.828936
$340$	0	0
$341$	7.71028e23	0.669735
$342$	0	0
$343$	−2.56026e22	−0.0210373
$344$	0	0
$345$	2.44719e23	0.190277
$346$	0	0
$347$	−6.69979e23	−0.493096	−0.246548	−	0.969131i	$-0.579296\pi$
$347$	−6.69979e23	−0.493096	−0.246548	+	0.969131i	$0.579296\pi$
$348$	0	0
$349$	1.12631e24	0.784904	0.392452	−	0.919772i	$-0.371627\pi$
$349$	1.12631e24	0.784904	0.392452	+	0.919772i	$0.371627\pi$
$350$	0	0
$351$	−3.00295e23	−0.198212
$352$	0	0
$353$	−2.01219e24	−1.25837	−0.629185	−	0.777255i	$-0.716611\pi$
$353$	−2.01219e24	−1.25837	−0.629185	+	0.777255i	$0.716611\pi$
$354$	0	0
$355$	2.11154e24	1.25150
$356$	0	0
$357$	2.21956e24	1.24715
$358$	0	0
$359$	−8.56528e22	−0.0456399	−0.0228199	−	0.999740i	$-0.507264\pi$
$359$	−8.56528e22	−0.0456399	−0.0228199	+	0.999740i	$0.507264\pi$
$360$	0	0
$361$	−1.48057e24	−0.748359
$362$	0	0
$363$	9.42871e23	0.452209
$364$	0	0
$365$	2.51840e23	0.114642
$366$	0	0
$367$	1.67458e24	0.723733	0.361866	−	0.932230i	$-0.382140\pi$
$367$	1.67458e24	0.723733	0.361866	+	0.932230i	$0.382140\pi$
$368$	0	0
$369$	1.19039e23	0.0488585
$370$	0	0
$371$	6.99010e24	2.72541
$372$	0	0
$373$	2.44335e23	0.0905214	0.0452607	−	0.998975i	$-0.485588\pi$
$373$	2.44335e23	0.0905214	0.0452607	+	0.998975i	$0.485588\pi$
$374$	0	0
$375$	−1.78359e24	−0.628056
$376$	0	0
$377$	−4.51118e24	−1.51024
$378$	0	0
$379$	−2.40816e24	−0.766679	−0.383339	−	0.923608i	$-0.625226\pi$
$379$	−2.40816e24	−0.766679	−0.383339	+	0.923608i	$0.625226\pi$
$380$	0	0
$381$	−3.59428e24	−1.08849
$382$	0	0
$383$	1.82640e24	0.526267	0.263134	−	0.964759i	$-0.415244\pi$
$383$	1.82640e24	0.526267	0.263134	+	0.964759i	$0.415244\pi$
$384$	0	0
$385$	1.90858e24	0.523400
$386$	0	0
$387$	−2.62698e22	−0.00685809
$388$	0	0
$389$	−7.79371e23	−0.193742	−0.0968709	−	0.995297i	$-0.530883\pi$
$389$	−7.79371e23	−0.193742	−0.0968709	+	0.995297i	$0.530883\pi$
$390$	0	0
$391$	−2.67456e24	−0.633247
$392$	0	0
$393$	1.48571e24	0.335124
$394$	0	0
$395$	5.30080e24	1.13938
$396$	0	0
$397$	5.48900e23	0.112456	0.0562281	−	0.998418i	$-0.482093\pi$
$397$	5.48900e23	0.112456	0.0562281	+	0.998418i	$0.482093\pi$
$398$	0	0
$399$	−2.08911e24	−0.408054
$400$	0	0
$401$	−8.59612e24	−1.60115	−0.800573	−	0.599235i	$-0.795471\pi$
$401$	−8.59612e24	−1.60115	−0.800573	+	0.599235i	$0.795471\pi$
$402$	0	0
$403$	−8.33937e24	−1.48161
$404$	0	0
$405$	−5.23053e23	−0.0886590
$406$	0	0
$407$	3.75338e24	0.607122
$408$	0	0
$409$	−7.77270e24	−1.20005	−0.600027	−	0.799980i	$-0.704843\pi$
$409$	−7.77270e24	−1.20005	−0.600027	+	0.799980i	$0.704843\pi$
$410$	0	0
$411$	−6.62396e24	−0.976384
$412$	0	0
$413$	3.90627e24	0.549841
$414$	0	0
$415$	−3.95062e23	−0.0531139
$416$	0	0
$417$	4.59859e24	0.590653
$418$	0	0
$419$	1.01515e25	1.24594	0.622970	−	0.782246i	$-0.285926\pi$
$419$	1.01515e25	1.24594	0.622970	+	0.782246i	$0.285926\pi$
$420$	0	0
$421$	2.06444e24	0.242171	0.121086	−	0.992642i	$-0.461362\pi$
$421$	2.06444e24	0.242171	0.121086	+	0.992642i	$0.461362\pi$
$422$	0	0
$423$	−3.70049e24	−0.414979
$424$	0	0
$425$	5.19468e24	0.557012
$426$	0	0
$427$	−3.51879e21	−0.000360852 0
$428$	0	0
$429$	2.82213e24	0.276842
$430$	0	0
$431$	1.33785e25	1.25566	0.627831	−	0.778350i	$-0.283943\pi$
$431$	1.33785e25	1.25566	0.627831	+	0.778350i	$0.283943\pi$
$432$	0	0
$433$	1.64121e25	1.47411	0.737055	−	0.675832i	$-0.236216\pi$
$433$	1.64121e25	1.47411	0.737055	+	0.675832i	$0.236216\pi$
$434$	0	0
$435$	−7.85755e24	−0.675522
$436$	0	0
$437$	2.51737e24	0.207192
$438$	0	0
$439$	−1.78811e24	−0.140923	−0.0704614	−	0.997515i	$-0.522447\pi$
$439$	−1.78811e24	−0.140923	−0.0704614	+	0.997515i	$0.522447\pi$
$440$	0	0
$441$	4.35023e24	0.328356
$442$	0	0
$443$	−8.77340e24	−0.634355	−0.317177	−	0.948366i	$-0.602735\pi$
$443$	−8.77340e24	−0.634355	−0.317177	+	0.948366i	$0.602735\pi$
$444$	0	0
$445$	1.39346e25	0.965325
$446$	0	0
$447$	−1.12346e24	−0.0745821
$448$	0	0
$449$	−1.21361e25	−0.772216	−0.386108	−	0.922454i	$-0.626181\pi$
$449$	−1.21361e25	−0.772216	−0.386108	+	0.922454i	$0.626181\pi$
$450$	0	0
$451$	−1.11871e24	−0.0682404
$452$	0	0
$453$	1.84903e25	1.08146
$454$	0	0
$455$	−2.06430e25	−1.15788
$456$	0	0
$457$	−3.76550e24	−0.202590	−0.101295	−	0.994856i	$-0.532299\pi$
$457$	−3.76550e24	−0.202590	−0.101295	+	0.994856i	$0.532299\pi$
$458$	0	0
$459$	5.71649e24	0.295059
$460$	0	0
$461$	−3.66029e24	−0.181283	−0.0906413	−	0.995884i	$-0.528892\pi$
$461$	−3.66029e24	−0.181283	−0.0906413	+	0.995884i	$0.528892\pi$
$462$	0	0
$463$	−1.50609e25	−0.715868	−0.357934	−	0.933747i	$-0.616519\pi$
$463$	−1.50609e25	−0.715868	−0.357934	+	0.933747i	$0.616519\pi$
$464$	0	0
$465$	−1.45255e25	−0.662716
$466$	0	0
$467$	−1.77658e25	−0.778169	−0.389084	−	0.921202i	$-0.627209\pi$
$467$	−1.77658e25	−0.778169	−0.389084	+	0.921202i	$0.627209\pi$
$468$	0	0
$469$	1.69501e25	0.712903
$470$	0	0
$471$	5.73651e24	0.231712
$472$	0	0
$473$	2.46880e23	0.00957867
$474$	0	0
$475$	−4.88938e24	−0.182249
$476$	0	0
$477$	1.80031e25	0.644796
$478$	0	0
$479$	1.35886e24	0.0467720	0.0233860	−	0.999727i	$-0.492555\pi$
$479$	1.35886e24	0.0467720	0.0233860	+	0.999727i	$0.492555\pi$
$480$	0	0
$481$	−4.05963e25	−1.34310
$482$	0	0
$483$	−1.05635e25	−0.335977
$484$	0	0
$485$	−4.51327e25	−1.38021
$486$	0	0
$487$	−3.13668e25	−0.922454	−0.461227	−	0.887282i	$-0.652591\pi$
$487$	−3.13668e25	−0.922454	−0.461227	+	0.887282i	$0.652591\pi$
$488$	0	0
$489$	1.30727e25	0.369769
$490$	0	0
$491$	−6.33012e24	−0.172242	−0.0861208	−	0.996285i	$-0.527447\pi$
$491$	−6.33012e24	−0.172242	−0.0861208	+	0.996285i	$0.527447\pi$
$492$	0	0
$493$	8.58759e25	2.24815
$494$	0	0
$495$	4.91557e24	0.123830
$496$	0	0
$497$	−9.11461e25	−2.20980
$498$	0	0
$499$	6.25147e25	1.45890	0.729452	−	0.684032i	$-0.239775\pi$
$499$	6.25147e25	1.45890	0.729452	+	0.684032i	$0.239775\pi$
$500$	0	0
$501$	1.93769e25	0.435337
$502$	0	0
$503$	4.61997e25	0.999409	0.499704	−	0.866196i	$-0.333442\pi$
$503$	4.61997e25	0.999409	0.499704	+	0.866196i	$0.333442\pi$
$504$	0	0
$505$	1.66032e25	0.345878
$506$	0	0
$507$	−1.74890e24	−0.0350906
$508$	0	0
$509$	3.66140e25	0.707666	0.353833	−	0.935309i	$-0.384878\pi$
$509$	3.66140e25	0.707666	0.353833	+	0.935309i	$0.384878\pi$
$510$	0	0
$511$	−1.08709e25	−0.202426
$512$	0	0
$513$	−5.38052e24	−0.0965404
$514$	0	0
$515$	6.10153e25	1.05504
$516$	0	0
$517$	3.47766e25	0.579599
$518$	0	0
$519$	−2.46506e25	−0.396039
$520$	0	0
$521$	9.36344e24	0.145036	0.0725182	−	0.997367i	$-0.476896\pi$
$521$	9.36344e24	0.145036	0.0725182	+	0.997367i	$0.476896\pi$
$522$	0	0
$523$	−9.29742e25	−1.38866	−0.694331	−	0.719656i	$-0.744300\pi$
$523$	−9.29742e25	−1.38866	−0.694331	+	0.719656i	$0.744300\pi$
$524$	0	0
$525$	2.05170e25	0.295529
$526$	0	0
$527$	1.58750e26	2.20553
$528$	0	0
$529$	−6.18865e25	−0.829406
$530$	0	0
$531$	1.00607e25	0.130085
$532$	0	0
$533$	1.20999e25	0.150964
$534$	0	0
$535$	9.93230e25	1.19588
$536$	0	0
$537$	−8.63908e25	−1.00395
$538$	0	0
$539$	−4.08828e25	−0.458614
$540$	0	0
$541$	−8.39302e25	−0.908959	−0.454480	−	0.890757i	$-0.650175\pi$
$541$	−8.39302e25	−0.908959	−0.454480	+	0.890757i	$0.650175\pi$
$542$	0	0
$543$	1.34598e24	0.0140747
$544$	0	0
$545$	−8.50623e25	−0.858955
$546$	0	0
$547$	−4.28049e25	−0.417459	−0.208730	−	0.977973i	$-0.566933\pi$
$547$	−4.28049e25	−0.417459	−0.208730	+	0.977973i	$0.566933\pi$
$548$	0	0
$549$	−9.06269e21	−8.53729e−5 0
$550$	0	0
$551$	−8.08288e25	−0.735573
$552$	0	0
$553$	−2.28813e26	−2.01183
$554$	0	0
$555$	−7.07104e25	−0.600759
$556$	0	0
$557$	1.78800e26	1.46806	0.734030	−	0.679117i	$-0.237637\pi$
$557$	1.78800e26	1.46806	0.734030	+	0.679117i	$0.237637\pi$
$558$	0	0
$559$	−2.67023e24	−0.0211903
$560$	0	0
$561$	−5.37227e25	−0.412108
$562$	0	0
$563$	−4.03205e25	−0.299017	−0.149509	−	0.988760i	$-0.547769\pi$
$563$	−4.03205e25	−0.299017	−0.149509	+	0.988760i	$0.547769\pi$
$564$	0	0
$565$	1.59774e26	1.14564
$566$	0	0
$567$	2.25779e25	0.156547
$568$	0	0
$569$	1.12186e26	0.752266	0.376133	−	0.926566i	$-0.377254\pi$
$569$	1.12186e26	0.752266	0.376133	+	0.926566i	$0.377254\pi$
$570$	0	0
$571$	1.86838e26	1.21178	0.605888	−	0.795550i	$-0.292818\pi$
$571$	1.86838e26	1.21178	0.605888	+	0.795550i	$0.292818\pi$
$572$	0	0
$573$	−6.60663e25	−0.414487
$574$	0	0
$575$	−2.47229e25	−0.150057
$576$	0	0
$577$	−2.22426e26	−1.30622	−0.653109	−	0.757264i	$-0.726536\pi$
$577$	−2.22426e26	−1.30622	−0.653109	+	0.757264i	$0.726536\pi$
$578$	0	0
$579$	−3.33005e25	−0.189236
$580$	0	0
$581$	1.70531e25	0.0937845
$582$	0	0
$583$	−1.69190e26	−0.900583
$584$	0	0
$585$	−5.31664e25	−0.273941
$586$	0	0
$587$	3.13380e26	1.56318	0.781591	−	0.623791i	$-0.214408\pi$
$587$	3.13380e26	1.56318	0.781591	+	0.623791i	$0.214408\pi$
$588$	0	0
$589$	−1.49420e26	−0.721629
$590$	0	0
$591$	1.83139e26	0.856444
$592$	0	0
$593$	−5.31661e25	−0.240777	−0.120389	−	0.992727i	$-0.538414\pi$
$593$	−5.31661e25	−0.240777	−0.120389	+	0.992727i	$0.538414\pi$
$594$	0	0
$595$	3.92966e26	1.72363
$596$	0	0
$597$	1.49220e25	0.0633972
$598$	0	0
$599$	−7.41542e25	−0.305197	−0.152599	−	0.988288i	$-0.548764\pi$
$599$	−7.41542e25	−0.305197	−0.152599	+	0.988288i	$0.548764\pi$
$600$	0	0
$601$	2.94986e26	1.17624	0.588118	−	0.808775i	$-0.299869\pi$
$601$	2.94986e26	1.17624	0.588118	+	0.808775i	$0.299869\pi$
$602$	0	0
$603$	4.36552e25	0.168664
$604$	0	0
$605$	1.66933e26	0.624978
$606$	0	0
$607$	−1.04165e26	−0.377947	−0.188973	−	0.981982i	$-0.560516\pi$
$607$	−1.04165e26	−0.377947	−0.188973	+	0.981982i	$0.560516\pi$
$608$	0	0
$609$	3.39177e26	1.19278
$610$	0	0
$611$	−3.76141e26	−1.28221
$612$	0	0
$613$	3.83965e26	1.26887	0.634435	−	0.772976i	$-0.281233\pi$
$613$	3.83965e26	1.26887	0.634435	+	0.772976i	$0.281233\pi$
$614$	0	0
$615$	2.10756e25	0.0675252
$616$	0	0
$617$	−3.05500e26	−0.949078	−0.474539	−	0.880234i	$-0.657385\pi$
$617$	−3.05500e26	−0.949078	−0.474539	+	0.880234i	$0.657385\pi$
$618$	0	0
$619$	2.47201e25	0.0744713	0.0372356	−	0.999307i	$-0.488145\pi$
$619$	2.47201e25	0.0744713	0.0372356	+	0.999307i	$0.488145\pi$
$620$	0	0
$621$	−2.72063e25	−0.0794877
$622$	0	0
$623$	−6.01496e26	−1.70450
$624$	0	0
$625$	−1.83609e26	−0.504702
$626$	0	0
$627$	5.05653e25	0.134838
$628$	0	0
$629$	7.72801e26	1.99934
$630$	0	0
$631$	−1.82411e26	−0.457901	−0.228951	−	0.973438i	$-0.573529\pi$
$631$	−1.82411e26	−0.457901	−0.228951	+	0.973438i	$0.573529\pi$
$632$	0	0
$633$	−2.21428e26	−0.539384
$634$	0	0
$635$	−6.36357e26	−1.50436
$636$	0	0
$637$	4.42184e26	1.01456
$638$	0	0
$639$	−2.34748e26	−0.522809
$640$	0	0
$641$	−5.78052e26	−1.24973	−0.624864	−	0.780734i	$-0.714845\pi$
$641$	−5.78052e26	−1.24973	−0.624864	+	0.780734i	$0.714845\pi$
$642$	0	0
$643$	−6.43157e26	−1.34993	−0.674967	−	0.737848i	$-0.735842\pi$
$643$	−6.43157e26	−1.34993	−0.674967	+	0.737848i	$0.735842\pi$
$644$	0	0
$645$	−4.65100e24	−0.00947827
$646$	0	0
$647$	−6.12415e26	−1.21187	−0.605933	−	0.795515i	$-0.707200\pi$
$647$	−6.12415e26	−1.21187	−0.605933	+	0.795515i	$0.707200\pi$
$648$	0	0
$649$	−9.45485e25	−0.181690
$650$	0	0
$651$	6.27003e26	1.17017
$652$	0	0
$653$	2.44526e26	0.443252	0.221626	−	0.975132i	$-0.428864\pi$
$653$	2.44526e26	0.443252	0.221626	+	0.975132i	$0.428864\pi$
$654$	0	0
$655$	2.63041e26	0.463161
$656$	0	0
$657$	−2.79981e25	−0.0478913
$658$	0	0
$659$	2.43875e26	0.405280	0.202640	−	0.979253i	$-0.435048\pi$
$659$	2.43875e26	0.405280	0.202640	+	0.979253i	$0.435048\pi$
$660$	0	0
$661$	9.59905e25	0.154994	0.0774969	−	0.996993i	$-0.475307\pi$
$661$	9.59905e25	0.154994	0.0774969	+	0.996993i	$0.475307\pi$
$662$	0	0
$663$	5.81060e26	0.911681
$664$	0	0
$665$	−3.69870e26	−0.563954
$666$	0	0
$667$	−4.08707e26	−0.605643
$668$	0	0
$669$	−2.37011e26	−0.341366
$670$	0	0
$671$	8.51698e22	0.000119240 0
$672$	0	0
$673$	−6.57497e25	−0.0894851	−0.0447426	−	0.998999i	$-0.514247\pi$
$673$	−6.57497e25	−0.0894851	−0.0447426	+	0.998999i	$0.514247\pi$
$674$	0	0
$675$	5.28418e25	0.0699184
$676$	0	0
$677$	1.04424e27	1.34341	0.671705	−	0.740819i	$-0.265562\pi$
$677$	1.04424e27	1.34341	0.671705	+	0.740819i	$0.265562\pi$
$678$	0	0
$679$	1.94819e27	2.43707
$680$	0	0
$681$	−2.91959e26	−0.355160
$682$	0	0
$683$	−3.79246e26	−0.448667	−0.224334	−	0.974512i	$-0.572021\pi$
$683$	−3.79246e26	−0.448667	−0.224334	+	0.974512i	$0.572021\pi$
$684$	0	0
$685$	−1.17275e27	−1.34942
$686$	0	0
$687$	−6.41659e26	−0.718150
$688$	0	0
$689$	1.82995e27	1.99230
$690$	0	0
$691$	3.43094e26	0.363389	0.181694	−	0.983355i	$-0.441842\pi$
$691$	3.43094e26	0.363389	0.181694	+	0.983355i	$0.441842\pi$
$692$	0	0
$693$	−2.12184e26	−0.218649
$694$	0	0
$695$	8.14167e26	0.816315
$696$	0	0
$697$	−2.30337e26	−0.224725
$698$	0	0
$699$	4.12945e26	0.392065
$700$	0	0
$701$	1.20243e27	1.11106	0.555531	−	0.831496i	$-0.312515\pi$
$701$	1.20243e27	1.11106	0.555531	+	0.831496i	$0.312515\pi$
$702$	0	0
$703$	−7.27381e26	−0.654164
$704$	0	0
$705$	−6.55161e26	−0.573524
$706$	0	0
$707$	−7.16688e26	−0.610725
$708$	0	0
$709$	5.73298e26	0.475599	0.237799	−	0.971314i	$-0.423574\pi$
$709$	5.73298e26	0.475599	0.237799	+	0.971314i	$0.423574\pi$
$710$	0	0
$711$	−5.89310e26	−0.475973
$712$	0	0
$713$	−7.55536e26	−0.594162
$714$	0	0
$715$	4.99649e26	0.382611
$716$	0	0
$717$	−9.65263e25	−0.0719803
$718$	0	0
$719$	6.44410e26	0.467991	0.233996	−	0.972238i	$-0.424820\pi$
$719$	6.44410e26	0.467991	0.233996	+	0.972238i	$0.424820\pi$
$720$	0	0
$721$	−2.63377e27	−1.86291
$722$	0	0
$723$	−4.22454e26	−0.291048
$724$	0	0
$725$	7.93815e26	0.532731
$726$	0	0
$727$	1.56438e27	1.02274	0.511369	−	0.859361i	$-0.329138\pi$
$727$	1.56438e27	1.02274	0.511369	+	0.859361i	$0.329138\pi$
$728$	0	0
$729$	5.81497e25	0.0370370
$730$	0	0
$731$	5.08312e25	0.0315439
$732$	0	0
$733$	−2.02196e27	−1.22260	−0.611301	−	0.791398i	$-0.709354\pi$
$733$	−2.02196e27	−1.22260	−0.611301	+	0.791398i	$0.709354\pi$
$734$	0	0
$735$	7.70195e26	0.453807
$736$	0	0
$737$	−4.10265e26	−0.235572
$738$	0	0
$739$	−1.84307e27	−1.03138	−0.515692	−	0.856774i	$-0.672465\pi$
$739$	−1.84307e27	−1.03138	−0.515692	+	0.856774i	$0.672465\pi$
$740$	0	0
$741$	−5.46910e26	−0.298293
$742$	0	0
$743$	−1.30251e26	−0.0692450	−0.0346225	−	0.999400i	$-0.511023\pi$
$743$	−1.30251e26	−0.0692450	−0.0346225	+	0.999400i	$0.511023\pi$
$744$	0	0
$745$	−1.98905e26	−0.103077
$746$	0	0
$747$	4.39205e25	0.0221882
$748$	0	0
$749$	−4.28735e27	−2.11160
$750$	0	0
$751$	−2.86173e27	−1.37420	−0.687098	−	0.726565i	$-0.741116\pi$
$751$	−2.86173e27	−1.37420	−0.687098	+	0.726565i	$0.741116\pi$
$752$	0	0
$753$	−1.59232e27	−0.745554
$754$	0	0
$755$	3.27366e27	1.49464
$756$	0	0
$757$	2.87305e26	0.127918	0.0639592	−	0.997953i	$-0.479627\pi$
$757$	2.87305e26	0.127918	0.0639592	+	0.997953i	$0.479627\pi$
$758$	0	0
$759$	2.55681e26	0.111020
$760$	0	0
$761$	2.89209e27	1.22478	0.612389	−	0.790556i	$-0.290209\pi$
$761$	2.89209e27	1.22478	0.612389	+	0.790556i	$0.290209\pi$
$762$	0	0
$763$	3.67178e27	1.51668
$764$	0	0
$765$	1.01209e27	0.407788
$766$	0	0
$767$	1.02263e27	0.401941
$768$	0	0
$769$	4.27802e27	1.64037	0.820187	−	0.572095i	$-0.193869\pi$
$769$	4.27802e27	1.64037	0.820187	+	0.572095i	$0.193869\pi$
$770$	0	0
$771$	−1.73633e27	−0.649556
$772$	0	0
$773$	−2.84915e27	−1.03994	−0.519972	−	0.854183i	$-0.674058\pi$
$773$	−2.84915e27	−1.03994	−0.519972	+	0.854183i	$0.674058\pi$
$774$	0	0
$775$	1.46745e27	0.522632
$776$	0	0
$777$	3.05227e27	1.06077
$778$	0	0
$779$	2.16799e26	0.0735279
$780$	0	0
$781$	2.20612e27	0.730205
$782$	0	0
$783$	8.73554e26	0.282197
$784$	0	0
$785$	1.01563e27	0.320240
$786$	0	0
$787$	−1.26950e27	−0.390725	−0.195363	−	0.980731i	$-0.562588\pi$
$787$	−1.26950e27	−0.390725	−0.195363	+	0.980731i	$0.562588\pi$
$788$	0	0
$789$	1.48049e27	0.444809
$790$	0	0
$791$	−6.89677e27	−2.02288
$792$	0	0
$793$	−9.21189e23	−0.000263787 0
$794$	0	0
$795$	3.18739e27	0.891144
$796$	0	0
$797$	−5.99147e27	−1.63561	−0.817804	−	0.575497i	$-0.804809\pi$
$797$	−5.99147e27	−1.63561	−0.817804	+	0.575497i	$0.804809\pi$
$798$	0	0
$799$	7.16032e27	1.90870
$800$	0	0
$801$	−1.54916e27	−0.403262
$802$	0	0
$803$	2.63121e26	0.0668896
$804$	0	0
$805$	−1.87023e27	−0.464339
$806$	0	0
$807$	1.12591e27	0.273026
$808$	0	0
$809$	6.44881e27	1.52746	0.763728	−	0.645538i	$-0.223367\pi$
$809$	6.44881e27	1.52746	0.763728	+	0.645538i	$0.223367\pi$
$810$	0	0
$811$	−5.12832e27	−1.18652	−0.593262	−	0.805009i	$-0.702160\pi$
$811$	−5.12832e27	−1.18652	−0.593262	+	0.805009i	$0.702160\pi$
$812$	0	0
$813$	−6.20379e26	−0.140216
$814$	0	0
$815$	2.31448e27	0.511042
$816$	0	0
$817$	−4.78437e25	−0.0103209
$818$	0	0
$819$	2.29496e27	0.483703
$820$	0	0
$821$	−9.20521e26	−0.189572	−0.0947859	−	0.995498i	$-0.530217\pi$
$821$	−9.20521e26	−0.189572	−0.0947859	+	0.995498i	$0.530217\pi$
$822$	0	0
$823$	−9.68332e27	−1.94862	−0.974308	−	0.225222i	$-0.927689\pi$
$823$	−9.68332e27	−1.94862	−0.974308	+	0.225222i	$0.927689\pi$
$824$	0	0
$825$	−4.96599e26	−0.0976547
$826$	0	0
$827$	−8.08592e27	−1.55391	−0.776957	−	0.629554i	$-0.783238\pi$
$827$	−8.08592e27	−1.55391	−0.776957	+	0.629554i	$0.783238\pi$
$828$	0	0
$829$	3.94960e27	0.741798	0.370899	−	0.928673i	$-0.379050\pi$
$829$	3.94960e27	0.741798	0.370899	+	0.928673i	$0.379050\pi$
$830$	0	0
$831$	−3.35247e27	−0.615397
$832$	0	0
$833$	−8.41753e27	−1.51028
$834$	0	0
$835$	3.43062e27	0.601660
$836$	0	0
$837$	1.61485e27	0.276848
$838$	0	0
$839$	2.22172e27	0.372350	0.186175	−	0.982517i	$-0.440391\pi$
$839$	2.22172e27	0.372350	0.186175	+	0.982517i	$0.440391\pi$
$840$	0	0
$841$	7.01969e27	1.15015
$842$	0	0
$843$	−5.04413e27	−0.808024
$844$	0	0
$845$	−3.09638e26	−0.0484971
$846$	0	0
$847$	−7.20576e27	−1.10354
$848$	0	0
$849$	6.34651e27	0.950413
$850$	0	0
$851$	−3.67797e27	−0.538614
$852$	0	0
$853$	−9.39541e26	−0.134555	−0.0672775	−	0.997734i	$-0.521431\pi$
$853$	−9.39541e26	−0.134555	−0.0672775	+	0.997734i	$0.521431\pi$
$854$	0	0
$855$	−9.52605e26	−0.133424
$856$	0	0
$857$	−6.22114e26	−0.0852221	−0.0426111	−	0.999092i	$-0.513568\pi$
$857$	−6.22114e26	−0.0852221	−0.0426111	+	0.999092i	$0.513568\pi$
$858$	0	0
$859$	5.86979e26	0.0786479	0.0393240	−	0.999227i	$-0.487480\pi$
$859$	5.86979e26	0.0786479	0.0393240	+	0.999227i	$0.487480\pi$
$860$	0	0
$861$	−9.09742e26	−0.119231
$862$	0	0
$863$	4.38488e27	0.562154	0.281077	−	0.959685i	$-0.409308\pi$
$863$	4.38488e27	0.562154	0.281077	+	0.959685i	$0.409308\pi$
$864$	0	0
$865$	−4.36433e27	−0.547349
$866$	0	0
$867$	−6.35555e27	−0.779779
$868$	0	0
$869$	5.53825e27	0.664790
$870$	0	0
$871$	4.43739e27	0.521140
$872$	0	0
$873$	5.01757e27	0.576579
$874$	0	0
$875$	1.36309e28	1.53266
$876$	0	0
$877$	2.71993e27	0.299269	0.149635	−	0.988741i	$-0.452190\pi$
$877$	2.71993e27	0.299269	0.149635	+	0.988741i	$0.452190\pi$
$878$	0	0
$879$	−7.82917e27	−0.842988
$880$	0	0
$881$	1.03761e28	1.09336	0.546680	−	0.837342i	$-0.315892\pi$
$881$	1.03761e28	1.09336	0.546680	+	0.837342i	$0.315892\pi$
$882$	0	0
$883$	−1.02514e28	−1.05720	−0.528600	−	0.848871i	$-0.677283\pi$
$883$	−1.02514e28	−1.05720	−0.528600	+	0.848871i	$0.677283\pi$
$884$	0	0
$885$	1.78121e27	0.179785
$886$	0	0
$887$	1.45231e28	1.43478	0.717390	−	0.696671i	$-0.245336\pi$
$887$	1.45231e28	1.43478	0.717390	+	0.696671i	$0.245336\pi$
$888$	0	0
$889$	2.74688e28	2.65628
$890$	0	0
$891$	−5.46482e26	−0.0517295
$892$	0	0
$893$	−6.73949e27	−0.624508
$894$	0	0
$895$	−1.52952e28	−1.38751
$896$	0	0
$897$	−2.76542e27	−0.245603
$898$	0	0
$899$	2.42591e28	2.10939
$900$	0	0
$901$	−3.48353e28	−2.96575
$902$	0	0
$903$	2.00764e26	0.0167360
$904$	0	0
$905$	2.38302e26	0.0194521
$906$	0	0
$907$	2.26606e28	1.81135	0.905673	−	0.423976i	$-0.139366\pi$
$907$	2.26606e28	1.81135	0.905673	+	0.423976i	$0.139366\pi$
$908$	0	0
$909$	−1.84584e27	−0.144490
$910$	0	0
$911$	5.19037e27	0.397899	0.198950	−	0.980010i	$-0.436247\pi$
$911$	5.19037e27	0.397899	0.198950	+	0.980010i	$0.436247\pi$
$912$	0	0
$913$	−4.12758e26	−0.0309901
$914$	0	0
$915$	−1.60452e24	−0.000117990 0
$916$	0	0
$917$	−1.13544e28	−0.817814
$918$	0	0
$919$	−1.02922e28	−0.726122	−0.363061	−	0.931765i	$-0.618268\pi$
$919$	−1.02922e28	−0.726122	−0.363061	+	0.931765i	$0.618268\pi$
$920$	0	0
$921$	9.90478e26	0.0684508
$922$	0	0
$923$	−2.38612e28	−1.61539
$924$	0	0
$925$	7.14357e27	0.473771
$926$	0	0
$927$	−6.78330e27	−0.440741
$928$	0	0
$929$	−3.52284e27	−0.224256	−0.112128	−	0.993694i	$-0.535767\pi$
$929$	−3.52284e27	−0.224256	−0.112128	+	0.993694i	$0.535767\pi$
$930$	0	0
$931$	7.92281e27	0.494148
$932$	0	0
$933$	−1.05596e28	−0.645315
$934$	0	0
$935$	−9.51145e27	−0.569556
$936$	0	0
$937$	−1.81523e28	−1.06514	−0.532569	−	0.846386i	$-0.678773\pi$
$937$	−1.81523e28	−1.06514	−0.532569	+	0.846386i	$0.678773\pi$
$938$	0	0
$939$	−2.34747e27	−0.134982
$940$	0	0
$941$	−1.62201e28	−0.914014	−0.457007	−	0.889463i	$-0.651079\pi$
$941$	−1.62201e28	−0.914014	−0.457007	+	0.889463i	$0.651079\pi$
$942$	0	0
$943$	1.09624e27	0.0605401
$944$	0	0
$945$	3.99736e27	0.216357
$946$	0	0
$947$	1.59756e28	0.847486	0.423743	−	0.905782i	$-0.360716\pi$
$947$	1.59756e28	0.847486	0.423743	+	0.905782i	$0.360716\pi$
$948$	0	0
$949$	−2.84590e27	−0.147976
$950$	0	0
$951$	2.22133e28	1.13214
$952$	0	0
$953$	−1.81675e28	−0.907637	−0.453818	−	0.891094i	$-0.649938\pi$
$953$	−1.81675e28	−0.907637	−0.453818	+	0.891094i	$0.649938\pi$
$954$	0	0
$955$	−1.16968e28	−0.572844
$956$	0	0
$957$	−8.20952e27	−0.394144
$958$	0	0
$959$	5.06228e28	2.38270
$960$	0	0
$961$	2.31747e28	1.06941
$962$	0	0
$963$	−1.10421e28	−0.499577
$964$	0	0
$965$	−5.89576e27	−0.261535
$966$	0	0
$967$	3.32693e27	0.144708	0.0723538	−	0.997379i	$-0.476949\pi$
$967$	3.32693e27	0.144708	0.0723538	+	0.997379i	$0.476949\pi$
$968$	0	0
$969$	1.04111e28	0.444039
$970$	0	0
$971$	−3.09409e28	−1.29405	−0.647023	−	0.762470i	$-0.723986\pi$
$971$	−3.09409e28	−1.29405	−0.647023	+	0.762470i	$0.723986\pi$
$972$	0	0
$973$	−3.51441e28	−1.44139
$974$	0	0
$975$	5.37117e27	0.216035
$976$	0	0
$977$	4.85749e25	0.00191608	0.000958041	−	1.00000i	$-0.499695\pi$
$977$	4.85749e25	0.00191608	0.000958041		1.00000i	$0.499695\pi$
$978$	0	0
$979$	1.45588e28	0.563234
$980$	0	0
$981$	9.45670e27	0.358826
$982$	0	0
$983$	−4.50372e28	−1.67615	−0.838075	−	0.545555i	$-0.816319\pi$
$983$	−4.50372e28	−1.67615	−0.838075	+	0.545555i	$0.816319\pi$
$984$	0	0
$985$	3.24242e28	1.18365
$986$	0	0
$987$	2.82805e28	1.01268
$988$	0	0
$989$	−2.41920e26	−0.00849780
$990$	0	0
$991$	2.24922e28	0.775056	0.387528	−	0.921858i	$-0.373329\pi$
$991$	2.24922e28	0.775056	0.387528	+	0.921858i	$0.373329\pi$
$992$	0	0
$993$	1.62977e28	0.550945
$994$	0	0
$995$	2.64189e27	0.0876185
$996$	0	0
$997$	5.43448e28	1.76829	0.884146	−	0.467210i	$-0.154741\pi$
$997$	5.43448e28	1.76829	0.884146	+	0.467210i	$0.154741\pi$
$998$	0	0
$999$	7.86114e27	0.250965

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.20.a.a.1.1	✓	2
3.2	odd	2		36.20.a.c.1.2		2
4.3	odd	2		48.20.a.i.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.20.a.a.1.1	✓	2	1.1	even	1	trivial
36.20.a.c.1.2		2	3.2	odd	2
48.20.a.i.1.1		2	4.3	odd	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:	\( N \)	\(=\)	\( 12 = 2^{2} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 20 \)
Character orbit:	\([\chi]\)	\(=\)	12.a (trivial)

\(f(q)\)	\(=\)	\(q-19683.0 q^{3} -3.48482e6 q^{5} +1.50425e8 q^{7} +3.87420e8 q^{9} +O(q^{10})\)	\(q-19683.0 q^{3} -3.48482e6 q^{5} +1.50425e8 q^{7} +3.87420e8 q^{9} -3.64092e9 q^{11} +3.93799e10 q^{13} +6.85917e10 q^{15} -7.49645e11 q^{17} +7.05586e11 q^{19} -2.96081e12 q^{21} +3.56777e12 q^{23} -6.92952e12 q^{25} -7.62560e12 q^{27} -1.14555e14 q^{29} -2.11767e14 q^{31} +7.16642e13 q^{33} -5.24203e14 q^{35} -1.03089e15 q^{37} -7.75114e14 q^{39} +3.07261e14 q^{41} -6.78071e13 q^{43} -1.35009e15 q^{45} -9.55161e15 q^{47} +1.12287e16 q^{49} +1.47553e16 q^{51} +4.64691e16 q^{53} +1.26879e16 q^{55} -1.38881e16 q^{57} +2.59683e16 q^{59} -2.33924e13 q^{61} +5.82776e16 q^{63} -1.37232e17 q^{65} +1.12682e17 q^{67} -7.02243e16 q^{69} -6.05925e17 q^{71} -7.22679e16 q^{73} +1.36394e17 q^{75} -5.47684e17 q^{77} -1.52111e18 q^{79} +1.50095e17 q^{81} +1.13367e17 q^{83} +2.61238e18 q^{85} +2.25479e18 q^{87} -3.99865e18 q^{89} +5.92370e18 q^{91} +4.16822e18 q^{93} -2.45884e18 q^{95} +1.29512e19 q^{97} -1.41057e18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 39366 q^{3} + 624780 q^{5} + 4667104 q^{7} + 774840978 q^{9}+O(q^{10}) \) 2 * q - 39366 * q^3 + 624780 * q^5 + 4667104 * q^7 + 774840978 * q^9	\( 2 q - 39366 q^{3} + 624780 q^{5} + 4667104 q^{7} + 774840978 q^{9} + 9015783192 q^{11} + 25948151500 q^{13} - 12297544740 q^{15} - 330949868796 q^{17} - 1555799112920 q^{19} - 91862608032 q^{21} - 2955794188464 q^{23} - 9114209254450 q^{25} - 15251194969974 q^{27} - 171112962219588 q^{29} - 439686204047792 q^{31} - 177457660568136 q^{33} - 11\!\cdots\!40 q^{35}+ \cdots + 34\!\cdots\!88 q^{99}+O(q^{100}) \) 2 * q - 39366 * q^3 + 624780 * q^5 + 4667104 * q^7 + 774840978 * q^9 + 9015783192 * q^11 + 25948151500 * q^13 - 12297544740 * q^15 - 330949868796 * q^17 - 1555799112920 * q^19 - 91862608032 * q^21 - 2955794188464 * q^23 - 9114209254450 * q^25 - 15251194969974 * q^27 - 171112962219588 * q^29 - 439686204047792 * q^31 - 177457660568136 * q^33 - 1123207990107840 * q^35 - 912841755316868 * q^37 - 510737465974500 * q^39 - 2039868884446380 * q^41 - 5607123893986664 * q^43 + 242052573117420 * q^45 - 720086631286464 * q^47 + 21075072481456722 * q^49 + 6514086267511668 * q^51 + 56983015881991692 * q^53 + 64701905169678480 * q^55 + 30622793939604360 * q^57 + 17959203991017432 * q^59 + 29244471161814604 * q^61 + 1808131713893856 * q^63 - 192430646530745400 * q^65 - 37110594569246744 * q^67 + 58178897011536912 * q^69 - 604502474341905360 * q^71 - 659663813215165772 * q^73 + 179394980755339350 * q^75 - 2392494322775596416 * q^77 - 352041739978389776 * q^79 + 300189270593998242 * q^81 - 2492903589286330584 * q^83 + 4333046841673556760 * q^85 + 3368016435368150604 * q^87 + 2523728600070118740 * q^89 + 7881478354364513600 * q^91 + 8654343554272689936 * q^93 - 11752227437069091600 * q^95 + 17246738574828858628 * q^97 + 3492899132962620888 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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