LMFDB - Embedded newform 392.6.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [392,6,Mod(1,392)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(392, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("392.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 392 = 2^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	392.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:	$62.8704573667$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{177}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 44 $ x^2 - x - 44
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-6.15207$ of defining polynomial
Character	$\chi$	$=$	392.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+26.3041 q^{3} +97.5207 q^{5} +448.908 q^{9} +O(q^{10})$	$q+26.3041 q^{3} +97.5207 q^{5} +448.908 q^{9} +406.175 q^{11} +905.594 q^{13} +2565.20 q^{15} -359.825 q^{17} -1813.25 q^{19} -2163.43 q^{23} +6385.28 q^{25} +5416.22 q^{27} -4090.26 q^{29} -4451.54 q^{31} +10684.1 q^{33} +8935.69 q^{37} +23820.9 q^{39} +3415.04 q^{41} -8694.80 q^{43} +43777.8 q^{45} -14204.5 q^{47} -9464.88 q^{51} -14616.4 q^{53} +39610.5 q^{55} -47696.1 q^{57} -21396.8 q^{59} -54081.9 q^{61} +88314.1 q^{65} +44973.1 q^{67} -56907.0 q^{69} +5700.89 q^{71} -47650.8 q^{73} +167959. q^{75} +5809.20 q^{79} +33384.4 q^{81} -27058.2 q^{83} -35090.4 q^{85} -107591. q^{87} +96245.2 q^{89} -117094. q^{93} -176830. q^{95} -101968. q^{97} +182335. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 26 q^{3} + 62 q^{5} + 206 q^{9}+O(q^{10}) $ 2 * q + 26 * q^3 + 62 * q^5 + 206 * q^9	$ 2 q + 26 q^{3} + 62 q^{5} + 206 q^{9} + 972 q^{11} - 78 q^{13} + 2576 q^{15} - 560 q^{17} - 2642 q^{19} + 2272 q^{23} + 4522 q^{25} + 5564 q^{27} - 7808 q^{29} - 5444 q^{31} + 10512 q^{33} + 576 q^{37} + 24120 q^{39} + 16888 q^{41} - 8396 q^{43} + 52406 q^{45} + 4532 q^{47} - 9404 q^{51} + 1420 q^{53} + 19512 q^{55} - 47444 q^{57} - 34146 q^{59} - 19106 q^{61} + 123252 q^{65} + 56952 q^{67} - 58256 q^{69} - 7224 q^{71} - 128828 q^{73} + 168526 q^{75} + 52808 q^{79} + 92366 q^{81} + 84486 q^{83} - 27980 q^{85} - 106460 q^{87} + 130972 q^{89} - 116792 q^{93} - 147392 q^{95} - 194624 q^{97} + 44892 q^{99}+O(q^{100}) $ 2 * q + 26 * q^3 + 62 * q^5 + 206 * q^9 + 972 * q^11 - 78 * q^13 + 2576 * q^15 - 560 * q^17 - 2642 * q^19 + 2272 * q^23 + 4522 * q^25 + 5564 * q^27 - 7808 * q^29 - 5444 * q^31 + 10512 * q^33 + 576 * q^37 + 24120 * q^39 + 16888 * q^41 - 8396 * q^43 + 52406 * q^45 + 4532 * q^47 - 9404 * q^51 + 1420 * q^53 + 19512 * q^55 - 47444 * q^57 - 34146 * q^59 - 19106 * q^61 + 123252 * q^65 + 56952 * q^67 - 58256 * q^69 - 7224 * q^71 - 128828 * q^73 + 168526 * q^75 + 52808 * q^79 + 92366 * q^81 + 84486 * q^83 - 27980 * q^85 - 106460 * q^87 + 130972 * q^89 - 116792 * q^93 - 147392 * q^95 - 194624 * q^97 + 44892 * 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
$3$	26.3041	1.68741	0.843706	−	0.536806i	$-0.180369\pi$
$3$	26.3041	1.68741	0.843706	+	0.536806i	$0.180369\pi$
$4$	0	0
$5$	97.5207	1.74450	0.872251	−	0.489058i	$-0.162659\pi$
$5$	97.5207	1.74450	0.872251	+	0.489058i	$0.162659\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	448.908	1.84736
$10$	0	0
$11$	406.175	1.01212	0.506060	−	0.862498i	$-0.331102\pi$
$11$	406.175	1.01212	0.506060	+	0.862498i	$0.331102\pi$
$12$	0	0
$13$	905.594	1.48619	0.743096	−	0.669185i	$-0.233357\pi$
$13$	905.594	1.48619	0.743096	+	0.669185i	$0.233357\pi$
$14$	0	0
$15$	2565.20	2.94369
$16$	0	0
$17$	−359.825	−0.301973	−0.150987	−	0.988536i	$-0.548245\pi$
$17$	−359.825	−0.301973	−0.150987	+	0.988536i	$0.548245\pi$
$18$	0	0
$19$	−1813.25	−1.15232	−0.576162	−	0.817336i	$-0.695450\pi$
$19$	−1813.25	−1.15232	−0.576162	+	0.817336i	$0.695450\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2163.43	−0.852751	−0.426376	−	0.904546i	$-0.640210\pi$
$23$	−2163.43	−0.852751	−0.426376	+	0.904546i	$0.640210\pi$
$24$	0	0
$25$	6385.28	2.04329
$26$	0	0
$27$	5416.22	1.42984
$28$	0	0
$29$	−4090.26	−0.903141	−0.451571	−	0.892235i	$-0.649136\pi$
$29$	−4090.26	−0.903141	−0.451571	+	0.892235i	$0.649136\pi$
$30$	0	0
$31$	−4451.54	−0.831966	−0.415983	−	0.909372i	$-0.636562\pi$
$31$	−4451.54	−0.831966	−0.415983	+	0.909372i	$0.636562\pi$
$32$	0	0
$33$	10684.1	1.70786
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8935.69	1.07306	0.536530	−	0.843882i	$-0.319735\pi$
$37$	8935.69	1.07306	0.536530	+	0.843882i	$0.319735\pi$
$38$	0	0
$39$	23820.9	2.50782
$40$	0	0
$41$	3415.04	0.317275	0.158637	−	0.987337i	$-0.449290\pi$
$41$	3415.04	0.317275	0.158637	+	0.987337i	$0.449290\pi$
$42$	0	0
$43$	−8694.80	−0.717114	−0.358557	−	0.933508i	$-0.616731\pi$
$43$	−8694.80	−0.717114	−0.358557	+	0.933508i	$0.616731\pi$
$44$	0	0
$45$	43777.8	3.22272
$46$	0	0
$47$	−14204.5	−0.937955	−0.468977	−	0.883210i	$-0.655377\pi$
$47$	−14204.5	−0.937955	−0.468977	+	0.883210i	$0.655377\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−9464.88	−0.509553
$52$	0	0
$53$	−14616.4	−0.714743	−0.357371	−	0.933962i	$-0.616327\pi$
$53$	−14616.4	−0.714743	−0.357371	+	0.933962i	$0.616327\pi$
$54$	0	0
$55$	39610.5	1.76564
$56$	0	0
$57$	−47696.1	−1.94444
$58$	0	0
$59$	−21396.8	−0.800239	−0.400119	−	0.916463i	$-0.631031\pi$
$59$	−21396.8	−0.800239	−0.400119	+	0.916463i	$0.631031\pi$
$60$	0	0
$61$	−54081.9	−1.86092	−0.930460	−	0.366394i	$-0.880592\pi$
$61$	−54081.9	−1.86092	−0.930460	+	0.366394i	$0.880592\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	88314.1	2.59267
$66$	0	0
$67$	44973.1	1.22396	0.611978	−	0.790874i	$-0.290374\pi$
$67$	44973.1	1.22396	0.611978	+	0.790874i	$0.290374\pi$
$68$	0	0
$69$	−56907.0	−1.43894
$70$	0	0
$71$	5700.89	0.134214	0.0671069	−	0.997746i	$-0.478623\pi$
$71$	5700.89	0.134214	0.0671069	+	0.997746i	$0.478623\pi$
$72$	0	0
$73$	−47650.8	−1.04656	−0.523279	−	0.852162i	$-0.675291\pi$
$73$	−47650.8	−1.04656	−0.523279	+	0.852162i	$0.675291\pi$
$74$	0	0
$75$	167959.	3.44787
$76$	0	0
$77$	0	0
$78$	0	0
$79$	5809.20	0.104725	0.0523623	−	0.998628i	$-0.483325\pi$
$79$	5809.20	0.104725	0.0523623	+	0.998628i	$0.483325\pi$
$80$	0	0
$81$	33384.4	0.565368
$82$	0	0
$83$	−27058.2	−0.431126	−0.215563	−	0.976490i	$-0.569159\pi$
$83$	−27058.2	−0.431126	−0.215563	+	0.976490i	$0.569159\pi$
$84$	0	0
$85$	−35090.4	−0.526794
$86$	0	0
$87$	−107591.	−1.52397
$88$	0	0
$89$	96245.2	1.28796	0.643982	−	0.765040i	$-0.277281\pi$
$89$	96245.2	1.28796	0.643982	+	0.765040i	$0.277281\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−117094.	−1.40387
$94$	0	0
$95$	−176830.	−2.01023
$96$	0	0
$97$	−101968.	−1.10036	−0.550182	−	0.835045i	$-0.685442\pi$
$97$	−101968.	−1.10036	−0.550182	+	0.835045i	$0.685442\pi$
$98$	0	0
$99$	182335.	1.86974
$100$	0	0
$101$	21856.8	0.213198	0.106599	−	0.994302i	$-0.466004\pi$
$101$	21856.8	0.213198	0.106599	+	0.994302i	$0.466004\pi$
$102$	0	0
$103$	136106.	1.26411	0.632056	−	0.774923i	$-0.282211\pi$
$103$	136106.	1.26411	0.632056	+	0.774923i	$0.282211\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	27394.7	0.231317	0.115658	−	0.993289i	$-0.463102\pi$
$107$	27394.7	0.231317	0.115658	+	0.993289i	$0.463102\pi$
$108$	0	0
$109$	88569.6	0.714033	0.357016	−	0.934098i	$-0.383794\pi$
$109$	88569.6	0.714033	0.357016	+	0.934098i	$0.383794\pi$
$110$	0	0
$111$	235046.	1.81069
$112$	0	0
$113$	18947.6	0.139591	0.0697955	−	0.997561i	$-0.477765\pi$
$113$	18947.6	0.139591	0.0697955	+	0.997561i	$0.477765\pi$
$114$	0	0
$115$	−210979.	−1.48763
$116$	0	0
$117$	406528.	2.74553
$118$	0	0
$119$	0	0
$120$	0	0
$121$	3927.29	0.0243854
$122$	0	0
$123$	89829.6	0.535373
$124$	0	0
$125$	317945.	1.82002
$126$	0	0
$127$	−188535.	−1.03725	−0.518625	−	0.855002i	$-0.673556\pi$
$127$	−188535.	−1.03725	−0.518625	+	0.855002i	$0.673556\pi$
$128$	0	0
$129$	−228709.	−1.21007
$130$	0	0
$131$	214161.	1.09034	0.545171	−	0.838325i	$-0.316465\pi$
$131$	214161.	1.09034	0.545171	+	0.838325i	$0.316465\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	528193.	2.49436
$136$	0	0
$137$	−44903.9	−0.204401	−0.102200	−	0.994764i	$-0.532588\pi$
$137$	−44903.9	−0.204401	−0.102200	+	0.994764i	$0.532588\pi$
$138$	0	0
$139$	−262924.	−1.15423	−0.577116	−	0.816662i	$-0.695822\pi$
$139$	−262924.	−1.15423	−0.577116	+	0.816662i	$0.695822\pi$
$140$	0	0
$141$	−373638.	−1.58272
$142$	0	0
$143$	367830.	1.50420
$144$	0	0
$145$	−398885.	−1.57553
$146$	0	0
$147$	0	0
$148$	0	0
$149$	209764.	0.774044	0.387022	−	0.922070i	$-0.373504\pi$
$149$	209764.	0.774044	0.387022	+	0.922070i	$0.373504\pi$
$150$	0	0
$151$	330665.	1.18017	0.590087	−	0.807340i	$-0.299093\pi$
$151$	330665.	1.18017	0.590087	+	0.807340i	$0.299093\pi$
$152$	0	0
$153$	−161528.	−0.557853
$154$	0	0
$155$	−434117.	−1.45137
$156$	0	0
$157$	510079.	1.65154	0.825769	−	0.564008i	$-0.190741\pi$
$157$	510079.	1.65154	0.825769	+	0.564008i	$0.190741\pi$
$158$	0	0
$159$	−384471.	−1.20606
$160$	0	0
$161$	0	0
$162$	0	0
$163$	138.797	0.000409176 0	0.000204588	−	1.00000i	$-0.499935\pi$
$163$	138.797	0.000409176 0	0.000204588		1.00000i	$0.499935\pi$
$164$	0	0
$165$	1.04192e6	2.97937
$166$	0	0
$167$	−527469.	−1.46354	−0.731772	−	0.681549i	$-0.761307\pi$
$167$	−527469.	−1.46354	−0.731772	+	0.681549i	$0.761307\pi$
$168$	0	0
$169$	448807.	1.20877
$170$	0	0
$171$	−813983.	−2.12875
$172$	0	0
$173$	345616.	0.877968	0.438984	−	0.898495i	$-0.355338\pi$
$173$	345616.	0.877968	0.438984	+	0.898495i	$0.355338\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−562825.	−1.35033
$178$	0	0
$179$	349527.	0.815358	0.407679	−	0.913125i	$-0.366338\pi$
$179$	349527.	0.815358	0.407679	+	0.913125i	$0.366338\pi$
$180$	0	0
$181$	469274.	1.06471	0.532353	−	0.846522i	$-0.321308\pi$
$181$	469274.	1.06471	0.532353	+	0.846522i	$0.321308\pi$
$182$	0	0
$183$	−1.42258e6	−3.14014
$184$	0	0
$185$	871414.	1.87195
$186$	0	0
$187$	−146152.	−0.305633
$188$	0	0
$189$	0	0
$190$	0	0
$191$	851324.	1.68854	0.844270	−	0.535918i	$-0.180034\pi$
$191$	851324.	1.68854	0.844270	+	0.535918i	$0.180034\pi$
$192$	0	0
$193$	122886.	0.237470	0.118735	−	0.992926i	$-0.462116\pi$
$193$	122886.	0.237470	0.118735	+	0.992926i	$0.462116\pi$
$194$	0	0
$195$	2.32303e6	4.37489
$196$	0	0
$197$	744847.	1.36742	0.683709	−	0.729755i	$-0.260366\pi$
$197$	744847.	1.36742	0.683709	+	0.729755i	$0.260366\pi$
$198$	0	0
$199$	−326738.	−0.584880	−0.292440	−	0.956284i	$-0.594467\pi$
$199$	−326738.	−0.584880	−0.292440	+	0.956284i	$0.594467\pi$
$200$	0	0
$201$	1.18298e6	2.06532
$202$	0	0
$203$	0	0
$204$	0	0
$205$	333037.	0.553487
$206$	0	0
$207$	−971178.	−1.57534
$208$	0	0
$209$	−736498.	−1.16629
$210$	0	0
$211$	336421.	0.520209	0.260104	−	0.965581i	$-0.416243\pi$
$211$	336421.	0.520209	0.260104	+	0.965581i	$0.416243\pi$
$212$	0	0
$213$	149957.	0.226474
$214$	0	0
$215$	−847923.	−1.25101
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−1.25341e6	−1.76597
$220$	0	0
$221$	−325855.	−0.448791
$222$	0	0
$223$	−633862.	−0.853556	−0.426778	−	0.904356i	$-0.640351\pi$
$223$	−633862.	−0.853556	−0.426778	+	0.904356i	$0.640351\pi$
$224$	0	0
$225$	2.86640e6	3.77468
$226$	0	0
$227$	297408.	0.383079	0.191539	−	0.981485i	$-0.438652\pi$
$227$	297408.	0.383079	0.191539	+	0.981485i	$0.438652\pi$
$228$	0	0
$229$	−594953.	−0.749711	−0.374855	−	0.927083i	$-0.622308\pi$
$229$	−594953.	−0.749711	−0.374855	+	0.927083i	$0.622308\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	150894.	0.182088	0.0910442	−	0.995847i	$-0.470980\pi$
$233$	150894.	0.182088	0.0910442	+	0.995847i	$0.470980\pi$
$234$	0	0
$235$	−1.38523e6	−1.63626
$236$	0	0
$237$	152806.	0.176713
$238$	0	0
$239$	−826500.	−0.935941	−0.467970	−	0.883744i	$-0.655015\pi$
$239$	−826500.	−0.935941	−0.467970	+	0.883744i	$0.655015\pi$
$240$	0	0
$241$	−1.36773e6	−1.51690	−0.758450	−	0.651731i	$-0.774043\pi$
$241$	−1.36773e6	−1.51690	−0.758450	+	0.651731i	$0.774043\pi$
$242$	0	0
$243$	−437993.	−0.475829
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.64207e6	−1.71257
$248$	0	0
$249$	−711744.	−0.727487
$250$	0	0
$251$	−1.73585e6	−1.73911	−0.869557	−	0.493832i	$-0.835596\pi$
$251$	−1.73585e6	−1.73911	−0.869557	+	0.493832i	$0.835596\pi$
$252$	0	0
$253$	−878730.	−0.863086
$254$	0	0
$255$	−923021.	−0.888917
$256$	0	0
$257$	2.04674e6	1.93299	0.966496	−	0.256680i	$-0.0826287\pi$
$257$	2.04674e6	1.93299	0.966496	+	0.256680i	$0.0826287\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.83615e6	−1.66842
$262$	0	0
$263$	−405737.	−0.361705	−0.180853	−	0.983510i	$-0.557886\pi$
$263$	−405737.	−0.361705	−0.180853	+	0.983510i	$0.557886\pi$
$264$	0	0
$265$	−1.42540e6	−1.24687
$266$	0	0
$267$	2.53165e6	2.17333
$268$	0	0
$269$	1.48037e6	1.24736	0.623678	−	0.781681i	$-0.285638\pi$
$269$	1.48037e6	1.24736	0.623678	+	0.781681i	$0.285638\pi$
$270$	0	0
$271$	−457477.	−0.378395	−0.189198	−	0.981939i	$-0.560589\pi$
$271$	−457477.	−0.378395	−0.189198	+	0.981939i	$0.560589\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.59354e6	2.06805
$276$	0	0
$277$	1.45830e6	1.14195	0.570977	−	0.820966i	$-0.306565\pi$
$277$	1.45830e6	1.14195	0.570977	+	0.820966i	$0.306565\pi$
$278$	0	0
$279$	−1.99833e6	−1.53694
$280$	0	0
$281$	−1.90172e6	−1.43675	−0.718373	−	0.695658i	$-0.755113\pi$
$281$	−1.90172e6	−1.43675	−0.718373	+	0.695658i	$0.755113\pi$
$282$	0	0
$283$	−1.01187e6	−0.751032	−0.375516	−	0.926816i	$-0.622535\pi$
$283$	−1.01187e6	−0.751032	−0.375516	+	0.926816i	$0.622535\pi$
$284$	0	0
$285$	−4.65135e6	−3.39209
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.29038e6	−0.908812
$290$	0	0
$291$	−2.68219e6	−1.85677
$292$	0	0
$293$	−1.30915e6	−0.890882	−0.445441	−	0.895311i	$-0.646953\pi$
$293$	−1.30915e6	−0.890882	−0.445441	+	0.895311i	$0.646953\pi$
$294$	0	0
$295$	−2.08663e6	−1.39602
$296$	0	0
$297$	2.19993e6	1.44717
$298$	0	0
$299$	−1.95918e6	−1.26735
$300$	0	0
$301$	0	0
$302$	0	0
$303$	574925.	0.359753
$304$	0	0
$305$	−5.27411e6	−3.24638
$306$	0	0
$307$	1.83705e6	1.11244	0.556219	−	0.831036i	$-0.312252\pi$
$307$	1.83705e6	1.11244	0.556219	+	0.831036i	$0.312252\pi$
$308$	0	0
$309$	3.58016e6	2.13307
$310$	0	0
$311$	−1.41953e6	−0.832233	−0.416116	−	0.909311i	$-0.636609\pi$
$311$	−1.41953e6	−0.832233	−0.416116	+	0.909311i	$0.636609\pi$
$312$	0	0
$313$	−2.67261e6	−1.54197	−0.770983	−	0.636855i	$-0.780235\pi$
$313$	−2.67261e6	−1.54197	−0.770983	+	0.636855i	$0.780235\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	34066.4	0.0190405	0.00952024	−	0.999955i	$-0.496970\pi$
$317$	34066.4	0.0190405	0.00952024	+	0.999955i	$0.496970\pi$
$318$	0	0
$319$	−1.66136e6	−0.914087
$320$	0	0
$321$	720595.	0.390327
$322$	0	0
$323$	652453.	0.347971
$324$	0	0
$325$	5.78247e6	3.03672
$326$	0	0
$327$	2.32975e6	1.20487
$328$	0	0
$329$	0	0
$330$	0	0
$331$	1.55050e6	0.777862	0.388931	−	0.921267i	$-0.372844\pi$
$331$	1.55050e6	0.777862	0.388931	+	0.921267i	$0.372844\pi$
$332$	0	0
$333$	4.01130e6	1.98232
$334$	0	0
$335$	4.38581e6	2.13520
$336$	0	0
$337$	−3.49728e6	−1.67747	−0.838736	−	0.544539i	$-0.816705\pi$
$337$	−3.49728e6	−1.67747	−0.838736	+	0.544539i	$0.816705\pi$
$338$	0	0
$339$	498399.	0.235547
$340$	0	0
$341$	−1.80810e6	−0.842049
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−5.54961e6	−2.51024
$346$	0	0
$347$	3.25824e6	1.45265	0.726323	−	0.687354i	$-0.241228\pi$
$347$	3.25824e6	1.45265	0.726323	+	0.687354i	$0.241228\pi$
$348$	0	0
$349$	1.40547e6	0.617672	0.308836	−	0.951115i	$-0.400060\pi$
$349$	1.40547e6	0.617672	0.308836	+	0.951115i	$0.400060\pi$
$350$	0	0
$351$	4.90489e6	2.12501
$352$	0	0
$353$	−2.37361e6	−1.01385	−0.506924	−	0.861991i	$-0.669217\pi$
$353$	−2.37361e6	−1.01385	−0.506924	+	0.861991i	$0.669217\pi$
$354$	0	0
$355$	555955.	0.234136
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−785602.	−0.321711	−0.160856	−	0.986978i	$-0.551425\pi$
$359$	−785602.	−0.321711	−0.160856	+	0.986978i	$0.551425\pi$
$360$	0	0
$361$	811787.	0.327849
$362$	0	0
$363$	103304.	0.0411481
$364$	0	0
$365$	−4.64694e6	−1.82572
$366$	0	0
$367$	2.62687e6	1.01806	0.509031	−	0.860748i	$-0.330004\pi$
$367$	2.62687e6	1.01806	0.509031	+	0.860748i	$0.330004\pi$
$368$	0	0
$369$	1.53304e6	0.586120
$370$	0	0
$371$	0	0
$372$	0	0
$373$	2.43475e6	0.906113	0.453056	−	0.891482i	$-0.350334\pi$
$373$	2.43475e6	0.906113	0.453056	+	0.891482i	$0.350334\pi$
$374$	0	0
$375$	8.36326e6	3.07113
$376$	0	0
$377$	−3.70411e6	−1.34224
$378$	0	0
$379$	83774.3	0.0299580	0.0149790	−	0.999888i	$-0.495232\pi$
$379$	83774.3	0.0299580	0.0149790	+	0.999888i	$0.495232\pi$
$380$	0	0
$381$	−4.95926e6	−1.75027
$382$	0	0
$383$	−1.75410e6	−0.611021	−0.305511	−	0.952189i	$-0.598827\pi$
$383$	−1.75410e6	−0.611021	−0.305511	+	0.952189i	$0.598827\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3.90316e6	−1.32477
$388$	0	0
$389$	3.75594e6	1.25848	0.629238	−	0.777213i	$-0.283367\pi$
$389$	3.75594e6	1.25848	0.629238	+	0.777213i	$0.283367\pi$
$390$	0	0
$391$	778454.	0.257508
$392$	0	0
$393$	5.63333e6	1.83985
$394$	0	0
$395$	566517.	0.182692
$396$	0	0
$397$	1.00266e6	0.319286	0.159643	−	0.987175i	$-0.448966\pi$
$397$	1.00266e6	0.319286	0.159643	+	0.987175i	$0.448966\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5.31342e6	1.65011	0.825056	−	0.565051i	$-0.191144\pi$
$401$	5.31342e6	1.65011	0.825056	+	0.565051i	$0.191144\pi$
$402$	0	0
$403$	−4.03128e6	−1.23646
$404$	0	0
$405$	3.25567e6	0.986286
$406$	0	0
$407$	3.62945e6	1.08606
$408$	0	0
$409$	4.58663e6	1.35577	0.677884	−	0.735169i	$-0.262897\pi$
$409$	4.58663e6	1.35577	0.677884	+	0.735169i	$0.262897\pi$
$410$	0	0
$411$	−1.18116e6	−0.344908
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−2.63874e6	−0.752101
$416$	0	0
$417$	−6.91599e6	−1.94767
$418$	0	0
$419$	−2.29852e6	−0.639607	−0.319803	−	0.947484i	$-0.603617\pi$
$419$	−2.29852e6	−0.639607	−0.319803	+	0.947484i	$0.603617\pi$
$420$	0	0
$421$	−5554.63	−0.00152739	−0.000763695	−	1.00000i	$-0.500243\pi$
$421$	−5554.63	−0.00152739	−0.000763695		1.00000i	$0.500243\pi$
$422$	0	0
$423$	−6.37652e6	−1.73274
$424$	0	0
$425$	−2.29758e6	−0.617019
$426$	0	0
$427$	0	0
$428$	0	0
$429$	9.67544e6	2.53821
$430$	0	0
$431$	−4.17192e6	−1.08179	−0.540895	−	0.841090i	$-0.681915\pi$
$431$	−4.17192e6	−1.08179	−0.540895	+	0.841090i	$0.681915\pi$
$432$	0	0
$433$	−1.68030e6	−0.430693	−0.215347	−	0.976538i	$-0.569088\pi$
$433$	−1.68030e6	−0.430693	−0.215347	+	0.976538i	$0.569088\pi$
$434$	0	0
$435$	−1.04923e7	−2.65857
$436$	0	0
$437$	3.92284e6	0.982645
$438$	0	0
$439$	2.12899e6	0.527245	0.263623	−	0.964626i	$-0.415083\pi$
$439$	2.12899e6	0.527245	0.263623	+	0.964626i	$0.415083\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5.46150e6	−1.32222	−0.661108	−	0.750291i	$-0.729913\pi$
$443$	−5.46150e6	−1.32222	−0.661108	+	0.750291i	$0.729913\pi$
$444$	0	0
$445$	9.38589e6	2.24686
$446$	0	0
$447$	5.51767e6	1.30613
$448$	0	0
$449$	1.17382e6	0.274779	0.137390	−	0.990517i	$-0.456129\pi$
$449$	1.17382e6	0.274779	0.137390	+	0.990517i	$0.456129\pi$
$450$	0	0
$451$	1.38710e6	0.321120
$452$	0	0
$453$	8.69786e6	1.99144
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.88231e6	0.869561	0.434780	−	0.900537i	$-0.356826\pi$
$457$	3.88231e6	0.869561	0.434780	+	0.900537i	$0.356826\pi$
$458$	0	0
$459$	−1.94889e6	−0.431773
$460$	0	0
$461$	1.13112e6	0.247889	0.123945	−	0.992289i	$-0.460445\pi$
$461$	1.13112e6	0.247889	0.123945	+	0.992289i	$0.460445\pi$
$462$	0	0
$463$	−1.74204e6	−0.377664	−0.188832	−	0.982009i	$-0.560470\pi$
$463$	−1.74204e6	−0.377664	−0.188832	+	0.982009i	$0.560470\pi$
$464$	0	0
$465$	−1.14191e7	−2.44905
$466$	0	0
$467$	−4.96811e6	−1.05414	−0.527071	−	0.849821i	$-0.676710\pi$
$467$	−4.96811e6	−1.05414	−0.527071	+	0.849821i	$0.676710\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	1.34172e7	2.78682
$472$	0	0
$473$	−3.53161e6	−0.725805
$474$	0	0
$475$	−1.15781e7	−2.35453
$476$	0	0
$477$	−6.56140e6	−1.32038
$478$	0	0
$479$	−1.72949e6	−0.344412	−0.172206	−	0.985061i	$-0.555090\pi$
$479$	−1.72949e6	−0.344412	−0.172206	+	0.985061i	$0.555090\pi$
$480$	0	0
$481$	8.09210e6	1.59477
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−9.94403e6	−1.91959
$486$	0	0
$487$	−3.90261e6	−0.745647	−0.372824	−	0.927902i	$-0.621610\pi$
$487$	−3.90261e6	−0.745647	−0.372824	+	0.927902i	$0.621610\pi$
$488$	0	0
$489$	3650.93	0.000690448 0
$490$	0	0
$491$	−8.62067e6	−1.61375	−0.806877	−	0.590720i	$-0.798844\pi$
$491$	−8.62067e6	−1.61375	−0.806877	+	0.590720i	$0.798844\pi$
$492$	0	0
$493$	1.47178e6	0.272725
$494$	0	0
$495$	1.77814e7	3.26177
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−7.01787e6	−1.26169	−0.630846	−	0.775908i	$-0.717292\pi$
$499$	−7.01787e6	−1.26169	−0.630846	+	0.775908i	$0.717292\pi$
$500$	0	0
$501$	−1.38746e7	−2.46960
$502$	0	0
$503$	3.89602e6	0.686596	0.343298	−	0.939226i	$-0.388456\pi$
$503$	3.89602e6	0.686596	0.343298	+	0.939226i	$0.388456\pi$
$504$	0	0
$505$	2.13149e6	0.371925
$506$	0	0
$507$	1.18055e7	2.03969
$508$	0	0
$509$	−1.54805e6	−0.264843	−0.132422	−	0.991193i	$-0.542275\pi$
$509$	−1.54805e6	−0.264843	−0.132422	+	0.991193i	$0.542275\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−9.82097e6	−1.64764
$514$	0	0
$515$	1.32732e7	2.20525
$516$	0	0
$517$	−5.76952e6	−0.949322
$518$	0	0
$519$	9.09113e6	1.48149
$520$	0	0
$521$	−7.11069e6	−1.14767	−0.573836	−	0.818970i	$-0.694545\pi$
$521$	−7.11069e6	−1.14767	−0.573836	+	0.818970i	$0.694545\pi$
$522$	0	0
$523$	−5.37329e6	−0.858986	−0.429493	−	0.903070i	$-0.641308\pi$
$523$	−5.37329e6	−0.858986	−0.429493	+	0.903070i	$0.641308\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.60177e6	0.251232
$528$	0	0
$529$	−1.75593e6	−0.272815
$530$	0	0
$531$	−9.60520e6	−1.47833
$532$	0	0
$533$	3.09264e6	0.471532
$534$	0	0
$535$	2.67155e6	0.403533
$536$	0	0
$537$	9.19401e6	1.37584
$538$	0	0
$539$	0	0
$540$	0	0
$541$	5.26079e6	0.772784	0.386392	−	0.922335i	$-0.373721\pi$
$541$	5.26079e6	0.772784	0.386392	+	0.922335i	$0.373721\pi$
$542$	0	0
$543$	1.23438e7	1.79660
$544$	0	0
$545$	8.63736e6	1.24563
$546$	0	0
$547$	4.26833e6	0.609943	0.304972	−	0.952361i	$-0.401353\pi$
$547$	4.26833e6	0.609943	0.304972	+	0.952361i	$0.401353\pi$
$548$	0	0
$549$	−2.42778e7	−3.43778
$550$	0	0
$551$	7.41667e6	1.04071
$552$	0	0
$553$	0	0
$554$	0	0
$555$	2.29218e7	3.15876
$556$	0	0
$557$	−5.28071e6	−0.721198	−0.360599	−	0.932721i	$-0.617428\pi$
$557$	−5.28071e6	−0.721198	−0.360599	+	0.932721i	$0.617428\pi$
$558$	0	0
$559$	−7.87395e6	−1.06577
$560$	0	0
$561$	−3.84440e6	−0.515729
$562$	0	0
$563$	−8.98229e6	−1.19431	−0.597154	−	0.802127i	$-0.703702\pi$
$563$	−8.98229e6	−1.19431	−0.597154	+	0.802127i	$0.703702\pi$
$564$	0	0
$565$	1.84778e6	0.243517
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.06571e7	1.37993	0.689966	−	0.723841i	$-0.257625\pi$
$569$	1.06571e7	1.37993	0.689966	+	0.723841i	$0.257625\pi$
$570$	0	0
$571$	−3.52812e6	−0.452849	−0.226425	−	0.974029i	$-0.572704\pi$
$571$	−3.52812e6	−0.452849	−0.226425	+	0.974029i	$0.572704\pi$
$572$	0	0
$573$	2.23933e7	2.84926
$574$	0	0
$575$	−1.38141e7	−1.74242
$576$	0	0
$577$	−2.63312e6	−0.329254	−0.164627	−	0.986356i	$-0.552642\pi$
$577$	−2.63312e6	−0.329254	−0.164627	+	0.986356i	$0.552642\pi$
$578$	0	0
$579$	3.23241e6	0.400709
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−5.93680e6	−0.723405
$584$	0	0
$585$	3.96449e7	4.78958
$586$	0	0
$587$	1.04984e7	1.25755	0.628776	−	0.777586i	$-0.283556\pi$
$587$	1.04984e7	1.25755	0.628776	+	0.777586i	$0.283556\pi$
$588$	0	0
$589$	8.07176e6	0.958694
$590$	0	0
$591$	1.95925e7	2.30740
$592$	0	0
$593$	−4.46953e6	−0.521946	−0.260973	−	0.965346i	$-0.584043\pi$
$593$	−4.46953e6	−0.521946	−0.260973	+	0.965346i	$0.584043\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−8.59455e6	−0.986933
$598$	0	0
$599$	8.44304e6	0.961461	0.480731	−	0.876868i	$-0.340371\pi$
$599$	8.44304e6	0.961461	0.480731	+	0.876868i	$0.340371\pi$
$600$	0	0
$601$	1.38045e7	1.55896	0.779481	−	0.626426i	$-0.215483\pi$
$601$	1.38045e7	1.55896	0.779481	+	0.626426i	$0.215483\pi$
$602$	0	0
$603$	2.01888e7	2.26108
$604$	0	0
$605$	382992.	0.0425403
$606$	0	0
$607$	6.43837e6	0.709258	0.354629	−	0.935007i	$-0.384607\pi$
$607$	6.43837e6	0.709258	0.354629	+	0.935007i	$0.384607\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.28635e7	−1.39398
$612$	0	0
$613$	−1.29974e6	−0.139703	−0.0698513	−	0.997557i	$-0.522252\pi$
$613$	−1.29974e6	−0.139703	−0.0698513	+	0.997557i	$0.522252\pi$
$614$	0	0
$615$	8.76024e6	0.933960
$616$	0	0
$617$	−5.51795e6	−0.583532	−0.291766	−	0.956490i	$-0.594243\pi$
$617$	−5.51795e6	−0.583532	−0.291766	+	0.956490i	$0.594243\pi$
$618$	0	0
$619$	9.36773e6	0.982671	0.491335	−	0.870971i	$-0.336509\pi$
$619$	9.36773e6	0.982671	0.491335	+	0.870971i	$0.336509\pi$
$620$	0	0
$621$	−1.17176e7	−1.21930
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.10522e7	1.13174
$626$	0	0
$627$	−1.93730e7	−1.96801
$628$	0	0
$629$	−3.21528e6	−0.324035
$630$	0	0
$631$	7.50281e6	0.750154	0.375077	−	0.926994i	$-0.377616\pi$
$631$	7.50281e6	0.750154	0.375077	+	0.926994i	$0.377616\pi$
$632$	0	0
$633$	8.84927e6	0.877806
$634$	0	0
$635$	−1.83861e7	−1.80948
$636$	0	0
$637$	0	0
$638$	0	0
$639$	2.55917e6	0.247941
$640$	0	0
$641$	112701.	0.0108338	0.00541690	−	0.999985i	$-0.498276\pi$
$641$	112701.	0.0108338	0.00541690	+	0.999985i	$0.498276\pi$
$642$	0	0
$643$	−453508.	−0.0432572	−0.0216286	−	0.999766i	$-0.506885\pi$
$643$	−453508.	−0.0432572	−0.0216286	+	0.999766i	$0.506885\pi$
$644$	0	0
$645$	−2.23039e7	−2.11096
$646$	0	0
$647$	−7.71032e6	−0.724122	−0.362061	−	0.932154i	$-0.617927\pi$
$647$	−7.71032e6	−0.724122	−0.362061	+	0.932154i	$0.617927\pi$
$648$	0	0
$649$	−8.69087e6	−0.809937
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−5.98665e6	−0.549416	−0.274708	−	0.961528i	$-0.588581\pi$
$653$	−5.98665e6	−0.549416	−0.274708	+	0.961528i	$0.588581\pi$
$654$	0	0
$655$	2.08851e7	1.90210
$656$	0	0
$657$	−2.13908e7	−1.93336
$658$	0	0
$659$	−1.44274e7	−1.29412	−0.647058	−	0.762441i	$-0.724001\pi$
$659$	−1.44274e7	−1.29412	−0.647058	+	0.762441i	$0.724001\pi$
$660$	0	0
$661$	1.80369e7	1.60568	0.802838	−	0.596197i	$-0.203322\pi$
$661$	1.80369e7	1.60568	0.802838	+	0.596197i	$0.203322\pi$
$662$	0	0
$663$	−8.57133e6	−0.757294
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8.84897e6	0.770155
$668$	0	0
$669$	−1.66732e7	−1.44030
$670$	0	0
$671$	−2.19667e7	−1.88347
$672$	0	0
$673$	−1.58113e7	−1.34564	−0.672820	−	0.739806i	$-0.734917\pi$
$673$	−1.58113e7	−1.34564	−0.672820	+	0.739806i	$0.734917\pi$
$674$	0	0
$675$	3.45841e7	2.92157
$676$	0	0
$677$	1.27877e7	1.07231	0.536154	−	0.844120i	$-0.319876\pi$
$677$	1.27877e7	1.07231	0.536154	+	0.844120i	$0.319876\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	7.82306e6	0.646411
$682$	0	0
$683$	−4.17491e6	−0.342448	−0.171224	−	0.985232i	$-0.554772\pi$
$683$	−4.17491e6	−0.342448	−0.171224	+	0.985232i	$0.554772\pi$
$684$	0	0
$685$	−4.37906e6	−0.356578
$686$	0	0
$687$	−1.56497e7	−1.26507
$688$	0	0
$689$	−1.32365e7	−1.06224
$690$	0	0
$691$	8.31609e6	0.662558	0.331279	−	0.943533i	$-0.392520\pi$
$691$	8.31609e6	0.662558	0.331279	+	0.943533i	$0.392520\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.56405e7	−2.01356
$696$	0	0
$697$	−1.22882e6	−0.0958086
$698$	0	0
$699$	3.96914e6	0.307258
$700$	0	0
$701$	−1.67718e6	−0.128909	−0.0644546	−	0.997921i	$-0.520531\pi$
$701$	−1.67718e6	−0.128909	−0.0644546	+	0.997921i	$0.520531\pi$
$702$	0	0
$703$	−1.62027e7	−1.23651
$704$	0	0
$705$	−3.64374e7	−2.76105
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1.66093e7	1.24090	0.620450	−	0.784246i	$-0.286950\pi$
$709$	1.66093e7	1.24090	0.620450	+	0.784246i	$0.286950\pi$
$710$	0	0
$711$	2.60779e6	0.193464
$712$	0	0
$713$	9.63057e6	0.709460
$714$	0	0
$715$	3.58710e7	2.62409
$716$	0	0
$717$	−2.17404e7	−1.57932
$718$	0	0
$719$	2.32802e7	1.67944	0.839722	−	0.543017i	$-0.182718\pi$
$719$	2.32802e7	1.67944	0.839722	+	0.543017i	$0.182718\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−3.59769e7	−2.55963
$724$	0	0
$725$	−2.61174e7	−1.84538
$726$	0	0
$727$	1.74527e7	1.22469	0.612345	−	0.790591i	$-0.290226\pi$
$727$	1.74527e7	1.22469	0.612345	+	0.790591i	$0.290226\pi$
$728$	0	0
$729$	−1.96334e7	−1.36829
$730$	0	0
$731$	3.12860e6	0.216549
$732$	0	0
$733$	−940675.	−0.0646666	−0.0323333	−	0.999477i	$-0.510294\pi$
$733$	−940675.	−0.0646666	−0.0323333	+	0.999477i	$0.510294\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.82670e7	1.23879
$738$	0	0
$739$	1.13900e7	0.767204	0.383602	−	0.923498i	$-0.374683\pi$
$739$	1.13900e7	0.767204	0.383602	+	0.923498i	$0.374683\pi$
$740$	0	0
$741$	−4.31932e7	−2.88982
$742$	0	0
$743$	262087.	0.0174170	0.00870851	−	0.999962i	$-0.497228\pi$
$743$	262087.	0.0174170	0.00870851	+	0.999962i	$0.497228\pi$
$744$	0	0
$745$	2.04563e7	1.35032
$746$	0	0
$747$	−1.21466e7	−0.796443
$748$	0	0
$749$	0	0
$750$	0	0
$751$	1.10676e6	0.0716069	0.0358035	−	0.999359i	$-0.488601\pi$
$751$	1.10676e6	0.0716069	0.0358035	+	0.999359i	$0.488601\pi$
$752$	0	0
$753$	−4.56601e7	−2.93460
$754$	0	0
$755$	3.22467e7	2.05882
$756$	0	0
$757$	3.07218e7	1.94853	0.974264	−	0.225412i	$-0.0723728\pi$
$757$	3.07218e7	1.94853	0.974264	+	0.225412i	$0.0723728\pi$
$758$	0	0
$759$	−2.31142e7	−1.45638
$760$	0	0
$761$	−1.18635e7	−0.742593	−0.371297	−	0.928514i	$-0.621087\pi$
$761$	−1.18635e7	−0.742593	−0.371297	+	0.928514i	$0.621087\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−1.57523e7	−0.973175
$766$	0	0
$767$	−1.93768e7	−1.18931
$768$	0	0
$769$	−2.08006e7	−1.26841	−0.634204	−	0.773166i	$-0.718672\pi$
$769$	−2.08006e7	−1.26841	−0.634204	+	0.773166i	$0.718672\pi$
$770$	0	0
$771$	5.38378e7	3.26175
$772$	0	0
$773$	−1.12633e7	−0.677980	−0.338990	−	0.940790i	$-0.610085\pi$
$773$	−1.12633e7	−0.677980	−0.338990	+	0.940790i	$0.610085\pi$
$774$	0	0
$775$	−2.84243e7	−1.69995
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6.19233e6	−0.365603
$780$	0	0
$781$	2.31556e6	0.135840
$782$	0	0
$783$	−2.21537e7	−1.29135
$784$	0	0
$785$	4.97433e7	2.88111
$786$	0	0
$787$	−2.65941e7	−1.53056	−0.765278	−	0.643700i	$-0.777399\pi$
$787$	−2.65941e7	−1.53056	−0.765278	+	0.643700i	$0.777399\pi$
$788$	0	0
$789$	−1.06726e7	−0.610346
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−4.89763e7	−2.76568
$794$	0	0
$795$	−3.74938e7	−2.10398
$796$	0	0
$797$	−1.25159e7	−0.697938	−0.348969	−	0.937134i	$-0.613468\pi$
$797$	−1.25159e7	−0.697938	−0.348969	+	0.937134i	$0.613468\pi$
$798$	0	0
$799$	5.11114e6	0.283237
$800$	0	0
$801$	4.32052e7	2.37933
$802$	0	0
$803$	−1.93546e7	−1.05924
$804$	0	0
$805$	0	0
$806$	0	0
$807$	3.89399e7	2.10480
$808$	0	0
$809$	1.02591e7	0.551109	0.275555	−	0.961285i	$-0.411139\pi$
$809$	1.02591e7	0.551109	0.275555	+	0.961285i	$0.411139\pi$
$810$	0	0
$811$	−1.32031e7	−0.704892	−0.352446	−	0.935832i	$-0.614650\pi$
$811$	−1.32031e7	−0.704892	−0.352446	+	0.935832i	$0.614650\pi$
$812$	0	0
$813$	−1.20335e7	−0.638508
$814$	0	0
$815$	13535.5	0.000713809 0
$816$	0	0
$817$	1.57659e7	0.826348
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.88139e7	1.49192	0.745958	−	0.665993i	$-0.231992\pi$
$821$	2.88139e7	1.49192	0.745958	+	0.665993i	$0.231992\pi$
$822$	0	0
$823$	2.72097e7	1.40031	0.700156	−	0.713990i	$-0.253114\pi$
$823$	2.72097e7	1.40031	0.700156	+	0.713990i	$0.253114\pi$
$824$	0	0
$825$	6.82209e7	3.48966
$826$	0	0
$827$	3.36307e7	1.70991	0.854954	−	0.518705i	$-0.173586\pi$
$827$	3.36307e7	1.70991	0.854954	+	0.518705i	$0.173586\pi$
$828$	0	0
$829$	3.30835e7	1.67196	0.835978	−	0.548763i	$-0.184901\pi$
$829$	3.30835e7	1.67196	0.835978	+	0.548763i	$0.184901\pi$
$830$	0	0
$831$	3.83594e7	1.92695
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−5.14392e7	−2.55316
$836$	0	0
$837$	−2.41105e7	−1.18958
$838$	0	0
$839$	−1.85094e7	−0.907796	−0.453898	−	0.891054i	$-0.649967\pi$
$839$	−1.85094e7	−0.907796	−0.453898	+	0.891054i	$0.649967\pi$
$840$	0	0
$841$	−3.78094e6	−0.184336
$842$	0	0
$843$	−5.00230e7	−2.42438
$844$	0	0
$845$	4.37679e7	2.10870
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−2.66164e7	−1.26730
$850$	0	0
$851$	−1.93317e7	−0.915052
$852$	0	0
$853$	−1.52810e7	−0.719086	−0.359543	−	0.933129i	$-0.617067\pi$
$853$	−1.52810e7	−0.719086	−0.359543	+	0.933129i	$0.617067\pi$
$854$	0	0
$855$	−7.93802e7	−3.71361
$856$	0	0
$857$	−5.10629e6	−0.237494	−0.118747	−	0.992925i	$-0.537888\pi$
$857$	−5.10629e6	−0.237494	−0.118747	+	0.992925i	$0.537888\pi$
$858$	0	0
$859$	2.32935e7	1.07709	0.538546	−	0.842596i	$-0.318974\pi$
$859$	2.32935e7	1.07709	0.538546	+	0.842596i	$0.318974\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.42566e7	1.10867	0.554337	−	0.832292i	$-0.312972\pi$
$863$	2.42566e7	1.10867	0.554337	+	0.832292i	$0.312972\pi$
$864$	0	0
$865$	3.37047e7	1.53162
$866$	0	0
$867$	−3.39424e7	−1.53354
$868$	0	0
$869$	2.35955e6	0.105994
$870$	0	0
$871$	4.07274e7	1.81903
$872$	0	0
$873$	−4.57744e7	−2.03276
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−2.81339e7	−1.23518	−0.617592	−	0.786499i	$-0.711892\pi$
$877$	−2.81339e7	−1.23518	−0.617592	+	0.786499i	$0.711892\pi$
$878$	0	0
$879$	−3.44361e7	−1.50328
$880$	0	0
$881$	8.51907e6	0.369788	0.184894	−	0.982759i	$-0.440806\pi$
$881$	8.51907e6	0.369788	0.184894	+	0.982759i	$0.440806\pi$
$882$	0	0
$883$	−1.36512e7	−0.589210	−0.294605	−	0.955619i	$-0.595188\pi$
$883$	−1.36512e7	−0.589210	−0.294605	+	0.955619i	$0.595188\pi$
$884$	0	0
$885$	−5.48871e7	−2.35566
$886$	0	0
$887$	−1.02750e7	−0.438502	−0.219251	−	0.975668i	$-0.570361\pi$
$887$	−1.02750e7	−0.438502	−0.219251	+	0.975668i	$0.570361\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.35599e7	0.572220
$892$	0	0
$893$	2.57564e7	1.08083
$894$	0	0
$895$	3.40861e7	1.42239
$896$	0	0
$897$	−5.15346e7	−2.13854
$898$	0	0
$899$	1.82079e7	0.751383
$900$	0	0
$901$	5.25933e6	0.215833
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4.57639e7	1.85738
$906$	0	0
$907$	3.08516e7	1.24526	0.622630	−	0.782516i	$-0.286064\pi$
$907$	3.08516e7	1.24526	0.622630	+	0.782516i	$0.286064\pi$
$908$	0	0
$909$	9.81170e6	0.393853
$910$	0	0
$911$	1.48424e7	0.592525	0.296263	−	0.955107i	$-0.404260\pi$
$911$	1.48424e7	0.592525	0.296263	+	0.955107i	$0.404260\pi$
$912$	0	0
$913$	−1.09904e7	−0.436351
$914$	0	0
$915$	−1.38731e8	−5.47798
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−3.14065e7	−1.22668	−0.613340	−	0.789819i	$-0.710174\pi$
$919$	−3.14065e7	−1.22668	−0.613340	+	0.789819i	$0.710174\pi$
$920$	0	0
$921$	4.83221e7	1.87714
$922$	0	0
$923$	5.16269e6	0.199467
$924$	0	0
$925$	5.70569e7	2.19257
$926$	0	0
$927$	6.10991e7	2.33526
$928$	0	0
$929$	7.75661e6	0.294871	0.147436	−	0.989072i	$-0.452898\pi$
$929$	7.75661e6	0.294871	0.147436	+	0.989072i	$0.452898\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−3.73396e7	−1.40432
$934$	0	0
$935$	−1.42528e7	−0.533178
$936$	0	0
$937$	−2.51058e7	−0.934167	−0.467084	−	0.884213i	$-0.654695\pi$
$937$	−2.51058e7	−0.934167	−0.467084	+	0.884213i	$0.654695\pi$
$938$	0	0
$939$	−7.03007e7	−2.60193
$940$	0	0
$941$	4.07777e7	1.50123	0.750617	−	0.660738i	$-0.229756\pi$
$941$	4.07777e7	1.50123	0.750617	+	0.660738i	$0.229756\pi$
$942$	0	0
$943$	−7.38818e6	−0.270557
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.11789e7	1.12976	0.564879	−	0.825174i	$-0.308923\pi$
$947$	3.11789e7	1.12976	0.564879	+	0.825174i	$0.308923\pi$
$948$	0	0
$949$	−4.31522e7	−1.55538
$950$	0	0
$951$	896086.	0.0321291
$952$	0	0
$953$	−2.49371e7	−0.889433	−0.444716	−	0.895671i	$-0.646696\pi$
$953$	−2.49371e7	−0.889433	−0.444716	+	0.895671i	$0.646696\pi$
$954$	0	0
$955$	8.30217e7	2.94566
$956$	0	0
$957$	−4.37007e7	−1.54244
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−8.81296e6	−0.307832
$962$	0	0
$963$	1.22977e7	0.427325
$964$	0	0
$965$	1.19839e7	0.414267
$966$	0	0
$967$	−9.62902e6	−0.331143	−0.165572	−	0.986198i	$-0.552947\pi$
$967$	−9.62902e6	−0.331143	−0.165572	+	0.986198i	$0.552947\pi$
$968$	0	0
$969$	1.71622e7	0.587170
$970$	0	0
$971$	5.76987e7	1.96389	0.981946	−	0.189161i	$-0.0605767\pi$
$971$	5.76987e7	1.96389	0.981946	+	0.189161i	$0.0605767\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	1.52103e8	5.12420
$976$	0	0
$977$	3.58334e7	1.20102	0.600512	−	0.799616i	$-0.294963\pi$
$977$	3.58334e7	1.20102	0.600512	+	0.799616i	$0.294963\pi$
$978$	0	0
$979$	3.90924e7	1.30357
$980$	0	0
$981$	3.97595e7	1.31907
$982$	0	0
$983$	−5.26717e7	−1.73857	−0.869287	−	0.494307i	$-0.835422\pi$
$983$	−5.26717e7	−1.73857	−0.869287	+	0.494307i	$0.835422\pi$
$984$	0	0
$985$	7.26380e7	2.38547
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.88105e7	0.611520
$990$	0	0
$991$	1.36827e7	0.442577	0.221288	−	0.975208i	$-0.428974\pi$
$991$	1.36827e7	0.442577	0.221288	+	0.975208i	$0.428974\pi$
$992$	0	0
$993$	4.07846e7	1.31257
$994$	0	0
$995$	−3.18637e7	−1.02032
$996$	0	0
$997$	−3.43844e7	−1.09553	−0.547763	−	0.836633i	$-0.684521\pi$
$997$	−3.43844e7	−1.09553	−0.547763	+	0.836633i	$0.684521\pi$
$998$	0	0
$999$	4.83976e7	1.53430

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.6.a.f.1.2		2
4.3	odd	2		784.6.a.p.1.1		2
7.2	even	3		392.6.i.g.361.1		4
7.3	odd	6		392.6.i.l.177.2		4
7.4	even	3		392.6.i.g.177.1		4
7.5	odd	6		392.6.i.l.361.2		4
7.6	odd	2		56.6.a.c.1.1	✓	2
21.20	even	2		504.6.a.s.1.2		2
28.27	even	2		112.6.a.k.1.2		2
56.13	odd	2		448.6.a.z.1.2		2
56.27	even	2		448.6.a.q.1.1		2
84.83	odd	2		1008.6.a.bt.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.6.a.c.1.1	✓	2	7.6	odd	2
112.6.a.k.1.2		2	28.27	even	2
392.6.a.f.1.2		2	1.1	even	1	trivial
392.6.i.g.177.1		4	7.4	even	3
392.6.i.g.361.1		4	7.2	even	3
392.6.i.l.177.2		4	7.3	odd	6
392.6.i.l.361.2		4	7.5	odd	6
448.6.a.q.1.1		2	56.27	even	2
448.6.a.z.1.2		2	56.13	odd	2
504.6.a.s.1.2		2	21.20	even	2
784.6.a.p.1.1		2	4.3	odd	2
1008.6.a.bt.1.2		2	84.83	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:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	392.a (trivial)

\(f(q)\)	\(=\)	\(q+26.3041 q^{3} +97.5207 q^{5} +448.908 q^{9} +O(q^{10})\)	\(q+26.3041 q^{3} +97.5207 q^{5} +448.908 q^{9} +406.175 q^{11} +905.594 q^{13} +2565.20 q^{15} -359.825 q^{17} -1813.25 q^{19} -2163.43 q^{23} +6385.28 q^{25} +5416.22 q^{27} -4090.26 q^{29} -4451.54 q^{31} +10684.1 q^{33} +8935.69 q^{37} +23820.9 q^{39} +3415.04 q^{41} -8694.80 q^{43} +43777.8 q^{45} -14204.5 q^{47} -9464.88 q^{51} -14616.4 q^{53} +39610.5 q^{55} -47696.1 q^{57} -21396.8 q^{59} -54081.9 q^{61} +88314.1 q^{65} +44973.1 q^{67} -56907.0 q^{69} +5700.89 q^{71} -47650.8 q^{73} +167959. q^{75} +5809.20 q^{79} +33384.4 q^{81} -27058.2 q^{83} -35090.4 q^{85} -107591. q^{87} +96245.2 q^{89} -117094. q^{93} -176830. q^{95} -101968. q^{97} +182335. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 26 q^{3} + 62 q^{5} + 206 q^{9}+O(q^{10}) \) 2 * q + 26 * q^3 + 62 * q^5 + 206 * q^9	\( 2 q + 26 q^{3} + 62 q^{5} + 206 q^{9} + 972 q^{11} - 78 q^{13} + 2576 q^{15} - 560 q^{17} - 2642 q^{19} + 2272 q^{23} + 4522 q^{25} + 5564 q^{27} - 7808 q^{29} - 5444 q^{31} + 10512 q^{33} + 576 q^{37} + 24120 q^{39} + 16888 q^{41} - 8396 q^{43} + 52406 q^{45} + 4532 q^{47} - 9404 q^{51} + 1420 q^{53} + 19512 q^{55} - 47444 q^{57} - 34146 q^{59} - 19106 q^{61} + 123252 q^{65} + 56952 q^{67} - 58256 q^{69} - 7224 q^{71} - 128828 q^{73} + 168526 q^{75} + 52808 q^{79} + 92366 q^{81} + 84486 q^{83} - 27980 q^{85} - 106460 q^{87} + 130972 q^{89} - 116792 q^{93} - 147392 q^{95} - 194624 q^{97} + 44892 q^{99}+O(q^{100}) \) 2 * q + 26 * q^3 + 62 * q^5 + 206 * q^9 + 972 * q^11 - 78 * q^13 + 2576 * q^15 - 560 * q^17 - 2642 * q^19 + 2272 * q^23 + 4522 * q^25 + 5564 * q^27 - 7808 * q^29 - 5444 * q^31 + 10512 * q^33 + 576 * q^37 + 24120 * q^39 + 16888 * q^41 - 8396 * q^43 + 52406 * q^45 + 4532 * q^47 - 9404 * q^51 + 1420 * q^53 + 19512 * q^55 - 47444 * q^57 - 34146 * q^59 - 19106 * q^61 + 123252 * q^65 + 56952 * q^67 - 58256 * q^69 - 7224 * q^71 - 128828 * q^73 + 168526 * q^75 + 52808 * q^79 + 92366 * q^81 + 84486 * q^83 - 27980 * q^85 - 106460 * q^87 + 130972 * q^89 - 116792 * q^93 - 147392 * q^95 - 194624 * q^97 + 44892 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more