LMFDB - Embedded newform 1100.6.a.i.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1100,6,Mod(1,1100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1100, 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("1100.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1100 = 2^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1100.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:	$176.422201794$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 1155x^{6} + 1122x^{5} + 365994x^{4} - 476334x^{3} - 28296355x^{2} + 54177943x + 132029745 $ x^8 - x^7 - 1155x^6 + 1122x^5 + 365994x^4 - 476334x^3 - 28296355x^2 + 54177943x + 132029745
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{7}\cdot 3\cdot 5^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-19.5686$ of defining polynomial
Character	$\chi$	$=$	1100.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+18.5686 q^{3} +50.8505 q^{7} +101.795 q^{9} -121.000 q^{11} -720.548 q^{13} +1193.88 q^{17} -398.674 q^{19} +944.226 q^{21} -767.070 q^{23} -2621.99 q^{27} +5025.45 q^{29} -5505.12 q^{31} -2246.81 q^{33} +14879.8 q^{37} -13379.6 q^{39} +2817.50 q^{41} +1590.94 q^{43} -21304.2 q^{47} -14221.2 q^{49} +22168.7 q^{51} +56.7744 q^{53} -7402.83 q^{57} -2976.29 q^{59} -22398.3 q^{61} +5176.32 q^{63} -61580.9 q^{67} -14243.5 q^{69} -59346.2 q^{71} +85386.6 q^{73} -6152.92 q^{77} -48608.4 q^{79} -73423.0 q^{81} -22158.4 q^{83} +93315.9 q^{87} -28120.4 q^{89} -36640.2 q^{91} -102223. q^{93} -58273.8 q^{97} -12317.2 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 9 q^{3} - 175 q^{7} + 377 q^{9} - 968 q^{11} + 1072 q^{13} - 37 q^{17} - 1356 q^{19} + 5462 q^{21} - 1327 q^{23} - 2670 q^{27} - 11785 q^{29} + 3532 q^{31} + 1089 q^{33} + 8176 q^{37} - 16831 q^{39}+ \cdots - 45617 q^{99}+O(q^{100}) $ 8 * q - 9 * q^3 - 175 * q^7 + 377 * q^9 - 968 * q^11 + 1072 * q^13 - 37 * q^17 - 1356 * q^19 + 5462 * q^21 - 1327 * q^23 - 2670 * q^27 - 11785 * q^29 + 3532 * q^31 + 1089 * q^33 + 8176 * q^37 - 16831 * q^39 + 15846 * q^41 + 530 * q^43 + 9252 * q^47 + 45745 * q^49 + 47733 * q^51 + 24991 * q^53 - 41957 * q^57 + 1440 * q^59 - 54447 * q^61 - 108968 * q^63 - 109232 * q^67 + 134200 * q^69 - 45026 * q^71 + 2221 * q^73 + 21175 * q^77 + 74369 * q^79 - 1132 * q^81 - 5443 * q^83 - 18665 * q^87 + 106151 * q^89 - 230723 * q^91 + 111315 * q^93 - 75273 * q^97 - 45617 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	18.5686	1.19118	0.595590	−	0.803289i	$-0.296918\pi$
$3$	18.5686	1.19118	0.595590	+	0.803289i	$0.296918\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	50.8505	0.392239	0.196119	−	0.980580i	$-0.437166\pi$
$7$	50.8505	0.392239	0.196119	+	0.980580i	$0.437166\pi$
$8$	0	0
$9$	101.795	0.418908
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	−720.548	−1.18251	−0.591254	−	0.806485i	$-0.701367\pi$
$13$	−720.548	−1.18251	−0.591254	+	0.806485i	$0.701367\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1193.88	1.00193	0.500966	−	0.865467i	$-0.332978\pi$
$17$	1193.88	1.00193	0.500966	+	0.865467i	$0.332978\pi$
$18$	0	0
$19$	−398.674	−0.253357	−0.126679	−	0.991944i	$-0.540432\pi$
$19$	−398.674	−0.253357	−0.126679	+	0.991944i	$0.540432\pi$
$20$	0	0
$21$	944.226	0.467227
$22$	0	0
$23$	−767.070	−0.302354	−0.151177	−	0.988507i	$-0.548306\pi$
$23$	−767.070	−0.302354	−0.151177	+	0.988507i	$0.548306\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−2621.99	−0.692184
$28$	0	0
$29$	5025.45	1.10964	0.554818	−	0.831972i	$-0.312788\pi$
$29$	5025.45	1.10964	0.554818	+	0.831972i	$0.312788\pi$
$30$	0	0
$31$	−5505.12	−1.02888	−0.514438	−	0.857528i	$-0.671999\pi$
$31$	−5505.12	−1.02888	−0.514438	+	0.857528i	$0.671999\pi$
$32$	0	0
$33$	−2246.81	−0.359154
$34$	0	0
$35$	0	0
$36$	0	0
$37$	14879.8	1.78687	0.893435	−	0.449192i	$-0.148288\pi$
$37$	14879.8	1.78687	0.893435	+	0.449192i	$0.148288\pi$
$38$	0	0
$39$	−13379.6	−1.40858
$40$	0	0
$41$	2817.50	0.261761	0.130880	−	0.991398i	$-0.458220\pi$
$41$	2817.50	0.261761	0.130880	+	0.991398i	$0.458220\pi$
$42$	0	0
$43$	1590.94	0.131215	0.0656075	−	0.997846i	$-0.479101\pi$
$43$	1590.94	0.131215	0.0656075	+	0.997846i	$0.479101\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−21304.2	−1.40676	−0.703382	−	0.710812i	$-0.748328\pi$
$47$	−21304.2	−1.40676	−0.703382	+	0.710812i	$0.748328\pi$
$48$	0	0
$49$	−14221.2	−0.846149
$50$	0	0
$51$	22168.7	1.19348
$52$	0	0
$53$	56.7744	0.00277628	0.00138814	−	0.999999i	$-0.499558\pi$
$53$	56.7744	0.00277628	0.00138814	+	0.999999i	$0.499558\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−7402.83	−0.301794
$58$	0	0
$59$	−2976.29	−0.111313	−0.0556564	−	0.998450i	$-0.517725\pi$
$59$	−2976.29	−0.111313	−0.0556564	+	0.998450i	$0.517725\pi$
$60$	0	0
$61$	−22398.3	−0.770708	−0.385354	−	0.922769i	$-0.625921\pi$
$61$	−22398.3	−0.770708	−0.385354	+	0.922769i	$0.625921\pi$
$62$	0	0
$63$	5176.32	0.164312
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−61580.9	−1.67594	−0.837970	−	0.545716i	$-0.816258\pi$
$67$	−61580.9	−1.67594	−0.837970	+	0.545716i	$0.816258\pi$
$68$	0	0
$69$	−14243.5	−0.360158
$70$	0	0
$71$	−59346.2	−1.39716	−0.698582	−	0.715530i	$-0.746185\pi$
$71$	−59346.2	−1.39716	−0.698582	+	0.715530i	$0.746185\pi$
$72$	0	0
$73$	85386.6	1.87535	0.937676	−	0.347511i	$-0.112973\pi$
$73$	85386.6	1.87535	0.937676	+	0.347511i	$0.112973\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6152.92	−0.118264
$78$	0	0
$79$	−48608.4	−0.876281	−0.438140	−	0.898907i	$-0.644363\pi$
$79$	−48608.4	−0.876281	−0.438140	+	0.898907i	$0.644363\pi$
$80$	0	0
$81$	−73423.0	−1.24342
$82$	0	0
$83$	−22158.4	−0.353056	−0.176528	−	0.984296i	$-0.556487\pi$
$83$	−22158.4	−0.353056	−0.176528	+	0.984296i	$0.556487\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	93315.9	1.32177
$88$	0	0
$89$	−28120.4	−0.376311	−0.188155	−	0.982139i	$-0.560251\pi$
$89$	−28120.4	−0.376311	−0.188155	+	0.982139i	$0.560251\pi$
$90$	0	0
$91$	−36640.2	−0.463826
$92$	0	0
$93$	−102223.	−1.22557
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−58273.8	−0.628845	−0.314423	−	0.949283i	$-0.601811\pi$
$97$	−58273.8	−0.628845	−0.314423	+	0.949283i	$0.601811\pi$
$98$	0	0
$99$	−12317.2	−0.126306
$100$	0	0
$101$	−2227.90	−0.0217316	−0.0108658	−	0.999941i	$-0.503459\pi$
$101$	−2227.90	−0.0217316	−0.0108658	+	0.999941i	$0.503459\pi$
$102$	0	0
$103$	199626.	1.85406	0.927032	−	0.374981i	$-0.122351\pi$
$103$	199626.	1.85406	0.927032	+	0.374981i	$0.122351\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−199695.	−1.68620	−0.843099	−	0.537759i	$-0.819271\pi$
$107$	−199695.	−1.68620	−0.843099	+	0.537759i	$0.819271\pi$
$108$	0	0
$109$	−96516.4	−0.778099	−0.389050	−	0.921217i	$-0.627197\pi$
$109$	−96516.4	−0.778099	−0.389050	+	0.921217i	$0.627197\pi$
$110$	0	0
$111$	276298.	2.12848
$112$	0	0
$113$	27313.2	0.201222	0.100611	−	0.994926i	$-0.467920\pi$
$113$	27313.2	0.201222	0.100611	+	0.994926i	$0.467920\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−73348.0	−0.495363
$118$	0	0
$119$	60709.5	0.392997
$120$	0	0
$121$	14641.0	0.0909091
$122$	0	0
$123$	52317.2	0.311804
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−40224.7	−0.221301	−0.110650	−	0.993859i	$-0.535293\pi$
$127$	−40224.7	−0.221301	−0.110650	+	0.993859i	$0.535293\pi$
$128$	0	0
$129$	29541.6	0.156301
$130$	0	0
$131$	−181705.	−0.925099	−0.462549	−	0.886594i	$-0.653065\pi$
$131$	−181705.	−0.925099	−0.462549	+	0.886594i	$0.653065\pi$
$132$	0	0
$133$	−20272.8	−0.0993766
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−36992.7	−0.168389	−0.0841947	−	0.996449i	$-0.526832\pi$
$137$	−36992.7	−0.168389	−0.0841947	+	0.996449i	$0.526832\pi$
$138$	0	0
$139$	28386.9	0.124618	0.0623091	−	0.998057i	$-0.480154\pi$
$139$	28386.9	0.124618	0.0623091	+	0.998057i	$0.480154\pi$
$140$	0	0
$141$	−395591.	−1.67571
$142$	0	0
$143$	87186.3	0.356540
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−264069.	−1.00791
$148$	0	0
$149$	−411427.	−1.51819	−0.759097	−	0.650977i	$-0.774359\pi$
$149$	−411427.	−1.51819	−0.759097	+	0.650977i	$0.774359\pi$
$150$	0	0
$151$	−19605.7	−0.0699745	−0.0349872	−	0.999388i	$-0.511139\pi$
$151$	−19605.7	−0.0699745	−0.0349872	+	0.999388i	$0.511139\pi$
$152$	0	0
$153$	121531.	0.419718
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−32924.3	−0.106603	−0.0533013	−	0.998578i	$-0.516974\pi$
$157$	−32924.3	−0.106603	−0.0533013	+	0.998578i	$0.516974\pi$
$158$	0	0
$159$	1054.22	0.00330704
$160$	0	0
$161$	−39005.9	−0.118595
$162$	0	0
$163$	−338514.	−0.997948	−0.498974	−	0.866617i	$-0.666290\pi$
$163$	−338514.	−0.997948	−0.498974	+	0.866617i	$0.666290\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−476235.	−1.32139	−0.660693	−	0.750656i	$-0.729738\pi$
$167$	−476235.	−1.32139	−0.660693	+	0.750656i	$0.729738\pi$
$168$	0	0
$169$	147896.	0.398328
$170$	0	0
$171$	−40582.9	−0.106134
$172$	0	0
$173$	323796.	0.822538	0.411269	−	0.911514i	$-0.365086\pi$
$173$	323796.	0.822538	0.411269	+	0.911514i	$0.365086\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−55265.7	−0.132593
$178$	0	0
$179$	−462543.	−1.07899	−0.539497	−	0.841987i	$-0.681386\pi$
$179$	−462543.	−1.07899	−0.539497	+	0.841987i	$0.681386\pi$
$180$	0	0
$181$	412875.	0.936746	0.468373	−	0.883531i	$-0.344840\pi$
$181$	412875.	0.936746	0.468373	+	0.883531i	$0.344840\pi$
$182$	0	0
$183$	−415906.	−0.918052
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−144460.	−0.302094
$188$	0	0
$189$	−133330.	−0.271502
$190$	0	0
$191$	−438200.	−0.869139	−0.434569	−	0.900638i	$-0.643099\pi$
$191$	−438200.	−0.869139	−0.434569	+	0.900638i	$0.643099\pi$
$192$	0	0
$193$	−156251.	−0.301946	−0.150973	−	0.988538i	$-0.548241\pi$
$193$	−156251.	−0.301946	−0.150973	+	0.988538i	$0.548241\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−90207.3	−0.165606	−0.0828030	−	0.996566i	$-0.526387\pi$
$197$	−90207.3	−0.165606	−0.0828030	+	0.996566i	$0.526387\pi$
$198$	0	0
$199$	590876.	1.05770	0.528851	−	0.848714i	$-0.322623\pi$
$199$	590876.	1.05770	0.528851	+	0.848714i	$0.322623\pi$
$200$	0	0
$201$	−1.14347e6	−1.99635
$202$	0	0
$203$	255547.	0.435242
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−78083.7	−0.126659
$208$	0	0
$209$	48239.5	0.0763902
$210$	0	0
$211$	346838.	0.536316	0.268158	−	0.963375i	$-0.413585\pi$
$211$	346838.	0.536316	0.268158	+	0.963375i	$0.413585\pi$
$212$	0	0
$213$	−1.10198e6	−1.66427
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−279938.	−0.403565
$218$	0	0
$219$	1.58551e6	2.23388
$220$	0	0
$221$	−860248.	−1.18479
$222$	0	0
$223$	−346396.	−0.466457	−0.233228	−	0.972422i	$-0.574929\pi$
$223$	−346396.	−0.466457	−0.233228	+	0.972422i	$0.574929\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	476461.	0.613710	0.306855	−	0.951756i	$-0.400723\pi$
$227$	476461.	0.613710	0.306855	+	0.951756i	$0.400723\pi$
$228$	0	0
$229$	1.11372e6	1.40342	0.701709	−	0.712464i	$-0.252421\pi$
$229$	1.11372e6	1.40342	0.701709	+	0.712464i	$0.252421\pi$
$230$	0	0
$231$	−114251.	−0.140874
$232$	0	0
$233$	459032.	0.553928	0.276964	−	0.960880i	$-0.410672\pi$
$233$	459032.	0.553928	0.276964	+	0.960880i	$0.410672\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−902592.	−1.04381
$238$	0	0
$239$	1.21303e6	1.37365	0.686825	−	0.726823i	$-0.259004\pi$
$239$	1.21303e6	1.37365	0.686825	+	0.726823i	$0.259004\pi$
$240$	0	0
$241$	1.04688e6	1.16106	0.580531	−	0.814238i	$-0.302845\pi$
$241$	1.04688e6	1.16106	0.580531	+	0.814238i	$0.302845\pi$
$242$	0	0
$243$	−726221.	−0.788957
$244$	0	0
$245$	0	0
$246$	0	0
$247$	287264.	0.299597
$248$	0	0
$249$	−411452.	−0.420554
$250$	0	0
$251$	750630.	0.752041	0.376021	−	0.926611i	$-0.377292\pi$
$251$	750630.	0.752041	0.376021	+	0.926611i	$0.377292\pi$
$252$	0	0
$253$	92815.5	0.0911631
$254$	0	0
$255$	0	0
$256$	0	0
$257$	446538.	0.421721	0.210860	−	0.977516i	$-0.432373\pi$
$257$	446538.	0.421721	0.210860	+	0.977516i	$0.432373\pi$
$258$	0	0
$259$	756646.	0.700880
$260$	0	0
$261$	511565.	0.464836
$262$	0	0
$263$	−1.49215e6	−1.33022	−0.665112	−	0.746744i	$-0.731616\pi$
$263$	−1.49215e6	−1.33022	−0.665112	+	0.746744i	$0.731616\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−522158.	−0.448254
$268$	0	0
$269$	2.10034e6	1.76974	0.884871	−	0.465837i	$-0.154247\pi$
$269$	2.10034e6	1.76974	0.884871	+	0.465837i	$0.154247\pi$
$270$	0	0
$271$	−1.20287e6	−0.994934	−0.497467	−	0.867483i	$-0.665736\pi$
$271$	−1.20287e6	−0.994934	−0.497467	+	0.867483i	$0.665736\pi$
$272$	0	0
$273$	−680360.	−0.552500
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−1.73274e6	−1.35686	−0.678428	−	0.734667i	$-0.737338\pi$
$277$	−1.73274e6	−1.35686	−0.678428	+	0.734667i	$0.737338\pi$
$278$	0	0
$279$	−560392.	−0.431004
$280$	0	0
$281$	−753701.	−0.569421	−0.284710	−	0.958614i	$-0.591897\pi$
$281$	−753701.	−0.569421	−0.284710	+	0.958614i	$0.591897\pi$
$282$	0	0
$283$	1.67332e6	1.24198	0.620988	−	0.783820i	$-0.286732\pi$
$283$	1.67332e6	1.24198	0.620988	+	0.783820i	$0.286732\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	143271.	0.102673
$288$	0	0
$289$	5493.51	0.00386906
$290$	0	0
$291$	−1.08207e6	−0.749068
$292$	0	0
$293$	−1.62686e6	−1.10709	−0.553543	−	0.832821i	$-0.686725\pi$
$293$	−1.62686e6	−1.10709	−0.553543	+	0.832821i	$0.686725\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	317261.	0.208701
$298$	0	0
$299$	552711.	0.357536
$300$	0	0
$301$	80900.3	0.0514676
$302$	0	0
$303$	−41369.0	−0.0258862
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−2.57568e6	−1.55972	−0.779858	−	0.625957i	$-0.784709\pi$
$307$	−2.57568e6	−1.55972	−0.779858	+	0.625957i	$0.784709\pi$
$308$	0	0
$309$	3.70679e6	2.20852
$310$	0	0
$311$	175724.	0.103022	0.0515110	−	0.998672i	$-0.483596\pi$
$311$	175724.	0.103022	0.0515110	+	0.998672i	$0.483596\pi$
$312$	0	0
$313$	−1.54471e6	−0.891221	−0.445610	−	0.895227i	$-0.647013\pi$
$313$	−1.54471e6	−0.891221	−0.445610	+	0.895227i	$0.647013\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.30488e6	0.729328	0.364664	−	0.931139i	$-0.381184\pi$
$317$	1.30488e6	0.729328	0.364664	+	0.931139i	$0.381184\pi$
$318$	0	0
$319$	−608080.	−0.334568
$320$	0	0
$321$	−3.70807e6	−2.00856
$322$	0	0
$323$	−475969.	−0.253847
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1.79218e6	−0.926856
$328$	0	0
$329$	−1.08333e6	−0.551788
$330$	0	0
$331$	1.79313e6	0.899584	0.449792	−	0.893133i	$-0.351498\pi$
$331$	1.79313e6	0.899584	0.449792	+	0.893133i	$0.351498\pi$
$332$	0	0
$333$	1.51469e6	0.748535
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−1.05155e6	−0.504376	−0.252188	−	0.967678i	$-0.581150\pi$
$337$	−1.05155e6	−0.504376	−0.252188	+	0.967678i	$0.581150\pi$
$338$	0	0
$339$	507169.	0.239692
$340$	0	0
$341$	666120.	0.310218
$342$	0	0
$343$	−1.57780e6	−0.724131
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2.29833e6	−1.02468	−0.512340	−	0.858783i	$-0.671221\pi$
$347$	−2.29833e6	−1.02468	−0.512340	+	0.858783i	$0.671221\pi$
$348$	0	0
$349$	−1.16071e6	−0.510108	−0.255054	−	0.966927i	$-0.582093\pi$
$349$	−1.16071e6	−0.510108	−0.255054	+	0.966927i	$0.582093\pi$
$350$	0	0
$351$	1.88927e6	0.818514
$352$	0	0
$353$	−2.38698e6	−1.01956	−0.509779	−	0.860305i	$-0.670273\pi$
$353$	−2.38698e6	−1.01956	−0.509779	+	0.860305i	$0.670273\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.12729e6	0.468130
$358$	0	0
$359$	3.27842e6	1.34254	0.671272	−	0.741211i	$-0.265748\pi$
$359$	3.27842e6	1.34254	0.671272	+	0.741211i	$0.265748\pi$
$360$	0	0
$361$	−2.31716e6	−0.935810
$362$	0	0
$363$	271864.	0.108289
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−2.60391e6	−1.00916	−0.504581	−	0.863364i	$-0.668353\pi$
$367$	−2.60391e6	−1.00916	−0.504581	+	0.863364i	$0.668353\pi$
$368$	0	0
$369$	286807.	0.109654
$370$	0	0
$371$	2887.01	0.00108896
$372$	0	0
$373$	−1.58371e6	−0.589390	−0.294695	−	0.955591i	$-0.595218\pi$
$373$	−1.58371e6	−0.589390	−0.294695	+	0.955591i	$0.595218\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.62108e6	−1.31215
$378$	0	0
$379$	3.32092e6	1.18757	0.593786	−	0.804623i	$-0.297632\pi$
$379$	3.32092e6	1.18757	0.593786	+	0.804623i	$0.297632\pi$
$380$	0	0
$381$	−746918.	−0.263609
$382$	0	0
$383$	1.36707e6	0.476204	0.238102	−	0.971240i	$-0.423475\pi$
$383$	1.36707e6	0.476204	0.238102	+	0.971240i	$0.423475\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	161949.	0.0549670
$388$	0	0
$389$	4.32467e6	1.44903	0.724517	−	0.689257i	$-0.242063\pi$
$389$	4.32467e6	1.44903	0.724517	+	0.689257i	$0.242063\pi$
$390$	0	0
$391$	−915790.	−0.302938
$392$	0	0
$393$	−3.37401e6	−1.10196
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−2.40481e6	−0.765780	−0.382890	−	0.923794i	$-0.625071\pi$
$397$	−2.40481e6	−0.765780	−0.382890	+	0.923794i	$0.625071\pi$
$398$	0	0
$399$	−376438.	−0.118375
$400$	0	0
$401$	−4.42677e6	−1.37476	−0.687379	−	0.726299i	$-0.741239\pi$
$401$	−4.42677e6	−1.37476	−0.687379	+	0.726299i	$0.741239\pi$
$402$	0	0
$403$	3.96670e6	1.21665
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.80046e6	−0.538762
$408$	0	0
$409$	−4.89681e6	−1.44745	−0.723727	−	0.690086i	$-0.757572\pi$
$409$	−4.89681e6	−1.44745	−0.723727	+	0.690086i	$0.757572\pi$
$410$	0	0
$411$	−686905.	−0.200582
$412$	0	0
$413$	−151346.	−0.0436612
$414$	0	0
$415$	0	0
$416$	0	0
$417$	527107.	0.148443
$418$	0	0
$419$	−3.88074e6	−1.07989	−0.539945	−	0.841700i	$-0.681555\pi$
$419$	−3.88074e6	−1.07989	−0.539945	+	0.841700i	$0.681555\pi$
$420$	0	0
$421$	−2.65122e6	−0.729023	−0.364511	−	0.931199i	$-0.618764\pi$
$421$	−2.65122e6	−0.729023	−0.364511	+	0.931199i	$0.618764\pi$
$422$	0	0
$423$	−2.16866e6	−0.589305
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1.13896e6	−0.302302
$428$	0	0
$429$	1.61893e6	0.424703
$430$	0	0
$431$	3.18071e6	0.824766	0.412383	−	0.911011i	$-0.364697\pi$
$431$	3.18071e6	0.824766	0.412383	+	0.911011i	$0.364697\pi$
$432$	0	0
$433$	771535.	0.197759	0.0988795	−	0.995099i	$-0.468474\pi$
$433$	771535.	0.197759	0.0988795	+	0.995099i	$0.468474\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	305811.	0.0766036
$438$	0	0
$439$	3.58319e6	0.887377	0.443688	−	0.896181i	$-0.353670\pi$
$439$	3.58319e6	0.887377	0.443688	+	0.896181i	$0.353670\pi$
$440$	0	0
$441$	−1.44765e6	−0.354459
$442$	0	0
$443$	2.10127e6	0.508712	0.254356	−	0.967111i	$-0.418136\pi$
$443$	2.10127e6	0.508712	0.254356	+	0.967111i	$0.418136\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−7.63965e6	−1.80844
$448$	0	0
$449$	4.27634e6	1.00105	0.500526	−	0.865722i	$-0.333140\pi$
$449$	4.27634e6	1.00105	0.500526	+	0.865722i	$0.333140\pi$
$450$	0	0
$451$	−340917.	−0.0789238
$452$	0	0
$453$	−364051.	−0.0833521
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.78038e6	0.846730	0.423365	−	0.905959i	$-0.360849\pi$
$457$	3.78038e6	0.846730	0.423365	+	0.905959i	$0.360849\pi$
$458$	0	0
$459$	−3.13034e6	−0.693522
$460$	0	0
$461$	−4.78330e6	−1.04828	−0.524138	−	0.851633i	$-0.675612\pi$
$461$	−4.78330e6	−1.04828	−0.524138	+	0.851633i	$0.675612\pi$
$462$	0	0
$463$	1.83859e6	0.398595	0.199298	−	0.979939i	$-0.436134\pi$
$463$	1.83859e6	0.398595	0.199298	+	0.979939i	$0.436134\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−790769.	−0.167787	−0.0838934	−	0.996475i	$-0.526736\pi$
$467$	−790769.	−0.167787	−0.0838934	+	0.996475i	$0.526736\pi$
$468$	0	0
$469$	−3.13142e6	−0.657369
$470$	0	0
$471$	−611360.	−0.126983
$472$	0	0
$473$	−192504.	−0.0395628
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5779.33	0.00116301
$478$	0	0
$479$	−8.39109e6	−1.67101	−0.835506	−	0.549481i	$-0.814826\pi$
$479$	−8.39109e6	−1.67101	−0.835506	+	0.549481i	$0.814826\pi$
$480$	0	0
$481$	−1.07216e7	−2.11299
$482$	0	0
$483$	−724287.	−0.141268
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−3.60140e6	−0.688095	−0.344048	−	0.938952i	$-0.611798\pi$
$487$	−3.60140e6	−0.688095	−0.344048	+	0.938952i	$0.611798\pi$
$488$	0	0
$489$	−6.28575e6	−1.18874
$490$	0	0
$491$	1.19689e6	0.224052	0.112026	−	0.993705i	$-0.464266\pi$
$491$	1.19689e6	0.224052	0.112026	+	0.993705i	$0.464266\pi$
$492$	0	0
$493$	5.99979e6	1.11178
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.01779e6	−0.548022
$498$	0	0
$499$	−1.27672e6	−0.229532	−0.114766	−	0.993393i	$-0.536612\pi$
$499$	−1.27672e6	−0.229532	−0.114766	+	0.993393i	$0.536612\pi$
$500$	0	0
$501$	−8.84304e6	−1.57401
$502$	0	0
$503$	2.76461e6	0.487207	0.243604	−	0.969875i	$-0.421670\pi$
$503$	2.76461e6	0.487207	0.243604	+	0.969875i	$0.421670\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2.74623e6	0.474480
$508$	0	0
$509$	−2.12988e6	−0.364385	−0.182192	−	0.983263i	$-0.558319\pi$
$509$	−2.12988e6	−0.364385	−0.182192	+	0.983263i	$0.558319\pi$
$510$	0	0
$511$	4.34196e6	0.735586
$512$	0	0
$513$	1.04532e6	0.175370
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2.57781e6	0.424156
$518$	0	0
$519$	6.01245e6	0.979790
$520$	0	0
$521$	9.30638e6	1.50206	0.751029	−	0.660270i	$-0.229558\pi$
$521$	9.30638e6	1.50206	0.751029	+	0.660270i	$0.229558\pi$
$522$	0	0
$523$	−3.34864e6	−0.535321	−0.267661	−	0.963513i	$-0.586251\pi$
$523$	−3.34864e6	−0.535321	−0.267661	+	0.963513i	$0.586251\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−6.57246e6	−1.03086
$528$	0	0
$529$	−5.84795e6	−0.908582
$530$	0	0
$531$	−302970.	−0.0466298
$532$	0	0
$533$	−2.03014e6	−0.309534
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−8.58879e6	−1.28528
$538$	0	0
$539$	1.72077e6	0.255123
$540$	0	0
$541$	4.52581e6	0.664819	0.332410	−	0.943135i	$-0.392138\pi$
$541$	4.52581e6	0.664819	0.332410	+	0.943135i	$0.392138\pi$
$542$	0	0
$543$	7.66653e6	1.11583
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−7.07805e6	−1.01145	−0.505726	−	0.862694i	$-0.668775\pi$
$547$	−7.07805e6	−1.01145	−0.505726	+	0.862694i	$0.668775\pi$
$548$	0	0
$549$	−2.28003e6	−0.322856
$550$	0	0
$551$	−2.00352e6	−0.281134
$552$	0	0
$553$	−2.47176e6	−0.343711
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.37491e6	0.324347	0.162174	−	0.986762i	$-0.448150\pi$
$557$	2.37491e6	0.324347	0.162174	+	0.986762i	$0.448150\pi$
$558$	0	0
$559$	−1.14635e6	−0.155163
$560$	0	0
$561$	−2.68242e6	−0.359848
$562$	0	0
$563$	−2.29312e6	−0.304898	−0.152449	−	0.988311i	$-0.548716\pi$
$563$	−2.29312e6	−0.304898	−0.152449	+	0.988311i	$0.548716\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−3.73360e6	−0.487719
$568$	0	0
$569$	1.25347e6	0.162305	0.0811525	−	0.996702i	$-0.474140\pi$
$569$	1.25347e6	0.162305	0.0811525	+	0.996702i	$0.474140\pi$
$570$	0	0
$571$	4.51214e6	0.579152	0.289576	−	0.957155i	$-0.406486\pi$
$571$	4.51214e6	0.579152	0.289576	+	0.957155i	$0.406486\pi$
$572$	0	0
$573$	−8.13678e6	−1.03530
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−3.86833e6	−0.483709	−0.241854	−	0.970313i	$-0.577756\pi$
$577$	−3.86833e6	−0.483709	−0.241854	+	0.970313i	$0.577756\pi$
$578$	0	0
$579$	−2.90137e6	−0.359672
$580$	0	0
$581$	−1.12677e6	−0.138482
$582$	0	0
$583$	−6869.70	−0.000837079 0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3.16903e6	−0.379604	−0.189802	−	0.981822i	$-0.560785\pi$
$587$	−3.16903e6	−0.379604	−0.189802	+	0.981822i	$0.560785\pi$
$588$	0	0
$589$	2.19475e6	0.260673
$590$	0	0
$591$	−1.67503e6	−0.197266
$592$	0	0
$593$	1.54355e7	1.80253	0.901266	−	0.433266i	$-0.142639\pi$
$593$	1.54355e7	1.80253	0.901266	+	0.433266i	$0.142639\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.09718e7	1.25991
$598$	0	0
$599$	1.39482e7	1.58837	0.794185	−	0.607677i	$-0.207898\pi$
$599$	1.39482e7	1.58837	0.794185	+	0.607677i	$0.207898\pi$
$600$	0	0
$601$	2.87124e6	0.324252	0.162126	−	0.986770i	$-0.448165\pi$
$601$	2.87124e6	0.324252	0.162126	+	0.986770i	$0.448165\pi$
$602$	0	0
$603$	−6.26860e6	−0.702065
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.42913e7	1.57434	0.787172	−	0.616734i	$-0.211545\pi$
$607$	1.42913e7	1.57434	0.787172	+	0.616734i	$0.211545\pi$
$608$	0	0
$609$	4.74516e6	0.518451
$610$	0	0
$611$	1.53507e7	1.66351
$612$	0	0
$613$	8.69125e6	0.934181	0.467090	−	0.884210i	$-0.345302\pi$
$613$	8.69125e6	0.934181	0.467090	+	0.884210i	$0.345302\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.30610e6	0.666880	0.333440	−	0.942771i	$-0.391790\pi$
$617$	6.30610e6	0.666880	0.333440	+	0.942771i	$0.391790\pi$
$618$	0	0
$619$	−5.56254e6	−0.583507	−0.291754	−	0.956493i	$-0.594239\pi$
$619$	−5.56254e6	−0.583507	−0.291754	+	0.956493i	$0.594239\pi$
$620$	0	0
$621$	2.01125e6	0.209285
$622$	0	0
$623$	−1.42994e6	−0.147604
$624$	0	0
$625$	0	0
$626$	0	0
$627$	895743.	0.0909944
$628$	0	0
$629$	1.77647e7	1.79032
$630$	0	0
$631$	6.58408e6	0.658297	0.329148	−	0.944278i	$-0.393238\pi$
$631$	6.58408e6	0.658297	0.329148	+	0.944278i	$0.393238\pi$
$632$	0	0
$633$	6.44032e6	0.638849
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.02471e7	1.00058
$638$	0	0
$639$	−6.04113e6	−0.585283
$640$	0	0
$641$	3.30096e6	0.317319	0.158659	−	0.987333i	$-0.449283\pi$
$641$	3.30096e6	0.317319	0.158659	+	0.987333i	$0.449283\pi$
$642$	0	0
$643$	−9.64316e6	−0.919797	−0.459899	−	0.887971i	$-0.652114\pi$
$643$	−9.64316e6	−0.919797	−0.459899	+	0.887971i	$0.652114\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.02316e7	0.960909	0.480454	−	0.877020i	$-0.340472\pi$
$647$	1.02316e7	0.960909	0.480454	+	0.877020i	$0.340472\pi$
$648$	0	0
$649$	360131.	0.0335621
$650$	0	0
$651$	−5.19808e6	−0.480718
$652$	0	0
$653$	−5.52030e6	−0.506617	−0.253308	−	0.967386i	$-0.581519\pi$
$653$	−5.52030e6	−0.506617	−0.253308	+	0.967386i	$0.581519\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	8.69191e6	0.785600
$658$	0	0
$659$	−1.63583e7	−1.46732	−0.733659	−	0.679517i	$-0.762189\pi$
$659$	−1.63583e7	−1.46732	−0.733659	+	0.679517i	$0.762189\pi$
$660$	0	0
$661$	−2.12627e7	−1.89284	−0.946421	−	0.322934i	$-0.895331\pi$
$661$	−2.12627e7	−1.89284	−0.946421	+	0.322934i	$0.895331\pi$
$662$	0	0
$663$	−1.59736e7	−1.41130
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3.85488e6	−0.335503
$668$	0	0
$669$	−6.43211e6	−0.555634
$670$	0	0
$671$	2.71019e6	0.232377
$672$	0	0
$673$	−1.90604e7	−1.62216	−0.811082	−	0.584932i	$-0.801121\pi$
$673$	−1.90604e7	−1.62216	−0.811082	+	0.584932i	$0.801121\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−7.43909e6	−0.623804	−0.311902	−	0.950114i	$-0.600966\pi$
$677$	−7.43909e6	−0.623804	−0.311902	+	0.950114i	$0.600966\pi$
$678$	0	0
$679$	−2.96325e6	−0.246657
$680$	0	0
$681$	8.84724e6	0.731038
$682$	0	0
$683$	9.05701e6	0.742905	0.371452	−	0.928452i	$-0.378860\pi$
$683$	9.05701e6	0.742905	0.371452	+	0.928452i	$0.378860\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.06803e7	1.67172
$688$	0	0
$689$	−40908.7	−0.00328297
$690$	0	0
$691$	7.81970e6	0.623010	0.311505	−	0.950245i	$-0.399167\pi$
$691$	7.81970e6	0.623010	0.311505	+	0.950245i	$0.399167\pi$
$692$	0	0
$693$	−626334.	−0.0495419
$694$	0	0
$695$	0	0
$696$	0	0
$697$	3.36376e6	0.262266
$698$	0	0
$699$	8.52361e6	0.659828
$700$	0	0
$701$	9.97868e6	0.766969	0.383485	−	0.923547i	$-0.374724\pi$
$701$	9.97868e6	0.766969	0.383485	+	0.923547i	$0.374724\pi$
$702$	0	0
$703$	−5.93219e6	−0.452717
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−113290.	−0.00852397
$708$	0	0
$709$	1.18480e6	0.0885178	0.0442589	−	0.999020i	$-0.485907\pi$
$709$	1.18480e6	0.0885178	0.0442589	+	0.999020i	$0.485907\pi$
$710$	0	0
$711$	−4.94808e6	−0.367081
$712$	0	0
$713$	4.22281e6	0.311084
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.25243e7	1.63626
$718$	0	0
$719$	−845379.	−0.0609859	−0.0304929	−	0.999535i	$-0.509708\pi$
$719$	−845379.	−0.0609859	−0.0304929	+	0.999535i	$0.509708\pi$
$720$	0	0
$721$	1.01511e7	0.727236
$722$	0	0
$723$	1.94392e7	1.38303
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−1.30098e7	−0.912921	−0.456460	−	0.889744i	$-0.650883\pi$
$727$	−1.30098e7	−0.912921	−0.456460	+	0.889744i	$0.650883\pi$
$728$	0	0
$729$	4.35683e6	0.303635
$730$	0	0
$731$	1.89939e6	0.131469
$732$	0	0
$733$	1.44691e7	0.994677	0.497338	−	0.867557i	$-0.334311\pi$
$733$	1.44691e7	0.994677	0.497338	+	0.867557i	$0.334311\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.45128e6	0.505315
$738$	0	0
$739$	1.48541e7	1.00054	0.500271	−	0.865869i	$-0.333234\pi$
$739$	1.48541e7	1.00054	0.500271	+	0.865869i	$0.333234\pi$
$740$	0	0
$741$	5.33410e6	0.356874
$742$	0	0
$743$	1.18605e6	0.0788189	0.0394095	−	0.999223i	$-0.487452\pi$
$743$	1.18605e6	0.0788189	0.0394095	+	0.999223i	$0.487452\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−2.25561e6	−0.147898
$748$	0	0
$749$	−1.01546e7	−0.661392
$750$	0	0
$751$	−121692.	−0.00787337	−0.00393669	−	0.999992i	$-0.501253\pi$
$751$	−121692.	−0.00787337	−0.00393669	+	0.999992i	$0.501253\pi$
$752$	0	0
$753$	1.39382e7	0.895816
$754$	0	0
$755$	0	0
$756$	0	0
$757$	3.99574e6	0.253430	0.126715	−	0.991939i	$-0.459557\pi$
$757$	3.99574e6	0.253430	0.126715	+	0.991939i	$0.459557\pi$
$758$	0	0
$759$	1.72346e6	0.108592
$760$	0	0
$761$	5.44861e6	0.341054	0.170527	−	0.985353i	$-0.445453\pi$
$761$	5.44861e6	0.341054	0.170527	+	0.985353i	$0.445453\pi$
$762$	0	0
$763$	−4.90791e6	−0.305201
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.14456e6	0.131628
$768$	0	0
$769$	3.03277e7	1.84937	0.924686	−	0.380731i	$-0.124328\pi$
$769$	3.03277e7	1.84937	0.924686	+	0.380731i	$0.124328\pi$
$770$	0	0
$771$	8.29160e6	0.502345
$772$	0	0
$773$	1.85500e7	1.11660	0.558298	−	0.829640i	$-0.311454\pi$
$773$	1.85500e7	1.11660	0.558298	+	0.829640i	$0.311454\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.40499e7	0.834873
$778$	0	0
$779$	−1.12326e6	−0.0663190
$780$	0	0
$781$	7.18089e6	0.421261
$782$	0	0
$783$	−1.31767e7	−0.768073
$784$	0	0
$785$	0	0
$786$	0	0
$787$	2.23335e7	1.28535	0.642673	−	0.766140i	$-0.277825\pi$
$787$	2.23335e7	1.28535	0.642673	+	0.766140i	$0.277825\pi$
$788$	0	0
$789$	−2.77073e7	−1.58453
$790$	0	0
$791$	1.38889e6	0.0789273
$792$	0	0
$793$	1.61390e7	0.911369
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.98275e7	1.66330	0.831651	−	0.555298i	$-0.187396\pi$
$797$	2.98275e7	1.66330	0.831651	+	0.555298i	$0.187396\pi$
$798$	0	0
$799$	−2.54347e7	−1.40948
$800$	0	0
$801$	−2.86251e6	−0.157640
$802$	0	0
$803$	−1.03318e7	−0.565440
$804$	0	0
$805$	0	0
$806$	0	0
$807$	3.90005e7	2.10808
$808$	0	0
$809$	−3.63063e7	−1.95034	−0.975172	−	0.221448i	$-0.928922\pi$
$809$	−3.63063e7	−1.95034	−0.975172	+	0.221448i	$0.928922\pi$
$810$	0	0
$811$	3.15529e7	1.68456	0.842282	−	0.539038i	$-0.181212\pi$
$811$	3.15529e7	1.68456	0.842282	+	0.539038i	$0.181212\pi$
$812$	0	0
$813$	−2.23356e7	−1.18514
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−634267.	−0.0332443
$818$	0	0
$819$	−3.72978e6	−0.194300
$820$	0	0
$821$	3.81005e6	0.197275	0.0986376	−	0.995123i	$-0.468552\pi$
$821$	3.81005e6	0.197275	0.0986376	+	0.995123i	$0.468552\pi$
$822$	0	0
$823$	−4.71125e6	−0.242458	−0.121229	−	0.992625i	$-0.538684\pi$
$823$	−4.71125e6	−0.242458	−0.121229	+	0.992625i	$0.538684\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−9.70064e6	−0.493215	−0.246608	−	0.969115i	$-0.579316\pi$
$827$	−9.70064e6	−0.493215	−0.246608	+	0.969115i	$0.579316\pi$
$828$	0	0
$829$	1.72124e7	0.869874	0.434937	−	0.900461i	$-0.356771\pi$
$829$	1.72124e7	0.869874	0.434937	+	0.900461i	$0.356771\pi$
$830$	0	0
$831$	−3.21746e7	−1.61626
$832$	0	0
$833$	−1.69784e7	−0.847784
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.44344e7	0.712171
$838$	0	0
$839$	1.57939e7	0.774613	0.387307	−	0.921951i	$-0.373405\pi$
$839$	1.57939e7	0.774613	0.387307	+	0.921951i	$0.373405\pi$
$840$	0	0
$841$	4.74405e6	0.231291
$842$	0	0
$843$	−1.39952e7	−0.678282
$844$	0	0
$845$	0	0
$846$	0	0
$847$	744503.	0.0356581
$848$	0	0
$849$	3.10713e7	1.47942
$850$	0	0
$851$	−1.14139e7	−0.540267
$852$	0	0
$853$	2.09849e7	0.987492	0.493746	−	0.869606i	$-0.335627\pi$
$853$	2.09849e7	0.987492	0.493746	+	0.869606i	$0.335627\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.86989e7	−0.869688	−0.434844	−	0.900506i	$-0.643197\pi$
$857$	−1.86989e7	−0.869688	−0.434844	+	0.900506i	$0.643197\pi$
$858$	0	0
$859$	1.67571e6	0.0774848	0.0387424	−	0.999249i	$-0.487665\pi$
$859$	1.67571e6	0.0774848	0.0387424	+	0.999249i	$0.487665\pi$
$860$	0	0
$861$	2.66036e6	0.122302
$862$	0	0
$863$	−2.61307e7	−1.19433	−0.597164	−	0.802119i	$-0.703706\pi$
$863$	−2.61307e7	−1.19433	−0.597164	+	0.802119i	$0.703706\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	102007.	0.00460875
$868$	0	0
$869$	5.88161e6	0.264209
$870$	0	0
$871$	4.43720e7	1.98182
$872$	0	0
$873$	−5.93196e6	−0.263428
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−2.49393e7	−1.09493	−0.547464	−	0.836829i	$-0.684407\pi$
$877$	−2.49393e7	−1.09493	−0.547464	+	0.836829i	$0.684407\pi$
$878$	0	0
$879$	−3.02086e7	−1.31874
$880$	0	0
$881$	3.43259e7	1.48999	0.744993	−	0.667072i	$-0.232453\pi$
$881$	3.43259e7	1.48999	0.744993	+	0.667072i	$0.232453\pi$
$882$	0	0
$883$	3.05727e7	1.31957	0.659784	−	0.751455i	$-0.270648\pi$
$883$	3.05727e7	1.31957	0.659784	+	0.751455i	$0.270648\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.61328e7	0.688496	0.344248	−	0.938879i	$-0.388134\pi$
$887$	1.61328e7	0.688496	0.344248	+	0.938879i	$0.388134\pi$
$888$	0	0
$889$	−2.04545e6	−0.0868028
$890$	0	0
$891$	8.88418e6	0.374906
$892$	0	0
$893$	8.49344e6	0.356414
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.02631e7	0.425890
$898$	0	0
$899$	−2.76657e7	−1.14168
$900$	0	0
$901$	67781.8	0.00278164
$902$	0	0
$903$	1.50221e6	0.0613071
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.84427e7	0.744400	0.372200	−	0.928153i	$-0.378604\pi$
$907$	1.84427e7	0.744400	0.372200	+	0.928153i	$0.378604\pi$
$908$	0	0
$909$	−226788.	−0.00910354
$910$	0	0
$911$	1.20231e7	0.479976	0.239988	−	0.970776i	$-0.422856\pi$
$911$	1.20231e7	0.479976	0.239988	+	0.970776i	$0.422856\pi$
$912$	0	0
$913$	2.68117e6	0.106451
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−9.23979e6	−0.362860
$918$	0	0
$919$	−5.27736e6	−0.206124	−0.103062	−	0.994675i	$-0.532864\pi$
$919$	−5.27736e6	−0.206124	−0.103062	+	0.994675i	$0.532864\pi$
$920$	0	0
$921$	−4.78268e7	−1.85790
$922$	0	0
$923$	4.27618e7	1.65216
$924$	0	0
$925$	0	0
$926$	0	0
$927$	2.03209e7	0.776683
$928$	0	0
$929$	3.27326e7	1.24435	0.622174	−	0.782879i	$-0.286250\pi$
$929$	3.27326e7	1.24435	0.622174	+	0.782879i	$0.286250\pi$
$930$	0	0
$931$	5.66963e6	0.214378
$932$	0	0
$933$	3.26296e6	0.122718
$934$	0	0
$935$	0	0
$936$	0	0
$937$	3.10156e7	1.15407	0.577034	−	0.816720i	$-0.304210\pi$
$937$	3.10156e7	1.15407	0.577034	+	0.816720i	$0.304210\pi$
$938$	0	0
$939$	−2.86831e7	−1.06160
$940$	0	0
$941$	−1.59504e7	−0.587215	−0.293607	−	0.955926i	$-0.594856\pi$
$941$	−1.59504e7	−0.587215	−0.293607	+	0.955926i	$0.594856\pi$
$942$	0	0
$943$	−2.16122e6	−0.0791443
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.24106e7	−0.812042	−0.406021	−	0.913864i	$-0.633084\pi$
$947$	−2.24106e7	−0.812042	−0.406021	+	0.913864i	$0.633084\pi$
$948$	0	0
$949$	−6.15252e7	−2.21762
$950$	0	0
$951$	2.42299e7	0.868760
$952$	0	0
$953$	4.39636e7	1.56805	0.784027	−	0.620726i	$-0.213162\pi$
$953$	4.39636e7	1.56805	0.784027	+	0.620726i	$0.213162\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−1.12912e7	−0.398530
$958$	0	0
$959$	−1.88110e6	−0.0660488
$960$	0	0
$961$	1.67721e6	0.0585841
$962$	0	0
$963$	−2.03279e7	−0.706362
$964$	0	0
$965$	0	0
$966$	0	0
$967$	5.84126e6	0.200882	0.100441	−	0.994943i	$-0.467975\pi$
$967$	5.84126e6	0.200882	0.100441	+	0.994943i	$0.467975\pi$
$968$	0	0
$969$	−8.83810e6	−0.302377
$970$	0	0
$971$	−2.78862e7	−0.949163	−0.474582	−	0.880211i	$-0.657401\pi$
$971$	−2.78862e7	−0.949163	−0.474582	+	0.880211i	$0.657401\pi$
$972$	0	0
$973$	1.44349e6	0.0488801
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.86145e7	0.623901	0.311950	−	0.950098i	$-0.399018\pi$
$977$	1.86145e7	0.623901	0.311950	+	0.950098i	$0.399018\pi$
$978$	0	0
$979$	3.40257e6	0.113462
$980$	0	0
$981$	−9.82486e6	−0.325952
$982$	0	0
$983$	−2.61363e7	−0.862699	−0.431350	−	0.902185i	$-0.641962\pi$
$983$	−2.61363e7	−0.862699	−0.431350	+	0.902185i	$0.641962\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−2.01160e7	−0.657278
$988$	0	0
$989$	−1.22036e6	−0.0396733
$990$	0	0
$991$	3.41962e7	1.10610	0.553050	−	0.833148i	$-0.313464\pi$
$991$	3.41962e7	1.10610	0.553050	+	0.833148i	$0.313464\pi$
$992$	0	0
$993$	3.32960e7	1.07157
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−6.00850e7	−1.91438	−0.957189	−	0.289462i	$-0.906523\pi$
$997$	−6.00850e7	−1.91438	−0.957189	+	0.289462i	$0.906523\pi$
$998$	0	0
$999$	−3.90147e7	−1.23684

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1100.6.a.i.1.7	✓	8
5.2	odd	4		1100.6.b.h.749.4		16
5.3	odd	4		1100.6.b.h.749.13		16
5.4	even	2		1100.6.a.j.1.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1100.6.a.i.1.7	✓	8	1.1	even	1	trivial
1100.6.a.j.1.2	yes	8	5.4	even	2
1100.6.b.h.749.4		16	5.2	odd	4
1100.6.b.h.749.13		16	5.3	odd	4

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:	\( N \)	\(=\)	\( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1100.a (trivial)

\(f(q)\)	\(=\)	\(q+18.5686 q^{3} +50.8505 q^{7} +101.795 q^{9} -121.000 q^{11} -720.548 q^{13} +1193.88 q^{17} -398.674 q^{19} +944.226 q^{21} -767.070 q^{23} -2621.99 q^{27} +5025.45 q^{29} -5505.12 q^{31} -2246.81 q^{33} +14879.8 q^{37} -13379.6 q^{39} +2817.50 q^{41} +1590.94 q^{43} -21304.2 q^{47} -14221.2 q^{49} +22168.7 q^{51} +56.7744 q^{53} -7402.83 q^{57} -2976.29 q^{59} -22398.3 q^{61} +5176.32 q^{63} -61580.9 q^{67} -14243.5 q^{69} -59346.2 q^{71} +85386.6 q^{73} -6152.92 q^{77} -48608.4 q^{79} -73423.0 q^{81} -22158.4 q^{83} +93315.9 q^{87} -28120.4 q^{89} -36640.2 q^{91} -102223. q^{93} -58273.8 q^{97} -12317.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 9 q^{3} - 175 q^{7} + 377 q^{9} - 968 q^{11} + 1072 q^{13} - 37 q^{17} - 1356 q^{19} + 5462 q^{21} - 1327 q^{23} - 2670 q^{27} - 11785 q^{29} + 3532 q^{31} + 1089 q^{33} + 8176 q^{37} - 16831 q^{39}+ \cdots - 45617 q^{99}+O(q^{100}) \) 8 * q - 9 * q^3 - 175 * q^7 + 377 * q^9 - 968 * q^11 + 1072 * q^13 - 37 * q^17 - 1356 * q^19 + 5462 * q^21 - 1327 * q^23 - 2670 * q^27 - 11785 * q^29 + 3532 * q^31 + 1089 * q^33 + 8176 * q^37 - 16831 * q^39 + 15846 * q^41 + 530 * q^43 + 9252 * q^47 + 45745 * q^49 + 47733 * q^51 + 24991 * q^53 - 41957 * q^57 + 1440 * q^59 - 54447 * q^61 - 108968 * q^63 - 109232 * q^67 + 134200 * q^69 - 45026 * q^71 + 2221 * q^73 + 21175 * q^77 + 74369 * q^79 - 1132 * q^81 - 5443 * q^83 - 18665 * q^87 + 106151 * q^89 - 230723 * q^91 + 111315 * q^93 - 75273 * q^97 - 45617 * q^99

Introduction

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