LMFDB - Embedded newform 800.6.c.k.449.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,6,Mod(449,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("800.449");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	800.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$128.307055850$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.6140289600.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 32x^{4} + 116x^{3} + 256x^{2} + 2778x + 7605 $ x^6 - 2x^5 + 32x^4 + 116x^3 + 256x^2 + 2778*x + 7605
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.4
Root			$0.844467 - 4.86464i$ of defining polynomial
Character	$\chi$	$=$	800.449
Dual form			800.6.c.k.449.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.755735i q^{3} -172.424i q^{7} +242.429 q^{9} +O(q^{10})$	$q+0.755735i q^{3} -172.424i q^{7} +242.429 q^{9} -391.540 q^{11} +149.245i q^{13} -1187.74i q^{17} +685.033 q^{19} +130.307 q^{21} -996.236i q^{23} +366.856i q^{27} +8769.13 q^{29} -9525.61 q^{31} -295.901i q^{33} -10226.8i q^{37} -112.790 q^{39} +32.6496 q^{41} -10321.5i q^{43} +16931.7i q^{47} -12923.1 q^{49} +897.614 q^{51} -22287.0i q^{53} +517.704i q^{57} +44251.2 q^{59} -21743.1 q^{61} -41800.6i q^{63} +20709.0i q^{67} +752.891 q^{69} -15379.5 q^{71} +57995.4i q^{73} +67511.0i q^{77} -62466.5 q^{79} +58633.0 q^{81} +43578.7i q^{83} +6627.14i q^{87} -66297.9 q^{89} +25733.5 q^{91} -7198.84i q^{93} +12264.3i q^{97} -94920.6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 934 q^{9}+O(q^{10}) $ 6 * q - 934 * q^9	$ 6 q - 934 q^{9} + 792 q^{11} + 6384 q^{19} - 1640 q^{21} - 852 q^{29} - 6552 q^{31} + 42696 q^{39} + 24900 q^{41} + 7698 q^{49} - 142888 q^{51} + 70080 q^{59} - 48276 q^{61} + 54072 q^{69} - 176184 q^{71} - 185904 q^{79} + 302782 q^{81} - 345372 q^{89} + 213624 q^{91} - 702872 q^{99}+O(q^{100}) $ 6 * q - 934 * q^9 + 792 * q^11 + 6384 * q^19 - 1640 * q^21 - 852 * q^29 - 6552 * q^31 + 42696 * q^39 + 24900 * q^41 + 7698 * q^49 - 142888 * q^51 + 70080 * q^59 - 48276 * q^61 + 54072 * q^69 - 176184 * q^71 - 185904 * q^79 + 302782 * q^81 - 345372 * q^89 + 213624 * q^91 - 702872 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	0.755735i	0.0484804i	0.999706	+	0.0242402i	$0.00771666\pi$
$3$	0.755735i	0.0484804i	−0.999706	+	0.0242402i	$0.992283\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 172.424i	− 1.33001i	−0.746841	−	0.665003i	$-0.768430\pi$
$7$	− 172.424i	− 1.33001i	0.746841	−	0.665003i	$-0.231570\pi$
$8$	0	0
$9$	242.429	0.997650
$10$	0	0
$11$	−391.540	−0.975651	−0.487825	−	0.872941i	$-0.662210\pi$
$11$	−391.540	−0.975651	−0.487825	+	0.872941i	$0.662210\pi$
$12$	0	0
$13$	149.245i	0.244930i	0.992473	+	0.122465i	$0.0390800\pi$
$13$	149.245i	0.244930i	−0.992473	+	0.122465i	$0.960920\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1187.74i	− 0.996776i	−0.866954	−	0.498388i	$-0.833925\pi$
$17$	− 1187.74i	− 0.996776i	0.866954	−	0.498388i	$-0.166075\pi$
$18$	0	0
$19$	685.033	0.435339	0.217670	−	0.976023i	$-0.430155\pi$
$19$	685.033	0.435339	0.217670	+	0.976023i	$0.430155\pi$
$20$	0	0
$21$	130.307	0.0644792
$22$	0	0
$23$	− 996.236i	− 0.392684i	−0.980536	−	0.196342i	$-0.937094\pi$
$23$	− 996.236i	− 0.392684i	0.980536	−	0.196342i	$-0.0629062\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	366.856i	0.0968469i
$28$	0	0
$29$	8769.13	1.93625	0.968125	−	0.250466i	$-0.0805839\pi$
$29$	8769.13	1.93625	0.968125	+	0.250466i	$0.0805839\pi$
$30$	0	0
$31$	−9525.61	−1.78028	−0.890140	−	0.455686i	$-0.849394\pi$
$31$	−9525.61	−1.78028	−0.890140	+	0.455686i	$0.849394\pi$
$32$	0	0
$33$	− 295.901i	− 0.0473000i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 10226.8i	− 1.22811i	−0.789264	−	0.614054i	$-0.789538\pi$
$37$	− 10226.8i	− 1.22811i	0.789264	−	0.614054i	$-0.210462\pi$
$38$	0	0
$39$	−112.790	−0.0118743
$40$	0	0
$41$	32.6496	0.00303332	0.00151666	−	0.999999i	$-0.499517\pi$
$41$	32.6496	0.00303332	0.00151666	+	0.999999i	$0.499517\pi$
$42$	0	0
$43$	− 10321.5i	− 0.851279i	−0.904893	−	0.425639i	$-0.860049\pi$
$43$	− 10321.5i	− 0.851279i	0.904893	−	0.425639i	$-0.139951\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	16931.7i	1.11804i	0.829156	+	0.559018i	$0.188822\pi$
$47$	16931.7i	1.11804i	−0.829156	+	0.559018i	$0.811178\pi$
$48$	0	0
$49$	−12923.1	−0.768914
$50$	0	0
$51$	897.614	0.0483242
$52$	0	0
$53$	− 22287.0i	− 1.08984i	−0.838489	−	0.544919i	$-0.816561\pi$
$53$	− 22287.0i	− 1.08984i	0.838489	−	0.544919i	$-0.183439\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	517.704i	0.0211054i
$58$	0	0
$59$	44251.2	1.65499	0.827494	−	0.561474i	$-0.189766\pi$
$59$	44251.2	1.65499	0.827494	+	0.561474i	$0.189766\pi$
$60$	0	0
$61$	−21743.1	−0.748165	−0.374083	−	0.927395i	$-0.622042\pi$
$61$	−21743.1	−0.748165	−0.374083	+	0.927395i	$0.622042\pi$
$62$	0	0
$63$	− 41800.6i	− 1.32688i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	20709.0i	0.563600i	0.959473	+	0.281800i	$0.0909315\pi$
$67$	20709.0i	0.563600i	−0.959473	+	0.281800i	$0.909068\pi$
$68$	0	0
$69$	752.891	0.0190375
$70$	0	0
$71$	−15379.5	−0.362073	−0.181037	−	0.983476i	$-0.557945\pi$
$71$	−15379.5	−0.362073	−0.181037	+	0.983476i	$0.557945\pi$
$72$	0	0
$73$	57995.4i	1.27376i	0.770964	+	0.636878i	$0.219775\pi$
$73$	57995.4i	1.27376i	−0.770964	+	0.636878i	$0.780225\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	67511.0i	1.29762i
$78$	0	0
$79$	−62466.5	−1.12611	−0.563054	−	0.826420i	$-0.690374\pi$
$79$	−62466.5	−1.12611	−0.563054	+	0.826420i	$0.690374\pi$
$80$	0	0
$81$	58633.0	0.992954
$82$	0	0
$83$	43578.7i	0.694351i	0.937800	+	0.347175i	$0.112859\pi$
$83$	43578.7i	0.694351i	−0.937800	+	0.347175i	$0.887141\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6627.14i	0.0938703i
$88$	0	0
$89$	−66297.9	−0.887207	−0.443603	−	0.896223i	$-0.646300\pi$
$89$	−66297.9	−0.887207	−0.443603	+	0.896223i	$0.646300\pi$
$90$	0	0
$91$	25733.5	0.325758
$92$	0	0
$93$	− 7198.84i	− 0.0863088i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	12264.3i	0.132346i	0.997808	+	0.0661732i	$0.0210790\pi$
$97$	12264.3i	0.132346i	−0.997808	+	0.0661732i	$0.978921\pi$
$98$	0	0
$99$	−94920.6	−0.973358
$100$	0	0
$101$	−37376.8	−0.364584	−0.182292	−	0.983244i	$-0.558352\pi$
$101$	−37376.8	−0.364584	−0.182292	+	0.983244i	$0.558352\pi$
$102$	0	0
$103$	− 41642.9i	− 0.386765i	−0.981123	−	0.193383i	$-0.938054\pi$
$103$	− 41642.9i	− 0.386765i	0.981123	−	0.193383i	$-0.0619459\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 69664.7i	− 0.588239i	−0.955769	−	0.294119i	$-0.904974\pi$
$107$	− 69664.7i	− 0.588239i	0.955769	−	0.294119i	$-0.0950263\pi$
$108$	0	0
$109$	88554.2	0.713909	0.356954	−	0.934122i	$-0.383815\pi$
$109$	88554.2	0.713909	0.356954	+	0.934122i	$0.383815\pi$
$110$	0	0
$111$	7728.78	0.0595392
$112$	0	0
$113$	77809.3i	0.573239i	0.958045	+	0.286619i	$0.0925315\pi$
$113$	77809.3i	0.573239i	−0.958045	+	0.286619i	$0.907469\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	36181.4i	0.244355i
$118$	0	0
$119$	−204795.	−1.32572
$120$	0	0
$121$	−7747.38	−0.0481051
$122$	0	0
$123$	24.6744i	0 0.000147057i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 294580.i	− 1.62067i	−0.585970	−	0.810333i	$-0.699286\pi$
$127$	− 294580.i	− 1.62067i	0.585970	−	0.810333i	$-0.300714\pi$
$128$	0	0
$129$	7800.32	0.0412704
$130$	0	0
$131$	−173565.	−0.883658	−0.441829	−	0.897099i	$-0.645670\pi$
$131$	−173565.	−0.883658	−0.441829	+	0.897099i	$0.645670\pi$
$132$	0	0
$133$	− 118116.i	− 0.579003i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 346072.i	− 1.57531i	−0.616119	−	0.787653i	$-0.711296\pi$
$137$	− 346072.i	− 1.57531i	0.616119	−	0.787653i	$-0.288704\pi$
$138$	0	0
$139$	−329344.	−1.44581	−0.722907	−	0.690945i	$-0.757195\pi$
$139$	−329344.	−1.44581	−0.722907	+	0.690945i	$0.757195\pi$
$140$	0	0
$141$	−12795.9	−0.0542029
$142$	0	0
$143$	− 58435.5i	− 0.238966i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 9766.47i	− 0.0372773i
$148$	0	0
$149$	−83495.2	−0.308103	−0.154052	−	0.988063i	$-0.549232\pi$
$149$	−83495.2	−0.308103	−0.154052	+	0.988063i	$0.549232\pi$
$150$	0	0
$151$	155105.	0.553585	0.276792	−	0.960930i	$-0.410729\pi$
$151$	155105.	0.553585	0.276792	+	0.960930i	$0.410729\pi$
$152$	0	0
$153$	− 287941.i	− 0.994433i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	60634.6i	0.196323i	0.995170	+	0.0981615i	$0.0312962\pi$
$157$	60634.6i	0.196323i	−0.995170	+	0.0981615i	$0.968704\pi$
$158$	0	0
$159$	16843.1	0.0528358
$160$	0	0
$161$	−171775.	−0.522271
$162$	0	0
$163$	− 528292.i	− 1.55742i	−0.627387	−	0.778708i	$-0.715875\pi$
$163$	− 528292.i	− 1.55742i	0.627387	−	0.778708i	$-0.284125\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	681927.i	1.89211i	0.324002	+	0.946056i	$0.394971\pi$
$167$	681927.i	1.89211i	−0.324002	+	0.946056i	$0.605029\pi$
$168$	0	0
$169$	349019.	0.940009
$170$	0	0
$171$	166072.	0.434316
$172$	0	0
$173$	− 148111.i	− 0.376245i	−0.982146	−	0.188122i	$-0.939760\pi$
$173$	− 148111.i	− 0.376245i	0.982146	−	0.188122i	$-0.0602402\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	33442.2i	0.0802346i
$178$	0	0
$179$	−650289.	−1.51696	−0.758480	−	0.651697i	$-0.774058\pi$
$179$	−650289.	−1.51696	−0.758480	+	0.651697i	$0.774058\pi$
$180$	0	0
$181$	315052.	0.714803	0.357401	−	0.933951i	$-0.383663\pi$
$181$	315052.	0.714803	0.357401	+	0.933951i	$0.383663\pi$
$182$	0	0
$183$	− 16432.1i	− 0.0362714i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	465046.i	0.972506i
$188$	0	0
$189$	63254.8	0.128807
$190$	0	0
$191$	−851610.	−1.68911	−0.844554	−	0.535471i	$-0.820134\pi$
$191$	−851610.	−1.68911	−0.844554	+	0.535471i	$0.820134\pi$
$192$	0	0
$193$	916910.i	1.77188i	0.463804	+	0.885938i	$0.346484\pi$
$193$	916910.i	1.77188i	−0.463804	+	0.885938i	$0.653516\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 595345.i	− 1.09296i	−0.837473	−	0.546479i	$-0.815968\pi$
$197$	− 595345.i	− 1.09296i	0.837473	−	0.546479i	$-0.184032\pi$
$198$	0	0
$199$	−159504.	−0.285522	−0.142761	−	0.989757i	$-0.545598\pi$
$199$	−159504.	−0.285522	−0.142761	+	0.989757i	$0.545598\pi$
$200$	0	0
$201$	−15650.5	−0.0273236
$202$	0	0
$203$	− 1.51201e6i	− 2.57522i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 241516.i	− 0.391761i
$208$	0	0
$209$	−268218.	−0.424739
$210$	0	0
$211$	−776648.	−1.20093	−0.600466	−	0.799650i	$-0.705018\pi$
$211$	−776648.	−1.20093	−0.600466	+	0.799650i	$0.705018\pi$
$212$	0	0
$213$	− 11622.8i	− 0.0175535i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.64245e6i	2.36778i
$218$	0	0
$219$	−43829.2	−0.0617523
$220$	0	0
$221$	177264.	0.244141
$222$	0	0
$223$	− 969846.i	− 1.30599i	−0.757361	−	0.652996i	$-0.773512\pi$
$223$	− 969846.i	− 1.30599i	0.757361	−	0.652996i	$-0.226488\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	32926.6i	0.0424113i	0.999775	+	0.0212057i	$0.00675048\pi$
$227$	32926.6i	0.0424113i	−0.999775	+	0.0212057i	$0.993250\pi$
$228$	0	0
$229$	−730255.	−0.920208	−0.460104	−	0.887865i	$-0.652188\pi$
$229$	−730255.	−0.920208	−0.460104	+	0.887865i	$0.652188\pi$
$230$	0	0
$231$	−51020.5	−0.0629092
$232$	0	0
$233$	− 550888.i	− 0.664774i	−0.943143	−	0.332387i	$-0.892146\pi$
$233$	− 550888.i	− 0.664774i	0.943143	−	0.332387i	$-0.107854\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 47208.2i	− 0.0545942i
$238$	0	0
$239$	−996266.	−1.12819	−0.564093	−	0.825711i	$-0.690774\pi$
$239$	−996266.	−1.12819	−0.564093	+	0.825711i	$0.690774\pi$
$240$	0	0
$241$	−953235.	−1.05720	−0.528600	−	0.848871i	$-0.677283\pi$
$241$	−953235.	−1.05720	−0.528600	+	0.848871i	$0.677283\pi$
$242$	0	0
$243$	133457.i	0.144986i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	102238.i	0.106628i
$248$	0	0
$249$	−32933.9	−0.0336624
$250$	0	0
$251$	−780609.	−0.782077	−0.391038	−	0.920374i	$-0.627884\pi$
$251$	−780609.	−0.782077	−0.391038	+	0.920374i	$0.627884\pi$
$252$	0	0
$253$	390066.i	0.383122i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	340539.i	0.321614i	0.986986	+	0.160807i	$0.0514096\pi$
$257$	340539.i	0.321614i	−0.986986	+	0.160807i	$0.948590\pi$
$258$	0	0
$259$	−1.76335e6	−1.63339
$260$	0	0
$261$	2.12589e6	1.93170
$262$	0	0
$263$	− 2.05298e6i	− 1.83018i	−0.403246	−	0.915092i	$-0.632118\pi$
$263$	− 2.05298e6i	− 1.83018i	0.403246	−	0.915092i	$-0.367882\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 50103.7i	− 0.0430122i
$268$	0	0
$269$	−98629.8	−0.0831050	−0.0415525	−	0.999136i	$-0.513230\pi$
$269$	−98629.8	−0.0831050	−0.0415525	+	0.999136i	$0.513230\pi$
$270$	0	0
$271$	−1.17220e6	−0.969565	−0.484783	−	0.874635i	$-0.661101\pi$
$271$	−1.17220e6	−0.969565	−0.484783	+	0.874635i	$0.661101\pi$
$272$	0	0
$273$	19447.7i	0.0157929i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 1.13901e6i	− 0.891923i	−0.895052	−	0.445962i	$-0.852862\pi$
$277$	− 1.13901e6i	− 0.891923i	0.895052	−	0.445962i	$-0.147138\pi$
$278$	0	0
$279$	−2.30928e6	−1.77610
$280$	0	0
$281$	−1.33128e6	−1.00578	−0.502889	−	0.864351i	$-0.667730\pi$
$281$	−1.33128e6	−1.00578	−0.502889	+	0.864351i	$0.667730\pi$
$282$	0	0
$283$	921840.i	0.684210i	0.939662	+	0.342105i	$0.111140\pi$
$283$	921840.i	0.684210i	−0.939662	+	0.342105i	$0.888860\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 5629.58i	− 0.00403433i
$288$	0	0
$289$	9140.20	0.00643741
$290$	0	0
$291$	−9268.54	−0.00641621
$292$	0	0
$293$	− 1.64240e6i	− 1.11766i	−0.829281	−	0.558831i	$-0.811250\pi$
$293$	− 1.64240e6i	− 1.11766i	0.829281	−	0.558831i	$-0.188750\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 143639.i	− 0.0944888i
$298$	0	0
$299$	148684.	0.0961801
$300$	0	0
$301$	−1.77968e6	−1.13221
$302$	0	0
$303$	− 28246.9i	− 0.0176752i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 810866.i	− 0.491025i	−0.969393	−	0.245512i	$-0.921044\pi$
$307$	− 810866.i	− 0.491025i	0.969393	−	0.245512i	$-0.0789562\pi$
$308$	0	0
$309$	31471.0	0.0187506
$310$	0	0
$311$	1.67678e6	0.983050	0.491525	−	0.870864i	$-0.336440\pi$
$311$	1.67678e6	0.983050	0.491525	+	0.870864i	$0.336440\pi$
$312$	0	0
$313$	1.43902e6i	0.830242i	0.909766	+	0.415121i	$0.136261\pi$
$313$	1.43902e6i	0.830242i	−0.909766	+	0.415121i	$0.863739\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 783839.i	− 0.438105i	−0.975713	−	0.219053i	$-0.929703\pi$
$317$	− 783839.i	− 0.438105i	0.975713	−	0.219053i	$-0.0702967\pi$
$318$	0	0
$319$	−3.43347e6	−1.88910
$320$	0	0
$321$	52648.1	0.0285181
$322$	0	0
$323$	− 813639.i	− 0.433936i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	66923.5i	0.0346106i
$328$	0	0
$329$	2.91943e6	1.48699
$330$	0	0
$331$	3.30343e6	1.65728	0.828638	−	0.559785i	$-0.189116\pi$
$331$	3.30343e6	1.65728	0.828638	+	0.559785i	$0.189116\pi$
$332$	0	0
$333$	− 2.47928e6i	− 1.22522i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 2.17037e6i	− 1.04102i	−0.853856	−	0.520510i	$-0.825742\pi$
$337$	− 2.17037e6i	− 1.04102i	0.853856	−	0.520510i	$-0.174258\pi$
$338$	0	0
$339$	−58803.2	−0.0277909
$340$	0	0
$341$	3.72966e6	1.73693
$342$	0	0
$343$	− 669673.i	− 0.307346i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.26327e6i	0.563215i	0.959530	+	0.281607i	$0.0908676\pi$
$347$	1.26327e6i	0.563215i	−0.959530	+	0.281607i	$0.909132\pi$
$348$	0	0
$349$	1.84682e6	0.811636	0.405818	−	0.913954i	$-0.366987\pi$
$349$	1.84682e6	0.811636	0.405818	+	0.913954i	$0.366987\pi$
$350$	0	0
$351$	−54751.5	−0.0237207
$352$	0	0
$353$	3.82123e6i	1.63217i	0.577929	+	0.816087i	$0.303861\pi$
$353$	3.82123e6i	1.63217i	−0.577929	+	0.816087i	$0.696139\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 154770.i	− 0.0642714i
$358$	0	0
$359$	1.71131e6	0.700798	0.350399	−	0.936600i	$-0.386046\pi$
$359$	1.71131e6	0.700798	0.350399	+	0.936600i	$0.386046\pi$
$360$	0	0
$361$	−2.00683e6	−0.810480
$362$	0	0
$363$	− 5854.97i	− 0.00233216i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 1.72599e6i	− 0.668919i	−0.942410	−	0.334460i	$-0.891446\pi$
$367$	− 1.72599e6i	− 0.668919i	0.942410	−	0.334460i	$-0.108554\pi$
$368$	0	0
$369$	7915.20	0.00302619
$370$	0	0
$371$	−3.84282e6	−1.44949
$372$	0	0
$373$	732009.i	0.272423i	0.990680	+	0.136212i	$0.0434927\pi$
$373$	732009.i	0.272423i	−0.990680	+	0.136212i	$0.956507\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.30875e6i	0.474246i
$378$	0	0
$379$	−3.07751e6	−1.10053	−0.550264	−	0.834991i	$-0.685473\pi$
$379$	−3.07751e6	−1.10053	−0.550264	+	0.834991i	$0.685473\pi$
$380$	0	0
$381$	222624.	0.0785706
$382$	0	0
$383$	− 1.26468e6i	− 0.440540i	−0.975439	−	0.220270i	$-0.929306\pi$
$383$	− 1.26468e6i	− 0.440540i	0.975439	−	0.220270i	$-0.0706938\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 2.50223e6i	− 0.849278i
$388$	0	0
$389$	−3.65533e6	−1.22476	−0.612382	−	0.790562i	$-0.709788\pi$
$389$	−3.65533e6	−1.22476	−0.612382	+	0.790562i	$0.709788\pi$
$390$	0	0
$391$	−1.18327e6	−0.391418
$392$	0	0
$393$	− 131169.i	− 0.0428401i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	4.99465e6i	1.59048i	0.606294	+	0.795241i	$0.292655\pi$
$397$	4.99465e6i	1.59048i	−0.606294	+	0.795241i	$0.707345\pi$
$398$	0	0
$399$	89264.7	0.0280703
$400$	0	0
$401$	298037.	0.0925570	0.0462785	−	0.998929i	$-0.485264\pi$
$401$	298037.	0.0925570	0.0462785	+	0.998929i	$0.485264\pi$
$402$	0	0
$403$	− 1.42165e6i	− 0.436045i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.00421e6i	1.19821i
$408$	0	0
$409$	−3.26110e6	−0.963954	−0.481977	−	0.876184i	$-0.660081\pi$
$409$	−3.26110e6	−0.963954	−0.481977	+	0.876184i	$0.660081\pi$
$410$	0	0
$411$	261539.	0.0763715
$412$	0	0
$413$	− 7.62998e6i	− 2.20114i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 248897.i	− 0.0700937i
$418$	0	0
$419$	1.24623e6	0.346786	0.173393	−	0.984853i	$-0.444527\pi$
$419$	1.24623e6	0.346786	0.173393	+	0.984853i	$0.444527\pi$
$420$	0	0
$421$	−1.88625e6	−0.518674	−0.259337	−	0.965787i	$-0.583504\pi$
$421$	−1.88625e6	−0.518674	−0.259337	+	0.965787i	$0.583504\pi$
$422$	0	0
$423$	4.10473e6i	1.11541i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3.74905e6i	0.995064i
$428$	0	0
$429$	44161.8	0.0115852
$430$	0	0
$431$	−3.30534e6	−0.857082	−0.428541	−	0.903522i	$-0.640972\pi$
$431$	−3.30534e6	−0.857082	−0.428541	+	0.903522i	$0.640972\pi$
$432$	0	0
$433$	7.61270e6i	1.95128i	0.219384	+	0.975639i	$0.429595\pi$
$433$	7.61270e6i	1.95128i	−0.219384	+	0.975639i	$0.570405\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 682455.i	− 0.170951i
$438$	0	0
$439$	5.45916e6	1.35196	0.675982	−	0.736918i	$-0.263720\pi$
$439$	5.45916e6	1.35196	0.675982	+	0.736918i	$0.263720\pi$
$440$	0	0
$441$	−3.13294e6	−0.767107
$442$	0	0
$443$	− 5.09261e6i	− 1.23291i	−0.787391	−	0.616454i	$-0.788569\pi$
$443$	− 5.09261e6i	− 1.23291i	0.787391	−	0.616454i	$-0.211431\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 63100.3i	− 0.0149370i
$448$	0	0
$449$	−5.83842e6	−1.36672	−0.683360	−	0.730082i	$-0.739482\pi$
$449$	−5.83842e6	−1.36672	−0.683360	+	0.730082i	$0.739482\pi$
$450$	0	0
$451$	−12783.6	−0.00295946
$452$	0	0
$453$	117219.i	0.0268380i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.06917e6i	1.13539i	0.823238	+	0.567697i	$0.192165\pi$
$457$	5.06917e6i	1.13539i	−0.823238	+	0.567697i	$0.807835\pi$
$458$	0	0
$459$	435728.	0.0965347
$460$	0	0
$461$	6.10777e6	1.33854	0.669269	−	0.743021i	$-0.266608\pi$
$461$	6.10777e6	1.33854	0.669269	+	0.743021i	$0.266608\pi$
$462$	0	0
$463$	4.29238e6i	0.930563i	0.885163	+	0.465282i	$0.154047\pi$
$463$	4.29238e6i	0.930563i	−0.885163	+	0.465282i	$0.845953\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 1.64461e6i	− 0.348957i	−0.984661	−	0.174478i	$-0.944176\pi$
$467$	− 1.64461e6i	− 0.348957i	0.984661	−	0.174478i	$-0.0558239\pi$
$468$	0	0
$469$	3.57073e6	0.749591
$470$	0	0
$471$	−45823.7	−0.00951783
$472$	0	0
$473$	4.04128e6i	0.830551i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 5.40301e6i	− 1.08728i
$478$	0	0
$479$	8.11578e6	1.61619	0.808093	−	0.589055i	$-0.200500\pi$
$479$	8.11578e6	1.61619	0.808093	+	0.589055i	$0.200500\pi$
$480$	0	0
$481$	1.52631e6	0.300801
$482$	0	0
$483$	− 129817.i	− 0.0253199i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 6.11681e6i	− 1.16870i	−0.811502	−	0.584350i	$-0.801350\pi$
$487$	− 6.11681e6i	− 1.16870i	0.811502	−	0.584350i	$-0.198650\pi$
$488$	0	0
$489$	399249.	0.0755042
$490$	0	0
$491$	−6.78707e6	−1.27051	−0.635256	−	0.772301i	$-0.719105\pi$
$491$	−6.78707e6	−1.27051	−0.635256	+	0.772301i	$0.719105\pi$
$492$	0	0
$493$	− 1.04154e7i	− 1.93001i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.65180e6i	0.481560i
$498$	0	0
$499$	6.07689e6	1.09252	0.546260	−	0.837615i	$-0.316051\pi$
$499$	6.07689e6	1.09252	0.546260	+	0.837615i	$0.316051\pi$
$500$	0	0
$501$	−515357.	−0.0917305
$502$	0	0
$503$	− 1.89757e6i	− 0.334410i	−0.985922	−	0.167205i	$-0.946526\pi$
$503$	− 1.89757e6i	− 0.334410i	0.985922	−	0.167205i	$-0.0534741\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	263766.i	0.0455721i
$508$	0	0
$509$	7.07101e6	1.20973	0.604863	−	0.796330i	$-0.293228\pi$
$509$	7.07101e6	1.20973	0.604863	+	0.796330i	$0.293228\pi$
$510$	0	0
$511$	9.99982e6	1.69410
$512$	0	0
$513$	251308.i	0.0421613i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 6.62944e6i	− 1.09081i
$518$	0	0
$519$	111932.	0.0182405
$520$	0	0
$521$	−7.99273e6	−1.29003	−0.645017	−	0.764169i	$-0.723150\pi$
$521$	−7.99273e6	−1.29003	−0.645017	+	0.764169i	$0.723150\pi$
$522$	0	0
$523$	1.04897e7i	1.67691i	0.544974	+	0.838453i	$0.316540\pi$
$523$	1.04897e7i	1.67691i	−0.544974	+	0.838453i	$0.683460\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.13139e7i	1.77454i
$528$	0	0
$529$	5.44386e6	0.845800
$530$	0	0
$531$	1.07278e7	1.65110
$532$	0	0
$533$	4872.79i	0 0.000742951i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 491447.i	− 0.0735429i
$538$	0	0
$539$	5.05992e6	0.750191
$540$	0	0
$541$	6.56741e6	0.964719	0.482359	−	0.875973i	$-0.339780\pi$
$541$	6.56741e6	0.964719	0.482359	+	0.875973i	$0.339780\pi$
$542$	0	0
$543$	238096.i	0.0346540i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	5.90324e6i	0.843573i	0.906695	+	0.421786i	$0.138597\pi$
$547$	5.90324e6i	0.843573i	−0.906695	+	0.421786i	$0.861403\pi$
$548$	0	0
$549$	−5.27116e6	−0.746407
$550$	0	0
$551$	6.00715e6	0.842926
$552$	0	0
$553$	1.07707e7i	1.49773i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 3.34903e6i	− 0.457384i	−0.973499	−	0.228692i	$-0.926555\pi$
$557$	− 3.34903e6i	− 0.457384i	0.973499	−	0.228692i	$-0.0734448\pi$
$558$	0	0
$559$	1.54044e6	0.208504
$560$	0	0
$561$	−351452.	−0.0471475
$562$	0	0
$563$	1.34193e7i	1.78426i	0.451774	+	0.892132i	$0.350791\pi$
$563$	1.34193e7i	1.78426i	−0.451774	+	0.892132i	$0.649209\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 1.01097e7i	− 1.32063i
$568$	0	0
$569$	1.38493e7	1.79327	0.896635	−	0.442770i	$-0.146004\pi$
$569$	1.38493e7	1.79327	0.896635	+	0.442770i	$0.146004\pi$
$570$	0	0
$571$	8.24337e6	1.05807	0.529035	−	0.848600i	$-0.322554\pi$
$571$	8.24337e6	1.05807	0.529035	+	0.848600i	$0.322554\pi$
$572$	0	0
$573$	− 643592.i	− 0.0818887i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 9.90608e6i	− 1.23869i	−0.785119	−	0.619345i	$-0.787398\pi$
$577$	− 9.90608e6i	− 1.23869i	0.785119	−	0.619345i	$-0.212602\pi$
$578$	0	0
$579$	−692941.	−0.0859013
$580$	0	0
$581$	7.51402e6	0.923490
$582$	0	0
$583$	8.72625e6i	1.06330i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1.29572e7i	− 1.55209i	−0.630679	−	0.776044i	$-0.717224\pi$
$587$	− 1.29572e7i	− 1.55209i	0.630679	−	0.776044i	$-0.282776\pi$
$588$	0	0
$589$	−6.52536e6	−0.775026
$590$	0	0
$591$	449923.	0.0529871
$592$	0	0
$593$	− 5.95284e6i	− 0.695164i	−0.937650	−	0.347582i	$-0.887003\pi$
$593$	− 5.95284e6i	− 0.695164i	0.937650	−	0.347582i	$-0.112997\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 120543.i	− 0.0138423i
$598$	0	0
$599$	2.48855e6	0.283387	0.141694	−	0.989911i	$-0.454745\pi$
$599$	2.48855e6	0.283387	0.141694	+	0.989911i	$0.454745\pi$
$600$	0	0
$601$	−1.34152e7	−1.51499	−0.757497	−	0.652838i	$-0.773578\pi$
$601$	−1.34152e7	−1.51499	−0.757497	+	0.652838i	$0.773578\pi$
$602$	0	0
$603$	5.02045e6i	0.562276i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 8.49423e6i	− 0.935734i	−0.883799	−	0.467867i	$-0.845023\pi$
$607$	− 8.49423e6i	− 0.935734i	0.883799	−	0.467867i	$-0.154977\pi$
$608$	0	0
$609$	1.14268e6	0.124848
$610$	0	0
$611$	−2.52698e6	−0.273841
$612$	0	0
$613$	− 1.36612e7i	− 1.46837i	−0.678948	−	0.734187i	$-0.737564\pi$
$613$	− 1.36612e7i	− 1.46837i	0.678948	−	0.734187i	$-0.262436\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.43013e7i	1.51238i	0.654351	+	0.756191i	$0.272942\pi$
$617$	1.43013e7i	1.51238i	−0.654351	+	0.756191i	$0.727058\pi$
$618$	0	0
$619$	1.03774e7	1.08858	0.544292	−	0.838896i	$-0.316798\pi$
$619$	1.03774e7	1.08858	0.544292	+	0.838896i	$0.316798\pi$
$620$	0	0
$621$	365475.	0.0380302
$622$	0	0
$623$	1.14314e7i	1.17999i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 202702.i	− 0.0205915i
$628$	0	0
$629$	−1.21468e7	−1.22415
$630$	0	0
$631$	−7.91347e6	−0.791214	−0.395607	−	0.918420i	$-0.629466\pi$
$631$	−7.91347e6	−0.791214	−0.395607	+	0.918420i	$0.629466\pi$
$632$	0	0
$633$	− 586941.i	− 0.0582217i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 1.92872e6i	− 0.188330i
$638$	0	0
$639$	−3.72844e6	−0.361222
$640$	0	0
$641$	−5.70174e6	−0.548103	−0.274052	−	0.961715i	$-0.588364\pi$
$641$	−5.70174e6	−0.548103	−0.274052	+	0.961715i	$0.588364\pi$
$642$	0	0
$643$	3.06376e6i	0.292232i	0.989267	+	0.146116i	$0.0466773\pi$
$643$	3.06376e6i	0.292232i	−0.989267	+	0.146116i	$0.953323\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.36482e6i	0.128178i	0.997944	+	0.0640890i	$0.0204141\pi$
$647$	1.36482e6i	0.128178i	−0.997944	+	0.0640890i	$0.979586\pi$
$648$	0	0
$649$	−1.73261e7	−1.61469
$650$	0	0
$651$	−1.24125e6	−0.114791
$652$	0	0
$653$	8.58212e6i	0.787610i	0.919194	+	0.393805i	$0.128842\pi$
$653$	8.58212e6i	0.787610i	−0.919194	+	0.393805i	$0.871158\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	1.40598e7i	1.27076i
$658$	0	0
$659$	1.84222e7	1.65245	0.826224	−	0.563342i	$-0.190485\pi$
$659$	1.84222e7	1.65245	0.826224	+	0.563342i	$0.190485\pi$
$660$	0	0
$661$	1.69479e7	1.50873	0.754367	−	0.656453i	$-0.227944\pi$
$661$	1.69479e7	1.50873	0.754367	+	0.656453i	$0.227944\pi$
$662$	0	0
$663$	133965.i	0.0118360i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 8.73613e6i	− 0.760334i
$668$	0	0
$669$	732947.	0.0633151
$670$	0	0
$671$	8.51331e6	0.729948
$672$	0	0
$673$	1.18245e6i	0.100634i	0.998733	+	0.0503172i	$0.0160232\pi$
$673$	1.18245e6i	0.100634i	−0.998733	+	0.0503172i	$0.983977\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4.34424e6i	0.364285i	0.983272	+	0.182143i	$0.0583033\pi$
$677$	4.34424e6i	0.364285i	−0.983272	+	0.182143i	$0.941697\pi$
$678$	0	0
$679$	2.11466e6	0.176021
$680$	0	0
$681$	−24883.8	−0.00205612
$682$	0	0
$683$	1.03260e7i	0.846991i	0.905898	+	0.423495i	$0.139197\pi$
$683$	1.03260e7i	0.846991i	−0.905898	+	0.423495i	$0.860803\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 551880.i	− 0.0446121i
$688$	0	0
$689$	3.32623e6	0.266934
$690$	0	0
$691$	2.07013e7	1.64931	0.824656	−	0.565635i	$-0.191369\pi$
$691$	2.07013e7	1.64931	0.824656	+	0.565635i	$0.191369\pi$
$692$	0	0
$693$	1.63666e7i	1.29457i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 38779.1i	− 0.00302354i
$698$	0	0
$699$	416326.	0.0322285
$700$	0	0
$701$	−2.75116e6	−0.211457	−0.105728	−	0.994395i	$-0.533717\pi$
$701$	−2.75116e6	−0.211457	−0.105728	+	0.994395i	$0.533717\pi$
$702$	0	0
$703$	− 7.00572e6i	− 0.534643i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.44466e6i	0.484899i
$708$	0	0
$709$	4.07210e6	0.304230	0.152115	−	0.988363i	$-0.451392\pi$
$709$	4.07210e6	0.304230	0.152115	+	0.988363i	$0.451392\pi$
$710$	0	0
$711$	−1.51437e7	−1.12346
$712$	0	0
$713$	9.48976e6i	0.699087i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 752914.i	− 0.0546950i
$718$	0	0
$719$	1.75553e7	1.26645	0.633223	−	0.773970i	$-0.281732\pi$
$719$	1.75553e7	1.26645	0.633223	+	0.773970i	$0.281732\pi$
$720$	0	0
$721$	−7.18024e6	−0.514400
$722$	0	0
$723$	− 720394.i	− 0.0512536i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.46241e7i	1.02620i	0.858328	+	0.513101i	$0.171504\pi$
$727$	1.46241e7i	1.02620i	−0.858328	+	0.513101i	$0.828496\pi$
$728$	0	0
$729$	1.41470e7	0.985925
$730$	0	0
$731$	−1.22592e7	−0.848534
$732$	0	0
$733$	− 1.54961e7i	− 1.06528i	−0.846342	−	0.532639i	$-0.821200\pi$
$733$	− 1.54961e7i	− 1.06528i	0.846342	−	0.532639i	$-0.178800\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 8.10839e6i	− 0.549877i
$738$	0	0
$739$	3.39408e6	0.228619	0.114309	−	0.993445i	$-0.463535\pi$
$739$	3.39408e6	0.228619	0.114309	+	0.993445i	$0.463535\pi$
$740$	0	0
$741$	−77264.9	−0.00516936
$742$	0	0
$743$	3.94591e6i	0.262225i	0.991367	+	0.131113i	$0.0418550\pi$
$743$	3.94591e6i	0.262225i	−0.991367	+	0.131113i	$0.958145\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.05647e7i	0.692719i
$748$	0	0
$749$	−1.20119e7	−0.782360
$750$	0	0
$751$	824656.	0.0533547	0.0266774	−	0.999644i	$-0.491507\pi$
$751$	824656.	0.0533547	0.0266774	+	0.999644i	$0.491507\pi$
$752$	0	0
$753$	− 589934.i	− 0.0379154i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.53800e7i	0.975474i	0.872991	+	0.487737i	$0.162177\pi$
$757$	1.53800e7i	0.975474i	−0.872991	+	0.487737i	$0.837823\pi$
$758$	0	0
$759$	−294787.	−0.0185739
$760$	0	0
$761$	8.06295e6	0.504699	0.252349	−	0.967636i	$-0.418797\pi$
$761$	8.06295e6	0.504699	0.252349	+	0.967636i	$0.418797\pi$
$762$	0	0
$763$	− 1.52689e7i	− 0.949502i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6.60429e6i	0.405357i
$768$	0	0
$769$	1.12351e7	0.685110	0.342555	−	0.939498i	$-0.388708\pi$
$769$	1.12351e7	0.685110	0.342555	+	0.939498i	$0.388708\pi$
$770$	0	0
$771$	−257358.	−0.0155920
$772$	0	0
$773$	554517.i	0.0333784i	0.999861	+	0.0166892i	$0.00531259\pi$
$773$	554517.i	0.0333784i	−0.999861	+	0.0166892i	$0.994687\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 1.33263e6i	− 0.0791875i
$778$	0	0
$779$	22366.0	0.00132052
$780$	0	0
$781$	6.02169e6	0.353257
$782$	0	0
$783$	3.21701e6i	0.187520i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.56812e7i	0.902492i	0.892400	+	0.451246i	$0.149020\pi$
$787$	1.56812e7i	0.902492i	−0.892400	+	0.451246i	$0.850980\pi$
$788$	0	0
$789$	1.55151e6	0.0887281
$790$	0	0
$791$	1.34162e7	0.762410
$792$	0	0
$793$	− 3.24506e6i	− 0.183248i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 3.44267e7i	− 1.91977i	−0.280393	−	0.959885i	$-0.590465\pi$
$797$	− 3.44267e7i	− 1.91977i	0.280393	−	0.959885i	$-0.409535\pi$
$798$	0	0
$799$	2.01104e7	1.11443
$800$	0	0
$801$	−1.60725e7	−0.885122
$802$	0	0
$803$	− 2.27075e7i	− 1.24274i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 74538.0i	− 0.00402897i
$808$	0	0
$809$	−1.20856e7	−0.649230	−0.324615	−	0.945846i	$-0.605235\pi$
$809$	−1.20856e7	−0.649230	−0.324615	+	0.945846i	$0.605235\pi$
$810$	0	0
$811$	−6.20419e6	−0.331232	−0.165616	−	0.986190i	$-0.552961\pi$
$811$	−6.20419e6	−0.331232	−0.165616	+	0.986190i	$0.552961\pi$
$812$	0	0
$813$	− 885870.i	− 0.0470050i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 7.07057e6i	− 0.370595i
$818$	0	0
$819$	6.23855e6	0.324993
$820$	0	0
$821$	−2.59426e7	−1.34325	−0.671624	−	0.740892i	$-0.734403\pi$
$821$	−2.59426e7	−1.34325	−0.671624	+	0.740892i	$0.734403\pi$
$822$	0	0
$823$	7.30743e6i	0.376067i	0.982163	+	0.188034i	$0.0602114\pi$
$823$	7.30743e6i	0.376067i	−0.982163	+	0.188034i	$0.939789\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1.22800e7i	− 0.624357i	−0.950023	−	0.312179i	$-0.898941\pi$
$827$	− 1.22800e7i	− 0.624357i	0.950023	−	0.312179i	$-0.101059\pi$
$828$	0	0
$829$	3.21244e7	1.62349	0.811743	−	0.584014i	$-0.198519\pi$
$829$	3.21244e7	1.62349	0.811743	+	0.584014i	$0.198519\pi$
$830$	0	0
$831$	860789.	0.0432408
$832$	0	0
$833$	1.53493e7i	0.766435i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 3.49452e6i	− 0.172415i
$838$	0	0
$839$	−6.36956e6	−0.312395	−0.156198	−	0.987726i	$-0.549924\pi$
$839$	−6.36956e6	−0.312395	−0.156198	+	0.987726i	$0.549924\pi$
$840$	0	0
$841$	5.63865e7	2.74907
$842$	0	0
$843$	− 1.00609e6i	− 0.0487606i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.33584e6i	0.0639801i
$848$	0	0
$849$	−696667.	−0.0331708
$850$	0	0
$851$	−1.01883e7	−0.482258
$852$	0	0
$853$	− 4.05840e7i	− 1.90978i	−0.296964	−	0.954889i	$-0.595974\pi$
$853$	− 4.05840e7i	− 1.90978i	0.296964	−	0.954889i	$-0.404026\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 3.30412e6i	− 0.153675i	−0.997044	−	0.0768376i	$-0.975518\pi$
$857$	− 3.30412e6i	− 0.153675i	0.997044	−	0.0768376i	$-0.0244823\pi$
$858$	0	0
$859$	−1.45392e7	−0.672292	−0.336146	−	0.941810i	$-0.609124\pi$
$859$	−1.45392e7	−0.672292	−0.336146	+	0.941810i	$0.609124\pi$
$860$	0	0
$861$	4254.47	0.000195586 0
$862$	0	0
$863$	− 2.28020e7i	− 1.04219i	−0.853499	−	0.521094i	$-0.825524\pi$
$863$	− 2.28020e7i	− 1.04219i	0.853499	−	0.521094i	$-0.174476\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	6907.58i	0 0.000312089i
$868$	0	0
$869$	2.44582e7	1.09869
$870$	0	0
$871$	−3.09072e6	−0.138043
$872$	0	0
$873$	2.97321e6i	0.132035i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	7.93150e6i	0.348222i	0.984726	+	0.174111i	$0.0557052\pi$
$877$	7.93150e6i	0.348222i	−0.984726	+	0.174111i	$0.944295\pi$
$878$	0	0
$879$	1.24122e6	0.0541848
$880$	0	0
$881$	−3.31349e7	−1.43829	−0.719144	−	0.694861i	$-0.755466\pi$
$881$	−3.31349e7	−1.43829	−0.719144	+	0.694861i	$0.755466\pi$
$882$	0	0
$883$	2.25079e7i	0.971479i	0.874104	+	0.485740i	$0.161450\pi$
$883$	2.25079e7i	0.971479i	−0.874104	+	0.485740i	$0.838550\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 3.66717e7i	− 1.56503i	−0.622632	−	0.782515i	$-0.713937\pi$
$887$	− 3.66717e7i	− 1.56503i	0.622632	−	0.782515i	$-0.286063\pi$
$888$	0	0
$889$	−5.07927e7	−2.15549
$890$	0	0
$891$	−2.29572e7	−0.968777
$892$	0	0
$893$	1.15988e7i	0.486725i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	112365.i	0.00466285i
$898$	0	0
$899$	−8.35313e7	−3.44707
$900$	0	0
$901$	−2.64711e7	−1.08632
$902$	0	0
$903$	− 1.34497e6i	− 0.0548898i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 2.40388e6i	− 0.0970274i	−0.998823	−	0.0485137i	$-0.984552\pi$
$907$	− 2.40388e6i	− 0.0970274i	0.998823	−	0.0485137i	$-0.0154485\pi$
$908$	0	0
$909$	−9.06120e6	−0.363728
$910$	0	0
$911$	−467873.	−0.0186781	−0.00933904	−	0.999956i	$-0.502973\pi$
$911$	−467873.	−0.0186781	−0.00933904	+	0.999956i	$0.502973\pi$
$912$	0	0
$913$	− 1.70628e7i	− 0.677444i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.99268e7i	1.17527i
$918$	0	0
$919$	−2.59601e7	−1.01395	−0.506976	−	0.861960i	$-0.669237\pi$
$919$	−2.59601e7	−1.01395	−0.506976	+	0.861960i	$0.669237\pi$
$920$	0	0
$921$	612800.	0.0238051
$922$	0	0
$923$	− 2.29532e6i	− 0.0886827i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 1.00954e7i	− 0.385856i
$928$	0	0
$929$	4.52441e6	0.171998	0.0859989	−	0.996295i	$-0.472592\pi$
$929$	4.52441e6	0.171998	0.0859989	+	0.996295i	$0.472592\pi$
$930$	0	0
$931$	−8.85278e6	−0.334738
$932$	0	0
$933$	1.26720e6i	0.0476587i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 2.69269e7i	− 1.00193i	−0.865468	−	0.500964i	$-0.832979\pi$
$937$	− 2.69269e7i	− 1.00193i	0.865468	−	0.500964i	$-0.167021\pi$
$938$	0	0
$939$	−1.08751e6	−0.0402505
$940$	0	0
$941$	−1.38789e6	−0.0510953	−0.0255477	−	0.999674i	$-0.508133\pi$
$941$	−1.38789e6	−0.0510953	−0.0255477	+	0.999674i	$0.508133\pi$
$942$	0	0
$943$	− 32526.7i	− 0.00119113i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1.10208e7i	− 0.399335i	−0.979864	−	0.199667i	$-0.936014\pi$
$947$	− 1.10208e7i	− 0.399335i	0.979864	−	0.199667i	$-0.0639862\pi$
$948$	0	0
$949$	−8.65555e6	−0.311982
$950$	0	0
$951$	592375.	0.0212396
$952$	0	0
$953$	− 1.09536e7i	− 0.390683i	−0.980735	−	0.195342i	$-0.937418\pi$
$953$	− 1.09536e7i	− 0.390683i	0.980735	−	0.195342i	$-0.0625815\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 2.59479e6i	− 0.0915847i
$958$	0	0
$959$	−5.96712e7	−2.09517
$960$	0	0
$961$	6.21081e7	2.16940
$962$	0	0
$963$	− 1.68887e7i	− 0.586856i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	3.52685e7i	1.21289i	0.795127	+	0.606443i	$0.207404\pi$
$967$	3.52685e7i	1.21289i	−0.795127	+	0.606443i	$0.792596\pi$
$968$	0	0
$969$	614895.	0.0210374
$970$	0	0
$971$	3.58480e7	1.22016	0.610079	−	0.792340i	$-0.291138\pi$
$971$	3.58480e7	1.22016	0.610079	+	0.792340i	$0.291138\pi$
$972$	0	0
$973$	5.67869e7i	1.92294i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	8.95941e6i	0.300292i	0.988664	+	0.150146i	$0.0479743\pi$
$977$	8.95941e6i	0.300292i	−0.988664	+	0.150146i	$0.952026\pi$
$978$	0	0
$979$	2.59583e7	0.865604
$980$	0	0
$981$	2.14681e7	0.712231
$982$	0	0
$983$	− 6.94666e6i	− 0.229294i	−0.993406	−	0.114647i	$-0.963426\pi$
$983$	− 6.94666e6i	− 0.229294i	0.993406	−	0.114647i	$-0.0365737\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2.20632e6i	0.0720901i
$988$	0	0
$989$	−1.02827e7	−0.334283
$990$	0	0
$991$	7.72743e6	0.249949	0.124974	−	0.992160i	$-0.460115\pi$
$991$	7.72743e6	0.249949	0.124974	+	0.992160i	$0.460115\pi$
$992$	0	0
$993$	2.49652e6i	0.0803455i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	4.32625e7i	1.37840i	0.724573	+	0.689198i	$0.242037\pi$
$997$	4.32625e7i	1.37840i	−0.724573	+	0.689198i	$0.757963\pi$
$998$	0	0
$999$	3.75177e6	0.118939

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.6.c.k.449.4		6
4.3	odd	2		800.6.c.j.449.3		6
5.2	odd	4		160.6.a.f.1.2	✓	3
5.3	odd	4		800.6.a.o.1.2		3
5.4	even	2	inner	800.6.c.k.449.3		6
20.3	even	4		800.6.a.n.1.2		3
20.7	even	4		160.6.a.g.1.2	yes	3
20.19	odd	2		800.6.c.j.449.4		6
40.27	even	4		320.6.a.x.1.2		3
40.37	odd	4		320.6.a.y.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.6.a.f.1.2	✓	3	5.2	odd	4
160.6.a.g.1.2	yes	3	20.7	even	4
320.6.a.x.1.2		3	40.27	even	4
320.6.a.y.1.2		3	40.37	odd	4
800.6.a.n.1.2		3	20.3	even	4
800.6.a.o.1.2		3	5.3	odd	4
800.6.c.j.449.3		6	4.3	odd	2
800.6.c.j.449.4		6	20.19	odd	2
800.6.c.k.449.3		6	5.4	even	2	inner
800.6.c.k.449.4		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	800.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.755735i q^{3} -172.424i q^{7} +242.429 q^{9} +O(q^{10})\)	\(q+0.755735i q^{3} -172.424i q^{7} +242.429 q^{9} -391.540 q^{11} +149.245i q^{13} -1187.74i q^{17} +685.033 q^{19} +130.307 q^{21} -996.236i q^{23} +366.856i q^{27} +8769.13 q^{29} -9525.61 q^{31} -295.901i q^{33} -10226.8i q^{37} -112.790 q^{39} +32.6496 q^{41} -10321.5i q^{43} +16931.7i q^{47} -12923.1 q^{49} +897.614 q^{51} -22287.0i q^{53} +517.704i q^{57} +44251.2 q^{59} -21743.1 q^{61} -41800.6i q^{63} +20709.0i q^{67} +752.891 q^{69} -15379.5 q^{71} +57995.4i q^{73} +67511.0i q^{77} -62466.5 q^{79} +58633.0 q^{81} +43578.7i q^{83} +6627.14i q^{87} -66297.9 q^{89} +25733.5 q^{91} -7198.84i q^{93} +12264.3i q^{97} -94920.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 934 q^{9}+O(q^{10}) \) 6 * q - 934 * q^9	\( 6 q - 934 q^{9} + 792 q^{11} + 6384 q^{19} - 1640 q^{21} - 852 q^{29} - 6552 q^{31} + 42696 q^{39} + 24900 q^{41} + 7698 q^{49} - 142888 q^{51} + 70080 q^{59} - 48276 q^{61} + 54072 q^{69} - 176184 q^{71} - 185904 q^{79} + 302782 q^{81} - 345372 q^{89} + 213624 q^{91} - 702872 q^{99}+O(q^{100}) \) 6 * q - 934 * q^9 + 792 * q^11 + 6384 * q^19 - 1640 * q^21 - 852 * q^29 - 6552 * q^31 + 42696 * q^39 + 24900 * q^41 + 7698 * q^49 - 142888 * q^51 + 70080 * q^59 - 48276 * q^61 + 54072 * q^69 - 176184 * q^71 - 185904 * q^79 + 302782 * q^81 - 345372 * q^89 + 213624 * q^91 - 702872 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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