LMFDB - Embedded newform 400.6.n.h.207.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,6,Mod(143,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 3]))

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

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

chi := DirichletCharacter("400.143");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$400 = 2^{4} \cdot 5^{2}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	400.n (of order $4$ , degree $2$ , 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:	$64.1535279252$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			207.4
Character	$\chi$	$=$	400.207
Dual form			400.6.n.h.143.4

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-13.2732 - 13.2732i) q^{3} +(25.6490 - 25.6490i) q^{7} +109.358i q^{9} +16.4579i q^{11} +(212.988 - 212.988i) q^{13} +(642.159 + 642.159i) q^{17} -1286.91 q^{19} -680.891 q^{21} +(562.518 + 562.518i) q^{23} +(-1773.86 + 1773.86i) q^{27} +6977.71i q^{29} -5017.46i q^{31} +(218.449 - 218.449i) q^{33} +(10553.8 + 10553.8i) q^{37} -5654.09 q^{39} -10982.2 q^{41} +(5190.60 + 5190.60i) q^{43} +(42.6550 - 42.6550i) q^{47} +15491.3i q^{49} -17047.1i q^{51} +(-1518.50 + 1518.50i) q^{53} +(17081.5 + 17081.5i) q^{57} -28156.2 q^{59} +15802.2 q^{61} +(2804.92 + 2804.92i) q^{63} +(27461.7 - 27461.7i) q^{67} -14932.9i q^{69} -45708.4i q^{71} +(-5310.40 + 5310.40i) q^{73} +(422.128 + 422.128i) q^{77} +45332.5 q^{79} +73663.8 q^{81} +(66159.8 + 66159.8i) q^{83} +(92616.8 - 92616.8i) q^{87} -79364.6i q^{89} -10925.9i q^{91} +(-66598.0 + 66598.0i) q^{93} +(-22001.3 - 22001.3i) q^{97} -1799.80 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$24 q + 19936 q^{21} + 32568 q^{41} + 454944 q^{61} + 1059176 q^{81}+O(q^{100})$ 24 * q + 19936 * q^21 + 32568 * q^41 + 454944 * q^61 + 1059176 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$ .

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$	$-1$

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
$3$	−13.2732	−	13.2732i	−0.851479	−	0.851479i	0.138836	−	0.990315i	$-0.455664\pi$
$3$	−13.2732	−	13.2732i	−0.851479	−	0.851479i	−0.990315	+	0.138836i	$0.955664\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	25.6490	−	25.6490i	0.197845	−	0.197845i	−0.601231	−	0.799076i	$-0.705323\pi$
$7$	25.6490	−	25.6490i	0.197845	−	0.197845i	0.799076	+	0.601231i	$0.205323\pi$
$8$		0			0
$9$			109.358i			0.450033i
$10$		0			0
$11$			16.4579i			0.0410102i	0.999790	+	0.0205051i	$0.00652743\pi$
$11$			16.4579i			0.0410102i	−0.999790	+	0.0205051i	$0.993473\pi$
$12$		0			0
$13$	212.988	−	212.988i	0.349540	−	0.349540i	−0.510398	−	0.859938i	$-0.670502\pi$
$13$	212.988	−	212.988i	0.349540	−	0.349540i	0.859938	+	0.510398i	$0.170502\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	642.159	+	642.159i	0.538915	+	0.538915i	0.923210	−	0.384295i	$-0.125556\pi$
$17$	642.159	+	642.159i	0.538915	+	0.538915i	−0.384295	+	0.923210i	$0.625556\pi$
$18$		0			0
$19$	−1286.91			−0.817834			−0.408917	−	0.912572i	$-0.634094\pi$
$19$	−1286.91			−0.817834			−0.408917	+	0.912572i	$0.634094\pi$
$20$		0			0
$21$	−680.891			−0.336922
$22$		0			0
$23$	562.518	+	562.518i	0.221726	+	0.221726i	0.809225	−	0.587499i	$-0.199887\pi$
$23$	562.518	+	562.518i	0.221726	+	0.221726i	−0.587499	+	0.809225i	$0.699887\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−1773.86	+	1773.86i	−0.468286	+	0.468286i
$28$		0			0
$29$			6977.71i			1.54070i	0.637622	+	0.770349i	$0.279918\pi$
$29$			6977.71i			1.54070i	−0.637622	+	0.770349i	$0.720082\pi$
$30$		0			0
$31$		−	5017.46i		−	0.937734i	−0.883269	−	0.468867i	$-0.844662\pi$
$31$		−	5017.46i		−	0.937734i	0.883269	−	0.468867i	$-0.155338\pi$
$32$		0			0
$33$	218.449	−	218.449i	0.0349193	−	0.0349193i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	10553.8	+	10553.8i	1.26737	+	1.26737i	0.947440	+	0.319932i	$0.103660\pi$
$37$	10553.8	+	10553.8i	1.26737	+	1.26737i	0.319932	+	0.947440i	$0.396340\pi$
$38$		0			0
$39$	−5654.09			−0.595253
$40$		0			0
$41$	−10982.2			−1.02030			−0.510150	−	0.860085i	$-0.670410\pi$
$41$	−10982.2			−1.02030			−0.510150	+	0.860085i	$0.670410\pi$
$42$		0			0
$43$	5190.60	+	5190.60i	0.428101	+	0.428101i	0.887981	−	0.459880i	$-0.152108\pi$
$43$	5190.60	+	5190.60i	0.428101	+	0.428101i	−0.459880	+	0.887981i	$0.652108\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	42.6550	−	42.6550i	0.00281660	−	0.00281660i	−0.705697	−	0.708514i	$-0.749366\pi$
$47$	42.6550	−	42.6550i	0.00281660	−	0.00281660i	0.708514	+	0.705697i	$0.249366\pi$
$48$		0			0
$49$			15491.3i			0.921715i
$50$		0			0
$51$		−	17047.1i		−	0.917749i
$52$		0			0
$53$	−1518.50	+	1518.50i	−0.0742548	+	0.0742548i	−0.743259	−	0.669004i	$-0.766721\pi$
$53$	−1518.50	+	1518.50i	−0.0742548	+	0.0742548i	0.669004	+	0.743259i	$0.266721\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	17081.5	+	17081.5i	0.696369	+	0.696369i
$58$		0			0
$59$	−28156.2			−1.05304			−0.526520	−	0.850163i	$-0.676503\pi$
$59$	−28156.2			−1.05304			−0.526520	+	0.850163i	$0.676503\pi$
$60$		0			0
$61$	15802.2			0.543741			0.271870	−	0.962334i	$-0.412358\pi$
$61$	15802.2			0.543741			0.271870	+	0.962334i	$0.412358\pi$
$62$		0			0
$63$	2804.92	+	2804.92i	0.0890368	+	0.0890368i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	27461.7	−	27461.7i	0.747379	−	0.747379i	−0.226607	−	0.973986i	$-0.572763\pi$
$67$	27461.7	−	27461.7i	0.747379	−	0.747379i	0.973986	+	0.226607i	$0.0727634\pi$
$68$		0			0
$69$		−	14932.9i		−	0.377590i
$70$		0			0
$71$		−	45708.4i		−	1.07609i	−0.842915	−	0.538047i	$-0.819162\pi$
$71$		−	45708.4i		−	1.07609i	0.842915	−	0.538047i	$-0.180838\pi$
$72$		0			0
$73$	−5310.40	+	5310.40i	−0.116633	+	0.116633i	−0.763014	−	0.646382i	$-0.776281\pi$
$73$	−5310.40	+	5310.40i	−0.116633	+	0.116633i	0.646382	+	0.763014i	$0.276281\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	422.128	+	422.128i	0.00811366	+	0.00811366i
$78$		0			0
$79$	45332.5			0.817225			0.408613	−	0.912708i	$-0.366013\pi$
$79$	45332.5			0.817225			0.408613	+	0.912708i	$0.366013\pi$
$80$		0			0
$81$	73663.8			1.24750
$82$		0			0
$83$	66159.8	+	66159.8i	1.05414	+	1.05414i	0.998448	+	0.0556935i	$0.0177370\pi$
$83$	66159.8	+	66159.8i	1.05414	+	1.05414i	0.0556935	+	0.998448i	$0.482263\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	92616.8	−	92616.8i	1.31187	−	1.31187i
$88$		0			0
$89$		−	79364.6i		−	1.06207i	−0.847351	−	0.531033i	$-0.821804\pi$
$89$		−	79364.6i		−	1.06207i	0.847351	−	0.531033i	$-0.178196\pi$
$90$		0			0
$91$		−	10925.9i		−	0.138310i
$92$		0			0
$93$	−66598.0	+	66598.0i	−0.798461	+	0.798461i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−22001.3	−	22001.3i	−0.237421	−	0.237421i	0.578360	−	0.815781i	$-0.303693\pi$
$97$	−22001.3	−	22001.3i	−0.237421	−	0.237421i	−0.815781	+	0.578360i	$0.803693\pi$
$98$		0			0
$99$	−1799.80			−0.0184559
$100$		0			0
$101$	123331.			1.20301			0.601503	−	0.798871i	$-0.294569\pi$
$101$	123331.			1.20301			0.601503	+	0.798871i	$0.294569\pi$
$102$		0			0
$103$	−8217.57	−	8217.57i	−0.0763221	−	0.0763221i	0.667915	−	0.744237i	$-0.267187\pi$
$103$	−8217.57	−	8217.57i	−0.0763221	−	0.0763221i	−0.744237	+	0.667915i	$0.767187\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	149735.	−	149735.i	1.26434	−	1.26434i	0.315376	−	0.948967i	$-0.397869\pi$
$107$	149735.	−	149735.i	1.26434	−	1.26434i	0.948967	−	0.315376i	$-0.102131\pi$
$108$		0			0
$109$		−	133191.i		−	1.07376i	−0.843657	−	0.536882i	$-0.819602\pi$
$109$		−	133191.i		−	1.07376i	0.843657	−	0.536882i	$-0.180398\pi$
$110$		0			0
$111$		−	280166.i		−	2.15828i
$112$		0			0
$113$	−46530.7	+	46530.7i	−0.342802	+	0.342802i	−0.857420	−	0.514618i	$-0.827934\pi$
$113$	−46530.7	+	46530.7i	−0.342802	+	0.342802i	0.514618	+	0.857420i	$0.327934\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	23292.0	+	23292.0i	0.157305	+	0.157305i
$118$		0			0
$119$	32941.5			0.213243
$120$		0			0
$121$	160780.			0.998318
$122$		0			0
$123$	145769.	+	145769.i	0.868764	+	0.868764i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	144960.	−	144960.i	0.797516	−	0.797516i	−0.185187	−	0.982703i	$-0.559289\pi$
$127$	144960.	−	144960.i	0.797516	−	0.797516i	0.982703	+	0.185187i	$0.0592891\pi$
$128$		0			0
$129$		−	137792.i		−	0.729039i
$130$		0			0
$131$			8473.48i			0.0431403i	0.999767	+	0.0215702i	$0.00686653\pi$
$131$			8473.48i			0.0431403i	−0.999767	+	0.0215702i	$0.993133\pi$
$132$		0			0
$133$	−33008.0	+	33008.0i	−0.161804	+	0.161804i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	209097.	+	209097.i	0.951801	+	0.951801i	0.998891	−	0.0470895i	$-0.0149946\pi$
$137$	209097.	+	209097.i	0.951801	+	0.951801i	−0.0470895	+	0.998891i	$0.514995\pi$
$138$		0			0
$139$	−239888.			−1.05310			−0.526552	−	0.850143i	$-0.676515\pi$
$139$	−239888.			−1.05310			−0.526552	+	0.850143i	$0.676515\pi$
$140$		0			0
$141$	−1132.34			−0.00479656
$142$		0			0
$143$	3505.33	+	3505.33i	0.0143347	+	0.0143347i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	205619.	−	205619.i	0.784821	−	0.784821i
$148$		0			0
$149$			400371.i			1.47740i	0.674036	+	0.738699i	$0.264559\pi$
$149$			400371.i			1.47740i	−0.674036	+	0.738699i	$0.735441\pi$
$150$		0			0
$151$		−	496995.i		−	1.77382i	−0.461943	−	0.886910i	$-0.652848\pi$
$151$		−	496995.i		−	1.77382i	0.461943	−	0.886910i	$-0.347152\pi$
$152$		0			0
$153$	−70225.2	+	70225.2i	−0.242529	+	0.242529i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	165920.	+	165920.i	0.537217	+	0.537217i	0.922710	−	0.385494i	$-0.125969\pi$
$157$	165920.	+	165920.i	0.537217	+	0.537217i	−0.385494	+	0.922710i	$0.625969\pi$
$158$		0			0
$159$	40310.8			0.126453
$160$		0			0
$161$	28856.0			0.0877348
$162$		0			0
$163$	399841.	+	399841.i	1.17874	+	1.17874i	0.980066	+	0.198675i	$0.0636637\pi$
$163$	399841.	+	399841.i	1.17874	+	1.17874i	0.198675	+	0.980066i	$0.436336\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−152879.	+	152879.i	−0.424187	+	0.424187i	−0.886642	−	0.462456i	$-0.846968\pi$
$167$	−152879.	+	152879.i	−0.424187	+	0.424187i	0.462456	+	0.886642i	$0.346968\pi$
$168$		0			0
$169$			280565.i			0.755643i
$170$		0			0
$171$		−	140734.i		−	0.368052i
$172$		0			0
$173$	40227.0	−	40227.0i	0.102189	−	0.102189i	−0.654164	−	0.756353i	$-0.726979\pi$
$173$	40227.0	−	40227.0i	0.102189	−	0.102189i	0.756353	+	0.654164i	$0.226979\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	373725.	+	373725.i	0.896641	+	0.896641i
$178$		0			0
$179$	−418256.			−0.975685			−0.487843	−	0.872932i	$-0.662216\pi$
$179$	−418256.			−0.975685			−0.487843	+	0.872932i	$0.662216\pi$
$180$		0			0
$181$	63219.1			0.143434			0.0717170	−	0.997425i	$-0.477152\pi$
$181$	63219.1			0.143434			0.0717170	+	0.997425i	$0.477152\pi$
$182$		0			0
$183$	−209746.	−	209746.i	−0.462984	−	0.462984i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−10568.6	+	10568.6i	−0.0221010	+	0.0221010i
$188$		0			0
$189$			90995.6i			0.185296i
$190$		0			0
$191$		−	497355.i		−	0.986467i	−0.869897	−	0.493234i	$-0.835815\pi$
$191$		−	497355.i		−	0.986467i	0.869897	−	0.493234i	$-0.164185\pi$
$192$		0			0
$193$	238746.	−	238746.i	0.461364	−	0.461364i	−0.437739	−	0.899102i	$-0.644220\pi$
$193$	238746.	−	238746.i	0.461364	−	0.461364i	0.899102	+	0.437739i	$0.144220\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	64625.8	+	64625.8i	0.118643	+	0.118643i	0.763935	−	0.645293i	$-0.223265\pi$
$197$	64625.8	+	64625.8i	0.118643	+	0.118643i	−0.645293	+	0.763935i	$0.723265\pi$
$198$		0			0
$199$	−133934.			−0.239750			−0.119875	−	0.992789i	$-0.538249\pi$
$199$	−133934.			−0.239750			−0.119875	+	0.992789i	$0.538249\pi$
$200$		0			0
$201$	−729012.			−1.27275
$202$		0			0
$203$	178971.	+	178971.i	0.304820	+	0.304820i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−61515.8	+	61515.8i	−0.0997840	+	0.0997840i
$208$		0			0
$209$		−	21179.8i		−	0.0335395i
$210$		0			0
$211$			892854.i			1.38062i	0.723513	+	0.690310i	$0.242526\pi$
$211$			892854.i			1.38062i	−0.723513	+	0.690310i	$0.757474\pi$
$212$		0			0
$213$	−606698.	+	606698.i	−0.916271	+	0.916271i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−128693.	−	128693.i	−0.185526	−	0.185526i
$218$		0			0
$219$	140973.			0.198621
$220$		0			0
$221$	273545.			0.376745
$222$		0			0
$223$	−460278.	−	460278.i	−0.619809	−	0.619809i	0.325674	−	0.945482i	$-0.394409\pi$
$223$	−460278.	−	460278.i	−0.619809	−	0.619809i	−0.945482	+	0.325674i	$0.894409\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	936.286	−	936.286i	0.00120599	−	0.00120599i	−0.706504	−	0.707710i	$-0.749729\pi$
$227$	936.286	−	936.286i	0.00120599	−	0.00120599i	0.707710	+	0.706504i	$0.249729\pi$
$228$		0			0
$229$			94774.2i			0.119427i	0.998216	+	0.0597134i	$0.0190187\pi$
$229$			94774.2i			0.119427i	−0.998216	+	0.0597134i	$0.980981\pi$
$230$		0			0
$231$		−	11206.0i		−	0.0138172i
$232$		0			0
$233$	739323.	−	739323.i	0.892163	−	0.892163i	−0.102563	−	0.994727i	$-0.532704\pi$
$233$	739323.	−	739323.i	0.892163	−	0.892163i	0.994727	+	0.102563i	$0.0327044\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−601709.	−	601709.i	−0.695850	−	0.695850i
$238$		0			0
$239$	917906.			1.03945			0.519725	−	0.854334i	$-0.326035\pi$
$239$	917906.			1.03945			0.519725	+	0.854334i	$0.326035\pi$
$240$		0			0
$241$	481284.			0.533775			0.266888	−	0.963728i	$-0.414005\pi$
$241$	481284.			0.533775			0.266888	+	0.963728i	$0.414005\pi$
$242$		0			0
$243$	−546709.	−	546709.i	−0.593937	−	0.593937i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−274098.	+	274098.i	−0.285866	+	0.285866i
$248$		0			0
$249$		−	1.75631e6i		−	1.79516i
$250$		0			0
$251$			1.00677e6i			1.00866i	0.863510	+	0.504331i	$0.168261\pi$
$251$			1.00677e6i			1.00866i	−0.863510	+	0.504331i	$0.831739\pi$
$252$		0			0
$253$	−9257.84	+	9257.84i	−0.00909303	+	0.00909303i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.13074e6	−	1.13074e6i	−1.06789	−	1.06789i	−0.997521	−	0.0703733i	$-0.977581\pi$
$257$	−1.13074e6	−	1.13074e6i	−1.06789	−	1.06789i	−0.0703733	−	0.997521i	$-0.522419\pi$
$258$		0			0
$259$	541388.			0.501487
$260$		0			0
$261$	−763068.			−0.693365
$262$		0			0
$263$	232140.	+	232140.i	0.206948	+	0.206948i	0.802969	−	0.596021i	$-0.203252\pi$
$263$	232140.	+	232140.i	0.206948	+	0.206948i	−0.596021	+	0.802969i	$0.703252\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−1.05343e6	+	1.05343e6i	−0.904327	+	0.904327i
$268$		0			0
$269$			484347.i			0.408109i	0.978960	+	0.204054i	$0.0654119\pi$
$269$			484347.i			0.408109i	−0.978960	+	0.204054i	$0.934588\pi$
$270$		0			0
$271$			2.10660e6i			1.74244i	0.490893	+	0.871220i	$0.336671\pi$
$271$			2.10660e6i			1.74244i	−0.490893	+	0.871220i	$0.663329\pi$
$272$		0			0
$273$	−145022.	+	145022.i	−0.117768	+	0.117768i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.34816e6	−	1.34816e6i	−1.05571	−	1.05571i	−0.998354	−	0.0573510i	$-0.981735\pi$
$277$	−1.34816e6	−	1.34816e6i	−1.05571	−	1.05571i	−0.0573510	−	0.998354i	$-0.518265\pi$
$278$		0			0
$279$	548699.			0.422011
$280$		0			0
$281$	513974.			0.388307			0.194154	−	0.980971i	$-0.437804\pi$
$281$	513974.			0.388307			0.194154	+	0.980971i	$0.437804\pi$
$282$		0			0
$283$	−376382.	−	376382.i	−0.279359	−	0.279359i	0.553494	−	0.832853i	$-0.313294\pi$
$283$	−376382.	−	376382.i	−0.279359	−	0.279359i	−0.832853	+	0.553494i	$0.813294\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−281681.	+	281681.i	−0.201861	+	0.201861i
$288$		0			0
$289$		−	595121.i		−	0.419142i
$290$		0			0
$291$			584057.i			0.404318i
$292$		0			0
$293$	−124607.	+	124607.i	−0.0847958	+	0.0847958i	−0.748232	−	0.663437i	$-0.769097\pi$
$293$	−124607.	+	124607.i	−0.0847958	+	0.0847958i	0.663437	+	0.748232i	$0.269097\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−29194.0	−	29194.0i	−0.0192045	−	0.0192045i
$298$		0			0
$299$	239620.			0.155004
$300$		0			0
$301$	266267.			0.169395
$302$		0			0
$303$	−1.63700e6	−	1.63700e6i	−1.02433	−	1.02433i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−295939.	+	295939.i	−0.179208	+	0.179208i	−0.791010	−	0.611803i	$-0.790445\pi$
$307$	−295939.	+	295939.i	−0.179208	+	0.179208i	0.611803	+	0.791010i	$0.290445\pi$
$308$		0			0
$309$			218148.i			0.129973i
$310$		0			0
$311$			2.44556e6i			1.43376i	0.697194	+	0.716882i	$0.254432\pi$
$311$			2.44556e6i			1.43376i	−0.697194	+	0.716882i	$0.745568\pi$
$312$		0			0
$313$	2.10337e6	−	2.10337e6i	1.21354	−	1.21354i	0.243687	−	0.969854i	$-0.421643\pi$
$313$	2.10337e6	−	2.10337e6i	1.21354	−	1.21354i	0.969854	−	0.243687i	$-0.0783570\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−935356.	−	935356.i	−0.522792	−	0.522792i	0.395622	−	0.918414i	$-0.370529\pi$
$317$	−935356.	−	935356.i	−0.522792	−	0.522792i	−0.918414	+	0.395622i	$0.870529\pi$
$318$		0			0
$319$	−114838.			−0.0631843
$320$		0			0
$321$	−3.97495e6			−2.15312
$322$		0			0
$323$	−826403.	−	826403.i	−0.440743	−	0.440743i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−1.76788e6	+	1.76788e6i	−0.914287	+	0.914287i
$328$		0			0
$329$		−	2188.12i		−	0.00111450i
$330$		0			0
$331$		−	773777.i		−	0.388191i	−0.980983	−	0.194096i	$-0.937823\pi$
$331$		−	773777.i		−	0.388191i	0.980983	−	0.194096i	$-0.0621772\pi$
$332$		0			0
$333$	−1.15414e6	+	1.15414e6i	−0.570359	+	0.570359i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	1.75232e6	+	1.75232e6i	0.840502	+	0.840502i	0.988924	−	0.148422i	$-0.0474194\pi$
$337$	1.75232e6	+	1.75232e6i	0.840502	+	0.840502i	−0.148422	+	0.988924i	$0.547419\pi$
$338$		0			0
$339$	1.23523e6			0.583778
$340$		0			0
$341$	82576.7			0.0384567
$342$		0			0
$343$	828418.	+	828418.i	0.380202	+	0.380202i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−2.22463e6	+	2.22463e6i	−0.991821	+	0.991821i	−0.999967	−	0.00814557i	$-0.997407\pi$
$347$	−2.22463e6	+	2.22463e6i	−0.991821	+	0.991821i	0.00814557	+	0.999967i	$0.497407\pi$
$348$		0			0
$349$			4.01834e6i			1.76597i	0.469402	+	0.882985i	$0.344470\pi$
$349$			4.01834e6i			1.76597i	−0.469402	+	0.882985i	$0.655530\pi$
$350$		0			0
$351$			755624.i			0.327369i
$352$		0			0
$353$	1.24862e6	−	1.24862e6i	0.533327	−	0.533327i	−0.388234	−	0.921561i	$-0.626915\pi$
$353$	1.24862e6	−	1.24862e6i	0.533327	−	0.533327i	0.921561	+	0.388234i	$0.126915\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−437240.	−	437240.i	−0.181572	−	0.181572i
$358$		0			0
$359$	−3.10227e6			−1.27041			−0.635205	−	0.772344i	$-0.719084\pi$
$359$	−3.10227e6			−1.27041			−0.635205	+	0.772344i	$0.719084\pi$
$360$		0			0
$361$	−819953.			−0.331147
$362$		0			0
$363$	−2.13407e6	−	2.13407e6i	−0.850047	−	0.850047i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−288241.	+	288241.i	−0.111710	+	0.111710i	−0.760752	−	0.649042i	$-0.775170\pi$
$367$	−288241.	+	288241.i	−0.111710	+	0.111710i	0.649042	+	0.760752i	$0.275170\pi$
$368$		0			0
$369$		−	1.20099e6i		−	0.459168i
$370$		0			0
$371$			77895.9i			0.0293819i
$372$		0			0
$373$	−368683.	+	368683.i	−0.137208	+	0.137208i	−0.772375	−	0.635167i	$-0.780931\pi$
$373$	−368683.	+	368683.i	−0.137208	+	0.137208i	0.635167	+	0.772375i	$0.280931\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.48617e6	+	1.48617e6i	0.538536	+	0.538536i
$378$		0			0
$379$	−2.00113e6			−0.715612			−0.357806	−	0.933796i	$-0.616475\pi$
$379$	−2.00113e6			−0.715612			−0.357806	+	0.933796i	$0.616475\pi$
$380$		0			0
$381$	−3.84818e6			−1.35814
$382$		0			0
$383$	3.37971e6	+	3.37971e6i	1.17729	+	1.17729i	0.980433	+	0.196854i	$0.0630724\pi$
$383$	3.37971e6	+	3.37971e6i	1.17729	+	1.17729i	0.196854	+	0.980433i	$0.436928\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−567634.	+	567634.i	−0.192660	+	0.192660i
$388$		0			0
$389$			5.00291e6i			1.67629i	0.545449	+	0.838144i	$0.316359\pi$
$389$			5.00291e6i			1.67629i	−0.545449	+	0.838144i	$0.683641\pi$
$390$		0			0
$391$			722452.i			0.238983i
$392$		0			0
$393$	112471.	−	112471.i	0.0367331	−	0.0367331i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−594009.	−	594009.i	−0.189155	−	0.189155i	0.606176	−	0.795331i	$-0.292703\pi$
$397$	−594009.	−	594009.i	−0.189155	−	0.189155i	−0.795331	+	0.606176i	$0.792703\pi$
$398$		0			0
$399$	876247.			0.275546
$400$		0			0
$401$	2.05651e6			0.638661			0.319331	−	0.947643i	$-0.396542\pi$
$401$	2.05651e6			0.638661			0.319331	+	0.947643i	$0.396542\pi$
$402$		0			0
$403$	−1.06866e6	−	1.06866e6i	−0.327776	−	0.327776i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−173693.	+	173693.i	−0.0519752	+	0.0519752i
$408$		0			0
$409$		−	907188.i		−	0.268157i	−0.990971	−	0.134078i	$-0.957193\pi$
$409$		−	907188.i		−	0.268157i	0.990971	−	0.134078i	$-0.0428074\pi$
$410$		0			0
$411$		−	5.55079e6i		−	1.62088i
$412$		0			0
$413$	−722179.	+	722179.i	−0.208339	+	0.208339i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	3.18409e6	+	3.18409e6i	0.896695	+	0.896695i
$418$		0			0
$419$	5.24071e6			1.45833			0.729164	−	0.684339i	$-0.239909\pi$
$419$	5.24071e6			1.45833			0.729164	+	0.684339i	$0.239909\pi$
$420$		0			0
$421$	3.20555e6			0.881449			0.440724	−	0.897642i	$-0.354722\pi$
$421$	3.20555e6			0.881449			0.440724	+	0.897642i	$0.354722\pi$
$422$		0			0
$423$	4664.67	+	4664.67i	0.00126756	+	0.00126756i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	405309.	−	405309.i	0.107576	−	0.107576i
$428$		0			0
$429$		−	93054.3i		−	0.0244114i
$430$		0			0
$431$			2.71397e6i			0.703739i	0.936049	+	0.351870i	$0.114454\pi$
$431$			2.71397e6i			0.703739i	−0.936049	+	0.351870i	$0.885546\pi$
$432$		0			0
$433$	4.58234e6	−	4.58234e6i	1.17454	−	1.17454i	0.193423	−	0.981115i	$-0.438041\pi$
$433$	4.58234e6	−	4.58234e6i	1.17454	−	1.17454i	0.981115	−	0.193423i	$-0.0619590\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−723912.	−	723912.i	−0.181335	−	0.181335i
$438$		0			0
$439$	−6.01748e6			−1.49023			−0.745115	−	0.666936i	$-0.767606\pi$
$439$	−6.01748e6			−1.49023			−0.745115	+	0.666936i	$0.767606\pi$
$440$		0			0
$441$	−1.69409e6			−0.414802
$442$		0			0
$443$	960410.	+	960410.i	0.232513	+	0.232513i	0.813741	−	0.581228i	$-0.197427\pi$
$443$	960410.	+	960410.i	0.232513	+	0.232513i	−0.581228	+	0.813741i	$0.697427\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	5.31423e6	−	5.31423e6i	1.25797	−	1.25797i
$448$		0			0
$449$		−	4.48981e6i		−	1.05102i	−0.850786	−	0.525512i	$-0.823874\pi$
$449$		−	4.48981e6i		−	1.05102i	0.850786	−	0.525512i	$-0.176126\pi$
$450$		0			0
$451$		−	180743.i		−	0.0418427i
$452$		0			0
$453$	−6.59673e6	+	6.59673e6i	−1.51037	+	1.51037i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.58302e6	−	1.58302e6i	−0.354565	−	0.354565i	0.507240	−	0.861805i	$-0.330666\pi$
$457$	−1.58302e6	−	1.58302e6i	−0.354565	−	0.354565i	−0.861805	+	0.507240i	$0.830666\pi$
$458$		0			0
$459$	−2.27820e6			−0.504732
$460$		0			0
$461$	4.96001e6			1.08700			0.543501	−	0.839408i	$-0.317098\pi$
$461$	4.96001e6			1.08700			0.543501	+	0.839408i	$0.317098\pi$
$462$		0			0
$463$	1.57842e6	+	1.57842e6i	0.342191	+	0.342191i	0.857191	−	0.514999i	$-0.172208\pi$
$463$	1.57842e6	+	1.57842e6i	0.342191	+	0.342191i	−0.514999	+	0.857191i	$0.672208\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	4.96940e6	−	4.96940e6i	1.05441	−	1.05441i	0.0559831	−	0.998432i	$-0.482171\pi$
$467$	4.96940e6	−	4.96940e6i	1.05441	−	1.05441i	0.998432	−	0.0559831i	$-0.0178293\pi$
$468$		0			0
$469$		−	1.40873e6i		−	0.295730i
$470$		0			0
$471$		−	4.40459e6i		−	0.914857i
$472$		0			0
$473$	−85426.2	+	85426.2i	−0.0175565	+	0.0175565i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−166060.	−	166060.i	−0.0334171	−	0.0334171i
$478$		0			0
$479$	4.17103e6			0.830624			0.415312	−	0.909679i	$-0.363672\pi$
$479$	4.17103e6			0.830624			0.415312	+	0.909679i	$0.363672\pi$
$480$		0			0
$481$	4.49567e6			0.885996
$482$		0			0
$483$	−383013.	−	383013.i	−0.0747044	−	0.0747044i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	794960.	−	794960.i	0.151888	−	0.151888i	−0.627073	−	0.778961i	$-0.715747\pi$
$487$	794960.	−	794960.i	0.151888	−	0.151888i	0.778961	+	0.627073i	$0.215747\pi$
$488$		0			0
$489$		−	1.06144e7i		−	2.00734i
$490$		0			0
$491$		−	6.37894e6i		−	1.19411i	−0.802200	−	0.597056i	$-0.796337\pi$
$491$		−	6.37894e6i		−	1.19411i	0.802200	−	0.597056i	$-0.203663\pi$
$492$		0			0
$493$	−4.48080e6	+	4.48080e6i	−0.830305	+	0.830305i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.17237e6	−	1.17237e6i	−0.212900	−	0.212900i
$498$		0			0
$499$	4.38308e6			0.788004			0.394002	−	0.919110i	$-0.371090\pi$
$499$	4.38308e6			0.788004			0.394002	+	0.919110i	$0.371090\pi$
$500$		0			0
$501$	4.05840e6			0.722372
$502$		0			0
$503$	4.18594e6	+	4.18594e6i	0.737689	+	0.737689i	0.972130	−	0.234441i	$-0.0753261\pi$
$503$	4.18594e6	+	4.18594e6i	0.737689	+	0.737689i	−0.234441	+	0.972130i	$0.575326\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	3.72401e6	−	3.72401e6i	0.643414	−	0.643414i
$508$		0			0
$509$			3.23786e6i			0.553941i	0.960878	+	0.276971i	$0.0893305\pi$
$509$			3.23786e6i			0.553941i	−0.960878	+	0.276971i	$0.910670\pi$
$510$		0			0
$511$			272413.i			0.0461504i
$512$		0			0
$513$	2.28281e6	−	2.28281e6i	0.382980	−	0.382980i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	702.011	+	702.011i	0.000115509	+	0.000115509i
$518$		0			0
$519$	−1.06789e6			−0.174023
$520$		0			0
$521$	2.14080e6			0.345528			0.172764	−	0.984963i	$-0.444730\pi$
$521$	2.14080e6			0.345528			0.172764	+	0.984963i	$0.444730\pi$
$522$		0			0
$523$	1.54020e6	+	1.54020e6i	0.246220	+	0.246220i	0.819417	−	0.573197i	$-0.194297\pi$
$523$	1.54020e6	+	1.54020e6i	0.246220	+	0.246220i	−0.573197	+	0.819417i	$0.694297\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	3.22201e6	−	3.22201e6i	0.505359	−	0.505359i
$528$		0			0
$529$		−	5.80349e6i		−	0.901675i
$530$		0			0
$531$		−	3.07911e6i		−	0.473902i
$532$		0			0
$533$	−2.33907e6	+	2.33907e6i	−0.356636	+	0.356636i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	5.55162e6	+	5.55162e6i	0.830776	+	0.830776i
$538$		0			0
$539$	−254953.			−0.0377997
$540$		0			0
$541$	4.84343e6			0.711475			0.355737	−	0.934586i	$-0.384230\pi$
$541$	4.84343e6			0.711475			0.355737	+	0.934586i	$0.384230\pi$
$542$		0			0
$543$	−839123.	−	839123.i	−0.122131	−	0.122131i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	8.29695e6	−	8.29695e6i	1.18563	−	1.18563i	0.207371	−	0.978262i	$-0.433509\pi$
$547$	8.29695e6	−	8.29695e6i	1.18563	−	1.18563i	0.978262	−	0.207371i	$-0.0664907\pi$
$548$		0			0
$549$			1.72809e6i			0.244701i
$550$		0			0
$551$		−	8.97970e6i		−	1.26004i
$552$		0			0
$553$	1.16273e6	−	1.16273e6i	0.161684	−	0.161684i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−5.77544e6	−	5.77544e6i	−0.788764	−	0.788764i	0.192527	−	0.981292i	$-0.438332\pi$
$557$	−5.77544e6	−	5.77544e6i	−0.788764	−	0.788764i	−0.981292	+	0.192527i	$0.938332\pi$
$558$		0			0
$559$	2.21108e6			0.299278
$560$		0			0
$561$	280558.			0.0376371
$562$		0			0
$563$	6.21187e6	+	6.21187e6i	0.825945	+	0.825945i	0.986953	−	0.161008i	$-0.0514745\pi$
$563$	6.21187e6	+	6.21187e6i	0.825945	+	0.825945i	−0.161008	+	0.986953i	$0.551475\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	1.88940e6	−	1.88940e6i	0.246812	−	0.246812i
$568$		0			0
$569$			4.01294e6i			0.519615i	0.965660	+	0.259808i	$0.0836592\pi$
$569$			4.01294e6i			0.519615i	−0.965660	+	0.259808i	$0.916341\pi$
$570$		0			0
$571$			628249.i			0.0806383i	0.999187	+	0.0403191i	$0.0128375\pi$
$571$			628249.i			0.0806383i	−0.999187	+	0.0403191i	$0.987163\pi$
$572$		0			0
$573$	−6.60151e6	+	6.60151e6i	−0.839956	+	0.839956i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	1.02324e7	+	1.02324e7i	1.27949	+	1.27949i	0.940953	+	0.338537i	$0.109932\pi$
$577$	1.02324e7	+	1.02324e7i	1.27949	+	1.27949i	0.338537	+	0.940953i	$0.390068\pi$
$578$		0			0
$579$	−6.33787e6			−0.785683
$580$		0			0
$581$	3.39386e6			0.417113
$582$		0			0
$583$	−24991.2	−	24991.2i	−0.00304520	−	0.00304520i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−8.55551e6	+	8.55551e6i	−1.02483	+	1.02483i	−0.0251433	+	0.999684i	$0.508004\pi$
$587$	−8.55551e6	+	8.55551e6i	−1.02483	+	1.02483i	−0.999684	+	0.0251433i	$0.991996\pi$
$588$		0			0
$589$			6.45704e6i			0.766911i
$590$		0			0
$591$		−	1.71559e6i		−	0.202043i
$592$		0			0
$593$	863723.	−	863723.i	0.100864	−	0.100864i	−0.654874	−	0.755738i	$-0.727278\pi$
$593$	863723.	−	863723.i	0.100864	−	0.100864i	0.755738	+	0.654874i	$0.227278\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	1.77774e6	+	1.77774e6i	0.204142	+	0.204142i
$598$		0			0
$599$	−5.41077e6			−0.616158			−0.308079	−	0.951361i	$-0.599686\pi$
$599$	−5.41077e6			−0.616158			−0.308079	+	0.951361i	$0.599686\pi$
$600$		0			0
$601$	−6.50235e6			−0.734318			−0.367159	−	0.930158i	$-0.619670\pi$
$601$	−6.50235e6			−0.734318			−0.367159	+	0.930158i	$0.619670\pi$
$602$		0			0
$603$	3.00316e6	+	3.00316e6i	0.336345	+	0.336345i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	7.08603e6	−	7.08603e6i	0.780604	−	0.780604i	−0.199328	−	0.979933i	$-0.563876\pi$
$607$	7.08603e6	−	7.08603e6i	0.780604	−	0.780604i	0.979933	+	0.199328i	$0.0638760\pi$
$608$		0			0
$609$		−	4.75105e6i		−	0.519095i
$610$		0			0
$611$		−	18170.0i		−	0.00196903i
$612$		0			0
$613$	5.00318e6	−	5.00318e6i	0.537768	−	0.537768i	−0.385105	−	0.922873i	$-0.625835\pi$
$613$	5.00318e6	−	5.00318e6i	0.537768	−	0.537768i	0.922873	+	0.385105i	$0.125835\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	5.51301e6	+	5.51301e6i	0.583010	+	0.583010i	0.935729	−	0.352719i	$-0.114743\pi$
$617$	5.51301e6	+	5.51301e6i	0.583010	+	0.583010i	−0.352719	+	0.935729i	$0.614743\pi$
$618$		0			0
$619$	−1.61540e7			−1.69454			−0.847272	−	0.531159i	$-0.821757\pi$
$619$	−1.61540e7			−1.69454			−0.847272	+	0.531159i	$0.821757\pi$
$620$		0			0
$621$	−1.99566e6			−0.207662
$622$		0			0
$623$	−2.03562e6	−	2.03562e6i	−0.210125	−	0.210125i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−281125.	+	281125.i	−0.0285582	+	0.0285582i
$628$		0			0
$629$			1.35544e7i			1.36601i
$630$		0			0
$631$		−	1.40737e7i		−	1.40713i	−0.710630	−	0.703566i	$-0.751590\pi$
$631$		−	1.40737e7i		−	1.40713i	0.710630	−	0.703566i	$-0.248410\pi$
$632$		0			0
$633$	1.18511e7	−	1.18511e7i	1.17557	−	1.17557i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	3.29946e6	+	3.29946e6i	0.322177	+	0.322177i
$638$		0			0
$639$	4.99857e6			0.484277
$640$		0			0
$641$	−3.06380e6			−0.294521			−0.147260	−	0.989098i	$-0.547046\pi$
$641$	−3.06380e6			−0.294521			−0.147260	+	0.989098i	$0.547046\pi$
$642$		0			0
$643$	−4.04269e6	−	4.04269e6i	−0.385605	−	0.385605i	0.487511	−	0.873117i	$-0.337905\pi$
$643$	−4.04269e6	−	4.04269e6i	−0.385605	−	0.385605i	−0.873117	+	0.487511i	$0.837905\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.27161e7	+	1.27161e7i	−1.19424	+	1.19424i	−0.218376	+	0.975865i	$0.570076\pi$
$647$	−1.27161e7	+	1.27161e7i	−1.19424	+	1.19424i	−0.975865	+	0.218376i	$0.929924\pi$
$648$		0			0
$649$		−	463392.i		−	0.0431854i
$650$		0			0
$651$			3.41634e6i			0.315943i
$652$		0			0
$653$	−7.98493e6	+	7.98493e6i	−0.732804	+	0.732804i	−0.971174	−	0.238370i	$-0.923387\pi$
$653$	−7.98493e6	+	7.98493e6i	−0.732804	+	0.732804i	0.238370	+	0.971174i	$0.423387\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−580735.	−	580735.i	−0.0524885	−	0.0524885i
$658$		0			0
$659$	1.36581e7			1.22511			0.612556	−	0.790427i	$-0.290141\pi$
$659$	1.36581e7			1.22511			0.612556	+	0.790427i	$0.290141\pi$
$660$		0			0
$661$	−8.73171e6			−0.777312			−0.388656	−	0.921383i	$-0.627061\pi$
$661$	−8.73171e6			−0.777312			−0.388656	+	0.921383i	$0.627061\pi$
$662$		0			0
$663$	−3.63082e6	−	3.63082e6i	−0.320790	−	0.320790i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−3.92508e6	+	3.92508e6i	−0.341613	+	0.341613i
$668$		0			0
$669$			1.22188e7i			1.05551i
$670$		0			0
$671$			260070.i			0.0222989i
$672$		0			0
$673$	−6.67754e6	+	6.67754e6i	−0.568301	+	0.568301i	−0.931652	−	0.363351i	$-0.881633\pi$
$673$	−6.67754e6	+	6.67754e6i	−0.568301	+	0.568301i	0.363351	+	0.931652i	$0.381633\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	5.82288e6	+	5.82288e6i	0.488277	+	0.488277i	0.907762	−	0.419485i	$-0.137789\pi$
$677$	5.82288e6	+	5.82288e6i	0.488277	+	0.488277i	−0.419485	+	0.907762i	$0.637789\pi$
$678$		0			0
$679$	−1.12862e6			−0.0939451
$680$		0			0
$681$	−24855.1			−0.00205375
$682$		0			0
$683$	1.44034e7	+	1.44034e7i	1.18144	+	1.18144i	0.979371	+	0.202071i	$0.0647671\pi$
$683$	1.44034e7	+	1.44034e7i	1.18144	+	1.18144i	0.202071	+	0.979371i	$0.435233\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	1.25796e6	−	1.25796e6i	0.101689	−	0.101689i
$688$		0			0
$689$			646845.i			0.0519101i
$690$		0			0
$691$			1.83585e7i			1.46266i	0.682026	+	0.731328i	$0.261099\pi$
$691$			1.83585e7i			1.46266i	−0.682026	+	0.731328i	$0.738901\pi$
$692$		0			0
$693$	−46163.0	+	46163.0i	−0.00365141	+	0.00365141i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−7.05229e6	−	7.05229e6i	−0.549855	−	0.549855i
$698$		0			0
$699$	−1.96264e7			−1.51932
$700$		0			0
$701$	−1.22419e7			−0.940921			−0.470461	−	0.882421i	$-0.655912\pi$
$701$	−1.22419e7			−0.940921			−0.470461	+	0.882421i	$0.655912\pi$
$702$		0			0
$703$	−1.35818e7	−	1.35818e7i	−1.03650	−	1.03650i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	3.16331e6	−	3.16331e6i	0.238009	−	0.238009i
$708$		0			0
$709$		−	4.19642e6i		−	0.313519i	−0.987637	−	0.156759i	$-0.949895\pi$
$709$		−	4.19642e6i		−	0.313519i	0.987637	−	0.156759i	$-0.0501047\pi$
$710$		0			0
$711$			4.95747e6i			0.367778i
$712$		0			0
$713$	2.82241e6	−	2.82241e6i	0.207920	−	0.207920i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−1.21836e7	−	1.21836e7i	−0.885069	−	0.885069i
$718$		0			0
$719$	2.36234e7			1.70420			0.852098	−	0.523382i	$-0.175330\pi$
$719$	2.36234e7			1.70420			0.852098	+	0.523382i	$0.175330\pi$
$720$		0			0
$721$	−421545.			−0.0301999
$722$		0			0
$723$	−6.38819e6	−	6.38819e6i	−0.454498	−	0.454498i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−1.69585e7	+	1.69585e7i	−1.19001	+	1.19001i	−0.212950	+	0.977063i	$0.568307\pi$
$727$	−1.69585e7	+	1.69585e7i	−1.19001	+	1.19001i	−0.977063	+	0.212950i	$0.931693\pi$
$728$		0			0
$729$		−	3.38711e6i		−	0.236053i
$730$		0			0
$731$			6.66638e6i			0.461420i
$732$		0			0
$733$	1.02325e7	−	1.02325e7i	0.703429	−	0.703429i	−0.261716	−	0.965145i	$-0.584288\pi$
$733$	1.02325e7	−	1.02325e7i	0.703429	−	0.703429i	0.965145	+	0.261716i	$0.0842884\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	451961.	+	451961.i	0.0306502	+	0.0306502i
$738$		0			0
$739$	4.74454e6			0.319582			0.159791	−	0.987151i	$-0.448918\pi$
$739$	4.74454e6			0.319582			0.159791	+	0.987151i	$0.448918\pi$
$740$		0			0
$741$	7.27633e6			0.486818
$742$		0			0
$743$	−2.11501e6	−	2.11501e6i	−0.140553	−	0.140553i	0.633329	−	0.773882i	$-0.281688\pi$
$743$	−2.11501e6	−	2.11501e6i	−0.140553	−	0.140553i	−0.773882	+	0.633329i	$0.781688\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−7.23510e6	+	7.23510e6i	−0.474398	+	0.474398i
$748$		0			0
$749$		−	7.68112e6i		−	0.500288i
$750$		0			0
$751$			6.32400e6i			0.409159i	0.978850	+	0.204579i	$0.0655827\pi$
$751$			6.32400e6i			0.409159i	−0.978850	+	0.204579i	$0.934417\pi$
$752$		0			0
$753$	1.33631e7	−	1.33631e7i	0.858855	−	0.858855i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1.52793e7	−	1.52793e7i	−0.969087	−	0.969087i	0.0304496	−	0.999536i	$-0.490306\pi$
$757$	−1.52793e7	−	1.52793e7i	−0.969087	−	0.969087i	−0.999536	+	0.0304496i	$0.990306\pi$
$758$		0			0
$759$	245763.			0.0154850
$760$		0			0
$761$	−1.62820e6			−0.101917			−0.0509583	−	0.998701i	$-0.516228\pi$
$761$	−1.62820e6			−0.101917			−0.0509583	+	0.998701i	$0.516228\pi$
$762$		0			0
$763$	−3.41622e6	−	3.41622e6i	−0.212439	−	0.212439i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−5.99695e6	+	5.99695e6i	−0.368080	+	0.368080i
$768$		0			0
$769$		−	1.89046e7i		−	1.15280i	−0.817169	−	0.576398i	$-0.804458\pi$
$769$		−	1.89046e7i		−	1.15280i	0.817169	−	0.576398i	$-0.195542\pi$
$770$		0			0
$771$			3.00170e7i			1.81858i
$772$		0			0
$773$	−2.31771e7	+	2.31771e7i	−1.39511	+	1.39511i	−0.581738	+	0.813376i	$0.697627\pi$
$773$	−2.31771e7	+	2.31771e7i	−1.39511	+	1.39511i	−0.813376	+	0.581738i	$0.802373\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−7.18598e6	−	7.18598e6i	−0.427005	−	0.427005i
$778$		0			0
$779$	1.41331e7			0.834437
$780$		0			0
$781$	752262.			0.0441308
$782$		0			0
$783$	−1.23775e7	−	1.23775e7i	−0.721487	−	0.721487i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	4.39692e6	−	4.39692e6i	0.253053	−	0.253053i	−0.569168	−	0.822221i	$-0.692735\pi$
$787$	4.39692e6	−	4.39692e6i	0.253053	−	0.253053i	0.822221	+	0.569168i	$0.192735\pi$
$788$		0			0
$789$		−	6.16251e6i		−	0.352424i
$790$		0			0
$791$			2.38693e6i			0.135643i
$792$		0			0
$793$	3.36568e6	−	3.36568e6i	0.190059	−	0.190059i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−2.19165e7	−	2.19165e7i	−1.22215	−	1.22215i	−0.966865	−	0.255289i	$-0.917829\pi$
$797$	−2.19165e7	−	2.19165e7i	−1.22215	−	1.22215i	−0.255289	−	0.966865i	$-0.582171\pi$
$798$		0			0
$799$	54782.6			0.00303582
$800$		0			0
$801$	8.67915e6			0.477965
$802$		0			0
$803$	−87397.9	−	87397.9i	−0.00478313	−	0.00478313i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	6.42886e6	−	6.42886e6i	0.347496	−	0.347496i
$808$		0			0
$809$			8.04847e6i			0.432356i	0.976354	+	0.216178i	$0.0693592\pi$
$809$			8.04847e6i			0.432356i	−0.976354	+	0.216178i	$0.930641\pi$
$810$		0			0
$811$			7.60133e6i			0.405824i	0.979197	+	0.202912i	$0.0650405\pi$
$811$			7.60133e6i			0.405824i	−0.979197	+	0.202912i	$0.934959\pi$
$812$		0			0
$813$	2.79613e7	−	2.79613e7i	1.48365	−	1.48365i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−6.67986e6	−	6.67986e6i	−0.350116	−	0.350116i
$818$		0			0
$819$	1.19483e6			0.0622439
$820$		0			0
$821$	−503676.			−0.0260791			−0.0130396	−	0.999915i	$-0.504151\pi$
$821$	−503676.			−0.0260791			−0.0130396	+	0.999915i	$0.504151\pi$
$822$		0			0
$823$	−1.77317e7	−	1.77317e7i	−0.912536	−	0.912536i	0.0839356	−	0.996471i	$-0.473251\pi$
$823$	−1.77317e7	−	1.77317e7i	−0.912536	−	0.912536i	−0.996471	+	0.0839356i	$0.973251\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.80164e7	+	1.80164e7i	−0.916019	+	0.916019i	−0.996737	−	0.0807182i	$-0.974279\pi$
$827$	−1.80164e7	+	1.80164e7i	−0.916019	+	0.916019i	0.0807182	+	0.996737i	$0.474279\pi$
$828$		0			0
$829$		−	1.51603e7i		−	0.766163i	−0.923715	−	0.383081i	$-0.874863\pi$
$829$		−	1.51603e7i		−	0.766163i	0.923715	−	0.383081i	$-0.125137\pi$
$830$		0			0
$831$			3.57890e7i			1.79782i
$832$		0			0
$833$	−9.94785e6	+	9.94785e6i	−0.496726	+	0.496726i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	8.90029e6	+	8.90029e6i	0.439127	+	0.439127i
$838$		0			0
$839$	9.14635e6			0.448583			0.224292	−	0.974522i	$-0.427993\pi$
$839$	9.14635e6			0.448583			0.224292	+	0.974522i	$0.427993\pi$
$840$		0			0
$841$	−2.81772e7			−1.37375
$842$		0			0
$843$	−6.82210e6	−	6.82210e6i	−0.330635	−	0.330635i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	4.12385e6	−	4.12385e6i	0.197512	−	0.197512i
$848$		0			0
$849$			9.99162e6i			0.475737i
$850$		0			0
$851$			1.18734e7i			0.562019i
$852$		0			0
$853$	−9.93591e6	+	9.93591e6i	−0.467558	+	0.467558i	−0.901122	−	0.433565i	$-0.857256\pi$
$853$	−9.93591e6	+	9.93591e6i	−0.467558	+	0.467558i	0.433565	+	0.901122i	$0.357256\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−2.56795e7	−	2.56795e7i	−1.19436	−	1.19436i	−0.975831	−	0.218529i	$-0.929874\pi$
$857$	−2.56795e7	−	2.56795e7i	−1.19436	−	1.19436i	−0.218529	−	0.975831i	$-0.570126\pi$
$858$		0			0
$859$	2.67501e6			0.123692			0.0618462	−	0.998086i	$-0.480301\pi$
$859$	2.67501e6			0.123692			0.0618462	+	0.998086i	$0.480301\pi$
$860$		0			0
$861$	7.47765e6			0.343761
$862$		0			0
$863$	2.64990e7	+	2.64990e7i	1.21116	+	1.21116i	0.970644	+	0.240519i	$0.0773177\pi$
$863$	2.64990e7	+	2.64990e7i	1.21116	+	1.21116i	0.240519	+	0.970644i	$0.422682\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−7.89919e6	+	7.89919e6i	−0.356890	+	0.356890i
$868$		0			0
$869$			746076.i			0.0335146i
$870$		0			0
$871$		−	1.16981e7i		−	0.522478i
$872$		0			0
$873$	2.40602e6	−	2.40602e6i	0.106847	−	0.106847i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	2.38294e7	+	2.38294e7i	1.04620	+	1.04620i	0.998880	+	0.0473169i	$0.0150671\pi$
$877$	2.38294e7	+	2.38294e7i	1.04620	+	1.04620i	0.0473169	+	0.998880i	$0.484933\pi$
$878$		0			0
$879$	3.30789e6			0.144404
$880$		0			0
$881$	−1.24233e7			−0.539261			−0.269630	−	0.962964i	$-0.586902\pi$
$881$	−1.24233e7			−0.539261			−0.269630	+	0.962964i	$0.586902\pi$
$882$		0			0
$883$	−9.67869e6	−	9.67869e6i	−0.417749	−	0.417749i	0.466678	−	0.884427i	$-0.345451\pi$
$883$	−9.67869e6	−	9.67869e6i	−0.417749	−	0.417749i	−0.884427	+	0.466678i	$0.845451\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−2.87959e7	+	2.87959e7i	−1.22891	+	1.22891i	−0.264540	+	0.964375i	$0.585220\pi$
$887$	−2.87959e7	+	2.87959e7i	−1.22891	+	1.22891i	−0.964375	+	0.264540i	$0.914780\pi$
$888$		0			0
$889$		−	7.43617e6i		−	0.315569i
$890$		0			0
$891$			1.21235e6i			0.0511603i
$892$		0			0
$893$	−54893.3	+	54893.3i	−0.00230351	+	0.00230351i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−3.18053e6	−	3.18053e6i	−0.131983	−	0.131983i
$898$		0			0
$899$	3.50104e7			1.44477
$900$		0			0
$901$	−1.95023e6			−0.0800340
$902$		0			0
$903$	−3.53423e6	−	3.53423e6i	−0.144237	−	0.144237i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	3.03420e6	−	3.03420e6i	0.122469	−	0.122469i	−0.643216	−	0.765685i	$-0.722400\pi$
$907$	3.03420e6	−	3.03420e6i	0.122469	−	0.122469i	0.765685	+	0.643216i	$0.222400\pi$
$908$		0			0
$909$			1.34872e7i			0.541392i
$910$		0			0
$911$		−	2.56728e7i		−	1.02489i	−0.858720	−	0.512446i	$-0.828740\pi$
$911$		−	2.56728e7i		−	1.02489i	0.858720	−	0.512446i	$-0.171260\pi$
$912$		0			0
$913$	−1.08885e6	+	1.08885e6i	−0.0432305	+	0.0432305i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	217336.	+	217336.i	0.00853510	+	0.00853510i
$918$		0			0
$919$	−3.64653e7			−1.42427			−0.712134	−	0.702044i	$-0.752271\pi$
$919$	−3.64653e7			−1.42427			−0.712134	+	0.702044i	$0.752271\pi$
$920$		0			0
$921$	7.85615e6			0.305183
$922$		0			0
$923$	−9.73535e6	−	9.73535e6i	−0.376138	−	0.376138i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	898657.	−	898657.i	0.0343475	−	0.0343475i
$928$		0			0
$929$			3.44216e7i			1.30855i	0.756255	+	0.654277i	$0.227027\pi$
$929$			3.44216e7i			1.30855i	−0.756255	+	0.654277i	$0.772973\pi$
$930$		0			0
$931$		−	1.99359e7i		−	0.753810i
$932$		0			0
$933$	3.24606e7	−	3.24606e7i	1.22082	−	1.22082i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1.93428e7	+	1.93428e7i	0.719733	+	0.719733i	0.968550	−	0.248818i	$-0.0800420\pi$
$937$	1.93428e7	+	1.93428e7i	0.719733	+	0.719733i	−0.248818	+	0.968550i	$0.580042\pi$
$938$		0			0
$939$	−5.58370e7			−2.06661
$940$		0			0
$941$	−3.46051e7			−1.27399			−0.636996	−	0.770867i	$-0.719823\pi$
$941$	−3.46051e7			−1.27399			−0.636996	+	0.770867i	$0.719823\pi$
$942$		0			0
$943$	−6.17766e6	−	6.17766e6i	−0.226227	−	0.226227i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	2.98661e6	−	2.98661e6i	0.108219	−	0.108219i	−0.650924	−	0.759143i	$-0.725618\pi$
$947$	2.98661e6	−	2.98661e6i	0.108219	−	0.108219i	0.759143	+	0.650924i	$0.225618\pi$
$948$		0			0
$949$			2.26211e6i			0.0815357i
$950$		0			0
$951$			2.48304e7i			0.890292i
$952$		0			0
$953$	−2.44213e7	+	2.44213e7i	−0.871035	+	0.871035i	−0.992585	−	0.121550i	$-0.961214\pi$
$953$	−2.44213e7	+	2.44213e7i	−0.871035	+	0.871035i	0.121550	+	0.992585i	$0.461214\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	1.52427e6	+	1.52427e6i	0.0538001	+	0.0538001i
$958$		0			0
$959$	1.07263e7			0.376618
$960$		0			0
$961$	3.45424e6			0.120655
$962$		0			0
$963$	1.63747e7	+	1.63747e7i	0.568996	+	0.568996i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	6.51546e6	−	6.51546e6i	0.224067	−	0.224067i	−0.586141	−	0.810209i	$-0.699354\pi$
$967$	6.51546e6	−	6.51546e6i	0.224067	−	0.224067i	0.810209	+	0.586141i	$0.199354\pi$
$968$		0			0
$969$			2.19381e7i			0.750567i
$970$		0			0
$971$		−	3.00564e7i		−	1.02303i	−0.859274	−	0.511516i	$-0.829084\pi$
$971$		−	3.00564e7i		−	1.02303i	0.859274	−	0.511516i	$-0.170916\pi$
$972$		0			0
$973$	−6.15288e6	+	6.15288e6i	−0.208351	+	0.208351i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	3.00560e7	+	3.00560e7i	1.00738	+	1.00738i	0.999973	+	0.00740945i	$0.00235852\pi$
$977$	3.00560e7	+	3.00560e7i	1.00738	+	1.00738i	0.00740945	+	0.999973i	$0.497641\pi$
$978$		0			0
$979$	1.30617e6			0.0435556
$980$		0			0
$981$	1.45655e7			0.483229
$982$		0			0
$983$	3.74910e7	+	3.74910e7i	1.23749	+	1.23749i	0.961022	+	0.276473i	$0.0891656\pi$
$983$	3.74910e7	+	3.74910e7i	1.23749	+	1.23749i	0.276473	+	0.961022i	$0.410834\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−29043.4	+	29043.4i	−0.000948975	+	0.000948975i
$988$		0			0
$989$			5.83961e6i			0.189842i
$990$		0			0
$991$			2.70332e7i			0.874408i	0.899362	+	0.437204i	$0.144031\pi$
$991$			2.70332e7i			0.874408i	−0.899362	+	0.437204i	$0.855969\pi$
$992$		0			0
$993$	−1.02705e7	+	1.02705e7i	−0.330537	+	0.330537i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	2.60732e7	+	2.60732e7i	0.830725	+	0.830725i	0.987616	−	0.156891i	$-0.0501471\pi$
$997$	2.60732e7	+	2.60732e7i	0.830725	+	0.830725i	−0.156891	+	0.987616i	$0.550147\pi$
$998$		0			0
$999$	−3.74420e7			−1.18698

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.n.h.207.4	yes	24
4.3	odd	2	inner	400.6.n.h.207.9	yes	24
5.2	odd	4	inner	400.6.n.h.143.3	✓	24
5.3	odd	4	inner	400.6.n.h.143.9	yes	24
5.4	even	2	inner	400.6.n.h.207.10	yes	24
20.3	even	4	inner	400.6.n.h.143.4	yes	24
20.7	even	4	inner	400.6.n.h.143.10	yes	24
20.19	odd	2	inner	400.6.n.h.207.3	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.6.n.h.143.3	✓	24	5.2	odd	4	inner
400.6.n.h.143.4	yes	24	20.3	even	4	inner
400.6.n.h.143.9	yes	24	5.3	odd	4	inner
400.6.n.h.143.10	yes	24	20.7	even	4	inner
400.6.n.h.207.3	yes	24	20.19	odd	2	inner
400.6.n.h.207.4	yes	24	1.1	even	1	trivial
400.6.n.h.207.9	yes	24	4.3	odd	2	inner
400.6.n.h.207.10	yes	24	5.4	even	2	inner

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	400.6.n.h.207.4

Level	$400$
Weight	$6$
Character	400.207
Analytic conductor	$64.154$
Analytic rank	$0$
Dimension	$24$
Inner twists	$8$