LMFDB - Embedded newform 400.5.p.c.257.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,5,Mod(193,400)]

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([0, 0, 3]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("400.193");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	400.p (of order $4$, degree $2$, 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:	$41.3479852335$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			257.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	400.257
Dual form			400.5.p.c.193.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(9.00000 - 9.00000i) q^{3} +(29.0000 + 29.0000i) q^{7} -81.0000i q^{9} +118.000 q^{11} +(-69.0000 + 69.0000i) q^{13} +(271.000 + 271.000i) q^{17} -280.000i q^{19} +522.000 q^{21} +(269.000 - 269.000i) q^{23} +680.000i q^{29} -202.000 q^{31} +(1062.00 - 1062.00i) q^{33} +(651.000 + 651.000i) q^{37} +1242.00i q^{39} +1682.00 q^{41} +(1089.00 - 1089.00i) q^{43} +(1269.00 + 1269.00i) q^{47} -719.000i q^{49} +4878.00 q^{51} +(611.000 - 611.000i) q^{53} +(-2520.00 - 2520.00i) q^{57} -1160.00i q^{59} -5598.00 q^{61} +(2349.00 - 2349.00i) q^{63} +(-751.000 - 751.000i) q^{67} -4842.00i q^{69} -6442.00 q^{71} +(2951.00 - 2951.00i) q^{73} +(3422.00 + 3422.00i) q^{77} -10560.0i q^{79} +6561.00 q^{81} +(-6231.00 + 6231.00i) q^{83} +(6120.00 + 6120.00i) q^{87} -14480.0i q^{89} -4002.00 q^{91} +(-1818.00 + 1818.00i) q^{93} +(7311.00 + 7311.00i) q^{97} -9558.00i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 18 q^{3} + 58 q^{7} + 236 q^{11} - 138 q^{13} + 542 q^{17} + 1044 q^{21} + 538 q^{23} - 404 q^{31} + 2124 q^{33} + 1302 q^{37} + 3364 q^{41} + 2178 q^{43} + 2538 q^{47} + 9756 q^{51} + 1222 q^{53}+ \cdots + 14622 q^{97}+O(q^{100}) $ 2 * q + 18 * q^3 + 58 * q^7 + 236 * q^11 - 138 * q^13 + 542 * q^17 + 1044 * q^21 + 538 * q^23 - 404 * q^31 + 2124 * q^33 + 1302 * q^37 + 3364 * q^41 + 2178 * q^43 + 2538 * q^47 + 9756 * q^51 + 1222 * q^53 - 5040 * q^57 - 11196 * q^61 + 4698 * q^63 - 1502 * q^67 - 12884 * q^71 + 5902 * q^73 + 6844 * q^77 + 13122 * q^81 - 12462 * q^83 + 12240 * q^87 - 8004 * q^91 - 3636 * q^93 + 14622 * q^97

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
$3$	9.00000	−	9.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$3$	9.00000	−	9.00000i	1.00000	−	1.00000i	1.00000			$0$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	29.0000	+	29.0000i	0.591837	+	0.591837i	0.938127	−	0.346291i	$-0.112559\pi$
$7$	29.0000	+	29.0000i	0.591837	+	0.591837i	−0.346291	+	0.938127i	$0.612559\pi$
$8$		0			0
$9$		−	81.0000i		−	1.00000i
$10$		0			0
$11$	118.000			0.975207			0.487603	−	0.873065i	$-0.337871\pi$
$11$	118.000			0.975207			0.487603	+	0.873065i	$0.337871\pi$
$12$		0			0
$13$	−69.0000	+	69.0000i	−0.408284	+	0.408284i	−0.881140	−	0.472856i	$-0.843223\pi$
$13$	−69.0000	+	69.0000i	−0.408284	+	0.408284i	0.472856	+	0.881140i	$0.343223\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	271.000	+	271.000i	0.937716	+	0.937716i	0.998171	−	0.0604547i	$-0.0192551\pi$
$17$	271.000	+	271.000i	0.937716	+	0.937716i	−0.0604547	+	0.998171i	$0.519255\pi$
$18$		0			0
$19$		−	280.000i		−	0.775623i	−0.921739	−	0.387812i	$-0.873231\pi$
$19$		−	280.000i		−	0.775623i	0.921739	−	0.387812i	$-0.126769\pi$
$20$		0			0
$21$	522.000			1.18367
$22$		0			0
$23$	269.000	−	269.000i	0.508507	−	0.508507i	−0.405561	−	0.914068i	$-0.632924\pi$
$23$	269.000	−	269.000i	0.508507	−	0.508507i	0.914068	+	0.405561i	$0.132924\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$			680.000i			0.808561i	0.914635	+	0.404281i	$0.132478\pi$
$29$			680.000i			0.808561i	−0.914635	+	0.404281i	$0.867522\pi$
$30$		0			0
$31$	−202.000			−0.210198			−0.105099	−	0.994462i	$-0.533516\pi$
$31$	−202.000			−0.210198			−0.105099	+	0.994462i	$0.533516\pi$
$32$		0			0
$33$	1062.00	−	1062.00i	0.975207	−	0.975207i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	651.000	+	651.000i	0.475530	+	0.475530i	0.903699	−	0.428169i	$-0.140841\pi$
$37$	651.000	+	651.000i	0.475530	+	0.475530i	−0.428169	+	0.903699i	$0.640841\pi$
$38$		0			0
$39$			1242.00i			0.816568i
$40$		0			0
$41$	1682.00			1.00059			0.500297	−	0.865854i	$-0.333224\pi$
$41$	1682.00			1.00059			0.500297	+	0.865854i	$0.333224\pi$
$42$		0			0
$43$	1089.00	−	1089.00i	0.588967	−	0.588967i	−0.348385	−	0.937352i	$-0.613270\pi$
$43$	1089.00	−	1089.00i	0.588967	−	0.588967i	0.937352	+	0.348385i	$0.113270\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1269.00	+	1269.00i	0.574468	+	0.574468i	0.933374	−	0.358906i	$-0.116850\pi$
$47$	1269.00	+	1269.00i	0.574468	+	0.574468i	−0.358906	+	0.933374i	$0.616850\pi$
$48$		0			0
$49$		−	719.000i		−	0.299459i
$50$		0			0
$51$	4878.00			1.87543
$52$		0			0
$53$	611.000	−	611.000i	0.217515	−	0.217515i	−0.589935	−	0.807450i	$-0.700847\pi$
$53$	611.000	−	611.000i	0.217515	−	0.217515i	0.807450	+	0.589935i	$0.200847\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−2520.00	−	2520.00i	−0.775623	−	0.775623i
$58$		0			0
$59$		−	1160.00i		−	0.333238i	−0.986021	−	0.166619i	$-0.946715\pi$
$59$		−	1160.00i		−	0.333238i	0.986021	−	0.166619i	$-0.0532849\pi$
$60$		0			0
$61$	−5598.00			−1.50443			−0.752217	−	0.658915i	$-0.771016\pi$
$61$	−5598.00			−1.50443			−0.752217	+	0.658915i	$0.771016\pi$
$62$		0			0
$63$	2349.00	−	2349.00i	0.591837	−	0.591837i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−751.000	−	751.000i	−0.167298	−	0.167298i	0.618493	−	0.785791i	$-0.287744\pi$
$67$	−751.000	−	751.000i	−0.167298	−	0.167298i	−0.785791	+	0.618493i	$0.787744\pi$
$68$		0			0
$69$		−	4842.00i		−	1.01701i
$70$		0			0
$71$	−6442.00			−1.27792			−0.638961	−	0.769240i	$-0.720635\pi$
$71$	−6442.00			−1.27792			−0.638961	+	0.769240i	$0.720635\pi$
$72$		0			0
$73$	2951.00	−	2951.00i	0.553762	−	0.553762i	−0.373762	−	0.927525i	$-0.621932\pi$
$73$	2951.00	−	2951.00i	0.553762	−	0.553762i	0.927525	+	0.373762i	$0.121932\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3422.00	+	3422.00i	0.577163	+	0.577163i
$78$		0			0
$79$		−	10560.0i		−	1.69204i	−0.533154	−	0.846018i	$-0.678993\pi$
$79$		−	10560.0i		−	1.69204i	0.533154	−	0.846018i	$-0.321007\pi$
$80$		0			0
$81$	6561.00			1.00000
$82$		0			0
$83$	−6231.00	+	6231.00i	−0.904485	+	0.904485i	−0.995820	−	0.0913348i	$-0.970887\pi$
$83$	−6231.00	+	6231.00i	−0.904485	+	0.904485i	0.0913348	+	0.995820i	$0.470887\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	6120.00	+	6120.00i	0.808561	+	0.808561i
$88$		0			0
$89$		−	14480.0i		−	1.82805i	−0.405656	−	0.914026i	$-0.632957\pi$
$89$		−	14480.0i		−	1.82805i	0.405656	−	0.914026i	$-0.367043\pi$
$90$		0			0
$91$	−4002.00			−0.483275
$92$		0			0
$93$	−1818.00	+	1818.00i	−0.210198	+	0.210198i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	7311.00	+	7311.00i	0.777022	+	0.777022i	0.979323	−	0.202301i	$-0.0648420\pi$
$97$	7311.00	+	7311.00i	0.777022	+	0.777022i	−0.202301	+	0.979323i	$0.564842\pi$
$98$		0			0
$99$		−	9558.00i		−	0.975207i
$100$		0			0
$101$	−878.000			−0.0860700			−0.0430350	−	0.999074i	$-0.513703\pi$
$101$	−878.000			−0.0860700			−0.0430350	+	0.999074i	$0.513703\pi$
$102$		0			0
$103$	10429.0	−	10429.0i	0.983033	−	0.983033i	−0.0168252	−	0.999858i	$-0.505356\pi$
$103$	10429.0	−	10429.0i	0.983033	−	0.983033i	0.999858	+	0.0168252i	$0.00535587\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−4711.00	−	4711.00i	−0.411477	−	0.411477i	0.470776	−	0.882253i	$-0.343974\pi$
$107$	−4711.00	−	4711.00i	−0.411477	−	0.411477i	−0.882253	+	0.470776i	$0.843974\pi$
$108$		0			0
$109$			22040.0i			1.85506i	0.373745	+	0.927531i	$0.378073\pi$
$109$			22040.0i			1.85506i	−0.373745	+	0.927531i	$0.621927\pi$
$110$		0			0
$111$	11718.0			0.951059
$112$		0			0
$113$	2111.00	−	2111.00i	0.165322	−	0.165322i	−0.619597	−	0.784920i	$-0.712704\pi$
$113$	2111.00	−	2111.00i	0.165322	−	0.165322i	0.784920	+	0.619597i	$0.212704\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	5589.00	+	5589.00i	0.408284	+	0.408284i
$118$		0			0
$119$			15718.0i			1.10995i
$120$		0			0
$121$	−717.000			−0.0489721
$122$		0			0
$123$	15138.0	−	15138.0i	1.00059	−	1.00059i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	5909.00	+	5909.00i	0.366359	+	0.366359i	0.866147	−	0.499789i	$-0.166589\pi$
$127$	5909.00	+	5909.00i	0.366359	+	0.366359i	−0.499789	+	0.866147i	$0.666589\pi$
$128$		0			0
$129$		−	19602.0i		−	1.17793i
$130$		0			0
$131$	6358.00			0.370491			0.185246	−	0.982692i	$-0.440692\pi$
$131$	6358.00			0.370491			0.185246	+	0.982692i	$0.440692\pi$
$132$		0			0
$133$	8120.00	−	8120.00i	0.459042	−	0.459042i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−20409.0	−	20409.0i	−1.08738	−	1.08738i	−0.995798	−	0.0915804i	$-0.970808\pi$
$137$	−20409.0	−	20409.0i	−1.08738	−	1.08738i	−0.0915804	−	0.995798i	$-0.529192\pi$
$138$		0			0
$139$			9400.00i			0.486517i	0.969961	+	0.243259i	$0.0782164\pi$
$139$			9400.00i			0.486517i	−0.969961	+	0.243259i	$0.921784\pi$
$140$		0			0
$141$	22842.0			1.14894
$142$		0			0
$143$	−8142.00	+	8142.00i	−0.398161	+	0.398161i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−6471.00	−	6471.00i	−0.299459	−	0.299459i
$148$		0			0
$149$			13800.0i			0.621594i	0.950476	+	0.310797i	$0.100596\pi$
$149$			13800.0i			0.621594i	−0.950476	+	0.310797i	$0.899404\pi$
$150$		0			0
$151$	18998.0			0.833209			0.416605	−	0.909088i	$-0.363220\pi$
$151$	18998.0			0.833209			0.416605	+	0.909088i	$0.363220\pi$
$152$		0			0
$153$	21951.0	−	21951.0i	0.937716	−	0.937716i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	16371.0	+	16371.0i	0.664165	+	0.664165i	0.956359	−	0.292194i	$-0.0943853\pi$
$157$	16371.0	+	16371.0i	0.664165	+	0.664165i	−0.292194	+	0.956359i	$0.594385\pi$
$158$		0			0
$159$		−	10998.0i		−	0.435030i
$160$		0			0
$161$	15602.0			0.601906
$162$		0			0
$163$	20009.0	−	20009.0i	0.753096	−	0.753096i	−0.221960	−	0.975056i	$-0.571245\pi$
$163$	20009.0	−	20009.0i	0.753096	−	0.753096i	0.975056	+	0.221960i	$0.0712455\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1549.00	+	1549.00i	0.0555416	+	0.0555416i	0.734332	−	0.678790i	$-0.237496\pi$
$167$	1549.00	+	1549.00i	0.0555416	+	0.0555416i	−0.678790	+	0.734332i	$0.737496\pi$
$168$		0			0
$169$			19039.0i			0.666608i
$170$		0			0
$171$	−22680.0			−0.775623
$172$		0			0
$173$	−2789.00	+	2789.00i	−0.0931872	+	0.0931872i	−0.752164	−	0.658976i	$-0.770990\pi$
$173$	−2789.00	+	2789.00i	−0.0931872	+	0.0931872i	0.658976	+	0.752164i	$0.270990\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−10440.0	−	10440.0i	−0.333238	−	0.333238i
$178$		0			0
$179$			2600.00i			0.0811460i	0.999177	+	0.0405730i	$0.0129183\pi$
$179$			2600.00i			0.0811460i	−0.999177	+	0.0405730i	$0.987082\pi$
$180$		0			0
$181$	−44398.0			−1.35521			−0.677604	−	0.735427i	$-0.736982\pi$
$181$	−44398.0			−1.35521			−0.677604	+	0.735427i	$0.736982\pi$
$182$		0			0
$183$	−50382.0	+	50382.0i	−1.50443	+	1.50443i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	31978.0	+	31978.0i	0.914467	+	0.914467i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	14678.0			0.402346			0.201173	−	0.979556i	$-0.435525\pi$
$191$	14678.0			0.402346			0.201173	+	0.979556i	$0.435525\pi$
$192$		0			0
$193$	−42849.0	+	42849.0i	−1.15034	+	1.15034i	−0.163855	+	0.986484i	$0.552393\pi$
$193$	−42849.0	+	42849.0i	−1.15034	+	1.15034i	−0.986484	+	0.163855i	$0.947607\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	10971.0	+	10971.0i	0.282692	+	0.282692i	0.834182	−	0.551490i	$-0.185940\pi$
$197$	10971.0	+	10971.0i	0.282692	+	0.282692i	−0.551490	+	0.834182i	$0.685940\pi$
$198$		0			0
$199$			38160.0i			0.963612i	0.876278	+	0.481806i	$0.160019\pi$
$199$			38160.0i			0.963612i	−0.876278	+	0.481806i	$0.839981\pi$
$200$		0			0
$201$	−13518.0			−0.334596
$202$		0			0
$203$	−19720.0	+	19720.0i	−0.478536	+	0.478536i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−21789.0	−	21789.0i	−0.508507	−	0.508507i
$208$		0			0
$209$		−	33040.0i		−	0.756393i
$210$		0			0
$211$	−72842.0			−1.63613			−0.818063	−	0.575128i	$-0.804952\pi$
$211$	−72842.0			−1.63613			−0.818063	+	0.575128i	$0.804952\pi$
$212$		0			0
$213$	−57978.0	+	57978.0i	−1.27792	+	1.27792i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−5858.00	−	5858.00i	−0.124403	−	0.124403i
$218$		0			0
$219$		−	53118.0i		−	1.10752i
$220$		0			0
$221$	−37398.0			−0.765709
$222$		0			0
$223$	−30891.0	+	30891.0i	−0.621187	+	0.621187i	−0.945835	−	0.324648i	$-0.894754\pi$
$223$	−30891.0	+	30891.0i	−0.621187	+	0.621187i	0.324648	+	0.945835i	$0.394754\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−54911.0	−	54911.0i	−1.06563	−	1.06563i	−0.997689	−	0.0679438i	$-0.978356\pi$
$227$	−54911.0	−	54911.0i	−1.06563	−	1.06563i	−0.0679438	−	0.997689i	$-0.521644\pi$
$228$		0			0
$229$		−	50280.0i		−	0.958792i	−0.877599	−	0.479396i	$-0.840856\pi$
$229$		−	50280.0i		−	0.958792i	0.877599	−	0.479396i	$-0.159144\pi$
$230$		0			0
$231$	61596.0			1.15433
$232$		0			0
$233$	2391.00	−	2391.00i	0.0440421	−	0.0440421i	−0.684743	−	0.728785i	$-0.740085\pi$
$233$	2391.00	−	2391.00i	0.0440421	−	0.0440421i	0.728785	+	0.684743i	$0.240085\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−95040.0	−	95040.0i	−1.69204	−	1.69204i
$238$		0			0
$239$			17760.0i			0.310919i	0.987842	+	0.155459i	$0.0496858\pi$
$239$			17760.0i			0.310919i	−0.987842	+	0.155459i	$0.950314\pi$
$240$		0			0
$241$	−28238.0			−0.486183			−0.243092	−	0.970003i	$-0.578162\pi$
$241$	−28238.0			−0.486183			−0.243092	+	0.970003i	$0.578162\pi$
$242$		0			0
$243$	59049.0	−	59049.0i	1.00000	−	1.00000i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	19320.0	+	19320.0i	0.316675	+	0.316675i
$248$		0			0
$249$			112158.i			1.80897i
$250$		0			0
$251$	−121002.			−1.92064			−0.960318	−	0.278907i	$-0.910028\pi$
$251$	−121002.			−1.92064			−0.960318	+	0.278907i	$0.910028\pi$
$252$		0			0
$253$	31742.0	−	31742.0i	0.495899	−	0.495899i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	72431.0	+	72431.0i	1.09663	+	1.09663i	0.994803	+	0.101823i	$0.0324674\pi$
$257$	72431.0	+	72431.0i	1.09663	+	1.09663i	0.101823	+	0.994803i	$0.467533\pi$
$258$		0			0
$259$			37758.0i			0.562872i
$260$		0			0
$261$	55080.0			0.808561
$262$		0			0
$263$	−14771.0	+	14771.0i	−0.213549	+	0.213549i	−0.805773	−	0.592224i	$-0.798250\pi$
$263$	−14771.0	+	14771.0i	−0.213549	+	0.213549i	0.592224	+	0.805773i	$0.298250\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−130320.	−	130320.i	−1.82805	−	1.82805i
$268$		0			0
$269$			89720.0i			1.23989i	0.784644	+	0.619947i	$0.212846\pi$
$269$			89720.0i			1.23989i	−0.784644	+	0.619947i	$0.787154\pi$
$270$		0			0
$271$	−68202.0			−0.928664			−0.464332	−	0.885661i	$-0.653706\pi$
$271$	−68202.0			−0.928664			−0.464332	+	0.885661i	$0.653706\pi$
$272$		0			0
$273$	−36018.0	+	36018.0i	−0.483275	+	0.483275i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−18549.0	−	18549.0i	−0.241747	−	0.241747i	0.575826	−	0.817573i	$-0.304681\pi$
$277$	−18549.0	−	18549.0i	−0.241747	−	0.241747i	−0.817573	+	0.575826i	$0.804681\pi$
$278$		0			0
$279$			16362.0i			0.210198i
$280$		0			0
$281$	2322.00			0.0294069			0.0147035	−	0.999892i	$-0.495320\pi$
$281$	2322.00			0.0294069			0.0147035	+	0.999892i	$0.495320\pi$
$282$		0			0
$283$	−91711.0	+	91711.0i	−1.14511	+	1.14511i	−0.157613	+	0.987501i	$0.550380\pi$
$283$	−91711.0	+	91711.0i	−1.14511	+	1.14511i	−0.987501	+	0.157613i	$0.949620\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	48778.0	+	48778.0i	0.592189	+	0.592189i
$288$		0			0
$289$			63361.0i			0.758624i
$290$		0			0
$291$	131598.			1.55404
$292$		0			0
$293$	4851.00	−	4851.00i	0.0565062	−	0.0565062i	−0.678289	−	0.734795i	$-0.737278\pi$
$293$	4851.00	−	4851.00i	0.0565062	−	0.0565062i	0.734795	+	0.678289i	$0.237278\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$			37122.0i			0.415230i
$300$		0			0
$301$	63162.0			0.697145
$302$		0			0
$303$	−7902.00	+	7902.00i	−0.0860700	+	0.0860700i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	42849.0	+	42849.0i	0.454636	+	0.454636i	0.896890	−	0.442254i	$-0.145821\pi$
$307$	42849.0	+	42849.0i	0.454636	+	0.454636i	−0.442254	+	0.896890i	$0.645821\pi$
$308$		0			0
$309$		−	187722.i		−	1.96607i
$310$		0			0
$311$	72278.0			0.747283			0.373642	−	0.927573i	$-0.378109\pi$
$311$	72278.0			0.747283			0.373642	+	0.927573i	$0.378109\pi$
$312$		0			0
$313$	−18249.0	+	18249.0i	−0.186273	+	0.186273i	−0.794083	−	0.607810i	$-0.792048\pi$
$313$	−18249.0	+	18249.0i	−0.186273	+	0.186273i	0.607810	+	0.794083i	$0.292048\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−25149.0	−	25149.0i	−0.250266	−	0.250266i	0.570814	−	0.821080i	$-0.306628\pi$
$317$	−25149.0	−	25149.0i	−0.250266	−	0.250266i	−0.821080	+	0.570814i	$0.806628\pi$
$318$		0			0
$319$			80240.0i			0.788514i
$320$		0			0
$321$	−84798.0			−0.822954
$322$		0			0
$323$	75880.0	−	75880.0i	0.727315	−	0.727315i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	198360.	+	198360.i	1.85506	+	1.85506i
$328$		0			0
$329$			73602.0i			0.679983i
$330$		0			0
$331$	54038.0			0.493223			0.246611	−	0.969114i	$-0.420683\pi$
$331$	54038.0			0.493223			0.246611	+	0.969114i	$0.420683\pi$
$332$		0			0
$333$	52731.0	−	52731.0i	0.475530	−	0.475530i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−8529.00	−	8529.00i	−0.0750997	−	0.0750997i	0.668559	−	0.743659i	$-0.266911\pi$
$337$	−8529.00	−	8529.00i	−0.0750997	−	0.0750997i	−0.743659	+	0.668559i	$0.766911\pi$
$338$		0			0
$339$		−	37998.0i		−	0.330645i
$340$		0			0
$341$	−23836.0			−0.204986
$342$		0			0
$343$	90480.0	−	90480.0i	0.769067	−	0.769067i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−56551.0	−	56551.0i	−0.469658	−	0.469658i	0.432146	−	0.901804i	$-0.357756\pi$
$347$	−56551.0	−	56551.0i	−0.469658	−	0.469658i	−0.901804	+	0.432146i	$0.857756\pi$
$348$		0			0
$349$		−	22520.0i		−	0.184892i	−0.995718	−	0.0924459i	$-0.970531\pi$
$349$		−	22520.0i		−	0.184892i	0.995718	−	0.0924459i	$-0.0294685\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	44511.0	−	44511.0i	0.357205	−	0.357205i	−0.505576	−	0.862782i	$-0.668720\pi$
$353$	44511.0	−	44511.0i	0.357205	−	0.357205i	0.862782	+	0.505576i	$0.168720\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	141462.	+	141462.i	1.10995	+	1.10995i
$358$		0			0
$359$			9680.00i			0.0751080i	0.999295	+	0.0375540i	$0.0119566\pi$
$359$			9680.00i			0.0751080i	−0.999295	+	0.0375540i	$0.988043\pi$
$360$		0			0
$361$	51921.0			0.398409
$362$		0			0
$363$	−6453.00	+	6453.00i	−0.0489721	+	0.0489721i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−14971.0	−	14971.0i	−0.111152	−	0.111152i	0.649343	−	0.760496i	$-0.275044\pi$
$367$	−14971.0	−	14971.0i	−0.111152	−	0.111152i	−0.760496	+	0.649343i	$0.775044\pi$
$368$		0			0
$369$		−	136242.i		−	1.00059i
$370$		0			0
$371$	35438.0			0.257467
$372$		0			0
$373$	13811.0	−	13811.0i	0.0992676	−	0.0992676i	−0.655729	−	0.754996i	$-0.727639\pi$
$373$	13811.0	−	13811.0i	0.0992676	−	0.0992676i	0.754996	+	0.655729i	$0.227639\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−46920.0	−	46920.0i	−0.330123	−	0.330123i
$378$		0			0
$379$		−	251080.i		−	1.74797i	−0.485954	−	0.873984i	$-0.661528\pi$
$379$		−	251080.i		−	1.74797i	0.485954	−	0.873984i	$-0.338472\pi$
$380$		0			0
$381$	106362.			0.732717
$382$		0			0
$383$	−86091.0	+	86091.0i	−0.586895	+	0.586895i	−0.936789	−	0.349894i	$-0.886217\pi$
$383$	−86091.0	+	86091.0i	−0.586895	+	0.586895i	0.349894	+	0.936789i	$0.386217\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−88209.0	−	88209.0i	−0.588967	−	0.588967i
$388$		0			0
$389$		−	75000.0i		−	0.495635i	−0.968807	−	0.247818i	$-0.920287\pi$
$389$		−	75000.0i		−	0.495635i	0.968807	−	0.247818i	$-0.0797134\pi$
$390$		0			0
$391$	145798.			0.953670
$392$		0			0
$393$	57222.0	−	57222.0i	0.370491	−	0.370491i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−29149.0	−	29149.0i	−0.184945	−	0.184945i	0.608562	−	0.793507i	$-0.291747\pi$
$397$	−29149.0	−	29149.0i	−0.184945	−	0.184945i	−0.793507	+	0.608562i	$0.791747\pi$
$398$		0			0
$399$		−	146160.i		−	0.918085i
$400$		0			0
$401$	−45918.0			−0.285558			−0.142779	−	0.989755i	$-0.545604\pi$
$401$	−45918.0			−0.285558			−0.142779	+	0.989755i	$0.545604\pi$
$402$		0			0
$403$	13938.0	−	13938.0i	0.0858204	−	0.0858204i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	76818.0	+	76818.0i	0.463740	+	0.463740i
$408$		0			0
$409$			78720.0i			0.470585i	0.971925	+	0.235293i	$0.0756049\pi$
$409$			78720.0i			0.470585i	−0.971925	+	0.235293i	$0.924395\pi$
$410$		0			0
$411$	−367362.			−2.17476
$412$		0			0
$413$	33640.0	−	33640.0i	0.197222	−	0.197222i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	84600.0	+	84600.0i	0.486517	+	0.486517i
$418$		0			0
$419$			14760.0i			0.0840733i	0.999116	+	0.0420367i	$0.0133846\pi$
$419$			14760.0i			0.0840733i	−0.999116	+	0.0420367i	$0.986615\pi$
$420$		0			0
$421$	221282.			1.24848			0.624240	−	0.781232i	$-0.285409\pi$
$421$	221282.			1.24848			0.624240	+	0.781232i	$0.285409\pi$
$422$		0			0
$423$	102789.	−	102789.i	0.574468	−	0.574468i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−162342.	−	162342.i	−0.890379	−	0.890379i
$428$		0			0
$429$			146556.i			0.796323i
$430$		0			0
$431$	−212522.			−1.14406			−0.572031	−	0.820232i	$-0.693844\pi$
$431$	−212522.			−1.14406			−0.572031	+	0.820232i	$0.693844\pi$
$432$		0			0
$433$	−145409.	+	145409.i	−0.775560	+	0.775560i	−0.979072	−	0.203512i	$-0.934764\pi$
$433$	−145409.	+	145409.i	−0.775560	+	0.775560i	0.203512	+	0.979072i	$0.434764\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−75320.0	−	75320.0i	−0.394410	−	0.394410i
$438$		0			0
$439$			299440.i			1.55375i	0.629656	+	0.776874i	$0.283196\pi$
$439$			299440.i			1.55375i	−0.629656	+	0.776874i	$0.716804\pi$
$440$		0			0
$441$	−58239.0			−0.299459
$442$		0			0
$443$	240609.	−	240609.i	1.22604	−	1.22604i	0.260590	−	0.965450i	$-0.416083\pi$
$443$	240609.	−	240609.i	1.22604	−	1.22604i	0.965450	−	0.260590i	$-0.0839170\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	124200.	+	124200.i	0.621594	+	0.621594i
$448$		0			0
$449$		−	82480.0i		−	0.409125i	−0.978854	−	0.204562i	$-0.934423\pi$
$449$		−	82480.0i		−	0.409125i	0.978854	−	0.204562i	$-0.0655772\pi$
$450$		0			0
$451$	198476.			0.975787
$452$		0			0
$453$	170982.	−	170982.i	0.833209	−	0.833209i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	188151.	+	188151.i	0.900895	+	0.900895i	0.995514	−	0.0946187i	$-0.0301632\pi$
$457$	188151.	+	188151.i	0.900895	+	0.900895i	−0.0946187	+	0.995514i	$0.530163\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−326158.			−1.53471			−0.767355	−	0.641223i	$-0.778427\pi$
$461$	−326158.			−1.53471			−0.767355	+	0.641223i	$0.778427\pi$
$462$		0			0
$463$	−218731.	+	218731.i	−1.02035	+	1.02035i	−0.0205595	+	0.999789i	$0.506545\pi$
$463$	−218731.	+	218731.i	−1.02035	+	1.02035i	−0.999789	+	0.0205595i	$0.993455\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	59249.0	+	59249.0i	0.271673	+	0.271673i	0.829774	−	0.558100i	$-0.188470\pi$
$467$	59249.0	+	59249.0i	0.271673	+	0.271673i	−0.558100	+	0.829774i	$0.688470\pi$
$468$		0			0
$469$		−	43558.0i		−	0.198026i
$470$		0			0
$471$	294678.			1.32833
$472$		0			0
$473$	128502.	−	128502.i	0.574365	−	0.574365i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−49491.0	−	49491.0i	−0.217515	−	0.217515i
$478$		0			0
$479$		−	273440.i		−	1.19177i	−0.803071	−	0.595883i	$-0.796802\pi$
$479$		−	273440.i		−	1.19177i	0.803071	−	0.595883i	$-0.203198\pi$
$480$		0			0
$481$	−89838.0			−0.388302
$482$		0			0
$483$	140418.	−	140418.i	0.601906	−	0.601906i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−123651.	−	123651.i	−0.521362	−	0.521362i	0.396620	−	0.917983i	$-0.370183\pi$
$487$	−123651.	−	123651.i	−0.521362	−	0.521362i	−0.917983	+	0.396620i	$0.870183\pi$
$488$		0			0
$489$		−	360162.i		−	1.50619i
$490$		0			0
$491$	−198442.			−0.823134			−0.411567	−	0.911379i	$-0.635018\pi$
$491$	−198442.			−0.823134			−0.411567	+	0.911379i	$0.635018\pi$
$492$		0			0
$493$	−184280.	+	184280.i	−0.758201	+	0.758201i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−186818.	−	186818.i	−0.756321	−	0.756321i
$498$		0			0
$499$		−	269240.i		−	1.08128i	−0.841254	−	0.540640i	$-0.818182\pi$
$499$		−	269240.i		−	1.08128i	0.841254	−	0.540640i	$-0.181818\pi$
$500$		0			0
$501$	27882.0			0.111083
$502$		0			0
$503$	109869.	−	109869.i	0.434249	−	0.434249i	−0.455822	−	0.890071i	$-0.650655\pi$
$503$	109869.	−	109869.i	0.434249	−	0.434249i	0.890071	+	0.455822i	$0.150655\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	171351.	+	171351.i	0.666608	+	0.666608i
$508$		0			0
$509$		−	211000.i		−	0.814417i	−0.913335	−	0.407209i	$-0.866502\pi$
$509$		−	211000.i		−	0.814417i	0.913335	−	0.407209i	$-0.133498\pi$
$510$		0			0
$511$	171158.			0.655474
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	149742.	+	149742.i	0.560225	+	0.560225i
$518$		0			0
$519$			50202.0i			0.186374i
$520$		0			0
$521$	297282.			1.09520			0.547600	−	0.836740i	$-0.315542\pi$
$521$	297282.			1.09520			0.547600	+	0.836740i	$0.315542\pi$
$522$		0			0
$523$	−25071.0	+	25071.0i	−0.0916576	+	0.0916576i	−0.751449	−	0.659791i	$-0.770645\pi$
$523$	−25071.0	+	25071.0i	−0.0916576	+	0.0916576i	0.659791	+	0.751449i	$0.270645\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−54742.0	−	54742.0i	−0.197106	−	0.197106i
$528$		0			0
$529$			135119.i			0.482842i
$530$		0			0
$531$	−93960.0			−0.333238
$532$		0			0
$533$	−116058.	+	116058.i	−0.408527	+	0.408527i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	23400.0	+	23400.0i	0.0811460	+	0.0811460i
$538$		0			0
$539$		−	84842.0i		−	0.292034i
$540$		0			0
$541$	−142478.			−0.486803			−0.243402	−	0.969926i	$-0.578263\pi$
$541$	−142478.			−0.486803			−0.243402	+	0.969926i	$0.578263\pi$
$542$		0			0
$543$	−399582.	+	399582.i	−1.35521	+	1.35521i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	291009.	+	291009.i	0.972594	+	0.972594i	0.999634	−	0.0270399i	$-0.00860813\pi$
$547$	291009.	+	291009.i	0.972594	+	0.972594i	−0.0270399	+	0.999634i	$0.508608\pi$
$548$		0			0
$549$			453438.i			1.50443i
$550$		0			0
$551$	190400.			0.627139
$552$		0			0
$553$	306240.	−	306240.i	1.00141	−	1.00141i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	83091.0	+	83091.0i	0.267820	+	0.267820i	0.828221	−	0.560401i	$-0.189353\pi$
$557$	83091.0	+	83091.0i	0.267820	+	0.267820i	−0.560401	+	0.828221i	$0.689353\pi$
$558$		0			0
$559$			150282.i			0.480932i
$560$		0			0
$561$	575604.			1.82893
$562$		0			0
$563$	43449.0	−	43449.0i	0.137076	−	0.137076i	−0.635239	−	0.772316i	$-0.719098\pi$
$563$	43449.0	−	43449.0i	0.137076	−	0.137076i	0.772316	+	0.635239i	$0.219098\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	190269.	+	190269.i	0.591837	+	0.591837i
$568$		0			0
$569$		−	270560.i		−	0.835678i	−0.908521	−	0.417839i	$-0.862788\pi$
$569$		−	270560.i		−	0.835678i	0.908521	−	0.417839i	$-0.137212\pi$
$570$		0			0
$571$	−57482.0			−0.176303			−0.0881515	−	0.996107i	$-0.528096\pi$
$571$	−57482.0			−0.176303			−0.0881515	+	0.996107i	$0.528096\pi$
$572$		0			0
$573$	132102.	−	132102.i	0.402346	−	0.402346i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−195889.	−	195889.i	−0.588381	−	0.588381i	0.348812	−	0.937193i	$-0.386585\pi$
$577$	−195889.	−	195889.i	−0.588381	−	0.588381i	−0.937193	+	0.348812i	$0.886585\pi$
$578$		0			0
$579$			771282.i			2.30068i
$580$		0			0
$581$	−361398.			−1.07062
$582$		0			0
$583$	72098.0	−	72098.0i	0.212122	−	0.212122i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−404631.	−	404631.i	−1.17431	−	1.17431i	−0.981171	−	0.193139i	$-0.938133\pi$
$587$	−404631.	−	404631.i	−1.17431	−	1.17431i	−0.193139	−	0.981171i	$-0.561867\pi$
$588$		0			0
$589$			56560.0i			0.163034i
$590$		0			0
$591$	197478.			0.565384
$592$		0			0
$593$	210991.	−	210991.i	0.600005	−	0.600005i	−0.340309	−	0.940314i	$-0.610532\pi$
$593$	210991.	−	210991.i	0.600005	−	0.600005i	0.940314	+	0.340309i	$0.110532\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	343440.	+	343440.i	0.963612	+	0.963612i
$598$		0			0
$599$		−	300560.i		−	0.837679i	−0.908060	−	0.418839i	$-0.862437\pi$
$599$		−	300560.i		−	0.837679i	0.908060	−	0.418839i	$-0.137563\pi$
$600$		0			0
$601$	367442.			1.01728			0.508639	−	0.860980i	$-0.330149\pi$
$601$	367442.			1.01728			0.508639	+	0.860980i	$0.330149\pi$
$602$		0			0
$603$	−60831.0	+	60831.0i	−0.167298	+	0.167298i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	146469.	+	146469.i	0.397529	+	0.397529i	0.877360	−	0.479832i	$-0.159302\pi$
$607$	146469.	+	146469.i	0.397529	+	0.397529i	−0.479832	+	0.877360i	$0.659302\pi$
$608$		0			0
$609$			354960.i			0.957072i
$610$		0			0
$611$	−175122.			−0.469092
$612$		0			0
$613$	−160989.	+	160989.i	−0.428425	+	0.428425i	−0.888092	−	0.459666i	$-0.847969\pi$
$613$	−160989.	+	160989.i	−0.428425	+	0.428425i	0.459666	+	0.888092i	$0.347969\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−320409.	−	320409.i	−0.841656	−	0.841656i	0.147419	−	0.989074i	$-0.452904\pi$
$617$	−320409.	−	320409.i	−0.841656	−	0.841656i	−0.989074	+	0.147419i	$0.952904\pi$
$618$		0			0
$619$		−	341160.i		−	0.890383i	−0.895435	−	0.445191i	$-0.853136\pi$
$619$		−	341160.i		−	0.890383i	0.895435	−	0.445191i	$-0.146864\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	419920.	−	419920.i	1.08191	−	1.08191i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−297360.	−	297360.i	−0.756393	−	0.756393i
$628$		0			0
$629$			352842.i			0.891824i
$630$		0			0
$631$	390998.			0.982010			0.491005	−	0.871157i	$-0.336630\pi$
$631$	390998.			0.982010			0.491005	+	0.871157i	$0.336630\pi$
$632$		0			0
$633$	−655578.	+	655578.i	−1.63613	+	1.63613i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	49611.0	+	49611.0i	0.122264	+	0.122264i
$638$		0			0
$639$			521802.i			1.27792i
$640$		0			0
$641$	−585038.			−1.42386			−0.711931	−	0.702249i	$-0.752179\pi$
$641$	−585038.			−1.42386			−0.711931	+	0.702249i	$0.752179\pi$
$642$		0			0
$643$	−31911.0	+	31911.0i	−0.0771824	+	0.0771824i	−0.744644	−	0.667462i	$-0.767381\pi$
$643$	−31911.0	+	31911.0i	−0.0771824	+	0.0771824i	0.667462	+	0.744644i	$0.267381\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−280931.	−	280931.i	−0.671106	−	0.671106i	0.286865	−	0.957971i	$-0.407387\pi$
$647$	−280931.	−	280931.i	−0.671106	−	0.671106i	−0.957971	+	0.286865i	$0.907387\pi$
$648$		0			0
$649$		−	136880.i		−	0.324975i
$650$		0			0
$651$	−105444.			−0.248805
$652$		0			0
$653$	−523989.	+	523989.i	−1.22884	+	1.22884i	−0.264439	+	0.964402i	$0.585187\pi$
$653$	−523989.	+	523989.i	−1.22884	+	1.22884i	−0.964402	+	0.264439i	$0.914813\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−239031.	−	239031.i	−0.553762	−	0.553762i
$658$		0			0
$659$			404360.i			0.931102i	0.885021	+	0.465551i	$0.154144\pi$
$659$			404360.i			0.931102i	−0.885021	+	0.465551i	$0.845856\pi$
$660$		0			0
$661$	−5278.00			−0.0120800			−0.00603999	−	0.999982i	$-0.501923\pi$
$661$	−5278.00			−0.0120800			−0.00603999	+	0.999982i	$0.501923\pi$
$662$		0			0
$663$	−336582.	+	336582.i	−0.765709	+	0.765709i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	182920.	+	182920.i	0.411159	+	0.411159i
$668$		0			0
$669$			556038.i			1.24237i
$670$		0			0
$671$	−660564.			−1.46713
$672$		0			0
$673$	332111.	−	332111.i	0.733252	−	0.733252i	−0.238011	−	0.971263i	$-0.576495\pi$
$673$	332111.	−	332111.i	0.733252	−	0.733252i	0.971263	+	0.238011i	$0.0764953\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−578309.	−	578309.i	−1.26178	−	1.26178i	−0.950231	−	0.311546i	$-0.899153\pi$
$677$	−578309.	−	578309.i	−1.26178	−	1.26178i	−0.311546	−	0.950231i	$-0.600847\pi$
$678$		0			0
$679$			424038.i			0.919740i
$680$		0			0
$681$	−988398.			−2.13127
$682$		0			0
$683$	−349311.	+	349311.i	−0.748809	+	0.748809i	−0.974255	−	0.225447i	$-0.927616\pi$
$683$	−349311.	+	349311.i	−0.748809	+	0.748809i	0.225447	+	0.974255i	$0.427616\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−452520.	−	452520.i	−0.958792	−	0.958792i
$688$		0			0
$689$			84318.0i			0.177616i
$690$		0			0
$691$	−282762.			−0.592195			−0.296098	−	0.955158i	$-0.595685\pi$
$691$	−282762.			−0.592195			−0.296098	+	0.955158i	$0.595685\pi$
$692$		0			0
$693$	277182.	−	277182.i	0.577163	−	0.577163i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	455822.	+	455822.i	0.938274	+	0.938274i
$698$		0			0
$699$		−	43038.0i		−	0.0880841i
$700$		0			0
$701$	270242.			0.549942			0.274971	−	0.961453i	$-0.411332\pi$
$701$	270242.			0.549942			0.274971	+	0.961453i	$0.411332\pi$
$702$		0			0
$703$	182280.	−	182280.i	0.368832	−	0.368832i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−25462.0	−	25462.0i	−0.0509394	−	0.0509394i
$708$		0			0
$709$		−	297800.i		−	0.592423i	−0.955122	−	0.296212i	$-0.904277\pi$
$709$		−	297800.i		−	0.592423i	0.955122	−	0.296212i	$-0.0957234\pi$
$710$		0			0
$711$	−855360.			−1.69204
$712$		0			0
$713$	−54338.0	+	54338.0i	−0.106887	+	0.106887i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	159840.	+	159840.i	0.310919	+	0.310919i
$718$		0			0
$719$		−	913760.i		−	1.76756i	−0.467902	−	0.883780i	$-0.654990\pi$
$719$		−	913760.i		−	1.76756i	0.467902	−	0.883780i	$-0.345010\pi$
$720$		0			0
$721$	604882.			1.16359
$722$		0			0
$723$	−254142.	+	254142.i	−0.486183	+	0.486183i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−417651.	−	417651.i	−0.790214	−	0.790214i	0.191315	−	0.981529i	$-0.438725\pi$
$727$	−417651.	−	417651.i	−0.790214	−	0.790214i	−0.981529	+	0.191315i	$0.938725\pi$
$728$		0			0
$729$		−	531441.i		−	1.00000i
$730$		0			0
$731$	590238.			1.10457
$732$		0			0
$733$	−394549.	+	394549.i	−0.734333	+	0.734333i	−0.971475	−	0.237142i	$-0.923789\pi$
$733$	−394549.	+	394549.i	−0.734333	+	0.734333i	0.237142	+	0.971475i	$0.423789\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−88618.0	−	88618.0i	−0.163150	−	0.163150i
$738$		0			0
$739$		−	109880.i		−	0.201201i	−0.994927	−	0.100600i	$-0.967924\pi$
$739$		−	109880.i		−	0.201201i	0.994927	−	0.100600i	$-0.0320764\pi$
$740$		0			0
$741$	347760.			0.633349
$742$		0			0
$743$	−466451.	+	466451.i	−0.844945	+	0.844945i	−0.989497	−	0.144552i	$-0.953826\pi$
$743$	−466451.	+	466451.i	−0.844945	+	0.844945i	0.144552	+	0.989497i	$0.453826\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	504711.	+	504711.i	0.904485	+	0.904485i
$748$		0			0
$749$		−	273238.i		−	0.487054i
$750$		0			0
$751$	−1.01092e6			−1.79241			−0.896206	−	0.443638i	$-0.853687\pi$
$751$	−1.01092e6			−1.79241			−0.896206	+	0.443638i	$0.853687\pi$
$752$		0			0
$753$	−1.08902e6	+	1.08902e6i	−1.92064	+	1.92064i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−313269.	−	313269.i	−0.546671	−	0.546671i	0.378806	−	0.925476i	$-0.376335\pi$
$757$	−313269.	−	313269.i	−0.546671	−	0.546671i	−0.925476	+	0.378806i	$0.876335\pi$
$758$		0			0
$759$		−	571356.i		−	0.991798i
$760$		0			0
$761$	142082.			0.245341			0.122670	−	0.992447i	$-0.460854\pi$
$761$	142082.			0.245341			0.122670	+	0.992447i	$0.460854\pi$
$762$		0			0
$763$	−639160.	+	639160.i	−1.09789	+	1.09789i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	80040.0	+	80040.0i	0.136056	+	0.136056i
$768$		0			0
$769$			13280.0i			0.0224567i	0.999937	+	0.0112283i	$0.00357417\pi$
$769$			13280.0i			0.0224567i	−0.999937	+	0.0112283i	$0.996426\pi$
$770$		0			0
$771$	1.30376e6			2.19325
$772$		0			0
$773$	782211.	−	782211.i	1.30908	−	1.30908i	0.386994	−	0.922082i	$-0.373513\pi$
$773$	782211.	−	782211.i	1.30908	−	1.30908i	0.922082	−	0.386994i	$-0.126487\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	339822.	+	339822.i	0.562872	+	0.562872i
$778$		0			0
$779$		−	470960.i		−	0.776085i
$780$		0			0
$781$	−760156.			−1.24624
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	201409.	+	201409.i	0.325184	+	0.325184i	0.850752	−	0.525568i	$-0.176147\pi$
$787$	201409.	+	201409.i	0.325184	+	0.325184i	−0.525568	+	0.850752i	$0.676147\pi$
$788$		0			0
$789$			265878.i			0.427099i
$790$		0			0
$791$	122438.			0.195688
$792$		0			0
$793$	386262.	−	386262.i	0.614236	−	0.614236i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	36291.0	+	36291.0i	0.0571324	+	0.0571324i	0.735096	−	0.677963i	$-0.237137\pi$
$797$	36291.0	+	36291.0i	0.0571324	+	0.0571324i	−0.677963	+	0.735096i	$0.737137\pi$
$798$		0			0
$799$			687798.i			1.07738i
$800$		0			0
$801$	−1.17288e6			−1.82805
$802$		0			0
$803$	348218.	−	348218.i	0.540033	−	0.540033i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	807480.	+	807480.i	1.23989	+	1.23989i
$808$		0			0
$809$		−	71600.0i		−	0.109400i	−0.998503	−	0.0546998i	$-0.982580\pi$
$809$		−	71600.0i		−	0.109400i	0.998503	−	0.0546998i	$-0.0174202\pi$
$810$		0			0
$811$	103318.			0.157085			0.0785424	−	0.996911i	$-0.474973\pi$
$811$	103318.			0.157085			0.0785424	+	0.996911i	$0.474973\pi$
$812$		0			0
$813$	−613818.	+	613818.i	−0.928664	+	0.928664i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−304920.	−	304920.i	−0.456817	−	0.456817i
$818$		0			0
$819$			324162.i			0.483275i
$820$		0			0
$821$	−157438.			−0.233573			−0.116787	−	0.993157i	$-0.537259\pi$
$821$	−157438.			−0.233573			−0.116787	+	0.993157i	$0.537259\pi$
$822$		0			0
$823$	791309.	−	791309.i	1.16828	−	1.16828i	0.185666	−	0.982613i	$-0.440556\pi$
$823$	791309.	−	791309.i	1.16828	−	1.16828i	0.982613	−	0.185666i	$-0.0594441\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−889671.	−	889671.i	−1.30082	−	1.30082i	−0.927837	−	0.372987i	$-0.878334\pi$
$827$	−889671.	−	889671.i	−1.30082	−	1.30082i	−0.372987	−	0.927837i	$-0.621666\pi$
$828$		0			0
$829$		−	618280.i		−	0.899655i	−0.893115	−	0.449828i	$-0.851485\pi$
$829$		−	618280.i		−	0.899655i	0.893115	−	0.449828i	$-0.148515\pi$
$830$		0			0
$831$	−333882.			−0.483494
$832$		0			0
$833$	194849.	−	194849.i	0.280807	−	0.280807i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$			821360.i			1.16684i	0.812172	+	0.583418i	$0.198285\pi$
$839$			821360.i			1.16684i	−0.812172	+	0.583418i	$0.801715\pi$
$840$		0			0
$841$	244881.			0.346229
$842$		0			0
$843$	20898.0	−	20898.0i	0.0294069	−	0.0294069i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−20793.0	−	20793.0i	−0.0289835	−	0.0289835i
$848$		0			0
$849$			1.65080e6i			2.29023i
$850$		0			0
$851$	350238.			0.483620
$852$		0			0
$853$	698291.	−	698291.i	0.959706	−	0.959706i	−0.0395127	−	0.999219i	$-0.512581\pi$
$853$	698291.	−	698291.i	0.959706	−	0.959706i	0.999219	+	0.0395127i	$0.0125806\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−144489.	−	144489.i	−0.196731	−	0.196731i	0.601866	−	0.798597i	$-0.294424\pi$
$857$	−144489.	−	144489.i	−0.196731	−	0.196731i	−0.798597	+	0.601866i	$0.794424\pi$
$858$		0			0
$859$			943480.i			1.27863i	0.768943	+	0.639317i	$0.220783\pi$
$859$			943480.i			1.27863i	−0.768943	+	0.639317i	$0.779217\pi$
$860$		0			0
$861$	878004.			1.18438
$862$		0			0
$863$	438149.	−	438149.i	0.588302	−	0.588302i	−0.348869	−	0.937171i	$-0.613434\pi$
$863$	438149.	−	438149.i	0.588302	−	0.588302i	0.937171	+	0.348869i	$0.113434\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	570249.	+	570249.i	0.758624	+	0.758624i
$868$		0			0
$869$		−	1.24608e6i		−	1.65009i
$870$		0			0
$871$	103638.			0.136610
$872$		0			0
$873$	592191.	−	592191.i	0.777022	−	0.777022i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−281469.	−	281469.i	−0.365958	−	0.365958i	0.500043	−	0.866001i	$-0.333318\pi$
$877$	−281469.	−	281469.i	−0.365958	−	0.365958i	−0.866001	+	0.500043i	$0.833318\pi$
$878$		0			0
$879$		−	87318.0i		−	0.113012i
$880$		0			0
$881$	876722.			1.12956			0.564781	−	0.825241i	$-0.308961\pi$
$881$	876722.			1.12956			0.564781	+	0.825241i	$0.308961\pi$
$882$		0			0
$883$	−327431.	+	327431.i	−0.419951	+	0.419951i	−0.885187	−	0.465236i	$-0.845969\pi$
$883$	−327431.	+	327431.i	−0.419951	+	0.419951i	0.465236	+	0.885187i	$0.345969\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−477171.	−	477171.i	−0.606494	−	0.606494i	0.335534	−	0.942028i	$-0.391083\pi$
$887$	−477171.	−	477171.i	−0.606494	−	0.606494i	−0.942028	+	0.335534i	$0.891083\pi$
$888$		0			0
$889$			342722.i			0.433649i
$890$		0			0
$891$	774198.			0.975207
$892$		0			0
$893$	355320.	−	355320.i	0.445571	−	0.445571i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	334098.	+	334098.i	0.415230	+	0.415230i
$898$		0			0
$899$		−	137360.i		−	0.169958i
$900$		0			0
$901$	331162.			0.407935
$902$		0			0
$903$	568458.	−	568458.i	0.697145	−	0.697145i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1.11209e6	+	1.11209e6i	1.35184	+	1.35184i	0.883604	+	0.468235i	$0.155110\pi$
$907$	1.11209e6	+	1.11209e6i	1.35184	+	1.35184i	0.468235	+	0.883604i	$0.344890\pi$
$908$		0			0
$909$			71118.0i			0.0860700i
$910$		0			0
$911$	883958.			1.06511			0.532556	−	0.846395i	$-0.321232\pi$
$911$	883958.			1.06511			0.532556	+	0.846395i	$0.321232\pi$
$912$		0			0
$913$	−735258.	+	735258.i	−0.882060	+	0.882060i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	184382.	+	184382.i	0.219270	+	0.219270i
$918$		0			0
$919$		−	1.24040e6i		−	1.46869i	−0.678775	−	0.734346i	$-0.737489\pi$
$919$		−	1.24040e6i		−	1.46869i	0.678775	−	0.734346i	$-0.262511\pi$
$920$		0			0
$921$	771282.			0.909272
$922$		0			0
$923$	444498.	−	444498.i	0.521755	−	0.521755i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−844749.	−	844749.i	−0.983033	−	0.983033i
$928$		0			0
$929$			1.22744e6i			1.42223i	0.703077	+	0.711113i	$0.251809\pi$
$929$			1.22744e6i			1.42223i	−0.703077	+	0.711113i	$0.748191\pi$
$930$		0			0
$931$	−201320.			−0.232267
$932$		0			0
$933$	650502.	−	650502.i	0.747283	−	0.747283i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1.07047e6	+	1.07047e6i	1.21926	+	1.21926i	0.967892	+	0.251366i	$0.0808799\pi$
$937$	1.07047e6	+	1.07047e6i	1.21926	+	1.21926i	0.251366	+	0.967892i	$0.419120\pi$
$938$		0			0
$939$			328482.i			0.372546i
$940$		0			0
$941$	558642.			0.630891			0.315446	−	0.948944i	$-0.397846\pi$
$941$	558642.			0.630891			0.315446	+	0.948944i	$0.397846\pi$
$942$		0			0
$943$	452458.	−	452458.i	0.508809	−	0.508809i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−191711.	−	191711.i	−0.213770	−	0.213770i	0.592097	−	0.805867i	$-0.298300\pi$
$947$	−191711.	−	191711.i	−0.213770	−	0.213770i	−0.805867	+	0.592097i	$0.798300\pi$
$948$		0			0
$949$			407238.i			0.452185i
$950$		0			0
$951$	−452682.			−0.500532
$952$		0			0
$953$	630231.	−	630231.i	0.693927	−	0.693927i	−0.269166	−	0.963094i	$-0.586748\pi$
$953$	630231.	−	630231.i	0.693927	−	0.693927i	0.963094	+	0.269166i	$0.0867482\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	722160.	+	722160.i	0.788514	+	0.788514i
$958$		0			0
$959$		−	1.18372e6i		−	1.28710i
$960$		0			0
$961$	−882717.			−0.955817
$962$		0			0
$963$	−381591.	+	381591.i	−0.411477	+	0.411477i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−345491.	−	345491.i	−0.369474	−	0.369474i	0.497811	−	0.867285i	$-0.334137\pi$
$967$	−345491.	−	345491.i	−0.369474	−	0.369474i	−0.867285	+	0.497811i	$0.834137\pi$
$968$		0			0
$969$		−	1.36584e6i		−	1.45463i
$970$		0			0
$971$	−1.08308e6			−1.14874			−0.574372	−	0.818595i	$-0.694753\pi$
$971$	−1.08308e6			−1.14874			−0.574372	+	0.818595i	$0.694753\pi$
$972$		0			0
$973$	−272600.	+	272600.i	−0.287939	+	0.287939i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	146751.	+	146751.i	0.153742	+	0.153742i	0.779787	−	0.626045i	$-0.215327\pi$
$977$	146751.	+	146751.i	0.153742	+	0.153742i	−0.626045	+	0.779787i	$0.715327\pi$
$978$		0			0
$979$		−	1.70864e6i		−	1.78273i
$980$		0			0
$981$	1.78524e6			1.85506
$982$		0			0
$983$	466909.	−	466909.i	0.483198	−	0.483198i	−0.422953	−	0.906151i	$-0.639007\pi$
$983$	466909.	−	466909.i	0.483198	−	0.483198i	0.906151	+	0.422953i	$0.139007\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	662418.	+	662418.i	0.679983	+	0.679983i
$988$		0			0
$989$		−	585882.i		−	0.598987i
$990$		0			0
$991$	901238.			0.917682			0.458841	−	0.888518i	$-0.348265\pi$
$991$	901238.			0.917682			0.458841	+	0.888518i	$0.348265\pi$
$992$		0			0
$993$	486342.	−	486342.i	0.493223	−	0.493223i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−152149.	−	152149.i	−0.153066	−	0.153066i	0.626420	−	0.779486i	$-0.284520\pi$
$997$	−152149.	−	152149.i	−0.153066	−	0.153066i	−0.779486	+	0.626420i	$0.784520\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.5.p.c.257.1		2
4.3	odd	2		50.5.c.b.7.1		2
5.2	odd	4		80.5.p.b.33.1		2
5.3	odd	4	inner	400.5.p.c.193.1		2
5.4	even	2		80.5.p.b.17.1		2
12.11	even	2		450.5.g.a.307.1		2
20.3	even	4		50.5.c.b.43.1		2
20.7	even	4		10.5.c.a.3.1	✓	2
20.19	odd	2		10.5.c.a.7.1	yes	2
40.19	odd	2		320.5.p.b.257.1		2
40.27	even	4		320.5.p.b.193.1		2
40.29	even	2		320.5.p.i.257.1		2
40.37	odd	4		320.5.p.i.193.1		2
60.23	odd	4		450.5.g.a.343.1		2
60.47	odd	4		90.5.g.b.73.1		2
60.59	even	2		90.5.g.b.37.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.5.c.a.3.1	✓	2	20.7	even	4
10.5.c.a.7.1	yes	2	20.19	odd	2
50.5.c.b.7.1		2	4.3	odd	2
50.5.c.b.43.1		2	20.3	even	4
80.5.p.b.17.1		2	5.4	even	2
80.5.p.b.33.1		2	5.2	odd	4
90.5.g.b.37.1		2	60.59	even	2
90.5.g.b.73.1		2	60.47	odd	4
320.5.p.b.193.1		2	40.27	even	4
320.5.p.b.257.1		2	40.19	odd	2
320.5.p.i.193.1		2	40.37	odd	4
320.5.p.i.257.1		2	40.29	even	2
400.5.p.c.193.1		2	5.3	odd	4	inner
400.5.p.c.257.1		2	1.1	even	1	trivial
450.5.g.a.307.1		2	12.11	even	2
450.5.g.a.343.1		2	60.23	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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	400.p (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(9.00000 - 9.00000i) q^{3} +(29.0000 + 29.0000i) q^{7} -81.0000i q^{9} +118.000 q^{11} +(-69.0000 + 69.0000i) q^{13} +(271.000 + 271.000i) q^{17} -280.000i q^{19} +522.000 q^{21} +(269.000 - 269.000i) q^{23} +680.000i q^{29} -202.000 q^{31} +(1062.00 - 1062.00i) q^{33} +(651.000 + 651.000i) q^{37} +1242.00i q^{39} +1682.00 q^{41} +(1089.00 - 1089.00i) q^{43} +(1269.00 + 1269.00i) q^{47} -719.000i q^{49} +4878.00 q^{51} +(611.000 - 611.000i) q^{53} +(-2520.00 - 2520.00i) q^{57} -1160.00i q^{59} -5598.00 q^{61} +(2349.00 - 2349.00i) q^{63} +(-751.000 - 751.000i) q^{67} -4842.00i q^{69} -6442.00 q^{71} +(2951.00 - 2951.00i) q^{73} +(3422.00 + 3422.00i) q^{77} -10560.0i q^{79} +6561.00 q^{81} +(-6231.00 + 6231.00i) q^{83} +(6120.00 + 6120.00i) q^{87} -14480.0i q^{89} -4002.00 q^{91} +(-1818.00 + 1818.00i) q^{93} +(7311.00 + 7311.00i) q^{97} -9558.00i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 18 q^{3} + 58 q^{7} + 236 q^{11} - 138 q^{13} + 542 q^{17} + 1044 q^{21} + 538 q^{23} - 404 q^{31} + 2124 q^{33} + 1302 q^{37} + 3364 q^{41} + 2178 q^{43} + 2538 q^{47} + 9756 q^{51} + 1222 q^{53}+ \cdots + 14622 q^{97}+O(q^{100}) \) 2 * q + 18 * q^3 + 58 * q^7 + 236 * q^11 - 138 * q^13 + 542 * q^17 + 1044 * q^21 + 538 * q^23 - 404 * q^31 + 2124 * q^33 + 1302 * q^37 + 3364 * q^41 + 2178 * q^43 + 2538 * q^47 + 9756 * q^51 + 1222 * q^53 - 5040 * q^57 - 11196 * q^61 + 4698 * q^63 - 1502 * q^67 - 12884 * q^71 + 5902 * q^73 + 6844 * q^77 + 13122 * q^81 - 12462 * q^83 + 12240 * q^87 - 8004 * q^91 - 3636 * q^93 + 14622 * q^97

Introduction

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