LMFDB - Embedded newform 252.5.z.b.145.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,5,Mod(73,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("252.73");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	252.z (of order $6$, 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:	$26.0492306971$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} - 5x + 25 $ x^4 - x^3 - 4x^2 - 5x + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			145.2
Root			$2.13746 + 0.656712i$ of defining polynomial
Character	$\chi$	$=$	252.145
Dual form			252.5.z.b.73.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(7.23713 + 4.17836i) q^{5} +(-17.5000 - 45.7684i) q^{7} +(38.8866 + 67.3536i) q^{11} -94.1795i q^{13} +(-93.5257 + 53.9971i) q^{17} +(-253.304 - 146.245i) q^{19} +(-126.598 + 219.274i) q^{23} +(-277.583 - 480.787i) q^{25} -548.310 q^{29} +(354.232 - 204.516i) q^{31} +(64.5872 - 404.353i) q^{35} +(-135.232 + 234.229i) q^{37} -3063.45i q^{41} +553.062 q^{43} +(-2037.62 - 1176.42i) q^{47} +(-1788.50 + 1601.90i) q^{49} +(-2511.81 - 4350.58i) q^{53} +649.929i q^{55} +(-2913.62 + 1682.18i) q^{59} +(-2871.61 - 1657.92i) q^{61} +(393.516 - 681.589i) q^{65} +(-299.491 - 518.734i) q^{67} -5363.24 q^{71} +(-1241.26 + 716.640i) q^{73} +(2402.15 - 2958.47i) q^{77} +(5586.98 - 9676.93i) q^{79} +1326.79i q^{83} -902.477 q^{85} +(10705.2 + 6180.67i) q^{89} +(-4310.45 + 1648.14i) q^{91} +(-1222.13 - 2116.79i) q^{95} -5319.78i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 39 q^{5} - 70 q^{7} - 3 q^{11} - 510 q^{17} + 459 q^{19} - 144 q^{23} - 227 q^{25} + 570 q^{29} + 2640 q^{31} + 2478 q^{35} + 433 q^{37} - 98 q^{43} + 1770 q^{47} - 7154 q^{49} + 213 q^{53} + 4857 q^{59}+ \cdots - 19656 q^{95}+O(q^{100}) $ 4 * q - 39 * q^5 - 70 * q^7 - 3 * q^11 - 510 * q^17 + 459 * q^19 - 144 * q^23 - 227 * q^25 + 570 * q^29 + 2640 * q^31 + 2478 * q^35 + 433 * q^37 - 98 * q^43 + 1770 * q^47 - 7154 * q^49 + 213 * q^53 + 4857 * q^59 - 12936 * q^61 - 102 * q^65 + 7205 * q^67 - 19188 * q^71 + 14355 * q^73 + 12621 * q^77 + 13424 * q^79 + 9708 * q^85 + 36162 * q^89 - 5985 * q^91 - 19656 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$e\left(\frac{5}{6}\right)$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	7.23713	+	4.17836i	0.289485	+	0.167134i	0.637710	−	0.770277i	$-0.279882\pi$
$5$	7.23713	+	4.17836i	0.289485	+	0.167134i	−0.348225	+	0.937411i	$0.613215\pi$
$6$		0			0
$7$	−17.5000	−	45.7684i	−0.357143	−	0.934050i
$7$	−17.5000	−	45.7684i	−0.357143	−	0.934050i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	38.8866	+	67.3536i	0.321377	+	0.556641i	0.980772	−	0.195155i	$-0.0625210\pi$
$11$	38.8866	+	67.3536i	0.321377	+	0.556641i	−0.659395	+	0.751796i	$0.729188\pi$
$12$		0			0
$13$		−	94.1795i		−	0.557275i	−0.960396	−	0.278638i	$-0.910117\pi$
$13$		−	94.1795i		−	0.557275i	0.960396	−	0.278638i	$-0.0898829\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−93.5257	+	53.9971i	−0.323618	+	0.186841i	−0.653004	−	0.757354i	$-0.726492\pi$
$17$	−93.5257	+	53.9971i	−0.323618	+	0.186841i	0.329386	+	0.944195i	$0.393158\pi$
$18$		0			0
$19$	−253.304	−	146.245i	−0.701674	−	0.405112i	0.106296	−	0.994334i	$-0.466101\pi$
$19$	−253.304	−	146.245i	−0.701674	−	0.405112i	−0.807971	+	0.589223i	$0.799434\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−126.598	+	219.274i	−0.239316	+	0.414507i	−0.960518	−	0.278217i	$-0.910256\pi$
$23$	−126.598	+	219.274i	−0.239316	+	0.414507i	0.721202	+	0.692724i	$0.243590\pi$
$24$		0			0
$25$	−277.583	−	480.787i	−0.444132	−	0.769260i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−548.310			−0.651974			−0.325987	−	0.945374i	$-0.605697\pi$
$29$	−548.310			−0.651974			−0.325987	+	0.945374i	$0.605697\pi$
$30$		0			0
$31$	354.232	−	204.516i	0.368607	−	0.212816i	−0.304242	−	0.952595i	$-0.598403\pi$
$31$	354.232	−	204.516i	0.368607	−	0.212816i	0.672850	+	0.739779i	$0.265070\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	64.5872	−	404.353i	0.0527242	−	0.330084i
$36$		0			0
$37$	−135.232	+	234.229i	−0.0987817	+	0.171095i	−0.911181	−	0.412007i	$-0.864828\pi$
$37$	−135.232	+	234.229i	−0.0987817	+	0.171095i	0.812399	+	0.583102i	$0.198161\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	3063.45i		−	1.82239i	−0.411970	−	0.911197i	$-0.635159\pi$
$41$		−	3063.45i		−	1.82239i	0.411970	−	0.911197i	$-0.364841\pi$
$42$		0			0
$43$	553.062			0.299114			0.149557	−	0.988753i	$-0.452215\pi$
$43$	553.062			0.299114			0.149557	+	0.988753i	$0.452215\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2037.62	−	1176.42i	−0.922418	−	0.532558i	−0.0380121	−	0.999277i	$-0.512103\pi$
$47$	−2037.62	−	1176.42i	−0.922418	−	0.532558i	−0.884406	+	0.466719i	$0.845436\pi$
$48$		0			0
$49$	−1788.50	+	1601.90i	−0.744898	+	0.667178i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2511.81	−	4350.58i	−0.894199	−	1.54880i	−0.834792	−	0.550565i	$-0.814412\pi$
$53$	−2511.81	−	4350.58i	−0.894199	−	1.54880i	−0.0594072	−	0.998234i	$-0.518921\pi$
$54$		0			0
$55$			649.929i			0.214853i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2913.62	+	1682.18i	−0.837007	+	0.483246i	−0.856246	−	0.516569i	$-0.827209\pi$
$59$	−2913.62	+	1682.18i	−0.837007	+	0.483246i	0.0192386	+	0.999815i	$0.493876\pi$
$60$		0			0
$61$	−2871.61	−	1657.92i	−0.771730	−	0.445559i	0.0617613	−	0.998091i	$-0.480328\pi$
$61$	−2871.61	−	1657.92i	−0.771730	−	0.445559i	−0.833491	+	0.552532i	$0.813662\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	393.516	−	681.589i	0.0931398	−	0.161323i
$66$		0			0
$67$	−299.491	−	518.734i	−0.0667167	−	0.115557i	0.830737	−	0.556664i	$-0.187919\pi$
$67$	−299.491	−	518.734i	−0.0667167	−	0.115557i	−0.897454	+	0.441108i	$0.854586\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−5363.24			−1.06392			−0.531962	−	0.846768i	$-0.678545\pi$
$71$	−5363.24			−1.06392			−0.531962	+	0.846768i	$0.678545\pi$
$72$		0			0
$73$	−1241.26	+	716.640i	−0.232925	+	0.134479i	−0.611921	−	0.790919i	$-0.709603\pi$
$73$	−1241.26	+	716.640i	−0.232925	+	0.134479i	0.378996	+	0.925398i	$0.376269\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2402.15	−	2958.47i	0.405153	−	0.498983i
$78$		0			0
$79$	5586.98	−	9676.93i	0.895205	−	1.55054i	0.0616549	−	0.998098i	$-0.480362\pi$
$79$	5586.98	−	9676.93i	0.895205	−	1.55054i	0.833550	−	0.552443i	$-0.186304\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			1326.79i			0.192595i	0.995353	+	0.0962977i	$0.0307001\pi$
$83$			1326.79i			0.192595i	−0.995353	+	0.0962977i	$0.969300\pi$
$84$		0			0
$85$	−902.477			−0.124910
$86$		0			0
$87$		0			0
$88$		0			0
$89$	10705.2	+	6180.67i	1.35150	+	0.780289i	0.988460	−	0.151484i	$-0.0484051\pi$
$89$	10705.2	+	6180.67i	1.35150	+	0.780289i	0.363041	+	0.931773i	$0.381738\pi$
$90$		0			0
$91$	−4310.45	+	1648.14i	−0.520523	+	0.199027i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1222.13	−	2116.79i	−0.135416	−	0.234548i
$96$		0			0
$97$		−	5319.78i		−	0.565393i	−0.959209	−	0.282696i	$-0.908771\pi$
$97$		−	5319.78i		−	0.565393i	0.959209	−	0.282696i	$-0.0912289\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3965.54	−	2289.51i	0.388740	−	0.224439i	−0.292874	−	0.956151i	$-0.594612\pi$
$101$	3965.54	−	2289.51i	0.388740	−	0.224439i	0.681614	+	0.731712i	$0.261278\pi$
$102$		0			0
$103$	−6332.08	−	3655.83i	−0.596859	−	0.344597i	0.170946	−	0.985280i	$-0.445318\pi$
$103$	−6332.08	−	3655.83i	−0.596859	−	0.344597i	−0.767805	+	0.640684i	$0.778651\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−9698.96	+	16799.1i	−0.847145	+	1.46730i	0.0366010	+	0.999330i	$0.488347\pi$
$107$	−9698.96	+	16799.1i	−0.847145	+	1.46730i	−0.883746	+	0.467968i	$0.844986\pi$
$108$		0			0
$109$	6465.30	+	11198.2i	0.544172	+	0.942533i	0.998659	+	0.0517795i	$0.0164893\pi$
$109$	6465.30	+	11198.2i	0.544172	+	0.942533i	−0.454487	+	0.890753i	$0.650177\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−5153.13			−0.403566			−0.201783	−	0.979430i	$-0.564674\pi$
$113$	−5153.13			−0.403566			−0.201783	+	0.979430i	$0.564674\pi$
$114$		0			0
$115$	−1832.41	+	1057.94i	−0.138557	+	0.0799957i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	4108.06	+	3335.58i	0.290097	+	0.235547i
$120$		0			0
$121$	4296.16	−	7441.17i	0.293434	−	0.508242i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	9862.31i		−	0.631188i
$126$		0			0
$127$	10320.4			0.639865			0.319932	−	0.947440i	$-0.396340\pi$
$127$	10320.4			0.639865			0.319932	+	0.947440i	$0.396340\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	27221.9	+	15716.6i	1.58627	+	0.915832i	0.993915	+	0.110154i	$0.0351345\pi$
$131$	27221.9	+	15716.6i	1.58627	+	0.915832i	0.592354	+	0.805678i	$0.298199\pi$
$132$		0			0
$133$	−2260.60	+	14152.6i	−0.127797	+	0.800082i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6927.98	−	11999.6i	−0.369118	−	0.639332i	0.620310	−	0.784357i	$-0.287007\pi$
$137$	−6927.98	−	11999.6i	−0.369118	−	0.639332i	−0.989428	+	0.145025i	$0.953674\pi$
$138$		0			0
$139$			24750.2i			1.28100i	0.767959	+	0.640499i	$0.221273\pi$
$139$			24750.2i			1.28100i	−0.767959	+	0.640499i	$0.778727\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	6343.33	−	3662.33i	0.310203	−	0.179096i
$144$		0			0
$145$	−3968.19	−	2291.03i	−0.188737	−	0.108967i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	85.4878	−	148.069i	0.00385063	−	0.00666949i	−0.864094	−	0.503331i	$-0.832108\pi$
$149$	85.4878	−	148.069i	0.00385063	−	0.00666949i	0.867944	+	0.496662i	$0.165441\pi$
$150$		0			0
$151$	18574.4	+	32171.7i	0.814629	+	1.41098i	0.909594	+	0.415498i	$0.136393\pi$
$151$	18574.4	+	32171.7i	0.814629	+	1.41098i	−0.0949648	+	0.995481i	$0.530274\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3418.16			0.142275
$156$		0			0
$157$	13150.6	−	7592.49i	0.533514	−	0.308024i	−0.208932	−	0.977930i	$-0.566999\pi$
$157$	13150.6	−	7592.49i	0.533514	−	0.308024i	0.742446	+	0.669906i	$0.233665\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	12251.3	+	1956.90i	0.472640	+	0.0754946i
$162$		0			0
$163$	1890.00	−	3273.58i	0.0711357	−	0.123211i	−0.828264	−	0.560339i	$-0.810671\pi$
$163$	1890.00	−	3273.58i	0.0711357	−	0.123211i	0.899399	+	0.437128i	$0.144004\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	10590.5i		−	0.379739i	−0.981809	−	0.189869i	$-0.939194\pi$
$167$		−	10590.5i		−	0.379739i	0.981809	−	0.189869i	$-0.0608064\pi$
$168$		0			0
$169$	19691.2			0.689444
$170$		0			0
$171$		0			0
$172$		0			0
$173$	34763.8	+	20070.9i	1.16154	+	0.670616i	0.951673	−	0.307114i	$-0.0993632\pi$
$173$	34763.8	+	20070.9i	1.16154	+	0.670616i	0.209868	+	0.977730i	$0.432697\pi$
$174$		0			0
$175$	−17147.2	+	21118.3i	−0.559908	+	0.689577i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	18788.4	+	32542.4i	0.586385	+	1.01565i	0.994701	+	0.102808i	$0.0327826\pi$
$179$	18788.4	+	32542.4i	0.586385	+	1.01565i	−0.408317	+	0.912840i	$0.633884\pi$
$180$		0			0
$181$		−	39852.1i		−	1.21645i	−0.793765	−	0.608224i	$-0.791882\pi$
$181$		−	39852.1i		−	1.21645i	0.793765	−	0.608224i	$-0.208118\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1957.38	+	1130.10i	−0.0571917	+	0.0330196i
$186$		0			0
$187$	−7273.80	−	4199.53i	−0.208007	−	0.120093i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−22747.5	+	39399.8i	−0.623544	+	1.08001i	0.365277	+	0.930899i	$0.380974\pi$
$191$	−22747.5	+	39399.8i	−0.623544	+	1.08001i	−0.988821	+	0.149111i	$0.952359\pi$
$192$		0			0
$193$	−6045.20	−	10470.6i	−0.162292	−	0.281097i	0.773399	−	0.633920i	$-0.218555\pi$
$193$	−6045.20	−	10470.6i	−0.162292	−	0.281097i	−0.935690	+	0.352823i	$0.885222\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−43371.5			−1.11756			−0.558782	−	0.829315i	$-0.688731\pi$
$197$	−43371.5			−1.11756			−0.558782	+	0.829315i	$0.688731\pi$
$198$		0			0
$199$	19667.6	−	11355.1i	0.496645	−	0.286738i	−0.230682	−	0.973029i	$-0.574096\pi$
$199$	19667.6	−	11355.1i	0.496645	−	0.286738i	0.727327	+	0.686291i	$0.240762\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	9595.42	+	25095.3i	0.232848	+	0.608976i
$204$		0			0
$205$	12800.2	−	22170.5i	0.304585	−	0.527556i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	22748.0i		−	0.520775i
$210$		0			0
$211$	−75885.9			−1.70450			−0.852249	−	0.523137i	$-0.824762\pi$
$211$	−75885.9			−1.70450			−0.852249	+	0.523137i	$0.824762\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4002.58	+	2310.89i	0.0865891	+	0.0499923i
$216$		0			0
$217$	−15559.4	−	12633.6i	−0.330426	−	0.268292i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	5085.42	+	8808.21i	0.104122	+	0.180345i
$222$		0			0
$223$			18276.7i			0.367526i	0.982971	+	0.183763i	$0.0588279\pi$
$223$			18276.7i			0.367526i	−0.982971	+	0.183763i	$0.941172\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	10304.8	−	5949.48i	0.199981	−	0.115459i	−0.396666	−	0.917963i	$-0.629833\pi$
$227$	10304.8	−	5949.48i	0.199981	−	0.115459i	0.596646	+	0.802504i	$0.296500\pi$
$228$		0			0
$229$	−254.071	−	146.688i	−0.00484488	−	0.00279720i	0.497576	−	0.867421i	$-0.334224\pi$
$229$	−254.071	−	146.688i	−0.00484488	−	0.00279720i	−0.502420	+	0.864623i	$0.667557\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	35754.6	−	61928.8i	0.658598	−	1.14073i	−0.322381	−	0.946610i	$-0.604483\pi$
$233$	35754.6	−	61928.8i	0.658598	−	1.14073i	0.980979	−	0.194115i	$-0.0621835\pi$
$234$		0			0
$235$	−9831.01	−	17027.8i	−0.178017	−	0.308335i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−62869.3			−1.10063			−0.550316	−	0.834956i	$-0.685493\pi$
$239$	−62869.3			−1.10063			−0.550316	+	0.834956i	$0.685493\pi$
$240$		0			0
$241$	52680.1	−	30414.9i	0.907011	−	0.523663i	0.0275425	−	0.999621i	$-0.491232\pi$
$241$	52680.1	−	30414.9i	0.907011	−	0.523663i	0.879468	+	0.475958i	$0.157899\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−19636.9	+	4120.13i	−0.327145	+	0.0686402i
$246$		0			0
$247$	−13773.3	+	23856.1i	−0.225759	+	0.391026i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	71630.1i		−	1.13697i	−0.822695	−	0.568484i	$-0.807530\pi$
$251$		−	71630.1i		−	1.13697i	0.822695	−	0.568484i	$-0.192470\pi$
$252$		0			0
$253$	−19691.9			−0.307642
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−50347.3	−	29068.0i	−0.762271	−	0.440098i	0.0678393	−	0.997696i	$-0.478389\pi$
$257$	−50347.3	−	29068.0i	−0.762271	−	0.440098i	−0.830111	+	0.557599i	$0.811723\pi$
$258$		0			0
$259$	13086.9	+	2090.36i	0.195090	+	0.0311617i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−15899.6	−	27539.0i	−0.229866	−	0.398140i	0.727902	−	0.685681i	$-0.240496\pi$
$263$	−15899.6	−	27539.0i	−0.229866	−	0.398140i	−0.957768	+	0.287541i	$0.907162\pi$
$264$		0			0
$265$		−	41980.9i		−	0.597806i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	47568.2	−	27463.5i	0.657374	−	0.379535i	−0.133902	−	0.990995i	$-0.542751\pi$
$269$	47568.2	−	27463.5i	0.657374	−	0.379535i	0.791276	+	0.611460i	$0.209417\pi$
$270$		0			0
$271$	79439.7	+	45864.6i	1.08168	+	0.624509i	0.931349	−	0.364127i	$-0.118633\pi$
$271$	79439.7	+	45864.6i	1.08168	+	0.624509i	0.150332	+	0.988636i	$0.451966\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	21588.5	−	37392.4i	0.285468	−	0.494445i
$276$		0			0
$277$	−3667.78	−	6352.78i	−0.0478018	−	0.0827951i	0.841135	−	0.540826i	$-0.181888\pi$
$277$	−3667.78	−	6352.78i	−0.0478018	−	0.0827951i	−0.888936	+	0.458031i	$0.848555\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−133185.			−1.68671			−0.843356	−	0.537355i	$-0.819424\pi$
$281$	−133185.			−1.68671			−0.843356	+	0.537355i	$0.819424\pi$
$282$		0			0
$283$	−108302.	+	62528.0i	−1.35227	+	0.780731i	−0.988567	−	0.150785i	$-0.951820\pi$
$283$	−108302.	+	62528.0i	−1.35227	+	0.780731i	−0.363700	+	0.931516i	$0.618487\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−140209.	+	53610.3i	−1.70221	+	0.650855i
$288$		0			0
$289$	−35929.1	+	62231.1i	−0.430181	+	0.745095i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	137393.i		−	1.60040i	−0.599730	−	0.800202i	$-0.704725\pi$
$293$		−	137393.i		−	1.60040i	0.599730	−	0.800202i	$-0.295275\pi$
$294$		0			0
$295$	−28115.0			−0.323068
$296$		0			0
$297$		0			0
$298$		0			0
$299$	20651.1	+	11922.9i	0.230995	+	0.133365i
$300$		0			0
$301$	−9678.59	−	25312.8i	−0.106827	−	0.279388i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−13854.8	−	23997.2i	−0.148936	−	0.257965i
$306$		0			0
$307$		−	59400.1i		−	0.630247i	−0.949051	−	0.315123i	$-0.897954\pi$
$307$		−	59400.1i		−	0.630247i	0.949051	−	0.315123i	$-0.102046\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−39127.7	+	22590.4i	−0.404542	+	0.233562i	−0.688442	−	0.725291i	$-0.741705\pi$
$311$	−39127.7	+	22590.4i	−0.404542	+	0.233562i	0.283900	+	0.958854i	$0.408372\pi$
$312$		0			0
$313$	92305.3	+	53292.5i	0.942189	+	0.543973i	0.890646	−	0.454698i	$-0.150253\pi$
$313$	92305.3	+	53292.5i	0.942189	+	0.543973i	0.0515431	+	0.998671i	$0.483586\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	76611.8	−	132696.i	0.762390	−	1.32050i	−0.179225	−	0.983808i	$-0.557359\pi$
$317$	76611.8	−	132696.i	0.762390	−	1.32050i	0.941615	−	0.336691i	$-0.109308\pi$
$318$		0			0
$319$	−21321.9	−	36930.7i	−0.209529	−	0.362916i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	31587.3			0.302766
$324$		0			0
$325$	−45280.3	+	26142.6i	−0.428689	+	0.247504i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−18184.6	+	113846.i	−0.168001	+	1.05178i
$330$		0			0
$331$	56180.6	−	97307.7i	0.512779	−	0.888160i	−0.487111	−	0.873340i	$-0.661949\pi$
$331$	56180.6	−	97307.7i	0.512779	−	0.888160i	0.999890	−	0.0148198i	$-0.00471745\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	5005.53i		−	0.0446026i
$336$		0			0
$337$	−141074.			−1.24219			−0.621093	−	0.783737i	$-0.713311\pi$
$337$	−141074.			−1.24219			−0.621093	+	0.783737i	$0.713311\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	27549.8	+	15905.9i	0.236924	+	0.136788i
$342$		0			0
$343$	104615.	+	53823.7i	0.889213	+	0.457494i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	100247.	+	173633.i	0.832557	+	1.44203i	0.896004	+	0.444046i	$0.146457\pi$
$347$	100247.	+	173633.i	0.832557	+	1.44203i	−0.0634473	+	0.997985i	$0.520209\pi$
$348$		0			0
$349$		−	209647.i		−	1.72122i	−0.509263	−	0.860611i	$-0.670082\pi$
$349$		−	209647.i		−	1.72122i	0.509263	−	0.860611i	$-0.329918\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	138290.	−	79841.9i	1.10979	−	0.640739i	0.171017	−	0.985268i	$-0.445295\pi$
$353$	138290.	−	79841.9i	1.10979	−	0.640739i	0.938776	+	0.344529i	$0.111961\pi$
$354$		0			0
$355$	−38814.4	−	22409.5i	−0.307990	−	0.177818i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−35094.0	+	60784.5i	−0.272297	+	0.471633i	−0.969450	−	0.245290i	$-0.921117\pi$
$359$	−35094.0	+	60784.5i	−0.272297	+	0.471633i	0.697152	+	0.716923i	$0.254450\pi$
$360$		0			0
$361$	−22385.1	−	38772.1i	−0.171769	−	0.297512i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−11977.5			−0.0899044
$366$		0			0
$367$	205812.	−	118826.i	1.52806	−	0.882223i	0.528613	−	0.848863i	$-0.322712\pi$
$367$	205812.	−	118826.i	1.52806	−	0.882223i	0.999443	−	0.0333603i	$-0.0106209\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−155162.	+	191097.i	−1.12730	+	1.38837i
$372$		0			0
$373$	−122554.	+	212269.i	−0.880863	+	1.52570i	−0.0304807	+	0.999535i	$0.509704\pi$
$373$	−122554.	+	212269.i	−0.880863	+	1.52570i	−0.850383	+	0.526165i	$0.823630\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			51639.6i			0.363329i
$378$		0			0
$379$	−95402.6			−0.664174			−0.332087	−	0.943249i	$-0.607753\pi$
$379$	−95402.6			−0.664174			−0.332087	+	0.943249i	$0.607753\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	57753.3	+	33343.9i	0.393712	+	0.227310i	0.683767	−	0.729700i	$-0.260340\pi$
$383$	57753.3	+	33343.9i	0.393712	+	0.227310i	−0.290055	+	0.957010i	$0.593674\pi$
$384$		0			0
$385$	29746.2	−	11373.8i	0.200683	−	0.0767331i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	77439.5	+	134129.i	0.511757	+	0.886389i	0.999907	+	0.0136291i	$0.00433840\pi$
$389$	77439.5	+	134129.i	0.511757	+	0.886389i	−0.488150	+	0.872760i	$0.662328\pi$
$390$		0			0
$391$		−	27343.7i		−	0.178856i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	80867.3	−	46688.8i	0.518297	−	0.299239i
$396$		0			0
$397$	174546.	+	100774.i	1.10746	+	0.639392i	0.938170	−	0.346174i	$-0.112519\pi$
$397$	174546.	+	100774.i	1.10746	+	0.639392i	0.169290	+	0.985566i	$0.445853\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	16945.5	−	29350.5i	0.105382	−	0.182527i	−0.808512	−	0.588479i	$-0.799727\pi$
$401$	16945.5	−	29350.5i	0.105382	−	0.182527i	0.913894	+	0.405952i	$0.133060\pi$
$402$		0			0
$403$	−19261.2	−	33361.4i	−0.118597	−	0.205416i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−21034.9			−0.126985
$408$		0			0
$409$	−201013.	+	116055.i	−1.20165	+	0.693771i	−0.960921	−	0.276822i	$-0.910719\pi$
$409$	−201013.	+	116055.i	−1.20165	+	0.693771i	−0.240725	+	0.970593i	$0.577385\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	127979.	+	103914.i	0.750307	+	0.609218i
$414$		0			0
$415$	−5543.80	+	9602.15i	−0.0321893	+	0.0557535i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	162717.i		−	0.926838i	−0.886139	−	0.463419i	$-0.846622\pi$
$419$		−	162717.i		−	0.926838i	0.886139	−	0.463419i	$-0.153378\pi$
$420$		0			0
$421$	244086.			1.37714			0.688570	−	0.725170i	$-0.258239\pi$
$421$	244086.			1.37714			0.688570	+	0.725170i	$0.258239\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	51922.2	+	29977.3i	0.287459	+	0.165964i
$426$		0			0
$427$	−25627.4	+	160443.i	−0.140556	+	0.879962i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	57761.7	+	100046.i	0.310946	+	0.538575i	0.978567	−	0.205926i	$-0.0660208\pi$
$431$	57761.7	+	100046.i	0.310946	+	0.538575i	−0.667621	+	0.744501i	$0.732687\pi$
$432$		0			0
$433$			54296.2i			0.289597i	0.989461	+	0.144798i	$0.0462534\pi$
$433$			54296.2i			0.289597i	−0.989461	+	0.144798i	$0.953747\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	64135.7	−	37028.7i	0.335843	−	0.193899i
$438$		0			0
$439$	279367.	+	161293.i	1.44959	+	0.836923i	0.998457	−	0.0555326i	$-0.0176857\pi$
$439$	279367.	+	161293.i	1.44959	+	0.836923i	0.451136	+	0.892455i	$0.351019\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	105739.	−	183146.i	0.538802	−	0.933232i	−0.460167	−	0.887832i	$-0.652211\pi$
$443$	105739.	−	183146.i	0.538802	−	0.933232i	0.998969	−	0.0453997i	$-0.0144562\pi$
$444$		0			0
$445$	51650.1	+	89460.6i	0.260826	+	0.451764i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−223169.			−1.10699			−0.553493	−	0.832854i	$-0.686705\pi$
$449$	−223169.			−1.10699			−0.553493	+	0.832854i	$0.686705\pi$
$450$		0			0
$451$	206334.	−	119127.i	1.01442	−	0.585676i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−38081.8	−	6082.79i	−0.183948	−	0.0293819i
$456$		0			0
$457$	93.6414	−	162.192i	0.000448369	−	0.000776598i	−0.865801	−	0.500388i	$-0.833191\pi$
$457$	93.6414	−	162.192i	0.000448369	−	0.000776598i	0.866250	+	0.499612i	$0.166524\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			8842.64i			0.0416083i	0.999784	+	0.0208042i	$0.00662265\pi$
$461$			8842.64i			0.0416083i	−0.999784	+	0.0208042i	$0.993377\pi$
$462$		0			0
$463$	−363326.			−1.69486			−0.847431	−	0.530906i	$-0.821852\pi$
$463$	−363326.			−1.69486			−0.847431	+	0.530906i	$0.821852\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	211698.	+	122224.i	0.970698	+	0.560433i	0.899449	−	0.437026i	$-0.143968\pi$
$467$	211698.	+	122224.i	0.970698	+	0.560433i	0.0712488	+	0.997459i	$0.477302\pi$
$468$		0			0
$469$	−18500.6	+	22785.1i	−0.0841084	+	0.103587i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	21506.7	+	37250.8i	0.0961285	+	0.166499i
$474$		0			0
$475$			162381.i			0.719693i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	30752.3	−	17754.8i	0.134031	−	0.0773829i	−0.431485	−	0.902120i	$-0.642010\pi$
$479$	30752.3	−	17754.8i	0.134031	−	0.0773829i	0.565516	+	0.824737i	$0.308677\pi$
$480$		0			0
$481$	22059.6	+	12736.1i	0.0953470	+	0.0550486i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	22227.9	−	38499.9i	0.0944965	−	0.163673i
$486$		0			0
$487$	−27074.8	−	46894.9i	−0.114158	−	0.197728i	0.803285	−	0.595595i	$-0.203084\pi$
$487$	−27074.8	−	46894.9i	−0.114158	−	0.197728i	−0.917443	+	0.397867i	$0.869750\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−92603.2			−0.384117			−0.192058	−	0.981384i	$-0.561516\pi$
$491$	−92603.2			−0.384117			−0.192058	+	0.981384i	$0.561516\pi$
$492$		0			0
$493$	51281.1	−	29607.1i	0.210991	−	0.121816i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	93856.7	+	245467.i	0.379973	+	0.993757i
$498$		0			0
$499$	−3480.87	+	6029.05i	−0.0139794	+	0.0242130i	−0.872930	−	0.487845i	$-0.837783\pi$
$499$	−3480.87	+	6029.05i	−0.0139794	+	0.0242130i	0.858951	+	0.512058i	$0.171117\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	466372.i		−	1.84330i	−0.388021	−	0.921650i	$-0.626841\pi$
$503$		−	466372.i		−	1.84330i	0.388021	−	0.921650i	$-0.373159\pi$
$504$		0			0
$505$	38265.5			0.150046
$506$		0			0
$507$		0			0
$508$		0			0
$509$	174228.	+	100590.i	0.672483	+	0.388259i	0.797017	−	0.603957i	$-0.206410\pi$
$509$	174228.	+	100590.i	0.672483	+	0.388259i	−0.124533	+	0.992215i	$0.539743\pi$
$510$		0			0
$511$	54521.5	+	44269.2i	0.208798	+	0.169535i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−30550.7	−	52915.4i	−0.115188	−	0.199511i
$516$		0			0
$517$		−	182988.i		−	0.684608i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−300077.	+	173249.i	−1.10549	+	0.638258i	−0.937659	−	0.347557i	$-0.887011\pi$
$521$	−300077.	+	173249.i	−1.10549	+	0.638258i	−0.167836	+	0.985815i	$0.553678\pi$
$522$		0			0
$523$	−252492.	−	145776.i	−0.923090	−	0.532946i	−0.0384707	−	0.999260i	$-0.512249\pi$
$523$	−252492.	−	145776.i	−0.923090	−	0.532946i	−0.884620	+	0.466313i	$0.845582\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−22086.5	+	38255.0i	−0.0795254	+	0.137742i
$528$		0			0
$529$	107866.	+	186830.i	0.385456	+	0.667629i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−288514.			−1.01558
$534$		0			0
$535$	−140385.	+	81051.4i	−0.490471	+	0.283174i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−177442.	−	58169.6i	−0.610772	−	0.200225i
$540$		0			0
$541$	193007.	−	334298.i	0.659445	−	1.14219i	−0.321314	−	0.946973i	$-0.604125\pi$
$541$	193007.	−	334298.i	0.659445	−	1.14219i	0.980759	−	0.195220i	$-0.0625422\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			108057.i			0.363799i
$546$		0			0
$547$	122247.			0.408566			0.204283	−	0.978912i	$-0.434514\pi$
$547$	122247.			0.408566			0.204283	+	0.978912i	$0.434514\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	138889.	+	80187.8i	0.457473	+	0.264122i
$552$		0			0
$553$	−540670.	−	86361.0i	−1.76800	−	0.282402i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	53061.4	+	91905.1i	0.171029	+	0.296230i	0.938780	−	0.344518i	$-0.111958\pi$
$557$	53061.4	+	91905.1i	0.171029	+	0.296230i	−0.767751	+	0.640748i	$0.778624\pi$
$558$		0			0
$559$		−	52087.2i		−	0.166689i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	254823.	−	147122.i	0.803935	−	0.464152i	−0.0409102	−	0.999163i	$-0.513026\pi$
$563$	254823.	−	147122.i	0.803935	−	0.464152i	0.844845	+	0.535011i	$0.179692\pi$
$564$		0			0
$565$	−37293.9	−	21531.6i	−0.116826	−	0.0674497i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−237514.	+	411386.i	−0.733609	+	1.27065i	0.221722	+	0.975110i	$0.428832\pi$
$569$	−237514.	+	411386.i	−0.733609	+	1.27065i	−0.955331	+	0.295538i	$0.904501\pi$
$570$		0			0
$571$	−92589.2	−	160369.i	−0.283980	−	0.491868i	0.688381	−	0.725349i	$-0.258322\pi$
$571$	−92589.2	−	160369.i	−0.283980	−	0.491868i	−0.972361	+	0.233481i	$0.924988\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	140566.			0.425151
$576$		0			0
$577$	139857.	−	80746.6i	0.420081	−	0.242534i	−0.275031	−	0.961435i	$-0.588688\pi$
$577$	139857.	−	80746.6i	0.420081	−	0.242534i	0.695112	+	0.718901i	$0.255355\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	60725.1	−	23218.8i	0.179894	−	0.0687841i
$582$		0			0
$583$	195351.	−	338358.i	0.574750	−	0.995497i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	333168.i		−	0.966911i	−0.875369	−	0.483456i	$-0.839381\pi$
$587$		−	333168.i		−	0.966911i	0.875369	−	0.483456i	$-0.160619\pi$
$588$		0			0
$589$	−119638.			−0.344856
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−97432.1	−	56252.5i	−0.277072	−	0.159968i	0.355025	−	0.934857i	$-0.384472\pi$
$593$	−97432.1	−	56252.5i	−0.277072	−	0.159968i	−0.632097	+	0.774889i	$0.717806\pi$
$594$		0			0
$595$	15793.3	+	41305.0i	0.0446108	+	0.116672i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−202632.	−	350970.i	−0.564749	−	0.978174i	−0.997073	−	0.0764554i	$-0.975640\pi$
$599$	−202632.	−	350970.i	−0.564749	−	0.978174i	0.432324	−	0.901718i	$-0.357694\pi$
$600$		0			0
$601$		−	372831.i		−	1.03220i	−0.856529	−	0.516098i	$-0.827384\pi$
$601$		−	372831.i		−	1.03220i	0.856529	−	0.516098i	$-0.172616\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	62183.7	−	35901.8i	0.169889	−	0.0980856i
$606$		0			0
$607$	−472368.	−	272722.i	−1.28204	−	0.740188i	−0.304822	−	0.952409i	$-0.598597\pi$
$607$	−472368.	−	272722.i	−1.28204	−	0.740188i	−0.977222	+	0.212221i	$0.931930\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−110795.	+	191902.i	−0.296782	+	0.514041i
$612$		0			0
$613$	219300.	+	379838.i	0.583602	+	1.01083i	0.995048	+	0.0993946i	$0.0316906\pi$
$613$	219300.	+	379838.i	0.583602	+	1.01083i	−0.411446	+	0.911434i	$0.634976\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−663957.			−1.74409			−0.872046	−	0.489423i	$-0.837207\pi$
$617$	−663957.			−1.74409			−0.872046	+	0.489423i	$0.837207\pi$
$618$		0			0
$619$	237343.	−	137030.i	0.619434	−	0.357630i	−0.157215	−	0.987564i	$-0.550251\pi$
$619$	237343.	−	137030.i	0.619434	−	0.357630i	0.776649	+	0.629934i	$0.216918\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	95538.1	−	598124.i	0.246150	−	1.54104i
$624$		0			0
$625$	−132281.	+	229117.i	−0.338639	+	0.586540i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	29208.6i		−	0.0738260i
$630$		0			0
$631$	−217088.			−0.545227			−0.272614	−	0.962124i	$-0.587888\pi$
$631$	−217088.			−0.545227			−0.272614	+	0.962124i	$0.587888\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	74689.9	+	43122.2i	0.185231	+	0.106943i
$636$		0			0
$637$	150866.	+	168440.i	0.371802	+	0.415113i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−261433.	−	452815.i	−0.636274	−	1.10206i	−0.986244	−	0.165298i	$-0.947141\pi$
$641$	−261433.	−	452815.i	−0.636274	−	1.10206i	0.349969	−	0.936761i	$-0.386192\pi$
$642$		0			0
$643$			447328.i			1.08194i	0.841041	+	0.540971i	$0.181943\pi$
$643$			447328.i			1.08194i	−0.841041	+	0.540971i	$0.818057\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	168608.	−	97346.0i	0.402782	−	0.232546i	−0.284901	−	0.958557i	$-0.591961\pi$
$647$	168608.	−	97346.0i	0.402782	−	0.232546i	0.687684	+	0.726010i	$0.258628\pi$
$648$		0			0
$649$	−226602.	−	130829.i	−0.537990	−	0.310609i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	92320.8	−	159904.i	0.216508	−	0.375002i	−0.737230	−	0.675642i	$-0.763867\pi$
$653$	92320.8	−	159904.i	0.216508	−	0.375002i	0.953738	+	0.300639i	$0.0972001\pi$
$654$		0			0
$655$	131339.	+	227486.i	0.306134	+	0.530240i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−269400.			−0.620336			−0.310168	−	0.950682i	$-0.600385\pi$
$659$	−269400.			−0.620336			−0.310168	+	0.950682i	$0.600385\pi$
$660$		0			0
$661$	−273711.	+	158027.i	−0.626453	+	0.361683i	−0.779377	−	0.626555i	$-0.784464\pi$
$661$	−273711.	+	158027.i	−0.626453	+	0.361683i	0.152924	+	0.988238i	$0.451131\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−75495.0	+	92978.9i	−0.170716	+	0.210252i
$666$		0			0
$667$	69414.9	−	120230.i	0.156028	−	0.270248i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	257884.i		−	0.572769i
$672$		0			0
$673$	85940.2			0.189743			0.0948716	−	0.995490i	$-0.469756\pi$
$673$	85940.2			0.189743			0.0948716	+	0.995490i	$0.469756\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	43495.9	+	25112.3i	0.0949010	+	0.0547911i	0.546699	−	0.837329i	$-0.315884\pi$
$677$	43495.9	+	25112.3i	0.0949010	+	0.0547911i	−0.451798	+	0.892120i	$0.649217\pi$
$678$		0			0
$679$	−243478.	+	93096.1i	−0.528105	+	0.201926i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−373719.	−	647301.i	−0.801132	−	1.38760i	−0.918871	−	0.394557i	$-0.870898\pi$
$683$	−373719.	−	647301.i	−0.801132	−	1.38760i	0.117739	−	0.993045i	$-0.462435\pi$
$684$		0			0
$685$		−	115790.i		−	0.246769i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−409735.	+	236561.i	−0.863108	+	0.498315i
$690$		0			0
$691$	151644.	+	87551.7i	0.317592	+	0.183362i	0.650319	−	0.759662i	$-0.274635\pi$
$691$	151644.	+	87551.7i	0.317592	+	0.183362i	−0.332727	+	0.943023i	$0.607969\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−103415.	+	179120.i	−0.214099	+	0.370830i
$696$		0			0
$697$	165417.	+	286511.i	0.340498	+	0.589761i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	91696.3			0.186602			0.0933009	−	0.995638i	$-0.470258\pi$
$701$	91696.3			0.186602			0.0933009	+	0.995638i	$0.470258\pi$
$702$		0			0
$703$	68509.8	−	39554.2i	0.138625	−	0.0800353i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−174184.	−	141430.i	−0.348473	−	0.282946i
$708$		0			0
$709$	45597.2	−	78976.7i	0.0907081	−	0.157111i	−0.817101	−	0.576494i	$-0.804420\pi$
$709$	45597.2	−	78976.7i	0.0907081	−	0.157111i	0.907809	+	0.419383i	$0.137754\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			103565.i			0.203720i
$714$		0			0
$715$	61210.0			0.119732
$716$		0			0
$717$		0			0
$718$		0			0
$719$	143600.	+	82907.7i	0.277778	+	0.160375i	0.632417	−	0.774628i	$-0.282063\pi$
$719$	143600.	+	82907.7i	0.277778	+	0.160375i	−0.354639	+	0.935003i	$0.615396\pi$
$720$		0			0
$721$	−56510.1	+	353786.i	−0.108707	+	0.680566i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	152201.	+	263620.i	0.289563	+	0.501537i
$726$		0			0
$727$		−	90288.3i		−	0.170829i	−0.996345	−	0.0854147i	$-0.972778\pi$
$727$		−	90288.3i		−	0.170829i	0.996345	−	0.0854147i	$-0.0272215\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−51725.6	+	29863.8i	−0.0967989	+	0.0558869i
$732$		0			0
$733$	−272185.	−	157146.i	−0.506590	−	0.292480i	0.224841	−	0.974395i	$-0.427814\pi$
$733$	−272185.	−	157146.i	−0.506590	−	0.292480i	−0.731431	+	0.681916i	$0.761147\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	23292.4	−	40343.7i	0.0428825	−	0.0742746i
$738$		0			0
$739$	−56214.0	−	97365.4i	−0.102933	−	0.178285i	0.809959	−	0.586487i	$-0.199489\pi$
$739$	−56214.0	−	97365.4i	−0.102933	−	0.178285i	−0.912892	+	0.408201i	$0.866156\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	805662.			1.45940			0.729702	−	0.683765i	$-0.239659\pi$
$743$	805662.			1.45940			0.729702	+	0.683765i	$0.239659\pi$
$744$		0			0
$745$	1237.37	−	714.397i	0.00222940	−	0.00128714i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	938600.	+	149922.i	1.67308	+	0.267240i
$750$		0			0
$751$	184929.	−	320306.i	0.327887	−	0.567917i	−0.654205	−	0.756317i	$-0.726997\pi$
$751$	184929.	−	320306.i	0.327887	−	0.567917i	0.982092	+	0.188400i	$0.0603300\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			310441.i			0.544610i
$756$		0			0
$757$	736616.			1.28543			0.642716	−	0.766104i	$-0.277807\pi$
$757$	736616.			1.28543			0.642716	+	0.766104i	$0.277807\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	111743.	+	64515.1i	0.192954	+	0.111402i	0.593365	−	0.804934i	$-0.297799\pi$
$761$	111743.	+	64515.1i	0.192954	+	0.111402i	−0.400411	+	0.916336i	$0.631133\pi$
$762$		0			0
$763$	399383.	−	491876.i	0.686026	−	0.844902i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	158427.	+	274404.i	0.269301	+	0.466444i
$768$		0			0
$769$			485806.i			0.821506i	0.911747	+	0.410753i	$0.134734\pi$
$769$			485806.i			0.821506i	−0.911747	+	0.410753i	$0.865266\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	239185.	−	138094.i	0.400290	−	0.231108i	−0.286319	−	0.958134i	$-0.592432\pi$
$773$	239185.	−	138094.i	0.400290	−	0.231108i	0.686609	+	0.727027i	$0.259098\pi$
$774$		0			0
$775$	−196657.	−	113540.i	−0.327421	−	0.189037i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−448015.	+	775984.i	−0.738274	+	1.27873i
$780$		0			0
$781$	−208558.	−	361233.i	−0.341921	−	0.592224i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	126897.			0.205926
$786$		0			0
$787$	83633.8	−	48286.0i	0.135031	−	0.0779600i	−0.430963	−	0.902370i	$-0.641826\pi$
$787$	83633.8	−	48286.0i	0.135031	−	0.0779600i	0.565994	+	0.824410i	$0.308493\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	90179.8	+	235851.i	0.144131	+	0.376951i
$792$		0			0
$793$	−156142.	+	270447.i	−0.248299	+	0.430066i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			540869.i			0.851481i	0.904845	+	0.425741i	$0.139986\pi$
$797$			540869.i			0.851481i	−0.904845	+	0.425741i	$0.860014\pi$
$798$		0			0
$799$	254093.			0.398015
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−96536.6	−	55735.4i	−0.149713	−	0.0864371i
$804$		0			0
$805$	80487.6	+	65352.6i	0.124205	+	0.100849i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	477611.	+	827246.i	0.729755	+	1.26397i	0.956987	+	0.290132i	$0.0936992\pi$
$809$	477611.	+	827246.i	0.729755	+	1.26397i	−0.227232	+	0.973841i	$0.572967\pi$
$810$		0			0
$811$			352574.i			0.536055i	0.963411	+	0.268027i	$0.0863718\pi$
$811$			352574.i			0.536055i	−0.963411	+	0.268027i	$0.913628\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	27356.4	−	15794.2i	0.0411854	−	0.0237784i
$816$		0			0
$817$	−140093.	−	80882.8i	−0.209881	−	0.121175i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	393020.	−	680731.i	0.583080	−	1.00992i	−0.412031	−	0.911170i	$-0.635181\pi$
$821$	393020.	−	680731.i	0.583080	−	1.00992i	0.995112	−	0.0987551i	$-0.0314860\pi$
$822$		0			0
$823$	−366023.	−	633971.i	−0.540392	−	0.935987i	−0.998881	−	0.0472866i	$-0.984943\pi$
$823$	−366023.	−	633971.i	−0.540392	−	0.935987i	0.458489	−	0.888700i	$-0.348391\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	279220.			0.408258			0.204129	−	0.978944i	$-0.434564\pi$
$827$	279220.			0.408258			0.204129	+	0.978944i	$0.434564\pi$
$828$		0			0
$829$	−176164.	+	101708.i	−0.256335	+	0.147995i	−0.622662	−	0.782491i	$-0.713949\pi$
$829$	−176164.	+	101708.i	−0.256335	+	0.147995i	0.366327	+	0.930486i	$0.380615\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	80773.1	−	246392.i	0.116406	−	0.355089i
$834$		0			0
$835$	44251.0	−	76645.0i	0.0634674	−	0.109929i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			1.36972e6i			1.94584i	0.231134	+	0.972922i	$0.425756\pi$
$839$			1.36972e6i			1.94584i	−0.231134	+	0.972922i	$0.574244\pi$
$840$		0			0
$841$	−406637.			−0.574930
$842$		0			0
$843$		0			0
$844$		0			0
$845$	142508.	+	82276.9i	0.199584	+	0.115230i
$846$		0			0
$847$	−415753.	−	66408.1i	−0.579521	−	0.0925666i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−34240.2	−	59305.8i	−0.0472800	−	0.0818914i
$852$		0			0
$853$			110844.i			0.152340i	0.997095	+	0.0761702i	$0.0242692\pi$
$853$			110844.i			0.152340i	−0.997095	+	0.0761702i	$0.975731\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−917323.	+	529617.i	−1.24899	+	0.721107i	−0.970909	−	0.239449i	$-0.923033\pi$
$857$	−917323.	+	529617.i	−1.24899	+	0.721107i	−0.278086	+	0.960556i	$0.589700\pi$
$858$		0			0
$859$	1.00197e6	+	578485.i	1.35790	+	0.783982i	0.989340	−	0.145624i	$-0.0465188\pi$
$859$	1.00197e6	+	578485.i	1.35790	+	0.783982i	0.368556	+	0.929605i	$0.379852\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−39295.5	+	68061.8i	−0.0527620	+	0.0913864i	−0.891200	−	0.453610i	$-0.850136\pi$
$863$	−39295.5	+	68061.8i	−0.0527620	+	0.0913864i	0.838438	+	0.544997i	$0.183469\pi$
$864$		0			0
$865$	167726.	+	290511.i	0.224166	+	0.388267i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	869035.			1.15079
$870$		0			0
$871$	−48854.2	+	28206.0i	−0.0643970	+	0.0371796i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−451382.	+	172590.i	−0.589561	+	0.225424i
$876$		0			0
$877$	−32541.5	+	56363.6i	−0.0423096	+	0.0732824i	−0.886405	−	0.462911i	$-0.846805\pi$
$877$	−32541.5	+	56363.6i	−0.0423096	+	0.0732824i	0.844095	+	0.536193i	$0.180138\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	510450.i		−	0.657661i	−0.944389	−	0.328830i	$-0.893346\pi$
$881$		−	510450.i		−	0.657661i	0.944389	−	0.328830i	$-0.106654\pi$
$882$		0			0
$883$	684144.			0.877458			0.438729	−	0.898620i	$-0.355429\pi$
$883$	684144.			0.877458			0.438729	+	0.898620i	$0.355429\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−251921.	−	145447.i	−0.320198	−	0.184866i	0.331283	−	0.943531i	$-0.392518\pi$
$887$	−251921.	−	145447.i	−0.320198	−	0.184866i	−0.651481	+	0.758665i	$0.725852\pi$
$888$		0			0
$889$	−180607.	−	472348.i	−0.228523	−	0.597665i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	344092.	+	595985.i	0.431491	+	0.747365i
$894$		0			0
$895$			314018.i			0.392020i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−194229.	+	112138.i	−0.240322	+	0.138750i
$900$		0			0
$901$	469837.	+	271261.i	0.578759	+	0.334147i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	166516.	−	288414.i	0.203310	−	0.352144i
$906$		0			0
$907$	19708.0	+	34135.2i	0.0239567	+	0.0414942i	0.877755	−	0.479109i	$-0.159040\pi$
$907$	19708.0	+	34135.2i	0.0239567	+	0.0414942i	−0.853798	+	0.520604i	$0.825707\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.07089e6			1.29035			0.645173	−	0.764036i	$-0.276785\pi$
$911$	1.07089e6			1.29035			0.645173	+	0.764036i	$0.276785\pi$
$912$		0			0
$913$	−89364.1	+	51594.4i	−0.107207	+	0.0618958i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	242940.	−	1.52095e6i	0.288909	−	1.80874i
$918$		0			0
$919$	−607876.	+	1.05287e6i	−0.719754	+	1.24665i	0.241343	+	0.970440i	$0.422412\pi$
$919$	−607876.	+	1.05287e6i	−0.719754	+	1.24665i	−0.961097	+	0.276211i	$0.910921\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			505107.i			0.592898i
$924$		0			0
$925$	150152.			0.175489
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.29549e6	−	747949.i	−1.50107	−	0.866644i	−0.999999	−	0.00123791i	$-0.999606\pi$
$929$	−1.29549e6	−	747949.i	−1.50107	−	0.866644i	−0.501072	−	0.865406i	$-0.667061\pi$
$930$		0			0
$931$	687305.	−	144207.i	0.792958	−	0.166375i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−35094.3	−	60785.1i	−0.0401433	−	0.0695303i
$936$		0			0
$937$		−	254082.i		−	0.289398i	−0.989476	−	0.144699i	$-0.953779\pi$
$937$		−	254082.i		−	0.289398i	0.989476	−	0.144699i	$-0.0462213\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	642429.	−	370907.i	0.725514	−	0.418876i	−0.0912645	−	0.995827i	$-0.529091\pi$
$941$	642429.	−	370907.i	0.725514	−	0.418876i	0.816779	+	0.576951i	$0.195758\pi$
$942$		0			0
$943$	671735.	+	387826.i	0.755395	+	0.436128i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	477051.	−	826277.i	0.531943	−	0.921352i	−0.467362	−	0.884066i	$-0.654795\pi$
$947$	477051.	−	826277.i	0.531943	−	0.921352i	0.999305	−	0.0372859i	$-0.0118712\pi$
$948$		0			0
$949$	67492.8	+	116901.i	0.0749420	+	0.129803i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	792911.			0.873049			0.436525	−	0.899692i	$-0.356209\pi$
$953$	792911.			0.873049			0.436525	+	0.899692i	$0.356209\pi$
$954$		0			0
$955$	−329253.	+	190094.i	−0.361013	+	0.208431i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−427964.	+	527076.i	−0.465340	+	0.573108i
$960$		0			0
$961$	−378107.	+	654901.i	−0.409419	+	0.709135i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	101036.i		−	0.108498i
$966$		0			0
$967$	−23877.3			−0.0255348			−0.0127674	−	0.999918i	$-0.504064\pi$
$967$	−23877.3			−0.0255348			−0.0127674	+	0.999918i	$0.504064\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−969512.	−	559748.i	−1.02829	−	0.593682i	−0.111794	−	0.993731i	$-0.535660\pi$
$971$	−969512.	−	559748.i	−1.02829	−	0.593682i	−0.916494	+	0.400049i	$0.868993\pi$
$972$		0			0
$973$	1.13278e6	−	433128.i	1.19652	−	0.457500i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−522163.	−	904413.i	−0.547037	−	0.947497i	−0.998476	−	0.0551943i	$-0.982422\pi$
$977$	−522163.	−	904413.i	−0.547037	−	0.947497i	0.451438	−	0.892302i	$-0.350911\pi$
$978$		0			0
$979$			961382.i			1.00307i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	187897.	−	108482.i	0.194452	−	0.112267i	−0.399613	−	0.916684i	$-0.630855\pi$
$983$	187897.	−	108482.i	0.194452	−	0.112267i	0.594065	+	0.804417i	$0.297522\pi$
$984$		0			0
$985$	−313885.	−	181222.i	−0.323518	−	0.186783i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−70016.6	+	121272.i	−0.0715828	+	0.123985i
$990$		0			0
$991$	−496994.	−	860819.i	−0.506062	−	0.876525i	−0.999975	−	0.00701423i	$-0.997767\pi$
$991$	−496994.	−	860819.i	−0.506062	−	0.876525i	0.493913	−	0.869511i	$-0.335566\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	189783.			0.191695
$996$		0			0
$997$	−674463.	+	389402.i	−0.678528	+	0.391748i	−0.799300	−	0.600932i	$-0.794796\pi$
$997$	−674463.	+	389402.i	−0.678528	+	0.391748i	0.120772	+	0.992680i	$0.461463\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.5.z.b.145.2		4
3.2	odd	2		84.5.m.a.61.1	✓	4
7.3	odd	6	inner	252.5.z.b.73.2		4
12.11	even	2		336.5.bh.c.145.1		4
21.2	odd	6		588.5.d.a.97.4		4
21.5	even	6		588.5.d.a.97.1		4
21.11	odd	6		588.5.m.a.325.2		4
21.17	even	6		84.5.m.a.73.1	yes	4
21.20	even	2		588.5.m.a.313.2		4
84.59	odd	6		336.5.bh.c.241.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.5.m.a.61.1	✓	4	3.2	odd	2
84.5.m.a.73.1	yes	4	21.17	even	6
252.5.z.b.73.2		4	7.3	odd	6	inner
252.5.z.b.145.2		4	1.1	even	1	trivial
336.5.bh.c.145.1		4	12.11	even	2
336.5.bh.c.241.1		4	84.59	odd	6
588.5.d.a.97.1		4	21.5	even	6
588.5.d.a.97.4		4	21.2	odd	6
588.5.m.a.313.2		4	21.20	even	2
588.5.m.a.325.2		4	21.11	odd	6

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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	252.z (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(7.23713 + 4.17836i) q^{5} +(-17.5000 - 45.7684i) q^{7} +(38.8866 + 67.3536i) q^{11} -94.1795i q^{13} +(-93.5257 + 53.9971i) q^{17} +(-253.304 - 146.245i) q^{19} +(-126.598 + 219.274i) q^{23} +(-277.583 - 480.787i) q^{25} -548.310 q^{29} +(354.232 - 204.516i) q^{31} +(64.5872 - 404.353i) q^{35} +(-135.232 + 234.229i) q^{37} -3063.45i q^{41} +553.062 q^{43} +(-2037.62 - 1176.42i) q^{47} +(-1788.50 + 1601.90i) q^{49} +(-2511.81 - 4350.58i) q^{53} +649.929i q^{55} +(-2913.62 + 1682.18i) q^{59} +(-2871.61 - 1657.92i) q^{61} +(393.516 - 681.589i) q^{65} +(-299.491 - 518.734i) q^{67} -5363.24 q^{71} +(-1241.26 + 716.640i) q^{73} +(2402.15 - 2958.47i) q^{77} +(5586.98 - 9676.93i) q^{79} +1326.79i q^{83} -902.477 q^{85} +(10705.2 + 6180.67i) q^{89} +(-4310.45 + 1648.14i) q^{91} +(-1222.13 - 2116.79i) q^{95} -5319.78i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 39 q^{5} - 70 q^{7} - 3 q^{11} - 510 q^{17} + 459 q^{19} - 144 q^{23} - 227 q^{25} + 570 q^{29} + 2640 q^{31} + 2478 q^{35} + 433 q^{37} - 98 q^{43} + 1770 q^{47} - 7154 q^{49} + 213 q^{53} + 4857 q^{59}+ \cdots - 19656 q^{95}+O(q^{100}) \) 4 * q - 39 * q^5 - 70 * q^7 - 3 * q^11 - 510 * q^17 + 459 * q^19 - 144 * q^23 - 227 * q^25 + 570 * q^29 + 2640 * q^31 + 2478 * q^35 + 433 * q^37 - 98 * q^43 + 1770 * q^47 - 7154 * q^49 + 213 * q^53 + 4857 * q^59 - 12936 * q^61 - 102 * q^65 + 7205 * q^67 - 19188 * q^71 + 14355 * q^73 + 12621 * q^77 + 13424 * q^79 + 9708 * q^85 + 36162 * q^89 - 5985 * q^91 - 19656 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more