LMFDB - Embedded newform 1280.3.e.j.639.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,3,Mod(639,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1280.639");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1280.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$34.8774738381$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	no (minimal twist has level 80)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			639.4
Root			$-0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	1280.639
Dual form			1280.3.e.j.639.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{3} +(4.89898 + 1.00000i) q^{5} +8.48528 q^{7} +1.00000 q^{9} +13.8564 q^{11} -9.79796 q^{13} +(2.82843 - 13.8564i) q^{15} -19.5959i q^{17} +13.8564 q^{19} -24.0000i q^{21} -25.4558 q^{23} +(23.0000 + 9.79796i) q^{25} -28.2843i q^{27} -22.0000i q^{29} +55.4256i q^{31} -39.1918i q^{33} +(41.5692 + 8.48528i) q^{35} +48.9898 q^{37} +27.7128i q^{39} -22.0000 q^{41} +59.3970i q^{43} +(4.89898 + 1.00000i) q^{45} -8.48528 q^{47} +23.0000 q^{49} -55.4256 q^{51} +29.3939 q^{53} +(67.8823 + 13.8564i) q^{55} -39.1918i q^{57} +13.8564 q^{59} +46.0000i q^{61} +8.48528 q^{63} +(-48.0000 - 9.79796i) q^{65} -59.3970i q^{67} +72.0000i q^{69} +27.7128i q^{71} -78.3837i q^{73} +(27.7128 - 65.0538i) q^{75} +117.576 q^{77} -71.0000 q^{81} +76.3675i q^{83} +(19.5959 - 96.0000i) q^{85} -62.2254 q^{87} -146.000 q^{89} -83.1384 q^{91} +156.767 q^{93} +(67.8823 + 13.8564i) q^{95} -58.7878i q^{97} +13.8564 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{9} + 184 q^{25} - 176 q^{41} + 184 q^{49} - 384 q^{65} - 568 q^{81} - 1168 q^{89}+O(q^{100}) $ 8 * q + 8 * q^9 + 184 * q^25 - 176 * q^41 + 184 * q^49 - 384 * q^65 - 568 * q^81 - 1168 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$-1$	$-1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		−	2.82843i		−	0.942809i	−0.881917	−	0.471405i	$-0.843747\pi$
$3$		−	2.82843i		−	0.942809i	0.881917	−	0.471405i	$-0.156253\pi$
$4$		0			0
$5$	4.89898	+	1.00000i	0.979796	+	0.200000i
$5$	4.89898	+	1.00000i	0.979796	+	0.200000i
$6$		0			0
$7$	8.48528			1.21218			0.606092	−	0.795395i	$-0.292737\pi$
$7$	8.48528			1.21218			0.606092	+	0.795395i	$0.292737\pi$
$8$		0			0
$9$	1.00000			0.111111
$10$		0			0
$11$	13.8564			1.25967			0.629837	−	0.776728i	$-0.283122\pi$
$11$	13.8564			1.25967			0.629837	+	0.776728i	$0.283122\pi$
$12$		0			0
$13$	−9.79796			−0.753689			−0.376845	−	0.926277i	$-0.622991\pi$
$13$	−9.79796			−0.753689			−0.376845	+	0.926277i	$0.622991\pi$
$14$		0			0
$15$	2.82843	−	13.8564i	0.188562	−	0.923760i
$16$		0			0
$17$		−	19.5959i		−	1.15270i	−0.817203	−	0.576351i	$-0.804476\pi$
$17$		−	19.5959i		−	1.15270i	0.817203	−	0.576351i	$-0.195524\pi$
$18$		0			0
$19$	13.8564			0.729285			0.364642	−	0.931148i	$-0.381191\pi$
$19$	13.8564			0.729285			0.364642	+	0.931148i	$0.381191\pi$
$20$		0			0
$21$		−	24.0000i		−	1.14286i
$22$		0			0
$23$	−25.4558			−1.10678			−0.553388	−	0.832924i	$-0.686665\pi$
$23$	−25.4558			−1.10678			−0.553388	+	0.832924i	$0.686665\pi$
$24$		0			0
$25$	23.0000	+	9.79796i	0.920000	+	0.391918i
$26$		0			0
$27$		−	28.2843i		−	1.04757i
$28$		0			0
$29$		−	22.0000i		−	0.758621i	−0.925270	−	0.379310i	$-0.876161\pi$
$29$		−	22.0000i		−	0.758621i	0.925270	−	0.379310i	$-0.123839\pi$
$30$		0			0
$31$			55.4256i			1.78792i	0.448143	+	0.893962i	$0.352085\pi$
$31$			55.4256i			1.78792i	−0.448143	+	0.893962i	$0.647915\pi$
$32$		0			0
$33$		−	39.1918i		−	1.18763i
$34$		0			0
$35$	41.5692	+	8.48528i	1.18769	+	0.242437i
$36$		0			0
$37$	48.9898			1.32405			0.662024	−	0.749482i	$-0.269698\pi$
$37$	48.9898			1.32405			0.662024	+	0.749482i	$0.269698\pi$
$38$		0			0
$39$			27.7128i			0.710585i
$40$		0			0
$41$	−22.0000			−0.536585			−0.268293	−	0.963337i	$-0.586459\pi$
$41$	−22.0000			−0.536585			−0.268293	+	0.963337i	$0.586459\pi$
$42$		0			0
$43$			59.3970i			1.38132i	0.723177	+	0.690662i	$0.242681\pi$
$43$			59.3970i			1.38132i	−0.723177	+	0.690662i	$0.757319\pi$
$44$		0			0
$45$	4.89898	+	1.00000i	0.108866	+	0.0222222i
$46$		0			0
$47$	−8.48528			−0.180538			−0.0902690	−	0.995917i	$-0.528773\pi$
$47$	−8.48528			−0.180538			−0.0902690	+	0.995917i	$0.528773\pi$
$48$		0			0
$49$	23.0000			0.469388
$50$		0			0
$51$	−55.4256			−1.08678
$52$		0			0
$53$	29.3939			0.554601			0.277301	−	0.960783i	$-0.410560\pi$
$53$	29.3939			0.554601			0.277301	+	0.960783i	$0.410560\pi$
$54$		0			0
$55$	67.8823	+	13.8564i	1.23422	+	0.251935i
$56$		0			0
$57$		−	39.1918i		−	0.687576i
$58$		0			0
$59$	13.8564			0.234854			0.117427	−	0.993081i	$-0.462535\pi$
$59$	13.8564			0.234854			0.117427	+	0.993081i	$0.462535\pi$
$60$		0			0
$61$			46.0000i			0.754098i	0.926193	+	0.377049i	$0.123061\pi$
$61$			46.0000i			0.754098i	−0.926193	+	0.377049i	$0.876939\pi$
$62$		0			0
$63$	8.48528			0.134687
$64$		0			0
$65$	−48.0000	−	9.79796i	−0.738462	−	0.150738i
$66$		0			0
$67$		−	59.3970i		−	0.886522i	−0.896393	−	0.443261i	$-0.853822\pi$
$67$		−	59.3970i		−	0.886522i	0.896393	−	0.443261i	$-0.146178\pi$
$68$		0			0
$69$			72.0000i			1.04348i
$70$		0			0
$71$			27.7128i			0.390321i	0.980771	+	0.195161i	$0.0625228\pi$
$71$			27.7128i			0.390321i	−0.980771	+	0.195161i	$0.937477\pi$
$72$		0			0
$73$		−	78.3837i		−	1.07375i	−0.843662	−	0.536874i	$-0.819605\pi$
$73$		−	78.3837i		−	1.07375i	0.843662	−	0.536874i	$-0.180395\pi$
$74$		0			0
$75$	27.7128	−	65.0538i	0.369504	−	0.867384i
$76$		0			0
$77$	117.576			1.52695
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	−71.0000			−0.876543
$82$		0			0
$83$			76.3675i			0.920091i	0.887895	+	0.460045i	$0.152167\pi$
$83$			76.3675i			0.920091i	−0.887895	+	0.460045i	$0.847833\pi$
$84$		0			0
$85$	19.5959	−	96.0000i	0.230540	−	1.12941i
$86$		0			0
$87$	−62.2254			−0.715234
$88$		0			0
$89$	−146.000			−1.64045			−0.820225	−	0.572041i	$-0.806152\pi$
$89$	−146.000			−1.64045			−0.820225	+	0.572041i	$0.806152\pi$
$90$		0			0
$91$	−83.1384			−0.913609
$92$		0			0
$93$	156.767			1.68567
$94$		0			0
$95$	67.8823	+	13.8564i	0.714550	+	0.145857i
$96$		0			0
$97$		−	58.7878i		−	0.606059i	−0.952981	−	0.303030i	$-0.902002\pi$
$97$		−	58.7878i		−	0.606059i	0.952981	−	0.303030i	$-0.0979981\pi$
$98$		0			0
$99$	13.8564			0.139964
$100$		0			0
$101$			70.0000i			0.693069i	0.938037	+	0.346535i	$0.112642\pi$
$101$			70.0000i			0.693069i	−0.938037	+	0.346535i	$0.887358\pi$
$102$		0			0
$103$	−25.4558			−0.247144			−0.123572	−	0.992336i	$-0.539435\pi$
$103$	−25.4558			−0.247144			−0.123572	+	0.992336i	$0.539435\pi$
$104$		0			0
$105$	24.0000	−	117.576i	0.228571	−	1.11977i
$106$		0			0
$107$		−	42.4264i		−	0.396508i	−0.980151	−	0.198254i	$-0.936473\pi$
$107$		−	42.4264i		−	0.396508i	0.980151	−	0.198254i	$-0.0635272\pi$
$108$		0			0
$109$		−	146.000i		−	1.33945i	−0.742609	−	0.669725i	$-0.766412\pi$
$109$		−	146.000i		−	1.33945i	0.742609	−	0.669725i	$-0.233588\pi$
$110$		0			0
$111$		−	138.564i		−	1.24832i
$112$		0			0
$113$			39.1918i			0.346830i	0.984849	+	0.173415i	$0.0554803\pi$
$113$			39.1918i			0.346830i	−0.984849	+	0.173415i	$0.944520\pi$
$114$		0			0
$115$	−124.708	−	25.4558i	−1.08441	−	0.221355i
$116$		0			0
$117$	−9.79796			−0.0837432
$118$		0			0
$119$		−	166.277i		−	1.39728i
$120$		0			0
$121$	71.0000			0.586777
$122$		0			0
$123$			62.2254i			0.505898i
$124$		0			0
$125$	102.879	+	71.0000i	0.823029	+	0.568000i
$126$		0			0
$127$	−110.309			−0.868572			−0.434286	−	0.900775i	$-0.642999\pi$
$127$	−110.309			−0.868572			−0.434286	+	0.900775i	$0.642999\pi$
$128$		0			0
$129$	168.000			1.30233
$130$		0			0
$131$	180.133			1.37506			0.687532	−	0.726154i	$-0.258694\pi$
$131$	180.133			1.37506			0.687532	+	0.726154i	$0.258694\pi$
$132$		0			0
$133$	117.576			0.884026
$134$		0			0
$135$	28.2843	−	138.564i	0.209513	−	1.02640i
$136$		0			0
$137$		−	215.555i		−	1.57339i	−0.617339	−	0.786697i	$-0.711789\pi$
$137$		−	215.555i		−	1.57339i	0.617339	−	0.786697i	$-0.288211\pi$
$138$		0			0
$139$	13.8564			0.0996864			0.0498432	−	0.998757i	$-0.484128\pi$
$139$	13.8564			0.0996864			0.0498432	+	0.998757i	$0.484128\pi$
$140$		0			0
$141$			24.0000i			0.170213i
$142$		0			0
$143$	−135.765			−0.949402
$144$		0			0
$145$	22.0000	−	107.778i	0.151724	−	0.743293i
$146$		0			0
$147$		−	65.0538i		−	0.442543i
$148$		0			0
$149$			2.00000i			0.0134228i	0.999977	+	0.00671141i	$0.00213632\pi$
$149$			2.00000i			0.0134228i	−0.999977	+	0.00671141i	$0.997864\pi$
$150$		0			0
$151$			27.7128i			0.183529i	0.995781	+	0.0917643i	$0.0292506\pi$
$151$			27.7128i			0.183529i	−0.995781	+	0.0917643i	$0.970749\pi$
$152$		0			0
$153$		−	19.5959i		−	0.128078i
$154$		0			0
$155$	−55.4256	+	271.529i	−0.357585	+	1.75180i
$156$		0			0
$157$	−68.5857			−0.436852			−0.218426	−	0.975854i	$-0.570092\pi$
$157$	−68.5857			−0.436852			−0.218426	+	0.975854i	$0.570092\pi$
$158$		0			0
$159$		−	83.1384i		−	0.522883i
$160$		0			0
$161$	−216.000			−1.34161
$162$		0			0
$163$			110.309i			0.676740i	0.941013	+	0.338370i	$0.109876\pi$
$163$			110.309i			0.676740i	−0.941013	+	0.338370i	$0.890124\pi$
$164$		0			0
$165$	39.1918	−	192.000i	0.237526	−	1.16364i
$166$		0			0
$167$	−93.3381			−0.558911			−0.279455	−	0.960159i	$-0.590154\pi$
$167$	−93.3381			−0.558911			−0.279455	+	0.960159i	$0.590154\pi$
$168$		0			0
$169$	−73.0000			−0.431953
$170$		0			0
$171$	13.8564			0.0810316
$172$		0			0
$173$	48.9898			0.283178			0.141589	−	0.989926i	$-0.454779\pi$
$173$	48.9898			0.283178			0.141589	+	0.989926i	$0.454779\pi$
$174$		0			0
$175$	195.161	+	83.1384i	1.11521	+	0.475077i
$176$		0			0
$177$		−	39.1918i		−	0.221423i
$178$		0			0
$179$	−263.272			−1.47079			−0.735396	−	0.677638i	$-0.763004\pi$
$179$	−263.272			−1.47079			−0.735396	+	0.677638i	$0.763004\pi$
$180$		0			0
$181$		−	26.0000i		−	0.143646i	−0.997417	−	0.0718232i	$-0.977118\pi$
$181$		−	26.0000i		−	0.143646i	0.997417	−	0.0718232i	$-0.0228817\pi$
$182$		0			0
$183$	130.108			0.710971
$184$		0			0
$185$	240.000	+	48.9898i	1.29730	+	0.264810i
$186$		0			0
$187$		−	271.529i		−	1.45203i
$188$		0			0
$189$		−	240.000i		−	1.26984i
$190$		0			0
$191$		−	110.851i		−	0.580373i	−0.956970	−	0.290187i	$-0.906283\pi$
$191$		−	110.851i		−	0.580373i	0.956970	−	0.290187i	$-0.0937174\pi$
$192$		0			0
$193$		−	333.131i		−	1.72607i	−0.505148	−	0.863033i	$-0.668562\pi$
$193$		−	333.131i		−	1.72607i	0.505148	−	0.863033i	$-0.331438\pi$
$194$		0			0
$195$	−27.7128	+	135.765i	−0.142117	+	0.696228i
$196$		0			0
$197$	107.778			0.547094			0.273547	−	0.961859i	$-0.411803\pi$
$197$	107.778			0.547094			0.273547	+	0.961859i	$0.411803\pi$
$198$		0			0
$199$			249.415i			1.25334i	0.779283	+	0.626672i	$0.215583\pi$
$199$			249.415i			1.25334i	−0.779283	+	0.626672i	$0.784417\pi$
$200$		0			0
$201$	−168.000			−0.835821
$202$		0			0
$203$		−	186.676i		−	0.919587i
$204$		0			0
$205$	−107.778	−	22.0000i	−0.525744	−	0.107317i
$206$		0			0
$207$	−25.4558			−0.122975
$208$		0			0
$209$	192.000			0.918660
$210$		0			0
$211$	−96.9948			−0.459691			−0.229846	−	0.973227i	$-0.573822\pi$
$211$	−96.9948			−0.459691			−0.229846	+	0.973227i	$0.573822\pi$
$212$		0			0
$213$	78.3837			0.367998
$214$		0			0
$215$	−59.3970	+	290.985i	−0.276265	+	1.35342i
$216$		0			0
$217$			470.302i			2.16729i
$218$		0			0
$219$	−221.703			−1.01234
$220$		0			0
$221$			192.000i			0.868778i
$222$		0			0
$223$	229.103			1.02737			0.513683	−	0.857980i	$-0.328281\pi$
$223$	229.103			1.02737			0.513683	+	0.857980i	$0.328281\pi$
$224$		0			0
$225$	23.0000	+	9.79796i	0.102222	+	0.0435465i
$226$		0			0
$227$		−	296.985i		−	1.30830i	−0.756363	−	0.654152i	$-0.773026\pi$
$227$		−	296.985i		−	1.30830i	0.756363	−	0.654152i	$-0.226974\pi$
$228$		0			0
$229$			70.0000i			0.305677i	0.988251	+	0.152838i	$0.0488414\pi$
$229$			70.0000i			0.305677i	−0.988251	+	0.152838i	$0.951159\pi$
$230$		0			0
$231$		−	332.554i		−	1.43963i
$232$		0			0
$233$		−	156.767i		−	0.672821i	−0.941715	−	0.336411i	$-0.890787\pi$
$233$		−	156.767i		−	0.672821i	0.941715	−	0.336411i	$-0.109213\pi$
$234$		0			0
$235$	−41.5692	−	8.48528i	−0.176890	−	0.0361076i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			332.554i			1.39144i	0.718314	+	0.695719i	$0.244914\pi$
$239$			332.554i			1.39144i	−0.718314	+	0.695719i	$0.755086\pi$
$240$		0			0
$241$	−122.000			−0.506224			−0.253112	−	0.967437i	$-0.581454\pi$
$241$	−122.000			−0.506224			−0.253112	+	0.967437i	$0.581454\pi$
$242$		0			0
$243$		−	53.7401i		−	0.221153i
$244$		0			0
$245$	112.677	+	23.0000i	0.459904	+	0.0938776i
$246$		0			0
$247$	−135.765			−0.549654
$248$		0			0
$249$	216.000			0.867470
$250$		0			0
$251$	290.985			1.15930			0.579650	−	0.814865i	$-0.303189\pi$
$251$	290.985			1.15930			0.579650	+	0.814865i	$0.303189\pi$
$252$		0			0
$253$	−352.727			−1.39418
$254$		0			0
$255$	−271.529	−	55.4256i	−1.06482	−	0.217355i
$256$		0			0
$257$		−	78.3837i		−	0.304995i	−0.988304	−	0.152497i	$-0.951268\pi$
$257$		−	78.3837i		−	0.304995i	0.988304	−	0.152497i	$-0.0487316\pi$
$258$		0			0
$259$	415.692			1.60499
$260$		0			0
$261$		−	22.0000i		−	0.0842912i
$262$		0			0
$263$	347.897			1.32280			0.661400	−	0.750033i	$-0.269962\pi$
$263$	347.897			1.32280			0.661400	+	0.750033i	$0.269962\pi$
$264$		0			0
$265$	144.000	+	29.3939i	0.543396	+	0.110920i
$266$		0			0
$267$			412.950i			1.54663i
$268$		0			0
$269$			142.000i			0.527881i	0.964539	+	0.263941i	$0.0850223\pi$
$269$			142.000i			0.527881i	−0.964539	+	0.263941i	$0.914978\pi$
$270$		0			0
$271$			332.554i			1.22714i	0.789642	+	0.613568i	$0.210266\pi$
$271$			332.554i			1.22714i	−0.789642	+	0.613568i	$0.789734\pi$
$272$		0			0
$273$			235.151i			0.861359i
$274$		0			0
$275$	318.697	+	135.765i	1.15890	+	0.493689i
$276$		0			0
$277$	88.1816			0.318345			0.159173	−	0.987251i	$-0.449117\pi$
$277$	88.1816			0.318345			0.159173	+	0.987251i	$0.449117\pi$
$278$		0			0
$279$			55.4256i			0.198658i
$280$		0			0
$281$	−406.000			−1.44484			−0.722420	−	0.691455i	$-0.756970\pi$
$281$	−406.000			−1.44484			−0.722420	+	0.691455i	$0.756970\pi$
$282$		0			0
$283$			93.3381i			0.329817i	0.986309	+	0.164908i	$0.0527328\pi$
$283$			93.3381i			0.329817i	−0.986309	+	0.164908i	$0.947267\pi$
$284$		0			0
$285$	39.1918	−	192.000i	0.137515	−	0.673684i
$286$		0			0
$287$	−186.676			−0.650440
$288$		0			0
$289$	−95.0000			−0.328720
$290$		0			0
$291$	−166.277			−0.571398
$292$		0			0
$293$	−303.737			−1.03664			−0.518322	−	0.855186i	$-0.673443\pi$
$293$	−303.737			−1.03664			−0.518322	+	0.855186i	$0.673443\pi$
$294$		0			0
$295$	67.8823	+	13.8564i	0.230109	+	0.0469709i
$296$		0			0
$297$		−	391.918i		−	1.31959i
$298$		0			0
$299$	249.415			0.834165
$300$		0			0
$301$			504.000i			1.67442i
$302$		0			0
$303$	197.990			0.653432
$304$		0			0
$305$	−46.0000	+	225.353i	−0.150820	+	0.738862i
$306$		0			0
$307$			280.014i			0.912099i	0.889955	+	0.456049i	$0.150736\pi$
$307$			280.014i			0.912099i	−0.889955	+	0.456049i	$0.849264\pi$
$308$		0			0
$309$			72.0000i			0.233010i
$310$		0			0
$311$			138.564i			0.445544i	0.974871	+	0.222772i	$0.0715105\pi$
$311$			138.564i			0.445544i	−0.974871	+	0.222772i	$0.928490\pi$
$312$		0			0
$313$			607.473i			1.94081i	0.241484	+	0.970405i	$0.422366\pi$
$313$			607.473i			1.94081i	−0.241484	+	0.970405i	$0.577634\pi$
$314$		0			0
$315$	41.5692	+	8.48528i	0.131966	+	0.0269374i
$316$		0			0
$317$	68.5857			0.216359			0.108179	−	0.994131i	$-0.465498\pi$
$317$	68.5857			0.216359			0.108179	+	0.994131i	$0.465498\pi$
$318$		0			0
$319$		−	304.841i		−	0.955614i
$320$		0			0
$321$	−120.000			−0.373832
$322$		0			0
$323$		−	271.529i		−	0.840647i
$324$		0			0
$325$	−225.353	−	96.0000i	−0.693394	−	0.295385i
$326$		0			0
$327$	−412.950			−1.26285
$328$		0			0
$329$	−72.0000			−0.218845
$330$		0			0
$331$	180.133			0.544209			0.272105	−	0.962268i	$-0.412280\pi$
$331$	180.133			0.544209			0.272105	+	0.962268i	$0.412280\pi$
$332$		0			0
$333$	48.9898			0.147117
$334$		0			0
$335$	59.3970	−	290.985i	0.177304	−	0.868611i
$336$		0			0
$337$			470.302i			1.39555i	0.716315	+	0.697777i	$0.245828\pi$
$337$			470.302i			1.39555i	−0.716315	+	0.697777i	$0.754172\pi$
$338$		0			0
$339$	110.851			0.326995
$340$		0			0
$341$			768.000i			2.25220i
$342$		0			0
$343$	−220.617			−0.643199
$344$		0			0
$345$	−72.0000	+	352.727i	−0.208696	+	1.02240i
$346$		0			0
$347$		−	619.426i		−	1.78509i	−0.450960	−	0.892544i	$-0.648918\pi$
$347$		−	619.426i		−	1.78509i	0.450960	−	0.892544i	$-0.351082\pi$
$348$		0			0
$349$		−	214.000i		−	0.613181i	−0.951842	−	0.306590i	$-0.900812\pi$
$349$		−	214.000i		−	0.613181i	0.951842	−	0.306590i	$-0.0991881\pi$
$350$		0			0
$351$			277.128i			0.789539i
$352$		0			0
$353$			352.727i			0.999225i	0.866249	+	0.499613i	$0.166524\pi$
$353$			352.727i			0.999225i	−0.866249	+	0.499613i	$0.833476\pi$
$354$		0			0
$355$	−27.7128	+	135.765i	−0.0780643	+	0.382435i
$356$		0			0
$357$	−470.302			−1.31737
$358$		0			0
$359$		−	27.7128i		−	0.0771945i	−0.999255	−	0.0385972i	$-0.987711\pi$
$359$		−	27.7128i		−	0.0771945i	0.999255	−	0.0385972i	$-0.0122889\pi$
$360$		0			0
$361$	−169.000			−0.468144
$362$		0			0
$363$		−	200.818i		−	0.553219i
$364$		0			0
$365$	78.3837	−	384.000i	0.214750	−	1.05205i
$366$		0			0
$367$	−246.073			−0.670499			−0.335250	−	0.942129i	$-0.608821\pi$
$367$	−246.073			−0.670499			−0.335250	+	0.942129i	$0.608821\pi$
$368$		0			0
$369$	−22.0000			−0.0596206
$370$		0			0
$371$	249.415			0.672278
$372$		0			0
$373$	−382.120			−1.02445			−0.512226	−	0.858851i	$-0.671179\pi$
$373$	−382.120			−1.02445			−0.512226	+	0.858851i	$0.671179\pi$
$374$		0			0
$375$	200.818	−	290.985i	0.535516	−	0.775959i
$376$		0			0
$377$			215.555i			0.571764i
$378$		0			0
$379$	290.985			0.767769			0.383885	−	0.923381i	$-0.374586\pi$
$379$	290.985			0.767769			0.383885	+	0.923381i	$0.374586\pi$
$380$		0			0
$381$			312.000i			0.818898i
$382$		0			0
$383$	568.514			1.48437			0.742185	−	0.670195i	$-0.233789\pi$
$383$	568.514			1.48437			0.742185	+	0.670195i	$0.233789\pi$
$384$		0			0
$385$	576.000	+	117.576i	1.49610	+	0.305391i
$386$		0			0
$387$			59.3970i			0.153481i
$388$		0			0
$389$		−	670.000i		−	1.72237i	−0.508296	−	0.861183i	$-0.669724\pi$
$389$		−	670.000i		−	1.72237i	0.508296	−	0.861183i	$-0.330276\pi$
$390$		0			0
$391$			498.831i			1.27578i
$392$		0			0
$393$		−	509.494i		−	1.29642i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−362.524			−0.913160			−0.456580	−	0.889682i	$-0.650926\pi$
$397$	−362.524			−0.913160			−0.456580	+	0.889682i	$0.650926\pi$
$398$		0			0
$399$		−	332.554i		−	0.833468i
$400$		0			0
$401$	2.00000			0.00498753			0.00249377	−	0.999997i	$-0.499206\pi$
$401$	2.00000			0.00498753			0.00249377	+	0.999997i	$0.499206\pi$
$402$		0			0
$403$		−	543.058i		−	1.34754i
$404$		0			0
$405$	−347.828	−	71.0000i	−0.858833	−	0.175309i
$406$		0			0
$407$	678.823			1.66787
$408$		0			0
$409$	650.000			1.58924			0.794621	−	0.607106i	$-0.207670\pi$
$409$	650.000			1.58924			0.794621	+	0.607106i	$0.207670\pi$
$410$		0			0
$411$	−609.682			−1.48341
$412$		0			0
$413$	117.576			0.284686
$414$		0			0
$415$	−76.3675	+	374.123i	−0.184018	+	0.901501i
$416$		0			0
$417$		−	39.1918i		−	0.0939852i
$418$		0			0
$419$	−540.400			−1.28974			−0.644869	−	0.764293i	$-0.723088\pi$
$419$	−540.400			−1.28974			−0.644869	+	0.764293i	$0.723088\pi$
$420$		0			0
$421$			482.000i			1.14489i	0.819942	+	0.572447i	$0.194006\pi$
$421$			482.000i			1.14489i	−0.819942	+	0.572447i	$0.805994\pi$
$422$		0			0
$423$	−8.48528			−0.0200598
$424$		0			0
$425$	192.000	−	450.706i	0.451765	−	1.06048i
$426$		0			0
$427$			390.323i			0.914105i
$428$		0			0
$429$			384.000i			0.895105i
$430$		0			0
$431$		−	221.703i		−	0.514391i	−0.966359	−	0.257195i	$-0.917202\pi$
$431$		−	221.703i		−	0.514391i	0.966359	−	0.257195i	$-0.0827984\pi$
$432$		0			0
$433$			568.282i			1.31243i	0.754575	+	0.656214i	$0.227843\pi$
$433$			568.282i			1.31243i	−0.754575	+	0.656214i	$0.772157\pi$
$434$		0			0
$435$	−304.841	−	62.2254i	−0.700784	−	0.143047i
$436$		0			0
$437$	−352.727			−0.807155
$438$		0			0
$439$		−	526.543i		−	1.19942i	−0.800219	−	0.599708i	$-0.795283\pi$
$439$		−	526.543i		−	1.19942i	0.800219	−	0.599708i	$-0.204717\pi$
$440$		0			0
$441$	23.0000			0.0521542
$442$		0			0
$443$			432.749i			0.976861i	0.872603	+	0.488430i	$0.162430\pi$
$443$			432.749i			0.976861i	−0.872603	+	0.488430i	$0.837570\pi$
$444$		0			0
$445$	−715.251	−	146.000i	−1.60731	−	0.328090i
$446$		0			0
$447$	5.65685			0.0126552
$448$		0			0
$449$	−26.0000			−0.0579065			−0.0289532	−	0.999581i	$-0.509217\pi$
$449$	−26.0000			−0.0579065			−0.0289532	+	0.999581i	$0.509217\pi$
$450$		0			0
$451$	−304.841			−0.675922
$452$		0			0
$453$	78.3837			0.173032
$454$		0			0
$455$	−407.294	−	83.1384i	−0.895151	−	0.182722i
$456$		0			0
$457$			137.171i			0.300156i	0.988674	+	0.150078i	$0.0479525\pi$
$457$			137.171i			0.300156i	−0.988674	+	0.150078i	$0.952047\pi$
$458$		0			0
$459$	−554.256			−1.20753
$460$		0			0
$461$		−	502.000i		−	1.08894i	−0.838781	−	0.544469i	$-0.816731\pi$
$461$		−	502.000i		−	1.08894i	0.838781	−	0.544469i	$-0.183269\pi$
$462$		0			0
$463$	−823.072			−1.77769			−0.888847	−	0.458204i	$-0.848493\pi$
$463$	−823.072			−1.77769			−0.888847	+	0.458204i	$0.848493\pi$
$464$		0			0
$465$	768.000	+	156.767i	1.65161	+	0.337134i
$466$		0			0
$467$			178.191i			0.381565i	0.981632	+	0.190783i	$0.0611025\pi$
$467$			178.191i			0.381565i	−0.981632	+	0.190783i	$0.938897\pi$
$468$		0			0
$469$		−	504.000i		−	1.07463i
$470$		0			0
$471$			193.990i			0.411868i
$472$		0			0
$473$			823.029i			1.74002i
$474$		0			0
$475$	318.697	+	135.765i	0.670942	+	0.285820i
$476$		0			0
$477$	29.3939			0.0616224
$478$		0			0
$479$			775.959i			1.61996i	0.586460	+	0.809978i	$0.300521\pi$
$479$			775.959i			1.61996i	−0.586460	+	0.809978i	$0.699479\pi$
$480$		0			0
$481$	−480.000			−0.997921
$482$		0			0
$483$			610.940i			1.26489i
$484$		0			0
$485$	58.7878	−	288.000i	0.121212	−	0.593814i
$486$		0			0
$487$	347.897			0.714367			0.357183	−	0.934034i	$-0.383737\pi$
$487$	347.897			0.714367			0.357183	+	0.934034i	$0.383737\pi$
$488$		0			0
$489$	312.000			0.638037
$490$		0			0
$491$	−928.379			−1.89079			−0.945396	−	0.325923i	$-0.894325\pi$
$491$	−928.379			−1.89079			−0.945396	+	0.325923i	$0.894325\pi$
$492$		0			0
$493$	−431.110			−0.874463
$494$		0			0
$495$	67.8823	+	13.8564i	0.137136	+	0.0279927i
$496$		0			0
$497$			235.151i			0.473141i
$498$		0			0
$499$	512.687			1.02743			0.513714	−	0.857961i	$-0.328269\pi$
$499$	512.687			1.02743			0.513714	+	0.857961i	$0.328269\pi$
$500$		0			0
$501$			264.000i			0.526946i
$502$		0			0
$503$	−704.278			−1.40016			−0.700078	−	0.714066i	$-0.746851\pi$
$503$	−704.278			−1.40016			−0.700078	+	0.714066i	$0.746851\pi$
$504$		0			0
$505$	−70.0000	+	342.929i	−0.138614	+	0.679066i
$506$		0			0
$507$			206.475i			0.407249i
$508$		0			0
$509$			170.000i			0.333988i	0.985958	+	0.166994i	$0.0534061\pi$
$509$			170.000i			0.333988i	−0.985958	+	0.166994i	$0.946594\pi$
$510$		0			0
$511$		−	665.108i		−	1.30158i
$512$		0			0
$513$		−	391.918i		−	0.763973i
$514$		0			0
$515$	−124.708	−	25.4558i	−0.242151	−	0.0494288i
$516$		0			0
$517$	−117.576			−0.227419
$518$		0			0
$519$		−	138.564i		−	0.266983i
$520$		0			0
$521$	−722.000			−1.38580			−0.692898	−	0.721035i	$-0.743667\pi$
$521$	−722.000			−1.38580			−0.692898	+	0.721035i	$0.743667\pi$
$522$		0			0
$523$		−	246.073i		−	0.470503i	−0.971935	−	0.235252i	$-0.924409\pi$
$523$		−	246.073i		−	0.470503i	0.971935	−	0.235252i	$-0.0755914\pi$
$524$		0			0
$525$	235.151	−	552.000i	0.447907	−	1.05143i
$526$		0			0
$527$	1086.12			2.06094
$528$		0			0
$529$	119.000			0.224953
$530$		0			0
$531$	13.8564			0.0260949
$532$		0			0
$533$	215.555			0.404419
$534$		0			0
$535$	42.4264	−	207.846i	0.0793017	−	0.388497i
$536$		0			0
$537$			744.645i			1.38668i
$538$		0			0
$539$	318.697			0.591275
$540$		0			0
$541$			362.000i			0.669131i	0.942372	+	0.334566i	$0.108590\pi$
$541$			362.000i			0.669131i	−0.942372	+	0.334566i	$0.891410\pi$
$542$		0			0
$543$	−73.5391			−0.135431
$544$		0			0
$545$	146.000	−	715.251i	0.267890	−	1.31239i
$546$		0			0
$547$			42.4264i			0.0775620i	0.999248	+	0.0387810i	$0.0123475\pi$
$547$			42.4264i			0.0775620i	−0.999248	+	0.0387810i	$0.987653\pi$
$548$		0			0
$549$			46.0000i			0.0837887i
$550$		0			0
$551$		−	304.841i		−	0.553250i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	138.564	−	678.823i	0.249665	−	1.22310i
$556$		0			0
$557$	−656.463			−1.17857			−0.589285	−	0.807925i	$-0.700590\pi$
$557$	−656.463			−1.17857			−0.589285	+	0.807925i	$0.700590\pi$
$558$		0			0
$559$		−	581.969i		−	1.04109i
$560$		0			0
$561$	−768.000			−1.36898
$562$		0			0
$563$			517.602i			0.919364i	0.888083	+	0.459682i	$0.152037\pi$
$563$			517.602i			0.919364i	−0.888083	+	0.459682i	$0.847963\pi$
$564$		0			0
$565$	−39.1918	+	192.000i	−0.0693661	+	0.339823i
$566$		0			0
$567$	−602.455			−1.06253
$568$		0			0
$569$	842.000			1.47979			0.739895	−	0.672723i	$-0.234875\pi$
$569$	842.000			1.47979			0.739895	+	0.672723i	$0.234875\pi$
$570$		0			0
$571$	401.836			0.703740			0.351870	−	0.936049i	$-0.385546\pi$
$571$	401.836			0.703740			0.351870	+	0.936049i	$0.385546\pi$
$572$		0			0
$573$	−313.535			−0.547181
$574$		0			0
$575$	−585.484	−	249.415i	−1.01823	−	0.433766i
$576$		0			0
$577$			39.1918i			0.0679235i	0.999423	+	0.0339617i	$0.0108124\pi$
$577$			39.1918i			0.0679235i	−0.999423	+	0.0339617i	$0.989188\pi$
$578$		0			0
$579$	−942.236			−1.62735
$580$		0			0
$581$			648.000i			1.11532i
$582$		0			0
$583$	407.294			0.698617
$584$		0			0
$585$	−48.0000	−	9.79796i	−0.0820513	−	0.0167486i
$586$		0			0
$587$			59.3970i			0.101187i	0.998719	+	0.0505937i	$0.0161113\pi$
$587$			59.3970i			0.101187i	−0.998719	+	0.0505937i	$0.983889\pi$
$588$		0			0
$589$			768.000i			1.30390i
$590$		0			0
$591$		−	304.841i		−	0.515805i
$592$		0			0
$593$		−	509.494i		−	0.859180i	−0.903024	−	0.429590i	$-0.858658\pi$
$593$		−	509.494i		−	0.859180i	0.903024	−	0.429590i	$-0.141342\pi$
$594$		0			0
$595$	166.277	−	814.587i	0.279457	−	1.36905i
$596$		0			0
$597$	705.453			1.18166
$598$		0			0
$599$			859.097i			1.43422i	0.696961	+	0.717110i	$0.254535\pi$
$599$			859.097i			1.43422i	−0.696961	+	0.717110i	$0.745465\pi$
$600$		0			0
$601$	−598.000			−0.995008			−0.497504	−	0.867462i	$-0.665750\pi$
$601$	−598.000			−0.995008			−0.497504	+	0.867462i	$0.665750\pi$
$602$		0			0
$603$		−	59.3970i		−	0.0985024i
$604$		0			0
$605$	347.828	+	71.0000i	0.574922	+	0.117355i
$606$		0			0
$607$	127.279			0.209686			0.104843	−	0.994489i	$-0.466566\pi$
$607$	127.279			0.209686			0.104843	+	0.994489i	$0.466566\pi$
$608$		0			0
$609$	−528.000			−0.866995
$610$		0			0
$611$	83.1384			0.136069
$612$		0			0
$613$	−205.757			−0.335656			−0.167828	−	0.985816i	$-0.553675\pi$
$613$	−205.757			−0.335656			−0.167828	+	0.985816i	$0.553675\pi$
$614$		0			0
$615$	−62.2254	+	304.841i	−0.101180	+	0.495676i
$616$		0			0
$617$		−	431.110i		−	0.698720i	−0.936989	−	0.349360i	$-0.886399\pi$
$617$		−	431.110i		−	0.698720i	0.936989	−	0.349360i	$-0.113601\pi$
$618$		0			0
$619$	−41.5692			−0.0671554			−0.0335777	−	0.999436i	$-0.510690\pi$
$619$	−41.5692			−0.0671554			−0.0335777	+	0.999436i	$0.510690\pi$
$620$		0			0
$621$			720.000i			1.15942i
$622$		0			0
$623$	−1238.85			−1.98853
$624$		0			0
$625$	433.000	+	450.706i	0.692800	+	0.721130i
$626$		0			0
$627$		−	543.058i		−	0.866121i
$628$		0			0
$629$		−	960.000i		−	1.52623i
$630$		0			0
$631$		−	969.948i		−	1.53716i	−0.639753	−	0.768580i	$-0.720963\pi$
$631$		−	969.948i		−	1.53716i	0.639753	−	0.768580i	$-0.279037\pi$
$632$		0			0
$633$			274.343i			0.433401i
$634$		0			0
$635$	−540.400	−	110.309i	−0.851023	−	0.173714i
$636$		0			0
$637$	−225.353			−0.353772
$638$		0			0
$639$			27.7128i			0.0433690i
$640$		0			0
$641$	166.000			0.258970			0.129485	−	0.991581i	$-0.458668\pi$
$641$	166.000			0.258970			0.129485	+	0.991581i	$0.458668\pi$
$642$		0			0
$643$			144.250i			0.224339i	0.993689	+	0.112169i	$0.0357799\pi$
$643$			144.250i			0.224339i	−0.993689	+	0.112169i	$0.964220\pi$
$644$		0			0
$645$	823.029	+	168.000i	1.27601	+	0.260465i
$646$		0			0
$647$	687.308			1.06230			0.531150	−	0.847278i	$-0.321760\pi$
$647$	687.308			1.06230			0.531150	+	0.847278i	$0.321760\pi$
$648$		0			0
$649$	192.000			0.295840
$650$		0			0
$651$	1330.22			2.04334
$652$		0			0
$653$	29.3939			0.0450136			0.0225068	−	0.999747i	$-0.492835\pi$
$653$	29.3939			0.0450136			0.0225068	+	0.999747i	$0.492835\pi$
$654$		0			0
$655$	882.469	+	180.133i	1.34728	+	0.275013i
$656$		0			0
$657$		−	78.3837i		−	0.119305i
$658$		0			0
$659$	623.538			0.946189			0.473094	−	0.881012i	$-0.343137\pi$
$659$	623.538			0.946189			0.473094	+	0.881012i	$0.343137\pi$
$660$		0			0
$661$			98.0000i			0.148260i	0.997249	+	0.0741301i	$0.0236180\pi$
$661$			98.0000i			0.148260i	−0.997249	+	0.0741301i	$0.976382\pi$
$662$		0			0
$663$	543.058			0.819092
$664$		0			0
$665$	576.000	+	117.576i	0.866165	+	0.176805i
$666$		0			0
$667$			560.029i			0.839623i
$668$		0			0
$669$		−	648.000i		−	0.968610i
$670$		0			0
$671$			637.395i			0.949918i
$672$		0			0
$673$			489.898i			0.727932i	0.931412	+	0.363966i	$0.118577\pi$
$673$			489.898i			0.727932i	−0.931412	+	0.363966i	$0.881423\pi$
$674$		0			0
$675$	277.128	−	650.538i	0.410560	−	0.963760i
$676$		0			0
$677$	146.969			0.217089			0.108545	−	0.994092i	$-0.465381\pi$
$677$	146.969			0.217089			0.108545	+	0.994092i	$0.465381\pi$
$678$		0			0
$679$		−	498.831i		−	0.734655i
$680$		0			0
$681$	−840.000			−1.23348
$682$		0			0
$683$			398.808i			0.583907i	0.956433	+	0.291953i	$0.0943052\pi$
$683$			398.808i			0.583907i	−0.956433	+	0.291953i	$0.905695\pi$
$684$		0			0
$685$	215.555	−	1056.00i	0.314679	−	1.54161i
$686$		0			0
$687$	197.990			0.288195
$688$		0			0
$689$	−288.000			−0.417997
$690$		0			0
$691$	789.815			1.14300			0.571502	−	0.820601i	$-0.306361\pi$
$691$	789.815			1.14300			0.571502	+	0.820601i	$0.306361\pi$
$692$		0			0
$693$	117.576			0.169662
$694$		0			0
$695$	67.8823	+	13.8564i	0.0976723	+	0.0199373i
$696$		0			0
$697$			431.110i			0.618523i
$698$		0			0
$699$	−443.405			−0.634342
$700$		0			0
$701$			718.000i			1.02425i	0.858911	+	0.512126i	$0.171142\pi$
$701$			718.000i			1.02425i	−0.858911	+	0.512126i	$0.828858\pi$
$702$		0			0
$703$	678.823			0.965608
$704$		0			0
$705$	−24.0000	+	117.576i	−0.0340426	+	0.166774i
$706$		0			0
$707$			593.970i			0.840127i
$708$		0			0
$709$			838.000i			1.18195i	0.806691	+	0.590973i	$0.201256\pi$
$709$			838.000i			1.18195i	−0.806691	+	0.590973i	$0.798744\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	1410.91i		−	1.97883i
$714$		0			0
$715$	−665.108	−	135.765i	−0.930220	−	0.189880i
$716$		0			0
$717$	940.604			1.31186
$718$		0			0
$719$			221.703i			0.308348i	0.988044	+	0.154174i	$0.0492717\pi$
$719$			221.703i			0.308348i	−0.988044	+	0.154174i	$0.950728\pi$
$720$		0			0
$721$	−216.000			−0.299584
$722$		0			0
$723$			345.068i			0.477273i
$724$		0			0
$725$	215.555	−	506.000i	0.297317	−	0.697931i
$726$		0			0
$727$	347.897			0.478537			0.239269	−	0.970953i	$-0.423092\pi$
$727$	347.897			0.478537			0.239269	+	0.970953i	$0.423092\pi$
$728$		0			0
$729$	−791.000			−1.08505
$730$		0			0
$731$	1163.94			1.59225
$732$		0			0
$733$	676.059			0.922318			0.461159	−	0.887317i	$-0.347434\pi$
$733$	676.059			0.922318			0.461159	+	0.887317i	$0.347434\pi$
$734$		0			0
$735$	65.0538	−	318.697i	0.0885086	−	0.433602i
$736$		0			0
$737$		−	823.029i		−	1.11673i
$738$		0			0
$739$	235.559			0.318754			0.159377	−	0.987218i	$-0.449052\pi$
$739$	235.559			0.318754			0.159377	+	0.987218i	$0.449052\pi$
$740$		0			0
$741$			384.000i			0.518219i
$742$		0			0
$743$	−1349.16			−1.81583			−0.907914	−	0.419157i	$-0.862326\pi$
$743$	−1349.16			−1.81583			−0.907914	+	0.419157i	$0.862326\pi$
$744$		0			0
$745$	−2.00000	+	9.79796i	−0.00268456	+	0.0131516i
$746$		0			0
$747$			76.3675i			0.102232i
$748$		0			0
$749$		−	360.000i		−	0.480641i
$750$		0			0
$751$			1163.94i			1.54985i	0.632053	+	0.774926i	$0.282213\pi$
$751$			1163.94i			1.54985i	−0.632053	+	0.774926i	$0.717787\pi$
$752$		0			0
$753$		−	823.029i		−	1.09300i
$754$		0			0
$755$	−27.7128	+	135.765i	−0.0367057	+	0.179821i
$756$		0			0
$757$	−1205.15			−1.59201			−0.796003	−	0.605292i	$-0.793056\pi$
$757$	−1205.15			−1.59201			−0.796003	+	0.605292i	$0.793056\pi$
$758$		0			0
$759$			997.661i			1.31444i
$760$		0			0
$761$	−530.000			−0.696452			−0.348226	−	0.937411i	$-0.613216\pi$
$761$	−530.000			−0.696452			−0.348226	+	0.937411i	$0.613216\pi$
$762$		0			0
$763$		−	1238.85i		−	1.62366i
$764$		0			0
$765$	19.5959	−	96.0000i	0.0256156	−	0.125490i
$766$		0			0
$767$	−135.765			−0.177007
$768$		0			0
$769$	386.000			0.501951			0.250975	−	0.967993i	$-0.419249\pi$
$769$	386.000			0.501951			0.250975	+	0.967993i	$0.419249\pi$
$770$		0			0
$771$	−221.703			−0.287552
$772$		0			0
$773$	695.655			0.899942			0.449971	−	0.893043i	$-0.351434\pi$
$773$	695.655			0.899942			0.449971	+	0.893043i	$0.351434\pi$
$774$		0			0
$775$	−543.058	+	1274.79i	−0.700720	+	1.64489i
$776$		0			0
$777$		−	1175.76i		−	1.51320i
$778$		0			0
$779$	−304.841			−0.391323
$780$		0			0
$781$			384.000i			0.491677i
$782$		0			0
$783$	−622.254			−0.794705
$784$		0			0
$785$	−336.000	−	68.5857i	−0.428025	−	0.0873703i
$786$		0			0
$787$			1298.25i			1.64962i	0.565413	+	0.824808i	$0.308717\pi$
$787$			1298.25i			1.64962i	−0.565413	+	0.824808i	$0.691283\pi$
$788$		0			0
$789$		−	984.000i		−	1.24715i
$790$		0			0
$791$			332.554i			0.420422i
$792$		0			0
$793$		−	450.706i		−	0.568356i
$794$		0			0
$795$	83.1384	−	407.294i	0.104577	−	0.512319i
$796$		0			0
$797$	1361.92			1.70880			0.854402	−	0.519613i	$-0.173924\pi$
$797$	1361.92			1.70880			0.854402	+	0.519613i	$0.173924\pi$
$798$		0			0
$799$			166.277i			0.208106i
$800$		0			0
$801$	−146.000			−0.182272
$802$		0			0
$803$		−	1086.12i		−	1.35257i
$804$		0			0
$805$	−1058.18	−	216.000i	−1.31451	−	0.268323i
$806$		0			0
$807$	401.637			0.497691
$808$		0			0
$809$	1006.00			1.24351			0.621755	−	0.783212i	$-0.286420\pi$
$809$	1006.00			1.24351			0.621755	+	0.783212i	$0.286420\pi$
$810$		0			0
$811$	−651.251			−0.803022			−0.401511	−	0.915854i	$-0.631515\pi$
$811$	−651.251			−0.803022			−0.401511	+	0.915854i	$0.631515\pi$
$812$		0			0
$813$	940.604			1.15695
$814$		0			0
$815$	−110.309	+	540.400i	−0.135348	+	0.663067i
$816$		0			0
$817$			823.029i			1.00738i
$818$		0			0
$819$	−83.1384			−0.101512
$820$		0			0
$821$			482.000i			0.587089i	0.955945	+	0.293544i	$0.0948349\pi$
$821$			482.000i			0.587089i	−0.955945	+	0.293544i	$0.905165\pi$
$822$		0			0
$823$	−636.396			−0.773264			−0.386632	−	0.922234i	$-0.626362\pi$
$823$	−636.396			−0.773264			−0.386632	+	0.922234i	$0.626362\pi$
$824$		0			0
$825$	384.000	−	901.412i	0.465455	−	1.09262i
$826$		0			0
$827$			398.808i			0.482235i	0.970496	+	0.241117i	$0.0775139\pi$
$827$			398.808i			0.482235i	−0.970496	+	0.241117i	$0.922486\pi$
$828$		0			0
$829$		−	1106.00i		−	1.33414i	−0.744996	−	0.667069i	$-0.767549\pi$
$829$		−	1106.00i		−	1.33414i	0.744996	−	0.667069i	$-0.232451\pi$
$830$		0			0
$831$		−	249.415i		−	0.300139i
$832$		0			0
$833$		−	450.706i		−	0.541064i
$834$		0			0
$835$	−457.261	−	93.3381i	−0.547618	−	0.111782i
$836$		0			0
$837$	1567.67			1.87297
$838$		0			0
$839$		−	360.267i		−	0.429400i	−0.976680	−	0.214700i	$-0.931123\pi$
$839$		−	360.267i		−	0.429400i	0.976680	−	0.214700i	$-0.0688774\pi$
$840$		0			0
$841$	357.000			0.424495
$842$		0			0
$843$			1148.34i			1.36221i
$844$		0			0
$845$	−357.626	−	73.0000i	−0.423225	−	0.0863905i
$846$		0			0
$847$	602.455			0.711281
$848$		0			0
$849$	264.000			0.310954
$850$		0			0
$851$	−1247.08			−1.46542
$852$		0			0
$853$	−676.059			−0.792566			−0.396283	−	0.918128i	$-0.629700\pi$
$853$	−676.059			−0.792566			−0.396283	+	0.918128i	$0.629700\pi$
$854$		0			0
$855$	67.8823	+	13.8564i	0.0793944	+	0.0162063i
$856$		0			0
$857$		−	1077.78i		−	1.25761i	−0.777561	−	0.628807i	$-0.783543\pi$
$857$		−	1077.78i		−	1.25761i	0.777561	−	0.628807i	$-0.216457\pi$
$858$		0			0
$859$	568.113			0.661365			0.330683	−	0.943742i	$-0.392721\pi$
$859$	568.113			0.661365			0.330683	+	0.943742i	$0.392721\pi$
$860$		0			0
$861$			528.000i			0.613240i
$862$		0			0
$863$	195.161			0.226143			0.113072	−	0.993587i	$-0.463931\pi$
$863$	195.161			0.226143			0.113072	+	0.993587i	$0.463931\pi$
$864$		0			0
$865$	240.000	+	48.9898i	0.277457	+	0.0566356i
$866$		0			0
$867$			268.701i			0.309920i
$868$		0			0
$869$		0			0
$870$		0			0
$871$			581.969i			0.668162i
$872$		0			0
$873$		−	58.7878i		−	0.0673399i
$874$		0			0
$875$	872.954	+	602.455i	0.997661	+	0.688520i
$876$		0			0
$877$	244.949			0.279303			0.139652	−	0.990201i	$-0.455402\pi$
$877$	244.949			0.279303			0.139652	+	0.990201i	$0.455402\pi$
$878$		0			0
$879$			859.097i			0.977357i
$880$		0			0
$881$	550.000			0.624291			0.312145	−	0.950034i	$-0.398952\pi$
$881$	550.000			0.624291			0.312145	+	0.950034i	$0.398952\pi$
$882$		0			0
$883$		−	636.396i		−	0.720720i	−0.932813	−	0.360360i	$-0.882654\pi$
$883$		−	636.396i		−	0.720720i	0.932813	−	0.360360i	$-0.117346\pi$
$884$		0			0
$885$	39.1918	−	192.000i	0.0442846	−	0.216949i
$886$		0			0
$887$	1230.37			1.38711			0.693555	−	0.720404i	$-0.256044\pi$
$887$	1230.37			1.38711			0.693555	+	0.720404i	$0.256044\pi$
$888$		0			0
$889$	−936.000			−1.05287
$890$		0			0
$891$	−983.805			−1.10416
$892$		0			0
$893$	−117.576			−0.131664
$894$		0			0
$895$	−1289.76	−	263.272i	−1.44108	−	0.294158i
$896$		0			0
$897$		−	705.453i		−	0.786458i
$898$		0			0
$899$	1219.36			1.35636
$900$		0			0
$901$		−	576.000i		−	0.639290i
$902$		0			0
$903$	1425.53			1.57866
$904$		0			0
$905$	26.0000	−	127.373i	0.0287293	−	0.140744i
$906$		0			0
$907$		−	585.484i		−	0.645518i	−0.946481	−	0.322759i	$-0.895390\pi$
$907$		−	585.484i		−	0.645518i	0.946481	−	0.322759i	$-0.104610\pi$
$908$		0			0
$909$			70.0000i			0.0770077i
$910$		0			0
$911$		−	886.810i		−	0.973447i	−0.873556	−	0.486723i	$-0.838192\pi$
$911$		−	886.810i		−	0.973447i	0.873556	−	0.486723i	$-0.161808\pi$
$912$		0			0
$913$			1058.18i			1.15901i
$914$		0			0
$915$	637.395	+	130.108i	0.696606	+	0.142194i
$916$		0			0
$917$	1528.48			1.66683
$918$		0			0
$919$			193.990i			0.211088i	0.994415	+	0.105544i	$0.0336584\pi$
$919$			193.990i			0.211088i	−0.994415	+	0.105544i	$0.966342\pi$
$920$		0			0
$921$	792.000			0.859935
$922$		0			0
$923$		−	271.529i		−	0.294181i
$924$		0			0
$925$	1126.77	+	480.000i	1.21812	+	0.518919i
$926$		0			0
$927$	−25.4558			−0.0274605
$928$		0			0
$929$	1318.00			1.41873			0.709365	−	0.704841i	$-0.248982\pi$
$929$	1318.00			1.41873			0.709365	+	0.704841i	$0.248982\pi$
$930$		0			0
$931$	318.697			0.342317
$932$		0			0
$933$	391.918			0.420063
$934$		0			0
$935$	271.529	−	1330.22i	0.290405	−	1.42269i
$936$		0			0
$937$		−	548.686i		−	0.585577i	−0.956177	−	0.292789i	$-0.905417\pi$
$937$		−	548.686i		−	0.585577i	0.956177	−	0.292789i	$-0.0945832\pi$
$938$		0			0
$939$	1718.19			1.82981
$940$		0			0
$941$		−	1654.00i		−	1.75770i	−0.477094	−	0.878852i	$-0.658310\pi$
$941$		−	1654.00i		−	1.75770i	0.477094	−	0.878852i	$-0.341690\pi$
$942$		0			0
$943$	560.029			0.593880
$944$		0			0
$945$	240.000	−	1175.76i	0.253968	−	1.24419i
$946$		0			0
$947$		−	1417.04i		−	1.49635i	−0.663502	−	0.748174i	$-0.730931\pi$
$947$		−	1417.04i		−	1.49635i	0.663502	−	0.748174i	$-0.269069\pi$
$948$		0			0
$949$			768.000i			0.809273i
$950$		0			0
$951$		−	193.990i		−	0.203985i
$952$		0			0
$953$		−	137.171i		−	0.143936i	−0.997407	−	0.0719682i	$-0.977072\pi$
$953$		−	137.171i		−	0.143936i	0.997407	−	0.0719682i	$-0.0229280\pi$
$954$		0			0
$955$	110.851	−	543.058i	0.116075	−	0.568647i
$956$		0			0
$957$	−862.220			−0.900962
$958$		0			0
$959$		−	1829.05i		−	1.90724i
$960$		0			0
$961$	−2111.00			−2.19667
$962$		0			0
$963$		−	42.4264i		−	0.0440565i
$964$		0			0
$965$	333.131	−	1632.00i	0.345213	−	1.69119i
$966$		0			0
$967$	−1450.98			−1.50050			−0.750250	−	0.661154i	$-0.770067\pi$
$967$	−1450.98			−1.50050			−0.750250	+	0.661154i	$0.770067\pi$
$968$		0			0
$969$	−768.000			−0.792570
$970$		0			0
$971$	−817.528			−0.841944			−0.420972	−	0.907074i	$-0.638311\pi$
$971$	−817.528			−0.841944			−0.420972	+	0.907074i	$0.638311\pi$
$972$		0			0
$973$	117.576			0.120838
$974$		0			0
$975$	−271.529	+	637.395i	−0.278491	+	0.653738i
$976$		0			0
$977$		−	1704.84i		−	1.74498i	−0.488633	−	0.872490i	$-0.662504\pi$
$977$		−	1704.84i		−	1.74498i	0.488633	−	0.872490i	$-0.337496\pi$
$978$		0			0
$979$	−2023.04			−2.06643
$980$		0			0
$981$		−	146.000i		−	0.148828i
$982$		0			0
$983$	−738.219			−0.750986			−0.375493	−	0.926825i	$-0.622527\pi$
$983$	−738.219			−0.750986			−0.375493	+	0.926825i	$0.622527\pi$
$984$		0			0
$985$	528.000	+	107.778i	0.536041	+	0.109419i
$986$		0			0
$987$			203.647i			0.206329i
$988$		0			0
$989$		−	1512.00i		−	1.52882i
$990$		0			0
$991$		−	775.959i		−	0.783006i	−0.920177	−	0.391503i	$-0.871955\pi$
$991$		−	775.959i		−	0.783006i	0.920177	−	0.391503i	$-0.128045\pi$
$992$		0			0
$993$		−	509.494i		−	0.513085i
$994$		0			0
$995$	−249.415	+	1221.88i	−0.250669	+	1.22802i
$996$		0			0
$997$	−1420.70			−1.42498			−0.712489	−	0.701683i	$-0.752432\pi$
$997$	−1420.70			−1.42498			−0.712489	+	0.701683i	$0.752432\pi$
$998$		0			0
$999$		−	1385.64i		−	1.38703i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.3.e.j.639.4		8
4.3	odd	2	inner	1280.3.e.j.639.8		8
5.4	even	2	inner	1280.3.e.j.639.6		8
8.3	odd	2	inner	1280.3.e.j.639.1		8
8.5	even	2	inner	1280.3.e.j.639.5		8
16.3	odd	4		320.3.h.e.319.1		4
16.5	even	4		80.3.h.b.79.2	yes	4
16.11	odd	4		80.3.h.b.79.4	yes	4
16.13	even	4		320.3.h.e.319.3		4
20.19	odd	2	inner	1280.3.e.j.639.2		8
40.19	odd	2	inner	1280.3.e.j.639.7		8
40.29	even	2	inner	1280.3.e.j.639.3		8
48.5	odd	4		720.3.j.e.559.1		4
48.11	even	4		720.3.j.e.559.2		4
80.3	even	4		1600.3.b.t.1151.2		4
80.13	odd	4		1600.3.b.t.1151.3		4
80.19	odd	4		320.3.h.e.319.4		4
80.27	even	4		400.3.b.h.351.1		4
80.29	even	4		320.3.h.e.319.2		4
80.37	odd	4		400.3.b.h.351.4		4
80.43	even	4		400.3.b.h.351.3		4
80.53	odd	4		400.3.b.h.351.2		4
80.59	odd	4		80.3.h.b.79.1	✓	4
80.67	even	4		1600.3.b.t.1151.4		4
80.69	even	4		80.3.h.b.79.3	yes	4
80.77	odd	4		1600.3.b.t.1151.1		4
240.53	even	4		3600.3.e.bd.3151.3		4
240.59	even	4		720.3.j.e.559.3		4
240.107	odd	4		3600.3.e.bd.3151.4		4
240.149	odd	4		720.3.j.e.559.4		4
240.197	even	4		3600.3.e.bd.3151.1		4
240.203	odd	4		3600.3.e.bd.3151.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.3.h.b.79.1	✓	4	80.59	odd	4
80.3.h.b.79.2	yes	4	16.5	even	4
80.3.h.b.79.3	yes	4	80.69	even	4
80.3.h.b.79.4	yes	4	16.11	odd	4
320.3.h.e.319.1		4	16.3	odd	4
320.3.h.e.319.2		4	80.29	even	4
320.3.h.e.319.3		4	16.13	even	4
320.3.h.e.319.4		4	80.19	odd	4
400.3.b.h.351.1		4	80.27	even	4
400.3.b.h.351.2		4	80.53	odd	4
400.3.b.h.351.3		4	80.43	even	4
400.3.b.h.351.4		4	80.37	odd	4
720.3.j.e.559.1		4	48.5	odd	4
720.3.j.e.559.2		4	48.11	even	4
720.3.j.e.559.3		4	240.59	even	4
720.3.j.e.559.4		4	240.149	odd	4
1280.3.e.j.639.1		8	8.3	odd	2	inner
1280.3.e.j.639.2		8	20.19	odd	2	inner
1280.3.e.j.639.3		8	40.29	even	2	inner
1280.3.e.j.639.4		8	1.1	even	1	trivial
1280.3.e.j.639.5		8	8.5	even	2	inner
1280.3.e.j.639.6		8	5.4	even	2	inner
1280.3.e.j.639.7		8	40.19	odd	2	inner
1280.3.e.j.639.8		8	4.3	odd	2	inner
1600.3.b.t.1151.1		4	80.77	odd	4
1600.3.b.t.1151.2		4	80.3	even	4
1600.3.b.t.1151.3		4	80.13	odd	4
1600.3.b.t.1151.4		4	80.67	even	4
3600.3.e.bd.3151.1		4	240.197	even	4
3600.3.e.bd.3151.2		4	240.203	odd	4
3600.3.e.bd.3151.3		4	240.53	even	4
3600.3.e.bd.3151.4		4	240.107	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:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1280.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.82843i q^{3} +(4.89898 + 1.00000i) q^{5} +8.48528 q^{7} +1.00000 q^{9} +13.8564 q^{11} -9.79796 q^{13} +(2.82843 - 13.8564i) q^{15} -19.5959i q^{17} +13.8564 q^{19} -24.0000i q^{21} -25.4558 q^{23} +(23.0000 + 9.79796i) q^{25} -28.2843i q^{27} -22.0000i q^{29} +55.4256i q^{31} -39.1918i q^{33} +(41.5692 + 8.48528i) q^{35} +48.9898 q^{37} +27.7128i q^{39} -22.0000 q^{41} +59.3970i q^{43} +(4.89898 + 1.00000i) q^{45} -8.48528 q^{47} +23.0000 q^{49} -55.4256 q^{51} +29.3939 q^{53} +(67.8823 + 13.8564i) q^{55} -39.1918i q^{57} +13.8564 q^{59} +46.0000i q^{61} +8.48528 q^{63} +(-48.0000 - 9.79796i) q^{65} -59.3970i q^{67} +72.0000i q^{69} +27.7128i q^{71} -78.3837i q^{73} +(27.7128 - 65.0538i) q^{75} +117.576 q^{77} -71.0000 q^{81} +76.3675i q^{83} +(19.5959 - 96.0000i) q^{85} -62.2254 q^{87} -146.000 q^{89} -83.1384 q^{91} +156.767 q^{93} +(67.8823 + 13.8564i) q^{95} -58.7878i q^{97} +13.8564 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{9} + 184 q^{25} - 176 q^{41} + 184 q^{49} - 384 q^{65} - 568 q^{81} - 1168 q^{89}+O(q^{100}) \) 8 * q + 8 * q^9 + 184 * q^25 - 176 * q^41 + 184 * q^49 - 384 * q^65 - 568 * q^81 - 1168 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more