LMFDB - Embedded newform 720.3.j.e.559.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [720,3,Mod(559,720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(720, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("720.559");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	720.j (of order $2$, degree $1$, 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:	$19.6185790339$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7}\cdot 3 $
Twist minimal:	no (minimal twist has level 80)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			559.4
Root			$-0.707107 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	720.559
Dual form			720.3.j.e.559.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 + 4.89898i) q^{5} +8.48528 q^{7} -13.8564i q^{11} -9.79796i q^{13} -19.5959i q^{17} -13.8564i q^{19} +25.4558 q^{23} +(-23.0000 + 9.79796i) q^{25} +22.0000 q^{29} +55.4256i q^{31} +(8.48528 + 41.5692i) q^{35} -48.9898i q^{37} -22.0000 q^{41} +59.3970 q^{43} -8.48528 q^{47} +23.0000 q^{49} +29.3939i q^{53} +(67.8823 - 13.8564i) q^{55} -13.8564i q^{59} +46.0000 q^{61} +(48.0000 - 9.79796i) q^{65} +59.3970 q^{67} +27.7128i q^{71} -78.3837i q^{73} -117.576i q^{77} +76.3675 q^{83} +(96.0000 - 19.5959i) q^{85} -146.000 q^{89} -83.1384i q^{91} +(67.8823 - 13.8564i) q^{95} +58.7878i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} - 92 q^{25} + 88 q^{29} - 88 q^{41} + 92 q^{49} + 184 q^{61} + 192 q^{65} + 384 q^{85} - 584 q^{89}+O(q^{100}) $ 4 * q + 4 * q^5 - 92 * q^25 + 88 * q^29 - 88 * q^41 + 92 * q^49 + 184 * q^61 + 192 * q^65 + 384 * q^85 - 584 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$.

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$-1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.00000	+	4.89898i	0.200000	+	0.979796i
$5$	1.00000	+	4.89898i	0.200000	+	0.979796i
$6$		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
$9$		0			0
$10$		0			0
$11$		−	13.8564i		−	1.25967i	−0.776728	−	0.629837i	$-0.783122\pi$
$11$		−	13.8564i		−	1.25967i	0.776728	−	0.629837i	$-0.216878\pi$
$12$		0			0
$13$		−	9.79796i		−	0.753689i	−0.926277	−	0.376845i	$-0.877009\pi$
$13$		−	9.79796i		−	0.753689i	0.926277	−	0.376845i	$-0.122991\pi$
$14$		0			0
$15$		0			0
$16$		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
$19$		−	13.8564i		−	0.729285i	−0.931148	−	0.364642i	$-0.881191\pi$
$19$		−	13.8564i		−	0.729285i	0.931148	−	0.364642i	$-0.118809\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	25.4558			1.10678			0.553388	−	0.832924i	$-0.313335\pi$
$23$	25.4558			1.10678			0.553388	+	0.832924i	$0.313335\pi$
$24$		0			0
$25$	−23.0000	+	9.79796i	−0.920000	+	0.391918i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	22.0000			0.758621			0.379310	−	0.925270i	$-0.376161\pi$
$29$	22.0000			0.758621			0.379310	+	0.925270i	$0.376161\pi$
$30$		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
$33$		0			0
$34$		0			0
$35$	8.48528	+	41.5692i	0.242437	+	1.18769i
$36$		0			0
$37$		−	48.9898i		−	1.32405i	−0.749482	−	0.662024i	$-0.769698\pi$
$37$		−	48.9898i		−	1.32405i	0.749482	−	0.662024i	$-0.230302\pi$
$38$		0			0
$39$		0			0
$40$		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
$43$	59.3970			1.38132			0.690662	−	0.723177i	$-0.257319\pi$
$43$	59.3970			1.38132			0.690662	+	0.723177i	$0.257319\pi$
$44$		0			0
$45$		0			0
$46$		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
$51$		0			0
$52$		0			0
$53$			29.3939i			0.554601i	0.960783	+	0.277301i	$0.0894398\pi$
$53$			29.3939i			0.554601i	−0.960783	+	0.277301i	$0.910560\pi$
$54$		0			0
$55$	67.8823	−	13.8564i	1.23422	−	0.251935i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	13.8564i		−	0.234854i	−0.993081	−	0.117427i	$-0.962535\pi$
$59$		−	13.8564i		−	0.234854i	0.993081	−	0.117427i	$-0.0374647\pi$
$60$		0			0
$61$	46.0000			0.754098			0.377049	−	0.926193i	$-0.376939\pi$
$61$	46.0000			0.754098			0.377049	+	0.926193i	$0.376939\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	48.0000	−	9.79796i	0.738462	−	0.150738i
$66$		0			0
$67$	59.3970			0.886522			0.443261	−	0.896393i	$-0.353822\pi$
$67$	59.3970			0.886522			0.443261	+	0.896393i	$0.353822\pi$
$68$		0			0
$69$		0			0
$70$		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
$75$		0			0
$76$		0			0
$77$		−	117.576i		−	1.52695i
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	76.3675			0.920091			0.460045	−	0.887895i	$-0.347833\pi$
$83$	76.3675			0.920091			0.460045	+	0.887895i	$0.347833\pi$
$84$		0			0
$85$	96.0000	−	19.5959i	1.12941	−	0.230540i
$86$		0			0
$87$		0			0
$88$		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
$91$		−	83.1384i		−	0.913609i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	67.8823	−	13.8564i	0.714550	−	0.145857i
$96$		0			0
$97$			58.7878i			0.606059i	0.952981	+	0.303030i	$0.0979981\pi$
$97$			58.7878i			0.606059i	−0.952981	+	0.303030i	$0.902002\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	70.0000			0.693069			0.346535	−	0.938037i	$-0.387358\pi$
$101$	70.0000			0.693069			0.346535	+	0.938037i	$0.387358\pi$
$102$		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
$105$		0			0
$106$		0			0
$107$	42.4264			0.396508			0.198254	−	0.980151i	$-0.436473\pi$
$107$	42.4264			0.396508			0.198254	+	0.980151i	$0.436473\pi$
$108$		0			0
$109$	−146.000			−1.33945			−0.669725	−	0.742609i	$-0.733588\pi$
$109$	−146.000			−1.33945			−0.669725	+	0.742609i	$0.733588\pi$
$110$		0			0
$111$		0			0
$112$		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
$115$	25.4558	+	124.708i	0.221355	+	1.08441i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	166.277i		−	1.39728i
$120$		0			0
$121$	−71.0000			−0.586777
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−71.0000	−	102.879i	−0.568000	−	0.823029i
$126$		0			0
$127$	110.309			0.868572			0.434286	−	0.900775i	$-0.357001\pi$
$127$	110.309			0.868572			0.434286	+	0.900775i	$0.357001\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			180.133i			1.37506i	0.726154	+	0.687532i	$0.241306\pi$
$131$			180.133i			1.37506i	−0.726154	+	0.687532i	$0.758694\pi$
$132$		0			0
$133$		−	117.576i		−	0.884026i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			215.555i			1.57339i	0.617339	+	0.786697i	$0.288211\pi$
$137$			215.555i			1.57339i	−0.617339	+	0.786697i	$0.711789\pi$
$138$		0			0
$139$			13.8564i			0.0996864i	0.998757	+	0.0498432i	$0.0158722\pi$
$139$			13.8564i			0.0996864i	−0.998757	+	0.0498432i	$0.984128\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−135.765			−0.949402
$144$		0			0
$145$	22.0000	+	107.778i	0.151724	+	0.743293i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2.00000			0.0134228			0.00671141	−	0.999977i	$-0.497864\pi$
$149$	2.00000			0.0134228			0.00671141	+	0.999977i	$0.497864\pi$
$150$		0			0
$151$		−	27.7128i		−	0.183529i	−0.995781	−	0.0917643i	$-0.970749\pi$
$151$		−	27.7128i		−	0.183529i	0.995781	−	0.0917643i	$-0.0292506\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−271.529	+	55.4256i	−1.75180	+	0.357585i
$156$		0			0
$157$		−	68.5857i		−	0.436852i	−0.975854	−	0.218426i	$-0.929908\pi$
$157$		−	68.5857i		−	0.436852i	0.975854	−	0.218426i	$-0.0700922\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	216.000			1.34161
$162$		0			0
$163$	−110.309			−0.676740			−0.338370	−	0.941013i	$-0.609876\pi$
$163$	−110.309			−0.676740			−0.338370	+	0.941013i	$0.609876\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	93.3381			0.558911			0.279455	−	0.960159i	$-0.409846\pi$
$167$	93.3381			0.558911			0.279455	+	0.960159i	$0.409846\pi$
$168$		0			0
$169$	73.0000			0.431953
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	48.9898i		−	0.283178i	−0.989926	−	0.141589i	$-0.954779\pi$
$173$		−	48.9898i		−	0.283178i	0.989926	−	0.141589i	$-0.0452211\pi$
$174$		0			0
$175$	−195.161	+	83.1384i	−1.11521	+	0.475077i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	263.272i		−	1.47079i	−0.677638	−	0.735396i	$-0.736996\pi$
$179$		−	263.272i		−	1.47079i	0.677638	−	0.735396i	$-0.263004\pi$
$180$		0			0
$181$	26.0000			0.143646			0.0718232	−	0.997417i	$-0.477118\pi$
$181$	26.0000			0.143646			0.0718232	+	0.997417i	$0.477118\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	240.000	−	48.9898i	1.29730	−	0.264810i
$186$		0			0
$187$	−271.529			−1.45203
$188$		0			0
$189$		0			0
$190$		0			0
$191$			110.851i			0.580373i	0.956970	+	0.290187i	$0.0937174\pi$
$191$			110.851i			0.580373i	−0.956970	+	0.290187i	$0.906283\pi$
$192$		0			0
$193$			333.131i			1.72607i	0.505148	+	0.863033i	$0.331438\pi$
$193$			333.131i			1.72607i	−0.505148	+	0.863033i	$0.668562\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			107.778i			0.547094i	0.961859	+	0.273547i	$0.0881969\pi$
$197$			107.778i			0.547094i	−0.961859	+	0.273547i	$0.911803\pi$
$198$		0			0
$199$		−	249.415i		−	1.25334i	−0.779283	−	0.626672i	$-0.784417\pi$
$199$		−	249.415i		−	1.25334i	0.779283	−	0.626672i	$-0.215583\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	186.676			0.919587
$204$		0			0
$205$	−22.0000	−	107.778i	−0.107317	−	0.525744i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−192.000			−0.918660
$210$		0			0
$211$			96.9948i			0.459691i	0.973227	+	0.229846i	$0.0738221\pi$
$211$			96.9948i			0.459691i	−0.973227	+	0.229846i	$0.926178\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	59.3970	+	290.985i	0.276265	+	1.35342i
$216$		0			0
$217$			470.302i			2.16729i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−192.000			−0.868778
$222$		0			0
$223$	−229.103			−1.02737			−0.513683	−	0.857980i	$-0.671719\pi$
$223$	−229.103			−1.02737			−0.513683	+	0.857980i	$0.671719\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−296.985			−1.30830			−0.654152	−	0.756363i	$-0.726974\pi$
$227$	−296.985			−1.30830			−0.654152	+	0.756363i	$0.726974\pi$
$228$		0			0
$229$	−70.0000			−0.305677			−0.152838	−	0.988251i	$-0.548841\pi$
$229$	−70.0000			−0.305677			−0.152838	+	0.988251i	$0.548841\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			156.767i			0.672821i	0.941715	+	0.336411i	$0.109213\pi$
$233$			156.767i			0.672821i	−0.941715	+	0.336411i	$0.890787\pi$
$234$		0			0
$235$	−8.48528	−	41.5692i	−0.0361076	−	0.176890i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	332.554i		−	1.39144i	−0.718314	−	0.695719i	$-0.755086\pi$
$239$		−	332.554i		−	1.39144i	0.718314	−	0.695719i	$-0.244914\pi$
$240$		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
$243$		0			0
$244$		0			0
$245$	23.0000	+	112.677i	0.0938776	+	0.459904i
$246$		0			0
$247$	−135.765			−0.549654
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	290.985i		−	1.15930i	−0.814865	−	0.579650i	$-0.803189\pi$
$251$		−	290.985i		−	1.15930i	0.814865	−	0.579650i	$-0.196811\pi$
$252$		0			0
$253$		−	352.727i		−	1.39418i
$254$		0			0
$255$		0			0
$256$		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
$259$		−	415.692i		−	1.60499i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−347.897			−1.32280			−0.661400	−	0.750033i	$-0.730038\pi$
$263$	−347.897			−1.32280			−0.661400	+	0.750033i	$0.730038\pi$
$264$		0			0
$265$	−144.000	+	29.3939i	−0.543396	+	0.110920i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−142.000			−0.527881			−0.263941	−	0.964539i	$-0.585022\pi$
$269$	−142.000			−0.527881			−0.263941	+	0.964539i	$0.585022\pi$
$270$		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
$273$		0			0
$274$		0			0
$275$	135.765	+	318.697i	0.493689	+	1.15890i
$276$		0			0
$277$		−	88.1816i		−	0.318345i	−0.987251	−	0.159173i	$-0.949117\pi$
$277$		−	88.1816i		−	0.318345i	0.987251	−	0.159173i	$-0.0508826\pi$
$278$		0			0
$279$		0			0
$280$		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
$283$	93.3381			0.329817			0.164908	−	0.986309i	$-0.447267\pi$
$283$	93.3381			0.329817			0.164908	+	0.986309i	$0.447267\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−186.676			−0.650440
$288$		0			0
$289$	−95.0000			−0.328720
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	303.737i		−	1.03664i	−0.855186	−	0.518322i	$-0.826557\pi$
$293$		−	303.737i		−	1.03664i	0.855186	−	0.518322i	$-0.173443\pi$
$294$		0			0
$295$	67.8823	−	13.8564i	0.230109	−	0.0469709i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	249.415i		−	0.834165i
$300$		0			0
$301$	504.000			1.67442
$302$		0			0
$303$		0			0
$304$		0			0
$305$	46.0000	+	225.353i	0.150820	+	0.738862i
$306$		0			0
$307$	−280.014			−0.912099			−0.456049	−	0.889955i	$-0.650736\pi$
$307$	−280.014			−0.912099			−0.456049	+	0.889955i	$0.650736\pi$
$308$		0			0
$309$		0			0
$310$		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
$315$		0			0
$316$		0			0
$317$		−	68.5857i		−	0.216359i	−0.994131	−	0.108179i	$-0.965498\pi$
$317$		−	68.5857i		−	0.216359i	0.994131	−	0.108179i	$-0.0345021\pi$
$318$		0			0
$319$		−	304.841i		−	0.955614i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−271.529			−0.840647
$324$		0			0
$325$	96.0000	+	225.353i	0.295385	+	0.693394i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−72.0000			−0.218845
$330$		0			0
$331$			180.133i			0.544209i	0.962268	+	0.272105i	$0.0877197\pi$
$331$			180.133i			0.544209i	−0.962268	+	0.272105i	$0.912280\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	59.3970	+	290.985i	0.177304	+	0.868611i
$336$		0			0
$337$		−	470.302i		−	1.39555i	−0.716315	−	0.697777i	$-0.754172\pi$
$337$		−	470.302i		−	1.39555i	0.716315	−	0.697777i	$-0.245828\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	768.000			2.25220
$342$		0			0
$343$	−220.617			−0.643199
$344$		0			0
$345$		0			0
$346$		0			0
$347$	619.426			1.78509			0.892544	−	0.450960i	$-0.148918\pi$
$347$	619.426			1.78509			0.892544	+	0.450960i	$0.148918\pi$
$348$		0			0
$349$	−214.000			−0.613181			−0.306590	−	0.951842i	$-0.599188\pi$
$349$	−214.000			−0.613181			−0.306590	+	0.951842i	$0.599188\pi$
$350$		0			0
$351$		0			0
$352$		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
$355$	−135.765	+	27.7128i	−0.382435	+	0.0780643i
$356$		0			0
$357$		0			0
$358$		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
$363$		0			0
$364$		0			0
$365$	384.000	−	78.3837i	1.05205	−	0.214750i
$366$		0			0
$367$	246.073			0.670499			0.335250	−	0.942129i	$-0.391179\pi$
$367$	246.073			0.670499			0.335250	+	0.942129i	$0.391179\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			249.415i			0.672278i
$372$		0			0
$373$			382.120i			1.02445i	0.858851	+	0.512226i	$0.171179\pi$
$373$			382.120i			1.02445i	−0.858851	+	0.512226i	$0.828821\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	215.555i		−	0.571764i
$378$		0			0
$379$			290.985i			0.767769i	0.923381	+	0.383885i	$0.125414\pi$
$379$			290.985i			0.767769i	−0.923381	+	0.383885i	$0.874586\pi$
$380$		0			0
$381$		0			0
$382$		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
$387$		0			0
$388$		0			0
$389$	−670.000			−1.72237			−0.861183	−	0.508296i	$-0.830276\pi$
$389$	−670.000			−1.72237			−0.861183	+	0.508296i	$0.830276\pi$
$390$		0			0
$391$		−	498.831i		−	1.27578i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	362.524i		−	0.913160i	−0.889682	−	0.456580i	$-0.849074\pi$
$397$		−	362.524i		−	0.913160i	0.889682	−	0.456580i	$-0.150926\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−2.00000			−0.00498753			−0.00249377	−	0.999997i	$-0.500794\pi$
$401$	−2.00000			−0.00498753			−0.00249377	+	0.999997i	$0.500794\pi$
$402$		0			0
$403$	543.058			1.34754
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−678.823			−1.66787
$408$		0			0
$409$	−650.000			−1.58924			−0.794621	−	0.607106i	$-0.792330\pi$
$409$	−650.000			−1.58924			−0.794621	+	0.607106i	$0.792330\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	117.576i		−	0.284686i
$414$		0			0
$415$	76.3675	+	374.123i	0.184018	+	0.901501i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	540.400i		−	1.28974i	−0.764293	−	0.644869i	$-0.776912\pi$
$419$		−	540.400i		−	1.28974i	0.764293	−	0.644869i	$-0.223088\pi$
$420$		0			0
$421$	−482.000			−1.14489			−0.572447	−	0.819942i	$-0.694006\pi$
$421$	−482.000			−1.14489			−0.572447	+	0.819942i	$0.694006\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	192.000	+	450.706i	0.451765	+	1.06048i
$426$		0			0
$427$	390.323			0.914105
$428$		0			0
$429$		0			0
$430$		0			0
$431$			221.703i			0.514391i	0.966359	+	0.257195i	$0.0827984\pi$
$431$			221.703i			0.514391i	−0.966359	+	0.257195i	$0.917202\pi$
$432$		0			0
$433$		−	568.282i		−	1.31243i	−0.754575	−	0.656214i	$-0.772157\pi$
$433$		−	568.282i		−	1.31243i	0.754575	−	0.656214i	$-0.227843\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	352.727i		−	0.807155i
$438$		0			0
$439$			526.543i			1.19942i	0.800219	+	0.599708i	$0.204717\pi$
$439$			526.543i			1.19942i	−0.800219	+	0.599708i	$0.795283\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−432.749			−0.976861			−0.488430	−	0.872603i	$-0.662430\pi$
$443$	−432.749			−0.976861			−0.488430	+	0.872603i	$0.662430\pi$
$444$		0			0
$445$	−146.000	−	715.251i	−0.328090	−	1.60731i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	26.0000			0.0579065			0.0289532	−	0.999581i	$-0.490783\pi$
$449$	26.0000			0.0579065			0.0289532	+	0.999581i	$0.490783\pi$
$450$		0			0
$451$			304.841i			0.675922i
$452$		0			0
$453$		0			0
$454$		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
$459$		0			0
$460$		0			0
$461$	502.000			1.08894			0.544469	−	0.838781i	$-0.316731\pi$
$461$	502.000			1.08894			0.544469	+	0.838781i	$0.316731\pi$
$462$		0			0
$463$	823.072			1.77769			0.888847	−	0.458204i	$-0.151507\pi$
$463$	823.072			1.77769			0.888847	+	0.458204i	$0.151507\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	178.191			0.381565			0.190783	−	0.981632i	$-0.438897\pi$
$467$	178.191			0.381565			0.190783	+	0.981632i	$0.438897\pi$
$468$		0			0
$469$	504.000			1.07463
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	823.029i		−	1.74002i
$474$		0			0
$475$	135.765	+	318.697i	0.285820	+	0.670942i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	775.959i		−	1.61996i	−0.586460	−	0.809978i	$-0.699479\pi$
$479$		−	775.959i		−	1.61996i	0.586460	−	0.809978i	$-0.300521\pi$
$480$		0			0
$481$	−480.000			−0.997921
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−288.000	+	58.7878i	−0.593814	+	0.121212i
$486$		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
$489$		0			0
$490$		0			0
$491$			928.379i			1.89079i	0.325923	+	0.945396i	$0.394325\pi$
$491$			928.379i			1.89079i	−0.325923	+	0.945396i	$0.605675\pi$
$492$		0			0
$493$		−	431.110i		−	0.874463i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			235.151i			0.473141i
$498$		0			0
$499$		−	512.687i		−	1.02743i	−0.857961	−	0.513714i	$-0.828269\pi$
$499$		−	512.687i		−	1.02743i	0.857961	−	0.513714i	$-0.171731\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	704.278			1.40016			0.700078	−	0.714066i	$-0.253149\pi$
$503$	704.278			1.40016			0.700078	+	0.714066i	$0.253149\pi$
$504$		0			0
$505$	70.0000	+	342.929i	0.138614	+	0.679066i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−170.000			−0.333988			−0.166994	−	0.985958i	$-0.553406\pi$
$509$	−170.000			−0.333988			−0.166994	+	0.985958i	$0.553406\pi$
$510$		0			0
$511$		−	665.108i		−	1.30158i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−25.4558	−	124.708i	−0.0494288	−	0.242151i
$516$		0			0
$517$			117.576i			0.227419i
$518$		0			0
$519$		0			0
$520$		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
$523$	−246.073			−0.470503			−0.235252	−	0.971935i	$-0.575591\pi$
$523$	−246.073			−0.470503			−0.235252	+	0.971935i	$0.575591\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1086.12			2.06094
$528$		0			0
$529$	119.000			0.224953
$530$		0			0
$531$		0			0
$532$		0			0
$533$			215.555i			0.404419i
$534$		0			0
$535$	42.4264	+	207.846i	0.0793017	+	0.388497i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	318.697i		−	0.591275i
$540$		0			0
$541$	362.000			0.669131			0.334566	−	0.942372i	$-0.391410\pi$
$541$	362.000			0.669131			0.334566	+	0.942372i	$0.391410\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−146.000	−	715.251i	−0.267890	−	1.31239i
$546$		0			0
$547$	−42.4264			−0.0775620			−0.0387810	−	0.999248i	$-0.512347\pi$
$547$	−42.4264			−0.0775620			−0.0387810	+	0.999248i	$0.512347\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	304.841i		−	0.553250i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$			656.463i			1.17857i	0.807925	+	0.589285i	$0.200590\pi$
$557$			656.463i			1.17857i	−0.807925	+	0.589285i	$0.799410\pi$
$558$		0			0
$559$		−	581.969i		−	1.04109i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	517.602			0.919364			0.459682	−	0.888083i	$-0.347963\pi$
$563$	517.602			0.919364			0.459682	+	0.888083i	$0.347963\pi$
$564$		0			0
$565$	−192.000	+	39.1918i	−0.339823	+	0.0693661i
$566$		0			0
$567$		0			0
$568$		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
$571$			401.836i			0.703740i	0.936049	+	0.351870i	$0.114454\pi$
$571$			401.836i			0.703740i	−0.936049	+	0.351870i	$0.885546\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−585.484	+	249.415i	−1.01823	+	0.433766i
$576$		0			0
$577$		−	39.1918i		−	0.0679235i	−0.999423	−	0.0339617i	$-0.989188\pi$
$577$		−	39.1918i		−	0.0679235i	0.999423	−	0.0339617i	$-0.0108124\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	648.000			1.11532
$582$		0			0
$583$	407.294			0.698617
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−59.3970			−0.101187			−0.0505937	−	0.998719i	$-0.516111\pi$
$587$	−59.3970			−0.101187			−0.0505937	+	0.998719i	$0.516111\pi$
$588$		0			0
$589$	768.000			1.30390
$590$		0			0
$591$		0			0
$592$		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
$595$	814.587	−	166.277i	1.36905	−	0.279457i
$596$		0			0
$597$		0			0
$598$		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	$-0.334250\pi$
$601$	598.000			0.995008			0.497504	+	0.867462i	$0.334250\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−71.0000	−	347.828i	−0.117355	−	0.574922i
$606$		0			0
$607$	−127.279			−0.209686			−0.104843	−	0.994489i	$-0.533434\pi$
$607$	−127.279			−0.209686			−0.104843	+	0.994489i	$0.533434\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			83.1384i			0.136069i
$612$		0			0
$613$			205.757i			0.335656i	0.985816	+	0.167828i	$0.0536753\pi$
$613$			205.757i			0.335656i	−0.985816	+	0.167828i	$0.946325\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			431.110i			0.698720i	0.936989	+	0.349360i	$0.113601\pi$
$617$			431.110i			0.698720i	−0.936989	+	0.349360i	$0.886399\pi$
$618$		0			0
$619$		−	41.5692i		−	0.0671554i	−0.999436	−	0.0335777i	$-0.989310\pi$
$619$		−	41.5692i		−	0.0671554i	0.999436	−	0.0335777i	$-0.0106901\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1238.85			−1.98853
$624$		0			0
$625$	433.000	−	450.706i	0.692800	−	0.721130i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−960.000			−1.52623
$630$		0			0
$631$			969.948i			1.53716i	0.639753	+	0.768580i	$0.279037\pi$
$631$			969.948i			1.53716i	−0.639753	+	0.768580i	$0.720963\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	110.309	+	540.400i	0.173714	+	0.851023i
$636$		0			0
$637$		−	225.353i		−	0.353772i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−166.000			−0.258970			−0.129485	−	0.991581i	$-0.541332\pi$
$641$	−166.000			−0.258970			−0.129485	+	0.991581i	$0.541332\pi$
$642$		0			0
$643$	−144.250			−0.224339			−0.112169	−	0.993689i	$-0.535780\pi$
$643$	−144.250			−0.224339			−0.112169	+	0.993689i	$0.535780\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−687.308			−1.06230			−0.531150	−	0.847278i	$-0.678240\pi$
$647$	−687.308			−1.06230			−0.531150	+	0.847278i	$0.678240\pi$
$648$		0			0
$649$	−192.000			−0.295840
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	29.3939i		−	0.0450136i	−0.999747	−	0.0225068i	$-0.992835\pi$
$653$		−	29.3939i		−	0.0450136i	0.999747	−	0.0225068i	$-0.00716474\pi$
$654$		0			0
$655$	−882.469	+	180.133i	−1.34728	+	0.275013i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			623.538i			0.946189i	0.881012	+	0.473094i	$0.156863\pi$
$659$			623.538i			0.946189i	−0.881012	+	0.473094i	$0.843137\pi$
$660$		0			0
$661$	−98.0000			−0.148260			−0.0741301	−	0.997249i	$-0.523618\pi$
$661$	−98.0000			−0.148260			−0.0741301	+	0.997249i	$0.523618\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	576.000	−	117.576i	0.866165	−	0.176805i
$666$		0			0
$667$	560.029			0.839623
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	637.395i		−	0.949918i
$672$		0			0
$673$		−	489.898i		−	0.727932i	−0.931412	−	0.363966i	$-0.881423\pi$
$673$		−	489.898i		−	0.727932i	0.931412	−	0.363966i	$-0.118577\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			146.969i			0.217089i	0.994092	+	0.108545i	$0.0346190\pi$
$677$			146.969i			0.217089i	−0.994092	+	0.108545i	$0.965381\pi$
$678$		0			0
$679$			498.831i			0.734655i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−398.808			−0.583907			−0.291953	−	0.956433i	$-0.594305\pi$
$683$	−398.808			−0.583907			−0.291953	+	0.956433i	$0.594305\pi$
$684$		0			0
$685$	−1056.00	+	215.555i	−1.54161	+	0.314679i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	288.000			0.417997
$690$		0			0
$691$		−	789.815i		−	1.14300i	−0.820601	−	0.571502i	$-0.806361\pi$
$691$		−	789.815i		−	1.14300i	0.820601	−	0.571502i	$-0.193639\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−67.8823	+	13.8564i	−0.0976723	+	0.0199373i
$696$		0			0
$697$			431.110i			0.618523i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−718.000			−1.02425			−0.512126	−	0.858911i	$-0.671142\pi$
$701$	−718.000			−1.02425			−0.512126	+	0.858911i	$0.671142\pi$
$702$		0			0
$703$	−678.823			−0.965608
$704$		0			0
$705$		0			0
$706$		0			0
$707$	593.970			0.840127
$708$		0			0
$709$	−838.000			−1.18195			−0.590973	−	0.806691i	$-0.701256\pi$
$709$	−838.000			−1.18195			−0.590973	+	0.806691i	$0.701256\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			1410.91i			1.97883i
$714$		0			0
$715$	−135.765	−	665.108i	−0.189880	−	0.930220i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	221.703i		−	0.308348i	−0.988044	−	0.154174i	$-0.950728\pi$
$719$		−	221.703i		−	0.308348i	0.988044	−	0.154174i	$-0.0492717\pi$
$720$		0			0
$721$	−216.000			−0.299584
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−506.000	+	215.555i	−0.697931	+	0.297317i
$726$		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
$729$		0			0
$730$		0			0
$731$		−	1163.94i		−	1.59225i
$732$		0			0
$733$			676.059i			0.922318i	0.887317	+	0.461159i	$0.152566\pi$
$733$			676.059i			0.922318i	−0.887317	+	0.461159i	$0.847434\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	823.029i		−	1.11673i
$738$		0			0
$739$		−	235.559i		−	0.318754i	−0.987218	−	0.159377i	$-0.949052\pi$
$739$		−	235.559i		−	0.318754i	0.987218	−	0.159377i	$-0.0509485\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1349.16			1.81583			0.907914	−	0.419157i	$-0.137674\pi$
$743$	1349.16			1.81583			0.907914	+	0.419157i	$0.137674\pi$
$744$		0			0
$745$	2.00000	+	9.79796i	0.00268456	+	0.0131516i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	360.000			0.480641
$750$		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
$753$		0			0
$754$		0			0
$755$	135.765	−	27.7128i	0.179821	−	0.0367057i
$756$		0			0
$757$			1205.15i			1.59201i	0.605292	+	0.796003i	$0.293056\pi$
$757$			1205.15i			1.59201i	−0.605292	+	0.796003i	$0.706944\pi$
$758$		0			0
$759$		0			0
$760$		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
$763$	−1238.85			−1.62366
$764$		0			0
$765$		0			0
$766$		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
$771$		0			0
$772$		0			0
$773$			695.655i			0.899942i	0.893043	+	0.449971i	$0.148566\pi$
$773$			695.655i			0.899942i	−0.893043	+	0.449971i	$0.851434\pi$
$774$		0			0
$775$	−543.058	−	1274.79i	−0.700720	−	1.64489i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			304.841i			0.391323i
$780$		0			0
$781$	384.000			0.491677
$782$		0			0
$783$		0			0
$784$		0			0
$785$	336.000	−	68.5857i	0.428025	−	0.0873703i
$786$		0			0
$787$	−1298.25			−1.64962			−0.824808	−	0.565413i	$-0.808717\pi$
$787$	−1298.25			−1.64962			−0.824808	+	0.565413i	$0.808717\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			332.554i			0.420422i
$792$		0			0
$793$		−	450.706i		−	0.568356i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	1361.92i		−	1.70880i	−0.519613	−	0.854402i	$-0.673924\pi$
$797$		−	1361.92i		−	1.70880i	0.519613	−	0.854402i	$-0.326076\pi$
$798$		0			0
$799$			166.277i			0.208106i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1086.12			−1.35257
$804$		0			0
$805$	216.000	+	1058.18i	0.268323	+	1.31451i
$806$		0			0
$807$		0			0
$808$		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
$811$		−	651.251i		−	0.803022i	−0.915854	−	0.401511i	$-0.868485\pi$
$811$		−	651.251i		−	0.803022i	0.915854	−	0.401511i	$-0.131515\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−110.309	−	540.400i	−0.135348	−	0.663067i
$816$		0			0
$817$		−	823.029i		−	1.00738i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	482.000			0.587089			0.293544	−	0.955945i	$-0.405165\pi$
$821$	482.000			0.587089			0.293544	+	0.955945i	$0.405165\pi$
$822$		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
$825$		0			0
$826$		0			0
$827$	−398.808			−0.482235			−0.241117	−	0.970496i	$-0.577514\pi$
$827$	−398.808			−0.482235			−0.241117	+	0.970496i	$0.577514\pi$
$828$		0			0
$829$	−1106.00			−1.33414			−0.667069	−	0.744996i	$-0.732451\pi$
$829$	−1106.00			−1.33414			−0.667069	+	0.744996i	$0.732451\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	450.706i		−	0.541064i
$834$		0			0
$835$	93.3381	+	457.261i	0.111782	+	0.547618i
$836$		0			0
$837$		0			0
$838$		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
$843$		0			0
$844$		0			0
$845$	73.0000	+	357.626i	0.0863905	+	0.423225i
$846$		0			0
$847$	−602.455			−0.711281
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	1247.08i		−	1.46542i
$852$		0			0
$853$			676.059i			0.792566i	0.918128	+	0.396283i	$0.129700\pi$
$853$			676.059i			0.792566i	−0.918128	+	0.396283i	$0.870300\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			1077.78i			1.25761i	0.777561	+	0.628807i	$0.216457\pi$
$857$			1077.78i			1.25761i	−0.777561	+	0.628807i	$0.783543\pi$
$858$		0			0
$859$			568.113i			0.661365i	0.943742	+	0.330683i	$0.107279\pi$
$859$			568.113i			0.661365i	−0.943742	+	0.330683i	$0.892721\pi$
$860$		0			0
$861$		0			0
$862$		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
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		−	581.969i		−	0.668162i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−602.455	−	872.954i	−0.688520	−	0.997661i
$876$		0			0
$877$			244.949i			0.279303i	0.990201	+	0.139652i	$0.0445983\pi$
$877$			244.949i			0.279303i	−0.990201	+	0.139652i	$0.955402\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−550.000			−0.624291			−0.312145	−	0.950034i	$-0.601048\pi$
$881$	−550.000			−0.624291			−0.312145	+	0.950034i	$0.601048\pi$
$882$		0			0
$883$	636.396			0.720720			0.360360	−	0.932813i	$-0.382654\pi$
$883$	636.396			0.720720			0.360360	+	0.932813i	$0.382654\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1230.37			−1.38711			−0.693555	−	0.720404i	$-0.743956\pi$
$887$	−1230.37			−1.38711			−0.693555	+	0.720404i	$0.743956\pi$
$888$		0			0
$889$	936.000			1.05287
$890$		0			0
$891$		0			0
$892$		0			0
$893$			117.576i			0.131664i
$894$		0			0
$895$	1289.76	−	263.272i	1.44108	−	0.294158i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			1219.36i			1.35636i
$900$		0			0
$901$	576.000			0.639290
$902$		0			0
$903$		0			0
$904$		0			0
$905$	26.0000	+	127.373i	0.0287293	+	0.140744i
$906$		0			0
$907$	−585.484			−0.645518			−0.322759	−	0.946481i	$-0.604610\pi$
$907$	−585.484			−0.645518			−0.322759	+	0.946481i	$0.604610\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			886.810i			0.973447i	0.873556	+	0.486723i	$0.161808\pi$
$911$			886.810i			0.973447i	−0.873556	+	0.486723i	$0.838192\pi$
$912$		0			0
$913$		−	1058.18i		−	1.15901i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			1528.48i			1.66683i
$918$		0			0
$919$		−	193.990i		−	0.211088i	−0.994415	−	0.105544i	$-0.966342\pi$
$919$		−	193.990i		−	0.211088i	0.994415	−	0.105544i	$-0.0336584\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	271.529			0.294181
$924$		0			0
$925$	480.000	+	1126.77i	0.518919	+	1.21812i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1318.00			−1.41873			−0.709365	−	0.704841i	$-0.751018\pi$
$929$	−1318.00			−1.41873			−0.709365	+	0.704841i	$0.751018\pi$
$930$		0			0
$931$		−	318.697i		−	0.342317i
$932$		0			0
$933$		0			0
$934$		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
$939$		0			0
$940$		0			0
$941$	1654.00			1.75770			0.878852	−	0.477094i	$-0.158310\pi$
$941$	1654.00			1.75770			0.878852	+	0.477094i	$0.158310\pi$
$942$		0			0
$943$	−560.029			−0.593880
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1417.04			−1.49635			−0.748174	−	0.663502i	$-0.769069\pi$
$947$	−1417.04			−1.49635			−0.748174	+	0.663502i	$0.769069\pi$
$948$		0			0
$949$	−768.000			−0.809273
$950$		0			0
$951$		0			0
$952$		0			0
$953$			137.171i			0.143936i	0.997407	+	0.0719682i	$0.0229280\pi$
$953$			137.171i			0.143936i	−0.997407	+	0.0719682i	$0.977072\pi$
$954$		0			0
$955$	−543.058	+	110.851i	−0.568647	+	0.116075i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			1829.05i			1.90724i
$960$		0			0
$961$	−2111.00			−2.19667
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1632.00	+	333.131i	−1.69119	+	0.345213i
$966$		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
$969$		0			0
$970$		0			0
$971$			817.528i			0.841944i	0.907074	+	0.420972i	$0.138311\pi$
$971$			817.528i			0.841944i	−0.907074	+	0.420972i	$0.861689\pi$
$972$		0			0
$973$			117.576i			0.120838i
$974$		0			0
$975$		0			0
$976$		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
$979$			2023.04i			2.06643i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	738.219			0.750986			0.375493	−	0.926825i	$-0.377473\pi$
$983$	738.219			0.750986			0.375493	+	0.926825i	$0.377473\pi$
$984$		0			0
$985$	−528.000	+	107.778i	−0.536041	+	0.109419i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1512.00			1.52882
$990$		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
$993$		0			0
$994$		0			0
$995$	1221.88	−	249.415i	1.22802	−	0.250669i
$996$		0			0
$997$			1420.70i			1.42498i	0.701683	+	0.712489i	$0.252432\pi$
$997$			1420.70i			1.42498i	−0.701683	+	0.712489i	$0.747568\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.3.j.e.559.4		4
3.2	odd	2		80.3.h.b.79.3	yes	4
4.3	odd	2	inner	720.3.j.e.559.3		4
5.2	odd	4		3600.3.e.bd.3151.3		4
5.3	odd	4		3600.3.e.bd.3151.1		4
5.4	even	2	inner	720.3.j.e.559.1		4
12.11	even	2		80.3.h.b.79.1	✓	4
15.2	even	4		400.3.b.h.351.2		4
15.8	even	4		400.3.b.h.351.4		4
15.14	odd	2		80.3.h.b.79.2	yes	4
20.3	even	4		3600.3.e.bd.3151.4		4
20.7	even	4		3600.3.e.bd.3151.2		4
20.19	odd	2	inner	720.3.j.e.559.2		4
24.5	odd	2		320.3.h.e.319.2		4
24.11	even	2		320.3.h.e.319.4		4
48.5	odd	4		1280.3.e.j.639.3		8
48.11	even	4		1280.3.e.j.639.7		8
48.29	odd	4		1280.3.e.j.639.6		8
48.35	even	4		1280.3.e.j.639.2		8
60.23	odd	4		400.3.b.h.351.1		4
60.47	odd	4		400.3.b.h.351.3		4
60.59	even	2		80.3.h.b.79.4	yes	4
120.29	odd	2		320.3.h.e.319.3		4
120.53	even	4		1600.3.b.t.1151.1		4
120.59	even	2		320.3.h.e.319.1		4
120.77	even	4		1600.3.b.t.1151.3		4
120.83	odd	4		1600.3.b.t.1151.4		4
120.107	odd	4		1600.3.b.t.1151.2		4
240.29	odd	4		1280.3.e.j.639.4		8
240.59	even	4		1280.3.e.j.639.1		8
240.149	odd	4		1280.3.e.j.639.5		8
240.179	even	4		1280.3.e.j.639.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.3.h.b.79.1	✓	4	12.11	even	2
80.3.h.b.79.2	yes	4	15.14	odd	2
80.3.h.b.79.3	yes	4	3.2	odd	2
80.3.h.b.79.4	yes	4	60.59	even	2
320.3.h.e.319.1		4	120.59	even	2
320.3.h.e.319.2		4	24.5	odd	2
320.3.h.e.319.3		4	120.29	odd	2
320.3.h.e.319.4		4	24.11	even	2
400.3.b.h.351.1		4	60.23	odd	4
400.3.b.h.351.2		4	15.2	even	4
400.3.b.h.351.3		4	60.47	odd	4
400.3.b.h.351.4		4	15.8	even	4
720.3.j.e.559.1		4	5.4	even	2	inner
720.3.j.e.559.2		4	20.19	odd	2	inner
720.3.j.e.559.3		4	4.3	odd	2	inner
720.3.j.e.559.4		4	1.1	even	1	trivial
1280.3.e.j.639.1		8	240.59	even	4
1280.3.e.j.639.2		8	48.35	even	4
1280.3.e.j.639.3		8	48.5	odd	4
1280.3.e.j.639.4		8	240.29	odd	4
1280.3.e.j.639.5		8	240.149	odd	4
1280.3.e.j.639.6		8	48.29	odd	4
1280.3.e.j.639.7		8	48.11	even	4
1280.3.e.j.639.8		8	240.179	even	4
1600.3.b.t.1151.1		4	120.53	even	4
1600.3.b.t.1151.2		4	120.107	odd	4
1600.3.b.t.1151.3		4	120.77	even	4
1600.3.b.t.1151.4		4	120.83	odd	4
3600.3.e.bd.3151.1		4	5.3	odd	4
3600.3.e.bd.3151.2		4	20.7	even	4
3600.3.e.bd.3151.3		4	5.2	odd	4
3600.3.e.bd.3151.4		4	20.3	even	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:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	720.j (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 4.89898i) q^{5} +8.48528 q^{7} -13.8564i q^{11} -9.79796i q^{13} -19.5959i q^{17} -13.8564i q^{19} +25.4558 q^{23} +(-23.0000 + 9.79796i) q^{25} +22.0000 q^{29} +55.4256i q^{31} +(8.48528 + 41.5692i) q^{35} -48.9898i q^{37} -22.0000 q^{41} +59.3970 q^{43} -8.48528 q^{47} +23.0000 q^{49} +29.3939i q^{53} +(67.8823 - 13.8564i) q^{55} -13.8564i q^{59} +46.0000 q^{61} +(48.0000 - 9.79796i) q^{65} +59.3970 q^{67} +27.7128i q^{71} -78.3837i q^{73} -117.576i q^{77} +76.3675 q^{83} +(96.0000 - 19.5959i) q^{85} -146.000 q^{89} -83.1384i q^{91} +(67.8823 - 13.8564i) q^{95} +58.7878i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} - 92 q^{25} + 88 q^{29} - 88 q^{41} + 92 q^{49} + 184 q^{61} + 192 q^{65} + 384 q^{85} - 584 q^{89}+O(q^{100}) \) 4 * q + 4 * q^5 - 92 * q^25 + 88 * q^29 - 88 * q^41 + 92 * q^49 + 184 * q^61 + 192 * q^65 + 384 * q^85 - 584 * q^89

Introduction

L-functions

Modular forms

Varieties

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Database

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