LMFDB - Embedded newform 4032.2.p.a.1567.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(1567,4032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4032, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4032.1567");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.p (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:	$32.1956820950$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1344)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1567.2
Root			$1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	4032.1567
Dual form			4032.2.p.a.1567.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.44949 + 1.00000i) q^{7} +O(q^{10})$	$q+(-2.44949 + 1.00000i) q^{7} -4.89898 q^{11} -4.00000 q^{13} -4.89898i q^{17} +4.00000i q^{19} +6.00000i q^{23} -5.00000 q^{25} -4.89898i q^{29} +4.89898 q^{31} +9.79796i q^{37} -4.89898i q^{41} +9.79796 q^{43} +9.79796 q^{47} +(5.00000 - 4.89898i) q^{49} -4.89898i q^{53} -12.0000i q^{59} +4.00000 q^{61} -6.00000i q^{71} +(12.0000 - 4.89898i) q^{77} +2.00000i q^{79} +12.0000i q^{83} -4.89898i q^{89} +(9.79796 - 4.00000i) q^{91} -9.79796i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 16 q^{13} - 20 q^{25} + 20 q^{49} + 16 q^{61} + 48 q^{77}+O(q^{100}) $ 4 * q - 16 * q^13 - 20 * q^25 + 20 * q^49 + 16 * q^61 + 48 * q^77

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$		0			0
$7$	−2.44949	+	1.00000i	−0.925820	+	0.377964i
$7$	−2.44949	+	1.00000i	−0.925820	+	0.377964i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.89898			−1.47710			−0.738549	−	0.674200i	$-0.764489\pi$
$11$	−4.89898			−1.47710			−0.738549	+	0.674200i	$0.764489\pi$
$12$		0			0
$13$	−4.00000			−1.10940			−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000			−1.10940			−0.554700	+	0.832050i	$0.687167\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	4.89898i		−	1.18818i	−0.804400	−	0.594089i	$-0.797513\pi$
$17$		−	4.89898i		−	1.18818i	0.804400	−	0.594089i	$-0.202487\pi$
$18$		0			0
$19$			4.00000i			0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$			4.00000i			0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			6.00000i			1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$			6.00000i			1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$		0			0
$25$	−5.00000			−1.00000
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	4.89898i		−	0.909718i	−0.890564	−	0.454859i	$-0.849690\pi$
$29$		−	4.89898i		−	0.909718i	0.890564	−	0.454859i	$-0.150310\pi$
$30$		0			0
$31$	4.89898			0.879883			0.439941	−	0.898027i	$-0.354999\pi$
$31$	4.89898			0.879883			0.439941	+	0.898027i	$0.354999\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$			9.79796i			1.61077i	0.592749	+	0.805387i	$0.298043\pi$
$37$			9.79796i			1.61077i	−0.592749	+	0.805387i	$0.701957\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	4.89898i		−	0.765092i	−0.923936	−	0.382546i	$-0.875047\pi$
$41$		−	4.89898i		−	0.765092i	0.923936	−	0.382546i	$-0.124953\pi$
$42$		0			0
$43$	9.79796			1.49417			0.747087	−	0.664726i	$-0.231452\pi$
$43$	9.79796			1.49417			0.747087	+	0.664726i	$0.231452\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	9.79796			1.42918			0.714590	−	0.699544i	$-0.246613\pi$
$47$	9.79796			1.42918			0.714590	+	0.699544i	$0.246613\pi$
$48$		0			0
$49$	5.00000	−	4.89898i	0.714286	−	0.699854i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	4.89898i		−	0.672927i	−0.941697	−	0.336463i	$-0.890769\pi$
$53$		−	4.89898i		−	0.672927i	0.941697	−	0.336463i	$-0.109231\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	12.0000i		−	1.56227i	−0.624364	−	0.781133i	$-0.714642\pi$
$59$		−	12.0000i		−	1.56227i	0.624364	−	0.781133i	$-0.285358\pi$
$60$		0			0
$61$	4.00000			0.512148			0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000			0.512148			0.256074	+	0.966657i	$0.417571\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	6.00000i		−	0.712069i	−0.934473	−	0.356034i	$-0.884129\pi$
$71$		−	6.00000i		−	0.712069i	0.934473	−	0.356034i	$-0.115871\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	12.0000	−	4.89898i	1.36753	−	0.558291i
$78$		0			0
$79$			2.00000i			0.225018i	0.993651	+	0.112509i	$0.0358886\pi$
$79$			2.00000i			0.225018i	−0.993651	+	0.112509i	$0.964111\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			12.0000i			1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$			12.0000i			1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	4.89898i		−	0.519291i	−0.965704	−	0.259645i	$-0.916394\pi$
$89$		−	4.89898i		−	0.519291i	0.965704	−	0.259645i	$-0.0836057\pi$
$90$		0			0
$91$	9.79796	−	4.00000i	1.02711	−	0.419314i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	9.79796i		−	0.994832i	−0.867512	−	0.497416i	$-0.834282\pi$
$97$		−	9.79796i		−	0.994832i	0.867512	−	0.497416i	$-0.165718\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$	14.6969			1.44813			0.724066	−	0.689730i	$-0.242271\pi$
$103$	14.6969			1.44813			0.724066	+	0.689730i	$0.242271\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−14.6969			−1.42081			−0.710403	−	0.703795i	$-0.751487\pi$
$107$	−14.6969			−1.42081			−0.710403	+	0.703795i	$0.751487\pi$
$108$		0			0
$109$			9.79796i			0.938474i	0.883072	+	0.469237i	$0.155471\pi$
$109$			9.79796i			0.938474i	−0.883072	+	0.469237i	$0.844529\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	18.0000			1.69330			0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000			1.69330			0.846649	+	0.532152i	$0.178617\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	4.89898	+	12.0000i	0.449089	+	1.10004i
$120$		0			0
$121$	13.0000			1.18182
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	2.00000i		−	0.177471i	−0.996055	−	0.0887357i	$-0.971717\pi$
$127$		−	2.00000i		−	0.177471i	0.996055	−	0.0887357i	$-0.0282826\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	12.0000i		−	1.04844i	−0.851581	−	0.524222i	$-0.824356\pi$
$131$		−	12.0000i		−	1.04844i	0.851581	−	0.524222i	$-0.175644\pi$
$132$		0			0
$133$	−4.00000	−	9.79796i	−0.346844	−	0.849591i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	6.00000			0.512615			0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000			0.512615			0.256307	+	0.966595i	$0.417494\pi$
$138$		0			0
$139$		−	4.00000i		−	0.339276i	−0.985506	−	0.169638i	$-0.945740\pi$
$139$		−	4.00000i		−	0.339276i	0.985506	−	0.169638i	$-0.0542598\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	19.5959			1.63869
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$			4.89898i			0.401340i	0.979659	+	0.200670i	$0.0643119\pi$
$149$			4.89898i			0.401340i	−0.979659	+	0.200670i	$0.935688\pi$
$150$		0			0
$151$			14.0000i			1.13930i	0.821886	+	0.569652i	$0.192922\pi$
$151$			14.0000i			1.13930i	−0.821886	+	0.569652i	$0.807078\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	4.00000			0.319235			0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000			0.319235			0.159617	+	0.987179i	$0.448974\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−6.00000	−	14.6969i	−0.472866	−	1.15828i
$162$		0			0
$163$	−19.5959			−1.53487			−0.767435	−	0.641126i	$-0.778467\pi$
$163$	−19.5959			−1.53487			−0.767435	+	0.641126i	$0.778467\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	19.5959			1.51638			0.758189	−	0.652035i	$-0.226085\pi$
$167$	19.5959			1.51638			0.758189	+	0.652035i	$0.226085\pi$
$168$		0			0
$169$	3.00000			0.230769
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−24.0000			−1.82469			−0.912343	−	0.409426i	$-0.865729\pi$
$173$	−24.0000			−1.82469			−0.912343	+	0.409426i	$0.865729\pi$
$174$		0			0
$175$	12.2474	−	5.00000i	0.925820	−	0.377964i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	14.6969			1.09850			0.549250	−	0.835658i	$-0.314913\pi$
$179$	14.6969			1.09850			0.549250	+	0.835658i	$0.314913\pi$
$180$		0			0
$181$	−20.0000			−1.48659			−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−20.0000			−1.48659			−0.743294	+	0.668965i	$0.766738\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$			24.0000i			1.75505i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	6.00000i		−	0.434145i	−0.976156	−	0.217072i	$-0.930349\pi$
$191$		−	6.00000i		−	0.434145i	0.976156	−	0.217072i	$-0.0696508\pi$
$192$		0			0
$193$	−2.00000			−0.143963			−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000			−0.143963			−0.0719816	+	0.997406i	$0.522932\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			4.89898i			0.349038i	0.984654	+	0.174519i	$0.0558370\pi$
$197$			4.89898i			0.349038i	−0.984654	+	0.174519i	$0.944163\pi$
$198$		0			0
$199$	−4.89898			−0.347279			−0.173640	−	0.984809i	$-0.555553\pi$
$199$	−4.89898			−0.347279			−0.173640	+	0.984809i	$0.555553\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	4.89898	+	12.0000i	0.343841	+	0.842235i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	19.5959i		−	1.35548i
$210$		0			0
$211$	−9.79796			−0.674519			−0.337260	−	0.941412i	$-0.609500\pi$
$211$	−9.79796			−0.674519			−0.337260	+	0.941412i	$0.609500\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−12.0000	+	4.89898i	−0.814613	+	0.332564i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			19.5959i			1.31816i
$222$		0			0
$223$	4.89898			0.328060			0.164030	−	0.986455i	$-0.447551\pi$
$223$	4.89898			0.328060			0.164030	+	0.986455i	$0.447551\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			12.0000i			0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$			12.0000i			0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$		0			0
$229$	28.0000			1.85029			0.925146	−	0.379611i	$-0.123942\pi$
$229$	28.0000			1.85029			0.925146	+	0.379611i	$0.123942\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	6.00000			0.393073			0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000			0.393073			0.196537	+	0.980497i	$0.437031\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			6.00000i			0.388108i	0.980991	+	0.194054i	$0.0621637\pi$
$239$			6.00000i			0.388108i	−0.980991	+	0.194054i	$0.937836\pi$
$240$		0			0
$241$		−	19.5959i		−	1.26228i	−0.775667	−	0.631142i	$-0.782587\pi$
$241$		−	19.5959i		−	1.26228i	0.775667	−	0.631142i	$-0.217413\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	16.0000i		−	1.01806i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			12.0000i			0.757433i	0.925513	+	0.378717i	$0.123635\pi$
$251$			12.0000i			0.757433i	−0.925513	+	0.378717i	$0.876365\pi$
$252$		0			0
$253$		−	29.3939i		−	1.84798i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	24.4949i		−	1.52795i	−0.645246	−	0.763975i	$-0.723245\pi$
$257$		−	24.4949i		−	1.52795i	0.645246	−	0.763975i	$-0.276755\pi$
$258$		0			0
$259$	−9.79796	−	24.0000i	−0.608816	−	1.49129i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			18.0000i			1.10993i	0.831875	+	0.554964i	$0.187268\pi$
$263$			18.0000i			1.10993i	−0.831875	+	0.554964i	$0.812732\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	−14.6969			−0.892775			−0.446388	−	0.894840i	$-0.647290\pi$
$271$	−14.6969			−0.892775			−0.446388	+	0.894840i	$0.647290\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	24.4949			1.47710
$276$		0			0
$277$		−	19.5959i		−	1.17740i	−0.808350	−	0.588702i	$-0.799639\pi$
$277$		−	19.5959i		−	1.17740i	0.808350	−	0.588702i	$-0.200361\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	6.00000			0.357930			0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000			0.357930			0.178965	+	0.983855i	$0.442725\pi$
$282$		0			0
$283$			4.00000i			0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$			4.00000i			0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	4.89898	+	12.0000i	0.289178	+	0.708338i
$288$		0			0
$289$	−7.00000			−0.411765
$290$		0			0
$291$		0			0
$292$		0			0
$293$	24.0000			1.40209			0.701047	−	0.713115i	$-0.252716\pi$
$293$	24.0000			1.40209			0.701047	+	0.713115i	$0.252716\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	24.0000i		−	1.38796i
$300$		0			0
$301$	−24.0000	+	9.79796i	−1.38334	+	0.564745i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	28.0000i		−	1.59804i	−0.601302	−	0.799022i	$-0.705351\pi$
$307$		−	28.0000i		−	1.59804i	0.601302	−	0.799022i	$-0.294649\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		−	9.79796i		−	0.553813i	−0.960897	−	0.276907i	$-0.910691\pi$
$313$		−	9.79796i		−	0.553813i	0.960897	−	0.276907i	$-0.0893093\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			4.89898i			0.275154i	0.990491	+	0.137577i	$0.0439315\pi$
$317$			4.89898i			0.275154i	−0.990491	+	0.137577i	$0.956069\pi$
$318$		0			0
$319$			24.0000i			1.34374i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	19.5959			1.09035
$324$		0			0
$325$	20.0000			1.10940
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−24.0000	+	9.79796i	−1.32316	+	0.540179i
$330$		0			0
$331$	9.79796			0.538545			0.269272	−	0.963064i	$-0.413217\pi$
$331$	9.79796			0.538545			0.269272	+	0.963064i	$0.413217\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	22.0000			1.19842			0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000			1.19842			0.599208	+	0.800593i	$0.295482\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−24.0000			−1.29967
$342$		0			0
$343$	−7.34847	+	17.0000i	−0.396780	+	0.917914i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−14.6969			−0.788973			−0.394486	−	0.918902i	$-0.629078\pi$
$347$	−14.6969			−0.788973			−0.394486	+	0.918902i	$0.629078\pi$
$348$		0			0
$349$	20.0000			1.07058			0.535288	−	0.844670i	$-0.320203\pi$
$349$	20.0000			1.07058			0.535288	+	0.844670i	$0.320203\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	34.2929i		−	1.82522i	−0.408826	−	0.912612i	$-0.634062\pi$
$353$		−	34.2929i		−	1.82522i	0.408826	−	0.912612i	$-0.365938\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$			6.00000i			0.316668i	0.987386	+	0.158334i	$0.0506123\pi$
$359$			6.00000i			0.316668i	−0.987386	+	0.158334i	$0.949388\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	14.6969			0.767174			0.383587	−	0.923505i	$-0.374689\pi$
$367$	14.6969			0.767174			0.383587	+	0.923505i	$0.374689\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	4.89898	+	12.0000i	0.254342	+	0.623009i
$372$		0			0
$373$		−	19.5959i		−	1.01464i	−0.861758	−	0.507319i	$-0.830637\pi$
$373$		−	19.5959i		−	1.01464i	0.861758	−	0.507319i	$-0.169363\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			19.5959i			1.00924i
$378$		0			0
$379$	−9.79796			−0.503287			−0.251644	−	0.967820i	$-0.580971\pi$
$379$	−9.79796			−0.503287			−0.251644	+	0.967820i	$0.580971\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	4.89898i		−	0.248388i	−0.992258	−	0.124194i	$-0.960365\pi$
$389$		−	4.89898i		−	0.248388i	0.992258	−	0.124194i	$-0.0396345\pi$
$390$		0			0
$391$	29.3939			1.48651
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	20.0000			1.00377			0.501886	−	0.864934i	$-0.332640\pi$
$397$	20.0000			1.00377			0.501886	+	0.864934i	$0.332640\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	18.0000			0.898877			0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000			0.898877			0.449439	+	0.893311i	$0.351624\pi$
$402$		0			0
$403$	−19.5959			−0.976142
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	48.0000i		−	2.37927i
$408$		0			0
$409$			9.79796i			0.484478i	0.970217	+	0.242239i	$0.0778818\pi$
$409$			9.79796i			0.484478i	−0.970217	+	0.242239i	$0.922118\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	12.0000	+	29.3939i	0.590481	+	1.44638i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$			12.0000i			0.586238i	0.956076	+	0.293119i	$0.0946933\pi$
$419$			12.0000i			0.586238i	−0.956076	+	0.293119i	$0.905307\pi$
$420$		0			0
$421$			39.1918i			1.91009i	0.296456	+	0.955047i	$0.404195\pi$
$421$			39.1918i			1.91009i	−0.296456	+	0.955047i	$0.595805\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			24.4949i			1.18818i
$426$		0			0
$427$	−9.79796	+	4.00000i	−0.474156	+	0.193574i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			30.0000i			1.44505i	0.691345	+	0.722525i	$0.257018\pi$
$431$			30.0000i			1.44505i	−0.691345	+	0.722525i	$0.742982\pi$
$432$		0			0
$433$			19.5959i			0.941720i	0.882208	+	0.470860i	$0.156056\pi$
$433$			19.5959i			0.941720i	−0.882208	+	0.470860i	$0.843944\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−24.0000			−1.14808
$438$		0			0
$439$	−14.6969			−0.701447			−0.350723	−	0.936479i	$-0.614064\pi$
$439$	−14.6969			−0.701447			−0.350723	+	0.936479i	$0.614064\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	14.6969			0.698273			0.349136	−	0.937072i	$-0.386475\pi$
$443$	14.6969			0.698273			0.349136	+	0.937072i	$0.386475\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	30.0000			1.41579			0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000			1.41579			0.707894	+	0.706319i	$0.249646\pi$
$450$		0			0
$451$			24.0000i			1.13012i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−10.0000			−0.467780			−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000			−0.467780			−0.233890	+	0.972263i	$0.575146\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	24.0000			1.11779			0.558896	−	0.829238i	$-0.311225\pi$
$461$	24.0000			1.11779			0.558896	+	0.829238i	$0.311225\pi$
$462$		0			0
$463$		−	14.0000i		−	0.650635i	−0.945605	−	0.325318i	$-0.894529\pi$
$463$		−	14.0000i		−	0.650635i	0.945605	−	0.325318i	$-0.105471\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	36.0000i		−	1.66588i	−0.553362	−	0.832941i	$-0.686655\pi$
$467$		−	36.0000i		−	1.66588i	0.553362	−	0.832941i	$-0.313345\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−48.0000			−2.20704
$474$		0			0
$475$		−	20.0000i		−	0.917663i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−29.3939			−1.34304			−0.671520	−	0.740986i	$-0.734358\pi$
$479$	−29.3939			−1.34304			−0.671520	+	0.740986i	$0.734358\pi$
$480$		0			0
$481$		−	39.1918i		−	1.78699i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	34.0000i		−	1.54069i	−0.637629	−	0.770344i	$-0.720085\pi$
$487$		−	34.0000i		−	1.54069i	0.637629	−	0.770344i	$-0.279915\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−24.4949			−1.10544			−0.552720	−	0.833367i	$-0.686410\pi$
$491$	−24.4949			−1.10544			−0.552720	+	0.833367i	$0.686410\pi$
$492$		0			0
$493$	−24.0000			−1.08091
$494$		0			0
$495$		0			0
$496$		0			0
$497$	6.00000	+	14.6969i	0.269137	+	0.659248i
$498$		0			0
$499$	29.3939			1.31585			0.657925	−	0.753083i	$-0.271434\pi$
$499$	29.3939			1.31585			0.657925	+	0.753083i	$0.271434\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−9.79796			−0.436869			−0.218435	−	0.975852i	$-0.570095\pi$
$503$	−9.79796			−0.436869			−0.218435	+	0.975852i	$0.570095\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−24.0000			−1.06378			−0.531891	−	0.846813i	$-0.678518\pi$
$509$	−24.0000			−1.06378			−0.531891	+	0.846813i	$0.678518\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−48.0000			−2.11104
$518$		0			0
$519$		0			0
$520$		0			0
$521$			34.2929i			1.50240i	0.660076	+	0.751199i	$0.270524\pi$
$521$			34.2929i			1.50240i	−0.660076	+	0.751199i	$0.729476\pi$
$522$		0			0
$523$			28.0000i			1.22435i	0.790721	+	0.612177i	$0.209706\pi$
$523$			28.0000i			1.22435i	−0.790721	+	0.612177i	$0.790294\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	24.0000i		−	1.04546i
$528$		0			0
$529$	−13.0000			−0.565217
$530$		0			0
$531$		0			0
$532$		0			0
$533$			19.5959i			0.848793i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−24.4949	+	24.0000i	−1.05507	+	1.03375i
$540$		0			0
$541$		−	19.5959i		−	0.842494i	−0.906946	−	0.421247i	$-0.861592\pi$
$541$		−	19.5959i		−	0.842494i	0.906946	−	0.421247i	$-0.138408\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	19.5959			0.834814
$552$		0			0
$553$	−2.00000	−	4.89898i	−0.0850487	−	0.208326i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	44.0908i		−	1.86819i	−0.357027	−	0.934094i	$-0.616210\pi$
$557$		−	44.0908i		−	1.86819i	0.357027	−	0.934094i	$-0.383790\pi$
$558$		0			0
$559$	−39.1918			−1.65764
$560$		0			0
$561$		0			0
$562$		0			0
$563$			12.0000i			0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$			12.0000i			0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−6.00000			−0.251533			−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000			−0.251533			−0.125767	+	0.992060i	$0.540139\pi$
$570$		0			0
$571$	19.5959			0.820064			0.410032	−	0.912071i	$-0.365518\pi$
$571$	19.5959			0.820064			0.410032	+	0.912071i	$0.365518\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	30.0000i		−	1.25109i
$576$		0			0
$577$			29.3939i			1.22368i	0.790980	+	0.611842i	$0.209571\pi$
$577$			29.3939i			1.22368i	−0.790980	+	0.611842i	$0.790429\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−12.0000	−	29.3939i	−0.497844	−	1.21946i
$582$		0			0
$583$			24.0000i			0.993978i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	12.0000i		−	0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$		−	12.0000i		−	0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$		0			0
$589$			19.5959i			0.807436i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	14.6969i		−	0.603531i	−0.953382	−	0.301765i	$-0.902424\pi$
$593$		−	14.6969i		−	0.603531i	0.953382	−	0.301765i	$-0.0975760\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	6.00000i		−	0.245153i	−0.992459	−	0.122577i	$-0.960884\pi$
$599$		−	6.00000i		−	0.245153i	0.992459	−	0.122577i	$-0.0391157\pi$
$600$		0			0
$601$			9.79796i			0.399667i	0.979830	+	0.199834i	$0.0640401\pi$
$601$			9.79796i			0.399667i	−0.979830	+	0.199834i	$0.935960\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	24.4949			0.994217			0.497109	−	0.867688i	$-0.334395\pi$
$607$	24.4949			0.994217			0.497109	+	0.867688i	$0.334395\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−39.1918			−1.58553
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	42.0000			1.69086			0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000			1.69086			0.845428	+	0.534089i	$0.179345\pi$
$618$		0			0
$619$		−	4.00000i		−	0.160774i	−0.996764	−	0.0803868i	$-0.974384\pi$
$619$		−	4.00000i		−	0.160774i	0.996764	−	0.0803868i	$-0.0256155\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	4.89898	+	12.0000i	0.196273	+	0.480770i
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	48.0000			1.91389
$630$		0			0
$631$		−	38.0000i		−	1.51276i	−0.654135	−	0.756378i	$-0.726967\pi$
$631$		−	38.0000i		−	1.51276i	0.654135	−	0.756378i	$-0.273033\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−20.0000	+	19.5959i	−0.792429	+	0.776419i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−30.0000			−1.18493			−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000			−1.18493			−0.592464	+	0.805597i	$0.701845\pi$
$642$		0			0
$643$			4.00000i			0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$			4.00000i			0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	9.79796			0.385198			0.192599	−	0.981278i	$-0.438308\pi$
$647$	9.79796			0.385198			0.192599	+	0.981278i	$0.438308\pi$
$648$		0			0
$649$			58.7878i			2.30762i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	44.0908i		−	1.72541i	−0.505710	−	0.862703i	$-0.668769\pi$
$653$		−	44.0908i		−	1.72541i	0.505710	−	0.862703i	$-0.331231\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−24.4949			−0.954186			−0.477093	−	0.878853i	$-0.658309\pi$
$659$	−24.4949			−0.954186			−0.477093	+	0.878853i	$0.658309\pi$
$660$		0			0
$661$	20.0000			0.777910			0.388955	−	0.921257i	$-0.372836\pi$
$661$	20.0000			0.777910			0.388955	+	0.921257i	$0.372836\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	29.3939			1.13814
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−19.5959			−0.756492
$672$		0			0
$673$	26.0000			1.00223			0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000			1.00223			0.501113	+	0.865382i	$0.332924\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	24.0000			0.922395			0.461197	−	0.887298i	$-0.347420\pi$
$677$	24.0000			0.922395			0.461197	+	0.887298i	$0.347420\pi$
$678$		0			0
$679$	9.79796	+	24.0000i	0.376011	+	0.921035i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	34.2929			1.31218			0.656090	−	0.754683i	$-0.272209\pi$
$683$	34.2929			1.31218			0.656090	+	0.754683i	$0.272209\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$			19.5959i			0.746545i
$690$		0			0
$691$			44.0000i			1.67384i	0.547326	+	0.836919i	$0.315646\pi$
$691$			44.0000i			1.67384i	−0.547326	+	0.836919i	$0.684354\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−24.0000			−0.909065
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	14.6969i		−	0.555096i	−0.960712	−	0.277548i	$-0.910478\pi$
$701$		−	14.6969i		−	0.555096i	0.960712	−	0.277548i	$-0.0895217\pi$
$702$		0			0
$703$	−39.1918			−1.47815
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$			9.79796i			0.367970i	0.982929	+	0.183985i	$0.0588998\pi$
$709$			9.79796i			0.367970i	−0.982929	+	0.183985i	$0.941100\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			29.3939i			1.10081i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−48.9898			−1.82701			−0.913506	−	0.406826i	$-0.866635\pi$
$719$	−48.9898			−1.82701			−0.913506	+	0.406826i	$0.866635\pi$
$720$		0			0
$721$	−36.0000	+	14.6969i	−1.34071	+	0.547343i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			24.4949i			0.909718i
$726$		0			0
$727$	−14.6969			−0.545079			−0.272540	−	0.962145i	$-0.587864\pi$
$727$	−14.6969			−0.545079			−0.272540	+	0.962145i	$0.587864\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	48.0000i		−	1.77534i
$732$		0			0
$733$	44.0000			1.62518			0.812589	−	0.582838i	$-0.198058\pi$
$733$	44.0000			1.62518			0.812589	+	0.582838i	$0.198058\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	39.1918			1.44169			0.720847	−	0.693094i	$-0.243753\pi$
$739$	39.1918			1.44169			0.720847	+	0.693094i	$0.243753\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	18.0000i		−	0.660356i	−0.943919	−	0.330178i	$-0.892891\pi$
$743$		−	18.0000i		−	0.660356i	0.943919	−	0.330178i	$-0.107109\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	36.0000	−	14.6969i	1.31541	−	0.537014i
$750$		0			0
$751$			22.0000i			0.802791i	0.915905	+	0.401396i	$0.131475\pi$
$751$			22.0000i			0.802791i	−0.915905	+	0.401396i	$0.868525\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$			29.3939i			1.06834i	0.845378	+	0.534169i	$0.179376\pi$
$757$			29.3939i			1.06834i	−0.845378	+	0.534169i	$0.820624\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			44.0908i			1.59829i	0.601138	+	0.799145i	$0.294714\pi$
$761$			44.0908i			1.59829i	−0.601138	+	0.799145i	$0.705286\pi$
$762$		0			0
$763$	−9.79796	−	24.0000i	−0.354710	−	0.868858i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			48.0000i			1.73318i
$768$		0			0
$769$		−	48.9898i		−	1.76662i	−0.468792	−	0.883309i	$-0.655311\pi$
$769$		−	48.9898i		−	1.76662i	0.468792	−	0.883309i	$-0.344689\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−24.0000			−0.863220			−0.431610	−	0.902060i	$-0.642054\pi$
$773$	−24.0000			−0.863220			−0.431610	+	0.902060i	$0.642054\pi$
$774$		0			0
$775$	−24.4949			−0.879883
$776$		0			0
$777$		0			0
$778$		0			0
$779$	19.5959			0.702097
$780$		0			0
$781$			29.3939i			1.05180i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	4.00000i		−	0.142585i	−0.997455	−	0.0712923i	$-0.977288\pi$
$787$		−	4.00000i		−	0.142585i	0.997455	−	0.0712923i	$-0.0227123\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−44.0908	+	18.0000i	−1.56769	+	0.640006i
$792$		0			0
$793$	−16.0000			−0.568177
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−48.0000			−1.70025			−0.850124	−	0.526583i	$-0.823473\pi$
$797$	−48.0000			−1.70025			−0.850124	+	0.526583i	$0.823473\pi$
$798$		0			0
$799$		−	48.0000i		−	1.69812i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−54.0000			−1.89854			−0.949269	−	0.314464i	$-0.898175\pi$
$809$	−54.0000			−1.89854			−0.949269	+	0.314464i	$0.898175\pi$
$810$		0			0
$811$		−	20.0000i		−	0.702295i	−0.936320	−	0.351147i	$-0.885792\pi$
$811$		−	20.0000i		−	0.702295i	0.936320	−	0.351147i	$-0.114208\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$			39.1918i			1.37115i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	24.4949i		−	0.854878i	−0.904044	−	0.427439i	$-0.859416\pi$
$821$		−	24.4949i		−	0.854878i	0.904044	−	0.427439i	$-0.140584\pi$
$822$		0			0
$823$			10.0000i			0.348578i	0.984695	+	0.174289i	$0.0557627\pi$
$823$			10.0000i			0.348578i	−0.984695	+	0.174289i	$0.944237\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	4.89898			0.170354			0.0851771	−	0.996366i	$-0.472854\pi$
$827$	4.89898			0.170354			0.0851771	+	0.996366i	$0.472854\pi$
$828$		0			0
$829$	−20.0000			−0.694629			−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000			−0.694629			−0.347314	+	0.937749i	$0.612906\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−24.0000	−	24.4949i	−0.831551	−	0.848698i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−29.3939			−1.01479			−0.507395	−	0.861714i	$-0.669391\pi$
$839$	−29.3939			−1.01479			−0.507395	+	0.861714i	$0.669391\pi$
$840$		0			0
$841$	5.00000			0.172414
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−31.8434	+	13.0000i	−1.09415	+	0.446685i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−58.7878			−2.01522
$852$		0			0
$853$	4.00000			0.136957			0.0684787	−	0.997653i	$-0.478185\pi$
$853$	4.00000			0.136957			0.0684787	+	0.997653i	$0.478185\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			24.4949i			0.836730i	0.908279	+	0.418365i	$0.137397\pi$
$857$			24.4949i			0.836730i	−0.908279	+	0.418365i	$0.862603\pi$
$858$		0			0
$859$		−	28.0000i		−	0.955348i	−0.878537	−	0.477674i	$-0.841480\pi$
$859$		−	28.0000i		−	0.955348i	0.878537	−	0.477674i	$-0.158520\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	54.0000i		−	1.83818i	−0.394046	−	0.919091i	$-0.628925\pi$
$863$		−	54.0000i		−	1.83818i	0.394046	−	0.919091i	$-0.371075\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	9.79796i		−	0.332373i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$			19.5959i			0.661707i	0.943682	+	0.330854i	$0.107337\pi$
$877$			19.5959i			0.661707i	−0.943682	+	0.330854i	$0.892663\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	24.4949i		−	0.825254i	−0.910900	−	0.412627i	$-0.864611\pi$
$881$		−	24.4949i		−	0.825254i	0.910900	−	0.412627i	$-0.135389\pi$
$882$		0			0
$883$	9.79796			0.329728			0.164864	−	0.986316i	$-0.447282\pi$
$883$	9.79796			0.329728			0.164864	+	0.986316i	$0.447282\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−39.1918			−1.31593			−0.657967	−	0.753047i	$-0.728583\pi$
$887$	−39.1918			−1.31593			−0.657967	+	0.753047i	$0.728583\pi$
$888$		0			0
$889$	2.00000	+	4.89898i	0.0670778	+	0.164306i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			39.1918i			1.31150i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	24.0000i		−	0.800445i
$900$		0			0
$901$	−24.0000			−0.799556
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	9.79796			0.325336			0.162668	−	0.986681i	$-0.447990\pi$
$907$	9.79796			0.325336			0.162668	+	0.986681i	$0.447990\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	42.0000i		−	1.39152i	−0.718273	−	0.695761i	$-0.755067\pi$
$911$		−	42.0000i		−	1.39152i	0.718273	−	0.695761i	$-0.244933\pi$
$912$		0			0
$913$		−	58.7878i		−	1.94559i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	12.0000	+	29.3939i	0.396275	+	0.970671i
$918$		0			0
$919$		−	46.0000i		−	1.51740i	−0.651440	−	0.758700i	$-0.725835\pi$
$919$		−	46.0000i		−	1.51740i	0.651440	−	0.758700i	$-0.274165\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			24.0000i			0.789970i
$924$		0			0
$925$		−	48.9898i		−	1.61077i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			24.4949i			0.803652i	0.915716	+	0.401826i	$0.131624\pi$
$929$			24.4949i			0.803652i	−0.915716	+	0.401826i	$0.868376\pi$
$930$		0			0
$931$	19.5959	+	20.0000i	0.642230	+	0.655474i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			39.1918i			1.28034i	0.768233	+	0.640171i	$0.221136\pi$
$937$			39.1918i			1.28034i	−0.768233	+	0.640171i	$0.778864\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	48.0000			1.56476			0.782378	−	0.622804i	$-0.214007\pi$
$941$	48.0000			1.56476			0.782378	+	0.622804i	$0.214007\pi$
$942$		0			0
$943$	29.3939			0.957196
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−4.89898			−0.159195			−0.0795977	−	0.996827i	$-0.525364\pi$
$947$	−4.89898			−0.159195			−0.0795977	+	0.996827i	$0.525364\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	6.00000			0.194359			0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000			0.194359			0.0971795	+	0.995267i	$0.469018\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−14.6969	+	6.00000i	−0.474589	+	0.193750i
$960$		0			0
$961$	−7.00000			−0.225806
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$			38.0000i			1.22200i	0.791632	+	0.610999i	$0.209232\pi$
$967$			38.0000i			1.22200i	−0.791632	+	0.610999i	$0.790768\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			12.0000i			0.385098i	0.981287	+	0.192549i	$0.0616755\pi$
$971$			12.0000i			0.385098i	−0.981287	+	0.192549i	$0.938325\pi$
$972$		0			0
$973$	4.00000	+	9.79796i	0.128234	+	0.314108i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−18.0000			−0.575871			−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000			−0.575871			−0.287936	+	0.957650i	$0.592969\pi$
$978$		0			0
$979$			24.0000i			0.767043i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−39.1918			−1.25003			−0.625013	−	0.780615i	$-0.714906\pi$
$983$	−39.1918			−1.25003			−0.625013	+	0.780615i	$0.714906\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$			58.7878i			1.86934i
$990$		0			0
$991$			10.0000i			0.317660i	0.987306	+	0.158830i	$0.0507723\pi$
$991$			10.0000i			0.317660i	−0.987306	+	0.158830i	$0.949228\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	4.00000			0.126681			0.0633406	−	0.997992i	$-0.479825\pi$
$997$	4.00000			0.126681			0.0633406	+	0.997992i	$0.479825\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.p.a.1567.2		4
3.2	odd	2		1344.2.p.a.223.1	✓	4
4.3	odd	2	inner	4032.2.p.a.1567.3		4
7.6	odd	2		4032.2.p.d.1567.4		4
8.3	odd	2		4032.2.p.d.1567.3		4
8.5	even	2		4032.2.p.d.1567.2		4
12.11	even	2		1344.2.p.a.223.4	yes	4
21.20	even	2		1344.2.p.b.223.4	yes	4
24.5	odd	2		1344.2.p.b.223.3	yes	4
24.11	even	2		1344.2.p.b.223.2	yes	4
28.27	even	2		4032.2.p.d.1567.1		4
56.13	odd	2	inner	4032.2.p.a.1567.4		4
56.27	even	2	inner	4032.2.p.a.1567.1		4
84.83	odd	2		1344.2.p.b.223.1	yes	4
168.83	odd	2		1344.2.p.a.223.3	yes	4
168.125	even	2		1344.2.p.a.223.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1344.2.p.a.223.1	✓	4	3.2	odd	2
1344.2.p.a.223.2	yes	4	168.125	even	2
1344.2.p.a.223.3	yes	4	168.83	odd	2
1344.2.p.a.223.4	yes	4	12.11	even	2
1344.2.p.b.223.1	yes	4	84.83	odd	2
1344.2.p.b.223.2	yes	4	24.11	even	2
1344.2.p.b.223.3	yes	4	24.5	odd	2
1344.2.p.b.223.4	yes	4	21.20	even	2
4032.2.p.a.1567.1		4	56.27	even	2	inner
4032.2.p.a.1567.2		4	1.1	even	1	trivial
4032.2.p.a.1567.3		4	4.3	odd	2	inner
4032.2.p.a.1567.4		4	56.13	odd	2	inner
4032.2.p.d.1567.1		4	28.27	even	2
4032.2.p.d.1567.2		4	8.5	even	2
4032.2.p.d.1567.3		4	8.3	odd	2
4032.2.p.d.1567.4		4	7.6	odd	2

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:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4032.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.44949 + 1.00000i) q^{7} +O(q^{10})\)	\(q+(-2.44949 + 1.00000i) q^{7} -4.89898 q^{11} -4.00000 q^{13} -4.89898i q^{17} +4.00000i q^{19} +6.00000i q^{23} -5.00000 q^{25} -4.89898i q^{29} +4.89898 q^{31} +9.79796i q^{37} -4.89898i q^{41} +9.79796 q^{43} +9.79796 q^{47} +(5.00000 - 4.89898i) q^{49} -4.89898i q^{53} -12.0000i q^{59} +4.00000 q^{61} -6.00000i q^{71} +(12.0000 - 4.89898i) q^{77} +2.00000i q^{79} +12.0000i q^{83} -4.89898i q^{89} +(9.79796 - 4.00000i) q^{91} -9.79796i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 16 q^{13} - 20 q^{25} + 20 q^{49} + 16 q^{61} + 48 q^{77}+O(q^{100}) \) 4 * q - 16 * q^13 - 20 * q^25 + 20 * q^49 + 16 * q^61 + 48 * q^77

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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