LMFDB - Embedded newform 960.3.c.i.449.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,3,Mod(449,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("960.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	960.c (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:	$26.1581053786$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 480)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			449.4
Root			$-1.14412 - 1.14412i$ of defining polynomial
Character	$\chi$	$=$	960.449
Dual form			960.3.c.i.449.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.166925 + 2.99535i) q^{3} +5.00000i q^{5} -12.3153i q^{7} +(-8.94427 - 1.00000i) q^{9} +O(q^{10})$	$q+(-0.166925 + 2.99535i) q^{3} +5.00000i q^{5} -12.3153i q^{7} +(-8.94427 - 1.00000i) q^{9} +(-14.9768 - 0.834626i) q^{15} +(36.8885 + 2.05573i) q^{21} +40.5995 q^{23} -25.0000 q^{25} +(4.48838 - 26.6243i) q^{27} -53.6656i q^{29} +61.5763 q^{35} -62.0000i q^{41} +25.2796i q^{43} +(5.00000 - 44.7214i) q^{45} +69.2363 q^{47} -102.666 q^{49} +107.331 q^{61} +(-12.3153 + 110.151i) q^{63} +34.5902i q^{67} +(-6.77709 + 121.610i) q^{69} +(4.17313 - 74.8838i) q^{75} +(79.0000 + 17.8885i) q^{81} +158.095 q^{83} +(160.747 + 8.95815i) q^{87} +107.331i q^{89} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 152 q^{21} - 200 q^{25} + 40 q^{45} - 392 q^{49} + 232 q^{69} + 632 q^{81}+O(q^{100}) $ 8 * q + 152 * q^21 - 200 * q^25 + 40 * q^45 - 392 * q^49 + 232 * q^69 + 632 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$.

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$	−0.166925	+	2.99535i	−0.0556418	+	0.998451i
$3$	−0.166925	+	2.99535i	−0.0556418	+	0.998451i
$4$		0			0
$5$			5.00000i			1.00000i
$5$			5.00000i			1.00000i
$6$		0			0
$7$		−	12.3153i		−	1.75932i	−0.475600	−	0.879661i	$-0.657769\pi$
$7$		−	12.3153i		−	1.75932i	0.475600	−	0.879661i	$-0.342231\pi$
$8$		0			0
$9$	−8.94427	−	1.00000i	−0.993808	−	0.111111i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$	−14.9768	−	0.834626i	−0.998451	−	0.0556418i
$16$		0			0
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$	36.8885	+	2.05573i	1.75660	+	0.0978918i
$22$		0			0
$23$	40.5995			1.76520			0.882599	−	0.470128i	$-0.155792\pi$
$23$	40.5995			1.76520			0.882599	+	0.470128i	$0.155792\pi$
$24$		0			0
$25$	−25.0000			−1.00000
$26$		0			0
$27$	4.48838	−	26.6243i	0.166236	−	0.986086i
$28$		0			0
$29$		−	53.6656i		−	1.85054i	−0.379310	−	0.925270i	$-0.623839\pi$
$29$		−	53.6656i		−	1.85054i	0.379310	−	0.925270i	$-0.376161\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	61.5763			1.75932
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	62.0000i		−	1.51220i	−0.654459	−	0.756098i	$-0.727104\pi$
$41$		−	62.0000i		−	1.51220i	0.654459	−	0.756098i	$-0.272896\pi$
$42$		0			0
$43$			25.2796i			0.587898i	0.955821	+	0.293949i	$0.0949696\pi$
$43$			25.2796i			0.587898i	−0.955821	+	0.293949i	$0.905030\pi$
$44$		0			0
$45$	5.00000	−	44.7214i	0.111111	−	0.993808i
$46$		0			0
$47$	69.2363			1.47311			0.736556	−	0.676377i	$-0.236451\pi$
$47$	69.2363			1.47311			0.736556	+	0.676377i	$0.236451\pi$
$48$		0			0
$49$	−102.666			−2.09522
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$	107.331			1.75953			0.879764	−	0.475410i	$-0.157700\pi$
$61$	107.331			1.75953			0.879764	+	0.475410i	$0.157700\pi$
$62$		0			0
$63$	−12.3153	+	110.151i	−0.195480	+	1.74843i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			34.5902i			0.516272i	0.966109	+	0.258136i	$0.0831083\pi$
$67$			34.5902i			0.516272i	−0.966109	+	0.258136i	$0.916892\pi$
$68$		0			0
$69$	−6.77709	+	121.610i	−0.0982187	+	1.76246i
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	4.17313	−	74.8838i	0.0556418	−	0.998451i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	79.0000	+	17.8885i	0.975309	+	0.220846i
$82$		0			0
$83$	158.095			1.90476			0.952381	−	0.304910i	$-0.0986264\pi$
$83$	158.095			1.90476			0.952381	+	0.304910i	$0.0986264\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	160.747	+	8.95815i	1.84767	+	0.102967i
$88$		0			0
$89$			107.331i			1.20597i	0.797753	+	0.602985i	$0.206022\pi$
$89$			107.331i			1.20597i	−0.797753	+	0.602985i	$0.793978\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			160.997i			1.59403i	0.603960	+	0.797014i	$0.293589\pi$
$101$			160.997i			1.59403i	−0.603960	+	0.797014i	$0.706411\pi$
$102$		0			0
$103$		−	173.415i		−	1.68364i	−0.539756	−	0.841821i	$-0.681484\pi$
$103$		−	173.415i		−	1.68364i	0.539756	−	0.841821i	$-0.318516\pi$
$104$		0			0
$105$	−10.2786	+	184.443i	−0.0978918	+	1.75660i
$106$		0			0
$107$	−35.6476			−0.333155			−0.166578	−	0.986028i	$-0.553272\pi$
$107$	−35.6476			−0.333155			−0.166578	+	0.986028i	$0.553272\pi$
$108$		0			0
$109$	−214.663			−1.96938			−0.984690	−	0.174312i	$-0.944230\pi$
$109$	−214.663			−1.96938			−0.984690	+	0.174312i	$0.944230\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$			202.998i			1.76520i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	121.000			1.00000
$122$		0			0
$123$	185.712	+	10.3494i	1.50985	+	0.0841412i
$124$		0			0
$125$		−	125.000i		−	1.00000i
$126$		0			0
$127$		−	233.285i		−	1.83689i	−0.395549	−	0.918445i	$-0.629446\pi$
$127$		−	233.285i		−	1.83689i	0.395549	−	0.918445i	$-0.370554\pi$
$128$		0			0
$129$	−75.7214	−	4.21981i	−0.586987	−	0.0327117i
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	133.122	+	22.4419i	0.986086	+	0.166236i
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	−11.5573	+	207.387i	−0.0819665	+	1.47083i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	268.328			1.85054
$146$		0			0
$147$	17.1375	−	307.520i	0.116582	−	2.09197i
$148$		0			0
$149$		−	278.000i		−	1.86577i	−0.360172	−	0.932886i	$-0.617282\pi$
$149$		−	278.000i		−	1.86577i	0.360172	−	0.932886i	$-0.382718\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	499.994i		−	3.10555i
$162$		0			0
$163$		−	83.2580i		−	0.510785i	−0.966837	−	0.255393i	$-0.917795\pi$
$163$		−	83.2580i		−	0.510785i	0.966837	−	0.255393i	$-0.0822047\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	11.2579			0.0674125			0.0337063	−	0.999432i	$-0.489269\pi$
$167$	11.2579			0.0674125			0.0337063	+	0.999432i	$0.489269\pi$
$168$		0			0
$169$	169.000			1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$			307.882i			1.75932i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$	−358.000			−1.97790			−0.988950	−	0.148248i	$-0.952637\pi$
$181$	−358.000			−1.97790			−0.988950	+	0.148248i	$0.952637\pi$
$182$		0			0
$183$	−17.9163	+	321.495i	−0.0979033	+	1.75680i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−327.885	−	55.2755i	−1.73484	−	0.292463i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$	−103.610	−	5.77398i	−0.515472	−	0.0287263i
$202$		0			0
$203$	−660.906			−3.25570
$204$		0			0
$205$	310.000			1.51220
$206$		0			0
$207$	−363.133	−	40.5995i	−1.75427	−	0.196133i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−126.398			−0.587898
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	241.891i		−	1.08471i	−0.840149	−	0.542356i	$-0.817532\pi$
$223$		−	241.891i		−	1.08471i	0.840149	−	0.542356i	$-0.182468\pi$
$224$		0			0
$225$	223.607	+	25.0000i	0.993808	+	0.111111i
$226$		0			0
$227$	450.954			1.98658			0.993290	−	0.115653i	$-0.0368961\pi$
$227$	450.954			1.98658			0.993290	+	0.115653i	$0.0368961\pi$
$228$		0			0
$229$	262.000			1.14410			0.572052	−	0.820217i	$-0.306147\pi$
$229$	262.000			1.14410			0.572052	+	0.820217i	$0.306147\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$			346.181i			1.47311i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−375.659			−1.55875			−0.779376	−	0.626556i	$-0.784464\pi$
$241$	−375.659			−1.55875			−0.779376	+	0.626556i	$0.784464\pi$
$242$		0			0
$243$	−66.7696	+	233.647i	−0.274772	+	0.961509i
$244$		0			0
$245$		−	513.328i		−	2.09522i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−26.3901	+	473.551i	−0.105984	+	1.90181i
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−53.6656	+	480.000i	−0.205615	+	1.83908i
$262$		0			0
$263$	513.884			1.95393			0.976965	−	0.213398i	$-0.0684530\pi$
$263$	513.884			1.95393			0.976965	+	0.213398i	$0.0684530\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−321.495	−	17.9163i	−1.20410	−	0.0671022i
$268$		0			0
$269$			38.0000i			0.141264i	0.997502	+	0.0706320i	$0.0225016\pi$
$269$			38.0000i			0.141264i	−0.997502	+	0.0706320i	$0.977498\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	418.000i		−	1.48754i	−0.668433	−	0.743772i	$-0.733035\pi$
$281$		−	418.000i		−	1.48754i	0.668433	−	0.743772i	$-0.266965\pi$
$282$		0			0
$283$			108.593i			0.383722i	0.981422	+	0.191861i	$0.0614523\pi$
$283$			108.593i			0.383722i	−0.981422	+	0.191861i	$0.938548\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−763.546			−2.66044
$288$		0			0
$289$	−289.000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	311.325			1.03430
$302$		0			0
$303$	−482.242	−	26.8744i	−1.59156	−	0.0886946i
$304$		0			0
$305$			536.656i			1.75953i
$306$		0			0
$307$		−	525.791i		−	1.71267i	−0.516418	−	0.856337i	$-0.672735\pi$
$307$		−	525.791i		−	1.71267i	0.516418	−	0.856337i	$-0.327265\pi$
$308$		0			0
$309$	519.440	+	28.9474i	1.68103	+	0.0936808i
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$	−550.755	−	61.5763i	−1.74843	−	0.195480i
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	5.95048	−	106.777i	0.0185373	−	0.332639i
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$	35.8326	−	642.990i	0.109580	−	1.96633i
$328$		0			0
$329$		−	852.663i		−	2.59168i
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−172.951			−0.516272
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$			660.906i			1.92684i
$344$		0			0
$345$	−608.050	−	33.8854i	−1.76246	−	0.0982187i
$346$		0			0
$347$	−401.804			−1.15794			−0.578968	−	0.815350i	$-0.696545\pi$
$347$	−401.804			−1.15794			−0.578968	+	0.815350i	$0.696545\pi$
$348$		0			0
$349$	−22.0000			−0.0630372			−0.0315186	−	0.999503i	$-0.510034\pi$
$349$	−22.0000			−0.0630372			−0.0315186	+	0.999503i	$0.510034\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	−361.000			−1.00000
$362$		0			0
$363$	−20.1980	+	362.438i	−0.0556418	+	0.998451i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	597.327i		−	1.62759i	−0.581150	−	0.813797i	$-0.697397\pi$
$367$		−	597.327i		−	1.62759i	0.581150	−	0.813797i	$-0.302603\pi$
$368$		0			0
$369$	−62.0000	+	554.545i	−0.168022	+	1.50283i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$	374.419	+	20.8657i	0.998451	+	0.0556418i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$	698.771	+	38.9412i	1.83404	+	0.102208i
$382$		0			0
$383$	−571.862			−1.49311			−0.746556	−	0.665322i	$-0.768294\pi$
$383$	−571.862			−1.49311			−0.746556	+	0.665322i	$0.768294\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	25.2796	−	226.108i	0.0653220	−	0.584258i
$388$		0			0
$389$		−	202.000i		−	0.519280i	−0.965705	−	0.259640i	$-0.916396\pi$
$389$		−	202.000i		−	0.519280i	0.965705	−	0.259640i	$-0.0836039\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			643.988i			1.60595i	0.596010	+	0.802977i	$0.296752\pi$
$401$			643.988i			1.60595i	−0.596010	+	0.802977i	$0.703248\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−89.4427	+	395.000i	−0.220846	+	0.975309i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	160.997			0.393635			0.196818	−	0.980440i	$-0.436939\pi$
$409$	160.997			0.393635			0.196818	+	0.980440i	$0.436939\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$			790.476i			1.90476i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	−321.994			−0.764831			−0.382415	−	0.923990i	$-0.624908\pi$
$421$	−321.994			−0.764831			−0.382415	+	0.923990i	$0.624908\pi$
$422$		0			0
$423$	−619.268	−	69.2363i	−1.46399	−	0.163679i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	1321.81i		−	3.09558i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$	−44.7907	+	803.737i	−0.102967	+	1.84767i
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	918.269	+	102.666i	2.08224	+	0.232802i
$442$		0			0
$443$	−287.739			−0.649523			−0.324762	−	0.945796i	$-0.605284\pi$
$443$	−287.739			−0.649523			−0.324762	+	0.945796i	$0.605284\pi$
$444$		0			0
$445$	−536.656			−1.20597
$446$		0			0
$447$	832.708	+	46.4052i	1.86288	+	0.103815i
$448$		0			0
$449$			398.000i			0.886414i	0.896419	+	0.443207i	$0.146159\pi$
$449$			398.000i			0.886414i	−0.896419	+	0.443207i	$0.853841\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			375.659i			0.814879i	0.913232	+	0.407440i	$0.133578\pi$
$461$			375.659i			0.814879i	−0.913232	+	0.407440i	$0.866422\pi$
$462$		0			0
$463$			910.216i			1.96591i	0.183848	+	0.982955i	$0.441145\pi$
$463$			910.216i			1.96591i	−0.183848	+	0.982955i	$0.558855\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−566.910			−1.21394			−0.606970	−	0.794725i	$-0.707615\pi$
$467$	−566.910			−1.21394			−0.606970	+	0.794725i	$0.707615\pi$
$468$		0			0
$469$	425.988			0.908289
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	1497.66	+	83.4616i	3.10074	+	0.172798i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	255.431i		−	0.524499i	−0.965000	−	0.262249i	$-0.915536\pi$
$487$		−	255.431i		−	0.524499i	0.965000	−	0.262249i	$-0.0844643\pi$
$488$		0			0
$489$	249.387	+	13.8979i	0.509994	+	0.0284210i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$	−1.87923	+	33.7214i	−0.00375095	+	0.0673081i
$502$		0			0
$503$	941.802			1.87237			0.936185	−	0.351509i	$-0.114331\pi$
$503$	941.802			1.87237			0.936185	+	0.351509i	$0.114331\pi$
$504$		0			0
$505$	−804.984			−1.59403
$506$		0			0
$507$	−28.2104	+	506.215i	−0.0556418	+	0.998451i
$508$		0			0
$509$			268.328i			0.527167i	0.964637	+	0.263584i	$0.0849045\pi$
$509$			268.328i			0.527167i	−0.964637	+	0.263584i	$0.915095\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	867.076			1.68364
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	751.319i		−	1.44207i	−0.692898	−	0.721035i	$-0.743667\pi$
$521$		−	751.319i		−	1.44207i	0.692898	−	0.721035i	$-0.256333\pi$
$522$		0			0
$523$			614.521i			1.17499i	0.809227	+	0.587496i	$0.199886\pi$
$523$			614.521i			1.17499i	−0.809227	+	0.587496i	$0.800114\pi$
$524$		0			0
$525$	−922.214	−	51.3932i	−1.75660	−	0.0978918i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	1119.32			2.11592
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		−	178.238i		−	0.333155i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−362.000			−0.669131			−0.334566	−	0.942372i	$-0.608590\pi$
$541$	−362.000			−0.669131			−0.334566	+	0.942372i	$0.608590\pi$
$542$		0			0
$543$	59.7592	−	1072.34i	0.110054	−	1.97484i
$544$		0			0
$545$		−	1073.31i		−	1.96938i
$546$		0			0
$547$			662.149i			1.21051i	0.796032	+	0.605255i	$0.206929\pi$
$547$			662.149i			1.21051i	−0.796032	+	0.605255i	$0.793071\pi$
$548$		0			0
$549$	−960.000	−	107.331i	−1.74863	−	0.195503i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−842.222			−1.49595			−0.747977	−	0.663725i	$-0.768975\pi$
$563$	−842.222			−1.49595			−0.747977	+	0.663725i	$0.768975\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	220.302	−	972.906i	0.388540	−	1.71588i
$568$		0			0
$569$			158.000i			0.277680i	0.990315	+	0.138840i	$0.0443374\pi$
$569$			158.000i			0.277680i	−0.990315	+	0.138840i	$0.955663\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1014.99			−1.76520
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	1946.98i		−	3.35109i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	428.808			0.730507			0.365254	−	0.930908i	$-0.380982\pi$
$587$	428.808			0.730507			0.365254	+	0.930908i	$0.380982\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	1126.98			1.87517			0.937586	−	0.347754i	$-0.113055\pi$
$601$	1126.98			1.87517			0.937586	+	0.347754i	$0.113055\pi$
$602$		0			0
$603$	34.5902	−	309.384i	0.0573636	−	0.513075i
$604$		0			0
$605$			605.000i			1.00000i
$606$		0			0
$607$			159.875i			0.263386i	0.991291	+	0.131693i	$0.0420413\pi$
$607$			159.875i			0.263386i	−0.991291	+	0.131693i	$0.957959\pi$
$608$		0			0
$609$	110.322	−	1979.65i	0.181153	−	3.25065i
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$	−51.7468	+	928.559i	−0.0841412	+	1.50985i
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$	182.226	−	1080.93i	0.293440	−	1.74064i
$622$		0			0
$623$	1321.81			2.12169
$624$		0			0
$625$	625.000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1166.43			1.83689
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$			1138.00i			1.77535i	0.460470	+	0.887676i	$0.347681\pi$
$641$			1138.00i			1.77535i	−0.460470	+	0.887676i	$0.652319\pi$
$642$		0			0
$643$		−	1148.97i		−	1.78689i	−0.449168	−	0.893447i	$-0.648280\pi$
$643$		−	1148.97i		−	1.78689i	0.449168	−	0.893447i	$-0.351720\pi$
$644$		0			0
$645$	21.0990	−	378.607i	0.0327117	−	0.586987i
$646$		0			0
$647$	1292.64			1.99790			0.998948	−	0.0458655i	$-0.0146046\pi$
$647$	1292.64			1.99790			0.998948	+	0.0458655i	$0.0146046\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−1287.98			−1.94853			−0.974263	−	0.225416i	$-0.927626\pi$
$661$	−1287.98			−1.94853			−0.974263	+	0.225416i	$0.927626\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	2178.80i		−	3.26657i
$668$		0			0
$669$	724.548	+	40.3777i	1.08303	+	0.0603553i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	−112.209	+	665.608i	−0.166236	+	0.986086i
$676$		0			0
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−75.2755	+	1350.76i	−0.110537	+	1.98350i
$682$		0			0
$683$	489.124			0.716141			0.358070	−	0.933695i	$-0.383435\pi$
$683$	489.124			0.716141			0.358070	+	0.933695i	$0.383435\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−43.7344	+	784.782i	−0.0636600	+	1.14233i
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$			902.000i			1.28673i	0.765558	+	0.643367i	$0.222463\pi$
$701$			902.000i			1.28673i	−0.765558	+	0.643367i	$0.777537\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−1036.93	−	57.7864i	−1.47083	−	0.0819665i
$706$		0			0
$707$	1982.72			2.80441
$708$		0			0
$709$	698.000			0.984485			0.492243	−	0.870458i	$-0.336177\pi$
$709$	698.000			0.984485			0.492243	+	0.870458i	$0.336177\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$	−2135.65			−2.96207
$722$		0			0
$723$	62.7070	−	1125.23i	0.0867317	−	1.55634i
$724$		0			0
$725$			1341.64i			1.85054i
$726$		0			0
$727$		−	854.129i		−	1.17487i	−0.809272	−	0.587434i	$-0.800138\pi$
$727$		−	854.129i		−	1.17487i	0.809272	−	0.587434i	$-0.199862\pi$
$728$		0			0
$729$	−688.709	−	239.000i	−0.944731	−	0.327846i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$	1537.60	+	85.6874i	2.09197	+	0.116582i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	361.020			0.485894			0.242947	−	0.970040i	$-0.421886\pi$
$743$	361.020			0.485894			0.242947	+	0.970040i	$0.421886\pi$
$744$		0			0
$745$	1390.00			1.86577
$746$		0			0
$747$	−1414.05	−	158.095i	−1.89297	−	0.211640i
$748$		0			0
$749$			439.009i			0.586127i
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			1502.64i			1.97456i	0.159001	+	0.987278i	$0.449173\pi$
$761$			1502.64i			1.97456i	−0.159001	+	0.987278i	$0.550827\pi$
$762$		0			0
$763$			2643.62i			3.46478i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	1342.00			1.74512			0.872562	−	0.488504i	$-0.162457\pi$
$769$	1342.00			1.74512			0.872562	+	0.488504i	$0.162457\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−1428.81	−	240.872i	−1.82479	−	0.307627i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			1027.94i			1.30614i	0.757296	+	0.653072i	$0.226520\pi$
$787$			1027.94i			1.30614i	−0.757296	+	0.653072i	$0.773480\pi$
$788$		0			0
$789$	−85.7802	+	1539.26i	−0.108720	+	1.95090i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	107.331	−	960.000i	0.133997	−	1.19850i
$802$		0			0
$803$		0			0
$804$		0			0
$805$	2499.97			3.10555
$806$		0			0
$807$	−113.823	−	6.34316i	−0.141045	−	0.00786017i
$808$		0			0
$809$			965.981i			1.19404i	0.802225	+	0.597022i	$0.203649\pi$
$809$			965.981i			1.19404i	−0.802225	+	0.597022i	$0.796351\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	416.290			0.510785
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$			662.000i			0.806334i	0.915126	+	0.403167i	$0.132091\pi$
$821$			662.000i			0.806334i	−0.915126	+	0.403167i	$0.867909\pi$
$822$		0			0
$823$			1603.77i			1.94868i	0.225077	+	0.974341i	$0.427737\pi$
$823$			1603.77i			1.94868i	−0.225077	+	0.974341i	$0.572263\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	669.550			0.809613			0.404807	−	0.914402i	$-0.367339\pi$
$827$	669.550			0.809613			0.404807	+	0.914402i	$0.367339\pi$
$828$		0			0
$829$	751.319			0.906295			0.453148	−	0.891435i	$-0.350301\pi$
$829$	751.319			0.906295			0.453148	+	0.891435i	$0.350301\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$			56.2895i			0.0674125i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	−2039.00			−2.42449
$842$		0			0
$843$	1252.06	+	69.7748i	1.48524	+	0.0827696i
$844$		0			0
$845$			845.000i			1.00000i
$846$		0			0
$847$		−	1490.15i		−	1.75932i
$848$		0			0
$849$	−325.276	−	18.1270i	−0.383128	−	0.0213510i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$	127.455	−	2287.09i	0.148032	−	2.65632i
$862$		0			0
$863$	767.867			0.889764			0.444882	−	0.895589i	$-0.353246\pi$
$863$	767.867			0.889764			0.444882	+	0.895589i	$0.353246\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	48.2414	−	865.657i	0.0556418	−	0.998451i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1539.41			−1.75932
$876$		0			0
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			1618.00i			1.83655i	0.395944	+	0.918275i	$0.370417\pi$
$881$			1618.00i			1.83655i	−0.395944	+	0.918275i	$0.629583\pi$
$882$		0			0
$883$		−	38.9665i		−	0.0441296i	−0.999757	−	0.0220648i	$-0.992976\pi$
$883$		−	38.9665i		−	0.0441296i	0.999757	−	0.0220648i	$-0.00702402\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1734.69			−1.95568			−0.977841	−	0.209350i	$-0.932865\pi$
$887$	−1734.69			−1.95568			−0.977841	+	0.209350i	$0.932865\pi$
$888$		0			0
$889$	−2872.97			−3.23168
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−51.9680	+	932.528i	−0.0575504	+	1.03270i
$904$		0			0
$905$		−	1790.00i		−	1.97790i
$906$		0			0
$907$			1450.21i			1.59891i	0.600724	+	0.799456i	$0.294879\pi$
$907$			1450.21i			1.59891i	−0.600724	+	0.799456i	$0.705121\pi$
$908$		0			0
$909$	160.997	−	1440.00i	0.177114	−	1.58416i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	−1607.47	−	89.5815i	−1.75680	−	0.0979033i
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$	1574.93	+	87.7678i	1.71002	+	0.0952962i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−173.415	+	1551.07i	−0.187071	+	1.67322i
$928$		0			0
$929$			562.000i			0.604952i	0.953157	+	0.302476i	$0.0978131\pi$
$929$			562.000i			0.604952i	−0.953157	+	0.302476i	$0.902187\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	1878.30i		−	1.99606i	−0.0626993	−	0.998032i	$-0.519971\pi$
$941$		−	1878.30i		−	1.99606i	0.0626993	−	0.998032i	$-0.480029\pi$
$942$		0			0
$943$		−	2517.17i		−	2.66932i
$944$		0			0
$945$	276.378	−	1639.43i	0.292463	−	1.73484i
$946$		0			0
$947$	−1683.55			−1.77778			−0.888888	−	0.458124i	$-0.848522\pi$
$947$	−1683.55			−1.77778			−0.888888	+	0.458124i	$0.848522\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−961.000			−1.00000
$962$		0			0
$963$	318.842	+	35.6476i	0.331092	+	0.0370172i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			1184.08i			1.22449i	0.790668	+	0.612246i	$0.209734\pi$
$967$			1184.08i			1.22449i	−0.790668	+	0.612246i	$0.790266\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	1920.00	+	214.663i	1.95719	+	0.218820i
$982$		0			0
$983$	−1576.41			−1.60367			−0.801836	−	0.597544i	$-0.796143\pi$
$983$	−1576.41			−1.60367			−0.801836	+	0.597544i	$0.796143\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	2554.02	+	142.331i	2.58766	+	0.144206i
$988$		0			0
$989$			1026.34i			1.03776i
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.3.c.i.449.4		8
3.2	odd	2	inner	960.3.c.i.449.6		8
4.3	odd	2	inner	960.3.c.i.449.5		8
5.4	even	2	inner	960.3.c.i.449.5		8
8.3	odd	2		480.3.c.a.449.4	yes	8
8.5	even	2		480.3.c.a.449.5	yes	8
12.11	even	2	inner	960.3.c.i.449.3		8
15.14	odd	2	inner	960.3.c.i.449.3		8
20.19	odd	2	CM	960.3.c.i.449.4		8
24.5	odd	2		480.3.c.a.449.3	✓	8
24.11	even	2		480.3.c.a.449.6	yes	8
40.19	odd	2		480.3.c.a.449.5	yes	8
40.29	even	2		480.3.c.a.449.4	yes	8
60.59	even	2	inner	960.3.c.i.449.6		8
120.29	odd	2		480.3.c.a.449.6	yes	8
120.59	even	2		480.3.c.a.449.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.3.c.a.449.3	✓	8	24.5	odd	2
480.3.c.a.449.3	✓	8	120.59	even	2
480.3.c.a.449.4	yes	8	8.3	odd	2
480.3.c.a.449.4	yes	8	40.29	even	2
480.3.c.a.449.5	yes	8	8.5	even	2
480.3.c.a.449.5	yes	8	40.19	odd	2
480.3.c.a.449.6	yes	8	24.11	even	2
480.3.c.a.449.6	yes	8	120.29	odd	2
960.3.c.i.449.3		8	12.11	even	2	inner
960.3.c.i.449.3		8	15.14	odd	2	inner
960.3.c.i.449.4		8	1.1	even	1	trivial
960.3.c.i.449.4		8	20.19	odd	2	CM
960.3.c.i.449.5		8	4.3	odd	2	inner
960.3.c.i.449.5		8	5.4	even	2	inner
960.3.c.i.449.6		8	3.2	odd	2	inner
960.3.c.i.449.6		8	60.59	even	2	inner

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:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	960.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.166925 + 2.99535i) q^{3} +5.00000i q^{5} -12.3153i q^{7} +(-8.94427 - 1.00000i) q^{9} +O(q^{10})\)	\(q+(-0.166925 + 2.99535i) q^{3} +5.00000i q^{5} -12.3153i q^{7} +(-8.94427 - 1.00000i) q^{9} +(-14.9768 - 0.834626i) q^{15} +(36.8885 + 2.05573i) q^{21} +40.5995 q^{23} -25.0000 q^{25} +(4.48838 - 26.6243i) q^{27} -53.6656i q^{29} +61.5763 q^{35} -62.0000i q^{41} +25.2796i q^{43} +(5.00000 - 44.7214i) q^{45} +69.2363 q^{47} -102.666 q^{49} +107.331 q^{61} +(-12.3153 + 110.151i) q^{63} +34.5902i q^{67} +(-6.77709 + 121.610i) q^{69} +(4.17313 - 74.8838i) q^{75} +(79.0000 + 17.8885i) q^{81} +158.095 q^{83} +(160.747 + 8.95815i) q^{87} +107.331i q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 152 q^{21} - 200 q^{25} + 40 q^{45} - 392 q^{49} + 232 q^{69} + 632 q^{81}+O(q^{100}) \) 8 * q + 152 * q^21 - 200 * q^25 + 40 * q^45 - 392 * q^49 + 232 * q^69 + 632 * q^81

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