LMFDB - Embedded newform 4032.2.k.a.3905.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(3905,4032)]

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([0, 0, 1, 1]))

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

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

chi := DirichletCharacter("4032.3905");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.k (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:	$32.1956820950$
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} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3905.3
Root			$-1.93185i$ of defining polynomial
Character	$\chi$	$=$	4032.3905
Dual form			4032.2.k.a.3905.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.46410 q^{5} +(-1.00000 - 2.44949i) q^{7} +O(q^{10})$	$q+3.46410 q^{5} +(-1.00000 - 2.44949i) q^{7} +4.24264i q^{11} +4.89898i q^{13} -3.46410 q^{17} -4.89898i q^{19} +4.24264i q^{23} +7.00000 q^{25} +4.24264i q^{29} +(-3.46410 - 8.48528i) q^{35} +8.00000 q^{37} +3.46410 q^{41} +2.00000 q^{43} +6.92820 q^{47} +(-5.00000 + 4.89898i) q^{49} +12.7279i q^{53} +14.6969i q^{55} -13.8564 q^{59} +9.79796i q^{61} +16.9706i q^{65} -8.00000 q^{67} +4.24264i q^{71} +4.89898i q^{73} +(10.3923 - 4.24264i) q^{77} -4.00000 q^{79} +6.92820 q^{83} -12.0000 q^{85} +10.3923 q^{89} +(12.0000 - 4.89898i) q^{91} -16.9706i q^{95} -4.89898i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7}+O(q^{10}) $ 4 * q - 4 * q^7	$ 4 q - 4 q^{7} + 28 q^{25} + 32 q^{37} + 8 q^{43} - 20 q^{49} - 32 q^{67} - 16 q^{79} - 48 q^{85} + 48 q^{91}+O(q^{100}) $ 4 * q - 4 * q^7 + 28 * q^25 + 32 * q^37 + 8 * q^43 - 20 * q^49 - 32 * q^67 - 16 * q^79 - 48 * q^85 + 48 * q^91

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
$5$	3.46410			1.54919			0.774597	−	0.632456i	$-0.217953\pi$
$5$	3.46410			1.54919			0.774597	+	0.632456i	$0.217953\pi$
$6$		0			0
$7$	−1.00000	−	2.44949i	−0.377964	−	0.925820i
$7$	−1.00000	−	2.44949i	−0.377964	−	0.925820i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			4.24264i			1.27920i	0.768706	+	0.639602i	$0.220901\pi$
$11$			4.24264i			1.27920i	−0.768706	+	0.639602i	$0.779099\pi$
$12$		0			0
$13$			4.89898i			1.35873i	0.733799	+	0.679366i	$0.237745\pi$
$13$			4.89898i			1.35873i	−0.733799	+	0.679366i	$0.762255\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.46410			−0.840168			−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410			−0.840168			−0.420084	+	0.907485i	$0.637999\pi$
$18$		0			0
$19$		−	4.89898i		−	1.12390i	−0.827170	−	0.561951i	$-0.810051\pi$
$19$		−	4.89898i		−	1.12390i	0.827170	−	0.561951i	$-0.189949\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			4.24264i			0.884652i	0.896854	+	0.442326i	$0.145847\pi$
$23$			4.24264i			0.884652i	−0.896854	+	0.442326i	$0.854153\pi$
$24$		0			0
$25$	7.00000			1.40000
$26$		0			0
$27$		0			0
$28$		0			0
$29$			4.24264i			0.787839i	0.919145	+	0.393919i	$0.128881\pi$
$29$			4.24264i			0.787839i	−0.919145	+	0.393919i	$0.871119\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.46410	−	8.48528i	−0.585540	−	1.43427i
$36$		0			0
$37$	8.00000			1.31519			0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000			1.31519			0.657596	+	0.753371i	$0.271573\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.46410			0.541002			0.270501	−	0.962720i	$-0.412811\pi$
$41$	3.46410			0.541002			0.270501	+	0.962720i	$0.412811\pi$
$42$		0			0
$43$	2.00000			0.304997			0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000			0.304997			0.152499	+	0.988304i	$0.451268\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.92820			1.01058			0.505291	−	0.862949i	$-0.331385\pi$
$47$	6.92820			1.01058			0.505291	+	0.862949i	$0.331385\pi$
$48$		0			0
$49$	−5.00000	+	4.89898i	−0.714286	+	0.699854i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			12.7279i			1.74831i	0.485643	+	0.874157i	$0.338586\pi$
$53$			12.7279i			1.74831i	−0.485643	+	0.874157i	$0.661414\pi$
$54$		0			0
$55$			14.6969i			1.98173i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−13.8564			−1.80395			−0.901975	−	0.431788i	$-0.857883\pi$
$59$	−13.8564			−1.80395			−0.901975	+	0.431788i	$0.857883\pi$
$60$		0			0
$61$			9.79796i			1.25450i	0.778818	+	0.627250i	$0.215820\pi$
$61$			9.79796i			1.25450i	−0.778818	+	0.627250i	$0.784180\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			16.9706i			2.10494i
$66$		0			0
$67$	−8.00000			−0.977356			−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000			−0.977356			−0.488678	+	0.872464i	$0.662521\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			4.24264i			0.503509i	0.967791	+	0.251754i	$0.0810075\pi$
$71$			4.24264i			0.503509i	−0.967791	+	0.251754i	$0.918992\pi$
$72$		0			0
$73$			4.89898i			0.573382i	0.958023	+	0.286691i	$0.0925553\pi$
$73$			4.89898i			0.573382i	−0.958023	+	0.286691i	$0.907445\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	10.3923	−	4.24264i	1.18431	−	0.483494i
$78$		0			0
$79$	−4.00000			−0.450035			−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000			−0.450035			−0.225018	+	0.974355i	$0.572244\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	6.92820			0.760469			0.380235	−	0.924890i	$-0.375843\pi$
$83$	6.92820			0.760469			0.380235	+	0.924890i	$0.375843\pi$
$84$		0			0
$85$	−12.0000			−1.30158
$86$		0			0
$87$		0			0
$88$		0			0
$89$	10.3923			1.10158			0.550791	−	0.834643i	$-0.314326\pi$
$89$	10.3923			1.10158			0.550791	+	0.834643i	$0.314326\pi$
$90$		0			0
$91$	12.0000	−	4.89898i	1.25794	−	0.513553i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	16.9706i		−	1.74114i
$96$		0			0
$97$		−	4.89898i		−	0.497416i	−0.968579	−	0.248708i	$-0.919994\pi$
$97$		−	4.89898i		−	0.497416i	0.968579	−	0.248708i	$-0.0800060\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	10.3923			1.03407			0.517036	−	0.855963i	$-0.327035\pi$
$101$	10.3923			1.03407			0.517036	+	0.855963i	$0.327035\pi$
$102$		0			0
$103$		−	9.79796i		−	0.965422i	−0.875780	−	0.482711i	$-0.839652\pi$
$103$		−	9.79796i		−	0.965422i	0.875780	−	0.482711i	$-0.160348\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			4.24264i			0.410152i	0.978746	+	0.205076i	$0.0657441\pi$
$107$			4.24264i			0.410152i	−0.978746	+	0.205076i	$0.934256\pi$
$108$		0			0
$109$	−4.00000			−0.383131			−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000			−0.383131			−0.191565	+	0.981480i	$0.561356\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	12.7279i		−	1.19734i	−0.800995	−	0.598671i	$-0.795696\pi$
$113$		−	12.7279i		−	1.19734i	0.800995	−	0.598671i	$-0.204304\pi$
$114$		0			0
$115$			14.6969i			1.37050i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3.46410	+	8.48528i	0.317554	+	0.777844i
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$		0			0
$124$		0			0
$125$	6.92820			0.619677
$126$		0			0
$127$	−4.00000			−0.354943			−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000			−0.354943			−0.177471	+	0.984126i	$0.556792\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	20.7846			1.81596			0.907980	−	0.419014i	$-0.137624\pi$
$131$	20.7846			1.81596			0.907980	+	0.419014i	$0.137624\pi$
$132$		0			0
$133$	−12.0000	+	4.89898i	−1.04053	+	0.424795i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	4.24264i		−	0.362473i	−0.983440	−	0.181237i	$-0.941990\pi$
$137$		−	4.24264i		−	0.362473i	0.983440	−	0.181237i	$-0.0580100\pi$
$138$		0			0
$139$			9.79796i			0.831052i	0.909581	+	0.415526i	$0.136402\pi$
$139$			9.79796i			0.831052i	−0.909581	+	0.415526i	$0.863598\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−20.7846			−1.73810
$144$		0			0
$145$			14.6969i			1.22051i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	12.7279i		−	1.04271i	−0.853339	−	0.521356i	$-0.825426\pi$
$149$		−	12.7279i		−	1.04271i	0.853339	−	0.521356i	$-0.174574\pi$
$150$		0			0
$151$	22.0000			1.79033			0.895167	−	0.445730i	$-0.147056\pi$
$151$	22.0000			1.79033			0.895167	+	0.445730i	$0.147056\pi$
$152$		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
$161$	10.3923	−	4.24264i	0.819028	−	0.334367i
$162$		0			0
$163$	−8.00000			−0.626608			−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000			−0.626608			−0.313304	+	0.949653i	$0.601436\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.92820			0.536120			0.268060	−	0.963402i	$-0.413617\pi$
$167$	6.92820			0.536120			0.268060	+	0.963402i	$0.413617\pi$
$168$		0			0
$169$	−11.0000			−0.846154
$170$		0			0
$171$		0			0
$172$		0			0
$173$	10.3923			0.790112			0.395056	−	0.918657i	$-0.370725\pi$
$173$	10.3923			0.790112			0.395056	+	0.918657i	$0.370725\pi$
$174$		0			0
$175$	−7.00000	−	17.1464i	−0.529150	−	1.29615i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	4.24264i		−	0.317110i	−0.987350	−	0.158555i	$-0.949317\pi$
$179$		−	4.24264i		−	0.317110i	0.987350	−	0.158555i	$-0.0506835\pi$
$180$		0			0
$181$			4.89898i			0.364138i	0.983286	+	0.182069i	$0.0582795\pi$
$181$			4.89898i			0.364138i	−0.983286	+	0.182069i	$0.941721\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	27.7128			2.03749
$186$		0			0
$187$		−	14.6969i		−	1.07475i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	21.2132i		−	1.53493i	−0.641089	−	0.767467i	$-0.721517\pi$
$191$		−	21.2132i		−	1.53493i	0.641089	−	0.767467i	$-0.278483\pi$
$192$		0			0
$193$	−20.0000			−1.43963			−0.719816	−	0.694165i	$-0.755774\pi$
$193$	−20.0000			−1.43963			−0.719816	+	0.694165i	$0.755774\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			4.24264i			0.302276i	0.988513	+	0.151138i	$0.0482937\pi$
$197$			4.24264i			0.302276i	−0.988513	+	0.151138i	$0.951706\pi$
$198$		0			0
$199$			24.4949i			1.73640i	0.496217	+	0.868199i	$0.334722\pi$
$199$			24.4949i			1.73640i	−0.496217	+	0.868199i	$0.665278\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	10.3923	−	4.24264i	0.729397	−	0.297775i
$204$		0			0
$205$	12.0000			0.838116
$206$		0			0
$207$		0			0
$208$		0			0
$209$	20.7846			1.43770
$210$		0			0
$211$	−10.0000			−0.688428			−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000			−0.688428			−0.344214	+	0.938891i	$0.611855\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	6.92820			0.472500
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	16.9706i		−	1.14156i
$222$		0			0
$223$		−	14.6969i		−	0.984180i	−0.870544	−	0.492090i	$-0.836233\pi$
$223$		−	14.6969i		−	0.984180i	0.870544	−	0.492090i	$-0.163767\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$		−	14.6969i		−	0.971201i	−0.874181	−	0.485601i	$-0.838601\pi$
$229$		−	14.6969i		−	0.971201i	0.874181	−	0.485601i	$-0.161399\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			21.2132i			1.38972i	0.719144	+	0.694862i	$0.244534\pi$
$233$			21.2132i			1.38972i	−0.719144	+	0.694862i	$0.755466\pi$
$234$		0			0
$235$	24.0000			1.56559
$236$		0			0
$237$		0			0
$238$		0			0
$239$			29.6985i			1.92104i	0.278219	+	0.960518i	$0.410256\pi$
$239$			29.6985i			1.92104i	−0.278219	+	0.960518i	$0.589744\pi$
$240$		0			0
$241$			4.89898i			0.315571i	0.987473	+	0.157786i	$0.0504355\pi$
$241$			4.89898i			0.315571i	−0.987473	+	0.157786i	$0.949565\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−17.3205	+	16.9706i	−1.10657	+	1.08421i
$246$		0			0
$247$	24.0000			1.52708
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−6.92820			−0.437304			−0.218652	−	0.975803i	$-0.570166\pi$
$251$	−6.92820			−0.437304			−0.218652	+	0.975803i	$0.570166\pi$
$252$		0			0
$253$	−18.0000			−1.13165
$254$		0			0
$255$		0			0
$256$		0			0
$257$	10.3923			0.648254			0.324127	−	0.946014i	$-0.394929\pi$
$257$	10.3923			0.648254			0.324127	+	0.946014i	$0.394929\pi$
$258$		0			0
$259$	−8.00000	−	19.5959i	−0.497096	−	1.21763i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			4.24264i			0.261612i	0.991408	+	0.130806i	$0.0417566\pi$
$263$			4.24264i			0.261612i	−0.991408	+	0.130806i	$0.958243\pi$
$264$		0			0
$265$			44.0908i			2.70848i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3.46410			−0.211210			−0.105605	−	0.994408i	$-0.533678\pi$
$269$	−3.46410			−0.211210			−0.105605	+	0.994408i	$0.533678\pi$
$270$		0			0
$271$		−	19.5959i		−	1.19037i	−0.803590	−	0.595184i	$-0.797079\pi$
$271$		−	19.5959i		−	1.19037i	0.803590	−	0.595184i	$-0.202921\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			29.6985i			1.79089i
$276$		0			0
$277$	−2.00000			−0.120168			−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000			−0.120168			−0.0600842	+	0.998193i	$0.519137\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			4.24264i			0.253095i	0.991961	+	0.126547i	$0.0403896\pi$
$281$			4.24264i			0.253095i	−0.991961	+	0.126547i	$0.959610\pi$
$282$		0			0
$283$			14.6969i			0.873642i	0.899548	+	0.436821i	$0.143896\pi$
$283$			14.6969i			0.873642i	−0.899548	+	0.436821i	$0.856104\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3.46410	−	8.48528i	−0.204479	−	0.500870i
$288$		0			0
$289$	−5.00000			−0.294118
$290$		0			0
$291$		0			0
$292$		0			0
$293$	17.3205			1.01187			0.505937	−	0.862570i	$-0.331147\pi$
$293$	17.3205			1.01187			0.505937	+	0.862570i	$0.331147\pi$
$294$		0			0
$295$	−48.0000			−2.79467
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−20.7846			−1.20201
$300$		0			0
$301$	−2.00000	−	4.89898i	−0.115278	−	0.282372i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			33.9411i			1.94346i
$306$		0			0
$307$			14.6969i			0.838799i	0.907802	+	0.419399i	$0.137759\pi$
$307$			14.6969i			0.838799i	−0.907802	+	0.419399i	$0.862241\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−13.8564			−0.785725			−0.392862	−	0.919597i	$-0.628515\pi$
$311$	−13.8564			−0.785725			−0.392862	+	0.919597i	$0.628515\pi$
$312$		0			0
$313$		−	19.5959i		−	1.10763i	−0.832641	−	0.553813i	$-0.813172\pi$
$313$		−	19.5959i		−	1.10763i	0.832641	−	0.553813i	$-0.186828\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			12.7279i			0.714871i	0.933938	+	0.357436i	$0.116349\pi$
$317$			12.7279i			0.714871i	−0.933938	+	0.357436i	$0.883651\pi$
$318$		0			0
$319$	−18.0000			−1.00781
$320$		0			0
$321$		0			0
$322$		0			0
$323$			16.9706i			0.944267i
$324$		0			0
$325$			34.2929i			1.90223i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−6.92820	−	16.9706i	−0.381964	−	0.935617i
$330$		0			0
$331$	−22.0000			−1.20923			−0.604615	−	0.796518i	$-0.706673\pi$
$331$	−22.0000			−1.20923			−0.604615	+	0.796518i	$0.706673\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−27.7128			−1.51411
$336$		0			0
$337$	16.0000			0.871576			0.435788	−	0.900049i	$-0.356470\pi$
$337$	16.0000			0.871576			0.435788	+	0.900049i	$0.356470\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	17.0000	+	7.34847i	0.917914	+	0.396780i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	21.2132i		−	1.13878i	−0.822066	−	0.569392i	$-0.807179\pi$
$347$		−	21.2132i		−	1.13878i	0.822066	−	0.569392i	$-0.192821\pi$
$348$		0			0
$349$		−	19.5959i		−	1.04895i	−0.851427	−	0.524473i	$-0.824262\pi$
$349$		−	19.5959i		−	1.04895i	0.851427	−	0.524473i	$-0.175738\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−3.46410			−0.184376			−0.0921878	−	0.995742i	$-0.529386\pi$
$353$	−3.46410			−0.184376			−0.0921878	+	0.995742i	$0.529386\pi$
$354$		0			0
$355$			14.6969i			0.780033i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	21.2132i		−	1.11959i	−0.828631	−	0.559795i	$-0.810880\pi$
$359$		−	21.2132i		−	1.11959i	0.828631	−	0.559795i	$-0.189120\pi$
$360$		0			0
$361$	−5.00000			−0.263158
$362$		0			0
$363$		0			0
$364$		0			0
$365$			16.9706i			0.888280i
$366$		0			0
$367$			24.4949i			1.27862i	0.768948	+	0.639312i	$0.220781\pi$
$367$			24.4949i			1.27862i	−0.768948	+	0.639312i	$0.779219\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	31.1769	−	12.7279i	1.61862	−	0.660801i
$372$		0			0
$373$	−2.00000			−0.103556			−0.0517780	−	0.998659i	$-0.516489\pi$
$373$	−2.00000			−0.103556			−0.0517780	+	0.998659i	$0.516489\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−20.7846			−1.07046
$378$		0			0
$379$	−10.0000			−0.513665			−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000			−0.513665			−0.256833	+	0.966456i	$0.582679\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−6.92820			−0.354015			−0.177007	−	0.984210i	$-0.556642\pi$
$383$	−6.92820			−0.354015			−0.177007	+	0.984210i	$0.556642\pi$
$384$		0			0
$385$	36.0000	−	14.6969i	1.83473	−	0.749025i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	21.2132i		−	1.07555i	−0.843088	−	0.537776i	$-0.819265\pi$
$389$		−	21.2132i		−	1.07555i	0.843088	−	0.537776i	$-0.180735\pi$
$390$		0			0
$391$		−	14.6969i		−	0.743256i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−13.8564			−0.697191
$396$		0			0
$397$			29.3939i			1.47524i	0.675218	+	0.737618i	$0.264050\pi$
$397$			29.3939i			1.47524i	−0.675218	+	0.737618i	$0.735950\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	21.2132i		−	1.05934i	−0.848205	−	0.529668i	$-0.822316\pi$
$401$		−	21.2132i		−	1.05934i	0.848205	−	0.529668i	$-0.177684\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$			33.9411i			1.68240i
$408$		0			0
$409$			4.89898i			0.242239i	0.992638	+	0.121119i	$0.0386484\pi$
$409$			4.89898i			0.242239i	−0.992638	+	0.121119i	$0.961352\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	13.8564	+	33.9411i	0.681829	+	1.67013i
$414$		0			0
$415$	24.0000			1.17811
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−27.7128			−1.35386			−0.676930	−	0.736048i	$-0.736690\pi$
$419$	−27.7128			−1.35386			−0.676930	+	0.736048i	$0.736690\pi$
$420$		0			0
$421$	34.0000			1.65706			0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000			1.65706			0.828529	+	0.559946i	$0.189178\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−24.2487			−1.17624
$426$		0			0
$427$	24.0000	−	9.79796i	1.16144	−	0.474156i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	4.24264i		−	0.204361i	−0.994766	−	0.102180i	$-0.967418\pi$
$431$		−	4.24264i		−	0.204361i	0.994766	−	0.102180i	$-0.0325819\pi$
$432$		0			0
$433$		−	29.3939i		−	1.41258i	−0.707923	−	0.706290i	$-0.750368\pi$
$433$		−	29.3939i		−	1.41258i	0.707923	−	0.706290i	$-0.249632\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	20.7846			0.994263
$438$		0			0
$439$			14.6969i			0.701447i	0.936479	+	0.350723i	$0.114064\pi$
$439$			14.6969i			0.701447i	−0.936479	+	0.350723i	$0.885936\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			12.7279i			0.604722i	0.953194	+	0.302361i	$0.0977748\pi$
$443$			12.7279i			0.604722i	−0.953194	+	0.302361i	$0.902225\pi$
$444$		0			0
$445$	36.0000			1.70656
$446$		0			0
$447$		0			0
$448$		0			0
$449$			4.24264i			0.200223i	0.994976	+	0.100111i	$0.0319199\pi$
$449$			4.24264i			0.200223i	−0.994976	+	0.100111i	$0.968080\pi$
$450$		0			0
$451$			14.6969i			0.692052i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	41.5692	−	16.9706i	1.94880	−	0.795592i
$456$		0			0
$457$	−8.00000			−0.374224			−0.187112	−	0.982339i	$-0.559913\pi$
$457$	−8.00000			−0.374224			−0.187112	+	0.982339i	$0.559913\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−3.46410			−0.161339			−0.0806696	−	0.996741i	$-0.525706\pi$
$461$	−3.46410			−0.161339			−0.0806696	+	0.996741i	$0.525706\pi$
$462$		0			0
$463$	20.0000			0.929479			0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000			0.929479			0.464739	+	0.885448i	$0.346148\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	20.7846			0.961797			0.480899	−	0.876776i	$-0.340311\pi$
$467$	20.7846			0.961797			0.480899	+	0.876776i	$0.340311\pi$
$468$		0			0
$469$	8.00000	+	19.5959i	0.369406	+	0.904855i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			8.48528i			0.390154i
$474$		0			0
$475$		−	34.2929i		−	1.57346i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	13.8564			0.633115			0.316558	−	0.948573i	$-0.397473\pi$
$479$	13.8564			0.633115			0.316558	+	0.948573i	$0.397473\pi$
$480$		0			0
$481$			39.1918i			1.78699i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	16.9706i		−	0.770594i
$486$		0			0
$487$	−14.0000			−0.634401			−0.317200	−	0.948359i	$-0.602743\pi$
$487$	−14.0000			−0.634401			−0.317200	+	0.948359i	$0.602743\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	12.7279i		−	0.574403i	−0.957870	−	0.287202i	$-0.907275\pi$
$491$		−	12.7279i		−	0.574403i	0.957870	−	0.287202i	$-0.0927249\pi$
$492$		0			0
$493$		−	14.6969i		−	0.661917i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	10.3923	−	4.24264i	0.466159	−	0.190308i
$498$		0			0
$499$	−10.0000			−0.447661			−0.223831	−	0.974628i	$-0.571856\pi$
$499$	−10.0000			−0.447661			−0.223831	+	0.974628i	$0.571856\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	34.6410			1.54457			0.772283	−	0.635278i	$-0.219115\pi$
$503$	34.6410			1.54457			0.772283	+	0.635278i	$0.219115\pi$
$504$		0			0
$505$	36.0000			1.60198
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−24.2487			−1.07481			−0.537403	−	0.843326i	$-0.680594\pi$
$509$	−24.2487			−1.07481			−0.537403	+	0.843326i	$0.680594\pi$
$510$		0			0
$511$	12.0000	−	4.89898i	0.530849	−	0.216718i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	33.9411i		−	1.49562i
$516$		0			0
$517$			29.3939i			1.29274i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	3.46410			0.151765			0.0758825	−	0.997117i	$-0.475823\pi$
$521$	3.46410			0.151765			0.0758825	+	0.997117i	$0.475823\pi$
$522$		0			0
$523$		−	39.1918i		−	1.71374i	−0.515533	−	0.856870i	$-0.672406\pi$
$523$		−	39.1918i		−	1.71374i	0.515533	−	0.856870i	$-0.327594\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	5.00000			0.217391
$530$		0			0
$531$		0			0
$532$		0			0
$533$			16.9706i			0.735077i
$534$		0			0
$535$			14.6969i			0.635404i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−20.7846	−	21.2132i	−0.895257	−	0.913717i
$540$		0			0
$541$	−26.0000			−1.11783			−0.558914	−	0.829226i	$-0.688782\pi$
$541$	−26.0000			−1.11783			−0.558914	+	0.829226i	$0.688782\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−13.8564			−0.593543
$546$		0			0
$547$	40.0000			1.71028			0.855138	−	0.518400i	$-0.173472\pi$
$547$	40.0000			1.71028			0.855138	+	0.518400i	$0.173472\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	20.7846			0.885454
$552$		0			0
$553$	4.00000	+	9.79796i	0.170097	+	0.416652i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			4.24264i			0.179766i	0.995952	+	0.0898832i	$0.0286494\pi$
$557$			4.24264i			0.179766i	−0.995952	+	0.0898832i	$0.971351\pi$
$558$		0			0
$559$			9.79796i			0.414410i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	20.7846			0.875967			0.437983	−	0.898983i	$-0.355693\pi$
$563$	20.7846			0.875967			0.437983	+	0.898983i	$0.355693\pi$
$564$		0			0
$565$		−	44.0908i		−	1.85491i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			4.24264i			0.177861i	0.996038	+	0.0889304i	$0.0283449\pi$
$569$			4.24264i			0.177861i	−0.996038	+	0.0889304i	$0.971655\pi$
$570$		0			0
$571$	16.0000			0.669579			0.334790	−	0.942293i	$-0.391335\pi$
$571$	16.0000			0.669579			0.334790	+	0.942293i	$0.391335\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			29.6985i			1.23851i
$576$		0			0
$577$		−	9.79796i		−	0.407894i	−0.978982	−	0.203947i	$-0.934623\pi$
$577$		−	9.79796i		−	0.407894i	0.978982	−	0.203947i	$-0.0653771\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−6.92820	−	16.9706i	−0.287430	−	0.704058i
$582$		0			0
$583$	−54.0000			−2.23645
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−20.7846			−0.857873			−0.428936	−	0.903335i	$-0.641112\pi$
$587$	−20.7846			−0.857873			−0.428936	+	0.903335i	$0.641112\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−17.3205			−0.711268			−0.355634	−	0.934625i	$-0.615735\pi$
$593$	−17.3205			−0.711268			−0.355634	+	0.934625i	$0.615735\pi$
$594$		0			0
$595$	12.0000	+	29.3939i	0.491952	+	1.20503i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	4.24264i		−	0.173350i	−0.996237	−	0.0866748i	$-0.972376\pi$
$599$		−	4.24264i		−	0.173350i	0.996237	−	0.0866748i	$-0.0276241\pi$
$600$		0			0
$601$		−	29.3939i		−	1.19900i	−0.800374	−	0.599501i	$-0.795366\pi$
$601$		−	29.3939i		−	1.19900i	0.800374	−	0.599501i	$-0.204634\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−24.2487			−0.985850
$606$		0			0
$607$		−	4.89898i		−	0.198843i	−0.995045	−	0.0994217i	$-0.968301\pi$
$607$		−	4.89898i		−	0.198843i	0.995045	−	0.0994217i	$-0.0316993\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			33.9411i			1.37311i
$612$		0			0
$613$	−14.0000			−0.565455			−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000			−0.565455			−0.282727	+	0.959200i	$0.591239\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	12.7279i		−	0.512407i	−0.966623	−	0.256203i	$-0.917528\pi$
$617$		−	12.7279i		−	0.512407i	0.966623	−	0.256203i	$-0.0824717\pi$
$618$		0			0
$619$		−	9.79796i		−	0.393813i	−0.980422	−	0.196907i	$-0.936910\pi$
$619$		−	9.79796i		−	0.393813i	0.980422	−	0.196907i	$-0.0630896\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−10.3923	−	25.4558i	−0.416359	−	1.01987i
$624$		0			0
$625$	−11.0000			−0.440000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−27.7128			−1.10498
$630$		0			0
$631$	−28.0000			−1.11466			−0.557331	−	0.830290i	$-0.688175\pi$
$631$	−28.0000			−1.11466			−0.557331	+	0.830290i	$0.688175\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−13.8564			−0.549875
$636$		0			0
$637$	−24.0000	−	24.4949i	−0.950915	−	0.970523i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	29.6985i		−	1.17302i	−0.809942	−	0.586510i	$-0.800502\pi$
$641$		−	29.6985i		−	1.17302i	0.809942	−	0.586510i	$-0.199498\pi$
$642$		0			0
$643$		−	14.6969i		−	0.579591i	−0.957089	−	0.289795i	$-0.906413\pi$
$643$		−	14.6969i		−	0.579591i	0.957089	−	0.289795i	$-0.0935872\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	41.5692			1.63425			0.817127	−	0.576457i	$-0.195565\pi$
$647$	41.5692			1.63425			0.817127	+	0.576457i	$0.195565\pi$
$648$		0			0
$649$		−	58.7878i		−	2.30762i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	46.6690i		−	1.82630i	−0.407623	−	0.913150i	$-0.633642\pi$
$653$		−	46.6690i		−	1.82630i	0.407623	−	0.913150i	$-0.366358\pi$
$654$		0			0
$655$	72.0000			2.81327
$656$		0			0
$657$		0			0
$658$		0			0
$659$			46.6690i			1.81797i	0.416831	+	0.908984i	$0.363141\pi$
$659$			46.6690i			1.81797i	−0.416831	+	0.908984i	$0.636859\pi$
$660$		0			0
$661$		−	39.1918i		−	1.52439i	−0.647350	−	0.762193i	$-0.724123\pi$
$661$		−	39.1918i		−	1.52439i	0.647350	−	0.762193i	$-0.275877\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−41.5692	+	16.9706i	−1.61199	+	0.658090i
$666$		0			0
$667$	−18.0000			−0.696963
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−41.5692			−1.60476
$672$		0			0
$673$	−20.0000			−0.770943			−0.385472	−	0.922720i	$-0.625961\pi$
$673$	−20.0000			−0.770943			−0.385472	+	0.922720i	$0.625961\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	31.1769			1.19823			0.599113	−	0.800664i	$-0.295520\pi$
$677$	31.1769			1.19823			0.599113	+	0.800664i	$0.295520\pi$
$678$		0			0
$679$	−12.0000	+	4.89898i	−0.460518	+	0.188006i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	12.7279i		−	0.487020i	−0.969898	−	0.243510i	$-0.921701\pi$
$683$		−	12.7279i		−	0.487020i	0.969898	−	0.243510i	$-0.0782989\pi$
$684$		0			0
$685$		−	14.6969i		−	0.561541i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−62.3538			−2.37549
$690$		0			0
$691$			29.3939i			1.11820i	0.829102	+	0.559098i	$0.188852\pi$
$691$			29.3939i			1.11820i	−0.829102	+	0.559098i	$0.811148\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			33.9411i			1.28746i
$696$		0			0
$697$	−12.0000			−0.454532
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	12.7279i		−	0.480727i	−0.970683	−	0.240363i	$-0.922733\pi$
$701$		−	12.7279i		−	0.480727i	0.970683	−	0.240363i	$-0.0772666\pi$
$702$		0			0
$703$		−	39.1918i		−	1.47815i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−10.3923	−	25.4558i	−0.390843	−	0.957366i
$708$		0			0
$709$	−4.00000			−0.150223			−0.0751116	−	0.997175i	$-0.523931\pi$
$709$	−4.00000			−0.150223			−0.0751116	+	0.997175i	$0.523931\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−72.0000			−2.69265
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−27.7128			−1.03351			−0.516757	−	0.856132i	$-0.672861\pi$
$719$	−27.7128			−1.03351			−0.516757	+	0.856132i	$0.672861\pi$
$720$		0			0
$721$	−24.0000	+	9.79796i	−0.893807	+	0.364895i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			29.6985i			1.10297i
$726$		0			0
$727$			29.3939i			1.09016i	0.838385	+	0.545079i	$0.183500\pi$
$727$			29.3939i			1.09016i	−0.838385	+	0.545079i	$0.816500\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−6.92820			−0.256249
$732$		0			0
$733$			44.0908i			1.62853i	0.580492	+	0.814266i	$0.302860\pi$
$733$			44.0908i			1.62853i	−0.580492	+	0.814266i	$0.697140\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	33.9411i		−	1.25024i
$738$		0			0
$739$	16.0000			0.588570			0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000			0.588570			0.294285	+	0.955718i	$0.404919\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	4.24264i		−	0.155647i	−0.996967	−	0.0778237i	$-0.975203\pi$
$743$		−	4.24264i		−	0.155647i	0.996967	−	0.0778237i	$-0.0247971\pi$
$744$		0			0
$745$		−	44.0908i		−	1.61536i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	10.3923	−	4.24264i	0.379727	−	0.155023i
$750$		0			0
$751$	10.0000			0.364905			0.182453	−	0.983215i	$-0.441596\pi$
$751$	10.0000			0.364905			0.182453	+	0.983215i	$0.441596\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	76.2102			2.77357
$756$		0			0
$757$	−28.0000			−1.01768			−0.508839	−	0.860862i	$-0.669925\pi$
$757$	−28.0000			−1.01768			−0.508839	+	0.860862i	$0.669925\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	45.0333			1.63246			0.816228	−	0.577729i	$-0.196061\pi$
$761$	45.0333			1.63246			0.816228	+	0.577729i	$0.196061\pi$
$762$		0			0
$763$	4.00000	+	9.79796i	0.144810	+	0.354710i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	67.8823i		−	2.45109i
$768$		0			0
$769$		−	9.79796i		−	0.353323i	−0.984272	−	0.176662i	$-0.943470\pi$
$769$		−	9.79796i		−	0.353323i	0.984272	−	0.176662i	$-0.0565299\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−17.3205			−0.622975			−0.311488	−	0.950250i	$-0.600827\pi$
$773$	−17.3205			−0.622975			−0.311488	+	0.950250i	$0.600827\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	16.9706i		−	0.608034i
$780$		0			0
$781$	−18.0000			−0.644091
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$			19.5959i			0.698519i	0.937026	+	0.349260i	$0.113567\pi$
$787$			19.5959i			0.698519i	−0.937026	+	0.349260i	$0.886433\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−31.1769	+	12.7279i	−1.10852	+	0.452553i
$792$		0			0
$793$	−48.0000			−1.70453
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−10.3923			−0.368114			−0.184057	−	0.982916i	$-0.558923\pi$
$797$	−10.3923			−0.368114			−0.184057	+	0.982916i	$0.558923\pi$
$798$		0			0
$799$	−24.0000			−0.849059
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−20.7846			−0.733473
$804$		0			0
$805$	36.0000	−	14.6969i	1.26883	−	0.517999i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	38.1838i		−	1.34247i	−0.741245	−	0.671235i	$-0.765764\pi$
$809$		−	38.1838i		−	1.34247i	0.741245	−	0.671235i	$-0.234236\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−27.7128			−0.970737
$816$		0			0
$817$		−	9.79796i		−	0.342787i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	12.7279i		−	0.444208i	−0.975023	−	0.222104i	$-0.928708\pi$
$821$		−	12.7279i		−	0.444208i	0.975023	−	0.222104i	$-0.0712924\pi$
$822$		0			0
$823$	20.0000			0.697156			0.348578	−	0.937280i	$-0.386665\pi$
$823$	20.0000			0.697156			0.348578	+	0.937280i	$0.386665\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			12.7279i			0.442593i	0.975207	+	0.221297i	$0.0710289\pi$
$827$			12.7279i			0.442593i	−0.975207	+	0.221297i	$0.928971\pi$
$828$		0			0
$829$		−	24.4949i		−	0.850743i	−0.905019	−	0.425371i	$-0.860143\pi$
$829$		−	24.4949i		−	0.850743i	0.905019	−	0.425371i	$-0.139857\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	17.3205	−	16.9706i	0.600120	−	0.587995i
$834$		0			0
$835$	24.0000			0.830554
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−34.6410			−1.19594			−0.597970	−	0.801518i	$-0.704026\pi$
$839$	−34.6410			−1.19594			−0.597970	+	0.801518i	$0.704026\pi$
$840$		0			0
$841$	11.0000			0.379310
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−38.1051			−1.31086
$846$		0			0
$847$	7.00000	+	17.1464i	0.240523	+	0.589158i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			33.9411i			1.16349i
$852$		0			0
$853$		−	9.79796i		−	0.335476i	−0.985832	−	0.167738i	$-0.946354\pi$
$853$		−	9.79796i		−	0.335476i	0.985832	−	0.167738i	$-0.0536462\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−24.2487			−0.828320			−0.414160	−	0.910204i	$-0.635925\pi$
$857$	−24.2487			−0.828320			−0.414160	+	0.910204i	$0.635925\pi$
$858$		0			0
$859$			4.89898i			0.167151i	0.996501	+	0.0835755i	$0.0266340\pi$
$859$			4.89898i			0.167151i	−0.996501	+	0.0835755i	$0.973366\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			55.1543i			1.87748i	0.344633	+	0.938738i	$0.388003\pi$
$863$			55.1543i			1.87748i	−0.344633	+	0.938738i	$0.611997\pi$
$864$		0			0
$865$	36.0000			1.22404
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	16.9706i		−	0.575687i
$870$		0			0
$871$		−	39.1918i		−	1.32796i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−6.92820	−	16.9706i	−0.234216	−	0.573710i
$876$		0			0
$877$	−2.00000			−0.0675352			−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000			−0.0675352			−0.0337676	+	0.999430i	$0.510751\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	31.1769			1.05038			0.525188	−	0.850986i	$-0.323995\pi$
$881$	31.1769			1.05038			0.525188	+	0.850986i	$0.323995\pi$
$882$		0			0
$883$	−46.0000			−1.54802			−0.774012	−	0.633171i	$-0.781753\pi$
$883$	−46.0000			−1.54802			−0.774012	+	0.633171i	$0.781753\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	6.92820			0.232626			0.116313	−	0.993213i	$-0.462892\pi$
$887$	6.92820			0.232626			0.116313	+	0.993213i	$0.462892\pi$
$888$		0			0
$889$	4.00000	+	9.79796i	0.134156	+	0.328613i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	33.9411i		−	1.13580i
$894$		0			0
$895$		−	14.6969i		−	0.491264i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		−	44.0908i		−	1.46888i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			16.9706i			0.564121i
$906$		0			0
$907$	−22.0000			−0.730498			−0.365249	−	0.930910i	$-0.619016\pi$
$907$	−22.0000			−0.730498			−0.365249	+	0.930910i	$0.619016\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			21.2132i			0.702825i	0.936221	+	0.351412i	$0.114298\pi$
$911$			21.2132i			0.702825i	−0.936221	+	0.351412i	$0.885702\pi$
$912$		0			0
$913$			29.3939i			0.972795i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−20.7846	−	50.9117i	−0.686368	−	1.68125i
$918$		0			0
$919$	−14.0000			−0.461817			−0.230909	−	0.972975i	$-0.574170\pi$
$919$	−14.0000			−0.461817			−0.230909	+	0.972975i	$0.574170\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−20.7846			−0.684134
$924$		0			0
$925$	56.0000			1.84127
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−45.0333			−1.47750			−0.738748	−	0.673982i	$-0.764582\pi$
$929$	−45.0333			−1.47750			−0.738748	+	0.673982i	$0.764582\pi$
$930$		0			0
$931$	24.0000	+	24.4949i	0.786568	+	0.802788i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	50.9117i		−	1.66499i
$936$		0			0
$937$			19.5959i			0.640171i	0.947389	+	0.320085i	$0.103712\pi$
$937$			19.5959i			0.640171i	−0.947389	+	0.320085i	$0.896288\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−45.0333			−1.46804			−0.734022	−	0.679126i	$-0.762359\pi$
$941$	−45.0333			−1.46804			−0.734022	+	0.679126i	$0.762359\pi$
$942$		0			0
$943$			14.6969i			0.478598i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	29.6985i		−	0.965071i	−0.875876	−	0.482536i	$-0.839716\pi$
$947$		−	29.6985i		−	0.965071i	0.875876	−	0.482536i	$-0.160284\pi$
$948$		0			0
$949$	−24.0000			−0.779073
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	38.1838i		−	1.23689i	−0.785827	−	0.618447i	$-0.787762\pi$
$953$		−	38.1838i		−	1.23689i	0.785827	−	0.618447i	$-0.212238\pi$
$954$		0			0
$955$		−	73.4847i		−	2.37791i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−10.3923	+	4.24264i	−0.335585	+	0.137002i
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−69.2820			−2.23027
$966$		0			0
$967$	−28.0000			−0.900419			−0.450210	−	0.892923i	$-0.648651\pi$
$967$	−28.0000			−0.900419			−0.450210	+	0.892923i	$0.648651\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	27.7128			0.889346			0.444673	−	0.895693i	$-0.353320\pi$
$971$	27.7128			0.889346			0.444673	+	0.895693i	$0.353320\pi$
$972$		0			0
$973$	24.0000	−	9.79796i	0.769405	−	0.314108i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			4.24264i			0.135734i	0.997694	+	0.0678671i	$0.0216194\pi$
$977$			4.24264i			0.135734i	−0.997694	+	0.0678671i	$0.978381\pi$
$978$		0			0
$979$			44.0908i			1.40915i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	34.6410			1.10488			0.552438	−	0.833554i	$-0.313697\pi$
$983$	34.6410			1.10488			0.552438	+	0.833554i	$0.313697\pi$
$984$		0			0
$985$			14.6969i			0.468283i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			8.48528i			0.269816i
$990$		0			0
$991$	−26.0000			−0.825917			−0.412959	−	0.910750i	$-0.635505\pi$
$991$	−26.0000			−0.825917			−0.412959	+	0.910750i	$0.635505\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			84.8528i			2.69002i
$996$		0			0
$997$		−	29.3939i		−	0.930913i	−0.885071	−	0.465457i	$-0.845890\pi$
$997$		−	29.3939i		−	0.930913i	0.885071	−	0.465457i	$-0.154110\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.k.a.3905.3		4
3.2	odd	2	inner	4032.2.k.a.3905.1		4
4.3	odd	2		4032.2.k.d.3905.4		4
7.6	odd	2	inner	4032.2.k.a.3905.2		4
8.3	odd	2		1008.2.k.b.881.2		4
8.5	even	2		252.2.f.a.125.1	✓	4
12.11	even	2		4032.2.k.d.3905.2		4
21.20	even	2	inner	4032.2.k.a.3905.4		4
24.5	odd	2		252.2.f.a.125.3	yes	4
24.11	even	2		1008.2.k.b.881.4		4
28.27	even	2		4032.2.k.d.3905.1		4
40.13	odd	4		6300.2.f.b.3149.3		8
40.29	even	2		6300.2.d.c.3401.3		4
40.37	odd	4		6300.2.f.b.3149.5		8
56.5	odd	6		1764.2.t.b.521.2		8
56.13	odd	2		252.2.f.a.125.4	yes	4
56.27	even	2		1008.2.k.b.881.3		4
56.37	even	6		1764.2.t.b.521.4		8
56.45	odd	6		1764.2.t.b.1097.1		8
56.53	even	6		1764.2.t.b.1097.3		8
72.5	odd	6		2268.2.x.i.1889.2		8
72.13	even	6		2268.2.x.i.1889.4		8
72.29	odd	6		2268.2.x.i.377.1		8
72.61	even	6		2268.2.x.i.377.3		8
84.83	odd	2		4032.2.k.d.3905.3		4
120.29	odd	2		6300.2.d.c.3401.4		4
120.53	even	4		6300.2.f.b.3149.4		8
120.77	even	4		6300.2.f.b.3149.6		8
168.5	even	6		1764.2.t.b.521.3		8
168.53	odd	6		1764.2.t.b.1097.2		8
168.83	odd	2		1008.2.k.b.881.1		4
168.101	even	6		1764.2.t.b.1097.4		8
168.125	even	2		252.2.f.a.125.2	yes	4
168.149	odd	6		1764.2.t.b.521.1		8
280.13	even	4		6300.2.f.b.3149.7		8
280.69	odd	2		6300.2.d.c.3401.1		4
280.237	even	4		6300.2.f.b.3149.1		8
504.13	odd	6		2268.2.x.i.1889.1		8
504.293	even	6		2268.2.x.i.1889.3		8
504.349	odd	6		2268.2.x.i.377.2		8
504.461	even	6		2268.2.x.i.377.4		8
840.293	odd	4		6300.2.f.b.3149.8		8
840.629	even	2		6300.2.d.c.3401.2		4
840.797	odd	4		6300.2.f.b.3149.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.f.a.125.1	✓	4	8.5	even	2
252.2.f.a.125.2	yes	4	168.125	even	2
252.2.f.a.125.3	yes	4	24.5	odd	2
252.2.f.a.125.4	yes	4	56.13	odd	2
1008.2.k.b.881.1		4	168.83	odd	2
1008.2.k.b.881.2		4	8.3	odd	2
1008.2.k.b.881.3		4	56.27	even	2
1008.2.k.b.881.4		4	24.11	even	2
1764.2.t.b.521.1		8	168.149	odd	6
1764.2.t.b.521.2		8	56.5	odd	6
1764.2.t.b.521.3		8	168.5	even	6
1764.2.t.b.521.4		8	56.37	even	6
1764.2.t.b.1097.1		8	56.45	odd	6
1764.2.t.b.1097.2		8	168.53	odd	6
1764.2.t.b.1097.3		8	56.53	even	6
1764.2.t.b.1097.4		8	168.101	even	6
2268.2.x.i.377.1		8	72.29	odd	6
2268.2.x.i.377.2		8	504.349	odd	6
2268.2.x.i.377.3		8	72.61	even	6
2268.2.x.i.377.4		8	504.461	even	6
2268.2.x.i.1889.1		8	504.13	odd	6
2268.2.x.i.1889.2		8	72.5	odd	6
2268.2.x.i.1889.3		8	504.293	even	6
2268.2.x.i.1889.4		8	72.13	even	6
4032.2.k.a.3905.1		4	3.2	odd	2	inner
4032.2.k.a.3905.2		4	7.6	odd	2	inner
4032.2.k.a.3905.3		4	1.1	even	1	trivial
4032.2.k.a.3905.4		4	21.20	even	2	inner
4032.2.k.d.3905.1		4	28.27	even	2
4032.2.k.d.3905.2		4	12.11	even	2
4032.2.k.d.3905.3		4	84.83	odd	2
4032.2.k.d.3905.4		4	4.3	odd	2
6300.2.d.c.3401.1		4	280.69	odd	2
6300.2.d.c.3401.2		4	840.629	even	2
6300.2.d.c.3401.3		4	40.29	even	2
6300.2.d.c.3401.4		4	120.29	odd	2
6300.2.f.b.3149.1		8	280.237	even	4
6300.2.f.b.3149.2		8	840.797	odd	4
6300.2.f.b.3149.3		8	40.13	odd	4
6300.2.f.b.3149.4		8	120.53	even	4
6300.2.f.b.3149.5		8	40.37	odd	4
6300.2.f.b.3149.6		8	120.77	even	4
6300.2.f.b.3149.7		8	280.13	even	4
6300.2.f.b.3149.8		8	840.293	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4032.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.46410 q^{5} +(-1.00000 - 2.44949i) q^{7} +O(q^{10})\)	\(q+3.46410 q^{5} +(-1.00000 - 2.44949i) q^{7} +4.24264i q^{11} +4.89898i q^{13} -3.46410 q^{17} -4.89898i q^{19} +4.24264i q^{23} +7.00000 q^{25} +4.24264i q^{29} +(-3.46410 - 8.48528i) q^{35} +8.00000 q^{37} +3.46410 q^{41} +2.00000 q^{43} +6.92820 q^{47} +(-5.00000 + 4.89898i) q^{49} +12.7279i q^{53} +14.6969i q^{55} -13.8564 q^{59} +9.79796i q^{61} +16.9706i q^{65} -8.00000 q^{67} +4.24264i q^{71} +4.89898i q^{73} +(10.3923 - 4.24264i) q^{77} -4.00000 q^{79} +6.92820 q^{83} -12.0000 q^{85} +10.3923 q^{89} +(12.0000 - 4.89898i) q^{91} -16.9706i q^{95} -4.89898i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7}+O(q^{10}) \) 4 * q - 4 * q^7	\( 4 q - 4 q^{7} + 28 q^{25} + 32 q^{37} + 8 q^{43} - 20 q^{49} - 32 q^{67} - 16 q^{79} - 48 q^{85} + 48 q^{91}+O(q^{100}) \) 4 * q - 4 * q^7 + 28 * q^25 + 32 * q^37 + 8 * q^43 - 20 * q^49 - 32 * q^67 - 16 * q^79 - 48 * q^85 + 48 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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