LMFDB - Embedded newform 4032.2.j.f.2591.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(2591,4032)]

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, 1, 0]))

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

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

chi := DirichletCharacter("4032.2591");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$32.1956820950$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.11007531417600000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 7x^{12} + 48x^{8} - 7x^{4} + 1 $ x^16 - 7x^12 + 48x^8 - 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{18}\cdot 3^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.16
Root			$1.56290 - 0.418778i$ of defining polynomial
Character	$\chi$	$=$	4032.2591
Dual form			4032.2.j.f.2591.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.96336 q^{5} +1.00000i q^{7} +O(q^{10})$	$q+3.96336 q^{5} +1.00000i q^{7} -6.41285i q^{11} +5.60503i q^{13} -6.86474i q^{17} -3.46410 q^{19} -2.62210 q^{23} +10.7082 q^{25} +2.44949 q^{29} -2.00000i q^{31} +3.96336i q^{35} -5.60503i q^{37} -6.86474i q^{41} +7.74597 q^{43} -1.00000 q^{49} -0.578246 q^{53} -25.4164i q^{55} -9.79796i q^{59} +4.28187i q^{61} +22.2148i q^{65} -5.60503 q^{67} +11.1074 q^{71} +7.70820 q^{73} +6.41285 q^{77} -11.7082i q^{79} -12.8257i q^{83} -27.2074i q^{85} +6.86474i q^{89} -5.60503 q^{91} -13.7295 q^{95} +4.29180 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 64 q^{25} - 16 q^{49} + 16 q^{73} + 176 q^{97}+O(q^{100}) $ 16 * q + 64 * q^25 - 16 * q^49 + 16 * q^73 + 176 * q^97

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
$5$	3.96336	1.77247	0.886234	−	0.463238i	$-0.153313\pi$
$5$	3.96336	1.77247	0.886234	+	0.463238i	$0.153313\pi$
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 6.41285i	− 1.93355i	−0.255636	−	0.966773i	$-0.582285\pi$
$11$	− 6.41285i	− 1.93355i	0.255636	−	0.966773i	$-0.417715\pi$
$12$	0	0
$13$	5.60503i	1.55456i	0.629157	+	0.777278i	$0.283400\pi$
$13$	5.60503i	1.55456i	−0.629157	+	0.777278i	$0.716600\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 6.86474i	− 1.66494i	−0.554068	−	0.832472i	$-0.686925\pi$
$17$	− 6.86474i	− 1.66494i	0.554068	−	0.832472i	$-0.313075\pi$
$18$	0	0
$19$	−3.46410	−0.794719	−0.397360	−	0.917663i	$-0.630073\pi$
$19$	−3.46410	−0.794719	−0.397360	+	0.917663i	$0.630073\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.62210	−0.546745	−0.273372	−	0.961908i	$-0.588139\pi$
$23$	−2.62210	−0.546745	−0.273372	+	0.961908i	$0.588139\pi$
$24$	0	0
$25$	10.7082	2.14164
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.44949	0.454859	0.227429	−	0.973795i	$-0.426968\pi$
$29$	2.44949	0.454859	0.227429	+	0.973795i	$0.426968\pi$
$30$	0	0
$31$	− 2.00000i	− 0.359211i	−0.983739	−	0.179605i	$-0.942518\pi$
$31$	− 2.00000i	− 0.359211i	0.983739	−	0.179605i	$-0.0574821\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.96336i	0.669930i
$36$	0	0
$37$	− 5.60503i	− 0.921462i	−0.887540	−	0.460731i	$-0.847587\pi$
$37$	− 5.60503i	− 0.921462i	0.887540	−	0.460731i	$-0.152413\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 6.86474i	− 1.07209i	−0.844189	−	0.536046i	$-0.819917\pi$
$41$	− 6.86474i	− 1.07209i	0.844189	−	0.536046i	$-0.180083\pi$
$42$	0	0
$43$	7.74597	1.18125	0.590624	−	0.806947i	$-0.298881\pi$
$43$	7.74597	1.18125	0.590624	+	0.806947i	$0.298881\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−0.578246	−0.0794282	−0.0397141	−	0.999211i	$-0.512645\pi$
$53$	−0.578246	−0.0794282	−0.0397141	+	0.999211i	$0.512645\pi$
$54$	0	0
$55$	− 25.4164i	− 3.42715i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 9.79796i	− 1.27559i	−0.770208	−	0.637793i	$-0.779848\pi$
$59$	− 9.79796i	− 1.27559i	0.770208	−	0.637793i	$-0.220152\pi$
$60$	0	0
$61$	4.28187i	0.548237i	0.961696	+	0.274118i	$0.0883860\pi$
$61$	4.28187i	0.548237i	−0.961696	+	0.274118i	$0.911614\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	22.2148i	2.75540i
$66$	0	0
$67$	−5.60503	−0.684764	−0.342382	−	0.939561i	$-0.611234\pi$
$67$	−5.60503	−0.684764	−0.342382	+	0.939561i	$0.611234\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	11.1074	1.31820	0.659102	−	0.752054i	$-0.270937\pi$
$71$	11.1074	1.31820	0.659102	+	0.752054i	$0.270937\pi$
$72$	0	0
$73$	7.70820	0.902177	0.451089	−	0.892479i	$-0.351036\pi$
$73$	7.70820	0.902177	0.451089	+	0.892479i	$0.351036\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.41285	0.730812
$78$	0	0
$79$	− 11.7082i	− 1.31728i	−0.752460	−	0.658638i	$-0.771133\pi$
$79$	− 11.7082i	− 1.31728i	0.752460	−	0.658638i	$-0.228867\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 12.8257i	− 1.40780i	−0.710298	−	0.703901i	$-0.751440\pi$
$83$	− 12.8257i	− 1.40780i	0.710298	−	0.703901i	$-0.248560\pi$
$84$	0	0
$85$	− 27.2074i	− 2.95106i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.86474i	0.727661i	0.931465	+	0.363830i	$0.118531\pi$
$89$	6.86474i	0.727661i	−0.931465	+	0.363830i	$0.881469\pi$
$90$	0	0
$91$	−5.60503	−0.587567
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−13.7295	−1.40861
$96$	0	0
$97$	4.29180	0.435766	0.217883	−	0.975975i	$-0.430085\pi$
$97$	4.29180	0.435766	0.217883	+	0.975975i	$0.430085\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.86234	0.881836	0.440918	−	0.897548i	$-0.354653\pi$
$101$	8.86234	0.881836	0.440918	+	0.897548i	$0.354653\pi$
$102$	0	0
$103$	17.4164i	1.71609i	0.513575	+	0.858045i	$0.328321\pi$
$103$	17.4164i	1.71609i	−0.513575	+	0.858045i	$0.671679\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 3.38511i	− 0.327251i	−0.986523	−	0.163626i	$-0.947681\pi$
$107$	− 3.38511i	− 0.327251i	0.986523	−	0.163626i	$-0.0523189\pi$
$108$	0	0
$109$	7.74597i	0.741929i	0.928647	+	0.370965i	$0.120973\pi$
$109$	7.74597i	0.741929i	−0.928647	+	0.370965i	$0.879027\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	12.7279i	1.19734i	0.800995	+	0.598671i	$0.204304\pi$
$113$	12.7279i	1.19734i	−0.800995	+	0.598671i	$0.795696\pi$
$114$	0	0
$115$	−10.3923	−0.969087
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.86474	0.629289
$120$	0	0
$121$	−30.1246	−2.73860
$122$	0	0
$123$	0	0
$124$	0	0
$125$	22.6237	2.02352
$126$	0	0
$127$	11.7082i	1.03894i	0.854490	+	0.519468i	$0.173870\pi$
$127$	11.7082i	1.03894i	−0.854490	+	0.519468i	$0.826130\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 1.87124i	− 0.163491i	−0.996653	−	0.0817457i	$-0.973950\pi$
$131$	− 1.87124i	− 0.163491i	0.996653	−	0.0817457i	$-0.0260495\pi$
$132$	0	0
$133$	− 3.46410i	− 0.300376i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.24264i	0.362473i	0.983440	+	0.181237i	$0.0580100\pi$
$137$	4.24264i	0.362473i	−0.983440	+	0.181237i	$0.941990\pi$
$138$	0	0
$139$	11.2101	0.950826	0.475413	−	0.879763i	$-0.342299\pi$
$139$	11.2101	0.950826	0.475413	+	0.879763i	$0.342299\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	35.9442	3.00581
$144$	0	0
$145$	9.70820	0.806222
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−13.4039	−1.09809	−0.549047	−	0.835792i	$-0.685009\pi$
$149$	−13.4039	−1.09809	−0.549047	+	0.835792i	$0.685009\pi$
$150$	0	0
$151$	15.4164i	1.25457i	0.778790	+	0.627285i	$0.215834\pi$
$151$	15.4164i	1.25457i	−0.778790	+	0.627285i	$0.784166\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 7.92672i	− 0.636689i
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 2.62210i	− 0.206650i
$162$	0	0
$163$	−8.25137	−0.646297	−0.323149	−	0.946348i	$-0.604741\pi$
$163$	−8.25137	−0.646297	−0.323149	+	0.946348i	$0.604741\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−13.7295	−1.06242	−0.531209	−	0.847241i	$-0.678262\pi$
$167$	−13.7295	−1.06242	−0.531209	+	0.847241i	$0.678262\pi$
$168$	0	0
$169$	−18.4164	−1.41665
$170$	0	0
$171$	0	0
$172$	0	0
$173$	5.83460	0.443597	0.221798	−	0.975093i	$-0.428807\pi$
$173$	5.83460	0.443597	0.221798	+	0.975093i	$0.428807\pi$
$174$	0	0
$175$	10.7082i	0.809464i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.51387i	0.113152i	0.998398	+	0.0565759i	$0.0180183\pi$
$179$	1.51387i	0.113152i	−0.998398	+	0.0565759i	$0.981982\pi$
$180$	0	0
$181$	− 23.7433i	− 1.76483i	−0.470476	−	0.882413i	$-0.655918\pi$
$181$	− 23.7433i	− 1.76483i	0.470476	−	0.882413i	$-0.344082\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 22.2148i	− 1.63326i
$186$	0	0
$187$	−44.0225	−3.21924
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.86319	0.424245	0.212123	−	0.977243i	$-0.431962\pi$
$191$	5.86319	0.424245	0.212123	+	0.977243i	$0.431962\pi$
$192$	0	0
$193$	23.7082	1.70655	0.853277	−	0.521458i	$-0.174612\pi$
$193$	23.7082	1.70655	0.853277	+	0.521458i	$0.174612\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.29300	0.0921223	0.0460611	−	0.998939i	$-0.485333\pi$
$197$	1.29300	0.0921223	0.0460611	+	0.998939i	$0.485333\pi$
$198$	0	0
$199$	11.4164i	0.809288i	0.914474	+	0.404644i	$0.132605\pi$
$199$	11.4164i	0.809288i	−0.914474	+	0.404644i	$0.867395\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.44949i	0.171920i
$204$	0	0
$205$	− 27.2074i	− 1.90025i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	22.2148i	1.53663i
$210$	0	0
$211$	−18.9560	−1.30499	−0.652494	−	0.757794i	$-0.726277\pi$
$211$	−18.9560	−1.30499	−0.652494	+	0.757794i	$0.726277\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	30.7000	2.09373
$216$	0	0
$217$	2.00000	0.135769
$218$	0	0
$219$	0	0
$220$	0	0
$221$	38.4771	2.58825
$222$	0	0
$223$	− 4.00000i	− 0.267860i	−0.990991	−	0.133930i	$-0.957240\pi$
$223$	− 4.00000i	− 0.267860i	0.990991	−	0.133930i	$-0.0427597\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.18423i	0.277717i	0.990312	+	0.138858i	$0.0443433\pi$
$227$	4.18423i	0.277717i	−0.990312	+	0.138858i	$0.955657\pi$
$228$	0	0
$229$	1.32317i	0.0874375i	0.999044	+	0.0437187i	$0.0139205\pi$
$229$	1.32317i	0.0874375i	−0.999044	+	0.0437187i	$0.986079\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.7310i	0.965062i	0.875879	+	0.482531i	$0.160282\pi$
$233$	14.7310i	0.965062i	−0.875879	+	0.482531i	$0.839718\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−11.1074	−0.718477	−0.359238	−	0.933246i	$-0.616963\pi$
$239$	−11.1074	−0.718477	−0.359238	+	0.933246i	$0.616963\pi$
$240$	0	0
$241$	19.7082	1.26952	0.634759	−	0.772711i	$-0.281100\pi$
$241$	19.7082	1.26952	0.634759	+	0.772711i	$0.281100\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.96336	−0.253210
$246$	0	0
$247$	− 19.4164i	− 1.23544i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	18.8812i	1.19177i	0.803070	+	0.595884i	$0.203198\pi$
$251$	18.8812i	1.19177i	−0.803070	+	0.595884i	$0.796802\pi$
$252$	0	0
$253$	16.8151i	1.05716i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 3.62365i	− 0.226037i	−0.993593	−	0.113018i	$-0.963948\pi$
$257$	− 3.62365i	− 0.226037i	0.993593	−	0.113018i	$-0.0360519\pi$
$258$	0	0
$259$	5.60503	0.348280
$260$	0	0
$261$	0	0
$262$	0	0
$263$	30.0810	1.85488	0.927438	−	0.373976i	$-0.122006\pi$
$263$	30.0810	1.85488	0.927438	+	0.373976i	$0.122006\pi$
$264$	0	0
$265$	−2.29180	−0.140784
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−12.6048	−0.768530	−0.384265	−	0.923223i	$-0.625545\pi$
$269$	−12.6048	−0.768530	−0.384265	+	0.923223i	$0.625545\pi$
$270$	0	0
$271$	2.00000i	0.121491i	0.998153	+	0.0607457i	$0.0193479\pi$
$271$	2.00000i	0.121491i	−0.998153	+	0.0607457i	$0.980652\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 68.6701i	− 4.14096i
$276$	0	0
$277$	− 29.8537i	− 1.79374i	−0.442297	−	0.896869i	$-0.645836\pi$
$277$	− 29.8537i	− 1.79374i	0.442297	−	0.896869i	$-0.354164\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	9.48683i	0.565937i	0.959129	+	0.282969i	$0.0913192\pi$
$281$	9.48683i	0.565937i	−0.959129	+	0.282969i	$0.908681\pi$
$282$	0	0
$283$	−28.5306	−1.69597	−0.847983	−	0.530023i	$-0.822183\pi$
$283$	−28.5306	−1.69597	−0.847983	+	0.530023i	$0.822183\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.86474	0.405213
$288$	0	0
$289$	−30.1246	−1.77204
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−2.09211	−0.122223	−0.0611113	−	0.998131i	$-0.519464\pi$
$293$	−2.09211	−0.122223	−0.0611113	+	0.998131i	$0.519464\pi$
$294$	0	0
$295$	− 38.8328i	− 2.26093i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 14.6969i	− 0.849946i
$300$	0	0
$301$	7.74597i	0.446470i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	16.9706i	0.971732i
$306$	0	0
$307$	25.8842	1.47729	0.738646	−	0.674094i	$-0.235466\pi$
$307$	25.8842	1.47729	0.738646	+	0.674094i	$0.235466\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−10.4884	−0.594742	−0.297371	−	0.954762i	$-0.596110\pi$
$311$	−10.4884	−0.594742	−0.297371	+	0.954762i	$0.596110\pi$
$312$	0	0
$313$	17.4164	0.984434	0.492217	−	0.870473i	$-0.336187\pi$
$313$	17.4164	0.984434	0.492217	+	0.870473i	$0.336187\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−18.3029	−1.02799	−0.513997	−	0.857792i	$-0.671836\pi$
$317$	−18.3029	−1.02799	−0.513997	+	0.857792i	$0.671836\pi$
$318$	0	0
$319$	− 15.7082i	− 0.879491i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	23.7801i	1.32316i
$324$	0	0
$325$	60.0198i	3.32930i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	16.3097	0.896462	0.448231	−	0.893918i	$-0.352054\pi$
$331$	16.3097	0.896462	0.448231	+	0.893918i	$0.352054\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−22.2148	−1.21372
$336$	0	0
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−12.8257	−0.694550
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	16.2108i	0.870242i	0.900372	+	0.435121i	$0.143294\pi$
$347$	16.2108i	0.870242i	−0.900372	+	0.435121i	$0.856706\pi$
$348$	0	0
$349$	− 25.0665i	− 1.34178i	−0.741558	−	0.670889i	$-0.765913\pi$
$349$	− 25.0665i	− 1.34178i	0.741558	−	0.670889i	$-0.234087\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	15.3500i	0.816999i	0.912759	+	0.408500i	$0.133948\pi$
$353$	15.3500i	0.816999i	−0.912759	+	0.408500i	$0.866052\pi$
$354$	0	0
$355$	44.0225	2.33647
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−33.3221	−1.75867	−0.879337	−	0.476199i	$-0.842014\pi$
$359$	−33.3221	−1.75867	−0.879337	+	0.476199i	$0.842014\pi$
$360$	0	0
$361$	−7.00000	−0.368421
$362$	0	0
$363$	0	0
$364$	0	0
$365$	30.5504	1.59908
$366$	0	0
$367$	− 0.583592i	− 0.0304633i	−0.999884	−	0.0152316i	$-0.995151\pi$
$367$	− 0.583592i	− 0.0304633i	0.999884	−	0.0152316i	$-0.00484857\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 0.578246i	− 0.0300210i
$372$	0	0
$373$	13.8564i	0.717458i	0.933442	+	0.358729i	$0.116790\pi$
$373$	13.8564i	0.717458i	−0.933442	+	0.358729i	$0.883210\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	13.7295i	0.707104i
$378$	0	0
$379$	21.6024	1.10964	0.554820	−	0.831971i	$-0.312787\pi$
$379$	21.6024	1.10964	0.554820	+	0.831971i	$0.312787\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	13.7295	0.701543	0.350772	−	0.936461i	$-0.385919\pi$
$383$	13.7295	0.701543	0.350772	+	0.936461i	$0.385919\pi$
$384$	0	0
$385$	25.4164	1.29534
$386$	0	0
$387$	0	0
$388$	0	0
$389$	10.3762	0.526094	0.263047	−	0.964783i	$-0.415272\pi$
$389$	10.3762	0.526094	0.263047	+	0.964783i	$0.415272\pi$
$390$	0	0
$391$	18.0000i	0.910299i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 46.4038i	− 2.33483i
$396$	0	0
$397$	− 15.4919i	− 0.777518i	−0.921340	−	0.388759i	$-0.872904\pi$
$397$	− 15.4919i	− 0.777518i	0.921340	−	0.388759i	$-0.127096\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 31.7016i	− 1.58310i	−0.611103	−	0.791551i	$-0.709274\pi$
$401$	− 31.7016i	− 1.58310i	0.611103	−	0.791551i	$-0.290726\pi$
$402$	0	0
$403$	11.2101	0.558413
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−35.9442	−1.78169
$408$	0	0
$409$	−16.2918	−0.805577	−0.402789	−	0.915293i	$-0.631959\pi$
$409$	−16.2918	−0.805577	−0.402789	+	0.915293i	$0.631959\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	9.79796	0.482126
$414$	0	0
$415$	− 50.8328i	− 2.49528i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 32.8634i	− 1.60548i	−0.596329	−	0.802740i	$-0.703375\pi$
$419$	− 32.8634i	− 1.60548i	0.596329	−	0.802740i	$-0.296625\pi$
$420$	0	0
$421$	− 7.43361i	− 0.362292i	−0.983456	−	0.181146i	$-0.942019\pi$
$421$	− 7.43361i	− 0.362292i	0.983456	−	0.181146i	$-0.0579806\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 73.5090i	− 3.56571i
$426$	0	0
$427$	−4.28187	−0.207214
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2.62210	−0.126302	−0.0631510	−	0.998004i	$-0.520115\pi$
$431$	−2.62210	−0.126302	−0.0631510	+	0.998004i	$0.520115\pi$
$432$	0	0
$433$	−9.41641	−0.452524	−0.226262	−	0.974067i	$-0.572651\pi$
$433$	−9.41641	−0.452524	−0.226262	+	0.974067i	$0.572651\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	9.08321	0.434509
$438$	0	0
$439$	− 20.0000i	− 0.954548i	−0.878755	−	0.477274i	$-0.841625\pi$
$439$	− 20.0000i	− 0.954548i	0.878755	−	0.477274i	$-0.158375\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	23.4228i	1.11285i	0.830898	+	0.556425i	$0.187827\pi$
$443$	23.4228i	1.11285i	−0.830898	+	0.556425i	$0.812173\pi$
$444$	0	0
$445$	27.2074i	1.28975i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	21.2132i	1.00111i	0.865704	+	0.500556i	$0.166871\pi$
$449$	21.2132i	1.00111i	−0.865704	+	0.500556i	$0.833129\pi$
$450$	0	0
$451$	−44.0225	−2.07294
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−22.2148	−1.04144
$456$	0	0
$457$	11.4164	0.534037	0.267019	−	0.963691i	$-0.413962\pi$
$457$	11.4164	0.534037	0.267019	+	0.963691i	$0.413962\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−14.9178	−0.694792	−0.347396	−	0.937719i	$-0.612934\pi$
$461$	−14.9178	−0.694792	−0.347396	+	0.937719i	$0.612934\pi$
$462$	0	0
$463$	4.29180i	0.199457i	0.995015	+	0.0997283i	$0.0317974\pi$
$463$	4.29180i	0.199457i	−0.995015	+	0.0997283i	$0.968203\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 3.02774i	− 0.140107i	−0.997543	−	0.0700535i	$-0.977683\pi$
$467$	− 3.02774i	− 0.140107i	0.997543	−	0.0700535i	$-0.0223170\pi$
$468$	0	0
$469$	− 5.60503i	− 0.258816i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 49.6737i	− 2.28400i
$474$	0	0
$475$	−37.0943	−1.70200
$476$	0	0
$477$	0	0
$478$	0	0
$479$	25.4558	1.16311	0.581554	−	0.813508i	$-0.302445\pi$
$479$	25.4558	1.16311	0.581554	+	0.813508i	$0.302445\pi$
$480$	0	0
$481$	31.4164	1.43246
$482$	0	0
$483$	0	0
$484$	0	0
$485$	17.0099	0.772381
$486$	0	0
$487$	32.0000i	1.45006i	0.688718	+	0.725029i	$0.258174\pi$
$487$	32.0000i	1.45006i	−0.688718	+	0.725029i	$0.741826\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 5.69810i	− 0.257151i	−0.991700	−	0.128576i	$-0.958959\pi$
$491$	− 5.69810i	− 0.257151i	0.991700	−	0.128576i	$-0.0410405\pi$
$492$	0	0
$493$	− 16.8151i	− 0.757314i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	11.1074i	0.498234i
$498$	0	0
$499$	28.5306	1.27720	0.638602	−	0.769537i	$-0.279513\pi$
$499$	28.5306	1.27720	0.638602	+	0.769537i	$0.279513\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	22.2148	0.990507	0.495253	−	0.868749i	$-0.335075\pi$
$503$	22.2148	0.990507	0.495253	+	0.868749i	$0.335075\pi$
$504$	0	0
$505$	35.1246	1.56302
$506$	0	0
$507$	0	0
$508$	0	0
$509$	8.14759	0.361135	0.180568	−	0.983563i	$-0.442207\pi$
$509$	8.14759	0.361135	0.180568	+	0.983563i	$0.442207\pi$
$510$	0	0
$511$	7.70820i	0.340991i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	69.0275i	3.04171i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 27.0764i	− 1.18624i	−0.805115	−	0.593119i	$-0.797896\pi$
$521$	− 27.0764i	− 1.18624i	0.805115	−	0.593119i	$-0.202104\pi$
$522$	0	0
$523$	11.2101	0.490182	0.245091	−	0.969500i	$-0.421182\pi$
$523$	11.2101	0.490182	0.245091	+	0.969500i	$0.421182\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−13.7295	−0.598065
$528$	0	0
$529$	−16.1246	−0.701070
$530$	0	0
$531$	0	0
$532$	0	0
$533$	38.4771	1.66663
$534$	0	0
$535$	− 13.4164i	− 0.580042i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.41285i	0.276221i
$540$	0	0
$541$	0.505406i	0.0217291i	0.999941	+	0.0108645i	$0.00345836\pi$
$541$	0.505406i	0.0217291i	−0.999941	+	0.0108645i	$0.996542\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	30.7000i	1.31505i
$546$	0	0
$547$	−21.0970	−0.902041	−0.451021	−	0.892514i	$-0.648940\pi$
$547$	−21.0970	−0.902041	−0.451021	+	0.892514i	$0.648940\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8.48528	−0.361485
$552$	0	0
$553$	11.7082	0.497883
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−14.5604	−0.616945	−0.308473	−	0.951233i	$-0.599818\pi$
$557$	−14.5604	−0.616945	−0.308473	+	0.951233i	$0.599818\pi$
$558$	0	0
$559$	43.4164i	1.83632i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	2.31298i	0.0974807i	0.998811	+	0.0487403i	$0.0155207\pi$
$563$	2.31298i	0.0974807i	−0.998811	+	0.0487403i	$0.984479\pi$
$564$	0	0
$565$	50.4453i	2.12225i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 21.2132i	− 0.889304i	−0.895703	−	0.444652i	$-0.853327\pi$
$569$	− 21.2132i	− 0.889304i	0.895703	−	0.444652i	$-0.146673\pi$
$570$	0	0
$571$	10.8977	0.456055	0.228027	−	0.973655i	$-0.426772\pi$
$571$	10.8977	0.456055	0.228027	+	0.973655i	$0.426772\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−28.0779	−1.17093
$576$	0	0
$577$	5.41641	0.225488	0.112744	−	0.993624i	$-0.464036\pi$
$577$	5.41641	0.225488	0.112744	+	0.993624i	$0.464036\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	12.8257	0.532099
$582$	0	0
$583$	3.70820i	0.153578i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	21.4672i	0.886045i	0.896511	+	0.443022i	$0.146094\pi$
$587$	21.4672i	0.886045i	−0.896511	+	0.443022i	$0.853906\pi$
$588$	0	0
$589$	6.92820i	0.285472i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	34.3237i	1.40950i	0.709453	+	0.704752i	$0.248942\pi$
$593$	34.3237i	1.40950i	−0.709453	+	0.704752i	$0.751058\pi$
$594$	0	0
$595$	27.2074	1.11539
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−0.618993	−0.0252914	−0.0126457	−	0.999920i	$-0.504025\pi$
$599$	−0.618993	−0.0252914	−0.0126457	+	0.999920i	$0.504025\pi$
$600$	0	0
$601$	−29.4164	−1.19992	−0.599960	−	0.800030i	$-0.704817\pi$
$601$	−29.4164	−1.19992	−0.599960	+	0.800030i	$0.704817\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−119.395	−4.85408
$606$	0	0
$607$	8.00000i	0.324710i	0.986732	+	0.162355i	$0.0519090\pi$
$607$	8.00000i	0.324710i	−0.986732	+	0.162355i	$0.948091\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 29.3483i	− 1.18537i	−0.805435	−	0.592684i	$-0.798068\pi$
$613$	− 29.3483i	− 1.18537i	0.805435	−	0.592684i	$-0.201932\pi$
$614$	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
$619$	1.63553	0.0657374	0.0328687	−	0.999460i	$-0.489536\pi$
$619$	1.63553	0.0657374	0.0328687	+	0.999460i	$0.489536\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−6.86474	−0.275030
$624$	0	0
$625$	36.1246	1.44498
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−38.4771	−1.53418
$630$	0	0
$631$	38.5410i	1.53429i	0.641471	+	0.767147i	$0.278324\pi$
$631$	38.5410i	1.53429i	−0.641471	+	0.767147i	$0.721676\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	46.4038i	1.84148i
$636$	0	0
$637$	− 5.60503i	− 0.222080i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	15.9690i	0.630738i	0.948969	+	0.315369i	$0.102128\pi$
$641$	15.9690i	0.630738i	−0.948969	+	0.315369i	$0.897872\pi$
$642$	0	0
$643$	−7.74597	−0.305471	−0.152736	−	0.988267i	$-0.548808\pi$
$643$	−7.74597	−0.305471	−0.152736	+	0.988267i	$0.548808\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24.2179	0.952102	0.476051	−	0.879418i	$-0.342068\pi$
$647$	24.2179	0.952102	0.476051	+	0.879418i	$0.342068\pi$
$648$	0	0
$649$	−62.8328	−2.46640
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−37.8988	−1.48310	−0.741548	−	0.670900i	$-0.765908\pi$
$653$	−37.8988	−1.48310	−0.741548	+	0.670900i	$0.765908\pi$
$654$	0	0
$655$	− 7.41641i	− 0.289783i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	45.6047i	1.77651i	0.459354	+	0.888253i	$0.348081\pi$
$659$	45.6047i	1.77651i	−0.459354	+	0.888253i	$0.651919\pi$
$660$	0	0
$661$	48.4974i	1.88633i	0.332323	+	0.943166i	$0.392168\pi$
$661$	48.4974i	1.88633i	−0.332323	+	0.943166i	$0.607832\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 13.7295i	− 0.532406i
$666$	0	0
$667$	−6.42280	−0.248692
$668$	0	0
$669$	0	0
$670$	0	0
$671$	27.4589	1.06004
$672$	0	0
$673$	24.2918	0.936380	0.468190	−	0.883628i	$-0.344906\pi$
$673$	24.2918	0.936380	0.468190	+	0.883628i	$0.344906\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	25.8723	0.994352	0.497176	−	0.867650i	$-0.334370\pi$
$677$	25.8723	0.994352	0.497176	+	0.867650i	$0.334370\pi$
$678$	0	0
$679$	4.29180i	0.164704i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 23.4228i	− 0.896247i	−0.893972	−	0.448124i	$-0.852092\pi$
$683$	− 23.4228i	− 0.896247i	0.893972	−	0.448124i	$-0.147908\pi$
$684$	0	0
$685$	16.8151i	0.642472i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 3.24109i	− 0.123476i
$690$	0	0
$691$	−23.4309	−0.891355	−0.445678	−	0.895194i	$-0.647037\pi$
$691$	−23.4309	−0.891355	−0.445678	+	0.895194i	$0.647037\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	44.4295	1.68531
$696$	0	0
$697$	−47.1246	−1.78497
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−0.578246	−0.0218401	−0.0109200	−	0.999940i	$-0.503476\pi$
$701$	−0.578246	−0.0218401	−0.0109200	+	0.999940i	$0.503476\pi$
$702$	0	0
$703$	19.4164i	0.732304i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8.86234i	0.333302i
$708$	0	0
$709$	35.4588i	1.33168i	0.746093	+	0.665841i	$0.231927\pi$
$709$	35.4588i	1.33168i	−0.746093	+	0.665841i	$0.768073\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5.24419i	0.196397i
$714$	0	0
$715$	142.460	5.32770
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−47.6706	−1.77781	−0.888907	−	0.458088i	$-0.848534\pi$
$719$	−47.6706	−1.77781	−0.888907	+	0.458088i	$0.848534\pi$
$720$	0	0
$721$	−17.4164	−0.648621
$722$	0	0
$723$	0	0
$724$	0	0
$725$	26.2296	0.974144
$726$	0	0
$727$	28.8328i	1.06935i	0.845058	+	0.534675i	$0.179566\pi$
$727$	28.8328i	1.06935i	−0.845058	+	0.534675i	$0.820434\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 53.1740i	− 1.96671i
$732$	0	0
$733$	1.32317i	0.0488724i	0.999701	+	0.0244362i	$0.00777905\pi$
$733$	1.32317i	0.0488724i	−0.999701	+	0.0244362i	$0.992221\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	35.9442i	1.32402i
$738$	0	0
$739$	−11.5224	−0.423859	−0.211930	−	0.977285i	$-0.567975\pi$
$739$	−11.5224	−0.423859	−0.211930	+	0.977285i	$0.567975\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−5.86319	−0.215099	−0.107550	−	0.994200i	$-0.534300\pi$
$743$	−5.86319	−0.215099	−0.107550	+	0.994200i	$0.534300\pi$
$744$	0	0
$745$	−53.1246	−1.94634
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.38511	0.123689
$750$	0	0
$751$	11.4164i	0.416591i	0.978066	+	0.208295i	$0.0667915\pi$
$751$	11.4164i	0.416591i	−0.978066	+	0.208295i	$0.933208\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	61.1007i	2.22368i
$756$	0	0
$757$	− 18.9560i	− 0.688969i	−0.938792	−	0.344484i	$-0.888054\pi$
$757$	− 18.9560i	− 0.688969i	0.938792	−	0.344484i	$-0.111946\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	36.3268i	1.31685i	0.752649	+	0.658423i	$0.228776\pi$
$761$	36.3268i	1.31685i	−0.752649	+	0.658423i	$0.771224\pi$
$762$	0	0
$763$	−7.74597	−0.280423
$764$	0	0
$765$	0	0
$766$	0	0
$767$	54.9179	1.98297
$768$	0	0
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−32.2007	−1.15818	−0.579090	−	0.815264i	$-0.696592\pi$
$773$	−32.2007	−1.15818	−0.579090	+	0.815264i	$0.696592\pi$
$774$	0	0
$775$	− 21.4164i	− 0.769300i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	23.7801i	0.852012i
$780$	0	0
$781$	− 71.2299i	− 2.54881i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	4.28187	0.152632	0.0763160	−	0.997084i	$-0.475684\pi$
$787$	4.28187	0.152632	0.0763160	+	0.997084i	$0.475684\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−12.7279	−0.452553
$792$	0	0
$793$	−24.0000	−0.852265
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−13.7613	−0.487451	−0.243725	−	0.969844i	$-0.578370\pi$
$797$	−13.7613	−0.487451	−0.243725	+	0.969844i	$0.578370\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 49.4315i	− 1.74440i
$804$	0	0
$805$	− 10.3923i	− 0.366281i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	47.4342i	1.66770i	0.551994	+	0.833848i	$0.313867\pi$
$809$	47.4342i	1.66770i	−0.551994	+	0.833848i	$0.686133\pi$
$810$	0	0
$811$	7.93901	0.278777	0.139388	−	0.990238i	$-0.455486\pi$
$811$	7.93901	0.278777	0.139388	+	0.990238i	$0.455486\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−32.7031	−1.14554
$816$	0	0
$817$	−26.8328	−0.938761
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−3.16424	−0.110433	−0.0552164	−	0.998474i	$-0.517585\pi$
$821$	−3.16424	−0.110433	−0.0552164	+	0.998474i	$0.517585\pi$
$822$	0	0
$823$	11.7082i	0.408122i	0.978958	+	0.204061i	$0.0654141\pi$
$823$	11.7082i	0.408122i	−0.978958	+	0.204061i	$0.934586\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	15.0543i	0.523490i	0.965137	+	0.261745i	$0.0842979\pi$
$827$	15.0543i	0.523490i	−0.965137	+	0.261745i	$0.915702\pi$
$828$	0	0
$829$	− 15.8043i	− 0.548906i	−0.961601	−	0.274453i	$-0.911503\pi$
$829$	− 15.8043i	− 0.548906i	0.961601	−	0.274453i	$-0.0884967\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.86474i	0.237849i
$834$	0	0
$835$	−54.4148	−1.88310
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−39.1853	−1.35283	−0.676414	−	0.736522i	$-0.736467\pi$
$839$	−39.1853	−1.35283	−0.676414	+	0.736522i	$0.736467\pi$
$840$	0	0
$841$	−23.0000	−0.793103
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−72.9908	−2.51096
$846$	0	0
$847$	− 30.1246i	− 1.03509i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	14.6969i	0.503805i
$852$	0	0
$853$	5.91739i	0.202608i	0.994856	+	0.101304i	$0.0323014\pi$
$853$	5.91739i	0.202608i	−0.994856	+	0.101304i	$0.967699\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 35.5617i	− 1.21476i	−0.794410	−	0.607382i	$-0.792220\pi$
$857$	− 35.5617i	− 1.21476i	0.794410	−	0.607382i	$-0.207780\pi$
$858$	0	0
$859$	−31.8016	−1.08506	−0.542529	−	0.840037i	$-0.682533\pi$
$859$	−31.8016	−1.08506	−0.542529	+	0.840037i	$0.682533\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	22.8337	0.777270	0.388635	−	0.921392i	$-0.372947\pi$
$863$	22.8337	0.777270	0.388635	+	0.921392i	$0.372947\pi$
$864$	0	0
$865$	23.1246	0.786260
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−75.0829	−2.54701
$870$	0	0
$871$	− 31.4164i	− 1.06450i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	22.6237i	0.764819i
$876$	0	0
$877$	50.1329i	1.69287i	0.532493	+	0.846435i	$0.321255\pi$
$877$	50.1329i	1.69287i	−0.532493	+	0.846435i	$0.678745\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 14.1120i	− 0.475446i	−0.971333	−	0.237723i	$-0.923599\pi$
$881$	− 14.1120i	− 0.475446i	0.971333	−	0.237723i	$-0.0764011\pi$
$882$	0	0
$883$	−43.0117	−1.44746	−0.723729	−	0.690084i	$-0.757574\pi$
$883$	−43.0117	−1.44746	−0.723729	+	0.690084i	$0.757574\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	37.1822	1.24846	0.624228	−	0.781242i	$-0.285414\pi$
$887$	37.1822	1.24846	0.624228	+	0.781242i	$0.285414\pi$
$888$	0	0
$889$	−11.7082	−0.392681
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	6.00000i	0.200558i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 4.89898i	− 0.163390i
$900$	0	0
$901$	3.96951i	0.132243i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 94.1032i	− 3.12810i
$906$	0	0
$907$	34.4480	1.14383	0.571913	−	0.820314i	$-0.306201\pi$
$907$	34.4480	1.14383	0.571913	+	0.820314i	$0.306201\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−3.86008	−0.127890	−0.0639451	−	0.997953i	$-0.520368\pi$
$911$	−3.86008	−0.127890	−0.0639451	+	0.997953i	$0.520368\pi$
$912$	0	0
$913$	−82.2492	−2.72205
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.87124	0.0617939
$918$	0	0
$919$	27.4164i	0.904384i	0.891921	+	0.452192i	$0.149358\pi$
$919$	27.4164i	0.904384i	−0.891921	+	0.452192i	$0.850642\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	62.2572i	2.04922i
$924$	0	0
$925$	− 60.0198i	− 1.97344i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 51.2942i	− 1.68291i	−0.540327	−	0.841455i	$-0.681700\pi$
$929$	− 51.2942i	− 1.68291i	0.540327	−	0.841455i	$-0.318300\pi$
$930$	0	0
$931$	3.46410	0.113531
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−174.477	−5.70601
$936$	0	0
$937$	2.58359	0.0844023	0.0422011	−	0.999109i	$-0.486563\pi$
$937$	2.58359	0.0844023	0.0422011	+	0.999109i	$0.486563\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−6.54935	−0.213503	−0.106751	−	0.994286i	$-0.534045\pi$
$941$	−6.54935	−0.213503	−0.106751	+	0.994286i	$0.534045\pi$
$942$	0	0
$943$	18.0000i	0.586161i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.51387i	0.0491941i	0.999697	+	0.0245970i	$0.00783027\pi$
$947$	1.51387i	0.0491941i	−0.999697	+	0.0245970i	$0.992170\pi$
$948$	0	0
$949$	43.2047i	1.40249i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 28.4605i	− 0.921926i	−0.887419	−	0.460963i	$-0.847504\pi$
$953$	− 28.4605i	− 0.921926i	0.887419	−	0.460963i	$-0.152496\pi$
$954$	0	0
$955$	23.2379	0.751961
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−4.24264	−0.137002
$960$	0	0
$961$	27.0000	0.870968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	93.9641	3.02481
$966$	0	0
$967$	55.7082i	1.79146i	0.444603	+	0.895728i	$0.353345\pi$
$967$	55.7082i	1.79146i	−0.444603	+	0.895728i	$0.646655\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 30.5504i	− 0.980408i	−0.871608	−	0.490204i	$-0.836922\pi$
$971$	− 30.5504i	− 0.980408i	0.871608	−	0.490204i	$-0.163078\pi$
$972$	0	0
$973$	11.2101i	0.359378i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	26.4574i	0.846447i	0.906025	+	0.423224i	$0.139102\pi$
$977$	26.4574i	0.846447i	−0.906025	+	0.423224i	$0.860898\pi$
$978$	0	0
$979$	44.0225	1.40697
$980$	0	0
$981$	0	0
$982$	0	0
$983$	42.4264	1.35319	0.676596	−	0.736354i	$-0.263454\pi$
$983$	42.4264	1.35319	0.676596	+	0.736354i	$0.263454\pi$
$984$	0	0
$985$	5.12461	0.163284
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−20.3107	−0.645842
$990$	0	0
$991$	42.8328i	1.36063i	0.732920	+	0.680315i	$0.238157\pi$
$991$	42.8328i	1.36063i	−0.732920	+	0.680315i	$0.761843\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	45.2473i	1.43444i
$996$	0	0
$997$	2.64634i	0.0838104i	0.999122	+	0.0419052i	$0.0133427\pi$
$997$	2.64634i	0.0838104i	−0.999122	+	0.0419052i	$0.986657\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.j.f.2591.16	yes	16
3.2	odd	2	inner	4032.2.j.f.2591.4	yes	16
4.3	odd	2	inner	4032.2.j.f.2591.14	yes	16
8.3	odd	2	inner	4032.2.j.f.2591.1	✓	16
8.5	even	2	inner	4032.2.j.f.2591.3	yes	16
12.11	even	2	inner	4032.2.j.f.2591.2	yes	16
24.5	odd	2	inner	4032.2.j.f.2591.15	yes	16
24.11	even	2	inner	4032.2.j.f.2591.13	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4032.2.j.f.2591.1	✓	16	8.3	odd	2	inner
4032.2.j.f.2591.2	yes	16	12.11	even	2	inner
4032.2.j.f.2591.3	yes	16	8.5	even	2	inner
4032.2.j.f.2591.4	yes	16	3.2	odd	2	inner
4032.2.j.f.2591.13	yes	16	24.11	even	2	inner
4032.2.j.f.2591.14	yes	16	4.3	odd	2	inner
4032.2.j.f.2591.15	yes	16	24.5	odd	2	inner
4032.2.j.f.2591.16	yes	16	1.1	even	1	trivial

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.j (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.96336 q^{5} +1.00000i q^{7} +O(q^{10})\)	\(q+3.96336 q^{5} +1.00000i q^{7} -6.41285i q^{11} +5.60503i q^{13} -6.86474i q^{17} -3.46410 q^{19} -2.62210 q^{23} +10.7082 q^{25} +2.44949 q^{29} -2.00000i q^{31} +3.96336i q^{35} -5.60503i q^{37} -6.86474i q^{41} +7.74597 q^{43} -1.00000 q^{49} -0.578246 q^{53} -25.4164i q^{55} -9.79796i q^{59} +4.28187i q^{61} +22.2148i q^{65} -5.60503 q^{67} +11.1074 q^{71} +7.70820 q^{73} +6.41285 q^{77} -11.7082i q^{79} -12.8257i q^{83} -27.2074i q^{85} +6.86474i q^{89} -5.60503 q^{91} -13.7295 q^{95} +4.29180 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 64 q^{25} - 16 q^{49} + 16 q^{73} + 176 q^{97}+O(q^{100}) \) 16 * q + 64 * q^25 - 16 * q^49 + 16 * q^73 + 176 * q^97

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