LMFDB - Embedded newform 4032.2.c.q.2017.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(2017,4032)]

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, 1, 0, 0]))

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

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

chi := DirichletCharacter("4032.2017");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.c (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:	$8$
Coefficient field:	8.0.897122304.10
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 8x^{6} + 51x^{4} - 104x^{2} + 169 $ x^8 - 8x^6 + 51x^4 - 104*x^2 + 169
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2017.5
Root			$2.07341 + 1.19709i$ of defining polynomial
Character	$\chi$	$=$	4032.2017
Dual form			4032.2.c.q.2017.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.75265i q^{5} -1.00000 q^{7} +O(q^{10})$	$q+1.75265i q^{5} -1.00000 q^{7} -1.75265i q^{11} -1.46410i q^{13} -6.54099 q^{17} +3.46410i q^{19} +3.03569 q^{23} +1.92820 q^{25} -9.57668i q^{29} +2.00000 q^{31} -1.75265i q^{35} -4.53590i q^{37} +6.54099 q^{41} +4.92820i q^{43} +9.57668 q^{47} +1.00000 q^{49} +3.07180 q^{55} +9.57668i q^{59} +1.46410i q^{61} +2.56606 q^{65} +10.0000i q^{67} -10.0463 q^{71} +12.9282 q^{73} +1.75265i q^{77} +10.9282 q^{79} -6.07137i q^{83} -11.4641i q^{85} +0.469622 q^{89} +1.46410i q^{91} -6.07137 q^{95} +8.92820 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{7}+O(q^{10}) $ 8 * q - 8 * q^7	$ 8 q - 8 q^{7} - 40 q^{25} + 16 q^{31} + 8 q^{49} + 80 q^{55} + 48 q^{73} + 32 q^{79} + 16 q^{97}+O(q^{100}) $ 8 * q - 8 * q^7 - 40 * q^25 + 16 * q^31 + 8 * q^49 + 80 * q^55 + 48 * q^73 + 32 * q^79 + 16 * 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
$5$	1.75265i	0.783811i	0.920006	+	0.391905i	$0.128184\pi$
$5$	1.75265i	0.783811i	−0.920006	+	0.391905i	$0.871816\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 1.75265i	− 0.528445i	−0.964462	−	0.264223i	$-0.914885\pi$
$11$	− 1.75265i	− 0.528445i	0.964462	−	0.264223i	$-0.0851153\pi$
$12$	0	0
$13$	− 1.46410i	− 0.406069i	−0.979172	−	0.203034i	$-0.934920\pi$
$13$	− 1.46410i	− 0.406069i	0.979172	−	0.203034i	$-0.0650803\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.54099	−1.58642	−0.793212	−	0.608945i	$-0.791593\pi$
$17$	−6.54099	−1.58642	−0.793212	+	0.608945i	$0.791593\pi$
$18$	0	0
$19$	3.46410i	0.794719i	0.917663	+	0.397360i	$0.130073\pi$
$19$	3.46410i	0.794719i	−0.917663	+	0.397360i	$0.869927\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.03569	0.632984	0.316492	−	0.948595i	$-0.397495\pi$
$23$	3.03569	0.632984	0.316492	+	0.948595i	$0.397495\pi$
$24$	0	0
$25$	1.92820	0.385641
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 9.57668i	− 1.77834i	−0.457572	−	0.889172i	$-0.651281\pi$
$29$	− 9.57668i	− 1.77834i	0.457572	−	0.889172i	$-0.348719\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 1.75265i	− 0.296253i
$36$	0	0
$37$	− 4.53590i	− 0.745697i	−0.927892	−	0.372849i	$-0.878381\pi$
$37$	− 4.53590i	− 0.745697i	0.927892	−	0.372849i	$-0.121619\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.54099	1.02153	0.510766	−	0.859720i	$-0.329362\pi$
$41$	6.54099	1.02153	0.510766	+	0.859720i	$0.329362\pi$
$42$	0	0
$43$	4.92820i	0.751544i	0.926712	+	0.375772i	$0.122622\pi$
$43$	4.92820i	0.751544i	−0.926712	+	0.375772i	$0.877378\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.57668	1.39690	0.698451	−	0.715658i	$-0.253873\pi$
$47$	9.57668	1.39690	0.698451	+	0.715658i	$0.253873\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	3.07180	0.414201
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.57668i	1.24678i	0.781912	+	0.623389i	$0.214245\pi$
$59$	9.57668i	1.24678i	−0.781912	+	0.623389i	$0.785755\pi$
$60$	0	0
$61$	1.46410i	0.187459i	0.995598	+	0.0937295i	$0.0298789\pi$
$61$	1.46410i	0.187459i	−0.995598	+	0.0937295i	$0.970121\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.56606	0.318281
$66$	0	0
$67$	10.0000i	1.22169i	0.791748	+	0.610847i	$0.209171\pi$
$67$	10.0000i	1.22169i	−0.791748	+	0.610847i	$0.790829\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−10.0463	−1.19228	−0.596138	−	0.802882i	$-0.703299\pi$
$71$	−10.0463	−1.19228	−0.596138	+	0.802882i	$0.703299\pi$
$72$	0	0
$73$	12.9282	1.51313	0.756566	−	0.653917i	$-0.226876\pi$
$73$	12.9282	1.51313	0.756566	+	0.653917i	$0.226876\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.75265i	0.199733i
$78$	0	0
$79$	10.9282	1.22952	0.614759	−	0.788715i	$-0.289253\pi$
$79$	10.9282	1.22952	0.614759	+	0.788715i	$0.289253\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 6.07137i	− 0.666420i	−0.942853	−	0.333210i	$-0.891868\pi$
$83$	− 6.07137i	− 0.666420i	0.942853	−	0.333210i	$-0.108132\pi$
$84$	0	0
$85$	− 11.4641i	− 1.24346i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0.469622	0.0497799	0.0248899	−	0.999690i	$-0.492076\pi$
$89$	0.469622	0.0497799	0.0248899	+	0.999690i	$0.492076\pi$
$90$	0	0
$91$	1.46410i	0.153480i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.07137	−0.622910
$96$	0	0
$97$	8.92820	0.906522	0.453261	−	0.891378i	$-0.350261\pi$
$97$	8.92820	0.906522	0.453261	+	0.891378i	$0.350261\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.75265i	0.174396i	0.996191	+	0.0871978i	$0.0277912\pi$
$101$	1.75265i	0.174396i	−0.996191	+	0.0871978i	$0.972209\pi$
$102$	0	0
$103$	4.92820	0.485590	0.242795	−	0.970078i	$-0.421936\pi$
$103$	4.92820	0.485590	0.242795	+	0.970078i	$0.421936\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 17.4007i	− 1.68219i	−0.540888	−	0.841095i	$-0.681912\pi$
$107$	− 17.4007i	− 1.68219i	0.540888	−	0.841095i	$-0.318088\pi$
$108$	0	0
$109$	− 4.00000i	− 0.383131i	−0.981480	−	0.191565i	$-0.938644\pi$
$109$	− 4.00000i	− 0.383131i	0.981480	−	0.191565i	$-0.0613564\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−3.50531	−0.329752	−0.164876	−	0.986314i	$-0.552722\pi$
$113$	−3.50531	−0.329752	−0.164876	+	0.986314i	$0.552722\pi$
$114$	0	0
$115$	5.32051i	0.496140i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.54099	0.599612
$120$	0	0
$121$	7.92820	0.720746
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.1427i	1.08608i
$126$	0	0
$127$	1.07180	0.0951066	0.0475533	−	0.998869i	$-0.484858\pi$
$127$	1.07180	0.0951066	0.0475533	+	0.998869i	$0.484858\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	6.07137i	0.530458i	0.964185	+	0.265229i	$0.0854476\pi$
$131$	6.07137i	0.530458i	−0.964185	+	0.265229i	$0.914552\pi$
$132$	0	0
$133$	− 3.46410i	− 0.300376i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.07137	−0.518712	−0.259356	−	0.965782i	$-0.583510\pi$
$137$	−6.07137	−0.518712	−0.259356	+	0.965782i	$0.583510\pi$
$138$	0	0
$139$	9.85641i	0.836009i	0.908445	+	0.418005i	$0.137270\pi$
$139$	9.85641i	0.836009i	−0.908445	+	0.418005i	$0.862730\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.56606	−0.214585
$144$	0	0
$145$	16.7846	1.39389
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 7.01062i	− 0.574332i	−0.957881	−	0.287166i	$-0.907287\pi$
$149$	− 7.01062i	− 0.574332i	0.957881	−	0.287166i	$-0.0927132\pi$
$150$	0	0
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.50531i	0.281553i
$156$	0	0
$157$	12.3923i	0.989014i	0.869174	+	0.494507i	$0.164651\pi$
$157$	12.3923i	0.989014i	−0.869174	+	0.494507i	$0.835349\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.03569	−0.239246
$162$	0	0
$163$	11.8564i	0.928665i	0.885661	+	0.464333i	$0.153706\pi$
$163$	11.8564i	0.928665i	−0.885661	+	0.464333i	$0.846294\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.07137	−0.469817	−0.234908	−	0.972018i	$-0.575479\pi$
$167$	−6.07137	−0.469817	−0.234908	+	0.972018i	$0.575479\pi$
$168$	0	0
$169$	10.8564	0.835108
$170$	0	0
$171$	0	0
$172$	0	0
$173$	5.25796i	0.399755i	0.979821	+	0.199878i	$0.0640545\pi$
$173$	5.25796i	0.399755i	−0.979821	+	0.199878i	$0.935945\pi$
$174$	0	0
$175$	−1.92820	−0.145758
$176$	0	0
$177$	0	0
$178$	0	0
$179$	24.4113i	1.82459i	0.409536	+	0.912294i	$0.365691\pi$
$179$	24.4113i	1.82459i	−0.409536	+	0.912294i	$0.634309\pi$
$180$	0	0
$181$	− 21.4641i	− 1.59541i	−0.603045	−	0.797707i	$-0.706046\pi$
$181$	− 21.4641i	− 1.59541i	0.603045	−	0.797707i	$-0.293954\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.94986	0.584485
$186$	0	0
$187$	11.4641i	0.838338i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−17.0569	−1.23420	−0.617098	−	0.786887i	$-0.711692\pi$
$191$	−17.0569	−1.23420	−0.617098	+	0.786887i	$0.711692\pi$
$192$	0	0
$193$	−1.07180	−0.0771496	−0.0385748	−	0.999256i	$-0.512282\pi$
$193$	−1.07180	−0.0771496	−0.0385748	+	0.999256i	$0.512282\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.07137i	0.432567i	0.976331	+	0.216284i	$0.0693936\pi$
$197$	6.07137i	0.432567i	−0.976331	+	0.216284i	$0.930606\pi$
$198$	0	0
$199$	24.7846	1.75693	0.878467	−	0.477803i	$-0.158567\pi$
$199$	24.7846	1.75693	0.878467	+	0.477803i	$0.158567\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	9.57668i	0.672151i
$204$	0	0
$205$	11.4641i	0.800688i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6.07137	0.419966
$210$	0	0
$211$	10.0000i	0.688428i	0.938891	+	0.344214i	$0.111855\pi$
$211$	10.0000i	0.688428i	−0.938891	+	0.344214i	$0.888145\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.63744	−0.589068
$216$	0	0
$217$	−2.00000	−0.135769
$218$	0	0
$219$	0	0
$220$	0	0
$221$	9.57668i	0.644197i
$222$	0	0
$223$	−5.07180	−0.339633	−0.169816	−	0.985476i	$-0.554317\pi$
$223$	−5.07180	−0.339633	−0.169816	+	0.985476i	$0.554317\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 10.5159i	− 0.697966i	−0.937129	−	0.348983i	$-0.886527\pi$
$227$	− 10.5159i	− 0.697966i	0.937129	−	0.348983i	$-0.113473\pi$
$228$	0	0
$229$	− 10.5359i	− 0.696232i	−0.937452	−	0.348116i	$-0.886822\pi$
$229$	− 10.5359i	− 0.696232i	0.937452	−	0.348116i	$-0.113178\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	28.7300	1.88217	0.941084	−	0.338173i	$-0.109809\pi$
$233$	28.7300	1.88217	0.941084	+	0.338173i	$0.109809\pi$
$234$	0	0
$235$	16.7846i	1.09491i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	23.1283	1.49604	0.748022	−	0.663673i	$-0.231004\pi$
$239$	23.1283	1.49604	0.748022	+	0.663673i	$0.231004\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.75265i	0.111973i
$246$	0	0
$247$	5.07180	0.322711
$248$	0	0
$249$	0	0
$250$	0	0
$251$	10.5159i	0.663759i	0.943322	+	0.331880i	$0.107683\pi$
$251$	10.5159i	0.663759i	−0.943322	+	0.331880i	$0.892317\pi$
$252$	0	0
$253$	− 5.32051i	− 0.334497i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−19.6230	−1.22405	−0.612024	−	0.790839i	$-0.709644\pi$
$257$	−19.6230	−1.22405	−0.612024	+	0.790839i	$0.709644\pi$
$258$	0	0
$259$	4.53590i	0.281847i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	16.1177	0.993858	0.496929	−	0.867791i	$-0.334461\pi$
$263$	16.1177	0.993858	0.496929	+	0.867791i	$0.334461\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	5.25796i	0.320584i	0.987070	+	0.160292i	$0.0512435\pi$
$269$	5.25796i	0.320584i	−0.987070	+	0.160292i	$0.948756\pi$
$270$	0	0
$271$	15.0718	0.915546	0.457773	−	0.889069i	$-0.348647\pi$
$271$	15.0718	0.915546	0.457773	+	0.889069i	$0.348647\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 3.37947i	− 0.203790i
$276$	0	0
$277$	− 30.3923i	− 1.82610i	−0.407851	−	0.913048i	$-0.633722\pi$
$277$	− 30.3923i	− 1.82610i	0.407851	−	0.913048i	$-0.366278\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2.56606	0.153079	0.0765393	−	0.997067i	$-0.475613\pi$
$281$	2.56606	0.153079	0.0765393	+	0.997067i	$0.475613\pi$
$282$	0	0
$283$	− 14.3923i	− 0.855534i	−0.903889	−	0.427767i	$-0.859300\pi$
$283$	− 14.3923i	− 0.855534i	0.903889	−	0.427767i	$-0.140700\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.54099	−0.386103
$288$	0	0
$289$	25.7846	1.51674
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 20.9060i	− 1.22134i	−0.791884	−	0.610671i	$-0.790900\pi$
$293$	− 20.9060i	− 1.22134i	0.791884	−	0.610671i	$-0.209100\pi$
$294$	0	0
$295$	−16.7846	−0.977238
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 4.44455i	− 0.257035i
$300$	0	0
$301$	− 4.92820i	− 0.284057i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.56606	−0.146932
$306$	0	0
$307$	18.3923i	1.04970i	0.851193	+	0.524852i	$0.175879\pi$
$307$	18.3923i	1.04970i	−0.851193	+	0.524852i	$0.824121\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9.57668	0.543044	0.271522	−	0.962432i	$-0.412473\pi$
$311$	9.57668	0.543044	0.271522	+	0.962432i	$0.412473\pi$
$312$	0	0
$313$	−0.928203	−0.0524651	−0.0262326	−	0.999656i	$-0.508351\pi$
$313$	−0.928203	−0.0524651	−0.0262326	+	0.999656i	$0.508351\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 26.1640i	− 1.46952i	−0.678330	−	0.734758i	$-0.737296\pi$
$317$	− 26.1640i	− 1.46952i	0.678330	−	0.734758i	$-0.262704\pi$
$318$	0	0
$319$	−16.7846	−0.939758
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 22.6587i	− 1.26076i
$324$	0	0
$325$	− 2.82309i	− 0.156597i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−9.57668	−0.527979
$330$	0	0
$331$	− 6.00000i	− 0.329790i	−0.986311	−	0.164895i	$-0.947272\pi$
$331$	− 6.00000i	− 0.329790i	0.986311	−	0.164895i	$-0.0527285\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−17.5265	−0.957577
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 3.50531i	− 0.189823i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 27.9166i	− 1.49864i	−0.662206	−	0.749322i	$-0.730380\pi$
$347$	− 27.9166i	− 1.49864i	0.662206	−	0.749322i	$-0.269620\pi$
$348$	0	0
$349$	13.4641i	0.720717i	0.932814	+	0.360358i	$0.117346\pi$
$349$	13.4641i	0.720717i	−0.932814	+	0.360358i	$0.882654\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−13.5516	−0.721279	−0.360640	−	0.932705i	$-0.617442\pi$
$353$	−13.5516	−0.721279	−0.360640	+	0.932705i	$0.617442\pi$
$354$	0	0
$355$	− 17.6077i	− 0.934519i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.03569	0.160217	0.0801087	−	0.996786i	$-0.474473\pi$
$359$	3.03569	0.160217	0.0801087	+	0.996786i	$0.474473\pi$
$360$	0	0
$361$	7.00000	0.368421
$362$	0	0
$363$	0	0
$364$	0	0
$365$	22.6587i	1.18601i
$366$	0	0
$367$	32.7846	1.71134	0.855671	−	0.517520i	$-0.173145\pi$
$367$	32.7846	1.71134	0.855671	+	0.517520i	$0.173145\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	6.92820i	0.358729i	0.983783	+	0.179364i	$0.0574041\pi$
$373$	6.92820i	0.358729i	−0.983783	+	0.179364i	$0.942596\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−14.0212	−0.722130
$378$	0	0
$379$	27.8564i	1.43089i	0.698670	+	0.715444i	$0.253775\pi$
$379$	27.8564i	1.43089i	−0.698670	+	0.715444i	$0.746225\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−10.5159	−0.537339	−0.268669	−	0.963232i	$-0.586584\pi$
$383$	−10.5159	−0.537339	−0.268669	+	0.963232i	$0.586584\pi$
$384$	0	0
$385$	−3.07180	−0.156553
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4.44455i	0.225348i	0.993632	+	0.112674i	$0.0359415\pi$
$389$	4.44455i	0.225348i	−0.993632	+	0.112674i	$0.964058\pi$
$390$	0	0
$391$	−19.8564	−1.00418
$392$	0	0
$393$	0	0
$394$	0	0
$395$	19.1534i	0.963710i
$396$	0	0
$397$	18.5359i	0.930290i	0.885234	+	0.465145i	$0.153998\pi$
$397$	18.5359i	0.930290i	−0.885234	+	0.465145i	$0.846002\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−0.939245	−0.0469036	−0.0234518	−	0.999725i	$-0.507466\pi$
$401$	−0.939245	−0.0469036	−0.0234518	+	0.999725i	$0.507466\pi$
$402$	0	0
$403$	− 2.92820i	− 0.145864i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−7.94986	−0.394060
$408$	0	0
$409$	8.92820	0.441471	0.220736	−	0.975334i	$-0.429154\pi$
$409$	8.92820	0.441471	0.220736	+	0.975334i	$0.429154\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 9.57668i	− 0.471238i
$414$	0	0
$415$	10.6410	0.522347
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 23.5979i	− 1.15283i	−0.817156	−	0.576417i	$-0.804450\pi$
$419$	− 23.5979i	− 1.15283i	0.817156	−	0.576417i	$-0.195550\pi$
$420$	0	0
$421$	− 29.3205i	− 1.42899i	−0.699638	−	0.714497i	$-0.746656\pi$
$421$	− 29.3205i	− 1.42899i	0.699638	−	0.714497i	$-0.253344\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−12.6124	−0.611790
$426$	0	0
$427$	− 1.46410i	− 0.0708528i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	16.1177	0.776361	0.388181	−	0.921583i	$-0.373104\pi$
$431$	16.1177	0.776361	0.388181	+	0.921583i	$0.373104\pi$
$432$	0	0
$433$	−8.14359	−0.391356	−0.195678	−	0.980668i	$-0.562691\pi$
$433$	−8.14359	−0.391356	−0.195678	+	0.980668i	$0.562691\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	10.5159i	0.503045i
$438$	0	0
$439$	13.8564	0.661330	0.330665	−	0.943748i	$-0.392727\pi$
$439$	13.8564	0.661330	0.330665	+	0.943748i	$0.392727\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	5.25796i	0.249813i	0.992168	+	0.124907i	$0.0398631\pi$
$443$	5.25796i	0.249813i	−0.992168	+	0.124907i	$0.960137\pi$
$444$	0	0
$445$	0.823085i	0.0390180i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	36.6799	1.73103	0.865516	−	0.500882i	$-0.166991\pi$
$449$	36.6799	1.73103	0.865516	+	0.500882i	$0.166991\pi$
$450$	0	0
$451$	− 11.4641i	− 0.539823i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.56606	−0.120299
$456$	0	0
$457$	−3.85641	−0.180395	−0.0901975	−	0.995924i	$-0.528750\pi$
$457$	−3.85641	−0.180395	−0.0901975	+	0.995924i	$0.528750\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 36.5541i	− 1.70249i	−0.524766	−	0.851246i	$-0.675847\pi$
$461$	− 36.5541i	− 1.70249i	0.524766	−	0.851246i	$-0.324153\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	29.6693i	1.37293i	0.727162	+	0.686465i	$0.240839\pi$
$467$	29.6693i	1.37293i	−0.727162	+	0.686465i	$0.759161\pi$
$468$	0	0
$469$	− 10.0000i	− 0.461757i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	8.63744	0.397150
$474$	0	0
$475$	6.67949i	0.306476i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	35.7407	1.63303	0.816516	−	0.577323i	$-0.195902\pi$
$479$	35.7407	1.63303	0.816516	+	0.577323i	$0.195902\pi$
$480$	0	0
$481$	−6.64102	−0.302804
$482$	0	0
$483$	0	0
$484$	0	0
$485$	15.6481i	0.710541i
$486$	0	0
$487$	−24.7846	−1.12310	−0.561549	−	0.827444i	$-0.689794\pi$
$487$	−24.7846	−1.12310	−0.561549	+	0.827444i	$0.689794\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 5.25796i	− 0.237289i	−0.992937	−	0.118644i	$-0.962145\pi$
$491$	− 5.25796i	− 0.237289i	0.992937	−	0.118644i	$-0.0378548\pi$
$492$	0	0
$493$	62.6410i	2.82121i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10.0463	0.450638
$498$	0	0
$499$	− 16.9282i	− 0.757810i	−0.925435	−	0.378905i	$-0.876301\pi$
$499$	− 16.9282i	− 0.757810i	0.925435	−	0.378905i	$-0.123699\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	22.6587	1.01030	0.505150	−	0.863032i	$-0.331437\pi$
$503$	22.6587	1.01030	0.505150	+	0.863032i	$0.331437\pi$
$504$	0	0
$505$	−3.07180	−0.136693
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 17.4007i	− 0.771273i	−0.922651	−	0.385636i	$-0.873982\pi$
$509$	− 17.4007i	− 0.771273i	0.922651	−	0.385636i	$-0.126018\pi$
$510$	0	0
$511$	−12.9282	−0.571910
$512$	0	0
$513$	0	0
$514$	0	0
$515$	8.63744i	0.380611i
$516$	0	0
$517$	− 16.7846i	− 0.738186i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−19.6230	−0.859699	−0.429849	−	0.902901i	$-0.641433\pi$
$521$	−19.6230	−0.859699	−0.429849	+	0.902901i	$0.641433\pi$
$522$	0	0
$523$	8.00000i	0.349816i	0.984585	+	0.174908i	$0.0559627\pi$
$523$	8.00000i	0.349816i	−0.984585	+	0.174908i	$0.944037\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−13.0820	−0.569860
$528$	0	0
$529$	−13.7846	−0.599331
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 9.57668i	− 0.414812i
$534$	0	0
$535$	30.4974	1.31852
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 1.75265i	− 0.0754922i
$540$	0	0
$541$	35.1769i	1.51237i	0.654356	+	0.756187i	$0.272940\pi$
$541$	35.1769i	1.51237i	−0.654356	+	0.756187i	$0.727060\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7.01062	0.300302
$546$	0	0
$547$	− 8.92820i	− 0.381742i	−0.981615	−	0.190871i	$-0.938869\pi$
$547$	− 8.92820i	− 0.381742i	0.981615	−	0.190871i	$-0.0611313\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	33.1746	1.41329
$552$	0	0
$553$	−10.9282	−0.464714
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 19.1534i	− 0.811554i	−0.913972	−	0.405777i	$-0.867001\pi$
$557$	− 19.1534i	− 0.811554i	0.913972	−	0.405777i	$-0.132999\pi$
$558$	0	0
$559$	7.21539	0.305178
$560$	0	0
$561$	0	0
$562$	0	0
$563$	16.5873i	0.699071i	0.936923	+	0.349536i	$0.113661\pi$
$563$	16.5873i	0.699071i	−0.936923	+	0.349536i	$0.886339\pi$
$564$	0	0
$565$	− 6.14359i	− 0.258463i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−35.7407	−1.49833	−0.749163	−	0.662385i	$-0.769544\pi$
$569$	−35.7407	−1.49833	−0.749163	+	0.662385i	$0.769544\pi$
$570$	0	0
$571$	− 26.7846i	− 1.12090i	−0.828188	−	0.560451i	$-0.810628\pi$
$571$	− 26.7846i	− 1.12090i	0.828188	−	0.560451i	$-0.189372\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	5.85342	0.244104
$576$	0	0
$577$	16.9282	0.704730	0.352365	−	0.935863i	$-0.385378\pi$
$577$	16.9282	0.704730	0.352365	+	0.935863i	$0.385378\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6.07137i	0.251883i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 3.50531i	− 0.144680i	−0.997380	−	0.0723398i	$-0.976953\pi$
$587$	− 3.50531i	− 0.144680i	0.997380	−	0.0723398i	$-0.0230466\pi$
$588$	0	0
$589$	6.92820i	0.285472i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	20.5622	0.844390	0.422195	−	0.906505i	$-0.361260\pi$
$593$	20.5622	0.844390	0.422195	+	0.906505i	$0.361260\pi$
$594$	0	0
$595$	11.4641i	0.469982i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−36.2103	−1.47951	−0.739756	−	0.672875i	$-0.765059\pi$
$599$	−36.2103	−1.47951	−0.739756	+	0.672875i	$0.765059\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	13.8954i	0.564928i
$606$	0	0
$607$	−29.0718	−1.17999	−0.589994	−	0.807408i	$-0.700870\pi$
$607$	−29.0718	−1.17999	−0.589994	+	0.807408i	$0.700870\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 14.0212i	− 0.567238i
$612$	0	0
$613$	− 31.7128i	− 1.28087i	−0.768013	−	0.640434i	$-0.778754\pi$
$613$	− 31.7128i	− 1.28087i	0.768013	−	0.640434i	$-0.221246\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−13.0820	−0.526661	−0.263331	−	0.964706i	$-0.584821\pi$
$617$	−13.0820	−0.526661	−0.263331	+	0.964706i	$0.584821\pi$
$618$	0	0
$619$	− 32.0000i	− 1.28619i	−0.765787	−	0.643094i	$-0.777650\pi$
$619$	− 32.0000i	− 1.28619i	0.765787	−	0.643094i	$-0.222350\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−0.469622	−0.0188150
$624$	0	0
$625$	−11.6410	−0.465641
$626$	0	0
$627$	0	0
$628$	0	0
$629$	29.6693i	1.18299i
$630$	0	0
$631$	−12.7846	−0.508947	−0.254474	−	0.967080i	$-0.581902\pi$
$631$	−12.7846	−0.508947	−0.254474	+	0.967080i	$0.581902\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.87849i	0.0745456i
$636$	0	0
$637$	− 1.46410i	− 0.0580098i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0.939245	0.0370979	0.0185490	−	0.999828i	$-0.494095\pi$
$641$	0.939245	0.0370979	0.0185490	+	0.999828i	$0.494095\pi$
$642$	0	0
$643$	− 34.1051i	− 1.34497i	−0.740109	−	0.672487i	$-0.765226\pi$
$643$	− 34.1051i	− 1.34497i	0.740109	−	0.672487i	$-0.234774\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−27.1032	−1.06554	−0.532769	−	0.846261i	$-0.678848\pi$
$647$	−27.1032	−1.06554	−0.532769	+	0.846261i	$0.678848\pi$
$648$	0	0
$649$	16.7846	0.658854
$650$	0	0
$651$	0	0
$652$	0	0
$653$	34.8014i	1.36188i	0.732337	+	0.680942i	$0.238430\pi$
$653$	34.8014i	1.36188i	−0.732337	+	0.680942i	$0.761570\pi$
$654$	0	0
$655$	−10.6410	−0.415779
$656$	0	0
$657$	0	0
$658$	0	0
$659$	12.2686i	0.477916i	0.971030	+	0.238958i	$0.0768058\pi$
$659$	12.2686i	0.477916i	−0.971030	+	0.238958i	$0.923194\pi$
$660$	0	0
$661$	49.1769i	1.91276i	0.292125	+	0.956380i	$0.405638\pi$
$661$	49.1769i	1.91276i	−0.292125	+	0.956380i	$0.594362\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	6.07137	0.235438
$666$	0	0
$667$	− 29.0718i	− 1.12566i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.56606	0.0990618
$672$	0	0
$673$	−34.6410	−1.33531	−0.667657	−	0.744469i	$-0.732703\pi$
$673$	−34.6410	−1.33531	−0.667657	+	0.744469i	$0.732703\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	27.9166i	1.07292i	0.843925	+	0.536462i	$0.180239\pi$
$677$	27.9166i	1.07292i	−0.843925	+	0.536462i	$0.819761\pi$
$678$	0	0
$679$	−8.92820	−0.342633
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 1.75265i	− 0.0670634i	−0.999438	−	0.0335317i	$-0.989325\pi$
$683$	− 1.75265i	− 0.0670634i	0.999438	−	0.0335317i	$-0.0106755\pi$
$684$	0	0
$685$	− 10.6410i	− 0.406572i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 47.7128i	− 1.81508i	−0.419965	−	0.907540i	$-0.637958\pi$
$691$	− 47.7128i	− 1.81508i	0.419965	−	0.907540i	$-0.362042\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−17.2749	−0.655273
$696$	0	0
$697$	−42.7846	−1.62058
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 10.5159i	− 0.397181i	−0.980083	−	0.198591i	$-0.936364\pi$
$701$	− 10.5159i	− 0.397181i	0.980083	−	0.198591i	$-0.0636364\pi$
$702$	0	0
$703$	15.7128	0.592620
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.75265i	− 0.0659153i
$708$	0	0
$709$	− 14.1436i	− 0.531174i	−0.964087	−	0.265587i	$-0.914434\pi$
$709$	− 14.1436i	− 0.531174i	0.964087	−	0.265587i	$-0.0855657\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	6.07137	0.227375
$714$	0	0
$715$	− 4.49742i	− 0.168194i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−48.8226	−1.82078	−0.910389	−	0.413754i	$-0.864217\pi$
$719$	−48.8226	−1.82078	−0.910389	+	0.413754i	$0.864217\pi$
$720$	0	0
$721$	−4.92820	−0.183536
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 18.4658i	− 0.685802i
$726$	0	0
$727$	−16.9282	−0.627832	−0.313916	−	0.949451i	$-0.601641\pi$
$727$	−16.9282	−0.627832	−0.313916	+	0.949451i	$0.601641\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 32.2354i	− 1.19227i
$732$	0	0
$733$	50.2487i	1.85598i	0.372607	+	0.927989i	$0.378464\pi$
$733$	50.2487i	1.85598i	−0.372607	+	0.927989i	$0.621536\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	17.5265	0.645598
$738$	0	0
$739$	49.7128i	1.82872i	0.404907	+	0.914358i	$0.367304\pi$
$739$	49.7128i	1.82872i	−0.404907	+	0.914358i	$0.632696\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−37.1495	−1.36288	−0.681442	−	0.731872i	$-0.738647\pi$
$743$	−37.1495	−1.36288	−0.681442	+	0.731872i	$0.738647\pi$
$744$	0	0
$745$	12.2872	0.450168
$746$	0	0
$747$	0	0
$748$	0	0
$749$	17.4007i	0.635808i
$750$	0	0
$751$	27.7128	1.01125	0.505627	−	0.862752i	$-0.331261\pi$
$751$	27.7128	1.01125	0.505627	+	0.862752i	$0.331261\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	7.01062i	0.255142i
$756$	0	0
$757$	− 47.7128i	− 1.73415i	−0.498176	−	0.867076i	$-0.665997\pi$
$757$	− 47.7128i	− 1.73415i	0.498176	−	0.867076i	$-0.334003\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	31.7657	1.15151	0.575753	−	0.817623i	$-0.304709\pi$
$761$	31.7657	1.15151	0.575753	+	0.817623i	$0.304709\pi$
$762$	0	0
$763$	4.00000i	0.144810i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	14.0212	0.506277
$768$	0	0
$769$	−31.5692	−1.13842	−0.569208	−	0.822194i	$-0.692750\pi$
$769$	−31.5692	−1.13842	−0.569208	+	0.822194i	$0.692750\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 38.4326i	− 1.38232i	−0.722700	−	0.691162i	$-0.757099\pi$
$773$	− 38.4326i	− 1.38232i	0.722700	−	0.691162i	$-0.242901\pi$
$774$	0	0
$775$	3.85641	0.138526
$776$	0	0
$777$	0	0
$778$	0	0
$779$	22.6587i	0.811831i
$780$	0	0
$781$	17.6077i	0.630053i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−21.7194	−0.775200
$786$	0	0
$787$	− 37.5692i	− 1.33920i	−0.742723	−	0.669599i	$-0.766466\pi$
$787$	− 37.5692i	− 1.33920i	0.742723	−	0.669599i	$-0.233534\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	3.50531	0.124634
$792$	0	0
$793$	2.14359	0.0761212
$794$	0	0
$795$	0	0
$796$	0	0
$797$	31.4219i	1.11302i	0.830840	+	0.556511i	$0.187860\pi$
$797$	31.4219i	1.11302i	−0.830840	+	0.556511i	$0.812140\pi$
$798$	0	0
$799$	−62.6410	−2.21608
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 22.6587i	− 0.799607i
$804$	0	0
$805$	− 5.32051i	− 0.187523i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−43.6905	−1.53608	−0.768038	−	0.640404i	$-0.778767\pi$
$809$	−43.6905	−1.53608	−0.768038	+	0.640404i	$0.778767\pi$
$810$	0	0
$811$	10.1436i	0.356190i	0.984013	+	0.178095i	$0.0569934\pi$
$811$	10.1436i	0.356190i	−0.984013	+	0.178095i	$0.943007\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−20.7802	−0.727898
$816$	0	0
$817$	−17.0718	−0.597267
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 12.1427i	− 0.423785i	−0.977293	−	0.211892i	$-0.932037\pi$
$821$	− 12.1427i	− 0.423785i	0.977293	−	0.211892i	$-0.0679626\pi$
$822$	0	0
$823$	−41.8564	−1.45902	−0.729511	−	0.683969i	$-0.760252\pi$
$823$	−41.8564	−1.45902	−0.729511	+	0.683969i	$0.760252\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	38.4326i	1.33643i	0.743968	+	0.668215i	$0.232942\pi$
$827$	38.4326i	1.33643i	−0.743968	+	0.668215i	$0.767058\pi$
$828$	0	0
$829$	49.1769i	1.70798i	0.520285	+	0.853992i	$0.325826\pi$
$829$	49.1769i	1.70798i	−0.520285	+	0.853992i	$0.674174\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−6.54099	−0.226632
$834$	0	0
$835$	− 10.6410i	− 0.368248i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	27.1032	0.935707	0.467854	−	0.883806i	$-0.345027\pi$
$839$	27.1032	0.935707	0.467854	+	0.883806i	$0.345027\pi$
$840$	0	0
$841$	−62.7128	−2.16251
$842$	0	0
$843$	0	0
$844$	0	0
$845$	19.0275i	0.654567i
$846$	0	0
$847$	−7.92820	−0.272416
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 13.7696i	− 0.472015i
$852$	0	0
$853$	− 7.60770i	− 0.260483i	−0.991482	−	0.130241i	$-0.958425\pi$
$853$	− 7.60770i	− 0.260483i	0.991482	−	0.130241i	$-0.0415752\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	50.9191	1.73936	0.869681	−	0.493613i	$-0.164324\pi$
$857$	50.9191	1.73936	0.869681	+	0.493613i	$0.164324\pi$
$858$	0	0
$859$	− 1.32051i	− 0.0450552i	−0.999746	−	0.0225276i	$-0.992829\pi$
$859$	− 1.32051i	− 0.0450552i	0.999746	−	0.0225276i	$-0.00717136\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	48.3530	1.64596	0.822978	−	0.568073i	$-0.192311\pi$
$863$	48.3530	1.64596	0.822978	+	0.568073i	$0.192311\pi$
$864$	0	0
$865$	−9.21539	−0.313333
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 19.1534i	− 0.649733i
$870$	0	0
$871$	14.6410	0.496092
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 12.1427i	− 0.410500i
$876$	0	0
$877$	44.0000i	1.48577i	0.669417	+	0.742887i	$0.266544\pi$
$877$	44.0000i	1.48577i	−0.669417	+	0.742887i	$0.733456\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.40887	0.0474659	0.0237330	−	0.999718i	$-0.492445\pi$
$881$	1.40887	0.0474659	0.0237330	+	0.999718i	$0.492445\pi$
$882$	0	0
$883$	− 11.0718i	− 0.372596i	−0.982493	−	0.186298i	$-0.940351\pi$
$883$	− 11.0718i	− 0.372596i	0.982493	−	0.186298i	$-0.0596489\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	51.3887	1.72546	0.862732	−	0.505661i	$-0.168751\pi$
$887$	51.3887	1.72546	0.862732	+	0.505661i	$0.168751\pi$
$888$	0	0
$889$	−1.07180	−0.0359469
$890$	0	0
$891$	0	0
$892$	0	0
$893$	33.1746i	1.11015i
$894$	0	0
$895$	−42.7846	−1.43013
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 19.1534i	− 0.638800i
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	37.6191	1.25050
$906$	0	0
$907$	− 24.6410i	− 0.818192i	−0.912491	−	0.409096i	$-0.865844\pi$
$907$	− 24.6410i	− 0.818192i	0.912491	−	0.409096i	$-0.134156\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−8.16781	−0.270612	−0.135306	−	0.990804i	$-0.543202\pi$
$911$	−8.16781	−0.270612	−0.135306	+	0.990804i	$0.543202\pi$
$912$	0	0
$913$	−10.6410	−0.352166
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 6.07137i	− 0.200494i
$918$	0	0
$919$	48.7846	1.60926	0.804628	−	0.593779i	$-0.202365\pi$
$919$	48.7846	1.60926	0.804628	+	0.593779i	$0.202365\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	14.7088i	0.484146i
$924$	0	0
$925$	− 8.74613i	− 0.287571i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	12.6124	0.413798	0.206899	−	0.978362i	$-0.433663\pi$
$929$	12.6124	0.413798	0.206899	+	0.978362i	$0.433663\pi$
$930$	0	0
$931$	3.46410i	0.113531i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−20.0926	−0.657098
$936$	0	0
$937$	3.07180	0.100351	0.0501756	−	0.998740i	$-0.484022\pi$
$937$	3.07180	0.100351	0.0501756	+	0.998740i	$0.484022\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 36.5541i	− 1.19163i	−0.803122	−	0.595814i	$-0.796829\pi$
$941$	− 36.5541i	− 1.19163i	0.803122	−	0.595814i	$-0.203171\pi$
$942$	0	0
$943$	19.8564	0.646614
$944$	0	0
$945$	0	0
$946$	0	0
$947$	26.0381i	0.846126i	0.906100	+	0.423063i	$0.139045\pi$
$947$	26.0381i	0.846126i	−0.906100	+	0.423063i	$0.860955\pi$
$948$	0	0
$949$	− 18.9282i	− 0.614435i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−33.1746	−1.07463	−0.537315	−	0.843381i	$-0.680561\pi$
$953$	−33.1746	−1.07463	−0.537315	+	0.843381i	$0.680561\pi$
$954$	0	0
$955$	− 29.8949i	− 0.967376i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	6.07137	0.196055
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 1.87849i	− 0.0604707i
$966$	0	0
$967$	−36.7846	−1.18291	−0.591457	−	0.806337i	$-0.701447\pi$
$967$	−36.7846	−1.18291	−0.591457	+	0.806337i	$0.701447\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 33.1746i	− 1.06462i	−0.846548	−	0.532312i	$-0.821323\pi$
$971$	− 33.1746i	− 1.06462i	0.846548	−	0.532312i	$-0.178677\pi$
$972$	0	0
$973$	− 9.85641i	− 0.315982i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	40.8728	1.30764	0.653818	−	0.756652i	$-0.273166\pi$
$977$	40.8728	1.30764	0.653818	+	0.756652i	$0.273166\pi$
$978$	0	0
$979$	− 0.823085i	− 0.0263059i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−59.3386	−1.89261	−0.946303	−	0.323280i	$-0.895214\pi$
$983$	−59.3386	−1.89261	−0.946303	+	0.323280i	$0.895214\pi$
$984$	0	0
$985$	−10.6410	−0.339051
$986$	0	0
$987$	0	0
$988$	0	0
$989$	14.9605i	0.475716i
$990$	0	0
$991$	−52.0000	−1.65183	−0.825917	−	0.563791i	$-0.809342\pi$
$991$	−52.0000	−1.65183	−0.825917	+	0.563791i	$0.809342\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	43.4389i	1.37710i
$996$	0	0
$997$	31.0333i	0.982835i	0.870924	+	0.491418i	$0.163521\pi$
$997$	31.0333i	0.982835i	−0.870924	+	0.491418i	$0.836479\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.c.q.2017.5	yes	8
3.2	odd	2	inner	4032.2.c.q.2017.3	✓	8
4.3	odd	2		4032.2.c.r.2017.5	yes	8
8.3	odd	2		4032.2.c.r.2017.4	yes	8
8.5	even	2	inner	4032.2.c.q.2017.4	yes	8
12.11	even	2		4032.2.c.r.2017.3	yes	8
24.5	odd	2	inner	4032.2.c.q.2017.6	yes	8
24.11	even	2		4032.2.c.r.2017.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4032.2.c.q.2017.3	✓	8	3.2	odd	2	inner
4032.2.c.q.2017.4	yes	8	8.5	even	2	inner
4032.2.c.q.2017.5	yes	8	1.1	even	1	trivial
4032.2.c.q.2017.6	yes	8	24.5	odd	2	inner
4032.2.c.r.2017.3	yes	8	12.11	even	2
4032.2.c.r.2017.4	yes	8	8.3	odd	2
4032.2.c.r.2017.5	yes	8	4.3	odd	2
4032.2.c.r.2017.6	yes	8	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.75265i q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+1.75265i q^{5} -1.00000 q^{7} -1.75265i q^{11} -1.46410i q^{13} -6.54099 q^{17} +3.46410i q^{19} +3.03569 q^{23} +1.92820 q^{25} -9.57668i q^{29} +2.00000 q^{31} -1.75265i q^{35} -4.53590i q^{37} +6.54099 q^{41} +4.92820i q^{43} +9.57668 q^{47} +1.00000 q^{49} +3.07180 q^{55} +9.57668i q^{59} +1.46410i q^{61} +2.56606 q^{65} +10.0000i q^{67} -10.0463 q^{71} +12.9282 q^{73} +1.75265i q^{77} +10.9282 q^{79} -6.07137i q^{83} -11.4641i q^{85} +0.469622 q^{89} +1.46410i q^{91} -6.07137 q^{95} +8.92820 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{7}+O(q^{10}) \) 8 * q - 8 * q^7	\( 8 q - 8 q^{7} - 40 q^{25} + 16 q^{31} + 8 q^{49} + 80 q^{55} + 48 q^{73} + 32 q^{79} + 16 q^{97}+O(q^{100}) \) 8 * q - 8 * q^7 - 40 * q^25 + 16 * q^31 + 8 * q^49 + 80 * q^55 + 48 * q^73 + 32 * q^79 + 16 * q^97

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