LMFDB - Embedded newform 4032.2.j.e.2591.8

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:	16.0.162447943996702457856.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{12} - 15x^{8} - 16x^{4} + 256 $ x^16 - x^12 - 15x^8 - 16x^4 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.8
Root			$0.825348 + 1.14839i$ of defining polynomial
Character	$\chi$	$=$	4032.2591
Dual form			4032.2.j.e.2591.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.646084 q^{5} +1.00000i q^{7} +O(q^{10})$	$q-0.646084 q^{5} +1.00000i q^{7} +3.09557i q^{11} +0.913701i q^{13} -6.77590i q^{17} -5.29150 q^{19} -2.53326 q^{23} -4.58258 q^{25} +9.93280 q^{29} -9.16515i q^{31} -0.646084i q^{35} -7.84190i q^{37} +9.01400i q^{41} +12.2197 q^{43} +5.65685 q^{47} -1.00000 q^{49} +1.15732 q^{53} -2.00000i q^{55} -6.92820i q^{61} -0.590327i q^{65} +7.84190 q^{67} +5.36169 q^{71} -0.417424 q^{73} -3.09557 q^{77} +10.7477i q^{79} +15.9891i q^{83} +4.37780i q^{85} -9.01400i q^{89} -0.913701 q^{91} +3.41875 q^{95} -11.5826 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 16 q^{49} - 80 q^{73} - 112 q^{97}+O(q^{100}) $ 16 * q - 16 * q^49 - 80 * q^73 - 112 * 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
$5$	−0.646084	−0.288937	−0.144469	−	0.989509i	$-0.546147\pi$
$5$	−0.646084	−0.288937	−0.144469	+	0.989509i	$0.546147\pi$
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.09557i	0.933351i	0.884429	+	0.466675i	$0.154548\pi$
$11$	3.09557i	0.933351i	−0.884429	+	0.466675i	$0.845452\pi$
$12$	0	0
$13$	0.913701i	0.253415i	0.991940	+	0.126707i	$0.0404409\pi$
$13$	0.913701i	0.253415i	−0.991940	+	0.126707i	$0.959559\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 6.77590i	− 1.64340i	−0.569922	−	0.821699i	$-0.693026\pi$
$17$	− 6.77590i	− 1.64340i	0.569922	−	0.821699i	$-0.306974\pi$
$18$	0	0
$19$	−5.29150	−1.21395	−0.606977	−	0.794719i	$-0.707618\pi$
$19$	−5.29150	−1.21395	−0.606977	+	0.794719i	$0.707618\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.53326	−0.528222	−0.264111	−	0.964492i	$-0.585079\pi$
$23$	−2.53326	−0.528222	−0.264111	+	0.964492i	$0.585079\pi$
$24$	0	0
$25$	−4.58258	−0.916515
$26$	0	0
$27$	0	0
$28$	0	0
$29$	9.93280	1.84448	0.922238	−	0.386623i	$-0.126359\pi$
$29$	9.93280	1.84448	0.922238	+	0.386623i	$0.126359\pi$
$30$	0	0
$31$	− 9.16515i	− 1.64611i	−0.567962	−	0.823055i	$-0.692268\pi$
$31$	− 9.16515i	− 1.64611i	0.567962	−	0.823055i	$-0.307732\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 0.646084i	− 0.109208i
$36$	0	0
$37$	− 7.84190i	− 1.28920i	−0.764520	−	0.644601i	$-0.777024\pi$
$37$	− 7.84190i	− 1.28920i	0.764520	−	0.644601i	$-0.222976\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.01400i	1.40775i	0.710323	+	0.703875i	$0.248549\pi$
$41$	9.01400i	1.40775i	−0.710323	+	0.703875i	$0.751451\pi$
$42$	0	0
$43$	12.2197	1.86349	0.931744	−	0.363116i	$-0.118287\pi$
$43$	12.2197	1.86349	0.931744	+	0.363116i	$0.118287\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.65685	0.825137	0.412568	−	0.910927i	$-0.364632\pi$
$47$	5.65685	0.825137	0.412568	+	0.910927i	$0.364632\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.15732	0.158970	0.0794852	−	0.996836i	$-0.474672\pi$
$53$	1.15732	0.158970	0.0794852	+	0.996836i	$0.474672\pi$
$54$	0	0
$55$	− 2.00000i	− 0.269680i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	− 6.92820i	− 0.887066i	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	− 6.92820i	− 0.887066i	0.896258	−	0.443533i	$-0.146275\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 0.590327i	− 0.0732211i
$66$	0	0
$67$	7.84190	0.958041	0.479021	−	0.877804i	$-0.340992\pi$
$67$	7.84190	0.958041	0.479021	+	0.877804i	$0.340992\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.36169	0.636316	0.318158	−	0.948038i	$-0.396936\pi$
$71$	5.36169	0.636316	0.318158	+	0.948038i	$0.396936\pi$
$72$	0	0
$73$	−0.417424	−0.0488558	−0.0244279	−	0.999702i	$-0.507776\pi$
$73$	−0.417424	−0.0488558	−0.0244279	+	0.999702i	$0.507776\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.09557	−0.352773
$78$	0	0
$79$	10.7477i	1.20921i	0.796524	+	0.604607i	$0.206670\pi$
$79$	10.7477i	1.20921i	−0.796524	+	0.604607i	$0.793330\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	15.9891i	1.75503i	0.479547	+	0.877516i	$0.340801\pi$
$83$	15.9891i	1.75503i	−0.479547	+	0.877516i	$0.659199\pi$
$84$	0	0
$85$	4.37780i	0.474839i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 9.01400i	− 0.955483i	−0.878501	−	0.477741i	$-0.841456\pi$
$89$	− 9.01400i	− 0.955483i	0.878501	−	0.477741i	$-0.158544\pi$
$90$	0	0
$91$	−0.913701	−0.0957818
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.41875	0.350757
$96$	0	0
$97$	−11.5826	−1.17603	−0.588016	−	0.808849i	$-0.700091\pi$
$97$	−11.5826	−1.17603	−0.588016	+	0.808849i	$0.700091\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	14.0509	1.39811	0.699056	−	0.715067i	$-0.253604\pi$
$101$	14.0509	1.39811	0.699056	+	0.715067i	$0.253604\pi$
$102$	0	0
$103$	2.00000i	0.197066i	0.995134	+	0.0985329i	$0.0314150\pi$
$103$	2.00000i	0.197066i	−0.995134	+	0.0985329i	$0.968585\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.70239i	0.647944i	0.946067	+	0.323972i	$0.105018\pi$
$107$	6.70239i	0.647944i	−0.946067	+	0.323972i	$0.894982\pi$
$108$	0	0
$109$	5.29150i	0.506834i	0.967357	+	0.253417i	$0.0815545\pi$
$109$	5.29150i	0.506834i	−0.967357	+	0.253417i	$0.918446\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 9.89949i	− 0.931266i	−0.884978	−	0.465633i	$-0.845827\pi$
$113$	− 9.89949i	− 0.931266i	0.884978	−	0.465633i	$-0.154173\pi$
$114$	0	0
$115$	1.63670	0.152623
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.77590	0.621146
$120$	0	0
$121$	1.41742	0.128857
$122$	0	0
$123$	0	0
$124$	0	0
$125$	6.19115	0.553753
$126$	0	0
$127$	− 4.41742i	− 0.391983i	−0.980606	−	0.195992i	$-0.937207\pi$
$127$	− 4.41742i	− 0.391983i	0.980606	−	0.195992i	$-0.0627925\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	11.0901i	0.968949i	0.874805	+	0.484474i	$0.160989\pi$
$131$	11.0901i	0.968949i	−0.874805	+	0.484474i	$0.839011\pi$
$132$	0	0
$133$	− 5.29150i	− 0.458831i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 5.89041i	− 0.503252i	−0.967825	−	0.251626i	$-0.919035\pi$
$137$	− 5.89041i	− 0.503252i	0.967825	−	0.251626i	$-0.0809653\pi$
$138$	0	0
$139$	−1.82740	−0.154998	−0.0774991	−	0.996992i	$-0.524693\pi$
$139$	−1.82740	−0.154998	−0.0774991	+	0.996992i	$0.524693\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.82843	−0.236525
$144$	0	0
$145$	−6.41742	−0.532938
$146$	0	0
$147$	0	0
$148$	0	0
$149$	19.7308	1.61641	0.808204	−	0.588903i	$-0.200440\pi$
$149$	19.7308	1.61641	0.808204	+	0.588903i	$0.200440\pi$
$150$	0	0
$151$	− 23.1652i	− 1.88515i	−0.333990	−	0.942577i	$-0.608395\pi$
$151$	− 23.1652i	− 1.88515i	0.333990	−	0.942577i	$-0.391605\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.92146i	0.475623i
$156$	0	0
$157$	− 15.6838i	− 1.25170i	−0.779942	−	0.625852i	$-0.784751\pi$
$157$	− 15.6838i	− 1.25170i	0.779942	−	0.625852i	$-0.215249\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 2.53326i	− 0.199649i
$162$	0	0
$163$	13.3241	1.04362	0.521812	−	0.853060i	$-0.325256\pi$
$163$	13.3241	1.04362	0.521812	+	0.853060i	$0.325256\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.89495	0.610930	0.305465	−	0.952203i	$-0.401188\pi$
$167$	7.89495	0.610930	0.305465	+	0.952203i	$0.401188\pi$
$168$	0	0
$169$	12.1652	0.935781
$170$	0	0
$171$	0	0
$172$	0	0
$173$	7.85971	0.597562	0.298781	−	0.954322i	$-0.403420\pi$
$173$	7.85971	0.597562	0.298781	+	0.954322i	$0.403420\pi$
$174$	0	0
$175$	− 4.58258i	− 0.346410i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	11.6014i	0.867127i	0.901123	+	0.433563i	$0.142744\pi$
$179$	11.6014i	0.867127i	−0.901123	+	0.433563i	$0.857256\pi$
$180$	0	0
$181$	− 9.66930i	− 0.718714i	−0.933200	−	0.359357i	$-0.882996\pi$
$181$	− 9.66930i	− 0.718714i	0.933200	−	0.359357i	$-0.117004\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.06653i	0.372498i
$186$	0	0
$187$	20.9753	1.53387
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.5613	1.48776	0.743881	−	0.668312i	$-0.232983\pi$
$191$	20.5613	1.48776	0.743881	+	0.668312i	$0.232983\pi$
$192$	0	0
$193$	6.74773	0.485712	0.242856	−	0.970062i	$-0.421916\pi$
$193$	6.74773	0.485712	0.242856	+	0.970062i	$0.421916\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−24.8994	−1.77401	−0.887006	−	0.461759i	$-0.847219\pi$
$197$	−24.8994	−1.77401	−0.887006	+	0.461759i	$0.847219\pi$
$198$	0	0
$199$	19.1652i	1.35858i	0.733869	+	0.679291i	$0.237712\pi$
$199$	19.1652i	1.35858i	−0.733869	+	0.679291i	$0.762288\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	9.93280i	0.697146i
$204$	0	0
$205$	− 5.82380i	− 0.406752i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 16.3802i	− 1.13304i
$210$	0	0
$211$	−0.190700	−0.0131284	−0.00656418	−	0.999978i	$-0.502089\pi$
$211$	−0.190700	−0.0131284	−0.00656418	+	0.999978i	$0.502089\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−7.89495	−0.538431
$216$	0	0
$217$	9.16515	0.622171
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6.19115	0.416462
$222$	0	0
$223$	− 20.0000i	− 1.33930i	−0.742677	−	0.669650i	$-0.766444\pi$
$223$	− 20.0000i	− 1.33930i	0.742677	−	0.669650i	$-0.233556\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 11.0901i	− 0.736078i	−0.929810	−	0.368039i	$-0.880029\pi$
$227$	− 11.0901i	− 0.736078i	0.929810	−	0.368039i	$-0.119971\pi$
$228$	0	0
$229$	− 11.4967i	− 0.759724i	−0.925043	−	0.379862i	$-0.875972\pi$
$229$	− 11.4967i	− 0.759724i	0.925043	−	0.379862i	$-0.124028\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 7.07107i	− 0.463241i	−0.972806	−	0.231621i	$-0.925597\pi$
$233$	− 7.07107i	− 0.463241i	0.972806	−	0.231621i	$-0.0744028\pi$
$234$	0	0
$235$	−3.65480	−0.238413
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5.95202	0.385004	0.192502	−	0.981297i	$-0.438340\pi$
$239$	5.95202	0.385004	0.192502	+	0.981297i	$0.438340\pi$
$240$	0	0
$241$	−20.4174	−1.31520	−0.657601	−	0.753366i	$-0.728429\pi$
$241$	−20.4174	−1.31520	−0.657601	+	0.753366i	$0.728429\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.646084	0.0412768
$246$	0	0
$247$	− 4.83485i	− 0.307634i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.02248i	0.0645381i	0.999479	+	0.0322691i	$0.0102733\pi$
$251$	1.02248i	0.0645381i	−0.999479	+	0.0322691i	$0.989727\pi$
$252$	0	0
$253$	− 7.84190i	− 0.493016i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 10.1947i	− 0.635925i	−0.948103	−	0.317963i	$-0.897001\pi$
$257$	− 10.1947i	− 0.635925i	0.948103	−	0.317963i	$-0.102999\pi$
$258$	0	0
$259$	7.84190	0.487272
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1.94294	−0.119807	−0.0599033	−	0.998204i	$-0.519079\pi$
$263$	−1.94294	−0.119807	−0.0599033	+	0.998204i	$0.519079\pi$
$264$	0	0
$265$	−0.747727	−0.0459325
$266$	0	0
$267$	0	0
$268$	0	0
$269$	25.1410	1.53287	0.766436	−	0.642320i	$-0.222028\pi$
$269$	25.1410	1.53287	0.766436	+	0.642320i	$0.222028\pi$
$270$	0	0
$271$	− 5.16515i	− 0.313761i	−0.987618	−	0.156880i	$-0.949856\pi$
$271$	− 5.16515i	− 0.313761i	0.987618	−	0.156880i	$-0.0501437\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 14.1857i	− 0.855430i
$276$	0	0
$277$	− 14.9608i	− 0.898908i	−0.893304	−	0.449454i	$-0.851619\pi$
$277$	− 14.9608i	− 0.898908i	0.893304	−	0.449454i	$-0.148381\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	20.6229i	1.23026i	0.788427	+	0.615129i	$0.210896\pi$
$281$	20.6229i	1.23026i	−0.788427	+	0.615129i	$0.789104\pi$
$282$	0	0
$283$	−17.3205	−1.02960	−0.514799	−	0.857311i	$-0.672133\pi$
$283$	−17.3205	−1.02960	−0.514799	+	0.857311i	$0.672133\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−9.01400	−0.532080
$288$	0	0
$289$	−28.9129	−1.70076
$290$	0	0
$291$	0	0
$292$	0	0
$293$	14.3205	0.836615	0.418308	−	0.908305i	$-0.362623\pi$
$293$	14.3205	0.836615	0.418308	+	0.908305i	$0.362623\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 2.31464i	− 0.133859i
$300$	0	0
$301$	12.2197i	0.704332i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.47620i	0.256306i
$306$	0	0
$307$	22.8027	1.30142	0.650710	−	0.759327i	$-0.274471\pi$
$307$	22.8027	1.30142	0.650710	+	0.759327i	$0.274471\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4.47620	−0.253822	−0.126911	−	0.991914i	$-0.540506\pi$
$311$	−4.47620	−0.253822	−0.126911	+	0.991914i	$0.540506\pi$
$312$	0	0
$313$	9.16515	0.518045	0.259022	−	0.965871i	$-0.416600\pi$
$313$	9.16515	0.518045	0.259022	+	0.965871i	$0.416600\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.134846	−0.00757368	−0.00378684	−	0.999993i	$-0.501205\pi$
$317$	−0.134846	−0.00757368	−0.00378684	+	0.999993i	$0.501205\pi$
$318$	0	0
$319$	30.7477i	1.72154i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	35.8547i	1.99501i
$324$	0	0
$325$	− 4.18710i	− 0.232259i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	5.65685i	0.311872i
$330$	0	0
$331$	−8.94630	−0.491733	−0.245867	−	0.969304i	$-0.579073\pi$
$331$	−8.94630	−0.491733	−0.245867	+	0.969304i	$0.579073\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−5.06653	−0.276814
$336$	0	0
$337$	−22.3303	−1.21641	−0.608205	−	0.793780i	$-0.708110\pi$
$337$	−22.3303	−1.21641	−0.608205	+	0.793780i	$0.708110\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	28.3714	1.53640
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	31.4670i	1.68924i	0.535370	+	0.844618i	$0.320172\pi$
$347$	31.4670i	1.68924i	−0.535370	+	0.844618i	$0.679828\pi$
$348$	0	0
$349$	21.1660i	1.13299i	0.824065	+	0.566495i	$0.191701\pi$
$349$	21.1660i	1.13299i	−0.824065	+	0.566495i	$0.808299\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 34.4698i	− 1.83464i	−0.398145	−	0.917322i	$-0.630346\pi$
$353$	− 34.4698i	− 1.83464i	0.398145	−	0.917322i	$-0.369654\pi$
$354$	0	0
$355$	−3.46410	−0.183855
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.18104	0.220667	0.110333	−	0.993895i	$-0.464808\pi$
$359$	4.18104	0.220667	0.110333	+	0.993895i	$0.464808\pi$
$360$	0	0
$361$	9.00000	0.473684
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0.269691	0.0141163
$366$	0	0
$367$	0.834849i	0.0435787i	0.999763	+	0.0217894i	$0.00693632\pi$
$367$	0.834849i	0.0435787i	−0.999763	+	0.0217894i	$0.993064\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.15732i	0.0600852i
$372$	0	0
$373$	21.1660i	1.09593i	0.836500	+	0.547967i	$0.184598\pi$
$373$	21.1660i	1.09593i	−0.836500	+	0.547967i	$0.815402\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.07561i	0.467418i
$378$	0	0
$379$	2.01810	0.103663	0.0518315	−	0.998656i	$-0.483494\pi$
$379$	2.01810	0.103663	0.0518315	+	0.998656i	$0.483494\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	9.07561	0.463742	0.231871	−	0.972747i	$-0.425515\pi$
$383$	9.07561	0.463742	0.231871	+	0.972747i	$0.425515\pi$
$384$	0	0
$385$	2.00000	0.101929
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−23.3376	−1.18326	−0.591631	−	0.806209i	$-0.701516\pi$
$389$	−23.3376	−1.18326	−0.591631	+	0.806209i	$0.701516\pi$
$390$	0	0
$391$	17.1652i	0.868079i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 6.94393i	− 0.349387i
$396$	0	0
$397$	8.75560i	0.439431i	0.975564	+	0.219716i	$0.0705129\pi$
$397$	8.75560i	0.439431i	−0.975564	+	0.219716i	$0.929487\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 32.5269i	− 1.62432i	−0.583437	−	0.812158i	$-0.698293\pi$
$401$	− 32.5269i	− 1.62432i	0.583437	−	0.812158i	$-0.301707\pi$
$402$	0	0
$403$	8.37420	0.417149
$404$	0	0
$405$	0	0
$406$	0	0
$407$	24.2752	1.20328
$408$	0	0
$409$	7.58258	0.374934	0.187467	−	0.982271i	$-0.439972\pi$
$409$	7.58258	0.374934	0.187467	+	0.982271i	$0.439972\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 10.3303i	− 0.507095i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 19.8656i	− 0.970499i	−0.874376	−	0.485249i	$-0.838729\pi$
$419$	− 19.8656i	− 0.970499i	0.874376	−	0.485249i	$-0.161271\pi$
$420$	0	0
$421$	− 11.3060i	− 0.551021i	−0.961298	−	0.275510i	$-0.911153\pi$
$421$	− 11.3060i	− 0.551021i	0.961298	−	0.275510i	$-0.0888469\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	31.0511i	1.50620i
$426$	0	0
$427$	6.92820	0.335279
$428$	0	0
$429$	0	0
$430$	0	0
$431$	17.7328	0.854161	0.427081	−	0.904214i	$-0.359542\pi$
$431$	17.7328	0.854161	0.427081	+	0.904214i	$0.359542\pi$
$432$	0	0
$433$	21.1652	1.01713	0.508566	−	0.861023i	$-0.330176\pi$
$433$	21.1652	1.01713	0.508566	+	0.861023i	$0.330176\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	13.4048	0.641237
$438$	0	0
$439$	− 18.3303i	− 0.874858i	−0.899253	−	0.437429i	$-0.855889\pi$
$439$	− 18.3303i	− 0.874858i	0.899253	−	0.437429i	$-0.144111\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 22.9612i	− 1.09092i	−0.838137	−	0.545459i	$-0.816355\pi$
$443$	− 22.9612i	− 1.09092i	0.838137	−	0.545459i	$-0.183645\pi$
$444$	0	0
$445$	5.82380i	0.276075i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	9.89949i	0.467186i	0.972334	+	0.233593i	$0.0750483\pi$
$449$	9.89949i	0.467186i	−0.972334	+	0.233593i	$0.924952\pi$
$450$	0	0
$451$	−27.9035	−1.31393
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0.590327	0.0276750
$456$	0	0
$457$	−3.16515	−0.148060	−0.0740298	−	0.997256i	$-0.523586\pi$
$457$	−3.16515	−0.148060	−0.0740298	+	0.997256i	$0.523586\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−31.6018	−1.47184	−0.735922	−	0.677067i	$-0.763251\pi$
$461$	−31.6018	−1.47184	−0.735922	+	0.677067i	$0.763251\pi$
$462$	0	0
$463$	− 11.5826i	− 0.538288i	−0.963100	−	0.269144i	$-0.913259\pi$
$463$	− 11.5826i	− 0.538288i	0.963100	−	0.269144i	$-0.0867407\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 33.5401i	− 1.55205i	−0.630703	−	0.776025i	$-0.717233\pi$
$467$	− 33.5401i	− 1.55205i	0.630703	−	0.776025i	$-0.282767\pi$
$468$	0	0
$469$	7.84190i	0.362105i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	37.8270i	1.73929i
$474$	0	0
$475$	24.2487	1.11261
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−35.5889	−1.62610	−0.813049	−	0.582195i	$-0.802194\pi$
$479$	−35.5889	−1.62610	−0.813049	+	0.582195i	$0.802194\pi$
$480$	0	0
$481$	7.16515	0.326703
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.48331	0.339800
$486$	0	0
$487$	− 22.3303i	− 1.01188i	−0.862568	−	0.505941i	$-0.831145\pi$
$487$	− 22.3303i	− 1.01188i	0.862568	−	0.505941i	$-0.168855\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	16.5003i	0.744650i	0.928102	+	0.372325i	$0.121439\pi$
$491$	16.5003i	0.744650i	−0.928102	+	0.372325i	$0.878561\pi$
$492$	0	0
$493$	− 67.3037i	− 3.03121i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.36169i	0.240505i
$498$	0	0
$499$	−22.4213	−1.00372	−0.501858	−	0.864950i	$-0.667350\pi$
$499$	−22.4213	−1.00372	−0.501858	+	0.864950i	$0.667350\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	27.6939	1.23481	0.617406	−	0.786645i	$-0.288184\pi$
$503$	27.6939	1.23481	0.617406	+	0.786645i	$0.288184\pi$
$504$	0	0
$505$	−9.07803	−0.403967
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−24.1185	−1.06903	−0.534517	−	0.845158i	$-0.679506\pi$
$509$	−24.1185	−1.06903	−0.534517	+	0.845158i	$0.679506\pi$
$510$	0	0
$511$	− 0.417424i	− 0.0184658i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 1.29217i	− 0.0569397i
$516$	0	0
$517$	17.5112i	0.770142i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 13.4902i	− 0.591017i	−0.955340	−	0.295508i	$-0.904511\pi$
$521$	− 13.4902i	− 0.591017i	0.955340	−	0.295508i	$-0.0954890\pi$
$522$	0	0
$523$	−40.5046	−1.77114	−0.885572	−	0.464503i	$-0.846233\pi$
$523$	−40.5046	−1.77114	−0.885572	+	0.464503i	$0.846233\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−62.1022	−2.70521
$528$	0	0
$529$	−16.5826	−0.720982
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−8.23610	−0.356745
$534$	0	0
$535$	− 4.33030i	− 0.187215i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 3.09557i	− 0.133336i
$540$	0	0
$541$	− 16.7882i	− 0.721781i	−0.932608	−	0.360891i	$-0.882473\pi$
$541$	− 16.7882i	− 0.721781i	0.932608	−	0.360891i	$-0.117527\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 3.41875i	− 0.146443i
$546$	0	0
$547$	7.46050	0.318988	0.159494	−	0.987199i	$-0.449014\pi$
$547$	7.46050	0.318988	0.159494	+	0.987199i	$0.449014\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−52.5595	−2.23911
$552$	0	0
$553$	−10.7477	−0.457040
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−11.9778	−0.507514	−0.253757	−	0.967268i	$-0.581666\pi$
$557$	−11.9778	−0.507514	−0.253757	+	0.967268i	$0.581666\pi$
$558$	0	0
$559$	11.1652i	0.472236i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 14.4272i	− 0.608036i	−0.952666	−	0.304018i	$-0.901672\pi$
$563$	− 14.4272i	− 0.608036i	0.952666	−	0.304018i	$-0.0983283\pi$
$564$	0	0
$565$	6.39590i	0.269078i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	34.1747i	1.43268i	0.697753	+	0.716339i	$0.254183\pi$
$569$	34.1747i	1.43268i	−0.697753	+	0.716339i	$0.745817\pi$
$570$	0	0
$571$	34.4901	1.44337	0.721683	−	0.692223i	$-0.243369\pi$
$571$	34.4901	1.44337	0.721683	+	0.692223i	$0.243369\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	11.6089	0.484123
$576$	0	0
$577$	5.16515	0.215028	0.107514	−	0.994204i	$-0.465711\pi$
$577$	5.16515	0.215028	0.107514	+	0.994204i	$0.465711\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−15.9891	−0.663340
$582$	0	0
$583$	3.58258i	0.148375i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	28.6411i	1.18214i	0.806619	+	0.591072i	$0.201295\pi$
$587$	28.6411i	1.18214i	−0.806619	+	0.591072i	$0.798705\pi$
$588$	0	0
$589$	48.4974i	1.99830i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 31.6414i	− 1.29936i	−0.760209	−	0.649679i	$-0.774903\pi$
$593$	− 31.6414i	− 1.29936i	0.760209	−	0.649679i	$-0.225097\pi$
$594$	0	0
$595$	−4.37780	−0.179472
$596$	0	0
$597$	0	0
$598$	0	0
$599$	27.3988	1.11948	0.559742	−	0.828667i	$-0.310900\pi$
$599$	27.3988	1.11948	0.559742	+	0.828667i	$0.310900\pi$
$600$	0	0
$601$	25.1652	1.02651	0.513254	−	0.858237i	$-0.328440\pi$
$601$	25.1652	1.02651	0.513254	+	0.858237i	$0.328440\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−0.915775	−0.0372315
$606$	0	0
$607$	14.3303i	0.581649i	0.956776	+	0.290825i	$0.0939296\pi$
$607$	14.3303i	0.581649i	−0.956776	+	0.290825i	$0.906070\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.16867i	0.209102i
$612$	0	0
$613$	41.9506i	1.69437i	0.531298	+	0.847185i	$0.321704\pi$
$613$	41.9506i	1.69437i	−0.531298	+	0.847185i	$0.678296\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11.0801i	0.446070i	0.974810	+	0.223035i	$0.0715964\pi$
$617$	11.0801i	0.446070i	−0.974810	+	0.223035i	$0.928404\pi$
$618$	0	0
$619$	0.381401	0.0153298	0.00766490	−	0.999971i	$-0.497560\pi$
$619$	0.381401	0.0153298	0.00766490	+	0.999971i	$0.497560\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	9.01400	0.361138
$624$	0	0
$625$	18.9129	0.756515
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−53.1360	−2.11867
$630$	0	0
$631$	13.9129i	0.553863i	0.960890	+	0.276931i	$0.0893175\pi$
$631$	13.9129i	0.553863i	−0.960890	+	0.276931i	$0.910682\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2.85403i	0.113259i
$636$	0	0
$637$	− 0.913701i	− 0.0362021i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 38.0606i	− 1.50330i	−0.659561	−	0.751651i	$-0.729258\pi$
$641$	− 38.0606i	− 1.50330i	0.659561	−	0.751651i	$-0.270742\pi$
$642$	0	0
$643$	−20.5939	−0.812145	−0.406072	−	0.913841i	$-0.633102\pi$
$643$	−20.5939	−0.812145	−0.406072	+	0.913841i	$0.633102\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−20.3893	−0.801587	−0.400793	−	0.916168i	$-0.631265\pi$
$647$	−20.3893	−0.801587	−0.400793	+	0.916168i	$0.631265\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	37.0120	1.44839	0.724196	−	0.689594i	$-0.242211\pi$
$653$	37.0120	1.44839	0.724196	+	0.689594i	$0.242211\pi$
$654$	0	0
$655$	− 7.16515i	− 0.279966i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	2.07310i	0.0807564i	0.999184	+	0.0403782i	$0.0128563\pi$
$659$	2.07310i	0.0807564i	−0.999184	+	0.0403782i	$0.987144\pi$
$660$	0	0
$661$	12.4104i	0.482709i	0.970437	+	0.241354i	$0.0775916\pi$
$661$	12.4104i	0.482709i	−0.970437	+	0.241354i	$0.922408\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.41875i	0.132574i
$666$	0	0
$667$	−25.1624	−0.974292
$668$	0	0
$669$	0	0
$670$	0	0
$671$	21.4468	0.827943
$672$	0	0
$673$	5.91288	0.227925	0.113962	−	0.993485i	$-0.463646\pi$
$673$	5.91288	0.227925	0.113962	+	0.993485i	$0.463646\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	16.9049	0.649707	0.324854	−	0.945764i	$-0.394685\pi$
$677$	16.9049	0.649707	0.324854	+	0.945764i	$0.394685\pi$
$678$	0	0
$679$	− 11.5826i	− 0.444498i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 36.9053i	− 1.41214i	−0.708140	−	0.706072i	$-0.750466\pi$
$683$	− 36.9053i	− 1.41214i	0.708140	−	0.706072i	$-0.249534\pi$
$684$	0	0
$685$	3.80570i	0.145408i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.05745i	0.0402855i
$690$	0	0
$691$	16.0652	0.611149	0.305575	−	0.952168i	$-0.401151\pi$
$691$	16.0652	0.611149	0.305575	+	0.952168i	$0.401151\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.18065	0.0447848
$696$	0	0
$697$	61.0780	2.31350
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−30.8209	−1.16409	−0.582044	−	0.813157i	$-0.697747\pi$
$701$	−30.8209	−1.16409	−0.582044	+	0.813157i	$0.697747\pi$
$702$	0	0
$703$	41.4955i	1.56503i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	14.0509i	0.528437i
$708$	0	0
$709$	− 29.7309i	− 1.11657i	−0.829650	−	0.558284i	$-0.811460\pi$
$709$	− 29.7309i	− 1.11657i	0.829650	−	0.558284i	$-0.188540\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	23.2177i	0.869511i
$714$	0	0
$715$	1.82740	0.0683409
$716$	0	0
$717$	0	0
$718$	0	0
$719$	22.5042	0.839265	0.419633	−	0.907694i	$-0.362159\pi$
$719$	22.5042	0.839265	0.419633	+	0.907694i	$0.362159\pi$
$720$	0	0
$721$	−2.00000	−0.0744839
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−45.5178	−1.69049
$726$	0	0
$727$	19.4955i	0.723046i	0.932363	+	0.361523i	$0.117743\pi$
$727$	19.4955i	0.723046i	−0.932363	+	0.361523i	$0.882257\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 82.7996i	− 3.06245i
$732$	0	0
$733$	2.35970i	0.0871575i	0.999050	+	0.0435788i	$0.0138759\pi$
$733$	2.35970i	0.0871575i	−0.999050	+	0.0435788i	$0.986124\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	24.2752i	0.894188i
$738$	0	0
$739$	−7.84190	−0.288469	−0.144235	−	0.989544i	$-0.546072\pi$
$739$	−7.84190	−0.288469	−0.144235	+	0.989544i	$0.546072\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−43.1887	−1.58444	−0.792220	−	0.610236i	$-0.791075\pi$
$743$	−43.1887	−1.58444	−0.792220	+	0.610236i	$0.791075\pi$
$744$	0	0
$745$	−12.7477	−0.467041
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−6.70239	−0.244900
$750$	0	0
$751$	35.1652i	1.28319i	0.767042	+	0.641597i	$0.221728\pi$
$751$	35.1652i	1.28319i	−0.767042	+	0.641597i	$0.778272\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	14.9666i	0.544691i
$756$	0	0
$757$	41.7599i	1.51779i	0.651213	+	0.758895i	$0.274260\pi$
$757$	41.7599i	1.51779i	−0.651213	+	0.758895i	$0.725740\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 5.00492i	− 0.181428i	−0.995877	−	0.0907142i	$-0.971085\pi$
$761$	− 5.00492i	− 0.181428i	0.995877	−	0.0907142i	$-0.0289150\pi$
$762$	0	0
$763$	−5.29150	−0.191565
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	47.3212	1.70203	0.851013	−	0.525144i	$-0.175989\pi$
$773$	47.3212	1.70203	0.851013	+	0.525144i	$0.175989\pi$
$774$	0	0
$775$	42.0000i	1.50868i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 47.6976i	− 1.70894i
$780$	0	0
$781$	16.5975i	0.593906i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	10.1331i	0.361664i
$786$	0	0
$787$	−53.9796	−1.92417	−0.962083	−	0.272757i	$-0.912064\pi$
$787$	−53.9796	−1.92417	−0.962083	+	0.272757i	$0.912064\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	9.89949	0.351986
$792$	0	0
$793$	6.33030	0.224796
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1.39887	−0.0495505	−0.0247752	−	0.999693i	$-0.507887\pi$
$797$	−1.39887	−0.0495505	−0.0247752	+	0.999693i	$0.507887\pi$
$798$	0	0
$799$	− 38.3303i	− 1.35603i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 1.29217i	− 0.0455996i
$804$	0	0
$805$	1.63670i	0.0576861i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	47.7265i	1.67797i	0.544151	+	0.838987i	$0.316852\pi$
$809$	47.7265i	1.67797i	−0.544151	+	0.838987i	$0.683148\pi$
$810$	0	0
$811$	−5.48220	−0.192506	−0.0962531	−	0.995357i	$-0.530686\pi$
$811$	−5.48220	−0.192506	−0.0962531	+	0.995357i	$0.530686\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−8.60849	−0.301542
$816$	0	0
$817$	−64.6606	−2.26219
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−3.74166	−0.130585	−0.0652924	−	0.997866i	$-0.520798\pi$
$821$	−3.74166	−0.130585	−0.0652924	+	0.997866i	$0.520798\pi$
$822$	0	0
$823$	1.91288i	0.0666788i	0.999444	+	0.0333394i	$0.0106142\pi$
$823$	1.91288i	0.0666788i	−0.999444	+	0.0333394i	$0.989386\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 30.7142i	− 1.06804i	−0.845473	−	0.534018i	$-0.820681\pi$
$827$	− 30.7142i	− 1.06804i	0.845473	−	0.534018i	$-0.179319\pi$
$828$	0	0
$829$	30.8353i	1.07095i	0.844550	+	0.535477i	$0.179868\pi$
$829$	30.8353i	1.07095i	−0.844550	+	0.535477i	$0.820132\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.77590i	0.234771i
$834$	0	0
$835$	−5.10080	−0.176521
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−39.0077	−1.34669	−0.673347	−	0.739327i	$-0.735144\pi$
$839$	−39.0077	−1.34669	−0.673347	+	0.739327i	$0.735144\pi$
$840$	0	0
$841$	69.6606	2.40209
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−7.85971	−0.270382
$846$	0	0
$847$	1.41742i	0.0487033i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	19.8656i	0.680984i
$852$	0	0
$853$	− 35.0224i	− 1.19914i	−0.800321	−	0.599572i	$-0.795337\pi$
$853$	− 35.0224i	− 1.19914i	0.800321	−	0.599572i	$-0.204663\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 13.0231i	− 0.444860i	−0.974949	−	0.222430i	$-0.928601\pi$
$857$	− 13.0231i	− 0.444860i	0.974949	−	0.222430i	$-0.0713989\pi$
$858$	0	0
$859$	−48.6881	−1.66122	−0.830609	−	0.556857i	$-0.812007\pi$
$859$	−48.6881	−1.66122	−0.830609	+	0.556857i	$0.812007\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−16.6754	−0.567637	−0.283819	−	0.958878i	$-0.591601\pi$
$863$	−16.6754	−0.567637	−0.283819	+	0.958878i	$0.591601\pi$
$864$	0	0
$865$	−5.07803	−0.172658
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−33.2704	−1.12862
$870$	0	0
$871$	7.16515i	0.242782i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	6.19115i	0.209299i
$876$	0	0
$877$	− 7.30960i	− 0.246828i	−0.992355	−	0.123414i	$-0.960616\pi$
$877$	− 7.30960i	− 0.246828i	0.992355	−	0.123414i	$-0.0393843\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	18.0896i	0.609455i	0.952440	+	0.304727i	$0.0985653\pi$
$881$	18.0896i	0.609455i	−0.952440	+	0.304727i	$0.901435\pi$
$882$	0	0
$883$	24.2487	0.816034	0.408017	−	0.912974i	$-0.366220\pi$
$883$	24.2487	0.816034	0.408017	+	0.912974i	$0.366220\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	22.5042	0.755617	0.377809	−	0.925884i	$-0.376678\pi$
$887$	22.5042	0.755617	0.377809	+	0.925884i	$0.376678\pi$
$888$	0	0
$889$	4.41742	0.148156
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−29.9333	−1.00168
$894$	0	0
$895$	− 7.49545i	− 0.250545i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 91.0357i	− 3.03621i
$900$	0	0
$901$	− 7.84190i	− 0.261252i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6.24718i	0.207663i
$906$	0	0
$907$	13.6657	0.453762	0.226881	−	0.973922i	$-0.427147\pi$
$907$	13.6657	0.453762	0.226881	+	0.973922i	$0.427147\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−13.2566	−0.439212	−0.219606	−	0.975589i	$-0.570477\pi$
$911$	−13.2566	−0.439212	−0.219606	+	0.975589i	$0.570477\pi$
$912$	0	0
$913$	−49.4955	−1.63806
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−11.0901	−0.366228
$918$	0	0
$919$	12.8348i	0.423383i	0.977337	+	0.211691i	$0.0678971\pi$
$919$	12.8348i	0.423383i	−0.977337	+	0.211691i	$0.932103\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	4.89898i	0.161252i
$924$	0	0
$925$	35.9361i	1.18157i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1.11905i	− 0.0367148i	−0.999831	−	0.0183574i	$-0.994156\pi$
$929$	− 1.11905i	− 0.0367148i	0.999831	−	0.0183574i	$-0.00584368\pi$
$930$	0	0
$931$	5.29150	0.173422
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−13.5518	−0.443192
$936$	0	0
$937$	39.4955	1.29026	0.645130	−	0.764073i	$-0.276803\pi$
$937$	39.4955	1.29026	0.645130	+	0.764073i	$0.276803\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	12.7587	0.415921	0.207961	−	0.978137i	$-0.433317\pi$
$941$	12.7587	0.415921	0.207961	+	0.978137i	$0.433317\pi$
$942$	0	0
$943$	− 22.8348i	− 0.743605i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.34279i	0.0761304i	0.999275	+	0.0380652i	$0.0121195\pi$
$947$	2.34279i	0.0761304i	−0.999275	+	0.0380652i	$0.987881\pi$
$948$	0	0
$949$	− 0.381401i	− 0.0123808i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	27.4604i	0.889529i	0.895648	+	0.444765i	$0.146713\pi$
$953$	27.4604i	0.889529i	−0.895648	+	0.444765i	$0.853287\pi$
$954$	0	0
$955$	−13.2843	−0.429870
$956$	0	0
$957$	0	0
$958$	0	0
$959$	5.89041	0.190211
$960$	0	0
$961$	−53.0000	−1.70968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−4.35960	−0.140340
$966$	0	0
$967$	− 13.0780i	− 0.420561i	−0.977641	−	0.210281i	$-0.932562\pi$
$967$	− 13.0780i	− 0.420561i	0.977641	−	0.210281i	$-0.0674377\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 9.52827i	− 0.305777i	−0.988243	−	0.152888i	$-0.951143\pi$
$971$	− 9.52827i	− 0.305777i	0.988243	−	0.152888i	$-0.0488575\pi$
$972$	0	0
$973$	− 1.82740i	− 0.0585838i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.01362i	0.192393i	0.995362	+	0.0961964i	$0.0306677\pi$
$977$	6.01362i	0.192393i	−0.995362	+	0.0961964i	$0.969332\pi$
$978$	0	0
$979$	27.9035	0.891800
$980$	0	0
$981$	0	0
$982$	0	0
$983$	27.5707	0.879370	0.439685	−	0.898152i	$-0.355090\pi$
$983$	27.5707	0.879370	0.439685	+	0.898152i	$0.355090\pi$
$984$	0	0
$985$	16.0871	0.512578
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−30.9557	−0.984335
$990$	0	0
$991$	− 20.0000i	− 0.635321i	−0.948205	−	0.317660i	$-0.897103\pi$
$991$	− 20.0000i	− 0.635321i	0.948205	−	0.317660i	$-0.102897\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 12.3823i	− 0.392545i
$996$	0	0
$997$	− 6.54680i	− 0.207339i	−0.994612	−	0.103670i	$-0.966942\pi$
$997$	− 6.54680i	− 0.207339i	0.994612	−	0.103670i	$-0.0330585\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.j.e.2591.8	yes	16
3.2	odd	2	inner	4032.2.j.e.2591.12	yes	16
4.3	odd	2	inner	4032.2.j.e.2591.6	yes	16
8.3	odd	2	inner	4032.2.j.e.2591.9	yes	16
8.5	even	2	inner	4032.2.j.e.2591.11	yes	16
12.11	even	2	inner	4032.2.j.e.2591.10	yes	16
24.5	odd	2	inner	4032.2.j.e.2591.7	yes	16
24.11	even	2	inner	4032.2.j.e.2591.5	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4032.2.j.e.2591.5	✓	16	24.11	even	2	inner
4032.2.j.e.2591.6	yes	16	4.3	odd	2	inner
4032.2.j.e.2591.7	yes	16	24.5	odd	2	inner
4032.2.j.e.2591.8	yes	16	1.1	even	1	trivial
4032.2.j.e.2591.9	yes	16	8.3	odd	2	inner
4032.2.j.e.2591.10	yes	16	12.11	even	2	inner
4032.2.j.e.2591.11	yes	16	8.5	even	2	inner
4032.2.j.e.2591.12	yes	16	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.646084 q^{5} +1.00000i q^{7} +O(q^{10})\)	\(q-0.646084 q^{5} +1.00000i q^{7} +3.09557i q^{11} +0.913701i q^{13} -6.77590i q^{17} -5.29150 q^{19} -2.53326 q^{23} -4.58258 q^{25} +9.93280 q^{29} -9.16515i q^{31} -0.646084i q^{35} -7.84190i q^{37} +9.01400i q^{41} +12.2197 q^{43} +5.65685 q^{47} -1.00000 q^{49} +1.15732 q^{53} -2.00000i q^{55} -6.92820i q^{61} -0.590327i q^{65} +7.84190 q^{67} +5.36169 q^{71} -0.417424 q^{73} -3.09557 q^{77} +10.7477i q^{79} +15.9891i q^{83} +4.37780i q^{85} -9.01400i q^{89} -0.913701 q^{91} +3.41875 q^{95} -11.5826 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 16 q^{49} - 80 q^{73} - 112 q^{97}+O(q^{100}) \) 16 * q - 16 * q^49 - 80 * q^73 - 112 * q^97

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