LMFDB - Embedded newform 4032.2.b.r.3583.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(3583,4032)]

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, 0, 0, 1]))

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

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

chi := DirichletCharacter("4032.3583");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$32.1956820950$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.29960650073923649536.7
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 7x^{12} + 40x^{8} - 112x^{4} + 256 $ x^16 - 7x^12 + 40x^8 - 112*x^4 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{30} $
Twist minimal:	no (minimal twist has level 2016)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3583.3
Root			$-1.32968 - 0.481610i$ of defining polynomial
Character	$\chi$	$=$	4032.3583
Dual form			4032.2.b.r.3583.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.33513i q^{5} +(0.662153 - 2.56155i) q^{7} -5.03680i q^{11} +6.04090i q^{13} -5.20798i q^{17} -4.71659 q^{19} +2.20837i q^{23} -6.12311 q^{25} -7.24517 q^{29} +6.04090 q^{31} +(-8.54312 - 2.20837i) q^{35} -5.12311 q^{37} -5.20798i q^{41} -9.12311i q^{43} +3.74571 q^{47} +(-6.12311 - 3.39228i) q^{49} -1.58831 q^{53} -16.7984 q^{55} +13.3405 q^{59} -12.8255i q^{61} +20.1472 q^{65} +9.12311i q^{67} +12.2820i q^{71} +12.0818i q^{73} +(-12.9020 - 3.33513i) q^{77} +5.12311i q^{79} -3.74571 q^{83} -17.3693 q^{85} +1.46228i q^{89} +(15.4741 + 4.00000i) q^{91} +15.7304i q^{95} -6.78456i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{25} - 16 q^{37} - 32 q^{49} - 80 q^{85}+O(q^{100}) $ 16 * q - 32 * q^25 - 16 * q^37 - 32 * q^49 - 80 * q^85

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
$5$		−	3.33513i		−	1.49152i	−0.666217	−	0.745758i	$-0.732087\pi$
$5$		−	3.33513i		−	1.49152i	0.666217	−	0.745758i	$-0.267913\pi$
$6$		0			0
$7$	0.662153	−	2.56155i	0.250270	−	0.968176i
$7$	0.662153	−	2.56155i	0.250270	−	0.968176i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	5.03680i		−	1.51865i	−0.650711	−	0.759326i	$-0.725529\pi$
$11$		−	5.03680i		−	1.51865i	0.650711	−	0.759326i	$-0.274471\pi$
$12$		0			0
$13$			6.04090i			1.67544i	0.546098	+	0.837722i	$0.316113\pi$
$13$			6.04090i			1.67544i	−0.546098	+	0.837722i	$0.683887\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	5.20798i		−	1.26312i	−0.775326	−	0.631561i	$-0.782415\pi$
$17$		−	5.20798i		−	1.26312i	0.775326	−	0.631561i	$-0.217585\pi$
$18$		0			0
$19$	−4.71659			−1.08206			−0.541030	−	0.841003i	$-0.681965\pi$
$19$	−4.71659			−1.08206			−0.541030	+	0.841003i	$0.681965\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			2.20837i			0.460477i	0.973134	+	0.230238i	$0.0739506\pi$
$23$			2.20837i			0.460477i	−0.973134	+	0.230238i	$0.926049\pi$
$24$		0			0
$25$	−6.12311			−1.22462
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−7.24517			−1.34539			−0.672697	−	0.739918i	$-0.734864\pi$
$29$	−7.24517			−1.34539			−0.672697	+	0.739918i	$0.734864\pi$
$30$		0			0
$31$	6.04090			1.08498			0.542488	−	0.840063i	$-0.317482\pi$
$31$	6.04090			1.08498			0.542488	+	0.840063i	$0.317482\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−8.54312	−	2.20837i	−1.44405	−	0.373283i
$36$		0			0
$37$	−5.12311			−0.842233			−0.421117	−	0.907006i	$-0.638362\pi$
$37$	−5.12311			−0.842233			−0.421117	+	0.907006i	$0.638362\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	5.20798i		−	0.813351i	−0.913573	−	0.406675i	$-0.866688\pi$
$41$		−	5.20798i		−	0.813351i	0.913573	−	0.406675i	$-0.133312\pi$
$42$		0			0
$43$		−	9.12311i		−	1.39126i	−0.718400	−	0.695630i	$-0.755125\pi$
$43$		−	9.12311i		−	1.39126i	0.718400	−	0.695630i	$-0.244875\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.74571			0.546367			0.273184	−	0.961962i	$-0.411923\pi$
$47$	3.74571			0.546367			0.273184	+	0.961962i	$0.411923\pi$
$48$		0			0
$49$	−6.12311	−	3.39228i	−0.874729	−	0.484612i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.58831			−0.218171			−0.109086	−	0.994032i	$-0.534792\pi$
$53$	−1.58831			−0.218171			−0.109086	+	0.994032i	$0.534792\pi$
$54$		0			0
$55$	−16.7984			−2.26509
$56$		0			0
$57$		0			0
$58$		0			0
$59$	13.3405			1.73679			0.868394	−	0.495874i	$-0.165152\pi$
$59$	13.3405			1.73679			0.868394	+	0.495874i	$0.165152\pi$
$60$		0			0
$61$		−	12.8255i		−	1.64213i	−0.570833	−	0.821066i	$-0.693380\pi$
$61$		−	12.8255i		−	1.64213i	0.570833	−	0.821066i	$-0.306620\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	20.1472			2.49895
$66$		0			0
$67$			9.12311i			1.11456i	0.830323	+	0.557282i	$0.188156\pi$
$67$			9.12311i			1.11456i	−0.830323	+	0.557282i	$0.811844\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			12.2820i			1.45760i	0.684726	+	0.728800i	$0.259922\pi$
$71$			12.2820i			1.45760i	−0.684726	+	0.728800i	$0.740078\pi$
$72$		0			0
$73$			12.0818i			1.41407i	0.707180	+	0.707033i	$0.249967\pi$
$73$			12.0818i			1.41407i	−0.707180	+	0.707033i	$0.750033\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−12.9020	−	3.33513i	−1.47032	−	0.380074i
$78$		0			0
$79$			5.12311i			0.576394i	0.957571	+	0.288197i	$0.0930559\pi$
$79$			5.12311i			0.576394i	−0.957571	+	0.288197i	$0.906944\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3.74571			−0.411145			−0.205572	−	0.978642i	$-0.565906\pi$
$83$	−3.74571			−0.411145			−0.205572	+	0.978642i	$0.565906\pi$
$84$		0			0
$85$	−17.3693			−1.88397
$86$		0			0
$87$		0			0
$88$		0			0
$89$			1.46228i			0.155001i	0.996992	+	0.0775006i	$0.0246940\pi$
$89$			1.46228i			0.155001i	−0.996992	+	0.0775006i	$0.975306\pi$
$90$		0			0
$91$	15.4741	+	4.00000i	1.62212	+	0.419314i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			15.7304i			1.61391i
$96$		0			0
$97$		−	6.78456i		−	0.688868i	−0.938811	−	0.344434i	$-0.888071\pi$
$97$		−	6.78456i		−	0.688868i	0.938811	−	0.344434i	$-0.111929\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			7.08084i			0.704570i	0.935893	+	0.352285i	$0.114595\pi$
$101$			7.08084i			0.704570i	−0.935893	+	0.352285i	$0.885405\pi$
$102$		0			0
$103$	12.8255			1.26373			0.631865	−	0.775078i	$-0.282290\pi$
$103$	12.8255			1.26373			0.631865	+	0.775078i	$0.282290\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	3.79668i		−	0.367039i	−0.983016	−	0.183519i	$-0.941251\pi$
$107$		−	3.79668i		−	0.367039i	0.983016	−	0.183519i	$-0.0587490\pi$
$108$		0			0
$109$	−8.24621			−0.789844			−0.394922	−	0.918715i	$-0.629228\pi$
$109$	−8.24621			−0.789844			−0.394922	+	0.918715i	$0.629228\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−5.65685			−0.532152			−0.266076	−	0.963952i	$-0.585727\pi$
$113$	−5.65685			−0.532152			−0.266076	+	0.963952i	$0.585727\pi$
$114$		0			0
$115$	7.36520			0.686809
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−13.3405	−	3.44849i	−1.22292	−	0.316122i
$120$		0			0
$121$	−14.3693			−1.30630
$122$		0			0
$123$		0			0
$124$		0			0
$125$			3.74571i			0.335026i
$126$		0			0
$127$		−	2.87689i		−	0.255283i	−0.991820	−	0.127642i	$-0.959259\pi$
$127$		−	2.87689i		−	0.255283i	0.991820	−	0.127642i	$-0.0407407\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−13.3405			−1.16557			−0.582784	−	0.812627i	$-0.698037\pi$
$131$	−13.3405			−1.16557			−0.582784	+	0.812627i	$0.698037\pi$
$132$		0			0
$133$	−3.12311	+	12.0818i	−0.270808	+	1.04762i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	3.17662			0.271397			0.135699	−	0.990750i	$-0.456672\pi$
$137$	3.17662			0.271397			0.135699	+	0.990750i	$0.456672\pi$
$138$		0			0
$139$	−5.29723			−0.449305			−0.224652	−	0.974439i	$-0.572125\pi$
$139$	−5.29723			−0.449305			−0.224652	+	0.974439i	$0.572125\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	30.4268			2.54441
$144$		0			0
$145$			24.1636i			2.00668i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.58831			0.130120			0.0650598	−	0.997881i	$-0.479276\pi$
$149$	1.58831			0.130120			0.0650598	+	0.997881i	$0.479276\pi$
$150$		0			0
$151$			2.87689i			0.234118i	0.993125	+	0.117059i	$0.0373467\pi$
$151$			2.87689i			0.234118i	−0.993125	+	0.117059i	$0.962653\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	20.1472i		−	1.61826i
$156$		0			0
$157$			11.3381i			0.904881i	0.891795	+	0.452440i	$0.149446\pi$
$157$			11.3381i			0.904881i	−0.891795	+	0.452440i	$0.850554\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	5.65685	+	1.46228i	0.445823	+	0.115244i
$162$		0			0
$163$			1.12311i			0.0879684i	0.999032	+	0.0439842i	$0.0140051\pi$
$163$			1.12311i			0.0879684i	−0.999032	+	0.0439842i	$0.985995\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−23.4924			−1.80711
$170$		0			0
$171$		0			0
$172$		0			0
$173$			20.4214i			1.55261i	0.630359	+	0.776304i	$0.282908\pi$
$173$			20.4214i			1.55261i	−0.630359	+	0.776304i	$0.717092\pi$
$174$		0			0
$175$	−4.05444	+	15.6847i	−0.306487	+	1.18565i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	22.0074i		−	1.64491i	−0.568832	−	0.822454i	$-0.692605\pi$
$179$		−	22.0074i		−	1.64491i	0.568832	−	0.822454i	$-0.307395\pi$
$180$		0			0
$181$			6.04090i			0.449016i	0.974472	+	0.224508i	$0.0720775\pi$
$181$			6.04090i			0.449016i	−0.974472	+	0.224508i	$0.927922\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			17.0862i			1.25620i
$186$		0			0
$187$	−26.2316			−1.91824
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	9.10534i		−	0.658839i	−0.944184	−	0.329420i	$-0.893147\pi$
$191$		−	9.10534i		−	0.658839i	0.944184	−	0.329420i	$-0.106853\pi$
$192$		0			0
$193$	−11.3693			−0.818381			−0.409191	−	0.912449i	$-0.634189\pi$
$193$	−11.3693			−0.818381			−0.409191	+	0.912449i	$0.634189\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−7.24517			−0.516197			−0.258098	−	0.966119i	$-0.583096\pi$
$197$	−7.24517			−0.516197			−0.258098	+	0.966119i	$0.583096\pi$
$198$		0			0
$199$	8.10887			0.574823			0.287411	−	0.957807i	$-0.407205\pi$
$199$	8.10887			0.574823			0.287411	+	0.957807i	$0.407205\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−4.79741	+	18.5589i	−0.336712	+	1.30258i
$204$		0			0
$205$	−17.3693			−1.21313
$206$		0			0
$207$		0			0
$208$		0			0
$209$			23.7565i			1.64327i
$210$		0			0
$211$			3.36932i			0.231953i	0.993252	+	0.115977i	$0.0369998\pi$
$211$			3.36932i			0.231953i	−0.993252	+	0.115977i	$0.963000\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−30.4268			−2.07509
$216$		0			0
$217$	4.00000	−	15.4741i	0.271538	−	1.05045i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	31.4609			2.11629
$222$		0			0
$223$	3.97292			0.266046			0.133023	−	0.991113i	$-0.457532\pi$
$223$	3.97292			0.266046			0.133023	+	0.991113i	$0.457532\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	9.59482			0.636831			0.318415	−	0.947951i	$-0.396849\pi$
$227$	9.59482			0.636831			0.318415	+	0.947951i	$0.396849\pi$
$228$		0			0
$229$			11.3381i			0.749244i	0.927178	+	0.374622i	$0.122228\pi$
$229$			11.3381i			0.749244i	−0.927178	+	0.374622i	$0.877772\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	16.9706			1.11178			0.555889	−	0.831256i	$-0.312378\pi$
$233$	16.9706			1.11178			0.555889	+	0.831256i	$0.312378\pi$
$234$		0			0
$235$		−	12.4924i		−	0.814916i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	21.1154i		−	1.36584i	−0.730491	−	0.682922i	$-0.760709\pi$
$239$		−	21.1154i		−	1.36584i	0.730491	−	0.682922i	$-0.239291\pi$
$240$		0			0
$241$		−	18.8664i		−	1.21529i	−0.794209	−	0.607644i	$-0.792115\pi$
$241$		−	18.8664i		−	1.21529i	0.794209	−	0.607644i	$-0.207885\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−11.3137	+	20.4214i	−0.722806	+	1.30467i
$246$		0			0
$247$		−	28.4924i		−	1.81293i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	17.0862			1.07847			0.539237	−	0.842154i	$-0.318713\pi$
$251$	17.0862			1.07847			0.539237	+	0.842154i	$0.318713\pi$
$252$		0			0
$253$	11.1231			0.699304
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	14.8028i		−	0.923374i	−0.887043	−	0.461687i	$-0.847244\pi$
$257$		−	14.8028i		−	0.923374i	0.887043	−	0.461687i	$-0.152756\pi$
$258$		0			0
$259$	−3.39228	+	13.1231i	−0.210786	+	0.815430i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			14.7622i			0.910276i	0.890421	+	0.455138i	$0.150410\pi$
$263$			14.7622i			0.910276i	−0.890421	+	0.455138i	$0.849590\pi$
$264$		0			0
$265$			5.29723i			0.325406i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	17.4968i		−	1.06680i	−0.845863	−	0.533400i	$-0.820914\pi$
$269$		−	17.4968i		−	1.06680i	0.845863	−	0.533400i	$-0.179086\pi$
$270$		0			0
$271$	−0.743668			−0.0451746			−0.0225873	−	0.999745i	$-0.507190\pi$
$271$	−0.743668			−0.0451746			−0.0225873	+	0.999745i	$0.507190\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			30.8408i			1.85977i
$276$		0			0
$277$	−23.3693			−1.40413			−0.702063	−	0.712115i	$-0.747738\pi$
$277$	−23.3693			−1.40413			−0.702063	+	0.712115i	$0.747738\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−5.65685			−0.337460			−0.168730	−	0.985662i	$-0.553967\pi$
$281$	−5.65685			−0.337460			−0.168730	+	0.985662i	$0.553967\pi$
$282$		0			0
$283$	−7.36520			−0.437816			−0.218908	−	0.975746i	$-0.570249\pi$
$283$	−7.36520			−0.437816			−0.218908	+	0.975746i	$0.570249\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−13.3405	−	3.44849i	−0.787466	−	0.203558i
$288$		0			0
$289$	−10.1231			−0.595477
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	4.15628i		−	0.242812i	−0.992603	−	0.121406i	$-0.961260\pi$
$293$		−	4.15628i		−	0.242812i	0.992603	−	0.121406i	$-0.0387404\pi$
$294$		0			0
$295$		−	44.4924i		−	2.59045i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−13.3405			−0.771503
$300$		0			0
$301$	−23.3693	−	6.04090i	−1.34699	−	0.348191i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−42.7746			−2.44927
$306$		0			0
$307$	12.6624			0.722683			0.361342	−	0.932434i	$-0.382319\pi$
$307$	12.6624			0.722683			0.361342	+	0.932434i	$0.382319\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	22.9354			1.30054			0.650272	−	0.759701i	$-0.274655\pi$
$311$	22.9354			1.30054			0.650272	+	0.759701i	$0.274655\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−24.2157			−1.36009			−0.680045	−	0.733170i	$-0.738040\pi$
$317$	−24.2157			−1.36009			−0.680045	+	0.733170i	$0.738040\pi$
$318$		0			0
$319$			36.4924i			2.04318i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			24.5639i			1.36677i
$324$		0			0
$325$		−	36.9890i		−	2.05178i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	2.48023	−	9.59482i	0.136740	−	0.528980i
$330$		0			0
$331$		−	29.6155i		−	1.62782i	−0.580993	−	0.813908i	$-0.697336\pi$
$331$		−	29.6155i		−	1.62782i	0.580993	−	0.813908i	$-0.302664\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	30.4268			1.66239
$336$		0			0
$337$	−8.24621			−0.449200			−0.224600	−	0.974451i	$-0.572108\pi$
$337$	−8.24621			−0.449200			−0.224600	+	0.974451i	$0.572108\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	30.4268i		−	1.64770i
$342$		0			0
$343$	−12.7439	+	13.4384i	−0.688108	+	0.725608i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			6.97330i			0.374347i	0.982327	+	0.187173i	$0.0599326\pi$
$347$			6.97330i			0.374347i	−0.982327	+	0.187173i	$0.940067\pi$
$348$		0			0
$349$		−	0.743668i		−	0.0398076i	−0.999802	−	0.0199038i	$-0.993664\pi$
$349$		−	0.743668i		−	0.0398076i	0.999802	−	0.0199038i	$-0.00633600\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	11.8782i		−	0.632215i	−0.948723	−	0.316108i	$-0.897624\pi$
$353$		−	11.8782i		−	0.632215i	0.948723	−	0.316108i	$-0.102376\pi$
$354$		0			0
$355$	40.9620			2.17404
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	33.6693i		−	1.77700i	−0.458881	−	0.888498i	$-0.651749\pi$
$359$		−	33.6693i		−	1.77700i	0.458881	−	0.888498i	$-0.348251\pi$
$360$		0			0
$361$	3.24621			0.170853
$362$		0			0
$363$		0			0
$364$		0			0
$365$	40.2944			2.10910
$366$		0			0
$367$	−26.9752			−1.40810			−0.704048	−	0.710153i	$-0.748626\pi$
$367$	−26.9752			−1.40810			−0.704048	+	0.710153i	$0.748626\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1.05171	+	4.06854i	−0.0546018	+	0.211228i
$372$		0			0
$373$	10.0000			0.517780			0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000			0.517780			0.258890	+	0.965907i	$0.416643\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	43.7673i		−	2.25413i
$378$		0			0
$379$		−	25.1231i		−	1.29049i	−0.763977	−	0.645244i	$-0.776756\pi$
$379$		−	25.1231i		−	1.29049i	0.763977	−	0.645244i	$-0.223244\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−37.9182			−1.93753			−0.968764	−	0.247984i	$-0.920232\pi$
$383$	−37.9182			−1.93753			−0.968764	+	0.247984i	$0.920232\pi$
$384$		0			0
$385$	−11.1231	+	43.0299i	−0.566886	+	2.19301i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	29.8726			1.51460			0.757300	−	0.653067i	$-0.226518\pi$
$389$	29.8726			1.51460			0.757300	+	0.653067i	$0.226518\pi$
$390$		0			0
$391$	11.5012			0.581638
$392$		0			0
$393$		0			0
$394$		0			0
$395$	17.0862			0.859702
$396$		0			0
$397$			7.52823i			0.377831i	0.981993	+	0.188916i	$0.0604973\pi$
$397$			7.52823i			0.377831i	−0.981993	+	0.188916i	$0.939503\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−32.1573			−1.60586			−0.802929	−	0.596075i	$-0.796726\pi$
$401$	−32.1573			−1.60586			−0.802929	+	0.596075i	$0.796726\pi$
$402$		0			0
$403$			36.4924i			1.81782i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			25.8040i			1.27906i
$408$		0			0
$409$			17.3790i			0.859337i	0.902987	+	0.429669i	$0.141370\pi$
$409$			17.3790i			0.859337i	−0.902987	+	0.429669i	$0.858630\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	8.83348	−	34.1725i	0.434667	−	1.68152i
$414$		0			0
$415$			12.4924i			0.613229i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	11.2371			0.548969			0.274485	−	0.961592i	$-0.411493\pi$
$419$	11.2371			0.548969			0.274485	+	0.961592i	$0.411493\pi$
$420$		0			0
$421$	5.12311			0.249685			0.124842	−	0.992177i	$-0.460157\pi$
$421$	5.12311			0.249685			0.124842	+	0.992177i	$0.460157\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			31.8890i			1.54685i
$426$		0			0
$427$	−32.8531	−	8.49242i	−1.58987	−	0.410977i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			21.6592i			1.04329i	0.853164	+	0.521643i	$0.174681\pi$
$431$			21.6592i			1.04329i	−0.853164	+	0.521643i	$0.825319\pi$
$432$		0			0
$433$			1.48734i			0.0714768i	0.999361	+	0.0357384i	$0.0113783\pi$
$433$			1.48734i			0.0714768i	−0.999361	+	0.0357384i	$0.988622\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	10.4160i		−	0.498263i
$438$		0			0
$439$	−16.0547			−0.766250			−0.383125	−	0.923697i	$-0.625152\pi$
$439$	−16.0547			−0.766250			−0.383125	+	0.923697i	$0.625152\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			22.0074i			1.04560i	0.852455	+	0.522801i	$0.175113\pi$
$443$			22.0074i			1.04560i	−0.852455	+	0.522801i	$0.824887\pi$
$444$		0			0
$445$	4.87689			0.231187
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−8.83348			−0.416878			−0.208439	−	0.978035i	$-0.566838\pi$
$449$	−8.83348			−0.416878			−0.208439	+	0.978035i	$0.566838\pi$
$450$		0			0
$451$	−26.2316			−1.23520
$452$		0			0
$453$		0			0
$454$		0			0
$455$	13.3405	−	51.6081i	0.625414	−	2.41942i
$456$		0			0
$457$	20.2462			0.947078			0.473539	−	0.880773i	$-0.342976\pi$
$457$	20.2462			0.947078			0.473539	+	0.880773i	$0.342976\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	16.6757i		−	0.776663i	−0.921520	−	0.388331i	$-0.873052\pi$
$461$		−	16.6757i		−	0.776663i	0.921520	−	0.388331i	$-0.126948\pi$
$462$		0			0
$463$			1.61553i			0.0750800i	0.999295	+	0.0375400i	$0.0119522\pi$
$463$			1.61553i			0.0750800i	−0.999295	+	0.0375400i	$0.988048\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−17.0862			−0.790657			−0.395328	−	0.918540i	$-0.629369\pi$
$467$	−17.0862			−0.790657			−0.395328	+	0.918540i	$0.629369\pi$
$468$		0			0
$469$	23.3693	+	6.04090i	1.07909	+	0.278943i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−45.9512			−2.11284
$474$		0			0
$475$	28.8802			1.32511
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−37.9182			−1.73253			−0.866263	−	0.499589i	$-0.833485\pi$
$479$	−37.9182			−1.73253			−0.866263	+	0.499589i	$0.833485\pi$
$480$		0			0
$481$		−	30.9481i		−	1.41111i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−22.6274			−1.02746
$486$		0			0
$487$		−	9.61553i		−	0.435721i	−0.975980	−	0.217861i	$-0.930092\pi$
$487$		−	9.61553i		−	0.435721i	0.975980	−	0.217861i	$-0.0699078\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	3.10029i		−	0.139914i	−0.997550	−	0.0699571i	$-0.977714\pi$
$491$		−	3.10029i		−	0.139914i	0.997550	−	0.0699571i	$-0.0222862\pi$
$492$		0			0
$493$			37.7327i			1.69940i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	31.4609	+	8.13254i	1.41121	+	0.364794i
$498$		0			0
$499$			31.8617i			1.42633i	0.700998	+	0.713164i	$0.252738\pi$
$499$			31.8617i			1.42633i	−0.700998	+	0.713164i	$0.747262\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−3.74571			−0.167013			−0.0835064	−	0.996507i	$-0.526612\pi$
$503$	−3.74571			−0.167013			−0.0835064	+	0.996507i	$0.526612\pi$
$504$		0			0
$505$	23.6155			1.05088
$506$		0			0
$507$		0			0
$508$		0			0
$509$			5.43854i			0.241059i	0.992710	+	0.120530i	$0.0384593\pi$
$509$			5.43854i			0.241059i	−0.992710	+	0.120530i	$0.961541\pi$
$510$		0			0
$511$	30.9481	+	8.00000i	1.36907	+	0.353899i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	42.7746i		−	1.88487i
$516$		0			0
$517$		−	18.8664i		−	0.829741i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	8.13254i		−	0.356293i	−0.984004	−	0.178147i	$-0.942990\pi$
$521$		−	8.13254i		−	0.356293i	0.984004	−	0.178147i	$-0.0570101\pi$
$522$		0			0
$523$	20.3537			0.890005			0.445002	−	0.895529i	$-0.353203\pi$
$523$	20.3537			0.890005			0.445002	+	0.895529i	$0.353203\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	31.4609i		−	1.37046i
$528$		0			0
$529$	18.1231			0.787961
$530$		0			0
$531$		0			0
$532$		0			0
$533$	31.4609			1.36272
$534$		0			0
$535$	−12.6624			−0.547445
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−17.0862	+	30.8408i	−0.735956	+	1.32841i
$540$		0			0
$541$	9.61553			0.413404			0.206702	−	0.978404i	$-0.433727\pi$
$541$	9.61553			0.413404			0.206702	+	0.978404i	$0.433727\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			27.5022i			1.17806i
$546$		0			0
$547$		−	30.8769i		−	1.32020i	−0.751177	−	0.660100i	$-0.770514\pi$
$547$		−	30.8769i		−	1.32020i	0.751177	−	0.660100i	$-0.229486\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	34.1725			1.45580
$552$		0			0
$553$	13.1231	+	3.39228i	0.558051	+	0.144255i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	6.54877			0.277480			0.138740	−	0.990329i	$-0.455695\pi$
$557$	6.54877			0.277480			0.138740	+	0.990329i	$0.455695\pi$
$558$		0			0
$559$	55.1117			2.33098
$560$		0			0
$561$		0			0
$562$		0			0
$563$	37.9182			1.59806			0.799030	−	0.601291i	$-0.205347\pi$
$563$	37.9182			1.59806			0.799030	+	0.601291i	$0.205347\pi$
$564$		0			0
$565$			18.8664i			0.793714i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−8.83348			−0.370319			−0.185159	−	0.982709i	$-0.559280\pi$
$569$	−8.83348			−0.370319			−0.185159	+	0.982709i	$0.559280\pi$
$570$		0			0
$571$			33.1231i			1.38616i	0.720861	+	0.693079i	$0.243746\pi$
$571$			33.1231i			1.38616i	−0.720861	+	0.693079i	$0.756254\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	13.5221i		−	0.563910i
$576$		0			0
$577$		−	22.6762i		−	0.944025i	−0.881592	−	0.472012i	$-0.843528\pi$
$577$		−	22.6762i		−	0.944025i	0.881592	−	0.472012i	$-0.156472\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2.48023	+	9.59482i	−0.102897	+	0.398060i
$582$		0			0
$583$			8.00000i			0.331326i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−7.49141			−0.309204			−0.154602	−	0.987977i	$-0.549409\pi$
$587$	−7.49141			−0.309204			−0.154602	+	0.987977i	$0.549409\pi$
$588$		0			0
$589$	−28.4924			−1.17401
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	28.1433i		−	1.15571i	−0.816140	−	0.577854i	$-0.803890\pi$
$593$		−	28.1433i		−	1.15571i	0.816140	−	0.577854i	$-0.196110\pi$
$594$		0			0
$595$	−11.5012	+	44.4924i	−0.471501	+	1.82401i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	34.3657i		−	1.40414i	−0.712107	−	0.702071i	$-0.752259\pi$
$599$		−	34.3657i		−	1.40414i	0.712107	−	0.702071i	$-0.247741\pi$
$600$		0			0
$601$			6.78456i			0.276748i	0.990380	+	0.138374i	$0.0441876\pi$
$601$			6.78456i			0.276748i	−0.990380	+	0.138374i	$0.955812\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			47.9236i			1.94837i
$606$		0			0
$607$	−9.59621			−0.389498			−0.194749	−	0.980853i	$-0.562389\pi$
$607$	−9.59621			−0.389498			−0.194749	+	0.980853i	$0.562389\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			22.6274i			0.915407i
$612$		0			0
$613$	41.2311			1.66531			0.832653	−	0.553795i	$-0.186821\pi$
$613$	41.2311			1.66531			0.832653	+	0.553795i	$0.186821\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−11.3137			−0.455473			−0.227736	−	0.973723i	$-0.573132\pi$
$617$	−11.3137			−0.455473			−0.227736	+	0.973723i	$0.573132\pi$
$618$		0			0
$619$	8.27190			0.332476			0.166238	−	0.986086i	$-0.446838\pi$
$619$	8.27190			0.332476			0.166238	+	0.986086i	$0.446838\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	3.74571	+	0.968253i	0.150069	+	0.0387922i
$624$		0			0
$625$	−18.1231			−0.724924
$626$		0			0
$627$		0			0
$628$		0			0
$629$			26.6811i			1.06384i
$630$		0			0
$631$		−	39.3693i		−	1.56727i	−0.621223	−	0.783634i	$-0.713364\pi$
$631$		−	39.3693i		−	1.56727i	0.621223	−	0.783634i	$-0.286636\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−9.59482			−0.380759
$636$		0			0
$637$	20.4924	−	36.9890i	0.811939	−	1.46556i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−8.13709			−0.321396			−0.160698	−	0.987004i	$-0.551374\pi$
$641$	−8.13709			−0.321396			−0.160698	+	0.987004i	$0.551374\pi$
$642$		0			0
$643$	−33.0161			−1.30203			−0.651014	−	0.759065i	$-0.725656\pi$
$643$	−33.0161			−1.30203			−0.651014	+	0.759065i	$0.725656\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	7.49141			0.294518			0.147259	−	0.989098i	$-0.452955\pi$
$647$	7.49141			0.294518			0.147259	+	0.989098i	$0.452955\pi$
$648$		0			0
$649$		−	67.1935i		−	2.63758i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−21.0391			−0.823324			−0.411662	−	0.911337i	$-0.635052\pi$
$653$	−21.0391			−0.823324			−0.411662	+	0.911337i	$0.635052\pi$
$654$		0			0
$655$			44.4924i			1.73846i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			5.03680i			0.196206i	0.995176	+	0.0981029i	$0.0312774\pi$
$659$			5.03680i			0.196206i	−0.995176	+	0.0981029i	$0.968723\pi$
$660$		0			0
$661$			0.743668i			0.0289253i	0.999895	+	0.0144627i	$0.00460377\pi$
$661$			0.743668i			0.0289253i	−0.999895	+	0.0144627i	$0.995396\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	40.2944	+	10.4160i	1.56255	+	0.403914i
$666$		0			0
$667$		−	16.0000i		−	0.619522i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−64.5992			−2.49383
$672$		0			0
$673$	29.6155			1.14159			0.570797	−	0.821091i	$-0.306634\pi$
$673$	29.6155			1.14159			0.570797	+	0.821091i	$0.306634\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	26.2705i		−	1.00966i	−0.863220	−	0.504828i	$-0.831556\pi$
$677$		−	26.2705i		−	1.00966i	0.863220	−	0.504828i	$-0.168444\pi$
$678$		0			0
$679$	−17.3790	−	4.49242i	−0.666946	−	0.172403i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	23.2475i		−	0.889540i	−0.895645	−	0.444770i	$-0.853285\pi$
$683$		−	23.2475i		−	0.889540i	0.895645	−	0.444770i	$-0.146715\pi$
$684$		0			0
$685$		−	10.5945i		−	0.404793i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	9.59482i		−	0.365534i
$690$		0			0
$691$	−6.78456			−0.258097			−0.129048	−	0.991638i	$-0.541192\pi$
$691$	−6.78456			−0.258097			−0.129048	+	0.991638i	$0.541192\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			17.6670i			0.670146i
$696$		0			0
$697$	−27.1231			−1.02736
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−12.9020			−0.487303			−0.243651	−	0.969863i	$-0.578345\pi$
$701$	−12.9020			−0.487303			−0.243651	+	0.969863i	$0.578345\pi$
$702$		0			0
$703$	24.1636			0.911347
$704$		0			0
$705$		0			0
$706$		0			0
$707$	18.1379	+	4.68860i	0.682147	+	0.176333i
$708$		0			0
$709$	6.00000			0.225335			0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000			0.225335			0.112667	+	0.993633i	$0.464061\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			13.3405i			0.499607i
$714$		0			0
$715$		−	101.477i		−	3.79503i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	30.4268			1.13473			0.567363	−	0.823468i	$-0.307964\pi$
$719$	30.4268			1.13473			0.567363	+	0.823468i	$0.307964\pi$
$720$		0			0
$721$	8.49242	−	32.8531i	0.316274	−	1.22351i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	44.3629			1.64760
$726$		0			0
$727$	24.9073			0.923759			0.461879	−	0.886943i	$-0.347175\pi$
$727$	24.9073			0.923759			0.461879	+	0.886943i	$0.347175\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−47.5130			−1.75733
$732$		0			0
$733$			0.743668i			0.0274680i	0.999906	+	0.0137340i	$0.00437181\pi$
$733$			0.743668i			0.0274680i	−0.999906	+	0.0137340i	$0.995628\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	45.9512			1.69264
$738$		0			0
$739$			25.1231i			0.924168i	0.886836	+	0.462084i	$0.152898\pi$
$739$			25.1231i			0.924168i	−0.886836	+	0.462084i	$0.847102\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			16.0023i			0.587068i	0.955949	+	0.293534i	$0.0948314\pi$
$743$			16.0023i			0.587068i	−0.955949	+	0.293534i	$0.905169\pi$
$744$		0			0
$745$		−	5.29723i		−	0.194075i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−9.72540	−	2.51398i	−0.355358	−	0.0918590i
$750$		0			0
$751$		−	0.630683i		−	0.0230140i	−0.999934	−	0.0115070i	$-0.996337\pi$
$751$		−	0.630683i		−	0.0230140i	0.999934	−	0.0115070i	$-0.00366287\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	9.59482			0.349191
$756$		0			0
$757$	30.9848			1.12616			0.563082	−	0.826401i	$-0.309616\pi$
$757$	30.9848			1.12616			0.563082	+	0.826401i	$0.309616\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			42.3050i			1.53356i	0.641913	+	0.766778i	$0.278141\pi$
$761$			42.3050i			1.53356i	−0.641913	+	0.766778i	$0.721859\pi$
$762$		0			0
$763$	−5.46026	+	21.1231i	−0.197675	+	0.764708i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			80.5887i			2.90989i
$768$		0			0
$769$			43.0299i			1.55170i	0.630918	+	0.775850i	$0.282678\pi$
$769$			43.0299i			1.55170i	−0.630918	+	0.775850i	$0.717322\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			0.410574i			0.0147673i	0.999973	+	0.00738365i	$0.00235031\pi$
$773$			0.410574i			0.0147673i	−0.999973	+	0.00738365i	$0.997650\pi$
$774$		0			0
$775$	−36.9890			−1.32869
$776$		0			0
$777$		0			0
$778$		0			0
$779$			24.5639i			0.880094i
$780$		0			0
$781$	61.8617			2.21359
$782$		0			0
$783$		0			0
$784$		0			0
$785$	37.8141			1.34964
$786$		0			0
$787$	46.8398			1.66966			0.834830	−	0.550508i	$-0.185566\pi$
$787$	46.8398			1.66966			0.834830	+	0.550508i	$0.185566\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−3.74571	+	14.4903i	−0.133182	+	0.515217i
$792$		0			0
$793$	77.4773			2.75130
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	12.9300i		−	0.458002i	−0.973426	−	0.229001i	$-0.926454\pi$
$797$		−	12.9300i		−	0.458002i	0.973426	−	0.229001i	$-0.0735460\pi$
$798$		0			0
$799$		−	19.5076i		−	0.690128i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	60.8535			2.14747
$804$		0			0
$805$	4.87689	−	18.8664i	0.171888	−	0.664952i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−54.0883			−1.90164			−0.950822	−	0.309738i	$-0.899759\pi$
$809$	−54.0883			−1.90164			−0.950822	+	0.309738i	$0.899759\pi$
$810$		0			0
$811$	−41.5426			−1.45876			−0.729379	−	0.684110i	$-0.760191\pi$
$811$	−41.5426			−1.45876			−0.729379	+	0.684110i	$0.760191\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	3.74571			0.131206
$816$		0			0
$817$			43.0299i			1.50543i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−53.1964			−1.85657			−0.928283	−	0.371874i	$-0.878716\pi$
$821$	−53.1964			−1.85657			−0.928283	+	0.371874i	$0.878716\pi$
$822$		0			0
$823$		−	19.8617i		−	0.692337i	−0.938172	−	0.346168i	$-0.887483\pi$
$823$		−	19.8617i		−	0.692337i	0.938172	−	0.346168i	$-0.112517\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	37.1941i		−	1.29336i	−0.762759	−	0.646682i	$-0.776156\pi$
$827$		−	37.1941i		−	1.29336i	0.762759	−	0.646682i	$-0.223844\pi$
$828$		0			0
$829$		−	11.3381i		−	0.393789i	−0.980425	−	0.196895i	$-0.936914\pi$
$829$		−	11.3381i		−	0.393789i	0.980425	−	0.196895i	$-0.0630857\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−17.6670	+	31.8890i	−0.612124	+	1.10489i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	34.1725			1.17976			0.589882	−	0.807489i	$-0.299174\pi$
$839$	34.1725			1.17976			0.589882	+	0.807489i	$0.299174\pi$
$840$		0			0
$841$	23.4924			0.810084
$842$		0			0
$843$		0			0
$844$		0			0
$845$			78.3503i			2.69533i
$846$		0			0
$847$	−9.51469	+	36.8078i	−0.326929	+	1.26473i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	11.3137i		−	0.387829i
$852$		0			0
$853$		−	35.5017i		−	1.21556i	−0.794107	−	0.607778i	$-0.792061\pi$
$853$		−	35.5017i		−	1.21556i	0.794107	−	0.607778i	$-0.207939\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			27.3222i			0.933308i	0.884440	+	0.466654i	$0.154541\pi$
$857$			27.3222i			0.933308i	−0.884440	+	0.466654i	$0.845459\pi$
$858$		0			0
$859$	38.3134			1.30723			0.653617	−	0.756825i	$-0.273251\pi$
$859$	38.3134			1.30723			0.653617	+	0.756825i	$0.273251\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	20.4190i		−	0.695072i	−0.937667	−	0.347536i	$-0.887018\pi$
$863$		−	20.4190i		−	0.695072i	0.937667	−	0.347536i	$-0.112982\pi$
$864$		0			0
$865$	68.1080			2.31574
$866$		0			0
$867$		0			0
$868$		0			0
$869$	25.8040			0.875342
$870$		0			0
$871$	−55.1117			−1.86739
$872$		0			0
$873$		0			0
$874$		0			0
$875$	9.59482	+	2.48023i	0.324364	+	0.0838471i
$876$		0			0
$877$	8.24621			0.278455			0.139227	−	0.990260i	$-0.455538\pi$
$877$	8.24621			0.278455			0.139227	+	0.990260i	$0.455538\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			20.6519i			0.695781i	0.937535	+	0.347891i	$0.113102\pi$
$881$			20.6519i			0.695781i	−0.937535	+	0.347891i	$0.886898\pi$
$882$		0			0
$883$			7.86174i			0.264569i	0.991212	+	0.132284i	$0.0422312\pi$
$883$			7.86174i			0.264569i	−0.991212	+	0.132284i	$0.957769\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	34.1725			1.14740			0.573700	−	0.819066i	$-0.305508\pi$
$887$	34.1725			1.14740			0.573700	+	0.819066i	$0.305508\pi$
$888$		0			0
$889$	−7.36932	−	1.90495i	−0.247159	−	0.0638898i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−17.6670			−0.591202
$894$		0			0
$895$	−73.3974			−2.45341
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−43.7673			−1.45972
$900$		0			0
$901$			8.27190i			0.275577i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	20.1472			0.669715
$906$		0			0
$907$		−	18.1080i		−	0.601265i	−0.953740	−	0.300632i	$-0.902802\pi$
$907$		−	18.1080i		−	0.601265i	0.953740	−	0.300632i	$-0.0971977\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			12.2820i			0.406920i	0.979083	+	0.203460i	$0.0652186\pi$
$911$			12.2820i			0.406920i	−0.979083	+	0.203460i	$0.934781\pi$
$912$		0			0
$913$			18.8664i			0.624385i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−8.83348	+	34.1725i	−0.291707	+	1.12847i
$918$		0			0
$919$		−	49.6155i		−	1.63667i	−0.574745	−	0.818333i	$-0.694899\pi$
$919$		−	49.6155i		−	1.63667i	0.574745	−	0.818333i	$-0.305101\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−74.1941			−2.44213
$924$		0			0
$925$	31.3693			1.03142
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	52.7210i		−	1.72972i	−0.502014	−	0.864860i	$-0.667407\pi$
$929$		−	52.7210i		−	1.72972i	0.502014	−	0.864860i	$-0.332593\pi$
$930$		0			0
$931$	28.8802	+	16.0000i	0.946509	+	0.524379i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			87.4857i			2.86109i
$936$		0			0
$937$			8.27190i			0.270231i	0.990830	+	0.135116i	$0.0431406\pi$
$937$			8.27190i			0.270231i	−0.990830	+	0.135116i	$0.956859\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	27.9128i		−	0.909931i	−0.890509	−	0.454965i	$-0.849652\pi$
$941$		−	27.9128i		−	0.909931i	0.890509	−	0.454965i	$-0.150348\pi$
$942$		0			0
$943$	11.5012			0.374529
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	46.0276i		−	1.49569i	−0.663871	−	0.747847i	$-0.731088\pi$
$947$		−	46.0276i		−	1.49569i	0.663871	−	0.747847i	$-0.268912\pi$
$948$		0			0
$949$	−72.9848			−2.36919
$950$		0			0
$951$		0			0
$952$		0			0
$953$	25.8040			0.835875			0.417937	−	0.908476i	$-0.362753\pi$
$953$	25.8040			0.835875			0.417937	+	0.908476i	$0.362753\pi$
$954$		0			0
$955$	−30.3675			−0.982670
$956$		0			0
$957$		0			0
$958$		0			0
$959$	2.10341	−	8.13709i	0.0679227	−	0.262760i
$960$		0			0
$961$	5.49242			0.177175
$962$		0			0
$963$		0			0
$964$		0			0
$965$			37.9182i			1.22063i
$966$		0			0
$967$		−	47.3693i		−	1.52329i	−0.647992	−	0.761647i	$-0.724391\pi$
$967$		−	47.3693i		−	1.52329i	0.647992	−	0.761647i	$-0.275609\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	36.2759			1.16415			0.582074	−	0.813136i	$-0.302241\pi$
$971$	36.2759			1.16415			0.582074	+	0.813136i	$0.302241\pi$
$972$		0			0
$973$	−3.50758	+	13.5691i	−0.112448	+	0.435006i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	49.1278			1.57174			0.785870	−	0.618392i	$-0.212216\pi$
$977$	49.1278			1.57174			0.785870	+	0.618392i	$0.212216\pi$
$978$		0			0
$979$	7.36520			0.235393
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−45.8707			−1.46305			−0.731524	−	0.681816i	$-0.761191\pi$
$983$	−45.8707			−1.46305			−0.731524	+	0.681816i	$0.761191\pi$
$984$		0			0
$985$			24.1636i			0.769916i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	20.1472			0.640643
$990$		0			0
$991$			39.3693i			1.25061i	0.780381	+	0.625304i	$0.215025\pi$
$991$			39.3693i			1.25061i	−0.780381	+	0.625304i	$0.784975\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	27.0442i		−	0.857357i
$996$		0			0
$997$			50.5582i			1.60119i	0.599204	+	0.800597i	$0.295484\pi$
$997$			50.5582i			1.60119i	−0.599204	+	0.800597i	$0.704516\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.b.r.3583.3		16
3.2	odd	2	inner	4032.2.b.r.3583.15		16
4.3	odd	2	inner	4032.2.b.r.3583.2		16
7.6	odd	2	inner	4032.2.b.r.3583.13		16
8.3	odd	2		2016.2.b.d.1567.14	yes	16
8.5	even	2		2016.2.b.d.1567.15	yes	16
12.11	even	2	inner	4032.2.b.r.3583.14		16
21.20	even	2	inner	4032.2.b.r.3583.1		16
24.5	odd	2		2016.2.b.d.1567.3	yes	16
24.11	even	2		2016.2.b.d.1567.2	yes	16
28.27	even	2	inner	4032.2.b.r.3583.16		16
56.13	odd	2		2016.2.b.d.1567.1	✓	16
56.27	even	2		2016.2.b.d.1567.4	yes	16
84.83	odd	2	inner	4032.2.b.r.3583.4		16
168.83	odd	2		2016.2.b.d.1567.16	yes	16
168.125	even	2		2016.2.b.d.1567.13	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2016.2.b.d.1567.1	✓	16	56.13	odd	2
2016.2.b.d.1567.2	yes	16	24.11	even	2
2016.2.b.d.1567.3	yes	16	24.5	odd	2
2016.2.b.d.1567.4	yes	16	56.27	even	2
2016.2.b.d.1567.13	yes	16	168.125	even	2
2016.2.b.d.1567.14	yes	16	8.3	odd	2
2016.2.b.d.1567.15	yes	16	8.5	even	2
2016.2.b.d.1567.16	yes	16	168.83	odd	2
4032.2.b.r.3583.1		16	21.20	even	2	inner
4032.2.b.r.3583.2		16	4.3	odd	2	inner
4032.2.b.r.3583.3		16	1.1	even	1	trivial
4032.2.b.r.3583.4		16	84.83	odd	2	inner
4032.2.b.r.3583.13		16	7.6	odd	2	inner
4032.2.b.r.3583.14		16	12.11	even	2	inner
4032.2.b.r.3583.15		16	3.2	odd	2	inner
4032.2.b.r.3583.16		16	28.27	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-3.33513i q^{5} +(0.662153 - 2.56155i) q^{7} -5.03680i q^{11} +6.04090i q^{13} -5.20798i q^{17} -4.71659 q^{19} +2.20837i q^{23} -6.12311 q^{25} -7.24517 q^{29} +6.04090 q^{31} +(-8.54312 - 2.20837i) q^{35} -5.12311 q^{37} -5.20798i q^{41} -9.12311i q^{43} +3.74571 q^{47} +(-6.12311 - 3.39228i) q^{49} -1.58831 q^{53} -16.7984 q^{55} +13.3405 q^{59} -12.8255i q^{61} +20.1472 q^{65} +9.12311i q^{67} +12.2820i q^{71} +12.0818i q^{73} +(-12.9020 - 3.33513i) q^{77} +5.12311i q^{79} -3.74571 q^{83} -17.3693 q^{85} +1.46228i q^{89} +(15.4741 + 4.00000i) q^{91} +15.7304i q^{95} -6.78456i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{25} - 16 q^{37} - 32 q^{49} - 80 q^{85}+O(q^{100}) \) 16 * q - 32 * q^25 - 16 * q^37 - 32 * q^49 - 80 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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