LMFDB - Embedded newform 4032.2.b.r.3583.4

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			3583.4
Root			$0.481610 - 1.32968i$ of defining polynomial
Character	$\chi$	$=$	4032.3583
Dual form			4032.2.b.r.3583.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.33513i q^{5} +(0.662153 + 2.56155i) q^{7} +O(q^{10})$	$q-3.33513i q^{5} +(0.662153 + 2.56155i) q^{7} -5.03680i q^{11} -6.04090i q^{13} -5.20798i q^{17} -4.71659 q^{19} +2.20837i q^{23} -6.12311 q^{25} +7.24517 q^{29} +6.04090 q^{31} +(8.54312 - 2.20837i) q^{35} -5.12311 q^{37} -5.20798i q^{41} +9.12311i q^{43} -3.74571 q^{47} +(-6.12311 + 3.39228i) q^{49} +1.58831 q^{53} -16.7984 q^{55} -13.3405 q^{59} +12.8255i q^{61} -20.1472 q^{65} -9.12311i q^{67} +12.2820i q^{71} -12.0818i q^{73} +(12.9020 - 3.33513i) q^{77} -5.12311i q^{79} +3.74571 q^{83} -17.3693 q^{85} +1.46228i q^{89} +(15.4741 - 4.00000i) q^{91} +15.7304i q^{95} +6.78456i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * 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

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$-1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		−	3.33513i		−	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	$-0.683887\pi$
$13$		−	6.04090i		−	1.67544i	0.546098	−	0.837722i	$-0.316113\pi$
$14$		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	$-0.265136\pi$
$29$	7.24517			1.34539			0.672697	+	0.739918i	$0.265136\pi$
$30$		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	$0.244875\pi$
$43$			9.12311i			1.39126i	−0.718400	+	0.695630i	$0.755125\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.74571			−0.546367			−0.273184	−	0.961962i	$-0.588077\pi$
$47$	−3.74571			−0.546367			−0.273184	+	0.961962i	$0.588077\pi$
$48$		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	$-0.465208\pi$
$53$	1.58831			0.218171			0.109086	+	0.994032i	$0.465208\pi$
$54$		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	$-0.834848\pi$
$59$	−13.3405			−1.73679			−0.868394	+	0.495874i	$0.834848\pi$
$60$		0			0
$61$			12.8255i			1.64213i	0.570833	+	0.821066i	$0.306620\pi$
$61$			12.8255i			1.64213i	−0.570833	+	0.821066i	$0.693380\pi$
$62$		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	$-0.811844\pi$
$67$		−	9.12311i		−	1.11456i	0.830323	−	0.557282i	$-0.188156\pi$
$68$		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	$-0.750033\pi$
$73$		−	12.0818i		−	1.41407i	0.707180	−	0.707033i	$-0.249967\pi$
$74$		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	$-0.906944\pi$
$79$		−	5.12311i		−	0.576394i	0.957571	−	0.288197i	$-0.0930559\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	3.74571			0.411145			0.205572	−	0.978642i	$-0.434094\pi$
$83$	3.74571			0.411145			0.205572	+	0.978642i	$0.434094\pi$
$84$		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	$0.111929\pi$
$97$			6.78456i			0.688868i	−0.938811	+	0.344434i	$0.888071\pi$
$98$		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	$-0.414273\pi$
$113$	5.65685			0.532152			0.266076	+	0.963952i	$0.414273\pi$
$114$		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	$0.0407407\pi$
$127$			2.87689i			0.255283i	−0.991820	+	0.127642i	$0.959259\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	13.3405			1.16557			0.582784	−	0.812627i	$-0.301963\pi$
$131$	13.3405			1.16557			0.582784	+	0.812627i	$0.301963\pi$
$132$		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	$-0.543328\pi$
$137$	−3.17662			−0.271397			−0.135699	+	0.990750i	$0.543328\pi$
$138$		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	$-0.520724\pi$
$149$	−1.58831			−0.130120			−0.0650598	+	0.997881i	$0.520724\pi$
$150$		0			0
$151$		−	2.87689i		−	0.234118i	−0.993125	−	0.117059i	$-0.962653\pi$
$151$		−	2.87689i		−	0.234118i	0.993125	−	0.117059i	$-0.0373467\pi$
$152$		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	$-0.850554\pi$
$157$		−	11.3381i		−	0.904881i	0.891795	−	0.452440i	$-0.149446\pi$
$158$		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	$-0.985995\pi$
$163$		−	1.12311i		−	0.0879684i	0.999032	−	0.0439842i	$-0.0140051\pi$
$164$		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	$-0.927922\pi$
$181$		−	6.04090i		−	0.449016i	0.974472	−	0.224508i	$-0.0720775\pi$
$182$		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	$-0.416904\pi$
$197$	7.24517			0.516197			0.258098	+	0.966119i	$0.416904\pi$
$198$		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	$-0.963000\pi$
$211$		−	3.36932i		−	0.231953i	0.993252	−	0.115977i	$-0.0369998\pi$
$212$		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	$-0.603151\pi$
$227$	−9.59482			−0.636831			−0.318415	+	0.947951i	$0.603151\pi$
$228$		0			0
$229$		−	11.3381i		−	0.749244i	−0.927178	−	0.374622i	$-0.877772\pi$
$229$		−	11.3381i		−	0.749244i	0.927178	−	0.374622i	$-0.122228\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−16.9706			−1.11178			−0.555889	−	0.831256i	$-0.687622\pi$
$233$	−16.9706			−1.11178			−0.555889	+	0.831256i	$0.687622\pi$
$234$		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	$0.207885\pi$
$241$			18.8664i			1.21529i	−0.794209	+	0.607644i	$0.792115\pi$
$242$		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	$-0.681287\pi$
$251$	−17.0862			−1.07847			−0.539237	+	0.842154i	$0.681287\pi$
$252$		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	$-0.446033\pi$
$281$	5.65685			0.337460			0.168730	+	0.985662i	$0.446033\pi$
$282$		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	$-0.725345\pi$
$311$	−22.9354			−1.30054			−0.650272	+	0.759701i	$0.725345\pi$
$312$		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	$-0.261960\pi$
$317$	24.2157			1.36009			0.680045	+	0.733170i	$0.261960\pi$
$318$		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	$0.302664\pi$
$331$			29.6155i			1.62782i	−0.580993	+	0.813908i	$0.697336\pi$
$332$		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	$0.00633600\pi$
$349$			0.743668i			0.0398076i	−0.999802	+	0.0199038i	$0.993664\pi$
$350$		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	$0.223244\pi$
$379$			25.1231i			1.29049i	−0.763977	+	0.645244i	$0.776756\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	37.9182			1.93753			0.968764	−	0.247984i	$-0.0797681\pi$
$383$	37.9182			1.93753			0.968764	+	0.247984i	$0.0797681\pi$
$384$		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	$-0.773482\pi$
$389$	−29.8726			−1.51460			−0.757300	+	0.653067i	$0.773482\pi$
$390$		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	$-0.939503\pi$
$397$		−	7.52823i		−	0.377831i	0.981993	−	0.188916i	$-0.0604973\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	32.1573			1.60586			0.802929	−	0.596075i	$-0.203274\pi$
$401$	32.1573			1.60586			0.802929	+	0.596075i	$0.203274\pi$
$402$		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	$-0.858630\pi$
$409$		−	17.3790i		−	0.859337i	0.902987	−	0.429669i	$-0.141370\pi$
$410$		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	$-0.588507\pi$
$419$	−11.2371			−0.548969			−0.274485	+	0.961592i	$0.588507\pi$
$420$		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	$-0.988622\pi$
$433$		−	1.48734i		−	0.0714768i	0.999361	−	0.0357384i	$-0.0113783\pi$
$434$		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	$-0.433162\pi$
$449$	8.83348			0.416878			0.208439	+	0.978035i	$0.433162\pi$
$450$		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	$-0.988048\pi$
$463$		−	1.61553i		−	0.0750800i	0.999295	−	0.0375400i	$-0.0119522\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	17.0862			0.790657			0.395328	−	0.918540i	$-0.370631\pi$
$467$	17.0862			0.790657			0.395328	+	0.918540i	$0.370631\pi$
$468$		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	$-0.166515\pi$
$479$	37.9182			1.73253			0.866263	+	0.499589i	$0.166515\pi$
$480$		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	$0.0699078\pi$
$487$			9.61553i			0.435721i	−0.975980	+	0.217861i	$0.930092\pi$
$488$		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	$-0.747262\pi$
$499$		−	31.8617i		−	1.42633i	0.700998	−	0.713164i	$-0.252738\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	3.74571			0.167013			0.0835064	−	0.996507i	$-0.473388\pi$
$503$	3.74571			0.167013			0.0835064	+	0.996507i	$0.473388\pi$
$504$		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	$0.229486\pi$
$547$			30.8769i			1.32020i	−0.751177	+	0.660100i	$0.770514\pi$
$548$		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	$-0.544305\pi$
$557$	−6.54877			−0.277480			−0.138740	+	0.990329i	$0.544305\pi$
$558$		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	$-0.794653\pi$
$563$	−37.9182			−1.59806			−0.799030	+	0.601291i	$0.794653\pi$
$564$		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	$-0.440720\pi$
$569$	8.83348			0.370319			0.185159	+	0.982709i	$0.440720\pi$
$570$		0			0
$571$		−	33.1231i		−	1.38616i	−0.720861	−	0.693079i	$-0.756254\pi$
$571$		−	33.1231i		−	1.38616i	0.720861	−	0.693079i	$-0.243746\pi$
$572$		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	$0.156472\pi$
$577$			22.6762i			0.944025i	−0.881592	+	0.472012i	$0.843528\pi$
$578$		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	$-0.450591\pi$
$587$	7.49141			0.309204			0.154602	+	0.987977i	$0.450591\pi$
$588$		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	$-0.955812\pi$
$601$		−	6.78456i		−	0.276748i	0.990380	−	0.138374i	$-0.0441876\pi$
$602$		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	$-0.426868\pi$
$617$	11.3137			0.455473			0.227736	+	0.973723i	$0.426868\pi$
$618$		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	$0.286636\pi$
$631$			39.3693i			1.56727i	−0.621223	+	0.783634i	$0.713364\pi$
$632$		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	$-0.448626\pi$
$641$	8.13709			0.321396			0.160698	+	0.987004i	$0.448626\pi$
$642$		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	$-0.547045\pi$
$647$	−7.49141			−0.294518			−0.147259	+	0.989098i	$0.547045\pi$
$648$		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	$-0.364948\pi$
$653$	21.0391			0.823324			0.411662	+	0.911337i	$0.364948\pi$
$654$		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	$-0.995396\pi$
$661$		−	0.743668i		−	0.0289253i	0.999895	−	0.0144627i	$-0.00460377\pi$
$662$		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	$-0.421655\pi$
$701$	12.9020			0.487303			0.243651	+	0.969863i	$0.421655\pi$
$702$		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	$-0.692036\pi$
$719$	−30.4268			−1.13473			−0.567363	+	0.823468i	$0.692036\pi$
$720$		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	$-0.995628\pi$
$733$		−	0.743668i		−	0.0274680i	0.999906	−	0.0137340i	$-0.00437181\pi$
$734$		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	$-0.847102\pi$
$739$		−	25.1231i		−	0.924168i	0.886836	−	0.462084i	$-0.152898\pi$
$740$		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	$0.00366287\pi$
$751$			0.630683i			0.0230140i	−0.999934	+	0.0115070i	$0.996337\pi$
$752$		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	$-0.717322\pi$
$769$		−	43.0299i		−	1.55170i	0.630918	−	0.775850i	$-0.282678\pi$
$770$		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	$-0.100241\pi$
$809$	54.0883			1.90164			0.950822	+	0.309738i	$0.100241\pi$
$810$		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	$-0.121284\pi$
$821$	53.1964			1.85657			0.928283	+	0.371874i	$0.121284\pi$
$822$		0			0
$823$			19.8617i			0.692337i	0.938172	+	0.346168i	$0.112517\pi$
$823$			19.8617i			0.692337i	−0.938172	+	0.346168i	$0.887483\pi$
$824$		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	$0.0630857\pi$
$829$			11.3381i			0.393789i	−0.980425	+	0.196895i	$0.936914\pi$
$830$		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	$-0.700826\pi$
$839$	−34.1725			−1.17976			−0.589882	+	0.807489i	$0.700826\pi$
$840$		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	$0.207939\pi$
$853$			35.5017i			1.21556i	−0.794107	+	0.607778i	$0.792061\pi$
$854$		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	$-0.957769\pi$
$883$		−	7.86174i		−	0.264569i	0.991212	−	0.132284i	$-0.0422312\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−34.1725			−1.14740			−0.573700	−	0.819066i	$-0.694492\pi$
$887$	−34.1725			−1.14740			−0.573700	+	0.819066i	$0.694492\pi$
$888$		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	$0.0971977\pi$
$907$			18.1080i			0.601265i	−0.953740	+	0.300632i	$0.902802\pi$
$908$		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	$0.305101\pi$
$919$			49.6155i			1.63667i	−0.574745	+	0.818333i	$0.694899\pi$
$920$		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	$-0.956859\pi$
$937$		−	8.27190i		−	0.270231i	0.990830	−	0.135116i	$-0.0431406\pi$
$938$		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	$-0.637247\pi$
$953$	−25.8040			−0.835875			−0.417937	+	0.908476i	$0.637247\pi$
$954$		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	$0.275609\pi$
$967$			47.3693i			1.52329i	−0.647992	+	0.761647i	$0.724391\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−36.2759			−1.16415			−0.582074	−	0.813136i	$-0.697759\pi$
$971$	−36.2759			−1.16415			−0.582074	+	0.813136i	$0.697759\pi$
$972$		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	$-0.787784\pi$
$977$	−49.1278			−1.57174			−0.785870	+	0.618392i	$0.787784\pi$
$978$		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	$-0.238809\pi$
$983$	45.8707			1.46305			0.731524	+	0.681816i	$0.238809\pi$
$984$		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	$-0.784975\pi$
$991$		−	39.3693i		−	1.25061i	0.780381	−	0.625304i	$-0.215025\pi$
$992$		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	$-0.704516\pi$
$997$		−	50.5582i		−	1.60119i	0.599204	−	0.800597i	$-0.295484\pi$
$998$		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.3583.4		16
3.2	odd	2	inner	4032.2.b.r.3583.16		16
4.3	odd	2	inner	4032.2.b.r.3583.1		16
7.6	odd	2	inner	4032.2.b.r.3583.14		16
8.3	odd	2		2016.2.b.d.1567.13	yes	16
8.5	even	2		2016.2.b.d.1567.16	yes	16
12.11	even	2	inner	4032.2.b.r.3583.13		16
21.20	even	2	inner	4032.2.b.r.3583.2		16
24.5	odd	2		2016.2.b.d.1567.4	yes	16
24.11	even	2		2016.2.b.d.1567.1	✓	16
28.27	even	2	inner	4032.2.b.r.3583.15		16
56.13	odd	2		2016.2.b.d.1567.2	yes	16
56.27	even	2		2016.2.b.d.1567.3	yes	16
84.83	odd	2	inner	4032.2.b.r.3583.3		16
168.83	odd	2		2016.2.b.d.1567.15	yes	16
168.125	even	2		2016.2.b.d.1567.14	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2016.2.b.d.1567.1	✓	16	24.11	even	2
2016.2.b.d.1567.2	yes	16	56.13	odd	2
2016.2.b.d.1567.3	yes	16	56.27	even	2
2016.2.b.d.1567.4	yes	16	24.5	odd	2
2016.2.b.d.1567.13	yes	16	8.3	odd	2
2016.2.b.d.1567.14	yes	16	168.125	even	2
2016.2.b.d.1567.15	yes	16	168.83	odd	2
2016.2.b.d.1567.16	yes	16	8.5	even	2
4032.2.b.r.3583.1		16	4.3	odd	2	inner
4032.2.b.r.3583.2		16	21.20	even	2	inner
4032.2.b.r.3583.3		16	84.83	odd	2	inner
4032.2.b.r.3583.4		16	1.1	even	1	trivial
4032.2.b.r.3583.13		16	12.11	even	2	inner
4032.2.b.r.3583.14		16	7.6	odd	2	inner
4032.2.b.r.3583.15		16	28.27	even	2	inner
4032.2.b.r.3583.16		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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.33513i q^{5} +(0.662153 + 2.56155i) q^{7} +O(q^{10})\)	\(q-3.33513i q^{5} +(0.662153 + 2.56155i) q^{7} -5.03680i q^{11} -6.04090i q^{13} -5.20798i q^{17} -4.71659 q^{19} +2.20837i q^{23} -6.12311 q^{25} +7.24517 q^{29} +6.04090 q^{31} +(8.54312 - 2.20837i) q^{35} -5.12311 q^{37} -5.20798i q^{41} +9.12311i q^{43} -3.74571 q^{47} +(-6.12311 + 3.39228i) q^{49} +1.58831 q^{53} -16.7984 q^{55} -13.3405 q^{59} +12.8255i q^{61} -20.1472 q^{65} -9.12311i q^{67} +12.2820i q^{71} -12.0818i q^{73} +(12.9020 - 3.33513i) q^{77} -5.12311i q^{79} +3.74571 q^{83} -17.3693 q^{85} +1.46228i q^{89} +(15.4741 - 4.00000i) q^{91} +15.7304i q^{95} +6.78456i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * 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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