LMFDB - Embedded newform 4032.2.b.o.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:	$8$
Coefficient field:	8.0.836829184.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 14x^{6} + 61x^{4} + 84x^{2} + 4 $ x^8 + 14x^6 + 61x^4 + 84*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 672)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3583.3
Root			$0.222191i$ of defining polynomial
Character	$\chi$	$=$	4032.3583
Dual form			4032.2.b.o.3583.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.72844i q^{5} +(-2.63640 - 0.222191i) q^{7} +O(q^{10})$	$q-1.72844i q^{5} +(-2.63640 - 0.222191i) q^{7} +6.17282i q^{11} -2.82843i q^{13} +1.72844i q^{17} -5.90126 q^{19} +1.54437i q^{23} +2.01250 q^{25} +8.28531 q^{29} +4.62845 q^{31} +(-0.384044 + 4.55687i) q^{35} -2.24441 q^{37} -3.92841i q^{41} -10.1012i q^{43} +4.88877 q^{47} +(6.90126 + 1.17157i) q^{49} -9.17407 q^{53} +10.6694 q^{55} -9.65685 q^{59} -14.2853i q^{61} -4.88877 q^{65} -6.64436i q^{67} +5.00125i q^{71} -2.19998i q^{73} +(1.37155 - 16.2741i) q^{77} -9.55812i q^{79} +12.2025 q^{83} +2.98750 q^{85} -9.16282i q^{89} +(-0.628452 + 7.45688i) q^{91} +10.2000i q^{95} +17.1137i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{7}+O(q^{10}) $ 8 * q - 4 * q^7	$ 8 q - 4 q^{7} + 8 q^{19} - 16 q^{25} + 16 q^{31} + 8 q^{35} - 8 q^{37} + 16 q^{47} + 16 q^{53} + 8 q^{55} - 32 q^{59} - 16 q^{65} + 32 q^{77} - 16 q^{83} + 56 q^{85} + 16 q^{91}+O(q^{100}) $ 8 * q - 4 * q^7 + 8 * q^19 - 16 * q^25 + 16 * q^31 + 8 * q^35 - 8 * q^37 + 16 * q^47 + 16 * q^53 + 8 * q^55 - 32 * q^59 - 16 * q^65 + 32 * q^77 - 16 * q^83 + 56 * q^85 + 16 * q^91

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		−	1.72844i		−	0.772982i	−0.922293	−	0.386491i	$-0.873687\pi$
$5$		−	1.72844i		−	0.772982i	0.922293	−	0.386491i	$-0.126313\pi$
$6$		0			0
$7$	−2.63640	−	0.222191i	−0.996467	−	0.0839804i
$7$	−2.63640	−	0.222191i	−0.996467	−	0.0839804i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			6.17282i			1.86118i	0.366068	+	0.930588i	$0.380704\pi$
$11$			6.17282i			1.86118i	−0.366068	+	0.930588i	$0.619296\pi$
$12$		0			0
$13$		−	2.82843i		−	0.784465i	−0.919866	−	0.392232i	$-0.871703\pi$
$13$		−	2.82843i		−	0.784465i	0.919866	−	0.392232i	$-0.128297\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			1.72844i			0.419208i	0.977786	+	0.209604i	$0.0672175\pi$
$17$			1.72844i			0.419208i	−0.977786	+	0.209604i	$0.932782\pi$
$18$		0			0
$19$	−5.90126			−1.35384			−0.676921	−	0.736056i	$-0.736686\pi$
$19$	−5.90126			−1.35384			−0.676921	+	0.736056i	$0.736686\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			1.54437i			0.322023i	0.986952	+	0.161012i	$0.0514757\pi$
$23$			1.54437i			0.322023i	−0.986952	+	0.161012i	$0.948524\pi$
$24$		0			0
$25$	2.01250			0.402499
$26$		0			0
$27$		0			0
$28$		0			0
$29$	8.28531			1.53854			0.769271	−	0.638922i	$-0.220619\pi$
$29$	8.28531			1.53854			0.769271	+	0.638922i	$0.220619\pi$
$30$		0			0
$31$	4.62845			0.831295			0.415647	−	0.909526i	$-0.363555\pi$
$31$	4.62845			0.831295			0.415647	+	0.909526i	$0.363555\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.384044	+	4.55687i	−0.0649153	+	0.770251i
$36$		0			0
$37$	−2.24441			−0.368978			−0.184489	−	0.982835i	$-0.559063\pi$
$37$	−2.24441			−0.368978			−0.184489	+	0.982835i	$0.559063\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	3.92841i		−	0.613515i	−0.951788	−	0.306758i	$-0.900756\pi$
$41$		−	3.92841i		−	0.613515i	0.951788	−	0.306758i	$-0.0992441\pi$
$42$		0			0
$43$		−	10.1012i		−	1.54042i	−0.637788	−	0.770212i	$-0.720150\pi$
$43$		−	10.1012i		−	1.54042i	0.637788	−	0.770212i	$-0.279850\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.88877			0.713100			0.356550	−	0.934276i	$-0.383953\pi$
$47$	4.88877			0.713100			0.356550	+	0.934276i	$0.383953\pi$
$48$		0			0
$49$	6.90126	+	1.17157i	0.985895	+	0.167368i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−9.17407			−1.26016			−0.630078	−	0.776532i	$-0.716977\pi$
$53$	−9.17407			−1.26016			−0.630078	+	0.776532i	$0.716977\pi$
$54$		0			0
$55$	10.6694			1.43865
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−9.65685			−1.25722			−0.628608	−	0.777723i	$-0.716375\pi$
$59$	−9.65685			−1.25722			−0.628608	+	0.777723i	$0.716375\pi$
$60$		0			0
$61$		−	14.2853i		−	1.82905i	−0.404534	−	0.914523i	$-0.632566\pi$
$61$		−	14.2853i		−	1.82905i	0.404534	−	0.914523i	$-0.367434\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−4.88877			−0.606377
$66$		0			0
$67$		−	6.64436i		−	0.811737i	−0.913931	−	0.405869i	$-0.866969\pi$
$67$		−	6.64436i		−	0.811737i	0.913931	−	0.405869i	$-0.133031\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			5.00125i			0.593539i	0.954949	+	0.296770i	$0.0959093\pi$
$71$			5.00125i			0.593539i	−0.954949	+	0.296770i	$0.904091\pi$
$72$		0			0
$73$		−	2.19998i		−	0.257488i	−0.991678	−	0.128744i	$-0.958906\pi$
$73$		−	2.19998i		−	0.257488i	0.991678	−	0.128744i	$-0.0410945\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.37155	−	16.2741i	0.156302	−	1.85460i
$78$		0			0
$79$		−	9.55812i		−	1.07537i	−0.843145	−	0.537686i	$-0.819299\pi$
$79$		−	9.55812i		−	1.07537i	0.843145	−	0.537686i	$-0.180701\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	12.2025			1.33940			0.669698	−	0.742634i	$-0.266424\pi$
$83$	12.2025			1.33940			0.669698	+	0.742634i	$0.266424\pi$
$84$		0			0
$85$	2.98750			0.324040
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	9.16282i		−	0.971257i	−0.874165	−	0.485629i	$-0.838591\pi$
$89$		−	9.16282i		−	0.971257i	0.874165	−	0.485629i	$-0.161409\pi$
$90$		0			0
$91$	−0.628452	+	7.45688i	−0.0658797	+	0.781693i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			10.2000i			1.04650i
$96$		0			0
$97$			17.1137i			1.73764i	0.495131	+	0.868818i	$0.335120\pi$
$97$			17.1137i			1.73764i	−0.495131	+	0.868818i	$0.664880\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	9.58527i		−	0.953770i	−0.878966	−	0.476885i	$-0.841766\pi$
$101$		−	9.58527i		−	0.953770i	0.878966	−	0.476885i	$-0.158234\pi$
$102$		0			0
$103$	3.37155			0.332208			0.166104	−	0.986108i	$-0.446881\pi$
$103$	3.37155			0.332208			0.166104	+	0.986108i	$0.446881\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	5.28406i		−	0.510829i	−0.966832	−	0.255415i	$-0.917788\pi$
$107$		−	5.28406i		−	0.510829i	0.966832	−	0.255415i	$-0.0822119\pi$
$108$		0			0
$109$	−8.54562			−0.818522			−0.409261	−	0.912417i	$-0.634213\pi$
$109$	−8.54562			−0.818522			−0.409261	+	0.912417i	$0.634213\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	2.36814			0.222776			0.111388	−	0.993777i	$-0.464470\pi$
$113$	2.36814			0.222776			0.111388	+	0.993777i	$0.464470\pi$
$114$		0			0
$115$	2.66935			0.248918
$116$		0			0
$117$		0			0
$118$		0			0
$119$	0.384044	−	4.55687i	0.0352053	−	0.417727i
$120$		0			0
$121$	−27.1037			−2.46398
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	12.1207i		−	1.08411i
$126$		0			0
$127$		−	6.24441i		−	0.554102i	−0.960855	−	0.277051i	$-0.910643\pi$
$127$		−	6.24441i		−	0.554102i	0.960855	−	0.277051i	$-0.0893570\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−10.1457			−0.886431			−0.443216	−	0.896415i	$-0.646162\pi$
$131$	−10.1457			−0.886431			−0.443216	+	0.896415i	$0.646162\pi$
$132$		0			0
$133$	15.5581	+	1.31121i	1.34906	+	0.113696i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	6.76809			0.578237			0.289118	−	0.957293i	$-0.406638\pi$
$137$	6.76809			0.578237			0.289118	+	0.957293i	$0.406638\pi$
$138$		0			0
$139$	−15.3137			−1.29889			−0.649446	−	0.760408i	$-0.724999\pi$
$139$	−15.3137			−1.29889			−0.649446	+	0.760408i	$0.724999\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	17.4594			1.46003
$144$		0			0
$145$		−	14.3207i		−	1.18927i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−8.40598			−0.688645			−0.344322	−	0.938851i	$-0.611891\pi$
$149$	−8.40598			−0.688645			−0.344322	+	0.938851i	$0.611891\pi$
$150$		0			0
$151$		−	4.04443i		−	0.329131i	−0.986366	−	0.164566i	$-0.947378\pi$
$151$		−	4.04443i		−	0.329131i	0.986366	−	0.164566i	$-0.0526222\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	8.00000i		−	0.642575i
$156$		0			0
$157$		−	2.97160i		−	0.237159i	−0.992945	−	0.118580i	$-0.962166\pi$
$157$		−	2.97160i		−	0.237159i	0.992945	−	0.118580i	$-0.0378341\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	0.343146	−	4.07159i	0.0270437	−	0.320886i
$162$		0			0
$163$			12.6694i			0.992340i	0.868225	+	0.496170i	$0.165261\pi$
$163$			12.6694i			0.992340i	−0.868225	+	0.496170i	$0.834739\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−9.28872			−0.718782			−0.359391	−	0.933187i	$-0.617016\pi$
$167$	−9.28872			−0.718782			−0.359391	+	0.933187i	$0.617016\pi$
$168$		0			0
$169$	5.00000			0.384615
$170$		0			0
$171$		0			0
$172$		0			0
$173$			12.0716i			0.917786i	0.888492	+	0.458893i	$0.151754\pi$
$173$			12.0716i			0.917786i	−0.888492	+	0.458893i	$0.848246\pi$
$174$		0			0
$175$	−5.30576	−	0.447159i	−0.401077	−	0.0338021i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	8.17532i		−	0.611052i	−0.952184	−	0.305526i	$-0.901168\pi$
$179$		−	8.17532i		−	0.611052i	0.952184	−	0.305526i	$-0.0988323\pi$
$180$		0			0
$181$		−	22.8334i		−	1.69720i	−0.529039	−	0.848598i	$-0.677447\pi$
$181$		−	22.8334i		−	1.69720i	0.529039	−	0.848598i	$-0.322553\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			3.87932i			0.285214i
$186$		0			0
$187$	−10.6694			−0.780220
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	12.6331i		−	0.914100i	−0.889441	−	0.457050i	$-0.848906\pi$
$191$		−	12.6331i		−	0.914100i	0.889441	−	0.457050i	$-0.151094\pi$
$192$		0			0
$193$	8.30121			0.597534			0.298767	−	0.954326i	$-0.403425\pi$
$193$	8.30121			0.597534			0.298767	+	0.954326i	$0.403425\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−14.8309			−1.05666			−0.528330	−	0.849039i	$-0.677182\pi$
$197$	−14.8309			−1.05666			−0.528330	+	0.849039i	$0.677182\pi$
$198$		0			0
$199$	−12.0159			−0.851785			−0.425892	−	0.904774i	$-0.640040\pi$
$199$	−12.0159			−0.851785			−0.425892	+	0.904774i	$0.640040\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−21.8434	−	1.84092i	−1.53311	−	0.129208i
$204$		0			0
$205$	−6.79003			−0.474236
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	36.4274i		−	2.51974i
$210$		0			0
$211$			17.9038i			1.23255i	0.787533	+	0.616273i	$0.211358\pi$
$211$			17.9038i			1.23255i	−0.787533	+	0.616273i	$0.788642\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−17.4594			−1.19072
$216$		0			0
$217$	−12.2025	−	1.02840i	−0.828358	−	0.0698125i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	4.88877			0.328854
$222$		0			0
$223$	8.58652			0.574996			0.287498	−	0.957781i	$-0.407177\pi$
$223$	8.58652			0.574996			0.287498	+	0.957781i	$0.407177\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	13.2569			0.879892			0.439946	−	0.898024i	$-0.354998\pi$
$227$	13.2569			0.879892			0.439946	+	0.898024i	$0.354998\pi$
$228$		0			0
$229$		−	3.91467i		−	0.258689i	−0.991600	−	0.129344i	$-0.958713\pi$
$229$		−	3.91467i		−	0.258689i	0.991600	−	0.129344i	$-0.0412872\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	0.343146			0.0224802			0.0112401	−	0.999937i	$-0.496422\pi$
$233$	0.343146			0.0224802			0.0112401	+	0.999937i	$0.496422\pi$
$234$		0			0
$235$		−	8.44994i		−	0.551213i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	28.8038i		−	1.86316i	−0.363535	−	0.931581i	$-0.618430\pi$
$239$		−	28.8038i		−	1.86316i	0.363535	−	0.931581i	$-0.381570\pi$
$240$		0			0
$241$		−	24.5481i		−	1.58128i	−0.612279	−	0.790642i	$-0.709747\pi$
$241$		−	24.5481i		−	1.58128i	0.612279	−	0.790642i	$-0.290253\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.02499	−	11.9284i	0.129372	−	0.762078i
$246$		0			0
$247$			16.6913i			1.06204i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	12.7681			0.805915			0.402957	−	0.915219i	$-0.367982\pi$
$251$	12.7681			0.805915			0.402957	+	0.915219i	$0.367982\pi$
$252$		0			0
$253$	−9.53312			−0.599342
$254$		0			0
$255$		0			0
$256$		0			0
$257$			18.9853i			1.18427i	0.805838	+	0.592137i	$0.201715\pi$
$257$			18.9853i			1.18427i	−0.805838	+	0.592137i	$0.798285\pi$
$258$		0			0
$259$	5.91717	+	0.498688i	0.367675	+	0.0309870i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			19.9150i			1.22801i	0.789301	+	0.614006i	$0.210443\pi$
$263$			19.9150i			1.22801i	−0.789301	+	0.614006i	$0.789557\pi$
$264$		0			0
$265$			15.8568i			0.974077i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	12.6197i		−	0.769437i	−0.923034	−	0.384718i	$-0.874299\pi$
$269$		−	12.6197i		−	0.769437i	0.923034	−	0.384718i	$-0.125701\pi$
$270$		0			0
$271$	5.14908			0.312784			0.156392	−	0.987695i	$-0.450014\pi$
$271$	5.14908			0.312784			0.156392	+	0.987695i	$0.450014\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			12.4228i			0.749122i
$276$		0			0
$277$	−14.3262			−0.860778			−0.430389	−	0.902643i	$-0.641624\pi$
$277$	−14.3262			−0.860778			−0.430389	+	0.902643i	$0.641624\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	12.9456			0.772268			0.386134	−	0.922443i	$-0.373810\pi$
$281$	12.9456			0.772268			0.386134	+	0.922443i	$0.373810\pi$
$282$		0			0
$283$	26.4719			1.57359			0.786795	−	0.617215i	$-0.211739\pi$
$283$	26.4719			1.57359			0.786795	+	0.617215i	$0.211739\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.872860	+	10.3569i	−0.0515233	+	0.611348i
$288$		0			0
$289$	14.0125			0.824264
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	15.5285i		−	0.907183i	−0.891210	−	0.453591i	$-0.850143\pi$
$293$		−	15.5285i		−	0.907183i	0.891210	−	0.453591i	$-0.149857\pi$
$294$		0			0
$295$			16.6913i			0.971804i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4.36814			0.252616
$300$		0			0
$301$	−2.24441	+	26.6310i	−0.129366	+	1.53498i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−24.6913			−1.41382
$306$		0			0
$307$	13.1332			0.749550			0.374775	−	0.927116i	$-0.377720\pi$
$307$	13.1332			0.749550			0.374775	+	0.927116i	$0.377720\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3.87932			0.219976			0.109988	−	0.993933i	$-0.464919\pi$
$311$	3.87932			0.219976			0.109988	+	0.993933i	$0.464919\pi$
$312$		0			0
$313$		−	21.1362i		−	1.19469i	−0.801984	−	0.597345i	$-0.796222\pi$
$313$		−	21.1362i		−	1.19469i	0.801984	−	0.597345i	$-0.203778\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	13.9422			0.783070			0.391535	−	0.920163i	$-0.371944\pi$
$317$	13.9422			0.783070			0.391535	+	0.920163i	$0.371944\pi$
$318$		0			0
$319$			51.1437i			2.86350i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	10.2000i		−	0.567542i
$324$		0			0
$325$		−	5.69220i		−	0.315746i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−12.8888	−	1.08624i	−0.710581	−	0.0598864i
$330$		0			0
$331$			14.3037i			0.786203i	0.919495	+	0.393102i	$0.128598\pi$
$331$			14.3037i			0.786203i	−0.919495	+	0.393102i	$0.871402\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−11.4844			−0.627458
$336$		0			0
$337$	17.1162			0.932381			0.466190	−	0.884684i	$-0.345626\pi$
$337$	17.1162			0.932381			0.466190	+	0.884684i	$0.345626\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			28.5706i			1.54719i
$342$		0			0
$343$	−17.9342	−	4.62214i	−0.968356	−	0.249572i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			4.11602i			0.220959i	0.993878	+	0.110480i	$0.0352387\pi$
$347$			4.11602i			0.220959i	−0.993878	+	0.110480i	$0.964761\pi$
$348$		0			0
$349$			34.5766i			1.85085i	0.378936	+	0.925423i	$0.376290\pi$
$349$			34.5766i			1.85085i	−0.378936	+	0.925423i	$0.623710\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	14.2991i		−	0.761062i	−0.924768	−	0.380531i	$-0.875741\pi$
$353$		−	14.2991i		−	0.761062i	0.924768	−	0.380531i	$-0.124259\pi$
$354$		0			0
$355$	8.64436			0.458795
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	26.8831i		−	1.41883i	−0.704789	−	0.709417i	$-0.748958\pi$
$359$		−	26.8831i		−	1.41883i	0.704789	−	0.709417i	$-0.251042\pi$
$360$		0			0
$361$	15.8249			0.832889
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−3.80252			−0.199033
$366$		0			0
$367$	16.3072			0.851231			0.425616	−	0.904904i	$-0.360058\pi$
$367$	16.3072			0.851231			0.425616	+	0.904904i	$0.360058\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	24.1866	+	2.03840i	1.25570	+	0.105828i
$372$		0			0
$373$	28.9188			1.49736			0.748678	−	0.662934i	$-0.230689\pi$
$373$	28.9188			1.49736			0.748678	+	0.662934i	$0.230689\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	23.4344i		−	1.20693i
$378$		0			0
$379$		−	28.6150i		−	1.46986i	−0.678145	−	0.734928i	$-0.737216\pi$
$379$		−	28.6150i		−	1.46986i	0.678145	−	0.734928i	$-0.262784\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	25.4594			1.30091			0.650457	−	0.759543i	$-0.274578\pi$
$383$	25.4594			1.30091			0.650457	+	0.759543i	$0.274578\pi$
$384$		0			0
$385$	−28.1287	−	2.37064i	−1.43357	−	0.120819i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	8.25350			0.418469			0.209234	−	0.977866i	$-0.432903\pi$
$389$	8.25350			0.418469			0.209234	+	0.977866i	$0.432903\pi$
$390$		0			0
$391$	−2.66935			−0.134995
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−16.5206			−0.831243
$396$		0			0
$397$		−	27.9422i		−	1.40238i	−0.712976	−	0.701188i	$-0.752653\pi$
$397$		−	27.9422i		−	1.40238i	0.712976	−	0.701188i	$-0.247347\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−23.8593			−1.19148			−0.595739	−	0.803178i	$-0.703141\pi$
$401$	−23.8593			−1.19148			−0.595739	+	0.803178i	$0.703141\pi$
$402$		0			0
$403$		−	13.0912i		−	0.652121i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	13.8543i		−	0.686734i
$408$		0			0
$409$		−	10.9363i		−	0.540763i	−0.962753	−	0.270381i	$-0.912850\pi$
$409$		−	10.9363i		−	0.540763i	0.962753	−	0.270381i	$-0.0871498\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	25.4594	+	2.14567i	1.25277	+	0.105582i
$414$		0			0
$415$		−	21.0912i		−	1.03533i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−5.08624			−0.248479			−0.124240	−	0.992252i	$-0.539649\pi$
$419$	−5.08624			−0.248479			−0.124240	+	0.992252i	$0.539649\pi$
$420$		0			0
$421$	10.8150			0.527092			0.263546	−	0.964647i	$-0.415108\pi$
$421$	10.8150			0.527092			0.263546	+	0.964647i	$0.415108\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			3.47848i			0.168731i
$426$		0			0
$427$	−3.17407	+	37.6619i	−0.153604	+	1.82258i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			2.10999i			0.101634i	0.998708	+	0.0508172i	$0.0161826\pi$
$431$			2.10999i			0.101634i	−0.998708	+	0.0508172i	$0.983817\pi$
$432$		0			0
$433$		−	21.2619i		−	1.02178i	−0.859646	−	0.510891i	$-0.829316\pi$
$433$		−	21.2619i		−	1.02178i	0.859646	−	0.510891i	$-0.170684\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	9.11373i		−	0.435969i
$438$		0			0
$439$	−35.6210			−1.70010			−0.850048	−	0.526706i	$-0.823427\pi$
$439$	−35.6210			−1.70010			−0.850048	+	0.526706i	$0.823427\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	2.14101i		−	0.101723i	−0.998706	−	0.0508613i	$-0.983803\pi$
$443$		−	2.14101i		−	0.101723i	0.998706	−	0.0508613i	$-0.0161966\pi$
$444$		0			0
$445$	−15.8374			−0.750764
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−9.40257			−0.443735			−0.221867	−	0.975077i	$-0.571215\pi$
$449$	−9.40257			−0.443735			−0.221867	+	0.975077i	$0.571215\pi$
$450$		0			0
$451$	24.2494			1.14186
$452$		0			0
$453$		0			0
$454$		0			0
$455$	12.8888	+	1.08624i	0.604235	+	0.0509238i
$456$		0			0
$457$	15.5801			0.728804			0.364402	−	0.931242i	$-0.381273\pi$
$457$	15.5801			0.728804			0.364402	+	0.931242i	$0.381273\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			29.8766i			1.39149i	0.718288	+	0.695746i	$0.244926\pi$
$461$			29.8766i			1.39149i	−0.718288	+	0.695746i	$0.755074\pi$
$462$		0			0
$463$		−	35.3356i		−	1.64219i	−0.570794	−	0.821093i	$-0.693364\pi$
$463$		−	35.3356i		−	1.64219i	0.570794	−	0.821093i	$-0.306636\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	24.3232			1.12554			0.562771	−	0.826613i	$-0.309735\pi$
$467$	24.3232			1.12554			0.562771	+	0.826613i	$0.309735\pi$
$468$		0			0
$469$	−1.47632	+	17.5172i	−0.0681701	+	0.808870i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	62.3531			2.86700
$474$		0			0
$475$	−11.8763			−0.544921
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−19.6000			−0.895549			−0.447775	−	0.894146i	$-0.647783\pi$
$479$	−19.6000			−0.895549			−0.447775	+	0.894146i	$0.647783\pi$
$480$		0			0
$481$			6.34814i			0.289450i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	29.5801			1.34316
$486$		0			0
$487$			21.7331i			0.984821i	0.870363	+	0.492410i	$0.163884\pi$
$487$			21.7331i			0.984821i	−0.870363	+	0.492410i	$0.836116\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	34.9409i		−	1.57686i	−0.615124	−	0.788431i	$-0.710894\pi$
$491$		−	34.9409i		−	1.57686i	0.615124	−	0.788431i	$-0.289106\pi$
$492$		0			0
$493$			14.3207i			0.644970i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.11123	−	13.1853i	0.0498457	−	0.591442i
$498$		0			0
$499$		−	19.5288i		−	0.874229i	−0.899406	−	0.437115i	$-0.856000\pi$
$499$		−	19.5288i		−	0.874229i	0.899406	−	0.437115i	$-0.144000\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−30.4249			−1.35658			−0.678291	−	0.734794i	$-0.737279\pi$
$503$	−30.4249			−1.35658			−0.678291	+	0.734794i	$0.737279\pi$
$504$		0			0
$505$	−16.5676			−0.737247
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	13.8766i		−	0.615070i	−0.951537	−	0.307535i	$-0.900496\pi$
$509$		−	13.8766i		−	0.615070i	0.951537	−	0.307535i	$-0.0995041\pi$
$510$		0			0
$511$	−0.488815	+	5.80002i	−0.0216239	+	0.256578i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	5.82752i		−	0.256791i
$516$		0			0
$517$			30.1775i			1.32720i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			11.3628i			0.497813i	0.968527	+	0.248907i	$0.0800712\pi$
$521$			11.3628i			0.497813i	−0.968527	+	0.248907i	$0.919929\pi$
$522$		0			0
$523$	−4.52063			−0.197673			−0.0988366	−	0.995104i	$-0.531512\pi$
$523$	−4.52063			−0.197673			−0.0988366	+	0.995104i	$0.531512\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			8.00000i			0.348485i
$528$		0			0
$529$	20.6149			0.896301
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−11.1112			−0.481281
$534$		0			0
$535$	−9.13317			−0.394862
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−7.23191	+	42.6003i	−0.311500	+	1.83492i
$540$		0			0
$541$	46.1287			1.98323			0.991614	−	0.129231i	$-0.0412510\pi$
$541$	46.1287			1.98323			0.991614	+	0.129231i	$0.0412510\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			14.7706i			0.632702i
$546$		0			0
$547$		−	16.8718i		−	0.721387i	−0.932684	−	0.360694i	$-0.882540\pi$
$547$		−	16.8718i		−	0.721387i	0.932684	−	0.360694i	$-0.117460\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−48.8938			−2.08294
$552$		0			0
$553$	−2.12373	+	25.1991i	−0.0903102	+	1.07157i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−43.7197			−1.85246			−0.926231	−	0.376955i	$-0.876971\pi$
$557$	−43.7197			−1.85246			−0.926231	+	0.376955i	$0.876971\pi$
$558$		0			0
$559$	−28.5706			−1.20841
$560$		0			0
$561$		0			0
$562$		0			0
$563$	19.4662			0.820402			0.410201	−	0.911995i	$-0.365458\pi$
$563$	19.4662			0.820402			0.410201	+	0.911995i	$0.365458\pi$
$564$		0			0
$565$		−	4.09318i		−	0.172202i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	12.1775			0.510507			0.255253	−	0.966874i	$-0.417841\pi$
$569$	12.1775			0.510507			0.255253	+	0.966874i	$0.417841\pi$
$570$		0			0
$571$			6.04193i			0.252847i	0.991976	+	0.126424i	$0.0403498\pi$
$571$			6.04193i			0.252847i	−0.991976	+	0.126424i	$0.959650\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			3.10804i			0.129614i
$576$		0			0
$577$			3.03444i			0.126325i	0.998003	+	0.0631626i	$0.0201187\pi$
$577$			3.03444i			0.126325i	−0.998003	+	0.0631626i	$0.979881\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−32.1707	−	2.71128i	−1.33466	−	0.112483i
$582$		0			0
$583$		−	56.6299i		−	2.34537i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−32.2075			−1.32934			−0.664672	−	0.747135i	$-0.731429\pi$
$587$	−32.2075			−1.32934			−0.664672	+	0.747135i	$0.731429\pi$
$588$		0			0
$589$	−27.3137			−1.12544
$590$		0			0
$591$		0			0
$592$		0			0
$593$			35.2196i			1.44630i	0.690693	+	0.723148i	$0.257306\pi$
$593$			35.2196i			1.44630i	−0.690693	+	0.723148i	$0.742694\pi$
$594$		0			0
$595$	−7.87627	−	0.663798i	−0.322896	−	0.0272130i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			20.7149i			0.846388i	0.906039	+	0.423194i	$0.139091\pi$
$599$			20.7149i			0.846388i	−0.906039	+	0.423194i	$0.860909\pi$
$600$		0			0
$601$			24.8620i			1.01414i	0.861905	+	0.507070i	$0.169271\pi$
$601$			24.8620i			1.01414i	−0.861905	+	0.507070i	$0.830729\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			46.8472i			1.90461i
$606$		0			0
$607$	−1.19101			−0.0483417			−0.0241709	−	0.999708i	$-0.507695\pi$
$607$	−1.19101			−0.0483417			−0.0241709	+	0.999708i	$0.507695\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	13.8275i		−	0.559401i
$612$		0			0
$613$	−2.60242			−0.105111			−0.0525555	−	0.998618i	$-0.516737\pi$
$613$	−2.60242			−0.105111			−0.0525555	+	0.998618i	$0.516737\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−33.9161			−1.36541			−0.682706	−	0.730693i	$-0.739197\pi$
$617$	−33.9161			−1.36541			−0.682706	+	0.730693i	$0.739197\pi$
$618$		0			0
$619$	−18.8688			−0.758400			−0.379200	−	0.925315i	$-0.623801\pi$
$619$	−18.8688			−0.758400			−0.379200	+	0.925315i	$0.623801\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−2.03590	+	24.1569i	−0.0815666	+	0.967826i
$624$		0			0
$625$	−10.8874			−0.435495
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	3.87932i		−	0.154679i
$630$		0			0
$631$			2.44688i			0.0974088i	0.998813	+	0.0487044i	$0.0155092\pi$
$631$			2.44688i			0.0974088i	−0.998813	+	0.0487044i	$0.984491\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−10.7931			−0.428310
$636$		0			0
$637$	3.31371	−	19.5197i	0.131294	−	0.773399i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	23.6619			0.934587			0.467293	−	0.884102i	$-0.345229\pi$
$641$	23.6619			0.934587			0.467293	+	0.884102i	$0.345229\pi$
$642$		0			0
$643$	20.1237			0.793602			0.396801	−	0.917905i	$-0.370120\pi$
$643$	20.1237			0.793602			0.396801	+	0.917905i	$0.370120\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	5.94320			0.233651			0.116826	−	0.993152i	$-0.462728\pi$
$647$	5.94320			0.233651			0.116826	+	0.993152i	$0.462728\pi$
$648$		0			0
$649$		−	59.6100i		−	2.33990i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−5.33974			−0.208960			−0.104480	−	0.994527i	$-0.533318\pi$
$653$	−5.33974			−0.208960			−0.104480	+	0.994527i	$0.533318\pi$
$654$		0			0
$655$			17.5362i			0.685195i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	25.5754i		−	0.996276i	−0.867098	−	0.498138i	$-0.834017\pi$
$659$		−	25.5754i		−	0.996276i	0.867098	−	0.498138i	$-0.165983\pi$
$660$		0			0
$661$			21.7197i			0.844798i	0.906410	+	0.422399i	$0.138812\pi$
$661$			21.7197i			0.844798i	−0.906410	+	0.422399i	$0.861188\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	2.26635	−	26.8913i	0.0878851	−	1.04280i
$666$		0			0
$667$			12.7956i			0.495447i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	88.1807			3.40418
$672$		0			0
$673$	−42.6175			−1.64279			−0.821393	−	0.570363i	$-0.806803\pi$
$673$	−42.6175			−1.64279			−0.821393	+	0.570363i	$0.806803\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			33.7559i			1.29735i	0.761067	+	0.648673i	$0.224676\pi$
$677$			33.7559i			1.29735i	−0.761067	+	0.648673i	$0.775324\pi$
$678$		0			0
$679$	3.80252	−	45.1187i	0.145927	−	1.73150i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			38.1553i			1.45997i	0.683462	+	0.729986i	$0.260474\pi$
$683$			38.1553i			1.45997i	−0.683462	+	0.729986i	$0.739526\pi$
$684$		0			0
$685$		−	11.6982i		−	0.446966i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			25.9482i			0.988547i
$690$		0			0
$691$	32.8120			1.24823			0.624113	−	0.781334i	$-0.285460\pi$
$691$	32.8120			1.24823			0.624113	+	0.781334i	$0.285460\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			26.4688i			1.00402i
$696$		0			0
$697$	6.79003			0.257191
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−14.9077			−0.563057			−0.281529	−	0.959553i	$-0.590841\pi$
$701$	−14.9077			−0.563057			−0.281529	+	0.959553i	$0.590841\pi$
$702$		0			0
$703$	13.2448			0.499539
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−2.12976	+	25.2707i	−0.0800980	+	0.950401i
$708$		0			0
$709$	−5.47937			−0.205782			−0.102891	−	0.994693i	$-0.532809\pi$
$709$	−5.47937			−0.205782			−0.102891	+	0.994693i	$0.532809\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			7.14804i			0.267696i
$714$		0			0
$715$		−	30.1775i		−	1.12857i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−3.60005			−0.134259			−0.0671296	−	0.997744i	$-0.521384\pi$
$719$	−3.60005			−0.134259			−0.0671296	+	0.997744i	$0.521384\pi$
$720$		0			0
$721$	−8.88877	−	0.749129i	−0.331035	−	0.0278990i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	16.6741			0.619262
$726$		0			0
$727$	17.3147			0.642168			0.321084	−	0.947051i	$-0.395953\pi$
$727$	17.3147			0.642168			0.321084	+	0.947051i	$0.395953\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	17.4594			0.645758
$732$		0			0
$733$			15.7490i			0.581703i	0.956768	+	0.290851i	$0.0939385\pi$
$733$			15.7490i			0.581703i	−0.956768	+	0.290851i	$0.906061\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	41.0144			1.51079
$738$		0			0
$739$		−	8.87182i		−	0.326355i	−0.986597	−	0.163178i	$-0.947826\pi$
$739$		−	8.87182i		−	0.326355i	0.986597	−	0.163178i	$-0.0521744\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	2.90057i		−	0.106412i	−0.998584	−	0.0532058i	$-0.983056\pi$
$743$		−	2.90057i		−	0.106412i	0.998584	−	0.0532058i	$-0.0169439\pi$
$744$		0			0
$745$			14.5292i			0.532310i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.17407	+	13.9309i	−0.0428997	+	0.509025i
$750$		0			0
$751$		−	9.98056i		−	0.364196i	−0.983280	−	0.182098i	$-0.941711\pi$
$751$		−	9.98056i		−	0.364196i	0.983280	−	0.182098i	$-0.0582888\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−6.99056			−0.254412
$756$		0			0
$757$	−4.34814			−0.158036			−0.0790180	−	0.996873i	$-0.525178\pi$
$757$	−4.34814			−0.158036			−0.0790180	+	0.996873i	$0.525178\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			46.8197i			1.69721i	0.529025	+	0.848606i	$0.322558\pi$
$761$			46.8197i			1.69721i	−0.529025	+	0.848606i	$0.677442\pi$
$762$		0			0
$763$	22.5297	+	1.89876i	0.815630	+	0.0687398i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			27.3137i			0.986241i
$768$		0			0
$769$			2.86377i			0.103270i	0.998666	+	0.0516351i	$0.0164433\pi$
$769$			2.86377i			0.103270i	−0.998666	+	0.0516351i	$0.983557\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			52.5040i			1.88844i	0.329317	+	0.944219i	$0.393181\pi$
$773$			52.5040i			1.88844i	−0.329317	+	0.944219i	$0.606819\pi$
$774$		0			0
$775$	9.31474			0.334595
$776$		0			0
$777$		0			0
$778$		0			0
$779$			23.1826i			0.830603i
$780$		0			0
$781$	−30.8718			−1.10468
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−5.13623			−0.183320
$786$		0			0
$787$	−46.3981			−1.65391			−0.826957	−	0.562265i	$-0.809930\pi$
$787$	−46.3981			−1.65391			−0.826957	+	0.562265i	$0.809930\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−6.24337	−	0.526180i	−0.221989	−	0.0187088i
$792$		0			0
$793$	−40.4049			−1.43482
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	28.5215i		−	1.01028i	−0.863036	−	0.505142i	$-0.831440\pi$
$797$		−	28.5215i		−	1.01028i	0.863036	−	0.505142i	$-0.168560\pi$
$798$		0			0
$799$			8.44994i			0.298937i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	13.5801			0.479230
$804$		0			0
$805$	−7.03749	−	0.593107i	−0.248039	−	0.0209043i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−53.6100			−1.88483			−0.942414	−	0.334447i	$-0.891450\pi$
$809$	−53.6100			−1.88483			−0.942414	+	0.334447i	$0.891450\pi$
$810$		0			0
$811$	−27.7207			−0.973406			−0.486703	−	0.873567i	$-0.661801\pi$
$811$	−27.7207			−0.973406			−0.486703	+	0.873567i	$0.661801\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	21.8982			0.767061
$816$		0			0
$817$			59.6100i			2.08549i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.56902			−0.0547593			−0.0273797	−	0.999625i	$-0.508716\pi$
$821$	−1.56902			−0.0547593			−0.0273797	+	0.999625i	$0.508716\pi$
$822$		0			0
$823$		−	24.7132i		−	0.861449i	−0.902483	−	0.430724i	$-0.858258\pi$
$823$		−	24.7132i		−	0.861449i	0.902483	−	0.430724i	$-0.141742\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	31.4840i		−	1.09481i	−0.836869	−	0.547404i	$-0.815616\pi$
$827$		−	31.4840i		−	1.09481i	0.836869	−	0.547404i	$-0.184384\pi$
$828$		0			0
$829$		−	40.4403i		−	1.40455i	−0.711906	−	0.702275i	$-0.752168\pi$
$829$		−	40.4403i		−	1.40455i	0.711906	−	0.702275i	$-0.247832\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−2.02499	+	11.9284i	−0.0701618	+	0.413295i
$834$		0			0
$835$			16.0550i			0.555606i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−39.5732			−1.36622			−0.683110	−	0.730316i	$-0.739373\pi$
$839$	−39.5732			−1.36622			−0.683110	+	0.730316i	$0.739373\pi$
$840$		0			0
$841$	39.6463			1.36711
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	8.64220i		−	0.297301i
$846$		0			0
$847$	71.4564	+	6.02222i	2.45527	+	0.206926i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	3.46620i		−	0.118820i
$852$		0			0
$853$			33.5540i			1.14887i	0.818551	+	0.574434i	$0.194778\pi$
$853$			33.5540i			1.14887i	−0.818551	+	0.574434i	$0.805222\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			19.5060i			0.666311i	0.942872	+	0.333156i	$0.108113\pi$
$857$			19.5060i			0.666311i	−0.942872	+	0.333156i	$0.891887\pi$
$858$		0			0
$859$	8.72616			0.297733			0.148866	−	0.988857i	$-0.452438\pi$
$859$	8.72616			0.297733			0.148866	+	0.988857i	$0.452438\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			3.08055i			0.104863i	0.998625	+	0.0524315i	$0.0166971\pi$
$863$			3.08055i			0.104863i	−0.998625	+	0.0524315i	$0.983303\pi$
$864$		0			0
$865$	20.8650			0.709432
$866$		0			0
$867$		0			0
$868$		0			0
$869$	59.0006			2.00146
$870$		0			0
$871$	−18.7931			−0.636779
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.69311	+	31.9550i	−0.0910437	+	1.08028i
$876$		0			0
$877$	45.4712			1.53545			0.767727	−	0.640778i	$-0.221388\pi$
$877$	45.4712			1.53545			0.767727	+	0.640778i	$0.221388\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			38.6765i			1.30304i	0.758629	+	0.651522i	$0.225869\pi$
$881$			38.6765i			1.30304i	−0.758629	+	0.651522i	$0.774131\pi$
$882$		0			0
$883$		−	15.5874i		−	0.524559i	−0.964992	−	0.262279i	$-0.915526\pi$
$883$		−	15.5874i		−	0.524559i	0.964992	−	0.262279i	$-0.0844742\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	31.8525			1.06950			0.534751	−	0.845009i	$-0.320405\pi$
$887$	31.8525			1.06950			0.534751	+	0.845009i	$0.320405\pi$
$888$		0			0
$889$	−1.38745	+	16.4628i	−0.0465337	+	0.552144i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−28.8499			−0.965425
$894$		0			0
$895$	−14.1305			−0.472332
$896$		0			0
$897$		0			0
$898$		0			0
$899$	38.3481			1.27898
$900$		0			0
$901$		−	15.8568i		−	0.528268i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−39.4662			−1.31190
$906$		0			0
$907$			5.05387i			0.167811i	0.996474	+	0.0839056i	$0.0267394\pi$
$907$			5.05387i			0.167811i	−0.996474	+	0.0839056i	$0.973261\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	32.0900i		−	1.06319i	−0.846999	−	0.531594i	$-0.821593\pi$
$911$		−	32.0900i		−	1.06319i	0.846999	−	0.531594i	$-0.178407\pi$
$912$		0			0
$913$			75.3237i			2.49285i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	26.7481	+	2.25428i	0.883300	+	0.0744429i
$918$		0			0
$919$			20.9332i			0.690522i	0.938507	+	0.345261i	$0.112210\pi$
$919$			20.9332i			0.690522i	−0.938507	+	0.345261i	$0.887790\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	14.1457			0.465610
$924$		0			0
$925$	−4.51686			−0.148514
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	21.7059i		−	0.712149i	−0.934458	−	0.356074i	$-0.884115\pi$
$929$		−	21.7059i		−	0.712149i	0.934458	−	0.356074i	$-0.115885\pi$
$930$		0			0
$931$	−40.7262	−	6.91376i	−1.33475	−	0.226589i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			18.4413i			0.603096i
$936$		0			0
$937$		−	18.1825i		−	0.593996i	−0.954878	−	0.296998i	$-0.904015\pi$
$937$		−	18.1825i		−	0.593996i	0.954878	−	0.296998i	$-0.0959854\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	12.8079i		−	0.417524i	−0.977966	−	0.208762i	$-0.933057\pi$
$941$		−	12.8079i		−	0.417524i	0.977966	−	0.208762i	$-0.0669435\pi$
$942$		0			0
$943$	6.06693			0.197566
$944$		0			0
$945$		0			0
$946$		0			0
$947$			23.8797i			0.775985i	0.921663	+	0.387992i	$0.126831\pi$
$947$			23.8797i			0.775985i	−0.921663	+	0.387992i	$0.873169\pi$
$948$		0			0
$949$	−6.22247			−0.201990
$950$		0			0
$951$		0			0
$952$		0			0
$953$	37.0006			1.19857			0.599283	−	0.800537i	$-0.295453\pi$
$953$	37.0006			1.19857			0.599283	+	0.800537i	$0.295453\pi$
$954$		0			0
$955$	−21.8356			−0.706582
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−17.8434	−	1.50381i	−0.576194	−	0.0485606i
$960$		0			0
$961$	−9.57743			−0.308949
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	14.3481i		−	0.461883i
$966$		0			0
$967$			7.93307i			0.255110i	0.991831	+	0.127555i	$0.0407130\pi$
$967$			7.93307i			0.255110i	−0.991831	+	0.127555i	$0.959287\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−49.2169			−1.57945			−0.789723	−	0.613464i	$-0.789776\pi$
$971$	−49.2169			−1.57945			−0.789723	+	0.613464i	$0.789776\pi$
$972$		0			0
$973$	40.3731	+	3.40257i	1.29430	+	0.109082i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	21.8075			0.697684			0.348842	−	0.937181i	$-0.386575\pi$
$977$	21.8075			0.697684			0.348842	+	0.937181i	$0.386575\pi$
$978$		0			0
$979$	56.5605			1.80768
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−45.7436			−1.45900			−0.729498	−	0.683983i	$-0.760246\pi$
$983$	−45.7436			−1.45900			−0.729498	+	0.683983i	$0.760246\pi$
$984$		0			0
$985$			25.6344i			0.816779i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	15.6000			0.496053
$990$		0			0
$991$			41.0468i			1.30389i	0.758264	+	0.651947i	$0.226048\pi$
$991$			41.0468i			1.30389i	−0.758264	+	0.651947i	$0.773952\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			20.7688i			0.658414i
$996$		0			0
$997$		−	23.3009i		−	0.737946i	−0.929440	−	0.368973i	$-0.879710\pi$
$997$		−	23.3009i		−	0.737946i	0.929440	−	0.368973i	$-0.120290\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.b.o.3583.3		8
3.2	odd	2		1344.2.b.h.895.6		8
4.3	odd	2		4032.2.b.q.3583.3		8
7.6	odd	2		4032.2.b.q.3583.6		8
8.3	odd	2		2016.2.b.c.1567.6		8
8.5	even	2		2016.2.b.a.1567.6		8
12.11	even	2		1344.2.b.g.895.6		8
21.20	even	2		1344.2.b.g.895.3		8
24.5	odd	2		672.2.b.a.223.3	✓	8
24.11	even	2		672.2.b.b.223.3	yes	8
28.27	even	2	inner	4032.2.b.o.3583.6		8
56.13	odd	2		2016.2.b.c.1567.3		8
56.27	even	2		2016.2.b.a.1567.3		8
84.83	odd	2		1344.2.b.h.895.3		8
168.83	odd	2		672.2.b.a.223.6	yes	8
168.125	even	2		672.2.b.b.223.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.b.a.223.3	✓	8	24.5	odd	2
672.2.b.a.223.6	yes	8	168.83	odd	2
672.2.b.b.223.3	yes	8	24.11	even	2
672.2.b.b.223.6	yes	8	168.125	even	2
1344.2.b.g.895.3		8	21.20	even	2
1344.2.b.g.895.6		8	12.11	even	2
1344.2.b.h.895.3		8	84.83	odd	2
1344.2.b.h.895.6		8	3.2	odd	2
2016.2.b.a.1567.3		8	56.27	even	2
2016.2.b.a.1567.6		8	8.5	even	2
2016.2.b.c.1567.3		8	56.13	odd	2
2016.2.b.c.1567.6		8	8.3	odd	2
4032.2.b.o.3583.3		8	1.1	even	1	trivial
4032.2.b.o.3583.6		8	28.27	even	2	inner
4032.2.b.q.3583.3		8	4.3	odd	2
4032.2.b.q.3583.6		8	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.72844i q^{5} +(-2.63640 - 0.222191i) q^{7} +O(q^{10})\)	\(q-1.72844i q^{5} +(-2.63640 - 0.222191i) q^{7} +6.17282i q^{11} -2.82843i q^{13} +1.72844i q^{17} -5.90126 q^{19} +1.54437i q^{23} +2.01250 q^{25} +8.28531 q^{29} +4.62845 q^{31} +(-0.384044 + 4.55687i) q^{35} -2.24441 q^{37} -3.92841i q^{41} -10.1012i q^{43} +4.88877 q^{47} +(6.90126 + 1.17157i) q^{49} -9.17407 q^{53} +10.6694 q^{55} -9.65685 q^{59} -14.2853i q^{61} -4.88877 q^{65} -6.64436i q^{67} +5.00125i q^{71} -2.19998i q^{73} +(1.37155 - 16.2741i) q^{77} -9.55812i q^{79} +12.2025 q^{83} +2.98750 q^{85} -9.16282i q^{89} +(-0.628452 + 7.45688i) q^{91} +10.2000i q^{95} +17.1137i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{7}+O(q^{10}) \) 8 * q - 4 * q^7	\( 8 q - 4 q^{7} + 8 q^{19} - 16 q^{25} + 16 q^{31} + 8 q^{35} - 8 q^{37} + 16 q^{47} + 16 q^{53} + 8 q^{55} - 32 q^{59} - 16 q^{65} + 32 q^{77} - 16 q^{83} + 56 q^{85} + 16 q^{91}+O(q^{100}) \) 8 * q - 4 * q^7 + 8 * q^19 - 16 * q^25 + 16 * q^31 + 8 * q^35 - 8 * q^37 + 16 * q^47 + 16 * q^53 + 8 * q^55 - 32 * q^59 - 16 * q^65 + 32 * q^77 - 16 * q^83 + 56 * q^85 + 16 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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