LMFDB - Embedded newform 4032.2.h.e.575.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(575,4032)]

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, 1, 0]))

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

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

chi := DirichletCharacter("4032.575");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.h (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:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1008)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			575.3
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	4032.575
Dual form			4032.2.h.e.575.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{5} -1.00000i q^{7} +O(q^{10})$	$q+1.41421i q^{5} -1.00000i q^{7} +5.65685 q^{11} +4.00000 q^{13} +1.41421i q^{17} -8.00000i q^{19} -5.65685 q^{23} +3.00000 q^{25} -7.07107i q^{29} +1.41421 q^{35} -2.00000 q^{37} -7.07107i q^{41} -8.00000i q^{43} -1.00000 q^{49} +9.89949i q^{53} +8.00000i q^{55} +11.3137 q^{59} -6.00000 q^{61} +5.65685i q^{65} -5.65685 q^{71} -8.00000 q^{73} -5.65685i q^{77} +11.3137 q^{83} -2.00000 q^{85} +9.89949i q^{89} -4.00000i q^{91} +11.3137 q^{95} -16.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 16 q^{13} + 12 q^{25} - 8 q^{37} - 4 q^{49} - 24 q^{61} - 32 q^{73} - 8 q^{85} - 64 q^{97}+O(q^{100}) $ 4 * q + 16 * q^13 + 12 * q^25 - 8 * q^37 - 4 * q^49 - 24 * q^61 - 32 * q^73 - 8 * q^85 - 64 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.41421i	0.632456i	0.948683	+	0.316228i	$0.102416\pi$
$5$	1.41421i	0.632456i	−0.948683	+	0.316228i	$0.897584\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.65685	1.70561	0.852803	−	0.522233i	$-0.174901\pi$
$11$	5.65685	1.70561	0.852803	+	0.522233i	$0.174901\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.41421i	0.342997i	0.985184	+	0.171499i	$0.0548609\pi$
$17$	1.41421i	0.342997i	−0.985184	+	0.171499i	$0.945139\pi$
$18$	0	0
$19$	− 8.00000i	− 1.83533i	−0.397360	−	0.917663i	$-0.630073\pi$
$19$	− 8.00000i	− 1.83533i	0.397360	−	0.917663i	$-0.369927\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.65685	−1.17954	−0.589768	−	0.807573i	$-0.700781\pi$
$23$	−5.65685	−1.17954	−0.589768	+	0.807573i	$0.700781\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 7.07107i	− 1.31306i	−0.754298	−	0.656532i	$-0.772023\pi$
$29$	− 7.07107i	− 1.31306i	0.754298	−	0.656532i	$-0.227977\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.41421	0.239046
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 7.07107i	− 1.10432i	−0.833740	−	0.552158i	$-0.813805\pi$
$41$	− 7.07107i	− 1.10432i	0.833740	−	0.552158i	$-0.186195\pi$
$42$	0	0
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	9.89949i	1.35980i	0.733305	+	0.679900i	$0.237977\pi$
$53$	9.89949i	1.35980i	−0.733305	+	0.679900i	$0.762023\pi$
$54$	0	0
$55$	8.00000i	1.07872i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.3137	1.47292	0.736460	−	0.676481i	$-0.236496\pi$
$59$	11.3137	1.47292	0.736460	+	0.676481i	$0.236496\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.65685i	0.701646i
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.65685	−0.671345	−0.335673	−	0.941979i	$-0.608964\pi$
$71$	−5.65685	−0.671345	−0.335673	+	0.941979i	$0.608964\pi$
$72$	0	0
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 5.65685i	− 0.644658i
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	11.3137	1.24184	0.620920	−	0.783874i	$-0.286759\pi$
$83$	11.3137	1.24184	0.620920	+	0.783874i	$0.286759\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	0	0
$88$	0	0
$89$	9.89949i	1.04934i	0.851304	+	0.524672i	$0.175812\pi$
$89$	9.89949i	1.04934i	−0.851304	+	0.524672i	$0.824188\pi$
$90$	0	0
$91$	− 4.00000i	− 0.419314i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	11.3137	1.16076
$96$	0	0
$97$	−16.0000	−1.62455	−0.812277	−	0.583272i	$-0.801772\pi$
$97$	−16.0000	−1.62455	−0.812277	+	0.583272i	$0.801772\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	7.07107i	0.703598i	0.936076	+	0.351799i	$0.114430\pi$
$101$	7.07107i	0.703598i	−0.936076	+	0.351799i	$0.885570\pi$
$102$	0	0
$103$	− 16.0000i	− 1.57653i	−0.615338	−	0.788263i	$-0.710980\pi$
$103$	− 16.0000i	− 1.57653i	0.615338	−	0.788263i	$-0.289020\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.65685	−0.546869	−0.273434	−	0.961891i	$-0.588160\pi$
$107$	−5.65685	−0.546869	−0.273434	+	0.961891i	$0.588160\pi$
$108$	0	0
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 1.41421i	− 0.133038i	−0.997785	−	0.0665190i	$-0.978811\pi$
$113$	− 1.41421i	− 0.133038i	0.997785	−	0.0665190i	$-0.0211893\pi$
$114$	0	0
$115$	− 8.00000i	− 0.746004i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.41421	0.129641
$120$	0	0
$121$	21.0000	1.90909
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.3137i	1.01193i
$126$	0	0
$127$	− 16.0000i	− 1.41977i	−0.704317	−	0.709885i	$-0.748747\pi$
$127$	− 16.0000i	− 1.41977i	0.704317	−	0.709885i	$-0.251253\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	11.3137	0.988483	0.494242	−	0.869325i	$-0.335446\pi$
$131$	11.3137	0.988483	0.494242	+	0.869325i	$0.335446\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 9.89949i	− 0.845771i	−0.906183	−	0.422885i	$-0.861017\pi$
$137$	− 9.89949i	− 0.845771i	0.906183	−	0.422885i	$-0.138983\pi$
$138$	0	0
$139$	16.0000i	1.35710i	0.734553	+	0.678551i	$0.237392\pi$
$139$	16.0000i	1.35710i	−0.734553	+	0.678551i	$0.762608\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	22.6274	1.89220
$144$	0	0
$145$	10.0000	0.830455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 18.3848i	− 1.50614i	−0.657941	−	0.753070i	$-0.728572\pi$
$149$	− 18.3848i	− 1.50614i	0.657941	−	0.753070i	$-0.271428\pi$
$150$	0	0
$151$	8.00000i	0.651031i	0.945537	+	0.325515i	$0.105538\pi$
$151$	8.00000i	0.651031i	−0.945537	+	0.325515i	$0.894462\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	6.00000	0.478852	0.239426	−	0.970915i	$-0.423041\pi$
$157$	6.00000	0.478852	0.239426	+	0.970915i	$0.423041\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.65685i	0.445823i
$162$	0	0
$163$	− 16.0000i	− 1.25322i	−0.779334	−	0.626608i	$-0.784443\pi$
$163$	− 16.0000i	− 1.25322i	0.779334	−	0.626608i	$-0.215557\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	22.6274	1.75096	0.875481	−	0.483252i	$-0.160545\pi$
$167$	22.6274	1.75096	0.875481	+	0.483252i	$0.160545\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 4.24264i	− 0.322562i	−0.986909	−	0.161281i	$-0.948437\pi$
$173$	− 4.24264i	− 0.322562i	0.986909	−	0.161281i	$-0.0515625\pi$
$174$	0	0
$175$	− 3.00000i	− 0.226779i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	5.65685	0.422813	0.211407	−	0.977398i	$-0.432196\pi$
$179$	5.65685	0.422813	0.211407	+	0.977398i	$0.432196\pi$
$180$	0	0
$181$	12.0000	0.891953	0.445976	−	0.895045i	$-0.352856\pi$
$181$	12.0000	0.891953	0.445976	+	0.895045i	$0.352856\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 2.82843i	− 0.207950i
$186$	0	0
$187$	8.00000i	0.585018i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	16.9706	1.22795	0.613973	−	0.789327i	$-0.289570\pi$
$191$	16.9706	1.22795	0.613973	+	0.789327i	$0.289570\pi$
$192$	0	0
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 21.2132i	− 1.51138i	−0.654931	−	0.755689i	$-0.727302\pi$
$197$	− 21.2132i	− 1.51138i	0.654931	−	0.755689i	$-0.272698\pi$
$198$	0	0
$199$	− 8.00000i	− 0.567105i	−0.958957	−	0.283552i	$-0.908487\pi$
$199$	− 8.00000i	− 0.567105i	0.958957	−	0.283552i	$-0.0915130\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−7.07107	−0.496292
$204$	0	0
$205$	10.0000	0.698430
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 45.2548i	− 3.13034i
$210$	0	0
$211$	8.00000i	0.550743i	0.961338	+	0.275371i	$0.0888008\pi$
$211$	8.00000i	0.550743i	−0.961338	+	0.275371i	$0.911199\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	11.3137	0.771589
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5.65685i	0.380521i
$222$	0	0
$223$	− 8.00000i	− 0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$	− 8.00000i	− 0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−11.3137	−0.750917	−0.375459	−	0.926839i	$-0.622515\pi$
$227$	−11.3137	−0.750917	−0.375459	+	0.926839i	$0.622515\pi$
$228$	0	0
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	18.3848i	1.20443i	0.798335	+	0.602213i	$0.205714\pi$
$233$	18.3848i	1.20443i	−0.798335	+	0.602213i	$0.794286\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	28.2843	1.82956	0.914779	−	0.403955i	$-0.132365\pi$
$239$	28.2843	1.82956	0.914779	+	0.403955i	$0.132365\pi$
$240$	0	0
$241$	16.0000	1.03065	0.515325	−	0.856995i	$-0.327671\pi$
$241$	16.0000	1.03065	0.515325	+	0.856995i	$0.327671\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 1.41421i	− 0.0903508i
$246$	0	0
$247$	− 32.0000i	− 2.03611i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	−32.0000	−2.01182
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.3848i	1.14681i	0.819272	+	0.573405i	$0.194378\pi$
$257$	18.3848i	1.14681i	−0.819272	+	0.573405i	$0.805622\pi$
$258$	0	0
$259$	2.00000i	0.124274i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	5.65685	0.348817	0.174408	−	0.984673i	$-0.444199\pi$
$263$	5.65685	0.348817	0.174408	+	0.984673i	$0.444199\pi$
$264$	0	0
$265$	−14.0000	−0.860013
$266$	0	0
$267$	0	0
$268$	0	0
$269$	24.0416i	1.46584i	0.680313	+	0.732922i	$0.261844\pi$
$269$	24.0416i	1.46584i	−0.680313	+	0.732922i	$0.738156\pi$
$270$	0	0
$271$	16.0000i	0.971931i	0.873978	+	0.485965i	$0.161532\pi$
$271$	16.0000i	0.971931i	−0.873978	+	0.485965i	$0.838468\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	16.9706	1.02336
$276$	0	0
$277$	4.00000	0.240337	0.120168	−	0.992754i	$-0.461657\pi$
$277$	4.00000	0.240337	0.120168	+	0.992754i	$0.461657\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	26.8701i	1.60293i	0.598040	+	0.801467i	$0.295947\pi$
$281$	26.8701i	1.60293i	−0.598040	+	0.801467i	$0.704053\pi$
$282$	0	0
$283$	24.0000i	1.42665i	0.700832	+	0.713326i	$0.252812\pi$
$283$	24.0000i	1.42665i	−0.700832	+	0.713326i	$0.747188\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−7.07107	−0.417392
$288$	0	0
$289$	15.0000	0.882353
$290$	0	0
$291$	0	0
$292$	0	0
$293$	26.8701i	1.56977i	0.619644	+	0.784883i	$0.287277\pi$
$293$	26.8701i	1.56977i	−0.619644	+	0.784883i	$0.712723\pi$
$294$	0	0
$295$	16.0000i	0.931556i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−22.6274	−1.30858
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 8.48528i	− 0.485866i
$306$	0	0
$307$	− 24.0000i	− 1.36975i	−0.728659	−	0.684876i	$-0.759856\pi$
$307$	− 24.0000i	− 1.36975i	0.728659	−	0.684876i	$-0.240144\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−33.9411	−1.92462	−0.962312	−	0.271947i	$-0.912333\pi$
$311$	−33.9411	−1.92462	−0.962312	+	0.271947i	$0.912333\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 4.24264i	− 0.238290i	−0.992877	−	0.119145i	$-0.961985\pi$
$317$	− 4.24264i	− 0.238290i	0.992877	−	0.119145i	$-0.0380154\pi$
$318$	0	0
$319$	− 40.0000i	− 2.23957i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	11.3137	0.629512
$324$	0	0
$325$	12.0000	0.665640
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	8.00000i	0.439720i	0.975531	+	0.219860i	$0.0705600\pi$
$331$	8.00000i	0.439720i	−0.975531	+	0.219860i	$0.929440\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	16.0000	0.871576	0.435788	−	0.900049i	$-0.356470\pi$
$337$	16.0000	0.871576	0.435788	+	0.900049i	$0.356470\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−16.9706	−0.911028	−0.455514	−	0.890229i	$-0.650544\pi$
$347$	−16.9706	−0.911028	−0.455514	+	0.890229i	$0.650544\pi$
$348$	0	0
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	26.8701i	1.43015i	0.699048	+	0.715074i	$0.253607\pi$
$353$	26.8701i	1.43015i	−0.699048	+	0.715074i	$0.746393\pi$
$354$	0	0
$355$	− 8.00000i	− 0.424596i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	5.65685	0.298557	0.149279	−	0.988795i	$-0.452305\pi$
$359$	5.65685	0.298557	0.149279	+	0.988795i	$0.452305\pi$
$360$	0	0
$361$	−45.0000	−2.36842
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 11.3137i	− 0.592187i
$366$	0	0
$367$	8.00000i	0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$	8.00000i	0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	9.89949	0.513956
$372$	0	0
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 28.2843i	− 1.45671i
$378$	0	0
$379$	− 24.0000i	− 1.23280i	−0.787434	−	0.616399i	$-0.788591\pi$
$379$	− 24.0000i	− 1.23280i	0.787434	−	0.616399i	$-0.211409\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	11.3137	0.578103	0.289052	−	0.957313i	$-0.406660\pi$
$383$	11.3137	0.578103	0.289052	+	0.957313i	$0.406660\pi$
$384$	0	0
$385$	8.00000	0.407718
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 18.3848i	− 0.932145i	−0.884746	−	0.466073i	$-0.845669\pi$
$389$	− 18.3848i	− 0.932145i	0.884746	−	0.466073i	$-0.154331\pi$
$390$	0	0
$391$	− 8.00000i	− 0.404577i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	18.3848i	0.918092i	0.888413	+	0.459046i	$0.151809\pi$
$401$	18.3848i	0.918092i	−0.888413	+	0.459046i	$0.848191\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−11.3137	−0.560800
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 11.3137i	− 0.556711i
$414$	0	0
$415$	16.0000i	0.785409i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	11.3137	0.552711	0.276355	−	0.961056i	$-0.410873\pi$
$419$	11.3137	0.552711	0.276355	+	0.961056i	$0.410873\pi$
$420$	0	0
$421$	4.00000	0.194948	0.0974740	−	0.995238i	$-0.468924\pi$
$421$	4.00000	0.194948	0.0974740	+	0.995238i	$0.468924\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.24264i	0.205798i
$426$	0	0
$427$	6.00000i	0.290360i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−28.2843	−1.36241	−0.681203	−	0.732095i	$-0.738543\pi$
$431$	−28.2843	−1.36241	−0.681203	+	0.732095i	$0.738543\pi$
$432$	0	0
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	45.2548i	2.16483i
$438$	0	0
$439$	24.0000i	1.14546i	0.819745	+	0.572729i	$0.194115\pi$
$439$	24.0000i	1.14546i	−0.819745	+	0.572729i	$0.805885\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	16.9706	0.806296	0.403148	−	0.915135i	$-0.367916\pi$
$443$	16.9706	0.806296	0.403148	+	0.915135i	$0.367916\pi$
$444$	0	0
$445$	−14.0000	−0.663664
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 29.6985i	− 1.40156i	−0.713378	−	0.700779i	$-0.752836\pi$
$449$	− 29.6985i	− 1.40156i	0.713378	−	0.700779i	$-0.247164\pi$
$450$	0	0
$451$	− 40.0000i	− 1.88353i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5.65685	0.265197
$456$	0	0
$457$	24.0000	1.12267	0.561336	−	0.827588i	$-0.310287\pi$
$457$	24.0000	1.12267	0.561336	+	0.827588i	$0.310287\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	15.5563i	0.724531i	0.932075	+	0.362266i	$0.117997\pi$
$461$	15.5563i	0.724531i	−0.932075	+	0.362266i	$0.882003\pi$
$462$	0	0
$463$	− 16.0000i	− 0.743583i	−0.928316	−	0.371792i	$-0.878744\pi$
$463$	− 16.0000i	− 0.743583i	0.928316	−	0.371792i	$-0.121256\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	22.6274	1.04707	0.523536	−	0.852004i	$-0.324613\pi$
$467$	22.6274	1.04707	0.523536	+	0.852004i	$0.324613\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 45.2548i	− 2.08082i
$474$	0	0
$475$	− 24.0000i	− 1.10120i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	11.3137	0.516937	0.258468	−	0.966020i	$-0.416782\pi$
$479$	11.3137	0.516937	0.258468	+	0.966020i	$0.416782\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 22.6274i	− 1.02746i
$486$	0	0
$487$	40.0000i	1.81257i	0.422664	+	0.906287i	$0.361095\pi$
$487$	40.0000i	1.81257i	−0.422664	+	0.906287i	$0.638905\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	5.65685	0.255290	0.127645	−	0.991820i	$-0.459258\pi$
$491$	5.65685	0.255290	0.127645	+	0.991820i	$0.459258\pi$
$492$	0	0
$493$	10.0000	0.450377
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.65685i	0.253745i
$498$	0	0
$499$	− 8.00000i	− 0.358129i	−0.983837	−	0.179065i	$-0.942693\pi$
$499$	− 8.00000i	− 0.358129i	0.983837	−	0.179065i	$-0.0573071\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−33.9411	−1.51336	−0.756680	−	0.653785i	$-0.773180\pi$
$503$	−33.9411	−1.51336	−0.756680	+	0.653785i	$0.773180\pi$
$504$	0	0
$505$	−10.0000	−0.444994
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 35.3553i	− 1.56710i	−0.621330	−	0.783549i	$-0.713407\pi$
$509$	− 35.3553i	− 1.56710i	0.621330	−	0.783549i	$-0.286593\pi$
$510$	0	0
$511$	8.00000i	0.353899i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	22.6274	0.997083
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	18.3848i	0.805452i	0.915321	+	0.402726i	$0.131937\pi$
$521$	18.3848i	0.805452i	−0.915321	+	0.402726i	$0.868063\pi$
$522$	0	0
$523$	16.0000i	0.699631i	0.936819	+	0.349816i	$0.113756\pi$
$523$	16.0000i	0.699631i	−0.936819	+	0.349816i	$0.886244\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 28.2843i	− 1.22513i
$534$	0	0
$535$	− 8.00000i	− 0.345870i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−5.65685	−0.243658
$540$	0	0
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5.65685i	0.242313i
$546$	0	0
$547$	32.0000i	1.36822i	0.729378	+	0.684111i	$0.239809\pi$
$547$	32.0000i	1.36822i	−0.729378	+	0.684111i	$0.760191\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−56.5685	−2.40990
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 1.41421i	− 0.0599222i	−0.999551	−	0.0299611i	$-0.990462\pi$
$557$	− 1.41421i	− 0.0599222i	0.999551	−	0.0299611i	$-0.00953833\pi$
$558$	0	0
$559$	− 32.0000i	− 1.35346i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	2.00000	0.0841406
$566$	0	0
$567$	0	0
$568$	0	0
$569$	12.7279i	0.533582i	0.963754	+	0.266791i	$0.0859634\pi$
$569$	12.7279i	0.533582i	−0.963754	+	0.266791i	$0.914037\pi$
$570$	0	0
$571$	− 32.0000i	− 1.33916i	−0.742741	−	0.669579i	$-0.766474\pi$
$571$	− 32.0000i	− 1.33916i	0.742741	−	0.669579i	$-0.233526\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−16.9706	−0.707721
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 11.3137i	− 0.469372i
$582$	0	0
$583$	56.0000i	2.31928i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−45.2548	−1.86787	−0.933933	−	0.357447i	$-0.883647\pi$
$587$	−45.2548	−1.86787	−0.933933	+	0.357447i	$0.883647\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	7.07107i	0.290374i	0.989404	+	0.145187i	$0.0463784\pi$
$593$	7.07107i	0.290374i	−0.989404	+	0.145187i	$0.953622\pi$
$594$	0	0
$595$	2.00000i	0.0819920i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−16.9706	−0.693398	−0.346699	−	0.937976i	$-0.612698\pi$
$599$	−16.9706	−0.693398	−0.346699	+	0.937976i	$0.612698\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	29.6985i	1.20742i
$606$	0	0
$607$	− 40.0000i	− 1.62355i	−0.583970	−	0.811775i	$-0.698502\pi$
$607$	− 40.0000i	− 1.62355i	0.583970	−	0.811775i	$-0.301498\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 7.07107i	− 0.284670i	−0.989819	−	0.142335i	$-0.954539\pi$
$617$	− 7.07107i	− 0.284670i	0.989819	−	0.142335i	$-0.0454611\pi$
$618$	0	0
$619$	16.0000i	0.643094i	0.946894	+	0.321547i	$0.104203\pi$
$619$	16.0000i	0.643094i	−0.946894	+	0.321547i	$0.895797\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	9.89949	0.396615
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 2.82843i	− 0.112777i
$630$	0	0
$631$	48.0000i	1.91085i	0.295234	+	0.955425i	$0.404602\pi$
$631$	48.0000i	1.91085i	−0.295234	+	0.955425i	$0.595398\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	22.6274	0.897942
$636$	0	0
$637$	−4.00000	−0.158486
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 26.8701i	− 1.06130i	−0.847590	−	0.530652i	$-0.821947\pi$
$641$	− 26.8701i	− 1.06130i	0.847590	−	0.530652i	$-0.178053\pi$
$642$	0	0
$643$	8.00000i	0.315489i	0.987480	+	0.157745i	$0.0504223\pi$
$643$	8.00000i	0.315489i	−0.987480	+	0.157745i	$0.949578\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−11.3137	−0.444788	−0.222394	−	0.974957i	$-0.571387\pi$
$647$	−11.3137	−0.444788	−0.222394	+	0.974957i	$0.571387\pi$
$648$	0	0
$649$	64.0000	2.51222
$650$	0	0
$651$	0	0
$652$	0	0
$653$	24.0416i	0.940822i	0.882448	+	0.470411i	$0.155894\pi$
$653$	24.0416i	0.940822i	−0.882448	+	0.470411i	$0.844106\pi$
$654$	0	0
$655$	16.0000i	0.625172i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	5.65685	0.220360	0.110180	−	0.993912i	$-0.464857\pi$
$659$	5.65685	0.220360	0.110180	+	0.993912i	$0.464857\pi$
$660$	0	0
$661$	30.0000	1.16686	0.583432	−	0.812162i	$-0.301709\pi$
$661$	30.0000	1.16686	0.583432	+	0.812162i	$0.301709\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 11.3137i	− 0.438727i
$666$	0	0
$667$	40.0000i	1.54881i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−33.9411	−1.31028
$672$	0	0
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	46.6690i	1.79364i	0.442398	+	0.896819i	$0.354128\pi$
$677$	46.6690i	1.79364i	−0.442398	+	0.896819i	$0.645872\pi$
$678$	0	0
$679$	16.0000i	0.614024i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−39.5980	−1.51517	−0.757587	−	0.652734i	$-0.773622\pi$
$683$	−39.5980	−1.51517	−0.757587	+	0.652734i	$0.773622\pi$
$684$	0	0
$685$	14.0000	0.534913
$686$	0	0
$687$	0	0
$688$	0	0
$689$	39.5980i	1.50856i
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−22.6274	−0.858307
$696$	0	0
$697$	10.0000	0.378777
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 18.3848i	− 0.694383i	−0.937794	−	0.347192i	$-0.887135\pi$
$701$	− 18.3848i	− 0.694383i	0.937794	−	0.347192i	$-0.112865\pi$
$702$	0	0
$703$	16.0000i	0.603451i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7.07107	0.265935
$708$	0	0
$709$	−36.0000	−1.35201	−0.676004	−	0.736898i	$-0.736290\pi$
$709$	−36.0000	−1.35201	−0.676004	+	0.736898i	$0.736290\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	32.0000i	1.19673i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	22.6274	0.843860	0.421930	−	0.906628i	$-0.361353\pi$
$719$	22.6274	0.843860	0.421930	+	0.906628i	$0.361353\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 21.2132i	− 0.787839i
$726$	0	0
$727$	32.0000i	1.18681i	0.804902	+	0.593407i	$0.202218\pi$
$727$	32.0000i	1.18681i	−0.804902	+	0.593407i	$0.797782\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	11.3137	0.418453
$732$	0	0
$733$	20.0000	0.738717	0.369358	−	0.929287i	$-0.379577\pi$
$733$	20.0000	0.738717	0.369358	+	0.929287i	$0.379577\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	16.0000i	0.588570i	0.955718	+	0.294285i	$0.0950814\pi$
$739$	16.0000i	0.588570i	−0.955718	+	0.294285i	$0.904919\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−5.65685	−0.207530	−0.103765	−	0.994602i	$-0.533089\pi$
$743$	−5.65685	−0.207530	−0.103765	+	0.994602i	$0.533089\pi$
$744$	0	0
$745$	26.0000	0.952566
$746$	0	0
$747$	0	0
$748$	0	0
$749$	5.65685i	0.206697i
$750$	0	0
$751$	8.00000i	0.291924i	0.989290	+	0.145962i	$0.0466277\pi$
$751$	8.00000i	0.291924i	−0.989290	+	0.145962i	$0.953372\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−11.3137	−0.411748
$756$	0	0
$757$	−36.0000	−1.30844	−0.654221	−	0.756303i	$-0.727003\pi$
$757$	−36.0000	−1.30844	−0.654221	+	0.756303i	$0.727003\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 7.07107i	− 0.256326i	−0.991753	−	0.128163i	$-0.959092\pi$
$761$	− 7.07107i	− 0.256326i	0.991753	−	0.128163i	$-0.0409081\pi$
$762$	0	0
$763$	− 4.00000i	− 0.144810i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	45.2548	1.63406
$768$	0	0
$769$	−46.0000	−1.65880	−0.829401	−	0.558653i	$-0.811318\pi$
$769$	−46.0000	−1.65880	−0.829401	+	0.558653i	$0.811318\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 7.07107i	− 0.254329i	−0.991882	−	0.127164i	$-0.959412\pi$
$773$	− 7.07107i	− 0.254329i	0.991882	−	0.127164i	$-0.0405876\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−56.5685	−2.02678
$780$	0	0
$781$	−32.0000	−1.14505
$782$	0	0
$783$	0	0
$784$	0	0
$785$	8.48528i	0.302853i
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.41421	−0.0502836
$792$	0	0
$793$	−24.0000	−0.852265
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 26.8701i	− 0.951786i	−0.879503	−	0.475893i	$-0.842125\pi$
$797$	− 26.8701i	− 0.951786i	0.879503	−	0.475893i	$-0.157875\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−45.2548	−1.59701
$804$	0	0
$805$	−8.00000	−0.281963
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 1.41421i	− 0.0497211i	−0.999691	−	0.0248606i	$-0.992086\pi$
$809$	− 1.41421i	− 0.0497211i	0.999691	−	0.0248606i	$-0.00791417\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	22.6274	0.792604
$816$	0	0
$817$	−64.0000	−2.23908
$818$	0	0
$819$	0	0
$820$	0	0
$821$	35.3553i	1.23391i	0.786998	+	0.616955i	$0.211634\pi$
$821$	35.3553i	1.23391i	−0.786998	+	0.616955i	$0.788366\pi$
$822$	0	0
$823$	− 48.0000i	− 1.67317i	−0.547833	−	0.836587i	$-0.684547\pi$
$823$	− 48.0000i	− 1.67317i	0.547833	−	0.836587i	$-0.315453\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−50.9117	−1.77037	−0.885186	−	0.465236i	$-0.845969\pi$
$827$	−50.9117	−1.77037	−0.885186	+	0.465236i	$0.845969\pi$
$828$	0	0
$829$	28.0000	0.972480	0.486240	−	0.873825i	$-0.338368\pi$
$829$	28.0000	0.972480	0.486240	+	0.873825i	$0.338368\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 1.41421i	− 0.0489996i
$834$	0	0
$835$	32.0000i	1.10741i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−22.6274	−0.781185	−0.390593	−	0.920564i	$-0.627730\pi$
$839$	−22.6274	−0.781185	−0.390593	+	0.920564i	$0.627730\pi$
$840$	0	0
$841$	−21.0000	−0.724138
$842$	0	0
$843$	0	0
$844$	0	0
$845$	4.24264i	0.145951i
$846$	0	0
$847$	− 21.0000i	− 0.721569i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	11.3137	0.387829
$852$	0	0
$853$	−34.0000	−1.16414	−0.582069	−	0.813139i	$-0.697757\pi$
$853$	−34.0000	−1.16414	−0.582069	+	0.813139i	$0.697757\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	26.8701i	0.917864i	0.888471	+	0.458932i	$0.151768\pi$
$857$	26.8701i	0.917864i	−0.888471	+	0.458932i	$0.848232\pi$
$858$	0	0
$859$	− 8.00000i	− 0.272956i	−0.990643	−	0.136478i	$-0.956422\pi$
$859$	− 8.00000i	− 0.272956i	0.990643	−	0.136478i	$-0.0435784\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−5.65685	−0.192562	−0.0962808	−	0.995354i	$-0.530695\pi$
$863$	−5.65685	−0.192562	−0.0962808	+	0.995354i	$0.530695\pi$
$864$	0	0
$865$	6.00000	0.204006
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	11.3137	0.382473
$876$	0	0
$877$	26.0000	0.877958	0.438979	−	0.898497i	$-0.355340\pi$
$877$	26.0000	0.877958	0.438979	+	0.898497i	$0.355340\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 4.24264i	− 0.142938i	−0.997443	−	0.0714691i	$-0.977231\pi$
$881$	− 4.24264i	− 0.142938i	0.997443	−	0.0714691i	$-0.0227687\pi$
$882$	0	0
$883$	− 24.0000i	− 0.807664i	−0.914833	−	0.403832i	$-0.867678\pi$
$883$	− 24.0000i	− 0.807664i	0.914833	−	0.403832i	$-0.132322\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	45.2548	1.51951	0.759754	−	0.650210i	$-0.225319\pi$
$887$	45.2548	1.51951	0.759754	+	0.650210i	$0.225319\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	8.00000i	0.267411i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−14.0000	−0.466408
$902$	0	0
$903$	0	0
$904$	0	0
$905$	16.9706i	0.564121i
$906$	0	0
$907$	56.0000i	1.85945i	0.368255	+	0.929725i	$0.379955\pi$
$907$	56.0000i	1.85945i	−0.368255	+	0.929725i	$0.620045\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−5.65685	−0.187420	−0.0937100	−	0.995600i	$-0.529873\pi$
$911$	−5.65685	−0.187420	−0.0937100	+	0.995600i	$0.529873\pi$
$912$	0	0
$913$	64.0000	2.11809
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 11.3137i	− 0.373612i
$918$	0	0
$919$	− 8.00000i	− 0.263896i	−0.991257	−	0.131948i	$-0.957877\pi$
$919$	− 8.00000i	− 0.263896i	0.991257	−	0.131948i	$-0.0421231\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−22.6274	−0.744791
$924$	0	0
$925$	−6.00000	−0.197279
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 9.89949i	− 0.324792i	−0.986726	−	0.162396i	$-0.948078\pi$
$929$	− 9.89949i	− 0.324792i	0.986726	−	0.162396i	$-0.0519222\pi$
$930$	0	0
$931$	8.00000i	0.262189i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−11.3137	−0.369998
$936$	0	0
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	4.24264i	0.138306i	0.997606	+	0.0691531i	$0.0220297\pi$
$941$	4.24264i	0.138306i	−0.997606	+	0.0691531i	$0.977970\pi$
$942$	0	0
$943$	40.0000i	1.30258i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−16.9706	−0.551469	−0.275735	−	0.961234i	$-0.588921\pi$
$947$	−16.9706	−0.551469	−0.275735	+	0.961234i	$0.588921\pi$
$948$	0	0
$949$	−32.0000	−1.03876
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 21.2132i	− 0.687163i	−0.939123	−	0.343582i	$-0.888360\pi$
$953$	− 21.2132i	− 0.687163i	0.939123	−	0.343582i	$-0.111640\pi$
$954$	0	0
$955$	24.0000i	0.776622i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−9.89949	−0.319671
$960$	0	0
$961$	31.0000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 25.4558i	− 0.819453i
$966$	0	0
$967$	− 16.0000i	− 0.514525i	−0.966342	−	0.257263i	$-0.917179\pi$
$967$	− 16.0000i	− 0.514525i	0.966342	−	0.257263i	$-0.0828206\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−22.6274	−0.726148	−0.363074	−	0.931760i	$-0.618273\pi$
$971$	−22.6274	−0.726148	−0.363074	+	0.931760i	$0.618273\pi$
$972$	0	0
$973$	16.0000	0.512936
$974$	0	0
$975$	0	0
$976$	0	0
$977$	26.8701i	0.859649i	0.902912	+	0.429825i	$0.141425\pi$
$977$	26.8701i	0.859649i	−0.902912	+	0.429825i	$0.858575\pi$
$978$	0	0
$979$	56.0000i	1.78977i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−11.3137	−0.360851	−0.180426	−	0.983589i	$-0.557748\pi$
$983$	−11.3137	−0.360851	−0.180426	+	0.983589i	$0.557748\pi$
$984$	0	0
$985$	30.0000	0.955879
$986$	0	0
$987$	0	0
$988$	0	0
$989$	45.2548i	1.43902i
$990$	0	0
$991$	− 8.00000i	− 0.254128i	−0.991894	−	0.127064i	$-0.959445\pi$
$991$	− 8.00000i	− 0.254128i	0.991894	−	0.127064i	$-0.0405554\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	11.3137	0.358669
$996$	0	0
$997$	50.0000	1.58352	0.791758	−	0.610835i	$-0.209166\pi$
$997$	50.0000	1.58352	0.791758	+	0.610835i	$0.209166\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.h.e.575.3		4
3.2	odd	2	inner	4032.2.h.e.575.1		4
4.3	odd	2	inner	4032.2.h.e.575.4		4
8.3	odd	2		1008.2.h.a.575.2	yes	4
8.5	even	2		1008.2.h.a.575.1	✓	4
12.11	even	2	inner	4032.2.h.e.575.2		4
24.5	odd	2		1008.2.h.a.575.3	yes	4
24.11	even	2		1008.2.h.a.575.4	yes	4
56.13	odd	2		7056.2.h.j.4607.3		4
56.27	even	2		7056.2.h.j.4607.4		4
168.83	odd	2		7056.2.h.j.4607.1		4
168.125	even	2		7056.2.h.j.4607.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.2.h.a.575.1	✓	4	8.5	even	2
1008.2.h.a.575.2	yes	4	8.3	odd	2
1008.2.h.a.575.3	yes	4	24.5	odd	2
1008.2.h.a.575.4	yes	4	24.11	even	2
4032.2.h.e.575.1		4	3.2	odd	2	inner
4032.2.h.e.575.2		4	12.11	even	2	inner
4032.2.h.e.575.3		4	1.1	even	1	trivial
4032.2.h.e.575.4		4	4.3	odd	2	inner
7056.2.h.j.4607.1		4	168.83	odd	2
7056.2.h.j.4607.2		4	168.125	even	2
7056.2.h.j.4607.3		4	56.13	odd	2
7056.2.h.j.4607.4		4	56.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.41421i q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q+1.41421i q^{5} -1.00000i q^{7} +5.65685 q^{11} +4.00000 q^{13} +1.41421i q^{17} -8.00000i q^{19} -5.65685 q^{23} +3.00000 q^{25} -7.07107i q^{29} +1.41421 q^{35} -2.00000 q^{37} -7.07107i q^{41} -8.00000i q^{43} -1.00000 q^{49} +9.89949i q^{53} +8.00000i q^{55} +11.3137 q^{59} -6.00000 q^{61} +5.65685i q^{65} -5.65685 q^{71} -8.00000 q^{73} -5.65685i q^{77} +11.3137 q^{83} -2.00000 q^{85} +9.89949i q^{89} -4.00000i q^{91} +11.3137 q^{95} -16.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 16 q^{13} + 12 q^{25} - 8 q^{37} - 4 q^{49} - 24 q^{61} - 32 q^{73} - 8 q^{85} - 64 q^{97}+O(q^{100}) \) 4 * q + 16 * q^13 + 12 * q^25 - 8 * q^37 - 4 * q^49 - 24 * q^61 - 32 * q^73 - 8 * q^85 - 64 * q^97

Introduction

L-functions

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