LMFDB - Embedded newform 4032.2.h.b.575.2

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 2016)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{5} +1.00000i q^{7} +O(q^{10})$	$q-1.41421i q^{5} +1.00000i q^{7} -2.82843 q^{11} -7.07107i q^{17} +4.00000i q^{19} +2.82843 q^{23} +3.00000 q^{25} +4.24264i q^{29} +4.00000i q^{31} +1.41421 q^{35} -2.00000 q^{37} -4.24264i q^{41} -12.0000i q^{43} +5.65685 q^{47} -1.00000 q^{49} -7.07107i q^{53} +4.00000i q^{55} -5.65685 q^{59} -14.0000 q^{61} -8.00000i q^{67} -8.48528 q^{71} -4.00000 q^{73} -2.82843i q^{77} +4.00000i q^{79} -10.0000 q^{85} -9.89949i q^{89} +5.65685 q^{95} +12.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 12 q^{25} - 8 q^{37} - 4 q^{49} - 56 q^{61} - 16 q^{73} - 40 q^{85} + 48 q^{97}+O(q^{100}) $ 4 * q + 12 * q^25 - 8 * q^37 - 4 * q^49 - 56 * q^61 - 16 * q^73 - 40 * q^85 + 48 * 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	$-0.897584\pi$
$5$	− 1.41421i	− 0.632456i	0.948683	−	0.316228i	$-0.102416\pi$
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.82843	−0.852803	−0.426401	−	0.904534i	$-0.640219\pi$
$11$	−2.82843	−0.852803	−0.426401	+	0.904534i	$0.640219\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 7.07107i	− 1.71499i	−0.514496	−	0.857493i	$-0.672021\pi$
$17$	− 7.07107i	− 1.71499i	0.514496	−	0.857493i	$-0.327979\pi$
$18$	0	0
$19$	4.00000i	0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$	4.00000i	0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.82843	0.589768	0.294884	−	0.955533i	$-0.404719\pi$
$23$	2.82843	0.589768	0.294884	+	0.955533i	$0.404719\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.24264i	0.787839i	0.919145	+	0.393919i	$0.128881\pi$
$29$	4.24264i	0.787839i	−0.919145	+	0.393919i	$0.871119\pi$
$30$	0	0
$31$	4.00000i	0.718421i	0.933257	+	0.359211i	$0.116954\pi$
$31$	4.00000i	0.718421i	−0.933257	+	0.359211i	$0.883046\pi$
$32$	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
$41$	− 4.24264i	− 0.662589i	−0.943527	−	0.331295i	$-0.892515\pi$
$41$	− 4.24264i	− 0.662589i	0.943527	−	0.331295i	$-0.107485\pi$
$42$	0	0
$43$	− 12.0000i	− 1.82998i	−0.403473	−	0.914991i	$-0.632197\pi$
$43$	− 12.0000i	− 1.82998i	0.403473	−	0.914991i	$-0.367803\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.65685	0.825137	0.412568	−	0.910927i	$-0.364632\pi$
$47$	5.65685	0.825137	0.412568	+	0.910927i	$0.364632\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 7.07107i	− 0.971286i	−0.874157	−	0.485643i	$-0.838586\pi$
$53$	− 7.07107i	− 0.971286i	0.874157	−	0.485643i	$-0.161414\pi$
$54$	0	0
$55$	4.00000i	0.539360i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−5.65685	−0.736460	−0.368230	−	0.929735i	$-0.620036\pi$
$59$	−5.65685	−0.736460	−0.368230	+	0.929735i	$0.620036\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.48528	−1.00702	−0.503509	−	0.863990i	$-0.667958\pi$
$71$	−8.48528	−1.00702	−0.503509	+	0.863990i	$0.667958\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 2.82843i	− 0.322329i
$78$	0	0
$79$	4.00000i	0.450035i	0.974355	+	0.225018i	$0.0722440\pi$
$79$	4.00000i	0.450035i	−0.974355	+	0.225018i	$0.927756\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−10.0000	−1.08465
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 9.89949i	− 1.04934i	−0.851304	−	0.524672i	$-0.824188\pi$
$89$	− 9.89949i	− 1.04934i	0.851304	−	0.524672i	$-0.175812\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5.65685	0.580381
$96$	0	0
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 1.41421i	− 0.140720i	−0.997522	−	0.0703598i	$-0.977585\pi$
$101$	− 1.41421i	− 0.140720i	0.997522	−	0.0703598i	$-0.0224147\pi$
$102$	0	0
$103$	4.00000i	0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$	4.00000i	0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−14.1421	−1.36717	−0.683586	−	0.729870i	$-0.739581\pi$
$107$	−14.1421	−1.36717	−0.683586	+	0.729870i	$0.739581\pi$
$108$	0	0
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 12.7279i	− 1.19734i	−0.800995	−	0.598671i	$-0.795696\pi$
$113$	− 12.7279i	− 1.19734i	0.800995	−	0.598671i	$-0.204304\pi$
$114$	0	0
$115$	− 4.00000i	− 0.373002i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	7.07107	0.648204
$120$	0	0
$121$	−3.00000	−0.272727
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 11.3137i	− 1.01193i
$126$	0	0
$127$	− 12.0000i	− 1.06483i	−0.846484	−	0.532414i	$-0.821285\pi$
$127$	− 12.0000i	− 1.06483i	0.846484	−	0.532414i	$-0.178715\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	5.65685	0.494242	0.247121	−	0.968985i	$-0.420516\pi$
$131$	5.65685	0.494242	0.247121	+	0.968985i	$0.420516\pi$
$132$	0	0
$133$	−4.00000	−0.346844
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 15.5563i	− 1.32907i	−0.747258	−	0.664534i	$-0.768630\pi$
$137$	− 15.5563i	− 1.32907i	0.747258	−	0.664534i	$-0.231370\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	6.00000	0.498273
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 1.41421i	− 0.115857i	−0.998321	−	0.0579284i	$-0.981550\pi$
$149$	− 1.41421i	− 0.115857i	0.998321	−	0.0579284i	$-0.0184495\pi$
$150$	0	0
$151$	16.0000i	1.30206i	0.759051	+	0.651031i	$0.225663\pi$
$151$	16.0000i	1.30206i	−0.759051	+	0.651031i	$0.774337\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.65685	0.454369
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.82843i	0.222911i
$162$	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
$167$	11.3137	0.875481	0.437741	−	0.899101i	$-0.355779\pi$
$167$	11.3137	0.875481	0.437741	+	0.899101i	$0.355779\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	9.89949i	0.752645i	0.926489	+	0.376322i	$0.122811\pi$
$173$	9.89949i	0.752645i	−0.926489	+	0.376322i	$0.877189\pi$
$174$	0	0
$175$	3.00000i	0.226779i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	14.1421	1.05703	0.528516	−	0.848923i	$-0.322748\pi$
$179$	14.1421	1.05703	0.528516	+	0.848923i	$0.322748\pi$
$180$	0	0
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.82843i	0.207950i
$186$	0	0
$187$	20.0000i	1.46254i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2.82843	−0.204658	−0.102329	−	0.994751i	$-0.532629\pi$
$191$	−2.82843	−0.204658	−0.102329	+	0.994751i	$0.532629\pi$
$192$	0	0
$193$	−26.0000	−1.87152	−0.935760	−	0.352636i	$-0.885285\pi$
$193$	−26.0000	−1.87152	−0.935760	+	0.352636i	$0.885285\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 4.24264i	− 0.302276i	−0.988513	−	0.151138i	$-0.951706\pi$
$197$	− 4.24264i	− 0.302276i	0.988513	−	0.151138i	$-0.0482937\pi$
$198$	0	0
$199$	16.0000i	1.13421i	0.823646	+	0.567105i	$0.191937\pi$
$199$	16.0000i	1.13421i	−0.823646	+	0.567105i	$0.808063\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−4.24264	−0.297775
$204$	0	0
$205$	−6.00000	−0.419058
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 11.3137i	− 0.782586i
$210$	0	0
$211$	12.0000i	0.826114i	0.910705	+	0.413057i	$0.135539\pi$
$211$	12.0000i	0.826114i	−0.910705	+	0.413057i	$0.864461\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−16.9706	−1.15738
$216$	0	0
$217$	−4.00000	−0.271538
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 24.0000i	− 1.60716i	−0.595198	−	0.803579i	$-0.702926\pi$
$223$	− 24.0000i	− 1.60716i	0.595198	−	0.803579i	$-0.297074\pi$
$224$	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
$229$	−16.0000	−1.05731	−0.528655	−	0.848837i	$-0.677303\pi$
$229$	−16.0000	−1.05731	−0.528655	+	0.848837i	$0.677303\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.41421i	0.0926482i	0.998926	+	0.0463241i	$0.0147507\pi$
$233$	1.41421i	0.0926482i	−0.998926	+	0.0463241i	$0.985249\pi$
$234$	0	0
$235$	− 8.00000i	− 0.521862i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	14.1421	0.914779	0.457389	−	0.889267i	$-0.348785\pi$
$239$	14.1421	0.914779	0.457389	+	0.889267i	$0.348785\pi$
$240$	0	0
$241$	−4.00000	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.41421i	0.0903508i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−5.65685	−0.357057	−0.178529	−	0.983935i	$-0.557134\pi$
$251$	−5.65685	−0.357057	−0.178529	+	0.983935i	$0.557134\pi$
$252$	0	0
$253$	−8.00000	−0.502956
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 7.07107i	− 0.441081i	−0.975378	−	0.220541i	$-0.929218\pi$
$257$	− 7.07107i	− 0.441081i	0.975378	−	0.220541i	$-0.0707821\pi$
$258$	0	0
$259$	− 2.00000i	− 0.124274i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	14.1421	0.872041	0.436021	−	0.899937i	$-0.356387\pi$
$263$	14.1421	0.872041	0.436021	+	0.899937i	$0.356387\pi$
$264$	0	0
$265$	−10.0000	−0.614295
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 24.0416i	− 1.46584i	−0.680313	−	0.732922i	$-0.738156\pi$
$269$	− 24.0416i	− 1.46584i	0.680313	−	0.732922i	$-0.261844\pi$
$270$	0	0
$271$	− 20.0000i	− 1.21491i	−0.794353	−	0.607457i	$-0.792190\pi$
$271$	− 20.0000i	− 1.21491i	0.794353	−	0.607457i	$-0.207810\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−8.48528	−0.511682
$276$	0	0
$277$	28.0000	1.68236	0.841178	−	0.540758i	$-0.181862\pi$
$277$	28.0000	1.68236	0.841178	+	0.540758i	$0.181862\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 12.7279i	− 0.759284i	−0.925133	−	0.379642i	$-0.876047\pi$
$281$	− 12.7279i	− 0.759284i	0.925133	−	0.379642i	$-0.123953\pi$
$282$	0	0
$283$	20.0000i	1.18888i	0.804141	+	0.594438i	$0.202626\pi$
$283$	20.0000i	1.18888i	−0.804141	+	0.594438i	$0.797374\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.24264	0.250435
$288$	0	0
$289$	−33.0000	−1.94118
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.41421i	0.0826192i	0.999146	+	0.0413096i	$0.0131530\pi$
$293$	1.41421i	0.0826192i	−0.999146	+	0.0413096i	$0.986847\pi$
$294$	0	0
$295$	8.00000i	0.465778i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	12.0000	0.691669
$302$	0	0
$303$	0	0
$304$	0	0
$305$	19.7990i	1.13369i
$306$	0	0
$307$	− 12.0000i	− 0.684876i	−0.939540	−	0.342438i	$-0.888747\pi$
$307$	− 12.0000i	− 0.684876i	0.939540	−	0.342438i	$-0.111253\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−28.2843	−1.60385	−0.801927	−	0.597422i	$-0.796192\pi$
$311$	−28.2843	−1.60385	−0.801927	+	0.597422i	$0.796192\pi$
$312$	0	0
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 32.5269i	− 1.82689i	−0.406958	−	0.913447i	$-0.633411\pi$
$317$	− 32.5269i	− 1.82689i	0.406958	−	0.913447i	$-0.366589\pi$
$318$	0	0
$319$	− 12.0000i	− 0.671871i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	28.2843	1.57378
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	5.65685i	0.311872i
$330$	0	0
$331$	− 4.00000i	− 0.219860i	−0.993939	−	0.109930i	$-0.964937\pi$
$331$	− 4.00000i	− 0.219860i	0.993939	−	0.109930i	$-0.0350627\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−11.3137	−0.618134
$336$	0	0
$337$	24.0000	1.30736	0.653682	−	0.756770i	$-0.273224\pi$
$337$	24.0000	1.30736	0.653682	+	0.756770i	$0.273224\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 11.3137i	− 0.612672i
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−14.1421	−0.759190	−0.379595	−	0.925153i	$-0.623937\pi$
$347$	−14.1421	−0.759190	−0.379595	+	0.925153i	$0.623937\pi$
$348$	0	0
$349$	18.0000	0.963518	0.481759	−	0.876304i	$-0.339998\pi$
$349$	18.0000	0.963518	0.481759	+	0.876304i	$0.339998\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.41421i	0.0752710i	0.999292	+	0.0376355i	$0.0119826\pi$
$353$	1.41421i	0.0752710i	−0.999292	+	0.0376355i	$0.988017\pi$
$354$	0	0
$355$	12.0000i	0.636894i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−36.7696	−1.94062	−0.970311	−	0.241859i	$-0.922243\pi$
$359$	−36.7696	−1.94062	−0.970311	+	0.241859i	$0.922243\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	0	0
$364$	0	0
$365$	5.65685i	0.296093i
$366$	0	0
$367$	32.0000i	1.67039i	0.549957	+	0.835193i	$0.314644\pi$
$367$	32.0000i	1.67039i	−0.549957	+	0.835193i	$0.685356\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	7.07107	0.367112
$372$	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
$377$	0	0
$378$	0	0
$379$	20.0000i	1.02733i	0.857991	+	0.513665i	$0.171713\pi$
$379$	20.0000i	1.02733i	−0.857991	+	0.513665i	$0.828287\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−28.2843	−1.44526	−0.722629	−	0.691236i	$-0.757067\pi$
$383$	−28.2843	−1.44526	−0.722629	+	0.691236i	$0.757067\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	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
$391$	− 20.0000i	− 1.01144i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5.65685	0.284627
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.41421i	0.0706225i	0.999376	+	0.0353112i	$0.0112422\pi$
$401$	1.41421i	0.0706225i	−0.999376	+	0.0353112i	$0.988758\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5.65685	0.280400
$408$	0	0
$409$	−20.0000	−0.988936	−0.494468	−	0.869196i	$-0.664637\pi$
$409$	−20.0000	−0.988936	−0.494468	+	0.869196i	$0.664637\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 5.65685i	− 0.278356i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	33.9411	1.65813	0.829066	−	0.559150i	$-0.188873\pi$
$419$	33.9411	1.65813	0.829066	+	0.559150i	$0.188873\pi$
$420$	0	0
$421$	12.0000	0.584844	0.292422	−	0.956289i	$-0.405539\pi$
$421$	12.0000	0.584844	0.292422	+	0.956289i	$0.405539\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 21.2132i	− 1.02899i
$426$	0	0
$427$	− 14.0000i	− 0.677507i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	36.7696	1.77113	0.885564	−	0.464518i	$-0.153773\pi$
$431$	36.7696	1.77113	0.885564	+	0.464518i	$0.153773\pi$
$432$	0	0
$433$	−18.0000	−0.865025	−0.432512	−	0.901628i	$-0.642373\pi$
$433$	−18.0000	−0.865025	−0.432512	+	0.901628i	$0.642373\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	11.3137i	0.541208i
$438$	0	0
$439$	− 40.0000i	− 1.90910i	−0.298057	−	0.954548i	$-0.596339\pi$
$439$	− 40.0000i	− 1.90910i	0.298057	−	0.954548i	$-0.403661\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−31.1127	−1.47821	−0.739104	−	0.673591i	$-0.764751\pi$
$443$	−31.1127	−1.47821	−0.739104	+	0.673591i	$0.764751\pi$
$444$	0	0
$445$	−14.0000	−0.663664
$446$	0	0
$447$	0	0
$448$	0	0
$449$	4.24264i	0.200223i	0.994976	+	0.100111i	$0.0319199\pi$
$449$	4.24264i	0.200223i	−0.994976	+	0.100111i	$0.968080\pi$
$450$	0	0
$451$	12.0000i	0.565058i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	12.7279i	0.592798i	0.955064	+	0.296399i	$0.0957859\pi$
$461$	12.7279i	0.592798i	−0.955064	+	0.296399i	$0.904214\pi$
$462$	0	0
$463$	− 36.0000i	− 1.67306i	−0.547920	−	0.836531i	$-0.684580\pi$
$463$	− 36.0000i	− 1.67306i	0.547920	−	0.836531i	$-0.315420\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	39.5980	1.83238	0.916188	−	0.400749i	$-0.131250\pi$
$467$	39.5980	1.83238	0.916188	+	0.400749i	$0.131250\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	0	0
$471$	0	0
$472$	0	0
$473$	33.9411i	1.56061i
$474$	0	0
$475$	12.0000i	0.550598i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	22.6274	1.03387	0.516937	−	0.856024i	$-0.327072\pi$
$479$	22.6274	1.03387	0.516937	+	0.856024i	$0.327072\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 16.9706i	− 0.770594i
$486$	0	0
$487$	8.00000i	0.362515i	0.983436	+	0.181257i	$0.0580167\pi$
$487$	8.00000i	0.362515i	−0.983436	+	0.181257i	$0.941983\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	14.1421	0.638226	0.319113	−	0.947717i	$-0.396615\pi$
$491$	14.1421	0.638226	0.319113	+	0.947717i	$0.396615\pi$
$492$	0	0
$493$	30.0000	1.35113
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 8.48528i	− 0.380617i
$498$	0	0
$499$	− 4.00000i	− 0.179065i	−0.995984	−	0.0895323i	$-0.971463\pi$
$499$	− 4.00000i	− 0.179065i	0.995984	−	0.0895323i	$-0.0285372\pi$
$500$	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
$505$	−2.00000	−0.0889988
$506$	0	0
$507$	0	0
$508$	0	0
$509$	24.0416i	1.06563i	0.846233	+	0.532813i	$0.178865\pi$
$509$	24.0416i	1.06563i	−0.846233	+	0.532813i	$0.821135\pi$
$510$	0	0
$511$	− 4.00000i	− 0.176950i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5.65685	0.249271
$516$	0	0
$517$	−16.0000	−0.703679
$518$	0	0
$519$	0	0
$520$	0	0
$521$	32.5269i	1.42503i	0.701657	+	0.712515i	$0.252444\pi$
$521$	32.5269i	1.42503i	−0.701657	+	0.712515i	$0.747556\pi$
$522$	0	0
$523$	40.0000i	1.74908i	0.484955	+	0.874539i	$0.338836\pi$
$523$	40.0000i	1.74908i	−0.484955	+	0.874539i	$0.661164\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	28.2843	1.23208
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	20.0000i	0.864675i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.82843	0.121829
$540$	0	0
$541$	44.0000	1.89171	0.945854	−	0.324593i	$-0.105227\pi$
$541$	44.0000	1.89171	0.945854	+	0.324593i	$0.105227\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5.65685i	0.242313i
$546$	0	0
$547$	− 8.00000i	− 0.342055i	−0.985266	−	0.171028i	$-0.945291\pi$
$547$	− 8.00000i	− 0.342055i	0.985266	−	0.171028i	$-0.0547087\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−16.9706	−0.722970
$552$	0	0
$553$	−4.00000	−0.170097
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 18.3848i	− 0.778988i	−0.921029	−	0.389494i	$-0.872650\pi$
$557$	− 18.3848i	− 0.778988i	0.921029	−	0.389494i	$-0.127350\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−39.5980	−1.66886	−0.834428	−	0.551117i	$-0.814202\pi$
$563$	−39.5980	−1.66886	−0.834428	+	0.551117i	$0.814202\pi$
$564$	0	0
$565$	−18.0000	−0.757266
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7.07107i	0.296435i	0.988955	+	0.148217i	$0.0473535\pi$
$569$	7.07107i	0.296435i	−0.988955	+	0.148217i	$0.952646\pi$
$570$	0	0
$571$	− 16.0000i	− 0.669579i	−0.942293	−	0.334790i	$-0.891335\pi$
$571$	− 16.0000i	− 0.669579i	0.942293	−	0.334790i	$-0.108665\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8.48528	0.353861
$576$	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
$581$	0	0
$582$	0	0
$583$	20.0000i	0.828315i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	−16.0000	−0.659269
$590$	0	0
$591$	0	0
$592$	0	0
$593$	9.89949i	0.406524i	0.979124	+	0.203262i	$0.0651542\pi$
$593$	9.89949i	0.406524i	−0.979124	+	0.203262i	$0.934846\pi$
$594$	0	0
$595$	− 10.0000i	− 0.409960i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−2.82843	−0.115566	−0.0577832	−	0.998329i	$-0.518403\pi$
$599$	−2.82843	−0.115566	−0.0577832	+	0.998329i	$0.518403\pi$
$600$	0	0
$601$	6.00000	0.244745	0.122373	−	0.992484i	$-0.460950\pi$
$601$	6.00000	0.244745	0.122373	+	0.992484i	$0.460950\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	4.24264i	0.172488i
$606$	0	0
$607$	8.00000i	0.324710i	0.986732	+	0.162355i	$0.0519090\pi$
$607$	8.00000i	0.324710i	−0.986732	+	0.162355i	$0.948091\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	21.2132i	0.854011i	0.904249	+	0.427006i	$0.140432\pi$
$617$	21.2132i	0.854011i	−0.904249	+	0.427006i	$0.859568\pi$
$618$	0	0
$619$	40.0000i	1.60774i	0.594808	+	0.803868i	$0.297228\pi$
$619$	40.0000i	1.60774i	−0.594808	+	0.803868i	$0.702772\pi$
$620$	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
$629$	14.1421i	0.563884i
$630$	0	0
$631$	− 36.0000i	− 1.43314i	−0.697517	−	0.716569i	$-0.745712\pi$
$631$	− 36.0000i	− 1.43314i	0.697517	−	0.716569i	$-0.254288\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−16.9706	−0.673456
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 9.89949i	− 0.391007i	−0.980703	−	0.195503i	$-0.937366\pi$
$641$	− 9.89949i	− 0.391007i	0.980703	−	0.195503i	$-0.0626340\pi$
$642$	0	0
$643$	− 12.0000i	− 0.473234i	−0.971603	−	0.236617i	$-0.923961\pi$
$643$	− 12.0000i	− 0.473234i	0.971603	−	0.236617i	$-0.0760386\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−45.2548	−1.77915	−0.889576	−	0.456788i	$-0.849000\pi$
$647$	−45.2548	−1.77915	−0.889576	+	0.456788i	$0.849000\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	0	0
$651$	0	0
$652$	0	0
$653$	35.3553i	1.38356i	0.722108	+	0.691781i	$0.243173\pi$
$653$	35.3553i	1.38356i	−0.722108	+	0.691781i	$0.756827\pi$
$654$	0	0
$655$	− 8.00000i	− 0.312586i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	25.4558	0.991619	0.495809	−	0.868431i	$-0.334871\pi$
$659$	25.4558	0.991619	0.495809	+	0.868431i	$0.334871\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5.65685i	0.219363i
$666$	0	0
$667$	12.0000i	0.464642i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	39.5980	1.52866
$672$	0	0
$673$	42.0000	1.61898	0.809491	−	0.587133i	$-0.199743\pi$
$673$	42.0000	1.61898	0.809491	+	0.587133i	$0.199743\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	49.4975i	1.90234i	0.308664	+	0.951171i	$0.400118\pi$
$677$	49.4975i	1.90234i	−0.308664	+	0.951171i	$0.599882\pi$
$678$	0	0
$679$	12.0000i	0.460518i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	31.1127	1.19049	0.595247	−	0.803543i	$-0.297054\pi$
$683$	31.1127	1.19049	0.595247	+	0.803543i	$0.297054\pi$
$684$	0	0
$685$	−22.0000	−0.840577
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	8.00000i	0.304334i	0.988355	+	0.152167i	$0.0486252\pi$
$691$	8.00000i	0.304334i	−0.988355	+	0.152167i	$0.951375\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−30.0000	−1.13633
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 7.07107i	− 0.267071i	−0.991044	−	0.133535i	$-0.957367\pi$
$701$	− 7.07107i	− 0.267071i	0.991044	−	0.133535i	$-0.0426329\pi$
$702$	0	0
$703$	− 8.00000i	− 0.301726i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.41421	0.0531870
$708$	0	0
$709$	52.0000	1.95290	0.976450	−	0.215742i	$-0.0692169\pi$
$709$	52.0000	1.95290	0.976450	+	0.215742i	$0.0692169\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	11.3137i	0.423702i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−5.65685	−0.210965	−0.105483	−	0.994421i	$-0.533639\pi$
$719$	−5.65685	−0.210965	−0.105483	+	0.994421i	$0.533639\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	0	0
$723$	0	0
$724$	0	0
$725$	12.7279i	0.472703i
$726$	0	0
$727$	12.0000i	0.445055i	0.974926	+	0.222528i	$0.0714308\pi$
$727$	12.0000i	0.445055i	−0.974926	+	0.222528i	$0.928569\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−84.8528	−3.13839
$732$	0	0
$733$	−8.00000	−0.295487	−0.147743	−	0.989026i	$-0.547201\pi$
$733$	−8.00000	−0.295487	−0.147743	+	0.989026i	$0.547201\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	22.6274i	0.833492i
$738$	0	0
$739$	− 40.0000i	− 1.47142i	−0.677295	−	0.735712i	$-0.736848\pi$
$739$	− 40.0000i	− 1.47142i	0.677295	−	0.735712i	$-0.263152\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	8.48528	0.311295	0.155647	−	0.987813i	$-0.450254\pi$
$743$	8.48528	0.311295	0.155647	+	0.987813i	$0.450254\pi$
$744$	0	0
$745$	−2.00000	−0.0732743
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 14.1421i	− 0.516742i
$750$	0	0
$751$	32.0000i	1.16770i	0.811863	+	0.583848i	$0.198454\pi$
$751$	32.0000i	1.16770i	−0.811863	+	0.583848i	$0.801546\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	22.6274	0.823496
$756$	0	0
$757$	−12.0000	−0.436147	−0.218074	−	0.975932i	$-0.569977\pi$
$757$	−12.0000	−0.436147	−0.218074	+	0.975932i	$0.569977\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 49.4975i	− 1.79428i	−0.441744	−	0.897141i	$-0.645640\pi$
$761$	− 49.4975i	− 1.79428i	0.441744	−	0.897141i	$-0.354360\pi$
$762$	0	0
$763$	− 4.00000i	− 0.144810i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	18.0000	0.649097	0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000	0.649097	0.324548	+	0.945869i	$0.394788\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 9.89949i	− 0.356060i	−0.984025	−	0.178030i	$-0.943028\pi$
$773$	− 9.89949i	− 0.356060i	0.984025	−	0.178030i	$-0.0569724\pi$
$774$	0	0
$775$	12.0000i	0.431053i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	16.9706	0.608034
$780$	0	0
$781$	24.0000	0.858788
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.82843i	0.100951i
$786$	0	0
$787$	− 32.0000i	− 1.14068i	−0.821410	−	0.570338i	$-0.806812\pi$
$787$	− 32.0000i	− 1.14068i	0.821410	−	0.570338i	$-0.193188\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	12.7279	0.452553
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	21.2132i	0.751410i	0.926739	+	0.375705i	$0.122599\pi$
$797$	21.2132i	0.751410i	−0.926739	+	0.375705i	$0.877401\pi$
$798$	0	0
$799$	− 40.0000i	− 1.41510i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	11.3137	0.399252
$804$	0	0
$805$	4.00000	0.140981
$806$	0	0
$807$	0	0
$808$	0	0
$809$	32.5269i	1.14359i	0.820398	+	0.571793i	$0.193752\pi$
$809$	32.5269i	1.14359i	−0.820398	+	0.571793i	$0.806248\pi$
$810$	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
$817$	48.0000	1.67931
$818$	0	0
$819$	0	0
$820$	0	0
$821$	41.0122i	1.43134i	0.698441	+	0.715668i	$0.253877\pi$
$821$	41.0122i	1.43134i	−0.698441	+	0.715668i	$0.746123\pi$
$822$	0	0
$823$	20.0000i	0.697156i	0.937280	+	0.348578i	$0.113335\pi$
$823$	20.0000i	0.697156i	−0.937280	+	0.348578i	$0.886665\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−25.4558	−0.885186	−0.442593	−	0.896723i	$-0.645941\pi$
$827$	−25.4558	−0.885186	−0.442593	+	0.896723i	$0.645941\pi$
$828$	0	0
$829$	−40.0000	−1.38926	−0.694629	−	0.719368i	$-0.744431\pi$
$829$	−40.0000	−1.38926	−0.694629	+	0.719368i	$0.744431\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	7.07107i	0.244998i
$834$	0	0
$835$	− 16.0000i	− 0.553703i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	16.9706	0.585889	0.292944	−	0.956129i	$-0.405365\pi$
$839$	16.9706	0.585889	0.292944	+	0.956129i	$0.405365\pi$
$840$	0	0
$841$	11.0000	0.379310
$842$	0	0
$843$	0	0
$844$	0	0
$845$	18.3848i	0.632456i
$846$	0	0
$847$	− 3.00000i	− 0.103081i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−5.65685	−0.193914
$852$	0	0
$853$	6.00000	0.205436	0.102718	−	0.994711i	$-0.467246\pi$
$853$	6.00000	0.205436	0.102718	+	0.994711i	$0.467246\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	7.07107i	0.241543i	0.992680	+	0.120772i	$0.0385368\pi$
$857$	7.07107i	0.241543i	−0.992680	+	0.120772i	$0.961463\pi$
$858$	0	0
$859$	28.0000i	0.955348i	0.878537	+	0.477674i	$0.158520\pi$
$859$	28.0000i	0.955348i	−0.878537	+	0.477674i	$0.841480\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−14.1421	−0.481404	−0.240702	−	0.970599i	$-0.577378\pi$
$863$	−14.1421	−0.481404	−0.240702	+	0.970599i	$0.577378\pi$
$864$	0	0
$865$	14.0000	0.476014
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 11.3137i	− 0.383791i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	11.3137	0.382473
$876$	0	0
$877$	−46.0000	−1.55331	−0.776655	−	0.629926i	$-0.783085\pi$
$877$	−46.0000	−1.55331	−0.776655	+	0.629926i	$0.783085\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	43.8406i	1.47703i	0.674238	+	0.738514i	$0.264472\pi$
$881$	43.8406i	1.47703i	−0.674238	+	0.738514i	$0.735528\pi$
$882$	0	0
$883$	− 36.0000i	− 1.21150i	−0.795656	−	0.605748i	$-0.792874\pi$
$883$	− 36.0000i	− 1.21150i	0.795656	−	0.605748i	$-0.207126\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−45.2548	−1.51951	−0.759754	−	0.650210i	$-0.774681\pi$
$887$	−45.2548	−1.51951	−0.759754	+	0.650210i	$0.774681\pi$
$888$	0	0
$889$	12.0000	0.402467
$890$	0	0
$891$	0	0
$892$	0	0
$893$	22.6274i	0.757198i
$894$	0	0
$895$	− 20.0000i	− 0.668526i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−16.9706	−0.566000
$900$	0	0
$901$	−50.0000	−1.66574
$902$	0	0
$903$	0	0
$904$	0	0
$905$	22.6274i	0.752161i
$906$	0	0
$907$	− 44.0000i	− 1.46100i	−0.682915	−	0.730498i	$-0.739288\pi$
$907$	− 44.0000i	− 1.46100i	0.682915	−	0.730498i	$-0.260712\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	8.48528	0.281130	0.140565	−	0.990071i	$-0.455108\pi$
$911$	8.48528	0.281130	0.140565	+	0.990071i	$0.455108\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.65685i	0.186806i
$918$	0	0
$919$	56.0000i	1.84727i	0.383274	+	0.923635i	$0.374797\pi$
$919$	56.0000i	1.84727i	−0.383274	+	0.923635i	$0.625203\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−6.00000	−0.197279
$926$	0	0
$927$	0	0
$928$	0	0
$929$	32.5269i	1.06717i	0.845745	+	0.533587i	$0.179156\pi$
$929$	32.5269i	1.06717i	−0.845745	+	0.533587i	$0.820844\pi$
$930$	0	0
$931$	− 4.00000i	− 0.131095i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	28.2843	0.924995
$936$	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
$941$	18.3848i	0.599327i	0.954045	+	0.299663i	$0.0968743\pi$
$941$	18.3848i	0.599327i	−0.954045	+	0.299663i	$0.903126\pi$
$942$	0	0
$943$	− 12.0000i	− 0.390774i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−53.7401	−1.74632	−0.873160	−	0.487435i	$-0.837933\pi$
$947$	−53.7401	−1.74632	−0.873160	+	0.487435i	$0.837933\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	12.7279i	0.412298i	0.978521	+	0.206149i	$0.0660931\pi$
$953$	12.7279i	0.412298i	−0.978521	+	0.206149i	$0.933907\pi$
$954$	0	0
$955$	4.00000i	0.129437i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	15.5563	0.502341
$960$	0	0
$961$	15.0000	0.483871
$962$	0	0
$963$	0	0
$964$	0	0
$965$	36.7696i	1.18365i
$966$	0	0
$967$	12.0000i	0.385894i	0.981209	+	0.192947i	$0.0618045\pi$
$967$	12.0000i	0.385894i	−0.981209	+	0.192947i	$0.938195\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	39.5980	1.27076	0.635380	−	0.772200i	$-0.280844\pi$
$971$	39.5980	1.27076	0.635380	+	0.772200i	$0.280844\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 24.0416i	− 0.769160i	−0.923092	−	0.384580i	$-0.874346\pi$
$977$	− 24.0416i	− 0.769160i	0.923092	−	0.384580i	$-0.125654\pi$
$978$	0	0
$979$	28.0000i	0.894884i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−5.65685	−0.180426	−0.0902128	−	0.995923i	$-0.528755\pi$
$983$	−5.65685	−0.180426	−0.0902128	+	0.995923i	$0.528755\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 33.9411i	− 1.07927i
$990$	0	0
$991$	8.00000i	0.254128i	0.991894	+	0.127064i	$0.0405554\pi$
$991$	8.00000i	0.254128i	−0.991894	+	0.127064i	$0.959445\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	22.6274	0.717337
$996$	0	0
$997$	10.0000	0.316703	0.158352	−	0.987383i	$-0.449382\pi$
$997$	10.0000	0.316703	0.158352	+	0.987383i	$0.449382\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.h.b.575.2		4
3.2	odd	2	inner	4032.2.h.b.575.4		4
4.3	odd	2	inner	4032.2.h.b.575.1		4
8.3	odd	2		2016.2.h.c.575.3	yes	4
8.5	even	2		2016.2.h.c.575.4	yes	4
12.11	even	2	inner	4032.2.h.b.575.3		4
24.5	odd	2		2016.2.h.c.575.2	yes	4
24.11	even	2		2016.2.h.c.575.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2016.2.h.c.575.1	✓	4	24.11	even	2
2016.2.h.c.575.2	yes	4	24.5	odd	2
2016.2.h.c.575.3	yes	4	8.3	odd	2
2016.2.h.c.575.4	yes	4	8.5	even	2
4032.2.h.b.575.1		4	4.3	odd	2	inner
4032.2.h.b.575.2		4	1.1	even	1	trivial
4032.2.h.b.575.3		4	12.11	even	2	inner
4032.2.h.b.575.4		4	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.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{5} +1.00000i q^{7} +O(q^{10})\)	\(q-1.41421i q^{5} +1.00000i q^{7} -2.82843 q^{11} -7.07107i q^{17} +4.00000i q^{19} +2.82843 q^{23} +3.00000 q^{25} +4.24264i q^{29} +4.00000i q^{31} +1.41421 q^{35} -2.00000 q^{37} -4.24264i q^{41} -12.0000i q^{43} +5.65685 q^{47} -1.00000 q^{49} -7.07107i q^{53} +4.00000i q^{55} -5.65685 q^{59} -14.0000 q^{61} -8.00000i q^{67} -8.48528 q^{71} -4.00000 q^{73} -2.82843i q^{77} +4.00000i q^{79} -10.0000 q^{85} -9.89949i q^{89} +5.65685 q^{95} +12.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 12 q^{25} - 8 q^{37} - 4 q^{49} - 56 q^{61} - 16 q^{73} - 40 q^{85} + 48 q^{97}+O(q^{100}) \) 4 * q + 12 * q^25 - 8 * q^37 - 4 * q^49 - 56 * q^61 - 16 * q^73 - 40 * q^85 + 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more