LMFDB - Embedded newform 384.2.d.c.193.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [384,2,Mod(193,384)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("384.193"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(384, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	384.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$3.06625543762$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			193.4
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.2.d.c.193.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +2.82843i q^{5} +2.82843 q^{7} -1.00000 q^{9} -4.00000i q^{11} +5.65685i q^{13} -2.82843 q^{15} -2.00000 q^{17} +4.00000i q^{19} +2.82843i q^{21} -5.65685 q^{23} -3.00000 q^{25} -1.00000i q^{27} +2.82843i q^{29} +8.48528 q^{31} +4.00000 q^{33} +8.00000i q^{35} -5.65685 q^{39} +10.0000 q^{41} -12.0000i q^{43} -2.82843i q^{45} -5.65685 q^{47} +1.00000 q^{49} -2.00000i q^{51} -2.82843i q^{53} +11.3137 q^{55} -4.00000 q^{57} +4.00000i q^{59} -11.3137i q^{61} -2.82843 q^{63} -16.0000 q^{65} +4.00000i q^{67} -5.65685i q^{69} +5.65685 q^{71} -2.00000 q^{73} -3.00000i q^{75} -11.3137i q^{77} +8.48528 q^{79} +1.00000 q^{81} -4.00000i q^{83} -5.65685i q^{85} -2.82843 q^{87} +6.00000 q^{89} +16.0000i q^{91} +8.48528i q^{93} -11.3137 q^{95} +14.0000 q^{97} +4.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9} - 8 q^{17} - 12 q^{25} + 16 q^{33} + 40 q^{41} + 4 q^{49} - 16 q^{57} - 64 q^{65} - 8 q^{73} + 4 q^{81} + 24 q^{89} + 56 q^{97}+O(q^{100}) $ 4 * q - 4 * q^9 - 8 * q^17 - 12 * q^25 + 16 * q^33 + 40 * q^41 + 4 * q^49 - 16 * q^57 - 64 * q^65 - 8 * q^73 + 4 * q^81 + 24 * q^89 + 56 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	2.82843i	1.26491i	0.774597	+	0.632456i	$0.217953\pi$
$5$	2.82843i	1.26491i	−0.774597	+	0.632456i	$0.782047\pi$
$6$	0	0
$7$	2.82843	1.06904	0.534522	−	0.845154i	$-0.320491\pi$
$7$	2.82843	1.06904	0.534522	+	0.845154i	$0.320491\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 4.00000i	− 1.20605i	−0.797724	−	0.603023i	$-0.793963\pi$
$11$	− 4.00000i	− 1.20605i	0.797724	−	0.603023i	$-0.206037\pi$
$12$	0	0
$13$	5.65685i	1.56893i	0.620174	+	0.784465i	$0.287062\pi$
$13$	5.65685i	1.56893i	−0.620174	+	0.784465i	$0.712938\pi$
$14$	0	0
$15$	−2.82843	−0.730297
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	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
$21$	2.82843i	0.617213i
$22$	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
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	2.82843i	0.525226i	0.964901	+	0.262613i	$0.0845842\pi$
$29$	2.82843i	0.525226i	−0.964901	+	0.262613i	$0.915416\pi$
$30$	0	0
$31$	8.48528	1.52400	0.762001	−	0.647576i	$-0.224217\pi$
$31$	8.48528	1.52400	0.762001	+	0.647576i	$0.224217\pi$
$32$	0	0
$33$	4.00000	0.696311
$34$	0	0
$35$	8.00000i	1.35225i
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	−5.65685	−0.905822
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	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
$45$	− 2.82843i	− 0.421637i
$46$	0	0
$47$	−5.65685	−0.825137	−0.412568	−	0.910927i	$-0.635368\pi$
$47$	−5.65685	−0.825137	−0.412568	+	0.910927i	$0.635368\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	− 2.00000i	− 0.280056i
$52$	0	0
$53$	− 2.82843i	− 0.388514i	−0.980951	−	0.194257i	$-0.937770\pi$
$53$	− 2.82843i	− 0.388514i	0.980951	−	0.194257i	$-0.0622296\pi$
$54$	0	0
$55$	11.3137	1.52554
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	4.00000i	0.520756i	0.965507	+	0.260378i	$0.0838471\pi$
$59$	4.00000i	0.520756i	−0.965507	+	0.260378i	$0.916153\pi$
$60$	0	0
$61$	− 11.3137i	− 1.44857i	−0.689500	−	0.724286i	$-0.742170\pi$
$61$	− 11.3137i	− 1.44857i	0.689500	−	0.724286i	$-0.257830\pi$
$62$	0	0
$63$	−2.82843	−0.356348
$64$	0	0
$65$	−16.0000	−1.98456
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	0	0
$69$	− 5.65685i	− 0.681005i
$70$	0	0
$71$	5.65685	0.671345	0.335673	−	0.941979i	$-0.391036\pi$
$71$	5.65685	0.671345	0.335673	+	0.941979i	$0.391036\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	− 3.00000i	− 0.346410i
$76$	0	0
$77$	− 11.3137i	− 1.28932i
$78$	0	0
$79$	8.48528	0.954669	0.477334	−	0.878722i	$-0.341603\pi$
$79$	8.48528	0.954669	0.477334	+	0.878722i	$0.341603\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 4.00000i	− 0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$	− 4.00000i	− 0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$	0	0
$85$	− 5.65685i	− 0.613572i
$86$	0	0
$87$	−2.82843	−0.303239
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	16.0000i	1.67726i
$92$	0	0
$93$	8.48528i	0.879883i
$94$	0	0
$95$	−11.3137	−1.16076
$96$	0	0
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	0	0
$99$	4.00000i	0.402015i
$100$	0	0
$101$	− 14.1421i	− 1.40720i	−0.710599	−	0.703598i	$-0.751576\pi$
$101$	− 14.1421i	− 1.40720i	0.710599	−	0.703598i	$-0.248424\pi$
$102$	0	0
$103$	−2.82843	−0.278693	−0.139347	−	0.990244i	$-0.544500\pi$
$103$	−2.82843	−0.278693	−0.139347	+	0.990244i	$0.544500\pi$
$104$	0	0
$105$	−8.00000	−0.780720
$106$	0	0
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	0	0
$109$	− 5.65685i	− 0.541828i	−0.962604	−	0.270914i	$-0.912674\pi$
$109$	− 5.65685i	− 0.541828i	0.962604	−	0.270914i	$-0.0873260\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	− 16.0000i	− 1.49201i
$116$	0	0
$117$	− 5.65685i	− 0.522976i
$118$	0	0
$119$	−5.65685	−0.518563
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	10.0000i	0.901670i
$124$	0	0
$125$	5.65685i	0.505964i
$126$	0	0
$127$	−8.48528	−0.752947	−0.376473	−	0.926427i	$-0.622863\pi$
$127$	−8.48528	−0.752947	−0.376473	+	0.926427i	$0.622863\pi$
$128$	0	0
$129$	12.0000	1.05654
$130$	0	0
$131$	− 12.0000i	− 1.04844i	−0.851581	−	0.524222i	$-0.824356\pi$
$131$	− 12.0000i	− 1.04844i	0.851581	−	0.524222i	$-0.175644\pi$
$132$	0	0
$133$	11.3137i	0.981023i
$134$	0	0
$135$	2.82843	0.243432
$136$	0	0
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	0	0
$139$	4.00000i	0.339276i	0.985506	+	0.169638i	$0.0542598\pi$
$139$	4.00000i	0.339276i	−0.985506	+	0.169638i	$0.945740\pi$
$140$	0	0
$141$	− 5.65685i	− 0.476393i
$142$	0	0
$143$	22.6274	1.89220
$144$	0	0
$145$	−8.00000	−0.664364
$146$	0	0
$147$	1.00000i	0.0824786i
$148$	0	0
$149$	14.1421i	1.15857i	0.815125	+	0.579284i	$0.196668\pi$
$149$	14.1421i	1.15857i	−0.815125	+	0.579284i	$0.803332\pi$
$150$	0	0
$151$	−8.48528	−0.690522	−0.345261	−	0.938507i	$-0.612210\pi$
$151$	−8.48528	−0.690522	−0.345261	+	0.938507i	$0.612210\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	24.0000i	1.92773i
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	2.82843	0.224309
$160$	0	0
$161$	−16.0000	−1.26098
$162$	0	0
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	0	0
$165$	11.3137i	0.880771i
$166$	0	0
$167$	−11.3137	−0.875481	−0.437741	−	0.899101i	$-0.644221\pi$
$167$	−11.3137	−0.875481	−0.437741	+	0.899101i	$0.644221\pi$
$168$	0	0
$169$	−19.0000	−1.46154
$170$	0	0
$171$	− 4.00000i	− 0.305888i
$172$	0	0
$173$	19.7990i	1.50529i	0.658427	+	0.752645i	$0.271222\pi$
$173$	19.7990i	1.50529i	−0.658427	+	0.752645i	$0.728778\pi$
$174$	0	0
$175$	−8.48528	−0.641427
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	− 12.0000i	− 0.896922i	−0.893802	−	0.448461i	$-0.851972\pi$
$179$	− 12.0000i	− 0.896922i	0.893802	−	0.448461i	$-0.148028\pi$
$180$	0	0
$181$	5.65685i	0.420471i	0.977651	+	0.210235i	$0.0674230\pi$
$181$	5.65685i	0.420471i	−0.977651	+	0.210235i	$0.932577\pi$
$182$	0	0
$183$	11.3137	0.836333
$184$	0	0
$185$	0	0
$186$	0	0
$187$	8.00000i	0.585018i
$188$	0	0
$189$	− 2.82843i	− 0.205738i
$190$	0	0
$191$	11.3137	0.818631	0.409316	−	0.912393i	$-0.365768\pi$
$191$	11.3137	0.818631	0.409316	+	0.912393i	$0.365768\pi$
$192$	0	0
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	0	0
$195$	− 16.0000i	− 1.14578i
$196$	0	0
$197$	8.48528i	0.604551i	0.953221	+	0.302276i	$0.0977463\pi$
$197$	8.48528i	0.604551i	−0.953221	+	0.302276i	$0.902254\pi$
$198$	0	0
$199$	−2.82843	−0.200502	−0.100251	−	0.994962i	$-0.531965\pi$
$199$	−2.82843	−0.200502	−0.100251	+	0.994962i	$0.531965\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	8.00000i	0.561490i
$204$	0	0
$205$	28.2843i	1.97546i
$206$	0	0
$207$	5.65685	0.393179
$208$	0	0
$209$	16.0000	1.10674
$210$	0	0
$211$	− 12.0000i	− 0.826114i	−0.910705	−	0.413057i	$-0.864461\pi$
$211$	− 12.0000i	− 0.826114i	0.910705	−	0.413057i	$-0.135539\pi$
$212$	0	0
$213$	5.65685i	0.387601i
$214$	0	0
$215$	33.9411	2.31477
$216$	0	0
$217$	24.0000	1.62923
$218$	0	0
$219$	− 2.00000i	− 0.135147i
$220$	0	0
$221$	− 11.3137i	− 0.761042i
$222$	0	0
$223$	−19.7990	−1.32584	−0.662919	−	0.748691i	$-0.730683\pi$
$223$	−19.7990	−1.32584	−0.662919	+	0.748691i	$0.730683\pi$
$224$	0	0
$225$	3.00000	0.200000
$226$	0	0
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	0	0
$229$	− 5.65685i	− 0.373815i	−0.982377	−	0.186908i	$-0.940153\pi$
$229$	− 5.65685i	− 0.373815i	0.982377	−	0.186908i	$-0.0598465\pi$
$230$	0	0
$231$	11.3137	0.744387
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	− 16.0000i	− 1.04372i
$236$	0	0
$237$	8.48528i	0.551178i
$238$	0	0
$239$	−22.6274	−1.46365	−0.731823	−	0.681495i	$-0.761330\pi$
$239$	−22.6274	−1.46365	−0.731823	+	0.681495i	$0.761330\pi$
$240$	0	0
$241$	6.00000	0.386494	0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000	0.386494	0.193247	+	0.981150i	$0.438098\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	2.82843i	0.180702i
$246$	0	0
$247$	−22.6274	−1.43975
$248$	0	0
$249$	4.00000	0.253490
$250$	0	0
$251$	12.0000i	0.757433i	0.925513	+	0.378717i	$0.123635\pi$
$251$	12.0000i	0.757433i	−0.925513	+	0.378717i	$0.876365\pi$
$252$	0	0
$253$	22.6274i	1.42257i
$254$	0	0
$255$	5.65685	0.354246
$256$	0	0
$257$	−30.0000	−1.87135	−0.935674	−	0.352865i	$-0.885208\pi$
$257$	−30.0000	−1.87135	−0.935674	+	0.352865i	$0.885208\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	− 2.82843i	− 0.175075i
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	8.00000	0.491436
$266$	0	0
$267$	6.00000i	0.367194i
$268$	0	0
$269$	− 8.48528i	− 0.517357i	−0.965964	−	0.258678i	$-0.916713\pi$
$269$	− 8.48528i	− 0.517357i	0.965964	−	0.258678i	$-0.0832870\pi$
$270$	0	0
$271$	−19.7990	−1.20270	−0.601351	−	0.798985i	$-0.705371\pi$
$271$	−19.7990	−1.20270	−0.601351	+	0.798985i	$0.705371\pi$
$272$	0	0
$273$	−16.0000	−0.968364
$274$	0	0
$275$	12.0000i	0.723627i
$276$	0	0
$277$	− 16.9706i	− 1.01966i	−0.860274	−	0.509831i	$-0.829708\pi$
$277$	− 16.9706i	− 1.01966i	0.860274	−	0.509831i	$-0.170292\pi$
$278$	0	0
$279$	−8.48528	−0.508001
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	0	0
$285$	− 11.3137i	− 0.670166i
$286$	0	0
$287$	28.2843	1.66957
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	14.0000i	0.820695i
$292$	0	0
$293$	19.7990i	1.15667i	0.815800	+	0.578335i	$0.196297\pi$
$293$	19.7990i	1.15667i	−0.815800	+	0.578335i	$0.803703\pi$
$294$	0	0
$295$	−11.3137	−0.658710
$296$	0	0
$297$	−4.00000	−0.232104
$298$	0	0
$299$	− 32.0000i	− 1.85061i
$300$	0	0
$301$	− 33.9411i	− 1.95633i
$302$	0	0
$303$	14.1421	0.812444
$304$	0	0
$305$	32.0000	1.83231
$306$	0	0
$307$	4.00000i	0.228292i	0.993464	+	0.114146i	$0.0364132\pi$
$307$	4.00000i	0.228292i	−0.993464	+	0.114146i	$0.963587\pi$
$308$	0	0
$309$	− 2.82843i	− 0.160904i
$310$	0	0
$311$	−11.3137	−0.641542	−0.320771	−	0.947157i	$-0.603942\pi$
$311$	−11.3137	−0.641542	−0.320771	+	0.947157i	$0.603942\pi$
$312$	0	0
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	0	0
$315$	− 8.00000i	− 0.450749i
$316$	0	0
$317$	− 8.48528i	− 0.476581i	−0.971194	−	0.238290i	$-0.923413\pi$
$317$	− 8.48528i	− 0.476581i	0.971194	−	0.238290i	$-0.0765870\pi$
$318$	0	0
$319$	11.3137	0.633446
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	− 8.00000i	− 0.445132i
$324$	0	0
$325$	− 16.9706i	− 0.941357i
$326$	0	0
$327$	5.65685	0.312825
$328$	0	0
$329$	−16.0000	−0.882109
$330$	0	0
$331$	20.0000i	1.09930i	0.835395	+	0.549650i	$0.185239\pi$
$331$	20.0000i	1.09930i	−0.835395	+	0.549650i	$0.814761\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−11.3137	−0.618134
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	0	0
$339$	2.00000i	0.108625i
$340$	0	0
$341$	− 33.9411i	− 1.83801i
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	16.0000	0.861411
$346$	0	0
$347$	− 36.0000i	− 1.93258i	−0.257454	−	0.966291i	$-0.582883\pi$
$347$	− 36.0000i	− 1.93258i	0.257454	−	0.966291i	$-0.417117\pi$
$348$	0	0
$349$	− 11.3137i	− 0.605609i	−0.953053	−	0.302804i	$-0.902077\pi$
$349$	− 11.3137i	− 0.605609i	0.953053	−	0.302804i	$-0.0979229\pi$
$350$	0	0
$351$	5.65685	0.301941
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	16.0000i	0.849192i
$356$	0	0
$357$	− 5.65685i	− 0.299392i
$358$	0	0
$359$	28.2843	1.49279	0.746393	−	0.665505i	$-0.231784\pi$
$359$	28.2843	1.49279	0.746393	+	0.665505i	$0.231784\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	− 5.00000i	− 0.262432i
$364$	0	0
$365$	− 5.65685i	− 0.296093i
$366$	0	0
$367$	8.48528	0.442928	0.221464	−	0.975169i	$-0.428916\pi$
$367$	8.48528	0.442928	0.221464	+	0.975169i	$0.428916\pi$
$368$	0	0
$369$	−10.0000	−0.520579
$370$	0	0
$371$	− 8.00000i	− 0.415339i
$372$	0	0
$373$	33.9411i	1.75740i	0.477370	+	0.878702i	$0.341590\pi$
$373$	33.9411i	1.75740i	−0.477370	+	0.878702i	$0.658410\pi$
$374$	0	0
$375$	−5.65685	−0.292119
$376$	0	0
$377$	−16.0000	−0.824042
$378$	0	0
$379$	− 28.0000i	− 1.43826i	−0.694874	−	0.719132i	$-0.744540\pi$
$379$	− 28.0000i	− 1.43826i	0.694874	−	0.719132i	$-0.255460\pi$
$380$	0	0
$381$	− 8.48528i	− 0.434714i
$382$	0	0
$383$	−11.3137	−0.578103	−0.289052	−	0.957313i	$-0.593340\pi$
$383$	−11.3137	−0.578103	−0.289052	+	0.957313i	$0.593340\pi$
$384$	0	0
$385$	32.0000	1.63087
$386$	0	0
$387$	12.0000i	0.609994i
$388$	0	0
$389$	− 31.1127i	− 1.57748i	−0.614729	−	0.788738i	$-0.710735\pi$
$389$	− 31.1127i	− 1.57748i	0.614729	−	0.788738i	$-0.289265\pi$
$390$	0	0
$391$	11.3137	0.572159
$392$	0	0
$393$	12.0000	0.605320
$394$	0	0
$395$	24.0000i	1.20757i
$396$	0	0
$397$	− 33.9411i	− 1.70346i	−0.523984	−	0.851728i	$-0.675555\pi$
$397$	− 33.9411i	− 1.70346i	0.523984	−	0.851728i	$-0.324445\pi$
$398$	0	0
$399$	−11.3137	−0.566394
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	48.0000i	2.39105i
$404$	0	0
$405$	2.82843i	0.140546i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	10.0000i	0.493264i
$412$	0	0
$413$	11.3137i	0.556711i
$414$	0	0
$415$	11.3137	0.555368
$416$	0	0
$417$	−4.00000	−0.195881
$418$	0	0
$419$	28.0000i	1.36789i	0.729534	+	0.683945i	$0.239737\pi$
$419$	28.0000i	1.36789i	−0.729534	+	0.683945i	$0.760263\pi$
$420$	0	0
$421$	− 16.9706i	− 0.827095i	−0.910483	−	0.413547i	$-0.864290\pi$
$421$	− 16.9706i	− 0.827095i	0.910483	−	0.413547i	$-0.135710\pi$
$422$	0	0
$423$	5.65685	0.275046
$424$	0	0
$425$	6.00000	0.291043
$426$	0	0
$427$	− 32.0000i	− 1.54859i
$428$	0	0
$429$	22.6274i	1.09246i
$430$	0	0
$431$	28.2843	1.36241	0.681203	−	0.732095i	$-0.261457\pi$
$431$	28.2843	1.36241	0.681203	+	0.732095i	$0.261457\pi$
$432$	0	0
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	0	0
$435$	− 8.00000i	− 0.383571i
$436$	0	0
$437$	− 22.6274i	− 1.08242i
$438$	0	0
$439$	31.1127	1.48493	0.742464	−	0.669886i	$-0.233657\pi$
$439$	31.1127	1.48493	0.742464	+	0.669886i	$0.233657\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	− 20.0000i	− 0.950229i	−0.879924	−	0.475114i	$-0.842407\pi$
$443$	− 20.0000i	− 0.950229i	0.879924	−	0.475114i	$-0.157593\pi$
$444$	0	0
$445$	16.9706i	0.804482i
$446$	0	0
$447$	−14.1421	−0.668900
$448$	0	0
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	− 40.0000i	− 1.88353i
$452$	0	0
$453$	− 8.48528i	− 0.398673i
$454$	0	0
$455$	−45.2548	−2.12158
$456$	0	0
$457$	−6.00000	−0.280668	−0.140334	−	0.990104i	$-0.544818\pi$
$457$	−6.00000	−0.280668	−0.140334	+	0.990104i	$0.544818\pi$
$458$	0	0
$459$	2.00000i	0.0933520i
$460$	0	0
$461$	8.48528i	0.395199i	0.980283	+	0.197599i	$0.0633145\pi$
$461$	8.48528i	0.395199i	−0.980283	+	0.197599i	$0.936685\pi$
$462$	0	0
$463$	−19.7990	−0.920137	−0.460069	−	0.887883i	$-0.652175\pi$
$463$	−19.7990	−0.920137	−0.460069	+	0.887883i	$0.652175\pi$
$464$	0	0
$465$	−24.0000	−1.11297
$466$	0	0
$467$	28.0000i	1.29569i	0.761774	+	0.647843i	$0.224329\pi$
$467$	28.0000i	1.29569i	−0.761774	+	0.647843i	$0.775671\pi$
$468$	0	0
$469$	11.3137i	0.522419i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−48.0000	−2.20704
$474$	0	0
$475$	− 12.0000i	− 0.550598i
$476$	0	0
$477$	2.82843i	0.129505i
$478$	0	0
$479$	−28.2843	−1.29234	−0.646171	−	0.763193i	$-0.723631\pi$
$479$	−28.2843	−1.29234	−0.646171	+	0.763193i	$0.723631\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 16.0000i	− 0.728025i
$484$	0	0
$485$	39.5980i	1.79805i
$486$	0	0
$487$	14.1421	0.640841	0.320421	−	0.947275i	$-0.396176\pi$
$487$	14.1421	0.640841	0.320421	+	0.947275i	$0.396176\pi$
$488$	0	0
$489$	−4.00000	−0.180886
$490$	0	0
$491$	20.0000i	0.902587i	0.892375	+	0.451294i	$0.149037\pi$
$491$	20.0000i	0.902587i	−0.892375	+	0.451294i	$0.850963\pi$
$492$	0	0
$493$	− 5.65685i	− 0.254772i
$494$	0	0
$495$	−11.3137	−0.508513
$496$	0	0
$497$	16.0000	0.717698
$498$	0	0
$499$	4.00000i	0.179065i	0.995984	+	0.0895323i	$0.0285372\pi$
$499$	4.00000i	0.179065i	−0.995984	+	0.0895323i	$0.971463\pi$
$500$	0	0
$501$	− 11.3137i	− 0.505459i
$502$	0	0
$503$	5.65685	0.252227	0.126113	−	0.992016i	$-0.459750\pi$
$503$	5.65685	0.252227	0.126113	+	0.992016i	$0.459750\pi$
$504$	0	0
$505$	40.0000	1.77998
$506$	0	0
$507$	− 19.0000i	− 0.843820i
$508$	0	0
$509$	2.82843i	0.125368i	0.998033	+	0.0626839i	$0.0199660\pi$
$509$	2.82843i	0.125368i	−0.998033	+	0.0626839i	$0.980034\pi$
$510$	0	0
$511$	−5.65685	−0.250244
$512$	0	0
$513$	4.00000	0.176604
$514$	0	0
$515$	− 8.00000i	− 0.352522i
$516$	0	0
$517$	22.6274i	0.995153i
$518$	0	0
$519$	−19.7990	−0.869079
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	0	0
$525$	− 8.48528i	− 0.370328i
$526$	0	0
$527$	−16.9706	−0.739249
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	− 4.00000i	− 0.173585i
$532$	0	0
$533$	56.5685i	2.45026i
$534$	0	0
$535$	33.9411	1.46740
$536$	0	0
$537$	12.0000	0.517838
$538$	0	0
$539$	− 4.00000i	− 0.172292i
$540$	0	0
$541$	16.9706i	0.729621i	0.931082	+	0.364811i	$0.118866\pi$
$541$	16.9706i	0.729621i	−0.931082	+	0.364811i	$0.881134\pi$
$542$	0	0
$543$	−5.65685	−0.242759
$544$	0	0
$545$	16.0000	0.685365
$546$	0	0
$547$	20.0000i	0.855138i	0.903983	+	0.427569i	$0.140630\pi$
$547$	20.0000i	0.855138i	−0.903983	+	0.427569i	$0.859370\pi$
$548$	0	0
$549$	11.3137i	0.482857i
$550$	0	0
$551$	−11.3137	−0.481980
$552$	0	0
$553$	24.0000	1.02058
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8.48528i	0.359533i	0.983709	+	0.179766i	$0.0575342\pi$
$557$	8.48528i	0.359533i	−0.983709	+	0.179766i	$0.942466\pi$
$558$	0	0
$559$	67.8823	2.87111
$560$	0	0
$561$	−8.00000	−0.337760
$562$	0	0
$563$	− 4.00000i	− 0.168580i	−0.996441	−	0.0842900i	$-0.973138\pi$
$563$	− 4.00000i	− 0.168580i	0.996441	−	0.0842900i	$-0.0268622\pi$
$564$	0	0
$565$	5.65685i	0.237986i
$566$	0	0
$567$	2.82843	0.118783
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	4.00000i	0.167395i	0.996491	+	0.0836974i	$0.0266729\pi$
$571$	4.00000i	0.167395i	−0.996491	+	0.0836974i	$0.973327\pi$
$572$	0	0
$573$	11.3137i	0.472637i
$574$	0	0
$575$	16.9706	0.707721
$576$	0	0
$577$	−22.0000	−0.915872	−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000	−0.915872	−0.457936	+	0.888985i	$0.651411\pi$
$578$	0	0
$579$	6.00000i	0.249351i
$580$	0	0
$581$	− 11.3137i	− 0.469372i
$582$	0	0
$583$	−11.3137	−0.468566
$584$	0	0
$585$	16.0000	0.661519
$586$	0	0
$587$	36.0000i	1.48588i	0.669359	+	0.742940i	$0.266569\pi$
$587$	36.0000i	1.48588i	−0.669359	+	0.742940i	$0.733431\pi$
$588$	0	0
$589$	33.9411i	1.39852i
$590$	0	0
$591$	−8.48528	−0.349038
$592$	0	0
$593$	34.0000	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$594$	0	0
$595$	− 16.0000i	− 0.655936i
$596$	0	0
$597$	− 2.82843i	− 0.115760i
$598$	0	0
$599$	−39.5980	−1.61793	−0.808965	−	0.587857i	$-0.799972\pi$
$599$	−39.5980	−1.61793	−0.808965	+	0.587857i	$0.799972\pi$
$600$	0	0
$601$	−34.0000	−1.38689	−0.693444	−	0.720510i	$-0.743908\pi$
$601$	−34.0000	−1.38689	−0.693444	+	0.720510i	$0.743908\pi$
$602$	0	0
$603$	− 4.00000i	− 0.162893i
$604$	0	0
$605$	− 14.1421i	− 0.574960i
$606$	0	0
$607$	14.1421	0.574012	0.287006	−	0.957929i	$-0.407340\pi$
$607$	14.1421	0.574012	0.287006	+	0.957929i	$0.407340\pi$
$608$	0	0
$609$	−8.00000	−0.324176
$610$	0	0
$611$	− 32.0000i	− 1.29458i
$612$	0	0
$613$	11.3137i	0.456956i	0.973549	+	0.228478i	$0.0733750\pi$
$613$	11.3137i	0.456956i	−0.973549	+	0.228478i	$0.926625\pi$
$614$	0	0
$615$	−28.2843	−1.14053
$616$	0	0
$617$	38.0000	1.52982	0.764911	−	0.644136i	$-0.222783\pi$
$617$	38.0000	1.52982	0.764911	+	0.644136i	$0.222783\pi$
$618$	0	0
$619$	36.0000i	1.44696i	0.690344	+	0.723481i	$0.257459\pi$
$619$	36.0000i	1.44696i	−0.690344	+	0.723481i	$0.742541\pi$
$620$	0	0
$621$	5.65685i	0.227002i
$622$	0	0
$623$	16.9706	0.679911
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	16.0000i	0.638978i
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−8.48528	−0.337794	−0.168897	−	0.985634i	$-0.554020\pi$
$631$	−8.48528	−0.337794	−0.168897	+	0.985634i	$0.554020\pi$
$632$	0	0
$633$	12.0000	0.476957
$634$	0	0
$635$	− 24.0000i	− 0.952411i
$636$	0	0
$637$	5.65685i	0.224133i
$638$	0	0
$639$	−5.65685	−0.223782
$640$	0	0
$641$	−34.0000	−1.34292	−0.671460	−	0.741041i	$-0.734332\pi$
$641$	−34.0000	−1.34292	−0.671460	+	0.741041i	$0.734332\pi$
$642$	0	0
$643$	− 44.0000i	− 1.73519i	−0.497271	−	0.867595i	$-0.665665\pi$
$643$	− 44.0000i	− 1.73519i	0.497271	−	0.867595i	$-0.334335\pi$
$644$	0	0
$645$	33.9411i	1.33643i
$646$	0	0
$647$	39.5980	1.55676	0.778379	−	0.627795i	$-0.216042\pi$
$647$	39.5980	1.55676	0.778379	+	0.627795i	$0.216042\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	0	0
$651$	24.0000i	0.940634i
$652$	0	0
$653$	42.4264i	1.66027i	0.557560	+	0.830137i	$0.311738\pi$
$653$	42.4264i	1.66027i	−0.557560	+	0.830137i	$0.688262\pi$
$654$	0	0
$655$	33.9411	1.32619
$656$	0	0
$657$	2.00000	0.0780274
$658$	0	0
$659$	4.00000i	0.155818i	0.996960	+	0.0779089i	$0.0248243\pi$
$659$	4.00000i	0.155818i	−0.996960	+	0.0779089i	$0.975176\pi$
$660$	0	0
$661$	− 45.2548i	− 1.76021i	−0.474780	−	0.880105i	$-0.657472\pi$
$661$	− 45.2548i	− 1.76021i	0.474780	−	0.880105i	$-0.342528\pi$
$662$	0	0
$663$	11.3137	0.439388
$664$	0	0
$665$	−32.0000	−1.24091
$666$	0	0
$667$	− 16.0000i	− 0.619522i
$668$	0	0
$669$	− 19.7990i	− 0.765473i
$670$	0	0
$671$	−45.2548	−1.74704
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	0	0
$675$	3.00000i	0.115470i
$676$	0	0
$677$	− 19.7990i	− 0.760937i	−0.924794	−	0.380468i	$-0.875763\pi$
$677$	− 19.7990i	− 0.760937i	0.924794	−	0.380468i	$-0.124237\pi$
$678$	0	0
$679$	39.5980	1.51963
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	− 20.0000i	− 0.765279i	−0.923898	−	0.382639i	$-0.875015\pi$
$683$	− 20.0000i	− 0.765279i	0.923898	−	0.382639i	$-0.124985\pi$
$684$	0	0
$685$	28.2843i	1.08069i
$686$	0	0
$687$	5.65685	0.215822
$688$	0	0
$689$	16.0000	0.609551
$690$	0	0
$691$	− 28.0000i	− 1.06517i	−0.846376	−	0.532585i	$-0.821221\pi$
$691$	− 28.0000i	− 1.06517i	0.846376	−	0.532585i	$-0.178779\pi$
$692$	0	0
$693$	11.3137i	0.429772i
$694$	0	0
$695$	−11.3137	−0.429153
$696$	0	0
$697$	−20.0000	−0.757554
$698$	0	0
$699$	6.00000i	0.226941i
$700$	0	0
$701$	− 19.7990i	− 0.747798i	−0.927470	−	0.373899i	$-0.878021\pi$
$701$	− 19.7990i	− 0.747798i	0.927470	−	0.373899i	$-0.121979\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	16.0000	0.602595
$706$	0	0
$707$	− 40.0000i	− 1.50435i
$708$	0	0
$709$	16.9706i	0.637343i	0.947865	+	0.318671i	$0.103237\pi$
$709$	16.9706i	0.637343i	−0.947865	+	0.318671i	$0.896763\pi$
$710$	0	0
$711$	−8.48528	−0.318223
$712$	0	0
$713$	−48.0000	−1.79761
$714$	0	0
$715$	64.0000i	2.39346i
$716$	0	0
$717$	− 22.6274i	− 0.845036i
$718$	0	0
$719$	−16.9706	−0.632895	−0.316448	−	0.948610i	$-0.602490\pi$
$719$	−16.9706	−0.632895	−0.316448	+	0.948610i	$0.602490\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	6.00000i	0.223142i
$724$	0	0
$725$	− 8.48528i	− 0.315135i
$726$	0	0
$727$	25.4558	0.944105	0.472052	−	0.881570i	$-0.343513\pi$
$727$	25.4558	0.944105	0.472052	+	0.881570i	$0.343513\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	24.0000i	0.887672i
$732$	0	0
$733$	− 16.9706i	− 0.626822i	−0.949618	−	0.313411i	$-0.898528\pi$
$733$	− 16.9706i	− 0.626822i	0.949618	−	0.313411i	$-0.101472\pi$
$734$	0	0
$735$	−2.82843	−0.104328
$736$	0	0
$737$	16.0000	0.589368
$738$	0	0
$739$	36.0000i	1.32428i	0.749380	+	0.662141i	$0.230352\pi$
$739$	36.0000i	1.32428i	−0.749380	+	0.662141i	$0.769648\pi$
$740$	0	0
$741$	− 22.6274i	− 0.831239i
$742$	0	0
$743$	22.6274	0.830119	0.415060	−	0.909794i	$-0.363761\pi$
$743$	22.6274	0.830119	0.415060	+	0.909794i	$0.363761\pi$
$744$	0	0
$745$	−40.0000	−1.46549
$746$	0	0
$747$	4.00000i	0.146352i
$748$	0	0
$749$	− 33.9411i	− 1.24018i
$750$	0	0
$751$	−48.0833	−1.75458	−0.877292	−	0.479958i	$-0.840652\pi$
$751$	−48.0833	−1.75458	−0.877292	+	0.479958i	$0.840652\pi$
$752$	0	0
$753$	−12.0000	−0.437304
$754$	0	0
$755$	− 24.0000i	− 0.873449i
$756$	0	0
$757$	− 28.2843i	− 1.02801i	−0.857787	−	0.514005i	$-0.828161\pi$
$757$	− 28.2843i	− 1.02801i	0.857787	−	0.514005i	$-0.171839\pi$
$758$	0	0
$759$	−22.6274	−0.821323
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	0	0
$763$	− 16.0000i	− 0.579239i
$764$	0	0
$765$	5.65685i	0.204524i
$766$	0	0
$767$	−22.6274	−0.817029
$768$	0	0
$769$	38.0000	1.37032	0.685158	−	0.728395i	$-0.259733\pi$
$769$	38.0000	1.37032	0.685158	+	0.728395i	$0.259733\pi$
$770$	0	0
$771$	− 30.0000i	− 1.08042i
$772$	0	0
$773$	− 2.82843i	− 0.101731i	−0.998706	−	0.0508657i	$-0.983802\pi$
$773$	− 2.82843i	− 0.101731i	0.998706	−	0.0508657i	$-0.0161981\pi$
$774$	0	0
$775$	−25.4558	−0.914401
$776$	0	0
$777$	0	0
$778$	0	0
$779$	40.0000i	1.43315i
$780$	0	0
$781$	− 22.6274i	− 0.809673i
$782$	0	0
$783$	2.82843	0.101080
$784$	0	0
$785$	0	0
$786$	0	0
$787$	20.0000i	0.712923i	0.934310	+	0.356462i	$0.116017\pi$
$787$	20.0000i	0.712923i	−0.934310	+	0.356462i	$0.883983\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	5.65685	0.201135
$792$	0	0
$793$	64.0000	2.27271
$794$	0	0
$795$	8.00000i	0.283731i
$796$	0	0
$797$	19.7990i	0.701316i	0.936504	+	0.350658i	$0.114042\pi$
$797$	19.7990i	0.701316i	−0.936504	+	0.350658i	$0.885958\pi$
$798$	0	0
$799$	11.3137	0.400250
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	8.00000i	0.282314i
$804$	0	0
$805$	− 45.2548i	− 1.59502i
$806$	0	0
$807$	8.48528	0.298696
$808$	0	0
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	0	0
$811$	20.0000i	0.702295i	0.936320	+	0.351147i	$0.114208\pi$
$811$	20.0000i	0.702295i	−0.936320	+	0.351147i	$0.885792\pi$
$812$	0	0
$813$	− 19.7990i	− 0.694381i
$814$	0	0
$815$	−11.3137	−0.396302
$816$	0	0
$817$	48.0000	1.67931
$818$	0	0
$819$	− 16.0000i	− 0.559085i
$820$	0	0
$821$	31.1127i	1.08584i	0.839784	+	0.542920i	$0.182681\pi$
$821$	31.1127i	1.08584i	−0.839784	+	0.542920i	$0.817319\pi$
$822$	0	0
$823$	−19.7990	−0.690149	−0.345075	−	0.938575i	$-0.612146\pi$
$823$	−19.7990	−0.690149	−0.345075	+	0.938575i	$0.612146\pi$
$824$	0	0
$825$	−12.0000	−0.417786
$826$	0	0
$827$	− 12.0000i	− 0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$	− 12.0000i	− 0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$	0	0
$829$	− 50.9117i	− 1.76824i	−0.467264	−	0.884118i	$-0.654760\pi$
$829$	− 50.9117i	− 1.76824i	0.467264	−	0.884118i	$-0.345240\pi$
$830$	0	0
$831$	16.9706	0.588702
$832$	0	0
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	− 32.0000i	− 1.10741i
$836$	0	0
$837$	− 8.48528i	− 0.293294i
$838$	0	0
$839$	−5.65685	−0.195296	−0.0976481	−	0.995221i	$-0.531132\pi$
$839$	−5.65685	−0.195296	−0.0976481	+	0.995221i	$0.531132\pi$
$840$	0	0
$841$	21.0000	0.724138
$842$	0	0
$843$	− 10.0000i	− 0.344418i
$844$	0	0
$845$	− 53.7401i	− 1.84872i
$846$	0	0
$847$	−14.1421	−0.485930
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	0	0
$852$	0	0
$853$	22.6274i	0.774748i	0.921923	+	0.387374i	$0.126618\pi$
$853$	22.6274i	0.774748i	−0.921923	+	0.387374i	$0.873382\pi$
$854$	0	0
$855$	11.3137	0.386921
$856$	0	0
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	0	0
$859$	− 44.0000i	− 1.50126i	−0.660722	−	0.750630i	$-0.729750\pi$
$859$	− 44.0000i	− 1.50126i	0.660722	−	0.750630i	$-0.270250\pi$
$860$	0	0
$861$	28.2843i	0.963925i
$862$	0	0
$863$	56.5685	1.92562	0.962808	−	0.270187i	$-0.0870856\pi$
$863$	56.5685	1.92562	0.962808	+	0.270187i	$0.0870856\pi$
$864$	0	0
$865$	−56.0000	−1.90406
$866$	0	0
$867$	− 13.0000i	− 0.441503i
$868$	0	0
$869$	− 33.9411i	− 1.15137i
$870$	0	0
$871$	−22.6274	−0.766701
$872$	0	0
$873$	−14.0000	−0.473828
$874$	0	0
$875$	16.0000i	0.540899i
$876$	0	0
$877$	11.3137i	0.382037i	0.981586	+	0.191018i	$0.0611790\pi$
$877$	11.3137i	0.382037i	−0.981586	+	0.191018i	$0.938821\pi$
$878$	0	0
$879$	−19.7990	−0.667803
$880$	0	0
$881$	−46.0000	−1.54978	−0.774890	−	0.632096i	$-0.782195\pi$
$881$	−46.0000	−1.54978	−0.774890	+	0.632096i	$0.782195\pi$
$882$	0	0
$883$	− 12.0000i	− 0.403832i	−0.979403	−	0.201916i	$-0.935283\pi$
$883$	− 12.0000i	− 0.403832i	0.979403	−	0.201916i	$-0.0647168\pi$
$884$	0	0
$885$	− 11.3137i	− 0.380306i
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	−24.0000	−0.804934
$890$	0	0
$891$	− 4.00000i	− 0.134005i
$892$	0	0
$893$	− 22.6274i	− 0.757198i
$894$	0	0
$895$	33.9411	1.13453
$896$	0	0
$897$	32.0000	1.06845
$898$	0	0
$899$	24.0000i	0.800445i
$900$	0	0
$901$	5.65685i	0.188457i
$902$	0	0
$903$	33.9411	1.12949
$904$	0	0
$905$	−16.0000	−0.531858
$906$	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
$909$	14.1421i	0.469065i
$910$	0	0
$911$	22.6274	0.749680	0.374840	−	0.927090i	$-0.377698\pi$
$911$	22.6274	0.749680	0.374840	+	0.927090i	$0.377698\pi$
$912$	0	0
$913$	−16.0000	−0.529523
$914$	0	0
$915$	32.0000i	1.05789i
$916$	0	0
$917$	− 33.9411i	− 1.12083i
$918$	0	0
$919$	2.82843	0.0933012	0.0466506	−	0.998911i	$-0.485145\pi$
$919$	2.82843	0.0933012	0.0466506	+	0.998911i	$0.485145\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	0	0
$923$	32.0000i	1.05329i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	2.82843	0.0928977
$928$	0	0
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	4.00000i	0.131095i
$932$	0	0
$933$	− 11.3137i	− 0.370394i
$934$	0	0
$935$	−22.6274	−0.739996
$936$	0	0
$937$	−42.0000	−1.37208	−0.686040	−	0.727564i	$-0.740653\pi$
$937$	−42.0000	−1.37208	−0.686040	+	0.727564i	$0.740653\pi$
$938$	0	0
$939$	− 14.0000i	− 0.456873i
$940$	0	0
$941$	− 14.1421i	− 0.461020i	−0.973070	−	0.230510i	$-0.925960\pi$
$941$	− 14.1421i	− 0.461020i	0.973070	−	0.230510i	$-0.0740395\pi$
$942$	0	0
$943$	−56.5685	−1.84213
$944$	0	0
$945$	8.00000	0.260240
$946$	0	0
$947$	36.0000i	1.16984i	0.811090	+	0.584921i	$0.198875\pi$
$947$	36.0000i	1.16984i	−0.811090	+	0.584921i	$0.801125\pi$
$948$	0	0
$949$	− 11.3137i	− 0.367259i
$950$	0	0
$951$	8.48528	0.275154
$952$	0	0
$953$	42.0000	1.36051	0.680257	−	0.732974i	$-0.261868\pi$
$953$	42.0000	1.36051	0.680257	+	0.732974i	$0.261868\pi$
$954$	0	0
$955$	32.0000i	1.03550i
$956$	0	0
$957$	11.3137i	0.365720i
$958$	0	0
$959$	28.2843	0.913347
$960$	0	0
$961$	41.0000	1.32258
$962$	0	0
$963$	12.0000i	0.386695i
$964$	0	0
$965$	16.9706i	0.546302i
$966$	0	0
$967$	−14.1421	−0.454780	−0.227390	−	0.973804i	$-0.573019\pi$
$967$	−14.1421	−0.454780	−0.227390	+	0.973804i	$0.573019\pi$
$968$	0	0
$969$	8.00000	0.256997
$970$	0	0
$971$	12.0000i	0.385098i	0.981287	+	0.192549i	$0.0616755\pi$
$971$	12.0000i	0.385098i	−0.981287	+	0.192549i	$0.938325\pi$
$972$	0	0
$973$	11.3137i	0.362701i
$974$	0	0
$975$	16.9706	0.543493
$976$	0	0
$977$	−2.00000	−0.0639857	−0.0319928	−	0.999488i	$-0.510185\pi$
$977$	−2.00000	−0.0639857	−0.0319928	+	0.999488i	$0.510185\pi$
$978$	0	0
$979$	− 24.0000i	− 0.767043i
$980$	0	0
$981$	5.65685i	0.180609i
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	−24.0000	−0.764704
$986$	0	0
$987$	− 16.0000i	− 0.509286i
$988$	0	0
$989$	67.8823i	2.15853i
$990$	0	0
$991$	−25.4558	−0.808632	−0.404316	−	0.914619i	$-0.632490\pi$
$991$	−25.4558	−0.808632	−0.404316	+	0.914619i	$0.632490\pi$
$992$	0	0
$993$	−20.0000	−0.634681
$994$	0	0
$995$	− 8.00000i	− 0.253617i
$996$	0	0
$997$	33.9411i	1.07493i	0.843287	+	0.537463i	$0.180617\pi$
$997$	33.9411i	1.07493i	−0.843287	+	0.537463i	$0.819383\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.d.c.193.4	yes	4
3.2	odd	2		1152.2.d.h.577.2		4
4.3	odd	2	inner	384.2.d.c.193.2	yes	4
8.3	odd	2	inner	384.2.d.c.193.3	yes	4
8.5	even	2	inner	384.2.d.c.193.1	✓	4
12.11	even	2		1152.2.d.h.577.1		4
16.3	odd	4		768.2.a.l.1.2		2
16.5	even	4		768.2.a.l.1.1		2
16.11	odd	4		768.2.a.i.1.1		2
16.13	even	4		768.2.a.i.1.2		2
24.5	odd	2		1152.2.d.h.577.4		4
24.11	even	2		1152.2.d.h.577.3		4
48.5	odd	4		2304.2.a.r.1.2		2
48.11	even	4		2304.2.a.x.1.2		2
48.29	odd	4		2304.2.a.x.1.1		2
48.35	even	4		2304.2.a.r.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.d.c.193.1	✓	4	8.5	even	2	inner
384.2.d.c.193.2	yes	4	4.3	odd	2	inner
384.2.d.c.193.3	yes	4	8.3	odd	2	inner
384.2.d.c.193.4	yes	4	1.1	even	1	trivial
768.2.a.i.1.1		2	16.11	odd	4
768.2.a.i.1.2		2	16.13	even	4
768.2.a.l.1.1		2	16.5	even	4
768.2.a.l.1.2		2	16.3	odd	4
1152.2.d.h.577.1		4	12.11	even	2
1152.2.d.h.577.2		4	3.2	odd	2
1152.2.d.h.577.3		4	24.11	even	2
1152.2.d.h.577.4		4	24.5	odd	2
2304.2.a.r.1.1		2	48.35	even	4
2304.2.a.r.1.2		2	48.5	odd	4
2304.2.a.x.1.1		2	48.29	odd	4
2304.2.a.x.1.2		2	48.11	even	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	384.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +2.82843i q^{5} +2.82843 q^{7} -1.00000 q^{9} -4.00000i q^{11} +5.65685i q^{13} -2.82843 q^{15} -2.00000 q^{17} +4.00000i q^{19} +2.82843i q^{21} -5.65685 q^{23} -3.00000 q^{25} -1.00000i q^{27} +2.82843i q^{29} +8.48528 q^{31} +4.00000 q^{33} +8.00000i q^{35} -5.65685 q^{39} +10.0000 q^{41} -12.0000i q^{43} -2.82843i q^{45} -5.65685 q^{47} +1.00000 q^{49} -2.00000i q^{51} -2.82843i q^{53} +11.3137 q^{55} -4.00000 q^{57} +4.00000i q^{59} -11.3137i q^{61} -2.82843 q^{63} -16.0000 q^{65} +4.00000i q^{67} -5.65685i q^{69} +5.65685 q^{71} -2.00000 q^{73} -3.00000i q^{75} -11.3137i q^{77} +8.48528 q^{79} +1.00000 q^{81} -4.00000i q^{83} -5.65685i q^{85} -2.82843 q^{87} +6.00000 q^{89} +16.0000i q^{91} +8.48528i q^{93} -11.3137 q^{95} +14.0000 q^{97} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9} - 8 q^{17} - 12 q^{25} + 16 q^{33} + 40 q^{41} + 4 q^{49} - 16 q^{57} - 64 q^{65} - 8 q^{73} + 4 q^{81} + 24 q^{89} + 56 q^{97}+O(q^{100}) \) 4 * q - 4 * q^9 - 8 * q^17 - 12 * q^25 + 16 * q^33 + 40 * q^41 + 4 * q^49 - 16 * q^57 - 64 * q^65 - 8 * q^73 + 4 * q^81 + 24 * q^89 + 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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