LMFDB - Embedded newform 3072.2.d.a.1537.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3072,2,Mod(1537,3072)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3072, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3072.1537");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3072 = 2^{10} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3072.d (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:	$24.5300435009$
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 768)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1537.2
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	3072.1537
Dual form			3072.2.d.a.1537.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +1.41421 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +1.41421 q^{7} -1.00000 q^{9} -1.41421i q^{13} -6.00000 q^{17} +6.00000i q^{19} -1.41421i q^{21} -8.48528 q^{23} +5.00000 q^{25} +1.00000i q^{27} +8.48528i q^{29} +1.41421 q^{31} +7.07107i q^{37} -1.41421 q^{39} +6.00000 q^{41} -6.00000i q^{43} -8.48528 q^{47} -5.00000 q^{49} +6.00000i q^{51} +8.48528i q^{53} +6.00000 q^{57} +7.07107i q^{61} -1.41421 q^{63} -4.00000i q^{67} +8.48528i q^{69} +8.48528 q^{71} -5.00000i q^{75} +1.41421 q^{79} +1.00000 q^{81} -12.0000i q^{83} +8.48528 q^{87} -6.00000 q^{89} -2.00000i q^{91} -1.41421i q^{93} -12.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 4 q - 4 q^{9} - 24 q^{17} + 20 q^{25} + 24 q^{41} - 20 q^{49} + 24 q^{57} + 4 q^{81} - 24 q^{89} - 48 q^{97}+O(q^{100}) $ 4 * q - 4 * q^9 - 24 * q^17 + 20 * q^25 + 24 * q^41 - 20 * q^49 + 24 * q^57 + 4 * q^81 - 24 * q^89 - 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3072\mathbb{Z}\right)^\times$.

$n$	$1025$	$2047$	$2053$
$\chi(n)$	$1$	$1$	$-1$

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
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	1.41421	0.534522	0.267261	−	0.963624i	$-0.413881\pi$
$7$	1.41421	0.534522	0.267261	+	0.963624i	$0.413881\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	− 1.41421i	− 0.392232i	−0.980581	−	0.196116i	$-0.937167\pi$
$13$	− 1.41421i	− 0.392232i	0.980581	−	0.196116i	$-0.0628330\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	6.00000i	1.37649i	0.725476	+	0.688247i	$0.241620\pi$
$19$	6.00000i	1.37649i	−0.725476	+	0.688247i	$0.758380\pi$
$20$	0	0
$21$	− 1.41421i	− 0.308607i
$22$	0	0
$23$	−8.48528	−1.76930	−0.884652	−	0.466252i	$-0.845604\pi$
$23$	−8.48528	−1.76930	−0.884652	+	0.466252i	$0.845604\pi$
$24$	0	0
$25$	5.00000	1.00000
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	8.48528i	1.57568i	0.615882	+	0.787839i	$0.288800\pi$
$29$	8.48528i	1.57568i	−0.615882	+	0.787839i	$0.711200\pi$
$30$	0	0
$31$	1.41421	0.254000	0.127000	−	0.991903i	$-0.459465\pi$
$31$	1.41421	0.254000	0.127000	+	0.991903i	$0.459465\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.07107i	1.16248i	0.813733	+	0.581238i	$0.197432\pi$
$37$	7.07107i	1.16248i	−0.813733	+	0.581238i	$0.802568\pi$
$38$	0	0
$39$	−1.41421	−0.226455
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	− 6.00000i	− 0.914991i	−0.889212	−	0.457496i	$-0.848747\pi$
$43$	− 6.00000i	− 0.914991i	0.889212	−	0.457496i	$-0.151253\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.48528	−1.23771	−0.618853	−	0.785507i	$-0.712402\pi$
$47$	−8.48528	−1.23771	−0.618853	+	0.785507i	$0.712402\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	0	0
$51$	6.00000i	0.840168i
$52$	0	0
$53$	8.48528i	1.16554i	0.812636	+	0.582772i	$0.198032\pi$
$53$	8.48528i	1.16554i	−0.812636	+	0.582772i	$0.801968\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	6.00000	0.794719
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	7.07107i	0.905357i	0.891674	+	0.452679i	$0.149532\pi$
$61$	7.07107i	0.905357i	−0.891674	+	0.452679i	$0.850468\pi$
$62$	0	0
$63$	−1.41421	−0.178174
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	0	0
$69$	8.48528i	1.02151i
$70$	0	0
$71$	8.48528	1.00702	0.503509	−	0.863990i	$-0.332042\pi$
$71$	8.48528	1.00702	0.503509	+	0.863990i	$0.332042\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	− 5.00000i	− 0.577350i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1.41421	0.159111	0.0795557	−	0.996830i	$-0.474650\pi$
$79$	1.41421	0.159111	0.0795557	+	0.996830i	$0.474650\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	8.48528	0.909718
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	− 2.00000i	− 0.209657i
$92$	0	0
$93$	− 1.41421i	− 0.146647i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−12.0000	−1.21842	−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000	−1.21842	−0.609208	+	0.793011i	$0.708512\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 8.48528i	− 0.844317i	−0.906522	−	0.422159i	$-0.861273\pi$
$101$	− 8.48528i	− 0.844317i	0.906522	−	0.422159i	$-0.138727\pi$
$102$	0	0
$103$	−9.89949	−0.975426	−0.487713	−	0.873004i	$-0.662169\pi$
$103$	−9.89949	−0.975426	−0.487713	+	0.873004i	$0.662169\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	0	0
$109$	18.3848i	1.76094i	0.474100	+	0.880471i	$0.342774\pi$
$109$	18.3848i	1.76094i	−0.474100	+	0.880471i	$0.657226\pi$
$110$	0	0
$111$	7.07107	0.671156
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.41421i	0.130744i
$118$	0	0
$119$	−8.48528	−0.777844
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	− 6.00000i	− 0.541002i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−9.89949	−0.878438	−0.439219	−	0.898380i	$-0.644745\pi$
$127$	−9.89949	−0.878438	−0.439219	+	0.898380i	$0.644745\pi$
$128$	0	0
$129$	−6.00000	−0.528271
$130$	0	0
$131$	12.0000i	1.04844i	0.851581	+	0.524222i	$0.175644\pi$
$131$	12.0000i	1.04844i	−0.851581	+	0.524222i	$0.824356\pi$
$132$	0	0
$133$	8.48528i	0.735767i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	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
$141$	8.48528i	0.714590i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	5.00000i	0.412393i
$148$	0	0
$149$	16.9706i	1.39028i	0.718873	+	0.695141i	$0.244658\pi$
$149$	16.9706i	1.39028i	−0.718873	+	0.695141i	$0.755342\pi$
$150$	0	0
$151$	18.3848	1.49613	0.748066	−	0.663624i	$-0.230983\pi$
$151$	18.3848	1.49613	0.748066	+	0.663624i	$0.230983\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 7.07107i	− 0.564333i	−0.959366	−	0.282166i	$-0.908947\pi$
$157$	− 7.07107i	− 0.564333i	0.959366	−	0.282166i	$-0.0910530\pi$
$158$	0	0
$159$	8.48528	0.672927
$160$	0	0
$161$	−12.0000	−0.945732
$162$	0	0
$163$	6.00000i	0.469956i	0.972001	+	0.234978i	$0.0755019\pi$
$163$	6.00000i	0.469956i	−0.972001	+	0.234978i	$0.924498\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−16.9706	−1.31322	−0.656611	−	0.754230i	$-0.728011\pi$
$167$	−16.9706	−1.31322	−0.656611	+	0.754230i	$0.728011\pi$
$168$	0	0
$169$	11.0000	0.846154
$170$	0	0
$171$	− 6.00000i	− 0.458831i
$172$	0	0
$173$	16.9706i	1.29025i	0.764078	+	0.645124i	$0.223194\pi$
$173$	16.9706i	1.29025i	−0.764078	+	0.645124i	$0.776806\pi$
$174$	0	0
$175$	7.07107	0.534522
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 24.0000i	− 1.79384i	−0.442189	−	0.896922i	$-0.645798\pi$
$179$	− 24.0000i	− 1.79384i	0.442189	−	0.896922i	$-0.354202\pi$
$180$	0	0
$181$	15.5563i	1.15629i	0.815933	+	0.578147i	$0.196224\pi$
$181$	15.5563i	1.15629i	−0.815933	+	0.578147i	$0.803776\pi$
$182$	0	0
$183$	7.07107	0.522708
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	1.41421i	0.102869i
$190$	0	0
$191$	−16.9706	−1.22795	−0.613973	−	0.789327i	$-0.710430\pi$
$191$	−16.9706	−1.22795	−0.613973	+	0.789327i	$0.710430\pi$
$192$	0	0
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	0	0
$195$	0	0
$196$	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
$199$	9.89949	0.701757	0.350878	−	0.936421i	$-0.385883\pi$
$199$	9.89949	0.701757	0.350878	+	0.936421i	$0.385883\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	12.0000i	0.842235i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	8.48528	0.589768
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 4.00000i	− 0.275371i	−0.990476	−	0.137686i	$-0.956034\pi$
$211$	− 4.00000i	− 0.275371i	0.990476	−	0.137686i	$-0.0439664\pi$
$212$	0	0
$213$	− 8.48528i	− 0.581402i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.00000	0.135769
$218$	0	0
$219$	0	0
$220$	0	0
$221$	8.48528i	0.570782i
$222$	0	0
$223$	26.8701	1.79935	0.899676	−	0.436558i	$-0.143803\pi$
$223$	26.8701	1.79935	0.899676	+	0.436558i	$0.143803\pi$
$224$	0	0
$225$	−5.00000	−0.333333
$226$	0	0
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	0	0
$229$	1.41421i	0.0934539i	0.998908	+	0.0467269i	$0.0148791\pi$
$229$	1.41421i	0.0934539i	−0.998908	+	0.0467269i	$0.985121\pi$
$230$	0	0
$231$	0	0
$232$	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
$235$	0	0
$236$	0	0
$237$	− 1.41421i	− 0.0918630i
$238$	0	0
$239$	−16.9706	−1.09773	−0.548867	−	0.835910i	$-0.684941\pi$
$239$	−16.9706	−1.09773	−0.548867	+	0.835910i	$0.684941\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.48528	0.539906
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	24.0000i	1.51487i	0.652913	+	0.757433i	$0.273547\pi$
$251$	24.0000i	1.51487i	−0.652913	+	0.757433i	$0.726453\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	10.0000i	0.621370i
$260$	0	0
$261$	− 8.48528i	− 0.525226i
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	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
$271$	26.8701	1.63224	0.816120	−	0.577883i	$-0.196121\pi$
$271$	26.8701	1.63224	0.816120	+	0.577883i	$0.196121\pi$
$272$	0	0
$273$	−2.00000	−0.121046
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 15.5563i	− 0.934690i	−0.884075	−	0.467345i	$-0.845211\pi$
$277$	− 15.5563i	− 0.934690i	0.884075	−	0.467345i	$-0.154789\pi$
$278$	0	0
$279$	−1.41421	−0.0846668
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.48528	0.500870
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	12.0000i	0.703452i
$292$	0	0
$293$	8.48528i	0.495715i	0.968796	+	0.247858i	$0.0797265\pi$
$293$	8.48528i	0.495715i	−0.968796	+	0.247858i	$0.920273\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	12.0000i	0.693978i
$300$	0	0
$301$	− 8.48528i	− 0.489083i
$302$	0	0
$303$	−8.48528	−0.487467
$304$	0	0
$305$	0	0
$306$	0	0
$307$	20.0000i	1.14146i	0.821138	+	0.570730i	$0.193340\pi$
$307$	20.0000i	1.14146i	−0.821138	+	0.570730i	$0.806660\pi$
$308$	0	0
$309$	9.89949i	0.563163i
$310$	0	0
$311$	−16.9706	−0.962312	−0.481156	−	0.876635i	$-0.659783\pi$
$311$	−16.9706	−0.962312	−0.481156	+	0.876635i	$0.659783\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	0	0
$315$	0	0
$316$	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
$319$	0	0
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	− 36.0000i	− 2.00309i
$324$	0	0
$325$	− 7.07107i	− 0.392232i
$326$	0	0
$327$	18.3848	1.01668
$328$	0	0
$329$	−12.0000	−0.661581
$330$	0	0
$331$	− 28.0000i	− 1.53902i	−0.638635	−	0.769510i	$-0.720501\pi$
$331$	− 28.0000i	− 1.53902i	0.638635	−	0.769510i	$-0.279499\pi$
$332$	0	0
$333$	− 7.07107i	− 0.387492i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−24.0000	−1.30736	−0.653682	−	0.756770i	$-0.726776\pi$
$337$	−24.0000	−1.30736	−0.653682	+	0.756770i	$0.726776\pi$
$338$	0	0
$339$	6.00000i	0.325875i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.0000i	0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$	12.0000i	0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$	0	0
$349$	26.8701i	1.43832i	0.694844	+	0.719161i	$0.255473\pi$
$349$	26.8701i	1.43832i	−0.694844	+	0.719161i	$0.744527\pi$
$350$	0	0
$351$	1.41421	0.0754851
$352$	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
$355$	0	0
$356$	0	0
$357$	8.48528i	0.449089i
$358$	0	0
$359$	−25.4558	−1.34351	−0.671754	−	0.740774i	$-0.734459\pi$
$359$	−25.4558	−1.34351	−0.671754	+	0.740774i	$0.734459\pi$
$360$	0	0
$361$	−17.0000	−0.894737
$362$	0	0
$363$	− 11.0000i	− 0.577350i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−32.5269	−1.69789	−0.848945	−	0.528480i	$-0.822762\pi$
$367$	−32.5269	−1.69789	−0.848945	+	0.528480i	$0.822762\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	12.0000i	0.623009i
$372$	0	0
$373$	7.07107i	0.366126i	0.983101	+	0.183063i	$0.0586012\pi$
$373$	7.07107i	0.366126i	−0.983101	+	0.183063i	$0.941399\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	− 30.0000i	− 1.54100i	−0.637442	−	0.770498i	$-0.720007\pi$
$379$	− 30.0000i	− 1.54100i	0.637442	−	0.770498i	$-0.279993\pi$
$380$	0	0
$381$	9.89949i	0.507166i
$382$	0	0
$383$	−33.9411	−1.73431	−0.867155	−	0.498038i	$-0.834054\pi$
$383$	−33.9411	−1.73431	−0.867155	+	0.498038i	$0.834054\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.00000i	0.304997i
$388$	0	0
$389$	16.9706i	0.860442i	0.902724	+	0.430221i	$0.141564\pi$
$389$	16.9706i	0.860442i	−0.902724	+	0.430221i	$0.858436\pi$
$390$	0	0
$391$	50.9117	2.57471
$392$	0	0
$393$	12.0000	0.605320
$394$	0	0
$395$	0	0
$396$	0	0
$397$	9.89949i	0.496841i	0.968652	+	0.248421i	$0.0799115\pi$
$397$	9.89949i	0.496841i	−0.968652	+	0.248421i	$0.920088\pi$
$398$	0	0
$399$	8.48528	0.424795
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	− 2.00000i	− 0.0996271i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−12.0000	−0.593362	−0.296681	−	0.954977i	$-0.595880\pi$
$409$	−12.0000	−0.593362	−0.296681	+	0.954977i	$0.595880\pi$
$410$	0	0
$411$	− 18.0000i	− 0.887875i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.00000	0.195881
$418$	0	0
$419$	− 36.0000i	− 1.75872i	−0.476162	−	0.879358i	$-0.657972\pi$
$419$	− 36.0000i	− 1.75872i	0.476162	−	0.879358i	$-0.342028\pi$
$420$	0	0
$421$	− 15.5563i	− 0.758170i	−0.925362	−	0.379085i	$-0.876239\pi$
$421$	− 15.5563i	− 0.758170i	0.925362	−	0.379085i	$-0.123761\pi$
$422$	0	0
$423$	8.48528	0.412568
$424$	0	0
$425$	−30.0000	−1.45521
$426$	0	0
$427$	10.0000i	0.483934i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−8.48528	−0.408722	−0.204361	−	0.978896i	$-0.565512\pi$
$431$	−8.48528	−0.408722	−0.204361	+	0.978896i	$0.565512\pi$
$432$	0	0
$433$	−24.0000	−1.15337	−0.576683	−	0.816968i	$-0.695653\pi$
$433$	−24.0000	−1.15337	−0.576683	+	0.816968i	$0.695653\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 50.9117i	− 2.43544i
$438$	0	0
$439$	−7.07107	−0.337484	−0.168742	−	0.985660i	$-0.553970\pi$
$439$	−7.07107	−0.337484	−0.168742	+	0.985660i	$0.553970\pi$
$440$	0	0
$441$	5.00000	0.238095
$442$	0	0
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	16.9706	0.802680
$448$	0	0
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	− 18.3848i	− 0.863792i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	12.0000	0.561336	0.280668	−	0.959805i	$-0.409444\pi$
$457$	12.0000	0.561336	0.280668	+	0.959805i	$0.409444\pi$
$458$	0	0
$459$	− 6.00000i	− 0.280056i
$460$	0	0
$461$	− 33.9411i	− 1.58080i	−0.612594	−	0.790398i	$-0.709874\pi$
$461$	− 33.9411i	− 1.58080i	0.612594	−	0.790398i	$-0.290126\pi$
$462$	0	0
$463$	7.07107	0.328620	0.164310	−	0.986409i	$-0.447460\pi$
$463$	7.07107	0.328620	0.164310	+	0.986409i	$0.447460\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.0000i	0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$	12.0000i	0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$	0	0
$469$	− 5.65685i	− 0.261209i
$470$	0	0
$471$	−7.07107	−0.325818
$472$	0	0
$473$	0	0
$474$	0	0
$475$	30.0000i	1.37649i
$476$	0	0
$477$	− 8.48528i	− 0.388514i
$478$	0	0
$479$	8.48528	0.387702	0.193851	−	0.981031i	$-0.437902\pi$
$479$	8.48528	0.387702	0.193851	+	0.981031i	$0.437902\pi$
$480$	0	0
$481$	10.0000	0.455961
$482$	0	0
$483$	12.0000i	0.546019i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−18.3848	−0.833094	−0.416547	−	0.909114i	$-0.636760\pi$
$487$	−18.3848	−0.833094	−0.416547	+	0.909114i	$0.636760\pi$
$488$	0	0
$489$	6.00000	0.271329
$490$	0	0
$491$	− 24.0000i	− 1.08310i	−0.840667	−	0.541552i	$-0.817837\pi$
$491$	− 24.0000i	− 1.08310i	0.840667	−	0.541552i	$-0.182163\pi$
$492$	0	0
$493$	− 50.9117i	− 2.29295i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12.0000	0.538274
$498$	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
$501$	16.9706i	0.758189i
$502$	0	0
$503$	−8.48528	−0.378340	−0.189170	−	0.981944i	$-0.560580\pi$
$503$	−8.48528	−0.378340	−0.189170	+	0.981944i	$0.560580\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 11.0000i	− 0.488527i
$508$	0	0
$509$	− 25.4558i	− 1.12831i	−0.825669	−	0.564155i	$-0.809202\pi$
$509$	− 25.4558i	− 1.12831i	0.825669	−	0.564155i	$-0.190798\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−6.00000	−0.264906
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	16.9706	0.744925
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	0	0
$523$	18.0000i	0.787085i	0.919306	+	0.393543i	$0.128751\pi$
$523$	18.0000i	0.787085i	−0.919306	+	0.393543i	$0.871249\pi$
$524$	0	0
$525$	− 7.07107i	− 0.308607i
$526$	0	0
$527$	−8.48528	−0.369625
$528$	0	0
$529$	49.0000	2.13043
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 8.48528i	− 0.367538i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−24.0000	−1.03568
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 35.3553i	− 1.52004i	−0.649897	−	0.760022i	$-0.725188\pi$
$541$	− 35.3553i	− 1.52004i	0.649897	−	0.760022i	$-0.274812\pi$
$542$	0	0
$543$	15.5563	0.667587
$544$	0	0
$545$	0	0
$546$	0	0
$547$	18.0000i	0.769624i	0.922995	+	0.384812i	$0.125734\pi$
$547$	18.0000i	0.769624i	−0.922995	+	0.384812i	$0.874266\pi$
$548$	0	0
$549$	− 7.07107i	− 0.301786i
$550$	0	0
$551$	−50.9117	−2.16891
$552$	0	0
$553$	2.00000	0.0850487
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16.9706i	0.719066i	0.933132	+	0.359533i	$0.117064\pi$
$557$	16.9706i	0.719066i	−0.933132	+	0.359533i	$0.882936\pi$
$558$	0	0
$559$	−8.48528	−0.358889
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.41421	0.0593914
$568$	0	0
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	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
$573$	16.9706i	0.708955i
$574$	0	0
$575$	−42.4264	−1.76930
$576$	0	0
$577$	36.0000	1.49870	0.749350	−	0.662174i	$-0.230366\pi$
$577$	36.0000	1.49870	0.749350	+	0.662174i	$0.230366\pi$
$578$	0	0
$579$	14.0000i	0.581820i
$580$	0	0
$581$	− 16.9706i	− 0.704058i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	8.48528i	0.349630i
$590$	0	0
$591$	8.48528	0.349038
$592$	0	0
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 9.89949i	− 0.405159i
$598$	0	0
$599$	25.4558	1.04010	0.520049	−	0.854137i	$-0.325914\pi$
$599$	25.4558	1.04010	0.520049	+	0.854137i	$0.325914\pi$
$600$	0	0
$601$	−12.0000	−0.489490	−0.244745	−	0.969587i	$-0.578704\pi$
$601$	−12.0000	−0.489490	−0.244745	+	0.969587i	$0.578704\pi$
$602$	0	0
$603$	4.00000i	0.162893i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	24.0416	0.975820	0.487910	−	0.872894i	$-0.337759\pi$
$607$	24.0416	0.975820	0.487910	+	0.872894i	$0.337759\pi$
$608$	0	0
$609$	12.0000	0.486265
$610$	0	0
$611$	12.0000i	0.485468i
$612$	0	0
$613$	− 26.8701i	− 1.08527i	−0.839968	−	0.542636i	$-0.817426\pi$
$613$	− 26.8701i	− 1.08527i	0.839968	−	0.542636i	$-0.182574\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	− 28.0000i	− 1.12542i	−0.826656	−	0.562708i	$-0.809760\pi$
$619$	− 28.0000i	− 1.12542i	0.826656	−	0.562708i	$-0.190240\pi$
$620$	0	0
$621$	− 8.48528i	− 0.340503i
$622$	0	0
$623$	−8.48528	−0.339956
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 42.4264i	− 1.69165i
$630$	0	0
$631$	−18.3848	−0.731886	−0.365943	−	0.930637i	$-0.619254\pi$
$631$	−18.3848	−0.731886	−0.365943	+	0.930637i	$0.619254\pi$
$632$	0	0
$633$	−4.00000	−0.158986
$634$	0	0
$635$	0	0
$636$	0	0
$637$	7.07107i	0.280166i
$638$	0	0
$639$	−8.48528	−0.335673
$640$	0	0
$641$	−42.0000	−1.65890	−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000	−1.65890	−0.829450	+	0.558581i	$0.811346\pi$
$642$	0	0
$643$	18.0000i	0.709851i	0.934895	+	0.354925i	$0.115494\pi$
$643$	18.0000i	0.709851i	−0.934895	+	0.354925i	$0.884506\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−8.48528	−0.333591	−0.166795	−	0.985992i	$-0.553342\pi$
$647$	−8.48528	−0.333591	−0.166795	+	0.985992i	$0.553342\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	− 2.00000i	− 0.0783862i
$652$	0	0
$653$	16.9706i	0.664109i	0.943260	+	0.332055i	$0.107742\pi$
$653$	16.9706i	0.664109i	−0.943260	+	0.332055i	$0.892258\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	9.89949i	0.385046i	0.981292	+	0.192523i	$0.0616670\pi$
$661$	9.89949i	0.385046i	−0.981292	+	0.192523i	$0.938333\pi$
$662$	0	0
$663$	8.48528	0.329541
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 72.0000i	− 2.78785i
$668$	0	0
$669$	− 26.8701i	− 1.03886i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−24.0000	−0.925132	−0.462566	−	0.886585i	$-0.653071\pi$
$673$	−24.0000	−0.925132	−0.462566	+	0.886585i	$0.653071\pi$
$674$	0	0
$675$	5.00000i	0.192450i
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	−16.9706	−0.651270
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	36.0000i	1.37750i	0.724998	+	0.688751i	$0.241841\pi$
$683$	36.0000i	1.37750i	−0.724998	+	0.688751i	$0.758159\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.41421	0.0539556
$688$	0	0
$689$	12.0000	0.457164
$690$	0	0
$691$	− 42.0000i	− 1.59776i	−0.601494	−	0.798878i	$-0.705427\pi$
$691$	− 42.0000i	− 1.59776i	0.601494	−	0.798878i	$-0.294573\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−36.0000	−1.36360
$698$	0	0
$699$	− 6.00000i	− 0.226941i
$700$	0	0
$701$	− 25.4558i	− 0.961454i	−0.876870	−	0.480727i	$-0.840373\pi$
$701$	− 25.4558i	− 0.961454i	0.876870	−	0.480727i	$-0.159627\pi$
$702$	0	0
$703$	−42.4264	−1.60014
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 12.0000i	− 0.451306i
$708$	0	0
$709$	− 1.41421i	− 0.0531119i	−0.999647	−	0.0265560i	$-0.991546\pi$
$709$	− 1.41421i	− 0.0531119i	0.999647	−	0.0265560i	$-0.00845402\pi$
$710$	0	0
$711$	−1.41421	−0.0530372
$712$	0	0
$713$	−12.0000	−0.449404
$714$	0	0
$715$	0	0
$716$	0	0
$717$	16.9706i	0.633777i
$718$	0	0
$719$	8.48528	0.316448	0.158224	−	0.987403i	$-0.449423\pi$
$719$	8.48528	0.316448	0.158224	+	0.987403i	$0.449423\pi$
$720$	0	0
$721$	−14.0000	−0.521387
$722$	0	0
$723$	− 10.0000i	− 0.371904i
$724$	0	0
$725$	42.4264i	1.57568i
$726$	0	0
$727$	32.5269	1.20636	0.603178	−	0.797606i	$-0.293901\pi$
$727$	32.5269	1.20636	0.603178	+	0.797606i	$0.293901\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	36.0000i	1.33151i
$732$	0	0
$733$	− 49.4975i	− 1.82823i	−0.405454	−	0.914115i	$-0.632887\pi$
$733$	− 49.4975i	− 1.82823i	0.405454	−	0.914115i	$-0.367113\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	20.0000i	0.735712i	0.929883	+	0.367856i	$0.119908\pi$
$739$	20.0000i	0.735712i	−0.929883	+	0.367856i	$0.880092\pi$
$740$	0	0
$741$	− 8.48528i	− 0.311715i
$742$	0	0
$743$	−16.9706	−0.622590	−0.311295	−	0.950313i	$-0.600763\pi$
$743$	−16.9706	−0.622590	−0.311295	+	0.950313i	$0.600763\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12.0000i	0.439057i
$748$	0	0
$749$	16.9706i	0.620091i
$750$	0	0
$751$	49.4975	1.80619	0.903094	−	0.429442i	$-0.141290\pi$
$751$	49.4975	1.80619	0.903094	+	0.429442i	$0.141290\pi$
$752$	0	0
$753$	24.0000	0.874609
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 1.41421i	− 0.0514005i	−0.999670	−	0.0257002i	$-0.991818\pi$
$757$	− 1.41421i	− 0.0514005i	0.999670	−	0.0257002i	$-0.00818154\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	0	0
$763$	26.0000i	0.941263i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	6.00000i	0.216085i
$772$	0	0
$773$	− 25.4558i	− 0.915583i	−0.889060	−	0.457792i	$-0.848641\pi$
$773$	− 25.4558i	− 0.915583i	0.889060	−	0.457792i	$-0.151359\pi$
$774$	0	0
$775$	7.07107	0.254000
$776$	0	0
$777$	10.0000	0.358748
$778$	0	0
$779$	36.0000i	1.28983i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−8.48528	−0.303239
$784$	0	0
$785$	0	0
$786$	0	0
$787$	18.0000i	0.641631i	0.947142	+	0.320815i	$0.103957\pi$
$787$	18.0000i	0.641631i	−0.947142	+	0.320815i	$0.896043\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−8.48528	−0.301702
$792$	0	0
$793$	10.0000	0.355110
$794$	0	0
$795$	0	0
$796$	0	0
$797$	16.9706i	0.601128i	0.953762	+	0.300564i	$0.0971749\pi$
$797$	16.9706i	0.601128i	−0.953762	+	0.300564i	$0.902825\pi$
$798$	0	0
$799$	50.9117	1.80113
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−8.48528	−0.298696
$808$	0	0
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	0	0
$811$	6.00000i	0.210688i	0.994436	+	0.105344i	$0.0335944\pi$
$811$	6.00000i	0.210688i	−0.994436	+	0.105344i	$0.966406\pi$
$812$	0	0
$813$	− 26.8701i	− 0.942374i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	36.0000	1.25948
$818$	0	0
$819$	2.00000i	0.0698857i
$820$	0	0
$821$	− 8.48528i	− 0.296138i	−0.988977	−	0.148069i	$-0.952694\pi$
$821$	− 8.48528i	− 0.296138i	0.988977	−	0.148069i	$-0.0473058\pi$
$822$	0	0
$823$	32.5269	1.13382	0.566908	−	0.823781i	$-0.308139\pi$
$823$	32.5269	1.13382	0.566908	+	0.823781i	$0.308139\pi$
$824$	0	0
$825$	0	0
$826$	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
$829$	− 15.5563i	− 0.540294i	−0.962819	−	0.270147i	$-0.912928\pi$
$829$	− 15.5563i	− 0.540294i	0.962819	−	0.270147i	$-0.0870724\pi$
$830$	0	0
$831$	−15.5563	−0.539644
$832$	0	0
$833$	30.0000	1.03944
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.41421i	0.0488824i
$838$	0	0
$839$	25.4558	0.878833	0.439417	−	0.898283i	$-0.355185\pi$
$839$	25.4558	0.878833	0.439417	+	0.898283i	$0.355185\pi$
$840$	0	0
$841$	−43.0000	−1.48276
$842$	0	0
$843$	− 18.0000i	− 0.619953i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	15.5563	0.534522
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	− 60.0000i	− 2.05677i
$852$	0	0
$853$	− 43.8406i	− 1.50107i	−0.660828	−	0.750537i	$-0.729795\pi$
$853$	− 43.8406i	− 1.50107i	0.660828	−	0.750537i	$-0.270205\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	0	0
$859$	6.00000i	0.204717i	0.994748	+	0.102359i	$0.0326389\pi$
$859$	6.00000i	0.204717i	−0.994748	+	0.102359i	$0.967361\pi$
$860$	0	0
$861$	− 8.48528i	− 0.289178i
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 19.0000i	− 0.645274i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−5.65685	−0.191675
$872$	0	0
$873$	12.0000	0.406138
$874$	0	0
$875$	0	0
$876$	0	0
$877$	7.07107i	0.238773i	0.992848	+	0.119386i	$0.0380928\pi$
$877$	7.07107i	0.238773i	−0.992848	+	0.119386i	$0.961907\pi$
$878$	0	0
$879$	8.48528	0.286201
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	30.0000i	1.00958i	0.863242	+	0.504790i	$0.168430\pi$
$883$	30.0000i	1.00958i	−0.863242	+	0.504790i	$0.831570\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	16.9706	0.569816	0.284908	−	0.958555i	$-0.408037\pi$
$887$	16.9706	0.569816	0.284908	+	0.958555i	$0.408037\pi$
$888$	0	0
$889$	−14.0000	−0.469545
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 50.9117i	− 1.70369i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	12.0000	0.400668
$898$	0	0
$899$	12.0000i	0.400222i
$900$	0	0
$901$	− 50.9117i	− 1.69611i
$902$	0	0
$903$	−8.48528	−0.282372
$904$	0	0
$905$	0	0
$906$	0	0
$907$	6.00000i	0.199227i	0.995026	+	0.0996134i	$0.0317606\pi$
$907$	6.00000i	0.199227i	−0.995026	+	0.0996134i	$0.968239\pi$
$908$	0	0
$909$	8.48528i	0.281439i
$910$	0	0
$911$	16.9706	0.562260	0.281130	−	0.959670i	$-0.409291\pi$
$911$	16.9706	0.562260	0.281130	+	0.959670i	$0.409291\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16.9706i	0.560417i
$918$	0	0
$919$	−1.41421	−0.0466506	−0.0233253	−	0.999728i	$-0.507425\pi$
$919$	−1.41421	−0.0466506	−0.0233253	+	0.999728i	$0.507425\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	0	0
$923$	− 12.0000i	− 0.394985i
$924$	0	0
$925$	35.3553i	1.16248i
$926$	0	0
$927$	9.89949	0.325142
$928$	0	0
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	− 30.0000i	− 0.983210i
$932$	0	0
$933$	16.9706i	0.555591i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	0	0
$939$	10.0000i	0.326338i
$940$	0	0
$941$	− 50.9117i	− 1.65967i	−0.558006	−	0.829837i	$-0.688433\pi$
$941$	− 50.9117i	− 1.65967i	0.558006	−	0.829837i	$-0.311567\pi$
$942$	0	0
$943$	−50.9117	−1.65791
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 24.0000i	− 0.779895i	−0.920837	−	0.389948i	$-0.872493\pi$
$947$	− 24.0000i	− 0.779895i	0.920837	−	0.389948i	$-0.127507\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−8.48528	−0.275154
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	25.4558	0.822012
$960$	0	0
$961$	−29.0000	−0.935484
$962$	0	0
$963$	− 12.0000i	− 0.386695i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	41.0122	1.31886	0.659432	−	0.751765i	$-0.270797\pi$
$967$	41.0122	1.31886	0.659432	+	0.751765i	$0.270797\pi$
$968$	0	0
$969$	−36.0000	−1.15649
$970$	0	0
$971$	− 60.0000i	− 1.92549i	−0.270408	−	0.962746i	$-0.587159\pi$
$971$	− 60.0000i	− 1.92549i	0.270408	−	0.962746i	$-0.412841\pi$
$972$	0	0
$973$	5.65685i	0.181350i
$974$	0	0
$975$	−7.07107	−0.226455
$976$	0	0
$977$	6.00000	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	− 18.3848i	− 0.586981i
$982$	0	0
$983$	−33.9411	−1.08255	−0.541277	−	0.840844i	$-0.682059\pi$
$983$	−33.9411	−1.08255	−0.541277	+	0.840844i	$0.682059\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	12.0000i	0.381964i
$988$	0	0
$989$	50.9117i	1.61890i
$990$	0	0
$991$	−52.3259	−1.66219	−0.831094	−	0.556133i	$-0.812285\pi$
$991$	−52.3259	−1.66219	−0.831094	+	0.556133i	$0.812285\pi$
$992$	0	0
$993$	−28.0000	−0.888553
$994$	0	0
$995$	0	0
$996$	0	0
$997$	57.9828i	1.83633i	0.396196	+	0.918166i	$0.370330\pi$
$997$	57.9828i	1.83633i	−0.396196	+	0.918166i	$0.629670\pi$
$998$	0	0
$999$	−7.07107	−0.223719

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3072.2.d.a.1537.2		4
4.3	odd	2	inner	3072.2.d.a.1537.3		4
8.3	odd	2	inner	3072.2.d.a.1537.1		4
8.5	even	2	inner	3072.2.d.a.1537.4		4
16.3	odd	4		3072.2.a.a.1.2		2
16.5	even	4		3072.2.a.a.1.1		2
16.11	odd	4		3072.2.a.g.1.2		2
16.13	even	4		3072.2.a.g.1.1		2
32.3	odd	8		768.2.j.b.193.1	✓	4
32.5	even	8		768.2.j.b.577.2	yes	4
32.11	odd	8		768.2.j.c.577.2	yes	4
32.13	even	8		768.2.j.c.193.1	yes	4
32.19	odd	8		768.2.j.c.193.2	yes	4
32.21	even	8		768.2.j.c.577.1	yes	4
32.27	odd	8		768.2.j.b.577.1	yes	4
32.29	even	8		768.2.j.b.193.2	yes	4
48.5	odd	4		9216.2.a.o.1.1		2
48.11	even	4		9216.2.a.n.1.2		2
48.29	odd	4		9216.2.a.n.1.1		2
48.35	even	4		9216.2.a.o.1.2		2
96.5	odd	8		2304.2.k.b.577.2		4
96.11	even	8		2304.2.k.c.577.1		4
96.29	odd	8		2304.2.k.b.1729.1		4
96.35	even	8		2304.2.k.b.1729.2		4
96.53	odd	8		2304.2.k.c.577.2		4
96.59	even	8		2304.2.k.b.577.1		4
96.77	odd	8		2304.2.k.c.1729.1		4
96.83	even	8		2304.2.k.c.1729.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
768.2.j.b.193.1	✓	4	32.3	odd	8
768.2.j.b.193.2	yes	4	32.29	even	8
768.2.j.b.577.1	yes	4	32.27	odd	8
768.2.j.b.577.2	yes	4	32.5	even	8
768.2.j.c.193.1	yes	4	32.13	even	8
768.2.j.c.193.2	yes	4	32.19	odd	8
768.2.j.c.577.1	yes	4	32.21	even	8
768.2.j.c.577.2	yes	4	32.11	odd	8
2304.2.k.b.577.1		4	96.59	even	8
2304.2.k.b.577.2		4	96.5	odd	8
2304.2.k.b.1729.1		4	96.29	odd	8
2304.2.k.b.1729.2		4	96.35	even	8
2304.2.k.c.577.1		4	96.11	even	8
2304.2.k.c.577.2		4	96.53	odd	8
2304.2.k.c.1729.1		4	96.77	odd	8
2304.2.k.c.1729.2		4	96.83	even	8
3072.2.a.a.1.1		2	16.5	even	4
3072.2.a.a.1.2		2	16.3	odd	4
3072.2.a.g.1.1		2	16.13	even	4
3072.2.a.g.1.2		2	16.11	odd	4
3072.2.d.a.1537.1		4	8.3	odd	2	inner
3072.2.d.a.1537.2		4	1.1	even	1	trivial
3072.2.d.a.1537.3		4	4.3	odd	2	inner
3072.2.d.a.1537.4		4	8.5	even	2	inner
9216.2.a.n.1.1		2	48.29	odd	4
9216.2.a.n.1.2		2	48.11	even	4
9216.2.a.o.1.1		2	48.5	odd	4
9216.2.a.o.1.2		2	48.35	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3072 = 2^{10} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3072.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +1.41421 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +1.41421 q^{7} -1.00000 q^{9} -1.41421i q^{13} -6.00000 q^{17} +6.00000i q^{19} -1.41421i q^{21} -8.48528 q^{23} +5.00000 q^{25} +1.00000i q^{27} +8.48528i q^{29} +1.41421 q^{31} +7.07107i q^{37} -1.41421 q^{39} +6.00000 q^{41} -6.00000i q^{43} -8.48528 q^{47} -5.00000 q^{49} +6.00000i q^{51} +8.48528i q^{53} +6.00000 q^{57} +7.07107i q^{61} -1.41421 q^{63} -4.00000i q^{67} +8.48528i q^{69} +8.48528 q^{71} -5.00000i q^{75} +1.41421 q^{79} +1.00000 q^{81} -12.0000i q^{83} +8.48528 q^{87} -6.00000 q^{89} -2.00000i q^{91} -1.41421i q^{93} -12.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} - 24 q^{17} + 20 q^{25} + 24 q^{41} - 20 q^{49} + 24 q^{57} + 4 q^{81} - 24 q^{89} - 48 q^{97}+O(q^{100}) \) 4 * q - 4 * q^9 - 24 * q^17 + 20 * q^25 + 24 * q^41 - 20 * q^49 + 24 * q^57 + 4 * q^81 - 24 * q^89 - 48 * q^97

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