LMFDB - Embedded newform 2736.2.d.a.2015.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(2015,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, 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("2736.2015");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.d (of order $2$, degree $1$, 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:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 12x^{10} + 47x^{8} - 44x^{6} + 81x^{4} + 848x^{2} + 1600 $ x^12 - 12x^10 + 47x^8 - 44x^6 + 81x^4 + 848*x^2 + 1600
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2015.5
Root			$-2.57272 + 0.525570i$ of defining polynomial
Character	$\chi$	$=$	2736.2015
Dual form			2736.2.d.a.2015.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.363074i q^{5} -4.38164i q^{7} +O(q^{10})$	$q-0.363074i q^{5} -4.38164i q^{7} +5.87159 q^{11} +5.79021 q^{13} +6.55965i q^{17} +1.00000i q^{19} -1.26587 q^{23} +4.86818 q^{25} -5.50851i q^{29} +1.79021i q^{31} -1.59086 q^{35} -6.76328 q^{37} +7.61079i q^{41} +3.92204i q^{43} +9.53642 q^{47} -12.1988 q^{49} +4.13238i q^{53} -2.13182i q^{55} +12.3932 q^{59} -0.895107 q^{61} -2.10228i q^{65} -6.97307i q^{67} -2.53174 q^{71} -8.38164 q^{73} -25.7272i q^{77} -16.6074i q^{79} -1.26587 q^{83} +2.38164 q^{85} +8.04026i q^{89} -25.3706i q^{91} +0.363074 q^{95} +6.81714 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q + 4 q^{25} + 8 q^{37} - 52 q^{49} + 24 q^{61} - 56 q^{73} - 16 q^{85} + 32 q^{97}+O(q^{100}) $ 12 * q + 4 * q^25 + 8 * q^37 - 52 * q^49 + 24 * q^61 - 56 * q^73 - 16 * q^85 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$5$	− 0.363074i	− 0.162372i	−0.996699	−	0.0811859i	$-0.974129\pi$
$5$	− 0.363074i	− 0.162372i	0.996699	−	0.0811859i	$-0.0258708\pi$
$6$	0	0
$7$	− 4.38164i	− 1.65610i	−0.560651	−	0.828052i	$-0.689449\pi$
$7$	− 4.38164i	− 1.65610i	0.560651	−	0.828052i	$-0.310551\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.87159	1.77035	0.885175	−	0.465258i	$-0.154039\pi$
$11$	5.87159	1.77035	0.885175	+	0.465258i	$0.154039\pi$
$12$	0	0
$13$	5.79021	1.60592	0.802958	−	0.596036i	$-0.203258\pi$
$13$	5.79021	1.60592	0.802958	+	0.596036i	$0.203258\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.55965i	1.59095i	0.605987	+	0.795475i	$0.292778\pi$
$17$	6.55965i	1.59095i	−0.605987	+	0.795475i	$0.707222\pi$
$18$	0	0
$19$	1.00000i	0.229416i
$19$	1.00000i	0.229416i
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.26587	−0.263953	−0.131976	−	0.991253i	$-0.542132\pi$
$23$	−1.26587	−0.263953	−0.131976	+	0.991253i	$0.542132\pi$
$24$	0	0
$25$	4.86818	0.973635
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 5.50851i	− 1.02291i	−0.859312	−	0.511453i	$-0.829108\pi$
$29$	− 5.50851i	− 1.02291i	0.859312	−	0.511453i	$-0.170892\pi$
$30$	0	0
$31$	1.79021i	0.321532i	0.986993	+	0.160766i	$0.0513964\pi$
$31$	1.79021i	0.321532i	−0.986993	+	0.160766i	$0.948604\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.59086	−0.268905
$36$	0	0
$37$	−6.76328	−1.11188	−0.555938	−	0.831223i	$-0.687641\pi$
$37$	−6.76328	−1.11188	−0.555938	+	0.831223i	$0.687641\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.61079i	1.18861i	0.804241	+	0.594303i	$0.202572\pi$
$41$	7.61079i	1.18861i	−0.804241	+	0.594303i	$0.797428\pi$
$42$	0	0
$43$	3.92204i	0.598105i	0.954237	+	0.299052i	$0.0966706\pi$
$43$	3.92204i	0.598105i	−0.954237	+	0.299052i	$0.903329\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.53642	1.39103	0.695515	−	0.718512i	$-0.255176\pi$
$47$	9.53642	1.39103	0.695515	+	0.718512i	$0.255176\pi$
$48$	0	0
$49$	−12.1988	−1.74268
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.13238i	0.567626i	0.958880	+	0.283813i	$0.0915996\pi$
$53$	4.13238i	0.567626i	−0.958880	+	0.283813i	$0.908400\pi$
$54$	0	0
$55$	− 2.13182i	− 0.287455i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.3932	1.61345	0.806726	−	0.590926i	$-0.201238\pi$
$59$	12.3932	1.61345	0.806726	+	0.590926i	$0.201238\pi$
$60$	0	0
$61$	−0.895107	−0.114607	−0.0573033	−	0.998357i	$-0.518250\pi$
$61$	−0.895107	−0.114607	−0.0573033	+	0.998357i	$0.518250\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 2.10228i	− 0.260756i
$66$	0	0
$67$	− 6.97307i	− 0.851896i	−0.904748	−	0.425948i	$-0.859941\pi$
$67$	− 6.97307i	− 0.851896i	0.904748	−	0.425948i	$-0.140059\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−2.53174	−0.300463	−0.150231	−	0.988651i	$-0.548002\pi$
$71$	−2.53174	−0.300463	−0.150231	+	0.988651i	$0.548002\pi$
$72$	0	0
$73$	−8.38164	−0.980997	−0.490498	−	0.871442i	$-0.663185\pi$
$73$	−8.38164	−0.980997	−0.490498	+	0.871442i	$0.663185\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 25.7272i	− 2.93189i
$78$	0	0
$79$	− 16.6074i	− 1.86847i	−0.356653	−	0.934237i	$-0.616082\pi$
$79$	− 16.6074i	− 1.86847i	0.356653	−	0.934237i	$-0.383918\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1.26587	−0.138947	−0.0694737	−	0.997584i	$-0.522132\pi$
$83$	−1.26587	−0.138947	−0.0694737	+	0.997584i	$0.522132\pi$
$84$	0	0
$85$	2.38164	0.258325
$86$	0	0
$87$	0	0
$88$	0	0
$89$	8.04026i	0.852265i	0.904661	+	0.426133i	$0.140124\pi$
$89$	8.04026i	0.852265i	−0.904661	+	0.426133i	$0.859876\pi$
$90$	0	0
$91$	− 25.3706i	− 2.65957i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0.363074	0.0372507
$96$	0	0
$97$	6.81714	0.692176	0.346088	−	0.938202i	$-0.387510\pi$
$97$	6.81714	0.692176	0.346088	+	0.938202i	$0.387510\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 2.86651i	− 0.285229i	−0.989778	−	0.142614i	$-0.954449\pi$
$101$	− 2.86651i	− 0.285229i	0.989778	−	0.142614i	$-0.0455508\pi$
$102$	0	0
$103$	12.3437i	1.21626i	0.793837	+	0.608131i	$0.208080\pi$
$103$	12.3437i	1.21626i	−0.793837	+	0.608131i	$0.791920\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.7432	1.13526	0.567628	−	0.823285i	$-0.307861\pi$
$107$	11.7432	1.13526	0.567628	+	0.823285i	$0.307861\pi$
$108$	0	0
$109$	−17.5266	−1.67874	−0.839370	−	0.543560i	$-0.817076\pi$
$109$	−17.5266	−1.67874	−0.839370	+	0.543560i	$0.817076\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 10.8687i	− 1.02244i	−0.859450	−	0.511220i	$-0.829194\pi$
$113$	− 10.8687i	− 1.02244i	0.859450	−	0.511220i	$-0.170806\pi$
$114$	0	0
$115$	0.459606i	0.0428584i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	28.7420	2.63478
$120$	0	0
$121$	23.4755	2.13414
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 3.58288i	− 0.320463i
$126$	0	0
$127$	− 14.8171i	− 1.31481i	−0.753538	−	0.657404i	$-0.771654\pi$
$127$	− 14.8171i	− 1.31481i	0.753538	−	0.657404i	$-0.228346\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−14.5999	−1.27560	−0.637800	−	0.770202i	$-0.720155\pi$
$131$	−14.5999	−1.27560	−0.637800	+	0.770202i	$0.720155\pi$
$132$	0	0
$133$	4.38164	0.379937
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 11.8096i	− 1.00896i	−0.863424	−	0.504480i	$-0.831684\pi$
$137$	− 11.8096i	− 1.00896i	0.863424	−	0.504480i	$-0.168316\pi$
$138$	0	0
$139$	4.38164i	0.371646i	0.982583	+	0.185823i	$0.0594951\pi$
$139$	4.38164i	0.371646i	−0.982583	+	0.185823i	$0.940505\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	33.9977	2.84303
$144$	0	0
$145$	−2.00000	−0.166091
$146$	0	0
$147$	0	0
$148$	0	0
$149$	20.8346i	1.70683i	0.521228	+	0.853417i	$0.325474\pi$
$149$	20.8346i	1.70683i	−0.521228	+	0.853417i	$0.674526\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.649981	0.0522077
$156$	0	0
$157$	12.8171	1.02292	0.511460	−	0.859307i	$-0.329105\pi$
$157$	12.8171	1.02292	0.511460	+	0.859307i	$0.329105\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.54660i	0.437133i
$162$	0	0
$163$	9.63429i	0.754615i	0.926088	+	0.377308i	$0.123150\pi$
$163$	9.63429i	0.754615i	−0.926088	+	0.377308i	$0.876850\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.5114	0.813396	0.406698	−	0.913563i	$-0.366680\pi$
$167$	10.5114	0.813396	0.406698	+	0.913563i	$0.366680\pi$
$168$	0	0
$169$	20.5266	1.57897
$170$	0	0
$171$	0	0
$172$	0	0
$173$	6.58796i	0.500873i	0.968133	+	0.250437i	$0.0805742\pi$
$173$	6.58796i	0.500873i	−0.968133	+	0.250437i	$0.919426\pi$
$174$	0	0
$175$	− 21.3306i	− 1.61244i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−0.649981	−0.0485818	−0.0242909	−	0.999705i	$-0.507733\pi$
$179$	−0.649981	−0.0485818	−0.0242909	+	0.999705i	$0.507733\pi$
$180$	0	0
$181$	−7.84407	−0.583045	−0.291523	−	0.956564i	$-0.594162\pi$
$181$	−7.84407	−0.583045	−0.291523	+	0.956564i	$0.594162\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.45558i	0.180537i
$186$	0	0
$187$	38.5156i	2.81654i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−15.2499	−1.10344	−0.551722	−	0.834028i	$-0.686029\pi$
$191$	−15.2499	−1.10344	−0.551722	+	0.834028i	$0.686029\pi$
$192$	0	0
$193$	−24.3976	−1.75618	−0.878088	−	0.478500i	$-0.841181\pi$
$193$	−24.3976	−1.75618	−0.878088	+	0.478500i	$0.841181\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.79034i	0.198804i	0.995047	+	0.0994019i	$0.0316929\pi$
$197$	2.79034i	0.198804i	−0.995047	+	0.0994019i	$0.968307\pi$
$198$	0	0
$199$	− 13.6045i	− 0.964400i	−0.876061	−	0.482200i	$-0.839838\pi$
$199$	− 13.6045i	− 0.964400i	0.876061	−	0.482200i	$-0.160162\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−24.1363	−1.69404
$204$	0	0
$205$	2.76328	0.192996
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.87159i	0.406146i
$210$	0	0
$211$	− 24.4996i	− 1.68662i	−0.537424	−	0.843312i	$-0.680602\pi$
$211$	− 24.4996i	− 1.68662i	0.537424	−	0.843312i	$-0.319398\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.42399	0.0971154
$216$	0	0
$217$	7.84407	0.532490
$218$	0	0
$219$	0	0
$220$	0	0
$221$	37.9818i	2.55493i
$222$	0	0
$223$	− 16.6074i	− 1.11211i	−0.831145	−	0.556055i	$-0.812314\pi$
$223$	− 16.6074i	− 1.11211i	0.831145	−	0.556055i	$-0.187686\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3.18172	−0.211178	−0.105589	−	0.994410i	$-0.533673\pi$
$227$	−3.18172	−0.211178	−0.105589	+	0.994410i	$0.533673\pi$
$228$	0	0
$229$	−7.76611	−0.513199	−0.256599	−	0.966518i	$-0.582602\pi$
$229$	−7.76611	−0.513199	−0.256599	+	0.966518i	$0.582602\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	12.2165i	0.800330i	0.916443	+	0.400165i	$0.131047\pi$
$233$	12.2165i	0.800330i	−0.916443	+	0.400165i	$0.868953\pi$
$234$	0	0
$235$	− 3.46243i	− 0.225864i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	24.4613	1.58227	0.791136	−	0.611641i	$-0.209490\pi$
$239$	24.4613	1.58227	0.791136	+	0.611641i	$0.209490\pi$
$240$	0	0
$241$	2.55350	0.164485	0.0822426	−	0.996612i	$-0.473792\pi$
$241$	2.55350	0.164485	0.0822426	+	0.996612i	$0.473792\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4.42907i	0.282963i
$246$	0	0
$247$	5.79021i	0.368422i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−27.4762	−1.73428	−0.867140	−	0.498064i	$-0.834044\pi$
$251$	−27.4762	−1.73428	−0.867140	+	0.498064i	$0.834044\pi$
$252$	0	0
$253$	−7.43268	−0.467288
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 25.6608i	− 1.60068i	−0.599549	−	0.800338i	$-0.704653\pi$
$257$	− 25.6608i	− 1.60068i	0.599549	−	0.800338i	$-0.295347\pi$
$258$	0	0
$259$	29.6343i	1.84138i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−18.9147	−1.16633	−0.583166	−	0.812353i	$-0.698186\pi$
$263$	−18.9147	−1.16633	−0.583166	+	0.812353i	$0.698186\pi$
$264$	0	0
$265$	1.50036	0.0921665
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 9.41639i	− 0.574127i	−0.957912	−	0.287064i	$-0.907321\pi$
$269$	− 9.41639i	− 0.574127i	0.957912	−	0.287064i	$-0.0926791\pi$
$270$	0	0
$271$	11.4245i	0.693989i	0.937867	+	0.346994i	$0.112798\pi$
$271$	11.4245i	0.693989i	−0.937867	+	0.346994i	$0.887202\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	28.5839	1.72368
$276$	0	0
$277$	−1.97590	−0.118720	−0.0593600	−	0.998237i	$-0.518906\pi$
$277$	−1.97590	−0.118720	−0.0593600	+	0.998237i	$0.518906\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	9.71307i	0.579433i	0.957112	+	0.289717i	$0.0935610\pi$
$281$	9.71307i	0.579433i	−0.957112	+	0.289717i	$0.906439\pi$
$282$	0	0
$283$	− 7.50246i	− 0.445975i	−0.974821	−	0.222988i	$-0.928419\pi$
$283$	− 7.50246i	− 0.445975i	0.974821	−	0.222988i	$-0.0715809\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	33.3478	1.96846
$288$	0	0
$289$	−26.0290	−1.53112
$290$	0	0
$291$	0	0
$292$	0	0
$293$	8.98692i	0.525021i	0.964929	+	0.262511i	$0.0845505\pi$
$293$	8.98692i	0.525021i	−0.964929	+	0.262511i	$0.915449\pi$
$294$	0	0
$295$	− 4.49964i	− 0.261979i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−7.32967	−0.423886
$300$	0	0
$301$	17.1850	0.990524
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0.324990i	0.0186089i
$306$	0	0
$307$	− 17.5266i	− 1.00029i	−0.865941	−	0.500147i	$-0.833279\pi$
$307$	− 17.5266i	− 1.00029i	0.865941	−	0.500147i	$-0.166721\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	24.4613	1.38707	0.693537	−	0.720421i	$-0.256051\pi$
$311$	24.4613	1.38707	0.693537	+	0.720421i	$0.256051\pi$
$312$	0	0
$313$	−8.81714	−0.498374	−0.249187	−	0.968455i	$-0.580163\pi$
$313$	−8.81714	−0.498374	−0.249187	+	0.968455i	$0.580163\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 25.9575i	− 1.45792i	−0.684558	−	0.728959i	$-0.740005\pi$
$317$	− 25.9575i	− 1.45792i	0.684558	−	0.728959i	$-0.259995\pi$
$318$	0	0
$319$	− 32.3437i	− 1.81090i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−6.55965	−0.364989
$324$	0	0
$325$	28.1878	1.56358
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 41.7852i	− 2.30369i
$330$	0	0
$331$	9.44650i	0.519227i	0.965713	+	0.259613i	$0.0835951\pi$
$331$	9.44650i	0.519227i	−0.965713	+	0.259613i	$0.916405\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2.53174	−0.138324
$336$	0	0
$337$	12.4514	0.678272	0.339136	−	0.940737i	$-0.389865\pi$
$337$	12.4514	0.678272	0.339136	+	0.940737i	$0.389865\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	10.5114i	0.569224i
$342$	0	0
$343$	22.7792i	1.22996i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	14.5999	0.783764	0.391882	−	0.920015i	$-0.371824\pi$
$347$	14.5999	0.783764	0.391882	+	0.920015i	$0.371824\pi$
$348$	0	0
$349$	−20.6715	−1.10652	−0.553260	−	0.833009i	$-0.686616\pi$
$349$	−20.6715	−1.10652	−0.553260	+	0.833009i	$0.686616\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 16.5596i	− 0.881380i	−0.897659	−	0.440690i	$-0.854734\pi$
$353$	− 16.5596i	− 0.881380i	0.897659	−	0.440690i	$-0.145266\pi$
$354$	0	0
$355$	0.919211i	0.0487867i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−20.0478	−1.05808	−0.529042	−	0.848596i	$-0.677448\pi$
$359$	−20.0478	−1.05808	−0.529042	+	0.848596i	$0.677448\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.04316i	0.159286i
$366$	0	0
$367$	25.3706i	1.32434i	0.749355	+	0.662168i	$0.230364\pi$
$367$	25.3706i	1.32434i	−0.749355	+	0.662168i	$0.769636\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	18.1066	0.940049
$372$	0	0
$373$	−7.79021	−0.403362	−0.201681	−	0.979451i	$-0.564640\pi$
$373$	−7.79021	−0.403362	−0.201681	+	0.979451i	$0.564640\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 31.8955i	− 1.64270i
$378$	0	0
$379$	26.2899i	1.35042i	0.737626	+	0.675209i	$0.235947\pi$
$379$	26.2899i	1.35042i	−0.737626	+	0.675209i	$0.764053\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−29.1998	−1.49204	−0.746020	−	0.665923i	$-0.768038\pi$
$383$	−29.1998	−1.49204	−0.746020	+	0.665923i	$0.768038\pi$
$384$	0	0
$385$	−9.34088	−0.476056
$386$	0	0
$387$	0	0
$388$	0	0
$389$	19.3257i	0.979850i	0.871765	+	0.489925i	$0.162976\pi$
$389$	19.3257i	0.979850i	−0.871765	+	0.489925i	$0.837024\pi$
$390$	0	0
$391$	− 8.30368i	− 0.419935i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−6.02971	−0.303387
$396$	0	0
$397$	−3.90821	−0.196147	−0.0980737	−	0.995179i	$-0.531268\pi$
$397$	−3.90821	−0.196147	−0.0980737	+	0.995179i	$0.531268\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.33007i	0.166296i	0.996537	+	0.0831478i	$0.0264974\pi$
$401$	3.33007i	0.166296i	−0.996537	+	0.0831478i	$0.973503\pi$
$402$	0	0
$403$	10.3657i	0.516353i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−39.7112	−1.96841
$408$	0	0
$409$	5.37064	0.265561	0.132781	−	0.991145i	$-0.457609\pi$
$409$	5.37064	0.265561	0.132781	+	0.991145i	$0.457609\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 54.3024i	− 2.67204i
$414$	0	0
$415$	0.459606i	0.0225612i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−32.0819	−1.56730	−0.783651	−	0.621201i	$-0.786645\pi$
$419$	−32.0819	−1.56730	−0.783651	+	0.621201i	$0.786645\pi$
$420$	0	0
$421$	15.8441	0.772193	0.386096	−	0.922458i	$-0.373823\pi$
$421$	15.8441	0.772193	0.386096	+	0.922458i	$0.373823\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	31.9335i	1.54900i
$426$	0	0
$427$	3.92204i	0.189801i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.88176	−0.0906413	−0.0453207	−	0.998972i	$-0.514431\pi$
$431$	−1.88176	−0.0906413	−0.0453207	+	0.998972i	$0.514431\pi$
$432$	0	0
$433$	24.4458	1.17479	0.587395	−	0.809301i	$-0.300154\pi$
$433$	24.4458	1.17479	0.587395	+	0.809301i	$0.300154\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1.26587i	− 0.0605549i
$438$	0	0
$439$	4.26365i	0.203493i	0.994810	+	0.101746i	$0.0324430\pi$
$439$	4.26365i	0.203493i	−0.994810	+	0.101746i	$0.967557\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−27.9593	−1.32839	−0.664193	−	0.747561i	$-0.731225\pi$
$443$	−27.9593	−1.32839	−0.664193	+	0.747561i	$0.731225\pi$
$444$	0	0
$445$	2.91921	0.138384
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 25.3641i	− 1.19701i	−0.801120	−	0.598503i	$-0.795762\pi$
$449$	− 25.3641i	− 1.19701i	0.801120	−	0.598503i	$-0.204238\pi$
$450$	0	0
$451$	44.6874i	2.10425i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−9.21143	−0.431839
$456$	0	0
$457$	−18.4355	−0.862376	−0.431188	−	0.902262i	$-0.641905\pi$
$457$	−18.4355	−0.862376	−0.431188	+	0.902262i	$0.641905\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 9.16756i	− 0.426976i	−0.976946	−	0.213488i	$-0.931518\pi$
$461$	− 9.16756i	− 0.426976i	0.976946	−	0.213488i	$-0.0684824\pi$
$462$	0	0
$463$	− 10.0241i	− 0.465860i	−0.972494	−	0.232930i	$-0.925169\pi$
$463$	− 10.0241i	− 0.465860i	0.972494	−	0.232930i	$-0.0748312\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−20.7965	−0.962347	−0.481173	−	0.876626i	$-0.659789\pi$
$467$	−20.7965	−0.962347	−0.481173	+	0.876626i	$0.659789\pi$
$468$	0	0
$469$	−30.5535	−1.41083
$470$	0	0
$471$	0	0
$472$	0	0
$473$	23.0286i	1.05886i
$474$	0	0
$475$	4.86818i	0.223367i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	13.6590	0.624097	0.312048	−	0.950066i	$-0.398985\pi$
$479$	13.6590	0.624097	0.312048	+	0.950066i	$0.398985\pi$
$480$	0	0
$481$	−39.1609	−1.78558
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 2.47513i	− 0.112390i
$486$	0	0
$487$	− 22.6612i	− 1.02688i	−0.858126	−	0.513439i	$-0.828371\pi$
$487$	− 22.6612i	− 1.02688i	0.858126	−	0.513439i	$-0.171629\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−30.8501	−1.39225	−0.696123	−	0.717923i	$-0.745093\pi$
$491$	−30.8501	−1.39225	−0.696123	+	0.717923i	$0.745093\pi$
$492$	0	0
$493$	36.1339	1.62739
$494$	0	0
$495$	0	0
$496$	0	0
$497$	11.0932i	0.497598i
$498$	0	0
$499$	− 11.5425i	− 0.516713i	−0.966050	−	0.258356i	$-0.916819\pi$
$499$	− 11.5425i	− 0.516713i	0.966050	−	0.258356i	$-0.0831809\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	16.1908	0.721911	0.360955	−	0.932583i	$-0.382451\pi$
$503$	16.1908	0.721911	0.360955	+	0.932583i	$0.382451\pi$
$504$	0	0
$505$	−1.04076	−0.0463131
$506$	0	0
$507$	0	0
$508$	0	0
$509$	30.7438i	1.36270i	0.731959	+	0.681348i	$0.238606\pi$
$509$	30.7438i	1.36270i	−0.731959	+	0.681348i	$0.761394\pi$
$510$	0	0
$511$	36.7254i	1.62463i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4.48169	0.197487
$516$	0	0
$517$	55.9939	2.46261
$518$	0	0
$519$	0	0
$520$	0	0
$521$	17.1755i	0.752473i	0.926524	+	0.376237i	$0.122782\pi$
$521$	17.1755i	0.752473i	−0.926524	+	0.376237i	$0.877218\pi$
$522$	0	0
$523$	− 35.9241i	− 1.57085i	−0.618955	−	0.785426i	$-0.712444\pi$
$523$	− 35.9241i	− 1.57085i	0.618955	−	0.785426i	$-0.287556\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−11.7432	−0.511541
$528$	0	0
$529$	−21.3976	−0.930329
$530$	0	0
$531$	0	0
$532$	0	0
$533$	44.0681i	1.90880i
$534$	0	0
$535$	− 4.26365i	− 0.184333i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−71.6262	−3.08516
$540$	0	0
$541$	2.63964	0.113487	0.0567435	−	0.998389i	$-0.481928\pi$
$541$	2.63964	0.113487	0.0567435	+	0.998389i	$0.481928\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	6.36345i	0.272580i
$546$	0	0
$547$	21.7902i	0.931682i	0.884868	+	0.465841i	$0.154248\pi$
$547$	21.7902i	0.931682i	−0.884868	+	0.465841i	$0.845752\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	5.50851	0.234671
$552$	0	0
$553$	−72.7675	−3.09439
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14.6946i	0.622630i	0.950307	+	0.311315i	$0.100769\pi$
$557$	14.6946i	0.622630i	−0.950307	+	0.311315i	$0.899231\pi$
$558$	0	0
$559$	22.7094i	0.960506i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−7.32967	−0.308909	−0.154454	−	0.988000i	$-0.549362\pi$
$563$	−7.32967	−0.308909	−0.154454	+	0.988000i	$0.549362\pi$
$564$	0	0
$565$	−3.94614	−0.166015
$566$	0	0
$567$	0	0
$568$	0	0
$569$	33.3438i	1.39784i	0.715198	+	0.698922i	$0.246336\pi$
$569$	33.3438i	1.39784i	−0.715198	+	0.698922i	$0.753664\pi$
$570$	0	0
$571$	− 11.4245i	− 0.478100i	−0.971007	−	0.239050i	$-0.923164\pi$
$571$	− 11.4245i	− 0.478100i	0.971007	−	0.239050i	$-0.0768360\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−6.16249	−0.256994
$576$	0	0
$577$	11.1049	0.462303	0.231151	−	0.972918i	$-0.425751\pi$
$577$	11.1049	0.462303	0.231151	+	0.972918i	$0.425751\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	5.54660i	0.230112i
$582$	0	0
$583$	24.2636i	1.00490i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−14.4330	−0.595715	−0.297858	−	0.954610i	$-0.596272\pi$
$587$	−14.4330	−0.595715	−0.297858	+	0.954610i	$0.596272\pi$
$588$	0	0
$589$	−1.79021	−0.0737644
$590$	0	0
$591$	0	0
$592$	0	0
$593$	28.0257i	1.15088i	0.817845	+	0.575438i	$0.195168\pi$
$593$	28.0257i	1.15088i	−0.817845	+	0.575438i	$0.804832\pi$
$594$	0	0
$595$	− 10.4355i	− 0.427814i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	2.53174	0.103444	0.0517221	−	0.998662i	$-0.483529\pi$
$599$	2.53174	0.103444	0.0517221	+	0.998662i	$0.483529\pi$
$600$	0	0
$601$	43.2629	1.76473	0.882366	−	0.470564i	$-0.155950\pi$
$601$	43.2629	1.76473	0.882366	+	0.470564i	$0.155950\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 8.52337i	− 0.346524i
$606$	0	0
$607$	− 12.1559i	− 0.493394i	−0.969093	−	0.246697i	$-0.920655\pi$
$607$	− 12.1559i	− 0.493394i	0.969093	−	0.246697i	$-0.0793452\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	55.2179	2.23388
$612$	0	0
$613$	36.2119	1.46258	0.731292	−	0.682064i	$-0.238918\pi$
$613$	36.2119	1.46258	0.731292	+	0.682064i	$0.238918\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 12.1062i	− 0.487379i	−0.969853	−	0.243690i	$-0.921642\pi$
$617$	− 12.1062i	− 0.487379i	0.969853	−	0.243690i	$-0.0783578\pi$
$618$	0	0
$619$	33.9461i	1.36441i	0.731161	+	0.682205i	$0.238979\pi$
$619$	33.9461i	1.36441i	−0.731161	+	0.682205i	$0.761021\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	35.2295	1.41144
$624$	0	0
$625$	23.0400	0.921601
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 44.3648i	− 1.76894i
$630$	0	0
$631$	25.2526i	1.00529i	0.864492	+	0.502646i	$0.167640\pi$
$631$	25.2526i	1.00529i	−0.864492	+	0.502646i	$0.832360\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−5.37973	−0.213488
$636$	0	0
$637$	−70.6336	−2.79860
$638$	0	0
$639$	0	0
$640$	0	0
$641$	10.2753i	0.405851i	0.979194	+	0.202925i	$0.0650449\pi$
$641$	10.2753i	0.405851i	−0.979194	+	0.202925i	$0.934955\pi$
$642$	0	0
$643$	25.0290i	0.987049i	0.869732	+	0.493524i	$0.164292\pi$
$643$	25.0290i	0.987049i	−0.869732	+	0.493524i	$0.835708\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	19.4965	0.766487	0.383244	−	0.923647i	$-0.374807\pi$
$647$	19.4965	0.766487	0.383244	+	0.923647i	$0.374807\pi$
$648$	0	0
$649$	72.7675	2.85637
$650$	0	0
$651$	0	0
$652$	0	0
$653$	5.88715i	0.230382i	0.993343	+	0.115191i	$0.0367480\pi$
$653$	5.88715i	0.230382i	−0.993343	+	0.115191i	$0.963252\pi$
$654$	0	0
$655$	5.30085i	0.207121i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−20.3728	−0.793612	−0.396806	−	0.917903i	$-0.629881\pi$
$659$	−20.3728	−0.793612	−0.396806	+	0.917903i	$0.629881\pi$
$660$	0	0
$661$	2.15593	0.0838559	0.0419279	−	0.999121i	$-0.486650\pi$
$661$	2.15593	0.0838559	0.0419279	+	0.999121i	$0.486650\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 1.59086i	− 0.0616910i
$666$	0	0
$667$	6.97307i	0.269998i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−5.25570	−0.202894
$672$	0	0
$673$	−1.73635	−0.0669315	−0.0334658	−	0.999440i	$-0.510654\pi$
$673$	−1.73635	−0.0669315	−0.0334658	+	0.999440i	$0.510654\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 39.6390i	− 1.52345i	−0.647899	−	0.761726i	$-0.724352\pi$
$677$	− 39.6390i	− 1.52345i	0.647899	−	0.761726i	$-0.275648\pi$
$678$	0	0
$679$	− 29.8703i	− 1.14632i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−40.0274	−1.53161	−0.765804	−	0.643074i	$-0.777659\pi$
$683$	−40.0274	−1.53161	−0.765804	+	0.643074i	$0.777659\pi$
$684$	0	0
$685$	−4.28775	−0.163827
$686$	0	0
$687$	0	0
$688$	0	0
$689$	23.9274i	0.911561i
$690$	0	0
$691$	22.1318i	0.841934i	0.907076	+	0.420967i	$0.138309\pi$
$691$	22.1318i	0.841934i	−0.907076	+	0.420967i	$0.861691\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.59086	0.0603448
$696$	0	0
$697$	−49.9241	−1.89101
$698$	0	0
$699$	0	0
$700$	0	0
$701$	39.5599i	1.49416i	0.664736	+	0.747078i	$0.268544\pi$
$701$	39.5599i	1.49416i	−0.664736	+	0.747078i	$0.731456\pi$
$702$	0	0
$703$	− 6.76328i	− 0.255082i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−12.5600	−0.472368
$708$	0	0
$709$	−5.60243	−0.210404	−0.105202	−	0.994451i	$-0.533549\pi$
$709$	−5.60243	−0.210404	−0.105202	+	0.994451i	$0.533549\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 2.26618i	− 0.0848691i
$714$	0	0
$715$	− 12.3437i	− 0.461629i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−19.4965	−0.727098	−0.363549	−	0.931575i	$-0.618435\pi$
$719$	−19.4965	−0.727098	−0.363549	+	0.931575i	$0.618435\pi$
$720$	0	0
$721$	54.0857	2.01426
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 26.8164i	− 0.995937i
$726$	0	0
$727$	− 25.0290i	− 0.928275i	−0.885763	−	0.464138i	$-0.846364\pi$
$727$	− 25.0290i	− 0.928275i	0.885763	−	0.464138i	$-0.153636\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−25.7272	−0.951555
$732$	0	0
$733$	12.7633	0.471423	0.235711	−	0.971823i	$-0.424258\pi$
$733$	12.7633	0.471423	0.235711	+	0.971823i	$0.424258\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 40.9430i	− 1.50815i
$738$	0	0
$739$	− 9.56450i	− 0.351836i	−0.984405	−	0.175918i	$-0.943711\pi$
$739$	− 9.56450i	− 0.351836i	0.984405	−	0.175918i	$-0.0562893\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−43.2092	−1.58519	−0.792595	−	0.609748i	$-0.791271\pi$
$743$	−43.2092	−1.58519	−0.792595	+	0.609748i	$0.791271\pi$
$744$	0	0
$745$	7.56450	0.277142
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 51.4544i	− 1.88010i
$750$	0	0
$751$	7.16085i	0.261303i	0.991428	+	0.130652i	$0.0417069\pi$
$751$	7.16085i	0.261303i	−0.991428	+	0.130652i	$0.958293\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−27.6584	−1.00526	−0.502631	−	0.864501i	$-0.667634\pi$
$757$	−27.6584	−1.00526	−0.502631	+	0.864501i	$0.667634\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	7.12190i	0.258169i	0.991634	+	0.129084i	$0.0412038\pi$
$761$	7.12190i	0.258169i	−0.991634	+	0.129084i	$0.958796\pi$
$762$	0	0
$763$	76.7951i	2.78017i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	71.7590	2.59107
$768$	0	0
$769$	37.6902	1.35914	0.679572	−	0.733608i	$-0.262165\pi$
$769$	37.6902	1.35914	0.679572	+	0.733608i	$0.262165\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 54.8040i	− 1.97116i	−0.169202	−	0.985581i	$-0.554119\pi$
$773$	− 54.8040i	− 1.97116i	0.169202	−	0.985581i	$-0.445881\pi$
$774$	0	0
$775$	8.71507i	0.313055i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−7.61079	−0.272685
$780$	0	0
$781$	−14.8654	−0.531924
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 4.65358i	− 0.166093i
$786$	0	0
$787$	− 33.0269i	− 1.17728i	−0.808394	−	0.588641i	$-0.799663\pi$
$787$	− 33.0269i	− 1.17728i	0.808394	−	0.588641i	$-0.200337\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−47.6227	−1.69327
$792$	0	0
$793$	−5.18286	−0.184049
$794$	0	0
$795$	0	0
$796$	0	0
$797$	37.9858i	1.34553i	0.739858	+	0.672763i	$0.234892\pi$
$797$	37.9858i	1.34553i	−0.739858	+	0.672763i	$0.765108\pi$
$798$	0	0
$799$	62.5556i	2.21306i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−49.2135	−1.73671
$804$	0	0
$805$	2.01383	0.0709781
$806$	0	0
$807$	0	0
$808$	0	0
$809$	42.5719i	1.49675i	0.663276	+	0.748375i	$0.269166\pi$
$809$	42.5719i	1.49675i	−0.663276	+	0.748375i	$0.730834\pi$
$810$	0	0
$811$	24.4514i	0.858606i	0.903161	+	0.429303i	$0.141241\pi$
$811$	24.4514i	0.858606i	−0.903161	+	0.429303i	$0.858759\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3.49796	0.122528
$816$	0	0
$817$	−3.92204	−0.137215
$818$	0	0
$819$	0	0
$820$	0	0
$821$	32.3913i	1.13046i	0.824932	+	0.565232i	$0.191213\pi$
$821$	32.3913i	1.13046i	−0.824932	+	0.565232i	$0.808787\pi$
$822$	0	0
$823$	− 4.38164i	− 0.152734i	−0.997080	−	0.0763672i	$-0.975668\pi$
$823$	− 4.38164i	− 0.152734i	0.997080	−	0.0763672i	$-0.0243321\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−40.9430	−1.42373	−0.711864	−	0.702318i	$-0.752149\pi$
$827$	−40.9430	−1.42373	−0.711864	+	0.702318i	$0.752149\pi$
$828$	0	0
$829$	13.0269	0.452444	0.226222	−	0.974076i	$-0.427363\pi$
$829$	13.0269	0.452444	0.226222	+	0.974076i	$0.427363\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 80.0198i	− 2.77252i
$834$	0	0
$835$	− 3.81642i	− 0.132073i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−25.1025	−0.866636	−0.433318	−	0.901241i	$-0.642657\pi$
$839$	−25.1025	−0.866636	−0.433318	+	0.901241i	$0.642657\pi$
$840$	0	0
$841$	−1.34371	−0.0463348
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 7.45267i	− 0.256380i
$846$	0	0
$847$	− 102.861i	− 3.53436i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	8.56145	0.293483
$852$	0	0
$853$	−16.7413	−0.573210	−0.286605	−	0.958049i	$-0.592527\pi$
$853$	−16.7413	−0.573210	−0.286605	+	0.958049i	$0.592527\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 18.0344i	− 0.616045i	−0.951379	−	0.308023i	$-0.900333\pi$
$857$	− 18.0344i	− 0.616045i	0.951379	−	0.308023i	$-0.0996672\pi$
$858$	0	0
$859$	35.9939i	1.22810i	0.789268	+	0.614048i	$0.210460\pi$
$859$	35.9939i	1.22810i	−0.789268	+	0.614048i	$0.789540\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	23.4863	0.799485	0.399742	−	0.916628i	$-0.369100\pi$
$863$	23.4863	0.799485	0.399742	+	0.916628i	$0.369100\pi$
$864$	0	0
$865$	2.39192	0.0813277
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 97.5115i	− 3.30785i
$870$	0	0
$871$	− 40.3756i	− 1.36807i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−15.6989	−0.530720
$876$	0	0
$877$	−27.8221	−0.939484	−0.469742	−	0.882804i	$-0.655653\pi$
$877$	−27.8221	−0.939484	−0.469742	+	0.882804i	$0.655653\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	3.70870i	0.124949i	0.998047	+	0.0624746i	$0.0198993\pi$
$881$	3.70870i	0.124949i	−0.998047	+	0.0624746i	$0.980101\pi$
$882$	0	0
$883$	− 30.6715i	− 1.03218i	−0.856535	−	0.516089i	$-0.827387\pi$
$883$	− 30.6715i	− 1.03218i	0.856535	−	0.516089i	$-0.172613\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	56.8341	1.90830	0.954151	−	0.299325i	$-0.0967614\pi$
$887$	56.8341	1.90830	0.954151	+	0.299325i	$0.0967614\pi$
$888$	0	0
$889$	−64.9234	−2.17746
$890$	0	0
$891$	0	0
$892$	0	0
$893$	9.53642i	0.319124i
$894$	0	0
$895$	0.235991i	0.00788832i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	9.86141	0.328896
$900$	0	0
$901$	−27.1070	−0.903065
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.84798i	0.0946701i
$906$	0	0
$907$	30.5053i	1.01291i	0.862266	+	0.506456i	$0.169045\pi$
$907$	30.5053i	1.01291i	−0.862266	+	0.506456i	$0.830955\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	30.1660	0.999445	0.499723	−	0.866185i	$-0.333435\pi$
$911$	30.1660	0.999445	0.499723	+	0.866185i	$0.333435\pi$
$912$	0	0
$913$	−7.43268	−0.245986
$914$	0	0
$915$	0	0
$916$	0	0
$917$	63.9716i	2.11253i
$918$	0	0
$919$	23.5804i	0.777846i	0.921270	+	0.388923i	$0.127153\pi$
$919$	23.5804i	0.777846i	−0.921270	+	0.388923i	$0.872847\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−14.6593	−0.482518
$924$	0	0
$925$	−32.9249	−1.08256
$926$	0	0
$927$	0	0
$928$	0	0
$929$	33.9030i	1.11232i	0.831074	+	0.556161i	$0.187726\pi$
$929$	33.9030i	1.11232i	−0.831074	+	0.556161i	$0.812274\pi$
$930$	0	0
$931$	− 12.1988i	− 0.399799i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	13.9840	0.457326
$936$	0	0
$937$	−0.0917913	−0.00299869	−0.00149935	−	0.999999i	$-0.500477\pi$
$937$	−0.0917913	−0.00299869	−0.00149935	+	0.999999i	$0.500477\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	18.8368i	0.614061i	0.951700	+	0.307031i	$0.0993354\pi$
$941$	18.8368i	0.614061i	−0.951700	+	0.307031i	$0.900665\pi$
$942$	0	0
$943$	− 9.63429i	− 0.313735i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	23.5204	0.764312	0.382156	−	0.924098i	$-0.375182\pi$
$947$	23.5204	0.764312	0.382156	+	0.924098i	$0.375182\pi$
$948$	0	0
$949$	−48.5315	−1.57540
$950$	0	0
$951$	0	0
$952$	0	0
$953$	47.3982i	1.53538i	0.640823	+	0.767689i	$0.278593\pi$
$953$	47.3982i	1.53538i	−0.640823	+	0.767689i	$0.721407\pi$
$954$	0	0
$955$	5.53684i	0.179168i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−51.7453	−1.67094
$960$	0	0
$961$	27.7951	0.896617
$962$	0	0
$963$	0	0
$964$	0	0
$965$	8.85813i	0.285153i
$966$	0	0
$967$	30.7413i	0.988573i	0.869299	+	0.494286i	$0.164571\pi$
$967$	30.7413i	0.988573i	−0.869299	+	0.494286i	$0.835429\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	49.5044	1.58867	0.794337	−	0.607478i	$-0.207819\pi$
$971$	49.5044	1.58867	0.794337	+	0.607478i	$0.207819\pi$
$972$	0	0
$973$	19.1988	0.615485
$974$	0	0
$975$	0	0
$976$	0	0
$977$	48.3448i	1.54669i	0.633987	+	0.773344i	$0.281417\pi$
$977$	48.3448i	1.54669i	−0.633987	+	0.773344i	$0.718583\pi$
$978$	0	0
$979$	47.2091i	1.50881i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	9.86141	0.314530	0.157265	−	0.987556i	$-0.449732\pi$
$983$	9.86141	0.314530	0.157265	+	0.987556i	$0.449732\pi$
$984$	0	0
$985$	1.01310	0.0322801
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 4.96479i	− 0.157871i
$990$	0	0
$991$	6.97307i	0.221507i	0.993848	+	0.110753i	$0.0353264\pi$
$991$	6.97307i	0.221507i	−0.993848	+	0.110753i	$0.964674\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−4.93946	−0.156591
$996$	0	0
$997$	9.31468	0.294999	0.147499	−	0.989062i	$-0.452878\pi$
$997$	9.31468	0.294999	0.147499	+	0.989062i	$0.452878\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.d.a.2015.5	✓	12
3.2	odd	2	inner	2736.2.d.a.2015.7	yes	12
4.3	odd	2	inner	2736.2.d.a.2015.6	yes	12
12.11	even	2	inner	2736.2.d.a.2015.8	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2736.2.d.a.2015.5	✓	12	1.1	even	1	trivial
2736.2.d.a.2015.6	yes	12	4.3	odd	2	inner
2736.2.d.a.2015.7	yes	12	3.2	odd	2	inner
2736.2.d.a.2015.8	yes	12	12.11	even	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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.363074i q^{5} -4.38164i q^{7} +O(q^{10})\)	\(q-0.363074i q^{5} -4.38164i q^{7} +5.87159 q^{11} +5.79021 q^{13} +6.55965i q^{17} +1.00000i q^{19} -1.26587 q^{23} +4.86818 q^{25} -5.50851i q^{29} +1.79021i q^{31} -1.59086 q^{35} -6.76328 q^{37} +7.61079i q^{41} +3.92204i q^{43} +9.53642 q^{47} -12.1988 q^{49} +4.13238i q^{53} -2.13182i q^{55} +12.3932 q^{59} -0.895107 q^{61} -2.10228i q^{65} -6.97307i q^{67} -2.53174 q^{71} -8.38164 q^{73} -25.7272i q^{77} -16.6074i q^{79} -1.26587 q^{83} +2.38164 q^{85} +8.04026i q^{89} -25.3706i q^{91} +0.363074 q^{95} +6.81714 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q + 4 q^{25} + 8 q^{37} - 52 q^{49} + 24 q^{61} - 56 q^{73} - 16 q^{85} + 32 q^{97}+O(q^{100}) \) 12 * q + 4 * q^25 + 8 * q^37 - 52 * q^49 + 24 * q^61 - 56 * q^73 - 16 * q^85 + 32 * q^97

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