LMFDB - Embedded newform 5184.2.d.o.2593.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5184,2,Mod(2593,5184)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5184, 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("5184.2593");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5184 = 2^{6} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5184.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:	$41.3944484078$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2593.8
Root			$-0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	5184.2593
Dual form			5184.2.d.o.2593.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.18154i q^{5} +2.44949 q^{7} +O(q^{10})$	$q+4.18154i q^{5} +2.44949 q^{7} +4.24264i q^{11} +0.717439i q^{13} -3.00000 q^{17} -6.24264i q^{19} +1.01461 q^{23} -12.4853 q^{25} +4.18154i q^{29} -8.36308 q^{31} +10.2426i q^{35} -7.64564i q^{37} -8.48528 q^{41} +12.2426i q^{43} +8.36308 q^{47} -1.00000 q^{49} -2.02922i q^{53} -17.7408 q^{55} +5.61642i q^{61} -3.00000 q^{65} -2.24264i q^{67} -11.4069 q^{71} +7.48528 q^{73} +10.3923i q^{77} -5.91359 q^{79} +6.00000i q^{83} -12.5446i q^{85} -17.4853 q^{89} +1.75736i q^{91} +26.1039 q^{95} +0.485281 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 24 q^{17} - 32 q^{25} - 8 q^{49} - 24 q^{65} - 8 q^{73} - 72 q^{89} - 64 q^{97}+O(q^{100}) $ 8 * q - 24 * q^17 - 32 * q^25 - 8 * q^49 - 24 * q^65 - 8 * q^73 - 72 * q^89 - 64 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5184\mathbb{Z}\right)^\times$.

$n$	$325$	$1217$	$2431$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	4.18154i	1.87004i	0.354593	+	0.935021i	$0.384620\pi$
$5$	4.18154i	1.87004i	−0.354593	+	0.935021i	$0.615380\pi$
$6$	0	0
$7$	2.44949	0.925820	0.462910	−	0.886405i	$-0.346805\pi$
$7$	2.44949	0.925820	0.462910	+	0.886405i	$0.346805\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.24264i	1.27920i	0.768706	+	0.639602i	$0.220901\pi$
$11$	4.24264i	1.27920i	−0.768706	+	0.639602i	$0.779099\pi$
$12$	0	0
$13$	0.717439i	0.198982i	0.995038	+	0.0994909i	$0.0317214\pi$
$13$	0.717439i	0.198982i	−0.995038	+	0.0994909i	$0.968279\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	− 6.24264i	− 1.43216i	−0.698018	−	0.716080i	$-0.745935\pi$
$19$	− 6.24264i	− 1.43216i	0.698018	−	0.716080i	$-0.254065\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.01461	0.211561	0.105781	−	0.994389i	$-0.466266\pi$
$23$	1.01461	0.211561	0.105781	+	0.994389i	$0.466266\pi$
$24$	0	0
$25$	−12.4853	−2.49706
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.18154i	0.776493i	0.921556	+	0.388246i	$0.126919\pi$
$29$	4.18154i	0.776493i	−0.921556	+	0.388246i	$0.873081\pi$
$30$	0	0
$31$	−8.36308	−1.50205	−0.751027	−	0.660272i	$-0.770441\pi$
$31$	−8.36308	−1.50205	−0.751027	+	0.660272i	$0.770441\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	10.2426i	1.73132i
$36$	0	0
$37$	− 7.64564i	− 1.25694i	−0.777836	−	0.628468i	$-0.783682\pi$
$37$	− 7.64564i	− 1.25694i	0.777836	−	0.628468i	$-0.216318\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.48528	−1.32518	−0.662589	−	0.748983i	$-0.730542\pi$
$41$	−8.48528	−1.32518	−0.662589	+	0.748983i	$0.730542\pi$
$42$	0	0
$43$	12.2426i	1.86699i	0.358597	+	0.933493i	$0.383255\pi$
$43$	12.2426i	1.86699i	−0.358597	+	0.933493i	$0.616745\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.36308	1.21988	0.609940	−	0.792447i	$-0.291193\pi$
$47$	8.36308	1.21988	0.609940	+	0.792447i	$0.291193\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 2.02922i	− 0.278735i	−0.990241	−	0.139368i	$-0.955493\pi$
$53$	− 2.02922i	− 0.278735i	0.990241	−	0.139368i	$-0.0445070\pi$
$54$	0	0
$55$	−17.7408	−2.39217
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	5.61642i	0.719109i	0.933124	+	0.359554i	$0.117071\pi$
$61$	5.61642i	0.719109i	−0.933124	+	0.359554i	$0.882929\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.00000	−0.372104
$66$	0	0
$67$	− 2.24264i	− 0.273982i	−0.990572	−	0.136991i	$-0.956257\pi$
$67$	− 2.24264i	− 0.273982i	0.990572	−	0.136991i	$-0.0437432\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−11.4069	−1.35375	−0.676876	−	0.736097i	$-0.736667\pi$
$71$	−11.4069	−1.35375	−0.676876	+	0.736097i	$0.736667\pi$
$72$	0	0
$73$	7.48528	0.876086	0.438043	−	0.898954i	$-0.355672\pi$
$73$	7.48528	0.876086	0.438043	+	0.898954i	$0.355672\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	10.3923i	1.18431i
$78$	0	0
$79$	−5.91359	−0.665331	−0.332666	−	0.943045i	$-0.607948\pi$
$79$	−5.91359	−0.665331	−0.332666	+	0.943045i	$0.607948\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	0	0
$85$	− 12.5446i	− 1.36066i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−17.4853	−1.85344	−0.926718	−	0.375757i	$-0.877383\pi$
$89$	−17.4853	−1.85344	−0.926718	+	0.375757i	$0.877383\pi$
$90$	0	0
$91$	1.75736i	0.184221i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	26.1039	2.67820
$96$	0	0
$97$	0.485281	0.0492729	0.0246364	−	0.999696i	$-0.492157\pi$
$97$	0.485281	0.0492729	0.0246364	+	0.999696i	$0.492157\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 6.33386i	− 0.630242i	−0.949051	−	0.315121i	$-0.897955\pi$
$101$	− 6.33386i	− 0.630242i	0.949051	−	0.315121i	$-0.102045\pi$
$102$	0	0
$103$	11.2328	1.10680	0.553402	−	0.832914i	$-0.313329\pi$
$103$	11.2328	1.10680	0.553402	+	0.832914i	$0.313329\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.00000i	0.580042i	0.957020	+	0.290021i	$0.0936623\pi$
$107$	6.00000i	0.580042i	−0.957020	+	0.290021i	$0.906338\pi$
$108$	0	0
$109$	− 6.21076i	− 0.594883i	−0.954740	−	0.297442i	$-0.903867\pi$
$109$	− 6.21076i	− 0.594883i	0.954740	−	0.297442i	$-0.0961334\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	17.4853	1.64488	0.822438	−	0.568854i	$-0.192613\pi$
$113$	17.4853	1.64488	0.822438	+	0.568854i	$0.192613\pi$
$114$	0	0
$115$	4.24264i	0.395628i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−7.34847	−0.673633
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 31.3000i	− 2.79956i
$126$	0	0
$127$	15.7116	1.39417	0.697087	−	0.716986i	$-0.254479\pi$
$127$	15.7116	1.39417	0.697087	+	0.716986i	$0.254479\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 13.7574i	− 1.20199i	−0.799254	−	0.600993i	$-0.794772\pi$
$131$	− 13.7574i	− 1.20199i	0.799254	−	0.600993i	$-0.205228\pi$
$132$	0	0
$133$	− 15.2913i	− 1.32592i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.48528	−0.468639	−0.234320	−	0.972160i	$-0.575286\pi$
$137$	−5.48528	−0.468639	−0.234320	+	0.972160i	$0.575286\pi$
$138$	0	0
$139$	10.0000i	0.848189i	0.905618	+	0.424094i	$0.139408\pi$
$139$	10.0000i	0.848189i	−0.905618	+	0.424094i	$0.860592\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3.04384	−0.254538
$144$	0	0
$145$	−17.4853	−1.45207
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.23999i	0.675046i	0.941317	+	0.337523i	$0.109589\pi$
$149$	8.23999i	0.675046i	−0.941317	+	0.337523i	$0.890411\pi$
$150$	0	0
$151$	−13.8564	−1.12762	−0.563809	−	0.825905i	$-0.690665\pi$
$151$	−13.8564	−1.12762	−0.563809	+	0.825905i	$0.690665\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 34.9706i	− 2.80890i
$156$	0	0
$157$	− 14.5738i	− 1.16312i	−0.813504	−	0.581560i	$-0.802443\pi$
$157$	− 14.5738i	− 1.16312i	0.813504	−	0.581560i	$-0.197557\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.48528	0.195868
$162$	0	0
$163$	20.9706i	1.64254i	0.570539	+	0.821271i	$0.306734\pi$
$163$	20.9706i	1.64254i	−0.570539	+	0.821271i	$0.693266\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.31925	−0.411616	−0.205808	−	0.978592i	$-0.565982\pi$
$167$	−5.31925	−0.411616	−0.205808	+	0.978592i	$0.565982\pi$
$168$	0	0
$169$	12.4853	0.960406
$170$	0	0
$171$	0	0
$172$	0	0
$173$	18.8785i	1.43530i	0.696402	+	0.717652i	$0.254783\pi$
$173$	18.8785i	1.43530i	−0.696402	+	0.717652i	$0.745217\pi$
$174$	0	0
$175$	−30.5826	−2.31182
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 10.9706i	− 0.819978i	−0.912090	−	0.409989i	$-0.865532\pi$
$179$	− 10.9706i	− 0.819978i	0.912090	−	0.409989i	$-0.134468\pi$
$180$	0	0
$181$	− 4.89898i	− 0.364138i	−0.983286	−	0.182069i	$-0.941721\pi$
$181$	− 4.89898i	− 0.364138i	0.983286	−	0.182069i	$-0.0582795\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	31.9706	2.35052
$186$	0	0
$187$	− 12.7279i	− 0.930758i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9.37769	0.678546	0.339273	−	0.940688i	$-0.389819\pi$
$191$	9.37769	0.678546	0.339273	+	0.940688i	$0.389819\pi$
$192$	0	0
$193$	−7.00000	−0.503871	−0.251936	−	0.967744i	$-0.581067\pi$
$193$	−7.00000	−0.503871	−0.251936	+	0.967744i	$0.581067\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 8.23999i	− 0.587075i	−0.955948	−	0.293537i	$-0.905167\pi$
$197$	− 8.23999i	− 0.587075i	0.955948	−	0.293537i	$-0.0948325\pi$
$198$	0	0
$199$	−22.2195	−1.57510	−0.787549	−	0.616252i	$-0.788650\pi$
$199$	−22.2195	−1.57510	−0.787549	+	0.616252i	$0.788650\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	10.2426i	0.718892i
$204$	0	0
$205$	− 35.4815i	− 2.47814i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	26.4853	1.83203
$210$	0	0
$211$	17.2132i	1.18501i	0.805568	+	0.592503i	$0.201860\pi$
$211$	17.2132i	1.18501i	−0.805568	+	0.592503i	$0.798140\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−51.1931	−3.49134
$216$	0	0
$217$	−20.4853	−1.39063
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 2.15232i	− 0.144780i
$222$	0	0
$223$	−19.7700	−1.32390	−0.661948	−	0.749549i	$-0.730270\pi$
$223$	−19.7700	−1.32390	−0.661948	+	0.749549i	$0.730270\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.2426i	0.679828i	0.940457	+	0.339914i	$0.110398\pi$
$227$	10.2426i	0.679828i	−0.940457	+	0.339914i	$0.889602\pi$
$228$	0	0
$229$	0.123093i	0.00813422i	0.999992	+	0.00406711i	$0.00129460\pi$
$229$	0.123093i	0.00813422i	−0.999992	+	0.00406711i	$0.998705\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	13.9706	0.915242	0.457621	−	0.889147i	$-0.348702\pi$
$233$	13.9706	0.915242	0.457621	+	0.889147i	$0.348702\pi$
$234$	0	0
$235$	34.9706i	2.28123i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	12.4215	0.803482	0.401741	−	0.915753i	$-0.368405\pi$
$239$	12.4215	0.803482	0.401741	+	0.915753i	$0.368405\pi$
$240$	0	0
$241$	11.0000	0.708572	0.354286	−	0.935137i	$-0.384724\pi$
$241$	11.0000	0.708572	0.354286	+	0.935137i	$0.384724\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 4.18154i	− 0.267149i
$246$	0	0
$247$	4.47871	0.284974
$248$	0	0
$249$	0	0
$250$	0	0
$251$	16.2426i	1.02523i	0.858620	+	0.512613i	$0.171323\pi$
$251$	16.2426i	1.02523i	−0.858620	+	0.512613i	$0.828677\pi$
$252$	0	0
$253$	4.30463i	0.270630i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5.48528	−0.342162	−0.171081	−	0.985257i	$-0.554726\pi$
$257$	−5.48528	−0.342162	−0.171081	+	0.985257i	$0.554726\pi$
$258$	0	0
$259$	− 18.7279i	− 1.16370i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−25.0892	−1.54707	−0.773535	−	0.633754i	$-0.781513\pi$
$263$	−25.0892	−1.54707	−0.773535	+	0.633754i	$0.781513\pi$
$264$	0	0
$265$	8.48528	0.521247
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 8.23999i	− 0.502401i	−0.967935	−	0.251200i	$-0.919175\pi$
$269$	− 8.23999i	− 0.502401i	0.967935	−	0.251200i	$-0.0808253\pi$
$270$	0	0
$271$	18.5813	1.12873	0.564367	−	0.825524i	$-0.309120\pi$
$271$	18.5813	1.12873	0.564367	+	0.825524i	$0.309120\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 52.9706i	− 3.19425i
$276$	0	0
$277$	− 16.7262i	− 1.00498i	−0.864584	−	0.502489i	$-0.832418\pi$
$277$	− 16.7262i	− 1.00498i	0.864584	−	0.502489i	$-0.167582\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5.48528	0.327224	0.163612	−	0.986525i	$-0.447685\pi$
$281$	5.48528	0.327224	0.163612	+	0.986525i	$0.447685\pi$
$282$	0	0
$283$	22.0000i	1.30776i	0.756596	+	0.653882i	$0.226861\pi$
$283$	22.0000i	1.30776i	−0.756596	+	0.653882i	$0.773139\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−20.7846	−1.22688
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	0	0
$292$	0	0
$293$	20.9077i	1.22144i	0.791846	+	0.610721i	$0.209120\pi$
$293$	20.9077i	1.22144i	−0.791846	+	0.610721i	$0.790880\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.727922i	0.0420968i
$300$	0	0
$301$	29.9882i	1.72849i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−23.4853	−1.34476
$306$	0	0
$307$	− 14.9706i	− 0.854415i	−0.904154	−	0.427208i	$-0.859497\pi$
$307$	− 14.9706i	− 0.854415i	0.904154	−	0.427208i	$-0.140503\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	16.7262	0.948454	0.474227	−	0.880403i	$-0.342728\pi$
$311$	16.7262	0.948454	0.474227	+	0.880403i	$0.342728\pi$
$312$	0	0
$313$	5.97056	0.337476	0.168738	−	0.985661i	$-0.446031\pi$
$313$	5.97056	0.337476	0.168738	+	0.985661i	$0.446031\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 27.2416i	− 1.53004i	−0.644007	−	0.765019i	$-0.722729\pi$
$317$	− 27.2416i	− 1.53004i	0.644007	−	0.765019i	$-0.277271\pi$
$318$	0	0
$319$	−17.7408	−0.993293
$320$	0	0
$321$	0	0
$322$	0	0
$323$	18.7279i	1.04205i
$324$	0	0
$325$	− 8.95743i	− 0.496869i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	20.4853	1.12939
$330$	0	0
$331$	5.75736i	0.316453i	0.987403	+	0.158227i	$0.0505776\pi$
$331$	5.75736i	0.316453i	−0.987403	+	0.158227i	$0.949422\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	9.37769	0.512358
$336$	0	0
$337$	−16.4853	−0.898010	−0.449005	−	0.893529i	$-0.648222\pi$
$337$	−16.4853	−0.898010	−0.449005	+	0.893529i	$0.648222\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 35.4815i	− 1.92143i
$342$	0	0
$343$	−19.5959	−1.05808
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.27208i	0.283020i	0.989937	+	0.141510i	$0.0451957\pi$
$347$	5.27208i	0.283020i	−0.989937	+	0.141510i	$0.954804\pi$
$348$	0	0
$349$	29.9882i	1.60523i	0.596496	+	0.802616i	$0.296559\pi$
$349$	29.9882i	1.60523i	−0.596496	+	0.802616i	$0.703441\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	− 47.6985i	− 2.53157i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	13.6823	0.722126	0.361063	−	0.932542i	$-0.382414\pi$
$359$	13.6823	0.722126	0.361063	+	0.932542i	$0.382414\pi$
$360$	0	0
$361$	−19.9706	−1.05108
$362$	0	0
$363$	0	0
$364$	0	0
$365$	31.3000i	1.63832i
$366$	0	0
$367$	−13.8564	−0.723299	−0.361649	−	0.932314i	$-0.617786\pi$
$367$	−13.8564	−0.723299	−0.361649	+	0.932314i	$0.617786\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 4.97056i	− 0.258059i
$372$	0	0
$373$	− 26.5241i	− 1.37337i	−0.726957	−	0.686683i	$-0.759066\pi$
$373$	− 26.5241i	− 1.37337i	0.726957	−	0.686683i	$-0.240934\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.00000	−0.154508
$378$	0	0
$379$	− 2.97056i	− 0.152588i	−0.997085	−	0.0762938i	$-0.975691\pi$
$379$	− 2.97056i	− 0.152588i	0.997085	−	0.0762938i	$-0.0243087\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−5.31925	−0.271801	−0.135900	−	0.990723i	$-0.543393\pi$
$383$	−5.31925	−0.271801	−0.135900	+	0.990723i	$0.543393\pi$
$384$	0	0
$385$	−43.4558	−2.21471
$386$	0	0
$387$	0	0
$388$	0	0
$389$	18.7554i	0.950936i	0.879733	+	0.475468i	$0.157721\pi$
$389$	18.7554i	0.950936i	−0.879733	+	0.475468i	$0.842279\pi$
$390$	0	0
$391$	−3.04384	−0.153933
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 24.7279i	− 1.24420i
$396$	0	0
$397$	− 8.48617i	− 0.425909i	−0.977062	−	0.212954i	$-0.931691\pi$
$397$	− 8.48617i	− 0.425909i	0.977062	−	0.212954i	$-0.0683086\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−22.4558	−1.12139	−0.560696	−	0.828022i	$-0.689466\pi$
$401$	−22.4558	−1.12139	−0.560696	+	0.828022i	$0.689466\pi$
$402$	0	0
$403$	− 6.00000i	− 0.298881i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	32.4377	1.60788
$408$	0	0
$409$	−17.9706	−0.888587	−0.444294	−	0.895881i	$-0.646545\pi$
$409$	−17.9706	−0.888587	−0.444294	+	0.895881i	$0.646545\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−25.0892	−1.23158
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4.97056i	0.242828i	0.992602	+	0.121414i	$0.0387428\pi$
$419$	4.97056i	0.242828i	−0.992602	+	0.121414i	$0.961257\pi$
$420$	0	0
$421$	− 3.34101i	− 0.162831i	−0.996680	−	0.0814154i	$-0.974056\pi$
$421$	− 3.34101i	− 0.162831i	0.996680	−	0.0814154i	$-0.0259440\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	37.4558	1.81688
$426$	0	0
$427$	13.7574i	0.665765i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	21.0308	1.01302	0.506509	−	0.862235i	$-0.330936\pi$
$431$	21.0308	1.01302	0.506509	+	0.862235i	$0.330936\pi$
$432$	0	0
$433$	32.4558	1.55973	0.779864	−	0.625949i	$-0.215288\pi$
$433$	32.4558	1.55973	0.779864	+	0.625949i	$0.215288\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 6.33386i	− 0.302989i
$438$	0	0
$439$	−4.30463	−0.205449	−0.102724	−	0.994710i	$-0.532756\pi$
$439$	−4.30463	−0.205449	−0.102724	+	0.994710i	$0.532756\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	10.9706i	0.521227i	0.965443	+	0.260614i	$0.0839249\pi$
$443$	10.9706i	0.521227i	−0.965443	+	0.260614i	$0.916075\pi$
$444$	0	0
$445$	− 73.1154i	− 3.46600i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−4.97056	−0.234575	−0.117288	−	0.993098i	$-0.537420\pi$
$449$	−4.97056	−0.234575	−0.117288	+	0.993098i	$0.537420\pi$
$450$	0	0
$451$	− 36.0000i	− 1.69517i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−7.34847	−0.344502
$456$	0	0
$457$	−9.48528	−0.443703	−0.221851	−	0.975080i	$-0.571210\pi$
$457$	−9.48528	−0.443703	−0.221851	+	0.975080i	$0.571210\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	20.7846i	0.968036i	0.875058	+	0.484018i	$0.160823\pi$
$461$	20.7846i	0.968036i	−0.875058	+	0.484018i	$0.839177\pi$
$462$	0	0
$463$	26.5241	1.23268	0.616340	−	0.787480i	$-0.288615\pi$
$463$	26.5241	1.23268	0.616340	+	0.787480i	$0.288615\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	40.9706i	1.89589i	0.318432	+	0.947946i	$0.396844\pi$
$467$	40.9706i	1.89589i	−0.318432	+	0.947946i	$0.603156\pi$
$468$	0	0
$469$	− 5.49333i	− 0.253658i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−51.9411	−2.38826
$474$	0	0
$475$	77.9411i	3.57618i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	7.34847	0.335760	0.167880	−	0.985807i	$-0.446308\pi$
$479$	7.34847	0.335760	0.167880	+	0.985807i	$0.446308\pi$
$480$	0	0
$481$	5.48528	0.250107
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.02922i	0.0921423i
$486$	0	0
$487$	1.43488	0.0650205	0.0325103	−	0.999471i	$-0.489650\pi$
$487$	1.43488	0.0650205	0.0325103	+	0.999471i	$0.489650\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	19.7574i	0.891637i	0.895123	+	0.445819i	$0.147087\pi$
$491$	19.7574i	0.891637i	−0.895123	+	0.445819i	$0.852913\pi$
$492$	0	0
$493$	− 12.5446i	− 0.564981i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−27.9411	−1.25333
$498$	0	0
$499$	− 8.24264i	− 0.368991i	−0.982833	−	0.184496i	$-0.940935\pi$
$499$	− 8.24264i	− 0.368991i	0.982833	−	0.184496i	$-0.0590652\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	4.05845	0.180957	0.0904786	−	0.995898i	$-0.471160\pi$
$503$	4.05845	0.180957	0.0904786	+	0.995898i	$0.471160\pi$
$504$	0	0
$505$	26.4853	1.17858
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 16.7262i	− 0.741374i	−0.928758	−	0.370687i	$-0.879122\pi$
$509$	− 16.7262i	− 0.741374i	0.928758	−	0.370687i	$-0.120878\pi$
$510$	0	0
$511$	18.3351	0.811098
$512$	0	0
$513$	0	0
$514$	0	0
$515$	46.9706i	2.06977i
$516$	0	0
$517$	35.4815i	1.56048i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−40.9706	−1.79495	−0.897476	−	0.441062i	$-0.854602\pi$
$521$	−40.9706	−1.79495	−0.897476	+	0.441062i	$0.854602\pi$
$522$	0	0
$523$	3.75736i	0.164298i	0.996620	+	0.0821489i	$0.0261783\pi$
$523$	3.75736i	0.164298i	−0.996620	+	0.0821489i	$0.973822\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	25.0892	1.09290
$528$	0	0
$529$	−21.9706	−0.955242
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 6.08767i	− 0.263686i
$534$	0	0
$535$	−25.0892	−1.08470
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 4.24264i	− 0.182743i
$540$	0	0
$541$	− 22.3426i	− 0.960583i	−0.877109	−	0.480291i	$-0.840531\pi$
$541$	− 22.3426i	− 0.960583i	0.877109	−	0.480291i	$-0.159469\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	25.9706	1.11246
$546$	0	0
$547$	2.97056i	0.127012i	0.997981	+	0.0635060i	$0.0202282\pi$
$547$	2.97056i	0.127012i	−0.997981	+	0.0635060i	$0.979772\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	26.1039	1.11206
$552$	0	0
$553$	−14.4853	−0.615977
$554$	0	0
$555$	0	0
$556$	0	0
$557$	37.6339i	1.59460i	0.603585	+	0.797299i	$0.293739\pi$
$557$	37.6339i	1.59460i	−0.603585	+	0.797299i	$0.706261\pi$
$558$	0	0
$559$	−8.78335	−0.371496
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 18.0000i	− 0.758610i	−0.925272	−	0.379305i	$-0.876163\pi$
$563$	− 18.0000i	− 0.758610i	0.925272	−	0.379305i	$-0.123837\pi$
$564$	0	0
$565$	73.1154i	3.07599i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−40.4558	−1.69600	−0.847999	−	0.529998i	$-0.822192\pi$
$569$	−40.4558	−1.69600	−0.847999	+	0.529998i	$0.822192\pi$
$570$	0	0
$571$	8.97056i	0.375406i	0.982226	+	0.187703i	$0.0601043\pi$
$571$	8.97056i	0.375406i	−0.982226	+	0.187703i	$0.939896\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−12.6677	−0.528280
$576$	0	0
$577$	17.9706	0.748124	0.374062	−	0.927404i	$-0.377965\pi$
$577$	17.9706	0.748124	0.374062	+	0.927404i	$0.377965\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	14.6969i	0.609732i
$582$	0	0
$583$	8.60927	0.356559
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 40.2426i	− 1.66099i	−0.557025	−	0.830496i	$-0.688057\pi$
$587$	− 40.2426i	− 1.66099i	0.557025	−	0.830496i	$-0.311943\pi$
$588$	0	0
$589$	52.2077i	2.15118i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−6.51472	−0.267527	−0.133764	−	0.991013i	$-0.542706\pi$
$593$	−6.51472	−0.267527	−0.133764	+	0.991013i	$0.542706\pi$
$594$	0	0
$595$	− 30.7279i	− 1.25972i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	37.7570	1.54271	0.771354	−	0.636407i	$-0.219580\pi$
$599$	37.7570	1.54271	0.771354	+	0.636407i	$0.219580\pi$
$600$	0	0
$601$	−1.00000	−0.0407909	−0.0203954	−	0.999792i	$-0.506493\pi$
$601$	−1.00000	−0.0407909	−0.0203954	+	0.999792i	$0.506493\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 29.2708i	− 1.19003i
$606$	0	0
$607$	2.44949	0.0994217	0.0497109	−	0.998764i	$-0.484170\pi$
$607$	2.44949	0.0994217	0.0497109	+	0.998764i	$0.484170\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6.00000i	0.242734i
$612$	0	0
$613$	28.5533i	1.15326i	0.817006	+	0.576629i	$0.195632\pi$
$613$	28.5533i	1.15326i	−0.817006	+	0.576629i	$0.804368\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.97056	0.0793319	0.0396659	−	0.999213i	$-0.487371\pi$
$617$	1.97056	0.0793319	0.0396659	+	0.999213i	$0.487371\pi$
$618$	0	0
$619$	4.00000i	0.160774i	0.996764	+	0.0803868i	$0.0256155\pi$
$619$	4.00000i	0.160774i	−0.996764	+	0.0803868i	$0.974384\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−42.8300	−1.71595
$624$	0	0
$625$	68.4558	2.73823
$626$	0	0
$627$	0	0
$628$	0	0
$629$	22.9369i	0.914555i
$630$	0	0
$631$	19.5959	0.780101	0.390051	−	0.920793i	$-0.372458\pi$
$631$	19.5959	0.780101	0.390051	+	0.920793i	$0.372458\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	65.6985i	2.60716i
$636$	0	0
$637$	− 0.717439i	− 0.0284260i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−23.4853	−0.927613	−0.463806	−	0.885937i	$-0.653517\pi$
$641$	−23.4853	−0.927613	−0.463806	+	0.885937i	$0.653517\pi$
$642$	0	0
$643$	2.97056i	0.117148i	0.998283	+	0.0585738i	$0.0186553\pi$
$643$	2.97056i	0.117148i	−0.998283	+	0.0585738i	$0.981345\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24.0746	0.946471	0.473236	−	0.880936i	$-0.343086\pi$
$647$	24.0746	0.946471	0.473236	+	0.880936i	$0.343086\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	14.6969i	0.575136i	0.957760	+	0.287568i	$0.0928467\pi$
$653$	14.6969i	0.575136i	−0.957760	+	0.287568i	$0.907153\pi$
$654$	0	0
$655$	57.5270	2.24776
$656$	0	0
$657$	0	0
$658$	0	0
$659$	34.2426i	1.33390i	0.745101	+	0.666952i	$0.232401\pi$
$659$	34.2426i	1.33390i	−0.745101	+	0.666952i	$0.767599\pi$
$660$	0	0
$661$	9.92105i	0.385884i	0.981210	+	0.192942i	$0.0618030\pi$
$661$	9.92105i	0.385884i	−0.981210	+	0.192942i	$0.938197\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	63.9411	2.47953
$666$	0	0
$667$	4.24264i	0.164276i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−23.8284	−0.919887
$672$	0	0
$673$	9.48528	0.365631	0.182815	−	0.983147i	$-0.441479\pi$
$673$	9.48528	0.365631	0.182815	+	0.983147i	$0.441479\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	46.1200i	1.77254i	0.463172	+	0.886269i	$0.346711\pi$
$677$	46.1200i	1.77254i	−0.463172	+	0.886269i	$0.653289\pi$
$678$	0	0
$679$	1.18869	0.0456178
$680$	0	0
$681$	0	0
$682$	0	0
$683$	27.9411i	1.06914i	0.845125	+	0.534569i	$0.179526\pi$
$683$	27.9411i	1.06914i	−0.845125	+	0.534569i	$0.820474\pi$
$684$	0	0
$685$	− 22.9369i	− 0.876375i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.45584	0.0554632
$690$	0	0
$691$	− 13.2721i	− 0.504894i	−0.967611	−	0.252447i	$-0.918765\pi$
$691$	− 13.2721i	− 0.504894i	0.967611	−	0.252447i	$-0.0812353\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−41.8154	−1.58615
$696$	0	0
$697$	25.4558	0.964209
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 12.5446i	− 0.473804i	−0.971534	−	0.236902i	$-0.923868\pi$
$701$	− 12.5446i	− 0.473804i	0.971534	−	0.236902i	$-0.0761320\pi$
$702$	0	0
$703$	−47.7290	−1.80013
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 15.5147i	− 0.583491i
$708$	0	0
$709$	− 18.2841i	− 0.686675i	−0.939212	−	0.343338i	$-0.888443\pi$
$709$	− 18.2841i	− 0.686675i	0.939212	−	0.343338i	$-0.111557\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−8.48528	−0.317776
$714$	0	0
$715$	− 12.7279i	− 0.475997i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−26.3500	−0.982691	−0.491345	−	0.870965i	$-0.663495\pi$
$719$	−26.3500	−0.982691	−0.491345	+	0.870965i	$0.663495\pi$
$720$	0	0
$721$	27.5147	1.02470
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 52.2077i	− 1.93895i
$726$	0	0
$727$	13.4361	0.498319	0.249159	−	0.968462i	$-0.419846\pi$
$727$	13.4361	0.498319	0.249159	+	0.968462i	$0.419846\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 36.7279i	− 1.35843i
$732$	0	0
$733$	− 27.3647i	− 1.01074i	−0.862904	−	0.505368i	$-0.831357\pi$
$733$	− 27.3647i	− 1.01074i	0.862904	−	0.505368i	$-0.168643\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.51472	0.350479
$738$	0	0
$739$	18.9706i	0.697843i	0.937152	+	0.348922i	$0.113452\pi$
$739$	18.9706i	0.697843i	−0.937152	+	0.348922i	$0.886548\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−37.7570	−1.38517	−0.692584	−	0.721337i	$-0.743528\pi$
$743$	−37.7570	−1.38517	−0.692584	+	0.721337i	$0.743528\pi$
$744$	0	0
$745$	−34.4558	−1.26236
$746$	0	0
$747$	0	0
$748$	0	0
$749$	14.6969i	0.537014i
$750$	0	0
$751$	44.0187	1.60627	0.803133	−	0.595800i	$-0.203165\pi$
$751$	44.0187	1.60627	0.803133	+	0.595800i	$0.203165\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 57.9411i	− 2.10869i
$756$	0	0
$757$	− 7.76874i	− 0.282359i	−0.989984	−	0.141180i	$-0.954910\pi$
$757$	− 7.76874i	− 0.282359i	0.989984	−	0.141180i	$-0.0450895\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	3.00000	0.108750	0.0543750	−	0.998521i	$-0.482683\pi$
$761$	3.00000	0.108750	0.0543750	+	0.998521i	$0.482683\pi$
$762$	0	0
$763$	− 15.2132i	− 0.550755i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−39.4853	−1.42388	−0.711938	−	0.702242i	$-0.752182\pi$
$769$	−39.4853	−1.42388	−0.711938	+	0.702242i	$0.752182\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	26.9954i	0.970956i	0.874249	+	0.485478i	$0.161354\pi$
$773$	26.9954i	0.970956i	−0.874249	+	0.485478i	$0.838646\pi$
$774$	0	0
$775$	104.415	3.75071
$776$	0	0
$777$	0	0
$778$	0	0
$779$	52.9706i	1.89787i
$780$	0	0
$781$	− 48.3954i	− 1.73173i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	60.9411	2.17508
$786$	0	0
$787$	9.69848i	0.345714i	0.984947	+	0.172857i	$0.0552998\pi$
$787$	9.69848i	0.345714i	−0.984947	+	0.172857i	$0.944700\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	42.8300	1.52286
$792$	0	0
$793$	−4.02944	−0.143090
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 26.9954i	− 0.956225i	−0.878299	−	0.478113i	$-0.841321\pi$
$797$	− 26.9954i	− 0.956225i	0.878299	−	0.478113i	$-0.158679\pi$
$798$	0	0
$799$	−25.0892	−0.887594
$800$	0	0
$801$	0	0
$802$	0	0
$803$	31.7574i	1.12069i
$804$	0	0
$805$	10.3923i	0.366281i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−33.4264	−1.17521	−0.587605	−	0.809148i	$-0.699929\pi$
$809$	−33.4264	−1.17521	−0.587605	+	0.809148i	$0.699929\pi$
$810$	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
$813$	0	0
$814$	0	0
$815$	−87.6893	−3.07162
$816$	0	0
$817$	76.4264	2.67382
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 0.123093i	− 0.00429598i	−0.999998	−	0.00214799i	$-0.999316\pi$
$821$	− 0.123093i	− 0.00429598i	0.999998	−	0.00214799i	$-0.000683727\pi$
$822$	0	0
$823$	−43.0041	−1.49903	−0.749514	−	0.661988i	$-0.769713\pi$
$823$	−43.0041	−1.49903	−0.749514	+	0.661988i	$0.769713\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 40.2426i	− 1.39937i	−0.714450	−	0.699687i	$-0.753323\pi$
$827$	− 40.2426i	− 1.39937i	0.714450	−	0.699687i	$-0.246677\pi$
$828$	0	0
$829$	53.3964i	1.85453i	0.374401	+	0.927267i	$0.377848\pi$
$829$	53.3964i	1.85453i	−0.374401	+	0.927267i	$0.622152\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.00000	0.103944
$834$	0	0
$835$	− 22.2426i	− 0.769738i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	36.4962	1.25999	0.629994	−	0.776600i	$-0.283057\pi$
$839$	36.4962	1.25999	0.629994	+	0.776600i	$0.283057\pi$
$840$	0	0
$841$	11.5147	0.397059
$842$	0	0
$843$	0	0
$844$	0	0
$845$	52.2077i	1.79600i
$846$	0	0
$847$	−17.1464	−0.589158
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 7.75736i	− 0.265919i
$852$	0	0
$853$	− 34.6410i	− 1.18609i	−0.805171	−	0.593043i	$-0.797926\pi$
$853$	− 34.6410i	− 1.18609i	0.805171	−	0.593043i	$-0.202074\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	13.9706	0.477225	0.238613	−	0.971115i	$-0.423307\pi$
$857$	13.9706	0.477225	0.238613	+	0.971115i	$0.423307\pi$
$858$	0	0
$859$	9.02944i	0.308080i	0.988065	+	0.154040i	$0.0492285\pi$
$859$	9.02944i	0.308080i	−0.988065	+	0.154040i	$0.950771\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	40.5546	1.38049	0.690247	−	0.723573i	$-0.257502\pi$
$863$	40.5546	1.38049	0.690247	+	0.723573i	$0.257502\pi$
$864$	0	0
$865$	−78.9411	−2.68408
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 25.0892i	− 0.851094i
$870$	0	0
$871$	1.60896	0.0545175
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 76.6690i	− 2.59189i
$876$	0	0
$877$	27.8359i	0.939952i	0.882679	+	0.469976i	$0.155737\pi$
$877$	27.8359i	0.939952i	−0.882679	+	0.469976i	$0.844263\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	16.9706	0.571753	0.285876	−	0.958267i	$-0.407715\pi$
$881$	16.9706	0.571753	0.285876	+	0.958267i	$0.407715\pi$
$882$	0	0
$883$	− 21.0294i	− 0.707697i	−0.935303	−	0.353848i	$-0.884873\pi$
$883$	− 21.0294i	− 0.707697i	0.935303	−	0.353848i	$-0.115127\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	11.6531	0.391273	0.195636	−	0.980676i	$-0.437323\pi$
$887$	11.6531	0.391273	0.195636	+	0.980676i	$0.437323\pi$
$888$	0	0
$889$	38.4853	1.29075
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 52.2077i	− 1.74706i
$894$	0	0
$895$	45.8739	1.53339
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 34.9706i	− 1.16633i
$900$	0	0
$901$	6.08767i	0.202810i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	20.4853	0.680954
$906$	0	0
$907$	− 25.9411i	− 0.861361i	−0.902504	−	0.430680i	$-0.858274\pi$
$907$	− 25.9411i	− 0.861361i	0.902504	−	0.430680i	$-0.141726\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−13.6823	−0.453316	−0.226658	−	0.973974i	$-0.572780\pi$
$911$	−13.6823	−0.453316	−0.226658	+	0.973974i	$0.572780\pi$
$912$	0	0
$913$	−25.4558	−0.842465
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 33.6985i	− 1.11282i
$918$	0	0
$919$	38.5254	1.27084	0.635418	−	0.772169i	$-0.280828\pi$
$919$	38.5254	1.27084	0.635418	+	0.772169i	$0.280828\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 8.18377i	− 0.269372i
$924$	0	0
$925$	95.4580i	3.13864i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−10.0294	−0.329055	−0.164528	−	0.986372i	$-0.552610\pi$
$929$	−10.0294	−0.329055	−0.164528	+	0.986372i	$0.552610\pi$
$930$	0	0
$931$	6.24264i	0.204594i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	53.2223	1.74056
$936$	0	0
$937$	31.0000	1.01273	0.506363	−	0.862320i	$-0.330990\pi$
$937$	31.0000	1.01273	0.506363	+	0.862320i	$0.330990\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 16.6031i	− 0.541245i	−0.962686	−	0.270622i	$-0.912771\pi$
$941$	− 16.6031i	− 0.541245i	0.962686	−	0.270622i	$-0.0872294\pi$
$942$	0	0
$943$	−8.60927	−0.280356
$944$	0	0
$945$	0	0
$946$	0	0
$947$	13.7574i	0.447054i	0.974698	+	0.223527i	$0.0717571\pi$
$947$	13.7574i	0.447054i	−0.974698	+	0.223527i	$0.928243\pi$
$948$	0	0
$949$	5.37023i	0.174325i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	53.4853	1.73256	0.866279	−	0.499560i	$-0.166505\pi$
$953$	53.4853	1.73256	0.866279	+	0.499560i	$0.166505\pi$
$954$	0	0
$955$	39.2132i	1.26891i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−13.4361	−0.433876
$960$	0	0
$961$	38.9411	1.25617
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 29.2708i	− 0.942260i
$966$	0	0
$967$	21.2049	0.681903	0.340951	−	0.940081i	$-0.389251\pi$
$967$	21.2049	0.681903	0.340951	+	0.940081i	$0.389251\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 10.2426i	− 0.328702i	−0.986402	−	0.164351i	$-0.947447\pi$
$971$	− 10.2426i	− 0.328702i	0.986402	−	0.164351i	$-0.0525530\pi$
$972$	0	0
$973$	24.4949i	0.785270i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−40.9706	−1.31076	−0.655382	−	0.755297i	$-0.727492\pi$
$977$	−40.9706	−1.31076	−0.655382	+	0.755297i	$0.727492\pi$
$978$	0	0
$979$	− 74.1838i	− 2.37092i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	8.60927	0.274593	0.137296	−	0.990530i	$-0.456159\pi$
$983$	8.60927	0.274593	0.137296	+	0.990530i	$0.456159\pi$
$984$	0	0
$985$	34.4558	1.09785
$986$	0	0
$987$	0	0
$988$	0	0
$989$	12.4215i	0.394982i
$990$	0	0
$991$	41.1490	1.30714	0.653570	−	0.756866i	$-0.273271\pi$
$991$	41.1490	1.30714	0.653570	+	0.756866i	$0.273271\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 92.9117i	− 2.94550i
$996$	0	0
$997$	18.6323i	0.590091i	0.955483	+	0.295045i	$0.0953347\pi$
$997$	18.6323i	0.590091i	−0.955483	+	0.295045i	$0.904665\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5184.2.d.o.2593.8	yes	8
3.2	odd	2		5184.2.d.p.2593.2	yes	8
4.3	odd	2	inner	5184.2.d.o.2593.7	yes	8
8.3	odd	2	inner	5184.2.d.o.2593.1	✓	8
8.5	even	2	inner	5184.2.d.o.2593.2	yes	8
12.11	even	2		5184.2.d.p.2593.1	yes	8
24.5	odd	2		5184.2.d.p.2593.8	yes	8
24.11	even	2		5184.2.d.p.2593.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5184.2.d.o.2593.1	✓	8	8.3	odd	2	inner
5184.2.d.o.2593.2	yes	8	8.5	even	2	inner
5184.2.d.o.2593.7	yes	8	4.3	odd	2	inner
5184.2.d.o.2593.8	yes	8	1.1	even	1	trivial
5184.2.d.p.2593.1	yes	8	12.11	even	2
5184.2.d.p.2593.2	yes	8	3.2	odd	2
5184.2.d.p.2593.7	yes	8	24.11	even	2
5184.2.d.p.2593.8	yes	8	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5184 = 2^{6} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5184.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.18154i q^{5} +2.44949 q^{7} +O(q^{10})\)	\(q+4.18154i q^{5} +2.44949 q^{7} +4.24264i q^{11} +0.717439i q^{13} -3.00000 q^{17} -6.24264i q^{19} +1.01461 q^{23} -12.4853 q^{25} +4.18154i q^{29} -8.36308 q^{31} +10.2426i q^{35} -7.64564i q^{37} -8.48528 q^{41} +12.2426i q^{43} +8.36308 q^{47} -1.00000 q^{49} -2.02922i q^{53} -17.7408 q^{55} +5.61642i q^{61} -3.00000 q^{65} -2.24264i q^{67} -11.4069 q^{71} +7.48528 q^{73} +10.3923i q^{77} -5.91359 q^{79} +6.00000i q^{83} -12.5446i q^{85} -17.4853 q^{89} +1.75736i q^{91} +26.1039 q^{95} +0.485281 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 24 q^{17} - 32 q^{25} - 8 q^{49} - 24 q^{65} - 8 q^{73} - 72 q^{89} - 64 q^{97}+O(q^{100}) \) 8 * q - 24 * q^17 - 32 * q^25 - 8 * q^49 - 24 * q^65 - 8 * q^73 - 72 * q^89 - 64 * q^97

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