LMFDB - Embedded newform 5184.2.d.q.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:	$12$
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} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 $ x^12 - 16x^8 - 24x^7 + 96x^5 + 304x^4 + 384x^3 + 288x^2 + 144*x + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 576)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2593.8
Root			$2.17840 + 0.583700i$ of defining polynomial
Character	$\chi$	$=$	5184.2593
Dual form			5184.2.d.q.2593.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{5} -3.61328 q^{7} +O(q^{10})$	$q+1.73205i q^{5} -3.61328 q^{7} -0.734191i q^{11} -0.609577i q^{13} -5.52420 q^{17} -2.00000i q^{19} +4.73576 q^{23} +2.00000 q^{25} -7.83614i q^{29} +9.41901 q^{31} -6.25839i q^{35} +2.34163i q^{37} -8.52420 q^{41} -10.2584i q^{43} +11.7606 q^{47} +6.05582 q^{49} +13.0323i q^{53} +1.27166 q^{55} +1.20999i q^{59} +11.3002i q^{61} +1.05582 q^{65} +6.31421i q^{67} -2.63999 q^{71} -2.05582 q^{73} +2.65284i q^{77} -2.49081 q^{79} +12.2584i q^{83} -9.56819i q^{85} -1.94418 q^{89} +2.20257i q^{91} +3.46410 q^{95} -15.5726 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q + 24 q^{25} - 36 q^{41} + 48 q^{49} - 12 q^{65} - 48 q^{89} + 12 q^{97}+O(q^{100}) $ 12 * q + 24 * q^25 - 36 * q^41 + 48 * q^49 - 12 * q^65 - 48 * q^89 + 12 * 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
$5$	1.73205i	0.774597i	0.921954	+	0.387298i	$0.126592\pi$
$5$	1.73205i	0.774597i	−0.921954	+	0.387298i	$0.873408\pi$
$6$	0	0
$7$	−3.61328	−1.36569	−0.682846	−	0.730562i	$-0.739258\pi$
$7$	−3.61328	−1.36569	−0.682846	+	0.730562i	$0.739258\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 0.734191i	− 0.221367i	−0.993856	−	0.110683i	$-0.964696\pi$
$11$	− 0.734191i	− 0.221367i	0.993856	−	0.110683i	$-0.0353040\pi$
$12$	0	0
$13$	− 0.609577i	− 0.169066i	−0.996421	−	0.0845331i	$-0.973060\pi$
$13$	− 0.609577i	− 0.169066i	0.996421	−	0.0845331i	$-0.0269399\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.52420	−1.33982	−0.669908	−	0.742444i	$-0.733666\pi$
$17$	−5.52420	−1.33982	−0.669908	+	0.742444i	$0.733666\pi$
$18$	0	0
$19$	− 2.00000i	− 0.458831i	−0.973329	−	0.229416i	$-0.926318\pi$
$19$	− 2.00000i	− 0.458831i	0.973329	−	0.229416i	$-0.0736815\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.73576	0.987474	0.493737	−	0.869611i	$-0.335631\pi$
$23$	4.73576	0.987474	0.493737	+	0.869611i	$0.335631\pi$
$24$	0	0
$25$	2.00000	0.400000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 7.83614i	− 1.45514i	−0.686036	−	0.727568i	$-0.740651\pi$
$29$	− 7.83614i	− 1.45514i	0.686036	−	0.727568i	$-0.259349\pi$
$30$	0	0
$31$	9.41901	1.69170	0.845852	−	0.533417i	$-0.179092\pi$
$31$	9.41901	1.69170	0.845852	+	0.533417i	$0.179092\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 6.25839i	− 1.05786i
$36$	0	0
$37$	2.34163i	0.384961i	0.981301	+	0.192481i	$0.0616532\pi$
$37$	2.34163i	0.384961i	−0.981301	+	0.192481i	$0.938347\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.52420	−1.33126	−0.665628	−	0.746284i	$-0.731836\pi$
$41$	−8.52420	−1.33126	−0.665628	+	0.746284i	$0.731836\pi$
$42$	0	0
$43$	− 10.2584i	− 1.56439i	−0.623034	−	0.782195i	$-0.714100\pi$
$43$	− 10.2584i	− 1.56439i	0.623034	−	0.782195i	$-0.285900\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.7606	1.71547	0.857733	−	0.514096i	$-0.171872\pi$
$47$	11.7606	1.71547	0.857733	+	0.514096i	$0.171872\pi$
$48$	0	0
$49$	6.05582	0.865117
$50$	0	0
$51$	0	0
$52$	0	0
$53$	13.0323i	1.79012i	0.445942	+	0.895062i	$0.352869\pi$
$53$	13.0323i	1.79012i	−0.445942	+	0.895062i	$0.647131\pi$
$54$	0	0
$55$	1.27166	0.171470
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.20999i	0.157527i	0.996893	+	0.0787637i	$0.0250973\pi$
$59$	1.20999i	0.157527i	−0.996893	+	0.0787637i	$0.974903\pi$
$60$	0	0
$61$	11.3002i	1.44685i	0.690404	+	0.723424i	$0.257433\pi$
$61$	11.3002i	1.44685i	−0.690404	+	0.723424i	$0.742567\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.05582	0.130958
$66$	0	0
$67$	6.31421i	0.771403i	0.922624	+	0.385702i	$0.126041\pi$
$67$	6.31421i	0.771403i	−0.922624	+	0.385702i	$0.873959\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−2.63999	−0.313309	−0.156655	−	0.987653i	$-0.550071\pi$
$71$	−2.63999	−0.313309	−0.156655	+	0.987653i	$0.550071\pi$
$72$	0	0
$73$	−2.05582	−0.240615	−0.120308	−	0.992737i	$-0.538388\pi$
$73$	−2.05582	−0.240615	−0.120308	+	0.992737i	$0.538388\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.65284i	0.302319i
$78$	0	0
$79$	−2.49081	−0.280238	−0.140119	−	0.990135i	$-0.544749\pi$
$79$	−2.49081	−0.280238	−0.140119	+	0.990135i	$0.544749\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	12.2584i	1.34553i	0.739855	+	0.672767i	$0.234894\pi$
$83$	12.2584i	1.34553i	−0.739855	+	0.672767i	$0.765106\pi$
$84$	0	0
$85$	− 9.56819i	− 1.03782i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1.94418	−0.206083	−0.103041	−	0.994677i	$-0.532857\pi$
$89$	−1.94418	−0.206083	−0.103041	+	0.994677i	$0.532857\pi$
$90$	0	0
$91$	2.20257i	0.230892i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.46410	0.355409
$96$	0	0
$97$	−15.5726	−1.58116	−0.790579	−	0.612360i	$-0.790220\pi$
$97$	−15.5726	−1.58116	−0.790579	+	0.612360i	$0.790220\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	11.2036i	1.11480i	0.830245	+	0.557398i	$0.188200\pi$
$101$	11.2036i	1.11480i	−0.830245	+	0.557398i	$0.811800\pi$
$102$	0	0
$103$	1.36834	0.134826	0.0674130	−	0.997725i	$-0.478525\pi$
$103$	1.36834	0.134826	0.0674130	+	0.997725i	$0.478525\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.4684i	1.30204i	0.759062	+	0.651019i	$0.225658\pi$
$107$	13.4684i	1.30204i	−0.759062	+	0.651019i	$0.774342\pi$
$108$	0	0
$109$	− 1.42084i	− 0.136092i	−0.997682	−	0.0680458i	$-0.978324\pi$
$109$	− 1.42084i	− 0.136092i	0.997682	−	0.0680458i	$-0.0216764\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	9.51678	0.895263	0.447632	−	0.894218i	$-0.352268\pi$
$113$	9.51678	0.895263	0.447632	+	0.894218i	$0.352268\pi$
$114$	0	0
$115$	8.20257i	0.764894i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	19.9605	1.82978
$120$	0	0
$121$	10.4610	0.950997
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.1244i	1.08444i
$126$	0	0
$127$	2.24495	0.199207	0.0996035	−	0.995027i	$-0.468243\pi$
$127$	2.24495	0.199207	0.0996035	+	0.995027i	$0.468243\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0.258391i	0.0225757i	0.999936	+	0.0112878i	$0.00359311\pi$
$131$	0.258391i	0.0225757i	−0.999936	+	0.0112878i	$0.996407\pi$
$132$	0	0
$133$	7.22657i	0.626623i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.00742	−0.171505	−0.0857527	−	0.996316i	$-0.527329\pi$
$137$	−2.00742	−0.171505	−0.0857527	+	0.996316i	$0.527329\pi$
$138$	0	0
$139$	− 2.25839i	− 0.191554i	−0.995403	−	0.0957771i	$-0.969466\pi$
$139$	− 2.25839i	− 0.191554i	0.995403	−	0.0957771i	$-0.0305336\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.447546	−0.0374256
$144$	0	0
$145$	13.5726	1.12714
$146$	0	0
$147$	0	0
$148$	0	0
$149$	14.7643i	1.20954i	0.796399	+	0.604771i	$0.206735\pi$
$149$	14.7643i	1.20954i	−0.796399	+	0.604771i	$0.793265\pi$
$150$	0	0
$151$	−5.65655	−0.460323	−0.230162	−	0.973152i	$-0.573925\pi$
$151$	−5.65655	−0.460323	−0.230162	+	0.973152i	$0.573925\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	16.3142i	1.31039i
$156$	0	0
$157$	− 15.8868i	− 1.26791i	−0.773371	−	0.633953i	$-0.781431\pi$
$157$	− 15.8868i	− 1.26791i	0.773371	−	0.633953i	$-0.218569\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−17.1116	−1.34859
$162$	0	0
$163$	− 8.51678i	− 0.667086i	−0.942735	−	0.333543i	$-0.891756\pi$
$163$	− 8.51678i	− 0.667086i	0.942735	−	0.333543i	$-0.108244\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.91165	−0.302692	−0.151346	−	0.988481i	$-0.548361\pi$
$167$	−3.91165	−0.302692	−0.151346	+	0.988481i	$0.548361\pi$
$168$	0	0
$169$	12.6284	0.971417
$170$	0	0
$171$	0	0
$172$	0	0
$173$	12.1244i	0.921798i	0.887453	+	0.460899i	$0.152473\pi$
$173$	12.1244i	0.921798i	−0.887453	+	0.460899i	$0.847527\pi$
$174$	0	0
$175$	−7.22657	−0.546277
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 6.00000i	− 0.448461i	−0.974536	−	0.224231i	$-0.928013\pi$
$179$	− 6.00000i	− 0.448461i	0.974536	−	0.224231i	$-0.0719869\pi$
$180$	0	0
$181$	− 2.34163i	− 0.174052i	−0.996206	−	0.0870259i	$-0.972264\pi$
$181$	− 2.34163i	− 0.174052i	0.996206	−	0.0870259i	$-0.0277363\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−4.05582	−0.298190
$186$	0	0
$187$	4.05582i	0.296591i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.36834	−0.0990092	−0.0495046	−	0.998774i	$-0.515764\pi$
$191$	−1.36834	−0.0990092	−0.0495046	+	0.998774i	$0.515764\pi$
$192$	0	0
$193$	−15.4610	−1.11290	−0.556452	−	0.830880i	$-0.687838\pi$
$193$	−15.4610	−1.11290	−0.556452	+	0.830880i	$0.687838\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.9356i	0.921625i	0.887498	+	0.460812i	$0.152442\pi$
$197$	12.9356i	0.921625i	−0.887498	+	0.460812i	$0.847558\pi$
$198$	0	0
$199$	−18.7413	−1.32854	−0.664269	−	0.747493i	$-0.731257\pi$
$199$	−18.7413	−1.32854	−0.664269	+	0.747493i	$0.731257\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	28.3142i	1.98727i
$204$	0	0
$205$	− 14.7643i	− 1.03119i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.46838	−0.101570
$210$	0	0
$211$	− 8.31421i	− 0.572374i	−0.958174	−	0.286187i	$-0.907612\pi$
$211$	− 8.31421i	− 0.572374i	0.958174	−	0.286187i	$-0.0923878\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	17.7681	1.21177
$216$	0	0
$217$	−34.0336	−2.31035
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3.36742i	0.226517i
$222$	0	0
$223$	7.99817	0.535597	0.267799	−	0.963475i	$-0.413704\pi$
$223$	7.99817	0.535597	0.267799	+	0.963475i	$0.413704\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 5.26581i	− 0.349504i	−0.984612	−	0.174752i	$-0.944088\pi$
$227$	− 5.26581i	− 0.349504i	0.984612	−	0.174752i	$-0.0559124\pi$
$228$	0	0
$229$	16.1852i	1.06955i	0.844995	+	0.534774i	$0.179603\pi$
$229$	16.1852i	1.06955i	−0.844995	+	0.534774i	$0.820397\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	28.0968	1.84068	0.920341	−	0.391116i	$-0.127911\pi$
$233$	28.0968	1.84068	0.920341	+	0.391116i	$0.127911\pi$
$234$	0	0
$235$	20.3700i	1.32879i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	18.5922	1.20263	0.601314	−	0.799013i	$-0.294644\pi$
$239$	18.5922	1.20263	0.601314	+	0.799013i	$0.294644\pi$
$240$	0	0
$241$	−9.11164	−0.586932	−0.293466	−	0.955970i	$-0.594809\pi$
$241$	−9.11164	−0.586932	−0.293466	+	0.955970i	$0.594809\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	10.4890i	0.670117i
$246$	0	0
$247$	−1.21915	−0.0775729
$248$	0	0
$249$	0	0
$250$	0	0
$251$	21.5800i	1.36212i	0.732228	+	0.681059i	$0.238480\pi$
$251$	21.5800i	1.36212i	−0.732228	+	0.681059i	$0.761520\pi$
$252$	0	0
$253$	− 3.47695i	− 0.218594i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5.41998	−0.338089	−0.169045	−	0.985608i	$-0.554068\pi$
$257$	−5.41998	−0.338089	−0.169045	+	0.985608i	$0.554068\pi$
$258$	0	0
$259$	− 8.46096i	− 0.525739i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−10.0157	−0.617597	−0.308798	−	0.951127i	$-0.599927\pi$
$263$	−10.0157	−0.617597	−0.308798	+	0.951127i	$0.599927\pi$
$264$	0	0
$265$	−22.5726	−1.38662
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 12.1115i	− 0.738452i	−0.929340	−	0.369226i	$-0.879623\pi$
$269$	− 12.1115i	− 0.738452i	0.929340	−	0.369226i	$-0.120377\pi$
$270$	0	0
$271$	20.0572	1.21839	0.609193	−	0.793022i	$-0.291493\pi$
$271$	20.0572	1.21839	0.609193	+	0.793022i	$0.291493\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1.46838i	− 0.0885468i
$276$	0	0
$277$	− 5.49452i	− 0.330133i	−0.986282	−	0.165067i	$-0.947216\pi$
$277$	− 5.49452i	− 0.330133i	0.986282	−	0.165067i	$-0.0527840\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−9.16745	−0.546884	−0.273442	−	0.961888i	$-0.588162\pi$
$281$	−9.16745	−0.546884	−0.273442	+	0.961888i	$0.588162\pi$
$282$	0	0
$283$	22.7752i	1.35384i	0.736055	+	0.676922i	$0.236686\pi$
$283$	22.7752i	1.35384i	−0.736055	+	0.676922i	$0.763314\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	30.8003	1.81809
$288$	0	0
$289$	13.5168	0.795105
$290$	0	0
$291$	0	0
$292$	0	0
$293$	21.5959i	1.26164i	0.775927	+	0.630822i	$0.217282\pi$
$293$	21.5959i	1.26164i	−0.775927	+	0.630822i	$0.782718\pi$
$294$	0	0
$295$	−2.09577	−0.122020
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 2.88681i	− 0.166948i
$300$	0	0
$301$	37.0665i	2.13648i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−19.5726	−1.12072
$306$	0	0
$307$	8.51678i	0.486078i	0.970016	+	0.243039i	$0.0781444\pi$
$307$	8.51678i	0.486078i	−0.970016	+	0.243039i	$0.921856\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	24.7929	1.40588	0.702939	−	0.711250i	$-0.251871\pi$
$311$	24.7929	1.40588	0.702939	+	0.711250i	$0.251871\pi$
$312$	0	0
$313$	15.1116	0.854160	0.427080	−	0.904214i	$-0.359542\pi$
$313$	15.1116	0.854160	0.427080	+	0.904214i	$0.359542\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.2210i	0.686402i	0.939262	+	0.343201i	$0.111511\pi$
$317$	12.2210i	0.686402i	−0.939262	+	0.343201i	$0.888489\pi$
$318$	0	0
$319$	−5.75323	−0.322119
$320$	0	0
$321$	0	0
$322$	0	0
$323$	11.0484i	0.614749i
$324$	0	0
$325$	− 1.21915i	− 0.0676264i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−42.4945	−2.34280
$330$	0	0
$331$	24.7194i	1.35870i	0.733815	+	0.679349i	$0.237738\pi$
$331$	24.7194i	1.35870i	−0.733815	+	0.679349i	$0.762262\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−10.9365	−0.597526
$336$	0	0
$337$	13.5168	0.736306	0.368153	−	0.929765i	$-0.379990\pi$
$337$	13.5168	0.736306	0.368153	+	0.929765i	$0.379990\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 6.91535i	− 0.374487i
$342$	0	0
$343$	3.41160	0.184209
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 15.3626i	− 0.824708i	−0.911024	−	0.412354i	$-0.864707\pi$
$347$	− 15.3626i	− 0.824708i	0.911024	−	0.412354i	$-0.135293\pi$
$348$	0	0
$349$	− 31.5591i	− 1.68932i	−0.535303	−	0.844660i	$-0.679803\pi$
$349$	− 31.5591i	− 1.68932i	0.535303	−	0.844660i	$-0.320197\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−27.9926	−1.48989	−0.744947	−	0.667123i	$-0.767525\pi$
$353$	−27.9926	−1.48989	−0.744947	+	0.667123i	$0.767525\pi$
$354$	0	0
$355$	− 4.57260i	− 0.242688i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	13.1290	0.692921	0.346460	−	0.938065i	$-0.387384\pi$
$359$	13.1290	0.692921	0.346460	+	0.938065i	$0.387384\pi$
$360$	0	0
$361$	15.0000	0.789474
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 3.56078i	− 0.186380i
$366$	0	0
$367$	13.4798	0.703642	0.351821	−	0.936067i	$-0.385563\pi$
$367$	13.4798	0.703642	0.351821	+	0.936067i	$0.385563\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 47.0894i	− 2.44476i
$372$	0	0
$373$	5.59120i	0.289501i	0.989468	+	0.144751i	$0.0462380\pi$
$373$	5.59120i	0.289501i	−0.989468	+	0.144751i	$0.953762\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.77673	−0.246014
$378$	0	0
$379$	− 33.0894i	− 1.69969i	−0.527035	−	0.849844i	$-0.676696\pi$
$379$	− 33.0894i	− 1.69969i	0.527035	−	0.849844i	$-0.323304\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	15.2247	0.777948	0.388974	−	0.921249i	$-0.372830\pi$
$383$	15.2247	0.777948	0.388974	+	0.921249i	$0.372830\pi$
$384$	0	0
$385$	−4.59485	−0.234175
$386$	0	0
$387$	0	0
$388$	0	0
$389$	6.11694i	0.310141i	0.987903	+	0.155071i	$0.0495605\pi$
$389$	6.11694i	0.310141i	−0.987903	+	0.155071i	$0.950439\pi$
$390$	0	0
$391$	−26.1613	−1.32303
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 4.31421i	− 0.217071i
$396$	0	0
$397$	22.2054i	1.11446i	0.830358	+	0.557230i	$0.188136\pi$
$397$	22.2054i	1.11446i	−0.830358	+	0.557230i	$0.811864\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	30.1042	1.50333	0.751666	−	0.659543i	$-0.229250\pi$
$401$	30.1042	1.50333	0.751666	+	0.659543i	$0.229250\pi$
$402$	0	0
$403$	− 5.74161i	− 0.286010i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.71920	0.0852177
$408$	0	0
$409$	−9.11164	−0.450541	−0.225271	−	0.974296i	$-0.572327\pi$
$409$	−9.11164	−0.450541	−0.225271	+	0.974296i	$0.572327\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 4.37204i	− 0.215134i
$414$	0	0
$415$	−21.2322	−1.04225
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 22.8310i	− 1.11537i	−0.830054	−	0.557683i	$-0.811690\pi$
$419$	− 22.8310i	− 1.11537i	0.830054	−	0.557683i	$-0.188310\pi$
$420$	0	0
$421$	− 24.5342i	− 1.19573i	−0.801599	−	0.597863i	$-0.796017\pi$
$421$	− 24.5342i	− 1.19573i	0.801599	−	0.597863i	$-0.203983\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−11.0484	−0.535926
$426$	0	0
$427$	− 40.8310i	− 1.97595i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−0.920789	−0.0443529	−0.0221764	−	0.999754i	$-0.507060\pi$
$431$	−0.920789	−0.0443529	−0.0221764	+	0.999754i	$0.507060\pi$
$432$	0	0
$433$	2.05582	0.0987963	0.0493981	−	0.998779i	$-0.484270\pi$
$433$	2.05582	0.0987963	0.0493981	+	0.998779i	$0.484270\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 9.47152i	− 0.453084i
$438$	0	0
$439$	22.0563	1.05269	0.526344	−	0.850272i	$-0.323562\pi$
$439$	22.0563	1.05269	0.526344	+	0.850272i	$0.323562\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	27.1952i	1.29208i	0.763303	+	0.646040i	$0.223576\pi$
$443$	27.1952i	1.29208i	−0.763303	+	0.646040i	$0.776424\pi$
$444$	0	0
$445$	− 3.36742i	− 0.159631i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−9.10422	−0.429655	−0.214827	−	0.976652i	$-0.568919\pi$
$449$	−9.10422	−0.429655	−0.214827	+	0.976652i	$0.568919\pi$
$450$	0	0
$451$	6.25839i	0.294696i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−3.81497	−0.178848
$456$	0	0
$457$	29.6842	1.38857	0.694285	−	0.719700i	$-0.255721\pi$
$457$	29.6842	1.38857	0.694285	+	0.719700i	$0.255721\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	23.5084i	1.09490i	0.836839	+	0.547448i	$0.184401\pi$
$461$	23.5084i	1.09490i	−0.836839	+	0.547448i	$0.815599\pi$
$462$	0	0
$463$	7.90150	0.367214	0.183607	−	0.983000i	$-0.441223\pi$
$463$	7.90150	0.367214	0.183607	+	0.983000i	$0.441223\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 3.06324i	− 0.141750i	−0.997485	−	0.0708748i	$-0.977421\pi$
$467$	− 3.06324i	− 0.141750i	0.997485	−	0.0708748i	$-0.0225791\pi$
$468$	0	0
$469$	− 22.8150i	− 1.05350i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−7.53162	−0.346304
$474$	0	0
$475$	− 4.00000i	− 0.183533i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	14.2073	0.649147	0.324573	−	0.945860i	$-0.394779\pi$
$479$	14.2073	0.649147	0.324573	+	0.945860i	$0.394779\pi$
$480$	0	0
$481$	1.42740	0.0650839
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 26.9725i	− 1.22476i
$486$	0	0
$487$	22.1088	1.00184	0.500922	−	0.865492i	$-0.332994\pi$
$487$	22.1088	1.00184	0.500922	+	0.865492i	$0.332994\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	39.5301i	1.78397i	0.452068	+	0.891983i	$0.350686\pi$
$491$	39.5301i	1.78397i	−0.452068	+	0.891983i	$0.649314\pi$
$492$	0	0
$493$	43.2884i	1.94961i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.53904	0.427884
$498$	0	0
$499$	− 11.8532i	− 0.530624i	−0.964163	−	0.265312i	$-0.914525\pi$
$499$	− 11.8532i	− 0.530624i	0.964163	−	0.265312i	$-0.0854750\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	23.4246	1.04445	0.522226	−	0.852807i	$-0.325102\pi$
$503$	23.4246	1.04445	0.522226	+	0.852807i	$0.325102\pi$
$504$	0	0
$505$	−19.4051	−0.863518
$506$	0	0
$507$	0	0
$508$	0	0
$509$	7.83614i	0.347331i	0.984805	+	0.173665i	$0.0555611\pi$
$509$	7.83614i	0.347331i	−0.984805	+	0.173665i	$0.944439\pi$
$510$	0	0
$511$	7.42825	0.328607
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2.37003i	0.104436i
$516$	0	0
$517$	− 8.63456i	− 0.379747i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	27.4535	1.20276	0.601381	−	0.798963i	$-0.294617\pi$
$521$	27.4535	1.20276	0.601381	+	0.798963i	$0.294617\pi$
$522$	0	0
$523$	− 3.83255i	− 0.167586i	−0.996483	−	0.0837928i	$-0.973297\pi$
$523$	− 3.83255i	− 0.167586i	0.996483	−	0.0837928i	$-0.0267034\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−52.0325	−2.26657
$528$	0	0
$529$	−0.572599	−0.0248956
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5.19615i	0.225070i
$534$	0	0
$535$	−23.3279	−1.00855
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 4.44613i	− 0.191508i
$540$	0	0
$541$	− 15.9707i	− 0.686632i	−0.939220	−	0.343316i	$-0.888450\pi$
$541$	− 15.9707i	− 0.686632i	0.939220	−	0.343316i	$-0.111550\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.46096	0.105416
$546$	0	0
$547$	10.4258i	0.445777i	0.974844	+	0.222888i	$0.0715486\pi$
$547$	10.4258i	0.445777i	−0.974844	+	0.222888i	$0.928451\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−15.6723	−0.667662
$552$	0	0
$553$	9.00000	0.382719
$554$	0	0
$555$	0	0
$556$	0	0
$557$	25.0471i	1.06128i	0.847597	+	0.530640i	$0.178049\pi$
$557$	25.0471i	1.06128i	−0.847597	+	0.530640i	$0.821951\pi$
$558$	0	0
$559$	−6.25327	−0.264485
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 2.63739i	− 0.111153i	−0.998454	−	0.0555764i	$-0.982300\pi$
$563$	− 2.63739i	− 0.111153i	0.998454	−	0.0555764i	$-0.0176996\pi$
$564$	0	0
$565$	16.4835i	0.693468i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−20.0484	−0.840473	−0.420236	−	0.907415i	$-0.638053\pi$
$569$	−20.0484	−0.840473	−0.420236	+	0.907415i	$0.638053\pi$
$570$	0	0
$571$	2.31421i	0.0968466i	0.998827	+	0.0484233i	$0.0154196\pi$
$571$	2.31421i	0.0968466i	−0.998827	+	0.0484233i	$0.984580\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	9.47152	0.394989
$576$	0	0
$577$	24.4610	1.01832	0.509162	−	0.860671i	$-0.329956\pi$
$577$	24.4610	1.01832	0.509162	+	0.860671i	$0.329956\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 44.2930i	− 1.83758i
$582$	0	0
$583$	9.56819	0.396274
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19.7268i	0.814211i	0.913381	+	0.407106i	$0.133462\pi$
$587$	19.7268i	0.814211i	−0.913381	+	0.407106i	$0.866538\pi$
$588$	0	0
$589$	− 18.8380i	− 0.776207i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	8.46096	0.347450	0.173725	−	0.984794i	$-0.444420\pi$
$593$	8.46096	0.347450	0.173725	+	0.984794i	$0.444420\pi$
$594$	0	0
$595$	34.5726i	1.41734i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	6.45496	0.263742	0.131871	−	0.991267i	$-0.457901\pi$
$599$	6.45496	0.263742	0.131871	+	0.991267i	$0.457901\pi$
$600$	0	0
$601$	−5.05582	−0.206231	−0.103116	−	0.994669i	$-0.532881\pi$
$601$	−5.05582	−0.206231	−0.103116	+	0.994669i	$0.532881\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	18.1189i	0.736639i
$606$	0	0
$607$	0.0441760	0.00179305	0.000896524	−	1.00000i	$-0.499715\pi$
$607$	0.0441760	0.00179305	0.000896524		1.00000i	$0.499715\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 7.16901i	− 0.290027i
$612$	0	0
$613$	26.8887i	1.08602i	0.839725	+	0.543012i	$0.182716\pi$
$613$	26.8887i	1.08602i	−0.839725	+	0.543012i	$0.817284\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18.4126	0.741262	0.370631	−	0.928780i	$-0.379141\pi$
$617$	18.4126	0.741262	0.370631	+	0.928780i	$0.379141\pi$
$618$	0	0
$619$	− 1.90906i	− 0.0767317i	−0.999264	−	0.0383658i	$-0.987785\pi$
$619$	− 1.90906i	− 0.0767317i	0.999264	−	0.0383658i	$-0.0122152\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	7.02488	0.281446
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 12.9356i	− 0.515777i
$630$	0	0
$631$	1.51752	0.0604114	0.0302057	−	0.999544i	$-0.490384\pi$
$631$	1.51752	0.0604114	0.0302057	+	0.999544i	$0.490384\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	3.88836i	0.154305i
$636$	0	0
$637$	− 3.69148i	− 0.146262i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	16.9852	0.670874	0.335437	−	0.942063i	$-0.391116\pi$
$641$	16.9852	0.670874	0.335437	+	0.942063i	$0.391116\pi$
$642$	0	0
$643$	10.4258i	0.411155i	0.978641	+	0.205578i	$0.0659073\pi$
$643$	10.4258i	0.411155i	−0.978641	+	0.205578i	$0.934093\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−44.2802	−1.74083	−0.870417	−	0.492315i	$-0.836151\pi$
$647$	−44.2802	−1.74083	−0.870417	+	0.492315i	$0.836151\pi$
$648$	0	0
$649$	0.888365	0.0348714
$650$	0	0
$651$	0	0
$652$	0	0
$653$	33.0766i	1.29439i	0.762325	+	0.647194i	$0.224058\pi$
$653$	33.0766i	1.29439i	−0.762325	+	0.647194i	$0.775942\pi$
$654$	0	0
$655$	−0.447546	−0.0174871
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 32.2026i	− 1.25443i	−0.778844	−	0.627217i	$-0.784194\pi$
$659$	− 32.2026i	− 1.25443i	0.778844	−	0.627217i	$-0.215806\pi$
$660$	0	0
$661$	13.0194i	0.506398i	0.967414	+	0.253199i	$0.0814827\pi$
$661$	13.0194i	0.506398i	−0.967414	+	0.253199i	$0.918517\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−12.5168	−0.485380
$666$	0	0
$667$	− 37.1101i	− 1.43691i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	8.29654	0.320284
$672$	0	0
$673$	−7.46096	−0.287599	−0.143800	−	0.989607i	$-0.545932\pi$
$673$	−7.46096	−0.287599	−0.143800	+	0.989607i	$0.545932\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 35.6456i	− 1.36997i	−0.728556	−	0.684987i	$-0.759808\pi$
$677$	− 35.6456i	− 1.36997i	0.728556	−	0.684987i	$-0.240192\pi$
$678$	0	0
$679$	56.2682	2.15938
$680$	0	0
$681$	0	0
$682$	0	0
$683$	27.4535i	1.05048i	0.850954	+	0.525240i	$0.176025\pi$
$683$	27.4535i	1.05048i	−0.850954	+	0.525240i	$0.823975\pi$
$684$	0	0
$685$	− 3.47695i	− 0.132847i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	7.94418	0.302649
$690$	0	0
$691$	− 22.3700i	− 0.850996i	−0.904959	−	0.425498i	$-0.860099\pi$
$691$	− 22.3700i	− 0.850996i	0.904959	−	0.425498i	$-0.139901\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.91165	0.148377
$696$	0	0
$697$	47.0894	1.78364
$698$	0	0
$699$	0	0
$700$	0	0
$701$	10.4633i	0.395193i	0.980283	+	0.197596i	$0.0633136\pi$
$701$	10.4633i	0.395193i	−0.980283	+	0.197596i	$0.936686\pi$
$702$	0	0
$703$	4.68325	0.176632
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 40.4817i	− 1.52247i
$708$	0	0
$709$	23.5425i	0.884155i	0.896977	+	0.442078i	$0.145758\pi$
$709$	23.5425i	0.884155i	−0.896977	+	0.442078i	$0.854242\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	44.6062	1.67051
$714$	0	0
$715$	− 0.775172i	− 0.0289898i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−45.0076	−1.67850	−0.839251	−	0.543745i	$-0.817006\pi$
$719$	−45.0076	−1.67850	−0.839251	+	0.543745i	$0.817006\pi$
$720$	0	0
$721$	−4.94418	−0.184131
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 15.6723i	− 0.582054i
$726$	0	0
$727$	−16.7506	−0.621245	−0.310622	−	0.950533i	$-0.600537\pi$
$727$	−16.7506	−0.621245	−0.310622	+	0.950533i	$0.600537\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	56.6694i	2.09599i
$732$	0	0
$733$	24.3409i	0.899050i	0.893268	+	0.449525i	$0.148407\pi$
$733$	24.3409i	0.899050i	−0.893268	+	0.449525i	$0.851593\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.63583	0.170763
$738$	0	0
$739$	− 32.0558i	− 1.17919i	−0.807698	−	0.589596i	$-0.799287\pi$
$739$	− 32.0558i	− 1.17919i	0.807698	−	0.589596i	$-0.200713\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−17.9614	−0.658940	−0.329470	−	0.944166i	$-0.606870\pi$
$743$	−17.9614	−0.658940	−0.329470	+	0.944166i	$0.606870\pi$
$744$	0	0
$745$	−25.5726	−0.936908
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 48.6651i	− 1.77818i
$750$	0	0
$751$	−21.8546	−0.797485	−0.398742	−	0.917063i	$-0.630553\pi$
$751$	−21.8546	−0.797485	−0.398742	+	0.917063i	$0.630553\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 9.79743i	− 0.356565i
$756$	0	0
$757$	52.1292i	1.89467i	0.320248	+	0.947334i	$0.396234\pi$
$757$	52.1292i	1.89467i	−0.320248	+	0.947334i	$0.603766\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	32.0894	1.16324	0.581620	−	0.813461i	$-0.302419\pi$
$761$	32.0894	1.16324	0.581620	+	0.813461i	$0.302419\pi$
$762$	0	0
$763$	5.13389i	0.185859i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0.737582	0.0266326
$768$	0	0
$769$	3.22327	0.116234	0.0581171	−	0.998310i	$-0.481490\pi$
$769$	3.22327	0.116234	0.0581171	+	0.998310i	$0.481490\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	19.2331i	0.691765i	0.938278	+	0.345883i	$0.112420\pi$
$773$	19.2331i	0.691765i	−0.938278	+	0.345883i	$0.887580\pi$
$774$	0	0
$775$	18.8380	0.676682
$776$	0	0
$777$	0	0
$778$	0	0
$779$	17.0484i	0.610822i
$780$	0	0
$781$	1.93826i	0.0693563i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	27.5168	0.982116
$786$	0	0
$787$	36.3700i	1.29645i	0.761448	+	0.648226i	$0.224489\pi$
$787$	36.3700i	1.29645i	−0.761448	+	0.648226i	$0.775511\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−34.3868	−1.22265
$792$	0	0
$793$	6.88836	0.244613
$794$	0	0
$795$	0	0
$796$	0	0
$797$	14.9577i	0.529829i	0.964272	+	0.264915i	$0.0853437\pi$
$797$	14.9577i	0.529829i	−0.964272	+	0.264915i	$0.914656\pi$
$798$	0	0
$799$	−64.9681	−2.29841
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.50936i	0.0532643i
$804$	0	0
$805$	− 29.6382i	− 1.04461i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−0.0409808	−0.00144081	−0.000720404	−	1.00000i	$-0.500229\pi$
$809$	−0.0409808	−0.00144081	−0.000720404		1.00000i	$0.500229\pi$
$810$	0	0
$811$	43.2568i	1.51895i	0.650535	+	0.759476i	$0.274545\pi$
$811$	43.2568i	1.51895i	−0.650535	+	0.759476i	$0.725455\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	14.7515	0.516722
$816$	0	0
$817$	−20.5168	−0.717791
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 28.6208i	− 0.998871i	−0.866351	−	0.499436i	$-0.833541\pi$
$821$	− 28.6208i	− 0.998871i	0.866351	−	0.499436i	$-0.166459\pi$
$822$	0	0
$823$	42.4118	1.47838	0.739191	−	0.673495i	$-0.235208\pi$
$823$	42.4118	1.47838	0.739191	+	0.673495i	$0.235208\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	14.1116i	0.490710i	0.969433	+	0.245355i	$0.0789045\pi$
$827$	14.1116i	0.490710i	−0.969433	+	0.245355i	$0.921096\pi$
$828$	0	0
$829$	− 22.6088i	− 0.785237i	−0.919701	−	0.392618i	$-0.871569\pi$
$829$	− 22.6088i	− 0.785237i	0.919701	−	0.392618i	$-0.128431\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−33.4535	−1.15910
$834$	0	0
$835$	− 6.77517i	− 0.234464i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−14.1106	−0.487152	−0.243576	−	0.969882i	$-0.578320\pi$
$839$	−14.1106	−0.487152	−0.243576	+	0.969882i	$0.578320\pi$
$840$	0	0
$841$	−32.4051	−1.11742
$842$	0	0
$843$	0	0
$844$	0	0
$845$	21.8731i	0.752456i
$846$	0	0
$847$	−37.7984	−1.29877
$848$	0	0
$849$	0	0
$850$	0	0
$851$	11.0894i	0.380139i
$852$	0	0
$853$	− 42.4772i	− 1.45439i	−0.686431	−	0.727195i	$-0.740824\pi$
$853$	− 42.4772i	− 1.45439i	0.686431	−	0.727195i	$-0.259176\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	33.6842	1.15063	0.575316	−	0.817931i	$-0.304879\pi$
$857$	33.6842	1.15063	0.575316	+	0.817931i	$0.304879\pi$
$858$	0	0
$859$	45.9762i	1.56869i	0.620327	+	0.784344i	$0.287000\pi$
$859$	45.9762i	1.56869i	−0.620327	+	0.784344i	$0.713000\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−31.8786	−1.08516	−0.542581	−	0.840004i	$-0.682553\pi$
$863$	−31.8786	−1.08516	−0.542581	+	0.840004i	$0.682553\pi$
$864$	0	0
$865$	−21.0000	−0.714021
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.82873i	0.0620354i
$870$	0	0
$871$	3.84899	0.130418
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 43.8087i	− 1.48101i
$876$	0	0
$877$	− 52.4147i	− 1.76992i	−0.465668	−	0.884959i	$-0.654186\pi$
$877$	− 52.4147i	− 1.76992i	0.465668	−	0.884959i	$-0.345814\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	34.0558	1.14737	0.573685	−	0.819076i	$-0.305513\pi$
$881$	34.0558	1.14737	0.573685	+	0.819076i	$0.305513\pi$
$882$	0	0
$883$	10.0000i	0.336527i	0.985742	+	0.168263i	$0.0538159\pi$
$883$	10.0000i	0.336527i	−0.985742	+	0.168263i	$0.946184\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−36.1060	−1.21232	−0.606161	−	0.795342i	$-0.707291\pi$
$887$	−36.1060	−1.21232	−0.606161	+	0.795342i	$0.707291\pi$
$888$	0	0
$889$	−8.11164	−0.272056
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 23.5213i	− 0.787110i
$894$	0	0
$895$	10.3923	0.347376
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 73.8087i	− 2.46166i
$900$	0	0
$901$	− 71.9930i	− 2.39843i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4.05582	0.134820
$906$	0	0
$907$	− 42.3700i	− 1.40687i	−0.710758	−	0.703437i	$-0.751648\pi$
$907$	− 42.3700i	− 1.40687i	0.710758	−	0.703437i	$-0.248352\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−50.6642	−1.67858	−0.839289	−	0.543685i	$-0.817029\pi$
$911$	−50.6642	−1.67858	−0.839289	+	0.543685i	$0.817029\pi$
$912$	0	0
$913$	9.00000	0.297857
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 0.933638i	− 0.0308315i
$918$	0	0
$919$	−25.3372	−0.835796	−0.417898	−	0.908494i	$-0.637233\pi$
$919$	−25.3372	−0.835796	−0.417898	+	0.908494i	$0.637233\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.60928i	0.0529700i
$924$	0	0
$925$	4.68325i	0.153984i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−15.3493	−0.503595	−0.251798	−	0.967780i	$-0.581022\pi$
$929$	−15.3493	−0.503595	−0.251798	+	0.967780i	$0.581022\pi$
$930$	0	0
$931$	− 12.1116i	− 0.396943i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−7.02488	−0.229738
$936$	0	0
$937$	−11.5390	−0.376964	−0.188482	−	0.982077i	$-0.560357\pi$
$937$	−11.5390	−0.376964	−0.188482	+	0.982077i	$0.560357\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 33.1023i	− 1.07910i	−0.841952	−	0.539552i	$-0.818593\pi$
$941$	− 33.1023i	− 1.07910i	0.841952	−	0.539552i	$-0.181407\pi$
$942$	0	0
$943$	−40.3685	−1.31458
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 17.2658i	− 0.561063i	−0.959845	−	0.280532i	$-0.909489\pi$
$947$	− 17.2658i	− 0.561063i	0.959845	−	0.280532i	$-0.0905108\pi$
$948$	0	0
$949$	1.25318i	0.0406799i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−29.2159	−0.946394	−0.473197	−	0.880957i	$-0.656900\pi$
$953$	−29.2159	−0.946394	−0.473197	+	0.880957i	$0.656900\pi$
$954$	0	0
$955$	− 2.37003i	− 0.0766922i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	7.25337	0.234224
$960$	0	0
$961$	57.7178	1.86186
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 26.7792i	− 0.862052i
$966$	0	0
$967$	4.00833	0.128899	0.0644495	−	0.997921i	$-0.479471\pi$
$967$	4.00833	0.128899	0.0644495	+	0.997921i	$0.479471\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 20.2791i	− 0.650787i	−0.945579	−	0.325393i	$-0.894503\pi$
$971$	− 20.2791i	− 0.650787i	0.945579	−	0.325393i	$-0.105497\pi$
$972$	0	0
$973$	8.16021i	0.261604i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.30835	0.105843	0.0529217	−	0.998599i	$-0.483147\pi$
$977$	3.30835	0.105843	0.0529217	+	0.998599i	$0.483147\pi$
$978$	0	0
$979$	1.42740i	0.0456199i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	41.1927	1.31384	0.656921	−	0.753960i	$-0.271859\pi$
$983$	41.1927	1.31384	0.656921	+	0.753960i	$0.271859\pi$
$984$	0	0
$985$	−22.4051	−0.713887
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 48.5813i	− 1.54479i
$990$	0	0
$991$	44.2802	1.40661	0.703303	−	0.710890i	$-0.251708\pi$
$991$	44.2802	1.40661	0.703303	+	0.710890i	$0.251708\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 32.4610i	− 1.02908i
$996$	0	0
$997$	35.6199i	1.12809i	0.825742	+	0.564047i	$0.190756\pi$
$997$	35.6199i	1.12809i	−0.825742	+	0.564047i	$0.809244\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5184.2.d.q.2593.8		12
3.2	odd	2		5184.2.d.r.2593.2		12
4.3	odd	2	inner	5184.2.d.q.2593.11		12
8.3	odd	2	inner	5184.2.d.q.2593.5		12
8.5	even	2	inner	5184.2.d.q.2593.2		12
9.2	odd	6		1728.2.r.f.1441.5		12
9.4	even	3		576.2.r.f.97.6	yes	12
9.5	odd	6		1728.2.r.e.289.5		12
9.7	even	3		576.2.r.e.481.1	yes	12
12.11	even	2		5184.2.d.r.2593.5		12
24.5	odd	2		5184.2.d.r.2593.8		12
24.11	even	2		5184.2.d.r.2593.11		12
36.7	odd	6		576.2.r.e.481.6	yes	12
36.11	even	6		1728.2.r.f.1441.2		12
36.23	even	6		1728.2.r.e.289.2		12
36.31	odd	6		576.2.r.f.97.1	yes	12
72.5	odd	6		1728.2.r.f.289.5		12
72.11	even	6		1728.2.r.e.1441.2		12
72.13	even	6		576.2.r.e.97.1	✓	12
72.29	odd	6		1728.2.r.e.1441.5		12
72.43	odd	6		576.2.r.f.481.1	yes	12
72.59	even	6		1728.2.r.f.289.2		12
72.61	even	6		576.2.r.f.481.6	yes	12
72.67	odd	6		576.2.r.e.97.6	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.r.e.97.1	✓	12	72.13	even	6
576.2.r.e.97.6	yes	12	72.67	odd	6
576.2.r.e.481.1	yes	12	9.7	even	3
576.2.r.e.481.6	yes	12	36.7	odd	6
576.2.r.f.97.1	yes	12	36.31	odd	6
576.2.r.f.97.6	yes	12	9.4	even	3
576.2.r.f.481.1	yes	12	72.43	odd	6
576.2.r.f.481.6	yes	12	72.61	even	6
1728.2.r.e.289.2		12	36.23	even	6
1728.2.r.e.289.5		12	9.5	odd	6
1728.2.r.e.1441.2		12	72.11	even	6
1728.2.r.e.1441.5		12	72.29	odd	6
1728.2.r.f.289.2		12	72.59	even	6
1728.2.r.f.289.5		12	72.5	odd	6
1728.2.r.f.1441.2		12	36.11	even	6
1728.2.r.f.1441.5		12	9.2	odd	6
5184.2.d.q.2593.2		12	8.5	even	2	inner
5184.2.d.q.2593.5		12	8.3	odd	2	inner
5184.2.d.q.2593.8		12	1.1	even	1	trivial
5184.2.d.q.2593.11		12	4.3	odd	2	inner
5184.2.d.r.2593.2		12	3.2	odd	2
5184.2.d.r.2593.5		12	12.11	even	2
5184.2.d.r.2593.8		12	24.5	odd	2
5184.2.d.r.2593.11		12	24.11	even	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)

\(f(q)\)	\(=\)	\(q+1.73205i q^{5} -3.61328 q^{7} +O(q^{10})\)	\(q+1.73205i q^{5} -3.61328 q^{7} -0.734191i q^{11} -0.609577i q^{13} -5.52420 q^{17} -2.00000i q^{19} +4.73576 q^{23} +2.00000 q^{25} -7.83614i q^{29} +9.41901 q^{31} -6.25839i q^{35} +2.34163i q^{37} -8.52420 q^{41} -10.2584i q^{43} +11.7606 q^{47} +6.05582 q^{49} +13.0323i q^{53} +1.27166 q^{55} +1.20999i q^{59} +11.3002i q^{61} +1.05582 q^{65} +6.31421i q^{67} -2.63999 q^{71} -2.05582 q^{73} +2.65284i q^{77} -2.49081 q^{79} +12.2584i q^{83} -9.56819i q^{85} -1.94418 q^{89} +2.20257i q^{91} +3.46410 q^{95} -15.5726 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q + 24 q^{25} - 36 q^{41} + 48 q^{49} - 12 q^{65} - 48 q^{89} + 12 q^{97}+O(q^{100}) \) 12 * q + 24 * q^25 - 36 * q^41 + 48 * q^49 - 12 * q^65 - 48 * q^89 + 12 * q^97

Introduction

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