LMFDB - Embedded newform 5184.2.d.r.2593.6

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			2593.6
Root			$-0.180407 - 0.673288i$ of defining polynomial
Character	$\chi$	$=$	5184.2593
Dual form			5184.2.d.r.2593.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{5} +4.35461 q^{7} +O(q^{10})$	$q-1.73205i q^{5} +4.35461 q^{7} -5.83502i q^{11} -4.01989i q^{13} -1.70739 q^{17} -2.00000i q^{19} +6.64245 q^{23} +2.00000 q^{25} -4.68934i q^{29} +4.86143 q^{31} -7.54241i q^{35} +5.75194i q^{37} +1.29261 q^{41} +3.54241i q^{43} -10.6134 q^{47} +11.9627 q^{49} -0.506816i q^{53} -10.1066 q^{55} -1.87237i q^{59} -1.22523i q^{61} -6.96265 q^{65} -1.57976i q^{67} -9.88549 q^{71} -7.96265 q^{73} -25.4093i q^{77} +2.06677 q^{79} +1.54241i q^{83} +2.95729i q^{85} -3.96265 q^{89} -17.5051i q^{91} -3.46410 q^{95} +6.12217 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.873408\pi$
$5$	− 1.73205i	− 0.774597i	0.921954	−	0.387298i	$-0.126592\pi$
$6$	0	0
$7$	4.35461	1.64589	0.822944	−	0.568122i	$-0.192330\pi$
$7$	4.35461	1.64589	0.822944	+	0.568122i	$0.192330\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 5.83502i	− 1.75933i	−0.475598	−	0.879663i	$-0.657768\pi$
$11$	− 5.83502i	− 1.75933i	0.475598	−	0.879663i	$-0.342232\pi$
$12$	0	0
$13$	− 4.01989i	− 1.11492i	−0.830205	−	0.557458i	$-0.811777\pi$
$13$	− 4.01989i	− 1.11492i	0.830205	−	0.557458i	$-0.188223\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.70739	−0.414103	−0.207051	−	0.978330i	$-0.566387\pi$
$17$	−1.70739	−0.414103	−0.207051	+	0.978330i	$0.566387\pi$
$18$	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
$23$	6.64245	1.38505	0.692523	−	0.721395i	$-0.256499\pi$
$23$	6.64245	1.38505	0.692523	+	0.721395i	$0.256499\pi$
$24$	0	0
$25$	2.00000	0.400000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 4.68934i	− 0.870788i	−0.900240	−	0.435394i	$-0.856609\pi$
$29$	− 4.68934i	− 0.870788i	0.900240	−	0.435394i	$-0.143391\pi$
$30$	0	0
$31$	4.86143	0.873138	0.436569	−	0.899671i	$-0.356193\pi$
$31$	4.86143	0.873138	0.436569	+	0.899671i	$0.356193\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 7.54241i	− 1.27490i
$36$	0	0
$37$	5.75194i	0.945613i	0.881166	+	0.472807i	$0.156759\pi$
$37$	5.75194i	0.945613i	−0.881166	+	0.472807i	$0.843241\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.29261	0.201872	0.100936	−	0.994893i	$-0.467816\pi$
$41$	1.29261	0.201872	0.100936	+	0.994893i	$0.467816\pi$
$42$	0	0
$43$	3.54241i	0.540213i	0.962831	+	0.270106i	$0.0870588\pi$
$43$	3.54241i	0.540213i	−0.962831	+	0.270106i	$0.912941\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.6134	−1.54812	−0.774060	−	0.633113i	$-0.781777\pi$
$47$	−10.6134	−1.54812	−0.774060	+	0.633113i	$0.781777\pi$
$48$	0	0
$49$	11.9627	1.70895
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 0.506816i	− 0.0696166i	−0.999394	−	0.0348083i	$-0.988918\pi$
$53$	− 0.506816i	− 0.0696166i	0.999394	−	0.0348083i	$-0.0110821\pi$
$54$	0	0
$55$	−10.1066	−1.36277
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 1.87237i	− 0.243762i	−0.992545	−	0.121881i	$-0.961107\pi$
$59$	− 1.87237i	− 0.243762i	0.992545	−	0.121881i	$-0.0388926\pi$
$60$	0	0
$61$	− 1.22523i	− 0.156875i	−0.996919	−	0.0784376i	$-0.975007\pi$
$61$	− 1.22523i	− 0.156875i	0.996919	−	0.0784376i	$-0.0249931\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.96265	−0.863611
$66$	0	0
$67$	− 1.57976i	− 0.192998i	−0.995333	−	0.0964990i	$-0.969236\pi$
$67$	− 1.57976i	− 0.192998i	0.995333	−	0.0964990i	$-0.0307645\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9.88549	−1.17319	−0.586596	−	0.809880i	$-0.699532\pi$
$71$	−9.88549	−1.17319	−0.586596	+	0.809880i	$0.699532\pi$
$72$	0	0
$73$	−7.96265	−0.931958	−0.465979	−	0.884796i	$-0.654298\pi$
$73$	−7.96265	−0.931958	−0.465979	+	0.884796i	$0.654298\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 25.4093i	− 2.89565i
$78$	0	0
$79$	2.06677	0.232530	0.116265	−	0.993218i	$-0.462908\pi$
$79$	2.06677	0.232530	0.116265	+	0.993218i	$0.462908\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.54241i	0.169302i	0.996411	+	0.0846508i	$0.0269775\pi$
$83$	1.54241i	0.169302i	−0.996411	+	0.0846508i	$0.973023\pi$
$84$	0	0
$85$	2.95729i	0.320763i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3.96265	−0.420040	−0.210020	−	0.977697i	$-0.567353\pi$
$89$	−3.96265	−0.420040	−0.210020	+	0.977697i	$0.567353\pi$
$90$	0	0
$91$	− 17.5051i	− 1.83503i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.46410	−0.355409
$96$	0	0
$97$	6.12217	0.621612	0.310806	−	0.950473i	$-0.399401\pi$
$97$	6.12217	0.621612	0.310806	+	0.950473i	$0.399401\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	11.5529i	1.14955i	0.818311	+	0.574776i	$0.194911\pi$
$101$	11.5529i	1.14955i	−0.818311	+	0.574776i	$0.805089\pi$
$102$	0	0
$103$	0.221065	0.0217822	0.0108911	−	0.999941i	$-0.496533\pi$
$103$	0.221065	0.0217822	0.0108911	+	0.999941i	$0.496533\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 0.329957i	− 0.0318982i	−0.999873	−	0.0159491i	$-0.994923\pi$
$107$	− 0.329957i	− 0.0318982i	0.999873	−	0.0159491i	$-0.00507697\pi$
$108$	0	0
$109$	17.9253i	1.71693i	0.512873	+	0.858465i	$0.328581\pi$
$109$	17.9253i	1.71693i	−0.512873	+	0.858465i	$0.671419\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	18.0848	1.70128	0.850638	−	0.525751i	$-0.176216\pi$
$113$	18.0848	1.70128	0.850638	+	0.525751i	$0.176216\pi$
$114$	0	0
$115$	− 11.5051i	− 1.07285i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−7.43502	−0.681567
$120$	0	0
$121$	−23.0475	−2.09523
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 12.1244i	− 1.08444i
$126$	0	0
$127$	−4.57568	−0.406026	−0.203013	−	0.979176i	$-0.565073\pi$
$127$	−4.57568	−0.406026	−0.203013	+	0.979176i	$0.565073\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	13.5424i	1.18321i	0.806229	+	0.591603i	$0.201505\pi$
$131$	13.5424i	1.18321i	−0.806229	+	0.591603i	$0.798495\pi$
$132$	0	0
$133$	− 8.70923i	− 0.755186i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	22.3774	1.91183	0.955917	−	0.293639i	$-0.0948662\pi$
$137$	22.3774	1.91183	0.955917	+	0.293639i	$0.0948662\pi$
$138$	0	0
$139$	11.5424i	0.979015i	0.871999	+	0.489507i	$0.162823\pi$
$139$	11.5424i	0.979015i	−0.871999	+	0.489507i	$0.837177\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−23.4561	−1.96150
$144$	0	0
$145$	−8.12217	−0.674509
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 2.23887i	− 0.183415i	−0.995786	−	0.0917076i	$-0.970767\pi$
$149$	− 2.23887i	− 0.183415i	0.995786	−	0.0917076i	$-0.0292325\pi$
$150$	0	0
$151$	−17.0348	−1.38627	−0.693134	−	0.720808i	$-0.743771\pi$
$151$	−17.0348	−1.38627	−0.693134	+	0.720808i	$0.743771\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 8.42024i	− 0.676330i
$156$	0	0
$157$	0.0489718i	0.00390837i	0.999998	+	0.00195419i	$0.000622037\pi$
$157$	0.0489718i	0.00390837i	−0.999998	+	0.00195419i	$0.999378\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	28.9253	2.27963
$162$	0	0
$163$	19.0848i	1.49484i	0.664353	+	0.747419i	$0.268707\pi$
$163$	19.0848i	1.49484i	−0.664353	+	0.747419i	$0.731293\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−19.9920	−1.54703	−0.773515	−	0.633778i	$-0.781503\pi$
$167$	−19.9920	−1.54703	−0.773515	+	0.633778i	$0.781503\pi$
$168$	0	0
$169$	−3.15951	−0.243040
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 12.1244i	− 0.921798i	−0.887453	−	0.460899i	$-0.847527\pi$
$173$	− 12.1244i	− 0.921798i	0.887453	−	0.460899i	$-0.152473\pi$
$174$	0	0
$175$	8.70923	0.658356
$176$	0	0
$177$	0	0
$178$	0	0
$179$	6.00000i	0.448461i	0.974536	+	0.224231i	$0.0719869\pi$
$179$	6.00000i	0.448461i	−0.974536	+	0.224231i	$0.928013\pi$
$180$	0	0
$181$	− 5.75194i	− 0.427538i	−0.976884	−	0.213769i	$-0.931426\pi$
$181$	− 5.75194i	− 0.427538i	0.976884	−	0.213769i	$-0.0685740\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	9.96265	0.732469
$186$	0	0
$187$	9.96265i	0.728541i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0.221065	0.0159957	0.00799786	−	0.999968i	$-0.497454\pi$
$191$	0.221065	0.0159957	0.00799786	+	0.999968i	$0.497454\pi$
$192$	0	0
$193$	18.0475	1.29909	0.649543	−	0.760325i	$-0.274960\pi$
$193$	18.0475	1.29909	0.649543	+	0.760325i	$0.274960\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.82080i	0.699703i	0.936805	+	0.349852i	$0.113768\pi$
$197$	9.82080i	0.699703i	−0.936805	+	0.349852i	$0.886232\pi$
$198$	0	0
$199$	0.604760	0.0428703	0.0214352	−	0.999770i	$-0.493176\pi$
$199$	0.604760	0.0428703	0.0214352	+	0.999770i	$0.493176\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 20.4202i	− 1.43322i
$204$	0	0
$205$	− 2.23887i	− 0.156369i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−11.6700	−0.807234
$210$	0	0
$211$	− 0.420243i	− 0.0289307i	−0.999895	−	0.0144654i	$-0.995395\pi$
$211$	− 0.420243i	− 0.0289307i	0.999895	−	0.0144654i	$-0.00460462\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	6.13564	0.418447
$216$	0	0
$217$	21.1696	1.43709
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6.86352i	0.461690i
$222$	0	0
$223$	22.7867	1.52591	0.762955	−	0.646452i	$-0.223748\pi$
$223$	22.7867	1.52591	0.762955	+	0.646452i	$0.223748\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.8350i	0.785518i	0.919641	+	0.392759i	$0.128479\pi$
$227$	11.8350i	0.785518i	−0.919641	+	0.392759i	$0.871521\pi$
$228$	0	0
$229$	− 15.6864i	− 1.03659i	−0.855203	−	0.518293i	$-0.826568\pi$
$229$	− 15.6864i	− 1.03659i	0.855203	−	0.518293i	$-0.173432\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0.829557	0.0543461	0.0271731	−	0.999631i	$-0.491349\pi$
$233$	0.829557	0.0543461	0.0271731	+	0.999631i	$0.491349\pi$
$234$	0	0
$235$	18.3829i	1.19917i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−7.21395	−0.466632	−0.233316	−	0.972401i	$-0.574958\pi$
$239$	−7.21395	−0.466632	−0.233316	+	0.972401i	$0.574958\pi$
$240$	0	0
$241$	−20.9253	−1.34792	−0.673959	−	0.738769i	$-0.735407\pi$
$241$	−20.9253	−1.34792	−0.673959	+	0.738769i	$0.735407\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 20.7199i	− 1.32375i
$246$	0	0
$247$	−8.03978	−0.511559
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 20.2553i	− 1.27850i	−0.768999	−	0.639250i	$-0.779245\pi$
$251$	− 20.2553i	− 1.27850i	0.768999	−	0.639250i	$-0.220755\pi$
$252$	0	0
$253$	− 38.7588i	− 2.43675i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6.74474	0.420725	0.210363	−	0.977623i	$-0.432536\pi$
$257$	6.74474	0.420725	0.210363	+	0.977623i	$0.432536\pi$
$258$	0	0
$259$	25.0475i	1.55637i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−26.4134	−1.62872	−0.814361	−	0.580359i	$-0.802912\pi$
$263$	−26.4134	−1.62872	−0.814361	+	0.580359i	$0.802912\pi$
$264$	0	0
$265$	−0.877832	−0.0539248
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 23.1704i	− 1.41272i	−0.707851	−	0.706362i	$-0.750335\pi$
$269$	− 23.1704i	− 1.41272i	0.707851	−	0.706362i	$-0.249665\pi$
$270$	0	0
$271$	17.7626	1.07900	0.539502	−	0.841984i	$-0.318613\pi$
$271$	17.7626	1.07900	0.539502	+	0.841984i	$0.318613\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 11.6700i	− 0.703730i
$276$	0	0
$277$	10.4413i	0.627356i	0.949529	+	0.313678i	$0.101561\pi$
$277$	10.4413i	0.627356i	−0.949529	+	0.313678i	$0.898439\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	26.8880	1.60400	0.802001	−	0.597323i	$-0.203769\pi$
$281$	26.8880	1.60400	0.802001	+	0.597323i	$0.203769\pi$
$282$	0	0
$283$	− 18.6272i	− 1.10727i	−0.832758	−	0.553637i	$-0.813240\pi$
$283$	− 18.6272i	− 1.10727i	0.832758	−	0.553637i	$-0.186760\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.62882	0.332259
$288$	0	0
$289$	−14.0848	−0.828519
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.16055i	0.0678000i	0.999425	+	0.0339000i	$0.0107928\pi$
$293$	1.16055i	0.0678000i	−0.999425	+	0.0339000i	$0.989207\pi$
$294$	0	0
$295$	−3.24304	−0.188817
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 26.7019i	− 1.54421i
$300$	0	0
$301$	15.4258i	0.889130i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.12217	−0.121515
$306$	0	0
$307$	− 19.0848i	− 1.08923i	−0.838687	−	0.544614i	$-0.816676\pi$
$307$	− 19.0848i	− 1.08923i	0.838687	−	0.544614i	$-0.183324\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−11.1202	−0.630568	−0.315284	−	0.948997i	$-0.602100\pi$
$311$	−11.1202	−0.630568	−0.315284	+	0.948997i	$0.602100\pi$
$312$	0	0
$313$	26.9253	1.52191	0.760954	−	0.648806i	$-0.224731\pi$
$313$	26.9253	1.52191	0.760954	+	0.648806i	$0.224731\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 22.4520i	− 1.26103i	−0.776178	−	0.630514i	$-0.782844\pi$
$317$	− 22.4520i	− 1.26103i	0.776178	−	0.630514i	$-0.217156\pi$
$318$	0	0
$319$	−27.3624	−1.53200
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3.41478i	0.190003i
$324$	0	0
$325$	− 8.03978i	− 0.445967i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−46.2171	−2.54803
$330$	0	0
$331$	− 22.5899i	− 1.24165i	−0.783948	−	0.620826i	$-0.786797\pi$
$331$	− 22.5899i	− 1.24165i	0.783948	−	0.620826i	$-0.213203\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2.73622	−0.149496
$336$	0	0
$337$	−14.0848	−0.767249	−0.383625	−	0.923489i	$-0.625324\pi$
$337$	−14.0848	−0.767249	−0.383625	+	0.923489i	$0.625324\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 28.3665i	− 1.53613i
$342$	0	0
$343$	21.6104	1.16685
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 6.99454i	− 0.375486i	−0.982218	−	0.187743i	$-0.939883\pi$
$347$	− 6.99454i	− 0.375486i	0.982218	−	0.187743i	$-0.0601173\pi$
$348$	0	0
$349$	9.42764i	0.504650i	0.967643	+	0.252325i	$0.0811952\pi$
$349$	9.42764i	0.504650i	−0.967643	+	0.252325i	$0.918805\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	7.62257	0.405708	0.202854	−	0.979209i	$-0.434978\pi$
$353$	7.62257	0.405708	0.202854	+	0.979209i	$0.434978\pi$
$354$	0	0
$355$	17.1222i	0.908750i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−10.8344	−0.571820	−0.285910	−	0.958257i	$-0.592296\pi$
$359$	−10.8344	−0.571820	−0.285910	+	0.958257i	$0.592296\pi$
$360$	0	0
$361$	15.0000	0.789474
$362$	0	0
$363$	0	0
$364$	0	0
$365$	13.7917i	0.721892i
$366$	0	0
$367$	−22.9493	−1.19795	−0.598973	−	0.800770i	$-0.704424\pi$
$367$	−22.9493	−1.19795	−0.598973	+	0.800770i	$0.704424\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 2.20699i	− 0.114581i
$372$	0	0
$373$	− 0.113658i	− 0.00588498i	−0.999996	−	0.00294249i	$-0.999063\pi$
$373$	− 0.113658i	− 0.00588498i	0.999996	−	0.00294249i	$-0.000936625\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−18.8506	−0.970856
$378$	0	0
$379$	16.2070i	0.832497i	0.909251	+	0.416249i	$0.136655\pi$
$379$	16.2070i	0.832497i	−0.909251	+	0.416249i	$0.863345\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−14.0775	−0.719325	−0.359663	−	0.933082i	$-0.617108\pi$
$383$	−14.0775	−0.719325	−0.359663	+	0.933082i	$0.617108\pi$
$384$	0	0
$385$	−44.0101	−2.24296
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 28.8734i	− 1.46394i	−0.681338	−	0.731969i	$-0.738602\pi$
$389$	− 28.8734i	− 1.46394i	0.681338	−	0.731969i	$-0.261398\pi$
$390$	0	0
$391$	−11.3413	−0.573552
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 3.57976i	− 0.180117i
$396$	0	0
$397$	2.85934i	0.143506i	0.997422	+	0.0717531i	$0.0228594\pi$
$397$	2.85934i	0.143506i	−0.997422	+	0.0717531i	$0.977141\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−21.5479	−1.07605	−0.538025	−	0.842929i	$-0.680829\pi$
$401$	−21.5479	−1.07605	−0.538025	+	0.842929i	$0.680829\pi$
$402$	0	0
$403$	− 19.5424i	− 0.973477i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	33.5627	1.66364
$408$	0	0
$409$	−20.9253	−1.03469	−0.517345	−	0.855777i	$-0.673080\pi$
$409$	−20.9253	−1.03469	−0.517345	+	0.855777i	$0.673080\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 8.15344i	− 0.401204i
$414$	0	0
$415$	2.67153	0.131140
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 12.6646i	− 0.618705i	−0.950947	−	0.309353i	$-0.899888\pi$
$419$	− 12.6646i	− 0.618705i	0.950947	−	0.309353i	$-0.100112\pi$
$420$	0	0
$421$	26.6835i	1.30047i	0.759732	+	0.650236i	$0.225330\pi$
$421$	26.6835i	1.30047i	−0.759732	+	0.650236i	$0.774670\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−3.41478	−0.165641
$426$	0	0
$427$	− 5.33542i	− 0.258199i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	23.6772	1.14049	0.570246	−	0.821474i	$-0.306848\pi$
$431$	23.6772	1.14049	0.570246	+	0.821474i	$0.306848\pi$
$432$	0	0
$433$	7.96265	0.382661	0.191330	−	0.981526i	$-0.438720\pi$
$433$	7.96265	0.382661	0.191330	+	0.981526i	$0.438720\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 13.2849i	− 0.635503i
$438$	0	0
$439$	10.6781	0.509636	0.254818	−	0.966989i	$-0.417984\pi$
$439$	10.6781	0.509636	0.254818	+	0.966989i	$0.417984\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12.8825i	0.612066i	0.952021	+	0.306033i	$0.0990018\pi$
$443$	12.8825i	0.612066i	−0.952021	+	0.306033i	$0.900998\pi$
$444$	0	0
$445$	6.86352i	0.325362i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0.547875	0.0258558	0.0129279	−	0.999916i	$-0.495885\pi$
$449$	0.547875	0.0258558	0.0129279	+	0.999916i	$0.495885\pi$
$450$	0	0
$451$	− 7.54241i	− 0.355158i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−30.3197	−1.42141
$456$	0	0
$457$	19.8031	0.926352	0.463176	−	0.886266i	$-0.346710\pi$
$457$	19.8031	0.926352	0.463176	+	0.886266i	$0.346710\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	14.0680i	0.655212i	0.944814	+	0.327606i	$0.106242\pi$
$461$	14.0680i	0.655212i	−0.944814	+	0.327606i	$0.893758\pi$
$462$	0	0
$463$	12.4591	0.579022	0.289511	−	0.957175i	$-0.406507\pi$
$463$	12.4591	0.579022	0.289511	+	0.957175i	$0.406507\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	29.3401i	1.35770i	0.734278	+	0.678849i	$0.237521\pi$
$467$	29.3401i	1.35770i	−0.734278	+	0.678849i	$0.762479\pi$
$468$	0	0
$469$	− 6.87923i	− 0.317653i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	20.6700	0.950410
$474$	0	0
$475$	− 4.00000i	− 0.183533i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	19.9274	0.910504	0.455252	−	0.890363i	$-0.349549\pi$
$479$	19.9274	0.910504	0.455252	+	0.890363i	$0.349549\pi$
$480$	0	0
$481$	23.1222	1.05428
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 10.6039i	− 0.481499i
$486$	0	0
$487$	−7.46828	−0.338420	−0.169210	−	0.985580i	$-0.554122\pi$
$487$	−7.46828	−0.338420	−0.169210	+	0.985580i	$0.554122\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 34.8934i	− 1.57472i	−0.616495	−	0.787359i	$-0.711448\pi$
$491$	− 34.8934i	− 1.57472i	0.616495	−	0.787359i	$-0.288552\pi$
$492$	0	0
$493$	8.00652i	0.360596i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−43.0475	−1.93094
$498$	0	0
$499$	− 37.4677i	− 1.67729i	−0.544682	−	0.838643i	$-0.683350\pi$
$499$	− 37.4677i	− 1.67729i	0.544682	−	0.838643i	$-0.316650\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−10.8991	−0.485968	−0.242984	−	0.970030i	$-0.578126\pi$
$503$	−10.8991	−0.485968	−0.242984	+	0.970030i	$0.578126\pi$
$504$	0	0
$505$	20.0101	0.890439
$506$	0	0
$507$	0	0
$508$	0	0
$509$	4.68934i	0.207851i	0.994585	+	0.103926i	$0.0331404\pi$
$509$	4.68934i	0.207851i	−0.994585	+	0.103926i	$0.966860\pi$
$510$	0	0
$511$	−34.6743	−1.53390
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 0.382896i	− 0.0168724i
$516$	0	0
$517$	61.9292i	2.72364i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	26.4249	1.15770	0.578848	−	0.815435i	$-0.303502\pi$
$521$	26.4249	1.15770	0.578848	+	0.815435i	$0.303502\pi$
$522$	0	0
$523$	13.8880i	0.607278i	0.952787	+	0.303639i	$0.0982017\pi$
$523$	13.8880i	0.607278i	−0.952787	+	0.303639i	$0.901798\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.30035	−0.361569
$528$	0	0
$529$	21.1222	0.918355
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 5.19615i	− 0.225070i
$534$	0	0
$535$	−0.571503	−0.0247082
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 69.8023i	− 3.00660i
$540$	0	0
$541$	25.0161i	1.07553i	0.843096	+	0.537763i	$0.180731\pi$
$541$	25.0161i	1.07553i	−0.843096	+	0.537763i	$0.819269\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	31.0475	1.32993
$546$	0	0
$547$	14.3455i	0.613371i	0.951811	+	0.306686i	$0.0992200\pi$
$547$	14.3455i	0.613371i	−0.951811	+	0.306686i	$0.900780\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−9.37867	−0.399545
$552$	0	0
$553$	9.00000	0.382719
$554$	0	0
$555$	0	0
$556$	0	0
$557$	32.9912i	1.39788i	0.715179	+	0.698941i	$0.246345\pi$
$557$	32.9912i	1.39788i	−0.715179	+	0.698941i	$0.753655\pi$
$558$	0	0
$559$	14.2401	0.602292
$560$	0	0
$561$	0	0
$562$	0	0
$563$	24.9945i	1.05339i	0.850053	+	0.526697i	$0.176570\pi$
$563$	24.9945i	1.05339i	−0.850053	+	0.526697i	$0.823430\pi$
$564$	0	0
$565$	− 31.3238i	− 1.31780i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5.58522	0.234145	0.117072	−	0.993123i	$-0.462649\pi$
$569$	5.58522	0.234145	0.117072	+	0.993123i	$0.462649\pi$
$570$	0	0
$571$	− 5.57976i	− 0.233506i	−0.993161	−	0.116753i	$-0.962751\pi$
$571$	− 5.57976i	− 0.233506i	0.993161	−	0.116753i	$-0.0372485\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	13.2849	0.554019
$576$	0	0
$577$	−9.04748	−0.376651	−0.188326	−	0.982107i	$-0.560306\pi$
$577$	−9.04748	−0.376651	−0.188326	+	0.982107i	$0.560306\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6.71660i	0.278652i
$582$	0	0
$583$	−2.95729	−0.122478
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.21245i	0.297690i	0.988861	+	0.148845i	$0.0475555\pi$
$587$	7.21245i	0.297690i	−0.988861	+	0.148845i	$0.952444\pi$
$588$	0	0
$589$	− 9.72286i	− 0.400623i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	25.0475	1.02858	0.514288	−	0.857617i	$-0.328056\pi$
$593$	25.0475	1.02858	0.514288	+	0.857617i	$0.328056\pi$
$594$	0	0
$595$	12.8778i	0.527940i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	40.2051	1.64274	0.821369	−	0.570397i	$-0.193211\pi$
$599$	40.2051	1.64274	0.821369	+	0.570397i	$0.193211\pi$
$600$	0	0
$601$	−10.9627	−0.447176	−0.223588	−	0.974684i	$-0.571777\pi$
$601$	−10.9627	−0.447176	−0.223588	+	0.974684i	$0.571777\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	39.9194i	1.62295i
$606$	0	0
$607$	28.4740	1.15572	0.577861	−	0.816135i	$-0.303888\pi$
$607$	28.4740	1.15572	0.577861	+	0.816135i	$0.303888\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	42.6646i	1.72602i
$612$	0	0
$613$	14.3632i	0.580125i	0.957008	+	0.290063i	$0.0936761\pi$
$613$	14.3632i	0.580125i	−0.957008	+	0.290063i	$0.906324\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0.632696	0.0254714	0.0127357	−	0.999919i	$-0.495946\pi$
$617$	0.632696	0.0254714	0.0127357	+	0.999919i	$0.495946\pi$
$618$	0	0
$619$	− 33.4304i	− 1.34368i	−0.740696	−	0.671840i	$-0.765504\pi$
$619$	− 33.4304i	− 1.34368i	0.740696	−	0.671840i	$-0.234496\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−17.2558	−0.691340
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 9.82080i	− 0.391581i
$630$	0	0
$631$	−7.59765	−0.302458	−0.151229	−	0.988499i	$-0.548323\pi$
$631$	−7.59765	−0.302458	−0.151229	+	0.988499i	$0.548323\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	7.92531i	0.314506i
$636$	0	0
$637$	− 48.0885i	− 1.90534i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	23.7549	0.938261	0.469130	−	0.883129i	$-0.344567\pi$
$641$	23.7549	0.938261	0.469130	+	0.883129i	$0.344567\pi$
$642$	0	0
$643$	14.3455i	0.565733i	0.959159	+	0.282867i	$0.0912854\pi$
$643$	14.3455i	0.565733i	−0.959159	+	0.282867i	$0.908715\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−28.5781	−1.12352	−0.561761	−	0.827299i	$-0.689876\pi$
$647$	−28.5781	−1.12352	−0.561761	+	0.827299i	$0.689876\pi$
$648$	0	0
$649$	−10.9253	−0.428856
$650$	0	0
$651$	0	0
$652$	0	0
$653$	17.0253i	0.666251i	0.942882	+	0.333126i	$0.108103\pi$
$653$	17.0253i	0.666251i	−0.942882	+	0.333126i	$0.891897\pi$
$654$	0	0
$655$	23.4561	0.916507
$656$	0	0
$657$	0	0
$658$	0	0
$659$	12.4949i	0.486734i	0.969934	+	0.243367i	$0.0782519\pi$
$659$	12.4949i	0.486734i	−0.969934	+	0.243367i	$0.921748\pi$
$660$	0	0
$661$	− 34.7879i	− 1.35309i	−0.736400	−	0.676547i	$-0.763476\pi$
$661$	− 34.7879i	− 1.35309i	0.736400	−	0.676547i	$-0.236524\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−15.0848	−0.584964
$666$	0	0
$667$	− 31.1487i	− 1.20608i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−7.14927	−0.275994
$672$	0	0
$673$	26.0475	1.00406	0.502028	−	0.864851i	$-0.332587\pi$
$673$	26.0475	1.00406	0.502028	+	0.864851i	$0.332587\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	33.3511i	1.28179i	0.767630	+	0.640893i	$0.221436\pi$
$677$	33.3511i	1.28179i	−0.767630	+	0.640893i	$0.778564\pi$
$678$	0	0
$679$	26.6597	1.02310
$680$	0	0
$681$	0	0
$682$	0	0
$683$	26.4249i	1.01112i	0.862791	+	0.505560i	$0.168714\pi$
$683$	26.4249i	1.01112i	−0.862791	+	0.505560i	$0.831286\pi$
$684$	0	0
$685$	− 38.7588i	− 1.48090i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2.03735	−0.0776167
$690$	0	0
$691$	− 20.3829i	− 0.775402i	−0.921785	−	0.387701i	$-0.873269\pi$
$691$	− 20.3829i	− 0.775402i	0.921785	−	0.387701i	$-0.126731\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	19.9920	0.758341
$696$	0	0
$697$	−2.20699	−0.0835957
$698$	0	0
$699$	0	0
$700$	0	0
$701$	49.8696i	1.88355i	0.336247	+	0.941774i	$0.390842\pi$
$701$	49.8696i	1.88355i	−0.336247	+	0.941774i	$0.609158\pi$
$702$	0	0
$703$	11.5039	0.433877
$704$	0	0
$705$	0	0
$706$	0	0
$707$	50.3082i	1.89203i
$708$	0	0
$709$	9.90120i	0.371847i	0.982564	+	0.185924i	$0.0595277\pi$
$709$	9.90120i	0.371847i	−0.982564	+	0.185924i	$0.940472\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	32.2918	1.20934
$714$	0	0
$715$	40.6272i	1.51937i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−25.5562	−0.953084	−0.476542	−	0.879152i	$-0.658110\pi$
$719$	−25.5562	−0.953084	−0.476542	+	0.879152i	$0.658110\pi$
$720$	0	0
$721$	0.962653	0.0358511
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 9.37867i	− 0.348315i
$726$	0	0
$727$	40.1405	1.48873	0.744364	−	0.667774i	$-0.232753\pi$
$727$	40.1405	1.48873	0.744364	+	0.667774i	$0.232753\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 6.04827i	− 0.223704i
$732$	0	0
$733$	− 47.3387i	− 1.74849i	−0.485481	−	0.874247i	$-0.661355\pi$
$733$	− 47.3387i	− 1.74849i	0.485481	−	0.874247i	$-0.338645\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9.21792	−0.339546
$738$	0	0
$739$	− 37.9627i	− 1.39648i	−0.715864	−	0.698239i	$-0.753967\pi$
$739$	− 37.9627i	− 1.39648i	0.715864	−	0.698239i	$-0.246033\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	14.5196	0.532673	0.266336	−	0.963880i	$-0.414187\pi$
$743$	14.5196	0.532673	0.266336	+	0.963880i	$0.414187\pi$
$744$	0	0
$745$	−3.87783	−0.142073
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 1.43684i	− 0.0525008i
$750$	0	0
$751$	−36.6431	−1.33713	−0.668563	−	0.743656i	$-0.733090\pi$
$751$	−36.6431	−1.33713	−0.668563	+	0.743656i	$0.733090\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	29.5051i	1.07380i
$756$	0	0
$757$	2.02727i	0.0736822i	0.999321	+	0.0368411i	$0.0117295\pi$
$757$	2.02727i	0.0736822i	−0.999321	+	0.0368411i	$0.988270\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	17.2070	0.623753	0.311876	−	0.950123i	$-0.399042\pi$
$761$	17.2070	0.623753	0.311876	+	0.950123i	$0.399042\pi$
$762$	0	0
$763$	78.0576i	2.82587i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−7.52671	−0.271774
$768$	0	0
$769$	26.8506	0.968258	0.484129	−	0.874997i	$-0.339137\pi$
$769$	26.8506	0.968258	0.484129	+	0.874997i	$0.339137\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 4.41305i	− 0.158726i	−0.996846	−	0.0793632i	$-0.974711\pi$
$773$	− 4.41305i	− 0.158726i	0.996846	−	0.0793632i	$-0.0252887\pi$
$774$	0	0
$775$	9.72286	0.349255
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 2.58522i	− 0.0926252i
$780$	0	0
$781$	57.6820i	2.06403i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0.0848216	0.00302741
$786$	0	0
$787$	34.3829i	1.22562i	0.790231	+	0.612809i	$0.209960\pi$
$787$	34.3829i	1.22562i	−0.790231	+	0.612809i	$0.790040\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	78.7524	2.80011
$792$	0	0
$793$	−4.92531	−0.174903
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 22.8941i	− 0.810951i	−0.914106	−	0.405475i	$-0.867106\pi$
$797$	− 22.8941i	− 0.810951i	0.914106	−	0.405475i	$-0.132894\pi$
$798$	0	0
$799$	18.1212	0.641080
$800$	0	0
$801$	0	0
$802$	0	0
$803$	46.4623i	1.63962i
$804$	0	0
$805$	− 50.1001i	− 1.76580i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−34.7922	−1.22323	−0.611614	−	0.791156i	$-0.709480\pi$
$809$	−34.7922	−1.22323	−0.611614	+	0.791156i	$0.709480\pi$
$810$	0	0
$811$	11.6810i	0.410174i	0.978744	+	0.205087i	$0.0657478\pi$
$811$	11.6810i	0.410174i	−0.978744	+	0.205087i	$0.934252\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	33.0559	1.15790
$816$	0	0
$817$	7.08482	0.247867
$818$	0	0
$819$	0	0
$820$	0	0
$821$	16.0953i	0.561729i	0.959747	+	0.280864i	$0.0906211\pi$
$821$	16.0953i	0.561729i	−0.959747	+	0.280864i	$0.909379\pi$
$822$	0	0
$823$	12.8033	0.446294	0.223147	−	0.974785i	$-0.428367\pi$
$823$	12.8033	0.446294	0.223147	+	0.974785i	$0.428367\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 25.9253i	− 0.901511i	−0.892647	−	0.450756i	$-0.851155\pi$
$827$	− 25.9253i	− 0.901511i	0.892647	−	0.450756i	$-0.148845\pi$
$828$	0	0
$829$	49.0708i	1.70430i	0.523299	+	0.852149i	$0.324701\pi$
$829$	49.0708i	1.70430i	−0.523299	+	0.852149i	$0.675299\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−20.4249	−0.707681
$834$	0	0
$835$	34.6272i	1.19832i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−30.2550	−1.04452	−0.522259	−	0.852787i	$-0.674911\pi$
$839$	−30.2550	−1.04452	−0.522259	+	0.852787i	$0.674911\pi$
$840$	0	0
$841$	7.01013	0.241729
$842$	0	0
$843$	0	0
$844$	0	0
$845$	5.47244i	0.188258i
$846$	0	0
$847$	−100.363	−3.44851
$848$	0	0
$849$	0	0
$850$	0	0
$851$	38.2070i	1.30972i
$852$	0	0
$853$	− 29.9517i	− 1.02553i	−0.858530	−	0.512763i	$-0.828622\pi$
$853$	− 29.9517i	− 1.02553i	0.858530	−	0.512763i	$-0.171378\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−23.8031	−0.813100	−0.406550	−	0.913629i	$-0.633268\pi$
$857$	−23.8031	−0.813100	−0.406550	+	0.913629i	$0.633268\pi$
$858$	0	0
$859$	− 32.9089i	− 1.12284i	−0.827532	−	0.561419i	$-0.810256\pi$
$859$	− 32.9089i	− 1.12284i	0.827532	−	0.561419i	$-0.189744\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−36.3906	−1.23875	−0.619375	−	0.785095i	$-0.712614\pi$
$863$	−36.3906	−1.23875	−0.619375	+	0.785095i	$0.712614\pi$
$864$	0	0
$865$	−21.0000	−0.714021
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 12.0597i	− 0.409096i
$870$	0	0
$871$	−6.35045	−0.215177
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 52.7969i	− 1.78486i
$876$	0	0
$877$	48.9049i	1.65140i	0.564108	+	0.825701i	$0.309220\pi$
$877$	48.9049i	1.65140i	−0.564108	+	0.825701i	$0.690780\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−39.9627	−1.34638	−0.673188	−	0.739471i	$-0.735076\pi$
$881$	−39.9627	−1.34638	−0.673188	+	0.739471i	$0.735076\pi$
$882$	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
$887$	45.1897	1.51732	0.758661	−	0.651486i	$-0.225854\pi$
$887$	45.1897	1.51732	0.758661	+	0.651486i	$0.225854\pi$
$888$	0	0
$889$	−19.9253	−0.668273
$890$	0	0
$891$	0	0
$892$	0	0
$893$	21.2267i	0.710326i
$894$	0	0
$895$	10.3923	0.347376
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 22.7969i	− 0.760318i
$900$	0	0
$901$	0.865333i	0.0288284i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−9.96265	−0.331170
$906$	0	0
$907$	− 40.3829i	− 1.34089i	−0.741958	−	0.670446i	$-0.766103\pi$
$907$	− 40.3829i	− 1.34089i	0.741958	−	0.670446i	$-0.233897\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−8.52142	−0.282327	−0.141164	−	0.989986i	$-0.545084\pi$
$911$	−8.52142	−0.282327	−0.141164	+	0.989986i	$0.545084\pi$
$912$	0	0
$913$	9.00000	0.297857
$914$	0	0
$915$	0	0
$916$	0	0
$917$	58.9720i	1.94743i
$918$	0	0
$919$	2.00834	0.0662490	0.0331245	−	0.999451i	$-0.489454\pi$
$919$	2.00834	0.0662490	0.0331245	+	0.999451i	$0.489454\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	39.7386i	1.30801i
$924$	0	0
$925$	11.5039i	0.378245i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−29.9728	−0.983375	−0.491688	−	0.870772i	$-0.663620\pi$
$929$	−29.9728	−0.983375	−0.491688	+	0.870772i	$0.663620\pi$
$930$	0	0
$931$	− 23.9253i	− 0.784120i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	17.2558	0.564326
$936$	0	0
$937$	−45.0475	−1.47164	−0.735818	−	0.677179i	$-0.763202\pi$
$937$	−45.0475	−1.47164	−0.735818	+	0.677179i	$0.763202\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	53.5642i	1.74614i	0.487592	+	0.873072i	$0.337875\pi$
$941$	53.5642i	1.74614i	−0.487592	+	0.873072i	$0.662125\pi$
$942$	0	0
$943$	8.58610	0.279602
$944$	0	0
$945$	0	0
$946$	0	0
$947$	23.8350i	0.774534i	0.921968	+	0.387267i	$0.126581\pi$
$947$	23.8350i	0.774534i	−0.921968	+	0.387267i	$0.873419\pi$
$948$	0	0
$949$	32.0090i	1.03906i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	32.4732	1.05191	0.525955	−	0.850513i	$-0.323708\pi$
$953$	32.4732	1.05191	0.525955	+	0.850513i	$0.323708\pi$
$954$	0	0
$955$	− 0.382896i	− 0.0123902i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	97.4451	3.14666
$960$	0	0
$961$	−7.36650	−0.237629
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 31.2591i	− 1.00627i
$966$	0	0
$967$	−9.66442	−0.310787	−0.155393	−	0.987853i	$-0.549665\pi$
$967$	−9.66442	−0.310787	−0.155393	+	0.987853i	$0.549665\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	49.8133i	1.59858i	0.600943	+	0.799292i	$0.294792\pi$
$971$	49.8133i	1.59858i	−0.600943	+	0.799292i	$0.705208\pi$
$972$	0	0
$973$	50.2627i	1.61135i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	7.18057	0.229727	0.114863	−	0.993381i	$-0.463357\pi$
$977$	7.18057	0.229727	0.114863	+	0.993381i	$0.463357\pi$
$978$	0	0
$979$	23.1222i	0.738988i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−4.76349	−0.151932	−0.0759658	−	0.997110i	$-0.524204\pi$
$983$	−4.76349	−0.151932	−0.0759658	+	0.997110i	$0.524204\pi$
$984$	0	0
$985$	17.0101	0.541988
$986$	0	0
$987$	0	0
$988$	0	0
$989$	23.5303i	0.748220i
$990$	0	0
$991$	−28.5781	−0.907815	−0.453907	−	0.891049i	$-0.649970\pi$
$991$	−28.5781	−0.907815	−0.453907	+	0.891049i	$0.649970\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 1.04748i	− 0.0332072i
$996$	0	0
$997$	− 37.2384i	− 1.17935i	−0.807640	−	0.589676i	$-0.799255\pi$
$997$	− 37.2384i	− 1.17935i	0.807640	−	0.589676i	$-0.200745\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5184.2.d.r.2593.6		12
3.2	odd	2		5184.2.d.q.2593.12		12
4.3	odd	2	inner	5184.2.d.r.2593.1		12
8.3	odd	2	inner	5184.2.d.r.2593.7		12
8.5	even	2	inner	5184.2.d.r.2593.12		12
9.2	odd	6		576.2.r.e.481.4	yes	12
9.4	even	3		1728.2.r.e.289.1		12
9.5	odd	6		576.2.r.f.97.3	yes	12
9.7	even	3		1728.2.r.f.1441.1		12
12.11	even	2		5184.2.d.q.2593.7		12
24.5	odd	2		5184.2.d.q.2593.6		12
24.11	even	2		5184.2.d.q.2593.1		12
36.7	odd	6		1728.2.r.f.1441.6		12
36.11	even	6		576.2.r.e.481.3	yes	12
36.23	even	6		576.2.r.f.97.4	yes	12
36.31	odd	6		1728.2.r.e.289.6		12
72.5	odd	6		576.2.r.e.97.4	yes	12
72.11	even	6		576.2.r.f.481.4	yes	12
72.13	even	6		1728.2.r.f.289.1		12
72.29	odd	6		576.2.r.f.481.3	yes	12
72.43	odd	6		1728.2.r.e.1441.6		12
72.59	even	6		576.2.r.e.97.3	✓	12
72.61	even	6		1728.2.r.e.1441.1		12
72.67	odd	6		1728.2.r.f.289.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.r.e.97.3	✓	12	72.59	even	6
576.2.r.e.97.4	yes	12	72.5	odd	6
576.2.r.e.481.3	yes	12	36.11	even	6
576.2.r.e.481.4	yes	12	9.2	odd	6
576.2.r.f.97.3	yes	12	9.5	odd	6
576.2.r.f.97.4	yes	12	36.23	even	6
576.2.r.f.481.3	yes	12	72.29	odd	6
576.2.r.f.481.4	yes	12	72.11	even	6
1728.2.r.e.289.1		12	9.4	even	3
1728.2.r.e.289.6		12	36.31	odd	6
1728.2.r.e.1441.1		12	72.61	even	6
1728.2.r.e.1441.6		12	72.43	odd	6
1728.2.r.f.289.1		12	72.13	even	6
1728.2.r.f.289.6		12	72.67	odd	6
1728.2.r.f.1441.1		12	9.7	even	3
1728.2.r.f.1441.6		12	36.7	odd	6
5184.2.d.q.2593.1		12	24.11	even	2
5184.2.d.q.2593.6		12	24.5	odd	2
5184.2.d.q.2593.7		12	12.11	even	2
5184.2.d.q.2593.12		12	3.2	odd	2
5184.2.d.r.2593.1		12	4.3	odd	2	inner
5184.2.d.r.2593.6		12	1.1	even	1	trivial
5184.2.d.r.2593.7		12	8.3	odd	2	inner
5184.2.d.r.2593.12		12	8.5	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:	\( N \)	\(=\)	\( 5184 = 2^{6} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5184.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{5} +4.35461 q^{7} +O(q^{10})\)	\(q-1.73205i q^{5} +4.35461 q^{7} -5.83502i q^{11} -4.01989i q^{13} -1.70739 q^{17} -2.00000i q^{19} +6.64245 q^{23} +2.00000 q^{25} -4.68934i q^{29} +4.86143 q^{31} -7.54241i q^{35} +5.75194i q^{37} +1.29261 q^{41} +3.54241i q^{43} -10.6134 q^{47} +11.9627 q^{49} -0.506816i q^{53} -10.1066 q^{55} -1.87237i q^{59} -1.22523i q^{61} -6.96265 q^{65} -1.57976i q^{67} -9.88549 q^{71} -7.96265 q^{73} -25.4093i q^{77} +2.06677 q^{79} +1.54241i q^{83} +2.95729i q^{85} -3.96265 q^{89} -17.5051i q^{91} -3.46410 q^{95} +6.12217 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 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

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