LMFDB - Embedded newform 5328.2.e.e.2591.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5328,2,Mod(2591,5328)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5328, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5328.2591");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5328 = 2^{4} \cdot 3^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5328.e (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:	$42.5442941969$
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} + 4x^{10} + 2x^{8} - 70x^{6} + 37x^{4} + 116x^{2} + 81 $ x^12 + 4x^10 + 2x^8 - 70x^6 + 37x^4 + 116*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{41}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.10
Root			$0.819567 - 2.05926i$ of defining polynomial
Character	$\chi$	$=$	5328.2591
Dual form			5328.2.e.e.2591.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.70430i q^{5} +4.43271i q^{7} +O(q^{10})$	$q+2.70430i q^{5} +4.43271i q^{7} -5.82446 q^{13} +4.56150i q^{17} +2.91061i q^{19} -5.46762 q^{23} -2.31324 q^{25} +0.847097i q^{29} -4.04365i q^{31} -11.9874 q^{35} +1.00000 q^{37} -4.24264i q^{41} -1.77757i q^{43} -7.87116 q^{47} -12.6489 q^{49} +1.16596i q^{53} +6.51976 q^{59} +4.62648 q^{61} -15.7511i q^{65} +12.5200i q^{67} +16.1036 q^{71} -9.31324 q^{73} +5.95481i q^{79} +7.87116 q^{83} -12.3357 q^{85} -11.1896i q^{89} -25.8181i q^{91} -7.87116 q^{95} -3.19798 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q - 16 q^{13} - 12 q^{25} + 12 q^{37} - 44 q^{49} + 24 q^{61} - 96 q^{73} - 56 q^{85} - 16 q^{97}+O(q^{100}) $ 12 * q - 16 * q^13 - 12 * q^25 + 12 * q^37 - 44 * q^49 + 24 * q^61 - 96 * q^73 - 56 * q^85 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5328\mathbb{Z}\right)^\times$.

$n$	$1297$	$1333$	$1999$	$2369$
$\chi(n)$	$1$	$1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	2.70430i	1.20940i	0.796453	+	0.604700i	$0.206707\pi$
$5$	2.70430i	1.20940i	−0.796453	+	0.604700i	$0.793293\pi$
$6$	0	0
$7$	4.43271i	1.67541i	0.546125	+	0.837703i	$0.316102\pi$
$7$	4.43271i	1.67541i	−0.546125	+	0.837703i	$0.683898\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−5.82446	−1.61541	−0.807707	−	0.589584i	$-0.799292\pi$
$13$	−5.82446	−1.61541	−0.807707	+	0.589584i	$0.799292\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.56150i	1.10633i	0.833073	+	0.553164i	$0.186580\pi$
$17$	4.56150i	1.10633i	−0.833073	+	0.553164i	$0.813420\pi$
$18$	0	0
$19$	2.91061i	0.667739i	0.942619	+	0.333870i	$0.108355\pi$
$19$	2.91061i	0.667739i	−0.942619	+	0.333870i	$0.891645\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.46762	−1.14008	−0.570039	−	0.821618i	$-0.693072\pi$
$23$	−5.46762	−1.14008	−0.570039	+	0.821618i	$0.693072\pi$
$24$	0	0
$25$	−2.31324	−0.462648
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.847097i	0.157302i	0.996902	+	0.0786510i	$0.0250613\pi$
$29$	0.847097i	0.157302i	−0.996902	+	0.0786510i	$0.974939\pi$
$30$	0	0
$31$	− 4.04365i	− 0.726260i	−0.931738	−	0.363130i	$-0.881708\pi$
$31$	− 4.04365i	− 0.726260i	0.931738	−	0.363130i	$-0.118292\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−11.9874	−2.02624
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 4.24264i	− 0.662589i	−0.943527	−	0.331295i	$-0.892515\pi$
$41$	− 4.24264i	− 0.662589i	0.943527	−	0.331295i	$-0.107485\pi$
$42$	0	0
$43$	− 1.77757i	− 0.271077i	−0.990772	−	0.135538i	$-0.956724\pi$
$43$	− 1.77757i	− 0.271077i	0.990772	−	0.135538i	$-0.0432764\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.87116	−1.14813	−0.574063	−	0.818811i	$-0.694634\pi$
$47$	−7.87116	−1.14813	−0.574063	+	0.818811i	$0.694634\pi$
$48$	0	0
$49$	−12.6489	−1.80699
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.16596i	0.160157i	0.996789	+	0.0800785i	$0.0255171\pi$
$53$	1.16596i	0.160157i	−0.996789	+	0.0800785i	$0.974483\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.51976	0.848801	0.424400	−	0.905475i	$-0.360485\pi$
$59$	6.51976	0.848801	0.424400	+	0.905475i	$0.360485\pi$
$60$	0	0
$61$	4.62648	0.592360	0.296180	−	0.955132i	$-0.404287\pi$
$61$	4.62648	0.592360	0.296180	+	0.955132i	$0.404287\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 15.7511i	− 1.95368i
$66$	0	0
$67$	12.5200i	1.52956i	0.644290	+	0.764781i	$0.277153\pi$
$67$	12.5200i	1.52956i	−0.644290	+	0.764781i	$0.722847\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	16.1036	1.91115	0.955573	−	0.294755i	$-0.0952380\pi$
$71$	16.1036	1.91115	0.955573	+	0.294755i	$0.0952380\pi$
$72$	0	0
$73$	−9.31324	−1.09003	−0.545016	−	0.838426i	$-0.683477\pi$
$73$	−9.31324	−1.09003	−0.545016	+	0.838426i	$0.683477\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	5.95481i	0.669969i	0.942224	+	0.334984i	$0.108731\pi$
$79$	5.95481i	0.669969i	−0.942224	+	0.334984i	$0.891269\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	7.87116	0.863972	0.431986	−	0.901880i	$-0.357813\pi$
$83$	7.87116	0.863972	0.431986	+	0.901880i	$0.357813\pi$
$84$	0	0
$85$	−12.3357	−1.33799
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 11.1896i	− 1.18609i	−0.805168	−	0.593047i	$-0.797925\pi$
$89$	− 11.1896i	− 1.18609i	0.805168	−	0.593047i	$-0.202075\pi$
$90$	0	0
$91$	− 25.8181i	− 2.70648i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−7.87116	−0.807564
$96$	0	0
$97$	−3.19798	−0.324705	−0.162353	−	0.986733i	$-0.551908\pi$
$97$	−3.19798	−0.324705	−0.162353	+	0.986733i	$0.551908\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 6.09984i	− 0.606957i	−0.952838	−	0.303479i	$-0.901852\pi$
$101$	− 6.09984i	− 0.606957i	0.952838	−	0.303479i	$-0.0981481\pi$
$102$	0	0
$103$	19.8633i	1.95719i	0.205793	+	0.978596i	$0.434023\pi$
$103$	19.8633i	1.95719i	−0.205793	+	0.978596i	$0.565977\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.9352	1.05715	0.528575	−	0.848887i	$-0.322727\pi$
$107$	10.9352	1.05715	0.528575	+	0.848887i	$0.322727\pi$
$108$	0	0
$109$	4.45094	0.426323	0.213161	−	0.977017i	$-0.431624\pi$
$109$	4.45094	0.426323	0.213161	+	0.977017i	$0.431624\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 18.2924i	− 1.72080i	−0.509618	−	0.860401i	$-0.670213\pi$
$113$	− 18.2924i	− 1.72080i	0.509618	−	0.860401i	$-0.329787\pi$
$114$	0	0
$115$	− 14.7861i	− 1.37881i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−20.2198	−1.85355
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	7.26580i	0.649873i
$126$	0	0
$127$	− 2.26608i	− 0.201082i	−0.994933	−	0.100541i	$-0.967943\pi$
$127$	− 2.26608i	− 0.201082i	0.994933	−	0.100541i	$-0.0320574\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	9.22256	0.805778	0.402889	−	0.915249i	$-0.368006\pi$
$131$	9.22256	0.805778	0.402889	+	0.915249i	$0.368006\pi$
$132$	0	0
$133$	−12.9019	−1.11873
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 0.691243i	− 0.0590569i	−0.999564	−	0.0295284i	$-0.990599\pi$
$137$	− 0.691243i	− 0.0590569i	0.999564	−	0.0295284i	$-0.00940056\pi$
$138$	0	0
$139$	1.38851i	0.117772i	0.998265	+	0.0588858i	$0.0187548\pi$
$139$	1.38851i	0.117772i	−0.998265	+	0.0588858i	$0.981245\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−2.29081	−0.190241
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 2.38544i	− 0.195423i	−0.995215	−	0.0977113i	$-0.968848\pi$
$149$	− 2.38544i	− 0.195423i	0.995215	−	0.0977113i	$-0.0311522\pi$
$150$	0	0
$151$	− 6.59934i	− 0.537047i	−0.963273	−	0.268523i	$-0.913464\pi$
$151$	− 6.59934i	− 0.537047i	0.963273	−	0.268523i	$-0.0865357\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	10.9352	0.878339
$156$	0	0
$157$	8.33568	0.665259	0.332630	−	0.943058i	$-0.392064\pi$
$157$	8.33568	0.665259	0.332630	+	0.943058i	$0.392064\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 24.2364i	− 1.91009i
$162$	0	0
$163$	6.30973i	0.494216i	0.968988	+	0.247108i	$0.0794802\pi$
$163$	6.30973i	0.494216i	−0.968988	+	0.247108i	$0.920520\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.3388	1.03219	0.516093	−	0.856533i	$-0.327386\pi$
$167$	13.3388	1.03219	0.516093	+	0.856533i	$0.327386\pi$
$168$	0	0
$169$	20.9243	1.60956
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3.02316i	0.229847i	0.993374	+	0.114923i	$0.0366623\pi$
$173$	3.02316i	0.229847i	−0.993374	+	0.114923i	$0.963338\pi$
$174$	0	0
$175$	− 10.2539i	− 0.775124i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.35140	0.101008	0.0505041	−	0.998724i	$-0.483917\pi$
$179$	1.35140	0.101008	0.0505041	+	0.998724i	$0.483917\pi$
$180$	0	0
$181$	−22.2151	−1.65124	−0.825618	−	0.564229i	$-0.809174\pi$
$181$	−22.2151	−1.65124	−0.825618	+	0.564229i	$0.809174\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.70430i	0.198824i
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	10.6360	0.769593	0.384796	−	0.923002i	$-0.374272\pi$
$191$	10.6360	0.769593	0.384796	+	0.923002i	$0.374272\pi$
$192$	0	0
$193$	−2.35108	−0.169235	−0.0846173	−	0.996414i	$-0.526967\pi$
$193$	−2.35108	−0.169235	−0.0846173	+	0.996414i	$0.526967\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.6315i	1.46993i	0.678105	+	0.734965i	$0.262801\pi$
$197$	20.6315i	1.46993i	−0.678105	+	0.734965i	$0.737199\pi$
$198$	0	0
$199$	− 22.5526i	− 1.59871i	−0.600858	−	0.799356i	$-0.705174\pi$
$199$	− 22.5526i	− 1.59871i	0.600858	−	0.799356i	$-0.294826\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−3.75494	−0.263545
$204$	0	0
$205$	11.4734	0.801335
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	6.59934i	0.454317i	0.973858	+	0.227159i	$0.0729436\pi$
$211$	6.59934i	0.454317i	−0.973858	+	0.227159i	$0.927056\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.80708	0.327840
$216$	0	0
$217$	17.9243	1.21678
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 26.5683i	− 1.78718i
$222$	0	0
$223$	25.8181i	1.72891i	0.502710	+	0.864455i	$0.332336\pi$
$223$	25.8181i	1.72891i	−0.502710	+	0.864455i	$0.667664\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	17.8163	1.18251	0.591254	−	0.806485i	$-0.298633\pi$
$227$	17.8163	1.18251	0.591254	+	0.806485i	$0.298633\pi$
$228$	0	0
$229$	−15.6489	−1.03411	−0.517055	−	0.855952i	$-0.672972\pi$
$229$	−15.6489	−1.03411	−0.517055	+	0.855952i	$0.672972\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 25.4023i	− 1.66416i	−0.554654	−	0.832081i	$-0.687149\pi$
$233$	− 25.4023i	− 1.66416i	0.554654	−	0.832081i	$-0.312851\pi$
$234$	0	0
$235$	− 21.2860i	− 1.38854i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−29.4424	−1.90447	−0.952235	−	0.305368i	$-0.901221\pi$
$239$	−29.4424	−1.90447	−0.952235	+	0.305368i	$0.901221\pi$
$240$	0	0
$241$	19.7488	1.27213	0.636065	−	0.771635i	$-0.280561\pi$
$241$	19.7488	1.27213	0.636065	+	0.771635i	$0.280561\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 34.2065i	− 2.18537i
$246$	0	0
$247$	− 16.9527i	− 1.07868i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−9.58384	−0.604927	−0.302463	−	0.953161i	$-0.597809\pi$
$251$	−9.58384	−0.604927	−0.302463	+	0.953161i	$0.597809\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 24.0270i	− 1.49876i	−0.662139	−	0.749381i	$-0.730351\pi$
$257$	− 24.0270i	− 1.49876i	0.662139	−	0.749381i	$-0.269649\pi$
$258$	0	0
$259$	4.43271i	0.275435i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−3.15311	−0.193694
$266$	0	0
$267$	0	0
$268$	0	0
$269$	11.3454i	0.691743i	0.938282	+	0.345872i	$0.112417\pi$
$269$	11.3454i	0.691743i	−0.938282	+	0.345872i	$0.887583\pi$
$270$	0	0
$271$	18.3412i	1.11415i	0.830462	+	0.557075i	$0.188076\pi$
$271$	18.3412i	1.11415i	−0.830462	+	0.557075i	$0.811924\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−8.22041	−0.493917	−0.246958	−	0.969026i	$-0.579431\pi$
$277$	−8.22041	−0.493917	−0.246958	+	0.969026i	$0.579431\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3.34203i	0.199369i	0.995019	+	0.0996843i	$0.0317833\pi$
$281$	3.34203i	0.199369i	−0.995019	+	0.0996843i	$0.968217\pi$
$282$	0	0
$283$	− 19.0852i	− 1.13450i	−0.823547	−	0.567249i	$-0.808008\pi$
$283$	− 19.0852i	− 1.13450i	0.823547	−	0.567249i	$-0.191992\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	18.8064	1.11011
$288$	0	0
$289$	−3.80731	−0.223960
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 2.22243i	− 0.129836i	−0.997891	−	0.0649179i	$-0.979321\pi$
$293$	− 2.22243i	− 0.129836i	0.997891	−	0.0649179i	$-0.0206785\pi$
$294$	0	0
$295$	17.6314i	1.02654i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	31.8459	1.84170
$300$	0	0
$301$	7.87945	0.454164
$302$	0	0
$303$	0	0
$304$	0	0
$305$	12.5114i	0.716400i
$306$	0	0
$307$	− 14.6866i	− 0.838211i	−0.907938	−	0.419105i	$-0.862344\pi$
$307$	− 14.6866i	− 0.838211i	0.907938	−	0.419105i	$-0.137656\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−6.15848	−0.349215	−0.174608	−	0.984638i	$-0.555866\pi$
$311$	−6.15848	−0.349215	−0.174608	+	0.984638i	$0.555866\pi$
$312$	0	0
$313$	1.76947	0.100016	0.0500082	−	0.998749i	$-0.484075\pi$
$313$	1.76947	0.100016	0.0500082	+	0.998749i	$0.484075\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.4221i	0.810027i	0.914311	+	0.405013i	$0.132733\pi$
$317$	14.4221i	0.810027i	−0.914311	+	0.405013i	$0.867267\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−13.2767	−0.738738
$324$	0	0
$325$	13.4734	0.747368
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 34.8906i	− 1.92358i
$330$	0	0
$331$	28.7287i	1.57907i	0.613703	+	0.789537i	$0.289679\pi$
$331$	28.7287i	1.57907i	−0.613703	+	0.789537i	$0.710321\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−33.8579	−1.84985
$336$	0	0
$337$	−10.6265	−0.578861	−0.289431	−	0.957199i	$-0.593466\pi$
$337$	−10.6265	−0.578861	−0.289431	+	0.957199i	$0.593466\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 25.0400i	− 1.35203i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10.6360	−0.570969	−0.285485	−	0.958383i	$-0.592155\pi$
$347$	−10.6360	−0.570969	−0.285485	+	0.958383i	$0.592155\pi$
$348$	0	0
$349$	−17.2376	−0.922705	−0.461353	−	0.887217i	$-0.652636\pi$
$349$	−17.2376	−0.922705	−0.461353	+	0.887217i	$0.652636\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 11.0266i	− 0.586885i	−0.955977	−	0.293443i	$-0.905199\pi$
$353$	− 11.0266i	− 0.586885i	0.955977	−	0.293443i	$-0.0948010\pi$
$354$	0	0
$355$	43.5490i	2.31134i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−21.2720	−1.12269	−0.561346	−	0.827581i	$-0.689716\pi$
$359$	−21.2720	−1.12269	−0.561346	+	0.827581i	$0.689716\pi$
$360$	0	0
$361$	10.5284	0.554124
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 25.1858i	− 1.31829i
$366$	0	0
$367$	31.1284i	1.62489i	0.583038	+	0.812445i	$0.301864\pi$
$367$	31.1284i	1.62489i	−0.583038	+	0.812445i	$0.698136\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−5.16836	−0.268328
$372$	0	0
$373$	−34.2600	−1.77392	−0.886958	−	0.461851i	$-0.847185\pi$
$373$	−34.2600	−1.77392	−0.886958	+	0.461851i	$0.847185\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 4.93388i	− 0.254108i
$378$	0	0
$379$	− 5.98896i	− 0.307632i	−0.988099	−	0.153816i	$-0.950844\pi$
$379$	− 5.98896i	− 0.307632i	0.988099	−	0.153816i	$-0.0491563\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−17.4550	−0.891909	−0.445954	−	0.895056i	$-0.647136\pi$
$383$	−17.4550	−0.891909	−0.445954	+	0.895056i	$0.647136\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 16.5982i	− 0.841561i	−0.907162	−	0.420781i	$-0.861756\pi$
$389$	− 16.5982i	− 0.841561i	0.907162	−	0.420781i	$-0.138244\pi$
$390$	0	0
$391$	− 24.9406i	− 1.26130i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−16.1036	−0.810260
$396$	0	0
$397$	15.9846	0.802244	0.401122	−	0.916025i	$-0.368620\pi$
$397$	15.9846	0.802244	0.401122	+	0.916025i	$0.368620\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 26.9407i	− 1.34535i	−0.739937	−	0.672676i	$-0.765145\pi$
$401$	− 26.9407i	− 1.34535i	0.739937	−	0.672676i	$-0.234855\pi$
$402$	0	0
$403$	23.5521i	1.17321i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−38.3753	−1.89753	−0.948767	−	0.315976i	$-0.897668\pi$
$409$	−38.3753	−1.89753	−0.948767	+	0.315976i	$0.897668\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	28.9002i	1.42209i
$414$	0	0
$415$	21.2860i	1.04489i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−19.1677	−0.936402	−0.468201	−	0.883622i	$-0.655098\pi$
$419$	−19.1677	−0.936402	−0.468201	+	0.883622i	$0.655098\pi$
$420$	0	0
$421$	−11.0774	−0.539881	−0.269940	−	0.962877i	$-0.587004\pi$
$421$	−11.0774	−0.539881	−0.269940	+	0.962877i	$0.587004\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 10.5519i	− 0.511840i
$426$	0	0
$427$	20.5079i	0.992444i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6.88104	0.331448	0.165724	−	0.986172i	$-0.447004\pi$
$431$	6.88104	0.331448	0.165724	+	0.986172i	$0.447004\pi$
$432$	0	0
$433$	−29.9089	−1.43733	−0.718665	−	0.695356i	$-0.755247\pi$
$433$	−29.9089	−1.43733	−0.718665	+	0.695356i	$0.755247\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 15.9141i	− 0.761274i
$438$	0	0
$439$	− 18.2193i	− 0.869562i	−0.900536	−	0.434781i	$-0.856826\pi$
$439$	− 18.2193i	− 0.869562i	0.900536	−	0.434781i	$-0.143174\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−18.8064	−0.893519	−0.446759	−	0.894654i	$-0.647422\pi$
$443$	−18.8064	−0.893519	−0.446759	+	0.894654i	$0.647422\pi$
$444$	0	0
$445$	30.2600	1.43446
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 18.4554i	− 0.870964i	−0.900198	−	0.435482i	$-0.856578\pi$
$449$	− 18.4554i	− 0.870964i	0.900198	−	0.435482i	$-0.143422\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	69.8200	3.27321
$456$	0	0
$457$	−30.7263	−1.43732	−0.718659	−	0.695363i	$-0.755244\pi$
$457$	−30.7263	−1.43732	−0.718659	+	0.695363i	$0.755244\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	14.9040i	0.694148i	0.937838	+	0.347074i	$0.112825\pi$
$461$	14.9040i	0.694148i	−0.937838	+	0.347074i	$0.887175\pi$
$462$	0	0
$463$	− 3.26552i	− 0.151762i	−0.997117	−	0.0758808i	$-0.975823\pi$
$463$	− 3.26552i	− 0.151762i	0.997117	−	0.0758808i	$-0.0241769\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−36.6227	−1.69470	−0.847348	−	0.531038i	$-0.821802\pi$
$467$	−36.6227	−1.69470	−0.847348	+	0.531038i	$0.821802\pi$
$468$	0	0
$469$	−55.4975	−2.56264
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 6.73294i	− 0.308928i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	23.6755	1.08176	0.540881	−	0.841099i	$-0.318091\pi$
$479$	23.6755	1.08176	0.540881	+	0.841099i	$0.318091\pi$
$480$	0	0
$481$	−5.82446	−0.265572
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 8.64829i	− 0.392699i
$486$	0	0
$487$	2.91061i	0.131892i	0.997823	+	0.0659461i	$0.0210065\pi$
$487$	2.91061i	0.131892i	−0.997823	+	0.0659461i	$0.978993\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	31.8459	1.43719	0.718593	−	0.695431i	$-0.244786\pi$
$491$	31.8459	1.43719	0.718593	+	0.695431i	$0.244786\pi$
$492$	0	0
$493$	−3.86404	−0.174028
$494$	0	0
$495$	0	0
$496$	0	0
$497$	71.3826i	3.20195i
$498$	0	0
$499$	15.9533i	0.714166i	0.934073	+	0.357083i	$0.116229\pi$
$499$	15.9533i	0.714166i	−0.934073	+	0.357083i	$0.883771\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−35.9001	−1.60071	−0.800353	−	0.599529i	$-0.795355\pi$
$503$	−35.9001	−1.60071	−0.800353	+	0.599529i	$0.795355\pi$
$504$	0	0
$505$	16.4958	0.734054
$506$	0	0
$507$	0	0
$508$	0	0
$509$	15.8046i	0.700527i	0.936651	+	0.350263i	$0.113908\pi$
$509$	15.8046i	0.700527i	−0.936651	+	0.350263i	$0.886092\pi$
$510$	0	0
$511$	− 41.2829i	− 1.82625i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−53.7164	−2.36703
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	27.7342i	1.21506i	0.794297	+	0.607530i	$0.207839\pi$
$521$	27.7342i	1.21506i	−0.794297	+	0.607530i	$0.792161\pi$
$522$	0	0
$523$	− 9.08674i	− 0.397335i	−0.980067	−	0.198668i	$-0.936339\pi$
$523$	− 9.08674i	− 0.397335i	0.980067	−	0.198668i	$-0.0636614\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	18.4451	0.803482
$528$	0	0
$529$	6.89485	0.299776
$530$	0	0
$531$	0	0
$532$	0	0
$533$	24.7111i	1.07036i
$534$	0	0
$535$	29.5722i	1.27852i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−25.2428	−1.08527	−0.542637	−	0.839967i	$-0.682574\pi$
$541$	−25.2428	−1.08527	−0.542637	+	0.839967i	$0.682574\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	12.0367i	0.515595i
$546$	0	0
$547$	− 26.8176i	− 1.14664i	−0.819333	−	0.573318i	$-0.805656\pi$
$547$	− 26.8176i	− 1.14664i	0.819333	−	0.573318i	$-0.194344\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2.46557	−0.105037
$552$	0	0
$553$	−26.3960	−1.12247
$554$	0	0
$555$	0	0
$556$	0	0
$557$	18.4554i	0.781980i	0.920395	+	0.390990i	$0.127867\pi$
$557$	18.4554i	0.781980i	−0.920395	+	0.390990i	$0.872133\pi$
$558$	0	0
$559$	10.3534i	0.437901i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−43.0804	−1.81562	−0.907811	−	0.419379i	$-0.862248\pi$
$563$	−43.0804	−1.81562	−0.907811	+	0.419379i	$0.862248\pi$
$564$	0	0
$565$	49.4681	2.08114
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 26.4659i	− 1.10951i	−0.832014	−	0.554755i	$-0.812812\pi$
$569$	− 26.4659i	− 1.10951i	0.832014	−	0.554755i	$-0.187188\pi$
$570$	0	0
$571$	6.69879i	0.280336i	0.990128	+	0.140168i	$0.0447642\pi$
$571$	6.69879i	0.280336i	−0.990128	+	0.140168i	$0.955236\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	12.6479	0.527455
$576$	0	0
$577$	−10.9019	−0.453851	−0.226926	−	0.973912i	$-0.572867\pi$
$577$	−10.9019	−0.453851	−0.226926	+	0.973912i	$0.572867\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	34.8906i	1.44750i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	25.3262	1.04532	0.522661	−	0.852541i	$-0.324939\pi$
$587$	25.3262	1.04532	0.522661	+	0.852541i	$0.324939\pi$
$588$	0	0
$589$	11.7695	0.484953
$590$	0	0
$591$	0	0
$592$	0	0
$593$	30.6479i	1.25856i	0.777179	+	0.629280i	$0.216650\pi$
$593$	30.6479i	1.25856i	−0.777179	+	0.629280i	$0.783350\pi$
$594$	0	0
$595$	− 54.6805i	− 2.24168i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	43.1424	1.76275	0.881376	−	0.472415i	$-0.156618\pi$
$599$	43.1424	1.76275	0.881376	+	0.472415i	$0.156618\pi$
$600$	0	0
$601$	20.3357	0.829510	0.414755	−	0.909933i	$-0.363867\pi$
$601$	20.3357	0.829510	0.414755	+	0.909933i	$0.363867\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 29.7473i	− 1.20940i
$606$	0	0
$607$	− 36.8160i	− 1.49432i	−0.664646	−	0.747158i	$-0.731418\pi$
$607$	− 36.8160i	− 1.49432i	0.664646	−	0.747158i	$-0.268582\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	45.8452	1.85470
$612$	0	0
$613$	29.5284	1.19264	0.596320	−	0.802747i	$-0.296629\pi$
$613$	29.5284	1.19264	0.596320	+	0.802747i	$0.296629\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 16.7540i	− 0.674492i	−0.941417	−	0.337246i	$-0.890505\pi$
$617$	− 16.7540i	− 0.674492i	0.941417	−	0.337246i	$-0.109495\pi$
$618$	0	0
$619$	28.8623i	1.16008i	0.814590	+	0.580038i	$0.196962\pi$
$619$	28.8623i	1.16008i	−0.814590	+	0.580038i	$0.803038\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	49.6002	1.98719
$624$	0	0
$625$	−31.2151	−1.24860
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4.56150i	0.181879i
$630$	0	0
$631$	16.3082i	0.649219i	0.945848	+	0.324609i	$0.105233\pi$
$631$	16.3082i	0.649219i	−0.945848	+	0.324609i	$0.894767\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	6.12816	0.243189
$636$	0	0
$637$	73.6731	2.91903
$638$	0	0
$639$	0	0
$640$	0	0
$641$	15.5346i	0.613578i	0.951778	+	0.306789i	$0.0992546\pi$
$641$	15.5346i	0.613578i	−0.951778	+	0.306789i	$0.900745\pi$
$642$	0	0
$643$	− 43.4154i	− 1.71214i	−0.516864	−	0.856068i	$-0.672901\pi$
$643$	− 43.4154i	− 1.71214i	0.516864	−	0.856068i	$-0.327099\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−10.6360	−0.418144	−0.209072	−	0.977900i	$-0.567044\pi$
$647$	−10.6360	−0.418144	−0.209072	+	0.977900i	$0.567044\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	22.5885i	0.883958i	0.897025	+	0.441979i	$0.145723\pi$
$653$	22.5885i	0.883958i	−0.897025	+	0.441979i	$0.854277\pi$
$654$	0	0
$655$	24.9406i	0.974508i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−39.7171	−1.54716	−0.773579	−	0.633700i	$-0.781535\pi$
$659$	−39.7171	−1.54716	−0.773579	+	0.633700i	$0.781535\pi$
$660$	0	0
$661$	−12.3960	−0.482147	−0.241073	−	0.970507i	$-0.577499\pi$
$661$	−12.3960	−0.482147	−0.241073	+	0.970507i	$0.577499\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 34.8906i	− 1.35300i
$666$	0	0
$667$	− 4.63161i	− 0.179336i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−4.33568	−0.167128	−0.0835640	−	0.996502i	$-0.526630\pi$
$673$	−4.33568	−0.167128	−0.0835640	+	0.996502i	$0.526630\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	38.8215i	1.49203i	0.665929	+	0.746016i	$0.268036\pi$
$677$	38.8215i	1.49203i	−0.665929	+	0.746016i	$0.731964\pi$
$678$	0	0
$679$	− 14.1757i	− 0.544014i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	3.81697	0.146052	0.0730261	−	0.997330i	$-0.476734\pi$
$683$	3.81697	0.146052	0.0730261	+	0.997330i	$0.476734\pi$
$684$	0	0
$685$	1.86933	0.0714234
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 6.79109i	− 0.258720i
$690$	0	0
$691$	− 0.877573i	− 0.0333844i	−0.999861	−	0.0166922i	$-0.994686\pi$
$691$	− 0.877573i	− 0.0333844i	0.999861	−	0.0166922i	$-0.00531355\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3.75494	−0.142433
$696$	0	0
$697$	19.3528	0.733040
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 45.4984i	− 1.71845i	−0.511597	−	0.859225i	$-0.670946\pi$
$701$	− 45.4984i	− 1.71845i	0.511597	−	0.859225i	$-0.329054\pi$
$702$	0	0
$703$	2.91061i	0.109776i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	27.0388	1.01690
$708$	0	0
$709$	−4.77121	−0.179186	−0.0895932	−	0.995978i	$-0.528557\pi$
$709$	−4.77121	−0.179186	−0.0895932	+	0.995978i	$0.528557\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	22.1091i	0.827993i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−0.361284	−0.0134736	−0.00673681	−	0.999977i	$-0.502144\pi$
$719$	−0.361284	−0.0134736	−0.00673681	+	0.999977i	$0.502144\pi$
$720$	0	0
$721$	−88.0483	−3.27909
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 1.95954i	− 0.0727755i
$726$	0	0
$727$	20.9964i	0.778712i	0.921087	+	0.389356i	$0.127302\pi$
$727$	20.9964i	0.778712i	−0.921087	+	0.389356i	$0.872698\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	8.10838	0.299899
$732$	0	0
$733$	51.6181	1.90656	0.953279	−	0.302091i	$-0.0976847\pi$
$733$	51.6181	1.90656	0.953279	+	0.302091i	$0.0976847\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 0.510933i	− 0.0187950i	−0.999956	−	0.00939749i	$-0.997009\pi$
$739$	− 0.510933i	− 0.0187950i	0.999956	−	0.00939749i	$-0.00299136\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	8.46967	0.310722	0.155361	−	0.987858i	$-0.450346\pi$
$743$	8.46967	0.310722	0.155361	+	0.987858i	$0.450346\pi$
$744$	0	0
$745$	6.45094	0.236344
$746$	0	0
$747$	0	0
$748$	0	0
$749$	48.4727i	1.77116i
$750$	0	0
$751$	− 14.0763i	− 0.513650i	−0.966458	−	0.256825i	$-0.917324\pi$
$751$	− 14.0763i	− 0.513650i	0.966458	−	0.256825i	$-0.0826764\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	17.8466	0.649504
$756$	0	0
$757$	42.0242	1.52739	0.763697	−	0.645575i	$-0.223382\pi$
$757$	42.0242	1.52739	0.763697	+	0.645575i	$0.223382\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	16.2793i	0.590125i	0.955478	+	0.295062i	$0.0953405\pi$
$761$	16.2793i	0.590125i	−0.955478	+	0.295062i	$0.904660\pi$
$762$	0	0
$763$	19.7297i	0.714264i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−37.9741	−1.37116
$768$	0	0
$769$	−15.6938	−0.565932	−0.282966	−	0.959130i	$-0.591318\pi$
$769$	−15.6938	−0.565932	−0.282966	+	0.959130i	$0.591318\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	39.9339i	1.43632i	0.695876	+	0.718162i	$0.255016\pi$
$773$	39.9339i	1.43632i	−0.695876	+	0.718162i	$0.744984\pi$
$774$	0	0
$775$	9.35393i	0.336003i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	12.3487	0.442437
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	22.5422i	0.804565i
$786$	0	0
$787$	50.0800i	1.78516i	0.450889	+	0.892580i	$0.351107\pi$
$787$	50.0800i	1.78516i	−0.450889	+	0.892580i	$0.648893\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	81.0848	2.88304
$792$	0	0
$793$	−26.9468	−0.956907
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1.95954i	− 0.0694105i	−0.999398	−	0.0347052i	$-0.988951\pi$
$797$	− 1.95954i	− 0.0694105i	0.999398	−	0.0347052i	$-0.0110492\pi$
$798$	0	0
$799$	− 35.9043i	− 1.27020i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	65.5424	2.31007
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 37.1761i	− 1.30704i	−0.756908	−	0.653521i	$-0.773291\pi$
$809$	− 37.1761i	− 1.30704i	0.756908	−	0.653521i	$-0.226709\pi$
$810$	0	0
$811$	− 20.0964i	− 0.705679i	−0.935684	−	0.352839i	$-0.885216\pi$
$811$	− 20.0964i	− 0.705679i	0.935684	−	0.352839i	$-0.114784\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−17.0634	−0.597705
$816$	0	0
$817$	5.17380	0.181009
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 27.0965i	− 0.945675i	−0.881150	−	0.472838i	$-0.843230\pi$
$821$	− 27.0965i	− 0.945675i	0.881150	−	0.472838i	$-0.156770\pi$
$822$	0	0
$823$	− 23.0411i	− 0.803163i	−0.915823	−	0.401582i	$-0.868461\pi$
$823$	− 23.0411i	− 0.803163i	0.915823	−	0.401582i	$-0.131539\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	45.1847	1.57123	0.785613	−	0.618718i	$-0.212348\pi$
$827$	45.1847	1.57123	0.785613	+	0.618718i	$0.212348\pi$
$828$	0	0
$829$	19.5182	0.677897	0.338948	−	0.940805i	$-0.389929\pi$
$829$	19.5182	0.677897	0.338948	+	0.940805i	$0.389929\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 57.6981i	− 1.99912i
$834$	0	0
$835$	36.0721i	1.24833i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	54.0777	1.86697	0.933484	−	0.358618i	$-0.116752\pi$
$839$	54.0777	1.86697	0.933484	+	0.358618i	$0.116752\pi$
$840$	0	0
$841$	28.2824	0.975256
$842$	0	0
$843$	0	0
$844$	0	0
$845$	56.5856i	1.94661i
$846$	0	0
$847$	− 48.7598i	− 1.67541i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−5.46762	−0.187428
$852$	0	0
$853$	−11.5732	−0.396260	−0.198130	−	0.980176i	$-0.563487\pi$
$853$	−11.5732	−0.396260	−0.198130	+	0.980176i	$0.563487\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.01601i	0.103025i	0.998672	+	0.0515125i	$0.0164042\pi$
$857$	3.01601i	0.103025i	−0.998672	+	0.0515125i	$0.983596\pi$
$858$	0	0
$859$	41.7714i	1.42522i	0.701559	+	0.712611i	$0.252488\pi$
$859$	41.7714i	1.42522i	−0.701559	+	0.712611i	$0.747512\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−40.0784	−1.36428	−0.682142	−	0.731220i	$-0.738951\pi$
$863$	−40.0784	−1.36428	−0.682142	+	0.731220i	$0.738951\pi$
$864$	0	0
$865$	−8.17554	−0.277977
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	− 72.9222i	− 2.47088i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−32.2072	−1.08880
$876$	0	0
$877$	−12.8865	−0.435145	−0.217573	−	0.976044i	$-0.569814\pi$
$877$	−12.8865	−0.435145	−0.217573	+	0.976044i	$0.569814\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 27.0965i	− 0.912905i	−0.889748	−	0.456453i	$-0.849120\pi$
$881$	− 27.0965i	− 0.912905i	0.889748	−	0.456453i	$-0.150880\pi$
$882$	0	0
$883$	47.4366i	1.59637i	0.602413	+	0.798184i	$0.294206\pi$
$883$	47.4366i	1.59637i	−0.602413	+	0.798184i	$0.705794\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−45.8452	−1.53933	−0.769666	−	0.638447i	$-0.779577\pi$
$887$	−45.8452	−1.53933	−0.769666	+	0.638447i	$0.779577\pi$
$888$	0	0
$889$	10.0449	0.336894
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 22.9099i	− 0.766649i
$894$	0	0
$895$	3.65458i	0.122159i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3.42536	0.114242
$900$	0	0
$901$	−5.31853	−0.177186
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 60.0764i	− 1.99701i
$906$	0	0
$907$	− 46.1047i	− 1.53088i	−0.643507	−	0.765440i	$-0.722521\pi$
$907$	− 46.1047i	− 1.53088i	0.643507	−	0.765440i	$-0.277479\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−35.9001	−1.18942	−0.594712	−	0.803939i	$-0.702734\pi$
$911$	−35.9001	−1.18942	−0.594712	+	0.803939i	$0.702734\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	40.8809i	1.35001i
$918$	0	0
$919$	− 24.3955i	− 0.804733i	−0.915479	−	0.402366i	$-0.868188\pi$
$919$	− 24.3955i	− 0.804733i	0.915479	−	0.402366i	$-0.131812\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−93.7947	−3.08729
$924$	0	0
$925$	−2.31324	−0.0760589
$926$	0	0
$927$	0	0
$928$	0	0
$929$	54.9403i	1.80253i	0.433267	+	0.901266i	$0.357361\pi$
$929$	54.9403i	1.80253i	−0.433267	+	0.901266i	$0.642639\pi$
$930$	0	0
$931$	− 36.8160i	− 1.20660i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−25.9692	−0.848376	−0.424188	−	0.905574i	$-0.639440\pi$
$937$	−25.9692	−0.848376	−0.424188	+	0.905574i	$0.639440\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	10.1260i	0.330097i	0.986285	+	0.165048i	$0.0527780\pi$
$941$	10.1260i	0.330097i	−0.986285	+	0.165048i	$0.947222\pi$
$942$	0	0
$943$	23.1971i	0.755403i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−23.9127	−0.777059	−0.388530	−	0.921436i	$-0.627017\pi$
$947$	−23.9127	−0.777059	−0.388530	+	0.921436i	$0.627017\pi$
$948$	0	0
$949$	54.2446	1.76085
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 24.6576i	− 0.798737i	−0.916791	−	0.399368i	$-0.869229\pi$
$953$	− 24.6576i	− 0.798737i	0.916791	−	0.399368i	$-0.130771\pi$
$954$	0	0
$955$	28.7629i	0.930745i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3.06408	0.0989443
$960$	0	0
$961$	14.6489	0.472546
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 6.35804i	− 0.204672i
$966$	0	0
$967$	7.59878i	0.244360i	0.992508	+	0.122180i	$0.0389886\pi$
$967$	7.59878i	0.244360i	−0.992508	+	0.122180i	$0.961011\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	13.9993	0.449259	0.224630	−	0.974444i	$-0.427883\pi$
$971$	13.9993	0.449259	0.224630	+	0.974444i	$0.427883\pi$
$972$	0	0
$973$	−6.15484	−0.197315
$974$	0	0
$975$	0	0
$976$	0	0
$977$	16.1235i	0.515835i	0.966167	+	0.257918i	$0.0830363\pi$
$977$	16.1235i	0.515835i	−0.966167	+	0.257918i	$0.916964\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−8.46967	−0.270140	−0.135070	−	0.990836i	$-0.543126\pi$
$983$	−8.46967	−0.270140	−0.135070	+	0.990836i	$0.543126\pi$
$984$	0	0
$985$	−55.7936	−1.77773
$986$	0	0
$987$	0	0
$988$	0	0
$989$	9.71907i	0.309048i
$990$	0	0
$991$	14.4653i	0.459506i	0.973249	+	0.229753i	$0.0737918\pi$
$991$	14.4653i	0.459506i	−0.973249	+	0.229753i	$0.926208\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	60.9890	1.93348
$996$	0	0
$997$	31.0224	0.982490	0.491245	−	0.871021i	$-0.336542\pi$
$997$	31.0224	0.982490	0.491245	+	0.871021i	$0.336542\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5328.2.e.e.2591.10	yes	12
3.2	odd	2	inner	5328.2.e.e.2591.4	yes	12
4.3	odd	2	inner	5328.2.e.e.2591.9	yes	12
12.11	even	2	inner	5328.2.e.e.2591.3	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5328.2.e.e.2591.3	✓	12	12.11	even	2	inner
5328.2.e.e.2591.4	yes	12	3.2	odd	2	inner
5328.2.e.e.2591.9	yes	12	4.3	odd	2	inner
5328.2.e.e.2591.10	yes	12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 5328 = 2^{4} \cdot 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5328.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.70430i q^{5} +4.43271i q^{7} +O(q^{10})\)	\(q+2.70430i q^{5} +4.43271i q^{7} -5.82446 q^{13} +4.56150i q^{17} +2.91061i q^{19} -5.46762 q^{23} -2.31324 q^{25} +0.847097i q^{29} -4.04365i q^{31} -11.9874 q^{35} +1.00000 q^{37} -4.24264i q^{41} -1.77757i q^{43} -7.87116 q^{47} -12.6489 q^{49} +1.16596i q^{53} +6.51976 q^{59} +4.62648 q^{61} -15.7511i q^{65} +12.5200i q^{67} +16.1036 q^{71} -9.31324 q^{73} +5.95481i q^{79} +7.87116 q^{83} -12.3357 q^{85} -11.1896i q^{89} -25.8181i q^{91} -7.87116 q^{95} -3.19798 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q - 16 q^{13} - 12 q^{25} + 12 q^{37} - 44 q^{49} + 24 q^{61} - 96 q^{73} - 56 q^{85} - 16 q^{97}+O(q^{100}) \) 12 * q - 16 * q^13 - 12 * q^25 + 12 * q^37 - 44 * q^49 + 24 * q^61 - 96 * q^73 - 56 * q^85 - 16 * q^97

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