LMFDB - Embedded newform 2592.2.f.c.1295.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2592,2,Mod(1295,2592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2592, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2592.1295");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2592 = 2^{5} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2592.f (of order $2$, degree $1$, not 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:	$20.6972242039$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 648)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1295.12
Character	$\chi$	$=$	2592.1295
Dual form			2592.2.f.c.1295.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.949666 q^{5} -4.71000i q^{7} +O(q^{10})$	$q-0.949666 q^{5} -4.71000i q^{7} +4.29556i q^{11} +4.06857i q^{13} +1.19178i q^{17} +3.17693 q^{19} -0.750651 q^{23} -4.09813 q^{25} -7.74362 q^{29} +0.573791i q^{31} +4.47293i q^{35} -4.87320i q^{37} +9.75877i q^{41} +5.31540 q^{43} +9.82024 q^{47} -15.1841 q^{49} +0.877682 q^{53} -4.07935i q^{55} +1.75441i q^{59} +9.92114i q^{61} -3.86379i q^{65} -10.9667 q^{67} +9.91048 q^{71} +8.74944 q^{73} +20.2321 q^{77} +5.74545i q^{79} +12.5035i q^{83} -1.13179i q^{85} +10.1440i q^{89} +19.1630 q^{91} -3.01702 q^{95} +7.83023 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 24 q^{25} - 24 q^{49} - 48 q^{67}+O(q^{100}) $ 24 * q + 24 * q^25 - 24 * q^49 - 48 * q^67

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times$.

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

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
$5$	−0.949666	−0.424704	−0.212352	−	0.977193i	$-0.568112\pi$
$5$	−0.949666	−0.424704	−0.212352	+	0.977193i	$0.568112\pi$
$6$	0	0
$7$	− 4.71000i	− 1.78021i	−0.455754	−	0.890106i	$-0.650630\pi$
$7$	− 4.71000i	− 1.78021i	0.455754	−	0.890106i	$-0.349370\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.29556i	1.29516i	0.761997	+	0.647581i	$0.224219\pi$
$11$	4.29556i	1.29516i	−0.761997	+	0.647581i	$0.775781\pi$
$12$	0	0
$13$	4.06857i	1.12842i	0.825632	+	0.564210i	$0.190819\pi$
$13$	4.06857i	1.12842i	−0.825632	+	0.564210i	$0.809181\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.19178i	0.289049i	0.989501	+	0.144524i	$0.0461652\pi$
$17$	1.19178i	0.289049i	−0.989501	+	0.144524i	$0.953835\pi$
$18$	0	0
$19$	3.17693	0.728837	0.364418	−	0.931235i	$-0.381268\pi$
$19$	3.17693	0.728837	0.364418	+	0.931235i	$0.381268\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.750651	−0.156522	−0.0782608	−	0.996933i	$-0.524937\pi$
$23$	−0.750651	−0.156522	−0.0782608	+	0.996933i	$0.524937\pi$
$24$	0	0
$25$	−4.09813	−0.819627
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−7.74362	−1.43795	−0.718977	−	0.695034i	$-0.755389\pi$
$29$	−7.74362	−1.43795	−0.718977	+	0.695034i	$0.755389\pi$
$30$	0	0
$31$	0.573791i	0.103056i	0.998672	+	0.0515279i	$0.0164091\pi$
$31$	0.573791i	0.103056i	−0.998672	+	0.0515279i	$0.983591\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.47293i	0.756062i
$36$	0	0
$37$	− 4.87320i	− 0.801148i	−0.916264	−	0.400574i	$-0.868811\pi$
$37$	− 4.87320i	− 0.801148i	0.916264	−	0.400574i	$-0.131189\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.75877i	1.52406i	0.647539	+	0.762032i	$0.275798\pi$
$41$	9.75877i	1.52406i	−0.647539	+	0.762032i	$0.724202\pi$
$42$	0	0
$43$	5.31540	0.810591	0.405295	−	0.914186i	$-0.367169\pi$
$43$	5.31540	0.810591	0.405295	+	0.914186i	$0.367169\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.82024	1.43243	0.716214	−	0.697880i	$-0.245873\pi$
$47$	9.82024	1.43243	0.716214	+	0.697880i	$0.245873\pi$
$48$	0	0
$49$	−15.1841	−2.16915
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0.877682	0.120559	0.0602794	−	0.998182i	$-0.480801\pi$
$53$	0.877682	0.120559	0.0602794	+	0.998182i	$0.480801\pi$
$54$	0	0
$55$	− 4.07935i	− 0.550060i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.75441i	0.228405i	0.993458	+	0.114202i	$0.0364312\pi$
$59$	1.75441i	0.228405i	−0.993458	+	0.114202i	$0.963569\pi$
$60$	0	0
$61$	9.92114i	1.27027i	0.772400	+	0.635136i	$0.219056\pi$
$61$	9.92114i	1.27027i	−0.772400	+	0.635136i	$0.780944\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 3.86379i	− 0.479244i
$66$	0	0
$67$	−10.9667	−1.33980	−0.669898	−	0.742453i	$-0.733662\pi$
$67$	−10.9667	−1.33980	−0.669898	+	0.742453i	$0.733662\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	9.91048	1.17616	0.588079	−	0.808804i	$-0.299884\pi$
$71$	9.91048	1.17616	0.588079	+	0.808804i	$0.299884\pi$
$72$	0	0
$73$	8.74944	1.02404	0.512022	−	0.858972i	$-0.328897\pi$
$73$	8.74944	1.02404	0.512022	+	0.858972i	$0.328897\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	20.2321	2.30566
$78$	0	0
$79$	5.74545i	0.646414i	0.946328	+	0.323207i	$0.104761\pi$
$79$	5.74545i	0.646414i	−0.946328	+	0.323207i	$0.895239\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	12.5035i	1.37243i	0.727397	+	0.686217i	$0.240730\pi$
$83$	12.5035i	1.37243i	−0.727397	+	0.686217i	$0.759270\pi$
$84$	0	0
$85$	− 1.13179i	− 0.122760i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	10.1440i	1.07527i	0.843179	+	0.537633i	$0.180681\pi$
$89$	10.1440i	1.07527i	−0.843179	+	0.537633i	$0.819319\pi$
$90$	0	0
$91$	19.1630	2.00882
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.01702	−0.309540
$96$	0	0
$97$	7.83023	0.795039	0.397520	−	0.917594i	$-0.369871\pi$
$97$	7.83023	0.795039	0.397520	+	0.917594i	$0.369871\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	17.8531	1.77645	0.888226	−	0.459407i	$-0.151938\pi$
$101$	17.8531	1.77645	0.888226	+	0.459407i	$0.151938\pi$
$102$	0	0
$103$	− 2.45412i	− 0.241812i	−0.992664	−	0.120906i	$-0.961420\pi$
$103$	− 2.45412i	− 0.241812i	0.992664	−	0.120906i	$-0.0385800\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.283208i	0.0273788i	0.999906	+	0.0136894i	$0.00435760\pi$
$107$	0.283208i	0.0273788i	−0.999906	+	0.0136894i	$0.995642\pi$
$108$	0	0
$109$	9.74716i	0.933609i	0.884361	+	0.466805i	$0.154595\pi$
$109$	9.74716i	0.933609i	−0.884361	+	0.466805i	$0.845405\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	4.72734i	0.444711i	0.974966	+	0.222356i	$0.0713746\pi$
$113$	4.72734i	0.444711i	−0.974966	+	0.222356i	$0.928625\pi$
$114$	0	0
$115$	0.712868	0.0664753
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.61328	0.514568
$120$	0	0
$121$	−7.45187	−0.677443
$122$	0	0
$123$	0	0
$124$	0	0
$125$	8.64019	0.772802
$126$	0	0
$127$	− 13.8042i	− 1.22492i	−0.790500	−	0.612462i	$-0.790179\pi$
$127$	− 13.8042i	− 1.22492i	0.790500	−	0.612462i	$-0.209821\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 4.73463i	− 0.413666i	−0.978376	−	0.206833i	$-0.933684\pi$
$131$	− 4.73463i	− 0.413666i	0.978376	−	0.206833i	$-0.0663157\pi$
$132$	0	0
$133$	− 14.9633i	− 1.29748i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	15.5543i	1.32890i	0.747334	+	0.664448i	$0.231333\pi$
$137$	15.5543i	1.32890i	−0.747334	+	0.664448i	$0.768667\pi$
$138$	0	0
$139$	−7.88979	−0.669204	−0.334602	−	0.942360i	$-0.608602\pi$
$139$	−7.88979	−0.669204	−0.334602	+	0.942360i	$0.608602\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−17.4768	−1.46148
$144$	0	0
$145$	7.35385	0.610704
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−16.0069	−1.31133	−0.655667	−	0.755050i	$-0.727613\pi$
$149$	−16.0069	−1.31133	−0.655667	+	0.755050i	$0.727613\pi$
$150$	0	0
$151$	1.82348i	0.148392i	0.997244	+	0.0741962i	$0.0236391\pi$
$151$	1.82348i	0.148392i	−0.997244	+	0.0741962i	$0.976361\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 0.544909i	− 0.0437682i
$156$	0	0
$157$	20.4887i	1.63518i	0.575802	+	0.817589i	$0.304690\pi$
$157$	20.4887i	1.63518i	−0.575802	+	0.817589i	$0.695310\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.53556i	0.278641i
$162$	0	0
$163$	7.59374	0.594787	0.297394	−	0.954755i	$-0.403883\pi$
$163$	7.59374	0.594787	0.297394	+	0.954755i	$0.403883\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.61328	0.434368	0.217184	−	0.976131i	$-0.430313\pi$
$167$	5.61328	0.434368	0.217184	+	0.976131i	$0.430313\pi$
$168$	0	0
$169$	−3.55328	−0.273329
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−12.6324	−0.960427	−0.480214	−	0.877152i	$-0.659441\pi$
$173$	−12.6324	−0.960427	−0.480214	+	0.877152i	$0.659441\pi$
$174$	0	0
$175$	19.3022i	1.45911i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	20.5282i	1.53435i	0.641439	+	0.767174i	$0.278338\pi$
$179$	20.5282i	1.53435i	−0.641439	+	0.767174i	$0.721662\pi$
$180$	0	0
$181$	− 9.96062i	− 0.740367i	−0.928959	−	0.370184i	$-0.879295\pi$
$181$	− 9.96062i	− 0.740367i	0.928959	−	0.370184i	$-0.120705\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4.62791i	0.340251i
$186$	0	0
$187$	−5.11936	−0.374365
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−22.9998	−1.66421	−0.832105	−	0.554618i	$-0.812865\pi$
$191$	−22.9998	−1.66421	−0.832105	+	0.554618i	$0.812865\pi$
$192$	0	0
$193$	−1.00000	−0.0719816	−0.0359908	−	0.999352i	$-0.511459\pi$
$193$	−1.00000	−0.0719816	−0.0359908	+	0.999352i	$0.511459\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.519639	0.0370228	0.0185114	−	0.999829i	$-0.494107\pi$
$197$	0.519639	0.0370228	0.0185114	+	0.999829i	$0.494107\pi$
$198$	0	0
$199$	0.630646i	0.0447053i	0.999750	+	0.0223527i	$0.00711567\pi$
$199$	0.630646i	0.0447053i	−0.999750	+	0.0223527i	$0.992884\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	36.4724i	2.55986i
$204$	0	0
$205$	− 9.26758i	− 0.647276i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	13.6467i	0.943961i
$210$	0	0
$211$	−9.91793	−0.682778	−0.341389	−	0.939922i	$-0.610897\pi$
$211$	−9.91793	−0.682778	−0.341389	+	0.939922i	$0.610897\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5.04785	−0.344261
$216$	0	0
$217$	2.70255	0.183461
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4.84884	−0.326168
$222$	0	0
$223$	− 7.10727i	− 0.475938i	−0.971273	−	0.237969i	$-0.923518\pi$
$223$	− 7.10727i	− 0.475938i	0.971273	−	0.237969i	$-0.0764816\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 12.4621i	− 0.827138i	−0.910473	−	0.413569i	$-0.864282\pi$
$227$	− 12.4621i	− 0.827138i	0.910473	−	0.413569i	$-0.135718\pi$
$228$	0	0
$229$	− 9.76872i	− 0.645535i	−0.946478	−	0.322768i	$-0.895387\pi$
$229$	− 9.76872i	− 0.645535i	0.946478	−	0.322768i	$-0.104613\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4.66388i	0.305541i	0.988262	+	0.152771i	$0.0488196\pi$
$233$	4.66388i	0.305541i	−0.988262	+	0.152771i	$0.951180\pi$
$234$	0	0
$235$	−9.32595	−0.608358
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3.42532	0.221565	0.110783	−	0.993845i	$-0.464664\pi$
$239$	3.42532	0.221565	0.110783	+	0.993845i	$0.464664\pi$
$240$	0	0
$241$	1.25572	0.0808880	0.0404440	−	0.999182i	$-0.487123\pi$
$241$	1.25572	0.0808880	0.0404440	+	0.999182i	$0.487123\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	14.4198	0.921248
$246$	0	0
$247$	12.9256i	0.822434i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 20.2340i	− 1.27716i	−0.769556	−	0.638579i	$-0.779523\pi$
$251$	− 20.2340i	− 1.27716i	0.769556	−	0.638579i	$-0.220477\pi$
$252$	0	0
$253$	− 3.22447i	− 0.202721i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1.17614i	− 0.0733655i	−0.999327	−	0.0366828i	$-0.988321\pi$
$257$	− 1.17614i	− 0.0733655i	0.999327	−	0.0366828i	$-0.0116791\pi$
$258$	0	0
$259$	−22.9527	−1.42621
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−7.02434	−0.433139	−0.216570	−	0.976267i	$-0.569487\pi$
$263$	−7.02434	−0.433139	−0.216570	+	0.976267i	$0.569487\pi$
$264$	0	0
$265$	−0.833505	−0.0512018
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−11.4172	−0.696119	−0.348060	−	0.937472i	$-0.613159\pi$
$269$	−11.4172	−0.696119	−0.348060	+	0.937472i	$0.613159\pi$
$270$	0	0
$271$	10.9905i	0.667625i	0.942639	+	0.333813i	$0.108335\pi$
$271$	10.9905i	0.667625i	−0.942639	+	0.333813i	$0.891665\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 17.6038i	− 1.06155i
$276$	0	0
$277$	3.60170i	0.216405i	0.994129	+	0.108203i	$0.0345096\pi$
$277$	3.60170i	0.216405i	−0.994129	+	0.108203i	$0.965490\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 12.2741i	− 0.732209i	−0.930574	−	0.366104i	$-0.880691\pi$
$281$	− 12.2741i	− 0.732209i	0.930574	−	0.366104i	$-0.119309\pi$
$282$	0	0
$283$	2.54474	0.151269	0.0756344	−	0.997136i	$-0.475902\pi$
$283$	2.54474	0.151269	0.0756344	+	0.997136i	$0.475902\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	45.9638	2.71316
$288$	0	0
$289$	15.5797	0.916451
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.85934	−0.108624	−0.0543120	−	0.998524i	$-0.517297\pi$
$293$	−1.85934	−0.108624	−0.0543120	+	0.998524i	$0.517297\pi$
$294$	0	0
$295$	− 1.66610i	− 0.0970042i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 3.05408i	− 0.176622i
$300$	0	0
$301$	− 25.0355i	− 1.44302i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 9.42177i	− 0.539489i
$306$	0	0
$307$	−29.0809	−1.65973	−0.829867	−	0.557961i	$-0.811584\pi$
$307$	−29.0809	−1.65973	−0.829867	+	0.557961i	$0.811584\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	18.7260	1.06185	0.530927	−	0.847417i	$-0.321844\pi$
$311$	18.7260	1.06185	0.530927	+	0.847417i	$0.321844\pi$
$312$	0	0
$313$	16.6393	0.940511	0.470256	−	0.882530i	$-0.344162\pi$
$313$	16.6393	0.940511	0.470256	+	0.882530i	$0.344162\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.1121	0.736449	0.368224	−	0.929737i	$-0.379966\pi$
$317$	13.1121	0.736449	0.368224	+	0.929737i	$0.379966\pi$
$318$	0	0
$319$	− 33.2632i	− 1.86238i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3.78619i	0.210669i
$324$	0	0
$325$	− 16.6736i	− 0.924883i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 46.2533i	− 2.55003i
$330$	0	0
$331$	32.7952	1.80258	0.901292	−	0.433211i	$-0.142620\pi$
$331$	32.7952	1.80258	0.901292	+	0.433211i	$0.142620\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	10.4147	0.569016
$336$	0	0
$337$	28.2701	1.53997	0.769986	−	0.638060i	$-0.220263\pi$
$337$	28.2701	1.53997	0.769986	+	0.638060i	$0.220263\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.46475	−0.133474
$342$	0	0
$343$	38.5470i	2.08134i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6.84376i	0.367392i	0.982983	+	0.183696i	$0.0588062\pi$
$347$	6.84376i	0.367392i	−0.982983	+	0.183696i	$0.941194\pi$
$348$	0	0
$349$	14.4443i	0.773183i	0.922251	+	0.386592i	$0.126348\pi$
$349$	14.4443i	0.773183i	−0.922251	+	0.386592i	$0.873652\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	15.8826i	0.845343i	0.906283	+	0.422671i	$0.138908\pi$
$353$	15.8826i	0.845343i	−0.906283	+	0.422671i	$0.861092\pi$
$354$	0	0
$355$	−9.41165	−0.499518
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.84545	−0.202955	−0.101477	−	0.994838i	$-0.532357\pi$
$359$	−3.84545	−0.202955	−0.101477	+	0.994838i	$0.532357\pi$
$360$	0	0
$361$	−8.90714	−0.468797
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−8.30904	−0.434915
$366$	0	0
$367$	27.5746i	1.43938i	0.694294	+	0.719692i	$0.255717\pi$
$367$	27.5746i	1.43938i	−0.694294	+	0.719692i	$0.744283\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 4.13388i	− 0.214620i
$372$	0	0
$373$	− 8.09190i	− 0.418982i	−0.977811	−	0.209491i	$-0.932819\pi$
$373$	− 8.09190i	− 0.418982i	0.977811	−	0.209491i	$-0.0671808\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 31.5055i	− 1.62261i
$378$	0	0
$379$	17.9143	0.920195	0.460098	−	0.887868i	$-0.347814\pi$
$379$	17.9143	0.920195	0.460098	+	0.887868i	$0.347814\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	5.77515	0.295096	0.147548	−	0.989055i	$-0.452862\pi$
$383$	5.77515	0.295096	0.147548	+	0.989055i	$0.452862\pi$
$384$	0	0
$385$	−19.2137	−0.979223
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−22.0326	−1.11709	−0.558547	−	0.829473i	$-0.688641\pi$
$389$	−22.0326	−1.11709	−0.558547	+	0.829473i	$0.688641\pi$
$390$	0	0
$391$	− 0.894610i	− 0.0452424i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 5.45626i	− 0.274534i
$396$	0	0
$397$	− 26.3815i	− 1.32405i	−0.749481	−	0.662026i	$-0.769697\pi$
$397$	− 26.3815i	− 1.32405i	0.749481	−	0.662026i	$-0.230303\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 1.29121i	− 0.0644801i	−0.999480	−	0.0322401i	$-0.989736\pi$
$401$	− 1.29121i	− 0.0644801i	0.999480	−	0.0322401i	$-0.0102641\pi$
$402$	0	0
$403$	−2.33451	−0.116290
$404$	0	0
$405$	0	0
$406$	0	0
$407$	20.9331	1.03762
$408$	0	0
$409$	−1.54485	−0.0763880	−0.0381940	−	0.999270i	$-0.512160\pi$
$409$	−1.54485	−0.0763880	−0.0381940	+	0.999270i	$0.512160\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8.26326	0.406608
$414$	0	0
$415$	− 11.8741i	− 0.582878i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 14.9313i	− 0.729444i	−0.931116	−	0.364722i	$-0.881164\pi$
$419$	− 14.9313i	− 0.729444i	0.931116	−	0.364722i	$-0.118836\pi$
$420$	0	0
$421$	14.8896i	0.725673i	0.931853	+	0.362836i	$0.118192\pi$
$421$	14.8896i	0.725673i	−0.931853	+	0.362836i	$0.881808\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 4.88407i	− 0.236912i
$426$	0	0
$427$	46.7286	2.26135
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−30.8436	−1.48568	−0.742842	−	0.669467i	$-0.766523\pi$
$431$	−30.8436	−1.48568	−0.742842	+	0.669467i	$0.766523\pi$
$432$	0	0
$433$	−29.1892	−1.40275	−0.701373	−	0.712795i	$-0.747429\pi$
$433$	−29.1892	−1.40275	−0.701373	+	0.712795i	$0.747429\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.38476	−0.114079
$438$	0	0
$439$	− 1.24969i	− 0.0596443i	−0.999555	−	0.0298221i	$-0.990506\pi$
$439$	− 1.24969i	− 0.0596443i	0.999555	−	0.0298221i	$-0.00949409\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 27.4266i	− 1.30308i	−0.758616	−	0.651538i	$-0.774124\pi$
$443$	− 27.4266i	− 1.30308i	0.758616	−	0.651538i	$-0.225876\pi$
$444$	0	0
$445$	− 9.63345i	− 0.456669i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	16.2552i	0.767132i	0.923513	+	0.383566i	$0.125304\pi$
$449$	16.2552i	0.767132i	−0.923513	+	0.383566i	$0.874696\pi$
$450$	0	0
$451$	−41.9194	−1.97391
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−18.1984	−0.853155
$456$	0	0
$457$	18.8861	0.883456	0.441728	−	0.897149i	$-0.354366\pi$
$457$	18.8861	0.883456	0.441728	+	0.897149i	$0.354366\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	19.3048	0.899114	0.449557	−	0.893252i	$-0.351582\pi$
$461$	19.3048	0.899114	0.449557	+	0.893252i	$0.351582\pi$
$462$	0	0
$463$	2.34200i	0.108842i	0.998518	+	0.0544209i	$0.0173313\pi$
$463$	2.34200i	0.108842i	−0.998518	+	0.0544209i	$0.982669\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.2034i	0.564708i	0.959310	+	0.282354i	$0.0911153\pi$
$467$	12.2034i	0.564708i	−0.959310	+	0.282354i	$0.908885\pi$
$468$	0	0
$469$	51.6531i	2.38512i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	22.8326i	1.04985i
$474$	0	0
$475$	−13.0195	−0.597374
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−35.4473	−1.61963	−0.809815	−	0.586685i	$-0.800433\pi$
$479$	−35.4473	−1.61963	−0.809815	+	0.586685i	$0.800433\pi$
$480$	0	0
$481$	19.8270	0.904031
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−7.43610	−0.337656
$486$	0	0
$487$	− 18.7853i	− 0.851242i	−0.904902	−	0.425621i	$-0.860056\pi$
$487$	− 18.7853i	− 0.851242i	0.904902	−	0.425621i	$-0.139944\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.22936i	0.0554803i	0.999615	+	0.0277402i	$0.00883110\pi$
$491$	1.22936i	0.0554803i	−0.999615	+	0.0277402i	$0.991169\pi$
$492$	0	0
$493$	− 9.22868i	− 0.415639i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 46.6783i	− 2.09381i
$498$	0	0
$499$	3.47322	0.155483	0.0777413	−	0.996974i	$-0.475229\pi$
$499$	3.47322	0.155483	0.0777413	+	0.996974i	$0.475229\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−31.5709	−1.40768	−0.703838	−	0.710360i	$-0.748532\pi$
$503$	−31.5709	−1.40768	−0.703838	+	0.710360i	$0.748532\pi$
$504$	0	0
$505$	−16.9545	−0.754465
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−31.5773	−1.39964	−0.699820	−	0.714319i	$-0.746736\pi$
$509$	−31.5773	−1.39964	−0.699820	+	0.714319i	$0.746736\pi$
$510$	0	0
$511$	− 41.2098i	− 1.82302i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2.33060i	0.102698i
$516$	0	0
$517$	42.1835i	1.85523i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 8.99232i	− 0.393961i	−0.980407	−	0.196980i	$-0.936886\pi$
$521$	− 8.99232i	− 0.393961i	0.980407	−	0.196980i	$-0.0631136\pi$
$522$	0	0
$523$	−2.78978	−0.121988	−0.0609942	−	0.998138i	$-0.519427\pi$
$523$	−2.78978	−0.121988	−0.0609942	+	0.998138i	$0.519427\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.683832	−0.0297882
$528$	0	0
$529$	−22.4365	−0.975501
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−39.7043	−1.71978
$534$	0	0
$535$	− 0.268953i	− 0.0116279i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 65.2242i	− 2.80940i
$540$	0	0
$541$	38.5456i	1.65721i	0.559836	+	0.828603i	$0.310864\pi$
$541$	38.5456i	1.65721i	−0.559836	+	0.828603i	$0.689136\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 9.25655i	− 0.396507i
$546$	0	0
$547$	−21.7781	−0.931163	−0.465581	−	0.885005i	$-0.654155\pi$
$547$	−21.7781	−0.931163	−0.465581	+	0.885005i	$0.654155\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−24.6009	−1.04803
$552$	0	0
$553$	27.0611	1.15075
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−22.9871	−0.973995	−0.486998	−	0.873403i	$-0.661908\pi$
$557$	−22.9871	−0.973995	−0.486998	+	0.873403i	$0.661908\pi$
$558$	0	0
$559$	21.6261i	0.914686i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 41.2641i	− 1.73907i	−0.493867	−	0.869537i	$-0.664417\pi$
$563$	− 41.2641i	− 1.73907i	0.493867	−	0.869537i	$-0.335583\pi$
$564$	0	0
$565$	− 4.48940i	− 0.188870i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 28.4760i	− 1.19378i	−0.802325	−	0.596888i	$-0.796404\pi$
$569$	− 28.4760i	− 1.19378i	0.802325	−	0.596888i	$-0.203596\pi$
$570$	0	0
$571$	35.5206	1.48649	0.743245	−	0.669019i	$-0.233286\pi$
$571$	35.5206	1.48649	0.743245	+	0.669019i	$0.233286\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.07627	0.128289
$576$	0	0
$577$	−30.5676	−1.27255	−0.636273	−	0.771464i	$-0.719525\pi$
$577$	−30.5676	−1.27255	−0.636273	+	0.771464i	$0.719525\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	58.8913	2.44322
$582$	0	0
$583$	3.77014i	0.156143i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18.6609i	0.770217i	0.922871	+	0.385109i	$0.125836\pi$
$587$	18.6609i	0.770217i	−0.922871	+	0.385109i	$0.874164\pi$
$588$	0	0
$589$	1.82289i	0.0751109i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 0.676531i	− 0.0277818i	−0.999904	−	0.0138909i	$-0.995578\pi$
$593$	− 0.676531i	− 0.0277818i	0.999904	−	0.0138909i	$-0.00442175\pi$
$594$	0	0
$595$	−5.33074	−0.218539
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−37.7371	−1.54189	−0.770947	−	0.636899i	$-0.780217\pi$
$599$	−37.7371	−1.54189	−0.770947	+	0.636899i	$0.780217\pi$
$600$	0	0
$601$	−23.1123	−0.942771	−0.471386	−	0.881927i	$-0.656246\pi$
$601$	−23.1123	−0.942771	−0.471386	+	0.881927i	$0.656246\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	7.07679	0.287712
$606$	0	0
$607$	2.35798i	0.0957075i	0.998854	+	0.0478537i	$0.0152381\pi$
$607$	2.35798i	0.0957075i	−0.998854	+	0.0478537i	$0.984762\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	39.9543i	1.61638i
$612$	0	0
$613$	23.7808i	0.960497i	0.877132	+	0.480249i	$0.159454\pi$
$613$	23.7808i	0.960497i	−0.877132	+	0.480249i	$0.840546\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3.66608i	0.147591i	0.997273	+	0.0737954i	$0.0235112\pi$
$617$	3.66608i	0.147591i	−0.997273	+	0.0737954i	$0.976489\pi$
$618$	0	0
$619$	17.5872	0.706889	0.353444	−	0.935456i	$-0.385010\pi$
$619$	17.5872	0.706889	0.353444	+	0.935456i	$0.385010\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	47.7784	1.91420
$624$	0	0
$625$	12.2854	0.491415
$626$	0	0
$627$	0	0
$628$	0	0
$629$	5.80777	0.231571
$630$	0	0
$631$	− 21.9248i	− 0.872811i	−0.899750	−	0.436405i	$-0.856251\pi$
$631$	− 21.9248i	− 0.872811i	0.899750	−	0.436405i	$-0.143749\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	13.1094i	0.520229i
$636$	0	0
$637$	− 61.7775i	− 2.44772i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	20.8495i	0.823507i	0.911295	+	0.411754i	$0.135084\pi$
$641$	20.8495i	0.823507i	−0.911295	+	0.411754i	$0.864916\pi$
$642$	0	0
$643$	25.7081	1.01383	0.506914	−	0.861997i	$-0.330786\pi$
$643$	25.7081	1.01383	0.506914	+	0.861997i	$0.330786\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	26.0025	1.02226	0.511131	−	0.859503i	$-0.329227\pi$
$647$	26.0025	1.02226	0.511131	+	0.859503i	$0.329227\pi$
$648$	0	0
$649$	−7.53617	−0.295821
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−21.5340	−0.842691	−0.421346	−	0.906900i	$-0.638442\pi$
$653$	−21.5340	−0.842691	−0.421346	+	0.906900i	$0.638442\pi$
$654$	0	0
$655$	4.49631i	0.175686i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 13.3884i	− 0.521538i	−0.965401	−	0.260769i	$-0.916024\pi$
$659$	− 13.3884i	− 0.521538i	0.965401	−	0.260769i	$-0.0839760\pi$
$660$	0	0
$661$	− 16.7021i	− 0.649635i	−0.945777	−	0.324818i	$-0.894697\pi$
$661$	− 16.7021i	− 0.649635i	0.945777	−	0.324818i	$-0.105303\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	14.2102i	0.551046i
$666$	0	0
$667$	5.81275	0.225071
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−42.6169	−1.64521
$672$	0	0
$673$	10.9578	0.422391	0.211196	−	0.977444i	$-0.432264\pi$
$673$	10.9578	0.422391	0.211196	+	0.977444i	$0.432264\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−22.5484	−0.866604	−0.433302	−	0.901249i	$-0.642652\pi$
$677$	−22.5484	−0.866604	−0.433302	+	0.901249i	$0.642652\pi$
$678$	0	0
$679$	− 36.8804i	− 1.41534i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 36.4666i	− 1.39536i	−0.716412	−	0.697678i	$-0.754217\pi$
$683$	− 36.4666i	− 1.39536i	0.716412	−	0.697678i	$-0.245783\pi$
$684$	0	0
$685$	− 14.7714i	− 0.564387i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3.57091i	0.136041i
$690$	0	0
$691$	−5.00515	−0.190405	−0.0952025	−	0.995458i	$-0.530350\pi$
$691$	−5.00515	−0.190405	−0.0952025	+	0.995458i	$0.530350\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7.49267	0.284213
$696$	0	0
$697$	−11.6303	−0.440529
$698$	0	0
$699$	0	0
$700$	0	0
$701$	11.4972	0.434243	0.217122	−	0.976145i	$-0.430333\pi$
$701$	11.4972	0.434243	0.217122	+	0.976145i	$0.430333\pi$
$702$	0	0
$703$	− 15.4818i	− 0.583907i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 84.0881i	− 3.16246i
$708$	0	0
$709$	34.3370i	1.28955i	0.764371	+	0.644776i	$0.223049\pi$
$709$	34.3370i	1.28955i	−0.764371	+	0.644776i	$0.776951\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 0.430716i	− 0.0161305i
$714$	0	0
$715$	16.5971	0.620698
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−40.5660	−1.51286	−0.756429	−	0.654076i	$-0.773058\pi$
$719$	−40.5660	−1.51286	−0.756429	+	0.654076i	$0.773058\pi$
$720$	0	0
$721$	−11.5589	−0.430477
$722$	0	0
$723$	0	0
$724$	0	0
$725$	31.7344	1.17859
$726$	0	0
$727$	38.9815i	1.44575i	0.690981	+	0.722873i	$0.257179\pi$
$727$	38.9815i	1.44575i	−0.690981	+	0.722873i	$0.742821\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	6.33478i	0.234300i
$732$	0	0
$733$	− 36.3971i	− 1.34436i	−0.740388	−	0.672179i	$-0.765358\pi$
$733$	− 36.3971i	− 1.34436i	0.740388	−	0.672179i	$-0.234642\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 47.1082i	− 1.73525i
$738$	0	0
$739$	19.8657	0.730770	0.365385	−	0.930857i	$-0.380937\pi$
$739$	19.8657	0.730770	0.365385	+	0.930857i	$0.380937\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	12.2738	0.450282	0.225141	−	0.974326i	$-0.427716\pi$
$743$	12.2738	0.450282	0.225141	+	0.974326i	$0.427716\pi$
$744$	0	0
$745$	15.2012	0.556929
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.33391	0.0487400
$750$	0	0
$751$	4.95743i	0.180899i	0.995901	+	0.0904496i	$0.0288304\pi$
$751$	4.95743i	0.180899i	−0.995901	+	0.0904496i	$0.971170\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 1.73169i	− 0.0630228i
$756$	0	0
$757$	0.135856i	0.00493778i	0.999997	+	0.00246889i	$0.000785872\pi$
$757$	0.135856i	0.00493778i	−0.999997	+	0.00246889i	$0.999214\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.14729i	0.0778390i	0.999242	+	0.0389195i	$0.0123916\pi$
$761$	2.14729i	0.0778390i	−0.999242	+	0.0389195i	$0.987608\pi$
$762$	0	0
$763$	45.9091	1.66202
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−7.13794	−0.257736
$768$	0	0
$769$	−13.7117	−0.494457	−0.247228	−	0.968957i	$-0.579520\pi$
$769$	−13.7117	−0.494457	−0.247228	+	0.968957i	$0.579520\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	21.2269	0.763477	0.381738	−	0.924270i	$-0.375326\pi$
$773$	21.2269	0.763477	0.381738	+	0.924270i	$0.375326\pi$
$774$	0	0
$775$	− 2.35147i	− 0.0844673i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	31.0029i	1.11079i
$780$	0	0
$781$	42.5711i	1.52331i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 19.4574i	− 0.694466i
$786$	0	0
$787$	−21.2066	−0.755933	−0.377966	−	0.925819i	$-0.623377\pi$
$787$	−21.2066	−0.755933	−0.377966	+	0.925819i	$0.623377\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	22.2658	0.791680
$792$	0	0
$793$	−40.3649	−1.43340
$794$	0	0
$795$	0	0
$796$	0	0
$797$	35.2871	1.24993	0.624967	−	0.780651i	$-0.285113\pi$
$797$	35.2871	1.24993	0.624967	+	0.780651i	$0.285113\pi$
$798$	0	0
$799$	11.7036i	0.414042i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	37.5838i	1.32630i
$804$	0	0
$805$	− 3.35761i	− 0.118340i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 26.8216i	− 0.942996i	−0.881867	−	0.471498i	$-0.843714\pi$
$809$	− 26.8216i	− 0.942996i	0.881867	−	0.471498i	$-0.156286\pi$
$810$	0	0
$811$	−7.68860	−0.269983	−0.134992	−	0.990847i	$-0.543101\pi$
$811$	−7.68860	−0.269983	−0.134992	+	0.990847i	$0.543101\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−7.21151	−0.252608
$816$	0	0
$817$	16.8866	0.590789
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−24.7379	−0.863360	−0.431680	−	0.902027i	$-0.642079\pi$
$821$	−24.7379	−0.863360	−0.431680	+	0.902027i	$0.642079\pi$
$822$	0	0
$823$	− 31.5126i	− 1.09846i	−0.835671	−	0.549230i	$-0.814921\pi$
$823$	− 31.5126i	− 1.09846i	0.835671	−	0.549230i	$-0.185079\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	15.9087i	0.553200i	0.960985	+	0.276600i	$0.0892077\pi$
$827$	15.9087i	0.553200i	−0.960985	+	0.276600i	$0.910792\pi$
$828$	0	0
$829$	− 40.2678i	− 1.39856i	−0.714848	−	0.699280i	$-0.753504\pi$
$829$	− 40.2678i	− 1.39856i	0.714848	−	0.699280i	$-0.246496\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 18.0961i	− 0.626992i
$834$	0	0
$835$	−5.33074	−0.184478
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−5.33781	−0.184282	−0.0921408	−	0.995746i	$-0.529371\pi$
$839$	−5.33781	−0.184282	−0.0921408	+	0.995746i	$0.529371\pi$
$840$	0	0
$841$	30.9637	1.06771
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3.37443	0.116084
$846$	0	0
$847$	35.0983i	1.20599i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	3.65807i	0.125397i
$852$	0	0
$853$	32.4923i	1.11252i	0.831009	+	0.556258i	$0.187763\pi$
$853$	32.4923i	1.11252i	−0.831009	+	0.556258i	$0.812237\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 24.3799i	− 0.832801i	−0.909181	−	0.416400i	$-0.863291\pi$
$857$	− 24.3799i	− 0.832801i	0.909181	−	0.416400i	$-0.136709\pi$
$858$	0	0
$859$	12.0962	0.412719	0.206359	−	0.978476i	$-0.433838\pi$
$859$	12.0962	0.412719	0.206359	+	0.978476i	$0.433838\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	39.4505	1.34291	0.671456	−	0.741044i	$-0.265669\pi$
$863$	39.4505	1.34291	0.671456	+	0.741044i	$0.265669\pi$
$864$	0	0
$865$	11.9966	0.407897
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−24.6800	−0.837210
$870$	0	0
$871$	− 44.6188i	− 1.51185i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 40.6953i	− 1.37575i
$876$	0	0
$877$	− 6.88935i	− 0.232637i	−0.993212	−	0.116318i	$-0.962891\pi$
$877$	− 6.88935i	− 0.232637i	0.993212	−	0.116318i	$-0.0371093\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	44.3192i	1.49315i	0.665300	+	0.746576i	$0.268304\pi$
$881$	44.3192i	1.49315i	−0.665300	+	0.746576i	$0.731696\pi$
$882$	0	0
$883$	−48.0579	−1.61728	−0.808639	−	0.588305i	$-0.799795\pi$
$883$	−48.0579	−1.61728	−0.808639	+	0.588305i	$0.799795\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	27.0292	0.907550	0.453775	−	0.891116i	$-0.350077\pi$
$887$	27.0292	0.907550	0.453775	+	0.891116i	$0.350077\pi$
$888$	0	0
$889$	−65.0177	−2.18062
$890$	0	0
$891$	0	0
$892$	0	0
$893$	31.1982	1.04401
$894$	0	0
$895$	− 19.4949i	− 0.651643i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 4.44322i	− 0.148190i
$900$	0	0
$901$	1.04600i	0.0348474i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	9.45927i	0.314437i
$906$	0	0
$907$	−5.43993	−0.180630	−0.0903149	−	0.995913i	$-0.528787\pi$
$907$	−5.43993	−0.180630	−0.0903149	+	0.995913i	$0.528787\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	39.5517	1.31041	0.655203	−	0.755453i	$-0.272583\pi$
$911$	39.5517	1.31041	0.655203	+	0.755453i	$0.272583\pi$
$912$	0	0
$913$	−53.7094	−1.77752
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−22.3001	−0.736414
$918$	0	0
$919$	− 22.7057i	− 0.748991i	−0.927229	−	0.374495i	$-0.877816\pi$
$919$	− 22.7057i	− 0.748991i	0.927229	−	0.374495i	$-0.122184\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	40.3215i	1.32720i
$924$	0	0
$925$	19.9710i	0.656643i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 41.6457i	− 1.36635i	−0.730254	−	0.683176i	$-0.760598\pi$
$929$	− 41.6457i	− 1.36635i	0.730254	−	0.683176i	$-0.239402\pi$
$930$	0	0
$931$	−48.2387	−1.58096
$932$	0	0
$933$	0	0
$934$	0	0
$935$	4.86169	0.158994
$936$	0	0
$937$	54.4421	1.77855	0.889274	−	0.457376i	$-0.151211\pi$
$937$	54.4421	1.77855	0.889274	+	0.457376i	$0.151211\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−43.6434	−1.42274	−0.711368	−	0.702820i	$-0.751924\pi$
$941$	−43.6434	−1.42274	−0.711368	+	0.702820i	$0.751924\pi$
$942$	0	0
$943$	− 7.32543i	− 0.238549i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	13.3781i	0.434730i	0.976090	+	0.217365i	$0.0697462\pi$
$947$	13.3781i	0.434730i	−0.976090	+	0.217365i	$0.930254\pi$
$948$	0	0
$949$	35.5977i	1.15555i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	36.5239i	1.18313i	0.806259	+	0.591563i	$0.201489\pi$
$953$	36.5239i	1.18313i	−0.806259	+	0.591563i	$0.798511\pi$
$954$	0	0
$955$	21.8422	0.706796
$956$	0	0
$957$	0	0
$958$	0	0
$959$	73.2609	2.36572
$960$	0	0
$961$	30.6708	0.989379
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0.949666	0.0305708
$966$	0	0
$967$	22.4030i	0.720430i	0.932869	+	0.360215i	$0.117297\pi$
$967$	22.4030i	0.720430i	−0.932869	+	0.360215i	$0.882703\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 33.7841i	− 1.08418i	−0.840320	−	0.542091i	$-0.817633\pi$
$971$	− 33.7841i	− 1.08418i	0.840320	−	0.542091i	$-0.182367\pi$
$972$	0	0
$973$	37.1609i	1.19132i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	14.6312i	0.468094i	0.972225	+	0.234047i	$0.0751969\pi$
$977$	14.6312i	0.468094i	−0.972225	+	0.234047i	$0.924803\pi$
$978$	0	0
$979$	−43.5744	−1.39264
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−37.6759	−1.20167	−0.600837	−	0.799372i	$-0.705166\pi$
$983$	−37.6759	−1.20167	−0.600837	+	0.799372i	$0.705166\pi$
$984$	0	0
$985$	−0.493484	−0.0157237
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3.99001	−0.126875
$990$	0	0
$991$	25.7352i	0.817504i	0.912645	+	0.408752i	$0.134036\pi$
$991$	25.7352i	0.817504i	−0.912645	+	0.408752i	$0.865964\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 0.598903i	− 0.0189865i
$996$	0	0
$997$	− 38.5688i	− 1.22149i	−0.791829	−	0.610743i	$-0.790871\pi$
$997$	− 38.5688i	− 1.22149i	0.791829	−	0.610743i	$-0.209129\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2592.2.f.c.1295.12		24
3.2	odd	2	inner	2592.2.f.c.1295.14		24
4.3	odd	2		648.2.f.c.323.10	yes	24
8.3	odd	2	inner	2592.2.f.c.1295.13		24
8.5	even	2		648.2.f.c.323.16	yes	24
9.2	odd	6		2592.2.p.g.2159.12		48
9.4	even	3		2592.2.p.g.431.13		48
9.5	odd	6		2592.2.p.g.431.11		48
9.7	even	3		2592.2.p.g.2159.14		48
12.11	even	2		648.2.f.c.323.15	yes	24
24.5	odd	2		648.2.f.c.323.9	✓	24
24.11	even	2	inner	2592.2.f.c.1295.11		24
36.7	odd	6		648.2.l.g.539.23		48
36.11	even	6		648.2.l.g.539.2		48
36.23	even	6		648.2.l.g.107.18		48
36.31	odd	6		648.2.l.g.107.7		48
72.5	odd	6		648.2.l.g.107.23		48
72.11	even	6		2592.2.p.g.2159.13		48
72.13	even	6		648.2.l.g.107.2		48
72.29	odd	6		648.2.l.g.539.7		48
72.43	odd	6		2592.2.p.g.2159.11		48
72.59	even	6		2592.2.p.g.431.14		48
72.61	even	6		648.2.l.g.539.18		48
72.67	odd	6		2592.2.p.g.431.12		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.f.c.323.9	✓	24	24.5	odd	2
648.2.f.c.323.10	yes	24	4.3	odd	2
648.2.f.c.323.15	yes	24	12.11	even	2
648.2.f.c.323.16	yes	24	8.5	even	2
648.2.l.g.107.2		48	72.13	even	6
648.2.l.g.107.7		48	36.31	odd	6
648.2.l.g.107.18		48	36.23	even	6
648.2.l.g.107.23		48	72.5	odd	6
648.2.l.g.539.2		48	36.11	even	6
648.2.l.g.539.7		48	72.29	odd	6
648.2.l.g.539.18		48	72.61	even	6
648.2.l.g.539.23		48	36.7	odd	6
2592.2.f.c.1295.11		24	24.11	even	2	inner
2592.2.f.c.1295.12		24	1.1	even	1	trivial
2592.2.f.c.1295.13		24	8.3	odd	2	inner
2592.2.f.c.1295.14		24	3.2	odd	2	inner
2592.2.p.g.431.11		48	9.5	odd	6
2592.2.p.g.431.12		48	72.67	odd	6
2592.2.p.g.431.13		48	9.4	even	3
2592.2.p.g.431.14		48	72.59	even	6
2592.2.p.g.2159.11		48	72.43	odd	6
2592.2.p.g.2159.12		48	9.2	odd	6
2592.2.p.g.2159.13		48	72.11	even	6
2592.2.p.g.2159.14		48	9.7	even	3

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:	\( N \)	\(=\)	\( 2592 = 2^{5} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2592.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.949666 q^{5} -4.71000i q^{7} +O(q^{10})\)	\(q-0.949666 q^{5} -4.71000i q^{7} +4.29556i q^{11} +4.06857i q^{13} +1.19178i q^{17} +3.17693 q^{19} -0.750651 q^{23} -4.09813 q^{25} -7.74362 q^{29} +0.573791i q^{31} +4.47293i q^{35} -4.87320i q^{37} +9.75877i q^{41} +5.31540 q^{43} +9.82024 q^{47} -15.1841 q^{49} +0.877682 q^{53} -4.07935i q^{55} +1.75441i q^{59} +9.92114i q^{61} -3.86379i q^{65} -10.9667 q^{67} +9.91048 q^{71} +8.74944 q^{73} +20.2321 q^{77} +5.74545i q^{79} +12.5035i q^{83} -1.13179i q^{85} +10.1440i q^{89} +19.1630 q^{91} -3.01702 q^{95} +7.83023 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 24 q^{25} - 24 q^{49} - 48 q^{67}+O(q^{100}) \) 24 * q + 24 * q^25 - 24 * q^49 - 48 * q^67

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