LMFDB - Embedded newform 1521.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,2,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1521.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1521.a (trivial)

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:	yes
Analytic conductor:	$12.1452461474$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.56155 q^{2} +0.438447 q^{4} -3.56155 q^{5} -0.561553 q^{7} +2.43845 q^{8} +5.56155 q^{10} -2.00000 q^{11} +0.876894 q^{14} -4.68466 q^{16} +1.56155 q^{17} -7.12311 q^{19} -1.56155 q^{20} +3.12311 q^{22} -2.00000 q^{23} +7.68466 q^{25} -0.246211 q^{28} -6.68466 q^{29} -2.56155 q^{31} +2.43845 q^{32} -2.43845 q^{34} +2.00000 q^{35} -7.56155 q^{37} +11.1231 q^{38} -8.68466 q^{40} -1.56155 q^{41} +4.56155 q^{43} -0.876894 q^{44} +3.12311 q^{46} +8.24621 q^{47} -6.68466 q^{49} -12.0000 q^{50} +0.684658 q^{53} +7.12311 q^{55} -1.36932 q^{56} +10.4384 q^{58} -2.87689 q^{59} +3.87689 q^{61} +4.00000 q^{62} +5.56155 q^{64} -4.56155 q^{67} +0.684658 q^{68} -3.12311 q^{70} +14.0000 q^{71} +10.1231 q^{73} +11.8078 q^{74} -3.12311 q^{76} +1.12311 q^{77} +5.43845 q^{79} +16.6847 q^{80} +2.43845 q^{82} -0.876894 q^{83} -5.56155 q^{85} -7.12311 q^{86} -4.87689 q^{88} +4.87689 q^{89} -0.876894 q^{92} -12.8769 q^{94} +25.3693 q^{95} +8.56155 q^{97} +10.4384 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 5 q^{4} - 3 q^{5} + 3 q^{7} + 9 q^{8} + 7 q^{10} - 4 q^{11} + 10 q^{14} + 3 q^{16} - q^{17} - 6 q^{19} + q^{20} - 2 q^{22} - 4 q^{23} + 3 q^{25} + 16 q^{28} - q^{29} - q^{31} + 9 q^{32}+ \cdots + 25 q^{98}+O(q^{100}) $ 2 * q + q^2 + 5 * q^4 - 3 * q^5 + 3 * q^7 + 9 * q^8 + 7 * q^10 - 4 * q^11 + 10 * q^14 + 3 * q^16 - q^17 - 6 * q^19 + q^20 - 2 * q^22 - 4 * q^23 + 3 * q^25 + 16 * q^28 - q^29 - q^31 + 9 * q^32 - 9 * q^34 + 4 * q^35 - 11 * q^37 + 14 * q^38 - 5 * q^40 + q^41 + 5 * q^43 - 10 * q^44 - 2 * q^46 - q^49 - 24 * q^50 - 11 * q^53 + 6 * q^55 + 22 * q^56 + 25 * q^58 - 14 * q^59 + 16 * q^61 + 8 * q^62 + 7 * q^64 - 5 * q^67 - 11 * q^68 + 2 * q^70 + 28 * q^71 + 12 * q^73 + 3 * q^74 + 2 * q^76 - 6 * q^77 + 15 * q^79 + 21 * q^80 + 9 * q^82 - 10 * q^83 - 7 * q^85 - 6 * q^86 - 18 * q^88 + 18 * q^89 - 10 * q^92 - 34 * q^94 + 26 * q^95 + 13 * q^97 + 25 * q^98

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 $

$2$	−1.56155	−1.10418	−0.552092	−	0.833783i	$-0.686170\pi$
$2$	−1.56155	−1.10418	−0.552092	+	0.833783i	$0.686170\pi$
$3$	0	0
$3$	0	0
$4$	0.438447	0.219224
$5$	−3.56155	−1.59277	−0.796387	−	0.604787i	$-0.793258\pi$
$5$	−3.56155	−1.59277	−0.796387	+	0.604787i	$0.793258\pi$
$6$	0	0
$7$	−0.561553	−0.212247	−0.106124	−	0.994353i	$-0.533844\pi$
$7$	−0.561553	−0.212247	−0.106124	+	0.994353i	$0.533844\pi$
$8$	2.43845	0.862121
$9$	0	0
$10$	5.56155	1.75872
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0.876894	0.234360
$15$	0	0
$16$	−4.68466	−1.17116
$17$	1.56155	0.378732	0.189366	−	0.981907i	$-0.439357\pi$
$17$	1.56155	0.378732	0.189366	+	0.981907i	$0.439357\pi$
$18$	0	0
$19$	−7.12311	−1.63415	−0.817076	−	0.576530i	$-0.804407\pi$
$19$	−7.12311	−1.63415	−0.817076	+	0.576530i	$0.804407\pi$
$20$	−1.56155	−0.349174
$21$	0	0
$22$	3.12311	0.665848
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	0	0
$25$	7.68466	1.53693
$26$	0	0
$27$	0	0
$28$	−0.246211	−0.0465296
$29$	−6.68466	−1.24131	−0.620655	−	0.784084i	$-0.713133\pi$
$29$	−6.68466	−1.24131	−0.620655	+	0.784084i	$0.713133\pi$
$30$	0	0
$31$	−2.56155	−0.460068	−0.230034	−	0.973183i	$-0.573884\pi$
$31$	−2.56155	−0.460068	−0.230034	+	0.973183i	$0.573884\pi$
$32$	2.43845	0.431061
$33$	0	0
$34$	−2.43845	−0.418190
$35$	2.00000	0.338062
$36$	0	0
$37$	−7.56155	−1.24311	−0.621556	−	0.783370i	$-0.713499\pi$
$37$	−7.56155	−1.24311	−0.621556	+	0.783370i	$0.713499\pi$
$38$	11.1231	1.80441
$39$	0	0
$40$	−8.68466	−1.37317
$41$	−1.56155	−0.243874	−0.121937	−	0.992538i	$-0.538911\pi$
$41$	−1.56155	−0.243874	−0.121937	+	0.992538i	$0.538911\pi$
$42$	0	0
$43$	4.56155	0.695630	0.347815	−	0.937563i	$-0.386924\pi$
$43$	4.56155	0.695630	0.347815	+	0.937563i	$0.386924\pi$
$44$	−0.876894	−0.132197
$45$	0	0
$46$	3.12311	0.460477
$47$	8.24621	1.20283	0.601417	−	0.798935i	$-0.294603\pi$
$47$	8.24621	1.20283	0.601417	+	0.798935i	$0.294603\pi$
$48$	0	0
$49$	−6.68466	−0.954951
$50$	−12.0000	−1.69706
$51$	0	0
$52$	0	0
$53$	0.684658	0.0940451	0.0470225	−	0.998894i	$-0.485027\pi$
$53$	0.684658	0.0940451	0.0470225	+	0.998894i	$0.485027\pi$
$54$	0	0
$55$	7.12311	0.960479
$56$	−1.36932	−0.182983
$57$	0	0
$58$	10.4384	1.37064
$59$	−2.87689	−0.374540	−0.187270	−	0.982309i	$-0.559964\pi$
$59$	−2.87689	−0.374540	−0.187270	+	0.982309i	$0.559964\pi$
$60$	0	0
$61$	3.87689	0.496385	0.248193	−	0.968711i	$-0.420163\pi$
$61$	3.87689	0.496385	0.248193	+	0.968711i	$0.420163\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	5.56155	0.695194
$65$	0	0
$66$	0	0
$67$	−4.56155	−0.557282	−0.278641	−	0.960395i	$-0.589884\pi$
$67$	−4.56155	−0.557282	−0.278641	+	0.960395i	$0.589884\pi$
$68$	0.684658	0.0830270
$69$	0	0
$70$	−3.12311	−0.373283
$71$	14.0000	1.66149	0.830747	−	0.556650i	$-0.187914\pi$
$71$	14.0000	1.66149	0.830747	+	0.556650i	$0.187914\pi$
$72$	0	0
$73$	10.1231	1.18482	0.592410	−	0.805637i	$-0.298177\pi$
$73$	10.1231	1.18482	0.592410	+	0.805637i	$0.298177\pi$
$74$	11.8078	1.37262
$75$	0	0
$76$	−3.12311	−0.358245
$77$	1.12311	0.127990
$78$	0	0
$79$	5.43845	0.611873	0.305937	−	0.952052i	$-0.401030\pi$
$79$	5.43845	0.611873	0.305937	+	0.952052i	$0.401030\pi$
$80$	16.6847	1.86540
$81$	0	0
$82$	2.43845	0.269281
$83$	−0.876894	−0.0962517	−0.0481258	−	0.998841i	$-0.515325\pi$
$83$	−0.876894	−0.0962517	−0.0481258	+	0.998841i	$0.515325\pi$
$84$	0	0
$85$	−5.56155	−0.603235
$86$	−7.12311	−0.768104
$87$	0	0
$88$	−4.87689	−0.519879
$89$	4.87689	0.516950	0.258475	−	0.966018i	$-0.416780\pi$
$89$	4.87689	0.516950	0.258475	+	0.966018i	$0.416780\pi$
$90$	0	0
$91$	0	0
$92$	−0.876894	−0.0914226
$93$	0	0
$94$	−12.8769	−1.32815
$95$	25.3693	2.60284
$96$	0	0
$97$	8.56155	0.869294	0.434647	−	0.900601i	$-0.356873\pi$
$97$	8.56155	0.869294	0.434647	+	0.900601i	$0.356873\pi$
$98$	10.4384	1.05444
$99$	0	0
$100$	3.36932	0.336932
$101$	7.56155	0.752403	0.376201	−	0.926538i	$-0.377230\pi$
$101$	7.56155	0.752403	0.376201	+	0.926538i	$0.377230\pi$
$102$	0	0
$103$	−3.43845	−0.338800	−0.169400	−	0.985547i	$-0.554183\pi$
$103$	−3.43845	−0.338800	−0.169400	+	0.985547i	$0.554183\pi$
$104$	0	0
$105$	0	0
$106$	−1.06913	−0.103843
$107$	8.24621	0.797191	0.398596	−	0.917127i	$-0.369498\pi$
$107$	8.24621	0.797191	0.398596	+	0.917127i	$0.369498\pi$
$108$	0	0
$109$	2.80776	0.268935	0.134468	−	0.990918i	$-0.457068\pi$
$109$	2.80776	0.268935	0.134468	+	0.990918i	$0.457068\pi$
$110$	−11.1231	−1.06055
$111$	0	0
$112$	2.63068	0.248576
$113$	−5.80776	−0.546348	−0.273174	−	0.961965i	$-0.588074\pi$
$113$	−5.80776	−0.546348	−0.273174	+	0.961965i	$0.588074\pi$
$114$	0	0
$115$	7.12311	0.664233
$116$	−2.93087	−0.272124
$117$	0	0
$118$	4.49242	0.413561
$119$	−0.876894	−0.0803848
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−6.05398	−0.548101
$123$	0	0
$124$	−1.12311	−0.100858
$125$	−9.56155	−0.855211
$126$	0	0
$127$	5.43845	0.482584	0.241292	−	0.970453i	$-0.422429\pi$
$127$	5.43845	0.482584	0.241292	+	0.970453i	$0.422429\pi$
$128$	−13.5616	−1.19868
$129$	0	0
$130$	0	0
$131$	−7.36932	−0.643860	−0.321930	−	0.946763i	$-0.604332\pi$
$131$	−7.36932	−0.643860	−0.321930	+	0.946763i	$0.604332\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	7.12311	0.615343
$135$	0	0
$136$	3.80776	0.326513
$137$	−5.56155	−0.475156	−0.237578	−	0.971369i	$-0.576353\pi$
$137$	−5.56155	−0.475156	−0.237578	+	0.971369i	$0.576353\pi$
$138$	0	0
$139$	−17.9309	−1.52088	−0.760438	−	0.649410i	$-0.775016\pi$
$139$	−17.9309	−1.52088	−0.760438	+	0.649410i	$0.775016\pi$
$140$	0.876894	0.0741111
$141$	0	0
$142$	−21.8617	−1.83460
$143$	0	0
$144$	0	0
$145$	23.8078	1.97713
$146$	−15.8078	−1.30826
$147$	0	0
$148$	−3.31534	−0.272519
$149$	−2.43845	−0.199765	−0.0998827	−	0.994999i	$-0.531847\pi$
$149$	−2.43845	−0.199765	−0.0998827	+	0.994999i	$0.531847\pi$
$150$	0	0
$151$	9.36932	0.762464	0.381232	−	0.924479i	$-0.375500\pi$
$151$	9.36932	0.762464	0.381232	+	0.924479i	$0.375500\pi$
$152$	−17.3693	−1.40884
$153$	0	0
$154$	−1.75379	−0.141324
$155$	9.12311	0.732785
$156$	0	0
$157$	20.3693	1.62565	0.812824	−	0.582509i	$-0.197929\pi$
$157$	20.3693	1.62565	0.812824	+	0.582509i	$0.197929\pi$
$158$	−8.49242	−0.675621
$159$	0	0
$160$	−8.68466	−0.686583
$161$	1.12311	0.0885131
$162$	0	0
$163$	4.80776	0.376573	0.188287	−	0.982114i	$-0.439707\pi$
$163$	4.80776	0.376573	0.188287	+	0.982114i	$0.439707\pi$
$164$	−0.684658	−0.0534628
$165$	0	0
$166$	1.36932	0.106280
$167$	10.2462	0.792876	0.396438	−	0.918062i	$-0.370246\pi$
$167$	10.2462	0.792876	0.396438	+	0.918062i	$0.370246\pi$
$168$	0	0
$169$	0	0
$170$	8.68466	0.666083
$171$	0	0
$172$	2.00000	0.152499
$173$	−20.2462	−1.53929	−0.769645	−	0.638471i	$-0.779567\pi$
$173$	−20.2462	−1.53929	−0.769645	+	0.638471i	$0.779567\pi$
$174$	0	0
$175$	−4.31534	−0.326209
$176$	9.36932	0.706239
$177$	0	0
$178$	−7.61553	−0.570808
$179$	4.87689	0.364516	0.182258	−	0.983251i	$-0.441659\pi$
$179$	4.87689	0.364516	0.182258	+	0.983251i	$0.441659\pi$
$180$	0	0
$181$	−2.68466	−0.199549	−0.0997745	−	0.995010i	$-0.531812\pi$
$181$	−2.68466	−0.199549	−0.0997745	+	0.995010i	$0.531812\pi$
$182$	0	0
$183$	0	0
$184$	−4.87689	−0.359529
$185$	26.9309	1.98000
$186$	0	0
$187$	−3.12311	−0.228384
$188$	3.61553	0.263689
$189$	0	0
$190$	−39.6155	−2.87401
$191$	9.12311	0.660125	0.330062	−	0.943959i	$-0.392930\pi$
$191$	9.12311	0.660125	0.330062	+	0.943959i	$0.392930\pi$
$192$	0	0
$193$	13.4924	0.971206	0.485603	−	0.874180i	$-0.338600\pi$
$193$	13.4924	0.971206	0.485603	+	0.874180i	$0.338600\pi$
$194$	−13.3693	−0.959861
$195$	0	0
$196$	−2.93087	−0.209348
$197$	13.3693	0.952524	0.476262	−	0.879303i	$-0.341991\pi$
$197$	13.3693	0.952524	0.476262	+	0.879303i	$0.341991\pi$
$198$	0	0
$199$	22.1771	1.57209	0.786046	−	0.618168i	$-0.212125\pi$
$199$	22.1771	1.57209	0.786046	+	0.618168i	$0.212125\pi$
$200$	18.7386	1.32502
$201$	0	0
$202$	−11.8078	−0.830791
$203$	3.75379	0.263464
$204$	0	0
$205$	5.56155	0.388436
$206$	5.36932	0.374098
$207$	0	0
$208$	0	0
$209$	14.2462	0.985431
$210$	0	0
$211$	19.6847	1.35515	0.677574	−	0.735455i	$-0.263031\pi$
$211$	19.6847	1.35515	0.677574	+	0.735455i	$0.263031\pi$
$212$	0.300187	0.0206169
$213$	0	0
$214$	−12.8769	−0.880246
$215$	−16.2462	−1.10798
$216$	0	0
$217$	1.43845	0.0976482
$218$	−4.38447	−0.296954
$219$	0	0
$220$	3.12311	0.210560
$221$	0	0
$222$	0	0
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	−1.36932	−0.0914913
$225$	0	0
$226$	9.06913	0.603270
$227$	7.12311	0.472777	0.236389	−	0.971659i	$-0.424036\pi$
$227$	7.12311	0.472777	0.236389	+	0.971659i	$0.424036\pi$
$228$	0	0
$229$	16.2462	1.07358	0.536790	−	0.843716i	$-0.319637\pi$
$229$	16.2462	1.07358	0.536790	+	0.843716i	$0.319637\pi$
$230$	−11.1231	−0.733436
$231$	0	0
$232$	−16.3002	−1.07016
$233$	−26.0000	−1.70332	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−26.0000	−1.70332	−0.851658	+	0.524097i	$0.824403\pi$
$234$	0	0
$235$	−29.3693	−1.91584
$236$	−1.26137	−0.0821079
$237$	0	0
$238$	1.36932	0.0887596
$239$	−25.3693	−1.64100	−0.820502	−	0.571643i	$-0.806306\pi$
$239$	−25.3693	−1.64100	−0.820502	+	0.571643i	$0.806306\pi$
$240$	0	0
$241$	17.8078	1.14710	0.573549	−	0.819171i	$-0.305566\pi$
$241$	17.8078	1.14710	0.573549	+	0.819171i	$0.305566\pi$
$242$	10.9309	0.702663
$243$	0	0
$244$	1.69981	0.108819
$245$	23.8078	1.52102
$246$	0	0
$247$	0	0
$248$	−6.24621	−0.396635
$249$	0	0
$250$	14.9309	0.944311
$251$	−18.7386	−1.18277	−0.591386	−	0.806389i	$-0.701419\pi$
$251$	−18.7386	−1.18277	−0.591386	+	0.806389i	$0.701419\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	−8.49242	−0.532862
$255$	0	0
$256$	10.0540	0.628373
$257$	−29.1771	−1.82002	−0.910008	−	0.414590i	$-0.863925\pi$
$257$	−29.1771	−1.82002	−0.910008	+	0.414590i	$0.863925\pi$
$258$	0	0
$259$	4.24621	0.263847
$260$	0	0
$261$	0	0
$262$	11.5076	0.710941
$263$	−9.36932	−0.577737	−0.288868	−	0.957369i	$-0.593279\pi$
$263$	−9.36932	−0.577737	−0.288868	+	0.957369i	$0.593279\pi$
$264$	0	0
$265$	−2.43845	−0.149793
$266$	−6.24621	−0.382980
$267$	0	0
$268$	−2.00000	−0.122169
$269$	21.3693	1.30291	0.651455	−	0.758687i	$-0.274159\pi$
$269$	21.3693	1.30291	0.651455	+	0.758687i	$0.274159\pi$
$270$	0	0
$271$	29.9309	1.81817	0.909085	−	0.416610i	$-0.136782\pi$
$271$	29.9309	1.81817	0.909085	+	0.416610i	$0.136782\pi$
$272$	−7.31534	−0.443558
$273$	0	0
$274$	8.68466	0.524659
$275$	−15.3693	−0.926805
$276$	0	0
$277$	5.31534	0.319368	0.159684	−	0.987168i	$-0.448952\pi$
$277$	5.31534	0.319368	0.159684	+	0.987168i	$0.448952\pi$
$278$	28.0000	1.67933
$279$	0	0
$280$	4.87689	0.291450
$281$	17.8078	1.06232	0.531161	−	0.847271i	$-0.321756\pi$
$281$	17.8078	1.06232	0.531161	+	0.847271i	$0.321756\pi$
$282$	0	0
$283$	−13.6847	−0.813469	−0.406734	−	0.913547i	$-0.633333\pi$
$283$	−13.6847	−0.813469	−0.406734	+	0.913547i	$0.633333\pi$
$284$	6.13826	0.364239
$285$	0	0
$286$	0	0
$287$	0.876894	0.0517614
$288$	0	0
$289$	−14.5616	−0.856562
$290$	−37.1771	−2.18311
$291$	0	0
$292$	4.43845	0.259740
$293$	20.4384	1.19403	0.597013	−	0.802231i	$-0.296354\pi$
$293$	20.4384	1.19403	0.597013	+	0.802231i	$0.296354\pi$
$294$	0	0
$295$	10.2462	0.596557
$296$	−18.4384	−1.07171
$297$	0	0
$298$	3.80776	0.220578
$299$	0	0
$300$	0	0
$301$	−2.56155	−0.147645
$302$	−14.6307	−0.841901
$303$	0	0
$304$	33.3693	1.91386
$305$	−13.8078	−0.790630
$306$	0	0
$307$	−30.8078	−1.75829	−0.879146	−	0.476553i	$-0.841886\pi$
$307$	−30.8078	−1.75829	−0.879146	+	0.476553i	$0.841886\pi$
$308$	0.492423	0.0280584
$309$	0	0
$310$	−14.2462	−0.809130
$311$	19.1231	1.08437	0.542186	−	0.840259i	$-0.317597\pi$
$311$	19.1231	1.08437	0.542186	+	0.840259i	$0.317597\pi$
$312$	0	0
$313$	−13.6847	−0.773503	−0.386751	−	0.922184i	$-0.626403\pi$
$313$	−13.6847	−0.773503	−0.386751	+	0.922184i	$0.626403\pi$
$314$	−31.8078	−1.79502
$315$	0	0
$316$	2.38447	0.134137
$317$	14.0540	0.789350	0.394675	−	0.918821i	$-0.370857\pi$
$317$	14.0540	0.789350	0.394675	+	0.918821i	$0.370857\pi$
$318$	0	0
$319$	13.3693	0.748538
$320$	−19.8078	−1.10729
$321$	0	0
$322$	−1.75379	−0.0977348
$323$	−11.1231	−0.618906
$324$	0	0
$325$	0	0
$326$	−7.50758	−0.415806
$327$	0	0
$328$	−3.80776	−0.210249
$329$	−4.63068	−0.255298
$330$	0	0
$331$	3.19224	0.175461	0.0877306	−	0.996144i	$-0.472039\pi$
$331$	3.19224	0.175461	0.0877306	+	0.996144i	$0.472039\pi$
$332$	−0.384472	−0.0211006
$333$	0	0
$334$	−16.0000	−0.875481
$335$	16.2462	0.887625
$336$	0	0
$337$	−6.12311	−0.333547	−0.166773	−	0.985995i	$-0.553335\pi$
$337$	−6.12311	−0.333547	−0.166773	+	0.985995i	$0.553335\pi$
$338$	0	0
$339$	0	0
$340$	−2.43845	−0.132243
$341$	5.12311	0.277432
$342$	0	0
$343$	7.68466	0.414933
$344$	11.1231	0.599718
$345$	0	0
$346$	31.6155	1.69966
$347$	−27.6155	−1.48248	−0.741240	−	0.671241i	$-0.765762\pi$
$347$	−27.6155	−1.48248	−0.741240	+	0.671241i	$0.765762\pi$
$348$	0	0
$349$	−6.80776	−0.364411	−0.182206	−	0.983260i	$-0.558324\pi$
$349$	−6.80776	−0.364411	−0.182206	+	0.983260i	$0.558324\pi$
$350$	6.73863	0.360195
$351$	0	0
$352$	−4.87689	−0.259939
$353$	5.31534	0.282907	0.141454	−	0.989945i	$-0.454822\pi$
$353$	5.31534	0.282907	0.141454	+	0.989945i	$0.454822\pi$
$354$	0	0
$355$	−49.8617	−2.64639
$356$	2.13826	0.113328
$357$	0	0
$358$	−7.61553	−0.402493
$359$	−9.36932	−0.494494	−0.247247	−	0.968953i	$-0.579526\pi$
$359$	−9.36932	−0.494494	−0.247247	+	0.968953i	$0.579526\pi$
$360$	0	0
$361$	31.7386	1.67045
$362$	4.19224	0.220339
$363$	0	0
$364$	0	0
$365$	−36.0540	−1.88715
$366$	0	0
$367$	−17.0540	−0.890210	−0.445105	−	0.895478i	$-0.646834\pi$
$367$	−17.0540	−0.890210	−0.445105	+	0.895478i	$0.646834\pi$
$368$	9.36932	0.488409
$369$	0	0
$370$	−42.0540	−2.18628
$371$	−0.384472	−0.0199608
$372$	0	0
$373$	28.3693	1.46891	0.734454	−	0.678659i	$-0.237438\pi$
$373$	28.3693	1.46891	0.734454	+	0.678659i	$0.237438\pi$
$374$	4.87689	0.252178
$375$	0	0
$376$	20.1080	1.03699
$377$	0	0
$378$	0	0
$379$	−23.6847	−1.21660	−0.608300	−	0.793708i	$-0.708148\pi$
$379$	−23.6847	−1.21660	−0.608300	+	0.793708i	$0.708148\pi$
$380$	11.1231	0.570603
$381$	0	0
$382$	−14.2462	−0.728900
$383$	−22.7386	−1.16189	−0.580945	−	0.813943i	$-0.697317\pi$
$383$	−22.7386	−1.16189	−0.580945	+	0.813943i	$0.697317\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	−21.0691	−1.07239
$387$	0	0
$388$	3.75379	0.190570
$389$	−34.0540	−1.72661	−0.863303	−	0.504687i	$-0.831608\pi$
$389$	−34.0540	−1.72661	−0.863303	+	0.504687i	$0.831608\pi$
$390$	0	0
$391$	−3.12311	−0.157942
$392$	−16.3002	−0.823284
$393$	0	0
$394$	−20.8769	−1.05176
$395$	−19.3693	−0.974576
$396$	0	0
$397$	−25.0540	−1.25742	−0.628711	−	0.777639i	$-0.716417\pi$
$397$	−25.0540	−1.25742	−0.628711	+	0.777639i	$0.716417\pi$
$398$	−34.6307	−1.73588
$399$	0	0
$400$	−36.0000	−1.80000
$401$	14.4384	0.721022	0.360511	−	0.932755i	$-0.382602\pi$
$401$	14.4384	0.721022	0.360511	+	0.932755i	$0.382602\pi$
$402$	0	0
$403$	0	0
$404$	3.31534	0.164944
$405$	0	0
$406$	−5.86174	−0.290913
$407$	15.1231	0.749625
$408$	0	0
$409$	−6.36932	−0.314942	−0.157471	−	0.987524i	$-0.550334\pi$
$409$	−6.36932	−0.314942	−0.157471	+	0.987524i	$0.550334\pi$
$410$	−8.68466	−0.428905
$411$	0	0
$412$	−1.50758	−0.0742730
$413$	1.61553	0.0794949
$414$	0	0
$415$	3.12311	0.153307
$416$	0	0
$417$	0	0
$418$	−22.2462	−1.08810
$419$	34.2462	1.67304	0.836518	−	0.547939i	$-0.184587\pi$
$419$	34.2462	1.67304	0.836518	+	0.547939i	$0.184587\pi$
$420$	0	0
$421$	−31.2462	−1.52285	−0.761424	−	0.648255i	$-0.775499\pi$
$421$	−31.2462	−1.52285	−0.761424	+	0.648255i	$0.775499\pi$
$422$	−30.7386	−1.49633
$423$	0	0
$424$	1.66950	0.0810783
$425$	12.0000	0.582086
$426$	0	0
$427$	−2.17708	−0.105356
$428$	3.61553	0.174763
$429$	0	0
$430$	25.3693	1.22342
$431$	−11.1231	−0.535781	−0.267891	−	0.963449i	$-0.586327\pi$
$431$	−11.1231	−0.535781	−0.267891	+	0.963449i	$0.586327\pi$
$432$	0	0
$433$	8.75379	0.420680	0.210340	−	0.977628i	$-0.432543\pi$
$433$	8.75379	0.420680	0.210340	+	0.977628i	$0.432543\pi$
$434$	−2.24621	−0.107822
$435$	0	0
$436$	1.23106	0.0589569
$437$	14.2462	0.681489
$438$	0	0
$439$	−13.6847	−0.653133	−0.326567	−	0.945174i	$-0.605892\pi$
$439$	−13.6847	−0.653133	−0.326567	+	0.945174i	$0.605892\pi$
$440$	17.3693	0.828050
$441$	0	0
$442$	0	0
$443$	34.7386	1.65048	0.825241	−	0.564781i	$-0.191039\pi$
$443$	34.7386	1.65048	0.825241	+	0.564781i	$0.191039\pi$
$444$	0	0
$445$	−17.3693	−0.823385
$446$	12.4924	0.591533
$447$	0	0
$448$	−3.12311	−0.147553
$449$	−8.24621	−0.389163	−0.194581	−	0.980886i	$-0.562335\pi$
$449$	−8.24621	−0.389163	−0.194581	+	0.980886i	$0.562335\pi$
$450$	0	0
$451$	3.12311	0.147061
$452$	−2.54640	−0.119772
$453$	0	0
$454$	−11.1231	−0.522033
$455$	0	0
$456$	0	0
$457$	−12.6155	−0.590130	−0.295065	−	0.955477i	$-0.595341\pi$
$457$	−12.6155	−0.590130	−0.295065	+	0.955477i	$0.595341\pi$
$458$	−25.3693	−1.18543
$459$	0	0
$460$	3.12311	0.145616
$461$	−16.1922	−0.754148	−0.377074	−	0.926183i	$-0.623070\pi$
$461$	−16.1922	−0.754148	−0.377074	+	0.926183i	$0.623070\pi$
$462$	0	0
$463$	14.3153	0.665290	0.332645	−	0.943052i	$-0.392059\pi$
$463$	14.3153	0.665290	0.332645	+	0.943052i	$0.392059\pi$
$464$	31.3153	1.45378
$465$	0	0
$466$	40.6004	1.88078
$467$	26.0000	1.20314	0.601568	−	0.798821i	$-0.294543\pi$
$467$	26.0000	1.20314	0.601568	+	0.798821i	$0.294543\pi$
$468$	0	0
$469$	2.56155	0.118282
$470$	45.8617	2.11544
$471$	0	0
$472$	−7.01515	−0.322899
$473$	−9.12311	−0.419481
$474$	0	0
$475$	−54.7386	−2.51158
$476$	−0.384472	−0.0176222
$477$	0	0
$478$	39.6155	1.81197
$479$	10.2462	0.468161	0.234081	−	0.972217i	$-0.424792\pi$
$479$	10.2462	0.468161	0.234081	+	0.972217i	$0.424792\pi$
$480$	0	0
$481$	0	0
$482$	−27.8078	−1.26661
$483$	0	0
$484$	−3.06913	−0.139506
$485$	−30.4924	−1.38459
$486$	0	0
$487$	−7.12311	−0.322779	−0.161389	−	0.986891i	$-0.551597\pi$
$487$	−7.12311	−0.322779	−0.161389	+	0.986891i	$0.551597\pi$
$488$	9.45360	0.427944
$489$	0	0
$490$	−37.1771	−1.67949
$491$	36.2462	1.63577	0.817884	−	0.575383i	$-0.195147\pi$
$491$	36.2462	1.63577	0.817884	+	0.575383i	$0.195147\pi$
$492$	0	0
$493$	−10.4384	−0.470124
$494$	0	0
$495$	0	0
$496$	12.0000	0.538816
$497$	−7.86174	−0.352647
$498$	0	0
$499$	−4.49242	−0.201108	−0.100554	−	0.994932i	$-0.532062\pi$
$499$	−4.49242	−0.201108	−0.100554	+	0.994932i	$0.532062\pi$
$500$	−4.19224	−0.187482
$501$	0	0
$502$	29.2614	1.30600
$503$	28.2462	1.25944	0.629718	−	0.776824i	$-0.283170\pi$
$503$	28.2462	1.25944	0.629718	+	0.776824i	$0.283170\pi$
$504$	0	0
$505$	−26.9309	−1.19841
$506$	−6.24621	−0.277678
$507$	0	0
$508$	2.38447	0.105794
$509$	−13.8078	−0.612018	−0.306009	−	0.952029i	$-0.598994\pi$
$509$	−13.8078	−0.612018	−0.306009	+	0.952029i	$0.598994\pi$
$510$	0	0
$511$	−5.68466	−0.251474
$512$	11.4233	0.504843
$513$	0	0
$514$	45.5616	2.00963
$515$	12.2462	0.539633
$516$	0	0
$517$	−16.4924	−0.725336
$518$	−6.63068	−0.291335
$519$	0	0
$520$	0	0
$521$	9.06913	0.397326	0.198663	−	0.980068i	$-0.436340\pi$
$521$	9.06913	0.397326	0.198663	+	0.980068i	$0.436340\pi$
$522$	0	0
$523$	33.8617	1.48067	0.740335	−	0.672238i	$-0.234667\pi$
$523$	33.8617	1.48067	0.740335	+	0.672238i	$0.234667\pi$
$524$	−3.23106	−0.141149
$525$	0	0
$526$	14.6307	0.637928
$527$	−4.00000	−0.174243
$528$	0	0
$529$	−19.0000	−0.826087
$530$	3.80776	0.165399
$531$	0	0
$532$	1.75379	0.0760364
$533$	0	0
$534$	0	0
$535$	−29.3693	−1.26975
$536$	−11.1231	−0.480445
$537$	0	0
$538$	−33.3693	−1.43865
$539$	13.3693	0.575857
$540$	0	0
$541$	−19.7386	−0.848630	−0.424315	−	0.905515i	$-0.639485\pi$
$541$	−19.7386	−0.848630	−0.424315	+	0.905515i	$0.639485\pi$
$542$	−46.7386	−2.00760
$543$	0	0
$544$	3.80776	0.163257
$545$	−10.0000	−0.428353
$546$	0	0
$547$	3.93087	0.168072	0.0840359	−	0.996463i	$-0.473219\pi$
$547$	3.93087	0.168072	0.0840359	+	0.996463i	$0.473219\pi$
$548$	−2.43845	−0.104165
$549$	0	0
$550$	24.0000	1.02336
$551$	47.6155	2.02849
$552$	0	0
$553$	−3.05398	−0.129868
$554$	−8.30019	−0.352641
$555$	0	0
$556$	−7.86174	−0.333412
$557$	−42.9309	−1.81904	−0.909520	−	0.415661i	$-0.863550\pi$
$557$	−42.9309	−1.81904	−0.909520	+	0.415661i	$0.863550\pi$
$558$	0	0
$559$	0	0
$560$	−9.36932	−0.395926
$561$	0	0
$562$	−27.8078	−1.17300
$563$	23.3693	0.984899	0.492450	−	0.870341i	$-0.336102\pi$
$563$	23.3693	0.984899	0.492450	+	0.870341i	$0.336102\pi$
$564$	0	0
$565$	20.6847	0.870210
$566$	21.3693	0.898219
$567$	0	0
$568$	34.1383	1.43241
$569$	8.73863	0.366343	0.183171	−	0.983081i	$-0.441364\pi$
$569$	8.73863	0.366343	0.183171	+	0.983081i	$0.441364\pi$
$570$	0	0
$571$	−5.36932	−0.224699	−0.112349	−	0.993669i	$-0.535838\pi$
$571$	−5.36932	−0.224699	−0.112349	+	0.993669i	$0.535838\pi$
$572$	0	0
$573$	0	0
$574$	−1.36932	−0.0571542
$575$	−15.3693	−0.640945
$576$	0	0
$577$	17.3153	0.720847	0.360424	−	0.932789i	$-0.382632\pi$
$577$	17.3153	0.720847	0.360424	+	0.932789i	$0.382632\pi$
$578$	22.7386	0.945802
$579$	0	0
$580$	10.4384	0.433433
$581$	0.492423	0.0204291
$582$	0	0
$583$	−1.36932	−0.0567113
$584$	24.6847	1.02146
$585$	0	0
$586$	−31.9157	−1.31843
$587$	39.3693	1.62495	0.812473	−	0.582999i	$-0.198121\pi$
$587$	39.3693	1.62495	0.812473	+	0.582999i	$0.198121\pi$
$588$	0	0
$589$	18.2462	0.751822
$590$	−16.0000	−0.658710
$591$	0	0
$592$	35.4233	1.45589
$593$	−17.4233	−0.715489	−0.357744	−	0.933820i	$-0.616454\pi$
$593$	−17.4233	−0.715489	−0.357744	+	0.933820i	$0.616454\pi$
$594$	0	0
$595$	3.12311	0.128035
$596$	−1.06913	−0.0437933
$597$	0	0
$598$	0	0
$599$	41.6155	1.70036	0.850182	−	0.526489i	$-0.176492\pi$
$599$	41.6155	1.70036	0.850182	+	0.526489i	$0.176492\pi$
$600$	0	0
$601$	−7.06913	−0.288356	−0.144178	−	0.989552i	$-0.546054\pi$
$601$	−7.06913	−0.288356	−0.144178	+	0.989552i	$0.546054\pi$
$602$	4.00000	0.163028
$603$	0	0
$604$	4.10795	0.167150
$605$	24.9309	1.01358
$606$	0	0
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	−17.3693	−0.704419
$609$	0	0
$610$	21.5616	0.873002
$611$	0	0
$612$	0	0
$613$	34.8617	1.40805	0.704026	−	0.710174i	$-0.251384\pi$
$613$	34.8617	1.40805	0.704026	+	0.710174i	$0.251384\pi$
$614$	48.1080	1.94148
$615$	0	0
$616$	2.73863	0.110343
$617$	9.80776	0.394846	0.197423	−	0.980318i	$-0.436743\pi$
$617$	9.80776	0.394846	0.197423	+	0.980318i	$0.436743\pi$
$618$	0	0
$619$	−29.3002	−1.17767	−0.588837	−	0.808252i	$-0.700414\pi$
$619$	−29.3002	−1.17767	−0.588837	+	0.808252i	$0.700414\pi$
$620$	4.00000	0.160644
$621$	0	0
$622$	−29.8617	−1.19735
$623$	−2.73863	−0.109721
$624$	0	0
$625$	−4.36932	−0.174773
$626$	21.3693	0.854090
$627$	0	0
$628$	8.93087	0.356380
$629$	−11.8078	−0.470806
$630$	0	0
$631$	18.5616	0.738924	0.369462	−	0.929246i	$-0.379542\pi$
$631$	18.5616	0.738924	0.369462	+	0.929246i	$0.379542\pi$
$632$	13.2614	0.527509
$633$	0	0
$634$	−21.9460	−0.871588
$635$	−19.3693	−0.768648
$636$	0	0
$637$	0	0
$638$	−20.8769	−0.826524
$639$	0	0
$640$	48.3002	1.90923
$641$	19.1771	0.757449	0.378725	−	0.925509i	$-0.376363\pi$
$641$	19.1771	0.757449	0.378725	+	0.925509i	$0.376363\pi$
$642$	0	0
$643$	31.5464	1.24407	0.622034	−	0.782990i	$-0.286306\pi$
$643$	31.5464	1.24407	0.622034	+	0.782990i	$0.286306\pi$
$644$	0.492423	0.0194042
$645$	0	0
$646$	17.3693	0.683387
$647$	6.38447	0.250999	0.125500	−	0.992094i	$-0.459947\pi$
$647$	6.38447	0.250999	0.125500	+	0.992094i	$0.459947\pi$
$648$	0	0
$649$	5.75379	0.225856
$650$	0	0
$651$	0	0
$652$	2.10795	0.0825537
$653$	−23.1231	−0.904877	−0.452439	−	0.891796i	$-0.649446\pi$
$653$	−23.1231	−0.904877	−0.452439	+	0.891796i	$0.649446\pi$
$654$	0	0
$655$	26.2462	1.02552
$656$	7.31534	0.285616
$657$	0	0
$658$	7.23106	0.281896
$659$	2.24621	0.0875000	0.0437500	−	0.999043i	$-0.486070\pi$
$659$	2.24621	0.0875000	0.0437500	+	0.999043i	$0.486070\pi$
$660$	0	0
$661$	−5.63068	−0.219008	−0.109504	−	0.993986i	$-0.534926\pi$
$661$	−5.63068	−0.219008	−0.109504	+	0.993986i	$0.534926\pi$
$662$	−4.98485	−0.193742
$663$	0	0
$664$	−2.13826	−0.0829806
$665$	−14.2462	−0.552444
$666$	0	0
$667$	13.3693	0.517662
$668$	4.49242	0.173817
$669$	0	0
$670$	−25.3693	−0.980102
$671$	−7.75379	−0.299332
$672$	0	0
$673$	−23.2462	−0.896076	−0.448038	−	0.894015i	$-0.647877\pi$
$673$	−23.2462	−0.896076	−0.448038	+	0.894015i	$0.647877\pi$
$674$	9.56155	0.368297
$675$	0	0
$676$	0	0
$677$	15.6155	0.600153	0.300077	−	0.953915i	$-0.402988\pi$
$677$	15.6155	0.600153	0.300077	+	0.953915i	$0.402988\pi$
$678$	0	0
$679$	−4.80776	−0.184505
$680$	−13.5616	−0.520062
$681$	0	0
$682$	−8.00000	−0.306336
$683$	−38.1080	−1.45816	−0.729080	−	0.684428i	$-0.760052\pi$
$683$	−38.1080	−1.45816	−0.729080	+	0.684428i	$0.760052\pi$
$684$	0	0
$685$	19.8078	0.756816
$686$	−12.0000	−0.458162
$687$	0	0
$688$	−21.3693	−0.814698
$689$	0	0
$690$	0	0
$691$	51.3002	1.95155	0.975776	−	0.218774i	$-0.0702058\pi$
$691$	51.3002	1.95155	0.975776	+	0.218774i	$0.0702058\pi$
$692$	−8.87689	−0.337449
$693$	0	0
$694$	43.1231	1.63693
$695$	63.8617	2.42241
$696$	0	0
$697$	−2.43845	−0.0923628
$698$	10.6307	0.402377
$699$	0	0
$700$	−1.89205	−0.0715127
$701$	5.36932	0.202796	0.101398	−	0.994846i	$-0.467668\pi$
$701$	5.36932	0.202796	0.101398	+	0.994846i	$0.467668\pi$
$702$	0	0
$703$	53.8617	2.03143
$704$	−11.1231	−0.419218
$705$	0	0
$706$	−8.30019	−0.312382
$707$	−4.24621	−0.159695
$708$	0	0
$709$	7.49242	0.281384	0.140692	−	0.990053i	$-0.455067\pi$
$709$	7.49242	0.281384	0.140692	+	0.990053i	$0.455067\pi$
$710$	77.8617	2.92210
$711$	0	0
$712$	11.8920	0.445673
$713$	5.12311	0.191862
$714$	0	0
$715$	0	0
$716$	2.13826	0.0799106
$717$	0	0
$718$	14.6307	0.546012
$719$	23.3693	0.871528	0.435764	−	0.900061i	$-0.356478\pi$
$719$	23.3693	0.871528	0.435764	+	0.900061i	$0.356478\pi$
$720$	0	0
$721$	1.93087	0.0719093
$722$	−49.5616	−1.84449
$723$	0	0
$724$	−1.17708	−0.0437459
$725$	−51.3693	−1.90781
$726$	0	0
$727$	−38.6695	−1.43417	−0.717086	−	0.696984i	$-0.754525\pi$
$727$	−38.6695	−1.43417	−0.717086	+	0.696984i	$0.754525\pi$
$728$	0	0
$729$	0	0
$730$	56.3002	2.08376
$731$	7.12311	0.263458
$732$	0	0
$733$	−20.5076	−0.757465	−0.378732	−	0.925506i	$-0.623640\pi$
$733$	−20.5076	−0.757465	−0.378732	+	0.925506i	$0.623640\pi$
$734$	26.6307	0.982956
$735$	0	0
$736$	−4.87689	−0.179765
$737$	9.12311	0.336054
$738$	0	0
$739$	−10.2462	−0.376913	−0.188456	−	0.982082i	$-0.560348\pi$
$739$	−10.2462	−0.376913	−0.188456	+	0.982082i	$0.560348\pi$
$740$	11.8078	0.434062
$741$	0	0
$742$	0.600373	0.0220404
$743$	−12.6307	−0.463375	−0.231687	−	0.972790i	$-0.574425\pi$
$743$	−12.6307	−0.463375	−0.231687	+	0.972790i	$0.574425\pi$
$744$	0	0
$745$	8.68466	0.318181
$746$	−44.3002	−1.62195
$747$	0	0
$748$	−1.36932	−0.0500672
$749$	−4.63068	−0.169201
$750$	0	0
$751$	44.1080	1.60952	0.804761	−	0.593599i	$-0.202293\pi$
$751$	44.1080	1.60952	0.804761	+	0.593599i	$0.202293\pi$
$752$	−38.6307	−1.40872
$753$	0	0
$754$	0	0
$755$	−33.3693	−1.21443
$756$	0	0
$757$	30.0000	1.09037	0.545184	−	0.838316i	$-0.316460\pi$
$757$	30.0000	1.09037	0.545184	+	0.838316i	$0.316460\pi$
$758$	36.9848	1.34335
$759$	0	0
$760$	61.8617	2.24396
$761$	9.36932	0.339637	0.169819	−	0.985475i	$-0.445682\pi$
$761$	9.36932	0.339637	0.169819	+	0.985475i	$0.445682\pi$
$762$	0	0
$763$	−1.57671	−0.0570807
$764$	4.00000	0.144715
$765$	0	0
$766$	35.5076	1.28294
$767$	0	0
$768$	0	0
$769$	18.0000	0.649097	0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000	0.649097	0.324548	+	0.945869i	$0.394788\pi$
$770$	6.24621	0.225098
$771$	0	0
$772$	5.91571	0.212911
$773$	−24.2462	−0.872076	−0.436038	−	0.899928i	$-0.643619\pi$
$773$	−24.2462	−0.872076	−0.436038	+	0.899928i	$0.643619\pi$
$774$	0	0
$775$	−19.6847	−0.707094
$776$	20.8769	0.749437
$777$	0	0
$778$	53.1771	1.90649
$779$	11.1231	0.398527
$780$	0	0
$781$	−28.0000	−1.00192
$782$	4.87689	0.174397
$783$	0	0
$784$	31.3153	1.11841
$785$	−72.5464	−2.58929
$786$	0	0
$787$	−44.1771	−1.57474	−0.787371	−	0.616479i	$-0.788559\pi$
$787$	−44.1771	−1.57474	−0.787371	+	0.616479i	$0.788559\pi$
$788$	5.86174	0.208816
$789$	0	0
$790$	30.2462	1.07611
$791$	3.26137	0.115961
$792$	0	0
$793$	0	0
$794$	39.1231	1.38843
$795$	0	0
$796$	9.72348	0.344640
$797$	0.384472	0.0136187	0.00680935	−	0.999977i	$-0.497833\pi$
$797$	0.384472	0.0136187	0.00680935	+	0.999977i	$0.497833\pi$
$798$	0	0
$799$	12.8769	0.455552
$800$	18.7386	0.662511
$801$	0	0
$802$	−22.5464	−0.796141
$803$	−20.2462	−0.714473
$804$	0	0
$805$	−4.00000	−0.140981
$806$	0	0
$807$	0	0
$808$	18.4384	0.648662
$809$	16.3002	0.573084	0.286542	−	0.958068i	$-0.407494\pi$
$809$	16.3002	0.573084	0.286542	+	0.958068i	$0.407494\pi$
$810$	0	0
$811$	−2.56155	−0.0899483	−0.0449741	−	0.998988i	$-0.514321\pi$
$811$	−2.56155	−0.0899483	−0.0449741	+	0.998988i	$0.514321\pi$
$812$	1.64584	0.0577576
$813$	0	0
$814$	−23.6155	−0.827724
$815$	−17.1231	−0.599796
$816$	0	0
$817$	−32.4924	−1.13677
$818$	9.94602	0.347755
$819$	0	0
$820$	2.43845	0.0851543
$821$	6.49242	0.226587	0.113294	−	0.993562i	$-0.463860\pi$
$821$	6.49242	0.226587	0.113294	+	0.993562i	$0.463860\pi$
$822$	0	0
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	−8.38447	−0.292087
$825$	0	0
$826$	−2.52273	−0.0877771
$827$	−14.7386	−0.512513	−0.256256	−	0.966609i	$-0.582489\pi$
$827$	−14.7386	−0.512513	−0.256256	+	0.966609i	$0.582489\pi$
$828$	0	0
$829$	13.4924	0.468611	0.234306	−	0.972163i	$-0.424718\pi$
$829$	13.4924	0.468611	0.234306	+	0.972163i	$0.424718\pi$
$830$	−4.87689	−0.169279
$831$	0	0
$832$	0	0
$833$	−10.4384	−0.361671
$834$	0	0
$835$	−36.4924	−1.26287
$836$	6.24621	0.216030
$837$	0	0
$838$	−53.4773	−1.84734
$839$	21.6155	0.746251	0.373125	−	0.927781i	$-0.378286\pi$
$839$	21.6155	0.746251	0.373125	+	0.927781i	$0.378286\pi$
$840$	0	0
$841$	15.6847	0.540850
$842$	48.7926	1.68150
$843$	0	0
$844$	8.63068	0.297080
$845$	0	0
$846$	0	0
$847$	3.93087	0.135066
$848$	−3.20739	−0.110142
$849$	0	0
$850$	−18.7386	−0.642730
$851$	15.1231	0.518413
$852$	0	0
$853$	−2.12311	−0.0726938	−0.0363469	−	0.999339i	$-0.511572\pi$
$853$	−2.12311	−0.0726938	−0.0363469	+	0.999339i	$0.511572\pi$
$854$	3.39963	0.116333
$855$	0	0
$856$	20.1080	0.687276
$857$	−35.5616	−1.21476	−0.607380	−	0.794412i	$-0.707779\pi$
$857$	−35.5616	−1.21476	−0.607380	+	0.794412i	$0.707779\pi$
$858$	0	0
$859$	24.5616	0.838029	0.419015	−	0.907979i	$-0.362376\pi$
$859$	24.5616	0.838029	0.419015	+	0.907979i	$0.362376\pi$
$860$	−7.12311	−0.242896
$861$	0	0
$862$	17.3693	0.591601
$863$	30.4924	1.03797	0.518987	−	0.854782i	$-0.326309\pi$
$863$	30.4924	1.03797	0.518987	+	0.854782i	$0.326309\pi$
$864$	0	0
$865$	72.1080	2.45174
$866$	−13.6695	−0.464509
$867$	0	0
$868$	0.630683	0.0214068
$869$	−10.8769	−0.368973
$870$	0	0
$871$	0	0
$872$	6.84658	0.231855
$873$	0	0
$874$	−22.2462	−0.752489
$875$	5.36932	0.181516
$876$	0	0
$877$	−23.5616	−0.795617	−0.397809	−	0.917468i	$-0.630229\pi$
$877$	−23.5616	−0.795617	−0.397809	+	0.917468i	$0.630229\pi$
$878$	21.3693	0.721180
$879$	0	0
$880$	−33.3693	−1.12488
$881$	9.06913	0.305547	0.152773	−	0.988261i	$-0.451180\pi$
$881$	9.06913	0.305547	0.152773	+	0.988261i	$0.451180\pi$
$882$	0	0
$883$	8.80776	0.296405	0.148202	−	0.988957i	$-0.452651\pi$
$883$	8.80776	0.296405	0.148202	+	0.988957i	$0.452651\pi$
$884$	0	0
$885$	0	0
$886$	−54.2462	−1.82244
$887$	−24.6307	−0.827017	−0.413509	−	0.910500i	$-0.635697\pi$
$887$	−24.6307	−0.827017	−0.413509	+	0.910500i	$0.635697\pi$
$888$	0	0
$889$	−3.05398	−0.102427
$890$	27.1231	0.909169
$891$	0	0
$892$	−3.50758	−0.117442
$893$	−58.7386	−1.96561
$894$	0	0
$895$	−17.3693	−0.580592
$896$	7.61553	0.254417
$897$	0	0
$898$	12.8769	0.429708
$899$	17.1231	0.571088
$900$	0	0
$901$	1.06913	0.0356179
$902$	−4.87689	−0.162383
$903$	0	0
$904$	−14.1619	−0.471019
$905$	9.56155	0.317837
$906$	0	0
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	3.12311	0.103644
$909$	0	0
$910$	0	0
$911$	−38.7386	−1.28347	−0.641734	−	0.766927i	$-0.721785\pi$
$911$	−38.7386	−1.28347	−0.641734	+	0.766927i	$0.721785\pi$
$912$	0	0
$913$	1.75379	0.0580419
$914$	19.6998	0.651612
$915$	0	0
$916$	7.12311	0.235354
$917$	4.13826	0.136657
$918$	0	0
$919$	11.5076	0.379600	0.189800	−	0.981823i	$-0.439216\pi$
$919$	11.5076	0.379600	0.189800	+	0.981823i	$0.439216\pi$
$920$	17.3693	0.572649
$921$	0	0
$922$	25.2850	0.832718
$923$	0	0
$924$	0	0
$925$	−58.1080	−1.91058
$926$	−22.3542	−0.734603
$927$	0	0
$928$	−16.3002	−0.535080
$929$	−7.80776	−0.256164	−0.128082	−	0.991764i	$-0.540882\pi$
$929$	−7.80776	−0.256164	−0.128082	+	0.991764i	$0.540882\pi$
$930$	0	0
$931$	47.6155	1.56054
$932$	−11.3996	−0.373407
$933$	0	0
$934$	−40.6004	−1.32848
$935$	11.1231	0.363764
$936$	0	0
$937$	−7.56155	−0.247025	−0.123513	−	0.992343i	$-0.539416\pi$
$937$	−7.56155	−0.247025	−0.123513	+	0.992343i	$0.539416\pi$
$938$	−4.00000	−0.130605
$939$	0	0
$940$	−12.8769	−0.419998
$941$	30.4924	0.994025	0.497012	−	0.867744i	$-0.334430\pi$
$941$	30.4924	0.994025	0.497012	+	0.867744i	$0.334430\pi$
$942$	0	0
$943$	3.12311	0.101702
$944$	13.4773	0.438648
$945$	0	0
$946$	14.2462	0.463184
$947$	−38.7386	−1.25884	−0.629418	−	0.777067i	$-0.716707\pi$
$947$	−38.7386	−1.25884	−0.629418	+	0.777067i	$0.716707\pi$
$948$	0	0
$949$	0	0
$950$	85.4773	2.77325
$951$	0	0
$952$	−2.13826	−0.0693014
$953$	−30.9848	−1.00370	−0.501849	−	0.864955i	$-0.667347\pi$
$953$	−30.9848	−1.00370	−0.501849	+	0.864955i	$0.667347\pi$
$954$	0	0
$955$	−32.4924	−1.05143
$956$	−11.1231	−0.359747
$957$	0	0
$958$	−16.0000	−0.516937
$959$	3.12311	0.100850
$960$	0	0
$961$	−24.4384	−0.788337
$962$	0	0
$963$	0	0
$964$	7.80776	0.251471
$965$	−48.0540	−1.54691
$966$	0	0
$967$	0.876894	0.0281990	0.0140995	−	0.999901i	$-0.495512\pi$
$967$	0.876894	0.0281990	0.0140995	+	0.999901i	$0.495512\pi$
$968$	−17.0691	−0.548623
$969$	0	0
$970$	47.6155	1.52884
$971$	12.9848	0.416704	0.208352	−	0.978054i	$-0.433190\pi$
$971$	12.9848	0.416704	0.208352	+	0.978054i	$0.433190\pi$
$972$	0	0
$973$	10.0691	0.322801
$974$	11.1231	0.356407
$975$	0	0
$976$	−18.1619	−0.581349
$977$	61.1771	1.95723	0.978614	−	0.205705i	$-0.0659486\pi$
$977$	61.1771	1.95723	0.978614	+	0.205705i	$0.0659486\pi$
$978$	0	0
$979$	−9.75379	−0.311732
$980$	10.4384	0.333444
$981$	0	0
$982$	−56.6004	−1.80619
$983$	13.6155	0.434268	0.217134	−	0.976142i	$-0.430329\pi$
$983$	13.6155	0.434268	0.217134	+	0.976142i	$0.430329\pi$
$984$	0	0
$985$	−47.6155	−1.51716
$986$	16.3002	0.519104
$987$	0	0
$988$	0	0
$989$	−9.12311	−0.290098
$990$	0	0
$991$	−50.3542	−1.59955	−0.799776	−	0.600298i	$-0.795049\pi$
$991$	−50.3542	−1.59955	−0.799776	+	0.600298i	$0.795049\pi$
$992$	−6.24621	−0.198317
$993$	0	0
$994$	12.2765	0.389388
$995$	−78.9848	−2.50399
$996$	0	0
$997$	−20.6155	−0.652900	−0.326450	−	0.945214i	$-0.605853\pi$
$997$	−20.6155	−0.652900	−0.326450	+	0.945214i	$0.605853\pi$
$998$	7.01515	0.222061
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.2.a.m.1.1		2
3.2	odd	2		507.2.a.d.1.2		2
12.11	even	2		8112.2.a.bo.1.2		2
13.4	even	6		117.2.g.c.55.1		4
13.5	odd	4		1521.2.b.h.1351.3		4
13.8	odd	4		1521.2.b.h.1351.2		4
13.10	even	6		117.2.g.c.100.1		4
13.12	even	2		1521.2.a.g.1.2		2
39.2	even	12		507.2.j.g.316.3		8
39.5	even	4		507.2.b.d.337.2		4
39.8	even	4		507.2.b.d.337.3		4
39.11	even	12		507.2.j.g.316.2		8
39.17	odd	6		39.2.e.b.16.2	✓	4
39.20	even	12		507.2.j.g.361.3		8
39.23	odd	6		39.2.e.b.22.2	yes	4
39.29	odd	6		507.2.e.g.22.1		4
39.32	even	12		507.2.j.g.361.2		8
39.35	odd	6		507.2.e.g.484.1		4
39.38	odd	2		507.2.a.g.1.1		2
52.23	odd	6		1872.2.t.r.1153.2		4
52.43	odd	6		1872.2.t.r.289.2		4
156.23	even	6		624.2.q.h.529.1		4
156.95	even	6		624.2.q.h.289.1		4
156.155	even	2		8112.2.a.bk.1.1		2
195.17	even	12		975.2.bb.i.874.2		8
195.23	even	12		975.2.bb.i.724.2		8
195.62	even	12		975.2.bb.i.724.3		8
195.134	odd	6		975.2.i.k.601.1		4
195.173	even	12		975.2.bb.i.874.3		8
195.179	odd	6		975.2.i.k.451.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.e.b.16.2	✓	4	39.17	odd	6
39.2.e.b.22.2	yes	4	39.23	odd	6
117.2.g.c.55.1		4	13.4	even	6
117.2.g.c.100.1		4	13.10	even	6
507.2.a.d.1.2		2	3.2	odd	2
507.2.a.g.1.1		2	39.38	odd	2
507.2.b.d.337.2		4	39.5	even	4
507.2.b.d.337.3		4	39.8	even	4
507.2.e.g.22.1		4	39.29	odd	6
507.2.e.g.484.1		4	39.35	odd	6
507.2.j.g.316.2		8	39.11	even	12
507.2.j.g.316.3		8	39.2	even	12
507.2.j.g.361.2		8	39.32	even	12
507.2.j.g.361.3		8	39.20	even	12
624.2.q.h.289.1		4	156.95	even	6
624.2.q.h.529.1		4	156.23	even	6
975.2.i.k.451.1		4	195.179	odd	6
975.2.i.k.601.1		4	195.134	odd	6
975.2.bb.i.724.2		8	195.23	even	12
975.2.bb.i.724.3		8	195.62	even	12
975.2.bb.i.874.2		8	195.17	even	12
975.2.bb.i.874.3		8	195.173	even	12
1521.2.a.g.1.2		2	13.12	even	2
1521.2.a.m.1.1		2	1.1	even	1	trivial
1521.2.b.h.1351.2		4	13.8	odd	4
1521.2.b.h.1351.3		4	13.5	odd	4
1872.2.t.r.289.2		4	52.43	odd	6
1872.2.t.r.1153.2		4	52.23	odd	6
8112.2.a.bk.1.1		2	156.155	even	2
8112.2.a.bo.1.2		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q-1.56155 q^{2} +0.438447 q^{4} -3.56155 q^{5} -0.561553 q^{7} +2.43845 q^{8} +5.56155 q^{10} -2.00000 q^{11} +0.876894 q^{14} -4.68466 q^{16} +1.56155 q^{17} -7.12311 q^{19} -1.56155 q^{20} +3.12311 q^{22} -2.00000 q^{23} +7.68466 q^{25} -0.246211 q^{28} -6.68466 q^{29} -2.56155 q^{31} +2.43845 q^{32} -2.43845 q^{34} +2.00000 q^{35} -7.56155 q^{37} +11.1231 q^{38} -8.68466 q^{40} -1.56155 q^{41} +4.56155 q^{43} -0.876894 q^{44} +3.12311 q^{46} +8.24621 q^{47} -6.68466 q^{49} -12.0000 q^{50} +0.684658 q^{53} +7.12311 q^{55} -1.36932 q^{56} +10.4384 q^{58} -2.87689 q^{59} +3.87689 q^{61} +4.00000 q^{62} +5.56155 q^{64} -4.56155 q^{67} +0.684658 q^{68} -3.12311 q^{70} +14.0000 q^{71} +10.1231 q^{73} +11.8078 q^{74} -3.12311 q^{76} +1.12311 q^{77} +5.43845 q^{79} +16.6847 q^{80} +2.43845 q^{82} -0.876894 q^{83} -5.56155 q^{85} -7.12311 q^{86} -4.87689 q^{88} +4.87689 q^{89} -0.876894 q^{92} -12.8769 q^{94} +25.3693 q^{95} +8.56155 q^{97} +10.4384 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 5 q^{4} - 3 q^{5} + 3 q^{7} + 9 q^{8} + 7 q^{10} - 4 q^{11} + 10 q^{14} + 3 q^{16} - q^{17} - 6 q^{19} + q^{20} - 2 q^{22} - 4 q^{23} + 3 q^{25} + 16 q^{28} - q^{29} - q^{31} + 9 q^{32}+ \cdots + 25 q^{98}+O(q^{100}) \) 2 * q + q^2 + 5 * q^4 - 3 * q^5 + 3 * q^7 + 9 * q^8 + 7 * q^10 - 4 * q^11 + 10 * q^14 + 3 * q^16 - q^17 - 6 * q^19 + q^20 - 2 * q^22 - 4 * q^23 + 3 * q^25 + 16 * q^28 - q^29 - q^31 + 9 * q^32 - 9 * q^34 + 4 * q^35 - 11 * q^37 + 14 * q^38 - 5 * q^40 + q^41 + 5 * q^43 - 10 * q^44 - 2 * q^46 - q^49 - 24 * q^50 - 11 * q^53 + 6 * q^55 + 22 * q^56 + 25 * q^58 - 14 * q^59 + 16 * q^61 + 8 * q^62 + 7 * q^64 - 5 * q^67 - 11 * q^68 + 2 * q^70 + 28 * q^71 + 12 * q^73 + 3 * q^74 + 2 * q^76 - 6 * q^77 + 15 * q^79 + 21 * q^80 + 9 * q^82 - 10 * q^83 - 7 * q^85 - 6 * q^86 - 18 * q^88 + 18 * q^89 - 10 * q^92 - 34 * q^94 + 26 * q^95 + 13 * q^97 + 25 * q^98

Introduction

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