LMFDB - Embedded newform 1521.2.a.g.1.2

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			1.2
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)

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

Display 100 coefficients 10 coefficients

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	$-0.313830\pi$
$2$	1.56155	1.10418	0.552092	+	0.833783i	$0.313830\pi$
$3$	0	0
$3$	0	0
$4$	0.438447	0.219224
$5$	3.56155	1.59277	0.796387	−	0.604787i	$-0.206742\pi$
$5$	3.56155	1.59277	0.796387	+	0.604787i	$0.206742\pi$
$6$	0	0
$7$	0.561553	0.212247	0.106124	−	0.994353i	$-0.466156\pi$
$7$	0.561553	0.212247	0.106124	+	0.994353i	$0.466156\pi$
$8$	−2.43845	−0.862121
$9$	0	0
$10$	5.56155	1.75872
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	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	$-0.195593\pi$
$19$	7.12311	1.63415	0.817076	+	0.576530i	$0.195593\pi$
$20$	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	$-0.426116\pi$
$31$	2.56155	0.460068	0.230034	+	0.973183i	$0.426116\pi$
$32$	−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	$-0.286501\pi$
$37$	7.56155	1.24311	0.621556	+	0.783370i	$0.286501\pi$
$38$	11.1231	1.80441
$39$	0	0
$40$	−8.68466	−1.37317
$41$	1.56155	0.243874	0.121937	−	0.992538i	$-0.461089\pi$
$41$	1.56155	0.243874	0.121937	+	0.992538i	$0.461089\pi$
$42$	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	$-0.705397\pi$
$47$	−8.24621	−1.20283	−0.601417	+	0.798935i	$0.705397\pi$
$48$	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	$-0.440036\pi$
$59$	2.87689	0.374540	0.187270	+	0.982309i	$0.440036\pi$
$60$	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	$-0.410116\pi$
$67$	4.56155	0.557282	0.278641	+	0.960395i	$0.410116\pi$
$68$	0.684658	0.0830270
$69$	0	0
$70$	3.12311	0.373283
$71$	−14.0000	−1.66149	−0.830747	−	0.556650i	$-0.812086\pi$
$71$	−14.0000	−1.66149	−0.830747	+	0.556650i	$0.812086\pi$
$72$	0	0
$73$	−10.1231	−1.18482	−0.592410	−	0.805637i	$-0.701823\pi$
$73$	−10.1231	−1.18482	−0.592410	+	0.805637i	$0.701823\pi$
$74$	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	$-0.484675\pi$
$83$	0.876894	0.0962517	0.0481258	+	0.998841i	$0.484675\pi$
$84$	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	$-0.583220\pi$
$89$	−4.87689	−0.516950	−0.258475	+	0.966018i	$0.583220\pi$
$90$	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	$-0.643127\pi$
$97$	−8.56155	−0.869294	−0.434647	+	0.900601i	$0.643127\pi$
$98$	−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	$-0.542932\pi$
$109$	−2.80776	−0.268935	−0.134468	+	0.990918i	$0.542932\pi$
$110$	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	$-0.423647\pi$
$137$	5.56155	0.475156	0.237578	+	0.971369i	$0.423647\pi$
$138$	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	$-0.468153\pi$
$149$	2.43845	0.199765	0.0998827	+	0.994999i	$0.468153\pi$
$150$	0	0
$151$	−9.36932	−0.762464	−0.381232	−	0.924479i	$-0.624500\pi$
$151$	−9.36932	−0.762464	−0.381232	+	0.924479i	$0.624500\pi$
$152$	−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	$-0.560293\pi$
$163$	−4.80776	−0.376573	−0.188287	+	0.982114i	$0.560293\pi$
$164$	0.684658	0.0534628
$165$	0	0
$166$	1.36932	0.106280
$167$	−10.2462	−0.792876	−0.396438	−	0.918062i	$-0.629754\pi$
$167$	−10.2462	−0.792876	−0.396438	+	0.918062i	$0.629754\pi$
$168$	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	$-0.661400\pi$
$193$	−13.4924	−0.971206	−0.485603	+	0.874180i	$0.661400\pi$
$194$	−13.3693	−0.959861
$195$	0	0
$196$	−2.93087	−0.209348
$197$	−13.3693	−0.952524	−0.476262	−	0.879303i	$-0.658009\pi$
$197$	−13.3693	−0.952524	−0.476262	+	0.879303i	$0.658009\pi$
$198$	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	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	−1.36932	−0.0914913
$225$	0	0
$226$	−9.06913	−0.603270
$227$	−7.12311	−0.472777	−0.236389	−	0.971659i	$-0.575964\pi$
$227$	−7.12311	−0.472777	−0.236389	+	0.971659i	$0.575964\pi$
$228$	0	0
$229$	−16.2462	−1.07358	−0.536790	−	0.843716i	$-0.680363\pi$
$229$	−16.2462	−1.07358	−0.536790	+	0.843716i	$0.680363\pi$
$230$	−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	$-0.193694\pi$
$239$	25.3693	1.64100	0.820502	+	0.571643i	$0.193694\pi$
$240$	0	0
$241$	−17.8078	−1.14710	−0.573549	−	0.819171i	$-0.694434\pi$
$241$	−17.8078	−1.14710	−0.573549	+	0.819171i	$0.694434\pi$
$242$	−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	$-0.863218\pi$
$271$	−29.9309	−1.81817	−0.909085	+	0.416610i	$0.863218\pi$
$272$	−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	$-0.678244\pi$
$281$	−17.8078	−1.06232	−0.531161	+	0.847271i	$0.678244\pi$
$282$	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	$-0.703646\pi$
$293$	−20.4384	−1.19403	−0.597013	+	0.802231i	$0.703646\pi$
$294$	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	$-0.158114\pi$
$307$	30.8078	1.75829	0.879146	+	0.476553i	$0.158114\pi$
$308$	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	$-0.629143\pi$
$317$	−14.0540	−0.789350	−0.394675	+	0.918821i	$0.629143\pi$
$318$	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	$-0.527961\pi$
$331$	−3.19224	−0.175461	−0.0877306	+	0.996144i	$0.527961\pi$
$332$	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	$-0.441676\pi$
$349$	6.80776	0.364411	0.182206	+	0.983260i	$0.441676\pi$
$350$	6.73863	0.360195
$351$	0	0
$352$	−4.87689	−0.259939
$353$	−5.31534	−0.282907	−0.141454	−	0.989945i	$-0.545178\pi$
$353$	−5.31534	−0.282907	−0.141454	+	0.989945i	$0.545178\pi$
$354$	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	$-0.420474\pi$
$359$	9.36932	0.494494	0.247247	+	0.968953i	$0.420474\pi$
$360$	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	$-0.291852\pi$
$379$	23.6847	1.21660	0.608300	+	0.793708i	$0.291852\pi$
$380$	11.1231	0.570603
$381$	0	0
$382$	14.2462	0.728900
$383$	22.7386	1.16189	0.580945	−	0.813943i	$-0.302683\pi$
$383$	22.7386	1.16189	0.580945	+	0.813943i	$0.302683\pi$
$384$	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	$-0.283583\pi$
$397$	25.0540	1.25742	0.628711	+	0.777639i	$0.283583\pi$
$398$	34.6307	1.73588
$399$	0	0
$400$	−36.0000	−1.80000
$401$	−14.4384	−0.721022	−0.360511	−	0.932755i	$-0.617398\pi$
$401$	−14.4384	−0.721022	−0.360511	+	0.932755i	$0.617398\pi$
$402$	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	$-0.449666\pi$
$409$	6.36932	0.314942	0.157471	+	0.987524i	$0.449666\pi$
$410$	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	$-0.224501\pi$
$421$	31.2462	1.52285	0.761424	+	0.648255i	$0.224501\pi$
$422$	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	$-0.413673\pi$
$431$	11.1231	0.535781	0.267891	+	0.963449i	$0.413673\pi$
$432$	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	$-0.437665\pi$
$449$	8.24621	0.389163	0.194581	+	0.980886i	$0.437665\pi$
$450$	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	$-0.404659\pi$
$457$	12.6155	0.590130	0.295065	+	0.955477i	$0.404659\pi$
$458$	−25.3693	−1.18543
$459$	0	0
$460$	−3.12311	−0.145616
$461$	16.1922	0.754148	0.377074	−	0.926183i	$-0.376930\pi$
$461$	16.1922	0.754148	0.377074	+	0.926183i	$0.376930\pi$
$462$	0	0
$463$	−14.3153	−0.665290	−0.332645	−	0.943052i	$-0.607941\pi$
$463$	−14.3153	−0.665290	−0.332645	+	0.943052i	$0.607941\pi$
$464$	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	$-0.575208\pi$
$479$	−10.2462	−0.468161	−0.234081	+	0.972217i	$0.575208\pi$
$480$	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	$-0.448403\pi$
$487$	7.12311	0.322779	0.161389	+	0.986891i	$0.448403\pi$
$488$	−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	$-0.467938\pi$
$499$	4.49242	0.201108	0.100554	+	0.994932i	$0.467938\pi$
$500$	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	$-0.401006\pi$
$509$	13.8078	0.612018	0.306009	+	0.952029i	$0.401006\pi$
$510$	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	$-0.360515\pi$
$541$	19.7386	0.848630	0.424315	+	0.905515i	$0.360515\pi$
$542$	−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	$-0.136450\pi$
$557$	42.9309	1.81904	0.909520	+	0.415661i	$0.136450\pi$
$558$	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	$-0.617368\pi$
$577$	−17.3153	−0.720847	−0.360424	+	0.932789i	$0.617368\pi$
$578$	−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	$-0.801879\pi$
$587$	−39.3693	−1.62495	−0.812473	+	0.582999i	$0.801879\pi$
$588$	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	$-0.383546\pi$
$593$	17.4233	0.715489	0.357744	+	0.933820i	$0.383546\pi$
$594$	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	$-0.748616\pi$
$613$	−34.8617	−1.40805	−0.704026	+	0.710174i	$0.748616\pi$
$614$	48.1080	1.94148
$615$	0	0
$616$	−2.73863	−0.110343
$617$	−9.80776	−0.394846	−0.197423	−	0.980318i	$-0.563257\pi$
$617$	−9.80776	−0.394846	−0.197423	+	0.980318i	$0.563257\pi$
$618$	0	0
$619$	29.3002	1.17767	0.588837	−	0.808252i	$-0.299586\pi$
$619$	29.3002	1.17767	0.588837	+	0.808252i	$0.299586\pi$
$620$	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	$-0.620458\pi$
$631$	−18.5616	−0.738924	−0.369462	+	0.929246i	$0.620458\pi$
$632$	−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	$-0.713694\pi$
$643$	−31.5464	−1.24407	−0.622034	+	0.782990i	$0.713694\pi$
$644$	−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	$-0.465074\pi$
$661$	5.63068	0.219008	0.109504	+	0.993986i	$0.465074\pi$
$662$	−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	$-0.239948\pi$
$683$	38.1080	1.45816	0.729080	+	0.684428i	$0.239948\pi$
$684$	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	$-0.929794\pi$
$691$	−51.3002	−1.95155	−0.975776	+	0.218774i	$0.929794\pi$
$692$	−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	$-0.544933\pi$
$709$	−7.49242	−0.281384	−0.140692	+	0.990053i	$0.544933\pi$
$710$	−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	$-0.376360\pi$
$733$	20.5076	0.757465	0.378732	+	0.925506i	$0.376360\pi$
$734$	−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	$-0.439652\pi$
$739$	10.2462	0.376913	0.188456	+	0.982082i	$0.439652\pi$
$740$	11.8078	0.434062
$741$	0	0
$742$	0.600373	0.0220404
$743$	12.6307	0.463375	0.231687	−	0.972790i	$-0.425575\pi$
$743$	12.6307	0.463375	0.231687	+	0.972790i	$0.425575\pi$
$744$	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	$-0.554318\pi$
$761$	−9.36932	−0.339637	−0.169819	+	0.985475i	$0.554318\pi$
$762$	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	$-0.605212\pi$
$769$	−18.0000	−0.649097	−0.324548	+	0.945869i	$0.605212\pi$
$770$	6.24621	0.225098
$771$	0	0
$772$	−5.91571	−0.212911
$773$	24.2462	0.872076	0.436038	−	0.899928i	$-0.356381\pi$
$773$	24.2462	0.872076	0.436038	+	0.899928i	$0.356381\pi$
$774$	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	$-0.211441\pi$
$787$	44.1771	1.57474	0.787371	+	0.616479i	$0.211441\pi$
$788$	−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	$-0.485679\pi$
$811$	2.56155	0.0899483	0.0449741	+	0.998988i	$0.485679\pi$
$812$	−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	$-0.536140\pi$
$821$	−6.49242	−0.226587	−0.113294	+	0.993562i	$0.536140\pi$
$822$	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	$-0.417511\pi$
$827$	14.7386	0.512513	0.256256	+	0.966609i	$0.417511\pi$
$828$	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	$-0.621714\pi$
$839$	−21.6155	−0.746251	−0.373125	+	0.927781i	$0.621714\pi$
$840$	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	$-0.488428\pi$
$853$	2.12311	0.0726938	0.0363469	+	0.999339i	$0.488428\pi$
$854$	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	$-0.673691\pi$
$863$	−30.4924	−1.03797	−0.518987	+	0.854782i	$0.673691\pi$
$864$	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	$-0.369771\pi$
$877$	23.5616	0.795617	0.397809	+	0.917468i	$0.369771\pi$
$878$	−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	$-0.459118\pi$
$929$	7.80776	0.256164	0.128082	+	0.991764i	$0.459118\pi$
$930$	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	$-0.665570\pi$
$941$	−30.4924	−0.994025	−0.497012	+	0.867744i	$0.665570\pi$
$942$	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	$-0.283293\pi$
$947$	38.7386	1.25884	0.629418	+	0.777067i	$0.283293\pi$
$948$	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	$-0.504488\pi$
$967$	−0.876894	−0.0281990	−0.0140995	+	0.999901i	$0.504488\pi$
$968$	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	$-0.934051\pi$
$977$	−61.1771	−1.95723	−0.978614	+	0.205705i	$0.934051\pi$
$978$	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	$-0.569671\pi$
$983$	−13.6155	−0.434268	−0.217134	+	0.976142i	$0.569671\pi$
$984$	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.g.1.2		2
3.2	odd	2		507.2.a.g.1.1		2
12.11	even	2		8112.2.a.bk.1.1		2
13.3	even	3		117.2.g.c.100.1		4
13.5	odd	4		1521.2.b.h.1351.2		4
13.8	odd	4		1521.2.b.h.1351.3		4
13.9	even	3		117.2.g.c.55.1		4
13.12	even	2		1521.2.a.m.1.1		2
39.2	even	12		507.2.j.g.316.2		8
39.5	even	4		507.2.b.d.337.3		4
39.8	even	4		507.2.b.d.337.2		4
39.11	even	12		507.2.j.g.316.3		8
39.17	odd	6		507.2.e.g.484.1		4
39.20	even	12		507.2.j.g.361.2		8
39.23	odd	6		507.2.e.g.22.1		4
39.29	odd	6		39.2.e.b.22.2	yes	4
39.32	even	12		507.2.j.g.361.3		8
39.35	odd	6		39.2.e.b.16.2	✓	4
39.38	odd	2		507.2.a.d.1.2		2
52.3	odd	6		1872.2.t.r.1153.2		4
52.35	odd	6		1872.2.t.r.289.2		4
156.35	even	6		624.2.q.h.289.1		4
156.107	even	6		624.2.q.h.529.1		4
156.155	even	2		8112.2.a.bo.1.2		2
195.29	odd	6		975.2.i.k.451.1		4
195.68	even	12		975.2.bb.i.724.2		8
195.74	odd	6		975.2.i.k.601.1		4
195.107	even	12		975.2.bb.i.724.3		8
195.113	even	12		975.2.bb.i.874.3		8
195.152	even	12		975.2.bb.i.874.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.e.b.16.2	✓	4	39.35	odd	6
39.2.e.b.22.2	yes	4	39.29	odd	6
117.2.g.c.55.1		4	13.9	even	3
117.2.g.c.100.1		4	13.3	even	3
507.2.a.d.1.2		2	39.38	odd	2
507.2.a.g.1.1		2	3.2	odd	2
507.2.b.d.337.2		4	39.8	even	4
507.2.b.d.337.3		4	39.5	even	4
507.2.e.g.22.1		4	39.23	odd	6
507.2.e.g.484.1		4	39.17	odd	6
507.2.j.g.316.2		8	39.2	even	12
507.2.j.g.316.3		8	39.11	even	12
507.2.j.g.361.2		8	39.20	even	12
507.2.j.g.361.3		8	39.32	even	12
624.2.q.h.289.1		4	156.35	even	6
624.2.q.h.529.1		4	156.107	even	6
975.2.i.k.451.1		4	195.29	odd	6
975.2.i.k.601.1		4	195.74	odd	6
975.2.bb.i.724.2		8	195.68	even	12
975.2.bb.i.724.3		8	195.107	even	12
975.2.bb.i.874.2		8	195.152	even	12
975.2.bb.i.874.3		8	195.113	even	12
1521.2.a.g.1.2		2	1.1	even	1	trivial
1521.2.a.m.1.1		2	13.12	even	2
1521.2.b.h.1351.2		4	13.5	odd	4
1521.2.b.h.1351.3		4	13.8	odd	4
1872.2.t.r.289.2		4	52.35	odd	6
1872.2.t.r.1153.2		4	52.3	odd	6
8112.2.a.bk.1.1		2	12.11	even	2
8112.2.a.bo.1.2		2	156.155	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)

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

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