LMFDB - Embedded newform 4815.2.a.u.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4815,2,Mod(1,4815)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4815, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4815.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4815 = 3^{2} \cdot 5 \cdot 107 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4815.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:	$38.4479685732$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 15 x^{10} + 49 x^{9} + 71 x^{8} - 278 x^{7} - 92 x^{6} + 649 x^{5} - 127 x^{4} + \cdots - 6 $ x^12 - 3x^11 - 15x^10 + 49x^9 + 71x^8 - 278x^7 - 92x^6 + 649x^5 - 127x^4 - 529x^3 + 267x^2 + 15*x - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1605)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-1.73551$ of defining polynomial
Character	$\chi$	$=$	4815.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73551 q^{2} +1.01200 q^{4} -1.00000 q^{5} -4.69063 q^{7} -1.71469 q^{8} +O(q^{10})$	\(q+1.73551 q^{2} +1.01200 q^{4} -1.00000 q^{5} -4.69063 q^{7} -1.71469 q^{8} -1.73551 q^{10} -2.19066 q^{11} +2.63307 q^{13} -8.14063 q^{14} -4.99986 q^{16} -1.38890 q^{17} +0.215496 q^{19} -1.01200 q^{20} -3.80191 q^{22} -2.79261 q^{23} +1.00000 q^{25} +4.56973 q^{26} -4.74690 q^{28} +7.70002 q^{29} -4.81186 q^{31} -5.24792 q^{32} -2.41045 q^{34} +4.69063 q^{35} +7.41620 q^{37} +0.373996 q^{38} +1.71469 q^{40} +7.32360 q^{41} +9.15677 q^{43} -2.21694 q^{44} -4.84660 q^{46} -1.99062 q^{47} +15.0020 q^{49} +1.73551 q^{50} +2.66466 q^{52} +5.83030 q^{53} +2.19066 q^{55} +8.04298 q^{56} +13.3635 q^{58} -10.7619 q^{59} -0.552233 q^{61} -8.35102 q^{62} +0.891893 q^{64} -2.63307 q^{65} +8.35384 q^{67} -1.40556 q^{68} +8.14063 q^{70} -4.85111 q^{71} +5.48768 q^{73} +12.8709 q^{74} +0.218081 q^{76} +10.2756 q^{77} +5.30027 q^{79} +4.99986 q^{80} +12.7102 q^{82} -8.87948 q^{83} +1.38890 q^{85} +15.8917 q^{86} +3.75630 q^{88} +3.68952 q^{89} -12.3508 q^{91} -2.82611 q^{92} -3.45475 q^{94} -0.215496 q^{95} +3.01913 q^{97} +26.0361 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{2} + 15 q^{4} - 12 q^{5} + 7 q^{7} - 3 q^{8}+O(q^{10}) $ 12 * q - 3 * q^2 + 15 * q^4 - 12 * q^5 + 7 * q^7 - 3 * q^8	$ 12 q - 3 q^{2} + 15 q^{4} - 12 q^{5} + 7 q^{7} - 3 q^{8} + 3 q^{10} - 4 q^{11} + 13 q^{13} - 4 q^{14} + 13 q^{16} + 4 q^{17} + 14 q^{19} - 15 q^{20} + 15 q^{22} - 11 q^{23} + 12 q^{25} + 8 q^{26} + 16 q^{28} + 7 q^{29} + 4 q^{31} - 4 q^{32} + q^{34} - 7 q^{35} + 24 q^{37} + 11 q^{38} + 3 q^{40} - 13 q^{41} + 25 q^{43} - 10 q^{44} - 22 q^{46} - 19 q^{47} + 9 q^{49} - 3 q^{50} + 20 q^{52} - 11 q^{53} + 4 q^{55} + 37 q^{56} - 2 q^{58} - 8 q^{59} + 7 q^{61} + 11 q^{62} - 19 q^{64} - 13 q^{65} + 33 q^{67} + 24 q^{68} + 4 q^{70} + 34 q^{73} + 27 q^{74} - 9 q^{76} + 29 q^{77} - 13 q^{80} + q^{82} + 24 q^{83} - 4 q^{85} + 36 q^{86} - 6 q^{88} + 10 q^{89} + 30 q^{91} + 28 q^{92} - 8 q^{94} - 14 q^{95} + 16 q^{97} + 36 q^{98}+O(q^{100}) $ 12 * q - 3 * q^2 + 15 * q^4 - 12 * q^5 + 7 * q^7 - 3 * q^8 + 3 * q^10 - 4 * q^11 + 13 * q^13 - 4 * q^14 + 13 * q^16 + 4 * q^17 + 14 * q^19 - 15 * q^20 + 15 * q^22 - 11 * q^23 + 12 * q^25 + 8 * q^26 + 16 * q^28 + 7 * q^29 + 4 * q^31 - 4 * q^32 + q^34 - 7 * q^35 + 24 * q^37 + 11 * q^38 + 3 * q^40 - 13 * q^41 + 25 * q^43 - 10 * q^44 - 22 * q^46 - 19 * q^47 + 9 * q^49 - 3 * q^50 + 20 * q^52 - 11 * q^53 + 4 * q^55 + 37 * q^56 - 2 * q^58 - 8 * q^59 + 7 * q^61 + 11 * q^62 - 19 * q^64 - 13 * q^65 + 33 * q^67 + 24 * q^68 + 4 * q^70 + 34 * q^73 + 27 * q^74 - 9 * q^76 + 29 * q^77 - 13 * q^80 + q^82 + 24 * q^83 - 4 * q^85 + 36 * q^86 - 6 * q^88 + 10 * q^89 + 30 * q^91 + 28 * q^92 - 8 * q^94 - 14 * q^95 + 16 * q^97 + 36 * 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 $

$2$	1.73551	1.22719	0.613596	−	0.789620i	$-0.289723\pi$
$2$	1.73551	1.22719	0.613596	+	0.789620i	$0.289723\pi$
$3$	0	0
$3$	0	0
$4$	1.01200	0.505998
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.69063	−1.77289	−0.886445	−	0.462834i	$-0.846833\pi$
$7$	−4.69063	−1.77289	−0.886445	+	0.462834i	$0.846833\pi$
$8$	−1.71469	−0.606235
$9$	0	0
$10$	−1.73551	−0.548817
$11$	−2.19066	−0.660508	−0.330254	−	0.943892i	$-0.607134\pi$
$11$	−2.19066	−0.660508	−0.330254	+	0.943892i	$0.607134\pi$
$12$	0	0
$13$	2.63307	0.730283	0.365142	−	0.930952i	$-0.381021\pi$
$13$	2.63307	0.730283	0.365142	+	0.930952i	$0.381021\pi$
$14$	−8.14063	−2.17568
$15$	0	0
$16$	−4.99986	−1.24996
$17$	−1.38890	−0.336858	−0.168429	−	0.985714i	$-0.553869\pi$
$17$	−1.38890	−0.336858	−0.168429	+	0.985714i	$0.553869\pi$
$18$	0	0
$19$	0.215496	0.0494382	0.0247191	−	0.999694i	$-0.492131\pi$
$19$	0.215496	0.0494382	0.0247191	+	0.999694i	$0.492131\pi$
$20$	−1.01200	−0.226289
$21$	0	0
$22$	−3.80191	−0.810569
$23$	−2.79261	−0.582299	−0.291150	−	0.956678i	$-0.594038\pi$
$23$	−2.79261	−0.582299	−0.291150	+	0.956678i	$0.594038\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	4.56973	0.896197
$27$	0	0
$28$	−4.74690	−0.897079
$29$	7.70002	1.42986	0.714929	−	0.699198i	$-0.246459\pi$
$29$	7.70002	1.42986	0.714929	+	0.699198i	$0.246459\pi$
$30$	0	0
$31$	−4.81186	−0.864235	−0.432117	−	0.901817i	$-0.642233\pi$
$31$	−4.81186	−0.864235	−0.432117	+	0.901817i	$0.642233\pi$
$32$	−5.24792	−0.927710
$33$	0	0
$34$	−2.41045	−0.413389
$35$	4.69063	0.792861
$36$	0	0
$37$	7.41620	1.21922	0.609608	−	0.792703i	$-0.291327\pi$
$37$	7.41620	1.21922	0.609608	+	0.792703i	$0.291327\pi$
$38$	0.373996	0.0606701
$39$	0	0
$40$	1.71469	0.271116
$41$	7.32360	1.14375	0.571877	−	0.820339i	$-0.306215\pi$
$41$	7.32360	1.14375	0.571877	+	0.820339i	$0.306215\pi$
$42$	0	0
$43$	9.15677	1.39639	0.698197	−	0.715906i	$-0.253986\pi$
$43$	9.15677	1.39639	0.698197	+	0.715906i	$0.253986\pi$
$44$	−2.21694	−0.334216
$45$	0	0
$46$	−4.84660	−0.714592
$47$	−1.99062	−0.290362	−0.145181	−	0.989405i	$-0.546376\pi$
$47$	−1.99062	−0.290362	−0.145181	+	0.989405i	$0.546376\pi$
$48$	0	0
$49$	15.0020	2.14314
$50$	1.73551	0.245438
$51$	0	0
$52$	2.66466	0.369522
$53$	5.83030	0.800853	0.400427	−	0.916329i	$-0.368862\pi$
$53$	5.83030	0.800853	0.400427	+	0.916329i	$0.368862\pi$
$54$	0	0
$55$	2.19066	0.295388
$56$	8.04298	1.07479
$57$	0	0
$58$	13.3635	1.75471
$59$	−10.7619	−1.40108	−0.700542	−	0.713611i	$-0.747058\pi$
$59$	−10.7619	−1.40108	−0.700542	+	0.713611i	$0.747058\pi$
$60$	0	0
$61$	−0.552233	−0.0707062	−0.0353531	−	0.999375i	$-0.511256\pi$
$61$	−0.552233	−0.0707062	−0.0353531	+	0.999375i	$0.511256\pi$
$62$	−8.35102	−1.06058
$63$	0	0
$64$	0.891893	0.111487
$65$	−2.63307	−0.326593
$66$	0	0
$67$	8.35384	1.02058	0.510292	−	0.860001i	$-0.329537\pi$
$67$	8.35384	1.02058	0.510292	+	0.860001i	$0.329537\pi$
$68$	−1.40556	−0.170449
$69$	0	0
$70$	8.14063	0.972992
$71$	−4.85111	−0.575721	−0.287860	−	0.957672i	$-0.592944\pi$
$71$	−4.85111	−0.575721	−0.287860	+	0.957672i	$0.592944\pi$
$72$	0	0
$73$	5.48768	0.642284	0.321142	−	0.947031i	$-0.395933\pi$
$73$	5.48768	0.642284	0.321142	+	0.947031i	$0.395933\pi$
$74$	12.8709	1.49621
$75$	0	0
$76$	0.218081	0.0250156
$77$	10.2756	1.17101
$78$	0	0
$79$	5.30027	0.596327	0.298164	−	0.954515i	$-0.403626\pi$
$79$	5.30027	0.596327	0.298164	+	0.954515i	$0.403626\pi$
$80$	4.99986	0.559001
$81$	0	0
$82$	12.7102	1.40360
$83$	−8.87948	−0.974649	−0.487325	−	0.873221i	$-0.662027\pi$
$83$	−8.87948	−0.974649	−0.487325	+	0.873221i	$0.662027\pi$
$84$	0	0
$85$	1.38890	0.150647
$86$	15.8917	1.71364
$87$	0	0
$88$	3.75630	0.400423
$89$	3.68952	0.391089	0.195544	−	0.980695i	$-0.437353\pi$
$89$	3.68952	0.391089	0.195544	+	0.980695i	$0.437353\pi$
$90$	0	0
$91$	−12.3508	−1.29471
$92$	−2.82611	−0.294642
$93$	0	0
$94$	−3.45475	−0.356330
$95$	−0.215496	−0.0221094
$96$	0	0
$97$	3.01913	0.306546	0.153273	−	0.988184i	$-0.451019\pi$
$97$	3.01913	0.306546	0.153273	+	0.988184i	$0.451019\pi$
$98$	26.0361	2.63004
$99$	0	0
$100$	1.01200	0.101200
$101$	5.05765	0.503255	0.251628	−	0.967824i	$-0.419034\pi$
$101$	5.05765	0.503255	0.251628	+	0.967824i	$0.419034\pi$
$102$	0	0
$103$	−1.01788	−0.100294	−0.0501471	−	0.998742i	$-0.515969\pi$
$103$	−1.01788	−0.100294	−0.0501471	+	0.998742i	$0.515969\pi$
$104$	−4.51491	−0.442723
$105$	0	0
$106$	10.1185	0.982800
$107$	−1.00000	−0.0966736
$107$	−1.00000	−0.0966736
$108$	0	0
$109$	−3.13292	−0.300079	−0.150040	−	0.988680i	$-0.547940\pi$
$109$	−3.13292	−0.300079	−0.150040	+	0.988680i	$0.547940\pi$
$110$	3.80191	0.362498
$111$	0	0
$112$	23.4525	2.21605
$113$	−18.5160	−1.74184	−0.870920	−	0.491426i	$-0.836476\pi$
$113$	−18.5160	−1.74184	−0.870920	+	0.491426i	$0.836476\pi$
$114$	0	0
$115$	2.79261	0.260412
$116$	7.79239	0.723505
$117$	0	0
$118$	−18.6774	−1.71940
$119$	6.51481	0.597212
$120$	0	0
$121$	−6.20102	−0.563729
$122$	−0.958406	−0.0867700
$123$	0	0
$124$	−4.86958	−0.437301
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−9.53839	−0.846395	−0.423198	−	0.906037i	$-0.639092\pi$
$127$	−9.53839	−0.846395	−0.423198	+	0.906037i	$0.639092\pi$
$128$	12.0437	1.06453
$129$	0	0
$130$	−4.56973	−0.400791
$131$	−15.1827	−1.32652	−0.663258	−	0.748391i	$-0.730827\pi$
$131$	−15.1827	−1.32652	−0.663258	+	0.748391i	$0.730827\pi$
$132$	0	0
$133$	−1.01081	−0.0876485
$134$	14.4982	1.25245
$135$	0	0
$136$	2.38153	0.204215
$137$	13.1819	1.12620	0.563102	−	0.826387i	$-0.309608\pi$
$137$	13.1819	1.12620	0.563102	+	0.826387i	$0.309608\pi$
$138$	0	0
$139$	15.6248	1.32528	0.662641	−	0.748938i	$-0.269436\pi$
$139$	15.6248	1.32528	0.662641	+	0.748938i	$0.269436\pi$
$140$	4.74690	0.401186
$141$	0	0
$142$	−8.41915	−0.706519
$143$	−5.76816	−0.482358
$144$	0	0
$145$	−7.70002	−0.639452
$146$	9.52392	0.788206
$147$	0	0
$148$	7.50517	0.616921
$149$	0.363978	0.0298182	0.0149091	−	0.999889i	$-0.495254\pi$
$149$	0.363978	0.0298182	0.0149091	+	0.999889i	$0.495254\pi$
$150$	0	0
$151$	21.8364	1.77702	0.888509	−	0.458859i	$-0.151742\pi$
$151$	21.8364	1.77702	0.888509	+	0.458859i	$0.151742\pi$
$152$	−0.369509	−0.0299712
$153$	0	0
$154$	17.8333	1.43705
$155$	4.81186	0.386498
$156$	0	0
$157$	16.3597	1.30565	0.652824	−	0.757510i	$-0.273584\pi$
$157$	16.3597	1.30565	0.652824	+	0.757510i	$0.273584\pi$
$158$	9.19868	0.731808
$159$	0	0
$160$	5.24792	0.414885
$161$	13.0991	1.03235
$162$	0	0
$163$	4.38626	0.343558	0.171779	−	0.985136i	$-0.445048\pi$
$163$	4.38626	0.343558	0.171779	+	0.985136i	$0.445048\pi$
$164$	7.41145	0.578737
$165$	0	0
$166$	−15.4104	−1.19608
$167$	−19.5427	−1.51226	−0.756131	−	0.654421i	$-0.772913\pi$
$167$	−19.5427	−1.51226	−0.756131	+	0.654421i	$0.772913\pi$
$168$	0	0
$169$	−6.06693	−0.466687
$170$	2.41045	0.184873
$171$	0	0
$172$	9.26661	0.706573
$173$	14.4078	1.09540	0.547702	−	0.836674i	$-0.315503\pi$
$173$	14.4078	1.09540	0.547702	+	0.836674i	$0.315503\pi$
$174$	0	0
$175$	−4.69063	−0.354578
$176$	10.9530	0.825611
$177$	0	0
$178$	6.40321	0.479941
$179$	−19.7077	−1.47303	−0.736513	−	0.676423i	$-0.763529\pi$
$179$	−19.7077	−1.47303	−0.736513	+	0.676423i	$0.763529\pi$
$180$	0	0
$181$	4.55282	0.338409	0.169204	−	0.985581i	$-0.445880\pi$
$181$	4.55282	0.338409	0.169204	+	0.985581i	$0.445880\pi$
$182$	−21.4349	−1.58886
$183$	0	0
$184$	4.78846	0.353010
$185$	−7.41620	−0.545250
$186$	0	0
$187$	3.04260	0.222497
$188$	−2.01450	−0.146923
$189$	0	0
$190$	−0.373996	−0.0271325
$191$	−8.23672	−0.595988	−0.297994	−	0.954568i	$-0.596318\pi$
$191$	−8.23672	−0.595988	−0.297994	+	0.954568i	$0.596318\pi$
$192$	0	0
$193$	17.3735	1.25057	0.625286	−	0.780396i	$-0.284982\pi$
$193$	17.3735	1.25057	0.625286	+	0.780396i	$0.284982\pi$
$194$	5.23973	0.376190
$195$	0	0
$196$	15.1819	1.08442
$197$	3.81741	0.271979	0.135989	−	0.990710i	$-0.456579\pi$
$197$	3.81741	0.271979	0.135989	+	0.990710i	$0.456579\pi$
$198$	0	0
$199$	−0.987724	−0.0700179	−0.0350090	−	0.999387i	$-0.511146\pi$
$199$	−0.987724	−0.0700179	−0.0350090	+	0.999387i	$0.511146\pi$
$200$	−1.71469	−0.121247
$201$	0	0
$202$	8.77761	0.617590
$203$	−36.1179	−2.53498
$204$	0	0
$205$	−7.32360	−0.511502
$206$	−1.76653	−0.123080
$207$	0	0
$208$	−13.1650	−0.912828
$209$	−0.472078	−0.0326543
$210$	0	0
$211$	−6.14274	−0.422883	−0.211442	−	0.977391i	$-0.567816\pi$
$211$	−6.14274	−0.422883	−0.211442	+	0.977391i	$0.567816\pi$
$212$	5.90024	0.405230
$213$	0	0
$214$	−1.73551	−0.118637
$215$	−9.15677	−0.624486
$216$	0	0
$217$	22.5706	1.53219
$218$	−5.43721	−0.368255
$219$	0	0
$220$	2.21694	0.149466
$221$	−3.65708	−0.246002
$222$	0	0
$223$	24.4314	1.63605	0.818024	−	0.575184i	$-0.195069\pi$
$223$	24.4314	1.63605	0.818024	+	0.575184i	$0.195069\pi$
$224$	24.6160	1.64473
$225$	0	0
$226$	−32.1347	−2.13757
$227$	25.5518	1.69593	0.847965	−	0.530052i	$-0.177827\pi$
$227$	25.5518	1.69593	0.847965	+	0.530052i	$0.177827\pi$
$228$	0	0
$229$	6.28372	0.415240	0.207620	−	0.978210i	$-0.433428\pi$
$229$	6.28372	0.415240	0.207620	+	0.978210i	$0.433428\pi$
$230$	4.84660	0.319575
$231$	0	0
$232$	−13.2031	−0.866829
$233$	−7.63191	−0.499983	−0.249992	−	0.968248i	$-0.580428\pi$
$233$	−7.63191	−0.499983	−0.249992	+	0.968248i	$0.580428\pi$
$234$	0	0
$235$	1.99062	0.129854
$236$	−10.8910	−0.708946
$237$	0	0
$238$	11.3065	0.732893
$239$	18.9375	1.22496	0.612481	−	0.790485i	$-0.290172\pi$
$239$	18.9375	1.22496	0.612481	+	0.790485i	$0.290172\pi$
$240$	0	0
$241$	−6.01729	−0.387608	−0.193804	−	0.981040i	$-0.562083\pi$
$241$	−6.01729	−0.387608	−0.193804	+	0.981040i	$0.562083\pi$
$242$	−10.7619	−0.691804
$243$	0	0
$244$	−0.558857	−0.0357772
$245$	−15.0020	−0.958442
$246$	0	0
$247$	0.567417	0.0361039
$248$	8.25084	0.523929
$249$	0	0
$250$	−1.73551	−0.109763
$251$	23.3493	1.47380	0.736898	−	0.676004i	$-0.236290\pi$
$251$	23.3493	1.47380	0.736898	+	0.676004i	$0.236290\pi$
$252$	0	0
$253$	6.11765	0.384613
$254$	−16.5540	−1.03869
$255$	0	0
$256$	19.1182	1.19489
$257$	−9.80843	−0.611833	−0.305916	−	0.952058i	$-0.598963\pi$
$257$	−9.80843	−0.611833	−0.305916	+	0.952058i	$0.598963\pi$
$258$	0	0
$259$	−34.7866	−2.16154
$260$	−2.66466	−0.165255
$261$	0	0
$262$	−26.3497	−1.62789
$263$	−5.59190	−0.344811	−0.172406	−	0.985026i	$-0.555154\pi$
$263$	−5.59190	−0.344811	−0.172406	+	0.985026i	$0.555154\pi$
$264$	0	0
$265$	−5.83030	−0.358153
$266$	−1.75427	−0.107561
$267$	0	0
$268$	8.45405	0.516413
$269$	0.393247	0.0239767	0.0119884	−	0.999928i	$-0.496184\pi$
$269$	0.393247	0.0239767	0.0119884	+	0.999928i	$0.496184\pi$
$270$	0	0
$271$	−5.34982	−0.324979	−0.162489	−	0.986710i	$-0.551952\pi$
$271$	−5.34982	−0.324979	−0.162489	+	0.986710i	$0.551952\pi$
$272$	6.94430	0.421060
$273$	0	0
$274$	22.8773	1.38207
$275$	−2.19066	−0.132102
$276$	0	0
$277$	2.03649	0.122361	0.0611804	−	0.998127i	$-0.480514\pi$
$277$	2.03649	0.122361	0.0611804	+	0.998127i	$0.480514\pi$
$278$	27.1171	1.62637
$279$	0	0
$280$	−8.04298	−0.480660
$281$	30.9269	1.84494	0.922472	−	0.386063i	$-0.126165\pi$
$281$	30.9269	1.84494	0.922472	+	0.386063i	$0.126165\pi$
$282$	0	0
$283$	11.0298	0.655651	0.327826	−	0.944738i	$-0.393684\pi$
$283$	11.0298	0.655651	0.327826	+	0.944738i	$0.393684\pi$
$284$	−4.90930	−0.291313
$285$	0	0
$286$	−10.0107	−0.591945
$287$	−34.3523	−2.02775
$288$	0	0
$289$	−15.0710	−0.886527
$290$	−13.3635	−0.784729
$291$	0	0
$292$	5.55351	0.324995
$293$	15.0271	0.877893	0.438947	−	0.898513i	$-0.355352\pi$
$293$	15.0271	0.877893	0.438947	+	0.898513i	$0.355352\pi$
$294$	0	0
$295$	10.7619	0.626584
$296$	−12.7165	−0.739131
$297$	0	0
$298$	0.631687	0.0365926
$299$	−7.35314	−0.425243
$300$	0	0
$301$	−42.9510	−2.47565
$302$	37.8972	2.18074
$303$	0	0
$304$	−1.07745	−0.0617960
$305$	0.552233	0.0316208
$306$	0	0
$307$	23.3647	1.33349	0.666746	−	0.745285i	$-0.267687\pi$
$307$	23.3647	1.33349	0.666746	+	0.745285i	$0.267687\pi$
$308$	10.3988	0.592528
$309$	0	0
$310$	8.35102	0.474306
$311$	−11.9330	−0.676656	−0.338328	−	0.941028i	$-0.609861\pi$
$311$	−11.9330	−0.676656	−0.338328	+	0.941028i	$0.609861\pi$
$312$	0	0
$313$	−28.5301	−1.61262	−0.806308	−	0.591495i	$-0.798538\pi$
$313$	−28.5301	−1.61262	−0.806308	+	0.591495i	$0.798538\pi$
$314$	28.3925	1.60228
$315$	0	0
$316$	5.36386	0.301741
$317$	13.5617	0.761699	0.380849	−	0.924637i	$-0.375632\pi$
$317$	13.5617	0.761699	0.380849	+	0.924637i	$0.375632\pi$
$318$	0	0
$319$	−16.8681	−0.944432
$320$	−0.891893	−0.0498583
$321$	0	0
$322$	22.7336	1.26689
$323$	−0.299303	−0.0166536
$324$	0	0
$325$	2.63307	0.146057
$326$	7.61239	0.421611
$327$	0	0
$328$	−12.5577	−0.693383
$329$	9.33727	0.514780
$330$	0	0
$331$	29.4602	1.61928	0.809639	−	0.586928i	$-0.199663\pi$
$331$	29.4602	1.61928	0.809639	+	0.586928i	$0.199663\pi$
$332$	−8.98600	−0.493171
$333$	0	0
$334$	−33.9166	−1.85583
$335$	−8.35384	−0.456419
$336$	0	0
$337$	23.5737	1.28414	0.642072	−	0.766645i	$-0.278075\pi$
$337$	23.5737	1.28414	0.642072	+	0.766645i	$0.278075\pi$
$338$	−10.5292	−0.572714
$339$	0	0
$340$	1.40556	0.0762273
$341$	10.5411	0.570834
$342$	0	0
$343$	−37.5343	−2.02666
$344$	−15.7010	−0.846543
$345$	0	0
$346$	25.0049	1.34427
$347$	34.8687	1.87185	0.935926	−	0.352196i	$-0.114565\pi$
$347$	34.8687	1.87185	0.935926	+	0.352196i	$0.114565\pi$
$348$	0	0
$349$	−8.99851	−0.481679	−0.240840	−	0.970565i	$-0.577423\pi$
$349$	−8.99851	−0.481679	−0.240840	+	0.970565i	$0.577423\pi$
$350$	−8.14063	−0.435135
$351$	0	0
$352$	11.4964	0.612760
$353$	−24.6248	−1.31065	−0.655324	−	0.755348i	$-0.727468\pi$
$353$	−24.6248	−1.31065	−0.655324	+	0.755348i	$0.727468\pi$
$354$	0	0
$355$	4.85111	0.257470
$356$	3.73378	0.197890
$357$	0	0
$358$	−34.2030	−1.80768
$359$	3.55702	0.187733	0.0938663	−	0.995585i	$-0.470077\pi$
$359$	3.55702	0.187733	0.0938663	+	0.995585i	$0.470077\pi$
$360$	0	0
$361$	−18.9536	−0.997556
$362$	7.90147	0.415292
$363$	0	0
$364$	−12.4989	−0.655122
$365$	−5.48768	−0.287238
$366$	0	0
$367$	−2.41794	−0.126215	−0.0631077	−	0.998007i	$-0.520101\pi$
$367$	−2.41794	−0.126215	−0.0631077	+	0.998007i	$0.520101\pi$
$368$	13.9626	0.727853
$369$	0	0
$370$	−12.8709	−0.669126
$371$	−27.3478	−1.41983
$372$	0	0
$373$	−23.6293	−1.22348	−0.611739	−	0.791060i	$-0.709530\pi$
$373$	−23.6293	−1.22348	−0.611739	+	0.791060i	$0.709530\pi$
$374$	5.28047	0.273047
$375$	0	0
$376$	3.41330	0.176028
$377$	20.2747	1.04420
$378$	0	0
$379$	−16.7055	−0.858101	−0.429051	−	0.903280i	$-0.641152\pi$
$379$	−16.7055	−0.858101	−0.429051	+	0.903280i	$0.641152\pi$
$380$	−0.218081	−0.0111873
$381$	0	0
$382$	−14.2949	−0.731392
$383$	3.46608	0.177109	0.0885543	−	0.996071i	$-0.471775\pi$
$383$	3.46608	0.177109	0.0885543	+	0.996071i	$0.471775\pi$
$384$	0	0
$385$	−10.2756	−0.523691
$386$	30.1519	1.53469
$387$	0	0
$388$	3.05534	0.155112
$389$	−14.7372	−0.747206	−0.373603	−	0.927589i	$-0.621878\pi$
$389$	−14.7372	−0.747206	−0.373603	+	0.927589i	$0.621878\pi$
$390$	0	0
$391$	3.87865	0.196152
$392$	−25.7238	−1.29925
$393$	0	0
$394$	6.62515	0.333770
$395$	−5.30027	−0.266686
$396$	0	0
$397$	15.4823	0.777035	0.388518	−	0.921441i	$-0.372987\pi$
$397$	15.4823	0.777035	0.388518	+	0.921441i	$0.372987\pi$
$398$	−1.71421	−0.0859254
$399$	0	0
$400$	−4.99986	−0.249993
$401$	−22.6681	−1.13199	−0.565995	−	0.824408i	$-0.691508\pi$
$401$	−22.6681	−1.13199	−0.565995	+	0.824408i	$0.691508\pi$
$402$	0	0
$403$	−12.6700	−0.631136
$404$	5.11832	0.254646
$405$	0	0
$406$	−62.6830	−3.11091
$407$	−16.2464	−0.805302
$408$	0	0
$409$	−22.5448	−1.11477	−0.557385	−	0.830254i	$-0.688195\pi$
$409$	−22.5448	−1.11477	−0.557385	+	0.830254i	$0.688195\pi$
$410$	−12.7102	−0.627711
$411$	0	0
$412$	−1.03009	−0.0507487
$413$	50.4802	2.48397
$414$	0	0
$415$	8.87948	0.435876
$416$	−13.8182	−0.677491
$417$	0	0
$418$	−0.819296	−0.0400731
$419$	25.0781	1.22515	0.612573	−	0.790414i	$-0.290134\pi$
$419$	25.0781	1.22515	0.612573	+	0.790414i	$0.290134\pi$
$420$	0	0
$421$	−14.4964	−0.706512	−0.353256	−	0.935527i	$-0.614926\pi$
$421$	−14.4964	−0.706512	−0.353256	+	0.935527i	$0.614926\pi$
$422$	−10.6608	−0.518959
$423$	0	0
$424$	−9.99716	−0.485505
$425$	−1.38890	−0.0673716
$426$	0	0
$427$	2.59032	0.125354
$428$	−1.01200	−0.0489167
$429$	0	0
$430$	−15.8917	−0.766364
$431$	−4.27036	−0.205696	−0.102848	−	0.994697i	$-0.532796\pi$
$431$	−4.27036	−0.205696	−0.102848	+	0.994697i	$0.532796\pi$
$432$	0	0
$433$	5.78696	0.278104	0.139052	−	0.990285i	$-0.455595\pi$
$433$	5.78696	0.278104	0.139052	+	0.990285i	$0.455595\pi$
$434$	39.1715	1.88029
$435$	0	0
$436$	−3.17050	−0.151840
$437$	−0.601796	−0.0287878
$438$	0	0
$439$	−33.3602	−1.59220	−0.796098	−	0.605168i	$-0.793106\pi$
$439$	−33.3602	−1.59220	−0.796098	+	0.605168i	$0.793106\pi$
$440$	−3.75630	−0.179075
$441$	0	0
$442$	−6.34689	−0.301891
$443$	19.4605	0.924597	0.462299	−	0.886724i	$-0.347025\pi$
$443$	19.4605	0.924597	0.462299	+	0.886724i	$0.347025\pi$
$444$	0	0
$445$	−3.68952	−0.174900
$446$	42.4010	2.00774
$447$	0	0
$448$	−4.18354	−0.197654
$449$	7.12147	0.336083	0.168041	−	0.985780i	$-0.446256\pi$
$449$	7.12147	0.336083	0.168041	+	0.985780i	$0.446256\pi$
$450$	0	0
$451$	−16.0435	−0.755458
$452$	−18.7381	−0.881367
$453$	0	0
$454$	44.3454	2.08123
$455$	12.3508	0.579013
$456$	0	0
$457$	14.9036	0.697163	0.348581	−	0.937279i	$-0.386663\pi$
$457$	14.9036	0.697163	0.348581	+	0.937279i	$0.386663\pi$
$458$	10.9055	0.509579
$459$	0	0
$460$	2.82611	0.131768
$461$	−0.409261	−0.0190612	−0.00953060	−	0.999955i	$-0.503034\pi$
$461$	−0.409261	−0.0190612	−0.00953060	+	0.999955i	$0.503034\pi$
$462$	0	0
$463$	−16.4368	−0.763883	−0.381942	−	0.924186i	$-0.624744\pi$
$463$	−16.4368	−0.763883	−0.381942	+	0.924186i	$0.624744\pi$
$464$	−38.4990	−1.78727
$465$	0	0
$466$	−13.2453	−0.613575
$467$	32.9064	1.52273	0.761364	−	0.648325i	$-0.224530\pi$
$467$	32.9064	1.52273	0.761364	+	0.648325i	$0.224530\pi$
$468$	0	0
$469$	−39.1847	−1.80938
$470$	3.45475	0.159356
$471$	0	0
$472$	18.4534	0.849386
$473$	−20.0593	−0.922329
$474$	0	0
$475$	0.215496	0.00988764
$476$	6.59297	0.302188
$477$	0	0
$478$	32.8662	1.50326
$479$	−32.5555	−1.48750	−0.743749	−	0.668459i	$-0.766954\pi$
$479$	−32.5555	−1.48750	−0.743749	+	0.668459i	$0.766954\pi$
$480$	0	0
$481$	19.5274	0.890373
$482$	−10.4431	−0.475669
$483$	0	0
$484$	−6.27541	−0.285246
$485$	−3.01913	−0.137091
$486$	0	0
$487$	0.578014	0.0261923	0.0130962	−	0.999914i	$-0.495831\pi$
$487$	0.578014	0.0261923	0.0130962	+	0.999914i	$0.495831\pi$
$488$	0.946908	0.0428645
$489$	0	0
$490$	−26.0361	−1.17619
$491$	14.0182	0.632631	0.316316	−	0.948654i	$-0.397554\pi$
$491$	14.0182	0.632631	0.316316	+	0.948654i	$0.397554\pi$
$492$	0	0
$493$	−10.6946	−0.481658
$494$	0.984758	0.0443064
$495$	0	0
$496$	24.0586	1.08026
$497$	22.7547	1.02069
$498$	0	0
$499$	25.7061	1.15076	0.575382	−	0.817885i	$-0.304854\pi$
$499$	25.7061	1.15076	0.575382	+	0.817885i	$0.304854\pi$
$500$	−1.01200	−0.0452578
$501$	0	0
$502$	40.5230	1.80863
$503$	−10.0865	−0.449733	−0.224866	−	0.974390i	$-0.572195\pi$
$503$	−10.0865	−0.449733	−0.224866	+	0.974390i	$0.572195\pi$
$504$	0	0
$505$	−5.05765	−0.225063
$506$	10.6172	0.471994
$507$	0	0
$508$	−9.65281	−0.428274
$509$	−13.6288	−0.604087	−0.302044	−	0.953294i	$-0.597669\pi$
$509$	−13.6288	−0.604087	−0.302044	+	0.953294i	$0.597669\pi$
$510$	0	0
$511$	−25.7407	−1.13870
$512$	9.09243	0.401832
$513$	0	0
$514$	−17.0226	−0.750836
$515$	1.01788	0.0448529
$516$	0	0
$517$	4.36077	0.191787
$518$	−60.3726	−2.65262
$519$	0	0
$520$	4.51491	0.197992
$521$	−10.0578	−0.440639	−0.220319	−	0.975428i	$-0.570710\pi$
$521$	−10.0578	−0.440639	−0.220319	+	0.975428i	$0.570710\pi$
$522$	0	0
$523$	−30.2988	−1.32487	−0.662437	−	0.749118i	$-0.730478\pi$
$523$	−30.2988	−1.32487	−0.662437	+	0.749118i	$0.730478\pi$
$524$	−15.3648	−0.671214
$525$	0	0
$526$	−9.70480	−0.423149
$527$	6.68319	0.291124
$528$	0	0
$529$	−15.2013	−0.660928
$530$	−10.1185	−0.439522
$531$	0	0
$532$	−1.02294	−0.0443500
$533$	19.2836	0.835264
$534$	0	0
$535$	1.00000	0.0432338
$536$	−14.3242	−0.618713
$537$	0	0
$538$	0.682484	0.0294240
$539$	−32.8642	−1.41556
$540$	0	0
$541$	−18.1896	−0.782030	−0.391015	−	0.920384i	$-0.627876\pi$
$541$	−18.1896	−0.782030	−0.391015	+	0.920384i	$0.627876\pi$
$542$	−9.28467	−0.398811
$543$	0	0
$544$	7.28884	0.312506
$545$	3.13292	0.134200
$546$	0	0
$547$	11.3421	0.484953	0.242476	−	0.970157i	$-0.422040\pi$
$547$	11.3421	0.484953	0.242476	+	0.970157i	$0.422040\pi$
$548$	13.3400	0.569857
$549$	0	0
$550$	−3.80191	−0.162114
$551$	1.65932	0.0706896
$552$	0	0
$553$	−24.8616	−1.05722
$554$	3.53435	0.150160
$555$	0	0
$556$	15.8123	0.670590
$557$	11.1633	0.473002	0.236501	−	0.971631i	$-0.423999\pi$
$557$	11.1633	0.473002	0.236501	+	0.971631i	$0.423999\pi$
$558$	0	0
$559$	24.1104	1.01976
$560$	−23.4525	−0.991047
$561$	0	0
$562$	53.6740	2.26410
$563$	−4.91211	−0.207021	−0.103510	−	0.994628i	$-0.533007\pi$
$563$	−4.91211	−0.207021	−0.103510	+	0.994628i	$0.533007\pi$
$564$	0	0
$565$	18.5160	0.778974
$566$	19.1423	0.804610
$567$	0	0
$568$	8.31815	0.349022
$569$	16.6229	0.696867	0.348433	−	0.937334i	$-0.386714\pi$
$569$	16.6229	0.696867	0.348433	+	0.937334i	$0.386714\pi$
$570$	0	0
$571$	0.524040	0.0219304	0.0109652	−	0.999940i	$-0.496510\pi$
$571$	0.524040	0.0219304	0.0109652	+	0.999940i	$0.496510\pi$
$572$	−5.83736	−0.244072
$573$	0	0
$574$	−59.6187	−2.48844
$575$	−2.79261	−0.116460
$576$	0	0
$577$	30.5562	1.27207	0.636035	−	0.771660i	$-0.280573\pi$
$577$	30.5562	1.27207	0.636035	+	0.771660i	$0.280573\pi$
$578$	−26.1558	−1.08794
$579$	0	0
$580$	−7.79239	−0.323561
$581$	41.6503	1.72795
$582$	0	0
$583$	−12.7722	−0.528970
$584$	−9.40967	−0.389375
$585$	0	0
$586$	26.0797	1.07734
$587$	29.5470	1.21953	0.609767	−	0.792580i	$-0.291263\pi$
$587$	29.5470	1.21953	0.609767	+	0.792580i	$0.291263\pi$
$588$	0	0
$589$	−1.03694	−0.0427262
$590$	18.6774	0.768938
$591$	0	0
$592$	−37.0799	−1.52398
$593$	30.0003	1.23196	0.615982	−	0.787760i	$-0.288759\pi$
$593$	30.0003	1.23196	0.615982	+	0.787760i	$0.288759\pi$
$594$	0	0
$595$	−6.51481	−0.267081
$596$	0.368344	0.0150880
$597$	0	0
$598$	−12.7615	−0.521855
$599$	6.95391	0.284129	0.142065	−	0.989857i	$-0.454626\pi$
$599$	6.95391	0.284129	0.142065	+	0.989857i	$0.454626\pi$
$600$	0	0
$601$	27.3021	1.11368	0.556839	−	0.830621i	$-0.312014\pi$
$601$	27.3021	1.11368	0.556839	+	0.830621i	$0.312014\pi$
$602$	−74.5419	−3.03810
$603$	0	0
$604$	22.0983	0.899168
$605$	6.20102	0.252107
$606$	0	0
$607$	−40.7525	−1.65409	−0.827047	−	0.562132i	$-0.809981\pi$
$607$	−40.7525	−1.65409	−0.827047	+	0.562132i	$0.809981\pi$
$608$	−1.13091	−0.0458643
$609$	0	0
$610$	0.958406	0.0388047
$611$	−5.24146	−0.212047
$612$	0	0
$613$	38.1944	1.54266	0.771328	−	0.636438i	$-0.219593\pi$
$613$	38.1944	1.54266	0.771328	+	0.636438i	$0.219593\pi$
$614$	40.5496	1.63645
$615$	0	0
$616$	−17.6194	−0.709906
$617$	−22.4393	−0.903373	−0.451686	−	0.892177i	$-0.649177\pi$
$617$	−22.4393	−0.903373	−0.451686	+	0.892177i	$0.649177\pi$
$618$	0	0
$619$	39.5413	1.58930	0.794650	−	0.607068i	$-0.207654\pi$
$619$	39.5413	1.58930	0.794650	+	0.607068i	$0.207654\pi$
$620$	4.86958	0.195567
$621$	0	0
$622$	−20.7098	−0.830386
$623$	−17.3062	−0.693358
$624$	0	0
$625$	1.00000	0.0400000
$626$	−49.5143	−1.97899
$627$	0	0
$628$	16.5560	0.660655
$629$	−10.3004	−0.410702
$630$	0	0
$631$	−2.14648	−0.0854498	−0.0427249	−	0.999087i	$-0.513604\pi$
$631$	−2.14648	−0.0854498	−0.0427249	+	0.999087i	$0.513604\pi$
$632$	−9.08833	−0.361514
$633$	0	0
$634$	23.5364	0.934750
$635$	9.53839	0.378519
$636$	0	0
$637$	39.5013	1.56510
$638$	−29.2747	−1.15900
$639$	0	0
$640$	−12.0437	−0.476070
$641$	22.8115	0.901000	0.450500	−	0.892777i	$-0.351246\pi$
$641$	22.8115	0.901000	0.450500	+	0.892777i	$0.351246\pi$
$642$	0	0
$643$	−20.5494	−0.810390	−0.405195	−	0.914230i	$-0.632796\pi$
$643$	−20.5494	−0.810390	−0.405195	+	0.914230i	$0.632796\pi$
$644$	13.2562	0.522368
$645$	0	0
$646$	−0.519443	−0.0204372
$647$	36.3986	1.43098	0.715489	−	0.698624i	$-0.246204\pi$
$647$	36.3986	1.43098	0.715489	+	0.698624i	$0.246204\pi$
$648$	0	0
$649$	23.5757	0.925427
$650$	4.56973	0.179239
$651$	0	0
$652$	4.43887	0.173840
$653$	−20.4785	−0.801387	−0.400694	−	0.916212i	$-0.631231\pi$
$653$	−20.4785	−0.801387	−0.400694	+	0.916212i	$0.631231\pi$
$654$	0	0
$655$	15.1827	0.593236
$656$	−36.6169	−1.42965
$657$	0	0
$658$	16.2049	0.631734
$659$	31.3294	1.22042	0.610209	−	0.792240i	$-0.291085\pi$
$659$	31.3294	1.22042	0.610209	+	0.792240i	$0.291085\pi$
$660$	0	0
$661$	35.4116	1.37735	0.688676	−	0.725069i	$-0.258192\pi$
$661$	35.4116	1.37735	0.688676	+	0.725069i	$0.258192\pi$
$662$	51.1285	1.98716
$663$	0	0
$664$	15.2256	0.590866
$665$	1.01081	0.0391976
$666$	0	0
$667$	−21.5031	−0.832604
$668$	−19.7772	−0.765201
$669$	0	0
$670$	−14.4982	−0.560113
$671$	1.20975	0.0467020
$672$	0	0
$673$	46.9118	1.80832	0.904158	−	0.427198i	$-0.140499\pi$
$673$	46.9118	1.80832	0.904158	+	0.427198i	$0.140499\pi$
$674$	40.9125	1.57589
$675$	0	0
$676$	−6.13970	−0.236142
$677$	24.8631	0.955567	0.477783	−	0.878478i	$-0.341440\pi$
$677$	24.8631	0.955567	0.477783	+	0.878478i	$0.341440\pi$
$678$	0	0
$679$	−14.1616	−0.543472
$680$	−2.38153	−0.0913277
$681$	0	0
$682$	18.2942	0.700522
$683$	7.09417	0.271451	0.135725	−	0.990746i	$-0.456664\pi$
$683$	7.09417	0.271451	0.135725	+	0.990746i	$0.456664\pi$
$684$	0	0
$685$	−13.1819	−0.503654
$686$	−65.1412	−2.48710
$687$	0	0
$688$	−45.7825	−1.74544
$689$	15.3516	0.584850
$690$	0	0
$691$	−2.59409	−0.0986838	−0.0493419	−	0.998782i	$-0.515712\pi$
$691$	−2.59409	−0.0986838	−0.0493419	+	0.998782i	$0.515712\pi$
$692$	14.5806	0.554272
$693$	0	0
$694$	60.5151	2.29712
$695$	−15.6248	−0.592684
$696$	0	0
$697$	−10.1717	−0.385282
$698$	−15.6170	−0.591112
$699$	0	0
$700$	−4.74690	−0.179416
$701$	2.71769	0.102646	0.0513229	−	0.998682i	$-0.483656\pi$
$701$	2.71769	0.102646	0.0513229	+	0.998682i	$0.483656\pi$
$702$	0	0
$703$	1.59816	0.0602758
$704$	−1.95383	−0.0736378
$705$	0	0
$706$	−42.7367	−1.60842
$707$	−23.7236	−0.892216
$708$	0	0
$709$	−4.33561	−0.162827	−0.0814136	−	0.996680i	$-0.525943\pi$
$709$	−4.33561	−0.162827	−0.0814136	+	0.996680i	$0.525943\pi$
$710$	8.41915	0.315965
$711$	0	0
$712$	−6.32639	−0.237092
$713$	13.4376	0.503243
$714$	0	0
$715$	5.76816	0.215717
$716$	−19.9442	−0.745348
$717$	0	0
$718$	6.17325	0.230384
$719$	40.4200	1.50741	0.753705	−	0.657213i	$-0.228265\pi$
$719$	40.4200	1.50741	0.753705	+	0.657213i	$0.228265\pi$
$720$	0	0
$721$	4.77447	0.177811
$722$	−32.8941	−1.22419
$723$	0	0
$724$	4.60744	0.171234
$725$	7.70002	0.285971
$726$	0	0
$727$	12.6228	0.468155	0.234078	−	0.972218i	$-0.424793\pi$
$727$	12.6228	0.468155	0.234078	+	0.972218i	$0.424793\pi$
$728$	21.1777	0.784899
$729$	0	0
$730$	−9.52392	−0.352496
$731$	−12.7178	−0.470386
$732$	0	0
$733$	−37.4973	−1.38500	−0.692498	−	0.721420i	$-0.743490\pi$
$733$	−37.4973	−1.38500	−0.692498	+	0.721420i	$0.743490\pi$
$734$	−4.19636	−0.154891
$735$	0	0
$736$	14.6554	0.540205
$737$	−18.3004	−0.674103
$738$	0	0
$739$	−9.57866	−0.352357	−0.176178	−	0.984358i	$-0.556374\pi$
$739$	−9.57866	−0.352357	−0.176178	+	0.984358i	$0.556374\pi$
$740$	−7.50517	−0.275895
$741$	0	0
$742$	−47.4623	−1.74240
$743$	−11.3735	−0.417252	−0.208626	−	0.977995i	$-0.566899\pi$
$743$	−11.3735	−0.417252	−0.208626	+	0.977995i	$0.566899\pi$
$744$	0	0
$745$	−0.363978	−0.0133351
$746$	−41.0089	−1.50144
$747$	0	0
$748$	3.07910	0.112583
$749$	4.69063	0.171392
$750$	0	0
$751$	−0.721272	−0.0263196	−0.0131598	−	0.999913i	$-0.504189\pi$
$751$	−0.721272	−0.0263196	−0.0131598	+	0.999913i	$0.504189\pi$
$752$	9.95283	0.362942
$753$	0	0
$754$	35.1870	1.28143
$755$	−21.8364	−0.794707
$756$	0	0
$757$	−46.0256	−1.67283	−0.836415	−	0.548097i	$-0.815352\pi$
$757$	−46.0256	−1.67283	−0.836415	+	0.548097i	$0.815352\pi$
$758$	−28.9925	−1.05305
$759$	0	0
$760$	0.369509	0.0134035
$761$	8.05330	0.291932	0.145966	−	0.989290i	$-0.453371\pi$
$761$	8.05330	0.291932	0.145966	+	0.989290i	$0.453371\pi$
$762$	0	0
$763$	14.6954	0.532008
$764$	−8.33553	−0.301569
$765$	0	0
$766$	6.01543	0.217346
$767$	−28.3370	−1.02319
$768$	0	0
$769$	24.8950	0.897738	0.448869	−	0.893598i	$-0.351827\pi$
$769$	24.8950	0.897738	0.448869	+	0.893598i	$0.351827\pi$
$770$	−17.8333	−0.642669
$771$	0	0
$772$	17.5819	0.632787
$773$	−50.9293	−1.83180	−0.915899	−	0.401408i	$-0.868521\pi$
$773$	−50.9293	−1.83180	−0.915899	+	0.401408i	$0.868521\pi$
$774$	0	0
$775$	−4.81186	−0.172847
$776$	−5.17687	−0.185839
$777$	0	0
$778$	−25.5766	−0.916965
$779$	1.57821	0.0565451
$780$	0	0
$781$	10.6271	0.380268
$782$	6.73144	0.240716
$783$	0	0
$784$	−75.0078	−2.67885
$785$	−16.3597	−0.583903
$786$	0	0
$787$	22.7639	0.811446	0.405723	−	0.913996i	$-0.367020\pi$
$787$	22.7639	0.811446	0.405723	+	0.913996i	$0.367020\pi$
$788$	3.86320	0.137621
$789$	0	0
$790$	−9.19868	−0.327274
$791$	86.8517	3.08809
$792$	0	0
$793$	−1.45407	−0.0516355
$794$	26.8697	0.953571
$795$	0	0
$796$	−0.999573	−0.0354289
$797$	53.0229	1.87817	0.939083	−	0.343690i	$-0.111677\pi$
$797$	53.0229	1.87817	0.939083	+	0.343690i	$0.111677\pi$
$798$	0	0
$799$	2.76478	0.0978108
$800$	−5.24792	−0.185542
$801$	0	0
$802$	−39.3407	−1.38917
$803$	−12.0216	−0.424234
$804$	0	0
$805$	−13.0991	−0.461682
$806$	−21.9889	−0.774525
$807$	0	0
$808$	−8.67231	−0.305091
$809$	−24.4401	−0.859267	−0.429634	−	0.903003i	$-0.641357\pi$
$809$	−24.4401	−0.859267	−0.429634	+	0.903003i	$0.641357\pi$
$810$	0	0
$811$	16.9695	0.595879	0.297940	−	0.954585i	$-0.403701\pi$
$811$	16.9695	0.595879	0.297940	+	0.954585i	$0.403701\pi$
$812$	−36.5512	−1.28269
$813$	0	0
$814$	−28.1957	−0.988259
$815$	−4.38626	−0.153644
$816$	0	0
$817$	1.97325	0.0690352
$818$	−39.1267	−1.36803
$819$	0	0
$820$	−7.41145	−0.258819
$821$	50.2029	1.75209	0.876047	−	0.482226i	$-0.160172\pi$
$821$	50.2029	1.75209	0.876047	+	0.482226i	$0.160172\pi$
$822$	0	0
$823$	−14.9669	−0.521714	−0.260857	−	0.965378i	$-0.584005\pi$
$823$	−14.9669	−0.521714	−0.260857	+	0.965378i	$0.584005\pi$
$824$	1.74534	0.0608018
$825$	0	0
$826$	87.6089	3.04831
$827$	20.0549	0.697377	0.348688	−	0.937239i	$-0.386627\pi$
$827$	20.0549	0.697377	0.348688	+	0.937239i	$0.386627\pi$
$828$	0	0
$829$	25.6887	0.892204	0.446102	−	0.894982i	$-0.352812\pi$
$829$	25.6887	0.892204	0.446102	+	0.894982i	$0.352812\pi$
$830$	15.4104	0.534904
$831$	0	0
$832$	2.34842	0.0814168
$833$	−20.8363	−0.721934
$834$	0	0
$835$	19.5427	0.676304
$836$	−0.477741	−0.0165230
$837$	0	0
$838$	43.5233	1.50349
$839$	7.79708	0.269185	0.134593	−	0.990901i	$-0.457027\pi$
$839$	7.79708	0.269185	0.134593	+	0.990901i	$0.457027\pi$
$840$	0	0
$841$	30.2903	1.04449
$842$	−25.1587	−0.867025
$843$	0	0
$844$	−6.21642	−0.213978
$845$	6.06693	0.208709
$846$	0	0
$847$	29.0867	0.999430
$848$	−29.1507	−1.00104
$849$	0	0
$850$	−2.41045	−0.0826778
$851$	−20.7105	−0.709948
$852$	0	0
$853$	17.2109	0.589291	0.294646	−	0.955607i	$-0.404798\pi$
$853$	17.2109	0.589291	0.294646	+	0.955607i	$0.404798\pi$
$854$	4.49552	0.153834
$855$	0	0
$856$	1.71469	0.0586069
$857$	−46.6226	−1.59260	−0.796300	−	0.604903i	$-0.793212\pi$
$857$	−46.6226	−1.59260	−0.796300	+	0.604903i	$0.793212\pi$
$858$	0	0
$859$	−11.7077	−0.399460	−0.199730	−	0.979851i	$-0.564007\pi$
$859$	−11.7077	−0.399460	−0.199730	+	0.979851i	$0.564007\pi$
$860$	−9.26661	−0.315989
$861$	0	0
$862$	−7.41125	−0.252428
$863$	−55.0794	−1.87492	−0.937462	−	0.348086i	$-0.886832\pi$
$863$	−55.0794	−1.87492	−0.937462	+	0.348086i	$0.886832\pi$
$864$	0	0
$865$	−14.4078	−0.489879
$866$	10.0433	0.341286
$867$	0	0
$868$	22.8414	0.775287
$869$	−11.6111	−0.393879
$870$	0	0
$871$	21.9963	0.745315
$872$	5.37199	0.181919
$873$	0	0
$874$	−1.04442	−0.0353281
$875$	4.69063	0.158572
$876$	0	0
$877$	−15.2004	−0.513280	−0.256640	−	0.966507i	$-0.582615\pi$
$877$	−15.2004	−0.513280	−0.256640	+	0.966507i	$0.582615\pi$
$878$	−57.8970	−1.95393
$879$	0	0
$880$	−10.9530	−0.369225
$881$	−55.4683	−1.86878	−0.934388	−	0.356257i	$-0.884053\pi$
$881$	−55.4683	−1.86878	−0.934388	+	0.356257i	$0.884053\pi$
$882$	0	0
$883$	55.6704	1.87346	0.936730	−	0.350054i	$-0.113837\pi$
$883$	55.6704	1.87346	0.936730	+	0.350054i	$0.113837\pi$
$884$	−3.70095	−0.124476
$885$	0	0
$886$	33.7739	1.13466
$887$	−7.96895	−0.267571	−0.133786	−	0.991010i	$-0.542713\pi$
$887$	−7.96895	−0.267571	−0.133786	+	0.991010i	$0.542713\pi$
$888$	0	0
$889$	44.7410	1.50057
$890$	−6.40321	−0.214636
$891$	0	0
$892$	24.7245	0.827837
$893$	−0.428972	−0.0143550
$894$	0	0
$895$	19.7077	0.658757
$896$	−56.4926	−1.88729
$897$	0	0
$898$	12.3594	0.412438
$899$	−37.0514	−1.23573
$900$	0	0
$901$	−8.09771	−0.269774
$902$	−27.8436	−0.927092
$903$	0	0
$904$	31.7492	1.05596
$905$	−4.55282	−0.151341
$906$	0	0
$907$	−24.9363	−0.827997	−0.413999	−	0.910277i	$-0.635868\pi$
$907$	−24.9363	−0.827997	−0.413999	+	0.910277i	$0.635868\pi$
$908$	25.8583	0.858138
$909$	0	0
$910$	21.4349	0.710559
$911$	−8.02446	−0.265862	−0.132931	−	0.991125i	$-0.542439\pi$
$911$	−8.02446	−0.265862	−0.132931	+	0.991125i	$0.542439\pi$
$912$	0	0
$913$	19.4519	0.643764
$914$	25.8654	0.855552
$915$	0	0
$916$	6.35910	0.210111
$917$	71.2162	2.35177
$918$	0	0
$919$	3.47063	0.114485	0.0572427	−	0.998360i	$-0.481769\pi$
$919$	3.47063	0.114485	0.0572427	+	0.998360i	$0.481769\pi$
$920$	−4.78846	−0.157871
$921$	0	0
$922$	−0.710277	−0.0233917
$923$	−12.7733	−0.420439
$924$	0	0
$925$	7.41620	0.243843
$926$	−28.5263	−0.937431
$927$	0	0
$928$	−40.4091	−1.32649
$929$	−40.0950	−1.31547	−0.657737	−	0.753248i	$-0.728486\pi$
$929$	−40.0950	−1.31547	−0.657737	+	0.753248i	$0.728486\pi$
$930$	0	0
$931$	3.23287	0.105953
$932$	−7.72346	−0.252990
$933$	0	0
$934$	57.1095	1.86868
$935$	−3.04260	−0.0995038
$936$	0	0
$937$	53.8498	1.75920	0.879599	−	0.475717i	$-0.157811\pi$
$937$	53.8498	1.75920	0.879599	+	0.475717i	$0.157811\pi$
$938$	−68.0055	−2.22046
$939$	0	0
$940$	2.01450	0.0657058
$941$	−11.8771	−0.387181	−0.193590	−	0.981082i	$-0.562013\pi$
$941$	−11.8771	−0.387181	−0.193590	+	0.981082i	$0.562013\pi$
$942$	0	0
$943$	−20.4519	−0.666007
$944$	53.8081	1.75131
$945$	0	0
$946$	−34.8132	−1.13187
$947$	44.1478	1.43461	0.717306	−	0.696759i	$-0.245375\pi$
$947$	44.1478	1.43461	0.717306	+	0.696759i	$0.245375\pi$
$948$	0	0
$949$	14.4495	0.469049
$950$	0.373996	0.0121340
$951$	0	0
$952$	−11.1709	−0.362051
$953$	20.1072	0.651337	0.325668	−	0.945484i	$-0.394411\pi$
$953$	20.1072	0.651337	0.325668	+	0.945484i	$0.394411\pi$
$954$	0	0
$955$	8.23672	0.266534
$956$	19.1646	0.619829
$957$	0	0
$958$	−56.5004	−1.82544
$959$	−61.8313	−1.99664
$960$	0	0
$961$	−7.84605	−0.253098
$962$	33.8900	1.09266
$963$	0	0
$964$	−6.08947	−0.196129
$965$	−17.3735	−0.559272
$966$	0	0
$967$	−48.0211	−1.54425	−0.772127	−	0.635468i	$-0.780807\pi$
$967$	−48.0211	−1.54425	−0.772127	+	0.635468i	$0.780807\pi$
$968$	10.6328	0.341752
$969$	0	0
$970$	−5.23973	−0.168237
$971$	28.6487	0.919381	0.459690	−	0.888079i	$-0.347960\pi$
$971$	28.6487	0.919381	0.459690	+	0.888079i	$0.347960\pi$
$972$	0	0
$973$	−73.2903	−2.34958
$974$	1.00315	0.0321430
$975$	0	0
$976$	2.76108	0.0883801
$977$	17.7864	0.569039	0.284519	−	0.958670i	$-0.408166\pi$
$977$	17.7864	0.569039	0.284519	+	0.958670i	$0.408166\pi$
$978$	0	0
$979$	−8.08248	−0.258317
$980$	−15.1819	−0.484970
$981$	0	0
$982$	24.3287	0.776360
$983$	−28.4687	−0.908011	−0.454005	−	0.890999i	$-0.650005\pi$
$983$	−28.4687	−0.908011	−0.454005	+	0.890999i	$0.650005\pi$
$984$	0	0
$985$	−3.81741	−0.121633
$986$	−18.5605	−0.591087
$987$	0	0
$988$	0.574224	0.0182685
$989$	−25.5713	−0.813119
$990$	0	0
$991$	22.6326	0.718947	0.359474	−	0.933155i	$-0.382956\pi$
$991$	22.6326	0.718947	0.359474	+	0.933155i	$0.382956\pi$
$992$	25.2522	0.801759
$993$	0	0
$994$	39.4911	1.25258
$995$	0.987724	0.0313130
$996$	0	0
$997$	−18.0862	−0.572795	−0.286398	−	0.958111i	$-0.592458\pi$
$997$	−18.0862	−0.572795	−0.286398	+	0.958111i	$0.592458\pi$
$998$	44.6132	1.41221
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4815.2.a.u.1.10		12
3.2	odd	2		1605.2.a.n.1.3	✓	12
15.14	odd	2		8025.2.a.bf.1.10		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1605.2.a.n.1.3	✓	12	3.2	odd	2
4815.2.a.u.1.10		12	1.1	even	1	trivial
8025.2.a.bf.1.10		12	15.14	odd	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:	\( N \)	\(=\)	\( 4815 = 3^{2} \cdot 5 \cdot 107 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4815.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73551 q^{2} +1.01200 q^{4} -1.00000 q^{5} -4.69063 q^{7} -1.71469 q^{8} +O(q^{10})\)	\(q+1.73551 q^{2} +1.01200 q^{4} -1.00000 q^{5} -4.69063 q^{7} -1.71469 q^{8} -1.73551 q^{10} -2.19066 q^{11} +2.63307 q^{13} -8.14063 q^{14} -4.99986 q^{16} -1.38890 q^{17} +0.215496 q^{19} -1.01200 q^{20} -3.80191 q^{22} -2.79261 q^{23} +1.00000 q^{25} +4.56973 q^{26} -4.74690 q^{28} +7.70002 q^{29} -4.81186 q^{31} -5.24792 q^{32} -2.41045 q^{34} +4.69063 q^{35} +7.41620 q^{37} +0.373996 q^{38} +1.71469 q^{40} +7.32360 q^{41} +9.15677 q^{43} -2.21694 q^{44} -4.84660 q^{46} -1.99062 q^{47} +15.0020 q^{49} +1.73551 q^{50} +2.66466 q^{52} +5.83030 q^{53} +2.19066 q^{55} +8.04298 q^{56} +13.3635 q^{58} -10.7619 q^{59} -0.552233 q^{61} -8.35102 q^{62} +0.891893 q^{64} -2.63307 q^{65} +8.35384 q^{67} -1.40556 q^{68} +8.14063 q^{70} -4.85111 q^{71} +5.48768 q^{73} +12.8709 q^{74} +0.218081 q^{76} +10.2756 q^{77} +5.30027 q^{79} +4.99986 q^{80} +12.7102 q^{82} -8.87948 q^{83} +1.38890 q^{85} +15.8917 q^{86} +3.75630 q^{88} +3.68952 q^{89} -12.3508 q^{91} -2.82611 q^{92} -3.45475 q^{94} -0.215496 q^{95} +3.01913 q^{97} +26.0361 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} + 15 q^{4} - 12 q^{5} + 7 q^{7} - 3 q^{8}+O(q^{10}) \) 12 * q - 3 * q^2 + 15 * q^4 - 12 * q^5 + 7 * q^7 - 3 * q^8	\( 12 q - 3 q^{2} + 15 q^{4} - 12 q^{5} + 7 q^{7} - 3 q^{8} + 3 q^{10} - 4 q^{11} + 13 q^{13} - 4 q^{14} + 13 q^{16} + 4 q^{17} + 14 q^{19} - 15 q^{20} + 15 q^{22} - 11 q^{23} + 12 q^{25} + 8 q^{26} + 16 q^{28} + 7 q^{29} + 4 q^{31} - 4 q^{32} + q^{34} - 7 q^{35} + 24 q^{37} + 11 q^{38} + 3 q^{40} - 13 q^{41} + 25 q^{43} - 10 q^{44} - 22 q^{46} - 19 q^{47} + 9 q^{49} - 3 q^{50} + 20 q^{52} - 11 q^{53} + 4 q^{55} + 37 q^{56} - 2 q^{58} - 8 q^{59} + 7 q^{61} + 11 q^{62} - 19 q^{64} - 13 q^{65} + 33 q^{67} + 24 q^{68} + 4 q^{70} + 34 q^{73} + 27 q^{74} - 9 q^{76} + 29 q^{77} - 13 q^{80} + q^{82} + 24 q^{83} - 4 q^{85} + 36 q^{86} - 6 q^{88} + 10 q^{89} + 30 q^{91} + 28 q^{92} - 8 q^{94} - 14 q^{95} + 16 q^{97} + 36 q^{98}+O(q^{100}) \) 12 * q - 3 * q^2 + 15 * q^4 - 12 * q^5 + 7 * q^7 - 3 * q^8 + 3 * q^10 - 4 * q^11 + 13 * q^13 - 4 * q^14 + 13 * q^16 + 4 * q^17 + 14 * q^19 - 15 * q^20 + 15 * q^22 - 11 * q^23 + 12 * q^25 + 8 * q^26 + 16 * q^28 + 7 * q^29 + 4 * q^31 - 4 * q^32 + q^34 - 7 * q^35 + 24 * q^37 + 11 * q^38 + 3 * q^40 - 13 * q^41 + 25 * q^43 - 10 * q^44 - 22 * q^46 - 19 * q^47 + 9 * q^49 - 3 * q^50 + 20 * q^52 - 11 * q^53 + 4 * q^55 + 37 * q^56 - 2 * q^58 - 8 * q^59 + 7 * q^61 + 11 * q^62 - 19 * q^64 - 13 * q^65 + 33 * q^67 + 24 * q^68 + 4 * q^70 + 34 * q^73 + 27 * q^74 - 9 * q^76 + 29 * q^77 - 13 * q^80 + q^82 + 24 * q^83 - 4 * q^85 + 36 * q^86 - 6 * q^88 + 10 * q^89 + 30 * q^91 + 28 * q^92 - 8 * q^94 - 14 * q^95 + 16 * q^97 + 36 * q^98

Introduction

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