LMFDB - Embedded newform 3872.2.c.i.1937.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3872,2,Mod(1937,3872)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3872, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3872.1937");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3872 = 2^{5} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3872.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$30.9180756626$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{18} - 2 x^{16} - 2 x^{15} - 4 x^{14} - 4 x^{13} + 12 x^{12} + 16 x^{11} + 32 x^{9} + \cdots + 1024 $ x^20 - x^18 - 2x^16 - 2x^15 - 4x^14 - 4x^13 + 12x^12 + 16x^11 + 32x^9 + 48x^8 - 32x^7 - 64x^6 - 64x^5 - 128x^4 - 256*x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 88)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1937.17
Root			$0.651763 + 1.25507i$ of defining polynomial
Character	$\chi$	$=$	3872.1937
Dual form			3872.2.c.i.1937.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.44793i q^{3} +0.200474i q^{5} -2.34615 q^{7} -2.99235 q^{9} -1.63159i q^{13} -0.490746 q^{15} +3.33639 q^{17} +4.92320i q^{19} -5.74321i q^{21} +9.25109 q^{23} +4.95981 q^{25} +0.0187332i q^{27} -6.19167i q^{29} +8.25919 q^{31} -0.470343i q^{35} -1.28006i q^{37} +3.99402 q^{39} +6.03055 q^{41} +2.47184i q^{43} -0.599888i q^{45} -4.75897 q^{47} -1.49556 q^{49} +8.16724i q^{51} +8.76165i q^{53} -12.0516 q^{57} +1.90497i q^{59} -6.39500i q^{61} +7.02051 q^{63} +0.327092 q^{65} +8.27349i q^{67} +22.6460i q^{69} +1.24836 q^{71} -6.89517 q^{73} +12.1413i q^{75} -14.6219 q^{79} -9.02290 q^{81} -9.37255i q^{83} +0.668860i q^{85} +15.1568 q^{87} +5.11129 q^{89} +3.82797i q^{91} +20.2179i q^{93} -0.986974 q^{95} +0.299741 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 10 q^{7} - 10 q^{9} + 4 q^{15} - 2 q^{17} + 4 q^{23} - 2 q^{25} - 2 q^{31} + 28 q^{39} + 2 q^{41} - 2 q^{47} - 2 q^{49} - 22 q^{57} - 30 q^{63} + 18 q^{65} + 34 q^{71} - 2 q^{73} - 58 q^{79} - 12 q^{81}+ \cdots + 10 q^{97}+O(q^{100}) $ 20 * q + 10 * q^7 - 10 * q^9 + 4 * q^15 - 2 * q^17 + 4 * q^23 - 2 * q^25 - 2 * q^31 + 28 * q^39 + 2 * q^41 - 2 * q^47 - 2 * q^49 - 22 * q^57 - 30 * q^63 + 18 * q^65 + 34 * q^71 - 2 * q^73 - 58 * q^79 - 12 * q^81 + 34 * q^87 - 8 * q^89 - 92 * q^95 + 10 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3872\mathbb{Z}\right)^\times$.

$n$	$485$	$1695$	$2785$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.44793i	1.41331i	0.707558	+	0.706656i	$0.249797\pi$
$3$	2.44793i	1.41331i	−0.707558	+	0.706656i	$0.750203\pi$
$4$	0	0
$5$	0.200474i	0.0896548i	0.998995	+	0.0448274i	$0.0142738\pi$
$5$	0.200474i	0.0896548i	−0.998995	+	0.0448274i	$0.985726\pi$
$6$	0	0
$7$	−2.34615	−0.886763	−0.443381	−	0.896333i	$-0.646221\pi$
$7$	−2.34615	−0.886763	−0.443381	+	0.896333i	$0.646221\pi$
$8$	0	0
$9$	−2.99235	−0.997449
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	− 1.63159i	− 0.452522i	−0.974067	−	0.226261i	$-0.927350\pi$
$13$	− 1.63159i	− 0.452522i	0.974067	−	0.226261i	$-0.0726503\pi$
$14$	0	0
$15$	−0.490746	−0.126710
$16$	0	0
$17$	3.33639	0.809193	0.404597	−	0.914495i	$-0.367412\pi$
$17$	3.33639	0.809193	0.404597	+	0.914495i	$0.367412\pi$
$18$	0	0
$19$	4.92320i	1.12946i	0.825276	+	0.564729i	$0.191019\pi$
$19$	4.92320i	1.12946i	−0.825276	+	0.564729i	$0.808981\pi$
$20$	0	0
$21$	− 5.74321i	− 1.25327i
$22$	0	0
$23$	9.25109	1.92899	0.964493	−	0.264108i	$-0.0850776\pi$
$23$	9.25109	1.92899	0.964493	+	0.264108i	$0.0850776\pi$
$24$	0	0
$25$	4.95981	0.991962
$26$	0	0
$27$	0.0187332i	0.00360520i
$28$	0	0
$29$	− 6.19167i	− 1.14977i	−0.818236	−	0.574883i	$-0.805048\pi$
$29$	− 6.19167i	− 1.14977i	0.818236	−	0.574883i	$-0.194952\pi$
$30$	0	0
$31$	8.25919	1.48339	0.741697	−	0.670735i	$-0.234021\pi$
$31$	8.25919	1.48339	0.741697	+	0.670735i	$0.234021\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 0.470343i	− 0.0795025i
$36$	0	0
$37$	− 1.28006i	− 0.210440i	−0.994449	−	0.105220i	$-0.966445\pi$
$37$	− 1.28006i	− 0.210440i	0.994449	−	0.105220i	$-0.0335546\pi$
$38$	0	0
$39$	3.99402	0.639555
$40$	0	0
$41$	6.03055	0.941814	0.470907	−	0.882183i	$-0.343927\pi$
$41$	6.03055	0.941814	0.470907	+	0.882183i	$0.343927\pi$
$42$	0	0
$43$	2.47184i	0.376952i	0.982078	+	0.188476i	$0.0603549\pi$
$43$	2.47184i	0.376952i	−0.982078	+	0.188476i	$0.939645\pi$
$44$	0	0
$45$	− 0.599888i	− 0.0894261i
$46$	0	0
$47$	−4.75897	−0.694167	−0.347084	−	0.937834i	$-0.612828\pi$
$47$	−4.75897	−0.694167	−0.347084	+	0.937834i	$0.612828\pi$
$48$	0	0
$49$	−1.49556	−0.213652
$50$	0	0
$51$	8.16724i	1.14364i
$52$	0	0
$53$	8.76165i	1.20351i	0.798682	+	0.601753i	$0.205531\pi$
$53$	8.76165i	1.20351i	−0.798682	+	0.601753i	$0.794469\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−12.0516	−1.59628
$58$	0	0
$59$	1.90497i	0.248006i	0.992282	+	0.124003i	$0.0395732\pi$
$59$	1.90497i	0.248006i	−0.992282	+	0.124003i	$0.960427\pi$
$60$	0	0
$61$	− 6.39500i	− 0.818795i	−0.912356	−	0.409398i	$-0.865739\pi$
$61$	− 6.39500i	− 0.818795i	0.912356	−	0.409398i	$-0.134261\pi$
$62$	0	0
$63$	7.02051	0.884501
$64$	0	0
$65$	0.327092	0.0405708
$66$	0	0
$67$	8.27349i	1.01077i	0.862895	+	0.505384i	$0.168649\pi$
$67$	8.27349i	1.01077i	−0.862895	+	0.505384i	$0.831351\pi$
$68$	0	0
$69$	22.6460i	2.72626i
$70$	0	0
$71$	1.24836	0.148153	0.0740763	−	0.997253i	$-0.476399\pi$
$71$	1.24836	0.148153	0.0740763	+	0.997253i	$0.476399\pi$
$72$	0	0
$73$	−6.89517	−0.807019	−0.403510	−	0.914975i	$-0.632210\pi$
$73$	−6.89517	−0.807019	−0.403510	+	0.914975i	$0.632210\pi$
$74$	0	0
$75$	12.1413i	1.40195i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−14.6219	−1.64509	−0.822544	−	0.568702i	$-0.807446\pi$
$79$	−14.6219	−1.64509	−0.822544	+	0.568702i	$0.807446\pi$
$80$	0	0
$81$	−9.02290	−1.00254
$82$	0	0
$83$	− 9.37255i	− 1.02877i	−0.857559	−	0.514386i	$-0.828020\pi$
$83$	− 9.37255i	− 1.02877i	0.857559	−	0.514386i	$-0.171980\pi$
$84$	0	0
$85$	0.668860i	0.0725480i
$86$	0	0
$87$	15.1568	1.62498
$88$	0	0
$89$	5.11129	0.541795	0.270898	−	0.962608i	$-0.412680\pi$
$89$	5.11129	0.541795	0.270898	+	0.962608i	$0.412680\pi$
$90$	0	0
$91$	3.82797i	0.401280i
$92$	0	0
$93$	20.2179i	2.09650i
$94$	0	0
$95$	−0.986974	−0.101261
$96$	0	0
$97$	0.299741	0.0304341	0.0152170	−	0.999884i	$-0.495156\pi$
$97$	0.299741	0.0304341	0.0152170	+	0.999884i	$0.495156\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	15.0198i	1.49453i	0.664527	+	0.747265i	$0.268633\pi$
$101$	15.0198i	1.49453i	−0.664527	+	0.747265i	$0.731367\pi$
$102$	0	0
$103$	−2.82480	−0.278336	−0.139168	−	0.990269i	$-0.544443\pi$
$103$	−2.82480	−0.278336	−0.139168	+	0.990269i	$0.544443\pi$
$104$	0	0
$105$	1.15137	0.112362
$106$	0	0
$107$	0.893354i	0.0863638i	0.999067	+	0.0431819i	$0.0137495\pi$
$107$	0.893354i	0.0863638i	−0.999067	+	0.0431819i	$0.986250\pi$
$108$	0	0
$109$	1.35318i	0.129611i	0.997898	+	0.0648057i	$0.0206428\pi$
$109$	1.35318i	0.129611i	−0.997898	+	0.0648057i	$0.979357\pi$
$110$	0	0
$111$	3.13348	0.297417
$112$	0	0
$113$	−3.76875	−0.354534	−0.177267	−	0.984163i	$-0.556726\pi$
$113$	−3.76875	−0.354534	−0.177267	+	0.984163i	$0.556726\pi$
$114$	0	0
$115$	1.85460i	0.172943i
$116$	0	0
$117$	4.88229i	0.451368i
$118$	0	0
$119$	−7.82768	−0.717562
$120$	0	0
$121$	0	0
$122$	0	0
$123$	14.7624i	1.33108i
$124$	0	0
$125$	1.99668i	0.178589i
$126$	0	0
$127$	7.35273	0.652450	0.326225	−	0.945292i	$-0.394223\pi$
$127$	7.35273	0.652450	0.326225	+	0.945292i	$0.394223\pi$
$128$	0	0
$129$	−6.05089	−0.532751
$130$	0	0
$131$	− 6.85311i	− 0.598759i	−0.954134	−	0.299380i	$-0.903220\pi$
$131$	− 6.85311i	− 0.598759i	0.954134	−	0.299380i	$-0.0967797\pi$
$132$	0	0
$133$	− 11.5506i	− 1.00156i
$134$	0	0
$135$	−0.00375551	−0.000323223 0
$136$	0	0
$137$	2.80303	0.239479	0.119740	−	0.992805i	$-0.461794\pi$
$137$	2.80303	0.239479	0.119740	+	0.992805i	$0.461794\pi$
$138$	0	0
$139$	8.90509i	0.755320i	0.925944	+	0.377660i	$0.123271\pi$
$139$	8.90509i	0.755320i	−0.925944	+	0.377660i	$0.876729\pi$
$140$	0	0
$141$	− 11.6496i	− 0.981074i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	1.24127	0.103082
$146$	0	0
$147$	− 3.66103i	− 0.301957i
$148$	0	0
$149$	12.0459i	0.986837i	0.869792	+	0.493418i	$0.164253\pi$
$149$	12.0459i	0.986837i	−0.869792	+	0.493418i	$0.835747\pi$
$150$	0	0
$151$	5.29676	0.431044	0.215522	−	0.976499i	$-0.430855\pi$
$151$	5.29676	0.431044	0.215522	+	0.976499i	$0.430855\pi$
$152$	0	0
$153$	−9.98363	−0.807129
$154$	0	0
$155$	1.65575i	0.132993i
$156$	0	0
$157$	3.95049i	0.315284i	0.987496	+	0.157642i	$0.0503892\pi$
$157$	3.95049i	0.315284i	−0.987496	+	0.157642i	$0.949611\pi$
$158$	0	0
$159$	−21.4479	−1.70093
$160$	0	0
$161$	−21.7045	−1.71055
$162$	0	0
$163$	16.0588i	1.25782i	0.777477	+	0.628911i	$0.216499\pi$
$163$	16.0588i	1.25782i	−0.777477	+	0.628911i	$0.783501\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.27935	0.718057	0.359029	−	0.933327i	$-0.383108\pi$
$167$	9.27935	0.718057	0.359029	+	0.933327i	$0.383108\pi$
$168$	0	0
$169$	10.3379	0.795224
$170$	0	0
$171$	− 14.7319i	− 1.12658i
$172$	0	0
$173$	− 8.84221i	− 0.672261i	−0.941815	−	0.336130i	$-0.890882\pi$
$173$	− 8.84221i	− 0.672261i	0.941815	−	0.336130i	$-0.109118\pi$
$174$	0	0
$175$	−11.6365	−0.879635
$176$	0	0
$177$	−4.66323	−0.350510
$178$	0	0
$179$	10.6539i	0.796309i	0.917318	+	0.398154i	$0.130349\pi$
$179$	10.6539i	0.796309i	−0.917318	+	0.398154i	$0.869651\pi$
$180$	0	0
$181$	− 13.4165i	− 0.997238i	−0.866821	−	0.498619i	$-0.833841\pi$
$181$	− 13.4165i	− 0.997238i	0.866821	−	0.498619i	$-0.166159\pi$
$182$	0	0
$183$	15.6545	1.15721
$184$	0	0
$185$	0.256618	0.0188669
$186$	0	0
$187$	0	0
$188$	0	0
$189$	− 0.0439509i	− 0.00319696i
$190$	0	0
$191$	14.8189	1.07226	0.536129	−	0.844136i	$-0.319886\pi$
$191$	14.8189	1.07226	0.536129	+	0.844136i	$0.319886\pi$
$192$	0	0
$193$	4.47394	0.322041	0.161021	−	0.986951i	$-0.448521\pi$
$193$	4.47394	0.322041	0.161021	+	0.986951i	$0.448521\pi$
$194$	0	0
$195$	0.800698i	0.0573391i
$196$	0	0
$197$	− 20.3350i	− 1.44881i	−0.689373	−	0.724406i	$-0.742114\pi$
$197$	− 20.3350i	− 1.44881i	0.689373	−	0.724406i	$-0.257886\pi$
$198$	0	0
$199$	−10.4142	−0.738245	−0.369122	−	0.929381i	$-0.620342\pi$
$199$	−10.4142	−0.738245	−0.369122	+	0.929381i	$0.620342\pi$
$200$	0	0
$201$	−20.2529	−1.42853
$202$	0	0
$203$	14.5266i	1.01957i
$204$	0	0
$205$	1.20897i	0.0844381i
$206$	0	0
$207$	−27.6825	−1.92407
$208$	0	0
$209$	0	0
$210$	0	0
$211$	18.7278i	1.28927i	0.764489	+	0.644637i	$0.222992\pi$
$211$	18.7278i	1.28927i	−0.764489	+	0.644637i	$0.777008\pi$
$212$	0	0
$213$	3.05588i	0.209386i
$214$	0	0
$215$	−0.495541	−0.0337956
$216$	0	0
$217$	−19.3773	−1.31542
$218$	0	0
$219$	− 16.8789i	− 1.14057i
$220$	0	0
$221$	− 5.44363i	− 0.366178i
$222$	0	0
$223$	−14.5773	−0.976172	−0.488086	−	0.872796i	$-0.662305\pi$
$223$	−14.5773	−0.976172	−0.488086	+	0.872796i	$0.662305\pi$
$224$	0	0
$225$	−14.8415	−0.989432
$226$	0	0
$227$	− 4.18195i	− 0.277566i	−0.990323	−	0.138783i	$-0.955681\pi$
$227$	− 4.18195i	− 0.277566i	0.990323	−	0.138783i	$-0.0443190\pi$
$228$	0	0
$229$	23.2412i	1.53582i	0.640557	+	0.767911i	$0.278704\pi$
$229$	23.2412i	1.53582i	−0.640557	+	0.767911i	$0.721296\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	13.9825	0.916021	0.458011	−	0.888947i	$-0.348562\pi$
$233$	13.9825	0.916021	0.458011	+	0.888947i	$0.348562\pi$
$234$	0	0
$235$	− 0.954051i	− 0.0622354i
$236$	0	0
$237$	− 35.7932i	− 2.32502i
$238$	0	0
$239$	−23.9363	−1.54831	−0.774155	−	0.632996i	$-0.781825\pi$
$239$	−23.9363	−1.54831	−0.774155	+	0.632996i	$0.781825\pi$
$240$	0	0
$241$	26.1842	1.68667	0.843336	−	0.537387i	$-0.180588\pi$
$241$	26.1842	1.68667	0.843336	+	0.537387i	$0.180588\pi$
$242$	0	0
$243$	− 22.0312i	− 1.41330i
$244$	0	0
$245$	− 0.299822i	− 0.0191549i
$246$	0	0
$247$	8.03265	0.511105
$248$	0	0
$249$	22.9433	1.45397
$250$	0	0
$251$	− 19.2762i	− 1.21671i	−0.793667	−	0.608353i	$-0.791831\pi$
$251$	− 19.2762i	− 1.21671i	0.793667	−	0.608353i	$-0.208169\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−1.63732	−0.102533
$256$	0	0
$257$	7.36823	0.459618	0.229809	−	0.973236i	$-0.426190\pi$
$257$	7.36823	0.459618	0.229809	+	0.973236i	$0.426190\pi$
$258$	0	0
$259$	3.00321i	0.186610i
$260$	0	0
$261$	18.5276i	1.14683i
$262$	0	0
$263$	2.69244	0.166023	0.0830114	−	0.996549i	$-0.473546\pi$
$263$	2.69244	0.166023	0.0830114	+	0.996549i	$0.473546\pi$
$264$	0	0
$265$	−1.75649	−0.107900
$266$	0	0
$267$	12.5121i	0.765725i
$268$	0	0
$269$	15.5571i	0.948532i	0.880382	+	0.474266i	$0.157286\pi$
$269$	15.5571i	0.948532i	−0.880382	+	0.474266i	$0.842714\pi$
$270$	0	0
$271$	−0.934048	−0.0567393	−0.0283697	−	0.999597i	$-0.509032\pi$
$271$	−0.934048	−0.0567393	−0.0283697	+	0.999597i	$0.509032\pi$
$272$	0	0
$273$	−9.37058	−0.567133
$274$	0	0
$275$	0	0
$276$	0	0
$277$	13.5603i	0.814760i	0.913259	+	0.407380i	$0.133558\pi$
$277$	13.5603i	0.814760i	−0.913259	+	0.407380i	$0.866442\pi$
$278$	0	0
$279$	−24.7144	−1.47961
$280$	0	0
$281$	15.5472	0.927470	0.463735	−	0.885974i	$-0.346509\pi$
$281$	15.5472	0.927470	0.463735	+	0.885974i	$0.346509\pi$
$282$	0	0
$283$	− 2.34360i	− 0.139313i	−0.997571	−	0.0696564i	$-0.977810\pi$
$283$	− 2.34360i	− 0.139313i	0.997571	−	0.0696564i	$-0.0221903\pi$
$284$	0	0
$285$	− 2.41604i	− 0.143114i
$286$	0	0
$287$	−14.1486	−0.835166
$288$	0	0
$289$	−5.86851	−0.345207
$290$	0	0
$291$	0.733744i	0.0430129i
$292$	0	0
$293$	18.3947i	1.07463i	0.843381	+	0.537316i	$0.180562\pi$
$293$	18.3947i	1.07463i	−0.843381	+	0.537316i	$0.819438\pi$
$294$	0	0
$295$	−0.381897	−0.0222349
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 15.0940i	− 0.872909i
$300$	0	0
$301$	− 5.79932i	− 0.334267i
$302$	0	0
$303$	−36.7675	−2.11224
$304$	0	0
$305$	1.28203	0.0734089
$306$	0	0
$307$	20.3704i	1.16260i	0.813690	+	0.581299i	$0.197455\pi$
$307$	20.3704i	1.16260i	−0.813690	+	0.581299i	$0.802545\pi$
$308$	0	0
$309$	− 6.91490i	− 0.393375i
$310$	0	0
$311$	−21.7086	−1.23098	−0.615492	−	0.788143i	$-0.711043\pi$
$311$	−21.7086	−1.23098	−0.615492	+	0.788143i	$0.711043\pi$
$312$	0	0
$313$	26.8274	1.51637	0.758186	−	0.652038i	$-0.226086\pi$
$313$	26.8274	1.51637	0.758186	+	0.652038i	$0.226086\pi$
$314$	0	0
$315$	1.40743i	0.0792997i
$316$	0	0
$317$	3.81008i	0.213995i	0.994259	+	0.106998i	$0.0341237\pi$
$317$	3.81008i	0.213995i	−0.994259	+	0.106998i	$0.965876\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−2.18687	−0.122059
$322$	0	0
$323$	16.4257i	0.913950i
$324$	0	0
$325$	− 8.09239i	− 0.448885i
$326$	0	0
$327$	−3.31249	−0.183181
$328$	0	0
$329$	11.1653	0.615562
$330$	0	0
$331$	− 28.2300i	− 1.55166i	−0.630941	−	0.775831i	$-0.717331\pi$
$331$	− 28.2300i	− 1.55166i	0.630941	−	0.775831i	$-0.282669\pi$
$332$	0	0
$333$	3.83037i	0.209903i
$334$	0	0
$335$	−1.65862	−0.0906201
$336$	0	0
$337$	20.5487	1.11936	0.559678	−	0.828710i	$-0.310925\pi$
$337$	20.5487	1.11936	0.559678	+	0.828710i	$0.310925\pi$
$338$	0	0
$339$	− 9.22563i	− 0.501067i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	19.9319	1.07622
$344$	0	0
$345$	−4.53994	−0.244422
$346$	0	0
$347$	3.76822i	0.202289i	0.994872	+	0.101144i	$0.0322504\pi$
$347$	3.76822i	0.202289i	−0.994872	+	0.101144i	$0.967750\pi$
$348$	0	0
$349$	− 31.4661i	− 1.68434i	−0.539211	−	0.842170i	$-0.681278\pi$
$349$	− 31.4661i	− 1.68434i	0.539211	−	0.842170i	$-0.318722\pi$
$350$	0	0
$351$	0.0305649	0.00163143
$352$	0	0
$353$	−30.7625	−1.63732	−0.818662	−	0.574276i	$-0.805284\pi$
$353$	−30.7625	−1.63732	−0.818662	+	0.574276i	$0.805284\pi$
$354$	0	0
$355$	0.250263i	0.0132826i
$356$	0	0
$357$	− 19.1616i	− 1.01414i
$358$	0	0
$359$	−12.4538	−0.657288	−0.328644	−	0.944454i	$-0.606592\pi$
$359$	−12.4538	−0.657288	−0.328644	+	0.944454i	$0.606592\pi$
$360$	0	0
$361$	−5.23787	−0.275677
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 1.38230i	− 0.0723531i
$366$	0	0
$367$	4.64806	0.242627	0.121313	−	0.992614i	$-0.461289\pi$
$367$	4.64806	0.242627	0.121313	+	0.992614i	$0.461289\pi$
$368$	0	0
$369$	−18.0455	−0.939412
$370$	0	0
$371$	− 20.5562i	− 1.06722i
$372$	0	0
$373$	18.0242i	0.933256i	0.884454	+	0.466628i	$0.154531\pi$
$373$	18.0242i	0.933256i	−0.884454	+	0.466628i	$0.845469\pi$
$374$	0	0
$375$	−4.88774	−0.252402
$376$	0	0
$377$	−10.1023	−0.520294
$378$	0	0
$379$	− 13.9070i	− 0.714355i	−0.934037	−	0.357177i	$-0.883739\pi$
$379$	− 13.9070i	− 0.714355i	0.934037	−	0.357177i	$-0.116261\pi$
$380$	0	0
$381$	17.9990i	0.922114i
$382$	0	0
$383$	22.5740	1.15348	0.576739	−	0.816928i	$-0.304325\pi$
$383$	22.5740	1.15348	0.576739	+	0.816928i	$0.304325\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 7.39661i	− 0.375991i
$388$	0	0
$389$	23.4650i	1.18972i	0.803829	+	0.594861i	$0.202793\pi$
$389$	23.4650i	1.18972i	−0.803829	+	0.594861i	$0.797207\pi$
$390$	0	0
$391$	30.8652	1.56092
$392$	0	0
$393$	16.7759	0.846233
$394$	0	0
$395$	− 2.93130i	− 0.147490i
$396$	0	0
$397$	29.9901i	1.50516i	0.658500	+	0.752581i	$0.271191\pi$
$397$	29.9901i	1.50516i	−0.658500	+	0.752581i	$0.728809\pi$
$398$	0	0
$399$	28.2750	1.41552
$400$	0	0
$401$	−20.2298	−1.01023	−0.505114	−	0.863052i	$-0.668550\pi$
$401$	−20.2298	−1.01023	−0.505114	+	0.863052i	$0.668550\pi$
$402$	0	0
$403$	− 13.4756i	− 0.671269i
$404$	0	0
$405$	− 1.80886i	− 0.0898829i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−4.43720	−0.219405	−0.109703	−	0.993964i	$-0.534990\pi$
$409$	−4.43720	−0.219405	−0.109703	+	0.993964i	$0.534990\pi$
$410$	0	0
$411$	6.86162i	0.338459i
$412$	0	0
$413$	− 4.46935i	− 0.219922i
$414$	0	0
$415$	1.87895	0.0922342
$416$	0	0
$417$	−21.7990	−1.06750
$418$	0	0
$419$	6.01641i	0.293921i	0.989142	+	0.146960i	$0.0469490\pi$
$419$	6.01641i	0.293921i	−0.989142	+	0.146960i	$0.953051\pi$
$420$	0	0
$421$	− 27.4541i	− 1.33803i	−0.743248	−	0.669016i	$-0.766716\pi$
$421$	− 27.4541i	− 1.33803i	0.743248	−	0.669016i	$-0.233284\pi$
$422$	0	0
$423$	14.2405	0.692397
$424$	0	0
$425$	16.5479	0.802689
$426$	0	0
$427$	15.0036i	0.726077i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−24.9012	−1.19945	−0.599723	−	0.800207i	$-0.704723\pi$
$431$	−24.9012	−1.19945	−0.599723	+	0.800207i	$0.704723\pi$
$432$	0	0
$433$	13.5442	0.650892	0.325446	−	0.945561i	$-0.394486\pi$
$433$	13.5442	0.650892	0.325446	+	0.945561i	$0.394486\pi$
$434$	0	0
$435$	3.03854i	0.145687i
$436$	0	0
$437$	45.5449i	2.17871i
$438$	0	0
$439$	−30.1967	−1.44121	−0.720606	−	0.693345i	$-0.756136\pi$
$439$	−30.1967	−1.44121	−0.720606	+	0.693345i	$0.756136\pi$
$440$	0	0
$441$	4.47525	0.213107
$442$	0	0
$443$	13.9387i	0.662247i	0.943587	+	0.331124i	$0.107428\pi$
$443$	13.9387i	0.662247i	−0.943587	+	0.331124i	$0.892572\pi$
$444$	0	0
$445$	1.02468i	0.0485745i
$446$	0	0
$447$	−29.4874	−1.39471
$448$	0	0
$449$	7.05742	0.333060	0.166530	−	0.986036i	$-0.446744\pi$
$449$	7.05742	0.333060	0.166530	+	0.986036i	$0.446744\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	12.9661i	0.609200i
$454$	0	0
$455$	−0.767408	−0.0359767
$456$	0	0
$457$	−6.26928	−0.293265	−0.146632	−	0.989191i	$-0.546843\pi$
$457$	−6.26928	−0.293265	−0.146632	+	0.989191i	$0.546843\pi$
$458$	0	0
$459$	0.0625011i	0.00291730i
$460$	0	0
$461$	18.0857i	0.842334i	0.906983	+	0.421167i	$0.138379\pi$
$461$	18.0857i	0.842334i	−0.906983	+	0.421167i	$0.861621\pi$
$462$	0	0
$463$	−9.95458	−0.462628	−0.231314	−	0.972879i	$-0.574303\pi$
$463$	−9.95458	−0.462628	−0.231314	+	0.972879i	$0.574303\pi$
$464$	0	0
$465$	−4.05316	−0.187961
$466$	0	0
$467$	32.6180i	1.50938i	0.656082	+	0.754690i	$0.272213\pi$
$467$	32.6180i	1.50938i	−0.656082	+	0.754690i	$0.727787\pi$
$468$	0	0
$469$	− 19.4109i	− 0.896311i
$470$	0	0
$471$	−9.67052	−0.445594
$472$	0	0
$473$	0	0
$474$	0	0
$475$	24.4181i	1.12038i
$476$	0	0
$477$	− 26.2179i	− 1.20044i
$478$	0	0
$479$	8.51155	0.388903	0.194451	−	0.980912i	$-0.437707\pi$
$479$	8.51155	0.388903	0.194451	+	0.980912i	$0.437707\pi$
$480$	0	0
$481$	−2.08853	−0.0952287
$482$	0	0
$483$	− 53.1310i	− 2.41754i
$484$	0	0
$485$	0.0600903i	0.00272856i
$486$	0	0
$487$	−7.13504	−0.323319	−0.161660	−	0.986847i	$-0.551685\pi$
$487$	−7.13504	−0.323319	−0.161660	+	0.986847i	$0.551685\pi$
$488$	0	0
$489$	−39.3108	−1.77770
$490$	0	0
$491$	− 12.1001i	− 0.546069i	−0.962004	−	0.273035i	$-0.911973\pi$
$491$	− 12.1001i	− 0.546069i	0.962004	−	0.273035i	$-0.0880274\pi$
$492$	0	0
$493$	− 20.6578i	− 0.930382i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.92883	−0.131376
$498$	0	0
$499$	− 14.0553i	− 0.629200i	−0.949224	−	0.314600i	$-0.898130\pi$
$499$	− 14.0553i	− 0.629200i	0.949224	−	0.314600i	$-0.101870\pi$
$500$	0	0
$501$	22.7152i	1.01484i
$502$	0	0
$503$	−27.5369	−1.22781	−0.613906	−	0.789379i	$-0.710403\pi$
$503$	−27.5369	−1.22781	−0.613906	+	0.789379i	$0.710403\pi$
$504$	0	0
$505$	−3.01109	−0.133992
$506$	0	0
$507$	25.3064i	1.12390i
$508$	0	0
$509$	0.301098i	0.0133459i	0.999978	+	0.00667297i	$0.00212409\pi$
$509$	0.301098i	0.0133459i	−0.999978	+	0.00667297i	$0.997876\pi$
$510$	0	0
$511$	16.1771	0.715634
$512$	0	0
$513$	−0.0922270	−0.00407192
$514$	0	0
$515$	− 0.566299i	− 0.0249541i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	21.6451	0.950114
$520$	0	0
$521$	−16.6902	−0.731210	−0.365605	−	0.930770i	$-0.619138\pi$
$521$	−16.6902	−0.731210	−0.365605	+	0.930770i	$0.619138\pi$
$522$	0	0
$523$	5.73532i	0.250788i	0.992107	+	0.125394i	$0.0400195\pi$
$523$	5.73532i	0.250788i	−0.992107	+	0.125394i	$0.959981\pi$
$524$	0	0
$525$	− 28.4852i	− 1.24320i
$526$	0	0
$527$	27.5559	1.20035
$528$	0	0
$529$	62.5827	2.72099
$530$	0	0
$531$	− 5.70033i	− 0.247373i
$532$	0	0
$533$	− 9.83940i	− 0.426192i
$534$	0	0
$535$	−0.179094	−0.00774293
$536$	0	0
$537$	−26.0799	−1.12543
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 24.1196i	− 1.03698i	−0.855083	−	0.518491i	$-0.826494\pi$
$541$	− 24.1196i	− 1.03698i	0.855083	−	0.518491i	$-0.173506\pi$
$542$	0	0
$543$	32.8425	1.40941
$544$	0	0
$545$	−0.271278	−0.0116203
$546$	0	0
$547$	− 7.39628i	− 0.316242i	−0.987420	−	0.158121i	$-0.949456\pi$
$547$	− 7.39628i	− 0.316242i	0.987420	−	0.158121i	$-0.0505436\pi$
$548$	0	0
$549$	19.1360i	0.816707i
$550$	0	0
$551$	30.4828	1.29861
$552$	0	0
$553$	34.3051	1.45880
$554$	0	0
$555$	0.628182i	0.0266648i
$556$	0	0
$557$	11.4190i	0.483839i	0.970296	+	0.241920i	$0.0777770\pi$
$557$	11.4190i	0.483839i	−0.970296	+	0.241920i	$0.922223\pi$
$558$	0	0
$559$	4.03304	0.170579
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 9.71580i	− 0.409472i	−0.978817	−	0.204736i	$-0.934366\pi$
$563$	− 9.71580i	− 0.409472i	0.978817	−	0.204736i	$-0.0656336\pi$
$564$	0	0
$565$	− 0.755537i	− 0.0317857i
$566$	0	0
$567$	21.1691	0.889019
$568$	0	0
$569$	24.1045	1.01051	0.505257	−	0.862969i	$-0.331398\pi$
$569$	24.1045	1.01051	0.505257	+	0.862969i	$0.331398\pi$
$570$	0	0
$571$	12.8180i	0.536415i	0.963361	+	0.268207i	$0.0864312\pi$
$571$	12.8180i	0.536415i	−0.963361	+	0.268207i	$0.913569\pi$
$572$	0	0
$573$	36.2756i	1.51543i
$574$	0	0
$575$	45.8837	1.91348
$576$	0	0
$577$	22.8507	0.951288	0.475644	−	0.879638i	$-0.342215\pi$
$577$	22.8507	0.951288	0.475644	+	0.879638i	$0.342215\pi$
$578$	0	0
$579$	10.9519i	0.455145i
$580$	0	0
$581$	21.9894i	0.912276i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−0.978773	−0.0404673
$586$	0	0
$587$	− 29.3027i	− 1.20945i	−0.796433	−	0.604726i	$-0.793283\pi$
$587$	− 29.3027i	− 1.20945i	0.796433	−	0.604726i	$-0.206717\pi$
$588$	0	0
$589$	40.6616i	1.67543i
$590$	0	0
$591$	49.7787	2.04762
$592$	0	0
$593$	−10.3694	−0.425820	−0.212910	−	0.977072i	$-0.568294\pi$
$593$	−10.3694	−0.425820	−0.212910	+	0.977072i	$0.568294\pi$
$594$	0	0
$595$	− 1.56925i	− 0.0643329i
$596$	0	0
$597$	− 25.4933i	− 1.04337i
$598$	0	0
$599$	20.8591	0.852279	0.426140	−	0.904657i	$-0.359873\pi$
$599$	20.8591	0.852279	0.426140	+	0.904657i	$0.359873\pi$
$600$	0	0
$601$	−39.2013	−1.59906	−0.799528	−	0.600629i	$-0.794917\pi$
$601$	−39.2013	−1.59906	−0.799528	+	0.600629i	$0.794917\pi$
$602$	0	0
$603$	− 24.7571i	− 1.00819i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−26.6834	−1.08305	−0.541523	−	0.840686i	$-0.682152\pi$
$607$	−26.6834	−1.08305	−0.541523	+	0.840686i	$0.682152\pi$
$608$	0	0
$609$	−35.5601	−1.44097
$610$	0	0
$611$	7.76470i	0.314126i
$612$	0	0
$613$	30.7684i	1.24272i	0.783523	+	0.621362i	$0.213420\pi$
$613$	30.7684i	1.24272i	−0.783523	+	0.621362i	$0.786580\pi$
$614$	0	0
$615$	−2.95947	−0.119337
$616$	0	0
$617$	10.0183	0.403323	0.201662	−	0.979455i	$-0.435366\pi$
$617$	10.0183	0.403323	0.201662	+	0.979455i	$0.435366\pi$
$618$	0	0
$619$	− 32.0971i	− 1.29009i	−0.764144	−	0.645046i	$-0.776838\pi$
$619$	− 32.0971i	− 1.29009i	0.764144	−	0.645046i	$-0.223162\pi$
$620$	0	0
$621$	0.173302i	0.00695438i
$622$	0	0
$623$	−11.9919	−0.480444
$624$	0	0
$625$	24.3988	0.975951
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 4.27076i	− 0.170286i
$630$	0	0
$631$	2.80016	0.111472	0.0557362	−	0.998446i	$-0.482249\pi$
$631$	2.80016	0.111472	0.0557362	+	0.998446i	$0.482249\pi$
$632$	0	0
$633$	−45.8443	−1.82215
$634$	0	0
$635$	1.47403i	0.0584952i
$636$	0	0
$637$	2.44015i	0.0966823i
$638$	0	0
$639$	−3.73551	−0.147775
$640$	0	0
$641$	16.8783	0.666652	0.333326	−	0.942812i	$-0.391829\pi$
$641$	16.8783	0.666652	0.333326	+	0.942812i	$0.391829\pi$
$642$	0	0
$643$	− 49.0739i	− 1.93528i	−0.252326	−	0.967642i	$-0.581196\pi$
$643$	− 49.0739i	− 1.93528i	0.252326	−	0.967642i	$-0.418804\pi$
$644$	0	0
$645$	− 1.21305i	− 0.0477637i
$646$	0	0
$647$	35.3413	1.38941	0.694705	−	0.719295i	$-0.255535\pi$
$647$	35.3413	1.38941	0.694705	+	0.719295i	$0.255535\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	− 47.4343i	− 1.85910i
$652$	0	0
$653$	− 0.192213i	− 0.00752186i	−0.999993	−	0.00376093i	$-0.998803\pi$
$653$	− 0.192213i	− 0.00752186i	0.999993	−	0.00376093i	$-0.00119714\pi$
$654$	0	0
$655$	1.37387	0.0536816
$656$	0	0
$657$	20.6328	0.804960
$658$	0	0
$659$	46.4826i	1.81071i	0.424660	+	0.905353i	$0.360394\pi$
$659$	46.4826i	1.81071i	−0.424660	+	0.905353i	$0.639606\pi$
$660$	0	0
$661$	− 24.0904i	− 0.937007i	−0.883462	−	0.468503i	$-0.844793\pi$
$661$	− 24.0904i	− 0.937007i	0.883462	−	0.468503i	$-0.155207\pi$
$662$	0	0
$663$	13.3256	0.517523
$664$	0	0
$665$	2.31559	0.0897948
$666$	0	0
$667$	− 57.2798i	− 2.21788i
$668$	0	0
$669$	− 35.6843i	− 1.37963i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	7.57085	0.291835	0.145917	−	0.989297i	$-0.453387\pi$
$673$	7.57085	0.291835	0.145917	+	0.989297i	$0.453387\pi$
$674$	0	0
$675$	0.0929129i	0.00357622i
$676$	0	0
$677$	20.6521i	0.793725i	0.917878	+	0.396862i	$0.129901\pi$
$677$	20.6521i	0.793725i	−0.917878	+	0.396862i	$0.870099\pi$
$678$	0	0
$679$	−0.703239	−0.0269878
$680$	0	0
$681$	10.2371	0.392287
$682$	0	0
$683$	14.6272i	0.559694i	0.960045	+	0.279847i	$0.0902837\pi$
$683$	14.6272i	0.559694i	−0.960045	+	0.279847i	$0.909716\pi$
$684$	0	0
$685$	0.561935i	0.0214704i
$686$	0	0
$687$	−56.8928	−2.17059
$688$	0	0
$689$	14.2954	0.544613
$690$	0	0
$691$	− 22.9427i	− 0.872781i	−0.899757	−	0.436390i	$-0.856257\pi$
$691$	− 22.9427i	− 0.872781i	0.899757	−	0.436390i	$-0.143743\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.78524	−0.0677181
$696$	0	0
$697$	20.1203	0.762109
$698$	0	0
$699$	34.2280i	1.29462i
$700$	0	0
$701$	− 0.700778i	− 0.0264680i	−0.999912	−	0.0132340i	$-0.995787\pi$
$701$	− 0.700778i	− 0.0264680i	0.999912	−	0.0132340i	$-0.00421264\pi$
$702$	0	0
$703$	6.30196	0.237683
$704$	0	0
$705$	2.33545	0.0879580
$706$	0	0
$707$	− 35.2388i	− 1.32529i
$708$	0	0
$709$	4.05276i	0.152205i	0.997100	+	0.0761023i	$0.0242476\pi$
$709$	4.05276i	0.152205i	−0.997100	+	0.0761023i	$0.975752\pi$
$710$	0	0
$711$	43.7537	1.64089
$712$	0	0
$713$	76.4065	2.86145
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 58.5943i	− 2.18824i
$718$	0	0
$719$	−10.0908	−0.376323	−0.188161	−	0.982138i	$-0.560253\pi$
$719$	−10.0908	−0.376323	−0.188161	+	0.982138i	$0.560253\pi$
$720$	0	0
$721$	6.62741	0.246818
$722$	0	0
$723$	64.0970i	2.38379i
$724$	0	0
$725$	− 30.7095i	− 1.14052i
$726$	0	0
$727$	13.6615	0.506675	0.253338	−	0.967378i	$-0.418472\pi$
$727$	13.6615	0.506675	0.253338	+	0.967378i	$0.418472\pi$
$728$	0	0
$729$	26.8621	0.994892
$730$	0	0
$731$	8.24703i	0.305027i
$732$	0	0
$733$	− 17.9204i	− 0.661906i	−0.943647	−	0.330953i	$-0.892630\pi$
$733$	− 17.9204i	− 0.661906i	0.943647	−	0.330953i	$-0.107370\pi$
$734$	0	0
$735$	0.733942	0.0270719
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 23.5590i	− 0.866631i	−0.901242	−	0.433316i	$-0.857344\pi$
$739$	− 23.5590i	− 0.866631i	0.901242	−	0.433316i	$-0.142656\pi$
$740$	0	0
$741$	19.6633i	0.722351i
$742$	0	0
$743$	−13.6295	−0.500018	−0.250009	−	0.968244i	$-0.580434\pi$
$743$	−13.6295	−0.500018	−0.250009	+	0.968244i	$0.580434\pi$
$744$	0	0
$745$	−2.41489	−0.0884746
$746$	0	0
$747$	28.0459i	1.02615i
$748$	0	0
$749$	− 2.09595i	− 0.0765842i
$750$	0	0
$751$	12.2314	0.446331	0.223165	−	0.974781i	$-0.428361\pi$
$751$	12.2314	0.446331	0.223165	+	0.974781i	$0.428361\pi$
$752$	0	0
$753$	47.1868	1.71958
$754$	0	0
$755$	1.06186i	0.0386452i
$756$	0	0
$757$	21.3405i	0.775634i	0.921736	+	0.387817i	$0.126771\pi$
$757$	21.3405i	0.775634i	−0.921736	+	0.387817i	$0.873229\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−11.8300	−0.428839	−0.214419	−	0.976742i	$-0.568786\pi$
$761$	−11.8300	−0.428839	−0.214419	+	0.976742i	$0.568786\pi$
$762$	0	0
$763$	− 3.17478i	− 0.114935i
$764$	0	0
$765$	− 2.00146i	− 0.0723629i
$766$	0	0
$767$	3.10813	0.112228
$768$	0	0
$769$	−2.94327	−0.106137	−0.0530685	−	0.998591i	$-0.516900\pi$
$769$	−2.94327	−0.106137	−0.0530685	+	0.998591i	$0.516900\pi$
$770$	0	0
$771$	18.0369i	0.649583i
$772$	0	0
$773$	7.95495i	0.286120i	0.989714	+	0.143060i	$0.0456942\pi$
$773$	7.95495i	0.286120i	−0.989714	+	0.143060i	$0.954306\pi$
$774$	0	0
$775$	40.9640	1.47147
$776$	0	0
$777$	−7.35163	−0.263738
$778$	0	0
$779$	29.6896i	1.06374i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0.115990	0.00414513
$784$	0	0
$785$	−0.791972	−0.0282667
$786$	0	0
$787$	2.72311i	0.0970685i	0.998822	+	0.0485342i	$0.0154550\pi$
$787$	2.72311i	0.0970685i	−0.998822	+	0.0485342i	$0.984545\pi$
$788$	0	0
$789$	6.59089i	0.234642i
$790$	0	0
$791$	8.84207	0.314388
$792$	0	0
$793$	−10.4340	−0.370523
$794$	0	0
$795$	− 4.29975i	− 0.152496i
$796$	0	0
$797$	− 28.0588i	− 0.993893i	−0.867781	−	0.496947i	$-0.834454\pi$
$797$	− 28.0588i	− 0.993893i	0.867781	−	0.496947i	$-0.165546\pi$
$798$	0	0
$799$	−15.8778	−0.561715
$800$	0	0
$801$	−15.2947	−0.540413
$802$	0	0
$803$	0	0
$804$	0	0
$805$	− 4.35119i	− 0.153359i
$806$	0	0
$807$	−38.0826	−1.34057
$808$	0	0
$809$	32.5279	1.14362	0.571810	−	0.820386i	$-0.306242\pi$
$809$	32.5279	1.14362	0.571810	+	0.820386i	$0.306242\pi$
$810$	0	0
$811$	− 8.90533i	− 0.312708i	−0.987701	−	0.156354i	$-0.950026\pi$
$811$	− 8.90533i	− 0.312708i	0.987701	−	0.156354i	$-0.0499741\pi$
$812$	0	0
$813$	− 2.28648i	− 0.0801904i
$814$	0	0
$815$	−3.21938	−0.112770
$816$	0	0
$817$	−12.1694	−0.425752
$818$	0	0
$819$	− 11.4546i	− 0.400256i
$820$	0	0
$821$	− 14.3875i	− 0.502126i	−0.967971	−	0.251063i	$-0.919220\pi$
$821$	− 14.3875i	− 0.502126i	0.967971	−	0.251063i	$-0.0807801\pi$
$822$	0	0
$823$	35.3163	1.23105	0.615525	−	0.788117i	$-0.288944\pi$
$823$	35.3163	1.23105	0.615525	+	0.788117i	$0.288944\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 12.8333i	− 0.446257i	−0.974789	−	0.223129i	$-0.928373\pi$
$827$	− 12.8333i	− 0.446257i	0.974789	−	0.223129i	$-0.0716270\pi$
$828$	0	0
$829$	− 23.8928i	− 0.829831i	−0.909860	−	0.414916i	$-0.863811\pi$
$829$	− 23.8928i	− 0.829831i	0.909860	−	0.414916i	$-0.136189\pi$
$830$	0	0
$831$	−33.1947	−1.15151
$832$	0	0
$833$	−4.98978	−0.172886
$834$	0	0
$835$	1.86027i	0.0643773i
$836$	0	0
$837$	0.154721i	0.00534793i
$838$	0	0
$839$	−23.6213	−0.815497	−0.407748	−	0.913094i	$-0.633686\pi$
$839$	−23.6213	−0.815497	−0.407748	+	0.913094i	$0.633686\pi$
$840$	0	0
$841$	−9.33683	−0.321960
$842$	0	0
$843$	38.0585i	1.31080i
$844$	0	0
$845$	2.07248i	0.0712956i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	5.73697	0.196892
$850$	0	0
$851$	− 11.8419i	− 0.405935i
$852$	0	0
$853$	− 41.7017i	− 1.42784i	−0.700228	−	0.713919i	$-0.746918\pi$
$853$	− 41.7017i	− 1.42784i	0.700228	−	0.713919i	$-0.253082\pi$
$854$	0	0
$855$	2.95337	0.101003
$856$	0	0
$857$	4.35256	0.148680	0.0743402	−	0.997233i	$-0.476315\pi$
$857$	4.35256	0.148680	0.0743402	+	0.997233i	$0.476315\pi$
$858$	0	0
$859$	7.91648i	0.270107i	0.990838	+	0.135053i	$0.0431206\pi$
$859$	7.91648i	0.270107i	−0.990838	+	0.135053i	$0.956879\pi$
$860$	0	0
$861$	− 34.6347i	− 1.18035i
$862$	0	0
$863$	12.3382	0.419996	0.209998	−	0.977702i	$-0.432654\pi$
$863$	12.3382	0.419996	0.209998	+	0.977702i	$0.432654\pi$
$864$	0	0
$865$	1.77263	0.0602714
$866$	0	0
$867$	− 14.3657i	− 0.487884i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	13.4990	0.457395
$872$	0	0
$873$	−0.896929	−0.0303565
$874$	0	0
$875$	− 4.68453i	− 0.158366i
$876$	0	0
$877$	− 58.3261i	− 1.96953i	−0.173887	−	0.984766i	$-0.555633\pi$
$877$	− 58.3261i	− 1.96953i	0.173887	−	0.984766i	$-0.444367\pi$
$878$	0	0
$879$	−45.0289	−1.51879
$880$	0	0
$881$	−25.8106	−0.869581	−0.434791	−	0.900532i	$-0.643178\pi$
$881$	−25.8106	−0.869581	−0.434791	+	0.900532i	$0.643178\pi$
$882$	0	0
$883$	36.6024i	1.23177i	0.787836	+	0.615885i	$0.211201\pi$
$883$	36.6024i	1.23177i	−0.787836	+	0.615885i	$0.788799\pi$
$884$	0	0
$885$	− 0.934857i	− 0.0314249i
$886$	0	0
$887$	21.8922	0.735067	0.367534	−	0.930010i	$-0.380202\pi$
$887$	21.8922	0.735067	0.367534	+	0.930010i	$0.380202\pi$
$888$	0	0
$889$	−17.2506	−0.578568
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 23.4294i	− 0.784033i
$894$	0	0
$895$	−2.13583	−0.0713929
$896$	0	0
$897$	36.9490	1.23369
$898$	0	0
$899$	− 51.1382i	− 1.70555i
$900$	0	0
$901$	29.2323i	0.973869i
$902$	0	0
$903$	14.1963	0.472424
$904$	0	0
$905$	2.68965	0.0894071
$906$	0	0
$907$	− 28.8065i	− 0.956503i	−0.878223	−	0.478252i	$-0.841271\pi$
$907$	− 28.8065i	− 0.956503i	0.878223	−	0.478252i	$-0.158729\pi$
$908$	0	0
$909$	− 44.9446i	− 1.49072i
$910$	0	0
$911$	−44.2181	−1.46501	−0.732506	−	0.680760i	$-0.761649\pi$
$911$	−44.2181	−1.46501	−0.732506	+	0.680760i	$0.761649\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	3.13832i	0.103750i
$916$	0	0
$917$	16.0785i	0.530957i
$918$	0	0
$919$	−18.9997	−0.626743	−0.313372	−	0.949631i	$-0.601459\pi$
$919$	−18.9997	−0.626743	−0.313372	+	0.949631i	$0.601459\pi$
$920$	0	0
$921$	−49.8652	−1.64311
$922$	0	0
$923$	− 2.03681i	− 0.0670423i
$924$	0	0
$925$	− 6.34883i	− 0.208748i
$926$	0	0
$927$	8.45277	0.277626
$928$	0	0
$929$	28.0670	0.920850	0.460425	−	0.887699i	$-0.347697\pi$
$929$	28.0670	0.920850	0.460425	+	0.887699i	$0.347697\pi$
$930$	0	0
$931$	− 7.36295i	− 0.241311i
$932$	0	0
$933$	− 53.1411i	− 1.73976i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−0.293629	−0.00959245	−0.00479623	−	0.999988i	$-0.501527\pi$
$937$	−0.293629	−0.00959245	−0.00479623	+	0.999988i	$0.501527\pi$
$938$	0	0
$939$	65.6715i	2.14311i
$940$	0	0
$941$	16.1292i	0.525797i	0.964823	+	0.262899i	$0.0846785\pi$
$941$	16.1292i	0.525797i	−0.964823	+	0.262899i	$0.915322\pi$
$942$	0	0
$943$	55.7892	1.81675
$944$	0	0
$945$	0.00881101	0.000286622 0
$946$	0	0
$947$	29.0812i	0.945011i	0.881328	+	0.472505i	$0.156650\pi$
$947$	29.0812i	0.945011i	−0.881328	+	0.472505i	$0.843350\pi$
$948$	0	0
$949$	11.2501i	0.365194i
$950$	0	0
$951$	−9.32680	−0.302442
$952$	0	0
$953$	−40.5658	−1.31405	−0.657027	−	0.753867i	$-0.728186\pi$
$953$	−40.5658	−1.31405	−0.657027	+	0.753867i	$0.728186\pi$
$954$	0	0
$955$	2.97081i	0.0961330i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6.57634	−0.212361
$960$	0	0
$961$	37.2141	1.20046
$962$	0	0
$963$	− 2.67323i	− 0.0861435i
$964$	0	0
$965$	0.896909i	0.0288725i
$966$	0	0
$967$	6.42732	0.206689	0.103344	−	0.994646i	$-0.467046\pi$
$967$	6.42732	0.206689	0.103344	+	0.994646i	$0.467046\pi$
$968$	0	0
$969$	−40.2089	−1.29170
$970$	0	0
$971$	− 50.7088i	− 1.62732i	−0.581338	−	0.813662i	$-0.697471\pi$
$971$	− 50.7088i	− 1.62732i	0.581338	−	0.813662i	$-0.302529\pi$
$972$	0	0
$973$	− 20.8927i	− 0.669790i
$974$	0	0
$975$	19.8096	0.634414
$976$	0	0
$977$	−0.100545	−0.00321673	−0.00160837	−	0.999999i	$-0.500512\pi$
$977$	−0.100545	−0.00321673	−0.00160837	+	0.999999i	$0.500512\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	− 4.04919i	− 0.129281i
$982$	0	0
$983$	−26.3392	−0.840091	−0.420046	−	0.907503i	$-0.637986\pi$
$983$	−26.3392	−0.840091	−0.420046	+	0.907503i	$0.637986\pi$
$984$	0	0
$985$	4.07665	0.129893
$986$	0	0
$987$	27.3318i	0.869980i
$988$	0	0
$989$	22.8672i	0.727136i
$990$	0	0
$991$	27.9640	0.888306	0.444153	−	0.895951i	$-0.353505\pi$
$991$	27.9640	0.888306	0.444153	+	0.895951i	$0.353505\pi$
$992$	0	0
$993$	69.1050	2.19298
$994$	0	0
$995$	− 2.08778i	− 0.0661871i
$996$	0	0
$997$	− 57.2928i	− 1.81448i	−0.420612	−	0.907241i	$-0.638185\pi$
$997$	− 57.2928i	− 1.81448i	0.420612	−	0.907241i	$-0.361815\pi$
$998$	0	0
$999$	0.0239795	0.000758677 0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3872.2.c.i.1937.17		20
4.3	odd	2		968.2.c.i.485.7		20
8.3	odd	2		968.2.c.i.485.8		20
8.5	even	2	inner	3872.2.c.i.1937.4		20
11.2	odd	10		352.2.w.a.81.9		40
11.6	odd	10		352.2.w.a.113.2		40
11.10	odd	2		3872.2.c.h.1937.17		20
44.3	odd	10		968.2.o.d.493.1		40
44.7	even	10		968.2.o.j.269.2		40
44.15	odd	10		968.2.o.d.269.9		40
44.19	even	10		968.2.o.j.493.10		40
44.27	odd	10		968.2.o.i.245.9		40
44.31	odd	10		968.2.o.i.565.4		40
44.35	even	10		88.2.o.a.37.7	yes	40
44.39	even	10		88.2.o.a.69.2	yes	40
44.43	even	2		968.2.c.h.485.14		20
88.3	odd	10		968.2.o.d.493.9		40
88.13	odd	10		352.2.w.a.81.2		40
88.19	even	10		968.2.o.j.493.2		40
88.21	odd	2		3872.2.c.h.1937.4		20
88.27	odd	10		968.2.o.i.245.4		40
88.35	even	10		88.2.o.a.37.2	✓	40
88.43	even	2		968.2.c.h.485.13		20
88.51	even	10		968.2.o.j.269.10		40
88.59	odd	10		968.2.o.d.269.1		40
88.61	odd	10		352.2.w.a.113.9		40
88.75	odd	10		968.2.o.i.565.9		40
88.83	even	10		88.2.o.a.69.7	yes	40
132.35	odd	10		792.2.br.b.37.4		40
132.83	odd	10		792.2.br.b.685.9		40
264.35	odd	10		792.2.br.b.37.9		40
264.83	odd	10		792.2.br.b.685.4		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.o.a.37.2	✓	40	88.35	even	10
88.2.o.a.37.7	yes	40	44.35	even	10
88.2.o.a.69.2	yes	40	44.39	even	10
88.2.o.a.69.7	yes	40	88.83	even	10
352.2.w.a.81.2		40	88.13	odd	10
352.2.w.a.81.9		40	11.2	odd	10
352.2.w.a.113.2		40	11.6	odd	10
352.2.w.a.113.9		40	88.61	odd	10
792.2.br.b.37.4		40	132.35	odd	10
792.2.br.b.37.9		40	264.35	odd	10
792.2.br.b.685.4		40	264.83	odd	10
792.2.br.b.685.9		40	132.83	odd	10
968.2.c.h.485.13		20	88.43	even	2
968.2.c.h.485.14		20	44.43	even	2
968.2.c.i.485.7		20	4.3	odd	2
968.2.c.i.485.8		20	8.3	odd	2
968.2.o.d.269.1		40	88.59	odd	10
968.2.o.d.269.9		40	44.15	odd	10
968.2.o.d.493.1		40	44.3	odd	10
968.2.o.d.493.9		40	88.3	odd	10
968.2.o.i.245.4		40	88.27	odd	10
968.2.o.i.245.9		40	44.27	odd	10
968.2.o.i.565.4		40	44.31	odd	10
968.2.o.i.565.9		40	88.75	odd	10
968.2.o.j.269.2		40	44.7	even	10
968.2.o.j.269.10		40	88.51	even	10
968.2.o.j.493.2		40	88.19	even	10
968.2.o.j.493.10		40	44.19	even	10
3872.2.c.h.1937.4		20	88.21	odd	2
3872.2.c.h.1937.17		20	11.10	odd	2
3872.2.c.i.1937.4		20	8.5	even	2	inner
3872.2.c.i.1937.17		20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3872 = 2^{5} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3872.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.44793i q^{3} +0.200474i q^{5} -2.34615 q^{7} -2.99235 q^{9} -1.63159i q^{13} -0.490746 q^{15} +3.33639 q^{17} +4.92320i q^{19} -5.74321i q^{21} +9.25109 q^{23} +4.95981 q^{25} +0.0187332i q^{27} -6.19167i q^{29} +8.25919 q^{31} -0.470343i q^{35} -1.28006i q^{37} +3.99402 q^{39} +6.03055 q^{41} +2.47184i q^{43} -0.599888i q^{45} -4.75897 q^{47} -1.49556 q^{49} +8.16724i q^{51} +8.76165i q^{53} -12.0516 q^{57} +1.90497i q^{59} -6.39500i q^{61} +7.02051 q^{63} +0.327092 q^{65} +8.27349i q^{67} +22.6460i q^{69} +1.24836 q^{71} -6.89517 q^{73} +12.1413i q^{75} -14.6219 q^{79} -9.02290 q^{81} -9.37255i q^{83} +0.668860i q^{85} +15.1568 q^{87} +5.11129 q^{89} +3.82797i q^{91} +20.2179i q^{93} -0.986974 q^{95} +0.299741 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 10 q^{7} - 10 q^{9} + 4 q^{15} - 2 q^{17} + 4 q^{23} - 2 q^{25} - 2 q^{31} + 28 q^{39} + 2 q^{41} - 2 q^{47} - 2 q^{49} - 22 q^{57} - 30 q^{63} + 18 q^{65} + 34 q^{71} - 2 q^{73} - 58 q^{79} - 12 q^{81}+ \cdots + 10 q^{97}+O(q^{100}) \) 20 * q + 10 * q^7 - 10 * q^9 + 4 * q^15 - 2 * q^17 + 4 * q^23 - 2 * q^25 - 2 * q^31 + 28 * q^39 + 2 * q^41 - 2 * q^47 - 2 * q^49 - 22 * q^57 - 30 * q^63 + 18 * q^65 + 34 * q^71 - 2 * q^73 - 58 * q^79 - 12 * q^81 + 34 * q^87 - 8 * q^89 - 92 * q^95 + 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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