LMFDB - Embedded newform 7935.2.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7935,2,Mod(1,7935)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7935, 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("7935.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7935 = 3 \cdot 5 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7935.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:	$63.3612940039$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{73}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 18 $ x^2 - x - 18
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.77200$ of defining polynomial
Character	$\chi$	$=$	7935.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.00000 q^{4} +1.00000 q^{5} -3.77200 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.00000 q^{4} +1.00000 q^{5} -3.77200 q^{7} +1.00000 q^{9} +4.77200 q^{11} -2.00000 q^{12} +3.77200 q^{13} +1.00000 q^{15} +4.00000 q^{16} +6.00000 q^{17} +2.77200 q^{19} -2.00000 q^{20} -3.77200 q^{21} +1.00000 q^{25} +1.00000 q^{27} +7.54400 q^{28} +6.00000 q^{29} -2.77200 q^{31} +4.77200 q^{33} -3.77200 q^{35} -2.00000 q^{36} -9.77200 q^{37} +3.77200 q^{39} -1.22800 q^{41} +5.77200 q^{43} -9.54400 q^{44} +1.00000 q^{45} -9.54400 q^{47} +4.00000 q^{48} +7.22800 q^{49} +6.00000 q^{51} -7.54400 q^{52} +3.54400 q^{53} +4.77200 q^{55} +2.77200 q^{57} -2.00000 q^{60} -8.54400 q^{61} -3.77200 q^{63} -8.00000 q^{64} +3.77200 q^{65} +11.7720 q^{67} -12.0000 q^{68} +1.22800 q^{71} +2.00000 q^{73} +1.00000 q^{75} -5.54400 q^{76} -18.0000 q^{77} +5.22800 q^{79} +4.00000 q^{80} +1.00000 q^{81} -3.54400 q^{83} +7.54400 q^{84} +6.00000 q^{85} +6.00000 q^{87} +9.54400 q^{89} -14.2280 q^{91} -2.77200 q^{93} +2.77200 q^{95} +0.455996 q^{97} +4.77200 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 4 q^{4} + 2 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 4 * q^4 + 2 * q^5 + q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 4 q^{4} + 2 q^{5} + q^{7} + 2 q^{9} + q^{11} - 4 q^{12} - q^{13} + 2 q^{15} + 8 q^{16} + 12 q^{17} - 3 q^{19} - 4 q^{20} + q^{21} + 2 q^{25} + 2 q^{27} - 2 q^{28} + 12 q^{29} + 3 q^{31} + q^{33} + q^{35} - 4 q^{36} - 11 q^{37} - q^{39} - 11 q^{41} + 3 q^{43} - 2 q^{44} + 2 q^{45} - 2 q^{47} + 8 q^{48} + 23 q^{49} + 12 q^{51} + 2 q^{52} - 10 q^{53} + q^{55} - 3 q^{57} - 4 q^{60} + q^{63} - 16 q^{64} - q^{65} + 15 q^{67} - 24 q^{68} + 11 q^{71} + 4 q^{73} + 2 q^{75} + 6 q^{76} - 36 q^{77} + 19 q^{79} + 8 q^{80} + 2 q^{81} + 10 q^{83} - 2 q^{84} + 12 q^{85} + 12 q^{87} + 2 q^{89} - 37 q^{91} + 3 q^{93} - 3 q^{95} + 18 q^{97} + q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 4 * q^4 + 2 * q^5 + q^7 + 2 * q^9 + q^11 - 4 * q^12 - q^13 + 2 * q^15 + 8 * q^16 + 12 * q^17 - 3 * q^19 - 4 * q^20 + q^21 + 2 * q^25 + 2 * q^27 - 2 * q^28 + 12 * q^29 + 3 * q^31 + q^33 + q^35 - 4 * q^36 - 11 * q^37 - q^39 - 11 * q^41 + 3 * q^43 - 2 * q^44 + 2 * q^45 - 2 * q^47 + 8 * q^48 + 23 * q^49 + 12 * q^51 + 2 * q^52 - 10 * q^53 + q^55 - 3 * q^57 - 4 * q^60 + q^63 - 16 * q^64 - q^65 + 15 * q^67 - 24 * q^68 + 11 * q^71 + 4 * q^73 + 2 * q^75 + 6 * q^76 - 36 * q^77 + 19 * q^79 + 8 * q^80 + 2 * q^81 + 10 * q^83 - 2 * q^84 + 12 * q^85 + 12 * q^87 + 2 * q^89 - 37 * q^91 + 3 * q^93 - 3 * q^95 + 18 * q^97 + q^99

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 $

$2$	0	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−2.00000	−1.00000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.77200	−1.42568	−0.712841	−	0.701325i	$-0.752592\pi$
$7$	−3.77200	−1.42568	−0.712841	+	0.701325i	$0.752592\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.77200	1.43881	0.719406	−	0.694589i	$-0.244414\pi$
$11$	4.77200	1.43881	0.719406	+	0.694589i	$0.244414\pi$
$12$	−2.00000	−0.577350
$13$	3.77200	1.04617	0.523083	−	0.852282i	$-0.324782\pi$
$13$	3.77200	1.04617	0.523083	+	0.852282i	$0.324782\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	4.00000	1.00000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	2.77200	0.635941	0.317970	−	0.948101i	$-0.396999\pi$
$19$	2.77200	0.635941	0.317970	+	0.948101i	$0.396999\pi$
$20$	−2.00000	−0.447214
$21$	−3.77200	−0.823118
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	7.54400	1.42568
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−2.77200	−0.497866	−0.248933	−	0.968521i	$-0.580080\pi$
$31$	−2.77200	−0.497866	−0.248933	+	0.968521i	$0.580080\pi$
$32$	0	0
$33$	4.77200	0.830699
$34$	0	0
$35$	−3.77200	−0.637585
$36$	−2.00000	−0.333333
$37$	−9.77200	−1.60651	−0.803254	−	0.595637i	$-0.796900\pi$
$37$	−9.77200	−1.60651	−0.803254	+	0.595637i	$0.796900\pi$
$38$	0	0
$39$	3.77200	0.604004
$40$	0	0
$41$	−1.22800	−0.191781	−0.0958905	−	0.995392i	$-0.530570\pi$
$41$	−1.22800	−0.191781	−0.0958905	+	0.995392i	$0.530570\pi$
$42$	0	0
$43$	5.77200	0.880222	0.440111	−	0.897943i	$-0.354939\pi$
$43$	5.77200	0.880222	0.440111	+	0.897943i	$0.354939\pi$
$44$	−9.54400	−1.43881
$45$	1.00000	0.149071
$46$	0	0
$47$	−9.54400	−1.39214	−0.696068	−	0.717976i	$-0.745069\pi$
$47$	−9.54400	−1.39214	−0.696068	+	0.717976i	$0.745069\pi$
$48$	4.00000	0.577350
$49$	7.22800	1.03257
$50$	0	0
$51$	6.00000	0.840168
$52$	−7.54400	−1.04617
$53$	3.54400	0.486806	0.243403	−	0.969925i	$-0.421736\pi$
$53$	3.54400	0.486806	0.243403	+	0.969925i	$0.421736\pi$
$54$	0	0
$55$	4.77200	0.643457
$56$	0	0
$57$	2.77200	0.367161
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	−2.00000	−0.258199
$61$	−8.54400	−1.09395	−0.546974	−	0.837150i	$-0.684220\pi$
$61$	−8.54400	−1.09395	−0.546974	+	0.837150i	$0.684220\pi$
$62$	0	0
$63$	−3.77200	−0.475228
$64$	−8.00000	−1.00000
$65$	3.77200	0.467859
$66$	0	0
$67$	11.7720	1.43818	0.719089	−	0.694918i	$-0.244559\pi$
$67$	11.7720	1.43818	0.719089	+	0.694918i	$0.244559\pi$
$68$	−12.0000	−1.45521
$69$	0	0
$70$	0	0
$71$	1.22800	0.145737	0.0728683	−	0.997342i	$-0.476785\pi$
$71$	1.22800	0.145737	0.0728683	+	0.997342i	$0.476785\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	−5.54400	−0.635941
$77$	−18.0000	−2.05129
$78$	0	0
$79$	5.22800	0.588196	0.294098	−	0.955775i	$-0.404981\pi$
$79$	5.22800	0.588196	0.294098	+	0.955775i	$0.404981\pi$
$80$	4.00000	0.447214
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.54400	−0.389005	−0.194502	−	0.980902i	$-0.562309\pi$
$83$	−3.54400	−0.389005	−0.194502	+	0.980902i	$0.562309\pi$
$84$	7.54400	0.823118
$85$	6.00000	0.650791
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	9.54400	1.01166	0.505831	−	0.862632i	$-0.331186\pi$
$89$	9.54400	1.01166	0.505831	+	0.862632i	$0.331186\pi$
$90$	0	0
$91$	−14.2280	−1.49150
$92$	0	0
$93$	−2.77200	−0.287443
$94$	0	0
$95$	2.77200	0.284401
$96$	0	0
$97$	0.455996	0.0462994	0.0231497	−	0.999732i	$-0.492631\pi$
$97$	0.455996	0.0462994	0.0231497	+	0.999732i	$0.492631\pi$
$98$	0	0
$99$	4.77200	0.479604
$100$	−2.00000	−0.200000
$101$	−14.3160	−1.42450	−0.712248	−	0.701928i	$-0.752323\pi$
$101$	−14.3160	−1.42450	−0.712248	+	0.701928i	$0.752323\pi$
$102$	0	0
$103$	−19.3160	−1.90326	−0.951631	−	0.307242i	$-0.900594\pi$
$103$	−19.3160	−1.90326	−0.951631	+	0.307242i	$0.900594\pi$
$104$	0	0
$105$	−3.77200	−0.368110
$106$	0	0
$107$	−3.54400	−0.342612	−0.171306	−	0.985218i	$-0.554799\pi$
$107$	−3.54400	−0.342612	−0.171306	+	0.985218i	$0.554799\pi$
$108$	−2.00000	−0.192450
$109$	18.3160	1.75436	0.877178	−	0.480166i	$-0.159424\pi$
$109$	18.3160	1.75436	0.877178	+	0.480166i	$0.159424\pi$
$110$	0	0
$111$	−9.77200	−0.927517
$112$	−15.0880	−1.42568
$113$	15.5440	1.46226	0.731128	−	0.682240i	$-0.238994\pi$
$113$	15.5440	1.46226	0.731128	+	0.682240i	$0.238994\pi$
$114$	0	0
$115$	0	0
$116$	−12.0000	−1.11417
$117$	3.77200	0.348722
$118$	0	0
$119$	−22.6320	−2.07467
$120$	0	0
$121$	11.7720	1.07018
$122$	0	0
$123$	−1.22800	−0.110725
$124$	5.54400	0.497866
$125$	1.00000	0.0894427
$126$	0	0
$127$	9.77200	0.867125	0.433562	−	0.901124i	$-0.357256\pi$
$127$	9.77200	0.867125	0.433562	+	0.901124i	$0.357256\pi$
$128$	0	0
$129$	5.77200	0.508196
$130$	0	0
$131$	−10.7720	−0.941154	−0.470577	−	0.882359i	$-0.655954\pi$
$131$	−10.7720	−0.941154	−0.470577	+	0.882359i	$0.655954\pi$
$132$	−9.54400	−0.830699
$133$	−10.4560	−0.906650
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−9.54400	−0.815399	−0.407700	−	0.913116i	$-0.633669\pi$
$137$	−9.54400	−0.815399	−0.407700	+	0.913116i	$0.633669\pi$
$138$	0	0
$139$	18.0880	1.53420	0.767102	−	0.641525i	$-0.221698\pi$
$139$	18.0880	1.53420	0.767102	+	0.641525i	$0.221698\pi$
$140$	7.54400	0.637585
$141$	−9.54400	−0.803750
$142$	0	0
$143$	18.0000	1.50524
$144$	4.00000	0.333333
$145$	6.00000	0.498273
$146$	0	0
$147$	7.22800	0.596155
$148$	19.5440	1.60651
$149$	−16.7720	−1.37402	−0.687008	−	0.726650i	$-0.741076\pi$
$149$	−16.7720	−1.37402	−0.687008	+	0.726650i	$0.741076\pi$
$150$	0	0
$151$	8.54400	0.695301	0.347651	−	0.937624i	$-0.386980\pi$
$151$	8.54400	0.695301	0.347651	+	0.937624i	$0.386980\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	−2.77200	−0.222653
$156$	−7.54400	−0.604004
$157$	19.5440	1.55978	0.779891	−	0.625916i	$-0.215275\pi$
$157$	19.5440	1.55978	0.779891	+	0.625916i	$0.215275\pi$
$158$	0	0
$159$	3.54400	0.281058
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−11.7720	−0.922054	−0.461027	−	0.887386i	$-0.652519\pi$
$163$	−11.7720	−0.922054	−0.461027	+	0.887386i	$0.652519\pi$
$164$	2.45600	0.191781
$165$	4.77200	0.371500
$166$	0	0
$167$	15.5440	1.20283	0.601416	−	0.798936i	$-0.294604\pi$
$167$	15.5440	1.20283	0.601416	+	0.798936i	$0.294604\pi$
$168$	0	0
$169$	1.22800	0.0944614
$170$	0	0
$171$	2.77200	0.211980
$172$	−11.5440	−0.880222
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	−3.77200	−0.285137
$176$	19.0880	1.43881
$177$	0	0
$178$	0	0
$179$	−14.3160	−1.07003	−0.535014	−	0.844843i	$-0.679694\pi$
$179$	−14.3160	−1.07003	−0.535014	+	0.844843i	$0.679694\pi$
$180$	−2.00000	−0.149071
$181$	−0.772002	−0.0573824	−0.0286912	−	0.999588i	$-0.509134\pi$
$181$	−0.772002	−0.0573824	−0.0286912	+	0.999588i	$0.509134\pi$
$182$	0	0
$183$	−8.54400	−0.631591
$184$	0	0
$185$	−9.77200	−0.718452
$186$	0	0
$187$	28.6320	2.09378
$188$	19.0880	1.39214
$189$	−3.77200	−0.274373
$190$	0	0
$191$	27.5440	1.99301	0.996507	−	0.0835083i	$-0.0266125\pi$
$191$	27.5440	1.99301	0.996507	+	0.0835083i	$0.0266125\pi$
$192$	−8.00000	−0.577350
$193$	0.227998	0.0164117	0.00820583	−	0.999966i	$-0.497388\pi$
$193$	0.227998	0.0164117	0.00820583	+	0.999966i	$0.497388\pi$
$194$	0	0
$195$	3.77200	0.270119
$196$	−14.4560	−1.03257
$197$	−19.0880	−1.35996	−0.679982	−	0.733229i	$-0.738012\pi$
$197$	−19.0880	−1.35996	−0.679982	+	0.733229i	$0.738012\pi$
$198$	0	0
$199$	4.54400	0.322116	0.161058	−	0.986945i	$-0.448509\pi$
$199$	4.54400	0.322116	0.161058	+	0.986945i	$0.448509\pi$
$200$	0	0
$201$	11.7720	0.830333
$202$	0	0
$203$	−22.6320	−1.58846
$204$	−12.0000	−0.840168
$205$	−1.22800	−0.0857671
$206$	0	0
$207$	0	0
$208$	15.0880	1.04617
$209$	13.2280	0.915000
$210$	0	0
$211$	−3.45600	−0.237921	−0.118960	−	0.992899i	$-0.537956\pi$
$211$	−3.45600	−0.237921	−0.118960	+	0.992899i	$0.537956\pi$
$212$	−7.08801	−0.486806
$213$	1.22800	0.0841410
$214$	0	0
$215$	5.77200	0.393647
$216$	0	0
$217$	10.4560	0.709799
$218$	0	0
$219$	2.00000	0.135147
$220$	−9.54400	−0.643457
$221$	22.6320	1.52239
$222$	0	0
$223$	0.227998	0.0152679	0.00763394	−	0.999971i	$-0.497570\pi$
$223$	0.227998	0.0152679	0.00763394	+	0.999971i	$0.497570\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−21.5440	−1.42993	−0.714963	−	0.699162i	$-0.753556\pi$
$227$	−21.5440	−1.42993	−0.714963	+	0.699162i	$0.753556\pi$
$228$	−5.54400	−0.367161
$229$	26.0880	1.72394	0.861972	−	0.506956i	$-0.169229\pi$
$229$	26.0880	1.72394	0.861972	+	0.506956i	$0.169229\pi$
$230$	0	0
$231$	−18.0000	−1.18431
$232$	0	0
$233$	−9.54400	−0.625248	−0.312624	−	0.949877i	$-0.601208\pi$
$233$	−9.54400	−0.625248	−0.312624	+	0.949877i	$0.601208\pi$
$234$	0	0
$235$	−9.54400	−0.622582
$236$	0	0
$237$	5.22800	0.339595
$238$	0	0
$239$	22.7720	1.47300	0.736499	−	0.676438i	$-0.236478\pi$
$239$	22.7720	1.47300	0.736499	+	0.676438i	$0.236478\pi$
$240$	4.00000	0.258199
$241$	19.0000	1.22390	0.611949	−	0.790897i	$-0.290386\pi$
$241$	19.0000	1.22390	0.611949	+	0.790897i	$0.290386\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	17.0880	1.09395
$245$	7.22800	0.461780
$246$	0	0
$247$	10.4560	0.665299
$248$	0	0
$249$	−3.54400	−0.224592
$250$	0	0
$251$	26.3160	1.66105	0.830526	−	0.556980i	$-0.188040\pi$
$251$	26.3160	1.66105	0.830526	+	0.556980i	$0.188040\pi$
$252$	7.54400	0.475228
$253$	0	0
$254$	0	0
$255$	6.00000	0.375735
$256$	16.0000	1.00000
$257$	22.6320	1.41175	0.705873	−	0.708338i	$-0.250555\pi$
$257$	22.6320	1.41175	0.705873	+	0.708338i	$0.250555\pi$
$258$	0	0
$259$	36.8600	2.29037
$260$	−7.54400	−0.467859
$261$	6.00000	0.371391
$262$	0	0
$263$	−13.0880	−0.807041	−0.403521	−	0.914971i	$-0.632214\pi$
$263$	−13.0880	−0.807041	−0.403521	+	0.914971i	$0.632214\pi$
$264$	0	0
$265$	3.54400	0.217706
$266$	0	0
$267$	9.54400	0.584084
$268$	−23.5440	−1.43818
$269$	4.77200	0.290954	0.145477	−	0.989362i	$-0.453528\pi$
$269$	4.77200	0.290954	0.145477	+	0.989362i	$0.453528\pi$
$270$	0	0
$271$	−29.7720	−1.80852	−0.904260	−	0.426982i	$-0.859577\pi$
$271$	−29.7720	−1.80852	−0.904260	+	0.426982i	$0.859577\pi$
$272$	24.0000	1.45521
$273$	−14.2280	−0.861118
$274$	0	0
$275$	4.77200	0.287763
$276$	0	0
$277$	−24.8600	−1.49369	−0.746847	−	0.664996i	$-0.768433\pi$
$277$	−24.8600	−1.49369	−0.746847	+	0.664996i	$0.768433\pi$
$278$	0	0
$279$	−2.77200	−0.165955
$280$	0	0
$281$	20.3160	1.21195	0.605976	−	0.795483i	$-0.292783\pi$
$281$	20.3160	1.21195	0.605976	+	0.795483i	$0.292783\pi$
$282$	0	0
$283$	−6.22800	−0.370216	−0.185108	−	0.982718i	$-0.559264\pi$
$283$	−6.22800	−0.370216	−0.185108	+	0.982718i	$0.559264\pi$
$284$	−2.45600	−0.145737
$285$	2.77200	0.164199
$286$	0	0
$287$	4.63201	0.273419
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	0.455996	0.0267310
$292$	−4.00000	−0.234082
$293$	3.54400	0.207043	0.103521	−	0.994627i	$-0.466989\pi$
$293$	3.54400	0.207043	0.103521	+	0.994627i	$0.466989\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.77200	0.276900
$298$	0	0
$299$	0	0
$300$	−2.00000	−0.115470
$301$	−21.7720	−1.25492
$302$	0	0
$303$	−14.3160	−0.822433
$304$	11.0880	0.635941
$305$	−8.54400	−0.489228
$306$	0	0
$307$	27.7720	1.58503	0.792516	−	0.609851i	$-0.208771\pi$
$307$	27.7720	1.58503	0.792516	+	0.609851i	$0.208771\pi$
$308$	36.0000	2.05129
$309$	−19.3160	−1.09885
$310$	0	0
$311$	−27.5440	−1.56188	−0.780939	−	0.624608i	$-0.785259\pi$
$311$	−27.5440	−1.56188	−0.780939	+	0.624608i	$0.785259\pi$
$312$	0	0
$313$	23.7720	1.34367	0.671836	−	0.740699i	$-0.265506\pi$
$313$	23.7720	1.34367	0.671836	+	0.740699i	$0.265506\pi$
$314$	0	0
$315$	−3.77200	−0.212528
$316$	−10.4560	−0.588196
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	0	0
$319$	28.6320	1.60308
$320$	−8.00000	−0.447214
$321$	−3.54400	−0.197807
$322$	0	0
$323$	16.6320	0.925430
$324$	−2.00000	−0.111111
$325$	3.77200	0.209233
$326$	0	0
$327$	18.3160	1.01288
$328$	0	0
$329$	36.0000	1.98474
$330$	0	0
$331$	14.5440	0.799411	0.399705	−	0.916644i	$-0.369112\pi$
$331$	14.5440	0.799411	0.399705	+	0.916644i	$0.369112\pi$
$332$	7.08801	0.389005
$333$	−9.77200	−0.535502
$334$	0	0
$335$	11.7720	0.643173
$336$	−15.0880	−0.823118
$337$	−18.2280	−0.992942	−0.496471	−	0.868053i	$-0.665371\pi$
$337$	−18.2280	−0.992942	−0.496471	+	0.868053i	$0.665371\pi$
$338$	0	0
$339$	15.5440	0.844234
$340$	−12.0000	−0.650791
$341$	−13.2280	−0.716336
$342$	0	0
$343$	−0.860009	−0.0464361
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−8.45600	−0.453942	−0.226971	−	0.973902i	$-0.572882\pi$
$347$	−8.45600	−0.453942	−0.226971	+	0.973902i	$0.572882\pi$
$348$	−12.0000	−0.643268
$349$	31.3160	1.67631	0.838154	−	0.545434i	$-0.183635\pi$
$349$	31.3160	1.67631	0.838154	+	0.545434i	$0.183635\pi$
$350$	0	0
$351$	3.77200	0.201335
$352$	0	0
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	1.22800	0.0651754
$356$	−19.0880	−1.01166
$357$	−22.6320	−1.19781
$358$	0	0
$359$	−31.0880	−1.64076	−0.820381	−	0.571817i	$-0.806239\pi$
$359$	−31.0880	−1.64076	−0.820381	+	0.571817i	$0.806239\pi$
$360$	0	0
$361$	−11.3160	−0.595579
$362$	0	0
$363$	11.7720	0.617870
$364$	28.4560	1.49150
$365$	2.00000	0.104685
$366$	0	0
$367$	−5.54400	−0.289395	−0.144697	−	0.989476i	$-0.546221\pi$
$367$	−5.54400	−0.289395	−0.144697	+	0.989476i	$0.546221\pi$
$368$	0	0
$369$	−1.22800	−0.0639270
$370$	0	0
$371$	−13.3680	−0.694031
$372$	5.54400	0.287443
$373$	14.2280	0.736698	0.368349	−	0.929688i	$-0.379923\pi$
$373$	14.2280	0.736698	0.368349	+	0.929688i	$0.379923\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	22.6320	1.16561
$378$	0	0
$379$	−6.22800	−0.319911	−0.159955	−	0.987124i	$-0.551135\pi$
$379$	−6.22800	−0.319911	−0.159955	+	0.987124i	$0.551135\pi$
$380$	−5.54400	−0.284401
$381$	9.77200	0.500635
$382$	0	0
$383$	33.5440	1.71402	0.857009	−	0.515301i	$-0.172320\pi$
$383$	33.5440	1.71402	0.857009	+	0.515301i	$0.172320\pi$
$384$	0	0
$385$	−18.0000	−0.917365
$386$	0	0
$387$	5.77200	0.293407
$388$	−0.911993	−0.0462994
$389$	3.68399	0.186786	0.0933930	−	0.995629i	$-0.470229\pi$
$389$	3.68399	0.186786	0.0933930	+	0.995629i	$0.470229\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−10.7720	−0.543376
$394$	0	0
$395$	5.22800	0.263049
$396$	−9.54400	−0.479604
$397$	2.68399	0.134706	0.0673529	−	0.997729i	$-0.478545\pi$
$397$	2.68399	0.134706	0.0673529	+	0.997729i	$0.478545\pi$
$398$	0	0
$399$	−10.4560	−0.523455
$400$	4.00000	0.200000
$401$	3.68399	0.183970	0.0919850	−	0.995760i	$-0.470679\pi$
$401$	3.68399	0.183970	0.0919850	+	0.995760i	$0.470679\pi$
$402$	0	0
$403$	−10.4560	−0.520850
$404$	28.6320	1.42450
$405$	1.00000	0.0496904
$406$	0	0
$407$	−46.6320	−2.31146
$408$	0	0
$409$	−21.4560	−1.06093	−0.530465	−	0.847707i	$-0.677983\pi$
$409$	−21.4560	−1.06093	−0.530465	+	0.847707i	$0.677983\pi$
$410$	0	0
$411$	−9.54400	−0.470771
$412$	38.6320	1.90326
$413$	0	0
$414$	0	0
$415$	−3.54400	−0.173968
$416$	0	0
$417$	18.0880	0.885774
$418$	0	0
$419$	20.3160	0.992502	0.496251	−	0.868179i	$-0.334710\pi$
$419$	20.3160	0.992502	0.496251	+	0.868179i	$0.334710\pi$
$420$	7.54400	0.368110
$421$	38.0880	1.85630	0.928148	−	0.372211i	$-0.121400\pi$
$421$	38.0880	1.85630	0.928148	+	0.372211i	$0.121400\pi$
$422$	0	0
$423$	−9.54400	−0.464045
$424$	0	0
$425$	6.00000	0.291043
$426$	0	0
$427$	32.2280	1.55962
$428$	7.08801	0.342612
$429$	18.0000	0.869048
$430$	0	0
$431$	−17.8600	−0.860286	−0.430143	−	0.902761i	$-0.641537\pi$
$431$	−17.8600	−0.860286	−0.430143	+	0.902761i	$0.641537\pi$
$432$	4.00000	0.192450
$433$	21.3160	1.02438	0.512191	−	0.858872i	$-0.328834\pi$
$433$	21.3160	1.02438	0.512191	+	0.858872i	$0.328834\pi$
$434$	0	0
$435$	6.00000	0.287678
$436$	−36.6320	−1.75436
$437$	0	0
$438$	0	0
$439$	−2.22800	−0.106337	−0.0531683	−	0.998586i	$-0.516932\pi$
$439$	−2.22800	−0.106337	−0.0531683	+	0.998586i	$0.516932\pi$
$440$	0	0
$441$	7.22800	0.344190
$442$	0	0
$443$	−6.00000	−0.285069	−0.142534	−	0.989790i	$-0.545525\pi$
$443$	−6.00000	−0.285069	−0.142534	+	0.989790i	$0.545525\pi$
$444$	19.5440	0.927517
$445$	9.54400	0.452429
$446$	0	0
$447$	−16.7720	−0.793288
$448$	30.1760	1.42568
$449$	37.0880	1.75029	0.875146	−	0.483860i	$-0.160765\pi$
$449$	37.0880	1.75029	0.875146	+	0.483860i	$0.160765\pi$
$450$	0	0
$451$	−5.86001	−0.275937
$452$	−31.0880	−1.46226
$453$	8.54400	0.401432
$454$	0	0
$455$	−14.2280	−0.667019
$456$	0	0
$457$	29.7720	1.39268	0.696338	−	0.717714i	$-0.254812\pi$
$457$	29.7720	1.39268	0.696338	+	0.717714i	$0.254812\pi$
$458$	0	0
$459$	6.00000	0.280056
$460$	0	0
$461$	−22.7720	−1.06060	−0.530299	−	0.847811i	$-0.677920\pi$
$461$	−22.7720	−1.06060	−0.530299	+	0.847811i	$0.677920\pi$
$462$	0	0
$463$	−11.7720	−0.547091	−0.273546	−	0.961859i	$-0.588196\pi$
$463$	−11.7720	−0.547091	−0.273546	+	0.961859i	$0.588196\pi$
$464$	24.0000	1.11417
$465$	−2.77200	−0.128549
$466$	0	0
$467$	−32.1760	−1.48893	−0.744464	−	0.667662i	$-0.767295\pi$
$467$	−32.1760	−1.48893	−0.744464	+	0.667662i	$0.767295\pi$
$468$	−7.54400	−0.348722
$469$	−44.4040	−2.05039
$470$	0	0
$471$	19.5440	0.900540
$472$	0	0
$473$	27.5440	1.26647
$474$	0	0
$475$	2.77200	0.127188
$476$	45.2640	2.07467
$477$	3.54400	0.162269
$478$	0	0
$479$	38.3160	1.75070	0.875351	−	0.483487i	$-0.160630\pi$
$479$	38.3160	1.75070	0.875351	+	0.483487i	$0.160630\pi$
$480$	0	0
$481$	−36.8600	−1.68067
$482$	0	0
$483$	0	0
$484$	−23.5440	−1.07018
$485$	0.455996	0.0207057
$486$	0	0
$487$	−27.3160	−1.23781	−0.618903	−	0.785467i	$-0.712423\pi$
$487$	−27.3160	−1.23781	−0.618903	+	0.785467i	$0.712423\pi$
$488$	0	0
$489$	−11.7720	−0.532348
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	2.45600	0.110725
$493$	36.0000	1.62136
$494$	0	0
$495$	4.77200	0.214486
$496$	−11.0880	−0.497866
$497$	−4.63201	−0.207774
$498$	0	0
$499$	−28.5440	−1.27781	−0.638903	−	0.769288i	$-0.720611\pi$
$499$	−28.5440	−1.27781	−0.638903	+	0.769288i	$0.720611\pi$
$500$	−2.00000	−0.0894427
$501$	15.5440	0.694455
$502$	0	0
$503$	−28.6320	−1.27664	−0.638319	−	0.769772i	$-0.720370\pi$
$503$	−28.6320	−1.27664	−0.638319	+	0.769772i	$0.720370\pi$
$504$	0	0
$505$	−14.3160	−0.637054
$506$	0	0
$507$	1.22800	0.0545373
$508$	−19.5440	−0.867125
$509$	−28.6320	−1.26909	−0.634546	−	0.772885i	$-0.718813\pi$
$509$	−28.6320	−1.26909	−0.634546	+	0.772885i	$0.718813\pi$
$510$	0	0
$511$	−7.54400	−0.333727
$512$	0	0
$513$	2.77200	0.122387
$514$	0	0
$515$	−19.3160	−0.851165
$516$	−11.5440	−0.508196
$517$	−45.5440	−2.00302
$518$	0	0
$519$	0	0
$520$	0	0
$521$	14.3160	0.627196	0.313598	−	0.949556i	$-0.398466\pi$
$521$	14.3160	0.627196	0.313598	+	0.949556i	$0.398466\pi$
$522$	0	0
$523$	−5.54400	−0.242422	−0.121211	−	0.992627i	$-0.538678\pi$
$523$	−5.54400	−0.242422	−0.121211	+	0.992627i	$0.538678\pi$
$524$	21.5440	0.941154
$525$	−3.77200	−0.164624
$526$	0	0
$527$	−16.6320	−0.724502
$528$	19.0880	0.830699
$529$	0	0
$530$	0	0
$531$	0	0
$532$	20.9120	0.906650
$533$	−4.63201	−0.200635
$534$	0	0
$535$	−3.54400	−0.153221
$536$	0	0
$537$	−14.3160	−0.617781
$538$	0	0
$539$	34.4920	1.48568
$540$	−2.00000	−0.0860663
$541$	−42.3160	−1.81931	−0.909654	−	0.415368i	$-0.863653\pi$
$541$	−42.3160	−1.81931	−0.909654	+	0.415368i	$0.863653\pi$
$542$	0	0
$543$	−0.772002	−0.0331298
$544$	0	0
$545$	18.3160	0.784571
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	19.0880	0.815399
$549$	−8.54400	−0.364649
$550$	0	0
$551$	16.6320	0.708548
$552$	0	0
$553$	−19.7200	−0.838580
$554$	0	0
$555$	−9.77200	−0.414798
$556$	−36.1760	−1.53420
$557$	−6.00000	−0.254228	−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000	−0.254228	−0.127114	+	0.991888i	$0.540571\pi$
$558$	0	0
$559$	21.7720	0.920858
$560$	−15.0880	−0.637585
$561$	28.6320	1.20884
$562$	0	0
$563$	−1.08801	−0.0458540	−0.0229270	−	0.999737i	$-0.507299\pi$
$563$	−1.08801	−0.0458540	−0.0229270	+	0.999737i	$0.507299\pi$
$564$	19.0880	0.803750
$565$	15.5440	0.653941
$566$	0	0
$567$	−3.77200	−0.158409
$568$	0	0
$569$	−9.68399	−0.405974	−0.202987	−	0.979181i	$-0.565065\pi$
$569$	−9.68399	−0.405974	−0.202987	+	0.979181i	$0.565065\pi$
$570$	0	0
$571$	7.00000	0.292941	0.146470	−	0.989215i	$-0.453209\pi$
$571$	7.00000	0.292941	0.146470	+	0.989215i	$0.453209\pi$
$572$	−36.0000	−1.50524
$573$	27.5440	1.15067
$574$	0	0
$575$	0	0
$576$	−8.00000	−0.333333
$577$	−12.4560	−0.518550	−0.259275	−	0.965804i	$-0.583484\pi$
$577$	−12.4560	−0.518550	−0.259275	+	0.965804i	$0.583484\pi$
$578$	0	0
$579$	0.227998	0.00947528
$580$	−12.0000	−0.498273
$581$	13.3680	0.554598
$582$	0	0
$583$	16.9120	0.700423
$584$	0	0
$585$	3.77200	0.155953
$586$	0	0
$587$	−8.45600	−0.349016	−0.174508	−	0.984656i	$-0.555834\pi$
$587$	−8.45600	−0.349016	−0.174508	+	0.984656i	$0.555834\pi$
$588$	−14.4560	−0.596155
$589$	−7.68399	−0.316613
$590$	0	0
$591$	−19.0880	−0.785176
$592$	−39.0880	−1.60651
$593$	−4.63201	−0.190214	−0.0951070	−	0.995467i	$-0.530319\pi$
$593$	−4.63201	−0.190214	−0.0951070	+	0.995467i	$0.530319\pi$
$594$	0	0
$595$	−22.6320	−0.927822
$596$	33.5440	1.37402
$597$	4.54400	0.185974
$598$	0	0
$599$	−16.7720	−0.685285	−0.342643	−	0.939466i	$-0.611322\pi$
$599$	−16.7720	−0.685285	−0.342643	+	0.939466i	$0.611322\pi$
$600$	0	0
$601$	−48.7200	−1.98733	−0.993666	−	0.112378i	$-0.964153\pi$
$601$	−48.7200	−1.98733	−0.993666	+	0.112378i	$0.964153\pi$
$602$	0	0
$603$	11.7720	0.479393
$604$	−17.0880	−0.695301
$605$	11.7720	0.478600
$606$	0	0
$607$	8.68399	0.352472	0.176236	−	0.984348i	$-0.443608\pi$
$607$	8.68399	0.352472	0.176236	+	0.984348i	$0.443608\pi$
$608$	0	0
$609$	−22.6320	−0.917095
$610$	0	0
$611$	−36.0000	−1.45640
$612$	−12.0000	−0.485071
$613$	21.3160	0.860945	0.430473	−	0.902604i	$-0.358347\pi$
$613$	21.3160	0.860945	0.430473	+	0.902604i	$0.358347\pi$
$614$	0	0
$615$	−1.22800	−0.0495177
$616$	0	0
$617$	12.0000	0.483102	0.241551	−	0.970388i	$-0.422344\pi$
$617$	12.0000	0.483102	0.241551	+	0.970388i	$0.422344\pi$
$618$	0	0
$619$	−35.0000	−1.40677	−0.703384	−	0.710810i	$-0.748329\pi$
$619$	−35.0000	−1.40677	−0.703384	+	0.710810i	$0.748329\pi$
$620$	5.54400	0.222653
$621$	0	0
$622$	0	0
$623$	−36.0000	−1.44231
$624$	15.0880	0.604004
$625$	1.00000	0.0400000
$626$	0	0
$627$	13.2280	0.528275
$628$	−39.0880	−1.55978
$629$	−58.6320	−2.33781
$630$	0	0
$631$	17.6320	0.701919	0.350960	−	0.936391i	$-0.385855\pi$
$631$	17.6320	0.701919	0.350960	+	0.936391i	$0.385855\pi$
$632$	0	0
$633$	−3.45600	−0.137364
$634$	0	0
$635$	9.77200	0.387790
$636$	−7.08801	−0.281058
$637$	27.2640	1.08024
$638$	0	0
$639$	1.22800	0.0485789
$640$	0	0
$641$	−0.139991	−0.00552930	−0.00276465	−	0.999996i	$-0.500880\pi$
$641$	−0.139991	−0.00552930	−0.00276465	+	0.999996i	$0.500880\pi$
$642$	0	0
$643$	−6.22800	−0.245608	−0.122804	−	0.992431i	$-0.539189\pi$
$643$	−6.22800	−0.245608	−0.122804	+	0.992431i	$0.539189\pi$
$644$	0	0
$645$	5.77200	0.227272
$646$	0	0
$647$	−22.6320	−0.889756	−0.444878	−	0.895591i	$-0.646753\pi$
$647$	−22.6320	−0.889756	−0.444878	+	0.895591i	$0.646753\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	10.4560	0.409803
$652$	23.5440	0.922054
$653$	43.0880	1.68616	0.843082	−	0.537785i	$-0.180739\pi$
$653$	43.0880	1.68616	0.843082	+	0.537785i	$0.180739\pi$
$654$	0	0
$655$	−10.7720	−0.420897
$656$	−4.91199	−0.191781
$657$	2.00000	0.0780274
$658$	0	0
$659$	15.6840	0.610962	0.305481	−	0.952198i	$-0.401183\pi$
$659$	15.6840	0.610962	0.305481	+	0.952198i	$0.401183\pi$
$660$	−9.54400	−0.371500
$661$	8.77200	0.341191	0.170596	−	0.985341i	$-0.445431\pi$
$661$	8.77200	0.341191	0.170596	+	0.985341i	$0.445431\pi$
$662$	0	0
$663$	22.6320	0.878954
$664$	0	0
$665$	−10.4560	−0.405466
$666$	0	0
$667$	0	0
$668$	−31.0880	−1.20283
$669$	0.227998	0.00881492
$670$	0	0
$671$	−40.7720	−1.57399
$672$	0	0
$673$	19.3160	0.744577	0.372289	−	0.928117i	$-0.378573\pi$
$673$	19.3160	0.744577	0.372289	+	0.928117i	$0.378573\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	−2.45600	−0.0944614
$677$	12.0000	0.461197	0.230599	−	0.973049i	$-0.425932\pi$
$677$	12.0000	0.461197	0.230599	+	0.973049i	$0.425932\pi$
$678$	0	0
$679$	−1.72002	−0.0660083
$680$	0	0
$681$	−21.5440	−0.825568
$682$	0	0
$683$	45.5440	1.74269	0.871346	−	0.490668i	$-0.163247\pi$
$683$	45.5440	1.74269	0.871346	+	0.490668i	$0.163247\pi$
$684$	−5.54400	−0.211980
$685$	−9.54400	−0.364658
$686$	0	0
$687$	26.0880	0.995320
$688$	23.0880	0.880222
$689$	13.3680	0.509280
$690$	0	0
$691$	9.63201	0.366419	0.183209	−	0.983074i	$-0.441351\pi$
$691$	9.63201	0.366419	0.183209	+	0.983074i	$0.441351\pi$
$692$	0	0
$693$	−18.0000	−0.683763
$694$	0	0
$695$	18.0880	0.686117
$696$	0	0
$697$	−7.36799	−0.279082
$698$	0	0
$699$	−9.54400	−0.360987
$700$	7.54400	0.285137
$701$	−16.6320	−0.628182	−0.314091	−	0.949393i	$-0.601700\pi$
$701$	−16.6320	−0.628182	−0.314091	+	0.949393i	$0.601700\pi$
$702$	0	0
$703$	−27.0880	−1.02164
$704$	−38.1760	−1.43881
$705$	−9.54400	−0.359448
$706$	0	0
$707$	54.0000	2.03088
$708$	0	0
$709$	−22.8600	−0.858526	−0.429263	−	0.903180i	$-0.641227\pi$
$709$	−22.8600	−0.858526	−0.429263	+	0.903180i	$0.641227\pi$
$710$	0	0
$711$	5.22800	0.196065
$712$	0	0
$713$	0	0
$714$	0	0
$715$	18.0000	0.673162
$716$	28.6320	1.07003
$717$	22.7720	0.850436
$718$	0	0
$719$	16.9120	0.630711	0.315355	−	0.948974i	$-0.397876\pi$
$719$	16.9120	0.630711	0.315355	+	0.948974i	$0.397876\pi$
$720$	4.00000	0.149071
$721$	72.8600	2.71345
$722$	0	0
$723$	19.0000	0.706618
$724$	1.54400	0.0573824
$725$	6.00000	0.222834
$726$	0	0
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	34.6320	1.28091
$732$	17.0880	0.631591
$733$	−32.4040	−1.19687	−0.598435	−	0.801172i	$-0.704210\pi$
$733$	−32.4040	−1.19687	−0.598435	+	0.801172i	$0.704210\pi$
$734$	0	0
$735$	7.22800	0.266609
$736$	0	0
$737$	56.1760	2.06927
$738$	0	0
$739$	42.0880	1.54823	0.774116	−	0.633044i	$-0.218195\pi$
$739$	42.0880	1.54823	0.774116	+	0.633044i	$0.218195\pi$
$740$	19.5440	0.718452
$741$	10.4560	0.384111
$742$	0	0
$743$	−40.6320	−1.49064	−0.745322	−	0.666705i	$-0.767704\pi$
$743$	−40.6320	−1.49064	−0.745322	+	0.666705i	$0.767704\pi$
$744$	0	0
$745$	−16.7720	−0.614479
$746$	0	0
$747$	−3.54400	−0.129668
$748$	−57.2640	−2.09378
$749$	13.3680	0.488456
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	−38.1760	−1.39214
$753$	26.3160	0.959009
$754$	0	0
$755$	8.54400	0.310948
$756$	7.54400	0.274373
$757$	−33.0880	−1.20260	−0.601302	−	0.799022i	$-0.705351\pi$
$757$	−33.0880	−1.20260	−0.601302	+	0.799022i	$0.705351\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	27.4040	0.993395	0.496697	−	0.867924i	$-0.334546\pi$
$761$	27.4040	0.993395	0.496697	+	0.867924i	$0.334546\pi$
$762$	0	0
$763$	−69.0880	−2.50115
$764$	−55.0880	−1.99301
$765$	6.00000	0.216930
$766$	0	0
$767$	0	0
$768$	16.0000	0.577350
$769$	10.5440	0.380226	0.190113	−	0.981762i	$-0.439114\pi$
$769$	10.5440	0.380226	0.190113	+	0.981762i	$0.439114\pi$
$770$	0	0
$771$	22.6320	0.815072
$772$	−0.455996	−0.0164117
$773$	37.0880	1.33396	0.666981	−	0.745074i	$-0.267586\pi$
$773$	37.0880	1.33396	0.666981	+	0.745074i	$0.267586\pi$
$774$	0	0
$775$	−2.77200	−0.0995732
$776$	0	0
$777$	36.8600	1.32235
$778$	0	0
$779$	−3.40401	−0.121961
$780$	−7.54400	−0.270119
$781$	5.86001	0.209688
$782$	0	0
$783$	6.00000	0.214423
$784$	28.9120	1.03257
$785$	19.5440	0.697555
$786$	0	0
$787$	1.54400	0.0550378	0.0275189	−	0.999621i	$-0.491239\pi$
$787$	1.54400	0.0550378	0.0275189	+	0.999621i	$0.491239\pi$
$788$	38.1760	1.35996
$789$	−13.0880	−0.465945
$790$	0	0
$791$	−58.6320	−2.08471
$792$	0	0
$793$	−32.2280	−1.14445
$794$	0	0
$795$	3.54400	0.125693
$796$	−9.08801	−0.322116
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	0	0
$799$	−57.2640	−2.02586
$800$	0	0
$801$	9.54400	0.337221
$802$	0	0
$803$	9.54400	0.336801
$804$	−23.5440	−0.830333
$805$	0	0
$806$	0	0
$807$	4.77200	0.167982
$808$	0	0
$809$	36.9480	1.29902	0.649512	−	0.760352i	$-0.274973\pi$
$809$	36.9480	1.29902	0.649512	+	0.760352i	$0.274973\pi$
$810$	0	0
$811$	11.0000	0.386262	0.193131	−	0.981173i	$-0.438136\pi$
$811$	11.0000	0.386262	0.193131	+	0.981173i	$0.438136\pi$
$812$	45.2640	1.58846
$813$	−29.7720	−1.04415
$814$	0	0
$815$	−11.7720	−0.412355
$816$	24.0000	0.840168
$817$	16.0000	0.559769
$818$	0	0
$819$	−14.2280	−0.497166
$820$	2.45600	0.0857671
$821$	−27.4040	−0.956407	−0.478203	−	0.878249i	$-0.658712\pi$
$821$	−27.4040	−0.956407	−0.478203	+	0.878249i	$0.658712\pi$
$822$	0	0
$823$	−32.6320	−1.13748	−0.568740	−	0.822517i	$-0.692569\pi$
$823$	−32.6320	−1.13748	−0.568740	+	0.822517i	$0.692569\pi$
$824$	0	0
$825$	4.77200	0.166140
$826$	0	0
$827$	9.54400	0.331878	0.165939	−	0.986136i	$-0.446935\pi$
$827$	9.54400	0.331878	0.165939	+	0.986136i	$0.446935\pi$
$828$	0	0
$829$	24.7720	0.860367	0.430184	−	0.902741i	$-0.358449\pi$
$829$	24.7720	0.860367	0.430184	+	0.902741i	$0.358449\pi$
$830$	0	0
$831$	−24.8600	−0.862384
$832$	−30.1760	−1.04617
$833$	43.3680	1.50261
$834$	0	0
$835$	15.5440	0.537922
$836$	−26.4560	−0.915000
$837$	−2.77200	−0.0958144
$838$	0	0
$839$	20.3160	0.701386	0.350693	−	0.936490i	$-0.385946\pi$
$839$	20.3160	0.701386	0.350693	+	0.936490i	$0.385946\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	20.3160	0.699720
$844$	6.91199	0.237921
$845$	1.22800	0.0422444
$846$	0	0
$847$	−44.4040	−1.52574
$848$	14.1760	0.486806
$849$	−6.22800	−0.213744
$850$	0	0
$851$	0	0
$852$	−2.45600	−0.0841410
$853$	47.9480	1.64171	0.820854	−	0.571137i	$-0.193498\pi$
$853$	47.9480	1.64171	0.820854	+	0.571137i	$0.193498\pi$
$854$	0	0
$855$	2.77200	0.0948005
$856$	0	0
$857$	16.6320	0.568139	0.284069	−	0.958804i	$-0.408315\pi$
$857$	16.6320	0.568139	0.284069	+	0.958804i	$0.408315\pi$
$858$	0	0
$859$	−17.7720	−0.606373	−0.303186	−	0.952931i	$-0.598050\pi$
$859$	−17.7720	−0.606373	−0.303186	+	0.952931i	$0.598050\pi$
$860$	−11.5440	−0.393647
$861$	4.63201	0.157859
$862$	0	0
$863$	22.6320	0.770403	0.385201	−	0.922833i	$-0.374132\pi$
$863$	22.6320	0.770403	0.385201	+	0.922833i	$0.374132\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	19.0000	0.645274
$868$	−20.9120	−0.709799
$869$	24.9480	0.846304
$870$	0	0
$871$	44.4040	1.50457
$872$	0	0
$873$	0.455996	0.0154331
$874$	0	0
$875$	−3.77200	−0.127517
$876$	−4.00000	−0.135147
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	0	0
$879$	3.54400	0.119536
$880$	19.0880	0.643457
$881$	−28.6320	−0.964637	−0.482318	−	0.875996i	$-0.660205\pi$
$881$	−28.6320	−0.964637	−0.482318	+	0.875996i	$0.660205\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	−45.2640	−1.52239
$885$	0	0
$886$	0	0
$887$	−2.45600	−0.0824643	−0.0412321	−	0.999150i	$-0.513128\pi$
$887$	−2.45600	−0.0824643	−0.0412321	+	0.999150i	$0.513128\pi$
$888$	0	0
$889$	−36.8600	−1.23625
$890$	0	0
$891$	4.77200	0.159868
$892$	−0.455996	−0.0152679
$893$	−26.4560	−0.885316
$894$	0	0
$895$	−14.3160	−0.478531
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−16.6320	−0.554709
$900$	−2.00000	−0.0666667
$901$	21.2640	0.708407
$902$	0	0
$903$	−21.7720	−0.724527
$904$	0	0
$905$	−0.772002	−0.0256622
$906$	0	0
$907$	−37.3160	−1.23906	−0.619529	−	0.784974i	$-0.712676\pi$
$907$	−37.3160	−1.23906	−0.619529	+	0.784974i	$0.712676\pi$
$908$	43.0880	1.42993
$909$	−14.3160	−0.474832
$910$	0	0
$911$	9.40401	0.311569	0.155784	−	0.987791i	$-0.450209\pi$
$911$	9.40401	0.311569	0.155784	+	0.987791i	$0.450209\pi$
$912$	11.0880	0.367161
$913$	−16.9120	−0.559705
$914$	0	0
$915$	−8.54400	−0.282456
$916$	−52.1760	−1.72394
$917$	40.6320	1.34179
$918$	0	0
$919$	−24.0880	−0.794590	−0.397295	−	0.917691i	$-0.630051\pi$
$919$	−24.0880	−0.794590	−0.397295	+	0.917691i	$0.630051\pi$
$920$	0	0
$921$	27.7720	0.915119
$922$	0	0
$923$	4.63201	0.152465
$924$	36.0000	1.18431
$925$	−9.77200	−0.321301
$926$	0	0
$927$	−19.3160	−0.634421
$928$	0	0
$929$	−15.6840	−0.514575	−0.257288	−	0.966335i	$-0.582829\pi$
$929$	−15.6840	−0.514575	−0.257288	+	0.966335i	$0.582829\pi$
$930$	0	0
$931$	20.0360	0.656654
$932$	19.0880	0.625248
$933$	−27.5440	−0.901750
$934$	0	0
$935$	28.6320	0.936367
$936$	0	0
$937$	16.4040	0.535896	0.267948	−	0.963433i	$-0.413655\pi$
$937$	16.4040	0.535896	0.267948	+	0.963433i	$0.413655\pi$
$938$	0	0
$939$	23.7720	0.775770
$940$	19.0880	0.622582
$941$	41.8600	1.36460	0.682299	−	0.731074i	$-0.260980\pi$
$941$	41.8600	1.36460	0.682299	+	0.731074i	$0.260980\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−3.77200	−0.122703
$946$	0	0
$947$	14.4560	0.469757	0.234878	−	0.972025i	$-0.424531\pi$
$947$	14.4560	0.469757	0.234878	+	0.972025i	$0.424531\pi$
$948$	−10.4560	−0.339595
$949$	7.54400	0.244889
$950$	0	0
$951$	−6.00000	−0.194563
$952$	0	0
$953$	−22.9120	−0.742192	−0.371096	−	0.928594i	$-0.621018\pi$
$953$	−22.9120	−0.742192	−0.371096	+	0.928594i	$0.621018\pi$
$954$	0	0
$955$	27.5440	0.891303
$956$	−45.5440	−1.47300
$957$	28.6320	0.925541
$958$	0	0
$959$	36.0000	1.16250
$960$	−8.00000	−0.258199
$961$	−23.3160	−0.752129
$962$	0	0
$963$	−3.54400	−0.114204
$964$	−38.0000	−1.22390
$965$	0.227998	0.00733952
$966$	0	0
$967$	10.8600	0.349234	0.174617	−	0.984636i	$-0.444131\pi$
$967$	10.8600	0.349234	0.174617	+	0.984636i	$0.444131\pi$
$968$	0	0
$969$	16.6320	0.534297
$970$	0	0
$971$	10.6320	0.341197	0.170599	−	0.985341i	$-0.445430\pi$
$971$	10.6320	0.341197	0.170599	+	0.985341i	$0.445430\pi$
$972$	−2.00000	−0.0641500
$973$	−68.2280	−2.18729
$974$	0	0
$975$	3.77200	0.120801
$976$	−34.1760	−1.09395
$977$	−9.54400	−0.305340	−0.152670	−	0.988277i	$-0.548787\pi$
$977$	−9.54400	−0.305340	−0.152670	+	0.988277i	$0.548787\pi$
$978$	0	0
$979$	45.5440	1.45559
$980$	−14.4560	−0.461780
$981$	18.3160	0.584785
$982$	0	0
$983$	40.6320	1.29596	0.647980	−	0.761657i	$-0.275614\pi$
$983$	40.6320	1.29596	0.647980	+	0.761657i	$0.275614\pi$
$984$	0	0
$985$	−19.0880	−0.608194
$986$	0	0
$987$	36.0000	1.14589
$988$	−20.9120	−0.665299
$989$	0	0
$990$	0	0
$991$	15.0880	0.479286	0.239643	−	0.970861i	$-0.422970\pi$
$991$	15.0880	0.479286	0.239643	+	0.970861i	$0.422970\pi$
$992$	0	0
$993$	14.5440	0.461540
$994$	0	0
$995$	4.54400	0.144055
$996$	7.08801	0.224592
$997$	−16.6840	−0.528387	−0.264194	−	0.964470i	$-0.585106\pi$
$997$	−16.6840	−0.528387	−0.264194	+	0.964470i	$0.585106\pi$
$998$	0	0
$999$	−9.77200	−0.309172

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7935.2.a.s.1.1	yes	2
23.22	odd	2		7935.2.a.r.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7935.2.a.r.1.2	✓	2	23.22	odd	2
7935.2.a.s.1.1	yes	2	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:	\( N \)	\(=\)	\( 7935 = 3 \cdot 5 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7935.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.00000 q^{4} +1.00000 q^{5} -3.77200 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.00000 q^{4} +1.00000 q^{5} -3.77200 q^{7} +1.00000 q^{9} +4.77200 q^{11} -2.00000 q^{12} +3.77200 q^{13} +1.00000 q^{15} +4.00000 q^{16} +6.00000 q^{17} +2.77200 q^{19} -2.00000 q^{20} -3.77200 q^{21} +1.00000 q^{25} +1.00000 q^{27} +7.54400 q^{28} +6.00000 q^{29} -2.77200 q^{31} +4.77200 q^{33} -3.77200 q^{35} -2.00000 q^{36} -9.77200 q^{37} +3.77200 q^{39} -1.22800 q^{41} +5.77200 q^{43} -9.54400 q^{44} +1.00000 q^{45} -9.54400 q^{47} +4.00000 q^{48} +7.22800 q^{49} +6.00000 q^{51} -7.54400 q^{52} +3.54400 q^{53} +4.77200 q^{55} +2.77200 q^{57} -2.00000 q^{60} -8.54400 q^{61} -3.77200 q^{63} -8.00000 q^{64} +3.77200 q^{65} +11.7720 q^{67} -12.0000 q^{68} +1.22800 q^{71} +2.00000 q^{73} +1.00000 q^{75} -5.54400 q^{76} -18.0000 q^{77} +5.22800 q^{79} +4.00000 q^{80} +1.00000 q^{81} -3.54400 q^{83} +7.54400 q^{84} +6.00000 q^{85} +6.00000 q^{87} +9.54400 q^{89} -14.2280 q^{91} -2.77200 q^{93} +2.77200 q^{95} +0.455996 q^{97} +4.77200 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 4 q^{4} + 2 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 4 * q^4 + 2 * q^5 + q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 4 q^{4} + 2 q^{5} + q^{7} + 2 q^{9} + q^{11} - 4 q^{12} - q^{13} + 2 q^{15} + 8 q^{16} + 12 q^{17} - 3 q^{19} - 4 q^{20} + q^{21} + 2 q^{25} + 2 q^{27} - 2 q^{28} + 12 q^{29} + 3 q^{31} + q^{33} + q^{35} - 4 q^{36} - 11 q^{37} - q^{39} - 11 q^{41} + 3 q^{43} - 2 q^{44} + 2 q^{45} - 2 q^{47} + 8 q^{48} + 23 q^{49} + 12 q^{51} + 2 q^{52} - 10 q^{53} + q^{55} - 3 q^{57} - 4 q^{60} + q^{63} - 16 q^{64} - q^{65} + 15 q^{67} - 24 q^{68} + 11 q^{71} + 4 q^{73} + 2 q^{75} + 6 q^{76} - 36 q^{77} + 19 q^{79} + 8 q^{80} + 2 q^{81} + 10 q^{83} - 2 q^{84} + 12 q^{85} + 12 q^{87} + 2 q^{89} - 37 q^{91} + 3 q^{93} - 3 q^{95} + 18 q^{97} + q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 4 * q^4 + 2 * q^5 + q^7 + 2 * q^9 + q^11 - 4 * q^12 - q^13 + 2 * q^15 + 8 * q^16 + 12 * q^17 - 3 * q^19 - 4 * q^20 + q^21 + 2 * q^25 + 2 * q^27 - 2 * q^28 + 12 * q^29 + 3 * q^31 + q^33 + q^35 - 4 * q^36 - 11 * q^37 - q^39 - 11 * q^41 + 3 * q^43 - 2 * q^44 + 2 * q^45 - 2 * q^47 + 8 * q^48 + 23 * q^49 + 12 * q^51 + 2 * q^52 - 10 * q^53 + q^55 - 3 * q^57 - 4 * q^60 + q^63 - 16 * q^64 - q^65 + 15 * q^67 - 24 * q^68 + 11 * q^71 + 4 * q^73 + 2 * q^75 + 6 * q^76 - 36 * q^77 + 19 * q^79 + 8 * q^80 + 2 * q^81 + 10 * q^83 - 2 * q^84 + 12 * q^85 + 12 * q^87 + 2 * q^89 - 37 * q^91 + 3 * q^93 - 3 * q^95 + 18 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more