LMFDB - Embedded newform 2394.2.a.bc.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2394,2,Mod(1,2394)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2394, 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("2394.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2394.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:	$19.1161862439$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	2394.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +0.622216 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +0.622216 q^{5} -1.00000 q^{7} +1.00000 q^{8} +0.622216 q^{10} -2.42864 q^{11} +2.62222 q^{13} -1.00000 q^{14} +1.00000 q^{16} +4.42864 q^{17} -1.00000 q^{19} +0.622216 q^{20} -2.42864 q^{22} +1.37778 q^{23} -4.61285 q^{25} +2.62222 q^{26} -1.00000 q^{28} +9.61285 q^{29} +3.80642 q^{31} +1.00000 q^{32} +4.42864 q^{34} -0.622216 q^{35} +5.80642 q^{37} -1.00000 q^{38} +0.622216 q^{40} +0.755569 q^{41} +7.61285 q^{43} -2.42864 q^{44} +1.37778 q^{46} +0.949145 q^{47} +1.00000 q^{49} -4.61285 q^{50} +2.62222 q^{52} +2.00000 q^{53} -1.51114 q^{55} -1.00000 q^{56} +9.61285 q^{58} -5.61285 q^{61} +3.80642 q^{62} +1.00000 q^{64} +1.63158 q^{65} -4.42864 q^{67} +4.42864 q^{68} -0.622216 q^{70} -0.857279 q^{71} +6.85728 q^{73} +5.80642 q^{74} -1.00000 q^{76} +2.42864 q^{77} -14.7239 q^{79} +0.622216 q^{80} +0.755569 q^{82} +4.75557 q^{83} +2.75557 q^{85} +7.61285 q^{86} -2.42864 q^{88} +14.8573 q^{89} -2.62222 q^{91} +1.37778 q^{92} +0.949145 q^{94} -0.622216 q^{95} +3.67307 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + 2 q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 + 2 * q^5 - 3 * q^7 + 3 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} + 2 q^{5} - 3 q^{7} + 3 q^{8} + 2 q^{10} + 6 q^{11} + 8 q^{13} - 3 q^{14} + 3 q^{16} - 3 q^{19} + 2 q^{20} + 6 q^{22} + 4 q^{23} + 13 q^{25} + 8 q^{26} - 3 q^{28} + 2 q^{29} - 2 q^{31} + 3 q^{32} - 2 q^{35} + 4 q^{37} - 3 q^{38} + 2 q^{40} + 2 q^{41} - 4 q^{43} + 6 q^{44} + 4 q^{46} + 16 q^{47} + 3 q^{49} + 13 q^{50} + 8 q^{52} + 6 q^{53} - 4 q^{55} - 3 q^{56} + 2 q^{58} + 10 q^{61} - 2 q^{62} + 3 q^{64} + 32 q^{65} - 2 q^{70} + 24 q^{71} - 6 q^{73} + 4 q^{74} - 3 q^{76} - 6 q^{77} - 18 q^{79} + 2 q^{80} + 2 q^{82} + 14 q^{83} + 8 q^{85} - 4 q^{86} + 6 q^{88} + 18 q^{89} - 8 q^{91} + 4 q^{92} + 16 q^{94} - 2 q^{95} - 2 q^{97} + 3 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 2 * q^5 - 3 * q^7 + 3 * q^8 + 2 * q^10 + 6 * q^11 + 8 * q^13 - 3 * q^14 + 3 * q^16 - 3 * q^19 + 2 * q^20 + 6 * q^22 + 4 * q^23 + 13 * q^25 + 8 * q^26 - 3 * q^28 + 2 * q^29 - 2 * q^31 + 3 * q^32 - 2 * q^35 + 4 * q^37 - 3 * q^38 + 2 * q^40 + 2 * q^41 - 4 * q^43 + 6 * q^44 + 4 * q^46 + 16 * q^47 + 3 * q^49 + 13 * q^50 + 8 * q^52 + 6 * q^53 - 4 * q^55 - 3 * q^56 + 2 * q^58 + 10 * q^61 - 2 * q^62 + 3 * q^64 + 32 * q^65 - 2 * q^70 + 24 * q^71 - 6 * q^73 + 4 * q^74 - 3 * q^76 - 6 * q^77 - 18 * q^79 + 2 * q^80 + 2 * q^82 + 14 * q^83 + 8 * q^85 - 4 * q^86 + 6 * q^88 + 18 * q^89 - 8 * q^91 + 4 * q^92 + 16 * q^94 - 2 * q^95 - 2 * q^97 + 3 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0.622216	0.278263	0.139132	−	0.990274i	$-0.455569\pi$
$5$	0.622216	0.278263	0.139132	+	0.990274i	$0.455569\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	0.622216	0.196762
$11$	−2.42864	−0.732262	−0.366131	−	0.930563i	$-0.619318\pi$
$11$	−2.42864	−0.732262	−0.366131	+	0.930563i	$0.619318\pi$
$12$	0	0
$13$	2.62222	0.727272	0.363636	−	0.931541i	$-0.381535\pi$
$13$	2.62222	0.727272	0.363636	+	0.931541i	$0.381535\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	4.42864	1.07410	0.537051	−	0.843550i	$-0.319538\pi$
$17$	4.42864	1.07410	0.537051	+	0.843550i	$0.319538\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0.622216	0.139132
$21$	0	0
$22$	−2.42864	−0.517788
$23$	1.37778	0.287288	0.143644	−	0.989629i	$-0.454118\pi$
$23$	1.37778	0.287288	0.143644	+	0.989629i	$0.454118\pi$
$24$	0	0
$25$	−4.61285	−0.922570
$26$	2.62222	0.514259
$27$	0	0
$28$	−1.00000	−0.188982
$29$	9.61285	1.78506	0.892531	−	0.450987i	$-0.148928\pi$
$29$	9.61285	1.78506	0.892531	+	0.450987i	$0.148928\pi$
$30$	0	0
$31$	3.80642	0.683654	0.341827	−	0.939763i	$-0.388954\pi$
$31$	3.80642	0.683654	0.341827	+	0.939763i	$0.388954\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	4.42864	0.759505
$35$	−0.622216	−0.105174
$36$	0	0
$37$	5.80642	0.954570	0.477285	−	0.878749i	$-0.341621\pi$
$37$	5.80642	0.954570	0.477285	+	0.878749i	$0.341621\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	0.622216	0.0983809
$41$	0.755569	0.118000	0.0590000	−	0.998258i	$-0.481209\pi$
$41$	0.755569	0.118000	0.0590000	+	0.998258i	$0.481209\pi$
$42$	0	0
$43$	7.61285	1.16095	0.580474	−	0.814279i	$-0.302867\pi$
$43$	7.61285	1.16095	0.580474	+	0.814279i	$0.302867\pi$
$44$	−2.42864	−0.366131
$45$	0	0
$46$	1.37778	0.203143
$47$	0.949145	0.138447	0.0692235	−	0.997601i	$-0.477948\pi$
$47$	0.949145	0.138447	0.0692235	+	0.997601i	$0.477948\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−4.61285	−0.652355
$51$	0	0
$52$	2.62222	0.363636
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	−1.51114	−0.203762
$56$	−1.00000	−0.133631
$57$	0	0
$58$	9.61285	1.26223
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−5.61285	−0.718652	−0.359326	−	0.933212i	$-0.616993\pi$
$61$	−5.61285	−0.718652	−0.359326	+	0.933212i	$0.616993\pi$
$62$	3.80642	0.483416
$63$	0	0
$64$	1.00000	0.125000
$65$	1.63158	0.202373
$66$	0	0
$67$	−4.42864	−0.541044	−0.270522	−	0.962714i	$-0.587196\pi$
$67$	−4.42864	−0.541044	−0.270522	+	0.962714i	$0.587196\pi$
$68$	4.42864	0.537051
$69$	0	0
$70$	−0.622216	−0.0743690
$71$	−0.857279	−0.101740	−0.0508701	−	0.998705i	$-0.516199\pi$
$71$	−0.857279	−0.101740	−0.0508701	+	0.998705i	$0.516199\pi$
$72$	0	0
$73$	6.85728	0.802584	0.401292	−	0.915950i	$-0.368561\pi$
$73$	6.85728	0.802584	0.401292	+	0.915950i	$0.368561\pi$
$74$	5.80642	0.674983
$75$	0	0
$76$	−1.00000	−0.114708
$77$	2.42864	0.276769
$78$	0	0
$79$	−14.7239	−1.65657	−0.828286	−	0.560306i	$-0.810683\pi$
$79$	−14.7239	−1.65657	−0.828286	+	0.560306i	$0.810683\pi$
$80$	0.622216	0.0695658
$81$	0	0
$82$	0.755569	0.0834386
$83$	4.75557	0.521991	0.260996	−	0.965340i	$-0.415949\pi$
$83$	4.75557	0.521991	0.260996	+	0.965340i	$0.415949\pi$
$84$	0	0
$85$	2.75557	0.298883
$86$	7.61285	0.820914
$87$	0	0
$88$	−2.42864	−0.258894
$89$	14.8573	1.57487	0.787434	−	0.616399i	$-0.211409\pi$
$89$	14.8573	1.57487	0.787434	+	0.616399i	$0.211409\pi$
$90$	0	0
$91$	−2.62222	−0.274883
$92$	1.37778	0.143644
$93$	0	0
$94$	0.949145	0.0978968
$95$	−0.622216	−0.0638380
$96$	0	0
$97$	3.67307	0.372944	0.186472	−	0.982460i	$-0.440295\pi$
$97$	3.67307	0.372944	0.186472	+	0.982460i	$0.440295\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	−4.61285	−0.461285
$101$	−3.37778	−0.336102	−0.168051	−	0.985778i	$-0.553747\pi$
$101$	−3.37778	−0.336102	−0.168051	+	0.985778i	$0.553747\pi$
$102$	0	0
$103$	−10.2953	−1.01442	−0.507212	−	0.861821i	$-0.669324\pi$
$103$	−10.2953	−1.01442	−0.507212	+	0.861821i	$0.669324\pi$
$104$	2.62222	0.257129
$105$	0	0
$106$	2.00000	0.194257
$107$	16.8573	1.62965	0.814827	−	0.579704i	$-0.196832\pi$
$107$	16.8573	1.62965	0.814827	+	0.579704i	$0.196832\pi$
$108$	0	0
$109$	9.41927	0.902203	0.451101	−	0.892473i	$-0.351031\pi$
$109$	9.41927	0.902203	0.451101	+	0.892473i	$0.351031\pi$
$110$	−1.51114	−0.144081
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	4.10171	0.385856	0.192928	−	0.981213i	$-0.438202\pi$
$113$	4.10171	0.385856	0.192928	+	0.981213i	$0.438202\pi$
$114$	0	0
$115$	0.857279	0.0799417
$116$	9.61285	0.892531
$117$	0	0
$118$	0	0
$119$	−4.42864	−0.405973
$120$	0	0
$121$	−5.10171	−0.463792
$122$	−5.61285	−0.508163
$123$	0	0
$124$	3.80642	0.341827
$125$	−5.98126	−0.534981
$126$	0	0
$127$	−1.47949	−0.131284	−0.0656420	−	0.997843i	$-0.520910\pi$
$127$	−1.47949	−0.131284	−0.0656420	+	0.997843i	$0.520910\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	1.63158	0.143099
$131$	−8.10171	−0.707850	−0.353925	−	0.935274i	$-0.615153\pi$
$131$	−8.10171	−0.707850	−0.353925	+	0.935274i	$0.615153\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	−4.42864	−0.382576
$135$	0	0
$136$	4.42864	0.379753
$137$	2.75557	0.235424	0.117712	−	0.993048i	$-0.462444\pi$
$137$	2.75557	0.235424	0.117712	+	0.993048i	$0.462444\pi$
$138$	0	0
$139$	−10.1017	−0.856816	−0.428408	−	0.903585i	$-0.640925\pi$
$139$	−10.1017	−0.856816	−0.428408	+	0.903585i	$0.640925\pi$
$140$	−0.622216	−0.0525868
$141$	0	0
$142$	−0.857279	−0.0719413
$143$	−6.36842	−0.532554
$144$	0	0
$145$	5.98126	0.496717
$146$	6.85728	0.567512
$147$	0	0
$148$	5.80642	0.477285
$149$	1.43801	0.117806	0.0589031	−	0.998264i	$-0.481240\pi$
$149$	1.43801	0.117806	0.0589031	+	0.998264i	$0.481240\pi$
$150$	0	0
$151$	−9.74620	−0.793135	−0.396567	−	0.918006i	$-0.629799\pi$
$151$	−9.74620	−0.793135	−0.396567	+	0.918006i	$0.629799\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	2.42864	0.195705
$155$	2.36842	0.190236
$156$	0	0
$157$	−7.71456	−0.615689	−0.307844	−	0.951437i	$-0.599608\pi$
$157$	−7.71456	−0.615689	−0.307844	+	0.951437i	$0.599608\pi$
$158$	−14.7239	−1.17137
$159$	0	0
$160$	0.622216	0.0491905
$161$	−1.37778	−0.108585
$162$	0	0
$163$	−9.51114	−0.744970	−0.372485	−	0.928038i	$-0.621494\pi$
$163$	−9.51114	−0.744970	−0.372485	+	0.928038i	$0.621494\pi$
$164$	0.755569	0.0590000
$165$	0	0
$166$	4.75557	0.369104
$167$	−5.24443	−0.405826	−0.202913	−	0.979197i	$-0.565041\pi$
$167$	−5.24443	−0.405826	−0.202913	+	0.979197i	$0.565041\pi$
$168$	0	0
$169$	−6.12399	−0.471076
$170$	2.75557	0.211342
$171$	0	0
$172$	7.61285	0.580474
$173$	−23.7146	−1.80298	−0.901492	−	0.432795i	$-0.857527\pi$
$173$	−23.7146	−1.80298	−0.901492	+	0.432795i	$0.857527\pi$
$174$	0	0
$175$	4.61285	0.348699
$176$	−2.42864	−0.183066
$177$	0	0
$178$	14.8573	1.11360
$179$	24.2034	1.80905	0.904524	−	0.426422i	$-0.140226\pi$
$179$	24.2034	1.80905	0.904524	+	0.426422i	$0.140226\pi$
$180$	0	0
$181$	−4.13335	−0.307230	−0.153615	−	0.988131i	$-0.549092\pi$
$181$	−4.13335	−0.307230	−0.153615	+	0.988131i	$0.549092\pi$
$182$	−2.62222	−0.194372
$183$	0	0
$184$	1.37778	0.101572
$185$	3.61285	0.265622
$186$	0	0
$187$	−10.7556	−0.786525
$188$	0.949145	0.0692235
$189$	0	0
$190$	−0.622216	−0.0451403
$191$	20.6035	1.49082	0.745408	−	0.666609i	$-0.232255\pi$
$191$	20.6035	1.49082	0.745408	+	0.666609i	$0.232255\pi$
$192$	0	0
$193$	1.61285	0.116095	0.0580477	−	0.998314i	$-0.481512\pi$
$193$	1.61285	0.116095	0.0580477	+	0.998314i	$0.481512\pi$
$194$	3.67307	0.263711
$195$	0	0
$196$	1.00000	0.0714286
$197$	5.43801	0.387442	0.193721	−	0.981057i	$-0.437944\pi$
$197$	5.43801	0.387442	0.193721	+	0.981057i	$0.437944\pi$
$198$	0	0
$199$	−27.2257	−1.92998	−0.964989	−	0.262290i	$-0.915522\pi$
$199$	−27.2257	−1.92998	−0.964989	+	0.262290i	$0.915522\pi$
$200$	−4.61285	−0.326178
$201$	0	0
$202$	−3.37778	−0.237660
$203$	−9.61285	−0.674690
$204$	0	0
$205$	0.470127	0.0328351
$206$	−10.2953	−0.717307
$207$	0	0
$208$	2.62222	0.181818
$209$	2.42864	0.167993
$210$	0	0
$211$	20.8988	1.43873	0.719365	−	0.694632i	$-0.244433\pi$
$211$	20.8988	1.43873	0.719365	+	0.694632i	$0.244433\pi$
$212$	2.00000	0.137361
$213$	0	0
$214$	16.8573	1.15234
$215$	4.73683	0.323049
$216$	0	0
$217$	−3.80642	−0.258397
$218$	9.41927	0.637954
$219$	0	0
$220$	−1.51114	−0.101881
$221$	11.6128	0.781165
$222$	0	0
$223$	11.1526	0.746831	0.373416	−	0.927664i	$-0.378187\pi$
$223$	11.1526	0.746831	0.373416	+	0.927664i	$0.378187\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	4.10171	0.272842
$227$	−17.1240	−1.13656	−0.568279	−	0.822836i	$-0.692391\pi$
$227$	−17.1240	−1.13656	−0.568279	+	0.822836i	$0.692391\pi$
$228$	0	0
$229$	−15.2444	−1.00738	−0.503690	−	0.863884i	$-0.668025\pi$
$229$	−15.2444	−1.00738	−0.503690	+	0.863884i	$0.668025\pi$
$230$	0.857279	0.0565273
$231$	0	0
$232$	9.61285	0.631114
$233$	16.8573	1.10436	0.552179	−	0.833726i	$-0.313797\pi$
$233$	16.8573	1.10436	0.552179	+	0.833726i	$0.313797\pi$
$234$	0	0
$235$	0.590573	0.0385247
$236$	0	0
$237$	0	0
$238$	−4.42864	−0.287066
$239$	4.99063	0.322817	0.161409	−	0.986888i	$-0.448396\pi$
$239$	4.99063	0.322817	0.161409	+	0.986888i	$0.448396\pi$
$240$	0	0
$241$	0.326929	0.0210594	0.0105297	−	0.999945i	$-0.496648\pi$
$241$	0.326929	0.0210594	0.0105297	+	0.999945i	$0.496648\pi$
$242$	−5.10171	−0.327950
$243$	0	0
$244$	−5.61285	−0.359326
$245$	0.622216	0.0397519
$246$	0	0
$247$	−2.62222	−0.166848
$248$	3.80642	0.241708
$249$	0	0
$250$	−5.98126	−0.378288
$251$	6.38715	0.403153	0.201577	−	0.979473i	$-0.435393\pi$
$251$	6.38715	0.403153	0.201577	+	0.979473i	$0.435393\pi$
$252$	0	0
$253$	−3.34614	−0.210370
$254$	−1.47949	−0.0928317
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	0	0
$259$	−5.80642	−0.360794
$260$	1.63158	0.101187
$261$	0	0
$262$	−8.10171	−0.500525
$263$	−6.88892	−0.424789	−0.212395	−	0.977184i	$-0.568126\pi$
$263$	−6.88892	−0.424789	−0.212395	+	0.977184i	$0.568126\pi$
$264$	0	0
$265$	1.24443	0.0764448
$266$	1.00000	0.0613139
$267$	0	0
$268$	−4.42864	−0.270522
$269$	−15.7146	−0.958134	−0.479067	−	0.877778i	$-0.659025\pi$
$269$	−15.7146	−0.958134	−0.479067	+	0.877778i	$0.659025\pi$
$270$	0	0
$271$	−18.1017	−1.09960	−0.549800	−	0.835296i	$-0.685296\pi$
$271$	−18.1017	−1.09960	−0.549800	+	0.835296i	$0.685296\pi$
$272$	4.42864	0.268526
$273$	0	0
$274$	2.75557	0.166470
$275$	11.2029	0.675563
$276$	0	0
$277$	6.59057	0.395989	0.197995	−	0.980203i	$-0.436557\pi$
$277$	6.59057	0.395989	0.197995	+	0.980203i	$0.436557\pi$
$278$	−10.1017	−0.605860
$279$	0	0
$280$	−0.622216	−0.0371845
$281$	−17.6128	−1.05069	−0.525347	−	0.850888i	$-0.676065\pi$
$281$	−17.6128	−1.05069	−0.525347	+	0.850888i	$0.676065\pi$
$282$	0	0
$283$	7.61285	0.452537	0.226268	−	0.974065i	$-0.427347\pi$
$283$	7.61285	0.452537	0.226268	+	0.974065i	$0.427347\pi$
$284$	−0.857279	−0.0508701
$285$	0	0
$286$	−6.36842	−0.376572
$287$	−0.755569	−0.0445998
$288$	0	0
$289$	2.61285	0.153697
$290$	5.98126	0.351232
$291$	0	0
$292$	6.85728	0.401292
$293$	17.2257	1.00634	0.503168	−	0.864189i	$-0.332168\pi$
$293$	17.2257	1.00634	0.503168	+	0.864189i	$0.332168\pi$
$294$	0	0
$295$	0	0
$296$	5.80642	0.337492
$297$	0	0
$298$	1.43801	0.0833015
$299$	3.61285	0.208936
$300$	0	0
$301$	−7.61285	−0.438797
$302$	−9.74620	−0.560831
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	−3.49240	−0.199974
$306$	0	0
$307$	−9.24443	−0.527608	−0.263804	−	0.964576i	$-0.584977\pi$
$307$	−9.24443	−0.527608	−0.263804	+	0.964576i	$0.584977\pi$
$308$	2.42864	0.138385
$309$	0	0
$310$	2.36842	0.134517
$311$	17.0321	0.965803	0.482901	−	0.875675i	$-0.339583\pi$
$311$	17.0321	0.965803	0.482901	+	0.875675i	$0.339583\pi$
$312$	0	0
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	−7.71456	−0.435358
$315$	0	0
$316$	−14.7239	−0.828286
$317$	−5.22570	−0.293504	−0.146752	−	0.989173i	$-0.546882\pi$
$317$	−5.22570	−0.293504	−0.146752	+	0.989173i	$0.546882\pi$
$318$	0	0
$319$	−23.3461	−1.30713
$320$	0.622216	0.0347829
$321$	0	0
$322$	−1.37778	−0.0767809
$323$	−4.42864	−0.246416
$324$	0	0
$325$	−12.0959	−0.670959
$326$	−9.51114	−0.526773
$327$	0	0
$328$	0.755569	0.0417193
$329$	−0.949145	−0.0523281
$330$	0	0
$331$	−19.7748	−1.08692	−0.543460	−	0.839435i	$-0.682886\pi$
$331$	−19.7748	−1.08692	−0.543460	+	0.839435i	$0.682886\pi$
$332$	4.75557	0.260996
$333$	0	0
$334$	−5.24443	−0.286963
$335$	−2.75557	−0.150553
$336$	0	0
$337$	13.1427	0.715930	0.357965	−	0.933735i	$-0.383471\pi$
$337$	13.1427	0.715930	0.357965	+	0.933735i	$0.383471\pi$
$338$	−6.12399	−0.333101
$339$	0	0
$340$	2.75557	0.149442
$341$	−9.24443	−0.500614
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	7.61285	0.410457
$345$	0	0
$346$	−23.7146	−1.27490
$347$	−22.3082	−1.19757	−0.598783	−	0.800911i	$-0.704349\pi$
$347$	−22.3082	−1.19757	−0.598783	+	0.800911i	$0.704349\pi$
$348$	0	0
$349$	−3.24443	−0.173670	−0.0868352	−	0.996223i	$-0.527675\pi$
$349$	−3.24443	−0.173670	−0.0868352	+	0.996223i	$0.527675\pi$
$350$	4.61285	0.246567
$351$	0	0
$352$	−2.42864	−0.129447
$353$	−11.9585	−0.636487	−0.318244	−	0.948009i	$-0.603093\pi$
$353$	−11.9585	−0.636487	−0.318244	+	0.948009i	$0.603093\pi$
$354$	0	0
$355$	−0.533412	−0.0283106
$356$	14.8573	0.787434
$357$	0	0
$358$	24.2034	1.27919
$359$	27.7462	1.46439	0.732194	−	0.681096i	$-0.238496\pi$
$359$	27.7462	1.46439	0.732194	+	0.681096i	$0.238496\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−4.13335	−0.217244
$363$	0	0
$364$	−2.62222	−0.137441
$365$	4.26671	0.223330
$366$	0	0
$367$	21.1240	1.10266	0.551332	−	0.834286i	$-0.314120\pi$
$367$	21.1240	1.10266	0.551332	+	0.834286i	$0.314120\pi$
$368$	1.37778	0.0718220
$369$	0	0
$370$	3.61285	0.187823
$371$	−2.00000	−0.103835
$372$	0	0
$373$	12.1748	0.630389	0.315195	−	0.949027i	$-0.397930\pi$
$373$	12.1748	0.630389	0.315195	+	0.949027i	$0.397930\pi$
$374$	−10.7556	−0.556157
$375$	0	0
$376$	0.949145	0.0489484
$377$	25.2070	1.29822
$378$	0	0
$379$	−24.5116	−1.25908	−0.629539	−	0.776969i	$-0.716756\pi$
$379$	−24.5116	−1.25908	−0.629539	+	0.776969i	$0.716756\pi$
$380$	−0.622216	−0.0319190
$381$	0	0
$382$	20.6035	1.05417
$383$	−30.1017	−1.53813	−0.769063	−	0.639173i	$-0.779277\pi$
$383$	−30.1017	−1.53813	−0.769063	+	0.639173i	$0.779277\pi$
$384$	0	0
$385$	1.51114	0.0770147
$386$	1.61285	0.0820918
$387$	0	0
$388$	3.67307	0.186472
$389$	9.70471	0.492049	0.246024	−	0.969264i	$-0.420876\pi$
$389$	9.70471	0.492049	0.246024	+	0.969264i	$0.420876\pi$
$390$	0	0
$391$	6.10171	0.308577
$392$	1.00000	0.0505076
$393$	0	0
$394$	5.43801	0.273963
$395$	−9.16146	−0.460963
$396$	0	0
$397$	5.61285	0.281701	0.140850	−	0.990031i	$-0.455016\pi$
$397$	5.61285	0.281701	0.140850	+	0.990031i	$0.455016\pi$
$398$	−27.2257	−1.36470
$399$	0	0
$400$	−4.61285	−0.230642
$401$	−18.0830	−0.903021	−0.451510	−	0.892266i	$-0.649115\pi$
$401$	−18.0830	−0.903021	−0.451510	+	0.892266i	$0.649115\pi$
$402$	0	0
$403$	9.98126	0.497202
$404$	−3.37778	−0.168051
$405$	0	0
$406$	−9.61285	−0.477078
$407$	−14.1017	−0.698996
$408$	0	0
$409$	32.4099	1.60257	0.801283	−	0.598285i	$-0.204151\pi$
$409$	32.4099	1.60257	0.801283	+	0.598285i	$0.204151\pi$
$410$	0.470127	0.0232179
$411$	0	0
$412$	−10.2953	−0.507212
$413$	0	0
$414$	0	0
$415$	2.95899	0.145251
$416$	2.62222	0.128565
$417$	0	0
$418$	2.42864	0.118789
$419$	−22.9403	−1.12070	−0.560352	−	0.828254i	$-0.689334\pi$
$419$	−22.9403	−1.12070	−0.560352	+	0.828254i	$0.689334\pi$
$420$	0	0
$421$	18.6637	0.909613	0.454807	−	0.890590i	$-0.349708\pi$
$421$	18.6637	0.909613	0.454807	+	0.890590i	$0.349708\pi$
$422$	20.8988	1.01734
$423$	0	0
$424$	2.00000	0.0971286
$425$	−20.4286	−0.990935
$426$	0	0
$427$	5.61285	0.271625
$428$	16.8573	0.814827
$429$	0	0
$430$	4.73683	0.228430
$431$	−10.4889	−0.505231	−0.252615	−	0.967567i	$-0.581291\pi$
$431$	−10.4889	−0.505231	−0.252615	+	0.967567i	$0.581291\pi$
$432$	0	0
$433$	−38.2449	−1.83793	−0.918966	−	0.394336i	$-0.870975\pi$
$433$	−38.2449	−1.83793	−0.918966	+	0.394336i	$0.870975\pi$
$434$	−3.80642	−0.182714
$435$	0	0
$436$	9.41927	0.451101
$437$	−1.37778	−0.0659084
$438$	0	0
$439$	17.5210	0.836231	0.418115	−	0.908394i	$-0.362691\pi$
$439$	17.5210	0.836231	0.418115	+	0.908394i	$0.362691\pi$
$440$	−1.51114	−0.0720407
$441$	0	0
$442$	11.6128	0.552367
$443$	−12.9175	−0.613729	−0.306865	−	0.951753i	$-0.599280\pi$
$443$	−12.9175	−0.613729	−0.306865	+	0.951753i	$0.599280\pi$
$444$	0	0
$445$	9.24443	0.438228
$446$	11.1526	0.528089
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	6.97773	0.329299	0.164650	−	0.986352i	$-0.447351\pi$
$449$	6.97773	0.329299	0.164650	+	0.986352i	$0.447351\pi$
$450$	0	0
$451$	−1.83500	−0.0864070
$452$	4.10171	0.192928
$453$	0	0
$454$	−17.1240	−0.803668
$455$	−1.63158	−0.0764898
$456$	0	0
$457$	−32.8385	−1.53612	−0.768061	−	0.640377i	$-0.778778\pi$
$457$	−32.8385	−1.53612	−0.768061	+	0.640377i	$0.778778\pi$
$458$	−15.2444	−0.712325
$459$	0	0
$460$	0.857279	0.0399708
$461$	−34.2163	−1.59361	−0.796807	−	0.604234i	$-0.793479\pi$
$461$	−34.2163	−1.59361	−0.796807	+	0.604234i	$0.793479\pi$
$462$	0	0
$463$	17.2444	0.801417	0.400708	−	0.916206i	$-0.368764\pi$
$463$	17.2444	0.801417	0.400708	+	0.916206i	$0.368764\pi$
$464$	9.61285	0.446265
$465$	0	0
$466$	16.8573	0.780898
$467$	−1.52987	−0.0707941	−0.0353970	−	0.999373i	$-0.511270\pi$
$467$	−1.52987	−0.0707941	−0.0353970	+	0.999373i	$0.511270\pi$
$468$	0	0
$469$	4.42864	0.204496
$470$	0.590573	0.0272411
$471$	0	0
$472$	0	0
$473$	−18.4889	−0.850119
$474$	0	0
$475$	4.61285	0.211652
$476$	−4.42864	−0.202986
$477$	0	0
$478$	4.99063	0.228266
$479$	−19.5210	−0.891936	−0.445968	−	0.895049i	$-0.647141\pi$
$479$	−19.5210	−0.891936	−0.445968	+	0.895049i	$0.647141\pi$
$480$	0	0
$481$	15.2257	0.694232
$482$	0.326929	0.0148912
$483$	0	0
$484$	−5.10171	−0.231896
$485$	2.28544	0.103777
$486$	0	0
$487$	−30.0701	−1.36260	−0.681302	−	0.732002i	$-0.738586\pi$
$487$	−30.0701	−1.36260	−0.681302	+	0.732002i	$0.738586\pi$
$488$	−5.61285	−0.254082
$489$	0	0
$490$	0.622216	0.0281088
$491$	−8.40990	−0.379534	−0.189767	−	0.981829i	$-0.560773\pi$
$491$	−8.40990	−0.379534	−0.189767	+	0.981829i	$0.560773\pi$
$492$	0	0
$493$	42.5718	1.91734
$494$	−2.62222	−0.117979
$495$	0	0
$496$	3.80642	0.170913
$497$	0.857279	0.0384542
$498$	0	0
$499$	−4.26671	−0.191004	−0.0955020	−	0.995429i	$-0.530446\pi$
$499$	−4.26671	−0.191004	−0.0955020	+	0.995429i	$0.530446\pi$
$500$	−5.98126	−0.267490
$501$	0	0
$502$	6.38715	0.285073
$503$	34.0098	1.51642	0.758212	−	0.652008i	$-0.226073\pi$
$503$	34.0098	1.51642	0.758212	+	0.652008i	$0.226073\pi$
$504$	0	0
$505$	−2.10171	−0.0935249
$506$	−3.34614	−0.148754
$507$	0	0
$508$	−1.47949	−0.0656420
$509$	−32.9590	−1.46088	−0.730441	−	0.682976i	$-0.760685\pi$
$509$	−32.9590	−1.46088	−0.730441	+	0.682976i	$0.760685\pi$
$510$	0	0
$511$	−6.85728	−0.303348
$512$	1.00000	0.0441942
$513$	0	0
$514$	2.00000	0.0882162
$515$	−6.40589	−0.282277
$516$	0	0
$517$	−2.30513	−0.101380
$518$	−5.80642	−0.255120
$519$	0	0
$520$	1.63158	0.0715497
$521$	−39.9180	−1.74884	−0.874419	−	0.485171i	$-0.838757\pi$
$521$	−39.9180	−1.74884	−0.874419	+	0.485171i	$0.838757\pi$
$522$	0	0
$523$	−33.9813	−1.48590	−0.742948	−	0.669349i	$-0.766573\pi$
$523$	−33.9813	−1.48590	−0.742948	+	0.669349i	$0.766573\pi$
$524$	−8.10171	−0.353925
$525$	0	0
$526$	−6.88892	−0.300371
$527$	16.8573	0.734315
$528$	0	0
$529$	−21.1017	−0.917466
$530$	1.24443	0.0540546
$531$	0	0
$532$	1.00000	0.0433555
$533$	1.98126	0.0858181
$534$	0	0
$535$	10.4889	0.453473
$536$	−4.42864	−0.191288
$537$	0	0
$538$	−15.7146	−0.677503
$539$	−2.42864	−0.104609
$540$	0	0
$541$	−12.3684	−0.531760	−0.265880	−	0.964006i	$-0.585662\pi$
$541$	−12.3684	−0.531760	−0.265880	+	0.964006i	$0.585662\pi$
$542$	−18.1017	−0.777535
$543$	0	0
$544$	4.42864	0.189876
$545$	5.86082	0.251050
$546$	0	0
$547$	−6.71408	−0.287073	−0.143537	−	0.989645i	$-0.545848\pi$
$547$	−6.71408	−0.287073	−0.143537	+	0.989645i	$0.545848\pi$
$548$	2.75557	0.117712
$549$	0	0
$550$	11.2029	0.477695
$551$	−9.61285	−0.409521
$552$	0	0
$553$	14.7239	0.626125
$554$	6.59057	0.280007
$555$	0	0
$556$	−10.1017	−0.428408
$557$	−30.5620	−1.29495	−0.647477	−	0.762085i	$-0.724176\pi$
$557$	−30.5620	−1.29495	−0.647477	+	0.762085i	$0.724176\pi$
$558$	0	0
$559$	19.9625	0.844325
$560$	−0.622216	−0.0262934
$561$	0	0
$562$	−17.6128	−0.742953
$563$	−7.02227	−0.295954	−0.147977	−	0.988991i	$-0.547276\pi$
$563$	−7.02227	−0.295954	−0.147977	+	0.988991i	$0.547276\pi$
$564$	0	0
$565$	2.55215	0.107370
$566$	7.61285	0.319992
$567$	0	0
$568$	−0.857279	−0.0359706
$569$	24.1847	1.01387	0.506937	−	0.861983i	$-0.330778\pi$
$569$	24.1847	1.01387	0.506937	+	0.861983i	$0.330778\pi$
$570$	0	0
$571$	−6.22216	−0.260389	−0.130195	−	0.991488i	$-0.541560\pi$
$571$	−6.22216	−0.260389	−0.130195	+	0.991488i	$0.541560\pi$
$572$	−6.36842	−0.266277
$573$	0	0
$574$	−0.755569	−0.0315368
$575$	−6.35551	−0.265043
$576$	0	0
$577$	39.7975	1.65679	0.828396	−	0.560142i	$-0.189254\pi$
$577$	39.7975	1.65679	0.828396	+	0.560142i	$0.189254\pi$
$578$	2.61285	0.108680
$579$	0	0
$580$	5.98126	0.248358
$581$	−4.75557	−0.197294
$582$	0	0
$583$	−4.85728	−0.201168
$584$	6.85728	0.283756
$585$	0	0
$586$	17.2257	0.711587
$587$	24.5718	1.01419	0.507094	−	0.861891i	$-0.330720\pi$
$587$	24.5718	1.01419	0.507094	+	0.861891i	$0.330720\pi$
$588$	0	0
$589$	−3.80642	−0.156841
$590$	0	0
$591$	0	0
$592$	5.80642	0.238643
$593$	−8.81579	−0.362021	−0.181011	−	0.983481i	$-0.557937\pi$
$593$	−8.81579	−0.362021	−0.181011	+	0.983481i	$0.557937\pi$
$594$	0	0
$595$	−2.75557	−0.112967
$596$	1.43801	0.0589031
$597$	0	0
$598$	3.61285	0.147740
$599$	4.47013	0.182644	0.0913222	−	0.995821i	$-0.470891\pi$
$599$	4.47013	0.182644	0.0913222	+	0.995821i	$0.470891\pi$
$600$	0	0
$601$	−12.1432	−0.495331	−0.247666	−	0.968846i	$-0.579663\pi$
$601$	−12.1432	−0.495331	−0.247666	+	0.968846i	$0.579663\pi$
$602$	−7.61285	−0.310277
$603$	0	0
$604$	−9.74620	−0.396567
$605$	−3.17436	−0.129056
$606$	0	0
$607$	35.7431	1.45077	0.725385	−	0.688344i	$-0.241662\pi$
$607$	35.7431	1.45077	0.725385	+	0.688344i	$0.241662\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	−3.49240	−0.141403
$611$	2.48886	0.100689
$612$	0	0
$613$	15.5111	0.626489	0.313244	−	0.949673i	$-0.398584\pi$
$613$	15.5111	0.626489	0.313244	+	0.949673i	$0.398584\pi$
$614$	−9.24443	−0.373075
$615$	0	0
$616$	2.42864	0.0978527
$617$	16.6539	0.670459	0.335230	−	0.942136i	$-0.391186\pi$
$617$	16.6539	0.670459	0.335230	+	0.942136i	$0.391186\pi$
$618$	0	0
$619$	−29.9813	−1.20505	−0.602524	−	0.798100i	$-0.705838\pi$
$619$	−29.9813	−1.20505	−0.602524	+	0.798100i	$0.705838\pi$
$620$	2.36842	0.0951179
$621$	0	0
$622$	17.0321	0.682926
$623$	−14.8573	−0.595244
$624$	0	0
$625$	19.3426	0.773704
$626$	−14.0000	−0.559553
$627$	0	0
$628$	−7.71456	−0.307844
$629$	25.7146	1.02531
$630$	0	0
$631$	32.4701	1.29262	0.646308	−	0.763077i	$-0.276312\pi$
$631$	32.4701	1.29262	0.646308	+	0.763077i	$0.276312\pi$
$632$	−14.7239	−0.585686
$633$	0	0
$634$	−5.22570	−0.207539
$635$	−0.920565	−0.0365315
$636$	0	0
$637$	2.62222	0.103896
$638$	−23.3461	−0.924283
$639$	0	0
$640$	0.622216	0.0245952
$641$	−17.2257	−0.680374	−0.340187	−	0.940358i	$-0.610490\pi$
$641$	−17.2257	−0.680374	−0.340187	+	0.940358i	$0.610490\pi$
$642$	0	0
$643$	−29.7975	−1.17510	−0.587550	−	0.809188i	$-0.699907\pi$
$643$	−29.7975	−1.17510	−0.587550	+	0.809188i	$0.699907\pi$
$644$	−1.37778	−0.0542923
$645$	0	0
$646$	−4.42864	−0.174242
$647$	−42.0098	−1.65158	−0.825789	−	0.563980i	$-0.809270\pi$
$647$	−42.0098	−1.65158	−0.825789	+	0.563980i	$0.809270\pi$
$648$	0	0
$649$	0	0
$650$	−12.0959	−0.474440
$651$	0	0
$652$	−9.51114	−0.372485
$653$	25.5210	0.998713	0.499357	−	0.866397i	$-0.333570\pi$
$653$	25.5210	0.998713	0.499357	+	0.866397i	$0.333570\pi$
$654$	0	0
$655$	−5.04101	−0.196969
$656$	0.755569	0.0295000
$657$	0	0
$658$	−0.949145	−0.0370015
$659$	−29.0607	−1.13204	−0.566022	−	0.824390i	$-0.691518\pi$
$659$	−29.0607	−1.13204	−0.566022	+	0.824390i	$0.691518\pi$
$660$	0	0
$661$	28.9906	1.12760	0.563802	−	0.825910i	$-0.309338\pi$
$661$	28.9906	1.12760	0.563802	+	0.825910i	$0.309338\pi$
$662$	−19.7748	−0.768569
$663$	0	0
$664$	4.75557	0.184552
$665$	0.622216	0.0241285
$666$	0	0
$667$	13.2444	0.512826
$668$	−5.24443	−0.202913
$669$	0	0
$670$	−2.75557	−0.106457
$671$	13.6316	0.526241
$672$	0	0
$673$	31.9813	1.23279	0.616394	−	0.787438i	$-0.288593\pi$
$673$	31.9813	1.23279	0.616394	+	0.787438i	$0.288593\pi$
$674$	13.1427	0.506239
$675$	0	0
$676$	−6.12399	−0.235538
$677$	−4.01874	−0.154453	−0.0772263	−	0.997014i	$-0.524606\pi$
$677$	−4.01874	−0.154453	−0.0772263	+	0.997014i	$0.524606\pi$
$678$	0	0
$679$	−3.67307	−0.140960
$680$	2.75557	0.105671
$681$	0	0
$682$	−9.24443	−0.353988
$683$	−4.53341	−0.173466	−0.0867331	−	0.996232i	$-0.527643\pi$
$683$	−4.53341	−0.173466	−0.0867331	+	0.996232i	$0.527643\pi$
$684$	0	0
$685$	1.71456	0.0655099
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	7.61285	0.290237
$689$	5.24443	0.199797
$690$	0	0
$691$	−38.6548	−1.47050	−0.735249	−	0.677797i	$-0.762935\pi$
$691$	−38.6548	−1.47050	−0.735249	+	0.677797i	$0.762935\pi$
$692$	−23.7146	−0.901492
$693$	0	0
$694$	−22.3082	−0.846807
$695$	−6.28544	−0.238420
$696$	0	0
$697$	3.34614	0.126744
$698$	−3.24443	−0.122804
$699$	0	0
$700$	4.61285	0.174349
$701$	26.6824	1.00778	0.503891	−	0.863767i	$-0.331901\pi$
$701$	26.6824	1.00778	0.503891	+	0.863767i	$0.331901\pi$
$702$	0	0
$703$	−5.80642	−0.218993
$704$	−2.42864	−0.0915328
$705$	0	0
$706$	−11.9585	−0.450065
$707$	3.37778	0.127035
$708$	0	0
$709$	30.6735	1.15197	0.575985	−	0.817461i	$-0.304619\pi$
$709$	30.6735	1.15197	0.575985	+	0.817461i	$0.304619\pi$
$710$	−0.533412	−0.0200186
$711$	0	0
$712$	14.8573	0.556800
$713$	5.24443	0.196405
$714$	0	0
$715$	−3.96253	−0.148190
$716$	24.2034	0.904524
$717$	0	0
$718$	27.7462	1.03548
$719$	2.58073	0.0962449	0.0481225	−	0.998841i	$-0.484676\pi$
$719$	2.58073	0.0962449	0.0481225	+	0.998841i	$0.484676\pi$
$720$	0	0
$721$	10.2953	0.383417
$722$	1.00000	0.0372161
$723$	0	0
$724$	−4.13335	−0.153615
$725$	−44.3426	−1.64684
$726$	0	0
$727$	−4.20342	−0.155896	−0.0779481	−	0.996957i	$-0.524837\pi$
$727$	−4.20342	−0.155896	−0.0779481	+	0.996957i	$0.524837\pi$
$728$	−2.62222	−0.0971858
$729$	0	0
$730$	4.26671	0.157918
$731$	33.7146	1.24698
$732$	0	0
$733$	0.0187359	0.000692025 0	0.000346012	−	1.00000i	$-0.499890\pi$
$733$	0.0187359	0.000692025 0	0.000346012		1.00000i	$0.499890\pi$
$734$	21.1240	0.779701
$735$	0	0
$736$	1.37778	0.0507858
$737$	10.7556	0.396186
$738$	0	0
$739$	−36.0830	−1.32733	−0.663667	−	0.748028i	$-0.731001\pi$
$739$	−36.0830	−1.32733	−0.663667	+	0.748028i	$0.731001\pi$
$740$	3.61285	0.132811
$741$	0	0
$742$	−2.00000	−0.0734223
$743$	16.8573	0.618434	0.309217	−	0.950992i	$-0.399933\pi$
$743$	16.8573	0.618434	0.309217	+	0.950992i	$0.399933\pi$
$744$	0	0
$745$	0.894751	0.0327811
$746$	12.1748	0.445753
$747$	0	0
$748$	−10.7556	−0.393263
$749$	−16.8573	−0.615952
$750$	0	0
$751$	30.8702	1.12647	0.563235	−	0.826297i	$-0.309557\pi$
$751$	30.8702	1.12647	0.563235	+	0.826297i	$0.309557\pi$
$752$	0.949145	0.0346118
$753$	0	0
$754$	25.2070	0.917983
$755$	−6.06424	−0.220700
$756$	0	0
$757$	31.9180	1.16008	0.580039	−	0.814588i	$-0.303037\pi$
$757$	31.9180	1.16008	0.580039	+	0.814588i	$0.303037\pi$
$758$	−24.5116	−0.890302
$759$	0	0
$760$	−0.622216	−0.0225701
$761$	28.5116	1.03355	0.516773	−	0.856123i	$-0.327133\pi$
$761$	28.5116	1.03355	0.516773	+	0.856123i	$0.327133\pi$
$762$	0	0
$763$	−9.41927	−0.341001
$764$	20.6035	0.745408
$765$	0	0
$766$	−30.1017	−1.08762
$767$	0	0
$768$	0	0
$769$	24.4514	0.881740	0.440870	−	0.897571i	$-0.354670\pi$
$769$	24.4514	0.881740	0.440870	+	0.897571i	$0.354670\pi$
$770$	1.51114	0.0544576
$771$	0	0
$772$	1.61285	0.0580477
$773$	−0.0187359	−0.000673882 0	−0.000336941	−	1.00000i	$-0.500107\pi$
$773$	−0.0187359	−0.000673882 0	−0.000336941		1.00000i	$0.500107\pi$
$774$	0	0
$775$	−17.5585	−0.630718
$776$	3.67307	0.131856
$777$	0	0
$778$	9.70471	0.347931
$779$	−0.755569	−0.0270711
$780$	0	0
$781$	2.08202	0.0745006
$782$	6.10171	0.218197
$783$	0	0
$784$	1.00000	0.0357143
$785$	−4.80012	−0.171324
$786$	0	0
$787$	9.83500	0.350580	0.175290	−	0.984517i	$-0.443914\pi$
$787$	9.83500	0.350580	0.175290	+	0.984517i	$0.443914\pi$
$788$	5.43801	0.193721
$789$	0	0
$790$	−9.16146	−0.325950
$791$	−4.10171	−0.145840
$792$	0	0
$793$	−14.7181	−0.522655
$794$	5.61285	0.199193
$795$	0	0
$796$	−27.2257	−0.964989
$797$	−21.9367	−0.777038	−0.388519	−	0.921441i	$-0.627013\pi$
$797$	−21.9367	−0.777038	−0.388519	+	0.921441i	$0.627013\pi$
$798$	0	0
$799$	4.20342	0.148706
$800$	−4.61285	−0.163089
$801$	0	0
$802$	−18.0830	−0.638532
$803$	−16.6539	−0.587702
$804$	0	0
$805$	−0.857279	−0.0302151
$806$	9.98126	0.351575
$807$	0	0
$808$	−3.37778	−0.118830
$809$	−43.0420	−1.51327	−0.756637	−	0.653835i	$-0.773159\pi$
$809$	−43.0420	−1.51327	−0.756637	+	0.653835i	$0.773159\pi$
$810$	0	0
$811$	41.4104	1.45412	0.727058	−	0.686577i	$-0.240887\pi$
$811$	41.4104	1.45412	0.727058	+	0.686577i	$0.240887\pi$
$812$	−9.61285	−0.337345
$813$	0	0
$814$	−14.1017	−0.494265
$815$	−5.91798	−0.207298
$816$	0	0
$817$	−7.61285	−0.266340
$818$	32.4099	1.13319
$819$	0	0
$820$	0.470127	0.0164175
$821$	50.7654	1.77173	0.885863	−	0.463948i	$-0.153567\pi$
$821$	50.7654	1.77173	0.885863	+	0.463948i	$0.153567\pi$
$822$	0	0
$823$	−13.9813	−0.487356	−0.243678	−	0.969856i	$-0.578354\pi$
$823$	−13.9813	−0.487356	−0.243678	+	0.969856i	$0.578354\pi$
$824$	−10.2953	−0.358653
$825$	0	0
$826$	0	0
$827$	34.6923	1.20637	0.603184	−	0.797602i	$-0.293898\pi$
$827$	34.6923	1.20637	0.603184	+	0.797602i	$0.293898\pi$
$828$	0	0
$829$	−1.96836	−0.0683639	−0.0341819	−	0.999416i	$-0.510883\pi$
$829$	−1.96836	−0.0683639	−0.0341819	+	0.999416i	$0.510883\pi$
$830$	2.95899	0.102708
$831$	0	0
$832$	2.62222	0.0909090
$833$	4.42864	0.153443
$834$	0	0
$835$	−3.26317	−0.112927
$836$	2.42864	0.0839963
$837$	0	0
$838$	−22.9403	−0.792458
$839$	−2.16500	−0.0747440	−0.0373720	−	0.999301i	$-0.511899\pi$
$839$	−2.16500	−0.0747440	−0.0373720	+	0.999301i	$0.511899\pi$
$840$	0	0
$841$	63.4068	2.18644
$842$	18.6637	0.643194
$843$	0	0
$844$	20.8988	0.719365
$845$	−3.81044	−0.131083
$846$	0	0
$847$	5.10171	0.175297
$848$	2.00000	0.0686803
$849$	0	0
$850$	−20.4286	−0.700697
$851$	8.00000	0.274236
$852$	0	0
$853$	55.7975	1.91047	0.955236	−	0.295846i	$-0.0956016\pi$
$853$	55.7975	1.91047	0.955236	+	0.295846i	$0.0956016\pi$
$854$	5.61285	0.192068
$855$	0	0
$856$	16.8573	0.576170
$857$	−52.5718	−1.79582	−0.897910	−	0.440179i	$-0.854915\pi$
$857$	−52.5718	−1.79582	−0.897910	+	0.440179i	$0.854915\pi$
$858$	0	0
$859$	2.95899	0.100959	0.0504797	−	0.998725i	$-0.483925\pi$
$859$	2.95899	0.100959	0.0504797	+	0.998725i	$0.483925\pi$
$860$	4.73683	0.161525
$861$	0	0
$862$	−10.4889	−0.357252
$863$	−13.0607	−0.444591	−0.222296	−	0.974979i	$-0.571355\pi$
$863$	−13.0607	−0.444591	−0.222296	+	0.974979i	$0.571355\pi$
$864$	0	0
$865$	−14.7556	−0.501704
$866$	−38.2449	−1.29961
$867$	0	0
$868$	−3.80642	−0.129198
$869$	35.7591	1.21304
$870$	0	0
$871$	−11.6128	−0.393486
$872$	9.41927	0.318977
$873$	0	0
$874$	−1.37778	−0.0466043
$875$	5.98126	0.202204
$876$	0	0
$877$	−24.1116	−0.814189	−0.407095	−	0.913386i	$-0.633458\pi$
$877$	−24.1116	−0.814189	−0.407095	+	0.913386i	$0.633458\pi$
$878$	17.5210	0.591304
$879$	0	0
$880$	−1.51114	−0.0509404
$881$	44.0415	1.48380	0.741898	−	0.670513i	$-0.233926\pi$
$881$	44.0415	1.48380	0.741898	+	0.670513i	$0.233926\pi$
$882$	0	0
$883$	−9.24443	−0.311100	−0.155550	−	0.987828i	$-0.549715\pi$
$883$	−9.24443	−0.311100	−0.155550	+	0.987828i	$0.549715\pi$
$884$	11.6128	0.390582
$885$	0	0
$886$	−12.9175	−0.433972
$887$	−8.77430	−0.294612	−0.147306	−	0.989091i	$-0.547060\pi$
$887$	−8.77430	−0.294612	−0.147306	+	0.989091i	$0.547060\pi$
$888$	0	0
$889$	1.47949	0.0496207
$890$	9.24443	0.309874
$891$	0	0
$892$	11.1526	0.373416
$893$	−0.949145	−0.0317619
$894$	0	0
$895$	15.0597	0.503392
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	6.97773	0.232850
$899$	36.5906	1.22036
$900$	0	0
$901$	8.85728	0.295079
$902$	−1.83500	−0.0610990
$903$	0	0
$904$	4.10171	0.136421
$905$	−2.57184	−0.0854908
$906$	0	0
$907$	−14.0602	−0.466862	−0.233431	−	0.972373i	$-0.574995\pi$
$907$	−14.0602	−0.466862	−0.233431	+	0.972373i	$0.574995\pi$
$908$	−17.1240	−0.568279
$909$	0	0
$910$	−1.63158	−0.0540865
$911$	−30.9590	−1.02572	−0.512859	−	0.858473i	$-0.671413\pi$
$911$	−30.9590	−1.02572	−0.512859	+	0.858473i	$0.671413\pi$
$912$	0	0
$913$	−11.5496	−0.382235
$914$	−32.8385	−1.08620
$915$	0	0
$916$	−15.2444	−0.503690
$917$	8.10171	0.267542
$918$	0	0
$919$	29.9813	0.988991	0.494495	−	0.869180i	$-0.335353\pi$
$919$	29.9813	0.988991	0.494495	+	0.869180i	$0.335353\pi$
$920$	0.857279	0.0282637
$921$	0	0
$922$	−34.2163	−1.12685
$923$	−2.24797	−0.0739928
$924$	0	0
$925$	−26.7841	−0.880657
$926$	17.2444	0.566687
$927$	0	0
$928$	9.61285	0.315557
$929$	26.1432	0.857730	0.428865	−	0.903368i	$-0.358914\pi$
$929$	26.1432	0.857730	0.428865	+	0.903368i	$0.358914\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	16.8573	0.552179
$933$	0	0
$934$	−1.52987	−0.0500590
$935$	−6.69228	−0.218861
$936$	0	0
$937$	9.02227	0.294745	0.147372	−	0.989081i	$-0.452918\pi$
$937$	9.02227	0.294745	0.147372	+	0.989081i	$0.452918\pi$
$938$	4.42864	0.144600
$939$	0	0
$940$	0.590573	0.0192624
$941$	49.1624	1.60265	0.801324	−	0.598230i	$-0.204129\pi$
$941$	49.1624	1.60265	0.801324	+	0.598230i	$0.204129\pi$
$942$	0	0
$943$	1.04101	0.0339000
$944$	0	0
$945$	0	0
$946$	−18.4889	−0.601125
$947$	7.59010	0.246645	0.123322	−	0.992367i	$-0.460645\pi$
$947$	7.59010	0.246645	0.123322	+	0.992367i	$0.460645\pi$
$948$	0	0
$949$	17.9813	0.583697
$950$	4.61285	0.149661
$951$	0	0
$952$	−4.42864	−0.143533
$953$	−10.3872	−0.336473	−0.168236	−	0.985747i	$-0.553807\pi$
$953$	−10.3872	−0.336473	−0.168236	+	0.985747i	$0.553807\pi$
$954$	0	0
$955$	12.8198	0.414839
$956$	4.99063	0.161409
$957$	0	0
$958$	−19.5210	−0.630694
$959$	−2.75557	−0.0889820
$960$	0	0
$961$	−16.5111	−0.532617
$962$	15.2257	0.490896
$963$	0	0
$964$	0.326929	0.0105297
$965$	1.00354	0.0323051
$966$	0	0
$967$	−16.0000	−0.514525	−0.257263	−	0.966342i	$-0.582821\pi$
$967$	−16.0000	−0.514525	−0.257263	+	0.966342i	$0.582821\pi$
$968$	−5.10171	−0.163975
$969$	0	0
$970$	2.28544	0.0733811
$971$	28.7940	0.924043	0.462022	−	0.886869i	$-0.347124\pi$
$971$	28.7940	0.924043	0.462022	+	0.886869i	$0.347124\pi$
$972$	0	0
$973$	10.1017	0.323846
$974$	−30.0701	−0.963507
$975$	0	0
$976$	−5.61285	−0.179663
$977$	5.14272	0.164530	0.0822651	−	0.996610i	$-0.473785\pi$
$977$	5.14272	0.164530	0.0822651	+	0.996610i	$0.473785\pi$
$978$	0	0
$979$	−36.0830	−1.15322
$980$	0.622216	0.0198759
$981$	0	0
$982$	−8.40990	−0.268371
$983$	−10.7556	−0.343049	−0.171525	−	0.985180i	$-0.554869\pi$
$983$	−10.7556	−0.343049	−0.171525	+	0.985180i	$0.554869\pi$
$984$	0	0
$985$	3.38361	0.107811
$986$	42.5718	1.35576
$987$	0	0
$988$	−2.62222	−0.0834238
$989$	10.4889	0.333526
$990$	0	0
$991$	−1.15563	−0.0367097	−0.0183549	−	0.999832i	$-0.505843\pi$
$991$	−1.15563	−0.0367097	−0.0183549	+	0.999832i	$0.505843\pi$
$992$	3.80642	0.120854
$993$	0	0
$994$	0.857279	0.0271912
$995$	−16.9403	−0.537042
$996$	0	0
$997$	−26.2034	−0.829871	−0.414935	−	0.909851i	$-0.636196\pi$
$997$	−26.2034	−0.829871	−0.414935	+	0.909851i	$0.636196\pi$
$998$	−4.26671	−0.135060
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2394.2.a.bc.1.2	yes	3
3.2	odd	2		2394.2.a.bb.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2394.2.a.bb.1.2	✓	3	3.2	odd	2
2394.2.a.bc.1.2	yes	3	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:	\( N \)	\(=\)	\( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2394.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +0.622216 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +0.622216 q^{5} -1.00000 q^{7} +1.00000 q^{8} +0.622216 q^{10} -2.42864 q^{11} +2.62222 q^{13} -1.00000 q^{14} +1.00000 q^{16} +4.42864 q^{17} -1.00000 q^{19} +0.622216 q^{20} -2.42864 q^{22} +1.37778 q^{23} -4.61285 q^{25} +2.62222 q^{26} -1.00000 q^{28} +9.61285 q^{29} +3.80642 q^{31} +1.00000 q^{32} +4.42864 q^{34} -0.622216 q^{35} +5.80642 q^{37} -1.00000 q^{38} +0.622216 q^{40} +0.755569 q^{41} +7.61285 q^{43} -2.42864 q^{44} +1.37778 q^{46} +0.949145 q^{47} +1.00000 q^{49} -4.61285 q^{50} +2.62222 q^{52} +2.00000 q^{53} -1.51114 q^{55} -1.00000 q^{56} +9.61285 q^{58} -5.61285 q^{61} +3.80642 q^{62} +1.00000 q^{64} +1.63158 q^{65} -4.42864 q^{67} +4.42864 q^{68} -0.622216 q^{70} -0.857279 q^{71} +6.85728 q^{73} +5.80642 q^{74} -1.00000 q^{76} +2.42864 q^{77} -14.7239 q^{79} +0.622216 q^{80} +0.755569 q^{82} +4.75557 q^{83} +2.75557 q^{85} +7.61285 q^{86} -2.42864 q^{88} +14.8573 q^{89} -2.62222 q^{91} +1.37778 q^{92} +0.949145 q^{94} -0.622216 q^{95} +3.67307 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 2 q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 + 2 * q^5 - 3 * q^7 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} + 2 q^{5} - 3 q^{7} + 3 q^{8} + 2 q^{10} + 6 q^{11} + 8 q^{13} - 3 q^{14} + 3 q^{16} - 3 q^{19} + 2 q^{20} + 6 q^{22} + 4 q^{23} + 13 q^{25} + 8 q^{26} - 3 q^{28} + 2 q^{29} - 2 q^{31} + 3 q^{32} - 2 q^{35} + 4 q^{37} - 3 q^{38} + 2 q^{40} + 2 q^{41} - 4 q^{43} + 6 q^{44} + 4 q^{46} + 16 q^{47} + 3 q^{49} + 13 q^{50} + 8 q^{52} + 6 q^{53} - 4 q^{55} - 3 q^{56} + 2 q^{58} + 10 q^{61} - 2 q^{62} + 3 q^{64} + 32 q^{65} - 2 q^{70} + 24 q^{71} - 6 q^{73} + 4 q^{74} - 3 q^{76} - 6 q^{77} - 18 q^{79} + 2 q^{80} + 2 q^{82} + 14 q^{83} + 8 q^{85} - 4 q^{86} + 6 q^{88} + 18 q^{89} - 8 q^{91} + 4 q^{92} + 16 q^{94} - 2 q^{95} - 2 q^{97} + 3 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 2 * q^5 - 3 * q^7 + 3 * q^8 + 2 * q^10 + 6 * q^11 + 8 * q^13 - 3 * q^14 + 3 * q^16 - 3 * q^19 + 2 * q^20 + 6 * q^22 + 4 * q^23 + 13 * q^25 + 8 * q^26 - 3 * q^28 + 2 * q^29 - 2 * q^31 + 3 * q^32 - 2 * q^35 + 4 * q^37 - 3 * q^38 + 2 * q^40 + 2 * q^41 - 4 * q^43 + 6 * q^44 + 4 * q^46 + 16 * q^47 + 3 * q^49 + 13 * q^50 + 8 * q^52 + 6 * q^53 - 4 * q^55 - 3 * q^56 + 2 * q^58 + 10 * q^61 - 2 * q^62 + 3 * q^64 + 32 * q^65 - 2 * q^70 + 24 * q^71 - 6 * q^73 + 4 * q^74 - 3 * q^76 - 6 * q^77 - 18 * q^79 + 2 * q^80 + 2 * q^82 + 14 * q^83 + 8 * q^85 - 4 * q^86 + 6 * q^88 + 18 * q^89 - 8 * q^91 + 4 * q^92 + 16 * q^94 - 2 * q^95 - 2 * q^97 + 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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