LMFDB - Embedded newform 2394.2.a.t.1.1

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:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{10} -0.828427 q^{11} -5.65685 q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.17157 q^{17} -1.00000 q^{19} +2.00000 q^{20} +0.828427 q^{22} +4.00000 q^{23} -1.00000 q^{25} +5.65685 q^{26} +1.00000 q^{28} +2.00000 q^{29} +0.828427 q^{31} -1.00000 q^{32} -1.17157 q^{34} +2.00000 q^{35} +8.48528 q^{37} +1.00000 q^{38} -2.00000 q^{40} +7.65685 q^{41} -1.65685 q^{43} -0.828427 q^{44} -4.00000 q^{46} +12.4853 q^{47} +1.00000 q^{49} +1.00000 q^{50} -5.65685 q^{52} +3.65685 q^{53} -1.65685 q^{55} -1.00000 q^{56} -2.00000 q^{58} +8.00000 q^{59} +9.31371 q^{61} -0.828427 q^{62} +1.00000 q^{64} -11.3137 q^{65} -1.17157 q^{67} +1.17157 q^{68} -2.00000 q^{70} +5.65685 q^{71} +3.65685 q^{73} -8.48528 q^{74} -1.00000 q^{76} -0.828427 q^{77} -3.65685 q^{79} +2.00000 q^{80} -7.65685 q^{82} +0.343146 q^{83} +2.34315 q^{85} +1.65685 q^{86} +0.828427 q^{88} -9.31371 q^{89} -5.65685 q^{91} +4.00000 q^{92} -12.4853 q^{94} -2.00000 q^{95} -1.51472 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 4 q^{5} + 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^5 + 2 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} + 4 q^{5} + 2 q^{7} - 2 q^{8} - 4 q^{10} + 4 q^{11} - 2 q^{14} + 2 q^{16} + 8 q^{17} - 2 q^{19} + 4 q^{20} - 4 q^{22} + 8 q^{23} - 2 q^{25} + 2 q^{28} + 4 q^{29} - 4 q^{31} - 2 q^{32} - 8 q^{34} + 4 q^{35} + 2 q^{38} - 4 q^{40} + 4 q^{41} + 8 q^{43} + 4 q^{44} - 8 q^{46} + 8 q^{47} + 2 q^{49} + 2 q^{50} - 4 q^{53} + 8 q^{55} - 2 q^{56} - 4 q^{58} + 16 q^{59} - 4 q^{61} + 4 q^{62} + 2 q^{64} - 8 q^{67} + 8 q^{68} - 4 q^{70} - 4 q^{73} - 2 q^{76} + 4 q^{77} + 4 q^{79} + 4 q^{80} - 4 q^{82} + 12 q^{83} + 16 q^{85} - 8 q^{86} - 4 q^{88} + 4 q^{89} + 8 q^{92} - 8 q^{94} - 4 q^{95} - 20 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^5 + 2 * q^7 - 2 * q^8 - 4 * q^10 + 4 * q^11 - 2 * q^14 + 2 * q^16 + 8 * q^17 - 2 * q^19 + 4 * q^20 - 4 * q^22 + 8 * q^23 - 2 * q^25 + 2 * q^28 + 4 * q^29 - 4 * q^31 - 2 * q^32 - 8 * q^34 + 4 * q^35 + 2 * q^38 - 4 * q^40 + 4 * q^41 + 8 * q^43 + 4 * q^44 - 8 * q^46 + 8 * q^47 + 2 * q^49 + 2 * q^50 - 4 * q^53 + 8 * q^55 - 2 * q^56 - 4 * q^58 + 16 * q^59 - 4 * q^61 + 4 * q^62 + 2 * q^64 - 8 * q^67 + 8 * q^68 - 4 * q^70 - 4 * q^73 - 2 * q^76 + 4 * q^77 + 4 * q^79 + 4 * q^80 - 4 * q^82 + 12 * q^83 + 16 * q^85 - 8 * q^86 - 4 * q^88 + 4 * q^89 + 8 * q^92 - 8 * q^94 - 4 * q^95 - 20 * q^97 - 2 * 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
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−2.00000	−0.632456
$11$	−0.828427	−0.249780	−0.124890	−	0.992171i	$-0.539858\pi$
$11$	−0.828427	−0.249780	−0.124890	+	0.992171i	$0.539858\pi$
$12$	0	0
$13$	−5.65685	−1.56893	−0.784465	−	0.620174i	$-0.787062\pi$
$13$	−5.65685	−1.56893	−0.784465	+	0.620174i	$0.787062\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	1.17157	0.284148	0.142074	−	0.989856i	$-0.454623\pi$
$17$	1.17157	0.284148	0.142074	+	0.989856i	$0.454623\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	2.00000	0.447214
$21$	0	0
$22$	0.828427	0.176621
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	5.65685	1.10940
$27$	0	0
$28$	1.00000	0.188982
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	0.828427	0.148790	0.0743950	−	0.997229i	$-0.476297\pi$
$31$	0.828427	0.148790	0.0743950	+	0.997229i	$0.476297\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−1.17157	−0.200923
$35$	2.00000	0.338062
$36$	0	0
$37$	8.48528	1.39497	0.697486	−	0.716599i	$-0.254302\pi$
$37$	8.48528	1.39497	0.697486	+	0.716599i	$0.254302\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	−2.00000	−0.316228
$41$	7.65685	1.19580	0.597900	−	0.801571i	$-0.296002\pi$
$41$	7.65685	1.19580	0.597900	+	0.801571i	$0.296002\pi$
$42$	0	0
$43$	−1.65685	−0.252668	−0.126334	−	0.991988i	$-0.540321\pi$
$43$	−1.65685	−0.252668	−0.126334	+	0.991988i	$0.540321\pi$
$44$	−0.828427	−0.124890
$45$	0	0
$46$	−4.00000	−0.589768
$47$	12.4853	1.82117	0.910583	−	0.413327i	$-0.135633\pi$
$47$	12.4853	1.82117	0.910583	+	0.413327i	$0.135633\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	0	0
$52$	−5.65685	−0.784465
$53$	3.65685	0.502308	0.251154	−	0.967947i	$-0.419190\pi$
$53$	3.65685	0.502308	0.251154	+	0.967947i	$0.419190\pi$
$54$	0	0
$55$	−1.65685	−0.223410
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−2.00000	−0.262613
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	9.31371	1.19250	0.596249	−	0.802799i	$-0.296657\pi$
$61$	9.31371	1.19250	0.596249	+	0.802799i	$0.296657\pi$
$62$	−0.828427	−0.105210
$63$	0	0
$64$	1.00000	0.125000
$65$	−11.3137	−1.40329
$66$	0	0
$67$	−1.17157	−0.143130	−0.0715652	−	0.997436i	$-0.522799\pi$
$67$	−1.17157	−0.143130	−0.0715652	+	0.997436i	$0.522799\pi$
$68$	1.17157	0.142074
$69$	0	0
$70$	−2.00000	−0.239046
$71$	5.65685	0.671345	0.335673	−	0.941979i	$-0.391036\pi$
$71$	5.65685	0.671345	0.335673	+	0.941979i	$0.391036\pi$
$72$	0	0
$73$	3.65685	0.428002	0.214001	−	0.976833i	$-0.431350\pi$
$73$	3.65685	0.428002	0.214001	+	0.976833i	$0.431350\pi$
$74$	−8.48528	−0.986394
$75$	0	0
$76$	−1.00000	−0.114708
$77$	−0.828427	−0.0944080
$78$	0	0
$79$	−3.65685	−0.411428	−0.205714	−	0.978612i	$-0.565952\pi$
$79$	−3.65685	−0.411428	−0.205714	+	0.978612i	$0.565952\pi$
$80$	2.00000	0.223607
$81$	0	0
$82$	−7.65685	−0.845558
$83$	0.343146	0.0376651	0.0188326	−	0.999823i	$-0.494005\pi$
$83$	0.343146	0.0376651	0.0188326	+	0.999823i	$0.494005\pi$
$84$	0	0
$85$	2.34315	0.254150
$86$	1.65685	0.178663
$87$	0	0
$88$	0.828427	0.0883106
$89$	−9.31371	−0.987251	−0.493626	−	0.869675i	$-0.664329\pi$
$89$	−9.31371	−0.987251	−0.493626	+	0.869675i	$0.664329\pi$
$90$	0	0
$91$	−5.65685	−0.592999
$92$	4.00000	0.417029
$93$	0	0
$94$	−12.4853	−1.28776
$95$	−2.00000	−0.205196
$96$	0	0
$97$	−1.51472	−0.153796	−0.0768982	−	0.997039i	$-0.524502\pi$
$97$	−1.51472	−0.153796	−0.0768982	+	0.997039i	$0.524502\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	−1.00000	−0.100000
$101$	17.3137	1.72278	0.861389	−	0.507946i	$-0.169595\pi$
$101$	17.3137	1.72278	0.861389	+	0.507946i	$0.169595\pi$
$102$	0	0
$103$	−12.8284	−1.26402	−0.632011	−	0.774959i	$-0.717770\pi$
$103$	−12.8284	−1.26402	−0.632011	+	0.774959i	$0.717770\pi$
$104$	5.65685	0.554700
$105$	0	0
$106$	−3.65685	−0.355185
$107$	−5.65685	−0.546869	−0.273434	−	0.961891i	$-0.588160\pi$
$107$	−5.65685	−0.546869	−0.273434	+	0.961891i	$0.588160\pi$
$108$	0	0
$109$	−5.17157	−0.495347	−0.247673	−	0.968844i	$-0.579666\pi$
$109$	−5.17157	−0.495347	−0.247673	+	0.968844i	$0.579666\pi$
$110$	1.65685	0.157975
$111$	0	0
$112$	1.00000	0.0944911
$113$	−0.343146	−0.0322804	−0.0161402	−	0.999870i	$-0.505138\pi$
$113$	−0.343146	−0.0322804	−0.0161402	+	0.999870i	$0.505138\pi$
$114$	0	0
$115$	8.00000	0.746004
$116$	2.00000	0.185695
$117$	0	0
$118$	−8.00000	−0.736460
$119$	1.17157	0.107398
$120$	0	0
$121$	−10.3137	−0.937610
$122$	−9.31371	−0.843224
$123$	0	0
$124$	0.828427	0.0743950
$125$	−12.0000	−1.07331
$126$	0	0
$127$	15.6569	1.38932	0.694661	−	0.719338i	$-0.255555\pi$
$127$	15.6569	1.38932	0.694661	+	0.719338i	$0.255555\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	11.3137	0.992278
$131$	13.3137	1.16322	0.581612	−	0.813466i	$-0.302422\pi$
$131$	13.3137	1.16322	0.581612	+	0.813466i	$0.302422\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	1.17157	0.101208
$135$	0	0
$136$	−1.17157	−0.100462
$137$	−3.31371	−0.283109	−0.141555	−	0.989930i	$-0.545210\pi$
$137$	−3.31371	−0.283109	−0.141555	+	0.989930i	$0.545210\pi$
$138$	0	0
$139$	−6.34315	−0.538019	−0.269009	−	0.963138i	$-0.586696\pi$
$139$	−6.34315	−0.538019	−0.269009	+	0.963138i	$0.586696\pi$
$140$	2.00000	0.169031
$141$	0	0
$142$	−5.65685	−0.474713
$143$	4.68629	0.391887
$144$	0	0
$145$	4.00000	0.332182
$146$	−3.65685	−0.302643
$147$	0	0
$148$	8.48528	0.697486
$149$	6.48528	0.531295	0.265647	−	0.964070i	$-0.414414\pi$
$149$	6.48528	0.531295	0.265647	+	0.964070i	$0.414414\pi$
$150$	0	0
$151$	−3.65685	−0.297591	−0.148795	−	0.988868i	$-0.547540\pi$
$151$	−3.65685	−0.297591	−0.148795	+	0.988868i	$0.547540\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	0.828427	0.0667566
$155$	1.65685	0.133082
$156$	0	0
$157$	−3.65685	−0.291849	−0.145924	−	0.989296i	$-0.546616\pi$
$157$	−3.65685	−0.291849	−0.145924	+	0.989296i	$0.546616\pi$
$158$	3.65685	0.290924
$159$	0	0
$160$	−2.00000	−0.158114
$161$	4.00000	0.315244
$162$	0	0
$163$	23.3137	1.82607	0.913035	−	0.407881i	$-0.133732\pi$
$163$	23.3137	1.82607	0.913035	+	0.407881i	$0.133732\pi$
$164$	7.65685	0.597900
$165$	0	0
$166$	−0.343146	−0.0266333
$167$	−5.65685	−0.437741	−0.218870	−	0.975754i	$-0.570237\pi$
$167$	−5.65685	−0.437741	−0.218870	+	0.975754i	$0.570237\pi$
$168$	0	0
$169$	19.0000	1.46154
$170$	−2.34315	−0.179711
$171$	0	0
$172$	−1.65685	−0.126334
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	−0.828427	−0.0624450
$177$	0	0
$178$	9.31371	0.698092
$179$	7.31371	0.546652	0.273326	−	0.961921i	$-0.411876\pi$
$179$	7.31371	0.546652	0.273326	+	0.961921i	$0.411876\pi$
$180$	0	0
$181$	−1.65685	−0.123153	−0.0615765	−	0.998102i	$-0.519613\pi$
$181$	−1.65685	−0.123153	−0.0615765	+	0.998102i	$0.519613\pi$
$182$	5.65685	0.419314
$183$	0	0
$184$	−4.00000	−0.294884
$185$	16.9706	1.24770
$186$	0	0
$187$	−0.970563	−0.0709746
$188$	12.4853	0.910583
$189$	0	0
$190$	2.00000	0.145095
$191$	3.31371	0.239772	0.119886	−	0.992788i	$-0.461747\pi$
$191$	3.31371	0.239772	0.119886	+	0.992788i	$0.461747\pi$
$192$	0	0
$193$	−26.9706	−1.94138	−0.970692	−	0.240328i	$-0.922745\pi$
$193$	−26.9706	−1.94138	−0.970692	+	0.240328i	$0.922745\pi$
$194$	1.51472	0.108750
$195$	0	0
$196$	1.00000	0.0714286
$197$	7.17157	0.510953	0.255477	−	0.966815i	$-0.417768\pi$
$197$	7.17157	0.510953	0.255477	+	0.966815i	$0.417768\pi$
$198$	0	0
$199$	17.6569	1.25166	0.625831	−	0.779959i	$-0.284760\pi$
$199$	17.6569	1.25166	0.625831	+	0.779959i	$0.284760\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−17.3137	−1.21819
$203$	2.00000	0.140372
$204$	0	0
$205$	15.3137	1.06956
$206$	12.8284	0.893799
$207$	0	0
$208$	−5.65685	−0.392232
$209$	0.828427	0.0573035
$210$	0	0
$211$	16.4853	1.13489	0.567447	−	0.823410i	$-0.307931\pi$
$211$	16.4853	1.13489	0.567447	+	0.823410i	$0.307931\pi$
$212$	3.65685	0.251154
$213$	0	0
$214$	5.65685	0.386695
$215$	−3.31371	−0.225993
$216$	0	0
$217$	0.828427	0.0562373
$218$	5.17157	0.350263
$219$	0	0
$220$	−1.65685	−0.111705
$221$	−6.62742	−0.445808
$222$	0	0
$223$	−8.82843	−0.591195	−0.295598	−	0.955313i	$-0.595519\pi$
$223$	−8.82843	−0.591195	−0.295598	+	0.955313i	$0.595519\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	0.343146	0.0228257
$227$	−11.3137	−0.750917	−0.375459	−	0.926839i	$-0.622515\pi$
$227$	−11.3137	−0.750917	−0.375459	+	0.926839i	$0.622515\pi$
$228$	0	0
$229$	3.65685	0.241652	0.120826	−	0.992674i	$-0.461446\pi$
$229$	3.65685	0.241652	0.120826	+	0.992674i	$0.461446\pi$
$230$	−8.00000	−0.527504
$231$	0	0
$232$	−2.00000	−0.131306
$233$	−10.3431	−0.677602	−0.338801	−	0.940858i	$-0.610021\pi$
$233$	−10.3431	−0.677602	−0.338801	+	0.940858i	$0.610021\pi$
$234$	0	0
$235$	24.9706	1.62890
$236$	8.00000	0.520756
$237$	0	0
$238$	−1.17157	−0.0759418
$239$	−20.9706	−1.35647	−0.678236	−	0.734844i	$-0.737255\pi$
$239$	−20.9706	−1.35647	−0.678236	+	0.734844i	$0.737255\pi$
$240$	0	0
$241$	5.51472	0.355234	0.177617	−	0.984100i	$-0.443161\pi$
$241$	5.51472	0.355234	0.177617	+	0.984100i	$0.443161\pi$
$242$	10.3137	0.662990
$243$	0	0
$244$	9.31371	0.596249
$245$	2.00000	0.127775
$246$	0	0
$247$	5.65685	0.359937
$248$	−0.828427	−0.0526052
$249$	0	0
$250$	12.0000	0.758947
$251$	25.3137	1.59779	0.798894	−	0.601472i	$-0.205419\pi$
$251$	25.3137	1.59779	0.798894	+	0.601472i	$0.205419\pi$
$252$	0	0
$253$	−3.31371	−0.208331
$254$	−15.6569	−0.982398
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.9706	−1.18335	−0.591676	−	0.806176i	$-0.701533\pi$
$257$	−18.9706	−1.18335	−0.591676	+	0.806176i	$0.701533\pi$
$258$	0	0
$259$	8.48528	0.527250
$260$	−11.3137	−0.701646
$261$	0	0
$262$	−13.3137	−0.822524
$263$	18.6274	1.14862	0.574308	−	0.818639i	$-0.305271\pi$
$263$	18.6274	1.14862	0.574308	+	0.818639i	$0.305271\pi$
$264$	0	0
$265$	7.31371	0.449278
$266$	1.00000	0.0613139
$267$	0	0
$268$	−1.17157	−0.0715652
$269$	−5.31371	−0.323983	−0.161991	−	0.986792i	$-0.551792\pi$
$269$	−5.31371	−0.323983	−0.161991	+	0.986792i	$0.551792\pi$
$270$	0	0
$271$	−12.9706	−0.787906	−0.393953	−	0.919131i	$-0.628893\pi$
$271$	−12.9706	−0.787906	−0.393953	+	0.919131i	$0.628893\pi$
$272$	1.17157	0.0710370
$273$	0	0
$274$	3.31371	0.200188
$275$	0.828427	0.0499560
$276$	0	0
$277$	16.3431	0.981964	0.490982	−	0.871170i	$-0.336638\pi$
$277$	16.3431	0.981964	0.490982	+	0.871170i	$0.336638\pi$
$278$	6.34315	0.380437
$279$	0	0
$280$	−2.00000	−0.119523
$281$	−18.9706	−1.13169	−0.565844	−	0.824512i	$-0.691450\pi$
$281$	−18.9706	−1.13169	−0.565844	+	0.824512i	$0.691450\pi$
$282$	0	0
$283$	25.6569	1.52514	0.762571	−	0.646905i	$-0.223937\pi$
$283$	25.6569	1.52514	0.762571	+	0.646905i	$0.223937\pi$
$284$	5.65685	0.335673
$285$	0	0
$286$	−4.68629	−0.277106
$287$	7.65685	0.451970
$288$	0	0
$289$	−15.6274	−0.919260
$290$	−4.00000	−0.234888
$291$	0	0
$292$	3.65685	0.214001
$293$	−29.3137	−1.71253	−0.856263	−	0.516541i	$-0.827219\pi$
$293$	−29.3137	−1.71253	−0.856263	+	0.516541i	$0.827219\pi$
$294$	0	0
$295$	16.0000	0.931556
$296$	−8.48528	−0.493197
$297$	0	0
$298$	−6.48528	−0.375682
$299$	−22.6274	−1.30858
$300$	0	0
$301$	−1.65685	−0.0954995
$302$	3.65685	0.210428
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	18.6274	1.06660
$306$	0	0
$307$	23.3137	1.33058	0.665292	−	0.746583i	$-0.268307\pi$
$307$	23.3137	1.33058	0.665292	+	0.746583i	$0.268307\pi$
$308$	−0.828427	−0.0472040
$309$	0	0
$310$	−1.65685	−0.0941030
$311$	26.1421	1.48238	0.741192	−	0.671293i	$-0.234261\pi$
$311$	26.1421	1.48238	0.741192	+	0.671293i	$0.234261\pi$
$312$	0	0
$313$	5.31371	0.300349	0.150174	−	0.988660i	$-0.452017\pi$
$313$	5.31371	0.300349	0.150174	+	0.988660i	$0.452017\pi$
$314$	3.65685	0.206368
$315$	0	0
$316$	−3.65685	−0.205714
$317$	−6.68629	−0.375540	−0.187770	−	0.982213i	$-0.560126\pi$
$317$	−6.68629	−0.375540	−0.187770	+	0.982213i	$0.560126\pi$
$318$	0	0
$319$	−1.65685	−0.0927660
$320$	2.00000	0.111803
$321$	0	0
$322$	−4.00000	−0.222911
$323$	−1.17157	−0.0651881
$324$	0	0
$325$	5.65685	0.313786
$326$	−23.3137	−1.29123
$327$	0	0
$328$	−7.65685	−0.422779
$329$	12.4853	0.688336
$330$	0	0
$331$	16.4853	0.906113	0.453057	−	0.891482i	$-0.350334\pi$
$331$	16.4853	0.906113	0.453057	+	0.891482i	$0.350334\pi$
$332$	0.343146	0.0188326
$333$	0	0
$334$	5.65685	0.309529
$335$	−2.34315	−0.128020
$336$	0	0
$337$	−12.3431	−0.672374	−0.336187	−	0.941795i	$-0.609137\pi$
$337$	−12.3431	−0.672374	−0.336187	+	0.941795i	$0.609137\pi$
$338$	−19.0000	−1.03346
$339$	0	0
$340$	2.34315	0.127075
$341$	−0.686292	−0.0371648
$342$	0	0
$343$	1.00000	0.0539949
$344$	1.65685	0.0893316
$345$	0	0
$346$	2.00000	0.107521
$347$	−14.4853	−0.777611	−0.388805	−	0.921320i	$-0.627112\pi$
$347$	−14.4853	−0.777611	−0.388805	+	0.921320i	$0.627112\pi$
$348$	0	0
$349$	−22.9706	−1.22959	−0.614793	−	0.788688i	$-0.710760\pi$
$349$	−22.9706	−1.22959	−0.614793	+	0.788688i	$0.710760\pi$
$350$	1.00000	0.0534522
$351$	0	0
$352$	0.828427	0.0441553
$353$	−1.85786	−0.0988841	−0.0494421	−	0.998777i	$-0.515744\pi$
$353$	−1.85786	−0.0988841	−0.0494421	+	0.998777i	$0.515744\pi$
$354$	0	0
$355$	11.3137	0.600469
$356$	−9.31371	−0.493626
$357$	0	0
$358$	−7.31371	−0.386542
$359$	−16.9706	−0.895672	−0.447836	−	0.894116i	$-0.647805\pi$
$359$	−16.9706	−0.895672	−0.447836	+	0.894116i	$0.647805\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	1.65685	0.0870823
$363$	0	0
$364$	−5.65685	−0.296500
$365$	7.31371	0.382817
$366$	0	0
$367$	−1.65685	−0.0864871	−0.0432435	−	0.999065i	$-0.513769\pi$
$367$	−1.65685	−0.0864871	−0.0432435	+	0.999065i	$0.513769\pi$
$368$	4.00000	0.208514
$369$	0	0
$370$	−16.9706	−0.882258
$371$	3.65685	0.189854
$372$	0	0
$373$	−29.1716	−1.51045	−0.755223	−	0.655467i	$-0.772472\pi$
$373$	−29.1716	−1.51045	−0.755223	+	0.655467i	$0.772472\pi$
$374$	0.970563	0.0501866
$375$	0	0
$376$	−12.4853	−0.643879
$377$	−11.3137	−0.582686
$378$	0	0
$379$	29.1716	1.49844	0.749222	−	0.662319i	$-0.230428\pi$
$379$	29.1716	1.49844	0.749222	+	0.662319i	$0.230428\pi$
$380$	−2.00000	−0.102598
$381$	0	0
$382$	−3.31371	−0.169544
$383$	−13.6569	−0.697833	−0.348916	−	0.937154i	$-0.613450\pi$
$383$	−13.6569	−0.697833	−0.348916	+	0.937154i	$0.613450\pi$
$384$	0	0
$385$	−1.65685	−0.0844411
$386$	26.9706	1.37277
$387$	0	0
$388$	−1.51472	−0.0768982
$389$	25.7990	1.30806	0.654030	−	0.756468i	$-0.273077\pi$
$389$	25.7990	1.30806	0.654030	+	0.756468i	$0.273077\pi$
$390$	0	0
$391$	4.68629	0.236996
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−7.17157	−0.361299
$395$	−7.31371	−0.367993
$396$	0	0
$397$	13.3137	0.668196	0.334098	−	0.942538i	$-0.391568\pi$
$397$	13.3137	0.668196	0.334098	+	0.942538i	$0.391568\pi$
$398$	−17.6569	−0.885058
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−9.31371	−0.465104	−0.232552	−	0.972584i	$-0.574708\pi$
$401$	−9.31371	−0.465104	−0.232552	+	0.972584i	$0.574708\pi$
$402$	0	0
$403$	−4.68629	−0.233441
$404$	17.3137	0.861389
$405$	0	0
$406$	−2.00000	−0.0992583
$407$	−7.02944	−0.348436
$408$	0	0
$409$	−1.51472	−0.0748980	−0.0374490	−	0.999299i	$-0.511923\pi$
$409$	−1.51472	−0.0748980	−0.0374490	+	0.999299i	$0.511923\pi$
$410$	−15.3137	−0.756290
$411$	0	0
$412$	−12.8284	−0.632011
$413$	8.00000	0.393654
$414$	0	0
$415$	0.686292	0.0336887
$416$	5.65685	0.277350
$417$	0	0
$418$	−0.828427	−0.0405197
$419$	−18.9706	−0.926773	−0.463386	−	0.886156i	$-0.653366\pi$
$419$	−18.9706	−0.926773	−0.463386	+	0.886156i	$0.653366\pi$
$420$	0	0
$421$	10.1421	0.494297	0.247149	−	0.968978i	$-0.420506\pi$
$421$	10.1421	0.494297	0.247149	+	0.968978i	$0.420506\pi$
$422$	−16.4853	−0.802491
$423$	0	0
$424$	−3.65685	−0.177593
$425$	−1.17157	−0.0568296
$426$	0	0
$427$	9.31371	0.450722
$428$	−5.65685	−0.273434
$429$	0	0
$430$	3.31371	0.159801
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	0	0
$433$	13.7990	0.663137	0.331569	−	0.943431i	$-0.392422\pi$
$433$	13.7990	0.663137	0.331569	+	0.943431i	$0.392422\pi$
$434$	−0.828427	−0.0397658
$435$	0	0
$436$	−5.17157	−0.247673
$437$	−4.00000	−0.191346
$438$	0	0
$439$	−25.7990	−1.23132	−0.615659	−	0.788012i	$-0.711110\pi$
$439$	−25.7990	−1.23132	−0.615659	+	0.788012i	$0.711110\pi$
$440$	1.65685	0.0789874
$441$	0	0
$442$	6.62742	0.315234
$443$	−32.8284	−1.55973	−0.779863	−	0.625950i	$-0.784711\pi$
$443$	−32.8284	−1.55973	−0.779863	+	0.625950i	$0.784711\pi$
$444$	0	0
$445$	−18.6274	−0.883024
$446$	8.82843	0.418038
$447$	0	0
$448$	1.00000	0.0472456
$449$	25.3137	1.19463	0.597314	−	0.802008i	$-0.296235\pi$
$449$	25.3137	1.19463	0.597314	+	0.802008i	$0.296235\pi$
$450$	0	0
$451$	−6.34315	−0.298687
$452$	−0.343146	−0.0161402
$453$	0	0
$454$	11.3137	0.530979
$455$	−11.3137	−0.530395
$456$	0	0
$457$	−25.3137	−1.18413	−0.592063	−	0.805892i	$-0.701686\pi$
$457$	−25.3137	−1.18413	−0.592063	+	0.805892i	$0.701686\pi$
$458$	−3.65685	−0.170874
$459$	0	0
$460$	8.00000	0.373002
$461$	−8.34315	−0.388579	−0.194290	−	0.980944i	$-0.562240\pi$
$461$	−8.34315	−0.388579	−0.194290	+	0.980944i	$0.562240\pi$
$462$	0	0
$463$	−8.68629	−0.403686	−0.201843	−	0.979418i	$-0.564693\pi$
$463$	−8.68629	−0.403686	−0.201843	+	0.979418i	$0.564693\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	10.3431	0.479137
$467$	−1.02944	−0.0476367	−0.0238183	−	0.999716i	$-0.507582\pi$
$467$	−1.02944	−0.0476367	−0.0238183	+	0.999716i	$0.507582\pi$
$468$	0	0
$469$	−1.17157	−0.0540982
$470$	−24.9706	−1.15181
$471$	0	0
$472$	−8.00000	−0.368230
$473$	1.37258	0.0631114
$474$	0	0
$475$	1.00000	0.0458831
$476$	1.17157	0.0536990
$477$	0	0
$478$	20.9706	0.959171
$479$	20.4853	0.935996	0.467998	−	0.883729i	$-0.344975\pi$
$479$	20.4853	0.935996	0.467998	+	0.883729i	$0.344975\pi$
$480$	0	0
$481$	−48.0000	−2.18861
$482$	−5.51472	−0.251189
$483$	0	0
$484$	−10.3137	−0.468805
$485$	−3.02944	−0.137560
$486$	0	0
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	−9.31371	−0.421612
$489$	0	0
$490$	−2.00000	−0.0903508
$491$	−24.1421	−1.08952	−0.544760	−	0.838592i	$-0.683379\pi$
$491$	−24.1421	−1.08952	−0.544760	+	0.838592i	$0.683379\pi$
$492$	0	0
$493$	2.34315	0.105530
$494$	−5.65685	−0.254514
$495$	0	0
$496$	0.828427	0.0371975
$497$	5.65685	0.253745
$498$	0	0
$499$	−16.6863	−0.746981	−0.373490	−	0.927634i	$-0.621839\pi$
$499$	−16.6863	−0.746981	−0.373490	+	0.927634i	$0.621839\pi$
$500$	−12.0000	−0.536656
$501$	0	0
$502$	−25.3137	−1.12981
$503$	−3.79899	−0.169389	−0.0846943	−	0.996407i	$-0.526991\pi$
$503$	−3.79899	−0.169389	−0.0846943	+	0.996407i	$0.526991\pi$
$504$	0	0
$505$	34.6274	1.54090
$506$	3.31371	0.147312
$507$	0	0
$508$	15.6569	0.694661
$509$	35.9411	1.59306	0.796531	−	0.604597i	$-0.206666\pi$
$509$	35.9411	1.59306	0.796531	+	0.604597i	$0.206666\pi$
$510$	0	0
$511$	3.65685	0.161770
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	18.9706	0.836756
$515$	−25.6569	−1.13058
$516$	0	0
$517$	−10.3431	−0.454891
$518$	−8.48528	−0.372822
$519$	0	0
$520$	11.3137	0.496139
$521$	4.34315	0.190277	0.0951383	−	0.995464i	$-0.469671\pi$
$521$	4.34315	0.190277	0.0951383	+	0.995464i	$0.469671\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	13.3137	0.581612
$525$	0	0
$526$	−18.6274	−0.812194
$527$	0.970563	0.0422784
$528$	0	0
$529$	−7.00000	−0.304348
$530$	−7.31371	−0.317687
$531$	0	0
$532$	−1.00000	−0.0433555
$533$	−43.3137	−1.87612
$534$	0	0
$535$	−11.3137	−0.489134
$536$	1.17157	0.0506042
$537$	0	0
$538$	5.31371	0.229090
$539$	−0.828427	−0.0356829
$540$	0	0
$541$	−39.9411	−1.71720	−0.858602	−	0.512644i	$-0.828666\pi$
$541$	−39.9411	−1.71720	−0.858602	+	0.512644i	$0.828666\pi$
$542$	12.9706	0.557133
$543$	0	0
$544$	−1.17157	−0.0502308
$545$	−10.3431	−0.443052
$546$	0	0
$547$	−23.1127	−0.988228	−0.494114	−	0.869397i	$-0.664507\pi$
$547$	−23.1127	−0.988228	−0.494114	+	0.869397i	$0.664507\pi$
$548$	−3.31371	−0.141555
$549$	0	0
$550$	−0.828427	−0.0353243
$551$	−2.00000	−0.0852029
$552$	0	0
$553$	−3.65685	−0.155505
$554$	−16.3431	−0.694354
$555$	0	0
$556$	−6.34315	−0.269009
$557$	−38.7696	−1.64272	−0.821359	−	0.570411i	$-0.806784\pi$
$557$	−38.7696	−1.64272	−0.821359	+	0.570411i	$0.806784\pi$
$558$	0	0
$559$	9.37258	0.396418
$560$	2.00000	0.0845154
$561$	0	0
$562$	18.9706	0.800225
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	0	0
$565$	−0.686292	−0.0288725
$566$	−25.6569	−1.07844
$567$	0	0
$568$	−5.65685	−0.237356
$569$	−42.9706	−1.80142	−0.900710	−	0.434421i	$-0.856953\pi$
$569$	−42.9706	−1.80142	−0.900710	+	0.434421i	$0.856953\pi$
$570$	0	0
$571$	23.3137	0.975648	0.487824	−	0.872942i	$-0.337791\pi$
$571$	23.3137	0.975648	0.487824	+	0.872942i	$0.337791\pi$
$572$	4.68629	0.195944
$573$	0	0
$574$	−7.65685	−0.319591
$575$	−4.00000	−0.166812
$576$	0	0
$577$	−30.2843	−1.26075	−0.630375	−	0.776290i	$-0.717099\pi$
$577$	−30.2843	−1.26075	−0.630375	+	0.776290i	$0.717099\pi$
$578$	15.6274	0.650015
$579$	0	0
$580$	4.00000	0.166091
$581$	0.343146	0.0142361
$582$	0	0
$583$	−3.02944	−0.125466
$584$	−3.65685	−0.151322
$585$	0	0
$586$	29.3137	1.21094
$587$	5.31371	0.219320	0.109660	−	0.993969i	$-0.465024\pi$
$587$	5.31371	0.219320	0.109660	+	0.993969i	$0.465024\pi$
$588$	0	0
$589$	−0.828427	−0.0341347
$590$	−16.0000	−0.658710
$591$	0	0
$592$	8.48528	0.348743
$593$	36.4853	1.49827	0.749135	−	0.662417i	$-0.230469\pi$
$593$	36.4853	1.49827	0.749135	+	0.662417i	$0.230469\pi$
$594$	0	0
$595$	2.34315	0.0960596
$596$	6.48528	0.265647
$597$	0	0
$598$	22.6274	0.925304
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	7.45584	0.304130	0.152065	−	0.988370i	$-0.451408\pi$
$601$	7.45584	0.304130	0.152065	+	0.988370i	$0.451408\pi$
$602$	1.65685	0.0675283
$603$	0	0
$604$	−3.65685	−0.148795
$605$	−20.6274	−0.838624
$606$	0	0
$607$	−15.8579	−0.643651	−0.321825	−	0.946799i	$-0.604296\pi$
$607$	−15.8579	−0.643651	−0.321825	+	0.946799i	$0.604296\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	−18.6274	−0.754202
$611$	−70.6274	−2.85728
$612$	0	0
$613$	−26.6863	−1.07785	−0.538925	−	0.842354i	$-0.681169\pi$
$613$	−26.6863	−1.07785	−0.538925	+	0.842354i	$0.681169\pi$
$614$	−23.3137	−0.940865
$615$	0	0
$616$	0.828427	0.0333783
$617$	3.02944	0.121961	0.0609803	−	0.998139i	$-0.480577\pi$
$617$	3.02944	0.121961	0.0609803	+	0.998139i	$0.480577\pi$
$618$	0	0
$619$	30.3431	1.21959	0.609797	−	0.792558i	$-0.291251\pi$
$619$	30.3431	1.21959	0.609797	+	0.792558i	$0.291251\pi$
$620$	1.65685	0.0665409
$621$	0	0
$622$	−26.1421	−1.04820
$623$	−9.31371	−0.373146
$624$	0	0
$625$	−19.0000	−0.760000
$626$	−5.31371	−0.212379
$627$	0	0
$628$	−3.65685	−0.145924
$629$	9.94113	0.396379
$630$	0	0
$631$	−18.6274	−0.741546	−0.370773	−	0.928724i	$-0.620907\pi$
$631$	−18.6274	−0.741546	−0.370773	+	0.928724i	$0.620907\pi$
$632$	3.65685	0.145462
$633$	0	0
$634$	6.68629	0.265547
$635$	31.3137	1.24265
$636$	0	0
$637$	−5.65685	−0.224133
$638$	1.65685	0.0655955
$639$	0	0
$640$	−2.00000	−0.0790569
$641$	−17.3137	−0.683850	−0.341925	−	0.939727i	$-0.611079\pi$
$641$	−17.3137	−0.683850	−0.341925	+	0.939727i	$0.611079\pi$
$642$	0	0
$643$	24.6863	0.973532	0.486766	−	0.873532i	$-0.338176\pi$
$643$	24.6863	0.973532	0.486766	+	0.873532i	$0.338176\pi$
$644$	4.00000	0.157622
$645$	0	0
$646$	1.17157	0.0460949
$647$	9.85786	0.387553	0.193776	−	0.981046i	$-0.437926\pi$
$647$	9.85786	0.387553	0.193776	+	0.981046i	$0.437926\pi$
$648$	0	0
$649$	−6.62742	−0.260149
$650$	−5.65685	−0.221880
$651$	0	0
$652$	23.3137	0.913035
$653$	−10.4853	−0.410321	−0.205160	−	0.978728i	$-0.565772\pi$
$653$	−10.4853	−0.410321	−0.205160	+	0.978728i	$0.565772\pi$
$654$	0	0
$655$	26.6274	1.04042
$656$	7.65685	0.298950
$657$	0	0
$658$	−12.4853	−0.486727
$659$	−18.3431	−0.714548	−0.357274	−	0.934000i	$-0.616294\pi$
$659$	−18.3431	−0.714548	−0.357274	+	0.934000i	$0.616294\pi$
$660$	0	0
$661$	31.3137	1.21796	0.608981	−	0.793185i	$-0.291579\pi$
$661$	31.3137	1.21796	0.608981	+	0.793185i	$0.291579\pi$
$662$	−16.4853	−0.640719
$663$	0	0
$664$	−0.343146	−0.0133166
$665$	−2.00000	−0.0775567
$666$	0	0
$667$	8.00000	0.309761
$668$	−5.65685	−0.218870
$669$	0	0
$670$	2.34315	0.0905236
$671$	−7.71573	−0.297862
$672$	0	0
$673$	−6.68629	−0.257738	−0.128869	−	0.991662i	$-0.541135\pi$
$673$	−6.68629	−0.257738	−0.128869	+	0.991662i	$0.541135\pi$
$674$	12.3431	0.475440
$675$	0	0
$676$	19.0000	0.730769
$677$	9.31371	0.357955	0.178977	−	0.983853i	$-0.442721\pi$
$677$	9.31371	0.357955	0.178977	+	0.983853i	$0.442721\pi$
$678$	0	0
$679$	−1.51472	−0.0581296
$680$	−2.34315	−0.0898555
$681$	0	0
$682$	0.686292	0.0262795
$683$	29.9411	1.14567	0.572833	−	0.819672i	$-0.305844\pi$
$683$	29.9411	1.14567	0.572833	+	0.819672i	$0.305844\pi$
$684$	0	0
$685$	−6.62742	−0.253221
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	−1.65685	−0.0631670
$689$	−20.6863	−0.788085
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	−2.00000	−0.0760286
$693$	0	0
$694$	14.4853	0.549854
$695$	−12.6863	−0.481218
$696$	0	0
$697$	8.97056	0.339784
$698$	22.9706	0.869449
$699$	0	0
$700$	−1.00000	−0.0377964
$701$	41.1127	1.55281	0.776403	−	0.630237i	$-0.217042\pi$
$701$	41.1127	1.55281	0.776403	+	0.630237i	$0.217042\pi$
$702$	0	0
$703$	−8.48528	−0.320028
$704$	−0.828427	−0.0312225
$705$	0	0
$706$	1.85786	0.0699216
$707$	17.3137	0.651149
$708$	0	0
$709$	6.97056	0.261785	0.130892	−	0.991397i	$-0.458216\pi$
$709$	6.97056	0.261785	0.130892	+	0.991397i	$0.458216\pi$
$710$	−11.3137	−0.424596
$711$	0	0
$712$	9.31371	0.349046
$713$	3.31371	0.124099
$714$	0	0
$715$	9.37258	0.350515
$716$	7.31371	0.273326
$717$	0	0
$718$	16.9706	0.633336
$719$	−7.79899	−0.290853	−0.145427	−	0.989369i	$-0.546455\pi$
$719$	−7.79899	−0.290853	−0.145427	+	0.989369i	$0.546455\pi$
$720$	0	0
$721$	−12.8284	−0.477756
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	−1.65685	−0.0615765
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	3.31371	0.122899	0.0614493	−	0.998110i	$-0.480428\pi$
$727$	3.31371	0.122899	0.0614493	+	0.998110i	$0.480428\pi$
$728$	5.65685	0.209657
$729$	0	0
$730$	−7.31371	−0.270692
$731$	−1.94113	−0.0717951
$732$	0	0
$733$	−22.9706	−0.848437	−0.424219	−	0.905560i	$-0.639451\pi$
$733$	−22.9706	−0.848437	−0.424219	+	0.905560i	$0.639451\pi$
$734$	1.65685	0.0611556
$735$	0	0
$736$	−4.00000	−0.147442
$737$	0.970563	0.0357511
$738$	0	0
$739$	−1.65685	−0.0609484	−0.0304742	−	0.999536i	$-0.509702\pi$
$739$	−1.65685	−0.0609484	−0.0304742	+	0.999536i	$0.509702\pi$
$740$	16.9706	0.623850
$741$	0	0
$742$	−3.65685	−0.134247
$743$	−42.3431	−1.55342	−0.776710	−	0.629859i	$-0.783113\pi$
$743$	−42.3431	−1.55342	−0.776710	+	0.629859i	$0.783113\pi$
$744$	0	0
$745$	12.9706	0.475205
$746$	29.1716	1.06805
$747$	0	0
$748$	−0.970563	−0.0354873
$749$	−5.65685	−0.206697
$750$	0	0
$751$	−4.62742	−0.168857	−0.0844284	−	0.996430i	$-0.526906\pi$
$751$	−4.62742	−0.168857	−0.0844284	+	0.996430i	$0.526906\pi$
$752$	12.4853	0.455291
$753$	0	0
$754$	11.3137	0.412021
$755$	−7.31371	−0.266173
$756$	0	0
$757$	35.9411	1.30630	0.653151	−	0.757228i	$-0.273447\pi$
$757$	35.9411	1.30630	0.653151	+	0.757228i	$0.273447\pi$
$758$	−29.1716	−1.05956
$759$	0	0
$760$	2.00000	0.0725476
$761$	38.8284	1.40753	0.703765	−	0.710433i	$-0.251501\pi$
$761$	38.8284	1.40753	0.703765	+	0.710433i	$0.251501\pi$
$762$	0	0
$763$	−5.17157	−0.187224
$764$	3.31371	0.119886
$765$	0	0
$766$	13.6569	0.493442
$767$	−45.2548	−1.63406
$768$	0	0
$769$	29.3137	1.05708	0.528540	−	0.848909i	$-0.322740\pi$
$769$	29.3137	1.05708	0.528540	+	0.848909i	$0.322740\pi$
$770$	1.65685	0.0597089
$771$	0	0
$772$	−26.9706	−0.970692
$773$	10.0000	0.359675	0.179838	−	0.983696i	$-0.442443\pi$
$773$	10.0000	0.359675	0.179838	+	0.983696i	$0.442443\pi$
$774$	0	0
$775$	−0.828427	−0.0297580
$776$	1.51472	0.0543752
$777$	0	0
$778$	−25.7990	−0.924939
$779$	−7.65685	−0.274335
$780$	0	0
$781$	−4.68629	−0.167689
$782$	−4.68629	−0.167581
$783$	0	0
$784$	1.00000	0.0357143
$785$	−7.31371	−0.261037
$786$	0	0
$787$	−1.65685	−0.0590605	−0.0295302	−	0.999564i	$-0.509401\pi$
$787$	−1.65685	−0.0590605	−0.0295302	+	0.999564i	$0.509401\pi$
$788$	7.17157	0.255477
$789$	0	0
$790$	7.31371	0.260210
$791$	−0.343146	−0.0122009
$792$	0	0
$793$	−52.6863	−1.87095
$794$	−13.3137	−0.472486
$795$	0	0
$796$	17.6569	0.625831
$797$	−47.9411	−1.69816	−0.849081	−	0.528263i	$-0.822844\pi$
$797$	−47.9411	−1.69816	−0.849081	+	0.528263i	$0.822844\pi$
$798$	0	0
$799$	14.6274	0.517481
$800$	1.00000	0.0353553
$801$	0	0
$802$	9.31371	0.328878
$803$	−3.02944	−0.106907
$804$	0	0
$805$	8.00000	0.281963
$806$	4.68629	0.165068
$807$	0	0
$808$	−17.3137	−0.609094
$809$	−9.65685	−0.339517	−0.169758	−	0.985486i	$-0.554299\pi$
$809$	−9.65685	−0.339517	−0.169758	+	0.985486i	$0.554299\pi$
$810$	0	0
$811$	31.3137	1.09957	0.549787	−	0.835305i	$-0.314709\pi$
$811$	31.3137	1.09957	0.549787	+	0.835305i	$0.314709\pi$
$812$	2.00000	0.0701862
$813$	0	0
$814$	7.02944	0.246382
$815$	46.6274	1.63329
$816$	0	0
$817$	1.65685	0.0579660
$818$	1.51472	0.0529609
$819$	0	0
$820$	15.3137	0.534778
$821$	1.51472	0.0528640	0.0264320	−	0.999651i	$-0.491585\pi$
$821$	1.51472	0.0528640	0.0264320	+	0.999651i	$0.491585\pi$
$822$	0	0
$823$	−16.6863	−0.581648	−0.290824	−	0.956777i	$-0.593929\pi$
$823$	−16.6863	−0.581648	−0.290824	+	0.956777i	$0.593929\pi$
$824$	12.8284	0.446899
$825$	0	0
$826$	−8.00000	−0.278356
$827$	0.686292	0.0238647	0.0119323	−	0.999929i	$-0.496202\pi$
$827$	0.686292	0.0238647	0.0119323	+	0.999929i	$0.496202\pi$
$828$	0	0
$829$	−37.9411	−1.31775	−0.658875	−	0.752253i	$-0.728967\pi$
$829$	−37.9411	−1.31775	−0.658875	+	0.752253i	$0.728967\pi$
$830$	−0.686292	−0.0238215
$831$	0	0
$832$	−5.65685	−0.196116
$833$	1.17157	0.0405926
$834$	0	0
$835$	−11.3137	−0.391527
$836$	0.828427	0.0286518
$837$	0	0
$838$	18.9706	0.655327
$839$	3.31371	0.114402	0.0572010	−	0.998363i	$-0.481782\pi$
$839$	3.31371	0.114402	0.0572010	+	0.998363i	$0.481782\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−10.1421	−0.349521
$843$	0	0
$844$	16.4853	0.567447
$845$	38.0000	1.30724
$846$	0	0
$847$	−10.3137	−0.354383
$848$	3.65685	0.125577
$849$	0	0
$850$	1.17157	0.0401846
$851$	33.9411	1.16349
$852$	0	0
$853$	−21.3137	−0.729767	−0.364884	−	0.931053i	$-0.618891\pi$
$853$	−21.3137	−0.729767	−0.364884	+	0.931053i	$0.618891\pi$
$854$	−9.31371	−0.318709
$855$	0	0
$856$	5.65685	0.193347
$857$	6.68629	0.228399	0.114200	−	0.993458i	$-0.463570\pi$
$857$	6.68629	0.228399	0.114200	+	0.993458i	$0.463570\pi$
$858$	0	0
$859$	−36.9706	−1.26142	−0.630710	−	0.776019i	$-0.717236\pi$
$859$	−36.9706	−1.26142	−0.630710	+	0.776019i	$0.717236\pi$
$860$	−3.31371	−0.112997
$861$	0	0
$862$	−32.0000	−1.08992
$863$	28.2843	0.962808	0.481404	−	0.876499i	$-0.340127\pi$
$863$	28.2843	0.962808	0.481404	+	0.876499i	$0.340127\pi$
$864$	0	0
$865$	−4.00000	−0.136004
$866$	−13.7990	−0.468909
$867$	0	0
$868$	0.828427	0.0281186
$869$	3.02944	0.102767
$870$	0	0
$871$	6.62742	0.224561
$872$	5.17157	0.175132
$873$	0	0
$874$	4.00000	0.135302
$875$	−12.0000	−0.405674
$876$	0	0
$877$	26.4264	0.892356	0.446178	−	0.894944i	$-0.352785\pi$
$877$	26.4264	0.892356	0.446178	+	0.894944i	$0.352785\pi$
$878$	25.7990	0.870674
$879$	0	0
$880$	−1.65685	−0.0558525
$881$	−25.8579	−0.871174	−0.435587	−	0.900147i	$-0.643459\pi$
$881$	−25.8579	−0.871174	−0.435587	+	0.900147i	$0.643459\pi$
$882$	0	0
$883$	36.0000	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$884$	−6.62742	−0.222904
$885$	0	0
$886$	32.8284	1.10289
$887$	49.9411	1.67686	0.838429	−	0.545010i	$-0.183474\pi$
$887$	49.9411	1.67686	0.838429	+	0.545010i	$0.183474\pi$
$888$	0	0
$889$	15.6569	0.525114
$890$	18.6274	0.624392
$891$	0	0
$892$	−8.82843	−0.295598
$893$	−12.4853	−0.417804
$894$	0	0
$895$	14.6274	0.488941
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−25.3137	−0.844729
$899$	1.65685	0.0552592
$900$	0	0
$901$	4.28427	0.142730
$902$	6.34315	0.211204
$903$	0	0
$904$	0.343146	0.0114129
$905$	−3.31371	−0.110151
$906$	0	0
$907$	−36.0833	−1.19813	−0.599063	−	0.800702i	$-0.704460\pi$
$907$	−36.0833	−1.19813	−0.599063	+	0.800702i	$0.704460\pi$
$908$	−11.3137	−0.375459
$909$	0	0
$910$	11.3137	0.375046
$911$	−14.6274	−0.484628	−0.242314	−	0.970198i	$-0.577906\pi$
$911$	−14.6274	−0.484628	−0.242314	+	0.970198i	$0.577906\pi$
$912$	0	0
$913$	−0.284271	−0.00940801
$914$	25.3137	0.837303
$915$	0	0
$916$	3.65685	0.120826
$917$	13.3137	0.439657
$918$	0	0
$919$	34.6274	1.14225	0.571127	−	0.820862i	$-0.306507\pi$
$919$	34.6274	1.14225	0.571127	+	0.820862i	$0.306507\pi$
$920$	−8.00000	−0.263752
$921$	0	0
$922$	8.34315	0.274767
$923$	−32.0000	−1.05329
$924$	0	0
$925$	−8.48528	−0.278994
$926$	8.68629	0.285449
$927$	0	0
$928$	−2.00000	−0.0656532
$929$	−9.45584	−0.310236	−0.155118	−	0.987896i	$-0.549576\pi$
$929$	−9.45584	−0.310236	−0.155118	+	0.987896i	$0.549576\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	−10.3431	−0.338801
$933$	0	0
$934$	1.02944	0.0336842
$935$	−1.94113	−0.0634816
$936$	0	0
$937$	23.9411	0.782122	0.391061	−	0.920365i	$-0.372108\pi$
$937$	23.9411	0.782122	0.391061	+	0.920365i	$0.372108\pi$
$938$	1.17157	0.0382532
$939$	0	0
$940$	24.9706	0.814450
$941$	53.3137	1.73798	0.868989	−	0.494832i	$-0.164770\pi$
$941$	53.3137	1.73798	0.868989	+	0.494832i	$0.164770\pi$
$942$	0	0
$943$	30.6274	0.997366
$944$	8.00000	0.260378
$945$	0	0
$946$	−1.37258	−0.0446265
$947$	−6.20101	−0.201506	−0.100753	−	0.994911i	$-0.532125\pi$
$947$	−6.20101	−0.201506	−0.100753	+	0.994911i	$0.532125\pi$
$948$	0	0
$949$	−20.6863	−0.671505
$950$	−1.00000	−0.0324443
$951$	0	0
$952$	−1.17157	−0.0379709
$953$	3.65685	0.118457	0.0592286	−	0.998244i	$-0.481136\pi$
$953$	3.65685	0.118457	0.0592286	+	0.998244i	$0.481136\pi$
$954$	0	0
$955$	6.62742	0.214458
$956$	−20.9706	−0.678236
$957$	0	0
$958$	−20.4853	−0.661849
$959$	−3.31371	−0.107005
$960$	0	0
$961$	−30.3137	−0.977862
$962$	48.0000	1.54758
$963$	0	0
$964$	5.51472	0.177617
$965$	−53.9411	−1.73643
$966$	0	0
$967$	−20.6863	−0.665226	−0.332613	−	0.943063i	$-0.607930\pi$
$967$	−20.6863	−0.665226	−0.332613	+	0.943063i	$0.607930\pi$
$968$	10.3137	0.331495
$969$	0	0
$970$	3.02944	0.0972694
$971$	−22.6274	−0.726148	−0.363074	−	0.931760i	$-0.618273\pi$
$971$	−22.6274	−0.726148	−0.363074	+	0.931760i	$0.618273\pi$
$972$	0	0
$973$	−6.34315	−0.203352
$974$	2.00000	0.0640841
$975$	0	0
$976$	9.31371	0.298125
$977$	56.6274	1.81167	0.905836	−	0.423629i	$-0.139244\pi$
$977$	56.6274	1.81167	0.905836	+	0.423629i	$0.139244\pi$
$978$	0	0
$979$	7.71573	0.246596
$980$	2.00000	0.0638877
$981$	0	0
$982$	24.1421	0.770407
$983$	10.3431	0.329895	0.164948	−	0.986302i	$-0.447255\pi$
$983$	10.3431	0.329895	0.164948	+	0.986302i	$0.447255\pi$
$984$	0	0
$985$	14.3431	0.457011
$986$	−2.34315	−0.0746210
$987$	0	0
$988$	5.65685	0.179969
$989$	−6.62742	−0.210740
$990$	0	0
$991$	−54.0000	−1.71537	−0.857683	−	0.514178i	$-0.828097\pi$
$991$	−54.0000	−1.71537	−0.857683	+	0.514178i	$0.828097\pi$
$992$	−0.828427	−0.0263026
$993$	0	0
$994$	−5.65685	−0.179425
$995$	35.3137	1.11952
$996$	0	0
$997$	−18.2843	−0.579069	−0.289534	−	0.957168i	$-0.593500\pi$
$997$	−18.2843	−0.579069	−0.289534	+	0.957168i	$0.593500\pi$
$998$	16.6863	0.528195
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2394.2.a.t.1.1	✓	2	1.1	even	1	trivial
2394.2.a.u.1.2	yes	2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2394.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{10} -0.828427 q^{11} -5.65685 q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.17157 q^{17} -1.00000 q^{19} +2.00000 q^{20} +0.828427 q^{22} +4.00000 q^{23} -1.00000 q^{25} +5.65685 q^{26} +1.00000 q^{28} +2.00000 q^{29} +0.828427 q^{31} -1.00000 q^{32} -1.17157 q^{34} +2.00000 q^{35} +8.48528 q^{37} +1.00000 q^{38} -2.00000 q^{40} +7.65685 q^{41} -1.65685 q^{43} -0.828427 q^{44} -4.00000 q^{46} +12.4853 q^{47} +1.00000 q^{49} +1.00000 q^{50} -5.65685 q^{52} +3.65685 q^{53} -1.65685 q^{55} -1.00000 q^{56} -2.00000 q^{58} +8.00000 q^{59} +9.31371 q^{61} -0.828427 q^{62} +1.00000 q^{64} -11.3137 q^{65} -1.17157 q^{67} +1.17157 q^{68} -2.00000 q^{70} +5.65685 q^{71} +3.65685 q^{73} -8.48528 q^{74} -1.00000 q^{76} -0.828427 q^{77} -3.65685 q^{79} +2.00000 q^{80} -7.65685 q^{82} +0.343146 q^{83} +2.34315 q^{85} +1.65685 q^{86} +0.828427 q^{88} -9.31371 q^{89} -5.65685 q^{91} +4.00000 q^{92} -12.4853 q^{94} -2.00000 q^{95} -1.51472 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{5} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^5 + 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{5} + 2 q^{7} - 2 q^{8} - 4 q^{10} + 4 q^{11} - 2 q^{14} + 2 q^{16} + 8 q^{17} - 2 q^{19} + 4 q^{20} - 4 q^{22} + 8 q^{23} - 2 q^{25} + 2 q^{28} + 4 q^{29} - 4 q^{31} - 2 q^{32} - 8 q^{34} + 4 q^{35} + 2 q^{38} - 4 q^{40} + 4 q^{41} + 8 q^{43} + 4 q^{44} - 8 q^{46} + 8 q^{47} + 2 q^{49} + 2 q^{50} - 4 q^{53} + 8 q^{55} - 2 q^{56} - 4 q^{58} + 16 q^{59} - 4 q^{61} + 4 q^{62} + 2 q^{64} - 8 q^{67} + 8 q^{68} - 4 q^{70} - 4 q^{73} - 2 q^{76} + 4 q^{77} + 4 q^{79} + 4 q^{80} - 4 q^{82} + 12 q^{83} + 16 q^{85} - 8 q^{86} - 4 q^{88} + 4 q^{89} + 8 q^{92} - 8 q^{94} - 4 q^{95} - 20 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^5 + 2 * q^7 - 2 * q^8 - 4 * q^10 + 4 * q^11 - 2 * q^14 + 2 * q^16 + 8 * q^17 - 2 * q^19 + 4 * q^20 - 4 * q^22 + 8 * q^23 - 2 * q^25 + 2 * q^28 + 4 * q^29 - 4 * q^31 - 2 * q^32 - 8 * q^34 + 4 * q^35 + 2 * q^38 - 4 * q^40 + 4 * q^41 + 8 * q^43 + 4 * q^44 - 8 * q^46 + 8 * q^47 + 2 * q^49 + 2 * q^50 - 4 * q^53 + 8 * q^55 - 2 * q^56 - 4 * q^58 + 16 * q^59 - 4 * q^61 + 4 * q^62 + 2 * q^64 - 8 * q^67 + 8 * q^68 - 4 * q^70 - 4 * q^73 - 2 * q^76 + 4 * q^77 + 4 * q^79 + 4 * q^80 - 4 * q^82 + 12 * q^83 + 16 * q^85 - 8 * q^86 - 4 * q^88 + 4 * q^89 + 8 * q^92 - 8 * q^94 - 4 * q^95 - 20 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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