LMFDB - Embedded newform 2394.2.a.x.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:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 798)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +3.46410 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +3.46410 q^{5} -1.00000 q^{7} +1.00000 q^{8} +3.46410 q^{10} +5.46410 q^{11} -3.46410 q^{13} -1.00000 q^{14} +1.00000 q^{16} -7.46410 q^{17} +1.00000 q^{19} +3.46410 q^{20} +5.46410 q^{22} +5.46410 q^{23} +7.00000 q^{25} -3.46410 q^{26} -1.00000 q^{28} +6.00000 q^{29} +6.92820 q^{31} +1.00000 q^{32} -7.46410 q^{34} -3.46410 q^{35} +10.0000 q^{37} +1.00000 q^{38} +3.46410 q^{40} +2.00000 q^{41} -6.92820 q^{43} +5.46410 q^{44} +5.46410 q^{46} -10.9282 q^{47} +1.00000 q^{49} +7.00000 q^{50} -3.46410 q^{52} -2.00000 q^{53} +18.9282 q^{55} -1.00000 q^{56} +6.00000 q^{58} -4.00000 q^{59} -4.92820 q^{61} +6.92820 q^{62} +1.00000 q^{64} -12.0000 q^{65} +9.46410 q^{67} -7.46410 q^{68} -3.46410 q^{70} -2.92820 q^{71} +10.0000 q^{73} +10.0000 q^{74} +1.00000 q^{76} -5.46410 q^{77} -12.3923 q^{79} +3.46410 q^{80} +2.00000 q^{82} -8.00000 q^{83} -25.8564 q^{85} -6.92820 q^{86} +5.46410 q^{88} +2.00000 q^{89} +3.46410 q^{91} +5.46410 q^{92} -10.9282 q^{94} +3.46410 q^{95} +4.53590 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} + 4 q^{11} - 2 q^{14} + 2 q^{16} - 8 q^{17} + 2 q^{19} + 4 q^{22} + 4 q^{23} + 14 q^{25} - 2 q^{28} + 12 q^{29} + 2 q^{32} - 8 q^{34} + 20 q^{37} + 2 q^{38} + 4 q^{41} + 4 q^{44} + 4 q^{46} - 8 q^{47} + 2 q^{49} + 14 q^{50} - 4 q^{53} + 24 q^{55} - 2 q^{56} + 12 q^{58} - 8 q^{59} + 4 q^{61} + 2 q^{64} - 24 q^{65} + 12 q^{67} - 8 q^{68} + 8 q^{71} + 20 q^{73} + 20 q^{74} + 2 q^{76} - 4 q^{77} - 4 q^{79} + 4 q^{82} - 16 q^{83} - 24 q^{85} + 4 q^{88} + 4 q^{89} + 4 q^{92} - 8 q^{94} + 16 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8 + 4 * q^11 - 2 * q^14 + 2 * q^16 - 8 * q^17 + 2 * q^19 + 4 * q^22 + 4 * q^23 + 14 * q^25 - 2 * q^28 + 12 * q^29 + 2 * q^32 - 8 * q^34 + 20 * q^37 + 2 * q^38 + 4 * q^41 + 4 * q^44 + 4 * q^46 - 8 * q^47 + 2 * q^49 + 14 * q^50 - 4 * q^53 + 24 * q^55 - 2 * q^56 + 12 * q^58 - 8 * q^59 + 4 * q^61 + 2 * q^64 - 24 * q^65 + 12 * q^67 - 8 * q^68 + 8 * q^71 + 20 * q^73 + 20 * q^74 + 2 * q^76 - 4 * q^77 - 4 * q^79 + 4 * q^82 - 16 * q^83 - 24 * q^85 + 4 * q^88 + 4 * q^89 + 4 * q^92 - 8 * q^94 + 16 * 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
$5$	3.46410	1.54919	0.774597	−	0.632456i	$-0.217953\pi$
$5$	3.46410	1.54919	0.774597	+	0.632456i	$0.217953\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	3.46410	1.09545
$11$	5.46410	1.64749	0.823744	−	0.566961i	$-0.191881\pi$
$11$	5.46410	1.64749	0.823744	+	0.566961i	$0.191881\pi$
$12$	0	0
$13$	−3.46410	−0.960769	−0.480384	−	0.877058i	$-0.659503\pi$
$13$	−3.46410	−0.960769	−0.480384	+	0.877058i	$0.659503\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−7.46410	−1.81031	−0.905155	−	0.425081i	$-0.860246\pi$
$17$	−7.46410	−1.81031	−0.905155	+	0.425081i	$0.860246\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	3.46410	0.774597
$21$	0	0
$22$	5.46410	1.16495
$23$	5.46410	1.13934	0.569672	−	0.821872i	$-0.307070\pi$
$23$	5.46410	1.13934	0.569672	+	0.821872i	$0.307070\pi$
$24$	0	0
$25$	7.00000	1.40000
$26$	−3.46410	−0.679366
$27$	0	0
$28$	−1.00000	−0.188982
$29$	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
$31$	6.92820	1.24434	0.622171	−	0.782881i	$-0.286251\pi$
$31$	6.92820	1.24434	0.622171	+	0.782881i	$0.286251\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−7.46410	−1.28008
$35$	−3.46410	−0.585540
$36$	0	0
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	3.46410	0.547723
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−6.92820	−1.05654	−0.528271	−	0.849076i	$-0.677159\pi$
$43$	−6.92820	−1.05654	−0.528271	+	0.849076i	$0.677159\pi$
$44$	5.46410	0.823744
$45$	0	0
$46$	5.46410	0.805638
$47$	−10.9282	−1.59404	−0.797021	−	0.603951i	$-0.793592\pi$
$47$	−10.9282	−1.59404	−0.797021	+	0.603951i	$0.793592\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	7.00000	0.989949
$51$	0	0
$52$	−3.46410	−0.480384
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	18.9282	2.55228
$56$	−1.00000	−0.133631
$57$	0	0
$58$	6.00000	0.787839
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−4.92820	−0.630992	−0.315496	−	0.948927i	$-0.602171\pi$
$61$	−4.92820	−0.630992	−0.315496	+	0.948927i	$0.602171\pi$
$62$	6.92820	0.879883
$63$	0	0
$64$	1.00000	0.125000
$65$	−12.0000	−1.48842
$66$	0	0
$67$	9.46410	1.15622	0.578112	−	0.815957i	$-0.303790\pi$
$67$	9.46410	1.15622	0.578112	+	0.815957i	$0.303790\pi$
$68$	−7.46410	−0.905155
$69$	0	0
$70$	−3.46410	−0.414039
$71$	−2.92820	−0.347514	−0.173757	−	0.984789i	$-0.555591\pi$
$71$	−2.92820	−0.347514	−0.173757	+	0.984789i	$0.555591\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	10.0000	1.16248
$75$	0	0
$76$	1.00000	0.114708
$77$	−5.46410	−0.622692
$78$	0	0
$79$	−12.3923	−1.39424	−0.697122	−	0.716953i	$-0.745536\pi$
$79$	−12.3923	−1.39424	−0.697122	+	0.716953i	$0.745536\pi$
$80$	3.46410	0.387298
$81$	0	0
$82$	2.00000	0.220863
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	0	0
$85$	−25.8564	−2.80452
$86$	−6.92820	−0.747087
$87$	0	0
$88$	5.46410	0.582475
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	3.46410	0.363137
$92$	5.46410	0.569672
$93$	0	0
$94$	−10.9282	−1.12716
$95$	3.46410	0.355409
$96$	0	0
$97$	4.53590	0.460551	0.230275	−	0.973126i	$-0.426037\pi$
$97$	4.53590	0.460551	0.230275	+	0.973126i	$0.426037\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	7.00000	0.700000
$101$	−4.53590	−0.451339	−0.225669	−	0.974204i	$-0.572457\pi$
$101$	−4.53590	−0.451339	−0.225669	+	0.974204i	$0.572457\pi$
$102$	0	0
$103$	−6.92820	−0.682656	−0.341328	−	0.939944i	$-0.610877\pi$
$103$	−6.92820	−0.682656	−0.341328	+	0.939944i	$0.610877\pi$
$104$	−3.46410	−0.339683
$105$	0	0
$106$	−2.00000	−0.194257
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	18.9282	1.80473
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−11.8564	−1.11536	−0.557678	−	0.830057i	$-0.688308\pi$
$113$	−11.8564	−1.11536	−0.557678	+	0.830057i	$0.688308\pi$
$114$	0	0
$115$	18.9282	1.76506
$116$	6.00000	0.557086
$117$	0	0
$118$	−4.00000	−0.368230
$119$	7.46410	0.684233
$120$	0	0
$121$	18.8564	1.71422
$122$	−4.92820	−0.446179
$123$	0	0
$124$	6.92820	0.622171
$125$	6.92820	0.619677
$126$	0	0
$127$	−17.4641	−1.54969	−0.774844	−	0.632152i	$-0.782172\pi$
$127$	−17.4641	−1.54969	−0.774844	+	0.632152i	$0.782172\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−12.0000	−1.05247
$131$	5.07180	0.443125	0.221562	−	0.975146i	$-0.428884\pi$
$131$	5.07180	0.443125	0.221562	+	0.975146i	$0.428884\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	9.46410	0.817574
$135$	0	0
$136$	−7.46410	−0.640041
$137$	0.928203	0.0793018	0.0396509	−	0.999214i	$-0.487375\pi$
$137$	0.928203	0.0793018	0.0396509	+	0.999214i	$0.487375\pi$
$138$	0	0
$139$	9.85641	0.836009	0.418005	−	0.908445i	$-0.362730\pi$
$139$	9.85641	0.836009	0.418005	+	0.908445i	$0.362730\pi$
$140$	−3.46410	−0.292770
$141$	0	0
$142$	−2.92820	−0.245729
$143$	−18.9282	−1.58286
$144$	0	0
$145$	20.7846	1.72607
$146$	10.0000	0.827606
$147$	0	0
$148$	10.0000	0.821995
$149$	3.07180	0.251651	0.125826	−	0.992052i	$-0.459842\pi$
$149$	3.07180	0.251651	0.125826	+	0.992052i	$0.459842\pi$
$150$	0	0
$151$	1.46410	0.119147	0.0595734	−	0.998224i	$-0.481026\pi$
$151$	1.46410	0.119147	0.0595734	+	0.998224i	$0.481026\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	−5.46410	−0.440310
$155$	24.0000	1.92773
$156$	0	0
$157$	19.8564	1.58471	0.792357	−	0.610058i	$-0.208854\pi$
$157$	19.8564	1.58471	0.792357	+	0.610058i	$0.208854\pi$
$158$	−12.3923	−0.985879
$159$	0	0
$160$	3.46410	0.273861
$161$	−5.46410	−0.430632
$162$	0	0
$163$	1.07180	0.0839496	0.0419748	−	0.999119i	$-0.486635\pi$
$163$	1.07180	0.0839496	0.0419748	+	0.999119i	$0.486635\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	−8.00000	−0.620920
$167$	18.9282	1.46471	0.732354	−	0.680924i	$-0.238422\pi$
$167$	18.9282	1.46471	0.732354	+	0.680924i	$0.238422\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	−25.8564	−1.98310
$171$	0	0
$172$	−6.92820	−0.528271
$173$	0.928203	0.0705700	0.0352850	−	0.999377i	$-0.488766\pi$
$173$	0.928203	0.0705700	0.0352850	+	0.999377i	$0.488766\pi$
$174$	0	0
$175$	−7.00000	−0.529150
$176$	5.46410	0.411872
$177$	0	0
$178$	2.00000	0.149906
$179$	9.85641	0.736702	0.368351	−	0.929687i	$-0.379922\pi$
$179$	9.85641	0.736702	0.368351	+	0.929687i	$0.379922\pi$
$180$	0	0
$181$	−8.53590	−0.634468	−0.317234	−	0.948347i	$-0.602754\pi$
$181$	−8.53590	−0.634468	−0.317234	+	0.948347i	$0.602754\pi$
$182$	3.46410	0.256776
$183$	0	0
$184$	5.46410	0.402819
$185$	34.6410	2.54686
$186$	0	0
$187$	−40.7846	−2.98247
$188$	−10.9282	−0.797021
$189$	0	0
$190$	3.46410	0.251312
$191$	5.46410	0.395369	0.197684	−	0.980266i	$-0.436658\pi$
$191$	5.46410	0.395369	0.197684	+	0.980266i	$0.436658\pi$
$192$	0	0
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	4.53590	0.325659
$195$	0	0
$196$	1.00000	0.0714286
$197$	−4.92820	−0.351120	−0.175560	−	0.984469i	$-0.556174\pi$
$197$	−4.92820	−0.351120	−0.175560	+	0.984469i	$0.556174\pi$
$198$	0	0
$199$	−24.7846	−1.75693	−0.878467	−	0.477803i	$-0.841433\pi$
$199$	−24.7846	−1.75693	−0.878467	+	0.477803i	$0.841433\pi$
$200$	7.00000	0.494975
$201$	0	0
$202$	−4.53590	−0.319145
$203$	−6.00000	−0.421117
$204$	0	0
$205$	6.92820	0.483887
$206$	−6.92820	−0.482711
$207$	0	0
$208$	−3.46410	−0.240192
$209$	5.46410	0.377960
$210$	0	0
$211$	1.46410	0.100793	0.0503965	−	0.998729i	$-0.483952\pi$
$211$	1.46410	0.100793	0.0503965	+	0.998729i	$0.483952\pi$
$212$	−2.00000	−0.137361
$213$	0	0
$214$	4.00000	0.273434
$215$	−24.0000	−1.63679
$216$	0	0
$217$	−6.92820	−0.470317
$218$	2.00000	0.135457
$219$	0	0
$220$	18.9282	1.27614
$221$	25.8564	1.73929
$222$	0	0
$223$	−12.0000	−0.803579	−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000	−0.803579	−0.401790	+	0.915732i	$0.631612\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−11.8564	−0.788676
$227$	−14.9282	−0.990820	−0.495410	−	0.868659i	$-0.664982\pi$
$227$	−14.9282	−0.990820	−0.495410	+	0.868659i	$0.664982\pi$
$228$	0	0
$229$	3.07180	0.202990	0.101495	−	0.994836i	$-0.467637\pi$
$229$	3.07180	0.202990	0.101495	+	0.994836i	$0.467637\pi$
$230$	18.9282	1.24809
$231$	0	0
$232$	6.00000	0.393919
$233$	−26.7846	−1.75472	−0.877359	−	0.479834i	$-0.840697\pi$
$233$	−26.7846	−1.75472	−0.877359	+	0.479834i	$0.840697\pi$
$234$	0	0
$235$	−37.8564	−2.46948
$236$	−4.00000	−0.260378
$237$	0	0
$238$	7.46410	0.483826
$239$	−30.2487	−1.95663	−0.978313	−	0.207131i	$-0.933587\pi$
$239$	−30.2487	−1.95663	−0.978313	+	0.207131i	$0.933587\pi$
$240$	0	0
$241$	21.3205	1.37337	0.686687	−	0.726953i	$-0.259064\pi$
$241$	21.3205	1.37337	0.686687	+	0.726953i	$0.259064\pi$
$242$	18.8564	1.21214
$243$	0	0
$244$	−4.92820	−0.315496
$245$	3.46410	0.221313
$246$	0	0
$247$	−3.46410	−0.220416
$248$	6.92820	0.439941
$249$	0	0
$250$	6.92820	0.438178
$251$	−5.07180	−0.320129	−0.160064	−	0.987107i	$-0.551170\pi$
$251$	−5.07180	−0.320129	−0.160064	+	0.987107i	$0.551170\pi$
$252$	0	0
$253$	29.8564	1.87706
$254$	−17.4641	−1.09580
$255$	0	0
$256$	1.00000	0.0625000
$257$	−24.9282	−1.55498	−0.777489	−	0.628896i	$-0.783507\pi$
$257$	−24.9282	−1.55498	−0.777489	+	0.628896i	$0.783507\pi$
$258$	0	0
$259$	−10.0000	−0.621370
$260$	−12.0000	−0.744208
$261$	0	0
$262$	5.07180	0.313337
$263$	−11.3205	−0.698052	−0.349026	−	0.937113i	$-0.613488\pi$
$263$	−11.3205	−0.698052	−0.349026	+	0.937113i	$0.613488\pi$
$264$	0	0
$265$	−6.92820	−0.425596
$266$	−1.00000	−0.0613139
$267$	0	0
$268$	9.46410	0.578112
$269$	16.9282	1.03213	0.516065	−	0.856549i	$-0.327396\pi$
$269$	16.9282	1.03213	0.516065	+	0.856549i	$0.327396\pi$
$270$	0	0
$271$	−29.8564	−1.81365	−0.906824	−	0.421510i	$-0.861500\pi$
$271$	−29.8564	−1.81365	−0.906824	+	0.421510i	$0.861500\pi$
$272$	−7.46410	−0.452578
$273$	0	0
$274$	0.928203	0.0560748
$275$	38.2487	2.30648
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	9.85641	0.591148
$279$	0	0
$280$	−3.46410	−0.207020
$281$	26.0000	1.55103	0.775515	−	0.631329i	$-0.217490\pi$
$281$	26.0000	1.55103	0.775515	+	0.631329i	$0.217490\pi$
$282$	0	0
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	−2.92820	−0.173757
$285$	0	0
$286$	−18.9282	−1.11925
$287$	−2.00000	−0.118056
$288$	0	0
$289$	38.7128	2.27722
$290$	20.7846	1.22051
$291$	0	0
$292$	10.0000	0.585206
$293$	−7.07180	−0.413139	−0.206569	−	0.978432i	$-0.566230\pi$
$293$	−7.07180	−0.413139	−0.206569	+	0.978432i	$0.566230\pi$
$294$	0	0
$295$	−13.8564	−0.806751
$296$	10.0000	0.581238
$297$	0	0
$298$	3.07180	0.177944
$299$	−18.9282	−1.09465
$300$	0	0
$301$	6.92820	0.399335
$302$	1.46410	0.0842496
$303$	0	0
$304$	1.00000	0.0573539
$305$	−17.0718	−0.977528
$306$	0	0
$307$	20.7846	1.18624	0.593120	−	0.805114i	$-0.297896\pi$
$307$	20.7846	1.18624	0.593120	+	0.805114i	$0.297896\pi$
$308$	−5.46410	−0.311346
$309$	0	0
$310$	24.0000	1.36311
$311$	−2.14359	−0.121552	−0.0607760	−	0.998151i	$-0.519358\pi$
$311$	−2.14359	−0.121552	−0.0607760	+	0.998151i	$0.519358\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	19.8564	1.12056
$315$	0	0
$316$	−12.3923	−0.697122
$317$	−7.85641	−0.441260	−0.220630	−	0.975358i	$-0.570811\pi$
$317$	−7.85641	−0.441260	−0.220630	+	0.975358i	$0.570811\pi$
$318$	0	0
$319$	32.7846	1.83559
$320$	3.46410	0.193649
$321$	0	0
$322$	−5.46410	−0.304502
$323$	−7.46410	−0.415314
$324$	0	0
$325$	−24.2487	−1.34508
$326$	1.07180	0.0593613
$327$	0	0
$328$	2.00000	0.110432
$329$	10.9282	0.602491
$330$	0	0
$331$	14.5359	0.798965	0.399483	−	0.916741i	$-0.369190\pi$
$331$	14.5359	0.798965	0.399483	+	0.916741i	$0.369190\pi$
$332$	−8.00000	−0.439057
$333$	0	0
$334$	18.9282	1.03571
$335$	32.7846	1.79121
$336$	0	0
$337$	10.7846	0.587475	0.293738	−	0.955886i	$-0.405101\pi$
$337$	10.7846	0.587475	0.293738	+	0.955886i	$0.405101\pi$
$338$	−1.00000	−0.0543928
$339$	0	0
$340$	−25.8564	−1.40226
$341$	37.8564	2.05004
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−6.92820	−0.373544
$345$	0	0
$346$	0.928203	0.0499005
$347$	−27.3205	−1.46664	−0.733321	−	0.679883i	$-0.762031\pi$
$347$	−27.3205	−1.46664	−0.733321	+	0.679883i	$0.762031\pi$
$348$	0	0
$349$	35.8564	1.91935	0.959675	−	0.281113i	$-0.0907035\pi$
$349$	35.8564	1.91935	0.959675	+	0.281113i	$0.0907035\pi$
$350$	−7.00000	−0.374166
$351$	0	0
$352$	5.46410	0.291238
$353$	−5.32051	−0.283182	−0.141591	−	0.989925i	$-0.545222\pi$
$353$	−5.32051	−0.283182	−0.141591	+	0.989925i	$0.545222\pi$
$354$	0	0
$355$	−10.1436	−0.538366
$356$	2.00000	0.106000
$357$	0	0
$358$	9.85641	0.520927
$359$	−35.3205	−1.86415	−0.932073	−	0.362272i	$-0.882001\pi$
$359$	−35.3205	−1.86415	−0.932073	+	0.362272i	$0.882001\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−8.53590	−0.448637
$363$	0	0
$364$	3.46410	0.181568
$365$	34.6410	1.81319
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	5.46410	0.284836
$369$	0	0
$370$	34.6410	1.80090
$371$	2.00000	0.103835
$372$	0	0
$373$	10.7846	0.558406	0.279203	−	0.960232i	$-0.409930\pi$
$373$	10.7846	0.558406	0.279203	+	0.960232i	$0.409930\pi$
$374$	−40.7846	−2.10892
$375$	0	0
$376$	−10.9282	−0.563579
$377$	−20.7846	−1.07046
$378$	0	0
$379$	−1.46410	−0.0752058	−0.0376029	−	0.999293i	$-0.511972\pi$
$379$	−1.46410	−0.0752058	−0.0376029	+	0.999293i	$0.511972\pi$
$380$	3.46410	0.177705
$381$	0	0
$382$	5.46410	0.279568
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	0	0
$385$	−18.9282	−0.964671
$386$	−22.0000	−1.11977
$387$	0	0
$388$	4.53590	0.230275
$389$	−29.7128	−1.50650	−0.753250	−	0.657735i	$-0.771515\pi$
$389$	−29.7128	−1.50650	−0.753250	+	0.657735i	$0.771515\pi$
$390$	0	0
$391$	−40.7846	−2.06257
$392$	1.00000	0.0505076
$393$	0	0
$394$	−4.92820	−0.248279
$395$	−42.9282	−2.15995
$396$	0	0
$397$	19.8564	0.996564	0.498282	−	0.867015i	$-0.333964\pi$
$397$	19.8564	0.996564	0.498282	+	0.867015i	$0.333964\pi$
$398$	−24.7846	−1.24234
$399$	0	0
$400$	7.00000	0.350000
$401$	−19.8564	−0.991582	−0.495791	−	0.868442i	$-0.665122\pi$
$401$	−19.8564	−0.991582	−0.495791	+	0.868442i	$0.665122\pi$
$402$	0	0
$403$	−24.0000	−1.19553
$404$	−4.53590	−0.225669
$405$	0	0
$406$	−6.00000	−0.297775
$407$	54.6410	2.70845
$408$	0	0
$409$	−31.1769	−1.54160	−0.770800	−	0.637078i	$-0.780143\pi$
$409$	−31.1769	−1.54160	−0.770800	+	0.637078i	$0.780143\pi$
$410$	6.92820	0.342160
$411$	0	0
$412$	−6.92820	−0.341328
$413$	4.00000	0.196827
$414$	0	0
$415$	−27.7128	−1.36037
$416$	−3.46410	−0.169842
$417$	0	0
$418$	5.46410	0.267258
$419$	−18.9282	−0.924703	−0.462352	−	0.886697i	$-0.652994\pi$
$419$	−18.9282	−0.924703	−0.462352	+	0.886697i	$0.652994\pi$
$420$	0	0
$421$	18.7846	0.915506	0.457753	−	0.889079i	$-0.348654\pi$
$421$	18.7846	0.915506	0.457753	+	0.889079i	$0.348654\pi$
$422$	1.46410	0.0712714
$423$	0	0
$424$	−2.00000	−0.0971286
$425$	−52.2487	−2.53443
$426$	0	0
$427$	4.92820	0.238492
$428$	4.00000	0.193347
$429$	0	0
$430$	−24.0000	−1.15738
$431$	34.9282	1.68243	0.841216	−	0.540699i	$-0.181840\pi$
$431$	34.9282	1.68243	0.841216	+	0.540699i	$0.181840\pi$
$432$	0	0
$433$	6.67949	0.320996	0.160498	−	0.987036i	$-0.448690\pi$
$433$	6.67949	0.320996	0.160498	+	0.987036i	$0.448690\pi$
$434$	−6.92820	−0.332564
$435$	0	0
$436$	2.00000	0.0957826
$437$	5.46410	0.261383
$438$	0	0
$439$	−4.78461	−0.228357	−0.114178	−	0.993460i	$-0.536424\pi$
$439$	−4.78461	−0.228357	−0.114178	+	0.993460i	$0.536424\pi$
$440$	18.9282	0.902367
$441$	0	0
$442$	25.8564	1.22986
$443$	11.3205	0.537854	0.268927	−	0.963161i	$-0.413331\pi$
$443$	11.3205	0.537854	0.268927	+	0.963161i	$0.413331\pi$
$444$	0	0
$445$	6.92820	0.328428
$446$	−12.0000	−0.568216
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	7.85641	0.370767	0.185383	−	0.982666i	$-0.440647\pi$
$449$	7.85641	0.370767	0.185383	+	0.982666i	$0.440647\pi$
$450$	0	0
$451$	10.9282	0.514589
$452$	−11.8564	−0.557678
$453$	0	0
$454$	−14.9282	−0.700615
$455$	12.0000	0.562569
$456$	0	0
$457$	12.1436	0.568053	0.284027	−	0.958816i	$-0.408330\pi$
$457$	12.1436	0.568053	0.284027	+	0.958816i	$0.408330\pi$
$458$	3.07180	0.143536
$459$	0	0
$460$	18.9282	0.882532
$461$	−38.1051	−1.77473	−0.887366	−	0.461065i	$-0.847467\pi$
$461$	−38.1051	−1.77473	−0.887366	+	0.461065i	$0.847467\pi$
$462$	0	0
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−26.7846	−1.24077
$467$	−35.7128	−1.65259	−0.826296	−	0.563236i	$-0.809556\pi$
$467$	−35.7128	−1.65259	−0.826296	+	0.563236i	$0.809556\pi$
$468$	0	0
$469$	−9.46410	−0.437012
$470$	−37.8564	−1.74619
$471$	0	0
$472$	−4.00000	−0.184115
$473$	−37.8564	−1.74064
$474$	0	0
$475$	7.00000	0.321182
$476$	7.46410	0.342117
$477$	0	0
$478$	−30.2487	−1.38354
$479$	−27.7128	−1.26623	−0.633115	−	0.774057i	$-0.718224\pi$
$479$	−27.7128	−1.26623	−0.633115	+	0.774057i	$0.718224\pi$
$480$	0	0
$481$	−34.6410	−1.57949
$482$	21.3205	0.971123
$483$	0	0
$484$	18.8564	0.857109
$485$	15.7128	0.713482
$486$	0	0
$487$	−39.3205	−1.78178	−0.890891	−	0.454217i	$-0.849919\pi$
$487$	−39.3205	−1.78178	−0.890891	+	0.454217i	$0.849919\pi$
$488$	−4.92820	−0.223089
$489$	0	0
$490$	3.46410	0.156492
$491$	40.3923	1.82288	0.911440	−	0.411434i	$-0.134972\pi$
$491$	40.3923	1.82288	0.911440	+	0.411434i	$0.134972\pi$
$492$	0	0
$493$	−44.7846	−2.01700
$494$	−3.46410	−0.155857
$495$	0	0
$496$	6.92820	0.311086
$497$	2.92820	0.131348
$498$	0	0
$499$	−22.9282	−1.02641	−0.513204	−	0.858267i	$-0.671541\pi$
$499$	−22.9282	−1.02641	−0.513204	+	0.858267i	$0.671541\pi$
$500$	6.92820	0.309839
$501$	0	0
$502$	−5.07180	−0.226365
$503$	21.8564	0.974529	0.487264	−	0.873254i	$-0.337995\pi$
$503$	21.8564	0.974529	0.487264	+	0.873254i	$0.337995\pi$
$504$	0	0
$505$	−15.7128	−0.699211
$506$	29.8564	1.32728
$507$	0	0
$508$	−17.4641	−0.774844
$509$	44.6410	1.97868	0.989339	−	0.145630i	$-0.0465209\pi$
$509$	44.6410	1.97868	0.989339	+	0.145630i	$0.0465209\pi$
$510$	0	0
$511$	−10.0000	−0.442374
$512$	1.00000	0.0441942
$513$	0	0
$514$	−24.9282	−1.09954
$515$	−24.0000	−1.05757
$516$	0	0
$517$	−59.7128	−2.62617
$518$	−10.0000	−0.439375
$519$	0	0
$520$	−12.0000	−0.526235
$521$	−25.7128	−1.12650	−0.563249	−	0.826287i	$-0.690449\pi$
$521$	−25.7128	−1.12650	−0.563249	+	0.826287i	$0.690449\pi$
$522$	0	0
$523$	26.6410	1.16493	0.582465	−	0.812856i	$-0.302088\pi$
$523$	26.6410	1.16493	0.582465	+	0.812856i	$0.302088\pi$
$524$	5.07180	0.221562
$525$	0	0
$526$	−11.3205	−0.493598
$527$	−51.7128	−2.25265
$528$	0	0
$529$	6.85641	0.298105
$530$	−6.92820	−0.300942
$531$	0	0
$532$	−1.00000	−0.0433555
$533$	−6.92820	−0.300094
$534$	0	0
$535$	13.8564	0.599065
$536$	9.46410	0.408787
$537$	0	0
$538$	16.9282	0.729827
$539$	5.46410	0.235356
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	−29.8564	−1.28244
$543$	0	0
$544$	−7.46410	−0.320021
$545$	6.92820	0.296772
$546$	0	0
$547$	−10.2487	−0.438203	−0.219102	−	0.975702i	$-0.570313\pi$
$547$	−10.2487	−0.438203	−0.219102	+	0.975702i	$0.570313\pi$
$548$	0.928203	0.0396509
$549$	0	0
$550$	38.2487	1.63093
$551$	6.00000	0.255609
$552$	0	0
$553$	12.3923	0.526974
$554$	22.0000	0.934690
$555$	0	0
$556$	9.85641	0.418005
$557$	−4.92820	−0.208815	−0.104407	−	0.994535i	$-0.533295\pi$
$557$	−4.92820	−0.208815	−0.104407	+	0.994535i	$0.533295\pi$
$558$	0	0
$559$	24.0000	1.01509
$560$	−3.46410	−0.146385
$561$	0	0
$562$	26.0000	1.09674
$563$	−25.8564	−1.08972	−0.544859	−	0.838528i	$-0.683417\pi$
$563$	−25.8564	−1.08972	−0.544859	+	0.838528i	$0.683417\pi$
$564$	0	0
$565$	−41.0718	−1.72790
$566$	−20.0000	−0.840663
$567$	0	0
$568$	−2.92820	−0.122865
$569$	45.7128	1.91638	0.958190	−	0.286131i	$-0.0923694\pi$
$569$	45.7128	1.91638	0.958190	+	0.286131i	$0.0923694\pi$
$570$	0	0
$571$	37.5692	1.57222	0.786111	−	0.618085i	$-0.212091\pi$
$571$	37.5692	1.57222	0.786111	+	0.618085i	$0.212091\pi$
$572$	−18.9282	−0.791428
$573$	0	0
$574$	−2.00000	−0.0834784
$575$	38.2487	1.59508
$576$	0	0
$577$	−3.85641	−0.160544	−0.0802722	−	0.996773i	$-0.525579\pi$
$577$	−3.85641	−0.160544	−0.0802722	+	0.996773i	$0.525579\pi$
$578$	38.7128	1.61024
$579$	0	0
$580$	20.7846	0.863034
$581$	8.00000	0.331896
$582$	0	0
$583$	−10.9282	−0.452600
$584$	10.0000	0.413803
$585$	0	0
$586$	−7.07180	−0.292133
$587$	8.00000	0.330195	0.165098	−	0.986277i	$-0.447206\pi$
$587$	8.00000	0.330195	0.165098	+	0.986277i	$0.447206\pi$
$588$	0	0
$589$	6.92820	0.285472
$590$	−13.8564	−0.570459
$591$	0	0
$592$	10.0000	0.410997
$593$	−12.5359	−0.514788	−0.257394	−	0.966307i	$-0.582864\pi$
$593$	−12.5359	−0.514788	−0.257394	+	0.966307i	$0.582864\pi$
$594$	0	0
$595$	25.8564	1.06001
$596$	3.07180	0.125826
$597$	0	0
$598$	−18.9282	−0.774032
$599$	−13.0718	−0.534099	−0.267050	−	0.963683i	$-0.586049\pi$
$599$	−13.0718	−0.534099	−0.267050	+	0.963683i	$0.586049\pi$
$600$	0	0
$601$	6.67949	0.272462	0.136231	−	0.990677i	$-0.456501\pi$
$601$	6.67949	0.272462	0.136231	+	0.990677i	$0.456501\pi$
$602$	6.92820	0.282372
$603$	0	0
$604$	1.46410	0.0595734
$605$	65.3205	2.65566
$606$	0	0
$607$	−17.8564	−0.724769	−0.362385	−	0.932029i	$-0.618037\pi$
$607$	−17.8564	−0.724769	−0.362385	+	0.932029i	$0.618037\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	−17.0718	−0.691217
$611$	37.8564	1.53151
$612$	0	0
$613$	−23.8564	−0.963551	−0.481776	−	0.876295i	$-0.660008\pi$
$613$	−23.8564	−0.963551	−0.481776	+	0.876295i	$0.660008\pi$
$614$	20.7846	0.838799
$615$	0	0
$616$	−5.46410	−0.220155
$617$	−12.9282	−0.520470	−0.260235	−	0.965545i	$-0.583800\pi$
$617$	−12.9282	−0.520470	−0.260235	+	0.965545i	$0.583800\pi$
$618$	0	0
$619$	26.6410	1.07079	0.535396	−	0.844601i	$-0.320162\pi$
$619$	26.6410	1.07079	0.535396	+	0.844601i	$0.320162\pi$
$620$	24.0000	0.963863
$621$	0	0
$622$	−2.14359	−0.0859503
$623$	−2.00000	−0.0801283
$624$	0	0
$625$	−11.0000	−0.440000
$626$	−6.00000	−0.239808
$627$	0	0
$628$	19.8564	0.792357
$629$	−74.6410	−2.97613
$630$	0	0
$631$	16.7846	0.668185	0.334092	−	0.942540i	$-0.391570\pi$
$631$	16.7846	0.668185	0.334092	+	0.942540i	$0.391570\pi$
$632$	−12.3923	−0.492939
$633$	0	0
$634$	−7.85641	−0.312018
$635$	−60.4974	−2.40077
$636$	0	0
$637$	−3.46410	−0.137253
$638$	32.7846	1.29796
$639$	0	0
$640$	3.46410	0.136931
$641$	15.8564	0.626290	0.313145	−	0.949705i	$-0.398617\pi$
$641$	15.8564	0.626290	0.313145	+	0.949705i	$0.398617\pi$
$642$	0	0
$643$	−14.9282	−0.588711	−0.294355	−	0.955696i	$-0.595105\pi$
$643$	−14.9282	−0.588711	−0.294355	+	0.955696i	$0.595105\pi$
$644$	−5.46410	−0.215316
$645$	0	0
$646$	−7.46410	−0.293671
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	−21.8564	−0.857939
$650$	−24.2487	−0.951113
$651$	0	0
$652$	1.07180	0.0419748
$653$	33.7128	1.31928	0.659642	−	0.751580i	$-0.270708\pi$
$653$	33.7128	1.31928	0.659642	+	0.751580i	$0.270708\pi$
$654$	0	0
$655$	17.5692	0.686486
$656$	2.00000	0.0780869
$657$	0	0
$658$	10.9282	0.426026
$659$	−14.1436	−0.550956	−0.275478	−	0.961307i	$-0.588836\pi$
$659$	−14.1436	−0.550956	−0.275478	+	0.961307i	$0.588836\pi$
$660$	0	0
$661$	45.3205	1.76276	0.881382	−	0.472405i	$-0.156614\pi$
$661$	45.3205	1.76276	0.881382	+	0.472405i	$0.156614\pi$
$662$	14.5359	0.564954
$663$	0	0
$664$	−8.00000	−0.310460
$665$	−3.46410	−0.134332
$666$	0	0
$667$	32.7846	1.26943
$668$	18.9282	0.732354
$669$	0	0
$670$	32.7846	1.26658
$671$	−26.9282	−1.03955
$672$	0	0
$673$	−16.9282	−0.652534	−0.326267	−	0.945278i	$-0.605791\pi$
$673$	−16.9282	−0.652534	−0.326267	+	0.945278i	$0.605791\pi$
$674$	10.7846	0.415408
$675$	0	0
$676$	−1.00000	−0.0384615
$677$	19.8564	0.763144	0.381572	−	0.924339i	$-0.375383\pi$
$677$	19.8564	0.763144	0.381572	+	0.924339i	$0.375383\pi$
$678$	0	0
$679$	−4.53590	−0.174072
$680$	−25.8564	−0.991548
$681$	0	0
$682$	37.8564	1.44960
$683$	−30.9282	−1.18343	−0.591717	−	0.806145i	$-0.701550\pi$
$683$	−30.9282	−1.18343	−0.591717	+	0.806145i	$0.701550\pi$
$684$	0	0
$685$	3.21539	0.122854
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	−6.92820	−0.264135
$689$	6.92820	0.263944
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	0.928203	0.0352850
$693$	0	0
$694$	−27.3205	−1.03707
$695$	34.1436	1.29514
$696$	0	0
$697$	−14.9282	−0.565446
$698$	35.8564	1.35718
$699$	0	0
$700$	−7.00000	−0.264575
$701$	−13.7128	−0.517926	−0.258963	−	0.965887i	$-0.583381\pi$
$701$	−13.7128	−0.517926	−0.258963	+	0.965887i	$0.583381\pi$
$702$	0	0
$703$	10.0000	0.377157
$704$	5.46410	0.205936
$705$	0	0
$706$	−5.32051	−0.200240
$707$	4.53590	0.170590
$708$	0	0
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	−10.1436	−0.380682
$711$	0	0
$712$	2.00000	0.0749532
$713$	37.8564	1.41773
$714$	0	0
$715$	−65.5692	−2.45215
$716$	9.85641	0.368351
$717$	0	0
$718$	−35.3205	−1.31815
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	0	0
$721$	6.92820	0.258020
$722$	1.00000	0.0372161
$723$	0	0
$724$	−8.53590	−0.317234
$725$	42.0000	1.55984
$726$	0	0
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	3.46410	0.128388
$729$	0	0
$730$	34.6410	1.28212
$731$	51.7128	1.91267
$732$	0	0
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	5.46410	0.201409
$737$	51.7128	1.90487
$738$	0	0
$739$	−9.85641	−0.362574	−0.181287	−	0.983430i	$-0.558026\pi$
$739$	−9.85641	−0.362574	−0.181287	+	0.983430i	$0.558026\pi$
$740$	34.6410	1.27343
$741$	0	0
$742$	2.00000	0.0734223
$743$	−27.7128	−1.01668	−0.508342	−	0.861155i	$-0.669742\pi$
$743$	−27.7128	−1.01668	−0.508342	+	0.861155i	$0.669742\pi$
$744$	0	0
$745$	10.6410	0.389857
$746$	10.7846	0.394853
$747$	0	0
$748$	−40.7846	−1.49123
$749$	−4.00000	−0.146157
$750$	0	0
$751$	20.3923	0.744126	0.372063	−	0.928208i	$-0.378651\pi$
$751$	20.3923	0.744126	0.372063	+	0.928208i	$0.378651\pi$
$752$	−10.9282	−0.398511
$753$	0	0
$754$	−20.7846	−0.756931
$755$	5.07180	0.184582
$756$	0	0
$757$	3.85641	0.140163	0.0700817	−	0.997541i	$-0.477674\pi$
$757$	3.85641	0.140163	0.0700817	+	0.997541i	$0.477674\pi$
$758$	−1.46410	−0.0531786
$759$	0	0
$760$	3.46410	0.125656
$761$	−4.53590	−0.164426	−0.0822131	−	0.996615i	$-0.526199\pi$
$761$	−4.53590	−0.164426	−0.0822131	+	0.996615i	$0.526199\pi$
$762$	0	0
$763$	−2.00000	−0.0724049
$764$	5.46410	0.197684
$765$	0	0
$766$	−8.00000	−0.289052
$767$	13.8564	0.500326
$768$	0	0
$769$	−19.8564	−0.716040	−0.358020	−	0.933714i	$-0.616548\pi$
$769$	−19.8564	−0.716040	−0.358020	+	0.933714i	$0.616548\pi$
$770$	−18.9282	−0.682125
$771$	0	0
$772$	−22.0000	−0.791797
$773$	−2.78461	−0.100155	−0.0500777	−	0.998745i	$-0.515947\pi$
$773$	−2.78461	−0.100155	−0.0500777	+	0.998745i	$0.515947\pi$
$774$	0	0
$775$	48.4974	1.74208
$776$	4.53590	0.162829
$777$	0	0
$778$	−29.7128	−1.06526
$779$	2.00000	0.0716574
$780$	0	0
$781$	−16.0000	−0.572525
$782$	−40.7846	−1.45845
$783$	0	0
$784$	1.00000	0.0357143
$785$	68.7846	2.45503
$786$	0	0
$787$	−11.2154	−0.399785	−0.199893	−	0.979818i	$-0.564059\pi$
$787$	−11.2154	−0.399785	−0.199893	+	0.979818i	$0.564059\pi$
$788$	−4.92820	−0.175560
$789$	0	0
$790$	−42.9282	−1.52732
$791$	11.8564	0.421565
$792$	0	0
$793$	17.0718	0.606237
$794$	19.8564	0.704677
$795$	0	0
$796$	−24.7846	−0.878467
$797$	44.6410	1.58127	0.790633	−	0.612290i	$-0.209752\pi$
$797$	44.6410	1.58127	0.790633	+	0.612290i	$0.209752\pi$
$798$	0	0
$799$	81.5692	2.88571
$800$	7.00000	0.247487
$801$	0	0
$802$	−19.8564	−0.701154
$803$	54.6410	1.92824
$804$	0	0
$805$	−18.9282	−0.667132
$806$	−24.0000	−0.845364
$807$	0	0
$808$	−4.53590	−0.159572
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	33.0718	1.16131	0.580654	−	0.814150i	$-0.302797\pi$
$811$	33.0718	1.16131	0.580654	+	0.814150i	$0.302797\pi$
$812$	−6.00000	−0.210559
$813$	0	0
$814$	54.6410	1.91517
$815$	3.71281	0.130054
$816$	0	0
$817$	−6.92820	−0.242387
$818$	−31.1769	−1.09008
$819$	0	0
$820$	6.92820	0.241943
$821$	48.9282	1.70761	0.853803	−	0.520596i	$-0.174290\pi$
$821$	48.9282	1.70761	0.853803	+	0.520596i	$0.174290\pi$
$822$	0	0
$823$	−21.8564	−0.761866	−0.380933	−	0.924603i	$-0.624397\pi$
$823$	−21.8564	−0.761866	−0.380933	+	0.924603i	$0.624397\pi$
$824$	−6.92820	−0.241355
$825$	0	0
$826$	4.00000	0.139178
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	−5.60770	−0.194763	−0.0973817	−	0.995247i	$-0.531047\pi$
$829$	−5.60770	−0.194763	−0.0973817	+	0.995247i	$0.531047\pi$
$830$	−27.7128	−0.961926
$831$	0	0
$832$	−3.46410	−0.120096
$833$	−7.46410	−0.258616
$834$	0	0
$835$	65.5692	2.26912
$836$	5.46410	0.188980
$837$	0	0
$838$	−18.9282	−0.653864
$839$	5.07180	0.175098	0.0875489	−	0.996160i	$-0.472097\pi$
$839$	5.07180	0.175098	0.0875489	+	0.996160i	$0.472097\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	18.7846	0.647360
$843$	0	0
$844$	1.46410	0.0503965
$845$	−3.46410	−0.119169
$846$	0	0
$847$	−18.8564	−0.647914
$848$	−2.00000	−0.0686803
$849$	0	0
$850$	−52.2487	−1.79212
$851$	54.6410	1.87307
$852$	0	0
$853$	35.0718	1.20084	0.600418	−	0.799687i	$-0.295001\pi$
$853$	35.0718	1.20084	0.600418	+	0.799687i	$0.295001\pi$
$854$	4.92820	0.168640
$855$	0	0
$856$	4.00000	0.136717
$857$	−47.5692	−1.62493	−0.812467	−	0.583007i	$-0.801876\pi$
$857$	−47.5692	−1.62493	−0.812467	+	0.583007i	$0.801876\pi$
$858$	0	0
$859$	−28.7846	−0.982118	−0.491059	−	0.871126i	$-0.663390\pi$
$859$	−28.7846	−0.982118	−0.491059	+	0.871126i	$0.663390\pi$
$860$	−24.0000	−0.818393
$861$	0	0
$862$	34.9282	1.18966
$863$	−18.1436	−0.617615	−0.308808	−	0.951125i	$-0.599930\pi$
$863$	−18.1436	−0.617615	−0.308808	+	0.951125i	$0.599930\pi$
$864$	0	0
$865$	3.21539	0.109327
$866$	6.67949	0.226978
$867$	0	0
$868$	−6.92820	−0.235159
$869$	−67.7128	−2.29700
$870$	0	0
$871$	−32.7846	−1.11086
$872$	2.00000	0.0677285
$873$	0	0
$874$	5.46410	0.184826
$875$	−6.92820	−0.234216
$876$	0	0
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	−4.78461	−0.161473
$879$	0	0
$880$	18.9282	0.638070
$881$	−43.1769	−1.45467	−0.727334	−	0.686284i	$-0.759241\pi$
$881$	−43.1769	−1.45467	−0.727334	+	0.686284i	$0.759241\pi$
$882$	0	0
$883$	48.4974	1.63207	0.816034	−	0.578004i	$-0.196168\pi$
$883$	48.4974	1.63207	0.816034	+	0.578004i	$0.196168\pi$
$884$	25.8564	0.869645
$885$	0	0
$886$	11.3205	0.380320
$887$	27.7128	0.930505	0.465253	−	0.885178i	$-0.345963\pi$
$887$	27.7128	0.930505	0.465253	+	0.885178i	$0.345963\pi$
$888$	0	0
$889$	17.4641	0.585727
$890$	6.92820	0.232234
$891$	0	0
$892$	−12.0000	−0.401790
$893$	−10.9282	−0.365698
$894$	0	0
$895$	34.1436	1.14129
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	7.85641	0.262172
$899$	41.5692	1.38641
$900$	0	0
$901$	14.9282	0.497331
$902$	10.9282	0.363869
$903$	0	0
$904$	−11.8564	−0.394338
$905$	−29.5692	−0.982914
$906$	0	0
$907$	−41.4641	−1.37679	−0.688396	−	0.725335i	$-0.741685\pi$
$907$	−41.4641	−1.37679	−0.688396	+	0.725335i	$0.741685\pi$
$908$	−14.9282	−0.495410
$909$	0	0
$910$	12.0000	0.397796
$911$	−22.6410	−0.750130	−0.375065	−	0.926998i	$-0.622380\pi$
$911$	−22.6410	−0.750130	−0.375065	+	0.926998i	$0.622380\pi$
$912$	0	0
$913$	−43.7128	−1.44668
$914$	12.1436	0.401674
$915$	0	0
$916$	3.07180	0.101495
$917$	−5.07180	−0.167485
$918$	0	0
$919$	−49.5692	−1.63514	−0.817569	−	0.575831i	$-0.804679\pi$
$919$	−49.5692	−1.63514	−0.817569	+	0.575831i	$0.804679\pi$
$920$	18.9282	0.624044
$921$	0	0
$922$	−38.1051	−1.25493
$923$	10.1436	0.333880
$924$	0	0
$925$	70.0000	2.30159
$926$	24.0000	0.788689
$927$	0	0
$928$	6.00000	0.196960
$929$	−25.6077	−0.840161	−0.420081	−	0.907487i	$-0.637998\pi$
$929$	−25.6077	−0.840161	−0.420081	+	0.907487i	$0.637998\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	−26.7846	−0.877359
$933$	0	0
$934$	−35.7128	−1.16856
$935$	−141.282	−4.62042
$936$	0	0
$937$	20.9282	0.683695	0.341847	−	0.939756i	$-0.388947\pi$
$937$	20.9282	0.683695	0.341847	+	0.939756i	$0.388947\pi$
$938$	−9.46410	−0.309014
$939$	0	0
$940$	−37.8564	−1.23474
$941$	5.21539	0.170017	0.0850084	−	0.996380i	$-0.472908\pi$
$941$	5.21539	0.170017	0.0850084	+	0.996380i	$0.472908\pi$
$942$	0	0
$943$	10.9282	0.355871
$944$	−4.00000	−0.130189
$945$	0	0
$946$	−37.8564	−1.23082
$947$	28.1051	0.913294	0.456647	−	0.889648i	$-0.349050\pi$
$947$	28.1051	0.913294	0.456647	+	0.889648i	$0.349050\pi$
$948$	0	0
$949$	−34.6410	−1.12449
$950$	7.00000	0.227110
$951$	0	0
$952$	7.46410	0.241913
$953$	31.8564	1.03193	0.515965	−	0.856610i	$-0.327433\pi$
$953$	31.8564	1.03193	0.515965	+	0.856610i	$0.327433\pi$
$954$	0	0
$955$	18.9282	0.612502
$956$	−30.2487	−0.978313
$957$	0	0
$958$	−27.7128	−0.895360
$959$	−0.928203	−0.0299732
$960$	0	0
$961$	17.0000	0.548387
$962$	−34.6410	−1.11687
$963$	0	0
$964$	21.3205	0.686687
$965$	−76.2102	−2.45329
$966$	0	0
$967$	−18.9282	−0.608690	−0.304345	−	0.952562i	$-0.598438\pi$
$967$	−18.9282	−0.608690	−0.304345	+	0.952562i	$0.598438\pi$
$968$	18.8564	0.606068
$969$	0	0
$970$	15.7128	0.504508
$971$	10.6410	0.341486	0.170743	−	0.985316i	$-0.445383\pi$
$971$	10.6410	0.341486	0.170743	+	0.985316i	$0.445383\pi$
$972$	0	0
$973$	−9.85641	−0.315982
$974$	−39.3205	−1.25991
$975$	0	0
$976$	−4.92820	−0.157748
$977$	7.85641	0.251349	0.125674	−	0.992072i	$-0.459891\pi$
$977$	7.85641	0.251349	0.125674	+	0.992072i	$0.459891\pi$
$978$	0	0
$979$	10.9282	0.349267
$980$	3.46410	0.110657
$981$	0	0
$982$	40.3923	1.28897
$983$	32.7846	1.04567	0.522833	−	0.852435i	$-0.324875\pi$
$983$	32.7846	1.04567	0.522833	+	0.852435i	$0.324875\pi$
$984$	0	0
$985$	−17.0718	−0.543953
$986$	−44.7846	−1.42623
$987$	0	0
$988$	−3.46410	−0.110208
$989$	−37.8564	−1.20376
$990$	0	0
$991$	35.6077	1.13112	0.565558	−	0.824709i	$-0.308661\pi$
$991$	35.6077	1.13112	0.565558	+	0.824709i	$0.308661\pi$
$992$	6.92820	0.219971
$993$	0	0
$994$	2.92820	0.0928770
$995$	−85.8564	−2.72183
$996$	0	0
$997$	−47.8564	−1.51563	−0.757814	−	0.652471i	$-0.773732\pi$
$997$	−47.8564	−1.51563	−0.757814	+	0.652471i	$0.773732\pi$
$998$	−22.9282	−0.725780
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2394.2.a.x.1.2		2
3.2	odd	2		798.2.a.k.1.1	✓	2
12.11	even	2		6384.2.a.br.1.1		2
21.20	even	2		5586.2.a.bd.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
798.2.a.k.1.1	✓	2	3.2	odd	2
2394.2.a.x.1.2		2	1.1	even	1	trivial
5586.2.a.bd.1.2		2	21.20	even	2
6384.2.a.br.1.1		2	12.11	even	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.46410 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.46410 q^{5} -1.00000 q^{7} +1.00000 q^{8} +3.46410 q^{10} +5.46410 q^{11} -3.46410 q^{13} -1.00000 q^{14} +1.00000 q^{16} -7.46410 q^{17} +1.00000 q^{19} +3.46410 q^{20} +5.46410 q^{22} +5.46410 q^{23} +7.00000 q^{25} -3.46410 q^{26} -1.00000 q^{28} +6.00000 q^{29} +6.92820 q^{31} +1.00000 q^{32} -7.46410 q^{34} -3.46410 q^{35} +10.0000 q^{37} +1.00000 q^{38} +3.46410 q^{40} +2.00000 q^{41} -6.92820 q^{43} +5.46410 q^{44} +5.46410 q^{46} -10.9282 q^{47} +1.00000 q^{49} +7.00000 q^{50} -3.46410 q^{52} -2.00000 q^{53} +18.9282 q^{55} -1.00000 q^{56} +6.00000 q^{58} -4.00000 q^{59} -4.92820 q^{61} +6.92820 q^{62} +1.00000 q^{64} -12.0000 q^{65} +9.46410 q^{67} -7.46410 q^{68} -3.46410 q^{70} -2.92820 q^{71} +10.0000 q^{73} +10.0000 q^{74} +1.00000 q^{76} -5.46410 q^{77} -12.3923 q^{79} +3.46410 q^{80} +2.00000 q^{82} -8.00000 q^{83} -25.8564 q^{85} -6.92820 q^{86} +5.46410 q^{88} +2.00000 q^{89} +3.46410 q^{91} +5.46410 q^{92} -10.9282 q^{94} +3.46410 q^{95} +4.53590 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} + 4 q^{11} - 2 q^{14} + 2 q^{16} - 8 q^{17} + 2 q^{19} + 4 q^{22} + 4 q^{23} + 14 q^{25} - 2 q^{28} + 12 q^{29} + 2 q^{32} - 8 q^{34} + 20 q^{37} + 2 q^{38} + 4 q^{41} + 4 q^{44} + 4 q^{46} - 8 q^{47} + 2 q^{49} + 14 q^{50} - 4 q^{53} + 24 q^{55} - 2 q^{56} + 12 q^{58} - 8 q^{59} + 4 q^{61} + 2 q^{64} - 24 q^{65} + 12 q^{67} - 8 q^{68} + 8 q^{71} + 20 q^{73} + 20 q^{74} + 2 q^{76} - 4 q^{77} - 4 q^{79} + 4 q^{82} - 16 q^{83} - 24 q^{85} + 4 q^{88} + 4 q^{89} + 4 q^{92} - 8 q^{94} + 16 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8 + 4 * q^11 - 2 * q^14 + 2 * q^16 - 8 * q^17 + 2 * q^19 + 4 * q^22 + 4 * q^23 + 14 * q^25 - 2 * q^28 + 12 * q^29 + 2 * q^32 - 8 * q^34 + 20 * q^37 + 2 * q^38 + 4 * q^41 + 4 * q^44 + 4 * q^46 - 8 * q^47 + 2 * q^49 + 14 * q^50 - 4 * q^53 + 24 * q^55 - 2 * q^56 + 12 * q^58 - 8 * q^59 + 4 * q^61 + 2 * q^64 - 24 * q^65 + 12 * q^67 - 8 * q^68 + 8 * q^71 + 20 * q^73 + 20 * q^74 + 2 * q^76 - 4 * q^77 - 4 * q^79 + 4 * q^82 - 16 * q^83 - 24 * q^85 + 4 * q^88 + 4 * q^89 + 4 * q^92 - 8 * q^94 + 16 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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