LMFDB - Embedded newform 693.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,2,Mod(1,693)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, 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("693.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	693.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:	$5.53363286007$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.11491$ of defining polynomial
Character	$\chi$	$=$	693.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.47283 q^{2} +4.11491 q^{4} +2.58774 q^{5} -1.00000 q^{7} -5.22982 q^{8} +O(q^{10})$	\(q-2.47283 q^{2} +4.11491 q^{4} +2.58774 q^{5} -1.00000 q^{7} -5.22982 q^{8} -6.39905 q^{10} +1.00000 q^{11} -5.87189 q^{13} +2.47283 q^{14} +4.70265 q^{16} -7.51396 q^{17} -2.35793 q^{19} +10.6483 q^{20} -2.47283 q^{22} -6.94567 q^{23} +1.69641 q^{25} +14.5202 q^{26} -4.11491 q^{28} +5.87189 q^{29} -3.66152 q^{31} -1.16924 q^{32} +18.5808 q^{34} -2.58774 q^{35} +3.30359 q^{37} +5.83076 q^{38} -13.5334 q^{40} -5.28415 q^{41} +7.40530 q^{43} +4.11491 q^{44} +17.1755 q^{46} -7.53341 q^{47} +1.00000 q^{49} -4.19493 q^{50} -24.1623 q^{52} +4.22982 q^{53} +2.58774 q^{55} +5.22982 q^{56} -14.5202 q^{58} +0.926221 q^{59} -2.00000 q^{61} +9.05433 q^{62} -6.51396 q^{64} -15.1949 q^{65} -10.1017 q^{67} -30.9193 q^{68} +6.39905 q^{70} -4.45963 q^{71} -2.12811 q^{73} -8.16924 q^{74} -9.70265 q^{76} -1.00000 q^{77} -4.45963 q^{79} +12.1692 q^{80} +13.0668 q^{82} +10.6894 q^{83} -19.4442 q^{85} -18.3121 q^{86} -5.22982 q^{88} -15.8913 q^{89} +5.87189 q^{91} -28.5808 q^{92} +18.6289 q^{94} -6.10170 q^{95} +10.1212 q^{97} -2.47283 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} + 6 q^{4} - 4 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - 2 * q^2 + 6 * q^4 - 4 * q^5 - 3 * q^7 - 3 * q^8	$ 3 q - 2 q^{2} + 6 q^{4} - 4 q^{5} - 3 q^{7} - 3 q^{8} - 11 q^{10} + 3 q^{11} - 4 q^{13} + 2 q^{14} - 4 q^{16} - 8 q^{17} - 8 q^{19} + 3 q^{20} - 2 q^{22} - 10 q^{23} + 15 q^{25} + q^{26} - 6 q^{28} + 4 q^{29} - 2 q^{31} - 8 q^{32} - 4 q^{34} + 4 q^{35} + 13 q^{38} - 18 q^{40} - 14 q^{41} - 14 q^{43} + 6 q^{44} + 28 q^{46} + 3 q^{49} + 19 q^{50} - 29 q^{52} - 4 q^{55} + 3 q^{56} - q^{58} - 6 q^{61} + 38 q^{62} - 5 q^{64} - 14 q^{65} - 4 q^{67} - 42 q^{68} + 11 q^{70} + 12 q^{71} - 20 q^{73} - 29 q^{74} - 11 q^{76} - 3 q^{77} + 12 q^{79} + 41 q^{80} - 6 q^{82} - 6 q^{83} - 6 q^{85} - 24 q^{86} - 3 q^{88} - 26 q^{89} + 4 q^{91} - 26 q^{92} + 35 q^{94} + 8 q^{95} - 4 q^{97} - 2 q^{98}+O(q^{100}) $ 3 * q - 2 * q^2 + 6 * q^4 - 4 * q^5 - 3 * q^7 - 3 * q^8 - 11 * q^10 + 3 * q^11 - 4 * q^13 + 2 * q^14 - 4 * q^16 - 8 * q^17 - 8 * q^19 + 3 * q^20 - 2 * q^22 - 10 * q^23 + 15 * q^25 + q^26 - 6 * q^28 + 4 * q^29 - 2 * q^31 - 8 * q^32 - 4 * q^34 + 4 * q^35 + 13 * q^38 - 18 * q^40 - 14 * q^41 - 14 * q^43 + 6 * q^44 + 28 * q^46 + 3 * q^49 + 19 * q^50 - 29 * q^52 - 4 * q^55 + 3 * q^56 - q^58 - 6 * q^61 + 38 * q^62 - 5 * q^64 - 14 * q^65 - 4 * q^67 - 42 * q^68 + 11 * q^70 + 12 * q^71 - 20 * q^73 - 29 * q^74 - 11 * q^76 - 3 * q^77 + 12 * q^79 + 41 * q^80 - 6 * q^82 - 6 * q^83 - 6 * q^85 - 24 * q^86 - 3 * q^88 - 26 * q^89 + 4 * q^91 - 26 * q^92 + 35 * q^94 + 8 * q^95 - 4 * 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 $

$2$	−2.47283	−1.74856	−0.874279	−	0.485424i	$-0.838665\pi$
$2$	−2.47283	−1.74856	−0.874279	+	0.485424i	$0.838665\pi$
$3$	0	0
$3$	0	0
$4$	4.11491	2.05745
$5$	2.58774	1.15727	0.578637	−	0.815586i	$-0.303585\pi$
$5$	2.58774	1.15727	0.578637	+	0.815586i	$0.303585\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−5.22982	−1.84902
$9$	0	0
$10$	−6.39905	−2.02356
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−5.87189	−1.62857	−0.814284	−	0.580466i	$-0.802870\pi$
$13$	−5.87189	−1.62857	−0.814284	+	0.580466i	$0.802870\pi$
$14$	2.47283	0.660893
$15$	0	0
$16$	4.70265	1.17566
$17$	−7.51396	−1.82240	−0.911202	−	0.411960i	$-0.864844\pi$
$17$	−7.51396	−1.82240	−0.911202	+	0.411960i	$0.864844\pi$
$18$	0	0
$19$	−2.35793	−0.540945	−0.270473	−	0.962728i	$-0.587180\pi$
$19$	−2.35793	−0.540945	−0.270473	+	0.962728i	$0.587180\pi$
$20$	10.6483	2.38104
$21$	0	0
$22$	−2.47283	−0.527210
$23$	−6.94567	−1.44827	−0.724136	−	0.689657i	$-0.757761\pi$
$23$	−6.94567	−1.44827	−0.724136	+	0.689657i	$0.757761\pi$
$24$	0	0
$25$	1.69641	0.339281
$26$	14.5202	2.84765
$27$	0	0
$28$	−4.11491	−0.777644
$29$	5.87189	1.09038	0.545191	−	0.838312i	$-0.316457\pi$
$29$	5.87189	1.09038	0.545191	+	0.838312i	$0.316457\pi$
$30$	0	0
$31$	−3.66152	−0.657629	−0.328814	−	0.944395i	$-0.606649\pi$
$31$	−3.66152	−0.657629	−0.328814	+	0.944395i	$0.606649\pi$
$32$	−1.16924	−0.206694
$33$	0	0
$34$	18.5808	3.18658
$35$	−2.58774	−0.437408
$36$	0	0
$37$	3.30359	0.543108	0.271554	−	0.962423i	$-0.412463\pi$
$37$	3.30359	0.543108	0.271554	+	0.962423i	$0.412463\pi$
$38$	5.83076	0.945874
$39$	0	0
$40$	−13.5334	−2.13982
$41$	−5.28415	−0.825245	−0.412623	−	0.910902i	$-0.635387\pi$
$41$	−5.28415	−0.825245	−0.412623	+	0.910902i	$0.635387\pi$
$42$	0	0
$43$	7.40530	1.12930	0.564649	−	0.825331i	$-0.309012\pi$
$43$	7.40530	1.12930	0.564649	+	0.825331i	$0.309012\pi$
$44$	4.11491	0.620346
$45$	0	0
$46$	17.1755	2.53239
$47$	−7.53341	−1.09886	−0.549430	−	0.835540i	$-0.685155\pi$
$47$	−7.53341	−1.09886	−0.549430	+	0.835540i	$0.685155\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−4.19493	−0.593253
$51$	0	0
$52$	−24.1623	−3.35071
$53$	4.22982	0.581010	0.290505	−	0.956874i	$-0.406177\pi$
$53$	4.22982	0.581010	0.290505	+	0.956874i	$0.406177\pi$
$54$	0	0
$55$	2.58774	0.348931
$56$	5.22982	0.698863
$57$	0	0
$58$	−14.5202	−1.90660
$59$	0.926221	0.120584	0.0602918	−	0.998181i	$-0.480797\pi$
$59$	0.926221	0.120584	0.0602918	+	0.998181i	$0.480797\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	9.05433	1.14990
$63$	0	0
$64$	−6.51396	−0.814245
$65$	−15.1949	−1.88470
$66$	0	0
$67$	−10.1017	−1.23412	−0.617060	−	0.786916i	$-0.711676\pi$
$67$	−10.1017	−1.23412	−0.617060	+	0.786916i	$0.711676\pi$
$68$	−30.9193	−3.74951
$69$	0	0
$70$	6.39905	0.764833
$71$	−4.45963	−0.529261	−0.264630	−	0.964350i	$-0.585250\pi$
$71$	−4.45963	−0.529261	−0.264630	+	0.964350i	$0.585250\pi$
$72$	0	0
$73$	−2.12811	−0.249077	−0.124538	−	0.992215i	$-0.539745\pi$
$73$	−2.12811	−0.249077	−0.124538	+	0.992215i	$0.539745\pi$
$74$	−8.16924	−0.949655
$75$	0	0
$76$	−9.70265	−1.11297
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−4.45963	−0.501748	−0.250874	−	0.968020i	$-0.580718\pi$
$79$	−4.45963	−0.501748	−0.250874	+	0.968020i	$0.580718\pi$
$80$	12.1692	1.36056
$81$	0	0
$82$	13.0668	1.44299
$83$	10.6894	1.17332	0.586660	−	0.809834i	$-0.300443\pi$
$83$	10.6894	1.17332	0.586660	+	0.809834i	$0.300443\pi$
$84$	0	0
$85$	−19.4442	−2.10902
$86$	−18.3121	−1.97464
$87$	0	0
$88$	−5.22982	−0.557500
$89$	−15.8913	−1.68448	−0.842239	−	0.539104i	$-0.818763\pi$
$89$	−15.8913	−1.68448	−0.842239	+	0.539104i	$0.818763\pi$
$90$	0	0
$91$	5.87189	0.615541
$92$	−28.5808	−2.97975
$93$	0	0
$94$	18.6289	1.92142
$95$	−6.10170	−0.626022
$96$	0	0
$97$	10.1212	1.02765	0.513824	−	0.857896i	$-0.328229\pi$
$97$	10.1212	1.02765	0.513824	+	0.857896i	$0.328229\pi$
$98$	−2.47283	−0.249794
$99$	0	0
$100$	6.98055	0.698055
$101$	13.4053	1.33388	0.666939	−	0.745113i	$-0.267604\pi$
$101$	13.4053	1.33388	0.666939	+	0.745113i	$0.267604\pi$
$102$	0	0
$103$	−4.71585	−0.464667	−0.232333	−	0.972636i	$-0.574636\pi$
$103$	−4.71585	−0.464667	−0.232333	+	0.972636i	$0.574636\pi$
$104$	30.7089	3.01125
$105$	0	0
$106$	−10.4596	−1.01593
$107$	−7.07378	−0.683848	−0.341924	−	0.939728i	$-0.611079\pi$
$107$	−7.07378	−0.683848	−0.341924	+	0.939728i	$0.611079\pi$
$108$	0	0
$109$	18.8370	1.80426	0.902129	−	0.431467i	$-0.142004\pi$
$109$	18.8370	1.80426	0.902129	+	0.431467i	$0.142004\pi$
$110$	−6.39905	−0.610126
$111$	0	0
$112$	−4.70265	−0.444359
$113$	9.02792	0.849276	0.424638	−	0.905363i	$-0.360401\pi$
$113$	9.02792	0.849276	0.424638	+	0.905363i	$0.360401\pi$
$114$	0	0
$115$	−17.9736	−1.67605
$116$	24.1623	2.24341
$117$	0	0
$118$	−2.29039	−0.210848
$119$	7.51396	0.688804
$120$	0	0
$121$	1.00000	0.0909091
$122$	4.94567	0.447760
$123$	0	0
$124$	−15.0668	−1.35304
$125$	−8.54885	−0.764632
$126$	0	0
$127$	−10.2298	−0.907749	−0.453875	−	0.891066i	$-0.649959\pi$
$127$	−10.2298	−0.907749	−0.453875	+	0.891066i	$0.649959\pi$
$128$	18.4464	1.63045
$129$	0	0
$130$	37.5745	3.29550
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	2.35793	0.204458
$134$	24.9798	2.15793
$135$	0	0
$136$	39.2966	3.36966
$137$	10.8370	0.925868	0.462934	−	0.886393i	$-0.346797\pi$
$137$	10.8370	0.925868	0.462934	+	0.886393i	$0.346797\pi$
$138$	0	0
$139$	0.459630	0.0389853	0.0194927	−	0.999810i	$-0.493795\pi$
$139$	0.459630	0.0389853	0.0194927	+	0.999810i	$0.493795\pi$
$140$	−10.6483	−0.899947
$141$	0	0
$142$	11.0279	0.925443
$143$	−5.87189	−0.491032
$144$	0	0
$145$	15.1949	1.26187
$146$	5.26247	0.435525
$147$	0	0
$148$	13.5940	1.11742
$149$	−11.0474	−0.905036	−0.452518	−	0.891755i	$-0.649474\pi$
$149$	−11.0474	−0.905036	−0.452518	+	0.891755i	$0.649474\pi$
$150$	0	0
$151$	−16.1212	−1.31192	−0.655960	−	0.754795i	$-0.727736\pi$
$151$	−16.1212	−1.31192	−0.655960	+	0.754795i	$0.727736\pi$
$152$	12.3315	1.00022
$153$	0	0
$154$	2.47283	0.199267
$155$	−9.47507	−0.761056
$156$	0	0
$157$	13.0668	1.04285	0.521423	−	0.853298i	$-0.325401\pi$
$157$	13.0668	1.04285	0.521423	+	0.853298i	$0.325401\pi$
$158$	11.0279	0.877335
$159$	0	0
$160$	−3.02569	−0.239202
$161$	6.94567	0.547395
$162$	0	0
$163$	−10.1017	−0.791227	−0.395613	−	0.918417i	$-0.629468\pi$
$163$	−10.1017	−0.791227	−0.395613	+	0.918417i	$0.629468\pi$
$164$	−21.7438	−1.69790
$165$	0	0
$166$	−26.4332	−2.05162
$167$	10.4860	0.811434	0.405717	−	0.913999i	$-0.367022\pi$
$167$	10.4860	0.811434	0.405717	+	0.913999i	$0.367022\pi$
$168$	0	0
$169$	21.4791	1.65224
$170$	48.0823	3.68774
$171$	0	0
$172$	30.4721	2.32348
$173$	−14.4985	−1.10230	−0.551151	−	0.834405i	$-0.685811\pi$
$173$	−14.4985	−1.10230	−0.551151	+	0.834405i	$0.685811\pi$
$174$	0	0
$175$	−1.69641	−0.128236
$176$	4.70265	0.354476
$177$	0	0
$178$	39.2966	2.94541
$179$	−14.3510	−1.07264	−0.536321	−	0.844014i	$-0.680186\pi$
$179$	−14.3510	−1.07264	−0.536321	+	0.844014i	$0.680186\pi$
$180$	0	0
$181$	9.66152	0.718135	0.359068	−	0.933312i	$-0.383095\pi$
$181$	9.66152	0.718135	0.359068	+	0.933312i	$0.383095\pi$
$182$	−14.5202	−1.07631
$183$	0	0
$184$	36.3246	2.67788
$185$	8.54885	0.628524
$186$	0	0
$187$	−7.51396	−0.549475
$188$	−30.9993	−2.26086
$189$	0	0
$190$	15.0885	1.09463
$191$	9.55286	0.691220	0.345610	−	0.938378i	$-0.387672\pi$
$191$	9.55286	0.691220	0.345610	+	0.938378i	$0.387672\pi$
$192$	0	0
$193$	−22.1212	−1.59232	−0.796158	−	0.605089i	$-0.793137\pi$
$193$	−22.1212	−1.59232	−0.796158	+	0.605089i	$0.793137\pi$
$194$	−25.0279	−1.79690
$195$	0	0
$196$	4.11491	0.293922
$197$	26.2423	1.86969	0.934843	−	0.355061i	$-0.115540\pi$
$197$	26.2423	1.86969	0.934843	+	0.355061i	$0.115540\pi$
$198$	0	0
$199$	1.05433	0.0747396	0.0373698	−	0.999302i	$-0.488102\pi$
$199$	1.05433	0.0747396	0.0373698	+	0.999302i	$0.488102\pi$
$200$	−8.87189	−0.627337
$201$	0	0
$202$	−33.1491	−2.33236
$203$	−5.87189	−0.412126
$204$	0	0
$205$	−13.6740	−0.955034
$206$	11.6615	0.812497
$207$	0	0
$208$	−27.6134	−1.91465
$209$	−2.35793	−0.163101
$210$	0	0
$211$	−8.71585	−0.600024	−0.300012	−	0.953935i	$-0.596991\pi$
$211$	−8.71585	−0.600024	−0.300012	+	0.953935i	$0.596991\pi$
$212$	17.4053	1.19540
$213$	0	0
$214$	17.4923	1.19575
$215$	19.1630	1.30691
$216$	0	0
$217$	3.66152	0.248560
$218$	−46.5808	−3.15485
$219$	0	0
$220$	10.6483	0.717909
$221$	44.1212	2.96791
$222$	0	0
$223$	−23.7438	−1.59000	−0.795000	−	0.606609i	$-0.792529\pi$
$223$	−23.7438	−1.59000	−0.795000	+	0.606609i	$0.792529\pi$
$224$	1.16924	0.0781231
$225$	0	0
$226$	−22.3246	−1.48501
$227$	13.1755	0.874488	0.437244	−	0.899343i	$-0.355955\pi$
$227$	13.1755	0.874488	0.437244	+	0.899343i	$0.355955\pi$
$228$	0	0
$229$	13.3664	0.883277	0.441638	−	0.897193i	$-0.354397\pi$
$229$	13.3664	0.883277	0.441638	+	0.897193i	$0.354397\pi$
$230$	44.4457	2.93066
$231$	0	0
$232$	−30.7089	−2.01614
$233$	−7.13659	−0.467533	−0.233767	−	0.972293i	$-0.575105\pi$
$233$	−7.13659	−0.467533	−0.233767	+	0.972293i	$0.575105\pi$
$234$	0	0
$235$	−19.4945	−1.27168
$236$	3.81131	0.248095
$237$	0	0
$238$	−18.5808	−1.20441
$239$	19.9930	1.29324	0.646621	−	0.762811i	$-0.276182\pi$
$239$	19.9930	1.29324	0.646621	+	0.762811i	$0.276182\pi$
$240$	0	0
$241$	−9.19493	−0.592297	−0.296149	−	0.955142i	$-0.595702\pi$
$241$	−9.19493	−0.592297	−0.296149	+	0.955142i	$0.595702\pi$
$242$	−2.47283	−0.158960
$243$	0	0
$244$	−8.22982	−0.526860
$245$	2.58774	0.165325
$246$	0	0
$247$	13.8455	0.880967
$248$	19.1491	1.21597
$249$	0	0
$250$	21.1399	1.33700
$251$	9.42474	0.594885	0.297442	−	0.954740i	$-0.403866\pi$
$251$	9.42474	0.594885	0.297442	+	0.954740i	$0.403866\pi$
$252$	0	0
$253$	−6.94567	−0.436670
$254$	25.2966	1.58725
$255$	0	0
$256$	−32.5870	−2.03669
$257$	0.0194469	0.00121307	0.000606533	−	1.00000i	$-0.499807\pi$
$257$	0.0194469	0.00121307	0.000606533		1.00000i	$0.499807\pi$
$258$	0	0
$259$	−3.30359	−0.205275
$260$	−62.5257	−3.87768
$261$	0	0
$262$	9.89134	0.611089
$263$	−27.9930	−1.72612	−0.863062	−	0.505097i	$-0.831457\pi$
$263$	−27.9930	−1.72612	−0.863062	+	0.505097i	$0.831457\pi$
$264$	0	0
$265$	10.9457	0.672387
$266$	−5.83076	−0.357507
$267$	0	0
$268$	−41.5676	−2.53914
$269$	−30.9193	−1.88518	−0.942590	−	0.333951i	$-0.891618\pi$
$269$	−30.9193	−1.88518	−0.942590	+	0.333951i	$0.891618\pi$
$270$	0	0
$271$	−22.3579	−1.35815	−0.679074	−	0.734070i	$-0.737618\pi$
$271$	−22.3579	−1.35815	−0.679074	+	0.734070i	$0.737618\pi$
$272$	−35.3355	−2.14253
$273$	0	0
$274$	−26.7981	−1.61893
$275$	1.69641	0.102297
$276$	0	0
$277$	24.7717	1.48839	0.744194	−	0.667964i	$-0.232834\pi$
$277$	24.7717	1.48839	0.744194	+	0.667964i	$0.232834\pi$
$278$	−1.13659	−0.0681681
$279$	0	0
$280$	13.5334	0.808776
$281$	1.19493	0.0712835	0.0356418	−	0.999365i	$-0.488652\pi$
$281$	1.19493	0.0712835	0.0356418	+	0.999365i	$0.488652\pi$
$282$	0	0
$283$	−15.5723	−0.925677	−0.462839	−	0.886443i	$-0.653169\pi$
$283$	−15.5723	−0.925677	−0.462839	+	0.886443i	$0.653169\pi$
$284$	−18.3510	−1.08893
$285$	0	0
$286$	14.5202	0.858598
$287$	5.28415	0.311913
$288$	0	0
$289$	39.4596	2.32115
$290$	−37.5745	−2.20645
$291$	0	0
$292$	−8.75698	−0.512464
$293$	9.74378	0.569238	0.284619	−	0.958641i	$-0.408133\pi$
$293$	9.74378	0.569238	0.284619	+	0.958641i	$0.408133\pi$
$294$	0	0
$295$	2.39682	0.139548
$296$	−17.2772	−1.00422
$297$	0	0
$298$	27.3183	1.58251
$299$	40.7842	2.35861
$300$	0	0
$301$	−7.40530	−0.426834
$302$	39.8649	2.29397
$303$	0	0
$304$	−11.0885	−0.635969
$305$	−5.17548	−0.296347
$306$	0	0
$307$	6.35097	0.362469	0.181234	−	0.983440i	$-0.441991\pi$
$307$	6.35097	0.362469	0.181234	+	0.983440i	$0.441991\pi$
$308$	−4.11491	−0.234469
$309$	0	0
$310$	23.4303	1.33075
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	16.9457	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$313$	16.9457	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$314$	−32.3121	−1.82348
$315$	0	0
$316$	−18.3510	−1.03232
$317$	−20.0125	−1.12401	−0.562007	−	0.827133i	$-0.689970\pi$
$317$	−20.0125	−1.12401	−0.562007	+	0.827133i	$0.689970\pi$
$318$	0	0
$319$	5.87189	0.328763
$320$	−16.8565	−0.942304
$321$	0	0
$322$	−17.1755	−0.957152
$323$	17.7174	0.985821
$324$	0	0
$325$	−9.96111	−0.552543
$326$	24.9798	1.38351
$327$	0	0
$328$	27.6351	1.52589
$329$	7.53341	0.415330
$330$	0	0
$331$	7.54037	0.414456	0.207228	−	0.978293i	$-0.433556\pi$
$331$	7.54037	0.414456	0.207228	+	0.978293i	$0.433556\pi$
$332$	43.9861	2.41405
$333$	0	0
$334$	−25.9302	−1.41884
$335$	−26.1406	−1.42821
$336$	0	0
$337$	−12.6894	−0.691238	−0.345619	−	0.938375i	$-0.612331\pi$
$337$	−12.6894	−0.691238	−0.345619	+	0.938375i	$0.612331\pi$
$338$	−53.1142	−2.88903
$339$	0	0
$340$	−80.0111	−4.33921
$341$	−3.66152	−0.198282
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−38.7283	−2.08809
$345$	0	0
$346$	35.8524	1.92744
$347$	−1.13659	−0.0610153	−0.0305076	−	0.999535i	$-0.509712\pi$
$347$	−1.13659	−0.0610153	−0.0305076	+	0.999535i	$0.509712\pi$
$348$	0	0
$349$	−14.7911	−0.791752	−0.395876	−	0.918304i	$-0.629559\pi$
$349$	−14.7911	−0.791752	−0.395876	+	0.918304i	$0.629559\pi$
$350$	4.19493	0.224228
$351$	0	0
$352$	−1.16924	−0.0623207
$353$	7.52092	0.400298	0.200149	−	0.979765i	$-0.435857\pi$
$353$	7.52092	0.400298	0.200149	+	0.979765i	$0.435857\pi$
$354$	0	0
$355$	−11.5404	−0.612499
$356$	−65.3914	−3.46574
$357$	0	0
$358$	35.4876	1.87558
$359$	17.5962	0.928693	0.464346	−	0.885654i	$-0.346289\pi$
$359$	17.5962	0.928693	0.464346	+	0.885654i	$0.346289\pi$
$360$	0	0
$361$	−13.4402	−0.707378
$362$	−23.8913	−1.25570
$363$	0	0
$364$	24.1623	1.26645
$365$	−5.50700	−0.288250
$366$	0	0
$367$	14.3121	0.747084	0.373542	−	0.927613i	$-0.378143\pi$
$367$	14.3121	0.747084	0.373542	+	0.927613i	$0.378143\pi$
$368$	−32.6630	−1.70268
$369$	0	0
$370$	−21.1399	−1.09901
$371$	−4.22982	−0.219601
$372$	0	0
$373$	−16.4332	−0.850880	−0.425440	−	0.904987i	$-0.639881\pi$
$373$	−16.4332	−0.850880	−0.425440	+	0.904987i	$0.639881\pi$
$374$	18.5808	0.960789
$375$	0	0
$376$	39.3983	2.03181
$377$	−34.4791	−1.77576
$378$	0	0
$379$	19.5334	1.00336	0.501682	−	0.865052i	$-0.332715\pi$
$379$	19.5334	1.00336	0.501682	+	0.865052i	$0.332715\pi$
$380$	−25.1079	−1.28801
$381$	0	0
$382$	−23.6226	−1.20864
$383$	−2.35097	−0.120129	−0.0600644	−	0.998195i	$-0.519131\pi$
$383$	−2.35097	−0.120129	−0.0600644	+	0.998195i	$0.519131\pi$
$384$	0	0
$385$	−2.58774	−0.131884
$386$	54.7019	2.78426
$387$	0	0
$388$	41.6476	2.11434
$389$	1.74378	0.0884130	0.0442065	−	0.999022i	$-0.485924\pi$
$389$	1.74378	0.0884130	0.0442065	+	0.999022i	$0.485924\pi$
$390$	0	0
$391$	52.1895	2.63934
$392$	−5.22982	−0.264146
$393$	0	0
$394$	−64.8929	−3.26925
$395$	−11.5404	−0.580659
$396$	0	0
$397$	6.54189	0.328328	0.164164	−	0.986433i	$-0.447507\pi$
$397$	6.54189	0.328328	0.164164	+	0.986433i	$0.447507\pi$
$398$	−2.60719	−0.130687
$399$	0	0
$400$	7.97760	0.398880
$401$	−5.66152	−0.282723	−0.141361	−	0.989958i	$-0.545148\pi$
$401$	−5.66152	−0.282723	−0.141361	+	0.989958i	$0.545148\pi$
$402$	0	0
$403$	21.5000	1.07099
$404$	55.1616	2.74439
$405$	0	0
$406$	14.5202	0.720626
$407$	3.30359	0.163753
$408$	0	0
$409$	−5.48755	−0.271342	−0.135671	−	0.990754i	$-0.543319\pi$
$409$	−5.48755	−0.271342	−0.135671	+	0.990754i	$0.543319\pi$
$410$	33.8135	1.66993
$411$	0	0
$412$	−19.4053	−0.956030
$413$	−0.926221	−0.0455764
$414$	0	0
$415$	27.6615	1.35785
$416$	6.86565	0.336616
$417$	0	0
$418$	5.83076	0.285192
$419$	−8.24926	−0.403003	−0.201501	−	0.979488i	$-0.564582\pi$
$419$	−8.24926	−0.403003	−0.201501	+	0.979488i	$0.564582\pi$
$420$	0	0
$421$	32.2617	1.57234	0.786171	−	0.618009i	$-0.212061\pi$
$421$	32.2617	1.57234	0.786171	+	0.618009i	$0.212061\pi$
$422$	21.5529	1.04918
$423$	0	0
$424$	−22.1212	−1.07430
$425$	−12.7467	−0.618307
$426$	0	0
$427$	2.00000	0.0967868
$428$	−29.1079	−1.40699
$429$	0	0
$430$	−47.3869	−2.28520
$431$	23.2772	1.12122	0.560611	−	0.828079i	$-0.310566\pi$
$431$	23.2772	1.12122	0.560611	+	0.828079i	$0.310566\pi$
$432$	0	0
$433$	−15.9302	−0.765558	−0.382779	−	0.923840i	$-0.625033\pi$
$433$	−15.9302	−0.765558	−0.382779	+	0.923840i	$0.625033\pi$
$434$	−9.05433	−0.434622
$435$	0	0
$436$	77.5125	3.71218
$437$	16.3774	0.783436
$438$	0	0
$439$	22.8565	1.09088	0.545439	−	0.838150i	$-0.316363\pi$
$439$	22.8565	1.09088	0.545439	+	0.838150i	$0.316363\pi$
$440$	−13.5334	−0.645180
$441$	0	0
$442$	−109.104	−5.18956
$443$	−21.5962	−1.02607	−0.513034	−	0.858368i	$-0.671478\pi$
$443$	−21.5962	−1.02607	−0.513034	+	0.858368i	$0.671478\pi$
$444$	0	0
$445$	−41.1227	−1.94940
$446$	58.7144	2.78021
$447$	0	0
$448$	6.51396	0.307756
$449$	−18.6630	−0.880763	−0.440382	−	0.897811i	$-0.645157\pi$
$449$	−18.6630	−0.880763	−0.440382	+	0.897811i	$0.645157\pi$
$450$	0	0
$451$	−5.28415	−0.248821
$452$	37.1491	1.74735
$453$	0	0
$454$	−32.5808	−1.52909
$455$	15.1949	0.712349
$456$	0	0
$457$	36.4721	1.70609	0.853047	−	0.521834i	$-0.174752\pi$
$457$	36.4721	1.70609	0.853047	+	0.521834i	$0.174752\pi$
$458$	−33.0529	−1.54446
$459$	0	0
$460$	−73.9597	−3.44839
$461$	−7.17548	−0.334196	−0.167098	−	0.985940i	$-0.553440\pi$
$461$	−7.17548	−0.334196	−0.167098	+	0.985940i	$0.553440\pi$
$462$	0	0
$463$	−0.452670	−0.0210373	−0.0105187	−	0.999945i	$-0.503348\pi$
$463$	−0.452670	−0.0210373	−0.0105187	+	0.999945i	$0.503348\pi$
$464$	27.6134	1.28192
$465$	0	0
$466$	17.6476	0.817509
$467$	9.68097	0.447982	0.223991	−	0.974591i	$-0.428091\pi$
$467$	9.68097	0.447982	0.223991	+	0.974591i	$0.428091\pi$
$468$	0	0
$469$	10.1017	0.466453
$470$	48.2067	2.22361
$471$	0	0
$472$	−4.84396	−0.222962
$473$	7.40530	0.340496
$474$	0	0
$475$	−4.00000	−0.183533
$476$	30.9193	1.41718
$477$	0	0
$478$	−49.4395	−2.26131
$479$	−24.3635	−1.11319	−0.556597	−	0.830782i	$-0.687893\pi$
$479$	−24.3635	−1.11319	−0.556597	+	0.830782i	$0.687893\pi$
$480$	0	0
$481$	−19.3983	−0.884488
$482$	22.7375	1.03567
$483$	0	0
$484$	4.11491	0.187041
$485$	26.1909	1.18927
$486$	0	0
$487$	−19.2702	−0.873217	−0.436609	−	0.899652i	$-0.643821\pi$
$487$	−19.2702	−0.873217	−0.436609	+	0.899652i	$0.643821\pi$
$488$	10.4596	0.473485
$489$	0	0
$490$	−6.39905	−0.289080
$491$	−29.6421	−1.33773	−0.668864	−	0.743385i	$-0.733219\pi$
$491$	−29.6421	−1.33773	−0.668864	+	0.743385i	$0.733219\pi$
$492$	0	0
$493$	−44.1212	−1.98712
$494$	−34.2376	−1.54042
$495$	0	0
$496$	−17.2188	−0.773149
$497$	4.45963	0.200042
$498$	0	0
$499$	29.3719	1.31487	0.657434	−	0.753512i	$-0.271642\pi$
$499$	29.3719	1.31487	0.657434	+	0.753512i	$0.271642\pi$
$500$	−35.1777	−1.57320
$501$	0	0
$502$	−23.3058	−1.04019
$503$	−12.5947	−0.561570	−0.280785	−	0.959771i	$-0.590595\pi$
$503$	−12.5947	−0.561570	−0.280785	+	0.959771i	$0.590595\pi$
$504$	0	0
$505$	34.6894	1.54366
$506$	17.1755	0.763543
$507$	0	0
$508$	−42.0947	−1.86765
$509$	15.2144	0.674365	0.337183	−	0.941439i	$-0.390526\pi$
$509$	15.2144	0.674365	0.337183	+	0.941439i	$0.390526\pi$
$510$	0	0
$511$	2.12811	0.0941421
$512$	43.6894	1.93082
$513$	0	0
$514$	−0.0480890	−0.00212112
$515$	−12.2034	−0.537746
$516$	0	0
$517$	−7.53341	−0.331319
$518$	8.16924	0.358936
$519$	0	0
$520$	79.4667	3.48484
$521$	−20.5180	−0.898909	−0.449454	−	0.893303i	$-0.648382\pi$
$521$	−20.5180	−0.898909	−0.449454	+	0.893303i	$0.648382\pi$
$522$	0	0
$523$	−29.8844	−1.30675	−0.653376	−	0.757033i	$-0.726648\pi$
$523$	−29.8844	−1.30675	−0.653376	+	0.757033i	$0.726648\pi$
$524$	−16.4596	−0.719042
$525$	0	0
$526$	69.2221	3.01823
$527$	27.5125	1.19846
$528$	0	0
$529$	25.2423	1.09749
$530$	−27.0668	−1.17571
$531$	0	0
$532$	9.70265	0.420663
$533$	31.0279	1.34397
$534$	0	0
$535$	−18.3051	−0.791399
$536$	52.8300	2.28191
$537$	0	0
$538$	76.4582	3.29635
$539$	1.00000	0.0430730
$540$	0	0
$541$	26.9582	1.15902	0.579511	−	0.814965i	$-0.303244\pi$
$541$	26.9582	1.15902	0.579511	+	0.814965i	$0.303244\pi$
$542$	55.2874	2.37480
$543$	0	0
$544$	8.78562	0.376680
$545$	48.7453	2.08802
$546$	0	0
$547$	26.0558	1.11407	0.557034	−	0.830490i	$-0.311939\pi$
$547$	26.0558	1.11407	0.557034	+	0.830490i	$0.311939\pi$
$548$	44.5933	1.90493
$549$	0	0
$550$	−4.19493	−0.178872
$551$	−13.8455	−0.589837
$552$	0	0
$553$	4.45963	0.189643
$554$	−61.2563	−2.60253
$555$	0	0
$556$	1.89134	0.0802105
$557$	−8.01945	−0.339795	−0.169897	−	0.985462i	$-0.554344\pi$
$557$	−8.01945	−0.339795	−0.169897	+	0.985462i	$0.554344\pi$
$558$	0	0
$559$	−43.4831	−1.83914
$560$	−12.1692	−0.514244
$561$	0	0
$562$	−2.95486	−0.124643
$563$	−21.9736	−0.926077	−0.463038	−	0.886338i	$-0.653241\pi$
$563$	−21.9736	−0.926077	−0.463038	+	0.886338i	$0.653241\pi$
$564$	0	0
$565$	23.3619	0.982844
$566$	38.5077	1.61860
$567$	0	0
$568$	23.3230	0.978613
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	37.6740	1.57661	0.788304	−	0.615286i	$-0.210959\pi$
$571$	37.6740	1.57661	0.788304	+	0.615286i	$0.210959\pi$
$572$	−24.1623	−1.01028
$573$	0	0
$574$	−13.0668	−0.545398
$575$	−11.7827	−0.491371
$576$	0	0
$577$	−7.80908	−0.325096	−0.162548	−	0.986701i	$-0.551971\pi$
$577$	−7.80908	−0.325096	−0.162548	+	0.986701i	$0.551971\pi$
$578$	−97.5771	−4.05867
$579$	0	0
$580$	62.5257	2.59624
$581$	−10.6894	−0.443473
$582$	0	0
$583$	4.22982	0.175181
$584$	11.1296	0.460547
$585$	0	0
$586$	−24.0947	−0.995345
$587$	43.2633	1.78567	0.892833	−	0.450388i	$-0.148714\pi$
$587$	43.2633	1.78567	0.892833	+	0.450388i	$0.148714\pi$
$588$	0	0
$589$	8.63360	0.355741
$590$	−5.92694	−0.244008
$591$	0	0
$592$	15.5356	0.638511
$593$	29.7438	1.22143	0.610715	−	0.791850i	$-0.290882\pi$
$593$	29.7438	1.22143	0.610715	+	0.791850i	$0.290882\pi$
$594$	0	0
$595$	19.4442	0.797134
$596$	−45.4589	−1.86207
$597$	0	0
$598$	−100.853	−4.12417
$599$	−11.5404	−0.471527	−0.235763	−	0.971810i	$-0.575759\pi$
$599$	−11.5404	−0.471527	−0.235763	+	0.971810i	$0.575759\pi$
$600$	0	0
$601$	7.06129	0.288036	0.144018	−	0.989575i	$-0.453998\pi$
$601$	7.06129	0.288036	0.144018	+	0.989575i	$0.453998\pi$
$602$	18.3121	0.746344
$603$	0	0
$604$	−66.3370	−2.69922
$605$	2.58774	0.105207
$606$	0	0
$607$	−17.3859	−0.705670	−0.352835	−	0.935686i	$-0.614782\pi$
$607$	−17.3859	−0.705670	−0.352835	+	0.935686i	$0.614782\pi$
$608$	2.75698	0.111810
$609$	0	0
$610$	12.7981	0.518180
$611$	44.2353	1.78957
$612$	0	0
$613$	39.4611	1.59382	0.796910	−	0.604098i	$-0.206466\pi$
$613$	39.4611	1.59382	0.796910	+	0.604098i	$0.206466\pi$
$614$	−15.7049	−0.633798
$615$	0	0
$616$	5.22982	0.210715
$617$	4.40378	0.177290	0.0886448	−	0.996063i	$-0.471746\pi$
$617$	4.40378	0.177290	0.0886448	+	0.996063i	$0.471746\pi$
$618$	0	0
$619$	16.4163	0.659825	0.329913	−	0.944011i	$-0.392981\pi$
$619$	16.4163	0.659825	0.329913	+	0.944011i	$0.392981\pi$
$620$	−38.9890	−1.56584
$621$	0	0
$622$	19.7827	0.793213
$623$	15.8913	0.636673
$624$	0	0
$625$	−30.6042	−1.22417
$626$	−41.9038	−1.67481
$627$	0	0
$628$	53.7688	2.14561
$629$	−24.8231	−0.989761
$630$	0	0
$631$	44.4596	1.76991	0.884955	−	0.465677i	$-0.154189\pi$
$631$	44.4596	1.76991	0.884955	+	0.465677i	$0.154189\pi$
$632$	23.3230	0.927741
$633$	0	0
$634$	49.4876	1.96540
$635$	−26.4721	−1.05051
$636$	0	0
$637$	−5.87189	−0.232653
$638$	−14.5202	−0.574860
$639$	0	0
$640$	47.7346	1.88688
$641$	−10.5419	−0.416380	−0.208190	−	0.978088i	$-0.566757\pi$
$641$	−10.5419	−0.416380	−0.208190	+	0.978088i	$0.566757\pi$
$642$	0	0
$643$	−23.4053	−0.923015	−0.461507	−	0.887136i	$-0.652691\pi$
$643$	−23.4053	−0.923015	−0.461507	+	0.887136i	$0.652691\pi$
$644$	28.5808	1.12624
$645$	0	0
$646$	−43.8121	−1.72376
$647$	−11.4945	−0.451896	−0.225948	−	0.974139i	$-0.572548\pi$
$647$	−11.4945	−0.451896	−0.225948	+	0.974139i	$0.572548\pi$
$648$	0	0
$649$	0.926221	0.0363574
$650$	24.6322	0.966153
$651$	0	0
$652$	−41.5676	−1.62791
$653$	−19.8774	−0.777863	−0.388932	−	0.921267i	$-0.627156\pi$
$653$	−19.8774	−0.777863	−0.388932	+	0.921267i	$0.627156\pi$
$654$	0	0
$655$	−10.3510	−0.404446
$656$	−24.8495	−0.970210
$657$	0	0
$658$	−18.6289	−0.726229
$659$	−15.0738	−0.587191	−0.293596	−	0.955930i	$-0.594852\pi$
$659$	−15.0738	−0.587191	−0.293596	+	0.955930i	$0.594852\pi$
$660$	0	0
$661$	−4.74226	−0.184453	−0.0922263	−	0.995738i	$-0.529398\pi$
$661$	−4.74226	−0.184453	−0.0922263	+	0.995738i	$0.529398\pi$
$662$	−18.6461	−0.724701
$663$	0	0
$664$	−55.9038	−2.16949
$665$	6.10170	0.236614
$666$	0	0
$667$	−40.7842	−1.57917
$668$	43.1491	1.66949
$669$	0	0
$670$	64.6414	2.49731
$671$	−2.00000	−0.0772091
$672$	0	0
$673$	9.87885	0.380802	0.190401	−	0.981706i	$-0.439021\pi$
$673$	9.87885	0.380802	0.190401	+	0.981706i	$0.439021\pi$
$674$	31.3789	1.20867
$675$	0	0
$676$	88.3844	3.39940
$677$	29.0279	1.11563	0.557817	−	0.829964i	$-0.311639\pi$
$677$	29.0279	1.11563	0.557817	+	0.829964i	$0.311639\pi$
$678$	0	0
$679$	−10.1212	−0.388414
$680$	101.690	3.89962
$681$	0	0
$682$	9.05433	0.346708
$683$	37.9597	1.45249	0.726243	−	0.687438i	$-0.241265\pi$
$683$	37.9597	1.45249	0.726243	+	0.687438i	$0.241265\pi$
$684$	0	0
$685$	28.0434	1.07148
$686$	2.47283	0.0944132
$687$	0	0
$688$	34.8245	1.32767
$689$	−24.8370	−0.946214
$690$	0	0
$691$	−15.5264	−0.590654	−0.295327	−	0.955396i	$-0.595429\pi$
$691$	−15.5264	−0.590654	−0.295327	+	0.955396i	$0.595429\pi$
$692$	−59.6601	−2.26794
$693$	0	0
$694$	2.81060	0.106689
$695$	1.18940	0.0451167
$696$	0	0
$697$	39.7049	1.50393
$698$	36.5761	1.38442
$699$	0	0
$700$	−6.98055	−0.263840
$701$	27.4317	1.03608	0.518041	−	0.855356i	$-0.326662\pi$
$701$	27.4317	1.03608	0.518041	+	0.855356i	$0.326662\pi$
$702$	0	0
$703$	−7.78963	−0.293792
$704$	−6.51396	−0.245504
$705$	0	0
$706$	−18.5980	−0.699945
$707$	−13.4053	−0.504158
$708$	0	0
$709$	−44.5180	−1.67191	−0.835954	−	0.548800i	$-0.815085\pi$
$709$	−44.5180	−1.67191	−0.835954	+	0.548800i	$0.815085\pi$
$710$	28.5374	1.07099
$711$	0	0
$712$	83.1087	3.11463
$713$	25.4317	0.952425
$714$	0	0
$715$	−15.1949	−0.568258
$716$	−59.0529	−2.20691
$717$	0	0
$718$	−43.5125	−1.62387
$719$	−17.3859	−0.648383	−0.324191	−	0.945992i	$-0.605092\pi$
$719$	−17.3859	−0.648383	−0.324191	+	0.945992i	$0.605092\pi$
$720$	0	0
$721$	4.71585	0.175628
$722$	33.2353	1.23689
$723$	0	0
$724$	39.7563	1.47753
$725$	9.96111	0.369946
$726$	0	0
$727$	−18.6506	−0.691711	−0.345855	−	0.938288i	$-0.612411\pi$
$727$	−18.6506	−0.691711	−0.345855	+	0.938288i	$0.612411\pi$
$728$	−30.7089	−1.13815
$729$	0	0
$730$	13.6179	0.504021
$731$	−55.6431	−2.05804
$732$	0	0
$733$	16.3510	0.603937	0.301968	−	0.953318i	$-0.402356\pi$
$733$	16.3510	0.603937	0.301968	+	0.953318i	$0.402356\pi$
$734$	−35.3914	−1.30632
$735$	0	0
$736$	8.12115	0.299350
$737$	−10.1017	−0.372101
$738$	0	0
$739$	−32.8370	−1.20793	−0.603964	−	0.797011i	$-0.706413\pi$
$739$	−32.8370	−1.20793	−0.603964	+	0.797011i	$0.706413\pi$
$740$	35.1777	1.29316
$741$	0	0
$742$	10.4596	0.383985
$743$	−20.9123	−0.767198	−0.383599	−	0.923500i	$-0.625315\pi$
$743$	−20.9123	−0.767198	−0.383599	+	0.923500i	$0.625315\pi$
$744$	0	0
$745$	−28.5877	−1.04737
$746$	40.6366	1.48781
$747$	0	0
$748$	−30.9193	−1.13052
$749$	7.07378	0.258470
$750$	0	0
$751$	−39.2633	−1.43274	−0.716368	−	0.697722i	$-0.754197\pi$
$751$	−39.2633	−1.43274	−0.716368	+	0.697722i	$0.754197\pi$
$752$	−35.4270	−1.29189
$753$	0	0
$754$	85.2610	3.10502
$755$	−41.7174	−1.51825
$756$	0	0
$757$	−10.3454	−0.376011	−0.188006	−	0.982168i	$-0.560202\pi$
$757$	−10.3454	−0.376011	−0.188006	+	0.982168i	$0.560202\pi$
$758$	−48.3029	−1.75444
$759$	0	0
$760$	31.9108	1.15753
$761$	−30.9582	−1.12223	−0.561116	−	0.827737i	$-0.689628\pi$
$761$	−30.9582	−1.12223	−0.561116	+	0.827737i	$0.689628\pi$
$762$	0	0
$763$	−18.8370	−0.681945
$764$	39.3091	1.42215
$765$	0	0
$766$	5.81355	0.210052
$767$	−5.43867	−0.196379
$768$	0	0
$769$	−28.8998	−1.04215	−0.521077	−	0.853510i	$-0.674470\pi$
$769$	−28.8998	−1.04215	−0.521077	+	0.853510i	$0.674470\pi$
$770$	6.39905	0.230606
$771$	0	0
$772$	−91.0265	−3.27612
$773$	26.7523	0.962212	0.481106	−	0.876663i	$-0.340235\pi$
$773$	26.7523	0.962212	0.481106	+	0.876663i	$0.340235\pi$
$774$	0	0
$775$	−6.21142	−0.223121
$776$	−52.9317	−1.90014
$777$	0	0
$778$	−4.31207	−0.154595
$779$	12.4596	0.446413
$780$	0	0
$781$	−4.45963	−0.159578
$782$	−129.056	−4.61503
$783$	0	0
$784$	4.70265	0.167952
$785$	33.8135	1.20686
$786$	0	0
$787$	22.1406	0.789227	0.394614	−	0.918847i	$-0.370878\pi$
$787$	22.1406	0.789227	0.394614	+	0.918847i	$0.370878\pi$
$788$	107.985	3.84679
$789$	0	0
$790$	28.5374	1.01532
$791$	−9.02792	−0.320996
$792$	0	0
$793$	11.7438	0.417034
$794$	−16.1770	−0.574100
$795$	0	0
$796$	4.33848	0.153773
$797$	−8.18396	−0.289891	−0.144945	−	0.989440i	$-0.546301\pi$
$797$	−8.18396	−0.289891	−0.144945	+	0.989440i	$0.546301\pi$
$798$	0	0
$799$	56.6058	2.00257
$800$	−1.98351	−0.0701275
$801$	0	0
$802$	14.0000	0.494357
$803$	−2.12811	−0.0750994
$804$	0	0
$805$	17.9736	0.633486
$806$	−53.1660	−1.87269
$807$	0	0
$808$	−70.1072	−2.46636
$809$	−17.6685	−0.621191	−0.310595	−	0.950542i	$-0.600528\pi$
$809$	−17.6685	−0.621191	−0.310595	+	0.950542i	$0.600528\pi$
$810$	0	0
$811$	−43.0210	−1.51067	−0.755335	−	0.655339i	$-0.772526\pi$
$811$	−43.0210	−1.51067	−0.755335	+	0.655339i	$0.772526\pi$
$812$	−24.1623	−0.847930
$813$	0	0
$814$	−8.16924	−0.286332
$815$	−26.1406	−0.915665
$816$	0	0
$817$	−17.4611	−0.610888
$818$	13.5698	0.474457
$819$	0	0
$820$	−56.2673	−1.96494
$821$	−18.0364	−0.629475	−0.314737	−	0.949179i	$-0.601916\pi$
$821$	−18.0364	−0.629475	−0.314737	+	0.949179i	$0.601916\pi$
$822$	0	0
$823$	−8.21037	−0.286195	−0.143098	−	0.989709i	$-0.545706\pi$
$823$	−8.21037	−0.286195	−0.143098	+	0.989709i	$0.545706\pi$
$824$	24.6630	0.859178
$825$	0	0
$826$	2.29039	0.0796929
$827$	−6.83148	−0.237554	−0.118777	−	0.992921i	$-0.537897\pi$
$827$	−6.83148	−0.237554	−0.118777	+	0.992921i	$0.537897\pi$
$828$	0	0
$829$	−25.7438	−0.894118	−0.447059	−	0.894504i	$-0.647529\pi$
$829$	−25.7438	−0.894118	−0.447059	+	0.894504i	$0.647529\pi$
$830$	−68.4023	−2.37428
$831$	0	0
$832$	38.2493	1.32605
$833$	−7.51396	−0.260343
$834$	0	0
$835$	27.1352	0.939051
$836$	−9.70265	−0.335573
$837$	0	0
$838$	20.3991	0.704674
$839$	30.6002	1.05644	0.528219	−	0.849108i	$-0.322860\pi$
$839$	30.6002	1.05644	0.528219	+	0.849108i	$0.322860\pi$
$840$	0	0
$841$	5.47908	0.188934
$842$	−79.7779	−2.74933
$843$	0	0
$844$	−35.8649	−1.23452
$845$	55.5823	1.91209
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	19.8913	0.683071
$849$	0	0
$850$	31.5205	1.08115
$851$	−22.9457	−0.786567
$852$	0	0
$853$	−30.4596	−1.04292	−0.521459	−	0.853276i	$-0.674612\pi$
$853$	−30.4596	−1.04292	−0.521459	+	0.853276i	$0.674612\pi$
$854$	−4.94567	−0.169237
$855$	0	0
$856$	36.9946	1.26445
$857$	41.2702	1.40976	0.704882	−	0.709325i	$-0.251000\pi$
$857$	41.2702	1.40976	0.704882	+	0.709325i	$0.251000\pi$
$858$	0	0
$859$	−28.9193	−0.986712	−0.493356	−	0.869827i	$-0.664230\pi$
$859$	−28.9193	−0.986712	−0.493356	+	0.869827i	$0.664230\pi$
$860$	78.8540	2.68890
$861$	0	0
$862$	−57.5606	−1.96052
$863$	−37.0015	−1.25955	−0.629773	−	0.776779i	$-0.716852\pi$
$863$	−37.0015	−1.25955	−0.629773	+	0.776779i	$0.716852\pi$
$864$	0	0
$865$	−37.5184	−1.27566
$866$	39.3928	1.33862
$867$	0	0
$868$	15.0668	0.511401
$869$	−4.45963	−0.151283
$870$	0	0
$871$	59.3161	2.00985
$872$	−98.5140	−3.33611
$873$	0	0
$874$	−40.4985	−1.36988
$875$	8.54885	0.289004
$876$	0	0
$877$	−6.12562	−0.206847	−0.103424	−	0.994637i	$-0.532980\pi$
$877$	−6.12562	−0.206847	−0.103424	+	0.994637i	$0.532980\pi$
$878$	−56.5202	−1.90746
$879$	0	0
$880$	12.1692	0.410225
$881$	−25.4123	−0.856161	−0.428080	−	0.903741i	$-0.640810\pi$
$881$	−25.4123	−0.856161	−0.428080	+	0.903741i	$0.640810\pi$
$882$	0	0
$883$	−17.0040	−0.572230	−0.286115	−	0.958195i	$-0.592364\pi$
$883$	−17.0040	−0.572230	−0.286115	+	0.958195i	$0.592364\pi$
$884$	181.554	6.10634
$885$	0	0
$886$	53.4039	1.79414
$887$	−35.2269	−1.18280	−0.591401	−	0.806378i	$-0.701425\pi$
$887$	−35.2269	−1.18280	−0.591401	+	0.806378i	$0.701425\pi$
$888$	0	0
$889$	10.2298	0.343097
$890$	101.690	3.40864
$891$	0	0
$892$	−97.7034	−3.27135
$893$	17.7632	0.594424
$894$	0	0
$895$	−37.1366	−1.24134
$896$	−18.4464	−0.616252
$897$	0	0
$898$	46.1506	1.54007
$899$	−21.5000	−0.717067
$900$	0	0
$901$	−31.7827	−1.05883
$902$	13.0668	0.435077
$903$	0	0
$904$	−47.2144	−1.57033
$905$	25.0015	0.831079
$906$	0	0
$907$	−9.72682	−0.322974	−0.161487	−	0.986875i	$-0.551629\pi$
$907$	−9.72682	−0.322974	−0.161487	+	0.986875i	$0.551629\pi$
$908$	54.2159	1.79922
$909$	0	0
$910$	−37.5745	−1.24558
$911$	42.7717	1.41709	0.708545	−	0.705666i	$-0.249352\pi$
$911$	42.7717	1.41709	0.708545	+	0.705666i	$0.249352\pi$
$912$	0	0
$913$	10.6894	0.353769
$914$	−90.1895	−2.98320
$915$	0	0
$916$	55.0015	1.81730
$917$	4.00000	0.132092
$918$	0	0
$919$	6.26871	0.206786	0.103393	−	0.994641i	$-0.467030\pi$
$919$	6.26871	0.206786	0.103393	+	0.994641i	$0.467030\pi$
$920$	93.9986	3.09904
$921$	0	0
$922$	17.7438	0.584360
$923$	26.1865	0.861938
$924$	0	0
$925$	5.60424	0.184266
$926$	1.11938	0.0367850
$927$	0	0
$928$	−6.86565	−0.225376
$929$	−20.0194	−0.656817	−0.328408	−	0.944536i	$-0.606512\pi$
$929$	−20.0194	−0.656817	−0.328408	+	0.944536i	$0.606512\pi$
$930$	0	0
$931$	−2.35793	−0.0772779
$932$	−29.3664	−0.961929
$933$	0	0
$934$	−23.9394	−0.783322
$935$	−19.4442	−0.635893
$936$	0	0
$937$	−11.2144	−0.366358	−0.183179	−	0.983080i	$-0.558639\pi$
$937$	−11.2144	−0.366358	−0.183179	+	0.983080i	$0.558639\pi$
$938$	−24.9798	−0.815621
$939$	0	0
$940$	−80.2181	−2.61643
$941$	−31.3400	−1.02165	−0.510827	−	0.859683i	$-0.670661\pi$
$941$	−31.3400	−1.02165	−0.510827	+	0.859683i	$0.670661\pi$
$942$	0	0
$943$	36.7019	1.19518
$944$	4.35569	0.141766
$945$	0	0
$946$	−18.3121	−0.595377
$947$	23.2841	0.756633	0.378317	−	0.925676i	$-0.376503\pi$
$947$	23.2841	0.756633	0.378317	+	0.925676i	$0.376503\pi$
$948$	0	0
$949$	12.4960	0.405638
$950$	9.89134	0.320917
$951$	0	0
$952$	−39.2966	−1.27361
$953$	−26.8300	−0.869110	−0.434555	−	0.900645i	$-0.643094\pi$
$953$	−26.8300	−0.869110	−0.434555	+	0.900645i	$0.643094\pi$
$954$	0	0
$955$	24.7203	0.799931
$956$	82.2695	2.66079
$957$	0	0
$958$	60.2468	1.94648
$959$	−10.8370	−0.349945
$960$	0	0
$961$	−17.5933	−0.567525
$962$	47.9689	1.54658
$963$	0	0
$964$	−37.8363	−1.21862
$965$	−57.2438	−1.84274
$966$	0	0
$967$	36.2034	1.16422	0.582112	−	0.813109i	$-0.302227\pi$
$967$	36.2034	1.16422	0.582112	+	0.813109i	$0.302227\pi$
$968$	−5.22982	−0.168093
$969$	0	0
$970$	−64.7658	−2.07950
$971$	−6.06281	−0.194565	−0.0972824	−	0.995257i	$-0.531015\pi$
$971$	−6.06281	−0.194565	−0.0972824	+	0.995257i	$0.531015\pi$
$972$	0	0
$973$	−0.459630	−0.0147351
$974$	47.6521	1.52687
$975$	0	0
$976$	−9.40530	−0.301056
$977$	4.28263	0.137013	0.0685067	−	0.997651i	$-0.478177\pi$
$977$	4.28263	0.137013	0.0685067	+	0.997651i	$0.478177\pi$
$978$	0	0
$979$	−15.8913	−0.507889
$980$	10.6483	0.340148
$981$	0	0
$982$	73.2999	2.33909
$983$	17.1894	0.548257	0.274128	−	0.961693i	$-0.411611\pi$
$983$	17.1894	0.548257	0.274128	+	0.961693i	$0.411611\pi$
$984$	0	0
$985$	67.9083	2.16374
$986$	109.104	3.47459
$987$	0	0
$988$	56.9729	1.81255
$989$	−51.4347	−1.63553
$990$	0	0
$991$	35.4945	1.12752	0.563760	−	0.825939i	$-0.309354\pi$
$991$	35.4945	1.12752	0.563760	+	0.825939i	$0.309354\pi$
$992$	4.28120	0.135928
$993$	0	0
$994$	−11.0279	−0.349785
$995$	2.72834	0.0864942
$996$	0	0
$997$	30.4068	0.962993	0.481497	−	0.876448i	$-0.340093\pi$
$997$	30.4068	0.962993	0.481497	+	0.876448i	$0.340093\pi$
$998$	−72.6319	−2.29912
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.2.a.l.1.1		3
3.2	odd	2		231.2.a.e.1.3	✓	3
7.6	odd	2		4851.2.a.bi.1.1		3
11.10	odd	2		7623.2.a.cd.1.3		3
12.11	even	2		3696.2.a.bo.1.1		3
15.14	odd	2		5775.2.a.bp.1.1		3
21.20	even	2		1617.2.a.t.1.3		3
33.32	even	2		2541.2.a.bg.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.a.e.1.3	✓	3	3.2	odd	2
693.2.a.l.1.1		3	1.1	even	1	trivial
1617.2.a.t.1.3		3	21.20	even	2
2541.2.a.bg.1.1		3	33.32	even	2
3696.2.a.bo.1.1		3	12.11	even	2
4851.2.a.bi.1.1		3	7.6	odd	2
5775.2.a.bp.1.1		3	15.14	odd	2
7623.2.a.cd.1.3		3	11.10	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:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

\(f(q)\)	\(=\)	\(q-2.47283 q^{2} +4.11491 q^{4} +2.58774 q^{5} -1.00000 q^{7} -5.22982 q^{8} +O(q^{10})\)	\(q-2.47283 q^{2} +4.11491 q^{4} +2.58774 q^{5} -1.00000 q^{7} -5.22982 q^{8} -6.39905 q^{10} +1.00000 q^{11} -5.87189 q^{13} +2.47283 q^{14} +4.70265 q^{16} -7.51396 q^{17} -2.35793 q^{19} +10.6483 q^{20} -2.47283 q^{22} -6.94567 q^{23} +1.69641 q^{25} +14.5202 q^{26} -4.11491 q^{28} +5.87189 q^{29} -3.66152 q^{31} -1.16924 q^{32} +18.5808 q^{34} -2.58774 q^{35} +3.30359 q^{37} +5.83076 q^{38} -13.5334 q^{40} -5.28415 q^{41} +7.40530 q^{43} +4.11491 q^{44} +17.1755 q^{46} -7.53341 q^{47} +1.00000 q^{49} -4.19493 q^{50} -24.1623 q^{52} +4.22982 q^{53} +2.58774 q^{55} +5.22982 q^{56} -14.5202 q^{58} +0.926221 q^{59} -2.00000 q^{61} +9.05433 q^{62} -6.51396 q^{64} -15.1949 q^{65} -10.1017 q^{67} -30.9193 q^{68} +6.39905 q^{70} -4.45963 q^{71} -2.12811 q^{73} -8.16924 q^{74} -9.70265 q^{76} -1.00000 q^{77} -4.45963 q^{79} +12.1692 q^{80} +13.0668 q^{82} +10.6894 q^{83} -19.4442 q^{85} -18.3121 q^{86} -5.22982 q^{88} -15.8913 q^{89} +5.87189 q^{91} -28.5808 q^{92} +18.6289 q^{94} -6.10170 q^{95} +10.1212 q^{97} -2.47283 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} + 6 q^{4} - 4 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - 2 * q^2 + 6 * q^4 - 4 * q^5 - 3 * q^7 - 3 * q^8	\( 3 q - 2 q^{2} + 6 q^{4} - 4 q^{5} - 3 q^{7} - 3 q^{8} - 11 q^{10} + 3 q^{11} - 4 q^{13} + 2 q^{14} - 4 q^{16} - 8 q^{17} - 8 q^{19} + 3 q^{20} - 2 q^{22} - 10 q^{23} + 15 q^{25} + q^{26} - 6 q^{28} + 4 q^{29} - 2 q^{31} - 8 q^{32} - 4 q^{34} + 4 q^{35} + 13 q^{38} - 18 q^{40} - 14 q^{41} - 14 q^{43} + 6 q^{44} + 28 q^{46} + 3 q^{49} + 19 q^{50} - 29 q^{52} - 4 q^{55} + 3 q^{56} - q^{58} - 6 q^{61} + 38 q^{62} - 5 q^{64} - 14 q^{65} - 4 q^{67} - 42 q^{68} + 11 q^{70} + 12 q^{71} - 20 q^{73} - 29 q^{74} - 11 q^{76} - 3 q^{77} + 12 q^{79} + 41 q^{80} - 6 q^{82} - 6 q^{83} - 6 q^{85} - 24 q^{86} - 3 q^{88} - 26 q^{89} + 4 q^{91} - 26 q^{92} + 35 q^{94} + 8 q^{95} - 4 q^{97} - 2 q^{98}+O(q^{100}) \) 3 * q - 2 * q^2 + 6 * q^4 - 4 * q^5 - 3 * q^7 - 3 * q^8 - 11 * q^10 + 3 * q^11 - 4 * q^13 + 2 * q^14 - 4 * q^16 - 8 * q^17 - 8 * q^19 + 3 * q^20 - 2 * q^22 - 10 * q^23 + 15 * q^25 + q^26 - 6 * q^28 + 4 * q^29 - 2 * q^31 - 8 * q^32 - 4 * q^34 + 4 * q^35 + 13 * q^38 - 18 * q^40 - 14 * q^41 - 14 * q^43 + 6 * q^44 + 28 * q^46 + 3 * q^49 + 19 * q^50 - 29 * q^52 - 4 * q^55 + 3 * q^56 - q^58 - 6 * q^61 + 38 * q^62 - 5 * q^64 - 14 * q^65 - 4 * q^67 - 42 * q^68 + 11 * q^70 + 12 * q^71 - 20 * q^73 - 29 * q^74 - 11 * q^76 - 3 * q^77 + 12 * q^79 + 41 * q^80 - 6 * q^82 - 6 * q^83 - 6 * q^85 - 24 * q^86 - 3 * q^88 - 26 * q^89 + 4 * q^91 - 26 * q^92 + 35 * q^94 + 8 * q^95 - 4 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more