LMFDB - Embedded newform 693.2.a.m.1.2

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:	$0$
Dimension:	$3$
Coefficient field:	3.3.837.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 1 $ x^3 - 6*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			1.2
Root			$-0.167449$ 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)

$f(q)$	$=$	$q+0.167449 q^{2} -1.97196 q^{4} -3.80451 q^{5} -1.00000 q^{7} -0.665102 q^{8} +O(q^{10})$	\(q+0.167449 q^{2} -1.97196 q^{4} -3.80451 q^{5} -1.00000 q^{7} -0.665102 q^{8} -0.637062 q^{10} -1.00000 q^{11} +3.80451 q^{13} -0.167449 q^{14} +3.83255 q^{16} -0.334898 q^{17} +8.13941 q^{19} +7.50235 q^{20} -0.167449 q^{22} +1.66510 q^{23} +9.47431 q^{25} +0.637062 q^{26} +1.97196 q^{28} -0.195488 q^{29} -9.94392 q^{31} +1.97196 q^{32} -0.0560785 q^{34} +3.80451 q^{35} -4.47431 q^{37} +1.36294 q^{38} +2.53039 q^{40} +6.27882 q^{41} +2.33490 q^{43} +1.97196 q^{44} +0.278820 q^{46} +12.1394 q^{47} +1.00000 q^{49} +1.58647 q^{50} -7.50235 q^{52} -7.94392 q^{53} +3.80451 q^{55} +0.665102 q^{56} -0.0327344 q^{58} -3.74843 q^{59} +6.00000 q^{61} -1.66510 q^{62} -7.33490 q^{64} -14.4743 q^{65} -0.139410 q^{67} +0.660406 q^{68} +0.637062 q^{70} -4.66980 q^{71} +4.19549 q^{73} -0.749219 q^{74} -16.0506 q^{76} +1.00000 q^{77} +3.33020 q^{79} -14.5810 q^{80} +1.05138 q^{82} +13.9439 q^{83} +1.27412 q^{85} +0.390977 q^{86} +0.665102 q^{88} +9.88784 q^{89} -3.80451 q^{91} -3.28352 q^{92} +2.03273 q^{94} -30.9665 q^{95} +0.0560785 q^{97} +0.167449 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{4} - 3 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q + 6 * q^4 - 3 * q^7 - 3 * q^8	$ 3 q + 6 q^{4} - 3 q^{7} - 3 q^{8} + 9 q^{10} - 3 q^{11} + 12 q^{16} + 12 q^{19} + 21 q^{20} + 6 q^{23} + 15 q^{25} - 9 q^{26} - 6 q^{28} - 12 q^{29} - 6 q^{31} - 6 q^{32} - 24 q^{34} + 15 q^{38} + 18 q^{40} - 6 q^{41} + 6 q^{43} - 6 q^{44} - 24 q^{46} + 24 q^{47} + 3 q^{49} + 39 q^{50} - 21 q^{52} + 3 q^{56} - 9 q^{58} + 24 q^{59} + 18 q^{61} - 6 q^{62} - 21 q^{64} - 30 q^{65} + 12 q^{67} + 6 q^{68} - 9 q^{70} - 12 q^{71} + 24 q^{73} - 39 q^{74} - 3 q^{76} + 3 q^{77} + 12 q^{79} - 9 q^{80} + 30 q^{82} + 18 q^{83} - 18 q^{85} + 24 q^{86} + 3 q^{88} - 18 q^{89} + 18 q^{92} + 15 q^{94} - 12 q^{95} + 24 q^{97}+O(q^{100}) $ 3 * q + 6 * q^4 - 3 * q^7 - 3 * q^8 + 9 * q^10 - 3 * q^11 + 12 * q^16 + 12 * q^19 + 21 * q^20 + 6 * q^23 + 15 * q^25 - 9 * q^26 - 6 * q^28 - 12 * q^29 - 6 * q^31 - 6 * q^32 - 24 * q^34 + 15 * q^38 + 18 * q^40 - 6 * q^41 + 6 * q^43 - 6 * q^44 - 24 * q^46 + 24 * q^47 + 3 * q^49 + 39 * q^50 - 21 * q^52 + 3 * q^56 - 9 * q^58 + 24 * q^59 + 18 * q^61 - 6 * q^62 - 21 * q^64 - 30 * q^65 + 12 * q^67 + 6 * q^68 - 9 * q^70 - 12 * q^71 + 24 * q^73 - 39 * q^74 - 3 * q^76 + 3 * q^77 + 12 * q^79 - 9 * q^80 + 30 * q^82 + 18 * q^83 - 18 * q^85 + 24 * q^86 + 3 * q^88 - 18 * q^89 + 18 * q^92 + 15 * q^94 - 12 * q^95 + 24 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.167449	0.118404	0.0592022	−	0.998246i	$-0.481144\pi$
$2$	0.167449	0.118404	0.0592022	+	0.998246i	$0.481144\pi$
$3$	0	0
$3$	0	0
$4$	−1.97196	−0.985980
$5$	−3.80451	−1.70143	−0.850715	−	0.525628i	$-0.823830\pi$
$5$	−3.80451	−1.70143	−0.850715	+	0.525628i	$0.823830\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−0.665102	−0.235149
$9$	0	0
$10$	−0.637062	−0.201457
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	3.80451	1.05518	0.527591	−	0.849499i	$-0.323095\pi$
$13$	3.80451	1.05518	0.527591	+	0.849499i	$0.323095\pi$
$14$	−0.167449	−0.0447527
$15$	0	0
$16$	3.83255	0.958138
$17$	−0.334898	−0.0812248	−0.0406124	−	0.999175i	$-0.512931\pi$
$17$	−0.334898	−0.0812248	−0.0406124	+	0.999175i	$0.512931\pi$
$18$	0	0
$19$	8.13941	1.86731	0.933654	−	0.358175i	$-0.116601\pi$
$19$	8.13941	1.86731	0.933654	+	0.358175i	$0.116601\pi$
$20$	7.50235	1.67758
$21$	0	0
$22$	−0.167449	−0.0357003
$23$	1.66510	0.347198	0.173599	−	0.984816i	$-0.444460\pi$
$23$	1.66510	0.347198	0.173599	+	0.984816i	$0.444460\pi$
$24$	0	0
$25$	9.47431	1.89486
$26$	0.637062	0.124938
$27$	0	0
$28$	1.97196	0.372666
$29$	−0.195488	−0.0363013	−0.0181506	−	0.999835i	$-0.505778\pi$
$29$	−0.195488	−0.0363013	−0.0181506	+	0.999835i	$0.505778\pi$
$30$	0	0
$31$	−9.94392	−1.78598	−0.892991	−	0.450075i	$-0.851397\pi$
$31$	−9.94392	−1.78598	−0.892991	+	0.450075i	$0.851397\pi$
$32$	1.97196	0.348597
$33$	0	0
$34$	−0.0560785	−0.00961738
$35$	3.80451	0.643080
$36$	0	0
$37$	−4.47431	−0.735572	−0.367786	−	0.929911i	$-0.619884\pi$
$37$	−4.47431	−0.735572	−0.367786	+	0.929911i	$0.619884\pi$
$38$	1.36294	0.221098
$39$	0	0
$40$	2.53039	0.400089
$41$	6.27882	0.980587	0.490293	−	0.871557i	$-0.336890\pi$
$41$	6.27882	0.980587	0.490293	+	0.871557i	$0.336890\pi$
$42$	0	0
$43$	2.33490	0.356069	0.178034	−	0.984024i	$-0.443026\pi$
$43$	2.33490	0.356069	0.178034	+	0.984024i	$0.443026\pi$
$44$	1.97196	0.297284
$45$	0	0
$46$	0.278820	0.0411098
$47$	12.1394	1.77071	0.885357	−	0.464911i	$-0.153914\pi$
$47$	12.1394	1.77071	0.885357	+	0.464911i	$0.153914\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.58647	0.224360
$51$	0	0
$52$	−7.50235	−1.04039
$53$	−7.94392	−1.09118	−0.545591	−	0.838052i	$-0.683695\pi$
$53$	−7.94392	−1.09118	−0.545591	+	0.838052i	$0.683695\pi$
$54$	0	0
$55$	3.80451	0.513000
$56$	0.665102	0.0888779
$57$	0	0
$58$	−0.0327344	−0.00429823
$59$	−3.74843	−0.488004	−0.244002	−	0.969775i	$-0.578460\pi$
$59$	−3.74843	−0.488004	−0.244002	+	0.969775i	$0.578460\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	−1.66510	−0.211468
$63$	0	0
$64$	−7.33490	−0.916862
$65$	−14.4743	−1.79532
$66$	0	0
$67$	−0.139410	−0.0170316	−0.00851582	−	0.999964i	$-0.502711\pi$
$67$	−0.139410	−0.0170316	−0.00851582	+	0.999964i	$0.502711\pi$
$68$	0.660406	0.0800860
$69$	0	0
$70$	0.637062	0.0761435
$71$	−4.66980	−0.554203	−0.277101	−	0.960841i	$-0.589374\pi$
$71$	−4.66980	−0.554203	−0.277101	+	0.960841i	$0.589374\pi$
$72$	0	0
$73$	4.19549	0.491045	0.245522	−	0.969391i	$-0.421041\pi$
$73$	4.19549	0.491045	0.245522	+	0.969391i	$0.421041\pi$
$74$	−0.749219	−0.0870950
$75$	0	0
$76$	−16.0506	−1.84113
$77$	1.00000	0.113961
$78$	0	0
$79$	3.33020	0.374677	0.187339	−	0.982295i	$-0.440014\pi$
$79$	3.33020	0.374677	0.187339	+	0.982295i	$0.440014\pi$
$80$	−14.5810	−1.63020
$81$	0	0
$82$	1.05138	0.116106
$83$	13.9439	1.53054	0.765272	−	0.643707i	$-0.222604\pi$
$83$	13.9439	1.53054	0.765272	+	0.643707i	$0.222604\pi$
$84$	0	0
$85$	1.27412	0.138198
$86$	0.390977	0.0421601
$87$	0	0
$88$	0.665102	0.0709001
$89$	9.88784	1.04811	0.524055	−	0.851685i	$-0.324419\pi$
$89$	9.88784	1.04811	0.524055	+	0.851685i	$0.324419\pi$
$90$	0	0
$91$	−3.80451	−0.398821
$92$	−3.28352	−0.342330
$93$	0	0
$94$	2.03273	0.209661
$95$	−30.9665	−3.17709
$96$	0	0
$97$	0.0560785	0.00569391	0.00284695	−	0.999996i	$-0.499094\pi$
$97$	0.0560785	0.00569391	0.00284695	+	0.999996i	$0.499094\pi$
$98$	0.167449	0.0169149
$99$	0	0
$100$	−18.6830	−1.86830
$101$	18.8831	1.87894	0.939472	−	0.342626i	$-0.111317\pi$
$101$	18.8831	1.87894	0.939472	+	0.342626i	$0.111317\pi$
$102$	0	0
$103$	−8.27882	−0.815736	−0.407868	−	0.913041i	$-0.633728\pi$
$103$	−8.27882	−0.815736	−0.407868	+	0.913041i	$0.633728\pi$
$104$	−2.53039	−0.248125
$105$	0	0
$106$	−1.33020	−0.129201
$107$	8.13941	0.786866	0.393433	−	0.919353i	$-0.371287\pi$
$107$	8.13941	0.786866	0.393433	+	0.919353i	$0.371287\pi$
$108$	0	0
$109$	11.5529	1.10657	0.553286	−	0.832992i	$-0.313374\pi$
$109$	11.5529	1.10657	0.553286	+	0.832992i	$0.313374\pi$
$110$	0.637062	0.0607415
$111$	0	0
$112$	−3.83255	−0.362142
$113$	1.33020	0.125135	0.0625675	−	0.998041i	$-0.480071\pi$
$113$	1.33020	0.125135	0.0625675	+	0.998041i	$0.480071\pi$
$114$	0	0
$115$	−6.33490	−0.590732
$116$	0.385496	0.0357924
$117$	0	0
$118$	−0.627672	−0.0577819
$119$	0.334898	0.0307001
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.00470	0.0909608
$123$	0	0
$124$	19.6090	1.76094
$125$	−17.0226	−1.52254
$126$	0	0
$127$	14.6137	1.29676	0.648379	−	0.761318i	$-0.275447\pi$
$127$	14.6137	1.29676	0.648379	+	0.761318i	$0.275447\pi$
$128$	−5.17214	−0.457157
$129$	0	0
$130$	−2.42371	−0.212574
$131$	−20.5576	−1.79613	−0.898065	−	0.439863i	$-0.855027\pi$
$131$	−20.5576	−1.79613	−0.898065	+	0.439863i	$0.855027\pi$
$132$	0	0
$133$	−8.13941	−0.705776
$134$	−0.0233441	−0.00201662
$135$	0	0
$136$	0.222741	0.0190999
$137$	16.2227	1.38600	0.693001	−	0.720936i	$-0.256288\pi$
$137$	16.2227	1.38600	0.693001	+	0.720936i	$0.256288\pi$
$138$	0	0
$139$	22.5482	1.91252	0.956259	−	0.292522i	$-0.0944945\pi$
$139$	22.5482	1.91252	0.956259	+	0.292522i	$0.0944945\pi$
$140$	−7.50235	−0.634064
$141$	0	0
$142$	−0.781954	−0.0656201
$143$	−3.80451	−0.318149
$144$	0	0
$145$	0.743738	0.0617641
$146$	0.702531	0.0581419
$147$	0	0
$148$	8.82316	0.725259
$149$	−8.19549	−0.671401	−0.335700	−	0.941969i	$-0.608973\pi$
$149$	−8.19549	−0.671401	−0.335700	+	0.941969i	$0.608973\pi$
$150$	0	0
$151$	−13.2741	−1.08023	−0.540116	−	0.841590i	$-0.681620\pi$
$151$	−13.2741	−1.08023	−0.540116	+	0.841590i	$0.681620\pi$
$152$	−5.41353	−0.439096
$153$	0	0
$154$	0.167449	0.0134934
$155$	37.8318	3.03872
$156$	0	0
$157$	−16.9392	−1.35190	−0.675949	−	0.736949i	$-0.736266\pi$
$157$	−16.9392	−1.35190	−0.675949	+	0.736949i	$0.736266\pi$
$158$	0.557640	0.0443634
$159$	0	0
$160$	−7.50235	−0.593113
$161$	−1.66510	−0.131228
$162$	0	0
$163$	−6.79982	−0.532603	−0.266301	−	0.963890i	$-0.585802\pi$
$163$	−6.79982	−0.532603	−0.266301	+	0.963890i	$0.585802\pi$
$164$	−12.3816	−0.966839
$165$	0	0
$166$	2.33490	0.181223
$167$	18.2227	1.41012	0.705059	−	0.709149i	$-0.250920\pi$
$167$	18.2227	1.41012	0.705059	+	0.709149i	$0.250920\pi$
$168$	0	0
$169$	1.47431	0.113408
$170$	0.213351	0.0163633
$171$	0	0
$172$	−4.60433	−0.351077
$173$	1.72118	0.130859	0.0654294	−	0.997857i	$-0.479158\pi$
$173$	1.72118	0.130859	0.0654294	+	0.997857i	$0.479158\pi$
$174$	0	0
$175$	−9.47431	−0.716190
$176$	−3.83255	−0.288889
$177$	0	0
$178$	1.65571	0.124101
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	0.725875	0.0539539	0.0269769	−	0.999636i	$-0.491412\pi$
$181$	0.725875	0.0539539	0.0269769	+	0.999636i	$0.491412\pi$
$182$	−0.637062	−0.0472222
$183$	0	0
$184$	−1.10746	−0.0816432
$185$	17.0226	1.25152
$186$	0	0
$187$	0.334898	0.0244902
$188$	−23.9384	−1.74589
$189$	0	0
$190$	−5.18531	−0.376182
$191$	−5.27412	−0.381622	−0.190811	−	0.981627i	$-0.561112\pi$
$191$	−5.27412	−0.381622	−0.190811	+	0.981627i	$0.561112\pi$
$192$	0	0
$193$	−19.8318	−1.42752	−0.713761	−	0.700390i	$-0.753010\pi$
$193$	−19.8318	−1.42752	−0.713761	+	0.700390i	$0.753010\pi$
$194$	0.00939029	0.000674184 0
$195$	0	0
$196$	−1.97196	−0.140854
$197$	−2.66980	−0.190215	−0.0951076	−	0.995467i	$-0.530320\pi$
$197$	−2.66980	−0.190215	−0.0951076	+	0.995467i	$0.530320\pi$
$198$	0	0
$199$	13.5529	0.960743	0.480371	−	0.877065i	$-0.340502\pi$
$199$	13.5529	0.960743	0.480371	+	0.877065i	$0.340502\pi$
$200$	−6.30138	−0.445575
$201$	0	0
$202$	3.16197	0.222475
$203$	0.195488	0.0137206
$204$	0	0
$205$	−23.8878	−1.66840
$206$	−1.38628	−0.0965868
$207$	0	0
$208$	14.5810	1.01101
$209$	−8.13941	−0.563015
$210$	0	0
$211$	−4.27882	−0.294566	−0.147283	−	0.989094i	$-0.547053\pi$
$211$	−4.27882	−0.294566	−0.147283	+	0.989094i	$0.547053\pi$
$212$	15.6651	1.07588
$213$	0	0
$214$	1.36294	0.0931685
$215$	−8.88315	−0.605826
$216$	0	0
$217$	9.94392	0.675037
$218$	1.93453	0.131023
$219$	0	0
$220$	−7.50235	−0.505808
$221$	−1.27412	−0.0857069
$222$	0	0
$223$	−10.2694	−0.687692	−0.343846	−	0.939026i	$-0.611730\pi$
$223$	−10.2694	−0.687692	−0.343846	+	0.939026i	$0.611730\pi$
$224$	−1.97196	−0.131757
$225$	0	0
$226$	0.222741	0.0148165
$227$	0.390977	0.0259500	0.0129750	−	0.999916i	$-0.495870\pi$
$227$	0.390977	0.0259500	0.0129750	+	0.999916i	$0.495870\pi$
$228$	0	0
$229$	7.94392	0.524949	0.262475	−	0.964939i	$-0.415461\pi$
$229$	7.94392	0.524949	0.262475	+	0.964939i	$0.415461\pi$
$230$	−1.06077	−0.0699453
$231$	0	0
$232$	0.130020	0.00853621
$233$	−26.5576	−1.73985	−0.869924	−	0.493185i	$-0.835833\pi$
$233$	−26.5576	−1.73985	−0.869924	+	0.493185i	$0.835833\pi$
$234$	0	0
$235$	−46.1845	−3.01275
$236$	7.39176	0.481163
$237$	0	0
$238$	0.0560785	0.00363503
$239$	−10.7998	−0.698582	−0.349291	−	0.937014i	$-0.613578\pi$
$239$	−10.7998	−0.698582	−0.349291	+	0.937014i	$0.613578\pi$
$240$	0	0
$241$	−12.0833	−0.778356	−0.389178	−	0.921163i	$-0.627241\pi$
$241$	−12.0833	−0.778356	−0.389178	+	0.921163i	$0.627241\pi$
$242$	0.167449	0.0107640
$243$	0	0
$244$	−11.8318	−0.757451
$245$	−3.80451	−0.243061
$246$	0	0
$247$	30.9665	1.97035
$248$	6.61372	0.419972
$249$	0	0
$250$	−2.85041	−0.180276
$251$	−4.80921	−0.303554	−0.151777	−	0.988415i	$-0.548500\pi$
$251$	−4.80921	−0.303554	−0.151777	+	0.988415i	$0.548500\pi$
$252$	0	0
$253$	−1.66510	−0.104684
$254$	2.44706	0.153542
$255$	0	0
$256$	13.8037	0.862733
$257$	16.7531	1.04503	0.522516	−	0.852630i	$-0.324994\pi$
$257$	16.7531	1.04503	0.522516	+	0.852630i	$0.324994\pi$
$258$	0	0
$259$	4.47431	0.278020
$260$	28.5428	1.77015
$261$	0	0
$262$	−3.44236	−0.212670
$263$	12.1394	0.748548	0.374274	−	0.927318i	$-0.377892\pi$
$263$	12.1394	0.748548	0.374274	+	0.927318i	$0.377892\pi$
$264$	0	0
$265$	30.2227	1.85657
$266$	−1.36294	−0.0835671
$267$	0	0
$268$	0.274911	0.0167929
$269$	25.2180	1.53757	0.768786	−	0.639506i	$-0.220861\pi$
$269$	25.2180	1.53757	0.768786	+	0.639506i	$0.220861\pi$
$270$	0	0
$271$	4.13941	0.251451	0.125726	−	0.992065i	$-0.459874\pi$
$271$	4.13941	0.251451	0.125726	+	0.992065i	$0.459874\pi$
$272$	−1.28352	−0.0778245
$273$	0	0
$274$	2.71648	0.164109
$275$	−9.47431	−0.571322
$276$	0	0
$277$	−18.2788	−1.09827	−0.549134	−	0.835734i	$-0.685042\pi$
$277$	−18.2788	−1.09827	−0.549134	+	0.835734i	$0.685042\pi$
$278$	3.77569	0.226451
$279$	0	0
$280$	−2.53039	−0.151220
$281$	−6.74374	−0.402298	−0.201149	−	0.979561i	$-0.564467\pi$
$281$	−6.74374	−0.402298	−0.201149	+	0.979561i	$0.564467\pi$
$282$	0	0
$283$	23.3575	1.38846	0.694228	−	0.719755i	$-0.255746\pi$
$283$	23.3575	1.38846	0.694228	+	0.719755i	$0.255746\pi$
$284$	9.20866	0.546433
$285$	0	0
$286$	−0.637062	−0.0376703
$287$	−6.27882	−0.370627
$288$	0	0
$289$	−16.8878	−0.993403
$290$	0.124538	0.00731314
$291$	0	0
$292$	−8.27334	−0.484161
$293$	14.1667	0.827625	0.413813	−	0.910362i	$-0.364197\pi$
$293$	14.1667	0.827625	0.413813	+	0.910362i	$0.364197\pi$
$294$	0	0
$295$	14.2610	0.830305
$296$	2.97587	0.172969
$297$	0	0
$298$	−1.37233	−0.0794968
$299$	6.33490	0.366357
$300$	0	0
$301$	−2.33490	−0.134581
$302$	−2.22274	−0.127904
$303$	0	0
$304$	31.1947	1.78914
$305$	−22.8271	−1.30707
$306$	0	0
$307$	−12.5576	−0.716702	−0.358351	−	0.933587i	$-0.616661\pi$
$307$	−12.5576	−0.716702	−0.358351	+	0.933587i	$0.616661\pi$
$308$	−1.97196	−0.112363
$309$	0	0
$310$	6.33490	0.359798
$311$	−9.33959	−0.529600	−0.264800	−	0.964303i	$-0.585306\pi$
$311$	−9.33959	−0.529600	−0.264800	+	0.964303i	$0.585306\pi$
$312$	0	0
$313$	−2.99530	−0.169305	−0.0846523	−	0.996411i	$-0.526978\pi$
$313$	−2.99530	−0.169305	−0.0846523	+	0.996411i	$0.526978\pi$
$314$	−2.83646	−0.160071
$315$	0	0
$316$	−6.56703	−0.369424
$317$	−9.93453	−0.557979	−0.278989	−	0.960294i	$-0.589999\pi$
$317$	−9.93453	−0.557979	−0.278989	+	0.960294i	$0.589999\pi$
$318$	0	0
$319$	0.195488	0.0109453
$320$	27.9057	1.55998
$321$	0	0
$322$	−0.278820	−0.0155380
$323$	−2.72588	−0.151672
$324$	0	0
$325$	36.0451	1.99942
$326$	−1.13862	−0.0630625
$327$	0	0
$328$	−4.17605	−0.230584
$329$	−12.1394	−0.669267
$330$	0	0
$331$	22.5482	1.23936	0.619682	−	0.784853i	$-0.287262\pi$
$331$	22.5482	1.23936	0.619682	+	0.784853i	$0.287262\pi$
$332$	−27.4969	−1.50909
$333$	0	0
$334$	3.05138	0.166964
$335$	0.530387	0.0289781
$336$	0	0
$337$	13.2835	0.723599	0.361800	−	0.932256i	$-0.382162\pi$
$337$	13.2835	0.723599	0.361800	+	0.932256i	$0.382162\pi$
$338$	0.246872	0.0134281
$339$	0	0
$340$	−2.51252	−0.136261
$341$	9.94392	0.538494
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−1.55294	−0.0837292
$345$	0	0
$346$	0.288210	0.0154943
$347$	27.7757	1.49108	0.745538	−	0.666463i	$-0.232192\pi$
$347$	27.7757	1.49108	0.745538	+	0.666463i	$0.232192\pi$
$348$	0	0
$349$	−2.85589	−0.152873	−0.0764363	−	0.997074i	$-0.524354\pi$
$349$	−2.85589	−0.152873	−0.0764363	+	0.997074i	$0.524354\pi$
$350$	−1.58647	−0.0848001
$351$	0	0
$352$	−1.97196	−0.105106
$353$	−24.6410	−1.31151	−0.655753	−	0.754975i	$-0.727649\pi$
$353$	−24.6410	−1.31151	−0.655753	+	0.754975i	$0.727649\pi$
$354$	0	0
$355$	17.7663	0.942937
$356$	−19.4984	−1.03342
$357$	0	0
$358$	2.00939	0.106200
$359$	11.8878	0.627416	0.313708	−	0.949519i	$-0.398429\pi$
$359$	11.8878	0.627416	0.313708	+	0.949519i	$0.398429\pi$
$360$	0	0
$361$	47.2500	2.48684
$362$	0.121547	0.00638838
$363$	0	0
$364$	7.50235	0.393230
$365$	−15.9618	−0.835478
$366$	0	0
$367$	3.60902	0.188389	0.0941947	−	0.995554i	$-0.469972\pi$
$367$	3.60902	0.188389	0.0941947	+	0.995554i	$0.469972\pi$
$368$	6.38159	0.332663
$369$	0	0
$370$	2.85041	0.148186
$371$	7.94392	0.412428
$372$	0	0
$373$	12.3349	0.638677	0.319338	−	0.947641i	$-0.396539\pi$
$373$	12.3349	0.638677	0.319338	+	0.947641i	$0.396539\pi$
$374$	0.0560785	0.00289975
$375$	0	0
$376$	−8.07394	−0.416382
$377$	−0.743738	−0.0383045
$378$	0	0
$379$	23.3575	1.19979	0.599896	−	0.800078i	$-0.295209\pi$
$379$	23.3575	1.19979	0.599896	+	0.800078i	$0.295209\pi$
$380$	61.0647	3.13255
$381$	0	0
$382$	−0.883148	−0.0451858
$383$	15.2180	0.777606	0.388803	−	0.921321i	$-0.372889\pi$
$383$	15.2180	0.777606	0.388803	+	0.921321i	$0.372889\pi$
$384$	0	0
$385$	−3.80451	−0.193896
$386$	−3.32081	−0.169025
$387$	0	0
$388$	−0.110585	−0.00561408
$389$	−3.73057	−0.189147	−0.0945737	−	0.995518i	$-0.530149\pi$
$389$	−3.73057	−0.189147	−0.0945737	+	0.995518i	$0.530149\pi$
$390$	0	0
$391$	−0.557640	−0.0282011
$392$	−0.665102	−0.0335927
$393$	0	0
$394$	−0.447055	−0.0225223
$395$	−12.6698	−0.637487
$396$	0	0
$397$	20.8925	1.04857	0.524283	−	0.851544i	$-0.324333\pi$
$397$	20.8925	1.04857	0.524283	+	0.851544i	$0.324333\pi$
$398$	2.26943	0.113756
$399$	0	0
$400$	36.3108	1.81554
$401$	−23.2741	−1.16225	−0.581127	−	0.813813i	$-0.697388\pi$
$401$	−23.2741	−1.16225	−0.581127	+	0.813813i	$0.697388\pi$
$402$	0	0
$403$	−37.8318	−1.88453
$404$	−37.2368	−1.85260
$405$	0	0
$406$	0.0327344	0.00162458
$407$	4.47431	0.221783
$408$	0	0
$409$	−23.8972	−1.18164	−0.590821	−	0.806803i	$-0.701196\pi$
$409$	−23.8972	−1.18164	−0.590821	+	0.806803i	$0.701196\pi$
$410$	−4.00000	−0.197546
$411$	0	0
$412$	16.3255	0.804300
$413$	3.74843	0.184448
$414$	0	0
$415$	−53.0498	−2.60411
$416$	7.50235	0.367833
$417$	0	0
$418$	−1.36294	−0.0666635
$419$	30.2967	1.48009	0.740045	−	0.672557i	$-0.234804\pi$
$419$	30.2967	1.48009	0.740045	+	0.672557i	$0.234804\pi$
$420$	0	0
$421$	−15.5257	−0.756676	−0.378338	−	0.925668i	$-0.623504\pi$
$421$	−15.5257	−0.756676	−0.378338	+	0.925668i	$0.623504\pi$
$422$	−0.716485	−0.0348779
$423$	0	0
$424$	5.28352	0.256590
$425$	−3.17293	−0.153910
$426$	0	0
$427$	−6.00000	−0.290360
$428$	−16.0506	−0.775835
$429$	0	0
$430$	−1.48748	−0.0717325
$431$	−3.07864	−0.148293	−0.0741463	−	0.997247i	$-0.523623\pi$
$431$	−3.07864	−0.148293	−0.0741463	+	0.997247i	$0.523623\pi$
$432$	0	0
$433$	16.9392	0.814047	0.407024	−	0.913418i	$-0.366567\pi$
$433$	16.9392	0.814047	0.407024	+	0.913418i	$0.366567\pi$
$434$	1.66510	0.0799274
$435$	0	0
$436$	−22.7820	−1.09106
$437$	13.5529	0.648325
$438$	0	0
$439$	−11.0786	−0.528754	−0.264377	−	0.964419i	$-0.585166\pi$
$439$	−11.0786	−0.528754	−0.264377	+	0.964419i	$0.585166\pi$
$440$	−2.53039	−0.120631
$441$	0	0
$442$	−0.213351	−0.0101481
$443$	14.5482	0.691208	0.345604	−	0.938380i	$-0.387674\pi$
$443$	14.5482	0.691208	0.345604	+	0.938380i	$0.387674\pi$
$444$	0	0
$445$	−37.6184	−1.78328
$446$	−1.71961	−0.0814258
$447$	0	0
$448$	7.33490	0.346541
$449$	−27.4875	−1.29721	−0.648607	−	0.761123i	$-0.724648\pi$
$449$	−27.4875	−1.29721	−0.648607	+	0.761123i	$0.724648\pi$
$450$	0	0
$451$	−6.27882	−0.295658
$452$	−2.62311	−0.123381
$453$	0	0
$454$	0.0654688	0.00307260
$455$	14.4743	0.678566
$456$	0	0
$457$	−7.94392	−0.371601	−0.185800	−	0.982587i	$-0.559488\pi$
$457$	−7.94392	−0.371601	−0.185800	+	0.982587i	$0.559488\pi$
$458$	1.33020	0.0621563
$459$	0	0
$460$	12.4922	0.582450
$461$	−8.26943	−0.385146	−0.192573	−	0.981283i	$-0.561683\pi$
$461$	−8.26943	−0.385146	−0.192573	+	0.981283i	$0.561683\pi$
$462$	0	0
$463$	9.73904	0.452612	0.226306	−	0.974056i	$-0.427335\pi$
$463$	9.73904	0.452612	0.226306	+	0.974056i	$0.427335\pi$
$464$	−0.749219	−0.0347816
$465$	0	0
$466$	−4.44706	−0.206006
$467$	40.6970	1.88323	0.941617	−	0.336685i	$-0.109306\pi$
$467$	40.6970	1.88323	0.941617	+	0.336685i	$0.109306\pi$
$468$	0	0
$469$	0.139410	0.00643735
$470$	−7.73356	−0.356723
$471$	0	0
$472$	2.49309	0.114754
$473$	−2.33490	−0.107359
$474$	0	0
$475$	77.1153	3.53829
$476$	−0.660406	−0.0302697
$477$	0	0
$478$	−1.80842	−0.0827152
$479$	−37.8318	−1.72858	−0.864289	−	0.502996i	$-0.832231\pi$
$479$	−37.8318	−1.72858	−0.864289	+	0.502996i	$0.832231\pi$
$480$	0	0
$481$	−17.0226	−0.776162
$482$	−2.02334	−0.0921608
$483$	0	0
$484$	−1.97196	−0.0896346
$485$	−0.213351	−0.00968778
$486$	0	0
$487$	40.5576	1.83784	0.918921	−	0.394442i	$-0.129062\pi$
$487$	40.5576	1.83784	0.918921	+	0.394442i	$0.129062\pi$
$488$	−3.99061	−0.180646
$489$	0	0
$490$	−0.637062	−0.0287795
$491$	−31.0786	−1.40256	−0.701280	−	0.712886i	$-0.747388\pi$
$491$	−31.0786	−1.40256	−0.701280	+	0.712886i	$0.747388\pi$
$492$	0	0
$493$	0.0654688	0.00294856
$494$	5.18531	0.233298
$495$	0	0
$496$	−38.1106	−1.71122
$497$	4.66980	0.209469
$498$	0	0
$499$	15.3575	0.687494	0.343747	−	0.939062i	$-0.388304\pi$
$499$	15.3575	0.687494	0.343747	+	0.939062i	$0.388304\pi$
$500$	33.5678	1.50120
$501$	0	0
$502$	−0.805298	−0.0359422
$503$	−14.8925	−0.664025	−0.332013	−	0.943275i	$-0.607728\pi$
$503$	−14.8925	−0.664025	−0.332013	+	0.943275i	$0.607728\pi$
$504$	0	0
$505$	−71.8412	−3.19689
$506$	−0.278820	−0.0123951
$507$	0	0
$508$	−28.8177	−1.27858
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	−4.19549	−0.185597
$512$	12.6557	0.559309
$513$	0	0
$514$	2.80530	0.123736
$515$	31.4969	1.38792
$516$	0	0
$517$	−12.1394	−0.533891
$518$	0.749219	0.0329188
$519$	0	0
$520$	9.62689	0.422167
$521$	−27.6924	−1.21322	−0.606612	−	0.794998i	$-0.707472\pi$
$521$	−27.6924	−1.21322	−0.606612	+	0.794998i	$0.707472\pi$
$522$	0	0
$523$	18.4088	0.804962	0.402481	−	0.915428i	$-0.368148\pi$
$523$	18.4088	0.804962	0.402481	+	0.915428i	$0.368148\pi$
$524$	40.5389	1.77095
$525$	0	0
$526$	2.03273	0.0886314
$527$	3.33020	0.145066
$528$	0	0
$529$	−20.2274	−0.879454
$530$	5.06077	0.219826
$531$	0	0
$532$	16.0506	0.695882
$533$	23.8878	1.03470
$534$	0	0
$535$	−30.9665	−1.33880
$536$	0.0927218	0.00400497
$537$	0	0
$538$	4.22274	0.182055
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−4.26943	−0.183557	−0.0917786	−	0.995779i	$-0.529255\pi$
$541$	−4.26943	−0.183557	−0.0917786	+	0.995779i	$0.529255\pi$
$542$	0.693141	0.0297729
$543$	0	0
$544$	−0.660406	−0.0283147
$545$	−43.9533	−1.88275
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	−31.9906	−1.36657
$549$	0	0
$550$	−1.58647	−0.0676471
$551$	−1.59116	−0.0677857
$552$	0	0
$553$	−3.33020	−0.141615
$554$	−3.06077	−0.130040
$555$	0	0
$556$	−44.4643	−1.88570
$557$	12.4743	0.528553	0.264277	−	0.964447i	$-0.414867\pi$
$557$	12.4743	0.528553	0.264277	+	0.964447i	$0.414867\pi$
$558$	0	0
$559$	8.88315	0.375717
$560$	14.5810	0.616159
$561$	0	0
$562$	−1.12923	−0.0476338
$563$	32.7710	1.38113	0.690566	−	0.723269i	$-0.257361\pi$
$563$	32.7710	1.38113	0.690566	+	0.723269i	$0.257361\pi$
$564$	0	0
$565$	−5.06077	−0.212908
$566$	3.91119	0.164399
$567$	0	0
$568$	3.10589	0.130320
$569$	29.2180	1.22488	0.612442	−	0.790515i	$-0.290187\pi$
$569$	29.2180	1.22488	0.612442	+	0.790515i	$0.290187\pi$
$570$	0	0
$571$	32.4455	1.35780	0.678901	−	0.734230i	$-0.262457\pi$
$571$	32.4455	1.35780	0.678901	+	0.734230i	$0.262457\pi$
$572$	7.50235	0.313689
$573$	0	0
$574$	−1.05138	−0.0438839
$575$	15.7757	0.657892
$576$	0	0
$577$	−14.8831	−0.619594	−0.309797	−	0.950803i	$-0.600261\pi$
$577$	−14.8831	−0.619594	−0.309797	+	0.950803i	$0.600261\pi$
$578$	−2.82786	−0.117623
$579$	0	0
$580$	−1.46662	−0.0608982
$581$	−13.9439	−0.578491
$582$	0	0
$583$	7.94392	0.329004
$584$	−2.79043	−0.115469
$585$	0	0
$586$	2.37220	0.0979945
$587$	4.53039	0.186989	0.0934945	−	0.995620i	$-0.470196\pi$
$587$	4.53039	0.186989	0.0934945	+	0.995620i	$0.470196\pi$
$588$	0	0
$589$	−80.9377	−3.33498
$590$	2.38799	0.0983118
$591$	0	0
$592$	−17.1480	−0.704779
$593$	−8.28821	−0.340356	−0.170178	−	0.985413i	$-0.554434\pi$
$593$	−8.28821	−0.340356	−0.170178	+	0.985413i	$0.554434\pi$
$594$	0	0
$595$	−1.27412	−0.0522340
$596$	16.1612	0.661988
$597$	0	0
$598$	1.06077	0.0433783
$599$	−1.99061	−0.0813341	−0.0406671	−	0.999173i	$-0.512948\pi$
$599$	−1.99061	−0.0813341	−0.0406671	+	0.999173i	$0.512948\pi$
$600$	0	0
$601$	−8.47431	−0.345674	−0.172837	−	0.984950i	$-0.555293\pi$
$601$	−8.47431	−0.345674	−0.172837	+	0.984950i	$0.555293\pi$
$602$	−0.390977	−0.0159350
$603$	0	0
$604$	26.1761	1.06509
$605$	−3.80451	−0.154675
$606$	0	0
$607$	−22.9665	−0.932181	−0.466090	−	0.884737i	$-0.654338\pi$
$607$	−22.9665	−0.932181	−0.466090	+	0.884737i	$0.654338\pi$
$608$	16.0506	0.650938
$609$	0	0
$610$	−3.82237	−0.154763
$611$	46.1845	1.86843
$612$	0	0
$613$	3.55294	0.143502	0.0717510	−	0.997423i	$-0.477141\pi$
$613$	3.55294	0.143502	0.0717510	+	0.997423i	$0.477141\pi$
$614$	−2.10277	−0.0848608
$615$	0	0
$616$	−0.665102	−0.0267977
$617$	−22.6698	−0.912652	−0.456326	−	0.889813i	$-0.650835\pi$
$617$	−22.6698	−0.912652	−0.456326	+	0.889813i	$0.650835\pi$
$618$	0	0
$619$	8.71648	0.350345	0.175173	−	0.984538i	$-0.443952\pi$
$619$	8.71648	0.350345	0.175173	+	0.984538i	$0.443952\pi$
$620$	−74.6028	−2.99612
$621$	0	0
$622$	−1.56391	−0.0627070
$623$	−9.88784	−0.396148
$624$	0	0
$625$	17.3910	0.695639
$626$	−0.501561	−0.0200464
$627$	0	0
$628$	33.4035	1.33294
$629$	1.49844	0.0597467
$630$	0	0
$631$	−11.8878	−0.473248	−0.236624	−	0.971601i	$-0.576041\pi$
$631$	−11.8878	−0.473248	−0.236624	+	0.971601i	$0.576041\pi$
$632$	−2.21492	−0.0881049
$633$	0	0
$634$	−1.66353	−0.0660672
$635$	−55.5981	−2.20634
$636$	0	0
$637$	3.80451	0.150740
$638$	0.0327344	0.00129597
$639$	0	0
$640$	19.6775	0.777821
$641$	4.89254	0.193244	0.0966218	−	0.995321i	$-0.469196\pi$
$641$	4.89254	0.193244	0.0966218	+	0.995321i	$0.469196\pi$
$642$	0	0
$643$	22.7804	0.898371	0.449185	−	0.893439i	$-0.351714\pi$
$643$	22.7804	0.898371	0.449185	+	0.893439i	$0.351714\pi$
$644$	3.28352	0.129389
$645$	0	0
$646$	−0.456446	−0.0179586
$647$	−31.2453	−1.22838	−0.614190	−	0.789158i	$-0.710517\pi$
$647$	−31.2453	−1.22838	−0.614190	+	0.789158i	$0.710517\pi$
$648$	0	0
$649$	3.74843	0.147139
$650$	6.03573	0.236741
$651$	0	0
$652$	13.4090	0.525136
$653$	−28.2882	−1.10700	−0.553502	−	0.832848i	$-0.686709\pi$
$653$	−28.2882	−1.10700	−0.553502	+	0.832848i	$0.686709\pi$
$654$	0	0
$655$	78.2118	3.05599
$656$	24.0639	0.939537
$657$	0	0
$658$	−2.03273	−0.0792442
$659$	−30.2967	−1.18019	−0.590096	−	0.807333i	$-0.700910\pi$
$659$	−30.2967	−1.18019	−0.590096	+	0.807333i	$0.700910\pi$
$660$	0	0
$661$	−35.7196	−1.38933	−0.694666	−	0.719333i	$-0.744448\pi$
$661$	−35.7196	−1.38933	−0.694666	+	0.719333i	$0.744448\pi$
$662$	3.77569	0.146746
$663$	0	0
$664$	−9.27412	−0.359906
$665$	30.9665	1.20083
$666$	0	0
$667$	−0.325508	−0.0126037
$668$	−35.9345	−1.39035
$669$	0	0
$670$	0.0888128	0.00343114
$671$	−6.00000	−0.231627
$672$	0	0
$673$	34.9377	1.34675	0.673374	−	0.739302i	$-0.264844\pi$
$673$	34.9377	1.34675	0.673374	+	0.739302i	$0.264844\pi$
$674$	2.22431	0.0856774
$675$	0	0
$676$	−2.90728	−0.111818
$677$	−20.0094	−0.769023	−0.384512	−	0.923120i	$-0.625630\pi$
$677$	−20.0094	−0.769023	−0.384512	+	0.923120i	$0.625630\pi$
$678$	0	0
$679$	−0.0560785	−0.00215209
$680$	−0.847422	−0.0324972
$681$	0	0
$682$	1.66510	0.0637600
$683$	−25.0498	−0.958504	−0.479252	−	0.877677i	$-0.659092\pi$
$683$	−25.0498	−0.958504	−0.479252	+	0.877677i	$0.659092\pi$
$684$	0	0
$685$	−61.7196	−2.35818
$686$	−0.167449	−0.00639324
$687$	0	0
$688$	8.94862	0.341163
$689$	−30.2227	−1.15139
$690$	0	0
$691$	−38.8271	−1.47705	−0.738526	−	0.674225i	$-0.764478\pi$
$691$	−38.8271	−1.47705	−0.738526	+	0.674225i	$0.764478\pi$
$692$	−3.39410	−0.129024
$693$	0	0
$694$	4.65102	0.176550
$695$	−85.7851	−3.25401
$696$	0	0
$697$	−2.10277	−0.0796480
$698$	−0.478217	−0.0181008
$699$	0	0
$700$	18.6830	0.706150
$701$	41.7757	1.57785	0.788923	−	0.614492i	$-0.210639\pi$
$701$	41.7757	1.57785	0.788923	+	0.614492i	$0.210639\pi$
$702$	0	0
$703$	−36.4182	−1.37354
$704$	7.33490	0.276444
$705$	0	0
$706$	−4.12611	−0.155288
$707$	−18.8831	−0.710174
$708$	0	0
$709$	−13.4041	−0.503403	−0.251702	−	0.967805i	$-0.580990\pi$
$709$	−13.4041	−0.503403	−0.251702	+	0.967805i	$0.580990\pi$
$710$	2.97495	0.111648
$711$	0	0
$712$	−6.57642	−0.246462
$713$	−16.5576	−0.620088
$714$	0	0
$715$	14.4743	0.541308
$716$	−23.6635	−0.884348
$717$	0	0
$718$	1.99061	0.0742889
$719$	−5.47900	−0.204332	−0.102166	−	0.994767i	$-0.532577\pi$
$719$	−5.47900	−0.204332	−0.102166	+	0.994767i	$0.532577\pi$
$720$	0	0
$721$	8.27882	0.308319
$722$	7.91197	0.294453
$723$	0	0
$724$	−1.43140	−0.0531975
$725$	−1.85212	−0.0687859
$726$	0	0
$727$	−32.8831	−1.21957	−0.609784	−	0.792567i	$-0.708744\pi$
$727$	−32.8831	−1.21957	−0.609784	+	0.792567i	$0.708744\pi$
$728$	2.53039	0.0937824
$729$	0	0
$730$	−2.67279	−0.0989243
$731$	−0.781954	−0.0289216
$732$	0	0
$733$	−35.8972	−1.32589	−0.662947	−	0.748666i	$-0.730695\pi$
$733$	−35.8972	−1.32589	−0.662947	+	0.748666i	$0.730695\pi$
$734$	0.604328	0.0223062
$735$	0	0
$736$	3.28352	0.121032
$737$	0.139410	0.00513523
$738$	0	0
$739$	29.6651	1.09125	0.545624	−	0.838030i	$-0.316293\pi$
$739$	29.6651	1.09125	0.545624	+	0.838030i	$0.316293\pi$
$740$	−33.5678	−1.23398
$741$	0	0
$742$	1.33020	0.0488333
$743$	18.7998	0.689698	0.344849	−	0.938658i	$-0.387930\pi$
$743$	18.7998	0.689698	0.344849	+	0.938658i	$0.387930\pi$
$744$	0	0
$745$	31.1798	1.14234
$746$	2.06547	0.0756222
$747$	0	0
$748$	−0.660406	−0.0241469
$749$	−8.13941	−0.297408
$750$	0	0
$751$	−9.59116	−0.349986	−0.174993	−	0.984570i	$-0.555990\pi$
$751$	−9.59116	−0.349986	−0.174993	+	0.984570i	$0.555990\pi$
$752$	46.5249	1.69659
$753$	0	0
$754$	−0.124538	−0.00453542
$755$	50.5016	1.83794
$756$	0	0
$757$	2.74374	0.0997229	0.0498614	−	0.998756i	$-0.484122\pi$
$757$	2.74374	0.0997229	0.0498614	+	0.998756i	$0.484122\pi$
$758$	3.91119	0.142061
$759$	0	0
$760$	20.5959	0.747090
$761$	−6.39098	−0.231673	−0.115836	−	0.993268i	$-0.536955\pi$
$761$	−6.39098	−0.231673	−0.115836	+	0.993268i	$0.536955\pi$
$762$	0	0
$763$	−11.5529	−0.418245
$764$	10.4004	0.376272
$765$	0	0
$766$	2.54825	0.0920720
$767$	−14.2610	−0.514933
$768$	0	0
$769$	17.7951	0.641708	0.320854	−	0.947129i	$-0.396030\pi$
$769$	17.7951	0.641708	0.320854	+	0.947129i	$0.396030\pi$
$770$	−0.637062	−0.0229581
$771$	0	0
$772$	39.1075	1.40751
$773$	31.5257	1.13390	0.566950	−	0.823752i	$-0.308123\pi$
$773$	31.5257	1.13390	0.566950	+	0.823752i	$0.308123\pi$
$774$	0	0
$775$	−94.2118	−3.38419
$776$	−0.0372979	−0.00133892
$777$	0	0
$778$	−0.624681	−0.0223959
$779$	51.1059	1.83106
$780$	0	0
$781$	4.66980	0.167098
$782$	−0.0933763	−0.00333913
$783$	0	0
$784$	3.83255	0.136877
$785$	64.4455	2.30016
$786$	0	0
$787$	31.8606	1.13571	0.567854	−	0.823130i	$-0.307774\pi$
$787$	31.8606	1.13571	0.567854	+	0.823130i	$0.307774\pi$
$788$	5.26473	0.187548
$789$	0	0
$790$	−2.12155	−0.0754813
$791$	−1.33020	−0.0472966
$792$	0	0
$793$	22.8271	0.810613
$794$	3.49844	0.124155
$795$	0	0
$796$	−26.7259	−0.947274
$797$	−43.8590	−1.55357	−0.776783	−	0.629768i	$-0.783150\pi$
$797$	−43.8590	−1.55357	−0.776783	+	0.629768i	$0.783150\pi$
$798$	0	0
$799$	−4.06547	−0.143826
$800$	18.6830	0.660543
$801$	0	0
$802$	−3.89723	−0.137616
$803$	−4.19549	−0.148056
$804$	0	0
$805$	6.33490	0.223276
$806$	−6.33490	−0.223137
$807$	0	0
$808$	−12.5592	−0.441832
$809$	−6.74374	−0.237097	−0.118549	−	0.992948i	$-0.537824\pi$
$809$	−6.74374	−0.237097	−0.118549	+	0.992948i	$0.537824\pi$
$810$	0	0
$811$	3.97275	0.139502	0.0697510	−	0.997564i	$-0.477780\pi$
$811$	3.97275	0.139502	0.0697510	+	0.997564i	$0.477780\pi$
$812$	−0.385496	−0.0135282
$813$	0	0
$814$	0.749219	0.0262601
$815$	25.8700	0.906186
$816$	0	0
$817$	19.0047	0.664890
$818$	−4.00157	−0.139912
$819$	0	0
$820$	47.1059	1.64501
$821$	−49.5896	−1.73069	−0.865344	−	0.501178i	$-0.832900\pi$
$821$	−49.5896	−1.73069	−0.865344	+	0.501178i	$0.832900\pi$
$822$	0	0
$823$	23.1908	0.808380	0.404190	−	0.914675i	$-0.367553\pi$
$823$	23.1908	0.808380	0.404190	+	0.914675i	$0.367553\pi$
$824$	5.50626	0.191820
$825$	0	0
$826$	0.627672	0.0218395
$827$	−27.7484	−0.964908	−0.482454	−	0.875921i	$-0.660254\pi$
$827$	−27.7484	−0.964908	−0.482454	+	0.875921i	$0.660254\pi$
$828$	0	0
$829$	0.269430	0.00935768	0.00467884	−	0.999989i	$-0.498511\pi$
$829$	0.269430	0.00935768	0.00467884	+	0.999989i	$0.498511\pi$
$830$	−8.88315	−0.308339
$831$	0	0
$832$	−27.9057	−0.967456
$833$	−0.334898	−0.0116035
$834$	0	0
$835$	−69.3286	−2.39922
$836$	16.0506	0.555122
$837$	0	0
$838$	5.07316	0.175249
$839$	27.3575	0.944484	0.472242	−	0.881469i	$-0.343445\pi$
$839$	27.3575	0.944484	0.472242	+	0.881469i	$0.343445\pi$
$840$	0	0
$841$	−28.9618	−0.998682
$842$	−2.59976	−0.0895938
$843$	0	0
$844$	8.43767	0.290436
$845$	−5.60902	−0.192956
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	−30.4455	−1.04550
$849$	0	0
$850$	−0.531305	−0.0182236
$851$	−7.45018	−0.255389
$852$	0	0
$853$	−28.5482	−0.977473	−0.488737	−	0.872431i	$-0.662542\pi$
$853$	−28.5482	−0.977473	−0.488737	+	0.872431i	$0.662542\pi$
$854$	−1.00470	−0.0343800
$855$	0	0
$856$	−5.41353	−0.185031
$857$	39.8972	1.36286	0.681432	−	0.731882i	$-0.261358\pi$
$857$	39.8972	1.36286	0.681432	+	0.731882i	$0.261358\pi$
$858$	0	0
$859$	20.5576	0.701418	0.350709	−	0.936485i	$-0.385941\pi$
$859$	20.5576	0.701418	0.350709	+	0.936485i	$0.385941\pi$
$860$	17.5172	0.597332
$861$	0	0
$862$	−0.515515	−0.0175585
$863$	9.66510	0.329004	0.164502	−	0.986377i	$-0.447398\pi$
$863$	9.66510	0.329004	0.164502	+	0.986377i	$0.447398\pi$
$864$	0	0
$865$	−6.54825	−0.222647
$866$	2.83646	0.0963868
$867$	0	0
$868$	−19.6090	−0.665574
$869$	−3.33020	−0.112969
$870$	0	0
$871$	−0.530387	−0.0179715
$872$	−7.68388	−0.260209
$873$	0	0
$874$	2.26943	0.0767646
$875$	17.0226	0.575467
$876$	0	0
$877$	−25.7212	−0.868543	−0.434271	−	0.900782i	$-0.642994\pi$
$877$	−25.7212	−0.868543	−0.434271	+	0.900782i	$0.642994\pi$
$878$	−1.85511	−0.0626069
$879$	0	0
$880$	14.5810	0.491525
$881$	14.6316	0.492950	0.246475	−	0.969149i	$-0.420728\pi$
$881$	14.6316	0.492950	0.246475	+	0.969149i	$0.420728\pi$
$882$	0	0
$883$	18.6877	0.628890	0.314445	−	0.949276i	$-0.398182\pi$
$883$	18.6877	0.628890	0.314445	+	0.949276i	$0.398182\pi$
$884$	2.51252	0.0845053
$885$	0	0
$886$	2.43609	0.0818421
$887$	−38.6137	−1.29652	−0.648261	−	0.761418i	$-0.724503\pi$
$887$	−38.6137	−1.29652	−0.648261	+	0.761418i	$0.724503\pi$
$888$	0	0
$889$	−14.6137	−0.490128
$890$	−6.29917	−0.211149
$891$	0	0
$892$	20.2509	0.678051
$893$	98.8076	3.30647
$894$	0	0
$895$	−45.6541	−1.52605
$896$	5.17214	0.172789
$897$	0	0
$898$	−4.60276	−0.153596
$899$	1.94392	0.0648334
$900$	0	0
$901$	2.66041	0.0886310
$902$	−1.05138	−0.0350072
$903$	0	0
$904$	−0.884720	−0.0294254
$905$	−2.76160	−0.0917987
$906$	0	0
$907$	49.0965	1.63022	0.815111	−	0.579304i	$-0.196676\pi$
$907$	49.0965	1.63022	0.815111	+	0.579304i	$0.196676\pi$
$908$	−0.770991	−0.0255862
$909$	0	0
$910$	2.42371	0.0803452
$911$	−29.3753	−0.973248	−0.486624	−	0.873612i	$-0.661772\pi$
$911$	−29.3753	−0.973248	−0.486624	+	0.873612i	$0.661772\pi$
$912$	0	0
$913$	−13.9439	−0.461476
$914$	−1.33020	−0.0439992
$915$	0	0
$916$	−15.6651	−0.517590
$917$	20.5576	0.678873
$918$	0	0
$919$	−27.0047	−0.890803	−0.445401	−	0.895331i	$-0.646939\pi$
$919$	−27.0047	−0.890803	−0.445401	+	0.895331i	$0.646939\pi$
$920$	4.21335	0.138910
$921$	0	0
$922$	−1.38471	−0.0456030
$923$	−17.7663	−0.584785
$924$	0	0
$925$	−42.3910	−1.39381
$926$	1.63079	0.0535912
$927$	0	0
$928$	−0.385496	−0.0126545
$929$	−57.8496	−1.89798	−0.948992	−	0.315299i	$-0.897895\pi$
$929$	−57.8496	−1.89798	−0.948992	+	0.315299i	$0.897895\pi$
$930$	0	0
$931$	8.13941	0.266758
$932$	52.3706	1.71546
$933$	0	0
$934$	6.81469	0.222983
$935$	−1.27412	−0.0416683
$936$	0	0
$937$	25.2180	0.823838	0.411919	−	0.911221i	$-0.364859\pi$
$937$	25.2180	0.823838	0.411919	+	0.911221i	$0.364859\pi$
$938$	0.0233441	0.000762211 0
$939$	0	0
$940$	91.0741	2.97051
$941$	34.2788	1.11746	0.558729	−	0.829350i	$-0.311289\pi$
$941$	34.2788	1.11746	0.558729	+	0.829350i	$0.311289\pi$
$942$	0	0
$943$	10.4549	0.340458
$944$	−14.3661	−0.467575
$945$	0	0
$946$	−0.390977	−0.0127118
$947$	18.1573	0.590032	0.295016	−	0.955492i	$-0.404675\pi$
$947$	18.1573	0.590032	0.295016	+	0.955492i	$0.404675\pi$
$948$	0	0
$949$	15.9618	0.518141
$950$	12.9129	0.418950
$951$	0	0
$952$	−0.222741	−0.00721909
$953$	19.8045	0.641531	0.320766	−	0.947159i	$-0.396060\pi$
$953$	19.8045	0.641531	0.320766	+	0.947159i	$0.396060\pi$
$954$	0	0
$955$	20.0655	0.649303
$956$	21.2968	0.688788
$957$	0	0
$958$	−6.33490	−0.204671
$959$	−16.2227	−0.523860
$960$	0	0
$961$	67.8816	2.18973
$962$	−2.85041	−0.0919010
$963$	0	0
$964$	23.8279	0.767444
$965$	75.4502	2.42883
$966$	0	0
$967$	−0.278820	−0.00896624	−0.00448312	−	0.999990i	$-0.501427\pi$
$967$	−0.278820	−0.00896624	−0.00448312	+	0.999990i	$0.501427\pi$
$968$	−0.665102	−0.0213772
$969$	0	0
$970$	−0.0357255	−0.00114708
$971$	−25.3481	−0.813458	−0.406729	−	0.913549i	$-0.633331\pi$
$971$	−25.3481	−0.813458	−0.406729	+	0.913549i	$0.633331\pi$
$972$	0	0
$973$	−22.5482	−0.722864
$974$	6.79134	0.217609
$975$	0	0
$976$	22.9953	0.736062
$977$	−6.71648	−0.214879	−0.107440	−	0.994212i	$-0.534265\pi$
$977$	−6.71648	−0.214879	−0.107440	+	0.994212i	$0.534265\pi$
$978$	0	0
$979$	−9.88784	−0.316017
$980$	7.50235	0.239654
$981$	0	0
$982$	−5.20409	−0.166069
$983$	−9.99061	−0.318651	−0.159325	−	0.987226i	$-0.550932\pi$
$983$	−9.99061	−0.318651	−0.159325	+	0.987226i	$0.550932\pi$
$984$	0	0
$985$	10.1573	0.323638
$986$	0.0109627	0.000349123 0
$987$	0	0
$988$	−61.0647	−1.94273
$989$	3.88784	0.123626
$990$	0	0
$991$	−26.7064	−0.848358	−0.424179	−	0.905578i	$-0.639437\pi$
$991$	−26.7064	−0.848358	−0.424179	+	0.905578i	$0.639437\pi$
$992$	−19.6090	−0.622587
$993$	0	0
$994$	0.781954	0.0248021
$995$	−51.5623	−1.63464
$996$	0	0
$997$	54.3333	1.72075	0.860377	−	0.509658i	$-0.170228\pi$
$997$	54.3333	1.72075	0.860377	+	0.509658i	$0.170228\pi$
$998$	2.57159	0.0814024
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.2.a.m.1.2		3
3.2	odd	2		231.2.a.d.1.2	✓	3
7.6	odd	2		4851.2.a.bp.1.2		3
11.10	odd	2		7623.2.a.cb.1.2		3
12.11	even	2		3696.2.a.bp.1.3		3
15.14	odd	2		5775.2.a.bw.1.2		3
21.20	even	2		1617.2.a.s.1.2		3
33.32	even	2		2541.2.a.bi.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.a.d.1.2	✓	3	3.2	odd	2
693.2.a.m.1.2		3	1.1	even	1	trivial
1617.2.a.s.1.2		3	21.20	even	2
2541.2.a.bi.1.2		3	33.32	even	2
3696.2.a.bp.1.3		3	12.11	even	2
4851.2.a.bp.1.2		3	7.6	odd	2
5775.2.a.bw.1.2		3	15.14	odd	2
7623.2.a.cb.1.2		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)

\(f(q)\)	\(=\)	\(q+0.167449 q^{2} -1.97196 q^{4} -3.80451 q^{5} -1.00000 q^{7} -0.665102 q^{8} +O(q^{10})\)	\(q+0.167449 q^{2} -1.97196 q^{4} -3.80451 q^{5} -1.00000 q^{7} -0.665102 q^{8} -0.637062 q^{10} -1.00000 q^{11} +3.80451 q^{13} -0.167449 q^{14} +3.83255 q^{16} -0.334898 q^{17} +8.13941 q^{19} +7.50235 q^{20} -0.167449 q^{22} +1.66510 q^{23} +9.47431 q^{25} +0.637062 q^{26} +1.97196 q^{28} -0.195488 q^{29} -9.94392 q^{31} +1.97196 q^{32} -0.0560785 q^{34} +3.80451 q^{35} -4.47431 q^{37} +1.36294 q^{38} +2.53039 q^{40} +6.27882 q^{41} +2.33490 q^{43} +1.97196 q^{44} +0.278820 q^{46} +12.1394 q^{47} +1.00000 q^{49} +1.58647 q^{50} -7.50235 q^{52} -7.94392 q^{53} +3.80451 q^{55} +0.665102 q^{56} -0.0327344 q^{58} -3.74843 q^{59} +6.00000 q^{61} -1.66510 q^{62} -7.33490 q^{64} -14.4743 q^{65} -0.139410 q^{67} +0.660406 q^{68} +0.637062 q^{70} -4.66980 q^{71} +4.19549 q^{73} -0.749219 q^{74} -16.0506 q^{76} +1.00000 q^{77} +3.33020 q^{79} -14.5810 q^{80} +1.05138 q^{82} +13.9439 q^{83} +1.27412 q^{85} +0.390977 q^{86} +0.665102 q^{88} +9.88784 q^{89} -3.80451 q^{91} -3.28352 q^{92} +2.03273 q^{94} -30.9665 q^{95} +0.0560785 q^{97} +0.167449 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{4} - 3 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q + 6 * q^4 - 3 * q^7 - 3 * q^8	\( 3 q + 6 q^{4} - 3 q^{7} - 3 q^{8} + 9 q^{10} - 3 q^{11} + 12 q^{16} + 12 q^{19} + 21 q^{20} + 6 q^{23} + 15 q^{25} - 9 q^{26} - 6 q^{28} - 12 q^{29} - 6 q^{31} - 6 q^{32} - 24 q^{34} + 15 q^{38} + 18 q^{40} - 6 q^{41} + 6 q^{43} - 6 q^{44} - 24 q^{46} + 24 q^{47} + 3 q^{49} + 39 q^{50} - 21 q^{52} + 3 q^{56} - 9 q^{58} + 24 q^{59} + 18 q^{61} - 6 q^{62} - 21 q^{64} - 30 q^{65} + 12 q^{67} + 6 q^{68} - 9 q^{70} - 12 q^{71} + 24 q^{73} - 39 q^{74} - 3 q^{76} + 3 q^{77} + 12 q^{79} - 9 q^{80} + 30 q^{82} + 18 q^{83} - 18 q^{85} + 24 q^{86} + 3 q^{88} - 18 q^{89} + 18 q^{92} + 15 q^{94} - 12 q^{95} + 24 q^{97}+O(q^{100}) \) 3 * q + 6 * q^4 - 3 * q^7 - 3 * q^8 + 9 * q^10 - 3 * q^11 + 12 * q^16 + 12 * q^19 + 21 * q^20 + 6 * q^23 + 15 * q^25 - 9 * q^26 - 6 * q^28 - 12 * q^29 - 6 * q^31 - 6 * q^32 - 24 * q^34 + 15 * q^38 + 18 * q^40 - 6 * q^41 + 6 * q^43 - 6 * q^44 - 24 * q^46 + 24 * q^47 + 3 * q^49 + 39 * q^50 - 21 * q^52 + 3 * q^56 - 9 * q^58 + 24 * q^59 + 18 * q^61 - 6 * q^62 - 21 * q^64 - 30 * q^65 + 12 * q^67 + 6 * q^68 - 9 * q^70 - 12 * q^71 + 24 * q^73 - 39 * q^74 - 3 * q^76 + 3 * q^77 + 12 * q^79 - 9 * q^80 + 30 * q^82 + 18 * q^83 - 18 * q^85 + 24 * q^86 + 3 * q^88 - 18 * q^89 + 18 * q^92 + 15 * q^94 - 12 * q^95 + 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more