LMFDB - Embedded newform 693.4.a.o.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,4,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, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("693.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
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:	$40.8883236340$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 42x^{3} + 18x^{2} + 368x + 352 $ x^5 - x^4 - 42x^3 + 18x^2 + 368*x + 352
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.18888$ 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)

$f(q)$	$=$	$q+2.18888 q^{2} -3.20880 q^{4} -7.60736 q^{5} +7.00000 q^{7} -24.5347 q^{8} +O(q^{10})$	\(q+2.18888 q^{2} -3.20880 q^{4} -7.60736 q^{5} +7.00000 q^{7} -24.5347 q^{8} -16.6516 q^{10} -11.0000 q^{11} +0.174238 q^{13} +15.3222 q^{14} -28.0332 q^{16} -128.863 q^{17} +141.685 q^{19} +24.4105 q^{20} -24.0777 q^{22} +133.369 q^{23} -67.1280 q^{25} +0.381386 q^{26} -22.4616 q^{28} +177.002 q^{29} +48.2757 q^{31} +134.917 q^{32} -282.065 q^{34} -53.2515 q^{35} +161.625 q^{37} +310.131 q^{38} +186.645 q^{40} +195.689 q^{41} -488.447 q^{43} +35.2968 q^{44} +291.928 q^{46} +171.705 q^{47} +49.0000 q^{49} -146.935 q^{50} -0.559095 q^{52} +431.477 q^{53} +83.6810 q^{55} -171.743 q^{56} +387.437 q^{58} -194.176 q^{59} +585.008 q^{61} +105.670 q^{62} +519.582 q^{64} -1.32549 q^{65} +155.905 q^{67} +413.496 q^{68} -116.561 q^{70} +374.994 q^{71} +210.419 q^{73} +353.778 q^{74} -454.639 q^{76} -77.0000 q^{77} -7.00618 q^{79} +213.258 q^{80} +428.339 q^{82} +93.6417 q^{83} +980.307 q^{85} -1069.15 q^{86} +269.882 q^{88} +307.119 q^{89} +1.21967 q^{91} -427.954 q^{92} +375.842 q^{94} -1077.85 q^{95} +965.991 q^{97} +107.255 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{2} + 45 q^{4} + 24 q^{5} + 35 q^{7} - 57 q^{8}+O(q^{10}) $ 5 * q - q^2 + 45 * q^4 + 24 * q^5 + 35 * q^7 - 57 * q^8	$ 5 q - q^{2} + 45 q^{4} + 24 q^{5} + 35 q^{7} - 57 q^{8} - 10 q^{10} - 55 q^{11} - 50 q^{13} - 7 q^{14} + 433 q^{16} - 222 q^{17} + 160 q^{19} + 430 q^{20} + 11 q^{22} - 54 q^{23} + 125 q^{25} + 1026 q^{26} + 315 q^{28} - 14 q^{29} - 34 q^{31} + 583 q^{32} - 750 q^{34} + 168 q^{35} + 1044 q^{37} + 156 q^{38} + 1158 q^{40} + 114 q^{41} + 672 q^{43} - 495 q^{44} - 1224 q^{46} + 292 q^{47} + 245 q^{49} - 1143 q^{50} - 914 q^{52} + 710 q^{53} - 264 q^{55} - 399 q^{56} - 810 q^{58} - 270 q^{59} + 138 q^{61} + 480 q^{62} + 3001 q^{64} + 196 q^{65} + 1942 q^{67} - 3130 q^{68} - 70 q^{70} + 278 q^{71} - 338 q^{73} - 462 q^{74} - 1332 q^{76} - 385 q^{77} + 576 q^{79} + 1602 q^{80} + 1386 q^{82} - 1644 q^{83} + 360 q^{85} - 2020 q^{86} + 627 q^{88} + 3656 q^{89} - 350 q^{91} + 748 q^{92} + 976 q^{94} + 1276 q^{95} + 692 q^{97} - 49 q^{98}+O(q^{100}) $ 5 * q - q^2 + 45 * q^4 + 24 * q^5 + 35 * q^7 - 57 * q^8 - 10 * q^10 - 55 * q^11 - 50 * q^13 - 7 * q^14 + 433 * q^16 - 222 * q^17 + 160 * q^19 + 430 * q^20 + 11 * q^22 - 54 * q^23 + 125 * q^25 + 1026 * q^26 + 315 * q^28 - 14 * q^29 - 34 * q^31 + 583 * q^32 - 750 * q^34 + 168 * q^35 + 1044 * q^37 + 156 * q^38 + 1158 * q^40 + 114 * q^41 + 672 * q^43 - 495 * q^44 - 1224 * q^46 + 292 * q^47 + 245 * q^49 - 1143 * q^50 - 914 * q^52 + 710 * q^53 - 264 * q^55 - 399 * q^56 - 810 * q^58 - 270 * q^59 + 138 * q^61 + 480 * q^62 + 3001 * q^64 + 196 * q^65 + 1942 * q^67 - 3130 * q^68 - 70 * q^70 + 278 * q^71 - 338 * q^73 - 462 * q^74 - 1332 * q^76 - 385 * q^77 + 576 * q^79 + 1602 * q^80 + 1386 * q^82 - 1644 * q^83 + 360 * q^85 - 2020 * q^86 + 627 * q^88 + 3656 * q^89 - 350 * q^91 + 748 * q^92 + 976 * q^94 + 1276 * q^95 + 692 * q^97 - 49 * 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 $

$2$	2.18888	0.773886	0.386943	−	0.922104i	$-0.373531\pi$
$2$	2.18888	0.773886	0.386943	+	0.922104i	$0.373531\pi$
$3$	0	0
$3$	0	0
$4$	−3.20880	−0.401100
$5$	−7.60736	−0.680423	−0.340212	−	0.940349i	$-0.610499\pi$
$5$	−7.60736	−0.680423	−0.340212	+	0.940349i	$0.610499\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−24.5347	−1.08429
$9$	0	0
$10$	−16.6516	−0.526570
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	0.174238	0.00371730	0.00185865	−	0.999998i	$-0.499408\pi$
$13$	0.174238	0.00371730	0.00185865	+	0.999998i	$0.499408\pi$
$14$	15.3222	0.292501
$15$	0	0
$16$	−28.0332	−0.438018
$17$	−128.863	−1.83846	−0.919231	−	0.393719i	$-0.871188\pi$
$17$	−128.863	−1.83846	−0.919231	+	0.393719i	$0.871188\pi$
$18$	0	0
$19$	141.685	1.71078	0.855388	−	0.517988i	$-0.173319\pi$
$19$	141.685	1.71078	0.855388	+	0.517988i	$0.173319\pi$
$20$	24.4105	0.272918
$21$	0	0
$22$	−24.0777	−0.233335
$23$	133.369	1.20910	0.604550	−	0.796567i	$-0.293353\pi$
$23$	133.369	1.20910	0.604550	+	0.796567i	$0.293353\pi$
$24$	0	0
$25$	−67.1280	−0.537024
$26$	0.381386	0.00287677
$27$	0	0
$28$	−22.4616	−0.151602
$29$	177.002	1.13340	0.566698	−	0.823925i	$-0.308221\pi$
$29$	177.002	1.13340	0.566698	+	0.823925i	$0.308221\pi$
$30$	0	0
$31$	48.2757	0.279696	0.139848	−	0.990173i	$-0.455339\pi$
$31$	48.2757	0.279696	0.139848	+	0.990173i	$0.455339\pi$
$32$	134.917	0.745316
$33$	0	0
$34$	−282.065	−1.42276
$35$	−53.2515	−0.257176
$36$	0	0
$37$	161.625	0.718136	0.359068	−	0.933311i	$-0.383095\pi$
$37$	161.625	0.718136	0.359068	+	0.933311i	$0.383095\pi$
$38$	310.131	1.32394
$39$	0	0
$40$	186.645	0.737778
$41$	195.689	0.745402	0.372701	−	0.927952i	$-0.378432\pi$
$41$	195.689	0.745402	0.372701	+	0.927952i	$0.378432\pi$
$42$	0	0
$43$	−488.447	−1.73227	−0.866133	−	0.499814i	$-0.833402\pi$
$43$	−488.447	−1.73227	−0.866133	+	0.499814i	$0.833402\pi$
$44$	35.2968	0.120936
$45$	0	0
$46$	291.928	0.935705
$47$	171.705	0.532889	0.266444	−	0.963850i	$-0.414151\pi$
$47$	171.705	0.532889	0.266444	+	0.963850i	$0.414151\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−146.935	−0.415596
$51$	0	0
$52$	−0.559095	−0.00149101
$53$	431.477	1.11826	0.559131	−	0.829079i	$-0.311135\pi$
$53$	431.477	1.11826	0.559131	+	0.829079i	$0.311135\pi$
$54$	0	0
$55$	83.6810	0.205155
$56$	−171.743	−0.409824
$57$	0	0
$58$	387.437	0.877120
$59$	−194.176	−0.428468	−0.214234	−	0.976782i	$-0.568725\pi$
$59$	−194.176	−0.428468	−0.214234	+	0.976782i	$0.568725\pi$
$60$	0	0
$61$	585.008	1.22791	0.613955	−	0.789341i	$-0.289577\pi$
$61$	585.008	1.22791	0.613955	+	0.789341i	$0.289577\pi$
$62$	105.670	0.216453
$63$	0	0
$64$	519.582	1.01481
$65$	−1.32549	−0.00252934
$66$	0	0
$67$	155.905	0.284281	0.142140	−	0.989847i	$-0.454602\pi$
$67$	155.905	0.284281	0.142140	+	0.989847i	$0.454602\pi$
$68$	413.496	0.737408
$69$	0	0
$70$	−116.561	−0.199025
$71$	374.994	0.626811	0.313405	−	0.949619i	$-0.398530\pi$
$71$	374.994	0.626811	0.313405	+	0.949619i	$0.398530\pi$
$72$	0	0
$73$	210.419	0.337365	0.168683	−	0.985670i	$-0.446049\pi$
$73$	210.419	0.337365	0.168683	+	0.985670i	$0.446049\pi$
$74$	353.778	0.555755
$75$	0	0
$76$	−454.639	−0.686193
$77$	−77.0000	−0.113961
$78$	0	0
$79$	−7.00618	−0.00997793	−0.00498897	−	0.999988i	$-0.501588\pi$
$79$	−7.00618	−0.00997793	−0.00498897	+	0.999988i	$0.501588\pi$
$80$	213.258	0.298038
$81$	0	0
$82$	428.339	0.576856
$83$	93.6417	0.123837	0.0619187	−	0.998081i	$-0.480278\pi$
$83$	93.6417	0.123837	0.0619187	+	0.998081i	$0.480278\pi$
$84$	0	0
$85$	980.307	1.25093
$86$	−1069.15	−1.34058
$87$	0	0
$88$	269.882	0.326926
$89$	307.119	0.365781	0.182891	−	0.983133i	$-0.441455\pi$
$89$	307.119	0.365781	0.182891	+	0.983133i	$0.441455\pi$
$90$	0	0
$91$	1.21967	0.00140501
$92$	−427.954	−0.484970
$93$	0	0
$94$	375.842	0.412395
$95$	−1077.85	−1.16405
$96$	0	0
$97$	965.991	1.01115	0.505575	−	0.862783i	$-0.331280\pi$
$97$	965.991	1.01115	0.505575	+	0.862783i	$0.331280\pi$
$98$	107.255	0.110555
$99$	0	0
$100$	215.401	0.215401
$101$	577.986	0.569424	0.284712	−	0.958613i	$-0.408102\pi$
$101$	577.986	0.569424	0.284712	+	0.958613i	$0.408102\pi$
$102$	0	0
$103$	133.232	0.127453	0.0637267	−	0.997967i	$-0.479701\pi$
$103$	133.232	0.127453	0.0637267	+	0.997967i	$0.479701\pi$
$104$	−4.27488	−0.00403064
$105$	0	0
$106$	944.451	0.865408
$107$	−8.33627	−0.00753175	−0.00376587	−	0.999993i	$-0.501199\pi$
$107$	−8.33627	−0.00753175	−0.00376587	+	0.999993i	$0.501199\pi$
$108$	0	0
$109$	−317.052	−0.278606	−0.139303	−	0.990250i	$-0.544486\pi$
$109$	−317.052	−0.278606	−0.139303	+	0.990250i	$0.544486\pi$
$110$	183.168	0.158767
$111$	0	0
$112$	−196.232	−0.165555
$113$	2170.00	1.80652	0.903260	−	0.429093i	$-0.141167\pi$
$113$	2170.00	1.80652	0.903260	+	0.429093i	$0.141167\pi$
$114$	0	0
$115$	−1014.58	−0.822700
$116$	−567.966	−0.454606
$117$	0	0
$118$	−425.028	−0.331585
$119$	−902.040	−0.694873
$120$	0	0
$121$	121.000	0.0909091
$122$	1280.51	0.950263
$123$	0	0
$124$	−154.907	−0.112186
$125$	1461.59	1.04583
$126$	0	0
$127$	1301.24	0.909184	0.454592	−	0.890700i	$-0.349785\pi$
$127$	1301.24	0.909184	0.454592	+	0.890700i	$0.349785\pi$
$128$	57.9688	0.0400294
$129$	0	0
$130$	−2.90134	−0.00195742
$131$	−2522.22	−1.68220	−0.841098	−	0.540883i	$-0.818090\pi$
$131$	−2522.22	−1.68220	−0.841098	+	0.540883i	$0.818090\pi$
$132$	0	0
$133$	991.794	0.646612
$134$	341.257	0.220001
$135$	0	0
$136$	3161.62	1.99343
$137$	−2574.70	−1.60563	−0.802815	−	0.596228i	$-0.796666\pi$
$137$	−2574.70	−1.60563	−0.802815	+	0.596228i	$0.796666\pi$
$138$	0	0
$139$	−2741.49	−1.67288	−0.836438	−	0.548061i	$-0.815366\pi$
$139$	−2741.49	−1.67288	−0.836438	+	0.548061i	$0.815366\pi$
$140$	170.874	0.103153
$141$	0	0
$142$	820.816	0.485080
$143$	−1.91662	−0.00112081
$144$	0	0
$145$	−1346.52	−0.771189
$146$	460.582	0.261082
$147$	0	0
$148$	−518.624	−0.288045
$149$	1021.98	0.561907	0.280954	−	0.959721i	$-0.409349\pi$
$149$	1021.98	0.561907	0.280954	+	0.959721i	$0.409349\pi$
$150$	0	0
$151$	1016.40	0.547769	0.273885	−	0.961763i	$-0.411691\pi$
$151$	1016.40	0.547769	0.273885	+	0.961763i	$0.411691\pi$
$152$	−3476.20	−1.85498
$153$	0	0
$154$	−168.544	−0.0881925
$155$	−367.251	−0.190312
$156$	0	0
$157$	−1580.49	−0.803422	−0.401711	−	0.915767i	$-0.631584\pi$
$157$	−1580.49	−0.803422	−0.401711	+	0.915767i	$0.631584\pi$
$158$	−15.3357	−0.00772178
$159$	0	0
$160$	−1026.36	−0.507130
$161$	933.581	0.456997
$162$	0	0
$163$	768.358	0.369218	0.184609	−	0.982812i	$-0.440898\pi$
$163$	768.358	0.369218	0.184609	+	0.982812i	$0.440898\pi$
$164$	−627.927	−0.298981
$165$	0	0
$166$	204.970	0.0958361
$167$	−3092.73	−1.43307	−0.716534	−	0.697552i	$-0.754273\pi$
$167$	−3092.73	−1.43307	−0.716534	+	0.697552i	$0.754273\pi$
$168$	0	0
$169$	−2196.97	−0.999986
$170$	2145.77	0.968079
$171$	0	0
$172$	1567.33	0.694812
$173$	1327.46	0.583380	0.291690	−	0.956513i	$-0.405782\pi$
$173$	1327.46	0.583380	0.291690	+	0.956513i	$0.405782\pi$
$174$	0	0
$175$	−469.896	−0.202976
$176$	308.365	0.132067
$177$	0	0
$178$	672.247	0.283073
$179$	−3141.37	−1.31171	−0.655857	−	0.754885i	$-0.727693\pi$
$179$	−3141.37	−1.31171	−0.655857	+	0.754885i	$0.727693\pi$
$180$	0	0
$181$	−3683.36	−1.51261	−0.756303	−	0.654221i	$-0.772997\pi$
$181$	−3683.36	−1.51261	−0.756303	+	0.654221i	$0.772997\pi$
$182$	2.66970	0.00108732
$183$	0	0
$184$	−3272.16	−1.31102
$185$	−1229.54	−0.488636
$186$	0	0
$187$	1417.49	0.554317
$188$	−550.968	−0.213742
$189$	0	0
$190$	−2359.28	−0.900843
$191$	2862.38	1.08437	0.542185	−	0.840259i	$-0.317597\pi$
$191$	2862.38	1.08437	0.542185	+	0.840259i	$0.317597\pi$
$192$	0	0
$193$	2023.91	0.754839	0.377420	−	0.926042i	$-0.376811\pi$
$193$	2023.91	0.754839	0.377420	+	0.926042i	$0.376811\pi$
$194$	2114.44	0.782515
$195$	0	0
$196$	−157.231	−0.0573001
$197$	4767.82	1.72433	0.862165	−	0.506628i	$-0.169108\pi$
$197$	4767.82	1.72433	0.862165	+	0.506628i	$0.169108\pi$
$198$	0	0
$199$	3546.72	1.26342	0.631709	−	0.775205i	$-0.282354\pi$
$199$	3546.72	1.26342	0.631709	+	0.775205i	$0.282354\pi$
$200$	1646.97	0.582291
$201$	0	0
$202$	1265.14	0.440669
$203$	1239.02	0.428384
$204$	0	0
$205$	−1488.68	−0.507189
$206$	291.628	0.0986343
$207$	0	0
$208$	−4.88444	−0.00162824
$209$	−1558.53	−0.515818
$210$	0	0
$211$	1453.49	0.474230	0.237115	−	0.971482i	$-0.423798\pi$
$211$	1453.49	0.474230	0.237115	+	0.971482i	$0.423798\pi$
$212$	−1384.52	−0.448536
$213$	0	0
$214$	−18.2471	−0.00582872
$215$	3715.79	1.17867
$216$	0	0
$217$	337.930	0.105715
$218$	−693.989	−0.215610
$219$	0	0
$220$	−268.516	−0.0822879
$221$	−22.4528	−0.00683411
$222$	0	0
$223$	6440.10	1.93391	0.966953	−	0.254955i	$-0.0820606\pi$
$223$	6440.10	1.93391	0.966953	+	0.254955i	$0.0820606\pi$
$224$	944.416	0.281703
$225$	0	0
$226$	4749.88	1.39804
$227$	1313.21	0.383970	0.191985	−	0.981398i	$-0.438508\pi$
$227$	1313.21	0.383970	0.191985	+	0.981398i	$0.438508\pi$
$228$	0	0
$229$	−3780.00	−1.09078	−0.545392	−	0.838181i	$-0.683619\pi$
$229$	−3780.00	−1.09078	−0.545392	+	0.838181i	$0.683619\pi$
$230$	−2220.80	−0.636676
$231$	0	0
$232$	−4342.70	−1.22893
$233$	−2322.65	−0.653055	−0.326527	−	0.945188i	$-0.605879\pi$
$233$	−2322.65	−0.653055	−0.326527	+	0.945188i	$0.605879\pi$
$234$	0	0
$235$	−1306.22	−0.362590
$236$	623.073	0.171859
$237$	0	0
$238$	−1974.46	−0.537753
$239$	−2204.79	−0.596718	−0.298359	−	0.954454i	$-0.596439\pi$
$239$	−2204.79	−0.596718	−0.298359	+	0.954454i	$0.596439\pi$
$240$	0	0
$241$	−4610.39	−1.23229	−0.616144	−	0.787634i	$-0.711306\pi$
$241$	−4610.39	−1.23229	−0.616144	+	0.787634i	$0.711306\pi$
$242$	264.855	0.0703533
$243$	0	0
$244$	−1877.17	−0.492516
$245$	−372.761	−0.0972033
$246$	0	0
$247$	24.6869	0.00635946
$248$	−1184.43	−0.303272
$249$	0	0
$250$	3199.24	0.809351
$251$	−4981.12	−1.25261	−0.626306	−	0.779577i	$-0.715434\pi$
$251$	−4981.12	−1.25261	−0.626306	+	0.779577i	$0.715434\pi$
$252$	0	0
$253$	−1467.06	−0.364557
$254$	2848.26	0.703605
$255$	0	0
$256$	−4029.77	−0.983829
$257$	−2726.08	−0.661665	−0.330833	−	0.943689i	$-0.607330\pi$
$257$	−2726.08	−0.661665	−0.330833	+	0.943689i	$0.607330\pi$
$258$	0	0
$259$	1131.38	0.271430
$260$	4.25324	0.00101452
$261$	0	0
$262$	−5520.84	−1.30183
$263$	4228.84	0.991489	0.495744	−	0.868468i	$-0.334895\pi$
$263$	4228.84	0.991489	0.495744	+	0.868468i	$0.334895\pi$
$264$	0	0
$265$	−3282.40	−0.760892
$266$	2170.92	0.500404
$267$	0	0
$268$	−500.268	−0.114025
$269$	2396.04	0.543081	0.271541	−	0.962427i	$-0.412467\pi$
$269$	2396.04	0.543081	0.271541	+	0.962427i	$0.412467\pi$
$270$	0	0
$271$	−2461.68	−0.551795	−0.275897	−	0.961187i	$-0.588975\pi$
$271$	−2461.68	−0.551795	−0.275897	+	0.961187i	$0.588975\pi$
$272$	3612.43	0.805279
$273$	0	0
$274$	−5635.71	−1.24258
$275$	738.408	0.161919
$276$	0	0
$277$	5890.54	1.27772	0.638860	−	0.769323i	$-0.279406\pi$
$277$	5890.54	1.27772	0.638860	+	0.769323i	$0.279406\pi$
$278$	−6000.79	−1.29462
$279$	0	0
$280$	1306.51	0.278854
$281$	1964.26	0.417004	0.208502	−	0.978022i	$-0.433141\pi$
$281$	1964.26	0.417004	0.208502	+	0.978022i	$0.433141\pi$
$282$	0	0
$283$	8513.60	1.78827	0.894136	−	0.447796i	$-0.147791\pi$
$283$	8513.60	1.78827	0.894136	+	0.447796i	$0.147791\pi$
$284$	−1203.28	−0.251414
$285$	0	0
$286$	−4.19524	−0.000867378 0
$287$	1369.82	0.281735
$288$	0	0
$289$	11692.6	2.37994
$290$	−2947.37	−0.596813
$291$	0	0
$292$	−675.193	−0.135317
$293$	−5219.58	−1.04072	−0.520360	−	0.853947i	$-0.674202\pi$
$293$	−5219.58	−1.04072	−0.520360	+	0.853947i	$0.674202\pi$
$294$	0	0
$295$	1477.17	0.291539
$296$	−3965.43	−0.778669
$297$	0	0
$298$	2237.00	0.434852
$299$	23.2379	0.00449459
$300$	0	0
$301$	−3419.13	−0.654735
$302$	2224.77	0.423911
$303$	0	0
$304$	−3971.87	−0.749350
$305$	−4450.37	−0.835499
$306$	0	0
$307$	2887.29	0.536764	0.268382	−	0.963313i	$-0.413511\pi$
$307$	2887.29	0.536764	0.268382	+	0.963313i	$0.413511\pi$
$308$	247.078	0.0457096
$309$	0	0
$310$	−803.868	−0.147279
$311$	5876.90	1.07154	0.535769	−	0.844364i	$-0.320022\pi$
$311$	5876.90	1.07154	0.535769	+	0.844364i	$0.320022\pi$
$312$	0	0
$313$	−7683.14	−1.38747	−0.693733	−	0.720233i	$-0.744035\pi$
$313$	−7683.14	−1.38747	−0.693733	+	0.720233i	$0.744035\pi$
$314$	−3459.51	−0.621757
$315$	0	0
$316$	22.4814	0.00400215
$317$	9642.33	1.70841	0.854207	−	0.519933i	$-0.174043\pi$
$317$	9642.33	1.70841	0.854207	+	0.519933i	$0.174043\pi$
$318$	0	0
$319$	−1947.03	−0.341732
$320$	−3952.65	−0.690499
$321$	0	0
$322$	2043.50	0.353663
$323$	−18257.9	−3.14519
$324$	0	0
$325$	−11.6962	−0.00199628
$326$	1681.84	0.285732
$327$	0	0
$328$	−4801.17	−0.808233
$329$	1201.94	0.201413
$330$	0	0
$331$	−2069.87	−0.343717	−0.171859	−	0.985122i	$-0.554977\pi$
$331$	−2069.87	−0.343717	−0.171859	+	0.985122i	$0.554977\pi$
$332$	−300.478	−0.0496713
$333$	0	0
$334$	−6769.61	−1.10903
$335$	−1186.03	−0.193431
$336$	0	0
$337$	7547.59	1.22001	0.610005	−	0.792397i	$-0.291167\pi$
$337$	7547.59	1.22001	0.610005	+	0.792397i	$0.291167\pi$
$338$	−4808.90	−0.773875
$339$	0	0
$340$	−3145.61	−0.501749
$341$	−531.033	−0.0843315
$342$	0	0
$343$	343.000	0.0539949
$344$	11983.9	1.87828
$345$	0	0
$346$	2905.65	0.451469
$347$	6353.22	0.982877	0.491439	−	0.870912i	$-0.336471\pi$
$347$	6353.22	0.982877	0.491439	+	0.870912i	$0.336471\pi$
$348$	0	0
$349$	−1352.47	−0.207439	−0.103719	−	0.994607i	$-0.533074\pi$
$349$	−1352.47	−0.207439	−0.103719	+	0.994607i	$0.533074\pi$
$350$	−1028.55	−0.157080
$351$	0	0
$352$	−1484.08	−0.224721
$353$	−4203.80	−0.633840	−0.316920	−	0.948452i	$-0.602649\pi$
$353$	−4203.80	−0.633840	−0.316920	+	0.948452i	$0.602649\pi$
$354$	0	0
$355$	−2852.71	−0.426497
$356$	−985.484	−0.146715
$357$	0	0
$358$	−6876.08	−1.01512
$359$	11100.7	1.63195	0.815976	−	0.578086i	$-0.196200\pi$
$359$	11100.7	1.63195	0.815976	+	0.578086i	$0.196200\pi$
$360$	0	0
$361$	13215.6	1.92675
$362$	−8062.43	−1.17059
$363$	0	0
$364$	−3.91367	−0.000563549 0
$365$	−1600.73	−0.229551
$366$	0	0
$367$	−4963.35	−0.705953	−0.352977	−	0.935632i	$-0.614830\pi$
$367$	−4963.35	−0.705953	−0.352977	+	0.935632i	$0.614830\pi$
$368$	−3738.74	−0.529607
$369$	0	0
$370$	−2691.32	−0.378149
$371$	3020.34	0.422663
$372$	0	0
$373$	−8882.54	−1.23303	−0.616515	−	0.787343i	$-0.711456\pi$
$373$	−8882.54	−1.23303	−0.616515	+	0.787343i	$0.711456\pi$
$374$	3102.72	0.428978
$375$	0	0
$376$	−4212.74	−0.577807
$377$	30.8405	0.00421317
$378$	0	0
$379$	−1218.83	−0.165190	−0.0825952	−	0.996583i	$-0.526321\pi$
$379$	−1218.83	−0.165190	−0.0825952	+	0.996583i	$0.526321\pi$
$380$	3458.60	0.466901
$381$	0	0
$382$	6265.41	0.839179
$383$	−112.614	−0.0150243	−0.00751214	−	0.999972i	$-0.502391\pi$
$383$	−112.614	−0.0150243	−0.00751214	+	0.999972i	$0.502391\pi$
$384$	0	0
$385$	585.767	0.0775414
$386$	4430.09	0.584159
$387$	0	0
$388$	−3099.68	−0.405573
$389$	−12656.2	−1.64961	−0.824804	−	0.565419i	$-0.808714\pi$
$389$	−12656.2	−1.64961	−0.824804	+	0.565419i	$0.808714\pi$
$390$	0	0
$391$	−17186.3	−2.22288
$392$	−1202.20	−0.154899
$393$	0	0
$394$	10436.2	1.33443
$395$	53.2985	0.00678922
$396$	0	0
$397$	3707.14	0.468655	0.234327	−	0.972158i	$-0.424711\pi$
$397$	3707.14	0.468655	0.234327	+	0.972158i	$0.424711\pi$
$398$	7763.35	0.977742
$399$	0	0
$400$	1881.81	0.235226
$401$	−5671.06	−0.706232	−0.353116	−	0.935579i	$-0.614878\pi$
$401$	−5671.06	−0.706232	−0.353116	+	0.935579i	$0.614878\pi$
$402$	0	0
$403$	8.41146	0.00103971
$404$	−1854.64	−0.228396
$405$	0	0
$406$	2712.06	0.331520
$407$	−1777.88	−0.216526
$408$	0	0
$409$	260.465	0.0314894	0.0157447	−	0.999876i	$-0.494988\pi$
$409$	260.465	0.0314894	0.0157447	+	0.999876i	$0.494988\pi$
$410$	−3258.53	−0.392506
$411$	0	0
$412$	−427.514	−0.0511216
$413$	−1359.23	−0.161946
$414$	0	0
$415$	−712.366	−0.0842619
$416$	23.5076	0.00277056
$417$	0	0
$418$	−3411.44	−0.399184
$419$	−4153.96	−0.484330	−0.242165	−	0.970235i	$-0.577858\pi$
$419$	−4153.96	−0.484330	−0.242165	+	0.970235i	$0.577858\pi$
$420$	0	0
$421$	4240.11	0.490856	0.245428	−	0.969415i	$-0.421072\pi$
$421$	4240.11	0.490856	0.245428	+	0.969415i	$0.421072\pi$
$422$	3181.52	0.367000
$423$	0	0
$424$	−10586.2	−1.21252
$425$	8650.31	0.987298
$426$	0	0
$427$	4095.05	0.464107
$428$	26.7494	0.00302099
$429$	0	0
$430$	8133.42	0.912159
$431$	−1012.17	−0.113119	−0.0565595	−	0.998399i	$-0.518013\pi$
$431$	−1012.17	−0.113119	−0.0565595	+	0.998399i	$0.518013\pi$
$432$	0	0
$433$	1604.50	0.178077	0.0890386	−	0.996028i	$-0.471621\pi$
$433$	1604.50	0.178077	0.0890386	+	0.996028i	$0.471621\pi$
$434$	739.688	0.0818114
$435$	0	0
$436$	1017.36	0.111749
$437$	18896.3	2.06850
$438$	0	0
$439$	10671.6	1.16020	0.580099	−	0.814546i	$-0.303014\pi$
$439$	10671.6	1.16020	0.580099	+	0.814546i	$0.303014\pi$
$440$	−2053.09	−0.222448
$441$	0	0
$442$	−49.1465	−0.00528882
$443$	−2193.74	−0.235277	−0.117638	−	0.993056i	$-0.537532\pi$
$443$	−2193.74	−0.235277	−0.117638	+	0.993056i	$0.537532\pi$
$444$	0	0
$445$	−2336.37	−0.248886
$446$	14096.6	1.49662
$447$	0	0
$448$	3637.07	0.383561
$449$	7070.36	0.743143	0.371571	−	0.928404i	$-0.378819\pi$
$449$	7070.36	0.743143	0.371571	+	0.928404i	$0.378819\pi$
$450$	0	0
$451$	−2152.58	−0.224747
$452$	−6963.12	−0.724596
$453$	0	0
$454$	2874.47	0.297149
$455$	−9.27844	−0.000956000 0
$456$	0	0
$457$	16245.8	1.66290	0.831451	−	0.555599i	$-0.187511\pi$
$457$	16245.8	1.66290	0.831451	+	0.555599i	$0.187511\pi$
$458$	−8273.98	−0.844143
$459$	0	0
$460$	3255.60	0.329985
$461$	5235.18	0.528908	0.264454	−	0.964398i	$-0.414808\pi$
$461$	5235.18	0.528908	0.264454	+	0.964398i	$0.414808\pi$
$462$	0	0
$463$	−11093.9	−1.11355	−0.556777	−	0.830662i	$-0.687962\pi$
$463$	−11093.9	−1.11355	−0.556777	+	0.830662i	$0.687962\pi$
$464$	−4961.93	−0.496448
$465$	0	0
$466$	−5084.00	−0.505390
$467$	7813.14	0.774195	0.387097	−	0.922039i	$-0.373478\pi$
$467$	7813.14	0.774195	0.387097	+	0.922039i	$0.373478\pi$
$468$	0	0
$469$	1091.33	0.107448
$470$	−2859.17	−0.280603
$471$	0	0
$472$	4764.06	0.464584
$473$	5372.91	0.522298
$474$	0	0
$475$	−9511.02	−0.918728
$476$	2894.47	0.278714
$477$	0	0
$478$	−4826.01	−0.461792
$479$	−5298.54	−0.505420	−0.252710	−	0.967542i	$-0.581322\pi$
$479$	−5298.54	−0.505420	−0.252710	+	0.967542i	$0.581322\pi$
$480$	0	0
$481$	28.1612	0.00266953
$482$	−10091.6	−0.953650
$483$	0	0
$484$	−388.265	−0.0364637
$485$	−7348.65	−0.688010
$486$	0	0
$487$	5580.89	0.519290	0.259645	−	0.965704i	$-0.416394\pi$
$487$	5580.89	0.519290	0.259645	+	0.965704i	$0.416394\pi$
$488$	−14353.0	−1.33141
$489$	0	0
$490$	−815.929	−0.0752243
$491$	−16718.8	−1.53668	−0.768341	−	0.640041i	$-0.778917\pi$
$491$	−16718.8	−1.53668	−0.768341	+	0.640041i	$0.778917\pi$
$492$	0	0
$493$	−22809.0	−2.08371
$494$	54.0366	0.00492150
$495$	0	0
$496$	−1353.32	−0.122512
$497$	2624.96	0.236912
$498$	0	0
$499$	19594.4	1.75784	0.878922	−	0.476965i	$-0.158263\pi$
$499$	19594.4	1.75784	0.878922	+	0.476965i	$0.158263\pi$
$500$	−4689.95	−0.419482
$501$	0	0
$502$	−10903.1	−0.969379
$503$	−9008.04	−0.798506	−0.399253	−	0.916841i	$-0.630731\pi$
$503$	−9008.04	−0.798506	−0.399253	+	0.916841i	$0.630731\pi$
$504$	0	0
$505$	−4396.95	−0.387449
$506$	−3211.21	−0.282126
$507$	0	0
$508$	−4175.42	−0.364674
$509$	3379.88	0.294324	0.147162	−	0.989112i	$-0.452986\pi$
$509$	3379.88	0.294324	0.147162	+	0.989112i	$0.452986\pi$
$510$	0	0
$511$	1472.93	0.127512
$512$	−9284.42	−0.801401
$513$	0	0
$514$	−5967.05	−0.512053
$515$	−1013.54	−0.0867222
$516$	0	0
$517$	−1888.76	−0.160672
$518$	2476.45	0.210056
$519$	0	0
$520$	32.5206	0.00274254
$521$	736.886	0.0619646	0.0309823	−	0.999520i	$-0.490136\pi$
$521$	736.886	0.0619646	0.0309823	+	0.999520i	$0.490136\pi$
$522$	0	0
$523$	9541.00	0.797703	0.398852	−	0.917015i	$-0.369409\pi$
$523$	9541.00	0.797703	0.398852	+	0.917015i	$0.369409\pi$
$524$	8093.32	0.674729
$525$	0	0
$526$	9256.43	0.767299
$527$	−6220.95	−0.514210
$528$	0	0
$529$	5620.20	0.461922
$530$	−7184.78	−0.588843
$531$	0	0
$532$	−3182.47	−0.259356
$533$	34.0964	0.00277088
$534$	0	0
$535$	63.4170	0.00512478
$536$	−3825.08	−0.308243
$537$	0	0
$538$	5244.64	0.420283
$539$	−539.000	−0.0430730
$540$	0	0
$541$	16721.0	1.32882	0.664411	−	0.747368i	$-0.268683\pi$
$541$	16721.0	1.32882	0.664411	+	0.747368i	$0.268683\pi$
$542$	−5388.32	−0.427026
$543$	0	0
$544$	−17385.7	−1.37023
$545$	2411.93	0.189570
$546$	0	0
$547$	8958.59	0.700259	0.350129	−	0.936701i	$-0.386138\pi$
$547$	8958.59	0.700259	0.350129	+	0.936701i	$0.386138\pi$
$548$	8261.70	0.644019
$549$	0	0
$550$	1616.29	0.125307
$551$	25078.5	1.93899
$552$	0	0
$553$	−49.0432	−0.00377130
$554$	12893.7	0.988809
$555$	0	0
$556$	8796.89	0.670992
$557$	14329.6	1.09006	0.545031	−	0.838416i	$-0.316518\pi$
$557$	14329.6	1.09006	0.545031	+	0.838416i	$0.316518\pi$
$558$	0	0
$559$	−85.1059	−0.00643935
$560$	1492.81	0.112648
$561$	0	0
$562$	4299.54	0.322713
$563$	−15791.2	−1.18209	−0.591047	−	0.806637i	$-0.701285\pi$
$563$	−15791.2	−1.18209	−0.591047	+	0.806637i	$0.701285\pi$
$564$	0	0
$565$	−16508.0	−1.22920
$566$	18635.2	1.38392
$567$	0	0
$568$	−9200.37	−0.679646
$569$	13996.4	1.03121	0.515605	−	0.856826i	$-0.327567\pi$
$569$	13996.4	1.03121	0.515605	+	0.856826i	$0.327567\pi$
$570$	0	0
$571$	−6642.54	−0.486833	−0.243417	−	0.969922i	$-0.578268\pi$
$571$	−6642.54	−0.486833	−0.243417	+	0.969922i	$0.578268\pi$
$572$	6.15005	0.000449557 0
$573$	0	0
$574$	2998.38	0.218031
$575$	−8952.78	−0.649316
$576$	0	0
$577$	−85.5448	−0.00617206	−0.00308603	−	0.999995i	$-0.500982\pi$
$577$	−85.5448	−0.00617206	−0.00308603	+	0.999995i	$0.500982\pi$
$578$	25593.8	1.84180
$579$	0	0
$580$	4320.72	0.309324
$581$	655.492	0.0468062
$582$	0	0
$583$	−4746.25	−0.337169
$584$	−5162.57	−0.365802
$585$	0	0
$586$	−11425.0	−0.805399
$587$	−11205.0	−0.787869	−0.393934	−	0.919139i	$-0.628886\pi$
$587$	−11205.0	−0.787869	−0.393934	+	0.919139i	$0.628886\pi$
$588$	0	0
$589$	6839.93	0.478497
$590$	3233.35	0.225618
$591$	0	0
$592$	−4530.87	−0.314556
$593$	−11664.6	−0.807773	−0.403886	−	0.914809i	$-0.632341\pi$
$593$	−11664.6	−0.807773	−0.403886	+	0.914809i	$0.632341\pi$
$594$	0	0
$595$	6862.15	0.472808
$596$	−3279.34	−0.225381
$597$	0	0
$598$	50.8649	0.00347830
$599$	20545.1	1.40142	0.700711	−	0.713445i	$-0.252866\pi$
$599$	20545.1	1.40142	0.700711	+	0.713445i	$0.252866\pi$
$600$	0	0
$601$	3885.01	0.263682	0.131841	−	0.991271i	$-0.457911\pi$
$601$	3885.01	0.263682	0.131841	+	0.991271i	$0.457911\pi$
$602$	−7484.06	−0.506690
$603$	0	0
$604$	−3261.42	−0.219710
$605$	−920.491	−0.0618567
$606$	0	0
$607$	8439.80	0.564351	0.282175	−	0.959363i	$-0.408944\pi$
$607$	8439.80	0.564351	0.282175	+	0.959363i	$0.408944\pi$
$608$	19115.6	1.27507
$609$	0	0
$610$	−9741.32	−0.646581
$611$	29.9175	0.00198091
$612$	0	0
$613$	17997.8	1.18584	0.592922	−	0.805260i	$-0.297974\pi$
$613$	17997.8	1.18584	0.592922	+	0.805260i	$0.297974\pi$
$614$	6319.94	0.415394
$615$	0	0
$616$	1889.17	0.123567
$617$	26336.6	1.71843	0.859214	−	0.511616i	$-0.170953\pi$
$617$	26336.6	1.71843	0.859214	+	0.511616i	$0.170953\pi$
$618$	0	0
$619$	27875.5	1.81003	0.905016	−	0.425376i	$-0.139858\pi$
$619$	27875.5	1.81003	0.905016	+	0.425376i	$0.139858\pi$
$620$	1178.44	0.0763341
$621$	0	0
$622$	12863.8	0.829249
$623$	2149.83	0.138252
$624$	0	0
$625$	−2727.83	−0.174581
$626$	−16817.5	−1.07374
$627$	0	0
$628$	5071.50	0.322253
$629$	−20827.5	−1.32027
$630$	0	0
$631$	8242.15	0.519992	0.259996	−	0.965610i	$-0.416279\pi$
$631$	8242.15	0.519992	0.259996	+	0.965610i	$0.416279\pi$
$632$	171.895	0.0108190
$633$	0	0
$634$	21105.9	1.32212
$635$	−9899.00	−0.618630
$636$	0	0
$637$	8.53766	0.000531043 0
$638$	−4261.80	−0.264462
$639$	0	0
$640$	−440.990	−0.0272370
$641$	7001.97	0.431453	0.215726	−	0.976454i	$-0.430788\pi$
$641$	7001.97	0.431453	0.215726	+	0.976454i	$0.430788\pi$
$642$	0	0
$643$	2473.25	0.151688	0.0758442	−	0.997120i	$-0.475835\pi$
$643$	2473.25	0.151688	0.0758442	+	0.997120i	$0.475835\pi$
$644$	−2995.68	−0.183302
$645$	0	0
$646$	−39964.4	−2.43402
$647$	−9153.21	−0.556182	−0.278091	−	0.960555i	$-0.589702\pi$
$647$	−9153.21	−0.556182	−0.278091	+	0.960555i	$0.589702\pi$
$648$	0	0
$649$	2135.94	0.129188
$650$	−25.6017	−0.00154489
$651$	0	0
$652$	−2465.51	−0.148093
$653$	−725.254	−0.0434630	−0.0217315	−	0.999764i	$-0.506918\pi$
$653$	−725.254	−0.0434630	−0.0217315	+	0.999764i	$0.506918\pi$
$654$	0	0
$655$	19187.5	1.14461
$656$	−5485.78	−0.326499
$657$	0	0
$658$	2630.89	0.155871
$659$	−14332.9	−0.847242	−0.423621	−	0.905840i	$-0.639241\pi$
$659$	−14332.9	−0.847242	−0.423621	+	0.905840i	$0.639241\pi$
$660$	0	0
$661$	26773.3	1.57543	0.787715	−	0.616040i	$-0.211264\pi$
$661$	26773.3	1.57543	0.787715	+	0.616040i	$0.211264\pi$
$662$	−4530.70	−0.265998
$663$	0	0
$664$	−2297.47	−0.134276
$665$	−7544.94	−0.439970
$666$	0	0
$667$	23606.6	1.37039
$668$	9923.95	0.574804
$669$	0	0
$670$	−2596.07	−0.149694
$671$	−6435.08	−0.370229
$672$	0	0
$673$	−25423.9	−1.45619	−0.728096	−	0.685475i	$-0.759595\pi$
$673$	−25423.9	−1.45619	−0.728096	+	0.685475i	$0.759595\pi$
$674$	16520.8	0.944149
$675$	0	0
$676$	7049.64	0.401095
$677$	−9287.48	−0.527248	−0.263624	−	0.964625i	$-0.584918\pi$
$677$	−9287.48	−0.527248	−0.263624	+	0.964625i	$0.584918\pi$
$678$	0	0
$679$	6761.94	0.382179
$680$	−24051.6	−1.35638
$681$	0	0
$682$	−1162.37	−0.0652629
$683$	31888.8	1.78652	0.893258	−	0.449544i	$-0.148414\pi$
$683$	31888.8	1.78652	0.893258	+	0.449544i	$0.148414\pi$
$684$	0	0
$685$	19586.7	1.09251
$686$	750.786	0.0417859
$687$	0	0
$688$	13692.7	0.758763
$689$	75.1796	0.00415692
$690$	0	0
$691$	−18650.3	−1.02676	−0.513379	−	0.858162i	$-0.671606\pi$
$691$	−18650.3	−1.02676	−0.513379	+	0.858162i	$0.671606\pi$
$692$	−4259.55	−0.233994
$693$	0	0
$694$	13906.4	0.760635
$695$	20855.5	1.13826
$696$	0	0
$697$	−25217.0	−1.37039
$698$	−2960.40	−0.160534
$699$	0	0
$700$	1507.80	0.0814138
$701$	−8566.01	−0.461532	−0.230766	−	0.973009i	$-0.574123\pi$
$701$	−8566.01	−0.461532	−0.230766	+	0.973009i	$0.574123\pi$
$702$	0	0
$703$	22899.8	1.22857
$704$	−5715.40	−0.305976
$705$	0	0
$706$	−9201.61	−0.490520
$707$	4045.90	0.215222
$708$	0	0
$709$	680.116	0.0360258	0.0180129	−	0.999838i	$-0.494266\pi$
$709$	680.116	0.0360258	0.0180129	+	0.999838i	$0.494266\pi$
$710$	−6244.25	−0.330060
$711$	0	0
$712$	−7535.08	−0.396614
$713$	6438.47	0.338180
$714$	0	0
$715$	14.5804	0.000762624 0
$716$	10080.0	0.526129
$717$	0	0
$718$	24298.0	1.26294
$719$	−12557.6	−0.651346	−0.325673	−	0.945482i	$-0.605591\pi$
$719$	−12557.6	−0.651346	−0.325673	+	0.945482i	$0.605591\pi$
$720$	0	0
$721$	932.621	0.0481728
$722$	28927.3	1.49109
$723$	0	0
$724$	11819.2	0.606707
$725$	−11881.8	−0.608661
$726$	0	0
$727$	22253.6	1.13527	0.567635	−	0.823281i	$-0.307859\pi$
$727$	22253.6	1.13527	0.567635	+	0.823281i	$0.307859\pi$
$728$	−29.9242	−0.00152344
$729$	0	0
$730$	−3503.81	−0.177646
$731$	62942.6	3.18470
$732$	0	0
$733$	6852.72	0.345308	0.172654	−	0.984982i	$-0.444766\pi$
$733$	6852.72	0.345308	0.172654	+	0.984982i	$0.444766\pi$
$734$	−10864.2	−0.546327
$735$	0	0
$736$	17993.6	0.901161
$737$	−1714.95	−0.0857139
$738$	0	0
$739$	−9133.20	−0.454628	−0.227314	−	0.973821i	$-0.572994\pi$
$739$	−9133.20	−0.454628	−0.227314	+	0.973821i	$0.572994\pi$
$740$	3945.36	0.195992
$741$	0	0
$742$	6611.16	0.327093
$743$	33458.7	1.65206	0.826031	−	0.563624i	$-0.190593\pi$
$743$	33458.7	1.65206	0.826031	+	0.563624i	$0.190593\pi$
$744$	0	0
$745$	−7774.60	−0.382335
$746$	−19442.8	−0.954225
$747$	0	0
$748$	−4548.45	−0.222337
$749$	−58.3539	−0.00284673
$750$	0	0
$751$	−37530.0	−1.82355	−0.911776	−	0.410688i	$-0.865289\pi$
$751$	−37530.0	−1.82355	−0.911776	+	0.410688i	$0.865289\pi$
$752$	−4813.44	−0.233415
$753$	0	0
$754$	67.5062	0.00326052
$755$	−7732.10	−0.372715
$756$	0	0
$757$	32174.8	1.54480	0.772399	−	0.635138i	$-0.219057\pi$
$757$	32174.8	1.54480	0.772399	+	0.635138i	$0.219057\pi$
$758$	−2667.88	−0.127839
$759$	0	0
$760$	26444.7	1.26217
$761$	−4457.43	−0.212328	−0.106164	−	0.994349i	$-0.533857\pi$
$761$	−4457.43	−0.212328	−0.106164	+	0.994349i	$0.533857\pi$
$762$	0	0
$763$	−2219.37	−0.105303
$764$	−9184.82	−0.434941
$765$	0	0
$766$	−246.498	−0.0116271
$767$	−33.8329	−0.00159274
$768$	0	0
$769$	−38329.0	−1.79737	−0.898687	−	0.438591i	$-0.855478\pi$
$769$	−38329.0	−1.79737	−0.898687	+	0.438591i	$0.855478\pi$
$770$	1282.17	0.0600082
$771$	0	0
$772$	−6494.32	−0.302766
$773$	9529.23	0.443393	0.221696	−	0.975116i	$-0.428841\pi$
$773$	9529.23	0.443393	0.221696	+	0.975116i	$0.428841\pi$
$774$	0	0
$775$	−3240.65	−0.150203
$776$	−23700.3	−1.09638
$777$	0	0
$778$	−27703.0	−1.27661
$779$	27726.1	1.27521
$780$	0	0
$781$	−4124.93	−0.188991
$782$	−37618.7	−1.72026
$783$	0	0
$784$	−1373.62	−0.0625740
$785$	12023.4	0.546667
$786$	0	0
$787$	29705.1	1.34545	0.672726	−	0.739892i	$-0.265123\pi$
$787$	29705.1	1.34545	0.672726	+	0.739892i	$0.265123\pi$
$788$	−15299.0	−0.691629
$789$	0	0
$790$	116.664	0.00525408
$791$	15190.0	0.682801
$792$	0	0
$793$	101.931	0.00456451
$794$	8114.48	0.362685
$795$	0	0
$796$	−11380.7	−0.506758
$797$	−8596.99	−0.382084	−0.191042	−	0.981582i	$-0.561187\pi$
$797$	−8596.99	−0.382084	−0.191042	+	0.981582i	$0.561187\pi$
$798$	0	0
$799$	−22126.4	−0.979695
$800$	−9056.69	−0.400253
$801$	0	0
$802$	−12413.3	−0.546543
$803$	−2314.61	−0.101719
$804$	0	0
$805$	−7102.09	−0.310951
$806$	18.4117	0.000804620 0
$807$	0	0
$808$	−14180.7	−0.617422
$809$	−16630.2	−0.722727	−0.361364	−	0.932425i	$-0.617689\pi$
$809$	−16630.2	−0.722727	−0.361364	+	0.932425i	$0.617689\pi$
$810$	0	0
$811$	18584.2	0.804660	0.402330	−	0.915495i	$-0.368201\pi$
$811$	18584.2	0.804660	0.402330	+	0.915495i	$0.368201\pi$
$812$	−3975.76	−0.171825
$813$	0	0
$814$	−3891.56	−0.167567
$815$	−5845.18	−0.251224
$816$	0	0
$817$	−69205.5	−2.96352
$818$	570.127	0.0243692
$819$	0	0
$820$	4776.87	0.203434
$821$	4088.88	0.173816	0.0869079	−	0.996216i	$-0.472301\pi$
$821$	4088.88	0.173816	0.0869079	+	0.996216i	$0.472301\pi$
$822$	0	0
$823$	−12830.0	−0.543410	−0.271705	−	0.962381i	$-0.587587\pi$
$823$	−12830.0	−0.543410	−0.271705	+	0.962381i	$0.587587\pi$
$824$	−3268.80	−0.138197
$825$	0	0
$826$	−2975.20	−0.125327
$827$	13768.7	0.578942	0.289471	−	0.957187i	$-0.406521\pi$
$827$	13768.7	0.578942	0.289471	+	0.957187i	$0.406521\pi$
$828$	0	0
$829$	−10907.8	−0.456987	−0.228494	−	0.973545i	$-0.573380\pi$
$829$	−10907.8	−0.456987	−0.228494	+	0.973545i	$0.573380\pi$
$830$	−1559.28	−0.0652091
$831$	0	0
$832$	90.5308	0.00377234
$833$	−6314.28	−0.262637
$834$	0	0
$835$	23527.5	0.975093
$836$	5001.03	0.206895
$837$	0	0
$838$	−9092.53	−0.374817
$839$	27174.8	1.11821	0.559105	−	0.829097i	$-0.311145\pi$
$839$	27174.8	1.11821	0.559105	+	0.829097i	$0.311145\pi$
$840$	0	0
$841$	6940.81	0.284588
$842$	9281.09	0.379866
$843$	0	0
$844$	−4663.97	−0.190214
$845$	16713.1	0.680414
$846$	0	0
$847$	847.000	0.0343604
$848$	−12095.7	−0.489819
$849$	0	0
$850$	18934.5	0.764056
$851$	21555.7	0.868298
$852$	0	0
$853$	−33975.1	−1.36376	−0.681878	−	0.731466i	$-0.738837\pi$
$853$	−33975.1	−1.36376	−0.681878	+	0.731466i	$0.738837\pi$
$854$	8963.58	0.359166
$855$	0	0
$856$	204.528	0.00816662
$857$	−20423.7	−0.814073	−0.407037	−	0.913412i	$-0.633438\pi$
$857$	−20423.7	−0.814073	−0.407037	+	0.913412i	$0.633438\pi$
$858$	0	0
$859$	−36704.2	−1.45789	−0.728947	−	0.684570i	$-0.759990\pi$
$859$	−36704.2	−1.45789	−0.728947	+	0.684570i	$0.759990\pi$
$860$	−11923.2	−0.472766
$861$	0	0
$862$	−2215.51	−0.0875413
$863$	−26222.8	−1.03434	−0.517170	−	0.855883i	$-0.673015\pi$
$863$	−26222.8	−1.03434	−0.517170	+	0.855883i	$0.673015\pi$
$864$	0	0
$865$	−10098.5	−0.396945
$866$	3512.06	0.137811
$867$	0	0
$868$	−1084.35	−0.0424024
$869$	77.0680	0.00300846
$870$	0	0
$871$	27.1645	0.00105676
$872$	7778.79	0.302091
$873$	0	0
$874$	41361.8	1.60078
$875$	10231.1	0.395285
$876$	0	0
$877$	44950.8	1.73076	0.865382	−	0.501113i	$-0.167076\pi$
$877$	44950.8	1.73076	0.865382	+	0.501113i	$0.167076\pi$
$878$	23358.8	0.897861
$879$	0	0
$880$	−2345.84	−0.0898617
$881$	−3896.87	−0.149023	−0.0745113	−	0.997220i	$-0.523740\pi$
$881$	−3896.87	−0.149023	−0.0745113	+	0.997220i	$0.523740\pi$
$882$	0	0
$883$	6257.41	0.238481	0.119240	−	0.992865i	$-0.461954\pi$
$883$	6257.41	0.238481	0.119240	+	0.992865i	$0.461954\pi$
$884$	72.0466	0.00274117
$885$	0	0
$886$	−4801.83	−0.182077
$887$	−40626.2	−1.53787	−0.768937	−	0.639324i	$-0.779214\pi$
$887$	−40626.2	−1.53787	−0.768937	+	0.639324i	$0.779214\pi$
$888$	0	0
$889$	9108.68	0.343639
$890$	−5114.02	−0.192610
$891$	0	0
$892$	−20665.0	−0.775691
$893$	24328.0	0.911653
$894$	0	0
$895$	23897.5	0.892521
$896$	405.782	0.0151297
$897$	0	0
$898$	15476.2	0.575108
$899$	8544.91	0.317006
$900$	0	0
$901$	−55601.4	−2.05588
$902$	−4711.73	−0.173929
$903$	0	0
$904$	−53240.5	−1.95880
$905$	28020.6	1.02921
$906$	0	0
$907$	42366.9	1.55101	0.775506	−	0.631340i	$-0.217495\pi$
$907$	42366.9	1.55101	0.775506	+	0.631340i	$0.217495\pi$
$908$	−4213.85	−0.154010
$909$	0	0
$910$	−20.3094	−0.000739835 0
$911$	−26231.4	−0.953992	−0.476996	−	0.878906i	$-0.658274\pi$
$911$	−26231.4	−0.953992	−0.476996	+	0.878906i	$0.658274\pi$
$912$	0	0
$913$	−1030.06	−0.0373384
$914$	35560.1	1.28690
$915$	0	0
$916$	12129.3	0.437514
$917$	−17655.6	−0.635810
$918$	0	0
$919$	2542.24	0.0912522	0.0456261	−	0.998959i	$-0.485472\pi$
$919$	2542.24	0.0912522	0.0456261	+	0.998959i	$0.485472\pi$
$920$	24892.5	0.892047
$921$	0	0
$922$	11459.2	0.409314
$923$	65.3381	0.00233004
$924$	0	0
$925$	−10849.6	−0.385656
$926$	−24283.2	−0.861764
$927$	0	0
$928$	23880.5	0.844739
$929$	20097.8	0.709782	0.354891	−	0.934908i	$-0.384518\pi$
$929$	20097.8	0.709782	0.354891	+	0.934908i	$0.384518\pi$
$930$	0	0
$931$	6942.56	0.244396
$932$	7452.92	0.261941
$933$	0	0
$934$	17102.0	0.599138
$935$	−10783.4	−0.377170
$936$	0	0
$937$	−202.788	−0.00707023	−0.00353511	−	0.999994i	$-0.501125\pi$
$937$	−202.788	−0.00707023	−0.00353511	+	0.999994i	$0.501125\pi$
$938$	2388.80	0.0831525
$939$	0	0
$940$	4191.41	0.145435
$941$	−12419.1	−0.430235	−0.215118	−	0.976588i	$-0.569013\pi$
$941$	−12419.1	−0.430235	−0.215118	+	0.976588i	$0.569013\pi$
$942$	0	0
$943$	26098.8	0.901265
$944$	5443.37	0.187677
$945$	0	0
$946$	11760.7	0.404199
$947$	−369.282	−0.0126717	−0.00633583	−	0.999980i	$-0.502017\pi$
$947$	−369.282	−0.0126717	−0.00633583	+	0.999980i	$0.502017\pi$
$948$	0	0
$949$	36.6629	0.00125409
$950$	−20818.5	−0.710990
$951$	0	0
$952$	22131.3	0.753445
$953$	−25994.1	−0.883558	−0.441779	−	0.897124i	$-0.645652\pi$
$953$	−25994.1	−0.883558	−0.441779	+	0.897124i	$0.645652\pi$
$954$	0	0
$955$	−21775.2	−0.737830
$956$	7074.72	0.239344
$957$	0	0
$958$	−11597.9	−0.391138
$959$	−18022.9	−0.606871
$960$	0	0
$961$	−27460.5	−0.921770
$962$	61.6416	0.00206591
$963$	0	0
$964$	14793.8	0.494271
$965$	−15396.6	−0.513610
$966$	0	0
$967$	−53830.0	−1.79013	−0.895065	−	0.445937i	$-0.852871\pi$
$967$	−53830.0	−1.79013	−0.895065	+	0.445937i	$0.852871\pi$
$968$	−2968.70	−0.0985720
$969$	0	0
$970$	−16085.3	−0.532441
$971$	1349.54	0.0446022	0.0223011	−	0.999751i	$-0.492901\pi$
$971$	1349.54	0.0446022	0.0223011	+	0.999751i	$0.492901\pi$
$972$	0	0
$973$	−19190.4	−0.632288
$974$	12215.9	0.401872
$975$	0	0
$976$	−16399.6	−0.537847
$977$	−6611.71	−0.216507	−0.108253	−	0.994123i	$-0.534526\pi$
$977$	−6611.71	−0.216507	−0.108253	+	0.994123i	$0.534526\pi$
$978$	0	0
$979$	−3378.31	−0.110287
$980$	1196.12	0.0389883
$981$	0	0
$982$	−36595.5	−1.18922
$983$	−40595.9	−1.31720	−0.658601	−	0.752493i	$-0.728851\pi$
$983$	−40595.9	−1.31720	−0.658601	+	0.752493i	$0.728851\pi$
$984$	0	0
$985$	−36270.5	−1.17327
$986$	−49926.2	−1.61255
$987$	0	0
$988$	−79.2153	−0.00255078
$989$	−65143.5	−2.09448
$990$	0	0
$991$	−48877.7	−1.56675	−0.783376	−	0.621548i	$-0.786504\pi$
$991$	−48877.7	−1.56675	−0.783376	+	0.621548i	$0.786504\pi$
$992$	6513.19	0.208462
$993$	0	0
$994$	5745.71	0.183343
$995$	−26981.2	−0.859660
$996$	0	0
$997$	13127.4	0.417000	0.208500	−	0.978022i	$-0.433142\pi$
$997$	13127.4	0.417000	0.208500	+	0.978022i	$0.433142\pi$
$998$	42889.7	1.36037
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.o.1.4		5
3.2	odd	2		77.4.a.e.1.2	✓	5
12.11	even	2		1232.4.a.y.1.2		5
15.14	odd	2		1925.4.a.r.1.4		5
21.20	even	2		539.4.a.h.1.2		5
33.32	even	2		847.4.a.f.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.e.1.2	✓	5	3.2	odd	2
539.4.a.h.1.2		5	21.20	even	2
693.4.a.o.1.4		5	1.1	even	1	trivial
847.4.a.f.1.4		5	33.32	even	2
1232.4.a.y.1.2		5	12.11	even	2
1925.4.a.r.1.4		5	15.14	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

\(f(q)\)	\(=\)	\(q+2.18888 q^{2} -3.20880 q^{4} -7.60736 q^{5} +7.00000 q^{7} -24.5347 q^{8} +O(q^{10})\)	\(q+2.18888 q^{2} -3.20880 q^{4} -7.60736 q^{5} +7.00000 q^{7} -24.5347 q^{8} -16.6516 q^{10} -11.0000 q^{11} +0.174238 q^{13} +15.3222 q^{14} -28.0332 q^{16} -128.863 q^{17} +141.685 q^{19} +24.4105 q^{20} -24.0777 q^{22} +133.369 q^{23} -67.1280 q^{25} +0.381386 q^{26} -22.4616 q^{28} +177.002 q^{29} +48.2757 q^{31} +134.917 q^{32} -282.065 q^{34} -53.2515 q^{35} +161.625 q^{37} +310.131 q^{38} +186.645 q^{40} +195.689 q^{41} -488.447 q^{43} +35.2968 q^{44} +291.928 q^{46} +171.705 q^{47} +49.0000 q^{49} -146.935 q^{50} -0.559095 q^{52} +431.477 q^{53} +83.6810 q^{55} -171.743 q^{56} +387.437 q^{58} -194.176 q^{59} +585.008 q^{61} +105.670 q^{62} +519.582 q^{64} -1.32549 q^{65} +155.905 q^{67} +413.496 q^{68} -116.561 q^{70} +374.994 q^{71} +210.419 q^{73} +353.778 q^{74} -454.639 q^{76} -77.0000 q^{77} -7.00618 q^{79} +213.258 q^{80} +428.339 q^{82} +93.6417 q^{83} +980.307 q^{85} -1069.15 q^{86} +269.882 q^{88} +307.119 q^{89} +1.21967 q^{91} -427.954 q^{92} +375.842 q^{94} -1077.85 q^{95} +965.991 q^{97} +107.255 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{2} + 45 q^{4} + 24 q^{5} + 35 q^{7} - 57 q^{8}+O(q^{10}) \) 5 * q - q^2 + 45 * q^4 + 24 * q^5 + 35 * q^7 - 57 * q^8	\( 5 q - q^{2} + 45 q^{4} + 24 q^{5} + 35 q^{7} - 57 q^{8} - 10 q^{10} - 55 q^{11} - 50 q^{13} - 7 q^{14} + 433 q^{16} - 222 q^{17} + 160 q^{19} + 430 q^{20} + 11 q^{22} - 54 q^{23} + 125 q^{25} + 1026 q^{26} + 315 q^{28} - 14 q^{29} - 34 q^{31} + 583 q^{32} - 750 q^{34} + 168 q^{35} + 1044 q^{37} + 156 q^{38} + 1158 q^{40} + 114 q^{41} + 672 q^{43} - 495 q^{44} - 1224 q^{46} + 292 q^{47} + 245 q^{49} - 1143 q^{50} - 914 q^{52} + 710 q^{53} - 264 q^{55} - 399 q^{56} - 810 q^{58} - 270 q^{59} + 138 q^{61} + 480 q^{62} + 3001 q^{64} + 196 q^{65} + 1942 q^{67} - 3130 q^{68} - 70 q^{70} + 278 q^{71} - 338 q^{73} - 462 q^{74} - 1332 q^{76} - 385 q^{77} + 576 q^{79} + 1602 q^{80} + 1386 q^{82} - 1644 q^{83} + 360 q^{85} - 2020 q^{86} + 627 q^{88} + 3656 q^{89} - 350 q^{91} + 748 q^{92} + 976 q^{94} + 1276 q^{95} + 692 q^{97} - 49 q^{98}+O(q^{100}) \) 5 * q - q^2 + 45 * q^4 + 24 * q^5 + 35 * q^7 - 57 * q^8 - 10 * q^10 - 55 * q^11 - 50 * q^13 - 7 * q^14 + 433 * q^16 - 222 * q^17 + 160 * q^19 + 430 * q^20 + 11 * q^22 - 54 * q^23 + 125 * q^25 + 1026 * q^26 + 315 * q^28 - 14 * q^29 - 34 * q^31 + 583 * q^32 - 750 * q^34 + 168 * q^35 + 1044 * q^37 + 156 * q^38 + 1158 * q^40 + 114 * q^41 + 672 * q^43 - 495 * q^44 - 1224 * q^46 + 292 * q^47 + 245 * q^49 - 1143 * q^50 - 914 * q^52 + 710 * q^53 - 264 * q^55 - 399 * q^56 - 810 * q^58 - 270 * q^59 + 138 * q^61 + 480 * q^62 + 3001 * q^64 + 196 * q^65 + 1942 * q^67 - 3130 * q^68 - 70 * q^70 + 278 * q^71 - 338 * q^73 - 462 * q^74 - 1332 * q^76 - 385 * q^77 + 576 * q^79 + 1602 * q^80 + 1386 * q^82 - 1644 * q^83 + 360 * q^85 - 2020 * q^86 + 627 * q^88 + 3656 * q^89 - 350 * q^91 + 748 * q^92 + 976 * q^94 + 1276 * q^95 + 692 * q^97 - 49 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more