LMFDB - Embedded newform 1323.4.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,4,Mod(1,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1323.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1323.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:	$78.0595269376$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 189)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.64575$ of defining polynomial
Character	$\chi$	$=$	1323.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.64575 q^{2} -1.00000 q^{4} +2.64575 q^{5} -23.8118 q^{8} +7.00000 q^{10} -18.5203 q^{11} +26.0000 q^{13} -55.0000 q^{16} +63.4980 q^{17} +35.0000 q^{19} -2.64575 q^{20} -49.0000 q^{22} +103.184 q^{23} -118.000 q^{25} +68.7895 q^{26} +5.29150 q^{29} +75.0000 q^{31} +44.9778 q^{32} +168.000 q^{34} -111.000 q^{37} +92.6013 q^{38} -63.0000 q^{40} -478.881 q^{41} -328.000 q^{43} +18.5203 q^{44} +273.000 q^{46} -380.988 q^{47} -312.199 q^{50} -26.0000 q^{52} -126.996 q^{53} -49.0000 q^{55} +14.0000 q^{58} -126.996 q^{59} +152.000 q^{61} +198.431 q^{62} +559.000 q^{64} +68.7895 q^{65} +202.000 q^{67} -63.4980 q^{68} -1055.65 q^{71} -672.000 q^{73} -293.678 q^{74} -35.0000 q^{76} -988.000 q^{79} -145.516 q^{80} -1267.00 q^{82} +142.871 q^{83} +168.000 q^{85} -867.806 q^{86} +441.000 q^{88} -965.699 q^{89} -103.184 q^{92} -1008.00 q^{94} +92.6013 q^{95} +492.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 14 q^{10} + 52 q^{13} - 110 q^{16} + 70 q^{19} - 98 q^{22} - 236 q^{25} + 150 q^{31} + 336 q^{34} - 222 q^{37} - 126 q^{40} - 656 q^{43} + 546 q^{46} - 52 q^{52} - 98 q^{55} + 28 q^{58}+ \cdots + 984 q^{97}+O(q^{100}) $ 2 * q - 2 * q^4 + 14 * q^10 + 52 * q^13 - 110 * q^16 + 70 * q^19 - 98 * q^22 - 236 * q^25 + 150 * q^31 + 336 * q^34 - 222 * q^37 - 126 * q^40 - 656 * q^43 + 546 * q^46 - 52 * q^52 - 98 * q^55 + 28 * q^58 + 304 * q^61 + 1118 * q^64 + 404 * q^67 - 1344 * q^73 - 70 * q^76 - 1976 * q^79 - 2534 * q^82 + 336 * q^85 + 882 * q^88 - 2016 * q^94 + 984 * q^97

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 $

$2$	2.64575	0.935414	0.467707	−	0.883883i	$-0.345080\pi$
$2$	2.64575	0.935414	0.467707	+	0.883883i	$0.345080\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.125000
$5$	2.64575	0.236643	0.118322	−	0.992975i	$-0.462249\pi$
$5$	2.64575	0.236643	0.118322	+	0.992975i	$0.462249\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−23.8118	−1.05234
$9$	0	0
$10$	7.00000	0.221359
$11$	−18.5203	−0.507643	−0.253821	−	0.967251i	$-0.581688\pi$
$11$	−18.5203	−0.507643	−0.253821	+	0.967251i	$0.581688\pi$
$12$	0	0
$13$	26.0000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	26.0000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	0	0
$16$	−55.0000	−0.859375
$17$	63.4980	0.905914	0.452957	−	0.891532i	$-0.350369\pi$
$17$	63.4980	0.905914	0.452957	+	0.891532i	$0.350369\pi$
$18$	0	0
$19$	35.0000	0.422608	0.211304	−	0.977420i	$-0.432229\pi$
$19$	35.0000	0.422608	0.211304	+	0.977420i	$0.432229\pi$
$20$	−2.64575	−0.0295804
$21$	0	0
$22$	−49.0000	−0.474856
$23$	103.184	0.935453	0.467726	−	0.883873i	$-0.345073\pi$
$23$	103.184	0.935453	0.467726	+	0.883873i	$0.345073\pi$
$24$	0	0
$25$	−118.000	−0.944000
$26$	68.7895	0.518875
$27$	0	0
$28$	0	0
$29$	5.29150	0.0338830	0.0169415	−	0.999856i	$-0.494607\pi$
$29$	5.29150	0.0338830	0.0169415	+	0.999856i	$0.494607\pi$
$30$	0	0
$31$	75.0000	0.434529	0.217264	−	0.976113i	$-0.430287\pi$
$31$	75.0000	0.434529	0.217264	+	0.976113i	$0.430287\pi$
$32$	44.9778	0.248469
$33$	0	0
$34$	168.000	0.847405
$35$	0	0
$36$	0	0
$37$	−111.000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−111.000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	92.6013	0.395314
$39$	0	0
$40$	−63.0000	−0.249029
$41$	−478.881	−1.82411	−0.912057	−	0.410064i	$-0.865506\pi$
$41$	−478.881	−1.82411	−0.912057	+	0.410064i	$0.865506\pi$
$42$	0	0
$43$	−328.000	−1.16324	−0.581622	−	0.813459i	$-0.697582\pi$
$43$	−328.000	−1.16324	−0.581622	+	0.813459i	$0.697582\pi$
$44$	18.5203	0.0634553
$45$	0	0
$46$	273.000	0.875036
$47$	−380.988	−1.18240	−0.591200	−	0.806525i	$-0.701346\pi$
$47$	−380.988	−1.18240	−0.591200	+	0.806525i	$0.701346\pi$
$48$	0	0
$49$	0	0
$50$	−312.199	−0.883031
$51$	0	0
$52$	−26.0000	−0.0693375
$53$	−126.996	−0.329137	−0.164568	−	0.986366i	$-0.552623\pi$
$53$	−126.996	−0.329137	−0.164568	+	0.986366i	$0.552623\pi$
$54$	0	0
$55$	−49.0000	−0.120130
$56$	0	0
$57$	0	0
$58$	14.0000	0.0316947
$59$	−126.996	−0.280228	−0.140114	−	0.990135i	$-0.544747\pi$
$59$	−126.996	−0.280228	−0.140114	+	0.990135i	$0.544747\pi$
$60$	0	0
$61$	152.000	0.319043	0.159521	−	0.987194i	$-0.449005\pi$
$61$	152.000	0.319043	0.159521	+	0.987194i	$0.449005\pi$
$62$	198.431	0.406465
$63$	0	0
$64$	559.000	1.09180
$65$	68.7895	0.131266
$66$	0	0
$67$	202.000	0.368332	0.184166	−	0.982895i	$-0.441042\pi$
$67$	202.000	0.368332	0.184166	+	0.982895i	$0.441042\pi$
$68$	−63.4980	−0.113239
$69$	0	0
$70$	0	0
$71$	−1055.65	−1.76455	−0.882276	−	0.470733i	$-0.843990\pi$
$71$	−1055.65	−1.76455	−0.882276	+	0.470733i	$0.843990\pi$
$72$	0	0
$73$	−672.000	−1.07742	−0.538710	−	0.842491i	$-0.681088\pi$
$73$	−672.000	−1.07742	−0.538710	+	0.842491i	$0.681088\pi$
$74$	−293.678	−0.461344
$75$	0	0
$76$	−35.0000	−0.0528260
$77$	0	0
$78$	0	0
$79$	−988.000	−1.40707	−0.703536	−	0.710660i	$-0.748397\pi$
$79$	−988.000	−1.40707	−0.703536	+	0.710660i	$0.748397\pi$
$80$	−145.516	−0.203365
$81$	0	0
$82$	−1267.00	−1.70630
$83$	142.871	0.188941	0.0944704	−	0.995528i	$-0.469884\pi$
$83$	142.871	0.188941	0.0944704	+	0.995528i	$0.469884\pi$
$84$	0	0
$85$	168.000	0.214378
$86$	−867.806	−1.08812
$87$	0	0
$88$	441.000	0.534213
$89$	−965.699	−1.15016	−0.575078	−	0.818098i	$-0.695028\pi$
$89$	−965.699	−1.15016	−0.575078	+	0.818098i	$0.695028\pi$
$90$	0	0
$91$	0	0
$92$	−103.184	−0.116932
$93$	0	0
$94$	−1008.00	−1.10603
$95$	92.6013	0.100007
$96$	0	0
$97$	492.000	0.515000	0.257500	−	0.966278i	$-0.417101\pi$
$97$	492.000	0.515000	0.257500	+	0.966278i	$0.417101\pi$
$98$	0	0
$99$	0	0
$100$	118.000	0.118000
$101$	1703.86	1.67862	0.839311	−	0.543652i	$-0.182959\pi$
$101$	1703.86	1.67862	0.839311	+	0.543652i	$0.182959\pi$
$102$	0	0
$103$	−845.000	−0.808353	−0.404176	−	0.914681i	$-0.632442\pi$
$103$	−845.000	−0.808353	−0.404176	+	0.914681i	$0.632442\pi$
$104$	−619.106	−0.583734
$105$	0	0
$106$	−336.000	−0.307879
$107$	1370.50	1.23824	0.619118	−	0.785298i	$-0.287490\pi$
$107$	1370.50	1.23824	0.619118	+	0.785298i	$0.287490\pi$
$108$	0	0
$109$	477.000	0.419159	0.209579	−	0.977792i	$-0.432791\pi$
$109$	477.000	0.419159	0.209579	+	0.977792i	$0.432791\pi$
$110$	−129.642	−0.112371
$111$	0	0
$112$	0	0
$113$	−1709.16	−1.42287	−0.711433	−	0.702754i	$-0.751953\pi$
$113$	−1709.16	−1.42287	−0.711433	+	0.702754i	$0.751953\pi$
$114$	0	0
$115$	273.000	0.221369
$116$	−5.29150	−0.00423538
$117$	0	0
$118$	−336.000	−0.262130
$119$	0	0
$120$	0	0
$121$	−988.000	−0.742299
$122$	402.154	0.298437
$123$	0	0
$124$	−75.0000	−0.0543161
$125$	−642.918	−0.460034
$126$	0	0
$127$	−1590.00	−1.11094	−0.555471	−	0.831536i	$-0.687462\pi$
$127$	−1590.00	−1.11094	−0.555471	+	0.831536i	$0.687462\pi$
$128$	1119.15	0.772813
$129$	0	0
$130$	182.000	0.122788
$131$	−656.146	−0.437617	−0.218808	−	0.975768i	$-0.570217\pi$
$131$	−656.146	−0.437617	−0.218808	+	0.975768i	$0.570217\pi$
$132$	0	0
$133$	0	0
$134$	534.442	0.344543
$135$	0	0
$136$	−1512.00	−0.953330
$137$	428.612	0.267290	0.133645	−	0.991029i	$-0.457332\pi$
$137$	428.612	0.267290	0.133645	+	0.991029i	$0.457332\pi$
$138$	0	0
$139$	2340.00	1.42789	0.713943	−	0.700204i	$-0.246907\pi$
$139$	2340.00	1.42789	0.713943	+	0.700204i	$0.246907\pi$
$140$	0	0
$141$	0	0
$142$	−2793.00	−1.65059
$143$	−481.527	−0.281589
$144$	0	0
$145$	14.0000	0.00801818
$146$	−1777.94	−1.00783
$147$	0	0
$148$	111.000	0.0616496
$149$	1889.07	1.03865	0.519323	−	0.854578i	$-0.326184\pi$
$149$	1889.07	1.03865	0.519323	+	0.854578i	$0.326184\pi$
$150$	0	0
$151$	−370.000	−0.199405	−0.0997026	−	0.995017i	$-0.531789\pi$
$151$	−370.000	−0.199405	−0.0997026	+	0.995017i	$0.531789\pi$
$152$	−833.412	−0.444728
$153$	0	0
$154$	0	0
$155$	198.431	0.102828
$156$	0	0
$157$	−490.000	−0.249084	−0.124542	−	0.992214i	$-0.539746\pi$
$157$	−490.000	−0.249084	−0.124542	+	0.992214i	$0.539746\pi$
$158$	−2614.00	−1.31620
$159$	0	0
$160$	119.000	0.0587986
$161$	0	0
$162$	0	0
$163$	−3746.00	−1.80006	−0.900029	−	0.435831i	$-0.856455\pi$
$163$	−3746.00	−1.80006	−0.900029	+	0.435831i	$0.856455\pi$
$164$	478.881	0.228014
$165$	0	0
$166$	378.000	0.176738
$167$	1550.41	0.718409	0.359205	−	0.933259i	$-0.383048\pi$
$167$	1550.41	0.718409	0.359205	+	0.933259i	$0.383048\pi$
$168$	0	0
$169$	−1521.00	−0.692308
$170$	444.486	0.200533
$171$	0	0
$172$	328.000	0.145406
$173$	2172.16	0.954604	0.477302	−	0.878739i	$-0.341615\pi$
$173$	2172.16	0.954604	0.477302	+	0.878739i	$0.341615\pi$
$174$	0	0
$175$	0	0
$176$	1018.61	0.436255
$177$	0	0
$178$	−2555.00	−1.07587
$179$	−989.511	−0.413182	−0.206591	−	0.978427i	$-0.566237\pi$
$179$	−989.511	−0.413182	−0.206591	+	0.978427i	$0.566237\pi$
$180$	0	0
$181$	2198.00	0.902630	0.451315	−	0.892365i	$-0.350955\pi$
$181$	2198.00	0.902630	0.451315	+	0.892365i	$0.350955\pi$
$182$	0	0
$183$	0	0
$184$	−2457.00	−0.984415
$185$	−293.678	−0.116712
$186$	0	0
$187$	−1176.00	−0.459880
$188$	380.988	0.147800
$189$	0	0
$190$	245.000	0.0935483
$191$	−4431.63	−1.67886	−0.839429	−	0.543470i	$-0.817110\pi$
$191$	−4431.63	−1.67886	−0.839429	+	0.543470i	$0.817110\pi$
$192$	0	0
$193$	386.000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	386.000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	1301.71	0.481739
$195$	0	0
$196$	0	0
$197$	−3455.35	−1.24966	−0.624831	−	0.780760i	$-0.714832\pi$
$197$	−3455.35	−1.24966	−0.624831	+	0.780760i	$0.714832\pi$
$198$	0	0
$199$	1605.00	0.571736	0.285868	−	0.958269i	$-0.407718\pi$
$199$	1605.00	0.571736	0.285868	+	0.958269i	$0.407718\pi$
$200$	2809.79	0.993410
$201$	0	0
$202$	4508.00	1.57021
$203$	0	0
$204$	0	0
$205$	−1267.00	−0.431664
$206$	−2235.66	−0.756145
$207$	0	0
$208$	−1430.00	−0.476695
$209$	−648.209	−0.214534
$210$	0	0
$211$	−398.000	−0.129855	−0.0649276	−	0.997890i	$-0.520682\pi$
$211$	−398.000	−0.129855	−0.0649276	+	0.997890i	$0.520682\pi$
$212$	126.996	0.0411421
$213$	0	0
$214$	3626.00	1.15826
$215$	−867.806	−0.275274
$216$	0	0
$217$	0	0
$218$	1262.02	0.392087
$219$	0	0
$220$	49.0000	0.0150163
$221$	1650.95	0.502511
$222$	0	0
$223$	−3577.00	−1.07414	−0.537071	−	0.843537i	$-0.680469\pi$
$223$	−3577.00	−1.07414	−0.537071	+	0.843537i	$0.680469\pi$
$224$	0	0
$225$	0	0
$226$	−4522.00	−1.33097
$227$	3540.02	1.03506	0.517531	−	0.855664i	$-0.326851\pi$
$227$	3540.02	1.03506	0.517531	+	0.855664i	$0.326851\pi$
$228$	0	0
$229$	−5924.00	−1.70947	−0.854736	−	0.519064i	$-0.826281\pi$
$229$	−5924.00	−1.70947	−0.854736	+	0.519064i	$0.826281\pi$
$230$	722.290	0.207071
$231$	0	0
$232$	−126.000	−0.0356565
$233$	−2460.55	−0.691828	−0.345914	−	0.938266i	$-0.612431\pi$
$233$	−2460.55	−0.691828	−0.345914	+	0.938266i	$0.612431\pi$
$234$	0	0
$235$	−1008.00	−0.279807
$236$	126.996	0.0350286
$237$	0	0
$238$	0	0
$239$	5190.96	1.40492	0.702459	−	0.711724i	$-0.252085\pi$
$239$	5190.96	1.40492	0.702459	+	0.711724i	$0.252085\pi$
$240$	0	0
$241$	−6428.00	−1.71811	−0.859054	−	0.511885i	$-0.828947\pi$
$241$	−6428.00	−1.71811	−0.859054	+	0.511885i	$0.828947\pi$
$242$	−2614.00	−0.694357
$243$	0	0
$244$	−152.000	−0.0398803
$245$	0	0
$246$	0	0
$247$	910.000	0.234421
$248$	−1785.88	−0.457273
$249$	0	0
$250$	−1701.00	−0.430323
$251$	5915.90	1.48768	0.743841	−	0.668356i	$-0.233002\pi$
$251$	5915.90	1.48768	0.743841	+	0.668356i	$0.233002\pi$
$252$	0	0
$253$	−1911.00	−0.474876
$254$	−4206.74	−1.03919
$255$	0	0
$256$	−1511.00	−0.368896
$257$	−3098.17	−0.751980	−0.375990	−	0.926624i	$-0.622697\pi$
$257$	−3098.17	−0.751980	−0.375990	+	0.926624i	$0.622697\pi$
$258$	0	0
$259$	0	0
$260$	−68.7895	−0.0164083
$261$	0	0
$262$	−1736.00	−0.409353
$263$	8029.86	1.88267	0.941335	−	0.337474i	$-0.109573\pi$
$263$	8029.86	1.88267	0.941335	+	0.337474i	$0.109573\pi$
$264$	0	0
$265$	−336.000	−0.0778880
$266$	0	0
$267$	0	0
$268$	−202.000	−0.0460415
$269$	−8659.54	−1.96276	−0.981379	−	0.192083i	$-0.938476\pi$
$269$	−8659.54	−1.96276	−0.981379	+	0.192083i	$0.938476\pi$
$270$	0	0
$271$	380.000	0.0851784	0.0425892	−	0.999093i	$-0.486439\pi$
$271$	380.000	0.0851784	0.0425892	+	0.999093i	$0.486439\pi$
$272$	−3492.39	−0.778520
$273$	0	0
$274$	1134.00	0.250027
$275$	2185.39	0.479215
$276$	0	0
$277$	6733.00	1.46046	0.730229	−	0.683203i	$-0.239413\pi$
$277$	6733.00	1.46046	0.730229	+	0.683203i	$0.239413\pi$
$278$	6191.06	1.33567
$279$	0	0
$280$	0	0
$281$	7249.36	1.53901	0.769503	−	0.638644i	$-0.220504\pi$
$281$	7249.36	1.53901	0.769503	+	0.638644i	$0.220504\pi$
$282$	0	0
$283$	−2812.00	−0.590657	−0.295329	−	0.955396i	$-0.595429\pi$
$283$	−2812.00	−0.590657	−0.295329	+	0.955396i	$0.595429\pi$
$284$	1055.65	0.220569
$285$	0	0
$286$	−1274.00	−0.263403
$287$	0	0
$288$	0	0
$289$	−881.000	−0.179320
$290$	37.0405	0.00750032
$291$	0	0
$292$	672.000	0.134677
$293$	4497.78	0.896802	0.448401	−	0.893833i	$-0.351994\pi$
$293$	4497.78	0.896802	0.448401	+	0.893833i	$0.351994\pi$
$294$	0	0
$295$	−336.000	−0.0663142
$296$	2643.11	0.519011
$297$	0	0
$298$	4998.00	0.971565
$299$	2682.79	0.518896
$300$	0	0
$301$	0	0
$302$	−978.928	−0.186526
$303$	0	0
$304$	−1925.00	−0.363179
$305$	402.154	0.0754993
$306$	0	0
$307$	4079.00	0.758309	0.379154	−	0.925333i	$-0.376215\pi$
$307$	4079.00	0.758309	0.379154	+	0.925333i	$0.376215\pi$
$308$	0	0
$309$	0	0
$310$	525.000	0.0961871
$311$	−2206.56	−0.402323	−0.201161	−	0.979558i	$-0.564472\pi$
$311$	−2206.56	−0.402323	−0.201161	+	0.979558i	$0.564472\pi$
$312$	0	0
$313$	−5054.00	−0.912680	−0.456340	−	0.889805i	$-0.650840\pi$
$313$	−5054.00	−0.912680	−0.456340	+	0.889805i	$0.650840\pi$
$314$	−1296.42	−0.232997
$315$	0	0
$316$	988.000	0.175884
$317$	380.988	0.0675029	0.0337515	−	0.999430i	$-0.489255\pi$
$317$	380.988	0.0675029	0.0337515	+	0.999430i	$0.489255\pi$
$318$	0	0
$319$	−98.0000	−0.0172005
$320$	1478.97	0.258366
$321$	0	0
$322$	0	0
$323$	2222.43	0.382846
$324$	0	0
$325$	−3068.00	−0.523637
$326$	−9910.98	−1.68380
$327$	0	0
$328$	11403.0	1.91959
$329$	0	0
$330$	0	0
$331$	−6820.00	−1.13251	−0.566255	−	0.824230i	$-0.691608\pi$
$331$	−6820.00	−1.13251	−0.566255	+	0.824230i	$0.691608\pi$
$332$	−142.871	−0.0236176
$333$	0	0
$334$	4102.00	0.672010
$335$	534.442	0.0871632
$336$	0	0
$337$	4051.00	0.654813	0.327407	−	0.944884i	$-0.393825\pi$
$337$	4051.00	0.654813	0.327407	+	0.944884i	$0.393825\pi$
$338$	−4024.19	−0.647595
$339$	0	0
$340$	−168.000	−0.0267973
$341$	−1389.02	−0.220585
$342$	0	0
$343$	0	0
$344$	7810.26	1.22413
$345$	0	0
$346$	5747.00	0.892950
$347$	1351.98	0.209159	0.104579	−	0.994517i	$-0.466650\pi$
$347$	1351.98	0.209159	0.104579	+	0.994517i	$0.466650\pi$
$348$	0	0
$349$	−12406.0	−1.90280	−0.951401	−	0.307955i	$-0.900356\pi$
$349$	−12406.0	−1.90280	−0.951401	+	0.307955i	$0.900356\pi$
$350$	0	0
$351$	0	0
$352$	−833.000	−0.126134
$353$	12861.0	1.93916	0.969578	−	0.244784i	$-0.0787171\pi$
$353$	12861.0	1.93916	0.969578	+	0.244784i	$0.0787171\pi$
$354$	0	0
$355$	−2793.00	−0.417569
$356$	965.699	0.143770
$357$	0	0
$358$	−2618.00	−0.386496
$359$	6058.77	0.890723	0.445362	−	0.895351i	$-0.353075\pi$
$359$	6058.77	0.890723	0.445362	+	0.895351i	$0.353075\pi$
$360$	0	0
$361$	−5634.00	−0.821403
$362$	5815.36	0.844333
$363$	0	0
$364$	0	0
$365$	−1777.94	−0.254964
$366$	0	0
$367$	8085.00	1.14996	0.574978	−	0.818169i	$-0.305011\pi$
$367$	8085.00	1.14996	0.574978	+	0.818169i	$0.305011\pi$
$368$	−5675.14	−0.803905
$369$	0	0
$370$	−777.000	−0.109174
$371$	0	0
$372$	0	0
$373$	11657.0	1.61817	0.809084	−	0.587693i	$-0.199964\pi$
$373$	11657.0	1.61817	0.809084	+	0.587693i	$0.199964\pi$
$374$	−3111.40	−0.430179
$375$	0	0
$376$	9072.00	1.24429
$377$	137.579	0.0187949
$378$	0	0
$379$	−8868.00	−1.20190	−0.600948	−	0.799288i	$-0.705210\pi$
$379$	−8868.00	−1.20190	−0.600948	+	0.799288i	$0.705210\pi$
$380$	−92.6013	−0.0125009
$381$	0	0
$382$	−11725.0	−1.57043
$383$	10112.1	1.34909	0.674546	−	0.738233i	$-0.264339\pi$
$383$	10112.1	1.34909	0.674546	+	0.738233i	$0.264339\pi$
$384$	0	0
$385$	0	0
$386$	1021.26	0.134665
$387$	0	0
$388$	−492.000	−0.0643750
$389$	9175.47	1.19592	0.597962	−	0.801524i	$-0.295977\pi$
$389$	9175.47	1.19592	0.597962	+	0.801524i	$0.295977\pi$
$390$	0	0
$391$	6552.00	0.847440
$392$	0	0
$393$	0	0
$394$	−9142.00	−1.16895
$395$	−2614.00	−0.332974
$396$	0	0
$397$	−1986.00	−0.251069	−0.125535	−	0.992089i	$-0.540065\pi$
$397$	−1986.00	−0.251069	−0.125535	+	0.992089i	$0.540065\pi$
$398$	4246.43	0.534810
$399$	0	0
$400$	6490.00	0.811250
$401$	−3476.52	−0.432940	−0.216470	−	0.976289i	$-0.569454\pi$
$401$	−3476.52	−0.432940	−0.216470	+	0.976289i	$0.569454\pi$
$402$	0	0
$403$	1950.00	0.241033
$404$	−1703.86	−0.209828
$405$	0	0
$406$	0	0
$407$	2055.75	0.250368
$408$	0	0
$409$	5056.00	0.611255	0.305627	−	0.952151i	$-0.401134\pi$
$409$	5056.00	0.611255	0.305627	+	0.952151i	$0.401134\pi$
$410$	−3352.17	−0.403785
$411$	0	0
$412$	845.000	0.101044
$413$	0	0
$414$	0	0
$415$	378.000	0.0447115
$416$	1169.42	0.137826
$417$	0	0
$418$	−1715.00	−0.200678
$419$	−1412.83	−0.164729	−0.0823643	−	0.996602i	$-0.526247\pi$
$419$	−1412.83	−0.164729	−0.0823643	+	0.996602i	$0.526247\pi$
$420$	0	0
$421$	−9343.00	−1.08159	−0.540796	−	0.841154i	$-0.681877\pi$
$421$	−9343.00	−1.08159	−0.540796	+	0.841154i	$0.681877\pi$
$422$	−1053.01	−0.121468
$423$	0	0
$424$	3024.00	0.346364
$425$	−7492.77	−0.855183
$426$	0	0
$427$	0	0
$428$	−1370.50	−0.154779
$429$	0	0
$430$	−2296.00	−0.257495
$431$	2087.50	0.233298	0.116649	−	0.993173i	$-0.462785\pi$
$431$	2087.50	0.233298	0.116649	+	0.993173i	$0.462785\pi$
$432$	0	0
$433$	−15622.0	−1.73382	−0.866912	−	0.498462i	$-0.833898\pi$
$433$	−15622.0	−1.73382	−0.866912	+	0.498462i	$0.833898\pi$
$434$	0	0
$435$	0	0
$436$	−477.000	−0.0523949
$437$	3611.45	0.395330
$438$	0	0
$439$	11424.0	1.24200	0.621000	−	0.783811i	$-0.286727\pi$
$439$	11424.0	1.24200	0.621000	+	0.783811i	$0.286727\pi$
$440$	1166.78	0.126418
$441$	0	0
$442$	4368.00	0.470056
$443$	1722.38	0.184724	0.0923622	−	0.995725i	$-0.470558\pi$
$443$	1722.38	0.184724	0.0923622	+	0.995725i	$0.470558\pi$
$444$	0	0
$445$	−2555.00	−0.272177
$446$	−9463.85	−1.00477
$447$	0	0
$448$	0	0
$449$	1317.58	0.138487	0.0692435	−	0.997600i	$-0.477941\pi$
$449$	1317.58	0.138487	0.0692435	+	0.997600i	$0.477941\pi$
$450$	0	0
$451$	8869.00	0.925998
$452$	1709.16	0.177858
$453$	0	0
$454$	9366.00	0.968212
$455$	0	0
$456$	0	0
$457$	5041.00	0.515991	0.257996	−	0.966146i	$-0.416938\pi$
$457$	5041.00	0.515991	0.257996	+	0.966146i	$0.416938\pi$
$458$	−15673.4	−1.59906
$459$	0	0
$460$	−273.000	−0.0276711
$461$	5061.32	0.511343	0.255672	−	0.966764i	$-0.417703\pi$
$461$	5061.32	0.511343	0.255672	+	0.966764i	$0.417703\pi$
$462$	0	0
$463$	4426.00	0.444263	0.222131	−	0.975017i	$-0.428699\pi$
$463$	4426.00	0.444263	0.222131	+	0.975017i	$0.428699\pi$
$464$	−291.033	−0.0291182
$465$	0	0
$466$	−6510.00	−0.647146
$467$	−9556.45	−0.946938	−0.473469	−	0.880811i	$-0.656998\pi$
$467$	−9556.45	−0.946938	−0.473469	+	0.880811i	$0.656998\pi$
$468$	0	0
$469$	0	0
$470$	−2666.92	−0.261736
$471$	0	0
$472$	3024.00	0.294896
$473$	6074.65	0.590513
$474$	0	0
$475$	−4130.00	−0.398942
$476$	0	0
$477$	0	0
$478$	13734.0	1.31418
$479$	−1248.79	−0.119121	−0.0595604	−	0.998225i	$-0.518970\pi$
$479$	−1248.79	−0.119121	−0.0595604	+	0.998225i	$0.518970\pi$
$480$	0	0
$481$	−2886.00	−0.273576
$482$	−17006.9	−1.60714
$483$	0	0
$484$	988.000	0.0927874
$485$	1301.71	0.121871
$486$	0	0
$487$	1888.00	0.175674	0.0878372	−	0.996135i	$-0.472004\pi$
$487$	1888.00	0.175674	0.0878372	+	0.996135i	$0.472004\pi$
$488$	−3619.39	−0.335742
$489$	0	0
$490$	0	0
$491$	−18713.4	−1.72001	−0.860004	−	0.510287i	$-0.829539\pi$
$491$	−18713.4	−1.72001	−0.860004	+	0.510287i	$0.829539\pi$
$492$	0	0
$493$	336.000	0.0306951
$494$	2407.63	0.219280
$495$	0	0
$496$	−4125.00	−0.373423
$497$	0	0
$498$	0	0
$499$	21148.0	1.89722	0.948612	−	0.316442i	$-0.102488\pi$
$499$	21148.0	1.89722	0.948612	+	0.316442i	$0.102488\pi$
$500$	642.918	0.0575043
$501$	0	0
$502$	15652.0	1.39160
$503$	2968.53	0.263142	0.131571	−	0.991307i	$-0.457998\pi$
$503$	2968.53	0.263142	0.131571	+	0.991307i	$0.457998\pi$
$504$	0	0
$505$	4508.00	0.397234
$506$	−5056.03	−0.444206
$507$	0	0
$508$	1590.00	0.138868
$509$	−201.077	−0.0175100	−0.00875500	−	0.999962i	$-0.502787\pi$
$509$	−201.077	−0.0175100	−0.00875500	+	0.999962i	$0.502787\pi$
$510$	0	0
$511$	0	0
$512$	−12951.0	−1.11788
$513$	0	0
$514$	−8197.00	−0.703413
$515$	−2235.66	−0.191291
$516$	0	0
$517$	7056.00	0.600237
$518$	0	0
$519$	0	0
$520$	−1638.00	−0.138137
$521$	11056.6	0.929747	0.464874	−	0.885377i	$-0.346100\pi$
$521$	11056.6	0.929747	0.464874	+	0.885377i	$0.346100\pi$
$522$	0	0
$523$	593.000	0.0495795	0.0247898	−	0.999693i	$-0.492108\pi$
$523$	593.000	0.0495795	0.0247898	+	0.999693i	$0.492108\pi$
$524$	656.146	0.0547021
$525$	0	0
$526$	21245.0	1.76108
$527$	4762.35	0.393646
$528$	0	0
$529$	−1520.00	−0.124928
$530$	−888.972	−0.0728575
$531$	0	0
$532$	0	0
$533$	−12450.9	−1.01184
$534$	0	0
$535$	3626.00	0.293020
$536$	−4809.98	−0.387611
$537$	0	0
$538$	−22911.0	−1.83599
$539$	0	0
$540$	0	0
$541$	15689.0	1.24681	0.623404	−	0.781900i	$-0.285749\pi$
$541$	15689.0	1.24681	0.623404	+	0.781900i	$0.285749\pi$
$542$	1005.39	0.0796771
$543$	0	0
$544$	2856.00	0.225092
$545$	1262.02	0.0991911
$546$	0	0
$547$	−20840.0	−1.62898	−0.814492	−	0.580175i	$-0.802984\pi$
$547$	−20840.0	−1.62898	−0.814492	+	0.580175i	$0.802984\pi$
$548$	−428.612	−0.0334113
$549$	0	0
$550$	5782.00	0.448264
$551$	185.203	0.0143192
$552$	0	0
$553$	0	0
$554$	17813.8	1.36613
$555$	0	0
$556$	−2340.00	−0.178486
$557$	−15742.2	−1.19752	−0.598761	−	0.800928i	$-0.704340\pi$
$557$	−15742.2	−1.19752	−0.598761	+	0.800928i	$0.704340\pi$
$558$	0	0
$559$	−8528.00	−0.645252
$560$	0	0
$561$	0	0
$562$	19180.0	1.43961
$563$	11487.9	0.859956	0.429978	−	0.902839i	$-0.358521\pi$
$563$	11487.9	0.859956	0.429978	+	0.902839i	$0.358521\pi$
$564$	0	0
$565$	−4522.00	−0.336711
$566$	−7439.85	−0.552509
$567$	0	0
$568$	25137.0	1.85691
$569$	−216.952	−0.0159843	−0.00799217	−	0.999968i	$-0.502544\pi$
$569$	−216.952	−0.0159843	−0.00799217	+	0.999968i	$0.502544\pi$
$570$	0	0
$571$	−20494.0	−1.50201	−0.751005	−	0.660297i	$-0.770430\pi$
$571$	−20494.0	−1.50201	−0.751005	+	0.660297i	$0.770430\pi$
$572$	481.527	0.0351987
$573$	0	0
$574$	0	0
$575$	−12175.7	−0.883067
$576$	0	0
$577$	7814.00	0.563780	0.281890	−	0.959447i	$-0.409039\pi$
$577$	7814.00	0.563780	0.281890	+	0.959447i	$0.409039\pi$
$578$	−2330.91	−0.167739
$579$	0	0
$580$	−14.0000	−0.00100227
$581$	0	0
$582$	0	0
$583$	2352.00	0.167084
$584$	16001.5	1.13381
$585$	0	0
$586$	11900.0	0.838881
$587$	−18737.2	−1.31749	−0.658746	−	0.752366i	$-0.728913\pi$
$587$	−18737.2	−1.31749	−0.658746	+	0.752366i	$0.728913\pi$
$588$	0	0
$589$	2625.00	0.183635
$590$	−888.972	−0.0620312
$591$	0	0
$592$	6105.00	0.423841
$593$	−6341.87	−0.439172	−0.219586	−	0.975593i	$-0.570471\pi$
$593$	−6341.87	−0.439172	−0.219586	+	0.975593i	$0.570471\pi$
$594$	0	0
$595$	0	0
$596$	−1889.07	−0.129831
$597$	0	0
$598$	7098.00	0.485383
$599$	6500.61	0.443419	0.221709	−	0.975113i	$-0.428836\pi$
$599$	6500.61	0.443419	0.221709	+	0.975113i	$0.428836\pi$
$600$	0	0
$601$	−18594.0	−1.26201	−0.631003	−	0.775781i	$-0.717356\pi$
$601$	−18594.0	−1.26201	−0.631003	+	0.775781i	$0.717356\pi$
$602$	0	0
$603$	0	0
$604$	370.000	0.0249256
$605$	−2614.00	−0.175660
$606$	0	0
$607$	−13916.0	−0.930532	−0.465266	−	0.885171i	$-0.654041\pi$
$607$	−13916.0	−0.930532	−0.465266	+	0.885171i	$0.654041\pi$
$608$	1574.22	0.105005
$609$	0	0
$610$	1064.00	0.0706231
$611$	−9905.69	−0.655878
$612$	0	0
$613$	−953.000	−0.0627917	−0.0313958	−	0.999507i	$-0.509995\pi$
$613$	−953.000	−0.0627917	−0.0313958	+	0.999507i	$0.509995\pi$
$614$	10792.0	0.709333
$615$	0	0
$616$	0	0
$617$	−10921.7	−0.712625	−0.356312	−	0.934367i	$-0.615966\pi$
$617$	−10921.7	−0.712625	−0.356312	+	0.934367i	$0.615966\pi$
$618$	0	0
$619$	21467.0	1.39391	0.696956	−	0.717114i	$-0.254537\pi$
$619$	21467.0	1.39391	0.696956	+	0.717114i	$0.254537\pi$
$620$	−198.431	−0.0128535
$621$	0	0
$622$	−5838.00	−0.376338
$623$	0	0
$624$	0	0
$625$	13049.0	0.835136
$626$	−13371.6	−0.853734
$627$	0	0
$628$	490.000	0.0311356
$629$	−7048.28	−0.446794
$630$	0	0
$631$	−25012.0	−1.57799	−0.788995	−	0.614399i	$-0.789398\pi$
$631$	−25012.0	−1.57799	−0.788995	+	0.614399i	$0.789398\pi$
$632$	23526.0	1.48072
$633$	0	0
$634$	1008.00	0.0631432
$635$	−4206.74	−0.262897
$636$	0	0
$637$	0	0
$638$	−259.284	−0.0160896
$639$	0	0
$640$	2961.00	0.182881
$641$	−4111.50	−0.253345	−0.126673	−	0.991945i	$-0.540430\pi$
$641$	−4111.50	−0.253345	−0.126673	+	0.991945i	$0.540430\pi$
$642$	0	0
$643$	9571.00	0.587004	0.293502	−	0.955959i	$-0.405179\pi$
$643$	9571.00	0.587004	0.293502	+	0.955959i	$0.405179\pi$
$644$	0	0
$645$	0	0
$646$	5880.00	0.358120
$647$	22150.2	1.34593	0.672964	−	0.739675i	$-0.265021\pi$
$647$	22150.2	1.34593	0.672964	+	0.739675i	$0.265021\pi$
$648$	0	0
$649$	2352.00	0.142256
$650$	−8117.17	−0.489818
$651$	0	0
$652$	3746.00	0.225007
$653$	19652.6	1.17774	0.588872	−	0.808226i	$-0.299572\pi$
$653$	19652.6	1.17774	0.588872	+	0.808226i	$0.299572\pi$
$654$	0	0
$655$	−1736.00	−0.103559
$656$	26338.5	1.56760
$657$	0	0
$658$	0	0
$659$	−12421.8	−0.734271	−0.367136	−	0.930167i	$-0.619662\pi$
$659$	−12421.8	−0.734271	−0.367136	+	0.930167i	$0.619662\pi$
$660$	0	0
$661$	21350.0	1.25631	0.628153	−	0.778090i	$-0.283811\pi$
$661$	21350.0	1.25631	0.628153	+	0.778090i	$0.283811\pi$
$662$	−18044.0	−1.05937
$663$	0	0
$664$	−3402.00	−0.198830
$665$	0	0
$666$	0	0
$667$	546.000	0.0316960
$668$	−1550.41	−0.0898012
$669$	0	0
$670$	1414.00	0.0815337
$671$	−2815.08	−0.161960
$672$	0	0
$673$	10750.0	0.615724	0.307862	−	0.951431i	$-0.400387\pi$
$673$	10750.0	0.615724	0.307862	+	0.951431i	$0.400387\pi$
$674$	10717.9	0.612522
$675$	0	0
$676$	1521.00	0.0865385
$677$	−20438.4	−1.16028	−0.580142	−	0.814515i	$-0.697003\pi$
$677$	−20438.4	−1.16028	−0.580142	+	0.814515i	$0.697003\pi$
$678$	0	0
$679$	0	0
$680$	−4000.38	−0.225599
$681$	0	0
$682$	−3675.00	−0.206339
$683$	16041.2	0.898681	0.449340	−	0.893361i	$-0.351659\pi$
$683$	16041.2	0.898681	0.449340	+	0.893361i	$0.351659\pi$
$684$	0	0
$685$	1134.00	0.0632524
$686$	0	0
$687$	0	0
$688$	18040.0	0.999664
$689$	−3301.90	−0.182572
$690$	0	0
$691$	28072.0	1.54545	0.772727	−	0.634738i	$-0.218892\pi$
$691$	28072.0	1.54545	0.772727	+	0.634738i	$0.218892\pi$
$692$	−2172.16	−0.119325
$693$	0	0
$694$	3577.00	0.195650
$695$	6191.06	0.337900
$696$	0	0
$697$	−30408.0	−1.65249
$698$	−32823.2	−1.77991
$699$	0	0
$700$	0	0
$701$	15096.7	0.813399	0.406700	−	0.913562i	$-0.366680\pi$
$701$	15096.7	0.813399	0.406700	+	0.913562i	$0.366680\pi$
$702$	0	0
$703$	−3885.00	−0.208429
$704$	−10352.8	−0.554243
$705$	0	0
$706$	34027.0	1.81391
$707$	0	0
$708$	0	0
$709$	−7671.00	−0.406334	−0.203167	−	0.979144i	$-0.565123\pi$
$709$	−7671.00	−0.406334	−0.203167	+	0.979144i	$0.565123\pi$
$710$	−7389.58	−0.390600
$711$	0	0
$712$	22995.0	1.21036
$713$	7738.82	0.406481
$714$	0	0
$715$	−1274.00	−0.0666362
$716$	989.511	0.0516477
$717$	0	0
$718$	16030.0	0.833195
$719$	−22732.3	−1.17910	−0.589549	−	0.807733i	$-0.700695\pi$
$719$	−22732.3	−1.17910	−0.589549	+	0.807733i	$0.700695\pi$
$720$	0	0
$721$	0	0
$722$	−14906.2	−0.768352
$723$	0	0
$724$	−2198.00	−0.112829
$725$	−624.397	−0.0319856
$726$	0	0
$727$	−26360.0	−1.34476	−0.672378	−	0.740208i	$-0.734727\pi$
$727$	−26360.0	−1.34476	−0.672378	+	0.740208i	$0.734727\pi$
$728$	0	0
$729$	0	0
$730$	−4704.00	−0.238497
$731$	−20827.4	−1.05380
$732$	0	0
$733$	16956.0	0.854412	0.427206	−	0.904154i	$-0.359498\pi$
$733$	16956.0	0.854412	0.427206	+	0.904154i	$0.359498\pi$
$734$	21390.9	1.07568
$735$	0	0
$736$	4641.00	0.232431
$737$	−3741.09	−0.186981
$738$	0	0
$739$	−26538.0	−1.32100	−0.660498	−	0.750828i	$-0.729655\pi$
$739$	−26538.0	−1.32100	−0.660498	+	0.750828i	$0.729655\pi$
$740$	293.678	0.0145890
$741$	0	0
$742$	0	0
$743$	15739.6	0.777159	0.388579	−	0.921415i	$-0.372966\pi$
$743$	15739.6	0.777159	0.388579	+	0.921415i	$0.372966\pi$
$744$	0	0
$745$	4998.00	0.245789
$746$	30841.5	1.51366
$747$	0	0
$748$	1176.00	0.0574851
$749$	0	0
$750$	0	0
$751$	10792.0	0.524375	0.262188	−	0.965017i	$-0.415556\pi$
$751$	10792.0	0.524375	0.262188	+	0.965017i	$0.415556\pi$
$752$	20954.4	1.01613
$753$	0	0
$754$	364.000	0.0175810
$755$	−978.928	−0.0471879
$756$	0	0
$757$	−16330.0	−0.784047	−0.392024	−	0.919955i	$-0.628225\pi$
$757$	−16330.0	−0.784047	−0.392024	+	0.919955i	$0.628225\pi$
$758$	−23462.5	−1.12427
$759$	0	0
$760$	−2205.00	−0.105242
$761$	24415.0	1.16300	0.581500	−	0.813546i	$-0.302466\pi$
$761$	24415.0	1.16300	0.581500	+	0.813546i	$0.302466\pi$
$762$	0	0
$763$	0	0
$764$	4431.63	0.209857
$765$	0	0
$766$	26754.0	1.26196
$767$	−3301.90	−0.155443
$768$	0	0
$769$	9058.00	0.424759	0.212380	−	0.977187i	$-0.431879\pi$
$769$	9058.00	0.424759	0.212380	+	0.977187i	$0.431879\pi$
$770$	0	0
$771$	0	0
$772$	−386.000	−0.0179954
$773$	7236.13	0.336695	0.168348	−	0.985728i	$-0.446157\pi$
$773$	7236.13	0.336695	0.168348	+	0.985728i	$0.446157\pi$
$774$	0	0
$775$	−8850.00	−0.410195
$776$	−11715.4	−0.541956
$777$	0	0
$778$	24276.0	1.11868
$779$	−16760.8	−0.770885
$780$	0	0
$781$	19551.0	0.895762
$782$	17335.0	0.792707
$783$	0	0
$784$	0	0
$785$	−1296.42	−0.0589441
$786$	0	0
$787$	23732.0	1.07491	0.537455	−	0.843292i	$-0.319386\pi$
$787$	23732.0	1.07491	0.537455	+	0.843292i	$0.319386\pi$
$788$	3455.35	0.156208
$789$	0	0
$790$	−6916.00	−0.311469
$791$	0	0
$792$	0	0
$793$	3952.00	0.176973
$794$	−5254.46	−0.234854
$795$	0	0
$796$	−1605.00	−0.0714670
$797$	16020.0	0.711993	0.355996	−	0.934487i	$-0.384142\pi$
$797$	16020.0	0.711993	0.355996	+	0.934487i	$0.384142\pi$
$798$	0	0
$799$	−24192.0	−1.07115
$800$	−5307.38	−0.234555
$801$	0	0
$802$	−9198.00	−0.404978
$803$	12445.6	0.546944
$804$	0	0
$805$	0	0
$806$	5159.22	0.225466
$807$	0	0
$808$	−40572.0	−1.76648
$809$	6651.42	0.289062	0.144531	−	0.989500i	$-0.453833\pi$
$809$	6651.42	0.289062	0.144531	+	0.989500i	$0.453833\pi$
$810$	0	0
$811$	7903.00	0.342185	0.171092	−	0.985255i	$-0.445270\pi$
$811$	7903.00	0.342185	0.171092	+	0.985255i	$0.445270\pi$
$812$	0	0
$813$	0	0
$814$	5439.00	0.234198
$815$	−9910.98	−0.425971
$816$	0	0
$817$	−11480.0	−0.491597
$818$	13376.9	0.571776
$819$	0	0
$820$	1267.00	0.0539580
$821$	−4847.02	−0.206044	−0.103022	−	0.994679i	$-0.532851\pi$
$821$	−4847.02	−0.206044	−0.103022	+	0.994679i	$0.532851\pi$
$822$	0	0
$823$	11598.0	0.491228	0.245614	−	0.969368i	$-0.421010\pi$
$823$	11598.0	0.491228	0.245614	+	0.969368i	$0.421010\pi$
$824$	20120.9	0.850663
$825$	0	0
$826$	0	0
$827$	20406.7	0.858053	0.429026	−	0.903292i	$-0.358857\pi$
$827$	20406.7	0.858053	0.429026	+	0.903292i	$0.358857\pi$
$828$	0	0
$829$	−5348.00	−0.224058	−0.112029	−	0.993705i	$-0.535735\pi$
$829$	−5348.00	−0.224058	−0.112029	+	0.993705i	$0.535735\pi$
$830$	1000.09	0.0418238
$831$	0	0
$832$	14534.0	0.605620
$833$	0	0
$834$	0	0
$835$	4102.00	0.170007
$836$	648.209	0.0268167
$837$	0	0
$838$	−3738.00	−0.154090
$839$	36728.3	1.51133	0.755663	−	0.654961i	$-0.227315\pi$
$839$	36728.3	1.51133	0.755663	+	0.654961i	$0.227315\pi$
$840$	0	0
$841$	−24361.0	−0.998852
$842$	−24719.3	−1.01174
$843$	0	0
$844$	398.000	0.0162319
$845$	−4024.19	−0.163830
$846$	0	0
$847$	0	0
$848$	6984.78	0.282852
$849$	0	0
$850$	−19824.0	−0.799950
$851$	−11453.5	−0.461362
$852$	0	0
$853$	−40628.0	−1.63080	−0.815402	−	0.578895i	$-0.803484\pi$
$853$	−40628.0	−1.63080	−0.815402	+	0.578895i	$0.803484\pi$
$854$	0	0
$855$	0	0
$856$	−32634.0	−1.30305
$857$	−45065.1	−1.79626	−0.898129	−	0.439731i	$-0.855074\pi$
$857$	−45065.1	−1.79626	−0.898129	+	0.439731i	$0.855074\pi$
$858$	0	0
$859$	−30329.0	−1.20467	−0.602335	−	0.798243i	$-0.705763\pi$
$859$	−30329.0	−1.20467	−0.602335	+	0.798243i	$0.705763\pi$
$860$	867.806	0.0344092
$861$	0	0
$862$	5523.00	0.218230
$863$	−18176.3	−0.716951	−0.358476	−	0.933539i	$-0.616703\pi$
$863$	−18176.3	−0.716951	−0.358476	+	0.933539i	$0.616703\pi$
$864$	0	0
$865$	5747.00	0.225900
$866$	−41331.9	−1.62184
$867$	0	0
$868$	0	0
$869$	18298.0	0.714290
$870$	0	0
$871$	5252.00	0.204314
$872$	−11358.2	−0.441098
$873$	0	0
$874$	9555.00	0.369797
$875$	0	0
$876$	0	0
$877$	−3326.00	−0.128063	−0.0640314	−	0.997948i	$-0.520396\pi$
$877$	−3326.00	−0.128063	−0.0640314	+	0.997948i	$0.520396\pi$
$878$	30225.1	1.16178
$879$	0	0
$880$	2695.00	0.103237
$881$	−23438.7	−0.896334	−0.448167	−	0.893950i	$-0.647923\pi$
$881$	−23438.7	−0.896334	−0.448167	+	0.893950i	$0.647923\pi$
$882$	0	0
$883$	−10768.0	−0.410387	−0.205194	−	0.978721i	$-0.565782\pi$
$883$	−10768.0	−0.410387	−0.205194	+	0.978721i	$0.565782\pi$
$884$	−1650.95	−0.0628138
$885$	0	0
$886$	4557.00	0.172794
$887$	25087.0	0.949650	0.474825	−	0.880080i	$-0.342511\pi$
$887$	25087.0	0.949650	0.474825	+	0.880080i	$0.342511\pi$
$888$	0	0
$889$	0	0
$890$	−6759.89	−0.254598
$891$	0	0
$892$	3577.00	0.134268
$893$	−13334.6	−0.499692
$894$	0	0
$895$	−2618.00	−0.0977766
$896$	0	0
$897$	0	0
$898$	3486.00	0.129543
$899$	396.863	0.0147232
$900$	0	0
$901$	−8064.00	−0.298170
$902$	23465.2	0.866191
$903$	0	0
$904$	40698.0	1.49734
$905$	5815.36	0.213601
$906$	0	0
$907$	10550.0	0.386226	0.193113	−	0.981177i	$-0.438142\pi$
$907$	10550.0	0.386226	0.193113	+	0.981177i	$0.438142\pi$
$908$	−3540.02	−0.129383
$909$	0	0
$910$	0	0
$911$	−2492.30	−0.0906405	−0.0453203	−	0.998973i	$-0.514431\pi$
$911$	−2492.30	−0.0906405	−0.0453203	+	0.998973i	$0.514431\pi$
$912$	0	0
$913$	−2646.00	−0.0959144
$914$	13337.2	0.482666
$915$	0	0
$916$	5924.00	0.213684
$917$	0	0
$918$	0	0
$919$	27056.0	0.971159	0.485579	−	0.874193i	$-0.338609\pi$
$919$	27056.0	0.971159	0.485579	+	0.874193i	$0.338609\pi$
$920$	−6500.61	−0.232955
$921$	0	0
$922$	13391.0	0.478318
$923$	−27447.0	−0.978797
$924$	0	0
$925$	13098.0	0.465578
$926$	11710.1	0.415570
$927$	0	0
$928$	238.000	0.00841889
$929$	43326.8	1.53015	0.765074	−	0.643943i	$-0.222702\pi$
$929$	43326.8	1.53015	0.765074	+	0.643943i	$0.222702\pi$
$930$	0	0
$931$	0	0
$932$	2460.55	0.0864785
$933$	0	0
$934$	−25284.0	−0.885779
$935$	−3111.40	−0.108828
$936$	0	0
$937$	−31092.0	−1.08402	−0.542012	−	0.840370i	$-0.682337\pi$
$937$	−31092.0	−1.08402	−0.542012	+	0.840370i	$0.682337\pi$
$938$	0	0
$939$	0	0
$940$	1008.00	0.0349759
$941$	−24624.0	−0.853050	−0.426525	−	0.904476i	$-0.640262\pi$
$941$	−24624.0	−0.853050	−0.426525	+	0.904476i	$0.640262\pi$
$942$	0	0
$943$	−49413.0	−1.70637
$944$	6984.78	0.240821
$945$	0	0
$946$	16072.0	0.552374
$947$	35942.5	1.23334	0.616671	−	0.787221i	$-0.288481\pi$
$947$	35942.5	1.23334	0.616671	+	0.787221i	$0.288481\pi$
$948$	0	0
$949$	−17472.0	−0.597645
$950$	−10927.0	−0.373176
$951$	0	0
$952$	0	0
$953$	53301.3	1.81175	0.905875	−	0.423544i	$-0.139214\pi$
$953$	53301.3	1.81175	0.905875	+	0.423544i	$0.139214\pi$
$954$	0	0
$955$	−11725.0	−0.397290
$956$	−5190.96	−0.175615
$957$	0	0
$958$	−3304.00	−0.111427
$959$	0	0
$960$	0	0
$961$	−24166.0	−0.811185
$962$	−7635.64	−0.255907
$963$	0	0
$964$	6428.00	0.214763
$965$	1021.26	0.0340679
$966$	0	0
$967$	4066.00	0.135216	0.0676079	−	0.997712i	$-0.478463\pi$
$967$	4066.00	0.135216	0.0676079	+	0.997712i	$0.478463\pi$
$968$	23526.0	0.781152
$969$	0	0
$970$	3444.00	0.114000
$971$	28431.2	0.939652	0.469826	−	0.882759i	$-0.344317\pi$
$971$	28431.2	0.939652	0.469826	+	0.882759i	$0.344317\pi$
$972$	0	0
$973$	0	0
$974$	4995.18	0.164328
$975$	0	0
$976$	−8360.00	−0.274177
$977$	29590.1	0.968957	0.484478	−	0.874803i	$-0.339009\pi$
$977$	29590.1	0.968957	0.484478	+	0.874803i	$0.339009\pi$
$978$	0	0
$979$	17885.0	0.583868
$980$	0	0
$981$	0	0
$982$	−49511.0	−1.60892
$983$	−36797.1	−1.19394	−0.596971	−	0.802263i	$-0.703629\pi$
$983$	−36797.1	−1.19394	−0.596971	+	0.802263i	$0.703629\pi$
$984$	0	0
$985$	−9142.00	−0.295724
$986$	888.972	0.0287126
$987$	0	0
$988$	−910.000	−0.0293026
$989$	−33844.5	−1.08816
$990$	0	0
$991$	42302.0	1.35597	0.677986	−	0.735075i	$-0.262853\pi$
$991$	42302.0	1.35597	0.677986	+	0.735075i	$0.262853\pi$
$992$	3373.33	0.107967
$993$	0	0
$994$	0	0
$995$	4246.43	0.135297
$996$	0	0
$997$	−17218.0	−0.546940	−0.273470	−	0.961880i	$-0.588171\pi$
$997$	−17218.0	−0.546940	−0.273470	+	0.961880i	$0.588171\pi$
$998$	55952.3	1.77469
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.4.a.s.1.2		2
3.2	odd	2	inner	1323.4.a.s.1.1		2
7.6	odd	2		189.4.a.g.1.2	yes	2
21.20	even	2		189.4.a.g.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.4.a.g.1.1	✓	2	21.20	even	2
189.4.a.g.1.2	yes	2	7.6	odd	2
1323.4.a.s.1.1		2	3.2	odd	2	inner
1323.4.a.s.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1323.a (trivial)

\(f(q)\)	\(=\)	\(q+2.64575 q^{2} -1.00000 q^{4} +2.64575 q^{5} -23.8118 q^{8} +7.00000 q^{10} -18.5203 q^{11} +26.0000 q^{13} -55.0000 q^{16} +63.4980 q^{17} +35.0000 q^{19} -2.64575 q^{20} -49.0000 q^{22} +103.184 q^{23} -118.000 q^{25} +68.7895 q^{26} +5.29150 q^{29} +75.0000 q^{31} +44.9778 q^{32} +168.000 q^{34} -111.000 q^{37} +92.6013 q^{38} -63.0000 q^{40} -478.881 q^{41} -328.000 q^{43} +18.5203 q^{44} +273.000 q^{46} -380.988 q^{47} -312.199 q^{50} -26.0000 q^{52} -126.996 q^{53} -49.0000 q^{55} +14.0000 q^{58} -126.996 q^{59} +152.000 q^{61} +198.431 q^{62} +559.000 q^{64} +68.7895 q^{65} +202.000 q^{67} -63.4980 q^{68} -1055.65 q^{71} -672.000 q^{73} -293.678 q^{74} -35.0000 q^{76} -988.000 q^{79} -145.516 q^{80} -1267.00 q^{82} +142.871 q^{83} +168.000 q^{85} -867.806 q^{86} +441.000 q^{88} -965.699 q^{89} -103.184 q^{92} -1008.00 q^{94} +92.6013 q^{95} +492.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 14 q^{10} + 52 q^{13} - 110 q^{16} + 70 q^{19} - 98 q^{22} - 236 q^{25} + 150 q^{31} + 336 q^{34} - 222 q^{37} - 126 q^{40} - 656 q^{43} + 546 q^{46} - 52 q^{52} - 98 q^{55} + 28 q^{58}+ \cdots + 984 q^{97}+O(q^{100}) \) 2 * q - 2 * q^4 + 14 * q^10 + 52 * q^13 - 110 * q^16 + 70 * q^19 - 98 * q^22 - 236 * q^25 + 150 * q^31 + 336 * q^34 - 222 * q^37 - 126 * q^40 - 656 * q^43 + 546 * q^46 - 52 * q^52 - 98 * q^55 + 28 * q^58 + 304 * q^61 + 1118 * q^64 + 404 * q^67 - 1344 * q^73 - 70 * q^76 - 1976 * q^79 - 2534 * q^82 + 336 * q^85 + 882 * q^88 - 2016 * q^94 + 984 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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