LMFDB - Embedded newform 8019.2.a.i.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8019,2,Mod(1,8019)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8019, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8019.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8019 = 3^{6} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8019.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:	$64.0320373809$
Analytic rank:	$1$
Dimension:	$48$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	8019.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.93884 q^{2} +1.75908 q^{4} -0.334950 q^{5} -4.22786 q^{7} +0.467097 q^{8} +O(q^{10})$	\(q-1.93884 q^{2} +1.75908 q^{4} -0.334950 q^{5} -4.22786 q^{7} +0.467097 q^{8} +0.649413 q^{10} -1.00000 q^{11} -2.53258 q^{13} +8.19713 q^{14} -4.42379 q^{16} -4.24924 q^{17} -2.75235 q^{19} -0.589205 q^{20} +1.93884 q^{22} -3.21630 q^{23} -4.88781 q^{25} +4.91026 q^{26} -7.43716 q^{28} -1.62374 q^{29} +8.19813 q^{31} +7.64281 q^{32} +8.23858 q^{34} +1.41612 q^{35} +10.3663 q^{37} +5.33635 q^{38} -0.156454 q^{40} +8.33484 q^{41} +10.0613 q^{43} -1.75908 q^{44} +6.23587 q^{46} +4.64082 q^{47} +10.8748 q^{49} +9.47666 q^{50} -4.45502 q^{52} -0.00292171 q^{53} +0.334950 q^{55} -1.97482 q^{56} +3.14816 q^{58} +5.77673 q^{59} +4.22004 q^{61} -15.8948 q^{62} -5.97057 q^{64} +0.848288 q^{65} +9.13168 q^{67} -7.47477 q^{68} -2.74563 q^{70} -2.71195 q^{71} +7.04961 q^{73} -20.0986 q^{74} -4.84161 q^{76} +4.22786 q^{77} -17.1151 q^{79} +1.48175 q^{80} -16.1599 q^{82} +4.82727 q^{83} +1.42328 q^{85} -19.5072 q^{86} -0.467097 q^{88} -9.76855 q^{89} +10.7074 q^{91} -5.65773 q^{92} -8.99778 q^{94} +0.921899 q^{95} +12.7629 q^{97} -21.0845 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 6 q^{2} + 54 q^{4} - 24 q^{5} - 18 q^{8}+O(q^{10}) $ 48 * q - 6 * q^2 + 54 * q^4 - 24 * q^5 - 18 * q^8	$ 48 q - 6 q^{2} + 54 q^{4} - 24 q^{5} - 18 q^{8} - 48 q^{11} - 24 q^{14} + 66 q^{16} - 24 q^{17} - 48 q^{20} + 6 q^{22} - 12 q^{23} + 60 q^{25} - 36 q^{26} - 18 q^{28} - 60 q^{29} + 36 q^{31} - 42 q^{32} + 12 q^{34} - 24 q^{35} + 6 q^{37} - 24 q^{38} - 72 q^{41} - 12 q^{43} - 54 q^{44} - 30 q^{46} - 36 q^{47} + 60 q^{49} - 42 q^{50} - 48 q^{53} + 24 q^{55} - 72 q^{56} + 12 q^{58} - 60 q^{59} - 24 q^{61} - 36 q^{62} + 90 q^{64} - 48 q^{65} - 60 q^{68} - 30 q^{70} - 60 q^{71} - 18 q^{73} - 36 q^{74} - 42 q^{76} - 12 q^{79} - 96 q^{80} + 12 q^{82} - 36 q^{83} + 18 q^{85} - 48 q^{86} + 18 q^{88} - 96 q^{89} + 30 q^{91} - 36 q^{92} - 48 q^{94} - 48 q^{95} + 30 q^{97} - 54 q^{98}+O(q^{100}) $ 48 * q - 6 * q^2 + 54 * q^4 - 24 * q^5 - 18 * q^8 - 48 * q^11 - 24 * q^14 + 66 * q^16 - 24 * q^17 - 48 * q^20 + 6 * q^22 - 12 * q^23 + 60 * q^25 - 36 * q^26 - 18 * q^28 - 60 * q^29 + 36 * q^31 - 42 * q^32 + 12 * q^34 - 24 * q^35 + 6 * q^37 - 24 * q^38 - 72 * q^41 - 12 * q^43 - 54 * q^44 - 30 * q^46 - 36 * q^47 + 60 * q^49 - 42 * q^50 - 48 * q^53 + 24 * q^55 - 72 * q^56 + 12 * q^58 - 60 * q^59 - 24 * q^61 - 36 * q^62 + 90 * q^64 - 48 * q^65 - 60 * q^68 - 30 * q^70 - 60 * q^71 - 18 * q^73 - 36 * q^74 - 42 * q^76 - 12 * q^79 - 96 * q^80 + 12 * q^82 - 36 * q^83 + 18 * q^85 - 48 * q^86 + 18 * q^88 - 96 * q^89 + 30 * q^91 - 36 * q^92 - 48 * q^94 - 48 * q^95 + 30 * q^97 - 54 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.93884	−1.37096	−0.685482	−	0.728090i	$-0.740408\pi$
$2$	−1.93884	−1.37096	−0.685482	+	0.728090i	$0.740408\pi$
$3$	0	0
$3$	0	0
$4$	1.75908	0.879542
$5$	−0.334950	−0.149794	−0.0748971	−	0.997191i	$-0.523863\pi$
$5$	−0.334950	−0.149794	−0.0748971	+	0.997191i	$0.523863\pi$
$6$	0	0
$7$	−4.22786	−1.59798	−0.798991	−	0.601344i	$-0.794632\pi$
$7$	−4.22786	−1.59798	−0.798991	+	0.601344i	$0.794632\pi$
$8$	0.467097	0.165144
$9$	0	0
$10$	0.649413	0.205363
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−2.53258	−0.702412	−0.351206	−	0.936298i	$-0.614228\pi$
$13$	−2.53258	−0.702412	−0.351206	+	0.936298i	$0.614228\pi$
$14$	8.19713	2.19077
$15$	0	0
$16$	−4.42379	−1.10595
$17$	−4.24924	−1.03059	−0.515296	−	0.857012i	$-0.672318\pi$
$17$	−4.24924	−1.03059	−0.515296	+	0.857012i	$0.672318\pi$
$18$	0	0
$19$	−2.75235	−0.631432	−0.315716	−	0.948854i	$-0.602245\pi$
$19$	−2.75235	−0.631432	−0.315716	+	0.948854i	$0.602245\pi$
$20$	−0.589205	−0.131750
$21$	0	0
$22$	1.93884	0.413361
$23$	−3.21630	−0.670644	−0.335322	−	0.942104i	$-0.608845\pi$
$23$	−3.21630	−0.670644	−0.335322	+	0.942104i	$0.608845\pi$
$24$	0	0
$25$	−4.88781	−0.977562
$26$	4.91026	0.962981
$27$	0	0
$28$	−7.43716	−1.40549
$29$	−1.62374	−0.301521	−0.150760	−	0.988570i	$-0.548172\pi$
$29$	−1.62374	−0.301521	−0.150760	+	0.988570i	$0.548172\pi$
$30$	0	0
$31$	8.19813	1.47243	0.736214	−	0.676749i	$-0.236612\pi$
$31$	8.19813	1.47243	0.736214	+	0.676749i	$0.236612\pi$
$32$	7.64281	1.35107
$33$	0	0
$34$	8.23858	1.41290
$35$	1.41612	0.239368
$36$	0	0
$37$	10.3663	1.70422	0.852108	−	0.523366i	$-0.175324\pi$
$37$	10.3663	1.70422	0.852108	+	0.523366i	$0.175324\pi$
$38$	5.33635	0.865670
$39$	0	0
$40$	−0.156454	−0.0247376
$41$	8.33484	1.30168	0.650841	−	0.759214i	$-0.274416\pi$
$41$	8.33484	1.30168	0.650841	+	0.759214i	$0.274416\pi$
$42$	0	0
$43$	10.0613	1.53434	0.767168	−	0.641446i	$-0.221665\pi$
$43$	10.0613	1.53434	0.767168	+	0.641446i	$0.221665\pi$
$44$	−1.75908	−0.265192
$45$	0	0
$46$	6.23587	0.919429
$47$	4.64082	0.676933	0.338466	−	0.940978i	$-0.390092\pi$
$47$	4.64082	0.676933	0.338466	+	0.940978i	$0.390092\pi$
$48$	0	0
$49$	10.8748	1.55354
$50$	9.47666	1.34020
$51$	0	0
$52$	−4.45502	−0.617801
$53$	−0.00292171	−0.000401327 0	−0.000200664	−	1.00000i	$-0.500064\pi$
$53$	−0.00292171	−0.000401327 0	−0.000200664		1.00000i	$0.500064\pi$
$54$	0	0
$55$	0.334950	0.0451647
$56$	−1.97482	−0.263896
$57$	0	0
$58$	3.14816	0.413374
$59$	5.77673	0.752066	0.376033	−	0.926606i	$-0.377288\pi$
$59$	5.77673	0.752066	0.376033	+	0.926606i	$0.377288\pi$
$60$	0	0
$61$	4.22004	0.540321	0.270160	−	0.962815i	$-0.412923\pi$
$61$	4.22004	0.540321	0.270160	+	0.962815i	$0.412923\pi$
$62$	−15.8948	−2.01864
$63$	0	0
$64$	−5.97057	−0.746322
$65$	0.848288	0.105217
$66$	0	0
$67$	9.13168	1.11561	0.557806	−	0.829971i	$-0.311643\pi$
$67$	9.13168	1.11561	0.557806	+	0.829971i	$0.311643\pi$
$68$	−7.47477	−0.906449
$69$	0	0
$70$	−2.74563	−0.328165
$71$	−2.71195	−0.321849	−0.160925	−	0.986967i	$-0.551448\pi$
$71$	−2.71195	−0.321849	−0.160925	+	0.986967i	$0.551448\pi$
$72$	0	0
$73$	7.04961	0.825094	0.412547	−	0.910936i	$-0.364639\pi$
$73$	7.04961	0.825094	0.412547	+	0.910936i	$0.364639\pi$
$74$	−20.0986	−2.33642
$75$	0	0
$76$	−4.84161	−0.555371
$77$	4.22786	0.481809
$78$	0	0
$79$	−17.1151	−1.92560	−0.962800	−	0.270215i	$-0.912905\pi$
$79$	−17.1151	−1.92560	−0.962800	+	0.270215i	$0.912905\pi$
$80$	1.48175	0.165665
$81$	0	0
$82$	−16.1599	−1.78456
$83$	4.82727	0.529862	0.264931	−	0.964267i	$-0.414651\pi$
$83$	4.82727	0.529862	0.264931	+	0.964267i	$0.414651\pi$
$84$	0	0
$85$	1.42328	0.154377
$86$	−19.5072	−2.10352
$87$	0	0
$88$	−0.467097	−0.0497927
$89$	−9.76855	−1.03546	−0.517732	−	0.855543i	$-0.673224\pi$
$89$	−9.76855	−1.03546	−0.517732	+	0.855543i	$0.673224\pi$
$90$	0	0
$91$	10.7074	1.12244
$92$	−5.65773	−0.589859
$93$	0	0
$94$	−8.99778	−0.928050
$95$	0.921899	0.0945848
$96$	0	0
$97$	12.7629	1.29588	0.647938	−	0.761693i	$-0.275631\pi$
$97$	12.7629	1.29588	0.647938	+	0.761693i	$0.275631\pi$
$98$	−21.0845	−2.12985
$99$	0	0
$100$	−8.59807	−0.859807
$101$	−18.3906	−1.82993	−0.914967	−	0.403529i	$-0.867784\pi$
$101$	−18.3906	−1.82993	−0.914967	+	0.403529i	$0.867784\pi$
$102$	0	0
$103$	2.05791	0.202772	0.101386	−	0.994847i	$-0.467672\pi$
$103$	2.05791	0.202772	0.101386	+	0.994847i	$0.467672\pi$
$104$	−1.18296	−0.115999
$105$	0	0
$106$	0.00566471	0.000550205 0
$107$	4.44649	0.429858	0.214929	−	0.976630i	$-0.431048\pi$
$107$	4.44649	0.429858	0.214929	+	0.976630i	$0.431048\pi$
$108$	0	0
$109$	11.0089	1.05446	0.527231	−	0.849722i	$-0.323230\pi$
$109$	11.0089	1.05446	0.527231	+	0.849722i	$0.323230\pi$
$110$	−0.649413	−0.0619191
$111$	0	0
$112$	18.7032	1.76728
$113$	−17.1572	−1.61401	−0.807006	−	0.590544i	$-0.798913\pi$
$113$	−17.1572	−1.61401	−0.807006	+	0.590544i	$0.798913\pi$
$114$	0	0
$115$	1.07730	0.100459
$116$	−2.85629	−0.265200
$117$	0	0
$118$	−11.2001	−1.03106
$119$	17.9652	1.64687
$120$	0	0
$121$	1.00000	0.0909091
$122$	−8.18196	−0.740760
$123$	0	0
$124$	14.4212	1.29506
$125$	3.31192	0.296227
$126$	0	0
$127$	−16.2330	−1.44045	−0.720225	−	0.693741i	$-0.755961\pi$
$127$	−16.2330	−1.44045	−0.720225	+	0.693741i	$0.755961\pi$
$128$	−3.70966	−0.327891
$129$	0	0
$130$	−1.64469	−0.144249
$131$	−5.70709	−0.498631	−0.249315	−	0.968422i	$-0.580206\pi$
$131$	−5.70709	−0.498631	−0.249315	+	0.968422i	$0.580206\pi$
$132$	0	0
$133$	11.6365	1.00902
$134$	−17.7048	−1.52946
$135$	0	0
$136$	−1.98481	−0.170196
$137$	−14.6488	−1.25153	−0.625764	−	0.780013i	$-0.715213\pi$
$137$	−14.6488	−1.25153	−0.625764	+	0.780013i	$0.715213\pi$
$138$	0	0
$139$	3.73839	0.317086	0.158543	−	0.987352i	$-0.449320\pi$
$139$	3.73839	0.317086	0.158543	+	0.987352i	$0.449320\pi$
$140$	2.49108	0.210535
$141$	0	0
$142$	5.25802	0.441243
$143$	2.53258	0.211785
$144$	0	0
$145$	0.543872	0.0451661
$146$	−13.6680	−1.13117
$147$	0	0
$148$	18.2353	1.49893
$149$	12.3239	1.00961	0.504806	−	0.863233i	$-0.331564\pi$
$149$	12.3239	1.00961	0.504806	+	0.863233i	$0.331564\pi$
$150$	0	0
$151$	−2.42141	−0.197052	−0.0985258	−	0.995134i	$-0.531413\pi$
$151$	−2.42141	−0.197052	−0.0985258	+	0.995134i	$0.531413\pi$
$152$	−1.28561	−0.104277
$153$	0	0
$154$	−8.19713	−0.660543
$155$	−2.74596	−0.220561
$156$	0	0
$157$	20.5343	1.63882	0.819408	−	0.573210i	$-0.194302\pi$
$157$	20.5343	1.63882	0.819408	+	0.573210i	$0.194302\pi$
$158$	33.1834	2.63993
$159$	0	0
$160$	−2.55996	−0.202383
$161$	13.5980	1.07168
$162$	0	0
$163$	10.8378	0.848882	0.424441	−	0.905456i	$-0.360471\pi$
$163$	10.8378	0.848882	0.424441	+	0.905456i	$0.360471\pi$
$164$	14.6617	1.14488
$165$	0	0
$166$	−9.35929	−0.726422
$167$	−16.8859	−1.30667	−0.653334	−	0.757070i	$-0.726630\pi$
$167$	−16.8859	−1.30667	−0.653334	+	0.757070i	$0.726630\pi$
$168$	0	0
$169$	−6.58603	−0.506618
$170$	−2.75951	−0.211645
$171$	0	0
$172$	17.6987	1.34951
$173$	9.26664	0.704530	0.352265	−	0.935900i	$-0.385412\pi$
$173$	9.26664	0.704530	0.352265	+	0.935900i	$0.385412\pi$
$174$	0	0
$175$	20.6650	1.56213
$176$	4.42379	0.333456
$177$	0	0
$178$	18.9396	1.41958
$179$	4.92971	0.368464	0.184232	−	0.982883i	$-0.441020\pi$
$179$	4.92971	0.368464	0.184232	+	0.982883i	$0.441020\pi$
$180$	0	0
$181$	−8.99516	−0.668605	−0.334302	−	0.942466i	$-0.608501\pi$
$181$	−8.99516	−0.668605	−0.334302	+	0.942466i	$0.608501\pi$
$182$	−20.7599	−1.53883
$183$	0	0
$184$	−1.50232	−0.110753
$185$	−3.47221	−0.255282
$186$	0	0
$187$	4.24924	0.310735
$188$	8.16359	0.595391
$189$	0	0
$190$	−1.78741	−0.129672
$191$	1.82833	0.132293	0.0661465	−	0.997810i	$-0.478930\pi$
$191$	1.82833	0.132293	0.0661465	+	0.997810i	$0.478930\pi$
$192$	0	0
$193$	−6.98702	−0.502937	−0.251468	−	0.967866i	$-0.580913\pi$
$193$	−6.98702	−0.502937	−0.251468	+	0.967866i	$0.580913\pi$
$194$	−24.7452	−1.77660
$195$	0	0
$196$	19.1297	1.36641
$197$	14.5432	1.03616	0.518082	−	0.855331i	$-0.326646\pi$
$197$	14.5432	1.03616	0.518082	+	0.855331i	$0.326646\pi$
$198$	0	0
$199$	10.9904	0.779088	0.389544	−	0.921008i	$-0.372633\pi$
$199$	10.9904	0.779088	0.389544	+	0.921008i	$0.372633\pi$
$200$	−2.28308	−0.161438
$201$	0	0
$202$	35.6564	2.50877
$203$	6.86494	0.481824
$204$	0	0
$205$	−2.79175	−0.194985
$206$	−3.98995	−0.277993
$207$	0	0
$208$	11.2036	0.776831
$209$	2.75235	0.190384
$210$	0	0
$211$	−8.79277	−0.605319	−0.302660	−	0.953099i	$-0.597875\pi$
$211$	−8.79277	−0.605319	−0.302660	+	0.953099i	$0.597875\pi$
$212$	−0.00513953	−0.000352984 0
$213$	0	0
$214$	−8.62101	−0.589320
$215$	−3.37004	−0.229835
$216$	0	0
$217$	−34.6605	−2.35291
$218$	−21.3445	−1.44563
$219$	0	0
$220$	0.589205	0.0397242
$221$	10.7615	0.723900
$222$	0	0
$223$	−2.71367	−0.181721	−0.0908604	−	0.995864i	$-0.528962\pi$
$223$	−2.71367	−0.181721	−0.0908604	+	0.995864i	$0.528962\pi$
$224$	−32.3127	−2.15899
$225$	0	0
$226$	33.2649	2.21275
$227$	9.43460	0.626197	0.313098	−	0.949721i	$-0.398633\pi$
$227$	9.43460	0.626197	0.313098	+	0.949721i	$0.398633\pi$
$228$	0	0
$229$	3.85984	0.255065	0.127533	−	0.991834i	$-0.459294\pi$
$229$	3.85984	0.255065	0.127533	+	0.991834i	$0.459294\pi$
$230$	−2.08870	−0.137725
$231$	0	0
$232$	−0.758443	−0.0497942
$233$	−24.2549	−1.58899	−0.794496	−	0.607269i	$-0.792265\pi$
$233$	−24.2549	−1.58899	−0.794496	+	0.607269i	$0.792265\pi$
$234$	0	0
$235$	−1.55444	−0.101401
$236$	10.1618	0.661474
$237$	0	0
$238$	−34.8315	−2.25779
$239$	−27.4758	−1.77726	−0.888631	−	0.458622i	$-0.848343\pi$
$239$	−27.4758	−1.77726	−0.888631	+	0.458622i	$0.848343\pi$
$240$	0	0
$241$	−22.3202	−1.43777	−0.718885	−	0.695129i	$-0.755347\pi$
$241$	−22.3202	−1.43777	−0.718885	+	0.695129i	$0.755347\pi$
$242$	−1.93884	−0.124633
$243$	0	0
$244$	7.42341	0.475235
$245$	−3.64252	−0.232712
$246$	0	0
$247$	6.97054	0.443525
$248$	3.82932	0.243162
$249$	0	0
$250$	−6.42127	−0.406117
$251$	−19.8548	−1.25322	−0.626612	−	0.779331i	$-0.715559\pi$
$251$	−19.8548	−1.25322	−0.626612	+	0.779331i	$0.715559\pi$
$252$	0	0
$253$	3.21630	0.202207
$254$	31.4732	1.97480
$255$	0	0
$256$	19.1336	1.19585
$257$	23.0938	1.44055	0.720276	−	0.693688i	$-0.244015\pi$
$257$	23.0938	1.44055	0.720276	+	0.693688i	$0.244015\pi$
$258$	0	0
$259$	−43.8275	−2.72331
$260$	1.49221	0.0925430
$261$	0	0
$262$	11.0651	0.683605
$263$	5.61669	0.346340	0.173170	−	0.984892i	$-0.444599\pi$
$263$	5.61669	0.346340	0.173170	+	0.984892i	$0.444599\pi$
$264$	0	0
$265$	0.000978626 0	6.01165e−5 0
$266$	−22.5613	−1.38332
$267$	0	0
$268$	16.0634	0.981228
$269$	5.53066	0.337211	0.168605	−	0.985684i	$-0.446074\pi$
$269$	5.53066	0.337211	0.168605	+	0.985684i	$0.446074\pi$
$270$	0	0
$271$	22.7027	1.37909	0.689544	−	0.724244i	$-0.257811\pi$
$271$	22.7027	1.37909	0.689544	+	0.724244i	$0.257811\pi$
$272$	18.7977	1.13978
$273$	0	0
$274$	28.4015	1.71580
$275$	4.88781	0.294746
$276$	0	0
$277$	−22.7379	−1.36619	−0.683095	−	0.730330i	$-0.739366\pi$
$277$	−22.7379	−1.36619	−0.683095	+	0.730330i	$0.739366\pi$
$278$	−7.24812	−0.434713
$279$	0	0
$280$	0.661466	0.0395302
$281$	−2.45412	−0.146400	−0.0732002	−	0.997317i	$-0.523321\pi$
$281$	−2.45412	−0.146400	−0.0732002	+	0.997317i	$0.523321\pi$
$282$	0	0
$283$	13.4710	0.800766	0.400383	−	0.916348i	$-0.368877\pi$
$283$	13.4710	0.800766	0.400383	+	0.916348i	$0.368877\pi$
$284$	−4.77055	−0.283080
$285$	0	0
$286$	−4.91026	−0.290350
$287$	−35.2385	−2.08006
$288$	0	0
$289$	1.05603	0.0621194
$290$	−1.05448	−0.0619211
$291$	0	0
$292$	12.4009	0.725705
$293$	−21.8155	−1.27448	−0.637238	−	0.770667i	$-0.719923\pi$
$293$	−21.8155	−1.27448	−0.637238	+	0.770667i	$0.719923\pi$
$294$	0	0
$295$	−1.93492	−0.112655
$296$	4.84208	0.281440
$297$	0	0
$298$	−23.8940	−1.38414
$299$	8.14553	0.471068
$300$	0	0
$301$	−42.5378	−2.45184
$302$	4.69472	0.270151
$303$	0	0
$304$	12.1758	0.698330
$305$	−1.41350	−0.0809370
$306$	0	0
$307$	−4.84612	−0.276582	−0.138291	−	0.990392i	$-0.544161\pi$
$307$	−4.84612	−0.276582	−0.138291	+	0.990392i	$0.544161\pi$
$308$	7.43716	0.423772
$309$	0	0
$310$	5.32397	0.302381
$311$	−32.5114	−1.84355	−0.921777	−	0.387719i	$-0.873263\pi$
$311$	−32.5114	−1.84355	−0.921777	+	0.387719i	$0.873263\pi$
$312$	0	0
$313$	3.35724	0.189762	0.0948811	−	0.995489i	$-0.469753\pi$
$313$	3.35724	0.189762	0.0948811	+	0.995489i	$0.469753\pi$
$314$	−39.8127	−2.24676
$315$	0	0
$316$	−30.1069	−1.69365
$317$	26.5123	1.48908	0.744539	−	0.667579i	$-0.232669\pi$
$317$	26.5123	1.48908	0.744539	+	0.667579i	$0.232669\pi$
$318$	0	0
$319$	1.62374	0.0909119
$320$	1.99984	0.111795
$321$	0	0
$322$	−26.3644	−1.46923
$323$	11.6954	0.650748
$324$	0	0
$325$	12.3788	0.686651
$326$	−21.0127	−1.16379
$327$	0	0
$328$	3.89317	0.214965
$329$	−19.6207	−1.08173
$330$	0	0
$331$	3.03935	0.167058	0.0835290	−	0.996505i	$-0.473381\pi$
$331$	3.03935	0.167058	0.0835290	+	0.996505i	$0.473381\pi$
$332$	8.49158	0.466036
$333$	0	0
$334$	32.7389	1.79139
$335$	−3.05866	−0.167112
$336$	0	0
$337$	1.74351	0.0949750	0.0474875	−	0.998872i	$-0.484879\pi$
$337$	1.74351	0.0949750	0.0474875	+	0.998872i	$0.484879\pi$
$338$	12.7692	0.694555
$339$	0	0
$340$	2.50367	0.135781
$341$	−8.19813	−0.443954
$342$	0	0
$343$	−16.3821	−0.884552
$344$	4.69961	0.253386
$345$	0	0
$346$	−17.9665	−0.965885
$347$	29.9029	1.60527	0.802637	−	0.596468i	$-0.203430\pi$
$347$	29.9029	1.60527	0.802637	+	0.596468i	$0.203430\pi$
$348$	0	0
$349$	−8.16245	−0.436926	−0.218463	−	0.975845i	$-0.570104\pi$
$349$	−8.16245	−0.436926	−0.218463	+	0.975845i	$0.570104\pi$
$350$	−40.0660	−2.14162
$351$	0	0
$352$	−7.64281	−0.407363
$353$	24.5857	1.30856	0.654281	−	0.756251i	$-0.272971\pi$
$353$	24.5857	1.30856	0.654281	+	0.756251i	$0.272971\pi$
$354$	0	0
$355$	0.908368	0.0482111
$356$	−17.1837	−0.910734
$357$	0	0
$358$	−9.55789	−0.505150
$359$	−22.1875	−1.17101	−0.585506	−	0.810668i	$-0.699104\pi$
$359$	−22.1875	−1.17101	−0.585506	+	0.810668i	$0.699104\pi$
$360$	0	0
$361$	−11.4246	−0.601294
$362$	17.4401	0.916633
$363$	0	0
$364$	18.8352	0.987234
$365$	−2.36127	−0.123594
$366$	0	0
$367$	0.315846	0.0164871	0.00824353	−	0.999966i	$-0.497376\pi$
$367$	0.315846	0.0164871	0.00824353	+	0.999966i	$0.497376\pi$
$368$	14.2282	0.741697
$369$	0	0
$370$	6.73204	0.349982
$371$	0.0123526	0.000641314 0
$372$	0	0
$373$	6.22516	0.322327	0.161163	−	0.986928i	$-0.448475\pi$
$373$	6.22516	0.322327	0.161163	+	0.986928i	$0.448475\pi$
$374$	−8.23858	−0.426007
$375$	0	0
$376$	2.16771	0.111791
$377$	4.11225	0.211792
$378$	0	0
$379$	−0.294090	−0.0151064	−0.00755321	−	0.999971i	$-0.502404\pi$
$379$	−0.294090	−0.0151064	−0.00755321	+	0.999971i	$0.502404\pi$
$380$	1.62170	0.0831913
$381$	0	0
$382$	−3.54482	−0.181369
$383$	−17.4662	−0.892482	−0.446241	−	0.894913i	$-0.647238\pi$
$383$	−17.4662	−0.892482	−0.446241	+	0.894913i	$0.647238\pi$
$384$	0	0
$385$	−1.41612	−0.0721723
$386$	13.5467	0.689508
$387$	0	0
$388$	22.4510	1.13978
$389$	−29.7573	−1.50875	−0.754376	−	0.656442i	$-0.772061\pi$
$389$	−29.7573	−1.50875	−0.754376	+	0.656442i	$0.772061\pi$
$390$	0	0
$391$	13.6668	0.691160
$392$	5.07958	0.256558
$393$	0	0
$394$	−28.1970	−1.42054
$395$	5.73271	0.288444
$396$	0	0
$397$	2.42989	0.121953	0.0609763	−	0.998139i	$-0.480579\pi$
$397$	2.42989	0.121953	0.0609763	+	0.998139i	$0.480579\pi$
$398$	−21.3085	−1.06810
$399$	0	0
$400$	21.6226	1.08113
$401$	−3.87608	−0.193562	−0.0967811	−	0.995306i	$-0.530855\pi$
$401$	−3.87608	−0.193562	−0.0967811	+	0.995306i	$0.530855\pi$
$402$	0	0
$403$	−20.7624	−1.03425
$404$	−32.3506	−1.60950
$405$	0	0
$406$	−13.3100	−0.660564
$407$	−10.3663	−0.513841
$408$	0	0
$409$	5.17650	0.255962	0.127981	−	0.991777i	$-0.459150\pi$
$409$	5.17650	0.255962	0.127981	+	0.991777i	$0.459150\pi$
$410$	5.41275	0.267317
$411$	0	0
$412$	3.62003	0.178346
$413$	−24.4232	−1.20179
$414$	0	0
$415$	−1.61690	−0.0793703
$416$	−19.3560	−0.949008
$417$	0	0
$418$	−5.33635	−0.261009
$419$	7.66295	0.374360	0.187180	−	0.982326i	$-0.440065\pi$
$419$	7.66295	0.374360	0.187180	+	0.982326i	$0.440065\pi$
$420$	0	0
$421$	11.5501	0.562916	0.281458	−	0.959574i	$-0.409182\pi$
$421$	11.5501	0.562916	0.281458	+	0.959574i	$0.409182\pi$
$422$	17.0477	0.829871
$423$	0	0
$424$	−0.00136472	−6.62767e−5 0
$425$	20.7695	1.00747
$426$	0	0
$427$	−17.8417	−0.863422
$428$	7.82174	0.378078
$429$	0	0
$430$	6.53395	0.315095
$431$	32.5221	1.56654	0.783268	−	0.621684i	$-0.213551\pi$
$431$	32.5221	1.56654	0.783268	+	0.621684i	$0.213551\pi$
$432$	0	0
$433$	−10.5035	−0.504766	−0.252383	−	0.967627i	$-0.581214\pi$
$433$	−10.5035	−0.504766	−0.252383	+	0.967627i	$0.581214\pi$
$434$	67.2011	3.22576
$435$	0	0
$436$	19.3656	0.927443
$437$	8.85236	0.423466
$438$	0	0
$439$	−1.68900	−0.0806117	−0.0403058	−	0.999187i	$-0.512833\pi$
$439$	−1.68900	−0.0806117	−0.0403058	+	0.999187i	$0.512833\pi$
$440$	0.156454	0.00745866
$441$	0	0
$442$	−20.8649	−0.992440
$443$	18.3247	0.870631	0.435316	−	0.900278i	$-0.356637\pi$
$443$	18.3247	0.870631	0.435316	+	0.900278i	$0.356637\pi$
$444$	0	0
$445$	3.27198	0.155107
$446$	5.26136	0.249133
$447$	0	0
$448$	25.2428	1.19261
$449$	−14.2195	−0.671060	−0.335530	−	0.942029i	$-0.608915\pi$
$449$	−14.2195	−0.671060	−0.335530	+	0.942029i	$0.608915\pi$
$450$	0	0
$451$	−8.33484	−0.392472
$452$	−30.1809	−1.41959
$453$	0	0
$454$	−18.2921	−0.858493
$455$	−3.58645	−0.168135
$456$	0	0
$457$	20.2263	0.946147	0.473074	−	0.881023i	$-0.343144\pi$
$457$	20.2263	0.946147	0.473074	+	0.881023i	$0.343144\pi$
$458$	−7.48360	−0.349686
$459$	0	0
$460$	1.89506	0.0883576
$461$	−17.6136	−0.820346	−0.410173	−	0.912008i	$-0.634532\pi$
$461$	−17.6136	−0.820346	−0.410173	+	0.912008i	$0.634532\pi$
$462$	0	0
$463$	2.87689	0.133701	0.0668503	−	0.997763i	$-0.478705\pi$
$463$	2.87689	0.133701	0.0668503	+	0.997763i	$0.478705\pi$
$464$	7.18308	0.333466
$465$	0	0
$466$	47.0263	2.17845
$467$	2.49722	0.115557	0.0577787	−	0.998329i	$-0.481598\pi$
$467$	2.49722	0.115557	0.0577787	+	0.998329i	$0.481598\pi$
$468$	0	0
$469$	−38.6075	−1.78273
$470$	3.01381	0.139017
$471$	0	0
$472$	2.69829	0.124199
$473$	−10.0613	−0.462620
$474$	0	0
$475$	13.4529	0.617263
$476$	31.6023	1.44849
$477$	0	0
$478$	53.2711	2.43656
$479$	33.2596	1.51967	0.759834	−	0.650117i	$-0.225280\pi$
$479$	33.2596	1.51967	0.759834	+	0.650117i	$0.225280\pi$
$480$	0	0
$481$	−26.2536	−1.19706
$482$	43.2752	1.97113
$483$	0	0
$484$	1.75908	0.0799584
$485$	−4.27494	−0.194115
$486$	0	0
$487$	−1.38177	−0.0626140	−0.0313070	−	0.999510i	$-0.509967\pi$
$487$	−1.38177	−0.0626140	−0.0313070	+	0.999510i	$0.509967\pi$
$488$	1.97117	0.0892305
$489$	0	0
$490$	7.06224	0.319040
$491$	−3.59300	−0.162150	−0.0810749	−	0.996708i	$-0.525835\pi$
$491$	−3.59300	−0.162150	−0.0810749	+	0.996708i	$0.525835\pi$
$492$	0	0
$493$	6.89965	0.310745
$494$	−13.5147	−0.608057
$495$	0	0
$496$	−36.2668	−1.62843
$497$	11.4657	0.514309
$498$	0	0
$499$	−35.4934	−1.58890	−0.794450	−	0.607329i	$-0.792241\pi$
$499$	−35.4934	−1.58890	−0.794450	+	0.607329i	$0.792241\pi$
$500$	5.82595	0.260544
$501$	0	0
$502$	38.4952	1.71813
$503$	12.0144	0.535695	0.267848	−	0.963461i	$-0.413688\pi$
$503$	12.0144	0.535695	0.267848	+	0.963461i	$0.413688\pi$
$504$	0	0
$505$	6.15994	0.274114
$506$	−6.23587	−0.277218
$507$	0	0
$508$	−28.5553	−1.26694
$509$	20.4133	0.904804	0.452402	−	0.891814i	$-0.350567\pi$
$509$	20.4133	0.904804	0.452402	+	0.891814i	$0.350567\pi$
$510$	0	0
$511$	−29.8048	−1.31848
$512$	−29.6775	−1.31157
$513$	0	0
$514$	−44.7751	−1.97494
$515$	−0.689297	−0.0303740
$516$	0	0
$517$	−4.64082	−0.204103
$518$	84.9742	3.73355
$519$	0	0
$520$	0.396233	0.0173760
$521$	−1.75715	−0.0769823	−0.0384912	−	0.999259i	$-0.512255\pi$
$521$	−1.75715	−0.0769823	−0.0384912	+	0.999259i	$0.512255\pi$
$522$	0	0
$523$	−14.8013	−0.647214	−0.323607	−	0.946192i	$-0.604896\pi$
$523$	−14.8013	−0.647214	−0.323607	+	0.946192i	$0.604896\pi$
$524$	−10.0392	−0.438567
$525$	0	0
$526$	−10.8898	−0.474819
$527$	−34.8358	−1.51747
$528$	0	0
$529$	−12.6554	−0.550237
$530$	−0.00189740	−8.24176e−5 0
$531$	0	0
$532$	20.4696	0.887472
$533$	−21.1086	−0.914317
$534$	0	0
$535$	−1.48935	−0.0643903
$536$	4.26538	0.184236
$537$	0	0
$538$	−10.7231	−0.462304
$539$	−10.8748	−0.468411
$540$	0	0
$541$	−8.63769	−0.371363	−0.185682	−	0.982610i	$-0.559449\pi$
$541$	−8.63769	−0.371363	−0.185682	+	0.982610i	$0.559449\pi$
$542$	−44.0167	−1.89068
$543$	0	0
$544$	−32.4761	−1.39240
$545$	−3.68743	−0.157952
$546$	0	0
$547$	21.7879	0.931582	0.465791	−	0.884895i	$-0.345770\pi$
$547$	21.7879	0.931582	0.465791	+	0.884895i	$0.345770\pi$
$548$	−25.7684	−1.10077
$549$	0	0
$550$	−9.47666	−0.404086
$551$	4.46909	0.190390
$552$	0	0
$553$	72.3603	3.07707
$554$	44.0851	1.87300
$555$	0	0
$556$	6.57614	0.278890
$557$	−36.4394	−1.54399	−0.771994	−	0.635629i	$-0.780741\pi$
$557$	−36.4394	−1.54399	−0.771994	+	0.635629i	$0.780741\pi$
$558$	0	0
$559$	−25.4811	−1.07774
$560$	−6.26463	−0.264729
$561$	0	0
$562$	4.75813	0.200710
$563$	−19.4423	−0.819394	−0.409697	−	0.912222i	$-0.634366\pi$
$563$	−19.4423	−0.819394	−0.409697	+	0.912222i	$0.634366\pi$
$564$	0	0
$565$	5.74680	0.241770
$566$	−26.1180	−1.09782
$567$	0	0
$568$	−1.26674	−0.0531513
$569$	7.83650	0.328523	0.164262	−	0.986417i	$-0.447476\pi$
$569$	7.83650	0.328523	0.164262	+	0.986417i	$0.447476\pi$
$570$	0	0
$571$	13.4303	0.562041	0.281021	−	0.959702i	$-0.409327\pi$
$571$	13.4303	0.562041	0.281021	+	0.959702i	$0.409327\pi$
$572$	4.45502	0.186274
$573$	0	0
$574$	68.3217	2.85169
$575$	15.7206	0.655596
$576$	0	0
$577$	30.3645	1.26409	0.632045	−	0.774931i	$-0.282216\pi$
$577$	30.3645	1.26409	0.632045	+	0.774931i	$0.282216\pi$
$578$	−2.04747	−0.0851634
$579$	0	0
$580$	0.956716	0.0397255
$581$	−20.4090	−0.846710
$582$	0	0
$583$	0.00292171	0.000121005 0
$584$	3.29285	0.136259
$585$	0	0
$586$	42.2967	1.74726
$587$	38.8024	1.60155	0.800774	−	0.598967i	$-0.204422\pi$
$587$	38.8024	1.60155	0.800774	+	0.598967i	$0.204422\pi$
$588$	0	0
$589$	−22.5641	−0.929737
$590$	3.75149	0.154446
$591$	0	0
$592$	−45.8585	−1.88477
$593$	−43.7057	−1.79478	−0.897388	−	0.441242i	$-0.854538\pi$
$593$	−43.7057	−1.79478	−0.897388	+	0.441242i	$0.854538\pi$
$594$	0	0
$595$	−6.01744	−0.246691
$596$	21.6787	0.887995
$597$	0	0
$598$	−15.7928	−0.645817
$599$	−26.8789	−1.09824	−0.549121	−	0.835743i	$-0.685037\pi$
$599$	−26.8789	−1.09824	−0.549121	+	0.835743i	$0.685037\pi$
$600$	0	0
$601$	22.8053	0.930246	0.465123	−	0.885246i	$-0.346010\pi$
$601$	22.8053	0.930246	0.465123	+	0.885246i	$0.346010\pi$
$602$	82.4739	3.36138
$603$	0	0
$604$	−4.25947	−0.173315
$605$	−0.334950	−0.0136177
$606$	0	0
$607$	−30.3402	−1.23147	−0.615735	−	0.787953i	$-0.711141\pi$
$607$	−30.3402	−1.23147	−0.615735	+	0.787953i	$0.711141\pi$
$608$	−21.0357	−0.853109
$609$	0	0
$610$	2.74055	0.110962
$611$	−11.7532	−0.475486
$612$	0	0
$613$	10.3269	0.417099	0.208549	−	0.978012i	$-0.433126\pi$
$613$	10.3269	0.417099	0.208549	+	0.978012i	$0.433126\pi$
$614$	9.39582	0.379185
$615$	0	0
$616$	1.97482	0.0795677
$617$	−20.4144	−0.821852	−0.410926	−	0.911669i	$-0.634795\pi$
$617$	−20.4144	−0.821852	−0.410926	+	0.911669i	$0.634795\pi$
$618$	0	0
$619$	16.9641	0.681844	0.340922	−	0.940092i	$-0.389261\pi$
$619$	16.9641	0.681844	0.340922	+	0.940092i	$0.389261\pi$
$620$	−4.83038	−0.193993
$621$	0	0
$622$	63.0343	2.52745
$623$	41.3001	1.65465
$624$	0	0
$625$	23.3297	0.933189
$626$	−6.50913	−0.260157
$627$	0	0
$628$	36.1216	1.44141
$629$	−44.0491	−1.75635
$630$	0	0
$631$	47.1666	1.87767	0.938837	−	0.344362i	$-0.111905\pi$
$631$	47.1666	1.87767	0.938837	+	0.344362i	$0.111905\pi$
$632$	−7.99441	−0.318001
$633$	0	0
$634$	−51.4030	−2.04147
$635$	5.43726	0.215771
$636$	0	0
$637$	−27.5413	−1.09123
$638$	−3.14816	−0.124637
$639$	0	0
$640$	1.24255	0.0491162
$641$	29.8337	1.17836	0.589180	−	0.808002i	$-0.299451\pi$
$641$	29.8337	1.17836	0.589180	+	0.808002i	$0.299451\pi$
$642$	0	0
$643$	−21.1638	−0.834619	−0.417309	−	0.908764i	$-0.637027\pi$
$643$	−21.1638	−0.834619	−0.417309	+	0.908764i	$0.637027\pi$
$644$	23.9201	0.942584
$645$	0	0
$646$	−22.6754	−0.892152
$647$	4.17576	0.164166	0.0820831	−	0.996625i	$-0.473843\pi$
$647$	4.17576	0.164166	0.0820831	+	0.996625i	$0.473843\pi$
$648$	0	0
$649$	−5.77673	−0.226757
$650$	−24.0004	−0.941373
$651$	0	0
$652$	19.0646	0.746627
$653$	−0.184434	−0.00721747	−0.00360873	−	0.999993i	$-0.501149\pi$
$653$	−0.184434	−0.00721747	−0.00360873	+	0.999993i	$0.501149\pi$
$654$	0	0
$655$	1.91159	0.0746920
$656$	−36.8716	−1.43959
$657$	0	0
$658$	38.0414	1.48301
$659$	−32.0547	−1.24867	−0.624336	−	0.781156i	$-0.714630\pi$
$659$	−32.0547	−1.24867	−0.624336	+	0.781156i	$0.714630\pi$
$660$	0	0
$661$	7.29725	0.283830	0.141915	−	0.989879i	$-0.454674\pi$
$661$	7.29725	0.283830	0.141915	+	0.989879i	$0.454674\pi$
$662$	−5.89281	−0.229030
$663$	0	0
$664$	2.25480	0.0875033
$665$	−3.89766	−0.151145
$666$	0	0
$667$	5.22242	0.202213
$668$	−29.7037	−1.14927
$669$	0	0
$670$	5.93024	0.229105
$671$	−4.22004	−0.162913
$672$	0	0
$673$	−1.24281	−0.0479067	−0.0239534	−	0.999713i	$-0.507625\pi$
$673$	−1.24281	−0.0479067	−0.0239534	+	0.999713i	$0.507625\pi$
$674$	−3.38038	−0.130207
$675$	0	0
$676$	−11.5854	−0.445592
$677$	−12.1931	−0.468619	−0.234310	−	0.972162i	$-0.575283\pi$
$677$	−12.1931	−0.468619	−0.234310	+	0.972162i	$0.575283\pi$
$678$	0	0
$679$	−53.9598	−2.07079
$680$	0.664811	0.0254943
$681$	0	0
$682$	15.8948	0.608644
$683$	33.4483	1.27986	0.639932	−	0.768431i	$-0.278962\pi$
$683$	33.4483	1.27986	0.639932	+	0.768431i	$0.278962\pi$
$684$	0	0
$685$	4.90660	0.187472
$686$	31.7623	1.21269
$687$	0	0
$688$	−44.5092	−1.69690
$689$	0.00739946	0.000281897 0
$690$	0	0
$691$	−51.6418	−1.96455	−0.982273	−	0.187458i	$-0.939975\pi$
$691$	−51.6418	−1.96455	−0.982273	+	0.187458i	$0.939975\pi$
$692$	16.3008	0.619663
$693$	0	0
$694$	−57.9769	−2.20077
$695$	−1.25217	−0.0474976
$696$	0	0
$697$	−35.4167	−1.34150
$698$	15.8257	0.599010
$699$	0	0
$700$	36.3514	1.37395
$701$	1.44569	0.0546031	0.0273015	−	0.999627i	$-0.491309\pi$
$701$	1.44569	0.0546031	0.0273015	+	0.999627i	$0.491309\pi$
$702$	0	0
$703$	−28.5318	−1.07610
$704$	5.97057	0.225024
$705$	0	0
$706$	−47.6676	−1.79399
$707$	77.7529	2.92420
$708$	0	0
$709$	−51.6448	−1.93956	−0.969780	−	0.243979i	$-0.921547\pi$
$709$	−51.6448	−1.93956	−0.969780	+	0.243979i	$0.921547\pi$
$710$	−1.76118	−0.0660957
$711$	0	0
$712$	−4.56286	−0.171000
$713$	−26.3676	−0.987474
$714$	0	0
$715$	−0.848288	−0.0317242
$716$	8.67177	0.324079
$717$	0	0
$718$	43.0179	1.60541
$719$	44.8878	1.67403	0.837016	−	0.547178i	$-0.184298\pi$
$719$	44.8878	1.67403	0.837016	+	0.547178i	$0.184298\pi$
$720$	0	0
$721$	−8.70055	−0.324025
$722$	22.1504	0.824353
$723$	0	0
$724$	−15.8232	−0.588066
$725$	7.93652	0.294755
$726$	0	0
$727$	−22.2012	−0.823398	−0.411699	−	0.911320i	$-0.635065\pi$
$727$	−22.2012	−0.823398	−0.411699	+	0.911320i	$0.635065\pi$
$728$	5.00139	0.185364
$729$	0	0
$730$	4.57811	0.169443
$731$	−42.7529	−1.58127
$732$	0	0
$733$	−26.0567	−0.962425	−0.481212	−	0.876604i	$-0.659803\pi$
$733$	−26.0567	−0.962425	−0.481212	+	0.876604i	$0.659803\pi$
$734$	−0.612374	−0.0226032
$735$	0	0
$736$	−24.5815	−0.906088
$737$	−9.13168	−0.336370
$738$	0	0
$739$	10.2584	0.377362	0.188681	−	0.982038i	$-0.439579\pi$
$739$	10.2584	0.377362	0.188681	+	0.982038i	$0.439579\pi$
$740$	−6.10791	−0.224531
$741$	0	0
$742$	−0.0239496	−0.000879218 0
$743$	−11.5267	−0.422875	−0.211437	−	0.977392i	$-0.567814\pi$
$743$	−11.5267	−0.422875	−0.211437	+	0.977392i	$0.567814\pi$
$744$	0	0
$745$	−4.12788	−0.151234
$746$	−12.0696	−0.441898
$747$	0	0
$748$	7.47477	0.273305
$749$	−18.7991	−0.686905
$750$	0	0
$751$	26.5554	0.969019	0.484509	−	0.874786i	$-0.338998\pi$
$751$	26.5554	0.969019	0.484509	+	0.874786i	$0.338998\pi$
$752$	−20.5300	−0.748652
$753$	0	0
$754$	−7.97298	−0.290359
$755$	0.811052	0.0295172
$756$	0	0
$757$	27.3423	0.993772	0.496886	−	0.867816i	$-0.334477\pi$
$757$	27.3423	0.993772	0.496886	+	0.867816i	$0.334477\pi$
$758$	0.570193	0.0207103
$759$	0	0
$760$	0.430616	0.0156201
$761$	44.6848	1.61982	0.809911	−	0.586553i	$-0.199516\pi$
$761$	44.6848	1.61982	0.809911	+	0.586553i	$0.199516\pi$
$762$	0	0
$763$	−46.5441	−1.68501
$764$	3.21618	0.116357
$765$	0	0
$766$	33.8641	1.22356
$767$	−14.6300	−0.528260
$768$	0	0
$769$	−4.32431	−0.155939	−0.0779693	−	0.996956i	$-0.524844\pi$
$769$	−4.32431	−0.155939	−0.0779693	+	0.996956i	$0.524844\pi$
$770$	2.74563	0.0989456
$771$	0	0
$772$	−12.2908	−0.442354
$773$	9.11139	0.327714	0.163857	−	0.986484i	$-0.447606\pi$
$773$	9.11139	0.327714	0.163857	+	0.986484i	$0.447606\pi$
$774$	0	0
$775$	−40.0709	−1.43939
$776$	5.96151	0.214006
$777$	0	0
$778$	57.6944	2.06845
$779$	−22.9404	−0.821923
$780$	0	0
$781$	2.71195	0.0970412
$782$	−26.4977	−0.947556
$783$	0	0
$784$	−48.1079	−1.71814
$785$	−6.87797	−0.245485
$786$	0	0
$787$	41.3976	1.47567	0.737833	−	0.674983i	$-0.235849\pi$
$787$	41.3976	1.47567	0.737833	+	0.674983i	$0.235849\pi$
$788$	25.5828	0.911349
$789$	0	0
$790$	−11.1148	−0.395446
$791$	72.5382	2.57916
$792$	0	0
$793$	−10.6876	−0.379528
$794$	−4.71115	−0.167193
$795$	0	0
$796$	19.3330	0.685240
$797$	21.2783	0.753717	0.376858	−	0.926271i	$-0.377004\pi$
$797$	21.2783	0.753717	0.376858	+	0.926271i	$0.377004\pi$
$798$	0	0
$799$	−19.7199	−0.697641
$800$	−37.3566	−1.32076
$801$	0	0
$802$	7.51508	0.265367
$803$	−7.04961	−0.248775
$804$	0	0
$805$	−4.55467	−0.160531
$806$	40.2549	1.41792
$807$	0	0
$808$	−8.59019	−0.302202
$809$	−36.4608	−1.28189	−0.640946	−	0.767586i	$-0.721458\pi$
$809$	−36.4608	−1.28189	−0.640946	+	0.767586i	$0.721458\pi$
$810$	0	0
$811$	16.8820	0.592809	0.296404	−	0.955063i	$-0.404212\pi$
$811$	16.8820	0.592809	0.296404	+	0.955063i	$0.404212\pi$
$812$	12.0760	0.423785
$813$	0	0
$814$	20.0986	0.704457
$815$	−3.63012	−0.127158
$816$	0	0
$817$	−27.6922	−0.968828
$818$	−10.0364	−0.350914
$819$	0	0
$820$	−4.91093	−0.171497
$821$	−18.2008	−0.635212	−0.317606	−	0.948223i	$-0.602879\pi$
$821$	−18.2008	−0.635212	−0.317606	+	0.948223i	$0.602879\pi$
$822$	0	0
$823$	−36.4596	−1.27090	−0.635451	−	0.772141i	$-0.719186\pi$
$823$	−36.4596	−1.27090	−0.635451	+	0.772141i	$0.719186\pi$
$824$	0.961242	0.0334864
$825$	0	0
$826$	47.3526	1.64761
$827$	22.5582	0.784424	0.392212	−	0.919875i	$-0.371710\pi$
$827$	22.5582	0.784424	0.392212	+	0.919875i	$0.371710\pi$
$828$	0	0
$829$	39.7026	1.37893	0.689464	−	0.724320i	$-0.257846\pi$
$829$	39.7026	1.37893	0.689464	+	0.724320i	$0.257846\pi$
$830$	3.13490	0.108814
$831$	0	0
$832$	15.1210	0.524225
$833$	−46.2096	−1.60107
$834$	0	0
$835$	5.65592	0.195731
$836$	4.84161	0.167451
$837$	0	0
$838$	−14.8572	−0.513234
$839$	−43.0299	−1.48556	−0.742778	−	0.669538i	$-0.766492\pi$
$839$	−43.0299	−1.48556	−0.742778	+	0.669538i	$0.766492\pi$
$840$	0	0
$841$	−26.3635	−0.909085
$842$	−22.3937	−0.771737
$843$	0	0
$844$	−15.4672	−0.532404
$845$	2.20599	0.0758884
$846$	0	0
$847$	−4.22786	−0.145271
$848$	0.0129250	0.000443847 0
$849$	0	0
$850$	−40.2686	−1.38120
$851$	−33.3412	−1.14292
$852$	0	0
$853$	−54.2870	−1.85875	−0.929375	−	0.369137i	$-0.879653\pi$
$853$	−54.2870	−1.85875	−0.929375	+	0.369137i	$0.879653\pi$
$854$	34.5922	1.18372
$855$	0	0
$856$	2.07694	0.0709883
$857$	50.2040	1.71494	0.857468	−	0.514537i	$-0.172036\pi$
$857$	50.2040	1.71494	0.857468	+	0.514537i	$0.172036\pi$
$858$	0	0
$859$	−25.6674	−0.875761	−0.437880	−	0.899033i	$-0.644271\pi$
$859$	−25.6674	−0.875761	−0.437880	+	0.899033i	$0.644271\pi$
$860$	−5.92818	−0.202149
$861$	0	0
$862$	−63.0551	−2.14767
$863$	30.8645	1.05064	0.525319	−	0.850905i	$-0.323946\pi$
$863$	30.8645	1.05064	0.525319	+	0.850905i	$0.323946\pi$
$864$	0	0
$865$	−3.10386	−0.105535
$866$	20.3645	0.692016
$867$	0	0
$868$	−60.9708	−2.06948
$869$	17.1151	0.580590
$870$	0	0
$871$	−23.1267	−0.783619
$872$	5.14222	0.174138
$873$	0	0
$874$	−17.1633	−0.580556
$875$	−14.0023	−0.473366
$876$	0	0
$877$	35.3446	1.19350	0.596752	−	0.802426i	$-0.296458\pi$
$877$	35.3446	1.19350	0.596752	+	0.802426i	$0.296458\pi$
$878$	3.27470	0.110516
$879$	0	0
$880$	−1.48175	−0.0499498
$881$	−3.76266	−0.126767	−0.0633836	−	0.997989i	$-0.520189\pi$
$881$	−3.76266	−0.126767	−0.0633836	+	0.997989i	$0.520189\pi$
$882$	0	0
$883$	43.3263	1.45805	0.729023	−	0.684489i	$-0.239975\pi$
$883$	43.3263	1.45805	0.729023	+	0.684489i	$0.239975\pi$
$884$	18.9305	0.636700
$885$	0	0
$886$	−35.5285	−1.19360
$887$	−14.9335	−0.501418	−0.250709	−	0.968062i	$-0.580664\pi$
$887$	−14.9335	−0.501418	−0.250709	+	0.968062i	$0.580664\pi$
$888$	0	0
$889$	68.6311	2.30181
$890$	−6.34383	−0.212646
$891$	0	0
$892$	−4.77357	−0.159831
$893$	−12.7731	−0.427437
$894$	0	0
$895$	−1.65121	−0.0551937
$896$	15.6839	0.523964
$897$	0	0
$898$	27.5693	0.920000
$899$	−13.3116	−0.443967
$900$	0	0
$901$	0.0124150	0.000413605 0
$902$	16.1599	0.538065
$903$	0	0
$904$	−8.01406	−0.266544
$905$	3.01293	0.100153
$906$	0	0
$907$	59.7070	1.98254	0.991269	−	0.131852i	$-0.0420923\pi$
$907$	59.7070	1.98254	0.991269	+	0.131852i	$0.0420923\pi$
$908$	16.5963	0.550766
$909$	0	0
$910$	6.95353	0.230507
$911$	−10.8646	−0.359960	−0.179980	−	0.983670i	$-0.557603\pi$
$911$	−10.8646	−0.359960	−0.179980	+	0.983670i	$0.557603\pi$
$912$	0	0
$913$	−4.82727	−0.159759
$914$	−39.2155	−1.29713
$915$	0	0
$916$	6.78978	0.224341
$917$	24.1288	0.796802
$918$	0	0
$919$	15.2109	0.501763	0.250881	−	0.968018i	$-0.419280\pi$
$919$	15.2109	0.501763	0.250881	+	0.968018i	$0.419280\pi$
$920$	0.503203	0.0165901
$921$	0	0
$922$	34.1498	1.12466
$923$	6.86823	0.226071
$924$	0	0
$925$	−50.6687	−1.66598
$926$	−5.57782	−0.183299
$927$	0	0
$928$	−12.4099	−0.407376
$929$	−22.7927	−0.747804	−0.373902	−	0.927468i	$-0.621980\pi$
$929$	−22.7927	−0.747804	−0.373902	+	0.927468i	$0.621980\pi$
$930$	0	0
$931$	−29.9312	−0.980956
$932$	−42.6664	−1.39759
$933$	0	0
$934$	−4.84169	−0.158425
$935$	−1.42328	−0.0465463
$936$	0	0
$937$	−46.2715	−1.51162	−0.755812	−	0.654788i	$-0.772758\pi$
$937$	−46.2715	−1.51162	−0.755812	+	0.654788i	$0.772758\pi$
$938$	74.8536	2.44406
$939$	0	0
$940$	−2.73439	−0.0891861
$941$	−48.3193	−1.57516	−0.787582	−	0.616210i	$-0.788667\pi$
$941$	−48.3193	−1.57516	−0.787582	+	0.616210i	$0.788667\pi$
$942$	0	0
$943$	−26.8073	−0.872966
$944$	−25.5551	−0.831746
$945$	0	0
$946$	19.5072	0.634235
$947$	−51.5251	−1.67434	−0.837171	−	0.546942i	$-0.815792\pi$
$947$	−51.5251	−1.67434	−0.837171	+	0.546942i	$0.815792\pi$
$948$	0	0
$949$	−17.8537	−0.579556
$950$	−26.0830	−0.846246
$951$	0	0
$952$	8.39148	0.271969
$953$	−56.5728	−1.83257	−0.916287	−	0.400523i	$-0.868828\pi$
$953$	−56.5728	−1.83257	−0.916287	+	0.400523i	$0.868828\pi$
$954$	0	0
$955$	−0.612398	−0.0198167
$956$	−48.3323	−1.56318
$957$	0	0
$958$	−64.4848	−2.08341
$959$	61.9329	1.99992
$960$	0	0
$961$	36.2093	1.16804
$962$	50.9014	1.64113
$963$	0	0
$964$	−39.2631	−1.26458
$965$	2.34030	0.0753370
$966$	0	0
$967$	−30.4089	−0.977884	−0.488942	−	0.872316i	$-0.662617\pi$
$967$	−30.4089	−0.977884	−0.488942	+	0.872316i	$0.662617\pi$
$968$	0.467097	0.0150131
$969$	0	0
$970$	8.28840	0.266125
$971$	48.4531	1.55493	0.777467	−	0.628924i	$-0.216504\pi$
$971$	48.4531	1.55493	0.777467	+	0.628924i	$0.216504\pi$
$972$	0	0
$973$	−15.8054	−0.506697
$974$	2.67903	0.0858416
$975$	0	0
$976$	−18.6686	−0.597567
$977$	−6.69386	−0.214156	−0.107078	−	0.994251i	$-0.534149\pi$
$977$	−6.69386	−0.214156	−0.107078	+	0.994251i	$0.534149\pi$
$978$	0	0
$979$	9.76855	0.312204
$980$	−6.40749	−0.204680
$981$	0	0
$982$	6.96624	0.222302
$983$	−23.4574	−0.748175	−0.374087	−	0.927393i	$-0.622044\pi$
$983$	−23.4574	−0.748175	−0.374087	+	0.927393i	$0.622044\pi$
$984$	0	0
$985$	−4.87126	−0.155211
$986$	−13.3773	−0.426020
$987$	0	0
$988$	12.2618	0.390099
$989$	−32.3602	−1.02899
$990$	0	0
$991$	22.9614	0.729393	0.364697	−	0.931126i	$-0.381173\pi$
$991$	22.9614	0.729393	0.364697	+	0.931126i	$0.381173\pi$
$992$	62.6568	1.98935
$993$	0	0
$994$	−22.2302	−0.705099
$995$	−3.68123	−0.116703
$996$	0	0
$997$	57.7071	1.82760	0.913801	−	0.406163i	$-0.133134\pi$
$997$	57.7071	1.82760	0.913801	+	0.406163i	$0.133134\pi$
$998$	68.8158	2.17833
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8019.2.a.i.1.11	✓	48
3.2	odd	2		8019.2.a.j.1.38	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8019.2.a.i.1.11	✓	48	1.1	even	1	trivial
8019.2.a.j.1.38	yes	48	3.2	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:	\( N \)	\(=\)	\( 8019 = 3^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8019.a (trivial)

\(f(q)\)	\(=\)	\(q-1.93884 q^{2} +1.75908 q^{4} -0.334950 q^{5} -4.22786 q^{7} +0.467097 q^{8} +O(q^{10})\)	\(q-1.93884 q^{2} +1.75908 q^{4} -0.334950 q^{5} -4.22786 q^{7} +0.467097 q^{8} +0.649413 q^{10} -1.00000 q^{11} -2.53258 q^{13} +8.19713 q^{14} -4.42379 q^{16} -4.24924 q^{17} -2.75235 q^{19} -0.589205 q^{20} +1.93884 q^{22} -3.21630 q^{23} -4.88781 q^{25} +4.91026 q^{26} -7.43716 q^{28} -1.62374 q^{29} +8.19813 q^{31} +7.64281 q^{32} +8.23858 q^{34} +1.41612 q^{35} +10.3663 q^{37} +5.33635 q^{38} -0.156454 q^{40} +8.33484 q^{41} +10.0613 q^{43} -1.75908 q^{44} +6.23587 q^{46} +4.64082 q^{47} +10.8748 q^{49} +9.47666 q^{50} -4.45502 q^{52} -0.00292171 q^{53} +0.334950 q^{55} -1.97482 q^{56} +3.14816 q^{58} +5.77673 q^{59} +4.22004 q^{61} -15.8948 q^{62} -5.97057 q^{64} +0.848288 q^{65} +9.13168 q^{67} -7.47477 q^{68} -2.74563 q^{70} -2.71195 q^{71} +7.04961 q^{73} -20.0986 q^{74} -4.84161 q^{76} +4.22786 q^{77} -17.1151 q^{79} +1.48175 q^{80} -16.1599 q^{82} +4.82727 q^{83} +1.42328 q^{85} -19.5072 q^{86} -0.467097 q^{88} -9.76855 q^{89} +10.7074 q^{91} -5.65773 q^{92} -8.99778 q^{94} +0.921899 q^{95} +12.7629 q^{97} -21.0845 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 6 q^{2} + 54 q^{4} - 24 q^{5} - 18 q^{8}+O(q^{10}) \) 48 * q - 6 * q^2 + 54 * q^4 - 24 * q^5 - 18 * q^8	\( 48 q - 6 q^{2} + 54 q^{4} - 24 q^{5} - 18 q^{8} - 48 q^{11} - 24 q^{14} + 66 q^{16} - 24 q^{17} - 48 q^{20} + 6 q^{22} - 12 q^{23} + 60 q^{25} - 36 q^{26} - 18 q^{28} - 60 q^{29} + 36 q^{31} - 42 q^{32} + 12 q^{34} - 24 q^{35} + 6 q^{37} - 24 q^{38} - 72 q^{41} - 12 q^{43} - 54 q^{44} - 30 q^{46} - 36 q^{47} + 60 q^{49} - 42 q^{50} - 48 q^{53} + 24 q^{55} - 72 q^{56} + 12 q^{58} - 60 q^{59} - 24 q^{61} - 36 q^{62} + 90 q^{64} - 48 q^{65} - 60 q^{68} - 30 q^{70} - 60 q^{71} - 18 q^{73} - 36 q^{74} - 42 q^{76} - 12 q^{79} - 96 q^{80} + 12 q^{82} - 36 q^{83} + 18 q^{85} - 48 q^{86} + 18 q^{88} - 96 q^{89} + 30 q^{91} - 36 q^{92} - 48 q^{94} - 48 q^{95} + 30 q^{97} - 54 q^{98}+O(q^{100}) \) 48 * q - 6 * q^2 + 54 * q^4 - 24 * q^5 - 18 * q^8 - 48 * q^11 - 24 * q^14 + 66 * q^16 - 24 * q^17 - 48 * q^20 + 6 * q^22 - 12 * q^23 + 60 * q^25 - 36 * q^26 - 18 * q^28 - 60 * q^29 + 36 * q^31 - 42 * q^32 + 12 * q^34 - 24 * q^35 + 6 * q^37 - 24 * q^38 - 72 * q^41 - 12 * q^43 - 54 * q^44 - 30 * q^46 - 36 * q^47 + 60 * q^49 - 42 * q^50 - 48 * q^53 + 24 * q^55 - 72 * q^56 + 12 * q^58 - 60 * q^59 - 24 * q^61 - 36 * q^62 + 90 * q^64 - 48 * q^65 - 60 * q^68 - 30 * q^70 - 60 * q^71 - 18 * q^73 - 36 * q^74 - 42 * q^76 - 12 * q^79 - 96 * q^80 + 12 * q^82 - 36 * q^83 + 18 * q^85 - 48 * q^86 + 18 * q^88 - 96 * q^89 + 30 * q^91 - 36 * q^92 - 48 * q^94 - 48 * q^95 + 30 * q^97 - 54 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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