LMFDB - Embedded newform 8037.2.a.r.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8037,2,Mod(1,8037)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8037, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8037.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8037 = 3^{2} \cdot 19 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8037.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:	$64.1757681045$
Analytic rank:	$0$
Dimension:	$23$
Twist minimal:	no (minimal twist has level 893)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	8037.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73522 q^{2} +1.01098 q^{4} -2.41866 q^{5} +1.62998 q^{7} +1.71616 q^{8} +O(q^{10})$	\(q-1.73522 q^{2} +1.01098 q^{4} -2.41866 q^{5} +1.62998 q^{7} +1.71616 q^{8} +4.19690 q^{10} +3.81087 q^{11} +4.54338 q^{13} -2.82837 q^{14} -4.99988 q^{16} -2.26965 q^{17} +1.00000 q^{19} -2.44522 q^{20} -6.61270 q^{22} +6.63199 q^{23} +0.849923 q^{25} -7.88375 q^{26} +1.64788 q^{28} +6.58423 q^{29} +10.1003 q^{31} +5.24355 q^{32} +3.93834 q^{34} -3.94237 q^{35} +8.23358 q^{37} -1.73522 q^{38} -4.15082 q^{40} +7.74948 q^{41} +4.82751 q^{43} +3.85272 q^{44} -11.5079 q^{46} +1.00000 q^{47} -4.34317 q^{49} -1.47480 q^{50} +4.59327 q^{52} +5.74123 q^{53} -9.21721 q^{55} +2.79731 q^{56} -11.4251 q^{58} +5.42213 q^{59} -7.22382 q^{61} -17.5261 q^{62} +0.901055 q^{64} -10.9889 q^{65} +3.61716 q^{67} -2.29458 q^{68} +6.84086 q^{70} -7.41116 q^{71} -7.84253 q^{73} -14.2871 q^{74} +1.01098 q^{76} +6.21164 q^{77} +10.8345 q^{79} +12.0930 q^{80} -13.4470 q^{82} -7.22122 q^{83} +5.48952 q^{85} -8.37678 q^{86} +6.54008 q^{88} +12.3612 q^{89} +7.40561 q^{91} +6.70481 q^{92} -1.73522 q^{94} -2.41866 q^{95} +5.54205 q^{97} +7.53634 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - q^{2} + 31 q^{4} - q^{5} + 15 q^{7}+O(q^{10}) $ 23 * q - q^2 + 31 * q^4 - q^5 + 15 * q^7	$ 23 q - q^{2} + 31 q^{4} - q^{5} + 15 q^{7} + 5 q^{10} - 2 q^{11} + 13 q^{13} - 6 q^{14} + 35 q^{16} - 12 q^{17} + 23 q^{19} - 3 q^{20} - 4 q^{22} - 13 q^{23} + 46 q^{25} + 7 q^{26} + 11 q^{28} + 18 q^{29} + 22 q^{31} + 4 q^{32} - 20 q^{34} - 25 q^{35} + 8 q^{37} - q^{38} - 16 q^{40} + 16 q^{41} + 68 q^{43} + 18 q^{44} + 13 q^{46} + 23 q^{47} + 52 q^{49} + 21 q^{50} + 54 q^{52} + 7 q^{53} + 32 q^{55} - 33 q^{56} + 6 q^{58} - 10 q^{59} + 28 q^{61} + 24 q^{62} + 40 q^{64} + 18 q^{65} + 55 q^{67} - 41 q^{68} - 40 q^{70} + 3 q^{71} + 48 q^{73} + 55 q^{74} + 31 q^{76} + 14 q^{77} - 31 q^{79} + 3 q^{80} + 48 q^{82} - 47 q^{83} - 5 q^{85} + 71 q^{86} + 9 q^{88} + 6 q^{89} + 40 q^{91} - 35 q^{92} - q^{94} - q^{95} - 18 q^{97} + 68 q^{98}+O(q^{100}) $ 23 * q - q^2 + 31 * q^4 - q^5 + 15 * q^7 + 5 * q^10 - 2 * q^11 + 13 * q^13 - 6 * q^14 + 35 * q^16 - 12 * q^17 + 23 * q^19 - 3 * q^20 - 4 * q^22 - 13 * q^23 + 46 * q^25 + 7 * q^26 + 11 * q^28 + 18 * q^29 + 22 * q^31 + 4 * q^32 - 20 * q^34 - 25 * q^35 + 8 * q^37 - q^38 - 16 * q^40 + 16 * q^41 + 68 * q^43 + 18 * q^44 + 13 * q^46 + 23 * q^47 + 52 * q^49 + 21 * q^50 + 54 * q^52 + 7 * q^53 + 32 * q^55 - 33 * q^56 + 6 * q^58 - 10 * q^59 + 28 * q^61 + 24 * q^62 + 40 * q^64 + 18 * q^65 + 55 * q^67 - 41 * q^68 - 40 * q^70 + 3 * q^71 + 48 * q^73 + 55 * q^74 + 31 * q^76 + 14 * q^77 - 31 * q^79 + 3 * q^80 + 48 * q^82 - 47 * q^83 - 5 * q^85 + 71 * q^86 + 9 * q^88 + 6 * q^89 + 40 * q^91 - 35 * q^92 - q^94 - q^95 - 18 * q^97 + 68 * 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 $

$2$	−1.73522	−1.22698	−0.613492	−	0.789701i	$-0.710236\pi$
$2$	−1.73522	−1.22698	−0.613492	+	0.789701i	$0.710236\pi$
$3$	0	0
$3$	0	0
$4$	1.01098	0.505490
$5$	−2.41866	−1.08166	−0.540829	−	0.841132i	$-0.681889\pi$
$5$	−2.41866	−1.08166	−0.540829	+	0.841132i	$0.681889\pi$
$6$	0	0
$7$	1.62998	0.616074	0.308037	−	0.951374i	$-0.400328\pi$
$7$	1.62998	0.616074	0.308037	+	0.951374i	$0.400328\pi$
$8$	1.71616	0.606756
$9$	0	0
$10$	4.19690	1.32718
$11$	3.81087	1.14902	0.574511	−	0.818497i	$-0.305192\pi$
$11$	3.81087	1.14902	0.574511	+	0.818497i	$0.305192\pi$
$12$	0	0
$13$	4.54338	1.26011	0.630054	−	0.776552i	$-0.283033\pi$
$13$	4.54338	1.26011	0.630054	+	0.776552i	$0.283033\pi$
$14$	−2.82837	−0.755913
$15$	0	0
$16$	−4.99988	−1.24997
$17$	−2.26965	−0.550472	−0.275236	−	0.961377i	$-0.588756\pi$
$17$	−2.26965	−0.550472	−0.275236	+	0.961377i	$0.588756\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−2.44522	−0.546768
$21$	0	0
$22$	−6.61270	−1.40983
$23$	6.63199	1.38286	0.691432	−	0.722441i	$-0.256980\pi$
$23$	6.63199	1.38286	0.691432	+	0.722441i	$0.256980\pi$
$24$	0	0
$25$	0.849923	0.169985
$26$	−7.88375	−1.54613
$27$	0	0
$28$	1.64788	0.311419
$29$	6.58423	1.22266	0.611330	−	0.791376i	$-0.290635\pi$
$29$	6.58423	1.22266	0.611330	+	0.791376i	$0.290635\pi$
$30$	0	0
$31$	10.1003	1.81406	0.907030	−	0.421066i	$-0.138344\pi$
$31$	10.1003	1.81406	0.907030	+	0.421066i	$0.138344\pi$
$32$	5.24355	0.926938
$33$	0	0
$34$	3.93834	0.675420
$35$	−3.94237	−0.666382
$36$	0	0
$37$	8.23358	1.35359	0.676796	−	0.736171i	$-0.263368\pi$
$37$	8.23358	1.35359	0.676796	+	0.736171i	$0.263368\pi$
$38$	−1.73522	−0.281489
$39$	0	0
$40$	−4.15082	−0.656302
$41$	7.74948	1.21026	0.605132	−	0.796125i	$-0.293120\pi$
$41$	7.74948	1.21026	0.605132	+	0.796125i	$0.293120\pi$
$42$	0	0
$43$	4.82751	0.736189	0.368094	−	0.929788i	$-0.380010\pi$
$43$	4.82751	0.736189	0.368094	+	0.929788i	$0.380010\pi$
$44$	3.85272	0.580819
$45$	0	0
$46$	−11.5079	−1.69675
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	−4.34317	−0.620453
$50$	−1.47480	−0.208568
$51$	0	0
$52$	4.59327	0.636972
$53$	5.74123	0.788618	0.394309	−	0.918978i	$-0.370984\pi$
$53$	5.74123	0.788618	0.394309	+	0.918978i	$0.370984\pi$
$54$	0	0
$55$	−9.21721	−1.24285
$56$	2.79731	0.373806
$57$	0	0
$58$	−11.4251	−1.50018
$59$	5.42213	0.705901	0.352950	−	0.935642i	$-0.385178\pi$
$59$	5.42213	0.705901	0.352950	+	0.935642i	$0.385178\pi$
$60$	0	0
$61$	−7.22382	−0.924915	−0.462458	−	0.886641i	$-0.653032\pi$
$61$	−7.22382	−0.924915	−0.462458	+	0.886641i	$0.653032\pi$
$62$	−17.5261	−2.22582
$63$	0	0
$64$	0.901055	0.112632
$65$	−10.9889	−1.36301
$66$	0	0
$67$	3.61716	0.441906	0.220953	−	0.975284i	$-0.429083\pi$
$67$	3.61716	0.441906	0.220953	+	0.975284i	$0.429083\pi$
$68$	−2.29458	−0.278258
$69$	0	0
$70$	6.84086	0.817640
$71$	−7.41116	−0.879543	−0.439772	−	0.898110i	$-0.644941\pi$
$71$	−7.41116	−0.879543	−0.439772	+	0.898110i	$0.644941\pi$
$72$	0	0
$73$	−7.84253	−0.917898	−0.458949	−	0.888463i	$-0.651774\pi$
$73$	−7.84253	−0.917898	−0.458949	+	0.888463i	$0.651774\pi$
$74$	−14.2871	−1.66084
$75$	0	0
$76$	1.01098	0.115967
$77$	6.21164	0.707882
$78$	0	0
$79$	10.8345	1.21897	0.609486	−	0.792796i	$-0.291376\pi$
$79$	10.8345	1.21897	0.609486	+	0.792796i	$0.291376\pi$
$80$	12.0930	1.35204
$81$	0	0
$82$	−13.4470	−1.48498
$83$	−7.22122	−0.792632	−0.396316	−	0.918114i	$-0.629712\pi$
$83$	−7.22122	−0.792632	−0.396316	+	0.918114i	$0.629712\pi$
$84$	0	0
$85$	5.48952	0.595422
$86$	−8.37678	−0.903292
$87$	0	0
$88$	6.54008	0.697175
$89$	12.3612	1.31028	0.655141	−	0.755507i	$-0.272609\pi$
$89$	12.3612	1.31028	0.655141	+	0.755507i	$0.272609\pi$
$90$	0	0
$91$	7.40561	0.776319
$92$	6.70481	0.699025
$93$	0	0
$94$	−1.73522	−0.178974
$95$	−2.41866	−0.248149
$96$	0	0
$97$	5.54205	0.562710	0.281355	−	0.959604i	$-0.409216\pi$
$97$	5.54205	0.562710	0.281355	+	0.959604i	$0.409216\pi$
$98$	7.53634	0.761286
$99$	0	0
$100$	0.859255	0.0859255
$101$	2.20082	0.218989	0.109495	−	0.993987i	$-0.465077\pi$
$101$	2.20082	0.218989	0.109495	+	0.993987i	$0.465077\pi$
$102$	0	0
$103$	15.3245	1.50997	0.754983	−	0.655744i	$-0.227645\pi$
$103$	15.3245	1.50997	0.754983	+	0.655744i	$0.227645\pi$
$104$	7.79719	0.764577
$105$	0	0
$106$	−9.96228	−0.967622
$107$	11.5577	1.11732	0.558662	−	0.829395i	$-0.311315\pi$
$107$	11.5577	1.11732	0.558662	+	0.829395i	$0.311315\pi$
$108$	0	0
$109$	8.94316	0.856600	0.428300	−	0.903637i	$-0.359113\pi$
$109$	8.94316	0.856600	0.428300	+	0.903637i	$0.359113\pi$
$110$	15.9939	1.52496
$111$	0	0
$112$	−8.14970	−0.770074
$113$	−12.7517	−1.19958	−0.599788	−	0.800159i	$-0.704749\pi$
$113$	−12.7517	−1.19958	−0.599788	+	0.800159i	$0.704749\pi$
$114$	0	0
$115$	−16.0405	−1.49579
$116$	6.65653	0.618043
$117$	0	0
$118$	−9.40857	−0.866129
$119$	−3.69949	−0.339131
$120$	0	0
$121$	3.52276	0.320251
$122$	12.5349	1.13486
$123$	0	0
$124$	10.2112	0.916990
$125$	10.0376	0.897793
$126$	0	0
$127$	−8.28208	−0.734916	−0.367458	−	0.930040i	$-0.619772\pi$
$127$	−8.28208	−0.734916	−0.367458	+	0.930040i	$0.619772\pi$
$128$	−12.0506	−1.06514
$129$	0	0
$130$	19.0681	1.67239
$131$	−13.8120	−1.20676	−0.603379	−	0.797455i	$-0.706179\pi$
$131$	−13.8120	−1.20676	−0.603379	+	0.797455i	$0.706179\pi$
$132$	0	0
$133$	1.62998	0.141337
$134$	−6.27656	−0.542212
$135$	0	0
$136$	−3.89510	−0.334002
$137$	13.7094	1.17128	0.585638	−	0.810573i	$-0.300844\pi$
$137$	13.7094	1.17128	0.585638	+	0.810573i	$0.300844\pi$
$138$	0	0
$139$	−11.9878	−1.01679	−0.508397	−	0.861123i	$-0.669762\pi$
$139$	−11.9878	−1.01679	−0.508397	+	0.861123i	$0.669762\pi$
$140$	−3.98566	−0.336849
$141$	0	0
$142$	12.8600	1.07919
$143$	17.3142	1.44789
$144$	0	0
$145$	−15.9250	−1.32250
$146$	13.6085	1.12625
$147$	0	0
$148$	8.32399	0.684228
$149$	−18.7123	−1.53297	−0.766485	−	0.642263i	$-0.777996\pi$
$149$	−18.7123	−1.53297	−0.766485	+	0.642263i	$0.777996\pi$
$150$	0	0
$151$	16.5450	1.34641	0.673206	−	0.739455i	$-0.264917\pi$
$151$	16.5450	1.34641	0.673206	+	0.739455i	$0.264917\pi$
$152$	1.71616	0.139199
$153$	0	0
$154$	−10.7786	−0.868561
$155$	−24.4291	−1.96219
$156$	0	0
$157$	11.7021	0.933926	0.466963	−	0.884277i	$-0.345348\pi$
$157$	11.7021	0.933926	0.466963	+	0.884277i	$0.345348\pi$
$158$	−18.8002	−1.49566
$159$	0	0
$160$	−12.6824	−1.00263
$161$	10.8100	0.851947
$162$	0	0
$163$	−12.5589	−0.983687	−0.491844	−	0.870683i	$-0.663677\pi$
$163$	−12.5589	−0.983687	−0.491844	+	0.870683i	$0.663677\pi$
$164$	7.83457	0.611777
$165$	0	0
$166$	12.5304	0.972547
$167$	−15.9061	−1.23085	−0.615427	−	0.788194i	$-0.711016\pi$
$167$	−15.9061	−1.23085	−0.615427	+	0.788194i	$0.711016\pi$
$168$	0	0
$169$	7.64231	0.587870
$170$	−9.52552	−0.730574
$171$	0	0
$172$	4.88052	0.372136
$173$	5.23871	0.398292	0.199146	−	0.979970i	$-0.436183\pi$
$173$	5.23871	0.398292	0.199146	+	0.979970i	$0.436183\pi$
$174$	0	0
$175$	1.38536	0.104723
$176$	−19.0539	−1.43624
$177$	0	0
$178$	−21.4493	−1.60769
$179$	−6.75418	−0.504831	−0.252416	−	0.967619i	$-0.581225\pi$
$179$	−6.75418	−0.504831	−0.252416	+	0.967619i	$0.581225\pi$
$180$	0	0
$181$	−14.0881	−1.04716	−0.523580	−	0.851976i	$-0.675404\pi$
$181$	−14.0881	−1.04716	−0.523580	+	0.851976i	$0.675404\pi$
$182$	−12.8504	−0.952532
$183$	0	0
$184$	11.3816	0.839061
$185$	−19.9142	−1.46412
$186$	0	0
$187$	−8.64936	−0.632504
$188$	1.01098	0.0737333
$189$	0	0
$190$	4.19690	0.304475
$191$	19.0944	1.38162	0.690810	−	0.723036i	$-0.257254\pi$
$191$	19.0944	1.38162	0.690810	+	0.723036i	$0.257254\pi$
$192$	0	0
$193$	−27.1007	−1.95075	−0.975376	−	0.220549i	$-0.929215\pi$
$193$	−27.1007	−1.95075	−0.975376	+	0.220549i	$0.929215\pi$
$194$	−9.61666	−0.690436
$195$	0	0
$196$	−4.39086	−0.313633
$197$	−5.37964	−0.383283	−0.191642	−	0.981465i	$-0.561381\pi$
$197$	−5.37964	−0.383283	−0.191642	+	0.981465i	$0.561381\pi$
$198$	0	0
$199$	4.64307	0.329138	0.164569	−	0.986366i	$-0.447377\pi$
$199$	4.64307	0.329138	0.164569	+	0.986366i	$0.447377\pi$
$200$	1.45861	0.103139
$201$	0	0
$202$	−3.81890	−0.268697
$203$	10.7321	0.753249
$204$	0	0
$205$	−18.7434	−1.30909
$206$	−26.5913	−1.85270
$207$	0	0
$208$	−22.7164	−1.57510
$209$	3.81087	0.263604
$210$	0	0
$211$	−21.6128	−1.48788	−0.743942	−	0.668244i	$-0.767046\pi$
$211$	−21.6128	−1.48788	−0.743942	+	0.668244i	$0.767046\pi$
$212$	5.80427	0.398639
$213$	0	0
$214$	−20.0551	−1.37094
$215$	−11.6761	−0.796305
$216$	0	0
$217$	16.4632	1.11760
$218$	−15.5183	−1.05103
$219$	0	0
$220$	−9.31842	−0.628248
$221$	−10.3119	−0.693654
$222$	0	0
$223$	−19.4410	−1.30186	−0.650932	−	0.759136i	$-0.725622\pi$
$223$	−19.4410	−1.30186	−0.650932	+	0.759136i	$0.725622\pi$
$224$	8.54688	0.571062
$225$	0	0
$226$	22.1269	1.47186
$227$	−25.5633	−1.69670	−0.848349	−	0.529438i	$-0.822403\pi$
$227$	−25.5633	−1.69670	−0.848349	+	0.529438i	$0.822403\pi$
$228$	0	0
$229$	23.6498	1.56282	0.781411	−	0.624016i	$-0.214500\pi$
$229$	23.6498	1.56282	0.781411	+	0.624016i	$0.214500\pi$
$230$	27.8338	1.83531
$231$	0	0
$232$	11.2996	0.741856
$233$	−6.91587	−0.453073	−0.226537	−	0.974003i	$-0.572740\pi$
$233$	−6.91587	−0.453073	−0.226537	+	0.974003i	$0.572740\pi$
$234$	0	0
$235$	−2.41866	−0.157776
$236$	5.48166	0.356826
$237$	0	0
$238$	6.41942	0.416109
$239$	−17.1239	−1.10765	−0.553826	−	0.832632i	$-0.686833\pi$
$239$	−17.1239	−1.10765	−0.553826	+	0.832632i	$0.686833\pi$
$240$	0	0
$241$	5.77925	0.372274	0.186137	−	0.982524i	$-0.440403\pi$
$241$	5.77925	0.372274	0.186137	+	0.982524i	$0.440403\pi$
$242$	−6.11275	−0.392942
$243$	0	0
$244$	−7.30314	−0.467536
$245$	10.5047	0.671118
$246$	0	0
$247$	4.54338	0.289088
$248$	17.3337	1.10069
$249$	0	0
$250$	−17.4175	−1.10158
$251$	−2.71034	−0.171075	−0.0855375	−	0.996335i	$-0.527261\pi$
$251$	−2.71034	−0.171075	−0.0855375	+	0.996335i	$0.527261\pi$
$252$	0	0
$253$	25.2737	1.58894
$254$	14.3712	0.901730
$255$	0	0
$256$	19.1084	1.19427
$257$	21.6807	1.35240	0.676202	−	0.736716i	$-0.263625\pi$
$257$	21.6807	1.35240	0.676202	+	0.736716i	$0.263625\pi$
$258$	0	0
$259$	13.4206	0.833913
$260$	−11.1096	−0.688986
$261$	0	0
$262$	23.9668	1.48067
$263$	−5.69454	−0.351140	−0.175570	−	0.984467i	$-0.556177\pi$
$263$	−5.69454	−0.351140	−0.175570	+	0.984467i	$0.556177\pi$
$264$	0	0
$265$	−13.8861	−0.853016
$266$	−2.82837	−0.173418
$267$	0	0
$268$	3.65688	0.223379
$269$	−2.03165	−0.123872	−0.0619358	−	0.998080i	$-0.519727\pi$
$269$	−2.03165	−0.123872	−0.0619358	+	0.998080i	$0.519727\pi$
$270$	0	0
$271$	9.81491	0.596213	0.298107	−	0.954533i	$-0.403645\pi$
$271$	9.81491	0.596213	0.298107	+	0.954533i	$0.403645\pi$
$272$	11.3480	0.688073
$273$	0	0
$274$	−23.7889	−1.43714
$275$	3.23895	0.195316
$276$	0	0
$277$	−10.1265	−0.608440	−0.304220	−	0.952602i	$-0.598396\pi$
$277$	−10.1265	−0.608440	−0.304220	+	0.952602i	$0.598396\pi$
$278$	20.8015	1.24759
$279$	0	0
$280$	−6.76575	−0.404331
$281$	−20.7012	−1.23493	−0.617466	−	0.786598i	$-0.711841\pi$
$281$	−20.7012	−1.23493	−0.617466	+	0.786598i	$0.711841\pi$
$282$	0	0
$283$	−31.9719	−1.90053	−0.950266	−	0.311439i	$-0.899189\pi$
$283$	−31.9719	−1.90053	−0.950266	+	0.311439i	$0.899189\pi$
$284$	−7.49254	−0.444601
$285$	0	0
$286$	−30.0440	−1.77654
$287$	12.6315	0.745613
$288$	0	0
$289$	−11.8487	−0.696981
$290$	27.6334	1.62269
$291$	0	0
$292$	−7.92864	−0.463989
$293$	−6.09074	−0.355825	−0.177912	−	0.984046i	$-0.556934\pi$
$293$	−6.09074	−0.355825	−0.177912	+	0.984046i	$0.556934\pi$
$294$	0	0
$295$	−13.1143	−0.763543
$296$	14.1302	0.821299
$297$	0	0
$298$	32.4699	1.88093
$299$	30.1316	1.74256
$300$	0	0
$301$	7.86874	0.453547
$302$	−28.7092	−1.65203
$303$	0	0
$304$	−4.99988	−0.286763
$305$	17.4720	1.00044
$306$	0	0
$307$	31.2669	1.78450	0.892249	−	0.451544i	$-0.149126\pi$
$307$	31.2669	1.78450	0.892249	+	0.451544i	$0.149126\pi$
$308$	6.27985	0.357828
$309$	0	0
$310$	42.3898	2.40758
$311$	−22.2186	−1.25990	−0.629951	−	0.776635i	$-0.716925\pi$
$311$	−22.2186	−1.25990	−0.629951	+	0.776635i	$0.716925\pi$
$312$	0	0
$313$	−15.2019	−0.859261	−0.429631	−	0.903005i	$-0.641356\pi$
$313$	−15.2019	−0.859261	−0.429631	+	0.903005i	$0.641356\pi$
$314$	−20.3056	−1.14591
$315$	0	0
$316$	10.9534	0.616179
$317$	−4.57169	−0.256772	−0.128386	−	0.991724i	$-0.540980\pi$
$317$	−4.57169	−0.256772	−0.128386	+	0.991724i	$0.540980\pi$
$318$	0	0
$319$	25.0917	1.40486
$320$	−2.17935	−0.121829
$321$	0	0
$322$	−18.7577	−1.04533
$323$	−2.26965	−0.126287
$324$	0	0
$325$	3.86152	0.214199
$326$	21.7924	1.20697
$327$	0	0
$328$	13.2994	0.734335
$329$	1.62998	0.0898636
$330$	0	0
$331$	16.1447	0.887394	0.443697	−	0.896177i	$-0.353667\pi$
$331$	16.1447	0.887394	0.443697	+	0.896177i	$0.353667\pi$
$332$	−7.30052	−0.400668
$333$	0	0
$334$	27.6006	1.51024
$335$	−8.74868	−0.477991
$336$	0	0
$337$	6.14872	0.334942	0.167471	−	0.985877i	$-0.446440\pi$
$337$	6.14872	0.334942	0.167471	+	0.985877i	$0.446440\pi$
$338$	−13.2611	−0.721307
$339$	0	0
$340$	5.54980	0.300980
$341$	38.4908	2.08439
$342$	0	0
$343$	−18.4891	−0.998319
$344$	8.28480	0.446687
$345$	0	0
$346$	−9.09030	−0.488698
$347$	−13.4004	−0.719369	−0.359685	−	0.933074i	$-0.617116\pi$
$347$	−13.4004	−0.719369	−0.359685	+	0.933074i	$0.617116\pi$
$348$	0	0
$349$	−3.26894	−0.174982	−0.0874910	−	0.996165i	$-0.527885\pi$
$349$	−3.26894	−0.174982	−0.0874910	+	0.996165i	$0.527885\pi$
$350$	−2.40389	−0.128494
$351$	0	0
$352$	19.9825	1.06507
$353$	−22.7222	−1.20938	−0.604691	−	0.796460i	$-0.706704\pi$
$353$	−22.7222	−1.20938	−0.604691	+	0.796460i	$0.706704\pi$
$354$	0	0
$355$	17.9251	0.951365
$356$	12.4969	0.662334
$357$	0	0
$358$	11.7200	0.619420
$359$	−7.78202	−0.410719	−0.205360	−	0.978687i	$-0.565836\pi$
$359$	−7.78202	−0.410719	−0.205360	+	0.978687i	$0.565836\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	24.4459	1.28485
$363$	0	0
$364$	7.48693	0.392422
$365$	18.9684	0.992852
$366$	0	0
$367$	15.7502	0.822154	0.411077	−	0.911601i	$-0.365153\pi$
$367$	15.7502	0.822154	0.411077	+	0.911601i	$0.365153\pi$
$368$	−33.1591	−1.72854
$369$	0	0
$370$	34.5555	1.79646
$371$	9.35808	0.485847
$372$	0	0
$373$	−19.7796	−1.02415	−0.512075	−	0.858941i	$-0.671123\pi$
$373$	−19.7796	−1.02415	−0.512075	+	0.858941i	$0.671123\pi$
$374$	15.0085	0.776072
$375$	0	0
$376$	1.71616	0.0885044
$377$	29.9146	1.54068
$378$	0	0
$379$	2.51510	0.129192	0.0645960	−	0.997911i	$-0.479424\pi$
$379$	2.51510	0.129192	0.0645960	+	0.997911i	$0.479424\pi$
$380$	−2.44522	−0.125437
$381$	0	0
$382$	−33.1329	−1.69523
$383$	4.42708	0.226213	0.113106	−	0.993583i	$-0.463920\pi$
$383$	4.42708	0.226213	0.113106	+	0.993583i	$0.463920\pi$
$384$	0	0
$385$	−15.0239	−0.765687
$386$	47.0256	2.39354
$387$	0	0
$388$	5.60290	0.284444
$389$	−7.58347	−0.384497	−0.192249	−	0.981346i	$-0.561578\pi$
$389$	−7.58347	−0.384497	−0.192249	+	0.981346i	$0.561578\pi$
$390$	0	0
$391$	−15.0523	−0.761228
$392$	−7.45359	−0.376463
$393$	0	0
$394$	9.33485	0.470283
$395$	−26.2049	−1.31851
$396$	0	0
$397$	25.1695	1.26322	0.631611	−	0.775285i	$-0.282394\pi$
$397$	25.1695	1.26322	0.631611	+	0.775285i	$0.282394\pi$
$398$	−8.05674	−0.403848
$399$	0	0
$400$	−4.24951	−0.212476
$401$	−30.7754	−1.53685	−0.768424	−	0.639941i	$-0.778959\pi$
$401$	−30.7754	−1.53685	−0.768424	+	0.639941i	$0.778959\pi$
$402$	0	0
$403$	45.8893	2.28591
$404$	2.22498	0.110697
$405$	0	0
$406$	−18.6226	−0.924225
$407$	31.3771	1.55531
$408$	0	0
$409$	−3.47289	−0.171723	−0.0858616	−	0.996307i	$-0.527364\pi$
$409$	−3.47289	−0.171723	−0.0858616	+	0.996307i	$0.527364\pi$
$410$	32.5238	1.60624
$411$	0	0
$412$	15.4928	0.763273
$413$	8.83795	0.434887
$414$	0	0
$415$	17.4657	0.857357
$416$	23.8235	1.16804
$417$	0	0
$418$	−6.61270	−0.323437
$419$	−35.6906	−1.74360	−0.871799	−	0.489863i	$-0.837047\pi$
$419$	−35.6906	−1.74360	−0.871799	+	0.489863i	$0.837047\pi$
$420$	0	0
$421$	−6.06604	−0.295641	−0.147820	−	0.989014i	$-0.547226\pi$
$421$	−6.06604	−0.295641	−0.147820	+	0.989014i	$0.547226\pi$
$422$	37.5029	1.82561
$423$	0	0
$424$	9.85289	0.478499
$425$	−1.92903	−0.0935717
$426$	0	0
$427$	−11.7747	−0.569816
$428$	11.6846	0.564797
$429$	0	0
$430$	20.2606	0.977053
$431$	5.45664	0.262837	0.131418	−	0.991327i	$-0.458047\pi$
$431$	5.45664	0.262837	0.131418	+	0.991327i	$0.458047\pi$
$432$	0	0
$433$	26.3539	1.26649	0.633243	−	0.773953i	$-0.281723\pi$
$433$	26.3539	1.26649	0.633243	+	0.773953i	$0.281723\pi$
$434$	−28.5672	−1.37127
$435$	0	0
$436$	9.04137	0.433003
$437$	6.63199	0.317251
$438$	0	0
$439$	7.66324	0.365746	0.182873	−	0.983137i	$-0.441460\pi$
$439$	7.66324	0.365746	0.182873	+	0.983137i	$0.441460\pi$
$440$	−15.8182	−0.754105
$441$	0	0
$442$	17.8934	0.851102
$443$	13.8103	0.656145	0.328073	−	0.944653i	$-0.393601\pi$
$443$	13.8103	0.656145	0.328073	+	0.944653i	$0.393601\pi$
$444$	0	0
$445$	−29.8975	−1.41728
$446$	33.7343	1.59737
$447$	0	0
$448$	1.46870	0.0693896
$449$	0.352484	0.0166347	0.00831737	−	0.999965i	$-0.497352\pi$
$449$	0.352484	0.0166347	0.00831737	+	0.999965i	$0.497352\pi$
$450$	0	0
$451$	29.5323	1.39062
$452$	−12.8917	−0.606374
$453$	0	0
$454$	44.3579	2.08182
$455$	−17.9117	−0.839712
$456$	0	0
$457$	39.5187	1.84861	0.924304	−	0.381657i	$-0.124646\pi$
$457$	39.5187	1.84861	0.924304	+	0.381657i	$0.124646\pi$
$458$	−41.0376	−1.91756
$459$	0	0
$460$	−16.2167	−0.756106
$461$	28.2674	1.31654	0.658271	−	0.752781i	$-0.271288\pi$
$461$	28.2674	1.31654	0.658271	+	0.752781i	$0.271288\pi$
$462$	0	0
$463$	−0.883630	−0.0410658	−0.0205329	−	0.999789i	$-0.506536\pi$
$463$	−0.883630	−0.0410658	−0.0205329	+	0.999789i	$0.506536\pi$
$464$	−32.9203	−1.52829
$465$	0	0
$466$	12.0005	0.555914
$467$	1.34380	0.0621835	0.0310917	−	0.999517i	$-0.490102\pi$
$467$	1.34380	0.0621835	0.0310917	+	0.999517i	$0.490102\pi$
$468$	0	0
$469$	5.89589	0.272247
$470$	4.19690	0.193589
$471$	0	0
$472$	9.30526	0.428309
$473$	18.3970	0.845897
$474$	0	0
$475$	0.849923	0.0389971
$476$	−3.74011	−0.171428
$477$	0	0
$478$	29.7137	1.35907
$479$	23.3665	1.06764	0.533820	−	0.845598i	$-0.320756\pi$
$479$	23.3665	1.06764	0.533820	+	0.845598i	$0.320756\pi$
$480$	0	0
$481$	37.4083	1.70567
$482$	−10.0283	−0.456775
$483$	0	0
$484$	3.56144	0.161884
$485$	−13.4043	−0.608660
$486$	0	0
$487$	33.5657	1.52101	0.760504	−	0.649333i	$-0.224952\pi$
$487$	33.5657	1.52101	0.760504	+	0.649333i	$0.224952\pi$
$488$	−12.3973	−0.561198
$489$	0	0
$490$	−18.2279	−0.823451
$491$	−12.3391	−0.556855	−0.278428	−	0.960457i	$-0.589813\pi$
$491$	−12.3391	−0.556855	−0.278428	+	0.960457i	$0.589813\pi$
$492$	0	0
$493$	−14.9439	−0.673040
$494$	−7.88375	−0.354707
$495$	0	0
$496$	−50.5001	−2.26752
$497$	−12.0800	−0.541864
$498$	0	0
$499$	23.7960	1.06526	0.532628	−	0.846349i	$-0.321204\pi$
$499$	23.7960	1.06526	0.532628	+	0.846349i	$0.321204\pi$
$500$	10.1479	0.453826
$501$	0	0
$502$	4.70302	0.209906
$503$	32.4311	1.44603	0.723016	−	0.690831i	$-0.242755\pi$
$503$	32.4311	1.44603	0.723016	+	0.690831i	$0.242755\pi$
$504$	0	0
$505$	−5.32303	−0.236872
$506$	−43.8553	−1.94961
$507$	0	0
$508$	−8.37302	−0.371493
$509$	37.0975	1.64432	0.822159	−	0.569258i	$-0.192770\pi$
$509$	37.0975	1.64432	0.822159	+	0.569258i	$0.192770\pi$
$510$	0	0
$511$	−12.7832	−0.565493
$512$	−9.05590	−0.400218
$513$	0	0
$514$	−37.6207	−1.65938
$515$	−37.0647	−1.63327
$516$	0	0
$517$	3.81087	0.167602
$518$	−23.2876	−1.02320
$519$	0	0
$520$	−18.8588	−0.827011
$521$	33.7786	1.47987	0.739934	−	0.672680i	$-0.234857\pi$
$521$	33.7786	1.47987	0.739934	+	0.672680i	$0.234857\pi$
$522$	0	0
$523$	−26.6693	−1.16617	−0.583083	−	0.812412i	$-0.698154\pi$
$523$	−26.6693	−1.16617	−0.583083	+	0.812412i	$0.698154\pi$
$524$	−13.9636	−0.610004
$525$	0	0
$526$	9.88127	0.430844
$527$	−22.9241	−0.998589
$528$	0	0
$529$	20.9832	0.912315
$530$	24.0954	1.04664
$531$	0	0
$532$	1.64788	0.0714445
$533$	35.2088	1.52506
$534$	0	0
$535$	−27.9541	−1.20856
$536$	6.20764	0.268129
$537$	0	0
$538$	3.52535	0.151989
$539$	−16.5513	−0.712914
$540$	0	0
$541$	−1.49584	−0.0643114	−0.0321557	−	0.999483i	$-0.510237\pi$
$541$	−1.49584	−0.0643114	−0.0321557	+	0.999483i	$0.510237\pi$
$542$	−17.0310	−0.731544
$543$	0	0
$544$	−11.9010	−0.510253
$545$	−21.6305	−0.926548
$546$	0	0
$547$	−14.4484	−0.617767	−0.308884	−	0.951100i	$-0.599955\pi$
$547$	−14.4484	−0.617767	−0.308884	+	0.951100i	$0.599955\pi$
$548$	13.8600	0.592069
$549$	0	0
$550$	−5.62028	−0.239650
$551$	6.58423	0.280497
$552$	0	0
$553$	17.6600	0.750978
$554$	17.5716	0.746546
$555$	0	0
$556$	−12.1195	−0.513979
$557$	4.40071	0.186464	0.0932320	−	0.995644i	$-0.470280\pi$
$557$	4.40071	0.186464	0.0932320	+	0.995644i	$0.470280\pi$
$558$	0	0
$559$	21.9332	0.927677
$560$	19.7114	0.832957
$561$	0	0
$562$	35.9211	1.51524
$563$	14.5571	0.613507	0.306754	−	0.951789i	$-0.400757\pi$
$563$	14.5571	0.613507	0.306754	+	0.951789i	$0.400757\pi$
$564$	0	0
$565$	30.8420	1.29753
$566$	55.4782	2.33192
$567$	0	0
$568$	−12.7188	−0.533668
$569$	−36.6503	−1.53646	−0.768231	−	0.640173i	$-0.778863\pi$
$569$	−36.6503	−1.53646	−0.768231	+	0.640173i	$0.778863\pi$
$570$	0	0
$571$	−42.9435	−1.79713	−0.898565	−	0.438841i	$-0.855389\pi$
$571$	−42.9435	−1.79713	−0.898565	+	0.438841i	$0.855389\pi$
$572$	17.5044	0.731894
$573$	0	0
$574$	−21.9184	−0.914855
$575$	5.63668	0.235066
$576$	0	0
$577$	4.77363	0.198729	0.0993644	−	0.995051i	$-0.468319\pi$
$577$	4.77363	0.198729	0.0993644	+	0.995051i	$0.468319\pi$
$578$	20.5600	0.855184
$579$	0	0
$580$	−16.0999	−0.668511
$581$	−11.7704	−0.488320
$582$	0	0
$583$	21.8791	0.906140
$584$	−13.4591	−0.556940
$585$	0	0
$586$	10.5688	0.436592
$587$	16.5166	0.681714	0.340857	−	0.940115i	$-0.389283\pi$
$587$	16.5166	0.681714	0.340857	+	0.940115i	$0.389283\pi$
$588$	0	0
$589$	10.1003	0.416174
$590$	22.7561	0.936856
$591$	0	0
$592$	−41.1669	−1.69195
$593$	−3.80261	−0.156154	−0.0780771	−	0.996947i	$-0.524878\pi$
$593$	−3.80261	−0.156154	−0.0780771	+	0.996947i	$0.524878\pi$
$594$	0	0
$595$	8.94781	0.366824
$596$	−18.9177	−0.774901
$597$	0	0
$598$	−52.2850	−2.13809
$599$	11.2625	0.460174	0.230087	−	0.973170i	$-0.426099\pi$
$599$	11.2625	0.460174	0.230087	+	0.973170i	$0.426099\pi$
$600$	0	0
$601$	25.4306	1.03734	0.518669	−	0.854975i	$-0.326428\pi$
$601$	25.4306	1.03734	0.518669	+	0.854975i	$0.326428\pi$
$602$	−13.6540	−0.556495
$603$	0	0
$604$	16.7267	0.680598
$605$	−8.52036	−0.346402
$606$	0	0
$607$	−26.2559	−1.06569	−0.532847	−	0.846211i	$-0.678878\pi$
$607$	−26.2559	−1.06569	−0.532847	+	0.846211i	$0.678878\pi$
$608$	5.24355	0.212654
$609$	0	0
$610$	−30.3177	−1.22753
$611$	4.54338	0.183806
$612$	0	0
$613$	−16.4972	−0.666314	−0.333157	−	0.942871i	$-0.608114\pi$
$613$	−16.4972	−0.666314	−0.333157	+	0.942871i	$0.608114\pi$
$614$	−54.2549	−2.18955
$615$	0	0
$616$	10.6602	0.429512
$617$	24.7949	0.998206	0.499103	−	0.866543i	$-0.333663\pi$
$617$	24.7949	0.998206	0.499103	+	0.866543i	$0.333663\pi$
$618$	0	0
$619$	34.1230	1.37152	0.685759	−	0.727828i	$-0.259470\pi$
$619$	34.1230	1.37152	0.685759	+	0.727828i	$0.259470\pi$
$620$	−24.6974	−0.991869
$621$	0	0
$622$	38.5541	1.54588
$623$	20.1484	0.807230
$624$	0	0
$625$	−28.5272	−1.14109
$626$	26.3786	1.05430
$627$	0	0
$628$	11.8305	0.472090
$629$	−18.6874	−0.745114
$630$	0	0
$631$	16.6215	0.661690	0.330845	−	0.943685i	$-0.392666\pi$
$631$	16.6215	0.661690	0.330845	+	0.943685i	$0.392666\pi$
$632$	18.5937	0.739619
$633$	0	0
$634$	7.93287	0.315055
$635$	20.0315	0.794928
$636$	0	0
$637$	−19.7327	−0.781837
$638$	−43.5395	−1.72374
$639$	0	0
$640$	29.1464	1.15211
$641$	−0.367731	−0.0145245	−0.00726226	−	0.999974i	$-0.502312\pi$
$641$	−0.367731	−0.0145245	−0.00726226	+	0.999974i	$0.502312\pi$
$642$	0	0
$643$	−29.7163	−1.17190	−0.585949	−	0.810348i	$-0.699278\pi$
$643$	−29.7163	−1.17190	−0.585949	+	0.810348i	$0.699278\pi$
$644$	10.9287	0.430651
$645$	0	0
$646$	3.93834	0.154952
$647$	−29.2575	−1.15023	−0.575115	−	0.818073i	$-0.695043\pi$
$647$	−29.2575	−1.15023	−0.575115	+	0.818073i	$0.695043\pi$
$648$	0	0
$649$	20.6630	0.811095
$650$	−6.70058	−0.262818
$651$	0	0
$652$	−12.6968	−0.497244
$653$	−32.0853	−1.25560	−0.627798	−	0.778376i	$-0.716044\pi$
$653$	−32.0853	−1.25560	−0.627798	+	0.778376i	$0.716044\pi$
$654$	0	0
$655$	33.4065	1.30530
$656$	−38.7464	−1.51279
$657$	0	0
$658$	−2.82837	−0.110261
$659$	−42.8644	−1.66976	−0.834880	−	0.550432i	$-0.814463\pi$
$659$	−42.8644	−1.66976	−0.834880	+	0.550432i	$0.814463\pi$
$660$	0	0
$661$	33.2849	1.29463	0.647316	−	0.762222i	$-0.275891\pi$
$661$	33.2849	1.29463	0.647316	+	0.762222i	$0.275891\pi$
$662$	−28.0146	−1.08882
$663$	0	0
$664$	−12.3928	−0.480934
$665$	−3.94237	−0.152878
$666$	0	0
$667$	43.6665	1.69077
$668$	−16.0808	−0.622184
$669$	0	0
$670$	15.1809	0.586488
$671$	−27.5291	−1.06275
$672$	0	0
$673$	−44.6254	−1.72018	−0.860091	−	0.510141i	$-0.829593\pi$
$673$	−44.6254	−1.72018	−0.860091	+	0.510141i	$0.829593\pi$
$674$	−10.6694	−0.410969
$675$	0	0
$676$	7.72623	0.297163
$677$	20.4784	0.787047	0.393524	−	0.919315i	$-0.371256\pi$
$677$	20.4784	0.787047	0.393524	+	0.919315i	$0.371256\pi$
$678$	0	0
$679$	9.03342	0.346671
$680$	9.42092	0.361276
$681$	0	0
$682$	−66.7899	−2.55752
$683$	10.4041	0.398100	0.199050	−	0.979989i	$-0.436214\pi$
$683$	10.4041	0.398100	0.199050	+	0.979989i	$0.436214\pi$
$684$	0	0
$685$	−33.1585	−1.26692
$686$	32.0827	1.22492
$687$	0	0
$688$	−24.1370	−0.920214
$689$	26.0846	0.993744
$690$	0	0
$691$	−12.9951	−0.494357	−0.247178	−	0.968970i	$-0.579503\pi$
$691$	−12.9951	−0.494357	−0.247178	+	0.968970i	$0.579503\pi$
$692$	5.29624	0.201333
$693$	0	0
$694$	23.2526	0.882655
$695$	28.9945	1.09982
$696$	0	0
$697$	−17.5886	−0.666217
$698$	5.67231	0.214700
$699$	0	0
$700$	1.40057	0.0529365
$701$	9.82810	0.371202	0.185601	−	0.982625i	$-0.440577\pi$
$701$	9.82810	0.371202	0.185601	+	0.982625i	$0.440577\pi$
$702$	0	0
$703$	8.23358	0.310535
$704$	3.43381	0.129417
$705$	0	0
$706$	39.4280	1.48389
$707$	3.58728	0.134914
$708$	0	0
$709$	−12.4996	−0.469431	−0.234716	−	0.972064i	$-0.575416\pi$
$709$	−12.4996	−0.469431	−0.234716	+	0.972064i	$0.575416\pi$
$710$	−31.1039	−1.16731
$711$	0	0
$712$	21.2138	0.795020
$713$	66.9848	2.50860
$714$	0	0
$715$	−41.8773	−1.56612
$716$	−6.82835	−0.255187
$717$	0	0
$718$	13.5035	0.503946
$719$	−15.8043	−0.589402	−0.294701	−	0.955589i	$-0.595220\pi$
$719$	−15.8043	−0.589402	−0.294701	+	0.955589i	$0.595220\pi$
$720$	0	0
$721$	24.9786	0.930251
$722$	−1.73522	−0.0645781
$723$	0	0
$724$	−14.2428	−0.529330
$725$	5.59608	0.207833
$726$	0	0
$727$	−52.0152	−1.92914	−0.964568	−	0.263836i	$-0.915012\pi$
$727$	−52.0152	−1.92914	−0.964568	+	0.263836i	$0.915012\pi$
$728$	12.7092	0.471036
$729$	0	0
$730$	−32.9143	−1.21821
$731$	−10.9568	−0.405251
$732$	0	0
$733$	−2.98108	−0.110109	−0.0550544	−	0.998483i	$-0.517533\pi$
$733$	−2.98108	−0.110109	−0.0550544	+	0.998483i	$0.517533\pi$
$734$	−27.3300	−1.00877
$735$	0	0
$736$	34.7752	1.28183
$737$	13.7845	0.507760
$738$	0	0
$739$	9.53144	0.350620	0.175310	−	0.984513i	$-0.443907\pi$
$739$	9.53144	0.350620	0.175310	+	0.984513i	$0.443907\pi$
$740$	−20.1329	−0.740100
$741$	0	0
$742$	−16.2383	−0.596127
$743$	28.0682	1.02972	0.514862	−	0.857273i	$-0.327843\pi$
$743$	28.0682	1.02972	0.514862	+	0.857273i	$0.327843\pi$
$744$	0	0
$745$	45.2587	1.65815
$746$	34.3220	1.25662
$747$	0	0
$748$	−8.74434	−0.319725
$749$	18.8388	0.688355
$750$	0	0
$751$	38.6776	1.41137	0.705683	−	0.708527i	$-0.250640\pi$
$751$	38.6776	1.41137	0.705683	+	0.708527i	$0.250640\pi$
$752$	−4.99988	−0.182327
$753$	0	0
$754$	−51.9084	−1.89039
$755$	−40.0167	−1.45636
$756$	0	0
$757$	−35.7717	−1.30015	−0.650073	−	0.759872i	$-0.725262\pi$
$757$	−35.7717	−1.30015	−0.650073	+	0.759872i	$0.725262\pi$
$758$	−4.36425	−0.158517
$759$	0	0
$760$	−4.15082	−0.150566
$761$	4.45692	0.161563	0.0807817	−	0.996732i	$-0.474258\pi$
$761$	4.45692	0.161563	0.0807817	+	0.996732i	$0.474258\pi$
$762$	0	0
$763$	14.5772	0.527729
$764$	19.3040	0.698396
$765$	0	0
$766$	−7.68194	−0.277560
$767$	24.6348	0.889511
$768$	0	0
$769$	22.2720	0.803149	0.401575	−	0.915826i	$-0.368463\pi$
$769$	22.2720	0.803149	0.401575	+	0.915826i	$0.368463\pi$
$770$	26.0697	0.939486
$771$	0	0
$772$	−27.3983	−0.986086
$773$	1.09820	0.0394996	0.0197498	−	0.999805i	$-0.493713\pi$
$773$	1.09820	0.0394996	0.0197498	+	0.999805i	$0.493713\pi$
$774$	0	0
$775$	8.58444	0.308362
$776$	9.51107	0.341427
$777$	0	0
$778$	13.1590	0.471772
$779$	7.74948	0.277654
$780$	0	0
$781$	−28.2430	−1.01061
$782$	26.1190	0.934015
$783$	0	0
$784$	21.7153	0.775547
$785$	−28.3033	−1.01019
$786$	0	0
$787$	36.0835	1.28624	0.643119	−	0.765766i	$-0.277640\pi$
$787$	36.0835	1.28624	0.643119	+	0.765766i	$0.277640\pi$
$788$	−5.43871	−0.193746
$789$	0	0
$790$	45.4712	1.61779
$791$	−20.7850	−0.739028
$792$	0	0
$793$	−32.8206	−1.16549
$794$	−43.6746	−1.54995
$795$	0	0
$796$	4.69405	0.166376
$797$	49.9833	1.77050	0.885250	−	0.465116i	$-0.153987\pi$
$797$	49.9833	1.77050	0.885250	+	0.465116i	$0.153987\pi$
$798$	0	0
$799$	−2.26965	−0.0802946
$800$	4.45661	0.157565
$801$	0	0
$802$	53.4019	1.88569
$803$	−29.8869	−1.05469
$804$	0	0
$805$	−26.1457	−0.921516
$806$	−79.6280	−2.80478
$807$	0	0
$808$	3.77696	0.132873
$809$	−10.6653	−0.374970	−0.187485	−	0.982267i	$-0.560034\pi$
$809$	−10.6653	−0.374970	−0.187485	+	0.982267i	$0.560034\pi$
$810$	0	0
$811$	−38.2881	−1.34448	−0.672238	−	0.740335i	$-0.734667\pi$
$811$	−38.2881	−1.34448	−0.672238	+	0.740335i	$0.734667\pi$
$812$	10.8500	0.380760
$813$	0	0
$814$	−54.4461	−1.90834
$815$	30.3757	1.06401
$816$	0	0
$817$	4.82751	0.168893
$818$	6.02621	0.210702
$819$	0	0
$820$	−18.9492	−0.661734
$821$	1.50950	0.0526820	0.0263410	−	0.999653i	$-0.491614\pi$
$821$	1.50950	0.0526820	0.0263410	+	0.999653i	$0.491614\pi$
$822$	0	0
$823$	−17.4467	−0.608154	−0.304077	−	0.952648i	$-0.598348\pi$
$823$	−17.4467	−0.608154	−0.304077	+	0.952648i	$0.598348\pi$
$824$	26.2993	0.916181
$825$	0	0
$826$	−15.3358	−0.533600
$827$	−10.1883	−0.354281	−0.177140	−	0.984186i	$-0.556685\pi$
$827$	−10.1883	−0.354281	−0.177140	+	0.984186i	$0.556685\pi$
$828$	0	0
$829$	−31.3153	−1.08763	−0.543813	−	0.839207i	$-0.683020\pi$
$829$	−31.3153	−1.08763	−0.543813	+	0.839207i	$0.683020\pi$
$830$	−30.3068	−1.05196
$831$	0	0
$832$	4.09384	0.141928
$833$	9.85749	0.341542
$834$	0	0
$835$	38.4715	1.33136
$836$	3.85272	0.133249
$837$	0	0
$838$	61.9309	2.13937
$839$	13.0170	0.449397	0.224698	−	0.974428i	$-0.427860\pi$
$839$	13.0170	0.449397	0.224698	+	0.974428i	$0.427860\pi$
$840$	0	0
$841$	14.3520	0.494898
$842$	10.5259	0.362747
$843$	0	0
$844$	−21.8501	−0.752111
$845$	−18.4842	−0.635874
$846$	0	0
$847$	5.74202	0.197298
$848$	−28.7055	−0.985749
$849$	0	0
$850$	3.34729	0.114811
$851$	54.6050	1.87183
$852$	0	0
$853$	10.9964	0.376511	0.188255	−	0.982120i	$-0.439717\pi$
$853$	10.9964	0.376511	0.188255	+	0.982120i	$0.439717\pi$
$854$	20.4316	0.699156
$855$	0	0
$856$	19.8349	0.677943
$857$	36.0993	1.23313	0.616565	−	0.787304i	$-0.288524\pi$
$857$	36.0993	1.23313	0.616565	+	0.787304i	$0.288524\pi$
$858$	0	0
$859$	−29.6412	−1.01134	−0.505672	−	0.862726i	$-0.668756\pi$
$859$	−29.6412	−1.01134	−0.505672	+	0.862726i	$0.668756\pi$
$860$	−11.8043	−0.402524
$861$	0	0
$862$	−9.46845	−0.322497
$863$	41.7816	1.42226	0.711131	−	0.703060i	$-0.248183\pi$
$863$	41.7816	1.42226	0.711131	+	0.703060i	$0.248183\pi$
$864$	0	0
$865$	−12.6707	−0.430816
$866$	−45.7297	−1.55396
$867$	0	0
$868$	16.6440	0.564934
$869$	41.2888	1.40063
$870$	0	0
$871$	16.4341	0.556849
$872$	15.3479	0.519747
$873$	0	0
$874$	−11.5079	−0.389262
$875$	16.3611	0.553107
$876$	0	0
$877$	12.2023	0.412044	0.206022	−	0.978547i	$-0.433948\pi$
$877$	12.2023	0.412044	0.206022	+	0.978547i	$0.433948\pi$
$878$	−13.2974	−0.448765
$879$	0	0
$880$	46.0850	1.55352
$881$	40.6290	1.36883	0.684413	−	0.729094i	$-0.260058\pi$
$881$	40.6290	1.36883	0.684413	+	0.729094i	$0.260058\pi$
$882$	0	0
$883$	−16.0985	−0.541756	−0.270878	−	0.962614i	$-0.587314\pi$
$883$	−16.0985	−0.541756	−0.270878	+	0.962614i	$0.587314\pi$
$884$	−10.4251	−0.350635
$885$	0	0
$886$	−23.9638	−0.805080
$887$	−48.1647	−1.61721	−0.808606	−	0.588351i	$-0.799777\pi$
$887$	−48.1647	−1.61721	−0.808606	+	0.588351i	$0.799777\pi$
$888$	0	0
$889$	−13.4996	−0.452762
$890$	51.8786	1.73898
$891$	0	0
$892$	−19.6544	−0.658080
$893$	1.00000	0.0334637
$894$	0	0
$895$	16.3361	0.546055
$896$	−19.6423	−0.656202
$897$	0	0
$898$	−0.611636	−0.0204106
$899$	66.5024	2.21798
$900$	0	0
$901$	−13.0306	−0.434112
$902$	−51.2449	−1.70627
$903$	0	0
$904$	−21.8840	−0.727850
$905$	34.0744	1.13267
$906$	0	0
$907$	−9.04711	−0.300405	−0.150202	−	0.988655i	$-0.547992\pi$
$907$	−9.04711	−0.300405	−0.150202	+	0.988655i	$0.547992\pi$
$908$	−25.8440	−0.857664
$909$	0	0
$910$	31.0807	1.03031
$911$	−20.6155	−0.683023	−0.341511	−	0.939878i	$-0.610939\pi$
$911$	−20.6155	−0.683023	−0.341511	+	0.939878i	$0.610939\pi$
$912$	0	0
$913$	−27.5192	−0.910751
$914$	−68.5736	−2.26821
$915$	0	0
$916$	23.9095	0.789992
$917$	−22.5132	−0.743452
$918$	0	0
$919$	−44.6138	−1.47167	−0.735837	−	0.677159i	$-0.763211\pi$
$919$	−44.6138	−1.47167	−0.735837	+	0.677159i	$0.763211\pi$
$920$	−27.5282	−0.907577
$921$	0	0
$922$	−49.0500	−1.61538
$923$	−33.6717	−1.10832
$924$	0	0
$925$	6.99790	0.230090
$926$	1.53329	0.0503871
$927$	0	0
$928$	34.5247	1.13333
$929$	1.48740	0.0488001	0.0244001	−	0.999702i	$-0.492232\pi$
$929$	1.48740	0.0488001	0.0244001	+	0.999702i	$0.492232\pi$
$930$	0	0
$931$	−4.34317	−0.142342
$932$	−6.99181	−0.229024
$933$	0	0
$934$	−2.33178	−0.0762981
$935$	20.9199	0.684153
$936$	0	0
$937$	2.00661	0.0655532	0.0327766	−	0.999463i	$-0.489565\pi$
$937$	2.00661	0.0655532	0.0327766	+	0.999463i	$0.489565\pi$
$938$	−10.2307	−0.334043
$939$	0	0
$940$	−2.44522	−0.0797543
$941$	2.76541	0.0901496	0.0450748	−	0.998984i	$-0.485647\pi$
$941$	2.76541	0.0901496	0.0450748	+	0.998984i	$0.485647\pi$
$942$	0	0
$943$	51.3944	1.67363
$944$	−27.1100	−0.882355
$945$	0	0
$946$	−31.9229	−1.03790
$947$	9.36034	0.304170	0.152085	−	0.988367i	$-0.451401\pi$
$947$	9.36034	0.304170	0.152085	+	0.988367i	$0.451401\pi$
$948$	0	0
$949$	−35.6316	−1.15665
$950$	−1.47480	−0.0478489
$951$	0	0
$952$	−6.34893	−0.205770
$953$	46.1540	1.49507	0.747537	−	0.664220i	$-0.231236\pi$
$953$	46.1540	1.49507	0.747537	+	0.664220i	$0.231236\pi$
$954$	0	0
$955$	−46.1828	−1.49444
$956$	−17.3119	−0.559908
$957$	0	0
$958$	−40.5459	−1.30998
$959$	22.3461	0.721593
$960$	0	0
$961$	71.0152	2.29081
$962$	−64.9115	−2.09283
$963$	0	0
$964$	5.84271	0.188181
$965$	65.5474	2.11005
$966$	0	0
$967$	−19.8036	−0.636842	−0.318421	−	0.947949i	$-0.603153\pi$
$967$	−19.8036	−0.636842	−0.318421	+	0.947949i	$0.603153\pi$
$968$	6.04563	0.194314
$969$	0	0
$970$	23.2594	0.746816
$971$	2.07149	0.0664774	0.0332387	−	0.999447i	$-0.489418\pi$
$971$	2.07149	0.0664774	0.0332387	+	0.999447i	$0.489418\pi$
$972$	0	0
$973$	−19.5399	−0.626420
$974$	−58.2438	−1.86625
$975$	0	0
$976$	36.1182	1.15612
$977$	−5.02140	−0.160649	−0.0803244	−	0.996769i	$-0.525596\pi$
$977$	−5.02140	−0.160649	−0.0803244	+	0.996769i	$0.525596\pi$
$978$	0	0
$979$	47.1068	1.50554
$980$	10.6200	0.339244
$981$	0	0
$982$	21.4110	0.683253
$983$	2.93113	0.0934884	0.0467442	−	0.998907i	$-0.485115\pi$
$983$	2.93113	0.0934884	0.0467442	+	0.998907i	$0.485115\pi$
$984$	0	0
$985$	13.0115	0.414582
$986$	25.9309	0.825809
$987$	0	0
$988$	4.59327	0.146131
$989$	32.0160	1.01805
$990$	0	0
$991$	−18.7712	−0.596286	−0.298143	−	0.954521i	$-0.596367\pi$
$991$	−18.7712	−0.596286	−0.298143	+	0.954521i	$0.596367\pi$
$992$	52.9612	1.68152
$993$	0	0
$994$	20.9615	0.664858
$995$	−11.2300	−0.356015
$996$	0	0
$997$	−6.19859	−0.196311	−0.0981557	−	0.995171i	$-0.531294\pi$
$997$	−6.19859	−0.196311	−0.0981557	+	0.995171i	$0.531294\pi$
$998$	−41.2913	−1.30705
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.r.1.6		23
3.2	odd	2		893.2.a.d.1.18	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
893.2.a.d.1.18	✓	23	3.2	odd	2
8037.2.a.r.1.6		23	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:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q-1.73522 q^{2} +1.01098 q^{4} -2.41866 q^{5} +1.62998 q^{7} +1.71616 q^{8} +O(q^{10})\)	\(q-1.73522 q^{2} +1.01098 q^{4} -2.41866 q^{5} +1.62998 q^{7} +1.71616 q^{8} +4.19690 q^{10} +3.81087 q^{11} +4.54338 q^{13} -2.82837 q^{14} -4.99988 q^{16} -2.26965 q^{17} +1.00000 q^{19} -2.44522 q^{20} -6.61270 q^{22} +6.63199 q^{23} +0.849923 q^{25} -7.88375 q^{26} +1.64788 q^{28} +6.58423 q^{29} +10.1003 q^{31} +5.24355 q^{32} +3.93834 q^{34} -3.94237 q^{35} +8.23358 q^{37} -1.73522 q^{38} -4.15082 q^{40} +7.74948 q^{41} +4.82751 q^{43} +3.85272 q^{44} -11.5079 q^{46} +1.00000 q^{47} -4.34317 q^{49} -1.47480 q^{50} +4.59327 q^{52} +5.74123 q^{53} -9.21721 q^{55} +2.79731 q^{56} -11.4251 q^{58} +5.42213 q^{59} -7.22382 q^{61} -17.5261 q^{62} +0.901055 q^{64} -10.9889 q^{65} +3.61716 q^{67} -2.29458 q^{68} +6.84086 q^{70} -7.41116 q^{71} -7.84253 q^{73} -14.2871 q^{74} +1.01098 q^{76} +6.21164 q^{77} +10.8345 q^{79} +12.0930 q^{80} -13.4470 q^{82} -7.22122 q^{83} +5.48952 q^{85} -8.37678 q^{86} +6.54008 q^{88} +12.3612 q^{89} +7.40561 q^{91} +6.70481 q^{92} -1.73522 q^{94} -2.41866 q^{95} +5.54205 q^{97} +7.53634 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - q^{2} + 31 q^{4} - q^{5} + 15 q^{7}+O(q^{10}) \) 23 * q - q^2 + 31 * q^4 - q^5 + 15 * q^7	\( 23 q - q^{2} + 31 q^{4} - q^{5} + 15 q^{7} + 5 q^{10} - 2 q^{11} + 13 q^{13} - 6 q^{14} + 35 q^{16} - 12 q^{17} + 23 q^{19} - 3 q^{20} - 4 q^{22} - 13 q^{23} + 46 q^{25} + 7 q^{26} + 11 q^{28} + 18 q^{29} + 22 q^{31} + 4 q^{32} - 20 q^{34} - 25 q^{35} + 8 q^{37} - q^{38} - 16 q^{40} + 16 q^{41} + 68 q^{43} + 18 q^{44} + 13 q^{46} + 23 q^{47} + 52 q^{49} + 21 q^{50} + 54 q^{52} + 7 q^{53} + 32 q^{55} - 33 q^{56} + 6 q^{58} - 10 q^{59} + 28 q^{61} + 24 q^{62} + 40 q^{64} + 18 q^{65} + 55 q^{67} - 41 q^{68} - 40 q^{70} + 3 q^{71} + 48 q^{73} + 55 q^{74} + 31 q^{76} + 14 q^{77} - 31 q^{79} + 3 q^{80} + 48 q^{82} - 47 q^{83} - 5 q^{85} + 71 q^{86} + 9 q^{88} + 6 q^{89} + 40 q^{91} - 35 q^{92} - q^{94} - q^{95} - 18 q^{97} + 68 q^{98}+O(q^{100}) \) 23 * q - q^2 + 31 * q^4 - q^5 + 15 * q^7 + 5 * q^10 - 2 * q^11 + 13 * q^13 - 6 * q^14 + 35 * q^16 - 12 * q^17 + 23 * q^19 - 3 * q^20 - 4 * q^22 - 13 * q^23 + 46 * q^25 + 7 * q^26 + 11 * q^28 + 18 * q^29 + 22 * q^31 + 4 * q^32 - 20 * q^34 - 25 * q^35 + 8 * q^37 - q^38 - 16 * q^40 + 16 * q^41 + 68 * q^43 + 18 * q^44 + 13 * q^46 + 23 * q^47 + 52 * q^49 + 21 * q^50 + 54 * q^52 + 7 * q^53 + 32 * q^55 - 33 * q^56 + 6 * q^58 - 10 * q^59 + 28 * q^61 + 24 * q^62 + 40 * q^64 + 18 * q^65 + 55 * q^67 - 41 * q^68 - 40 * q^70 + 3 * q^71 + 48 * q^73 + 55 * q^74 + 31 * q^76 + 14 * q^77 - 31 * q^79 + 3 * q^80 + 48 * q^82 - 47 * q^83 - 5 * q^85 + 71 * q^86 + 9 * q^88 + 6 * q^89 + 40 * q^91 - 35 * q^92 - q^94 - q^95 - 18 * q^97 + 68 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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