LMFDB - Embedded newform 1127.2.a.j.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1127,2,Mod(1,1127)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1127, 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("1127.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1127 = 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1127.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:	$8.99914030780$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.89672832.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 9x^{4} + 21x^{2} - 8 $ x^6 - 9x^4 + 21x^2 - 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.77973$ of defining polynomial
Character	$\chi$	$=$	1127.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.167449 q^{2} -2.79366 q^{3} -1.97196 q^{4} +4.27537 q^{5} +0.467795 q^{6} +0.665102 q^{8} +4.80451 q^{9} +O(q^{10})$	\(q-0.167449 q^{2} -2.79366 q^{3} -1.97196 q^{4} +4.27537 q^{5} +0.467795 q^{6} +0.665102 q^{8} +4.80451 q^{9} -0.715908 q^{10} +2.00000 q^{11} +5.50898 q^{12} +4.10559 q^{13} -11.9439 q^{15} +3.83255 q^{16} -4.87140 q^{17} -0.804512 q^{18} -6.52290 q^{19} -8.43087 q^{20} -0.334898 q^{22} -1.00000 q^{23} -1.85806 q^{24} +13.2788 q^{25} -0.687478 q^{26} -5.04118 q^{27} +5.80451 q^{29} +2.00000 q^{30} -4.32528 q^{31} -1.97196 q^{32} -5.58731 q^{33} +0.815713 q^{34} -9.47431 q^{36} -3.94392 q^{37} +1.09225 q^{38} -11.4696 q^{39} +2.84356 q^{40} +1.48172 q^{41} +4.00000 q^{43} -3.94392 q^{44} +20.5411 q^{45} +0.167449 q^{46} +4.32528 q^{47} -10.7068 q^{48} -2.22353 q^{50} +13.6090 q^{51} -8.09607 q^{52} +11.2741 q^{53} +0.844142 q^{54} +8.55075 q^{55} +18.2227 q^{57} -0.971961 q^{58} +3.77915 q^{59} +23.5529 q^{60} -7.83484 q^{61} +0.724264 q^{62} -7.33490 q^{64} +17.5529 q^{65} +0.935591 q^{66} +8.61372 q^{67} +9.60622 q^{68} +2.79366 q^{69} +4.13941 q^{71} +3.19549 q^{72} +10.0325 q^{73} +0.660406 q^{74} -37.0964 q^{75} +12.8629 q^{76} +1.92058 q^{78} +0.0560785 q^{79} +16.3856 q^{80} -0.330203 q^{81} -0.248112 q^{82} +4.49506 q^{83} -20.8271 q^{85} -0.669797 q^{86} -16.2158 q^{87} +1.33020 q^{88} +8.17440 q^{89} -3.43959 q^{90} +1.97196 q^{92} +12.0833 q^{93} -0.724264 q^{94} -27.8878 q^{95} +5.50898 q^{96} -10.3021 q^{97} +9.60902 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 12 q^{4} + 6 q^{8} + 6 q^{9}+O(q^{10}) $ 6 * q + 12 * q^4 + 6 * q^8 + 6 * q^9	$ 6 q + 12 q^{4} + 6 q^{8} + 6 q^{9} + 12 q^{11} - 24 q^{15} + 24 q^{16} + 18 q^{18} - 6 q^{23} + 30 q^{25} + 12 q^{29} + 12 q^{30} + 12 q^{32} - 30 q^{36} + 24 q^{37} - 48 q^{39} + 24 q^{43} + 24 q^{44} - 60 q^{50} + 36 q^{51} + 24 q^{53} + 12 q^{57} + 18 q^{58} + 48 q^{60} - 42 q^{64} + 12 q^{65} + 42 q^{72} + 12 q^{74} - 66 q^{78} + 48 q^{79} - 6 q^{81} + 12 q^{85} + 12 q^{88} - 12 q^{92} - 72 q^{95} + 12 q^{99}+O(q^{100}) $ 6 * q + 12 * q^4 + 6 * q^8 + 6 * q^9 + 12 * q^11 - 24 * q^15 + 24 * q^16 + 18 * q^18 - 6 * q^23 + 30 * q^25 + 12 * q^29 + 12 * q^30 + 12 * q^32 - 30 * q^36 + 24 * q^37 - 48 * q^39 + 24 * q^43 + 24 * q^44 - 60 * q^50 + 36 * q^51 + 24 * q^53 + 12 * q^57 + 18 * q^58 + 48 * q^60 - 42 * q^64 + 12 * q^65 + 42 * q^72 + 12 * q^74 - 66 * q^78 + 48 * q^79 - 6 * q^81 + 12 * q^85 + 12 * q^88 - 12 * q^92 - 72 * q^95 + 12 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.167449	−0.118404	−0.0592022	−	0.998246i	$-0.518856\pi$
$2$	−0.167449	−0.118404	−0.0592022	+	0.998246i	$0.518856\pi$
$3$	−2.79366	−1.61292	−0.806459	−	0.591290i	$-0.798619\pi$
$3$	−2.79366	−1.61292	−0.806459	+	0.591290i	$0.798619\pi$
$4$	−1.97196	−0.985980
$5$	4.27537	1.91201	0.956003	−	0.293358i	$-0.0947728\pi$
$5$	4.27537	1.91201	0.956003	+	0.293358i	$0.0947728\pi$
$6$	0.467795	0.190977
$7$	0	0
$7$	0	0
$8$	0.665102	0.235149
$9$	4.80451	1.60150
$10$	−0.715908	−0.226390
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	5.50898	1.59031
$13$	4.10559	1.13869	0.569343	−	0.822100i	$-0.307198\pi$
$13$	4.10559	1.13869	0.569343	+	0.822100i	$0.307198\pi$
$14$	0	0
$15$	−11.9439	−3.08391
$16$	3.83255	0.958138
$17$	−4.87140	−1.18149	−0.590744	−	0.806859i	$-0.701166\pi$
$17$	−4.87140	−1.18149	−0.590744	+	0.806859i	$0.701166\pi$
$18$	−0.804512	−0.189625
$19$	−6.52290	−1.49646	−0.748228	−	0.663441i	$-0.769095\pi$
$19$	−6.52290	−1.49646	−0.748228	+	0.663441i	$0.769095\pi$
$20$	−8.43087	−1.88520
$21$	0	0
$22$	−0.334898	−0.0714006
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.85806	−0.379276
$25$	13.2788	2.65576
$26$	−0.687478	−0.134826
$27$	−5.04118	−0.970176
$28$	0	0
$29$	5.80451	1.07787	0.538935	−	0.842347i	$-0.318827\pi$
$29$	5.80451	1.07787	0.538935	+	0.842347i	$0.318827\pi$
$30$	2.00000	0.365148
$31$	−4.32528	−0.776843	−0.388421	−	0.921482i	$-0.626979\pi$
$31$	−4.32528	−0.776843	−0.388421	+	0.921482i	$0.626979\pi$
$32$	−1.97196	−0.348597
$33$	−5.58731	−0.972626
$34$	0.815713	0.139894
$35$	0	0
$36$	−9.47431	−1.57905
$37$	−3.94392	−0.648377	−0.324188	−	0.945993i	$-0.605091\pi$
$37$	−3.94392	−0.648377	−0.324188	+	0.945993i	$0.605091\pi$
$38$	1.09225	0.177187
$39$	−11.4696	−1.83661
$40$	2.84356	0.449606
$41$	1.48172	0.231405	0.115703	−	0.993284i	$-0.463088\pi$
$41$	1.48172	0.231405	0.115703	+	0.993284i	$0.463088\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−3.94392	−0.594569
$45$	20.5411	3.06208
$46$	0.167449	0.0246890
$47$	4.32528	0.630906	0.315453	−	0.948941i	$-0.397843\pi$
$47$	4.32528	0.630906	0.315453	+	0.948941i	$0.397843\pi$
$48$	−10.7068	−1.54540
$49$	0	0
$50$	−2.22353	−0.314454
$51$	13.6090	1.90564
$52$	−8.09607	−1.12272
$53$	11.2741	1.54862	0.774310	−	0.632806i	$-0.218097\pi$
$53$	11.2741	1.54862	0.774310	+	0.632806i	$0.218097\pi$
$54$	0.844142	0.114873
$55$	8.55075	1.15298
$56$	0	0
$57$	18.2227	2.41366
$58$	−0.971961	−0.127625
$59$	3.77915	0.492003	0.246002	−	0.969269i	$-0.420883\pi$
$59$	3.77915	0.492003	0.246002	+	0.969269i	$0.420883\pi$
$60$	23.5529	3.04067
$61$	−7.83484	−1.00315	−0.501574	−	0.865115i	$-0.667246\pi$
$61$	−7.83484	−1.00315	−0.501574	+	0.865115i	$0.667246\pi$
$62$	0.724264	0.0919816
$63$	0	0
$64$	−7.33490	−0.916862
$65$	17.5529	2.17717
$66$	0.935591	0.115163
$67$	8.61372	1.05233	0.526167	−	0.850382i	$-0.323629\pi$
$67$	8.61372	1.05233	0.526167	+	0.850382i	$0.323629\pi$
$68$	9.60622	1.16492
$69$	2.79366	0.336317
$70$	0	0
$71$	4.13941	0.491258	0.245629	−	0.969364i	$-0.421006\pi$
$71$	4.13941	0.491258	0.245629	+	0.969364i	$0.421006\pi$
$72$	3.19549	0.376592
$73$	10.0325	1.17421	0.587106	−	0.809510i	$-0.300267\pi$
$73$	10.0325	1.17421	0.587106	+	0.809510i	$0.300267\pi$
$74$	0.660406	0.0767707
$75$	−37.0964	−4.28353
$76$	12.8629	1.47548
$77$	0	0
$78$	1.92058	0.217463
$79$	0.0560785	0.00630932	0.00315466	−	0.999995i	$-0.498996\pi$
$79$	0.0560785	0.00630932	0.00315466	+	0.999995i	$0.498996\pi$
$80$	16.3856	1.83196
$81$	−0.330203	−0.0366892
$82$	−0.248112	−0.0273994
$83$	4.49506	0.493397	0.246698	−	0.969092i	$-0.420654\pi$
$83$	4.49506	0.493397	0.246698	+	0.969092i	$0.420654\pi$
$84$	0	0
$85$	−20.8271	−2.25901
$86$	−0.669797	−0.0722260
$87$	−16.2158	−1.73852
$88$	1.33020	0.141800
$89$	8.17440	0.866485	0.433242	−	0.901277i	$-0.357369\pi$
$89$	8.17440	0.866485	0.433242	+	0.901277i	$0.357369\pi$
$90$	−3.43959	−0.362564
$91$	0	0
$92$	1.97196	0.205591
$93$	12.0833	1.25298
$94$	−0.724264	−0.0747021
$95$	−27.8878	−2.86123
$96$	5.50898	0.562258
$97$	−10.3021	−1.04601	−0.523007	−	0.852328i	$-0.675190\pi$
$97$	−10.3021	−1.04601	−0.523007	+	0.852328i	$0.675190\pi$
$98$	0	0
$99$	9.60902	0.965743
$100$	−26.1853	−2.61853
$101$	2.62388	0.261085	0.130543	−	0.991443i	$-0.458328\pi$
$101$	2.62388	0.261085	0.130543	+	0.991443i	$0.458328\pi$
$102$	−2.27882	−0.225637
$103$	−0.496225	−0.0488945	−0.0244473	−	0.999701i	$-0.507783\pi$
$103$	−0.496225	−0.0488945	−0.0244473	+	0.999701i	$0.507783\pi$
$104$	2.73064	0.267761
$105$	0	0
$106$	−1.88784	−0.183364
$107$	10.3349	0.999112	0.499556	−	0.866281i	$-0.333496\pi$
$107$	10.3349	0.999112	0.499556	+	0.866281i	$0.333496\pi$
$108$	9.94102	0.956575
$109$	10.2788	0.984532	0.492266	−	0.870445i	$-0.336169\pi$
$109$	10.2788	0.984532	0.492266	+	0.870445i	$0.336169\pi$
$110$	−1.43182	−0.136518
$111$	11.0180	1.04578
$112$	0	0
$113$	−4.93923	−0.464643	−0.232322	−	0.972639i	$-0.574632\pi$
$113$	−4.93923	−0.464643	−0.232322	+	0.972639i	$0.574632\pi$
$114$	−3.05138	−0.285788
$115$	−4.27537	−0.398681
$116$	−11.4463	−1.06276
$117$	19.7254	1.82361
$118$	−0.632815	−0.0582554
$119$	0	0
$120$	−7.94392	−0.725178
$121$	−7.00000	−0.636364
$122$	1.31194	0.118777
$123$	−4.13941	−0.373238
$124$	8.52927	0.765951
$125$	35.3950	3.16583
$126$	0	0
$127$	12.8092	1.13663	0.568317	−	0.822810i	$-0.307595\pi$
$127$	12.8092	1.13663	0.568317	+	0.822810i	$0.307595\pi$
$128$	5.17214	0.457157
$129$	−11.1746	−0.983871
$130$	−2.93923	−0.257787
$131$	14.4076	1.25880	0.629401	−	0.777081i	$-0.283300\pi$
$131$	14.4076	1.25880	0.629401	+	0.777081i	$0.283300\pi$
$132$	11.0180	0.958990
$133$	0	0
$134$	−1.44236	−0.124601
$135$	−21.5529	−1.85498
$136$	−3.23998	−0.277826
$137$	8.33490	0.712098	0.356049	−	0.934467i	$-0.384124\pi$
$137$	8.33490	0.712098	0.356049	+	0.934467i	$0.384124\pi$
$138$	−0.467795	−0.0398214
$139$	−1.26204	−0.107044	−0.0535222	−	0.998567i	$-0.517045\pi$
$139$	−1.26204	−0.107044	−0.0535222	+	0.998567i	$0.517045\pi$
$140$	0	0
$141$	−12.0833	−1.01760
$142$	−0.693141	−0.0581671
$143$	8.21119	0.686654
$144$	18.4135	1.53446
$145$	24.8165	2.06089
$146$	−1.67993	−0.139032
$147$	0	0
$148$	7.77726	0.639287
$149$	−16.6137	−1.36105	−0.680524	−	0.732725i	$-0.738248\pi$
$149$	−16.6137	−1.36105	−0.680524	+	0.732725i	$0.738248\pi$
$150$	6.21177	0.507189
$151$	−11.4696	−0.933384	−0.466692	−	0.884420i	$-0.654554\pi$
$151$	−11.4696	−0.933384	−0.466692	+	0.884420i	$0.654554\pi$
$152$	−4.33839	−0.351890
$153$	−23.4047	−1.89216
$154$	0	0
$155$	−18.4922	−1.48533
$156$	22.6176	1.81086
$157$	−19.3490	−1.54422	−0.772110	−	0.635489i	$-0.780799\pi$
$157$	−19.3490	−1.54422	−0.772110	+	0.635489i	$0.780799\pi$
$158$	−0.00939029	−0.000747052 0
$159$	−31.4960	−2.49780
$160$	−8.43087	−0.666519
$161$	0	0
$162$	0.0552923	0.00434417
$163$	20.4182	1.59928	0.799640	−	0.600480i	$-0.205024\pi$
$163$	20.4182	1.59928	0.799640	+	0.600480i	$0.205024\pi$
$164$	−2.92189	−0.228161
$165$	−23.8878	−1.85967
$166$	−0.752694	−0.0584204
$167$	−12.3299	−0.954116	−0.477058	−	0.878872i	$-0.658297\pi$
$167$	−12.3299	−0.954116	−0.477058	+	0.878872i	$0.658297\pi$
$168$	0	0
$169$	3.85589	0.296607
$170$	3.48748	0.267477
$171$	−31.3394	−2.39658
$172$	−7.88784	−0.601442
$173$	21.1572	1.60855	0.804276	−	0.594257i	$-0.202554\pi$
$173$	21.1572	1.60855	0.804276	+	0.594257i	$0.202554\pi$
$174$	2.71532	0.205848
$175$	0	0
$176$	7.66510	0.577779
$177$	−10.5576	−0.793561
$178$	−1.36880	−0.102596
$179$	−7.74843	−0.579145	−0.289573	−	0.957156i	$-0.593513\pi$
$179$	−7.74843	−0.579145	−0.289573	+	0.957156i	$0.593513\pi$
$180$	−40.5062	−3.01915
$181$	−0.715908	−0.0532130	−0.0266065	−	0.999646i	$-0.508470\pi$
$181$	−0.715908	−0.0532130	−0.0266065	+	0.999646i	$0.508470\pi$
$182$	0	0
$183$	21.8878	1.61800
$184$	−0.665102	−0.0490319
$185$	−16.8617	−1.23970
$186$	−2.02334	−0.148359
$187$	−9.74281	−0.712465
$188$	−8.52927	−0.622061
$189$	0	0
$190$	4.66980	0.338783
$191$	13.5529	0.980657	0.490328	−	0.871538i	$-0.336877\pi$
$191$	13.5529	0.980657	0.490328	+	0.871538i	$0.336877\pi$
$192$	20.4912	1.47882
$193$	0.195488	0.0140716	0.00703578	−	0.999975i	$-0.497760\pi$
$193$	0.195488	0.0140716	0.00703578	+	0.999975i	$0.497760\pi$
$194$	1.72507	0.123853
$195$	−49.0369	−3.51160
$196$	0	0
$197$	0.195488	0.0139280	0.00696399	−	0.999976i	$-0.497783\pi$
$197$	0.195488	0.0139280	0.00696399	+	0.999976i	$0.497783\pi$
$198$	−1.60902	−0.114348
$199$	15.3301	1.08672	0.543362	−	0.839499i	$-0.317151\pi$
$199$	15.3301	1.08672	0.543362	+	0.839499i	$0.317151\pi$
$200$	8.83176	0.624500
$201$	−24.0638	−1.69733
$202$	−0.439366	−0.0309137
$203$	0	0
$204$	−26.8365	−1.87893
$205$	6.33490	0.442448
$206$	0.0830925	0.00578933
$207$	−4.80451	−0.333937
$208$	15.7349	1.09102
$209$	−13.0458	−0.902397
$210$	0	0
$211$	15.2180	1.04765	0.523827	−	0.851825i	$-0.324504\pi$
$211$	15.2180	1.04765	0.523827	+	0.851825i	$0.324504\pi$
$212$	−22.2321	−1.52691
$213$	−11.5641	−0.792358
$214$	−1.73057	−0.118299
$215$	17.1015	1.16631
$216$	−3.35290	−0.228136
$217$	0	0
$218$	−1.72118	−0.116573
$219$	−28.0273	−1.89391
$220$	−16.8617	−1.13682
$221$	−20.0000	−1.34535
$222$	−1.84495	−0.123825
$223$	−20.2015	−1.35279	−0.676397	−	0.736537i	$-0.736460\pi$
$223$	−20.2015	−1.35279	−0.676397	+	0.736537i	$0.736460\pi$
$224$	0	0
$225$	63.7982	4.25322
$226$	0.827069	0.0550158
$227$	3.06324	0.203314	0.101657	−	0.994819i	$-0.467586\pi$
$227$	3.06324	0.203314	0.101657	+	0.994819i	$0.467586\pi$
$228$	−35.9345	−2.37982
$229$	15.1104	0.998526	0.499263	−	0.866451i	$-0.333604\pi$
$229$	15.1104	0.998526	0.499263	+	0.866451i	$0.333604\pi$
$230$	0.715908	0.0472056
$231$	0	0
$232$	3.86059	0.253460
$233$	−18.3622	−1.20294	−0.601472	−	0.798894i	$-0.705419\pi$
$233$	−18.3622	−1.20294	−0.601472	+	0.798894i	$0.705419\pi$
$234$	−3.30300	−0.215924
$235$	18.4922	1.20630
$236$	−7.45233	−0.485106
$237$	−0.156664	−0.0101764
$238$	0	0
$239$	0.418230	0.0270530	0.0135265	−	0.999909i	$-0.495694\pi$
$239$	0.418230	0.0270530	0.0135265	+	0.999909i	$0.495694\pi$
$240$	−45.7757	−2.95481
$241$	−0.0367857	−0.00236958	−0.00118479	−	0.999999i	$-0.500377\pi$
$241$	−0.0367857	−0.00236958	−0.00118479	+	0.999999i	$0.500377\pi$
$242$	1.17214	0.0753483
$243$	16.0460	1.02935
$244$	15.4500	0.989085
$245$	0	0
$246$	0.693141	0.0441930
$247$	−26.7804	−1.70399
$248$	−2.87675	−0.182674
$249$	−12.5576	−0.795808
$250$	−5.92687	−0.374848
$251$	−1.68828	−0.106564	−0.0532818	−	0.998580i	$-0.516968\pi$
$251$	−1.68828	−0.106564	−0.0532818	+	0.998580i	$0.516968\pi$
$252$	0	0
$253$	−2.00000	−0.125739
$254$	−2.14489	−0.134582
$255$	58.1837	3.64360
$256$	13.8037	0.862733
$257$	−12.6563	−0.789481	−0.394740	−	0.918793i	$-0.629165\pi$
$257$	−12.6563	−0.789481	−0.394740	+	0.918793i	$0.629165\pi$
$258$	1.87118	0.116495
$259$	0	0
$260$	−34.6137	−2.14665
$261$	27.8878	1.72621
$262$	−2.41255	−0.149048
$263$	25.1059	1.54810	0.774048	−	0.633127i	$-0.218229\pi$
$263$	25.1059	1.54810	0.774048	+	0.633127i	$0.218229\pi$
$264$	−3.71613	−0.228712
$265$	48.2011	2.96097
$266$	0	0
$267$	−22.8365	−1.39757
$268$	−16.9859	−1.03758
$269$	−1.58152	−0.0964271	−0.0482136	−	0.998837i	$-0.515353\pi$
$269$	−1.58152	−0.0964271	−0.0482136	+	0.998837i	$0.515353\pi$
$270$	3.60902	0.219638
$271$	−28.7523	−1.74658	−0.873288	−	0.487204i	$-0.838017\pi$
$271$	−28.7523	−1.74658	−0.873288	+	0.487204i	$0.838017\pi$
$272$	−18.6699	−1.13203
$273$	0	0
$274$	−1.39567	−0.0843156
$275$	26.5576	1.60149
$276$	−5.50898	−0.331602
$277$	14.8653	0.893168	0.446584	−	0.894742i	$-0.352640\pi$
$277$	14.8653	0.893168	0.446584	+	0.894742i	$0.352640\pi$
$278$	0.211327	0.0126745
$279$	−20.7808	−1.24412
$280$	0	0
$281$	−7.55294	−0.450571	−0.225285	−	0.974293i	$-0.572331\pi$
$281$	−7.55294	−0.450571	−0.225285	+	0.974293i	$0.572331\pi$
$282$	2.02334	0.120488
$283$	5.14795	0.306014	0.153007	−	0.988225i	$-0.451104\pi$
$283$	5.14795	0.306014	0.153007	+	0.988225i	$0.451104\pi$
$284$	−8.16275	−0.484370
$285$	77.9090	4.61493
$286$	−1.37496	−0.0813029
$287$	0	0
$288$	−9.47431	−0.558279
$289$	6.73057	0.395916
$290$	−4.15550	−0.244019
$291$	28.7804	1.68714
$292$	−19.7836	−1.15775
$293$	12.4866	0.729473	0.364736	−	0.931111i	$-0.381159\pi$
$293$	12.4866	0.729473	0.364736	+	0.931111i	$0.381159\pi$
$294$	0	0
$295$	16.1573	0.940713
$296$	−2.62311	−0.152465
$297$	−10.0824	−0.585038
$298$	2.78195	0.161154
$299$	−4.10559	−0.237433
$300$	73.1527	4.22348
$301$	0	0
$302$	1.92058	0.110517
$303$	−7.33020	−0.421109
$304$	−24.9994	−1.43381
$305$	−33.4969	−1.91802
$306$	3.91910	0.224040
$307$	0.715908	0.0408590	0.0204295	−	0.999791i	$-0.493497\pi$
$307$	0.715908	0.0408590	0.0204295	+	0.999791i	$0.493497\pi$
$308$	0	0
$309$	1.38628	0.0788628
$310$	3.09650	0.175869
$311$	−28.7855	−1.63227	−0.816137	−	0.577859i	$-0.803888\pi$
$311$	−28.7855	−1.63227	−0.816137	+	0.577859i	$0.803888\pi$
$312$	−7.62846	−0.431876
$313$	20.0449	1.13300	0.566501	−	0.824061i	$-0.308297\pi$
$313$	20.0449	1.13300	0.566501	+	0.824061i	$0.308297\pi$
$314$	3.23998	0.182843
$315$	0	0
$316$	−0.110585	−0.00622087
$317$	−30.9392	−1.73772	−0.868860	−	0.495058i	$-0.835147\pi$
$317$	−30.9392	−1.73772	−0.868860	+	0.495058i	$0.835147\pi$
$318$	5.27398	0.295750
$319$	11.6090	0.649981
$320$	−31.3594	−1.75305
$321$	−28.8721	−1.61149
$322$	0	0
$323$	31.7757	1.76805
$324$	0.651148	0.0361749
$325$	54.5174	3.02408
$326$	−3.41902	−0.189362
$327$	−28.7155	−1.58797
$328$	0.985493	0.0544148
$329$	0	0
$330$	4.00000	0.220193
$331$	−9.20018	−0.505688	−0.252844	−	0.967507i	$-0.581366\pi$
$331$	−9.20018	−0.505688	−0.252844	+	0.967507i	$0.581366\pi$
$332$	−8.86408	−0.486479
$333$	−18.9486	−1.03838
$334$	2.06463	0.112972
$335$	36.8269	2.01207
$336$	0	0
$337$	30.8831	1.68231	0.841156	−	0.540792i	$-0.181876\pi$
$337$	30.8831	1.68231	0.841156	+	0.540792i	$0.181876\pi$
$338$	−0.645666	−0.0351196
$339$	13.7985	0.749432
$340$	41.0702	2.22734
$341$	−8.65055	−0.468454
$342$	5.24775	0.283766
$343$	0	0
$344$	2.66041	0.143440
$345$	11.9439	0.643039
$346$	−3.54275	−0.190460
$347$	−23.2180	−1.24641	−0.623205	−	0.782059i	$-0.714170\pi$
$347$	−23.2180	−1.24641	−0.623205	+	0.782059i	$0.714170\pi$
$348$	31.9769	1.71414
$349$	12.5565	0.672136	0.336068	−	0.941838i	$-0.390903\pi$
$349$	12.5565	0.672136	0.336068	+	0.941838i	$0.390903\pi$
$350$	0	0
$351$	−20.6970	−1.10473
$352$	−3.94392	−0.210212
$353$	23.1781	1.23364	0.616822	−	0.787103i	$-0.288420\pi$
$353$	23.1781	1.23364	0.616822	+	0.787103i	$0.288420\pi$
$354$	1.76787	0.0939611
$355$	17.6975	0.939287
$356$	−16.1196	−0.854337
$357$	0	0
$358$	1.29747	0.0685734
$359$	−23.8318	−1.25779	−0.628896	−	0.777489i	$-0.716493\pi$
$359$	−23.8318	−1.25779	−0.628896	+	0.777489i	$0.716493\pi$
$360$	13.6619	0.720046
$361$	23.5482	1.23938
$362$	0.119878	0.00630066
$363$	19.5556	1.02640
$364$	0	0
$365$	42.8925	2.24510
$366$	−3.66510	−0.191578
$367$	−0.439366	−0.0229347	−0.0114674	−	0.999934i	$-0.503650\pi$
$367$	−0.439366	−0.0229347	−0.0114674	+	0.999934i	$0.503650\pi$
$368$	−3.83255	−0.199786
$369$	7.11893	0.370597
$370$	2.82348	0.146786
$371$	0	0
$372$	−23.8279	−1.23542
$373$	−6.66980	−0.345349	−0.172674	−	0.984979i	$-0.555241\pi$
$373$	−6.66980	−0.345349	−0.172674	+	0.984979i	$0.555241\pi$
$374$	1.63143	0.0843590
$375$	−98.8816	−5.10622
$376$	2.87675	0.148357
$377$	23.8310	1.22736
$378$	0	0
$379$	−10.9953	−0.564791	−0.282395	−	0.959298i	$-0.591129\pi$
$379$	−10.9953	−0.564791	−0.282395	+	0.959298i	$0.591129\pi$
$380$	54.9937	2.82112
$381$	−35.7845	−1.83330
$382$	−2.26943	−0.116114
$383$	−6.86246	−0.350655	−0.175328	−	0.984510i	$-0.556099\pi$
$383$	−6.86246	−0.350655	−0.175328	+	0.984510i	$0.556099\pi$
$384$	−14.4492	−0.737357
$385$	0	0
$386$	−0.0327344	−0.00166614
$387$	19.2180	0.976908
$388$	20.3152	1.03135
$389$	−2.94862	−0.149501	−0.0747504	−	0.997202i	$-0.523816\pi$
$389$	−2.94862	−0.149501	−0.0747504	+	0.997202i	$0.523816\pi$
$390$	8.21119	0.415790
$391$	4.87140	0.246357
$392$	0	0
$393$	−40.2500	−2.03034
$394$	−0.0327344	−0.00164913
$395$	0.239756	0.0120635
$396$	−18.9486	−0.952204
$397$	−14.0882	−0.707064	−0.353532	−	0.935422i	$-0.615019\pi$
$397$	−14.0882	−0.707064	−0.353532	+	0.935422i	$0.615019\pi$
$398$	−2.56702	−0.128673
$399$	0	0
$400$	50.8918	2.54459
$401$	9.88784	0.493775	0.246888	−	0.969044i	$-0.420592\pi$
$401$	9.88784	0.493775	0.246888	+	0.969044i	$0.420592\pi$
$402$	4.02946	0.200971
$403$	−17.7578	−0.884580
$404$	−5.17418	−0.257425
$405$	−1.41174	−0.0701500
$406$	0	0
$407$	−7.88784	−0.390986
$408$	9.05138	0.448110
$409$	21.2071	1.04862	0.524312	−	0.851527i	$-0.324323\pi$
$409$	21.2071	1.04862	0.524312	+	0.851527i	$0.324323\pi$
$410$	−1.06077	−0.0523879
$411$	−23.2848	−1.14856
$412$	0.978536	0.0482090
$413$	0	0
$414$	0.804512	0.0395396
$415$	19.2180	0.943377
$416$	−8.09607	−0.396942
$417$	3.52569	0.172654
$418$	2.18451	0.106848
$419$	−11.8537	−0.579093	−0.289547	−	0.957164i	$-0.593505\pi$
$419$	−11.8537	−0.579093	−0.289547	+	0.957164i	$0.593505\pi$
$420$	0	0
$421$	−21.6090	−1.05316	−0.526580	−	0.850126i	$-0.676526\pi$
$421$	−21.6090	−1.05316	−0.526580	+	0.850126i	$0.676526\pi$
$422$	−2.54825	−0.124047
$423$	20.7808	1.01040
$424$	7.49844	0.364156
$425$	−64.6865	−3.13776
$426$	1.93640	0.0938187
$427$	0	0
$428$	−20.3800	−0.985105
$429$	−22.9392	−1.10752
$430$	−2.86363	−0.138097
$431$	39.8224	1.91818	0.959088	−	0.283108i	$-0.0913654\pi$
$431$	39.8224	1.91818	0.959088	+	0.283108i	$0.0913654\pi$
$432$	−19.3206	−0.929562
$433$	−1.90797	−0.0916911	−0.0458455	−	0.998949i	$-0.514598\pi$
$433$	−1.90797	−0.0916911	−0.0458455	+	0.998949i	$0.514598\pi$
$434$	0	0
$435$	−69.3286	−3.32405
$436$	−20.2694	−0.970730
$437$	6.52290	0.312033
$438$	4.69314	0.224247
$439$	20.3345	0.970513	0.485257	−	0.874372i	$-0.338726\pi$
$439$	20.3345	0.970513	0.485257	+	0.874372i	$0.338726\pi$
$440$	5.68712	0.271123
$441$	0	0
$442$	3.34898	0.159295
$443$	18.7998	0.893206	0.446603	−	0.894732i	$-0.352634\pi$
$443$	18.7998	0.893206	0.446603	+	0.894732i	$0.352634\pi$
$444$	−21.7270	−1.03112
$445$	34.9486	1.65672
$446$	3.38273	0.160177
$447$	46.4130	2.19526
$448$	0	0
$449$	−24.1667	−1.14050	−0.570248	−	0.821473i	$-0.693153\pi$
$449$	−24.1667	−1.14050	−0.570248	+	0.821473i	$0.693153\pi$
$450$	−10.6830	−0.503600
$451$	2.96344	0.139543
$452$	9.73996	0.458129
$453$	32.0421	1.50547
$454$	−0.512937	−0.0240733
$455$	0	0
$456$	12.1200	0.567570
$457$	32.8271	1.53559	0.767793	−	0.640698i	$-0.221355\pi$
$457$	32.8271	1.53559	0.767793	+	0.640698i	$0.221355\pi$
$458$	−2.53023	−0.118230
$459$	24.5576	1.14625
$460$	8.43087	0.393091
$461$	5.53741	0.257903	0.128951	−	0.991651i	$-0.458839\pi$
$461$	5.53741	0.257903	0.128951	+	0.991651i	$0.458839\pi$
$462$	0	0
$463$	−1.33959	−0.0622562	−0.0311281	−	0.999515i	$-0.509910\pi$
$463$	−1.33959	−0.0622562	−0.0311281	+	0.999515i	$0.509910\pi$
$464$	22.2461	1.03275
$465$	51.6608	2.39571
$466$	3.07473	0.142434
$467$	7.53206	0.348542	0.174271	−	0.984698i	$-0.444243\pi$
$467$	7.53206	0.348542	0.174271	+	0.984698i	$0.444243\pi$
$468$	−38.8977	−1.79804
$469$	0	0
$470$	−3.09650	−0.142831
$471$	54.0545	2.49070
$472$	2.51352	0.115694
$473$	8.00000	0.367840
$474$	0.0262332	0.00120493
$475$	−86.6164	−3.97423
$476$	0	0
$477$	54.1667	2.48012
$478$	−0.0700323	−0.00320320
$479$	−8.29428	−0.378975	−0.189488	−	0.981883i	$-0.560683\pi$
$479$	−8.29428	−0.378975	−0.189488	+	0.981883i	$0.560683\pi$
$480$	23.5529	1.07504
$481$	−16.1921	−0.738298
$482$	0.00615974	0.000280568 0
$483$	0	0
$484$	13.8037	0.627442
$485$	−44.0451	−1.99999
$486$	−2.68689	−0.121880
$487$	−12.1394	−0.550089	−0.275045	−	0.961431i	$-0.588693\pi$
$487$	−12.1394	−0.550089	−0.275045	+	0.961431i	$0.588693\pi$
$488$	−5.21096	−0.235889
$489$	−57.0415	−2.57951
$490$	0	0
$491$	−36.0273	−1.62589	−0.812944	−	0.582342i	$-0.802136\pi$
$491$	−36.0273	−1.62589	−0.812944	+	0.582342i	$0.802136\pi$
$492$	8.16275	0.368005
$493$	−28.2761	−1.27349
$494$	4.48435	0.201761
$495$	41.0822	1.84651
$496$	−16.5768	−0.744322
$497$	0	0
$498$	2.10277	0.0942272
$499$	−30.0179	−1.34378	−0.671892	−	0.740649i	$-0.734518\pi$
$499$	−30.0179	−1.34378	−0.671892	+	0.740649i	$0.734518\pi$
$500$	−69.7976	−3.12145
$501$	34.4455	1.53891
$502$	0.282702	0.0126176
$503$	14.8339	0.661411	0.330705	−	0.943734i	$-0.392713\pi$
$503$	14.8339	0.661411	0.330705	+	0.943734i	$0.392713\pi$
$504$	0	0
$505$	11.2180	0.499197
$506$	0.334898	0.0148880
$507$	−10.7720	−0.478403
$508$	−25.2593	−1.12070
$509$	−10.6854	−0.473620	−0.236810	−	0.971556i	$-0.576102\pi$
$509$	−10.6854	−0.473620	−0.236810	+	0.971556i	$0.576102\pi$
$510$	−9.74281	−0.431419
$511$	0	0
$512$	−12.6557	−0.559309
$513$	32.8831	1.45183
$514$	2.11929	0.0934780
$515$	−2.12155	−0.0934865
$516$	22.0359	0.970077
$517$	8.65055	0.380451
$518$	0	0
$519$	−59.1059	−2.59446
$520$	11.6745	0.511960
$521$	−8.92709	−0.391103	−0.195552	−	0.980693i	$-0.562650\pi$
$521$	−8.92709	−0.391103	−0.195552	+	0.980693i	$0.562650\pi$
$522$	−4.66980	−0.204391
$523$	−15.7265	−0.687673	−0.343837	−	0.939029i	$-0.611727\pi$
$523$	−15.7265	−0.687673	−0.343837	+	0.939029i	$0.611727\pi$
$524$	−28.4113	−1.24115
$525$	0	0
$526$	−4.20396	−0.183301
$527$	21.0702	0.917831
$528$	−21.4137	−0.931910
$529$	1.00000	0.0434783
$530$	−8.07123	−0.350592
$531$	18.1570	0.787945
$532$	0	0
$533$	6.08333	0.263498
$534$	3.82395	0.165478
$535$	44.1856	1.91031
$536$	5.72900	0.247455
$537$	21.6465	0.934113
$538$	0.264825	0.0114174
$539$	0	0
$540$	42.5016	1.82898
$541$	−35.6830	−1.53413	−0.767065	−	0.641569i	$-0.778284\pi$
$541$	−35.6830	−1.53413	−0.767065	+	0.641569i	$0.778284\pi$
$542$	4.81454	0.206802
$543$	2.00000	0.0858282
$544$	9.60622	0.411863
$545$	43.9458	1.88243
$546$	0	0
$547$	−7.19079	−0.307456	−0.153728	−	0.988113i	$-0.549128\pi$
$547$	−7.19079	−0.307456	−0.153728	+	0.988113i	$0.549128\pi$
$548$	−16.4361	−0.702115
$549$	−37.6426	−1.60655
$550$	−4.44706	−0.189623
$551$	−37.8623	−1.61299
$552$	1.85806	0.0790845
$553$	0	0
$554$	−2.48918	−0.105755
$555$	47.1059	1.99953
$556$	2.48868	0.105544
$557$	−40.1106	−1.69954	−0.849770	−	0.527154i	$-0.823259\pi$
$557$	−40.1106	−1.69954	−0.849770	+	0.527154i	$0.823259\pi$
$558$	3.47973	0.147309
$559$	16.4224	0.694592
$560$	0	0
$561$	27.2180	1.14915
$562$	1.26473	0.0533496
$563$	30.1640	1.27126	0.635631	−	0.771993i	$-0.280740\pi$
$563$	30.1640	1.27126	0.635631	+	0.771993i	$0.280740\pi$
$564$	23.8279	1.00333
$565$	−21.1170	−0.888400
$566$	−0.862019	−0.0362334
$567$	0	0
$568$	2.75313	0.115519
$569$	−17.9345	−0.751855	−0.375927	−	0.926649i	$-0.622676\pi$
$569$	−17.9345	−0.751855	−0.375927	+	0.926649i	$0.622676\pi$
$570$	−13.0458	−0.546429
$571$	−24.4922	−1.02497	−0.512483	−	0.858698i	$-0.671274\pi$
$571$	−24.4922	−1.02497	−0.512483	+	0.858698i	$0.671274\pi$
$572$	−16.1921	−0.677027
$573$	−37.8623	−1.58172
$574$	0	0
$575$	−13.2788	−0.553765
$576$	−35.2406	−1.46836
$577$	−24.2703	−1.01039	−0.505193	−	0.863006i	$-0.668579\pi$
$577$	−24.2703	−1.01039	−0.505193	+	0.863006i	$0.668579\pi$
$578$	−1.12703	−0.0468782
$579$	−0.546127	−0.0226963
$580$	−48.9371	−2.03200
$581$	0	0
$582$	−4.81925	−0.199764
$583$	22.5482	0.933853
$584$	6.67261	0.276115
$585$	84.3333	3.48675
$586$	−2.09086	−0.0863728
$587$	−20.7476	−0.856347	−0.428174	−	0.903697i	$-0.640843\pi$
$587$	−20.7476	−0.856347	−0.428174	+	0.903697i	$0.640843\pi$
$588$	0	0
$589$	28.2134	1.16251
$590$	−2.70552	−0.111385
$591$	−0.546127	−0.0224647
$592$	−15.1153	−0.621234
$593$	19.1723	0.787311	0.393656	−	0.919258i	$-0.371210\pi$
$593$	19.1723	0.787311	0.393656	+	0.919258i	$0.371210\pi$
$594$	1.68828	0.0692711
$595$	0	0
$596$	32.7616	1.34197
$597$	−42.8271	−1.75280
$598$	0.687478	0.0281131
$599$	−18.0094	−0.735844	−0.367922	−	0.929857i	$-0.619931\pi$
$599$	−18.0094	−0.735844	−0.367922	+	0.929857i	$0.619931\pi$
$600$	−24.6729	−1.00727
$601$	−27.9129	−1.13859	−0.569295	−	0.822133i	$-0.692784\pi$
$601$	−27.9129	−1.13859	−0.569295	+	0.822133i	$0.692784\pi$
$602$	0	0
$603$	41.3847	1.68532
$604$	22.6176	0.920298
$605$	−29.9276	−1.21673
$606$	1.22744	0.0498612
$607$	−20.2015	−0.819954	−0.409977	−	0.912096i	$-0.634463\pi$
$607$	−20.2015	−0.819954	−0.409977	+	0.912096i	$0.634463\pi$
$608$	12.8629	0.521660
$609$	0	0
$610$	5.60902	0.227103
$611$	17.7578	0.718405
$612$	46.1532	1.86563
$613$	11.6651	0.471149	0.235575	−	0.971856i	$-0.424303\pi$
$613$	11.6651	0.471149	0.235575	+	0.971856i	$0.424303\pi$
$614$	−0.119878	−0.00483789
$615$	−17.6975	−0.713633
$616$	0	0
$617$	−2.71648	−0.109362	−0.0546808	−	0.998504i	$-0.517414\pi$
$617$	−2.71648	−0.109362	−0.0546808	+	0.998504i	$0.517414\pi$
$618$	−0.232132	−0.00933771
$619$	−42.7704	−1.71909	−0.859545	−	0.511061i	$-0.829253\pi$
$619$	−42.7704	−1.71909	−0.859545	+	0.511061i	$0.829253\pi$
$620$	36.4658	1.46450
$621$	5.04118	0.202296
$622$	4.82010	0.193268
$623$	0	0
$624$	−43.9579	−1.75972
$625$	84.9330	3.39732
$626$	−3.35650	−0.134153
$627$	36.4455	1.45549
$628$	38.1555	1.52257
$629$	19.2124	0.766050
$630$	0	0
$631$	42.0545	1.67416	0.837082	−	0.547078i	$-0.184260\pi$
$631$	42.0545	1.67416	0.837082	+	0.547078i	$0.184260\pi$
$632$	0.0372979	0.00148363
$633$	−42.5140	−1.68978
$634$	5.18075	0.205754
$635$	54.7641	2.17325
$636$	62.1089	2.46278
$637$	0	0
$638$	−1.94392	−0.0769606
$639$	19.8878	0.786751
$640$	22.1128	0.874087
$641$	−30.8365	−1.21797	−0.608983	−	0.793183i	$-0.708422\pi$
$641$	−30.8365	−1.21797	−0.608983	+	0.793183i	$0.708422\pi$
$642$	4.83462	0.190807
$643$	−9.38653	−0.370169	−0.185084	−	0.982723i	$-0.559256\pi$
$643$	−9.38653	−0.370169	−0.185084	+	0.982723i	$0.559256\pi$
$644$	0	0
$645$	−47.7757	−1.88117
$646$	−5.32081	−0.209345
$647$	−22.6188	−0.889238	−0.444619	−	0.895720i	$-0.646661\pi$
$647$	−22.6188	−0.889238	−0.444619	+	0.895720i	$0.646661\pi$
$648$	−0.219619	−0.00862744
$649$	7.55830	0.296689
$650$	−9.12890	−0.358065
$651$	0	0
$652$	−40.2639	−1.57686
$653$	31.3014	1.22492	0.612459	−	0.790503i	$-0.290181\pi$
$653$	31.3014	1.22492	0.612459	+	0.790503i	$0.290181\pi$
$654$	4.80838	0.188023
$655$	61.5981	2.40684
$656$	5.67876	0.221718
$657$	48.2011	1.88050
$658$	0	0
$659$	−23.2835	−0.906997	−0.453498	−	0.891257i	$-0.649824\pi$
$659$	−23.2835	−0.906997	−0.453498	+	0.891257i	$0.649824\pi$
$660$	47.1059	1.83359
$661$	42.2943	1.64506	0.822529	−	0.568724i	$-0.192563\pi$
$661$	42.2943	1.64506	0.822529	+	0.568724i	$0.192563\pi$
$662$	1.54056	0.0598757
$663$	55.8731	2.16993
$664$	2.98967	0.116022
$665$	0	0
$666$	3.17293	0.122949
$667$	−5.80451	−0.224752
$668$	24.3141	0.940740
$669$	56.4361	2.18195
$670$	−6.16663	−0.238238
$671$	−15.6697	−0.604921
$672$	0	0
$673$	49.3014	1.90043	0.950214	−	0.311597i	$-0.100864\pi$
$673$	49.3014	1.90043	0.950214	+	0.311597i	$0.100864\pi$
$674$	−5.17136	−0.199193
$675$	−66.9410	−2.57656
$676$	−7.60367	−0.292449
$677$	8.77043	0.337075	0.168538	−	0.985695i	$-0.446096\pi$
$677$	8.77043	0.337075	0.168538	+	0.985695i	$0.446096\pi$
$678$	−2.31055	−0.0887360
$679$	0	0
$680$	−13.8521	−0.531204
$681$	−8.55764	−0.327929
$682$	1.44853	0.0554670
$683$	−49.3668	−1.88897	−0.944485	−	0.328555i	$-0.893438\pi$
$683$	−49.3668	−1.88897	−0.944485	+	0.328555i	$0.893438\pi$
$684$	61.8000	2.36298
$685$	35.6348	1.36154
$686$	0	0
$687$	−42.2134	−1.61054
$688$	15.3302	0.584459
$689$	46.2870	1.76339
$690$	−2.00000	−0.0761387
$691$	39.1742	1.49026	0.745128	−	0.666921i	$-0.232388\pi$
$691$	39.1742	1.49026	0.745128	+	0.666921i	$0.232388\pi$
$692$	−41.7211	−1.58600
$693$	0	0
$694$	3.88784	0.147581
$695$	−5.39567	−0.204670
$696$	−10.7852	−0.408810
$697$	−7.21805	−0.273403
$698$	−2.10258	−0.0795839
$699$	51.2975	1.94025
$700$	0	0
$701$	−10.8831	−0.411051	−0.205525	−	0.978652i	$-0.565890\pi$
$701$	−10.8831	−0.411051	−0.205525	+	0.978652i	$0.565890\pi$
$702$	3.46570	0.130805
$703$	25.7258	0.970267
$704$	−14.6698	−0.552889
$705$	−51.6608	−1.94566
$706$	−3.88115	−0.146069
$707$	0	0
$708$	20.8193	0.782435
$709$	−40.1761	−1.50884	−0.754422	−	0.656390i	$-0.772083\pi$
$709$	−40.1761	−1.50884	−0.754422	+	0.656390i	$0.772083\pi$
$710$	−2.96344	−0.111216
$711$	0.269430	0.0101044
$712$	5.43681	0.203753
$713$	4.32528	0.161983
$714$	0	0
$715$	35.1059	1.31289
$716$	15.2796	0.571026
$717$	−1.16839	−0.0436343
$718$	3.99061	0.148928
$719$	−34.6791	−1.29331	−0.646657	−	0.762781i	$-0.723833\pi$
$719$	−34.6791	−1.29331	−0.646657	+	0.762781i	$0.723833\pi$
$720$	78.7247	2.93390
$721$	0	0
$722$	−3.94314	−0.146748
$723$	0.102767	0.00382193
$724$	1.41174	0.0524670
$725$	77.0771	2.86257
$726$	−3.27457	−0.121531
$727$	17.0184	0.631178	0.315589	−	0.948896i	$-0.397798\pi$
$727$	17.0184	0.631178	0.315589	+	0.948896i	$0.397798\pi$
$728$	0	0
$729$	−43.8365	−1.62357
$730$	−7.18232	−0.265830
$731$	−19.4856	−0.720701
$732$	−43.1620	−1.59531
$733$	−6.81616	−0.251760	−0.125880	−	0.992045i	$-0.540176\pi$
$733$	−6.81616	−0.251760	−0.125880	+	0.992045i	$0.540176\pi$
$734$	0.0735714	0.00271557
$735$	0	0
$736$	1.97196	0.0726874
$737$	17.2274	0.634581
$738$	−1.19206	−0.0438803
$739$	−18.6877	−0.687437	−0.343718	−	0.939073i	$-0.611687\pi$
$739$	−18.6877	−0.687437	−0.343718	+	0.939073i	$0.611687\pi$
$740$	33.2507	1.22232
$741$	74.8152	2.74840
$742$	0	0
$743$	11.0608	0.405780	0.202890	−	0.979201i	$-0.434967\pi$
$743$	11.0608	0.405780	0.202890	+	0.979201i	$0.434967\pi$
$744$	8.03664	0.294638
$745$	−71.0299	−2.60233
$746$	1.11685	0.0408909
$747$	21.5966	0.790176
$748$	19.2124	0.702476
$749$	0	0
$750$	16.5576	0.604600
$751$	10.9486	0.399521	0.199760	−	0.979845i	$-0.435984\pi$
$751$	10.9486	0.399521	0.199760	+	0.979845i	$0.435984\pi$
$752$	16.5768	0.604495
$753$	4.71648	0.171878
$754$	−3.99048	−0.145325
$755$	−49.0369	−1.78463
$756$	0	0
$757$	−10.5576	−0.383724	−0.191862	−	0.981422i	$-0.561453\pi$
$757$	−10.5576	−0.383724	−0.191862	+	0.981422i	$0.561453\pi$
$758$	1.84115	0.0668738
$759$	5.58731	0.202807
$760$	−18.5482	−0.672816
$761$	−48.7043	−1.76553	−0.882764	−	0.469816i	$-0.844320\pi$
$761$	−48.7043	−1.76553	−0.882764	+	0.469816i	$0.844320\pi$
$762$	5.99209	0.217070
$763$	0	0
$764$	−26.7259	−0.966908
$765$	−100.064	−3.61782
$766$	1.14911	0.0415192
$767$	15.5156	0.560238
$768$	−38.5629	−1.39152
$769$	−32.5515	−1.17384	−0.586918	−	0.809646i	$-0.699659\pi$
$769$	−32.5515	−1.17384	−0.586918	+	0.809646i	$0.699659\pi$
$770$	0	0
$771$	35.3575	1.27337
$772$	−0.385496	−0.0138743
$773$	−2.06463	−0.0742596	−0.0371298	−	0.999310i	$-0.511822\pi$
$773$	−2.06463	−0.0742596	−0.0371298	+	0.999310i	$0.511822\pi$
$774$	−3.21805	−0.115670
$775$	−57.4346	−2.06311
$776$	−6.85191	−0.245969
$777$	0	0
$778$	0.493743	0.0177016
$779$	−9.66510	−0.346288
$780$	96.6988	3.46237
$781$	8.27882	0.296239
$782$	−0.815713	−0.0291698
$783$	−29.2616	−1.04572
$784$	0	0
$785$	−82.7243	−2.95256
$786$	6.73983	0.240402
$787$	−34.9557	−1.24604	−0.623018	−	0.782208i	$-0.714094\pi$
$787$	−34.9557	−1.24604	−0.623018	+	0.782208i	$0.714094\pi$
$788$	−0.385496	−0.0137327
$789$	−70.1372	−2.49695
$790$	−0.0401470	−0.00142837
$791$	0	0
$792$	6.39098	0.227093
$793$	−32.1667	−1.14227
$794$	2.35905	0.0837196
$795$	−134.657	−4.77580
$796$	−30.2304	−1.07149
$797$	−4.95450	−0.175497	−0.0877486	−	0.996143i	$-0.527967\pi$
$797$	−4.95450	−0.175497	−0.0877486	+	0.996143i	$0.527967\pi$
$798$	0	0
$799$	−21.0702	−0.745409
$800$	−26.1853	−0.925791
$801$	39.2740	1.38768
$802$	−1.65571	−0.0584652
$803$	20.0649	0.708076
$804$	47.4528	1.67353
$805$	0	0
$806$	2.97353	0.104738
$807$	4.41823	0.155529
$808$	1.74514	0.0613939
$809$	−17.7757	−0.624960	−0.312480	−	0.949924i	$-0.601160\pi$
$809$	−17.7757	−0.624960	−0.312480	+	0.949924i	$0.601160\pi$
$810$	0.236395	0.00830608
$811$	33.3803	1.17214	0.586071	−	0.810260i	$-0.300674\pi$
$811$	33.3803	1.17214	0.586071	+	0.810260i	$0.300674\pi$
$812$	0	0
$813$	80.3239	2.81708
$814$	1.32081	0.0462945
$815$	87.2956	3.05783
$816$	52.1573	1.82587
$817$	−26.0916	−0.912830
$818$	−3.55111	−0.124162
$819$	0	0
$820$	−12.4922	−0.436246
$821$	−19.7212	−0.688274	−0.344137	−	0.938919i	$-0.611828\pi$
$821$	−19.7212	−0.688274	−0.344137	+	0.938919i	$0.611828\pi$
$822$	3.89903	0.135994
$823$	2.26096	0.0788120	0.0394060	−	0.999223i	$-0.487453\pi$
$823$	2.26096	0.0788120	0.0394060	+	0.999223i	$0.487453\pi$
$824$	−0.330040	−0.0114975
$825$	−74.1929	−2.58307
$826$	0	0
$827$	17.2274	0.599057	0.299528	−	0.954087i	$-0.403171\pi$
$827$	17.2274	0.599057	0.299528	+	0.954087i	$0.403171\pi$
$828$	9.47431	0.329255
$829$	−28.2761	−0.982070	−0.491035	−	0.871140i	$-0.663381\pi$
$829$	−28.2761	−0.982070	−0.491035	+	0.871140i	$0.663381\pi$
$830$	−3.21805	−0.111700
$831$	−41.5285	−1.44061
$832$	−30.1141	−1.04402
$833$	0	0
$834$	−0.590374	−0.0204430
$835$	−52.7149	−1.82427
$836$	25.7258	0.889746
$837$	21.8045	0.753674
$838$	1.98490	0.0685672
$839$	6.18334	0.213473	0.106736	−	0.994287i	$-0.465960\pi$
$839$	6.18334	0.213473	0.106736	+	0.994287i	$0.465960\pi$
$840$	0	0
$841$	4.69235	0.161805
$842$	3.61841	0.124699
$843$	21.1003	0.726734
$844$	−30.0094	−1.03297
$845$	16.4854	0.567115
$846$	−3.47973	−0.119636
$847$	0	0
$848$	43.2087	1.48379
$849$	−14.3816	−0.493575
$850$	10.8317	0.371524
$851$	3.94392	0.135196
$852$	22.8039	0.781250
$853$	11.9273	0.408384	0.204192	−	0.978931i	$-0.434543\pi$
$853$	11.9273	0.408384	0.204192	+	0.978931i	$0.434543\pi$
$854$	0	0
$855$	−133.987	−4.58227
$856$	6.87376	0.234940
$857$	30.3706	1.03744	0.518720	−	0.854944i	$-0.326409\pi$
$857$	30.3706	1.03744	0.518720	+	0.854944i	$0.326409\pi$
$858$	3.84115	0.131135
$859$	−0.269585	−0.00919813	−0.00459906	−	0.999989i	$-0.501464\pi$
$859$	−0.269585	−0.00919813	−0.00459906	+	0.999989i	$0.501464\pi$
$860$	−33.7235	−1.14996
$861$	0	0
$862$	−6.66822	−0.227121
$863$	11.6363	0.396103	0.198052	−	0.980192i	$-0.436539\pi$
$863$	11.6363	0.396103	0.198052	+	0.980192i	$0.436539\pi$
$864$	9.94102	0.338200
$865$	90.4549	3.07556
$866$	0.319488	0.0108566
$867$	−18.8029	−0.638580
$868$	0	0
$869$	0.112157	0.00380466
$870$	11.6090	0.393583
$871$	35.3644	1.19828
$872$	6.83646	0.231512
$873$	−49.4963	−1.67520
$874$	−1.09225	−0.0369461
$875$	0	0
$876$	55.2686	1.86735
$877$	32.1479	1.08556	0.542778	−	0.839876i	$-0.317372\pi$
$877$	32.1479	1.08556	0.542778	+	0.839876i	$0.317372\pi$
$878$	−3.40500	−0.114913
$879$	−34.8831	−1.17658
$880$	32.7712	1.10472
$881$	−51.4979	−1.73501	−0.867505	−	0.497429i	$-0.834278\pi$
$881$	−51.4979	−1.73501	−0.867505	+	0.497429i	$0.834278\pi$
$882$	0	0
$883$	−41.1153	−1.38364	−0.691820	−	0.722070i	$-0.743191\pi$
$883$	−41.1153	−1.38364	−0.691820	+	0.722070i	$0.743191\pi$
$884$	39.4392	1.32648
$885$	−45.1379	−1.51729
$886$	−3.14801	−0.105760
$887$	54.7371	1.83789	0.918946	−	0.394383i	$-0.129042\pi$
$887$	54.7371	1.83789	0.918946	+	0.394383i	$0.129042\pi$
$888$	7.32806	0.245914
$889$	0	0
$890$	−5.85212	−0.196163
$891$	−0.660406	−0.0221244
$892$	39.8366	1.33383
$893$	−28.2134	−0.944124
$894$	−7.77182	−0.259929
$895$	−33.1274	−1.10733
$896$	0	0
$897$	11.4696	0.382959
$898$	4.04669	0.135040
$899$	−25.1061	−0.837336
$900$	−125.808	−4.19359
$901$	−54.9208	−1.82968
$902$	−0.496225	−0.0165225
$903$	0	0
$904$	−3.28509	−0.109260
$905$	−3.06077	−0.101744
$906$	−5.36543	−0.178255
$907$	−18.1106	−0.601352	−0.300676	−	0.953726i	$-0.597212\pi$
$907$	−18.1106	−0.601352	−0.300676	+	0.953726i	$0.597212\pi$
$908$	−6.04059	−0.200464
$909$	12.6064	0.418129
$910$	0	0
$911$	37.6635	1.24785	0.623924	−	0.781485i	$-0.285537\pi$
$911$	37.6635	1.24785	0.623924	+	0.781485i	$0.285537\pi$
$912$	69.8396	2.31262
$913$	8.99011	0.297529
$914$	−5.49687	−0.181820
$915$	93.5787	3.09362
$916$	−29.7972	−0.984527
$917$	0	0
$918$	−4.11216	−0.135721
$919$	20.3349	0.670786	0.335393	−	0.942078i	$-0.391131\pi$
$919$	20.3349	0.670786	0.335393	+	0.942078i	$0.391131\pi$
$920$	−2.84356	−0.0937493
$921$	−2.00000	−0.0659022
$922$	−0.927235	−0.0305368
$923$	16.9947	0.559388
$924$	0	0
$925$	−52.3706	−1.72194
$926$	0.224314	0.00737141
$927$	−2.38412	−0.0783047
$928$	−11.4463	−0.375742
$929$	−47.6383	−1.56296	−0.781480	−	0.623930i	$-0.785535\pi$
$929$	−47.6383	−1.56296	−0.781480	+	0.623930i	$0.785535\pi$
$930$	−8.65055	−0.283663
$931$	0	0
$932$	36.2094	1.18608
$933$	80.4167	2.63272
$934$	−1.26124	−0.0412690
$935$	−41.6541	−1.36224
$936$	13.1194	0.428820
$937$	19.7053	0.643744	0.321872	−	0.946783i	$-0.395688\pi$
$937$	19.7053	0.643744	0.321872	+	0.946783i	$0.395688\pi$
$938$	0	0
$939$	−55.9984	−1.82744
$940$	−36.4658	−1.18938
$941$	16.6683	0.543371	0.271685	−	0.962386i	$-0.412419\pi$
$941$	16.6683	0.543371	0.271685	+	0.962386i	$0.412419\pi$
$942$	−9.05138	−0.294910
$943$	−1.48172	−0.0482514
$944$	14.4838	0.471407
$945$	0	0
$946$	−1.33959	−0.0435539
$947$	−31.3575	−1.01898	−0.509490	−	0.860476i	$-0.670166\pi$
$947$	−31.3575	−1.01898	−0.509490	+	0.860476i	$0.670166\pi$
$948$	0.308935	0.0100337
$949$	41.1892	1.33706
$950$	14.5039	0.470567
$951$	86.4335	2.80280
$952$	0	0
$953$	−25.2180	−0.816893	−0.408446	−	0.912782i	$-0.633929\pi$
$953$	−25.2180	−0.816893	−0.408446	+	0.912782i	$0.633929\pi$
$954$	−9.07016	−0.293657
$955$	57.9439	1.87502
$956$	−0.824733	−0.0266738
$957$	−32.4316	−1.04837
$958$	1.38887	0.0448724
$959$	0	0
$960$	87.6075	2.82752
$961$	−12.2920	−0.396516
$962$	2.71136	0.0874178
$963$	49.6541	1.60008
$964$	0.0725400	0.00233636
$965$	0.835786	0.0269049
$966$	0	0
$967$	−2.13002	−0.0684968	−0.0342484	−	0.999413i	$-0.510904\pi$
$967$	−2.13002	−0.0684968	−0.0342484	+	0.999413i	$0.510904\pi$
$968$	−4.65571	−0.149640
$969$	−88.7703	−2.85171
$970$	7.37532	0.236807
$971$	55.2035	1.77156	0.885782	−	0.464101i	$-0.153622\pi$
$971$	55.2035	1.77156	0.885782	+	0.464101i	$0.153622\pi$
$972$	−31.6421	−1.01492
$973$	0	0
$974$	2.03273	0.0651330
$975$	−152.303	−4.87760
$976$	−30.0274	−0.961154
$977$	20.0561	0.641651	0.320825	−	0.947138i	$-0.396040\pi$
$977$	20.0561	0.641651	0.320825	+	0.947138i	$0.396040\pi$
$978$	9.55155	0.305425
$979$	16.3488	0.522510
$980$	0	0
$981$	49.3847	1.57673
$982$	6.03273	0.192512
$983$	9.46010	0.301730	0.150865	−	0.988554i	$-0.451794\pi$
$983$	9.46010	0.301730	0.150865	+	0.988554i	$0.451794\pi$
$984$	−2.75313	−0.0877665
$985$	0.835786	0.0266304
$986$	4.73481	0.150787
$987$	0	0
$988$	52.8099	1.68011
$989$	−4.00000	−0.127193
$990$	−6.87918	−0.218635
$991$	17.8972	0.568524	0.284262	−	0.958747i	$-0.408251\pi$
$991$	17.8972	0.568524	0.284262	+	0.958747i	$0.408251\pi$
$992$	8.52927	0.270805
$993$	25.7021	0.815633
$994$	0	0
$995$	65.5420	2.07782
$996$	24.7632	0.784651
$997$	−45.0643	−1.42720	−0.713600	−	0.700553i	$-0.752937\pi$
$997$	−45.0643	−1.42720	−0.713600	+	0.700553i	$0.752937\pi$
$998$	5.02647	0.159110
$999$	19.8820	0.629040

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1127.2.a.j.1.3	✓	6
7.6	odd	2	inner	1127.2.a.j.1.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1127.2.a.j.1.3	✓	6	1.1	even	1	trivial
1127.2.a.j.1.4	yes	6	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 1127 = 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1127.a (trivial)

\(f(q)\)	\(=\)	\(q-0.167449 q^{2} -2.79366 q^{3} -1.97196 q^{4} +4.27537 q^{5} +0.467795 q^{6} +0.665102 q^{8} +4.80451 q^{9} +O(q^{10})\)	\(q-0.167449 q^{2} -2.79366 q^{3} -1.97196 q^{4} +4.27537 q^{5} +0.467795 q^{6} +0.665102 q^{8} +4.80451 q^{9} -0.715908 q^{10} +2.00000 q^{11} +5.50898 q^{12} +4.10559 q^{13} -11.9439 q^{15} +3.83255 q^{16} -4.87140 q^{17} -0.804512 q^{18} -6.52290 q^{19} -8.43087 q^{20} -0.334898 q^{22} -1.00000 q^{23} -1.85806 q^{24} +13.2788 q^{25} -0.687478 q^{26} -5.04118 q^{27} +5.80451 q^{29} +2.00000 q^{30} -4.32528 q^{31} -1.97196 q^{32} -5.58731 q^{33} +0.815713 q^{34} -9.47431 q^{36} -3.94392 q^{37} +1.09225 q^{38} -11.4696 q^{39} +2.84356 q^{40} +1.48172 q^{41} +4.00000 q^{43} -3.94392 q^{44} +20.5411 q^{45} +0.167449 q^{46} +4.32528 q^{47} -10.7068 q^{48} -2.22353 q^{50} +13.6090 q^{51} -8.09607 q^{52} +11.2741 q^{53} +0.844142 q^{54} +8.55075 q^{55} +18.2227 q^{57} -0.971961 q^{58} +3.77915 q^{59} +23.5529 q^{60} -7.83484 q^{61} +0.724264 q^{62} -7.33490 q^{64} +17.5529 q^{65} +0.935591 q^{66} +8.61372 q^{67} +9.60622 q^{68} +2.79366 q^{69} +4.13941 q^{71} +3.19549 q^{72} +10.0325 q^{73} +0.660406 q^{74} -37.0964 q^{75} +12.8629 q^{76} +1.92058 q^{78} +0.0560785 q^{79} +16.3856 q^{80} -0.330203 q^{81} -0.248112 q^{82} +4.49506 q^{83} -20.8271 q^{85} -0.669797 q^{86} -16.2158 q^{87} +1.33020 q^{88} +8.17440 q^{89} -3.43959 q^{90} +1.97196 q^{92} +12.0833 q^{93} -0.724264 q^{94} -27.8878 q^{95} +5.50898 q^{96} -10.3021 q^{97} +9.60902 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 12 q^{4} + 6 q^{8} + 6 q^{9}+O(q^{10}) \) 6 * q + 12 * q^4 + 6 * q^8 + 6 * q^9	\( 6 q + 12 q^{4} + 6 q^{8} + 6 q^{9} + 12 q^{11} - 24 q^{15} + 24 q^{16} + 18 q^{18} - 6 q^{23} + 30 q^{25} + 12 q^{29} + 12 q^{30} + 12 q^{32} - 30 q^{36} + 24 q^{37} - 48 q^{39} + 24 q^{43} + 24 q^{44} - 60 q^{50} + 36 q^{51} + 24 q^{53} + 12 q^{57} + 18 q^{58} + 48 q^{60} - 42 q^{64} + 12 q^{65} + 42 q^{72} + 12 q^{74} - 66 q^{78} + 48 q^{79} - 6 q^{81} + 12 q^{85} + 12 q^{88} - 12 q^{92} - 72 q^{95} + 12 q^{99}+O(q^{100}) \) 6 * q + 12 * q^4 + 6 * q^8 + 6 * q^9 + 12 * q^11 - 24 * q^15 + 24 * q^16 + 18 * q^18 - 6 * q^23 + 30 * q^25 + 12 * q^29 + 12 * q^30 + 12 * q^32 - 30 * q^36 + 24 * q^37 - 48 * q^39 + 24 * q^43 + 24 * q^44 - 60 * q^50 + 36 * q^51 + 24 * q^53 + 12 * q^57 + 18 * q^58 + 48 * q^60 - 42 * q^64 + 12 * q^65 + 42 * q^72 + 12 * q^74 - 66 * q^78 + 48 * q^79 - 6 * q^81 + 12 * q^85 + 12 * q^88 - 12 * q^92 - 72 * q^95 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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