LMFDB - Embedded newform 6223.2.a.s.1.33

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6223,2,Mod(1,6223)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6223, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6223.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 6223 = 7^{2} \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6223.a (trivial)

Newform invariants

comment:select newform

sage:traces = [54,-14,0,46,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$49.6909051778$
Analytic rank:	$1$
Dimension:	$54$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.33
Character	$\chi$	$=$	6223.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.117634 q^{2} -1.63542 q^{3} -1.98616 q^{4} -2.28766 q^{5} -0.192381 q^{6} -0.468908 q^{8} -0.325386 q^{9} -0.269106 q^{10} -3.49315 q^{11} +3.24822 q^{12} +0.309784 q^{13} +3.74129 q^{15} +3.91717 q^{16} +1.81047 q^{17} -0.0382764 q^{18} -7.89879 q^{19} +4.54366 q^{20} -0.410913 q^{22} -2.28467 q^{23} +0.766864 q^{24} +0.233383 q^{25} +0.0364411 q^{26} +5.43842 q^{27} +5.11516 q^{29} +0.440103 q^{30} +6.98025 q^{31} +1.39861 q^{32} +5.71278 q^{33} +0.212973 q^{34} +0.646269 q^{36} -8.44681 q^{37} -0.929166 q^{38} -0.506628 q^{39} +1.07270 q^{40} +7.91169 q^{41} -1.00193 q^{43} +6.93796 q^{44} +0.744372 q^{45} -0.268754 q^{46} -7.30561 q^{47} -6.40623 q^{48} +0.0274538 q^{50} -2.96089 q^{51} -0.615281 q^{52} +4.42705 q^{53} +0.639743 q^{54} +7.99113 q^{55} +12.9179 q^{57} +0.601716 q^{58} +13.3546 q^{59} -7.43082 q^{60} +3.90611 q^{61} +0.821115 q^{62} -7.66981 q^{64} -0.708680 q^{65} +0.672017 q^{66} +6.28582 q^{67} -3.59590 q^{68} +3.73640 q^{69} -2.05298 q^{71} +0.152576 q^{72} -1.56116 q^{73} -0.993632 q^{74} -0.381681 q^{75} +15.6883 q^{76} -0.0595967 q^{78} +10.5077 q^{79} -8.96114 q^{80} -7.91797 q^{81} +0.930683 q^{82} +3.56441 q^{83} -4.14175 q^{85} -0.117861 q^{86} -8.36545 q^{87} +1.63796 q^{88} +3.06147 q^{89} +0.0875634 q^{90} +4.53772 q^{92} -11.4157 q^{93} -0.859387 q^{94} +18.0697 q^{95} -2.28732 q^{96} -11.7154 q^{97} +1.13662 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 54 q - 14 q^{2} + 46 q^{4} - 42 q^{8} + 14 q^{9} - 20 q^{11} - 16 q^{15} + 38 q^{16} - 42 q^{18} - 28 q^{22} - 64 q^{23} - 10 q^{25} - 44 q^{29} - 12 q^{30} - 78 q^{32} - 38 q^{36} - 32 q^{37} - 40 q^{39}+ \cdots - 72 q^{99}+O(q^{100}) $ 54 * q - 14 * q^2 + 46 * q^4 - 42 * q^8 + 14 * q^9 - 20 * q^11 - 16 * q^15 + 38 * q^16 - 42 * q^18 - 28 * q^22 - 64 * q^23 - 10 * q^25 - 44 * q^29 - 12 * q^30 - 78 * q^32 - 38 * q^36 - 32 * q^37 - 40 * q^39 - 52 * q^43 - 48 * q^44 - 8 * q^46 - 66 * q^50 - 24 * q^51 - 136 * q^53 - 80 * q^57 - 4 * q^58 - 28 * q^60 - 10 * q^64 - 88 * q^65 - 48 * q^67 - 52 * q^71 - 22 * q^72 - 28 * q^74 - 36 * q^78 + 24 * q^79 - 66 * q^81 - 48 * q^85 + 64 * q^86 - 68 * q^88 - 192 * q^92 - 88 * q^93 - 112 * q^95 - 72 * q^99

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 $

$2$	0.117634	0.0831798	0.0415899	−	0.999135i	$-0.486758\pi$
$2$	0.117634	0.0831798	0.0415899	+	0.999135i	$0.486758\pi$
$3$	−1.63542	−0.944213	−0.472106	−	0.881542i	$-0.656506\pi$
$3$	−1.63542	−0.944213	−0.472106	+	0.881542i	$0.656506\pi$
$4$	−1.98616	−0.993081
$5$	−2.28766	−1.02307	−0.511536	−	0.859262i	$-0.670923\pi$
$5$	−2.28766	−1.02307	−0.511536	+	0.859262i	$0.670923\pi$
$6$	−0.192381	−0.0785394
$7$	0	0
$7$	0	0
$8$	−0.468908	−0.165784
$9$	−0.325386	−0.108462
$10$	−0.269106	−0.0850989
$11$	−3.49315	−1.05322	−0.526612	−	0.850106i	$-0.676538\pi$
$11$	−3.49315	−1.05322	−0.526612	+	0.850106i	$0.676538\pi$
$12$	3.24822	0.937680
$13$	0.309784	0.0859186	0.0429593	−	0.999077i	$-0.486321\pi$
$13$	0.309784	0.0859186	0.0429593	+	0.999077i	$0.486321\pi$
$14$	0	0
$15$	3.74129	0.965998
$16$	3.91717	0.979291
$17$	1.81047	0.439105	0.219552	−	0.975601i	$-0.429540\pi$
$17$	1.81047	0.439105	0.219552	+	0.975601i	$0.429540\pi$
$18$	−0.0382764	−0.00902184
$19$	−7.89879	−1.81211	−0.906054	−	0.423162i	$-0.860920\pi$
$19$	−7.89879	−1.81211	−0.906054	+	0.423162i	$0.860920\pi$
$20$	4.54366	1.01599
$21$	0	0
$22$	−0.410913	−0.0876069
$23$	−2.28467	−0.476386	−0.238193	−	0.971218i	$-0.576555\pi$
$23$	−2.28467	−0.476386	−0.238193	+	0.971218i	$0.576555\pi$
$24$	0.766864	0.156535
$25$	0.233383	0.0466767
$26$	0.0364411	0.00714669
$27$	5.43842	1.04662
$28$	0	0
$29$	5.11516	0.949861	0.474930	−	0.880023i	$-0.342473\pi$
$29$	5.11516	0.949861	0.474930	+	0.880023i	$0.342473\pi$
$30$	0.440103	0.0803515
$31$	6.98025	1.25369	0.626845	−	0.779144i	$-0.284346\pi$
$31$	6.98025	1.25369	0.626845	+	0.779144i	$0.284346\pi$
$32$	1.39861	0.247241
$33$	5.71278	0.994467
$34$	0.212973	0.0365246
$35$	0	0
$36$	0.646269	0.107712
$37$	−8.44681	−1.38865	−0.694324	−	0.719663i	$-0.744296\pi$
$37$	−8.44681	−1.38865	−0.694324	+	0.719663i	$0.744296\pi$
$38$	−0.929166	−0.150731
$39$	−0.506628	−0.0811254
$40$	1.07270	0.169609
$41$	7.91169	1.23560	0.617799	−	0.786336i	$-0.288024\pi$
$41$	7.91169	1.23560	0.617799	+	0.786336i	$0.288024\pi$
$42$	0	0
$43$	−1.00193	−0.152793	−0.0763967	−	0.997078i	$-0.524342\pi$
$43$	−1.00193	−0.152793	−0.0763967	+	0.997078i	$0.524342\pi$
$44$	6.93796	1.04594
$45$	0.744372	0.110964
$46$	−0.268754	−0.0396257
$47$	−7.30561	−1.06563	−0.532816	−	0.846231i	$-0.678866\pi$
$47$	−7.30561	−1.06563	−0.532816	+	0.846231i	$0.678866\pi$
$48$	−6.40623	−0.924659
$49$	0	0
$50$	0.0274538	0.00388255
$51$	−2.96089	−0.414608
$52$	−0.615281	−0.0853241
$53$	4.42705	0.608102	0.304051	−	0.952656i	$-0.401661\pi$
$53$	4.42705	0.608102	0.304051	+	0.952656i	$0.401661\pi$
$54$	0.639743	0.0870579
$55$	7.99113	1.07752
$56$	0	0
$57$	12.9179	1.71102
$58$	0.601716	0.0790092
$59$	13.3546	1.73862	0.869308	−	0.494272i	$-0.164565\pi$
$59$	13.3546	1.73862	0.869308	+	0.494272i	$0.164565\pi$
$60$	−7.43082	−0.959314
$61$	3.90611	0.500126	0.250063	−	0.968230i	$-0.419549\pi$
$61$	3.90611	0.500126	0.250063	+	0.968230i	$0.419549\pi$
$62$	0.821115	0.104282
$63$	0	0
$64$	−7.66981	−0.958726
$65$	−0.708680	−0.0879009
$66$	0.672017	0.0827195
$67$	6.28582	0.767935	0.383967	−	0.923347i	$-0.374558\pi$
$67$	6.28582	0.767935	0.383967	+	0.923347i	$0.374558\pi$
$68$	−3.59590	−0.436066
$69$	3.73640	0.449810
$70$	0	0
$71$	−2.05298	−0.243644	−0.121822	−	0.992552i	$-0.538874\pi$
$71$	−2.05298	−0.243644	−0.121822	+	0.992552i	$0.538874\pi$
$72$	0.152576	0.0179813
$73$	−1.56116	−0.182720	−0.0913598	−	0.995818i	$-0.529121\pi$
$73$	−1.56116	−0.182720	−0.0913598	+	0.995818i	$0.529121\pi$
$74$	−0.993632	−0.115507
$75$	−0.381681	−0.0440727
$76$	15.6883	1.79957
$77$	0	0
$78$	−0.0595967	−0.00674799
$79$	10.5077	1.18221	0.591106	−	0.806594i	$-0.298692\pi$
$79$	10.5077	1.18221	0.591106	+	0.806594i	$0.298692\pi$
$80$	−8.96114	−1.00189
$81$	−7.91797	−0.879774
$82$	0.930683	0.102777
$83$	3.56441	0.391245	0.195622	−	0.980679i	$-0.437327\pi$
$83$	3.56441	0.391245	0.195622	+	0.980679i	$0.437327\pi$
$84$	0	0
$85$	−4.14175	−0.449236
$86$	−0.117861	−0.0127093
$87$	−8.36545	−0.896871
$88$	1.63796	0.174608
$89$	3.06147	0.324515	0.162257	−	0.986748i	$-0.448123\pi$
$89$	3.06147	0.324515	0.162257	+	0.986748i	$0.448123\pi$
$90$	0.0875634	0.00922999
$91$	0	0
$92$	4.53772	0.473090
$93$	−11.4157	−1.18375
$94$	−0.859387	−0.0886390
$95$	18.0697	1.85392
$96$	−2.28732	−0.233448
$97$	−11.7154	−1.18951	−0.594757	−	0.803905i	$-0.702752\pi$
$97$	−11.7154	−1.18951	−0.594757	+	0.803905i	$0.702752\pi$
$98$	0	0
$99$	1.13662	0.114235
$100$	−0.463537	−0.0463537
$101$	5.08551	0.506027	0.253014	−	0.967463i	$-0.418578\pi$
$101$	5.08551	0.506027	0.253014	+	0.967463i	$0.418578\pi$
$102$	−0.348302	−0.0344870
$103$	15.5195	1.52918	0.764591	−	0.644516i	$-0.222941\pi$
$103$	15.5195	1.52918	0.764591	+	0.644516i	$0.222941\pi$
$104$	−0.145260	−0.0142439
$105$	0	0
$106$	0.520771	0.0505818
$107$	−6.83685	−0.660943	−0.330472	−	0.943816i	$-0.607208\pi$
$107$	−6.83685	−0.660943	−0.330472	+	0.943816i	$0.607208\pi$
$108$	−10.8016	−1.03938
$109$	−3.46724	−0.332101	−0.166051	−	0.986117i	$-0.553101\pi$
$109$	−3.46724	−0.332101	−0.166051	+	0.986117i	$0.553101\pi$
$110$	0.940028	0.0896281
$111$	13.8141	1.31118
$112$	0	0
$113$	−0.196498	−0.0184850	−0.00924248	−	0.999957i	$-0.502942\pi$
$113$	−0.196498	−0.0184850	−0.00924248	+	0.999957i	$0.502942\pi$
$114$	1.51958	0.142322
$115$	5.22654	0.487377
$116$	−10.1595	−0.943289
$117$	−0.100799	−0.00931889
$118$	1.57095	0.144618
$119$	0	0
$120$	−1.75432	−0.160147
$121$	1.20207	0.109279
$122$	0.459491	0.0416004
$123$	−12.9390	−1.16667
$124$	−13.8639	−1.24502
$125$	10.9044	0.975319
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−3.69944	−0.326988
$129$	1.63859	0.144270
$130$	−0.0833648	−0.00731158
$131$	9.85159	0.860737	0.430369	−	0.902653i	$-0.358384\pi$
$131$	9.85159	0.860737	0.430369	+	0.902653i	$0.358384\pi$
$132$	−11.3465	−0.987586
$133$	0	0
$134$	0.739425	0.0638766
$135$	−12.4412	−1.07077
$136$	−0.848946	−0.0727965
$137$	−11.5555	−0.987250	−0.493625	−	0.869675i	$-0.664328\pi$
$137$	−11.5555	−0.987250	−0.493625	+	0.869675i	$0.664328\pi$
$138$	0.439528	0.0374151
$139$	8.57613	0.727418	0.363709	−	0.931513i	$-0.381510\pi$
$139$	8.57613	0.727418	0.363709	+	0.931513i	$0.381510\pi$
$140$	0	0
$141$	11.9478	1.00618
$142$	−0.241501	−0.0202663
$143$	−1.08212	−0.0904914
$144$	−1.27459	−0.106216
$145$	−11.7017	−0.971776
$146$	−0.183645	−0.0151986
$147$	0	0
$148$	16.7767	1.37904
$149$	11.2269	0.919747	0.459873	−	0.887984i	$-0.347895\pi$
$149$	11.2269	0.919747	0.459873	+	0.887984i	$0.347895\pi$
$150$	−0.0448986	−0.00366596
$151$	−18.4540	−1.50176	−0.750882	−	0.660437i	$-0.770371\pi$
$151$	−18.4540	−1.50176	−0.750882	+	0.660437i	$0.770371\pi$
$152$	3.70381	0.300419
$153$	−0.589103	−0.0476261
$154$	0	0
$155$	−15.9684	−1.28262
$156$	1.00625	0.0805641
$157$	−10.5588	−0.842681	−0.421341	−	0.906902i	$-0.638440\pi$
$157$	−10.5588	−0.842681	−0.421341	+	0.906902i	$0.638440\pi$
$158$	1.23606	0.0983360
$159$	−7.24010	−0.574178
$160$	−3.19954	−0.252946
$161$	0	0
$162$	−0.931422	−0.0731794
$163$	21.9787	1.72150	0.860752	−	0.509024i	$-0.169994\pi$
$163$	21.9787	1.72150	0.860752	+	0.509024i	$0.169994\pi$
$164$	−15.7139	−1.22705
$165$	−13.0689	−1.01741
$166$	0.419295	0.0325436
$167$	4.43098	0.342880	0.171440	−	0.985195i	$-0.445158\pi$
$167$	4.43098	0.342880	0.171440	+	0.985195i	$0.445158\pi$
$168$	0	0
$169$	−12.9040	−0.992618
$170$	−0.487210	−0.0373673
$171$	2.57016	0.196545
$172$	1.99000	0.151736
$173$	19.6340	1.49274	0.746372	−	0.665529i	$-0.231794\pi$
$173$	19.6340	1.49274	0.746372	+	0.665529i	$0.231794\pi$
$174$	−0.984061	−0.0746015
$175$	0	0
$176$	−13.6832	−1.03141
$177$	−21.8404	−1.64162
$178$	0.360132	0.0269931
$179$	−2.80666	−0.209779	−0.104890	−	0.994484i	$-0.533449\pi$
$179$	−2.80666	−0.209779	−0.104890	+	0.994484i	$0.533449\pi$
$180$	−1.47844	−0.110197
$181$	−0.404035	−0.0300317	−0.0150158	−	0.999887i	$-0.504780\pi$
$181$	−0.404035	−0.0300317	−0.0150158	+	0.999887i	$0.504780\pi$
$182$	0	0
$183$	−6.38815	−0.472225
$184$	1.07130	0.0789772
$185$	19.3234	1.42069
$186$	−1.34287	−0.0984641
$187$	−6.32425	−0.462475
$188$	14.5101	1.05826
$189$	0	0
$190$	2.12562	0.154208
$191$	−16.2081	−1.17278	−0.586388	−	0.810030i	$-0.699451\pi$
$191$	−16.2081	−1.17278	−0.586388	+	0.810030i	$0.699451\pi$
$192$	12.5434	0.905241
$193$	−0.371031	−0.0267074	−0.0133537	−	0.999911i	$-0.504251\pi$
$193$	−0.371031	−0.0267074	−0.0133537	+	0.999911i	$0.504251\pi$
$194$	−1.37812	−0.0989436
$195$	1.15899	0.0829972
$196$	0	0
$197$	−16.5695	−1.18053	−0.590263	−	0.807211i	$-0.700976\pi$
$197$	−16.5695	−1.18053	−0.590263	+	0.807211i	$0.700976\pi$
$198$	0.133705	0.00950201
$199$	0.493767	0.0350022	0.0175011	−	0.999847i	$-0.494429\pi$
$199$	0.493767	0.0350022	0.0175011	+	0.999847i	$0.494429\pi$
$200$	−0.109435	−0.00773825
$201$	−10.2800	−0.725094
$202$	0.598229	0.0420912
$203$	0	0
$204$	5.88082	0.411740
$205$	−18.0992	−1.26411
$206$	1.82562	0.127197
$207$	0.743398	0.0516698
$208$	1.21347	0.0841393
$209$	27.5916	1.90855
$210$	0	0
$211$	9.83931	0.677366	0.338683	−	0.940901i	$-0.390019\pi$
$211$	9.83931	0.677366	0.338683	+	0.940901i	$0.390019\pi$
$212$	−8.79283	−0.603894
$213$	3.35750	0.230052
$214$	−0.804246	−0.0549771
$215$	2.29208	0.156319
$216$	−2.55012	−0.173514
$217$	0	0
$218$	−0.407865	−0.0276241
$219$	2.55315	0.172526
$220$	−15.8717	−1.07007
$221$	0.560856	0.0377272
$222$	1.62501	0.109064
$223$	14.9203	0.999138	0.499569	−	0.866274i	$-0.333492\pi$
$223$	14.9203	0.999138	0.499569	+	0.866274i	$0.333492\pi$
$224$	0	0
$225$	−0.0759396	−0.00506264
$226$	−0.0231148	−0.00153758
$227$	−19.9131	−1.32168	−0.660840	−	0.750527i	$-0.729800\pi$
$227$	−19.9131	−1.32168	−0.660840	+	0.750527i	$0.729800\pi$
$228$	−25.6570	−1.69918
$229$	7.44013	0.491658	0.245829	−	0.969313i	$-0.420940\pi$
$229$	7.44013	0.491658	0.245829	+	0.969313i	$0.420940\pi$
$230$	0.614818	0.0405399
$231$	0	0
$232$	−2.39854	−0.157472
$233$	−6.57658	−0.430846	−0.215423	−	0.976521i	$-0.569113\pi$
$233$	−6.57658	−0.430846	−0.215423	+	0.976521i	$0.569113\pi$
$234$	−0.0118574	−0.000775143 0
$235$	16.7127	1.09022
$236$	−26.5243	−1.72659
$237$	−17.1846	−1.11626
$238$	0	0
$239$	−20.3289	−1.31497	−0.657484	−	0.753469i	$-0.728379\pi$
$239$	−20.3289	−1.31497	−0.657484	+	0.753469i	$0.728379\pi$
$240$	14.6553	0.945993
$241$	−4.98939	−0.321395	−0.160697	−	0.987004i	$-0.551374\pi$
$241$	−4.98939	−0.321395	−0.160697	+	0.987004i	$0.551374\pi$
$242$	0.141404	0.00908982
$243$	−3.36602	−0.215930
$244$	−7.75817	−0.496666
$245$	0	0
$246$	−1.52206	−0.0970432
$247$	−2.44692	−0.155694
$248$	−3.27310	−0.207842
$249$	−5.82932	−0.369418
$250$	1.28273	0.0811268
$251$	1.30825	0.0825763	0.0412882	−	0.999147i	$-0.486854\pi$
$251$	1.30825	0.0825763	0.0412882	+	0.999147i	$0.486854\pi$
$252$	0	0
$253$	7.98068	0.501741
$254$	−0.117634	−0.00738101
$255$	6.77352	0.424174
$256$	14.9044	0.931527
$257$	−23.2218	−1.44853	−0.724267	−	0.689520i	$-0.757822\pi$
$257$	−23.2218	−1.44853	−0.724267	+	0.689520i	$0.757822\pi$
$258$	0.192753	0.0120003
$259$	0	0
$260$	1.40755	0.0872927
$261$	−1.66440	−0.103024
$262$	1.15888	0.0715959
$263$	15.4914	0.955239	0.477619	−	0.878567i	$-0.341500\pi$
$263$	15.4914	0.955239	0.477619	+	0.878567i	$0.341500\pi$
$264$	−2.67877	−0.164867
$265$	−10.1276	−0.622132
$266$	0	0
$267$	−5.00680	−0.306411
$268$	−12.4847	−0.762621
$269$	−21.2075	−1.29304	−0.646521	−	0.762896i	$-0.723777\pi$
$269$	−21.2075	−1.29304	−0.646521	+	0.762896i	$0.723777\pi$
$270$	−1.46351	−0.0890666
$271$	−14.7960	−0.898791	−0.449395	−	0.893333i	$-0.648361\pi$
$271$	−14.7960	−0.898791	−0.449395	+	0.893333i	$0.648361\pi$
$272$	7.09193	0.430011
$273$	0	0
$274$	−1.35932	−0.0821192
$275$	−0.815242	−0.0491610
$276$	−7.42110	−0.446698
$277$	7.66953	0.460818	0.230409	−	0.973094i	$-0.425994\pi$
$277$	7.66953	0.460818	0.230409	+	0.973094i	$0.425994\pi$
$278$	1.00884	0.0605064
$279$	−2.27128	−0.135978
$280$	0	0
$281$	−3.63180	−0.216655	−0.108327	−	0.994115i	$-0.534549\pi$
$281$	−3.63180	−0.216655	−0.108327	+	0.994115i	$0.534549\pi$
$282$	1.40546	0.0836941
$283$	19.0692	1.13354	0.566772	−	0.823874i	$-0.308192\pi$
$283$	19.0692	1.13354	0.566772	+	0.823874i	$0.308192\pi$
$284$	4.07756	0.241959
$285$	−29.5517	−1.75049
$286$	−0.127294	−0.00752706
$287$	0	0
$288$	−0.455087	−0.0268163
$289$	−13.7222	−0.807187
$290$	−1.37652	−0.0808321
$291$	19.1596	1.12316
$292$	3.10071	0.181455
$293$	−9.54674	−0.557727	−0.278863	−	0.960331i	$-0.589958\pi$
$293$	−9.54674	−0.557727	−0.278863	+	0.960331i	$0.589958\pi$
$294$	0	0
$295$	−30.5507	−1.77873
$296$	3.96078	0.230216
$297$	−18.9972	−1.10233
$298$	1.32067	0.0765043
$299$	−0.707753	−0.0409304
$300$	0.758080	0.0437678
$301$	0	0
$302$	−2.17081	−0.124916
$303$	−8.31697	−0.477797
$304$	−30.9409	−1.77458
$305$	−8.93584	−0.511665
$306$	−0.0692985	−0.00396153
$307$	30.2979	1.72919	0.864595	−	0.502469i	$-0.167575\pi$
$307$	30.2979	1.72919	0.864595	+	0.502469i	$0.167575\pi$
$308$	0	0
$309$	−25.3810	−1.44387
$310$	−1.87843	−0.106688
$311$	−31.3207	−1.77604	−0.888018	−	0.459808i	$-0.847918\pi$
$311$	−31.3207	−1.77604	−0.888018	+	0.459808i	$0.847918\pi$
$312$	0.237562	0.0134493
$313$	−31.5568	−1.78369	−0.891847	−	0.452336i	$-0.850591\pi$
$313$	−31.5568	−1.78369	−0.891847	+	0.452336i	$0.850591\pi$
$314$	−1.24207	−0.0700940
$315$	0	0
$316$	−20.8700	−1.17403
$317$	11.8895	0.667784	0.333892	−	0.942611i	$-0.391638\pi$
$317$	11.8895	0.667784	0.333892	+	0.942611i	$0.391638\pi$
$318$	−0.851682	−0.0477599
$319$	−17.8680	−1.00042
$320$	17.5459	0.980846
$321$	11.1812	0.624071
$322$	0	0
$323$	−14.3006	−0.795705
$324$	15.7264	0.873687
$325$	0.0722984	0.00401039
$326$	2.58544	0.143194
$327$	5.67041	0.313574
$328$	−3.70985	−0.204842
$329$	0	0
$330$	−1.53734	−0.0846280
$331$	30.0225	1.65019	0.825093	−	0.564998i	$-0.191123\pi$
$331$	30.0225	1.65019	0.825093	+	0.564998i	$0.191123\pi$
$332$	−7.07949	−0.388538
$333$	2.74847	0.150615
$334$	0.521234	0.0285207
$335$	−14.3798	−0.785653
$336$	0	0
$337$	−27.8584	−1.51754	−0.758772	−	0.651356i	$-0.774201\pi$
$337$	−27.8584	−1.51754	−0.758772	+	0.651356i	$0.774201\pi$
$338$	−1.51795	−0.0825657
$339$	0.321358	0.0174537
$340$	8.22618	0.446127
$341$	−24.3830	−1.32042
$342$	0.302338	0.0163485
$343$	0	0
$344$	0.469815	0.0253307
$345$	−8.54761	−0.460188
$346$	2.30962	0.124166
$347$	−23.1186	−1.24107	−0.620535	−	0.784179i	$-0.713084\pi$
$347$	−23.1186	−1.24107	−0.620535	+	0.784179i	$0.713084\pi$
$348$	16.6151	0.890665
$349$	2.86945	0.153598	0.0767991	−	0.997047i	$-0.475530\pi$
$349$	2.86945	0.153598	0.0767991	+	0.997047i	$0.475530\pi$
$350$	0	0
$351$	1.68473	0.0899244
$352$	−4.88554	−0.260400
$353$	−3.23557	−0.172212	−0.0861059	−	0.996286i	$-0.527442\pi$
$353$	−3.23557	−0.172212	−0.0861059	+	0.996286i	$0.527442\pi$
$354$	−2.56917	−0.136550
$355$	4.69653	0.249266
$356$	−6.08057	−0.322270
$357$	0	0
$358$	−0.330158	−0.0174494
$359$	8.22817	0.434266	0.217133	−	0.976142i	$-0.430329\pi$
$359$	8.22817	0.434266	0.217133	+	0.976142i	$0.430329\pi$
$360$	−0.349042	−0.0183961
$361$	43.3910	2.28373
$362$	−0.0475282	−0.00249803
$363$	−1.96590	−0.103183
$364$	0	0
$365$	3.57139	0.186935
$366$	−0.751463	−0.0392796
$367$	5.35818	0.279695	0.139847	−	0.990173i	$-0.455339\pi$
$367$	5.35818	0.279695	0.139847	+	0.990173i	$0.455339\pi$
$368$	−8.94942	−0.466521
$369$	−2.57435	−0.134015
$370$	2.27309	0.118172
$371$	0	0
$372$	22.6734	1.17556
$373$	6.99935	0.362413	0.181206	−	0.983445i	$-0.442000\pi$
$373$	6.99935	0.362413	0.181206	+	0.983445i	$0.442000\pi$
$374$	−0.743947	−0.0384686
$375$	−17.8333	−0.920908
$376$	3.42566	0.176665
$377$	1.58459	0.0816107
$378$	0	0
$379$	34.4246	1.76827	0.884137	−	0.467227i	$-0.154747\pi$
$379$	34.4246	1.76827	0.884137	+	0.467227i	$0.154747\pi$
$380$	−35.8895	−1.84109
$381$	1.63542	0.0837853
$382$	−1.90662	−0.0975512
$383$	24.4691	1.25031	0.625157	−	0.780499i	$-0.285035\pi$
$383$	24.4691	1.25031	0.625157	+	0.780499i	$0.285035\pi$
$384$	6.05016	0.308746
$385$	0	0
$386$	−0.0436458	−0.00222151
$387$	0.326015	0.0165723
$388$	23.2686	1.18128
$389$	−12.6458	−0.641170	−0.320585	−	0.947220i	$-0.603879\pi$
$389$	−12.6458	−0.641170	−0.320585	+	0.947220i	$0.603879\pi$
$390$	0.136337	0.00690368
$391$	−4.13633	−0.209183
$392$	0	0
$393$	−16.1115	−0.812719
$394$	−1.94913	−0.0981959
$395$	−24.0381	−1.20949
$396$	−2.25751	−0.113444
$397$	−5.83699	−0.292950	−0.146475	−	0.989214i	$-0.546793\pi$
$397$	−5.83699	−0.292950	−0.146475	+	0.989214i	$0.546793\pi$
$398$	0.0580837	0.00291147
$399$	0	0
$400$	0.914201	0.0457101
$401$	5.13604	0.256481	0.128241	−	0.991743i	$-0.459067\pi$
$401$	5.13604	0.256481	0.128241	+	0.991743i	$0.459067\pi$
$402$	−1.20927	−0.0603131
$403$	2.16237	0.107715
$404$	−10.1007	−0.502526
$405$	18.1136	0.900072
$406$	0	0
$407$	29.5059	1.46256
$408$	1.38839	0.0687354
$409$	36.8806	1.82363	0.911815	−	0.410602i	$-0.134681\pi$
$409$	36.8806	1.82363	0.911815	+	0.410602i	$0.134681\pi$
$410$	−2.12909	−0.105148
$411$	18.8981	0.932174
$412$	−30.8242	−1.51860
$413$	0	0
$414$	0.0874489	0.00429788
$415$	−8.15415	−0.400271
$416$	0.433266	0.0212426
$417$	−14.0256	−0.686837
$418$	3.24571	0.158753
$419$	−30.1959	−1.47517	−0.737584	−	0.675255i	$-0.764033\pi$
$419$	−30.1959	−1.47517	−0.737584	+	0.675255i	$0.764033\pi$
$420$	0	0
$421$	−2.43091	−0.118475	−0.0592377	−	0.998244i	$-0.518867\pi$
$421$	−2.43091	−0.118475	−0.0592377	+	0.998244i	$0.518867\pi$
$422$	1.15744	0.0563431
$423$	2.37714	0.115581
$424$	−2.07588	−0.100814
$425$	0.422535	0.0204959
$426$	0.394956	0.0191357
$427$	0	0
$428$	13.5791	0.656370
$429$	1.76973	0.0854432
$430$	0.269627	0.0130026
$431$	−40.3288	−1.94257	−0.971286	−	0.237916i	$-0.923536\pi$
$431$	−40.3288	−1.94257	−0.971286	+	0.237916i	$0.923536\pi$
$432$	21.3032	1.02495
$433$	30.2222	1.45239	0.726193	−	0.687491i	$-0.241288\pi$
$433$	30.2222	1.45239	0.726193	+	0.687491i	$0.241288\pi$
$434$	0	0
$435$	19.1373	0.917563
$436$	6.88649	0.329803
$437$	18.0461	0.863263
$438$	0.300338	0.0143507
$439$	3.99077	0.190469	0.0952345	−	0.995455i	$-0.469640\pi$
$439$	3.99077	0.190469	0.0952345	+	0.995455i	$0.469640\pi$
$440$	−3.74710	−0.178636
$441$	0	0
$442$	0.0659757	0.00313814
$443$	7.31386	0.347492	0.173746	−	0.984791i	$-0.444413\pi$
$443$	7.31386	0.347492	0.173746	+	0.984791i	$0.444413\pi$
$444$	−27.4371	−1.30211
$445$	−7.00359	−0.332002
$446$	1.75514	0.0831081
$447$	−18.3608	−0.868437
$448$	0	0
$449$	−32.0626	−1.51313	−0.756563	−	0.653921i	$-0.773123\pi$
$449$	−32.0626	−1.51313	−0.756563	+	0.653921i	$0.773123\pi$
$450$	−0.00893308	−0.000421109 0
$451$	−27.6367	−1.30136
$452$	0.390277	0.0183571
$453$	30.1801	1.41798
$454$	−2.34246	−0.109937
$455$	0	0
$456$	−6.05730	−0.283659
$457$	7.32535	0.342665	0.171333	−	0.985213i	$-0.445193\pi$
$457$	7.32535	0.342665	0.171333	+	0.985213i	$0.445193\pi$
$458$	0.875212	0.0408960
$459$	9.84612	0.459577
$460$	−10.3808	−0.484005
$461$	0.281945	0.0131315	0.00656574	−	0.999978i	$-0.497910\pi$
$461$	0.281945	0.0131315	0.00656574	+	0.999978i	$0.497910\pi$
$462$	0	0
$463$	−19.1133	−0.888270	−0.444135	−	0.895960i	$-0.646489\pi$
$463$	−19.1133	−0.888270	−0.444135	+	0.895960i	$0.646489\pi$
$464$	20.0369	0.930190
$465$	26.1152	1.21106
$466$	−0.773629	−0.0358377
$467$	33.8343	1.56566	0.782831	−	0.622234i	$-0.213775\pi$
$467$	33.8343	1.56566	0.782831	+	0.622234i	$0.213775\pi$
$468$	0.200204	0.00925442
$469$	0	0
$470$	1.96598	0.0906841
$471$	17.2681	0.795671
$472$	−6.26206	−0.288235
$473$	3.49990	0.160926
$474$	−2.02149	−0.0928502
$475$	−1.84345	−0.0845832
$476$	0	0
$477$	−1.44050	−0.0659559
$478$	−2.39137	−0.109379
$479$	−18.0017	−0.822519	−0.411259	−	0.911518i	$-0.634911\pi$
$479$	−18.0017	−0.822519	−0.411259	+	0.911518i	$0.634911\pi$
$480$	5.23260	0.238835
$481$	−2.61669	−0.119311
$482$	−0.586922	−0.0267335
$483$	0	0
$484$	−2.38751	−0.108523
$485$	26.8008	1.21696
$486$	−0.395958	−0.0179610
$487$	1.40651	0.0637351	0.0318676	−	0.999492i	$-0.489855\pi$
$487$	1.40651	0.0637351	0.0318676	+	0.999492i	$0.489855\pi$
$488$	−1.83161	−0.0829129
$489$	−35.9445	−1.62547
$490$	0	0
$491$	−16.3078	−0.735961	−0.367980	−	0.929834i	$-0.619951\pi$
$491$	−16.3078	−0.735961	−0.367980	+	0.929834i	$0.619951\pi$
$492$	25.6989	1.15860
$493$	9.26086	0.417088
$494$	−0.287841	−0.0129506
$495$	−2.60020	−0.116870
$496$	27.3428	1.22773
$497$	0	0
$498$	−0.685726	−0.0307281
$499$	7.88273	0.352879	0.176440	−	0.984311i	$-0.443542\pi$
$499$	7.88273	0.352879	0.176440	+	0.984311i	$0.443542\pi$
$500$	−21.6579	−0.968570
$501$	−7.24654	−0.323751
$502$	0.153895	0.00686868
$503$	−4.73607	−0.211171	−0.105585	−	0.994410i	$-0.533672\pi$
$503$	−4.73607	−0.211171	−0.105585	+	0.994410i	$0.533672\pi$
$504$	0	0
$505$	−11.6339	−0.517702
$506$	0.938798	0.0417347
$507$	21.1036	0.937243
$508$	1.98616	0.0881217
$509$	34.6521	1.53593	0.767964	−	0.640493i	$-0.221270\pi$
$509$	34.6521	1.53593	0.767964	+	0.640493i	$0.221270\pi$
$510$	0.796796	0.0352827
$511$	0	0
$512$	9.15216	0.404472
$513$	−42.9570	−1.89660
$514$	−2.73167	−0.120489
$515$	−35.5033	−1.56446
$516$	−3.25450	−0.143271
$517$	25.5195	1.12235
$518$	0	0
$519$	−32.1099	−1.40947
$520$	0.332306	0.0145726
$521$	−31.3197	−1.37214	−0.686071	−	0.727534i	$-0.740666\pi$
$521$	−31.3197	−1.37214	−0.686071	+	0.727534i	$0.740666\pi$
$522$	−0.195790	−0.00856949
$523$	−25.5072	−1.11535	−0.557677	−	0.830058i	$-0.688307\pi$
$523$	−25.5072	−1.11535	−0.557677	+	0.830058i	$0.688307\pi$
$524$	−19.5669	−0.854782
$525$	0	0
$526$	1.82231	0.0794565
$527$	12.6376	0.550501
$528$	22.3779	0.973873
$529$	−17.7803	−0.773056
$530$	−1.19135	−0.0517488
$531$	−4.34538	−0.188574
$532$	0	0
$533$	2.45091	0.106161
$534$	−0.588969	−0.0254872
$535$	15.6404	0.676193
$536$	−2.94747	−0.127311
$537$	4.59008	0.198076
$538$	−2.49472	−0.107555
$539$	0	0
$540$	24.7103	1.06336
$541$	7.49876	0.322397	0.161198	−	0.986922i	$-0.448464\pi$
$541$	7.49876	0.322397	0.161198	+	0.986922i	$0.448464\pi$
$542$	−1.74051	−0.0747612
$543$	0.660769	0.0283563
$544$	2.53214	0.108565
$545$	7.93186	0.339763
$546$	0	0
$547$	15.8592	0.678090	0.339045	−	0.940770i	$-0.389896\pi$
$547$	15.8592	0.678090	0.339045	+	0.940770i	$0.389896\pi$
$548$	22.9510	0.980419
$549$	−1.27099	−0.0542446
$550$	−0.0959002	−0.00408920
$551$	−40.4036	−1.72125
$552$	−1.75203	−0.0745713
$553$	0	0
$554$	0.902197	0.0383307
$555$	−31.6020	−1.34143
$556$	−17.0336	−0.722385
$557$	−3.30361	−0.139978	−0.0699892	−	0.997548i	$-0.522296\pi$
$557$	−3.30361	−0.139978	−0.0699892	+	0.997548i	$0.522296\pi$
$558$	−0.267179	−0.0113106
$559$	−0.310383	−0.0131278
$560$	0	0
$561$	10.3428	0.436675
$562$	−0.427223	−0.0180213
$563$	28.6290	1.20657	0.603285	−	0.797526i	$-0.293858\pi$
$563$	28.6290	1.20657	0.603285	+	0.797526i	$0.293858\pi$
$564$	−23.7302	−0.999222
$565$	0.449520	0.0189115
$566$	2.24318	0.0942880
$567$	0	0
$568$	0.962661	0.0403923
$569$	18.9825	0.795789	0.397894	−	0.917431i	$-0.369741\pi$
$569$	18.9825	0.795789	0.397894	+	0.917431i	$0.369741\pi$
$570$	−3.47628	−0.145606
$571$	−43.6174	−1.82533	−0.912666	−	0.408707i	$-0.865980\pi$
$571$	−43.6174	−1.82533	−0.912666	+	0.408707i	$0.865980\pi$
$572$	2.14927	0.0898653
$573$	26.5071	1.10735
$574$	0	0
$575$	−0.533203	−0.0222361
$576$	2.49565	0.103985
$577$	−3.14395	−0.130884	−0.0654422	−	0.997856i	$-0.520846\pi$
$577$	−3.14395	−0.130884	−0.0654422	+	0.997856i	$0.520846\pi$
$578$	−1.61419	−0.0671416
$579$	0.606793	0.0252175
$580$	23.2415	0.965052
$581$	0	0
$582$	2.25382	0.0934238
$583$	−15.4643	−0.640467
$584$	0.732039	0.0302920
$585$	0.230594	0.00953390
$586$	−1.12302	−0.0463916
$587$	−36.1059	−1.49025	−0.745125	−	0.666925i	$-0.767610\pi$
$587$	−36.1059	−1.49025	−0.745125	+	0.666925i	$0.767610\pi$
$588$	0	0
$589$	−55.1356	−2.27182
$590$	−3.59380	−0.147954
$591$	27.0981	1.11467
$592$	−33.0876	−1.35989
$593$	−6.83095	−0.280514	−0.140257	−	0.990115i	$-0.544793\pi$
$593$	−6.83095	−0.280514	−0.140257	+	0.990115i	$0.544793\pi$
$594$	−2.23471	−0.0916914
$595$	0	0
$596$	−22.2985	−0.913383
$597$	−0.807518	−0.0330495
$598$	−0.0832557	−0.00340458
$599$	−7.00453	−0.286198	−0.143099	−	0.989708i	$-0.545707\pi$
$599$	−7.00453	−0.286198	−0.143099	+	0.989708i	$0.545707\pi$
$600$	0.178973	0.00730655
$601$	−7.51669	−0.306612	−0.153306	−	0.988179i	$-0.548992\pi$
$601$	−7.51669	−0.306612	−0.153306	+	0.988179i	$0.548992\pi$
$602$	0	0
$603$	−2.04532	−0.0832917
$604$	36.6526	1.49137
$605$	−2.74993	−0.111800
$606$	−0.978358	−0.0397431
$607$	−28.5901	−1.16044	−0.580218	−	0.814461i	$-0.697033\pi$
$607$	−28.5901	−1.16044	−0.580218	+	0.814461i	$0.697033\pi$
$608$	−11.0473	−0.448028
$609$	0	0
$610$	−1.05116	−0.0425602
$611$	−2.26316	−0.0915576
$612$	1.17005	0.0472966
$613$	22.6255	0.913834	0.456917	−	0.889509i	$-0.348954\pi$
$613$	22.6255	0.913834	0.456917	+	0.889509i	$0.348954\pi$
$614$	3.56406	0.143834
$615$	29.6000	1.19359
$616$	0	0
$617$	12.5516	0.505310	0.252655	−	0.967556i	$-0.418696\pi$
$617$	12.5516	0.505310	0.252655	+	0.967556i	$0.418696\pi$
$618$	−2.98566	−0.120101
$619$	−0.905356	−0.0363893	−0.0181947	−	0.999834i	$-0.505792\pi$
$619$	−0.905356	−0.0363893	−0.0181947	+	0.999834i	$0.505792\pi$
$620$	31.7159	1.27374
$621$	−12.4250	−0.498597
$622$	−3.68438	−0.147730
$623$	0	0
$624$	−1.98455	−0.0794454
$625$	−26.1124	−1.04450
$626$	−3.71215	−0.148367
$627$	−45.1241	−1.80208
$628$	20.9714	0.836851
$629$	−15.2927	−0.609761
$630$	0	0
$631$	29.6232	1.17928	0.589641	−	0.807666i	$-0.299269\pi$
$631$	29.6232	1.17928	0.589641	+	0.807666i	$0.299269\pi$
$632$	−4.92716	−0.195992
$633$	−16.0914	−0.639578
$634$	1.39861	0.0555461
$635$	2.28766	0.0907830
$636$	14.3800	0.570205
$637$	0	0
$638$	−2.10188	−0.0832143
$639$	0.668012	0.0264261
$640$	8.46307	0.334532
$641$	48.2219	1.90465	0.952326	−	0.305083i	$-0.0986843\pi$
$641$	48.2219	1.90465	0.952326	+	0.305083i	$0.0986843\pi$
$642$	1.31528	0.0519101
$643$	22.8599	0.901506	0.450753	−	0.892649i	$-0.351156\pi$
$643$	22.8599	0.901506	0.450753	+	0.892649i	$0.351156\pi$
$644$	0	0
$645$	−3.74853	−0.147598
$646$	−1.68223	−0.0661865
$647$	−19.3916	−0.762363	−0.381181	−	0.924500i	$-0.624483\pi$
$647$	−19.3916	−0.762363	−0.381181	+	0.924500i	$0.624483\pi$
$648$	3.71280	0.145852
$649$	−46.6494	−1.83115
$650$	0.00850474	0.000333583 0
$651$	0	0
$652$	−43.6533	−1.70959
$653$	−30.4775	−1.19268	−0.596339	−	0.802732i	$-0.703379\pi$
$653$	−30.4775	−1.19268	−0.596339	+	0.802732i	$0.703379\pi$
$654$	0.667032	0.0260830
$655$	−22.5371	−0.880597
$656$	30.9914	1.21001
$657$	0.507978	0.0198181
$658$	0	0
$659$	13.5899	0.529387	0.264693	−	0.964333i	$-0.414729\pi$
$659$	13.5899	0.529387	0.264693	+	0.964333i	$0.414729\pi$
$660$	25.9569	1.01037
$661$	41.1524	1.60064	0.800321	−	0.599572i	$-0.204663\pi$
$661$	41.1524	1.60064	0.800321	+	0.599572i	$0.204663\pi$
$662$	3.53166	0.137262
$663$	−0.917237	−0.0356225
$664$	−1.67138	−0.0648621
$665$	0	0
$666$	0.323314	0.0125282
$667$	−11.6864	−0.452500
$668$	−8.80065	−0.340507
$669$	−24.4011	−0.943399
$670$	−1.69155	−0.0653504
$671$	−13.6446	−0.526744
$672$	0	0
$673$	−28.8349	−1.11150	−0.555752	−	0.831348i	$-0.687570\pi$
$673$	−28.8349	−1.11150	−0.555752	+	0.831348i	$0.687570\pi$
$674$	−3.27710	−0.126229
$675$	1.26924	0.0488529
$676$	25.6295	0.985750
$677$	33.2881	1.27937	0.639683	−	0.768639i	$-0.279065\pi$
$677$	33.2881	1.27937	0.639683	+	0.768639i	$0.279065\pi$
$678$	0.0378026	0.00145180
$679$	0	0
$680$	1.94210	0.0744761
$681$	32.5664	1.24795
$682$	−2.86827	−0.109832
$683$	−15.0093	−0.574315	−0.287158	−	0.957883i	$-0.592710\pi$
$683$	−15.0093	−0.574315	−0.287158	+	0.957883i	$0.592710\pi$
$684$	−5.10475	−0.195185
$685$	26.4350	1.01003
$686$	0	0
$687$	−12.1678	−0.464230
$688$	−3.92474	−0.149629
$689$	1.37143	0.0522472
$690$	−1.00549	−0.0382783
$691$	27.6312	1.05114	0.525570	−	0.850750i	$-0.323852\pi$
$691$	27.6312	1.05114	0.525570	+	0.850750i	$0.323852\pi$
$692$	−38.9963	−1.48242
$693$	0	0
$694$	−2.71953	−0.103232
$695$	−19.6193	−0.744201
$696$	3.92263	0.148687
$697$	14.3239	0.542557
$698$	0.337545	0.0127763
$699$	10.7555	0.406811
$700$	0	0
$701$	3.11329	0.117587	0.0587936	−	0.998270i	$-0.481275\pi$
$701$	3.11329	0.117587	0.0587936	+	0.998270i	$0.481275\pi$
$702$	0.198182	0.00747989
$703$	66.7196	2.51638
$704$	26.7918	1.00975
$705$	−27.3324	−1.02940
$706$	−0.380613	−0.0143245
$707$	0	0
$708$	43.3785	1.63026
$709$	34.8890	1.31028	0.655141	−	0.755507i	$-0.272609\pi$
$709$	34.8890	1.31028	0.655141	+	0.755507i	$0.272609\pi$
$710$	0.552471	0.0207339
$711$	−3.41906	−0.128225
$712$	−1.43555	−0.0537994
$713$	−15.9476	−0.597240
$714$	0	0
$715$	2.47552	0.0925793
$716$	5.57448	0.208328
$717$	33.2464	1.24161
$718$	0.967912	0.0361222
$719$	25.7784	0.961373	0.480687	−	0.876892i	$-0.340387\pi$
$719$	25.7784	0.961373	0.480687	+	0.876892i	$0.340387\pi$
$720$	2.91583	0.108666
$721$	0	0
$722$	5.10425	0.189961
$723$	8.15977	0.303465
$724$	0.802479	0.0298239
$725$	1.19379	0.0443363
$726$	−0.231256	−0.00858272
$727$	−14.1393	−0.524398	−0.262199	−	0.965014i	$-0.584448\pi$
$727$	−14.1393	−0.524398	−0.262199	+	0.965014i	$0.584448\pi$
$728$	0	0
$729$	29.2588	1.08366
$730$	0.420117	0.0155492
$731$	−1.81397	−0.0670923
$732$	12.6879	0.468958
$733$	−37.6113	−1.38921	−0.694603	−	0.719394i	$-0.744420\pi$
$733$	−37.6113	−1.38921	−0.694603	+	0.719394i	$0.744420\pi$
$734$	0.630304	0.0232650
$735$	0	0
$736$	−3.19535	−0.117782
$737$	−21.9573	−0.808807
$738$	−0.302831	−0.0111474
$739$	2.14016	0.0787271	0.0393635	−	0.999225i	$-0.487467\pi$
$739$	2.14016	0.0787271	0.0393635	+	0.999225i	$0.487467\pi$
$740$	−38.3795	−1.41086
$741$	4.00175	0.147008
$742$	0	0
$743$	31.1326	1.14214	0.571072	−	0.820900i	$-0.306528\pi$
$743$	31.1326	1.14214	0.571072	+	0.820900i	$0.306528\pi$
$744$	5.35290	0.196247
$745$	−25.6834	−0.940967
$746$	0.823362	0.0301454
$747$	−1.15981	−0.0424352
$748$	12.5610	0.459275
$749$	0	0
$750$	−2.09780	−0.0766009
$751$	18.8104	0.686402	0.343201	−	0.939262i	$-0.388489\pi$
$751$	18.8104	0.686402	0.343201	+	0.939262i	$0.388489\pi$
$752$	−28.6173	−1.04356
$753$	−2.13955	−0.0779696
$754$	0.186402	0.00678835
$755$	42.2164	1.53641
$756$	0	0
$757$	6.53602	0.237556	0.118778	−	0.992921i	$-0.462102\pi$
$757$	6.53602	0.237556	0.118778	+	0.992921i	$0.462102\pi$
$758$	4.04951	0.147085
$759$	−13.0518	−0.473750
$760$	−8.47305	−0.307350
$761$	−19.3016	−0.699682	−0.349841	−	0.936809i	$-0.613764\pi$
$761$	−19.3016	−0.699682	−0.349841	+	0.936809i	$0.613764\pi$
$762$	0.192381	0.00696925
$763$	0	0
$764$	32.1919	1.16466
$765$	1.34767	0.0487250
$766$	2.87840	0.104001
$767$	4.13702	0.149379
$768$	−24.3751	−0.879560
$769$	6.53725	0.235739	0.117870	−	0.993029i	$-0.462394\pi$
$769$	6.53725	0.235739	0.117870	+	0.993029i	$0.462394\pi$
$770$	0	0
$771$	37.9774	1.36772
$772$	0.736927	0.0265226
$773$	−7.22321	−0.259801	−0.129900	−	0.991527i	$-0.541466\pi$
$773$	−7.22321	−0.259801	−0.129900	+	0.991527i	$0.541466\pi$
$774$	0.0383504	0.00137848
$775$	1.62907	0.0585181
$776$	5.49343	0.197203
$777$	0	0
$778$	−1.48758	−0.0533324
$779$	−62.4928	−2.23904
$780$	−2.30195	−0.0824229
$781$	7.17137	0.256612
$782$	−0.486573	−0.0173998
$783$	27.8184	0.994147
$784$	0	0
$785$	24.1549	0.862124
$786$	−1.89526	−0.0676018
$787$	−15.7689	−0.562099	−0.281050	−	0.959693i	$-0.590683\pi$
$787$	−15.7689	−0.562099	−0.281050	+	0.959693i	$0.590683\pi$
$788$	32.9097	1.17236
$789$	−25.3350	−0.901949
$790$	−2.82769	−0.100605
$791$	0	0
$792$	−0.532970	−0.0189383
$793$	1.21005	0.0429701
$794$	−0.686628	−0.0243675
$795$	16.5629	0.587425
$796$	−0.980700	−0.0347600
$797$	−33.7298	−1.19477	−0.597386	−	0.801954i	$-0.703794\pi$
$797$	−33.7298	−1.19477	−0.597386	+	0.801954i	$0.703794\pi$
$798$	0	0
$799$	−13.2266	−0.467924
$800$	0.326412	0.0115404
$801$	−0.996158	−0.0351975
$802$	0.604172	0.0213341
$803$	5.45335	0.192445
$804$	20.4177	0.720077
$805$	0	0
$806$	0.254368	0.00895973
$807$	34.6832	1.22091
$808$	−2.38464	−0.0838912
$809$	46.5214	1.63561	0.817803	−	0.575498i	$-0.195192\pi$
$809$	46.5214	1.63561	0.817803	+	0.575498i	$0.195192\pi$
$810$	2.13078	0.0748678
$811$	−10.8343	−0.380443	−0.190221	−	0.981741i	$-0.560921\pi$
$811$	−10.8343	−0.380443	−0.190221	+	0.981741i	$0.560921\pi$
$812$	0	0
$813$	24.1977	0.848650
$814$	3.47090	0.121655
$815$	−50.2798	−1.76122
$816$	−11.5983	−0.406022
$817$	7.91407	0.276878
$818$	4.33841	0.151689
$819$	0	0
$820$	35.9480	1.25536
$821$	15.5281	0.541934	0.270967	−	0.962589i	$-0.412657\pi$
$821$	15.5281	0.541934	0.270967	+	0.962589i	$0.412657\pi$
$822$	2.22306	0.0775380
$823$	53.3811	1.86075	0.930374	−	0.366611i	$-0.119482\pi$
$823$	53.3811	1.86075	0.930374	+	0.366611i	$0.119482\pi$
$824$	−7.27722	−0.253514
$825$	1.33327	0.0464184
$826$	0	0
$827$	20.2963	0.705772	0.352886	−	0.935666i	$-0.385200\pi$
$827$	20.2963	0.705772	0.352886	+	0.935666i	$0.385200\pi$
$828$	−1.47651	−0.0513123
$829$	28.8465	1.00188	0.500940	−	0.865482i	$-0.332988\pi$
$829$	28.8465	1.00188	0.500940	+	0.865482i	$0.332988\pi$
$830$	−0.959205	−0.0332945
$831$	−12.5429	−0.435110
$832$	−2.37598	−0.0823723
$833$	0	0
$834$	−1.64989	−0.0571309
$835$	−10.1366	−0.350791
$836$	−54.8015	−1.89535
$837$	37.9615	1.31214
$838$	−3.55207	−0.122704
$839$	19.0230	0.656745	0.328373	−	0.944548i	$-0.393500\pi$
$839$	19.0230	0.656745	0.328373	+	0.944548i	$0.393500\pi$
$840$	0	0
$841$	−2.83518	−0.0977648
$842$	−0.285958	−0.00985475
$843$	5.93953	0.204568
$844$	−19.5425	−0.672679
$845$	29.5200	1.01552
$846$	0.279632	0.00961396
$847$	0	0
$848$	17.3415	0.595509
$849$	−31.1862	−1.07031
$850$	0.0497044	0.00170485
$851$	19.2982	0.661532
$852$	−6.66854	−0.228460
$853$	−5.20728	−0.178294	−0.0891469	−	0.996018i	$-0.528414\pi$
$853$	−5.20728	−0.178294	−0.0891469	+	0.996018i	$0.528414\pi$
$854$	0	0
$855$	−5.87964	−0.201079
$856$	3.20585	0.109574
$857$	44.9487	1.53542	0.767710	−	0.640798i	$-0.221396\pi$
$857$	44.9487	1.53542	0.767710	+	0.640798i	$0.221396\pi$
$858$	0.208180	0.00710714
$859$	−10.6438	−0.363161	−0.181580	−	0.983376i	$-0.558121\pi$
$859$	−10.6438	−0.363161	−0.181580	+	0.983376i	$0.558121\pi$
$860$	−4.55245	−0.155237
$861$	0	0
$862$	−4.74404	−0.161583
$863$	52.4727	1.78619	0.893096	−	0.449866i	$-0.148528\pi$
$863$	52.4727	1.78619	0.893096	+	0.449866i	$0.148528\pi$
$864$	7.60621	0.258769
$865$	−44.9159	−1.52719
$866$	3.55516	0.120809
$867$	22.4416	0.762157
$868$	0	0
$869$	−36.7050	−1.24513
$870$	2.25120	0.0763227
$871$	1.94724	0.0659798
$872$	1.62582	0.0550570
$873$	3.81201	0.129017
$874$	2.12284	0.0718060
$875$	0	0
$876$	−5.07098	−0.171333
$877$	−34.6023	−1.16844	−0.584218	−	0.811597i	$-0.698599\pi$
$877$	−34.6023	−1.16844	−0.584218	+	0.811597i	$0.698599\pi$
$878$	0.469450	0.0158432
$879$	15.6130	0.526613
$880$	31.3026	1.05521
$881$	−14.8015	−0.498674	−0.249337	−	0.968417i	$-0.580213\pi$
$881$	−14.8015	−0.498674	−0.249337	+	0.968417i	$0.580213\pi$
$882$	0	0
$883$	55.3107	1.86135	0.930677	−	0.365841i	$-0.119219\pi$
$883$	55.3107	1.86135	0.930677	+	0.365841i	$0.119219\pi$
$884$	−1.11395	−0.0374662
$885$	49.9633	1.67950
$886$	0.860358	0.0289043
$887$	15.2860	0.513253	0.256626	−	0.966511i	$-0.417389\pi$
$887$	15.2860	0.513253	0.256626	+	0.966511i	$0.417389\pi$
$888$	−6.47755	−0.217372
$889$	0	0
$890$	−0.823860	−0.0276159
$891$	27.6586	0.926598
$892$	−29.6342	−0.992225
$893$	57.7055	1.93104
$894$	−2.15986	−0.0722364
$895$	6.42067	0.214619
$896$	0	0
$897$	1.15748	0.0386470
$898$	−3.77165	−0.125861
$899$	35.7051	1.19083
$900$	0.150828	0.00502761
$901$	8.01506	0.267020
$902$	−3.25101	−0.108247
$903$	0	0
$904$	0.0921394	0.00306451
$905$	0.924294	0.0307246
$906$	3.55020	0.117948
$907$	17.0436	0.565924	0.282962	−	0.959131i	$-0.408683\pi$
$907$	17.0436	0.565924	0.282962	+	0.959131i	$0.408683\pi$
$908$	39.5507	1.31254
$909$	−1.65475	−0.0548847
$910$	0	0
$911$	1.20681	0.0399835	0.0199917	−	0.999800i	$-0.493636\pi$
$911$	1.20681	0.0399835	0.0199917	+	0.999800i	$0.493636\pi$
$912$	50.6015	1.67558
$913$	−12.4510	−0.412068
$914$	0.861710	0.0285028
$915$	14.6139	0.483121
$916$	−14.7773	−0.488256
$917$	0	0
$918$	1.15824	0.0382275
$919$	−25.2862	−0.834113	−0.417057	−	0.908881i	$-0.636938\pi$
$919$	−25.2862	−0.834113	−0.417057	+	0.908881i	$0.636938\pi$
$920$	−2.45077	−0.0807994
$921$	−49.5499	−1.63272
$922$	0.0331663	0.00109227
$923$	−0.635981	−0.0209336
$924$	0	0
$925$	−1.97135	−0.0648174
$926$	−2.24837	−0.0738861
$927$	−5.04983	−0.165858
$928$	7.15410	0.234845
$929$	40.6238	1.33282	0.666412	−	0.745584i	$-0.267829\pi$
$929$	40.6238	1.33282	0.666412	+	0.745584i	$0.267829\pi$
$930$	3.07203	0.100736
$931$	0	0
$932$	13.0622	0.427865
$933$	51.2227	1.67696
$934$	3.98006	0.130231
$935$	14.4677	0.473145
$936$	0.0472656	0.00154492
$937$	−19.0315	−0.621733	−0.310866	−	0.950454i	$-0.600619\pi$
$937$	−19.0315	−0.621733	−0.310866	+	0.950454i	$0.600619\pi$
$938$	0	0
$939$	51.6087	1.68419
$940$	−33.1942	−1.08268
$941$	−57.2741	−1.86708	−0.933541	−	0.358470i	$-0.883299\pi$
$941$	−57.2741	−1.86708	−0.933541	+	0.358470i	$0.883299\pi$
$942$	2.03131	0.0661837
$943$	−18.0756	−0.588622
$944$	52.3120	1.70261
$945$	0	0
$946$	0.411707	0.0133858
$947$	−20.1712	−0.655477	−0.327739	−	0.944768i	$-0.606287\pi$
$947$	−20.1712	−0.655477	−0.327739	+	0.944768i	$0.606287\pi$
$948$	34.1314	1.10854
$949$	−0.483621	−0.0156990
$950$	−0.216852	−0.00703561
$951$	−19.4445	−0.630530
$952$	0	0
$953$	0.775304	0.0251146	0.0125573	−	0.999921i	$-0.496003\pi$
$953$	0.775304	0.0251146	0.0125573	+	0.999921i	$0.496003\pi$
$954$	−0.169452	−0.00548620
$955$	37.0786	1.19983
$956$	40.3765	1.30587
$957$	29.2217	0.944605
$958$	−2.11761	−0.0684169
$959$	0	0
$960$	−28.6950	−0.926127
$961$	17.7239	0.571739
$962$	−0.307811	−0.00992423
$963$	2.22461	0.0716872
$964$	9.90974	0.319171
$965$	0.848792	0.0273236
$966$	0	0
$967$	−25.5917	−0.822973	−0.411486	−	0.911416i	$-0.634990\pi$
$967$	−25.5917	−0.822973	−0.411486	+	0.911416i	$0.634990\pi$
$968$	−0.563661	−0.0181167
$969$	23.3875	0.751315
$970$	3.15268	0.101226
$971$	−22.8004	−0.731701	−0.365851	−	0.930674i	$-0.619222\pi$
$971$	−22.8004	−0.731701	−0.365851	+	0.930674i	$0.619222\pi$
$972$	6.68546	0.214436
$973$	0	0
$974$	0.165454	0.00530147
$975$	−0.118239	−0.00378666
$976$	15.3009	0.489769
$977$	−26.4348	−0.845723	−0.422861	−	0.906194i	$-0.638974\pi$
$977$	−26.4348	−0.845723	−0.422861	+	0.906194i	$0.638974\pi$
$978$	−4.22829	−0.135206
$979$	−10.6942	−0.341787
$980$	0	0
$981$	1.12819	0.0360203
$982$	−1.91835	−0.0612171
$983$	−19.5759	−0.624375	−0.312187	−	0.950021i	$-0.601062\pi$
$983$	−19.5759	−0.624375	−0.312187	+	0.950021i	$0.601062\pi$
$984$	6.06719	0.193415
$985$	37.9053	1.20776
$986$	1.08939	0.0346933
$987$	0	0
$988$	4.85998	0.154616
$989$	2.28908	0.0727886
$990$	−0.305872	−0.00972124
$991$	−44.5489	−1.41514	−0.707572	−	0.706641i	$-0.750209\pi$
$991$	−44.5489	−1.41514	−0.707572	+	0.706641i	$0.750209\pi$
$992$	9.76263	0.309964
$993$	−49.0995	−1.55813
$994$	0	0
$995$	−1.12957	−0.0358098
$996$	11.5780	0.366862
$997$	54.7584	1.73422	0.867108	−	0.498121i	$-0.165976\pi$
$997$	54.7584	1.73422	0.867108	+	0.498121i	$0.165976\pi$
$998$	0.927276	0.0293524
$999$	−45.9373	−1.45339

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6223.2.a.s.1.33	✓	54
7.6	odd	2	inner	6223.2.a.s.1.34	yes	54

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6223.2.a.s.1.33	✓	54	1.1	even	1	trivial
6223.2.a.s.1.34	yes	54	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6223 = 7^{2} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6223.a (trivial)

\(f(q)\)	\(=\)	\(q+0.117634 q^{2} -1.63542 q^{3} -1.98616 q^{4} -2.28766 q^{5} -0.192381 q^{6} -0.468908 q^{8} -0.325386 q^{9} -0.269106 q^{10} -3.49315 q^{11} +3.24822 q^{12} +0.309784 q^{13} +3.74129 q^{15} +3.91717 q^{16} +1.81047 q^{17} -0.0382764 q^{18} -7.89879 q^{19} +4.54366 q^{20} -0.410913 q^{22} -2.28467 q^{23} +0.766864 q^{24} +0.233383 q^{25} +0.0364411 q^{26} +5.43842 q^{27} +5.11516 q^{29} +0.440103 q^{30} +6.98025 q^{31} +1.39861 q^{32} +5.71278 q^{33} +0.212973 q^{34} +0.646269 q^{36} -8.44681 q^{37} -0.929166 q^{38} -0.506628 q^{39} +1.07270 q^{40} +7.91169 q^{41} -1.00193 q^{43} +6.93796 q^{44} +0.744372 q^{45} -0.268754 q^{46} -7.30561 q^{47} -6.40623 q^{48} +0.0274538 q^{50} -2.96089 q^{51} -0.615281 q^{52} +4.42705 q^{53} +0.639743 q^{54} +7.99113 q^{55} +12.9179 q^{57} +0.601716 q^{58} +13.3546 q^{59} -7.43082 q^{60} +3.90611 q^{61} +0.821115 q^{62} -7.66981 q^{64} -0.708680 q^{65} +0.672017 q^{66} +6.28582 q^{67} -3.59590 q^{68} +3.73640 q^{69} -2.05298 q^{71} +0.152576 q^{72} -1.56116 q^{73} -0.993632 q^{74} -0.381681 q^{75} +15.6883 q^{76} -0.0595967 q^{78} +10.5077 q^{79} -8.96114 q^{80} -7.91797 q^{81} +0.930683 q^{82} +3.56441 q^{83} -4.14175 q^{85} -0.117861 q^{86} -8.36545 q^{87} +1.63796 q^{88} +3.06147 q^{89} +0.0875634 q^{90} +4.53772 q^{92} -11.4157 q^{93} -0.859387 q^{94} +18.0697 q^{95} -2.28732 q^{96} -11.7154 q^{97} +1.13662 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 14 q^{2} + 46 q^{4} - 42 q^{8} + 14 q^{9} - 20 q^{11} - 16 q^{15} + 38 q^{16} - 42 q^{18} - 28 q^{22} - 64 q^{23} - 10 q^{25} - 44 q^{29} - 12 q^{30} - 78 q^{32} - 38 q^{36} - 32 q^{37} - 40 q^{39}+ \cdots - 72 q^{99}+O(q^{100}) \) 54 * q - 14 * q^2 + 46 * q^4 - 42 * q^8 + 14 * q^9 - 20 * q^11 - 16 * q^15 + 38 * q^16 - 42 * q^18 - 28 * q^22 - 64 * q^23 - 10 * q^25 - 44 * q^29 - 12 * q^30 - 78 * q^32 - 38 * q^36 - 32 * q^37 - 40 * q^39 - 52 * q^43 - 48 * q^44 - 8 * q^46 - 66 * q^50 - 24 * q^51 - 136 * q^53 - 80 * q^57 - 4 * q^58 - 28 * q^60 - 10 * q^64 - 88 * q^65 - 48 * q^67 - 52 * q^71 - 22 * q^72 - 28 * q^74 - 36 * q^78 + 24 * q^79 - 66 * q^81 - 48 * q^85 + 64 * q^86 - 68 * q^88 - 192 * q^92 - 88 * q^93 - 112 * q^95 - 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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