LMFDB - Embedded newform 6223.2.a.o.1.11

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)

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);

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")

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

Newform invariants

comment:select newform

sage:traces = [38,-2,-11,38,-16] 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:	$38$
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
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)

$f(q)$	$=$	\(q-1.45409 q^{2} +1.33511 q^{3} +0.114369 q^{4} -3.16583 q^{5} -1.94137 q^{6} +2.74187 q^{8} -1.21748 q^{9} +4.60340 q^{10} +2.94763 q^{11} +0.152695 q^{12} -3.25559 q^{13} -4.22674 q^{15} -4.21566 q^{16} +4.63744 q^{17} +1.77032 q^{18} -3.26400 q^{19} -0.362072 q^{20} -4.28611 q^{22} +8.26045 q^{23} +3.66071 q^{24} +5.02251 q^{25} +4.73391 q^{26} -5.63080 q^{27} -0.659879 q^{29} +6.14605 q^{30} -3.38402 q^{31} +0.646188 q^{32} +3.93541 q^{33} -6.74323 q^{34} -0.139241 q^{36} -1.21539 q^{37} +4.74614 q^{38} -4.34658 q^{39} -8.68031 q^{40} -3.84925 q^{41} -5.49619 q^{43} +0.337116 q^{44} +3.85433 q^{45} -12.0114 q^{46} -1.93728 q^{47} -5.62837 q^{48} -7.30316 q^{50} +6.19150 q^{51} -0.372338 q^{52} +1.07878 q^{53} +8.18768 q^{54} -9.33170 q^{55} -4.35781 q^{57} +0.959521 q^{58} +4.73108 q^{59} -0.483407 q^{60} +11.1301 q^{61} +4.92066 q^{62} +7.49170 q^{64} +10.3067 q^{65} -5.72243 q^{66} +8.20739 q^{67} +0.530377 q^{68} +11.0286 q^{69} +10.5442 q^{71} -3.33816 q^{72} -9.44300 q^{73} +1.76728 q^{74} +6.70561 q^{75} -0.373299 q^{76} +6.32031 q^{78} +9.81748 q^{79} +13.3461 q^{80} -3.86533 q^{81} +5.59715 q^{82} +4.61978 q^{83} -14.6814 q^{85} +7.99194 q^{86} -0.881012 q^{87} +8.08202 q^{88} -11.4358 q^{89} -5.60453 q^{90} +0.944736 q^{92} -4.51805 q^{93} +2.81697 q^{94} +10.3333 q^{95} +0.862733 q^{96} +10.2372 q^{97} -3.58867 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 38 q - 2 q^{2} - 11 q^{3} + 38 q^{4} - 16 q^{5} - 11 q^{6} + 47 q^{9} - 12 q^{10} - 2 q^{11} - 30 q^{12} - 21 q^{13} + 7 q^{15} + 46 q^{16} - 58 q^{17} - 13 q^{18} - 17 q^{19} - 44 q^{20} + 21 q^{22} + 7 q^{23}+ \cdots + 16 q^{99}+O(q^{100}) $ 38 * q - 2 * q^2 - 11 * q^3 + 38 * q^4 - 16 * q^5 - 11 * q^6 + 47 * q^9 - 12 * q^10 - 2 * q^11 - 30 * q^12 - 21 * q^13 + 7 * q^15 + 46 * q^16 - 58 * q^17 - 13 * q^18 - 17 * q^19 - 44 * q^20 + 21 * q^22 + 7 * q^23 - 21 * q^24 + 52 * q^25 - 49 * q^26 - 35 * q^27 - 2 * q^29 + 6 * q^30 - 2 * q^31 - 22 * q^32 - 29 * q^33 - 37 * q^34 + 25 * q^36 + 11 * q^37 - 40 * q^38 + 2 * q^39 + 8 * q^40 - 68 * q^41 - 38 * q^43 + 43 * q^44 - 60 * q^45 - 28 * q^46 - 35 * q^47 - 66 * q^48 - 25 * q^50 - 3 * q^51 - 32 * q^52 - 18 * q^53 + 68 * q^54 - 19 * q^55 - 31 * q^57 + 55 * q^58 - 21 * q^59 - 11 * q^60 - 5 * q^61 - 38 * q^62 + 18 * q^64 - 9 * q^65 - 20 * q^66 + 6 * q^67 - 74 * q^68 - 53 * q^69 - 26 * q^71 + 24 * q^72 - 22 * q^73 - 2 * q^74 - 38 * q^75 - 39 * q^76 - 114 * q^78 - 14 * q^79 - 140 * q^80 + 58 * q^81 + 8 * q^82 - 46 * q^83 - 11 * q^85 + 51 * q^86 - 59 * q^87 - 4 * q^88 - 74 * q^89 - q^90 - 81 * q^92 + 92 * q^93 + 36 * q^94 + 20 * q^95 + 108 * q^96 - 66 * q^97 + 16 * 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 $

$2$	−1.45409	−1.02819	−0.514097	−	0.857732i	$-0.671873\pi$
$2$	−1.45409	−1.02819	−0.514097	+	0.857732i	$0.671873\pi$
$3$	1.33511	0.770827	0.385414	−	0.922744i	$-0.374059\pi$
$3$	1.33511	0.770827	0.385414	+	0.922744i	$0.374059\pi$
$4$	0.114369	0.0571843
$5$	−3.16583	−1.41580	−0.707902	−	0.706311i	$-0.750358\pi$
$5$	−3.16583	−1.41580	−0.707902	+	0.706311i	$0.750358\pi$
$6$	−1.94137	−0.792561
$7$	0	0
$7$	0	0
$8$	2.74187	0.969398
$9$	−1.21748	−0.405825
$10$	4.60340	1.45572
$11$	2.94763	0.888743	0.444372	−	0.895843i	$-0.353427\pi$
$11$	2.94763	0.888743	0.444372	+	0.895843i	$0.353427\pi$
$12$	0.152695	0.0440792
$13$	−3.25559	−0.902939	−0.451469	−	0.892287i	$-0.649100\pi$
$13$	−3.25559	−0.902939	−0.451469	+	0.892287i	$0.649100\pi$
$14$	0	0
$15$	−4.22674	−1.09134
$16$	−4.21566	−1.05391
$17$	4.63744	1.12474	0.562372	−	0.826885i	$-0.309889\pi$
$17$	4.63744	1.12474	0.562372	+	0.826885i	$0.309889\pi$
$18$	1.77032	0.417267
$19$	−3.26400	−0.748813	−0.374407	−	0.927265i	$-0.622154\pi$
$19$	−3.26400	−0.748813	−0.374407	+	0.927265i	$0.622154\pi$
$20$	−0.362072	−0.0809618
$21$	0	0
$22$	−4.28611	−0.913801
$23$	8.26045	1.72242	0.861212	−	0.508247i	$-0.169706\pi$
$23$	8.26045	1.72242	0.861212	+	0.508247i	$0.169706\pi$
$24$	3.66071	0.747239
$25$	5.02251	1.00450
$26$	4.73391	0.928397
$27$	−5.63080	−1.08365
$28$	0	0
$29$	−0.659879	−0.122536	−0.0612682	−	0.998121i	$-0.519514\pi$
$29$	−0.659879	−0.122536	−0.0612682	+	0.998121i	$0.519514\pi$
$30$	6.14605	1.12211
$31$	−3.38402	−0.607788	−0.303894	−	0.952706i	$-0.598287\pi$
$31$	−3.38402	−0.607788	−0.303894	+	0.952706i	$0.598287\pi$
$32$	0.646188	0.114231
$33$	3.93541	0.685068
$34$	−6.74323	−1.15646
$35$	0	0
$36$	−0.139241	−0.0232068
$37$	−1.21539	−0.199809	−0.0999044	−	0.994997i	$-0.531854\pi$
$37$	−1.21539	−0.199809	−0.0999044	+	0.994997i	$0.531854\pi$
$38$	4.74614	0.769926
$39$	−4.34658	−0.696010
$40$	−8.68031	−1.37248
$41$	−3.84925	−0.601152	−0.300576	−	0.953758i	$-0.597179\pi$
$41$	−3.84925	−0.601152	−0.300576	+	0.953758i	$0.597179\pi$
$42$	0	0
$43$	−5.49619	−0.838162	−0.419081	−	0.907949i	$-0.637648\pi$
$43$	−5.49619	−0.838162	−0.419081	+	0.907949i	$0.637648\pi$
$44$	0.337116	0.0508222
$45$	3.85433	0.574569
$46$	−12.0114	−1.77099
$47$	−1.93728	−0.282581	−0.141291	−	0.989968i	$-0.545125\pi$
$47$	−1.93728	−0.282581	−0.141291	+	0.989968i	$0.545125\pi$
$48$	−5.62837	−0.812386
$49$	0	0
$50$	−7.30316	−1.03282
$51$	6.19150	0.866983
$52$	−0.372338	−0.0516339
$53$	1.07878	0.148181	0.0740907	−	0.997252i	$-0.476395\pi$
$53$	1.07878	0.148181	0.0740907	+	0.997252i	$0.476395\pi$
$54$	8.18768	1.11420
$55$	−9.33170	−1.25829
$56$	0	0
$57$	−4.35781	−0.577206
$58$	0.959521	0.125991
$59$	4.73108	0.615935	0.307967	−	0.951397i	$-0.400351\pi$
$59$	4.73108	0.615935	0.307967	+	0.951397i	$0.400351\pi$
$60$	−0.483407	−0.0624075
$61$	11.1301	1.42507	0.712534	−	0.701637i	$-0.247547\pi$
$61$	11.1301	1.42507	0.712534	+	0.701637i	$0.247547\pi$
$62$	4.92066	0.624925
$63$	0	0
$64$	7.49170	0.936463
$65$	10.3067	1.27838
$66$	−5.72243	−0.704383
$67$	8.20739	1.00269	0.501346	−	0.865247i	$-0.332838\pi$
$67$	8.20739	1.00269	0.501346	+	0.865247i	$0.332838\pi$
$68$	0.530377	0.0643177
$69$	11.0286	1.32769
$70$	0	0
$71$	10.5442	1.25136	0.625681	−	0.780079i	$-0.284821\pi$
$71$	10.5442	1.25136	0.625681	+	0.780079i	$0.284821\pi$
$72$	−3.33816	−0.393406
$73$	−9.44300	−1.10522	−0.552610	−	0.833440i	$-0.686368\pi$
$73$	−9.44300	−1.10522	−0.552610	+	0.833440i	$0.686368\pi$
$74$	1.76728	0.205442
$75$	6.70561	0.774297
$76$	−0.373299	−0.0428204
$77$	0	0
$78$	6.32031	0.715634
$79$	9.81748	1.10455	0.552277	−	0.833661i	$-0.313759\pi$
$79$	9.81748	1.10455	0.552277	+	0.833661i	$0.313759\pi$
$80$	13.3461	1.49214
$81$	−3.86533	−0.429481
$82$	5.59715	0.618101
$83$	4.61978	0.507086	0.253543	−	0.967324i	$-0.418404\pi$
$83$	4.61978	0.507086	0.253543	+	0.967324i	$0.418404\pi$
$84$	0	0
$85$	−14.6814	−1.59242
$86$	7.99194	0.861793
$87$	−0.881012	−0.0944544
$88$	8.08202	0.861546
$89$	−11.4358	−1.21219	−0.606094	−	0.795393i	$-0.707264\pi$
$89$	−11.4358	−1.21219	−0.606094	+	0.795393i	$0.707264\pi$
$90$	−5.60453	−0.590769
$91$	0	0
$92$	0.944736	0.0984956
$93$	−4.51805	−0.468500
$94$	2.81697	0.290548
$95$	10.3333	1.06017
$96$	0.862733	0.0880523
$97$	10.2372	1.03943	0.519713	−	0.854341i	$-0.326039\pi$
$97$	10.2372	1.03943	0.519713	+	0.854341i	$0.326039\pi$
$98$	0	0
$99$	−3.58867	−0.360674
$100$	0.574417	0.0574417
$101$	−12.6942	−1.26312	−0.631561	−	0.775326i	$-0.717586\pi$
$101$	−12.6942	−1.26312	−0.631561	+	0.775326i	$0.717586\pi$
$102$	−9.00297	−0.891427
$103$	−13.2391	−1.30448	−0.652242	−	0.758011i	$-0.726171\pi$
$103$	−13.2391	−1.30448	−0.652242	+	0.758011i	$0.726171\pi$
$104$	−8.92642	−0.875307
$105$	0	0
$106$	−1.56863	−0.152359
$107$	−1.55325	−0.150159	−0.0750793	−	0.997178i	$-0.523921\pi$
$107$	−1.55325	−0.150159	−0.0750793	+	0.997178i	$0.523921\pi$
$108$	−0.643987	−0.0619677
$109$	11.9851	1.14796	0.573981	−	0.818868i	$-0.305398\pi$
$109$	11.9851	1.14796	0.573981	+	0.818868i	$0.305398\pi$
$110$	13.5691	1.29376
$111$	−1.62268	−0.154018
$112$	0	0
$113$	−2.68377	−0.252468	−0.126234	−	0.992001i	$-0.540289\pi$
$113$	−2.68377	−0.252468	−0.126234	+	0.992001i	$0.540289\pi$
$114$	6.33663	0.593480
$115$	−26.1512	−2.43861
$116$	−0.0754694	−0.00700716
$117$	3.96360	0.366435
$118$	−6.87941	−0.633301
$119$	0	0
$120$	−11.5892	−1.05794
$121$	−2.31149	−0.210135
$122$	−16.1842	−1.46525
$123$	−5.13918	−0.463385
$124$	−0.387026	−0.0347560
$125$	−0.0712523	−0.00637300
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−12.1860	−1.07710
$129$	−7.33803	−0.646078
$130$	−14.9868	−1.31443
$131$	19.9814	1.74579	0.872894	−	0.487911i	$-0.162241\pi$
$131$	19.9814	1.74579	0.872894	+	0.487911i	$0.162241\pi$
$132$	0.450088	0.0391751
$133$	0	0
$134$	−11.9343	−1.03096
$135$	17.8262	1.53423
$136$	12.7153	1.09032
$137$	14.1582	1.20961	0.604806	−	0.796373i	$-0.293251\pi$
$137$	14.1582	1.20961	0.604806	+	0.796373i	$0.293251\pi$
$138$	−16.0366	−1.36512
$139$	−1.17991	−0.100079	−0.0500395	−	0.998747i	$-0.515935\pi$
$139$	−1.17991	−0.100079	−0.0500395	+	0.998747i	$0.515935\pi$
$140$	0	0
$141$	−2.58648	−0.217821
$142$	−15.3321	−1.28664
$143$	−9.59627	−0.802481
$144$	5.13246	0.427705
$145$	2.08907	0.173488
$146$	13.7309	1.13638
$147$	0	0
$148$	−0.139002	−0.0114259
$149$	−19.3077	−1.58175	−0.790876	−	0.611976i	$-0.790375\pi$
$149$	−19.3077	−1.58175	−0.790876	+	0.611976i	$0.790375\pi$
$150$	−9.75054	−0.796128
$151$	−1.19916	−0.0975866	−0.0487933	−	0.998809i	$-0.515538\pi$
$151$	−1.19916	−0.0975866	−0.0487933	+	0.998809i	$0.515538\pi$
$152$	−8.94947	−0.725898
$153$	−5.64596	−0.456449
$154$	0	0
$155$	10.7133	0.860509
$156$	−0.497112	−0.0398008
$157$	13.2375	1.05647	0.528233	−	0.849099i	$-0.322855\pi$
$157$	13.2375	1.05647	0.528233	+	0.849099i	$0.322855\pi$
$158$	−14.2755	−1.13570
$159$	1.44029	0.114222
$160$	−2.04572	−0.161729
$161$	0	0
$162$	5.62052	0.441590
$163$	−12.5688	−0.984462	−0.492231	−	0.870465i	$-0.663818\pi$
$163$	−12.5688	−0.984462	−0.492231	+	0.870465i	$0.663818\pi$
$164$	−0.440234	−0.0343765
$165$	−12.4589	−0.969922
$166$	−6.71756	−0.521384
$167$	14.7279	1.13968	0.569839	−	0.821757i	$-0.307006\pi$
$167$	14.7279	1.13968	0.569839	+	0.821757i	$0.307006\pi$
$168$	0	0
$169$	−2.40112	−0.184701
$170$	21.3480	1.63731
$171$	3.97384	0.303887
$172$	−0.628592	−0.0479297
$173$	2.82875	0.215066	0.107533	−	0.994202i	$-0.465705\pi$
$173$	2.82875	0.215066	0.107533	+	0.994202i	$0.465705\pi$
$174$	1.28107	0.0971175
$175$	0	0
$176$	−12.4262	−0.936659
$177$	6.31653	0.474779
$178$	16.6286	1.24636
$179$	−7.15827	−0.535034	−0.267517	−	0.963553i	$-0.586203\pi$
$179$	−7.15827	−0.535034	−0.267517	+	0.963553i	$0.586203\pi$
$180$	0.440814	0.0328563
$181$	−16.9849	−1.26248	−0.631240	−	0.775587i	$-0.717454\pi$
$181$	−16.9849	−1.26248	−0.631240	+	0.775587i	$0.717454\pi$
$182$	0	0
$183$	14.8600	1.09848
$184$	22.6491	1.66971
$185$	3.84772	0.282890
$186$	6.56964	0.481709
$187$	13.6694	0.999608
$188$	−0.221564	−0.0161592
$189$	0	0
$190$	−15.0255	−1.09006
$191$	−9.77084	−0.706993	−0.353497	−	0.935436i	$-0.615007\pi$
$191$	−9.77084	−0.706993	−0.353497	+	0.935436i	$0.615007\pi$
$192$	10.0023	0.721851
$193$	8.89861	0.640536	0.320268	−	0.947327i	$-0.396227\pi$
$193$	8.89861	0.640536	0.320268	+	0.947327i	$0.396227\pi$
$194$	−14.8857	−1.06873
$195$	13.7606	0.985414
$196$	0	0
$197$	−5.85715	−0.417305	−0.208652	−	0.977990i	$-0.566908\pi$
$197$	−5.85715	−0.417305	−0.208652	+	0.977990i	$0.566908\pi$
$198$	5.21823	0.370844
$199$	9.32457	0.661001	0.330501	−	0.943806i	$-0.392782\pi$
$199$	9.32457	0.661001	0.330501	+	0.943806i	$0.392782\pi$
$200$	13.7711	0.973762
$201$	10.9578	0.772903
$202$	18.4585	1.29874
$203$	0	0
$204$	0.708113	0.0495778
$205$	12.1861	0.851114
$206$	19.2508	1.34126
$207$	−10.0569	−0.699003
$208$	13.7245	0.951620
$209$	−9.62106	−0.665503
$210$	0	0
$211$	−25.8432	−1.77912	−0.889560	−	0.456818i	$-0.848989\pi$
$211$	−25.8432	−1.77912	−0.889560	+	0.456818i	$0.848989\pi$
$212$	0.123378	0.00847365
$213$	14.0776	0.964583
$214$	2.25856	0.154392
$215$	17.4000	1.18667
$216$	−15.4389	−1.05049
$217$	0	0
$218$	−17.4274	−1.18033
$219$	−12.6075	−0.851934
$220$	−1.06725	−0.0719542
$221$	−15.0976	−1.01557
$222$	2.35952	0.158361
$223$	−21.8729	−1.46472	−0.732360	−	0.680918i	$-0.761581\pi$
$223$	−21.8729	−1.46472	−0.732360	+	0.680918i	$0.761581\pi$
$224$	0	0
$225$	−6.11478	−0.407652
$226$	3.90243	0.259586
$227$	17.9966	1.19448	0.597240	−	0.802063i	$-0.296264\pi$
$227$	17.9966	1.19448	0.597240	+	0.802063i	$0.296264\pi$
$228$	−0.498396	−0.0330071
$229$	−17.2428	−1.13944	−0.569718	−	0.821840i	$-0.692948\pi$
$229$	−17.2428	−1.13944	−0.569718	+	0.821840i	$0.692948\pi$
$230$	38.0261	2.50737
$231$	0	0
$232$	−1.80930	−0.118787
$233$	−17.7700	−1.16415	−0.582074	−	0.813136i	$-0.697759\pi$
$233$	−17.7700	−1.16415	−0.582074	+	0.813136i	$0.697759\pi$
$234$	−5.76343	−0.376767
$235$	6.13310	0.400079
$236$	0.541087	0.0352218
$237$	13.1074	0.851420
$238$	0	0
$239$	−17.2737	−1.11734	−0.558671	−	0.829389i	$-0.688689\pi$
$239$	−17.2737	−1.11734	−0.558671	+	0.829389i	$0.688689\pi$
$240$	17.8185	1.15018
$241$	−22.4518	−1.44625	−0.723123	−	0.690719i	$-0.757294\pi$
$241$	−22.4518	−1.44625	−0.723123	+	0.690719i	$0.757294\pi$
$242$	3.36111	0.216060
$243$	11.7318	0.752593
$244$	1.27294	0.0814916
$245$	0	0
$246$	7.47282	0.476450
$247$	10.6263	0.676133
$248$	−9.27856	−0.589189
$249$	6.16792	0.390876
$250$	0.103607	0.00655268
$251$	−14.2713	−0.900795	−0.450397	−	0.892828i	$-0.648718\pi$
$251$	−14.2713	−0.900795	−0.450397	+	0.892828i	$0.648718\pi$
$252$	0	0
$253$	24.3487	1.53079
$254$	−1.45409	−0.0912375
$255$	−19.6013	−1.22748
$256$	2.73604	0.171003
$257$	−26.0467	−1.62475	−0.812374	−	0.583137i	$-0.801825\pi$
$257$	−26.0467	−1.62475	−0.812374	+	0.583137i	$0.801825\pi$
$258$	10.6701	0.664294
$259$	0	0
$260$	1.17876	0.0731035
$261$	0.803386	0.0497284
$262$	−29.0548	−1.79501
$263$	−25.2766	−1.55862	−0.779312	−	0.626637i	$-0.784431\pi$
$263$	−25.2766	−1.55862	−0.779312	+	0.626637i	$0.784431\pi$
$264$	10.7904	0.664103
$265$	−3.41523	−0.209796
$266$	0	0
$267$	−15.2680	−0.934387
$268$	0.938668	0.0573383
$269$	−15.4733	−0.943425	−0.471712	−	0.881753i	$-0.656364\pi$
$269$	−15.4733	−0.943425	−0.471712	+	0.881753i	$0.656364\pi$
$270$	−25.9208	−1.57749
$271$	4.27569	0.259729	0.129865	−	0.991532i	$-0.458546\pi$
$271$	4.27569	0.259729	0.129865	+	0.991532i	$0.458546\pi$
$272$	−19.5498	−1.18538
$273$	0	0
$274$	−20.5872	−1.24372
$275$	14.8045	0.892744
$276$	1.26133	0.0759231
$277$	20.8137	1.25058	0.625288	−	0.780394i	$-0.284981\pi$
$277$	20.8137	1.25058	0.625288	+	0.780394i	$0.284981\pi$
$278$	1.71570	0.102901
$279$	4.11996	0.246656
$280$	0	0
$281$	−3.69664	−0.220523	−0.110261	−	0.993903i	$-0.535169\pi$
$281$	−3.69664	−0.220523	−0.110261	+	0.993903i	$0.535169\pi$
$282$	3.76097	0.223963
$283$	−18.3119	−1.08853	−0.544266	−	0.838913i	$-0.683192\pi$
$283$	−18.3119	−1.08853	−0.544266	+	0.838913i	$0.683192\pi$
$284$	1.20592	0.0715582
$285$	13.7961	0.817210
$286$	13.9538	0.825107
$287$	0	0
$288$	−0.786718	−0.0463578
$289$	4.50581	0.265047
$290$	−3.03768	−0.178379
$291$	13.6678	0.801218
$292$	−1.07998	−0.0632012
$293$	4.09252	0.239088	0.119544	−	0.992829i	$-0.461857\pi$
$293$	4.09252	0.239088	0.119544	+	0.992829i	$0.461857\pi$
$294$	0	0
$295$	−14.9778	−0.872043
$296$	−3.33244	−0.193694
$297$	−16.5975	−0.963085
$298$	28.0751	1.62635
$299$	−26.8927	−1.55524
$300$	0.766911	0.0442776
$301$	0	0
$302$	1.74369	0.100338
$303$	−16.9482	−0.973650
$304$	13.7599	0.789185
$305$	−35.2362	−2.01762
$306$	8.20972	0.469319
$307$	24.3046	1.38714	0.693568	−	0.720391i	$-0.256038\pi$
$307$	24.3046	1.38714	0.693568	+	0.720391i	$0.256038\pi$
$308$	0	0
$309$	−17.6756	−1.00553
$310$	−15.5780	−0.884771
$311$	−17.7040	−1.00390	−0.501951	−	0.864896i	$-0.667384\pi$
$311$	−17.7040	−1.00390	−0.501951	+	0.864896i	$0.667384\pi$
$312$	−11.9178	−0.674711
$313$	−26.9681	−1.52433	−0.762164	−	0.647384i	$-0.775863\pi$
$313$	−26.9681	−1.52433	−0.762164	+	0.647384i	$0.775863\pi$
$314$	−19.2484	−1.08625
$315$	0	0
$316$	1.12281	0.0631631
$317$	18.0931	1.01621	0.508105	−	0.861295i	$-0.330346\pi$
$317$	18.0931	1.01621	0.508105	+	0.861295i	$0.330346\pi$
$318$	−2.09430	−0.117443
$319$	−1.94508	−0.108903
$320$	−23.7175	−1.32585
$321$	−2.07377	−0.115746
$322$	0	0
$323$	−15.1366	−0.842223
$324$	−0.442072	−0.0245596
$325$	−16.3512	−0.907003
$326$	18.2761	1.01222
$327$	16.0014	0.884881
$328$	−10.5542	−0.582756
$329$	0	0
$330$	18.1163	0.997268
$331$	−10.0570	−0.552782	−0.276391	−	0.961045i	$-0.589139\pi$
$331$	−10.0570	−0.552782	−0.276391	+	0.961045i	$0.589139\pi$
$332$	0.528358	0.0289974
$333$	1.47971	0.0810874
$334$	−21.4156	−1.17181
$335$	−25.9832	−1.41962
$336$	0	0
$337$	32.0911	1.74812	0.874058	−	0.485822i	$-0.161480\pi$
$337$	32.0911	1.74812	0.874058	+	0.485822i	$0.161480\pi$
$338$	3.49143	0.189909
$339$	−3.58313	−0.194609
$340$	−1.67909	−0.0910612
$341$	−9.97484	−0.540168
$342$	−5.77831	−0.312455
$343$	0	0
$344$	−15.0699	−0.812512
$345$	−34.9148	−1.87975
$346$	−4.11325	−0.221130
$347$	−26.6402	−1.43012	−0.715060	−	0.699063i	$-0.753601\pi$
$347$	−26.6402	−1.43012	−0.715060	+	0.699063i	$0.753601\pi$
$348$	−0.100760	−0.00540131
$349$	6.31366	0.337963	0.168981	−	0.985619i	$-0.445952\pi$
$349$	6.31366	0.337963	0.168981	+	0.985619i	$0.445952\pi$
$350$	0	0
$351$	18.3316	0.978468
$352$	1.90472	0.101522
$353$	7.83647	0.417093	0.208547	−	0.978012i	$-0.433127\pi$
$353$	7.83647	0.417093	0.208547	+	0.978012i	$0.433127\pi$
$354$	−9.18478	−0.488165
$355$	−33.3811	−1.77168
$356$	−1.30789	−0.0693181
$357$	0	0
$358$	10.4088	0.550120
$359$	35.5778	1.87773	0.938863	−	0.344291i	$-0.111881\pi$
$359$	35.5778	1.87773	0.938863	+	0.344291i	$0.111881\pi$
$360$	10.5681	0.556986
$361$	−8.34629	−0.439279
$362$	24.6976	1.29808
$363$	−3.08610	−0.161978
$364$	0	0
$365$	29.8950	1.56477
$366$	−21.6077	−1.12945
$367$	21.4899	1.12176	0.560882	−	0.827896i	$-0.310462\pi$
$367$	21.4899	1.12176	0.560882	+	0.827896i	$0.310462\pi$
$368$	−34.8232	−1.81529
$369$	4.68637	0.243963
$370$	−5.59492	−0.290866
$371$	0	0
$372$	−0.516723	−0.0267908
$373$	−17.1602	−0.888522	−0.444261	−	0.895897i	$-0.646534\pi$
$373$	−17.1602	−0.888522	−0.444261	+	0.895897i	$0.646534\pi$
$374$	−19.8765	−1.02779
$375$	−0.0951298	−0.00491248
$376$	−5.31177	−0.273934
$377$	2.14830	0.110643
$378$	0	0
$379$	−0.793301	−0.0407491	−0.0203746	−	0.999792i	$-0.506486\pi$
$379$	−0.793301	−0.0407491	−0.0203746	+	0.999792i	$0.506486\pi$
$380$	1.18180	0.0606253
$381$	1.33511	0.0683999
$382$	14.2077	0.726927
$383$	−32.7388	−1.67287	−0.836437	−	0.548064i	$-0.815365\pi$
$383$	−32.7388	−1.67287	−0.836437	+	0.548064i	$0.815365\pi$
$384$	−16.2696	−0.830256
$385$	0	0
$386$	−12.9394	−0.658596
$387$	6.69148	0.340147
$388$	1.17081	0.0594389
$389$	−1.72700	−0.0875622	−0.0437811	−	0.999041i	$-0.513940\pi$
$389$	−1.72700	−0.0875622	−0.0437811	+	0.999041i	$0.513940\pi$
$390$	−20.0090	−1.01320
$391$	38.3073	1.93728
$392$	0	0
$393$	26.6775	1.34570
$394$	8.51681	0.429071
$395$	−31.0805	−1.56383
$396$	−0.410431	−0.0206249
$397$	25.9689	1.30334	0.651671	−	0.758501i	$-0.274068\pi$
$397$	25.9689	1.30334	0.651671	+	0.758501i	$0.274068\pi$
$398$	−13.5587	−0.679638
$399$	0	0
$400$	−21.1732	−1.05866
$401$	−11.3301	−0.565797	−0.282898	−	0.959150i	$-0.591296\pi$
$401$	−11.3301	−0.565797	−0.282898	+	0.959150i	$0.591296\pi$
$402$	−15.9336	−0.794695
$403$	11.0170	0.548796
$404$	−1.45182	−0.0722308
$405$	12.2370	0.608060
$406$	0	0
$407$	−3.58251	−0.177579
$408$	16.9763	0.840452
$409$	−1.52148	−0.0752325	−0.0376163	−	0.999292i	$-0.511976\pi$
$409$	−1.52148	−0.0752325	−0.0376163	+	0.999292i	$0.511976\pi$
$410$	−17.7196	−0.875111
$411$	18.9027	0.932402
$412$	−1.51413	−0.0745960
$413$	0	0
$414$	14.6236	0.718711
$415$	−14.6255	−0.717935
$416$	−2.10372	−0.103144
$417$	−1.57532	−0.0771436
$418$	13.9899	0.684266
$419$	5.70243	0.278582	0.139291	−	0.990252i	$-0.455518\pi$
$419$	5.70243	0.278582	0.139291	+	0.990252i	$0.455518\pi$
$420$	0	0
$421$	17.4907	0.852443	0.426221	−	0.904619i	$-0.359844\pi$
$421$	17.4907	0.852443	0.426221	+	0.904619i	$0.359844\pi$
$422$	37.5783	1.82928
$423$	2.35859	0.114679
$424$	2.95787	0.143647
$425$	23.2916	1.12981
$426$	−20.4701	−0.991780
$427$	0	0
$428$	−0.177643	−0.00858672
$429$	−12.8121	−0.618574
$430$	−25.3012	−1.22013
$431$	−24.8936	−1.19908	−0.599541	−	0.800344i	$-0.704650\pi$
$431$	−24.8936	−1.19908	−0.599541	+	0.800344i	$0.704650\pi$
$432$	23.7375	1.14207
$433$	−18.6350	−0.895542	−0.447771	−	0.894148i	$-0.647782\pi$
$433$	−18.6350	−0.895542	−0.447771	+	0.894148i	$0.647782\pi$
$434$	0	0
$435$	2.78914	0.133729
$436$	1.37072	0.0656454
$437$	−26.9621	−1.28977
$438$	18.3323	0.875954
$439$	−10.5952	−0.505683	−0.252842	−	0.967508i	$-0.581365\pi$
$439$	−10.5952	−0.505683	−0.252842	+	0.967508i	$0.581365\pi$
$440$	−25.5863	−1.21978
$441$	0	0
$442$	21.9532	1.04421
$443$	28.2410	1.34177	0.670884	−	0.741562i	$-0.265915\pi$
$443$	28.2410	1.34177	0.670884	+	0.741562i	$0.265915\pi$
$444$	−0.185584	−0.00880741
$445$	36.2037	1.71622
$446$	31.8051	1.50602
$447$	−25.7780	−1.21926
$448$	0	0
$449$	−14.5768	−0.687920	−0.343960	−	0.938984i	$-0.611768\pi$
$449$	−14.5768	−0.687920	−0.343960	+	0.938984i	$0.611768\pi$
$450$	8.89142	0.419146
$451$	−11.3462	−0.534270
$452$	−0.306939	−0.0144372
$453$	−1.60102	−0.0752224
$454$	−26.1687	−1.22816
$455$	0	0
$456$	−11.9486	−0.559542
$457$	8.80603	0.411929	0.205964	−	0.978560i	$-0.433967\pi$
$457$	8.80603	0.411929	0.205964	+	0.978560i	$0.433967\pi$
$458$	25.0725	1.17156
$459$	−26.1125	−1.21883
$460$	−2.99088	−0.139450
$461$	−40.8213	−1.90124	−0.950619	−	0.310360i	$-0.899551\pi$
$461$	−40.8213	−1.90124	−0.950619	+	0.310360i	$0.899551\pi$
$462$	0	0
$463$	4.79086	0.222650	0.111325	−	0.993784i	$-0.464491\pi$
$463$	4.79086	0.222650	0.111325	+	0.993784i	$0.464491\pi$
$464$	2.78182	0.129143
$465$	14.3034	0.663304
$466$	25.8391	1.19697
$467$	−17.9632	−0.831240	−0.415620	−	0.909538i	$-0.636435\pi$
$467$	−17.9632	−0.831240	−0.415620	+	0.909538i	$0.636435\pi$
$468$	0.453312	0.0209544
$469$	0	0
$470$	−8.91806	−0.411360
$471$	17.6735	0.814353
$472$	12.9720	0.597086
$473$	−16.2007	−0.744911
$474$	−19.0594	−0.875425
$475$	−16.3935	−0.752184
$476$	0	0
$477$	−1.31338	−0.0601357
$478$	25.1175	1.14885
$479$	−22.6657	−1.03562	−0.517811	−	0.855495i	$-0.673253\pi$
$479$	−22.6657	−1.03562	−0.517811	+	0.855495i	$0.673253\pi$
$480$	−2.73127	−0.124665
$481$	3.95681	0.180415
$482$	32.6468	1.48702
$483$	0	0
$484$	−0.264362	−0.0120164
$485$	−32.4092	−1.47162
$486$	−17.0590	−0.773812
$487$	−31.0757	−1.40817	−0.704087	−	0.710114i	$-0.748643\pi$
$487$	−31.0757	−1.40817	−0.704087	+	0.710114i	$0.748643\pi$
$488$	30.5174	1.38146
$489$	−16.7807	−0.758850
$490$	0	0
$491$	−8.03230	−0.362492	−0.181246	−	0.983438i	$-0.558013\pi$
$491$	−8.03230	−0.362492	−0.181246	+	0.983438i	$0.558013\pi$
$492$	−0.587761	−0.0264983
$493$	−3.06014	−0.137822
$494$	−15.4515	−0.695196
$495$	11.3611	0.510644
$496$	14.2659	0.640557
$497$	0	0
$498$	−8.96869	−0.401897
$499$	−11.3248	−0.506967	−0.253484	−	0.967340i	$-0.581576\pi$
$499$	−11.3248	−0.506967	−0.253484	+	0.967340i	$0.581576\pi$
$500$	−0.00814902	−0.000364435 0
$501$	19.6634	0.878494
$502$	20.7517	0.926193
$503$	2.74320	0.122313	0.0611565	−	0.998128i	$-0.480521\pi$
$503$	2.74320	0.122313	0.0611565	+	0.998128i	$0.480521\pi$
$504$	0	0
$505$	40.1878	1.78833
$506$	−35.4052	−1.57395
$507$	−3.20576	−0.142373
$508$	0.114369	0.00507429
$509$	19.9495	0.884247	0.442123	−	0.896954i	$-0.354225\pi$
$509$	19.9495	0.884247	0.442123	+	0.896954i	$0.354225\pi$
$510$	28.5019	1.26209
$511$	0	0
$512$	20.3935	0.901273
$513$	18.3790	0.811450
$514$	37.8741	1.67056
$515$	41.9127	1.84689
$516$	−0.839241	−0.0369455
$517$	−5.71037	−0.251142
$518$	0	0
$519$	3.77670	0.165779
$520$	28.2596	1.23926
$521$	1.66032	0.0727400	0.0363700	−	0.999338i	$-0.488421\pi$
$521$	1.66032	0.0727400	0.0363700	+	0.999338i	$0.488421\pi$
$522$	−1.16819	−0.0511304
$523$	−26.3175	−1.15079	−0.575393	−	0.817877i	$-0.695151\pi$
$523$	−26.3175	−1.15079	−0.575393	+	0.817877i	$0.695151\pi$
$524$	2.28525	0.0998316
$525$	0	0
$526$	36.7544	1.60257
$527$	−15.6932	−0.683606
$528$	−16.5904	−0.722002
$529$	45.2351	1.96674
$530$	4.96604	0.215711
$531$	−5.75998	−0.249962
$532$	0	0
$533$	12.5316	0.542804
$534$	22.2010	0.960732
$535$	4.91734	0.212595
$536$	22.5036	0.972008
$537$	−9.55710	−0.412419
$538$	22.4995	0.970024
$539$	0	0
$540$	2.03876	0.0877341
$541$	−18.1396	−0.779882	−0.389941	−	0.920840i	$-0.627505\pi$
$541$	−18.1396	−0.779882	−0.389941	+	0.920840i	$0.627505\pi$
$542$	−6.21722	−0.267052
$543$	−22.6768	−0.973155
$544$	2.99665	0.128480
$545$	−37.9428	−1.62529
$546$	0	0
$547$	−29.7917	−1.27380	−0.636901	−	0.770946i	$-0.719784\pi$
$547$	−29.7917	−1.27380	−0.636901	+	0.770946i	$0.719784\pi$
$548$	1.61925	0.0691709
$549$	−13.5507	−0.578329
$550$	−21.5270	−0.917914
$551$	2.15384	0.0917569
$552$	30.2391	1.28706
$553$	0	0
$554$	−30.2650	−1.28584
$555$	5.13714	0.218059
$556$	−0.134945	−0.00572294
$557$	12.6998	0.538108	0.269054	−	0.963125i	$-0.413289\pi$
$557$	12.6998	0.538108	0.269054	+	0.963125i	$0.413289\pi$
$558$	−5.99079	−0.253610
$559$	17.8934	0.756809
$560$	0	0
$561$	18.2502	0.770525
$562$	5.37523	0.226740
$563$	−4.56232	−0.192279	−0.0961393	−	0.995368i	$-0.530649\pi$
$563$	−4.56232	−0.192279	−0.0961393	+	0.995368i	$0.530649\pi$
$564$	−0.295812	−0.0124560
$565$	8.49637	0.357445
$566$	26.6271	1.11922
$567$	0	0
$568$	28.9107	1.21307
$569$	15.5944	0.653750	0.326875	−	0.945068i	$-0.394004\pi$
$569$	15.5944	0.653750	0.326875	+	0.945068i	$0.394004\pi$
$570$	−20.0607	−0.840251
$571$	−17.7970	−0.744782	−0.372391	−	0.928076i	$-0.621462\pi$
$571$	−17.7970	−0.744782	−0.372391	+	0.928076i	$0.621462\pi$
$572$	−1.09751	−0.0458893
$573$	−13.0452	−0.544970
$574$	0	0
$575$	41.4882	1.73018
$576$	−9.12096	−0.380040
$577$	43.3194	1.80341	0.901704	−	0.432353i	$-0.142317\pi$
$577$	43.3194	1.80341	0.901704	+	0.432353i	$0.142317\pi$
$578$	−6.55184	−0.272520
$579$	11.8806	0.493743
$580$	0.238924	0.00992076
$581$	0	0
$582$	−19.8741	−0.823809
$583$	3.17983	0.131695
$584$	−25.8915	−1.07140
$585$	−12.5481	−0.518801
$586$	−5.95088	−0.245828
$587$	−34.5184	−1.42472	−0.712362	−	0.701812i	$-0.752375\pi$
$587$	−34.5184	−1.42472	−0.712362	+	0.701812i	$0.752375\pi$
$588$	0	0
$589$	11.0455	0.455120
$590$	21.7791	0.896630
$591$	−7.81995	−0.321670
$592$	5.12366	0.210581
$593$	−28.8753	−1.18577	−0.592884	−	0.805288i	$-0.702011\pi$
$593$	−28.8753	−1.18577	−0.592884	+	0.805288i	$0.702011\pi$
$594$	24.1342	0.990239
$595$	0	0
$596$	−2.20820	−0.0904514
$597$	12.4493	0.509518
$598$	39.1043	1.59909
$599$	−14.4388	−0.589954	−0.294977	−	0.955504i	$-0.595312\pi$
$599$	−14.4388	−0.589954	−0.294977	+	0.955504i	$0.595312\pi$
$600$	18.3859	0.750602
$601$	−15.8226	−0.645416	−0.322708	−	0.946499i	$-0.604593\pi$
$601$	−15.8226	−0.645416	−0.322708	+	0.946499i	$0.604593\pi$
$602$	0	0
$603$	−9.99230	−0.406918
$604$	−0.137147	−0.00558042
$605$	7.31779	0.297511
$606$	24.6442	1.00110
$607$	−41.2059	−1.67249	−0.836247	−	0.548353i	$-0.815255\pi$
$607$	−41.2059	−1.67249	−0.836247	+	0.548353i	$0.815255\pi$
$608$	−2.10916	−0.0855377
$609$	0	0
$610$	51.2365	2.07450
$611$	6.30699	0.255153
$612$	−0.645721	−0.0261017
$613$	47.2254	1.90742	0.953709	−	0.300731i	$-0.0972308\pi$
$613$	47.2254	1.90742	0.953709	+	0.300731i	$0.0972308\pi$
$614$	−35.3410	−1.42625
$615$	16.2698	0.656062
$616$	0	0
$617$	−34.7515	−1.39904	−0.699522	−	0.714611i	$-0.746604\pi$
$617$	−34.7515	−1.39904	−0.699522	+	0.714611i	$0.746604\pi$
$618$	25.7019	1.03388
$619$	18.0458	0.725322	0.362661	−	0.931921i	$-0.381868\pi$
$619$	18.0458	0.725322	0.362661	+	0.931921i	$0.381868\pi$
$620$	1.22526	0.0492076
$621$	−46.5130	−1.86650
$622$	25.7432	1.03221
$623$	0	0
$624$	18.3237	0.733535
$625$	−24.8870	−0.995478
$626$	39.2140	1.56731
$627$	−12.8452	−0.512988
$628$	1.51395	0.0604133
$629$	−5.63629	−0.224734
$630$	0	0
$631$	23.2552	0.925773	0.462887	−	0.886418i	$-0.346814\pi$
$631$	23.2552	0.925773	0.462887	+	0.886418i	$0.346814\pi$
$632$	26.9183	1.07075
$633$	−34.5036	−1.37139
$634$	−26.3089	−1.04486
$635$	−3.16583	−0.125632
$636$	0.164724	0.00653172
$637$	0	0
$638$	2.82831	0.111974
$639$	−12.8373	−0.507834
$640$	38.5787	1.52496
$641$	35.1576	1.38864	0.694320	−	0.719667i	$-0.255705\pi$
$641$	35.1576	1.38864	0.694320	+	0.719667i	$0.255705\pi$
$642$	3.01544	0.119010
$643$	3.69229	0.145610	0.0728049	−	0.997346i	$-0.476805\pi$
$643$	3.69229	0.145610	0.0728049	+	0.997346i	$0.476805\pi$
$644$	0	0
$645$	23.2310	0.914720
$646$	22.0099	0.865969
$647$	−32.1683	−1.26467	−0.632334	−	0.774696i	$-0.717903\pi$
$647$	−32.1683	−1.26467	−0.632334	+	0.774696i	$0.717903\pi$
$648$	−10.5982	−0.416338
$649$	13.9455	0.547408
$650$	23.7761	0.932576
$651$	0	0
$652$	−1.43747	−0.0562958
$653$	36.2533	1.41870	0.709350	−	0.704856i	$-0.248988\pi$
$653$	36.2533	1.41870	0.709350	+	0.704856i	$0.248988\pi$
$654$	−23.2675	−0.909830
$655$	−63.2580	−2.47169
$656$	16.2271	0.633563
$657$	11.4966	0.448526
$658$	0	0
$659$	27.4080	1.06767	0.533833	−	0.845590i	$-0.320751\pi$
$659$	27.4080	1.06767	0.533833	+	0.845590i	$0.320751\pi$
$660$	−1.42490	−0.0554643
$661$	9.38996	0.365227	0.182614	−	0.983185i	$-0.441544\pi$
$661$	9.38996	0.365227	0.182614	+	0.983185i	$0.441544\pi$
$662$	14.6237	0.568368
$663$	−20.1570	−0.782833
$664$	12.6668	0.491569
$665$	0	0
$666$	−2.15162	−0.0833737
$667$	−5.45090	−0.211060
$668$	1.68441	0.0651716
$669$	−29.2028	−1.12905
$670$	37.7819	1.45964
$671$	32.8075	1.26652
$672$	0	0
$673$	7.25667	0.279724	0.139862	−	0.990171i	$-0.455334\pi$
$673$	7.25667	0.279724	0.139862	+	0.990171i	$0.455334\pi$
$674$	−46.6633	−1.79740
$675$	−28.2807	−1.08853
$676$	−0.274613	−0.0105620
$677$	7.99835	0.307401	0.153701	−	0.988117i	$-0.450881\pi$
$677$	7.99835	0.307401	0.153701	+	0.988117i	$0.450881\pi$
$678$	5.21019	0.200096
$679$	0	0
$680$	−40.2544	−1.54369
$681$	24.0275	0.920737
$682$	14.5043	0.555398
$683$	32.2856	1.23537	0.617687	−	0.786424i	$-0.288070\pi$
$683$	32.2856	1.23537	0.617687	+	0.786424i	$0.288070\pi$
$684$	0.454483	0.0173776
$685$	−44.8224	−1.71257
$686$	0	0
$687$	−23.0211	−0.878309
$688$	23.1701	0.883351
$689$	−3.51206	−0.133799
$690$	50.7692	1.93275
$691$	−10.1259	−0.385206	−0.192603	−	0.981277i	$-0.561693\pi$
$691$	−10.1259	−0.385206	−0.192603	+	0.981277i	$0.561693\pi$
$692$	0.323521	0.0122984
$693$	0	0
$694$	38.7371	1.47044
$695$	3.73541	0.141692
$696$	−2.41562	−0.0915639
$697$	−17.8507	−0.676142
$698$	−9.18061	−0.347491
$699$	−23.7249	−0.897358
$700$	0	0
$701$	−30.3871	−1.14771	−0.573853	−	0.818958i	$-0.694552\pi$
$701$	−30.3871	−1.14771	−0.573853	+	0.818958i	$0.694552\pi$
$702$	−26.6557	−1.00606
$703$	3.96703	0.149619
$704$	22.0827	0.832275
$705$	8.18838	0.308392
$706$	−11.3949	−0.428853
$707$	0	0
$708$	0.722412	0.0271499
$709$	44.2150	1.66053	0.830265	−	0.557369i	$-0.188189\pi$
$709$	44.2150	1.66053	0.830265	+	0.557369i	$0.188189\pi$
$710$	48.5389	1.82163
$711$	−11.9525	−0.448256
$712$	−31.3554	−1.17509
$713$	−27.9536	−1.04687
$714$	0	0
$715$	30.3802	1.13616
$716$	−0.818682	−0.0305956
$717$	−23.0623	−0.861278
$718$	−51.7332	−1.93067
$719$	−31.7052	−1.18240	−0.591201	−	0.806524i	$-0.701346\pi$
$719$	−31.7052	−1.18240	−0.591201	+	0.806524i	$0.701346\pi$
$720$	−16.2485	−0.605547
$721$	0	0
$722$	12.1362	0.451664
$723$	−29.9756	−1.11481
$724$	−1.94254	−0.0721941
$725$	−3.31424	−0.123088
$726$	4.48745	0.166545
$727$	−12.7541	−0.473022	−0.236511	−	0.971629i	$-0.576004\pi$
$727$	−12.7541	−0.473022	−0.236511	+	0.971629i	$0.576004\pi$
$728$	0	0
$729$	27.2592	1.00960
$730$	−43.4699	−1.60889
$731$	−25.4882	−0.942717
$732$	1.69952	0.0628159
$733$	27.6964	1.02299	0.511494	−	0.859287i	$-0.329092\pi$
$733$	27.6964	1.02299	0.511494	+	0.859287i	$0.329092\pi$
$734$	−31.2482	−1.15339
$735$	0	0
$736$	5.33780	0.196754
$737$	24.1923	0.891136
$738$	−6.81439	−0.250841
$739$	9.10696	0.335005	0.167502	−	0.985872i	$-0.446430\pi$
$739$	9.10696	0.335005	0.167502	+	0.985872i	$0.446430\pi$
$740$	0.440058	0.0161769
$741$	14.1872	0.521182
$742$	0	0
$743$	20.1476	0.739145	0.369572	−	0.929202i	$-0.379504\pi$
$743$	20.1476	0.739145	0.369572	+	0.929202i	$0.379504\pi$
$744$	−12.3879	−0.454163
$745$	61.1251	2.23945
$746$	24.9524	0.913574
$747$	−5.62447	−0.205788
$748$	1.56335	0.0571619
$749$	0	0
$750$	0.138327	0.00505099
$751$	−31.2534	−1.14045	−0.570226	−	0.821488i	$-0.693144\pi$
$751$	−31.2534	−1.14045	−0.570226	+	0.821488i	$0.693144\pi$
$752$	8.16690	0.297816
$753$	−19.0538	−0.694357
$754$	−3.12381	−0.113762
$755$	3.79636	0.138164
$756$	0	0
$757$	39.4793	1.43490	0.717449	−	0.696611i	$-0.245309\pi$
$757$	39.4793	1.43490	0.717449	+	0.696611i	$0.245309\pi$
$758$	1.15353	0.0418980
$759$	32.5083	1.17998
$760$	28.3326	1.02773
$761$	−18.9290	−0.686177	−0.343088	−	0.939303i	$-0.611473\pi$
$761$	−18.9290	−0.686177	−0.343088	+	0.939303i	$0.611473\pi$
$762$	−1.94137	−0.0703284
$763$	0	0
$764$	−1.11748	−0.0404289
$765$	17.8742	0.646243
$766$	47.6050	1.72004
$767$	−15.4025	−0.556151
$768$	3.65292	0.131813
$769$	20.9761	0.756418	0.378209	−	0.925720i	$-0.376540\pi$
$769$	20.9761	0.756418	0.378209	+	0.925720i	$0.376540\pi$
$770$	0	0
$771$	−34.7752	−1.25240
$772$	1.01772	0.0366286
$773$	−14.4654	−0.520284	−0.260142	−	0.965570i	$-0.583769\pi$
$773$	−14.4654	−0.520284	−0.260142	+	0.965570i	$0.583769\pi$
$774$	−9.73000	−0.349738
$775$	−16.9963	−0.610524
$776$	28.0690	1.00762
$777$	0	0
$778$	2.51120	0.0900310
$779$	12.5640	0.450151
$780$	1.57378	0.0563502
$781$	31.0802	1.11214
$782$	−55.7022	−1.99191
$783$	3.71565	0.132786
$784$	0	0
$785$	−41.9077	−1.49575
$786$	−38.7914	−1.38364
$787$	−8.55312	−0.304886	−0.152443	−	0.988312i	$-0.548714\pi$
$787$	−8.55312	−0.304886	−0.152443	+	0.988312i	$0.548714\pi$
$788$	−0.669874	−0.0238633
$789$	−33.7471	−1.20143
$790$	45.1938	1.60792
$791$	0	0
$792$	−9.83966	−0.349637
$793$	−36.2352	−1.28675
$794$	−37.7611	−1.34009
$795$	−4.55971	−0.161716
$796$	1.06644	0.0377989
$797$	−12.6558	−0.448292	−0.224146	−	0.974556i	$-0.571959\pi$
$797$	−12.6558	−0.448292	−0.224146	+	0.974556i	$0.571959\pi$
$798$	0	0
$799$	−8.98400	−0.317831
$800$	3.24548	0.114745
$801$	13.9228	0.491936
$802$	16.4749	0.581749
$803$	−27.8345	−0.982257
$804$	1.25323	0.0441979
$805$	0	0
$806$	−16.0197	−0.564269
$807$	−20.6586	−0.727217
$808$	−34.8059	−1.22447
$809$	10.3831	0.365049	0.182525	−	0.983201i	$-0.441573\pi$
$809$	10.3831	0.365049	0.182525	+	0.983201i	$0.441573\pi$
$810$	−17.7936	−0.625205
$811$	13.5938	0.477342	0.238671	−	0.971101i	$-0.423288\pi$
$811$	13.5938	0.477342	0.238671	+	0.971101i	$0.423288\pi$
$812$	0	0
$813$	5.70852	0.200207
$814$	5.20929	0.182585
$815$	39.7906	1.39381
$816$	−26.1012	−0.913726
$817$	17.9396	0.627627
$818$	2.21237	0.0773537
$819$	0	0
$820$	1.39371	0.0486703
$821$	21.8912	0.764008	0.382004	−	0.924161i	$-0.375234\pi$
$821$	21.8912	0.764008	0.382004	+	0.924161i	$0.375234\pi$
$822$	−27.4862	−0.958691
$823$	5.31624	0.185312	0.0926562	−	0.995698i	$-0.470464\pi$
$823$	5.31624	0.185312	0.0926562	+	0.995698i	$0.470464\pi$
$824$	−36.2998	−1.26456
$825$	19.7656	0.688151
$826$	0	0
$827$	−46.5790	−1.61971	−0.809855	−	0.586630i	$-0.800454\pi$
$827$	−46.5790	−1.61971	−0.809855	+	0.586630i	$0.800454\pi$
$828$	−1.15019	−0.0399720
$829$	−8.30663	−0.288501	−0.144251	−	0.989541i	$-0.546077\pi$
$829$	−8.30663	−0.288501	−0.144251	+	0.989541i	$0.546077\pi$
$830$	21.2667	0.738177
$831$	27.7887	0.963978
$832$	−24.3899	−0.845568
$833$	0	0
$834$	2.29065	0.0793186
$835$	−46.6260	−1.61356
$836$	−1.10035	−0.0380563
$837$	19.0548	0.658629
$838$	−8.29183	−0.286436
$839$	−22.4382	−0.774654	−0.387327	−	0.921942i	$-0.626602\pi$
$839$	−22.4382	−0.774654	−0.387327	+	0.921942i	$0.626602\pi$
$840$	0	0
$841$	−28.5646	−0.984985
$842$	−25.4329	−0.876477
$843$	−4.93542	−0.169985
$844$	−2.95565	−0.101738
$845$	7.60154	0.261501
$846$	−3.42959	−0.117912
$847$	0	0
$848$	−4.54775	−0.156170
$849$	−24.4485	−0.839070
$850$	−33.8679	−1.16166
$851$	−10.0397	−0.344155
$852$	1.61004	0.0551590
$853$	−39.4484	−1.35069	−0.675344	−	0.737503i	$-0.736005\pi$
$853$	−39.4484	−1.35069	−0.675344	+	0.737503i	$0.736005\pi$
$854$	0	0
$855$	−12.5805	−0.430245
$856$	−4.25882	−0.145563
$857$	−5.88195	−0.200923	−0.100462	−	0.994941i	$-0.532032\pi$
$857$	−5.88195	−0.200923	−0.100462	+	0.994941i	$0.532032\pi$
$858$	18.6299	0.636015
$859$	4.69697	0.160258	0.0801292	−	0.996784i	$-0.474467\pi$
$859$	4.69697	0.160258	0.0801292	+	0.996784i	$0.474467\pi$
$860$	1.99002	0.0678591
$861$	0	0
$862$	36.1974	1.23289
$863$	−24.2095	−0.824100	−0.412050	−	0.911161i	$-0.635187\pi$
$863$	−24.2095	−0.824100	−0.412050	+	0.911161i	$0.635187\pi$
$864$	−3.63856	−0.123786
$865$	−8.95536	−0.304491
$866$	27.0970	0.920792
$867$	6.01576	0.204306
$868$	0	0
$869$	28.9383	0.981664
$870$	−4.05565	−0.137499
$871$	−26.7199	−0.905370
$872$	32.8616	1.11283
$873$	−12.4635	−0.421826
$874$	39.2053	1.32614
$875$	0	0
$876$	−1.44190	−0.0487172
$877$	−20.8421	−0.703787	−0.351894	−	0.936040i	$-0.614462\pi$
$877$	−20.8421	−0.703787	−0.351894	+	0.936040i	$0.614462\pi$
$878$	15.4064	0.519941
$879$	5.46397	0.184295
$880$	39.3393	1.32613
$881$	13.1316	0.442416	0.221208	−	0.975227i	$-0.429000\pi$
$881$	13.1316	0.442416	0.221208	+	0.975227i	$0.429000\pi$
$882$	0	0
$883$	−40.4925	−1.36268	−0.681341	−	0.731966i	$-0.738603\pi$
$883$	−40.4925	−1.36268	−0.681341	+	0.731966i	$0.738603\pi$
$884$	−1.72669	−0.0580749
$885$	−19.9971	−0.672194
$886$	−41.0648	−1.37960
$887$	23.9662	0.804706	0.402353	−	0.915485i	$-0.368192\pi$
$887$	23.9662	0.804706	0.402353	+	0.915485i	$0.368192\pi$
$888$	−4.44918	−0.149305
$889$	0	0
$890$	−52.6433	−1.76461
$891$	−11.3935	−0.381698
$892$	−2.50158	−0.0837590
$893$	6.32328	0.211600
$894$	37.4835	1.25363
$895$	22.6619	0.757504
$896$	0	0
$897$	−35.9047	−1.19882
$898$	21.1959	0.707316
$899$	2.23304	0.0744762
$900$	−0.699339	−0.0233113
$901$	5.00276	0.166666
$902$	16.4983	0.549334
$903$	0	0
$904$	−7.35855	−0.244742
$905$	53.7715	1.78743
$906$	2.32802	0.0773433
$907$	45.7912	1.52047	0.760235	−	0.649648i	$-0.225084\pi$
$907$	45.7912	1.52047	0.760235	+	0.649648i	$0.225084\pi$
$908$	2.05825	0.0683055
$909$	15.4549	0.512607
$910$	0	0
$911$	53.5661	1.77472	0.887362	−	0.461073i	$-0.152535\pi$
$911$	53.5661	1.77472	0.887362	+	0.461073i	$0.152535\pi$
$912$	18.3710	0.608325
$913$	13.6174	0.450670
$914$	−12.8047	−0.423543
$915$	−47.0443	−1.55524
$916$	−1.97204	−0.0651579
$917$	0	0
$918$	37.9698	1.25319
$919$	5.37163	0.177194	0.0885969	−	0.996068i	$-0.471762\pi$
$919$	5.37163	0.177194	0.0885969	+	0.996068i	$0.471762\pi$
$920$	−71.7033	−2.36399
$921$	32.4493	1.06924
$922$	59.3577	1.95484
$923$	−34.3275	−1.12990
$924$	0	0
$925$	−6.10430	−0.200708
$926$	−6.96632	−0.228928
$927$	16.1182	0.529393
$928$	−0.426406	−0.0139974
$929$	−20.0345	−0.657312	−0.328656	−	0.944450i	$-0.606596\pi$
$929$	−20.0345	−0.657312	−0.328656	+	0.944450i	$0.606596\pi$
$930$	−20.7984	−0.682006
$931$	0	0
$932$	−2.03233	−0.0665710
$933$	−23.6368	−0.773835
$934$	26.1201	0.854676
$935$	−43.2752	−1.41525
$936$	10.8677	0.355222
$937$	−42.0847	−1.37485	−0.687424	−	0.726257i	$-0.741258\pi$
$937$	−42.0847	−1.37485	−0.687424	+	0.726257i	$0.741258\pi$
$938$	0	0
$939$	−36.0055	−1.17499
$940$	0.701434	0.0228783
$941$	25.0121	0.815372	0.407686	−	0.913122i	$-0.366336\pi$
$941$	25.0121	0.815372	0.407686	+	0.913122i	$0.366336\pi$
$942$	−25.6988	−0.837313
$943$	−31.7966	−1.03544
$944$	−19.9446	−0.649142
$945$	0	0
$946$	23.5573	0.765913
$947$	−20.1076	−0.653408	−0.326704	−	0.945127i	$-0.605938\pi$
$947$	−20.1076	−0.653408	−0.326704	+	0.945127i	$0.605938\pi$
$948$	1.49908	0.0486878
$949$	30.7426	0.997946
$950$	23.8375	0.773392
$951$	24.1563	0.783322
$952$	0	0
$953$	−19.3053	−0.625360	−0.312680	−	0.949859i	$-0.601227\pi$
$953$	−19.3053	−0.625360	−0.312680	+	0.949859i	$0.601227\pi$
$954$	1.90977	0.0618312
$955$	30.9329	1.00096
$956$	−1.97557	−0.0638945
$957$	−2.59690	−0.0839457
$958$	32.9579	1.06482
$959$	0	0
$960$	−31.6655	−1.02200
$961$	−19.5484	−0.630593
$962$	−5.75355	−0.185502
$963$	1.89105	0.0609382
$964$	−2.56778	−0.0827025
$965$	−28.1715	−0.906873
$966$	0	0
$967$	51.0868	1.64284	0.821420	−	0.570324i	$-0.193182\pi$
$967$	51.0868	1.64284	0.821420	+	0.570324i	$0.193182\pi$
$968$	−6.33781	−0.203705
$969$	−20.2091	−0.649208
$970$	47.1257	1.51312
$971$	−19.4338	−0.623661	−0.311831	−	0.950138i	$-0.600942\pi$
$971$	−19.4338	−0.623661	−0.311831	+	0.950138i	$0.600942\pi$
$972$	1.34175	0.0430365
$973$	0	0
$974$	45.1868	1.44788
$975$	−21.8307	−0.699143
$976$	−46.9209	−1.50190
$977$	0.106157	0.00339628	0.00169814	−	0.999999i	$-0.499459\pi$
$977$	0.106157	0.00339628	0.00169814	+	0.999999i	$0.499459\pi$
$978$	24.4006	0.780246
$979$	−33.7084	−1.07732
$980$	0	0
$981$	−14.5915	−0.465872
$982$	11.6797	0.372713
$983$	−33.5457	−1.06994	−0.534971	−	0.844870i	$-0.679678\pi$
$983$	−33.5457	−1.06994	−0.534971	+	0.844870i	$0.679678\pi$
$984$	−14.0910	−0.449204
$985$	18.5428	0.590822
$986$	4.44972	0.141708
$987$	0	0
$988$	1.21531	0.0386642
$989$	−45.4010	−1.44367
$990$	−16.5201	−0.525042
$991$	−45.5223	−1.44606	−0.723032	−	0.690814i	$-0.757252\pi$
$991$	−45.5223	−1.44606	−0.723032	+	0.690814i	$0.757252\pi$
$992$	−2.18671	−0.0694282
$993$	−13.4272	−0.426100
$994$	0	0
$995$	−29.5200	−0.935848
$996$	0.705417	0.0223520
$997$	55.6680	1.76302	0.881511	−	0.472163i	$-0.156527\pi$
$997$	55.6680	1.76302	0.881511	+	0.472163i	$0.156527\pi$
$998$	16.4672	0.521261
$999$	6.84362	0.216522

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6223.2.a.o.1.11		38
7.3	odd	6		889.2.f.c.128.28	✓	76
7.5	odd	6		889.2.f.c.382.28	yes	76
7.6	odd	2		6223.2.a.p.1.11		38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.f.c.128.28	✓	76	7.3	odd	6
889.2.f.c.382.28	yes	76	7.5	odd	6
6223.2.a.o.1.11		38	1.1	even	1	trivial
6223.2.a.p.1.11		38	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-1.45409 q^{2} +1.33511 q^{3} +0.114369 q^{4} -3.16583 q^{5} -1.94137 q^{6} +2.74187 q^{8} -1.21748 q^{9} +4.60340 q^{10} +2.94763 q^{11} +0.152695 q^{12} -3.25559 q^{13} -4.22674 q^{15} -4.21566 q^{16} +4.63744 q^{17} +1.77032 q^{18} -3.26400 q^{19} -0.362072 q^{20} -4.28611 q^{22} +8.26045 q^{23} +3.66071 q^{24} +5.02251 q^{25} +4.73391 q^{26} -5.63080 q^{27} -0.659879 q^{29} +6.14605 q^{30} -3.38402 q^{31} +0.646188 q^{32} +3.93541 q^{33} -6.74323 q^{34} -0.139241 q^{36} -1.21539 q^{37} +4.74614 q^{38} -4.34658 q^{39} -8.68031 q^{40} -3.84925 q^{41} -5.49619 q^{43} +0.337116 q^{44} +3.85433 q^{45} -12.0114 q^{46} -1.93728 q^{47} -5.62837 q^{48} -7.30316 q^{50} +6.19150 q^{51} -0.372338 q^{52} +1.07878 q^{53} +8.18768 q^{54} -9.33170 q^{55} -4.35781 q^{57} +0.959521 q^{58} +4.73108 q^{59} -0.483407 q^{60} +11.1301 q^{61} +4.92066 q^{62} +7.49170 q^{64} +10.3067 q^{65} -5.72243 q^{66} +8.20739 q^{67} +0.530377 q^{68} +11.0286 q^{69} +10.5442 q^{71} -3.33816 q^{72} -9.44300 q^{73} +1.76728 q^{74} +6.70561 q^{75} -0.373299 q^{76} +6.32031 q^{78} +9.81748 q^{79} +13.3461 q^{80} -3.86533 q^{81} +5.59715 q^{82} +4.61978 q^{83} -14.6814 q^{85} +7.99194 q^{86} -0.881012 q^{87} +8.08202 q^{88} -11.4358 q^{89} -5.60453 q^{90} +0.944736 q^{92} -4.51805 q^{93} +2.81697 q^{94} +10.3333 q^{95} +0.862733 q^{96} +10.2372 q^{97} -3.58867 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 2 q^{2} - 11 q^{3} + 38 q^{4} - 16 q^{5} - 11 q^{6} + 47 q^{9} - 12 q^{10} - 2 q^{11} - 30 q^{12} - 21 q^{13} + 7 q^{15} + 46 q^{16} - 58 q^{17} - 13 q^{18} - 17 q^{19} - 44 q^{20} + 21 q^{22} + 7 q^{23}+ \cdots + 16 q^{99}+O(q^{100}) \) 38 * q - 2 * q^2 - 11 * q^3 + 38 * q^4 - 16 * q^5 - 11 * q^6 + 47 * q^9 - 12 * q^10 - 2 * q^11 - 30 * q^12 - 21 * q^13 + 7 * q^15 + 46 * q^16 - 58 * q^17 - 13 * q^18 - 17 * q^19 - 44 * q^20 + 21 * q^22 + 7 * q^23 - 21 * q^24 + 52 * q^25 - 49 * q^26 - 35 * q^27 - 2 * q^29 + 6 * q^30 - 2 * q^31 - 22 * q^32 - 29 * q^33 - 37 * q^34 + 25 * q^36 + 11 * q^37 - 40 * q^38 + 2 * q^39 + 8 * q^40 - 68 * q^41 - 38 * q^43 + 43 * q^44 - 60 * q^45 - 28 * q^46 - 35 * q^47 - 66 * q^48 - 25 * q^50 - 3 * q^51 - 32 * q^52 - 18 * q^53 + 68 * q^54 - 19 * q^55 - 31 * q^57 + 55 * q^58 - 21 * q^59 - 11 * q^60 - 5 * q^61 - 38 * q^62 + 18 * q^64 - 9 * q^65 - 20 * q^66 + 6 * q^67 - 74 * q^68 - 53 * q^69 - 26 * q^71 + 24 * q^72 - 22 * q^73 - 2 * q^74 - 38 * q^75 - 39 * q^76 - 114 * q^78 - 14 * q^79 - 140 * q^80 + 58 * q^81 + 8 * q^82 - 46 * q^83 - 11 * q^85 + 51 * q^86 - 59 * q^87 - 4 * q^88 - 74 * q^89 - q^90 - 81 * q^92 + 92 * q^93 + 36 * q^94 + 20 * q^95 + 108 * q^96 - 66 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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