LMFDB - Embedded newform 8967.2.a.bl.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8967,2,Mod(1,8967)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8967, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8967.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8967 = 3 \cdot 7^{2} \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8967.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:	$71.6018554925$
Analytic rank:	$0$
Dimension:	$21$
Twist minimal:	no (minimal twist has level 1281)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	8967.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.11936 q^{2} +1.00000 q^{3} -0.747042 q^{4} +4.03875 q^{5} +1.11936 q^{6} -3.07492 q^{8} +1.00000 q^{9} +4.52079 q^{10} -6.13738 q^{11} -0.747042 q^{12} +3.32325 q^{13} +4.03875 q^{15} -1.94784 q^{16} -1.38850 q^{17} +1.11936 q^{18} -3.74887 q^{19} -3.01711 q^{20} -6.86992 q^{22} +1.62146 q^{23} -3.07492 q^{24} +11.3115 q^{25} +3.71990 q^{26} +1.00000 q^{27} +7.09219 q^{29} +4.52079 q^{30} +2.02771 q^{31} +3.96951 q^{32} -6.13738 q^{33} -1.55422 q^{34} -0.747042 q^{36} +7.24458 q^{37} -4.19632 q^{38} +3.32325 q^{39} -12.4188 q^{40} -4.88816 q^{41} +3.81967 q^{43} +4.58488 q^{44} +4.03875 q^{45} +1.81500 q^{46} +1.13923 q^{47} -1.94784 q^{48} +12.6616 q^{50} -1.38850 q^{51} -2.48261 q^{52} -0.678837 q^{53} +1.11936 q^{54} -24.7873 q^{55} -3.74887 q^{57} +7.93869 q^{58} +11.1464 q^{59} -3.01711 q^{60} +1.00000 q^{61} +2.26973 q^{62} +8.33898 q^{64} +13.4218 q^{65} -6.86992 q^{66} +9.92296 q^{67} +1.03727 q^{68} +1.62146 q^{69} -4.32247 q^{71} -3.07492 q^{72} +9.71269 q^{73} +8.10927 q^{74} +11.3115 q^{75} +2.80056 q^{76} +3.71990 q^{78} -8.65346 q^{79} -7.86685 q^{80} +1.00000 q^{81} -5.47160 q^{82} -15.1426 q^{83} -5.60778 q^{85} +4.27557 q^{86} +7.09219 q^{87} +18.8719 q^{88} +13.7938 q^{89} +4.52079 q^{90} -1.21130 q^{92} +2.02771 q^{93} +1.27520 q^{94} -15.1407 q^{95} +3.96951 q^{96} +18.9768 q^{97} -6.13738 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 21 q^{3} + 32 q^{4} + 11 q^{5} + 21 q^{9} + 6 q^{10} + q^{11} + 32 q^{12} + 14 q^{13} + 11 q^{15} + 30 q^{16} + 21 q^{17} + 4 q^{19} + 26 q^{20} + 4 q^{22} + 8 q^{23} + 20 q^{25} + 38 q^{26} + 21 q^{27}+ \cdots + q^{99}+O(q^{100}) $ 21 * q + 21 * q^3 + 32 * q^4 + 11 * q^5 + 21 * q^9 + 6 * q^10 + q^11 + 32 * q^12 + 14 * q^13 + 11 * q^15 + 30 * q^16 + 21 * q^17 + 4 * q^19 + 26 * q^20 + 4 * q^22 + 8 * q^23 + 20 * q^25 + 38 * q^26 + 21 * q^27 + 9 * q^29 + 6 * q^30 + 8 * q^31 - 20 * q^32 + q^33 - 2 * q^34 + 32 * q^36 + 13 * q^37 - 7 * q^38 + 14 * q^39 + 18 * q^40 + 35 * q^41 + 2 * q^43 - q^44 + 11 * q^45 + 7 * q^46 + 42 * q^47 + 30 * q^48 + 19 * q^50 + 21 * q^51 + 11 * q^52 - 8 * q^53 + 24 * q^55 + 4 * q^57 + 6 * q^58 + 13 * q^59 + 26 * q^60 + 21 * q^61 + 28 * q^62 + 104 * q^64 + 14 * q^65 + 4 * q^66 + 14 * q^67 + 97 * q^68 + 8 * q^69 + 16 * q^71 + 46 * q^73 - 63 * q^74 + 20 * q^75 + 40 * q^76 + 38 * q^78 + 18 * q^79 + 32 * q^80 + 21 * q^81 + 46 * q^82 + 52 * q^85 - 47 * q^86 + 9 * q^87 - 41 * q^88 + 56 * q^89 + 6 * q^90 + 16 * q^92 + 8 * q^93 + 27 * q^94 - 20 * q^95 - 20 * q^96 + 62 * q^97 + 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 $

$2$	1.11936	0.791504	0.395752	−	0.918357i	$-0.370484\pi$
$2$	1.11936	0.791504	0.395752	+	0.918357i	$0.370484\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−0.747042	−0.373521
$5$	4.03875	1.80618	0.903091	−	0.429449i	$-0.141292\pi$
$5$	4.03875	1.80618	0.903091	+	0.429449i	$0.141292\pi$
$6$	1.11936	0.456975
$7$	0	0
$7$	0	0
$8$	−3.07492	−1.08715
$9$	1.00000	0.333333
$10$	4.52079	1.42960
$11$	−6.13738	−1.85049	−0.925245	−	0.379369i	$-0.876141\pi$
$11$	−6.13738	−1.85049	−0.925245	+	0.379369i	$0.876141\pi$
$12$	−0.747042	−0.215652
$13$	3.32325	0.921703	0.460852	−	0.887477i	$-0.347544\pi$
$13$	3.32325	0.921703	0.460852	+	0.887477i	$0.347544\pi$
$14$	0	0
$15$	4.03875	1.04280
$16$	−1.94784	−0.486961
$17$	−1.38850	−0.336760	−0.168380	−	0.985722i	$-0.553854\pi$
$17$	−1.38850	−0.336760	−0.168380	+	0.985722i	$0.553854\pi$
$18$	1.11936	0.263835
$19$	−3.74887	−0.860049	−0.430025	−	0.902817i	$-0.641495\pi$
$19$	−3.74887	−0.860049	−0.430025	+	0.902817i	$0.641495\pi$
$20$	−3.01711	−0.674647
$21$	0	0
$22$	−6.86992	−1.46467
$23$	1.62146	0.338099	0.169049	−	0.985608i	$-0.445930\pi$
$23$	1.62146	0.338099	0.169049	+	0.985608i	$0.445930\pi$
$24$	−3.07492	−0.627665
$25$	11.3115	2.26229
$26$	3.71990	0.729532
$27$	1.00000	0.192450
$28$	0	0
$29$	7.09219	1.31699	0.658494	−	0.752586i	$-0.271194\pi$
$29$	7.09219	1.31699	0.658494	+	0.752586i	$0.271194\pi$
$30$	4.52079	0.825380
$31$	2.02771	0.364188	0.182094	−	0.983281i	$-0.441712\pi$
$31$	2.02771	0.364188	0.182094	+	0.983281i	$0.441712\pi$
$32$	3.96951	0.701716
$33$	−6.13738	−1.06838
$34$	−1.55422	−0.266547
$35$	0	0
$36$	−0.747042	−0.124507
$37$	7.24458	1.19100	0.595501	−	0.803355i	$-0.296954\pi$
$37$	7.24458	1.19100	0.595501	+	0.803355i	$0.296954\pi$
$38$	−4.19632	−0.680733
$39$	3.32325	0.532146
$40$	−12.4188	−1.96359
$41$	−4.88816	−0.763403	−0.381702	−	0.924286i	$-0.624662\pi$
$41$	−4.88816	−0.763403	−0.381702	+	0.924286i	$0.624662\pi$
$42$	0	0
$43$	3.81967	0.582494	0.291247	−	0.956648i	$-0.405930\pi$
$43$	3.81967	0.582494	0.291247	+	0.956648i	$0.405930\pi$
$44$	4.58488	0.691197
$45$	4.03875	0.602061
$46$	1.81500	0.267606
$47$	1.13923	0.166173	0.0830865	−	0.996542i	$-0.473522\pi$
$47$	1.13923	0.166173	0.0830865	+	0.996542i	$0.473522\pi$
$48$	−1.94784	−0.281147
$49$	0	0
$50$	12.6616	1.79061
$51$	−1.38850	−0.194428
$52$	−2.48261	−0.344276
$53$	−0.678837	−0.0932454	−0.0466227	−	0.998913i	$-0.514846\pi$
$53$	−0.678837	−0.0932454	−0.0466227	+	0.998913i	$0.514846\pi$
$54$	1.11936	0.152325
$55$	−24.7873	−3.34232
$56$	0	0
$57$	−3.74887	−0.496550
$58$	7.93869	1.04240
$59$	11.1464	1.45113	0.725566	−	0.688152i	$-0.241578\pi$
$59$	11.1464	1.45113	0.725566	+	0.688152i	$0.241578\pi$
$60$	−3.01711	−0.389508
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	2.26973	0.288257
$63$	0	0
$64$	8.33898	1.04237
$65$	13.4218	1.66476
$66$	−6.86992	−0.845628
$67$	9.92296	1.21228	0.606141	−	0.795357i	$-0.292717\pi$
$67$	9.92296	1.21228	0.606141	+	0.795357i	$0.292717\pi$
$68$	1.03727	0.125787
$69$	1.62146	0.195201
$70$	0	0
$71$	−4.32247	−0.512983	−0.256492	−	0.966546i	$-0.582567\pi$
$71$	−4.32247	−0.512983	−0.256492	+	0.966546i	$0.582567\pi$
$72$	−3.07492	−0.362383
$73$	9.71269	1.13678	0.568392	−	0.822758i	$-0.307566\pi$
$73$	9.71269	1.13678	0.568392	+	0.822758i	$0.307566\pi$
$74$	8.10927	0.942683
$75$	11.3115	1.30614
$76$	2.80056	0.321247
$77$	0	0
$78$	3.71990	0.421196
$79$	−8.65346	−0.973591	−0.486795	−	0.873516i	$-0.661834\pi$
$79$	−8.65346	−0.973591	−0.486795	+	0.873516i	$0.661834\pi$
$80$	−7.86685	−0.879540
$81$	1.00000	0.111111
$82$	−5.47160	−0.604237
$83$	−15.1426	−1.66212	−0.831059	−	0.556184i	$-0.812265\pi$
$83$	−15.1426	−1.66212	−0.831059	+	0.556184i	$0.812265\pi$
$84$	0	0
$85$	−5.60778	−0.608250
$86$	4.27557	0.461046
$87$	7.09219	0.760363
$88$	18.8719	2.01176
$89$	13.7938	1.46214	0.731068	−	0.682304i	$-0.239022\pi$
$89$	13.7938	1.46214	0.731068	+	0.682304i	$0.239022\pi$
$90$	4.52079	0.476534
$91$	0	0
$92$	−1.21130	−0.126287
$93$	2.02771	0.210264
$94$	1.27520	0.131527
$95$	−15.1407	−1.55341
$96$	3.96951	0.405136
$97$	18.9768	1.92680	0.963399	−	0.268071i	$-0.0863863\pi$
$97$	18.9768	1.92680	0.963399	+	0.268071i	$0.0863863\pi$
$98$	0	0
$99$	−6.13738	−0.616830
$100$	−8.45014	−0.845014
$101$	−2.65103	−0.263788	−0.131894	−	0.991264i	$-0.542106\pi$
$101$	−2.65103	−0.263788	−0.131894	+	0.991264i	$0.542106\pi$
$102$	−1.55422	−0.153891
$103$	15.8109	1.55790	0.778949	−	0.627087i	$-0.215753\pi$
$103$	15.8109	1.55790	0.778949	+	0.627087i	$0.215753\pi$
$104$	−10.2187	−1.00203
$105$	0	0
$106$	−0.759860	−0.0738041
$107$	2.26187	0.218663	0.109332	−	0.994005i	$-0.465129\pi$
$107$	2.26187	0.218663	0.109332	+	0.994005i	$0.465129\pi$
$108$	−0.747042	−0.0718842
$109$	16.6307	1.59293	0.796467	−	0.604681i	$-0.206700\pi$
$109$	16.6307	1.59293	0.796467	+	0.604681i	$0.206700\pi$
$110$	−27.7458	−2.64546
$111$	7.24458	0.687625
$112$	0	0
$113$	−10.4132	−0.979594	−0.489797	−	0.871837i	$-0.662929\pi$
$113$	−10.4132	−0.979594	−0.489797	+	0.871837i	$0.662929\pi$
$114$	−4.19632	−0.393021
$115$	6.54868	0.610668
$116$	−5.29817	−0.491923
$117$	3.32325	0.307234
$118$	12.4767	1.14858
$119$	0	0
$120$	−12.4188	−1.13368
$121$	26.6675	2.42432
$122$	1.11936	0.101342
$123$	−4.88816	−0.440751
$124$	−1.51479	−0.136032
$125$	25.4904	2.27993
$126$	0	0
$127$	−13.0847	−1.16108	−0.580540	−	0.814232i	$-0.697159\pi$
$127$	−13.0847	−1.16108	−0.580540	+	0.814232i	$0.697159\pi$
$128$	1.39527	0.123326
$129$	3.81967	0.336303
$130$	15.0237	1.31767
$131$	−5.39891	−0.471705	−0.235852	−	0.971789i	$-0.575788\pi$
$131$	−5.39891	−0.471705	−0.235852	+	0.971789i	$0.575788\pi$
$132$	4.58488	0.399063
$133$	0	0
$134$	11.1073	0.959526
$135$	4.03875	0.347600
$136$	4.26951	0.366108
$137$	5.37225	0.458982	0.229491	−	0.973311i	$-0.426294\pi$
$137$	5.37225	0.458982	0.229491	+	0.973311i	$0.426294\pi$
$138$	1.81500	0.154503
$139$	−12.1706	−1.03230	−0.516148	−	0.856499i	$-0.672635\pi$
$139$	−12.1706	−1.03230	−0.516148	+	0.856499i	$0.672635\pi$
$140$	0	0
$141$	1.13923	0.0959401
$142$	−4.83839	−0.406028
$143$	−20.3960	−1.70560
$144$	−1.94784	−0.162320
$145$	28.6436	2.37872
$146$	10.8720	0.899769
$147$	0	0
$148$	−5.41201	−0.444864
$149$	−13.5495	−1.11002	−0.555009	−	0.831844i	$-0.687285\pi$
$149$	−13.5495	−1.11002	−0.555009	+	0.831844i	$0.687285\pi$
$150$	12.6616	1.03381
$151$	13.5508	1.10275	0.551375	−	0.834258i	$-0.314103\pi$
$151$	13.5508	1.10275	0.551375	+	0.834258i	$0.314103\pi$
$152$	11.5275	0.935001
$153$	−1.38850	−0.112253
$154$	0	0
$155$	8.18942	0.657790
$156$	−2.48261	−0.198768
$157$	6.75670	0.539243	0.269622	−	0.962966i	$-0.413101\pi$
$157$	6.75670	0.539243	0.269622	+	0.962966i	$0.413101\pi$
$158$	−9.68631	−0.770601
$159$	−0.678837	−0.0538352
$160$	16.0318	1.26743
$161$	0	0
$162$	1.11936	0.0879449
$163$	18.4035	1.44147	0.720737	−	0.693208i	$-0.243803\pi$
$163$	18.4035	1.44147	0.720737	+	0.693208i	$0.243803\pi$
$164$	3.65166	0.285147
$165$	−24.7873	−1.92969
$166$	−16.9500	−1.31557
$167$	−9.60542	−0.743290	−0.371645	−	0.928375i	$-0.621206\pi$
$167$	−9.60542	−0.743290	−0.371645	+	0.928375i	$0.621206\pi$
$168$	0	0
$169$	−1.95602	−0.150463
$170$	−6.27711	−0.481432
$171$	−3.74887	−0.286683
$172$	−2.85345	−0.217574
$173$	6.22115	0.472985	0.236493	−	0.971633i	$-0.424002\pi$
$173$	6.22115	0.472985	0.236493	+	0.971633i	$0.424002\pi$
$174$	7.93869	0.601831
$175$	0	0
$176$	11.9547	0.901117
$177$	11.1464	0.837812
$178$	15.4401	1.15729
$179$	−3.58780	−0.268165	−0.134082	−	0.990970i	$-0.542809\pi$
$179$	−3.58780	−0.268165	−0.134082	+	0.990970i	$0.542809\pi$
$180$	−3.01711	−0.224882
$181$	−18.8370	−1.40014	−0.700071	−	0.714073i	$-0.746848\pi$
$181$	−18.8370	−1.40014	−0.700071	+	0.714073i	$0.746848\pi$
$182$	0	0
$183$	1.00000	0.0739221
$184$	−4.98587	−0.367563
$185$	29.2590	2.15117
$186$	2.26973	0.166425
$187$	8.52174	0.623171
$188$	−0.851049	−0.0620691
$189$	0	0
$190$	−16.9479	−1.22953
$191$	15.8542	1.14717	0.573585	−	0.819146i	$-0.305552\pi$
$191$	15.8542	1.14717	0.573585	+	0.819146i	$0.305552\pi$
$192$	8.33898	0.601814
$193$	8.27509	0.595654	0.297827	−	0.954620i	$-0.403738\pi$
$193$	8.27509	0.595654	0.297827	+	0.954620i	$0.403738\pi$
$194$	21.2418	1.52507
$195$	13.4218	0.961152
$196$	0	0
$197$	7.03870	0.501487	0.250743	−	0.968054i	$-0.419325\pi$
$197$	7.03870	0.501487	0.250743	+	0.968054i	$0.419325\pi$
$198$	−6.86992	−0.488224
$199$	4.12697	0.292553	0.146277	−	0.989244i	$-0.453271\pi$
$199$	4.12697	0.292553	0.146277	+	0.989244i	$0.453271\pi$
$200$	−34.7818	−2.45945
$201$	9.92296	0.699911
$202$	−2.96745	−0.208789
$203$	0	0
$204$	1.03727	0.0726231
$205$	−19.7421	−1.37884
$206$	17.6981	1.23308
$207$	1.62146	0.112700
$208$	−6.47317	−0.448834
$209$	23.0082	1.59151
$210$	0	0
$211$	17.2211	1.18555	0.592774	−	0.805369i	$-0.298033\pi$
$211$	17.2211	1.18555	0.592774	+	0.805369i	$0.298033\pi$
$212$	0.507120	0.0348291
$213$	−4.32247	−0.296171
$214$	2.53184	0.173073
$215$	15.4267	1.05209
$216$	−3.07492	−0.209222
$217$	0	0
$218$	18.6157	1.26081
$219$	9.71269	0.656322
$220$	18.5172	1.24843
$221$	−4.61432	−0.310393
$222$	8.10927	0.544258
$223$	−20.0385	−1.34188	−0.670938	−	0.741513i	$-0.734109\pi$
$223$	−20.0385	−1.34188	−0.670938	+	0.741513i	$0.734109\pi$
$224$	0	0
$225$	11.3115	0.754098
$226$	−11.6561	−0.775353
$227$	−15.2445	−1.01181	−0.505907	−	0.862588i	$-0.668842\pi$
$227$	−15.2445	−1.01181	−0.505907	+	0.862588i	$0.668842\pi$
$228$	2.80056	0.185472
$229$	−12.6049	−0.832956	−0.416478	−	0.909146i	$-0.636736\pi$
$229$	−12.6049	−0.832956	−0.416478	+	0.909146i	$0.636736\pi$
$230$	7.33030	0.483346
$231$	0	0
$232$	−21.8079	−1.43176
$233$	−16.5702	−1.08555	−0.542774	−	0.839879i	$-0.682626\pi$
$233$	−16.5702	−1.08555	−0.542774	+	0.839879i	$0.682626\pi$
$234$	3.71990	0.243177
$235$	4.60104	0.300139
$236$	−8.32680	−0.542029
$237$	−8.65346	−0.562103
$238$	0	0
$239$	−3.05639	−0.197701	−0.0988506	−	0.995102i	$-0.531517\pi$
$239$	−3.05639	−0.197701	−0.0988506	+	0.995102i	$0.531517\pi$
$240$	−7.86685	−0.507803
$241$	−28.8097	−1.85580	−0.927899	−	0.372832i	$-0.878387\pi$
$241$	−28.8097	−1.85580	−0.927899	+	0.372832i	$0.878387\pi$
$242$	29.8504	1.91886
$243$	1.00000	0.0641500
$244$	−0.747042	−0.0478245
$245$	0	0
$246$	−5.47160	−0.348856
$247$	−12.4584	−0.792710
$248$	−6.23506	−0.395927
$249$	−15.1426	−0.959624
$250$	28.5328	1.80457
$251$	17.3307	1.09390	0.546952	−	0.837164i	$-0.315788\pi$
$251$	17.3307	1.09390	0.546952	+	0.837164i	$0.315788\pi$
$252$	0	0
$253$	−9.95154	−0.625648
$254$	−14.6465	−0.919000
$255$	−5.60778	−0.351173
$256$	−15.1161	−0.944759
$257$	14.9887	0.934967	0.467483	−	0.884002i	$-0.345161\pi$
$257$	14.9887	0.934967	0.467483	+	0.884002i	$0.345161\pi$
$258$	4.27557	0.266185
$259$	0	0
$260$	−10.0266	−0.621824
$261$	7.09219	0.438996
$262$	−6.04330	−0.373356
$263$	8.40696	0.518395	0.259198	−	0.965824i	$-0.416542\pi$
$263$	8.40696	0.518395	0.259198	+	0.965824i	$0.416542\pi$
$264$	18.8719	1.16149
$265$	−2.74165	−0.168418
$266$	0	0
$267$	13.7938	0.844165
$268$	−7.41286	−0.452813
$269$	24.7936	1.51169	0.755847	−	0.654749i	$-0.227226\pi$
$269$	24.7936	1.51169	0.755847	+	0.654749i	$0.227226\pi$
$270$	4.52079	0.275127
$271$	−4.91574	−0.298610	−0.149305	−	0.988791i	$-0.547704\pi$
$271$	−4.91574	−0.298610	−0.149305	+	0.988791i	$0.547704\pi$
$272$	2.70457	0.163989
$273$	0	0
$274$	6.01346	0.363286
$275$	−69.4228	−4.18635
$276$	−1.21130	−0.0729118
$277$	7.86600	0.472622	0.236311	−	0.971677i	$-0.424062\pi$
$277$	7.86600	0.472622	0.236311	+	0.971677i	$0.424062\pi$
$278$	−13.6232	−0.817067
$279$	2.02771	0.121396
$280$	0	0
$281$	−25.6585	−1.53066	−0.765329	−	0.643639i	$-0.777424\pi$
$281$	−25.6585	−1.53066	−0.765329	+	0.643639i	$0.777424\pi$
$282$	1.27520	0.0759370
$283$	−18.1005	−1.07596	−0.537982	−	0.842956i	$-0.680813\pi$
$283$	−18.1005	−1.07596	−0.537982	+	0.842956i	$0.680813\pi$
$284$	3.22907	0.191610
$285$	−15.1407	−0.896859
$286$	−22.8304	−1.34999
$287$	0	0
$288$	3.96951	0.233905
$289$	−15.0721	−0.886593
$290$	32.0624	1.88277
$291$	18.9768	1.11244
$292$	−7.25578	−0.424613
$293$	−18.3637	−1.07282	−0.536410	−	0.843958i	$-0.680220\pi$
$293$	−18.3637	−1.07282	−0.536410	+	0.843958i	$0.680220\pi$
$294$	0	0
$295$	45.0173	2.62101
$296$	−22.2765	−1.29480
$297$	−6.13738	−0.356127
$298$	−15.1667	−0.878584
$299$	5.38853	0.311627
$300$	−8.45014	−0.487869
$301$	0	0
$302$	15.1682	0.872831
$303$	−2.65103	−0.152298
$304$	7.30221	0.418811
$305$	4.03875	0.231258
$306$	−1.55422	−0.0888490
$307$	8.65273	0.493837	0.246919	−	0.969036i	$-0.420582\pi$
$307$	8.65273	0.493837	0.246919	+	0.969036i	$0.420582\pi$
$308$	0	0
$309$	15.8109	0.899453
$310$	9.16688	0.520644
$311$	−13.3478	−0.756885	−0.378443	−	0.925625i	$-0.623540\pi$
$311$	−13.3478	−0.756885	−0.378443	+	0.925625i	$0.623540\pi$
$312$	−10.2187	−0.578521
$313$	−14.4104	−0.814525	−0.407263	−	0.913311i	$-0.633517\pi$
$313$	−14.4104	−0.814525	−0.407263	+	0.913311i	$0.633517\pi$
$314$	7.56315	0.426813
$315$	0	0
$316$	6.46450	0.363657
$317$	−26.5176	−1.48938	−0.744689	−	0.667412i	$-0.767402\pi$
$317$	−26.5176	−1.48938	−0.744689	+	0.667412i	$0.767402\pi$
$318$	−0.759860	−0.0426108
$319$	−43.5275	−2.43707
$320$	33.6790	1.88271
$321$	2.26187	0.126245
$322$	0	0
$323$	5.20529	0.289630
$324$	−0.747042	−0.0415023
$325$	37.5908	2.08516
$326$	20.6001	1.14093
$327$	16.6307	0.919681
$328$	15.0307	0.829932
$329$	0	0
$330$	−27.7458	−1.52736
$331$	−11.2768	−0.619830	−0.309915	−	0.950764i	$-0.600301\pi$
$331$	−11.2768	−0.619830	−0.309915	+	0.950764i	$0.600301\pi$
$332$	11.3122	0.620836
$333$	7.24458	0.397001
$334$	−10.7519	−0.588317
$335$	40.0763	2.18960
$336$	0	0
$337$	−1.94483	−0.105942	−0.0529709	−	0.998596i	$-0.516869\pi$
$337$	−1.94483	−0.105942	−0.0529709	+	0.998596i	$0.516869\pi$
$338$	−2.18948	−0.119092
$339$	−10.4132	−0.565569
$340$	4.18925	0.227194
$341$	−12.4449	−0.673927
$342$	−4.19632	−0.226911
$343$	0	0
$344$	−11.7452	−0.633257
$345$	6.54868	0.352569
$346$	6.96368	0.374370
$347$	2.48630	0.133471	0.0667357	−	0.997771i	$-0.478742\pi$
$347$	2.48630	0.133471	0.0667357	+	0.997771i	$0.478742\pi$
$348$	−5.29817	−0.284012
$349$	−1.23136	−0.0659131	−0.0329565	−	0.999457i	$-0.510492\pi$
$349$	−1.23136	−0.0659131	−0.0329565	+	0.999457i	$0.510492\pi$
$350$	0	0
$351$	3.32325	0.177382
$352$	−24.3624	−1.29852
$353$	−10.4378	−0.555546	−0.277773	−	0.960647i	$-0.589596\pi$
$353$	−10.4378	−0.555546	−0.277773	+	0.960647i	$0.589596\pi$
$354$	12.4767	0.663132
$355$	−17.4574	−0.926541
$356$	−10.3045	−0.546139
$357$	0	0
$358$	−4.01603	−0.212254
$359$	−4.26160	−0.224919	−0.112459	−	0.993656i	$-0.535873\pi$
$359$	−4.26160	−0.224919	−0.112459	+	0.993656i	$0.535873\pi$
$360$	−12.4188	−0.654529
$361$	−4.94599	−0.260315
$362$	−21.0853	−1.10822
$363$	26.6675	1.39968
$364$	0	0
$365$	39.2271	2.05324
$366$	1.11936	0.0585097
$367$	8.23430	0.429827	0.214913	−	0.976633i	$-0.431053\pi$
$367$	8.23430	0.429827	0.214913	+	0.976633i	$0.431053\pi$
$368$	−3.15836	−0.164641
$369$	−4.88816	−0.254468
$370$	32.7513	1.70266
$371$	0	0
$372$	−1.51479	−0.0785381
$373$	9.76653	0.505692	0.252846	−	0.967507i	$-0.418633\pi$
$373$	9.76653	0.505692	0.252846	+	0.967507i	$0.418633\pi$
$374$	9.53886	0.493242
$375$	25.4904	1.31632
$376$	−3.50302	−0.180655
$377$	23.5691	1.21387
$378$	0	0
$379$	−29.0210	−1.49071	−0.745353	−	0.666670i	$-0.767719\pi$
$379$	−29.0210	−1.49071	−0.745353	+	0.666670i	$0.767719\pi$
$380$	11.3108	0.580230
$381$	−13.0847	−0.670350
$382$	17.7465	0.907989
$383$	1.19631	0.0611284	0.0305642	−	0.999533i	$-0.490270\pi$
$383$	1.19631	0.0611284	0.0305642	+	0.999533i	$0.490270\pi$
$384$	1.39527	0.0712023
$385$	0	0
$386$	9.26278	0.471463
$387$	3.81967	0.194165
$388$	−14.1764	−0.719700
$389$	15.6753	0.794767	0.397384	−	0.917653i	$-0.369918\pi$
$389$	15.6753	0.794767	0.397384	+	0.917653i	$0.369918\pi$
$390$	15.0237	0.760756
$391$	−2.25140	−0.113858
$392$	0	0
$393$	−5.39891	−0.272339
$394$	7.87881	0.396929
$395$	−34.9491	−1.75848
$396$	4.58488	0.230399
$397$	−12.7490	−0.639854	−0.319927	−	0.947442i	$-0.603658\pi$
$397$	−12.7490	−0.639854	−0.319927	+	0.947442i	$0.603658\pi$
$398$	4.61955	0.231557
$399$	0	0
$400$	−22.0330	−1.10165
$401$	−27.2192	−1.35926	−0.679630	−	0.733555i	$-0.737860\pi$
$401$	−27.2192	−1.35926	−0.679630	+	0.733555i	$0.737860\pi$
$402$	11.1073	0.553983
$403$	6.73860	0.335674
$404$	1.98043	0.0985303
$405$	4.03875	0.200687
$406$	0	0
$407$	−44.4628	−2.20394
$408$	4.26951	0.211372
$409$	19.1477	0.946794	0.473397	−	0.880849i	$-0.343028\pi$
$409$	19.1477	0.946794	0.473397	+	0.880849i	$0.343028\pi$
$410$	−22.0984	−1.09136
$411$	5.37225	0.264993
$412$	−11.8114	−0.581908
$413$	0	0
$414$	1.81500	0.0892021
$415$	−61.1572	−3.00209
$416$	13.1917	0.646774
$417$	−12.1706	−0.595997
$418$	25.7544	1.25969
$419$	13.7980	0.674075	0.337037	−	0.941491i	$-0.390575\pi$
$419$	13.7980	0.674075	0.337037	+	0.941491i	$0.390575\pi$
$420$	0	0
$421$	5.90717	0.287898	0.143949	−	0.989585i	$-0.454020\pi$
$421$	5.90717	0.287898	0.143949	+	0.989585i	$0.454020\pi$
$422$	19.2765	0.938367
$423$	1.13923	0.0553910
$424$	2.08737	0.101372
$425$	−15.7059	−0.761850
$426$	−4.83839	−0.234421
$427$	0	0
$428$	−1.68971	−0.0816753
$429$	−20.3960	−0.984730
$430$	17.2679	0.832733
$431$	7.55215	0.363774	0.181887	−	0.983319i	$-0.441779\pi$
$431$	7.55215	0.363774	0.181887	+	0.983319i	$0.441779\pi$
$432$	−1.94784	−0.0937157
$433$	−22.4792	−1.08028	−0.540140	−	0.841575i	$-0.681629\pi$
$433$	−22.4792	−1.08028	−0.540140	+	0.841575i	$0.681629\pi$
$434$	0	0
$435$	28.6436	1.37335
$436$	−12.4239	−0.594995
$437$	−6.07865	−0.290781
$438$	10.8720	0.519482
$439$	15.6329	0.746115	0.373058	−	0.927808i	$-0.378309\pi$
$439$	15.6329	0.746115	0.373058	+	0.927808i	$0.378309\pi$
$440$	76.2190	3.63360
$441$	0	0
$442$	−5.16507	−0.245677
$443$	−22.1604	−1.05287	−0.526437	−	0.850214i	$-0.676472\pi$
$443$	−22.1604	−1.05287	−0.526437	+	0.850214i	$0.676472\pi$
$444$	−5.41201	−0.256843
$445$	55.7095	2.64088
$446$	−22.4302	−1.06210
$447$	−13.5495	−0.640869
$448$	0	0
$449$	37.9813	1.79245	0.896225	−	0.443600i	$-0.146299\pi$
$449$	37.9813	1.79245	0.896225	+	0.443600i	$0.146299\pi$
$450$	12.6616	0.596871
$451$	30.0005	1.41267
$452$	7.77912	0.365899
$453$	13.5508	0.636673
$454$	−17.0640	−0.800855
$455$	0	0
$456$	11.5275	0.539823
$457$	41.2012	1.92731	0.963655	−	0.267151i	$-0.0860822\pi$
$457$	41.2012	1.92731	0.963655	+	0.267151i	$0.0860822\pi$
$458$	−14.1094	−0.659288
$459$	−1.38850	−0.0648095
$460$	−4.89214	−0.228097
$461$	−4.95495	−0.230775	−0.115387	−	0.993321i	$-0.536811\pi$
$461$	−4.95495	−0.230775	−0.115387	+	0.993321i	$0.536811\pi$
$462$	0	0
$463$	33.6873	1.56558	0.782792	−	0.622284i	$-0.213795\pi$
$463$	33.6873	1.56558	0.782792	+	0.622284i	$0.213795\pi$
$464$	−13.8145	−0.641322
$465$	8.18942	0.379775
$466$	−18.5479	−0.859216
$467$	20.7522	0.960299	0.480149	−	0.877187i	$-0.340582\pi$
$467$	20.7522	0.960299	0.480149	+	0.877187i	$0.340582\pi$
$468$	−2.48261	−0.114759
$469$	0	0
$470$	5.15020	0.237561
$471$	6.75670	0.311332
$472$	−34.2741	−1.57760
$473$	−23.4428	−1.07790
$474$	−9.68631	−0.444907
$475$	−42.4052	−1.94568
$476$	0	0
$477$	−0.678837	−0.0310818
$478$	−3.42119	−0.156481
$479$	8.31585	0.379961	0.189981	−	0.981788i	$-0.439158\pi$
$479$	8.31585	0.379961	0.189981	+	0.981788i	$0.439158\pi$
$480$	16.0318	0.731749
$481$	24.0755	1.09775
$482$	−32.2483	−1.46887
$483$	0	0
$484$	−19.9217	−0.905533
$485$	76.6423	3.48015
$486$	1.11936	0.0507750
$487$	−13.0439	−0.591073	−0.295537	−	0.955331i	$-0.595498\pi$
$487$	−13.0439	−0.591073	−0.295537	+	0.955331i	$0.595498\pi$
$488$	−3.07492	−0.139195
$489$	18.4035	0.832236
$490$	0	0
$491$	−3.79658	−0.171337	−0.0856686	−	0.996324i	$-0.527303\pi$
$491$	−3.79658	−0.171337	−0.0856686	+	0.996324i	$0.527303\pi$
$492$	3.65166	0.164630
$493$	−9.84749	−0.443509
$494$	−13.9454	−0.627434
$495$	−24.7873	−1.11411
$496$	−3.94967	−0.177346
$497$	0	0
$498$	−16.9500	−0.759547
$499$	23.6318	1.05790	0.528951	−	0.848652i	$-0.322585\pi$
$499$	23.6318	1.05790	0.528951	+	0.848652i	$0.322585\pi$
$500$	−19.0424	−0.851602
$501$	−9.60542	−0.429138
$502$	19.3992	0.865829
$503$	−15.4986	−0.691050	−0.345525	−	0.938410i	$-0.612299\pi$
$503$	−15.4986	−0.691050	−0.345525	+	0.938410i	$0.612299\pi$
$504$	0	0
$505$	−10.7068	−0.476449
$506$	−11.1393	−0.495203
$507$	−1.95602	−0.0868699
$508$	9.77483	0.433688
$509$	19.9556	0.884515	0.442258	−	0.896888i	$-0.354178\pi$
$509$	19.9556	0.884515	0.442258	+	0.896888i	$0.354178\pi$
$510$	−6.27711	−0.277955
$511$	0	0
$512$	−19.7109	−0.871107
$513$	−3.74887	−0.165517
$514$	16.7776	0.740030
$515$	63.8564	2.81385
$516$	−2.85345	−0.125616
$517$	−6.99186	−0.307502
$518$	0	0
$519$	6.22115	0.273078
$520$	−41.2708	−1.80984
$521$	19.9862	0.875611	0.437806	−	0.899070i	$-0.355756\pi$
$521$	19.9862	0.875611	0.437806	+	0.899070i	$0.355756\pi$
$522$	7.93869	0.347467
$523$	35.5672	1.55524	0.777622	−	0.628733i	$-0.216426\pi$
$523$	35.5672	1.55524	0.777622	+	0.628733i	$0.216426\pi$
$524$	4.03321	0.176192
$525$	0	0
$526$	9.41038	0.410312
$527$	−2.81548	−0.122644
$528$	11.9547	0.520260
$529$	−20.3709	−0.885689
$530$	−3.06888	−0.133304
$531$	11.1464	0.483711
$532$	0	0
$533$	−16.2446	−0.703631
$534$	15.4401	0.668160
$535$	9.13511	0.394945
$536$	−30.5123	−1.31793
$537$	−3.58780	−0.154825
$538$	27.7529	1.19651
$539$	0	0
$540$	−3.01711	−0.129836
$541$	−0.636558	−0.0273677	−0.0136839	−	0.999906i	$-0.504356\pi$
$541$	−0.636558	−0.0273677	−0.0136839	+	0.999906i	$0.504356\pi$
$542$	−5.50246	−0.236351
$543$	−18.8370	−0.808372
$544$	−5.51164	−0.236310
$545$	67.1673	2.87713
$546$	0	0
$547$	−23.9282	−1.02309	−0.511547	−	0.859255i	$-0.670927\pi$
$547$	−23.9282	−1.02309	−0.511547	+	0.859255i	$0.670927\pi$
$548$	−4.01329	−0.171439
$549$	1.00000	0.0426790
$550$	−77.7088	−3.31352
$551$	−26.5877	−1.13267
$552$	−4.98587	−0.212213
$553$	0	0
$554$	8.80485	0.374082
$555$	29.2590	1.24198
$556$	9.09195	0.385584
$557$	19.8641	0.841668	0.420834	−	0.907138i	$-0.361737\pi$
$557$	19.8641	0.841668	0.420834	+	0.907138i	$0.361737\pi$
$558$	2.26973	0.0960855
$559$	12.6937	0.536886
$560$	0	0
$561$	8.52174	0.359788
$562$	−28.7210	−1.21152
$563$	−37.1065	−1.56385	−0.781926	−	0.623371i	$-0.785763\pi$
$563$	−37.1065	−1.56385	−0.781926	+	0.623371i	$0.785763\pi$
$564$	−0.851049	−0.0358356
$565$	−42.0564	−1.76932
$566$	−20.2609	−0.851630
$567$	0	0
$568$	13.2913	0.557689
$569$	−16.9384	−0.710096	−0.355048	−	0.934848i	$-0.615535\pi$
$569$	−16.9384	−0.710096	−0.355048	+	0.934848i	$0.615535\pi$
$570$	−16.9479	−0.709868
$571$	−30.1546	−1.26193	−0.630966	−	0.775811i	$-0.717341\pi$
$571$	−30.1546	−1.26193	−0.630966	+	0.775811i	$0.717341\pi$
$572$	15.2367	0.637079
$573$	15.8542	0.662318
$574$	0	0
$575$	18.3411	0.764878
$576$	8.33898	0.347457
$577$	6.42792	0.267598	0.133799	−	0.991009i	$-0.457282\pi$
$577$	6.42792	0.267598	0.133799	+	0.991009i	$0.457282\pi$
$578$	−16.8710	−0.701742
$579$	8.27509	0.343901
$580$	−21.3980	−0.888502
$581$	0	0
$582$	21.2418	0.880499
$583$	4.16628	0.172550
$584$	−29.8657	−1.23585
$585$	13.4218	0.554921
$586$	−20.5555	−0.849142
$587$	−9.57038	−0.395012	−0.197506	−	0.980302i	$-0.563284\pi$
$587$	−9.57038	−0.395012	−0.197506	+	0.980302i	$0.563284\pi$
$588$	0	0
$589$	−7.60164	−0.313220
$590$	50.3904	2.07454
$591$	7.03870	0.289534
$592$	−14.1113	−0.579972
$593$	15.4903	0.636111	0.318055	−	0.948072i	$-0.396970\pi$
$593$	15.4903	0.636111	0.318055	+	0.948072i	$0.396970\pi$
$594$	−6.86992	−0.281876
$595$	0	0
$596$	10.1220	0.414615
$597$	4.12697	0.168906
$598$	6.03168	0.246654
$599$	−40.2855	−1.64602	−0.823010	−	0.568027i	$-0.807707\pi$
$599$	−40.2855	−1.64602	−0.823010	+	0.568027i	$0.807707\pi$
$600$	−34.7818	−1.41996
$601$	−28.0663	−1.14485	−0.572424	−	0.819958i	$-0.693997\pi$
$601$	−28.0663	−1.14485	−0.572424	+	0.819958i	$0.693997\pi$
$602$	0	0
$603$	9.92296	0.404094
$604$	−10.1230	−0.411900
$605$	107.703	4.37875
$606$	−2.96745	−0.120544
$607$	−30.8453	−1.25197	−0.625987	−	0.779834i	$-0.715304\pi$
$607$	−30.8453	−1.25197	−0.625987	+	0.779834i	$0.715304\pi$
$608$	−14.8812	−0.603510
$609$	0	0
$610$	4.52079	0.183042
$611$	3.78593	0.153162
$612$	1.03727	0.0419290
$613$	11.9284	0.481783	0.240891	−	0.970552i	$-0.422560\pi$
$613$	11.9284	0.481783	0.240891	+	0.970552i	$0.422560\pi$
$614$	9.68548	0.390874
$615$	−19.7421	−0.796076
$616$	0	0
$617$	−22.3208	−0.898601	−0.449301	−	0.893381i	$-0.648327\pi$
$617$	−22.3208	−0.898601	−0.449301	+	0.893381i	$0.648327\pi$
$618$	17.6981	0.711921
$619$	−30.4804	−1.22511	−0.612555	−	0.790428i	$-0.709858\pi$
$619$	−30.4804	−1.22511	−0.612555	+	0.790428i	$0.709858\pi$
$620$	−6.11784	−0.245699
$621$	1.62146	0.0650671
$622$	−14.9410	−0.599078
$623$	0	0
$624$	−6.47317	−0.259134
$625$	46.3919	1.85568
$626$	−16.1304	−0.644700
$627$	23.0082	0.918861
$628$	−5.04754	−0.201419
$629$	−10.0591	−0.401082
$630$	0	0
$631$	8.02405	0.319433	0.159716	−	0.987163i	$-0.448942\pi$
$631$	8.02405	0.319433	0.159716	+	0.987163i	$0.448942\pi$
$632$	26.6087	1.05844
$633$	17.2211	0.684477
$634$	−29.6827	−1.17885
$635$	−52.8458	−2.09712
$636$	0.507120	0.0201086
$637$	0	0
$638$	−48.7228	−1.92895
$639$	−4.32247	−0.170994
$640$	5.63516	0.222749
$641$	28.1678	1.11256	0.556281	−	0.830995i	$-0.312228\pi$
$641$	28.1678	1.11256	0.556281	+	0.830995i	$0.312228\pi$
$642$	2.53184	0.0999236
$643$	19.8552	0.783014	0.391507	−	0.920175i	$-0.371954\pi$
$643$	19.8552	0.783014	0.391507	+	0.920175i	$0.371954\pi$
$644$	0	0
$645$	15.4267	0.607424
$646$	5.82657	0.229243
$647$	16.0991	0.632920	0.316460	−	0.948606i	$-0.397506\pi$
$647$	16.0991	0.632920	0.316460	+	0.948606i	$0.397506\pi$
$648$	−3.07492	−0.120794
$649$	−68.4095	−2.68531
$650$	42.0775	1.65042
$651$	0	0
$652$	−13.7482	−0.538421
$653$	0.761519	0.0298006	0.0149003	−	0.999889i	$-0.495257\pi$
$653$	0.761519	0.0298006	0.0149003	+	0.999889i	$0.495257\pi$
$654$	18.6157	0.727932
$655$	−21.8048	−0.851985
$656$	9.52138	0.371748
$657$	9.71269	0.378928
$658$	0	0
$659$	−11.8946	−0.463347	−0.231673	−	0.972794i	$-0.574420\pi$
$659$	−11.8946	−0.463347	−0.231673	+	0.972794i	$0.574420\pi$
$660$	18.5172	0.720780
$661$	−12.6464	−0.491889	−0.245945	−	0.969284i	$-0.579098\pi$
$661$	−12.6464	−0.491889	−0.245945	+	0.969284i	$0.579098\pi$
$662$	−12.6228	−0.490598
$663$	−4.61432	−0.179205
$664$	46.5623	1.80697
$665$	0	0
$666$	8.10927	0.314228
$667$	11.4997	0.445272
$668$	7.17565	0.277634
$669$	−20.0385	−0.774733
$670$	44.8596	1.73308
$671$	−6.13738	−0.236931
$672$	0	0
$673$	−29.7186	−1.14557	−0.572784	−	0.819706i	$-0.694137\pi$
$673$	−29.7186	−1.14557	−0.572784	+	0.819706i	$0.694137\pi$
$674$	−2.17696	−0.0838535
$675$	11.3115	0.435378
$676$	1.46123	0.0562011
$677$	44.9283	1.72673	0.863367	−	0.504577i	$-0.168351\pi$
$677$	44.9283	1.72673	0.863367	+	0.504577i	$0.168351\pi$
$678$	−11.6561	−0.447650
$679$	0	0
$680$	17.2435	0.661257
$681$	−15.2445	−0.584171
$682$	−13.9302	−0.533416
$683$	−11.9795	−0.458384	−0.229192	−	0.973381i	$-0.573608\pi$
$683$	−11.9795	−0.458384	−0.229192	+	0.973381i	$0.573608\pi$
$684$	2.80056	0.107082
$685$	21.6971	0.829005
$686$	0	0
$687$	−12.6049	−0.480908
$688$	−7.44012	−0.283652
$689$	−2.25594	−0.0859446
$690$	7.33030	0.279060
$691$	1.48309	0.0564194	0.0282097	−	0.999602i	$-0.491019\pi$
$691$	1.48309	0.0564194	0.0282097	+	0.999602i	$0.491019\pi$
$692$	−4.64746	−0.176670
$693$	0	0
$694$	2.78305	0.105643
$695$	−49.1539	−1.86451
$696$	−21.8079	−0.826627
$697$	6.78720	0.257084
$698$	−1.37833	−0.0521705
$699$	−16.5702	−0.626742
$700$	0	0
$701$	10.3093	0.389376	0.194688	−	0.980865i	$-0.437631\pi$
$701$	10.3093	0.389376	0.194688	+	0.980865i	$0.437631\pi$
$702$	3.71990	0.140399
$703$	−27.1590	−1.02432
$704$	−51.1795	−1.92890
$705$	4.60104	0.173285
$706$	−11.6836	−0.439717
$707$	0	0
$708$	−8.32680	−0.312940
$709$	−4.06701	−0.152740	−0.0763699	−	0.997080i	$-0.524333\pi$
$709$	−4.06701	−0.152740	−0.0763699	+	0.997080i	$0.524333\pi$
$710$	−19.5410	−0.733361
$711$	−8.65346	−0.324530
$712$	−42.4147	−1.58956
$713$	3.28787	0.123132
$714$	0	0
$715$	−82.3744	−3.08063
$716$	2.68024	0.100165
$717$	−3.05639	−0.114143
$718$	−4.77025	−0.178024
$719$	−3.04710	−0.113638	−0.0568189	−	0.998385i	$-0.518096\pi$
$719$	−3.04710	−0.113638	−0.0568189	+	0.998385i	$0.518096\pi$
$720$	−7.86685	−0.293180
$721$	0	0
$722$	−5.53632	−0.206040
$723$	−28.8097	−1.07145
$724$	14.0720	0.522982
$725$	80.2231	2.97941
$726$	29.8504	1.10785
$727$	18.6606	0.692082	0.346041	−	0.938219i	$-0.387526\pi$
$727$	18.6606	0.692082	0.346041	+	0.938219i	$0.387526\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	43.9091	1.62515
$731$	−5.30359	−0.196161
$732$	−0.747042	−0.0276115
$733$	27.1448	1.00262	0.501308	−	0.865269i	$-0.332852\pi$
$733$	27.1448	1.00262	0.501308	+	0.865269i	$0.332852\pi$
$734$	9.21711	0.340210
$735$	0	0
$736$	6.43641	0.237249
$737$	−60.9010	−2.24332
$738$	−5.47160	−0.201412
$739$	2.53735	0.0933379	0.0466689	−	0.998910i	$-0.485139\pi$
$739$	2.53735	0.0933379	0.0466689	+	0.998910i	$0.485139\pi$
$740$	−21.8577	−0.803506
$741$	−12.4584	−0.457672
$742$	0	0
$743$	−27.8242	−1.02077	−0.510385	−	0.859946i	$-0.670497\pi$
$743$	−27.8242	−1.02077	−0.510385	+	0.859946i	$0.670497\pi$
$744$	−6.23506	−0.228588
$745$	−54.7230	−2.00489
$746$	10.9322	0.400257
$747$	−15.1426	−0.554039
$748$	−6.36609	−0.232767
$749$	0	0
$750$	28.5328	1.04187
$751$	35.9268	1.31099	0.655494	−	0.755200i	$-0.272460\pi$
$751$	35.9268	1.31099	0.655494	+	0.755200i	$0.272460\pi$
$752$	−2.21903	−0.0809198
$753$	17.3307	0.631565
$754$	26.3822	0.960785
$755$	54.7283	1.99177
$756$	0	0
$757$	−53.7172	−1.95238	−0.976192	−	0.216908i	$-0.930403\pi$
$757$	−53.7172	−1.95238	−0.976192	+	0.216908i	$0.930403\pi$
$758$	−32.4848	−1.17990
$759$	−9.95154	−0.361218
$760$	46.5565	1.68878
$761$	11.0521	0.400637	0.200318	−	0.979731i	$-0.435802\pi$
$761$	11.0521	0.400637	0.200318	+	0.979731i	$0.435802\pi$
$762$	−14.6465	−0.530585
$763$	0	0
$764$	−11.8437	−0.428492
$765$	−5.60778	−0.202750
$766$	1.33909	0.0483834
$767$	37.0421	1.33751
$768$	−15.1161	−0.545457
$769$	42.9706	1.54956	0.774781	−	0.632230i	$-0.217860\pi$
$769$	42.9706	1.54956	0.774781	+	0.632230i	$0.217860\pi$
$770$	0	0
$771$	14.9887	0.539803
$772$	−6.18184	−0.222489
$773$	38.7386	1.39333	0.696664	−	0.717397i	$-0.254667\pi$
$773$	38.7386	1.39333	0.696664	+	0.717397i	$0.254667\pi$
$774$	4.27557	0.153682
$775$	22.9364	0.823901
$776$	−58.3520	−2.09471
$777$	0	0
$778$	17.5462	0.629062
$779$	18.3251	0.656564
$780$	−10.0266	−0.359010
$781$	26.5287	0.949271
$782$	−2.52011	−0.0901191
$783$	7.09219	0.253454
$784$	0	0
$785$	27.2886	0.973971
$786$	−6.04330	−0.215557
$787$	36.7717	1.31077	0.655385	−	0.755295i	$-0.272506\pi$
$787$	36.7717	1.31077	0.655385	+	0.755295i	$0.272506\pi$
$788$	−5.25821	−0.187316
$789$	8.40696	0.299296
$790$	−39.1205	−1.39185
$791$	0	0
$792$	18.8719	0.670586
$793$	3.32325	0.118012
$794$	−14.2707	−0.506447
$795$	−2.74165	−0.0972363
$796$	−3.08302	−0.109275
$797$	16.3361	0.578653	0.289327	−	0.957230i	$-0.406569\pi$
$797$	16.3361	0.578653	0.289327	+	0.957230i	$0.406569\pi$
$798$	0	0
$799$	−1.58181	−0.0559604
$800$	44.9009	1.58749
$801$	13.7938	0.487379
$802$	−30.4679	−1.07586
$803$	−59.6105	−2.10361
$804$	−7.41286	−0.261432
$805$	0	0
$806$	7.54289	0.265687
$807$	24.7936	0.872776
$808$	8.15171	0.286776
$809$	−31.0850	−1.09289	−0.546445	−	0.837495i	$-0.684019\pi$
$809$	−31.0850	−1.09289	−0.546445	+	0.837495i	$0.684019\pi$
$810$	4.52079	0.158845
$811$	34.5832	1.21438	0.607191	−	0.794556i	$-0.292296\pi$
$811$	34.5832	1.21438	0.607191	+	0.794556i	$0.292296\pi$
$812$	0	0
$813$	−4.91574	−0.172402
$814$	−49.7697	−1.74443
$815$	74.3271	2.60357
$816$	2.70457	0.0946791
$817$	−14.3194	−0.500973
$818$	21.4331	0.749391
$819$	0	0
$820$	14.7481	0.515028
$821$	4.85028	0.169276	0.0846379	−	0.996412i	$-0.473027\pi$
$821$	4.85028	0.169276	0.0846379	+	0.996412i	$0.473027\pi$
$822$	6.01346	0.209743
$823$	−23.4599	−0.817760	−0.408880	−	0.912588i	$-0.634080\pi$
$823$	−23.4599	−0.817760	−0.408880	+	0.912588i	$0.634080\pi$
$824$	−48.6173	−1.69367
$825$	−69.4228	−2.41699
$826$	0	0
$827$	−37.9723	−1.32043	−0.660213	−	0.751079i	$-0.729534\pi$
$827$	−37.9723	−1.32043	−0.660213	+	0.751079i	$0.729534\pi$
$828$	−1.21130	−0.0420956
$829$	22.5075	0.781718	0.390859	−	0.920451i	$-0.372178\pi$
$829$	22.5075	0.781718	0.390859	+	0.920451i	$0.372178\pi$
$830$	−68.4567	−2.37617
$831$	7.86600	0.272868
$832$	27.7125	0.960758
$833$	0	0
$834$	−13.6232	−0.471734
$835$	−38.7938	−1.34252
$836$	−17.1881	−0.594464
$837$	2.02771	0.0700881
$838$	15.4448	0.533533
$839$	−7.41216	−0.255896	−0.127948	−	0.991781i	$-0.540839\pi$
$839$	−7.41216	−0.255896	−0.127948	+	0.991781i	$0.540839\pi$
$840$	0	0
$841$	21.2992	0.734456
$842$	6.61223	0.227873
$843$	−25.6585	−0.883726
$844$	−12.8649	−0.442827
$845$	−7.89987	−0.271764
$846$	1.27520	0.0438422
$847$	0	0
$848$	1.32227	0.0454069
$849$	−18.1005	−0.621208
$850$	−17.5805	−0.603007
$851$	11.7468	0.402676
$852$	3.22907	0.110626
$853$	−10.7030	−0.366464	−0.183232	−	0.983070i	$-0.558656\pi$
$853$	−10.7030	−0.366464	−0.183232	+	0.983070i	$0.558656\pi$
$854$	0	0
$855$	−15.1407	−0.517802
$856$	−6.95506	−0.237719
$857$	−56.9070	−1.94391	−0.971954	−	0.235172i	$-0.924435\pi$
$857$	−56.9070	−1.94391	−0.971954	+	0.235172i	$0.924435\pi$
$858$	−22.8304	−0.779418
$859$	−47.8453	−1.63246	−0.816231	−	0.577726i	$-0.803940\pi$
$859$	−47.8453	−1.63246	−0.816231	+	0.577726i	$0.803940\pi$
$860$	−11.5244	−0.392978
$861$	0	0
$862$	8.45354	0.287929
$863$	56.5642	1.92547	0.962734	−	0.270451i	$-0.0871727\pi$
$863$	56.5642	1.92547	0.962734	+	0.270451i	$0.0871727\pi$
$864$	3.96951	0.135045
$865$	25.1257	0.854298
$866$	−25.1622	−0.855046
$867$	−15.0721	−0.511875
$868$	0	0
$869$	53.1096	1.80162
$870$	32.0624	1.08702
$871$	32.9764	1.11736
$872$	−51.1381	−1.73176
$873$	18.9768	0.642266
$874$	−6.80418	−0.230155
$875$	0	0
$876$	−7.25578	−0.245150
$877$	−40.8403	−1.37908	−0.689540	−	0.724248i	$-0.742187\pi$
$877$	−40.8403	−1.37908	−0.689540	+	0.724248i	$0.742187\pi$
$878$	17.4987	0.590553
$879$	−18.3637	−0.619393
$880$	48.2818	1.62758
$881$	25.9048	0.872756	0.436378	−	0.899764i	$-0.356261\pi$
$881$	25.9048	0.872756	0.436378	+	0.899764i	$0.356261\pi$
$882$	0	0
$883$	−0.591065	−0.0198909	−0.00994546	−	0.999951i	$-0.503166\pi$
$883$	−0.591065	−0.0198909	−0.00994546	+	0.999951i	$0.503166\pi$
$884$	3.44709	0.115938
$885$	45.0173	1.51324
$886$	−24.8054	−0.833354
$887$	25.4331	0.853959	0.426979	−	0.904261i	$-0.359578\pi$
$887$	25.4331	0.853959	0.426979	+	0.904261i	$0.359578\pi$
$888$	−22.2765	−0.747550
$889$	0	0
$890$	62.3588	2.09027
$891$	−6.13738	−0.205610
$892$	14.9696	0.501219
$893$	−4.27080	−0.142917
$894$	−15.1667	−0.507251
$895$	−14.4902	−0.484354
$896$	0	0
$897$	5.38853	0.179918
$898$	42.5146	1.41873
$899$	14.3809	0.479631
$900$	−8.45014	−0.281671
$901$	0.942562	0.0314013
$902$	33.5813	1.11813
$903$	0	0
$904$	32.0198	1.06496
$905$	−76.0778	−2.52891
$906$	15.1682	0.503929
$907$	38.6361	1.28289	0.641445	−	0.767169i	$-0.278335\pi$
$907$	38.6361	1.28289	0.641445	+	0.767169i	$0.278335\pi$
$908$	11.3883	0.377934
$909$	−2.65103	−0.0879292
$910$	0	0
$911$	39.1663	1.29764	0.648818	−	0.760944i	$-0.275264\pi$
$911$	39.1663	1.29764	0.648818	+	0.760944i	$0.275264\pi$
$912$	7.30221	0.241800
$913$	92.9360	3.07573
$914$	46.1188	1.52547
$915$	4.03875	0.133517
$916$	9.41640	0.311127
$917$	0	0
$918$	−1.55422	−0.0512970
$919$	−5.46667	−0.180329	−0.0901643	−	0.995927i	$-0.528739\pi$
$919$	−5.46667	−0.180329	−0.0901643	+	0.995927i	$0.528739\pi$
$920$	−20.1367	−0.663886
$921$	8.65273	0.285117
$922$	−5.54635	−0.182659
$923$	−14.3647	−0.472818
$924$	0	0
$925$	81.9468	2.69440
$926$	37.7081	1.23917
$927$	15.8109	0.519299
$928$	28.1525	0.924151
$929$	−50.5753	−1.65932	−0.829661	−	0.558268i	$-0.811466\pi$
$929$	−50.5753	−1.65932	−0.829661	+	0.558268i	$0.811466\pi$
$930$	9.16688	0.300594
$931$	0	0
$932$	12.3786	0.405475
$933$	−13.3478	−0.436988
$934$	23.2291	0.760080
$935$	34.4171	1.12556
$936$	−10.2187	−0.334009
$937$	15.3809	0.502472	0.251236	−	0.967926i	$-0.419163\pi$
$937$	15.3809	0.502472	0.251236	+	0.967926i	$0.419163\pi$
$938$	0	0
$939$	−14.4104	−0.470266
$940$	−3.43717	−0.112108
$941$	1.25770	0.0409999	0.0204999	−	0.999790i	$-0.493474\pi$
$941$	1.25770	0.0409999	0.0204999	+	0.999790i	$0.493474\pi$
$942$	7.56315	0.246421
$943$	−7.92598	−0.258105
$944$	−21.7114	−0.706645
$945$	0	0
$946$	−26.2408	−0.853162
$947$	16.2496	0.528040	0.264020	−	0.964517i	$-0.414952\pi$
$947$	16.2496	0.528040	0.264020	+	0.964517i	$0.414952\pi$
$948$	6.46450	0.209957
$949$	32.2777	1.04778
$950$	−47.4665	−1.54002
$951$	−26.5176	−0.859893
$952$	0	0
$953$	−1.28220	−0.0415345	−0.0207672	−	0.999784i	$-0.506611\pi$
$953$	−1.28220	−0.0415345	−0.0207672	+	0.999784i	$0.506611\pi$
$954$	−0.759860	−0.0246014
$955$	64.0310	2.07200
$956$	2.28325	0.0738456
$957$	−43.5275	−1.40704
$958$	9.30840	0.300741
$959$	0	0
$960$	33.6790	1.08699
$961$	−26.8884	−0.867367
$962$	26.9491	0.868874
$963$	2.26187	0.0728877
$964$	21.5221	0.693179
$965$	33.4210	1.07586
$966$	0	0
$967$	26.1594	0.841229	0.420615	−	0.907239i	$-0.361814\pi$
$967$	26.1594	0.841229	0.420615	+	0.907239i	$0.361814\pi$
$968$	−82.0003	−2.63559
$969$	5.20529	0.167218
$970$	85.7900	2.75455
$971$	33.9488	1.08947	0.544735	−	0.838608i	$-0.316630\pi$
$971$	33.9488	1.08947	0.544735	+	0.838608i	$0.316630\pi$
$972$	−0.747042	−0.0239614
$973$	0	0
$974$	−14.6007	−0.467837
$975$	37.5908	1.20387
$976$	−1.94784	−0.0623490
$977$	−1.81397	−0.0580342	−0.0290171	−	0.999579i	$-0.509238\pi$
$977$	−1.81397	−0.0580342	−0.0290171	+	0.999579i	$0.509238\pi$
$978$	20.6001	0.658718
$979$	−84.6576	−2.70567
$980$	0	0
$981$	16.6307	0.530978
$982$	−4.24972	−0.135614
$983$	41.1875	1.31368	0.656838	−	0.754032i	$-0.271893\pi$
$983$	41.1875	1.31368	0.656838	+	0.754032i	$0.271893\pi$
$984$	15.0307	0.479161
$985$	28.4275	0.905776
$986$	−11.0228	−0.351039
$987$	0	0
$988$	9.30696	0.296094
$989$	6.19345	0.196940
$990$	−27.7458	−0.881821
$991$	8.20643	0.260686	0.130343	−	0.991469i	$-0.458392\pi$
$991$	8.20643	0.260686	0.130343	+	0.991469i	$0.458392\pi$
$992$	8.04902	0.255557
$993$	−11.2768	−0.357859
$994$	0	0
$995$	16.6678	0.528404
$996$	11.3122	0.358440
$997$	5.35382	0.169557	0.0847785	−	0.996400i	$-0.472982\pi$
$997$	5.35382	0.169557	0.0847785	+	0.996400i	$0.472982\pi$
$998$	26.4523	0.837334
$999$	7.24458	0.229208

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8967.2.a.bl.1.14		21
7.3	odd	6		1281.2.j.i.184.8	✓	42
7.5	odd	6		1281.2.j.i.550.8	yes	42
7.6	odd	2		8967.2.a.bk.1.14		21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1281.2.j.i.184.8	✓	42	7.3	odd	6
1281.2.j.i.550.8	yes	42	7.5	odd	6
8967.2.a.bk.1.14		21	7.6	odd	2
8967.2.a.bl.1.14		21	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8967 = 3 \cdot 7^{2} \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8967.a (trivial)

\(f(q)\)	\(=\)	\(q+1.11936 q^{2} +1.00000 q^{3} -0.747042 q^{4} +4.03875 q^{5} +1.11936 q^{6} -3.07492 q^{8} +1.00000 q^{9} +4.52079 q^{10} -6.13738 q^{11} -0.747042 q^{12} +3.32325 q^{13} +4.03875 q^{15} -1.94784 q^{16} -1.38850 q^{17} +1.11936 q^{18} -3.74887 q^{19} -3.01711 q^{20} -6.86992 q^{22} +1.62146 q^{23} -3.07492 q^{24} +11.3115 q^{25} +3.71990 q^{26} +1.00000 q^{27} +7.09219 q^{29} +4.52079 q^{30} +2.02771 q^{31} +3.96951 q^{32} -6.13738 q^{33} -1.55422 q^{34} -0.747042 q^{36} +7.24458 q^{37} -4.19632 q^{38} +3.32325 q^{39} -12.4188 q^{40} -4.88816 q^{41} +3.81967 q^{43} +4.58488 q^{44} +4.03875 q^{45} +1.81500 q^{46} +1.13923 q^{47} -1.94784 q^{48} +12.6616 q^{50} -1.38850 q^{51} -2.48261 q^{52} -0.678837 q^{53} +1.11936 q^{54} -24.7873 q^{55} -3.74887 q^{57} +7.93869 q^{58} +11.1464 q^{59} -3.01711 q^{60} +1.00000 q^{61} +2.26973 q^{62} +8.33898 q^{64} +13.4218 q^{65} -6.86992 q^{66} +9.92296 q^{67} +1.03727 q^{68} +1.62146 q^{69} -4.32247 q^{71} -3.07492 q^{72} +9.71269 q^{73} +8.10927 q^{74} +11.3115 q^{75} +2.80056 q^{76} +3.71990 q^{78} -8.65346 q^{79} -7.86685 q^{80} +1.00000 q^{81} -5.47160 q^{82} -15.1426 q^{83} -5.60778 q^{85} +4.27557 q^{86} +7.09219 q^{87} +18.8719 q^{88} +13.7938 q^{89} +4.52079 q^{90} -1.21130 q^{92} +2.02771 q^{93} +1.27520 q^{94} -15.1407 q^{95} +3.96951 q^{96} +18.9768 q^{97} -6.13738 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 21 q^{3} + 32 q^{4} + 11 q^{5} + 21 q^{9} + 6 q^{10} + q^{11} + 32 q^{12} + 14 q^{13} + 11 q^{15} + 30 q^{16} + 21 q^{17} + 4 q^{19} + 26 q^{20} + 4 q^{22} + 8 q^{23} + 20 q^{25} + 38 q^{26} + 21 q^{27}+ \cdots + q^{99}+O(q^{100}) \) 21 * q + 21 * q^3 + 32 * q^4 + 11 * q^5 + 21 * q^9 + 6 * q^10 + q^11 + 32 * q^12 + 14 * q^13 + 11 * q^15 + 30 * q^16 + 21 * q^17 + 4 * q^19 + 26 * q^20 + 4 * q^22 + 8 * q^23 + 20 * q^25 + 38 * q^26 + 21 * q^27 + 9 * q^29 + 6 * q^30 + 8 * q^31 - 20 * q^32 + q^33 - 2 * q^34 + 32 * q^36 + 13 * q^37 - 7 * q^38 + 14 * q^39 + 18 * q^40 + 35 * q^41 + 2 * q^43 - q^44 + 11 * q^45 + 7 * q^46 + 42 * q^47 + 30 * q^48 + 19 * q^50 + 21 * q^51 + 11 * q^52 - 8 * q^53 + 24 * q^55 + 4 * q^57 + 6 * q^58 + 13 * q^59 + 26 * q^60 + 21 * q^61 + 28 * q^62 + 104 * q^64 + 14 * q^65 + 4 * q^66 + 14 * q^67 + 97 * q^68 + 8 * q^69 + 16 * q^71 + 46 * q^73 - 63 * q^74 + 20 * q^75 + 40 * q^76 + 38 * q^78 + 18 * q^79 + 32 * q^80 + 21 * q^81 + 46 * q^82 + 52 * q^85 - 47 * q^86 + 9 * q^87 - 41 * q^88 + 56 * q^89 + 6 * q^90 + 16 * q^92 + 8 * q^93 + 27 * q^94 - 20 * q^95 - 20 * q^96 + 62 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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