LMFDB - Embedded newform 7168.2.a.bj.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7168,2,Mod(1,7168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7168, 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("7168.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7168 = 2^{10} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7168.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:	$57.2367681689$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 24x^{10} + 221x^{8} - 968x^{6} + 2008x^{4} - 1640x^{2} + 196 $ x^12 - 24x^10 + 221x^8 - 968x^6 + 2008x^4 - 1640*x^2 + 196
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 112)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$1.33191$ of defining polynomial
Character	$\chi$	$=$	7168.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.13347 q^{3} +0.555963 q^{5} +1.00000 q^{7} +6.81864 q^{9} +O(q^{10})$	$q+3.13347 q^{3} +0.555963 q^{5} +1.00000 q^{7} +6.81864 q^{9} +3.14807 q^{11} +4.47339 q^{13} +1.74209 q^{15} -0.980951 q^{17} +7.44484 q^{19} +3.13347 q^{21} -1.25951 q^{23} -4.69090 q^{25} +11.9656 q^{27} +4.48799 q^{29} -4.43140 q^{31} +9.86440 q^{33} +0.555963 q^{35} -0.912373 q^{37} +14.0172 q^{39} +1.21375 q^{41} +1.36686 q^{43} +3.79091 q^{45} -9.97147 q^{47} +1.00000 q^{49} -3.07378 q^{51} -11.4242 q^{53} +1.75021 q^{55} +23.3282 q^{57} -2.56517 q^{59} -3.65974 q^{61} +6.81864 q^{63} +2.48704 q^{65} -2.25229 q^{67} -3.94663 q^{69} -0.934634 q^{71} +0.710511 q^{73} -14.6988 q^{75} +3.14807 q^{77} -11.8644 q^{79} +17.0379 q^{81} +9.58105 q^{83} -0.545372 q^{85} +14.0630 q^{87} -10.2082 q^{89} +4.47339 q^{91} -13.8856 q^{93} +4.13906 q^{95} -3.03684 q^{97} +21.4656 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q + 12 * q^7 + 12 * q^9	$ 12 q + 12 q^{7} + 12 q^{9} + 24 q^{15} + 8 q^{17} + 16 q^{23} + 20 q^{25} - 8 q^{31} + 16 q^{39} + 32 q^{41} - 16 q^{47} + 12 q^{49} + 24 q^{55} + 64 q^{57} + 12 q^{63} + 32 q^{65} + 8 q^{71} - 24 q^{79} + 44 q^{81} - 32 q^{87} + 24 q^{89} + 48 q^{97}+O(q^{100}) $ 12 * q + 12 * q^7 + 12 * q^9 + 24 * q^15 + 8 * q^17 + 16 * q^23 + 20 * q^25 - 8 * q^31 + 16 * q^39 + 32 * q^41 - 16 * q^47 + 12 * q^49 + 24 * q^55 + 64 * q^57 + 12 * q^63 + 32 * q^65 + 8 * q^71 - 24 * q^79 + 44 * q^81 - 32 * q^87 + 24 * q^89 + 48 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	3.13347	1.80911	0.904555	−	0.426357i	$-0.140203\pi$
$3$	3.13347	1.80911	0.904555	+	0.426357i	$0.140203\pi$
$4$	0	0
$5$	0.555963	0.248634	0.124317	−	0.992243i	$-0.460326\pi$
$5$	0.555963	0.248634	0.124317	+	0.992243i	$0.460326\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	6.81864	2.27288
$10$	0	0
$11$	3.14807	0.949180	0.474590	−	0.880207i	$-0.342596\pi$
$11$	3.14807	0.949180	0.474590	+	0.880207i	$0.342596\pi$
$12$	0	0
$13$	4.47339	1.24070	0.620348	−	0.784327i	$-0.286992\pi$
$13$	4.47339	1.24070	0.620348	+	0.784327i	$0.286992\pi$
$14$	0	0
$15$	1.74209	0.449807
$16$	0	0
$17$	−0.980951	−0.237915	−0.118958	−	0.992899i	$-0.537955\pi$
$17$	−0.980951	−0.237915	−0.118958	+	0.992899i	$0.537955\pi$
$18$	0	0
$19$	7.44484	1.70796	0.853981	−	0.520304i	$-0.174181\pi$
$19$	7.44484	1.70796	0.853981	+	0.520304i	$0.174181\pi$
$20$	0	0
$21$	3.13347	0.683779
$22$	0	0
$23$	−1.25951	−0.262626	−0.131313	−	0.991341i	$-0.541919\pi$
$23$	−1.25951	−0.262626	−0.131313	+	0.991341i	$0.541919\pi$
$24$	0	0
$25$	−4.69090	−0.938181
$26$	0	0
$27$	11.9656	2.30278
$28$	0	0
$29$	4.48799	0.833399	0.416700	−	0.909044i	$-0.363187\pi$
$29$	4.48799	0.833399	0.416700	+	0.909044i	$0.363187\pi$
$30$	0	0
$31$	−4.43140	−0.795902	−0.397951	−	0.917407i	$-0.630279\pi$
$31$	−4.43140	−0.795902	−0.397951	+	0.917407i	$0.630279\pi$
$32$	0	0
$33$	9.86440	1.71717
$34$	0	0
$35$	0.555963	0.0939749
$36$	0	0
$37$	−0.912373	−0.149993	−0.0749966	−	0.997184i	$-0.523895\pi$
$37$	−0.912373	−0.149993	−0.0749966	+	0.997184i	$0.523895\pi$
$38$	0	0
$39$	14.0172	2.24455
$40$	0	0
$41$	1.21375	0.189556	0.0947779	−	0.995498i	$-0.469786\pi$
$41$	1.21375	0.189556	0.0947779	+	0.995498i	$0.469786\pi$
$42$	0	0
$43$	1.36686	0.208444	0.104222	−	0.994554i	$-0.466765\pi$
$43$	1.36686	0.208444	0.104222	+	0.994554i	$0.466765\pi$
$44$	0	0
$45$	3.79091	0.565116
$46$	0	0
$47$	−9.97147	−1.45449	−0.727244	−	0.686379i	$-0.759199\pi$
$47$	−9.97147	−1.45449	−0.727244	+	0.686379i	$0.759199\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.07378	−0.430415
$52$	0	0
$53$	−11.4242	−1.56924	−0.784619	−	0.619979i	$-0.787141\pi$
$53$	−11.4242	−1.56924	−0.784619	+	0.619979i	$0.787141\pi$
$54$	0	0
$55$	1.75021	0.235999
$56$	0	0
$57$	23.3282	3.08989
$58$	0	0
$59$	−2.56517	−0.333957	−0.166978	−	0.985961i	$-0.553401\pi$
$59$	−2.56517	−0.333957	−0.166978	+	0.985961i	$0.553401\pi$
$60$	0	0
$61$	−3.65974	−0.468581	−0.234291	−	0.972167i	$-0.575277\pi$
$61$	−3.65974	−0.468581	−0.234291	+	0.972167i	$0.575277\pi$
$62$	0	0
$63$	6.81864	0.859067
$64$	0	0
$65$	2.48704	0.308479
$66$	0	0
$67$	−2.25229	−0.275161	−0.137581	−	0.990491i	$-0.543933\pi$
$67$	−2.25229	−0.275161	−0.137581	+	0.990491i	$0.543933\pi$
$68$	0	0
$69$	−3.94663	−0.475119
$70$	0	0
$71$	−0.934634	−0.110921	−0.0554603	−	0.998461i	$-0.517663\pi$
$71$	−0.934634	−0.110921	−0.0554603	+	0.998461i	$0.517663\pi$
$72$	0	0
$73$	0.710511	0.0831591	0.0415795	−	0.999135i	$-0.486761\pi$
$73$	0.710511	0.0831591	0.0415795	+	0.999135i	$0.486761\pi$
$74$	0	0
$75$	−14.6988	−1.69727
$76$	0	0
$77$	3.14807	0.358756
$78$	0	0
$79$	−11.8644	−1.33485	−0.667424	−	0.744678i	$-0.732603\pi$
$79$	−11.8644	−1.33485	−0.667424	+	0.744678i	$0.732603\pi$
$80$	0	0
$81$	17.0379	1.89310
$82$	0	0
$83$	9.58105	1.05166	0.525828	−	0.850591i	$-0.323755\pi$
$83$	9.58105	1.05166	0.525828	+	0.850591i	$0.323755\pi$
$84$	0	0
$85$	−0.545372	−0.0591539
$86$	0	0
$87$	14.0630	1.50771
$88$	0	0
$89$	−10.2082	−1.08206	−0.541032	−	0.841002i	$-0.681966\pi$
$89$	−10.2082	−1.08206	−0.541032	+	0.841002i	$0.681966\pi$
$90$	0	0
$91$	4.47339	0.468939
$92$	0	0
$93$	−13.8856	−1.43987
$94$	0	0
$95$	4.13906	0.424658
$96$	0	0
$97$	−3.03684	−0.308344	−0.154172	−	0.988044i	$-0.549271\pi$
$97$	−3.03684	−0.308344	−0.154172	+	0.988044i	$0.549271\pi$
$98$	0	0
$99$	21.4656	2.15737
$100$	0	0
$101$	14.3752	1.43038	0.715191	−	0.698930i	$-0.246340\pi$
$101$	14.3752	1.43038	0.715191	+	0.698930i	$0.246340\pi$
$102$	0	0
$103$	−6.05520	−0.596636	−0.298318	−	0.954466i	$-0.596426\pi$
$103$	−6.05520	−0.596636	−0.298318	+	0.954466i	$0.596426\pi$
$104$	0	0
$105$	1.74209	0.170011
$106$	0	0
$107$	17.6738	1.70859	0.854295	−	0.519788i	$-0.173989\pi$
$107$	17.6738	1.70859	0.854295	+	0.519788i	$0.173989\pi$
$108$	0	0
$109$	−14.2234	−1.36236	−0.681179	−	0.732117i	$-0.738533\pi$
$109$	−14.2234	−1.36236	−0.681179	+	0.732117i	$0.738533\pi$
$110$	0	0
$111$	−2.85889	−0.271354
$112$	0	0
$113$	−5.01929	−0.472175	−0.236088	−	0.971732i	$-0.575865\pi$
$113$	−5.01929	−0.472175	−0.236088	+	0.971732i	$0.575865\pi$
$114$	0	0
$115$	−0.700241	−0.0652978
$116$	0	0
$117$	30.5024	2.81995
$118$	0	0
$119$	−0.980951	−0.0899236
$120$	0	0
$121$	−1.08963	−0.0990575
$122$	0	0
$123$	3.80325	0.342927
$124$	0	0
$125$	−5.38779	−0.481898
$126$	0	0
$127$	−19.3869	−1.72031	−0.860156	−	0.510031i	$-0.829634\pi$
$127$	−19.3869	−1.72031	−0.860156	+	0.510031i	$0.829634\pi$
$128$	0	0
$129$	4.28301	0.377098
$130$	0	0
$131$	11.7270	1.02459	0.512296	−	0.858809i	$-0.328795\pi$
$131$	11.7270	1.02459	0.512296	+	0.858809i	$0.328795\pi$
$132$	0	0
$133$	7.44484	0.645549
$134$	0	0
$135$	6.65242	0.572549
$136$	0	0
$137$	−10.3723	−0.886168	−0.443084	−	0.896480i	$-0.646116\pi$
$137$	−10.3723	−0.886168	−0.443084	+	0.896480i	$0.646116\pi$
$138$	0	0
$139$	−6.83475	−0.579716	−0.289858	−	0.957070i	$-0.593608\pi$
$139$	−6.83475	−0.579716	−0.289858	+	0.957070i	$0.593608\pi$
$140$	0	0
$141$	−31.2453	−2.63133
$142$	0	0
$143$	14.0826	1.17764
$144$	0	0
$145$	2.49516	0.207212
$146$	0	0
$147$	3.13347	0.258444
$148$	0	0
$149$	2.46143	0.201648	0.100824	−	0.994904i	$-0.467852\pi$
$149$	2.46143	0.201648	0.100824	+	0.994904i	$0.467852\pi$
$150$	0	0
$151$	20.8131	1.69374	0.846871	−	0.531798i	$-0.178483\pi$
$151$	20.8131	1.69374	0.846871	+	0.531798i	$0.178483\pi$
$152$	0	0
$153$	−6.68874	−0.540753
$154$	0	0
$155$	−2.46369	−0.197889
$156$	0	0
$157$	−14.3459	−1.14493	−0.572466	−	0.819929i	$-0.694013\pi$
$157$	−14.3459	−1.14493	−0.572466	+	0.819929i	$0.694013\pi$
$158$	0	0
$159$	−35.7975	−2.83892
$160$	0	0
$161$	−1.25951	−0.0992632
$162$	0	0
$163$	−16.8124	−1.31685	−0.658424	−	0.752647i	$-0.728777\pi$
$163$	−16.8124	−1.31685	−0.658424	+	0.752647i	$0.728777\pi$
$164$	0	0
$165$	5.48424	0.426948
$166$	0	0
$167$	−1.10868	−0.0857923	−0.0428962	−	0.999080i	$-0.513658\pi$
$167$	−1.10868	−0.0857923	−0.0428962	+	0.999080i	$0.513658\pi$
$168$	0	0
$169$	7.01121	0.539324
$170$	0	0
$171$	50.7636	3.88199
$172$	0	0
$173$	10.2385	0.778418	0.389209	−	0.921149i	$-0.372748\pi$
$173$	10.2385	0.778418	0.389209	+	0.921149i	$0.372748\pi$
$174$	0	0
$175$	−4.69090	−0.354599
$176$	0	0
$177$	−8.03789	−0.604164
$178$	0	0
$179$	−14.0567	−1.05065	−0.525324	−	0.850902i	$-0.676056\pi$
$179$	−14.0567	−1.05065	−0.525324	+	0.850902i	$0.676056\pi$
$180$	0	0
$181$	10.3993	0.772975	0.386487	−	0.922295i	$-0.373688\pi$
$181$	10.3993	0.772975	0.386487	+	0.922295i	$0.373688\pi$
$182$	0	0
$183$	−11.4677	−0.847715
$184$	0	0
$185$	−0.507246	−0.0372935
$186$	0	0
$187$	−3.08810	−0.225825
$188$	0	0
$189$	11.9656	0.870368
$190$	0	0
$191$	3.54684	0.256640	0.128320	−	0.991733i	$-0.459042\pi$
$191$	3.54684	0.256640	0.128320	+	0.991733i	$0.459042\pi$
$192$	0	0
$193$	2.99394	0.215509	0.107754	−	0.994178i	$-0.465634\pi$
$193$	2.99394	0.215509	0.107754	+	0.994178i	$0.465634\pi$
$194$	0	0
$195$	7.79307	0.558073
$196$	0	0
$197$	−5.02534	−0.358041	−0.179020	−	0.983845i	$-0.557293\pi$
$197$	−5.02534	−0.358041	−0.179020	+	0.983845i	$0.557293\pi$
$198$	0	0
$199$	1.80109	0.127676	0.0638380	−	0.997960i	$-0.479666\pi$
$199$	1.80109	0.127676	0.0638380	+	0.997960i	$0.479666\pi$
$200$	0	0
$201$	−7.05749	−0.497797
$202$	0	0
$203$	4.48799	0.314995
$204$	0	0
$205$	0.674800	0.0471301
$206$	0	0
$207$	−8.58813	−0.596916
$208$	0	0
$209$	23.4369	1.62116
$210$	0	0
$211$	21.3577	1.47032	0.735161	−	0.677892i	$-0.237106\pi$
$211$	21.3577	1.47032	0.735161	+	0.677892i	$0.237106\pi$
$212$	0	0
$213$	−2.92865	−0.200668
$214$	0	0
$215$	0.759924	0.0518264
$216$	0	0
$217$	−4.43140	−0.300823
$218$	0	0
$219$	2.22637	0.150444
$220$	0	0
$221$	−4.38817	−0.295181
$222$	0	0
$223$	7.11258	0.476294	0.238147	−	0.971229i	$-0.423460\pi$
$223$	7.11258	0.476294	0.238147	+	0.971229i	$0.423460\pi$
$224$	0	0
$225$	−31.9856	−2.13237
$226$	0	0
$227$	−17.2091	−1.14221	−0.571106	−	0.820877i	$-0.693485\pi$
$227$	−17.2091	−1.14221	−0.571106	+	0.820877i	$0.693485\pi$
$228$	0	0
$229$	0.878911	0.0580800	0.0290400	−	0.999578i	$-0.490755\pi$
$229$	0.878911	0.0580800	0.0290400	+	0.999578i	$0.490755\pi$
$230$	0	0
$231$	9.86440	0.649030
$232$	0	0
$233$	4.85688	0.318185	0.159093	−	0.987264i	$-0.449143\pi$
$233$	4.85688	0.318185	0.159093	+	0.987264i	$0.449143\pi$
$234$	0	0
$235$	−5.54377	−0.361636
$236$	0	0
$237$	−37.1767	−2.41489
$238$	0	0
$239$	−4.91033	−0.317623	−0.158811	−	0.987309i	$-0.550766\pi$
$239$	−4.91033	−0.317623	−0.158811	+	0.987309i	$0.550766\pi$
$240$	0	0
$241$	−10.8591	−0.699498	−0.349749	−	0.936843i	$-0.613733\pi$
$241$	−10.8591	−0.699498	−0.349749	+	0.936843i	$0.613733\pi$
$242$	0	0
$243$	17.4910	1.12205
$244$	0	0
$245$	0.555963	0.0355192
$246$	0	0
$247$	33.3037	2.11906
$248$	0	0
$249$	30.0219	1.90256
$250$	0	0
$251$	−11.5756	−0.730643	−0.365322	−	0.930881i	$-0.619041\pi$
$251$	−11.5756	−0.730643	−0.365322	+	0.930881i	$0.619041\pi$
$252$	0	0
$253$	−3.96503	−0.249279
$254$	0	0
$255$	−1.70891	−0.107016
$256$	0	0
$257$	20.6130	1.28580	0.642902	−	0.765948i	$-0.277730\pi$
$257$	20.6130	1.28580	0.642902	+	0.765948i	$0.277730\pi$
$258$	0	0
$259$	−0.912373	−0.0566921
$260$	0	0
$261$	30.6020	1.89422
$262$	0	0
$263$	13.6297	0.840444	0.420222	−	0.907421i	$-0.361952\pi$
$263$	13.6297	0.840444	0.420222	+	0.907421i	$0.361952\pi$
$264$	0	0
$265$	−6.35145	−0.390166
$266$	0	0
$267$	−31.9870	−1.95757
$268$	0	0
$269$	−6.26495	−0.381981	−0.190990	−	0.981592i	$-0.561170\pi$
$269$	−6.26495	−0.381981	−0.190990	+	0.981592i	$0.561170\pi$
$270$	0	0
$271$	−12.6851	−0.770564	−0.385282	−	0.922799i	$-0.625896\pi$
$271$	−12.6851	−0.770564	−0.385282	+	0.922799i	$0.625896\pi$
$272$	0	0
$273$	14.0172	0.848362
$274$	0	0
$275$	−14.7673	−0.890503
$276$	0	0
$277$	24.1607	1.45168	0.725838	−	0.687865i	$-0.241452\pi$
$277$	24.1607	1.45168	0.725838	+	0.687865i	$0.241452\pi$
$278$	0	0
$279$	−30.2161	−1.80899
$280$	0	0
$281$	12.8628	0.767330	0.383665	−	0.923472i	$-0.374662\pi$
$281$	12.8628	0.767330	0.383665	+	0.923472i	$0.374662\pi$
$282$	0	0
$283$	1.24322	0.0739019	0.0369509	−	0.999317i	$-0.488235\pi$
$283$	1.24322	0.0739019	0.0369509	+	0.999317i	$0.488235\pi$
$284$	0	0
$285$	12.9696	0.768253
$286$	0	0
$287$	1.21375	0.0716454
$288$	0	0
$289$	−16.0377	−0.943396
$290$	0	0
$291$	−9.51585	−0.557829
$292$	0	0
$293$	25.4467	1.48661	0.743306	−	0.668952i	$-0.233257\pi$
$293$	25.4467	1.48661	0.743306	+	0.668952i	$0.233257\pi$
$294$	0	0
$295$	−1.42614	−0.0830331
$296$	0	0
$297$	37.6685	2.18575
$298$	0	0
$299$	−5.63427	−0.325838
$300$	0	0
$301$	1.36686	0.0787845
$302$	0	0
$303$	45.0441	2.58772
$304$	0	0
$305$	−2.03468	−0.116505
$306$	0	0
$307$	2.47469	0.141238	0.0706189	−	0.997503i	$-0.477503\pi$
$307$	2.47469	0.141238	0.0706189	+	0.997503i	$0.477503\pi$
$308$	0	0
$309$	−18.9738	−1.07938
$310$	0	0
$311$	−17.4288	−0.988296	−0.494148	−	0.869378i	$-0.664520\pi$
$311$	−17.4288	−0.988296	−0.494148	+	0.869378i	$0.664520\pi$
$312$	0	0
$313$	28.3430	1.60204	0.801021	−	0.598637i	$-0.204291\pi$
$313$	28.3430	1.60204	0.801021	+	0.598637i	$0.204291\pi$
$314$	0	0
$315$	3.79091	0.213594
$316$	0	0
$317$	11.6587	0.654820	0.327410	−	0.944882i	$-0.393824\pi$
$317$	11.6587	0.654820	0.327410	+	0.944882i	$0.393824\pi$
$318$	0	0
$319$	14.1285	0.791046
$320$	0	0
$321$	55.3803	3.09103
$322$	0	0
$323$	−7.30302	−0.406351
$324$	0	0
$325$	−20.9842	−1.16400
$326$	0	0
$327$	−44.5687	−2.46466
$328$	0	0
$329$	−9.97147	−0.549745
$330$	0	0
$331$	0.332994	0.0183030	0.00915150	−	0.999958i	$-0.497087\pi$
$331$	0.332994	0.0183030	0.00915150	+	0.999958i	$0.497087\pi$
$332$	0	0
$333$	−6.22114	−0.340916
$334$	0	0
$335$	−1.25219	−0.0684146
$336$	0	0
$337$	−9.22099	−0.502299	−0.251150	−	0.967948i	$-0.580809\pi$
$337$	−9.22099	−0.502299	−0.251150	+	0.967948i	$0.580809\pi$
$338$	0	0
$339$	−15.7278	−0.854217
$340$	0	0
$341$	−13.9504	−0.755454
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−2.19418	−0.118131
$346$	0	0
$347$	11.3522	0.609418	0.304709	−	0.952446i	$-0.401441\pi$
$347$	11.3522	0.609418	0.304709	+	0.952446i	$0.401441\pi$
$348$	0	0
$349$	16.3878	0.877219	0.438609	−	0.898678i	$-0.355471\pi$
$349$	16.3878	0.877219	0.438609	+	0.898678i	$0.355471\pi$
$350$	0	0
$351$	53.5267	2.85704
$352$	0	0
$353$	18.9481	1.00850	0.504252	−	0.863557i	$-0.331768\pi$
$353$	18.9481	1.00850	0.504252	+	0.863557i	$0.331768\pi$
$354$	0	0
$355$	−0.519622	−0.0275787
$356$	0	0
$357$	−3.07378	−0.162682
$358$	0	0
$359$	−3.54172	−0.186925	−0.0934624	−	0.995623i	$-0.529793\pi$
$359$	−3.54172	−0.186925	−0.0934624	+	0.995623i	$0.529793\pi$
$360$	0	0
$361$	36.4256	1.91714
$362$	0	0
$363$	−3.41433	−0.179206
$364$	0	0
$365$	0.395018	0.0206762
$366$	0	0
$367$	−11.1874	−0.583979	−0.291989	−	0.956422i	$-0.594317\pi$
$367$	−11.1874	−0.583979	−0.291989	+	0.956422i	$0.594317\pi$
$368$	0	0
$369$	8.27611	0.430837
$370$	0	0
$371$	−11.4242	−0.593116
$372$	0	0
$373$	19.5263	1.01103	0.505516	−	0.862817i	$-0.331302\pi$
$373$	19.5263	1.01103	0.505516	+	0.862817i	$0.331302\pi$
$374$	0	0
$375$	−16.8825	−0.871807
$376$	0	0
$377$	20.0765	1.03399
$378$	0	0
$379$	15.5387	0.798169	0.399084	−	0.916914i	$-0.369328\pi$
$379$	15.5387	0.798169	0.399084	+	0.916914i	$0.369328\pi$
$380$	0	0
$381$	−60.7484	−3.11223
$382$	0	0
$383$	−4.83945	−0.247284	−0.123642	−	0.992327i	$-0.539458\pi$
$383$	−4.83945	−0.247284	−0.123642	+	0.992327i	$0.539458\pi$
$384$	0	0
$385$	1.75021	0.0891991
$386$	0	0
$387$	9.32012	0.473768
$388$	0	0
$389$	−17.6503	−0.894907	−0.447454	−	0.894307i	$-0.647669\pi$
$389$	−17.6503	−0.894907	−0.447454	+	0.894307i	$0.647669\pi$
$390$	0	0
$391$	1.23552	0.0624827
$392$	0	0
$393$	36.7462	1.85360
$394$	0	0
$395$	−6.59617	−0.331889
$396$	0	0
$397$	16.6231	0.834288	0.417144	−	0.908840i	$-0.363031\pi$
$397$	16.6231	0.834288	0.417144	+	0.908840i	$0.363031\pi$
$398$	0	0
$399$	23.3282	1.16787
$400$	0	0
$401$	25.5823	1.27752	0.638760	−	0.769406i	$-0.279448\pi$
$401$	25.5823	1.27752	0.638760	+	0.769406i	$0.279448\pi$
$402$	0	0
$403$	−19.8234	−0.987472
$404$	0	0
$405$	9.47244	0.470689
$406$	0	0
$407$	−2.87222	−0.142371
$408$	0	0
$409$	−19.2126	−0.950001	−0.475001	−	0.879985i	$-0.657552\pi$
$409$	−19.2126	−0.950001	−0.475001	+	0.879985i	$0.657552\pi$
$410$	0	0
$411$	−32.5014	−1.60318
$412$	0	0
$413$	−2.56517	−0.126224
$414$	0	0
$415$	5.32671	0.261478
$416$	0	0
$417$	−21.4165	−1.04877
$418$	0	0
$419$	15.9472	0.779073	0.389537	−	0.921011i	$-0.372635\pi$
$419$	15.9472	0.779073	0.389537	+	0.921011i	$0.372635\pi$
$420$	0	0
$421$	−38.3299	−1.86809	−0.934043	−	0.357161i	$-0.883745\pi$
$421$	−38.3299	−1.86809	−0.934043	+	0.357161i	$0.883745\pi$
$422$	0	0
$423$	−67.9918	−3.30588
$424$	0	0
$425$	4.60155	0.223208
$426$	0	0
$427$	−3.65974	−0.177107
$428$	0	0
$429$	44.1273	2.13049
$430$	0	0
$431$	14.7618	0.711053	0.355526	−	0.934666i	$-0.384302\pi$
$431$	14.7618	0.711053	0.355526	+	0.934666i	$0.384302\pi$
$432$	0	0
$433$	30.0057	1.44198	0.720991	−	0.692944i	$-0.243687\pi$
$433$	30.0057	1.44198	0.720991	+	0.692944i	$0.243687\pi$
$434$	0	0
$435$	7.81851	0.374869
$436$	0	0
$437$	−9.37684	−0.448555
$438$	0	0
$439$	−33.7523	−1.61091	−0.805454	−	0.592659i	$-0.798078\pi$
$439$	−33.7523	−1.61091	−0.805454	+	0.592659i	$0.798078\pi$
$440$	0	0
$441$	6.81864	0.324697
$442$	0	0
$443$	−7.38248	−0.350752	−0.175376	−	0.984502i	$-0.556114\pi$
$443$	−7.38248	−0.350752	−0.175376	+	0.984502i	$0.556114\pi$
$444$	0	0
$445$	−5.67537	−0.269038
$446$	0	0
$447$	7.71281	0.364803
$448$	0	0
$449$	−1.59006	−0.0750395	−0.0375197	−	0.999296i	$-0.511946\pi$
$449$	−1.59006	−0.0750395	−0.0375197	+	0.999296i	$0.511946\pi$
$450$	0	0
$451$	3.82097	0.179923
$452$	0	0
$453$	65.2171	3.06417
$454$	0	0
$455$	2.48704	0.116594
$456$	0	0
$457$	2.14917	0.100534	0.0502671	−	0.998736i	$-0.483993\pi$
$457$	2.14917	0.100534	0.0502671	+	0.998736i	$0.483993\pi$
$458$	0	0
$459$	−11.7376	−0.547866
$460$	0	0
$461$	37.8169	1.76131	0.880655	−	0.473759i	$-0.157103\pi$
$461$	37.8169	1.76131	0.880655	+	0.473759i	$0.157103\pi$
$462$	0	0
$463$	−16.2686	−0.756065	−0.378033	−	0.925792i	$-0.623399\pi$
$463$	−16.2686	−0.756065	−0.378033	+	0.925792i	$0.623399\pi$
$464$	0	0
$465$	−7.71991	−0.358002
$466$	0	0
$467$	−25.3547	−1.17328	−0.586638	−	0.809849i	$-0.699549\pi$
$467$	−25.3547	−1.17328	−0.586638	+	0.809849i	$0.699549\pi$
$468$	0	0
$469$	−2.25229	−0.104001
$470$	0	0
$471$	−44.9526	−2.07131
$472$	0	0
$473$	4.30297	0.197851
$474$	0	0
$475$	−34.9230	−1.60238
$476$	0	0
$477$	−77.8976	−3.56669
$478$	0	0
$479$	−16.9860	−0.776112	−0.388056	−	0.921636i	$-0.626853\pi$
$479$	−16.9860	−0.776112	−0.388056	+	0.921636i	$0.626853\pi$
$480$	0	0
$481$	−4.08140	−0.186096
$482$	0	0
$483$	−3.94663	−0.179578
$484$	0	0
$485$	−1.68837	−0.0766650
$486$	0	0
$487$	39.0342	1.76881	0.884404	−	0.466722i	$-0.154565\pi$
$487$	39.0342	1.76881	0.884404	+	0.466722i	$0.154565\pi$
$488$	0	0
$489$	−52.6811	−2.38232
$490$	0	0
$491$	−38.0181	−1.71573	−0.857865	−	0.513875i	$-0.828210\pi$
$491$	−38.0181	−1.71573	−0.857865	+	0.513875i	$0.828210\pi$
$492$	0	0
$493$	−4.40250	−0.198279
$494$	0	0
$495$	11.9341	0.536396
$496$	0	0
$497$	−0.934634	−0.0419241
$498$	0	0
$499$	−6.40595	−0.286770	−0.143385	−	0.989667i	$-0.545799\pi$
$499$	−6.40595	−0.286770	−0.143385	+	0.989667i	$0.545799\pi$
$500$	0	0
$501$	−3.47402	−0.155208
$502$	0	0
$503$	35.6215	1.58829	0.794143	−	0.607731i	$-0.207920\pi$
$503$	35.6215	1.58829	0.794143	+	0.607731i	$0.207920\pi$
$504$	0	0
$505$	7.99206	0.355642
$506$	0	0
$507$	21.9694	0.975697
$508$	0	0
$509$	25.4988	1.13022	0.565108	−	0.825017i	$-0.308835\pi$
$509$	25.4988	1.13022	0.565108	+	0.825017i	$0.308835\pi$
$510$	0	0
$511$	0.710511	0.0314312
$512$	0	0
$513$	89.0818	3.93306
$514$	0	0
$515$	−3.36647	−0.148344
$516$	0	0
$517$	−31.3909	−1.38057
$518$	0	0
$519$	32.0820	1.40824
$520$	0	0
$521$	15.5300	0.680380	0.340190	−	0.940357i	$-0.389509\pi$
$521$	15.5300	0.680380	0.340190	+	0.940357i	$0.389509\pi$
$522$	0	0
$523$	10.7208	0.468787	0.234394	−	0.972142i	$-0.424690\pi$
$523$	10.7208	0.468787	0.234394	+	0.972142i	$0.424690\pi$
$524$	0	0
$525$	−14.6988	−0.641509
$526$	0	0
$527$	4.34698	0.189357
$528$	0	0
$529$	−21.4136	−0.931028
$530$	0	0
$531$	−17.4910	−0.759043
$532$	0	0
$533$	5.42957	0.235181
$534$	0	0
$535$	9.82598	0.424814
$536$	0	0
$537$	−44.0463	−1.90074
$538$	0	0
$539$	3.14807	0.135597
$540$	0	0
$541$	23.7475	1.02098	0.510492	−	0.859883i	$-0.329463\pi$
$541$	23.7475	1.02098	0.510492	+	0.859883i	$0.329463\pi$
$542$	0	0
$543$	32.5859	1.39840
$544$	0	0
$545$	−7.90771	−0.338729
$546$	0	0
$547$	−40.8933	−1.74847	−0.874235	−	0.485503i	$-0.838637\pi$
$547$	−40.8933	−1.74847	−0.874235	+	0.485503i	$0.838637\pi$
$548$	0	0
$549$	−24.9544	−1.06503
$550$	0	0
$551$	33.4124	1.42342
$552$	0	0
$553$	−11.8644	−0.504525
$554$	0	0
$555$	−1.58944	−0.0674680
$556$	0	0
$557$	−16.7946	−0.711610	−0.355805	−	0.934560i	$-0.615793\pi$
$557$	−16.7946	−0.711610	−0.355805	+	0.934560i	$0.615793\pi$
$558$	0	0
$559$	6.11449	0.258616
$560$	0	0
$561$	−9.67648	−0.408541
$562$	0	0
$563$	3.56290	0.150158	0.0750791	−	0.997178i	$-0.476079\pi$
$563$	3.56290	0.150158	0.0750791	+	0.997178i	$0.476079\pi$
$564$	0	0
$565$	−2.79054	−0.117399
$566$	0	0
$567$	17.0379	0.715524
$568$	0	0
$569$	−1.90395	−0.0798176	−0.0399088	−	0.999203i	$-0.512707\pi$
$569$	−1.90395	−0.0798176	−0.0399088	+	0.999203i	$0.512707\pi$
$570$	0	0
$571$	−2.75129	−0.115138	−0.0575689	−	0.998342i	$-0.518335\pi$
$571$	−2.75129	−0.115138	−0.0575689	+	0.998342i	$0.518335\pi$
$572$	0	0
$573$	11.1139	0.464290
$574$	0	0
$575$	5.90824	0.246390
$576$	0	0
$577$	−0.270046	−0.0112422	−0.00562108	−	0.999984i	$-0.501789\pi$
$577$	−0.270046	−0.0112422	−0.00562108	+	0.999984i	$0.501789\pi$
$578$	0	0
$579$	9.38143	0.389879
$580$	0	0
$581$	9.58105	0.397489
$582$	0	0
$583$	−35.9643	−1.48949
$584$	0	0
$585$	16.9582	0.701136
$586$	0	0
$587$	−31.7570	−1.31075	−0.655377	−	0.755302i	$-0.727490\pi$
$587$	−31.7570	−1.31075	−0.655377	+	0.755302i	$0.727490\pi$
$588$	0	0
$589$	−32.9910	−1.35937
$590$	0	0
$591$	−15.7468	−0.647735
$592$	0	0
$593$	36.0626	1.48091	0.740456	−	0.672105i	$-0.234610\pi$
$593$	36.0626	1.48091	0.740456	+	0.672105i	$0.234610\pi$
$594$	0	0
$595$	−0.545372	−0.0223581
$596$	0	0
$597$	5.64366	0.230980
$598$	0	0
$599$	10.7292	0.438381	0.219191	−	0.975682i	$-0.429658\pi$
$599$	10.7292	0.438381	0.219191	+	0.975682i	$0.429658\pi$
$600$	0	0
$601$	−36.0677	−1.47123	−0.735616	−	0.677399i	$-0.763107\pi$
$601$	−36.0677	−1.47123	−0.735616	+	0.677399i	$0.763107\pi$
$602$	0	0
$603$	−15.3576	−0.625408
$604$	0	0
$605$	−0.605795	−0.0246291
$606$	0	0
$607$	6.98617	0.283560	0.141780	−	0.989898i	$-0.454717\pi$
$607$	6.98617	0.283560	0.141780	+	0.989898i	$0.454717\pi$
$608$	0	0
$609$	14.0630	0.569861
$610$	0	0
$611$	−44.6063	−1.80458
$612$	0	0
$613$	33.0304	1.33408	0.667042	−	0.745020i	$-0.267560\pi$
$613$	33.0304	1.33408	0.667042	+	0.745020i	$0.267560\pi$
$614$	0	0
$615$	2.11447	0.0852635
$616$	0	0
$617$	41.5107	1.67116	0.835579	−	0.549370i	$-0.185132\pi$
$617$	41.5107	1.67116	0.835579	+	0.549370i	$0.185132\pi$
$618$	0	0
$619$	−27.3716	−1.10016	−0.550079	−	0.835113i	$-0.685402\pi$
$619$	−27.3716	−1.10016	−0.550079	+	0.835113i	$0.685402\pi$
$620$	0	0
$621$	−15.0708	−0.604769
$622$	0	0
$623$	−10.2082	−0.408982
$624$	0	0
$625$	20.4591	0.818365
$626$	0	0
$627$	73.4388	2.93286
$628$	0	0
$629$	0.894993	0.0356857
$630$	0	0
$631$	−2.56032	−0.101925	−0.0509624	−	0.998701i	$-0.516229\pi$
$631$	−2.56032	−0.101925	−0.0509624	+	0.998701i	$0.516229\pi$
$632$	0	0
$633$	66.9236	2.65997
$634$	0	0
$635$	−10.7784	−0.427729
$636$	0	0
$637$	4.47339	0.177242
$638$	0	0
$639$	−6.37293	−0.252109
$640$	0	0
$641$	−0.654189	−0.0258389	−0.0129195	−	0.999917i	$-0.504113\pi$
$641$	−0.654189	−0.0258389	−0.0129195	+	0.999917i	$0.504113\pi$
$642$	0	0
$643$	4.54964	0.179420	0.0897102	−	0.995968i	$-0.471406\pi$
$643$	4.54964	0.179420	0.0897102	+	0.995968i	$0.471406\pi$
$644$	0	0
$645$	2.38120	0.0937596
$646$	0	0
$647$	39.4069	1.54925	0.774623	−	0.632423i	$-0.217939\pi$
$647$	39.4069	1.54925	0.774623	+	0.632423i	$0.217939\pi$
$648$	0	0
$649$	−8.07535	−0.316985
$650$	0	0
$651$	−13.8856	−0.544221
$652$	0	0
$653$	−35.9592	−1.40719	−0.703596	−	0.710601i	$-0.748423\pi$
$653$	−35.9592	−1.40719	−0.703596	+	0.710601i	$0.748423\pi$
$654$	0	0
$655$	6.51978	0.254749
$656$	0	0
$657$	4.84472	0.189010
$658$	0	0
$659$	−44.9728	−1.75189	−0.875946	−	0.482410i	$-0.839762\pi$
$659$	−44.9728	−1.75189	−0.875946	+	0.482410i	$0.839762\pi$
$660$	0	0
$661$	−7.71462	−0.300064	−0.150032	−	0.988681i	$-0.547938\pi$
$661$	−7.71462	−0.300064	−0.150032	+	0.988681i	$0.547938\pi$
$662$	0	0
$663$	−13.7502	−0.534014
$664$	0	0
$665$	4.13906	0.160506
$666$	0	0
$667$	−5.65267	−0.218872
$668$	0	0
$669$	22.2871	0.861667
$670$	0	0
$671$	−11.5211	−0.444768
$672$	0	0
$673$	−28.3929	−1.09446	−0.547232	−	0.836981i	$-0.684319\pi$
$673$	−28.3929	−1.09446	−0.547232	+	0.836981i	$0.684319\pi$
$674$	0	0
$675$	−56.1294	−2.16042
$676$	0	0
$677$	−10.7285	−0.412331	−0.206166	−	0.978517i	$-0.566099\pi$
$677$	−10.7285	−0.412331	−0.206166	+	0.978517i	$0.566099\pi$
$678$	0	0
$679$	−3.03684	−0.116543
$680$	0	0
$681$	−53.9243	−2.06639
$682$	0	0
$683$	−15.0604	−0.576270	−0.288135	−	0.957590i	$-0.593035\pi$
$683$	−15.0604	−0.576270	−0.288135	+	0.957590i	$0.593035\pi$
$684$	0	0
$685$	−5.76664	−0.220332
$686$	0	0
$687$	2.75404	0.105073
$688$	0	0
$689$	−51.1050	−1.94695
$690$	0	0
$691$	−1.86400	−0.0709100	−0.0354550	−	0.999371i	$-0.511288\pi$
$691$	−1.86400	−0.0709100	−0.0354550	+	0.999371i	$0.511288\pi$
$692$	0	0
$693$	21.4656	0.815409
$694$	0	0
$695$	−3.79987	−0.144137
$696$	0	0
$697$	−1.19063	−0.0450983
$698$	0	0
$699$	15.2189	0.575632
$700$	0	0
$701$	8.17265	0.308677	0.154338	−	0.988018i	$-0.450675\pi$
$701$	8.17265	0.308677	0.154338	+	0.988018i	$0.450675\pi$
$702$	0	0
$703$	−6.79247	−0.256183
$704$	0	0
$705$	−17.3712	−0.654239
$706$	0	0
$707$	14.3752	0.540633
$708$	0	0
$709$	38.9463	1.46266	0.731330	−	0.682024i	$-0.238900\pi$
$709$	38.9463	1.46266	0.731330	+	0.682024i	$0.238900\pi$
$710$	0	0
$711$	−80.8990	−3.03395
$712$	0	0
$713$	5.58138	0.209024
$714$	0	0
$715$	7.82939	0.292802
$716$	0	0
$717$	−15.3864	−0.574615
$718$	0	0
$719$	3.79808	0.141645	0.0708223	−	0.997489i	$-0.477438\pi$
$719$	3.79808	0.141645	0.0708223	+	0.997489i	$0.477438\pi$
$720$	0	0
$721$	−6.05520	−0.225507
$722$	0	0
$723$	−34.0268	−1.26547
$724$	0	0
$725$	−21.0527	−0.781879
$726$	0	0
$727$	−25.5053	−0.945937	−0.472969	−	0.881079i	$-0.656818\pi$
$727$	−25.5053	−0.945937	−0.472969	+	0.881079i	$0.656818\pi$
$728$	0	0
$729$	3.69376	0.136806
$730$	0	0
$731$	−1.34082	−0.0495921
$732$	0	0
$733$	−41.0330	−1.51559	−0.757794	−	0.652494i	$-0.773723\pi$
$733$	−41.0330	−1.51559	−0.757794	+	0.652494i	$0.773723\pi$
$734$	0	0
$735$	1.74209	0.0642581
$736$	0	0
$737$	−7.09038	−0.261178
$738$	0	0
$739$	2.36133	0.0868628	0.0434314	−	0.999056i	$-0.486171\pi$
$739$	2.36133	0.0868628	0.0434314	+	0.999056i	$0.486171\pi$
$740$	0	0
$741$	104.356	3.83361
$742$	0	0
$743$	−24.1373	−0.885513	−0.442756	−	0.896642i	$-0.645999\pi$
$743$	−24.1373	−0.885513	−0.442756	+	0.896642i	$0.645999\pi$
$744$	0	0
$745$	1.36846	0.0501366
$746$	0	0
$747$	65.3297	2.39029
$748$	0	0
$749$	17.6738	0.645787
$750$	0	0
$751$	−17.6703	−0.644797	−0.322398	−	0.946604i	$-0.604489\pi$
$751$	−17.6703	−0.644797	−0.322398	+	0.946604i	$0.604489\pi$
$752$	0	0
$753$	−36.2717	−1.32181
$754$	0	0
$755$	11.5713	0.421123
$756$	0	0
$757$	−20.2849	−0.737268	−0.368634	−	0.929575i	$-0.620174\pi$
$757$	−20.2849	−0.737268	−0.368634	+	0.929575i	$0.620174\pi$
$758$	0	0
$759$	−12.4243	−0.450973
$760$	0	0
$761$	−13.4406	−0.487220	−0.243610	−	0.969873i	$-0.578332\pi$
$761$	−13.4406	−0.487220	−0.243610	+	0.969873i	$0.578332\pi$
$762$	0	0
$763$	−14.2234	−0.514923
$764$	0	0
$765$	−3.71870	−0.134450
$766$	0	0
$767$	−11.4750	−0.414338
$768$	0	0
$769$	−42.7134	−1.54029	−0.770143	−	0.637871i	$-0.779815\pi$
$769$	−42.7134	−1.54029	−0.770143	+	0.637871i	$0.779815\pi$
$770$	0	0
$771$	64.5902	2.32616
$772$	0	0
$773$	−24.9239	−0.896451	−0.448225	−	0.893921i	$-0.647944\pi$
$773$	−24.9239	−0.896451	−0.448225	+	0.893921i	$0.647944\pi$
$774$	0	0
$775$	20.7873	0.746700
$776$	0	0
$777$	−2.85889	−0.102562
$778$	0	0
$779$	9.03617	0.323754
$780$	0	0
$781$	−2.94230	−0.105284
$782$	0	0
$783$	53.7014	1.91913
$784$	0	0
$785$	−7.97582	−0.284669
$786$	0	0
$787$	−38.8329	−1.38424	−0.692121	−	0.721781i	$-0.743324\pi$
$787$	−38.8329	−1.38424	−0.692121	+	0.721781i	$0.743324\pi$
$788$	0	0
$789$	42.7083	1.52046
$790$	0	0
$791$	−5.01929	−0.178466
$792$	0	0
$793$	−16.3714	−0.581367
$794$	0	0
$795$	−19.9021	−0.705854
$796$	0	0
$797$	52.7407	1.86817	0.934085	−	0.357051i	$-0.116218\pi$
$797$	52.7407	1.86817	0.934085	+	0.357051i	$0.116218\pi$
$798$	0	0
$799$	9.78152	0.346045
$800$	0	0
$801$	−69.6058	−2.45940
$802$	0	0
$803$	2.23674	0.0789329
$804$	0	0
$805$	−0.700241	−0.0246802
$806$	0	0
$807$	−19.6310	−0.691045
$808$	0	0
$809$	−44.3124	−1.55794	−0.778970	−	0.627061i	$-0.784258\pi$
$809$	−44.3124	−1.55794	−0.778970	+	0.627061i	$0.784258\pi$
$810$	0	0
$811$	13.3564	0.469008	0.234504	−	0.972115i	$-0.424653\pi$
$811$	13.3564	0.469008	0.234504	+	0.972115i	$0.424653\pi$
$812$	0	0
$813$	−39.7483	−1.39403
$814$	0	0
$815$	−9.34707	−0.327414
$816$	0	0
$817$	10.1760	0.356015
$818$	0	0
$819$	30.5024	1.06584
$820$	0	0
$821$	49.9789	1.74427	0.872137	−	0.489262i	$-0.162734\pi$
$821$	49.9789	1.74427	0.872137	+	0.489262i	$0.162734\pi$
$822$	0	0
$823$	−23.7136	−0.826605	−0.413302	−	0.910594i	$-0.635625\pi$
$823$	−23.7136	−0.826605	−0.413302	+	0.910594i	$0.635625\pi$
$824$	0	0
$825$	−46.2729	−1.61102
$826$	0	0
$827$	−49.2787	−1.71359	−0.856794	−	0.515659i	$-0.827547\pi$
$827$	−49.2787	−1.71359	−0.856794	+	0.515659i	$0.827547\pi$
$828$	0	0
$829$	16.9516	0.588755	0.294377	−	0.955689i	$-0.404888\pi$
$829$	16.9516	0.588755	0.294377	+	0.955689i	$0.404888\pi$
$830$	0	0
$831$	75.7069	2.62624
$832$	0	0
$833$	−0.980951	−0.0339879
$834$	0	0
$835$	−0.616386	−0.0213309
$836$	0	0
$837$	−53.0242	−1.83279
$838$	0	0
$839$	1.32175	0.0456320	0.0228160	−	0.999740i	$-0.492737\pi$
$839$	1.32175	0.0456320	0.0228160	+	0.999740i	$0.492737\pi$
$840$	0	0
$841$	−8.85792	−0.305445
$842$	0	0
$843$	40.3052	1.38818
$844$	0	0
$845$	3.89798	0.134095
$846$	0	0
$847$	−1.08963	−0.0374402
$848$	0	0
$849$	3.89560	0.133697
$850$	0	0
$851$	1.14914	0.0393921
$852$	0	0
$853$	47.7977	1.63656	0.818281	−	0.574819i	$-0.194928\pi$
$853$	47.7977	1.63656	0.818281	+	0.574819i	$0.194928\pi$
$854$	0	0
$855$	28.2227	0.965196
$856$	0	0
$857$	12.2024	0.416826	0.208413	−	0.978041i	$-0.433170\pi$
$857$	12.2024	0.416826	0.208413	+	0.978041i	$0.433170\pi$
$858$	0	0
$859$	13.6056	0.464218	0.232109	−	0.972690i	$-0.425437\pi$
$859$	13.6056	0.464218	0.232109	+	0.972690i	$0.425437\pi$
$860$	0	0
$861$	3.80325	0.129614
$862$	0	0
$863$	−48.5092	−1.65127	−0.825636	−	0.564204i	$-0.809183\pi$
$863$	−48.5092	−1.65127	−0.825636	+	0.564204i	$0.809183\pi$
$864$	0	0
$865$	5.69222	0.193541
$866$	0	0
$867$	−50.2538	−1.70671
$868$	0	0
$869$	−37.3500	−1.26701
$870$	0	0
$871$	−10.0754	−0.341391
$872$	0	0
$873$	−20.7071	−0.700829
$874$	0	0
$875$	−5.38779	−0.182140
$876$	0	0
$877$	22.0915	0.745979	0.372989	−	0.927836i	$-0.378333\pi$
$877$	22.0915	0.745979	0.372989	+	0.927836i	$0.378333\pi$
$878$	0	0
$879$	79.7365	2.68944
$880$	0	0
$881$	−22.7269	−0.765688	−0.382844	−	0.923813i	$-0.625055\pi$
$881$	−22.7269	−0.765688	−0.382844	+	0.923813i	$0.625055\pi$
$882$	0	0
$883$	42.8271	1.44125	0.720624	−	0.693327i	$-0.243856\pi$
$883$	42.8271	1.44125	0.720624	+	0.693327i	$0.243856\pi$
$884$	0	0
$885$	−4.46877	−0.150216
$886$	0	0
$887$	38.7161	1.29996	0.649980	−	0.759951i	$-0.274777\pi$
$887$	38.7161	1.29996	0.649980	+	0.759951i	$0.274777\pi$
$888$	0	0
$889$	−19.3869	−0.650217
$890$	0	0
$891$	53.6365	1.79689
$892$	0	0
$893$	−74.2360	−2.48421
$894$	0	0
$895$	−7.81501	−0.261227
$896$	0	0
$897$	−17.6548	−0.589478
$898$	0	0
$899$	−19.8881	−0.663304
$900$	0	0
$901$	11.2066	0.373346
$902$	0	0
$903$	4.28301	0.142530
$904$	0	0
$905$	5.78163	0.192188
$906$	0	0
$907$	−36.0235	−1.19614	−0.598070	−	0.801444i	$-0.704065\pi$
$907$	−36.0235	−1.19614	−0.598070	+	0.801444i	$0.704065\pi$
$908$	0	0
$909$	98.0189	3.25108
$910$	0	0
$911$	52.6922	1.74577	0.872886	−	0.487924i	$-0.162246\pi$
$911$	52.6922	1.74577	0.872886	+	0.487924i	$0.162246\pi$
$912$	0	0
$913$	30.1618	0.998212
$914$	0	0
$915$	−6.37561	−0.210771
$916$	0	0
$917$	11.7270	0.387260
$918$	0	0
$919$	25.5116	0.841549	0.420775	−	0.907165i	$-0.361758\pi$
$919$	25.5116	0.841549	0.420775	+	0.907165i	$0.361758\pi$
$920$	0	0
$921$	7.75436	0.255515
$922$	0	0
$923$	−4.18098	−0.137619
$924$	0	0
$925$	4.27986	0.140721
$926$	0	0
$927$	−41.2882	−1.35608
$928$	0	0
$929$	49.1998	1.61419	0.807097	−	0.590419i	$-0.201038\pi$
$929$	49.1998	1.61419	0.807097	+	0.590419i	$0.201038\pi$
$930$	0	0
$931$	7.44484	0.243995
$932$	0	0
$933$	−54.6126	−1.78794
$934$	0	0
$935$	−1.71687	−0.0561477
$936$	0	0
$937$	44.3522	1.44892	0.724462	−	0.689315i	$-0.242088\pi$
$937$	44.3522	1.44892	0.724462	+	0.689315i	$0.242088\pi$
$938$	0	0
$939$	88.8120	2.89827
$940$	0	0
$941$	−19.8946	−0.648545	−0.324273	−	0.945964i	$-0.605120\pi$
$941$	−19.8946	−0.648545	−0.324273	+	0.945964i	$0.605120\pi$
$942$	0	0
$943$	−1.52873	−0.0497822
$944$	0	0
$945$	6.65242	0.216403
$946$	0	0
$947$	31.5128	1.02403	0.512014	−	0.858977i	$-0.328899\pi$
$947$	31.5128	1.02403	0.512014	+	0.858977i	$0.328899\pi$
$948$	0	0
$949$	3.17839	0.103175
$950$	0	0
$951$	36.5323	1.18464
$952$	0	0
$953$	52.5006	1.70066	0.850330	−	0.526250i	$-0.176402\pi$
$953$	52.5006	1.70066	0.850330	+	0.526250i	$0.176402\pi$
$954$	0	0
$955$	1.97191	0.0638095
$956$	0	0
$957$	44.2713	1.43109
$958$	0	0
$959$	−10.3723	−0.334940
$960$	0	0
$961$	−11.3627	−0.366540
$962$	0	0
$963$	120.511	3.88342
$964$	0	0
$965$	1.66452	0.0535829
$966$	0	0
$967$	22.7183	0.730572	0.365286	−	0.930895i	$-0.380971\pi$
$967$	22.7183	0.730572	0.365286	+	0.930895i	$0.380971\pi$
$968$	0	0
$969$	−22.8838	−0.735133
$970$	0	0
$971$	−12.0402	−0.386387	−0.193193	−	0.981161i	$-0.561884\pi$
$971$	−12.0402	−0.386387	−0.193193	+	0.981161i	$0.561884\pi$
$972$	0	0
$973$	−6.83475	−0.219112
$974$	0	0
$975$	−65.7535	−2.10580
$976$	0	0
$977$	37.7274	1.20701	0.603503	−	0.797361i	$-0.293771\pi$
$977$	37.7274	1.20701	0.603503	+	0.797361i	$0.293771\pi$
$978$	0	0
$979$	−32.1361	−1.02707
$980$	0	0
$981$	−96.9845	−3.09648
$982$	0	0
$983$	−45.1569	−1.44028	−0.720141	−	0.693828i	$-0.755923\pi$
$983$	−45.1569	−1.44028	−0.720141	+	0.693828i	$0.755923\pi$
$984$	0	0
$985$	−2.79390	−0.0890212
$986$	0	0
$987$	−31.2453	−0.994549
$988$	0	0
$989$	−1.72157	−0.0547428
$990$	0	0
$991$	−46.2280	−1.46848	−0.734240	−	0.678890i	$-0.762462\pi$
$991$	−46.2280	−1.46848	−0.734240	+	0.678890i	$0.762462\pi$
$992$	0	0
$993$	1.04343	0.0331121
$994$	0	0
$995$	1.00134	0.0317446
$996$	0	0
$997$	38.5314	1.22030	0.610152	−	0.792285i	$-0.291109\pi$
$997$	38.5314	1.22030	0.610152	+	0.792285i	$0.291109\pi$
$998$	0	0
$999$	−10.9171	−0.345401

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7168.2.a.bj.1.12		12
4.3	odd	2		7168.2.a.bi.1.1		12
8.3	odd	2		7168.2.a.bi.1.12		12
8.5	even	2	inner	7168.2.a.bj.1.1		12
32.3	odd	8		448.2.m.d.337.1		12
32.5	even	8		896.2.m.g.225.1		12
32.11	odd	8		448.2.m.d.113.1		12
32.13	even	8		896.2.m.g.673.1		12
32.19	odd	8		896.2.m.h.673.6		12
32.21	even	8		112.2.m.d.85.2	yes	12
32.27	odd	8		896.2.m.h.225.6		12
32.29	even	8		112.2.m.d.29.2	✓	12
224.53	even	24		784.2.x.l.373.3		24
224.61	odd	24		784.2.x.m.557.3		24
224.93	even	24		784.2.x.l.557.3		24
224.117	odd	24		784.2.x.m.165.6		24
224.125	odd	8		784.2.m.h.589.2		12
224.149	even	24		784.2.x.l.165.6		24
224.157	odd	24		784.2.x.m.765.6		24
224.181	odd	8		784.2.m.h.197.2		12
224.213	odd	24		784.2.x.m.373.3		24
224.221	even	24		784.2.x.l.765.6		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.m.d.29.2	✓	12	32.29	even	8
112.2.m.d.85.2	yes	12	32.21	even	8
448.2.m.d.113.1		12	32.11	odd	8
448.2.m.d.337.1		12	32.3	odd	8
784.2.m.h.197.2		12	224.181	odd	8
784.2.m.h.589.2		12	224.125	odd	8
784.2.x.l.165.6		24	224.149	even	24
784.2.x.l.373.3		24	224.53	even	24
784.2.x.l.557.3		24	224.93	even	24
784.2.x.l.765.6		24	224.221	even	24
784.2.x.m.165.6		24	224.117	odd	24
784.2.x.m.373.3		24	224.213	odd	24
784.2.x.m.557.3		24	224.61	odd	24
784.2.x.m.765.6		24	224.157	odd	24
896.2.m.g.225.1		12	32.5	even	8
896.2.m.g.673.1		12	32.13	even	8
896.2.m.h.225.6		12	32.27	odd	8
896.2.m.h.673.6		12	32.19	odd	8
7168.2.a.bi.1.1		12	4.3	odd	2
7168.2.a.bi.1.12		12	8.3	odd	2
7168.2.a.bj.1.1		12	8.5	even	2	inner
7168.2.a.bj.1.12		12	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:	\( N \)	\(=\)	\( 7168 = 2^{10} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7168.a (trivial)

\(f(q)\)	\(=\)	\(q+3.13347 q^{3} +0.555963 q^{5} +1.00000 q^{7} +6.81864 q^{9} +O(q^{10})\)	\(q+3.13347 q^{3} +0.555963 q^{5} +1.00000 q^{7} +6.81864 q^{9} +3.14807 q^{11} +4.47339 q^{13} +1.74209 q^{15} -0.980951 q^{17} +7.44484 q^{19} +3.13347 q^{21} -1.25951 q^{23} -4.69090 q^{25} +11.9656 q^{27} +4.48799 q^{29} -4.43140 q^{31} +9.86440 q^{33} +0.555963 q^{35} -0.912373 q^{37} +14.0172 q^{39} +1.21375 q^{41} +1.36686 q^{43} +3.79091 q^{45} -9.97147 q^{47} +1.00000 q^{49} -3.07378 q^{51} -11.4242 q^{53} +1.75021 q^{55} +23.3282 q^{57} -2.56517 q^{59} -3.65974 q^{61} +6.81864 q^{63} +2.48704 q^{65} -2.25229 q^{67} -3.94663 q^{69} -0.934634 q^{71} +0.710511 q^{73} -14.6988 q^{75} +3.14807 q^{77} -11.8644 q^{79} +17.0379 q^{81} +9.58105 q^{83} -0.545372 q^{85} +14.0630 q^{87} -10.2082 q^{89} +4.47339 q^{91} -13.8856 q^{93} +4.13906 q^{95} -3.03684 q^{97} +21.4656 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q + 12 * q^7 + 12 * q^9	\( 12 q + 12 q^{7} + 12 q^{9} + 24 q^{15} + 8 q^{17} + 16 q^{23} + 20 q^{25} - 8 q^{31} + 16 q^{39} + 32 q^{41} - 16 q^{47} + 12 q^{49} + 24 q^{55} + 64 q^{57} + 12 q^{63} + 32 q^{65} + 8 q^{71} - 24 q^{79} + 44 q^{81} - 32 q^{87} + 24 q^{89} + 48 q^{97}+O(q^{100}) \) 12 * q + 12 * q^7 + 12 * q^9 + 24 * q^15 + 8 * q^17 + 16 * q^23 + 20 * q^25 - 8 * q^31 + 16 * q^39 + 32 * q^41 - 16 * q^47 + 12 * q^49 + 24 * q^55 + 64 * q^57 + 12 * q^63 + 32 * q^65 + 8 * q^71 - 24 * q^79 + 44 * q^81 - 32 * q^87 + 24 * q^89 + 48 * q^97

Introduction

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