LMFDB - Embedded newform 3584.2.b.f.1793.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3584,2,Mod(1793,3584)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3584, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("3584.1793");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3584 = 2^{9} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3584.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$28.6183840844$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.46091337728.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 14x^{6} + 35x^{4} + 16x^{2} + 2 $ x^8 + 14x^6 + 35x^4 + 16*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1793.8
Root			$-0.481620i$ of defining polynomial
Character	$\chi$	$=$	3584.1793
Dual form			3584.2.b.f.1793.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.93637i q^{3} +2.25526i q^{5} +1.00000 q^{7} -5.62226 q^{9} +O(q^{10})$	$q+2.93637i q^{3} +2.25526i q^{5} +1.00000 q^{7} -5.62226 q^{9} -3.47154i q^{11} -5.44467i q^{13} -6.62226 q^{15} -6.53686 q^{17} -3.54175i q^{19} +2.93637i q^{21} +0.0853971 q^{23} -0.0861758 q^{25} -7.69992i q^{27} -2.32547i q^{29} -1.17157 q^{31} +10.1937 q^{33} +2.25526i q^{35} -4.05336i q^{37} +15.9876 q^{39} +11.2445 q^{41} -6.76025i q^{43} -12.6796i q^{45} +6.41609 q^{47} +1.00000 q^{49} -19.1946i q^{51} +13.2148i q^{53} +7.82921 q^{55} +10.3999 q^{57} -1.10918i q^{59} -4.83929i q^{61} -5.62226 q^{63} +12.2791 q^{65} -2.40120i q^{67} +0.250757i q^{69} -14.6098 q^{71} +6.53686 q^{73} -0.253044i q^{75} -3.47154i q^{77} -14.5369 q^{79} +5.74303 q^{81} -11.7478i q^{83} -14.7423i q^{85} +6.82843 q^{87} +12.6061 q^{89} -5.44467i q^{91} -3.44017i q^{93} +7.98755 q^{95} -14.1245 q^{97} +19.5179i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{7} - 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^7 - 8 * q^9	$ 8 q + 8 q^{7} - 8 q^{9} - 16 q^{15} - 8 q^{17} + 8 q^{23} - 16 q^{25} - 32 q^{31} - 8 q^{33} + 24 q^{39} + 16 q^{41} + 8 q^{49} + 48 q^{55} + 8 q^{57} - 8 q^{63} + 16 q^{65} + 24 q^{71} + 8 q^{73} - 72 q^{79} + 16 q^{81} + 32 q^{87} + 40 q^{89} - 40 q^{95} - 40 q^{97}+O(q^{100}) $ 8 * q + 8 * q^7 - 8 * q^9 - 16 * q^15 - 8 * q^17 + 8 * q^23 - 16 * q^25 - 32 * q^31 - 8 * q^33 + 24 * q^39 + 16 * q^41 + 8 * q^49 + 48 * q^55 + 8 * q^57 - 8 * q^63 + 16 * q^65 + 24 * q^71 + 8 * q^73 - 72 * q^79 + 16 * q^81 + 32 * q^87 + 40 * q^89 - 40 * q^95 - 40 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3584\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	2.93637i	1.69531i	0.530545	+	0.847657i	$0.321987\pi$
$3$	2.93637i	1.69531i	−0.530545	+	0.847657i	$0.678013\pi$
$4$	0	0
$5$	2.25526i	1.00858i	0.863534	+	0.504290i	$0.168246\pi$
$5$	2.25526i	1.00858i	−0.863534	+	0.504290i	$0.831754\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−5.62226	−1.87409
$10$	0	0
$11$	− 3.47154i	− 1.04671i	−0.852115	−	0.523354i	$-0.824680\pi$
$11$	− 3.47154i	− 1.04671i	0.852115	−	0.523354i	$-0.175320\pi$
$12$	0	0
$13$	− 5.44467i	− 1.51008i	−0.655679	−	0.755040i	$-0.727618\pi$
$13$	− 5.44467i	− 1.51008i	0.655679	−	0.755040i	$-0.272382\pi$
$14$	0	0
$15$	−6.62226	−1.70986
$16$	0	0
$17$	−6.53686	−1.58542	−0.792711	−	0.609597i	$-0.791331\pi$
$17$	−6.53686	−1.58542	−0.792711	+	0.609597i	$0.791331\pi$
$18$	0	0
$19$	− 3.54175i	− 0.812533i	−0.913755	−	0.406267i	$-0.866830\pi$
$19$	− 3.54175i	− 0.812533i	0.913755	−	0.406267i	$-0.133170\pi$
$20$	0	0
$21$	2.93637i	0.640768i
$22$	0	0
$23$	0.0853971	0.0178065	0.00890326	−	0.999960i	$-0.497166\pi$
$23$	0.0853971	0.0178065	0.00890326	+	0.999960i	$0.497166\pi$
$24$	0	0
$25$	−0.0861758	−0.0172352
$26$	0	0
$27$	− 7.69992i	− 1.48185i
$28$	0	0
$29$	− 2.32547i	− 0.431828i	−0.976412	−	0.215914i	$-0.930727\pi$
$29$	− 2.32547i	− 0.431828i	0.976412	−	0.215914i	$-0.0692731\pi$
$30$	0	0
$31$	−1.17157	−0.210421	−0.105210	−	0.994450i	$-0.533552\pi$
$31$	−1.17157	−0.210421	−0.105210	+	0.994450i	$0.533552\pi$
$32$	0	0
$33$	10.1937	1.77450
$34$	0	0
$35$	2.25526i	0.381208i
$36$	0	0
$37$	− 4.05336i	− 0.666368i	−0.942862	−	0.333184i	$-0.891877\pi$
$37$	− 4.05336i	− 0.666368i	0.942862	−	0.333184i	$-0.108123\pi$
$38$	0	0
$39$	15.9876	2.56006
$40$	0	0
$41$	11.2445	1.75610	0.878050	−	0.478570i	$-0.158845\pi$
$41$	11.2445	1.75610	0.878050	+	0.478570i	$0.158845\pi$
$42$	0	0
$43$	− 6.76025i	− 1.03093i	−0.856911	−	0.515464i	$-0.827620\pi$
$43$	− 6.76025i	− 1.03093i	0.856911	−	0.515464i	$-0.172380\pi$
$44$	0	0
$45$	− 12.6796i	− 1.89017i
$46$	0	0
$47$	6.41609	0.935883	0.467942	−	0.883759i	$-0.344996\pi$
$47$	6.41609	0.935883	0.467942	+	0.883759i	$0.344996\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	− 19.1946i	− 2.68779i
$52$	0	0
$53$	13.2148i	1.81519i	0.419844	+	0.907596i	$0.362085\pi$
$53$	13.2148i	1.81519i	−0.419844	+	0.907596i	$0.637915\pi$
$54$	0	0
$55$	7.82921	1.05569
$56$	0	0
$57$	10.3999	1.37750
$58$	0	0
$59$	− 1.10918i	− 0.144403i	−0.997390	−	0.0722017i	$-0.976997\pi$
$59$	− 1.10918i	− 0.144403i	0.997390	−	0.0722017i	$-0.0230025\pi$
$60$	0	0
$61$	− 4.83929i	− 0.619607i	−0.950801	−	0.309804i	$-0.899737\pi$
$61$	− 4.83929i	− 0.619607i	0.950801	−	0.309804i	$-0.100263\pi$
$62$	0	0
$63$	−5.62226	−0.708338
$64$	0	0
$65$	12.2791	1.52304
$66$	0	0
$67$	− 2.40120i	− 0.293353i	−0.989185	−	0.146677i	$-0.953142\pi$
$67$	− 2.40120i	− 0.293353i	0.989185	−	0.146677i	$-0.0468576\pi$
$68$	0	0
$69$	0.250757i	0.0301876i
$70$	0	0
$71$	−14.6098	−1.73387	−0.866933	−	0.498425i	$-0.833912\pi$
$71$	−14.6098	−1.73387	−0.866933	+	0.498425i	$0.833912\pi$
$72$	0	0
$73$	6.53686	0.765082	0.382541	−	0.923939i	$-0.375049\pi$
$73$	6.53686	0.765082	0.382541	+	0.923939i	$0.375049\pi$
$74$	0	0
$75$	− 0.253044i	− 0.0292190i
$76$	0	0
$77$	− 3.47154i	− 0.395619i
$78$	0	0
$79$	−14.5369	−1.63552	−0.817762	−	0.575556i	$-0.804786\pi$
$79$	−14.5369	−1.63552	−0.817762	+	0.575556i	$0.804786\pi$
$80$	0	0
$81$	5.74303	0.638114
$82$	0	0
$83$	− 11.7478i	− 1.28948i	−0.764400	−	0.644742i	$-0.776965\pi$
$83$	− 11.7478i	− 1.28948i	0.764400	−	0.644742i	$-0.223035\pi$
$84$	0	0
$85$	− 14.7423i	− 1.59903i
$86$	0	0
$87$	6.82843	0.732084
$88$	0	0
$89$	12.6061	1.33624	0.668119	−	0.744054i	$-0.267100\pi$
$89$	12.6061	1.33624	0.668119	+	0.744054i	$0.267100\pi$
$90$	0	0
$91$	− 5.44467i	− 0.570756i
$92$	0	0
$93$	− 3.44017i	− 0.356729i
$94$	0	0
$95$	7.98755	0.819505
$96$	0	0
$97$	−14.1245	−1.43413	−0.717064	−	0.697007i	$-0.754515\pi$
$97$	−14.1245	−1.43413	−0.717064	+	0.697007i	$0.754515\pi$
$98$	0	0
$99$	19.5179i	1.96162i
$100$	0	0
$101$	− 6.30081i	− 0.626954i	−0.949596	−	0.313477i	$-0.898506\pi$
$101$	− 6.30081i	− 0.626954i	0.949596	−	0.313477i	$-0.101494\pi$
$102$	0	0
$103$	−17.5590	−1.73014	−0.865070	−	0.501651i	$-0.832726\pi$
$103$	−17.5590	−1.73014	−0.865070	+	0.501651i	$0.832726\pi$
$104$	0	0
$105$	−6.62226	−0.646266
$106$	0	0
$107$	− 12.4344i	− 1.20208i	−0.799220	−	0.601039i	$-0.794754\pi$
$107$	− 12.4344i	− 1.20208i	0.799220	−	0.601039i	$-0.205246\pi$
$108$	0	0
$109$	14.2851i	1.36827i	0.729356	+	0.684134i	$0.239820\pi$
$109$	14.2851i	1.36827i	−0.729356	+	0.684134i	$0.760180\pi$
$110$	0	0
$111$	11.9022	1.12970
$112$	0	0
$113$	15.6444	1.47170	0.735851	−	0.677144i	$-0.236782\pi$
$113$	15.6444	1.47170	0.735851	+	0.677144i	$0.236782\pi$
$114$	0	0
$115$	0.192592i	0.0179593i
$116$	0	0
$117$	30.6113i	2.83002i
$118$	0	0
$119$	−6.53686	−0.599233
$120$	0	0
$121$	−1.05158	−0.0955983
$122$	0	0
$123$	33.0181i	2.97714i
$124$	0	0
$125$	11.0819i	0.991198i
$126$	0	0
$127$	−3.15536	−0.279993	−0.139997	−	0.990152i	$-0.544709\pi$
$127$	−3.15536	−0.279993	−0.139997	+	0.990152i	$0.544709\pi$
$128$	0	0
$129$	19.8506	1.74775
$130$	0	0
$131$	− 11.8992i	− 1.03964i	−0.854276	−	0.519820i	$-0.825999\pi$
$131$	− 11.8992i	− 1.03964i	0.854276	−	0.519820i	$-0.174001\pi$
$132$	0	0
$133$	− 3.54175i	− 0.307109i
$134$	0	0
$135$	17.3653	1.49457
$136$	0	0
$137$	−11.3653	−0.971002	−0.485501	−	0.874236i	$-0.661363\pi$
$137$	−11.3653	−0.971002	−0.485501	+	0.874236i	$0.661363\pi$
$138$	0	0
$139$	− 15.7933i	− 1.33957i	−0.742555	−	0.669786i	$-0.766386\pi$
$139$	− 15.7933i	− 1.33957i	0.742555	−	0.669786i	$-0.233614\pi$
$140$	0	0
$141$	18.8400i	1.58662i
$142$	0	0
$143$	−18.9014	−1.58061
$144$	0	0
$145$	5.24452	0.435534
$146$	0	0
$147$	2.93637i	0.242188i
$148$	0	0
$149$	5.12370i	0.419750i	0.977728	+	0.209875i	$0.0673057\pi$
$149$	5.12370i	0.419750i	−0.977728	+	0.209875i	$0.932694\pi$
$150$	0	0
$151$	8.08540	0.657980	0.328990	−	0.944333i	$-0.393292\pi$
$151$	8.08540	0.657980	0.328990	+	0.944333i	$0.393292\pi$
$152$	0	0
$153$	36.7519	2.97122
$154$	0	0
$155$	− 2.64220i	− 0.212226i
$156$	0	0
$157$	− 9.23946i	− 0.737389i	−0.929551	−	0.368695i	$-0.879805\pi$
$157$	− 9.23946i	− 0.737389i	0.929551	−	0.368695i	$-0.120195\pi$
$158$	0	0
$159$	−38.8035	−3.07732
$160$	0	0
$161$	0.0853971	0.00673023
$162$	0	0
$163$	0.829343i	0.0649592i	0.999472	+	0.0324796i	$0.0103404\pi$
$163$	0.829343i	0.0649592i	−0.999472	+	0.0324796i	$0.989660\pi$
$164$	0	0
$165$	22.9894i	1.78972i
$166$	0	0
$167$	12.0729	0.934233	0.467116	−	0.884196i	$-0.345293\pi$
$167$	12.0729	0.934233	0.467116	+	0.884196i	$0.345293\pi$
$168$	0	0
$169$	−16.6444	−1.28034
$170$	0	0
$171$	19.9126i	1.52276i
$172$	0	0
$173$	− 3.57635i	− 0.271905i	−0.990715	−	0.135953i	$-0.956591\pi$
$173$	− 3.57635i	− 0.271905i	0.990715	−	0.135953i	$-0.0434095\pi$
$174$	0	0
$175$	−0.0861758	−0.00651428
$176$	0	0
$177$	3.25697	0.244809
$178$	0	0
$179$	− 11.0566i	− 0.826406i	−0.910639	−	0.413203i	$-0.864410\pi$
$179$	− 11.0566i	− 0.826406i	0.910639	−	0.413203i	$-0.135590\pi$
$180$	0	0
$181$	− 4.97971i	− 0.370139i	−0.982725	−	0.185069i	$-0.940749\pi$
$181$	− 4.97971i	− 0.370139i	0.982725	−	0.185069i	$-0.0592510\pi$
$182$	0	0
$183$	14.2099	1.05043
$184$	0	0
$185$	9.14136	0.672086
$186$	0	0
$187$	22.6930i	1.65947i
$188$	0	0
$189$	− 7.69992i	− 0.560087i
$190$	0	0
$191$	−4.75924	−0.344366	−0.172183	−	0.985065i	$-0.555082\pi$
$191$	−4.75924	−0.344366	−0.172183	+	0.985065i	$0.555082\pi$
$192$	0	0
$193$	−6.03913	−0.434706	−0.217353	−	0.976093i	$-0.569742\pi$
$193$	−6.03913	−0.434706	−0.217353	+	0.976093i	$0.569742\pi$
$194$	0	0
$195$	36.0560i	2.58202i
$196$	0	0
$197$	5.40454i	0.385058i	0.981291	+	0.192529i	$0.0616689\pi$
$197$	5.40454i	0.385058i	−0.981291	+	0.192529i	$0.938331\pi$
$198$	0	0
$199$	27.9022	1.97793	0.988966	−	0.148145i	$-0.0473303\pi$
$199$	27.9022	1.97793	0.988966	+	0.148145i	$0.0473303\pi$
$200$	0	0
$201$	7.05080	0.497325
$202$	0	0
$203$	− 2.32547i	− 0.163216i
$204$	0	0
$205$	25.3593i	1.77117i
$206$	0	0
$207$	−0.480125	−0.0333710
$208$	0	0
$209$	−12.2953	−0.850485
$210$	0	0
$211$	8.78002i	0.604442i	0.953238	+	0.302221i	$0.0977280\pi$
$211$	8.78002i	0.604442i	−0.953238	+	0.302221i	$0.902272\pi$
$212$	0	0
$213$	− 42.8998i	− 2.93945i
$214$	0	0
$215$	15.2461	1.03977
$216$	0	0
$217$	−1.17157	−0.0795315
$218$	0	0
$219$	19.1946i	1.29705i
$220$	0	0
$221$	35.5910i	2.39411i
$222$	0	0
$223$	−13.8276	−0.925968	−0.462984	−	0.886367i	$-0.653221\pi$
$223$	−13.8276	−0.925968	−0.462984	+	0.886367i	$0.653221\pi$
$224$	0	0
$225$	0.484503	0.0323002
$226$	0	0
$227$	19.7977i	1.31402i	0.753881	+	0.657011i	$0.228179\pi$
$227$	19.7977i	1.31402i	−0.753881	+	0.657011i	$0.771821\pi$
$228$	0	0
$229$	− 21.8411i	− 1.44330i	−0.692259	−	0.721649i	$-0.743384\pi$
$229$	− 21.8411i	− 1.44330i	0.692259	−	0.721649i	$-0.256616\pi$
$230$	0	0
$231$	10.1937	0.670697
$232$	0	0
$233$	−4.29156	−0.281150	−0.140575	−	0.990070i	$-0.544895\pi$
$233$	−4.29156	−0.281150	−0.140575	+	0.990070i	$0.544895\pi$
$234$	0	0
$235$	14.4699i	0.943914i
$236$	0	0
$237$	− 42.6856i	− 2.77273i
$238$	0	0
$239$	−10.3307	−0.668237	−0.334119	−	0.942531i	$-0.608439\pi$
$239$	−10.3307	−0.668237	−0.334119	+	0.942531i	$0.608439\pi$
$240$	0	0
$241$	−4.36451	−0.281143	−0.140571	−	0.990071i	$-0.544894\pi$
$241$	−4.36451	−0.281143	−0.140571	+	0.990071i	$0.544894\pi$
$242$	0	0
$243$	− 6.23612i	− 0.400047i
$244$	0	0
$245$	2.25526i	0.144083i
$246$	0	0
$247$	−19.2837	−1.22699
$248$	0	0
$249$	34.4958	2.18608
$250$	0	0
$251$	− 18.6281i	− 1.17580i	−0.808935	−	0.587898i	$-0.799956\pi$
$251$	− 18.6281i	− 1.17580i	0.808935	−	0.587898i	$-0.200044\pi$
$252$	0	0
$253$	− 0.296459i	− 0.0186382i
$254$	0	0
$255$	43.2888	2.71085
$256$	0	0
$257$	−17.4169	−1.08643	−0.543217	−	0.839592i	$-0.682794\pi$
$257$	−17.4169	−1.08643	−0.543217	+	0.839592i	$0.682794\pi$
$258$	0	0
$259$	− 4.05336i	− 0.251863i
$260$	0	0
$261$	13.0744i	0.809284i
$262$	0	0
$263$	15.4345	0.951731	0.475865	−	0.879518i	$-0.342135\pi$
$263$	15.4345	0.951731	0.475865	+	0.879518i	$0.342135\pi$
$264$	0	0
$265$	−29.8027	−1.83077
$266$	0	0
$267$	37.0160i	2.26534i
$268$	0	0
$269$	18.0463i	1.10030i	0.835065	+	0.550151i	$0.185430\pi$
$269$	18.0463i	1.10030i	−0.835065	+	0.550151i	$0.814570\pi$
$270$	0	0
$271$	−19.4845	−1.18360	−0.591800	−	0.806085i	$-0.701582\pi$
$271$	−19.4845	−1.18360	−0.591800	+	0.806085i	$0.701582\pi$
$272$	0	0
$273$	15.9876	0.967611
$274$	0	0
$275$	0.299163i	0.0180402i
$276$	0	0
$277$	− 21.8230i	− 1.31122i	−0.755100	−	0.655609i	$-0.772412\pi$
$277$	− 21.8230i	− 1.31122i	0.755100	−	0.655609i	$-0.227588\pi$
$278$	0	0
$279$	6.58689	0.394347
$280$	0	0
$281$	6.23998	0.372246	0.186123	−	0.982526i	$-0.440408\pi$
$281$	6.23998	0.372246	0.186123	+	0.982526i	$0.440408\pi$
$282$	0	0
$283$	− 17.1255i	− 1.01800i	−0.860766	−	0.509001i	$-0.830015\pi$
$283$	− 17.1255i	− 1.01800i	0.860766	−	0.509001i	$-0.169985\pi$
$284$	0	0
$285$	23.4544i	1.38932i
$286$	0	0
$287$	11.2445	0.663743
$288$	0	0
$289$	25.7306	1.51356
$290$	0	0
$291$	− 41.4748i	− 2.43130i
$292$	0	0
$293$	− 30.5592i	− 1.78529i	−0.450765	−	0.892643i	$-0.648849\pi$
$293$	− 30.5592i	− 1.78529i	0.450765	−	0.892643i	$-0.351151\pi$
$294$	0	0
$295$	2.50149	0.145642
$296$	0	0
$297$	−26.7306	−1.55107
$298$	0	0
$299$	− 0.464959i	− 0.0268893i
$300$	0	0
$301$	− 6.76025i	− 0.389654i
$302$	0	0
$303$	18.5015	1.06288
$304$	0	0
$305$	10.9138	0.624924
$306$	0	0
$307$	17.3130i	0.988105i	0.869432	+	0.494053i	$0.164485\pi$
$307$	17.3130i	0.988105i	−0.869432	+	0.494053i	$0.835515\pi$
$308$	0	0
$309$	− 51.5597i	− 2.93313i
$310$	0	0
$311$	−7.41687	−0.420572	−0.210286	−	0.977640i	$-0.567440\pi$
$311$	−7.41687	−0.420572	−0.210286	+	0.977640i	$0.567440\pi$
$312$	0	0
$313$	11.8276	0.668538	0.334269	−	0.942478i	$-0.391511\pi$
$313$	11.8276	0.668538	0.334269	+	0.942478i	$0.391511\pi$
$314$	0	0
$315$	− 12.6796i	− 0.714416i
$316$	0	0
$317$	3.39581i	0.190728i	0.995442	+	0.0953638i	$0.0304014\pi$
$317$	3.39581i	0.190728i	−0.995442	+	0.0953638i	$0.969599\pi$
$318$	0	0
$319$	−8.07295	−0.451998
$320$	0	0
$321$	36.5120	2.03790
$322$	0	0
$323$	23.1519i	1.28821i
$324$	0	0
$325$	0.469198i	0.0260264i
$326$	0	0
$327$	−41.9464	−2.31964
$328$	0	0
$329$	6.41609	0.353731
$330$	0	0
$331$	4.33872i	0.238478i	0.992866	+	0.119239i	$0.0380454\pi$
$331$	4.33872i	0.238478i	−0.992866	+	0.119239i	$0.961955\pi$
$332$	0	0
$333$	22.7890i	1.24883i
$334$	0	0
$335$	5.41531	0.295870
$336$	0	0
$337$	−17.7136	−0.964921	−0.482460	−	0.875918i	$-0.660257\pi$
$337$	−17.7136	−0.964921	−0.482460	+	0.875918i	$0.660257\pi$
$338$	0	0
$339$	45.9377i	2.49500i
$340$	0	0
$341$	4.06716i	0.220249i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−0.565522	−0.0304467
$346$	0	0
$347$	6.19599i	0.332618i	0.986074	+	0.166309i	$0.0531850\pi$
$347$	6.19599i	0.332618i	−0.986074	+	0.166309i	$0.946815\pi$
$348$	0	0
$349$	− 26.9726i	− 1.44381i	−0.691993	−	0.721905i	$-0.743267\pi$
$349$	− 26.9726i	− 1.44381i	0.691993	−	0.721905i	$-0.256733\pi$
$350$	0	0
$351$	−41.9235	−2.23771
$352$	0	0
$353$	15.4861	0.824240	0.412120	−	0.911130i	$-0.364788\pi$
$353$	15.4861	0.824240	0.412120	+	0.911130i	$0.364788\pi$
$354$	0	0
$355$	− 32.9489i	− 1.74874i
$356$	0	0
$357$	− 19.1946i	− 1.01589i
$358$	0	0
$359$	−2.42854	−0.128174	−0.0640868	−	0.997944i	$-0.520413\pi$
$359$	−2.42854	−0.128174	−0.0640868	+	0.997944i	$0.520413\pi$
$360$	0	0
$361$	6.45600	0.339790
$362$	0	0
$363$	− 3.08783i	− 0.162069i
$364$	0	0
$365$	14.7423i	0.771647i
$366$	0	0
$367$	−0.412334	−0.0215236	−0.0107618	−	0.999942i	$-0.503426\pi$
$367$	−0.412334	−0.0215236	−0.0107618	+	0.999942i	$0.503426\pi$
$368$	0	0
$369$	−63.2196	−3.29108
$370$	0	0
$371$	13.2148i	0.686078i
$372$	0	0
$373$	25.1589i	1.30268i	0.758787	+	0.651338i	$0.225792\pi$
$373$	25.1589i	1.30268i	−0.758787	+	0.651338i	$0.774208\pi$
$374$	0	0
$375$	−32.5406	−1.68039
$376$	0	0
$377$	−12.6614	−0.652095
$378$	0	0
$379$	1.75926i	0.0903671i	0.998979	+	0.0451836i	$0.0143873\pi$
$379$	1.75926i	0.0903671i	−0.998979	+	0.0451836i	$0.985613\pi$
$380$	0	0
$381$	− 9.26531i	− 0.474676i
$382$	0	0
$383$	11.1024	0.567305	0.283653	−	0.958927i	$-0.408454\pi$
$383$	11.1024	0.567305	0.283653	+	0.958927i	$0.408454\pi$
$384$	0	0
$385$	7.82921	0.399013
$386$	0	0
$387$	38.0079i	1.93205i
$388$	0	0
$389$	35.5467i	1.80229i	0.433519	+	0.901144i	$0.357272\pi$
$389$	35.5467i	1.80229i	−0.433519	+	0.901144i	$0.642728\pi$
$390$	0	0
$391$	−0.558229	−0.0282309
$392$	0	0
$393$	34.9405	1.76252
$394$	0	0
$395$	− 32.7843i	− 1.64956i
$396$	0	0
$397$	12.0742i	0.605989i	0.952992	+	0.302995i	$0.0979864\pi$
$397$	12.0742i	0.605989i	−0.952992	+	0.302995i	$0.902014\pi$
$398$	0	0
$399$	10.3999	0.520645
$400$	0	0
$401$	12.2791	0.613190	0.306595	−	0.951840i	$-0.400810\pi$
$401$	12.2791	0.613190	0.306595	+	0.951840i	$0.400810\pi$
$402$	0	0
$403$	6.37882i	0.317752i
$404$	0	0
$405$	12.9520i	0.643590i
$406$	0	0
$407$	−14.0714	−0.697493
$408$	0	0
$409$	5.65841	0.279790	0.139895	−	0.990166i	$-0.455323\pi$
$409$	5.65841	0.279790	0.139895	+	0.990166i	$0.455323\pi$
$410$	0	0
$411$	− 33.3727i	− 1.64615i
$412$	0	0
$413$	− 1.10918i	− 0.0545793i
$414$	0	0
$415$	26.4942	1.30055
$416$	0	0
$417$	46.3750	2.27099
$418$	0	0
$419$	− 10.7983i	− 0.527533i	−0.964587	−	0.263766i	$-0.915035\pi$
$419$	− 10.7983i	− 0.527533i	0.964587	−	0.263766i	$-0.0849648\pi$
$420$	0	0
$421$	25.5873i	1.24705i	0.781805	+	0.623524i	$0.214299\pi$
$421$	25.5873i	1.24705i	−0.781805	+	0.623524i	$0.785701\pi$
$422$	0	0
$423$	−36.0729	−1.75393
$424$	0	0
$425$	0.563319	0.0273250
$426$	0	0
$427$	− 4.83929i	− 0.234189i
$428$	0	0
$429$	− 55.5014i	− 2.67963i
$430$	0	0
$431$	16.9830	0.818043	0.409021	−	0.912525i	$-0.365870\pi$
$431$	16.9830	0.818043	0.409021	+	0.912525i	$0.365870\pi$
$432$	0	0
$433$	19.1275	0.919210	0.459605	−	0.888124i	$-0.347991\pi$
$433$	19.1275	0.919210	0.459605	+	0.888124i	$0.347991\pi$
$434$	0	0
$435$	15.3998i	0.738366i
$436$	0	0
$437$	− 0.302455i	− 0.0144684i
$438$	0	0
$439$	41.4936	1.98038	0.990190	−	0.139726i	$-0.0446221\pi$
$439$	41.4936	1.98038	0.990190	+	0.139726i	$0.0446221\pi$
$440$	0	0
$441$	−5.62226	−0.267727
$442$	0	0
$443$	7.20359i	0.342253i	0.985249	+	0.171127i	$0.0547407\pi$
$443$	7.20359i	0.342253i	−0.985249	+	0.171127i	$0.945259\pi$
$444$	0	0
$445$	28.4299i	1.34770i
$446$	0	0
$447$	−15.0451	−0.711607
$448$	0	0
$449$	−13.7069	−0.646868	−0.323434	−	0.946251i	$-0.604837\pi$
$449$	−13.7069	−0.646868	−0.323434	+	0.946251i	$0.604837\pi$
$450$	0	0
$451$	− 39.0358i	− 1.83812i
$452$	0	0
$453$	23.7417i	1.11548i
$454$	0	0
$455$	12.2791	0.575654
$456$	0	0
$457$	−3.22455	−0.150838	−0.0754191	−	0.997152i	$-0.524029\pi$
$457$	−3.22455	−0.150838	−0.0754191	+	0.997152i	$0.524029\pi$
$458$	0	0
$459$	50.3333i	2.34936i
$460$	0	0
$461$	− 0.0258489i	− 0.00120390i	−1.00000		0.000601952i	$-0.999808\pi$
$461$	− 0.0258489i	− 0.00120390i	1.00000		0.000601952i	$-0.000191607\pi$
$462$	0	0
$463$	34.2204	1.59036	0.795178	−	0.606375i	$-0.207377\pi$
$463$	34.2204	1.59036	0.795178	+	0.606375i	$0.207377\pi$
$464$	0	0
$465$	7.75846	0.359790
$466$	0	0
$467$	18.5915i	0.860314i	0.902754	+	0.430157i	$0.141542\pi$
$467$	18.5915i	0.860314i	−0.902754	+	0.430157i	$0.858458\pi$
$468$	0	0
$469$	− 2.40120i	− 0.110877i
$470$	0	0
$471$	27.1305	1.25011
$472$	0	0
$473$	−23.4685	−1.07908
$474$	0	0
$475$	0.305213i	0.0140041i
$476$	0	0
$477$	− 74.2971i	− 3.40183i
$478$	0	0
$479$	8.38212	0.382989	0.191494	−	0.981494i	$-0.438667\pi$
$479$	8.38212	0.382989	0.191494	+	0.981494i	$0.438667\pi$
$480$	0	0
$481$	−22.0692	−1.00627
$482$	0	0
$483$	0.250757i	0.0114099i
$484$	0	0
$485$	− 31.8544i	− 1.44643i
$486$	0	0
$487$	−15.2607	−0.691530	−0.345765	−	0.938321i	$-0.612381\pi$
$487$	−15.2607	−0.691530	−0.345765	+	0.938321i	$0.612381\pi$
$488$	0	0
$489$	−2.43526	−0.110126
$490$	0	0
$491$	− 38.9111i	− 1.75603i	−0.478629	−	0.878017i	$-0.658866\pi$
$491$	− 38.9111i	− 1.75603i	0.478629	−	0.878017i	$-0.341134\pi$
$492$	0	0
$493$	15.2013i	0.684630i
$494$	0	0
$495$	−44.0178	−1.97845
$496$	0	0
$497$	−14.6098	−0.655340
$498$	0	0
$499$	− 20.8912i	− 0.935217i	−0.883936	−	0.467608i	$-0.845116\pi$
$499$	− 20.8912i	− 0.935217i	0.883936	−	0.467608i	$-0.154884\pi$
$500$	0	0
$501$	35.4506i	1.58382i
$502$	0	0
$503$	−22.0692	−0.984016	−0.492008	−	0.870591i	$-0.663737\pi$
$503$	−22.0692	−0.984016	−0.492008	+	0.870591i	$0.663737\pi$
$504$	0	0
$505$	14.2099	0.632333
$506$	0	0
$507$	− 48.8741i	− 2.17058i
$508$	0	0
$509$	9.98527i	0.442589i	0.975207	+	0.221295i	$0.0710282\pi$
$509$	9.98527i	0.442589i	−0.975207	+	0.221295i	$0.928972\pi$
$510$	0	0
$511$	6.53686	0.289174
$512$	0	0
$513$	−27.2712	−1.20405
$514$	0	0
$515$	− 39.6000i	− 1.74499i
$516$	0	0
$517$	− 22.2737i	− 0.979597i
$518$	0	0
$519$	10.5015	0.460964
$520$	0	0
$521$	−9.89684	−0.433588	−0.216794	−	0.976217i	$-0.569560\pi$
$521$	−9.89684	−0.433588	−0.216794	+	0.976217i	$0.569560\pi$
$522$	0	0
$523$	6.27724i	0.274485i	0.990537	+	0.137242i	$0.0438239\pi$
$523$	6.27724i	0.274485i	−0.990537	+	0.137242i	$0.956176\pi$
$524$	0	0
$525$	− 0.253044i	− 0.0110437i
$526$	0	0
$527$	7.65841	0.333606
$528$	0	0
$529$	−22.9927	−0.999683
$530$	0	0
$531$	6.23612i	0.270624i
$532$	0	0
$533$	− 61.2227i	− 2.65185i
$534$	0	0
$535$	28.0427	1.21239
$536$	0	0
$537$	32.4661	1.40102
$538$	0	0
$539$	− 3.47154i	− 0.149530i
$540$	0	0
$541$	− 6.85159i	− 0.294573i	−0.989094	−	0.147286i	$-0.952946\pi$
$541$	− 6.85159i	− 0.294573i	0.989094	−	0.147286i	$-0.0470539\pi$
$542$	0	0
$543$	14.6223	0.627501
$544$	0	0
$545$	−32.2166	−1.38001
$546$	0	0
$547$	− 23.4557i	− 1.00289i	−0.865189	−	0.501446i	$-0.832802\pi$
$547$	− 23.4557i	− 1.00289i	0.865189	−	0.501446i	$-0.167198\pi$
$548$	0	0
$549$	27.2077i	1.16120i
$550$	0	0
$551$	−8.23622	−0.350875
$552$	0	0
$553$	−14.5369	−0.618170
$554$	0	0
$555$	26.8424i	1.13940i
$556$	0	0
$557$	− 13.7163i	− 0.581179i	−0.956848	−	0.290589i	$-0.906149\pi$
$557$	− 13.7163i	− 0.581179i	0.956848	−	0.290589i	$-0.0938514\pi$
$558$	0	0
$559$	−36.8073	−1.55678
$560$	0	0
$561$	−66.6349	−2.81333
$562$	0	0
$563$	32.8473i	1.38435i	0.721731	+	0.692174i	$0.243347\pi$
$563$	32.8473i	1.38435i	−0.721731	+	0.692174i	$0.756653\pi$
$564$	0	0
$565$	35.2821i	1.48433i
$566$	0	0
$567$	5.74303	0.241185
$568$	0	0
$569$	36.3058	1.52202	0.761009	−	0.648741i	$-0.224704\pi$
$569$	36.3058	1.52202	0.761009	+	0.648741i	$0.224704\pi$
$570$	0	0
$571$	16.2828i	0.681413i	0.940170	+	0.340707i	$0.110666\pi$
$571$	16.2828i	0.681413i	−0.940170	+	0.340707i	$0.889334\pi$
$572$	0	0
$573$	− 13.9749i	− 0.583809i
$574$	0	0
$575$	−0.00735916	−0.000306898 0
$576$	0	0
$577$	−7.72760	−0.321704	−0.160852	−	0.986979i	$-0.551424\pi$
$577$	−7.72760	−0.321704	−0.160852	+	0.986979i	$0.551424\pi$
$578$	0	0
$579$	− 17.7331i	− 0.736963i
$580$	0	0
$581$	− 11.7478i	− 0.487379i
$582$	0	0
$583$	45.8757	1.89998
$584$	0	0
$585$	−69.0364	−2.85430
$586$	0	0
$587$	7.61995i	0.314509i	0.987558	+	0.157255i	$0.0502643\pi$
$587$	7.61995i	0.314509i	−0.987558	+	0.157255i	$0.949736\pi$
$588$	0	0
$589$	4.14942i	0.170974i
$590$	0	0
$591$	−15.8697	−0.652794
$592$	0	0
$593$	12.7306	0.522782	0.261391	−	0.965233i	$-0.415819\pi$
$593$	12.7306	0.522782	0.261391	+	0.965233i	$0.415819\pi$
$594$	0	0
$595$	− 14.7423i	− 0.604375i
$596$	0	0
$597$	81.9310i	3.35321i
$598$	0	0
$599$	−7.65841	−0.312914	−0.156457	−	0.987685i	$-0.550007\pi$
$599$	−7.65841	−0.312914	−0.156457	+	0.987685i	$0.550007\pi$
$600$	0	0
$601$	−36.7519	−1.49914	−0.749572	−	0.661923i	$-0.769740\pi$
$601$	−36.7519	−1.49914	−0.749572	+	0.661923i	$0.769740\pi$
$602$	0	0
$603$	13.5002i	0.549769i
$604$	0	0
$605$	− 2.37159i	− 0.0964187i
$606$	0	0
$607$	−16.5831	−0.673088	−0.336544	−	0.941668i	$-0.609258\pi$
$607$	−16.5831	−0.673088	−0.336544	+	0.941668i	$0.609258\pi$
$608$	0	0
$609$	6.82843	0.276702
$610$	0	0
$611$	− 34.9335i	− 1.41326i
$612$	0	0
$613$	2.73826i	0.110597i	0.998470	+	0.0552986i	$0.0176111\pi$
$613$	2.73826i	0.110597i	−0.998470	+	0.0552986i	$0.982389\pi$
$614$	0	0
$615$	−74.4641	−3.00268
$616$	0	0
$617$	−37.4987	−1.50964	−0.754821	−	0.655931i	$-0.772276\pi$
$617$	−37.4987	−1.50964	−0.754821	+	0.655931i	$0.772276\pi$
$618$	0	0
$619$	− 33.1608i	− 1.33284i	−0.745575	−	0.666422i	$-0.767825\pi$
$619$	− 33.1608i	− 1.33284i	0.745575	−	0.666422i	$-0.232175\pi$
$620$	0	0
$621$	− 0.657551i	− 0.0263866i
$622$	0	0
$623$	12.6061	0.505051
$624$	0	0
$625$	−25.4235	−1.01694
$626$	0	0
$627$	− 36.1036i	− 1.44184i
$628$	0	0
$629$	26.4962i	1.05647i
$630$	0	0
$631$	−16.9706	−0.675587	−0.337794	−	0.941220i	$-0.609681\pi$
$631$	−16.9706	−0.675587	−0.337794	+	0.941220i	$0.609681\pi$
$632$	0	0
$633$	−25.7814	−1.02472
$634$	0	0
$635$	− 7.11615i	− 0.282396i
$636$	0	0
$637$	− 5.44467i	− 0.215726i
$638$	0	0
$639$	82.1402	3.24941
$640$	0	0
$641$	10.8962	0.430375	0.215187	−	0.976573i	$-0.430964\pi$
$641$	10.8962	0.430375	0.215187	+	0.976573i	$0.430964\pi$
$642$	0	0
$643$	− 26.2347i	− 1.03460i	−0.855805	−	0.517298i	$-0.826938\pi$
$643$	− 26.2347i	− 1.03460i	0.855805	−	0.517298i	$-0.173062\pi$
$644$	0	0
$645$	44.7681i	1.76274i
$646$	0	0
$647$	8.93003	0.351076	0.175538	−	0.984473i	$-0.443834\pi$
$647$	8.93003	0.351076	0.175538	+	0.984473i	$0.443834\pi$
$648$	0	0
$649$	−3.85057	−0.151148
$650$	0	0
$651$	− 3.44017i	− 0.134831i
$652$	0	0
$653$	12.2764i	0.480413i	0.970722	+	0.240206i	$0.0772151\pi$
$653$	12.2764i	0.480413i	−0.970722	+	0.240206i	$0.922785\pi$
$654$	0	0
$655$	26.8358	1.04856
$656$	0	0
$657$	−36.7519	−1.43383
$658$	0	0
$659$	− 15.3841i	− 0.599279i	−0.954052	−	0.299640i	$-0.903134\pi$
$659$	− 15.3841i	− 0.599279i	0.954052	−	0.299640i	$-0.0968664\pi$
$660$	0	0
$661$	− 39.3660i	− 1.53116i	−0.643341	−	0.765580i	$-0.722452\pi$
$661$	− 39.3660i	− 1.53116i	0.643341	−	0.765580i	$-0.277548\pi$
$662$	0	0
$663$	−104.508	−4.05877
$664$	0	0
$665$	7.98755	0.309744
$666$	0	0
$667$	− 0.198588i	− 0.00768936i
$668$	0	0
$669$	− 40.6031i	− 1.56981i
$670$	0	0
$671$	−16.7998	−0.648548
$672$	0	0
$673$	−9.65685	−0.372244	−0.186122	−	0.982527i	$-0.559592\pi$
$673$	−9.65685	−0.372244	−0.186122	+	0.982527i	$0.559592\pi$
$674$	0	0
$675$	0.663547i	0.0255399i
$676$	0	0
$677$	− 11.8451i	− 0.455244i	−0.973750	−	0.227622i	$-0.926905\pi$
$677$	− 11.8451i	− 0.455244i	0.973750	−	0.227622i	$-0.0730951\pi$
$678$	0	0
$679$	−14.1245	−0.542050
$680$	0	0
$681$	−58.1334	−2.22768
$682$	0	0
$683$	− 8.11143i	− 0.310375i	−0.987885	−	0.155188i	$-0.950402\pi$
$683$	− 8.11143i	− 0.310375i	0.987885	−	0.155188i	$-0.0495982\pi$
$684$	0	0
$685$	− 25.6316i	− 0.979334i
$686$	0	0
$687$	64.1334	2.44684
$688$	0	0
$689$	71.9502	2.74108
$690$	0	0
$691$	− 17.3130i	− 0.658617i	−0.944222	−	0.329309i	$-0.893184\pi$
$691$	− 17.3130i	− 0.658617i	0.944222	−	0.329309i	$-0.106816\pi$
$692$	0	0
$693$	19.5179i	0.741424i
$694$	0	0
$695$	35.6179	1.35107
$696$	0	0
$697$	−73.5039	−2.78416
$698$	0	0
$699$	− 12.6016i	− 0.476637i
$700$	0	0
$701$	10.1513i	0.383411i	0.981453	+	0.191705i	$0.0614018\pi$
$701$	10.1513i	0.383411i	−0.981453	+	0.191705i	$0.938598\pi$
$702$	0	0
$703$	−14.3560	−0.541446
$704$	0	0
$705$	−42.4890	−1.60023
$706$	0	0
$707$	− 6.30081i	− 0.236966i
$708$	0	0
$709$	3.34411i	0.125591i	0.998026	+	0.0627953i	$0.0200015\pi$
$709$	3.34411i	0.125591i	−0.998026	+	0.0627953i	$0.979998\pi$
$710$	0	0
$711$	81.7300	3.06512
$712$	0	0
$713$	−0.100049	−0.00374686
$714$	0	0
$715$	− 42.6274i	− 1.59418i
$716$	0	0
$717$	− 30.3347i	− 1.13287i
$718$	0	0
$719$	10.2777	0.383294	0.191647	−	0.981464i	$-0.438617\pi$
$719$	10.2777	0.383294	0.191647	+	0.981464i	$0.438617\pi$
$720$	0	0
$721$	−17.5590	−0.653932
$722$	0	0
$723$	− 12.8158i	− 0.476625i
$724$	0	0
$725$	0.200399i	0.00744263i
$726$	0	0
$727$	−49.2926	−1.82816	−0.914080	−	0.405534i	$-0.867086\pi$
$727$	−49.2926	−1.82816	−0.914080	+	0.405534i	$0.867086\pi$
$728$	0	0
$729$	35.5406	1.31632
$730$	0	0
$731$	44.1908i	1.63446i
$732$	0	0
$733$	44.6600i	1.64955i	0.565458	+	0.824777i	$0.308700\pi$
$733$	44.6600i	1.64955i	−0.565458	+	0.824777i	$0.691300\pi$
$734$	0	0
$735$	−6.62226	−0.244266
$736$	0	0
$737$	−8.33585	−0.307055
$738$	0	0
$739$	− 7.04109i	− 0.259011i	−0.991579	−	0.129505i	$-0.958661\pi$
$739$	− 7.04109i	− 0.259011i	0.991579	−	0.129505i	$-0.0413389\pi$
$740$	0	0
$741$	− 56.6239i	− 2.08013i
$742$	0	0
$743$	−24.2562	−0.889873	−0.444937	−	0.895562i	$-0.646774\pi$
$743$	−24.2562	−0.889873	−0.444937	+	0.895562i	$0.646774\pi$
$744$	0	0
$745$	−11.5552	−0.423352
$746$	0	0
$747$	66.0490i	2.41661i
$748$	0	0
$749$	− 12.4344i	− 0.454343i
$750$	0	0
$751$	−31.1629	−1.13715	−0.568575	−	0.822632i	$-0.692505\pi$
$751$	−31.1629	−1.13715	−0.568575	+	0.822632i	$0.692505\pi$
$752$	0	0
$753$	54.6990	1.99334
$754$	0	0
$755$	18.2346i	0.663626i
$756$	0	0
$757$	34.1438i	1.24098i	0.784215	+	0.620489i	$0.213066\pi$
$757$	34.1438i	1.24098i	−0.784215	+	0.620489i	$0.786934\pi$
$758$	0	0
$759$	0.870514	0.0315977
$760$	0	0
$761$	45.6335	1.65421	0.827107	−	0.562045i	$-0.189985\pi$
$761$	45.6335	1.65421	0.827107	+	0.562045i	$0.189985\pi$
$762$	0	0
$763$	14.2851i	0.517157i
$764$	0	0
$765$	82.8850i	2.99671i
$766$	0	0
$767$	−6.03913	−0.218060
$768$	0	0
$769$	24.6136	0.887588	0.443794	−	0.896129i	$-0.353632\pi$
$769$	24.6136	0.887588	0.443794	+	0.896129i	$0.353632\pi$
$770$	0	0
$771$	− 51.1424i	− 1.84185i
$772$	0	0
$773$	25.3029i	0.910080i	0.890471	+	0.455040i	$0.150375\pi$
$773$	25.3029i	0.910080i	−0.890471	+	0.455040i	$0.849625\pi$
$774$	0	0
$775$	0.100961	0.00362663
$776$	0	0
$777$	11.9022	0.426987
$778$	0	0
$779$	− 39.8253i	− 1.42689i
$780$	0	0
$781$	50.7185i	1.81485i
$782$	0	0
$783$	−17.9059	−0.639905
$784$	0	0
$785$	20.8373	0.743717
$786$	0	0
$787$	− 1.55253i	− 0.0553418i	−0.999617	−	0.0276709i	$-0.991191\pi$
$787$	− 1.55253i	− 0.0553418i	0.999617	−	0.0276709i	$-0.00880905\pi$
$788$	0	0
$789$	45.3213i	1.61348i
$790$	0	0
$791$	15.6444	0.556251
$792$	0	0
$793$	−26.3483	−0.935656
$794$	0	0
$795$	− 87.5119i	− 3.10373i
$796$	0	0
$797$	30.7955i	1.09083i	0.838166	+	0.545415i	$0.183628\pi$
$797$	30.7955i	1.09083i	−0.838166	+	0.545415i	$0.816372\pi$
$798$	0	0
$799$	−41.9411	−1.48377
$800$	0	0
$801$	−70.8745	−2.50423
$802$	0	0
$803$	− 22.6930i	− 0.800818i
$804$	0	0
$805$	0.192592i	0.00678798i
$806$	0	0
$807$	−52.9905	−1.86536
$808$	0	0
$809$	13.3337	0.468787	0.234394	−	0.972142i	$-0.424690\pi$
$809$	13.3337	0.468787	0.234394	+	0.972142i	$0.424690\pi$
$810$	0	0
$811$	− 54.9310i	− 1.92889i	−0.264289	−	0.964443i	$-0.585137\pi$
$811$	− 54.9310i	− 1.92889i	0.264289	−	0.964443i	$-0.414863\pi$
$812$	0	0
$813$	− 57.2137i	− 2.00657i
$814$	0	0
$815$	−1.87038	−0.0655166
$816$	0	0
$817$	−23.9431	−0.837663
$818$	0	0
$819$	30.6113i	1.06965i
$820$	0	0
$821$	− 55.2649i	− 1.92876i	−0.264523	−	0.964379i	$-0.585214\pi$
$821$	− 55.2649i	− 1.92876i	0.264523	−	0.964379i	$-0.414786\pi$
$822$	0	0
$823$	35.6835	1.24385	0.621925	−	0.783077i	$-0.286351\pi$
$823$	35.6835	1.24385	0.621925	+	0.783077i	$0.286351\pi$
$824$	0	0
$825$	−0.878452	−0.0305838
$826$	0	0
$827$	− 33.2831i	− 1.15737i	−0.815552	−	0.578684i	$-0.803566\pi$
$827$	− 33.2831i	− 1.15737i	0.815552	−	0.578684i	$-0.196434\pi$
$828$	0	0
$829$	− 34.6509i	− 1.20348i	−0.798694	−	0.601738i	$-0.794475\pi$
$829$	− 34.6509i	− 1.20348i	0.798694	−	0.601738i	$-0.205525\pi$
$830$	0	0
$831$	64.0805	2.22293
$832$	0	0
$833$	−6.53686	−0.226489
$834$	0	0
$835$	27.2276i	0.942249i
$836$	0	0
$837$	9.02102i	0.311812i
$838$	0	0
$839$	−10.9316	−0.377400	−0.188700	−	0.982035i	$-0.560427\pi$
$839$	−10.9316	−0.377400	−0.188700	+	0.982035i	$0.560427\pi$
$840$	0	0
$841$	23.5922	0.813524
$842$	0	0
$843$	18.3229i	0.631074i
$844$	0	0
$845$	− 37.5374i	− 1.29133i
$846$	0	0
$847$	−1.05158	−0.0361328
$848$	0	0
$849$	50.2866	1.72583
$850$	0	0
$851$	− 0.346145i	− 0.0118657i
$852$	0	0
$853$	41.9075i	1.43488i	0.696618	+	0.717442i	$0.254687\pi$
$853$	41.9075i	1.43488i	−0.696618	+	0.717442i	$0.745313\pi$
$854$	0	0
$855$	−44.9081	−1.53582
$856$	0	0
$857$	−3.51096	−0.119932	−0.0599660	−	0.998200i	$-0.519099\pi$
$857$	−3.51096	−0.119932	−0.0599660	+	0.998200i	$0.519099\pi$
$858$	0	0
$859$	15.4493i	0.527122i	0.964643	+	0.263561i	$0.0848970\pi$
$859$	15.4493i	0.527122i	−0.964643	+	0.263561i	$0.915103\pi$
$860$	0	0
$861$	33.0181i	1.12525i
$862$	0	0
$863$	27.7282	0.943880	0.471940	−	0.881631i	$-0.343554\pi$
$863$	27.7282	0.943880	0.471940	+	0.881631i	$0.343554\pi$
$864$	0	0
$865$	8.06559	0.274238
$866$	0	0
$867$	75.5545i	2.56596i
$868$	0	0
$869$	50.4653i	1.71192i
$870$	0	0
$871$	−13.0737	−0.442986
$872$	0	0
$873$	79.4118	2.68768
$874$	0	0
$875$	11.0819i	0.374638i
$876$	0	0
$877$	18.9361i	0.639426i	0.947514	+	0.319713i	$0.103586\pi$
$877$	18.9361i	0.639426i	−0.947514	+	0.319713i	$0.896414\pi$
$878$	0	0
$879$	89.7330	3.02662
$880$	0	0
$881$	41.0473	1.38292	0.691459	−	0.722416i	$-0.256968\pi$
$881$	41.0473	1.38292	0.691459	+	0.722416i	$0.256968\pi$
$882$	0	0
$883$	36.4073i	1.22520i	0.790391	+	0.612602i	$0.209877\pi$
$883$	36.4073i	1.22520i	−0.790391	+	0.612602i	$0.790123\pi$
$884$	0	0
$885$	7.34530i	0.246910i
$886$	0	0
$887$	−45.2850	−1.52052	−0.760262	−	0.649617i	$-0.774929\pi$
$887$	−45.2850	−1.52052	−0.760262	+	0.649617i	$0.774929\pi$
$888$	0	0
$889$	−3.15536	−0.105828
$890$	0	0
$891$	− 19.9372i	− 0.667920i
$892$	0	0
$893$	− 22.7242i	− 0.760436i
$894$	0	0
$895$	24.9354	0.833497
$896$	0	0
$897$	1.36529	0.0455857
$898$	0	0
$899$	2.72445i	0.0908656i
$900$	0	0
$901$	− 86.3834i	− 2.87785i
$902$	0	0
$903$	19.8506	0.660586
$904$	0	0
$905$	11.2305	0.373315
$906$	0	0
$907$	49.6129i	1.64737i	0.567048	+	0.823685i	$0.308085\pi$
$907$	49.6129i	1.64737i	−0.567048	+	0.823685i	$0.691915\pi$
$908$	0	0
$909$	35.4248i	1.17497i
$910$	0	0
$911$	−6.67540	−0.221166	−0.110583	−	0.993867i	$-0.535272\pi$
$911$	−6.67540	−0.221166	−0.110583	+	0.993867i	$0.535272\pi$
$912$	0	0
$913$	−40.7828	−1.34971
$914$	0	0
$915$	32.0470i	1.05944i
$916$	0	0
$917$	− 11.8992i	− 0.392947i
$918$	0	0
$919$	42.1651	1.39090	0.695448	−	0.718576i	$-0.255206\pi$
$919$	42.1651	1.39090	0.695448	+	0.718576i	$0.255206\pi$
$920$	0	0
$921$	−50.8373	−1.67515
$922$	0	0
$923$	79.5456i	2.61827i
$924$	0	0
$925$	0.349301i	0.0114850i
$926$	0	0
$927$	98.7213	3.24243
$928$	0	0
$929$	−10.0463	−0.329607	−0.164804	−	0.986326i	$-0.552699\pi$
$929$	−10.0463	−0.329607	−0.164804	+	0.986326i	$0.552699\pi$
$930$	0	0
$931$	− 3.54175i	− 0.116076i
$932$	0	0
$933$	− 21.7787i	− 0.713002i
$934$	0	0
$935$	−51.1784	−1.67371
$936$	0	0
$937$	54.0996	1.76736	0.883679	−	0.468093i	$-0.155059\pi$
$937$	54.0996	1.76736	0.883679	+	0.468093i	$0.155059\pi$
$938$	0	0
$939$	34.7303i	1.13338i
$940$	0	0
$941$	− 2.38120i	− 0.0776250i	−0.999247	−	0.0388125i	$-0.987642\pi$
$941$	− 2.38120i	− 0.0776250i	0.999247	−	0.0388125i	$-0.0123575\pi$
$942$	0	0
$943$	0.960249	0.0312700
$944$	0	0
$945$	17.3653	0.564893
$946$	0	0
$947$	− 12.5748i	− 0.408627i	−0.978906	−	0.204313i	$-0.934504\pi$
$947$	− 12.5748i	− 0.408627i	0.978906	−	0.204313i	$-0.0654961\pi$
$948$	0	0
$949$	− 35.5910i	− 1.15533i
$950$	0	0
$951$	−9.97134	−0.323343
$952$	0	0
$953$	−12.4890	−0.404560	−0.202280	−	0.979328i	$-0.564835\pi$
$953$	−12.4890	−0.404560	−0.202280	+	0.979328i	$0.564835\pi$
$954$	0	0
$955$	− 10.7333i	− 0.347321i
$956$	0	0
$957$	− 23.7051i	− 0.766279i
$958$	0	0
$959$	−11.3653	−0.367004
$960$	0	0
$961$	−29.6274	−0.955723
$962$	0	0
$963$	69.9094i	2.25280i
$964$	0	0
$965$	− 13.6198i	− 0.438436i
$966$	0	0
$967$	42.0621	1.35262	0.676312	−	0.736615i	$-0.263577\pi$
$967$	42.0621	1.35262	0.676312	+	0.736615i	$0.263577\pi$
$968$	0	0
$969$	−67.9826	−2.18392
$970$	0	0
$971$	2.57575i	0.0826597i	0.999146	+	0.0413298i	$0.0131594\pi$
$971$	2.57575i	0.0826597i	−0.999146	+	0.0413298i	$0.986841\pi$
$972$	0	0
$973$	− 15.7933i	− 0.506310i
$974$	0	0
$975$	−1.37774	−0.0441230
$976$	0	0
$977$	−22.6422	−0.724389	−0.362195	−	0.932103i	$-0.617972\pi$
$977$	−22.6422	−0.724389	−0.362195	+	0.932103i	$0.617972\pi$
$978$	0	0
$979$	− 43.7624i	− 1.39865i
$980$	0	0
$981$	− 80.3148i	− 2.56425i
$982$	0	0
$983$	−14.0038	−0.446651	−0.223325	−	0.974744i	$-0.571691\pi$
$983$	−14.0038	−0.446651	−0.223325	+	0.974744i	$0.571691\pi$
$984$	0	0
$985$	−12.1886	−0.388362
$986$	0	0
$987$	18.8400i	0.599684i
$988$	0	0
$989$	− 0.577305i	− 0.0183572i
$990$	0	0
$991$	−55.5074	−1.76325	−0.881626	−	0.471949i	$-0.843551\pi$
$991$	−55.5074	−1.76325	−0.881626	+	0.471949i	$0.843551\pi$
$992$	0	0
$993$	−12.7401	−0.404294
$994$	0	0
$995$	62.9265i	1.99490i
$996$	0	0
$997$	32.8258i	1.03960i	0.854287	+	0.519802i	$0.173994\pi$
$997$	32.8258i	1.03960i	−0.854287	+	0.519802i	$0.826006\pi$
$998$	0	0
$999$	−31.2105	−0.987458

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3584.2.b.f.1793.8		8
4.3	odd	2		3584.2.b.e.1793.1		8
8.3	odd	2		3584.2.b.e.1793.8		8
8.5	even	2	inner	3584.2.b.f.1793.1		8
16.3	odd	4		3584.2.a.n.1.8	yes	8
16.5	even	4		3584.2.a.m.1.8	yes	8
16.11	odd	4		3584.2.a.n.1.1	yes	8
16.13	even	4		3584.2.a.m.1.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3584.2.a.m.1.1	✓	8	16.13	even	4
3584.2.a.m.1.8	yes	8	16.5	even	4
3584.2.a.n.1.1	yes	8	16.11	odd	4
3584.2.a.n.1.8	yes	8	16.3	odd	4
3584.2.b.e.1793.1		8	4.3	odd	2
3584.2.b.e.1793.8		8	8.3	odd	2
3584.2.b.f.1793.1		8	8.5	even	2	inner
3584.2.b.f.1793.8		8	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:	\( N \)	\(=\)	\( 3584 = 2^{9} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3584.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.93637i q^{3} +2.25526i q^{5} +1.00000 q^{7} -5.62226 q^{9} +O(q^{10})\)	\(q+2.93637i q^{3} +2.25526i q^{5} +1.00000 q^{7} -5.62226 q^{9} -3.47154i q^{11} -5.44467i q^{13} -6.62226 q^{15} -6.53686 q^{17} -3.54175i q^{19} +2.93637i q^{21} +0.0853971 q^{23} -0.0861758 q^{25} -7.69992i q^{27} -2.32547i q^{29} -1.17157 q^{31} +10.1937 q^{33} +2.25526i q^{35} -4.05336i q^{37} +15.9876 q^{39} +11.2445 q^{41} -6.76025i q^{43} -12.6796i q^{45} +6.41609 q^{47} +1.00000 q^{49} -19.1946i q^{51} +13.2148i q^{53} +7.82921 q^{55} +10.3999 q^{57} -1.10918i q^{59} -4.83929i q^{61} -5.62226 q^{63} +12.2791 q^{65} -2.40120i q^{67} +0.250757i q^{69} -14.6098 q^{71} +6.53686 q^{73} -0.253044i q^{75} -3.47154i q^{77} -14.5369 q^{79} +5.74303 q^{81} -11.7478i q^{83} -14.7423i q^{85} +6.82843 q^{87} +12.6061 q^{89} -5.44467i q^{91} -3.44017i q^{93} +7.98755 q^{95} -14.1245 q^{97} +19.5179i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{7} - 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^7 - 8 * q^9	\( 8 q + 8 q^{7} - 8 q^{9} - 16 q^{15} - 8 q^{17} + 8 q^{23} - 16 q^{25} - 32 q^{31} - 8 q^{33} + 24 q^{39} + 16 q^{41} + 8 q^{49} + 48 q^{55} + 8 q^{57} - 8 q^{63} + 16 q^{65} + 24 q^{71} + 8 q^{73} - 72 q^{79} + 16 q^{81} + 32 q^{87} + 40 q^{89} - 40 q^{95} - 40 q^{97}+O(q^{100}) \) 8 * q + 8 * q^7 - 8 * q^9 - 16 * q^15 - 8 * q^17 + 8 * q^23 - 16 * q^25 - 32 * q^31 - 8 * q^33 + 24 * q^39 + 16 * q^41 + 8 * q^49 + 48 * q^55 + 8 * q^57 - 8 * q^63 + 16 * q^65 + 24 * q^71 + 8 * q^73 - 72 * q^79 + 16 * q^81 + 32 * q^87 + 40 * q^89 - 40 * q^95 - 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more