LMFDB - Embedded newform 3584.2.b.g.1793.10

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4x^{9} + 12x^{8} - 12x^{7} + 3x^{6} - 48x^{5} + 158x^{4} - 80x^{3} - 66x^{2} + 72x + 36 $ x^10 - 4x^9 + 12x^8 - 12x^7 + 3x^6 - 48x^5 + 158x^4 - 80x^3 - 66x^2 + 72*x + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1793.10
Root			$0.875338 - 0.673961i$ of defining polynomial
Character	$\chi$	$=$	3584.1793
Dual form			3584.2.b.g.1793.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.17593i q^{3} -4.23755i q^{5} -1.00000 q^{7} -7.08655 q^{9} +O(q^{10})$	$q+3.17593i q^{3} -4.23755i q^{5} -1.00000 q^{7} -7.08655 q^{9} +4.99251i q^{11} +0.439239i q^{13} +13.4582 q^{15} +3.50135 q^{17} -3.45080i q^{19} -3.17593i q^{21} +4.39770 q^{23} -12.9568 q^{25} -12.9786i q^{27} -0.132582i q^{29} -5.55912 q^{31} -15.8559 q^{33} +4.23755i q^{35} +0.572613i q^{37} -1.39499 q^{39} -5.42377 q^{41} -5.69770i q^{43} +30.0296i q^{45} -9.37162 q^{47} +1.00000 q^{49} +11.1201i q^{51} +3.65602i q^{53} +21.1560 q^{55} +10.9595 q^{57} -2.20605i q^{59} -9.89440i q^{61} +7.08655 q^{63} +1.86130 q^{65} +8.51595i q^{67} +13.9668i q^{69} -3.11263 q^{71} -9.87027 q^{73} -41.1500i q^{75} -4.99251i q^{77} +7.11263 q^{79} +19.9595 q^{81} -2.48092i q^{83} -14.8371i q^{85} +0.421071 q^{87} +5.06047 q^{89} -0.439239i q^{91} -17.6554i q^{93} -14.6229 q^{95} -16.6573 q^{97} -35.3796i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{7} - 22 q^{9}+O(q^{10}) $ 10 * q - 10 * q^7 - 22 * q^9	$ 10 q - 10 q^{7} - 22 q^{9} + 16 q^{15} + 16 q^{17} - 8 q^{23} - 30 q^{25} - 8 q^{31} + 12 q^{33} - 20 q^{41} - 24 q^{47} + 10 q^{49} + 32 q^{55} - 28 q^{57} + 22 q^{63} + 32 q^{65} - 48 q^{73} + 40 q^{79} + 62 q^{81} + 8 q^{87} - 16 q^{89} - 32 q^{95} + 32 q^{97}+O(q^{100}) $ 10 * q - 10 * q^7 - 22 * q^9 + 16 * q^15 + 16 * q^17 - 8 * q^23 - 30 * q^25 - 8 * q^31 + 12 * q^33 - 20 * q^41 - 24 * q^47 + 10 * q^49 + 32 * q^55 - 28 * q^57 + 22 * q^63 + 32 * q^65 - 48 * q^73 + 40 * q^79 + 62 * q^81 + 8 * q^87 - 16 * q^89 - 32 * q^95 + 32 * 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
$3$	3.17593i	1.83363i	0.399317	+	0.916813i	$0.369247\pi$
$3$	3.17593i	1.83363i	−0.399317	+	0.916813i	$0.630753\pi$
$4$	0	0
$5$	− 4.23755i	− 1.89509i	−0.319623	−	0.947545i	$-0.603556\pi$
$5$	− 4.23755i	− 1.89509i	0.319623	−	0.947545i	$-0.396444\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−7.08655	−2.36218
$10$	0	0
$11$	4.99251i	1.50530i	0.658422	+	0.752649i	$0.271224\pi$
$11$	4.99251i	1.50530i	−0.658422	+	0.752649i	$0.728776\pi$
$12$	0	0
$13$	0.439239i	0.121823i	0.998143	+	0.0609115i	$0.0194008\pi$
$13$	0.439239i	0.121823i	−0.998143	+	0.0609115i	$0.980599\pi$
$14$	0	0
$15$	13.4582	3.47488
$16$	0	0
$17$	3.50135	0.849202	0.424601	−	0.905380i	$-0.360414\pi$
$17$	3.50135	0.849202	0.424601	+	0.905380i	$0.360414\pi$
$18$	0	0
$19$	− 3.45080i	− 0.791669i	−0.918322	−	0.395834i	$-0.870455\pi$
$19$	− 3.45080i	− 0.791669i	0.918322	−	0.395834i	$-0.129545\pi$
$20$	0	0
$21$	− 3.17593i	− 0.693045i
$22$	0	0
$23$	4.39770	0.916983	0.458492	−	0.888699i	$-0.348390\pi$
$23$	4.39770	0.916983	0.458492	+	0.888699i	$0.348390\pi$
$24$	0	0
$25$	−12.9568	−2.59136
$26$	0	0
$27$	− 12.9786i	− 2.49773i
$28$	0	0
$29$	− 0.132582i	− 0.0246199i	−0.999924	−	0.0123099i	$-0.996082\pi$
$29$	− 0.132582i	− 0.0246199i	0.999924	−	0.0123099i	$-0.00391847\pi$
$30$	0	0
$31$	−5.55912	−0.998447	−0.499224	−	0.866473i	$-0.666381\pi$
$31$	−5.55912	−0.998447	−0.499224	+	0.866473i	$0.666381\pi$
$32$	0	0
$33$	−15.8559	−2.76015
$34$	0	0
$35$	4.23755i	0.716276i
$36$	0	0
$37$	0.572613i	0.0941371i	0.998892	+	0.0470685i	$0.0149879\pi$
$37$	0.572613i	0.0941371i	−0.998892	+	0.0470685i	$0.985012\pi$
$38$	0	0
$39$	−1.39499	−0.223378
$40$	0	0
$41$	−5.42377	−0.847051	−0.423526	−	0.905884i	$-0.639208\pi$
$41$	−5.42377	−0.847051	−0.423526	+	0.905884i	$0.639208\pi$
$42$	0	0
$43$	− 5.69770i	− 0.868891i	−0.900698	−	0.434446i	$-0.856944\pi$
$43$	− 5.69770i	− 0.868891i	0.900698	−	0.434446i	$-0.143056\pi$
$44$	0	0
$45$	30.0296i	4.47655i
$46$	0	0
$47$	−9.37162	−1.36699	−0.683496	−	0.729955i	$-0.739541\pi$
$47$	−9.37162	−1.36699	−0.683496	+	0.729955i	$0.739541\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	11.1201i	1.55712i
$52$	0	0
$53$	3.65602i	0.502193i	0.967962	+	0.251096i	$0.0807911\pi$
$53$	3.65602i	0.502193i	−0.967962	+	0.251096i	$0.919209\pi$
$54$	0	0
$55$	21.1560	2.85267
$56$	0	0
$57$	10.9595	1.45162
$58$	0	0
$59$	− 2.20605i	− 0.287203i	−0.989636	−	0.143602i	$-0.954132\pi$
$59$	− 2.20605i	− 0.287203i	0.989636	−	0.143602i	$-0.0458684\pi$
$60$	0	0
$61$	− 9.89440i	− 1.26685i	−0.773805	−	0.633424i	$-0.781649\pi$
$61$	− 9.89440i	− 1.26685i	0.773805	−	0.633424i	$-0.218351\pi$
$62$	0	0
$63$	7.08655	0.892821
$64$	0	0
$65$	1.86130	0.230866
$66$	0	0
$67$	8.51595i	1.04039i	0.854048	+	0.520194i	$0.174140\pi$
$67$	8.51595i	1.04039i	−0.854048	+	0.520194i	$0.825860\pi$
$68$	0	0
$69$	13.9668i	1.68140i
$70$	0	0
$71$	−3.11263	−0.369401	−0.184700	−	0.982795i	$-0.559131\pi$
$71$	−3.11263	−0.369401	−0.184700	+	0.982795i	$0.559131\pi$
$72$	0	0
$73$	−9.87027	−1.15523	−0.577614	−	0.816310i	$-0.696016\pi$
$73$	−9.87027	−1.15523	−0.577614	+	0.816310i	$0.696016\pi$
$74$	0	0
$75$	− 41.1500i	− 4.75159i
$76$	0	0
$77$	− 4.99251i	− 0.568949i
$78$	0	0
$79$	7.11263	0.800233	0.400116	−	0.916464i	$-0.368970\pi$
$79$	7.11263	0.800233	0.400116	+	0.916464i	$0.368970\pi$
$80$	0	0
$81$	19.9595	2.21772
$82$	0	0
$83$	− 2.48092i	− 0.272317i	−0.990687	−	0.136158i	$-0.956524\pi$
$83$	− 2.48092i	− 0.272317i	0.990687	−	0.136158i	$-0.0434756\pi$
$84$	0	0
$85$	− 14.8371i	− 1.60931i
$86$	0	0
$87$	0.421071	0.0451436
$88$	0	0
$89$	5.06047	0.536409	0.268204	−	0.963362i	$-0.413570\pi$
$89$	5.06047	0.536409	0.268204	+	0.963362i	$0.413570\pi$
$90$	0	0
$91$	− 0.439239i	− 0.0460448i
$92$	0	0
$93$	− 17.6554i	− 1.83078i
$94$	0	0
$95$	−14.6229	−1.50028
$96$	0	0
$97$	−16.6573	−1.69130	−0.845648	−	0.533740i	$-0.820786\pi$
$97$	−16.6573	−1.69130	−0.845648	+	0.533740i	$0.820786\pi$
$98$	0	0
$99$	− 35.3796i	− 3.55579i
$100$	0	0
$101$	− 10.2408i	− 1.01900i	−0.860472	−	0.509498i	$-0.829831\pi$
$101$	− 10.2408i	− 1.01900i	0.860472	−	0.509498i	$-0.170169\pi$
$102$	0	0
$103$	−10.5420	−1.03874	−0.519368	−	0.854551i	$-0.673832\pi$
$103$	−10.5420	−1.03874	−0.519368	+	0.854551i	$0.673832\pi$
$104$	0	0
$105$	−13.4582	−1.31338
$106$	0	0
$107$	0.214131i	0.0207009i	0.999946	+	0.0103504i	$0.00329470\pi$
$107$	0.214131i	0.0207009i	−0.999946	+	0.0103504i	$0.996705\pi$
$108$	0	0
$109$	− 19.7277i	− 1.88957i	−0.327685	−	0.944787i	$-0.606268\pi$
$109$	− 19.7277i	− 1.88957i	0.327685	−	0.944787i	$-0.393732\pi$
$110$	0	0
$111$	−1.81858	−0.172612
$112$	0	0
$113$	−4.95411	−0.466044	−0.233022	−	0.972471i	$-0.574861\pi$
$113$	−4.95411	−0.466044	−0.233022	+	0.972471i	$0.574861\pi$
$114$	0	0
$115$	− 18.6355i	− 1.73777i
$116$	0	0
$117$	− 3.11269i	− 0.287768i
$118$	0	0
$119$	−3.50135	−0.320968
$120$	0	0
$121$	−13.9251	−1.26592
$122$	0	0
$123$	− 17.2255i	− 1.55317i
$124$	0	0
$125$	33.7174i	3.01578i
$126$	0	0
$127$	7.77540	0.689955	0.344978	−	0.938611i	$-0.387887\pi$
$127$	7.77540	0.689955	0.344978	+	0.938611i	$0.387887\pi$
$128$	0	0
$129$	18.0955	1.59322
$130$	0	0
$131$	− 8.21947i	− 0.718139i	−0.933311	−	0.359069i	$-0.883094\pi$
$131$	− 8.21947i	− 0.718139i	0.933311	−	0.359069i	$-0.116906\pi$
$132$	0	0
$133$	3.45080i	0.299223i
$134$	0	0
$135$	−54.9975	−4.73343
$136$	0	0
$137$	12.0434	1.02893	0.514467	−	0.857510i	$-0.327990\pi$
$137$	12.0434	1.02893	0.514467	+	0.857510i	$0.327990\pi$
$138$	0	0
$139$	− 6.45251i	− 0.547295i	−0.961830	−	0.273648i	$-0.911770\pi$
$139$	− 6.45251i	− 0.547295i	0.961830	−	0.273648i	$-0.0882302\pi$
$140$	0	0
$141$	− 29.7636i	− 2.50655i
$142$	0	0
$143$	−2.19291	−0.183380
$144$	0	0
$145$	−0.561823	−0.0466568
$146$	0	0
$147$	3.17593i	0.261947i
$148$	0	0
$149$	− 14.2645i	− 1.16860i	−0.811539	−	0.584298i	$-0.801370\pi$
$149$	− 14.2645i	− 1.16860i	0.811539	−	0.584298i	$-0.198630\pi$
$150$	0	0
$151$	−2.44985	−0.199366	−0.0996830	−	0.995019i	$-0.531783\pi$
$151$	−2.44985	−0.199366	−0.0996830	+	0.995019i	$0.531783\pi$
$152$	0	0
$153$	−24.8125	−2.00597
$154$	0	0
$155$	23.5570i	1.89215i
$156$	0	0
$157$	− 21.2893i	− 1.69907i	−0.527529	−	0.849537i	$-0.676882\pi$
$157$	− 21.2893i	− 1.69907i	0.527529	−	0.849537i	$-0.323118\pi$
$158$	0	0
$159$	−11.6113	−0.920834
$160$	0	0
$161$	−4.39770	−0.346587
$162$	0	0
$163$	7.11574i	0.557348i	0.960386	+	0.278674i	$0.0898948\pi$
$163$	7.11574i	0.557348i	−0.960386	+	0.278674i	$0.910105\pi$
$164$	0	0
$165$	67.1900i	5.23073i
$166$	0	0
$167$	9.74662	0.754216	0.377108	−	0.926169i	$-0.376919\pi$
$167$	9.74662	0.754216	0.377108	+	0.926169i	$0.376919\pi$
$168$	0	0
$169$	12.8071	0.985159
$170$	0	0
$171$	24.4543i	1.87007i
$172$	0	0
$173$	− 18.7998i	− 1.42932i	−0.699470	−	0.714662i	$-0.746580\pi$
$173$	− 18.7998i	− 1.42932i	0.699470	−	0.714662i	$-0.253420\pi$
$174$	0	0
$175$	12.9568	0.979443
$176$	0	0
$177$	7.00627	0.526623
$178$	0	0
$179$	− 14.1728i	− 1.05932i	−0.848209	−	0.529662i	$-0.822319\pi$
$179$	− 14.1728i	− 1.05932i	0.848209	−	0.529662i	$-0.177681\pi$
$180$	0	0
$181$	− 2.56453i	− 0.190620i	−0.995448	−	0.0953101i	$-0.969616\pi$
$181$	− 2.56453i	− 0.190620i	0.995448	−	0.0953101i	$-0.0303843\pi$
$182$	0	0
$183$	31.4240	2.32293
$184$	0	0
$185$	2.42648	0.178398
$186$	0	0
$187$	17.4805i	1.27830i
$188$	0	0
$189$	12.9786i	0.944054i
$190$	0	0
$191$	−12.9685	−0.938367	−0.469184	−	0.883101i	$-0.655452\pi$
$191$	−12.9685	−0.938367	−0.469184	+	0.883101i	$0.655452\pi$
$192$	0	0
$193$	10.1387	0.729800	0.364900	−	0.931047i	$-0.381103\pi$
$193$	10.1387	0.729800	0.364900	+	0.931047i	$0.381103\pi$
$194$	0	0
$195$	5.91136i	0.423321i
$196$	0	0
$197$	− 2.84111i	− 0.202421i	−0.994865	−	0.101211i	$-0.967728\pi$
$197$	− 2.84111i	− 0.202421i	0.994865	−	0.101211i	$-0.0322716\pi$
$198$	0	0
$199$	11.4094	0.808789	0.404394	−	0.914585i	$-0.367482\pi$
$199$	11.4094	0.808789	0.404394	+	0.914585i	$0.367482\pi$
$200$	0	0
$201$	−27.0461	−1.90768
$202$	0	0
$203$	0.132582i	0.00930543i
$204$	0	0
$205$	22.9835i	1.60524i
$206$	0	0
$207$	−31.1645	−2.16608
$208$	0	0
$209$	17.2282	1.19170
$210$	0	0
$211$	− 7.70104i	− 0.530161i	−0.964226	−	0.265081i	$-0.914601\pi$
$211$	− 7.70104i	− 0.530161i	0.964226	−	0.265081i	$-0.0853986\pi$
$212$	0	0
$213$	− 9.88549i	− 0.677342i
$214$	0	0
$215$	−24.1443	−1.64663
$216$	0	0
$217$	5.55912	0.377378
$218$	0	0
$219$	− 31.3473i	− 2.11825i
$220$	0	0
$221$	1.53793i	0.103452i
$222$	0	0
$223$	−21.9334	−1.46877	−0.734386	−	0.678732i	$-0.762530\pi$
$223$	−21.9334	−1.46877	−0.734386	+	0.678732i	$0.762530\pi$
$224$	0	0
$225$	91.8191	6.12127
$226$	0	0
$227$	− 4.75913i	− 0.315875i	−0.987449	−	0.157937i	$-0.949516\pi$
$227$	− 4.75913i	− 0.315875i	0.987449	−	0.157937i	$-0.0504844\pi$
$228$	0	0
$229$	13.1430i	0.868512i	0.900790	+	0.434256i	$0.142989\pi$
$229$	13.1430i	0.868512i	−0.900790	+	0.434256i	$0.857011\pi$
$230$	0	0
$231$	15.8559	1.04324
$232$	0	0
$233$	−21.0461	−1.37877	−0.689387	−	0.724393i	$-0.742120\pi$
$233$	−21.0461	−1.37877	−0.689387	+	0.724393i	$0.742120\pi$
$234$	0	0
$235$	39.7127i	2.59057i
$236$	0	0
$237$	22.5892i	1.46733i
$238$	0	0
$239$	−25.3482	−1.63964	−0.819821	−	0.572620i	$-0.805927\pi$
$239$	−25.3482	−1.63964	−0.819821	+	0.572620i	$0.805927\pi$
$240$	0	0
$241$	13.4492	0.866339	0.433169	−	0.901312i	$-0.357395\pi$
$241$	13.4492	0.866339	0.433169	+	0.901312i	$0.357395\pi$
$242$	0	0
$243$	24.4543i	1.56874i
$244$	0	0
$245$	− 4.23755i	− 0.270727i
$246$	0	0
$247$	1.51573	0.0964435
$248$	0	0
$249$	7.87924	0.499327
$250$	0	0
$251$	10.2228i	0.645258i	0.946526	+	0.322629i	$0.104567\pi$
$251$	10.2228i	0.645258i	−0.946526	+	0.322629i	$0.895433\pi$
$252$	0	0
$253$	21.9555i	1.38033i
$254$	0	0
$255$	47.1218	2.95088
$256$	0	0
$257$	3.76035	0.234564	0.117282	−	0.993099i	$-0.462582\pi$
$257$	3.76035	0.234564	0.117282	+	0.993099i	$0.462582\pi$
$258$	0	0
$259$	− 0.572613i	− 0.0355805i
$260$	0	0
$261$	0.939549i	0.0581566i
$262$	0	0
$263$	−22.8532	−1.40919	−0.704593	−	0.709611i	$-0.748870\pi$
$263$	−22.8532	−1.40919	−0.704593	+	0.709611i	$0.748870\pi$
$264$	0	0
$265$	15.4926	0.951701
$266$	0	0
$267$	16.0717i	0.983573i
$268$	0	0
$269$	18.7998i	1.14625i	0.819470	+	0.573123i	$0.194268\pi$
$269$	18.7998i	1.14625i	−0.819470	+	0.573123i	$0.805732\pi$
$270$	0	0
$271$	13.9334	0.846396	0.423198	−	0.906037i	$-0.360907\pi$
$271$	13.9334	0.846396	0.423198	+	0.906037i	$0.360907\pi$
$272$	0	0
$273$	1.39499	0.0844289
$274$	0	0
$275$	− 64.6870i	− 3.90077i
$276$	0	0
$277$	21.0462i	1.26455i	0.774746	+	0.632273i	$0.217878\pi$
$277$	21.0462i	1.26455i	−0.774746	+	0.632273i	$0.782122\pi$
$278$	0	0
$279$	39.3950	2.35852
$280$	0	0
$281$	2.38040	0.142003	0.0710015	−	0.997476i	$-0.477380\pi$
$281$	2.38040	0.142003	0.0710015	+	0.997476i	$0.477380\pi$
$282$	0	0
$283$	− 10.4422i	− 0.620726i	−0.950618	−	0.310363i	$-0.899549\pi$
$283$	− 10.4422i	− 0.620726i	0.950618	−	0.310363i	$-0.100451\pi$
$284$	0	0
$285$	− 46.4415i	− 2.75096i
$286$	0	0
$287$	5.42377	0.320155
$288$	0	0
$289$	−4.74054	−0.278855
$290$	0	0
$291$	− 52.9026i	− 3.10121i
$292$	0	0
$293$	− 15.3657i	− 0.897675i	−0.893613	−	0.448838i	$-0.851838\pi$
$293$	− 15.3657i	− 0.897675i	0.893613	−	0.448838i	$-0.148162\pi$
$294$	0	0
$295$	−9.34825	−0.544276
$296$	0	0
$297$	64.7958	3.75983
$298$	0	0
$299$	1.93164i	0.111710i
$300$	0	0
$301$	5.69770i	0.328410i
$302$	0	0
$303$	32.5240	1.86846
$304$	0	0
$305$	−41.9280	−2.40079
$306$	0	0
$307$	25.0880i	1.43185i	0.698179	+	0.715923i	$0.253994\pi$
$307$	25.0880i	1.43185i	−0.698179	+	0.715923i	$0.746006\pi$
$308$	0	0
$309$	− 33.4807i	− 1.90465i
$310$	0	0
$311$	−20.6079	−1.16857	−0.584283	−	0.811550i	$-0.698624\pi$
$311$	−20.6079	−1.16857	−0.584283	+	0.811550i	$0.698624\pi$
$312$	0	0
$313$	−29.5501	−1.67027	−0.835136	−	0.550044i	$-0.814611\pi$
$313$	−29.5501	−1.67027	−0.835136	+	0.550044i	$0.814611\pi$
$314$	0	0
$315$	− 30.0296i	− 1.69198i
$316$	0	0
$317$	− 6.41069i	− 0.360060i	−0.983661	−	0.180030i	$-0.942380\pi$
$317$	− 6.41069i	− 0.360060i	0.983661	−	0.180030i	$-0.0576195\pi$
$318$	0	0
$319$	0.661916	0.0370602
$320$	0	0
$321$	−0.680067	−0.0379576
$322$	0	0
$323$	− 12.0825i	− 0.672287i
$324$	0	0
$325$	− 5.69114i	− 0.315688i
$326$	0	0
$327$	62.6540	3.46477
$328$	0	0
$329$	9.37162	0.516674
$330$	0	0
$331$	1.45124i	0.0797673i	0.999204	+	0.0398837i	$0.0126987\pi$
$331$	1.45124i	0.0797673i	−0.999204	+	0.0398837i	$0.987301\pi$
$332$	0	0
$333$	− 4.05785i	− 0.222369i
$334$	0	0
$335$	36.0867	1.97163
$336$	0	0
$337$	−11.1794	−0.608979	−0.304489	−	0.952516i	$-0.598486\pi$
$337$	−11.1794	−0.608979	−0.304489	+	0.952516i	$0.598486\pi$
$338$	0	0
$339$	− 15.7339i	− 0.854550i
$340$	0	0
$341$	− 27.7539i	− 1.50296i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	59.1850	3.18641
$346$	0	0
$347$	− 20.4430i	− 1.09744i	−0.836007	−	0.548718i	$-0.815116\pi$
$347$	− 20.4430i	− 1.09744i	0.836007	−	0.548718i	$-0.184884\pi$
$348$	0	0
$349$	− 24.2915i	− 1.30030i	−0.759808	−	0.650148i	$-0.774707\pi$
$349$	− 24.2915i	− 1.30030i	0.759808	−	0.650148i	$-0.225293\pi$
$350$	0	0
$351$	5.70071	0.304281
$352$	0	0
$353$	14.7810	0.786713	0.393356	−	0.919386i	$-0.371314\pi$
$353$	14.7810	0.786713	0.393356	+	0.919386i	$0.371314\pi$
$354$	0	0
$355$	13.1899i	0.700047i
$356$	0	0
$357$	− 11.1201i	− 0.588536i
$358$	0	0
$359$	0.340137	0.0179518	0.00897588	−	0.999960i	$-0.497143\pi$
$359$	0.340137	0.0179518	0.00897588	+	0.999960i	$0.497143\pi$
$360$	0	0
$361$	7.09195	0.373261
$362$	0	0
$363$	− 44.2253i	− 2.32122i
$364$	0	0
$365$	41.8257i	2.18926i
$366$	0	0
$367$	18.9163	0.987425	0.493712	−	0.869625i	$-0.335640\pi$
$367$	18.9163	0.987425	0.493712	+	0.869625i	$0.335640\pi$
$368$	0	0
$369$	38.4358	2.00089
$370$	0	0
$371$	− 3.65602i	− 0.189811i
$372$	0	0
$373$	34.0151i	1.76124i	0.473826	+	0.880619i	$0.342873\pi$
$373$	34.0151i	1.76124i	−0.473826	+	0.880619i	$0.657127\pi$
$374$	0	0
$375$	−107.084	−5.52980
$376$	0	0
$377$	0.0582352	0.00299927
$378$	0	0
$379$	30.1628i	1.54936i	0.632354	+	0.774680i	$0.282089\pi$
$379$	30.1628i	1.54936i	−0.632354	+	0.774680i	$0.717911\pi$
$380$	0	0
$381$	24.6941i	1.26512i
$382$	0	0
$383$	−23.3518	−1.19322	−0.596611	−	0.802531i	$-0.703486\pi$
$383$	−23.3518	−1.19322	−0.596611	+	0.802531i	$0.703486\pi$
$384$	0	0
$385$	−21.1560	−1.07821
$386$	0	0
$387$	40.3770i	2.05248i
$388$	0	0
$389$	− 7.26880i	− 0.368543i	−0.982875	−	0.184271i	$-0.941007\pi$
$389$	− 7.26880i	− 0.368543i	0.982875	−	0.184271i	$-0.0589925\pi$
$390$	0	0
$391$	15.3979	0.778704
$392$	0	0
$393$	26.1045	1.31680
$394$	0	0
$395$	− 30.1401i	− 1.51651i
$396$	0	0
$397$	11.5674i	0.580552i	0.956943	+	0.290276i	$0.0937471\pi$
$397$	11.5674i	0.580552i	−0.956943	+	0.290276i	$0.906253\pi$
$398$	0	0
$399$	−10.9595	−0.548662
$400$	0	0
$401$	−31.0029	−1.54821	−0.774105	−	0.633057i	$-0.781800\pi$
$401$	−31.0029	−1.54821	−0.774105	+	0.633057i	$0.781800\pi$
$402$	0	0
$403$	− 2.44178i	− 0.121634i
$404$	0	0
$405$	− 84.5794i	− 4.20279i
$406$	0	0
$407$	−2.85878	−0.141704
$408$	0	0
$409$	−33.5825	−1.66055	−0.830273	−	0.557357i	$-0.811816\pi$
$409$	−33.5825	−1.66055	−0.830273	+	0.557357i	$0.811816\pi$
$410$	0	0
$411$	38.2489i	1.88668i
$412$	0	0
$413$	2.20605i	0.108553i
$414$	0	0
$415$	−10.5130	−0.516064
$416$	0	0
$417$	20.4927	1.00353
$418$	0	0
$419$	− 12.4659i	− 0.609001i	−0.952512	−	0.304500i	$-0.901511\pi$
$419$	− 12.4659i	− 0.609001i	0.952512	−	0.304500i	$-0.0984895\pi$
$420$	0	0
$421$	21.0462i	1.02573i	0.858469	+	0.512866i	$0.171416\pi$
$421$	21.0462i	1.02573i	−0.858469	+	0.512866i	$0.828584\pi$
$422$	0	0
$423$	66.4124	3.22908
$424$	0	0
$425$	−45.3664	−2.20059
$426$	0	0
$427$	9.89440i	0.478824i
$428$	0	0
$429$	− 6.96452i	− 0.336250i
$430$	0	0
$431$	8.55285	0.411976	0.205988	−	0.978554i	$-0.433959\pi$
$431$	8.55285	0.411976	0.205988	+	0.978554i	$0.433959\pi$
$432$	0	0
$433$	9.10654	0.437633	0.218816	−	0.975766i	$-0.429780\pi$
$433$	9.10654	0.437633	0.218816	+	0.975766i	$0.429780\pi$
$434$	0	0
$435$	− 1.78431i	− 0.0855511i
$436$	0	0
$437$	− 15.1756i	− 0.725947i
$438$	0	0
$439$	−0.0179421	−0.000856331 0	−0.000428166	−	1.00000i	$-0.500136\pi$
$439$	−0.0179421	−0.000856331 0	−0.000428166		1.00000i	$0.500136\pi$
$440$	0	0
$441$	−7.08655	−0.337455
$442$	0	0
$443$	− 8.69941i	− 0.413322i	−0.978413	−	0.206661i	$-0.933740\pi$
$443$	− 8.69941i	− 0.413322i	0.978413	−	0.206661i	$-0.0662597\pi$
$444$	0	0
$445$	− 21.4440i	− 1.01654i
$446$	0	0
$447$	45.3032	2.14277
$448$	0	0
$449$	−4.85503	−0.229123	−0.114562	−	0.993416i	$-0.536546\pi$
$449$	−4.85503	−0.229123	−0.114562	+	0.993416i	$0.536546\pi$
$450$	0	0
$451$	− 27.0782i	− 1.27506i
$452$	0	0
$453$	− 7.78056i	− 0.365563i
$454$	0	0
$455$	−1.86130	−0.0872590
$456$	0	0
$457$	1.27114	0.0594613	0.0297306	−	0.999558i	$-0.490535\pi$
$457$	1.27114	0.0594613	0.0297306	+	0.999558i	$0.490535\pi$
$458$	0	0
$459$	− 45.4426i	− 2.12108i
$460$	0	0
$461$	8.38224i	0.390400i	0.980763	+	0.195200i	$0.0625356\pi$
$461$	8.38224i	0.390400i	−0.980763	+	0.195200i	$0.937464\pi$
$462$	0	0
$463$	39.7117	1.84556	0.922780	−	0.385326i	$-0.125911\pi$
$463$	39.7117	1.84556	0.922780	+	0.385326i	$0.125911\pi$
$464$	0	0
$465$	−74.8156	−3.46949
$466$	0	0
$467$	43.0897i	1.99396i	0.0776879	+	0.996978i	$0.475246\pi$
$467$	43.0897i	1.99396i	−0.0776879	+	0.996978i	$0.524754\pi$
$468$	0	0
$469$	− 8.51595i	− 0.393230i
$470$	0	0
$471$	67.6135	3.11546
$472$	0	0
$473$	28.4458	1.30794
$474$	0	0
$475$	44.7114i	2.05150i
$476$	0	0
$477$	− 25.9086i	− 1.18627i
$478$	0	0
$479$	16.9224	0.773205	0.386603	−	0.922246i	$-0.373649\pi$
$479$	16.9224	0.773205	0.386603	+	0.922246i	$0.373649\pi$
$480$	0	0
$481$	−0.251514	−0.0114681
$482$	0	0
$483$	− 13.9668i	− 0.635511i
$484$	0	0
$485$	70.5863i	3.20516i
$486$	0	0
$487$	−12.7727	−0.578786	−0.289393	−	0.957210i	$-0.593453\pi$
$487$	−12.7727	−0.578786	−0.289393	+	0.957210i	$0.593453\pi$
$488$	0	0
$489$	−22.5991	−1.02197
$490$	0	0
$491$	17.4413i	0.787113i	0.919300	+	0.393556i	$0.128755\pi$
$491$	17.4413i	0.787113i	−0.919300	+	0.393556i	$0.871245\pi$
$492$	0	0
$493$	− 0.464216i	− 0.0209072i
$494$	0	0
$495$	−149.923	−6.73854
$496$	0	0
$497$	3.11263	0.139620
$498$	0	0
$499$	26.0896i	1.16793i	0.811778	+	0.583966i	$0.198500\pi$
$499$	26.0896i	1.16793i	−0.811778	+	0.583966i	$0.801500\pi$
$500$	0	0
$501$	30.9546i	1.38295i
$502$	0	0
$503$	−13.4299	−0.598808	−0.299404	−	0.954126i	$-0.596788\pi$
$503$	−13.4299	−0.598808	−0.299404	+	0.954126i	$0.596788\pi$
$504$	0	0
$505$	−43.3958	−1.93109
$506$	0	0
$507$	40.6744i	1.80641i
$508$	0	0
$509$	0.235880i	0.0104552i	0.999986	+	0.00522760i	$0.00166401\pi$
$509$	0.235880i	0.0104552i	−0.999986	+	0.00522760i	$0.998336\pi$
$510$	0	0
$511$	9.87027	0.436635
$512$	0	0
$513$	−44.7866	−1.97738
$514$	0	0
$515$	44.6723i	1.96850i
$516$	0	0
$517$	− 46.7879i	− 2.05773i
$518$	0	0
$519$	59.7070	2.62085
$520$	0	0
$521$	−13.7124	−0.600751	−0.300376	−	0.953821i	$-0.597112\pi$
$521$	−13.7124	−0.600751	−0.300376	+	0.953821i	$0.597112\pi$
$522$	0	0
$523$	− 0.594362i	− 0.0259897i	−0.999916	−	0.0129948i	$-0.995864\pi$
$523$	− 0.594362i	− 0.0259897i	0.999916	−	0.0129948i	$-0.00413650\pi$
$524$	0	0
$525$	41.1500i	1.79593i
$526$	0	0
$527$	−19.4644	−0.847884
$528$	0	0
$529$	−3.66026	−0.159142
$530$	0	0
$531$	15.6333i	0.678427i
$532$	0	0
$533$	− 2.38233i	− 0.103190i
$534$	0	0
$535$	0.907392	0.0392300
$536$	0	0
$537$	45.0119	1.94240
$538$	0	0
$539$	4.99251i	0.215042i
$540$	0	0
$541$	23.2258i	0.998555i	0.866442	+	0.499278i	$0.166401\pi$
$541$	23.2258i	0.998555i	−0.866442	+	0.499278i	$0.833599\pi$
$542$	0	0
$543$	8.14478	0.349526
$544$	0	0
$545$	−83.5972	−3.58091
$546$	0	0
$547$	29.6868i	1.26932i	0.772793	+	0.634659i	$0.218859\pi$
$547$	29.6868i	1.26932i	−0.772793	+	0.634659i	$0.781141\pi$
$548$	0	0
$549$	70.1172i	2.99253i
$550$	0	0
$551$	−0.457514	−0.0194908
$552$	0	0
$553$	−7.11263	−0.302460
$554$	0	0
$555$	7.70633i	0.327115i
$556$	0	0
$557$	18.0090i	0.763065i	0.924355	+	0.381532i	$0.124604\pi$
$557$	18.0090i	0.763065i	−0.924355	+	0.381532i	$0.875396\pi$
$558$	0	0
$559$	2.50265	0.105851
$560$	0	0
$561$	−55.5170	−2.34393
$562$	0	0
$563$	− 35.6985i	− 1.50451i	−0.658870	−	0.752257i	$-0.728965\pi$
$563$	− 35.6985i	− 1.50451i	0.658870	−	0.752257i	$-0.271035\pi$
$564$	0	0
$565$	20.9933i	0.883195i
$566$	0	0
$567$	−19.9595	−0.838221
$568$	0	0
$569$	−40.4761	−1.69685	−0.848423	−	0.529318i	$-0.822448\pi$
$569$	−40.4761	−1.69685	−0.848423	+	0.529318i	$0.822448\pi$
$570$	0	0
$571$	6.39496i	0.267621i	0.991007	+	0.133810i	$0.0427213\pi$
$571$	6.39496i	0.267621i	−0.991007	+	0.133810i	$0.957279\pi$
$572$	0	0
$573$	− 41.1871i	− 1.72061i
$574$	0	0
$575$	−56.9802	−2.37624
$576$	0	0
$577$	25.6686	1.06860	0.534298	−	0.845296i	$-0.320576\pi$
$577$	25.6686	1.06860	0.534298	+	0.845296i	$0.320576\pi$
$578$	0	0
$579$	32.1998i	1.33818i
$580$	0	0
$581$	2.48092i	0.102926i
$582$	0	0
$583$	−18.2527	−0.755950
$584$	0	0
$585$	−13.1902	−0.545347
$586$	0	0
$587$	30.4700i	1.25763i	0.777555	+	0.628815i	$0.216460\pi$
$587$	30.4700i	1.25763i	−0.777555	+	0.628815i	$0.783540\pi$
$588$	0	0
$589$	19.1834i	0.790439i
$590$	0	0
$591$	9.02319	0.371164
$592$	0	0
$593$	14.9541	0.614091	0.307045	−	0.951695i	$-0.400660\pi$
$593$	14.9541	0.614091	0.307045	+	0.951695i	$0.400660\pi$
$594$	0	0
$595$	14.8371i	0.608264i
$596$	0	0
$597$	36.2354i	1.48302i
$598$	0	0
$599$	−27.9316	−1.14125	−0.570627	−	0.821210i	$-0.693300\pi$
$599$	−27.9316	−1.14125	−0.570627	+	0.821210i	$0.693300\pi$
$600$	0	0
$601$	−33.5110	−1.36694	−0.683471	−	0.729978i	$-0.739530\pi$
$601$	−33.5110	−1.36694	−0.683471	+	0.729978i	$0.739530\pi$
$602$	0	0
$603$	− 60.3487i	− 2.45759i
$604$	0	0
$605$	59.0084i	2.39903i
$606$	0	0
$607$	16.0723	0.652356	0.326178	−	0.945308i	$-0.394239\pi$
$607$	16.0723	0.652356	0.326178	+	0.945308i	$0.394239\pi$
$608$	0	0
$609$	−0.421071	−0.0170627
$610$	0	0
$611$	− 4.11638i	− 0.166531i
$612$	0	0
$613$	− 27.4341i	− 1.10805i	−0.832500	−	0.554026i	$-0.813091\pi$
$613$	− 27.4341i	− 1.10805i	0.832500	−	0.554026i	$-0.186909\pi$
$614$	0	0
$615$	−72.9941	−2.94340
$616$	0	0
$617$	16.0290	0.645303	0.322651	−	0.946518i	$-0.395426\pi$
$617$	16.0290	0.645303	0.322651	+	0.946518i	$0.395426\pi$
$618$	0	0
$619$	− 6.58966i	− 0.264861i	−0.991192	−	0.132430i	$-0.957722\pi$
$619$	− 6.58966i	− 0.264861i	0.991192	−	0.132430i	$-0.0422781\pi$
$620$	0	0
$621$	− 57.0760i	− 2.29038i
$622$	0	0
$623$	−5.06047	−0.202743
$624$	0	0
$625$	78.0950	3.12380
$626$	0	0
$627$	54.7155i	2.18513i
$628$	0	0
$629$	2.00492i	0.0799414i
$630$	0	0
$631$	−15.2225	−0.606000	−0.303000	−	0.952991i	$-0.597988\pi$
$631$	−15.2225	−0.606000	−0.303000	+	0.952991i	$0.597988\pi$
$632$	0	0
$633$	24.4580	0.972117
$634$	0	0
$635$	− 32.9486i	− 1.30753i
$636$	0	0
$637$	0.439239i	0.0174033i
$638$	0	0
$639$	22.0578	0.872592
$640$	0	0
$641$	13.0371	0.514934	0.257467	−	0.966287i	$-0.417112\pi$
$641$	13.0371	0.514934	0.257467	+	0.966287i	$0.417112\pi$
$642$	0	0
$643$	− 15.6242i	− 0.616159i	−0.951361	−	0.308079i	$-0.900314\pi$
$643$	− 15.6242i	− 0.616159i	0.951361	−	0.308079i	$-0.0996862\pi$
$644$	0	0
$645$	− 76.6806i	− 3.01930i
$646$	0	0
$647$	14.5276	0.571139	0.285570	−	0.958358i	$-0.407817\pi$
$647$	14.5276	0.571139	0.285570	+	0.958358i	$0.407817\pi$
$648$	0	0
$649$	11.0137	0.432326
$650$	0	0
$651$	17.6554i	0.691969i
$652$	0	0
$653$	− 34.8479i	− 1.36370i	−0.731491	−	0.681851i	$-0.761175\pi$
$653$	− 34.8479i	− 1.36370i	0.731491	−	0.681851i	$-0.238825\pi$
$654$	0	0
$655$	−34.8304	−1.36094
$656$	0	0
$657$	69.9461	2.72886
$658$	0	0
$659$	48.8820i	1.90417i	0.305828	+	0.952087i	$0.401067\pi$
$659$	48.8820i	1.90417i	−0.305828	+	0.952087i	$0.598933\pi$
$660$	0	0
$661$	− 1.21595i	− 0.0472948i	−0.999720	−	0.0236474i	$-0.992472\pi$
$661$	− 1.21595i	− 0.0472948i	0.999720	−	0.0236474i	$-0.00752791\pi$
$662$	0	0
$663$	−4.88437	−0.189693
$664$	0	0
$665$	14.6229	0.567054
$666$	0	0
$667$	− 0.583055i	− 0.0225760i
$668$	0	0
$669$	− 69.6591i	− 2.69318i
$670$	0	0
$671$	49.3979	1.90698
$672$	0	0
$673$	15.3381	0.591240	0.295620	−	0.955306i	$-0.404474\pi$
$673$	15.3381	0.591240	0.295620	+	0.955306i	$0.404474\pi$
$674$	0	0
$675$	168.161i	6.47253i
$676$	0	0
$677$	34.1184i	1.31128i	0.755074	+	0.655639i	$0.227601\pi$
$677$	34.1184i	1.31128i	−0.755074	+	0.655639i	$0.772399\pi$
$678$	0	0
$679$	16.6573	0.639250
$680$	0	0
$681$	15.1147	0.579196
$682$	0	0
$683$	21.4966i	0.822545i	0.911512	+	0.411272i	$0.134916\pi$
$683$	21.4966i	0.822545i	−0.911512	+	0.411272i	$0.865084\pi$
$684$	0	0
$685$	− 51.0343i	− 1.94992i
$686$	0	0
$687$	−41.7412	−1.59253
$688$	0	0
$689$	−1.60587	−0.0611787
$690$	0	0
$691$	− 47.8380i	− 1.81984i	−0.414778	−	0.909922i	$-0.636141\pi$
$691$	− 47.8380i	− 1.81984i	0.414778	−	0.909922i	$-0.363859\pi$
$692$	0	0
$693$	35.3796i	1.34396i
$694$	0	0
$695$	−27.3428	−1.03717
$696$	0	0
$697$	−18.9905	−0.719318
$698$	0	0
$699$	− 66.8409i	− 2.52815i
$700$	0	0
$701$	− 25.1274i	− 0.949047i	−0.880243	−	0.474524i	$-0.842620\pi$
$701$	− 25.1274i	− 0.949047i	0.880243	−	0.474524i	$-0.157380\pi$
$702$	0	0
$703$	1.97598	0.0745254
$704$	0	0
$705$	−126.125	−4.75014
$706$	0	0
$707$	10.2408i	0.385144i
$708$	0	0
$709$	47.8088i	1.79550i	0.440507	+	0.897749i	$0.354799\pi$
$709$	47.8088i	1.79550i	−0.440507	+	0.897749i	$0.645201\pi$
$710$	0	0
$711$	−50.4040	−1.89030
$712$	0	0
$713$	−24.4473	−0.915560
$714$	0	0
$715$	9.29254i	0.347521i
$716$	0	0
$717$	− 80.5043i	− 3.00649i
$718$	0	0
$719$	53.1845	1.98345	0.991723	−	0.128393i	$-0.0409820\pi$
$719$	53.1845	1.98345	0.991723	+	0.128393i	$0.0409820\pi$
$720$	0	0
$721$	10.5420	0.392605
$722$	0	0
$723$	42.7137i	1.58854i
$724$	0	0
$725$	1.71784i	0.0637990i
$726$	0	0
$727$	−35.4638	−1.31528	−0.657639	−	0.753333i	$-0.728445\pi$
$727$	−35.4638	−1.31528	−0.657639	+	0.753333i	$0.728445\pi$
$728$	0	0
$729$	−17.7866	−0.658763
$730$	0	0
$731$	− 19.9497i	− 0.737865i
$732$	0	0
$733$	− 13.6131i	− 0.502810i	−0.967882	−	0.251405i	$-0.919107\pi$
$733$	− 13.6131i	− 0.502810i	0.967882	−	0.251405i	$-0.0808927\pi$
$734$	0	0
$735$	13.4582	0.496412
$736$	0	0
$737$	−42.5159	−1.56609
$738$	0	0
$739$	− 17.4117i	− 0.640498i	−0.947333	−	0.320249i	$-0.896233\pi$
$739$	− 17.4117i	− 0.640498i	0.947333	−	0.320249i	$-0.103767\pi$
$740$	0	0
$741$	4.81385i	0.176841i
$742$	0	0
$743$	14.7947	0.542767	0.271383	−	0.962471i	$-0.412519\pi$
$743$	14.7947	0.542767	0.271383	+	0.962471i	$0.412519\pi$
$744$	0	0
$745$	−60.4467	−2.21459
$746$	0	0
$747$	17.5812i	0.643261i
$748$	0	0
$749$	− 0.214131i	− 0.00782419i
$750$	0	0
$751$	−3.48677	−0.127234	−0.0636170	−	0.997974i	$-0.520264\pi$
$751$	−3.48677	−0.127234	−0.0636170	+	0.997974i	$0.520264\pi$
$752$	0	0
$753$	−32.4670	−1.18316
$754$	0	0
$755$	10.3814i	0.377817i
$756$	0	0
$757$	− 35.4628i	− 1.28892i	−0.764640	−	0.644458i	$-0.777083\pi$
$757$	− 35.4628i	− 1.28892i	0.764640	−	0.644458i	$-0.222917\pi$
$758$	0	0
$759$	−69.7293	−2.53101
$760$	0	0
$761$	47.1553	1.70938	0.854689	−	0.519140i	$-0.173748\pi$
$761$	47.1553	1.70938	0.854689	+	0.519140i	$0.173748\pi$
$762$	0	0
$763$	19.7277i	0.714192i
$764$	0	0
$765$	105.144i	3.80149i
$766$	0	0
$767$	0.968984	0.0349880
$768$	0	0
$769$	22.9478	0.827520	0.413760	−	0.910386i	$-0.364215\pi$
$769$	22.9478	0.827520	0.413760	+	0.910386i	$0.364215\pi$
$770$	0	0
$771$	11.9426i	0.430103i
$772$	0	0
$773$	− 32.9936i	− 1.18670i	−0.804946	−	0.593348i	$-0.797806\pi$
$773$	− 32.9936i	− 1.18670i	0.804946	−	0.593348i	$-0.202194\pi$
$774$	0	0
$775$	72.0285	2.58734
$776$	0	0
$777$	1.81858	0.0652413
$778$	0	0
$779$	18.7164i	0.670584i
$780$	0	0
$781$	− 15.5398i	− 0.556058i
$782$	0	0
$783$	−1.72073	−0.0614938
$784$	0	0
$785$	−90.2146	−3.21990
$786$	0	0
$787$	11.0916i	0.395372i	0.980265	+	0.197686i	$0.0633427\pi$
$787$	11.0916i	0.395372i	−0.980265	+	0.197686i	$0.936657\pi$
$788$	0	0
$789$	− 72.5801i	− 2.58392i
$790$	0	0
$791$	4.95411	0.176148
$792$	0	0
$793$	4.34601	0.154331
$794$	0	0
$795$	49.2033i	1.74506i
$796$	0	0
$797$	8.23745i	0.291785i	0.989300	+	0.145893i	$0.0466054\pi$
$797$	8.23745i	0.291785i	−0.989300	+	0.145893i	$0.953395\pi$
$798$	0	0
$799$	−32.8133	−1.16085
$800$	0	0
$801$	−35.8613	−1.26710
$802$	0	0
$803$	− 49.2774i	− 1.73896i
$804$	0	0
$805$	18.6355i	0.656814i
$806$	0	0
$807$	−59.7070	−2.10179
$808$	0	0
$809$	−12.9074	−0.453799	−0.226899	−	0.973918i	$-0.572859\pi$
$809$	−12.9074	−0.453799	−0.226899	+	0.973918i	$0.572859\pi$
$810$	0	0
$811$	19.4515i	0.683034i	0.939876	+	0.341517i	$0.110941\pi$
$811$	19.4515i	0.683034i	−0.939876	+	0.341517i	$0.889059\pi$
$812$	0	0
$813$	44.2517i	1.55197i
$814$	0	0
$815$	30.1533	1.05622
$816$	0	0
$817$	−19.6617	−0.687874
$818$	0	0
$819$	3.11269i	0.108766i
$820$	0	0
$821$	− 21.7594i	− 0.759408i	−0.925108	−	0.379704i	$-0.876026\pi$
$821$	− 21.7594i	− 0.759408i	0.925108	−	0.379704i	$-0.123974\pi$
$822$	0	0
$823$	−13.8847	−0.483989	−0.241995	−	0.970278i	$-0.577802\pi$
$823$	−13.8847	−0.483989	−0.241995	+	0.970278i	$0.577802\pi$
$824$	0	0
$825$	205.442	7.15256
$826$	0	0
$827$	− 32.2565i	− 1.12167i	−0.827929	−	0.560833i	$-0.810481\pi$
$827$	− 32.2565i	− 1.12167i	0.827929	−	0.560833i	$-0.189519\pi$
$828$	0	0
$829$	41.1781i	1.43018i	0.699035	+	0.715088i	$0.253613\pi$
$829$	41.1781i	1.43018i	−0.699035	+	0.715088i	$0.746387\pi$
$830$	0	0
$831$	−66.8415	−2.31870
$832$	0	0
$833$	3.50135	0.121315
$834$	0	0
$835$	− 41.3018i	− 1.42931i
$836$	0	0
$837$	72.1496i	2.49385i
$838$	0	0
$839$	22.9026	0.790686	0.395343	−	0.918534i	$-0.370626\pi$
$839$	22.9026	0.790686	0.395343	+	0.918534i	$0.370626\pi$
$840$	0	0
$841$	28.9824	0.999394
$842$	0	0
$843$	7.56000i	0.260380i
$844$	0	0
$845$	− 54.2706i	− 1.86696i
$846$	0	0
$847$	13.9251	0.478473
$848$	0	0
$849$	33.1638	1.13818
$850$	0	0
$851$	2.51818i	0.0863221i
$852$	0	0
$853$	37.8914i	1.29738i	0.761054	+	0.648688i	$0.224682\pi$
$853$	37.8914i	1.29738i	−0.761054	+	0.648688i	$0.775318\pi$
$854$	0	0
$855$	103.626	3.54394
$856$	0	0
$857$	4.47323	0.152802	0.0764012	−	0.997077i	$-0.475657\pi$
$857$	4.47323	0.152802	0.0764012	+	0.997077i	$0.475657\pi$
$858$	0	0
$859$	− 40.5244i	− 1.38268i	−0.722532	−	0.691338i	$-0.757022\pi$
$859$	− 40.5244i	− 1.38268i	0.722532	−	0.691338i	$-0.242978\pi$
$860$	0	0
$861$	17.2255i	0.587045i
$862$	0	0
$863$	−43.2050	−1.47071	−0.735357	−	0.677680i	$-0.762986\pi$
$863$	−43.2050	−1.47071	−0.735357	+	0.677680i	$0.762986\pi$
$864$	0	0
$865$	−79.6652	−2.70870
$866$	0	0
$867$	− 15.0556i	− 0.511316i
$868$	0	0
$869$	35.5098i	1.20459i
$870$	0	0
$871$	−3.74054	−0.126743
$872$	0	0
$873$	118.043	3.99515
$874$	0	0
$875$	− 33.7174i	− 1.13986i
$876$	0	0
$877$	28.0420i	0.946910i	0.880818	+	0.473455i	$0.156993\pi$
$877$	28.0420i	0.946910i	−0.880818	+	0.473455i	$0.843007\pi$
$878$	0	0
$879$	48.8005	1.64600
$880$	0	0
$881$	−49.2769	−1.66018	−0.830091	−	0.557628i	$-0.811712\pi$
$881$	−49.2769	−1.66018	−0.830091	+	0.557628i	$0.811712\pi$
$882$	0	0
$883$	15.5295i	0.522610i	0.965256	+	0.261305i	$0.0841528\pi$
$883$	15.5295i	0.522610i	−0.965256	+	0.261305i	$0.915847\pi$
$884$	0	0
$885$	− 29.6894i	− 0.997998i
$886$	0	0
$887$	31.0779	1.04350	0.521748	−	0.853100i	$-0.325280\pi$
$887$	31.0779	1.04350	0.521748	+	0.853100i	$0.325280\pi$
$888$	0	0
$889$	−7.77540	−0.260779
$890$	0	0
$891$	99.6480i	3.33834i
$892$	0	0
$893$	32.3396i	1.08220i
$894$	0	0
$895$	−60.0579	−2.00752
$896$	0	0
$897$	−6.13476	−0.204834
$898$	0	0
$899$	0.737039i	0.0245816i
$900$	0	0
$901$	12.8010i	0.426464i
$902$	0	0
$903$	−18.0955	−0.602181
$904$	0	0
$905$	−10.8673	−0.361242
$906$	0	0
$907$	21.3972i	0.710482i	0.934775	+	0.355241i	$0.115601\pi$
$907$	21.3972i	0.710482i	−0.934775	+	0.355241i	$0.884399\pi$
$908$	0	0
$909$	72.5718i	2.40706i
$910$	0	0
$911$	−22.0861	−0.731745	−0.365872	−	0.930665i	$-0.619229\pi$
$911$	−22.0861	−0.731745	−0.365872	+	0.930665i	$0.619229\pi$
$912$	0	0
$913$	12.3860	0.409917
$914$	0	0
$915$	− 133.161i	− 4.40215i
$916$	0	0
$917$	8.21947i	0.271431i
$918$	0	0
$919$	6.21964	0.205167	0.102583	−	0.994724i	$-0.467289\pi$
$919$	6.21964	0.205167	0.102583	+	0.994724i	$0.467289\pi$
$920$	0	0
$921$	−79.6777	−2.62547
$922$	0	0
$923$	− 1.36719i	− 0.0450015i
$924$	0	0
$925$	− 7.41925i	− 0.243943i
$926$	0	0
$927$	74.7065	2.45368
$928$	0	0
$929$	11.2163	0.367993	0.183997	−	0.982927i	$-0.441096\pi$
$929$	11.2163	0.367993	0.183997	+	0.982927i	$0.441096\pi$
$930$	0	0
$931$	− 3.45080i	− 0.113096i
$932$	0	0
$933$	− 65.4493i	− 2.14271i
$934$	0	0
$935$	74.0746	2.42250
$936$	0	0
$937$	1.58518	0.0517854	0.0258927	−	0.999665i	$-0.491757\pi$
$937$	1.58518	0.0517854	0.0258927	+	0.999665i	$0.491757\pi$
$938$	0	0
$939$	− 93.8492i	− 3.06265i
$940$	0	0
$941$	− 49.6193i	− 1.61754i	−0.588122	−	0.808772i	$-0.700133\pi$
$941$	− 49.6193i	− 1.61754i	0.588122	−	0.808772i	$-0.299867\pi$
$942$	0	0
$943$	−23.8521	−0.776732
$944$	0	0
$945$	54.9975	1.78907
$946$	0	0
$947$	− 11.2446i	− 0.365399i	−0.983169	−	0.182699i	$-0.941516\pi$
$947$	− 11.2446i	− 0.365399i	0.983169	−	0.182699i	$-0.0584835\pi$
$948$	0	0
$949$	− 4.33541i	− 0.140733i
$950$	0	0
$951$	20.3599	0.660216
$952$	0	0
$953$	−22.3174	−0.722931	−0.361466	−	0.932385i	$-0.617723\pi$
$953$	−22.3174	−0.722931	−0.361466	+	0.932385i	$0.617723\pi$
$954$	0	0
$955$	54.9546i	1.77829i
$956$	0	0
$957$	2.10220i	0.0679545i
$958$	0	0
$959$	−12.0434	−0.388901
$960$	0	0
$961$	−0.0961895	−0.00310289
$962$	0	0
$963$	− 1.51745i	− 0.0488992i
$964$	0	0
$965$	− 42.9632i	− 1.38304i
$966$	0	0
$967$	−23.9831	−0.771244	−0.385622	−	0.922657i	$-0.626013\pi$
$967$	−23.9831	−0.771244	−0.385622	+	0.922657i	$0.626013\pi$
$968$	0	0
$969$	38.3731	1.23272
$970$	0	0
$971$	4.04470i	0.129801i	0.997892	+	0.0649004i	$0.0206730\pi$
$971$	4.04470i	0.129801i	−0.997892	+	0.0649004i	$0.979327\pi$
$972$	0	0
$973$	6.45251i	0.206858i
$974$	0	0
$975$	18.0747	0.578853
$976$	0	0
$977$	38.0146	1.21619	0.608097	−	0.793863i	$-0.291933\pi$
$977$	38.0146	1.21619	0.608097	+	0.793863i	$0.291933\pi$
$978$	0	0
$979$	25.2644i	0.807455i
$980$	0	0
$981$	139.802i	4.46352i
$982$	0	0
$983$	21.6742	0.691301	0.345650	−	0.938363i	$-0.387658\pi$
$983$	21.6742	0.691301	0.345650	+	0.938363i	$0.387658\pi$
$984$	0	0
$985$	−12.0394	−0.383606
$986$	0	0
$987$	29.7636i	0.947387i
$988$	0	0
$989$	− 25.0568i	− 0.796759i
$990$	0	0
$991$	26.3460	0.836908	0.418454	−	0.908238i	$-0.362572\pi$
$991$	26.3460	0.836908	0.418454	+	0.908238i	$0.362572\pi$
$992$	0	0
$993$	−4.60904	−0.146263
$994$	0	0
$995$	− 48.3478i	− 1.53273i
$996$	0	0
$997$	6.87457i	0.217720i	0.994057	+	0.108860i	$0.0347200\pi$
$997$	6.87457i	0.217720i	−0.994057	+	0.108860i	$0.965280\pi$
$998$	0	0
$999$	7.43172	0.235129

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3584.2.b.g.1793.10		10
4.3	odd	2		3584.2.b.h.1793.1		10
8.3	odd	2		3584.2.b.h.1793.10		10
8.5	even	2	inner	3584.2.b.g.1793.1		10
16.3	odd	4		3584.2.a.o.1.10	yes	10
16.5	even	4		3584.2.a.p.1.10	yes	10
16.11	odd	4		3584.2.a.o.1.1	✓	10
16.13	even	4		3584.2.a.p.1.1	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3584.2.a.o.1.1	✓	10	16.11	odd	4
3584.2.a.o.1.10	yes	10	16.3	odd	4
3584.2.a.p.1.1	yes	10	16.13	even	4
3584.2.a.p.1.10	yes	10	16.5	even	4
3584.2.b.g.1793.1		10	8.5	even	2	inner
3584.2.b.g.1793.10		10	1.1	even	1	trivial
3584.2.b.h.1793.1		10	4.3	odd	2
3584.2.b.h.1793.10		10	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+3.17593i q^{3} -4.23755i q^{5} -1.00000 q^{7} -7.08655 q^{9} +O(q^{10})\)	\(q+3.17593i q^{3} -4.23755i q^{5} -1.00000 q^{7} -7.08655 q^{9} +4.99251i q^{11} +0.439239i q^{13} +13.4582 q^{15} +3.50135 q^{17} -3.45080i q^{19} -3.17593i q^{21} +4.39770 q^{23} -12.9568 q^{25} -12.9786i q^{27} -0.132582i q^{29} -5.55912 q^{31} -15.8559 q^{33} +4.23755i q^{35} +0.572613i q^{37} -1.39499 q^{39} -5.42377 q^{41} -5.69770i q^{43} +30.0296i q^{45} -9.37162 q^{47} +1.00000 q^{49} +11.1201i q^{51} +3.65602i q^{53} +21.1560 q^{55} +10.9595 q^{57} -2.20605i q^{59} -9.89440i q^{61} +7.08655 q^{63} +1.86130 q^{65} +8.51595i q^{67} +13.9668i q^{69} -3.11263 q^{71} -9.87027 q^{73} -41.1500i q^{75} -4.99251i q^{77} +7.11263 q^{79} +19.9595 q^{81} -2.48092i q^{83} -14.8371i q^{85} +0.421071 q^{87} +5.06047 q^{89} -0.439239i q^{91} -17.6554i q^{93} -14.6229 q^{95} -16.6573 q^{97} -35.3796i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{7} - 22 q^{9}+O(q^{10}) \) 10 * q - 10 * q^7 - 22 * q^9	\( 10 q - 10 q^{7} - 22 q^{9} + 16 q^{15} + 16 q^{17} - 8 q^{23} - 30 q^{25} - 8 q^{31} + 12 q^{33} - 20 q^{41} - 24 q^{47} + 10 q^{49} + 32 q^{55} - 28 q^{57} + 22 q^{63} + 32 q^{65} - 48 q^{73} + 40 q^{79} + 62 q^{81} + 8 q^{87} - 16 q^{89} - 32 q^{95} + 32 q^{97}+O(q^{100}) \) 10 * q - 10 * q^7 - 22 * q^9 + 16 * q^15 + 16 * q^17 - 8 * q^23 - 30 * q^25 - 8 * q^31 + 12 * q^33 - 20 * q^41 - 24 * q^47 + 10 * q^49 + 32 * q^55 - 28 * q^57 + 22 * q^63 + 32 * q^65 - 48 * q^73 + 40 * q^79 + 62 * q^81 + 8 * q^87 - 16 * q^89 - 32 * q^95 + 32 * q^97

Introduction

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