LMFDB - Embedded newform 3584.2.b.i.1793.12

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:	$12$
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} - 4 x^{11} + 4 x^{10} + 8 x^{9} - 24 x^{8} + 24 x^{7} + 176 x^{6} + 160 x^{5} + 145 x^{4} + \cdots + 8 $ x^12 - 4x^11 + 4x^10 + 8x^9 - 24x^8 + 24x^7 + 176x^6 + 160x^5 + 145x^4 + 108x^3 + 68x^2 + 32*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1793.12
Root			$0.960396 + 2.31860i$ of defining polynomial
Character	$\chi$	$=$	3584.1793
Dual form			3584.2.b.i.1793.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.22299i q^{3} -0.725063i q^{5} -1.00000 q^{7} -7.38766 q^{9} +O(q^{10})$	$q+3.22299i q^{3} -0.725063i q^{5} -1.00000 q^{7} -7.38766 q^{9} -2.21848i q^{11} -1.75046i q^{13} +2.33687 q^{15} +1.69102 q^{17} +1.80241i q^{19} -3.22299i q^{21} +8.47013 q^{23} +4.47428 q^{25} -14.1414i q^{27} -0.704174i q^{29} +6.28756 q^{31} +7.15015 q^{33} +0.725063i q^{35} -6.75496i q^{37} +5.64171 q^{39} -8.72453 q^{41} +7.33448i q^{43} +5.35652i q^{45} +10.1709 q^{47} +1.00000 q^{49} +5.45013i q^{51} +13.2440i q^{53} -1.60854 q^{55} -5.80913 q^{57} -8.86761i q^{59} +10.4417i q^{61} +7.38766 q^{63} -1.26919 q^{65} -1.71940i q^{67} +27.2992i q^{69} +1.47808 q^{71} -8.52472 q^{73} +14.4206i q^{75} +2.21848i q^{77} -10.6421 q^{79} +23.4146 q^{81} -10.8440i q^{83} -1.22609i q^{85} +2.26954 q^{87} +2.13326 q^{89} +1.75046i q^{91} +20.2648i q^{93} +1.30686 q^{95} -5.73933 q^{97} +16.3894i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{7} - 20 q^{9}+O(q^{10}) $ 12 * q - 12 * q^7 - 20 * q^9	$ 12 q - 12 q^{7} - 20 q^{9} - 16 q^{15} - 4 q^{25} - 16 q^{31} + 8 q^{33} + 8 q^{41} + 16 q^{47} + 12 q^{49} - 32 q^{55} + 8 q^{57} + 20 q^{63} - 16 q^{65} + 16 q^{71} - 32 q^{73} - 48 q^{79} + 20 q^{81} + 16 q^{87} - 32 q^{89} - 32 q^{95} + 32 q^{97}+O(q^{100}) $ 12 * q - 12 * q^7 - 20 * q^9 - 16 * q^15 - 4 * q^25 - 16 * q^31 + 8 * q^33 + 8 * q^41 + 16 * q^47 + 12 * q^49 - 32 * q^55 + 8 * q^57 + 20 * q^63 - 16 * q^65 + 16 * q^71 - 32 * q^73 - 48 * q^79 + 20 * q^81 + 16 * q^87 - 32 * 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
$3$	3.22299i	1.86079i	0.366554	+	0.930397i	$0.380538\pi$
$3$	3.22299i	1.86079i	−0.366554	+	0.930397i	$0.619462\pi$
$4$	0	0
$5$	− 0.725063i	− 0.324258i	−0.986770	−	0.162129i	$-0.948164\pi$
$5$	− 0.725063i	− 0.324258i	0.986770	−	0.162129i	$-0.0518361\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−7.38766	−2.46255
$10$	0	0
$11$	− 2.21848i	− 0.668898i	−0.942414	−	0.334449i	$-0.891450\pi$
$11$	− 2.21848i	− 0.668898i	0.942414	−	0.334449i	$-0.108550\pi$
$12$	0	0
$13$	− 1.75046i	− 0.485490i	−0.970090	−	0.242745i	$-0.921952\pi$
$13$	− 1.75046i	− 0.485490i	0.970090	−	0.242745i	$-0.0780478\pi$
$14$	0	0
$15$	2.33687	0.603378
$16$	0	0
$17$	1.69102	0.410132	0.205066	−	0.978748i	$-0.434259\pi$
$17$	1.69102	0.410132	0.205066	+	0.978748i	$0.434259\pi$
$18$	0	0
$19$	1.80241i	0.413500i	0.978394	+	0.206750i	$0.0662887\pi$
$19$	1.80241i	0.413500i	−0.978394	+	0.206750i	$0.933711\pi$
$20$	0	0
$21$	− 3.22299i	− 0.703314i
$22$	0	0
$23$	8.47013	1.76615	0.883073	−	0.469236i	$-0.155471\pi$
$23$	8.47013	1.76615	0.883073	+	0.469236i	$0.155471\pi$
$24$	0	0
$25$	4.47428	0.894857
$26$	0	0
$27$	− 14.1414i	− 2.72151i
$28$	0	0
$29$	− 0.704174i	− 0.130762i	−0.997860	−	0.0653809i	$-0.979174\pi$
$29$	− 0.704174i	− 0.130762i	0.997860	−	0.0653809i	$-0.0208262\pi$
$30$	0	0
$31$	6.28756	1.12928	0.564640	−	0.825337i	$-0.309015\pi$
$31$	6.28756	1.12928	0.564640	+	0.825337i	$0.309015\pi$
$32$	0	0
$33$	7.15015	1.24468
$34$	0	0
$35$	0.725063i	0.122558i
$36$	0	0
$37$	− 6.75496i	− 1.11051i	−0.831680	−	0.555255i	$-0.812621\pi$
$37$	− 6.75496i	− 1.11051i	0.831680	−	0.555255i	$-0.187379\pi$
$38$	0	0
$39$	5.64171	0.903396
$40$	0	0
$41$	−8.72453	−1.36254	−0.681272	−	0.732031i	$-0.738573\pi$
$41$	−8.72453	−1.36254	−0.681272	+	0.732031i	$0.738573\pi$
$42$	0	0
$43$	7.33448i	1.11850i	0.829000	+	0.559249i	$0.188910\pi$
$43$	7.33448i	1.11850i	−0.829000	+	0.559249i	$0.811090\pi$
$44$	0	0
$45$	5.35652i	0.798503i
$46$	0	0
$47$	10.1709	1.48358	0.741791	−	0.670631i	$-0.233977\pi$
$47$	10.1709	1.48358	0.741791	+	0.670631i	$0.233977\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.45013i	0.763170i
$52$	0	0
$53$	13.2440i	1.81920i	0.415485	+	0.909600i	$0.363612\pi$
$53$	13.2440i	1.81920i	−0.415485	+	0.909600i	$0.636388\pi$
$54$	0	0
$55$	−1.60854	−0.216896
$56$	0	0
$57$	−5.80913	−0.769438
$58$	0	0
$59$	− 8.86761i	− 1.15446i	−0.816580	−	0.577232i	$-0.804133\pi$
$59$	− 8.86761i	− 1.15446i	0.816580	−	0.577232i	$-0.195867\pi$
$60$	0	0
$61$	10.4417i	1.33692i	0.743746	+	0.668462i	$0.233047\pi$
$61$	10.4417i	1.33692i	−0.743746	+	0.668462i	$0.766953\pi$
$62$	0	0
$63$	7.38766	0.930758
$64$	0	0
$65$	−1.26919	−0.157424
$66$	0	0
$67$	− 1.71940i	− 0.210058i	−0.994469	−	0.105029i	$-0.966506\pi$
$67$	− 1.71940i	− 0.210058i	0.994469	−	0.105029i	$-0.0334936\pi$
$68$	0	0
$69$	27.2992i	3.28643i
$70$	0	0
$71$	1.47808	0.175416	0.0877079	−	0.996146i	$-0.472046\pi$
$71$	1.47808	0.175416	0.0877079	+	0.996146i	$0.472046\pi$
$72$	0	0
$73$	−8.52472	−0.997743	−0.498872	−	0.866676i	$-0.666252\pi$
$73$	−8.52472	−0.997743	−0.498872	+	0.866676i	$0.666252\pi$
$74$	0	0
$75$	14.4206i	1.66514i
$76$	0	0
$77$	2.21848i	0.252820i
$78$	0	0
$79$	−10.6421	−1.19733	−0.598663	−	0.801001i	$-0.704301\pi$
$79$	−10.6421	−1.19733	−0.598663	+	0.801001i	$0.704301\pi$
$80$	0	0
$81$	23.4146	2.60162
$82$	0	0
$83$	− 10.8440i	− 1.19029i	−0.803619	−	0.595144i	$-0.797095\pi$
$83$	− 10.8440i	− 1.19029i	0.803619	−	0.595144i	$-0.202905\pi$
$84$	0	0
$85$	− 1.22609i	− 0.132989i
$86$	0	0
$87$	2.26954	0.243321
$88$	0	0
$89$	2.13326	0.226125	0.113063	−	0.993588i	$-0.463934\pi$
$89$	2.13326	0.226125	0.113063	+	0.993588i	$0.463934\pi$
$90$	0	0
$91$	1.75046i	0.183498i
$92$	0	0
$93$	20.2648i	2.10136i
$94$	0	0
$95$	1.30686	0.134081
$96$	0	0
$97$	−5.73933	−0.582740	−0.291370	−	0.956610i	$-0.594111\pi$
$97$	−5.73933	−0.582740	−0.291370	+	0.956610i	$0.594111\pi$
$98$	0	0
$99$	16.3894i	1.64720i
$100$	0	0
$101$	7.40731i	0.737055i	0.929617	+	0.368528i	$0.120138\pi$
$101$	7.40731i	0.737055i	−0.929617	+	0.368528i	$0.879862\pi$
$102$	0	0
$103$	16.6714	1.64268	0.821342	−	0.570436i	$-0.193226\pi$
$103$	16.6714	1.64268	0.821342	+	0.570436i	$0.193226\pi$
$104$	0	0
$105$	−2.33687	−0.228055
$106$	0	0
$107$	12.4492i	1.20351i	0.798680	+	0.601755i	$0.205532\pi$
$107$	12.4492i	1.20351i	−0.798680	+	0.601755i	$0.794468\pi$
$108$	0	0
$109$	− 2.53093i	− 0.242419i	−0.992627	−	0.121210i	$-0.961323\pi$
$109$	− 2.53093i	− 0.242419i	0.992627	−	0.121210i	$-0.0386773\pi$
$110$	0	0
$111$	21.7712	2.06643
$112$	0	0
$113$	1.16568	0.109658	0.0548291	−	0.998496i	$-0.482539\pi$
$113$	1.16568	0.109658	0.0548291	+	0.998496i	$0.482539\pi$
$114$	0	0
$115$	− 6.14139i	− 0.572687i
$116$	0	0
$117$	12.9318i	1.19554i
$118$	0	0
$119$	−1.69102	−0.155015
$120$	0	0
$121$	6.07833	0.552575
$122$	0	0
$123$	− 28.1191i	− 2.53541i
$124$	0	0
$125$	− 6.86946i	− 0.614423i
$126$	0	0
$127$	0.248418	0.0220435	0.0110218	−	0.999939i	$-0.496492\pi$
$127$	0.248418	0.0220435	0.0110218	+	0.999939i	$0.496492\pi$
$128$	0	0
$129$	−23.6389	−2.08129
$130$	0	0
$131$	18.0997i	1.58138i	0.612216	+	0.790690i	$0.290278\pi$
$131$	18.0997i	1.58138i	−0.612216	+	0.790690i	$0.709722\pi$
$132$	0	0
$133$	− 1.80241i	− 0.156288i
$134$	0	0
$135$	−10.2534	−0.882472
$136$	0	0
$137$	17.7633	1.51762	0.758812	−	0.651310i	$-0.225780\pi$
$137$	17.7633	1.51762	0.758812	+	0.651310i	$0.225780\pi$
$138$	0	0
$139$	− 12.7027i	− 1.07743i	−0.842490	−	0.538713i	$-0.818911\pi$
$139$	− 12.7027i	− 1.07743i	0.842490	−	0.538713i	$-0.181089\pi$
$140$	0	0
$141$	32.7808i	2.76064i
$142$	0	0
$143$	−3.88336	−0.324743
$144$	0	0
$145$	−0.510571	−0.0424006
$146$	0	0
$147$	3.22299i	0.265828i
$148$	0	0
$149$	− 7.98106i	− 0.653834i	−0.945053	−	0.326917i	$-0.893990\pi$
$149$	− 7.98106i	− 0.653834i	0.945053	−	0.326917i	$-0.106010\pi$
$150$	0	0
$151$	18.6055	1.51409	0.757047	−	0.653361i	$-0.226641\pi$
$151$	18.6055	1.51409	0.757047	+	0.653361i	$0.226641\pi$
$152$	0	0
$153$	−12.4927	−1.00997
$154$	0	0
$155$	− 4.55888i	− 0.366178i
$156$	0	0
$157$	− 12.3166i	− 0.982970i	−0.870886	−	0.491485i	$-0.836454\pi$
$157$	− 12.3166i	− 0.982970i	0.870886	−	0.491485i	$-0.163546\pi$
$158$	0	0
$159$	−42.6852	−3.38516
$160$	0	0
$161$	−8.47013	−0.667540
$162$	0	0
$163$	24.8444i	1.94596i	0.230882	+	0.972982i	$0.425839\pi$
$163$	24.8444i	1.94596i	−0.230882	+	0.972982i	$0.574161\pi$
$164$	0	0
$165$	− 5.18431i	− 0.403598i
$166$	0	0
$167$	−10.4042	−0.805101	−0.402551	−	0.915398i	$-0.631876\pi$
$167$	−10.4042	−0.805101	−0.402551	+	0.915398i	$0.631876\pi$
$168$	0	0
$169$	9.93590	0.764300
$170$	0	0
$171$	− 13.3156i	− 1.01827i
$172$	0	0
$173$	− 13.1678i	− 1.00113i	−0.865699	−	0.500565i	$-0.833126\pi$
$173$	− 13.1678i	− 1.00113i	0.865699	−	0.500565i	$-0.166874\pi$
$174$	0	0
$175$	−4.47428	−0.338224
$176$	0	0
$177$	28.5802	2.14822
$178$	0	0
$179$	− 1.59431i	− 0.119164i	−0.998223	−	0.0595821i	$-0.981023\pi$
$179$	− 1.59431i	− 0.119164i	0.998223	−	0.0595821i	$-0.0189768\pi$
$180$	0	0
$181$	5.93385i	0.441060i	0.975380	+	0.220530i	$0.0707786\pi$
$181$	5.93385i	0.441060i	−0.975380	+	0.220530i	$0.929221\pi$
$182$	0	0
$183$	−33.6535	−2.48774
$184$	0	0
$185$	−4.89778	−0.360092
$186$	0	0
$187$	− 3.75149i	− 0.274336i
$188$	0	0
$189$	14.1414i	1.02863i
$190$	0	0
$191$	16.2217	1.17376	0.586881	−	0.809673i	$-0.300356\pi$
$191$	16.2217	1.17376	0.586881	+	0.809673i	$0.300356\pi$
$192$	0	0
$193$	0.805434	0.0579764	0.0289882	−	0.999580i	$-0.490771\pi$
$193$	0.805434	0.0579764	0.0289882	+	0.999580i	$0.490771\pi$
$194$	0	0
$195$	− 4.09060i	− 0.292934i
$196$	0	0
$197$	− 9.93905i	− 0.708128i	−0.935221	−	0.354064i	$-0.884800\pi$
$197$	− 9.93905i	− 0.708128i	0.935221	−	0.354064i	$-0.115200\pi$
$198$	0	0
$199$	−19.4280	−1.37722	−0.688608	−	0.725134i	$-0.741778\pi$
$199$	−19.4280	−1.37722	−0.688608	+	0.725134i	$0.741778\pi$
$200$	0	0
$201$	5.54161	0.390875
$202$	0	0
$203$	0.704174i	0.0494233i
$204$	0	0
$205$	6.32584i	0.441816i
$206$	0	0
$207$	−62.5745	−4.34923
$208$	0	0
$209$	3.99861	0.276590
$210$	0	0
$211$	18.4252i	1.26844i	0.773151	+	0.634222i	$0.218679\pi$
$211$	18.4252i	1.26844i	−0.773151	+	0.634222i	$0.781321\pi$
$212$	0	0
$213$	4.76384i	0.326413i
$214$	0	0
$215$	5.31796	0.362682
$216$	0	0
$217$	−6.28756	−0.426828
$218$	0	0
$219$	− 27.4751i	− 1.85659i
$220$	0	0
$221$	− 2.96005i	− 0.199115i
$222$	0	0
$223$	3.60025	0.241090	0.120545	−	0.992708i	$-0.461536\pi$
$223$	3.60025	0.241090	0.120545	+	0.992708i	$0.461536\pi$
$224$	0	0
$225$	−33.0545	−2.20363
$226$	0	0
$227$	− 10.9011i	− 0.723531i	−0.932269	−	0.361765i	$-0.882174\pi$
$227$	− 10.9011i	− 0.723531i	0.932269	−	0.361765i	$-0.117826\pi$
$228$	0	0
$229$	13.3147i	0.879863i	0.898031	+	0.439932i	$0.144997\pi$
$229$	13.3147i	0.879863i	−0.898031	+	0.439932i	$0.855003\pi$
$230$	0	0
$231$	−7.15015	−0.470446
$232$	0	0
$233$	22.3624	1.46501	0.732506	−	0.680760i	$-0.238350\pi$
$233$	22.3624	1.46501	0.732506	+	0.680760i	$0.238350\pi$
$234$	0	0
$235$	− 7.37457i	− 0.481064i
$236$	0	0
$237$	− 34.2992i	− 2.22798i
$238$	0	0
$239$	18.1505	1.17406	0.587029	−	0.809566i	$-0.300297\pi$
$239$	18.1505	1.17406	0.587029	+	0.809566i	$0.300297\pi$
$240$	0	0
$241$	−19.7278	−1.27078	−0.635388	−	0.772193i	$-0.719160\pi$
$241$	−19.7278	−1.27078	−0.635388	+	0.772193i	$0.719160\pi$
$242$	0	0
$243$	33.0407i	2.11956i
$244$	0	0
$245$	− 0.725063i	− 0.0463226i
$246$	0	0
$247$	3.15503	0.200750
$248$	0	0
$249$	34.9502	2.21488
$250$	0	0
$251$	7.38794i	0.466322i	0.972438	+	0.233161i	$0.0749070\pi$
$251$	7.38794i	0.466322i	−0.972438	+	0.233161i	$0.925093\pi$
$252$	0	0
$253$	− 18.7909i	− 1.18137i
$254$	0	0
$255$	3.95169	0.247464
$256$	0	0
$257$	−8.13863	−0.507674	−0.253837	−	0.967247i	$-0.581693\pi$
$257$	−8.13863	−0.507674	−0.253837	+	0.967247i	$0.581693\pi$
$258$	0	0
$259$	6.75496i	0.419733i
$260$	0	0
$261$	5.20220i	0.322008i
$262$	0	0
$263$	18.0689	1.11418	0.557088	−	0.830453i	$-0.311919\pi$
$263$	18.0689	1.11418	0.557088	+	0.830453i	$0.311919\pi$
$264$	0	0
$265$	9.60272	0.589891
$266$	0	0
$267$	6.87548i	0.420773i
$268$	0	0
$269$	28.5831i	1.74274i	0.490624	+	0.871371i	$0.336769\pi$
$269$	28.5831i	1.74274i	−0.490624	+	0.871371i	$0.663231\pi$
$270$	0	0
$271$	19.6073	1.19106	0.595528	−	0.803335i	$-0.296943\pi$
$271$	19.6073	1.19106	0.595528	+	0.803335i	$0.296943\pi$
$272$	0	0
$273$	−5.64171	−0.341452
$274$	0	0
$275$	− 9.92613i	− 0.598568i
$276$	0	0
$277$	− 23.1932i	− 1.39354i	−0.717293	−	0.696772i	$-0.754619\pi$
$277$	− 23.1932i	− 1.39354i	0.717293	−	0.696772i	$-0.245381\pi$
$278$	0	0
$279$	−46.4504	−2.78091
$280$	0	0
$281$	9.51739	0.567760	0.283880	−	0.958860i	$-0.408378\pi$
$281$	9.51739	0.567760	0.283880	+	0.958860i	$0.408378\pi$
$282$	0	0
$283$	− 5.09510i	− 0.302872i	−0.988467	−	0.151436i	$-0.951610\pi$
$283$	− 5.09510i	− 0.302872i	0.988467	−	0.151436i	$-0.0483897\pi$
$284$	0	0
$285$	4.21199i	0.249497i
$286$	0	0
$287$	8.72453	0.514993
$288$	0	0
$289$	−14.1405	−0.831792
$290$	0	0
$291$	− 18.4978i	− 1.08436i
$292$	0	0
$293$	− 2.71074i	− 0.158363i	−0.996860	−	0.0791815i	$-0.974769\pi$
$293$	− 2.71074i	− 0.158363i	0.996860	−	0.0791815i	$-0.0252307\pi$
$294$	0	0
$295$	−6.42958	−0.374344
$296$	0	0
$297$	−31.3724	−1.82041
$298$	0	0
$299$	− 14.8266i	− 0.857445i
$300$	0	0
$301$	− 7.33448i	− 0.422752i
$302$	0	0
$303$	−23.8737	−1.37151
$304$	0	0
$305$	7.57091	0.433509
$306$	0	0
$307$	− 1.04975i	− 0.0599121i	−0.999551	−	0.0299561i	$-0.990463\pi$
$307$	− 1.04975i	− 0.0599121i	0.999551	−	0.0299561i	$-0.00953674\pi$
$308$	0	0
$309$	53.7318i	3.05670i
$310$	0	0
$311$	−1.41330	−0.0801410	−0.0400705	−	0.999197i	$-0.512758\pi$
$311$	−1.41330	−0.0801410	−0.0400705	+	0.999197i	$0.512758\pi$
$312$	0	0
$313$	26.5463	1.50049	0.750244	−	0.661161i	$-0.229936\pi$
$313$	26.5463	1.50049	0.750244	+	0.661161i	$0.229936\pi$
$314$	0	0
$315$	− 5.35652i	− 0.301806i
$316$	0	0
$317$	24.5350i	1.37802i	0.724751	+	0.689011i	$0.241955\pi$
$317$	24.5350i	1.37802i	−0.724751	+	0.689011i	$0.758045\pi$
$318$	0	0
$319$	−1.56220	−0.0874663
$320$	0	0
$321$	−40.1237	−2.23949
$322$	0	0
$323$	3.04790i	0.169589i
$324$	0	0
$325$	− 7.83204i	− 0.434444i
$326$	0	0
$327$	8.15716	0.451092
$328$	0	0
$329$	−10.1709	−0.560741
$330$	0	0
$331$	− 21.6727i	− 1.19124i	−0.803267	−	0.595619i	$-0.796907\pi$
$331$	− 21.6727i	− 1.19124i	0.803267	−	0.595619i	$-0.203093\pi$
$332$	0	0
$333$	49.9034i	2.73469i
$334$	0	0
$335$	−1.24667	−0.0681131
$336$	0	0
$337$	−8.42780	−0.459092	−0.229546	−	0.973298i	$-0.573724\pi$
$337$	−8.42780	−0.459092	−0.229546	+	0.973298i	$0.573724\pi$
$338$	0	0
$339$	3.75698i	0.204051i
$340$	0	0
$341$	− 13.9489i	− 0.755373i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	19.7936	1.06565
$346$	0	0
$347$	32.4126i	1.74000i	0.493053	+	0.869999i	$0.335881\pi$
$347$	32.4126i	1.74000i	−0.493053	+	0.869999i	$0.664119\pi$
$348$	0	0
$349$	27.3672i	1.46493i	0.680803	+	0.732467i	$0.261631\pi$
$349$	27.3672i	1.46493i	−0.680803	+	0.732467i	$0.738369\pi$
$350$	0	0
$351$	−24.7539	−1.32127
$352$	0	0
$353$	−8.81238	−0.469035	−0.234518	−	0.972112i	$-0.575351\pi$
$353$	−8.81238	−0.469035	−0.234518	+	0.972112i	$0.575351\pi$
$354$	0	0
$355$	− 1.07170i	− 0.0568800i
$356$	0	0
$357$	− 5.45013i	− 0.288451i
$358$	0	0
$359$	4.54142	0.239687	0.119844	−	0.992793i	$-0.461761\pi$
$359$	4.54142	0.239687	0.119844	+	0.992793i	$0.461761\pi$
$360$	0	0
$361$	15.7513	0.829018
$362$	0	0
$363$	19.5904i	1.02823i
$364$	0	0
$365$	6.18096i	0.323526i
$366$	0	0
$367$	15.7472	0.821996	0.410998	−	0.911636i	$-0.365180\pi$
$367$	15.7472	0.821996	0.410998	+	0.911636i	$0.365180\pi$
$368$	0	0
$369$	64.4539	3.35534
$370$	0	0
$371$	− 13.2440i	− 0.687593i
$372$	0	0
$373$	10.3575i	0.536289i	0.963379	+	0.268144i	$0.0864104\pi$
$373$	10.3575i	0.536289i	−0.963379	+	0.268144i	$0.913590\pi$
$374$	0	0
$375$	22.1402	1.14331
$376$	0	0
$377$	−1.23263	−0.0634835
$378$	0	0
$379$	17.7152i	0.909967i	0.890500	+	0.454983i	$0.150355\pi$
$379$	17.7152i	0.909967i	−0.890500	+	0.454983i	$0.849645\pi$
$380$	0	0
$381$	0.800648i	0.0410184i
$382$	0	0
$383$	−27.1645	−1.38804	−0.694020	−	0.719956i	$-0.744162\pi$
$383$	−27.1645	−1.38804	−0.694020	+	0.719956i	$0.744162\pi$
$384$	0	0
$385$	1.60854	0.0819789
$386$	0	0
$387$	− 54.1846i	− 2.75436i
$388$	0	0
$389$	20.7629i	1.05272i	0.850261	+	0.526361i	$0.176444\pi$
$389$	20.7629i	1.05272i	−0.850261	+	0.526361i	$0.823556\pi$
$390$	0	0
$391$	14.3231	0.724352
$392$	0	0
$393$	−58.3352	−2.94262
$394$	0	0
$395$	7.71617i	0.388243i
$396$	0	0
$397$	− 31.0965i	− 1.56069i	−0.625350	−	0.780345i	$-0.715044\pi$
$397$	− 31.0965i	− 1.56069i	0.625350	−	0.780345i	$-0.284956\pi$
$398$	0	0
$399$	5.80913	0.290820
$400$	0	0
$401$	−21.6415	−1.08072	−0.540362	−	0.841433i	$-0.681713\pi$
$401$	−21.6415	−1.08072	−0.540362	+	0.841433i	$0.681713\pi$
$402$	0	0
$403$	− 11.0061i	− 0.548254i
$404$	0	0
$405$	− 16.9770i	− 0.843596i
$406$	0	0
$407$	−14.9858	−0.742818
$408$	0	0
$409$	18.1632	0.898111	0.449056	−	0.893504i	$-0.351761\pi$
$409$	18.1632	0.898111	0.449056	+	0.893504i	$0.351761\pi$
$410$	0	0
$411$	57.2510i	2.82398i
$412$	0	0
$413$	8.86761i	0.436346i
$414$	0	0
$415$	−7.86262	−0.385961
$416$	0	0
$417$	40.9405	2.00487
$418$	0	0
$419$	24.1668i	1.18062i	0.807175	+	0.590312i	$0.200995\pi$
$419$	24.1668i	1.18062i	−0.807175	+	0.590312i	$0.799005\pi$
$420$	0	0
$421$	− 23.7262i	− 1.15635i	−0.815914	−	0.578173i	$-0.803766\pi$
$421$	− 23.7262i	− 1.15635i	0.815914	−	0.578173i	$-0.196234\pi$
$422$	0	0
$423$	−75.1394	−3.65340
$424$	0	0
$425$	7.56608	0.367009
$426$	0	0
$427$	− 10.4417i	− 0.505310i
$428$	0	0
$429$	− 12.5160i	− 0.604280i
$430$	0	0
$431$	−21.8343	−1.05172	−0.525861	−	0.850571i	$-0.676257\pi$
$431$	−21.8343	−1.05172	−0.525861	+	0.850571i	$0.676257\pi$
$432$	0	0
$433$	−35.6444	−1.71296	−0.856480	−	0.516181i	$-0.827353\pi$
$433$	−35.6444	−1.71296	−0.856480	+	0.516181i	$0.827353\pi$
$434$	0	0
$435$	− 1.64556i	− 0.0788987i
$436$	0	0
$437$	15.2666i	0.730301i
$438$	0	0
$439$	0.0557759	0.00266204	0.00133102	−	0.999999i	$-0.499576\pi$
$439$	0.0557759	0.00266204	0.00133102	+	0.999999i	$0.499576\pi$
$440$	0	0
$441$	−7.38766	−0.351793
$442$	0	0
$443$	14.7541i	0.700991i	0.936564	+	0.350495i	$0.113987\pi$
$443$	14.7541i	0.700991i	−0.936564	+	0.350495i	$0.886013\pi$
$444$	0	0
$445$	− 1.54675i	− 0.0733230i
$446$	0	0
$447$	25.7229	1.21665
$448$	0	0
$449$	−11.0183	−0.519986	−0.259993	−	0.965611i	$-0.583720\pi$
$449$	−11.0183	−0.519986	−0.259993	+	0.965611i	$0.583720\pi$
$450$	0	0
$451$	19.3552i	0.911403i
$452$	0	0
$453$	59.9653i	2.81742i
$454$	0	0
$455$	1.26919	0.0595007
$456$	0	0
$457$	−30.5056	−1.42699	−0.713495	−	0.700660i	$-0.752889\pi$
$457$	−30.5056	−1.42699	−0.713495	+	0.700660i	$0.752889\pi$
$458$	0	0
$459$	− 23.9133i	− 1.11618i
$460$	0	0
$461$	9.76890i	0.454983i	0.973780	+	0.227492i	$0.0730524\pi$
$461$	9.76890i	0.454983i	−0.973780	+	0.227492i	$0.926948\pi$
$462$	0	0
$463$	21.0301	0.977352	0.488676	−	0.872465i	$-0.337480\pi$
$463$	21.0301	0.977352	0.488676	+	0.872465i	$0.337480\pi$
$464$	0	0
$465$	14.6932	0.681382
$466$	0	0
$467$	− 5.67899i	− 0.262792i	−0.991330	−	0.131396i	$-0.958054\pi$
$467$	− 5.67899i	− 0.262792i	0.991330	−	0.131396i	$-0.0419460\pi$
$468$	0	0
$469$	1.71940i	0.0793945i
$470$	0	0
$471$	39.6962	1.82910
$472$	0	0
$473$	16.2714	0.748161
$474$	0	0
$475$	8.06447i	0.370023i
$476$	0	0
$477$	− 97.8420i	− 4.47988i
$478$	0	0
$479$	30.3538	1.38690	0.693449	−	0.720505i	$-0.256090\pi$
$479$	30.3538	1.38690	0.693449	+	0.720505i	$0.256090\pi$
$480$	0	0
$481$	−11.8243	−0.539141
$482$	0	0
$483$	− 27.2992i	− 1.24215i
$484$	0	0
$485$	4.16138i	0.188958i
$486$	0	0
$487$	12.6275	0.572206	0.286103	−	0.958199i	$-0.407640\pi$
$487$	12.6275	0.572206	0.286103	+	0.958199i	$0.407640\pi$
$488$	0	0
$489$	−80.0732	−3.62104
$490$	0	0
$491$	− 27.5915i	− 1.24519i	−0.782545	−	0.622594i	$-0.786079\pi$
$491$	− 27.5915i	− 1.24519i	0.782545	−	0.622594i	$-0.213921\pi$
$492$	0	0
$493$	− 1.19077i	− 0.0536295i
$494$	0	0
$495$	11.8834	0.534118
$496$	0	0
$497$	−1.47808	−0.0663009
$498$	0	0
$499$	− 8.82017i	− 0.394845i	−0.980318	−	0.197423i	$-0.936743\pi$
$499$	− 8.82017i	− 0.394845i	0.980318	−	0.197423i	$-0.0632571\pi$
$500$	0	0
$501$	− 33.5326i	− 1.49813i
$502$	0	0
$503$	30.2244	1.34764	0.673819	−	0.738896i	$-0.264653\pi$
$503$	30.2244	1.34764	0.673819	+	0.738896i	$0.264653\pi$
$504$	0	0
$505$	5.37077	0.238996
$506$	0	0
$507$	32.0233i	1.42220i
$508$	0	0
$509$	− 6.45130i	− 0.285949i	−0.989726	−	0.142974i	$-0.954333\pi$
$509$	− 6.45130i	− 0.285949i	0.989726	−	0.142974i	$-0.0456667\pi$
$510$	0	0
$511$	8.52472	0.377111
$512$	0	0
$513$	25.4885	1.12535
$514$	0	0
$515$	− 12.0878i	− 0.532654i
$516$	0	0
$517$	− 22.5640i	− 0.992366i
$518$	0	0
$519$	42.4397	1.86290
$520$	0	0
$521$	29.9742	1.31319	0.656596	−	0.754242i	$-0.271996\pi$
$521$	29.9742	1.31319	0.656596	+	0.754242i	$0.271996\pi$
$522$	0	0
$523$	− 7.07881i	− 0.309535i	−0.987951	−	0.154767i	$-0.950537\pi$
$523$	− 7.07881i	− 0.309535i	0.987951	−	0.154767i	$-0.0494628\pi$
$524$	0	0
$525$	− 14.4206i	− 0.629365i
$526$	0	0
$527$	10.6324	0.463153
$528$	0	0
$529$	48.7432	2.11927
$530$	0	0
$531$	65.5109i	2.84293i
$532$	0	0
$533$	15.2719i	0.661501i
$534$	0	0
$535$	9.02647	0.390248
$536$	0	0
$537$	5.13844	0.221740
$538$	0	0
$539$	− 2.21848i	− 0.0955569i
$540$	0	0
$541$	− 21.6394i	− 0.930351i	−0.885218	−	0.465175i	$-0.845991\pi$
$541$	− 21.6394i	− 0.930351i	0.885218	−	0.465175i	$-0.154009\pi$
$542$	0	0
$543$	−19.1247	−0.820721
$544$	0	0
$545$	−1.83508	−0.0786064
$546$	0	0
$547$	− 14.8782i	− 0.636145i	−0.948066	−	0.318073i	$-0.896964\pi$
$547$	− 14.8782i	− 0.636145i	0.948066	−	0.318073i	$-0.103036\pi$
$548$	0	0
$549$	− 77.1399i	− 3.29225i
$550$	0	0
$551$	1.26921	0.0540700
$552$	0	0
$553$	10.6421	0.452546
$554$	0	0
$555$	− 15.7855i	− 0.670056i
$556$	0	0
$557$	18.1723i	0.769986i	0.922920	+	0.384993i	$0.125796\pi$
$557$	18.1723i	0.769986i	−0.922920	+	0.384993i	$0.874204\pi$
$558$	0	0
$559$	12.8387	0.543019
$560$	0	0
$561$	12.0910	0.510483
$562$	0	0
$563$	5.41059i	0.228029i	0.993479	+	0.114015i	$0.0363711\pi$
$563$	5.41059i	0.228029i	−0.993479	+	0.114015i	$0.963629\pi$
$564$	0	0
$565$	− 0.845193i	− 0.0355575i
$566$	0	0
$567$	−23.4146	−0.983319
$568$	0	0
$569$	−23.6309	−0.990660	−0.495330	−	0.868705i	$-0.664953\pi$
$569$	−23.6309	−0.990660	−0.495330	+	0.868705i	$0.664953\pi$
$570$	0	0
$571$	− 15.9091i	− 0.665773i	−0.942967	−	0.332887i	$-0.891977\pi$
$571$	− 15.9091i	− 0.665773i	0.942967	−	0.332887i	$-0.108023\pi$
$572$	0	0
$573$	52.2824i	2.18413i
$574$	0	0
$575$	37.8978	1.58045
$576$	0	0
$577$	10.9906	0.457544	0.228772	−	0.973480i	$-0.426529\pi$
$577$	10.9906	0.457544	0.228772	+	0.973480i	$0.426529\pi$
$578$	0	0
$579$	2.59591i	0.107882i
$580$	0	0
$581$	10.8440i	0.449887i
$582$	0	0
$583$	29.3816	1.21686
$584$	0	0
$585$	9.37637	0.387665
$586$	0	0
$587$	19.7045i	0.813291i	0.913586	+	0.406646i	$0.133302\pi$
$587$	19.7045i	0.813291i	−0.913586	+	0.406646i	$0.866698\pi$
$588$	0	0
$589$	11.3327i	0.466957i
$590$	0	0
$591$	32.0334	1.31768
$592$	0	0
$593$	7.05691	0.289792	0.144896	−	0.989447i	$-0.453715\pi$
$593$	7.05691	0.289792	0.144896	+	0.989447i	$0.453715\pi$
$594$	0	0
$595$	1.22609i	0.0502650i
$596$	0	0
$597$	− 62.6163i	− 2.56272i
$598$	0	0
$599$	6.03378	0.246534	0.123267	−	0.992374i	$-0.460663\pi$
$599$	6.03378	0.246534	0.123267	+	0.992374i	$0.460663\pi$
$600$	0	0
$601$	−26.3735	−1.07580	−0.537900	−	0.843009i	$-0.680782\pi$
$601$	−26.3735	−1.07580	−0.537900	+	0.843009i	$0.680782\pi$
$602$	0	0
$603$	12.7023i	0.517280i
$604$	0	0
$605$	− 4.40717i	− 0.179177i
$606$	0	0
$607$	−7.38950	−0.299930	−0.149965	−	0.988691i	$-0.547916\pi$
$607$	−7.38950	−0.299930	−0.149965	+	0.988691i	$0.547916\pi$
$608$	0	0
$609$	−2.26954	−0.0919666
$610$	0	0
$611$	− 17.8038i	− 0.720264i
$612$	0	0
$613$	− 38.9532i	− 1.57330i	−0.617397	−	0.786652i	$-0.711813\pi$
$613$	− 38.9532i	− 1.57330i	0.617397	−	0.786652i	$-0.288187\pi$
$614$	0	0
$615$	−20.3881	−0.822128
$616$	0	0
$617$	12.2102	0.491563	0.245781	−	0.969325i	$-0.420955\pi$
$617$	12.2102	0.491563	0.245781	+	0.969325i	$0.420955\pi$
$618$	0	0
$619$	− 47.9001i	− 1.92527i	−0.270804	−	0.962634i	$-0.587290\pi$
$619$	− 47.9001i	− 1.92527i	0.270804	−	0.962634i	$-0.412710\pi$
$620$	0	0
$621$	− 119.779i	− 4.80658i
$622$	0	0
$623$	−2.13326	−0.0854674
$624$	0	0
$625$	17.3906	0.695625
$626$	0	0
$627$	12.8875i	0.514676i
$628$	0	0
$629$	− 11.4228i	− 0.455455i
$630$	0	0
$631$	13.8111	0.549810	0.274905	−	0.961471i	$-0.411354\pi$
$631$	13.8111	0.549810	0.274905	+	0.961471i	$0.411354\pi$
$632$	0	0
$633$	−59.3843	−2.36031
$634$	0	0
$635$	− 0.180119i	− 0.00714779i
$636$	0	0
$637$	− 1.75046i	− 0.0693557i
$638$	0	0
$639$	−10.9196	−0.431971
$640$	0	0
$641$	5.42340	0.214212	0.107106	−	0.994248i	$-0.465842\pi$
$641$	5.42340	0.214212	0.107106	+	0.994248i	$0.465842\pi$
$642$	0	0
$643$	29.1749i	1.15054i	0.817962	+	0.575272i	$0.195104\pi$
$643$	29.1749i	1.15054i	−0.817962	+	0.575272i	$0.804896\pi$
$644$	0	0
$645$	17.1397i	0.674876i
$646$	0	0
$647$	9.72718	0.382415	0.191207	−	0.981550i	$-0.438760\pi$
$647$	9.72718	0.382415	0.191207	+	0.981550i	$0.438760\pi$
$648$	0	0
$649$	−19.6726	−0.772219
$650$	0	0
$651$	− 20.2648i	− 0.794238i
$652$	0	0
$653$	− 22.5984i	− 0.884346i	−0.896930	−	0.442173i	$-0.854208\pi$
$653$	− 22.5984i	− 0.884346i	0.896930	−	0.442173i	$-0.145792\pi$
$654$	0	0
$655$	13.1235	0.512776
$656$	0	0
$657$	62.9777	2.45700
$658$	0	0
$659$	− 21.0888i	− 0.821502i	−0.911748	−	0.410751i	$-0.865267\pi$
$659$	− 21.0888i	− 0.821502i	0.911748	−	0.410751i	$-0.134733\pi$
$660$	0	0
$661$	− 21.3835i	− 0.831722i	−0.909428	−	0.415861i	$-0.863480\pi$
$661$	− 21.3835i	− 0.831722i	0.909428	−	0.415861i	$-0.136520\pi$
$662$	0	0
$663$	9.54022	0.370511
$664$	0	0
$665$	−1.30686	−0.0506778
$666$	0	0
$667$	− 5.96445i	− 0.230944i
$668$	0	0
$669$	11.6036i	0.448619i
$670$	0	0
$671$	23.1648	0.894267
$672$	0	0
$673$	−9.99170	−0.385152	−0.192576	−	0.981282i	$-0.561684\pi$
$673$	−9.99170	−0.385152	−0.192576	+	0.981282i	$0.561684\pi$
$674$	0	0
$675$	− 63.2726i	− 2.43536i
$676$	0	0
$677$	4.11871i	0.158295i	0.996863	+	0.0791475i	$0.0252198\pi$
$677$	4.11871i	0.158295i	−0.996863	+	0.0791475i	$0.974780\pi$
$678$	0	0
$679$	5.73933	0.220255
$680$	0	0
$681$	35.1341	1.34634
$682$	0	0
$683$	2.28787i	0.0875429i	0.999042	+	0.0437714i	$0.0139373\pi$
$683$	2.28787i	0.0875429i	−0.999042	+	0.0437714i	$0.986063\pi$
$684$	0	0
$685$	− 12.8795i	− 0.492102i
$686$	0	0
$687$	−42.9133	−1.63724
$688$	0	0
$689$	23.1830	0.883203
$690$	0	0
$691$	9.67099i	0.367902i	0.982935	+	0.183951i	$0.0588887\pi$
$691$	9.67099i	0.367902i	−0.982935	+	0.183951i	$0.941111\pi$
$692$	0	0
$693$	− 16.3894i	− 0.622582i
$694$	0	0
$695$	−9.21023	−0.349364
$696$	0	0
$697$	−14.7533	−0.558822
$698$	0	0
$699$	72.0739i	2.72609i
$700$	0	0
$701$	− 51.4462i	− 1.94310i	−0.236840	−	0.971549i	$-0.576112\pi$
$701$	− 51.4462i	− 1.94310i	0.236840	−	0.971549i	$-0.423888\pi$
$702$	0	0
$703$	12.1752	0.459196
$704$	0	0
$705$	23.7682	0.895161
$706$	0	0
$707$	− 7.40731i	− 0.278581i
$708$	0	0
$709$	− 9.65849i	− 0.362732i	−0.983416	−	0.181366i	$-0.941948\pi$
$709$	− 9.65849i	− 0.362732i	0.983416	−	0.181366i	$-0.0580519\pi$
$710$	0	0
$711$	78.6199	2.94848
$712$	0	0
$713$	53.2565	1.99447
$714$	0	0
$715$	2.81569i	0.105301i
$716$	0	0
$717$	58.4989i	2.18468i
$718$	0	0
$719$	−19.5159	−0.727820	−0.363910	−	0.931434i	$-0.618559\pi$
$719$	−19.5159	−0.727820	−0.363910	+	0.931434i	$0.618559\pi$
$720$	0	0
$721$	−16.6714	−0.620876
$722$	0	0
$723$	− 63.5824i	− 2.36465i
$724$	0	0
$725$	− 3.15067i	− 0.117013i
$726$	0	0
$727$	−15.5192	−0.575575	−0.287787	−	0.957694i	$-0.592920\pi$
$727$	−15.5192	−0.575575	−0.287787	+	0.957694i	$0.592920\pi$
$728$	0	0
$729$	−36.2462	−1.34245
$730$	0	0
$731$	12.4027i	0.458731i
$732$	0	0
$733$	− 3.95395i	− 0.146042i	−0.997330	−	0.0730212i	$-0.976736\pi$
$733$	− 3.95395i	− 0.146042i	0.997330	−	0.0730212i	$-0.0232641\pi$
$734$	0	0
$735$	2.33687	0.0861968
$736$	0	0
$737$	−3.81446	−0.140508
$738$	0	0
$739$	45.5632i	1.67607i	0.545617	+	0.838034i	$0.316295\pi$
$739$	45.5632i	1.67607i	−0.545617	+	0.838034i	$0.683705\pi$
$740$	0	0
$741$	10.1686i	0.373554i
$742$	0	0
$743$	−22.6814	−0.832100	−0.416050	−	0.909342i	$-0.636586\pi$
$743$	−22.6814	−0.832100	−0.416050	+	0.909342i	$0.636586\pi$
$744$	0	0
$745$	−5.78677	−0.212011
$746$	0	0
$747$	80.1121i	2.93115i
$748$	0	0
$749$	− 12.4492i	− 0.454884i
$750$	0	0
$751$	39.5036	1.44151	0.720753	−	0.693192i	$-0.243796\pi$
$751$	39.5036	1.44151	0.720753	+	0.693192i	$0.243796\pi$
$752$	0	0
$753$	−23.8112	−0.867730
$754$	0	0
$755$	− 13.4902i	− 0.490957i
$756$	0	0
$757$	2.75169i	0.100012i	0.998749	+	0.0500059i	$0.0159240\pi$
$757$	2.75169i	0.100012i	−0.998749	+	0.0500059i	$0.984076\pi$
$758$	0	0
$759$	60.5628	2.19829
$760$	0	0
$761$	−15.0780	−0.546576	−0.273288	−	0.961932i	$-0.588111\pi$
$761$	−15.0780	−0.546576	−0.273288	+	0.961932i	$0.588111\pi$
$762$	0	0
$763$	2.53093i	0.0916258i
$764$	0	0
$765$	9.05797i	0.327491i
$766$	0	0
$767$	−15.5224	−0.560480
$768$	0	0
$769$	−23.1596	−0.835158	−0.417579	−	0.908641i	$-0.637121\pi$
$769$	−23.1596	−0.835158	−0.417579	+	0.908641i	$0.637121\pi$
$770$	0	0
$771$	− 26.2307i	− 0.944677i
$772$	0	0
$773$	14.7575i	0.530790i	0.964140	+	0.265395i	$0.0855023\pi$
$773$	14.7575i	0.530790i	−0.964140	+	0.265395i	$0.914498\pi$
$774$	0	0
$775$	28.1323	1.01054
$776$	0	0
$777$	−21.7712	−0.781037
$778$	0	0
$779$	− 15.7251i	− 0.563412i
$780$	0	0
$781$	− 3.27910i	− 0.117335i
$782$	0	0
$783$	−9.95799	−0.355870
$784$	0	0
$785$	−8.93030	−0.318736
$786$	0	0
$787$	− 12.3963i	− 0.441879i	−0.975287	−	0.220940i	$-0.929088\pi$
$787$	− 12.3963i	− 0.441879i	0.975287	−	0.220940i	$-0.0709124\pi$
$788$	0	0
$789$	58.2359i	2.07325i
$790$	0	0
$791$	−1.16568	−0.0414469
$792$	0	0
$793$	18.2778	0.649063
$794$	0	0
$795$	30.9495i	1.09766i
$796$	0	0
$797$	18.9843i	0.672457i	0.941780	+	0.336229i	$0.109152\pi$
$797$	18.9843i	0.672457i	−0.941780	+	0.336229i	$0.890848\pi$
$798$	0	0
$799$	17.1992	0.608464
$800$	0	0
$801$	−15.7598	−0.556846
$802$	0	0
$803$	18.9120i	0.667389i
$804$	0	0
$805$	6.14139i	0.216455i
$806$	0	0
$807$	−92.1230	−3.24288
$808$	0	0
$809$	−6.90560	−0.242788	−0.121394	−	0.992604i	$-0.538736\pi$
$809$	−6.90560	−0.242788	−0.121394	+	0.992604i	$0.538736\pi$
$810$	0	0
$811$	13.8566i	0.486572i	0.969955	+	0.243286i	$0.0782253\pi$
$811$	13.8566i	0.486572i	−0.969955	+	0.243286i	$0.921775\pi$
$812$	0	0
$813$	63.1940i	2.21631i
$814$	0	0
$815$	18.0138	0.630995
$816$	0	0
$817$	−13.2197	−0.462499
$818$	0	0
$819$	− 12.9318i	− 0.451873i
$820$	0	0
$821$	30.6552i	1.06987i	0.844892	+	0.534936i	$0.179664\pi$
$821$	30.6552i	1.06987i	−0.844892	+	0.534936i	$0.820336\pi$
$822$	0	0
$823$	−22.9379	−0.799564	−0.399782	−	0.916610i	$-0.630914\pi$
$823$	−22.9379	−0.799564	−0.399782	+	0.916610i	$0.630914\pi$
$824$	0	0
$825$	31.9918	1.11381
$826$	0	0
$827$	12.7310i	0.442701i	0.975194	+	0.221350i	$0.0710464\pi$
$827$	12.7310i	0.442701i	−0.975194	+	0.221350i	$0.928954\pi$
$828$	0	0
$829$	− 19.3666i	− 0.672630i	−0.941750	−	0.336315i	$-0.890819\pi$
$829$	− 19.3666i	− 0.672630i	0.941750	−	0.336315i	$-0.109181\pi$
$830$	0	0
$831$	74.7514	2.59310
$832$	0	0
$833$	1.69102	0.0585902
$834$	0	0
$835$	7.54370i	0.261061i
$836$	0	0
$837$	− 88.9149i	− 3.07335i
$838$	0	0
$839$	26.7513	0.923558	0.461779	−	0.886995i	$-0.347211\pi$
$839$	26.7513	0.923558	0.461779	+	0.886995i	$0.347211\pi$
$840$	0	0
$841$	28.5041	0.982901
$842$	0	0
$843$	30.6744i	1.05648i
$844$	0	0
$845$	− 7.20416i	− 0.247830i
$846$	0	0
$847$	−6.07833	−0.208854
$848$	0	0
$849$	16.4214	0.563582
$850$	0	0
$851$	− 57.2154i	− 1.96132i
$852$	0	0
$853$	− 48.1783i	− 1.64959i	−0.565430	−	0.824796i	$-0.691290\pi$
$853$	− 48.1783i	− 1.64959i	0.565430	−	0.824796i	$-0.308710\pi$
$854$	0	0
$855$	−9.65462	−0.330181
$856$	0	0
$857$	−26.2828	−0.897802	−0.448901	−	0.893581i	$-0.648184\pi$
$857$	−26.2828	−0.897802	−0.448901	+	0.893581i	$0.648184\pi$
$858$	0	0
$859$	15.3098i	0.522363i	0.965290	+	0.261181i	$0.0841121\pi$
$859$	15.3098i	0.522363i	−0.965290	+	0.261181i	$0.915888\pi$
$860$	0	0
$861$	28.1191i	0.958296i
$862$	0	0
$863$	−25.5692	−0.870386	−0.435193	−	0.900337i	$-0.643320\pi$
$863$	−25.5692	−0.870386	−0.435193	+	0.900337i	$0.643320\pi$
$864$	0	0
$865$	−9.54750	−0.324625
$866$	0	0
$867$	− 45.5746i	− 1.54779i
$868$	0	0
$869$	23.6092i	0.800889i
$870$	0	0
$871$	−3.00974	−0.101981
$872$	0	0
$873$	42.4002	1.43503
$874$	0	0
$875$	6.86946i	0.232230i
$876$	0	0
$877$	− 47.6643i	− 1.60951i	−0.593607	−	0.804755i	$-0.702297\pi$
$877$	− 47.6643i	− 1.60951i	0.593607	−	0.804755i	$-0.297703\pi$
$878$	0	0
$879$	8.73668	0.294681
$880$	0	0
$881$	−57.2434	−1.92858	−0.964290	−	0.264850i	$-0.914678\pi$
$881$	−57.2434	−1.92858	−0.964290	+	0.264850i	$0.914678\pi$
$882$	0	0
$883$	− 25.8202i	− 0.868919i	−0.900691	−	0.434460i	$-0.856939\pi$
$883$	− 25.8202i	− 0.868919i	0.900691	−	0.434460i	$-0.143061\pi$
$884$	0	0
$885$	− 20.7225i	− 0.696578i
$886$	0	0
$887$	34.1175	1.14555	0.572777	−	0.819711i	$-0.305866\pi$
$887$	34.1175	1.14555	0.572777	+	0.819711i	$0.305866\pi$
$888$	0	0
$889$	−0.248418	−0.00833167
$890$	0	0
$891$	− 51.9448i	− 1.74022i
$892$	0	0
$893$	18.3321i	0.613461i
$894$	0	0
$895$	−1.15597	−0.0386400
$896$	0	0
$897$	47.7860	1.59553
$898$	0	0
$899$	− 4.42754i	− 0.147667i
$900$	0	0
$901$	22.3958i	0.746111i
$902$	0	0
$903$	23.6389	0.786655
$904$	0	0
$905$	4.30242	0.143017
$906$	0	0
$907$	− 8.87794i	− 0.294787i	−0.989078	−	0.147394i	$-0.952912\pi$
$907$	− 8.87794i	− 0.294787i	0.989078	−	0.147394i	$-0.0470884\pi$
$908$	0	0
$909$	− 54.7227i	− 1.81504i
$910$	0	0
$911$	−4.33134	−0.143504	−0.0717519	−	0.997423i	$-0.522859\pi$
$911$	−4.33134	−0.143504	−0.0717519	+	0.997423i	$0.522859\pi$
$912$	0	0
$913$	−24.0573	−0.796182
$914$	0	0
$915$	24.4010i	0.806671i
$916$	0	0
$917$	− 18.0997i	− 0.597706i
$918$	0	0
$919$	8.08794	0.266796	0.133398	−	0.991063i	$-0.457411\pi$
$919$	8.08794	0.266796	0.133398	+	0.991063i	$0.457411\pi$
$920$	0	0
$921$	3.38332	0.111484
$922$	0	0
$923$	− 2.58732i	− 0.0851626i
$924$	0	0
$925$	− 30.2236i	− 0.993746i
$926$	0	0
$927$	−123.163	−4.04520
$928$	0	0
$929$	−21.9875	−0.721387	−0.360694	−	0.932684i	$-0.617460\pi$
$929$	−21.9875	−0.721387	−0.360694	+	0.932684i	$0.617460\pi$
$930$	0	0
$931$	1.80241i	0.0590714i
$932$	0	0
$933$	− 4.55506i	− 0.149126i
$934$	0	0
$935$	−2.72007	−0.0889558
$936$	0	0
$937$	−37.5410	−1.22641	−0.613205	−	0.789924i	$-0.710120\pi$
$937$	−37.5410	−1.22641	−0.613205	+	0.789924i	$0.710120\pi$
$938$	0	0
$939$	85.5586i	2.79210i
$940$	0	0
$941$	4.15697i	0.135513i	0.997702	+	0.0677567i	$0.0215842\pi$
$941$	4.15697i	0.135513i	−0.997702	+	0.0677567i	$0.978416\pi$
$942$	0	0
$943$	−73.8980	−2.40645
$944$	0	0
$945$	10.2534	0.333543
$946$	0	0
$947$	− 0.495081i	− 0.0160880i	−0.999968	−	0.00804399i	$-0.997439\pi$
$947$	− 0.495081i	− 0.0160880i	0.999968	−	0.00804399i	$-0.00256051\pi$
$948$	0	0
$949$	14.9222i	0.484394i
$950$	0	0
$951$	−79.0760	−2.56421
$952$	0	0
$953$	−3.51461	−0.113849	−0.0569246	−	0.998378i	$-0.518129\pi$
$953$	−3.51461	−0.113849	−0.0569246	+	0.998378i	$0.518129\pi$
$954$	0	0
$955$	− 11.7618i	− 0.380602i
$956$	0	0
$957$	− 5.03495i	− 0.162757i
$958$	0	0
$959$	−17.7633	−0.573608
$960$	0	0
$961$	8.53346	0.275273
$962$	0	0
$963$	− 91.9706i	− 2.96371i
$964$	0	0
$965$	− 0.583991i	− 0.0187993i
$966$	0	0
$967$	−15.3243	−0.492795	−0.246398	−	0.969169i	$-0.579247\pi$
$967$	−15.3243	−0.492795	−0.246398	+	0.969169i	$0.579247\pi$
$968$	0	0
$969$	−9.82334	−0.315571
$970$	0	0
$971$	18.7915i	0.603049i	0.953458	+	0.301524i	$0.0974955\pi$
$971$	18.7915i	0.603049i	−0.953458	+	0.301524i	$0.902505\pi$
$972$	0	0
$973$	12.7027i	0.407229i
$974$	0	0
$975$	25.2426	0.808410
$976$	0	0
$977$	1.16324	0.0372154	0.0186077	−	0.999827i	$-0.494077\pi$
$977$	1.16324	0.0372154	0.0186077	+	0.999827i	$0.494077\pi$
$978$	0	0
$979$	− 4.73261i	− 0.151255i
$980$	0	0
$981$	18.6977i	0.596970i
$982$	0	0
$983$	0.537086	0.0171304	0.00856519	−	0.999963i	$-0.497274\pi$
$983$	0.537086	0.0171304	0.00856519	+	0.999963i	$0.497274\pi$
$984$	0	0
$985$	−7.20644	−0.229616
$986$	0	0
$987$	− 32.7808i	− 1.04342i
$988$	0	0
$989$	62.1240i	1.97543i
$990$	0	0
$991$	−7.63276	−0.242463	−0.121231	−	0.992624i	$-0.538684\pi$
$991$	−7.63276	−0.242463	−0.121231	+	0.992624i	$0.538684\pi$
$992$	0	0
$993$	69.8508	2.21665
$994$	0	0
$995$	14.0866i	0.446574i
$996$	0	0
$997$	33.0138i	1.04556i	0.852468	+	0.522779i	$0.175105\pi$
$997$	33.0138i	1.04556i	−0.852468	+	0.522779i	$0.824895\pi$
$998$	0	0
$999$	−95.5245	−3.02226

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3584.2.b.i.1793.12		12
4.3	odd	2		3584.2.b.k.1793.1		12
8.3	odd	2		3584.2.b.k.1793.12		12
8.5	even	2	inner	3584.2.b.i.1793.1		12
16.3	odd	4		3584.2.a.k.1.6	yes	6
16.5	even	4		3584.2.a.l.1.6	yes	6
16.11	odd	4		3584.2.a.e.1.1	✓	6
16.13	even	4		3584.2.a.f.1.1	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3584.2.a.e.1.1	✓	6	16.11	odd	4
3584.2.a.f.1.1	yes	6	16.13	even	4
3584.2.a.k.1.6	yes	6	16.3	odd	4
3584.2.a.l.1.6	yes	6	16.5	even	4
3584.2.b.i.1793.1		12	8.5	even	2	inner
3584.2.b.i.1793.12		12	1.1	even	1	trivial
3584.2.b.k.1793.1		12	4.3	odd	2
3584.2.b.k.1793.12		12	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)

\(f(q)\)	\(=\)	\(q+3.22299i q^{3} -0.725063i q^{5} -1.00000 q^{7} -7.38766 q^{9} +O(q^{10})\)	\(q+3.22299i q^{3} -0.725063i q^{5} -1.00000 q^{7} -7.38766 q^{9} -2.21848i q^{11} -1.75046i q^{13} +2.33687 q^{15} +1.69102 q^{17} +1.80241i q^{19} -3.22299i q^{21} +8.47013 q^{23} +4.47428 q^{25} -14.1414i q^{27} -0.704174i q^{29} +6.28756 q^{31} +7.15015 q^{33} +0.725063i q^{35} -6.75496i q^{37} +5.64171 q^{39} -8.72453 q^{41} +7.33448i q^{43} +5.35652i q^{45} +10.1709 q^{47} +1.00000 q^{49} +5.45013i q^{51} +13.2440i q^{53} -1.60854 q^{55} -5.80913 q^{57} -8.86761i q^{59} +10.4417i q^{61} +7.38766 q^{63} -1.26919 q^{65} -1.71940i q^{67} +27.2992i q^{69} +1.47808 q^{71} -8.52472 q^{73} +14.4206i q^{75} +2.21848i q^{77} -10.6421 q^{79} +23.4146 q^{81} -10.8440i q^{83} -1.22609i q^{85} +2.26954 q^{87} +2.13326 q^{89} +1.75046i q^{91} +20.2648i q^{93} +1.30686 q^{95} -5.73933 q^{97} +16.3894i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{7} - 20 q^{9}+O(q^{10}) \) 12 * q - 12 * q^7 - 20 * q^9	\( 12 q - 12 q^{7} - 20 q^{9} - 16 q^{15} - 4 q^{25} - 16 q^{31} + 8 q^{33} + 8 q^{41} + 16 q^{47} + 12 q^{49} - 32 q^{55} + 8 q^{57} + 20 q^{63} - 16 q^{65} + 16 q^{71} - 32 q^{73} - 48 q^{79} + 20 q^{81} + 16 q^{87} - 32 q^{89} - 32 q^{95} + 32 q^{97}+O(q^{100}) \) 12 * q - 12 * q^7 - 20 * q^9 - 16 * q^15 - 4 * q^25 - 16 * q^31 + 8 * q^33 + 8 * q^41 + 16 * q^47 + 12 * q^49 - 32 * q^55 + 8 * q^57 + 20 * q^63 - 16 * q^65 + 16 * q^71 - 32 * q^73 - 48 * q^79 + 20 * q^81 + 16 * q^87 - 32 * q^89 - 32 * q^95 + 32 * q^97

Introduction

L-functions

Modular forms

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Database

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