LMFDB - Embedded newform 3840.2.d.bl.2689.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3840,2,Mod(2689,3840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3840, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3840.2689");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3840 = 2^{8} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3840.d (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:	$30.6625543762$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 1920)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2689.4
Root			$-0.854638 - 0.854638i$ of defining polynomial
Character	$\chi$	$=$	3840.2689
Dual form			3840.2.d.bl.2689.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +(-0.539189 + 2.17009i) q^{5} +2.34017i q^{7} +1.00000 q^{9} -3.07838i q^{11} +0.921622 q^{13} +(-0.539189 + 2.17009i) q^{15} -7.75872i q^{17} -4.00000i q^{19} +2.34017i q^{21} -2.15676i q^{23} +(-4.41855 - 2.34017i) q^{25} +1.00000 q^{27} -6.49693i q^{29} -2.00000 q^{31} -3.07838i q^{33} +(-5.07838 - 1.26180i) q^{35} +3.07838 q^{37} +0.921622 q^{39} -10.6803 q^{41} +2.15676 q^{43} +(-0.539189 + 2.17009i) q^{45} -8.68035i q^{47} +1.52359 q^{49} -7.75872i q^{51} -2.92162 q^{53} +(6.68035 + 1.65983i) q^{55} -4.00000i q^{57} +11.7587i q^{59} -6.00000i q^{61} +2.34017i q^{63} +(-0.496928 + 2.00000i) q^{65} -6.83710 q^{67} -2.15676i q^{69} +11.5174 q^{71} -6.52359i q^{73} +(-4.41855 - 2.34017i) q^{75} +7.20394 q^{77} -15.3607 q^{79} +1.00000 q^{81} +1.84324 q^{83} +(16.8371 + 4.18342i) q^{85} -6.49693i q^{87} +6.00000 q^{89} +2.15676i q^{91} -2.00000 q^{93} +(8.68035 + 2.15676i) q^{95} -3.07838i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} + 6 q^{9} + 12 q^{13} + 2 q^{25} + 6 q^{27} - 12 q^{31} - 24 q^{35} + 12 q^{37} + 12 q^{39} - 20 q^{41} - 22 q^{49} - 24 q^{53} - 4 q^{55} + 32 q^{65} + 16 q^{67} - 32 q^{71} + 2 q^{75} - 32 q^{77}+ \cdots + 8 q^{95}+O(q^{100}) $ 6 * q + 6 * q^3 + 6 * q^9 + 12 * q^13 + 2 * q^25 + 6 * q^27 - 12 * q^31 - 24 * q^35 + 12 * q^37 + 12 * q^39 - 20 * q^41 - 22 * q^49 - 24 * q^53 - 4 * q^55 + 32 * q^65 + 16 * q^67 - 32 * q^71 + 2 * q^75 - 32 * q^77 - 4 * q^79 + 6 * q^81 + 24 * q^83 + 44 * q^85 + 36 * q^89 - 12 * q^93 + 8 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times$.

$n$	$511$	$1537$	$2561$	$2821$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$	−0.539189	+	2.17009i	−0.241133	+	0.970492i
$5$	−0.539189	+	2.17009i	−0.241133	+	0.970492i
$6$		0			0
$7$			2.34017i			0.884502i	0.896891	+	0.442251i	$0.145820\pi$
$7$			2.34017i			0.884502i	−0.896891	+	0.442251i	$0.854180\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	3.07838i		−	0.928166i	−0.885792	−	0.464083i	$-0.846384\pi$
$11$		−	3.07838i		−	0.928166i	0.885792	−	0.464083i	$-0.153616\pi$
$12$		0			0
$13$	0.921622			0.255612			0.127806	−	0.991799i	$-0.459207\pi$
$13$	0.921622			0.255612			0.127806	+	0.991799i	$0.459207\pi$
$14$		0			0
$15$	−0.539189	+	2.17009i	−0.139218	+	0.560314i
$16$		0			0
$17$		−	7.75872i		−	1.88177i	−0.338730	−	0.940883i	$-0.609997\pi$
$17$		−	7.75872i		−	1.88177i	0.338730	−	0.940883i	$-0.390003\pi$
$18$		0			0
$19$		−	4.00000i		−	0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$		−	4.00000i		−	0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$		0			0
$21$			2.34017i			0.510668i
$22$		0			0
$23$		−	2.15676i		−	0.449715i	−0.974392	−	0.224857i	$-0.927808\pi$
$23$		−	2.15676i		−	0.449715i	0.974392	−	0.224857i	$-0.0721916\pi$
$24$		0			0
$25$	−4.41855	−	2.34017i	−0.883710	−	0.468035i
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$		−	6.49693i		−	1.20645i	−0.797571	−	0.603225i	$-0.793882\pi$
$29$		−	6.49693i		−	1.20645i	0.797571	−	0.603225i	$-0.206118\pi$
$30$		0			0
$31$	−2.00000			−0.359211			−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000			−0.359211			−0.179605	+	0.983739i	$0.557482\pi$
$32$		0			0
$33$		−	3.07838i		−	0.535877i
$34$		0			0
$35$	−5.07838	−	1.26180i	−0.858403	−	0.213282i
$36$		0			0
$37$	3.07838			0.506082			0.253041	−	0.967456i	$-0.418569\pi$
$37$	3.07838			0.506082			0.253041	+	0.967456i	$0.418569\pi$
$38$		0			0
$39$	0.921622			0.147578
$40$		0			0
$41$	−10.6803			−1.66799			−0.833995	−	0.551772i	$-0.813952\pi$
$41$	−10.6803			−1.66799			−0.833995	+	0.551772i	$0.813952\pi$
$42$		0			0
$43$	2.15676			0.328902			0.164451	−	0.986385i	$-0.447415\pi$
$43$	2.15676			0.328902			0.164451	+	0.986385i	$0.447415\pi$
$44$		0			0
$45$	−0.539189	+	2.17009i	−0.0803775	+	0.323497i
$46$		0			0
$47$		−	8.68035i		−	1.26616i	−0.774087	−	0.633079i	$-0.781791\pi$
$47$		−	8.68035i		−	1.26616i	0.774087	−	0.633079i	$-0.218209\pi$
$48$		0			0
$49$	1.52359			0.217656
$50$		0			0
$51$		−	7.75872i		−	1.08644i
$52$		0			0
$53$	−2.92162			−0.401316			−0.200658	−	0.979661i	$-0.564308\pi$
$53$	−2.92162			−0.401316			−0.200658	+	0.979661i	$0.564308\pi$
$54$		0			0
$55$	6.68035	+	1.65983i	0.900778	+	0.223811i
$56$		0			0
$57$		−	4.00000i		−	0.529813i
$58$		0			0
$59$			11.7587i			1.53086i	0.643522	+	0.765428i	$0.277473\pi$
$59$			11.7587i			1.53086i	−0.643522	+	0.765428i	$0.722527\pi$
$60$		0			0
$61$		−	6.00000i		−	0.768221i	−0.923287	−	0.384111i	$-0.874508\pi$
$61$		−	6.00000i		−	0.768221i	0.923287	−	0.384111i	$-0.125492\pi$
$62$		0			0
$63$			2.34017i			0.294834i
$64$		0			0
$65$	−0.496928	+	2.00000i	−0.0616364	+	0.248069i
$66$		0			0
$67$	−6.83710			−0.835285			−0.417642	−	0.908611i	$-0.637144\pi$
$67$	−6.83710			−0.835285			−0.417642	+	0.908611i	$0.637144\pi$
$68$		0			0
$69$		−	2.15676i		−	0.259643i
$70$		0			0
$71$	11.5174			1.36687			0.683435	−	0.730012i	$-0.260485\pi$
$71$	11.5174			1.36687			0.683435	+	0.730012i	$0.260485\pi$
$72$		0			0
$73$		−	6.52359i		−	0.763529i	−0.924260	−	0.381764i	$-0.875317\pi$
$73$		−	6.52359i		−	0.763529i	0.924260	−	0.381764i	$-0.124683\pi$
$74$		0			0
$75$	−4.41855	−	2.34017i	−0.510210	−	0.270220i
$76$		0			0
$77$	7.20394			0.820965
$78$		0			0
$79$	−15.3607			−1.72821			−0.864106	−	0.503309i	$-0.832116\pi$
$79$	−15.3607			−1.72821			−0.864106	+	0.503309i	$0.832116\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	1.84324			0.202322			0.101161	−	0.994870i	$-0.467744\pi$
$83$	1.84324			0.202322			0.101161	+	0.994870i	$0.467744\pi$
$84$		0			0
$85$	16.8371	+	4.18342i	1.82624	+	0.453755i
$86$		0			0
$87$		−	6.49693i		−	0.696544i
$88$		0			0
$89$	6.00000			0.635999			0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000			0.635999			0.317999	+	0.948091i	$0.396989\pi$
$90$		0			0
$91$			2.15676i			0.226089i
$92$		0			0
$93$	−2.00000			−0.207390
$94$		0			0
$95$	8.68035	+	2.15676i	0.890585	+	0.221278i
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$		−	3.07838i		−	0.309389i
$100$		0			0
$101$			8.34017i			0.829878i	0.909849	+	0.414939i	$0.136197\pi$
$101$			8.34017i			0.829878i	−0.909849	+	0.414939i	$0.863803\pi$
$102$		0			0
$103$		−	10.3402i		−	1.01885i	−0.860516	−	0.509424i	$-0.829859\pi$
$103$		−	10.3402i		−	1.01885i	0.860516	−	0.509424i	$-0.170141\pi$
$104$		0			0
$105$	−5.07838	−	1.26180i	−0.495599	−	0.123139i
$106$		0			0
$107$	13.3607			1.29163			0.645813	−	0.763495i	$-0.276518\pi$
$107$	13.3607			1.29163			0.645813	+	0.763495i	$0.276518\pi$
$108$		0			0
$109$			5.31965i			0.509530i	0.967003	+	0.254765i	$0.0819982\pi$
$109$			5.31965i			0.509530i	−0.967003	+	0.254765i	$0.918002\pi$
$110$		0			0
$111$	3.07838			0.292187
$112$		0			0
$113$		−	19.7587i		−	1.85874i	−0.369144	−	0.929372i	$-0.620349\pi$
$113$		−	19.7587i		−	1.85874i	0.369144	−	0.929372i	$-0.379651\pi$
$114$		0			0
$115$	4.68035	+	1.16290i	0.436445	+	0.108441i
$116$		0			0
$117$	0.921622			0.0852040
$118$		0			0
$119$	18.1568			1.66443
$120$		0			0
$121$	1.52359			0.138508
$122$		0			0
$123$	−10.6803			−0.963014
$124$		0			0
$125$	7.46081	−	8.32684i	0.667315	−	0.744775i
$126$		0			0
$127$			19.3340i			1.71562i	0.513969	+	0.857809i	$0.328175\pi$
$127$			19.3340i			1.71562i	−0.513969	+	0.857809i	$0.671825\pi$
$128$		0			0
$129$	2.15676			0.189892
$130$		0			0
$131$		−	7.44521i		−	0.650491i	−0.945630	−	0.325246i	$-0.894553\pi$
$131$		−	7.44521i		−	0.650491i	0.945630	−	0.325246i	$-0.105447\pi$
$132$		0			0
$133$	9.36069			0.811675
$134$		0			0
$135$	−0.539189	+	2.17009i	−0.0464060	+	0.186771i
$136$		0			0
$137$		−	3.44521i		−	0.294344i	−0.989111	−	0.147172i	$-0.952983\pi$
$137$		−	3.44521i		−	0.294344i	0.989111	−	0.147172i	$-0.0470171\pi$
$138$		0			0
$139$			19.2039i			1.62886i	0.580264	+	0.814428i	$0.302949\pi$
$139$			19.2039i			1.62886i	−0.580264	+	0.814428i	$0.697051\pi$
$140$		0			0
$141$		−	8.68035i		−	0.731017i
$142$		0			0
$143$		−	2.83710i		−	0.237250i
$144$		0			0
$145$	14.0989	+	3.50307i	1.17085	+	0.290914i
$146$		0			0
$147$	1.52359			0.125664
$148$		0			0
$149$		−	9.81658i		−	0.804206i	−0.915594	−	0.402103i	$-0.868279\pi$
$149$		−	9.81658i		−	0.804206i	0.915594	−	0.402103i	$-0.131721\pi$
$150$		0			0
$151$	−23.6742			−1.92658			−0.963290	−	0.268464i	$-0.913484\pi$
$151$	−23.6742			−1.92658			−0.963290	+	0.268464i	$0.913484\pi$
$152$		0			0
$153$		−	7.75872i		−	0.627256i
$154$		0			0
$155$	1.07838	−	4.34017i	0.0866174	−	0.348611i
$156$		0			0
$157$	11.0784			0.884151			0.442075	−	0.896978i	$-0.354242\pi$
$157$	11.0784			0.884151			0.442075	+	0.896978i	$0.354242\pi$
$158$		0			0
$159$	−2.92162			−0.231700
$160$		0			0
$161$	5.04718			0.397774
$162$		0			0
$163$	12.9939			1.01776			0.508879	−	0.860838i	$-0.330060\pi$
$163$	12.9939			1.01776			0.508879	+	0.860838i	$0.330060\pi$
$164$		0			0
$165$	6.68035	+	1.65983i	0.520064	+	0.129217i
$166$		0			0
$167$			12.0000i			0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$			12.0000i			0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$		0			0
$169$	−12.1506			−0.934662
$170$		0			0
$171$		−	4.00000i		−	0.305888i
$172$		0			0
$173$	18.4391			1.40190			0.700948	−	0.713212i	$-0.252760\pi$
$173$	18.4391			1.40190			0.700948	+	0.713212i	$0.252760\pi$
$174$		0			0
$175$	5.47641	−	10.3402i	0.413978	−	0.781644i
$176$		0			0
$177$			11.7587i			0.883840i
$178$		0			0
$179$			1.91548i			0.143170i	0.997435	+	0.0715848i	$0.0228057\pi$
$179$			1.91548i			0.143170i	−0.997435	+	0.0715848i	$0.977194\pi$
$180$		0			0
$181$		−	6.99386i		−	0.519849i	−0.965629	−	0.259925i	$-0.916302\pi$
$181$		−	6.99386i		−	0.519849i	0.965629	−	0.259925i	$-0.0836977\pi$
$182$		0			0
$183$		−	6.00000i		−	0.443533i
$184$		0			0
$185$	−1.65983	+	6.68035i	−0.122033	+	0.491149i
$186$		0			0
$187$	−23.8843			−1.74659
$188$		0			0
$189$			2.34017i			0.170223i
$190$		0			0
$191$	−6.15676			−0.445487			−0.222744	−	0.974877i	$-0.571501\pi$
$191$	−6.15676			−0.445487			−0.222744	+	0.974877i	$0.571501\pi$
$192$		0			0
$193$		−	14.5236i		−	1.04543i	−0.852507	−	0.522715i	$-0.824919\pi$
$193$		−	14.5236i		−	1.04543i	0.852507	−	0.522715i	$-0.175081\pi$
$194$		0			0
$195$	−0.496928	+	2.00000i	−0.0355858	+	0.143223i
$196$		0			0
$197$	24.5958			1.75238			0.876190	−	0.481966i	$-0.160077\pi$
$197$	24.5958			1.75238			0.876190	+	0.481966i	$0.160077\pi$
$198$		0			0
$199$	−3.36069			−0.238233			−0.119117	−	0.992880i	$-0.538006\pi$
$199$	−3.36069			−0.238233			−0.119117	+	0.992880i	$0.538006\pi$
$200$		0			0
$201$	−6.83710			−0.482252
$202$		0			0
$203$	15.2039			1.06711
$204$		0			0
$205$	5.75872	−	23.1773i	0.402207	−	1.61877i
$206$		0			0
$207$		−	2.15676i		−	0.149905i
$208$		0			0
$209$	−12.3135			−0.851743
$210$		0			0
$211$			11.2039i			0.771311i	0.922643	+	0.385655i	$0.126025\pi$
$211$			11.2039i			0.771311i	−0.922643	+	0.385655i	$0.873975\pi$
$212$		0			0
$213$	11.5174			0.789162
$214$		0			0
$215$	−1.16290	+	4.68035i	−0.0793090	+	0.319197i
$216$		0			0
$217$		−	4.68035i		−	0.317723i
$218$		0			0
$219$		−	6.52359i		−	0.440823i
$220$		0			0
$221$		−	7.15061i		−	0.481002i
$222$		0			0
$223$			6.65368i			0.445564i	0.974868	+	0.222782i	$0.0715138\pi$
$223$			6.65368i			0.445564i	−0.974868	+	0.222782i	$0.928486\pi$
$224$		0			0
$225$	−4.41855	−	2.34017i	−0.294570	−	0.156012i
$226$		0			0
$227$	−9.84324			−0.653319			−0.326660	−	0.945142i	$-0.605923\pi$
$227$	−9.84324			−0.653319			−0.326660	+	0.945142i	$0.605923\pi$
$228$		0			0
$229$		−	1.31965i		−	0.0872052i	−0.999049	−	0.0436026i	$-0.986116\pi$
$229$		−	1.31965i		−	0.0872052i	0.999049	−	0.0436026i	$-0.0138835\pi$
$230$		0			0
$231$	7.20394			0.473984
$232$		0			0
$233$		−	2.39803i		−	0.157100i	−0.996910	−	0.0785501i	$-0.974971\pi$
$233$		−	2.39803i		−	0.157100i	0.996910	−	0.0785501i	$-0.0250291\pi$
$234$		0			0
$235$	18.8371	+	4.68035i	1.22880	+	0.305312i
$236$		0			0
$237$	−15.3607			−0.997784
$238$		0			0
$239$	−6.15676			−0.398247			−0.199124	−	0.979974i	$-0.563810\pi$
$239$	−6.15676			−0.398247			−0.199124	+	0.979974i	$0.563810\pi$
$240$		0			0
$241$	0.639308			0.0411815			0.0205907	−	0.999788i	$-0.493445\pi$
$241$	0.639308			0.0411815			0.0205907	+	0.999788i	$0.493445\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$	−0.821503	+	3.30632i	−0.0524839	+	0.211233i
$246$		0			0
$247$		−	3.68649i		−	0.234566i
$248$		0			0
$249$	1.84324			0.116811
$250$		0			0
$251$		−	12.9216i		−	0.815606i	−0.913070	−	0.407803i	$-0.866295\pi$
$251$		−	12.9216i		−	0.815606i	0.913070	−	0.407803i	$-0.133705\pi$
$252$		0			0
$253$	−6.63931			−0.417410
$254$		0			0
$255$	16.8371	+	4.18342i	1.05438	+	0.261976i
$256$		0			0
$257$		−	14.9627i		−	0.933345i	−0.884430	−	0.466673i	$-0.845453\pi$
$257$		−	14.9627i		−	0.933345i	0.884430	−	0.466673i	$-0.154547\pi$
$258$		0			0
$259$			7.20394i			0.447631i
$260$		0			0
$261$		−	6.49693i		−	0.402150i
$262$		0			0
$263$			13.3607i			0.823856i	0.911216	+	0.411928i	$0.135144\pi$
$263$			13.3607i			0.823856i	−0.911216	+	0.411928i	$0.864856\pi$
$264$		0			0
$265$	1.57531	−	6.34017i	0.0967703	−	0.389474i
$266$		0			0
$267$	6.00000			0.367194
$268$		0			0
$269$			27.5441i			1.67939i	0.543055	+	0.839697i	$0.317267\pi$
$269$			27.5441i			1.67939i	−0.543055	+	0.839697i	$0.682733\pi$
$270$		0			0
$271$	−10.0000			−0.607457			−0.303728	−	0.952759i	$-0.598232\pi$
$271$	−10.0000			−0.607457			−0.303728	+	0.952759i	$0.598232\pi$
$272$		0			0
$273$			2.15676i			0.130533i
$274$		0			0
$275$	−7.20394	+	13.6020i	−0.434414	+	0.820230i
$276$		0			0
$277$	4.12556			0.247881			0.123940	−	0.992290i	$-0.460447\pi$
$277$	4.12556			0.247881			0.123940	+	0.992290i	$0.460447\pi$
$278$		0			0
$279$	−2.00000			−0.119737
$280$		0			0
$281$	11.3607			0.677722			0.338861	−	0.940836i	$-0.389958\pi$
$281$	11.3607			0.677722			0.338861	+	0.940836i	$0.389958\pi$
$282$		0			0
$283$	−10.5236			−0.625563			−0.312781	−	0.949825i	$-0.601261\pi$
$283$	−10.5236			−0.625563			−0.312781	+	0.949825i	$0.601261\pi$
$284$		0			0
$285$	8.68035	+	2.15676i	0.514179	+	0.127755i
$286$		0			0
$287$		−	24.9939i		−	1.47534i
$288$		0			0
$289$	−43.1978			−2.54105
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−32.5958			−1.90427			−0.952134	−	0.305680i	$-0.901116\pi$
$293$	−32.5958			−1.90427			−0.952134	+	0.305680i	$0.901116\pi$
$294$		0			0
$295$	−25.5174	−	6.34017i	−1.48568	−	0.369139i
$296$		0			0
$297$		−	3.07838i		−	0.178626i
$298$		0			0
$299$		−	1.98771i		−	0.114952i
$300$		0			0
$301$			5.04718i			0.290915i
$302$		0			0
$303$			8.34017i			0.479130i
$304$		0			0
$305$	13.0205	+	3.23513i	0.745553	+	0.185243i
$306$		0			0
$307$	29.9877			1.71149			0.855745	−	0.517398i	$-0.173099\pi$
$307$	29.9877			1.71149			0.855745	+	0.517398i	$0.173099\pi$
$308$		0			0
$309$		−	10.3402i		−	0.588232i
$310$		0			0
$311$	−27.2039			−1.54259			−0.771297	−	0.636476i	$-0.780392\pi$
$311$	−27.2039			−1.54259			−0.771297	+	0.636476i	$0.780392\pi$
$312$		0			0
$313$		−	26.8371i		−	1.51692i	−0.651718	−	0.758461i	$-0.725951\pi$
$313$		−	26.8371i		−	1.51692i	0.651718	−	0.758461i	$-0.274049\pi$
$314$		0			0
$315$	−5.07838	−	1.26180i	−0.286134	−	0.0710941i
$316$		0			0
$317$	24.5958			1.38144			0.690720	−	0.723123i	$-0.257294\pi$
$317$	24.5958			1.38144			0.690720	+	0.723123i	$0.257294\pi$
$318$		0			0
$319$	−20.0000			−1.11979
$320$		0			0
$321$	13.3607			0.745721
$322$		0			0
$323$	−31.0349			−1.72683
$324$		0			0
$325$	−4.07223	−	2.15676i	−0.225887	−	0.119635i
$326$		0			0
$327$			5.31965i			0.294178i
$328$		0			0
$329$	20.3135			1.11992
$330$		0			0
$331$		−	12.3135i		−	0.676812i	−0.941000	−	0.338406i	$-0.890112\pi$
$331$		−	12.3135i		−	0.676812i	0.941000	−	0.338406i	$-0.109888\pi$
$332$		0			0
$333$	3.07838			0.168694
$334$		0			0
$335$	3.68649	−	14.8371i	0.201414	−	0.810637i
$336$		0			0
$337$		−	9.47641i		−	0.516213i	−0.966116	−	0.258106i	$-0.916901\pi$
$337$		−	9.47641i		−	0.516213i	0.966116	−	0.258106i	$-0.0830985\pi$
$338$		0			0
$339$		−	19.7587i		−	1.07315i
$340$		0			0
$341$			6.15676i			0.333407i
$342$		0			0
$343$			19.9467i			1.07702i
$344$		0			0
$345$	4.68035	+	1.16290i	0.251981	+	0.0626084i
$346$		0			0
$347$	−7.51745			−0.403558			−0.201779	−	0.979431i	$-0.564672\pi$
$347$	−7.51745			−0.403558			−0.201779	+	0.979431i	$0.564672\pi$
$348$		0			0
$349$			27.6742i			1.48137i	0.671855	+	0.740683i	$0.265498\pi$
$349$			27.6742i			1.48137i	−0.671855	+	0.740683i	$0.734502\pi$
$350$		0			0
$351$	0.921622			0.0491926
$352$		0			0
$353$		−	21.6020i		−	1.14976i	−0.818239	−	0.574878i	$-0.805049\pi$
$353$		−	21.6020i		−	1.14976i	0.818239	−	0.574878i	$-0.194951\pi$
$354$		0			0
$355$	−6.21008	+	24.9939i	−0.329597	+	1.32654i
$356$		0			0
$357$	18.1568			0.960957
$358$		0			0
$359$	12.0000			0.633336			0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000			0.633336			0.316668	+	0.948536i	$0.397436\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$		0			0
$363$	1.52359			0.0799678
$364$		0			0
$365$	14.1568	+	3.51745i	0.740998	+	0.184112i
$366$		0			0
$367$			23.0205i			1.20166i	0.799376	+	0.600831i	$0.205163\pi$
$367$			23.0205i			1.20166i	−0.799376	+	0.600831i	$0.794837\pi$
$368$		0			0
$369$	−10.6803			−0.555997
$370$		0			0
$371$		−	6.83710i		−	0.354965i
$372$		0			0
$373$	−26.5958			−1.37708			−0.688540	−	0.725199i	$-0.741748\pi$
$373$	−26.5958			−1.37708			−0.688540	+	0.725199i	$0.741748\pi$
$374$		0			0
$375$	7.46081	−	8.32684i	0.385275	−	0.429996i
$376$		0			0
$377$		−	5.98771i		−	0.308383i
$378$		0			0
$379$			36.1445i			1.85662i	0.371811	+	0.928308i	$0.378737\pi$
$379$			36.1445i			1.85662i	−0.371811	+	0.928308i	$0.621263\pi$
$380$		0			0
$381$			19.3340i			0.990512i
$382$		0			0
$383$		−	6.83710i		−	0.349360i	−0.984625	−	0.174680i	$-0.944111\pi$
$383$		−	6.83710i		−	0.349360i	0.984625	−	0.174680i	$-0.0558890\pi$
$384$		0			0
$385$	−3.88428	+	15.6332i	−0.197961	+	0.796740i
$386$		0			0
$387$	2.15676			0.109634
$388$		0			0
$389$		−	16.7070i		−	0.847079i	−0.905878	−	0.423539i	$-0.860787\pi$
$389$		−	16.7070i		−	0.847079i	0.905878	−	0.423539i	$-0.139213\pi$
$390$		0			0
$391$	−16.7337			−0.846258
$392$		0			0
$393$		−	7.44521i		−	0.375561i
$394$		0			0
$395$	8.28231	−	33.3340i	0.416728	−	1.67722i
$396$		0			0
$397$	3.56093			0.178718			0.0893590	−	0.995999i	$-0.471518\pi$
$397$	3.56093			0.178718			0.0893590	+	0.995999i	$0.471518\pi$
$398$		0			0
$399$	9.36069			0.468621
$400$		0			0
$401$	−25.7152			−1.28416			−0.642079	−	0.766639i	$-0.721928\pi$
$401$	−25.7152			−1.28416			−0.642079	+	0.766639i	$0.721928\pi$
$402$		0			0
$403$	−1.84324			−0.0918185
$404$		0			0
$405$	−0.539189	+	2.17009i	−0.0267925	+	0.107832i
$406$		0			0
$407$		−	9.47641i		−	0.469728i
$408$		0			0
$409$	28.8371			1.42590			0.712951	−	0.701213i	$-0.247358\pi$
$409$	28.8371			1.42590			0.712951	+	0.701213i	$0.247358\pi$
$410$		0			0
$411$		−	3.44521i		−	0.169940i
$412$		0			0
$413$	−27.5174			−1.35405
$414$		0			0
$415$	−0.993857	+	4.00000i	−0.0487865	+	0.196352i
$416$		0			0
$417$			19.2039i			0.940421i
$418$		0			0
$419$		−	6.28231i		−	0.306911i	−0.988156	−	0.153456i	$-0.950960\pi$
$419$		−	6.28231i		−	0.306911i	0.988156	−	0.153456i	$-0.0490402\pi$
$420$		0			0
$421$		−	10.9939i		−	0.535808i	−0.963446	−	0.267904i	$-0.913669\pi$
$421$		−	10.9939i		−	0.535808i	0.963446	−	0.267904i	$-0.0863310\pi$
$422$		0			0
$423$		−	8.68035i		−	0.422053i
$424$		0			0
$425$	−18.1568	+	34.2823i	−0.880732	+	1.66294i
$426$		0			0
$427$	14.0410			0.679493
$428$		0			0
$429$		−	2.83710i		−	0.136977i
$430$		0			0
$431$	21.3607			1.02891			0.514454	−	0.857518i	$-0.327995\pi$
$431$	21.3607			1.02891			0.514454	+	0.857518i	$0.327995\pi$
$432$		0			0
$433$		−	18.4703i		−	0.887624i	−0.896120	−	0.443812i	$-0.853626\pi$
$433$		−	18.4703i		−	0.887624i	0.896120	−	0.443812i	$-0.146374\pi$
$434$		0			0
$435$	14.0989	+	3.50307i	0.675990	+	0.167959i
$436$		0			0
$437$	−8.62702			−0.412686
$438$		0			0
$439$	−6.31351			−0.301327			−0.150664	−	0.988585i	$-0.548141\pi$
$439$	−6.31351			−0.301327			−0.150664	+	0.988585i	$0.548141\pi$
$440$		0			0
$441$	1.52359			0.0725519
$442$		0			0
$443$	−27.8310			−1.32229			−0.661144	−	0.750259i	$-0.729929\pi$
$443$	−27.8310			−1.32229			−0.661144	+	0.750259i	$0.729929\pi$
$444$		0			0
$445$	−3.23513	+	13.0205i	−0.153360	+	0.617232i
$446$		0			0
$447$		−	9.81658i		−	0.464308i
$448$		0			0
$449$	−14.6803			−0.692808			−0.346404	−	0.938085i	$-0.612597\pi$
$449$	−14.6803			−0.692808			−0.346404	+	0.938085i	$0.612597\pi$
$450$		0			0
$451$			32.8781i			1.54817i
$452$		0			0
$453$	−23.6742			−1.11231
$454$		0			0
$455$	−4.68035	−	1.16290i	−0.219418	−	0.0545175i
$456$		0			0
$457$			19.6865i			0.920895i	0.887687	+	0.460448i	$0.152311\pi$
$457$			19.6865i			0.920895i	−0.887687	+	0.460448i	$0.847689\pi$
$458$		0			0
$459$		−	7.75872i		−	0.362146i
$460$		0			0
$461$		−	13.5031i		−	0.628901i	−0.949274	−	0.314450i	$-0.898180\pi$
$461$		−	13.5031i		−	0.628901i	0.949274	−	0.314450i	$-0.101820\pi$
$462$		0			0
$463$		−	6.02666i		−	0.280083i	−0.990146	−	0.140041i	$-0.955276\pi$
$463$		−	6.02666i		−	0.280083i	0.990146	−	0.140041i	$-0.0447236\pi$
$464$		0			0
$465$	1.07838	−	4.34017i	0.0500086	−	0.201271i
$466$		0			0
$467$	−19.8310			−0.917667			−0.458834	−	0.888522i	$-0.651733\pi$
$467$	−19.8310			−0.917667			−0.458834	+	0.888522i	$0.651733\pi$
$468$		0			0
$469$		−	16.0000i		−	0.738811i
$470$		0			0
$471$	11.0784			0.510465
$472$		0			0
$473$		−	6.63931i		−	0.305276i
$474$		0			0
$475$	−9.36069	+	17.6742i	−0.429498	+	0.810948i
$476$		0			0
$477$	−2.92162			−0.133772
$478$		0			0
$479$	23.2039			1.06021			0.530107	−	0.847930i	$-0.322152\pi$
$479$	23.2039			1.06021			0.530107	+	0.847930i	$0.322152\pi$
$480$		0			0
$481$	2.83710			0.129361
$482$		0			0
$483$	5.04718			0.229655
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	23.3874i		−	1.05978i	−0.848066	−	0.529891i	$-0.822233\pi$
$487$		−	23.3874i		−	1.05978i	0.848066	−	0.529891i	$-0.177767\pi$
$488$		0			0
$489$	12.9939			0.587603
$490$		0			0
$491$		−	30.2290i		−	1.36422i	−0.731252	−	0.682108i	$-0.761064\pi$
$491$		−	30.2290i		−	1.36422i	0.731252	−	0.682108i	$-0.238936\pi$
$492$		0			0
$493$	−50.4079			−2.27026
$494$		0			0
$495$	6.68035	+	1.65983i	0.300259	+	0.0746037i
$496$		0			0
$497$			26.9528i			1.20900i
$498$		0			0
$499$			1.36069i			0.0609129i	0.999536	+	0.0304565i	$0.00969609\pi$
$499$			1.36069i			0.0609129i	−0.999536	+	0.0304565i	$0.990304\pi$
$500$		0			0
$501$			12.0000i			0.536120i
$502$		0			0
$503$			16.6803i			0.743740i	0.928285	+	0.371870i	$0.121283\pi$
$503$			16.6803i			0.743740i	−0.928285	+	0.371870i	$0.878717\pi$
$504$		0			0
$505$	−18.0989	−	4.49693i	−0.805390	−	0.200111i
$506$		0			0
$507$	−12.1506			−0.539628
$508$		0			0
$509$		−	14.0144i		−	0.621176i	−0.950545	−	0.310588i	$-0.899474\pi$
$509$		−	14.0144i		−	0.621176i	0.950545	−	0.310588i	$-0.100526\pi$
$510$		0			0
$511$	15.2663			0.675343
$512$		0			0
$513$		−	4.00000i		−	0.176604i
$514$		0			0
$515$	22.4391	+	5.57531i	0.988784	+	0.245677i
$516$		0			0
$517$	−26.7214			−1.17521
$518$		0			0
$519$	18.4391			0.809385
$520$		0			0
$521$	−6.68035			−0.292671			−0.146336	−	0.989235i	$-0.546748\pi$
$521$	−6.68035			−0.292671			−0.146336	+	0.989235i	$0.546748\pi$
$522$		0			0
$523$	26.0410			1.13870			0.569348	−	0.822097i	$-0.307196\pi$
$523$	26.0410			1.13870			0.569348	+	0.822097i	$0.307196\pi$
$524$		0			0
$525$	5.47641	−	10.3402i	0.239010	−	0.451282i
$526$		0			0
$527$			15.5174i			0.675951i
$528$		0			0
$529$	18.3484			0.797757
$530$		0			0
$531$			11.7587i			0.510285i
$532$		0			0
$533$	−9.84324			−0.426358
$534$		0			0
$535$	−7.20394	+	28.9939i	−0.311453	+	1.25351i
$536$		0			0
$537$			1.91548i			0.0826590i
$538$		0			0
$539$		−	4.69019i		−	0.202021i
$540$		0			0
$541$			14.9939i			0.644636i	0.946631	+	0.322318i	$0.104462\pi$
$541$			14.9939i			0.644636i	−0.946631	+	0.322318i	$0.895538\pi$
$542$		0			0
$543$		−	6.99386i		−	0.300135i
$544$		0			0
$545$	−11.5441	−	2.86830i	−0.494495	−	0.122864i
$546$		0			0
$547$	26.1568			1.11838			0.559191	−	0.829039i	$-0.311112\pi$
$547$	26.1568			1.11838			0.559191	+	0.829039i	$0.311112\pi$
$548$		0			0
$549$		−	6.00000i		−	0.256074i
$550$		0			0
$551$	−25.9877			−1.10711
$552$		0			0
$553$		−	35.9467i		−	1.52861i
$554$		0			0
$555$	−1.65983	+	6.68035i	−0.0704557	+	0.283565i
$556$		0			0
$557$	14.7526			0.625087			0.312543	−	0.949903i	$-0.398819\pi$
$557$	14.7526			0.625087			0.312543	+	0.949903i	$0.398819\pi$
$558$		0			0
$559$	1.98771			0.0840713
$560$		0			0
$561$	−23.8843			−1.00840
$562$		0			0
$563$	38.5523			1.62479			0.812394	−	0.583109i	$-0.198164\pi$
$563$	38.5523			1.62479			0.812394	+	0.583109i	$0.198164\pi$
$564$		0			0
$565$	42.8781	+	10.6537i	1.80390	+	0.448204i
$566$		0			0
$567$			2.34017i			0.0982780i
$568$		0			0
$569$	14.6803			0.615432			0.307716	−	0.951478i	$-0.400435\pi$
$569$	14.6803			0.615432			0.307716	+	0.951478i	$0.400435\pi$
$570$		0			0
$571$			13.6742i			0.572248i	0.958193	+	0.286124i	$0.0923669\pi$
$571$			13.6742i			0.572248i	−0.958193	+	0.286124i	$0.907633\pi$
$572$		0			0
$573$	−6.15676			−0.257202
$574$		0			0
$575$	−5.04718	+	9.52973i	−0.210482	+	0.397417i
$576$		0			0
$577$			22.6681i			0.943684i	0.881683	+	0.471842i	$0.156411\pi$
$577$			22.6681i			0.943684i	−0.881683	+	0.471842i	$0.843589\pi$
$578$		0			0
$579$		−	14.5236i		−	0.603580i
$580$		0			0
$581$			4.31351i			0.178955i
$582$		0			0
$583$			8.99386i			0.372487i
$584$		0			0
$585$	−0.496928	+	2.00000i	−0.0205455	+	0.0826898i
$586$		0			0
$587$	37.3607			1.54204			0.771020	−	0.636810i	$-0.219747\pi$
$587$	37.3607			1.54204			0.771020	+	0.636810i	$0.219747\pi$
$588$		0			0
$589$			8.00000i			0.329634i
$590$		0			0
$591$	24.5958			1.01174
$592$		0			0
$593$			15.5897i			0.640192i	0.947385	+	0.320096i	$0.103715\pi$
$593$			15.5897i			0.640192i	−0.947385	+	0.320096i	$0.896285\pi$
$594$		0			0
$595$	−9.78992	+	39.4017i	−0.401348	+	1.61531i
$596$		0			0
$597$	−3.36069			−0.137544
$598$		0			0
$599$	45.3607			1.85339			0.926694	−	0.375817i	$-0.122638\pi$
$599$	45.3607			1.85339			0.926694	+	0.375817i	$0.122638\pi$
$600$		0			0
$601$	−16.5236			−0.674011			−0.337006	−	0.941503i	$-0.609414\pi$
$601$	−16.5236			−0.674011			−0.337006	+	0.941503i	$0.609414\pi$
$602$		0			0
$603$	−6.83710			−0.278428
$604$		0			0
$605$	−0.821503	+	3.30632i	−0.0333988	+	0.134421i
$606$		0			0
$607$		−	30.0554i		−	1.21991i	−0.792435	−	0.609956i	$-0.791187\pi$
$607$		−	30.0554i		−	1.21991i	0.792435	−	0.609956i	$-0.208813\pi$
$608$		0			0
$609$	15.2039			0.616095
$610$		0			0
$611$		−	8.00000i		−	0.323645i
$612$		0			0
$613$	24.2700			0.980257			0.490129	−	0.871650i	$-0.336950\pi$
$613$	24.2700			0.980257			0.490129	+	0.871650i	$0.336950\pi$
$614$		0			0
$615$	5.75872	−	23.1773i	0.232214	−	0.934598i
$616$		0			0
$617$			0.0722347i			0.00290806i	0.999999	+	0.00145403i	$0.000462832\pi$
$617$			0.0722347i			0.00290806i	−0.999999	+	0.00145403i	$0.999537\pi$
$618$		0			0
$619$		−	29.1917i		−	1.17331i	−0.809836	−	0.586656i	$-0.800444\pi$
$619$		−	29.1917i		−	1.17331i	0.809836	−	0.586656i	$-0.199556\pi$
$620$		0			0
$621$		−	2.15676i		−	0.0865476i
$622$		0			0
$623$			14.0410i			0.562542i
$624$		0			0
$625$	14.0472	+	20.6803i	0.561887	+	0.827214i
$626$		0			0
$627$	−12.3135			−0.491754
$628$		0			0
$629$		−	23.8843i		−	0.952329i
$630$		0			0
$631$	3.36069			0.133787			0.0668935	−	0.997760i	$-0.478691\pi$
$631$	3.36069			0.133787			0.0668935	+	0.997760i	$0.478691\pi$
$632$		0			0
$633$			11.2039i			0.445316i
$634$		0			0
$635$	−41.9565	−	10.4247i	−1.66499	−	0.413691i
$636$		0			0
$637$	1.40417			0.0556354
$638$		0			0
$639$	11.5174			0.455623
$640$		0			0
$641$	36.6681			1.44830			0.724151	−	0.689642i	$-0.242232\pi$
$641$	36.6681			1.44830			0.724151	+	0.689642i	$0.242232\pi$
$642$		0			0
$643$	−40.6803			−1.60428			−0.802138	−	0.597139i	$-0.796304\pi$
$643$	−40.6803			−1.60428			−0.802138	+	0.597139i	$0.796304\pi$
$644$		0			0
$645$	−1.16290	+	4.68035i	−0.0457891	+	0.184288i
$646$		0			0
$647$			37.7275i			1.48322i	0.670830	+	0.741611i	$0.265938\pi$
$647$			37.7275i			1.48322i	−0.670830	+	0.741611i	$0.734062\pi$
$648$		0			0
$649$	36.1978			1.42089
$650$		0			0
$651$		−	4.68035i		−	0.183437i
$652$		0			0
$653$	−38.9216			−1.52312			−0.761560	−	0.648094i	$-0.775566\pi$
$653$	−38.9216			−1.52312			−0.761560	+	0.648094i	$0.775566\pi$
$654$		0			0
$655$	16.1568	+	4.01438i	0.631297	+	0.156855i
$656$		0			0
$657$		−	6.52359i		−	0.254510i
$658$		0			0
$659$		−	32.9504i		−	1.28356i	−0.766887	−	0.641782i	$-0.778195\pi$
$659$		−	32.9504i		−	1.28356i	0.766887	−	0.641782i	$-0.221805\pi$
$660$		0			0
$661$			36.3012i			1.41195i	0.708235	+	0.705977i	$0.249492\pi$
$661$			36.3012i			1.41195i	−0.708235	+	0.705977i	$0.750508\pi$
$662$		0			0
$663$		−	7.15061i		−	0.277707i
$664$		0			0
$665$	−5.04718	+	20.3135i	−0.195721	+	0.787724i
$666$		0			0
$667$	−14.0123			−0.542558
$668$		0			0
$669$			6.65368i			0.257246i
$670$		0			0
$671$	−18.4703			−0.713037
$672$		0			0
$673$			37.1917i			1.43363i	0.697262	+	0.716816i	$0.254401\pi$
$673$			37.1917i			1.43363i	−0.697262	+	0.716816i	$0.745599\pi$
$674$		0			0
$675$	−4.41855	−	2.34017i	−0.170070	−	0.0900733i
$676$		0			0
$677$	9.07838			0.348910			0.174455	−	0.984665i	$-0.444184\pi$
$677$	9.07838			0.348910			0.174455	+	0.984665i	$0.444184\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−9.84324			−0.377194
$682$		0			0
$683$	9.84324			0.376641			0.188321	−	0.982108i	$-0.439696\pi$
$683$	9.84324			0.376641			0.188321	+	0.982108i	$0.439696\pi$
$684$		0			0
$685$	7.47641	+	1.85762i	0.285659	+	0.0709760i
$686$		0			0
$687$		−	1.31965i		−	0.0503479i
$688$		0			0
$689$	−2.69263			−0.102581
$690$		0			0
$691$		−	1.67420i		−	0.0636897i	−0.999493	−	0.0318448i	$-0.989862\pi$
$691$		−	1.67420i		−	0.0636897i	0.999493	−	0.0318448i	$-0.0101382\pi$
$692$		0			0
$693$	7.20394			0.273655
$694$		0			0
$695$	−41.6742	−	10.3545i	−1.58079	−	0.392770i
$696$		0			0
$697$			82.8659i			3.13877i
$698$		0			0
$699$		−	2.39803i		−	0.0907019i
$700$		0			0
$701$			8.96719i			0.338686i	0.985557	+	0.169343i	$0.0541646\pi$
$701$			8.96719i			0.338686i	−0.985557	+	0.169343i	$0.945835\pi$
$702$		0			0
$703$		−	12.3135i		−	0.464413i
$704$		0			0
$705$	18.8371	+	4.68035i	0.709446	+	0.176272i
$706$		0			0
$707$	−19.5174			−0.734029
$708$		0			0
$709$		−	25.7152i		−	0.965756i	−0.875688	−	0.482878i	$-0.839591\pi$
$709$		−	25.7152i		−	0.965756i	0.875688	−	0.482878i	$-0.160409\pi$
$710$		0			0
$711$	−15.3607			−0.576071
$712$		0			0
$713$			4.31351i			0.161542i
$714$		0			0
$715$	6.15676	+	1.52973i	0.230250	+	0.0572088i
$716$		0			0
$717$	−6.15676			−0.229928
$718$		0			0
$719$	18.4079			0.686498			0.343249	−	0.939244i	$-0.388473\pi$
$719$	18.4079			0.686498			0.343249	+	0.939244i	$0.388473\pi$
$720$		0			0
$721$	24.1978			0.901173
$722$		0			0
$723$	0.639308			0.0237761
$724$		0			0
$725$	−15.2039	+	28.7070i	−0.564660	+	1.06615i
$726$		0			0
$727$		−	42.3402i		−	1.57031i	−0.619299	−	0.785155i	$-0.712583\pi$
$727$		−	42.3402i		−	1.57031i	0.619299	−	0.785155i	$-0.287417\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$		−	16.7337i		−	0.618917i
$732$		0			0
$733$	4.75258			0.175541			0.0877703	−	0.996141i	$-0.472026\pi$
$733$	4.75258			0.175541			0.0877703	+	0.996141i	$0.472026\pi$
$734$		0			0
$735$	−0.821503	+	3.30632i	−0.0303016	+	0.121956i
$736$		0			0
$737$			21.0472i			0.775283i
$738$		0			0
$739$		−	21.0472i		−	0.774233i	−0.922031	−	0.387117i	$-0.873471\pi$
$739$		−	21.0472i		−	0.774233i	0.922031	−	0.387117i	$-0.126529\pi$
$740$		0			0
$741$		−	3.68649i		−	0.135427i
$742$		0			0
$743$		−	49.5585i		−	1.81812i	−0.416660	−	0.909062i	$-0.636800\pi$
$743$		−	49.5585i		−	1.81812i	0.416660	−	0.909062i	$-0.363200\pi$
$744$		0			0
$745$	21.3028	+	5.29299i	0.780475	+	0.193920i
$746$		0			0
$747$	1.84324			0.0674408
$748$		0			0
$749$			31.2663i			1.14245i
$750$		0			0
$751$	−41.6619			−1.52026			−0.760132	−	0.649768i	$-0.774866\pi$
$751$	−41.6619			−1.52026			−0.760132	+	0.649768i	$0.774866\pi$
$752$		0			0
$753$		−	12.9216i		−	0.470890i
$754$		0			0
$755$	12.7649	−	51.3751i	0.464561	−	1.86973i
$756$		0			0
$757$	−21.9688			−0.798470			−0.399235	−	0.916849i	$-0.630724\pi$
$757$	−21.9688			−0.798470			−0.399235	+	0.916849i	$0.630724\pi$
$758$		0			0
$759$	−6.63931			−0.240992
$760$		0			0
$761$	11.3074			0.409892			0.204946	−	0.978773i	$-0.434298\pi$
$761$	11.3074			0.409892			0.204946	+	0.978773i	$0.434298\pi$
$762$		0			0
$763$	−12.4489			−0.450681
$764$		0			0
$765$	16.8371	+	4.18342i	0.608747	+	0.151252i
$766$		0			0
$767$			10.8371i			0.391305i
$768$		0			0
$769$	−6.19779			−0.223498			−0.111749	−	0.993736i	$-0.535645\pi$
$769$	−6.19779			−0.223498			−0.111749	+	0.993736i	$0.535645\pi$
$770$		0			0
$771$		−	14.9627i		−	0.538867i
$772$		0			0
$773$	−30.4391			−1.09482			−0.547409	−	0.836865i	$-0.684386\pi$
$773$	−30.4391			−1.09482			−0.547409	+	0.836865i	$0.684386\pi$
$774$		0			0
$775$	8.83710	+	4.68035i	0.317438	+	0.168123i
$776$		0			0
$777$			7.20394i			0.258440i
$778$		0			0
$779$			42.7214i			1.53065i
$780$		0			0
$781$		−	35.4551i		−	1.26868i
$782$		0			0
$783$		−	6.49693i		−	0.232181i
$784$		0			0
$785$	−5.97334	+	24.0410i	−0.213198	+	0.858061i
$786$		0			0
$787$	36.9939			1.31869			0.659344	−	0.751841i	$-0.270834\pi$
$787$	36.9939			1.31869			0.659344	+	0.751841i	$0.270834\pi$
$788$		0			0
$789$			13.3607i			0.475653i
$790$		0			0
$791$	46.2388			1.64406
$792$		0			0
$793$		−	5.52973i		−	0.196367i
$794$		0			0
$795$	1.57531	−	6.34017i	0.0558704	−	0.224863i
$796$		0			0
$797$	−30.7526			−1.08931			−0.544656	−	0.838659i	$-0.683340\pi$
$797$	−30.7526			−1.08931			−0.544656	+	0.838659i	$0.683340\pi$
$798$		0			0
$799$	−67.3484			−2.38262
$800$		0			0
$801$	6.00000			0.212000
$802$		0			0
$803$	−20.0821			−0.708681
$804$		0			0
$805$	−2.72138	+	10.9528i	−0.0959162	+	0.386036i
$806$		0			0
$807$			27.5441i			0.969599i
$808$		0			0
$809$	−5.31965			−0.187029			−0.0935145	−	0.995618i	$-0.529810\pi$
$809$	−5.31965			−0.187029			−0.0935145	+	0.995618i	$0.529810\pi$
$810$		0			0
$811$			49.9253i			1.75312i	0.481297	+	0.876558i	$0.340166\pi$
$811$			49.9253i			1.75312i	−0.481297	+	0.876558i	$0.659834\pi$
$812$		0			0
$813$	−10.0000			−0.350715
$814$		0			0
$815$	−7.00614	+	28.1978i	−0.245414	+	0.987726i
$816$		0			0
$817$		−	8.62702i		−	0.301821i
$818$		0			0
$819$			2.15676i			0.0753631i
$820$		0			0
$821$		−	44.1711i		−	1.54158i	−0.637087	−	0.770792i	$-0.719861\pi$
$821$		−	44.1711i		−	1.54158i	0.637087	−	0.770792i	$-0.280139\pi$
$822$		0			0
$823$			19.0738i			0.664872i	0.943126	+	0.332436i	$0.107871\pi$
$823$			19.0738i			0.664872i	−0.943126	+	0.332436i	$0.892129\pi$
$824$		0			0
$825$	−7.20394	+	13.6020i	−0.250809	+	0.473560i
$826$		0			0
$827$	−43.0349			−1.49647			−0.748235	−	0.663434i	$-0.769098\pi$
$827$	−43.0349			−1.49647			−0.748235	+	0.663434i	$0.769098\pi$
$828$		0			0
$829$		−	36.0410i		−	1.25176i	−0.779921	−	0.625878i	$-0.784741\pi$
$829$		−	36.0410i		−	1.25176i	0.779921	−	0.625878i	$-0.215259\pi$
$830$		0			0
$831$	4.12556			0.143114
$832$		0			0
$833$		−	11.8211i		−	0.409577i
$834$		0			0
$835$	−26.0410	−	6.47027i	−0.901187	−	0.223913i
$836$		0			0
$837$	−2.00000			−0.0691301
$838$		0			0
$839$	43.0349			1.48573			0.742865	−	0.669441i	$-0.233467\pi$
$839$	43.0349			1.48573			0.742865	+	0.669441i	$0.233467\pi$
$840$		0			0
$841$	−13.2101			−0.455520
$842$		0			0
$843$	11.3607			0.391283
$844$		0			0
$845$	6.55148	−	26.3679i	0.225378	−	0.907083i
$846$		0			0
$847$			3.56547i			0.122511i
$848$		0			0
$849$	−10.5236			−0.361169
$850$		0			0
$851$		−	6.63931i		−	0.227593i
$852$		0			0
$853$	−47.1605			−1.61474			−0.807372	−	0.590043i	$-0.799111\pi$
$853$	−47.1605			−1.61474			−0.807372	+	0.590043i	$0.799111\pi$
$854$		0			0
$855$	8.68035	+	2.15676i	0.296862	+	0.0737595i
$856$		0			0
$857$			12.8059i			0.437441i	0.975788	+	0.218721i	$0.0701884\pi$
$857$			12.8059i			0.437441i	−0.975788	+	0.218721i	$0.929812\pi$
$858$		0			0
$859$		−	14.4703i		−	0.493719i	−0.969051	−	0.246860i	$-0.920601\pi$
$859$		−	14.4703i		−	0.493719i	0.969051	−	0.246860i	$-0.0793986\pi$
$860$		0			0
$861$		−	24.9939i		−	0.851788i
$862$		0			0
$863$		−	24.3135i		−	0.827642i	−0.910358	−	0.413821i	$-0.864194\pi$
$863$		−	24.3135i		−	0.827642i	0.910358	−	0.413821i	$-0.135806\pi$
$864$		0			0
$865$	−9.94214	+	40.0144i	−0.338043	+	1.36053i
$866$		0			0
$867$	−43.1978			−1.46707
$868$		0			0
$869$			47.2860i			1.60407i
$870$		0			0
$871$	−6.30122			−0.213509
$872$		0			0
$873$		0			0
$874$		0			0
$875$	19.4863	+	17.4596i	0.658756	+	0.590242i
$876$		0			0
$877$	44.4391			1.50060			0.750300	−	0.661097i	$-0.229909\pi$
$877$	44.4391			1.50060			0.750300	+	0.661097i	$0.229909\pi$
$878$		0			0
$879$	−32.5958			−1.09943
$880$		0			0
$881$	6.62702			0.223270			0.111635	−	0.993749i	$-0.464391\pi$
$881$	6.62702			0.223270			0.111635	+	0.993749i	$0.464391\pi$
$882$		0			0
$883$	7.31965			0.246326			0.123163	−	0.992386i	$-0.460696\pi$
$883$	7.31965			0.246326			0.123163	+	0.992386i	$0.460696\pi$
$884$		0			0
$885$	−25.5174	−	6.34017i	−0.857760	−	0.213123i
$886$		0			0
$887$		−	12.0000i		−	0.402921i	−0.979497	−	0.201460i	$-0.935431\pi$
$887$		−	12.0000i		−	0.402921i	0.979497	−	0.201460i	$-0.0645687\pi$
$888$		0			0
$889$	−45.2450			−1.51747
$890$		0			0
$891$		−	3.07838i		−	0.103130i
$892$		0			0
$893$	−34.7214			−1.16191
$894$		0			0
$895$	−4.15676	−	1.03281i	−0.138945	−	0.0345229i
$896$		0			0
$897$		−	1.98771i		−	0.0663678i
$898$		0			0
$899$			12.9939i			0.433369i
$900$		0			0
$901$			22.6681i			0.755183i
$902$		0			0
$903$			5.04718i			0.167960i
$904$		0			0
$905$	15.1773	+	3.77101i	0.504510	+	0.125353i
$906$		0			0
$907$	−4.48255			−0.148841			−0.0744204	−	0.997227i	$-0.523711\pi$
$907$	−4.48255			−0.148841			−0.0744204	+	0.997227i	$0.523711\pi$
$908$		0			0
$909$			8.34017i			0.276626i
$910$		0			0
$911$	−8.73367			−0.289359			−0.144680	−	0.989479i	$-0.546215\pi$
$911$	−8.73367			−0.289359			−0.144680	+	0.989479i	$0.546215\pi$
$912$		0			0
$913$		−	5.67420i		−	0.187789i
$914$		0			0
$915$	13.0205	+	3.23513i	0.430445	+	0.106950i
$916$		0			0
$917$	17.4231			0.575361
$918$		0			0
$919$	−18.7337			−0.617967			−0.308983	−	0.951067i	$-0.599989\pi$
$919$	−18.7337			−0.617967			−0.308983	+	0.951067i	$0.599989\pi$
$920$		0			0
$921$	29.9877			0.988129
$922$		0			0
$923$	10.6147			0.349388
$924$		0			0
$925$	−13.6020	−	7.20394i	−0.447230	−	0.236864i
$926$		0			0
$927$		−	10.3402i		−	0.339616i
$928$		0			0
$929$	20.0410			0.657525			0.328763	−	0.944413i	$-0.393368\pi$
$929$	20.0410			0.657525			0.328763	+	0.944413i	$0.393368\pi$
$930$		0			0
$931$		−	6.09436i		−	0.199735i
$932$		0			0
$933$	−27.2039			−0.890617
$934$		0			0
$935$	12.8781	−	51.8310i	0.421160	−	1.69505i
$936$		0			0
$937$			45.0472i			1.47163i	0.677184	+	0.735814i	$0.263200\pi$
$937$			45.0472i			1.47163i	−0.677184	+	0.735814i	$0.736800\pi$
$938$		0			0
$939$		−	26.8371i		−	0.875796i
$940$		0			0
$941$			19.3751i			0.631609i	0.948824	+	0.315805i	$0.102274\pi$
$941$			19.3751i			0.631609i	−0.948824	+	0.315805i	$0.897726\pi$
$942$		0			0
$943$			23.0349i			0.750119i
$944$		0			0
$945$	−5.07838	−	1.26180i	−0.165200	−	0.0410462i
$946$		0			0
$947$	−6.95282			−0.225936			−0.112968	−	0.993599i	$-0.536036\pi$
$947$	−6.95282			−0.225936			−0.112968	+	0.993599i	$0.536036\pi$
$948$		0			0
$949$		−	6.01229i		−	0.195167i
$950$		0			0
$951$	24.5958			0.797574
$952$		0			0
$953$		−	29.2885i		−	0.948746i	−0.880324	−	0.474373i	$-0.842675\pi$
$953$		−	29.2885i		−	0.948746i	0.880324	−	0.474373i	$-0.157325\pi$
$954$		0			0
$955$	3.31965	−	13.3607i	0.107421	−	0.432342i
$956$		0			0
$957$	−20.0000			−0.646508
$958$		0			0
$959$	8.06239			0.260348
$960$		0			0
$961$	−27.0000			−0.870968
$962$		0			0
$963$	13.3607			0.430542
$964$		0			0
$965$	31.5174	+	7.83096i	1.01458	+	0.252087i
$966$		0			0
$967$			47.3874i			1.52387i	0.647651	+	0.761937i	$0.275752\pi$
$967$			47.3874i			1.52387i	−0.647651	+	0.761937i	$0.724248\pi$
$968$		0			0
$969$	−31.0349			−0.996984
$970$		0			0
$971$		−	2.59583i		−	0.0833040i	−0.999132	−	0.0416520i	$-0.986738\pi$
$971$		−	2.59583i		−	0.0833040i	0.999132	−	0.0416520i	$-0.0132621\pi$
$972$		0			0
$973$	−44.9405			−1.44073
$974$		0			0
$975$	−4.07223	−	2.15676i	−0.130416	−	0.0690715i
$976$		0			0
$977$			28.3234i			0.906144i	0.891474	+	0.453072i	$0.149672\pi$
$977$			28.3234i			0.906144i	−0.891474	+	0.453072i	$0.850328\pi$
$978$		0			0
$979$		−	18.4703i		−	0.590312i
$980$		0			0
$981$			5.31965i			0.169843i
$982$		0			0
$983$		−	1.16290i		−	0.0370907i	−0.999828	−	0.0185454i	$-0.994096\pi$
$983$		−	1.16290i		−	0.0370907i	0.999828	−	0.0185454i	$-0.00590351\pi$
$984$		0			0
$985$	−13.2618	+	53.3751i	−0.422556	+	1.70067i
$986$		0			0
$987$	20.3135			0.646586
$988$		0			0
$989$		−	4.65159i		−	0.147912i
$990$		0			0
$991$	12.9528			0.411460			0.205730	−	0.978609i	$-0.434043\pi$
$991$	12.9528			0.411460			0.205730	+	0.978609i	$0.434043\pi$
$992$		0			0
$993$		−	12.3135i		−	0.390757i
$994$		0			0
$995$	1.81205	−	7.29299i	0.0574458	−	0.231203i
$996$		0			0
$997$	−19.0784			−0.604218			−0.302109	−	0.953273i	$-0.597691\pi$
$997$	−19.0784			−0.604218			−0.302109	+	0.953273i	$0.597691\pi$
$998$		0			0
$999$	3.07838			0.0973956

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3840.2.d.bl.2689.4		6
4.3	odd	2		3840.2.d.bj.2689.4		6
5.4	even	2		3840.2.d.bi.2689.4		6
8.3	odd	2		3840.2.d.bk.2689.3		6
8.5	even	2		3840.2.d.bi.2689.3		6
16.3	odd	4		1920.2.f.p.769.3	yes	6
16.5	even	4		1920.2.f.n.769.1	yes	6
16.11	odd	4		1920.2.f.m.769.4	yes	6
16.13	even	4		1920.2.f.o.769.6	yes	6
20.19	odd	2		3840.2.d.bk.2689.4		6
40.19	odd	2		3840.2.d.bj.2689.3		6
40.29	even	2	inner	3840.2.d.bl.2689.3		6
80.3	even	4		9600.2.a.dt.1.3		3
80.13	odd	4		9600.2.a.dq.1.1		3
80.19	odd	4		1920.2.f.p.769.6	yes	6
80.27	even	4		9600.2.a.du.1.1		3
80.29	even	4		1920.2.f.o.769.3	yes	6
80.37	odd	4		9600.2.a.dp.1.3		3
80.43	even	4		9600.2.a.do.1.3		3
80.53	odd	4		9600.2.a.dv.1.1		3
80.59	odd	4		1920.2.f.m.769.1	✓	6
80.67	even	4		9600.2.a.dr.1.1		3
80.69	even	4		1920.2.f.n.769.4	yes	6
80.77	odd	4		9600.2.a.ds.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.f.m.769.1	✓	6	80.59	odd	4
1920.2.f.m.769.4	yes	6	16.11	odd	4
1920.2.f.n.769.1	yes	6	16.5	even	4
1920.2.f.n.769.4	yes	6	80.69	even	4
1920.2.f.o.769.3	yes	6	80.29	even	4
1920.2.f.o.769.6	yes	6	16.13	even	4
1920.2.f.p.769.3	yes	6	16.3	odd	4
1920.2.f.p.769.6	yes	6	80.19	odd	4
3840.2.d.bi.2689.3		6	8.5	even	2
3840.2.d.bi.2689.4		6	5.4	even	2
3840.2.d.bj.2689.3		6	40.19	odd	2
3840.2.d.bj.2689.4		6	4.3	odd	2
3840.2.d.bk.2689.3		6	8.3	odd	2
3840.2.d.bk.2689.4		6	20.19	odd	2
3840.2.d.bl.2689.3		6	40.29	even	2	inner
3840.2.d.bl.2689.4		6	1.1	even	1	trivial
9600.2.a.do.1.3		3	80.43	even	4
9600.2.a.dp.1.3		3	80.37	odd	4
9600.2.a.dq.1.1		3	80.13	odd	4
9600.2.a.dr.1.1		3	80.67	even	4
9600.2.a.ds.1.3		3	80.77	odd	4
9600.2.a.dt.1.3		3	80.3	even	4
9600.2.a.du.1.1		3	80.27	even	4
9600.2.a.dv.1.1		3	80.53	odd	4

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:	\( N \)	\(=\)	\( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3840.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +(-0.539189 + 2.17009i) q^{5} +2.34017i q^{7} +1.00000 q^{9} -3.07838i q^{11} +0.921622 q^{13} +(-0.539189 + 2.17009i) q^{15} -7.75872i q^{17} -4.00000i q^{19} +2.34017i q^{21} -2.15676i q^{23} +(-4.41855 - 2.34017i) q^{25} +1.00000 q^{27} -6.49693i q^{29} -2.00000 q^{31} -3.07838i q^{33} +(-5.07838 - 1.26180i) q^{35} +3.07838 q^{37} +0.921622 q^{39} -10.6803 q^{41} +2.15676 q^{43} +(-0.539189 + 2.17009i) q^{45} -8.68035i q^{47} +1.52359 q^{49} -7.75872i q^{51} -2.92162 q^{53} +(6.68035 + 1.65983i) q^{55} -4.00000i q^{57} +11.7587i q^{59} -6.00000i q^{61} +2.34017i q^{63} +(-0.496928 + 2.00000i) q^{65} -6.83710 q^{67} -2.15676i q^{69} +11.5174 q^{71} -6.52359i q^{73} +(-4.41855 - 2.34017i) q^{75} +7.20394 q^{77} -15.3607 q^{79} +1.00000 q^{81} +1.84324 q^{83} +(16.8371 + 4.18342i) q^{85} -6.49693i q^{87} +6.00000 q^{89} +2.15676i q^{91} -2.00000 q^{93} +(8.68035 + 2.15676i) q^{95} -3.07838i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} + 6 q^{9} + 12 q^{13} + 2 q^{25} + 6 q^{27} - 12 q^{31} - 24 q^{35} + 12 q^{37} + 12 q^{39} - 20 q^{41} - 22 q^{49} - 24 q^{53} - 4 q^{55} + 32 q^{65} + 16 q^{67} - 32 q^{71} + 2 q^{75} - 32 q^{77}+ \cdots + 8 q^{95}+O(q^{100}) \) 6 * q + 6 * q^3 + 6 * q^9 + 12 * q^13 + 2 * q^25 + 6 * q^27 - 12 * q^31 - 24 * q^35 + 12 * q^37 + 12 * q^39 - 20 * q^41 - 22 * q^49 - 24 * q^53 - 4 * q^55 + 32 * q^65 + 16 * q^67 - 32 * q^71 + 2 * q^75 - 32 * q^77 - 4 * q^79 + 6 * q^81 + 24 * q^83 + 44 * q^85 + 36 * q^89 - 12 * q^93 + 8 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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