LMFDB - Embedded newform 4032.2.v.c.3599.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(1583,4032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4032, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 2, 0]))

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

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

chi := DirichletCharacter("4032.1583");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.v (of order $4$, degree $2$, 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:	$32.1956820950$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	12.0.653473922154496.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4x^{10} + 13x^{8} - 28x^{6} + 52x^{4} - 64x^{2} + 64 $ x^12 - 4x^10 + 13x^8 - 28x^6 + 52x^4 - 64*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 1008)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3599.2
Root			$-1.16947 + 0.795191i$ of defining polynomial
Character	$\chi$	$=$	4032.3599
Dual form			4032.2.v.c.1583.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.41421 + 1.41421i) q^{5} +1.00000 q^{7} +O(q^{10})$	$q+(-1.41421 + 1.41421i) q^{5} +1.00000 q^{7} +(0.748567 + 0.748567i) q^{11} +(4.24914 - 4.24914i) q^{13} -7.50633i q^{17} -4.59498i q^{23} +1.00000i q^{25} +(-5.26063 - 5.26063i) q^{29} +8.49828i q^{31} +(-1.41421 + 1.41421i) q^{35} +(-1.00000 - 1.00000i) q^{37} -0.978956 q^{41} +(-6.80605 + 6.80605i) q^{43} -6.36153 q^{47} +1.00000 q^{49} +(-8.17197 + 8.17197i) q^{53} -2.11727 q^{55} +(-4.51206 - 4.51206i) q^{59} +(0.249141 - 0.249141i) q^{61} +12.0184i q^{65} +(-0.750859 - 0.750859i) q^{67} -9.62521i q^{71} -8.61555i q^{73} +(0.748567 + 0.748567i) q^{77} -16.9966i q^{79} +(2.47609 - 2.47609i) q^{83} +(10.6155 + 10.6155i) q^{85} +2.55415 q^{89} +(4.24914 - 4.24914i) q^{91} +18.1104 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{7}+O(q^{10}) $ 12 * q + 12 * q^7	$ 12 q + 12 q^{7} + 16 q^{13} - 12 q^{37} + 20 q^{43} + 12 q^{49} - 32 q^{55} - 32 q^{61} - 44 q^{67} + 64 q^{85} + 16 q^{91} - 56 q^{97}+O(q^{100}) $ 12 * q + 12 * q^7 + 16 * q^13 - 12 * q^37 + 20 * q^43 + 12 * q^49 - 32 * q^55 - 32 * q^61 - 44 * q^67 + 64 * q^85 + 16 * q^91 - 56 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$-1$	$1$	$-1$	$e\left(\frac{1}{4}\right)$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.41421	+	1.41421i	−0.632456	+	0.632456i	−0.948683	−	0.316228i	$-0.897584\pi$
$5$	−1.41421	+	1.41421i	−0.632456	+	0.632456i	0.316228	+	0.948683i	$0.397584\pi$
$6$		0			0
$7$	1.00000			0.377964
$7$	1.00000			0.377964
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.748567	+	0.748567i	0.225701	+	0.225701i	0.810894	−	0.585193i	$-0.198981\pi$
$11$	0.748567	+	0.748567i	0.225701	+	0.225701i	−0.585193	+	0.810894i	$0.698981\pi$
$12$		0			0
$13$	4.24914	−	4.24914i	1.17850	−	1.17850i	0.198373	−	0.980127i	$-0.436434\pi$
$13$	4.24914	−	4.24914i	1.17850	−	1.17850i	0.980127	−	0.198373i	$-0.0635657\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	7.50633i		−	1.82055i	−0.414003	−	0.910276i	$-0.635870\pi$
$17$		−	7.50633i		−	1.82055i	0.414003	−	0.910276i	$-0.364130\pi$
$18$		0			0
$19$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$19$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	4.59498i		−	0.958119i	−0.877782	−	0.479060i	$-0.840978\pi$
$23$		−	4.59498i		−	0.958119i	0.877782	−	0.479060i	$-0.159022\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−5.26063	−	5.26063i	−0.976874	−	0.976874i	0.0228649	−	0.999739i	$-0.492721\pi$
$29$	−5.26063	−	5.26063i	−0.976874	−	0.976874i	−0.999739	+	0.0228649i	$0.992721\pi$
$30$		0			0
$31$			8.49828i			1.52634i	0.646200	+	0.763168i	$0.276357\pi$
$31$			8.49828i			1.52634i	−0.646200	+	0.763168i	$0.723643\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.41421	+	1.41421i	−0.239046	+	0.239046i
$36$		0			0
$37$	−1.00000	−	1.00000i	−0.164399	−	0.164399i	0.620113	−	0.784512i	$-0.287087\pi$
$37$	−1.00000	−	1.00000i	−0.164399	−	0.164399i	−0.784512	+	0.620113i	$0.787087\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.978956			−0.152887			−0.0764436	−	0.997074i	$-0.524357\pi$
$41$	−0.978956			−0.152887			−0.0764436	+	0.997074i	$0.524357\pi$
$42$		0			0
$43$	−6.80605	+	6.80605i	−1.03791	+	1.03791i	−0.0386612	+	0.999252i	$0.512309\pi$
$43$	−6.80605	+	6.80605i	−1.03791	+	1.03791i	−0.999252	+	0.0386612i	$0.987691\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−6.36153			−0.927925			−0.463962	−	0.885855i	$-0.653573\pi$
$47$	−6.36153			−0.927925			−0.463962	+	0.885855i	$0.653573\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−8.17197	+	8.17197i	−1.12251	+	1.12251i	−0.131143	+	0.991363i	$0.541865\pi$
$53$	−8.17197	+	8.17197i	−1.12251	+	1.12251i	−0.991363	+	0.131143i	$0.958135\pi$
$54$		0			0
$55$	−2.11727			−0.285492
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.51206	−	4.51206i	−0.587420	−	0.587420i	0.349512	−	0.936932i	$-0.386347\pi$
$59$	−4.51206	−	4.51206i	−0.587420	−	0.587420i	−0.936932	+	0.349512i	$0.886347\pi$
$60$		0			0
$61$	0.249141	−	0.249141i	0.0318992	−	0.0318992i	−0.690977	−	0.722876i	$-0.742820\pi$
$61$	0.249141	−	0.249141i	0.0318992	−	0.0318992i	0.722876	+	0.690977i	$0.242820\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			12.0184i			1.49070i
$66$		0			0
$67$	−0.750859	−	0.750859i	−0.0917321	−	0.0917321i	0.659752	−	0.751484i	$-0.270661\pi$
$67$	−0.750859	−	0.750859i	−0.0917321	−	0.0917321i	−0.751484	+	0.659752i	$0.770661\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	9.62521i		−	1.14230i	−0.820845	−	0.571151i	$-0.806497\pi$
$71$		−	9.62521i		−	1.14230i	0.820845	−	0.571151i	$-0.193503\pi$
$72$		0			0
$73$		−	8.61555i		−	1.00837i	−0.863595	−	0.504187i	$-0.831792\pi$
$73$		−	8.61555i		−	1.00837i	0.863595	−	0.504187i	$-0.168208\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.748567	+	0.748567i	0.0853071	+	0.0853071i
$78$		0			0
$79$		−	16.9966i		−	1.91226i	−0.292938	−	0.956131i	$-0.594633\pi$
$79$		−	16.9966i		−	1.91226i	0.292938	−	0.956131i	$-0.405367\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2.47609	−	2.47609i	0.271786	−	0.271786i	−0.558033	−	0.829819i	$-0.688444\pi$
$83$	2.47609	−	2.47609i	0.271786	−	0.271786i	0.829819	+	0.558033i	$0.188444\pi$
$84$		0			0
$85$	10.6155	+	10.6155i	1.15142	+	1.15142i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	2.55415			0.270739			0.135370	−	0.990795i	$-0.456778\pi$
$89$	2.55415			0.270739			0.135370	+	0.990795i	$0.456778\pi$
$90$		0			0
$91$	4.24914	−	4.24914i	0.445431	−	0.445431i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	18.1104			1.83883			0.919416	−	0.393287i	$-0.128662\pi$
$97$	18.1104			1.83883			0.919416	+	0.393287i	$0.128662\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−4.59498	+	4.59498i	−0.457217	+	0.457217i	−0.897741	−	0.440524i	$-0.854793\pi$
$101$	−4.59498	+	4.59498i	−0.457217	+	0.457217i	0.440524	+	0.897741i	$0.354793\pi$
$102$		0			0
$103$	−15.1138			−1.48921			−0.744605	−	0.667505i	$-0.767362\pi$
$103$	−15.1138			−1.48921			−0.744605	+	0.667505i	$0.767362\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	5.34355	+	5.34355i	0.516580	+	0.516580i	0.916535	−	0.399955i	$-0.130974\pi$
$107$	5.34355	+	5.34355i	0.516580	+	0.516580i	−0.399955	+	0.916535i	$0.630974\pi$
$108$		0			0
$109$	1.00000	−	1.00000i	0.0957826	−	0.0957826i	−0.657592	−	0.753374i	$-0.728425\pi$
$109$	1.00000	−	1.00000i	0.0957826	−	0.0957826i	0.753374	+	0.657592i	$0.228425\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	9.10704i		−	0.856718i	−0.903609	−	0.428359i	$-0.859092\pi$
$113$		−	9.10704i		−	0.856718i	0.903609	−	0.428359i	$-0.140908\pi$
$114$		0			0
$115$	6.49828	+	6.49828i	0.605968	+	0.605968i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	7.50633i		−	0.688104i
$120$		0			0
$121$		−	9.87930i		−	0.898118i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−8.48528	−	8.48528i	−0.758947	−	0.758947i
$126$		0			0
$127$		−	16.3810i		−	1.45358i	−0.686860	−	0.726790i	$-0.741011\pi$
$127$		−	16.3810i		−	1.45358i	0.686860	−	0.726790i	$-0.258989\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.18077	+	3.18077i	−0.277905	+	0.277905i	−0.832272	−	0.554367i	$-0.812960\pi$
$131$	−3.18077	+	3.18077i	−0.277905	+	0.277905i	0.554367	+	0.832272i	$0.312960\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	0.248759			0.0212530			0.0106265	−	0.999944i	$-0.496617\pi$
$137$	0.248759			0.0212530			0.0106265	+	0.999944i	$0.496617\pi$
$138$		0			0
$139$	10.9966	−	10.9966i	0.932716	−	0.932716i	−0.0651587	−	0.997875i	$-0.520755\pi$
$139$	10.9966	−	10.9966i	0.932716	−	0.932716i	0.997875	+	0.0651587i	$0.0207554\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	6.36153			0.531978
$144$		0			0
$145$	14.8793			1.23566
$146$		0			0
$147$		0			0
$148$		0			0
$149$	15.7819	−	15.7819i	1.29290	−	1.29290i	0.359918	−	0.932984i	$-0.382805\pi$
$149$	15.7819	−	15.7819i	1.29290	−	1.29290i	0.932984	−	0.359918i	$-0.117195\pi$
$150$		0			0
$151$	−6.73281			−0.547909			−0.273954	−	0.961743i	$-0.588332\pi$
$151$	−6.73281			−0.547909			−0.273954	+	0.961743i	$0.588332\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−12.0184	−	12.0184i	−0.965340	−	0.965340i
$156$		0			0
$157$	−8.86469	+	8.86469i	−0.707479	+	0.707479i	−0.966004	−	0.258525i	$-0.916763\pi$
$157$	−8.86469	+	8.86469i	−0.707479	+	0.707479i	0.258525	+	0.966004i	$0.416763\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	4.59498i		−	0.362135i
$162$		0			0
$163$	9.98195	+	9.98195i	0.781847	+	0.781847i	0.980142	−	0.198295i	$-0.0635405\pi$
$163$	9.98195	+	9.98195i	0.781847	+	0.781847i	−0.198295	+	0.980142i	$0.563541\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	21.2083i		−	1.64115i	−0.571538	−	0.820575i	$-0.693653\pi$
$167$		−	21.2083i		−	1.64115i	0.571538	−	0.820575i	$-0.306347\pi$
$168$		0			0
$169$		−	23.1104i		−	1.77772i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	8.56820	+	8.56820i	0.651428	+	0.651428i	0.953337	−	0.301909i	$-0.0976237\pi$
$173$	8.56820	+	8.56820i	0.651428	+	0.651428i	−0.301909	+	0.953337i	$0.597624\pi$
$174$		0			0
$175$			1.00000i			0.0755929i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1.64460	+	1.64460i	−0.122923	+	0.122923i	−0.765892	−	0.642969i	$-0.777702\pi$
$179$	−1.64460	+	1.64460i	−0.122923	+	0.122923i	0.642969	+	0.765892i	$0.277702\pi$
$180$		0			0
$181$	−6.13187	−	6.13187i	−0.455779	−	0.455779i	0.441488	−	0.897267i	$-0.354451\pi$
$181$	−6.13187	−	6.13187i	−0.455779	−	0.455779i	−0.897267	+	0.441488i	$0.854451\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.82843			0.207950
$186$		0			0
$187$	5.61899	−	5.61899i	0.410901	−	0.410901i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−18.9030			−1.36777			−0.683885	−	0.729590i	$-0.739711\pi$
$191$	−18.9030			−1.36777			−0.683885	+	0.729590i	$0.739711\pi$
$192$		0			0
$193$	−2.00000			−0.143963			−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000			−0.143963			−0.0719816	+	0.997406i	$0.522932\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−11.7880	+	11.7880i	−0.839860	+	0.839860i	−0.988840	−	0.148980i	$-0.952401\pi$
$197$	−11.7880	+	11.7880i	−0.839860	+	0.839860i	0.148980	+	0.988840i	$0.452401\pi$
$198$		0			0
$199$	1.61899			0.114767			0.0573834	−	0.998352i	$-0.481724\pi$
$199$	1.61899			0.114767			0.0573834	+	0.998352i	$0.481724\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−5.26063	−	5.26063i	−0.369224	−	0.369224i
$204$		0			0
$205$	1.38445	−	1.38445i	0.0966944	−	0.0966944i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−7.69223	−	7.69223i	−0.529555	−	0.529555i	0.390885	−	0.920440i	$-0.372169\pi$
$211$	−7.69223	−	7.69223i	−0.529555	−	0.529555i	−0.920440	+	0.390885i	$0.872169\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	19.2504i		−	1.31287i
$216$		0			0
$217$			8.49828i			0.576901i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−31.8954	−	31.8954i	−2.14552	−	2.14552i
$222$		0			0
$223$			2.85008i			0.190855i	0.995436	+	0.0954277i	$0.0304219\pi$
$223$			2.85008i			0.190855i	−0.995436	+	0.0954277i	$0.969578\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−13.5155	+	13.5155i	−0.897056	+	0.897056i	−0.995175	−	0.0981183i	$-0.968718\pi$
$227$	−13.5155	+	13.5155i	−0.897056	+	0.897056i	0.0981183	+	0.995175i	$0.468718\pi$
$228$		0			0
$229$	5.86813	+	5.86813i	0.387777	+	0.387777i	0.873894	−	0.486117i	$-0.161587\pi$
$229$	5.86813	+	5.86813i	0.387777	+	0.387777i	−0.486117	+	0.873894i	$0.661587\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	19.8819			1.30251			0.651254	−	0.758860i	$-0.274243\pi$
$233$	19.8819			1.30251			0.651254	+	0.758860i	$0.274243\pi$
$234$		0			0
$235$	8.99656	−	8.99656i	0.586871	−	0.586871i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.30577			0.0844634			0.0422317	−	0.999108i	$-0.486553\pi$
$239$	1.30577			0.0844634			0.0422317	+	0.999108i	$0.486553\pi$
$240$		0			0
$241$	3.88273			0.250109			0.125054	−	0.992150i	$-0.460089\pi$
$241$	3.88273			0.250109			0.125054	+	0.992150i	$0.460089\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.41421	+	1.41421i	−0.0903508	+	0.0903508i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	13.6814	+	13.6814i	0.863560	+	0.863560i	0.991750	−	0.128190i	$-0.0409167\pi$
$251$	13.6814	+	13.6814i	0.863560	+	0.863560i	−0.128190	+	0.991750i	$0.540917\pi$
$252$		0			0
$253$	3.43965	−	3.43965i	0.216249	−	0.216249i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			11.2053i			0.698965i	0.936943	+	0.349483i	$0.113643\pi$
$257$			11.2053i			0.698965i	−0.936943	+	0.349483i	$0.886357\pi$
$258$		0			0
$259$	−1.00000	−	1.00000i	−0.0621370	−	0.0621370i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			3.42952i			0.211474i	0.994394	+	0.105737i	$0.0337201\pi$
$263$			3.42952i			0.211474i	−0.994394	+	0.105737i	$0.966280\pi$
$264$		0			0
$265$		−	23.1138i		−	1.41987i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.435258	−	0.435258i	−0.0265381	−	0.0265381i	0.693713	−	0.720251i	$-0.255974\pi$
$269$	−0.435258	−	0.435258i	−0.0265381	−	0.0265381i	−0.720251	+	0.693713i	$0.755974\pi$
$270$		0			0
$271$			7.11383i			0.432134i	0.976379	+	0.216067i	$0.0693230\pi$
$271$			7.11383i			0.432134i	−0.976379	+	0.216067i	$0.930677\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.748567	+	0.748567i	−0.0451403	+	0.0451403i
$276$		0			0
$277$	3.05520	+	3.05520i	0.183569	+	0.183569i	0.792909	−	0.609340i	$-0.208566\pi$
$277$	3.05520	+	3.05520i	0.183569	+	0.183569i	−0.609340	+	0.792909i	$0.708566\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	10.1434			0.605104			0.302552	−	0.953133i	$-0.402161\pi$
$281$	10.1434			0.605104			0.302552	+	0.953133i	$0.402161\pi$
$282$		0			0
$283$	−1.26719	+	1.26719i	−0.0753264	+	0.0753264i	−0.743766	−	0.668440i	$-0.766962\pi$
$283$	−1.26719	+	1.26719i	−0.0753264	+	0.0753264i	0.668440	+	0.743766i	$0.266962\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.978956			−0.0577859
$288$		0			0
$289$	−39.3449			−2.31441
$290$		0			0
$291$		0			0
$292$		0			0
$293$	5.46549	−	5.46549i	0.319298	−	0.319298i	−0.529200	−	0.848497i	$-0.677508\pi$
$293$	5.46549	−	5.46549i	0.319298	−	0.319298i	0.848497	+	0.529200i	$0.177508\pi$
$294$		0			0
$295$	12.7620			0.743034
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−19.5247	−	19.5247i	−1.12914	−	1.12914i
$300$		0			0
$301$	−6.80605	+	6.80605i	−0.392294	+	0.392294i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			0.704676i			0.0403496i
$306$		0			0
$307$	−0.234533	−	0.234533i	−0.0133855	−	0.0133855i	0.700382	−	0.713768i	$-0.253013\pi$
$307$	−0.234533	−	0.234533i	−0.0133855	−	0.0133855i	−0.713768	+	0.700382i	$0.753013\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			15.3856i			0.872440i	0.899840	+	0.436220i	$0.143683\pi$
$311$			15.3856i			0.872440i	−0.899840	+	0.436220i	$0.856317\pi$
$312$		0			0
$313$			5.26719i			0.297719i	0.988858	+	0.148859i	$0.0475602\pi$
$313$			5.26719i			0.297719i	−0.988858	+	0.148859i	$0.952440\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2.51512	−	2.51512i	−0.141263	−	0.141263i	0.632939	−	0.774202i	$-0.281848\pi$
$317$	−2.51512	−	2.51512i	−0.141263	−	0.141263i	−0.774202	+	0.632939i	$0.781848\pi$
$318$		0			0
$319$		−	7.87586i		−	0.440963i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	4.24914	+	4.24914i	0.235700	+	0.235700i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−6.36153			−0.350723
$330$		0			0
$331$	24.0371	−	24.0371i	1.32120	−	1.32120i	0.408397	−	0.912805i	$-0.366088\pi$
$331$	24.0371	−	24.0371i	1.32120	−	1.32120i	0.912805	−	0.408397i	$-0.133912\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2.12375			0.116033
$336$		0			0
$337$	6.76203			0.368351			0.184176	−	0.982893i	$-0.441038\pi$
$337$	6.76203			0.368351			0.184176	+	0.982893i	$0.441038\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−6.36153	+	6.36153i	−0.344496	+	0.344496i
$342$		0			0
$343$	1.00000			0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−2.97590	−	2.97590i	−0.159754	−	0.159754i	0.622703	−	0.782458i	$-0.286034\pi$
$347$	−2.97590	−	2.97590i	−0.159754	−	0.159754i	−0.782458	+	0.622703i	$0.786034\pi$
$348$		0			0
$349$	−11.4802	+	11.4802i	−0.614523	+	0.614523i	−0.944121	−	0.329598i	$-0.893087\pi$
$349$	−11.4802	+	11.4802i	−0.614523	+	0.614523i	0.329598	+	0.944121i	$0.393087\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	2.01531i		−	0.107264i	−0.998561	−	0.0536321i	$-0.982920\pi$
$353$		−	2.01531i		−	0.107264i	0.998561	−	0.0536321i	$-0.0170798\pi$
$354$		0			0
$355$	13.6121	+	13.6121i	0.722456	+	0.722456i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			5.13381i			0.270952i	0.990781	+	0.135476i	$0.0432564\pi$
$359$			5.13381i			0.270952i	−0.990781	+	0.135476i	$0.956744\pi$
$360$		0			0
$361$		−	19.0000i		−	1.00000i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	12.1842	+	12.1842i	0.637751	+	0.637751i
$366$		0			0
$367$			13.3845i			0.698663i	0.936999	+	0.349331i	$0.113591\pi$
$367$			13.3845i			0.698663i	−0.936999	+	0.349331i	$0.886409\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−8.17197	+	8.17197i	−0.424268	+	0.424268i
$372$		0			0
$373$	3.17246	+	3.17246i	0.164264	+	0.164264i	0.784453	−	0.620189i	$-0.212944\pi$
$373$	3.17246	+	3.17246i	0.164264	+	0.164264i	−0.620189	+	0.784453i	$0.712944\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−44.7063			−2.30249
$378$		0			0
$379$	21.3043	−	21.3043i	1.09433	−	1.09433i	0.0992697	−	0.995061i	$-0.468349\pi$
$379$	21.3043	−	21.3043i	1.09433	−	1.09433i	0.995061	−	0.0992697i	$-0.0316506\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−11.8525			−0.605637			−0.302818	−	0.953048i	$-0.597928\pi$
$383$	−11.8525			−0.605637			−0.302818	+	0.953048i	$0.597928\pi$
$384$		0			0
$385$	−2.11727			−0.107906
$386$		0			0
$387$		0			0
$388$		0			0
$389$	24.4281	−	24.4281i	1.23856	−	1.23856i	0.277964	−	0.960591i	$-0.410340\pi$
$389$	24.4281	−	24.4281i	1.23856	−	1.23856i	0.960591	−	0.277964i	$-0.0896597\pi$
$390$		0			0
$391$	−34.4914			−1.74431
$392$		0			0
$393$		0			0
$394$		0			0
$395$	24.0368	+	24.0368i	1.20942	+	1.20942i
$396$		0			0
$397$	−1.51633	+	1.51633i	−0.0761023	+	0.0761023i	−0.744133	−	0.668031i	$-0.767137\pi$
$397$	−1.51633	+	1.51633i	−0.0761023	+	0.0761023i	0.668031	+	0.744133i	$0.267137\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	26.9481i		−	1.34572i	−0.739768	−	0.672862i	$-0.765065\pi$
$401$		−	26.9481i		−	1.34572i	0.739768	−	0.672862i	$-0.234935\pi$
$402$		0			0
$403$	36.1104	+	36.1104i	1.79879	+	1.79879i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	1.49713i		−	0.0742101i
$408$		0			0
$409$		−	0.351799i		−	0.0173953i	−0.999962	−	0.00869767i	$-0.997231\pi$
$409$		−	0.351799i		−	0.0173953i	0.999962	−	0.00869767i	$-0.00276859\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−4.51206	−	4.51206i	−0.222024	−	0.222024i
$414$		0			0
$415$			7.00344i			0.343785i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−7.67217	+	7.67217i	−0.374810	+	0.374810i	−0.869226	−	0.494416i	$-0.835382\pi$
$419$	−7.67217	+	7.67217i	−0.374810	+	0.374810i	0.494416	+	0.869226i	$0.335382\pi$
$420$		0			0
$421$	17.1725	+	17.1725i	0.836935	+	0.836935i	0.988454	−	0.151520i	$-0.0484167\pi$
$421$	17.1725	+	17.1725i	0.836935	+	0.836935i	−0.151520	+	0.988454i	$0.548417\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	7.50633			0.364110
$426$		0			0
$427$	0.249141	−	0.249141i	0.0120568	−	0.0120568i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−24.4720			−1.17878			−0.589388	−	0.807850i	$-0.700631\pi$
$431$	−24.4720			−1.17878			−0.589388	+	0.807850i	$0.700631\pi$
$432$		0			0
$433$	18.7328			0.900242			0.450121	−	0.892968i	$-0.351381\pi$
$433$	18.7328			0.900242			0.450121	+	0.892968i	$0.351381\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−30.4622			−1.45388			−0.726940	−	0.686700i	$-0.759058\pi$
$439$	−30.4622			−1.45388			−0.726940	+	0.686700i	$0.759058\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	4.11583	+	4.11583i	0.195549	+	0.195549i	0.798089	−	0.602540i	$-0.205845\pi$
$443$	4.11583	+	4.11583i	0.195549	+	0.195549i	−0.602540	+	0.798089i	$0.705845\pi$
$444$		0			0
$445$	−3.61211	+	3.61211i	−0.171230	+	0.171230i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			1.41421i			0.0667409i	0.999443	+	0.0333704i	$0.0106241\pi$
$449$			1.41421i			0.0667409i	−0.999443	+	0.0333704i	$0.989376\pi$
$450$		0			0
$451$	−0.732814	−	0.732814i	−0.0345069	−	0.0345069i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			12.0184i			0.563431i
$456$		0			0
$457$			10.8793i			0.508912i	0.967084	+	0.254456i	$0.0818964\pi$
$457$			10.8793i			0.508912i	−0.967084	+	0.254456i	$0.918104\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−9.97755	−	9.97755i	−0.464701	−	0.464701i	0.435492	−	0.900193i	$-0.356575\pi$
$461$	−9.97755	−	9.97755i	−0.464701	−	0.464701i	−0.900193	+	0.435492i	$0.856575\pi$
$462$		0			0
$463$		−	25.9931i		−	1.20800i	−0.796983	−	0.604001i	$-0.793572\pi$
$463$		−	25.9931i		−	1.20800i	0.796983	−	0.604001i	$-0.206428\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	13.5155	−	13.5155i	0.625424	−	0.625424i	−0.321490	−	0.946913i	$-0.604184\pi$
$467$	13.5155	−	13.5155i	0.625424	−	0.625424i	0.946913	+	0.321490i	$0.104184\pi$
$468$		0			0
$469$	−0.750859	−	0.750859i	−0.0346715	−	0.0346715i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−10.1896			−0.468517
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	38.7177			1.76906			0.884529	−	0.466485i	$-0.154480\pi$
$479$	38.7177			1.76906			0.884529	+	0.466485i	$0.154480\pi$
$480$		0			0
$481$	−8.49828			−0.387488
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−25.6120	+	25.6120i	−1.16298	+	1.16298i
$486$		0			0
$487$	30.4983			1.38201			0.691005	−	0.722850i	$-0.257168\pi$
$487$	30.4983			1.38201			0.691005	+	0.722850i	$0.257168\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−17.6313	−	17.6313i	−0.795692	−	0.795692i	0.186721	−	0.982413i	$-0.440214\pi$
$491$	−17.6313	−	17.6313i	−0.795692	−	0.795692i	−0.982413	+	0.186721i	$0.940214\pi$
$492$		0			0
$493$	−39.4880	+	39.4880i	−1.77845	+	1.77845i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	9.62521i		−	0.431750i
$498$		0			0
$499$	−6.19051	−	6.19051i	−0.277125	−	0.277125i	0.554835	−	0.831960i	$-0.312781\pi$
$499$	−6.19051	−	6.19051i	−0.277125	−	0.277125i	−0.831960	+	0.554835i	$0.812781\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			1.41908i			0.0632734i	0.999499	+	0.0316367i	$0.0100720\pi$
$503$			1.41908i			0.0632734i	−0.999499	+	0.0316367i	$0.989928\pi$
$504$		0			0
$505$		−	12.9966i		−	0.578339i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−17.5717	−	17.5717i	−0.778850	−	0.778850i	0.200785	−	0.979635i	$-0.435651\pi$
$509$	−17.5717	−	17.5717i	−0.778850	−	0.778850i	−0.979635	+	0.200785i	$0.935651\pi$
$510$		0			0
$511$		−	8.61555i		−	0.381129i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	21.3742	−	21.3742i	0.941859	−	0.941859i
$516$		0			0
$517$	−4.76203	−	4.76203i	−0.209434	−	0.209434i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	18.4884			0.809990			0.404995	−	0.914319i	$-0.367273\pi$
$521$	18.4884			0.809990			0.404995	+	0.914319i	$0.367273\pi$
$522$		0			0
$523$	−30.4914	+	30.4914i	−1.33330	+	1.33330i	−0.430893	+	0.902403i	$0.641801\pi$
$523$	−30.4914	+	30.4914i	−1.33330	+	1.33330i	−0.902403	+	0.430893i	$0.858199\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	63.7909			2.77877
$528$		0			0
$529$	1.88617			0.0820075
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−4.15972	+	4.15972i	−0.180178	+	0.180178i
$534$		0			0
$535$	−15.1138			−0.653428
$536$		0			0
$537$		0			0
$538$		0			0
$539$	0.748567	+	0.748567i	0.0322430	+	0.0322430i
$540$		0			0
$541$	−16.7034	+	16.7034i	−0.718135	+	0.718135i	−0.968223	−	0.250088i	$-0.919540\pi$
$541$	−16.7034	+	16.7034i	−0.718135	+	0.718135i	0.250088	+	0.968223i	$0.419540\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			2.82843i			0.121157i
$546$		0			0
$547$	26.3630	+	26.3630i	1.12720	+	1.12720i	0.990631	+	0.136569i	$0.0436074\pi$
$547$	26.3630	+	26.3630i	1.12720	+	1.12720i	0.136569	+	0.990631i	$0.456393\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		−	16.9966i		−	0.722767i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−26.1740	−	26.1740i	−1.10903	−	1.10903i	−0.993278	−	0.115751i	$-0.963073\pi$
$557$	−26.1740	−	26.1740i	−1.10903	−	1.10903i	−0.115751	−	0.993278i	$-0.536927\pi$
$558$		0			0
$559$			57.8398i			2.44636i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−18.2715	+	18.2715i	−0.770051	+	0.770051i	−0.978115	−	0.208064i	$-0.933284\pi$
$563$	−18.2715	+	18.2715i	−0.770051	+	0.770051i	0.208064	+	0.978115i	$0.433284\pi$
$564$		0			0
$565$	12.8793	+	12.8793i	0.541836	+	0.541836i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	23.1614			0.970976			0.485488	−	0.874243i	$-0.338642\pi$
$569$	23.1614			0.970976			0.485488	+	0.874243i	$0.338642\pi$
$570$		0			0
$571$	21.0958	−	21.0958i	0.882831	−	0.882831i	−0.110990	−	0.993821i	$-0.535402\pi$
$571$	21.0958	−	21.0958i	0.882831	−	0.882831i	0.993821	+	0.110990i	$0.0354023\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.59498			0.191624
$576$		0			0
$577$	−34.7620			−1.44716			−0.723581	−	0.690239i	$-0.757505\pi$
$577$	−34.7620			−1.44716			−0.723581	+	0.690239i	$0.757505\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	2.47609	−	2.47609i	0.102725	−	0.102725i
$582$		0			0
$583$	−12.2345			−0.506703
$584$		0			0
$585$		0			0
$586$		0			0
$587$	13.6814	+	13.6814i	0.564690	+	0.564690i	0.930636	−	0.365946i	$-0.119255\pi$
$587$	13.6814	+	13.6814i	0.564690	+	0.564690i	−0.365946	+	0.930636i	$0.619255\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			29.0877i			1.19449i	0.802060	+	0.597244i	$0.203737\pi$
$593$			29.0877i			1.19449i	−0.802060	+	0.597244i	$0.796263\pi$
$594$		0			0
$595$	10.6155	+	10.6155i	0.435195	+	0.435195i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	5.76043i		−	0.235365i	−0.993051	−	0.117682i	$-0.962454\pi$
$599$		−	5.76043i		−	0.235365i	0.993051	−	0.117682i	$-0.0375465\pi$
$600$		0			0
$601$		−	27.4656i		−	1.12035i	−0.828376	−	0.560173i	$-0.810735\pi$
$601$		−	27.4656i		−	1.12035i	0.828376	−	0.560173i	$-0.189265\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	13.9714	+	13.9714i	0.568020	+	0.568020i
$606$		0			0
$607$		−	35.8466i		−	1.45497i	−0.686123	−	0.727485i	$-0.740689\pi$
$607$		−	35.8466i		−	1.45497i	0.686123	−	0.727485i	$-0.259311\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−27.0310	+	27.0310i	−1.09356	+	1.09356i
$612$		0			0
$613$	−4.55691	−	4.55691i	−0.184052	−	0.184052i	0.609067	−	0.793119i	$-0.291544\pi$
$613$	−4.55691	−	4.55691i	−0.184052	−	0.184052i	−0.793119	+	0.609067i	$0.791544\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	7.60990			0.306363			0.153182	−	0.988198i	$-0.451048\pi$
$617$	7.60990			0.306363			0.153182	+	0.988198i	$0.451048\pi$
$618$		0			0
$619$	−16.4983	+	16.4983i	−0.663122	+	0.663122i	−0.956115	−	0.292993i	$-0.905349\pi$
$619$	−16.4983	+	16.4983i	−0.663122	+	0.663122i	0.292993	+	0.956115i	$0.405349\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	2.55415			0.102330
$624$		0			0
$625$	19.0000			0.760000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−7.50633	+	7.50633i	−0.299297	+	0.299297i
$630$		0			0
$631$	−33.3776			−1.32874			−0.664370	−	0.747404i	$-0.731300\pi$
$631$	−33.3776			−1.32874			−0.664370	+	0.747404i	$0.731300\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	23.1663	+	23.1663i	0.919325	+	0.919325i
$636$		0			0
$637$	4.24914	−	4.24914i	0.168357	−	0.168357i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	18.4628i		−	0.729238i	−0.931157	−	0.364619i	$-0.881199\pi$
$641$		−	18.4628i		−	0.729238i	0.931157	−	0.364619i	$-0.118801\pi$
$642$		0			0
$643$	−27.4948	−	27.4948i	−1.08429	−	1.08429i	−0.996104	−	0.0881868i	$-0.971893\pi$
$643$	−27.4948	−	27.4948i	−1.08429	−	1.08429i	−0.0881868	−	0.996104i	$-0.528107\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	6.52737i		−	0.256617i	−0.991734	−	0.128309i	$-0.959045\pi$
$647$		−	6.52737i		−	0.256617i	0.991734	−	0.128309i	$-0.0409548\pi$
$648$		0			0
$649$		−	6.75515i		−	0.265163i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	11.5441	+	11.5441i	0.451755	+	0.451755i	0.895937	−	0.444181i	$-0.146505\pi$
$653$	11.5441	+	11.5441i	0.451755	+	0.451755i	−0.444181	+	0.895937i	$0.646505\pi$
$654$		0			0
$655$		−	8.99656i		−	0.351525i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−27.4224	+	27.4224i	−1.06823	+	1.06823i	−0.0707296	+	0.997496i	$0.522533\pi$
$659$	−27.4224	+	27.4224i	−1.06823	+	1.06823i	−0.997496	+	0.0707296i	$0.977467\pi$
$660$		0			0
$661$	−23.6267	−	23.6267i	−0.918973	−	0.918973i	0.0779820	−	0.996955i	$-0.475152\pi$
$661$	−23.6267	−	23.6267i	−0.918973	−	0.918973i	−0.996955	+	0.0779820i	$0.975152\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−24.1725	+	24.1725i	−0.935961	+	0.935961i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	0.372997			0.0143994
$672$		0			0
$673$	0.762030			0.0293741			0.0146870	−	0.999892i	$-0.495325\pi$
$673$	0.762030			0.0293741			0.0146870	+	0.999892i	$0.495325\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−6.77613	+	6.77613i	−0.260428	+	0.260428i	−0.825228	−	0.564800i	$-0.808953\pi$
$677$	−6.77613	+	6.77613i	−0.260428	+	0.260428i	0.564800	+	0.825228i	$0.308953\pi$
$678$		0			0
$679$	18.1104			0.695013
$680$		0			0
$681$		0			0
$682$		0			0
$683$	31.1469	+	31.1469i	1.19180	+	1.19180i	0.976561	+	0.215241i	$0.0690536\pi$
$683$	31.1469	+	31.1469i	1.19180	+	1.19180i	0.215241	+	0.976561i	$0.430946\pi$
$684$		0			0
$685$	−0.351799	+	0.351799i	−0.0134415	+	0.0134415i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			69.4477i			2.64575i
$690$		0			0
$691$	9.11383	+	9.11383i	0.346706	+	0.346706i	0.858881	−	0.512175i	$-0.171160\pi$
$691$	9.11383	+	9.11383i	0.346706	+	0.346706i	−0.512175	+	0.858881i	$0.671160\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			31.1030i			1.17980i
$696$		0			0
$697$			7.34836i			0.278339i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	6.37990	+	6.37990i	0.240966	+	0.240966i	0.817250	−	0.576284i	$-0.195498\pi$
$701$	6.37990	+	6.37990i	0.240966	+	0.240966i	−0.576284	+	0.817250i	$0.695498\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−4.59498	+	4.59498i	−0.172812	+	0.172812i
$708$		0			0
$709$	8.88273	+	8.88273i	0.333598	+	0.333598i	0.853951	−	0.520353i	$-0.174200\pi$
$709$	8.88273	+	8.88273i	0.333598	+	0.333598i	−0.520353	+	0.853951i	$0.674200\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	39.0494			1.46241
$714$		0			0
$715$	−8.99656	+	8.99656i	−0.336452	+	0.336452i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	22.8346			0.851586			0.425793	−	0.904821i	$-0.359995\pi$
$719$	22.8346			0.851586			0.425793	+	0.904821i	$0.359995\pi$
$720$		0			0
$721$	−15.1138			−0.562868
$722$		0			0
$723$		0			0
$724$		0			0
$725$	5.26063	−	5.26063i	0.195375	−	0.195375i
$726$		0			0
$727$	38.1173			1.41369			0.706846	−	0.707368i	$-0.250118\pi$
$727$	38.1173			1.41369			0.706846	+	0.707368i	$0.250118\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	51.0885	+	51.0885i	1.88958	+	1.88958i
$732$		0			0
$733$	28.9751	−	28.9751i	1.07022	−	1.07022i	0.0728781	−	0.997341i	$-0.476782\pi$
$733$	28.9751	−	28.9751i	1.07022	−	1.07022i	0.997341	−	0.0728781i	$-0.0232184\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	1.12414i		−	0.0414081i
$738$		0			0
$739$	−11.6302	−	11.6302i	−0.427822	−	0.427822i	0.460064	−	0.887886i	$-0.347827\pi$
$739$	−11.6302	−	11.6302i	−0.427822	−	0.427822i	−0.887886	+	0.460064i	$0.847827\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	46.8045i		−	1.71709i	−0.512738	−	0.858545i	$-0.671369\pi$
$743$		−	46.8045i		−	1.71709i	0.512738	−	0.858545i	$-0.328631\pi$
$744$		0			0
$745$			44.6379i			1.63541i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	5.34355	+	5.34355i	0.195249	+	0.195249i
$750$		0			0
$751$		−	39.6933i		−	1.44843i	−0.689575	−	0.724214i	$-0.742203\pi$
$751$		−	39.6933i		−	1.44843i	0.689575	−	0.724214i	$-0.257797\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	9.52164	−	9.52164i	0.346528	−	0.346528i
$756$		0			0
$757$	−25.1173	−	25.1173i	−0.912903	−	0.912903i	0.0835971	−	0.996500i	$-0.473359\pi$
$757$	−25.1173	−	25.1173i	−0.912903	−	0.912903i	−0.996500	+	0.0835971i	$0.973359\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	38.7751			1.40560			0.702799	−	0.711389i	$-0.251933\pi$
$761$	38.7751			1.40560			0.702799	+	0.711389i	$0.251933\pi$
$762$		0			0
$763$	1.00000	−	1.00000i	0.0362024	−	0.0362024i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−38.3447			−1.38455
$768$		0			0
$769$	25.9570			0.936035			0.468017	−	0.883719i	$-0.344968\pi$
$769$	25.9570			0.936035			0.468017	+	0.883719i	$0.344968\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	12.2097	−	12.2097i	0.439154	−	0.439154i	−0.452573	−	0.891727i	$-0.649494\pi$
$773$	12.2097	−	12.2097i	0.439154	−	0.439154i	0.891727	+	0.452573i	$0.149494\pi$
$774$		0			0
$775$	−8.49828			−0.305267
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	7.20512	−	7.20512i	0.257819	−	0.257819i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	25.0731i		−	0.894898i
$786$		0			0
$787$	−11.9931	−	11.9931i	−0.427509	−	0.427509i	0.460270	−	0.887779i	$-0.347753\pi$
$787$	−11.9931	−	11.9931i	−0.427509	−	0.427509i	−0.887779	+	0.460270i	$0.847753\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	9.10704i		−	0.323809i
$792$		0			0
$793$		−	2.11727i		−	0.0751863i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	12.6401	+	12.6401i	0.447737	+	0.447737i	0.894602	−	0.446865i	$-0.147459\pi$
$797$	12.6401	+	12.6401i	0.447737	+	0.447737i	−0.446865	+	0.894602i	$0.647459\pi$
$798$		0			0
$799$			47.7517i			1.68933i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	6.44931	−	6.44931i	0.227591	−	0.227591i
$804$		0			0
$805$	6.49828	+	6.49828i	0.229034	+	0.229034i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	18.5506			0.652205			0.326102	−	0.945334i	$-0.394265\pi$
$809$	18.5506			0.652205			0.326102	+	0.945334i	$0.394265\pi$
$810$		0			0
$811$	20.1104	−	20.1104i	0.706171	−	0.706171i	−0.259557	−	0.965728i	$-0.583576\pi$
$811$	20.1104	−	20.1104i	0.706171	−	0.706171i	0.965728	+	0.259557i	$0.0835764\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−28.2332			−0.988967
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−20.8950	+	20.8950i	−0.729242	+	0.729242i	−0.970469	−	0.241227i	$-0.922450\pi$
$821$	−20.8950	+	20.8950i	−0.729242	+	0.729242i	0.241227	+	0.970469i	$0.422450\pi$
$822$		0			0
$823$	−31.3845			−1.09399			−0.546997	−	0.837135i	$-0.684229\pi$
$823$	−31.3845			−1.09399			−0.546997	+	0.837135i	$0.684229\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−22.0605	−	22.0605i	−0.767118	−	0.767118i	0.210480	−	0.977598i	$-0.432497\pi$
$827$	−22.0605	−	22.0605i	−0.767118	−	0.767118i	−0.977598	+	0.210480i	$0.932497\pi$
$828$		0			0
$829$	4.60094	−	4.60094i	0.159797	−	0.159797i	−0.622680	−	0.782477i	$-0.713956\pi$
$829$	4.60094	−	4.60094i	0.159797	−	0.159797i	0.782477	+	0.622680i	$0.213956\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	7.50633i		−	0.260079i
$834$		0			0
$835$	29.9931	+	29.9931i	1.03795	+	1.03795i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	2.12375i		−	0.0733200i	−0.999328	−	0.0366600i	$-0.988328\pi$
$839$		−	2.12375i		−	0.0733200i	0.999328	−	0.0366600i	$-0.0116719\pi$
$840$		0			0
$841$			26.3484i			0.908564i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	32.6830	+	32.6830i	1.12433	+	1.12433i
$846$		0			0
$847$		−	9.87930i		−	0.339457i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−4.59498	+	4.59498i	−0.157514	+	0.157514i
$852$		0			0
$853$	−29.2096	−	29.2096i	−1.00012	−	1.00012i	−1.00000		0.000118046i	$-0.999962\pi$
$853$	−29.2096	−	29.2096i	−1.00012	−	1.00012i	−0.000118046	−	1.00000i	$-0.500038\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	9.67139			0.330369			0.165184	−	0.986263i	$-0.447178\pi$
$857$	9.67139			0.330369			0.165184	+	0.986263i	$0.447178\pi$
$858$		0			0
$859$	−38.7259	+	38.7259i	−1.32131	+	1.32131i	−0.408597	+	0.912715i	$0.633982\pi$
$859$	−38.7259	+	38.7259i	−1.32131	+	1.32131i	−0.912715	+	0.408597i	$0.866018\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	21.5655			0.734100			0.367050	−	0.930201i	$-0.380368\pi$
$863$	21.5655			0.734100			0.367050	+	0.930201i	$0.380368\pi$
$864$		0			0
$865$	−24.2345			−0.823999
$866$		0			0
$867$		0			0
$868$		0			0
$869$	12.7231	−	12.7231i	0.431600	−	0.431600i
$870$		0			0
$871$	−6.38101			−0.216212
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−8.48528	−	8.48528i	−0.286855	−	0.286855i
$876$		0			0
$877$	−24.3224	+	24.3224i	−0.821308	+	0.821308i	−0.986296	−	0.164987i	$-0.947242\pi$
$877$	−24.3224	+	24.3224i	−0.821308	+	0.821308i	0.164987	+	0.986296i	$0.447242\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	26.9736i		−	0.908765i	−0.890807	−	0.454382i	$-0.849860\pi$
$881$		−	26.9736i		−	0.908765i	0.890807	−	0.454382i	$-0.150140\pi$
$882$		0			0
$883$	−29.0698	−	29.0698i	−0.978277	−	0.978277i	0.0214922	−	0.999769i	$-0.493158\pi$
$883$	−29.0698	−	29.0698i	−0.978277	−	0.978277i	−0.999769	+	0.0214922i	$0.993158\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			1.94819i			0.0654137i	0.999465	+	0.0327069i	$0.0104128\pi$
$887$			1.94819i			0.0654137i	−0.999465	+	0.0327069i	$0.989587\pi$
$888$		0			0
$889$		−	16.3810i		−	0.549402i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		−	4.65164i		−	0.155487i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	44.7063	−	44.7063i	1.49104	−	1.49104i
$900$		0			0
$901$	61.3415	+	61.3415i	2.04358	+	2.04358i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	17.3436			0.576519
$906$		0			0
$907$	12.3078	−	12.3078i	0.408673	−	0.408673i	−0.472603	−	0.881276i	$-0.656685\pi$
$907$	12.3078	−	12.3078i	0.408673	−	0.408673i	0.881276	+	0.472603i	$0.156685\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−49.9181			−1.65386			−0.826931	−	0.562303i	$-0.809915\pi$
$911$	−49.9181			−1.65386			−0.826931	+	0.562303i	$0.809915\pi$
$912$		0			0
$913$	3.70704			0.122685
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−3.18077	+	3.18077i	−0.105038	+	0.105038i
$918$		0			0
$919$	42.4622			1.40070			0.700349	−	0.713800i	$-0.253028\pi$
$919$	42.4622			1.40070			0.700349	+	0.713800i	$0.253028\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−40.8989	−	40.8989i	−1.34620	−	1.34620i
$924$		0			0
$925$	1.00000	−	1.00000i	0.0328798	−	0.0328798i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			12.9973i			0.426429i	0.977005	+	0.213214i	$0.0683933\pi$
$929$			12.9973i			0.426429i	−0.977005	+	0.213214i	$0.931607\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$			15.8929i			0.519753i
$936$		0			0
$937$			43.9931i			1.43719i	0.695427	+	0.718596i	$0.255215\pi$
$937$			43.9931i			1.43719i	−0.695427	+	0.718596i	$0.744785\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	39.2614	+	39.2614i	1.27989	+	1.27989i	0.940730	+	0.339156i	$0.110142\pi$
$941$	39.2614	+	39.2614i	1.27989	+	1.27989i	0.339156	+	0.940730i	$0.389858\pi$
$942$		0			0
$943$			4.49828i			0.146484i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	23.7235	−	23.7235i	0.770909	−	0.770909i	−0.207357	−	0.978265i	$-0.566486\pi$
$947$	23.7235	−	23.7235i	0.770909	−	0.770909i	0.978265	+	0.207357i	$0.0664860\pi$
$948$		0			0
$949$	−36.6087	−	36.6087i	−1.18837	−	1.18837i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−58.7655			−1.90360			−0.951800	−	0.306718i	$-0.900769\pi$
$953$	−58.7655			−1.90360			−0.951800	+	0.306718i	$0.900769\pi$
$954$		0			0
$955$	26.7328	−	26.7328i	0.865054	−	0.865054i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	0.248759			0.00803286
$960$		0			0
$961$	−41.2208			−1.32970
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2.82843	−	2.82843i	0.0910503	−	0.0910503i
$966$		0			0
$967$	−14.2277			−0.457531			−0.228765	−	0.973482i	$-0.573469\pi$
$967$	−14.2277			−0.457531			−0.228765	+	0.973482i	$0.573469\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	4.78634	+	4.78634i	0.153601	+	0.153601i	0.779724	−	0.626123i	$-0.215359\pi$
$971$	4.78634	+	4.78634i	0.153601	+	0.153601i	−0.626123	+	0.779724i	$0.715359\pi$
$972$		0			0
$973$	10.9966	−	10.9966i	0.352534	−	0.352534i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			50.8558i			1.62702i	0.581550	+	0.813510i	$0.302446\pi$
$977$			50.8558i			1.62702i	−0.581550	+	0.813510i	$0.697554\pi$
$978$		0			0
$979$	1.91195	+	1.91195i	0.0611062	+	0.0611062i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			7.28309i			0.232294i	0.993232	+	0.116147i	$0.0370544\pi$
$983$			7.28309i			0.232294i	−0.993232	+	0.116147i	$0.962946\pi$
$984$		0			0
$985$		−	33.3415i		−	1.06235i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	31.2737	+	31.2737i	0.994445	+	0.994445i
$990$		0			0
$991$			12.7259i			0.404253i	0.979359	+	0.202126i	$0.0647852\pi$
$991$			12.7259i			0.404253i	−0.979359	+	0.202126i	$0.935215\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−2.28959	+	2.28959i	−0.0725849	+	0.0725849i
$996$		0			0
$997$	38.6302	+	38.6302i	1.22343	+	1.22343i	0.966405	+	0.257024i	$0.0827421\pi$
$997$	38.6302	+	38.6302i	1.22343	+	1.22343i	0.257024	+	0.966405i	$0.417258\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.v.c.3599.2		12
3.2	odd	2	inner	4032.2.v.c.3599.5		12
4.3	odd	2		1008.2.v.c.827.5	yes	12
12.11	even	2		1008.2.v.c.827.2	yes	12
16.3	odd	4	inner	4032.2.v.c.1583.5		12
16.13	even	4		1008.2.v.c.323.2	✓	12
48.29	odd	4		1008.2.v.c.323.5	yes	12
48.35	even	4	inner	4032.2.v.c.1583.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.2.v.c.323.2	✓	12	16.13	even	4
1008.2.v.c.323.5	yes	12	48.29	odd	4
1008.2.v.c.827.2	yes	12	12.11	even	2
1008.2.v.c.827.5	yes	12	4.3	odd	2
4032.2.v.c.1583.2		12	48.35	even	4	inner
4032.2.v.c.1583.5		12	16.3	odd	4	inner
4032.2.v.c.3599.2		12	1.1	even	1	trivial
4032.2.v.c.3599.5		12	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4032.v (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.41421 + 1.41421i) q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+(-1.41421 + 1.41421i) q^{5} +1.00000 q^{7} +(0.748567 + 0.748567i) q^{11} +(4.24914 - 4.24914i) q^{13} -7.50633i q^{17} -4.59498i q^{23} +1.00000i q^{25} +(-5.26063 - 5.26063i) q^{29} +8.49828i q^{31} +(-1.41421 + 1.41421i) q^{35} +(-1.00000 - 1.00000i) q^{37} -0.978956 q^{41} +(-6.80605 + 6.80605i) q^{43} -6.36153 q^{47} +1.00000 q^{49} +(-8.17197 + 8.17197i) q^{53} -2.11727 q^{55} +(-4.51206 - 4.51206i) q^{59} +(0.249141 - 0.249141i) q^{61} +12.0184i q^{65} +(-0.750859 - 0.750859i) q^{67} -9.62521i q^{71} -8.61555i q^{73} +(0.748567 + 0.748567i) q^{77} -16.9966i q^{79} +(2.47609 - 2.47609i) q^{83} +(10.6155 + 10.6155i) q^{85} +2.55415 q^{89} +(4.24914 - 4.24914i) q^{91} +18.1104 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{7}+O(q^{10}) \) 12 * q + 12 * q^7	\( 12 q + 12 q^{7} + 16 q^{13} - 12 q^{37} + 20 q^{43} + 12 q^{49} - 32 q^{55} - 32 q^{61} - 44 q^{67} + 64 q^{85} + 16 q^{91} - 56 q^{97}+O(q^{100}) \) 12 * q + 12 * q^7 + 16 * q^13 - 12 * q^37 + 20 * q^43 + 12 * q^49 - 32 * q^55 - 32 * q^61 - 44 * q^67 + 64 * q^85 + 16 * q^91 - 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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