LMFDB - Embedded newform 4368.2.h.s.337.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(337,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 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("4368.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.h (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:	$34.8786556029$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 21x^{8} + 124x^{6} + 212x^{4} + 116x^{2} + 16 $ x^10 + 21x^8 + 124x^6 + 212x^4 + 116x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 2184)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.2
Root			$0.812474i$ of defining polynomial
Character	$\chi$	$=$	4368.337
Dual form			4368.2.h.s.337.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.46162i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.46162i q^{5} -1.00000i q^{7} +1.00000 q^{9} -0.836670i q^{11} +(3.51282 + 0.812474i) q^{13} +2.46162i q^{15} +2.16196 q^{17} -6.24516i q^{19} +1.00000i q^{21} -0.863673 q^{23} -1.05956 q^{25} -1.00000 q^{27} +4.16196 q^{29} +0.836670i q^{33} -2.46162 q^{35} +1.23873i q^{37} +(-3.51282 - 0.812474i) q^{39} -6.95024i q^{41} +5.32192 q^{43} -2.46162i q^{45} -4.99863i q^{47} -1.00000 q^{49} -2.16196 q^{51} +9.48388 q^{53} -2.05956 q^{55} +6.24516i q^{57} +10.0051i q^{59} +9.87011 q^{61} -1.00000i q^{63} +(2.00000 - 8.64721i) q^{65} +9.92967i q^{67} +0.863673 q^{69} +3.68114i q^{71} +8.62021i q^{73} +1.05956 q^{75} -0.836670 q^{77} +4.30472 q^{79} +1.00000 q^{81} -6.60438i q^{83} -5.32192i q^{85} -4.16196 q^{87} -1.10377i q^{89} +(0.812474 - 3.51282i) q^{91} -15.3732 q^{95} -13.9489i q^{97} -0.836670i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{3} + 10 q^{9}+O(q^{10}) $ 10 * q - 10 * q^3 + 10 * q^9	$ 10 q - 10 q^{3} + 10 q^{9} - 22 q^{17} + 18 q^{23} - 8 q^{25} - 10 q^{27} - 2 q^{29} - 10 q^{35} + 2 q^{43} - 10 q^{49} + 22 q^{51} - 18 q^{55} + 6 q^{61} + 20 q^{65} - 18 q^{69} + 8 q^{75} - 6 q^{77} - 40 q^{79} + 10 q^{81} + 2 q^{87} + 2 q^{91} + 38 q^{95}+O(q^{100}) $ 10 * q - 10 * q^3 + 10 * q^9 - 22 * q^17 + 18 * q^23 - 8 * q^25 - 10 * q^27 - 2 * q^29 - 10 * q^35 + 2 * q^43 - 10 * q^49 + 22 * q^51 - 18 * q^55 + 6 * q^61 + 20 * q^65 - 18 * q^69 + 8 * q^75 - 6 * q^77 - 40 * q^79 + 10 * q^81 + 2 * q^87 + 2 * q^91 + 38 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4368\mathbb{Z}\right)^\times$.

$n$	$1093$	$1249$	$1457$	$2017$	$3823$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

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
$5$		−	2.46162i		−	1.10087i	−0.834878	−	0.550434i	$-0.814462\pi$
$5$		−	2.46162i		−	1.10087i	0.834878	−	0.550434i	$-0.185538\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	0.836670i		−	0.252266i	−0.992013	−	0.126133i	$-0.959743\pi$
$11$		−	0.836670i		−	0.252266i	0.992013	−	0.126133i	$-0.0402565\pi$
$12$		0			0
$13$	3.51282	+	0.812474i	0.974280	+	0.225340i
$13$	3.51282	+	0.812474i	0.974280	+	0.225340i
$14$		0			0
$15$			2.46162i			0.635587i
$16$		0			0
$17$	2.16196			0.524353			0.262176	−	0.965020i	$-0.415560\pi$
$17$	2.16196			0.524353			0.262176	+	0.965020i	$0.415560\pi$
$18$		0			0
$19$		−	6.24516i		−	1.43274i	−0.697722	−	0.716369i	$-0.745803\pi$
$19$		−	6.24516i		−	1.43274i	0.697722	−	0.716369i	$-0.254197\pi$
$20$		0			0
$21$			1.00000i			0.218218i
$22$		0			0
$23$	−0.863673			−0.180088			−0.0900442	−	0.995938i	$-0.528701\pi$
$23$	−0.863673			−0.180088			−0.0900442	+	0.995938i	$0.528701\pi$
$24$		0			0
$25$	−1.05956			−0.211912
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	4.16196			0.772857			0.386428	−	0.922319i	$-0.373709\pi$
$29$	4.16196			0.772857			0.386428	+	0.922319i	$0.373709\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$			0.836670i			0.145646i
$34$		0			0
$35$	−2.46162			−0.416089
$36$		0			0
$37$			1.23873i			0.203645i	0.994803	+	0.101823i	$0.0324674\pi$
$37$			1.23873i			0.203645i	−0.994803	+	0.101823i	$0.967533\pi$
$38$		0			0
$39$	−3.51282	−	0.812474i	−0.562501	−	0.130100i
$40$		0			0
$41$		−	6.95024i		−	1.08544i	−0.839912	−	0.542722i	$-0.817394\pi$
$41$		−	6.95024i		−	1.08544i	0.839912	−	0.542722i	$-0.182606\pi$
$42$		0			0
$43$	5.32192			0.811586			0.405793	−	0.913965i	$-0.366995\pi$
$43$	5.32192			0.811586			0.405793	+	0.913965i	$0.366995\pi$
$44$		0			0
$45$		−	2.46162i		−	0.366956i
$46$		0			0
$47$		−	4.99863i		−	0.729125i	−0.931179	−	0.364563i	$-0.881218\pi$
$47$		−	4.99863i		−	0.729125i	0.931179	−	0.364563i	$-0.118782\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−2.16196			−0.302735
$52$		0			0
$53$	9.48388			1.30271			0.651356	−	0.758772i	$-0.274201\pi$
$53$	9.48388			1.30271			0.651356	+	0.758772i	$0.274201\pi$
$54$		0			0
$55$	−2.05956			−0.277711
$56$		0			0
$57$			6.24516i			0.827191i
$58$		0			0
$59$			10.0051i			1.30255i	0.758842	+	0.651274i	$0.225765\pi$
$59$			10.0051i			1.30255i	−0.758842	+	0.651274i	$0.774235\pi$
$60$		0			0
$61$	9.87011			1.26374			0.631869	−	0.775075i	$-0.282288\pi$
$61$	9.87011			1.26374			0.631869	+	0.775075i	$0.282288\pi$
$62$		0			0
$63$		−	1.00000i		−	0.125988i
$64$		0			0
$65$	2.00000	−	8.64721i	0.248069	−	1.07255i
$66$		0			0
$67$			9.92967i			1.21310i	0.795045	+	0.606551i	$0.207447\pi$
$67$			9.92967i			1.21310i	−0.795045	+	0.606551i	$0.792553\pi$
$68$		0			0
$69$	0.863673			0.103974
$70$		0			0
$71$			3.68114i			0.436871i	0.975851	+	0.218436i	$0.0700953\pi$
$71$			3.68114i			0.436871i	−0.975851	+	0.218436i	$0.929905\pi$
$72$		0			0
$73$			8.62021i			1.00892i	0.863435	+	0.504460i	$0.168308\pi$
$73$			8.62021i			1.00892i	−0.863435	+	0.504460i	$0.831692\pi$
$74$		0			0
$75$	1.05956			0.122348
$76$		0			0
$77$	−0.836670			−0.0953474
$78$		0			0
$79$	4.30472			0.484319			0.242159	−	0.970236i	$-0.422144\pi$
$79$	4.30472			0.484319			0.242159	+	0.970236i	$0.422144\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	6.60438i		−	0.724925i	−0.931998	−	0.362462i	$-0.881936\pi$
$83$		−	6.60438i		−	0.724925i	0.931998	−	0.362462i	$-0.118064\pi$
$84$		0			0
$85$		−	5.32192i		−	0.577244i
$86$		0			0
$87$	−4.16196			−0.446209
$88$		0			0
$89$		−	1.10377i		−	0.116999i	−0.998287	−	0.0584996i	$-0.981368\pi$
$89$		−	1.10377i		−	0.116999i	0.998287	−	0.0584996i	$-0.0186316\pi$
$90$		0			0
$91$	0.812474	−	3.51282i	0.0851704	−	0.368243i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−15.3732			−1.57726
$96$		0			0
$97$		−	13.9489i		−	1.41629i	−0.706065	−	0.708147i	$-0.749532\pi$
$97$		−	13.9489i		−	1.41629i	0.706065	−	0.708147i	$-0.250468\pi$
$98$		0			0
$99$		−	0.836670i		−	0.0840885i
$100$		0			0
$101$	−2.40986			−0.239790			−0.119895	−	0.992787i	$-0.538256\pi$
$101$	−2.40986			−0.239790			−0.119895	+	0.992787i	$0.538256\pi$
$102$		0			0
$103$	−12.7933			−1.26057			−0.630283	−	0.776366i	$-0.717061\pi$
$103$	−12.7933			−1.26057			−0.630283	+	0.776366i	$0.717061\pi$
$104$		0			0
$105$	2.46162			0.240229
$106$		0			0
$107$	−7.02563			−0.679194			−0.339597	−	0.940571i	$-0.610291\pi$
$107$	−7.02563			−0.679194			−0.339597	+	0.940571i	$0.610291\pi$
$108$		0			0
$109$			4.48862i			0.429932i	0.976622	+	0.214966i	$0.0689641\pi$
$109$			4.48862i			0.429932i	−0.976622	+	0.214966i	$0.931036\pi$
$110$		0			0
$111$		−	1.23873i		−	0.117575i
$112$		0			0
$113$	−8.76297			−0.824351			−0.412175	−	0.911105i	$-0.635231\pi$
$113$	−8.76297			−0.824351			−0.412175	+	0.911105i	$0.635231\pi$
$114$		0			0
$115$			2.12603i			0.198254i
$116$		0			0
$117$	3.51282	+	0.812474i	0.324760	+	0.0751132i
$118$		0			0
$119$		−	2.16196i		−	0.198187i
$120$		0			0
$121$	10.3000			0.936362
$122$		0			0
$123$			6.95024i			0.626682i
$124$		0			0
$125$		−	9.69985i		−	0.867581i
$126$		0			0
$127$	2.29186			0.203369			0.101685	−	0.994817i	$-0.467577\pi$
$127$	2.29186			0.203369			0.101685	+	0.994817i	$0.467577\pi$
$128$		0			0
$129$	−5.32192			−0.468569
$130$		0			0
$131$	−1.92797			−0.168448			−0.0842239	−	0.996447i	$-0.526841\pi$
$131$	−1.92797			−0.168448			−0.0842239	+	0.996447i	$0.526841\pi$
$132$		0			0
$133$	−6.24516			−0.541524
$134$		0			0
$135$			2.46162i			0.211862i
$136$		0			0
$137$		−	12.0838i		−	1.03239i	−0.856471	−	0.516196i	$-0.827348\pi$
$137$		−	12.0838i		−	1.03239i	0.856471	−	0.516196i	$-0.172652\pi$
$138$		0			0
$139$	22.2052			1.88342			0.941710	−	0.336425i	$-0.109218\pi$
$139$	22.2052			1.88342			0.941710	+	0.336425i	$0.109218\pi$
$140$		0			0
$141$			4.99863i			0.420961i
$142$		0			0
$143$	0.679773	−	2.93907i	0.0568454	−	0.245777i
$144$		0			0
$145$		−	10.2452i		−	0.850814i
$146$		0			0
$147$	1.00000			0.0824786
$148$		0			0
$149$		−	3.24209i		−	0.265603i	−0.991143	−	0.132801i	$-0.957603\pi$
$149$		−	3.24209i		−	0.265603i	0.991143	−	0.132801i	$-0.0423972\pi$
$150$		0			0
$151$		−	11.4923i		−	0.935231i	−0.883932	−	0.467616i	$-0.845113\pi$
$151$		−	11.4923i		−	0.935231i	0.883932	−	0.467616i	$-0.154887\pi$
$152$		0			0
$153$	2.16196			0.174784
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−12.4859			−0.996482			−0.498241	−	0.867039i	$-0.666021\pi$
$157$	−12.4859			−0.996482			−0.498241	+	0.867039i	$0.666021\pi$
$158$		0			0
$159$	−9.48388			−0.752121
$160$		0			0
$161$			0.863673i			0.0680670i
$162$		0			0
$163$		−	8.08320i		−	0.633125i	−0.948572	−	0.316562i	$-0.897471\pi$
$163$		−	8.08320i		−	0.633125i	0.948572	−	0.316562i	$-0.102529\pi$
$164$		0			0
$165$	2.05956			0.160337
$166$		0			0
$167$		−	7.72398i		−	0.597699i	−0.954300	−	0.298850i	$-0.903397\pi$
$167$		−	7.72398i		−	0.597699i	0.954300	−	0.298850i	$-0.0966029\pi$
$168$		0			0
$169$	11.6798	+	5.70814i	0.898444	+	0.439088i
$170$		0			0
$171$		−	6.24516i		−	0.477579i
$172$		0			0
$173$	−15.6030			−1.18627			−0.593137	−	0.805101i	$-0.702111\pi$
$173$	−15.6030			−1.18627			−0.593137	+	0.805101i	$0.702111\pi$
$174$		0			0
$175$			1.05956i			0.0800954i
$176$		0			0
$177$		−	10.0051i		−	0.752027i
$178$		0			0
$179$	−13.0385			−0.974543			−0.487272	−	0.873250i	$-0.662008\pi$
$179$	−13.0385			−0.974543			−0.487272	+	0.873250i	$0.662008\pi$
$180$		0			0
$181$	−15.9837			−1.18806			−0.594029	−	0.804444i	$-0.702463\pi$
$181$	−15.9837			−1.18806			−0.594029	+	0.804444i	$0.702463\pi$
$182$		0			0
$183$	−9.87011			−0.729619
$184$		0			0
$185$	3.04927			0.224187
$186$		0			0
$187$		−	1.80885i		−	0.132276i
$188$		0			0
$189$			1.00000i			0.0727393i
$190$		0			0
$191$	−20.6039			−1.49084			−0.745422	−	0.666592i	$-0.767752\pi$
$191$	−20.6039			−1.49084			−0.745422	+	0.666592i	$0.767752\pi$
$192$		0			0
$193$		−	1.55422i		−	0.111875i	−0.998434	−	0.0559375i	$-0.982185\pi$
$193$		−	1.55422i		−	0.111875i	0.998434	−	0.0559375i	$-0.0178147\pi$
$194$		0			0
$195$	−2.00000	+	8.64721i	−0.143223	+	0.619240i
$196$		0			0
$197$			2.31212i			0.164732i	0.996602	+	0.0823659i	$0.0262476\pi$
$197$			2.31212i			0.164732i	−0.996602	+	0.0823659i	$0.973752\pi$
$198$		0			0
$199$	4.63941			0.328879			0.164440	−	0.986387i	$-0.447418\pi$
$199$	4.63941			0.328879			0.164440	+	0.986387i	$0.447418\pi$
$200$		0			0
$201$		−	9.92967i		−	0.700385i
$202$		0			0
$203$		−	4.16196i		−	0.292112i
$204$		0			0
$205$	−17.1088			−1.19493
$206$		0			0
$207$	−0.863673			−0.0600295
$208$		0			0
$209$	−5.22514			−0.361430
$210$		0			0
$211$	−5.13633			−0.353599			−0.176800	−	0.984247i	$-0.556574\pi$
$211$	−5.13633			−0.353599			−0.176800	+	0.984247i	$0.556574\pi$
$212$		0			0
$213$		−	3.68114i		−	0.252228i
$214$		0			0
$215$		−	13.1005i		−	0.893449i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		−	8.62021i		−	0.582500i
$220$		0			0
$221$	7.59458	+	1.75654i	0.510866	+	0.118157i
$222$		0			0
$223$			9.99726i			0.669466i	0.942313	+	0.334733i	$0.108646\pi$
$223$			9.99726i			0.669466i	−0.942313	+	0.334733i	$0.891354\pi$
$224$		0			0
$225$	−1.05956			−0.0706375
$226$		0			0
$227$		−	11.7324i		−	0.778707i	−0.921088	−	0.389354i	$-0.872698\pi$
$227$		−	11.7324i		−	0.778707i	0.921088	−	0.389354i	$-0.127302\pi$
$228$		0			0
$229$			7.95161i			0.525457i	0.964870	+	0.262729i	$0.0846224\pi$
$229$			7.95161i			0.525457i	−0.964870	+	0.262729i	$0.915378\pi$
$230$		0			0
$231$	0.836670			0.0550489
$232$		0			0
$233$	2.75010			0.180165			0.0900827	−	0.995934i	$-0.471287\pi$
$233$	2.75010			0.180165			0.0900827	+	0.995934i	$0.471287\pi$
$234$		0			0
$235$	−12.3047			−0.802671
$236$		0			0
$237$	−4.30472			−0.279622
$238$		0			0
$239$		−	22.4893i		−	1.45471i	−0.686262	−	0.727354i	$-0.740750\pi$
$239$		−	22.4893i		−	1.45471i	0.686262	−	0.727354i	$-0.259250\pi$
$240$		0			0
$241$			11.4647i			0.738505i	0.929329	+	0.369253i	$0.120386\pi$
$241$			11.4647i			0.738505i	−0.929329	+	0.369253i	$0.879614\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$			2.46162i			0.157267i
$246$		0			0
$247$	5.07403	−	21.9381i	0.322853	−	1.39589i
$248$		0			0
$249$			6.60438i			0.418535i
$250$		0			0
$251$	11.0493			0.697424			0.348712	−	0.937230i	$-0.386619\pi$
$251$	11.0493			0.697424			0.348712	+	0.937230i	$0.386619\pi$
$252$		0			0
$253$			0.722610i			0.0454301i
$254$		0			0
$255$			5.32192i			0.333272i
$256$		0			0
$257$	−0.563385			−0.0351430			−0.0175715	−	0.999846i	$-0.505593\pi$
$257$	−0.563385			−0.0351430			−0.0175715	+	0.999846i	$0.505593\pi$
$258$		0			0
$259$	1.23873			0.0769707
$260$		0			0
$261$	4.16196			0.257619
$262$		0			0
$263$	−13.2691			−0.818208			−0.409104	−	0.912488i	$-0.634159\pi$
$263$	−13.2691			−0.818208			−0.409104	+	0.912488i	$0.634159\pi$
$264$		0			0
$265$		−	23.3457i		−	1.43411i
$266$		0			0
$267$			1.10377i			0.0675495i
$268$		0			0
$269$	22.2469			1.35641			0.678207	−	0.734871i	$-0.262757\pi$
$269$	22.2469			1.35641			0.678207	+	0.734871i	$0.262757\pi$
$270$		0			0
$271$		−	23.2984i		−	1.41528i	−0.706574	−	0.707639i	$-0.749760\pi$
$271$		−	23.2984i		−	1.41528i	0.706574	−	0.707639i	$-0.250240\pi$
$272$		0			0
$273$	−0.812474	+	3.51282i	−0.0491732	+	0.212605i
$274$		0			0
$275$			0.886504i			0.0534582i
$276$		0			0
$277$	−25.2917			−1.51963			−0.759815	−	0.650139i	$-0.774711\pi$
$277$	−25.2917			−1.51963			−0.759815	+	0.650139i	$0.774711\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			13.5620i			0.809042i	0.914529	+	0.404521i	$0.132562\pi$
$281$			13.5620i			0.809042i	−0.914529	+	0.404521i	$0.867438\pi$
$282$		0			0
$283$	27.7402			1.64898			0.824492	−	0.565873i	$-0.191461\pi$
$283$	27.7402			1.64898			0.824492	+	0.565873i	$0.191461\pi$
$284$		0			0
$285$	15.3732			0.910629
$286$		0			0
$287$	−6.95024			−0.410260
$288$		0			0
$289$	−12.3259			−0.725054
$290$		0			0
$291$			13.9489i			0.817697i
$292$		0			0
$293$		−	19.1910i		−	1.12115i	−0.828104	−	0.560574i	$-0.810581\pi$
$293$		−	19.1910i		−	1.12115i	0.828104	−	0.560574i	$-0.189419\pi$
$294$		0			0
$295$	24.6286			1.43394
$296$		0			0
$297$			0.836670i			0.0485485i
$298$		0			0
$299$	−3.03393	−	0.701712i	−0.175457	−	0.0405811i
$300$		0			0
$301$		−	5.32192i		−	0.306751i
$302$		0			0
$303$	2.40986			0.138443
$304$		0			0
$305$		−	24.2964i		−	1.39121i
$306$		0			0
$307$			23.2444i			1.32663i	0.748341	+	0.663315i	$0.230851\pi$
$307$			23.2444i			1.32663i	−0.748341	+	0.663315i	$0.769149\pi$
$308$		0			0
$309$	12.7933			0.727788
$310$		0			0
$311$	−12.3463			−0.700092			−0.350046	−	0.936732i	$-0.613834\pi$
$311$	−12.3463			−0.700092			−0.350046	+	0.936732i	$0.613834\pi$
$312$		0			0
$313$	15.9129			0.899453			0.449726	−	0.893166i	$-0.351522\pi$
$313$	15.9129			0.899453			0.449726	+	0.893166i	$0.351522\pi$
$314$		0			0
$315$	−2.46162			−0.138696
$316$		0			0
$317$			13.3741i			0.751163i	0.926789	+	0.375582i	$0.122557\pi$
$317$			13.3741i			0.751163i	−0.926789	+	0.375582i	$0.877443\pi$
$318$		0			0
$319$		−	3.48219i		−	0.194965i
$320$		0			0
$321$	7.02563			0.392133
$322$		0			0
$323$		−	13.5018i		−	0.751260i
$324$		0			0
$325$	−3.72205	−	0.860866i	−0.206462	−	0.0477523i
$326$		0			0
$327$		−	4.48862i		−	0.248221i
$328$		0			0
$329$	−4.99863			−0.275583
$330$		0			0
$331$			24.0933i			1.32429i	0.749376	+	0.662144i	$0.230353\pi$
$331$			24.0933i			1.32429i	−0.749376	+	0.662144i	$0.769647\pi$
$332$		0			0
$333$			1.23873i			0.0678818i
$334$		0			0
$335$	24.4430			1.33547
$336$		0			0
$337$	−14.1108			−0.768666			−0.384333	−	0.923195i	$-0.625569\pi$
$337$	−14.1108			−0.768666			−0.384333	+	0.923195i	$0.625569\pi$
$338$		0			0
$339$	8.76297			0.475939
$340$		0			0
$341$		0			0
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$		−	2.12603i		−	0.114462i
$346$		0			0
$347$	28.4075			1.52500			0.762498	−	0.646991i	$-0.223973\pi$
$347$	28.4075			1.52500			0.762498	+	0.646991i	$0.223973\pi$
$348$		0			0
$349$		−	17.2304i		−	0.922324i	−0.887316	−	0.461162i	$-0.847433\pi$
$349$		−	17.2304i		−	0.922324i	0.887316	−	0.461162i	$-0.152567\pi$
$350$		0			0
$351$	−3.51282	−	0.812474i	−0.187500	−	0.0433666i
$352$		0			0
$353$		−	9.54681i		−	0.508126i	−0.967188	−	0.254063i	$-0.918233\pi$
$353$		−	9.54681i		−	0.508126i	0.967188	−	0.254063i	$-0.0817670\pi$
$354$		0			0
$355$	9.06156			0.480938
$356$		0			0
$357$			2.16196i			0.114423i
$358$		0			0
$359$		−	9.77082i		−	0.515684i	−0.966187	−	0.257842i	$-0.916989\pi$
$359$		−	9.77082i		−	0.515684i	0.966187	−	0.257842i	$-0.0830114\pi$
$360$		0			0
$361$	−20.0020			−1.05274
$362$		0			0
$363$	−10.3000			−0.540609
$364$		0			0
$365$	21.2197			1.11069
$366$		0			0
$367$	−10.5414			−0.550259			−0.275129	−	0.961407i	$-0.588721\pi$
$367$	−10.5414			−0.550259			−0.275129	+	0.961407i	$0.588721\pi$
$368$		0			0
$369$		−	6.95024i		−	0.361815i
$370$		0			0
$371$		−	9.48388i		−	0.492379i
$372$		0			0
$373$	−9.35915			−0.484598			−0.242299	−	0.970202i	$-0.577902\pi$
$373$	−9.35915			−0.484598			−0.242299	+	0.970202i	$0.577902\pi$
$374$		0			0
$375$			9.69985i			0.500898i
$376$		0			0
$377$	14.6202	+	3.38148i	0.752979	+	0.174155i
$378$		0			0
$379$		−	3.93240i		−	0.201994i	−0.994887	−	0.100997i	$-0.967797\pi$
$379$		−	3.93240i		−	0.201994i	0.994887	−	0.100997i	$-0.0322033\pi$
$380$		0			0
$381$	−2.29186			−0.117415
$382$		0			0
$383$		−	7.23366i		−	0.369623i	−0.982774	−	0.184811i	$-0.940833\pi$
$383$		−	7.23366i		−	0.369623i	0.982774	−	0.184811i	$-0.0591674\pi$
$384$		0			0
$385$			2.05956i			0.104965i
$386$		0			0
$387$	5.32192			0.270529
$388$		0			0
$389$	−21.6638			−1.09840			−0.549198	−	0.835692i	$-0.685067\pi$
$389$	−21.6638			−1.09840			−0.549198	+	0.835692i	$0.685067\pi$
$390$		0			0
$391$	−1.86723			−0.0944298
$392$		0			0
$393$	1.92797			0.0972534
$394$		0			0
$395$		−	10.5966i		−	0.533172i
$396$		0			0
$397$			18.4776i			0.927364i	0.886002	+	0.463682i	$0.153472\pi$
$397$			18.4776i			0.927364i	−0.886002	+	0.463682i	$0.846528\pi$
$398$		0			0
$399$	6.24516			0.312649
$400$		0			0
$401$		−	13.3517i		−	0.666754i	−0.942794	−	0.333377i	$-0.891812\pi$
$401$		−	13.3517i		−	0.666754i	0.942794	−	0.333377i	$-0.108188\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		−	2.46162i		−	0.122319i
$406$		0			0
$407$	1.03640			0.0513727
$408$		0			0
$409$		−	28.1400i		−	1.39143i	−0.718316	−	0.695717i	$-0.755087\pi$
$409$		−	28.1400i		−	1.39143i	0.718316	−	0.695717i	$-0.244913\pi$
$410$		0			0
$411$			12.0838i			0.596051i
$412$		0			0
$413$	10.0051			0.492317
$414$		0			0
$415$	−16.2575			−0.798047
$416$		0			0
$417$	−22.2052			−1.08739
$418$		0			0
$419$	−1.90563			−0.0930963			−0.0465482	−	0.998916i	$-0.514822\pi$
$419$	−1.90563			−0.0930963			−0.0465482	+	0.998916i	$0.514822\pi$
$420$		0			0
$421$			12.8400i			0.625785i	0.949789	+	0.312893i	$0.101298\pi$
$421$			12.8400i			0.625785i	−0.949789	+	0.312893i	$0.898702\pi$
$422$		0			0
$423$		−	4.99863i		−	0.243042i
$424$		0			0
$425$	−2.29073			−0.111117
$426$		0			0
$427$		−	9.87011i		−	0.477648i
$428$		0			0
$429$	−0.679773	+	2.93907i	−0.0328197	+	0.141900i
$430$		0			0
$431$		−	8.87429i		−	0.427460i	−0.976893	−	0.213730i	$-0.931439\pi$
$431$		−	8.87429i		−	0.427460i	0.976893	−	0.213730i	$-0.0685612\pi$
$432$		0			0
$433$	−0.314195			−0.0150992			−0.00754961	−	0.999972i	$-0.502403\pi$
$433$	−0.314195			−0.0150992			−0.00754961	+	0.999972i	$0.502403\pi$
$434$		0			0
$435$			10.2452i			0.491218i
$436$		0			0
$437$			5.39378i			0.258019i
$438$		0			0
$439$	38.0722			1.81709			0.908543	−	0.417792i	$-0.137196\pi$
$439$	38.0722			1.81709			0.908543	+	0.417792i	$0.137196\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$		0			0
$443$	−21.0385			−0.999569			−0.499785	−	0.866150i	$-0.666588\pi$
$443$	−21.0385			−0.999569			−0.499785	+	0.866150i	$0.666588\pi$
$444$		0			0
$445$	−2.71705			−0.128801
$446$		0			0
$447$			3.24209i			0.153346i
$448$		0			0
$449$		−	12.6831i		−	0.598554i	−0.954166	−	0.299277i	$-0.903254\pi$
$449$		−	12.6831i		−	0.598554i	0.954166	−	0.299277i	$-0.0967455\pi$
$450$		0			0
$451$	−5.81506			−0.273820
$452$		0			0
$453$			11.4923i			0.539956i
$454$		0			0
$455$	−8.64721	−	2.00000i	−0.405388	−	0.0937614i
$456$		0			0
$457$			29.7982i			1.39390i	0.717119	+	0.696951i	$0.245460\pi$
$457$			29.7982i			1.39390i	−0.717119	+	0.696951i	$0.754540\pi$
$458$		0			0
$459$	−2.16196			−0.100912
$460$		0			0
$461$		−	40.3499i		−	1.87928i	−0.342163	−	0.939641i	$-0.611159\pi$
$461$		−	40.3499i		−	1.87928i	0.342163	−	0.939641i	$-0.388841\pi$
$462$		0			0
$463$			10.6713i			0.495939i	0.968768	+	0.247970i	$0.0797633\pi$
$463$			10.6713i			0.495939i	−0.968768	+	0.247970i	$0.920237\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	34.4601			1.59462			0.797311	−	0.603568i	$-0.206255\pi$
$467$	34.4601			1.59462			0.797311	+	0.603568i	$0.206255\pi$
$468$		0			0
$469$	9.92967			0.458509
$470$		0			0
$471$	12.4859			0.575319
$472$		0			0
$473$		−	4.45269i		−	0.204735i
$474$		0			0
$475$			6.61713i			0.303615i
$476$		0			0
$477$	9.48388			0.434237
$478$		0			0
$479$		−	9.30053i		−	0.424952i	−0.977166	−	0.212476i	$-0.931847\pi$
$479$		−	9.30053i		−	0.424952i	0.977166	−	0.212476i	$-0.0681528\pi$
$480$		0			0
$481$	−1.00643	+	4.35142i	−0.0458894	+	0.198408i
$482$		0			0
$483$		−	0.863673i		−	0.0392985i
$484$		0			0
$485$	−34.3368			−1.55915
$486$		0			0
$487$		−	24.1664i		−	1.09508i	−0.836778	−	0.547542i	$-0.815564\pi$
$487$		−	24.1664i		−	1.09508i	0.836778	−	0.547542i	$-0.184436\pi$
$488$		0			0
$489$			8.08320i			0.365535i
$490$		0			0
$491$	13.2983			0.600143			0.300072	−	0.953917i	$-0.402989\pi$
$491$	13.2983			0.600143			0.300072	+	0.953917i	$0.402989\pi$
$492$		0			0
$493$	8.99800			0.405250
$494$		0			0
$495$	−2.05956			−0.0925704
$496$		0			0
$497$	3.68114			0.165122
$498$		0			0
$499$			4.56339i			0.204285i	0.994770	+	0.102143i	$0.0325698\pi$
$499$			4.56339i			0.204285i	−0.994770	+	0.102143i	$0.967430\pi$
$500$		0			0
$501$			7.72398i			0.345082i
$502$		0			0
$503$	−7.36902			−0.328568			−0.164284	−	0.986413i	$-0.552531\pi$
$503$	−7.36902			−0.328568			−0.164284	+	0.986413i	$0.552531\pi$
$504$		0			0
$505$			5.93215i			0.263977i
$506$		0			0
$507$	−11.6798	−	5.70814i	−0.518717	−	0.253508i
$508$		0			0
$509$		−	20.0867i		−	0.890328i	−0.895449	−	0.445164i	$-0.853145\pi$
$509$		−	20.0867i		−	0.890328i	0.895449	−	0.445164i	$-0.146855\pi$
$510$		0			0
$511$	8.62021			0.381336
$512$		0			0
$513$			6.24516i			0.275730i
$514$		0			0
$515$			31.4923i			1.38772i
$516$		0			0
$517$	−4.18221			−0.183933
$518$		0			0
$519$	15.6030			0.684896
$520$		0			0
$521$	−36.1917			−1.58559			−0.792793	−	0.609490i	$-0.791374\pi$
$521$	−36.1917			−1.58559			−0.792793	+	0.609490i	$0.791374\pi$
$522$		0			0
$523$	24.3533			1.06489			0.532447	−	0.846464i	$-0.321273\pi$
$523$	24.3533			1.06489			0.532447	+	0.846464i	$0.321273\pi$
$524$		0			0
$525$		−	1.05956i		−	0.0462431i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−22.2541			−0.967568
$530$		0			0
$531$			10.0051i			0.434183i
$532$		0			0
$533$	5.64689	−	24.4149i	0.244594	−	1.05753i
$534$		0			0
$535$			17.2944i			0.747703i
$536$		0			0
$537$	13.0385			0.562653
$538$		0			0
$539$			0.836670i			0.0360379i
$540$		0			0
$541$			21.7830i			0.936526i	0.883589	+	0.468263i	$0.155120\pi$
$541$			21.7830i			0.936526i	−0.883589	+	0.468263i	$0.844880\pi$
$542$		0			0
$543$	15.9837			0.685925
$544$		0			0
$545$	11.0493			0.473299
$546$		0			0
$547$	1.30116			0.0556338			0.0278169	−	0.999613i	$-0.491144\pi$
$547$	1.30116			0.0556338			0.0278169	+	0.999613i	$0.491144\pi$
$548$		0			0
$549$	9.87011			0.421246
$550$		0			0
$551$		−	25.9921i		−	1.10730i
$552$		0			0
$553$		−	4.30472i		−	0.183055i
$554$		0			0
$555$	−3.04927			−0.129434
$556$		0			0
$557$		−	6.73957i		−	0.285565i	−0.989754	−	0.142782i	$-0.954395\pi$
$557$		−	6.73957i		−	0.285565i	0.989754	−	0.142782i	$-0.0456049\pi$
$558$		0			0
$559$	18.6949	+	4.32392i	0.790712	+	0.182882i
$560$		0			0
$561$			1.80885i			0.0763696i
$562$		0			0
$563$	−25.3861			−1.06989			−0.534947	−	0.844885i	$-0.679669\pi$
$563$	−25.3861			−1.06989			−0.534947	+	0.844885i	$0.679669\pi$
$564$		0			0
$565$			21.5711i			0.907502i
$566$		0			0
$567$		−	1.00000i		−	0.0419961i
$568$		0			0
$569$	−39.8410			−1.67022			−0.835111	−	0.550082i	$-0.814597\pi$
$569$	−39.8410			−1.67022			−0.835111	+	0.550082i	$0.814597\pi$
$570$		0			0
$571$	−9.54175			−0.399310			−0.199655	−	0.979866i	$-0.563982\pi$
$571$	−9.54175			−0.399310			−0.199655	+	0.979866i	$0.563982\pi$
$572$		0			0
$573$	20.6039			0.860740
$574$		0			0
$575$	0.915116			0.0381630
$576$		0			0
$577$			23.5187i			0.979096i	0.871976	+	0.489548i	$0.162838\pi$
$577$			23.5187i			0.979096i	−0.871976	+	0.489548i	$0.837162\pi$
$578$		0			0
$579$			1.55422i			0.0645910i
$580$		0			0
$581$	−6.60438			−0.273996
$582$		0			0
$583$		−	7.93488i		−	0.328629i
$584$		0			0
$585$	2.00000	−	8.64721i	0.0826898	−	0.357518i
$586$		0			0
$587$			42.7002i			1.76242i	0.472720	+	0.881212i	$0.343272\pi$
$587$			42.7002i			1.76242i	−0.472720	+	0.881212i	$0.656728\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		−	2.31212i		−	0.0951080i
$592$		0			0
$593$			26.1461i			1.07369i	0.843680	+	0.536846i	$0.180385\pi$
$593$			26.1461i			1.07369i	−0.843680	+	0.536846i	$0.819615\pi$
$594$		0			0
$595$	−5.32192			−0.218178
$596$		0			0
$597$	−4.63941			−0.189879
$598$		0			0
$599$	−10.5438			−0.430806			−0.215403	−	0.976525i	$-0.569107\pi$
$599$	−10.5438			−0.430806			−0.215403	+	0.976525i	$0.569107\pi$
$600$		0			0
$601$	14.9357			0.609240			0.304620	−	0.952474i	$-0.401471\pi$
$601$	14.9357			0.609240			0.304620	+	0.952474i	$0.401471\pi$
$602$		0			0
$603$			9.92967i			0.404367i
$604$		0			0
$605$		−	25.3546i		−	1.03081i
$606$		0			0
$607$	2.33743			0.0948733			0.0474367	−	0.998874i	$-0.484895\pi$
$607$	2.33743			0.0948733			0.0474367	+	0.998874i	$0.484895\pi$
$608$		0			0
$609$			4.16196i			0.168651i
$610$		0			0
$611$	4.06126	−	17.5593i	0.164301	−	0.710372i
$612$		0			0
$613$			38.7603i			1.56551i	0.622328	+	0.782757i	$0.286187\pi$
$613$			38.7603i			1.56551i	−0.622328	+	0.782757i	$0.713813\pi$
$614$		0			0
$615$	17.1088			0.689895
$616$		0			0
$617$			42.3184i			1.70367i	0.523808	+	0.851836i	$0.324511\pi$
$617$			42.3184i			1.70367i	−0.523808	+	0.851836i	$0.675489\pi$
$618$		0			0
$619$			38.7266i			1.55655i	0.627921	+	0.778277i	$0.283906\pi$
$619$			38.7266i			1.55655i	−0.627921	+	0.778277i	$0.716094\pi$
$620$		0			0
$621$	0.863673			0.0346580
$622$		0			0
$623$	−1.10377			−0.0442215
$624$		0			0
$625$	−29.1751			−1.16701
$626$		0			0
$627$	5.22514			0.208672
$628$		0			0
$629$			2.67808i			0.106782i
$630$		0			0
$631$		−	12.2964i		−	0.489513i	−0.969585	−	0.244757i	$-0.921292\pi$
$631$		−	12.2964i		−	0.489513i	0.969585	−	0.244757i	$-0.0787080\pi$
$632$		0			0
$633$	5.13633			0.204151
$634$		0			0
$635$		−	5.64167i		−	0.223883i
$636$		0			0
$637$	−3.51282	−	0.812474i	−0.139183	−	0.0321914i
$638$		0			0
$639$			3.68114i			0.145624i
$640$		0			0
$641$	−33.0285			−1.30455			−0.652274	−	0.757983i	$-0.726185\pi$
$641$	−33.0285			−1.30455			−0.652274	+	0.757983i	$0.726185\pi$
$642$		0			0
$643$		−	8.63020i		−	0.340342i	−0.985415	−	0.170171i	$-0.945568\pi$
$643$		−	8.63020i		−	0.340342i	0.985415	−	0.170171i	$-0.0544320\pi$
$644$		0			0
$645$			13.1005i			0.515833i
$646$		0			0
$647$	44.1833			1.73702			0.868512	−	0.495669i	$-0.165077\pi$
$647$	44.1833			1.73702			0.868512	+	0.495669i	$0.165077\pi$
$648$		0			0
$649$	8.37094			0.328588
$650$		0			0
$651$		0			0
$652$		0			0
$653$	39.2489			1.53593			0.767963	−	0.640494i	$-0.221270\pi$
$653$	39.2489			1.53593			0.767963	+	0.640494i	$0.221270\pi$
$654$		0			0
$655$			4.74593i			0.185439i
$656$		0			0
$657$			8.62021i			0.336306i
$658$		0			0
$659$	36.8666			1.43612			0.718060	−	0.695981i	$-0.245030\pi$
$659$	36.8666			1.43612			0.718060	+	0.695981i	$0.245030\pi$
$660$		0			0
$661$		−	0.253191i		−	0.00984800i	−0.999988	−	0.00492400i	$-0.998433\pi$
$661$		−	0.253191i		−	0.00984800i	0.999988	−	0.00492400i	$-0.00156736\pi$
$662$		0			0
$663$	−7.59458	−	1.75654i	−0.294949	−	0.0682182i
$664$		0			0
$665$			15.3732i			0.596147i
$666$		0			0
$667$	−3.59458			−0.139183
$668$		0			0
$669$		−	9.99726i		−	0.386517i
$670$		0			0
$671$		−	8.25802i		−	0.318797i
$672$		0			0
$673$	−5.85436			−0.225669			−0.112835	−	0.993614i	$-0.535993\pi$
$673$	−5.85436			−0.225669			−0.112835	+	0.993614i	$0.535993\pi$
$674$		0			0
$675$	1.05956			0.0407826
$676$		0			0
$677$	−10.2853			−0.395295			−0.197647	−	0.980273i	$-0.563330\pi$
$677$	−10.2853			−0.395295			−0.197647	+	0.980273i	$0.563330\pi$
$678$		0			0
$679$	−13.9489			−0.535309
$680$		0			0
$681$			11.7324i			0.449587i
$682$		0			0
$683$			25.9816i			0.994157i	0.867706	+	0.497079i	$0.165594\pi$
$683$			25.9816i			0.994157i	−0.867706	+	0.497079i	$0.834406\pi$
$684$		0			0
$685$	−29.7458			−1.13653
$686$		0			0
$687$		−	7.95161i		−	0.303373i
$688$		0			0
$689$	33.3152	+	7.70541i	1.26921	+	0.293553i
$690$		0			0
$691$		−	11.9133i		−	0.453205i	−0.973987	−	0.226602i	$-0.927238\pi$
$691$		−	11.9133i		−	0.453205i	0.973987	−	0.226602i	$-0.0727618\pi$
$692$		0			0
$693$	−0.836670			−0.0317825
$694$		0			0
$695$		−	54.6607i		−	2.07340i
$696$		0			0
$697$		−	15.0261i		−	0.569156i
$698$		0			0
$699$	−2.75010			−0.104019
$700$		0			0
$701$	14.8818			0.562079			0.281040	−	0.959696i	$-0.409321\pi$
$701$	14.8818			0.562079			0.281040	+	0.959696i	$0.409321\pi$
$702$		0			0
$703$	7.73604			0.291770
$704$		0			0
$705$	12.3047			0.463423
$706$		0			0
$707$			2.40986i			0.0906320i
$708$		0			0
$709$			29.0988i			1.09283i	0.837515	+	0.546415i	$0.184008\pi$
$709$			29.0988i			1.09283i	−0.837515	+	0.546415i	$0.815992\pi$
$710$		0			0
$711$	4.30472			0.161440
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−7.23487	−	1.67334i	−0.270569	−	0.0625794i
$716$		0			0
$717$			22.4893i			0.839877i
$718$		0			0
$719$	12.5119			0.466614			0.233307	−	0.972403i	$-0.425045\pi$
$719$	12.5119			0.466614			0.233307	+	0.972403i	$0.425045\pi$
$720$		0			0
$721$			12.7933i			0.476449i
$722$		0			0
$723$		−	11.4647i		−	0.426376i
$724$		0			0
$725$	−4.40986			−0.163778
$726$		0			0
$727$	−0.633659			−0.0235011			−0.0117506	−	0.999931i	$-0.503740\pi$
$727$	−0.633659			−0.0235011			−0.0117506	+	0.999931i	$0.503740\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	11.5058			0.425557
$732$		0			0
$733$			30.4101i			1.12322i	0.827401	+	0.561612i	$0.189818\pi$
$733$			30.4101i			1.12322i	−0.827401	+	0.561612i	$0.810182\pi$
$734$		0			0
$735$		−	2.46162i		−	0.0907981i
$736$		0			0
$737$	8.30786			0.306024
$738$		0			0
$739$			1.87129i			0.0688364i	0.999408	+	0.0344182i	$0.0109578\pi$
$739$			1.87129i			0.0688364i	−0.999408	+	0.0344182i	$0.989042\pi$
$740$		0			0
$741$	−5.07403	+	21.9381i	−0.186399	+	0.805916i
$742$		0			0
$743$		−	16.0975i		−	0.590559i	−0.955411	−	0.295280i	$-0.904587\pi$
$743$		−	16.0975i		−	0.590559i	0.955411	−	0.295280i	$-0.0954128\pi$
$744$		0			0
$745$	−7.98080			−0.292394
$746$		0			0
$747$		−	6.60438i		−	0.241642i
$748$		0			0
$749$			7.02563i			0.256711i
$750$		0			0
$751$	2.04552			0.0746419			0.0373210	−	0.999303i	$-0.488118\pi$
$751$	2.04552			0.0746419			0.0373210	+	0.999303i	$0.488118\pi$
$752$		0			0
$753$	−11.0493			−0.402658
$754$		0			0
$755$	−28.2897			−1.02957
$756$		0			0
$757$	25.4422			0.924714			0.462357	−	0.886694i	$-0.347004\pi$
$757$	25.4422			0.924714			0.462357	+	0.886694i	$0.347004\pi$
$758$		0			0
$759$		−	0.722610i		−	0.0262291i
$760$		0			0
$761$		−	19.3920i		−	0.702960i	−0.936195	−	0.351480i	$-0.885678\pi$
$761$		−	19.3920i		−	0.702960i	0.936195	−	0.351480i	$-0.114322\pi$
$762$		0			0
$763$	4.48862			0.162499
$764$		0			0
$765$		−	5.32192i		−	0.192415i
$766$		0			0
$767$	−8.12885	+	35.1460i	−0.293516	+	1.26905i
$768$		0			0
$769$		−	22.3610i		−	0.806359i	−0.915121	−	0.403180i	$-0.867905\pi$
$769$		−	22.3610i		−	0.806359i	0.915121	−	0.403180i	$-0.132095\pi$
$770$		0			0
$771$	0.563385			0.0202898
$772$		0			0
$773$		−	20.9547i		−	0.753687i	−0.926277	−	0.376844i	$-0.877009\pi$
$773$		−	20.9547i		−	0.753687i	0.926277	−	0.376844i	$-0.122991\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−1.23873			−0.0444390
$778$		0			0
$779$	−43.4053			−1.55516
$780$		0			0
$781$	3.07990			0.110208
$782$		0			0
$783$	−4.16196			−0.148736
$784$		0			0
$785$			30.7355i			1.09700i
$786$		0			0
$787$			44.3606i			1.58128i	0.612279	+	0.790642i	$0.290253\pi$
$787$			44.3606i			1.58128i	−0.612279	+	0.790642i	$0.709747\pi$
$788$		0			0
$789$	13.2691			0.472392
$790$		0			0
$791$			8.76297i			0.311575i
$792$		0			0
$793$	34.6719	+	8.01920i	1.23123	+	0.284770i
$794$		0			0
$795$			23.3457i			0.827987i
$796$		0			0
$797$	−43.1992			−1.53019			−0.765096	−	0.643916i	$-0.777309\pi$
$797$	−43.1992			−1.53019			−0.765096	+	0.643916i	$0.777309\pi$
$798$		0			0
$799$		−	10.8068i		−	0.382319i
$800$		0			0
$801$		−	1.10377i		−	0.0389997i
$802$		0			0
$803$	7.21227			0.254516
$804$		0			0
$805$	2.12603			0.0749329
$806$		0			0
$807$	−22.2469			−0.783126
$808$		0			0
$809$	13.3690			0.470030			0.235015	−	0.971992i	$-0.424486\pi$
$809$	13.3690			0.470030			0.235015	+	0.971992i	$0.424486\pi$
$810$		0			0
$811$			34.7333i			1.21965i	0.792535	+	0.609826i	$0.208761\pi$
$811$			34.7333i			1.21965i	−0.792535	+	0.609826i	$0.791239\pi$
$812$		0			0
$813$			23.2984i			0.817112i
$814$		0			0
$815$	−19.8977			−0.696987
$816$		0			0
$817$		−	33.2362i		−	1.16279i
$818$		0			0
$819$	0.812474	−	3.51282i	0.0283901	−	0.122748i
$820$		0			0
$821$		−	1.39901i		−	0.0488259i	−0.999702	−	0.0244130i	$-0.992228\pi$
$821$		−	1.39901i		−	0.0488259i	0.999702	−	0.0244130i	$-0.00777166\pi$
$822$		0			0
$823$	6.21191			0.216534			0.108267	−	0.994122i	$-0.465470\pi$
$823$	6.21191			0.216534			0.108267	+	0.994122i	$0.465470\pi$
$824$		0			0
$825$		−	0.886504i		−	0.0308641i
$826$		0			0
$827$			32.7661i			1.13939i	0.821857	+	0.569694i	$0.192938\pi$
$827$			32.7661i			1.13939i	−0.821857	+	0.569694i	$0.807062\pi$
$828$		0			0
$829$	−49.1841			−1.70824			−0.854118	−	0.520080i	$-0.825902\pi$
$829$	−49.1841			−1.70824			−0.854118	+	0.520080i	$0.825902\pi$
$830$		0			0
$831$	25.2917			0.877359
$832$		0			0
$833$	−2.16196			−0.0749075
$834$		0			0
$835$	−19.0135			−0.657989
$836$		0			0
$837$		0			0
$838$		0			0
$839$			41.9409i			1.44796i	0.689821	+	0.723980i	$0.257689\pi$
$839$			41.9409i			1.44796i	−0.689821	+	0.723980i	$0.742311\pi$
$840$		0			0
$841$	−11.6781			−0.402692
$842$		0			0
$843$		−	13.5620i		−	0.467101i
$844$		0			0
$845$	14.0513	−	28.7511i	0.483378	−	0.989069i
$846$		0			0
$847$		−	10.3000i		−	0.353912i
$848$		0			0
$849$	−27.7402			−0.952042
$850$		0			0
$851$		−	1.06985i		−	0.0366741i
$852$		0			0
$853$		−	20.1889i		−	0.691254i	−0.938372	−	0.345627i	$-0.887666\pi$
$853$		−	20.1889i		−	0.691254i	0.938372	−	0.345627i	$-0.112334\pi$
$854$		0			0
$855$	−15.3732			−0.525752
$856$		0			0
$857$	26.7561			0.913972			0.456986	−	0.889474i	$-0.348929\pi$
$857$	26.7561			0.913972			0.456986	+	0.889474i	$0.348929\pi$
$858$		0			0
$859$	−9.78573			−0.333885			−0.166942	−	0.985967i	$-0.553389\pi$
$859$	−9.78573			−0.333885			−0.166942	+	0.985967i	$0.553389\pi$
$860$		0			0
$861$	6.95024			0.236864
$862$		0			0
$863$			28.9756i			0.986340i	0.869933	+	0.493170i	$0.164162\pi$
$863$			28.9756i			0.986340i	−0.869933	+	0.493170i	$0.835838\pi$
$864$		0			0
$865$			38.4086i			1.30593i
$866$		0			0
$867$	12.3259			0.418610
$868$		0			0
$869$		−	3.60163i		−	0.122177i
$870$		0			0
$871$	−8.06760	+	34.8811i	−0.273360	+	1.18190i
$872$		0			0
$873$		−	13.9489i		−	0.472098i
$874$		0			0
$875$	−9.69985			−0.327915
$876$		0			0
$877$			20.4557i			0.690738i	0.938467	+	0.345369i	$0.112246\pi$
$877$			20.4557i			0.690738i	−0.938467	+	0.345369i	$0.887754\pi$
$878$		0			0
$879$			19.1910i			0.647295i
$880$		0			0
$881$	−5.93482			−0.199949			−0.0999746	−	0.994990i	$-0.531876\pi$
$881$	−5.93482			−0.199949			−0.0999746	+	0.994990i	$0.531876\pi$
$882$		0			0
$883$	−37.5075			−1.26223			−0.631114	−	0.775690i	$-0.717402\pi$
$883$	−37.5075			−1.26223			−0.631114	+	0.775690i	$0.717402\pi$
$884$		0			0
$885$	−24.6286			−0.827883
$886$		0			0
$887$	−9.28430			−0.311736			−0.155868	−	0.987778i	$-0.549817\pi$
$887$	−9.28430			−0.311736			−0.155868	+	0.987778i	$0.549817\pi$
$888$		0			0
$889$		−	2.29186i		−	0.0768664i
$890$		0			0
$891$		−	0.836670i		−	0.0280295i
$892$		0			0
$893$	−31.2172			−1.04465
$894$		0			0
$895$			32.0958i			1.07284i
$896$		0			0
$897$	3.03393	+	0.701712i	0.101300	+	0.0234295i
$898$		0			0
$899$		0			0
$900$		0			0
$901$	20.5038			0.683080
$902$		0			0
$903$			5.32192i			0.177102i
$904$		0			0
$905$			39.3457i			1.30790i
$906$		0			0
$907$	−17.9793			−0.596994			−0.298497	−	0.954411i	$-0.596485\pi$
$907$	−17.9793			−0.596994			−0.298497	+	0.954411i	$0.596485\pi$
$908$		0			0
$909$	−2.40986			−0.0799299
$910$		0			0
$911$	33.3869			1.10616			0.553079	−	0.833129i	$-0.313453\pi$
$911$	33.3869			1.10616			0.553079	+	0.833129i	$0.313453\pi$
$912$		0			0
$913$	−5.52568			−0.182873
$914$		0			0
$915$			24.2964i			0.803215i
$916$		0			0
$917$			1.92797i			0.0636673i
$918$		0			0
$919$	32.1417			1.06026			0.530129	−	0.847917i	$-0.322144\pi$
$919$	32.1417			1.06026			0.530129	+	0.847917i	$0.322144\pi$
$920$		0			0
$921$		−	23.2444i		−	0.765930i
$922$		0			0
$923$	−2.99083	+	12.9312i	−0.0984444	+	0.425635i
$924$		0			0
$925$		−	1.31251i		−	0.0431550i
$926$		0			0
$927$	−12.7933			−0.420188
$928$		0			0
$929$			55.2459i			1.81256i	0.422679	+	0.906279i	$0.361090\pi$
$929$			55.2459i			1.81256i	−0.422679	+	0.906279i	$0.638910\pi$
$930$		0			0
$931$			6.24516i			0.204677i
$932$		0			0
$933$	12.3463			0.404198
$934$		0			0
$935$	−4.45269			−0.145619
$936$		0			0
$937$	56.8598			1.85753			0.928764	−	0.370672i	$-0.120872\pi$
$937$	56.8598			1.85753			0.928764	+	0.370672i	$0.120872\pi$
$938$		0			0
$939$	−15.9129			−0.519299
$940$		0			0
$941$		−	40.1560i		−	1.30905i	−0.756041	−	0.654524i	$-0.772869\pi$
$941$		−	40.1560i		−	1.30905i	0.756041	−	0.654524i	$-0.227131\pi$
$942$		0			0
$943$			6.00274i			0.195476i
$944$		0			0
$945$	2.46162			0.0800764
$946$		0			0
$947$		−	8.98746i		−	0.292053i	−0.989281	−	0.146027i	$-0.953351\pi$
$947$		−	8.98746i		−	0.292053i	0.989281	−	0.146027i	$-0.0466485\pi$
$948$		0			0
$949$	−7.00370	+	30.2812i	−0.227350	+	0.982970i
$950$		0			0
$951$		−	13.3741i		−	0.433684i
$952$		0			0
$953$	12.3975			0.401595			0.200798	−	0.979633i	$-0.435647\pi$
$953$	12.3975			0.401595			0.200798	+	0.979633i	$0.435647\pi$
$954$		0			0
$955$			50.7189i			1.64122i
$956$		0			0
$957$			3.48219i			0.112563i
$958$		0			0
$959$	−12.0838			−0.390207
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$	−7.02563			−0.226398
$964$		0			0
$965$	−3.82589			−0.123160
$966$		0			0
$967$		−	3.54057i		−	0.113857i	−0.998378	−	0.0569285i	$-0.981869\pi$
$967$		−	3.54057i		−	0.113857i	0.998378	−	0.0569285i	$-0.0181307\pi$
$968$		0			0
$969$			13.5018i			0.433740i
$970$		0			0
$971$	14.1927			0.455466			0.227733	−	0.973724i	$-0.426869\pi$
$971$	14.1927			0.455466			0.227733	+	0.973724i	$0.426869\pi$
$972$		0			0
$973$		−	22.2052i		−	0.711866i
$974$		0			0
$975$	3.72205	+	0.860866i	0.119201	+	0.0275698i
$976$		0			0
$977$		−	36.3144i		−	1.16180i	−0.813975	−	0.580900i	$-0.802701\pi$
$977$		−	36.3144i		−	1.16180i	0.813975	−	0.580900i	$-0.197299\pi$
$978$		0			0
$979$	−0.923490			−0.0295149
$980$		0			0
$981$			4.48862i			0.143311i
$982$		0			0
$983$		−	0.544955i		−	0.0173814i	−0.999962	−	0.00869068i	$-0.997234\pi$
$983$		−	0.544955i		−	0.0173814i	0.999962	−	0.00869068i	$-0.00276636\pi$
$984$		0			0
$985$	5.69156			0.181348
$986$		0			0
$987$	4.99863			0.159108
$988$		0			0
$989$	−4.59640			−0.146157
$990$		0			0
$991$	−33.1348			−1.05256			−0.526280	−	0.850311i	$-0.676414\pi$
$991$	−33.1348			−1.05256			−0.526280	+	0.850311i	$0.676414\pi$
$992$		0			0
$993$		−	24.0933i		−	0.764579i
$994$		0			0
$995$		−	11.4205i		−	0.362053i
$996$		0			0
$997$	23.4967			0.744149			0.372075	−	0.928203i	$-0.378647\pi$
$997$	23.4967			0.744149			0.372075	+	0.928203i	$0.378647\pi$
$998$		0			0
$999$		−	1.23873i		−	0.0391916i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.h.s.337.2		10
4.3	odd	2		2184.2.h.g.337.2	✓	10
13.12	even	2	inner	4368.2.h.s.337.9		10
52.51	odd	2		2184.2.h.g.337.9	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2184.2.h.g.337.2	✓	10	4.3	odd	2
2184.2.h.g.337.9	yes	10	52.51	odd	2
4368.2.h.s.337.2		10	1.1	even	1	trivial
4368.2.h.s.337.9		10	13.12	even	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:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.46162i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.46162i q^{5} -1.00000i q^{7} +1.00000 q^{9} -0.836670i q^{11} +(3.51282 + 0.812474i) q^{13} +2.46162i q^{15} +2.16196 q^{17} -6.24516i q^{19} +1.00000i q^{21} -0.863673 q^{23} -1.05956 q^{25} -1.00000 q^{27} +4.16196 q^{29} +0.836670i q^{33} -2.46162 q^{35} +1.23873i q^{37} +(-3.51282 - 0.812474i) q^{39} -6.95024i q^{41} +5.32192 q^{43} -2.46162i q^{45} -4.99863i q^{47} -1.00000 q^{49} -2.16196 q^{51} +9.48388 q^{53} -2.05956 q^{55} +6.24516i q^{57} +10.0051i q^{59} +9.87011 q^{61} -1.00000i q^{63} +(2.00000 - 8.64721i) q^{65} +9.92967i q^{67} +0.863673 q^{69} +3.68114i q^{71} +8.62021i q^{73} +1.05956 q^{75} -0.836670 q^{77} +4.30472 q^{79} +1.00000 q^{81} -6.60438i q^{83} -5.32192i q^{85} -4.16196 q^{87} -1.10377i q^{89} +(0.812474 - 3.51282i) q^{91} -15.3732 q^{95} -13.9489i q^{97} -0.836670i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{3} + 10 q^{9}+O(q^{10}) \) 10 * q - 10 * q^3 + 10 * q^9	\( 10 q - 10 q^{3} + 10 q^{9} - 22 q^{17} + 18 q^{23} - 8 q^{25} - 10 q^{27} - 2 q^{29} - 10 q^{35} + 2 q^{43} - 10 q^{49} + 22 q^{51} - 18 q^{55} + 6 q^{61} + 20 q^{65} - 18 q^{69} + 8 q^{75} - 6 q^{77} - 40 q^{79} + 10 q^{81} + 2 q^{87} + 2 q^{91} + 38 q^{95}+O(q^{100}) \) 10 * q - 10 * q^3 + 10 * q^9 - 22 * q^17 + 18 * q^23 - 8 * q^25 - 10 * q^27 - 2 * q^29 - 10 * q^35 + 2 * q^43 - 10 * q^49 + 22 * q^51 - 18 * q^55 + 6 * q^61 + 20 * q^65 - 18 * q^69 + 8 * q^75 - 6 * q^77 - 40 * q^79 + 10 * q^81 + 2 * q^87 + 2 * q^91 + 38 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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