LMFDB - Embedded newform 1710.2.d.a.1369.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1710,2,Mod(1369,1710)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1710, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1710.1369");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1710.d (of order $2$, degree $1$, 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:	$13.6544187456$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 570)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1369.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1710.1369
Dual form			1710.2.d.a.1369.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +(-2.00000 + 1.00000i) q^{5} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} +(-2.00000 + 1.00000i) q^{5} -1.00000i q^{8} +(-1.00000 - 2.00000i) q^{10} +4.00000 q^{11} -4.00000i q^{13} +1.00000 q^{16} +6.00000i q^{17} +1.00000 q^{19} +(2.00000 - 1.00000i) q^{20} +4.00000i q^{22} +(3.00000 - 4.00000i) q^{25} +4.00000 q^{26} -2.00000 q^{29} +1.00000i q^{32} -6.00000 q^{34} +4.00000i q^{37} +1.00000i q^{38} +(1.00000 + 2.00000i) q^{40} +12.0000 q^{41} +6.00000i q^{43} -4.00000 q^{44} +7.00000 q^{49} +(4.00000 + 3.00000i) q^{50} +4.00000i q^{52} +14.0000i q^{53} +(-8.00000 + 4.00000i) q^{55} -2.00000i q^{58} -10.0000 q^{59} -6.00000 q^{61} -1.00000 q^{64} +(4.00000 + 8.00000i) q^{65} +4.00000i q^{67} -6.00000i q^{68} -12.0000 q^{71} +8.00000i q^{73} -4.00000 q^{74} -1.00000 q^{76} +8.00000 q^{79} +(-2.00000 + 1.00000i) q^{80} +12.0000i q^{82} -12.0000i q^{83} +(-6.00000 - 12.0000i) q^{85} -6.00000 q^{86} -4.00000i q^{88} -8.00000 q^{89} +(-2.00000 + 1.00000i) q^{95} +10.0000i q^{97} +7.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 4 q^{5}+O(q^{10}) $ 2 * q - 2 * q^4 - 4 * q^5	$ 2 q - 2 q^{4} - 4 q^{5} - 2 q^{10} + 8 q^{11} + 2 q^{16} + 2 q^{19} + 4 q^{20} + 6 q^{25} + 8 q^{26} - 4 q^{29} - 12 q^{34} + 2 q^{40} + 24 q^{41} - 8 q^{44} + 14 q^{49} + 8 q^{50} - 16 q^{55} - 20 q^{59} - 12 q^{61} - 2 q^{64} + 8 q^{65} - 24 q^{71} - 8 q^{74} - 2 q^{76} + 16 q^{79} - 4 q^{80} - 12 q^{85} - 12 q^{86} - 16 q^{89} - 4 q^{95}+O(q^{100}) $ 2 * q - 2 * q^4 - 4 * q^5 - 2 * q^10 + 8 * q^11 + 2 * q^16 + 2 * q^19 + 4 * q^20 + 6 * q^25 + 8 * q^26 - 4 * q^29 - 12 * q^34 + 2 * q^40 + 24 * q^41 - 8 * q^44 + 14 * q^49 + 8 * q^50 - 16 * q^55 - 20 * q^59 - 12 * q^61 - 2 * q^64 + 8 * q^65 - 24 * q^71 - 8 * q^74 - 2 * q^76 + 16 * q^79 - 4 * q^80 - 12 * q^85 - 12 * q^86 - 16 * q^89 - 4 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1710\mathbb{Z}\right)^\times$.

$n$	$191$	$1027$	$1351$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	−2.00000	+	1.00000i	−0.894427	+	0.447214i
$5$	−2.00000	+	1.00000i	−0.894427	+	0.447214i
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	−1.00000	−	2.00000i	−0.316228	−	0.632456i
$11$	4.00000			1.20605			0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000			1.20605			0.603023	+	0.797724i	$0.293963\pi$
$12$		0			0
$13$		−	4.00000i		−	1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$		−	4.00000i		−	1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$		0			0
$15$		0			0
$16$	1.00000			0.250000
$17$			6.00000i			1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$			6.00000i			1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$		0			0
$19$	1.00000			0.229416
$19$	1.00000			0.229416
$20$	2.00000	−	1.00000i	0.447214	−	0.223607i
$21$		0			0
$22$			4.00000i			0.852803i
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$	4.00000			0.784465
$27$		0			0
$28$		0			0
$29$	−2.00000			−0.371391			−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000			−0.371391			−0.185695	+	0.982607i	$0.559454\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$	−6.00000			−1.02899
$35$		0			0
$36$		0			0
$37$			4.00000i			0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$			4.00000i			0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$			1.00000i			0.162221i
$39$		0			0
$40$	1.00000	+	2.00000i	0.158114	+	0.316228i
$41$	12.0000			1.87409			0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000			1.87409			0.937043	+	0.349215i	$0.113552\pi$
$42$		0			0
$43$			6.00000i			0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$			6.00000i			0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$	−4.00000			−0.603023
$45$		0			0
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$	4.00000	+	3.00000i	0.565685	+	0.424264i
$51$		0			0
$52$			4.00000i			0.554700i
$53$			14.0000i			1.92305i	0.274721	+	0.961524i	$0.411414\pi$
$53$			14.0000i			1.92305i	−0.274721	+	0.961524i	$0.588586\pi$
$54$		0			0
$55$	−8.00000	+	4.00000i	−1.07872	+	0.539360i
$56$		0			0
$57$		0			0
$58$		−	2.00000i		−	0.262613i
$59$	−10.0000			−1.30189			−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000			−1.30189			−0.650945	+	0.759125i	$0.725627\pi$
$60$		0			0
$61$	−6.00000			−0.768221			−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000			−0.768221			−0.384111	+	0.923287i	$0.625492\pi$
$62$		0			0
$63$		0			0
$64$	−1.00000			−0.125000
$65$	4.00000	+	8.00000i	0.496139	+	0.992278i
$66$		0			0
$67$			4.00000i			0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$			4.00000i			0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$		−	6.00000i		−	0.727607i
$69$		0			0
$70$		0			0
$71$	−12.0000			−1.42414			−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000			−1.42414			−0.712069	+	0.702109i	$0.752242\pi$
$72$		0			0
$73$			8.00000i			0.936329i	0.883641	+	0.468165i	$0.155085\pi$
$73$			8.00000i			0.936329i	−0.883641	+	0.468165i	$0.844915\pi$
$74$	−4.00000			−0.464991
$75$		0			0
$76$	−1.00000			−0.114708
$77$		0			0
$78$		0			0
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$	−2.00000	+	1.00000i	−0.223607	+	0.111803i
$81$		0			0
$82$			12.0000i			1.32518i
$83$		−	12.0000i		−	1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$		−	12.0000i		−	1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$		0			0
$85$	−6.00000	−	12.0000i	−0.650791	−	1.30158i
$86$	−6.00000			−0.646997
$87$		0			0
$88$		−	4.00000i		−	0.426401i
$89$	−8.00000			−0.847998			−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000			−0.847998			−0.423999	+	0.905663i	$0.639374\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.00000	+	1.00000i	−0.205196	+	0.102598i
$96$		0			0
$97$			10.0000i			1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$			10.0000i			1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$			7.00000i			0.707107i
$99$		0			0
$100$	−3.00000	+	4.00000i	−0.300000	+	0.400000i
$101$	−12.0000			−1.19404			−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000			−1.19404			−0.597022	+	0.802225i	$0.703650\pi$
$102$		0			0
$103$			2.00000i			0.197066i	0.995134	+	0.0985329i	$0.0314150\pi$
$103$			2.00000i			0.197066i	−0.995134	+	0.0985329i	$0.968585\pi$
$104$	−4.00000			−0.392232
$105$		0			0
$106$	−14.0000			−1.35980
$107$			20.0000i			1.93347i	0.255774	+	0.966736i	$0.417670\pi$
$107$			20.0000i			1.93347i	−0.255774	+	0.966736i	$0.582330\pi$
$108$		0			0
$109$	14.0000			1.34096			0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000			1.34096			0.670478	+	0.741929i	$0.266089\pi$
$110$	−4.00000	−	8.00000i	−0.381385	−	0.762770i
$111$		0			0
$112$		0			0
$113$			10.0000i			0.940721i	0.882474	+	0.470360i	$0.155876\pi$
$113$			10.0000i			0.940721i	−0.882474	+	0.470360i	$0.844124\pi$
$114$		0			0
$115$		0			0
$116$	2.00000			0.185695
$117$		0			0
$118$		−	10.0000i		−	0.920575i
$119$		0			0
$120$		0			0
$121$	5.00000			0.454545
$122$		−	6.00000i		−	0.543214i
$123$		0			0
$124$		0			0
$125$	−2.00000	+	11.0000i	−0.178885	+	0.983870i
$126$		0			0
$127$			10.0000i			0.887357i	0.896186	+	0.443678i	$0.146327\pi$
$127$			10.0000i			0.887357i	−0.896186	+	0.443678i	$0.853673\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	−8.00000	+	4.00000i	−0.701646	+	0.350823i
$131$	20.0000			1.74741			0.873704	−	0.486458i	$-0.161711\pi$
$131$	20.0000			1.74741			0.873704	+	0.486458i	$0.161711\pi$
$132$		0			0
$133$		0			0
$134$	−4.00000			−0.345547
$135$		0			0
$136$	6.00000			0.514496
$137$		−	6.00000i		−	0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$		−	6.00000i		−	0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$		0			0
$139$	12.0000			1.01783			0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000			1.01783			0.508913	+	0.860818i	$0.330047\pi$
$140$		0			0
$141$		0			0
$142$		−	12.0000i		−	1.00702i
$143$		−	16.0000i		−	1.33799i
$144$		0			0
$145$	4.00000	−	2.00000i	0.332182	−	0.166091i
$146$	−8.00000			−0.662085
$147$		0			0
$148$		−	4.00000i		−	0.328798i
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$	16.0000			1.30206			0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000			1.30206			0.651031	+	0.759051i	$0.274337\pi$
$152$		−	1.00000i		−	0.0811107i
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	6.00000i		−	0.478852i	−0.970915	−	0.239426i	$-0.923041\pi$
$157$		−	6.00000i		−	0.478852i	0.970915	−	0.239426i	$-0.0769593\pi$
$158$			8.00000i			0.636446i
$159$		0			0
$160$	−1.00000	−	2.00000i	−0.0790569	−	0.158114i
$161$		0			0
$162$		0			0
$163$		−	10.0000i		−	0.783260i	−0.920123	−	0.391630i	$-0.871911\pi$
$163$		−	10.0000i		−	0.783260i	0.920123	−	0.391630i	$-0.128089\pi$
$164$	−12.0000			−0.937043
$165$		0			0
$166$	12.0000			0.931381
$167$			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
$169$	−3.00000			−0.230769
$170$	12.0000	−	6.00000i	0.920358	−	0.460179i
$171$		0			0
$172$		−	6.00000i		−	0.457496i
$173$			2.00000i			0.152057i	0.997106	+	0.0760286i	$0.0242240\pi$
$173$			2.00000i			0.152057i	−0.997106	+	0.0760286i	$0.975776\pi$
$174$		0			0
$175$		0			0
$176$	4.00000			0.301511
$177$		0			0
$178$		−	8.00000i		−	0.599625i
$179$	6.00000			0.448461			0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000			0.448461			0.224231	+	0.974536i	$0.428013\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4.00000	−	8.00000i	−0.294086	−	0.588172i
$186$		0			0
$187$			24.0000i			1.75505i
$188$		0			0
$189$		0			0
$190$	−1.00000	−	2.00000i	−0.0725476	−	0.145095i
$191$	−6.00000			−0.434145			−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000			−0.434145			−0.217072	+	0.976156i	$0.569651\pi$
$192$		0			0
$193$		−	10.0000i		−	0.719816i	−0.932988	−	0.359908i	$-0.882808\pi$
$193$		−	10.0000i		−	0.719816i	0.932988	−	0.359908i	$-0.117192\pi$
$194$	−10.0000			−0.717958
$195$		0			0
$196$	−7.00000			−0.500000
$197$		−	10.0000i		−	0.712470i	−0.934396	−	0.356235i	$-0.884060\pi$
$197$		−	10.0000i		−	0.712470i	0.934396	−	0.356235i	$-0.115940\pi$
$198$		0			0
$199$	4.00000			0.283552			0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000			0.283552			0.141776	+	0.989899i	$0.454719\pi$
$200$	−4.00000	−	3.00000i	−0.282843	−	0.212132i
$201$		0			0
$202$		−	12.0000i		−	0.844317i
$203$		0			0
$204$		0			0
$205$	−24.0000	+	12.0000i	−1.67623	+	0.838116i
$206$	−2.00000			−0.139347
$207$		0			0
$208$		−	4.00000i		−	0.277350i
$209$	4.00000			0.276686
$210$		0			0
$211$	−8.00000			−0.550743			−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000			−0.550743			−0.275371	+	0.961338i	$0.588801\pi$
$212$		−	14.0000i		−	0.961524i
$213$		0			0
$214$	−20.0000			−1.36717
$215$	−6.00000	−	12.0000i	−0.409197	−	0.818393i
$216$		0			0
$217$		0			0
$218$			14.0000i			0.948200i
$219$		0			0
$220$	8.00000	−	4.00000i	0.539360	−	0.269680i
$221$	24.0000			1.61441
$222$		0			0
$223$			10.0000i			0.669650i	0.942280	+	0.334825i	$0.108677\pi$
$223$			10.0000i			0.669650i	−0.942280	+	0.334825i	$0.891323\pi$
$224$		0			0
$225$		0			0
$226$	−10.0000			−0.665190
$227$		−	12.0000i		−	0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$		−	12.0000i		−	0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$		0			0
$229$	10.0000			0.660819			0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000			0.660819			0.330409	+	0.943838i	$0.392813\pi$
$230$		0			0
$231$		0			0
$232$			2.00000i			0.131306i
$233$		−	14.0000i		−	0.917170i	−0.888650	−	0.458585i	$-0.848356\pi$
$233$		−	14.0000i		−	0.917170i	0.888650	−	0.458585i	$-0.151644\pi$
$234$		0			0
$235$		0			0
$236$	10.0000			0.650945
$237$		0			0
$238$		0			0
$239$	10.0000			0.646846			0.323423	−	0.946254i	$-0.395166\pi$
$239$	10.0000			0.646846			0.323423	+	0.946254i	$0.395166\pi$
$240$		0			0
$241$	26.0000			1.67481			0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000			1.67481			0.837404	+	0.546585i	$0.184072\pi$
$242$			5.00000i			0.321412i
$243$		0			0
$244$	6.00000			0.384111
$245$	−14.0000	+	7.00000i	−0.894427	+	0.447214i
$246$		0			0
$247$		−	4.00000i		−	0.254514i
$248$		0			0
$249$		0			0
$250$	−11.0000	−	2.00000i	−0.695701	−	0.126491i
$251$	24.0000			1.51487			0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000			1.51487			0.757433	+	0.652913i	$0.226453\pi$
$252$		0			0
$253$		0			0
$254$	−10.0000			−0.627456
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	2.00000i		−	0.124757i	−0.998053	−	0.0623783i	$-0.980131\pi$
$257$		−	2.00000i		−	0.124757i	0.998053	−	0.0623783i	$-0.0198685\pi$
$258$		0			0
$259$		0			0
$260$	−4.00000	−	8.00000i	−0.248069	−	0.496139i
$261$		0			0
$262$			20.0000i			1.23560i
$263$		−	16.0000i		−	0.986602i	−0.869859	−	0.493301i	$-0.835790\pi$
$263$		−	16.0000i		−	0.986602i	0.869859	−	0.493301i	$-0.164210\pi$
$264$		0			0
$265$	−14.0000	−	28.0000i	−0.860013	−	1.72003i
$266$		0			0
$267$		0			0
$268$		−	4.00000i		−	0.244339i
$269$	−2.00000			−0.121942			−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000			−0.121942			−0.0609711	+	0.998140i	$0.519420\pi$
$270$		0			0
$271$	−28.0000			−1.70088			−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000			−1.70088			−0.850439	+	0.526073i	$0.823664\pi$
$272$			6.00000i			0.363803i
$273$		0			0
$274$	6.00000			0.362473
$275$	12.0000	−	16.0000i	0.723627	−	0.964836i
$276$		0			0
$277$			14.0000i			0.841178i	0.907251	+	0.420589i	$0.138177\pi$
$277$			14.0000i			0.841178i	−0.907251	+	0.420589i	$0.861823\pi$
$278$			12.0000i			0.719712i
$279$		0			0
$280$		0			0
$281$	−12.0000			−0.715860			−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000			−0.715860			−0.357930	+	0.933748i	$0.616517\pi$
$282$		0			0
$283$		−	2.00000i		−	0.118888i	−0.998232	−	0.0594438i	$-0.981067\pi$
$283$		−	2.00000i		−	0.118888i	0.998232	−	0.0594438i	$-0.0189327\pi$
$284$	12.0000			0.712069
$285$		0			0
$286$	16.0000			0.946100
$287$		0			0
$288$		0			0
$289$	−19.0000			−1.11765
$290$	2.00000	+	4.00000i	0.117444	+	0.234888i
$291$		0			0
$292$		−	8.00000i		−	0.468165i
$293$		−	14.0000i		−	0.817889i	−0.912559	−	0.408944i	$-0.865897\pi$
$293$		−	14.0000i		−	0.817889i	0.912559	−	0.408944i	$-0.134103\pi$
$294$		0			0
$295$	20.0000	−	10.0000i	1.16445	−	0.582223i
$296$	4.00000			0.232495
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$			16.0000i			0.920697i
$303$		0			0
$304$	1.00000			0.0573539
$305$	12.0000	−	6.00000i	0.687118	−	0.343559i
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	18.0000			1.02069			0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000			1.02069			0.510343	+	0.859971i	$0.329518\pi$
$312$		0			0
$313$			4.00000i			0.226093i	0.993590	+	0.113047i	$0.0360610\pi$
$313$			4.00000i			0.226093i	−0.993590	+	0.113047i	$0.963939\pi$
$314$	6.00000			0.338600
$315$		0			0
$316$	−8.00000			−0.450035
$317$			18.0000i			1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$			18.0000i			1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$		0			0
$319$	−8.00000			−0.447914
$320$	2.00000	−	1.00000i	0.111803	−	0.0559017i
$321$		0			0
$322$		0			0
$323$			6.00000i			0.333849i
$324$		0			0
$325$	−16.0000	−	12.0000i	−0.887520	−	0.665640i
$326$	10.0000			0.553849
$327$		0			0
$328$		−	12.0000i		−	0.662589i
$329$		0			0
$330$		0			0
$331$	8.00000			0.439720			0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000			0.439720			0.219860	+	0.975531i	$0.429440\pi$
$332$			12.0000i			0.658586i
$333$		0			0
$334$	−12.0000			−0.656611
$335$	−4.00000	−	8.00000i	−0.218543	−	0.437087i
$336$		0			0
$337$		−	18.0000i		−	0.980522i	−0.871576	−	0.490261i	$-0.836901\pi$
$337$		−	18.0000i		−	0.980522i	0.871576	−	0.490261i	$-0.163099\pi$
$338$		−	3.00000i		−	0.163178i
$339$		0			0
$340$	6.00000	+	12.0000i	0.325396	+	0.650791i
$341$		0			0
$342$		0			0
$343$		0			0
$344$	6.00000			0.323498
$345$		0			0
$346$	−2.00000			−0.107521
$347$			32.0000i			1.71785i	0.512101	+	0.858925i	$0.328867\pi$
$347$			32.0000i			1.71785i	−0.512101	+	0.858925i	$0.671133\pi$
$348$		0			0
$349$	−34.0000			−1.81998			−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000			−1.81998			−0.909989	+	0.414632i	$0.863910\pi$
$350$		0			0
$351$		0			0
$352$			4.00000i			0.213201i
$353$		−	22.0000i		−	1.17094i	−0.810693	−	0.585471i	$-0.800910\pi$
$353$		−	22.0000i		−	1.17094i	0.810693	−	0.585471i	$-0.199090\pi$
$354$		0			0
$355$	24.0000	−	12.0000i	1.27379	−	0.636894i
$356$	8.00000			0.423999
$357$		0			0
$358$			6.00000i			0.317110i
$359$	34.0000			1.79445			0.897226	−	0.441572i	$-0.145579\pi$
$359$	34.0000			1.79445			0.897226	+	0.441572i	$0.145579\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$		−	10.0000i		−	0.525588i
$363$		0			0
$364$		0			0
$365$	−8.00000	−	16.0000i	−0.418739	−	0.837478i
$366$		0			0
$367$			12.0000i			0.626395i	0.949688	+	0.313197i	$0.101400\pi$
$367$			12.0000i			0.626395i	−0.949688	+	0.313197i	$0.898600\pi$
$368$		0			0
$369$		0			0
$370$	8.00000	−	4.00000i	0.415900	−	0.207950i
$371$		0			0
$372$		0			0
$373$			12.0000i			0.621336i	0.950518	+	0.310668i	$0.100553\pi$
$373$			12.0000i			0.621336i	−0.950518	+	0.310668i	$0.899447\pi$
$374$	−24.0000			−1.24101
$375$		0			0
$376$		0			0
$377$			8.00000i			0.412021i
$378$		0			0
$379$	−12.0000			−0.616399			−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000			−0.616399			−0.308199	+	0.951322i	$0.599726\pi$
$380$	2.00000	−	1.00000i	0.102598	−	0.0512989i
$381$		0			0
$382$		−	6.00000i		−	0.306987i
$383$		−	36.0000i		−	1.83951i	−0.392488	−	0.919757i	$-0.628386\pi$
$383$		−	36.0000i		−	1.83951i	0.392488	−	0.919757i	$-0.371614\pi$
$384$		0			0
$385$		0			0
$386$	10.0000			0.508987
$387$		0			0
$388$		−	10.0000i		−	0.507673i
$389$	−4.00000			−0.202808			−0.101404	−	0.994845i	$-0.532333\pi$
$389$	−4.00000			−0.202808			−0.101404	+	0.994845i	$0.532333\pi$
$390$		0			0
$391$		0			0
$392$		−	7.00000i		−	0.353553i
$393$		0			0
$394$	10.0000			0.503793
$395$	−16.0000	+	8.00000i	−0.805047	+	0.402524i
$396$		0			0
$397$		−	18.0000i		−	0.903394i	−0.892171	−	0.451697i	$-0.850819\pi$
$397$		−	18.0000i		−	0.903394i	0.892171	−	0.451697i	$-0.149181\pi$
$398$			4.00000i			0.200502i
$399$		0			0
$400$	3.00000	−	4.00000i	0.150000	−	0.200000i
$401$	−28.0000			−1.39825			−0.699127	−	0.714998i	$-0.746428\pi$
$401$	−28.0000			−1.39825			−0.699127	+	0.714998i	$0.746428\pi$
$402$		0			0
$403$		0			0
$404$	12.0000			0.597022
$405$		0			0
$406$		0			0
$407$			16.0000i			0.793091i
$408$		0			0
$409$	34.0000			1.68119			0.840596	−	0.541663i	$-0.182205\pi$
$409$	34.0000			1.68119			0.840596	+	0.541663i	$0.182205\pi$
$410$	−12.0000	−	24.0000i	−0.592638	−	1.18528i
$411$		0			0
$412$		−	2.00000i		−	0.0985329i
$413$		0			0
$414$		0			0
$415$	12.0000	+	24.0000i	0.589057	+	1.17811i
$416$	4.00000			0.196116
$417$		0			0
$418$			4.00000i			0.195646i
$419$	8.00000			0.390826			0.195413	−	0.980721i	$-0.437395\pi$
$419$	8.00000			0.390826			0.195413	+	0.980721i	$0.437395\pi$
$420$		0			0
$421$	−14.0000			−0.682318			−0.341159	−	0.940006i	$-0.610819\pi$
$421$	−14.0000			−0.682318			−0.341159	+	0.940006i	$0.610819\pi$
$422$		−	8.00000i		−	0.389434i
$423$		0			0
$424$	14.0000			0.679900
$425$	24.0000	+	18.0000i	1.16417	+	0.873128i
$426$		0			0
$427$		0			0
$428$		−	20.0000i		−	0.966736i
$429$		0			0
$430$	12.0000	−	6.00000i	0.578691	−	0.289346i
$431$	12.0000			0.578020			0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000			0.578020			0.289010	+	0.957326i	$0.406674\pi$
$432$		0			0
$433$		−	18.0000i		−	0.865025i	−0.901628	−	0.432512i	$-0.857627\pi$
$433$		−	18.0000i		−	0.865025i	0.901628	−	0.432512i	$-0.142373\pi$
$434$		0			0
$435$		0			0
$436$	−14.0000			−0.670478
$437$		0			0
$438$		0			0
$439$	−16.0000			−0.763638			−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000			−0.763638			−0.381819	+	0.924237i	$0.624702\pi$
$440$	4.00000	+	8.00000i	0.190693	+	0.381385i
$441$		0			0
$442$			24.0000i			1.14156i
$443$			12.0000i			0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$			12.0000i			0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$		0			0
$445$	16.0000	−	8.00000i	0.758473	−	0.379236i
$446$	−10.0000			−0.473514
$447$		0			0
$448$		0			0
$449$	−32.0000			−1.51017			−0.755087	−	0.655625i	$-0.772405\pi$
$449$	−32.0000			−1.51017			−0.755087	+	0.655625i	$0.772405\pi$
$450$		0			0
$451$	48.0000			2.26023
$452$		−	10.0000i		−	0.470360i
$453$		0			0
$454$	12.0000			0.563188
$455$		0			0
$456$		0			0
$457$			32.0000i			1.49690i	0.663193	+	0.748448i	$0.269201\pi$
$457$			32.0000i			1.49690i	−0.663193	+	0.748448i	$0.730799\pi$
$458$			10.0000i			0.467269i
$459$		0			0
$460$		0			0
$461$	−24.0000			−1.11779			−0.558896	−	0.829238i	$-0.688775\pi$
$461$	−24.0000			−1.11779			−0.558896	+	0.829238i	$0.688775\pi$
$462$		0			0
$463$			16.0000i			0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$			16.0000i			0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$	−2.00000			−0.0928477
$465$		0			0
$466$	14.0000			0.648537
$467$		−	24.0000i		−	1.11059i	−0.831654	−	0.555294i	$-0.812606\pi$
$467$		−	24.0000i		−	1.11059i	0.831654	−	0.555294i	$-0.187394\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$			10.0000i			0.460287i
$473$			24.0000i			1.10352i
$474$		0			0
$475$	3.00000	−	4.00000i	0.137649	−	0.183533i
$476$		0			0
$477$		0			0
$478$			10.0000i			0.457389i
$479$	38.0000			1.73626			0.868132	−	0.496333i	$-0.165321\pi$
$479$	38.0000			1.73626			0.868132	+	0.496333i	$0.165321\pi$
$480$		0			0
$481$	16.0000			0.729537
$482$			26.0000i			1.18427i
$483$		0			0
$484$	−5.00000			−0.227273
$485$	−10.0000	−	20.0000i	−0.454077	−	0.908153i
$486$		0			0
$487$			18.0000i			0.815658i	0.913058	+	0.407829i	$0.133714\pi$
$487$			18.0000i			0.815658i	−0.913058	+	0.407829i	$0.866286\pi$
$488$			6.00000i			0.271607i
$489$		0			0
$490$	−7.00000	−	14.0000i	−0.316228	−	0.632456i
$491$	−36.0000			−1.62466			−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000			−1.62466			−0.812329	+	0.583200i	$0.801800\pi$
$492$		0			0
$493$		−	12.0000i		−	0.540453i
$494$	4.00000			0.179969
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	4.00000			0.179065			0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000			0.179065			0.0895323	+	0.995984i	$0.471463\pi$
$500$	2.00000	−	11.0000i	0.0894427	−	0.491935i
$501$		0			0
$502$			24.0000i			1.07117i
$503$		−	16.0000i		−	0.713405i	−0.934218	−	0.356702i	$-0.883901\pi$
$503$		−	16.0000i		−	0.713405i	0.934218	−	0.356702i	$-0.116099\pi$
$504$		0			0
$505$	24.0000	−	12.0000i	1.06799	−	0.533993i
$506$		0			0
$507$		0			0
$508$		−	10.0000i		−	0.443678i
$509$	18.0000			0.797836			0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000			0.797836			0.398918	+	0.916987i	$0.369386\pi$
$510$		0			0
$511$		0			0
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	2.00000			0.0882162
$515$	−2.00000	−	4.00000i	−0.0881305	−	0.176261i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	8.00000	−	4.00000i	0.350823	−	0.175412i
$521$	12.0000			0.525730			0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000			0.525730			0.262865	+	0.964833i	$0.415333\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$	−20.0000			−0.873704
$525$		0			0
$526$	16.0000			0.697633
$527$		0			0
$528$		0			0
$529$	23.0000			1.00000
$530$	28.0000	−	14.0000i	1.21624	−	0.608121i
$531$		0			0
$532$		0			0
$533$		−	48.0000i		−	2.07911i
$534$		0			0
$535$	−20.0000	−	40.0000i	−0.864675	−	1.72935i
$536$	4.00000			0.172774
$537$		0			0
$538$		−	2.00000i		−	0.0862261i
$539$	28.0000			1.20605
$540$		0			0
$541$	2.00000			0.0859867			0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000			0.0859867			0.0429934	+	0.999075i	$0.486311\pi$
$542$		−	28.0000i		−	1.20270i
$543$		0			0
$544$	−6.00000			−0.257248
$545$	−28.0000	+	14.0000i	−1.19939	+	0.599694i
$546$		0			0
$547$		−	24.0000i		−	1.02617i	−0.858339	−	0.513083i	$-0.828503\pi$
$547$		−	24.0000i		−	1.02617i	0.858339	−	0.513083i	$-0.171497\pi$
$548$			6.00000i			0.256307i
$549$		0			0
$550$	16.0000	+	12.0000i	0.682242	+	0.511682i
$551$	−2.00000			−0.0852029
$552$		0			0
$553$		0			0
$554$	−14.0000			−0.594803
$555$		0			0
$556$	−12.0000			−0.508913
$557$		−	30.0000i		−	1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$		−	30.0000i		−	1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$		0			0
$559$	24.0000			1.01509
$560$		0			0
$561$		0			0
$562$		−	12.0000i		−	0.506189i
$563$			36.0000i			1.51722i	0.651546	+	0.758610i	$0.274121\pi$
$563$			36.0000i			1.51722i	−0.651546	+	0.758610i	$0.725879\pi$
$564$		0			0
$565$	−10.0000	−	20.0000i	−0.420703	−	0.841406i
$566$	2.00000			0.0840663
$567$		0			0
$568$			12.0000i			0.503509i
$569$	32.0000			1.34151			0.670755	−	0.741679i	$-0.265970\pi$
$569$	32.0000			1.34151			0.670755	+	0.741679i	$0.265970\pi$
$570$		0			0
$571$	4.00000			0.167395			0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000			0.167395			0.0836974	+	0.996491i	$0.473327\pi$
$572$			16.0000i			0.668994i
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$			8.00000i			0.333044i	0.986038	+	0.166522i	$0.0532537\pi$
$577$			8.00000i			0.333044i	−0.986038	+	0.166522i	$0.946746\pi$
$578$		−	19.0000i		−	0.790296i
$579$		0			0
$580$	−4.00000	+	2.00000i	−0.166091	+	0.0830455i
$581$		0			0
$582$		0			0
$583$			56.0000i			2.31928i
$584$	8.00000			0.331042
$585$		0			0
$586$	14.0000			0.578335
$587$			32.0000i			1.32078i	0.750922	+	0.660391i	$0.229609\pi$
$587$			32.0000i			1.32078i	−0.750922	+	0.660391i	$0.770391\pi$
$588$		0			0
$589$		0			0
$590$	10.0000	+	20.0000i	0.411693	+	0.823387i
$591$		0			0
$592$			4.00000i			0.164399i
$593$			22.0000i			0.903432i	0.892162	+	0.451716i	$0.149188\pi$
$593$			22.0000i			0.903432i	−0.892162	+	0.451716i	$0.850812\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−12.0000			−0.490307			−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000			−0.490307			−0.245153	+	0.969484i	$0.578838\pi$
$600$		0			0
$601$	−42.0000			−1.71322			−0.856608	−	0.515968i	$-0.827432\pi$
$601$	−42.0000			−1.71322			−0.856608	+	0.515968i	$0.827432\pi$
$602$		0			0
$603$		0			0
$604$	−16.0000			−0.651031
$605$	−10.0000	+	5.00000i	−0.406558	+	0.203279i
$606$		0			0
$607$		−	30.0000i		−	1.21766i	−0.793300	−	0.608831i	$-0.791639\pi$
$607$		−	30.0000i		−	1.21766i	0.793300	−	0.608831i	$-0.208361\pi$
$608$			1.00000i			0.0405554i
$609$		0			0
$610$	6.00000	+	12.0000i	0.242933	+	0.485866i
$611$		0			0
$612$		0			0
$613$		−	46.0000i		−	1.85792i	−0.370177	−	0.928961i	$-0.620703\pi$
$613$		−	46.0000i		−	1.85792i	0.370177	−	0.928961i	$-0.379297\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			6.00000i			0.241551i	0.992680	+	0.120775i	$0.0385381\pi$
$617$			6.00000i			0.241551i	−0.992680	+	0.120775i	$0.961462\pi$
$618$		0			0
$619$	28.0000			1.12542			0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000			1.12542			0.562708	+	0.826656i	$0.309760\pi$
$620$		0			0
$621$		0			0
$622$			18.0000i			0.721734i
$623$		0			0
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$	−4.00000			−0.159872
$627$		0			0
$628$			6.00000i			0.239426i
$629$	−24.0000			−0.956943
$630$		0			0
$631$	28.0000			1.11466			0.557331	−	0.830290i	$-0.311825\pi$
$631$	28.0000			1.11466			0.557331	+	0.830290i	$0.311825\pi$
$632$		−	8.00000i		−	0.318223i
$633$		0			0
$634$	−18.0000			−0.714871
$635$	−10.0000	−	20.0000i	−0.396838	−	0.793676i
$636$		0			0
$637$		−	28.0000i		−	1.10940i
$638$		−	8.00000i		−	0.316723i
$639$		0			0
$640$	1.00000	+	2.00000i	0.0395285	+	0.0790569i
$641$	−36.0000			−1.42191			−0.710957	−	0.703235i	$-0.751738\pi$
$641$	−36.0000			−1.42191			−0.710957	+	0.703235i	$0.751738\pi$
$642$		0			0
$643$		−	26.0000i		−	1.02534i	−0.858586	−	0.512670i	$-0.828656\pi$
$643$		−	26.0000i		−	1.02534i	0.858586	−	0.512670i	$-0.171344\pi$
$644$		0			0
$645$		0			0
$646$	−6.00000			−0.236067
$647$		−	24.0000i		−	0.943537i	−0.881722	−	0.471769i	$-0.843616\pi$
$647$		−	24.0000i		−	0.943537i	0.881722	−	0.471769i	$-0.156384\pi$
$648$		0			0
$649$	−40.0000			−1.57014
$650$	12.0000	−	16.0000i	0.470679	−	0.627572i
$651$		0			0
$652$			10.0000i			0.391630i
$653$		−	18.0000i		−	0.704394i	−0.935926	−	0.352197i	$-0.885435\pi$
$653$		−	18.0000i		−	0.704394i	0.935926	−	0.352197i	$-0.114565\pi$
$654$		0			0
$655$	−40.0000	+	20.0000i	−1.56293	+	0.781465i
$656$	12.0000			0.468521
$657$		0			0
$658$		0			0
$659$	−26.0000			−1.01282			−0.506408	−	0.862294i	$-0.669027\pi$
$659$	−26.0000			−1.01282			−0.506408	+	0.862294i	$0.669027\pi$
$660$		0			0
$661$	−2.00000			−0.0777910			−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000			−0.0777910			−0.0388955	+	0.999243i	$0.512384\pi$
$662$			8.00000i			0.310929i
$663$		0			0
$664$	−12.0000			−0.465690
$665$		0			0
$666$		0			0
$667$		0			0
$668$		−	12.0000i		−	0.464294i
$669$		0			0
$670$	8.00000	−	4.00000i	0.309067	−	0.154533i
$671$	−24.0000			−0.926510
$672$		0			0
$673$			26.0000i			1.00223i	0.865382	+	0.501113i	$0.167076\pi$
$673$			26.0000i			1.00223i	−0.865382	+	0.501113i	$0.832924\pi$
$674$	18.0000			0.693334
$675$		0			0
$676$	3.00000			0.115385
$677$		−	30.0000i		−	1.15299i	−0.817099	−	0.576497i	$-0.804419\pi$
$677$		−	30.0000i		−	1.15299i	0.817099	−	0.576497i	$-0.195581\pi$
$678$		0			0
$679$		0			0
$680$	−12.0000	+	6.00000i	−0.460179	+	0.230089i
$681$		0			0
$682$		0			0
$683$			12.0000i			0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$			12.0000i			0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$		0			0
$685$	6.00000	+	12.0000i	0.229248	+	0.458496i
$686$		0			0
$687$		0			0
$688$			6.00000i			0.228748i
$689$	56.0000			2.13343
$690$		0			0
$691$	−4.00000			−0.152167			−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000			−0.152167			−0.0760836	+	0.997101i	$0.524242\pi$
$692$		−	2.00000i		−	0.0760286i
$693$		0			0
$694$	−32.0000			−1.21470
$695$	−24.0000	+	12.0000i	−0.910372	+	0.455186i
$696$		0			0
$697$			72.0000i			2.72719i
$698$		−	34.0000i		−	1.28692i
$699$		0			0
$700$		0			0
$701$	−20.0000			−0.755390			−0.377695	−	0.925930i	$-0.623283\pi$
$701$	−20.0000			−0.755390			−0.377695	+	0.925930i	$0.623283\pi$
$702$		0			0
$703$			4.00000i			0.150863i
$704$	−4.00000			−0.150756
$705$		0			0
$706$	22.0000			0.827981
$707$		0			0
$708$		0			0
$709$	−22.0000			−0.826227			−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000			−0.826227			−0.413114	+	0.910679i	$0.635559\pi$
$710$	12.0000	+	24.0000i	0.450352	+	0.900704i
$711$		0			0
$712$			8.00000i			0.299813i
$713$		0			0
$714$		0			0
$715$	16.0000	+	32.0000i	0.598366	+	1.19673i
$716$	−6.00000			−0.224231
$717$		0			0
$718$			34.0000i			1.26887i
$719$	6.00000			0.223762			0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000			0.223762			0.111881	+	0.993722i	$0.464312\pi$
$720$		0			0
$721$		0			0
$722$			1.00000i			0.0372161i
$723$		0			0
$724$	10.0000			0.371647
$725$	−6.00000	+	8.00000i	−0.222834	+	0.297113i
$726$		0			0
$727$			40.0000i			1.48352i	0.670667	+	0.741759i	$0.266008\pi$
$727$			40.0000i			1.48352i	−0.670667	+	0.741759i	$0.733992\pi$
$728$		0			0
$729$		0			0
$730$	16.0000	−	8.00000i	0.592187	−	0.296093i
$731$	−36.0000			−1.33151
$732$		0			0
$733$		−	26.0000i		−	0.960332i	−0.877178	−	0.480166i	$-0.840576\pi$
$733$		−	26.0000i		−	0.960332i	0.877178	−	0.480166i	$-0.159424\pi$
$734$	−12.0000			−0.442928
$735$		0			0
$736$		0			0
$737$			16.0000i			0.589368i
$738$		0			0
$739$	−20.0000			−0.735712			−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000			−0.735712			−0.367856	+	0.929883i	$0.619908\pi$
$740$	4.00000	+	8.00000i	0.147043	+	0.294086i
$741$		0			0
$742$		0			0
$743$			44.0000i			1.61420i	0.590412	+	0.807102i	$0.298965\pi$
$743$			44.0000i			1.61420i	−0.590412	+	0.807102i	$0.701035\pi$
$744$		0			0
$745$		0			0
$746$	−12.0000			−0.439351
$747$		0			0
$748$		−	24.0000i		−	0.877527i
$749$		0			0
$750$		0			0
$751$	−8.00000			−0.291924			−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000			−0.291924			−0.145962	+	0.989290i	$0.546628\pi$
$752$		0			0
$753$		0			0
$754$	−8.00000			−0.291343
$755$	−32.0000	+	16.0000i	−1.16460	+	0.582300i
$756$		0			0
$757$			2.00000i			0.0726912i	0.999339	+	0.0363456i	$0.0115717\pi$
$757$			2.00000i			0.0726912i	−0.999339	+	0.0363456i	$0.988428\pi$
$758$		−	12.0000i		−	0.435860i
$759$		0			0
$760$	1.00000	+	2.00000i	0.0362738	+	0.0725476i
$761$	30.0000			1.08750			0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000			1.08750			0.543750	+	0.839248i	$0.317004\pi$
$762$		0			0
$763$		0			0
$764$	6.00000			0.217072
$765$		0			0
$766$	36.0000			1.30073
$767$			40.0000i			1.44432i
$768$		0			0
$769$	−50.0000			−1.80305			−0.901523	−	0.432731i	$-0.857550\pi$
$769$	−50.0000			−1.80305			−0.901523	+	0.432731i	$0.857550\pi$
$770$		0			0
$771$		0			0
$772$			10.0000i			0.359908i
$773$			38.0000i			1.36677i	0.730061	+	0.683383i	$0.239492\pi$
$773$			38.0000i			1.36677i	−0.730061	+	0.683383i	$0.760508\pi$
$774$		0			0
$775$		0			0
$776$	10.0000			0.358979
$777$		0			0
$778$		−	4.00000i		−	0.143407i
$779$	12.0000			0.429945
$780$		0			0
$781$	−48.0000			−1.71758
$782$		0			0
$783$		0			0
$784$	7.00000			0.250000
$785$	6.00000	+	12.0000i	0.214149	+	0.428298i
$786$		0			0
$787$		−	48.0000i		−	1.71102i	−0.517790	−	0.855508i	$-0.673245\pi$
$787$		−	48.0000i		−	1.71102i	0.517790	−	0.855508i	$-0.326755\pi$
$788$			10.0000i			0.356235i
$789$		0			0
$790$	−8.00000	−	16.0000i	−0.284627	−	0.569254i
$791$		0			0
$792$		0			0
$793$			24.0000i			0.852265i
$794$	18.0000			0.638796

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:	\( N \)	\(=\)	\( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1710.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(-2.00000 + 1.00000i) q^{5} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(-2.00000 + 1.00000i) q^{5} -1.00000i q^{8} +(-1.00000 - 2.00000i) q^{10} +4.00000 q^{11} -4.00000i q^{13} +1.00000 q^{16} +6.00000i q^{17} +1.00000 q^{19} +(2.00000 - 1.00000i) q^{20} +4.00000i q^{22} +(3.00000 - 4.00000i) q^{25} +4.00000 q^{26} -2.00000 q^{29} +1.00000i q^{32} -6.00000 q^{34} +4.00000i q^{37} +1.00000i q^{38} +(1.00000 + 2.00000i) q^{40} +12.0000 q^{41} +6.00000i q^{43} -4.00000 q^{44} +7.00000 q^{49} +(4.00000 + 3.00000i) q^{50} +4.00000i q^{52} +14.0000i q^{53} +(-8.00000 + 4.00000i) q^{55} -2.00000i q^{58} -10.0000 q^{59} -6.00000 q^{61} -1.00000 q^{64} +(4.00000 + 8.00000i) q^{65} +4.00000i q^{67} -6.00000i q^{68} -12.0000 q^{71} +8.00000i q^{73} -4.00000 q^{74} -1.00000 q^{76} +8.00000 q^{79} +(-2.00000 + 1.00000i) q^{80} +12.0000i q^{82} -12.0000i q^{83} +(-6.00000 - 12.0000i) q^{85} -6.00000 q^{86} -4.00000i q^{88} -8.00000 q^{89} +(-2.00000 + 1.00000i) q^{95} +10.0000i q^{97} +7.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 4 q^{5}+O(q^{10}) \) 2 * q - 2 * q^4 - 4 * q^5	\( 2 q - 2 q^{4} - 4 q^{5} - 2 q^{10} + 8 q^{11} + 2 q^{16} + 2 q^{19} + 4 q^{20} + 6 q^{25} + 8 q^{26} - 4 q^{29} - 12 q^{34} + 2 q^{40} + 24 q^{41} - 8 q^{44} + 14 q^{49} + 8 q^{50} - 16 q^{55} - 20 q^{59} - 12 q^{61} - 2 q^{64} + 8 q^{65} - 24 q^{71} - 8 q^{74} - 2 q^{76} + 16 q^{79} - 4 q^{80} - 12 q^{85} - 12 q^{86} - 16 q^{89} - 4 q^{95}+O(q^{100}) \) 2 * q - 2 * q^4 - 4 * q^5 - 2 * q^10 + 8 * q^11 + 2 * q^16 + 2 * q^19 + 4 * q^20 + 6 * q^25 + 8 * q^26 - 4 * q^29 - 12 * q^34 + 2 * q^40 + 24 * q^41 - 8 * q^44 + 14 * q^49 + 8 * q^50 - 16 * q^55 - 20 * q^59 - 12 * q^61 - 2 * q^64 + 8 * q^65 - 24 * q^71 - 8 * q^74 - 2 * q^76 + 16 * q^79 - 4 * q^80 - 12 * q^85 - 12 * q^86 - 16 * q^89 - 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more