LMFDB - Embedded newform 4680.2.g.e.2521.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4680,2,Mod(2521,4680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4680, 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("4680.2521");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4680.g (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:	$37.3699881460$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1560)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2521.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	4680.2521
Dual form			4680.2.g.e.2521.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{5} -3.00000i q^{7} +O(q^{10})$	$q-1.00000i q^{5} -3.00000i q^{7} -3.00000i q^{11} +(-2.00000 + 3.00000i) q^{13} +1.00000 q^{17} +8.00000i q^{19} -3.00000 q^{23} -1.00000 q^{25} -6.00000 q^{29} +10.0000i q^{31} -3.00000 q^{35} -1.00000i q^{37} +5.00000i q^{41} +6.00000 q^{43} +8.00000i q^{47} -2.00000 q^{49} -5.00000 q^{53} -3.00000 q^{55} -4.00000i q^{59} +13.0000 q^{61} +(3.00000 + 2.00000i) q^{65} +8.00000i q^{67} +11.0000i q^{71} -2.00000i q^{73} -9.00000 q^{77} -5.00000 q^{79} -6.00000i q^{83} -1.00000i q^{85} -15.0000i q^{89} +(9.00000 + 6.00000i) q^{91} +8.00000 q^{95} -5.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q - 4 q^{13} + 2 q^{17} - 6 q^{23} - 2 q^{25} - 12 q^{29} - 6 q^{35} + 12 q^{43} - 4 q^{49} - 10 q^{53} - 6 q^{55} + 26 q^{61} + 6 q^{65} - 18 q^{77} - 10 q^{79} + 18 q^{91} + 16 q^{95}+O(q^{100}) $ 2 * q - 4 * q^13 + 2 * q^17 - 6 * q^23 - 2 * q^25 - 12 * q^29 - 6 * q^35 + 12 * q^43 - 4 * q^49 - 10 * q^53 - 6 * q^55 + 26 * q^61 + 6 * q^65 - 18 * q^77 - 10 * q^79 + 18 * q^91 + 16 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4680\mathbb{Z}\right)^\times$.

$n$	$937$	$1081$	$2081$	$2341$	$3511$
$\chi(n)$	$1$	$-1$	$1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$		−	3.00000i		−	1.13389i	−0.823754	−	0.566947i	$-0.808125\pi$
$7$		−	3.00000i		−	1.13389i	0.823754	−	0.566947i	$-0.191875\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	3.00000i		−	0.904534i	−0.891883	−	0.452267i	$-0.850615\pi$
$11$		−	3.00000i		−	0.904534i	0.891883	−	0.452267i	$-0.149385\pi$
$12$		0			0
$13$	−2.00000	+	3.00000i	−0.554700	+	0.832050i
$13$	−2.00000	+	3.00000i	−0.554700	+	0.832050i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.00000			0.242536			0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000			0.242536			0.121268	+	0.992620i	$0.461304\pi$
$18$		0			0
$19$			8.00000i			1.83533i	0.397360	+	0.917663i	$0.369927\pi$
$19$			8.00000i			1.83533i	−0.397360	+	0.917663i	$0.630073\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.00000			−0.625543			−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000			−0.625543			−0.312772	+	0.949828i	$0.601257\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$			10.0000i			1.79605i	0.439941	+	0.898027i	$0.354999\pi$
$31$			10.0000i			1.79605i	−0.439941	+	0.898027i	$0.645001\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.00000			−0.507093
$36$		0			0
$37$		−	1.00000i		−	0.164399i	−0.996616	−	0.0821995i	$-0.973806\pi$
$37$		−	1.00000i		−	0.164399i	0.996616	−	0.0821995i	$-0.0261945\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			5.00000i			0.780869i	0.920631	+	0.390434i	$0.127675\pi$
$41$			5.00000i			0.780869i	−0.920631	+	0.390434i	$0.872325\pi$
$42$		0			0
$43$	6.00000			0.914991			0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000			0.914991			0.457496	+	0.889212i	$0.348747\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			8.00000i			1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$			8.00000i			1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$		0			0
$49$	−2.00000			−0.285714
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−5.00000			−0.686803			−0.343401	−	0.939189i	$-0.611579\pi$
$53$	−5.00000			−0.686803			−0.343401	+	0.939189i	$0.611579\pi$
$54$		0			0
$55$	−3.00000			−0.404520
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	4.00000i		−	0.520756i	−0.965507	−	0.260378i	$-0.916153\pi$
$59$		−	4.00000i		−	0.520756i	0.965507	−	0.260378i	$-0.0838471\pi$
$60$		0			0
$61$	13.0000			1.66448			0.832240	−	0.554416i	$-0.187058\pi$
$61$	13.0000			1.66448			0.832240	+	0.554416i	$0.187058\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.00000	+	2.00000i	0.372104	+	0.248069i
$66$		0			0
$67$			8.00000i			0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$			8.00000i			0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			11.0000i			1.30546i	0.757591	+	0.652730i	$0.226376\pi$
$71$			11.0000i			1.30546i	−0.757591	+	0.652730i	$0.773624\pi$
$72$		0			0
$73$		−	2.00000i		−	0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$		−	2.00000i		−	0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−9.00000			−1.02565
$78$		0			0
$79$	−5.00000			−0.562544			−0.281272	−	0.959628i	$-0.590756\pi$
$79$	−5.00000			−0.562544			−0.281272	+	0.959628i	$0.590756\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	6.00000i		−	0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$		−	6.00000i		−	0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$		0			0
$85$		−	1.00000i		−	0.108465i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	15.0000i		−	1.59000i	−0.606612	−	0.794998i	$-0.707472\pi$
$89$		−	15.0000i		−	1.59000i	0.606612	−	0.794998i	$-0.292528\pi$
$90$		0			0
$91$	9.00000	+	6.00000i	0.943456	+	0.628971i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	8.00000			0.820783
$96$		0			0
$97$		−	5.00000i		−	0.507673i	−0.967247	−	0.253837i	$-0.918307\pi$
$97$		−	5.00000i		−	0.507673i	0.967247	−	0.253837i	$-0.0816925\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	14.0000			1.39305			0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000			1.39305			0.696526	+	0.717532i	$0.254728\pi$
$102$		0			0
$103$	−12.0000			−1.18240			−0.591198	−	0.806527i	$-0.701345\pi$
$103$	−12.0000			−1.18240			−0.591198	+	0.806527i	$0.701345\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	19.0000			1.83680			0.918400	−	0.395654i	$-0.129482\pi$
$107$	19.0000			1.83680			0.918400	+	0.395654i	$0.129482\pi$
$108$		0			0
$109$			6.00000i			0.574696i	0.957826	+	0.287348i	$0.0927736\pi$
$109$			6.00000i			0.574696i	−0.957826	+	0.287348i	$0.907226\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.00000			0.564433			0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000			0.564433			0.282216	+	0.959351i	$0.408930\pi$
$114$		0			0
$115$			3.00000i			0.279751i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	3.00000i		−	0.275010i
$120$		0			0
$121$	2.00000			0.181818
$122$		0			0
$123$		0			0
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$	14.0000			1.24230			0.621150	−	0.783692i	$-0.286666\pi$
$127$	14.0000			1.24230			0.621150	+	0.783692i	$0.286666\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−8.00000			−0.698963			−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000			−0.698963			−0.349482	+	0.936943i	$0.613642\pi$
$132$		0			0
$133$	24.0000			2.08106
$134$		0			0
$135$		0			0
$136$		0			0
$137$			6.00000i			0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$			6.00000i			0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$		0			0
$139$	−19.0000			−1.61156			−0.805779	−	0.592216i	$-0.798253\pi$
$139$	−19.0000			−1.61156			−0.805779	+	0.592216i	$0.798253\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	9.00000	+	6.00000i	0.752618	+	0.501745i
$144$		0			0
$145$			6.00000i			0.498273i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			15.0000i			1.22885i	0.788976	+	0.614424i	$0.210612\pi$
$149$			15.0000i			1.22885i	−0.788976	+	0.614424i	$0.789388\pi$
$150$		0			0
$151$			2.00000i			0.162758i	0.996683	+	0.0813788i	$0.0259324\pi$
$151$			2.00000i			0.162758i	−0.996683	+	0.0813788i	$0.974068\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	10.0000			0.803219
$156$		0			0
$157$	12.0000			0.957704			0.478852	−	0.877896i	$-0.341053\pi$
$157$	12.0000			0.957704			0.478852	+	0.877896i	$0.341053\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			9.00000i			0.709299i
$162$		0			0
$163$		−	7.00000i		−	0.548282i	−0.961689	−	0.274141i	$-0.911606\pi$
$163$		−	7.00000i		−	0.548282i	0.961689	−	0.274141i	$-0.0883936\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	20.0000i		−	1.54765i	−0.633402	−	0.773823i	$-0.718342\pi$
$167$		−	20.0000i		−	1.54765i	0.633402	−	0.773823i	$-0.281658\pi$
$168$		0			0
$169$	−5.00000	−	12.0000i	−0.384615	−	0.923077i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	6.00000			0.456172			0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000			0.456172			0.228086	+	0.973641i	$0.426753\pi$
$174$		0			0
$175$			3.00000i			0.226779i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	16.0000			1.19590			0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000			1.19590			0.597948	+	0.801535i	$0.295983\pi$
$180$		0			0
$181$	19.0000			1.41226			0.706129	−	0.708083i	$-0.250440\pi$
$181$	19.0000			1.41226			0.706129	+	0.708083i	$0.250440\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.00000			−0.0735215
$186$		0			0
$187$		−	3.00000i		−	0.219382i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−24.0000			−1.73658			−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000			−1.73658			−0.868290	+	0.496058i	$0.834780\pi$
$192$		0			0
$193$		−	1.00000i		−	0.0719816i	−0.999352	−	0.0359908i	$-0.988541\pi$
$193$		−	1.00000i		−	0.0719816i	0.999352	−	0.0359908i	$-0.0114587\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			18.0000i			1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$			18.0000i			1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$		0			0
$199$	−16.0000			−1.13421			−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000			−1.13421			−0.567105	+	0.823646i	$0.691937\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			18.0000i			1.26335i
$204$		0			0
$205$	5.00000			0.349215
$206$		0			0
$207$		0			0
$208$		0			0
$209$	24.0000			1.66011
$210$		0			0
$211$	−12.0000			−0.826114			−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000			−0.826114			−0.413057	+	0.910705i	$0.635539\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	6.00000i		−	0.409197i
$216$		0			0
$217$	30.0000			2.03653
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2.00000	+	3.00000i	−0.134535	+	0.201802i
$222$		0			0
$223$			16.0000i			1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$			16.0000i			1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		0			0
$229$			2.00000i			0.132164i	0.997814	+	0.0660819i	$0.0210498\pi$
$229$			2.00000i			0.132164i	−0.997814	+	0.0660819i	$0.978950\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−13.0000			−0.851658			−0.425829	−	0.904804i	$-0.640018\pi$
$233$	−13.0000			−0.851658			−0.425829	+	0.904804i	$0.640018\pi$
$234$		0			0
$235$	8.00000			0.521862
$236$		0			0
$237$		0			0
$238$		0			0
$239$			15.0000i			0.970269i	0.874439	+	0.485135i	$0.161229\pi$
$239$			15.0000i			0.970269i	−0.874439	+	0.485135i	$0.838771\pi$
$240$		0			0
$241$		−	14.0000i		−	0.901819i	−0.892570	−	0.450910i	$-0.851100\pi$
$241$		−	14.0000i		−	0.901819i	0.892570	−	0.450910i	$-0.148900\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			2.00000i			0.127775i
$246$		0			0
$247$	−24.0000	−	16.0000i	−1.52708	−	1.01806i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−6.00000			−0.378717			−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000			−0.378717			−0.189358	+	0.981908i	$0.560641\pi$
$252$		0			0
$253$			9.00000i			0.565825i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2.00000			−0.124757			−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000			−0.124757			−0.0623783	+	0.998053i	$0.519869\pi$
$258$		0			0
$259$	−3.00000			−0.186411
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−16.0000			−0.986602			−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000			−0.986602			−0.493301	+	0.869859i	$0.664210\pi$
$264$		0			0
$265$			5.00000i			0.307148i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$			18.0000i			1.09342i	0.837321	+	0.546711i	$0.184120\pi$
$271$			18.0000i			1.09342i	−0.837321	+	0.546711i	$0.815880\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			3.00000i			0.180907i
$276$		0			0
$277$	−20.0000			−1.20168			−0.600842	−	0.799368i	$-0.705168\pi$
$277$	−20.0000			−1.20168			−0.600842	+	0.799368i	$0.705168\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			14.0000i			0.835170i	0.908638	+	0.417585i	$0.137123\pi$
$281$			14.0000i			0.835170i	−0.908638	+	0.417585i	$0.862877\pi$
$282$		0			0
$283$	−14.0000			−0.832214			−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000			−0.832214			−0.416107	+	0.909316i	$0.636606\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	15.0000			0.885422
$288$		0			0
$289$	−16.0000			−0.941176
$290$		0			0
$291$		0			0
$292$		0			0
$293$			18.0000i			1.05157i	0.850617	+	0.525786i	$0.176229\pi$
$293$			18.0000i			1.05157i	−0.850617	+	0.525786i	$0.823771\pi$
$294$		0			0
$295$	−4.00000			−0.232889
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6.00000	−	9.00000i	0.346989	−	0.520483i
$300$		0			0
$301$		−	18.0000i		−	1.03750i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	13.0000i		−	0.744378i
$306$		0			0
$307$			5.00000i			0.285365i	0.989769	+	0.142683i	$0.0455728\pi$
$307$			5.00000i			0.285365i	−0.989769	+	0.142683i	$0.954427\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	14.0000			0.793867			0.396934	−	0.917847i	$-0.370074\pi$
$311$	14.0000			0.793867			0.396934	+	0.917847i	$0.370074\pi$
$312$		0			0
$313$	16.0000			0.904373			0.452187	−	0.891923i	$-0.350644\pi$
$313$	16.0000			0.904373			0.452187	+	0.891923i	$0.350644\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			2.00000i			0.112331i	0.998421	+	0.0561656i	$0.0178875\pi$
$317$			2.00000i			0.112331i	−0.998421	+	0.0561656i	$0.982113\pi$
$318$		0			0
$319$			18.0000i			1.00781i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			8.00000i			0.445132i
$324$		0			0
$325$	2.00000	−	3.00000i	0.110940	−	0.166410i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	24.0000			1.32316
$330$		0			0
$331$		−	4.00000i		−	0.219860i	−0.993939	−	0.109930i	$-0.964937\pi$
$331$		−	4.00000i		−	0.219860i	0.993939	−	0.109930i	$-0.0350627\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	8.00000			0.437087
$336$		0			0
$337$	−2.00000			−0.108947			−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000			−0.108947			−0.0544735	+	0.998515i	$0.517348\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	30.0000			1.62459
$342$		0			0
$343$		−	15.0000i		−	0.809924i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−21.0000			−1.12734			−0.563670	−	0.826000i	$-0.690611\pi$
$347$	−21.0000			−1.12734			−0.563670	+	0.826000i	$0.690611\pi$
$348$		0			0
$349$			10.0000i			0.535288i	0.963518	+	0.267644i	$0.0862451\pi$
$349$			10.0000i			0.535288i	−0.963518	+	0.267644i	$0.913755\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			28.0000i			1.49029i	0.666903	+	0.745145i	$0.267620\pi$
$353$			28.0000i			1.49029i	−0.666903	+	0.745145i	$0.732380\pi$
$354$		0			0
$355$	11.0000			0.583819
$356$		0			0
$357$		0			0
$358$		0			0
$359$			24.0000i			1.26667i	0.773877	+	0.633336i	$0.218315\pi$
$359$			24.0000i			1.26667i	−0.773877	+	0.633336i	$0.781685\pi$
$360$		0			0
$361$	−45.0000			−2.36842
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2.00000			−0.104685
$366$		0			0
$367$	26.0000			1.35719			0.678594	−	0.734513i	$-0.262589\pi$
$367$	26.0000			1.35719			0.678594	+	0.734513i	$0.262589\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			15.0000i			0.778761i
$372$		0			0
$373$	32.0000			1.65690			0.828449	−	0.560065i	$-0.189224\pi$
$373$	32.0000			1.65690			0.828449	+	0.560065i	$0.189224\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	12.0000	−	18.0000i	0.618031	−	0.927047i
$378$		0			0
$379$			34.0000i			1.74646i	0.487306	+	0.873231i	$0.337980\pi$
$379$			34.0000i			1.74646i	−0.487306	+	0.873231i	$0.662020\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			38.0000i			1.94171i	0.239669	+	0.970855i	$0.422961\pi$
$383$			38.0000i			1.94171i	−0.239669	+	0.970855i	$0.577039\pi$
$384$		0			0
$385$			9.00000i			0.458682i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−22.0000			−1.11544			−0.557722	−	0.830028i	$-0.688325\pi$
$389$	−22.0000			−1.11544			−0.557722	+	0.830028i	$0.688325\pi$
$390$		0			0
$391$	−3.00000			−0.151717
$392$		0			0
$393$		0			0
$394$		0			0
$395$			5.00000i			0.251577i
$396$		0			0
$397$		−	37.0000i		−	1.85698i	−0.371361	−	0.928488i	$-0.621109\pi$
$397$		−	37.0000i		−	1.85698i	0.371361	−	0.928488i	$-0.378891\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			18.0000i			0.898877i	0.893311	+	0.449439i	$0.148376\pi$
$401$			18.0000i			0.898877i	−0.893311	+	0.449439i	$0.851624\pi$
$402$		0			0
$403$	−30.0000	−	20.0000i	−1.49441	−	0.996271i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3.00000			−0.148704
$408$		0			0
$409$			28.0000i			1.38451i	0.721653	+	0.692255i	$0.243383\pi$
$409$			28.0000i			1.38451i	−0.721653	+	0.692255i	$0.756617\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−12.0000			−0.590481
$414$		0			0
$415$	−6.00000			−0.294528
$416$		0			0
$417$		0			0
$418$		0			0
$419$	22.0000			1.07477			0.537385	−	0.843337i	$-0.319412\pi$
$419$	22.0000			1.07477			0.537385	+	0.843337i	$0.319412\pi$
$420$		0			0
$421$		−	26.0000i		−	1.26716i	−0.773676	−	0.633581i	$-0.781584\pi$
$421$		−	26.0000i		−	1.26716i	0.773676	−	0.633581i	$-0.218416\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.00000			−0.0485071
$426$		0			0
$427$		−	39.0000i		−	1.88734i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			8.00000i			0.385346i	0.981263	+	0.192673i	$0.0617157\pi$
$431$			8.00000i			0.385346i	−0.981263	+	0.192673i	$0.938284\pi$
$432$		0			0
$433$	6.00000			0.288342			0.144171	−	0.989553i	$-0.453949\pi$
$433$	6.00000			0.288342			0.144171	+	0.989553i	$0.453949\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	24.0000i		−	1.14808i
$438$		0			0
$439$	−1.00000			−0.0477274			−0.0238637	−	0.999715i	$-0.507597\pi$
$439$	−1.00000			−0.0477274			−0.0238637	+	0.999715i	$0.507597\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−17.0000			−0.807694			−0.403847	−	0.914826i	$-0.632327\pi$
$443$	−17.0000			−0.807694			−0.403847	+	0.914826i	$0.632327\pi$
$444$		0			0
$445$	−15.0000			−0.711068
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	33.0000i		−	1.55737i	−0.627417	−	0.778683i	$-0.715888\pi$
$449$		−	33.0000i		−	1.55737i	0.627417	−	0.778683i	$-0.284112\pi$
$450$		0			0
$451$	15.0000			0.706322
$452$		0			0
$453$		0			0
$454$		0			0
$455$	6.00000	−	9.00000i	0.281284	−	0.421927i
$456$		0			0
$457$		−	13.0000i		−	0.608114i	−0.952654	−	0.304057i	$-0.901659\pi$
$457$		−	13.0000i		−	0.608114i	0.952654	−	0.304057i	$-0.0983414\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	3.00000i		−	0.139724i	−0.997557	−	0.0698620i	$-0.977744\pi$
$461$		−	3.00000i		−	0.139724i	0.997557	−	0.0698620i	$-0.0222559\pi$
$462$		0			0
$463$		−	7.00000i		−	0.325318i	−0.986682	−	0.162659i	$-0.947993\pi$
$463$		−	7.00000i		−	0.325318i	0.986682	−	0.162659i	$-0.0520070\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−3.00000			−0.138823			−0.0694117	−	0.997588i	$-0.522112\pi$
$467$	−3.00000			−0.138823			−0.0694117	+	0.997588i	$0.522112\pi$
$468$		0			0
$469$	24.0000			1.10822
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	18.0000i		−	0.827641i
$474$		0			0
$475$		−	8.00000i		−	0.367065i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			39.0000i			1.78196i	0.454047	+	0.890978i	$0.349980\pi$
$479$			39.0000i			1.78196i	−0.454047	+	0.890978i	$0.650020\pi$
$480$		0			0
$481$	3.00000	+	2.00000i	0.136788	+	0.0911922i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−5.00000			−0.227038
$486$		0			0
$487$			33.0000i			1.49537i	0.664052	+	0.747686i	$0.268835\pi$
$487$			33.0000i			1.49537i	−0.664052	+	0.747686i	$0.731165\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	8.00000			0.361035			0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000			0.361035			0.180517	+	0.983572i	$0.442223\pi$
$492$		0			0
$493$	−6.00000			−0.270226
$494$		0			0
$495$		0			0
$496$		0			0
$497$	33.0000			1.48025
$498$		0			0
$499$		−	8.00000i		−	0.358129i	−0.983837	−	0.179065i	$-0.942693\pi$
$499$		−	8.00000i		−	0.358129i	0.983837	−	0.179065i	$-0.0573071\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−12.0000			−0.535054			−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000			−0.535054			−0.267527	+	0.963550i	$0.586206\pi$
$504$		0			0
$505$		−	14.0000i		−	0.622992i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	5.00000i		−	0.221621i	−0.993842	−	0.110811i	$-0.964655\pi$
$509$		−	5.00000i		−	0.221621i	0.993842	−	0.110811i	$-0.0353447\pi$
$510$		0			0
$511$	−6.00000			−0.265424
$512$		0			0
$513$		0			0
$514$		0			0
$515$			12.0000i			0.528783i
$516$		0			0
$517$	24.0000			1.05552
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−28.0000			−1.22670			−0.613351	−	0.789810i	$-0.710179\pi$
$521$	−28.0000			−1.22670			−0.613351	+	0.789810i	$0.710179\pi$
$522$		0			0
$523$	−14.0000			−0.612177			−0.306089	−	0.952003i	$-0.599020\pi$
$523$	−14.0000			−0.612177			−0.306089	+	0.952003i	$0.599020\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			10.0000i			0.435607i
$528$		0			0
$529$	−14.0000			−0.608696
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−15.0000	−	10.0000i	−0.649722	−	0.433148i
$534$		0			0
$535$		−	19.0000i		−	0.821442i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			6.00000i			0.258438i
$540$		0			0
$541$			20.0000i			0.859867i	0.902861	+	0.429934i	$0.141463\pi$
$541$			20.0000i			0.859867i	−0.902861	+	0.429934i	$0.858537\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	6.00000			0.257012
$546$		0			0
$547$	6.00000			0.256541			0.128271	−	0.991739i	$-0.459057\pi$
$547$	6.00000			0.256541			0.128271	+	0.991739i	$0.459057\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	48.0000i		−	2.04487i
$552$		0			0
$553$			15.0000i			0.637865i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			30.0000i			1.27114i	0.772043	+	0.635570i	$0.219235\pi$
$557$			30.0000i			1.27114i	−0.772043	+	0.635570i	$0.780765\pi$
$558$		0			0
$559$	−12.0000	+	18.0000i	−0.507546	+	0.761319i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	21.0000			0.885044			0.442522	−	0.896758i	$-0.354084\pi$
$563$	21.0000			0.885044			0.442522	+	0.896758i	$0.354084\pi$
$564$		0			0
$565$		−	6.00000i		−	0.252422i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	26.0000			1.08998			0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000			1.08998			0.544988	+	0.838444i	$0.316534\pi$
$570$		0			0
$571$	33.0000			1.38101			0.690504	−	0.723329i	$-0.257389\pi$
$571$	33.0000			1.38101			0.690504	+	0.723329i	$0.257389\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3.00000			0.125109
$576$		0			0
$577$			7.00000i			0.291414i	0.989328	+	0.145707i	$0.0465456\pi$
$577$			7.00000i			0.291414i	−0.989328	+	0.145707i	$0.953454\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−18.0000			−0.746766
$582$		0			0
$583$			15.0000i			0.621237i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			18.0000i			0.742940i	0.928445	+	0.371470i	$0.121146\pi$
$587$			18.0000i			0.742940i	−0.928445	+	0.371470i	$0.878854\pi$
$588$		0			0
$589$	−80.0000			−3.29634
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	18.0000i		−	0.739171i	−0.929197	−	0.369586i	$-0.879500\pi$
$593$		−	18.0000i		−	0.739171i	0.929197	−	0.369586i	$-0.120500\pi$
$594$		0			0
$595$	−3.00000			−0.122988
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−24.0000			−0.980613			−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000			−0.980613			−0.490307	+	0.871550i	$0.663115\pi$
$600$		0			0
$601$	−9.00000			−0.367118			−0.183559	−	0.983009i	$-0.558762\pi$
$601$	−9.00000			−0.367118			−0.183559	+	0.983009i	$0.558762\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	2.00000i		−	0.0813116i
$606$		0			0
$607$	8.00000			0.324710			0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000			0.324710			0.162355	+	0.986732i	$0.448091\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−24.0000	−	16.0000i	−0.970936	−	0.647291i
$612$		0			0
$613$		−	21.0000i		−	0.848182i	−0.905620	−	0.424091i	$-0.860594\pi$
$613$		−	21.0000i		−	0.848182i	0.905620	−	0.424091i	$-0.139406\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	18.0000i		−	0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$		−	18.0000i		−	0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$		0			0
$619$			14.0000i			0.562708i	0.959604	+	0.281354i	$0.0907834\pi$
$619$			14.0000i			0.562708i	−0.959604	+	0.281354i	$0.909217\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−45.0000			−1.80289
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	1.00000i		−	0.0398726i
$630$		0			0
$631$			34.0000i			1.35352i	0.736204	+	0.676759i	$0.236616\pi$
$631$			34.0000i			1.35352i	−0.736204	+	0.676759i	$0.763384\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	14.0000i		−	0.555573i
$636$		0			0
$637$	4.00000	−	6.00000i	0.158486	−	0.237729i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	20.0000			0.789953			0.394976	−	0.918691i	$-0.370753\pi$
$641$	20.0000			0.789953			0.394976	+	0.918691i	$0.370753\pi$
$642$		0			0
$643$		−	41.0000i		−	1.61688i	−0.588577	−	0.808441i	$-0.700312\pi$
$643$		−	41.0000i		−	1.61688i	0.588577	−	0.808441i	$-0.299688\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.00000			−0.0393141			−0.0196570	−	0.999807i	$-0.506257\pi$
$647$	−1.00000			−0.0393141			−0.0196570	+	0.999807i	$0.506257\pi$
$648$		0			0
$649$	−12.0000			−0.471041
$650$		0			0
$651$		0			0
$652$		0			0
$653$	18.0000			0.704394			0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000			0.704394			0.352197	+	0.935926i	$0.385435\pi$
$654$		0			0
$655$			8.00000i			0.312586i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−38.0000			−1.48027			−0.740135	−	0.672458i	$-0.765238\pi$
$659$	−38.0000			−1.48027			−0.740135	+	0.672458i	$0.765238\pi$
$660$		0			0
$661$		−	36.0000i		−	1.40024i	−0.714026	−	0.700119i	$-0.753130\pi$
$661$		−	36.0000i		−	1.40024i	0.714026	−	0.700119i	$-0.246870\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	24.0000i		−	0.930680i
$666$		0			0
$667$	18.0000			0.696963
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	39.0000i		−	1.50558i
$672$		0			0
$673$	16.0000			0.616755			0.308377	−	0.951264i	$-0.400214\pi$
$673$	16.0000			0.616755			0.308377	+	0.951264i	$0.400214\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	39.0000			1.49889			0.749446	−	0.662066i	$-0.230320\pi$
$677$	39.0000			1.49889			0.749446	+	0.662066i	$0.230320\pi$
$678$		0			0
$679$	−15.0000			−0.575647
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	10.0000i		−	0.382639i	−0.981528	−	0.191320i	$-0.938723\pi$
$683$		−	10.0000i		−	0.382639i	0.981528	−	0.191320i	$-0.0612767\pi$
$684$		0			0
$685$	6.00000			0.229248
$686$		0			0
$687$		0			0
$688$		0			0
$689$	10.0000	−	15.0000i	0.380970	−	0.571454i
$690$		0			0
$691$			10.0000i			0.380418i	0.981744	+	0.190209i	$0.0609166\pi$
$691$			10.0000i			0.380418i	−0.981744	+	0.190209i	$0.939083\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			19.0000i			0.720711i
$696$		0			0
$697$			5.00000i			0.189389i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	12.0000			0.453234			0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000			0.453234			0.226617	+	0.973984i	$0.427233\pi$
$702$		0			0
$703$	8.00000			0.301726
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	42.0000i		−	1.57957i
$708$		0			0
$709$			32.0000i			1.20179i	0.799330	+	0.600893i	$0.205188\pi$
$709$			32.0000i			1.20179i	−0.799330	+	0.600893i	$0.794812\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	30.0000i		−	1.12351i
$714$		0			0
$715$	6.00000	−	9.00000i	0.224387	−	0.336581i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−2.00000			−0.0745874			−0.0372937	−	0.999304i	$-0.511874\pi$
$719$	−2.00000			−0.0745874			−0.0372937	+	0.999304i	$0.511874\pi$
$720$		0			0
$721$			36.0000i			1.34071i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	6.00000			0.222834
$726$		0			0
$727$	26.0000			0.964287			0.482143	−	0.876092i	$-0.339858\pi$
$727$	26.0000			0.964287			0.482143	+	0.876092i	$0.339858\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	6.00000			0.221918
$732$		0			0
$733$			1.00000i			0.0369358i	0.999829	+	0.0184679i	$0.00587886\pi$
$733$			1.00000i			0.0369358i	−0.999829	+	0.0184679i	$0.994121\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	24.0000			0.884051
$738$		0			0
$739$			26.0000i			0.956425i	0.878244	+	0.478213i	$0.158715\pi$
$739$			26.0000i			0.956425i	−0.878244	+	0.478213i	$0.841285\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	12.0000i		−	0.440237i	−0.975473	−	0.220119i	$-0.929356\pi$
$743$		−	12.0000i		−	0.440237i	0.975473	−	0.220119i	$-0.0706445\pi$
$744$		0			0
$745$	15.0000			0.549557
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	57.0000i		−	2.08273i
$750$		0			0
$751$	−49.0000			−1.78804			−0.894018	−	0.448032i	$-0.852125\pi$
$751$	−49.0000			−1.78804			−0.894018	+	0.448032i	$0.852125\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	2.00000			0.0727875
$756$		0			0
$757$	40.0000			1.45382			0.726912	−	0.686730i	$-0.240955\pi$
$757$	40.0000			1.45382			0.726912	+	0.686730i	$0.240955\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			10.0000i			0.362500i	0.983437	+	0.181250i	$0.0580143\pi$
$761$			10.0000i			0.362500i	−0.983437	+	0.181250i	$0.941986\pi$
$762$		0			0
$763$	18.0000			0.651644
$764$		0			0
$765$		0			0
$766$		0			0
$767$	12.0000	+	8.00000i	0.433295	+	0.288863i
$768$		0			0
$769$		−	32.0000i		−	1.15395i	−0.816762	−	0.576975i	$-0.804233\pi$
$769$		−	32.0000i		−	1.15395i	0.816762	−	0.576975i	$-0.195767\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	10.0000i		−	0.359675i	−0.983696	−	0.179838i	$-0.942443\pi$
$773$		−	10.0000i		−	0.359675i	0.983696	−	0.179838i	$-0.0575572\pi$
$774$		0			0
$775$		−	10.0000i		−	0.359211i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−40.0000			−1.43315
$780$		0			0
$781$	33.0000			1.18083
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	12.0000i		−	0.428298i
$786$		0			0
$787$			28.0000i			0.998092i	0.866575	+	0.499046i	$0.166316\pi$
$787$			28.0000i			0.998092i	−0.866575	+	0.499046i	$0.833684\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	18.0000i		−	0.640006i
$792$		0			0
$793$	−26.0000	+	39.0000i	−0.923287	+	1.38493i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−13.0000			−0.460484			−0.230242	−	0.973133i	$-0.573952\pi$
$797$	−13.0000			−0.460484			−0.230242	+	0.973133i	$0.573952\pi$
$798$		0			0
$799$			8.00000i			0.283020i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−6.00000			−0.211735
$804$		0			0
$805$	9.00000			0.317208
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−8.00000			−0.281265			−0.140633	−	0.990062i	$-0.544914\pi$
$809$	−8.00000			−0.281265			−0.140633	+	0.990062i	$0.544914\pi$
$810$		0			0
$811$		−	46.0000i		−	1.61528i	−0.589677	−	0.807639i	$-0.700745\pi$
$811$		−	46.0000i		−	1.61528i	0.589677	−	0.807639i	$-0.299255\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−7.00000			−0.245199
$816$		0			0
$817$			48.0000i			1.67931i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	3.00000i		−	0.104701i	−0.998629	−	0.0523504i	$-0.983329\pi$
$821$		−	3.00000i		−	0.104701i	0.998629	−	0.0523504i	$-0.0166713\pi$
$822$		0			0
$823$	−6.00000			−0.209147			−0.104573	−	0.994517i	$-0.533348\pi$
$823$	−6.00000			−0.209147			−0.104573	+	0.994517i	$0.533348\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	8.00000i		−	0.278187i	−0.990279	−	0.139094i	$-0.955581\pi$
$827$		−	8.00000i		−	0.278187i	0.990279	−	0.139094i	$-0.0444189\pi$
$828$		0			0
$829$	38.0000			1.31979			0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000			1.31979			0.659897	+	0.751356i	$0.270600\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−2.00000			−0.0692959
$834$		0			0
$835$	−20.0000			−0.692129
$836$		0			0
$837$		0			0
$838$		0			0
$839$			13.0000i			0.448810i	0.974496	+	0.224405i	$0.0720438\pi$
$839$			13.0000i			0.448810i	−0.974496	+	0.224405i	$0.927956\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−12.0000	+	5.00000i	−0.412813	+	0.172005i
$846$		0			0
$847$		−	6.00000i		−	0.206162i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			3.00000i			0.102839i
$852$		0			0
$853$			17.0000i			0.582069i	0.956713	+	0.291034i	$0.0939994\pi$
$853$			17.0000i			0.582069i	−0.956713	+	0.291034i	$0.906001\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−57.0000			−1.94708			−0.973541	−	0.228510i	$-0.926614\pi$
$857$	−57.0000			−1.94708			−0.973541	+	0.228510i	$0.926614\pi$
$858$		0			0
$859$	53.0000			1.80834			0.904168	−	0.427176i	$-0.140492\pi$
$859$	53.0000			1.80834			0.904168	+	0.427176i	$0.140492\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			42.0000i			1.42970i	0.699280	+	0.714848i	$0.253504\pi$
$863$			42.0000i			1.42970i	−0.699280	+	0.714848i	$0.746496\pi$
$864$		0			0
$865$		−	6.00000i		−	0.204006i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			15.0000i			0.508840i
$870$		0			0
$871$	−24.0000	−	16.0000i	−0.813209	−	0.542139i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	3.00000			0.101419
$876$		0			0
$877$		−	42.0000i		−	1.41824i	−0.705088	−	0.709120i	$-0.749093\pi$
$877$		−	42.0000i		−	1.41824i	0.705088	−	0.709120i	$-0.250907\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−36.0000			−1.21287			−0.606435	−	0.795133i	$-0.707401\pi$
$881$	−36.0000			−1.21287			−0.606435	+	0.795133i	$0.707401\pi$
$882$		0			0
$883$	32.0000			1.07689			0.538443	−	0.842662i	$-0.319013\pi$
$883$	32.0000			1.07689			0.538443	+	0.842662i	$0.319013\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−15.0000			−0.503651			−0.251825	−	0.967773i	$-0.581031\pi$
$887$	−15.0000			−0.503651			−0.251825	+	0.967773i	$0.581031\pi$
$888$		0			0
$889$		−	42.0000i		−	1.40863i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−64.0000			−2.14168
$894$		0			0
$895$		−	16.0000i		−	0.534821i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	60.0000i		−	2.00111i
$900$		0			0
$901$	−5.00000			−0.166574
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	19.0000i		−	0.631581i
$906$		0			0
$907$	32.0000			1.06254			0.531271	−	0.847202i	$-0.321714\pi$
$907$	32.0000			1.06254			0.531271	+	0.847202i	$0.321714\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−50.0000			−1.65657			−0.828287	−	0.560304i	$-0.810684\pi$
$911$	−50.0000			−1.65657			−0.828287	+	0.560304i	$0.810684\pi$
$912$		0			0
$913$	−18.0000			−0.595713
$914$		0			0
$915$		0			0
$916$		0			0
$917$			24.0000i			0.792550i
$918$		0			0
$919$	−49.0000			−1.61636			−0.808180	−	0.588935i	$-0.799547\pi$
$919$	−49.0000			−1.61636			−0.808180	+	0.588935i	$0.799547\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−33.0000	−	22.0000i	−1.08621	−	0.724139i
$924$		0			0
$925$			1.00000i			0.0328798i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			19.0000i			0.623370i	0.950186	+	0.311685i	$0.100893\pi$
$929$			19.0000i			0.623370i	−0.950186	+	0.311685i	$0.899107\pi$
$930$		0			0
$931$		−	16.0000i		−	0.524379i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−3.00000			−0.0981105
$936$		0			0
$937$	30.0000			0.980057			0.490029	−	0.871706i	$-0.336986\pi$
$937$	30.0000			0.980057			0.490029	+	0.871706i	$0.336986\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			35.0000i			1.14097i	0.821309	+	0.570484i	$0.193244\pi$
$941$			35.0000i			1.14097i	−0.821309	+	0.570484i	$0.806756\pi$
$942$		0			0
$943$		−	15.0000i		−	0.488467i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	48.0000i		−	1.55979i	−0.625910	−	0.779895i	$-0.715272\pi$
$947$		−	48.0000i		−	1.55979i	0.625910	−	0.779895i	$-0.284728\pi$
$948$		0			0
$949$	6.00000	+	4.00000i	0.194768	+	0.129845i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	41.0000			1.32812			0.664060	−	0.747679i	$-0.268832\pi$
$953$	41.0000			1.32812			0.664060	+	0.747679i	$0.268832\pi$
$954$		0			0
$955$			24.0000i			0.776622i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	18.0000			0.581250
$960$		0			0
$961$	−69.0000			−2.22581
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1.00000			−0.0321911
$966$		0			0
$967$			24.0000i			0.771788i	0.922543	+	0.385894i	$0.126107\pi$
$967$			24.0000i			0.771788i	−0.922543	+	0.385894i	$0.873893\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−38.0000			−1.21948			−0.609739	−	0.792602i	$-0.708726\pi$
$971$	−38.0000			−1.21948			−0.609739	+	0.792602i	$0.708726\pi$
$972$		0			0
$973$			57.0000i			1.82734i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	36.0000i		−	1.15174i	−0.817541	−	0.575871i	$-0.804663\pi$
$977$		−	36.0000i		−	1.15174i	0.817541	−	0.575871i	$-0.195337\pi$
$978$		0			0
$979$	−45.0000			−1.43821
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	28.0000i		−	0.893061i	−0.894768	−	0.446531i	$-0.852659\pi$
$983$		−	28.0000i		−	0.893061i	0.894768	−	0.446531i	$-0.147341\pi$
$984$		0			0
$985$	18.0000			0.573528
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−18.0000			−0.572367
$990$		0			0
$991$	53.0000			1.68360			0.841800	−	0.539789i	$-0.181496\pi$
$991$	53.0000			1.68360			0.841800	+	0.539789i	$0.181496\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			16.0000i			0.507234i
$996$		0			0
$997$	−24.0000			−0.760088			−0.380044	−	0.924968i	$-0.624091\pi$
$997$	−24.0000			−0.760088			−0.380044	+	0.924968i	$0.624091\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4680.2.g.e.2521.1		2
3.2	odd	2		1560.2.g.d.961.2	yes	2
12.11	even	2		3120.2.g.b.961.2		2
13.12	even	2	inner	4680.2.g.e.2521.2		2
39.38	odd	2		1560.2.g.d.961.1	✓	2
156.155	even	2		3120.2.g.b.961.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.g.d.961.1	✓	2	39.38	odd	2
1560.2.g.d.961.2	yes	2	3.2	odd	2
3120.2.g.b.961.1		2	156.155	even	2
3120.2.g.b.961.2		2	12.11	even	2
4680.2.g.e.2521.1		2	1.1	even	1	trivial
4680.2.g.e.2521.2		2	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:	\( N \)	\(=\)	\( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4680.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{5} -3.00000i q^{7} +O(q^{10})\)	\(q-1.00000i q^{5} -3.00000i q^{7} -3.00000i q^{11} +(-2.00000 + 3.00000i) q^{13} +1.00000 q^{17} +8.00000i q^{19} -3.00000 q^{23} -1.00000 q^{25} -6.00000 q^{29} +10.0000i q^{31} -3.00000 q^{35} -1.00000i q^{37} +5.00000i q^{41} +6.00000 q^{43} +8.00000i q^{47} -2.00000 q^{49} -5.00000 q^{53} -3.00000 q^{55} -4.00000i q^{59} +13.0000 q^{61} +(3.00000 + 2.00000i) q^{65} +8.00000i q^{67} +11.0000i q^{71} -2.00000i q^{73} -9.00000 q^{77} -5.00000 q^{79} -6.00000i q^{83} -1.00000i q^{85} -15.0000i q^{89} +(9.00000 + 6.00000i) q^{91} +8.00000 q^{95} -5.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q - 4 q^{13} + 2 q^{17} - 6 q^{23} - 2 q^{25} - 12 q^{29} - 6 q^{35} + 12 q^{43} - 4 q^{49} - 10 q^{53} - 6 q^{55} + 26 q^{61} + 6 q^{65} - 18 q^{77} - 10 q^{79} + 18 q^{91} + 16 q^{95}+O(q^{100}) \) 2 * q - 4 * q^13 + 2 * q^17 - 6 * q^23 - 2 * q^25 - 12 * q^29 - 6 * q^35 + 12 * q^43 - 4 * q^49 - 10 * q^53 - 6 * q^55 + 26 * q^61 + 6 * q^65 - 18 * q^77 - 10 * q^79 + 18 * q^91 + 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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