LMFDB - Embedded newform 980.2.e.e.589.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [980,2,Mod(589,980)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(980, 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("980.589");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 980 = 2^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	980.e (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:	$7.82533939809$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			589.1
Root			$0.707107 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	980.589
Dual form			980.2.e.e.589.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} +(-1.41421 + 1.73205i) q^{5} +O(q^{10})$	$q-1.73205i q^{3} +(-1.41421 + 1.73205i) q^{5} -1.00000 q^{11} -1.73205i q^{13} +(3.00000 + 2.44949i) q^{15} -5.19615i q^{17} -2.82843 q^{19} -2.44949i q^{23} +(-1.00000 - 4.89898i) q^{25} -5.19615i q^{27} +7.00000 q^{29} -7.07107 q^{31} +1.73205i q^{33} -7.34847i q^{37} -3.00000 q^{39} -7.07107 q^{41} +9.79796i q^{43} -12.1244i q^{47} -9.00000 q^{51} -12.2474i q^{53} +(1.41421 - 1.73205i) q^{55} +4.89898i q^{57} -7.07107 q^{59} +14.1421 q^{61} +(3.00000 + 2.44949i) q^{65} +12.2474i q^{67} -4.24264 q^{69} -10.0000 q^{71} +(-8.48528 + 1.73205i) q^{75} -3.00000 q^{79} -9.00000 q^{81} +(9.00000 + 7.34847i) q^{85} -12.1244i q^{87} +12.2474i q^{93} +(4.00000 - 4.89898i) q^{95} +5.19615i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 4 q^{11} + 12 q^{15} - 4 q^{25} + 28 q^{29} - 12 q^{39} - 36 q^{51} + 12 q^{65} - 40 q^{71} - 12 q^{79} - 36 q^{81} + 36 q^{85} + 16 q^{95}+O(q^{100}) $ 4 * q - 4 * q^11 + 12 * q^15 - 4 * q^25 + 28 * q^29 - 12 * q^39 - 36 * q^51 + 12 * q^65 - 40 * q^71 - 12 * q^79 - 36 * q^81 + 36 * q^85 + 16 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/980\mathbb{Z}\right)^\times$.

$n$	$101$	$197$	$491$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		−	1.73205i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$3$		−	1.73205i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$4$		0			0
$5$	−1.41421	+	1.73205i	−0.632456	+	0.774597i
$5$	−1.41421	+	1.73205i	−0.632456	+	0.774597i
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.00000			−0.301511			−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000			−0.301511			−0.150756	+	0.988571i	$0.548171\pi$
$12$		0			0
$13$		−	1.73205i		−	0.480384i	−0.970725	−	0.240192i	$-0.922790\pi$
$13$		−	1.73205i		−	0.480384i	0.970725	−	0.240192i	$-0.0772105\pi$
$14$		0			0
$15$	3.00000	+	2.44949i	0.774597	+	0.632456i
$16$		0			0
$17$		−	5.19615i		−	1.26025i	−0.776493	−	0.630126i	$-0.783003\pi$
$17$		−	5.19615i		−	1.26025i	0.776493	−	0.630126i	$-0.216997\pi$
$18$		0			0
$19$	−2.82843			−0.648886			−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843			−0.648886			−0.324443	+	0.945905i	$0.605177\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	2.44949i		−	0.510754i	−0.966842	−	0.255377i	$-0.917800\pi$
$23$		−	2.44949i		−	0.510754i	0.966842	−	0.255377i	$-0.0821996\pi$
$24$		0			0
$25$	−1.00000	−	4.89898i	−0.200000	−	0.979796i
$26$		0			0
$27$		−	5.19615i		−	1.00000i
$28$		0			0
$29$	7.00000			1.29987			0.649934	−	0.759991i	$-0.274797\pi$
$29$	7.00000			1.29987			0.649934	+	0.759991i	$0.274797\pi$
$30$		0			0
$31$	−7.07107			−1.27000			−0.635001	−	0.772512i	$-0.719000\pi$
$31$	−7.07107			−1.27000			−0.635001	+	0.772512i	$0.719000\pi$
$32$		0			0
$33$			1.73205i			0.301511i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	7.34847i		−	1.20808i	−0.796954	−	0.604040i	$-0.793557\pi$
$37$		−	7.34847i		−	1.20808i	0.796954	−	0.604040i	$-0.206443\pi$
$38$		0			0
$39$	−3.00000			−0.480384
$40$		0			0
$41$	−7.07107			−1.10432			−0.552158	−	0.833740i	$-0.686195\pi$
$41$	−7.07107			−1.10432			−0.552158	+	0.833740i	$0.686195\pi$
$42$		0			0
$43$			9.79796i			1.49417i	0.664726	+	0.747087i	$0.268548\pi$
$43$			9.79796i			1.49417i	−0.664726	+	0.747087i	$0.731452\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	12.1244i		−	1.76852i	−0.466996	−	0.884260i	$-0.654664\pi$
$47$		−	12.1244i		−	1.76852i	0.466996	−	0.884260i	$-0.345336\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−9.00000			−1.26025
$52$		0			0
$53$		−	12.2474i		−	1.68232i	−0.540789	−	0.841158i	$-0.681874\pi$
$53$		−	12.2474i		−	1.68232i	0.540789	−	0.841158i	$-0.318126\pi$
$54$		0			0
$55$	1.41421	−	1.73205i	0.190693	−	0.233550i
$56$		0			0
$57$			4.89898i			0.648886i
$58$		0			0
$59$	−7.07107			−0.920575			−0.460287	−	0.887770i	$-0.652254\pi$
$59$	−7.07107			−0.920575			−0.460287	+	0.887770i	$0.652254\pi$
$60$		0			0
$61$	14.1421			1.81071			0.905357	−	0.424650i	$-0.139603\pi$
$61$	14.1421			1.81071			0.905357	+	0.424650i	$0.139603\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.00000	+	2.44949i	0.372104	+	0.303822i
$66$		0			0
$67$			12.2474i			1.49626i	0.663550	+	0.748132i	$0.269049\pi$
$67$			12.2474i			1.49626i	−0.663550	+	0.748132i	$0.730951\pi$
$68$		0			0
$69$	−4.24264			−0.510754
$70$		0			0
$71$	−10.0000			−1.18678			−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000			−1.18678			−0.593391	+	0.804914i	$0.702211\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	−8.48528	+	1.73205i	−0.979796	+	0.200000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−3.00000			−0.337526			−0.168763	−	0.985657i	$-0.553977\pi$
$79$	−3.00000			−0.337526			−0.168763	+	0.985657i	$0.553977\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$	9.00000	+	7.34847i	0.976187	+	0.797053i
$86$		0			0
$87$		−	12.1244i		−	1.29987i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$			12.2474i			1.27000i
$94$		0			0
$95$	4.00000	−	4.89898i	0.410391	−	0.502625i
$96$		0			0
$97$			5.19615i			0.527589i	0.964579	+	0.263795i	$0.0849741\pi$
$97$			5.19615i			0.527589i	−0.964579	+	0.263795i	$0.915026\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−5.65685			−0.562878			−0.281439	−	0.959579i	$-0.590812\pi$
$101$	−5.65685			−0.562878			−0.281439	+	0.959579i	$0.590812\pi$
$102$		0			0
$103$			1.73205i			0.170664i	0.996353	+	0.0853320i	$0.0271951\pi$
$103$			1.73205i			0.170664i	−0.996353	+	0.0853320i	$0.972805\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	4.89898i		−	0.473602i	−0.971558	−	0.236801i	$-0.923901\pi$
$107$		−	4.89898i		−	0.473602i	0.971558	−	0.236801i	$-0.0760990\pi$
$108$		0			0
$109$	3.00000			0.287348			0.143674	−	0.989625i	$-0.454108\pi$
$109$	3.00000			0.287348			0.143674	+	0.989625i	$0.454108\pi$
$110$		0			0
$111$	−12.7279			−1.20808
$112$		0			0
$113$		−	2.44949i		−	0.230429i	−0.993341	−	0.115214i	$-0.963245\pi$
$113$		−	2.44949i		−	0.230429i	0.993341	−	0.115214i	$-0.0367555\pi$
$114$		0			0
$115$	4.24264	+	3.46410i	0.395628	+	0.323029i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−10.0000			−0.909091
$122$		0			0
$123$			12.2474i			1.10432i
$124$		0			0
$125$	9.89949	+	5.19615i	0.885438	+	0.464758i
$126$		0			0
$127$		−	12.2474i		−	1.08679i	−0.839479	−	0.543393i	$-0.817139\pi$
$127$		−	12.2474i		−	1.08679i	0.839479	−	0.543393i	$-0.182861\pi$
$128$		0			0
$129$	16.9706			1.49417
$130$		0			0
$131$	12.7279			1.11204			0.556022	−	0.831168i	$-0.312327\pi$
$131$	12.7279			1.11204			0.556022	+	0.831168i	$0.312327\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	9.00000	+	7.34847i	0.774597	+	0.632456i
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$	18.3848			1.55938			0.779688	−	0.626168i	$-0.215378\pi$
$139$	18.3848			1.55938			0.779688	+	0.626168i	$0.215378\pi$
$140$		0			0
$141$	−21.0000			−1.76852
$142$		0			0
$143$			1.73205i			0.144841i
$144$		0			0
$145$	−9.89949	+	12.1244i	−0.822108	+	1.00687i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	20.0000			1.63846			0.819232	−	0.573462i	$-0.194400\pi$
$149$	20.0000			1.63846			0.819232	+	0.573462i	$0.194400\pi$
$150$		0			0
$151$	11.0000			0.895167			0.447584	−	0.894242i	$-0.352285\pi$
$151$	11.0000			0.895167			0.447584	+	0.894242i	$0.352285\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	10.0000	−	12.2474i	0.803219	−	0.983739i
$156$		0			0
$157$			17.3205i			1.38233i	0.722698	+	0.691164i	$0.242902\pi$
$157$			17.3205i			1.38233i	−0.722698	+	0.691164i	$0.757098\pi$
$158$		0			0
$159$	−21.2132			−1.68232
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	22.0454i		−	1.72673i	−0.504580	−	0.863365i	$-0.668353\pi$
$163$		−	22.0454i		−	1.72673i	0.504580	−	0.863365i	$-0.331647\pi$
$164$		0			0
$165$	−3.00000	−	2.44949i	−0.233550	−	0.190693i
$166$		0			0
$167$		−	5.19615i		−	0.402090i	−0.979582	−	0.201045i	$-0.935566\pi$
$167$		−	5.19615i		−	0.402090i	0.979582	−	0.201045i	$-0.0644338\pi$
$168$		0			0
$169$	10.0000			0.769231
$170$		0			0
$171$		0			0
$172$		0			0
$173$			1.73205i			0.131685i	0.997830	+	0.0658427i	$0.0209736\pi$
$173$			1.73205i			0.131685i	−0.997830	+	0.0658427i	$0.979026\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			12.2474i			0.920575i
$178$		0			0
$179$	10.0000			0.747435			0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000			0.747435			0.373718	+	0.927543i	$0.378083\pi$
$180$		0			0
$181$	7.07107			0.525588			0.262794	−	0.964852i	$-0.415356\pi$
$181$	7.07107			0.525588			0.262794	+	0.964852i	$0.415356\pi$
$182$		0			0
$183$		−	24.4949i		−	1.81071i
$184$		0			0
$185$	12.7279	+	10.3923i	0.935775	+	0.764057i
$186$		0			0
$187$			5.19615i			0.379980i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1.00000			−0.0723575			−0.0361787	−	0.999345i	$-0.511519\pi$
$191$	−1.00000			−0.0723575			−0.0361787	+	0.999345i	$0.511519\pi$
$192$		0			0
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$	4.24264	−	5.19615i	0.303822	−	0.372104i
$196$		0			0
$197$		−	12.2474i		−	0.872595i	−0.899803	−	0.436297i	$-0.856290\pi$
$197$		−	12.2474i		−	0.872595i	0.899803	−	0.436297i	$-0.143710\pi$
$198$		0			0
$199$	7.07107			0.501255			0.250627	−	0.968084i	$-0.419363\pi$
$199$	7.07107			0.501255			0.250627	+	0.968084i	$0.419363\pi$
$200$		0			0
$201$	21.2132			1.49626
$202$		0			0
$203$		0			0
$204$		0			0
$205$	10.0000	−	12.2474i	0.698430	−	0.855399i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2.82843			0.195646
$210$		0			0
$211$	−21.0000			−1.44570			−0.722850	−	0.691005i	$-0.757168\pi$
$211$	−21.0000			−1.44570			−0.722850	+	0.691005i	$0.757168\pi$
$212$		0			0
$213$			17.3205i			1.18678i
$214$		0			0
$215$	−16.9706	−	13.8564i	−1.15738	−	0.944999i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−9.00000			−0.605406
$222$		0			0
$223$			19.0526i			1.27585i	0.770097	+	0.637927i	$0.220208\pi$
$223$			19.0526i			1.27585i	−0.770097	+	0.637927i	$0.779792\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			12.1244i			0.804722i	0.915481	+	0.402361i	$0.131810\pi$
$227$			12.1244i			0.804722i	−0.915481	+	0.402361i	$0.868190\pi$
$228$		0			0
$229$	7.07107			0.467269			0.233635	−	0.972324i	$-0.424938\pi$
$229$	7.07107			0.467269			0.233635	+	0.972324i	$0.424938\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			24.4949i			1.60471i	0.596844	+	0.802357i	$0.296421\pi$
$233$			24.4949i			1.60471i	−0.596844	+	0.802357i	$0.703579\pi$
$234$		0			0
$235$	21.0000	+	17.1464i	1.36989	+	1.11851i
$236$		0			0
$237$			5.19615i			0.337526i
$238$		0			0
$239$	−13.0000			−0.840900			−0.420450	−	0.907316i	$-0.638128\pi$
$239$	−13.0000			−0.840900			−0.420450	+	0.907316i	$0.638128\pi$
$240$		0			0
$241$	−1.41421			−0.0910975			−0.0455488	−	0.998962i	$-0.514504\pi$
$241$	−1.41421			−0.0910975			−0.0455488	+	0.998962i	$0.514504\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$			4.89898i			0.311715i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−15.5563			−0.981908			−0.490954	−	0.871185i	$-0.663352\pi$
$251$	−15.5563			−0.981908			−0.490954	+	0.871185i	$0.663352\pi$
$252$		0			0
$253$			2.44949i			0.153998i
$254$		0			0
$255$	12.7279	−	15.5885i	0.797053	−	0.976187i
$256$		0			0
$257$			17.3205i			1.08042i	0.841529	+	0.540212i	$0.181656\pi$
$257$			17.3205i			1.08042i	−0.841529	+	0.540212i	$0.818344\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$			9.79796i			0.604168i	0.953281	+	0.302084i	$0.0976823\pi$
$263$			9.79796i			0.604168i	−0.953281	+	0.302084i	$0.902318\pi$
$264$		0			0
$265$	21.2132	+	17.3205i	1.30312	+	1.06399i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	4.24264			0.258678			0.129339	−	0.991600i	$-0.458714\pi$
$269$	4.24264			0.258678			0.129339	+	0.991600i	$0.458714\pi$
$270$		0			0
$271$	28.2843			1.71815			0.859074	−	0.511852i	$-0.171040\pi$
$271$	28.2843			1.71815			0.859074	+	0.511852i	$0.171040\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.00000	+	4.89898i	0.0603023	+	0.295420i
$276$		0			0
$277$		−	7.34847i		−	0.441527i	−0.975327	−	0.220763i	$-0.929145\pi$
$277$		−	7.34847i		−	0.441527i	0.975327	−	0.220763i	$-0.0708548\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−29.0000			−1.72999			−0.864997	−	0.501776i	$-0.832680\pi$
$281$	−29.0000			−1.72999			−0.864997	+	0.501776i	$0.832680\pi$
$282$		0			0
$283$		−	1.73205i		−	0.102960i	−0.998674	−	0.0514799i	$-0.983606\pi$
$283$		−	1.73205i		−	0.102960i	0.998674	−	0.0514799i	$-0.0163938\pi$
$284$		0			0
$285$	−8.48528	−	6.92820i	−0.502625	−	0.410391i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−10.0000			−0.588235
$290$		0			0
$291$	9.00000			0.527589
$292$		0			0
$293$		−	19.0526i		−	1.11306i	−0.830827	−	0.556531i	$-0.812132\pi$
$293$		−	19.0526i		−	1.11306i	0.830827	−	0.556531i	$-0.187868\pi$
$294$		0			0
$295$	10.0000	−	12.2474i	0.582223	−	0.713074i
$296$		0			0
$297$			5.19615i			0.301511i
$298$		0			0
$299$	−4.24264			−0.245358
$300$		0			0
$301$		0			0
$302$		0			0
$303$			9.79796i			0.562878i
$304$		0			0
$305$	−20.0000	+	24.4949i	−1.14520	+	1.40257i
$306$		0			0
$307$			29.4449i			1.68051i	0.542194	+	0.840254i	$0.317594\pi$
$307$			29.4449i			1.68051i	−0.542194	+	0.840254i	$0.682406\pi$
$308$		0			0
$309$	3.00000			0.170664
$310$		0			0
$311$	−8.48528			−0.481156			−0.240578	−	0.970630i	$-0.577337\pi$
$311$	−8.48528			−0.481156			−0.240578	+	0.970630i	$0.577337\pi$
$312$		0			0
$313$		−	1.73205i		−	0.0979013i	−0.998801	−	0.0489506i	$-0.984412\pi$
$313$		−	1.73205i		−	0.0979013i	0.998801	−	0.0489506i	$-0.0155877\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			19.5959i			1.10062i	0.834962	+	0.550308i	$0.185490\pi$
$317$			19.5959i			1.10062i	−0.834962	+	0.550308i	$0.814510\pi$
$318$		0			0
$319$	−7.00000			−0.391925
$320$		0			0
$321$	−8.48528			−0.473602
$322$		0			0
$323$			14.6969i			0.817760i
$324$		0			0
$325$	−8.48528	+	1.73205i	−0.470679	+	0.0960769i
$326$		0			0
$327$		−	5.19615i		−	0.287348i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−30.0000			−1.64895			−0.824475	−	0.565899i	$-0.808529\pi$
$331$	−30.0000			−1.64895			−0.824475	+	0.565899i	$0.808529\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−21.2132	−	17.3205i	−1.15900	−	0.946320i
$336$		0			0
$337$			31.8434i			1.73462i	0.497770	+	0.867309i	$0.334153\pi$
$337$			31.8434i			1.73462i	−0.497770	+	0.867309i	$0.665847\pi$
$338$		0			0
$339$	−4.24264			−0.230429
$340$		0			0
$341$	7.07107			0.382920
$342$		0			0
$343$		0			0
$344$		0			0
$345$	6.00000	−	7.34847i	0.323029	−	0.395628i
$346$		0			0
$347$		−	17.1464i		−	0.920468i	−0.887798	−	0.460234i	$-0.847765\pi$
$347$		−	17.1464i		−	0.920468i	0.887798	−	0.460234i	$-0.152235\pi$
$348$		0			0
$349$	−2.82843			−0.151402			−0.0757011	−	0.997131i	$-0.524119\pi$
$349$	−2.82843			−0.151402			−0.0757011	+	0.997131i	$0.524119\pi$
$350$		0			0
$351$	−9.00000			−0.480384
$352$		0			0
$353$		−	15.5885i		−	0.829690i	−0.909892	−	0.414845i	$-0.863836\pi$
$353$		−	15.5885i		−	0.829690i	0.909892	−	0.414845i	$-0.136164\pi$
$354$		0			0
$355$	14.1421	−	17.3205i	0.750587	−	0.919277i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	20.0000			1.05556			0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000			1.05556			0.527780	+	0.849381i	$0.323025\pi$
$360$		0			0
$361$	−11.0000			−0.578947
$362$		0			0
$363$			17.3205i			0.909091i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			5.19615i			0.271237i	0.990761	+	0.135618i	$0.0433021\pi$
$367$			5.19615i			0.271237i	−0.990761	+	0.135618i	$0.956698\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	24.4949i		−	1.26830i	−0.773211	−	0.634149i	$-0.781351\pi$
$373$		−	24.4949i		−	1.26830i	0.773211	−	0.634149i	$-0.218649\pi$
$374$		0			0
$375$	9.00000	−	17.1464i	0.464758	−	0.885438i
$376$		0			0
$377$		−	12.1244i		−	0.624436i
$378$		0			0
$379$	10.0000			0.513665			0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000			0.513665			0.256833	+	0.966456i	$0.417321\pi$
$380$		0			0
$381$	−21.2132			−1.08679
$382$		0			0
$383$		−	20.7846i		−	1.06204i	−0.847358	−	0.531022i	$-0.821808\pi$
$383$		−	20.7846i		−	1.06204i	0.847358	−	0.531022i	$-0.178192\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	17.0000			0.861934			0.430967	−	0.902368i	$-0.358172\pi$
$389$	17.0000			0.861934			0.430967	+	0.902368i	$0.358172\pi$
$390$		0			0
$391$	−12.7279			−0.643679
$392$		0			0
$393$		−	22.0454i		−	1.11204i
$394$		0			0
$395$	4.24264	−	5.19615i	0.213470	−	0.261447i
$396$		0			0
$397$			29.4449i			1.47780i	0.673818	+	0.738898i	$0.264653\pi$
$397$			29.4449i			1.47780i	−0.673818	+	0.738898i	$0.735347\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−11.0000			−0.549314			−0.274657	−	0.961542i	$-0.588564\pi$
$401$	−11.0000			−0.549314			−0.274657	+	0.961542i	$0.588564\pi$
$402$		0			0
$403$			12.2474i			0.610089i
$404$		0			0
$405$	12.7279	−	15.5885i	0.632456	−	0.774597i
$406$		0			0
$407$			7.34847i			0.364250i
$408$		0			0
$409$	11.3137			0.559427			0.279713	−	0.960084i	$-0.409761\pi$
$409$	11.3137			0.559427			0.279713	+	0.960084i	$0.409761\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		−	31.8434i		−	1.55938i
$418$		0			0
$419$	21.2132			1.03633			0.518166	−	0.855280i	$-0.326615\pi$
$419$	21.2132			1.03633			0.518166	+	0.855280i	$0.326615\pi$
$420$		0			0
$421$	21.0000			1.02348			0.511739	−	0.859141i	$-0.329002\pi$
$421$	21.0000			1.02348			0.511739	+	0.859141i	$0.329002\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−25.4558	+	5.19615i	−1.23479	+	0.252050i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	3.00000			0.144841
$430$		0			0
$431$	−11.0000			−0.529851			−0.264926	−	0.964269i	$-0.585347\pi$
$431$	−11.0000			−0.529851			−0.264926	+	0.964269i	$0.585347\pi$
$432$		0			0
$433$		−	17.3205i		−	0.832370i	−0.909280	−	0.416185i	$-0.863367\pi$
$433$		−	17.3205i		−	0.832370i	0.909280	−	0.416185i	$-0.136633\pi$
$434$		0			0
$435$	21.0000	+	17.1464i	1.00687	+	0.822108i
$436$		0			0
$437$			6.92820i			0.331421i
$438$		0			0
$439$	−18.3848			−0.877457			−0.438729	−	0.898620i	$-0.644571\pi$
$439$	−18.3848			−0.877457			−0.438729	+	0.898620i	$0.644571\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	34.2929i		−	1.62930i	−0.579951	−	0.814651i	$-0.696928\pi$
$443$		−	34.2929i		−	1.62930i	0.579951	−	0.814651i	$-0.303072\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	34.6410i		−	1.63846i
$448$		0			0
$449$	23.0000			1.08544			0.542719	−	0.839915i	$-0.317395\pi$
$449$	23.0000			1.08544			0.542719	+	0.839915i	$0.317395\pi$
$450$		0			0
$451$	7.07107			0.332964
$452$		0			0
$453$		−	19.0526i		−	0.895167i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	36.7423i		−	1.71873i	−0.511359	−	0.859367i	$-0.670858\pi$
$457$		−	36.7423i		−	1.71873i	0.511359	−	0.859367i	$-0.329142\pi$
$458$		0			0
$459$	−27.0000			−1.26025
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$			24.4949i			1.13837i	0.822208	+	0.569187i	$0.192742\pi$
$463$			24.4949i			1.13837i	−0.822208	+	0.569187i	$0.807258\pi$
$464$		0			0
$465$	−21.2132	−	17.3205i	−0.983739	−	0.803219i
$466$		0			0
$467$			12.1244i			0.561048i	0.959847	+	0.280524i	$0.0905083\pi$
$467$			12.1244i			0.561048i	−0.959847	+	0.280524i	$0.909492\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	30.0000			1.38233
$472$		0			0
$473$		−	9.79796i		−	0.450511i
$474$		0			0
$475$	2.82843	+	13.8564i	0.129777	+	0.635776i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−38.1838			−1.74466			−0.872330	−	0.488917i	$-0.837392\pi$
$479$	−38.1838			−1.74466			−0.872330	+	0.488917i	$0.837392\pi$
$480$		0			0
$481$	−12.7279			−0.580343
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−9.00000	−	7.34847i	−0.408669	−	0.333677i
$486$		0			0
$487$		−	17.1464i		−	0.776979i	−0.921453	−	0.388489i	$-0.872997\pi$
$487$		−	17.1464i		−	0.776979i	0.921453	−	0.388489i	$-0.127003\pi$
$488$		0			0
$489$	−38.1838			−1.72673
$490$		0			0
$491$	29.0000			1.30875			0.654376	−	0.756169i	$-0.272931\pi$
$491$	29.0000			1.30875			0.654376	+	0.756169i	$0.272931\pi$
$492$		0			0
$493$		−	36.3731i		−	1.63816i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−37.0000			−1.65635			−0.828174	−	0.560471i	$-0.810620\pi$
$499$	−37.0000			−1.65635			−0.828174	+	0.560471i	$0.810620\pi$
$500$		0			0
$501$	−9.00000			−0.402090
$502$		0			0
$503$		−	32.9090i		−	1.46734i	−0.679507	−	0.733669i	$-0.737806\pi$
$503$		−	32.9090i		−	1.46734i	0.679507	−	0.733669i	$-0.262194\pi$
$504$		0			0
$505$	8.00000	−	9.79796i	0.355995	−	0.436003i
$506$		0			0
$507$		−	17.3205i		−	0.769231i
$508$		0			0
$509$	4.24264			0.188052			0.0940259	−	0.995570i	$-0.470026\pi$
$509$	4.24264			0.188052			0.0940259	+	0.995570i	$0.470026\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$			14.6969i			0.648886i
$514$		0			0
$515$	−3.00000	−	2.44949i	−0.132196	−	0.107937i
$516$		0			0
$517$			12.1244i			0.533229i
$518$		0			0
$519$	3.00000			0.131685
$520$		0			0
$521$	8.48528			0.371747			0.185873	−	0.982574i	$-0.440489\pi$
$521$	8.48528			0.371747			0.185873	+	0.982574i	$0.440489\pi$
$522$		0			0
$523$			34.6410i			1.51475i	0.652983	+	0.757373i	$0.273517\pi$
$523$			34.6410i			1.51475i	−0.652983	+	0.757373i	$0.726483\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			36.7423i			1.60052i
$528$		0			0
$529$	17.0000			0.739130
$530$		0			0
$531$		0			0
$532$		0			0
$533$			12.2474i			0.530496i
$534$		0			0
$535$	8.48528	+	6.92820i	0.366851	+	0.299532i
$536$		0			0
$537$		−	17.3205i		−	0.747435i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−9.00000			−0.386940			−0.193470	−	0.981106i	$-0.561974\pi$
$541$	−9.00000			−0.386940			−0.193470	+	0.981106i	$0.561974\pi$
$542$		0			0
$543$		−	12.2474i		−	0.525588i
$544$		0			0
$545$	−4.24264	+	5.19615i	−0.181735	+	0.222579i
$546$		0			0
$547$			24.4949i			1.04733i	0.851925	+	0.523663i	$0.175435\pi$
$547$			24.4949i			1.04733i	−0.851925	+	0.523663i	$0.824565\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−19.7990			−0.843465
$552$		0			0
$553$		0			0
$554$		0			0
$555$	18.0000	−	22.0454i	0.764057	−	0.935775i
$556$		0			0
$557$			24.4949i			1.03788i	0.854810	+	0.518941i	$0.173674\pi$
$557$			24.4949i			1.03788i	−0.854810	+	0.518941i	$0.826326\pi$
$558$		0			0
$559$	16.9706			0.717778
$560$		0			0
$561$	9.00000			0.379980
$562$		0			0
$563$		−	17.3205i		−	0.729972i	−0.931013	−	0.364986i	$-0.881074\pi$
$563$		−	17.3205i		−	0.729972i	0.931013	−	0.364986i	$-0.118926\pi$
$564$		0			0
$565$	4.24264	+	3.46410i	0.178489	+	0.145736i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	20.0000			0.838444			0.419222	−	0.907884i	$-0.362303\pi$
$569$	20.0000			0.838444			0.419222	+	0.907884i	$0.362303\pi$
$570$		0			0
$571$	10.0000			0.418487			0.209243	−	0.977864i	$-0.432900\pi$
$571$	10.0000			0.418487			0.209243	+	0.977864i	$0.432900\pi$
$572$		0			0
$573$			1.73205i			0.0723575i
$574$		0			0
$575$	−12.0000	+	2.44949i	−0.500435	+	0.102151i
$576$		0			0
$577$		−	46.7654i		−	1.94687i	−0.228968	−	0.973434i	$-0.573535\pi$
$577$		−	46.7654i		−	1.94687i	0.228968	−	0.973434i	$-0.426465\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$			12.2474i			0.507237i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	6.92820i		−	0.285958i	−0.989726	−	0.142979i	$-0.954332\pi$
$587$		−	6.92820i		−	0.285958i	0.989726	−	0.142979i	$-0.0456681\pi$
$588$		0			0
$589$	20.0000			0.824086
$590$		0			0
$591$	−21.2132			−0.872595
$592$		0			0
$593$		−	1.73205i		−	0.0711268i	−0.999367	−	0.0355634i	$-0.988677\pi$
$593$		−	1.73205i		−	0.0711268i	0.999367	−	0.0355634i	$-0.0113226\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	12.2474i		−	0.501255i
$598$		0			0
$599$	−43.0000			−1.75693			−0.878466	−	0.477805i	$-0.841433\pi$
$599$	−43.0000			−1.75693			−0.878466	+	0.477805i	$0.841433\pi$
$600$		0			0
$601$	19.7990			0.807618			0.403809	−	0.914843i	$-0.367686\pi$
$601$	19.7990			0.807618			0.403809	+	0.914843i	$0.367686\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	14.1421	−	17.3205i	0.574960	−	0.704179i
$606$		0			0
$607$		−	5.19615i		−	0.210905i	−0.994424	−	0.105453i	$-0.966371\pi$
$607$		−	5.19615i		−	0.210905i	0.994424	−	0.105453i	$-0.0336291\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−21.0000			−0.849569
$612$		0			0
$613$		−	9.79796i		−	0.395736i	−0.980229	−	0.197868i	$-0.936598\pi$
$613$		−	9.79796i		−	0.395736i	0.980229	−	0.197868i	$-0.0634017\pi$
$614$		0			0
$615$	−21.2132	−	17.3205i	−0.855399	−	0.698430i
$616$		0			0
$617$		−	7.34847i		−	0.295838i	−0.988999	−	0.147919i	$-0.952742\pi$
$617$		−	7.34847i		−	0.295838i	0.988999	−	0.147919i	$-0.0472575\pi$
$618$		0			0
$619$	9.89949			0.397894			0.198947	−	0.980010i	$-0.436248\pi$
$619$	9.89949			0.397894			0.198947	+	0.980010i	$0.436248\pi$
$620$		0			0
$621$	−12.7279			−0.510754
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−23.0000	+	9.79796i	−0.920000	+	0.391918i
$626$		0			0
$627$		−	4.89898i		−	0.195646i
$628$		0			0
$629$	−38.1838			−1.52249
$630$		0			0
$631$	9.00000			0.358284			0.179142	−	0.983823i	$-0.442668\pi$
$631$	9.00000			0.358284			0.179142	+	0.983823i	$0.442668\pi$
$632$		0			0
$633$			36.3731i			1.44570i
$634$		0			0
$635$	21.2132	+	17.3205i	0.841820	+	0.687343i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	8.00000			0.315981			0.157991	−	0.987441i	$-0.449498\pi$
$641$	8.00000			0.315981			0.157991	+	0.987441i	$0.449498\pi$
$642$		0			0
$643$			36.3731i			1.43441i	0.696860	+	0.717207i	$0.254580\pi$
$643$			36.3731i			1.43441i	−0.696860	+	0.717207i	$0.745420\pi$
$644$		0			0
$645$	−24.0000	+	29.3939i	−0.944999	+	1.15738i
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$	7.07107			0.277564
$650$		0			0
$651$		0			0
$652$		0			0
$653$			9.79796i			0.383424i	0.981451	+	0.191712i	$0.0614039\pi$
$653$			9.79796i			0.383424i	−0.981451	+	0.191712i	$0.938596\pi$
$654$		0			0
$655$	−18.0000	+	22.0454i	−0.703318	+	0.861385i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−17.0000			−0.662226			−0.331113	−	0.943591i	$-0.607424\pi$
$659$	−17.0000			−0.662226			−0.331113	+	0.943591i	$0.607424\pi$
$660$		0			0
$661$	−14.1421			−0.550065			−0.275033	−	0.961435i	$-0.588689\pi$
$661$	−14.1421			−0.550065			−0.275033	+	0.961435i	$0.588689\pi$
$662$		0			0
$663$			15.5885i			0.605406i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	17.1464i		−	0.663912i
$668$		0			0
$669$	33.0000			1.27585
$670$		0			0
$671$	−14.1421			−0.545951
$672$		0			0
$673$			24.4949i			0.944209i	0.881543	+	0.472104i	$0.156505\pi$
$673$			24.4949i			0.944209i	−0.881543	+	0.472104i	$0.843495\pi$
$674$		0			0
$675$	−25.4558	+	5.19615i	−0.979796	+	0.200000i
$676$		0			0
$677$		−	46.7654i		−	1.79734i	−0.438626	−	0.898670i	$-0.644535\pi$
$677$		−	46.7654i		−	1.79734i	0.438626	−	0.898670i	$-0.355465\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	21.0000			0.804722
$682$		0			0
$683$		−	24.4949i		−	0.937271i	−0.883392	−	0.468636i	$-0.844746\pi$
$683$		−	24.4949i		−	0.937271i	0.883392	−	0.468636i	$-0.155254\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		−	12.2474i		−	0.467269i
$688$		0			0
$689$	−21.2132			−0.808159
$690$		0			0
$691$	14.1421			0.537992			0.268996	−	0.963141i	$-0.413308\pi$
$691$	14.1421			0.537992			0.268996	+	0.963141i	$0.413308\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−26.0000	+	31.8434i	−0.986236	+	1.20789i
$696$		0			0
$697$			36.7423i			1.39172i
$698$		0			0
$699$	42.4264			1.60471
$700$		0			0
$701$	−11.0000			−0.415464			−0.207732	−	0.978186i	$-0.566608\pi$
$701$	−11.0000			−0.415464			−0.207732	+	0.978186i	$0.566608\pi$
$702$		0			0
$703$			20.7846i			0.783906i
$704$		0			0
$705$	29.6985	−	36.3731i	1.11851	−	1.36989i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−37.0000			−1.38956			−0.694782	−	0.719220i	$-0.744499\pi$
$709$	−37.0000			−1.38956			−0.694782	+	0.719220i	$0.744499\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			17.3205i			0.648658i
$714$		0			0
$715$	−3.00000	−	2.44949i	−0.112194	−	0.0916057i
$716$		0			0
$717$			22.5167i			0.840900i
$718$		0			0
$719$	24.0416			0.896602			0.448301	−	0.893883i	$-0.352029\pi$
$719$	24.0416			0.896602			0.448301	+	0.893883i	$0.352029\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$			2.44949i			0.0910975i
$724$		0			0
$725$	−7.00000	−	34.2929i	−0.259973	−	1.27360i
$726$		0			0
$727$		−	17.3205i		−	0.642382i	−0.947014	−	0.321191i	$-0.895917\pi$
$727$		−	17.3205i		−	0.642382i	0.947014	−	0.321191i	$-0.104083\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$	50.9117			1.88304
$732$		0			0
$733$		−	19.0526i		−	0.703722i	−0.936052	−	0.351861i	$-0.885549\pi$
$733$		−	19.0526i		−	0.703722i	0.936052	−	0.351861i	$-0.114451\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	12.2474i		−	0.451141i
$738$		0			0
$739$	−23.0000			−0.846069			−0.423034	−	0.906114i	$-0.639035\pi$
$739$	−23.0000			−0.846069			−0.423034	+	0.906114i	$0.639035\pi$
$740$		0			0
$741$	8.48528			0.311715
$742$		0			0
$743$			12.2474i			0.449315i	0.974438	+	0.224658i	$0.0721264\pi$
$743$			12.2474i			0.449315i	−0.974438	+	0.224658i	$0.927874\pi$
$744$		0			0
$745$	−28.2843	+	34.6410i	−1.03626	+	1.26915i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	51.0000			1.86102			0.930508	−	0.366271i	$-0.119366\pi$
$751$	51.0000			1.86102			0.930508	+	0.366271i	$0.119366\pi$
$752$		0			0
$753$			26.9444i			0.981908i
$754$		0			0
$755$	−15.5563	+	19.0526i	−0.566154	+	0.693394i
$756$		0			0
$757$		−	24.4949i		−	0.890282i	−0.895460	−	0.445141i	$-0.853154\pi$
$757$		−	24.4949i		−	0.890282i	0.895460	−	0.445141i	$-0.146846\pi$
$758$		0			0
$759$	4.24264			0.153998
$760$		0			0
$761$	−29.6985			−1.07657			−0.538285	−	0.842763i	$-0.680927\pi$
$761$	−29.6985			−1.07657			−0.538285	+	0.842763i	$0.680927\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$			12.2474i			0.442230i
$768$		0			0
$769$	−39.5980			−1.42794			−0.713970	−	0.700176i	$-0.753105\pi$
$769$	−39.5980			−1.42794			−0.713970	+	0.700176i	$0.753105\pi$
$770$		0			0
$771$	30.0000			1.08042
$772$		0			0
$773$		−	15.5885i		−	0.560678i	−0.959901	−	0.280339i	$-0.909553\pi$
$773$		−	15.5885i		−	0.560678i	0.959901	−	0.280339i	$-0.0904469\pi$
$774$		0			0
$775$	7.07107	+	34.6410i	0.254000	+	1.24434i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	20.0000			0.716574
$780$		0			0
$781$	10.0000			0.357828
$782$		0			0
$783$		−	36.3731i		−	1.29987i
$784$		0			0
$785$	−30.0000	−	24.4949i	−1.07075	−	0.874260i
$786$		0			0
$787$			22.5167i			0.802632i	0.915940	+	0.401316i	$0.131447\pi$
$787$			22.5167i			0.802632i	−0.915940	+	0.401316i	$0.868553\pi$
$788$		0			0
$789$	16.9706			0.604168
$790$		0			0
$791$		0			0
$792$		0			0
$793$		−	24.4949i		−	0.869839i
$794$		0			0
$795$	30.0000	−	36.7423i	1.06399	−	1.30312i
$796$		0			0
$797$			39.8372i			1.41110i	0.708658	+	0.705552i	$0.249301\pi$
$797$			39.8372i			1.41110i	−0.708658	+	0.705552i	$0.750699\pi$
$798$		0			0
$799$	−63.0000			−2.22878
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	7.34847i		−	0.258678i
$808$		0			0
$809$	53.0000			1.86338			0.931690	−	0.363253i	$-0.118334\pi$
$809$	53.0000			1.86338			0.931690	+	0.363253i	$0.118334\pi$
$810$		0			0
$811$	43.8406			1.53945			0.769726	−	0.638374i	$-0.220393\pi$
$811$	43.8406			1.53945			0.769726	+	0.638374i	$0.220393\pi$
$812$		0			0
$813$		−	48.9898i		−	1.71815i
$814$		0			0
$815$	38.1838	+	31.1769i	1.33752	+	1.09208i
$816$		0			0
$817$		−	27.7128i		−	0.969549i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.00000			0.0349002			0.0174501	−	0.999848i	$-0.494445\pi$
$821$	1.00000			0.0349002			0.0174501	+	0.999848i	$0.494445\pi$
$822$		0			0
$823$		−	12.2474i		−	0.426919i	−0.976952	−	0.213460i	$-0.931527\pi$
$823$		−	12.2474i		−	0.426919i	0.976952	−	0.213460i	$-0.0684732\pi$
$824$		0			0
$825$	8.48528	−	1.73205i	0.295420	−	0.0603023i
$826$		0			0
$827$		−	17.1464i		−	0.596240i	−0.954528	−	0.298120i	$-0.903640\pi$
$827$		−	17.1464i		−	0.596240i	0.954528	−	0.298120i	$-0.0963595\pi$
$828$		0			0
$829$	49.4975			1.71912			0.859559	−	0.511036i	$-0.170738\pi$
$829$	49.4975			1.71912			0.859559	+	0.511036i	$0.170738\pi$
$830$		0			0
$831$	−12.7279			−0.441527
$832$		0			0
$833$		0			0
$834$		0			0
$835$	9.00000	+	7.34847i	0.311458	+	0.254304i
$836$		0			0
$837$			36.7423i			1.27000i
$838$		0			0
$839$	18.3848			0.634713			0.317356	−	0.948306i	$-0.397205\pi$
$839$	18.3848			0.634713			0.317356	+	0.948306i	$0.397205\pi$
$840$		0			0
$841$	20.0000			0.689655
$842$		0			0
$843$			50.2295i			1.72999i
$844$		0			0
$845$	−14.1421	+	17.3205i	−0.486504	+	0.595844i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−3.00000			−0.102960
$850$		0			0
$851$	−18.0000			−0.617032
$852$		0			0
$853$			3.46410i			0.118609i	0.998240	+	0.0593043i	$0.0188882\pi$
$853$			3.46410i			0.118609i	−0.998240	+	0.0593043i	$0.981112\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	17.3205i		−	0.591657i	−0.955241	−	0.295829i	$-0.904404\pi$
$857$		−	17.3205i		−	0.591657i	0.955241	−	0.295829i	$-0.0955957\pi$
$858$		0			0
$859$	14.1421			0.482523			0.241262	−	0.970460i	$-0.422439\pi$
$859$	14.1421			0.482523			0.241262	+	0.970460i	$0.422439\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			36.7423i			1.25072i	0.780335	+	0.625362i	$0.215049\pi$
$863$			36.7423i			1.25072i	−0.780335	+	0.625362i	$0.784951\pi$
$864$		0			0
$865$	−3.00000	−	2.44949i	−0.102003	−	0.0832851i
$866$		0			0
$867$			17.3205i			0.588235i
$868$		0			0
$869$	3.00000			0.101768
$870$		0			0
$871$	21.2132			0.718782
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$			19.5959i			0.661707i	0.943682	+	0.330854i	$0.107337\pi$
$877$			19.5959i			0.661707i	−0.943682	+	0.330854i	$0.892663\pi$
$878$		0			0
$879$	−33.0000			−1.11306
$880$		0			0
$881$	28.2843			0.952921			0.476461	−	0.879196i	$-0.341919\pi$
$881$	28.2843			0.952921			0.476461	+	0.879196i	$0.341919\pi$
$882$		0			0
$883$		−	24.4949i		−	0.824319i	−0.911112	−	0.412159i	$-0.864775\pi$
$883$		−	24.4949i		−	0.824319i	0.911112	−	0.412159i	$-0.135225\pi$
$884$		0			0
$885$	−21.2132	−	17.3205i	−0.713074	−	0.582223i
$886$		0			0
$887$			17.3205i			0.581566i	0.956789	+	0.290783i	$0.0939157\pi$
$887$			17.3205i			0.581566i	−0.956789	+	0.290783i	$0.906084\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	9.00000			0.301511
$892$		0			0
$893$			34.2929i			1.14757i
$894$		0			0
$895$	−14.1421	+	17.3205i	−0.472719	+	0.578961i
$896$		0			0
$897$			7.34847i			0.245358i
$898$		0			0
$899$	−49.4975			−1.65083
$900$		0			0
$901$	−63.6396			−2.12014
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−10.0000	+	12.2474i	−0.332411	+	0.407119i
$906$		0			0
$907$			41.6413i			1.38268i	0.722531	+	0.691339i	$0.242979\pi$
$907$			41.6413i			1.38268i	−0.722531	+	0.691339i	$0.757021\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	20.0000			0.662630			0.331315	−	0.943520i	$-0.392508\pi$
$911$	20.0000			0.662630			0.331315	+	0.943520i	$0.392508\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	42.4264	+	34.6410i	1.40257	+	1.14520i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	3.00000			0.0989609			0.0494804	−	0.998775i	$-0.484243\pi$
$919$	3.00000			0.0989609			0.0494804	+	0.998775i	$0.484243\pi$
$920$		0			0
$921$	51.0000			1.68051
$922$		0			0
$923$			17.3205i			0.570111i
$924$		0			0
$925$	−36.0000	+	7.34847i	−1.18367	+	0.241616i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	21.2132			0.695983			0.347991	−	0.937498i	$-0.386864\pi$
$929$	21.2132			0.695983			0.347991	+	0.937498i	$0.386864\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$			14.6969i			0.481156i
$934$		0			0
$935$	−9.00000	−	7.34847i	−0.294331	−	0.240321i
$936$		0			0
$937$		−	39.8372i		−	1.30142i	−0.759325	−	0.650712i	$-0.774471\pi$
$937$		−	39.8372i		−	1.30142i	0.759325	−	0.650712i	$-0.225529\pi$
$938$		0			0
$939$	−3.00000			−0.0979013
$940$		0			0
$941$	56.5685			1.84408			0.922041	−	0.387092i	$-0.126521\pi$
$941$	56.5685			1.84408			0.922041	+	0.387092i	$0.126521\pi$
$942$		0			0
$943$			17.3205i			0.564033i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	36.7423i		−	1.19397i	−0.802254	−	0.596983i	$-0.796366\pi$
$947$		−	36.7423i		−	1.19397i	0.802254	−	0.596983i	$-0.203634\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	33.9411			1.10062
$952$		0			0
$953$		−	36.7423i		−	1.19020i	−0.803651	−	0.595101i	$-0.797112\pi$
$953$		−	36.7423i		−	1.19020i	0.803651	−	0.595101i	$-0.202888\pi$
$954$		0			0
$955$	1.41421	−	1.73205i	0.0457629	−	0.0560478i
$956$		0			0
$957$			12.1244i			0.391925i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	19.0000			0.612903
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	44.0908i		−	1.41787i	−0.705276	−	0.708933i	$-0.749177\pi$
$967$		−	44.0908i		−	1.41787i	0.705276	−	0.708933i	$-0.250823\pi$
$968$		0			0
$969$	25.4558			0.817760
$970$		0			0
$971$	−49.4975			−1.58845			−0.794225	−	0.607624i	$-0.792123\pi$
$971$	−49.4975			−1.58845			−0.794225	+	0.607624i	$0.792123\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	3.00000	+	14.6969i	0.0960769	+	0.470679i
$976$		0			0
$977$		−	4.89898i		−	0.156732i	−0.996925	−	0.0783661i	$-0.975030\pi$
$977$		−	4.89898i		−	0.156732i	0.996925	−	0.0783661i	$-0.0249703\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	32.9090i		−	1.04963i	−0.851215	−	0.524816i	$-0.824134\pi$
$983$		−	32.9090i		−	1.04963i	0.851215	−	0.524816i	$-0.175866\pi$
$984$		0			0
$985$	21.2132	+	17.3205i	0.675909	+	0.551877i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	24.0000			0.763156
$990$		0			0
$991$	30.0000			0.952981			0.476491	−	0.879180i	$-0.341909\pi$
$991$	30.0000			0.952981			0.476491	+	0.879180i	$0.341909\pi$
$992$		0			0
$993$			51.9615i			1.64895i
$994$		0			0
$995$	−10.0000	+	12.2474i	−0.317021	+	0.388270i
$996$		0			0
$997$		−	5.19615i		−	0.164564i	−0.996609	−	0.0822819i	$-0.973779\pi$
$997$		−	5.19615i		−	0.164564i	0.996609	−	0.0822819i	$-0.0262208\pi$
$998$		0			0
$999$	−38.1838			−1.20808

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	980.2.e.e.589.1	✓	4
5.2	odd	4		4900.2.a.bh.1.2		4
5.3	odd	4		4900.2.a.bh.1.4		4
5.4	even	2	inner	980.2.e.e.589.3	yes	4
7.2	even	3		980.2.q.h.949.2		4
7.3	odd	6		980.2.q.h.569.1		4
7.4	even	3		980.2.q.a.569.2		4
7.5	odd	6		980.2.q.a.949.1		4
7.6	odd	2	inner	980.2.e.e.589.4	yes	4
35.4	even	6		980.2.q.h.569.2		4
35.9	even	6		980.2.q.a.949.2		4
35.13	even	4		4900.2.a.bh.1.1		4
35.19	odd	6		980.2.q.h.949.1		4
35.24	odd	6		980.2.q.a.569.1		4
35.27	even	4		4900.2.a.bh.1.3		4
35.34	odd	2	inner	980.2.e.e.589.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
980.2.e.e.589.1	✓	4	1.1	even	1	trivial
980.2.e.e.589.2	yes	4	35.34	odd	2	inner
980.2.e.e.589.3	yes	4	5.4	even	2	inner
980.2.e.e.589.4	yes	4	7.6	odd	2	inner
980.2.q.a.569.1		4	35.24	odd	6
980.2.q.a.569.2		4	7.4	even	3
980.2.q.a.949.1		4	7.5	odd	6
980.2.q.a.949.2		4	35.9	even	6
980.2.q.h.569.1		4	7.3	odd	6
980.2.q.h.569.2		4	35.4	even	6
980.2.q.h.949.1		4	35.19	odd	6
980.2.q.h.949.2		4	7.2	even	3
4900.2.a.bh.1.1		4	35.13	even	4
4900.2.a.bh.1.2		4	5.2	odd	4
4900.2.a.bh.1.3		4	35.27	even	4
4900.2.a.bh.1.4		4	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	980.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} +(-1.41421 + 1.73205i) q^{5} +O(q^{10})\)	\(q-1.73205i q^{3} +(-1.41421 + 1.73205i) q^{5} -1.00000 q^{11} -1.73205i q^{13} +(3.00000 + 2.44949i) q^{15} -5.19615i q^{17} -2.82843 q^{19} -2.44949i q^{23} +(-1.00000 - 4.89898i) q^{25} -5.19615i q^{27} +7.00000 q^{29} -7.07107 q^{31} +1.73205i q^{33} -7.34847i q^{37} -3.00000 q^{39} -7.07107 q^{41} +9.79796i q^{43} -12.1244i q^{47} -9.00000 q^{51} -12.2474i q^{53} +(1.41421 - 1.73205i) q^{55} +4.89898i q^{57} -7.07107 q^{59} +14.1421 q^{61} +(3.00000 + 2.44949i) q^{65} +12.2474i q^{67} -4.24264 q^{69} -10.0000 q^{71} +(-8.48528 + 1.73205i) q^{75} -3.00000 q^{79} -9.00000 q^{81} +(9.00000 + 7.34847i) q^{85} -12.1244i q^{87} +12.2474i q^{93} +(4.00000 - 4.89898i) q^{95} +5.19615i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 4 q^{11} + 12 q^{15} - 4 q^{25} + 28 q^{29} - 12 q^{39} - 36 q^{51} + 12 q^{65} - 40 q^{71} - 12 q^{79} - 36 q^{81} + 36 q^{85} + 16 q^{95}+O(q^{100}) \) 4 * q - 4 * q^11 + 12 * q^15 - 4 * q^25 + 28 * q^29 - 12 * q^39 - 36 * q^51 + 12 * q^65 - 40 * q^71 - 12 * q^79 - 36 * q^81 + 36 * q^85 + 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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