LMFDB - Embedded newform 2160.2.f.k.1729.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,2,Mod(1729,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2160.1729");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2160.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$17.2476868366$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 135)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1729.2
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2160.1729
Dual form			2160.2.f.k.1729.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.12132 + 0.707107i) q^{5} -3.00000i q^{7} +O(q^{10})$	$q+(-2.12132 + 0.707107i) q^{5} -3.00000i q^{7} -4.24264 q^{11} -3.00000i q^{13} -2.82843i q^{17} +1.00000 q^{19} +7.07107i q^{23} +(4.00000 - 3.00000i) q^{25} +4.24264 q^{29} -2.00000 q^{31} +(2.12132 + 6.36396i) q^{35} +9.00000i q^{37} -4.24264 q^{41} +6.00000i q^{43} +2.82843i q^{47} -2.00000 q^{49} +9.89949i q^{53} +(9.00000 - 3.00000i) q^{55} -8.48528 q^{59} -13.0000 q^{61} +(2.12132 + 6.36396i) q^{65} +3.00000i q^{67} +12.7279 q^{71} -9.00000i q^{73} +12.7279i q^{77} -5.00000 q^{79} -1.41421i q^{83} +(2.00000 + 6.00000i) q^{85} -9.00000 q^{91} +(-2.12132 + 0.707107i) q^{95} +3.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 4 q^{19} + 16 q^{25} - 8 q^{31} - 8 q^{49} + 36 q^{55} - 52 q^{61} - 20 q^{79} + 8 q^{85} - 36 q^{91}+O(q^{100}) $ 4 * q + 4 * q^19 + 16 * q^25 - 8 * q^31 - 8 * q^49 + 36 * q^55 - 52 * q^61 - 20 * q^79 + 8 * q^85 - 36 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$.

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$5$	−2.12132	+	0.707107i	−0.948683	+	0.316228i
$5$	−2.12132	+	0.707107i	−0.948683	+	0.316228i
$6$		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
$11$	−4.24264			−1.27920			−0.639602	−	0.768706i	$-0.720901\pi$
$11$	−4.24264			−1.27920			−0.639602	+	0.768706i	$0.720901\pi$
$12$		0			0
$13$		−	3.00000i		−	0.832050i	−0.909353	−	0.416025i	$-0.863423\pi$
$13$		−	3.00000i		−	0.832050i	0.909353	−	0.416025i	$-0.136577\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	2.82843i		−	0.685994i	−0.939336	−	0.342997i	$-0.888558\pi$
$17$		−	2.82843i		−	0.685994i	0.939336	−	0.342997i	$-0.111442\pi$
$18$		0			0
$19$	1.00000			0.229416			0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000			0.229416			0.114708	+	0.993399i	$0.463407\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			7.07107i			1.47442i	0.675664	+	0.737210i	$0.263857\pi$
$23$			7.07107i			1.47442i	−0.675664	+	0.737210i	$0.736143\pi$
$24$		0			0
$25$	4.00000	−	3.00000i	0.800000	−	0.600000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.24264			0.787839			0.393919	−	0.919145i	$-0.371119\pi$
$29$	4.24264			0.787839			0.393919	+	0.919145i	$0.371119\pi$
$30$		0			0
$31$	−2.00000			−0.359211			−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000			−0.359211			−0.179605	+	0.983739i	$0.557482\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.12132	+	6.36396i	0.358569	+	1.07571i
$36$		0			0
$37$			9.00000i			1.47959i	0.672832	+	0.739795i	$0.265078\pi$
$37$			9.00000i			1.47959i	−0.672832	+	0.739795i	$0.734922\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.24264			−0.662589			−0.331295	−	0.943527i	$-0.607485\pi$
$41$	−4.24264			−0.662589			−0.331295	+	0.943527i	$0.607485\pi$
$42$		0			0
$43$			6.00000i			0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$			6.00000i			0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			2.82843i			0.412568i	0.978492	+	0.206284i	$0.0661372\pi$
$47$			2.82843i			0.412568i	−0.978492	+	0.206284i	$0.933863\pi$
$48$		0			0
$49$	−2.00000			−0.285714
$50$		0			0
$51$		0			0
$52$		0			0
$53$			9.89949i			1.35980i	0.733305	+	0.679900i	$0.237977\pi$
$53$			9.89949i			1.35980i	−0.733305	+	0.679900i	$0.762023\pi$
$54$		0			0
$55$	9.00000	−	3.00000i	1.21356	−	0.404520i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−8.48528			−1.10469			−0.552345	−	0.833616i	$-0.686267\pi$
$59$	−8.48528			−1.10469			−0.552345	+	0.833616i	$0.686267\pi$
$60$		0			0
$61$	−13.0000			−1.66448			−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000			−1.66448			−0.832240	+	0.554416i	$0.812942\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.12132	+	6.36396i	0.263117	+	0.789352i
$66$		0			0
$67$			3.00000i			0.366508i	0.983066	+	0.183254i	$0.0586631\pi$
$67$			3.00000i			0.366508i	−0.983066	+	0.183254i	$0.941337\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	12.7279			1.51053			0.755263	−	0.655422i	$-0.227509\pi$
$71$	12.7279			1.51053			0.755263	+	0.655422i	$0.227509\pi$
$72$		0			0
$73$		−	9.00000i		−	1.05337i	−0.850060	−	0.526685i	$-0.823435\pi$
$73$		−	9.00000i		−	1.05337i	0.850060	−	0.526685i	$-0.176565\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			12.7279i			1.45048i
$78$		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
$83$		−	1.41421i		−	0.155230i	−0.996983	−	0.0776151i	$-0.975269\pi$
$83$		−	1.41421i		−	0.155230i	0.996983	−	0.0776151i	$-0.0247305\pi$
$84$		0			0
$85$	2.00000	+	6.00000i	0.216930	+	0.650791i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	−9.00000			−0.943456
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.12132	+	0.707107i	−0.217643	+	0.0725476i
$96$		0			0
$97$			3.00000i			0.304604i	0.988334	+	0.152302i	$0.0486686\pi$
$97$			3.00000i			0.304604i	−0.988334	+	0.152302i	$0.951331\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−8.48528			−0.844317			−0.422159	−	0.906522i	$-0.638727\pi$
$101$	−8.48528			−0.844317			−0.422159	+	0.906522i	$0.638727\pi$
$102$		0			0
$103$			3.00000i			0.295599i	0.989017	+	0.147799i	$0.0472190\pi$
$103$			3.00000i			0.295599i	−0.989017	+	0.147799i	$0.952781\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			7.07107i			0.683586i	0.939775	+	0.341793i	$0.111034\pi$
$107$			7.07107i			0.683586i	−0.939775	+	0.341793i	$0.888966\pi$
$108$		0			0
$109$	8.00000			0.766261			0.383131	−	0.923694i	$-0.374846\pi$
$109$	8.00000			0.766261			0.383131	+	0.923694i	$0.374846\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			1.41421i			0.133038i	0.997785	+	0.0665190i	$0.0211893\pi$
$113$			1.41421i			0.133038i	−0.997785	+	0.0665190i	$0.978811\pi$
$114$		0			0
$115$	−5.00000	−	15.0000i	−0.466252	−	1.39876i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−8.48528			−0.777844
$120$		0			0
$121$	7.00000			0.636364
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−6.36396	+	9.19239i	−0.569210	+	0.822192i
$126$		0			0
$127$			18.0000i			1.59724i	0.601834	+	0.798621i	$0.294437\pi$
$127$			18.0000i			1.59724i	−0.601834	+	0.798621i	$0.705563\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−8.48528			−0.741362			−0.370681	−	0.928760i	$-0.620876\pi$
$131$	−8.48528			−0.741362			−0.370681	+	0.928760i	$0.620876\pi$
$132$		0			0
$133$		−	3.00000i		−	0.260133i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			1.41421i			0.120824i	0.998174	+	0.0604122i	$0.0192415\pi$
$137$			1.41421i			0.120824i	−0.998174	+	0.0604122i	$0.980758\pi$
$138$		0			0
$139$	13.0000			1.10265			0.551323	−	0.834292i	$-0.314123\pi$
$139$	13.0000			1.10265			0.551323	+	0.834292i	$0.314123\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			12.7279i			1.06436i
$144$		0			0
$145$	−9.00000	+	3.00000i	−0.747409	+	0.249136i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−16.9706			−1.39028			−0.695141	−	0.718873i	$-0.744658\pi$
$149$	−16.9706			−1.39028			−0.695141	+	0.718873i	$0.744658\pi$
$150$		0			0
$151$	1.00000			0.0813788			0.0406894	−	0.999172i	$-0.487045\pi$
$151$	1.00000			0.0813788			0.0406894	+	0.999172i	$0.487045\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	4.24264	−	1.41421i	0.340777	−	0.113592i
$156$		0			0
$157$			6.00000i			0.478852i	0.970915	+	0.239426i	$0.0769593\pi$
$157$			6.00000i			0.478852i	−0.970915	+	0.239426i	$0.923041\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	21.2132			1.67183
$162$		0			0
$163$		−	9.00000i		−	0.704934i	−0.935824	−	0.352467i	$-0.885343\pi$
$163$		−	9.00000i		−	0.704934i	0.935824	−	0.352467i	$-0.114657\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	5.65685i		−	0.437741i	−0.975754	−	0.218870i	$-0.929763\pi$
$167$		−	5.65685i		−	0.437741i	0.975754	−	0.218870i	$-0.0702371\pi$
$168$		0			0
$169$	4.00000			0.307692
$170$		0			0
$171$		0			0
$172$		0			0
$173$			14.1421i			1.07521i	0.843198	+	0.537603i	$0.180670\pi$
$173$			14.1421i			1.07521i	−0.843198	+	0.537603i	$0.819330\pi$
$174$		0			0
$175$	−9.00000	−	12.0000i	−0.680336	−	0.907115i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	12.7279			0.951330			0.475665	−	0.879627i	$-0.342208\pi$
$179$	12.7279			0.951330			0.475665	+	0.879627i	$0.342208\pi$
$180$		0			0
$181$	−1.00000			−0.0743294			−0.0371647	−	0.999309i	$-0.511833\pi$
$181$	−1.00000			−0.0743294			−0.0371647	+	0.999309i	$0.511833\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−6.36396	−	19.0919i	−0.467888	−	1.40366i
$186$		0			0
$187$			12.0000i			0.877527i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−16.9706			−1.22795			−0.613973	−	0.789327i	$-0.710430\pi$
$191$	−16.9706			−1.22795			−0.613973	+	0.789327i	$0.710430\pi$
$192$		0			0
$193$		−	3.00000i		−	0.215945i	−0.994154	−	0.107972i	$-0.965564\pi$
$193$		−	3.00000i		−	0.215945i	0.994154	−	0.107972i	$-0.0344358\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			22.6274i			1.61214i	0.591822	+	0.806068i	$0.298409\pi$
$197$			22.6274i			1.61214i	−0.591822	+	0.806068i	$0.701591\pi$
$198$		0			0
$199$	−11.0000			−0.779769			−0.389885	−	0.920864i	$-0.627485\pi$
$199$	−11.0000			−0.779769			−0.389885	+	0.920864i	$0.627485\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	12.7279i		−	0.893325i
$204$		0			0
$205$	9.00000	−	3.00000i	0.628587	−	0.209529i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−4.24264			−0.293470
$210$		0			0
$211$	7.00000			0.481900			0.240950	−	0.970538i	$-0.422541\pi$
$211$	7.00000			0.481900			0.240950	+	0.970538i	$0.422541\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−4.24264	−	12.7279i	−0.289346	−	0.868037i
$216$		0			0
$217$			6.00000i			0.407307i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−8.48528			−0.570782
$222$		0			0
$223$			6.00000i			0.401790i	0.979613	+	0.200895i	$0.0643850\pi$
$223$			6.00000i			0.401790i	−0.979613	+	0.200895i	$0.935615\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			24.0416i			1.59570i	0.602857	+	0.797850i	$0.294029\pi$
$227$			24.0416i			1.59570i	−0.602857	+	0.797850i	$0.705971\pi$
$228$		0			0
$229$	−16.0000			−1.05731			−0.528655	−	0.848837i	$-0.677303\pi$
$229$	−16.0000			−1.05731			−0.528655	+	0.848837i	$0.677303\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	19.7990i		−	1.29707i	−0.761183	−	0.648537i	$-0.775381\pi$
$233$		−	19.7990i		−	1.29707i	0.761183	−	0.648537i	$-0.224619\pi$
$234$		0			0
$235$	−2.00000	−	6.00000i	−0.130466	−	0.391397i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	4.24264			0.274434			0.137217	−	0.990541i	$-0.456184\pi$
$239$	4.24264			0.274434			0.137217	+	0.990541i	$0.456184\pi$
$240$		0			0
$241$	5.00000			0.322078			0.161039	−	0.986948i	$-0.448515\pi$
$241$	5.00000			0.322078			0.161039	+	0.986948i	$0.448515\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	4.24264	−	1.41421i	0.271052	−	0.0903508i
$246$		0			0
$247$		−	3.00000i		−	0.190885i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	12.7279			0.803379			0.401690	−	0.915776i	$-0.368423\pi$
$251$	12.7279			0.803379			0.401690	+	0.915776i	$0.368423\pi$
$252$		0			0
$253$		−	30.0000i		−	1.88608i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	19.7990i		−	1.23503i	−0.786560	−	0.617514i	$-0.788140\pi$
$257$		−	19.7990i		−	1.23503i	0.786560	−	0.617514i	$-0.211860\pi$
$258$		0			0
$259$	27.0000			1.67770
$260$		0			0
$261$		0			0
$262$		0			0
$263$			2.82843i			0.174408i	0.996190	+	0.0872041i	$0.0277932\pi$
$263$			2.82843i			0.174408i	−0.996190	+	0.0872041i	$0.972207\pi$
$264$		0			0
$265$	−7.00000	−	21.0000i	−0.430007	−	1.29002i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	25.4558			1.55207			0.776035	−	0.630690i	$-0.217228\pi$
$269$	25.4558			1.55207			0.776035	+	0.630690i	$0.217228\pi$
$270$		0			0
$271$	19.0000			1.15417			0.577084	−	0.816685i	$-0.304191\pi$
$271$	19.0000			1.15417			0.577084	+	0.816685i	$0.304191\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−16.9706	+	12.7279i	−1.02336	+	0.767523i
$276$		0			0
$277$			12.0000i			0.721010i	0.932757	+	0.360505i	$0.117396\pi$
$277$			12.0000i			0.721010i	−0.932757	+	0.360505i	$0.882604\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−8.48528			−0.506189			−0.253095	−	0.967442i	$-0.581448\pi$
$281$	−8.48528			−0.506189			−0.253095	+	0.967442i	$0.581448\pi$
$282$		0			0
$283$		−	6.00000i		−	0.356663i	−0.983970	−	0.178331i	$-0.942930\pi$
$283$		−	6.00000i		−	0.356663i	0.983970	−	0.178331i	$-0.0570699\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			12.7279i			0.751305i
$288$		0			0
$289$	9.00000			0.529412
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	7.07107i		−	0.413096i	−0.978436	−	0.206548i	$-0.933777\pi$
$293$		−	7.07107i		−	0.413096i	0.978436	−	0.206548i	$-0.0662230\pi$
$294$		0			0
$295$	18.0000	−	6.00000i	1.04800	−	0.349334i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	21.2132			1.22679
$300$		0			0
$301$	18.0000			1.03750
$302$		0			0
$303$		0			0
$304$		0			0
$305$	27.5772	−	9.19239i	1.57906	−	0.526355i
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−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
$313$			21.0000i			1.18699i	0.804838	+	0.593495i	$0.202252\pi$
$313$			21.0000i			1.18699i	−0.804838	+	0.593495i	$0.797748\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	11.3137i		−	0.635441i	−0.948184	−	0.317721i	$-0.897083\pi$
$317$		−	11.3137i		−	0.635441i	0.948184	−	0.317721i	$-0.102917\pi$
$318$		0			0
$319$	−18.0000			−1.00781
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	2.82843i		−	0.157378i
$324$		0			0
$325$	−9.00000	−	12.0000i	−0.499230	−	0.665640i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	8.48528			0.467809
$330$		0			0
$331$	−17.0000			−0.934405			−0.467202	−	0.884150i	$-0.654738\pi$
$331$	−17.0000			−0.934405			−0.467202	+	0.884150i	$0.654738\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−2.12132	−	6.36396i	−0.115900	−	0.347700i
$336$		0			0
$337$			33.0000i			1.79762i	0.438334	+	0.898812i	$0.355569\pi$
$337$			33.0000i			1.79762i	−0.438334	+	0.898812i	$0.644431\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	8.48528			0.459504
$342$		0			0
$343$		−	15.0000i		−	0.809924i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	26.8701i		−	1.44246i	−0.692696	−	0.721230i	$-0.743577\pi$
$347$		−	26.8701i		−	1.44246i	0.692696	−	0.721230i	$-0.256423\pi$
$348$		0			0
$349$	5.00000			0.267644			0.133822	−	0.991005i	$-0.457275\pi$
$349$	5.00000			0.267644			0.133822	+	0.991005i	$0.457275\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	28.2843i		−	1.50542i	−0.658352	−	0.752710i	$-0.728746\pi$
$353$		−	28.2843i		−	1.50542i	0.658352	−	0.752710i	$-0.271254\pi$
$354$		0			0
$355$	−27.0000	+	9.00000i	−1.43301	+	0.477670i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	12.7279			0.671754			0.335877	−	0.941906i	$-0.390967\pi$
$359$	12.7279			0.671754			0.335877	+	0.941906i	$0.390967\pi$
$360$		0			0
$361$	−18.0000			−0.947368
$362$		0			0
$363$		0			0
$364$		0			0
$365$	6.36396	+	19.0919i	0.333105	+	0.999315i
$366$		0			0
$367$			15.0000i			0.782994i	0.920179	+	0.391497i	$0.128043\pi$
$367$			15.0000i			0.782994i	−0.920179	+	0.391497i	$0.871957\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	29.6985			1.54187
$372$		0			0
$373$		−	21.0000i		−	1.08734i	−0.839299	−	0.543669i	$-0.817035\pi$
$373$		−	21.0000i		−	1.08734i	0.839299	−	0.543669i	$-0.182965\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	12.7279i		−	0.655521i
$378$		0			0
$379$	−35.0000			−1.79783			−0.898915	−	0.438124i	$-0.855643\pi$
$379$	−35.0000			−1.79783			−0.898915	+	0.438124i	$0.855643\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	5.65685i		−	0.289052i	−0.989501	−	0.144526i	$-0.953834\pi$
$383$		−	5.65685i		−	0.289052i	0.989501	−	0.144526i	$-0.0461657\pi$
$384$		0			0
$385$	−9.00000	−	27.0000i	−0.458682	−	1.37605i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−33.9411			−1.72088			−0.860442	−	0.509549i	$-0.829812\pi$
$389$	−33.9411			−1.72088			−0.860442	+	0.509549i	$0.829812\pi$
$390$		0			0
$391$	20.0000			1.01144
$392$		0			0
$393$		0			0
$394$		0			0
$395$	10.6066	−	3.53553i	0.533676	−	0.177892i
$396$		0			0
$397$			36.0000i			1.80679i	0.428811	+	0.903394i	$0.358933\pi$
$397$			36.0000i			1.80679i	−0.428811	+	0.903394i	$0.641067\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−4.24264			−0.211867			−0.105934	−	0.994373i	$-0.533783\pi$
$401$	−4.24264			−0.211867			−0.105934	+	0.994373i	$0.533783\pi$
$402$		0			0
$403$			6.00000i			0.298881i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	38.1838i		−	1.89270i
$408$		0			0
$409$	−25.0000			−1.23617			−0.618085	−	0.786111i	$-0.712091\pi$
$409$	−25.0000			−1.23617			−0.618085	+	0.786111i	$0.712091\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			25.4558i			1.25260i
$414$		0			0
$415$	1.00000	+	3.00000i	0.0490881	+	0.147264i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−33.9411			−1.65813			−0.829066	−	0.559150i	$-0.811127\pi$
$419$	−33.9411			−1.65813			−0.829066	+	0.559150i	$0.811127\pi$
$420$		0			0
$421$	23.0000			1.12095			0.560476	−	0.828171i	$-0.310618\pi$
$421$	23.0000			1.12095			0.560476	+	0.828171i	$0.310618\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−8.48528	−	11.3137i	−0.411597	−	0.548795i
$426$		0			0
$427$			39.0000i			1.88734i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−12.7279			−0.613082			−0.306541	−	0.951857i	$-0.599172\pi$
$431$	−12.7279			−0.613082			−0.306541	+	0.951857i	$0.599172\pi$
$432$		0			0
$433$			36.0000i			1.73005i	0.501729	+	0.865025i	$0.332697\pi$
$433$			36.0000i			1.73005i	−0.501729	+	0.865025i	$0.667303\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			7.07107i			0.338255i
$438$		0			0
$439$	−14.0000			−0.668184			−0.334092	−	0.942541i	$-0.608430\pi$
$439$	−14.0000			−0.668184			−0.334092	+	0.942541i	$0.608430\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	14.1421i		−	0.671913i	−0.941877	−	0.335957i	$-0.890940\pi$
$443$		−	14.1421i		−	0.671913i	0.941877	−	0.335957i	$-0.109060\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$	18.0000			0.847587
$452$		0			0
$453$		0			0
$454$		0			0
$455$	19.0919	−	6.36396i	0.895041	−	0.298347i
$456$		0			0
$457$			12.0000i			0.561336i	0.959805	+	0.280668i	$0.0905560\pi$
$457$			12.0000i			0.561336i	−0.959805	+	0.280668i	$0.909444\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	16.9706			0.790398			0.395199	−	0.918596i	$-0.370676\pi$
$461$	16.9706			0.790398			0.395199	+	0.918596i	$0.370676\pi$
$462$		0			0
$463$		−	15.0000i		−	0.697109i	−0.937288	−	0.348555i	$-0.886673\pi$
$463$		−	15.0000i		−	0.697109i	0.937288	−	0.348555i	$-0.113327\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	22.6274i		−	1.04707i	−0.852004	−	0.523536i	$-0.824613\pi$
$467$		−	22.6274i		−	1.04707i	0.852004	−	0.523536i	$-0.175387\pi$
$468$		0			0
$469$	9.00000			0.415581
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	25.4558i		−	1.17046i
$474$		0			0
$475$	4.00000	−	3.00000i	0.183533	−	0.137649i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−4.24264			−0.193851			−0.0969256	−	0.995292i	$-0.530901\pi$
$479$	−4.24264			−0.193851			−0.0969256	+	0.995292i	$0.530901\pi$
$480$		0			0
$481$	27.0000			1.23109
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.12132	−	6.36396i	−0.0963242	−	0.288973i
$486$		0			0
$487$			9.00000i			0.407829i	0.978989	+	0.203914i	$0.0653664\pi$
$487$			9.00000i			0.407829i	−0.978989	+	0.203914i	$0.934634\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−21.2132			−0.957338			−0.478669	−	0.877995i	$-0.658881\pi$
$491$	−21.2132			−0.957338			−0.478669	+	0.877995i	$0.658881\pi$
$492$		0			0
$493$		−	12.0000i		−	0.540453i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	38.1838i		−	1.71278i
$498$		0			0
$499$	16.0000			0.716258			0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000			0.716258			0.358129	+	0.933672i	$0.383415\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	35.3553i		−	1.57642i	−0.615409	−	0.788208i	$-0.711009\pi$
$503$		−	35.3553i		−	1.57642i	0.615409	−	0.788208i	$-0.288991\pi$
$504$		0			0
$505$	18.0000	−	6.00000i	0.800989	−	0.266996i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−29.6985			−1.31636			−0.658181	−	0.752860i	$-0.728674\pi$
$509$	−29.6985			−1.31636			−0.658181	+	0.752860i	$0.728674\pi$
$510$		0			0
$511$	−27.0000			−1.19441
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.12132	−	6.36396i	−0.0934765	−	0.280430i
$516$		0			0
$517$		−	12.0000i		−	0.527759i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	12.7279			0.557620			0.278810	−	0.960346i	$-0.410060\pi$
$521$	12.7279			0.557620			0.278810	+	0.960346i	$0.410060\pi$
$522$		0			0
$523$		−	9.00000i		−	0.393543i	−0.980449	−	0.196771i	$-0.936954\pi$
$523$		−	9.00000i		−	0.393543i	0.980449	−	0.196771i	$-0.0630456\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			5.65685i			0.246416i
$528$		0			0
$529$	−27.0000			−1.17391
$530$		0			0
$531$		0			0
$532$		0			0
$533$			12.7279i			0.551308i
$534$		0			0
$535$	−5.00000	−	15.0000i	−0.216169	−	0.648507i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	8.48528			0.365487
$540$		0			0
$541$	−25.0000			−1.07483			−0.537417	−	0.843317i	$-0.680600\pi$
$541$	−25.0000			−1.07483			−0.537417	+	0.843317i	$0.680600\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−16.9706	+	5.65685i	−0.726939	+	0.242313i
$546$		0			0
$547$		−	3.00000i		−	0.128271i	−0.997941	−	0.0641354i	$-0.979571\pi$
$547$		−	3.00000i		−	0.128271i	0.997941	−	0.0641354i	$-0.0204289\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	4.24264			0.180743
$552$		0			0
$553$			15.0000i			0.637865i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	19.7990i		−	0.838910i	−0.907776	−	0.419455i	$-0.862221\pi$
$557$		−	19.7990i		−	0.838910i	0.907776	−	0.419455i	$-0.137779\pi$
$558$		0			0
$559$	18.0000			0.761319
$560$		0			0
$561$		0			0
$562$		0			0
$563$			11.3137i			0.476816i	0.971165	+	0.238408i	$0.0766255\pi$
$563$			11.3137i			0.476816i	−0.971165	+	0.238408i	$0.923374\pi$
$564$		0			0
$565$	−1.00000	−	3.00000i	−0.0420703	−	0.126211i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	42.4264			1.77861			0.889304	−	0.457317i	$-0.151190\pi$
$569$	42.4264			1.77861			0.889304	+	0.457317i	$0.151190\pi$
$570$		0			0
$571$	25.0000			1.04622			0.523109	−	0.852266i	$-0.324772\pi$
$571$	25.0000			1.04622			0.523109	+	0.852266i	$0.324772\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	21.2132	+	28.2843i	0.884652	+	1.17954i
$576$		0			0
$577$		−	45.0000i		−	1.87337i	−0.350167	−	0.936687i	$-0.613875\pi$
$577$		−	45.0000i		−	1.87337i	0.350167	−	0.936687i	$-0.386125\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−4.24264			−0.176014
$582$		0			0
$583$		−	42.0000i		−	1.73946i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	26.8701i		−	1.10905i	−0.832168	−	0.554523i	$-0.812901\pi$
$587$		−	26.8701i		−	1.10905i	0.832168	−	0.554523i	$-0.187099\pi$
$588$		0			0
$589$	−2.00000			−0.0824086
$590$		0			0
$591$		0			0
$592$		0			0
$593$			9.89949i			0.406524i	0.979124	+	0.203262i	$0.0651542\pi$
$593$			9.89949i			0.406524i	−0.979124	+	0.203262i	$0.934846\pi$
$594$		0			0
$595$	18.0000	−	6.00000i	0.737928	−	0.245976i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	29.6985			1.21345			0.606724	−	0.794913i	$-0.292483\pi$
$599$	29.6985			1.21345			0.606724	+	0.794913i	$0.292483\pi$
$600$		0			0
$601$	−40.0000			−1.63163			−0.815817	−	0.578310i	$-0.803712\pi$
$601$	−40.0000			−1.63163			−0.815817	+	0.578310i	$0.803712\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−14.8492	+	4.94975i	−0.603708	+	0.201236i
$606$		0			0
$607$			39.0000i			1.58296i	0.611194	+	0.791481i	$0.290689\pi$
$607$			39.0000i			1.58296i	−0.611194	+	0.791481i	$0.709311\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	8.48528			0.343278
$612$		0			0
$613$		−	9.00000i		−	0.363507i	−0.983344	−	0.181753i	$-0.941823\pi$
$613$		−	9.00000i		−	0.363507i	0.983344	−	0.181753i	$-0.0581772\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	7.07107i		−	0.284670i	−0.989819	−	0.142335i	$-0.954539\pi$
$617$		−	7.07107i		−	0.284670i	0.989819	−	0.142335i	$-0.0454611\pi$
$618$		0			0
$619$	19.0000			0.763674			0.381837	−	0.924230i	$-0.375291\pi$
$619$	19.0000			0.763674			0.381837	+	0.924230i	$0.375291\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	7.00000	−	24.0000i	0.280000	−	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	25.4558			1.01499
$630$		0			0
$631$	−17.0000			−0.676759			−0.338380	−	0.941010i	$-0.609879\pi$
$631$	−17.0000			−0.676759			−0.338380	+	0.941010i	$0.609879\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−12.7279	−	38.1838i	−0.505092	−	1.51528i
$636$		0			0
$637$			6.00000i			0.237729i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	16.9706			0.670297			0.335148	−	0.942165i	$-0.391214\pi$
$641$	16.9706			0.670297			0.335148	+	0.942165i	$0.391214\pi$
$642$		0			0
$643$		−	42.0000i		−	1.65632i	−0.560493	−	0.828159i	$-0.689388\pi$
$643$		−	42.0000i		−	1.65632i	0.560493	−	0.828159i	$-0.310612\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	9.89949i		−	0.389189i	−0.980884	−	0.194595i	$-0.937661\pi$
$647$		−	9.89949i		−	0.389189i	0.980884	−	0.194595i	$-0.0623391\pi$
$648$		0			0
$649$	36.0000			1.41312
$650$		0			0
$651$		0			0
$652$		0			0
$653$			5.65685i			0.221370i	0.993856	+	0.110685i	$0.0353044\pi$
$653$			5.65685i			0.221370i	−0.993856	+	0.110685i	$0.964696\pi$
$654$		0			0
$655$	18.0000	−	6.00000i	0.703318	−	0.234439i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−16.9706			−0.661079			−0.330540	−	0.943792i	$-0.607231\pi$
$659$	−16.9706			−0.661079			−0.330540	+	0.943792i	$0.607231\pi$
$660$		0			0
$661$	23.0000			0.894596			0.447298	−	0.894385i	$-0.352386\pi$
$661$	23.0000			0.894596			0.447298	+	0.894385i	$0.352386\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	2.12132	+	6.36396i	0.0822613	+	0.246784i
$666$		0			0
$667$			30.0000i			1.16160i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	55.1543			2.12921
$672$		0			0
$673$		−	15.0000i		−	0.578208i	−0.957298	−	0.289104i	$-0.906643\pi$
$673$		−	15.0000i		−	0.578208i	0.957298	−	0.289104i	$-0.0933573\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			48.0833i			1.84799i	0.382405	+	0.923995i	$0.375096\pi$
$677$			48.0833i			1.84799i	−0.382405	+	0.923995i	$0.624904\pi$
$678$		0			0
$679$	9.00000			0.345388
$680$		0			0
$681$		0			0
$682$		0			0
$683$			15.5563i			0.595247i	0.954683	+	0.297624i	$0.0961940\pi$
$683$			15.5563i			0.595247i	−0.954683	+	0.297624i	$0.903806\pi$
$684$		0			0
$685$	−1.00000	−	3.00000i	−0.0382080	−	0.114624i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	29.6985			1.13142
$690$		0			0
$691$	−50.0000			−1.90209			−0.951045	−	0.309053i	$-0.899988\pi$
$691$	−50.0000			−1.90209			−0.951045	+	0.309053i	$0.899988\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−27.5772	+	9.19239i	−1.04606	+	0.348687i
$696$		0			0
$697$			12.0000i			0.454532i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−38.1838			−1.44218			−0.721090	−	0.692841i	$-0.756359\pi$
$701$	−38.1838			−1.44218			−0.721090	+	0.692841i	$0.756359\pi$
$702$		0			0
$703$			9.00000i			0.339441i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			25.4558i			0.957366i
$708$		0			0
$709$	17.0000			0.638448			0.319224	−	0.947679i	$-0.396578\pi$
$709$	17.0000			0.638448			0.319224	+	0.947679i	$0.396578\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	14.1421i		−	0.529627i
$714$		0			0
$715$	−9.00000	−	27.0000i	−0.336581	−	1.00974i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−12.7279			−0.474671			−0.237336	−	0.971428i	$-0.576274\pi$
$719$	−12.7279			−0.474671			−0.237336	+	0.971428i	$0.576274\pi$
$720$		0			0
$721$	9.00000			0.335178
$722$		0			0
$723$		0			0
$724$		0			0
$725$	16.9706	−	12.7279i	0.630271	−	0.472703i
$726$		0			0
$727$			42.0000i			1.55769i	0.627214	+	0.778847i	$0.284195\pi$
$727$			42.0000i			1.55769i	−0.627214	+	0.778847i	$0.715805\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	16.9706			0.627679
$732$		0			0
$733$			24.0000i			0.886460i	0.896408	+	0.443230i	$0.146168\pi$
$733$			24.0000i			0.886460i	−0.896408	+	0.443230i	$0.853832\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	12.7279i		−	0.468839i
$738$		0			0
$739$	10.0000			0.367856			0.183928	−	0.982940i	$-0.441119\pi$
$739$	10.0000			0.367856			0.183928	+	0.982940i	$0.441119\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	14.1421i		−	0.518825i	−0.965767	−	0.259412i	$-0.916471\pi$
$743$		−	14.1421i		−	0.518825i	0.965767	−	0.259412i	$-0.0835289\pi$
$744$		0			0
$745$	36.0000	−	12.0000i	1.31894	−	0.439646i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	21.2132			0.775114
$750$		0			0
$751$	7.00000			0.255434			0.127717	−	0.991811i	$-0.459235\pi$
$751$	7.00000			0.255434			0.127717	+	0.991811i	$0.459235\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−2.12132	+	0.707107i	−0.0772028	+	0.0257343i
$756$		0			0
$757$			27.0000i			0.981332i	0.871348	+	0.490666i	$0.163246\pi$
$757$			27.0000i			0.981332i	−0.871348	+	0.490666i	$0.836754\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−4.24264			−0.153796			−0.0768978	−	0.997039i	$-0.524502\pi$
$761$	−4.24264			−0.153796			−0.0768978	+	0.997039i	$0.524502\pi$
$762$		0			0
$763$		−	24.0000i		−	0.868858i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			25.4558i			0.919157i
$768$		0			0
$769$	11.0000			0.396670			0.198335	−	0.980134i	$-0.436447\pi$
$769$	11.0000			0.396670			0.198335	+	0.980134i	$0.436447\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	41.0122i		−	1.47511i	−0.675289	−	0.737553i	$-0.735981\pi$
$773$		−	41.0122i		−	1.47511i	0.675289	−	0.737553i	$-0.264019\pi$
$774$		0			0
$775$	−8.00000	+	6.00000i	−0.287368	+	0.215526i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4.24264			−0.152008
$780$		0			0
$781$	−54.0000			−1.93227
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−4.24264	−	12.7279i	−0.151426	−	0.454279i
$786$		0			0
$787$		−	15.0000i		−	0.534692i	−0.963601	−	0.267346i	$-0.913853\pi$
$787$		−	15.0000i		−	0.534692i	0.963601	−	0.267346i	$-0.0861467\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	4.24264			0.150851
$792$		0			0
$793$			39.0000i			1.38493i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	19.7990i		−	0.701316i	−0.936504	−	0.350658i	$-0.885958\pi$
$797$		−	19.7990i		−	0.701316i	0.936504	−	0.350658i	$-0.114042\pi$
$798$		0			0
$799$	8.00000			0.283020
$800$		0			0
$801$		0			0
$802$		0			0
$803$			38.1838i			1.34748i
$804$		0			0
$805$	−45.0000	+	15.0000i	−1.58604	+	0.528681i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−25.4558			−0.894980			−0.447490	−	0.894289i	$-0.647682\pi$
$809$	−25.4558			−0.894980			−0.447490	+	0.894289i	$0.647682\pi$
$810$		0			0
$811$	−2.00000			−0.0702295			−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000			−0.0702295			−0.0351147	+	0.999383i	$0.511180\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	6.36396	+	19.0919i	0.222920	+	0.668759i
$816$		0			0
$817$			6.00000i			0.209913i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−8.48528			−0.296138			−0.148069	−	0.988977i	$-0.547306\pi$
$821$	−8.48528			−0.296138			−0.148069	+	0.988977i	$0.547306\pi$
$822$		0			0
$823$			39.0000i			1.35945i	0.733465	+	0.679727i	$0.237902\pi$
$823$			39.0000i			1.35945i	−0.733465	+	0.679727i	$0.762098\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			2.82843i			0.0983540i	0.998790	+	0.0491770i	$0.0156598\pi$
$827$			2.82843i			0.0983540i	−0.998790	+	0.0491770i	$0.984340\pi$
$828$		0			0
$829$	−7.00000			−0.243120			−0.121560	−	0.992584i	$-0.538790\pi$
$829$	−7.00000			−0.243120			−0.121560	+	0.992584i	$0.538790\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			5.65685i			0.195998i
$834$		0			0
$835$	4.00000	+	12.0000i	0.138426	+	0.415277i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−42.4264			−1.46472			−0.732361	−	0.680916i	$-0.761582\pi$
$839$	−42.4264			−1.46472			−0.732361	+	0.680916i	$0.761582\pi$
$840$		0			0
$841$	−11.0000			−0.379310
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−8.48528	+	2.82843i	−0.291903	+	0.0973009i
$846$		0			0
$847$		−	21.0000i		−	0.721569i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−63.6396			−2.18154
$852$		0			0
$853$		−	33.0000i		−	1.12990i	−0.825126	−	0.564949i	$-0.808896\pi$
$853$		−	33.0000i		−	1.12990i	0.825126	−	0.564949i	$-0.191104\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			26.8701i			0.917864i	0.888471	+	0.458932i	$0.151768\pi$
$857$			26.8701i			0.917864i	−0.888471	+	0.458932i	$0.848232\pi$
$858$		0			0
$859$	−47.0000			−1.60362			−0.801810	−	0.597580i	$-0.796129\pi$
$859$	−47.0000			−1.60362			−0.801810	+	0.597580i	$0.796129\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			28.2843i			0.962808i	0.876499	+	0.481404i	$0.159873\pi$
$863$			28.2843i			0.962808i	−0.876499	+	0.481404i	$0.840127\pi$
$864$		0			0
$865$	−10.0000	−	30.0000i	−0.340010	−	1.02003i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	21.2132			0.719609
$870$		0			0
$871$	9.00000			0.304953
$872$		0			0
$873$		0			0
$874$		0			0
$875$	27.5772	+	19.0919i	0.932278	+	0.645423i
$876$		0			0
$877$			15.0000i			0.506514i	0.967399	+	0.253257i	$0.0815018\pi$
$877$			15.0000i			0.506514i	−0.967399	+	0.253257i	$0.918498\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−38.1838			−1.28644			−0.643222	−	0.765680i	$-0.722403\pi$
$881$	−38.1838			−1.28644			−0.643222	+	0.765680i	$0.722403\pi$
$882$		0			0
$883$			27.0000i			0.908622i	0.890843	+	0.454311i	$0.150115\pi$
$883$			27.0000i			0.908622i	−0.890843	+	0.454311i	$0.849885\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			45.2548i			1.51951i	0.650210	+	0.759754i	$0.274681\pi$
$887$			45.2548i			1.51951i	−0.650210	+	0.759754i	$0.725319\pi$
$888$		0			0
$889$	54.0000			1.81110
$890$		0			0
$891$		0			0
$892$		0			0
$893$			2.82843i			0.0946497i
$894$		0			0
$895$	−27.0000	+	9.00000i	−0.902510	+	0.300837i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−8.48528			−0.283000
$900$		0			0
$901$	28.0000			0.932815
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2.12132	−	0.707107i	0.0705151	−	0.0235050i
$906$		0			0
$907$		−	3.00000i		−	0.0996134i	−0.998759	−	0.0498067i	$-0.984139\pi$
$907$		−	3.00000i		−	0.0996134i	0.998759	−	0.0498067i	$-0.0158605\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−42.4264			−1.40565			−0.702825	−	0.711363i	$-0.748078\pi$
$911$	−42.4264			−1.40565			−0.702825	+	0.711363i	$0.748078\pi$
$912$		0			0
$913$			6.00000i			0.198571i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			25.4558i			0.840626i
$918$		0			0
$919$	−2.00000			−0.0659739			−0.0329870	−	0.999456i	$-0.510502\pi$
$919$	−2.00000			−0.0659739			−0.0329870	+	0.999456i	$0.510502\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	38.1838i		−	1.25683i
$924$		0			0
$925$	27.0000	+	36.0000i	0.887755	+	1.18367i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−46.6690			−1.53116			−0.765581	−	0.643340i	$-0.777548\pi$
$929$	−46.6690			−1.53116			−0.765581	+	0.643340i	$0.777548\pi$
$930$		0			0
$931$	−2.00000			−0.0655474
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−8.48528	−	25.4558i	−0.277498	−	0.832495i
$936$		0			0
$937$		−	9.00000i		−	0.294017i	−0.989135	−	0.147009i	$-0.953036\pi$
$937$		−	9.00000i		−	0.294017i	0.989135	−	0.147009i	$-0.0469645\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	21.2132			0.691531			0.345765	−	0.938321i	$-0.387619\pi$
$941$	21.2132			0.691531			0.345765	+	0.938321i	$0.387619\pi$
$942$		0			0
$943$		−	30.0000i		−	0.976934i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			36.7696i			1.19485i	0.801925	+	0.597425i	$0.203809\pi$
$947$			36.7696i			1.19485i	−0.801925	+	0.597425i	$0.796191\pi$
$948$		0			0
$949$	−27.0000			−0.876457
$950$		0			0
$951$		0			0
$952$		0			0
$953$			5.65685i			0.183243i	0.995794	+	0.0916217i	$0.0292051\pi$
$953$			5.65685i			0.183243i	−0.995794	+	0.0916217i	$0.970795\pi$
$954$		0			0
$955$	36.0000	−	12.0000i	1.16493	−	0.388311i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	4.24264			0.137002
$960$		0			0
$961$	−27.0000			−0.870968
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2.12132	+	6.36396i	0.0682877	+	0.204863i
$966$		0			0
$967$		−	33.0000i		−	1.06121i	−0.847620	−	0.530604i	$-0.821965\pi$
$967$		−	33.0000i		−	1.06121i	0.847620	−	0.530604i	$-0.178035\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−25.4558			−0.816917			−0.408458	−	0.912777i	$-0.633934\pi$
$971$	−25.4558			−0.816917			−0.408458	+	0.912777i	$0.633934\pi$
$972$		0			0
$973$		−	39.0000i		−	1.25028i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			1.41421i			0.0452447i	0.999744	+	0.0226224i	$0.00720153\pi$
$977$			1.41421i			0.0452447i	−0.999744	+	0.0226224i	$0.992798\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	52.3259i		−	1.66894i	−0.551056	−	0.834469i	$-0.685775\pi$
$983$		−	52.3259i		−	1.66894i	0.551056	−	0.834469i	$-0.314225\pi$
$984$		0			0
$985$	−16.0000	−	48.0000i	−0.509802	−	1.52941i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−42.4264			−1.34908
$990$		0			0
$991$	−53.0000			−1.68360			−0.841800	−	0.539789i	$-0.818504\pi$
$991$	−53.0000			−1.68360			−0.841800	+	0.539789i	$0.818504\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	23.3345	−	7.77817i	0.739754	−	0.246585i
$996$		0			0
$997$			48.0000i			1.52018i	0.649821	+	0.760088i	$0.274844\pi$
$997$			48.0000i			1.52018i	−0.649821	+	0.760088i	$0.725156\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.2.f.k.1729.2		4
3.2	odd	2	inner	2160.2.f.k.1729.3		4
4.3	odd	2		135.2.b.b.109.3	yes	4
5.4	even	2	inner	2160.2.f.k.1729.1		4
12.11	even	2		135.2.b.b.109.2	yes	4
15.14	odd	2	inner	2160.2.f.k.1729.4		4
20.3	even	4		675.2.a.m.1.2		2
20.7	even	4		675.2.a.l.1.1		2
20.19	odd	2		135.2.b.b.109.1	✓	4
36.7	odd	6		405.2.j.f.109.4		8
36.11	even	6		405.2.j.f.109.1		8
36.23	even	6		405.2.j.f.379.3		8
36.31	odd	6		405.2.j.f.379.2		8
60.23	odd	4		675.2.a.m.1.1		2
60.47	odd	4		675.2.a.l.1.2		2
60.59	even	2		135.2.b.b.109.4	yes	4
180.59	even	6		405.2.j.f.379.1		8
180.79	odd	6		405.2.j.f.109.2		8
180.119	even	6		405.2.j.f.109.3		8
180.139	odd	6		405.2.j.f.379.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.2.b.b.109.1	✓	4	20.19	odd	2
135.2.b.b.109.2	yes	4	12.11	even	2
135.2.b.b.109.3	yes	4	4.3	odd	2
135.2.b.b.109.4	yes	4	60.59	even	2
405.2.j.f.109.1		8	36.11	even	6
405.2.j.f.109.2		8	180.79	odd	6
405.2.j.f.109.3		8	180.119	even	6
405.2.j.f.109.4		8	36.7	odd	6
405.2.j.f.379.1		8	180.59	even	6
405.2.j.f.379.2		8	36.31	odd	6
405.2.j.f.379.3		8	36.23	even	6
405.2.j.f.379.4		8	180.139	odd	6
675.2.a.l.1.1		2	20.7	even	4
675.2.a.l.1.2		2	60.47	odd	4
675.2.a.m.1.1		2	60.23	odd	4
675.2.a.m.1.2		2	20.3	even	4
2160.2.f.k.1729.1		4	5.4	even	2	inner
2160.2.f.k.1729.2		4	1.1	even	1	trivial
2160.2.f.k.1729.3		4	3.2	odd	2	inner
2160.2.f.k.1729.4		4	15.14	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2160.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.12132 + 0.707107i) q^{5} -3.00000i q^{7} +O(q^{10})\)	\(q+(-2.12132 + 0.707107i) q^{5} -3.00000i q^{7} -4.24264 q^{11} -3.00000i q^{13} -2.82843i q^{17} +1.00000 q^{19} +7.07107i q^{23} +(4.00000 - 3.00000i) q^{25} +4.24264 q^{29} -2.00000 q^{31} +(2.12132 + 6.36396i) q^{35} +9.00000i q^{37} -4.24264 q^{41} +6.00000i q^{43} +2.82843i q^{47} -2.00000 q^{49} +9.89949i q^{53} +(9.00000 - 3.00000i) q^{55} -8.48528 q^{59} -13.0000 q^{61} +(2.12132 + 6.36396i) q^{65} +3.00000i q^{67} +12.7279 q^{71} -9.00000i q^{73} +12.7279i q^{77} -5.00000 q^{79} -1.41421i q^{83} +(2.00000 + 6.00000i) q^{85} -9.00000 q^{91} +(-2.12132 + 0.707107i) q^{95} +3.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 4 q^{19} + 16 q^{25} - 8 q^{31} - 8 q^{49} + 36 q^{55} - 52 q^{61} - 20 q^{79} + 8 q^{85} - 36 q^{91}+O(q^{100}) \) 4 * q + 4 * q^19 + 16 * q^25 - 8 * q^31 - 8 * q^49 + 36 * q^55 - 52 * q^61 - 20 * q^79 + 8 * q^85 - 36 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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