LMFDB - Embedded newform 1764.4.k.bb.361.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,4,Mod(361,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1764.361");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1764.k (of order $3$, degree $2$, 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:	$104.079369250$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 27x^{6} + 10x^{5} + 446x^{4} + 62x^{3} + 3061x^{2} + 2142x + 14161 $ x^8 - 2x^7 + 27x^6 + 10x^5 + 446x^4 + 62x^3 + 3061x^2 + 2142*x + 14161
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{8}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 588)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.2
Root			$-1.65506 + 2.86665i$ of defining polynomial
Character	$\chi$	$=$	1764.361
Dual form			1764.4.k.bb.1549.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.08470 - 7.07491i) q^{5} +O(q^{10})$	$q+(-4.08470 - 7.07491i) q^{5} +(18.9094 - 32.7521i) q^{11} -39.9319 q^{13} +(-4.96798 + 8.60480i) q^{17} +(-45.2229 - 78.3284i) q^{19} +(59.2973 + 102.706i) q^{23} +(29.1304 - 50.4554i) q^{25} +78.4061 q^{29} +(46.0055 - 79.6838i) q^{31} +(-166.218 - 287.897i) q^{37} +71.7451 q^{41} -115.947 q^{43} +(-153.964 - 266.673i) q^{47} +(-201.575 + 349.138i) q^{53} -308.958 q^{55} +(-296.855 + 514.168i) q^{59} +(166.585 + 288.534i) q^{61} +(163.110 + 282.515i) q^{65} +(371.756 - 643.900i) q^{67} +728.272 q^{71} +(-400.874 + 694.335i) q^{73} +(-533.953 - 924.833i) q^{79} +906.756 q^{83} +81.1709 q^{85} +(-556.634 - 964.119i) q^{89} +(-369.444 + 639.896i) q^{95} -1480.94 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 48 q^{17} + 192 q^{19} + 192 q^{23} - 324 q^{25} - 192 q^{29} + 48 q^{31} - 256 q^{37} + 2016 q^{41} - 224 q^{43} - 864 q^{47} - 648 q^{53} - 4704 q^{55} - 336 q^{59} + 960 q^{61} - 360 q^{65} - 720 q^{67} + 2688 q^{71} + 672 q^{73} + 1984 q^{79} + 6240 q^{83} + 1360 q^{85} - 2160 q^{89} - 3744 q^{95} - 4032 q^{97}+O(q^{100}) $ 8 * q - 48 * q^17 + 192 * q^19 + 192 * q^23 - 324 * q^25 - 192 * q^29 + 48 * q^31 - 256 * q^37 + 2016 * q^41 - 224 * q^43 - 864 * q^47 - 648 * q^53 - 4704 * q^55 - 336 * q^59 + 960 * q^61 - 360 * q^65 - 720 * q^67 + 2688 * q^71 + 672 * q^73 + 1984 * q^79 + 6240 * q^83 + 1360 * q^85 - 2160 * q^89 - 3744 * q^95 - 4032 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	−4.08470	−	7.07491i	−0.365347	−	0.632799i	0.623485	−	0.781835i	$-0.285716\pi$
$5$	−4.08470	−	7.07491i	−0.365347	−	0.632799i	−0.988832	+	0.149036i	$0.952383\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	18.9094	−	32.7521i	0.518310	−	0.897739i	−0.481464	−	0.876466i	$-0.659895\pi$
$11$	18.9094	−	32.7521i	0.518310	−	0.897739i	0.999774	−	0.0212729i	$-0.00677189\pi$
$12$		0			0
$13$	−39.9319			−0.851933			−0.425966	−	0.904739i	$-0.640066\pi$
$13$	−39.9319			−0.851933			−0.425966	+	0.904739i	$0.640066\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.96798	+	8.60480i	−0.0708772	+	0.122763i	−0.899286	−	0.437361i	$-0.855913\pi$
$17$	−4.96798	+	8.60480i	−0.0708772	+	0.122763i	0.828409	+	0.560124i	$0.189247\pi$
$18$		0			0
$19$	−45.2229	−	78.3284i	−0.546045	−	0.945777i	−0.998540	−	0.0540105i	$-0.982800\pi$
$19$	−45.2229	−	78.3284i	−0.546045	−	0.945777i	0.452496	−	0.891767i	$-0.350534\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	59.2973	+	102.706i	0.537580	+	0.931116i	0.999034	+	0.0439513i	$0.0139947\pi$
$23$	59.2973	+	102.706i	0.537580	+	0.931116i	−0.461454	+	0.887164i	$0.652672\pi$
$24$		0			0
$25$	29.1304	−	50.4554i	0.233043	−	0.403643i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	78.4061			0.502057			0.251028	−	0.967980i	$-0.419231\pi$
$29$	78.4061			0.502057			0.251028	+	0.967980i	$0.419231\pi$
$30$		0			0
$31$	46.0055	−	79.6838i	0.266543	−	0.461666i	−0.701424	−	0.712744i	$-0.747452\pi$
$31$	46.0055	−	79.6838i	0.266543	−	0.461666i	0.967967	+	0.251079i	$0.0807853\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−166.218	−	287.897i	−0.738541	−	1.27919i	−0.953152	−	0.302491i	$-0.902182\pi$
$37$	−166.218	−	287.897i	−0.738541	−	1.27919i	0.214611	−	0.976700i	$-0.431152\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	71.7451			0.273285			0.136643	−	0.990620i	$-0.456369\pi$
$41$	71.7451			0.273285			0.136643	+	0.990620i	$0.456369\pi$
$42$		0			0
$43$	−115.947			−0.411202			−0.205601	−	0.978636i	$-0.565915\pi$
$43$	−115.947			−0.411202			−0.205601	+	0.978636i	$0.565915\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−153.964	−	266.673i	−0.477828	−	0.827623i	0.521849	−	0.853038i	$-0.325242\pi$
$47$	−153.964	−	266.673i	−0.477828	−	0.827623i	−0.999677	+	0.0254154i	$0.991909\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−201.575	+	349.138i	−0.522424	+	0.904865i	0.477236	+	0.878775i	$0.341639\pi$
$53$	−201.575	+	349.138i	−0.522424	+	0.904865i	−0.999660	+	0.0260895i	$0.991695\pi$
$54$		0			0
$55$	−308.958			−0.757451
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−296.855	+	514.168i	−0.655038	+	1.13456i	0.326847	+	0.945077i	$0.394014\pi$
$59$	−296.855	+	514.168i	−0.655038	+	1.13456i	−0.981884	+	0.189481i	$0.939319\pi$
$60$		0			0
$61$	166.585	+	288.534i	0.349657	+	0.605623i	0.986188	−	0.165628i	$-0.0529649\pi$
$61$	166.585	+	288.534i	0.349657	+	0.605623i	−0.636532	+	0.771250i	$0.719632\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	163.110	+	282.515i	0.311251	+	0.539102i
$66$		0			0
$67$	371.756	−	643.900i	0.677869	−	1.17410i	−0.297753	−	0.954643i	$-0.596237\pi$
$67$	371.756	−	643.900i	0.677869	−	1.17410i	0.975622	−	0.219460i	$-0.0704295\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	728.272			1.21732			0.608662	−	0.793429i	$-0.291706\pi$
$71$	728.272			1.21732			0.608662	+	0.793429i	$0.291706\pi$
$72$		0			0
$73$	−400.874	+	694.335i	−0.642723	+	1.11323i	0.342099	+	0.939664i	$0.388862\pi$
$73$	−400.874	+	694.335i	−0.642723	+	1.11323i	−0.984822	+	0.173566i	$0.944471\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−533.953	−	924.833i	−0.760435	−	1.31711i	−0.942626	−	0.333849i	$-0.891652\pi$
$79$	−533.953	−	924.833i	−0.760435	−	1.31711i	0.182191	−	0.983263i	$-0.441681\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	906.756			1.19915			0.599575	−	0.800319i	$-0.295336\pi$
$83$	906.756			1.19915			0.599575	+	0.800319i	$0.295336\pi$
$84$		0			0
$85$	81.1709			0.103579
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−556.634	−	964.119i	−0.662957	−	1.14827i	−0.979835	−	0.199808i	$-0.935968\pi$
$89$	−556.634	−	964.119i	−0.662957	−	1.14827i	0.316878	−	0.948466i	$-0.397365\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−369.444	+	639.896i	−0.398991	+	0.691073i
$96$		0			0
$97$	−1480.94			−1.55017			−0.775084	−	0.631858i	$-0.782293\pi$
$97$	−1480.94			−1.55017			−0.775084	+	0.631858i	$0.782293\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	278.158	−	481.784i	0.274037	−	0.474647i	−0.695854	−	0.718183i	$-0.744974\pi$
$101$	278.158	−	481.784i	0.274037	−	0.474647i	0.969892	+	0.243536i	$0.0783075\pi$
$102$		0			0
$103$	276.217	+	478.423i	0.264238	+	0.457674i	0.967364	−	0.253392i	$-0.0815462\pi$
$103$	276.217	+	478.423i	0.264238	+	0.457674i	−0.703126	+	0.711066i	$0.748213\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	266.902	+	462.288i	0.241144	+	0.417674i	0.961040	−	0.276408i	$-0.0891440\pi$
$107$	266.902	+	462.288i	0.241144	+	0.417674i	−0.719896	+	0.694082i	$0.755811\pi$
$108$		0			0
$109$	−547.314	+	947.976i	−0.480947	+	0.833025i	−0.999761	−	0.0218626i	$-0.993040\pi$
$109$	−547.314	+	947.976i	−0.480947	+	0.833025i	0.518814	+	0.854887i	$0.326374\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1425.18			1.18646			0.593228	−	0.805035i	$-0.297853\pi$
$113$	1425.18			1.18646			0.593228	+	0.805035i	$0.297853\pi$
$114$		0			0
$115$	484.423	−	839.046i	0.392806	−	0.680360i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−49.6330	−	85.9668i	−0.0372900	−	0.0645882i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1497.13			−1.07126
$126$		0			0
$127$	−786.485			−0.549522			−0.274761	−	0.961513i	$-0.588599\pi$
$127$	−786.485			−0.549522			−0.274761	+	0.961513i	$0.588599\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	13.8651	+	24.0151i	0.00924735	+	0.0160169i	0.870612	−	0.491970i	$-0.163723\pi$
$131$	13.8651	+	24.0151i	0.00924735	+	0.0160169i	−0.861365	+	0.507987i	$0.830390\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1181.49	+	2046.40i	−0.736798	+	1.27617i	0.217132	+	0.976142i	$0.430330\pi$
$137$	−1181.49	+	2046.40i	−0.736798	+	1.27617i	−0.953930	+	0.300029i	$0.903004\pi$
$138$		0			0
$139$	−2513.28			−1.53362			−0.766811	−	0.641873i	$-0.778158\pi$
$139$	−2513.28			−1.53362			−0.766811	+	0.641873i	$0.778158\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−755.090	+	1307.85i	−0.441565	+	0.764813i
$144$		0			0
$145$	−320.266	−	554.716i	−0.183425	−	0.317701i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1132.92	+	1962.27i	0.622900	+	1.07889i	0.988943	+	0.148296i	$0.0473788\pi$
$149$	1132.92	+	1962.27i	0.622900	+	1.07889i	−0.366043	+	0.930598i	$0.619288\pi$
$150$		0			0
$151$	−141.573	+	245.212i	−0.0762984	+	0.132153i	−0.901650	−	0.432466i	$-0.857644\pi$
$151$	−141.573	+	245.212i	−0.0762984	+	0.132153i	0.825352	+	0.564619i	$0.190977\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−751.675			−0.389522
$156$		0			0
$157$	96.2904	−	166.780i	0.0489478	−	0.0847801i	−0.840513	−	0.541791i	$-0.817747\pi$
$157$	96.2904	−	166.780i	0.0489478	−	0.0847801i	0.889461	+	0.457011i	$0.151080\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	421.417	+	729.915i	0.202502	+	0.350744i	0.949334	−	0.314269i	$-0.101759\pi$
$163$	421.417	+	729.915i	0.202502	+	0.350744i	−0.746832	+	0.665013i	$0.768426\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−3859.21			−1.78823			−0.894116	−	0.447836i	$-0.852195\pi$
$167$	−3859.21			−1.78823			−0.894116	+	0.447836i	$0.852195\pi$
$168$		0			0
$169$	−602.441			−0.274211
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−755.329	−	1308.27i	−0.331946	−	0.574947i	0.650948	−	0.759123i	$-0.274372\pi$
$173$	−755.329	−	1308.27i	−0.331946	−	0.574947i	−0.982893	+	0.184176i	$0.941038\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1232.31	+	2134.43i	−0.514566	+	0.891255i	0.485291	+	0.874353i	$0.338714\pi$
$179$	−1232.31	+	2134.43i	−0.514566	+	0.891255i	−0.999857	+	0.0169022i	$0.994620\pi$
$180$		0			0
$181$	3297.36			1.35409			0.677047	−	0.735940i	$-0.263259\pi$
$181$	3297.36			1.35409			0.677047	+	0.735940i	$0.263259\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1357.90	+	2351.95i	−0.539647	+	0.934696i
$186$		0			0
$187$	187.883	+	325.424i	0.0734727	+	0.127258i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2364.85	−	4096.05i	−0.895889	−	1.55173i	−0.832700	−	0.553724i	$-0.813206\pi$
$191$	−2364.85	−	4096.05i	−0.895889	−	1.55173i	−0.0631890	−	0.998002i	$-0.520127\pi$
$192$		0			0
$193$	−2276.59	+	3943.17i	−0.849080	+	1.47065i	0.0329498	+	0.999457i	$0.489510\pi$
$193$	−2276.59	+	3943.17i	−0.849080	+	1.47065i	−0.882030	+	0.471193i	$0.843823\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	3109.06			1.12442			0.562212	−	0.826993i	$-0.309950\pi$
$197$	3109.06			1.12442			0.562212	+	0.826993i	$0.309950\pi$
$198$		0			0
$199$	−221.756	+	384.092i	−0.0789943	+	0.136822i	−0.902816	−	0.430027i	$-0.858504\pi$
$199$	−221.756	+	384.092i	−0.0789943	+	0.136822i	0.823822	+	0.566849i	$0.191838\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−293.057	−	507.590i	−0.0998439	−	0.172935i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−3420.56			−1.13208
$210$		0			0
$211$	4653.28			1.51822			0.759112	−	0.650960i	$-0.225633\pi$
$211$	4653.28			1.51822			0.759112	+	0.650960i	$0.225633\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	473.607	+	820.311i	0.150231	+	0.260208i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	198.381	−	343.606i	0.0603826	−	0.104586i
$222$		0			0
$223$	−2778.90			−0.834481			−0.417240	−	0.908796i	$-0.637003\pi$
$223$	−2778.90			−0.834481			−0.417240	+	0.908796i	$0.637003\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1608.61	+	2786.20i	−0.470340	+	0.814653i	−0.999425	−	0.0339159i	$-0.989202\pi$
$227$	−1608.61	+	2786.20i	−0.470340	+	0.814653i	0.529084	+	0.848569i	$0.322536\pi$
$228$		0			0
$229$	−1864.01	−	3228.56i	−0.537891	−	0.931655i	−0.999017	−	0.0443203i	$-0.985888\pi$
$229$	−1864.01	−	3228.56i	−0.537891	−	0.931655i	0.461126	−	0.887335i	$-0.347446\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1801.76	−	3120.73i	−0.506596	−	0.877451i	−0.999971	−	0.00763377i	$-0.997570\pi$
$233$	−1801.76	−	3120.73i	−0.506596	−	0.877451i	0.493374	−	0.869817i	$-0.335763\pi$
$234$		0			0
$235$	−1257.79	+	2178.56i	−0.349146	+	0.604739i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−2348.76			−0.635684			−0.317842	−	0.948144i	$-0.602958\pi$
$239$	−2348.76			−0.635684			−0.317842	+	0.948144i	$0.602958\pi$
$240$		0			0
$241$	−2546.70	+	4411.01i	−0.680693	+	1.17900i	0.294076	+	0.955782i	$0.404988\pi$
$241$	−2546.70	+	4411.01i	−0.680693	+	1.17900i	−0.974770	+	0.223213i	$0.928345\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	1805.84	+	3127.80i	0.465193	+	0.805738i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−5939.02			−1.49350			−0.746749	−	0.665106i	$-0.768386\pi$
$251$	−5939.02			−1.49350			−0.746749	+	0.665106i	$0.768386\pi$
$252$		0			0
$253$	4485.11			1.11453
$254$		0			0
$255$		0			0
$256$		0			0
$257$	757.944	+	1312.80i	0.183966	+	0.318638i	0.943228	−	0.332147i	$-0.107773\pi$
$257$	757.944	+	1312.80i	0.183966	+	0.318638i	−0.759262	+	0.650785i	$0.774440\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−3216.96	+	5571.94i	−0.754244	+	1.30639i	0.191505	+	0.981492i	$0.438663\pi$
$263$	−3216.96	+	5571.94i	−0.754244	+	1.30639i	−0.945749	+	0.324898i	$0.894670\pi$
$264$		0			0
$265$	3293.50			0.763464
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3474.90	−	6018.69i	0.787614	−	1.36419i	−0.139811	−	0.990178i	$-0.544650\pi$
$269$	3474.90	−	6018.69i	0.787614	−	1.36419i	0.927425	−	0.374009i	$-0.122017\pi$
$270$		0			0
$271$	−480.997	−	833.111i	−0.107817	−	0.186745i	0.807068	−	0.590458i	$-0.201053\pi$
$271$	−480.997	−	833.111i	−0.107817	−	0.186745i	−0.914886	+	0.403713i	$0.867719\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1101.68	−	1908.16i	−0.241577	−	0.418424i
$276$		0			0
$277$	−380.005	+	658.188i	−0.0824271	+	0.142768i	−0.904292	−	0.426915i	$-0.859600\pi$
$277$	−380.005	+	658.188i	−0.0824271	+	0.142768i	0.821865	+	0.569682i	$0.192934\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	4412.07			0.936662			0.468331	−	0.883553i	$-0.344855\pi$
$281$	4412.07			0.936662			0.468331	+	0.883553i	$0.344855\pi$
$282$		0			0
$283$	−1301.07	+	2253.53i	−0.273289	+	0.473351i	−0.969702	−	0.244291i	$-0.921445\pi$
$283$	−1301.07	+	2253.53i	−0.273289	+	0.473351i	0.696413	+	0.717641i	$0.254778\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	2407.14	+	4169.29i	0.489953	+	0.848623i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−9332.18			−1.86072			−0.930361	−	0.366644i	$-0.880507\pi$
$293$	−9332.18			−1.86072			−0.930361	+	0.366644i	$0.880507\pi$
$294$		0			0
$295$	4850.26			0.957264
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2367.85	−	4101.24i	−0.457982	−	0.793248i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1360.90	−	2357.15i	0.255492	−	0.442525i
$306$		0			0
$307$	−4895.51			−0.910103			−0.455051	−	0.890465i	$-0.650379\pi$
$307$	−4895.51			−0.910103			−0.455051	+	0.890465i	$0.650379\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−3752.64	+	6499.77i	−0.684221	+	1.18511i	0.289460	+	0.957190i	$0.406524\pi$
$311$	−3752.64	+	6499.77i	−0.684221	+	1.18511i	−0.973681	+	0.227916i	$0.926809\pi$
$312$		0			0
$313$	−3174.64	−	5498.63i	−0.573294	−	0.992974i	−0.996225	−	0.0868124i	$-0.972332\pi$
$313$	−3174.64	−	5498.63i	−0.573294	−	0.992974i	0.422931	−	0.906162i	$-0.361001\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3999.80	−	6927.86i	−0.708679	−	1.22747i	−0.965347	−	0.260968i	$-0.915958\pi$
$317$	−3999.80	−	6927.86i	−0.708679	−	1.22747i	0.256669	−	0.966499i	$-0.417375\pi$
$318$		0			0
$319$	1482.61	−	2567.96i	0.260221	−	0.450716i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	898.666			0.154808
$324$		0			0
$325$	−1163.23	+	2014.78i	−0.198537	+	0.343876i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1060.78	+	1837.32i	0.176150	+	0.305101i	0.940559	−	0.339631i	$-0.110302\pi$
$331$	1060.78	+	1837.32i	0.176150	+	0.305101i	−0.764409	+	0.644732i	$0.776969\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−6074.05			−0.990629
$336$		0			0
$337$	−9114.19			−1.47324			−0.736620	−	0.676307i	$-0.763579\pi$
$337$	−9114.19			−1.47324			−0.736620	+	0.676307i	$0.763579\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1739.87	−	3013.55i	−0.276303	−	0.478572i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−836.534	+	1448.92i	−0.129416	+	0.224156i	−0.923451	−	0.383717i	$-0.874644\pi$
$347$	−836.534	+	1448.92i	−0.129416	+	0.224156i	0.794034	+	0.607873i	$0.207977\pi$
$348$		0			0
$349$	3467.56			0.531845			0.265923	−	0.963994i	$-0.414323\pi$
$349$	3467.56			0.531845			0.265923	+	0.963994i	$0.414323\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1992.12	+	3450.45i	−0.300368	+	0.520252i	−0.976219	−	0.216785i	$-0.930443\pi$
$353$	−1992.12	+	3450.45i	−0.300368	+	0.520252i	0.675851	+	0.737038i	$0.263776\pi$
$354$		0			0
$355$	−2974.78	−	5152.46i	−0.444746	−	0.770322i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1085.47	+	1880.08i	0.159579	+	0.276398i	0.934717	−	0.355394i	$-0.115653\pi$
$359$	1085.47	+	1880.08i	0.159579	+	0.276398i	−0.775138	+	0.631792i	$0.782320\pi$
$360$		0			0
$361$	−660.724	+	1144.41i	−0.0963295	+	0.166848i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	6549.81			0.939268
$366$		0			0
$367$	1796.06	−	3110.86i	0.255459	−	0.442468i	−0.709561	−	0.704644i	$-0.751107\pi$
$367$	1796.06	−	3110.86i	0.255459	−	0.442468i	0.965020	+	0.262176i	$0.0844400\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−3719.90	−	6443.05i	−0.516378	−	0.894393i	−0.999819	−	0.0190163i	$-0.993947\pi$
$373$	−3719.90	−	6443.05i	−0.516378	−	0.894393i	0.483441	−	0.875377i	$-0.339387\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−3130.91			−0.427719
$378$		0			0
$379$	11243.9			1.52390			0.761951	−	0.647635i	$-0.224242\pi$
$379$	11243.9			1.52390			0.761951	+	0.647635i	$0.224242\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1770.62	−	3066.81i	−0.236226	−	0.409156i	0.723402	−	0.690427i	$-0.242577\pi$
$383$	−1770.62	−	3066.81i	−0.236226	−	0.409156i	−0.959628	+	0.281271i	$0.909244\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−5279.10	+	9143.66i	−0.688074	+	1.19178i	0.284386	+	0.958710i	$0.408210\pi$
$389$	−5279.10	+	9143.66i	−0.688074	+	1.19178i	−0.972460	+	0.233070i	$0.925123\pi$
$390$		0			0
$391$	−1178.35			−0.152409
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−4362.08	+	7555.34i	−0.555645	+	0.962406i
$396$		0			0
$397$	−270.287	−	468.151i	−0.0341696	−	0.0591834i	0.848435	−	0.529300i	$-0.177545\pi$
$397$	−270.287	−	468.151i	−0.0341696	−	0.0591834i	−0.882604	+	0.470116i	$0.844212\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1083.52	+	1876.70i	0.134933	+	0.233711i	0.925572	−	0.378572i	$-0.123585\pi$
$401$	1083.52	+	1876.70i	0.134933	+	0.233711i	−0.790639	+	0.612283i	$0.790251\pi$
$402$		0			0
$403$	−1837.09	+	3181.93i	−0.227077	+	0.393308i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−12572.3			−1.53117
$408$		0			0
$409$	−478.845	+	829.384i	−0.0578908	+	0.100270i	−0.893518	−	0.449027i	$-0.851771\pi$
$409$	−478.845	+	829.384i	−0.0578908	+	0.100270i	0.835628	+	0.549296i	$0.185104\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−3703.83	−	6415.22i	−0.438105	−	0.758821i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6464.42			0.753718			0.376859	−	0.926271i	$-0.377004\pi$
$419$	6464.42			0.753718			0.376859	+	0.926271i	$0.377004\pi$
$420$		0			0
$421$	−6201.23			−0.717885			−0.358943	−	0.933360i	$-0.616863\pi$
$421$	−6201.23			−0.717885			−0.358943	+	0.933360i	$0.616863\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	289.439	+	501.322i	0.0330349	+	0.0572181i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	707.796	−	1225.94i	0.0791029	−	0.137010i	−0.823760	−	0.566938i	$-0.808128\pi$
$431$	707.796	−	1225.94i	0.0791029	−	0.137010i	0.902863	+	0.429928i	$0.141461\pi$
$432$		0			0
$433$	−8905.73			−0.988411			−0.494206	−	0.869345i	$-0.664541\pi$
$433$	−8905.73			−0.988411			−0.494206	+	0.869345i	$0.664541\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	5363.19	−	9289.32i	0.587085	−	1.01686i
$438$		0			0
$439$	5438.25	+	9419.33i	0.591238	+	1.02405i	0.994066	+	0.108779i	$0.0346940\pi$
$439$	5438.25	+	9419.33i	0.591238	+	1.02405i	−0.402828	+	0.915276i	$0.631973\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	5701.48	+	9875.25i	0.611479	+	1.05911i	0.990991	+	0.133927i	$0.0427586\pi$
$443$	5701.48	+	9875.25i	0.611479	+	1.05911i	−0.379512	+	0.925187i	$0.623908\pi$
$444$		0			0
$445$	−4547.37	+	7876.28i	−0.484418	+	0.839037i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	12689.0			1.33370			0.666852	−	0.745190i	$-0.267641\pi$
$449$	12689.0			1.33370			0.666852	+	0.745190i	$0.267641\pi$
$450$		0			0
$451$	1356.66	−	2349.80i	0.141646	−	0.245339i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1135.02	−	1965.92i	−0.116180	−	0.201229i	0.802071	−	0.597229i	$-0.203732\pi$
$457$	−1135.02	−	1965.92i	−0.116180	−	0.201229i	−0.918251	+	0.395999i	$0.870398\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	10731.2			1.08417			0.542085	−	0.840324i	$-0.317635\pi$
$461$	10731.2			1.08417			0.542085	+	0.840324i	$0.317635\pi$
$462$		0			0
$463$	−3307.74			−0.332017			−0.166008	−	0.986124i	$-0.553088\pi$
$463$	−3307.74			−0.332017			−0.166008	+	0.986124i	$0.553088\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	4623.35	+	8007.87i	0.458122	+	0.793490i	0.998862	−	0.0476996i	$-0.0151890\pi$
$467$	4623.35	+	8007.87i	0.458122	+	0.793490i	−0.540740	+	0.841190i	$0.681856\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2192.48	+	3797.49i	−0.213130	+	0.369152i
$474$		0			0
$475$	−5269.45			−0.509008
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−7611.66	+	13183.8i	−0.726066	+	1.25758i	0.232468	+	0.972604i	$0.425320\pi$
$479$	−7611.66	+	13183.8i	−0.726066	+	1.25758i	−0.958534	+	0.284978i	$0.908014\pi$
$480$		0			0
$481$	6637.39	+	11496.3i	0.629187	+	1.08978i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	6049.19	+	10477.5i	0.566349	+	0.980946i
$486$		0			0
$487$	3879.36	−	6719.25i	0.360966	−	0.625212i	−0.627154	−	0.778895i	$-0.715780\pi$
$487$	3879.36	−	6719.25i	0.360966	−	0.625212i	0.988120	+	0.153683i	$0.0491136\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	8342.30			0.766767			0.383384	−	0.923589i	$-0.374759\pi$
$491$	8342.30			0.766767			0.383384	+	0.923589i	$0.374759\pi$
$492$		0			0
$493$	−389.520	+	674.668i	−0.0355844	+	0.0616340i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	1293.47	+	2240.36i	0.116040	+	0.200987i	0.918195	−	0.396129i	$-0.129647\pi$
$499$	1293.47	+	2240.36i	0.116040	+	0.200987i	−0.802155	+	0.597116i	$0.796313\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	409.682			0.0363157			0.0181578	−	0.999835i	$-0.494220\pi$
$503$	409.682			0.0363157			0.0181578	+	0.999835i	$0.494220\pi$
$504$		0			0
$505$	−4544.77			−0.400475
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−2195.25	−	3802.29i	−0.191165	−	0.331107i	0.754472	−	0.656333i	$-0.227893\pi$
$509$	−2195.25	−	3802.29i	−0.191165	−	0.331107i	−0.945637	+	0.325225i	$0.894560\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2256.53	−	3908.43i	0.193077	−	0.334419i
$516$		0			0
$517$	−11645.5			−0.990652
$518$		0			0
$519$		0			0
$520$		0			0
$521$	9714.76	−	16826.5i	0.816912	−	1.41493i	−0.0910341	−	0.995848i	$-0.529017\pi$
$521$	9714.76	−	16826.5i	0.816912	−	1.41493i	0.907947	−	0.419086i	$-0.137649\pi$
$522$		0			0
$523$	8974.35	+	15544.0i	0.750326	+	1.29960i	0.947664	+	0.319268i	$0.103437\pi$
$523$	8974.35	+	15544.0i	0.750326	+	1.29960i	−0.197338	+	0.980336i	$0.563230\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	457.109	+	791.735i	0.0377836	+	0.0654431i
$528$		0			0
$529$	−948.833	+	1643.43i	−0.0779841	+	0.135072i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−2864.92			−0.232821
$534$		0			0
$535$	2180.43	−	3776.62i	0.176202	−	0.305192i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−11547.0	−	20000.0i	−0.917641	−	1.58940i	−0.802989	−	0.595994i	$-0.796758\pi$
$541$	−11547.0	−	20000.0i	−0.917641	−	1.58940i	−0.114652	−	0.993406i	$-0.536575\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	8942.47			0.702850
$546$		0			0
$547$	−2266.68			−0.177178			−0.0885889	−	0.996068i	$-0.528236\pi$
$547$	−2266.68			−0.177178			−0.0885889	+	0.996068i	$0.528236\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−3545.75	−	6141.42i	−0.274145	−	0.474834i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−5519.27	+	9559.65i	−0.419854	+	0.727209i	−0.995924	−	0.0901913i	$-0.971252\pi$
$557$	−5519.27	+	9559.65i	−0.419854	+	0.727209i	0.576070	+	0.817400i	$0.304585\pi$
$558$		0			0
$559$	4629.97			0.350316
$560$		0			0
$561$		0			0
$562$		0			0
$563$	3529.68	−	6113.58i	0.264224	−	0.457650i	−0.703136	−	0.711055i	$-0.748217\pi$
$563$	3529.68	−	6113.58i	0.264224	−	0.457650i	0.967360	+	0.253406i	$0.0815508\pi$
$564$		0			0
$565$	−5821.43	−	10083.0i	−0.433468	−	0.750788i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−676.403	−	1171.56i	−0.0498353	−	0.0863172i	0.840032	−	0.542537i	$-0.182536\pi$
$569$	−676.403	−	1171.56i	−0.0498353	−	0.0863172i	−0.889867	+	0.456220i	$0.849203\pi$
$570$		0			0
$571$	10419.6	−	18047.3i	0.763655	−	1.32269i	−0.177300	−	0.984157i	$-0.556736\pi$
$571$	10419.6	−	18047.3i	0.763655	−	1.32269i	0.940955	−	0.338532i	$-0.109930\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	6909.42			0.501117
$576$		0			0
$577$	5007.32	−	8672.92i	0.361278	−	0.625751i	−0.626894	−	0.779105i	$-0.715674\pi$
$577$	5007.32	−	8672.92i	0.361278	−	0.625751i	0.988171	+	0.153353i	$0.0490073\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	7623.34	+	13204.0i	0.541555	+	0.938001i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	20864.8			1.46709			0.733546	−	0.679640i	$-0.237864\pi$
$587$	20864.8			1.46709			0.733546	+	0.679640i	$0.237864\pi$
$588$		0			0
$589$	−8322.01			−0.582177
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−8773.84	−	15196.7i	−0.607586	−	1.05237i	−0.991637	−	0.129058i	$-0.958805\pi$
$593$	−8773.84	−	15196.7i	−0.607586	−	1.05237i	0.384051	−	0.923312i	$-0.374529\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−65.3409	+	113.174i	−0.00445703	+	0.00771980i	−0.868245	−	0.496135i	$-0.834752\pi$
$599$	−65.3409	+	113.174i	−0.00445703	+	0.00771980i	0.863788	+	0.503855i	$0.168085\pi$
$600$		0			0
$601$	5964.47			0.404818			0.202409	−	0.979301i	$-0.435123\pi$
$601$	5964.47			0.404818			0.202409	+	0.979301i	$0.435123\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−405.472	+	702.298i	−0.0272476	+	0.0471942i
$606$		0			0
$607$	−4955.82	−	8583.73i	−0.331384	−	0.573975i	0.651399	−	0.758735i	$-0.274182\pi$
$607$	−4955.82	−	8583.73i	−0.331384	−	0.573975i	−0.982784	+	0.184761i	$0.940849\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	6148.07	+	10648.8i	0.407077	+	0.705079i
$612$		0			0
$613$	3241.26	−	5614.03i	0.213562	−	0.369900i	−0.739265	−	0.673415i	$-0.764827\pi$
$613$	3241.26	−	5614.03i	0.213562	−	0.369900i	0.952827	+	0.303515i	$0.0981602\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−7189.36			−0.469097			−0.234548	−	0.972104i	$-0.575361\pi$
$617$	−7189.36			−0.469097			−0.234548	+	0.972104i	$0.575361\pi$
$618$		0			0
$619$	930.696	−	1612.01i	0.0604327	−	0.104672i	−0.834226	−	0.551422i	$-0.814085\pi$
$619$	930.696	−	1612.01i	0.0604327	−	0.104672i	0.894659	+	0.446750i	$0.147419\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	2474.04	+	4285.16i	0.158338	+	0.274250i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	3303.06			0.209383
$630$		0			0
$631$	−29032.0			−1.83161			−0.915803	−	0.401627i	$-0.868445\pi$
$631$	−29032.0			−1.83161			−0.915803	+	0.401627i	$0.868445\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	3212.56	+	5564.32i	0.200766	+	0.347737i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	12228.9	−	21181.0i	0.753527	−	1.30515i	−0.192576	−	0.981282i	$-0.561684\pi$
$641$	12228.9	−	21181.0i	0.753527	−	1.30515i	0.946103	−	0.323866i	$-0.104983\pi$
$642$		0			0
$643$	−10968.2			−0.672695			−0.336348	−	0.941738i	$-0.609192\pi$
$643$	−10968.2			−0.672695			−0.336348	+	0.941738i	$0.609192\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−2742.37	+	4749.92i	−0.166636	+	0.288622i	−0.937235	−	0.348698i	$-0.886624\pi$
$647$	−2742.37	+	4749.92i	−0.166636	+	0.288622i	0.770599	+	0.637320i	$0.219957\pi$
$648$		0			0
$649$	11226.7	+	19445.2i	0.679025	+	1.17611i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−5164.98	−	8946.01i	−0.309527	−	0.536117i	0.668732	−	0.743504i	$-0.266837\pi$
$653$	−5164.98	−	8946.01i	−0.309527	−	0.536117i	−0.978259	+	0.207387i	$0.933504\pi$
$654$		0			0
$655$	113.270	−	196.189i	0.00675698	−	0.0117034i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−26822.2			−1.58550			−0.792751	−	0.609546i	$-0.791352\pi$
$659$	−26822.2			−1.58550			−0.792751	+	0.609546i	$0.791352\pi$
$660$		0			0
$661$	1249.38	−	2163.99i	0.0735177	−	0.127336i	−0.826923	−	0.562315i	$-0.809911\pi$
$661$	1249.38	−	2163.99i	0.0735177	−	0.127336i	0.900441	+	0.434979i	$0.143244\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	4649.27	+	8052.77i	0.269896	+	0.467473i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	12600.1			0.724922
$672$		0			0
$673$	10092.1			0.578044			0.289022	−	0.957322i	$-0.406670\pi$
$673$	10092.1			0.578044			0.289022	+	0.957322i	$0.406670\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	13183.2	+	22833.9i	0.748406	+	1.29628i	0.948586	+	0.316518i	$0.102514\pi$
$677$	13183.2	+	22833.9i	0.748406	+	1.29628i	−0.200181	+	0.979759i	$0.564153\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	11285.4	−	19546.9i	0.632245	−	1.09508i	−0.354847	−	0.934924i	$-0.615467\pi$
$683$	11285.4	−	19546.9i	0.632245	−	1.09508i	0.987092	−	0.160155i	$-0.0511996\pi$
$684$		0			0
$685$	19304.1			1.07675
$686$		0			0
$687$		0			0
$688$		0			0
$689$	8049.28	−	13941.8i	0.445070	−	0.770884i
$690$		0			0
$691$	−5965.75	−	10333.0i	−0.328434	−	0.568864i	0.653768	−	0.756695i	$-0.273187\pi$
$691$	−5965.75	−	10333.0i	−0.328434	−	0.568864i	−0.982201	+	0.187832i	$0.939854\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	10266.0	+	17781.2i	0.560304	+	0.970475i
$696$		0			0
$697$	−356.428	+	617.352i	−0.0193697	+	0.0335493i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−16592.6			−0.893997			−0.446999	−	0.894535i	$-0.647507\pi$
$701$	−16592.6			−0.893997			−0.446999	+	0.894535i	$0.647507\pi$
$702$		0			0
$703$	−15033.7	+	26039.1i	−0.806553	+	1.39699i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−9357.00	−	16206.8i	−0.495641	−	0.858476i	0.504346	−	0.863502i	$-0.331734\pi$
$709$	−9357.00	−	16206.8i	−0.495641	−	0.858476i	−0.999987	+	0.00502575i	$0.998400\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	10912.0			0.573152
$714$		0			0
$715$	12337.3			0.645298
$716$		0			0
$717$		0			0
$718$		0			0
$719$	12427.4	+	21524.9i	0.644594	+	1.11647i	0.984395	+	0.175972i	$0.0563070\pi$
$719$	12427.4	+	21524.9i	0.644594	+	1.11647i	−0.339801	+	0.940497i	$0.610360\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2284.00	−	3956.01i	0.117001	−	0.202652i
$726$		0			0
$727$	22506.0			1.14814			0.574071	−	0.818805i	$-0.305363\pi$
$727$	22506.0			1.14814			0.574071	+	0.818805i	$0.305363\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	576.020	−	997.696i	0.0291448	−	0.0504803i
$732$		0			0
$733$	−11896.0	−	20604.5i	−0.599440	−	1.03826i	−0.992904	−	0.118920i	$-0.962057\pi$
$733$	−11896.0	−	20604.5i	−0.599440	−	1.03826i	0.393464	−	0.919340i	$-0.371277\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−14059.4	−	24351.6i	−0.702692	−	1.21710i
$738$		0			0
$739$	−2401.62	+	4159.72i	−0.119547	+	0.207061i	−0.919588	−	0.392884i	$-0.871477\pi$
$739$	−2401.62	+	4159.72i	−0.119547	+	0.207061i	0.800041	+	0.599945i	$0.204811\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−5076.10			−0.250638			−0.125319	−	0.992116i	$-0.539995\pi$
$743$	−5076.10			−0.250638			−0.125319	+	0.992116i	$0.539995\pi$
$744$		0			0
$745$	9255.24	−	16030.6i	0.455149	−	0.788341i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	4579.36	+	7931.68i	0.222508	+	0.385394i	0.955569	−	0.294768i	$-0.0952425\pi$
$751$	4579.36	+	7931.68i	0.222508	+	0.385394i	−0.733061	+	0.680163i	$0.761909\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	2313.14			0.111502
$756$		0			0
$757$	33682.2			1.61717			0.808587	−	0.588377i	$-0.200233\pi$
$757$	33682.2			1.61717			0.808587	+	0.588377i	$0.200233\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−6673.14	−	11558.2i	−0.317873	−	0.550571i	0.662171	−	0.749352i	$-0.269635\pi$
$761$	−6673.14	−	11558.2i	−0.317873	−	0.550571i	−0.980044	+	0.198781i	$0.936302\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	11854.0	−	20531.7i	0.558048	−	0.966568i
$768$		0			0
$769$	15530.6			0.728281			0.364140	−	0.931344i	$-0.381363\pi$
$769$	15530.6			0.728281			0.364140	+	0.931344i	$0.381363\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	5507.62	−	9539.47i	0.256268	−	0.443869i	−0.708971	−	0.705238i	$-0.750840\pi$
$773$	5507.62	−	9539.47i	0.256268	−	0.443869i	0.965239	+	0.261368i	$0.0841737\pi$
$774$		0			0
$775$	−2680.32	−	4642.45i	−0.124232	−	0.215176i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−3244.52	−	5619.67i	−0.149226	−	0.258467i
$780$		0			0
$781$	13771.2	−	23852.4i	0.630951	−	1.09284i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1573.27			−0.0715317
$786$		0			0
$787$	10596.3	−	18353.3i	0.479946	−	0.831291i	−0.519789	−	0.854294i	$-0.673990\pi$
$787$	10596.3	−	18353.3i	0.479946	−	0.831291i	0.999735	+	0.0230036i	$0.00732292\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−6652.07	−	11521.7i	−0.297884	−	0.515950i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	18189.9			0.808432			0.404216	−	0.914663i	$-0.367544\pi$
$797$	18189.9			0.808432			0.404216	+	0.914663i	$0.367544\pi$
$798$		0			0
$799$	3059.56			0.135468
$800$		0			0
$801$		0			0
$802$		0			0
$803$	15160.6	+	26259.0i	0.666260	+	1.15400i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	22758.3	−	39418.5i	0.989045	−	1.71308i	0.366688	−	0.930344i	$-0.380492\pi$
$809$	22758.3	−	39418.5i	0.989045	−	1.71308i	0.622358	−	0.782733i	$-0.286175\pi$
$810$		0			0
$811$	42099.9			1.82285			0.911423	−	0.411472i	$-0.134985\pi$
$811$	42099.9			1.82285			0.911423	+	0.411472i	$0.134985\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	3442.72	−	5962.97i	0.147967	−	0.256287i
$816$		0			0
$817$	5243.44	+	9081.90i	0.224535	+	0.388905i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−13278.0	−	22998.1i	−0.564439	−	0.977637i	−0.997102	−	0.0760810i	$-0.975759\pi$
$821$	−13278.0	−	22998.1i	−0.564439	−	0.977637i	0.432663	−	0.901556i	$-0.357574\pi$
$822$		0			0
$823$	777.124	−	1346.02i	0.0329147	−	0.0570100i	−0.849099	−	0.528234i	$-0.822854\pi$
$823$	777.124	−	1346.02i	0.0329147	−	0.0570100i	0.882014	+	0.471224i	$0.156188\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	10548.3			0.443531			0.221766	−	0.975100i	$-0.428818\pi$
$827$	10548.3			0.443531			0.221766	+	0.975100i	$0.428818\pi$
$828$		0			0
$829$	2165.63	−	3750.98i	0.0907303	−	0.157150i	−0.817088	−	0.576513i	$-0.804413\pi$
$829$	2165.63	−	3750.98i	0.0907303	−	0.157150i	0.907819	+	0.419363i	$0.137747\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	15763.7	+	27303.6i	0.653325	+	1.13159i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−23557.7			−0.969369			−0.484685	−	0.874689i	$-0.661066\pi$
$839$	−23557.7			−0.969369			−0.484685	+	0.874689i	$0.661066\pi$
$840$		0			0
$841$	−18241.5			−0.747939
$842$		0			0
$843$		0			0
$844$		0			0
$845$	2460.79	+	4262.22i	0.100182	+	0.173520i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	19712.5	−	34143.1i	0.794049	−	1.37533i
$852$		0			0
$853$	11493.6			0.461350			0.230675	−	0.973031i	$-0.425907\pi$
$853$	11493.6			0.461350			0.230675	+	0.973031i	$0.425907\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1267.11	−	2194.71i	0.0505062	−	0.0874792i	−0.839667	−	0.543101i	$-0.817250\pi$
$857$	1267.11	−	2194.71i	0.0505062	−	0.0874792i	0.890173	+	0.455622i	$0.150583\pi$
$858$		0			0
$859$	1563.54	+	2708.14i	0.0621041	+	0.107567i	0.895406	−	0.445251i	$-0.146886\pi$
$859$	1563.54	+	2708.14i	0.0621041	+	0.107567i	−0.833302	+	0.552819i	$0.813552\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	7654.94	+	13258.7i	0.301943	+	0.522981i	0.976576	−	0.215172i	$-0.0690313\pi$
$863$	7654.94	+	13258.7i	0.301943	+	0.522981i	−0.674633	+	0.738153i	$0.735698\pi$
$864$		0			0
$865$	−6170.59	+	10687.8i	−0.242551	+	0.420110i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−40387.0			−1.57656
$870$		0			0
$871$	−14844.9	+	25712.2i	−0.577498	+	1.00026i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	21210.3	+	36737.4i	0.816673	+	1.41452i	0.908120	+	0.418710i	$0.137518\pi$
$877$	21210.3	+	36737.4i	0.816673	+	1.41452i	−0.0914467	+	0.995810i	$0.529149\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	10060.8			0.384740			0.192370	−	0.981322i	$-0.438383\pi$
$881$	10060.8			0.384740			0.192370	+	0.981322i	$0.438383\pi$
$882$		0			0
$883$	17437.8			0.664585			0.332293	−	0.943176i	$-0.392178\pi$
$883$	17437.8			0.664585			0.332293	+	0.943176i	$0.392178\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	8008.33	+	13870.8i	0.303149	+	0.525070i	0.976848	−	0.213937i	$-0.0686286\pi$
$887$	8008.33	+	13870.8i	0.303149	+	0.525070i	−0.673698	+	0.739007i	$0.735295\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−13925.4	+	24119.5i	−0.521831	+	0.903838i
$894$		0			0
$895$	20134.5			0.751981
$896$		0			0
$897$		0			0
$898$		0			0
$899$	3607.11	−	6247.70i	0.133820	−	0.231782i
$900$		0			0
$901$	−2002.84	−	3469.03i	−0.0740559	−	0.128269i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−13468.7	−	23328.5i	−0.494714	−	0.856870i
$906$		0			0
$907$	5065.50	−	8773.70i	0.185443	−	0.321197i	−0.758282	−	0.651926i	$-0.773961\pi$
$907$	5065.50	−	8773.70i	0.185443	−	0.321197i	0.943726	+	0.330729i	$0.107295\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1320.58			−0.0480273			−0.0240137	−	0.999712i	$-0.507645\pi$
$911$	−1320.58			−0.0480273			−0.0240137	+	0.999712i	$0.507645\pi$
$912$		0			0
$913$	17146.2	−	29698.2i	0.621531	−	1.07652i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	3642.88	+	6309.66i	0.130759	+	0.226481i	0.923969	−	0.382466i	$-0.124925\pi$
$919$	3642.88	+	6309.66i	0.130759	+	0.226481i	−0.793210	+	0.608948i	$0.791592\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−29081.3			−1.03708
$924$		0			0
$925$	−19368.0			−0.688448
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−11854.0	−	20531.7i	−0.418640	−	0.725106i	0.577163	−	0.816629i	$-0.304160\pi$
$929$	−11854.0	−	20531.7i	−0.418640	−	0.725106i	−0.995803	+	0.0915233i	$0.970826\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1534.90	−	2658.52i	0.0536860	−	0.0929869i
$936$		0			0
$937$	8853.60			0.308682			0.154341	−	0.988018i	$-0.450675\pi$
$937$	8853.60			0.308682			0.154341	+	0.988018i	$0.450675\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	26985.8	−	46740.7i	0.934869	−	1.61924i	0.160000	−	0.987117i	$-0.448850\pi$
$941$	26985.8	−	46740.7i	0.934869	−	1.61924i	0.774868	−	0.632123i	$-0.217816\pi$
$942$		0			0
$943$	4254.29	+	7368.64i	0.146913	+	0.254460i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	4009.90	+	6945.35i	0.137597	+	0.238325i	0.926586	−	0.376082i	$-0.122729\pi$
$947$	4009.90	+	6945.35i	0.137597	+	0.238325i	−0.788990	+	0.614407i	$0.789396\pi$
$948$		0			0
$949$	16007.7	−	27726.1i	0.547557	−	0.948397i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−42628.0			−1.44896			−0.724479	−	0.689297i	$-0.757920\pi$
$953$	−42628.0			−1.44896			−0.724479	+	0.689297i	$0.757920\pi$
$954$		0			0
$955$	−19319.5	+	33462.3i	−0.654621	+	1.13384i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	10662.5	+	18468.0i	0.357910	+	0.619918i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	37196.8			1.24084
$966$		0			0
$967$	−43161.7			−1.43535			−0.717676	−	0.696377i	$-0.754794\pi$
$967$	−43161.7			−1.43535			−0.717676	+	0.696377i	$0.754794\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	19887.7	+	34446.5i	0.657288	+	1.13846i	0.981315	+	0.192408i	$0.0616297\pi$
$971$	19887.7	+	34446.5i	0.657288	+	1.13846i	−0.324027	+	0.946048i	$0.605037\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−13329.0	+	23086.6i	−0.436472	+	0.755992i	−0.997415	−	0.0718626i	$-0.977106\pi$
$977$	−13329.0	+	23086.6i	−0.436472	+	0.755992i	0.560942	+	0.827855i	$0.310439\pi$
$978$		0			0
$979$	−42102.6			−1.37447
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1471.21	−	2548.22i	0.0477359	−	0.0826810i	−0.841170	−	0.540771i	$-0.818133\pi$
$983$	1471.21	−	2548.22i	0.0477359	−	0.0826810i	0.888906	+	0.458089i	$0.151466\pi$
$984$		0			0
$985$	−12699.6	−	21996.4i	−0.410805	−	0.711535i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−6875.31	−	11908.4i	−0.221054	−	0.382876i
$990$		0			0
$991$	−22484.9	+	38945.0i	−0.720743	+	1.24836i	0.239960	+	0.970783i	$0.422866\pi$
$991$	−22484.9	+	38945.0i	−0.720743	+	1.24836i	−0.960703	+	0.277580i	$0.910468\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	3623.23			0.115441
$996$		0			0
$997$	13558.2	−	23483.5i	0.430685	−	0.745968i	−0.566247	−	0.824235i	$-0.691605\pi$
$997$	13558.2	−	23483.5i	0.430685	−	0.745968i	0.996932	+	0.0782670i	$0.0249387\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.4.k.bb.361.2		8
3.2	odd	2		588.4.i.l.361.3		8
7.2	even	3	inner	1764.4.k.bb.1549.2		8
7.3	odd	6		1764.4.a.ba.1.2		4
7.4	even	3		1764.4.a.bc.1.3		4
7.5	odd	6		1764.4.k.bd.1549.3		8
7.6	odd	2		1764.4.k.bd.361.3		8
21.2	odd	6		588.4.i.l.373.3		8
21.5	even	6		588.4.i.k.373.2		8
21.11	odd	6		588.4.a.j.1.2	✓	4
21.17	even	6		588.4.a.k.1.3	yes	4
21.20	even	2		588.4.i.k.361.2		8
84.11	even	6		2352.4.a.cq.1.2		4
84.59	odd	6		2352.4.a.cl.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.4.a.j.1.2	✓	4	21.11	odd	6
588.4.a.k.1.3	yes	4	21.17	even	6
588.4.i.k.361.2		8	21.20	even	2
588.4.i.k.373.2		8	21.5	even	6
588.4.i.l.361.3		8	3.2	odd	2
588.4.i.l.373.3		8	21.2	odd	6
1764.4.a.ba.1.2		4	7.3	odd	6
1764.4.a.bc.1.3		4	7.4	even	3
1764.4.k.bb.361.2		8	1.1	even	1	trivial
1764.4.k.bb.1549.2		8	7.2	even	3	inner
1764.4.k.bd.361.3		8	7.6	odd	2
1764.4.k.bd.1549.3		8	7.5	odd	6
2352.4.a.cl.1.3		4	84.59	odd	6
2352.4.a.cq.1.2		4	84.11	even	6

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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.k (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.08470 - 7.07491i) q^{5} +O(q^{10})\)	\(q+(-4.08470 - 7.07491i) q^{5} +(18.9094 - 32.7521i) q^{11} -39.9319 q^{13} +(-4.96798 + 8.60480i) q^{17} +(-45.2229 - 78.3284i) q^{19} +(59.2973 + 102.706i) q^{23} +(29.1304 - 50.4554i) q^{25} +78.4061 q^{29} +(46.0055 - 79.6838i) q^{31} +(-166.218 - 287.897i) q^{37} +71.7451 q^{41} -115.947 q^{43} +(-153.964 - 266.673i) q^{47} +(-201.575 + 349.138i) q^{53} -308.958 q^{55} +(-296.855 + 514.168i) q^{59} +(166.585 + 288.534i) q^{61} +(163.110 + 282.515i) q^{65} +(371.756 - 643.900i) q^{67} +728.272 q^{71} +(-400.874 + 694.335i) q^{73} +(-533.953 - 924.833i) q^{79} +906.756 q^{83} +81.1709 q^{85} +(-556.634 - 964.119i) q^{89} +(-369.444 + 639.896i) q^{95} -1480.94 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 48 q^{17} + 192 q^{19} + 192 q^{23} - 324 q^{25} - 192 q^{29} + 48 q^{31} - 256 q^{37} + 2016 q^{41} - 224 q^{43} - 864 q^{47} - 648 q^{53} - 4704 q^{55} - 336 q^{59} + 960 q^{61} - 360 q^{65} - 720 q^{67} + 2688 q^{71} + 672 q^{73} + 1984 q^{79} + 6240 q^{83} + 1360 q^{85} - 2160 q^{89} - 3744 q^{95} - 4032 q^{97}+O(q^{100}) \) 8 * q - 48 * q^17 + 192 * q^19 + 192 * q^23 - 324 * q^25 - 192 * q^29 + 48 * q^31 - 256 * q^37 + 2016 * q^41 - 224 * q^43 - 864 * q^47 - 648 * q^53 - 4704 * q^55 - 336 * q^59 + 960 * q^61 - 360 * q^65 - 720 * q^67 + 2688 * q^71 + 672 * q^73 + 1984 * q^79 + 6240 * q^83 + 1360 * q^85 - 2160 * q^89 - 3744 * q^95 - 4032 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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