LMFDB - Embedded newform 324.4.b.a.323.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,4,Mod(323,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("324.323");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	324.b (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:	$19.1166188419$
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} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			323.4
Root			$0.517638i$ of defining polynomial
Character	$\chi$	$=$	324.323
Dual form			324.4.b.a.323.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843i q^{2} -8.00000 q^{4} +1.83032i q^{5} -22.6274i q^{8} -5.17691 q^{10} -30.4115 q^{13} +64.0000 q^{16} -117.714i q^{17} -14.6425i q^{20} +121.650 q^{25} -86.0168i q^{26} -199.075i q^{29} +181.019i q^{32} +332.946 q^{34} +449.946 q^{37} +41.4153 q^{40} +171.120i q^{41} +343.000 q^{49} +344.078i q^{50} +243.292 q^{52} -770.746i q^{53} +563.069 q^{58} -820.300 q^{61} -512.000 q^{64} -55.6627i q^{65} +941.714i q^{68} +1246.90 q^{73} +1272.64i q^{74} +117.140i q^{80} -484.000 q^{82} +215.454 q^{85} -262.225i q^{89} -1816.00 q^{97} +970.151i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{4} + 104 q^{10} - 184 q^{13} + 256 q^{16} - 324 q^{25} - 40 q^{34} + 428 q^{37} - 832 q^{40} + 1372 q^{49} + 1472 q^{52} - 616 q^{58} - 1660 q^{61} - 2048 q^{64} + 1184 q^{73} - 1936 q^{82}+ \cdots - 7264 q^{97}+O(q^{100}) $ 4 * q - 32 * q^4 + 104 * q^10 - 184 * q^13 + 256 * q^16 - 324 * q^25 - 40 * q^34 + 428 * q^37 - 832 * q^40 + 1372 * q^49 + 1472 * q^52 - 616 * q^58 - 1660 * q^61 - 2048 * q^64 + 1184 * q^73 - 1936 * q^82 + 5476 * q^85 - 7264 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$-1$	$-1$

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 $

$2$	2.82843i	1.00000i
$2$	2.82843i	1.00000i
$3$	0	0
$3$	0	0
$4$	−8.00000	−1.00000
$5$	1.83032i	0.163708i	0.996644	+	0.0818542i	$0.0260842\pi$
$5$	1.83032i	0.163708i	−0.996644	+	0.0818542i	$0.973916\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	− 22.6274i	− 1.00000i
$9$	0	0
$10$	−5.17691	−0.163708
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−30.4115	−0.648819	−0.324409	−	0.945917i	$-0.605166\pi$
$13$	−30.4115	−0.648819	−0.324409	+	0.945917i	$0.605166\pi$
$14$	0	0
$15$	0	0
$16$	64.0000	1.00000
$17$	− 117.714i	− 1.67941i	−0.543047	−	0.839703i	$-0.682729\pi$
$17$	− 117.714i	− 1.67941i	0.543047	−	0.839703i	$-0.317271\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	− 14.6425i	− 0.163708i
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	121.650	0.973200
$26$	− 86.0168i	− 0.648819i
$27$	0	0
$28$	0	0
$29$	− 199.075i	− 1.27473i	−0.770560	−	0.637367i	$-0.780023\pi$
$29$	− 199.075i	− 1.27473i	0.770560	−	0.637367i	$-0.219977\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	181.019i	1.00000i
$33$	0	0
$34$	332.946	1.67941
$35$	0	0
$36$	0	0
$37$	449.946	1.99921	0.999604	−	0.0281490i	$-0.00896130\pi$
$37$	449.946	1.99921	0.999604	+	0.0281490i	$0.00896130\pi$
$38$	0	0
$39$	0	0
$40$	41.4153	0.163708
$41$	171.120i	0.651815i	0.945402	+	0.325908i	$0.105670\pi$
$41$	171.120i	0.651815i	−0.945402	+	0.325908i	$0.894330\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	343.000	1.00000
$50$	344.078i	0.973200i
$51$	0	0
$52$	243.292	0.648819
$53$	− 770.746i	− 1.99755i	−0.0494806	−	0.998775i	$-0.515757\pi$
$53$	− 770.746i	− 1.99755i	0.0494806	−	0.998775i	$-0.484243\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	563.069	1.27473
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−820.300	−1.72178	−0.860890	−	0.508790i	$-0.830093\pi$
$61$	−820.300	−1.72178	−0.860890	+	0.508790i	$0.830093\pi$
$62$	0	0
$63$	0	0
$64$	−512.000	−1.00000
$65$	− 55.6627i	− 0.106217i
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	941.714i	1.67941i
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	1246.90	1.99915	0.999576	−	0.0291103i	$-0.00926742\pi$
$73$	1246.90	1.99915	0.999576	+	0.0291103i	$0.00926742\pi$
$74$	1272.64i	1.99921i
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	117.140i	0.163708i
$81$	0	0
$82$	−484.000	−0.651815
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	215.454	0.274933
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 262.225i	− 0.312312i	−0.987732	−	0.156156i	$-0.950090\pi$
$89$	− 262.225i	− 0.312312i	0.987732	−	0.156156i	$-0.0499103\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1816.00	−1.90090	−0.950448	−	0.310884i	$-0.899375\pi$
$97$	−1816.00	−1.90090	−0.950448	+	0.310884i	$0.899375\pi$
$98$	970.151i	1.00000i
$99$	0	0
$100$	−973.200	−0.973200
$101$	− 948.937i	− 0.934879i	−0.884025	−	0.467440i	$-0.845177\pi$
$101$	− 948.937i	− 0.934879i	0.884025	−	0.467440i	$-0.154823\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	688.135i	0.648819i
$105$	0	0
$106$	2180.00	1.99755
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	782.080	0.687245	0.343623	−	0.939108i	$-0.388346\pi$
$109$	782.080	0.687245	0.343623	+	0.939108i	$0.388346\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 2277.50i	− 1.89601i	−0.318261	−	0.948003i	$-0.603099\pi$
$113$	− 2277.50i	− 1.89601i	0.318261	−	0.948003i	$-0.396901\pi$
$114$	0	0
$115$	0	0
$116$	1592.60i	1.27473i
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1331.00	−1.00000
$122$	− 2320.16i	− 1.72178i
$123$	0	0
$124$	0	0
$125$	451.447i	0.323029i
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	− 1448.15i	− 1.00000i
$129$	0	0
$130$	157.438	0.106217
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	−2663.57	−1.67941
$137$	− 2265.75i	− 1.41296i	−0.707732	−	0.706481i	$-0.750282\pi$
$137$	− 2265.75i	− 1.41296i	0.707732	−	0.706481i	$-0.249718\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	364.370	0.208685
$146$	3526.75i	1.99915i
$147$	0	0
$148$	−3599.57	−1.99921
$149$	− 3150.97i	− 1.73247i	−0.499638	−	0.866234i	$-0.666534\pi$
$149$	− 3150.97i	− 1.73247i	0.499638	−	0.866234i	$-0.333466\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−3255.28	−1.65478	−0.827388	−	0.561630i	$-0.810174\pi$
$157$	−3255.28	−1.65478	−0.827388	+	0.561630i	$0.810174\pi$
$158$	0	0
$159$	0	0
$160$	−331.323	−0.163708
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	− 1368.96i	− 0.651815i
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−1272.14	−0.579034
$170$	609.396i	0.274933i
$171$	0	0
$172$	0	0
$173$	3422.17i	1.50394i	0.659195	+	0.751972i	$0.270897\pi$
$173$	3422.17i	1.50394i	−0.659195	+	0.751972i	$0.729103\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	741.684	0.312312
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−2860.00	−1.17449	−0.587243	−	0.809410i	$-0.699787\pi$
$181$	−2860.00	−1.17449	−0.587243	+	0.809410i	$0.699787\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	823.543i	0.327287i
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	2743.35	1.02317	0.511583	−	0.859234i	$-0.329059\pi$
$193$	2743.35	1.02317	0.511583	+	0.859234i	$0.329059\pi$
$194$	− 5136.42i	− 1.90090i
$195$	0	0
$196$	−2744.00	−1.00000
$197$	264.661i	0.0957174i	0.998854	+	0.0478587i	$0.0152397\pi$
$197$	264.661i	0.0957174i	−0.998854	+	0.0478587i	$0.984760\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	− 2752.62i	− 0.973200i
$201$	0	0
$202$	2684.00	0.934879
$203$	0	0
$204$	0	0
$205$	−313.203	−0.106708
$206$	0	0
$207$	0	0
$208$	−1946.34	−0.648819
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	6165.97i	1.99755i
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	2212.06i	0.687245i
$219$	0	0
$220$	0	0
$221$	3579.87i	1.08963i
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	0	0
$226$	6441.73	1.89601
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	4191.90	1.20964	0.604822	−	0.796360i	$-0.293244\pi$
$229$	4191.90	1.20964	0.604822	+	0.796360i	$0.293244\pi$
$230$	0	0
$231$	0	0
$232$	−4504.55	−1.27473
$233$	7001.26i	1.96853i	0.176698	+	0.984265i	$0.443458\pi$
$233$	7001.26i	1.96853i	−0.176698	+	0.984265i	$0.556542\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	7234.59	1.93370	0.966849	−	0.255349i	$-0.0821903\pi$
$241$	7234.59	1.93370	0.966849	+	0.255349i	$0.0821903\pi$
$242$	− 3764.64i	− 1.00000i
$243$	0	0
$244$	6562.40	1.72178
$245$	627.798i	0.163708i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	−1276.89	−0.323029
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	4096.00	1.00000
$257$	8217.16i	1.99445i	0.0744739	+	0.997223i	$0.476272\pi$
$257$	8217.16i	1.99445i	−0.0744739	+	0.997223i	$0.523728\pi$
$258$	0	0
$259$	0	0
$260$	445.302i	0.106217i
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	1410.71	0.327016
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 6981.10i	− 1.58232i	−0.611607	−	0.791162i	$-0.709477\pi$
$269$	− 6981.10i	− 1.58232i	0.611607	−	0.791162i	$-0.290523\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	− 7533.71i	− 1.67941i
$273$	0	0
$274$	6408.50	1.41296
$275$	0	0
$276$	0	0
$277$	1316.00	0.285454	0.142727	−	0.989762i	$-0.454413\pi$
$277$	1316.00	0.285454	0.142727	+	0.989762i	$0.454413\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5677.75i	1.20536i	0.797983	+	0.602680i	$0.205900\pi$
$281$	5677.75i	1.20536i	−0.797983	+	0.602680i	$0.794100\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−8943.63	−1.82040
$290$	1030.59i	0.208685i
$291$	0	0
$292$	−9975.17	−1.99915
$293$	5771.93i	1.15085i	0.817853	+	0.575427i	$0.195164\pi$
$293$	5771.93i	1.15085i	−0.817853	+	0.575427i	$0.804836\pi$
$294$	0	0
$295$	0	0
$296$	− 10181.1i	− 1.99921i
$297$	0	0
$298$	8912.30	1.73247
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 1501.41i	− 0.281870i
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	10963.8	1.97991	0.989954	−	0.141388i	$-0.0451565\pi$
$313$	10963.8	1.97991	0.989954	+	0.141388i	$0.0451565\pi$
$314$	− 9207.33i	− 1.65478i
$315$	0	0
$316$	0	0
$317$	1857.56i	0.329120i	0.986367	+	0.164560i	$0.0526204\pi$
$317$	1857.56i	0.329120i	−0.986367	+	0.164560i	$0.947380\pi$
$318$	0	0
$319$	0	0
$320$	− 937.122i	− 0.163708i
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−3699.56	−0.631430
$326$	0	0
$327$	0	0
$328$	3872.00	0.651815
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	416.000	0.0672432	0.0336216	−	0.999435i	$-0.489296\pi$
$337$	416.000	0.0672432	0.0336216	+	0.999435i	$0.489296\pi$
$338$	− 3598.15i	− 0.579034i
$339$	0	0
$340$	−1723.63	−0.274933
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−9679.35	−1.50394
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	9470.00	1.45249	0.726243	−	0.687438i	$-0.241265\pi$
$349$	9470.00	1.45249	0.726243	+	0.687438i	$0.241265\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 6752.87i	− 1.01818i	−0.860712	−	0.509092i	$-0.829981\pi$
$353$	− 6752.87i	− 1.01818i	0.860712	−	0.509092i	$-0.170019\pi$
$354$	0	0
$355$	0	0
$356$	2097.80i	0.312312i
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	6859.00	1.00000
$362$	− 8089.30i	− 1.17449i
$363$	0	0
$364$	0	0
$365$	2282.21i	0.327278i
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	−2329.33	−0.327287
$371$	0	0
$372$	0	0
$373$	−12922.0	−1.79377	−0.896884	−	0.442265i	$-0.854175\pi$
$373$	−12922.0	−1.79377	−0.896884	+	0.442265i	$0.854175\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6054.18i	0.827072i
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	7759.38i	1.02317i
$387$	0	0
$388$	14528.0	1.90090
$389$	10582.6i	1.37932i	0.724131	+	0.689662i	$0.242241\pi$
$389$	10582.6i	1.37932i	−0.724131	+	0.689662i	$0.757759\pi$
$390$	0	0
$391$	0	0
$392$	− 7761.20i	− 1.00000i
$393$	0	0
$394$	−748.575	−0.0957174
$395$	0	0
$396$	0	0
$397$	−15687.7	−1.98324	−0.991619	−	0.129199i	$-0.958759\pi$
$397$	−15687.7	−1.98324	−0.991619	+	0.129199i	$0.958759\pi$
$398$	0	0
$399$	0	0
$400$	7785.60	0.973200
$401$	− 14718.3i	− 1.83290i	−0.400145	−	0.916452i	$-0.631040\pi$
$401$	− 14718.3i	− 1.83290i	0.400145	−	0.916452i	$-0.368960\pi$
$402$	0	0
$403$	0	0
$404$	7591.50i	0.934879i
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	13648.6	1.65008	0.825038	−	0.565078i	$-0.191154\pi$
$409$	13648.6	1.65008	0.825038	+	0.565078i	$0.191154\pi$
$410$	− 885.873i	− 0.106708i
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	− 5505.08i	− 0.648819i
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−16137.0	−1.86810	−0.934050	−	0.357142i	$-0.883751\pi$
$421$	−16137.0	−1.86810	−0.934050	+	0.357142i	$0.883751\pi$
$422$	0	0
$423$	0	0
$424$	−17440.0	−1.99755
$425$	− 14319.9i	− 1.63440i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	12596.3	1.39801	0.699005	−	0.715117i	$-0.253627\pi$
$433$	12596.3	1.39801	0.699005	+	0.715117i	$0.253627\pi$
$434$	0	0
$435$	0	0
$436$	−6256.64	−0.687245
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	0	0
$442$	−10125.4	−1.08963
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	479.954	0.0511281
$446$	0	0
$447$	0	0
$448$	0	0
$449$	18550.2i	1.94975i	0.222742	+	0.974877i	$0.428499\pi$
$449$	18550.2i	1.94975i	−0.222742	+	0.974877i	$0.571501\pi$
$450$	0	0
$451$	0	0
$452$	18220.0i	1.89601i
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−9066.62	−0.928049	−0.464024	−	0.885822i	$-0.653595\pi$
$457$	−9066.62	−0.928049	−0.464024	+	0.885822i	$0.653595\pi$
$458$	11856.5i	1.20964i
$459$	0	0
$460$	0	0
$461$	12262.6i	1.23889i	0.785040	+	0.619445i	$0.212642\pi$
$461$	12262.6i	1.23889i	−0.785040	+	0.619445i	$0.787358\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	− 12740.8i	− 1.27473i
$465$	0	0
$466$	−19802.5	−1.96853
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−13683.6	−1.29712
$482$	20462.5i	1.93370i
$483$	0	0
$484$	10648.0	1.00000
$485$	− 3323.85i	− 0.311193i
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	18561.3i	1.72178i
$489$	0	0
$490$	−1775.68	−0.163708
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	−23434.0	−2.14080
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	− 3611.58i	− 0.323029i
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	1736.85	0.153048
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 22815.5i	− 1.98680i	−0.114715	−	0.993398i	$-0.536596\pi$
$509$	− 22815.5i	− 1.98680i	0.114715	−	0.993398i	$-0.463404\pi$
$510$	0	0
$511$	0	0
$512$	11585.2i	1.00000i
$513$	0	0
$514$	−23241.6	−1.99445
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	−1259.50	−0.106217
$521$	− 15738.8i	− 1.32347i	−0.749737	−	0.661736i	$-0.769820\pi$
$521$	− 15738.8i	− 1.32347i	0.749737	−	0.661736i	$-0.230180\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−12167.0	−1.00000
$530$	3990.09i	0.327016i
$531$	0	0
$532$	0	0
$533$	− 5204.02i	− 0.422910i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	19745.5	1.58232
$539$	0	0
$540$	0	0
$541$	7101.40	0.564349	0.282175	−	0.959363i	$-0.408944\pi$
$541$	7101.40	0.564349	0.282175	+	0.959363i	$0.408944\pi$
$542$	0	0
$543$	0	0
$544$	21308.5	1.67941
$545$	1431.45i	0.112508i
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	18126.0i	1.41296i
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	3722.21i	0.285454i
$555$	0	0
$556$	0	0
$557$	− 14760.1i	− 1.12281i	−0.827541	−	0.561406i	$-0.810261\pi$
$557$	− 14760.1i	− 1.12281i	0.827541	−	0.561406i	$-0.189739\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−16059.1	−1.20536
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	4168.54	0.310392
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 27000.4i	− 1.98930i	−0.103286	−	0.994652i	$-0.532936\pi$
$569$	− 27000.4i	− 1.98930i	0.103286	−	0.994652i	$-0.467064\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	25546.2	1.84316	0.921578	−	0.388194i	$-0.126901\pi$
$577$	25546.2	1.84316	0.921578	+	0.388194i	$0.126901\pi$
$578$	− 25296.4i	− 1.82040i
$579$	0	0
$580$	−2914.96	−0.208685
$581$	0	0
$582$	0	0
$583$	0	0
$584$	− 28214.0i	− 1.99915i
$585$	0	0
$586$	−16325.5	−1.15085
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	28796.5	1.99921
$593$	27549.9i	1.90782i	0.300089	+	0.953911i	$0.402983\pi$
$593$	27549.9i	1.90782i	−0.300089	+	0.953911i	$0.597017\pi$
$594$	0	0
$595$	0	0
$596$	25207.8i	1.73247i
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	12311.2	0.835580	0.417790	−	0.908544i	$-0.362805\pi$
$601$	12311.2	0.835580	0.417790	+	0.908544i	$0.362805\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 2436.15i	− 0.163708i
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	4246.62	0.281870
$611$	0	0
$612$	0	0
$613$	23222.0	1.53006	0.765031	−	0.643994i	$-0.222724\pi$
$613$	23222.0	1.53006	0.765031	+	0.643994i	$0.222724\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 21559.4i	− 1.40672i	−0.710833	−	0.703361i	$-0.751682\pi$
$617$	− 21559.4i	− 1.40672i	0.710833	−	0.703361i	$-0.248318\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	14380.0	0.920317
$626$	31010.3i	1.97991i
$627$	0	0
$628$	26042.3	1.65478
$629$	− 52965.0i	− 3.35748i
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	−5253.98	−0.329120
$635$	0	0
$636$	0	0
$637$	−10431.2	−0.648819
$638$	0	0
$639$	0	0
$640$	2650.58	0.163708
$641$	− 21832.2i	− 1.34527i	−0.739974	−	0.672636i	$-0.765162\pi$
$641$	− 21832.2i	− 1.34527i	0.739974	−	0.672636i	$-0.234838\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	− 10463.9i	− 0.631430i
$651$	0	0
$652$	0	0
$653$	24303.3i	1.45645i	0.685340	+	0.728224i	$0.259654\pi$
$653$	24303.3i	1.45645i	−0.685340	+	0.728224i	$0.740346\pi$
$654$	0	0
$655$	0	0
$656$	10951.7i	0.651815i
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−32036.4	−1.88513	−0.942567	−	0.334017i	$-0.891596\pi$
$661$	−32036.4	−1.88513	−0.942567	+	0.334017i	$0.891596\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	32223.2	1.84564	0.922818	−	0.385237i	$-0.125880\pi$
$673$	32223.2	1.84564	0.922818	+	0.385237i	$0.125880\pi$
$674$	1176.63i	0.0672432i
$675$	0	0
$676$	10177.1	0.579034
$677$	27612.5i	1.56756i	0.621041	+	0.783778i	$0.286710\pi$
$677$	27612.5i	1.56756i	−0.621041	+	0.783778i	$0.713290\pi$
$678$	0	0
$679$	0	0
$680$	− 4875.17i	− 0.274933i
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	4147.03	0.231314
$686$	0	0
$687$	0	0
$688$	0	0
$689$	23439.6i	1.29605i
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	− 27377.3i	− 1.50394i
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	20143.2	1.09466
$698$	26785.2i	1.45249i
$699$	0	0
$700$	0	0
$701$	− 35371.3i	− 1.90578i	−0.303311	−	0.952892i	$-0.598092\pi$
$701$	− 35371.3i	− 1.90578i	0.303311	−	0.952892i	$-0.401908\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	19100.0	1.01818
$707$	0	0
$708$	0	0
$709$	−27676.4	−1.46602	−0.733010	−	0.680217i	$-0.761885\pi$
$709$	−27676.4	−1.46602	−0.733010	+	0.680217i	$0.761885\pi$
$710$	0	0
$711$	0	0
$712$	−5933.47	−0.312312
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	19400.2i	1.00000i
$723$	0	0
$724$	22880.0	1.17449
$725$	− 24217.5i	− 1.24057i
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	0	0
$730$	−6455.07	−0.327278
$731$	0	0
$732$	0	0
$733$	8732.00	0.440005	0.220003	−	0.975499i	$-0.429393\pi$
$733$	8732.00	0.440005	0.220003	+	0.975499i	$0.429393\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	− 6588.35i	− 0.327287i
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	5767.28	0.283620
$746$	− 36548.9i	− 1.79377i
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	−17123.8	−0.827072
$755$	0	0
$756$	0	0
$757$	−22516.0	−1.08105	−0.540527	−	0.841327i	$-0.681775\pi$
$757$	−22516.0	−1.08105	−0.540527	+	0.841327i	$0.681775\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	34640.8i	1.65010i	0.565059	+	0.825050i	$0.308853\pi$
$761$	34640.8i	1.65010i	−0.565059	+	0.825050i	$0.691147\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	31153.2	1.46087	0.730437	−	0.682980i	$-0.239317\pi$
$769$	31153.2	1.46087	0.730437	+	0.682980i	$0.239317\pi$
$770$	0	0
$771$	0	0
$772$	−21946.8	−1.02317
$773$	42556.3i	1.98013i	0.140603	+	0.990066i	$0.455096\pi$
$773$	42556.3i	1.98013i	−0.140603	+	0.990066i	$0.544904\pi$
$774$	0	0
$775$	0	0
$776$	41091.4i	1.90090i
$777$	0	0
$778$	−29932.0	−1.37932
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	21952.0	1.00000
$785$	− 5958.20i	− 0.270901i
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	− 2117.29i	− 0.0957174i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	24946.6	1.11712
$794$	− 44371.6i	− 1.98324i
$795$	0	0
$796$	0	0
$797$	44737.0i	1.98829i	0.108064	+	0.994144i	$0.465535\pi$
$797$	44737.0i	1.98829i	−0.108064	+	0.994144i	$0.534465\pi$
$798$	0	0
$799$	0	0
$800$	22021.0i	0.973200i
$801$	0	0
$802$	41629.5	1.83290
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−21472.0	−0.934879
$809$	32328.4i	1.40495i	0.711708	+	0.702476i	$0.247922\pi$
$809$	32328.4i	1.40495i	−0.711708	+	0.702476i	$0.752078\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	38604.1i	1.65008i
$819$	0	0
$820$	2505.63	0.106708
$821$	44931.3i	1.91000i	0.296600	+	0.955002i	$0.404147\pi$
$821$	44931.3i	1.91000i	−0.296600	+	0.955002i	$0.595853\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	−41740.0	−1.74872	−0.874361	−	0.485276i	$-0.838719\pi$
$829$	−41740.0	−1.74872	−0.874361	+	0.485276i	$0.838719\pi$
$830$	0	0
$831$	0	0
$832$	15570.7	0.648819
$833$	− 40376.0i	− 1.67941i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−15241.8	−0.624947
$842$	− 45642.4i	− 1.86810i
$843$	0	0
$844$	0	0
$845$	− 2328.41i	− 0.0947928i
$846$	0	0
$847$	0	0
$848$	− 49327.8i	− 1.99755i
$849$	0	0
$850$	40502.9	1.63440
$851$	0	0
$852$	0	0
$853$	20378.0	0.817971	0.408986	−	0.912541i	$-0.365883\pi$
$853$	20378.0	0.817971	0.408986	+	0.912541i	$0.365883\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 7468.49i	− 0.297688i	−0.988861	−	0.148844i	$-0.952445\pi$
$857$	− 7468.49i	− 0.297688i	0.988861	−	0.148844i	$-0.0475552\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	−6263.64	−0.246208
$866$	35627.6i	1.39801i
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	− 17696.5i	− 0.687245i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−4588.66	−0.176680	−0.0883399	−	0.996090i	$-0.528156\pi$
$877$	−4588.66	−0.176680	−0.0883399	+	0.996090i	$0.528156\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	41689.6i	1.59428i	0.603796	+	0.797139i	$0.293654\pi$
$881$	41689.6i	1.59428i	−0.603796	+	0.797139i	$0.706346\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	− 28639.0i	− 1.08963i
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	1357.52i	0.0511281i
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−52468.0	−1.94975
$899$	0	0
$900$	0	0
$901$	−90727.8	−3.35470
$902$	0	0
$903$	0	0
$904$	−51533.8	−1.89601
$905$	− 5234.70i	− 0.192273i
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	− 25644.3i	− 0.928049i
$915$	0	0
$916$	−33535.2	−1.20964
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	−34684.0	−1.23889
$923$	0	0
$924$	0	0
$925$	54735.9	1.94563
$926$	0	0
$927$	0	0
$928$	36036.4	1.27473
$929$	17525.5i	0.618939i	0.950909	+	0.309469i	$0.100151\pi$
$929$	17525.5i	0.618939i	−0.950909	+	0.309469i	$0.899849\pi$
$930$	0	0
$931$	0	0
$932$	− 56010.0i	− 1.96853i
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	4897.41	0.170748	0.0853742	−	0.996349i	$-0.472791\pi$
$937$	4897.41	0.170748	0.0853742	+	0.996349i	$0.472791\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 17766.4i	− 0.615483i	−0.951470	−	0.307742i	$-0.900427\pi$
$941$	− 17766.4i	− 0.615483i	0.951470	−	0.307742i	$-0.0995732\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	−37920.0	−1.29709
$950$	0	0
$951$	0	0
$952$	0	0
$953$	29673.5i	1.00862i	0.863522	+	0.504312i	$0.168254\pi$
$953$	29673.5i	1.00862i	−0.863522	+	0.504312i	$0.831746\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	29791.0	1.00000
$962$	− 38702.9i	− 1.29712i
$963$	0	0
$964$	−57876.8	−1.93370
$965$	5021.20i	0.167501i
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	30117.1i	1.00000i
$969$	0	0
$970$	9401.28	0.311193
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	−52499.2	−1.72178
$977$	− 23808.3i	− 0.779626i	−0.920894	−	0.389813i	$-0.872540\pi$
$977$	− 23808.3i	− 0.779626i	0.920894	−	0.389813i	$-0.127460\pi$
$978$	0	0
$979$	0	0
$980$	− 5022.39i	− 0.163708i
$981$	0	0
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	−484.414	−0.0156697
$986$	− 66281.2i	− 2.14080i
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−56029.9	−1.77982	−0.889912	−	0.456132i	$-0.849234\pi$
$997$	−56029.9	−1.77982	−0.889912	+	0.456132i	$0.849234\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.4.b.a.323.4	yes	4
3.2	odd	2	inner	324.4.b.a.323.1	✓	4
4.3	odd	2	CM	324.4.b.a.323.4	yes	4
12.11	even	2	inner	324.4.b.a.323.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.4.b.a.323.1	✓	4	3.2	odd	2	inner
324.4.b.a.323.1	✓	4	12.11	even	2	inner
324.4.b.a.323.4	yes	4	1.1	even	1	trivial
324.4.b.a.323.4	yes	4	4.3	odd	2	CM

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:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	324.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.82843i q^{2} -8.00000 q^{4} +1.83032i q^{5} -22.6274i q^{8} -5.17691 q^{10} -30.4115 q^{13} +64.0000 q^{16} -117.714i q^{17} -14.6425i q^{20} +121.650 q^{25} -86.0168i q^{26} -199.075i q^{29} +181.019i q^{32} +332.946 q^{34} +449.946 q^{37} +41.4153 q^{40} +171.120i q^{41} +343.000 q^{49} +344.078i q^{50} +243.292 q^{52} -770.746i q^{53} +563.069 q^{58} -820.300 q^{61} -512.000 q^{64} -55.6627i q^{65} +941.714i q^{68} +1246.90 q^{73} +1272.64i q^{74} +117.140i q^{80} -484.000 q^{82} +215.454 q^{85} -262.225i q^{89} -1816.00 q^{97} +970.151i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{4} + 104 q^{10} - 184 q^{13} + 256 q^{16} - 324 q^{25} - 40 q^{34} + 428 q^{37} - 832 q^{40} + 1372 q^{49} + 1472 q^{52} - 616 q^{58} - 1660 q^{61} - 2048 q^{64} + 1184 q^{73} - 1936 q^{82}+ \cdots - 7264 q^{97}+O(q^{100}) \) 4 * q - 32 * q^4 + 104 * q^10 - 184 * q^13 + 256 * q^16 - 324 * q^25 - 40 * q^34 + 428 * q^37 - 832 * q^40 + 1372 * q^49 + 1472 * q^52 - 616 * q^58 - 1660 * q^61 - 2048 * q^64 + 1184 * q^73 - 1936 * q^82 + 5476 * q^85 - 7264 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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