LMFDB - Embedded newform 324.6.a.d.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,6,Mod(1,324)]

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([0, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("324.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	324.a (trivial)

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:	yes
Analytic conductor:	$51.9643576194$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 110x^{3} + 39x^{2} + 2214x - 1944 $ x^5 - 2x^4 - 110x^3 + 39x^2 + 2214x - 1944
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{7} $
Twist minimal:	no (minimal twist has level 36)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-6.24439$ of defining polynomial
Character	$\chi$	$=$	324.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+81.4540 q^{5} -179.262 q^{7} +O(q^{10})$	$q+81.4540 q^{5} -179.262 q^{7} +500.501 q^{11} -550.491 q^{13} -753.636 q^{17} -2570.83 q^{19} +2745.45 q^{23} +3509.75 q^{25} -3909.72 q^{29} -3104.84 q^{31} -14601.6 q^{35} -9568.10 q^{37} -2226.94 q^{41} +14286.8 q^{43} +6472.14 q^{47} +15328.0 q^{49} -13692.2 q^{53} +40767.8 q^{55} -5708.43 q^{59} -11799.2 q^{61} -44839.7 q^{65} -3543.32 q^{67} +58429.0 q^{71} -60181.3 q^{73} -89720.9 q^{77} -55623.4 q^{79} -39980.6 q^{83} -61386.6 q^{85} -103171. q^{89} +98682.3 q^{91} -209404. q^{95} -165994. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 21 q^{5} - 29 q^{7}+O(q^{10}) $ 5 * q - 21 * q^5 - 29 * q^7	$ 5 q - 21 q^{5} - 29 q^{7} + 177 q^{11} + 181 q^{13} - 1140 q^{17} - 416 q^{19} + 399 q^{23} + 4778 q^{25} - 6033 q^{29} - 2759 q^{31} - 18573 q^{35} - 7586 q^{37} - 18435 q^{41} - 1469 q^{43} - 25155 q^{47} + 4056 q^{49} - 58422 q^{53} + 7389 q^{55} - 90537 q^{59} - 1403 q^{61} - 148407 q^{65} - 13907 q^{67} - 114684 q^{71} + 7600 q^{73} - 211983 q^{77} - 29993 q^{79} - 228951 q^{83} + 49662 q^{85} - 299166 q^{89} + 62465 q^{91} - 394764 q^{95} - 40541 q^{97}+O(q^{100}) $ 5 * q - 21 * q^5 - 29 * q^7 + 177 * q^11 + 181 * q^13 - 1140 * q^17 - 416 * q^19 + 399 * q^23 + 4778 * q^25 - 6033 * q^29 - 2759 * q^31 - 18573 * q^35 - 7586 * q^37 - 18435 * q^41 - 1469 * q^43 - 25155 * q^47 + 4056 * q^49 - 58422 * q^53 + 7389 * q^55 - 90537 * q^59 - 1403 * q^61 - 148407 * q^65 - 13907 * q^67 - 114684 * q^71 + 7600 * q^73 - 211983 * q^77 - 29993 * q^79 - 228951 * q^83 + 49662 * q^85 - 299166 * q^89 + 62465 * q^91 - 394764 * q^95 - 40541 * q^97

Display 100 coefficients 10 coefficients

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
$5$	81.4540	1.45709	0.728546	−	0.684997i	$-0.240196\pi$
$5$	81.4540	1.45709	0.728546	+	0.684997i	$0.240196\pi$
$6$	0	0
$7$	−179.262	−1.38275	−0.691376	−	0.722496i	$-0.742995\pi$
$7$	−179.262	−1.38275	−0.691376	+	0.722496i	$0.742995\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	500.501	1.24716	0.623581	−	0.781759i	$-0.285677\pi$
$11$	500.501	1.24716	0.623581	+	0.781759i	$0.285677\pi$
$12$	0	0
$13$	−550.491	−0.903424	−0.451712	−	0.892164i	$-0.649187\pi$
$13$	−550.491	−0.903424	−0.451712	+	0.892164i	$0.649187\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−753.636	−0.632469	−0.316234	−	0.948681i	$-0.602419\pi$
$17$	−753.636	−0.632469	−0.316234	+	0.948681i	$0.602419\pi$
$18$	0	0
$19$	−2570.83	−1.63376	−0.816882	−	0.576805i	$-0.804299\pi$
$19$	−2570.83	−1.63376	−0.816882	+	0.576805i	$0.804299\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2745.45	1.08217	0.541083	−	0.840969i	$-0.318014\pi$
$23$	2745.45	1.08217	0.541083	+	0.840969i	$0.318014\pi$
$24$	0	0
$25$	3509.75	1.12312
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3909.72	−0.863278	−0.431639	−	0.902046i	$-0.642065\pi$
$29$	−3909.72	−0.863278	−0.431639	+	0.902046i	$0.642065\pi$
$30$	0	0
$31$	−3104.84	−0.580276	−0.290138	−	0.956985i	$-0.593701\pi$
$31$	−3104.84	−0.580276	−0.290138	+	0.956985i	$0.593701\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−14601.6	−2.01480
$36$	0	0
$37$	−9568.10	−1.14900	−0.574502	−	0.818503i	$-0.694804\pi$
$37$	−9568.10	−1.14900	−0.574502	+	0.818503i	$0.694804\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2226.94	−0.206894	−0.103447	−	0.994635i	$-0.532987\pi$
$41$	−2226.94	−0.206894	−0.103447	+	0.994635i	$0.532987\pi$
$42$	0	0
$43$	14286.8	1.17832	0.589162	−	0.808015i	$-0.299458\pi$
$43$	14286.8	1.17832	0.589162	+	0.808015i	$0.299458\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6472.14	0.427369	0.213685	−	0.976903i	$-0.431454\pi$
$47$	6472.14	0.427369	0.213685	+	0.976903i	$0.431454\pi$
$48$	0	0
$49$	15328.0	0.912001
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−13692.2	−0.669553	−0.334777	−	0.942298i	$-0.608661\pi$
$53$	−13692.2	−0.669553	−0.334777	+	0.942298i	$0.608661\pi$
$54$	0	0
$55$	40767.8	1.81723
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−5708.43	−0.213495	−0.106747	−	0.994286i	$-0.534044\pi$
$59$	−5708.43	−0.213495	−0.106747	+	0.994286i	$0.534044\pi$
$60$	0	0
$61$	−11799.2	−0.406001	−0.203000	−	0.979179i	$-0.565069\pi$
$61$	−11799.2	−0.406001	−0.203000	+	0.979179i	$0.565069\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−44839.7	−1.31637
$66$	0	0
$67$	−3543.32	−0.0964324	−0.0482162	−	0.998837i	$-0.515354\pi$
$67$	−3543.32	−0.0964324	−0.0482162	+	0.998837i	$0.515354\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	58429.0	1.37557	0.687785	−	0.725914i	$-0.258583\pi$
$71$	58429.0	1.37557	0.687785	+	0.725914i	$0.258583\pi$
$72$	0	0
$73$	−60181.3	−1.32176	−0.660882	−	0.750490i	$-0.729818\pi$
$73$	−60181.3	−1.32176	−0.660882	+	0.750490i	$0.729818\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−89720.9	−1.72452
$78$	0	0
$79$	−55623.4	−1.00274	−0.501372	−	0.865232i	$-0.667171\pi$
$79$	−55623.4	−1.00274	−0.501372	+	0.865232i	$0.667171\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−39980.6	−0.637022	−0.318511	−	0.947919i	$-0.603183\pi$
$83$	−39980.6	−0.637022	−0.318511	+	0.947919i	$0.603183\pi$
$84$	0	0
$85$	−61386.6	−0.921566
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−103171.	−1.38065	−0.690327	−	0.723498i	$-0.742533\pi$
$89$	−103171.	−1.38065	−0.690327	+	0.723498i	$0.742533\pi$
$90$	0	0
$91$	98682.3	1.24921
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−209404.	−2.38054
$96$	0	0
$97$	−165994.	−1.79128	−0.895638	−	0.444783i	$-0.853281\pi$
$97$	−165994.	−1.79128	−0.895638	+	0.444783i	$0.853281\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−165049.	−1.60994	−0.804969	−	0.593317i	$-0.797818\pi$
$101$	−165049.	−1.60994	−0.804969	+	0.593317i	$0.797818\pi$
$102$	0	0
$103$	72860.9	0.676709	0.338354	−	0.941019i	$-0.390130\pi$
$103$	72860.9	0.676709	0.338354	+	0.941019i	$0.390130\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−123024.	−1.03880	−0.519399	−	0.854532i	$-0.673844\pi$
$107$	−123024.	−1.03880	−0.519399	+	0.854532i	$0.673844\pi$
$108$	0	0
$109$	24274.5	0.195697	0.0978486	−	0.995201i	$-0.468804\pi$
$109$	24274.5	0.195697	0.0978486	+	0.995201i	$0.468804\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	201372.	1.48356	0.741778	−	0.670646i	$-0.233983\pi$
$113$	201372.	1.48356	0.741778	+	0.670646i	$0.233983\pi$
$114$	0	0
$115$	223628.	1.57682
$116$	0	0
$117$	0	0
$118$	0	0
$119$	135099.	0.874547
$120$	0	0
$121$	89449.9	0.555414
$122$	0	0
$123$	0	0
$124$	0	0
$125$	31339.2	0.179396
$126$	0	0
$127$	−104163.	−0.573067	−0.286534	−	0.958070i	$-0.592503\pi$
$127$	−104163.	−0.573067	−0.286534	+	0.958070i	$0.592503\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−178850.	−0.910563	−0.455282	−	0.890347i	$-0.650461\pi$
$131$	−178850.	−0.910563	−0.455282	+	0.890347i	$0.650461\pi$
$132$	0	0
$133$	460853.	2.25909
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−135303.	−0.615892	−0.307946	−	0.951404i	$-0.599642\pi$
$137$	−135303.	−0.615892	−0.307946	+	0.951404i	$0.599642\pi$
$138$	0	0
$139$	−226166.	−0.992867	−0.496434	−	0.868075i	$-0.665357\pi$
$139$	−226166.	−0.992867	−0.496434	+	0.868075i	$0.665357\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−275521.	−1.12672
$144$	0	0
$145$	−318462.	−1.25788
$146$	0	0
$147$	0	0
$148$	0	0
$149$	342897.	1.26531	0.632657	−	0.774432i	$-0.281964\pi$
$149$	342897.	1.26531	0.632657	+	0.774432i	$0.281964\pi$
$150$	0	0
$151$	−243940.	−0.870644	−0.435322	−	0.900275i	$-0.643366\pi$
$151$	−243940.	−0.870644	−0.435322	+	0.900275i	$0.643366\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−252901.	−0.845516
$156$	0	0
$157$	348876.	1.12959	0.564797	−	0.825230i	$-0.308954\pi$
$157$	348876.	1.12959	0.564797	+	0.825230i	$0.308954\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−492156.	−1.49637
$162$	0	0
$163$	303629.	0.895107	0.447553	−	0.894257i	$-0.352295\pi$
$163$	303629.	0.895107	0.447553	+	0.894257i	$0.352295\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	32775.5	0.0909405	0.0454703	−	0.998966i	$-0.485521\pi$
$167$	32775.5	0.0909405	0.0454703	+	0.998966i	$0.485521\pi$
$168$	0	0
$169$	−68252.8	−0.183825
$170$	0	0
$171$	0	0
$172$	0	0
$173$	355302.	0.902574	0.451287	−	0.892379i	$-0.350965\pi$
$173$	355302.	0.902574	0.451287	+	0.892379i	$0.350965\pi$
$174$	0	0
$175$	−629166.	−1.55299
$176$	0	0
$177$	0	0
$178$	0	0
$179$	459272.	1.07137	0.535683	−	0.844419i	$-0.320054\pi$
$179$	459272.	1.07137	0.535683	+	0.844419i	$0.320054\pi$
$180$	0	0
$181$	−190088.	−0.431279	−0.215640	−	0.976473i	$-0.569184\pi$
$181$	−190088.	−0.431279	−0.215640	+	0.976473i	$0.569184\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−779360.	−1.67421
$186$	0	0
$187$	−377195.	−0.788791
$188$	0	0
$189$	0	0
$190$	0	0
$191$	688559.	1.36571	0.682854	−	0.730555i	$-0.260738\pi$
$191$	688559.	1.36571	0.682854	+	0.730555i	$0.260738\pi$
$192$	0	0
$193$	344277.	0.665295	0.332648	−	0.943051i	$-0.392058\pi$
$193$	344277.	0.665295	0.332648	+	0.943051i	$0.392058\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	596238.	1.09460	0.547298	−	0.836938i	$-0.315656\pi$
$197$	596238.	1.09460	0.547298	+	0.836938i	$0.315656\pi$
$198$	0	0
$199$	−49436.6	−0.0884945	−0.0442473	−	0.999021i	$-0.514089\pi$
$199$	−49436.6	−0.0884945	−0.0442473	+	0.999021i	$0.514089\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	700866.	1.19370
$204$	0	0
$205$	−181393.	−0.301464
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.28670e6	−2.03757
$210$	0	0
$211$	−151998.	−0.235035	−0.117518	−	0.993071i	$-0.537494\pi$
$211$	−151998.	−0.235035	−0.117518	+	0.993071i	$0.537494\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.16372e6	1.71693
$216$	0	0
$217$	556580.	0.802377
$218$	0	0
$219$	0	0
$220$	0	0
$221$	414870.	0.571388
$222$	0	0
$223$	−562114.	−0.756942	−0.378471	−	0.925613i	$-0.623550\pi$
$223$	−562114.	−0.756942	−0.378471	+	0.925613i	$0.623550\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.26099e6	1.62422	0.812111	−	0.583503i	$-0.198318\pi$
$227$	1.26099e6	1.62422	0.812111	+	0.583503i	$0.198318\pi$
$228$	0	0
$229$	139607.	0.175922	0.0879609	−	0.996124i	$-0.471965\pi$
$229$	139607.	0.175922	0.0879609	+	0.996124i	$0.471965\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.18799e6	1.43358	0.716790	−	0.697289i	$-0.245610\pi$
$233$	1.18799e6	1.43358	0.716790	+	0.697289i	$0.245610\pi$
$234$	0	0
$235$	527181.	0.622716
$236$	0	0
$237$	0	0
$238$	0	0
$239$	329903.	0.373587	0.186794	−	0.982399i	$-0.440190\pi$
$239$	329903.	0.373587	0.186794	+	0.982399i	$0.440190\pi$
$240$	0	0
$241$	168745.	0.187149	0.0935745	−	0.995612i	$-0.470171\pi$
$241$	168745.	0.187149	0.0935745	+	0.995612i	$0.470171\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.24853e6	1.32887
$246$	0	0
$247$	1.41522e6	1.47598
$248$	0	0
$249$	0	0
$250$	0	0
$251$	224796.	0.225218	0.112609	−	0.993639i	$-0.464079\pi$
$251$	224796.	0.225218	0.112609	+	0.993639i	$0.464079\pi$
$252$	0	0
$253$	1.37410e6	1.34964
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−695936.	−0.657259	−0.328629	−	0.944459i	$-0.606587\pi$
$257$	−695936.	−0.657259	−0.328629	+	0.944459i	$0.606587\pi$
$258$	0	0
$259$	1.71520e6	1.58879
$260$	0	0
$261$	0	0
$262$	0	0
$263$	482731.	0.430344	0.215172	−	0.976576i	$-0.430969\pi$
$263$	482731.	0.430344	0.215172	+	0.976576i	$0.430969\pi$
$264$	0	0
$265$	−1.11529e6	−0.975601
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2.23733e6	−1.88516	−0.942581	−	0.333978i	$-0.891609\pi$
$269$	−2.23733e6	−1.88516	−0.942581	+	0.333978i	$0.891609\pi$
$270$	0	0
$271$	−1.74480e6	−1.44318	−0.721592	−	0.692318i	$-0.756589\pi$
$271$	−1.74480e6	−1.44318	−0.721592	+	0.692318i	$0.756589\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.75663e6	1.40071
$276$	0	0
$277$	−757942.	−0.593521	−0.296761	−	0.954952i	$-0.595906\pi$
$277$	−757942.	−0.593521	−0.296761	+	0.954952i	$0.595906\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−430431.	−0.325191	−0.162595	−	0.986693i	$-0.551986\pi$
$281$	−430431.	−0.325191	−0.162595	+	0.986693i	$0.551986\pi$
$282$	0	0
$283$	957298.	0.710528	0.355264	−	0.934766i	$-0.384391\pi$
$283$	957298.	0.710528	0.355264	+	0.934766i	$0.384391\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	399206.	0.286083
$288$	0	0
$289$	−851890.	−0.599983
$290$	0	0
$291$	0	0
$292$	0	0
$293$	566294.	0.385366	0.192683	−	0.981261i	$-0.438281\pi$
$293$	566294.	0.385366	0.192683	+	0.981261i	$0.438281\pi$
$294$	0	0
$295$	−464974.	−0.311081
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.51135e6	−0.977655
$300$	0	0
$301$	−2.56109e6	−1.62933
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−961090.	−0.591581
$306$	0	0
$307$	−2.00565e6	−1.21453	−0.607265	−	0.794499i	$-0.707733\pi$
$307$	−2.00565e6	−1.21453	−0.607265	+	0.794499i	$0.707733\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2.11670e6	1.24096	0.620481	−	0.784221i	$-0.286937\pi$
$311$	2.11670e6	1.24096	0.620481	+	0.784221i	$0.286937\pi$
$312$	0	0
$313$	1.21334e6	0.700037	0.350018	−	0.936743i	$-0.386175\pi$
$313$	1.21334e6	0.700037	0.350018	+	0.936743i	$0.386175\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.56756e6	1.43507	0.717533	−	0.696524i	$-0.245271\pi$
$317$	2.56756e6	1.43507	0.717533	+	0.696524i	$0.245271\pi$
$318$	0	0
$319$	−1.95682e6	−1.07665
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.93747e6	1.03330
$324$	0	0
$325$	−1.93208e6	−1.01465
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.16021e6	−0.590945
$330$	0	0
$331$	1.44226e6	0.723557	0.361778	−	0.932264i	$-0.382170\pi$
$331$	1.44226e6	0.723557	0.361778	+	0.932264i	$0.382170\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−288617.	−0.140511
$336$	0	0
$337$	1.58187e6	0.758744	0.379372	−	0.925244i	$-0.376140\pi$
$337$	1.58187e6	0.758744	0.379372	+	0.925244i	$0.376140\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.55397e6	−0.723698
$342$	0	0
$343$	265129.	0.121681
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.64997e6	0.735618	0.367809	−	0.929901i	$-0.380108\pi$
$347$	1.64997e6	0.735618	0.367809	+	0.929901i	$0.380108\pi$
$348$	0	0
$349$	650366.	0.285821	0.142911	−	0.989736i	$-0.454354\pi$
$349$	650366.	0.285821	0.142911	+	0.989736i	$0.454354\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1.18163e6	−0.504715	−0.252358	−	0.967634i	$-0.581206\pi$
$353$	−1.18163e6	−0.504715	−0.252358	+	0.967634i	$0.581206\pi$
$354$	0	0
$355$	4.75928e6	2.00433
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−985030.	−0.403379	−0.201690	−	0.979449i	$-0.564643\pi$
$359$	−985030.	−0.403379	−0.201690	+	0.979449i	$0.564643\pi$
$360$	0	0
$361$	4.13306e6	1.66918
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4.90200e6	−1.92593
$366$	0	0
$367$	−1.19687e6	−0.463855	−0.231928	−	0.972733i	$-0.574503\pi$
$367$	−1.19687e6	−0.463855	−0.231928	+	0.972733i	$0.574503\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.45450e6	0.925825
$372$	0	0
$373$	−2.46460e6	−0.917222	−0.458611	−	0.888637i	$-0.651653\pi$
$373$	−2.46460e6	−0.917222	−0.458611	+	0.888637i	$0.651653\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.15227e6	0.779906
$378$	0	0
$379$	5.33540e6	1.90796	0.953979	−	0.299875i	$-0.0969449\pi$
$379$	5.33540e6	1.90796	0.953979	+	0.299875i	$0.0969449\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.65223e6	−0.575539	−0.287770	−	0.957700i	$-0.592914\pi$
$383$	−1.65223e6	−0.575539	−0.287770	+	0.957700i	$0.592914\pi$
$384$	0	0
$385$	−7.30813e6	−2.51278
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−5.66887e6	−1.89942	−0.949712	−	0.313123i	$-0.898625\pi$
$389$	−5.66887e6	−1.89942	−0.949712	+	0.313123i	$0.898625\pi$
$390$	0	0
$391$	−2.06907e6	−0.684437
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4.53074e6	−1.46109
$396$	0	0
$397$	418875.	0.133385	0.0666927	−	0.997774i	$-0.478755\pi$
$397$	418875.	0.133385	0.0666927	+	0.997774i	$0.478755\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.86512e6	−0.889777	−0.444889	−	0.895586i	$-0.646757\pi$
$401$	−2.86512e6	−0.889777	−0.444889	+	0.895586i	$0.646757\pi$
$402$	0	0
$403$	1.70918e6	0.524235
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4.78884e6	−1.43299
$408$	0	0
$409$	5.30476e6	1.56804	0.784021	−	0.620735i	$-0.213166\pi$
$409$	5.30476e6	1.56804	0.784021	+	0.620735i	$0.213166\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.02331e6	0.295210
$414$	0	0
$415$	−3.25658e6	−0.928200
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4.84273e6	−1.34758	−0.673791	−	0.738922i	$-0.735335\pi$
$419$	−4.84273e6	−1.34758	−0.673791	+	0.738922i	$0.735335\pi$
$420$	0	0
$421$	4.57939e6	1.25922	0.629610	−	0.776911i	$-0.283215\pi$
$421$	4.57939e6	1.25922	0.629610	+	0.776911i	$0.283215\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.64507e6	−0.710338
$426$	0	0
$427$	2.11515e6	0.561398
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.09717e6	0.803105	0.401552	−	0.915836i	$-0.368471\pi$
$431$	3.09717e6	0.803105	0.401552	+	0.915836i	$0.368471\pi$
$432$	0	0
$433$	−992453.	−0.254384	−0.127192	−	0.991878i	$-0.540596\pi$
$433$	−992453.	−0.254384	−0.127192	+	0.991878i	$0.540596\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.05808e6	−1.76800
$438$	0	0
$439$	−885373.	−0.219263	−0.109631	−	0.993972i	$-0.534967\pi$
$439$	−885373.	−0.219263	−0.109631	+	0.993972i	$0.534967\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	540777.	0.130921	0.0654605	−	0.997855i	$-0.479148\pi$
$443$	540777.	0.130921	0.0654605	+	0.997855i	$0.479148\pi$
$444$	0	0
$445$	−8.40373e6	−2.01174
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2.22193e6	−0.520132	−0.260066	−	0.965591i	$-0.583744\pi$
$449$	−2.22193e6	−0.520132	−0.260066	+	0.965591i	$0.583744\pi$
$450$	0	0
$451$	−1.11458e6	−0.258031
$452$	0	0
$453$	0	0
$454$	0	0
$455$	8.03806e6	1.82022
$456$	0	0
$457$	−577784.	−0.129412	−0.0647061	−	0.997904i	$-0.520611\pi$
$457$	−577784.	−0.129412	−0.0647061	+	0.997904i	$0.520611\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	3.41438e6	0.748273	0.374137	−	0.927374i	$-0.377939\pi$
$461$	3.41438e6	0.748273	0.374137	+	0.927374i	$0.377939\pi$
$462$	0	0
$463$	−1.85555e6	−0.402272	−0.201136	−	0.979563i	$-0.564463\pi$
$463$	−1.85555e6	−0.402272	−0.201136	+	0.979563i	$0.564463\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.73338e6	0.792154	0.396077	−	0.918217i	$-0.370371\pi$
$467$	3.73338e6	0.792154	0.396077	+	0.918217i	$0.370371\pi$
$468$	0	0
$469$	635184.	0.133342
$470$	0	0
$471$	0	0
$472$	0	0
$473$	7.15057e6	1.46956
$474$	0	0
$475$	−9.02296e6	−1.83491
$476$	0	0
$477$	0	0
$478$	0	0
$479$	3.98493e6	0.793563	0.396782	−	0.917913i	$-0.370127\pi$
$479$	3.98493e6	0.793563	0.396782	+	0.917913i	$0.370127\pi$
$480$	0	0
$481$	5.26715e6	1.03804
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.35209e7	−2.61006
$486$	0	0
$487$	4.19007e6	0.800570	0.400285	−	0.916391i	$-0.368911\pi$
$487$	4.19007e6	0.800570	0.400285	+	0.916391i	$0.368911\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−4.19213e6	−0.784749	−0.392374	−	0.919806i	$-0.628346\pi$
$491$	−4.19213e6	−0.784749	−0.392374	+	0.919806i	$0.628346\pi$
$492$	0	0
$493$	2.94651e6	0.545997
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.04741e7	−1.90207
$498$	0	0
$499$	−3.70338e6	−0.665805	−0.332903	−	0.942961i	$-0.608028\pi$
$499$	−3.70338e6	−0.665805	−0.332903	+	0.942961i	$0.608028\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−4.03820e6	−0.711653	−0.355827	−	0.934552i	$-0.615801\pi$
$503$	−4.03820e6	−0.711653	−0.355827	+	0.934552i	$0.615801\pi$
$504$	0	0
$505$	−1.34439e7	−2.34583
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−167961.	−0.0287351	−0.0143676	−	0.999897i	$-0.504573\pi$
$509$	−167961.	−0.0287351	−0.0143676	+	0.999897i	$0.504573\pi$
$510$	0	0
$511$	1.07882e7	1.82767
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5.93481e6	0.986027
$516$	0	0
$517$	3.23931e6	0.532999
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−3.98731e6	−0.643555	−0.321777	−	0.946815i	$-0.604280\pi$
$521$	−3.98731e6	−0.643555	−0.321777	+	0.946815i	$0.604280\pi$
$522$	0	0
$523$	−4.41694e6	−0.706102	−0.353051	−	0.935604i	$-0.614856\pi$
$523$	−4.41694e6	−0.706102	−0.353051	+	0.935604i	$0.614856\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.33992e6	0.367007
$528$	0	0
$529$	1.10115e6	0.171084
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.22591e6	0.186913
$534$	0	0
$535$	−1.00208e7	−1.51362
$536$	0	0
$537$	0	0
$538$	0	0
$539$	7.67167e6	1.13741
$540$	0	0
$541$	9.24640e6	1.35825	0.679125	−	0.734023i	$-0.262360\pi$
$541$	9.24640e6	1.35825	0.679125	+	0.734023i	$0.262360\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1.97726e6	0.285149
$546$	0	0
$547$	1.52578e6	0.218033	0.109017	−	0.994040i	$-0.465230\pi$
$547$	1.52578e6	0.218033	0.109017	+	0.994040i	$0.465230\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.00512e7	1.41039
$552$	0	0
$553$	9.97118e6	1.38654
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−5.86988e6	−0.801662	−0.400831	−	0.916152i	$-0.631279\pi$
$557$	−5.86988e6	−0.801662	−0.400831	+	0.916152i	$0.631279\pi$
$558$	0	0
$559$	−7.86477e6	−1.06453
$560$	0	0
$561$	0	0
$562$	0	0
$563$	213372.	0.0283704	0.0141852	−	0.999899i	$-0.495485\pi$
$563$	213372.	0.0283704	0.0141852	+	0.999899i	$0.495485\pi$
$564$	0	0
$565$	1.64026e7	2.16168
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.12903e7	−1.46192	−0.730961	−	0.682419i	$-0.760928\pi$
$569$	−1.12903e7	−1.46192	−0.730961	+	0.682419i	$0.760928\pi$
$570$	0	0
$571$	−464683.	−0.0596440	−0.0298220	−	0.999555i	$-0.509494\pi$
$571$	−464683.	−0.0596440	−0.0298220	+	0.999555i	$0.509494\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	9.63583e6	1.21540
$576$	0	0
$577$	8.78227e6	1.09816	0.549082	−	0.835769i	$-0.314978\pi$
$577$	8.78227e6	1.09816	0.549082	+	0.835769i	$0.314978\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	7.16702e6	0.880843
$582$	0	0
$583$	−6.85298e6	−0.835041
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.00417e7	−1.20285	−0.601423	−	0.798931i	$-0.705399\pi$
$587$	−1.00417e7	−1.20285	−0.601423	+	0.798931i	$0.705399\pi$
$588$	0	0
$589$	7.98200e6	0.948034
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.15945e7	1.35399	0.676995	−	0.735988i	$-0.263282\pi$
$593$	1.15945e7	1.35399	0.676995	+	0.735988i	$0.263282\pi$
$594$	0	0
$595$	1.10043e7	1.27430
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.48884e6	0.169544	0.0847720	−	0.996400i	$-0.472984\pi$
$599$	1.48884e6	0.169544	0.0847720	+	0.996400i	$0.472984\pi$
$600$	0	0
$601$	−9.51007e6	−1.07398	−0.536992	−	0.843588i	$-0.680439\pi$
$601$	−9.51007e6	−1.07398	−0.536992	+	0.843588i	$0.680439\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	7.28605e6	0.809289
$606$	0	0
$607$	7.40405e6	0.815639	0.407819	−	0.913063i	$-0.366289\pi$
$607$	7.40405e6	0.815639	0.407819	+	0.913063i	$0.366289\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−3.56285e6	−0.386096
$612$	0	0
$613$	−7.14368e6	−0.767840	−0.383920	−	0.923366i	$-0.625426\pi$
$613$	−7.14368e6	−0.767840	−0.383920	+	0.923366i	$0.625426\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.00404e6	0.106179	0.0530895	−	0.998590i	$-0.483093\pi$
$617$	1.00404e6	0.106179	0.0530895	+	0.998590i	$0.483093\pi$
$618$	0	0
$619$	−7.48656e6	−0.785336	−0.392668	−	0.919680i	$-0.628448\pi$
$619$	−7.48656e6	−0.785336	−0.392668	+	0.919680i	$0.628448\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.84948e7	1.90910
$624$	0	0
$625$	−8.41526e6	−0.861723
$626$	0	0
$627$	0	0
$628$	0	0
$629$	7.21086e6	0.726709
$630$	0	0
$631$	1.34648e7	1.34626	0.673128	−	0.739526i	$-0.264950\pi$
$631$	1.34648e7	1.34626	0.673128	+	0.739526i	$0.264950\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−8.48452e6	−0.835012
$636$	0	0
$637$	−8.43792e6	−0.823924
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.62617e7	1.56322	0.781610	−	0.623768i	$-0.214399\pi$
$641$	1.62617e7	1.56322	0.781610	+	0.623768i	$0.214399\pi$
$642$	0	0
$643$	−1.13687e7	−1.08439	−0.542194	−	0.840253i	$-0.682406\pi$
$643$	−1.13687e7	−1.08439	−0.542194	+	0.840253i	$0.682406\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.18311e7	1.11113	0.555563	−	0.831474i	$-0.312503\pi$
$647$	1.18311e7	1.11113	0.555563	+	0.831474i	$0.312503\pi$
$648$	0	0
$649$	−2.85707e6	−0.266262
$650$	0	0
$651$	0	0
$652$	0	0
$653$	5.34599e6	0.490620	0.245310	−	0.969445i	$-0.421110\pi$
$653$	5.34599e6	0.490620	0.245310	+	0.969445i	$0.421110\pi$
$654$	0	0
$655$	−1.45680e7	−1.32677
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.23736e7	1.10990	0.554949	−	0.831884i	$-0.312738\pi$
$659$	1.23736e7	1.10990	0.554949	+	0.831884i	$0.312738\pi$
$660$	0	0
$661$	−2.09830e7	−1.86795	−0.933974	−	0.357342i	$-0.883683\pi$
$661$	−2.09830e7	−1.86795	−0.933974	+	0.357342i	$0.883683\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.75383e7	3.29170
$666$	0	0
$667$	−1.07339e7	−0.934210
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−5.90550e6	−0.506349
$672$	0	0
$673$	−1.26175e7	−1.07383	−0.536915	−	0.843637i	$-0.680410\pi$
$673$	−1.26175e7	−1.07383	−0.536915	+	0.843637i	$0.680410\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−7.99456e6	−0.670383	−0.335192	−	0.942150i	$-0.608801\pi$
$677$	−7.99456e6	−0.670383	−0.335192	+	0.942150i	$0.608801\pi$
$678$	0	0
$679$	2.97565e7	2.47689
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.23985e7	−1.01699	−0.508496	−	0.861064i	$-0.669798\pi$
$683$	−1.23985e7	−1.01699	−0.508496	+	0.861064i	$0.669798\pi$
$684$	0	0
$685$	−1.10209e7	−0.897412
$686$	0	0
$687$	0	0
$688$	0	0
$689$	7.53746e6	0.604891
$690$	0	0
$691$	2.06344e7	1.64398	0.821990	−	0.569502i	$-0.192864\pi$
$691$	2.06344e7	1.64398	0.821990	+	0.569502i	$0.192864\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.84222e7	−1.44670
$696$	0	0
$697$	1.67830e6	0.130854
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−2.28362e7	−1.75521	−0.877603	−	0.479388i	$-0.840859\pi$
$701$	−2.28362e7	−1.75521	−0.877603	+	0.479388i	$0.840859\pi$
$702$	0	0
$703$	2.45980e7	1.87720
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.95870e7	2.22614
$708$	0	0
$709$	−1.78247e7	−1.33170	−0.665852	−	0.746084i	$-0.731932\pi$
$709$	−1.78247e7	−1.33170	−0.665852	+	0.746084i	$0.731932\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−8.52418e6	−0.627955
$714$	0	0
$715$	−2.24423e7	−1.64173
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.10161e7	0.794702	0.397351	−	0.917667i	$-0.369929\pi$
$719$	1.10161e7	0.794702	0.397351	+	0.917667i	$0.369929\pi$
$720$	0	0
$721$	−1.30612e7	−0.935720
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.37221e7	−0.969564
$726$	0	0
$727$	2.60033e7	1.82470	0.912351	−	0.409410i	$-0.134265\pi$
$727$	2.60033e7	1.82470	0.912351	+	0.409410i	$0.134265\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.07671e7	−0.745254
$732$	0	0
$733$	2.66711e7	1.83350	0.916750	−	0.399462i	$-0.130803\pi$
$733$	2.66711e7	1.83350	0.916750	+	0.399462i	$0.130803\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.77343e6	−0.120267
$738$	0	0
$739$	1.31065e6	0.0882829	0.0441414	−	0.999025i	$-0.485945\pi$
$739$	1.31065e6	0.0882829	0.0441414	+	0.999025i	$0.485945\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−2.05004e7	−1.36236	−0.681178	−	0.732118i	$-0.738532\pi$
$743$	−2.05004e7	−1.36236	−0.681178	+	0.732118i	$0.738532\pi$
$744$	0	0
$745$	2.79303e7	1.84368
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2.20536e7	1.43640
$750$	0	0
$751$	−1.35740e7	−0.878230	−0.439115	−	0.898431i	$-0.644708\pi$
$751$	−1.35740e7	−0.878230	−0.439115	+	0.898431i	$0.644708\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1.98699e7	−1.26861
$756$	0	0
$757$	−470115.	−0.0298171	−0.0149085	−	0.999889i	$-0.504746\pi$
$757$	−470115.	−0.0298171	−0.0149085	+	0.999889i	$0.504746\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.05381e6	0.128558	0.0642789	−	0.997932i	$-0.479525\pi$
$761$	2.05381e6	0.128558	0.0642789	+	0.997932i	$0.479525\pi$
$762$	0	0
$763$	−4.35151e6	−0.270600
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.14244e6	0.192876
$768$	0	0
$769$	2.51201e6	0.153181	0.0765905	−	0.997063i	$-0.475597\pi$
$769$	2.51201e6	0.153181	0.0765905	+	0.997063i	$0.475597\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	8.03339e6	0.483560	0.241780	−	0.970331i	$-0.422269\pi$
$773$	8.03339e6	0.483560	0.241780	+	0.970331i	$0.422269\pi$
$774$	0	0
$775$	−1.08972e7	−0.651719
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5.72508e6	0.338016
$780$	0	0
$781$	2.92438e7	1.71556
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.84174e7	1.64592
$786$	0	0
$787$	3.15766e7	1.81731	0.908653	−	0.417553i	$-0.137112\pi$
$787$	3.15766e7	1.81731	0.908653	+	0.417553i	$0.137112\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−3.60985e7	−2.05139
$792$	0	0
$793$	6.49534e6	0.366791
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.67046e7	1.48915	0.744577	−	0.667536i	$-0.232651\pi$
$797$	2.67046e7	1.48915	0.744577	+	0.667536i	$0.232651\pi$
$798$	0	0
$799$	−4.87763e6	−0.270298
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−3.01208e7	−1.64846
$804$	0	0
$805$	−4.00880e7	−2.18034
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.27526e7	−0.685057	−0.342529	−	0.939507i	$-0.611283\pi$
$809$	−1.27526e7	−0.685057	−0.342529	+	0.939507i	$0.611283\pi$
$810$	0	0
$811$	−2.34690e7	−1.25297	−0.626487	−	0.779432i	$-0.715508\pi$
$811$	−2.34690e7	−1.25297	−0.626487	+	0.779432i	$0.715508\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2.47318e7	1.30425
$816$	0	0
$817$	−3.67290e7	−1.92510
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.58076e7	0.818481	0.409240	−	0.912427i	$-0.365794\pi$
$821$	1.58076e7	0.818481	0.409240	+	0.912427i	$0.365794\pi$
$822$	0	0
$823$	1.62543e7	0.836507	0.418254	−	0.908330i	$-0.362642\pi$
$823$	1.62543e7	0.836507	0.418254	+	0.908330i	$0.362642\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.39257e7	0.708031	0.354016	−	0.935239i	$-0.384816\pi$
$827$	1.39257e7	0.708031	0.354016	+	0.935239i	$0.384816\pi$
$828$	0	0
$829$	2.52491e7	1.27603	0.638013	−	0.770026i	$-0.279757\pi$
$829$	2.52491e7	1.27603	0.638013	+	0.770026i	$0.279757\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.15517e7	−0.576812
$834$	0	0
$835$	2.66969e6	0.132509
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−4.99108e6	−0.244788	−0.122394	−	0.992482i	$-0.539057\pi$
$839$	−4.99108e6	−0.244788	−0.122394	+	0.992482i	$0.539057\pi$
$840$	0	0
$841$	−5.22523e6	−0.254751
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−5.55946e6	−0.267850
$846$	0	0
$847$	−1.60350e7	−0.767999
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−2.62687e7	−1.24341
$852$	0	0
$853$	7.52571e6	0.354140	0.177070	−	0.984198i	$-0.443338\pi$
$853$	7.52571e6	0.354140	0.177070	+	0.984198i	$0.443338\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.73880e7	−1.73892	−0.869461	−	0.494001i	$-0.835534\pi$
$857$	−3.73880e7	−1.73892	−0.869461	+	0.494001i	$0.835534\pi$
$858$	0	0
$859$	−8.43595e6	−0.390078	−0.195039	−	0.980796i	$-0.562483\pi$
$859$	−8.43595e6	−0.390078	−0.195039	+	0.980796i	$0.562483\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.10157e7	0.503482	0.251741	−	0.967795i	$-0.418997\pi$
$863$	1.10157e7	0.503482	0.251741	+	0.967795i	$0.418997\pi$
$864$	0	0
$865$	2.89408e7	1.31513
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−2.78395e7	−1.25058
$870$	0	0
$871$	1.95056e6	0.0871194
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−5.61793e6	−0.248060
$876$	0	0
$877$	−1.48438e7	−0.651698	−0.325849	−	0.945422i	$-0.605650\pi$
$877$	−1.48438e7	−0.651698	−0.325849	+	0.945422i	$0.605650\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−2.90839e7	−1.26245	−0.631224	−	0.775600i	$-0.717447\pi$
$881$	−2.90839e7	−1.26245	−0.631224	+	0.775600i	$0.717447\pi$
$882$	0	0
$883$	−3.79255e7	−1.63693	−0.818464	−	0.574558i	$-0.805174\pi$
$883$	−3.79255e7	−1.63693	−0.818464	+	0.574558i	$0.805174\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.53153e7	−0.653607	−0.326803	−	0.945092i	$-0.605971\pi$
$887$	−1.53153e7	−0.653607	−0.326803	+	0.945092i	$0.605971\pi$
$888$	0	0
$889$	1.86726e7	0.792410
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.66388e7	−0.698220
$894$	0	0
$895$	3.74096e7	1.56108
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.21390e7	0.500940
$900$	0	0
$901$	1.03190e7	0.423472
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.54834e7	−0.628414
$906$	0	0
$907$	−2.68468e7	−1.08361	−0.541806	−	0.840504i	$-0.682259\pi$
$907$	−2.68468e7	−1.08361	−0.541806	+	0.840504i	$0.682259\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−4.76866e6	−0.190371	−0.0951854	−	0.995460i	$-0.530344\pi$
$911$	−4.76866e6	−0.190371	−0.0951854	+	0.995460i	$0.530344\pi$
$912$	0	0
$913$	−2.00103e7	−0.794470
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.20610e7	1.25908
$918$	0	0
$919$	−2.23134e7	−0.871518	−0.435759	−	0.900063i	$-0.643520\pi$
$919$	−2.23134e7	−0.871518	−0.435759	+	0.900063i	$0.643520\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−3.21646e7	−1.24272
$924$	0	0
$925$	−3.35816e7	−1.29047
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−2.31474e7	−0.879961	−0.439980	−	0.898007i	$-0.645015\pi$
$929$	−2.31474e7	−0.879961	−0.439980	+	0.898007i	$0.645015\pi$
$930$	0	0
$931$	−3.94057e7	−1.48999
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−3.07240e7	−1.14934
$936$	0	0
$937$	−3.45453e6	−0.128541	−0.0642703	−	0.997933i	$-0.520472\pi$
$937$	−3.45453e6	−0.128541	−0.0642703	+	0.997933i	$0.520472\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−3.54521e7	−1.30517	−0.652586	−	0.757715i	$-0.726316\pi$
$941$	−3.54521e7	−1.30517	−0.652586	+	0.757715i	$0.726316\pi$
$942$	0	0
$943$	−6.11395e6	−0.223894
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.90255e7	−1.41408	−0.707039	−	0.707174i	$-0.749970\pi$
$947$	−3.90255e7	−1.41408	−0.707039	+	0.707174i	$0.749970\pi$
$948$	0	0
$949$	3.31292e7	1.19411
$950$	0	0
$951$	0	0
$952$	0	0
$953$	2.07434e6	0.0739858	0.0369929	−	0.999316i	$-0.488222\pi$
$953$	2.07434e6	0.0739858	0.0369929	+	0.999316i	$0.488222\pi$
$954$	0	0
$955$	5.60859e7	1.98996
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.42547e7	0.851626
$960$	0	0
$961$	−1.89891e7	−0.663280
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2.80427e7	0.969397
$966$	0	0
$967$	−1.14931e7	−0.395251	−0.197625	−	0.980278i	$-0.563323\pi$
$967$	−1.14931e7	−0.395251	−0.197625	+	0.980278i	$0.563323\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−4.36234e7	−1.48481	−0.742405	−	0.669951i	$-0.766315\pi$
$971$	−4.36234e7	−1.48481	−0.742405	+	0.669951i	$0.766315\pi$
$972$	0	0
$973$	4.05431e7	1.37289
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−3.03599e7	−1.01757	−0.508785	−	0.860894i	$-0.669905\pi$
$977$	−3.03599e7	−1.01757	−0.508785	+	0.860894i	$0.669905\pi$
$978$	0	0
$979$	−5.16374e7	−1.72190
$980$	0	0
$981$	0	0
$982$	0	0
$983$	6.28069e6	0.207312	0.103656	−	0.994613i	$-0.466946\pi$
$983$	6.28069e6	0.207312	0.103656	+	0.994613i	$0.466946\pi$
$984$	0	0
$985$	4.85659e7	1.59493
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3.92238e7	1.27514
$990$	0	0
$991$	1.30907e7	0.423428	0.211714	−	0.977332i	$-0.432096\pi$
$991$	1.30907e7	0.423428	0.211714	+	0.977332i	$0.432096\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−4.02681e6	−0.128945
$996$	0	0
$997$	−3.38507e6	−0.107852	−0.0539262	−	0.998545i	$-0.517174\pi$
$997$	−3.38507e6	−0.107852	−0.0539262	+	0.998545i	$0.517174\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.6.a.d.1.5		5
3.2	odd	2		324.6.a.e.1.1		5
9.2	odd	6		36.6.e.a.13.3	✓	10
9.4	even	3		108.6.e.a.73.1		10
9.5	odd	6		36.6.e.a.25.3	yes	10
9.7	even	3		108.6.e.a.37.1		10
36.7	odd	6		432.6.i.d.145.1		10
36.11	even	6		144.6.i.d.49.3		10
36.23	even	6		144.6.i.d.97.3		10
36.31	odd	6		432.6.i.d.289.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.6.e.a.13.3	✓	10	9.2	odd	6
36.6.e.a.25.3	yes	10	9.5	odd	6
108.6.e.a.37.1		10	9.7	even	3
108.6.e.a.73.1		10	9.4	even	3
144.6.i.d.49.3		10	36.11	even	6
144.6.i.d.97.3		10	36.23	even	6
324.6.a.d.1.5		5	1.1	even	1	trivial
324.6.a.e.1.1		5	3.2	odd	2
432.6.i.d.145.1		10	36.7	odd	6
432.6.i.d.289.1		10	36.31	odd	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:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	324.a (trivial)

\(f(q)\)	\(=\)	\(q+81.4540 q^{5} -179.262 q^{7} +O(q^{10})\)	\(q+81.4540 q^{5} -179.262 q^{7} +500.501 q^{11} -550.491 q^{13} -753.636 q^{17} -2570.83 q^{19} +2745.45 q^{23} +3509.75 q^{25} -3909.72 q^{29} -3104.84 q^{31} -14601.6 q^{35} -9568.10 q^{37} -2226.94 q^{41} +14286.8 q^{43} +6472.14 q^{47} +15328.0 q^{49} -13692.2 q^{53} +40767.8 q^{55} -5708.43 q^{59} -11799.2 q^{61} -44839.7 q^{65} -3543.32 q^{67} +58429.0 q^{71} -60181.3 q^{73} -89720.9 q^{77} -55623.4 q^{79} -39980.6 q^{83} -61386.6 q^{85} -103171. q^{89} +98682.3 q^{91} -209404. q^{95} -165994. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 21 q^{5} - 29 q^{7}+O(q^{10}) \) 5 * q - 21 * q^5 - 29 * q^7	\( 5 q - 21 q^{5} - 29 q^{7} + 177 q^{11} + 181 q^{13} - 1140 q^{17} - 416 q^{19} + 399 q^{23} + 4778 q^{25} - 6033 q^{29} - 2759 q^{31} - 18573 q^{35} - 7586 q^{37} - 18435 q^{41} - 1469 q^{43} - 25155 q^{47} + 4056 q^{49} - 58422 q^{53} + 7389 q^{55} - 90537 q^{59} - 1403 q^{61} - 148407 q^{65} - 13907 q^{67} - 114684 q^{71} + 7600 q^{73} - 211983 q^{77} - 29993 q^{79} - 228951 q^{83} + 49662 q^{85} - 299166 q^{89} + 62465 q^{91} - 394764 q^{95} - 40541 q^{97}+O(q^{100}) \) 5 * q - 21 * q^5 - 29 * q^7 + 177 * q^11 + 181 * q^13 - 1140 * q^17 - 416 * q^19 + 399 * q^23 + 4778 * q^25 - 6033 * q^29 - 2759 * q^31 - 18573 * q^35 - 7586 * q^37 - 18435 * q^41 - 1469 * q^43 - 25155 * q^47 + 4056 * q^49 - 58422 * q^53 + 7389 * q^55 - 90537 * q^59 - 1403 * q^61 - 148407 * q^65 - 13907 * q^67 - 114684 * q^71 + 7600 * q^73 - 211983 * q^77 - 29993 * q^79 - 228951 * q^83 + 49662 * q^85 - 299166 * q^89 + 62465 * q^91 - 394764 * q^95 - 40541 * q^97

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