LMFDB - Embedded newform 324.6.a.d.1.4

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			1.4
Root			$0.900358$ 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)

$f(q)$	$=$	$q+26.3205 q^{5} +63.2575 q^{7} +O(q^{10})$	$q+26.3205 q^{5} +63.2575 q^{7} -98.2387 q^{11} -738.285 q^{13} -250.060 q^{17} +1102.41 q^{19} -4409.41 q^{23} -2432.23 q^{25} +7882.12 q^{29} -4611.29 q^{31} +1664.97 q^{35} +11896.3 q^{37} -10081.2 q^{41} +7037.98 q^{43} -14918.7 q^{47} -12805.5 q^{49} -22451.7 q^{53} -2585.69 q^{55} -10810.5 q^{59} -1189.29 q^{61} -19432.0 q^{65} +59180.5 q^{67} -14326.6 q^{71} -53098.2 q^{73} -6214.33 q^{77} -37391.1 q^{79} -120879. q^{83} -6581.71 q^{85} -97873.2 q^{89} -46702.1 q^{91} +29015.9 q^{95} +106713. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	26.3205	0.470835	0.235418	−	0.971894i	$-0.424354\pi$
$5$	26.3205	0.470835	0.235418	+	0.971894i	$0.424354\pi$
$6$	0	0
$7$	63.2575	0.487940	0.243970	−	0.969783i	$-0.421550\pi$
$7$	63.2575	0.487940	0.243970	+	0.969783i	$0.421550\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−98.2387	−0.244794	−0.122397	−	0.992481i	$-0.539058\pi$
$11$	−98.2387	−0.244794	−0.122397	+	0.992481i	$0.539058\pi$
$12$	0	0
$13$	−738.285	−1.21162	−0.605809	−	0.795610i	$-0.707151\pi$
$13$	−738.285	−1.21162	−0.605809	+	0.795610i	$0.707151\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−250.060	−0.209856	−0.104928	−	0.994480i	$-0.533461\pi$
$17$	−250.060	−0.209856	−0.104928	+	0.994480i	$0.533461\pi$
$18$	0	0
$19$	1102.41	0.700579	0.350290	−	0.936641i	$-0.386083\pi$
$19$	1102.41	0.700579	0.350290	+	0.936641i	$0.386083\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4409.41	−1.73804	−0.869022	−	0.494774i	$-0.835251\pi$
$23$	−4409.41	−1.73804	−0.869022	+	0.494774i	$0.835251\pi$
$24$	0	0
$25$	−2432.23	−0.778314
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7882.12	1.74039	0.870197	−	0.492703i	$-0.163991\pi$
$29$	7882.12	1.74039	0.870197	+	0.492703i	$0.163991\pi$
$30$	0	0
$31$	−4611.29	−0.861823	−0.430912	−	0.902394i	$-0.641808\pi$
$31$	−4611.29	−0.861823	−0.430912	+	0.902394i	$0.641808\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1664.97	0.229740
$36$	0	0
$37$	11896.3	1.42859	0.714297	−	0.699843i	$-0.246747\pi$
$37$	11896.3	1.42859	0.714297	+	0.699843i	$0.246747\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−10081.2	−0.936594	−0.468297	−	0.883571i	$-0.655132\pi$
$41$	−10081.2	−0.936594	−0.468297	+	0.883571i	$0.655132\pi$
$42$	0	0
$43$	7037.98	0.580466	0.290233	−	0.956956i	$-0.406267\pi$
$43$	7037.98	0.580466	0.290233	+	0.956956i	$0.406267\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−14918.7	−0.985111	−0.492555	−	0.870281i	$-0.663937\pi$
$47$	−14918.7	−0.985111	−0.492555	+	0.870281i	$0.663937\pi$
$48$	0	0
$49$	−12805.5	−0.761914
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−22451.7	−1.09789	−0.548947	−	0.835857i	$-0.684971\pi$
$53$	−22451.7	−1.09789	−0.548947	+	0.835857i	$0.684971\pi$
$54$	0	0
$55$	−2585.69	−0.115258
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10810.5	−0.404311	−0.202156	−	0.979353i	$-0.564795\pi$
$59$	−10810.5	−0.404311	−0.202156	+	0.979353i	$0.564795\pi$
$60$	0	0
$61$	−1189.29	−0.0409227	−0.0204614	−	0.999791i	$-0.506514\pi$
$61$	−1189.29	−0.0409227	−0.0204614	+	0.999791i	$0.506514\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−19432.0	−0.570473
$66$	0	0
$67$	59180.5	1.61061	0.805307	−	0.592858i	$-0.202000\pi$
$67$	59180.5	1.61061	0.805307	+	0.592858i	$0.202000\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−14326.6	−0.337284	−0.168642	−	0.985677i	$-0.553938\pi$
$71$	−14326.6	−0.337284	−0.168642	+	0.985677i	$0.553938\pi$
$72$	0	0
$73$	−53098.2	−1.16620	−0.583099	−	0.812401i	$-0.698160\pi$
$73$	−53098.2	−1.16620	−0.583099	+	0.812401i	$0.698160\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6214.33	−0.119445
$78$	0	0
$79$	−37391.1	−0.674063	−0.337032	−	0.941493i	$-0.609423\pi$
$79$	−37391.1	−0.674063	−0.337032	+	0.941493i	$0.609423\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−120879.	−1.92600	−0.962998	−	0.269509i	$-0.913139\pi$
$83$	−120879.	−1.92600	−0.962998	+	0.269509i	$0.913139\pi$
$84$	0	0
$85$	−6581.71	−0.0988078
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−97873.2	−1.30975	−0.654875	−	0.755737i	$-0.727279\pi$
$89$	−97873.2	−1.30975	−0.654875	+	0.755737i	$0.727279\pi$
$90$	0	0
$91$	−46702.1	−0.591198
$92$	0	0
$93$	0	0
$94$	0	0
$95$	29015.9	0.329858
$96$	0	0
$97$	106713.	1.15157	0.575784	−	0.817602i	$-0.304697\pi$
$97$	106713.	1.15157	0.575784	+	0.817602i	$0.304697\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−96472.1	−0.941019	−0.470510	−	0.882395i	$-0.655930\pi$
$101$	−96472.1	−0.941019	−0.470510	+	0.882395i	$0.655930\pi$
$102$	0	0
$103$	29944.4	0.278114	0.139057	−	0.990284i	$-0.455593\pi$
$103$	29944.4	0.278114	0.139057	+	0.990284i	$0.455593\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−22758.9	−0.192173	−0.0960865	−	0.995373i	$-0.530633\pi$
$107$	−22758.9	−0.192173	−0.0960865	+	0.995373i	$0.530633\pi$
$108$	0	0
$109$	−2671.93	−0.0215407	−0.0107703	−	0.999942i	$-0.503428\pi$
$109$	−2671.93	−0.0215407	−0.0107703	+	0.999942i	$0.503428\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−131339.	−0.967606	−0.483803	−	0.875177i	$-0.660745\pi$
$113$	−131339.	−0.967606	−0.483803	+	0.875177i	$0.660745\pi$
$114$	0	0
$115$	−116058.	−0.818332
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−15818.2	−0.102397
$120$	0	0
$121$	−151400.	−0.940076
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−146269.	−0.837293
$126$	0	0
$127$	236012.	1.29845	0.649223	−	0.760598i	$-0.275094\pi$
$127$	236012.	1.29845	0.649223	+	0.760598i	$0.275094\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−360366.	−1.83470	−0.917352	−	0.398078i	$-0.869677\pi$
$131$	−360366.	−1.83470	−0.917352	+	0.398078i	$0.869677\pi$
$132$	0	0
$133$	69735.4	0.341841
$134$	0	0
$135$	0	0
$136$	0	0
$137$	237666.	1.08185	0.540923	−	0.841072i	$-0.318075\pi$
$137$	237666.	1.08185	0.540923	+	0.841072i	$0.318075\pi$
$138$	0	0
$139$	163347.	0.717090	0.358545	−	0.933513i	$-0.383273\pi$
$139$	163347.	0.717090	0.358545	+	0.933513i	$0.383273\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	72528.2	0.296597
$144$	0	0
$145$	207461.	0.819440
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−335607.	−1.23841	−0.619205	−	0.785229i	$-0.712545\pi$
$149$	−335607.	−1.23841	−0.619205	+	0.785229i	$0.712545\pi$
$150$	0	0
$151$	−135577.	−0.483886	−0.241943	−	0.970290i	$-0.577785\pi$
$151$	−135577.	−0.483886	−0.241943	+	0.970290i	$0.577785\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−121371.	−0.405777
$156$	0	0
$157$	−425432.	−1.37747	−0.688733	−	0.725016i	$-0.741833\pi$
$157$	−425432.	−1.37747	−0.688733	+	0.725016i	$0.741833\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−278928.	−0.848062
$162$	0	0
$163$	−158679.	−0.467789	−0.233895	−	0.972262i	$-0.575147\pi$
$163$	−158679.	−0.467789	−0.233895	+	0.972262i	$0.575147\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	518446.	1.43851	0.719254	−	0.694747i	$-0.244484\pi$
$167$	518446.	1.43851	0.719254	+	0.694747i	$0.244484\pi$
$168$	0	0
$169$	173772.	0.468019
$170$	0	0
$171$	0	0
$172$	0	0
$173$	153178.	0.389118	0.194559	−	0.980891i	$-0.437672\pi$
$173$	153178.	0.389118	0.194559	+	0.980891i	$0.437672\pi$
$174$	0	0
$175$	−153857.	−0.379771
$176$	0	0
$177$	0	0
$178$	0	0
$179$	736845.	1.71887	0.859437	−	0.511242i	$-0.170815\pi$
$179$	736845.	1.71887	0.859437	+	0.511242i	$0.170815\pi$
$180$	0	0
$181$	28183.8	0.0639445	0.0319722	−	0.999489i	$-0.489821\pi$
$181$	28183.8	0.0639445	0.0319722	+	0.999489i	$0.489821\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	313118.	0.672633
$186$	0	0
$187$	24565.6	0.0513716
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−99321.0	−0.196996	−0.0984981	−	0.995137i	$-0.531404\pi$
$191$	−99321.0	−0.196996	−0.0984981	+	0.995137i	$0.531404\pi$
$192$	0	0
$193$	417272.	0.806355	0.403177	−	0.915122i	$-0.367906\pi$
$193$	417272.	0.806355	0.403177	+	0.915122i	$0.367906\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−254565.	−0.467340	−0.233670	−	0.972316i	$-0.575073\pi$
$197$	−254565.	−0.467340	−0.233670	+	0.972316i	$0.575073\pi$
$198$	0	0
$199$	−599702.	−1.07350	−0.536751	−	0.843741i	$-0.680348\pi$
$199$	−599702.	−1.07350	−0.536751	+	0.843741i	$0.680348\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	498603.	0.849209
$204$	0	0
$205$	−265341.	−0.440982
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−108299.	−0.171498
$210$	0	0
$211$	756584.	1.16991	0.584953	−	0.811067i	$-0.301113\pi$
$211$	756584.	1.16991	0.584953	+	0.811067i	$0.301113\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	185243.	0.273304
$216$	0	0
$217$	−291699.	−0.420518
$218$	0	0
$219$	0	0
$220$	0	0
$221$	184616.	0.254266
$222$	0	0
$223$	−111205.	−0.149748	−0.0748741	−	0.997193i	$-0.523855\pi$
$223$	−111205.	−0.149748	−0.0748741	+	0.997193i	$0.523855\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	490760.	0.632127	0.316063	−	0.948738i	$-0.397639\pi$
$227$	490760.	0.632127	0.316063	+	0.948738i	$0.397639\pi$
$228$	0	0
$229$	761021.	0.958977	0.479488	−	0.877548i	$-0.340822\pi$
$229$	761021.	0.958977	0.479488	+	0.877548i	$0.340822\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	356267.	0.429918	0.214959	−	0.976623i	$-0.431038\pi$
$233$	356267.	0.429918	0.214959	+	0.976623i	$0.431038\pi$
$234$	0	0
$235$	−392667.	−0.463825
$236$	0	0
$237$	0	0
$238$	0	0
$239$	906239.	1.02624	0.513119	−	0.858318i	$-0.328490\pi$
$239$	906239.	1.02624	0.513119	+	0.858318i	$0.328490\pi$
$240$	0	0
$241$	522576.	0.579571	0.289785	−	0.957092i	$-0.406416\pi$
$241$	522576.	0.579571	0.289785	+	0.957092i	$0.406416\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−337047.	−0.358736
$246$	0	0
$247$	−813890.	−0.848835
$248$	0	0
$249$	0	0
$250$	0	0
$251$	625287.	0.626462	0.313231	−	0.949677i	$-0.398589\pi$
$251$	625287.	0.626462	0.313231	+	0.949677i	$0.398589\pi$
$252$	0	0
$253$	433175.	0.425463
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.30221e6	1.22984	0.614921	−	0.788589i	$-0.289188\pi$
$257$	1.30221e6	1.22984	0.614921	+	0.788589i	$0.289188\pi$
$258$	0	0
$259$	752532.	0.697069
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.43352e6	1.27795	0.638977	−	0.769226i	$-0.279358\pi$
$263$	1.43352e6	1.27795	0.638977	+	0.769226i	$0.279358\pi$
$264$	0	0
$265$	−590941.	−0.516927
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.52329e6	1.28352	0.641758	−	0.766907i	$-0.278205\pi$
$269$	1.52329e6	1.28352	0.641758	+	0.766907i	$0.278205\pi$
$270$	0	0
$271$	1.60134e6	1.32453	0.662263	−	0.749271i	$-0.269596\pi$
$271$	1.60134e6	1.32453	0.662263	+	0.749271i	$0.269596\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	238939.	0.190527
$276$	0	0
$277$	−893943.	−0.700020	−0.350010	−	0.936746i	$-0.613822\pi$
$277$	−893943.	−0.700020	−0.350010	+	0.936746i	$0.613822\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	28400.1	0.0214563	0.0107281	−	0.999942i	$-0.496585\pi$
$281$	28400.1	0.0214563	0.0107281	+	0.999942i	$0.496585\pi$
$282$	0	0
$283$	−1.10957e6	−0.823544	−0.411772	−	0.911287i	$-0.635090\pi$
$283$	−1.10957e6	−0.823544	−0.411772	+	0.911287i	$0.635090\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−637709.	−0.457002
$288$	0	0
$289$	−1.35733e6	−0.955960
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−2.45768e6	−1.67246	−0.836230	−	0.548378i	$-0.815245\pi$
$293$	−2.45768e6	−1.67246	−0.836230	+	0.548378i	$0.815245\pi$
$294$	0	0
$295$	−284538.	−0.190364
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.25540e6	2.10585
$300$	0	0
$301$	445205.	0.283233
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−31302.8	−0.0192679
$306$	0	0
$307$	1.77336e6	1.07387	0.536934	−	0.843624i	$-0.319583\pi$
$307$	1.77336e6	1.07387	0.536934	+	0.843624i	$0.319583\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.50571e6	−0.882756	−0.441378	−	0.897321i	$-0.645510\pi$
$311$	−1.50571e6	−0.882756	−0.441378	+	0.897321i	$0.645510\pi$
$312$	0	0
$313$	−812705.	−0.468891	−0.234446	−	0.972129i	$-0.575327\pi$
$313$	−812705.	−0.468891	−0.234446	+	0.972129i	$0.575327\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.04760e6	−0.585525	−0.292763	−	0.956185i	$-0.594575\pi$
$317$	−1.04760e6	−0.585525	−0.292763	+	0.956185i	$0.594575\pi$
$318$	0	0
$319$	−774329.	−0.426038
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−275668.	−0.147021
$324$	0	0
$325$	1.79568e6	0.943020
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−943717.	−0.480675
$330$	0	0
$331$	−68005.3	−0.0341172	−0.0170586	−	0.999854i	$-0.505430\pi$
$331$	−68005.3	−0.0341172	−0.0170586	+	0.999854i	$0.505430\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.55766e6	0.758334
$336$	0	0
$337$	−416472.	−0.199761	−0.0998806	−	0.994999i	$-0.531846\pi$
$337$	−416472.	−0.199761	−0.0998806	+	0.994999i	$0.531846\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	453007.	0.210969
$342$	0	0
$343$	−1.87321e6	−0.859709
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.80740e6	1.25164	0.625822	−	0.779966i	$-0.284764\pi$
$347$	2.80740e6	1.25164	0.625822	+	0.779966i	$0.284764\pi$
$348$	0	0
$349$	−481973.	−0.211816	−0.105908	−	0.994376i	$-0.533775\pi$
$349$	−481973.	−0.211816	−0.105908	+	0.994376i	$0.533775\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−4.00037e6	−1.70869	−0.854346	−	0.519705i	$-0.826042\pi$
$353$	−4.00037e6	−1.70869	−0.854346	+	0.519705i	$0.826042\pi$
$354$	0	0
$355$	−377082.	−0.158805
$356$	0	0
$357$	0	0
$358$	0	0
$359$	800858.	0.327959	0.163979	−	0.986464i	$-0.447567\pi$
$359$	800858.	0.327959	0.163979	+	0.986464i	$0.447567\pi$
$360$	0	0
$361$	−1.26080e6	−0.509189
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1.39757e6	−0.549087
$366$	0	0
$367$	3.00972e6	1.16644	0.583218	−	0.812316i	$-0.301793\pi$
$367$	3.00972e6	1.16644	0.583218	+	0.812316i	$0.301793\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.42024e6	−0.535707
$372$	0	0
$373$	1.59331e6	0.592962	0.296481	−	0.955039i	$-0.404187\pi$
$373$	1.59331e6	0.592962	0.296481	+	0.955039i	$0.404187\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5.81925e6	−2.10869
$378$	0	0
$379$	−297930.	−0.106541	−0.0532703	−	0.998580i	$-0.516965\pi$
$379$	−297930.	−0.106541	−0.0532703	+	0.998580i	$0.516965\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1.29030e6	0.449464	0.224732	−	0.974421i	$-0.427849\pi$
$383$	1.29030e6	0.449464	0.224732	+	0.974421i	$0.427849\pi$
$384$	0	0
$385$	−163564.	−0.0562389
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−4.99873e6	−1.67489	−0.837444	−	0.546523i	$-0.815951\pi$
$389$	−4.99873e6	−1.67489	−0.837444	+	0.546523i	$0.815951\pi$
$390$	0	0
$391$	1.10262e6	0.364740
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−984153.	−0.317373
$396$	0	0
$397$	−558980.	−0.178000	−0.0890000	−	0.996032i	$-0.528367\pi$
$397$	−558980.	−0.178000	−0.0890000	+	0.996032i	$0.528367\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−94207.2	−0.0292565	−0.0146283	−	0.999893i	$-0.504656\pi$
$401$	−94207.2	−0.0292565	−0.0146283	+	0.999893i	$0.504656\pi$
$402$	0	0
$403$	3.40445e6	1.04420
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.16868e6	−0.349711
$408$	0	0
$409$	5.46230e6	1.61461	0.807304	−	0.590135i	$-0.200926\pi$
$409$	5.46230e6	1.61461	0.807304	+	0.590135i	$0.200926\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−683845.	−0.197280
$414$	0	0
$415$	−3.18159e6	−0.906827
$416$	0	0
$417$	0	0
$418$	0	0
$419$	836972.	0.232903	0.116452	−	0.993196i	$-0.462848\pi$
$419$	836972.	0.232903	0.116452	+	0.993196i	$0.462848\pi$
$420$	0	0
$421$	−6.42809e6	−1.76757	−0.883785	−	0.467894i	$-0.845013\pi$
$421$	−6.42809e6	−1.76757	−0.883785	+	0.467894i	$0.845013\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	608204.	0.163334
$426$	0	0
$427$	−75231.7	−0.0199678
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5.11509e6	1.32636	0.663178	−	0.748462i	$-0.269207\pi$
$431$	5.11509e6	1.32636	0.663178	+	0.748462i	$0.269207\pi$
$432$	0	0
$433$	−1.39089e6	−0.356512	−0.178256	−	0.983984i	$-0.557045\pi$
$433$	−1.39089e6	−0.356512	−0.178256	+	0.983984i	$0.557045\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.86095e6	−1.21764
$438$	0	0
$439$	−3.52532e6	−0.873045	−0.436523	−	0.899693i	$-0.643790\pi$
$439$	−3.52532e6	−0.873045	−0.436523	+	0.899693i	$0.643790\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2.04432e6	−0.494924	−0.247462	−	0.968898i	$-0.579597\pi$
$443$	−2.04432e6	−0.494924	−0.247462	+	0.968898i	$0.579597\pi$
$444$	0	0
$445$	−2.57607e6	−0.616677
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−7.70508e6	−1.80369	−0.901844	−	0.432062i	$-0.857786\pi$
$449$	−7.70508e6	−1.80369	−0.901844	+	0.432062i	$0.857786\pi$
$450$	0	0
$451$	990361.	0.229273
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.22922e6	−0.278357
$456$	0	0
$457$	8.57692e6	1.92106	0.960529	−	0.278179i	$-0.0897308\pi$
$457$	8.57692e6	1.92106	0.960529	+	0.278179i	$0.0897308\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−1.43263e6	−0.313966	−0.156983	−	0.987601i	$-0.550177\pi$
$461$	−1.43263e6	−0.313966	−0.156983	+	0.987601i	$0.550177\pi$
$462$	0	0
$463$	2.94447e6	0.638343	0.319172	−	0.947697i	$-0.396595\pi$
$463$	2.94447e6	0.638343	0.319172	+	0.947697i	$0.396595\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4.96160e6	−1.05276	−0.526380	−	0.850249i	$-0.676451\pi$
$467$	−4.96160e6	−1.05276	−0.526380	+	0.850249i	$0.676451\pi$
$468$	0	0
$469$	3.74361e6	0.785884
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−691402.	−0.142095
$474$	0	0
$475$	−2.68130e6	−0.545271
$476$	0	0
$477$	0	0
$478$	0	0
$479$	241513.	0.0480953	0.0240476	−	0.999711i	$-0.492345\pi$
$479$	241513.	0.0480953	0.0240476	+	0.999711i	$0.492345\pi$
$480$	0	0
$481$	−8.78289e6	−1.73091
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.80875e6	0.542199
$486$	0	0
$487$	325055.	0.0621061	0.0310530	−	0.999518i	$-0.490114\pi$
$487$	325055.	0.0621061	0.0310530	+	0.999518i	$0.490114\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−4.98047e6	−0.932323	−0.466162	−	0.884700i	$-0.654363\pi$
$491$	−4.98047e6	−0.932323	−0.466162	+	0.884700i	$0.654363\pi$
$492$	0	0
$493$	−1.97100e6	−0.365233
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−906262.	−0.164575
$498$	0	0
$499$	−5.86495e6	−1.05442	−0.527209	−	0.849736i	$-0.676761\pi$
$499$	−5.86495e6	−1.05442	−0.527209	+	0.849736i	$0.676761\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.36396e6	0.240370	0.120185	−	0.992751i	$-0.461651\pi$
$503$	1.36396e6	0.240370	0.120185	+	0.992751i	$0.461651\pi$
$504$	0	0
$505$	−2.53919e6	−0.443065
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−3.70047e6	−0.633085	−0.316542	−	0.948578i	$-0.602522\pi$
$509$	−3.70047e6	−0.633085	−0.316542	+	0.948578i	$0.602522\pi$
$510$	0	0
$511$	−3.35885e6	−0.569035
$512$	0	0
$513$	0	0
$514$	0	0
$515$	788153.	0.130946
$516$	0	0
$517$	1.46559e6	0.241149
$518$	0	0
$519$	0	0
$520$	0	0
$521$	285260.	0.0460412	0.0230206	−	0.999735i	$-0.492672\pi$
$521$	285260.	0.0460412	0.0230206	+	0.999735i	$0.492672\pi$
$522$	0	0
$523$	8.28809e6	1.32495	0.662476	−	0.749083i	$-0.269506\pi$
$523$	8.28809e6	1.32495	0.662476	+	0.749083i	$0.269506\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.15310e6	0.180859
$528$	0	0
$529$	1.30065e7	2.02080
$530$	0	0
$531$	0	0
$532$	0	0
$533$	7.44278e6	1.13479
$534$	0	0
$535$	−599027.	−0.0904819
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.25800e6	0.186512
$540$	0	0
$541$	1.29750e7	1.90596	0.952982	−	0.303028i	$-0.0979975\pi$
$541$	1.29750e7	1.90596	0.952982	+	0.303028i	$0.0979975\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−70326.5	−0.0101421
$546$	0	0
$547$	−9.31601e6	−1.33126	−0.665628	−	0.746284i	$-0.731836\pi$
$547$	−9.31601e6	−1.33126	−0.665628	+	0.746284i	$0.731836\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	8.68929e6	1.21928
$552$	0	0
$553$	−2.36527e6	−0.328903
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.18234e6	0.571192	0.285596	−	0.958350i	$-0.407809\pi$
$557$	4.18234e6	0.571192	0.285596	+	0.958350i	$0.407809\pi$
$558$	0	0
$559$	−5.19604e6	−0.703304
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.31319e6	0.440531	0.220265	−	0.975440i	$-0.429308\pi$
$563$	3.31319e6	0.440531	0.220265	+	0.975440i	$0.429308\pi$
$564$	0	0
$565$	−3.45692e6	−0.455583
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5.85391e6	0.757993	0.378996	−	0.925398i	$-0.376269\pi$
$569$	5.85391e6	0.757993	0.378996	+	0.925398i	$0.376269\pi$
$570$	0	0
$571$	1.35086e7	1.73388	0.866941	−	0.498411i	$-0.166083\pi$
$571$	1.35086e7	1.73388	0.866941	+	0.498411i	$0.166083\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.07247e7	1.35274
$576$	0	0
$577$	−1.12898e7	−1.41171	−0.705855	−	0.708356i	$-0.749437\pi$
$577$	−1.12898e7	−1.41171	−0.705855	+	0.708356i	$0.749437\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−7.64649e6	−0.939771
$582$	0	0
$583$	2.20563e6	0.268758
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.98280e6	−0.237510	−0.118755	−	0.992924i	$-0.537890\pi$
$587$	−1.98280e6	−0.237510	−0.118755	+	0.992924i	$0.537890\pi$
$588$	0	0
$589$	−5.08351e6	−0.603776
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−1.22629e7	−1.43204	−0.716021	−	0.698078i	$-0.754039\pi$
$593$	−1.22629e7	−1.43204	−0.716021	+	0.698078i	$0.754039\pi$
$594$	0	0
$595$	−416342.	−0.0482123
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.56995e7	1.78780	0.893902	−	0.448263i	$-0.147957\pi$
$599$	1.56995e7	1.78780	0.893902	+	0.448263i	$0.147957\pi$
$600$	0	0
$601$	−5.31944e6	−0.600731	−0.300366	−	0.953824i	$-0.597109\pi$
$601$	−5.31944e6	−0.600731	−0.300366	+	0.953824i	$0.597109\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.98493e6	−0.442621
$606$	0	0
$607$	4.36964e6	0.481365	0.240682	−	0.970604i	$-0.422629\pi$
$607$	4.36964e6	0.481365	0.240682	+	0.970604i	$0.422629\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.10142e7	1.19358
$612$	0	0
$613$	−7.62875e6	−0.819978	−0.409989	−	0.912091i	$-0.634467\pi$
$613$	−7.62875e6	−0.819978	−0.409989	+	0.912091i	$0.634467\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.43300e7	1.51542	0.757708	−	0.652594i	$-0.226319\pi$
$617$	1.43300e7	1.51542	0.757708	+	0.652594i	$0.226319\pi$
$618$	0	0
$619$	1.08181e7	1.13482	0.567408	−	0.823437i	$-0.307946\pi$
$619$	1.08181e7	1.13482	0.567408	+	0.823437i	$0.307946\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−6.19121e6	−0.639080
$624$	0	0
$625$	3.75085e6	0.384087
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−2.97480e6	−0.299800
$630$	0	0
$631$	7.96579e6	0.796444	0.398222	−	0.917289i	$-0.369627\pi$
$631$	7.96579e6	0.796444	0.398222	+	0.917289i	$0.369627\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	6.21194e6	0.611354
$636$	0	0
$637$	9.45411e6	0.923149
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−6.80448e6	−0.654108	−0.327054	−	0.945006i	$-0.606056\pi$
$641$	−6.80448e6	−0.654108	−0.327054	+	0.945006i	$0.606056\pi$
$642$	0	0
$643$	−5.32929e6	−0.508325	−0.254163	−	0.967161i	$-0.581800\pi$
$643$	−5.32929e6	−0.508325	−0.254163	+	0.967161i	$0.581800\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.03021e6	0.472417	0.236208	−	0.971702i	$-0.424095\pi$
$647$	5.03021e6	0.472417	0.236208	+	0.971702i	$0.424095\pi$
$648$	0	0
$649$	1.06201e6	0.0989731
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.13341e7	1.04017	0.520084	−	0.854115i	$-0.325901\pi$
$653$	1.13341e7	1.04017	0.520084	+	0.854115i	$0.325901\pi$
$654$	0	0
$655$	−9.48502e6	−0.863843
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−3.82007e6	−0.342655	−0.171328	−	0.985214i	$-0.554806\pi$
$659$	−3.82007e6	−0.342655	−0.171328	+	0.985214i	$0.554806\pi$
$660$	0	0
$661$	926573.	0.0824852	0.0412426	−	0.999149i	$-0.486868\pi$
$661$	926573.	0.0824852	0.0412426	+	0.999149i	$0.486868\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.83547e6	0.160951
$666$	0	0
$667$	−3.47555e7	−3.02488
$668$	0	0
$669$	0	0
$670$	0	0
$671$	116835.	0.0100176
$672$	0	0
$673$	4.90291e6	0.417269	0.208635	−	0.977994i	$-0.433098\pi$
$673$	4.90291e6	0.417269	0.208635	+	0.977994i	$0.433098\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.47440e7	1.23636	0.618179	−	0.786037i	$-0.287871\pi$
$677$	1.47440e7	1.23636	0.618179	+	0.786037i	$0.287871\pi$
$678$	0	0
$679$	6.75041e6	0.561896
$680$	0	0
$681$	0	0
$682$	0	0
$683$	6.53893e6	0.536358	0.268179	−	0.963369i	$-0.413578\pi$
$683$	6.53893e6	0.536358	0.268179	+	0.963369i	$0.413578\pi$
$684$	0	0
$685$	6.25548e6	0.509371
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.65758e7	1.33023
$690$	0	0
$691$	−1.09050e7	−0.868818	−0.434409	−	0.900716i	$-0.643043\pi$
$691$	−1.09050e7	−0.868818	−0.434409	+	0.900716i	$0.643043\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.29937e6	0.337631
$696$	0	0
$697$	2.52090e6	0.196550
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.89162e7	1.45391	0.726957	−	0.686683i	$-0.240934\pi$
$701$	1.89162e7	1.45391	0.726957	+	0.686683i	$0.240934\pi$
$702$	0	0
$703$	1.31146e7	1.00084
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6.10258e6	−0.459161
$708$	0	0
$709$	−1.92732e7	−1.43992	−0.719960	−	0.694015i	$-0.755840\pi$
$709$	−1.92732e7	−1.43992	−0.719960	+	0.694015i	$0.755840\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.03331e7	1.49789
$714$	0	0
$715$	1.90898e6	0.139648
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.42859e7	−1.03059	−0.515293	−	0.857014i	$-0.672317\pi$
$719$	−1.42859e7	−1.03059	−0.515293	+	0.857014i	$0.672317\pi$
$720$	0	0
$721$	1.89421e6	0.135703
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.91711e7	−1.35457
$726$	0	0
$727$	2.82771e7	1.98426	0.992132	−	0.125198i	$-0.0399567\pi$
$727$	2.82771e7	1.98426	0.992132	+	0.125198i	$0.0399567\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.75992e6	−0.121815
$732$	0	0
$733$	9.87699e6	0.678992	0.339496	−	0.940607i	$-0.389743\pi$
$733$	9.87699e6	0.678992	0.339496	+	0.940607i	$0.389743\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.81381e6	−0.394269
$738$	0	0
$739$	6.31081e6	0.425083	0.212542	−	0.977152i	$-0.431826\pi$
$739$	6.31081e6	0.425083	0.212542	+	0.977152i	$0.431826\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	63059.8	0.00419064	0.00209532	−	0.999998i	$-0.499333\pi$
$743$	63059.8	0.00419064	0.00209532	+	0.999998i	$0.499333\pi$
$744$	0	0
$745$	−8.83333e6	−0.583088
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1.43967e6	−0.0937690
$750$	0	0
$751$	−1.73837e7	−1.12471	−0.562356	−	0.826895i	$-0.690105\pi$
$751$	−1.73837e7	−1.12471	−0.562356	+	0.826895i	$0.690105\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−3.56845e6	−0.227831
$756$	0	0
$757$	9.43592e6	0.598473	0.299237	−	0.954179i	$-0.403268\pi$
$757$	9.43592e6	0.598473	0.299237	+	0.954179i	$0.403268\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.61282e7	−1.00954	−0.504771	−	0.863254i	$-0.668423\pi$
$761$	−1.61282e7	−1.00954	−0.504771	+	0.863254i	$0.668423\pi$
$762$	0	0
$763$	−169020.	−0.0105106
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7.98124e6	0.489871
$768$	0	0
$769$	−2.56863e7	−1.56634	−0.783169	−	0.621809i	$-0.786398\pi$
$769$	−2.56863e7	−1.56634	−0.783169	+	0.621809i	$0.786398\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	2.01235e7	1.21131	0.605654	−	0.795728i	$-0.292912\pi$
$773$	2.01235e7	1.21131	0.605654	+	0.795728i	$0.292912\pi$
$774$	0	0
$775$	1.12157e7	0.670769
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.11135e7	−0.656158
$780$	0	0
$781$	1.40742e6	0.0825652
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1.11976e7	−0.648559
$786$	0	0
$787$	5.75281e6	0.331088	0.165544	−	0.986202i	$-0.447062\pi$
$787$	5.75281e6	0.331088	0.165544	+	0.986202i	$0.447062\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−8.30819e6	−0.472134
$792$	0	0
$793$	878038.	0.0495827
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.01964e7	0.568595	0.284298	−	0.958736i	$-0.408240\pi$
$797$	1.01964e7	0.568595	0.284298	+	0.958736i	$0.408240\pi$
$798$	0	0
$799$	3.73056e6	0.206732
$800$	0	0
$801$	0	0
$802$	0	0
$803$	5.21629e6	0.285478
$804$	0	0
$805$	−7.34152e6	−0.399297
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−6.00420e6	−0.322540	−0.161270	−	0.986910i	$-0.551559\pi$
$809$	−6.00420e6	−0.322540	−0.161270	+	0.986910i	$0.551559\pi$
$810$	0	0
$811$	3.23880e7	1.72915	0.864574	−	0.502506i	$-0.167589\pi$
$811$	3.23880e7	1.72915	0.864574	+	0.502506i	$0.167589\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−4.17651e6	−0.220252
$816$	0	0
$817$	7.75871e6	0.406663
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.56747e7	−0.811600	−0.405800	−	0.913962i	$-0.633007\pi$
$821$	−1.56747e7	−0.811600	−0.405800	+	0.913962i	$0.633007\pi$
$822$	0	0
$823$	−2.71466e7	−1.39706	−0.698532	−	0.715579i	$-0.746163\pi$
$823$	−2.71466e7	−1.39706	−0.698532	+	0.715579i	$0.746163\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.28543e7	−1.16199	−0.580997	−	0.813906i	$-0.697337\pi$
$827$	−2.28543e7	−1.16199	−0.580997	+	0.813906i	$0.697337\pi$
$828$	0	0
$829$	−2.05418e7	−1.03813	−0.519066	−	0.854734i	$-0.673720\pi$
$829$	−2.05418e7	−1.03813	−0.519066	+	0.854734i	$0.673720\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.20214e6	0.159893
$834$	0	0
$835$	1.36458e7	0.677301
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.47887e7	0.725313	0.362657	−	0.931923i	$-0.381870\pi$
$839$	1.47887e7	0.725313	0.362657	+	0.931923i	$0.381870\pi$
$840$	0	0
$841$	4.16166e7	2.02897
$842$	0	0
$843$	0	0
$844$	0	0
$845$	4.57378e6	0.220360
$846$	0	0
$847$	−9.57719e6	−0.458701
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−5.24558e7	−2.48296
$852$	0	0
$853$	−1.05245e7	−0.495253	−0.247627	−	0.968856i	$-0.579651\pi$
$853$	−1.05245e7	−0.495253	−0.247627	+	0.968856i	$0.579651\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.48671e7	−0.691473	−0.345736	−	0.938332i	$-0.612371\pi$
$857$	−1.48671e7	−0.691473	−0.345736	+	0.938332i	$0.612371\pi$
$858$	0	0
$859$	−8.64679e6	−0.399827	−0.199913	−	0.979814i	$-0.564066\pi$
$859$	−8.64679e6	−0.399827	−0.199913	+	0.979814i	$0.564066\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−2.38401e7	−1.08964	−0.544818	−	0.838555i	$-0.683401\pi$
$863$	−2.38401e7	−1.08964	−0.544818	+	0.838555i	$0.683401\pi$
$864$	0	0
$865$	4.03172e6	0.183210
$866$	0	0
$867$	0	0
$868$	0	0
$869$	3.67325e6	0.165007
$870$	0	0
$871$	−4.36921e7	−1.95145
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−9.25261e6	−0.408549
$876$	0	0
$877$	−38315.7	−0.00168220	−0.000841100	−	1.00000i	$-0.500268\pi$
$877$	−38315.7	−0.00168220	−0.000841100		1.00000i	$0.500268\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2.55072e7	1.10719	0.553596	−	0.832785i	$-0.313255\pi$
$881$	2.55072e7	1.10719	0.553596	+	0.832785i	$0.313255\pi$
$882$	0	0
$883$	−3.37096e7	−1.45496	−0.727482	−	0.686127i	$-0.759309\pi$
$883$	−3.37096e7	−1.45496	−0.727482	+	0.686127i	$0.759309\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.64222e6	0.112761	0.0563806	−	0.998409i	$-0.482044\pi$
$887$	2.64222e6	0.112761	0.0563806	+	0.998409i	$0.482044\pi$
$888$	0	0
$889$	1.49295e7	0.633564
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.64464e7	−0.690148
$894$	0	0
$895$	1.93941e7	0.809306
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−3.63467e7	−1.49991
$900$	0	0
$901$	5.61429e6	0.230400
$902$	0	0
$903$	0	0
$904$	0	0
$905$	741812.	0.0301073
$906$	0	0
$907$	7.13018e6	0.287795	0.143897	−	0.989593i	$-0.454037\pi$
$907$	7.13018e6	0.287795	0.143897	+	0.989593i	$0.454037\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−3.13154e7	−1.25015	−0.625075	−	0.780564i	$-0.714932\pi$
$911$	−3.13154e7	−1.25015	−0.625075	+	0.780564i	$0.714932\pi$
$912$	0	0
$913$	1.18750e7	0.471472
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.27959e7	−0.895226
$918$	0	0
$919$	2.28042e6	0.0890687	0.0445344	−	0.999008i	$-0.485820\pi$
$919$	2.28042e6	0.0890687	0.0445344	+	0.999008i	$0.485820\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.05771e7	0.408660
$924$	0	0
$925$	−2.89346e7	−1.11189
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.05725e7	−0.401919	−0.200959	−	0.979600i	$-0.564406\pi$
$929$	−1.05725e7	−0.401919	−0.200959	+	0.979600i	$0.564406\pi$
$930$	0	0
$931$	−1.41168e7	−0.533781
$932$	0	0
$933$	0	0
$934$	0	0
$935$	646579.	0.0241876
$936$	0	0
$937$	−3.99327e7	−1.48587	−0.742934	−	0.669365i	$-0.766566\pi$
$937$	−3.99327e7	−1.48587	−0.742934	+	0.669365i	$0.766566\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	8.32632e6	0.306534	0.153267	−	0.988185i	$-0.451021\pi$
$941$	8.32632e6	0.306534	0.153267	+	0.988185i	$0.451021\pi$
$942$	0	0
$943$	4.44520e7	1.62784
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.75667e7	−0.998873	−0.499437	−	0.866350i	$-0.666460\pi$
$947$	−2.75667e7	−0.998873	−0.499437	+	0.866350i	$0.666460\pi$
$948$	0	0
$949$	3.92016e7	1.41299
$950$	0	0
$951$	0	0
$952$	0	0
$953$	3.81280e7	1.35992	0.679958	−	0.733251i	$-0.261998\pi$
$953$	3.81280e7	1.35992	0.679958	+	0.733251i	$0.261998\pi$
$954$	0	0
$955$	−2.61418e6	−0.0927528
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.50341e7	0.527876
$960$	0	0
$961$	−7.36516e6	−0.257261
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.09828e7	0.379660
$966$	0	0
$967$	−2.72711e6	−0.0937857	−0.0468928	−	0.998900i	$-0.514932\pi$
$967$	−2.72711e6	−0.0937857	−0.0468928	+	0.998900i	$0.514932\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−1.26521e7	−0.430641	−0.215320	−	0.976543i	$-0.569080\pi$
$971$	−1.26521e7	−0.430641	−0.215320	+	0.976543i	$0.569080\pi$
$972$	0	0
$973$	1.03329e7	0.349897
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.37252e7	1.13036	0.565182	−	0.824967i	$-0.308806\pi$
$977$	3.37252e7	1.13036	0.565182	+	0.824967i	$0.308806\pi$
$978$	0	0
$979$	9.61493e6	0.320619
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−1.18743e7	−0.391944	−0.195972	−	0.980610i	$-0.562786\pi$
$983$	−1.18743e7	−0.391944	−0.195972	+	0.980610i	$0.562786\pi$
$984$	0	0
$985$	−6.70027e6	−0.220040
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3.10333e7	−1.00888
$990$	0	0
$991$	−2.31910e7	−0.750128	−0.375064	−	0.926999i	$-0.622379\pi$
$991$	−2.31910e7	−0.750128	−0.375064	+	0.926999i	$0.622379\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1.57845e7	−0.505443
$996$	0	0
$997$	3.86264e7	1.23068	0.615342	−	0.788260i	$-0.289018\pi$
$997$	3.86264e7	1.23068	0.615342	+	0.788260i	$0.289018\pi$
$998$	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.1.4		5
3.2	odd	2		324.6.a.e.1.2		5
9.2	odd	6		36.6.e.a.13.4	✓	10
9.4	even	3		108.6.e.a.73.2		10
9.5	odd	6		36.6.e.a.25.4	yes	10
9.7	even	3		108.6.e.a.37.2		10
36.7	odd	6		432.6.i.d.145.2		10
36.11	even	6		144.6.i.d.49.2		10
36.23	even	6		144.6.i.d.97.2		10
36.31	odd	6		432.6.i.d.289.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.6.e.a.13.4	✓	10	9.2	odd	6
36.6.e.a.25.4	yes	10	9.5	odd	6
108.6.e.a.37.2		10	9.7	even	3
108.6.e.a.73.2		10	9.4	even	3
144.6.i.d.49.2		10	36.11	even	6
144.6.i.d.97.2		10	36.23	even	6
324.6.a.d.1.4		5	1.1	even	1	trivial
324.6.a.e.1.2		5	3.2	odd	2
432.6.i.d.145.2		10	36.7	odd	6
432.6.i.d.289.2		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)

\(f(q)\)	\(=\)	\(q+26.3205 q^{5} +63.2575 q^{7} +O(q^{10})\)	\(q+26.3205 q^{5} +63.2575 q^{7} -98.2387 q^{11} -738.285 q^{13} -250.060 q^{17} +1102.41 q^{19} -4409.41 q^{23} -2432.23 q^{25} +7882.12 q^{29} -4611.29 q^{31} +1664.97 q^{35} +11896.3 q^{37} -10081.2 q^{41} +7037.98 q^{43} -14918.7 q^{47} -12805.5 q^{49} -22451.7 q^{53} -2585.69 q^{55} -10810.5 q^{59} -1189.29 q^{61} -19432.0 q^{65} +59180.5 q^{67} -14326.6 q^{71} -53098.2 q^{73} -6214.33 q^{77} -37391.1 q^{79} -120879. q^{83} -6581.71 q^{85} -97873.2 q^{89} -46702.1 q^{91} +29015.9 q^{95} +106713. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 21 q^{5} - 29 q^{7}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more