LMFDB - Embedded newform 432.5.g.h.271.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,5,Mod(271,432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(432, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("432.271");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	432.g (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:	$44.6558240522$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.280120707.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 10x^{4} + 5x^{3} + 83x^{2} - 18x + 4 $ x^6 - x^5 + 10x^4 + 5x^3 + 83x^2 - 18x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{43}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			271.4
Root			$-1.33123 - 2.30576i$ of defining polynomial
Character	$\chi$	$=$	432.271
Dual form			432.5.g.h.271.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.46769 q^{5} +54.5282i q^{7} +O(q^{10})$	$q+2.46769 q^{5} +54.5282i q^{7} -173.055i q^{11} +108.655 q^{13} -228.611 q^{17} -297.252i q^{19} +66.9522i q^{23} -618.911 q^{25} +1309.16 q^{29} +36.6726i q^{31} +134.559i q^{35} +1420.48 q^{37} +1151.29 q^{41} +1476.75i q^{43} -642.355i q^{47} -572.328 q^{49} +3666.73 q^{53} -427.046i q^{55} -3352.76i q^{59} -2093.68 q^{61} +268.126 q^{65} -4232.98i q^{67} -9531.70i q^{71} +194.019 q^{73} +9436.38 q^{77} +1006.73i q^{79} -3474.64i q^{83} -564.140 q^{85} +12987.9 q^{89} +5924.76i q^{91} -733.525i q^{95} +14834.2 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 42 q^{5}+O(q^{10}) $ 6 * q + 42 * q^5	$ 6 q + 42 q^{5} + 168 q^{13} + 624 q^{17} + 1632 q^{25} + 1596 q^{29} - 2652 q^{37} + 9828 q^{41} - 4908 q^{49} + 13158 q^{53} + 3936 q^{61} + 38472 q^{65} + 3606 q^{73} + 47826 q^{77} - 5760 q^{85} + 52620 q^{89} + 13290 q^{97}+O(q^{100}) $ 6 * q + 42 * q^5 + 168 * q^13 + 624 * q^17 + 1632 * q^25 + 1596 * q^29 - 2652 * q^37 + 9828 * q^41 - 4908 * q^49 + 13158 * q^53 + 3936 * q^61 + 38472 * q^65 + 3606 * q^73 + 47826 * q^77 - 5760 * q^85 + 52620 * q^89 + 13290 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$.

$n$	$271$	$325$	$353$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	2.46769	0.0987075	0.0493537	−	0.998781i	$-0.484284\pi$
$5$	2.46769	0.0987075	0.0493537	+	0.998781i	$0.484284\pi$
$6$	0	0
$7$	54.5282i	1.11282i	0.830908	+	0.556411i	$0.187822\pi$
$7$	54.5282i	1.11282i	−0.830908	+	0.556411i	$0.812178\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 173.055i	− 1.43021i	−0.699019	−	0.715103i	$-0.746380\pi$
$11$	− 173.055i	− 1.43021i	0.699019	−	0.715103i	$-0.253620\pi$
$12$	0	0
$13$	108.655	0.642928	0.321464	−	0.946922i	$-0.395825\pi$
$13$	108.655	0.642928	0.321464	+	0.946922i	$0.395825\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−228.611	−0.791041	−0.395520	−	0.918457i	$-0.629436\pi$
$17$	−228.611	−0.791041	−0.395520	+	0.918457i	$0.629436\pi$
$18$	0	0
$19$	− 297.252i	− 0.823413i	−0.911316	−	0.411707i	$-0.864933\pi$
$19$	− 297.252i	− 0.823413i	0.911316	−	0.411707i	$-0.135067\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	66.9522i	0.126564i	0.997996	+	0.0632818i	$0.0201567\pi$
$23$	66.9522i	0.126564i	−0.997996	+	0.0632818i	$0.979843\pi$
$24$	0	0
$25$	−618.911	−0.990257
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1309.16	1.55667	0.778333	−	0.627852i	$-0.216066\pi$
$29$	1309.16	1.55667	0.778333	+	0.627852i	$0.216066\pi$
$30$	0	0
$31$	36.6726i	0.0381609i	0.999818	+	0.0190805i	$0.00607387\pi$
$31$	36.6726i	0.0381609i	−0.999818	+	0.0190805i	$0.993926\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	134.559i	0.109844i
$36$	0	0
$37$	1420.48	1.03760	0.518801	−	0.854895i	$-0.326379\pi$
$37$	1420.48	1.03760	0.518801	+	0.854895i	$0.326379\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1151.29	0.684884	0.342442	−	0.939539i	$-0.388746\pi$
$41$	1151.29	0.684884	0.342442	+	0.939539i	$0.388746\pi$
$42$	0	0
$43$	1476.75i	0.798675i	0.916804	+	0.399337i	$0.130760\pi$
$43$	1476.75i	0.798675i	−0.916804	+	0.399337i	$0.869240\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 642.355i	− 0.290790i	−0.989374	−	0.145395i	$-0.953555\pi$
$47$	− 642.355i	− 0.290790i	0.989374	−	0.145395i	$-0.0464453\pi$
$48$	0	0
$49$	−572.328	−0.238371
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3666.73	1.30535	0.652674	−	0.757638i	$-0.273647\pi$
$53$	3666.73	1.30535	0.652674	+	0.757638i	$0.273647\pi$
$54$	0	0
$55$	− 427.046i	− 0.141172i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 3352.76i	− 0.963161i	−0.876402	−	0.481580i	$-0.840063\pi$
$59$	− 3352.76i	− 0.963161i	0.876402	−	0.481580i	$-0.159937\pi$
$60$	0	0
$61$	−2093.68	−0.562666	−0.281333	−	0.959610i	$-0.590777\pi$
$61$	−2093.68	−0.562666	−0.281333	+	0.959610i	$0.590777\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	268.126	0.0634618
$66$	0	0
$67$	− 4232.98i	− 0.942968i	−0.881875	−	0.471484i	$-0.843719\pi$
$67$	− 4232.98i	− 0.942968i	0.881875	−	0.471484i	$-0.156281\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 9531.70i	− 1.89083i	−0.325863	−	0.945417i	$-0.605655\pi$
$71$	− 9531.70i	− 1.89083i	0.325863	−	0.945417i	$-0.394345\pi$
$72$	0	0
$73$	194.019	0.0364081	0.0182040	−	0.999834i	$-0.494205\pi$
$73$	194.019	0.0364081	0.0182040	+	0.999834i	$0.494205\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	9436.38	1.59156
$78$	0	0
$79$	1006.73i	0.161309i	0.996742	+	0.0806543i	$0.0257010\pi$
$79$	1006.73i	0.161309i	−0.996742	+	0.0806543i	$0.974299\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 3474.64i	− 0.504376i	−0.967678	−	0.252188i	$-0.918850\pi$
$83$	− 3474.64i	− 0.504376i	0.967678	−	0.252188i	$-0.0811501\pi$
$84$	0	0
$85$	−564.140	−0.0780817
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12987.9	1.63969	0.819843	−	0.572589i	$-0.194061\pi$
$89$	12987.9	1.63969	0.819843	+	0.572589i	$0.194061\pi$
$90$	0	0
$91$	5924.76i	0.715464i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 733.525i	− 0.0812771i
$96$	0	0
$97$	14834.2	1.57660	0.788300	−	0.615291i	$-0.210962\pi$
$97$	14834.2	1.57660	0.788300	+	0.615291i	$0.210962\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	15274.8	1.49738	0.748690	−	0.662920i	$-0.230683\pi$
$101$	15274.8	1.49738	0.748690	+	0.662920i	$0.230683\pi$
$102$	0	0
$103$	1802.18i	0.169872i	0.996386	+	0.0849362i	$0.0270686\pi$
$103$	1802.18i	0.169872i	−0.996386	+	0.0849362i	$0.972931\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8452.71i	0.738293i	0.929371	+	0.369146i	$0.120350\pi$
$107$	8452.71i	0.738293i	−0.929371	+	0.369146i	$0.879650\pi$
$108$	0	0
$109$	−11802.2	−0.993372	−0.496686	−	0.867930i	$-0.665450\pi$
$109$	−11802.2	−0.993372	−0.496686	+	0.867930i	$0.665450\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	9598.03	0.751667	0.375833	−	0.926687i	$-0.377357\pi$
$113$	9598.03	0.751667	0.375833	+	0.926687i	$0.377357\pi$
$114$	0	0
$115$	165.217i	0.0124928i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 12465.7i	− 0.880287i
$120$	0	0
$121$	−15307.0	−1.04549
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−3069.58	−0.196453
$126$	0	0
$127$	− 8354.48i	− 0.517979i	−0.965880	−	0.258989i	$-0.916611\pi$
$127$	− 8354.48i	− 0.517979i	0.965880	−	0.258989i	$-0.0833895\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 15872.4i	− 0.924911i	−0.886643	−	0.462455i	$-0.846969\pi$
$131$	− 15872.4i	− 0.924911i	0.886643	−	0.462455i	$-0.153031\pi$
$132$	0	0
$133$	16208.6	0.916312
$134$	0	0
$135$	0	0
$136$	0	0
$137$	29971.0	1.59683	0.798417	−	0.602105i	$-0.205671\pi$
$137$	29971.0	1.59683	0.798417	+	0.602105i	$0.205671\pi$
$138$	0	0
$139$	− 30956.1i	− 1.60220i	−0.598532	−	0.801099i	$-0.704249\pi$
$139$	− 30956.1i	− 1.60220i	0.598532	−	0.801099i	$-0.295751\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 18803.3i	− 0.919520i
$144$	0	0
$145$	3230.59	0.153655
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−6692.74	−0.301461	−0.150731	−	0.988575i	$-0.548163\pi$
$149$	−6692.74	−0.301461	−0.150731	+	0.988575i	$0.548163\pi$
$150$	0	0
$151$	30512.9i	1.33822i	0.743161	+	0.669112i	$0.233325\pi$
$151$	30512.9i	1.33822i	−0.743161	+	0.669112i	$0.766675\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	90.4966i	0.00376677i
$156$	0	0
$157$	−7817.22	−0.317141	−0.158571	−	0.987348i	$-0.550689\pi$
$157$	−7817.22	−0.317141	−0.158571	+	0.987348i	$0.550689\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3650.78	−0.140843
$162$	0	0
$163$	40886.2i	1.53887i	0.638726	+	0.769434i	$0.279462\pi$
$163$	40886.2i	1.53887i	−0.638726	+	0.769434i	$0.720538\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 2639.58i	− 0.0946458i	−0.998880	−	0.0473229i	$-0.984931\pi$
$167$	− 2639.58i	− 0.0946458i	0.998880	−	0.0473229i	$-0.0150690\pi$
$168$	0	0
$169$	−16755.1	−0.586643
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−48545.4	−1.62202	−0.811010	−	0.585033i	$-0.801082\pi$
$173$	−48545.4	−1.62202	−0.811010	+	0.585033i	$0.801082\pi$
$174$	0	0
$175$	− 33748.1i	− 1.10198i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	21723.4i	0.677988i	0.940789	+	0.338994i	$0.110087\pi$
$179$	21723.4i	0.677988i	−0.940789	+	0.338994i	$0.889913\pi$
$180$	0	0
$181$	43722.7	1.33460	0.667298	−	0.744791i	$-0.267451\pi$
$181$	43722.7	1.33460	0.667298	+	0.744791i	$0.267451\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3505.29	0.102419
$186$	0	0
$187$	39562.2i	1.13135i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4907.85i	0.134532i	0.997735	+	0.0672659i	$0.0214276\pi$
$191$	4907.85i	0.134532i	−0.997735	+	0.0672659i	$0.978572\pi$
$192$	0	0
$193$	−17533.2	−0.470703	−0.235352	−	0.971910i	$-0.575624\pi$
$193$	−17533.2	−0.470703	−0.235352	+	0.971910i	$0.575624\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	30835.0	0.794533	0.397267	−	0.917703i	$-0.369959\pi$
$197$	30835.0	0.794533	0.397267	+	0.917703i	$0.369959\pi$
$198$	0	0
$199$	− 47140.7i	− 1.19039i	−0.803580	−	0.595196i	$-0.797074\pi$
$199$	− 47140.7i	− 1.19039i	0.803580	−	0.595196i	$-0.202926\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	71385.9i	1.73229i
$204$	0	0
$205$	2841.02	0.0676032
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−51441.0	−1.17765
$210$	0	0
$211$	− 80619.1i	− 1.81081i	−0.424548	−	0.905405i	$-0.639567\pi$
$211$	− 80619.1i	− 1.81081i	0.424548	−	0.905405i	$-0.360433\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	3644.15i	0.0788352i
$216$	0	0
$217$	−1999.69	−0.0424663
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−24839.7	−0.508582
$222$	0	0
$223$	78774.1i	1.58407i	0.610477	+	0.792034i	$0.290978\pi$
$223$	78774.1i	1.58407i	−0.610477	+	0.792034i	$0.709022\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	99016.2i	1.92156i	0.277307	+	0.960782i	$0.410558\pi$
$227$	99016.2i	1.92156i	−0.277307	+	0.960782i	$0.589442\pi$
$228$	0	0
$229$	−77324.9	−1.47451	−0.737256	−	0.675613i	$-0.763879\pi$
$229$	−77324.9	−1.47451	−0.737256	+	0.675613i	$0.763879\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−73390.6	−1.35185	−0.675926	−	0.736970i	$-0.736256\pi$
$233$	−73390.6	−1.35185	−0.675926	+	0.736970i	$0.736256\pi$
$234$	0	0
$235$	− 1585.13i	− 0.0287031i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 75037.6i	− 1.31366i	−0.754038	−	0.656830i	$-0.771897\pi$
$239$	− 75037.6i	− 1.31366i	0.754038	−	0.656830i	$-0.228103\pi$
$240$	0	0
$241$	76243.6	1.31271	0.656356	−	0.754452i	$-0.272097\pi$
$241$	76243.6	1.31271	0.656356	+	0.754452i	$0.272097\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1412.33	−0.0235290
$246$	0	0
$247$	− 32297.9i	− 0.529396i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	16231.4i	0.257637i	0.991668	+	0.128818i	$0.0411184\pi$
$251$	16231.4i	0.257637i	−0.991668	+	0.128818i	$0.958882\pi$
$252$	0	0
$253$	11586.4	0.181012
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−124921.	−1.89134	−0.945668	−	0.325133i	$-0.894591\pi$
$257$	−124921.	−1.89134	−0.945668	+	0.325133i	$0.894591\pi$
$258$	0	0
$259$	77456.0i	1.15466i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 68578.3i	− 0.991460i	−0.868477	−	0.495730i	$-0.834901\pi$
$263$	− 68578.3i	− 0.991460i	0.868477	−	0.495730i	$-0.165099\pi$
$264$	0	0
$265$	9048.33	0.128848
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−129453.	−1.78899	−0.894496	−	0.447077i	$-0.852465\pi$
$269$	−129453.	−1.78899	−0.894496	+	0.447077i	$0.852465\pi$
$270$	0	0
$271$	81390.3i	1.10824i	0.832436	+	0.554121i	$0.186945\pi$
$271$	81390.3i	1.10824i	−0.832436	+	0.554121i	$0.813055\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	107106.i	1.41627i
$276$	0	0
$277$	23139.8	0.301578	0.150789	−	0.988566i	$-0.451819\pi$
$277$	23139.8	0.301578	0.150789	+	0.988566i	$0.451819\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−43205.8	−0.547178	−0.273589	−	0.961847i	$-0.588211\pi$
$281$	−43205.8	−0.547178	−0.273589	+	0.961847i	$0.588211\pi$
$282$	0	0
$283$	− 98595.1i	− 1.23107i	−0.788110	−	0.615534i	$-0.788940\pi$
$283$	− 98595.1i	− 1.23107i	0.788110	−	0.615534i	$-0.211060\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	62777.8i	0.762154i
$288$	0	0
$289$	−31258.1	−0.374254
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3191.31	0.0371735	0.0185868	−	0.999827i	$-0.494083\pi$
$293$	3191.31	0.0371735	0.0185868	+	0.999827i	$0.494083\pi$
$294$	0	0
$295$	− 8273.57i	− 0.0950712i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	7274.68i	0.0813713i
$300$	0	0
$301$	−80524.5	−0.888782
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−5166.55	−0.0555393
$306$	0	0
$307$	17818.2i	0.189054i	0.995522	+	0.0945270i	$0.0301339\pi$
$307$	17818.2i	0.189054i	−0.995522	+	0.0945270i	$0.969866\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 7104.24i	− 0.0734508i	−0.999325	−	0.0367254i	$-0.988307\pi$
$311$	− 7104.24i	− 0.0734508i	0.999325	−	0.0367254i	$-0.0116927\pi$
$312$	0	0
$313$	108876.	1.11133	0.555667	−	0.831405i	$-0.312463\pi$
$313$	108876.	1.11133	0.555667	+	0.831405i	$0.312463\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	43039.7	0.428303	0.214152	−	0.976800i	$-0.431301\pi$
$317$	43039.7	0.428303	0.214152	+	0.976800i	$0.431301\pi$
$318$	0	0
$319$	− 226556.i	− 2.22635i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	67955.1i	0.651354i
$324$	0	0
$325$	−67247.6	−0.636664
$326$	0	0
$327$	0	0
$328$	0	0
$329$	35026.5	0.323597
$330$	0	0
$331$	113395.i	1.03499i	0.855685	+	0.517497i	$0.173136\pi$
$331$	113395.i	1.03499i	−0.855685	+	0.517497i	$0.826864\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 10445.7i	− 0.0930779i
$336$	0	0
$337$	−199455.	−1.75625	−0.878123	−	0.478434i	$-0.841205\pi$
$337$	−199455.	−1.75625	−0.878123	+	0.478434i	$0.841205\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	6346.39	0.0545780
$342$	0	0
$343$	99714.2i	0.847557i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	141321.i	1.17367i	0.809706	+	0.586836i	$0.199627\pi$
$347$	141321.i	1.17367i	−0.809706	+	0.586836i	$0.800373\pi$
$348$	0	0
$349$	188455.	1.54723	0.773617	−	0.633654i	$-0.218446\pi$
$349$	188455.	1.54723	0.773617	+	0.633654i	$0.218446\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−34799.5	−0.279270	−0.139635	−	0.990203i	$-0.544593\pi$
$353$	−34799.5	−0.279270	−0.139635	+	0.990203i	$0.544593\pi$
$354$	0	0
$355$	− 23521.2i	− 0.186639i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 157540.i	− 1.22237i	−0.791490	−	0.611183i	$-0.790694\pi$
$359$	− 157540.i	− 1.22237i	0.791490	−	0.611183i	$-0.209306\pi$
$360$	0	0
$361$	41962.1	0.321991
$362$	0	0
$363$	0	0
$364$	0	0
$365$	478.777	0.00359375
$366$	0	0
$367$	117053.i	0.869064i	0.900656	+	0.434532i	$0.143086\pi$
$367$	117053.i	0.869064i	−0.900656	+	0.434532i	$0.856914\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	199940.i	1.45262i
$372$	0	0
$373$	81284.3	0.584237	0.292119	−	0.956382i	$-0.405640\pi$
$373$	81284.3	0.584237	0.292119	+	0.956382i	$0.405640\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	142246.	1.00082
$378$	0	0
$379$	− 47193.6i	− 0.328553i	−0.986414	−	0.164276i	$-0.947471\pi$
$379$	− 47193.6i	− 0.328553i	0.986414	−	0.164276i	$-0.0525289\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 254766.i	− 1.73678i	−0.495884	−	0.868389i	$-0.665156\pi$
$383$	− 254766.i	− 1.73678i	0.495884	−	0.868389i	$-0.334844\pi$
$384$	0	0
$385$	23286.0	0.157099
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−86513.4	−0.571721	−0.285861	−	0.958271i	$-0.592279\pi$
$389$	−86513.4	−0.571721	−0.285861	+	0.958271i	$0.592279\pi$
$390$	0	0
$391$	− 15306.0i	− 0.100117i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2484.29i	0.0159224i
$396$	0	0
$397$	162451.	1.03072	0.515362	−	0.856973i	$-0.327657\pi$
$397$	162451.	1.03072	0.515362	+	0.856973i	$0.327657\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	129689.	0.806519	0.403259	−	0.915086i	$-0.367877\pi$
$401$	129689.	0.806519	0.403259	+	0.915086i	$0.367877\pi$
$402$	0	0
$403$	3984.66i	0.0245347i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 245820.i	− 1.48398i
$408$	0	0
$409$	−98135.7	−0.586652	−0.293326	−	0.956013i	$-0.594762\pi$
$409$	−98135.7	−0.586652	−0.293326	+	0.956013i	$0.594762\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	182820.	1.07183
$414$	0	0
$415$	− 8574.33i	− 0.0497857i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 155660.i	− 0.886641i	−0.896363	−	0.443321i	$-0.853800\pi$
$419$	− 155660.i	− 0.886641i	0.896363	−	0.443321i	$-0.146200\pi$
$420$	0	0
$421$	−326132.	−1.84005	−0.920025	−	0.391859i	$-0.871832\pi$
$421$	−326132.	−1.84005	−0.920025	+	0.391859i	$0.871832\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	141490.	0.783334
$426$	0	0
$427$	− 114165.i	− 0.626146i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 258591.i	− 1.39206i	−0.718012	−	0.696030i	$-0.754948\pi$
$431$	− 258591.i	− 1.39206i	0.718012	−	0.696030i	$-0.245052\pi$
$432$	0	0
$433$	−148246.	−0.790694	−0.395347	−	0.918532i	$-0.629376\pi$
$433$	−148246.	−0.790694	−0.395347	+	0.918532i	$0.629376\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	19901.7	0.104214
$438$	0	0
$439$	146959.i	0.762550i	0.924462	+	0.381275i	$0.124515\pi$
$439$	146959.i	0.762550i	−0.924462	+	0.381275i	$0.875485\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 86636.1i	− 0.441460i	−0.975335	−	0.220730i	$-0.929156\pi$
$443$	− 86636.1i	− 0.441460i	0.975335	−	0.220730i	$-0.0708440\pi$
$444$	0	0
$445$	32050.2	0.161849
$446$	0	0
$447$	0	0
$448$	0	0
$449$	126353.	0.626747	0.313373	−	0.949630i	$-0.398541\pi$
$449$	126353.	0.626747	0.313373	+	0.949630i	$0.398541\pi$
$450$	0	0
$451$	− 199237.i	− 0.979526i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	14620.4i	0.0706216i
$456$	0	0
$457$	141344.	0.676777	0.338389	−	0.941006i	$-0.390118\pi$
$457$	141344.	0.676777	0.338389	+	0.941006i	$0.390118\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	173764.	0.817631	0.408815	−	0.912617i	$-0.365942\pi$
$461$	173764.	0.817631	0.408815	+	0.912617i	$0.365942\pi$
$462$	0	0
$463$	− 245068.i	− 1.14321i	−0.820530	−	0.571603i	$-0.806322\pi$
$463$	− 245068.i	− 1.14321i	0.820530	−	0.571603i	$-0.193678\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	334587.i	1.53417i	0.641543	+	0.767087i	$0.278295\pi$
$467$	334587.i	1.53417i	−0.641543	+	0.767087i	$0.721705\pi$
$468$	0	0
$469$	230817.	1.04935
$470$	0	0
$471$	0	0
$472$	0	0
$473$	255559.	1.14227
$474$	0	0
$475$	183973.i	0.815391i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 45931.7i	− 0.200190i	−0.994978	−	0.100095i	$-0.968085\pi$
$479$	− 45931.7i	− 0.200190i	0.994978	−	0.100095i	$-0.0319146\pi$
$480$	0	0
$481$	154342.	0.667103
$482$	0	0
$483$	0	0
$484$	0	0
$485$	36606.2	0.155622
$486$	0	0
$487$	59863.7i	0.252410i	0.992004	+	0.126205i	$0.0402796\pi$
$487$	59863.7i	0.252410i	−0.992004	+	0.126205i	$0.959720\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	100750.i	0.417908i	0.977925	+	0.208954i	$0.0670059\pi$
$491$	100750.i	0.417908i	−0.977925	+	0.208954i	$0.932994\pi$
$492$	0	0
$493$	−299287.	−1.23139
$494$	0	0
$495$	0	0
$496$	0	0
$497$	519747.	2.10416
$498$	0	0
$499$	392427.i	1.57600i	0.615673	+	0.788002i	$0.288884\pi$
$499$	392427.i	1.57600i	−0.615673	+	0.788002i	$0.711116\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 288135.i	− 1.13883i	−0.822049	−	0.569416i	$-0.807169\pi$
$503$	− 288135.i	− 1.13883i	0.822049	−	0.569416i	$-0.192831\pi$
$504$	0	0
$505$	37693.4	0.147803
$506$	0	0
$507$	0	0
$508$	0	0
$509$	124063.	0.478856	0.239428	−	0.970914i	$-0.423040\pi$
$509$	124063.	0.478856	0.239428	+	0.970914i	$0.423040\pi$
$510$	0	0
$511$	10579.5i	0.0405157i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4447.20i	0.0167677i
$516$	0	0
$517$	−111163.	−0.415889
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−221254.	−0.815107	−0.407554	−	0.913181i	$-0.633618\pi$
$521$	−221254.	−0.815107	−0.407554	+	0.913181i	$0.633618\pi$
$522$	0	0
$523$	73638.7i	0.269217i	0.990899	+	0.134609i	$0.0429777\pi$
$523$	73638.7i	0.269217i	−0.990899	+	0.134609i	$0.957022\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 8383.76i	− 0.0301869i
$528$	0	0
$529$	275358.	0.983982
$530$	0	0
$531$	0	0
$532$	0	0
$533$	125093.	0.440331
$534$	0	0
$535$	20858.7i	0.0728750i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	99044.3i	0.340920i
$540$	0	0
$541$	−63515.6	−0.217013	−0.108506	−	0.994096i	$-0.534607\pi$
$541$	−63515.6	−0.217013	−0.108506	+	0.994096i	$0.534607\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−29124.3	−0.0980532
$546$	0	0
$547$	− 410967.i	− 1.37351i	−0.726888	−	0.686756i	$-0.759034\pi$
$547$	− 410967.i	− 1.37351i	0.726888	−	0.686756i	$-0.240966\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 389149.i	− 1.28178i
$552$	0	0
$553$	−54895.1	−0.179508
$554$	0	0
$555$	0	0
$556$	0	0
$557$	242945.	0.783065	0.391532	−	0.920164i	$-0.371945\pi$
$557$	242945.	0.783065	0.391532	+	0.920164i	$0.371945\pi$
$558$	0	0
$559$	160456.i	0.513490i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 167628.i	− 0.528848i	−0.964407	−	0.264424i	$-0.914818\pi$
$563$	− 167628.i	− 0.528848i	0.964407	−	0.264424i	$-0.0851818\pi$
$564$	0	0
$565$	23684.9	0.0741951
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−434991.	−1.34356	−0.671778	−	0.740752i	$-0.734469\pi$
$569$	−434991.	−1.34356	−0.671778	+	0.740752i	$0.734469\pi$
$570$	0	0
$571$	− 85471.1i	− 0.262148i	−0.991373	−	0.131074i	$-0.958157\pi$
$571$	− 85471.1i	− 0.262148i	0.991373	−	0.131074i	$-0.0418426\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 41437.4i	− 0.125331i
$576$	0	0
$577$	−277454.	−0.833372	−0.416686	−	0.909050i	$-0.636809\pi$
$577$	−277454.	−0.833372	−0.416686	+	0.909050i	$0.636809\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	189466.	0.561280
$582$	0	0
$583$	− 634545.i	− 1.86692i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 66152.6i	− 0.191987i	−0.995382	−	0.0959933i	$-0.969397\pi$
$587$	− 66152.6i	− 0.191987i	0.995382	−	0.0959933i	$-0.0306027\pi$
$588$	0	0
$589$	10901.0	0.0314222
$590$	0	0
$591$	0	0
$592$	0	0
$593$	143078.	0.406877	0.203438	−	0.979088i	$-0.434788\pi$
$593$	143078.	0.406877	0.203438	+	0.979088i	$0.434788\pi$
$594$	0	0
$595$	− 30761.6i	− 0.0868909i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 98445.1i	− 0.274372i	−0.990545	−	0.137186i	$-0.956194\pi$
$599$	− 98445.1i	− 0.274372i	0.990545	−	0.137186i	$-0.0438059\pi$
$600$	0	0
$601$	23896.9	0.0661595	0.0330797	−	0.999453i	$-0.489468\pi$
$601$	23896.9	0.0661595	0.0330797	+	0.999453i	$0.489468\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−37773.0	−0.103198
$606$	0	0
$607$	444576.i	1.20661i	0.797509	+	0.603307i	$0.206151\pi$
$607$	444576.i	1.20661i	−0.797509	+	0.603307i	$0.793849\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 69794.9i	− 0.186957i
$612$	0	0
$613$	58183.0	0.154837	0.0774185	−	0.996999i	$-0.475332\pi$
$613$	58183.0	0.154837	0.0774185	+	0.996999i	$0.475332\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−696234.	−1.82888	−0.914439	−	0.404723i	$-0.867368\pi$
$617$	−696234.	−1.82888	−0.914439	+	0.404723i	$0.867368\pi$
$618$	0	0
$619$	16725.8i	0.0436521i	0.999762	+	0.0218260i	$0.00694800\pi$
$619$	16725.8i	0.0436521i	−0.999762	+	0.0218260i	$0.993052\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	708210.i	1.82468i
$624$	0	0
$625$	379244.	0.970865
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−324736.	−0.820785
$630$	0	0
$631$	− 157304.i	− 0.395076i	−0.980295	−	0.197538i	$-0.936705\pi$
$631$	− 157304.i	− 0.395076i	0.980295	−	0.197538i	$-0.0632946\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 20616.2i	− 0.0511284i
$636$	0	0
$637$	−62186.3	−0.153255
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−555317.	−1.35153	−0.675763	−	0.737119i	$-0.736186\pi$
$641$	−555317.	−1.35153	−0.675763	+	0.737119i	$0.736186\pi$
$642$	0	0
$643$	514474.i	1.24435i	0.782880	+	0.622173i	$0.213750\pi$
$643$	514474.i	1.24435i	−0.782880	+	0.622173i	$0.786250\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 143743.i	− 0.343382i	−0.985151	−	0.171691i	$-0.945077\pi$
$647$	− 143743.i	− 0.343382i	0.985151	−	0.171691i	$-0.0549230\pi$
$648$	0	0
$649$	−580212.	−1.37752
$650$	0	0
$651$	0	0
$652$	0	0
$653$	13021.8	0.0305383	0.0152692	−	0.999883i	$-0.495139\pi$
$653$	13021.8	0.0305383	0.0152692	+	0.999883i	$0.495139\pi$
$654$	0	0
$655$	− 39168.1i	− 0.0912956i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	573549.i	1.32069i	0.750964	+	0.660344i	$0.229589\pi$
$659$	573549.i	1.32069i	−0.750964	+	0.660344i	$0.770411\pi$
$660$	0	0
$661$	745737.	1.70680	0.853400	−	0.521257i	$-0.174537\pi$
$661$	745737.	1.70680	0.853400	+	0.521257i	$0.174537\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	39997.8	0.0904468
$666$	0	0
$667$	87650.8i	0.197017i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	362322.i	0.804728i
$672$	0	0
$673$	−68474.4	−0.151181	−0.0755906	−	0.997139i	$-0.524084\pi$
$673$	−68474.4	−0.151181	−0.0755906	+	0.997139i	$0.524084\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−201357.	−0.439329	−0.219665	−	0.975575i	$-0.570496\pi$
$677$	−201357.	−0.439329	−0.219665	+	0.975575i	$0.570496\pi$
$678$	0	0
$679$	808884.i	1.75447i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	30506.9i	0.0653968i	0.999465	+	0.0326984i	$0.0104101\pi$
$683$	30506.9i	0.0653968i	−0.999465	+	0.0326984i	$0.989590\pi$
$684$	0	0
$685$	73959.0	0.157619
$686$	0	0
$687$	0	0
$688$	0	0
$689$	398408.	0.839246
$690$	0	0
$691$	− 383968.i	− 0.804154i	−0.915606	−	0.402077i	$-0.868288\pi$
$691$	− 383968.i	− 0.804154i	0.915606	−	0.402077i	$-0.131712\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 76389.9i	− 0.158149i
$696$	0	0
$697$	−263197.	−0.541772
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−109762.	−0.223366	−0.111683	−	0.993744i	$-0.535624\pi$
$701$	−109762.	−0.223366	−0.111683	+	0.993744i	$0.535624\pi$
$702$	0	0
$703$	− 422240.i	− 0.854375i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	832906.i	1.66632i
$708$	0	0
$709$	−492345.	−0.979439	−0.489719	−	0.871880i	$-0.662901\pi$
$709$	−492345.	−0.979439	−0.489719	+	0.871880i	$0.662901\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2455.31	−0.00482979
$714$	0	0
$715$	− 46400.6i	− 0.0907635i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	915350.i	1.77064i	0.464985	+	0.885319i	$0.346060\pi$
$719$	915350.i	1.77064i	−0.464985	+	0.885319i	$0.653940\pi$
$720$	0	0
$721$	−98269.4	−0.189037
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−810250.	−1.54150
$726$	0	0
$727$	− 598176.i	− 1.13177i	−0.824483	−	0.565887i	$-0.808534\pi$
$727$	− 598176.i	− 1.13177i	0.824483	−	0.565887i	$-0.191466\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 337601.i	− 0.631784i
$732$	0	0
$733$	183356.	0.341261	0.170630	−	0.985335i	$-0.445420\pi$
$733$	183356.	0.341261	0.170630	+	0.985335i	$0.445420\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−732539.	−1.34864
$738$	0	0
$739$	− 193576.i	− 0.354456i	−0.984170	−	0.177228i	$-0.943287\pi$
$739$	− 193576.i	− 0.354456i	0.984170	−	0.177228i	$-0.0567130\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 98910.8i	− 0.179170i	−0.995979	−	0.0895852i	$-0.971446\pi$
$743$	− 98910.8i	− 0.179170i	0.995979	−	0.0895852i	$-0.0285541\pi$
$744$	0	0
$745$	−16515.6	−0.0297565
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−460912.	−0.821588
$750$	0	0
$751$	308879.i	0.547657i	0.961779	+	0.273828i	$0.0882901\pi$
$751$	308879.i	0.547657i	−0.961779	+	0.273828i	$0.911710\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	75296.2i	0.132093i
$756$	0	0
$757$	901639.	1.57341	0.786704	−	0.617331i	$-0.211786\pi$
$757$	901639.	1.57341	0.786704	+	0.617331i	$0.211786\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−240104.	−0.414601	−0.207300	−	0.978277i	$-0.566468\pi$
$761$	−240104.	−0.414601	−0.207300	+	0.978277i	$0.566468\pi$
$762$	0	0
$763$	− 643556.i	− 1.10545i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 364294.i	− 0.619243i
$768$	0	0
$769$	380007.	0.642598	0.321299	−	0.946978i	$-0.395881\pi$
$769$	380007.	0.642598	0.321299	+	0.946978i	$0.395881\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−770627.	−1.28969	−0.644845	−	0.764313i	$-0.723078\pi$
$773$	−770627.	−1.28969	−0.644845	+	0.764313i	$0.723078\pi$
$774$	0	0
$775$	− 22697.1i	− 0.0377891i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 342224.i	− 0.563943i
$780$	0	0
$781$	−1.64951e6	−2.70428
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−19290.4	−0.0313042
$786$	0	0
$787$	665650.i	1.07472i	0.843352	+	0.537361i	$0.180579\pi$
$787$	665650.i	1.07472i	−0.843352	+	0.537361i	$0.819421\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	523364.i	0.836470i
$792$	0	0
$793$	−227488.	−0.361754
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−773853.	−1.21826	−0.609132	−	0.793069i	$-0.708482\pi$
$797$	−773853.	−1.21826	−0.609132	+	0.793069i	$0.708482\pi$
$798$	0	0
$799$	146849.i	0.230027i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 33575.9i	− 0.0520711i
$804$	0	0
$805$	−9008.99	−0.0139022
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−322823.	−0.493250	−0.246625	−	0.969111i	$-0.579322\pi$
$809$	−322823.	−0.493250	−0.246625	+	0.969111i	$0.579322\pi$
$810$	0	0
$811$	424280.i	0.645076i	0.946557	+	0.322538i	$0.104536\pi$
$811$	424280.i	0.645076i	−0.946557	+	0.322538i	$0.895464\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	100894.i	0.151898i
$816$	0	0
$817$	438967.	0.657639
$818$	0	0
$819$	0	0
$820$	0	0
$821$	280362.	0.415943	0.207971	−	0.978135i	$-0.433314\pi$
$821$	280362.	0.415943	0.207971	+	0.978135i	$0.433314\pi$
$822$	0	0
$823$	945937.i	1.39657i	0.715820	+	0.698285i	$0.246053\pi$
$823$	945937.i	1.39657i	−0.715820	+	0.698285i	$0.753947\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1.27987e6i	− 1.87134i	−0.352870	−	0.935672i	$-0.614794\pi$
$827$	− 1.27987e6i	− 1.87134i	0.352870	−	0.935672i	$-0.385206\pi$
$828$	0	0
$829$	667047.	0.970616	0.485308	−	0.874343i	$-0.338707\pi$
$829$	667047.	0.970616	0.485308	+	0.874343i	$0.338707\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	130840.	0.188561
$834$	0	0
$835$	− 6513.65i	− 0.00934225i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	906336.i	1.28755i	0.765213	+	0.643777i	$0.222634\pi$
$839$	906336.i	1.28755i	−0.765213	+	0.643777i	$0.777366\pi$
$840$	0	0
$841$	1.00661e6	1.42321
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−41346.4	−0.0579061
$846$	0	0
$847$	− 834666.i	− 1.16344i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	95103.9i	0.131323i
$852$	0	0
$853$	−702831.	−0.965947	−0.482973	−	0.875635i	$-0.660443\pi$
$853$	−702831.	−0.965947	−0.482973	+	0.875635i	$0.660443\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	554001.	0.754308	0.377154	−	0.926151i	$-0.376903\pi$
$857$	554001.	0.754308	0.377154	+	0.926151i	$0.376903\pi$
$858$	0	0
$859$	− 100538.i	− 0.136253i	−0.997677	−	0.0681265i	$-0.978298\pi$
$859$	− 100538.i	− 0.136253i	0.997677	−	0.0681265i	$-0.0217021\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	938509.i	1.26013i	0.776541	+	0.630067i	$0.216973\pi$
$863$	938509.i	1.26013i	−0.776541	+	0.630067i	$0.783027\pi$
$864$	0	0
$865$	−119795.	−0.160105
$866$	0	0
$867$	0	0
$868$	0	0
$869$	174219.	0.230705
$870$	0	0
$871$	− 459934.i	− 0.606260i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 167379.i	− 0.218617i
$876$	0	0
$877$	639199.	0.831069	0.415534	−	0.909577i	$-0.363595\pi$
$877$	639199.	0.831069	0.415534	+	0.909577i	$0.363595\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	659338.	0.849486	0.424743	−	0.905314i	$-0.360365\pi$
$881$	659338.	0.849486	0.424743	+	0.905314i	$0.360365\pi$
$882$	0	0
$883$	342156.i	0.438836i	0.975631	+	0.219418i	$0.0704159\pi$
$883$	342156.i	0.438836i	−0.975631	+	0.219418i	$0.929584\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 187095.i	− 0.237801i	−0.992906	−	0.118901i	$-0.962063\pi$
$887$	− 187095.i	− 0.237801i	0.992906	−	0.118901i	$-0.0379370\pi$
$888$	0	0
$889$	455555.	0.576418
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−190941.	−0.239440
$894$	0	0
$895$	53606.6i	0.0669225i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	48010.2i	0.0594038i
$900$	0	0
$901$	−838253.	−1.03258
$902$	0	0
$903$	0	0
$904$	0	0
$905$	107894.	0.131735
$906$	0	0
$907$	− 284828.i	− 0.346232i	−0.984901	−	0.173116i	$-0.944616\pi$
$907$	− 284828.i	− 0.346232i	0.984901	−	0.173116i	$-0.0553836\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	756497.i	0.911529i	0.890100	+	0.455764i	$0.150634\pi$
$911$	756497.i	0.911529i	−0.890100	+	0.455764i	$0.849366\pi$
$912$	0	0
$913$	−601305.	−0.721361
$914$	0	0
$915$	0	0
$916$	0	0
$917$	865494.	1.02926
$918$	0	0
$919$	− 901399.i	− 1.06730i	−0.845706	−	0.533649i	$-0.820820\pi$
$919$	− 901399.i	− 1.06730i	0.845706	−	0.533649i	$-0.179180\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 1.03566e6i	− 1.21567i
$924$	0	0
$925$	−879147.	−1.02749
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.06031e6	−1.22857	−0.614284	−	0.789085i	$-0.710555\pi$
$929$	−1.06031e6	−1.22857	−0.614284	+	0.789085i	$0.710555\pi$
$930$	0	0
$931$	170126.i	0.196278i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	97627.2i	0.111673i
$936$	0	0
$937$	−1.29339e6	−1.47316	−0.736582	−	0.676348i	$-0.763562\pi$
$937$	−1.29339e6	−1.47316	−0.736582	+	0.676348i	$0.763562\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.39708e6	1.57776	0.788880	−	0.614547i	$-0.210661\pi$
$941$	1.39708e6	1.57776	0.788880	+	0.614547i	$0.210661\pi$
$942$	0	0
$943$	77081.4i	0.0866815i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1.07267e6i	− 1.19610i	−0.801459	−	0.598050i	$-0.795942\pi$
$947$	− 1.07267e6i	− 1.19610i	0.801459	−	0.598050i	$-0.204058\pi$
$948$	0	0
$949$	21081.1	0.0234078
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.51614e6	1.66937	0.834684	−	0.550729i	$-0.185650\pi$
$953$	1.51614e6	1.66937	0.834684	+	0.550729i	$0.185650\pi$
$954$	0	0
$955$	12111.0i	0.0132793i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.63426e6i	1.77699i
$960$	0	0
$961$	922176.	0.998544
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−43266.5	−0.0464619
$966$	0	0
$967$	1.23379e6i	1.31943i	0.751515	+	0.659716i	$0.229324\pi$
$967$	1.23379e6i	1.31943i	−0.751515	+	0.659716i	$0.770676\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1.04219e6i	− 1.10538i	−0.833388	−	0.552689i	$-0.813602\pi$
$971$	− 1.04219e6i	− 1.10538i	0.833388	−	0.552689i	$-0.186398\pi$
$972$	0	0
$973$	1.68798e6	1.78296
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−119284.	−0.124966	−0.0624830	−	0.998046i	$-0.519902\pi$
$977$	−119284.	−0.124966	−0.0624830	+	0.998046i	$0.519902\pi$
$978$	0	0
$979$	− 2.24763e6i	− 2.34509i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 246025.i	− 0.254608i	−0.991864	−	0.127304i	$-0.959368\pi$
$983$	− 246025.i	− 0.254608i	0.991864	−	0.127304i	$-0.0406323\pi$
$984$	0	0
$985$	76091.2	0.0784264
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−98871.6	−0.101083
$990$	0	0
$991$	1.46229e6i	1.48897i	0.667638	+	0.744486i	$0.267305\pi$
$991$	1.46229e6i	1.48897i	−0.667638	+	0.744486i	$0.732695\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 116329.i	− 0.117501i
$996$	0	0
$997$	236624.	0.238050	0.119025	−	0.992891i	$-0.462023\pi$
$997$	236624.	0.238050	0.119025	+	0.992891i	$0.462023\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.5.g.h.271.4	yes	6
3.2	odd	2		432.5.g.g.271.4	yes	6
4.3	odd	2	inner	432.5.g.h.271.3	yes	6
12.11	even	2		432.5.g.g.271.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.5.g.g.271.3	✓	6	12.11	even	2
432.5.g.g.271.4	yes	6	3.2	odd	2
432.5.g.h.271.3	yes	6	4.3	odd	2	inner
432.5.g.h.271.4	yes	6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	432.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.46769 q^{5} +54.5282i q^{7} +O(q^{10})\)	\(q+2.46769 q^{5} +54.5282i q^{7} -173.055i q^{11} +108.655 q^{13} -228.611 q^{17} -297.252i q^{19} +66.9522i q^{23} -618.911 q^{25} +1309.16 q^{29} +36.6726i q^{31} +134.559i q^{35} +1420.48 q^{37} +1151.29 q^{41} +1476.75i q^{43} -642.355i q^{47} -572.328 q^{49} +3666.73 q^{53} -427.046i q^{55} -3352.76i q^{59} -2093.68 q^{61} +268.126 q^{65} -4232.98i q^{67} -9531.70i q^{71} +194.019 q^{73} +9436.38 q^{77} +1006.73i q^{79} -3474.64i q^{83} -564.140 q^{85} +12987.9 q^{89} +5924.76i q^{91} -733.525i q^{95} +14834.2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 42 q^{5}+O(q^{10}) \) 6 * q + 42 * q^5	\( 6 q + 42 q^{5} + 168 q^{13} + 624 q^{17} + 1632 q^{25} + 1596 q^{29} - 2652 q^{37} + 9828 q^{41} - 4908 q^{49} + 13158 q^{53} + 3936 q^{61} + 38472 q^{65} + 3606 q^{73} + 47826 q^{77} - 5760 q^{85} + 52620 q^{89} + 13290 q^{97}+O(q^{100}) \) 6 * q + 42 * q^5 + 168 * q^13 + 624 * q^17 + 1632 * q^25 + 1596 * q^29 - 2652 * q^37 + 9828 * q^41 - 4908 * q^49 + 13158 * q^53 + 3936 * q^61 + 38472 * q^65 + 3606 * q^73 + 47826 * q^77 - 5760 * q^85 + 52620 * q^89 + 13290 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more