LMFDB - Embedded newform 1200.3.l.t.401.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,3,Mod(401,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1200.401");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1200.l (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$32.6976317232$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 16x^{2} + 81 $ x^4 - 16*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			401.3
Root			$-2.91548 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1200.401
Dual form			1200.3.l.t.401.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.91548 - 0.707107i) q^{3} +5.83095 q^{7} +(8.00000 - 4.12311i) q^{9} +O(q^{10})$	$q+(2.91548 - 0.707107i) q^{3} +5.83095 q^{7} +(8.00000 - 4.12311i) q^{9} +16.4924i q^{11} -11.3137i q^{17} +12.0000 q^{19} +(17.0000 - 4.12311i) q^{21} +24.0416i q^{23} +(20.4083 - 17.6777i) q^{27} +32.0000 q^{31} +(11.6619 + 48.0833i) q^{33} -23.3238 q^{37} +57.7235i q^{41} +40.8167 q^{43} +35.3553i q^{47} -15.0000 q^{49} +(-8.00000 - 32.9848i) q^{51} -67.8823i q^{53} +(34.9857 - 8.48528i) q^{57} -16.4924i q^{59} -16.0000 q^{61} +(46.6476 - 24.0416i) q^{63} -5.83095 q^{67} +(17.0000 + 70.0928i) q^{69} +116.619 q^{73} +96.1665i q^{77} -72.0000 q^{79} +(47.0000 - 65.9697i) q^{81} -43.8406i q^{83} -65.9697i q^{89} +(93.2952 - 22.6274i) q^{93} +163.267 q^{97} +(68.0000 + 131.939i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 32 q^{9}+O(q^{10}) $ 4 * q + 32 * q^9	$ 4 q + 32 q^{9} + 48 q^{19} + 68 q^{21} + 128 q^{31} - 60 q^{49} - 32 q^{51} - 64 q^{61} + 68 q^{69} - 288 q^{79} + 188 q^{81} + 272 q^{99}+O(q^{100}) $ 4 * q + 32 * q^9 + 48 * q^19 + 68 * q^21 + 128 * q^31 - 60 * q^49 - 32 * q^51 - 64 * q^61 + 68 * q^69 - 288 * q^79 + 188 * q^81 + 272 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	2.91548	−	0.707107i	0.971825	−	0.235702i
$3$	2.91548	−	0.707107i	0.971825	−	0.235702i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	5.83095			0.832993			0.416497	−	0.909137i	$-0.363258\pi$
$7$	5.83095			0.832993			0.416497	+	0.909137i	$0.363258\pi$
$8$		0			0
$9$	8.00000	−	4.12311i	0.888889	−	0.458123i
$10$		0			0
$11$			16.4924i			1.49931i	0.661828	+	0.749656i	$0.269781\pi$
$11$			16.4924i			1.49931i	−0.661828	+	0.749656i	$0.730219\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	11.3137i		−	0.665512i	−0.943013	−	0.332756i	$-0.892021\pi$
$17$		−	11.3137i		−	0.665512i	0.943013	−	0.332756i	$-0.107979\pi$
$18$		0			0
$19$	12.0000			0.631579			0.315789	−	0.948829i	$-0.397731\pi$
$19$	12.0000			0.631579			0.315789	+	0.948829i	$0.397731\pi$
$20$		0			0
$21$	17.0000	−	4.12311i	0.809524	−	0.196338i
$22$		0			0
$23$			24.0416i			1.04529i	0.852551	+	0.522644i	$0.175054\pi$
$23$			24.0416i			1.04529i	−0.852551	+	0.522644i	$0.824946\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	20.4083	−	17.6777i	0.755864	−	0.654729i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	32.0000			1.03226			0.516129	−	0.856511i	$-0.327372\pi$
$31$	32.0000			1.03226			0.516129	+	0.856511i	$0.327372\pi$
$32$		0			0
$33$	11.6619	+	48.0833i	0.353391	+	1.45707i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−23.3238			−0.630373			−0.315187	−	0.949030i	$-0.602067\pi$
$37$	−23.3238			−0.630373			−0.315187	+	0.949030i	$0.602067\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			57.7235i			1.40789i	0.710255	+	0.703945i	$0.248580\pi$
$41$			57.7235i			1.40789i	−0.710255	+	0.703945i	$0.751420\pi$
$42$		0			0
$43$	40.8167			0.949225			0.474612	−	0.880195i	$-0.342588\pi$
$43$	40.8167			0.949225			0.474612	+	0.880195i	$0.342588\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			35.3553i			0.752241i	0.926571	+	0.376121i	$0.122742\pi$
$47$			35.3553i			0.752241i	−0.926571	+	0.376121i	$0.877258\pi$
$48$		0			0
$49$	−15.0000			−0.306122
$50$		0			0
$51$	−8.00000	−	32.9848i	−0.156863	−	0.646762i
$52$		0			0
$53$		−	67.8823i		−	1.28080i	−0.768043	−	0.640399i	$-0.778769\pi$
$53$		−	67.8823i		−	1.28080i	0.768043	−	0.640399i	$-0.221231\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	34.9857	−	8.48528i	0.613784	−	0.148865i
$58$		0			0
$59$		−	16.4924i		−	0.279533i	−0.990185	−	0.139766i	$-0.955365\pi$
$59$		−	16.4924i		−	0.279533i	0.990185	−	0.139766i	$-0.0446351\pi$
$60$		0			0
$61$	−16.0000			−0.262295			−0.131148	−	0.991363i	$-0.541866\pi$
$61$	−16.0000			−0.262295			−0.131148	+	0.991363i	$0.541866\pi$
$62$		0			0
$63$	46.6476	−	24.0416i	0.740438	−	0.381613i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−5.83095			−0.0870291			−0.0435146	−	0.999053i	$-0.513855\pi$
$67$	−5.83095			−0.0870291			−0.0435146	+	0.999053i	$0.513855\pi$
$68$		0			0
$69$	17.0000	+	70.0928i	0.246377	+	1.01584i
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	116.619			1.59752			0.798761	−	0.601649i	$-0.205489\pi$
$73$	116.619			1.59752			0.798761	+	0.601649i	$0.205489\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			96.1665i			1.24892i
$78$		0			0
$79$	−72.0000			−0.911392			−0.455696	−	0.890135i	$-0.650610\pi$
$79$	−72.0000			−0.911392			−0.455696	+	0.890135i	$0.650610\pi$
$80$		0			0
$81$	47.0000	−	65.9697i	0.580247	−	0.814441i
$82$		0			0
$83$		−	43.8406i		−	0.528200i	−0.964495	−	0.264100i	$-0.914925\pi$
$83$		−	43.8406i		−	0.528200i	0.964495	−	0.264100i	$-0.0850749\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	65.9697i		−	0.741232i	−0.928786	−	0.370616i	$-0.879147\pi$
$89$		−	65.9697i		−	0.741232i	0.928786	−	0.370616i	$-0.120853\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	93.2952	−	22.6274i	1.00317	−	0.243306i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	163.267			1.68316			0.841581	−	0.540131i	$-0.181625\pi$
$97$	163.267			1.68316			0.841581	+	0.540131i	$0.181625\pi$
$98$		0			0
$99$	68.0000	+	131.939i	0.686869	+	1.33272i
$100$		0			0
$101$		−	131.939i		−	1.30633i	−0.757215	−	0.653165i	$-0.773441\pi$
$101$		−	131.939i		−	1.30633i	0.757215	−	0.653165i	$-0.226559\pi$
$102$		0			0
$103$	99.1262			0.962390			0.481195	−	0.876614i	$-0.340203\pi$
$103$	99.1262			0.962390			0.481195	+	0.876614i	$0.340203\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			55.1543i			0.515461i	0.966217	+	0.257731i	$0.0829747\pi$
$107$			55.1543i			0.515461i	−0.966217	+	0.257731i	$0.917025\pi$
$108$		0			0
$109$	−80.0000			−0.733945			−0.366972	−	0.930232i	$-0.619606\pi$
$109$	−80.0000			−0.733945			−0.366972	+	0.930232i	$0.619606\pi$
$110$		0			0
$111$	−68.0000	+	16.4924i	−0.612613	+	0.148580i
$112$		0			0
$113$			152.735i			1.35164i	0.737068	+	0.675819i	$0.236210\pi$
$113$			152.735i			1.35164i	−0.737068	+	0.675819i	$0.763790\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	65.9697i		−	0.554367i
$120$		0			0
$121$	−151.000			−1.24793
$122$		0			0
$123$	40.8167	+	168.291i	0.331843	+	1.36822i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	40.8167			0.321391			0.160696	−	0.987004i	$-0.448626\pi$
$127$	40.8167			0.321391			0.160696	+	0.987004i	$0.448626\pi$
$128$		0			0
$129$	119.000	−	28.8617i	0.922481	−	0.223734i
$130$		0			0
$131$			49.4773i			0.377689i	0.982007	+	0.188845i	$0.0604742\pi$
$131$			49.4773i			0.377689i	−0.982007	+	0.188845i	$0.939526\pi$
$132$		0			0
$133$	69.9714			0.526101
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	50.9117i		−	0.371618i	−0.982586	−	0.185809i	$-0.940509\pi$
$137$		−	50.9117i		−	0.371618i	0.982586	−	0.185809i	$-0.0594906\pi$
$138$		0			0
$139$	44.0000			0.316547			0.158273	−	0.987395i	$-0.449407\pi$
$139$	44.0000			0.316547			0.158273	+	0.987395i	$0.449407\pi$
$140$		0			0
$141$	25.0000	+	103.078i	0.177305	+	0.731047i
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−43.7321	+	10.6066i	−0.297498	+	0.0721538i
$148$		0			0
$149$		−	8.24621i		−	0.0553437i	−0.999617	−	0.0276718i	$-0.991191\pi$
$149$		−	8.24621i		−	0.0553437i	0.999617	−	0.0276718i	$-0.00880935\pi$
$150$		0			0
$151$	−136.000			−0.900662			−0.450331	−	0.892862i	$-0.648694\pi$
$151$	−136.000			−0.900662			−0.450331	+	0.892862i	$0.648694\pi$
$152$		0			0
$153$	−46.6476	−	90.5097i	−0.304886	−	0.591566i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−116.619			−0.742796			−0.371398	−	0.928474i	$-0.621122\pi$
$157$	−116.619			−0.742796			−0.371398	+	0.928474i	$0.621122\pi$
$158$		0			0
$159$	−48.0000	−	197.909i	−0.301887	−	1.24471i
$160$		0			0
$161$			140.186i			0.870718i
$162$		0			0
$163$	99.1262			0.608136			0.304068	−	0.952650i	$-0.401655\pi$
$163$	99.1262			0.608136			0.304068	+	0.952650i	$0.401655\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	292.742i		−	1.75295i	−0.481450	−	0.876474i	$-0.659890\pi$
$167$		−	292.742i		−	1.75295i	0.481450	−	0.876474i	$-0.340110\pi$
$168$		0			0
$169$	−169.000			−1.00000
$170$		0			0
$171$	96.0000	−	49.4773i	0.561404	−	0.289341i
$172$		0			0
$173$			164.049i			0.948259i	0.880455	+	0.474129i	$0.157237\pi$
$173$			164.049i			0.948259i	−0.880455	+	0.474129i	$0.842763\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−11.6619	−	48.0833i	−0.0658865	−	0.271657i
$178$		0			0
$179$		−	16.4924i		−	0.0921364i	−0.998938	−	0.0460682i	$-0.985331\pi$
$179$		−	16.4924i		−	0.0921364i	0.998938	−	0.0460682i	$-0.0146692\pi$
$180$		0			0
$181$	−82.0000			−0.453039			−0.226519	−	0.974007i	$-0.572735\pi$
$181$	−82.0000			−0.453039			−0.226519	+	0.974007i	$0.572735\pi$
$182$		0			0
$183$	−46.6476	+	11.3137i	−0.254905	+	0.0618235i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	186.590			0.997810
$188$		0			0
$189$	119.000	−	103.078i	0.629630	−	0.545384i
$190$		0			0
$191$		−	296.864i		−	1.55426i	−0.629340	−	0.777130i	$-0.716675\pi$
$191$		−	296.864i		−	1.55426i	0.629340	−	0.777130i	$-0.283325\pi$
$192$		0			0
$193$	116.619			0.604244			0.302122	−	0.953269i	$-0.402305\pi$
$193$	116.619			0.604244			0.302122	+	0.953269i	$0.402305\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	192.333i		−	0.976310i	−0.872757	−	0.488155i	$-0.837670\pi$
$197$		−	192.333i		−	0.976310i	0.872757	−	0.488155i	$-0.162330\pi$
$198$		0			0
$199$	−312.000			−1.56784			−0.783920	−	0.620862i	$-0.786783\pi$
$199$	−312.000			−1.56784			−0.783920	+	0.620862i	$0.786783\pi$
$200$		0			0
$201$	−17.0000	+	4.12311i	−0.0845771	+	0.0205130i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	99.1262	+	192.333i	0.478870	+	0.929145i
$208$		0			0
$209$			197.909i			0.946933i
$210$		0			0
$211$	12.0000			0.0568720			0.0284360	−	0.999596i	$-0.490947\pi$
$211$	12.0000			0.0568720			0.0284360	+	0.999596i	$0.490947\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	186.590			0.859864
$218$		0			0
$219$	340.000	−	82.4621i	1.55251	−	0.376539i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	40.8167			0.183034			0.0915172	−	0.995803i	$-0.470828\pi$
$223$	40.8167			0.183034			0.0915172	+	0.995803i	$0.470828\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			159.806i			0.703992i	0.936002	+	0.351996i	$0.114497\pi$
$227$			159.806i			0.703992i	−0.936002	+	0.351996i	$0.885503\pi$
$228$		0			0
$229$	−82.0000			−0.358079			−0.179039	−	0.983842i	$-0.557299\pi$
$229$	−82.0000			−0.358079			−0.179039	+	0.983842i	$0.557299\pi$
$230$		0			0
$231$	68.0000	+	280.371i	0.294372	+	1.21373i
$232$		0			0
$233$		−	192.333i		−	0.825464i	−0.910853	−	0.412732i	$-0.864575\pi$
$233$		−	192.333i		−	0.825464i	0.910853	−	0.412732i	$-0.135425\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−209.914	+	50.9117i	−0.885714	+	0.214817i
$238$		0			0
$239$			461.788i			1.93217i	0.258231	+	0.966083i	$0.416861\pi$
$239$			461.788i			1.93217i	−0.258231	+	0.966083i	$0.583139\pi$
$240$		0			0
$241$	304.000			1.26141			0.630705	−	0.776022i	$-0.282766\pi$
$241$	304.000			1.26141			0.630705	+	0.776022i	$0.282766\pi$
$242$		0			0
$243$	90.3798	−	225.567i	0.371933	−	0.928260i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−31.0000	−	127.816i	−0.124498	−	0.513318i
$250$		0			0
$251$			346.341i			1.37984i	0.723884	+	0.689922i	$0.242355\pi$
$251$			346.341i			1.37984i	−0.723884	+	0.689922i	$0.757645\pi$
$252$		0			0
$253$	−396.505			−1.56721
$254$		0			0
$255$		0			0
$256$		0			0
$257$			390.323i			1.51877i	0.650644	+	0.759383i	$0.274499\pi$
$257$			390.323i			1.51877i	−0.650644	+	0.759383i	$0.725501\pi$
$258$		0			0
$259$	−136.000			−0.525097
$260$		0			0
$261$		0			0
$262$		0			0
$263$			295.571i			1.12384i	0.827191	+	0.561921i	$0.189938\pi$
$263$			295.571i			1.12384i	−0.827191	+	0.561921i	$0.810062\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−46.6476	−	192.333i	−0.174710	−	0.720348i
$268$		0			0
$269$			74.2159i			0.275896i	0.990440	+	0.137948i	$0.0440506\pi$
$269$			74.2159i			0.275896i	−0.990440	+	0.137948i	$0.955949\pi$
$270$		0			0
$271$	−40.0000			−0.147601			−0.0738007	−	0.997273i	$-0.523513\pi$
$271$	−40.0000			−0.147601			−0.0738007	+	0.997273i	$0.523513\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−443.152			−1.59983			−0.799914	−	0.600115i	$-0.795122\pi$
$277$	−443.152			−1.59983			−0.799914	+	0.600115i	$0.795122\pi$
$278$		0			0
$279$	256.000	−	131.939i	0.917563	−	0.472901i
$280$		0			0
$281$		−	519.511i		−	1.84879i	−0.381431	−	0.924397i	$-0.624569\pi$
$281$		−	519.511i		−	1.84879i	0.381431	−	0.924397i	$-0.375431\pi$
$282$		0			0
$283$	−320.702			−1.13322			−0.566612	−	0.823985i	$-0.691746\pi$
$283$	−320.702			−1.13322			−0.566612	+	0.823985i	$0.691746\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			336.583i			1.17276i
$288$		0			0
$289$	161.000			0.557093
$290$		0			0
$291$	476.000	−	115.447i	1.63574	−	0.396725i
$292$		0			0
$293$		−	84.8528i		−	0.289600i	−0.989461	−	0.144800i	$-0.953746\pi$
$293$		−	84.8528i		−	0.289600i	0.989461	−	0.144800i	$-0.0462539\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	291.548	+	336.583i	0.981642	+	1.13328i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	238.000			0.790698
$302$		0			0
$303$	−93.2952	−	384.666i	−0.307905	−	1.26953i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−367.350			−1.19658			−0.598290	−	0.801280i	$-0.704153\pi$
$307$	−367.350			−1.19658			−0.598290	+	0.801280i	$0.704153\pi$
$308$		0			0
$309$	289.000	−	70.0928i	0.935275	−	0.226838i
$310$		0			0
$311$		−	98.9545i		−	0.318182i	−0.987264	−	0.159091i	$-0.949144\pi$
$311$		−	98.9545i		−	0.318182i	0.987264	−	0.159091i	$-0.0508563\pi$
$312$		0			0
$313$	186.590			0.596136			0.298068	−	0.954545i	$-0.403658\pi$
$313$	186.590			0.596136			0.298068	+	0.954545i	$0.403658\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	520.431i		−	1.64174i	−0.571117	−	0.820868i	$-0.693490\pi$
$317$		−	520.431i		−	1.64174i	0.571117	−	0.820868i	$-0.306510\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	39.0000	+	160.801i	0.121495	+	0.500938i
$322$		0			0
$323$		−	135.765i		−	0.420324i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−233.238	+	56.5685i	−0.713266	+	0.172992i
$328$		0			0
$329$			206.155i			0.626612i
$330$		0			0
$331$	−292.000			−0.882175			−0.441088	−	0.897464i	$-0.645407\pi$
$331$	−292.000			−0.882175			−0.441088	+	0.897464i	$0.645407\pi$
$332$		0			0
$333$	−186.590	+	96.1665i	−0.560332	+	0.288788i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	326.533			0.968942			0.484471	−	0.874807i	$-0.339012\pi$
$337$	326.533			0.968942			0.484471	+	0.874807i	$0.339012\pi$
$338$		0			0
$339$	108.000	+	445.295i	0.318584	+	1.31356i
$340$		0			0
$341$			527.758i			1.54768i
$342$		0			0
$343$	−373.181			−1.08799
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	394.566i		−	1.13708i	−0.822657	−	0.568538i	$-0.807509\pi$
$347$		−	394.566i		−	1.13708i	0.822657	−	0.568538i	$-0.192491\pi$
$348$		0			0
$349$	−254.000			−0.727794			−0.363897	−	0.931439i	$-0.618554\pi$
$349$	−254.000			−0.727794			−0.363897	+	0.931439i	$0.618554\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	345.068i		−	0.977530i	−0.872415	−	0.488765i	$-0.837448\pi$
$353$		−	345.068i		−	0.977530i	0.872415	−	0.488765i	$-0.162552\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−46.6476	−	192.333i	−0.130666	−	0.538748i
$358$		0			0
$359$		−	395.818i		−	1.10256i	−0.834321	−	0.551279i	$-0.814140\pi$
$359$		−	395.818i		−	1.10256i	0.834321	−	0.551279i	$-0.185860\pi$
$360$		0			0
$361$	−217.000			−0.601108
$362$		0			0
$363$	−440.237	+	106.773i	−1.21277	+	0.294141i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−413.998			−1.12806			−0.564029	−	0.825755i	$-0.690750\pi$
$367$	−413.998			−1.12806			−0.564029	+	0.825755i	$0.690750\pi$
$368$		0			0
$369$	238.000	+	461.788i	0.644986	+	1.25146i
$370$		0			0
$371$		−	395.818i		−	1.06690i
$372$		0			0
$373$	−629.743			−1.68832			−0.844159	−	0.536092i	$-0.819900\pi$
$373$	−629.743			−1.68832			−0.844159	+	0.536092i	$0.819900\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−572.000			−1.50923			−0.754617	−	0.656165i	$-0.772178\pi$
$379$	−572.000			−1.50923			−0.754617	+	0.656165i	$0.772178\pi$
$380$		0			0
$381$	119.000	−	28.8617i	0.312336	−	0.0757526i
$382$		0			0
$383$		−	193.747i		−	0.505868i	−0.967484	−	0.252934i	$-0.918605\pi$
$383$		−	193.747i		−	0.505868i	0.967484	−	0.252934i	$-0.0813955\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	326.533	−	168.291i	0.843755	−	0.434862i
$388$		0			0
$389$		−	387.572i		−	0.996329i	−0.867083	−	0.498164i	$-0.834008\pi$
$389$		−	387.572i		−	0.996329i	0.867083	−	0.498164i	$-0.165992\pi$
$390$		0			0
$391$	272.000			0.695652
$392$		0			0
$393$	34.9857	+	144.250i	0.0890222	+	0.367048i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−513.124			−1.29250			−0.646252	−	0.763124i	$-0.723664\pi$
$397$	−513.124			−1.29250			−0.646252	+	0.763124i	$0.723664\pi$
$398$		0			0
$399$	204.000	−	49.4773i	0.511278	−	0.124003i
$400$		0			0
$401$			65.9697i			0.164513i	0.996611	+	0.0822565i	$0.0262127\pi$
$401$			65.9697i			0.164513i	−0.996611	+	0.0822565i	$0.973787\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	384.666i		−	0.945126i
$408$		0			0
$409$	−640.000			−1.56479			−0.782396	−	0.622781i	$-0.786003\pi$
$409$	−640.000			−1.56479			−0.782396	+	0.622781i	$0.786003\pi$
$410$		0			0
$411$	−36.0000	−	148.432i	−0.0875912	−	0.361148i
$412$		0			0
$413$		−	96.1665i		−	0.232849i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	128.281	−	31.1127i	0.307628	−	0.0746108i
$418$		0			0
$419$		−	577.235i		−	1.37765i	−0.724928	−	0.688824i	$-0.758127\pi$
$419$		−	577.235i		−	1.37765i	0.724928	−	0.688824i	$-0.241873\pi$
$420$		0			0
$421$	−656.000			−1.55819			−0.779097	−	0.626903i	$-0.784322\pi$
$421$	−656.000			−1.55819			−0.779097	+	0.626903i	$0.784322\pi$
$422$		0			0
$423$	145.774	+	282.843i	0.344619	+	0.668659i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−93.2952			−0.218490
$428$		0			0
$429$		0			0
$430$		0			0
$431$			362.833i			0.841841i	0.907098	+	0.420920i	$0.138293\pi$
$431$			362.833i			0.841841i	−0.907098	+	0.420920i	$0.861707\pi$
$432$		0			0
$433$	−163.267			−0.377059			−0.188530	−	0.982068i	$-0.560372\pi$
$433$	−163.267			−0.377059			−0.188530	+	0.982068i	$0.560372\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			288.500i			0.660182i
$438$		0			0
$439$	432.000			0.984055			0.492027	−	0.870580i	$-0.336256\pi$
$439$	432.000			0.984055			0.492027	+	0.870580i	$0.336256\pi$
$440$		0			0
$441$	−120.000	+	61.8466i	−0.272109	+	0.140242i
$442$		0			0
$443$			123.037i			0.277735i	0.990311	+	0.138867i	$0.0443462\pi$
$443$			123.037i			0.277735i	−0.990311	+	0.138867i	$0.955654\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−5.83095	−	24.0416i	−0.0130446	−	0.0537844i
$448$		0			0
$449$		−	865.852i		−	1.92840i	−0.265174	−	0.964201i	$-0.585429\pi$
$449$		−	865.852i		−	1.92840i	0.265174	−	0.964201i	$-0.414571\pi$
$450$		0			0
$451$	−952.000			−2.11086
$452$		0			0
$453$	−396.505	+	96.1665i	−0.875286	+	0.212288i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	466.476			1.02074			0.510368	−	0.859956i	$-0.329509\pi$
$457$	466.476			1.02074			0.510368	+	0.859956i	$0.329509\pi$
$458$		0			0
$459$	−200.000	−	230.894i	−0.435730	−	0.503037i
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−612.250			−1.32235			−0.661177	−	0.750230i	$-0.729943\pi$
$463$	−612.250			−1.32235			−0.661177	+	0.750230i	$0.729943\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			767.918i			1.64436i	0.569225	+	0.822182i	$0.307243\pi$
$467$			767.918i			1.64436i	−0.569225	+	0.822182i	$0.692757\pi$
$468$		0			0
$469$	−34.0000			−0.0724947
$470$		0			0
$471$	−340.000	+	82.4621i	−0.721868	+	0.175079i
$472$		0			0
$473$			673.166i			1.42318i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−279.886	−	543.058i	−0.586762	−	1.13849i
$478$		0			0
$479$			560.742i			1.17065i	0.810798	+	0.585326i	$0.199033\pi$
$479$			560.742i			1.17065i	−0.810798	+	0.585326i	$0.800967\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	99.1262	+	408.708i	0.205230	+	0.846186i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	647.236			1.32903			0.664513	−	0.747277i	$-0.268639\pi$
$487$	647.236			1.32903			0.664513	+	0.747277i	$0.268639\pi$
$488$		0			0
$489$	289.000	−	70.0928i	0.591002	−	0.143339i
$490$		0			0
$491$			346.341i			0.705379i	0.935740	+	0.352689i	$0.114733\pi$
$491$			346.341i			0.705379i	−0.935740	+	0.352689i	$0.885267\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−660.000			−1.32265			−0.661323	−	0.750102i	$-0.730005\pi$
$499$	−660.000			−1.32265			−0.661323	+	0.750102i	$0.730005\pi$
$500$		0			0
$501$	−207.000	−	853.483i	−0.413174	−	1.70356i
$502$		0			0
$503$		−	182.434i		−	0.362691i	−0.983419	−	0.181345i	$-0.941955\pi$
$503$		−	182.434i		−	0.362691i	0.983419	−	0.181345i	$-0.0580452\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−492.715	+	119.501i	−0.971825	+	0.235702i
$508$		0			0
$509$			395.818i			0.777639i	0.921314	+	0.388819i	$0.127117\pi$
$509$			395.818i			0.777639i	−0.921314	+	0.388819i	$0.872883\pi$
$510$		0			0
$511$	680.000			1.33072
$512$		0			0
$513$	244.900	−	212.132i	0.477388	−	0.413513i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−583.095			−1.12784
$518$		0			0
$519$	116.000	+	478.280i	0.223507	+	0.921542i
$520$		0			0
$521$		−	131.939i		−	0.253243i	−0.991951	−	0.126621i	$-0.959587\pi$
$521$		−	131.939i		−	0.253243i	0.991951	−	0.126621i	$-0.0404133\pi$
$522$		0			0
$523$	−145.774			−0.278726			−0.139363	−	0.990241i	$-0.544506\pi$
$523$	−145.774			−0.278726			−0.139363	+	0.990241i	$0.544506\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	362.039i		−	0.686980i
$528$		0			0
$529$	−49.0000			−0.0926276
$530$		0			0
$531$	−68.0000	−	131.939i	−0.128060	−	0.248473i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−11.6619	−	48.0833i	−0.0217168	−	0.0895405i
$538$		0			0
$539$		−	247.386i		−	0.458973i
$540$		0			0
$541$	418.000			0.772643			0.386322	−	0.922364i	$-0.373745\pi$
$541$	418.000			0.772643			0.386322	+	0.922364i	$0.373745\pi$
$542$		0			0
$543$	−239.069	+	57.9828i	−0.440274	+	0.106782i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	285.717			0.522334			0.261167	−	0.965294i	$-0.415893\pi$
$547$	285.717			0.522334			0.261167	+	0.965294i	$0.415893\pi$
$548$		0			0
$549$	−128.000	+	65.9697i	−0.233151	+	0.120163i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−419.829			−0.759184
$554$		0			0
$555$		0			0
$556$		0			0
$557$			424.264i			0.761695i	0.924638	+	0.380847i	$0.124368\pi$
$557$			424.264i			0.761695i	−0.924638	+	0.380847i	$0.875632\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	544.000	−	131.939i	0.969697	−	0.235186i
$562$		0			0
$563$		−	813.173i		−	1.44436i	−0.691707	−	0.722178i	$-0.743141\pi$
$563$		−	813.173i		−	1.44436i	0.691707	−	0.722178i	$-0.256859\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	274.055	−	384.666i	0.483342	−	0.678423i
$568$		0			0
$569$			453.542i			0.797085i	0.917150	+	0.398543i	$0.130484\pi$
$569$			453.542i			0.797085i	−0.917150	+	0.398543i	$0.869516\pi$
$570$		0			0
$571$	−220.000			−0.385289			−0.192644	−	0.981269i	$-0.561706\pi$
$571$	−220.000			−0.385289			−0.192644	+	0.981269i	$0.561706\pi$
$572$		0			0
$573$	−209.914	−	865.499i	−0.366343	−	1.51047i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	46.6476			0.0808451			0.0404225	−	0.999183i	$-0.487130\pi$
$577$	46.6476			0.0808451			0.0404225	+	0.999183i	$0.487130\pi$
$578$		0			0
$579$	340.000	−	82.4621i	0.587219	−	0.142422i
$580$		0			0
$581$		−	255.633i		−	0.439987i
$582$		0			0
$583$	1119.54			1.92031
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	55.1543i		−	0.0939597i	−0.998896	−	0.0469798i	$-0.985040\pi$
$587$		−	55.1543i		−	0.0939597i	0.998896	−	0.0469798i	$-0.0149597\pi$
$588$		0			0
$589$	384.000			0.651952
$590$		0			0
$591$	−136.000	−	560.742i	−0.230118	−	0.948803i
$592$		0			0
$593$		−	390.323i		−	0.658217i	−0.944292	−	0.329109i	$-0.893252\pi$
$593$		−	390.323i		−	0.658217i	0.944292	−	0.329109i	$-0.106748\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−909.628	+	220.617i	−1.52367	+	0.369543i
$598$		0			0
$599$			98.9545i			0.165200i	0.996583	+	0.0825998i	$0.0263223\pi$
$599$			98.9545i			0.165200i	−0.996583	+	0.0825998i	$0.973678\pi$
$600$		0			0
$601$	−880.000			−1.46423			−0.732113	−	0.681183i	$-0.761466\pi$
$601$	−880.000			−1.46423			−0.732113	+	0.681183i	$0.761466\pi$
$602$		0			0
$603$	−46.6476	+	24.0416i	−0.0773592	+	0.0398700i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−425.659			−0.701251			−0.350626	−	0.936516i	$-0.614031\pi$
$607$	−425.659			−0.701251			−0.350626	+	0.936516i	$0.614031\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	606.419			0.989264			0.494632	−	0.869102i	$-0.335303\pi$
$613$	606.419			0.989264			0.494632	+	0.869102i	$0.335303\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	113.137i		−	0.183366i	−0.995788	−	0.0916832i	$-0.970775\pi$
$617$		−	113.137i		−	0.183366i	0.995788	−	0.0916832i	$-0.0292247\pi$
$618$		0			0
$619$	52.0000			0.0840065			0.0420032	−	0.999117i	$-0.486626\pi$
$619$	52.0000			0.0840065			0.0420032	+	0.999117i	$0.486626\pi$
$620$		0			0
$621$	425.000	+	490.650i	0.684380	+	0.790096i
$622$		0			0
$623$		−	384.666i		−	0.617442i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	139.943	+	576.999i	0.223194	+	0.920254i
$628$		0			0
$629$			263.879i			0.419521i
$630$		0			0
$631$	544.000			0.862124			0.431062	−	0.902322i	$-0.358139\pi$
$631$	544.000			0.862124			0.431062	+	0.902322i	$0.358139\pi$
$632$		0			0
$633$	34.9857	−	8.48528i	0.0552697	−	0.0134049i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$			849.360i			1.32505i	0.749038	+	0.662527i	$0.230516\pi$
$641$			849.360i			1.32505i	−0.749038	+	0.662527i	$0.769484\pi$
$642$		0			0
$643$	367.350			0.571306			0.285653	−	0.958333i	$-0.407789\pi$
$643$	367.350			0.571306			0.285653	+	0.958333i	$0.407789\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			971.565i			1.50165i	0.660504	+	0.750823i	$0.270343\pi$
$647$			971.565i			1.50165i	−0.660504	+	0.750823i	$0.729657\pi$
$648$		0			0
$649$	272.000			0.419106
$650$		0			0
$651$	544.000	−	131.939i	0.835637	−	0.202672i
$652$		0			0
$653$		−	350.725i		−	0.537098i	−0.963266	−	0.268549i	$-0.913456\pi$
$653$		−	350.725i		−	0.537098i	0.963266	−	0.268549i	$-0.0865441\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	932.952	−	480.833i	1.42002	−	0.731861i
$658$		0			0
$659$			577.235i			0.875925i	0.898993	+	0.437963i	$0.144300\pi$
$659$			577.235i			0.875925i	−0.898993	+	0.437963i	$0.855700\pi$
$660$		0			0
$661$	80.0000			0.121029			0.0605144	−	0.998167i	$-0.480726\pi$
$661$	80.0000			0.121029			0.0605144	+	0.998167i	$0.480726\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	119.000	−	28.8617i	0.177877	−	0.0431416i
$670$		0			0
$671$		−	263.879i		−	0.393262i
$672$		0			0
$673$	489.800			0.727786			0.363893	−	0.931441i	$-0.381447\pi$
$673$	489.800			0.727786			0.363893	+	0.931441i	$0.381447\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			192.333i			0.284096i	0.989860	+	0.142048i	$0.0453687\pi$
$677$			192.333i			0.284096i	−0.989860	+	0.142048i	$0.954631\pi$
$678$		0			0
$679$	952.000			1.40206
$680$		0			0
$681$	113.000	+	465.911i	0.165932	+	0.684157i
$682$		0			0
$683$			236.174i			0.345789i	0.984940	+	0.172894i	$0.0553119\pi$
$683$			236.174i			0.345789i	−0.984940	+	0.172894i	$0.944688\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−239.069	+	57.9828i	−0.347990	+	0.0843999i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	548.000			0.793054			0.396527	−	0.918023i	$-0.370215\pi$
$691$	548.000			0.793054			0.396527	+	0.918023i	$0.370215\pi$
$692$		0			0
$693$	396.505	+	769.332i	0.572157	+	1.11015i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	653.067			0.936968
$698$		0			0
$699$	−136.000	−	560.742i	−0.194564	−	0.802207i
$700$		0			0
$701$			57.7235i			0.0823445i	0.999152	+	0.0411722i	$0.0131092\pi$
$701$			57.7235i			0.0823445i	−0.999152	+	0.0411722i	$0.986891\pi$
$702$		0			0
$703$	−279.886			−0.398130
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	769.332i		−	1.08816i
$708$		0			0
$709$	−1230.00			−1.73484			−0.867419	−	0.497579i	$-0.834223\pi$
$709$	−1230.00			−1.73484			−0.867419	+	0.497579i	$0.834223\pi$
$710$		0			0
$711$	−576.000	+	296.864i	−0.810127	+	0.417530i
$712$		0			0
$713$			769.332i			1.07901i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	326.533	+	1346.33i	0.455416	+	1.87773i
$718$		0			0
$719$		−	626.712i		−	0.871644i	−0.900033	−	0.435822i	$-0.856458\pi$
$719$		−	626.712i		−	0.871644i	0.900033	−	0.435822i	$-0.143542\pi$
$720$		0			0
$721$	578.000			0.801664
$722$		0			0
$723$	886.305	−	214.960i	1.22587	−	0.297317i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	367.350			0.505296			0.252648	−	0.967558i	$-0.418699\pi$
$727$	367.350			0.505296			0.252648	+	0.967558i	$0.418699\pi$
$728$		0			0
$729$	104.000	−	721.543i	0.142661	−	0.989772i
$730$		0			0
$731$		−	461.788i		−	0.631721i
$732$		0			0
$733$	1002.92			1.36825			0.684123	−	0.729367i	$-0.260185\pi$
$733$	1002.92			1.36825			0.684123	+	0.729367i	$0.260185\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	96.1665i		−	0.130484i
$738$		0			0
$739$	−340.000			−0.460081			−0.230041	−	0.973181i	$-0.573886\pi$
$739$	−340.000			−0.460081			−0.230041	+	0.973181i	$0.573886\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			1243.09i			1.67307i	0.547911	+	0.836537i	$0.315423\pi$
$743$			1243.09i			1.67307i	−0.547911	+	0.836537i	$0.684577\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−180.760	−	350.725i	−0.241981	−	0.469511i
$748$		0			0
$749$			321.602i			0.429375i
$750$		0			0
$751$	520.000			0.692410			0.346205	−	0.938159i	$-0.387470\pi$
$751$	520.000			0.692410			0.346205	+	0.938159i	$0.387470\pi$
$752$		0			0
$753$	244.900	+	1009.75i	0.325232	+	1.34097i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	816.333			1.07838			0.539190	−	0.842184i	$-0.318731\pi$
$757$	816.333			1.07838			0.539190	+	0.842184i	$0.318731\pi$
$758$		0			0
$759$	−1156.00	+	280.371i	−1.52306	+	0.369395i
$760$		0			0
$761$		−	395.818i		−	0.520129i	−0.965591	−	0.260064i	$-0.916256\pi$
$761$		−	395.818i		−	0.520129i	0.965591	−	0.260064i	$-0.0837438\pi$
$762$		0			0
$763$	−466.476			−0.611371
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	306.000			0.397919			0.198960	−	0.980008i	$-0.436244\pi$
$769$	306.000			0.397919			0.198960	+	0.980008i	$0.436244\pi$
$770$		0			0
$771$	276.000	+	1137.98i	0.357977	+	1.47598i
$772$		0			0
$773$		−	305.470i		−	0.395175i	−0.980285	−	0.197587i	$-0.936689\pi$
$773$		−	305.470i		−	0.395175i	0.980285	−	0.197587i	$-0.0633106\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−396.505	+	96.1665i	−0.510302	+	0.123766i
$778$		0			0
$779$			692.682i			0.889194i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	565.602			0.718681			0.359341	−	0.933206i	$-0.383002\pi$
$787$	565.602			0.718681			0.359341	+	0.933206i	$0.383002\pi$
$788$		0			0
$789$	209.000	+	861.729i	0.264892	+	1.09218i
$790$		0			0
$791$			890.591i			1.12590i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	1063.49i		−	1.33436i	−0.744894	−	0.667182i	$-0.767500\pi$
$797$		−	1063.49i		−	1.33436i	0.744894	−	0.667182i	$-0.232500\pi$
$798$		0			0
$799$	400.000			0.500626
$800$		0			0
$801$	−272.000	−	527.758i	−0.339576	−	0.658873i
$802$		0			0
$803$			1923.33i			2.39518i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	52.4786	+	216.375i	0.0650292	+	0.268122i
$808$		0			0
$809$		−	791.636i		−	0.978537i	−0.872133	−	0.489268i	$-0.837264\pi$
$809$		−	791.636i		−	0.978537i	0.872133	−	0.489268i	$-0.162736\pi$
$810$		0			0
$811$	−436.000			−0.537608			−0.268804	−	0.963195i	$-0.586628\pi$
$811$	−436.000			−0.537608			−0.268804	+	0.963195i	$0.586628\pi$
$812$		0			0
$813$	−116.619	+	28.2843i	−0.143443	+	0.0347900i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	489.800			0.599510
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	799.882i		−	0.974278i	−0.873324	−	0.487139i	$-0.838040\pi$
$821$		−	799.882i		−	0.974278i	0.873324	−	0.487139i	$-0.161960\pi$
$822$		0			0
$823$	1311.96			1.59412			0.797062	−	0.603897i	$-0.206386\pi$
$823$	1311.96			1.59412			0.797062	+	0.603897i	$0.206386\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	985.707i		−	1.19191i	−0.803019	−	0.595953i	$-0.796774\pi$
$827$		−	985.707i		−	1.19191i	0.803019	−	0.595953i	$-0.203226\pi$
$828$		0			0
$829$	1488.00			1.79493			0.897467	−	0.441082i	$-0.145405\pi$
$829$	1488.00			1.79493			0.897467	+	0.441082i	$0.145405\pi$
$830$		0			0
$831$	−1292.00	+	313.356i	−1.55475	+	0.377083i
$832$		0			0
$833$			169.706i			0.203728i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	653.067	−	565.685i	0.780247	−	0.675849i
$838$		0			0
$839$		−	461.788i		−	0.550403i	−0.961387	−	0.275201i	$-0.911255\pi$
$839$		−	461.788i		−	0.550403i	0.961387	−	0.275201i	$-0.0887445\pi$
$840$		0			0
$841$	841.000			1.00000
$842$		0			0
$843$	−367.350	−	1514.62i	−0.435765	−	1.79671i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−880.474			−1.03952
$848$		0			0
$849$	−935.000	+	226.771i	−1.10130	+	0.267103i
$850$		0			0
$851$		−	560.742i		−	0.658922i
$852$		0			0
$853$	−326.533			−0.382806			−0.191403	−	0.981512i	$-0.561304\pi$
$853$	−326.533			−0.382806			−0.191403	+	0.981512i	$0.561304\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	288.500i		−	0.336639i	−0.985732	−	0.168319i	$-0.946166\pi$
$857$		−	288.500i		−	0.336639i	0.985732	−	0.168319i	$-0.0538340\pi$
$858$		0			0
$859$	796.000			0.926659			0.463329	−	0.886186i	$-0.346655\pi$
$859$	796.000			0.926659			0.463329	+	0.886186i	$0.346655\pi$
$860$		0			0
$861$	238.000	+	981.299i	0.276423	+	1.13972i
$862$		0			0
$863$		−	352.139i		−	0.408041i	−0.978967	−	0.204020i	$-0.934599\pi$
$863$		−	352.139i		−	0.408041i	0.978967	−	0.204020i	$-0.0654009\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	469.392	−	113.844i	0.541397	−	0.131308i
$868$		0			0
$869$		−	1187.45i		−	1.36646i
$870$		0			0
$871$		0			0
$872$		0			0
$873$	1306.13	−	673.166i	1.49614	−	0.771095i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−676.390			−0.771255			−0.385627	−	0.922655i	$-0.626015\pi$
$877$	−676.390			−0.771255			−0.385627	+	0.922655i	$0.626015\pi$
$878$		0			0
$879$	−60.0000	−	247.386i	−0.0682594	−	0.281441i
$880$		0			0
$881$			519.511i			0.589684i	0.955546	+	0.294842i	$0.0952670\pi$
$881$			519.511i			0.589684i	−0.955546	+	0.294842i	$0.904733\pi$
$882$		0			0
$883$	1591.85			1.80277			0.901387	−	0.433014i	$-0.142550\pi$
$883$	1591.85			1.80277			0.901387	+	0.433014i	$0.142550\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			767.918i			0.865747i	0.901455	+	0.432874i	$0.142500\pi$
$887$			767.918i			0.865747i	−0.901455	+	0.432874i	$0.857500\pi$
$888$		0			0
$889$	238.000			0.267717
$890$		0			0
$891$	1088.00	+	775.144i	1.22110	+	0.869971i
$892$		0			0
$893$			424.264i			0.475100i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−768.000			−0.852386
$902$		0			0
$903$	693.883	−	168.291i	0.768420	−	0.186369i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−332.364			−0.366444			−0.183222	−	0.983072i	$-0.558653\pi$
$907$	−332.364			−0.366444			−0.183222	+	0.983072i	$0.558653\pi$
$908$		0			0
$909$	−544.000	−	1055.52i	−0.598460	−	1.16118i
$910$		0			0
$911$			692.682i			0.760353i	0.924914	+	0.380177i	$0.124137\pi$
$911$			692.682i			0.760353i	−0.924914	+	0.380177i	$0.875863\pi$
$912$		0			0
$913$	723.038			0.791937
$914$		0			0
$915$		0			0
$916$		0			0
$917$			288.500i			0.314612i
$918$		0			0
$919$	−912.000			−0.992383			−0.496192	−	0.868213i	$-0.665269\pi$
$919$	−912.000			−0.992383			−0.496192	+	0.868213i	$0.665269\pi$
$920$		0			0
$921$	−1071.00	+	259.756i	−1.16287	+	0.282037i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	793.009	−	408.708i	0.855458	−	0.440893i
$928$		0			0
$929$			1261.67i			1.35810i	0.734094	+	0.679048i	$0.237607\pi$
$929$			1261.67i			1.35810i	−0.734094	+	0.679048i	$0.762393\pi$
$930$		0			0
$931$	−180.000			−0.193340
$932$		0			0
$933$	−69.9714	−	288.500i	−0.0749962	−	0.309217i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−816.333			−0.871220			−0.435610	−	0.900135i	$-0.643467\pi$
$937$	−816.333			−0.871220			−0.435610	+	0.900135i	$0.643467\pi$
$938$		0			0
$939$	544.000	−	131.939i	0.579340	−	0.140511i
$940$		0			0
$941$		−	1055.52i		−	1.12170i	−0.827919	−	0.560848i	$-0.810475\pi$
$941$		−	1055.52i		−	1.12170i	0.827919	−	0.560848i	$-0.189525\pi$
$942$		0			0
$943$	−1387.77			−1.47165
$944$		0			0
$945$		0			0
$946$		0			0
$947$			114.551i			0.120962i	0.998169	+	0.0604812i	$0.0192635\pi$
$947$			114.551i			0.120962i	−0.998169	+	0.0604812i	$0.980736\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−368.000	−	1517.30i	−0.386961	−	1.59548i
$952$		0			0
$953$		−	243.245i		−	0.255241i	−0.991823	−	0.127621i	$-0.959266\pi$
$953$		−	243.245i		−	0.255241i	0.991823	−	0.127621i	$-0.0407340\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	296.864i		−	0.309555i
$960$		0			0
$961$	63.0000			0.0655567
$962$		0			0
$963$	227.407	+	441.235i	0.236144	+	0.458188i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	938.783			0.970820			0.485410	−	0.874287i	$-0.338670\pi$
$967$	938.783			0.970820			0.485410	+	0.874287i	$0.338670\pi$
$968$		0			0
$969$	−96.0000	−	395.818i	−0.0990712	−	0.408481i
$970$		0			0
$971$		−	1335.89i		−	1.37578i	−0.725813	−	0.687892i	$-0.758536\pi$
$971$		−	1335.89i		−	1.37578i	0.725813	−	0.687892i	$-0.241464\pi$
$972$		0			0
$973$	256.562			0.263681
$974$		0			0
$975$		0			0
$976$		0			0
$977$			305.470i			0.312661i	0.987705	+	0.156331i	$0.0499665\pi$
$977$			305.470i			0.312661i	−0.987705	+	0.156331i	$0.950033\pi$
$978$		0			0
$979$	1088.00			1.11134
$980$		0			0
$981$	−640.000	+	329.848i	−0.652396	+	0.336237i
$982$		0			0
$983$		−	1110.16i		−	1.12936i	−0.825311	−	0.564678i	$-0.809000\pi$
$983$		−	1110.16i		−	1.12936i	0.825311	−	0.564678i	$-0.191000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	145.774	+	601.041i	0.147694	+	0.608957i
$988$		0			0
$989$			981.299i			0.992213i
$990$		0			0
$991$	−640.000			−0.645812			−0.322906	−	0.946431i	$-0.604660\pi$
$991$	−640.000			−0.645812			−0.322906	+	0.946431i	$0.604660\pi$
$992$		0			0
$993$	−851.319	+	206.475i	−0.857320	+	0.207931i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−279.886			−0.280728			−0.140364	−	0.990100i	$-0.544827\pi$
$997$	−279.886			−0.280728			−0.140364	+	0.990100i	$0.544827\pi$
$998$		0			0
$999$	−476.000	+	412.311i	−0.476476	+	0.412723i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.3.l.t.401.3		4
3.2	odd	2	inner	1200.3.l.t.401.4		4
4.3	odd	2		150.3.d.d.101.3		4
5.2	odd	4		240.3.c.c.209.1		4
5.3	odd	4		240.3.c.c.209.4		4
5.4	even	2	inner	1200.3.l.t.401.2		4
12.11	even	2		150.3.d.d.101.1		4
15.2	even	4		240.3.c.c.209.3		4
15.8	even	4		240.3.c.c.209.2		4
15.14	odd	2	inner	1200.3.l.t.401.1		4
20.3	even	4		30.3.b.a.29.3	yes	4
20.7	even	4		30.3.b.a.29.2	yes	4
20.19	odd	2		150.3.d.d.101.2		4
40.3	even	4		960.3.c.f.449.4		4
40.13	odd	4		960.3.c.e.449.1		4
40.27	even	4		960.3.c.f.449.1		4
40.37	odd	4		960.3.c.e.449.4		4
60.23	odd	4		30.3.b.a.29.1	✓	4
60.47	odd	4		30.3.b.a.29.4	yes	4
60.59	even	2		150.3.d.d.101.4		4
120.53	even	4		960.3.c.e.449.3		4
120.77	even	4		960.3.c.e.449.2		4
120.83	odd	4		960.3.c.f.449.2		4
120.107	odd	4		960.3.c.f.449.3		4
180.7	even	12		810.3.j.c.539.3		8
180.23	odd	12		810.3.j.c.269.3		8
180.43	even	12		810.3.j.c.539.1		8
180.47	odd	12		810.3.j.c.539.2		8
180.67	even	12		810.3.j.c.269.4		8
180.83	odd	12		810.3.j.c.539.4		8
180.103	even	12		810.3.j.c.269.2		8
180.167	odd	12		810.3.j.c.269.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.3.b.a.29.1	✓	4	60.23	odd	4
30.3.b.a.29.2	yes	4	20.7	even	4
30.3.b.a.29.3	yes	4	20.3	even	4
30.3.b.a.29.4	yes	4	60.47	odd	4
150.3.d.d.101.1		4	12.11	even	2
150.3.d.d.101.2		4	20.19	odd	2
150.3.d.d.101.3		4	4.3	odd	2
150.3.d.d.101.4		4	60.59	even	2
240.3.c.c.209.1		4	5.2	odd	4
240.3.c.c.209.2		4	15.8	even	4
240.3.c.c.209.3		4	15.2	even	4
240.3.c.c.209.4		4	5.3	odd	4
810.3.j.c.269.1		8	180.167	odd	12
810.3.j.c.269.2		8	180.103	even	12
810.3.j.c.269.3		8	180.23	odd	12
810.3.j.c.269.4		8	180.67	even	12
810.3.j.c.539.1		8	180.43	even	12
810.3.j.c.539.2		8	180.47	odd	12
810.3.j.c.539.3		8	180.7	even	12
810.3.j.c.539.4		8	180.83	odd	12
960.3.c.e.449.1		4	40.13	odd	4
960.3.c.e.449.2		4	120.77	even	4
960.3.c.e.449.3		4	120.53	even	4
960.3.c.e.449.4		4	40.37	odd	4
960.3.c.f.449.1		4	40.27	even	4
960.3.c.f.449.2		4	120.83	odd	4
960.3.c.f.449.3		4	120.107	odd	4
960.3.c.f.449.4		4	40.3	even	4
1200.3.l.t.401.1		4	15.14	odd	2	inner
1200.3.l.t.401.2		4	5.4	even	2	inner
1200.3.l.t.401.3		4	1.1	even	1	trivial
1200.3.l.t.401.4		4	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1200.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.91548 - 0.707107i) q^{3} +5.83095 q^{7} +(8.00000 - 4.12311i) q^{9} +O(q^{10})\)	\(q+(2.91548 - 0.707107i) q^{3} +5.83095 q^{7} +(8.00000 - 4.12311i) q^{9} +16.4924i q^{11} -11.3137i q^{17} +12.0000 q^{19} +(17.0000 - 4.12311i) q^{21} +24.0416i q^{23} +(20.4083 - 17.6777i) q^{27} +32.0000 q^{31} +(11.6619 + 48.0833i) q^{33} -23.3238 q^{37} +57.7235i q^{41} +40.8167 q^{43} +35.3553i q^{47} -15.0000 q^{49} +(-8.00000 - 32.9848i) q^{51} -67.8823i q^{53} +(34.9857 - 8.48528i) q^{57} -16.4924i q^{59} -16.0000 q^{61} +(46.6476 - 24.0416i) q^{63} -5.83095 q^{67} +(17.0000 + 70.0928i) q^{69} +116.619 q^{73} +96.1665i q^{77} -72.0000 q^{79} +(47.0000 - 65.9697i) q^{81} -43.8406i q^{83} -65.9697i q^{89} +(93.2952 - 22.6274i) q^{93} +163.267 q^{97} +(68.0000 + 131.939i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 32 q^{9}+O(q^{10}) \) 4 * q + 32 * q^9	\( 4 q + 32 q^{9} + 48 q^{19} + 68 q^{21} + 128 q^{31} - 60 q^{49} - 32 q^{51} - 64 q^{61} + 68 q^{69} - 288 q^{79} + 188 q^{81} + 272 q^{99}+O(q^{100}) \) 4 * q + 32 * q^9 + 48 * q^19 + 68 * q^21 + 128 * q^31 - 60 * q^49 - 32 * q^51 - 64 * q^61 + 68 * q^69 - 288 * q^79 + 188 * q^81 + 272 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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