LMFDB - Embedded newform 1200.3.c.l.449.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,3,Mod(449,1200)]

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, 1]))

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

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

chi := DirichletCharacter("1200.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1200.c (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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 18x^{10} + 75x^{8} + 1270x^{6} + 14397x^{4} - 7740x^{2} + 39204 $ x^12 + 18x^10 + 75x^8 + 1270x^6 + 14397x^4 - 7740*x^2 + 39204
Coefficient ring:	$\Z[a_1, \ldots, a_{37}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 600)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.6
Root			$2.54797 + 1.77752i$ of defining polynomial
Character	$\chi$	$=$	1200.449
Dual form			1200.3.c.l.449.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.13375 + 2.77752i) q^{3} -5.85843i q^{7} +(-6.42922 - 6.29803i) q^{9} +O(q^{10})$	$q+(-1.13375 + 2.77752i) q^{3} -5.85843i q^{7} +(-6.42922 - 6.29803i) q^{9} -12.4594i q^{11} +11.8584i q^{13} -29.2932 q^{17} +3.19332 q^{19} +(16.2719 + 6.64200i) q^{21} +19.5353 q^{23} +(24.7820 - 10.7169i) q^{27} +30.3022i q^{29} -3.57529 q^{31} +(34.6061 + 14.1258i) q^{33} +42.7639i q^{37} +(-32.9370 - 13.4445i) q^{39} +6.39241i q^{41} +62.6224i q^{43} +69.3534 q^{47} +14.6788 q^{49} +(33.2112 - 81.3625i) q^{51} +57.7856 q^{53} +(-3.62043 + 8.86951i) q^{57} -78.9226i q^{59} +68.5189 q^{61} +(-36.8966 + 37.6651i) q^{63} +90.5235i q^{67} +(-22.1481 + 54.2596i) q^{69} +26.5398i q^{71} +40.0851i q^{73} -72.9923 q^{77} -148.858 q^{79} +(1.66962 + 80.9828i) q^{81} -9.36722 q^{83} +(-84.1649 - 34.3551i) q^{87} +109.706i q^{89} +69.4718 q^{91} +(4.05349 - 9.93044i) q^{93} +161.849i q^{97} +(-78.4695 + 80.1039i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 32 q^{9}+O(q^{10}) $ 12 * q - 32 * q^9	$ 12 q - 32 q^{9} + 100 q^{19} - 36 q^{21} + 228 q^{31} - 12 q^{39} - 152 q^{49} + 12 q^{51} + 124 q^{61} - 312 q^{69} - 152 q^{79} - 448 q^{81} + 620 q^{91} - 500 q^{99}+O(q^{100}) $ 12 * q - 32 * q^9 + 100 * q^19 - 36 * q^21 + 228 * q^31 - 12 * q^39 - 152 * q^49 + 12 * q^51 + 124 * q^61 - 312 * q^69 - 152 * q^79 - 448 * q^81 + 620 * q^91 - 500 * 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
$3$	−1.13375	+	2.77752i	−0.377917	+	0.925839i
$3$	−1.13375	+	2.77752i	−0.377917	+	0.925839i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	5.85843i		−	0.836919i	−0.908236	−	0.418459i	$-0.862570\pi$
$7$		−	5.85843i		−	0.836919i	0.908236	−	0.418459i	$-0.137430\pi$
$8$		0			0
$9$	−6.42922	−	6.29803i	−0.714357	−	0.699781i
$10$		0			0
$11$		−	12.4594i		−	1.13267i	−0.824175	−	0.566335i	$-0.808361\pi$
$11$		−	12.4594i		−	1.13267i	0.824175	−	0.566335i	$-0.191639\pi$
$12$		0			0
$13$			11.8584i			0.912187i	0.889932	+	0.456093i	$0.150752\pi$
$13$			11.8584i			0.912187i	−0.889932	+	0.456093i	$0.849248\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−29.2932			−1.72313			−0.861566	−	0.507646i	$-0.830516\pi$
$17$	−29.2932			−1.72313			−0.861566	+	0.507646i	$0.830516\pi$
$18$		0			0
$19$	3.19332			0.168070			0.0840348	−	0.996463i	$-0.473219\pi$
$19$	3.19332			0.168070			0.0840348	+	0.996463i	$0.473219\pi$
$20$		0			0
$21$	16.2719	+	6.64200i	0.774852	+	0.316286i
$22$		0			0
$23$	19.5353			0.849360			0.424680	−	0.905344i	$-0.360387\pi$
$23$	19.5353			0.849360			0.424680	+	0.905344i	$0.360387\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	24.7820	−	10.7169i	0.917853	−	0.396921i
$28$		0			0
$29$			30.3022i			1.04490i	0.852669	+	0.522451i	$0.174982\pi$
$29$			30.3022i			1.04490i	−0.852669	+	0.522451i	$0.825018\pi$
$30$		0			0
$31$	−3.57529			−0.115332			−0.0576660	−	0.998336i	$-0.518366\pi$
$31$	−3.57529			−0.115332			−0.0576660	+	0.998336i	$0.518366\pi$
$32$		0			0
$33$	34.6061	+	14.1258i	1.04867	+	0.428055i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			42.7639i			1.15578i	0.816114	+	0.577891i	$0.196124\pi$
$37$			42.7639i			1.15578i	−0.816114	+	0.577891i	$0.803876\pi$
$38$		0			0
$39$	−32.9370	−	13.4445i	−0.844539	−	0.344731i
$40$		0			0
$41$			6.39241i			0.155913i	0.996957	+	0.0779563i	$0.0248395\pi$
$41$			6.39241i			0.155913i	−0.996957	+	0.0779563i	$0.975161\pi$
$42$		0			0
$43$			62.6224i			1.45633i	0.685400	+	0.728167i	$0.259628\pi$
$43$			62.6224i			1.45633i	−0.685400	+	0.728167i	$0.740372\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	69.3534			1.47560			0.737802	−	0.675017i	$-0.235864\pi$
$47$	69.3534			1.47560			0.737802	+	0.675017i	$0.235864\pi$
$48$		0			0
$49$	14.6788			0.299567
$50$		0			0
$51$	33.2112	−	81.3625i	0.651201	−	1.59534i
$52$		0			0
$53$	57.7856			1.09029			0.545147	−	0.838340i	$-0.316474\pi$
$53$	57.7856			1.09029			0.545147	+	0.838340i	$0.316474\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−3.62043	+	8.86951i	−0.0635164	+	0.155605i
$58$		0			0
$59$		−	78.9226i		−	1.33767i	−0.743411	−	0.668835i	$-0.766793\pi$
$59$		−	78.9226i		−	1.33767i	0.743411	−	0.668835i	$-0.233207\pi$
$60$		0			0
$61$	68.5189			1.12326			0.561630	−	0.827388i	$-0.310174\pi$
$61$	68.5189			1.12326			0.561630	+	0.827388i	$0.310174\pi$
$62$		0			0
$63$	−36.8966	+	37.6651i	−0.585660	+	0.597859i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			90.5235i			1.35110i	0.737315	+	0.675549i	$0.236093\pi$
$67$			90.5235i			1.35110i	−0.737315	+	0.675549i	$0.763907\pi$
$68$		0			0
$69$	−22.1481	+	54.2596i	−0.320988	+	0.786371i
$70$		0			0
$71$			26.5398i			0.373799i	0.982379	+	0.186900i	$0.0598439\pi$
$71$			26.5398i			0.373799i	−0.982379	+	0.186900i	$0.940156\pi$
$72$		0			0
$73$			40.0851i			0.549112i	0.961571	+	0.274556i	$0.0885308\pi$
$73$			40.0851i			0.549112i	−0.961571	+	0.274556i	$0.911469\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−72.9923			−0.947952
$78$		0			0
$79$	−148.858			−1.88428			−0.942140	−	0.335220i	$-0.891189\pi$
$79$	−148.858			−1.88428			−0.942140	+	0.335220i	$0.891189\pi$
$80$		0			0
$81$	1.66962	+	80.9828i	0.0206125	+	0.999788i
$82$		0			0
$83$	−9.36722			−0.112858			−0.0564290	−	0.998407i	$-0.517971\pi$
$83$	−9.36722			−0.112858			−0.0564290	+	0.998407i	$0.517971\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−84.1649	−	34.3551i	−0.967412	−	0.394887i
$88$		0			0
$89$			109.706i			1.23265i	0.787490	+	0.616327i	$0.211380\pi$
$89$			109.706i			1.23265i	−0.787490	+	0.616327i	$0.788620\pi$
$90$		0			0
$91$	69.4718			0.763426
$92$		0			0
$93$	4.05349	−	9.93044i	0.0435859	−	0.106779i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			161.849i			1.66855i	0.551351	+	0.834274i	$0.314113\pi$
$97$			161.849i			1.66855i	−0.551351	+	0.834274i	$0.685887\pi$
$98$		0			0
$99$	−78.4695	+	80.1039i	−0.792621	+	0.809131i
$100$		0			0
$101$		−	139.273i		−	1.37894i	−0.724315	−	0.689470i	$-0.757844\pi$
$101$		−	139.273i		−	1.37894i	0.724315	−	0.689470i	$-0.242156\pi$
$102$		0			0
$103$			1.22672i			0.0119099i	0.999982	+	0.00595493i	$0.00189553\pi$
$103$			1.22672i			0.0119099i	−0.999982	+	0.00595493i	$0.998104\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	199.298			1.86260			0.931298	−	0.364259i	$-0.118678\pi$
$107$	199.298			1.86260			0.931298	+	0.364259i	$0.118678\pi$
$108$		0			0
$109$	−209.480			−1.92184			−0.960920	−	0.276828i	$-0.910717\pi$
$109$	−209.480			−1.92184			−0.960920	+	0.276828i	$0.910717\pi$
$110$		0			0
$111$	−118.778	−	48.4837i	−1.07007	−	0.436790i
$112$		0			0
$113$	−126.007			−1.11510			−0.557551	−	0.830143i	$-0.688259\pi$
$113$	−126.007			−1.11510			−0.557551	+	0.830143i	$0.688259\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	74.6848	−	76.2404i	0.638331	−	0.651627i
$118$		0			0
$119$			171.612i			1.44212i
$120$		0			0
$121$	−34.2357			−0.282940
$122$		0			0
$123$	−17.7550	−	7.24741i	−0.144350	−	0.0589220i
$124$		0			0
$125$		0			0
$126$		0			0
$127$			49.4064i			0.389026i	0.980900	+	0.194513i	$0.0623127\pi$
$127$			49.4064i			0.389026i	−0.980900	+	0.194513i	$0.937687\pi$
$128$		0			0
$129$	−173.935	−	70.9982i	−1.34833	−	0.550374i
$130$		0			0
$131$			102.351i			0.781304i	0.920538	+	0.390652i	$0.127750\pi$
$131$			102.351i			0.781304i	−0.920538	+	0.390652i	$0.872250\pi$
$132$		0			0
$133$		−	18.7078i		−	0.140661i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	123.181			0.899129			0.449565	−	0.893248i	$-0.351579\pi$
$137$	123.181			0.899129			0.449565	+	0.893248i	$0.351579\pi$
$138$		0			0
$139$	35.9285			0.258479			0.129239	−	0.991613i	$-0.458746\pi$
$139$	35.9285			0.258479			0.129239	+	0.991613i	$0.458746\pi$
$140$		0			0
$141$	−78.6295	+	192.630i	−0.557656	+	1.36617i
$142$		0			0
$143$	147.748			1.03321
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−16.6421	+	40.7706i	−0.113212	+	0.277351i
$148$		0			0
$149$		−	19.2425i		−	0.129144i	−0.997913	−	0.0645722i	$-0.979432\pi$
$149$		−	19.2425i		−	0.129144i	0.997913	−	0.0645722i	$-0.0205683\pi$
$150$		0			0
$151$	67.6695			0.448142			0.224071	−	0.974573i	$-0.428065\pi$
$151$	67.6695			0.448142			0.224071	+	0.974573i	$0.428065\pi$
$152$		0			0
$153$	188.332	+	184.490i	1.23093	+	1.20581i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			27.0003i			0.171977i	0.996296	+	0.0859883i	$0.0274048\pi$
$157$			27.0003i			0.171977i	−0.996296	+	0.0859883i	$0.972595\pi$
$158$		0			0
$159$	−65.5145	+	160.501i	−0.412041	+	1.00944i
$160$		0			0
$161$		−	114.446i		−	0.710845i
$162$		0			0
$163$		−	150.362i		−	0.922466i	−0.887279	−	0.461233i	$-0.847407\pi$
$163$		−	150.362i		−	0.922466i	0.887279	−	0.461233i	$-0.152593\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−252.006			−1.50902			−0.754509	−	0.656290i	$-0.772125\pi$
$167$	−252.006			−1.50902			−0.754509	+	0.656290i	$0.772125\pi$
$168$		0			0
$169$	28.3776			0.167915
$170$		0			0
$171$	−20.5305	−	20.1116i	−0.120062	−	0.117612i
$172$		0			0
$173$	−295.790			−1.70977			−0.854884	−	0.518820i	$-0.826372\pi$
$173$	−295.790			−1.70977			−0.854884	+	0.518820i	$0.826372\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	219.209	+	89.4786i	1.23847	+	0.505529i
$178$		0			0
$179$			13.6377i			0.0761884i	0.999274	+	0.0380942i	$0.0121287\pi$
$179$			13.6377i			0.0761884i	−0.999274	+	0.0380942i	$0.987871\pi$
$180$		0			0
$181$	323.377			1.78661			0.893307	−	0.449448i	$-0.148379\pi$
$181$	323.377			1.78661			0.893307	+	0.449448i	$0.148379\pi$
$182$		0			0
$183$	−77.6834	+	190.312i	−0.424499	+	1.03996i
$184$		0			0
$185$		0			0
$186$		0			0
$187$			364.975i			1.95174i
$188$		0			0
$189$	−62.7840	−	145.184i	−0.332190	−	0.768168i
$190$		0			0
$191$			326.893i			1.71148i	0.517406	+	0.855740i	$0.326898\pi$
$191$			326.893i			1.71148i	−0.517406	+	0.855740i	$0.673102\pi$
$192$		0			0
$193$		−	23.8007i		−	0.123319i	−0.998097	−	0.0616597i	$-0.980361\pi$
$193$		−	23.8007i		−	0.123319i	0.998097	−	0.0616597i	$-0.0196394\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	8.71030			0.0442147			0.0221074	−	0.999756i	$-0.492962\pi$
$197$	8.71030			0.0442147			0.0221074	+	0.999756i	$0.492962\pi$
$198$		0			0
$199$	86.9997			0.437184			0.218592	−	0.975816i	$-0.429854\pi$
$199$	86.9997			0.437184			0.218592	+	0.975816i	$0.429854\pi$
$200$		0			0
$201$	−251.431	−	102.631i	−1.25090	−	0.510603i
$202$		0			0
$203$	177.523			0.874499
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−125.596	−	123.034i	−0.606746	−	0.594366i
$208$		0			0
$209$		−	39.7867i		−	0.190367i
$210$		0			0
$211$	248.258			1.17658			0.588290	−	0.808650i	$-0.299801\pi$
$211$	248.258			1.17658			0.588290	+	0.808650i	$0.299801\pi$
$212$		0			0
$213$	−73.7147	−	30.0895i	−0.346078	−	0.141265i
$214$		0			0
$215$		0			0
$216$		0			0
$217$			20.9456i			0.0965235i
$218$		0			0
$219$	−111.337	−	45.4466i	−0.508389	−	0.207519i
$220$		0			0
$221$		−	347.372i		−	1.57182i
$222$		0			0
$223$			8.54624i			0.0383239i	0.999816	+	0.0191620i	$0.00609982\pi$
$223$			8.54624i			0.0383239i	−0.999816	+	0.0191620i	$0.993900\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−79.4623			−0.350054			−0.175027	−	0.984564i	$-0.556001\pi$
$227$	−79.4623			−0.350054			−0.175027	+	0.984564i	$0.556001\pi$
$228$		0			0
$229$	167.301			0.730574			0.365287	−	0.930895i	$-0.380971\pi$
$229$	167.301			0.730574			0.365287	+	0.930895i	$0.380971\pi$
$230$		0			0
$231$	82.7551	−	202.737i	0.358247	−	0.877651i
$232$		0			0
$233$	227.399			0.975963			0.487982	−	0.872854i	$-0.337733\pi$
$233$	227.399			0.975963			0.487982	+	0.872854i	$0.337733\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	168.768	−	413.456i	0.712102	−	1.74454i
$238$		0			0
$239$		−	227.491i		−	0.951846i	−0.879487	−	0.475923i	$-0.842114\pi$
$239$		−	227.491i		−	0.951846i	0.879487	−	0.475923i	$-0.157886\pi$
$240$		0			0
$241$	−285.556			−1.18488			−0.592440	−	0.805614i	$-0.701835\pi$
$241$	−285.556			−1.18488			−0.592440	+	0.805614i	$0.701835\pi$
$242$		0			0
$243$	−226.824	−	87.1770i	−0.933433	−	0.358753i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			37.8678i			0.153311i
$248$		0			0
$249$	10.6201	−	26.0176i	0.0426510	−	0.104488i
$250$		0			0
$251$			63.3323i			0.252320i	0.992010	+	0.126160i	$0.0402653\pi$
$251$			63.3323i			0.252320i	−0.992010	+	0.126160i	$0.959735\pi$
$252$		0			0
$253$		−	243.397i		−	0.962044i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	37.9769			0.147770			0.0738851	−	0.997267i	$-0.476460\pi$
$257$	37.9769			0.147770			0.0738851	+	0.997267i	$0.476460\pi$
$258$		0			0
$259$	250.530			0.967296
$260$		0			0
$261$	190.844	−	194.819i	0.731203	−	0.746434i
$262$		0			0
$263$	112.740			0.428670			0.214335	−	0.976760i	$-0.431242\pi$
$263$	112.740			0.428670			0.214335	+	0.976760i	$0.431242\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−304.711	−	124.380i	−1.14124	−	0.465841i
$268$		0			0
$269$			95.8931i			0.356480i	0.983987	+	0.178240i	$0.0570403\pi$
$269$			95.8931i			0.356480i	−0.983987	+	0.178240i	$0.942960\pi$
$270$		0			0
$271$	73.8204			0.272400			0.136200	−	0.990681i	$-0.456511\pi$
$271$	73.8204			0.272400			0.136200	+	0.990681i	$0.456511\pi$
$272$		0			0
$273$	−78.7637	+	192.959i	−0.288512	+	0.706810i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	44.5583i		−	0.160860i	−0.996760	−	0.0804301i	$-0.974371\pi$
$277$		−	44.5583i		−	0.160860i	0.996760	−	0.0804301i	$-0.0256294\pi$
$278$		0			0
$279$	22.9863	+	22.5173i	0.0823882	+	0.0807071i
$280$		0			0
$281$			179.919i			0.640280i	0.947370	+	0.320140i	$0.103730\pi$
$281$			179.919i			0.640280i	−0.947370	+	0.320140i	$0.896270\pi$
$282$		0			0
$283$			106.070i			0.374806i	0.982283	+	0.187403i	$0.0600070\pi$
$283$			106.070i			0.374806i	−0.982283	+	0.187403i	$0.939993\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	37.4495			0.130486
$288$		0			0
$289$	569.093			1.96918
$290$		0			0
$291$	−449.539	−	183.497i	−1.54481	−	0.630573i
$292$		0			0
$293$	198.829			0.678599			0.339299	−	0.940678i	$-0.389810\pi$
$293$	198.829			0.678599			0.339299	+	0.940678i	$0.389810\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−133.525	−	308.768i	−0.449580	−	1.03962i
$298$		0			0
$299$			231.658i			0.774775i
$300$		0			0
$301$	366.869			1.21883
$302$		0			0
$303$	386.833	+	157.901i	1.27668	+	0.521125i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	245.107i		−	0.798395i	−0.916865	−	0.399197i	$-0.869289\pi$
$307$		−	245.107i		−	0.798395i	0.916865	−	0.399197i	$-0.130711\pi$
$308$		0			0
$309$	−3.40723	−	1.39079i	−0.0110266	−	0.00450094i
$310$		0			0
$311$		−	6.45769i		−	0.0207643i	−0.999946	−	0.0103821i	$-0.996695\pi$
$311$		−	6.45769i		−	0.0207643i	0.999946	−	0.0103821i	$-0.00330480\pi$
$312$		0			0
$313$			401.243i			1.28193i	0.767572	+	0.640963i	$0.221465\pi$
$313$			401.243i			1.28193i	−0.767572	+	0.640963i	$0.778535\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−321.613			−1.01455			−0.507277	−	0.861783i	$-0.669348\pi$
$317$	−321.613			−1.01455			−0.507277	+	0.861783i	$0.669348\pi$
$318$		0			0
$319$	377.546			1.18353
$320$		0			0
$321$	−225.954	+	553.553i	−0.703907	+	1.72446i
$322$		0			0
$323$	−93.5427			−0.289606
$324$		0			0
$325$		0			0
$326$		0			0
$327$	237.499	−	581.836i	0.726296	−	1.77931i
$328$		0			0
$329$		−	406.302i		−	1.23496i
$330$		0			0
$331$	−28.3052			−0.0855141			−0.0427571	−	0.999085i	$-0.513614\pi$
$331$	−28.3052			−0.0855141			−0.0427571	+	0.999085i	$0.513614\pi$
$332$		0			0
$333$	269.329	−	274.939i	0.808795	−	0.825641i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			453.867i			1.34679i	0.739285	+	0.673393i	$0.235164\pi$
$337$			453.867i			1.34679i	−0.739285	+	0.673393i	$0.764836\pi$
$338$		0			0
$339$	142.860	−	349.985i	0.421416	−	1.03241i
$340$		0			0
$341$			44.5458i			0.130633i
$342$		0			0
$343$		−	373.058i		−	1.08763i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	148.725			0.428602			0.214301	−	0.976768i	$-0.431253\pi$
$347$	148.725			0.428602			0.214301	+	0.976768i	$0.431253\pi$
$348$		0			0
$349$	−141.120			−0.404355			−0.202177	−	0.979349i	$-0.564802\pi$
$349$	−141.120			−0.404355			−0.202177	+	0.979349i	$0.564802\pi$
$350$		0			0
$351$	127.085	+	293.876i	0.362066	+	0.837253i
$352$		0			0
$353$	−209.069			−0.592263			−0.296132	−	0.955147i	$-0.595697\pi$
$353$	−209.069			−0.592263			−0.296132	+	0.955147i	$0.595697\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−476.656	−	194.566i	−1.33517	−	0.545002i
$358$		0			0
$359$		−	457.136i		−	1.27336i	−0.771128	−	0.636680i	$-0.780307\pi$
$359$		−	457.136i		−	1.27336i	0.771128	−	0.636680i	$-0.219693\pi$
$360$		0			0
$361$	−350.803			−0.971753
$362$		0			0
$363$	38.8148	−	95.0904i	0.106928	−	0.261957i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			343.509i			0.935990i	0.883731	+	0.467995i	$0.155024\pi$
$367$			343.509i			0.935990i	−0.883731	+	0.467995i	$0.844976\pi$
$368$		0			0
$369$	40.2596	−	41.0982i	0.109105	−	0.111377i
$370$		0			0
$371$		−	338.533i		−	0.912488i
$372$		0			0
$373$		−	385.038i		−	1.03227i	−0.856506	−	0.516137i	$-0.827369\pi$
$373$		−	385.038i		−	1.03227i	0.856506	−	0.516137i	$-0.172631\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−359.336			−0.953147
$378$		0			0
$379$	316.721			0.835676			0.417838	−	0.908522i	$-0.362788\pi$
$379$	316.721			0.835676			0.417838	+	0.908522i	$0.362788\pi$
$380$		0			0
$381$	−137.227	−	56.0145i	−0.360176	−	0.147020i
$382$		0			0
$383$	−576.465			−1.50513			−0.752565	−	0.658518i	$-0.771184\pi$
$383$	−576.465			−1.50513			−0.752565	+	0.658518i	$0.771184\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	394.398	−	402.613i	1.01912	−	1.04034i
$388$		0			0
$389$		−	54.0496i		−	0.138945i	−0.997584	−	0.0694725i	$-0.977868\pi$
$389$		−	54.0496i		−	0.138945i	0.997584	−	0.0694725i	$-0.0221316\pi$
$390$		0			0
$391$	−572.251			−1.46356
$392$		0			0
$393$	−284.281	−	116.040i	−0.723362	−	0.295268i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	27.0211i		−	0.0680631i	−0.999421	−	0.0340316i	$-0.989165\pi$
$397$		−	27.0211i		−	0.0680631i	0.999421	−	0.0340316i	$-0.0108347\pi$
$398$		0			0
$399$	51.9614	+	21.2101i	0.130229	+	0.0531580i
$400$		0			0
$401$			282.008i			0.703262i	0.936139	+	0.351631i	$0.114373\pi$
$401$			282.008i			0.703262i	−0.936139	+	0.351631i	$0.885627\pi$
$402$		0			0
$403$		−	42.3973i		−	0.105204i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	532.811			1.30912
$408$		0			0
$409$	704.603			1.72274			0.861372	−	0.507974i	$-0.169605\pi$
$409$	704.603			1.72274			0.861372	+	0.507974i	$0.169605\pi$
$410$		0			0
$411$	−139.656	+	342.137i	−0.339796	+	0.832449i
$412$		0			0
$413$	−462.362			−1.11952
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−40.7340	+	99.7922i	−0.0976835	+	0.239310i
$418$		0			0
$419$		−	345.764i		−	0.825212i	−0.910909	−	0.412606i	$-0.864619\pi$
$419$		−	345.764i		−	0.825212i	0.910909	−	0.412606i	$-0.135381\pi$
$420$		0			0
$421$	585.603			1.39098			0.695490	−	0.718536i	$-0.255187\pi$
$421$	585.603			1.39098			0.695490	+	0.718536i	$0.255187\pi$
$422$		0			0
$423$	−445.888	−	436.790i	−1.05411	−	1.03260i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	401.413i		−	0.940077i
$428$		0			0
$429$	−167.510	+	410.374i	−0.390466	+	0.956583i
$430$		0			0
$431$		−	824.304i		−	1.91254i	−0.292487	−	0.956269i	$-0.594483\pi$
$431$		−	824.304i		−	1.91254i	0.292487	−	0.956269i	$-0.405517\pi$
$432$		0			0
$433$		−	685.949i		−	1.58418i	−0.610406	−	0.792089i	$-0.708994\pi$
$433$		−	685.949i		−	1.58418i	0.610406	−	0.792089i	$-0.291006\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	62.3824			0.142751
$438$		0			0
$439$	166.154			0.378482			0.189241	−	0.981931i	$-0.439397\pi$
$439$	166.154			0.378482			0.189241	+	0.981931i	$0.439397\pi$
$440$		0			0
$441$	−94.3731	−	92.4475i	−0.213998	−	0.209632i
$442$		0			0
$443$	−241.838			−0.545910			−0.272955	−	0.962027i	$-0.588001\pi$
$443$	−241.838			−0.545910			−0.272955	+	0.962027i	$0.588001\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	53.4464	+	21.8162i	0.119567	+	0.0488059i
$448$		0			0
$449$			265.195i			0.590634i	0.955399	+	0.295317i	$0.0954253\pi$
$449$			265.195i			0.590634i	−0.955399	+	0.295317i	$0.904575\pi$
$450$		0			0
$451$	79.6454			0.176597
$452$		0			0
$453$	−76.7203	+	187.953i	−0.169361	+	0.414908i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			603.375i			1.32029i	0.751136	+	0.660147i	$0.229506\pi$
$457$			603.375i			1.32029i	−0.751136	+	0.660147i	$0.770494\pi$
$458$		0			0
$459$	−725.946	+	313.931i	−1.58158	+	0.683946i
$460$		0			0
$461$			772.098i			1.67483i	0.546566	+	0.837416i	$0.315935\pi$
$461$			772.098i			1.67483i	−0.546566	+	0.837416i	$0.684065\pi$
$462$		0			0
$463$		−	643.789i		−	1.39047i	−0.718781	−	0.695236i	$-0.755300\pi$
$463$		−	643.789i		−	1.39047i	0.718781	−	0.695236i	$-0.244700\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−503.823			−1.07885			−0.539425	−	0.842034i	$-0.681358\pi$
$467$	−503.823			−1.07885			−0.539425	+	0.842034i	$0.681358\pi$
$468$		0			0
$469$	530.326			1.13076
$470$		0			0
$471$	−74.9939	−	30.6117i	−0.159223	−	0.0649929i
$472$		0			0
$473$	780.235			1.64955
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−371.516	−	363.936i	−0.778860	−	0.762968i
$478$		0			0
$479$			248.720i			0.519249i	0.965710	+	0.259624i	$0.0835987\pi$
$479$			248.720i			0.519249i	−0.965710	+	0.259624i	$0.916401\pi$
$480$		0			0
$481$	−507.113			−1.05429
$482$		0			0
$483$	317.876	+	129.753i	0.658128	+	0.268640i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			586.988i			1.20531i	0.798001	+	0.602657i	$0.205891\pi$
$487$			586.988i			1.20531i	−0.798001	+	0.602657i	$0.794109\pi$
$488$		0			0
$489$	417.633	+	170.473i	0.854055	+	0.348616i
$490$		0			0
$491$			106.210i			0.216314i	0.994134	+	0.108157i	$0.0344949\pi$
$491$			106.210i			0.216314i	−0.994134	+	0.108157i	$0.965505\pi$
$492$		0			0
$493$		−	887.649i		−	1.80050i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	155.481			0.312840
$498$		0			0
$499$	404.643			0.810908			0.405454	−	0.914115i	$-0.367113\pi$
$499$	404.643			0.810908			0.405454	+	0.914115i	$0.367113\pi$
$500$		0			0
$501$	285.712	−	699.951i	0.570284	−	1.39711i
$502$		0			0
$503$	−457.971			−0.910478			−0.455239	−	0.890369i	$-0.650446\pi$
$503$	−457.971			−0.910478			−0.455239	+	0.890369i	$0.650446\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−32.1732	+	78.8194i	−0.0634579	+	0.155462i
$508$		0			0
$509$			913.401i			1.79450i	0.441522	+	0.897250i	$0.354439\pi$
$509$			913.401i			1.79450i	−0.441522	+	0.897250i	$0.645561\pi$
$510$		0			0
$511$	234.836			0.459562
$512$		0			0
$513$	79.1370	−	34.2224i	0.154263	−	0.0667103i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	864.099i		−	1.67137i
$518$		0			0
$519$	335.352	−	821.561i	0.646150	−	1.58297i
$520$		0			0
$521$			482.739i			0.926563i	0.886211	+	0.463282i	$0.153328\pi$
$521$			482.739i			0.926563i	−0.886211	+	0.463282i	$0.846672\pi$
$522$		0			0
$523$			334.636i			0.639839i	0.947445	+	0.319919i	$0.103656\pi$
$523$			334.636i			0.639839i	−0.947445	+	0.319919i	$0.896344\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	104.732			0.198732
$528$		0			0
$529$	−147.373			−0.278588
$530$		0			0
$531$	−497.057	+	507.410i	−0.936077	+	0.955575i
$532$		0			0
$533$	−75.8040			−0.142221
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−37.8790	−	15.4618i	−0.0705383	−	0.0287929i
$538$		0			0
$539$		−	182.888i		−	0.339311i
$540$		0			0
$541$	−82.4307			−0.152367			−0.0761836	−	0.997094i	$-0.524274\pi$
$541$	−82.4307			−0.152367			−0.0761836	+	0.997094i	$0.524274\pi$
$542$		0			0
$543$	−366.629	+	898.185i	−0.675192	+	1.65412i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	663.343i		−	1.21269i	−0.795201	−	0.606346i	$-0.792635\pi$
$547$		−	663.343i		−	1.21269i	0.795201	−	0.606346i	$-0.207365\pi$
$548$		0			0
$549$	−440.523	−	431.534i	−0.802409	−	0.786036i
$550$		0			0
$551$			96.7646i			0.175616i
$552$		0			0
$553$			872.075i			1.57699i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	487.179			0.874648			0.437324	−	0.899304i	$-0.355926\pi$
$557$	487.179			0.874648			0.437324	+	0.899304i	$0.355926\pi$
$558$		0			0
$559$	−742.603			−1.32845
$560$		0			0
$561$	−1013.72	−	413.791i	−1.80700	−	0.737595i
$562$		0			0
$563$	29.6248			0.0526196			0.0263098	−	0.999654i	$-0.491624\pi$
$563$	29.6248			0.0526196			0.0263098	+	0.999654i	$0.491624\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	474.432	−	9.78133i	0.836741	−	0.0172510i
$568$		0			0
$569$		−	946.809i		−	1.66399i	−0.554784	−	0.831994i	$-0.687199\pi$
$569$		−	946.809i		−	1.66399i	0.554784	−	0.831994i	$-0.312801\pi$
$570$		0			0
$571$	385.718			0.675514			0.337757	−	0.941233i	$-0.390332\pi$
$571$	385.718			0.675514			0.337757	+	0.941233i	$0.390332\pi$
$572$		0			0
$573$	−907.951	−	370.615i	−1.58456	−	0.646798i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			216.888i			0.375889i	0.982180	+	0.187944i	$0.0601825\pi$
$577$			216.888i			0.375889i	−0.982180	+	0.187944i	$0.939818\pi$
$578$		0			0
$579$	66.1067	+	26.9840i	0.114174	+	0.0466045i
$580$		0			0
$581$			54.8772i			0.0944530i
$582$		0			0
$583$		−	719.972i		−	1.23494i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−499.326			−0.850641			−0.425321	−	0.905043i	$-0.639839\pi$
$587$	−499.326			−0.850641			−0.425321	+	0.905043i	$0.639839\pi$
$588$		0			0
$589$	−11.4171			−0.0193838
$590$		0			0
$591$	−9.87531	+	24.1930i	−0.0167095	+	0.0409357i
$592$		0			0
$593$	339.358			0.572274			0.286137	−	0.958189i	$-0.407629\pi$
$593$	339.358			0.572274			0.286137	+	0.958189i	$0.407629\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−98.6360	+	241.643i	−0.165219	+	0.404762i
$598$		0			0
$599$			641.821i			1.07149i	0.844381	+	0.535744i	$0.179969\pi$
$599$			641.821i			1.07149i	−0.844381	+	0.535744i	$0.820031\pi$
$600$		0			0
$601$	121.025			0.201372			0.100686	−	0.994918i	$-0.467896\pi$
$601$	121.025			0.201372			0.100686	+	0.994918i	$0.467896\pi$
$602$		0			0
$603$	570.120	−	581.995i	0.945473	−	0.965166i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			218.456i			0.359895i	0.983676	+	0.179947i	$0.0575928\pi$
$607$			218.456i			0.359895i	−0.983676	+	0.179947i	$0.942407\pi$
$608$		0			0
$609$	−201.267	+	493.074i	−0.330488	+	0.809645i
$610$		0			0
$611$			822.422i			1.34603i
$612$		0			0
$613$			201.848i			0.329279i	0.986354	+	0.164639i	$0.0526460\pi$
$613$			201.848i			0.329279i	−0.986354	+	0.164639i	$0.947354\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	745.812			1.20877			0.604386	−	0.796692i	$-0.293418\pi$
$617$	745.812			1.20877			0.604386	+	0.796692i	$0.293418\pi$
$618$		0			0
$619$	−474.849			−0.767124			−0.383562	−	0.923515i	$-0.625303\pi$
$619$	−474.849			−0.767124			−0.383562	+	0.923515i	$0.625303\pi$
$620$		0			0
$621$	484.124	−	209.357i	0.779587	−	0.337128i
$622$		0			0
$623$	642.706			1.03163
$624$		0			0
$625$		0			0
$626$		0			0
$627$	110.508	+	45.1083i	0.176249	+	0.0719430i
$628$		0			0
$629$		−	1252.69i		−	1.99156i
$630$		0			0
$631$	−207.368			−0.328634			−0.164317	−	0.986408i	$-0.552542\pi$
$631$	−207.368			−0.328634			−0.164317	+	0.986408i	$0.552542\pi$
$632$		0			0
$633$	−281.463	+	689.542i	−0.444650	+	1.08932i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			174.067i			0.273261i
$638$		0			0
$639$	167.148	−	170.630i	0.261578	−	0.267026i
$640$		0			0
$641$			371.542i			0.579629i	0.957083	+	0.289815i	$0.0935937\pi$
$641$			371.542i			0.579629i	−0.957083	+	0.289815i	$0.906406\pi$
$642$		0			0
$643$			779.511i			1.21230i	0.795349	+	0.606151i	$0.207287\pi$
$643$			779.511i			1.21230i	−0.795349	+	0.606151i	$0.792713\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1129.46			−1.74568			−0.872840	−	0.488006i	$-0.837724\pi$
$647$	−1129.46			−1.74568			−0.872840	+	0.488006i	$0.837724\pi$
$648$		0			0
$649$	−983.325			−1.51514
$650$		0			0
$651$	−58.1768	−	23.7471i	−0.0893652	−	0.0364779i
$652$		0			0
$653$	327.316			0.501250			0.250625	−	0.968084i	$-0.419364\pi$
$653$	327.316			0.501250			0.250625	+	0.968084i	$0.419364\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	252.457	−	257.716i	0.384258	−	0.392262i
$658$		0			0
$659$			949.895i			1.44142i	0.693237	+	0.720710i	$0.256184\pi$
$659$			949.895i			1.44142i	−0.693237	+	0.720710i	$0.743816\pi$
$660$		0			0
$661$	736.462			1.11416			0.557082	−	0.830458i	$-0.311921\pi$
$661$	736.462			1.11416			0.557082	+	0.830458i	$0.311921\pi$
$662$		0			0
$663$	964.831	+	393.833i	1.45525	+	0.594017i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			591.961i			0.887498i
$668$		0			0
$669$	−23.7373	−	9.68931i	−0.0354818	−	0.0144833i
$670$		0			0
$671$		−	853.701i		−	1.27228i
$672$		0			0
$673$			602.296i			0.894943i	0.894298	+	0.447471i	$0.147675\pi$
$673$			602.296i			0.894943i	−0.894298	+	0.447471i	$0.852325\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−830.123			−1.22618			−0.613089	−	0.790014i	$-0.710073\pi$
$677$	−830.123			−1.22618			−0.613089	+	0.790014i	$0.710073\pi$
$678$		0			0
$679$	948.182			1.39644
$680$		0			0
$681$	90.0905	−	220.708i	0.132291	−	0.324094i
$682$		0			0
$683$	−276.640			−0.405037			−0.202518	−	0.979278i	$-0.564913\pi$
$683$	−276.640			−0.405037			−0.202518	+	0.979278i	$0.564913\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−189.678	+	464.683i	−0.276097	+	0.676394i
$688$		0			0
$689$			685.247i			0.994553i
$690$		0			0
$691$	247.973			0.358861			0.179431	−	0.983771i	$-0.442574\pi$
$691$	247.973			0.358861			0.179431	+	0.983771i	$0.442574\pi$
$692$		0			0
$693$	469.283	+	459.708i	0.677176	+	0.663359i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	187.254i		−	0.268658i
$698$		0			0
$699$	−257.814	+	631.606i	−0.368833	+	0.903585i
$700$		0			0
$701$		−	1213.96i		−	1.73175i	−0.500263	−	0.865874i	$-0.666763\pi$
$701$		−	1213.96i		−	1.73175i	0.500263	−	0.865874i	$-0.333237\pi$
$702$		0			0
$703$			136.559i			0.194252i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−815.920			−1.15406
$708$		0			0
$709$	−745.830			−1.05195			−0.525973	−	0.850501i	$-0.676299\pi$
$709$	−745.830			−1.05195			−0.525973	+	0.850501i	$0.676299\pi$
$710$		0			0
$711$	957.041	+	937.513i	1.34605	+	1.31858i
$712$		0			0
$713$	−69.8443			−0.0979583
$714$		0			0
$715$		0			0
$716$		0			0
$717$	631.861	+	257.919i	0.881257	+	0.359719i
$718$		0			0
$719$			551.198i			0.766618i	0.923620	+	0.383309i	$0.125216\pi$
$719$			551.198i			0.766618i	−0.923620	+	0.383309i	$0.874784\pi$
$720$		0			0
$721$	7.18663			0.00996759
$722$		0			0
$723$	323.750	−	793.138i	0.447787	−	1.09701i
$724$		0			0
$725$		0			0
$726$		0			0
$727$			63.5439i			0.0874056i	0.999045	+	0.0437028i	$0.0139155\pi$
$727$			63.5439i			0.0874056i	−0.999045	+	0.0437028i	$0.986085\pi$
$728$		0			0
$729$	499.298	−	531.171i	0.684908	−	0.728630i
$730$		0			0
$731$		−	1834.41i		−	2.50945i
$732$		0			0
$733$			37.5413i			0.0512159i	0.999672	+	0.0256080i	$0.00815216\pi$
$733$			37.5413i			0.0512159i	−0.999672	+	0.0256080i	$0.991848\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1127.87			1.53035
$738$		0			0
$739$	501.470			0.678579			0.339289	−	0.940682i	$-0.389813\pi$
$739$	501.470			0.678579			0.339289	+	0.940682i	$0.389813\pi$
$740$		0			0
$741$	−105.178	−	42.9326i	−0.141941	−	0.0579388i
$742$		0			0
$743$	150.710			0.202840			0.101420	−	0.994844i	$-0.467661\pi$
$743$	150.710			0.202840			0.101420	+	0.994844i	$0.467661\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	60.2238	+	58.9950i	0.0806209	+	0.0789759i
$748$		0			0
$749$		−	1167.57i		−	1.55884i
$750$		0			0
$751$	1349.71			1.79722			0.898609	−	0.438751i	$-0.144579\pi$
$751$	1349.71			1.79722			0.898609	+	0.438751i	$0.144579\pi$
$752$		0			0
$753$	−175.907	−	71.8031i	−0.233608	−	0.0953561i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	205.791i		−	0.271851i	−0.990719	−	0.135925i	$-0.956599\pi$
$757$		−	205.791i		−	0.271851i	0.990719	−	0.135925i	$-0.0434007\pi$
$758$		0			0
$759$	676.040	+	275.952i	0.890698	+	0.363573i
$760$		0			0
$761$		−	738.457i		−	0.970377i	−0.874410	−	0.485189i	$-0.838751\pi$
$761$		−	738.457i		−	0.970377i	0.874410	−	0.485189i	$-0.161249\pi$
$762$		0			0
$763$			1227.23i			1.60842i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	935.898			1.22021
$768$		0			0
$769$	−467.952			−0.608520			−0.304260	−	0.952589i	$-0.598409\pi$
$769$	−467.952			−0.608520			−0.304260	+	0.952589i	$0.598409\pi$
$770$		0			0
$771$	−43.0564	+	105.482i	−0.0558449	+	0.136812i
$772$		0			0
$773$	233.233			0.301724			0.150862	−	0.988555i	$-0.451795\pi$
$773$	233.233			0.301724			0.150862	+	0.988555i	$0.451795\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−284.038	+	695.850i	−0.365558	+	0.895560i
$778$		0			0
$779$			20.4130i			0.0262041i
$780$		0			0
$781$	330.668			0.423391
$782$		0			0
$783$	324.744	+	750.950i	0.414744	+	0.959067i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			254.788i			0.323746i	0.986812	+	0.161873i	$0.0517535\pi$
$787$			254.788i			0.323746i	−0.986812	+	0.161873i	$0.948247\pi$
$788$		0			0
$789$	−127.819	+	313.138i	−0.162002	+	0.396880i
$790$		0			0
$791$			738.201i			0.933250i
$792$		0			0
$793$			812.526i			1.02462i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1196.67			1.50147			0.750737	−	0.660602i	$-0.229699\pi$
$797$	1196.67			1.50147			0.750737	+	0.660602i	$0.229699\pi$
$798$		0			0
$799$	−2031.58			−2.54266
$800$		0			0
$801$	690.933	−	705.325i	0.862589	−	0.880556i
$802$		0			0
$803$	499.435			0.621962
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−266.345	−	108.719i	−0.330043	−	0.134720i
$808$		0			0
$809$			561.420i			0.693968i	0.937871	+	0.346984i	$0.112794\pi$
$809$			561.420i			0.693968i	−0.937871	+	0.346984i	$0.887206\pi$
$810$		0			0
$811$	−388.881			−0.479508			−0.239754	−	0.970834i	$-0.577067\pi$
$811$	−388.881			−0.479508			−0.239754	+	0.970834i	$0.577067\pi$
$812$		0			0
$813$	−83.6939	+	205.037i	−0.102945	+	0.252199i
$814$		0			0
$815$		0			0
$816$		0			0
$817$			199.973i			0.244765i
$818$		0			0
$819$	−446.649	−	437.535i	−0.545359	−	0.534231i
$820$		0			0
$821$		−	309.777i		−	0.377317i	−0.982043	−	0.188658i	$-0.939586\pi$
$821$		−	309.777i		−	0.377317i	0.982043	−	0.188658i	$-0.0604139\pi$
$822$		0			0
$823$			178.579i			0.216985i	0.994097	+	0.108493i	$0.0346024\pi$
$823$			178.579i			0.216985i	−0.994097	+	0.108493i	$0.965398\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	165.734			0.200404			0.100202	−	0.994967i	$-0.468051\pi$
$827$	165.734			0.200404			0.100202	+	0.994967i	$0.468051\pi$
$828$		0			0
$829$	−173.278			−0.209021			−0.104510	−	0.994524i	$-0.533328\pi$
$829$	−173.278			−0.209021			−0.104510	+	0.994524i	$0.533328\pi$
$830$		0			0
$831$	123.761	+	50.5180i	0.148931	+	0.0607919i
$832$		0			0
$833$	−429.989			−0.516194
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−88.6030	+	38.3159i	−0.105858	+	0.0457777i
$838$		0			0
$839$		−	815.931i		−	0.972504i	−0.873819	−	0.486252i	$-0.838364\pi$
$839$		−	815.931i		−	0.972504i	0.873819	−	0.486252i	$-0.161636\pi$
$840$		0			0
$841$	−77.2223			−0.0918220
$842$		0			0
$843$	−499.727	−	203.983i	−0.592796	−	0.241973i
$844$		0			0
$845$		0			0
$846$		0			0
$847$			200.568i			0.236798i
$848$		0			0
$849$	−294.612	−	120.257i	−0.347010	−	0.141646i
$850$		0			0
$851$			835.405i			0.981675i
$852$		0			0
$853$			631.549i			0.740386i	0.928955	+	0.370193i	$0.120708\pi$
$853$			631.549i			0.740386i	−0.928955	+	0.370193i	$0.879292\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−300.140			−0.350221			−0.175111	−	0.984549i	$-0.556028\pi$
$857$	−300.140			−0.350221			−0.175111	+	0.984549i	$0.556028\pi$
$858$		0			0
$859$	−1243.32			−1.44741			−0.723703	−	0.690111i	$-0.757562\pi$
$859$	−1243.32			−1.44741			−0.723703	+	0.690111i	$0.757562\pi$
$860$		0			0
$861$	−42.4584	+	104.017i	−0.0493129	+	0.120809i
$862$		0			0
$863$	134.903			0.156319			0.0781596	−	0.996941i	$-0.475096\pi$
$863$	134.903			0.156319			0.0781596	+	0.996941i	$0.475096\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−645.210	+	1580.67i	−0.744187	+	1.82314i
$868$		0			0
$869$			1854.68i			2.13427i
$870$		0			0
$871$	−1073.47			−1.23245
$872$		0			0
$873$	1019.33	−	1040.56i	1.16762	−	1.19194i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			14.4694i			0.0164988i	0.999966	+	0.00824940i	$0.00262589\pi$
$877$			14.4694i			0.0164988i	−0.999966	+	0.00824940i	$0.997374\pi$
$878$		0			0
$879$	−225.423	+	552.252i	−0.256454	+	0.628273i
$880$		0			0
$881$			1281.36i			1.45444i	0.686404	+	0.727221i	$0.259188\pi$
$881$			1281.36i			1.45444i	−0.686404	+	0.727221i	$0.740812\pi$
$882$		0			0
$883$		−	565.182i		−	0.640070i	−0.947406	−	0.320035i	$-0.896305\pi$
$883$		−	565.182i		−	0.640070i	0.947406	−	0.320035i	$-0.103695\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−517.280			−0.583179			−0.291590	−	0.956543i	$-0.594184\pi$
$887$	−517.280			−0.583179			−0.291590	+	0.956543i	$0.594184\pi$
$888$		0			0
$889$	289.444			0.325583
$890$		0			0
$891$	1008.99	−	20.8023i	1.13243	−	0.0233472i
$892$		0			0
$893$	221.468			0.248004
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−643.433	−	262.642i	−0.717317	−	0.292801i
$898$		0			0
$899$		−	108.339i		−	0.120511i
$900$		0			0
$901$	−1692.73			−1.87872
$902$		0			0
$903$	−415.938	+	1018.98i	−0.460618	+	1.12844i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			591.131i			0.651743i	0.945414	+	0.325872i	$0.105658\pi$
$907$			591.131i			0.651743i	−0.945414	+	0.325872i	$0.894342\pi$
$908$		0			0
$909$	−877.145	+	895.415i	−0.964956	+	0.985055i
$910$		0			0
$911$			328.905i			0.361038i	0.983572	+	0.180519i	$0.0577777\pi$
$911$			328.905i			0.361038i	−0.983572	+	0.180519i	$0.942222\pi$
$912$		0			0
$913$			116.710i			0.127831i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	599.615			0.653888
$918$		0			0
$919$	−739.366			−0.804533			−0.402266	−	0.915523i	$-0.631777\pi$
$919$	−739.366			−0.804533			−0.402266	+	0.915523i	$0.631777\pi$
$920$		0			0
$921$	680.790	+	277.891i	0.739185	+	0.301727i
$922$		0			0
$923$	−314.720			−0.340975
$924$		0			0
$925$		0			0
$926$		0			0
$927$	7.72590	−	7.88682i	0.00833430	−	0.00850790i
$928$		0			0
$929$		−	1163.63i		−	1.25256i	−0.779599	−	0.626279i	$-0.784577\pi$
$929$		−	1163.63i		−	1.25256i	0.779599	−	0.626279i	$-0.215423\pi$
$930$		0			0
$931$	46.8741			0.0503481
$932$		0			0
$933$	17.9364	+	7.32142i	0.0192244	+	0.00784718i
$934$		0			0
$935$		0			0
$936$		0			0
$937$			379.177i			0.404671i	0.979316	+	0.202336i	$0.0648532\pi$
$937$			379.177i			0.404671i	−0.979316	+	0.202336i	$0.935147\pi$
$938$		0			0
$939$	−1114.46	−	454.910i	−1.18686	−	0.484462i
$940$		0			0
$941$			1102.24i			1.17135i	0.810545	+	0.585676i	$0.199171\pi$
$941$			1102.24i			1.17135i	−0.810545	+	0.585676i	$0.800829\pi$
$942$		0			0
$943$			124.878i			0.132426i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1585.60			1.67434			0.837172	−	0.546940i	$-0.184207\pi$
$947$	1585.60			1.67434			0.837172	+	0.546940i	$0.184207\pi$
$948$		0			0
$949$	−475.347			−0.500892
$950$		0			0
$951$	364.630	−	893.287i	0.383417	−	0.939314i
$952$		0			0
$953$	23.3120			0.0244617			0.0122308	−	0.999925i	$-0.496107\pi$
$953$	23.3120			0.0244617			0.0122308	+	0.999925i	$0.496107\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−428.043	+	1048.64i	−0.447276	+	1.09576i
$958$		0			0
$959$		−	721.646i		−	0.752498i
$960$		0			0
$961$	−948.217			−0.986699
$962$		0			0
$963$	−1281.33	−	1255.18i	−1.33056	−	1.30341i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	1446.24i		−	1.49560i	−0.663926	−	0.747798i	$-0.731111\pi$
$967$		−	1446.24i		−	1.49560i	0.663926	−	0.747798i	$-0.268889\pi$
$968$		0			0
$969$	106.054	−	259.816i	0.109447	−	0.268128i
$970$		0			0
$971$			168.397i			0.173427i	0.996233	+	0.0867134i	$0.0276365\pi$
$971$			168.397i			0.173427i	−0.996233	+	0.0867134i	$0.972364\pi$
$972$		0			0
$973$		−	210.485i		−	0.216326i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−623.561			−0.638241			−0.319120	−	0.947714i	$-0.603387\pi$
$977$	−623.561			−0.638241			−0.319120	+	0.947714i	$0.603387\pi$
$978$		0			0
$979$	1366.87			1.39619
$980$		0			0
$981$	1346.80	+	1319.31i	1.37288	+	1.34487i
$982$		0			0
$983$	378.106			0.384645			0.192323	−	0.981332i	$-0.438398\pi$
$983$	378.106			0.384645			0.192323	+	0.981332i	$0.438398\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	1128.51	+	460.645i	1.14337	+	0.466713i
$988$		0			0
$989$			1223.34i			1.23695i
$990$		0			0
$991$	−1753.42			−1.76934			−0.884672	−	0.466215i	$-0.845617\pi$
$991$	−1753.42			−1.76934			−0.884672	+	0.466215i	$0.845617\pi$
$992$		0			0
$993$	32.0910	−	78.6181i	0.0323173	−	0.0791723i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	839.151i		−	0.841676i	−0.907136	−	0.420838i	$-0.861736\pi$
$997$		−	839.151i		−	0.841676i	0.907136	−	0.420838i	$-0.138264\pi$
$998$		0			0
$999$	458.295	+	1059.78i	0.458754	+	1.06084i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.3.c.l.449.6		12
3.2	odd	2	inner	1200.3.c.l.449.8		12
4.3	odd	2		600.3.c.c.449.7		12
5.2	odd	4		1200.3.l.w.401.6		6
5.3	odd	4		1200.3.l.v.401.1		6
5.4	even	2	inner	1200.3.c.l.449.7		12
12.11	even	2		600.3.c.c.449.5		12
15.2	even	4		1200.3.l.w.401.5		6
15.8	even	4		1200.3.l.v.401.2		6
15.14	odd	2	inner	1200.3.c.l.449.5		12
20.3	even	4		600.3.l.e.401.6	yes	6
20.7	even	4		600.3.l.d.401.1	✓	6
20.19	odd	2		600.3.c.c.449.6		12
60.23	odd	4		600.3.l.e.401.5	yes	6
60.47	odd	4		600.3.l.d.401.2	yes	6
60.59	even	2		600.3.c.c.449.8		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.3.c.c.449.5		12	12.11	even	2
600.3.c.c.449.6		12	20.19	odd	2
600.3.c.c.449.7		12	4.3	odd	2
600.3.c.c.449.8		12	60.59	even	2
600.3.l.d.401.1	✓	6	20.7	even	4
600.3.l.d.401.2	yes	6	60.47	odd	4
600.3.l.e.401.5	yes	6	60.23	odd	4
600.3.l.e.401.6	yes	6	20.3	even	4
1200.3.c.l.449.5		12	15.14	odd	2	inner
1200.3.c.l.449.6		12	1.1	even	1	trivial
1200.3.c.l.449.7		12	5.4	even	2	inner
1200.3.c.l.449.8		12	3.2	odd	2	inner
1200.3.l.v.401.1		6	5.3	odd	4
1200.3.l.v.401.2		6	15.8	even	4
1200.3.l.w.401.5		6	15.2	even	4
1200.3.l.w.401.6		6	5.2	odd	4

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.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.13375 + 2.77752i) q^{3} -5.85843i q^{7} +(-6.42922 - 6.29803i) q^{9} +O(q^{10})\)	\(q+(-1.13375 + 2.77752i) q^{3} -5.85843i q^{7} +(-6.42922 - 6.29803i) q^{9} -12.4594i q^{11} +11.8584i q^{13} -29.2932 q^{17} +3.19332 q^{19} +(16.2719 + 6.64200i) q^{21} +19.5353 q^{23} +(24.7820 - 10.7169i) q^{27} +30.3022i q^{29} -3.57529 q^{31} +(34.6061 + 14.1258i) q^{33} +42.7639i q^{37} +(-32.9370 - 13.4445i) q^{39} +6.39241i q^{41} +62.6224i q^{43} +69.3534 q^{47} +14.6788 q^{49} +(33.2112 - 81.3625i) q^{51} +57.7856 q^{53} +(-3.62043 + 8.86951i) q^{57} -78.9226i q^{59} +68.5189 q^{61} +(-36.8966 + 37.6651i) q^{63} +90.5235i q^{67} +(-22.1481 + 54.2596i) q^{69} +26.5398i q^{71} +40.0851i q^{73} -72.9923 q^{77} -148.858 q^{79} +(1.66962 + 80.9828i) q^{81} -9.36722 q^{83} +(-84.1649 - 34.3551i) q^{87} +109.706i q^{89} +69.4718 q^{91} +(4.05349 - 9.93044i) q^{93} +161.849i q^{97} +(-78.4695 + 80.1039i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 32 q^{9}+O(q^{10}) \) 12 * q - 32 * q^9	\( 12 q - 32 q^{9} + 100 q^{19} - 36 q^{21} + 228 q^{31} - 12 q^{39} - 152 q^{49} + 12 q^{51} + 124 q^{61} - 312 q^{69} - 152 q^{79} - 448 q^{81} + 620 q^{91} - 500 q^{99}+O(q^{100}) \) 12 * q - 32 * q^9 + 100 * q^19 - 36 * q^21 + 228 * q^31 - 12 * q^39 - 152 * q^49 + 12 * q^51 + 124 * q^61 - 312 * q^69 - 152 * q^79 - 448 * q^81 + 620 * q^91 - 500 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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