LMFDB - Embedded newform 1200.3.c.l.449.7

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			449.7
Root			$-2.54797 - 1.77752i$ of defining polynomial
Character	$\chi$	$=$	1200.449
Dual form			1200.3.c.l.449.8

$q$-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	$0.137430\pi$
$7$			5.85843i			0.836919i	−0.908236	+	0.418459i	$0.862570\pi$
$8$		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	$-0.849248\pi$
$13$		−	11.8584i		−	0.912187i	0.889932	−	0.456093i	$-0.150752\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	29.2932			1.72313			0.861566	−	0.507646i	$-0.169484\pi$
$17$	29.2932			1.72313			0.861566	+	0.507646i	$0.169484\pi$
$18$		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	$-0.639613\pi$
$23$	−19.5353			−0.849360			−0.424680	+	0.905344i	$0.639613\pi$
$24$		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	$-0.803876\pi$
$37$		−	42.7639i		−	1.15578i	0.816114	−	0.577891i	$-0.196124\pi$
$38$		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	$-0.740372\pi$
$43$		−	62.6224i		−	1.45633i	0.685400	−	0.728167i	$-0.259628\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−69.3534			−1.47560			−0.737802	−	0.675017i	$-0.764136\pi$
$47$	−69.3534			−1.47560			−0.737802	+	0.675017i	$0.764136\pi$
$48$		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	$-0.683526\pi$
$53$	−57.7856			−1.09029			−0.545147	+	0.838340i	$0.683526\pi$
$54$		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	$-0.763907\pi$
$67$		−	90.5235i		−	1.35110i	0.737315	−	0.675549i	$-0.236093\pi$
$68$		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	$-0.911469\pi$
$73$		−	40.0851i		−	0.549112i	0.961571	−	0.274556i	$-0.0885308\pi$
$74$		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	$-0.482029\pi$
$83$	9.36722			0.112858			0.0564290	+	0.998407i	$0.482029\pi$
$84$		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	$-0.685887\pi$
$97$		−	161.849i		−	1.66855i	0.551351	−	0.834274i	$-0.314113\pi$
$98$		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	$-0.998104\pi$
$103$		−	1.22672i		−	0.0119099i	0.999982	−	0.00595493i	$-0.00189553\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−199.298			−1.86260			−0.931298	−	0.364259i	$-0.881322\pi$
$107$	−199.298			−1.86260			−0.931298	+	0.364259i	$0.881322\pi$
$108$		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	$-0.311741\pi$
$113$	126.007			1.11510			0.557551	+	0.830143i	$0.311741\pi$
$114$		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	$-0.937687\pi$
$127$		−	49.4064i		−	0.389026i	0.980900	−	0.194513i	$-0.0623127\pi$
$128$		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	$-0.648421\pi$
$137$	−123.181			−0.899129			−0.449565	+	0.893248i	$0.648421\pi$
$138$		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	$-0.972595\pi$
$157$		−	27.0003i		−	0.171977i	0.996296	−	0.0859883i	$-0.0274048\pi$
$158$		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	$0.152593\pi$
$163$			150.362i			0.922466i	−0.887279	+	0.461233i	$0.847407\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	252.006			1.50902			0.754509	−	0.656290i	$-0.227875\pi$
$167$	252.006			1.50902			0.754509	+	0.656290i	$0.227875\pi$
$168$		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	$-0.173628\pi$
$173$	295.790			1.70977			0.854884	+	0.518820i	$0.173628\pi$
$174$		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	$0.0196394\pi$
$193$			23.8007i			0.123319i	−0.998097	+	0.0616597i	$0.980361\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−8.71030			−0.0442147			−0.0221074	−	0.999756i	$-0.507038\pi$
$197$	−8.71030			−0.0442147			−0.0221074	+	0.999756i	$0.507038\pi$
$198$		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	$-0.993900\pi$
$223$		−	8.54624i		−	0.0383239i	0.999816	−	0.0191620i	$-0.00609982\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	79.4623			0.350054			0.175027	−	0.984564i	$-0.443999\pi$
$227$	79.4623			0.350054			0.175027	+	0.984564i	$0.443999\pi$
$228$		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	$-0.662267\pi$
$233$	−227.399			−0.975963			−0.487982	+	0.872854i	$0.662267\pi$
$234$		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	$-0.523540\pi$
$257$	−37.9769			−0.147770			−0.0738851	+	0.997267i	$0.523540\pi$
$258$		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	$-0.568758\pi$
$263$	−112.740			−0.428670			−0.214335	+	0.976760i	$0.568758\pi$
$264$		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	$0.0256294\pi$
$277$			44.5583i			0.160860i	−0.996760	+	0.0804301i	$0.974371\pi$
$278$		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	$-0.939993\pi$
$283$		−	106.070i		−	0.374806i	0.982283	−	0.187403i	$-0.0600070\pi$
$284$		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	$-0.610190\pi$
$293$	−198.829			−0.678599			−0.339299	+	0.940678i	$0.610190\pi$
$294$		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	$0.130711\pi$
$307$			245.107i			0.798395i	−0.916865	+	0.399197i	$0.869289\pi$
$308$		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	$-0.778535\pi$
$313$		−	401.243i		−	1.28193i	0.767572	−	0.640963i	$-0.221465\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	321.613			1.01455			0.507277	−	0.861783i	$-0.330652\pi$
$317$	321.613			1.01455			0.507277	+	0.861783i	$0.330652\pi$
$318$		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	$-0.764836\pi$
$337$		−	453.867i		−	1.34679i	0.739285	−	0.673393i	$-0.235164\pi$
$338$		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	$-0.568747\pi$
$347$	−148.725			−0.428602			−0.214301	+	0.976768i	$0.568747\pi$
$348$		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	$-0.404303\pi$
$353$	209.069			0.592263			0.296132	+	0.955147i	$0.404303\pi$
$354$		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	$-0.844976\pi$
$367$		−	343.509i		−	0.935990i	0.883731	−	0.467995i	$-0.155024\pi$
$368$		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	$0.172631\pi$
$373$			385.038i			1.03227i	−0.856506	+	0.516137i	$0.827369\pi$
$374$		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	$-0.228816\pi$
$383$	576.465			1.50513			0.752565	+	0.658518i	$0.228816\pi$
$384$		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	$0.0108347\pi$
$397$			27.0211i			0.0680631i	−0.999421	+	0.0340316i	$0.989165\pi$
$398$		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	$0.291006\pi$
$433$			685.949i			1.58418i	−0.610406	+	0.792089i	$0.708994\pi$
$434$		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	$-0.411999\pi$
$443$	241.838			0.545910			0.272955	+	0.962027i	$0.411999\pi$
$444$		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	$-0.770494\pi$
$457$		−	603.375i		−	1.32029i	0.751136	−	0.660147i	$-0.229506\pi$
$458$		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	$0.244700\pi$
$463$			643.789i			1.39047i	−0.718781	+	0.695236i	$0.755300\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	503.823			1.07885			0.539425	−	0.842034i	$-0.318642\pi$
$467$	503.823			1.07885			0.539425	+	0.842034i	$0.318642\pi$
$468$		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	$-0.794109\pi$
$487$		−	586.988i		−	1.20531i	0.798001	−	0.602657i	$-0.205891\pi$
$488$		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	$-0.349554\pi$
$503$	457.971			0.910478			0.455239	+	0.890369i	$0.349554\pi$
$504$		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	$-0.896344\pi$
$523$		−	334.636i		−	0.639839i	0.947445	−	0.319919i	$-0.103656\pi$
$524$		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	$0.207365\pi$
$547$			663.343i			1.21269i	−0.795201	+	0.606346i	$0.792635\pi$
$548$		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	$-0.644074\pi$
$557$	−487.179			−0.874648			−0.437324	+	0.899304i	$0.644074\pi$
$558$		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	$-0.508376\pi$
$563$	−29.6248			−0.0526196			−0.0263098	+	0.999654i	$0.508376\pi$
$564$		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	$-0.939818\pi$
$577$		−	216.888i		−	0.375889i	0.982180	−	0.187944i	$-0.0601825\pi$
$578$		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	$-0.360161\pi$
$587$	499.326			0.850641			0.425321	+	0.905043i	$0.360161\pi$
$588$		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	$-0.592371\pi$
$593$	−339.358			−0.572274			−0.286137	+	0.958189i	$0.592371\pi$
$594$		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	$-0.942407\pi$
$607$		−	218.456i		−	0.359895i	0.983676	−	0.179947i	$-0.0575928\pi$
$608$		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	$-0.947354\pi$
$613$		−	201.848i		−	0.329279i	0.986354	−	0.164639i	$-0.0526460\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−745.812			−1.20877			−0.604386	−	0.796692i	$-0.706582\pi$
$617$	−745.812			−1.20877			−0.604386	+	0.796692i	$0.706582\pi$
$618$		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	$-0.792713\pi$
$643$		−	779.511i		−	1.21230i	0.795349	−	0.606151i	$-0.207287\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1129.46			1.74568			0.872840	−	0.488006i	$-0.162276\pi$
$647$	1129.46			1.74568			0.872840	+	0.488006i	$0.162276\pi$
$648$		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	$-0.580636\pi$
$653$	−327.316			−0.501250			−0.250625	+	0.968084i	$0.580636\pi$
$654$		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	$-0.852325\pi$
$673$		−	602.296i		−	0.894943i	0.894298	−	0.447471i	$-0.147675\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	830.123			1.22618			0.613089	−	0.790014i	$-0.289927\pi$
$677$	830.123			1.22618			0.613089	+	0.790014i	$0.289927\pi$
$678$		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	$-0.435087\pi$
$683$	276.640			0.405037			0.202518	+	0.979278i	$0.435087\pi$
$684$		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	$-0.986085\pi$
$727$		−	63.5439i		−	0.0874056i	0.999045	−	0.0437028i	$-0.0139155\pi$
$728$		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	$-0.991848\pi$
$733$		−	37.5413i		−	0.0512159i	0.999672	−	0.0256080i	$-0.00815216\pi$
$734$		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	$-0.532339\pi$
$743$	−150.710			−0.202840			−0.101420	+	0.994844i	$0.532339\pi$
$744$		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	$0.0434007\pi$
$757$			205.791i			0.271851i	−0.990719	+	0.135925i	$0.956599\pi$
$758$		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	$-0.548205\pi$
$773$	−233.233			−0.301724			−0.150862	+	0.988555i	$0.548205\pi$
$774$		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	$-0.948247\pi$
$787$		−	254.788i		−	0.323746i	0.986812	−	0.161873i	$-0.0517535\pi$
$788$		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	$-0.770301\pi$
$797$	−1196.67			−1.50147			−0.750737	+	0.660602i	$0.770301\pi$
$798$		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	$-0.965398\pi$
$823$		−	178.579i		−	0.216985i	0.994097	−	0.108493i	$-0.0346024\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−165.734			−0.200404			−0.100202	−	0.994967i	$-0.531949\pi$
$827$	−165.734			−0.200404			−0.100202	+	0.994967i	$0.531949\pi$
$828$		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	$-0.879292\pi$
$853$		−	631.549i		−	0.740386i	0.928955	−	0.370193i	$-0.120708\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	300.140			0.350221			0.175111	−	0.984549i	$-0.443972\pi$
$857$	300.140			0.350221			0.175111	+	0.984549i	$0.443972\pi$
$858$		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	$-0.524904\pi$
$863$	−134.903			−0.156319			−0.0781596	+	0.996941i	$0.524904\pi$
$864$		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	$-0.997374\pi$
$877$		−	14.4694i		−	0.0164988i	0.999966	−	0.00824940i	$-0.00262589\pi$
$878$		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	$0.103695\pi$
$883$			565.182i			0.640070i	−0.947406	+	0.320035i	$0.896305\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	517.280			0.583179			0.291590	−	0.956543i	$-0.405816\pi$
$887$	517.280			0.583179			0.291590	+	0.956543i	$0.405816\pi$
$888$		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	$-0.894342\pi$
$907$		−	591.131i		−	0.651743i	0.945414	−	0.325872i	$-0.105658\pi$
$908$		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	$-0.935147\pi$
$937$		−	379.177i		−	0.404671i	0.979316	−	0.202336i	$-0.0648532\pi$
$938$		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	$-0.815793\pi$
$947$	−1585.60			−1.67434			−0.837172	+	0.546940i	$0.815793\pi$
$948$		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	$-0.503893\pi$
$953$	−23.3120			−0.0244617			−0.0122308	+	0.999925i	$0.503893\pi$
$954$		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	$0.268889\pi$
$967$			1446.24i			1.49560i	−0.663926	+	0.747798i	$0.731111\pi$
$968$		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	$-0.396613\pi$
$977$	623.561			0.638241			0.319120	+	0.947714i	$0.396613\pi$
$978$		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	$-0.561602\pi$
$983$	−378.106			−0.384645			−0.192323	+	0.981332i	$0.561602\pi$
$984$		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	$0.138264\pi$
$997$			839.151i			0.841676i	−0.907136	+	0.420838i	$0.861736\pi$
$998$		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.449.7		12
3.2	odd	2	inner	1200.3.c.l.449.5		12
4.3	odd	2		600.3.c.c.449.6		12
5.2	odd	4		1200.3.l.v.401.1		6
5.3	odd	4		1200.3.l.w.401.6		6
5.4	even	2	inner	1200.3.c.l.449.6		12
12.11	even	2		600.3.c.c.449.8		12
15.2	even	4		1200.3.l.v.401.2		6
15.8	even	4		1200.3.l.w.401.5		6
15.14	odd	2	inner	1200.3.c.l.449.8		12
20.3	even	4		600.3.l.d.401.1	✓	6
20.7	even	4		600.3.l.e.401.6	yes	6
20.19	odd	2		600.3.c.c.449.7		12
60.23	odd	4		600.3.l.d.401.2	yes	6
60.47	odd	4		600.3.l.e.401.5	yes	6
60.59	even	2		600.3.c.c.449.5		12

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