LMFDB - Embedded newform 600.3.c.d.449.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,3,Mod(449,600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(600, 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("600.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	600.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:	$16.3488158616$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 138x^{12} + 3393x^{8} + 15208x^{4} + 1296 $ x^16 + 138x^12 + 3393x^8 + 15208*x^4 + 1296
Coefficient ring:	$\Z[a_1, \ldots, a_{37}]$
Coefficient ring index:	$ 2^{26}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.6
Root			$-1.57158 - 1.57158i$ of defining polynomial
Character	$\chi$	$=$	600.449
Dual form			600.3.c.d.449.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.864473 + 2.87275i) q^{3} +9.02416i q^{7} +(-7.50537 - 4.96683i) q^{9} +O(q^{10})$	$q+(-0.864473 + 2.87275i) q^{3} +9.02416i q^{7} +(-7.50537 - 4.96683i) q^{9} -21.8827i q^{11} -21.6599i q^{13} -12.1078 q^{17} -3.03757 q^{19} +(-25.9241 - 7.80114i) q^{21} +28.5735 q^{23} +(20.7566 - 17.2674i) q^{27} +12.0364i q^{29} +2.19085 q^{31} +(62.8636 + 18.9170i) q^{33} +0.839959i q^{37} +(62.2233 + 18.7244i) q^{39} -35.5690i q^{41} +12.7152i q^{43} +22.5481 q^{47} -32.4354 q^{49} +(10.4668 - 34.7826i) q^{51} -9.13775 q^{53} +(2.62590 - 8.72618i) q^{57} -80.4459i q^{59} -57.8816 q^{61} +(44.8214 - 67.7297i) q^{63} -63.0560i q^{67} +(-24.7011 + 82.0846i) q^{69} -17.0218i q^{71} -52.1181i q^{73} +197.473 q^{77} +7.46224 q^{79} +(31.6612 + 74.5558i) q^{81} +82.3758 q^{83} +(-34.5774 - 10.4051i) q^{87} -27.5850i q^{89} +195.462 q^{91} +(-1.89393 + 6.29376i) q^{93} +114.989i q^{97} +(-108.688 + 164.238i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 40 q^{9}+O(q^{10}) $ 16 * q - 40 * q^9	$ 16 q - 40 q^{9} + 16 q^{19} + 56 q^{21} + 240 q^{31} + 144 q^{39} - 128 q^{49} + 128 q^{51} + 16 q^{61} - 200 q^{69} - 176 q^{79} + 448 q^{81} + 1120 q^{91} - 64 q^{99}+O(q^{100}) $ 16 * q - 40 * q^9 + 16 * q^19 + 56 * q^21 + 240 * q^31 + 144 * q^39 - 128 * q^49 + 128 * q^51 + 16 * q^61 - 200 * q^69 - 176 * q^79 + 448 * q^81 + 1120 * q^91 - 64 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$.

$n$	$151$	$301$	$401$	$577$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$	−0.864473	+	2.87275i	−0.288158	+	0.957583i
$3$	−0.864473	+	2.87275i	−0.288158	+	0.957583i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$			9.02416i			1.28917i	0.764535	+	0.644583i	$0.222969\pi$
$7$			9.02416i			1.28917i	−0.764535	+	0.644583i	$0.777031\pi$
$8$		0			0
$9$	−7.50537	−	4.96683i	−0.833930	−	0.551870i
$10$		0			0
$11$		−	21.8827i		−	1.98934i	−0.103119	−	0.994669i	$-0.532882\pi$
$11$		−	21.8827i		−	1.98934i	0.103119	−	0.994669i	$-0.467118\pi$
$12$		0			0
$13$		−	21.6599i		−	1.66614i	−0.553166	−	0.833071i	$-0.686580\pi$
$13$		−	21.6599i		−	1.66614i	0.553166	−	0.833071i	$-0.313420\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−12.1078			−0.712221			−0.356111	−	0.934444i	$-0.615897\pi$
$17$	−12.1078			−0.712221			−0.356111	+	0.934444i	$0.615897\pi$
$18$		0			0
$19$	−3.03757			−0.159872			−0.0799361	−	0.996800i	$-0.525472\pi$
$19$	−3.03757			−0.159872			−0.0799361	+	0.996800i	$0.525472\pi$
$20$		0			0
$21$	−25.9241	−	7.80114i	−1.23448	−	0.371483i
$22$		0			0
$23$	28.5735			1.24233			0.621164	−	0.783681i	$-0.286660\pi$
$23$	28.5735			1.24233			0.621164	+	0.783681i	$0.286660\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	20.7566	−	17.2674i	0.768765	−	0.639532i
$28$		0			0
$29$			12.0364i			0.415047i	0.978230	+	0.207523i	$0.0665403\pi$
$29$			12.0364i			0.415047i	−0.978230	+	0.207523i	$0.933460\pi$
$30$		0			0
$31$	2.19085			0.0706725			0.0353363	−	0.999375i	$-0.488750\pi$
$31$	2.19085			0.0706725			0.0353363	+	0.999375i	$0.488750\pi$
$32$		0			0
$33$	62.8636	+	18.9170i	1.90496	+	0.573243i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			0.839959i			0.0227016i	0.999936	+	0.0113508i	$0.00361315\pi$
$37$			0.839959i			0.0227016i	−0.999936	+	0.0113508i	$0.996387\pi$
$38$		0			0
$39$	62.2233	+	18.7244i	1.59547	+	0.480112i
$40$		0			0
$41$		−	35.5690i		−	0.867537i	−0.901024	−	0.433769i	$-0.857184\pi$
$41$		−	35.5690i		−	0.867537i	0.901024	−	0.433769i	$-0.142816\pi$
$42$		0			0
$43$			12.7152i			0.295702i	0.989010	+	0.147851i	$0.0472356\pi$
$43$			12.7152i			0.295702i	−0.989010	+	0.147851i	$0.952764\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	22.5481			0.479746			0.239873	−	0.970804i	$-0.422894\pi$
$47$	22.5481			0.479746			0.239873	+	0.970804i	$0.422894\pi$
$48$		0			0
$49$	−32.4354			−0.661947
$50$		0			0
$51$	10.4668	−	34.7826i	0.205232	−	0.682011i
$52$		0			0
$53$	−9.13775			−0.172410			−0.0862052	−	0.996277i	$-0.527474\pi$
$53$	−9.13775			−0.172410			−0.0862052	+	0.996277i	$0.527474\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	2.62590	−	8.72618i	0.0460684	−	0.153091i
$58$		0			0
$59$		−	80.4459i		−	1.36349i	−0.731590	−	0.681745i	$-0.761221\pi$
$59$		−	80.4459i		−	1.36349i	0.731590	−	0.681745i	$-0.238779\pi$
$60$		0			0
$61$	−57.8816			−0.948878			−0.474439	−	0.880288i	$-0.657349\pi$
$61$	−57.8816			−0.948878			−0.474439	+	0.880288i	$0.657349\pi$
$62$		0			0
$63$	44.8214	−	67.7297i	0.711452	−	1.07507i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	63.0560i		−	0.941135i	−0.882364	−	0.470567i	$-0.844049\pi$
$67$		−	63.0560i		−	0.941135i	0.882364	−	0.470567i	$-0.155951\pi$
$68$		0			0
$69$	−24.7011	+	82.0846i	−0.357986	+	1.18963i
$70$		0			0
$71$		−	17.0218i		−	0.239743i	−0.992789	−	0.119872i	$-0.961752\pi$
$71$		−	17.0218i		−	0.239743i	0.992789	−	0.119872i	$-0.0382483\pi$
$72$		0			0
$73$		−	52.1181i		−	0.713947i	−0.934115	−	0.356973i	$-0.883809\pi$
$73$		−	52.1181i		−	0.713947i	0.934115	−	0.356973i	$-0.116191\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	197.473			2.56459
$78$		0			0
$79$	7.46224			0.0944588			0.0472294	−	0.998884i	$-0.484961\pi$
$79$	7.46224			0.0944588			0.0472294	+	0.998884i	$0.484961\pi$
$80$		0			0
$81$	31.6612	+	74.5558i	0.390879	+	0.920442i
$82$		0			0
$83$	82.3758			0.992480			0.496240	−	0.868185i	$-0.334714\pi$
$83$	82.3758			0.992480			0.496240	+	0.868185i	$0.334714\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−34.5774	−	10.4051i	−0.397442	−	0.119599i
$88$		0			0
$89$		−	27.5850i		−	0.309944i	−0.987919	−	0.154972i	$-0.950471\pi$
$89$		−	27.5850i		−	0.309944i	0.987919	−	0.154972i	$-0.0495287\pi$
$90$		0			0
$91$	195.462			2.14793
$92$		0			0
$93$	−1.89393	+	6.29376i	−0.0203648	+	0.0676748i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			114.989i			1.18545i	0.805404	+	0.592727i	$0.201949\pi$
$97$			114.989i			1.18545i	−0.805404	+	0.592727i	$0.798051\pi$
$98$		0			0
$99$	−108.688	+	164.238i	−1.09786	+	1.65897i
$100$		0			0
$101$		−	122.804i		−	1.21588i	−0.793982	−	0.607941i	$-0.791996\pi$
$101$		−	122.804i		−	1.21588i	0.793982	−	0.607941i	$-0.208004\pi$
$102$		0			0
$103$			46.8275i			0.454636i	0.973821	+	0.227318i	$0.0729957\pi$
$103$			46.8275i			0.454636i	−0.973821	+	0.227318i	$0.927004\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−105.086			−0.982113			−0.491056	−	0.871128i	$-0.663389\pi$
$107$	−105.086			−0.982113			−0.491056	+	0.871128i	$0.663389\pi$
$108$		0			0
$109$	116.777			1.07135			0.535673	−	0.844426i	$-0.320058\pi$
$109$	116.777			1.07135			0.535673	+	0.844426i	$0.320058\pi$
$110$		0			0
$111$	−2.41299	−	0.726122i	−0.0217387	−	0.00654164i
$112$		0			0
$113$	−10.8116			−0.0956779			−0.0478389	−	0.998855i	$-0.515233\pi$
$113$	−10.8116			−0.0956779			−0.0478389	+	0.998855i	$0.515233\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−107.581	+	162.565i	−0.919494	+	1.38945i
$118$		0			0
$119$		−	109.262i		−	0.918171i
$120$		0			0
$121$	−357.853			−2.95747
$122$		0			0
$123$	102.181	+	30.7485i	0.830739	+	0.249988i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	192.459i		−	1.51543i	−0.652587	−	0.757714i	$-0.726316\pi$
$127$		−	192.459i		−	1.51543i	0.652587	−	0.757714i	$-0.273684\pi$
$128$		0			0
$129$	−36.5276	−	10.9919i	−0.283159	−	0.0852089i
$130$		0			0
$131$		−	48.6360i		−	0.371267i	−0.982619	−	0.185633i	$-0.940566\pi$
$131$		−	48.6360i		−	0.371267i	0.982619	−	0.185633i	$-0.0594337\pi$
$132$		0			0
$133$		−	27.4115i		−	0.206102i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−157.869			−1.15233			−0.576163	−	0.817335i	$-0.695451\pi$
$137$	−157.869			−1.15233			−0.576163	+	0.817335i	$0.695451\pi$
$138$		0			0
$139$	164.752			1.18526			0.592632	−	0.805473i	$-0.298089\pi$
$139$	164.752			1.18526			0.592632	+	0.805473i	$0.298089\pi$
$140$		0			0
$141$	−19.4922	+	64.7749i	−0.138243	+	0.459397i
$142$		0			0
$143$	−473.977			−3.31452
$144$		0			0
$145$		0			0
$146$		0			0
$147$	28.0396	−	93.1788i	0.190745	−	0.633869i
$148$		0			0
$149$		−	262.935i		−	1.76467i	−0.470626	−	0.882333i	$-0.655972\pi$
$149$		−	262.935i		−	1.76467i	0.470626	−	0.882333i	$-0.344028\pi$
$150$		0			0
$151$	15.8171			0.104749			0.0523745	−	0.998628i	$-0.483321\pi$
$151$	15.8171			0.104749			0.0523745	+	0.998628i	$0.483321\pi$
$152$		0			0
$153$	90.8733	+	60.1372i	0.593943	+	0.393054i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	6.11941i		−	0.0389771i	−0.999810	−	0.0194886i	$-0.993796\pi$
$157$		−	6.11941i		−	0.0389771i	0.999810	−	0.0194886i	$-0.00620380\pi$
$158$		0			0
$159$	7.89934	−	26.2505i	0.0496814	−	0.165097i
$160$		0			0
$161$			257.852i			1.60157i
$162$		0			0
$163$			170.444i			1.04567i	0.852435	+	0.522833i	$0.175125\pi$
$163$			170.444i			1.04567i	−0.852435	+	0.522833i	$0.824875\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−61.2668			−0.366867			−0.183434	−	0.983032i	$-0.558721\pi$
$167$	−61.2668			−0.366867			−0.183434	+	0.983032i	$0.558721\pi$
$168$		0			0
$169$	−300.149			−1.77603
$170$		0			0
$171$	22.7981	+	15.0871i	0.133322	+	0.0882286i
$172$		0			0
$173$	−262.548			−1.51762			−0.758810	−	0.651312i	$-0.774219\pi$
$173$	−262.548			−1.51762			−0.758810	+	0.651312i	$0.774219\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	231.101	+	69.5434i	1.30566	+	0.392900i
$178$		0			0
$179$		−	6.88752i		−	0.0384778i	−0.999815	−	0.0192389i	$-0.993876\pi$
$179$		−	6.88752i		−	0.0384778i	0.999815	−	0.0192389i	$-0.00612431\pi$
$180$		0			0
$181$	218.536			1.20738			0.603691	−	0.797218i	$-0.293696\pi$
$181$	218.536			1.20738			0.603691	+	0.797218i	$0.293696\pi$
$182$		0			0
$183$	50.0371	−	166.279i	0.273427	−	0.908630i
$184$		0			0
$185$		0			0
$186$		0			0
$187$			264.951i			1.41685i
$188$		0			0
$189$	155.823	+	187.311i	0.824462	+	0.991065i
$190$		0			0
$191$		−	75.2506i		−	0.393982i	−0.980405	−	0.196991i	$-0.936883\pi$
$191$		−	75.2506i		−	0.393982i	0.980405	−	0.196991i	$-0.0631170\pi$
$192$		0			0
$193$			212.587i			1.10149i	0.834674	+	0.550744i	$0.185656\pi$
$193$			212.587i			1.10149i	−0.834674	+	0.550744i	$0.814344\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−190.640			−0.967718			−0.483859	−	0.875146i	$-0.660765\pi$
$197$	−190.640			−0.967718			−0.483859	+	0.875146i	$0.660765\pi$
$198$		0			0
$199$	−209.996			−1.05526			−0.527629	−	0.849475i	$-0.676919\pi$
$199$	−209.996			−1.05526			−0.527629	+	0.849475i	$0.676919\pi$
$200$		0			0
$201$	181.144	+	54.5102i	0.901214	+	0.271195i
$202$		0			0
$203$	−108.618			−0.535064
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−214.455	−	141.920i	−1.03601	−	0.685603i
$208$		0			0
$209$			66.4703i			0.318040i
$210$		0			0
$211$	176.419			0.836110			0.418055	−	0.908422i	$-0.362712\pi$
$211$	176.419			0.836110			0.418055	+	0.908422i	$0.362712\pi$
$212$		0			0
$213$	48.8993	+	14.7149i	0.229574	+	0.0690839i
$214$		0			0
$215$		0			0
$216$		0			0
$217$			19.7706i			0.0911086i
$218$		0			0
$219$	149.722	+	45.0547i	0.683663	+	0.205729i
$220$		0			0
$221$			262.252i			1.18666i
$222$		0			0
$223$		−	132.362i		−	0.593552i	−0.954947	−	0.296776i	$-0.904089\pi$
$223$		−	132.362i		−	0.593552i	0.954947	−	0.296776i	$-0.0959114\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	187.624			0.826537			0.413268	−	0.910609i	$-0.364387\pi$
$227$	187.624			0.826537			0.413268	+	0.910609i	$0.364387\pi$
$228$		0			0
$229$	178.571			0.779788			0.389894	−	0.920860i	$-0.372512\pi$
$229$	178.571			0.779788			0.389894	+	0.920860i	$0.372512\pi$
$230$		0			0
$231$	−170.710	+	567.291i	−0.739005	+	2.45580i
$232$		0			0
$233$	−296.711			−1.27344			−0.636718	−	0.771097i	$-0.719708\pi$
$233$	−296.711			−1.27344			−0.636718	+	0.771097i	$0.719708\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−6.45091	+	21.4371i	−0.0272190	+	0.0904521i
$238$		0			0
$239$			137.976i			0.577306i	0.957434	+	0.288653i	$0.0932074\pi$
$239$			137.976i			0.577306i	−0.957434	+	0.288653i	$0.906793\pi$
$240$		0			0
$241$	42.5687			0.176633			0.0883167	−	0.996092i	$-0.471851\pi$
$241$	42.5687			0.176633			0.0883167	+	0.996092i	$0.471851\pi$
$242$		0			0
$243$	−241.550	+	26.5032i	−0.994034	+	0.109067i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			65.7933i			0.266370i
$248$		0			0
$249$	−71.2117	+	236.645i	−0.285991	+	0.950382i
$250$		0			0
$251$			205.885i			0.820259i	0.912027	+	0.410130i	$0.134517\pi$
$251$			205.885i			0.820259i	−0.912027	+	0.410130i	$0.865483\pi$
$252$		0			0
$253$		−	625.267i		−	2.47141i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	188.382			0.733004			0.366502	−	0.930417i	$-0.380555\pi$
$257$	188.382			0.733004			0.366502	+	0.930417i	$0.380555\pi$
$258$		0			0
$259$	−7.57992			−0.0292661
$260$		0			0
$261$	59.7825	−	90.3373i	0.229052	−	0.346120i
$262$		0			0
$263$	188.745			0.717660			0.358830	−	0.933403i	$-0.383176\pi$
$263$	188.745			0.717660			0.358830	+	0.933403i	$0.383176\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	79.2447	+	23.8465i	0.296797	+	0.0893127i
$268$		0			0
$269$		−	333.372i		−	1.23930i	−0.784878	−	0.619651i	$-0.787274\pi$
$269$		−	333.372i		−	1.23930i	0.784878	−	0.619651i	$-0.212726\pi$
$270$		0			0
$271$	262.047			0.966964			0.483482	−	0.875354i	$-0.339372\pi$
$271$	262.047			0.966964			0.483482	+	0.875354i	$0.339372\pi$
$272$		0			0
$273$	−168.972	+	561.513i	−0.618944	+	2.05682i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	400.100i		−	1.44440i	−0.691682	−	0.722202i	$-0.743130\pi$
$277$		−	400.100i		−	1.44440i	0.691682	−	0.722202i	$-0.256870\pi$
$278$		0			0
$279$	−16.4431	−	10.8816i	−0.0589360	−	0.0390020i
$280$		0			0
$281$		−	350.698i		−	1.24804i	−0.781410	−	0.624018i	$-0.785499\pi$
$281$		−	350.698i		−	1.24804i	0.781410	−	0.624018i	$-0.214501\pi$
$282$		0			0
$283$			464.015i			1.63963i	0.572630	+	0.819814i	$0.305923\pi$
$283$			464.015i			1.63963i	−0.572630	+	0.819814i	$0.694077\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	320.981			1.11840
$288$		0			0
$289$	−142.402			−0.492741
$290$		0			0
$291$	−330.334	−	99.4049i	−1.13517	−	0.341598i
$292$		0			0
$293$	47.4080			0.161802			0.0809009	−	0.996722i	$-0.474220\pi$
$293$	47.4080			0.161802			0.0809009	+	0.996722i	$0.474220\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−377.857	−	454.212i	−1.27224	−	1.52933i
$298$		0			0
$299$		−	618.899i		−	2.06990i
$300$		0			0
$301$	−114.744			−0.381209
$302$		0			0
$303$	352.785	+	106.161i	1.16431	+	0.350366i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			461.894i			1.50454i	0.658855	+	0.752270i	$0.271041\pi$
$307$			461.894i			1.50454i	−0.658855	+	0.752270i	$0.728959\pi$
$308$		0			0
$309$	−134.524	−	40.4812i	−0.435352	−	0.131007i
$310$		0			0
$311$		−	123.057i		−	0.395681i	−0.980234	−	0.197841i	$-0.936607\pi$
$311$		−	123.057i		−	0.395681i	0.980234	−	0.197841i	$-0.0633929\pi$
$312$		0			0
$313$		−	97.4353i		−	0.311295i	−0.987813	−	0.155647i	$-0.950254\pi$
$313$		−	97.4353i		−	0.311295i	0.987813	−	0.155647i	$-0.0497464\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−252.388			−0.796175			−0.398088	−	0.917347i	$-0.630326\pi$
$317$	−252.388			−0.796175			−0.398088	+	0.917347i	$0.630326\pi$
$318$		0			0
$319$	263.388			0.825668
$320$		0			0
$321$	90.8441	−	301.886i	0.283003	−	0.940454i
$322$		0			0
$323$	36.7782			0.113864
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−100.950	+	335.470i	−0.308716	+	1.02590i
$328$		0			0
$329$			203.477i			0.618472i
$330$		0			0
$331$	−303.273			−0.916231			−0.458116	−	0.888893i	$-0.651475\pi$
$331$	−303.273			−0.916231			−0.458116	+	0.888893i	$0.651475\pi$
$332$		0			0
$333$	4.17193	−	6.30420i	0.0125283	−	0.0189315i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	352.738i		−	1.04670i	−0.852118	−	0.523350i	$-0.824682\pi$
$337$		−	352.738i		−	1.04670i	0.852118	−	0.523350i	$-0.175318\pi$
$338$		0			0
$339$	9.34634	−	31.0590i	0.0275703	−	0.0916195i
$340$		0			0
$341$		−	47.9417i		−	0.140592i
$342$		0			0
$343$			149.481i			0.435806i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−280.382			−0.808018			−0.404009	−	0.914755i	$-0.632384\pi$
$347$	−280.382			−0.808018			−0.404009	+	0.914755i	$0.632384\pi$
$348$		0			0
$349$	586.721			1.68115			0.840575	−	0.541696i	$-0.182217\pi$
$349$	586.721			1.68115			0.840575	+	0.541696i	$0.182217\pi$
$350$		0			0
$351$	−374.008	−	449.586i	−1.06555	−	1.28087i
$352$		0			0
$353$	558.927			1.58336			0.791681	−	0.610935i	$-0.209206\pi$
$353$	558.927			1.58336			0.791681	+	0.610935i	$0.209206\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	313.883	+	94.4544i	0.879225	+	0.264578i
$358$		0			0
$359$			323.554i			0.901264i	0.892710	+	0.450632i	$0.148801\pi$
$359$			323.554i			0.901264i	−0.892710	+	0.450632i	$0.851199\pi$
$360$		0			0
$361$	−351.773			−0.974441
$362$		0			0
$363$	309.355	−	1028.02i	0.852217	−	2.83202i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			176.636i			0.481296i	0.970612	+	0.240648i	$0.0773600\pi$
$367$			176.636i			0.481296i	−0.970612	+	0.240648i	$0.922640\pi$
$368$		0			0
$369$	−176.665	+	266.959i	−0.478768	+	0.723466i
$370$		0			0
$371$		−	82.4605i		−	0.222265i
$372$		0			0
$373$		−	367.327i		−	0.984790i	−0.870372	−	0.492395i	$-0.836122\pi$
$373$		−	367.327i		−	0.984790i	0.870372	−	0.492395i	$-0.163878\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	260.706			0.691527
$378$		0			0
$379$	−611.014			−1.61217			−0.806087	−	0.591798i	$-0.798418\pi$
$379$	−611.014			−1.61217			−0.806087	+	0.591798i	$0.798418\pi$
$380$		0			0
$381$	552.887	+	166.376i	1.45115	+	0.436682i
$382$		0			0
$383$	−13.6994			−0.0357687			−0.0178844	−	0.999840i	$-0.505693\pi$
$383$	−13.6994			−0.0357687			−0.0178844	+	0.999840i	$0.505693\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	63.1542	−	95.4322i	0.163189	−	0.246595i
$388$		0			0
$389$			379.601i			0.975838i	0.872889	+	0.487919i	$0.162244\pi$
$389$			379.601i			0.975838i	−0.872889	+	0.487919i	$0.837756\pi$
$390$		0			0
$391$	−345.962			−0.884812
$392$		0			0
$393$	139.719	+	42.0445i	0.355519	+	0.106983i
$394$		0			0
$395$		0			0
$396$		0			0
$397$			450.560i			1.13491i	0.823404	+	0.567456i	$0.192072\pi$
$397$			450.560i			1.13491i	−0.823404	+	0.567456i	$0.807928\pi$
$398$		0			0
$399$	78.7464	+	23.6965i	0.197359	+	0.0593898i
$400$		0			0
$401$		−	503.683i		−	1.25607i	−0.778186	−	0.628034i	$-0.783860\pi$
$401$		−	503.683i		−	1.25607i	0.778186	−	0.628034i	$-0.216140\pi$
$402$		0			0
$403$		−	47.4535i		−	0.117751i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	18.3806			0.0451611
$408$		0			0
$409$	−240.726			−0.588573			−0.294286	−	0.955717i	$-0.595082\pi$
$409$	−240.726			−0.588573			−0.294286	+	0.955717i	$0.595082\pi$
$410$		0			0
$411$	136.473	−	453.517i	0.332051	−	1.10345i
$412$		0			0
$413$	725.957			1.75776
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−142.423	+	473.290i	−0.341543	+	1.13499i
$418$		0			0
$419$			676.873i			1.61545i	0.589561	+	0.807724i	$0.299301\pi$
$419$			676.873i			1.61545i	−0.589561	+	0.807724i	$0.700699\pi$
$420$		0			0
$421$	683.755			1.62412			0.812061	−	0.583572i	$-0.198346\pi$
$421$	683.755			1.62412			0.812061	+	0.583572i	$0.198346\pi$
$422$		0			0
$423$	−169.232	−	111.992i	−0.400075	−	0.264757i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	522.332i		−	1.22326i
$428$		0			0
$429$	409.740	−	1361.62i	0.955105	−	3.17393i
$430$		0			0
$431$		−	213.608i		−	0.495610i	−0.968810	−	0.247805i	$-0.920291\pi$
$431$		−	213.608i		−	0.495610i	0.968810	−	0.247805i	$-0.0797092\pi$
$432$		0			0
$433$			383.579i			0.885864i	0.896555	+	0.442932i	$0.146062\pi$
$433$			383.579i			0.885864i	−0.896555	+	0.442932i	$0.853938\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−86.7941			−0.198614
$438$		0			0
$439$	−523.900			−1.19340			−0.596698	−	0.802466i	$-0.703521\pi$
$439$	−523.900			−1.19340			−0.596698	+	0.802466i	$0.703521\pi$
$440$		0			0
$441$	243.440	+	161.101i	0.552018	+	0.365309i
$442$		0			0
$443$	391.277			0.883243			0.441622	−	0.897201i	$-0.354403\pi$
$443$	391.277			0.883243			0.441622	+	0.897201i	$0.354403\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	755.347	+	227.301i	1.68981	+	0.508502i
$448$		0			0
$449$		−	375.014i		−	0.835220i	−0.908626	−	0.417610i	$-0.862868\pi$
$449$		−	375.014i		−	0.835220i	0.908626	−	0.417610i	$-0.137132\pi$
$450$		0			0
$451$	−778.347			−1.72582
$452$		0			0
$453$	−13.6735	+	45.4386i	−0.0301843	+	0.100306i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	734.032i		−	1.60620i	−0.595847	−	0.803098i	$-0.703183\pi$
$457$		−	734.032i		−	1.60620i	0.595847	−	0.803098i	$-0.296817\pi$
$458$		0			0
$459$	−251.317	+	209.069i	−0.547531	+	0.455488i
$460$		0			0
$461$		−	775.239i		−	1.68165i	−0.541310	−	0.840823i	$-0.682072\pi$
$461$		−	775.239i		−	1.68165i	0.541310	−	0.840823i	$-0.317928\pi$
$462$		0			0
$463$			323.967i			0.699714i	0.936803	+	0.349857i	$0.113770\pi$
$463$			323.967i			0.699714i	−0.936803	+	0.349857i	$0.886230\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	160.785			0.344294			0.172147	−	0.985071i	$-0.444930\pi$
$467$	160.785			0.344294			0.172147	+	0.985071i	$0.444930\pi$
$468$		0			0
$469$	569.027			1.21328
$470$		0			0
$471$	17.5795	+	5.29007i	0.0373239	+	0.0112316i
$472$		0			0
$473$	278.243			0.588252
$474$		0			0
$475$		0			0
$476$		0			0
$477$	68.5822	+	45.3856i	0.143778	+	0.0951481i
$478$		0			0
$479$			61.7565i			0.128928i	0.997920	+	0.0644640i	$0.0205338\pi$
$479$			61.7565i			0.128928i	−0.997920	+	0.0644640i	$0.979466\pi$
$480$		0			0
$481$	18.1934			0.0378241
$482$		0			0
$483$	−740.744	−	222.906i	−1.53363	−	0.461504i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			129.969i			0.266876i	0.991057	+	0.133438i	$0.0426017\pi$
$487$			129.969i			0.266876i	−0.991057	+	0.133438i	$0.957398\pi$
$488$		0			0
$489$	−489.642	−	147.344i	−1.00131	−	0.301317i
$490$		0			0
$491$			810.511i			1.65074i	0.564595	+	0.825368i	$0.309032\pi$
$491$			810.511i			1.65074i	−0.564595	+	0.825368i	$0.690968\pi$
$492$		0			0
$493$		−	145.733i		−	0.295605i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	153.607			0.309069
$498$		0			0
$499$	600.897			1.20420			0.602101	−	0.798420i	$-0.294330\pi$
$499$	600.897			1.20420			0.602101	+	0.798420i	$0.294330\pi$
$500$		0			0
$501$	52.9635	−	176.004i	0.105716	−	0.351306i
$502$		0			0
$503$	688.332			1.36845			0.684226	−	0.729270i	$-0.260140\pi$
$503$	688.332			1.36845			0.684226	+	0.729270i	$0.260140\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	259.471	−	862.254i	0.511777	−	1.70070i
$508$		0			0
$509$			823.791i			1.61845i	0.587498	+	0.809225i	$0.300113\pi$
$509$			823.791i			1.61845i	−0.587498	+	0.809225i	$0.699887\pi$
$510$		0			0
$511$	470.322			0.920395
$512$		0			0
$513$	−63.0498	+	52.4508i	−0.122904	+	0.102243i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	493.413i		−	0.954377i
$518$		0			0
$519$	226.966	−	754.235i	0.437314	−	1.45325i
$520$		0			0
$521$			964.525i			1.85130i	0.378386	+	0.925648i	$0.376479\pi$
$521$			964.525i			1.85130i	−0.378386	+	0.925648i	$0.623521\pi$
$522$		0			0
$523$		−	462.679i		−	0.884664i	−0.896851	−	0.442332i	$-0.854151\pi$
$523$		−	462.679i		−	0.884664i	0.896851	−	0.442332i	$-0.145849\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−26.5263			−0.0503345
$528$		0			0
$529$	287.447			0.543378
$530$		0			0
$531$	−399.561	+	603.777i	−0.752469	+	1.13706i
$532$		0			0
$533$	−770.420			−1.44544
$534$		0			0
$535$		0			0
$536$		0			0
$537$	19.7861	+	5.95408i	0.0368457	+	0.0110877i
$538$		0			0
$539$			709.775i			1.31684i
$540$		0			0
$541$	−516.752			−0.955180			−0.477590	−	0.878583i	$-0.658490\pi$
$541$	−516.752			−0.955180			−0.477590	+	0.878583i	$0.658490\pi$
$542$		0			0
$543$	−188.919	+	627.800i	−0.347917	+	1.15617i
$544$		0			0
$545$		0			0
$546$		0			0
$547$			478.953i			0.875599i	0.899073	+	0.437799i	$0.144242\pi$
$547$			478.953i			0.875599i	−0.899073	+	0.437799i	$0.855758\pi$
$548$		0			0
$549$	434.423	+	287.488i	0.791298	+	0.523657i
$550$		0			0
$551$		−	36.5613i		−	0.0663544i
$552$		0			0
$553$			67.3404i			0.121773i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	343.982			0.617561			0.308781	−	0.951133i	$-0.400079\pi$
$557$	343.982			0.617561			0.308781	+	0.951133i	$0.400079\pi$
$558$		0			0
$559$	275.409			0.492682
$560$		0			0
$561$	−761.137	−	229.043i	−1.35675	−	0.408276i
$562$		0			0
$563$	394.056			0.699922			0.349961	−	0.936764i	$-0.386195\pi$
$563$	394.056			0.699922			0.349961	+	0.936764i	$0.386195\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−672.803	+	285.716i	−1.18660	+	0.503908i
$568$		0			0
$569$			537.452i			0.944556i	0.881450	+	0.472278i	$0.156568\pi$
$569$			537.452i			0.944556i	−0.881450	+	0.472278i	$0.843432\pi$
$570$		0			0
$571$	710.555			1.24440			0.622202	−	0.782856i	$-0.286238\pi$
$571$	710.555			1.24440			0.622202	+	0.782856i	$0.286238\pi$
$572$		0			0
$573$	216.176	+	65.0521i	0.377271	+	0.113529i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			452.846i			0.784828i	0.919789	+	0.392414i	$0.128360\pi$
$577$			452.846i			0.784828i	−0.919789	+	0.392414i	$0.871640\pi$
$578$		0			0
$579$	−610.709	−	183.776i	−1.05477	−	0.317402i
$580$		0			0
$581$			743.372i			1.27947i
$582$		0			0
$583$			199.959i			0.342983i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1103.34			−1.87963			−0.939813	−	0.341690i	$-0.889001\pi$
$587$	−1103.34			−1.87963			−0.939813	+	0.341690i	$0.889001\pi$
$588$		0			0
$589$	−6.65486			−0.0112986
$590$		0			0
$591$	164.804	−	547.662i	0.278855	−	0.926670i
$592$		0			0
$593$	249.474			0.420698			0.210349	−	0.977626i	$-0.432540\pi$
$593$	249.474			0.420698			0.210349	+	0.977626i	$0.432540\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	181.536	−	603.267i	0.304081	−	1.01050i
$598$		0			0
$599$		−	596.120i		−	0.995191i	−0.867409	−	0.497596i	$-0.834216\pi$
$599$		−	596.120i		−	0.995191i	0.867409	−	0.497596i	$-0.165784\pi$
$600$		0			0
$601$	476.515			0.792871			0.396435	−	0.918063i	$-0.370247\pi$
$601$	476.515			0.792871			0.396435	+	0.918063i	$0.370247\pi$
$602$		0			0
$603$	−313.188	+	473.259i	−0.519384	+	0.784841i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			27.8813i			0.0459329i	0.999736	+	0.0229664i	$0.00731109\pi$
$607$			27.8813i			0.0459329i	−0.999736	+	0.0229664i	$0.992689\pi$
$608$		0			0
$609$	93.8973	−	312.032i	0.154183	−	0.512368i
$610$		0			0
$611$		−	488.388i		−	0.799325i
$612$		0			0
$613$		−	465.472i		−	0.759334i	−0.925123	−	0.379667i	$-0.876039\pi$
$613$		−	465.472i		−	0.759334i	0.925123	−	0.379667i	$-0.123961\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	48.6709			0.0788832			0.0394416	−	0.999222i	$-0.487442\pi$
$617$	48.6709			0.0788832			0.0394416	+	0.999222i	$0.487442\pi$
$618$		0			0
$619$	213.318			0.344617			0.172309	−	0.985043i	$-0.444877\pi$
$619$	213.318			0.344617			0.172309	+	0.985043i	$0.444877\pi$
$620$		0			0
$621$	593.091	−	493.389i	0.955058	−	0.794508i
$622$		0			0
$623$	248.931			0.399569
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−190.953	−	57.4618i	−0.304549	−	0.0916456i
$628$		0			0
$629$		−	10.1700i		−	0.0161686i
$630$		0			0
$631$	582.489			0.923121			0.461560	−	0.887109i	$-0.347290\pi$
$631$	582.489			0.923121			0.461560	+	0.887109i	$0.347290\pi$
$632$		0			0
$633$	−152.510	+	506.808i	−0.240931	+	0.800644i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			702.546i			1.10290i
$638$		0			0
$639$	−84.5442	+	127.755i	−0.132307	+	0.199929i
$640$		0			0
$641$			319.635i			0.498650i	0.968420	+	0.249325i	$0.0802088\pi$
$641$			319.635i			0.498650i	−0.968420	+	0.249325i	$0.919791\pi$
$642$		0			0
$643$		−	458.627i		−	0.713261i	−0.934246	−	0.356630i	$-0.883926\pi$
$643$		−	458.627i		−	0.713261i	0.934246	−	0.356630i	$-0.116074\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	507.599			0.784543			0.392271	−	0.919850i	$-0.371689\pi$
$647$	507.599			0.784543			0.392271	+	0.919850i	$0.371689\pi$
$648$		0			0
$649$	−1760.38			−2.71244
$650$		0			0
$651$	−56.7959	−	17.0911i	−0.0872440	−	0.0262536i
$652$		0			0
$653$	−1197.20			−1.83339			−0.916694	−	0.399591i	$-0.869152\pi$
$653$	−1197.20			−1.83339			−0.916694	+	0.399591i	$0.869152\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−258.862	+	391.166i	−0.394006	+	0.595382i
$658$		0			0
$659$		−	632.173i		−	0.959291i	−0.877462	−	0.479645i	$-0.840765\pi$
$659$		−	632.173i		−	0.959291i	0.877462	−	0.479645i	$-0.159235\pi$
$660$		0			0
$661$	−565.316			−0.855243			−0.427622	−	0.903958i	$-0.640648\pi$
$661$	−565.316			−0.855243			−0.427622	+	0.903958i	$0.640648\pi$
$662$		0			0
$663$	−753.385	−	226.710i	−1.13633	−	0.341946i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			343.921i			0.515624i
$668$		0			0
$669$	380.243	+	114.423i	0.568375	+	0.171036i
$670$		0			0
$671$			1266.61i			1.88764i
$672$		0			0
$673$			306.607i			0.455582i	0.973710	+	0.227791i	$0.0731503\pi$
$673$			306.607i			0.455582i	−0.973710	+	0.227791i	$0.926850\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−546.672			−0.807491			−0.403746	−	0.914871i	$-0.632292\pi$
$677$	−546.672			−0.807491			−0.403746	+	0.914871i	$0.632292\pi$
$678$		0			0
$679$	−1037.68			−1.52824
$680$		0			0
$681$	−162.196	+	538.996i	−0.238173	+	0.791478i
$682$		0			0
$683$	−812.204			−1.18917			−0.594586	−	0.804032i	$-0.702684\pi$
$683$	−812.204			−1.18917			−0.594586	+	0.804032i	$0.702684\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−154.370	+	512.991i	−0.224702	+	0.746712i
$688$		0			0
$689$			197.922i			0.287260i
$690$		0			0
$691$	−45.9358			−0.0664773			−0.0332387	−	0.999447i	$-0.510582\pi$
$691$	−45.9358			−0.0664773			−0.0332387	+	0.999447i	$0.510582\pi$
$692$		0			0
$693$	−1482.11	−	980.815i	−2.13869	−	1.41532i
$694$		0			0
$695$		0			0
$696$		0			0
$697$			430.661i			0.617879i
$698$		0			0
$699$	256.498	−	852.375i	0.366950	−	1.21942i
$700$		0			0
$701$		−	674.615i		−	0.962360i	−0.876622	−	0.481180i	$-0.840208\pi$
$701$		−	674.615i		−	0.962360i	0.876622	−	0.481180i	$-0.159792\pi$
$702$		0			0
$703$		−	2.55143i		−	0.00362935i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1108.20			1.56747
$708$		0			0
$709$	−656.626			−0.926129			−0.463065	−	0.886324i	$-0.653250\pi$
$709$	−656.626			−0.926129			−0.463065	+	0.886324i	$0.653250\pi$
$710$		0			0
$711$	−56.0069	−	37.0637i	−0.0787720	−	0.0521290i
$712$		0			0
$713$	62.6003			0.0877984
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−396.371	−	119.277i	−0.552819	−	0.166355i
$718$		0			0
$719$			644.279i			0.896076i	0.894014	+	0.448038i	$0.147877\pi$
$719$			644.279i			0.896076i	−0.894014	+	0.448038i	$0.852123\pi$
$720$		0			0
$721$	−422.579			−0.586101
$722$		0			0
$723$	−36.7995	+	122.289i	−0.0508983	+	0.169141i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	981.269i		−	1.34975i	−0.737932	−	0.674875i	$-0.764197\pi$
$727$		−	981.269i		−	1.34975i	0.737932	−	0.674875i	$-0.235803\pi$
$728$		0			0
$729$	132.677	−	716.825i	0.181998	−	0.983299i
$730$		0			0
$731$		−	153.953i		−	0.210605i
$732$		0			0
$733$			127.756i			0.174292i	0.996196	+	0.0871460i	$0.0277747\pi$
$733$			127.756i			0.174292i	−0.996196	+	0.0871460i	$0.972225\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1379.84			−1.87223
$738$		0			0
$739$	1401.59			1.89660			0.948302	−	0.317368i	$-0.102799\pi$
$739$	1401.59			1.89660			0.948302	+	0.317368i	$0.102799\pi$
$740$		0			0
$741$	−189.008	−	56.8766i	−0.255071	−	0.0767565i
$742$		0			0
$743$	−11.6734			−0.0157112			−0.00785561	−	0.999969i	$-0.502501\pi$
$743$	−11.6734			−0.0157112			−0.00785561	+	0.999969i	$0.502501\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−618.261	−	409.147i	−0.827659	−	0.547720i
$748$		0			0
$749$		−	948.313i		−	1.26611i
$750$		0			0
$751$	380.403			0.506529			0.253264	−	0.967397i	$-0.418496\pi$
$751$	380.403			0.506529			0.253264	+	0.967397i	$0.418496\pi$
$752$		0			0
$753$	−591.456	−	177.982i	−0.785466	−	0.236364i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			63.6621i			0.0840979i	0.999116	+	0.0420490i	$0.0133885\pi$
$757$			63.6621i			0.0840979i	−0.999116	+	0.0420490i	$0.986611\pi$
$758$		0			0
$759$	1796.23	+	540.526i	2.36658	+	0.712156i
$760$		0			0
$761$			377.891i			0.496571i	0.968687	+	0.248286i	$0.0798671\pi$
$761$			377.891i			0.496571i	−0.968687	+	0.248286i	$0.920133\pi$
$762$		0			0
$763$			1053.81i			1.38114i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1742.45			−2.27177
$768$		0			0
$769$	−231.920			−0.301586			−0.150793	−	0.988565i	$-0.548183\pi$
$769$	−231.920			−0.301586			−0.150793	+	0.988565i	$0.548183\pi$
$770$		0			0
$771$	−162.851	+	541.174i	−0.211221	+	0.701912i
$772$		0			0
$773$	1509.05			1.95220			0.976101	−	0.217320i	$-0.0697314\pi$
$773$	1509.05			1.95220			0.976101	+	0.217320i	$0.0697314\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	6.55264	−	21.7752i	0.00843326	−	0.0280247i
$778$		0			0
$779$			108.043i			0.138695i
$780$		0			0
$781$	−372.483			−0.476930
$782$		0			0
$783$	207.836	+	249.834i	0.265436	+	0.319073i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			1171.43i			1.48848i	0.667912	+	0.744240i	$0.267188\pi$
$787$			1171.43i			1.48848i	−0.667912	+	0.744240i	$0.732812\pi$
$788$		0			0
$789$	−163.165	+	542.216i	−0.206799	+	0.687219i
$790$		0			0
$791$		−	97.5656i		−	0.123345i
$792$		0			0
$793$			1253.71i			1.58097i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	988.589			1.24039			0.620194	−	0.784448i	$-0.287054\pi$
$797$	988.589			1.24039			0.620194	+	0.784448i	$0.287054\pi$
$798$		0			0
$799$	−273.007			−0.341685
$800$		0			0
$801$	−137.010	+	207.036i	−0.171049	+	0.258471i
$802$		0			0
$803$	−1140.49			−1.42028
$804$		0			0
$805$		0			0
$806$		0			0
$807$	957.694	+	288.191i	1.18673	+	0.357114i
$808$		0			0
$809$			1334.53i			1.64961i	0.565419	+	0.824804i	$0.308714\pi$
$809$			1334.53i			1.64961i	−0.565419	+	0.824804i	$0.691286\pi$
$810$		0			0
$811$	610.590			0.752885			0.376443	−	0.926440i	$-0.377147\pi$
$811$	610.590			0.752885			0.376443	+	0.926440i	$0.377147\pi$
$812$		0			0
$813$	−226.533	+	752.796i	−0.278638	+	0.925949i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	38.6233i		−	0.0472745i
$818$		0			0
$819$	−1467.01	−	970.826i	−1.79123	−	1.18538i
$820$		0			0
$821$			877.538i			1.06887i	0.845211	+	0.534433i	$0.179475\pi$
$821$			877.538i			1.06887i	−0.845211	+	0.534433i	$0.820525\pi$
$822$		0			0
$823$		−	230.855i		−	0.280504i	−0.990116	−	0.140252i	$-0.955209\pi$
$823$		−	230.855i		−	0.280504i	0.990116	−	0.140252i	$-0.0447913\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	969.434			1.17223			0.586115	−	0.810228i	$-0.300657\pi$
$827$	969.434			1.17223			0.586115	+	0.810228i	$0.300657\pi$
$828$		0			0
$829$	1049.09			1.26549			0.632746	−	0.774360i	$-0.281928\pi$
$829$	1049.09			1.26549			0.632746	+	0.774360i	$0.281928\pi$
$830$		0			0
$831$	1149.39	+	345.876i	1.38314	+	0.416216i
$832$		0			0
$833$	392.720			0.471453
$834$		0			0
$835$		0			0
$836$		0			0
$837$	45.4747	−	37.8302i	0.0543305	−	0.0451973i
$838$		0			0
$839$			1214.62i			1.44770i	0.689959	+	0.723849i	$0.257629\pi$
$839$			1214.62i			1.44770i	−0.689959	+	0.723849i	$0.742371\pi$
$840$		0			0
$841$	696.126			0.827736
$842$		0			0
$843$	1007.47	+	303.169i	1.19510	+	0.359631i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	3229.33i		−	3.81266i
$848$		0			0
$849$	−1333.00	−	401.128i	−1.57008	−	0.472472i
$850$		0			0
$851$			24.0006i			0.0282028i
$852$		0			0
$853$		−	190.704i		−	0.223569i	−0.993732	−	0.111784i	$-0.964343\pi$
$853$		−	190.704i		−	0.223569i	0.993732	−	0.111784i	$-0.0356566\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−351.238			−0.409846			−0.204923	−	0.978778i	$-0.565694\pi$
$857$	−351.238			−0.409846			−0.204923	+	0.978778i	$0.565694\pi$
$858$		0			0
$859$	1233.74			1.43625			0.718123	−	0.695916i	$-0.245002\pi$
$859$	1233.74			1.43625			0.718123	+	0.695916i	$0.245002\pi$
$860$		0			0
$861$	−277.479	+	922.096i	−0.322275	+	1.07096i
$862$		0			0
$863$	−463.404			−0.536968			−0.268484	−	0.963284i	$-0.586523\pi$
$863$	−463.404			−0.536968			−0.268484	+	0.963284i	$0.586523\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	123.103	−	409.085i	0.141987	−	0.471840i
$868$		0			0
$869$		−	163.294i		−	0.187910i
$870$		0			0
$871$	−1365.78			−1.56806
$872$		0			0
$873$	571.130	−	863.035i	0.654216	−	0.988585i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			777.195i			0.886197i	0.896473	+	0.443099i	$0.146121\pi$
$877$			777.195i			0.886197i	−0.896473	+	0.443099i	$0.853879\pi$
$878$		0			0
$879$	−40.9829	+	136.191i	−0.0466245	+	0.154939i
$880$		0			0
$881$			1609.50i			1.82690i	0.406951	+	0.913450i	$0.366592\pi$
$881$			1609.50i			1.82690i	−0.406951	+	0.913450i	$0.633408\pi$
$882$		0			0
$883$		−	331.153i		−	0.375032i	−0.982262	−	0.187516i	$-0.939956\pi$
$883$		−	331.153i		−	0.375032i	0.982262	−	0.187516i	$-0.0600437\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	365.778			0.412377			0.206188	−	0.978512i	$-0.433894\pi$
$887$	365.778			0.412377			0.206188	+	0.978512i	$0.433894\pi$
$888$		0			0
$889$	1736.78			1.95364
$890$		0			0
$891$	1631.48	−	692.833i	1.83107	−	0.777591i
$892$		0			0
$893$	−68.4913			−0.0766980
$894$		0			0
$895$		0			0
$896$		0			0
$897$	1777.94	+	535.021i	1.98210	+	0.596456i
$898$		0			0
$899$			26.3698i			0.0293324i
$900$		0			0
$901$	110.638			0.122794
$902$		0			0
$903$	99.1930	−	329.630i	0.109848	−	0.365039i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	270.791i		−	0.298556i	−0.988795	−	0.149278i	$-0.952305\pi$
$907$		−	270.791i		−	0.298556i	0.988795	−	0.149278i	$-0.0476950\pi$
$908$		0			0
$909$	−609.947	+	921.690i	−0.671008	+	1.01396i
$910$		0			0
$911$		−	1238.74i		−	1.35976i	−0.733324	−	0.679879i	$-0.762032\pi$
$911$		−	1238.74i		−	1.35976i	0.733324	−	0.679879i	$-0.237968\pi$
$912$		0			0
$913$		−	1802.61i		−	1.97438i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	438.898			0.478624
$918$		0			0
$919$	−578.894			−0.629918			−0.314959	−	0.949105i	$-0.601991\pi$
$919$	−578.894			−0.629918			−0.314959	+	0.949105i	$0.601991\pi$
$920$		0			0
$921$	−1326.91	−	399.295i	−1.44072	−	0.433545i
$922$		0			0
$923$	−368.689			−0.399447
$924$		0			0
$925$		0			0
$926$		0			0
$927$	232.584	−	351.458i	0.250900	−	0.379135i
$928$		0			0
$929$		−	1307.58i		−	1.40752i	−0.710440	−	0.703758i	$-0.751504\pi$
$929$		−	1307.58i		−	1.40752i	0.710440	−	0.703758i	$-0.248496\pi$
$930$		0			0
$931$	98.5249			0.105827
$932$		0			0
$933$	353.511	+	106.379i	0.378898	+	0.114019i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	693.314i		−	0.739929i	−0.929046	−	0.369965i	$-0.879370\pi$
$937$		−	693.314i		−	0.739929i	0.929046	−	0.369965i	$-0.120630\pi$
$938$		0			0
$939$	279.907	+	84.2302i	0.298091	+	0.0897020i
$940$		0			0
$941$			783.592i			0.832723i	0.909199	+	0.416362i	$0.136695\pi$
$941$			783.592i			0.832723i	−0.909199	+	0.416362i	$0.863305\pi$
$942$		0			0
$943$		−	1016.33i		−	1.07777i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1621.44			−1.71219			−0.856094	−	0.516820i	$-0.827116\pi$
$947$	−1621.44			−1.71219			−0.856094	+	0.516820i	$0.827116\pi$
$948$		0			0
$949$	−1128.87			−1.18954
$950$		0			0
$951$	218.182	−	725.046i	0.229424	−	0.762404i
$952$		0			0
$953$	−1105.90			−1.16044			−0.580218	−	0.814461i	$-0.697033\pi$
$953$	−1105.90			−1.16044			−0.580218	+	0.814461i	$0.697033\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−227.692	+	756.648i	−0.237923	+	0.790646i
$958$		0			0
$959$		−	1424.63i		−	1.48554i
$960$		0			0
$961$	−956.200			−0.995005
$962$		0			0
$963$	788.710	+	521.945i	0.819013	+	0.541998i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			1307.89i			1.35253i	0.736659	+	0.676264i	$0.236402\pi$
$967$			1307.89i			1.35253i	−0.736659	+	0.676264i	$0.763598\pi$
$968$		0			0
$969$	−31.7938	+	105.655i	−0.0328109	+	0.109035i
$970$		0			0
$971$		−	749.988i		−	0.772387i	−0.922418	−	0.386194i	$-0.873790\pi$
$971$		−	749.988i		−	0.772387i	0.922418	−	0.386194i	$-0.126210\pi$
$972$		0			0
$973$			1486.75i			1.52800i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	373.792			0.382591			0.191296	−	0.981532i	$-0.438731\pi$
$977$	373.792			0.382591			0.191296	+	0.981532i	$0.438731\pi$
$978$		0			0
$979$	−603.634			−0.616583
$980$		0			0
$981$	−876.452	−	580.009i	−0.893427	−	0.591243i
$982$		0			0
$983$	27.7025			0.0281815			0.0140908	−	0.999901i	$-0.495515\pi$
$983$	27.7025			0.0281815			0.0140908	+	0.999901i	$0.495515\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−584.539	−	175.901i	−0.592238	−	0.178218i
$988$		0			0
$989$			363.318i			0.367359i
$990$		0			0
$991$	1030.34			1.03970			0.519850	−	0.854258i	$-0.325988\pi$
$991$	1030.34			1.03970			0.519850	+	0.854258i	$0.325988\pi$
$992$		0			0
$993$	262.171	−	871.226i	0.264019	−	0.877367i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			1217.62i			1.22128i	0.791909	+	0.610640i	$0.209088\pi$
$997$			1217.62i			1.22128i	−0.791909	+	0.610640i	$0.790912\pi$
$998$		0			0
$999$	14.5039	+	17.4347i	0.0145184	+	0.0174522i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.3.c.d.449.6		16
3.2	odd	2	inner	600.3.c.d.449.12		16
4.3	odd	2		1200.3.c.m.449.11		16
5.2	odd	4		600.3.l.f.401.8		8
5.3	odd	4		120.3.l.a.41.1	✓	8
5.4	even	2	inner	600.3.c.d.449.11		16
12.11	even	2		1200.3.c.m.449.5		16
15.2	even	4		600.3.l.f.401.7		8
15.8	even	4		120.3.l.a.41.2	yes	8
15.14	odd	2	inner	600.3.c.d.449.5		16
20.3	even	4		240.3.l.d.161.8		8
20.7	even	4		1200.3.l.x.401.1		8
20.19	odd	2		1200.3.c.m.449.6		16
40.3	even	4		960.3.l.g.641.1		8
40.13	odd	4		960.3.l.h.641.8		8
60.23	odd	4		240.3.l.d.161.7		8
60.47	odd	4		1200.3.l.x.401.2		8
60.59	even	2		1200.3.c.m.449.12		16
120.53	even	4		960.3.l.h.641.7		8
120.83	odd	4		960.3.l.g.641.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.3.l.a.41.1	✓	8	5.3	odd	4
120.3.l.a.41.2	yes	8	15.8	even	4
240.3.l.d.161.7		8	60.23	odd	4
240.3.l.d.161.8		8	20.3	even	4
600.3.c.d.449.5		16	15.14	odd	2	inner
600.3.c.d.449.6		16	1.1	even	1	trivial
600.3.c.d.449.11		16	5.4	even	2	inner
600.3.c.d.449.12		16	3.2	odd	2	inner
600.3.l.f.401.7		8	15.2	even	4
600.3.l.f.401.8		8	5.2	odd	4
960.3.l.g.641.1		8	40.3	even	4
960.3.l.g.641.2		8	120.83	odd	4
960.3.l.h.641.7		8	120.53	even	4
960.3.l.h.641.8		8	40.13	odd	4
1200.3.c.m.449.5		16	12.11	even	2
1200.3.c.m.449.6		16	20.19	odd	2
1200.3.c.m.449.11		16	4.3	odd	2
1200.3.c.m.449.12		16	60.59	even	2
1200.3.l.x.401.1		8	20.7	even	4
1200.3.l.x.401.2		8	60.47	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:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	600.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.864473 + 2.87275i) q^{3} +9.02416i q^{7} +(-7.50537 - 4.96683i) q^{9} +O(q^{10})\)	\(q+(-0.864473 + 2.87275i) q^{3} +9.02416i q^{7} +(-7.50537 - 4.96683i) q^{9} -21.8827i q^{11} -21.6599i q^{13} -12.1078 q^{17} -3.03757 q^{19} +(-25.9241 - 7.80114i) q^{21} +28.5735 q^{23} +(20.7566 - 17.2674i) q^{27} +12.0364i q^{29} +2.19085 q^{31} +(62.8636 + 18.9170i) q^{33} +0.839959i q^{37} +(62.2233 + 18.7244i) q^{39} -35.5690i q^{41} +12.7152i q^{43} +22.5481 q^{47} -32.4354 q^{49} +(10.4668 - 34.7826i) q^{51} -9.13775 q^{53} +(2.62590 - 8.72618i) q^{57} -80.4459i q^{59} -57.8816 q^{61} +(44.8214 - 67.7297i) q^{63} -63.0560i q^{67} +(-24.7011 + 82.0846i) q^{69} -17.0218i q^{71} -52.1181i q^{73} +197.473 q^{77} +7.46224 q^{79} +(31.6612 + 74.5558i) q^{81} +82.3758 q^{83} +(-34.5774 - 10.4051i) q^{87} -27.5850i q^{89} +195.462 q^{91} +(-1.89393 + 6.29376i) q^{93} +114.989i q^{97} +(-108.688 + 164.238i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 40 q^{9}+O(q^{10}) \) 16 * q - 40 * q^9	\( 16 q - 40 q^{9} + 16 q^{19} + 56 q^{21} + 240 q^{31} + 144 q^{39} - 128 q^{49} + 128 q^{51} + 16 q^{61} - 200 q^{69} - 176 q^{79} + 448 q^{81} + 1120 q^{91} - 64 q^{99}+O(q^{100}) \) 16 * q - 40 * q^9 + 16 * q^19 + 56 * q^21 + 240 * q^31 + 144 * q^39 - 128 * q^49 + 128 * q^51 + 16 * q^61 - 200 * q^69 - 176 * q^79 + 448 * q^81 + 1120 * q^91 - 64 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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