LMFDB - Embedded newform 600.3.l.g.401.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,3,Mod(401,600)]

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, 0]))

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

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

chi := DirichletCharacter("600.401");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	600.l (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$16.3488158616$
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} - 4x^{10} + 30x^{8} - 216x^{6} + 1080x^{4} - 5184x^{2} + 46656 $ x^12 - 4x^10 + 30x^8 - 216x^6 + 1080x^4 - 5184*x^2 + 46656
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{18}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			401.4
Root			$1.79523 - 1.66648i$ of defining polynomial
Character	$\chi$	$=$	600.401
Dual form			600.3.l.g.401.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.67109 + 2.49147i) q^{3} -12.7692 q^{7} +(-3.41489 - 8.32698i) q^{9} +O(q^{10})$	$q+(-1.67109 + 2.49147i) q^{3} -12.7692 q^{7} +(-3.41489 - 8.32698i) q^{9} +12.6296i q^{11} -7.44085 q^{13} -14.0550i q^{17} +31.0176 q^{19} +(21.3386 - 31.8142i) q^{21} -7.50423i q^{23} +(26.4530 + 5.40707i) q^{27} -15.7298i q^{29} -20.4893 q^{31} +(-31.4663 - 21.1053i) q^{33} +12.9261 q^{37} +(12.4344 - 18.5387i) q^{39} -13.8451i q^{41} +30.0797 q^{43} +20.2570i q^{47} +114.053 q^{49} +(35.0176 + 23.4872i) q^{51} -29.1185i q^{53} +(-51.8333 + 77.2795i) q^{57} -47.6333i q^{59} +43.0176 q^{61} +(43.6054 + 106.329i) q^{63} -0.630153 q^{67} +(18.6966 + 12.5403i) q^{69} -90.4047i q^{71} +46.2193 q^{73} -161.270i q^{77} -37.9610 q^{79} +(-57.6771 + 56.8713i) q^{81} +80.2267i q^{83} +(39.1904 + 26.2860i) q^{87} -140.923i q^{89} +95.0138 q^{91} +(34.2396 - 51.0486i) q^{93} +10.3429 q^{97} +(105.166 - 43.1286i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 8 q^{9}+O(q^{10}) $ 12 * q - 8 * q^9	$ 12 q - 8 q^{9} + 4 q^{21} - 48 q^{31} + 128 q^{39} + 252 q^{49} + 48 q^{51} + 144 q^{61} - 268 q^{69} - 432 q^{79} - 188 q^{81} + 560 q^{99}+O(q^{100}) $ 12 * q - 8 * q^9 + 4 * q^21 - 48 * q^31 + 128 * q^39 + 252 * q^49 + 48 * q^51 + 144 * q^61 - 268 * q^69 - 432 * q^79 - 188 * q^81 + 560 * 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
$3$	−1.67109	+	2.49147i	−0.557032	+	0.830491i
$3$	−1.67109	+	2.49147i	−0.557032	+	0.830491i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−12.7692			−1.82417			−0.912086	−	0.409998i	$-0.865529\pi$
$7$	−12.7692			−1.82417			−0.912086	+	0.409998i	$0.865529\pi$
$8$		0			0
$9$	−3.41489	−	8.32698i	−0.379432	−	0.925220i
$10$		0			0
$11$			12.6296i			1.14815i	0.818804	+	0.574073i	$0.194637\pi$
$11$			12.6296i			1.14815i	−0.818804	+	0.574073i	$0.805363\pi$
$12$		0			0
$13$	−7.44085			−0.572373			−0.286187	−	0.958174i	$-0.592388\pi$
$13$	−7.44085			−0.572373			−0.286187	+	0.958174i	$0.592388\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	14.0550i		−	0.826763i	−0.910558	−	0.413381i	$-0.864348\pi$
$17$		−	14.0550i		−	0.826763i	0.910558	−	0.413381i	$-0.135652\pi$
$18$		0			0
$19$	31.0176			1.63250			0.816252	−	0.577696i	$-0.196048\pi$
$19$	31.0176			1.63250			0.816252	+	0.577696i	$0.196048\pi$
$20$		0			0
$21$	21.3386	−	31.8142i	1.01612	−	1.51496i
$22$		0			0
$23$		−	7.50423i		−	0.326271i	−0.986604	−	0.163135i	$-0.947839\pi$
$23$		−	7.50423i		−	0.326271i	0.986604	−	0.163135i	$-0.0521608\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	26.4530	+	5.40707i	0.979742	+	0.200262i
$28$		0			0
$29$		−	15.7298i		−	0.542407i	−0.962522	−	0.271204i	$-0.912578\pi$
$29$		−	15.7298i		−	0.542407i	0.962522	−	0.271204i	$-0.0874217\pi$
$30$		0			0
$31$	−20.4893			−0.660945			−0.330473	−	0.943816i	$-0.607208\pi$
$31$	−20.4893			−0.660945			−0.330473	+	0.943816i	$0.607208\pi$
$32$		0			0
$33$	−31.4663	−	21.1053i	−0.953525	−	0.639553i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	12.9261			0.349355			0.174677	−	0.984626i	$-0.444112\pi$
$37$	12.9261			0.349355			0.174677	+	0.984626i	$0.444112\pi$
$38$		0			0
$39$	12.4344	−	18.5387i	0.318830	−	0.475351i
$40$		0			0
$41$		−	13.8451i		−	0.337685i	−0.985643	−	0.168843i	$-0.945997\pi$
$41$		−	13.8451i		−	0.337685i	0.985643	−	0.168843i	$-0.0540029\pi$
$42$		0			0
$43$	30.0797			0.699528			0.349764	−	0.936838i	$-0.386262\pi$
$43$	30.0797			0.699528			0.349764	+	0.936838i	$0.386262\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			20.2570i			0.431000i	0.976504	+	0.215500i	$0.0691382\pi$
$47$			20.2570i			0.431000i	−0.976504	+	0.215500i	$0.930862\pi$
$48$		0			0
$49$	114.053			2.32761
$50$		0			0
$51$	35.0176	+	23.4872i	0.686619	+	0.460533i
$52$		0			0
$53$		−	29.1185i		−	0.549406i	−0.961529	−	0.274703i	$-0.911420\pi$
$53$		−	29.1185i		−	0.549406i	0.961529	−	0.274703i	$-0.0885796\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−51.8333	+	77.2795i	−0.909356	+	1.35578i
$58$		0			0
$59$		−	47.6333i		−	0.807344i	−0.914904	−	0.403672i	$-0.867734\pi$
$59$		−	47.6333i		−	0.807344i	0.914904	−	0.403672i	$-0.132266\pi$
$60$		0			0
$61$	43.0176			0.705206			0.352603	−	0.935773i	$-0.385297\pi$
$61$	43.0176			0.705206			0.352603	+	0.935773i	$0.385297\pi$
$62$		0			0
$63$	43.6054	+	106.329i	0.692149	+	1.68776i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−0.630153			−0.00940527			−0.00470264	−	0.999989i	$-0.501497\pi$
$67$	−0.630153			−0.00940527			−0.00470264	+	0.999989i	$0.501497\pi$
$68$		0			0
$69$	18.6966	+	12.5403i	0.270965	+	0.181743i
$70$		0			0
$71$		−	90.4047i		−	1.27330i	−0.771151	−	0.636652i	$-0.780319\pi$
$71$		−	90.4047i		−	1.27330i	0.771151	−	0.636652i	$-0.219681\pi$
$72$		0			0
$73$	46.2193			0.633140			0.316570	−	0.948569i	$-0.397469\pi$
$73$	46.2193			0.633140			0.316570	+	0.948569i	$0.397469\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	161.270i		−	2.09442i
$78$		0			0
$79$	−37.9610			−0.480519			−0.240260	−	0.970709i	$-0.577233\pi$
$79$	−37.9610			−0.480519			−0.240260	+	0.970709i	$0.577233\pi$
$80$		0			0
$81$	−57.6771	+	56.8713i	−0.712063	+	0.702115i
$82$		0			0
$83$			80.2267i			0.966587i	0.875458	+	0.483294i	$0.160560\pi$
$83$			80.2267i			0.966587i	−0.875458	+	0.483294i	$0.839440\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	39.1904	+	26.2860i	0.450465	+	0.302138i
$88$		0			0
$89$		−	140.923i		−	1.58341i	−0.610907	−	0.791703i	$-0.709195\pi$
$89$		−	140.923i		−	1.58341i	0.610907	−	0.791703i	$-0.290805\pi$
$90$		0			0
$91$	95.0138			1.04411
$92$		0			0
$93$	34.2396	−	51.0486i	0.368168	−	0.548909i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	10.3429			0.106628			0.0533138	−	0.998578i	$-0.483022\pi$
$97$	10.3429			0.106628			0.0533138	+	0.998578i	$0.483022\pi$
$98$		0			0
$99$	105.166	−	43.1286i	1.06229	−	0.435643i
$100$		0			0
$101$			19.2739i			0.190830i	0.995438	+	0.0954152i	$0.0304179\pi$
$101$			19.2739i			0.190830i	−0.995438	+	0.0954152i	$0.969582\pi$
$102$		0			0
$103$	6.97008			0.0676707			0.0338353	−	0.999427i	$-0.489228\pi$
$103$	6.97008			0.0676707			0.0338353	+	0.999427i	$0.489228\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	73.7731i		−	0.689468i	−0.938700	−	0.344734i	$-0.887969\pi$
$107$		−	73.7731i		−	0.689468i	0.938700	−	0.344734i	$-0.112031\pi$
$108$		0			0
$109$	−74.0314			−0.679187			−0.339593	−	0.940572i	$-0.610289\pi$
$109$	−74.0314			−0.679187			−0.339593	+	0.940572i	$0.610289\pi$
$110$		0			0
$111$	−21.6008	+	32.2051i	−0.194602	+	0.290136i
$112$		0			0
$113$		−	147.215i		−	1.30279i	−0.758739	−	0.651394i	$-0.774184\pi$
$113$		−	147.215i		−	1.30279i	0.758739	−	0.651394i	$-0.225816\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	25.4096	+	61.9598i	0.217176	+	0.529571i
$118$		0			0
$119$			179.471i			1.50816i
$120$		0			0
$121$	−38.5069			−0.318239
$122$		0			0
$123$	34.4947	+	23.1364i	0.280444	+	0.188101i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	90.0171			0.708796			0.354398	−	0.935095i	$-0.384686\pi$
$127$	90.0171			0.708796			0.354398	+	0.935095i	$0.384686\pi$
$128$		0			0
$129$	−50.2660	+	74.9428i	−0.389659	+	0.580952i
$130$		0			0
$131$		−	11.2911i		−	0.0861917i	−0.999071	−	0.0430958i	$-0.986278\pi$
$131$		−	11.2911i		−	0.0861917i	0.999071	−	0.0430958i	$-0.0137221\pi$
$132$		0			0
$133$	−396.070			−2.97797
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	22.4905i		−	0.164164i	−0.996626	−	0.0820822i	$-0.973843\pi$
$137$		−	22.4905i		−	0.164164i	0.996626	−	0.0820822i	$-0.0261570\pi$
$138$		0			0
$139$	−91.0955			−0.655363			−0.327682	−	0.944788i	$-0.606267\pi$
$139$	−91.0955			−0.655363			−0.327682	+	0.944788i	$0.606267\pi$
$140$		0			0
$141$	−50.4698	−	33.8514i	−0.357942	−	0.240081i
$142$		0			0
$143$		−	93.9750i		−	0.657168i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−190.593	+	284.159i	−1.29655	+	1.93306i
$148$		0			0
$149$			228.330i			1.53242i	0.642593	+	0.766208i	$0.277859\pi$
$149$			228.330i			1.53242i	−0.642593	+	0.766208i	$0.722141\pi$
$150$		0			0
$151$	−74.0390			−0.490324			−0.245162	−	0.969482i	$-0.578841\pi$
$151$	−74.0390			−0.490324			−0.245162	+	0.969482i	$0.578841\pi$
$152$		0			0
$153$	−117.035	+	47.9961i	−0.764937	+	0.313700i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−245.742			−1.56523			−0.782616	−	0.622504i	$-0.786115\pi$
$157$	−245.742			−1.56523			−0.782616	+	0.622504i	$0.786115\pi$
$158$		0			0
$159$	72.5481	+	48.6598i	0.456277	+	0.306037i
$160$		0			0
$161$			95.8231i			0.595175i
$162$		0			0
$163$	60.1570			0.369061			0.184531	−	0.982827i	$-0.440924\pi$
$163$	60.1570			0.369061			0.184531	+	0.982827i	$0.440924\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	81.5664i		−	0.488421i	−0.969722	−	0.244211i	$-0.921471\pi$
$167$		−	81.5664i		−	0.488421i	0.969722	−	0.244211i	$-0.0785288\pi$
$168$		0			0
$169$	−113.634			−0.672389
$170$		0			0
$171$	−105.921	−	258.283i	−0.619424	−	1.51043i
$172$		0			0
$173$		−	167.064i		−	0.965687i	−0.875707	−	0.482843i	$-0.839604\pi$
$173$		−	167.064i		−	0.965687i	0.875707	−	0.482843i	$-0.160396\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	118.677	+	79.5997i	0.670492	+	0.449716i
$178$		0			0
$179$		−	270.104i		−	1.50896i	−0.656322	−	0.754481i	$-0.727889\pi$
$179$		−	270.104i		−	1.50896i	0.656322	−	0.754481i	$-0.272111\pi$
$180$		0			0
$181$	86.9786			0.480545			0.240272	−	0.970705i	$-0.422763\pi$
$181$	86.9786			0.480545			0.240272	+	0.970705i	$0.422763\pi$
$182$		0			0
$183$	−71.8865	+	107.177i	−0.392822	+	0.585668i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	177.509			0.949244
$188$		0			0
$189$	−337.785	−	69.0440i	−1.78722	−	0.365312i
$190$		0			0
$191$			302.223i			1.58232i	0.611610	+	0.791159i	$0.290522\pi$
$191$			302.223i			1.58232i	−0.611610	+	0.791159i	$0.709478\pi$
$192$		0			0
$193$	306.780			1.58953			0.794766	−	0.606916i	$-0.207593\pi$
$193$	306.780			1.58953			0.794766	+	0.606916i	$0.207593\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	289.956i		−	1.47186i	−0.677058	−	0.735930i	$-0.736745\pi$
$197$		−	289.956i		−	1.47186i	0.677058	−	0.735930i	$-0.263255\pi$
$198$		0			0
$199$	−382.595			−1.92259			−0.961293	−	0.275527i	$-0.911148\pi$
$199$	−382.595			−1.92259			−0.961293	+	0.275527i	$0.911148\pi$
$200$		0			0
$201$	1.05305	−	1.57001i	0.00523903	−	0.00781100i
$202$		0			0
$203$			200.857i			0.989445i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−62.4876	+	25.6261i	−0.301872	+	0.123798i
$208$		0			0
$209$			391.740i			1.87435i
$210$		0			0
$211$	−321.115			−1.52187			−0.760937	−	0.648826i	$-0.775260\pi$
$211$	−321.115			−1.52187			−0.760937	+	0.648826i	$0.775260\pi$
$212$		0			0
$213$	225.241	+	151.075i	1.05747	+	0.709271i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	261.632			1.20568
$218$		0			0
$219$	−77.2368	+	115.154i	−0.352679	+	0.525818i
$220$		0			0
$221$			104.581i			0.473217i
$222$		0			0
$223$	292.432			1.31135			0.655676	−	0.755042i	$-0.272384\pi$
$223$	292.432			1.31135			0.655676	+	0.755042i	$0.272384\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	370.155i		−	1.63064i	−0.579013	−	0.815319i	$-0.696562\pi$
$227$		−	370.155i		−	1.63064i	0.579013	−	0.815319i	$-0.303438\pi$
$228$		0			0
$229$	381.985			1.66806			0.834028	−	0.551722i	$-0.186029\pi$
$229$	381.985			1.66806			0.834028	+	0.551722i	$0.186029\pi$
$230$		0			0
$231$	401.800	+	269.498i	1.73939	+	1.16666i
$232$		0			0
$233$			144.262i			0.619150i	0.950875	+	0.309575i	$0.100187\pi$
$233$			144.262i			0.619150i	−0.950875	+	0.309575i	$0.899813\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	63.4365	−	94.5789i	0.267665	−	0.399067i
$238$		0			0
$239$			249.478i			1.04384i	0.852994	+	0.521921i	$0.174784\pi$
$239$			249.478i			1.04384i	−0.852994	+	0.521921i	$0.825216\pi$
$240$		0			0
$241$	30.4541			0.126366			0.0631829	−	0.998002i	$-0.479875\pi$
$241$	30.4541			0.126366			0.0631829	+	0.998002i	$0.479875\pi$
$242$		0			0
$243$	−45.3095	−	238.738i	−0.186459	−	0.982463i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−230.797			−0.934401
$248$		0			0
$249$	−199.883	−	134.066i	−0.802742	−	0.538420i
$250$		0			0
$251$		−	68.9183i		−	0.274575i	−0.990531	−	0.137287i	$-0.956162\pi$
$251$		−	68.9183i		−	0.274575i	0.990531	−	0.137287i	$-0.0438384\pi$
$252$		0			0
$253$	94.7755			0.374607
$254$		0			0
$255$		0			0
$256$		0			0
$257$			362.374i			1.41002i	0.709199	+	0.705009i	$0.249057\pi$
$257$			362.374i			1.41002i	−0.709199	+	0.705009i	$0.750943\pi$
$258$		0			0
$259$	−165.057			−0.637284
$260$		0			0
$261$	−130.982	+	53.7155i	−0.501846	+	0.205807i
$262$		0			0
$263$			23.4485i			0.0891580i	0.999006	+	0.0445790i	$0.0141946\pi$
$263$			23.4485i			0.0891580i	−0.999006	+	0.0445790i	$0.985805\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	351.106	+	235.496i	1.31500	+	0.882007i
$268$		0			0
$269$		−	396.738i		−	1.47486i	−0.675421	−	0.737432i	$-0.736038\pi$
$269$		−	396.738i		−	1.47486i	0.675421	−	0.737432i	$-0.263962\pi$
$270$		0			0
$271$	−143.926			−0.531092			−0.265546	−	0.964098i	$-0.585552\pi$
$271$	−143.926			−0.531092			−0.265546	+	0.964098i	$0.585552\pi$
$272$		0			0
$273$	−158.777	+	236.724i	−0.581601	+	0.867122i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−360.884			−1.30283			−0.651415	−	0.758722i	$-0.725824\pi$
$277$	−360.884			−1.30283			−0.651415	+	0.758722i	$0.725824\pi$
$278$		0			0
$279$	69.9686	+	170.614i	0.250784	+	0.611520i
$280$		0			0
$281$		−	531.655i		−	1.89201i	−0.324149	−	0.946006i	$-0.605078\pi$
$281$		−	531.655i		−	1.89201i	0.324149	−	0.946006i	$-0.394922\pi$
$282$		0			0
$283$	−257.400			−0.909541			−0.454770	−	0.890609i	$-0.650279\pi$
$283$	−257.400			−0.909541			−0.454770	+	0.890609i	$0.650279\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			176.791i			0.615996i
$288$		0			0
$289$	91.4579			0.316463
$290$		0			0
$291$	−17.2839	+	25.7690i	−0.0593949	+	0.0885533i
$292$		0			0
$293$		−	310.456i		−	1.05958i	−0.848130	−	0.529789i	$-0.822271\pi$
$293$		−	310.456i		−	1.05958i	0.848130	−	0.529789i	$-0.177729\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−68.2892	+	334.091i	−0.229930	+	1.12489i
$298$		0			0
$299$			55.8379i			0.186749i
$300$		0			0
$301$	−384.094			−1.27606
$302$		0			0
$303$	−48.0204	−	32.2085i	−0.158483	−	0.106299i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−266.169			−0.866999			−0.433499	−	0.901154i	$-0.642721\pi$
$307$	−266.169			−0.866999			−0.433499	+	0.901154i	$0.642721\pi$
$308$		0			0
$309$	−11.6477	+	17.3658i	−0.0376947	+	0.0561999i
$310$		0			0
$311$		−	186.580i		−	0.599934i	−0.953950	−	0.299967i	$-0.903024\pi$
$311$		−	186.580i		−	0.599934i	0.953950	−	0.299967i	$-0.0969757\pi$
$312$		0			0
$313$	20.5021			0.0655019			0.0327510	−	0.999464i	$-0.489573\pi$
$313$	20.5021			0.0655019			0.0327510	+	0.999464i	$0.489573\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			18.9792i			0.0598711i	0.999552	+	0.0299356i	$0.00953021\pi$
$317$			18.9792i			0.0598711i	−0.999552	+	0.0299356i	$0.990470\pi$
$318$		0			0
$319$	198.661			0.622763
$320$		0			0
$321$	183.804	+	123.282i	0.572597	+	0.384055i
$322$		0			0
$323$		−	435.951i		−	1.34969i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	123.713	−	184.447i	0.378328	−	0.564059i
$328$		0			0
$329$		−	258.666i		−	0.786219i
$330$		0			0
$331$	413.193			1.24832			0.624159	−	0.781297i	$-0.285442\pi$
$331$	413.193			1.24832			0.624159	+	0.781297i	$0.285442\pi$
$332$		0			0
$333$	−44.1413	−	107.636i	−0.132556	−	0.323230i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	484.733			1.43838			0.719188	−	0.694816i	$-0.244514\pi$
$337$	484.733			1.43838			0.719188	+	0.694816i	$0.244514\pi$
$338$		0			0
$339$	366.783	+	246.010i	1.08195	+	0.725694i
$340$		0			0
$341$		−	258.772i		−	0.758862i
$342$		0			0
$343$	−830.672			−2.42178
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	336.214i		−	0.968915i	−0.874815	−	0.484457i	$-0.839017\pi$
$347$		−	336.214i		−	0.968915i	0.874815	−	0.484457i	$-0.160983\pi$
$348$		0			0
$349$	428.261			1.22711			0.613555	−	0.789652i	$-0.289739\pi$
$349$	428.261			1.22711			0.613555	+	0.789652i	$0.289739\pi$
$350$		0			0
$351$	−196.833	−	40.2332i	−0.560778	−	0.114625i
$352$		0			0
$353$			558.817i			1.58305i	0.611137	+	0.791525i	$0.290713\pi$
$353$			558.817i			1.58305i	−0.611137	+	0.791525i	$0.709287\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−447.147	−	299.913i	−1.25251	−	0.840092i
$358$		0			0
$359$		−	206.915i		−	0.576365i	−0.957576	−	0.288182i	$-0.906949\pi$
$359$		−	206.915i		−	0.576365i	0.957576	−	0.288182i	$-0.0930509\pi$
$360$		0			0
$361$	601.090			1.66507
$362$		0			0
$363$	64.3487	−	95.9389i	0.177269	−	0.264295i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	185.425			0.505246			0.252623	−	0.967565i	$-0.418707\pi$
$367$	185.425			0.505246			0.252623	+	0.967565i	$0.418707\pi$
$368$		0			0
$369$	−115.288	+	47.2794i	−0.312433	+	0.128128i
$370$		0			0
$371$			371.821i			1.00221i
$372$		0			0
$373$	427.345			1.14570			0.572848	−	0.819661i	$-0.305838\pi$
$373$	427.345			1.14570			0.572848	+	0.819661i	$0.305838\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			117.043i			0.310459i
$378$		0			0
$379$	−117.727			−0.310626			−0.155313	−	0.987865i	$-0.549639\pi$
$379$	−117.727			−0.310626			−0.155313	+	0.987865i	$0.549639\pi$
$380$		0			0
$381$	−150.427	+	224.275i	−0.394822	+	0.588649i
$382$		0			0
$383$		−	470.016i		−	1.22720i	−0.789619	−	0.613598i	$-0.789722\pi$
$383$		−	470.016i		−	1.22720i	0.789619	−	0.613598i	$-0.210278\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−102.719	−	250.473i	−0.265423	−	0.647217i
$388$		0			0
$389$		−	128.160i		−	0.329461i	−0.986339	−	0.164730i	$-0.947325\pi$
$389$		−	128.160i		−	0.329461i	0.986339	−	0.164730i	$-0.0526754\pi$
$390$		0			0
$391$	−105.472			−0.269749
$392$		0			0
$393$	28.1315	+	18.8685i	0.0715814	+	0.0480115i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	276.316			0.696011			0.348005	−	0.937493i	$-0.386859\pi$
$397$	276.316			0.696011			0.348005	+	0.937493i	$0.386859\pi$
$398$		0			0
$399$	661.871	−	986.798i	1.65882	−	2.47318i
$400$		0			0
$401$			549.912i			1.37135i	0.727907	+	0.685675i	$0.240493\pi$
$401$			549.912i			1.37135i	−0.727907	+	0.685675i	$0.759507\pi$
$402$		0			0
$403$	152.458			0.378307
$404$		0			0
$405$		0			0
$406$		0			0
$407$			163.252i			0.401110i
$408$		0			0
$409$	219.166			0.535858			0.267929	−	0.963439i	$-0.413661\pi$
$409$	219.166			0.535858			0.267929	+	0.963439i	$0.413661\pi$
$410$		0			0
$411$	56.0346	+	37.5838i	0.136337	+	0.0914448i
$412$		0			0
$413$			608.240i			1.47273i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	152.229	−	226.962i	0.365058	−	0.544274i
$418$		0			0
$419$		−	204.522i		−	0.488119i	−0.969760	−	0.244060i	$-0.921521\pi$
$419$		−	204.522i		−	0.488119i	0.969760	−	0.244060i	$-0.0784792\pi$
$420$		0			0
$421$	577.186			1.37099			0.685494	−	0.728078i	$-0.259586\pi$
$421$	577.186			1.37099			0.685494	+	0.728078i	$0.259586\pi$
$422$		0			0
$423$	168.680	−	69.1754i	0.398770	−	0.163535i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−549.301			−1.28642
$428$		0			0
$429$	234.136	+	157.041i	0.545772	+	0.366063i
$430$		0			0
$431$		−	663.363i		−	1.53913i	−0.638571	−	0.769563i	$-0.720474\pi$
$431$		−	663.363i		−	1.53913i	0.638571	−	0.769563i	$-0.279526\pi$
$432$		0			0
$433$	−226.876			−0.523964			−0.261982	−	0.965073i	$-0.584376\pi$
$433$	−226.876			−0.523964			−0.261982	+	0.965073i	$0.584376\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	232.763i		−	0.532639i
$438$		0			0
$439$	−282.524			−0.643564			−0.321782	−	0.946814i	$-0.604282\pi$
$439$	−282.524			−0.643564			−0.321782	+	0.946814i	$0.604282\pi$
$440$		0			0
$441$	−389.477	−	949.715i	−0.883168	−	2.15355i
$442$		0			0
$443$			455.605i			1.02845i	0.857654	+	0.514227i	$0.171921\pi$
$443$			455.605i			1.02845i	−0.857654	+	0.514227i	$0.828079\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−568.878	−	381.561i	−1.27266	−	0.853604i
$448$		0			0
$449$			157.206i			0.350124i	0.984557	+	0.175062i	$0.0560127\pi$
$449$			157.206i			0.350124i	−0.984557	+	0.175062i	$0.943987\pi$
$450$		0			0
$451$	174.858			0.387712
$452$		0			0
$453$	123.726	−	184.466i	0.273126	−	0.407210i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	397.152			0.869041			0.434520	−	0.900662i	$-0.356918\pi$
$457$	397.152			0.869041			0.434520	+	0.900662i	$0.356918\pi$
$458$		0			0
$459$	75.9962	−	371.797i	0.165569	−	0.810014i
$460$		0			0
$461$			350.730i			0.760803i	0.924821	+	0.380401i	$0.124214\pi$
$461$			350.730i			0.760803i	−0.924821	+	0.380401i	$0.875786\pi$
$462$		0			0
$463$	308.385			0.666058			0.333029	−	0.942917i	$-0.391929\pi$
$463$	308.385			0.666058			0.333029	+	0.942917i	$0.391929\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			800.129i			1.71334i	0.515866	+	0.856669i	$0.327470\pi$
$467$			800.129i			1.71334i	−0.515866	+	0.856669i	$0.672530\pi$
$468$		0			0
$469$	8.04656			0.0171568
$470$		0			0
$471$	410.657	−	612.259i	0.871884	−	1.29991i
$472$		0			0
$473$			379.895i			0.803160i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−242.469	+	99.4364i	−0.508321	+	0.208462i
$478$		0			0
$479$		−	630.135i		−	1.31552i	−0.753227	−	0.657761i	$-0.771504\pi$
$479$		−	630.135i		−	1.31552i	0.753227	−	0.657761i	$-0.228496\pi$
$480$		0			0
$481$	−96.1814			−0.199961
$482$		0			0
$483$	−238.741	−	160.130i	−0.494287	−	0.331531i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	103.952			0.213454			0.106727	−	0.994288i	$-0.465963\pi$
$487$	103.952			0.213454			0.106727	+	0.994288i	$0.465963\pi$
$488$		0			0
$489$	−100.528	+	149.879i	−0.205579	+	0.306502i
$490$		0			0
$491$			286.049i			0.582585i	0.956634	+	0.291292i	$0.0940853\pi$
$491$			286.049i			0.582585i	−0.956634	+	0.291292i	$0.905915\pi$
$492$		0			0
$493$	−221.082			−0.448442
$494$		0			0
$495$		0			0
$496$		0			0
$497$			1154.40i			2.32273i
$498$		0			0
$499$	528.285			1.05869			0.529344	−	0.848407i	$-0.322438\pi$
$499$	528.285			1.05869			0.529344	+	0.848407i	$0.322438\pi$
$500$		0			0
$501$	203.220	+	136.305i	0.405630	+	0.272066i
$502$		0			0
$503$		−	733.418i		−	1.45809i	−0.684467	−	0.729044i	$-0.739965\pi$
$503$		−	733.418i		−	1.45809i	0.684467	−	0.729044i	$-0.260035\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	189.893	−	283.116i	0.374542	−	0.558413i
$508$		0			0
$509$			312.619i			0.614183i	0.951680	+	0.307092i	$0.0993558\pi$
$509$			312.619i			0.614183i	−0.951680	+	0.307092i	$0.900644\pi$
$510$		0			0
$511$	−590.183			−1.15496
$512$		0			0
$513$	820.509	+	167.714i	1.59943	+	0.326928i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−255.838			−0.494851
$518$		0			0
$519$	416.235	+	279.179i	0.801995	+	0.537918i
$520$		0			0
$521$			715.719i			1.37374i	0.726780	+	0.686870i	$0.241016\pi$
$521$			715.719i			1.37374i	−0.726780	+	0.686870i	$0.758984\pi$
$522$		0			0
$523$	−109.080			−0.208567			−0.104283	−	0.994548i	$-0.533255\pi$
$523$	−109.080			−0.208567			−0.104283	+	0.994548i	$0.533255\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			287.977i			0.546445i
$528$		0			0
$529$	472.686			0.893547
$530$		0			0
$531$	−396.641	+	162.662i	−0.746971	+	0.306332i
$532$		0			0
$533$			103.019i			0.193282i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	672.958	+	451.370i	1.25318	+	0.840540i
$538$		0			0
$539$			1440.44i			2.67243i
$540$		0			0
$541$	−14.9710			−0.0276729			−0.0138364	−	0.999904i	$-0.504404\pi$
$541$	−14.9710			−0.0276729			−0.0138364	+	0.999904i	$0.504404\pi$
$542$		0			0
$543$	−145.350	+	216.705i	−0.267679	+	0.399088i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	842.765			1.54070			0.770352	−	0.637618i	$-0.220080\pi$
$547$	842.765			1.54070			0.770352	+	0.637618i	$0.220080\pi$
$548$		0			0
$549$	−146.900	−	358.206i	−0.267578	−	0.652471i
$550$		0			0
$551$		−	487.901i		−	0.885482i
$552$		0			0
$553$	484.733			0.876551
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	845.989i		−	1.51883i	−0.650606	−	0.759415i	$-0.725485\pi$
$557$		−	845.989i		−	1.51883i	0.650606	−	0.759415i	$-0.274515\pi$
$558$		0			0
$559$	−223.819			−0.400391
$560$		0			0
$561$	−296.634	+	442.258i	−0.528759	+	0.788339i
$562$		0			0
$563$		−	336.509i		−	0.597707i	−0.954299	−	0.298853i	$-0.903396\pi$
$563$		−	336.509i		−	0.597707i	0.954299	−	0.298853i	$-0.0966041\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	736.491	−	726.202i	1.29893	−	1.28078i
$568$		0			0
$569$		−	552.736i		−	0.971416i	−0.874121	−	0.485708i	$-0.838562\pi$
$569$		−	552.736i		−	0.971416i	0.874121	−	0.485708i	$-0.161438\pi$
$570$		0			0
$571$	772.222			1.35240			0.676202	−	0.736717i	$-0.263625\pi$
$571$	772.222			1.35240			0.676202	+	0.736717i	$0.263625\pi$
$572$		0			0
$573$	−752.980	−	505.043i	−1.31410	−	0.881401i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−848.557			−1.47064			−0.735318	−	0.677722i	$-0.762967\pi$
$577$	−848.557			−1.47064			−0.735318	+	0.677722i	$0.762967\pi$
$578$		0			0
$579$	−512.658	+	764.334i	−0.885420	+	1.32009i
$580$		0			0
$581$		−	1024.43i		−	1.76322i
$582$		0			0
$583$	367.755			0.630798
$584$		0			0
$585$		0			0
$586$		0			0
$587$			575.536i			0.980470i	0.871590	+	0.490235i	$0.163089\pi$
$587$			575.536i			0.980470i	−0.871590	+	0.490235i	$0.836911\pi$
$588$		0			0
$589$	−635.529			−1.07900
$590$		0			0
$591$	722.419	+	484.544i	1.22237	+	0.819872i
$592$		0			0
$593$		−	156.935i		−	0.264646i	−0.991207	−	0.132323i	$-0.957756\pi$
$593$		−	156.935i		−	0.264646i	0.991207	−	0.132323i	$-0.0422436\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	639.352	−	953.225i	1.07094	−	1.59669i
$598$		0			0
$599$		−	517.564i		−	0.864047i	−0.901862	−	0.432024i	$-0.857800\pi$
$599$		−	517.564i		−	0.864047i	0.901862	−	0.432024i	$-0.142200\pi$
$600$		0			0
$601$	−883.588			−1.47020			−0.735098	−	0.677961i	$-0.762864\pi$
$601$	−883.588			−1.47020			−0.735098	+	0.677961i	$0.762864\pi$
$602$		0			0
$603$	2.15190	+	5.24727i	0.00356866	+	0.00870194i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	242.929			0.400212			0.200106	−	0.979774i	$-0.435871\pi$
$607$	242.929			0.400212			0.200106	+	0.979774i	$0.435871\pi$
$608$		0			0
$609$	−500.431	−	335.652i	−0.821725	−	0.551152i
$610$		0			0
$611$		−	150.729i		−	0.246693i
$612$		0			0
$613$	157.263			0.256546			0.128273	−	0.991739i	$-0.459057\pi$
$613$	157.263			0.256546			0.128273	+	0.991739i	$0.459057\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			471.040i			0.763436i	0.924279	+	0.381718i	$0.124667\pi$
$617$			471.040i			0.763436i	−0.924279	+	0.381718i	$0.875333\pi$
$618$		0			0
$619$	−550.320			−0.889047			−0.444524	−	0.895767i	$-0.646627\pi$
$619$	−550.320			−0.889047			−0.444524	+	0.895767i	$0.646627\pi$
$620$		0			0
$621$	40.5759	−	198.510i	0.0653396	−	0.319662i
$622$		0			0
$623$			1799.48i			2.88841i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−976.009	−	654.634i	−1.55663	−	1.04407i
$628$		0			0
$629$		−	181.676i		−	0.288834i
$630$		0			0
$631$	−347.362			−0.550495			−0.275248	−	0.961373i	$-0.588760\pi$
$631$	−347.362			−0.550495			−0.275248	+	0.961373i	$0.588760\pi$
$632$		0			0
$633$	536.614	−	800.051i	0.847732	−	1.26390i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−848.649			−1.33226
$638$		0			0
$639$	−752.798	+	308.721i	−1.17809	+	0.483132i
$640$		0			0
$641$		−	333.802i		−	0.520752i	−0.965507	−	0.260376i	$-0.916153\pi$
$641$		−	333.802i		−	0.520752i	0.965507	−	0.260376i	$-0.0838465\pi$
$642$		0			0
$643$	−495.512			−0.770625			−0.385313	−	0.922786i	$-0.625906\pi$
$643$	−495.512			−0.770625			−0.385313	+	0.922786i	$0.625906\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	149.493i		−	0.231056i	−0.993304	−	0.115528i	$-0.963144\pi$
$647$		−	149.493i		−	0.231056i	0.993304	−	0.115528i	$-0.0368559\pi$
$648$		0			0
$649$	601.590			0.926948
$650$		0			0
$651$	−437.212	+	651.850i	−0.671601	+	1.00131i
$652$		0			0
$653$		−	261.846i		−	0.400990i	−0.979695	−	0.200495i	$-0.935745\pi$
$653$		−	261.846i		−	0.400990i	0.979695	−	0.200495i	$-0.0642550\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−157.833	−	384.867i	−0.240234	−	0.585794i
$658$		0			0
$659$		−	148.718i		−	0.225672i	−0.993614	−	0.112836i	$-0.964006\pi$
$659$		−	148.718i		−	0.225672i	0.993614	−	0.112836i	$-0.0359935\pi$
$660$		0			0
$661$	−535.548			−0.810209			−0.405104	−	0.914270i	$-0.632765\pi$
$661$	−535.548			−0.810209			−0.405104	+	0.914270i	$0.632765\pi$
$662$		0			0
$663$	−260.561	−	174.765i	−0.393002	−	0.263597i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−118.040			−0.176972
$668$		0			0
$669$	−488.681	+	728.586i	−0.730465	+	1.08907i
$670$		0			0
$671$			543.295i			0.809680i
$672$		0			0
$673$	−678.388			−1.00801			−0.504003	−	0.863702i	$-0.668140\pi$
$673$	−678.388			−1.00801			−0.504003	+	0.863702i	$0.668140\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			809.743i			1.19608i	0.801468	+	0.598038i	$0.204053\pi$
$677$			809.743i			1.19608i	−0.801468	+	0.598038i	$0.795947\pi$
$678$		0			0
$679$	−132.070			−0.194507
$680$		0			0
$681$	922.231	+	618.563i	1.35423	+	0.908316i
$682$		0			0
$683$			150.099i			0.219764i	0.993945	+	0.109882i	$0.0350473\pi$
$683$			150.099i			0.219764i	−0.993945	+	0.109882i	$0.964953\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−638.333	+	951.705i	−0.929160	+	1.38531i
$688$		0			0
$689$			216.667i			0.314465i
$690$		0			0
$691$	−334.001			−0.483359			−0.241679	−	0.970356i	$-0.577698\pi$
$691$	−334.001			−0.483359			−0.241679	+	0.970356i	$0.577698\pi$
$692$		0			0
$693$	−1342.89	+	550.719i	−1.93780	+	0.794688i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−194.592			−0.279185
$698$		0			0
$699$	−359.425	−	241.076i	−0.514199	−	0.344886i
$700$		0			0
$701$			924.471i			1.31879i	0.751797	+	0.659394i	$0.229187\pi$
$701$			924.471i			1.31879i	−0.751797	+	0.659394i	$0.770813\pi$
$702$		0			0
$703$	400.937			0.570324
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	246.112i		−	0.348108i
$708$		0			0
$709$	−797.459			−1.12477			−0.562383	−	0.826877i	$-0.690115\pi$
$709$	−797.459			−1.12477			−0.562383	+	0.826877i	$0.690115\pi$
$710$		0			0
$711$	129.633	+	316.101i	0.182324	+	0.444586i
$712$		0			0
$713$			153.757i			0.215647i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−621.568	−	416.902i	−0.866902	−	0.581453i
$718$		0			0
$719$			907.966i			1.26282i	0.775450	+	0.631409i	$0.217523\pi$
$719$			907.966i			1.26282i	−0.775450	+	0.631409i	$0.782477\pi$
$720$		0			0
$721$	−89.0024			−0.123443
$722$		0			0
$723$	−50.8918	+	75.8757i	−0.0703897	+	0.104946i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−1038.16			−1.42801			−0.714005	−	0.700140i	$-0.753121\pi$
$727$	−1038.16			−1.42801			−0.714005	+	0.700140i	$0.753121\pi$
$728$		0			0
$729$	670.527	+	286.067i	0.919790	+	0.392410i
$730$		0			0
$731$		−	422.769i		−	0.578344i
$732$		0			0
$733$	−399.107			−0.544485			−0.272242	−	0.962229i	$-0.587765\pi$
$733$	−399.107			−0.544485			−0.272242	+	0.962229i	$0.587765\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	7.95858i		−	0.0107986i
$738$		0			0
$739$	−452.504			−0.612319			−0.306159	−	0.951980i	$-0.599044\pi$
$739$	−452.504			−0.612319			−0.306159	+	0.951980i	$0.599044\pi$
$740$		0			0
$741$	385.684	−	575.025i	0.520491	−	0.776012i
$742$		0			0
$743$		−	486.909i		−	0.655328i	−0.944794	−	0.327664i	$-0.893739\pi$
$743$		−	486.909i		−	0.655328i	0.944794	−	0.327664i	$-0.106261\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	668.046	−	273.965i	0.894306	−	0.366754i
$748$		0			0
$749$			942.024i			1.25771i
$750$		0			0
$751$	−470.899			−0.627029			−0.313515	−	0.949583i	$-0.601506\pi$
$751$	−470.899			−0.627029			−0.313515	+	0.949583i	$0.601506\pi$
$752$		0			0
$753$	171.708	+	115.169i	0.228032	+	0.152947i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	886.264			1.17076			0.585379	−	0.810760i	$-0.300946\pi$
$757$	886.264			1.17076			0.585379	+	0.810760i	$0.300946\pi$
$758$		0			0
$759$	−158.379	+	236.131i	−0.208668	+	0.311108i
$760$		0			0
$761$		−	1022.30i		−	1.34336i	−0.740841	−	0.671680i	$-0.765573\pi$
$761$		−	1022.30i		−	1.34336i	0.740841	−	0.671680i	$-0.234427\pi$
$762$		0			0
$763$	945.322			1.23895
$764$		0			0
$765$		0			0
$766$		0			0
$767$			354.432i			0.462102i
$768$		0			0
$769$	−1051.96			−1.36796			−0.683982	−	0.729499i	$-0.739753\pi$
$769$	−1051.96			−1.36796			−0.683982	+	0.729499i	$0.739753\pi$
$770$		0			0
$771$	−902.847	−	605.562i	−1.17101	−	0.785424i
$772$		0			0
$773$		−	983.554i		−	1.27238i	−0.771530	−	0.636192i	$-0.780508\pi$
$773$		−	983.554i		−	1.27238i	0.771530	−	0.636192i	$-0.219492\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	275.825	−	411.234i	0.354987	−	0.529259i
$778$		0			0
$779$		−	429.441i		−	0.551272i
$780$		0			0
$781$	1141.77			1.46194
$782$		0			0
$783$	85.0522	−	416.101i	0.108624	−	0.531419i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	984.726			1.25124			0.625620	−	0.780128i	$-0.284846\pi$
$787$	984.726			1.25124			0.625620	+	0.780128i	$0.284846\pi$
$788$		0			0
$789$	−58.4214	−	39.1847i	−0.0740449	−	0.0496638i
$790$		0			0
$791$			1879.82i			2.37651i
$792$		0			0
$793$	−320.087			−0.403641
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	359.710i		−	0.451330i	−0.974205	−	0.225665i	$-0.927544\pi$
$797$		−	359.710i		−	0.451330i	0.974205	−	0.225665i	$-0.0724555\pi$
$798$		0			0
$799$	284.712			0.356335
$800$		0			0
$801$	−1173.46	+	481.236i	−1.46500	+	0.600794i
$802$		0			0
$803$			583.731i			0.726938i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	988.463	+	662.987i	1.22486	+	0.821546i
$808$		0			0
$809$			1174.75i			1.45211i	0.687638	+	0.726053i	$0.258648\pi$
$809$			1174.75i			1.45211i	−0.687638	+	0.726053i	$0.741352\pi$
$810$		0			0
$811$	−1000.55			−1.23373			−0.616863	−	0.787071i	$-0.711597\pi$
$811$	−1000.55			−1.23373			−0.616863	+	0.787071i	$0.711597\pi$
$812$		0			0
$813$	240.514	−	358.588i	0.295835	−	0.441067i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	933.000			1.14198
$818$		0			0
$819$	−324.461	−	791.178i	−0.396167	−	0.966029i
$820$		0			0
$821$		−	536.992i		−	0.654071i	−0.945012	−	0.327036i	$-0.893950\pi$
$821$		−	536.992i		−	0.654071i	0.945012	−	0.327036i	$-0.106050\pi$
$822$		0			0
$823$	−563.712			−0.684948			−0.342474	−	0.939527i	$-0.611265\pi$
$823$	−563.712			−0.684948			−0.342474	+	0.939527i	$0.611265\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			976.487i			1.18076i	0.807126	+	0.590379i	$0.201022\pi$
$827$			976.487i			1.18076i	−0.807126	+	0.590379i	$0.798978\pi$
$828$		0			0
$829$	358.250			0.432147			0.216074	−	0.976377i	$-0.430675\pi$
$829$	358.250			0.432147			0.216074	+	0.976377i	$0.430675\pi$
$830$		0			0
$831$	603.071	−	899.133i	0.725718	−	1.08199i
$832$		0			0
$833$		−	1603.01i		−	1.92438i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−542.005	−	110.787i	−0.647556	−	0.132362i
$838$		0			0
$839$		−	98.9138i		−	0.117895i	−0.998261	−	0.0589474i	$-0.981226\pi$
$839$		−	98.9138i		−	0.117895i	0.998261	−	0.0589474i	$-0.0187744\pi$
$840$		0			0
$841$	593.573			0.705794
$842$		0			0
$843$	1324.61	+	888.447i	1.57130	+	1.05391i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	491.703			0.580523
$848$		0			0
$849$	430.140	−	641.305i	0.506643	−	0.755366i
$850$		0			0
$851$		−	97.0007i		−	0.113984i
$852$		0			0
$853$	262.197			0.307383			0.153691	−	0.988119i	$-0.450884\pi$
$853$	262.197			0.307383			0.153691	+	0.988119i	$0.450884\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	41.1542i		−	0.0480212i	−0.999712	−	0.0240106i	$-0.992356\pi$
$857$		−	41.1542i		−	0.0480212i	0.999712	−	0.0240106i	$-0.00764354\pi$
$858$		0			0
$859$	414.736			0.482813			0.241406	−	0.970424i	$-0.422391\pi$
$859$	414.736			0.482813			0.241406	+	0.970424i	$0.422391\pi$
$860$		0			0
$861$	−440.470	−	295.434i	−0.511579	−	0.343129i
$862$		0			0
$863$			1253.21i			1.45216i	0.687613	+	0.726078i	$0.258659\pi$
$863$			1253.21i			1.45216i	−0.687613	+	0.726078i	$0.741341\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−152.835	+	227.865i	−0.176280	+	0.262820i
$868$		0			0
$869$		−	479.433i		−	0.551706i
$870$		0			0
$871$	4.68888			0.00538332
$872$		0			0
$873$	−35.3197	−	86.1249i	−0.0404579	−	0.0986539i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	861.957			0.982847			0.491424	−	0.870921i	$-0.336477\pi$
$877$	861.957			0.982847			0.491424	+	0.870921i	$0.336477\pi$
$878$		0			0
$879$	773.494	+	518.802i	0.879970	+	0.590218i
$880$		0			0
$881$		−	87.0009i		−	0.0987524i	−0.998780	−	0.0493762i	$-0.984277\pi$
$881$		−	87.0009i		−	0.0987524i	0.998780	−	0.0493762i	$-0.0157233\pi$
$882$		0			0
$883$	1022.42			1.15789			0.578945	−	0.815366i	$-0.303464\pi$
$883$	1022.42			1.15789			0.578945	+	0.815366i	$0.303464\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	669.924i		−	0.755269i	−0.925955	−	0.377634i	$-0.876738\pi$
$887$		−	669.924i		−	0.755269i	0.925955	−	0.377634i	$-0.123262\pi$
$888$		0			0
$889$	−1149.45			−1.29297
$890$		0			0
$891$	−718.263	−	728.439i	−0.806131	−	0.817552i
$892$		0			0
$893$			628.324i			0.703610i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−139.119	−	93.3104i	−0.155093	−	0.104025i
$898$		0			0
$899$			322.293i			0.358502i
$900$		0			0
$901$	−409.260			−0.454229
$902$		0			0
$903$	641.858	−	956.960i	0.710806	−	1.05976i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1296.52			−1.42946			−0.714729	−	0.699402i	$-0.753450\pi$
$907$	−1296.52			−1.42946			−0.714729	+	0.699402i	$0.753450\pi$
$908$		0			0
$909$	160.493	−	65.8181i	0.176560	−	0.0724071i
$910$		0			0
$911$			926.622i			1.01715i	0.861018	+	0.508574i	$0.169827\pi$
$911$			926.622i			1.01715i	−0.861018	+	0.508574i	$0.830173\pi$
$912$		0			0
$913$	−1013.23			−1.10978
$914$		0			0
$915$		0			0
$916$		0			0
$917$			144.179i			0.157229i
$918$		0			0
$919$	478.391			0.520556			0.260278	−	0.965534i	$-0.416186\pi$
$919$	478.391			0.520556			0.260278	+	0.965534i	$0.416186\pi$
$920$		0			0
$921$	444.793	−	663.152i	0.482946	−	0.720035i
$922$		0			0
$923$			672.687i			0.728805i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−23.8020	−	58.0397i	−0.0256764	−	0.0626102i
$928$		0			0
$929$		−	1452.00i		−	1.56297i	−0.623923	−	0.781486i	$-0.714462\pi$
$929$		−	1452.00i		−	1.56297i	0.623923	−	0.781486i	$-0.285538\pi$
$930$		0			0
$931$	3537.64			3.79983
$932$		0			0
$933$	464.858	+	311.792i	0.498240	+	0.334182i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1535.86			−1.63912			−0.819562	−	0.572991i	$-0.805783\pi$
$937$	−1535.86			−1.63912			−0.819562	+	0.572991i	$0.805783\pi$
$938$		0			0
$939$	−34.2610	+	51.0805i	−0.0364866	+	0.0543988i
$940$		0			0
$941$			1237.50i			1.31509i	0.753415	+	0.657545i	$0.228405\pi$
$941$			1237.50i			1.31509i	−0.753415	+	0.657545i	$0.771595\pi$
$942$		0			0
$943$	−103.897			−0.110177
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	603.800i		−	0.637593i	−0.947823	−	0.318796i	$-0.896721\pi$
$947$		−	603.800i		−	0.637593i	0.947823	−	0.318796i	$-0.103279\pi$
$948$		0			0
$949$	−343.911			−0.362393
$950$		0			0
$951$	−47.2861	−	31.7160i	−0.0497225	−	0.0333501i
$952$		0			0
$953$			534.553i			0.560916i	0.959866	+	0.280458i	$0.0904863\pi$
$953$			534.553i			0.560916i	−0.959866	+	0.280458i	$0.909514\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−331.982	+	494.959i	−0.346899	+	0.517199i
$958$		0			0
$959$			287.186i			0.299464i
$960$		0			0
$961$	−541.188			−0.563151
$962$		0			0
$963$	−614.307	+	251.927i	−0.637909	+	0.261606i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−271.601			−0.280870			−0.140435	−	0.990090i	$-0.544850\pi$
$967$	−271.601			−0.280870			−0.140435	+	0.990090i	$0.544850\pi$
$968$		0			0
$969$	1086.16	+	728.515i	1.12091	+	0.751822i
$970$		0			0
$971$		−	912.176i		−	0.939419i	−0.882821	−	0.469709i	$-0.844359\pi$
$971$		−	912.176i		−	0.939419i	0.882821	−	0.469709i	$-0.155641\pi$
$972$		0			0
$973$	1163.22			1.19550
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	333.628i		−	0.341482i	−0.985316	−	0.170741i	$-0.945384\pi$
$977$		−	333.628i		−	0.341482i	0.985316	−	0.170741i	$-0.0546162\pi$
$978$		0			0
$979$	1779.80			1.81798
$980$		0			0
$981$	252.809	+	616.458i	0.257705	+	0.628397i
$982$		0			0
$983$			1600.35i			1.62803i	0.580847	+	0.814013i	$0.302722\pi$
$983$			1600.35i			1.62803i	−0.580847	+	0.814013i	$0.697278\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	644.460	+	432.256i	0.652948	+	0.437949i
$988$		0			0
$989$		−	225.725i		−	0.228236i
$990$		0			0
$991$	−69.3757			−0.0700057			−0.0350029	−	0.999387i	$-0.511144\pi$
$991$	−69.3757			−0.0700057			−0.0350029	+	0.999387i	$0.511144\pi$
$992$		0			0
$993$	−690.485	+	1029.46i	−0.695353	+	1.03672i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−482.497			−0.483949			−0.241974	−	0.970283i	$-0.577795\pi$
$997$	−482.497			−0.483949			−0.241974	+	0.970283i	$0.577795\pi$
$998$		0			0
$999$	341.936	+	69.8925i	0.342278	+	0.0699625i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.3.l.g.401.4		12
3.2	odd	2	inner	600.3.l.g.401.3		12
4.3	odd	2		1200.3.l.y.401.9		12
5.2	odd	4		120.3.c.a.89.10	yes	12
5.3	odd	4		120.3.c.a.89.3	✓	12
5.4	even	2	inner	600.3.l.g.401.9		12
12.11	even	2		1200.3.l.y.401.10		12
15.2	even	4		120.3.c.a.89.4	yes	12
15.8	even	4		120.3.c.a.89.9	yes	12
15.14	odd	2	inner	600.3.l.g.401.10		12
20.3	even	4		240.3.c.e.209.10		12
20.7	even	4		240.3.c.e.209.3		12
20.19	odd	2		1200.3.l.y.401.4		12
40.3	even	4		960.3.c.j.449.3		12
40.13	odd	4		960.3.c.k.449.10		12
40.27	even	4		960.3.c.j.449.10		12
40.37	odd	4		960.3.c.k.449.3		12
60.23	odd	4		240.3.c.e.209.4		12
60.47	odd	4		240.3.c.e.209.9		12
60.59	even	2		1200.3.l.y.401.3		12
120.53	even	4		960.3.c.k.449.4		12
120.77	even	4		960.3.c.k.449.9		12
120.83	odd	4		960.3.c.j.449.9		12
120.107	odd	4		960.3.c.j.449.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.3.c.a.89.3	✓	12	5.3	odd	4
120.3.c.a.89.4	yes	12	15.2	even	4
120.3.c.a.89.9	yes	12	15.8	even	4
120.3.c.a.89.10	yes	12	5.2	odd	4
240.3.c.e.209.3		12	20.7	even	4
240.3.c.e.209.4		12	60.23	odd	4
240.3.c.e.209.9		12	60.47	odd	4
240.3.c.e.209.10		12	20.3	even	4
600.3.l.g.401.3		12	3.2	odd	2	inner
600.3.l.g.401.4		12	1.1	even	1	trivial
600.3.l.g.401.9		12	5.4	even	2	inner
600.3.l.g.401.10		12	15.14	odd	2	inner
960.3.c.j.449.3		12	40.3	even	4
960.3.c.j.449.4		12	120.107	odd	4
960.3.c.j.449.9		12	120.83	odd	4
960.3.c.j.449.10		12	40.27	even	4
960.3.c.k.449.3		12	40.37	odd	4
960.3.c.k.449.4		12	120.53	even	4
960.3.c.k.449.9		12	120.77	even	4
960.3.c.k.449.10		12	40.13	odd	4
1200.3.l.y.401.3		12	60.59	even	2
1200.3.l.y.401.4		12	20.19	odd	2
1200.3.l.y.401.9		12	4.3	odd	2
1200.3.l.y.401.10		12	12.11	even	2

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.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.67109 + 2.49147i) q^{3} -12.7692 q^{7} +(-3.41489 - 8.32698i) q^{9} +O(q^{10})\)	\(q+(-1.67109 + 2.49147i) q^{3} -12.7692 q^{7} +(-3.41489 - 8.32698i) q^{9} +12.6296i q^{11} -7.44085 q^{13} -14.0550i q^{17} +31.0176 q^{19} +(21.3386 - 31.8142i) q^{21} -7.50423i q^{23} +(26.4530 + 5.40707i) q^{27} -15.7298i q^{29} -20.4893 q^{31} +(-31.4663 - 21.1053i) q^{33} +12.9261 q^{37} +(12.4344 - 18.5387i) q^{39} -13.8451i q^{41} +30.0797 q^{43} +20.2570i q^{47} +114.053 q^{49} +(35.0176 + 23.4872i) q^{51} -29.1185i q^{53} +(-51.8333 + 77.2795i) q^{57} -47.6333i q^{59} +43.0176 q^{61} +(43.6054 + 106.329i) q^{63} -0.630153 q^{67} +(18.6966 + 12.5403i) q^{69} -90.4047i q^{71} +46.2193 q^{73} -161.270i q^{77} -37.9610 q^{79} +(-57.6771 + 56.8713i) q^{81} +80.2267i q^{83} +(39.1904 + 26.2860i) q^{87} -140.923i q^{89} +95.0138 q^{91} +(34.2396 - 51.0486i) q^{93} +10.3429 q^{97} +(105.166 - 43.1286i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8 q^{9}+O(q^{10}) \) 12 * q - 8 * q^9	\( 12 q - 8 q^{9} + 4 q^{21} - 48 q^{31} + 128 q^{39} + 252 q^{49} + 48 q^{51} + 144 q^{61} - 268 q^{69} - 432 q^{79} - 188 q^{81} + 560 q^{99}+O(q^{100}) \) 12 * q - 8 * q^9 + 4 * q^21 - 48 * q^31 + 128 * q^39 + 252 * q^49 + 48 * q^51 + 144 * q^61 - 268 * q^69 - 432 * q^79 - 188 * q^81 + 560 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more