LMFDB - Embedded newform 300.9.g.e.101.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,9,Mod(101,300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(300, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("300.101");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 300 = 2^{2} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	300.g (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:	$122.213583018$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} + 17581 x^{8} - 268094 x^{7} + 129938570 x^{6} - 2805075950 x^{5} + 497042336337 x^{4} + \cdots + 51\!\cdots\!56 $ x^10 - x^9 + 17581x^8 - 268094x^7 + 129938570x^6 - 2805075950x^5 + 497042336337x^4 - 12249078945417x^3 + 863119630222893x^2 - 20295663225220440x + 511594269195067956
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{15}\cdot 3^{17}\cdot 5^{7}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			101.8
Root			$-23.3576 - 73.0318i$ of defining polynomial
Character	$\chi$	$=$	300.101
Dual form			300.9.g.e.101.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(35.0336 + 73.0318i) q^{3} -4179.12 q^{7} +(-4106.29 + 5117.14i) q^{9} +8719.18i q^{11} -16396.6 q^{13} +89507.5i q^{17} -165647. q^{19} +(-146410. - 305209. i) q^{21} +446571. i q^{23} +(-517572. - 120618. i) q^{27} +1.40654e6i q^{29} +820045. q^{31} +(-636778. + 305465. i) q^{33} -219949. q^{37} +(-574432. - 1.19747e6i) q^{39} -2.78325e6i q^{41} -1.47284e6 q^{43} -3.99527e6i q^{47} +1.17003e7 q^{49} +(-6.53690e6 + 3.13577e6i) q^{51} -9.07653e6i q^{53} +(-5.80320e6 - 1.20975e7i) q^{57} -944914. i q^{59} -7.30286e6 q^{61} +(1.71607e7 - 2.13851e7i) q^{63} -1.58736e7 q^{67} +(-3.26139e7 + 1.56450e7i) q^{69} -2.16192e7i q^{71} -3.08890e7 q^{73} -3.64385e7i q^{77} +6.47687e7 q^{79} +(-9.32347e6 - 4.20249e7i) q^{81} +3.12999e7i q^{83} +(-1.02722e8 + 4.92762e7i) q^{87} +5.43004e7i q^{89} +6.85234e7 q^{91} +(2.87291e7 + 5.98894e7i) q^{93} +1.29617e8 q^{97} +(-4.46173e7 - 3.58035e7i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 137 q^{3} - 1048 q^{7} - 12455 q^{9} + 34472 q^{13} - 376030 q^{19} - 142540 q^{21} - 1458782 q^{27} - 410860 q^{31} - 523275 q^{33} - 110344 q^{37} + 3527870 q^{39} - 6252148 q^{43} + 13891530 q^{49}+ \cdots + 52292505 q^{99}+O(q^{100}) $ 10 * q - 137 * q^3 - 1048 * q^7 - 12455 * q^9 + 34472 * q^13 - 376030 * q^19 - 142540 * q^21 - 1458782 * q^27 - 410860 * q^31 - 523275 * q^33 - 110344 * q^37 + 3527870 * q^39 - 6252148 * q^43 + 13891530 * q^49 - 644745 * q^51 - 10355323 * q^57 - 51373540 * q^61 + 6641072 * q^63 - 82069258 * q^67 - 12562590 * q^69 - 33368314 * q^73 + 36211400 * q^79 - 75697775 * q^81 - 33748950 * q^87 + 64519480 * q^91 - 109348198 * q^93 + 248767112 * q^97 + 52292505 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$.

$n$	$101$	$151$	$277$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	35.0336	+	73.0318i	0.432514	+	0.901627i
$3$	35.0336	+	73.0318i	0.432514	+	0.901627i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−4179.12			−1.74058			−0.870288	−	0.492543i	$-0.836067\pi$
$7$	−4179.12			−1.74058			−0.870288	+	0.492543i	$0.836067\pi$
$8$		0			0
$9$	−4106.29	+	5117.14i	−0.625864	+	0.779933i
$10$		0			0
$11$			8719.18i			0.595532i	0.954639	+	0.297766i	$0.0962415\pi$
$11$			8719.18i			0.595532i	−0.954639	+	0.297766i	$0.903758\pi$
$12$		0			0
$13$	−16396.6			−0.574091			−0.287045	−	0.957917i	$-0.592673\pi$
$13$	−16396.6			−0.574091			−0.287045	+	0.957917i	$0.592673\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			89507.5i			1.07168i	0.844321	+	0.535838i	$0.180004\pi$
$17$			89507.5i			1.07168i	−0.844321	+	0.535838i	$0.819996\pi$
$18$		0			0
$19$	−165647.			−1.27107			−0.635533	−	0.772074i	$-0.719220\pi$
$19$	−165647.			−1.27107			−0.635533	+	0.772074i	$0.719220\pi$
$20$		0			0
$21$	−146410.	−	305209.i	−0.752823	−	1.56935i
$22$		0			0
$23$			446571.i			1.59580i	0.602788	+	0.797901i	$0.294056\pi$
$23$			446571.i			1.59580i	−0.602788	+	0.797901i	$0.705944\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−517572.	−	120618.i	−0.973903	−	0.226964i
$28$		0			0
$29$			1.40654e6i			1.98866i	0.106350	+	0.994329i	$0.466084\pi$
$29$			1.40654e6i			1.98866i	−0.106350	+	0.994329i	$0.533916\pi$
$30$		0			0
$31$	820045.			0.887955			0.443977	−	0.896038i	$-0.353567\pi$
$31$	820045.			0.887955			0.443977	+	0.896038i	$0.353567\pi$
$32$		0			0
$33$	−636778.	+	305465.i	−0.536948	+	0.257576i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−219949.			−0.117359			−0.0586794	−	0.998277i	$-0.518689\pi$
$37$	−219949.			−0.117359			−0.0586794	+	0.998277i	$0.518689\pi$
$38$		0			0
$39$	−574432.	−	1.19747e6i	−0.248302	−	0.517616i
$40$		0			0
$41$		−	2.78325e6i		−	0.984957i	−0.870325	−	0.492478i	$-0.836091\pi$
$41$		−	2.78325e6i		−	0.984957i	0.870325	−	0.492478i	$-0.163909\pi$
$42$		0			0
$43$	−1.47284e6			−0.430806			−0.215403	−	0.976525i	$-0.569106\pi$
$43$	−1.47284e6			−0.430806			−0.215403	+	0.976525i	$0.569106\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	3.99527e6i		−	0.818757i	−0.912365	−	0.409379i	$-0.865746\pi$
$47$		−	3.99527e6i		−	0.818757i	0.912365	−	0.409379i	$-0.134254\pi$
$48$		0			0
$49$	1.17003e7			2.02960
$50$		0			0
$51$	−6.53690e6	+	3.13577e6i	−0.966253	+	0.463515i
$52$		0			0
$53$		−	9.07653e6i		−	1.15031i	−0.818043	−	0.575157i	$-0.804941\pi$
$53$		−	9.07653e6i		−	1.15031i	0.818043	−	0.575157i	$-0.195059\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−5.80320e6	−	1.20975e7i	−0.549753	−	1.14603i
$58$		0			0
$59$		−	944914.i		−	0.0779802i	−0.999240	−	0.0389901i	$-0.987586\pi$
$59$		−	944914.i		−	0.0779802i	0.999240	−	0.0389901i	$-0.0124141\pi$
$60$		0			0
$61$	−7.30286e6			−0.527440			−0.263720	−	0.964599i	$-0.584950\pi$
$61$	−7.30286e6			−0.527440			−0.263720	+	0.964599i	$0.584950\pi$
$62$		0			0
$63$	1.71607e7	−	2.13851e7i	1.08936	−	1.35753i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−1.58736e7			−0.787728			−0.393864	−	0.919169i	$-0.628862\pi$
$67$	−1.58736e7			−0.787728			−0.393864	+	0.919169i	$0.628862\pi$
$68$		0			0
$69$	−3.26139e7	+	1.56450e7i	−1.43882	+	0.690207i
$70$		0			0
$71$		−	2.16192e7i		−	0.850760i	−0.905015	−	0.425380i	$-0.860140\pi$
$71$		−	2.16192e7i		−	0.850760i	0.905015	−	0.425380i	$-0.139860\pi$
$72$		0			0
$73$	−3.08890e7			−1.08771			−0.543853	−	0.839180i	$-0.683035\pi$
$73$	−3.08890e7			−1.08771			−0.543853	+	0.839180i	$0.683035\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	3.64385e7i		−	1.03657i
$78$		0			0
$79$	6.47687e7			1.66286			0.831432	−	0.555627i	$-0.187522\pi$
$79$	6.47687e7			1.66286			0.831432	+	0.555627i	$0.187522\pi$
$80$		0			0
$81$	−9.32347e6	−	4.20249e7i	−0.216590	−	0.976263i
$82$		0			0
$83$			3.12999e7i			0.659523i	0.944064	+	0.329762i	$0.106968\pi$
$83$			3.12999e7i			0.659523i	−0.944064	+	0.329762i	$0.893032\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−1.02722e8	+	4.92762e7i	−1.79303	+	0.860122i
$88$		0			0
$89$			5.43004e7i			0.865451i	0.901526	+	0.432726i	$0.142448\pi$
$89$			5.43004e7i			0.865451i	−0.901526	+	0.432726i	$0.857552\pi$
$90$		0			0
$91$	6.85234e7			0.999248
$92$		0			0
$93$	2.87291e7	+	5.98894e7i	0.384053	+	0.800604i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.29617e8			1.46411			0.732056	−	0.681244i	$-0.238561\pi$
$97$	1.29617e8			1.46411			0.732056	+	0.681244i	$0.238561\pi$
$98$		0			0
$99$	−4.46173e7	−	3.58035e7i	−0.464475	−	0.372722i
$100$		0			0
$101$		−	1.00294e8i		−	0.963803i	−0.876225	−	0.481902i	$-0.839946\pi$
$101$		−	1.00294e8i		−	0.963803i	0.876225	−	0.481902i	$-0.160054\pi$
$102$		0			0
$103$	−9.72620e7			−0.864160			−0.432080	−	0.901835i	$-0.642220\pi$
$103$	−9.72620e7			−0.864160			−0.432080	+	0.901835i	$0.642220\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			1.96584e8i			1.49973i	0.661592	+	0.749864i	$0.269881\pi$
$107$			1.96584e8i			1.49973i	−0.661592	+	0.749864i	$0.730119\pi$
$108$		0			0
$109$	4.66324e7			0.330355			0.165178	−	0.986264i	$-0.447180\pi$
$109$	4.66324e7			0.330355			0.165178	+	0.986264i	$0.447180\pi$
$110$		0			0
$111$	−7.70563e6	−	1.60633e7i	−0.0507593	−	0.105814i
$112$		0			0
$113$		−	2.87687e8i		−	1.76444i	−0.470839	−	0.882219i	$-0.656049\pi$
$113$		−	2.87687e8i		−	1.76444i	0.470839	−	0.882219i	$-0.343951\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	6.73292e7	−	8.39037e7i	0.359302	−	0.447752i
$118$		0			0
$119$		−	3.74063e8i		−	1.86534i
$120$		0			0
$121$	1.38335e8			0.645342
$122$		0			0
$123$	2.03266e8	−	9.75074e7i	0.888064	−	0.426007i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−1.08389e8			−0.416648			−0.208324	−	0.978060i	$-0.566801\pi$
$127$	−1.08389e8			−0.416648			−0.208324	+	0.978060i	$0.566801\pi$
$128$		0			0
$129$	−5.15989e7	−	1.07564e8i	−0.186329	−	0.388426i
$130$		0			0
$131$			1.57958e8i			0.536361i	0.963369	+	0.268181i	$0.0864224\pi$
$131$			1.57958e8i			0.536361i	−0.963369	+	0.268181i	$0.913578\pi$
$132$		0			0
$133$	6.92257e8			2.21239
$134$		0			0
$135$		0			0
$136$		0			0
$137$			4.74008e8i			1.34556i	0.739842	+	0.672781i	$0.234900\pi$
$137$			4.74008e8i			1.34556i	−0.739842	+	0.672781i	$0.765100\pi$
$138$		0			0
$139$	−2.05158e8			−0.549578			−0.274789	−	0.961505i	$-0.588608\pi$
$139$	−2.05158e8			−0.549578			−0.274789	+	0.961505i	$0.588608\pi$
$140$		0			0
$141$	2.91782e8	−	1.39969e8i	0.738214	−	0.354124i
$142$		0			0
$143$		−	1.42965e8i		−	0.341889i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	4.09903e8	+	8.54492e8i	0.877832	+	1.82995i
$148$		0			0
$149$		−	2.30368e8i		−	0.467387i	−0.972310	−	0.233694i	$-0.924919\pi$
$149$		−	2.30368e8i		−	0.467387i	0.972310	−	0.233694i	$-0.0750812\pi$
$150$		0			0
$151$	3.62887e8			0.698014			0.349007	−	0.937120i	$-0.386519\pi$
$151$	3.62887e8			0.698014			0.349007	+	0.937120i	$0.386519\pi$
$152$		0			0
$153$	−4.58022e8	−	3.67544e8i	−0.835836	−	0.670724i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−6.59642e8			−1.08570			−0.542850	−	0.839830i	$-0.682655\pi$
$157$	−6.59642e8			−1.08570			−0.542850	+	0.839830i	$0.682655\pi$
$158$		0			0
$159$	6.62876e8	−	3.17984e8i	1.03715	−	0.497527i
$160$		0			0
$161$		−	1.86627e9i		−	2.77761i
$162$		0			0
$163$	−3.13982e8			−0.444789			−0.222395	−	0.974957i	$-0.571387\pi$
$163$	−3.13982e8			−0.444789			−0.222395	+	0.974957i	$0.571387\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			2.82650e8i			0.363398i	0.983354	+	0.181699i	$0.0581597\pi$
$167$			2.82650e8i			0.363398i	−0.983354	+	0.181699i	$0.941840\pi$
$168$		0			0
$169$	−5.46882e8			−0.670420
$170$		0			0
$171$	6.80193e8	−	8.47636e8i	0.795514	−	0.991345i
$172$		0			0
$173$			7.66398e8i			0.855598i	0.903874	+	0.427799i	$0.140711\pi$
$173$			7.66398e8i			0.855598i	−0.903874	+	0.427799i	$0.859289\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	6.90088e7	−	3.31037e7i	0.0703090	−	0.0337275i
$178$		0			0
$179$			2.05804e8i			0.200466i	0.994964	+	0.100233i	$0.0319589\pi$
$179$			2.05804e8i			0.200466i	−0.994964	+	0.100233i	$0.968041\pi$
$180$		0			0
$181$	8.08809e8			0.753584			0.376792	−	0.926298i	$-0.377027\pi$
$181$	8.08809e8			0.753584			0.376792	+	0.926298i	$0.377027\pi$
$182$		0			0
$183$	−2.55846e8	−	5.33341e8i	−0.228125	−	0.475555i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−7.80433e8			−0.638218
$188$		0			0
$189$	2.16300e9	+	5.04077e8i	1.69515	+	0.395048i
$190$		0			0
$191$		−	2.34075e9i		−	1.75882i	−0.476065	−	0.879410i	$-0.657937\pi$
$191$		−	2.34075e9i		−	1.75882i	0.476065	−	0.879410i	$-0.342063\pi$
$192$		0			0
$193$	1.81941e9			1.31130			0.655648	−	0.755066i	$-0.272395\pi$
$193$	1.81941e9			1.31130			0.655648	+	0.755066i	$0.272395\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			2.35043e9i			1.56057i	0.625425	+	0.780284i	$0.284926\pi$
$197$			2.35043e9i			1.56057i	−0.625425	+	0.780284i	$0.715074\pi$
$198$		0			0
$199$	−8.79684e8			−0.560938			−0.280469	−	0.959863i	$-0.590490\pi$
$199$	−8.79684e8			−0.560938			−0.280469	+	0.959863i	$0.590490\pi$
$200$		0			0
$201$	−5.56110e8	−	1.15928e9i	−0.340703	−	0.710237i
$202$		0			0
$203$		−	5.87810e9i		−	3.46141i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−2.28516e9	−	1.83375e9i	−1.24462	−	0.998755i
$208$		0			0
$209$		−	1.44430e9i		−	0.756960i
$210$		0			0
$211$	−1.01342e9			−0.511281			−0.255641	−	0.966772i	$-0.582286\pi$
$211$	−1.01342e9			−0.511281			−0.255641	+	0.966772i	$0.582286\pi$
$212$		0			0
$213$	1.57889e9	−	7.57401e8i	0.767069	−	0.367966i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−3.42707e9			−1.54555
$218$		0			0
$219$	−1.08215e9	−	2.25588e9i	−0.470448	−	0.980706i
$220$		0			0
$221$		−	1.46762e9i		−	0.615240i
$222$		0			0
$223$	−5.79370e8			−0.234281			−0.117140	−	0.993115i	$-0.537373\pi$
$223$	−5.79370e8			−0.234281			−0.117140	+	0.993115i	$0.537373\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	3.33665e9i		−	1.25663i	−0.777959	−	0.628315i	$-0.783745\pi$
$227$		−	3.33665e9i		−	1.25663i	0.777959	−	0.628315i	$-0.216255\pi$
$228$		0			0
$229$	5.18028e9			1.88370			0.941850	−	0.336034i	$-0.109086\pi$
$229$	5.18028e9			1.88370			0.941850	+	0.336034i	$0.109086\pi$
$230$		0			0
$231$	2.66117e9	−	1.27657e9i	0.934599	−	0.448330i
$232$		0			0
$233$		−	1.08783e9i		−	0.369093i	−0.982824	−	0.184547i	$-0.940918\pi$
$233$		−	1.08783e9i		−	0.369093i	0.982824	−	0.184547i	$-0.0590816\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	2.26908e9	+	4.73017e9i	0.719211	+	1.49928i
$238$		0			0
$239$			4.07636e8i			0.124934i	0.998047	+	0.0624670i	$0.0198968\pi$
$239$			4.07636e8i			0.124934i	−0.998047	+	0.0624670i	$0.980103\pi$
$240$		0			0
$241$	2.69852e9			0.799939			0.399969	−	0.916528i	$-0.369021\pi$
$241$	2.69852e9			0.799939			0.399969	+	0.916528i	$0.369021\pi$
$242$		0			0
$243$	2.74252e9	−	2.15319e9i	0.786547	−	0.617530i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	2.71604e9			0.729707
$248$		0			0
$249$	−2.28589e9	+	1.09655e9i	−0.594644	+	0.285253i
$250$		0			0
$251$		−	4.51277e9i		−	1.13697i	−0.822694	−	0.568484i	$-0.807530\pi$
$251$		−	4.51277e9i		−	1.13697i	0.822694	−	0.568484i	$-0.192470\pi$
$252$		0			0
$253$	−3.89373e9			−0.950351
$254$		0			0
$255$		0			0
$256$		0			0
$257$			3.72596e9i			0.854093i	0.904230	+	0.427047i	$0.140446\pi$
$257$			3.72596e9i			0.854093i	−0.904230	+	0.427047i	$0.859554\pi$
$258$		0			0
$259$	9.19196e8			0.204272
$260$		0			0
$261$	−7.19746e9	−	5.77566e9i	−1.55102	−	1.24463i
$262$		0			0
$263$			2.85403e9i			0.596534i	0.954482	+	0.298267i	$0.0964086\pi$
$263$			2.85403e9i			0.596534i	−0.954482	+	0.298267i	$0.903591\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−3.96565e9	+	1.90234e9i	−0.780315	+	0.374320i
$268$		0			0
$269$		−	2.98523e9i		−	0.570123i	−0.958509	−	0.285062i	$-0.907986\pi$
$269$		−	2.98523e9i		−	0.570123i	0.958509	−	0.285062i	$-0.0920141\pi$
$270$		0			0
$271$	9.76314e9			1.81014			0.905071	−	0.425261i	$-0.139818\pi$
$271$	9.76314e9			1.81014			0.905071	+	0.425261i	$0.139818\pi$
$272$		0			0
$273$	2.40062e9	+	5.00439e9i	0.432189	+	0.900950i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	2.40876e9			0.409142			0.204571	−	0.978852i	$-0.434420\pi$
$277$	2.40876e9			0.409142			0.204571	+	0.978852i	$0.434420\pi$
$278$		0			0
$279$	−3.36734e9	+	4.19628e9i	−0.555738	+	0.692545i
$280$		0			0
$281$			9.46751e8i			0.151848i	0.997114	+	0.0759242i	$0.0241907\pi$
$281$			9.46751e8i			0.151848i	−0.997114	+	0.0759242i	$0.975809\pi$
$282$		0			0
$283$	−6.94381e9			−1.08256			−0.541280	−	0.840842i	$-0.682060\pi$
$283$	−6.94381e9			−1.08256			−0.541280	+	0.840842i	$0.682060\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			1.16316e10i			1.71439i
$288$		0			0
$289$	−1.03584e9			−0.148492
$290$		0			0
$291$	4.54095e9	+	9.46615e9i	0.633249	+	1.32008i
$292$		0			0
$293$			8.17789e9i			1.10961i	0.831980	+	0.554806i	$0.187207\pi$
$293$			8.17789e9i			1.10961i	−0.831980	+	0.554806i	$0.812793\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	1.05169e9	−	4.51281e9i	0.135164	−	0.579991i
$298$		0			0
$299$		−	7.32225e9i		−	0.916135i
$300$		0			0
$301$	6.15518e9			0.749850
$302$		0			0
$303$	7.32464e9	−	3.51365e9i	0.868991	−	0.416858i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	5.11330e9			0.575636			0.287818	−	0.957685i	$-0.407070\pi$
$307$	5.11330e9			0.575636			0.287818	+	0.957685i	$0.407070\pi$
$308$		0			0
$309$	−3.40744e9	−	7.10322e9i	−0.373761	−	0.779150i
$310$		0			0
$311$			9.69674e9i			1.03654i	0.855218	+	0.518268i	$0.173423\pi$
$311$			9.69674e9i			1.03654i	−0.855218	+	0.518268i	$0.826577\pi$
$312$		0			0
$313$	1.41774e10			1.47713			0.738565	−	0.674183i	$-0.235504\pi$
$313$	1.41774e10			1.47713			0.738565	+	0.674183i	$0.235504\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	6.22001e9i		−	0.615962i	−0.951392	−	0.307981i	$-0.900347\pi$
$317$		−	6.22001e9i		−	0.615962i	0.951392	−	0.307981i	$-0.0996534\pi$
$318$		0			0
$319$	−1.22639e10			−1.18431
$320$		0			0
$321$	−1.43569e10	+	6.88704e9i	−1.35220	+	0.648653i
$322$		0			0
$323$		−	1.48266e10i		−	1.36217i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	1.63370e9	+	3.40565e9i	0.142883	+	0.297857i
$328$		0			0
$329$			1.66967e10i			1.42511i
$330$		0			0
$331$	−2.17671e10			−1.81338			−0.906688	−	0.421801i	$-0.861398\pi$
$331$	−2.17671e10			−1.81338			−0.906688	+	0.421801i	$0.861398\pi$
$332$		0			0
$333$	9.03176e8	−	1.12551e9i	0.0734507	−	0.0915320i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	6.41409e9			0.497297			0.248648	−	0.968594i	$-0.420014\pi$
$337$	6.41409e9			0.497297			0.248648	+	0.968594i	$0.420014\pi$
$338$		0			0
$339$	2.10103e10	−	1.00787e10i	1.59087	−	0.763144i
$340$		0			0
$341$			7.15012e9i			0.528805i
$342$		0			0
$343$	−2.48050e10			−1.79210
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	1.74835e10i		−	1.20590i	−0.797780	−	0.602949i	$-0.793992\pi$
$347$		−	1.74835e10i		−	1.20590i	0.797780	−	0.602949i	$-0.206008\pi$
$348$		0			0
$349$	−1.08571e9			−0.0731831			−0.0365916	−	0.999330i	$-0.511650\pi$
$349$	−1.08571e9			−0.0731831			−0.0365916	+	0.999330i	$0.511650\pi$
$350$		0			0
$351$	8.48642e9	+	1.97773e9i	0.559109	+	0.130298i
$352$		0			0
$353$			1.37210e10i			0.883663i	0.897098	+	0.441831i	$0.145671\pi$
$353$			1.37210e10i			0.883663i	−0.897098	+	0.441831i	$0.854329\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	2.73185e10	−	1.31048e10i	1.68184	−	0.806783i
$358$		0			0
$359$			7.33089e8i			0.0441346i	0.999756	+	0.0220673i	$0.00702480\pi$
$359$			7.33089e8i			0.0441346i	−0.999756	+	0.0220673i	$0.992975\pi$
$360$		0			0
$361$	1.04552e10			0.615608
$362$		0			0
$363$	4.84637e9	+	1.01028e10i	0.279119	+	0.581858i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−3.00539e10			−1.65667			−0.828337	−	0.560230i	$-0.810713\pi$
$367$	−3.00539e10			−1.65667			−0.828337	+	0.560230i	$0.810713\pi$
$368$		0			0
$369$	1.42423e10	+	1.14288e10i	0.768200	+	0.616448i
$370$		0			0
$371$			3.79319e10i			2.00221i
$372$		0			0
$373$	−2.21686e10			−1.14526			−0.572628	−	0.819816i	$-0.694076\pi$
$373$	−2.21686e10			−1.14526			−0.572628	+	0.819816i	$0.694076\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	2.30625e10i		−	1.14167i
$378$		0			0
$379$	−3.02473e10			−1.46598			−0.732992	−	0.680237i	$-0.761877\pi$
$379$	−3.02473e10			−1.46598			−0.732992	+	0.680237i	$0.761877\pi$
$380$		0			0
$381$	−3.79725e9	−	7.91583e9i	−0.180206	−	0.375661i
$382$		0			0
$383$			4.18683e10i			1.94576i	0.231302	+	0.972882i	$0.425701\pi$
$383$			4.18683e10i			1.94576i	−0.231302	+	0.972882i	$0.574299\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	6.04791e9	−	7.53672e9i	0.269626	−	0.335999i
$388$		0			0
$389$			1.93799e10i			0.846357i	0.906046	+	0.423179i	$0.139086\pi$
$389$			1.93799e10i			0.846357i	−0.906046	+	0.423179i	$0.860914\pi$
$390$		0			0
$391$	−3.99715e10			−1.71018
$392$		0			0
$393$	−1.15360e10	+	5.53385e9i	−0.483598	+	0.231984i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−3.39887e10			−1.36827			−0.684136	−	0.729354i	$-0.739821\pi$
$397$	−3.39887e10			−1.36827			−0.684136	+	0.729354i	$0.739821\pi$
$398$		0			0
$399$	2.42523e10	+	5.05568e10i	0.956888	+	1.99475i
$400$		0			0
$401$		−	2.57572e10i		−	0.996142i	−0.867136	−	0.498071i	$-0.834042\pi$
$401$		−	2.57572e10i		−	0.996142i	0.867136	−	0.498071i	$-0.165958\pi$
$402$		0			0
$403$	−1.34460e10			−0.509767
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	1.91778e9i		−	0.0698910i
$408$		0			0
$409$	−2.43084e10			−0.868688			−0.434344	−	0.900747i	$-0.643020\pi$
$409$	−2.43084e10			−0.868688			−0.434344	+	0.900747i	$0.643020\pi$
$410$		0			0
$411$	−3.46177e10	+	1.66062e10i	−1.21319	+	0.581974i
$412$		0			0
$413$			3.94891e9i			0.135730i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−7.18743e9	−	1.49831e10i	−0.237700	−	0.495515i
$418$		0			0
$419$			3.38028e10i			1.09672i	0.836242	+	0.548361i	$0.184748\pi$
$419$			3.38028e10i			1.09672i	−0.836242	+	0.548361i	$0.815252\pi$
$420$		0			0
$421$	2.54840e10			0.811222			0.405611	−	0.914046i	$-0.367059\pi$
$421$	2.54840e10			0.811222			0.405611	+	0.914046i	$0.367059\pi$
$422$		0			0
$423$	2.04444e10	+	1.64058e10i	0.638575	+	0.512430i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	3.05195e10			0.918050
$428$		0			0
$429$	1.04410e10	−	5.00858e9i	0.308257	−	0.147872i
$430$		0			0
$431$			2.50760e10i			0.726691i	0.931654	+	0.363346i	$0.118366\pi$
$431$			2.50760e10i			0.726691i	−0.931654	+	0.363346i	$0.881634\pi$
$432$		0			0
$433$	−2.53682e9			−0.0721669			−0.0360835	−	0.999349i	$-0.511488\pi$
$433$	−2.53682e9			−0.0721669			−0.0360835	+	0.999349i	$0.511488\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	7.39729e10i		−	2.02837i
$438$		0			0
$439$	−4.93433e10			−1.32853			−0.664263	−	0.747499i	$-0.731255\pi$
$439$	−4.93433e10			−1.32853			−0.664263	+	0.747499i	$0.731255\pi$
$440$		0			0
$441$	−4.80447e10	+	5.98719e10i	−1.27026	+	1.58295i
$442$		0			0
$443$		−	3.86918e10i		−	1.00463i	−0.864686	−	0.502313i	$-0.832483\pi$
$443$		−	3.86918e10i		−	1.00463i	0.864686	−	0.502313i	$-0.167517\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	1.68242e10	−	8.07062e9i	0.421409	−	0.202151i
$448$		0			0
$449$		−	7.02127e10i		−	1.72755i	−0.503879	−	0.863774i	$-0.668094\pi$
$449$		−	7.02127e10i		−	1.72755i	0.503879	−	0.863774i	$-0.331906\pi$
$450$		0			0
$451$	2.42677e10			0.586573
$452$		0			0
$453$	1.27133e10	+	2.65023e10i	0.301901	+	0.629348i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	4.15932e9			0.0953582			0.0476791	−	0.998863i	$-0.484818\pi$
$457$	4.15932e9			0.0953582			0.0476791	+	0.998863i	$0.484818\pi$
$458$		0			0
$459$	1.07962e10	−	4.63266e10i	0.243232	−	1.04371i
$460$		0			0
$461$			8.40854e10i			1.86173i	0.365362	+	0.930865i	$0.380945\pi$
$461$			8.40854e10i			1.86173i	−0.365362	+	0.930865i	$0.619055\pi$
$462$		0			0
$463$	−6.80644e10			−1.48114			−0.740570	−	0.671979i	$-0.765444\pi$
$463$	−6.80644e10			−1.48114			−0.740570	+	0.671979i	$0.765444\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			1.79582e10i			0.377568i	0.982019	+	0.188784i	$0.0604546\pi$
$467$			1.79582e10i			0.377568i	−0.982019	+	0.188784i	$0.939545\pi$
$468$		0			0
$469$	6.63378e10			1.37110
$470$		0			0
$471$	−2.31096e10	−	4.81748e10i	−0.469580	−	0.978896i
$472$		0			0
$473$		−	1.28420e10i		−	0.256559i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	4.64459e10	+	3.72709e10i	0.897168	+	0.719940i
$478$		0			0
$479$			8.11511e10i			1.54153i	0.637119	+	0.770766i	$0.280126\pi$
$479$			8.11511e10i			1.54153i	−0.637119	+	0.770766i	$0.719874\pi$
$480$		0			0
$481$	3.60642e9			0.0673746
$482$		0			0
$483$	1.36297e11	−	6.53824e10i	2.50437	−	1.20136i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	2.81287e10			0.500074			0.250037	−	0.968236i	$-0.419557\pi$
$487$	2.81287e10			0.500074			0.250037	+	0.968236i	$0.419557\pi$
$488$		0			0
$489$	−1.09999e10	−	2.29307e10i	−0.192378	−	0.401034i
$490$		0			0
$491$			2.20521e10i			0.379423i	0.981840	+	0.189711i	$0.0607552\pi$
$491$			2.20521e10i			0.379423i	−0.981840	+	0.189711i	$0.939245\pi$
$492$		0			0
$493$	−1.25896e11			−2.13120
$494$		0			0
$495$		0			0
$496$		0			0
$497$			9.03495e10i			1.48081i
$498$		0			0
$499$	1.05438e10			0.170058			0.0850289	−	0.996378i	$-0.472902\pi$
$499$	1.05438e10			0.170058			0.0850289	+	0.996378i	$0.472902\pi$
$500$		0			0
$501$	−2.06424e10	+	9.90224e9i	−0.327650	+	0.157175i
$502$		0			0
$503$			1.19759e11i			1.87084i	0.353533	+	0.935422i	$0.384980\pi$
$503$			1.19759e11i			1.87084i	−0.353533	+	0.935422i	$0.615020\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1.91593e10	−	3.99398e10i	−0.289966	−	0.604469i
$508$		0			0
$509$		−	9.68848e10i		−	1.44339i	−0.692209	−	0.721697i	$-0.743363\pi$
$509$		−	9.68848e10i		−	1.44339i	0.692209	−	0.721697i	$-0.256637\pi$
$510$		0			0
$511$	1.29089e11			1.89324
$512$		0			0
$513$	8.57340e10	+	1.99800e10i	1.23789	+	0.288486i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	3.48355e10			0.487596
$518$		0			0
$519$	−5.59714e10	+	2.68497e10i	−0.771431	+	0.370058i
$520$		0			0
$521$		−	6.75974e9i		−	0.0917443i	−0.998947	−	0.0458721i	$-0.985393\pi$
$521$		−	6.75974e9i		−	0.0917443i	0.998947	−	0.0458721i	$-0.0146067\pi$
$522$		0			0
$523$	5.43309e10			0.726173			0.363086	−	0.931755i	$-0.381723\pi$
$523$	5.43309e10			0.726173			0.363086	+	0.931755i	$0.381723\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			7.34002e10i			0.951601i
$528$		0			0
$529$	−1.21115e11			−1.54658
$530$		0			0
$531$	4.83525e9	+	3.88009e9i	0.0608193	+	0.0488049i
$532$		0			0
$533$			4.56359e10i			0.565454i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−1.50302e10	+	7.21006e9i	−0.180746	+	0.0867045i
$538$		0			0
$539$			1.02017e11i			1.20869i
$540$		0			0
$541$	1.01482e11			1.18468			0.592338	−	0.805690i	$-0.298205\pi$
$541$	1.01482e11			1.18468			0.592338	+	0.805690i	$0.298205\pi$
$542$		0			0
$543$	2.83355e10	+	5.90688e10i	0.325935	+	0.679452i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−8.55692e10			−0.955803			−0.477901	−	0.878414i	$-0.658602\pi$
$547$	−8.55692e10			−0.955803			−0.477901	+	0.878414i	$0.658602\pi$
$548$		0			0
$549$	2.99877e10	−	3.73697e10i	0.330106	−	0.411368i
$550$		0			0
$551$		−	2.32988e11i		−	2.52771i
$552$		0			0
$553$	−2.70676e11			−2.89434
$554$		0			0
$555$		0			0
$556$		0			0
$557$			1.07634e10i			0.111823i	0.998436	+	0.0559114i	$0.0178064\pi$
$557$			1.07634e10i			0.111823i	−0.998436	+	0.0559114i	$0.982194\pi$
$558$		0			0
$559$	2.41496e10			0.247322
$560$		0			0
$561$	−2.73414e10	−	5.69964e10i	−0.276038	−	0.575435i
$562$		0			0
$563$		−	1.84933e11i		−	1.84069i	−0.391110	−	0.920344i	$-0.627909\pi$
$563$		−	1.84933e11i		−	1.84069i	0.391110	−	0.920344i	$-0.372091\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	3.89639e10	+	1.75627e11i	0.376991	+	1.69926i
$568$		0			0
$569$		−	1.25723e11i		−	1.19941i	−0.800222	−	0.599703i	$-0.795285\pi$
$569$		−	1.25723e11i		−	1.19941i	0.800222	−	0.599703i	$-0.204715\pi$
$570$		0			0
$571$	−7.61496e10			−0.716347			−0.358173	−	0.933655i	$-0.616600\pi$
$571$	−7.61496e10			−0.716347			−0.358173	+	0.933655i	$0.616600\pi$
$572$		0			0
$573$	1.70949e11	−	8.20049e10i	1.58580	−	0.760714i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−7.83948e10			−0.707268			−0.353634	−	0.935384i	$-0.615054\pi$
$577$	−7.83948e10			−0.707268			−0.353634	+	0.935384i	$0.615054\pi$
$578$		0			0
$579$	6.37405e10	+	1.32875e11i	0.567154	+	1.18230i
$580$		0			0
$581$		−	1.30806e11i		−	1.14795i
$582$		0			0
$583$	7.91400e10			0.685049
$584$		0			0
$585$		0			0
$586$		0			0
$587$			2.21499e10i			0.186560i	0.995640	+	0.0932802i	$0.0297353\pi$
$587$			2.21499e10i			0.186560i	−0.995640	+	0.0932802i	$0.970265\pi$
$588$		0			0
$589$	−1.35838e11			−1.12865
$590$		0			0
$591$	−1.71656e11	+	8.23441e10i	−1.40705	+	0.674967i
$592$		0			0
$593$		−	6.03267e10i		−	0.487855i	−0.969793	−	0.243927i	$-0.921564\pi$
$593$		−	6.03267e10i		−	0.487855i	0.969793	−	0.243927i	$-0.0784359\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−3.08185e10	−	6.42449e10i	−0.242613	−	0.505757i
$598$		0			0
$599$		−	9.18267e10i		−	0.713283i	−0.934241	−	0.356641i	$-0.883922\pi$
$599$		−	9.18267e10i		−	0.713283i	0.934241	−	0.356641i	$-0.116078\pi$
$600$		0			0
$601$	2.18686e11			1.67619			0.838093	−	0.545527i	$-0.183670\pi$
$601$	2.18686e11			1.67619			0.838093	+	0.545527i	$0.183670\pi$
$602$		0			0
$603$	6.51817e10	−	8.12274e10i	0.493010	−	0.614375i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−5.82890e10			−0.429370			−0.214685	−	0.976683i	$-0.568872\pi$
$607$	−5.82890e10			−0.429370			−0.214685	+	0.976683i	$0.568872\pi$
$608$		0			0
$609$	4.29288e11	−	2.05931e11i	3.12090	−	1.49711i
$610$		0			0
$611$			6.55089e10i			0.470041i
$612$		0			0
$613$	7.28755e10			0.516107			0.258053	−	0.966131i	$-0.416919\pi$
$613$	7.28755e10			0.516107			0.258053	+	0.966131i	$0.416919\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	1.77077e11i		−	1.22186i	−0.791686	−	0.610928i	$-0.790796\pi$
$617$		−	1.77077e11i		−	1.22186i	0.791686	−	0.610928i	$-0.209204\pi$
$618$		0			0
$619$	5.29155e10			0.360429			0.180215	−	0.983627i	$-0.442321\pi$
$619$	5.29155e10			0.360429			0.180215	+	0.983627i	$0.442321\pi$
$620$		0			0
$621$	5.38645e10	−	2.31133e11i	0.362190	−	1.55416i
$622$		0			0
$623$		−	2.26928e11i		−	1.50638i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	1.05480e11	−	5.05992e10i	0.682496	−	0.327396i
$628$		0			0
$629$		−	1.96871e10i		−	0.125771i
$630$		0			0
$631$	2.70775e11			1.70801			0.854007	−	0.520261i	$-0.174165\pi$
$631$	2.70775e11			1.70801			0.854007	+	0.520261i	$0.174165\pi$
$632$		0			0
$633$	−3.55038e10	−	7.40119e10i	−0.221136	−	0.460985i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−1.91845e11			−1.16518
$638$		0			0
$639$	1.10629e11	+	8.87749e10i	0.663536	+	0.532460i
$640$		0			0
$641$		−	1.61140e11i		−	0.954489i	−0.878771	−	0.477244i	$-0.841636\pi$
$641$		−	1.61140e11i		−	0.954489i	0.878771	−	0.477244i	$-0.158364\pi$
$642$		0			0
$643$	−1.55843e11			−0.911685			−0.455842	−	0.890061i	$-0.650662\pi$
$643$	−1.55843e11			−0.911685			−0.455842	+	0.890061i	$0.650662\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			1.91876e11i			1.09498i	0.836813	+	0.547488i	$0.184416\pi$
$647$			1.91876e11i			1.09498i	−0.836813	+	0.547488i	$0.815584\pi$
$648$		0			0
$649$	8.23888e9			0.0464397
$650$		0			0
$651$	−1.20063e11	−	2.50285e11i	−0.668473	−	1.39351i
$652$		0			0
$653$		−	1.53462e11i		−	0.844013i	−0.906593	−	0.422006i	$-0.861326\pi$
$653$		−	1.53462e11i		−	0.844013i	0.906593	−	0.422006i	$-0.138674\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1.26839e11	−	1.58063e11i	0.680756	−	0.848338i
$658$		0			0
$659$			4.15442e10i			0.220277i	0.993916	+	0.110139i	$0.0351294\pi$
$659$			4.15442e10i			0.220277i	−0.993916	+	0.110139i	$0.964871\pi$
$660$		0			0
$661$	−3.07506e11			−1.61082			−0.805411	−	0.592717i	$-0.798055\pi$
$661$	−3.07506e11			−1.61082			−0.805411	+	0.592717i	$0.798055\pi$
$662$		0			0
$663$	1.07183e11	−	5.14160e10i	0.554717	−	0.266100i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−6.28120e11			−3.17350
$668$		0			0
$669$	−2.02974e10	−	4.23124e10i	−0.101330	−	0.211234i
$670$		0			0
$671$		−	6.36750e10i		−	0.314108i
$672$		0			0
$673$	−4.50513e10			−0.219607			−0.109804	−	0.993953i	$-0.535022\pi$
$673$	−4.50513e10			−0.219607			−0.109804	+	0.993953i	$0.535022\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	1.78106e10i		−	0.0847860i	−0.999101	−	0.0423930i	$-0.986502\pi$
$677$		−	1.78106e10i		−	0.0847860i	0.999101	−	0.0423930i	$-0.0134982\pi$
$678$		0			0
$679$	−5.41684e11			−2.54840
$680$		0			0
$681$	2.43682e11	−	1.16895e11i	1.13301	−	0.543509i
$682$		0			0
$683$		−	8.63634e9i		−	0.0396869i	−0.999803	−	0.0198434i	$-0.993683\pi$
$683$		−	8.63634e9i		−	0.0396869i	0.999803	−	0.0198434i	$-0.00631678\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	1.81484e11	+	3.78326e11i	0.814726	+	1.69840i
$688$		0			0
$689$			1.48824e11i			0.660385i
$690$		0			0
$691$	1.18726e11			0.520757			0.260379	−	0.965507i	$-0.416153\pi$
$691$	1.18726e11			0.520757			0.260379	+	0.965507i	$0.416153\pi$
$692$		0			0
$693$	1.86461e11	+	1.49627e11i	0.808454	+	0.648751i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	2.49122e11			1.05556
$698$		0			0
$699$	7.94459e10	−	3.81105e10i	0.332784	−	0.159638i
$700$		0			0
$701$		−	3.79013e11i		−	1.56957i	−0.619765	−	0.784787i	$-0.712772\pi$
$701$		−	3.79013e11i		−	1.56957i	0.619765	−	0.784787i	$-0.287228\pi$
$702$		0			0
$703$	3.64339e10			0.149171
$704$		0			0
$705$		0			0
$706$		0			0
$707$			4.19140e11i			1.67757i
$708$		0			0
$709$	7.31698e10			0.289566			0.144783	−	0.989463i	$-0.453752\pi$
$709$	7.31698e10			0.289566			0.144783	+	0.989463i	$0.453752\pi$
$710$		0			0
$711$	−2.65959e11	+	3.31430e11i	−1.04073	+	1.29692i
$712$		0			0
$713$			3.66208e11i			1.41700i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−2.97704e10	+	1.42810e10i	−0.112644	+	0.0540357i
$718$		0			0
$719$			1.69806e11i			0.635384i	0.948194	+	0.317692i	$0.102908\pi$
$719$			1.69806e11i			0.635384i	−0.948194	+	0.317692i	$0.897092\pi$
$720$		0			0
$721$	4.06470e11			1.50414
$722$		0			0
$723$	9.45388e10	+	1.97077e11i	0.345985	+	0.721247i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−9.31171e10			−0.333343			−0.166672	−	0.986012i	$-0.553302\pi$
$727$	−9.31171e10			−0.333343			−0.166672	+	0.986012i	$0.553302\pi$
$728$		0			0
$729$	2.53332e11	+	1.24857e11i	0.896975	+	0.442082i
$730$		0			0
$731$		−	1.31830e11i		−	0.461685i
$732$		0			0
$733$	−4.98363e11			−1.72635			−0.863176	−	0.504903i	$-0.831528\pi$
$733$	−4.98363e11			−1.72635			−0.863176	+	0.504903i	$0.831528\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	1.38405e11i		−	0.469117i
$738$		0			0
$739$	−1.79121e11			−0.600578			−0.300289	−	0.953848i	$-0.597083\pi$
$739$	−1.79121e11			−0.600578			−0.300289	+	0.953848i	$0.597083\pi$
$740$		0			0
$741$	9.51527e10	+	1.98357e11i	0.315608	+	0.657924i
$742$		0			0
$743$		−	3.91726e10i		−	0.128537i	−0.997933	−	0.0642684i	$-0.979529\pi$
$743$		−	3.91726e10i		−	0.128537i	0.997933	−	0.0642684i	$-0.0204714\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−1.60166e11	−	1.28526e11i	−0.514384	−	0.412771i
$748$		0			0
$749$		−	8.21548e11i		−	2.61039i
$750$		0			0
$751$	9.54686e9			0.0300124			0.0150062	−	0.999887i	$-0.495223\pi$
$751$	9.54686e9			0.0300124			0.0150062	+	0.999887i	$0.495223\pi$
$752$		0			0
$753$	3.29576e11	−	1.58099e11i	1.02512	−	0.491755i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	5.13847e11			1.56477			0.782384	−	0.622796i	$-0.214003\pi$
$757$	5.13847e11			1.56477			0.782384	+	0.622796i	$0.214003\pi$
$758$		0			0
$759$	−1.36412e11	−	2.84366e11i	−0.411040	−	0.856863i
$760$		0			0
$761$			1.74758e11i			0.521073i	0.965464	+	0.260536i	$0.0838994\pi$
$761$			1.74758e11i			0.521073i	−0.965464	+	0.260536i	$0.916101\pi$
$762$		0			0
$763$	−1.94882e11			−0.575009
$764$		0			0
$765$		0			0
$766$		0			0
$767$			1.54934e10i			0.0447677i
$768$		0			0
$769$	7.33447e10			0.209731			0.104866	−	0.994486i	$-0.466559\pi$
$769$	7.33447e10			0.209731			0.104866	+	0.994486i	$0.466559\pi$
$770$		0			0
$771$	−2.72113e11	+	1.30534e11i	−0.770074	+	0.369407i
$772$		0			0
$773$			1.35597e11i			0.379780i	0.981805	+	0.189890i	$0.0608131\pi$
$773$			1.35597e11i			0.379780i	−0.981805	+	0.189890i	$0.939187\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	3.22028e10	+	6.71305e10i	0.0883505	+	0.184177i
$778$		0			0
$779$			4.61036e11i			1.25194i
$780$		0			0
$781$	1.88502e11			0.506655
$782$		0			0
$783$	1.69654e11	−	7.27986e11i	0.451354	−	1.93676i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	5.66612e11			1.47702			0.738510	−	0.674242i	$-0.235530\pi$
$787$	5.66612e11			1.47702			0.738510	+	0.674242i	$0.235530\pi$
$788$		0			0
$789$	−2.08435e11	+	9.99869e10i	−0.537851	+	0.258009i
$790$		0			0
$791$			1.20228e12i			3.07114i
$792$		0			0
$793$	1.19742e11			0.302799
$794$		0			0
$795$		0			0
$796$		0			0
$797$			4.99028e10i			0.123678i	0.998086	+	0.0618389i	$0.0196965\pi$
$797$			4.99028e10i			0.123678i	−0.998086	+	0.0618389i	$0.980303\pi$
$798$		0			0
$799$	3.57607e11			0.877443
$800$		0			0
$801$	−2.77862e11	−	2.22973e11i	−0.674994	−	0.541655i
$802$		0			0
$803$		−	2.69326e11i		−	0.647764i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	2.18017e11	−	1.04583e11i	0.514039	−	0.246586i
$808$		0			0
$809$		−	5.76756e11i		−	1.34647i	−0.739427	−	0.673237i	$-0.764903\pi$
$809$		−	5.76756e11i		−	1.34647i	0.739427	−	0.673237i	$-0.235097\pi$
$810$		0			0
$811$	4.93574e11			1.14096			0.570478	−	0.821313i	$-0.306758\pi$
$811$	4.93574e11			1.14096			0.570478	+	0.821313i	$0.306758\pi$
$812$		0			0
$813$	3.42038e11	+	7.13020e11i	0.782911	+	1.63207i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	2.43971e11			0.547582
$818$		0			0
$819$	−2.81377e11	+	3.50644e11i	−0.625393	+	0.779346i
$820$		0			0
$821$			5.16643e11i			1.13715i	0.822631	+	0.568575i	$0.192505\pi$
$821$			5.16643e11i			1.13715i	−0.822631	+	0.568575i	$0.807495\pi$
$822$		0			0
$823$	−7.98184e10			−0.173982			−0.0869909	−	0.996209i	$-0.527725\pi$
$823$	−7.98184e10			−0.173982			−0.0869909	+	0.996209i	$0.527725\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	5.20937e11i		−	1.11369i	−0.830618	−	0.556843i	$-0.812012\pi$
$827$		−	5.20937e11i		−	1.11369i	0.830618	−	0.556843i	$-0.187988\pi$
$828$		0			0
$829$	2.97950e11			0.630850			0.315425	−	0.948951i	$-0.397853\pi$
$829$	2.97950e11			0.630850			0.315425	+	0.948951i	$0.397853\pi$
$830$		0			0
$831$	8.43874e10	+	1.75916e11i	0.176959	+	0.368893i
$832$		0			0
$833$			1.04726e12i			2.17508i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−4.24432e11	−	9.89122e10i	−0.864782	−	0.201534i
$838$		0			0
$839$			6.40063e11i			1.29174i	0.763448	+	0.645870i	$0.223505\pi$
$839$			6.40063e11i			1.29174i	−0.763448	+	0.645870i	$0.776495\pi$
$840$		0			0
$841$	−1.47811e12			−2.95476
$842$		0			0
$843$	−6.91429e10	+	3.31681e10i	−0.136911	+	0.0656765i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−5.78118e11			−1.12327
$848$		0			0
$849$	−2.43267e11	−	5.07119e11i	−0.468222	−	0.976066i
$850$		0			0
$851$		−	9.82230e10i		−	0.187282i
$852$		0			0
$853$	−4.75342e11			−0.897863			−0.448932	−	0.893566i	$-0.648195\pi$
$853$	−4.75342e11			−0.897863			−0.448932	+	0.893566i	$0.648195\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	3.20049e11i		−	0.593326i	−0.954982	−	0.296663i	$-0.904126\pi$
$857$		−	3.20049e11i		−	0.593326i	0.954982	−	0.296663i	$-0.0958738\pi$
$858$		0			0
$859$	−6.92943e11			−1.27270			−0.636349	−	0.771402i	$-0.719556\pi$
$859$	−6.92943e11			−1.27270			−0.636349	+	0.771402i	$0.719556\pi$
$860$		0			0
$861$	−8.49473e11	+	4.07495e11i	−1.54574	+	0.741498i
$862$		0			0
$863$		−	7.38964e11i		−	1.33223i	−0.745848	−	0.666117i	$-0.767955\pi$
$863$		−	7.38964e11i		−	1.33223i	0.745848	−	0.666117i	$-0.232045\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−3.62893e10	−	7.56493e10i	−0.0642246	−	0.133884i
$868$		0			0
$869$			5.64730e11i			0.990288i
$870$		0			0
$871$	2.60273e11			0.452228
$872$		0			0
$873$	−5.32244e11	+	6.63267e11i	−0.916334	+	1.14191i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.59450e11			0.269543			0.134771	−	0.990877i	$-0.456970\pi$
$877$	1.59450e11			0.269543			0.134771	+	0.990877i	$0.456970\pi$
$878$		0			0
$879$	−5.97246e11	+	2.86501e11i	−1.00046	+	0.479922i
$880$		0			0
$881$		−	6.87039e11i		−	1.14045i	−0.821487	−	0.570227i	$-0.806855\pi$
$881$		−	6.87039e11i		−	1.14045i	0.821487	−	0.570227i	$-0.193145\pi$
$882$		0			0
$883$	−6.85979e11			−1.12841			−0.564207	−	0.825634i	$-0.690818\pi$
$883$	−6.85979e11			−1.12841			−0.564207	+	0.825634i	$0.690818\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	1.22765e11i		−	0.198326i	−0.995071	−	0.0991630i	$-0.968383\pi$
$887$		−	1.22765e11i		−	0.198326i	0.995071	−	0.0991630i	$-0.0316165\pi$
$888$		0			0
$889$	4.52970e11			0.725208
$890$		0			0
$891$	3.66423e11	−	8.12931e10i	0.581396	−	0.128986i
$892$		0			0
$893$			6.61803e11i			1.04069i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	5.34757e11	−	2.56525e11i	0.826013	−	0.396241i
$898$		0			0
$899$			1.15343e12i			1.76584i
$900$		0			0
$901$	8.12418e11			1.23277
$902$		0			0
$903$	2.15638e11	+	4.49524e11i	0.324321	+	0.676085i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−6.58589e11			−0.973162			−0.486581	−	0.873635i	$-0.661756\pi$
$907$	−6.58589e11			−0.973162			−0.486581	+	0.873635i	$0.661756\pi$
$908$		0			0
$909$	5.13217e11	+	4.11835e11i	0.751702	+	0.603209i
$910$		0			0
$911$			8.84636e11i			1.28437i	0.766549	+	0.642186i	$0.221972\pi$
$911$			8.84636e11i			1.28437i	−0.766549	+	0.642186i	$0.778028\pi$
$912$		0			0
$913$	−2.72909e11			−0.392767
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	6.60127e11i		−	0.933578i
$918$		0			0
$919$	−1.60467e11			−0.224970			−0.112485	−	0.993653i	$-0.535881\pi$
$919$	−1.60467e11			−0.224970			−0.112485	+	0.993653i	$0.535881\pi$
$920$		0			0
$921$	1.79137e11	+	3.73434e11i	0.248970	+	0.519009i
$922$		0			0
$923$			3.54482e11i			0.488414i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	3.99386e11	−	4.97703e11i	0.540846	−	0.673986i
$928$		0			0
$929$			3.79729e11i			0.509814i	0.966966	+	0.254907i	$0.0820448\pi$
$929$			3.79729e11i			0.509814i	−0.966966	+	0.254907i	$0.917955\pi$
$930$		0			0
$931$	−1.93811e12			−2.57976
$932$		0			0
$933$	−7.08170e11	+	3.39712e11i	−0.934569	+	0.448316i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−6.57277e11			−0.852688			−0.426344	−	0.904561i	$-0.640199\pi$
$937$	−6.57277e11			−0.852688			−0.426344	+	0.904561i	$0.640199\pi$
$938$		0			0
$939$	4.96685e11	+	1.03540e12i	0.638879	+	1.33182i
$940$		0			0
$941$			5.64330e11i			0.719738i	0.933003	+	0.359869i	$0.117179\pi$
$941$			5.64330e11i			0.719738i	−0.933003	+	0.359869i	$0.882821\pi$
$942$		0			0
$943$	1.24292e12			1.57180
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	1.05278e12i		−	1.30899i	−0.756067	−	0.654494i	$-0.772882\pi$
$947$		−	1.05278e12i		−	1.30899i	0.756067	−	0.654494i	$-0.227118\pi$
$948$		0			0
$949$	5.06474e11			0.624442
$950$		0			0
$951$	4.54259e11	−	2.17910e11i	0.555369	−	0.266412i
$952$		0			0
$953$			1.45077e11i			0.175884i	0.996126	+	0.0879420i	$0.0280290\pi$
$953$			1.45077e11i			0.175884i	−0.996126	+	0.0879420i	$0.971971\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−4.29648e11	−	8.95653e11i	−0.512230	−	1.06781i
$958$		0			0
$959$		−	1.98094e12i		−	2.34205i
$960$		0			0
$961$	−1.80418e11			−0.211536
$962$		0			0
$963$	−1.00595e12	−	8.07230e11i	−1.16969	−	0.938625i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	1.48562e12			1.69903			0.849515	−	0.527564i	$-0.176895\pi$
$967$	1.48562e12			1.69903			0.849515	+	0.527564i	$0.176895\pi$
$968$		0			0
$969$	1.08281e12	−	5.19430e11i	1.22817	−	0.589158i
$970$		0			0
$971$		−	6.58861e10i		−	0.0741169i	−0.999313	−	0.0370584i	$-0.988201\pi$
$971$		−	6.58861e10i		−	0.0741169i	0.999313	−	0.0370584i	$-0.0117988\pi$
$972$		0			0
$973$	8.57381e11			0.956583
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	4.29683e11i		−	0.471596i	−0.971802	−	0.235798i	$-0.924230\pi$
$977$		−	4.29683e11i		−	0.471596i	0.971802	−	0.235798i	$-0.0757703\pi$
$978$		0			0
$979$	−4.73455e11			−0.515404
$980$		0			0
$981$	−1.91486e11	+	2.38624e11i	−0.206757	+	0.257655i
$982$		0			0
$983$		−	3.39473e11i		−	0.363573i	−0.983338	−	0.181786i	$-0.941812\pi$
$983$		−	3.39473e11i		−	0.363573i	0.983338	−	0.181786i	$-0.0581879\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−1.21939e12	+	5.84947e11i	−1.28492	+	0.616379i
$988$		0			0
$989$		−	6.57727e11i		−	0.687481i
$990$		0			0
$991$	−1.40952e12			−1.46143			−0.730714	−	0.682684i	$-0.760813\pi$
$991$	−1.40952e12			−1.46143			−0.730714	+	0.682684i	$0.760813\pi$
$992$		0			0
$993$	−7.62579e11	−	1.58969e12i	−0.784311	−	1.63499i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1.13181e12			1.14549			0.572746	−	0.819733i	$-0.305878\pi$
$997$	1.13181e12			1.14549			0.572746	+	0.819733i	$0.305878\pi$
$998$		0			0
$999$	1.13840e11	+	2.65299e10i	0.114296	+	0.0266363i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.9.g.e.101.8	yes	10
3.2	odd	2	inner	300.9.g.e.101.7	✓	10
5.2	odd	4		300.9.b.e.149.17		20
5.3	odd	4		300.9.b.e.149.4		20
5.4	even	2		300.9.g.g.101.3	yes	10
15.2	even	4		300.9.b.e.149.3		20
15.8	even	4		300.9.b.e.149.18		20
15.14	odd	2		300.9.g.g.101.4	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.9.b.e.149.3		20	15.2	even	4
300.9.b.e.149.4		20	5.3	odd	4
300.9.b.e.149.17		20	5.2	odd	4
300.9.b.e.149.18		20	15.8	even	4
300.9.g.e.101.7	✓	10	3.2	odd	2	inner
300.9.g.e.101.8	yes	10	1.1	even	1	trivial
300.9.g.g.101.3	yes	10	5.4	even	2
300.9.g.g.101.4	yes	10	15.14	odd	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:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	300.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(35.0336 + 73.0318i) q^{3} -4179.12 q^{7} +(-4106.29 + 5117.14i) q^{9} +8719.18i q^{11} -16396.6 q^{13} +89507.5i q^{17} -165647. q^{19} +(-146410. - 305209. i) q^{21} +446571. i q^{23} +(-517572. - 120618. i) q^{27} +1.40654e6i q^{29} +820045. q^{31} +(-636778. + 305465. i) q^{33} -219949. q^{37} +(-574432. - 1.19747e6i) q^{39} -2.78325e6i q^{41} -1.47284e6 q^{43} -3.99527e6i q^{47} +1.17003e7 q^{49} +(-6.53690e6 + 3.13577e6i) q^{51} -9.07653e6i q^{53} +(-5.80320e6 - 1.20975e7i) q^{57} -944914. i q^{59} -7.30286e6 q^{61} +(1.71607e7 - 2.13851e7i) q^{63} -1.58736e7 q^{67} +(-3.26139e7 + 1.56450e7i) q^{69} -2.16192e7i q^{71} -3.08890e7 q^{73} -3.64385e7i q^{77} +6.47687e7 q^{79} +(-9.32347e6 - 4.20249e7i) q^{81} +3.12999e7i q^{83} +(-1.02722e8 + 4.92762e7i) q^{87} +5.43004e7i q^{89} +6.85234e7 q^{91} +(2.87291e7 + 5.98894e7i) q^{93} +1.29617e8 q^{97} +(-4.46173e7 - 3.58035e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 137 q^{3} - 1048 q^{7} - 12455 q^{9} + 34472 q^{13} - 376030 q^{19} - 142540 q^{21} - 1458782 q^{27} - 410860 q^{31} - 523275 q^{33} - 110344 q^{37} + 3527870 q^{39} - 6252148 q^{43} + 13891530 q^{49}+ \cdots + 52292505 q^{99}+O(q^{100}) \) 10 * q - 137 * q^3 - 1048 * q^7 - 12455 * q^9 + 34472 * q^13 - 376030 * q^19 - 142540 * q^21 - 1458782 * q^27 - 410860 * q^31 - 523275 * q^33 - 110344 * q^37 + 3527870 * q^39 - 6252148 * q^43 + 13891530 * q^49 - 644745 * q^51 - 10355323 * q^57 - 51373540 * q^61 + 6641072 * q^63 - 82069258 * q^67 - 12562590 * q^69 - 33368314 * q^73 + 36211400 * q^79 - 75697775 * q^81 - 33748950 * q^87 + 64519480 * q^91 - 109348198 * q^93 + 248767112 * q^97 + 52292505 * q^99

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