LMFDB - Embedded newform 300.9.b.e.149.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,9,Mod(149,300)]

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, 1]))

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

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

chi := DirichletCharacter("300.149");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 300 = 2^{2} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	300.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$122.213583018$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} + 35173 x^{18} - 571368 x^{17} + 569751077 x^{16} - 15299176388 x^{15} + \cdots + 26\!\cdots\!00 $ x^20 - 2x^19 + 35173x^18 - 571368x^17 + 569751077x^16 - 15299176388x^15 + 5642945868762x^14 - 193834541993788x^13 + 37628230630444322x^12 - 1468671293830303208x^11 + 175046187290190273666x^10 - 7148635183236871455416x^9 + 568911768318074275591549x^8 - 22560685367134629931805530x^7 + 1254439763408994086472638025x^6 - 44169029244134078527077970400x^5 + 1749494928504161864641030737325x^4 - 47541520663093761975076772545500x^3 + 1293487618191667123878564427143500x^2 - 20763700800917333495897555439005000*x + 261258231716584782083242023582462500
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{34}\cdot 3^{34}\cdot 5^{30}\cdot 7^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			149.8
Root			$-5.98515 - 55.3224i$ of defining polynomial
Character	$\chi$	$=$	300.149
Dual form			300.9.b.e.149.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-56.3224 + 58.2133i) q^{3} -2295.51i q^{7} +(-216.575 - 6557.42i) q^{9} +17396.4i q^{11} -16463.9i q^{13} -13497.4 q^{17} -195706. q^{19} +(133629. + 129288. i) q^{21} -206229. q^{23} +(393927. + 356722. i) q^{27} -367170. i q^{29} -933775. q^{31} +(-1.01270e6 - 979805. i) q^{33} -410.325i q^{37} +(958418. + 927287. i) q^{39} -3.34234e6i q^{41} +1.94035e6i q^{43} +5.83328e6 q^{47} +495447. q^{49} +(760207. - 785730. i) q^{51} +7.57320e6 q^{53} +(1.10226e7 - 1.13927e7i) q^{57} -2.32367e7i q^{59} +8.94563e6 q^{61} +(-1.50526e7 + 497149. i) q^{63} +2.59768e7i q^{67} +(1.16153e7 - 1.20053e7i) q^{69} -2.66873e7i q^{71} +1.77737e7i q^{73} +3.99335e7 q^{77} -3.48684e7 q^{79} +(-4.29529e7 + 2.84034e6i) q^{81} -4.67421e7 q^{83} +(2.13742e7 + 2.06799e7i) q^{87} +2.42212e7i q^{89} -3.77930e7 q^{91} +(5.25925e7 - 5.43581e7i) q^{93} -2.59451e7i q^{97} +(1.14075e8 - 3.76761e6i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 24910 q^{9} + 752060 q^{19} - 285080 q^{21} - 821720 q^{31} - 7055740 q^{39} - 27783060 q^{49} - 1289490 q^{51} - 102747080 q^{61} + 25125180 q^{69} - 72422800 q^{79} - 151395550 q^{81} + 129038960 q^{91}+ \cdots - 104585010 q^{99}+O(q^{100}) $ 20 * q + 24910 * q^9 + 752060 * q^19 - 285080 * q^21 - 821720 * q^31 - 7055740 * q^39 - 27783060 * q^49 - 1289490 * q^51 - 102747080 * q^61 + 25125180 * q^69 - 72422800 * q^79 - 151395550 * q^81 + 129038960 * q^91 - 104585010 * 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
$3$	−56.3224	+	58.2133i	−0.695338	+	0.718683i
$3$	−56.3224	+	58.2133i	−0.695338	+	0.718683i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	2295.51i		−	0.956063i	−0.878343	−	0.478032i	$-0.841350\pi$
$7$		−	2295.51i		−	0.956063i	0.878343	−	0.478032i	$-0.158650\pi$
$8$		0			0
$9$	−216.575	−	6557.42i	−0.0330094	−	0.999455i
$10$		0			0
$11$			17396.4i			1.18819i	0.804393	+	0.594097i	$0.202491\pi$
$11$			17396.4i			1.18819i	−0.804393	+	0.594097i	$0.797509\pi$
$12$		0			0
$13$		−	16463.9i		−	0.576447i	−0.957563	−	0.288224i	$-0.906935\pi$
$13$		−	16463.9i		−	0.576447i	0.957563	−	0.288224i	$-0.0930646\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−13497.4			−0.161605			−0.0808026	−	0.996730i	$-0.525748\pi$
$17$	−13497.4			−0.161605			−0.0808026	+	0.996730i	$0.525748\pi$
$18$		0			0
$19$	−195706.			−1.50172			−0.750860	−	0.660461i	$-0.770361\pi$
$19$	−195706.			−1.50172			−0.750860	+	0.660461i	$0.770361\pi$
$20$		0			0
$21$	133629.	+	129288.i	0.687106	+	0.664787i
$22$		0			0
$23$	−206229.			−0.736952			−0.368476	−	0.929637i	$-0.620120\pi$
$23$	−206229.			−0.736952			−0.368476	+	0.929637i	$0.620120\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	393927.	+	356722.i	0.741244	+	0.671236i
$28$		0			0
$29$		−	367170.i		−	0.519128i	−0.965726	−	0.259564i	$-0.916421\pi$
$29$		−	367170.i		−	0.519128i	0.965726	−	0.259564i	$-0.0835789\pi$
$30$		0			0
$31$	−933775.			−1.01110			−0.505552	−	0.862796i	$-0.668711\pi$
$31$	−933775.			−1.01110			−0.505552	+	0.862796i	$0.668711\pi$
$32$		0			0
$33$	−1.01270e6	−	979805.i	−0.853935	−	0.826197i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	410.325i		−	0.000218938i	−1.00000		0.000109469i	$-0.999965\pi$
$37$		−	410.325i		−	0.000218938i	1.00000		0.000109469i	$-3.48450e-5\pi$
$38$		0			0
$39$	958418.	+	927287.i	0.414283	+	0.400826i
$40$		0			0
$41$		−	3.34234e6i		−	1.18281i	−0.806375	−	0.591405i	$-0.798573\pi$
$41$		−	3.34234e6i		−	1.18281i	0.806375	−	0.591405i	$-0.201427\pi$
$42$		0			0
$43$			1.94035e6i			0.567553i	0.958891	+	0.283776i	$0.0915873\pi$
$43$			1.94035e6i			0.567553i	−0.958891	+	0.283776i	$0.908413\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.83328e6			1.19542			0.597712	−	0.801711i	$-0.296077\pi$
$47$	5.83328e6			1.19542			0.597712	+	0.801711i	$0.296077\pi$
$48$		0			0
$49$	495447.			0.0859435
$50$		0			0
$51$	760207.	−	785730.i	0.112370	−	0.116143i
$52$		0			0
$53$	7.57320e6			0.959790			0.479895	−	0.877326i	$-0.340675\pi$
$53$	7.57320e6			0.959790			0.479895	+	0.877326i	$0.340675\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	1.10226e7	−	1.13927e7i	1.04420	−	1.07926i
$58$		0			0
$59$		−	2.32367e7i		−	1.91764i	−0.284021	−	0.958818i	$-0.591669\pi$
$59$		−	2.32367e7i		−	1.91764i	0.284021	−	0.958818i	$-0.408331\pi$
$60$		0			0
$61$	8.94563e6			0.646088			0.323044	−	0.946384i	$-0.395294\pi$
$61$	8.94563e6			0.646088			0.323044	+	0.946384i	$0.395294\pi$
$62$		0			0
$63$	−1.50526e7	+	497149.i	−0.955542	+	0.0315591i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			2.59768e7i			1.28910i	0.764562	+	0.644551i	$0.222956\pi$
$67$			2.59768e7i			1.28910i	−0.764562	+	0.644551i	$0.777044\pi$
$68$		0			0
$69$	1.16153e7	−	1.20053e7i	0.512431	−	0.529634i
$70$		0			0
$71$		−	2.66873e7i		−	1.05020i	−0.851041	−	0.525100i	$-0.824028\pi$
$71$		−	2.66873e7i		−	1.05020i	0.851041	−	0.525100i	$-0.175972\pi$
$72$		0			0
$73$			1.77737e7i			0.625875i	0.949774	+	0.312937i	$0.101313\pi$
$73$			1.77737e7i			0.625875i	−0.949774	+	0.312937i	$0.898687\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.99335e7			1.13599
$78$		0			0
$79$	−3.48684e7			−0.895209			−0.447604	−	0.894232i	$-0.647723\pi$
$79$	−3.48684e7			−0.895209			−0.447604	+	0.894232i	$0.647723\pi$
$80$		0			0
$81$	−4.29529e7	+	2.84034e6i	−0.997821	+	0.0659828i
$82$		0			0
$83$	−4.67421e7			−0.984909			−0.492454	−	0.870338i	$-0.663900\pi$
$83$	−4.67421e7			−0.984909			−0.492454	+	0.870338i	$0.663900\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	2.13742e7	+	2.06799e7i	0.373088	+	0.360970i
$88$		0			0
$89$			2.42212e7i			0.386043i	0.981195	+	0.193021i	$0.0618286\pi$
$89$			2.42212e7i			0.386043i	−0.981195	+	0.193021i	$0.938171\pi$
$90$		0			0
$91$	−3.77930e7			−0.551120
$92$		0			0
$93$	5.25925e7	−	5.43581e7i	0.703059	−	0.726662i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	2.59451e7i		−	0.293068i	−0.989206	−	0.146534i	$-0.953188\pi$
$97$		−	2.59451e7i		−	0.293068i	0.989206	−	0.146534i	$-0.0468117\pi$
$98$		0			0
$99$	1.14075e8	−	3.76761e6i	1.18755	−	0.0392216i
$100$		0			0
$101$			1.57296e8i			1.51158i	0.654814	+	0.755790i	$0.272747\pi$
$101$			1.57296e8i			1.51158i	−0.654814	+	0.755790i	$0.727253\pi$
$102$		0			0
$103$			2.36343e7i			0.209987i	0.994473	+	0.104994i	$0.0334822\pi$
$103$			2.36343e7i			0.209987i	−0.994473	+	0.104994i	$0.966518\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	5.24020e7			0.399773			0.199886	−	0.979819i	$-0.435943\pi$
$107$	5.24020e7			0.399773			0.199886	+	0.979819i	$0.435943\pi$
$108$		0			0
$109$	−4.49101e7			−0.318154			−0.159077	−	0.987266i	$-0.550852\pi$
$109$	−4.49101e7			−0.318154			−0.159077	+	0.987266i	$0.550852\pi$
$110$		0			0
$111$	23886.4	+	23110.5i	0.000157347	+	0.000152236i
$112$		0			0
$113$	1.45262e8			0.890921			0.445460	−	0.895302i	$-0.353040\pi$
$113$	1.45262e8			0.890921			0.445460	+	0.895302i	$0.353040\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−1.07961e8	+	3.56566e6i	−0.576133	+	0.0190282i
$118$		0			0
$119$			3.09834e7i			0.154505i
$120$		0			0
$121$	−8.82745e7			−0.411807
$122$		0			0
$123$	1.94568e8	+	1.88248e8i	0.850065	+	0.822453i
$124$		0			0
$125$		0			0
$126$		0			0
$127$			3.12436e8i			1.20101i	0.799622	+	0.600504i	$0.205033\pi$
$127$			3.12436e8i			1.20101i	−0.799622	+	0.600504i	$0.794967\pi$
$128$		0			0
$129$	−1.12954e8	−	1.09285e8i	−0.407890	−	0.394641i
$130$		0			0
$131$			4.47775e8i			1.52046i	0.649656	+	0.760229i	$0.274913\pi$
$131$			4.47775e8i			1.52046i	−0.649656	+	0.760229i	$0.725087\pi$
$132$		0			0
$133$			4.49244e8i			1.43574i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.98335e8			−1.41462			−0.707309	−	0.706904i	$-0.750091\pi$
$137$	−4.98335e8			−1.41462			−0.707309	+	0.706904i	$0.750091\pi$
$138$		0			0
$139$	5.33850e8			1.43008			0.715039	−	0.699084i	$-0.246409\pi$
$139$	5.33850e8			1.43008			0.715039	+	0.699084i	$0.246409\pi$
$140$		0			0
$141$	−3.28545e8	+	3.39575e8i	−0.831224	+	0.859130i
$142$		0			0
$143$	2.86412e8			0.684932
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−2.79048e7	+	2.88416e7i	−0.0597598	+	0.0617661i
$148$		0			0
$149$		−	2.63916e7i		−	0.0535452i	−0.999642	−	0.0267726i	$-0.991477\pi$
$149$		−	2.63916e7i		−	0.0535452i	0.999642	−	0.0267726i	$-0.00852301\pi$
$150$		0			0
$151$	−8.51062e8			−1.63702			−0.818509	−	0.574493i	$-0.805199\pi$
$151$	−8.51062e8			−1.63702			−0.818509	+	0.574493i	$0.805199\pi$
$152$		0			0
$153$	2.92320e6	+	8.85084e7i	0.00533449	+	0.161517i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			2.48170e8i			0.408462i	0.978923	+	0.204231i	$0.0654693\pi$
$157$			2.48170e8i			0.408462i	−0.978923	+	0.204231i	$0.934531\pi$
$158$		0			0
$159$	−4.26541e8	+	4.40861e8i	−0.667379	+	0.689784i
$160$		0			0
$161$			4.73401e8i			0.704572i
$162$		0			0
$163$		−	3.20988e8i		−	0.454713i	−0.973812	−	0.227357i	$-0.926992\pi$
$163$		−	3.20988e8i		−	0.454713i	0.973812	−	0.227357i	$-0.0730083\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.46565e9			1.88436			0.942180	−	0.335108i	$-0.108773\pi$
$167$	1.46565e9			1.88436			0.942180	+	0.335108i	$0.108773\pi$
$168$		0			0
$169$	5.44670e8			0.667709
$170$		0			0
$171$	4.23849e7	+	1.28333e9i	0.0495709	+	1.50090i
$172$		0			0
$173$	1.44526e9			1.61347			0.806734	−	0.590915i	$-0.201233\pi$
$173$	1.44526e9			1.61347			0.806734	+	0.590915i	$0.201233\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	1.35268e9	+	1.30875e9i	1.37817	+	1.33341i
$178$		0			0
$179$		−	7.19759e8i		−	0.701092i	−0.936546	−	0.350546i	$-0.885996\pi$
$179$		−	7.19759e8i		−	0.701092i	0.936546	−	0.350546i	$-0.114004\pi$
$180$		0			0
$181$	−1.56467e9			−1.45784			−0.728918	−	0.684601i	$-0.759977\pi$
$181$	−1.56467e9			−1.45784			−0.728918	+	0.684601i	$0.759977\pi$
$182$		0			0
$183$	−5.03839e8	+	5.20755e8i	−0.449250	+	0.464332i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	2.34806e8i		−	0.192018i
$188$		0			0
$189$	8.18859e8	−	9.04263e8i	0.641744	−	0.708676i
$190$		0			0
$191$			1.31338e9i			0.986864i	0.869784	+	0.493432i	$0.164258\pi$
$191$			1.31338e9i			0.986864i	−0.869784	+	0.493432i	$0.835742\pi$
$192$		0			0
$193$			1.12320e9i			0.809523i	0.914422	+	0.404761i	$0.132645\pi$
$193$			1.12320e9i			0.809523i	−0.914422	+	0.404761i	$0.867355\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1.07566e9			0.714185			0.357092	−	0.934069i	$-0.383768\pi$
$197$	1.07566e9			0.714185			0.357092	+	0.934069i	$0.383768\pi$
$198$		0			0
$199$	6.55984e8			0.418293			0.209147	−	0.977884i	$-0.432931\pi$
$199$	6.55984e8			0.418293			0.209147	+	0.977884i	$0.432931\pi$
$200$		0			0
$201$	−1.51220e9	−	1.46308e9i	−0.926455	−	0.896361i
$202$		0			0
$203$	−8.42840e8			−0.496319
$204$		0			0
$205$		0			0
$206$		0			0
$207$	4.46640e7	+	1.35233e9i	0.0243263	+	0.736550i
$208$		0			0
$209$		−	3.40457e9i		−	1.78434i
$210$		0			0
$211$	−1.48749e9			−0.750453			−0.375227	−	0.926933i	$-0.622435\pi$
$211$	−1.48749e9			−0.750453			−0.375227	+	0.926933i	$0.622435\pi$
$212$		0			0
$213$	1.55356e9	+	1.50309e9i	0.754760	+	0.730244i
$214$		0			0
$215$		0			0
$216$		0			0
$217$			2.14349e9i			0.966679i
$218$		0			0
$219$	−1.03467e9	−	1.00106e9i	−0.449805	−	0.435195i
$220$		0			0
$221$			2.22220e8i			0.0931569i
$222$		0			0
$223$			4.80438e9i			1.94276i	0.237540	+	0.971378i	$0.423659\pi$
$223$			4.80438e9i			1.94276i	−0.237540	+	0.971378i	$0.576341\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	4.19807e9			1.58105			0.790525	−	0.612429i	$-0.209808\pi$
$227$	4.19807e9			1.58105			0.790525	+	0.612429i	$0.209808\pi$
$228$		0			0
$229$	1.93560e9			0.703841			0.351920	−	0.936030i	$-0.385529\pi$
$229$	1.93560e9			0.703841			0.351920	+	0.936030i	$0.385529\pi$
$230$		0			0
$231$	−2.24915e9	+	2.32466e9i	−0.789897	+	0.816416i
$232$		0			0
$233$	2.52861e9			0.857942			0.428971	−	0.903318i	$-0.358876\pi$
$233$	2.52861e9			0.857942			0.428971	+	0.903318i	$0.358876\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	1.96387e9	−	2.02981e9i	0.622473	−	0.643371i
$238$		0			0
$239$			1.76815e9i			0.541910i	0.962592	+	0.270955i	$0.0873395\pi$
$239$			1.76815e9i			0.541910i	−0.962592	+	0.270955i	$0.912661\pi$
$240$		0			0
$241$	1.55220e9			0.460129			0.230064	−	0.973175i	$-0.426106\pi$
$241$	1.55220e9			0.460129			0.230064	+	0.973175i	$0.426106\pi$
$242$		0			0
$243$	2.25387e9	−	2.66041e9i	0.646402	−	0.762997i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			3.22208e9i			0.865663i
$248$		0			0
$249$	2.63263e9	−	2.72101e9i	0.684845	−	0.707837i
$250$		0			0
$251$			4.84318e9i			1.22021i	0.792319	+	0.610107i	$0.208873\pi$
$251$			4.84318e9i			1.22021i	−0.792319	+	0.610107i	$0.791127\pi$
$252$		0			0
$253$		−	3.58764e9i		−	0.875642i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	6.97195e8			0.159816			0.0799082	−	0.996802i	$-0.474537\pi$
$257$	6.97195e8			0.159816			0.0799082	+	0.996802i	$0.474537\pi$
$258$		0			0
$259$	−941903.			−0.000209318
$260$		0			0
$261$	−2.40769e9	+	7.95196e7i	−0.518845	+	0.0171361i
$262$		0			0
$263$	−4.25701e9			−0.889777			−0.444889	−	0.895586i	$-0.646757\pi$
$263$	−4.25701e9			−0.889777			−0.444889	+	0.895586i	$0.646757\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−1.40999e9	−	1.36419e9i	−0.277442	−	0.268430i
$268$		0			0
$269$			3.37356e9i			0.644288i	0.946691	+	0.322144i	$0.104403\pi$
$269$			3.37356e9i			0.644288i	−0.946691	+	0.322144i	$0.895597\pi$
$270$		0			0
$271$	−6.57981e9			−1.21993			−0.609966	−	0.792427i	$-0.708817\pi$
$271$	−6.57981e9			−1.21993			−0.609966	+	0.792427i	$0.708817\pi$
$272$		0			0
$273$	2.12859e9	−	2.20006e9i	0.383215	−	0.396080i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	5.05227e7i		−	0.00858158i	−0.999991	−	0.00429079i	$-0.998634\pi$
$277$		−	5.05227e7i		−	0.00858158i	0.999991	−	0.00429079i	$-0.00136581\pi$
$278$		0			0
$279$	2.02232e8	+	6.12316e9i	0.0333759	+	1.01055i
$280$		0			0
$281$			5.00839e9i			0.803290i	0.915795	+	0.401645i	$0.131561\pi$
$281$			5.00839e9i			0.803290i	−0.915795	+	0.401645i	$0.868439\pi$
$282$		0			0
$283$			1.00231e10i			1.56263i	0.624139	+	0.781314i	$0.285450\pi$
$283$			1.00231e10i			1.56263i	−0.624139	+	0.781314i	$0.714550\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−7.67236e9			−1.13084
$288$		0			0
$289$	−6.79358e9			−0.973884
$290$		0			0
$291$	1.51035e9	+	1.46129e9i	0.210623	+	0.203781i
$292$		0			0
$293$	8.52756e9			1.15706			0.578528	−	0.815663i	$-0.303627\pi$
$293$	8.52756e9			1.15706			0.578528	+	0.815663i	$0.303627\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−6.20567e9	+	6.85290e9i	−0.797559	+	0.880742i
$298$		0			0
$299$			3.39534e9i			0.424814i
$300$		0			0
$301$	4.45409e9			0.542616
$302$		0			0
$303$	−9.15670e9	−	8.85927e9i	−1.08635	−	1.05106i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	1.69426e9i		−	0.190733i	−0.995442	−	0.0953665i	$-0.969598\pi$
$307$		−	1.69426e9i		−	0.190733i	0.995442	−	0.0953665i	$-0.0304023\pi$
$308$		0			0
$309$	−1.37583e9	−	1.33114e9i	−0.150914	−	0.146012i
$310$		0			0
$311$		−	5.14916e9i		−	0.550420i	−0.961384	−	0.275210i	$-0.911253\pi$
$311$		−	5.14916e9i		−	0.550420i	0.961384	−	0.275210i	$-0.0887474\pi$
$312$		0			0
$313$			8.32301e9i			0.867168i	0.901113	+	0.433584i	$0.142751\pi$
$313$			8.32301e9i			0.867168i	−0.901113	+	0.433584i	$0.857249\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.67633e10			−1.66006			−0.830029	−	0.557721i	$-0.811676\pi$
$317$	−1.67633e10			−1.66006			−0.830029	+	0.557721i	$0.811676\pi$
$318$		0			0
$319$	6.38742e9			0.616826
$320$		0			0
$321$	−2.95141e9	+	3.05050e9i	−0.277977	+	0.287310i
$322$		0			0
$323$	2.64152e9			0.242686
$324$		0			0
$325$		0			0
$326$		0			0
$327$	2.52944e9	−	2.61436e9i	0.221225	−	0.228652i
$328$		0			0
$329$		−	1.33903e10i		−	1.14290i
$330$		0			0
$331$	2.17827e10			1.81468			0.907341	−	0.420396i	$-0.138109\pi$
$331$	2.17827e10			1.81468			0.907341	+	0.420396i	$0.138109\pi$
$332$		0			0
$333$	−2.69067e6	+	88865.9i	−0.000218819	+	7.22700e-6i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	2.33242e10i		−	1.80837i	−0.427138	−	0.904186i	$-0.640478\pi$
$337$		−	2.33242e10i		−	1.80837i	0.427138	−	0.904186i	$-0.359522\pi$
$338$		0			0
$339$	−8.18152e9	+	8.45619e9i	−0.619491	+	0.640289i
$340$		0			0
$341$		−	1.62443e10i		−	1.20139i
$342$		0			0
$343$		−	1.43704e10i		−	1.03823i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.42872e10			0.985437			0.492718	−	0.870189i	$-0.336003\pi$
$347$	1.42872e10			0.985437			0.492718	+	0.870189i	$0.336003\pi$
$348$		0			0
$349$	1.25048e10			0.842897			0.421449	−	0.906852i	$-0.361522\pi$
$349$	1.25048e10			0.842897			0.421449	+	0.906852i	$0.361522\pi$
$350$		0			0
$351$	5.87304e9	−	6.48558e9i	0.386932	−	0.427288i
$352$		0			0
$353$	−3.48520e9			−0.224454			−0.112227	−	0.993683i	$-0.535798\pi$
$353$	−3.48520e9			−0.224454			−0.112227	+	0.993683i	$0.535798\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−1.80365e9	−	1.74506e9i	−0.111040	−	0.107433i
$358$		0			0
$359$			1.70468e10i			1.02628i	0.858305	+	0.513139i	$0.171518\pi$
$359$			1.70468e10i			1.02628i	−0.858305	+	0.513139i	$0.828482\pi$
$360$		0			0
$361$	2.13172e10			1.25517
$362$		0			0
$363$	4.97183e9	−	5.13875e9i	0.286345	−	0.295959i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			5.41628e9i			0.298563i	0.988795	+	0.149282i	$0.0476961\pi$
$367$			5.41628e9i			0.298563i	−0.988795	+	0.149282i	$0.952304\pi$
$368$		0			0
$369$	−2.19171e10	+	7.23865e8i	−1.18217	+	0.0390438i
$370$		0			0
$371$		−	1.73843e10i		−	0.917620i
$372$		0			0
$373$		−	1.31731e10i		−	0.680539i	−0.940328	−	0.340269i	$-0.889482\pi$
$373$		−	1.31731e10i		−	0.680539i	0.940328	−	0.340269i	$-0.110518\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−6.04505e9			−0.299250
$378$		0			0
$379$	−7.75719e8			−0.0375965			−0.0187983	−	0.999823i	$-0.505984\pi$
$379$	−7.75719e8			−0.0375965			−0.0187983	+	0.999823i	$0.505984\pi$
$380$		0			0
$381$	−1.81879e10	−	1.75971e10i	−0.863143	−	0.835106i
$382$		0			0
$383$	−3.66582e10			−1.70363			−0.851816	−	0.523841i	$-0.824498\pi$
$383$	−3.66582e10			−1.70363			−0.851816	+	0.523841i	$0.824498\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	1.27237e10	−	4.20230e8i	0.567243	−	0.0187346i
$388$		0			0
$389$			1.07813e10i			0.470839i	0.971894	+	0.235419i	$0.0756463\pi$
$389$			1.07813e10i			0.470839i	−0.971894	+	0.235419i	$0.924354\pi$
$390$		0			0
$391$	2.78356e9			0.119095
$392$		0			0
$393$	−2.60664e10	−	2.52197e10i	−1.09273	−	1.05723i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	3.48948e10i		−	1.40475i	−0.711807	−	0.702375i	$-0.752123\pi$
$397$		−	3.48948e10i		−	1.40475i	0.711807	−	0.702375i	$-0.247877\pi$
$398$		0			0
$399$	−2.61520e10	−	2.53025e10i	−1.03184	−	0.998325i
$400$		0			0
$401$		−	3.53994e10i		−	1.36905i	−0.728991	−	0.684524i	$-0.760010\pi$
$401$		−	3.53994e10i		−	1.36905i	0.728991	−	0.684524i	$-0.239990\pi$
$402$		0			0
$403$			1.53736e10i			0.582848i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	7.13816e6			0.000260141
$408$		0			0
$409$	2.49105e10			0.890205			0.445102	−	0.895480i	$-0.353167\pi$
$409$	2.49105e10			0.890205			0.445102	+	0.895480i	$0.353167\pi$
$410$		0			0
$411$	2.80674e10	−	2.90097e10i	0.983638	−	1.01666i
$412$		0			0
$413$	−5.33400e10			−1.83338
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−3.00677e10	+	3.10772e10i	−0.994388	+	1.02777i
$418$		0			0
$419$		−	2.06624e10i		−	0.670385i	−0.942150	−	0.335192i	$-0.891199\pi$
$419$		−	2.06624e10i		−	0.670385i	0.942150	−	0.335192i	$-0.108801\pi$
$420$		0			0
$421$	4.40984e10			1.40377			0.701883	−	0.712292i	$-0.252343\pi$
$421$	4.40984e10			1.40377			0.701883	+	0.712292i	$0.252343\pi$
$422$		0			0
$423$	−1.26334e9	−	3.82513e10i	−0.0394602	−	1.19477i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	2.05348e10i		−	0.617701i
$428$		0			0
$429$	−1.61314e10	+	1.66730e10i	−0.476259	+	0.492248i
$430$		0			0
$431$			6.54249e9i			0.189598i	0.995496	+	0.0947990i	$0.0302208\pi$
$431$			6.54249e9i			0.189598i	−0.995496	+	0.0947990i	$0.969779\pi$
$432$		0			0
$433$			1.78927e10i			0.509008i	0.967072	+	0.254504i	$0.0819123\pi$
$433$			1.78927e10i			0.509008i	−0.967072	+	0.254504i	$0.918088\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	4.03603e10			1.10670
$438$		0			0
$439$	−4.08114e10			−1.09881			−0.549406	−	0.835556i	$-0.685146\pi$
$439$	−4.08114e10			−1.09881			−0.549406	+	0.835556i	$0.685146\pi$
$440$		0			0
$441$	−1.07301e8	−	3.24886e9i	−0.00283694	−	0.0858967i
$442$		0			0
$443$	4.66125e10			1.21028			0.605142	−	0.796117i	$-0.293116\pi$
$443$	4.66125e10			1.21028			0.605142	+	0.796117i	$0.293116\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	1.53634e9	+	1.48644e9i	0.0384820	+	0.0372320i
$448$		0			0
$449$			1.18770e9i			0.0292229i	0.999893	+	0.0146114i	$0.00465113\pi$
$449$			1.18770e9i			0.0292229i	−0.999893	+	0.0146114i	$0.995349\pi$
$450$		0			0
$451$	5.81445e10			1.40541
$452$		0			0
$453$	4.79339e10	−	4.95431e10i	1.13828	−	1.17650i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	3.89499e10i		−	0.892979i	−0.894789	−	0.446489i	$-0.852674\pi$
$457$		−	3.89499e10i		−	0.892979i	0.894789	−	0.446489i	$-0.147326\pi$
$458$		0			0
$459$	−5.31700e9	−	4.81483e9i	−0.119789	−	0.108475i
$460$		0			0
$461$			4.68231e8i			0.0103671i	0.999987	+	0.00518354i	$0.00164998\pi$
$461$			4.68231e8i			0.0103671i	−0.999987	+	0.00518354i	$0.998350\pi$
$462$		0			0
$463$			6.51315e10i			1.41732i	0.705552	+	0.708658i	$0.250699\pi$
$463$			6.51315e10i			1.41732i	−0.705552	+	0.708658i	$0.749301\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−9.36135e10			−1.96821			−0.984104	−	0.177594i	$-0.943169\pi$
$467$	−9.36135e10			−1.96821			−0.984104	+	0.177594i	$0.943169\pi$
$468$		0			0
$469$	5.96300e10			1.23246
$470$		0			0
$471$	−1.44468e10	−	1.39776e10i	−0.293554	−	0.284019i
$472$		0			0
$473$	−3.37550e10			−0.674363
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1.64016e9	−	4.96607e10i	−0.0316821	−	0.959267i
$478$		0			0
$479$			8.18125e10i			1.55410i	0.629442	+	0.777048i	$0.283284\pi$
$479$			8.18125e10i			1.55410i	−0.629442	+	0.777048i	$0.716716\pi$
$480$		0			0
$481$	−6.75555e6			−0.000126206
$482$		0			0
$483$	−2.75582e10	−	2.66631e10i	−0.506364	−	0.489916i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	1.53575e10i		−	0.273026i	−0.990638	−	0.136513i	$-0.956410\pi$
$487$		−	1.53575e10i		−	0.273026i	0.990638	−	0.136513i	$-0.0435896\pi$
$488$		0			0
$489$	1.86857e10	+	1.80788e10i	0.326795	+	0.316180i
$490$		0			0
$491$		−	7.76212e10i		−	1.33553i	−0.744371	−	0.667766i	$-0.767251\pi$
$491$		−	7.76212e10i		−	1.33553i	0.744371	−	0.667766i	$-0.232749\pi$
$492$		0			0
$493$			4.95584e9i			0.0838938i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6.12610e10			−1.00406
$498$		0			0
$499$	3.55417e10			0.573239			0.286620	−	0.958044i	$-0.407468\pi$
$499$	3.55417e10			0.573239			0.286620	+	0.958044i	$0.407468\pi$
$500$		0			0
$501$	−8.25488e10	+	8.53202e10i	−1.31027	+	1.35426i
$502$		0			0
$503$	1.19756e11			1.87079			0.935396	−	0.353603i	$-0.115044\pi$
$503$	1.19756e11			1.87079			0.935396	+	0.353603i	$0.115044\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−3.06771e10	+	3.17071e10i	−0.464283	+	0.479871i
$508$		0			0
$509$		−	4.51523e10i		−	0.672680i	−0.941741	−	0.336340i	$-0.890811\pi$
$509$		−	4.51523e10i		−	0.672680i	0.941741	−	0.336340i	$-0.109189\pi$
$510$		0			0
$511$	4.07997e10			0.598376
$512$		0			0
$513$	−7.70938e10	−	6.98126e10i	−1.11314	−	1.00801i
$514$		0			0
$515$		0			0
$516$		0			0
$517$			1.01478e11i			1.42040i
$518$		0			0
$519$	−8.14003e10	+	8.41331e10i	−1.12191	+	1.15957i
$520$		0			0
$521$			1.25898e11i			1.70871i	0.519687	+	0.854357i	$0.326049\pi$
$521$			1.25898e11i			1.70871i	−0.519687	+	0.854357i	$0.673951\pi$
$522$		0			0
$523$			1.27416e11i			1.70301i	0.524343	+	0.851507i	$0.324311\pi$
$523$			1.27416e11i			1.70301i	−0.524343	+	0.851507i	$0.675689\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1.26036e10			0.163400
$528$		0			0
$529$	−3.57805e10			−0.456902
$530$		0			0
$531$	−1.52373e11	+	5.03248e9i	−1.91659	+	0.0633000i
$532$		0			0
$533$	−5.50279e10			−0.681827
$534$		0			0
$535$		0			0
$536$		0			0
$537$	4.18995e10	+	4.05385e10i	0.503862	+	0.487496i
$538$		0			0
$539$			8.61898e9i			0.102118i
$540$		0			0
$541$	1.05813e11			1.23524			0.617620	−	0.786476i	$-0.288097\pi$
$541$	1.05813e11			1.23524			0.617620	+	0.786476i	$0.288097\pi$
$542$		0			0
$543$	8.81260e10	−	9.10847e10i	1.01369	−	1.04772i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	1.17218e10i		−	0.130932i	−0.997855	−	0.0654661i	$-0.979147\pi$
$547$		−	1.17218e10i		−	0.130932i	0.997855	−	0.0654661i	$-0.0208534\pi$
$548$		0			0
$549$	−1.93740e9	−	5.86603e10i	−0.0213270	−	0.645736i
$550$		0			0
$551$			7.18572e10i			0.779586i
$552$		0			0
$553$			8.00408e10i			0.855876i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1.90665e10			0.198085			0.0990423	−	0.995083i	$-0.468422\pi$
$557$	1.90665e10			0.198085			0.0990423	+	0.995083i	$0.468422\pi$
$558$		0			0
$559$	3.19457e10			0.327164
$560$		0			0
$561$	1.36688e10	+	1.32248e10i	0.138000	+	0.133518i
$562$		0			0
$563$	4.94309e10			0.492000			0.246000	−	0.969270i	$-0.420884\pi$
$563$	4.94309e10			0.492000			0.246000	+	0.969270i	$0.420884\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	6.52003e9	+	9.85987e10i	0.0630837	+	0.953980i
$568$		0			0
$569$		−	7.53303e10i		−	0.718655i	−0.933211	−	0.359328i	$-0.883006\pi$
$569$		−	7.53303e10i		−	0.718655i	0.933211	−	0.359328i	$-0.116994\pi$
$570$		0			0
$571$	−7.43397e10			−0.699321			−0.349661	−	0.936876i	$-0.613703\pi$
$571$	−7.43397e10			−0.699321			−0.349661	+	0.936876i	$0.613703\pi$
$572$		0			0
$573$	−7.64562e10	−	7.39728e10i	−0.709242	−	0.686204i
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	1.46246e11i		−	1.31941i	−0.751524	−	0.659706i	$-0.770681\pi$
$577$		−	1.46246e11i		−	1.31941i	0.751524	−	0.659706i	$-0.229319\pi$
$578$		0			0
$579$	−6.53854e10	−	6.32615e10i	−0.581790	−	0.562892i
$580$		0			0
$581$			1.07297e11i			0.941635i
$582$		0			0
$583$			1.31746e11i			1.14042i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1.88952e11			1.59147			0.795735	−	0.605646i	$-0.207085\pi$
$587$	1.88952e11			1.59147			0.795735	+	0.605646i	$0.207085\pi$
$588$		0			0
$589$	1.82745e11			1.51840
$590$		0			0
$591$	−6.05838e10	+	6.26178e10i	−0.496600	+	0.513272i
$592$		0			0
$593$	5.67120e10			0.458624			0.229312	−	0.973353i	$-0.426352\pi$
$593$	5.67120e10			0.458624			0.229312	+	0.973353i	$0.426352\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−3.69466e10	+	3.81870e10i	−0.290855	+	0.300620i
$598$		0			0
$599$		−	4.00737e10i		−	0.311281i	−0.987814	−	0.155640i	$-0.950256\pi$
$599$		−	4.00737e10i		−	0.311281i	0.987814	−	0.155640i	$-0.0497441\pi$
$600$		0			0
$601$	−2.27069e11			−1.74044			−0.870220	−	0.492663i	$-0.836024\pi$
$601$	−2.27069e11			−1.74044			−0.870220	+	0.492663i	$0.836024\pi$
$602$		0			0
$603$	1.70341e11	−	5.62592e9i	1.28840	−	0.0425524i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			6.87812e10i			0.506658i	0.967380	+	0.253329i	$0.0815255\pi$
$607$			6.87812e10i			0.506658i	−0.967380	+	0.253329i	$0.918474\pi$
$608$		0			0
$609$	4.74708e10	−	4.90645e10i	0.345110	−	0.356696i
$610$		0			0
$611$		−	9.60387e10i		−	0.689098i
$612$		0			0
$613$			2.22523e11i			1.57592i	0.615728	+	0.787959i	$0.288862\pi$
$613$			2.22523e11i			1.57592i	−0.615728	+	0.787959i	$0.711138\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.41332e11			0.975215			0.487608	−	0.873063i	$-0.337870\pi$
$617$	1.41332e11			0.975215			0.487608	+	0.873063i	$0.337870\pi$
$618$		0			0
$619$	1.28936e11			0.878236			0.439118	−	0.898429i	$-0.355291\pi$
$619$	1.28936e11			0.878236			0.439118	+	0.898429i	$0.355291\pi$
$620$		0			0
$621$	−8.12393e10	−	7.35666e10i	−0.546261	−	0.494668i
$622$		0			0
$623$	5.55999e10			0.369081
$624$		0			0
$625$		0			0
$626$		0			0
$627$	1.98191e11	+	1.91753e11i	1.28237	+	1.24072i
$628$		0			0
$629$			5.53833e6i			3.53815e-5i
$630$		0			0
$631$	1.64177e11			1.03561			0.517804	−	0.855499i	$-0.326750\pi$
$631$	1.64177e11			1.03561			0.517804	+	0.855499i	$0.326750\pi$
$632$		0			0
$633$	8.37789e10	−	8.65916e10i	0.521819	−	0.539338i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	8.15700e9i		−	0.0495419i
$638$		0			0
$639$	−1.75000e11	+	5.77980e9i	−1.04963	+	0.0346664i
$640$		0			0
$641$			3.26564e10i			0.193436i	0.995312	+	0.0967178i	$0.0308344\pi$
$641$			3.26564e10i			0.193436i	−0.995312	+	0.0967178i	$0.969166\pi$
$642$		0			0
$643$			2.64540e11i			1.54756i	0.633454	+	0.773781i	$0.281637\pi$
$643$			2.64540e11i			1.54756i	−0.633454	+	0.773781i	$0.718363\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.87714e11			1.07122			0.535610	−	0.844466i	$-0.320082\pi$
$647$	1.87714e11			1.07122			0.535610	+	0.844466i	$0.320082\pi$
$648$		0			0
$649$	4.04234e11			2.27853
$650$		0			0
$651$	−1.24779e11	−	1.20726e11i	−0.694735	−	0.672169i
$652$		0			0
$653$	−1.62529e11			−0.893879			−0.446939	−	0.894564i	$-0.647486\pi$
$653$	−1.62529e11			−0.893879			−0.446939	+	0.894564i	$0.647486\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1.16550e11	−	3.84934e9i	0.625533	−	0.0206597i
$658$		0			0
$659$		−	1.36472e11i		−	0.723608i	−0.932254	−	0.361804i	$-0.882161\pi$
$659$		−	1.36472e11i		−	0.723608i	0.932254	−	0.361804i	$-0.117839\pi$
$660$		0			0
$661$	−2.89793e11			−1.51804			−0.759019	−	0.651069i	$-0.774321\pi$
$661$	−2.89793e11			−1.51804			−0.759019	+	0.651069i	$0.774321\pi$
$662$		0			0
$663$	−1.29362e10	−	1.25160e10i	−0.0669502	−	0.0647755i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			7.57211e10i			0.382572i
$668$		0			0
$669$	−2.79679e11	−	2.70594e11i	−1.39622	−	1.35087i
$670$		0			0
$671$			1.55621e11i			0.767678i
$672$		0			0
$673$		−	1.95738e10i		−	0.0954145i	−0.998861	−	0.0477072i	$-0.984809\pi$
$673$		−	1.95738e10i		−	0.0954145i	0.998861	−	0.0477072i	$-0.0151915\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.54461e11			−0.735297			−0.367649	−	0.929965i	$-0.619837\pi$
$677$	−1.54461e11			−0.735297			−0.367649	+	0.929965i	$0.619837\pi$
$678$		0			0
$679$	−5.95571e10			−0.280191
$680$		0			0
$681$	−2.36445e11	+	2.44383e11i	−1.09937	+	1.13627i
$682$		0			0
$683$	−3.32703e10			−0.152888			−0.0764440	−	0.997074i	$-0.524357\pi$
$683$	−3.32703e10			−0.152888			−0.0764440	+	0.997074i	$0.524357\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−1.09018e11	+	1.12678e11i	−0.489407	+	0.505838i
$688$		0			0
$689$		−	1.24685e11i		−	0.553268i
$690$		0			0
$691$	3.41253e11			1.49680			0.748401	−	0.663246i	$-0.230822\pi$
$691$	3.41253e11			1.49680			0.748401	+	0.663246i	$0.230822\pi$
$692$		0			0
$693$	−8.64858e9	−	2.61861e11i	−0.0374983	−	1.13537i
$694$		0			0
$695$		0			0
$696$		0			0
$697$			4.51130e10i			0.191148i
$698$		0			0
$699$	−1.42417e11	+	1.47199e11i	−0.596560	+	0.616588i
$700$		0			0
$701$			3.40014e11i			1.40807i	0.710165	+	0.704036i	$0.248620\pi$
$701$			3.40014e11i			1.40807i	−0.710165	+	0.704036i	$0.751380\pi$
$702$		0			0
$703$			8.03029e7i			0.000328783i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	3.61073e11			1.44517
$708$		0			0
$709$	2.31712e11			0.916987			0.458493	−	0.888698i	$-0.348389\pi$
$709$	2.31712e11			0.916987			0.458493	+	0.888698i	$0.348389\pi$
$710$		0			0
$711$	7.55162e9	+	2.28647e11i	0.0295503	+	0.894721i
$712$		0			0
$713$	1.92572e11			0.745134
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−1.02930e11	−	9.95864e10i	−0.389461	−	0.376811i
$718$		0			0
$719$			1.94401e11i			0.727417i	0.931513	+	0.363709i	$0.118490\pi$
$719$			1.94401e11i			0.727417i	−0.931513	+	0.363709i	$0.881510\pi$
$720$		0			0
$721$	5.42526e10			0.200761
$722$		0			0
$723$	−8.74236e10	+	9.03586e10i	−0.319945	+	0.330687i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	5.06699e10i		−	0.181390i	−0.995879	−	0.0906948i	$-0.971091\pi$
$727$		−	5.06699e10i		−	0.181390i	0.995879	−	0.0906948i	$-0.0289088\pi$
$728$		0			0
$729$	2.79278e10	+	2.81045e11i	0.0988843	+	0.995099i
$730$		0			0
$731$		−	2.61897e10i		−	0.0917195i
$732$		0			0
$733$			3.08608e11i			1.06903i	0.845158	+	0.534517i	$0.179506\pi$
$733$			3.08608e11i			1.06903i	−0.845158	+	0.534517i	$0.820494\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−4.51902e11			−1.53170
$738$		0			0
$739$	−5.24612e11			−1.75898			−0.879489	−	0.475920i	$-0.842115\pi$
$739$	−5.24612e11			−1.75898			−0.879489	+	0.475920i	$0.842115\pi$
$740$		0			0
$741$	−1.87568e11	−	1.81475e11i	−0.622137	−	0.601928i
$742$		0			0
$743$	4.07793e11			1.33809			0.669043	−	0.743223i	$-0.266704\pi$
$743$	4.07793e11			1.33809			0.669043	+	0.743223i	$0.266704\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	1.01232e10	+	3.06508e11i	0.0325112	+	0.984372i
$748$		0			0
$749$		−	1.20289e11i		−	0.382208i
$750$		0			0
$751$	−2.27808e11			−0.716160			−0.358080	−	0.933691i	$-0.616568\pi$
$751$	−2.27808e11			−0.716160			−0.358080	+	0.933691i	$0.616568\pi$
$752$		0			0
$753$	−2.81938e11	−	2.72780e11i	−0.876946	−	0.848461i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	3.82063e11i		−	1.16346i	−0.813382	−	0.581730i	$-0.802376\pi$
$757$		−	3.82063e11i		−	1.16346i	0.813382	−	0.581730i	$-0.197624\pi$
$758$		0			0
$759$	2.08848e11	+	2.02064e11i	0.629309	+	0.608867i
$760$		0			0
$761$		−	3.24968e11i		−	0.968950i	−0.874805	−	0.484475i	$-0.839011\pi$
$761$		−	3.24968e11i		−	0.968950i	0.874805	−	0.484475i	$-0.160989\pi$
$762$		0			0
$763$			1.03091e11i			0.304175i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−3.82567e11			−1.10542
$768$		0			0
$769$	5.24436e11			1.49964			0.749821	−	0.661641i	$-0.230140\pi$
$769$	5.24436e11			1.49964			0.749821	+	0.661641i	$0.230140\pi$
$770$		0			0
$771$	−3.92677e10	+	4.05860e10i	−0.111127	+	0.114857i
$772$		0			0
$773$	3.98964e10			0.111742			0.0558710	−	0.998438i	$-0.482206\pi$
$773$	3.98964e10			0.111742			0.0558710	+	0.998438i	$0.482206\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	5.30503e7	−	5.48313e7i	0.000145547	−	0.000150433i
$778$		0			0
$779$			6.54115e11i			1.77625i
$780$		0			0
$781$	4.64262e11			1.24784
$782$		0			0
$783$	1.30978e11	−	1.44638e11i	0.348458	−	0.384801i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			1.80418e10i			0.0470307i	0.999723	+	0.0235154i	$0.00748586\pi$
$787$			1.80418e10i			0.0470307i	−0.999723	+	0.0235154i	$0.992514\pi$
$788$		0			0
$789$	2.39765e11	−	2.47814e11i	0.618696	−	0.639468i
$790$		0			0
$791$		−	3.33451e11i		−	0.851776i
$792$		0			0
$793$		−	1.47280e11i		−	0.372436i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−6.64874e11			−1.64781			−0.823903	−	0.566730i	$-0.808208\pi$
$797$	−6.64874e11			−1.64781			−0.823903	+	0.566730i	$0.808208\pi$
$798$		0			0
$799$	−7.87343e10			−0.193187
$800$		0			0
$801$	1.58829e11	−	5.24569e9i	0.385832	−	0.0127430i
$802$		0			0
$803$	−3.09198e11			−0.743661
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−1.96386e11	−	1.90007e11i	−0.463038	−	0.447998i
$808$		0			0
$809$		−	5.10440e11i		−	1.19165i	−0.803113	−	0.595827i	$-0.796824\pi$
$809$		−	5.10440e11i		−	1.19165i	0.803113	−	0.595827i	$-0.203176\pi$
$810$		0			0
$811$	5.30337e11			1.22594			0.612969	−	0.790107i	$-0.289975\pi$
$811$	5.30337e11			1.22594			0.612969	+	0.790107i	$0.289975\pi$
$812$		0			0
$813$	3.70590e11	−	3.83032e11i	0.848266	−	0.876744i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	3.79738e11i		−	0.852306i
$818$		0			0
$819$	8.18501e9	+	2.47825e11i	0.0181921	+	0.550820i
$820$		0			0
$821$			1.20378e11i			0.264956i	0.991186	+	0.132478i	$0.0422934\pi$
$821$			1.20378e11i			0.264956i	−0.991186	+	0.132478i	$0.957707\pi$
$822$		0			0
$823$			8.09679e11i			1.76487i	0.470432	+	0.882436i	$0.344098\pi$
$823$			8.09679e11i			1.76487i	−0.470432	+	0.882436i	$0.655902\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	5.02238e11			1.07371			0.536856	−	0.843674i	$-0.319612\pi$
$827$	5.02238e11			1.07371			0.536856	+	0.843674i	$0.319612\pi$
$828$		0			0
$829$	−5.45793e11			−1.15561			−0.577803	−	0.816176i	$-0.696090\pi$
$829$	−5.45793e11			−1.15561			−0.577803	+	0.816176i	$0.696090\pi$
$830$		0			0
$831$	2.94109e9	+	2.84556e9i	0.00616743	+	0.00596710i
$832$		0			0
$833$	−6.68726e9			−0.0138889
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−3.67840e11	−	3.33099e11i	−0.749474	−	0.678689i
$838$		0			0
$839$			7.36835e11i			1.48704i	0.668715	+	0.743519i	$0.266845\pi$
$839$			7.36835e11i			1.48704i	−0.668715	+	0.743519i	$0.733155\pi$
$840$		0			0
$841$	3.65433e11			0.730506
$842$		0			0
$843$	−2.91555e11	−	2.82084e11i	−0.577311	−	0.558558i
$844$		0			0
$845$		0			0
$846$		0			0
$847$			2.02635e11i			0.393714i
$848$		0			0
$849$	−5.83476e11	−	5.64524e11i	−1.12303	−	1.08655i
$850$		0			0
$851$			8.46210e7i			0.000161347i
$852$		0			0
$853$		−	4.95665e11i		−	0.936250i	−0.883662	−	0.468125i	$-0.844930\pi$
$853$		−	4.95665e11i		−	0.936250i	0.883662	−	0.468125i	$-0.155070\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.67085e11			0.309753			0.154876	−	0.987934i	$-0.450502\pi$
$857$	1.67085e11			0.309753			0.154876	+	0.987934i	$0.450502\pi$
$858$		0			0
$859$	−6.89277e11			−1.26596			−0.632982	−	0.774167i	$-0.718169\pi$
$859$	−6.89277e11			−1.26596			−0.632982	+	0.774167i	$0.718169\pi$
$860$		0			0
$861$	4.32126e11	−	4.46633e11i	0.786317	−	0.812716i
$862$		0			0
$863$	−6.64209e11			−1.19746			−0.598731	−	0.800950i	$-0.704328\pi$
$863$	−6.64209e11			−1.19746			−0.598731	+	0.800950i	$0.704328\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	3.82631e11	−	3.95476e11i	0.677179	−	0.699913i
$868$		0			0
$869$		−	6.06584e11i		−	1.06368i
$870$		0			0
$871$	4.27680e11			0.743099
$872$		0			0
$873$	−1.70133e11	+	5.61904e9i	−0.292908	+	0.00967398i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			3.97348e11i			0.671695i	0.941916	+	0.335848i	$0.109023\pi$
$877$			3.97348e11i			0.671695i	−0.941916	+	0.335848i	$0.890977\pi$
$878$		0			0
$879$	−4.80292e11	+	4.96417e11i	−0.804545	+	0.831556i
$880$		0			0
$881$		−	7.33038e11i		−	1.21681i	−0.793627	−	0.608405i	$-0.791810\pi$
$881$		−	7.33038e11i		−	1.21681i	0.793627	−	0.608405i	$-0.208190\pi$
$882$		0			0
$883$		−	1.10665e12i		−	1.82040i	−0.414170	−	0.910199i	$-0.635928\pi$
$883$		−	1.10665e12i		−	1.82040i	0.414170	−	0.910199i	$-0.364072\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−7.21287e11			−1.16524			−0.582618	−	0.812746i	$-0.697972\pi$
$887$	−7.21287e11			−1.16524			−0.582618	+	0.812746i	$0.697972\pi$
$888$		0			0
$889$	7.17198e11			1.14824
$890$		0			0
$891$	−4.94116e10	−	7.47224e11i	−0.0784004	−	1.18561i
$892$		0			0
$893$	−1.14161e12			−1.79519
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−1.97654e11	−	1.91234e11i	−0.305306	−	0.295389i
$898$		0			0
$899$			3.42854e11i			0.524892i
$900$		0			0
$901$	−1.02219e11			−0.155107
$902$		0			0
$903$	−2.50865e11	+	2.59287e11i	−0.377302	+	0.389969i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	2.73405e11i		−	0.403996i	−0.979386	−	0.201998i	$-0.935257\pi$
$907$		−	2.73405e11i		−	0.403996i	0.979386	−	0.201998i	$-0.0647434\pi$
$908$		0			0
$909$	1.03145e12	−	3.40662e10i	1.51076	−	0.0498963i
$910$		0			0
$911$		−	5.29336e11i		−	0.768525i	−0.923224	−	0.384263i	$-0.874456\pi$
$911$		−	5.29336e11i		−	0.768525i	0.923224	−	0.384263i	$-0.125544\pi$
$912$		0			0
$913$		−	8.13143e11i		−	1.17026i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.02787e12			1.45365
$918$		0			0
$919$	−3.37368e11			−0.472978			−0.236489	−	0.971634i	$-0.575997\pi$
$919$	−3.37368e11			−0.472978			−0.236489	+	0.971634i	$0.575997\pi$
$920$		0			0
$921$	9.86283e10	+	9.54246e10i	0.137076	+	0.132624i
$922$		0			0
$923$	−4.39378e11			−0.605384
$924$		0			0
$925$		0			0
$926$		0			0
$927$	1.54980e11	−	5.11858e9i	0.209873	−	0.00693156i
$928$		0			0
$929$		−	3.33858e11i		−	0.448229i	−0.974563	−	0.224114i	$-0.928051\pi$
$929$		−	3.33858e11i		−	0.448229i	0.974563	−	0.224114i	$-0.0719489\pi$
$930$		0			0
$931$	−9.69619e10			−0.129063
$932$		0			0
$933$	2.99749e11	+	2.90013e11i	0.395577	+	0.382728i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	5.13902e11i		−	0.666688i	−0.942805	−	0.333344i	$-0.891823\pi$
$937$		−	5.13902e11i		−	0.666688i	0.942805	−	0.333344i	$-0.108177\pi$
$938$		0			0
$939$	−4.84510e11	−	4.68772e11i	−0.623218	−	0.602975i
$940$		0			0
$941$		−	6.37307e11i		−	0.812812i	−0.913693	−	0.406406i	$-0.866782\pi$
$941$		−	6.37307e11i		−	0.812812i	0.913693	−	0.406406i	$-0.133218\pi$
$942$		0			0
$943$			6.89288e11i			0.871674i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1.13296e12			1.40869			0.704346	−	0.709857i	$-0.251240\pi$
$947$	1.13296e12			1.40869			0.704346	+	0.709857i	$0.251240\pi$
$948$		0			0
$949$	2.92625e11			0.360784
$950$		0			0
$951$	9.44150e11	−	9.75848e11i	1.15430	−	1.19305i
$952$		0			0
$953$	1.04326e11			0.126480			0.0632400	−	0.997998i	$-0.479857\pi$
$953$	1.04326e11			0.126480			0.0632400	+	0.997998i	$0.479857\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−3.59755e11	+	3.71832e11i	−0.428902	+	0.443302i
$958$		0			0
$959$			1.14393e12i			1.35246i
$960$		0			0
$961$	1.90453e10			0.0223303
$962$		0			0
$963$	−1.13489e10	−	3.43622e11i	−0.0131962	−	0.399555i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	4.71576e11i		−	0.539319i	−0.962956	−	0.269659i	$-0.913089\pi$
$967$		−	4.71576e11i		−	0.539319i	0.962956	−	0.269659i	$-0.0869111\pi$
$968$		0			0
$969$	−1.48777e11	+	1.53772e11i	−0.168749	+	0.174414i
$970$		0			0
$971$		−	4.60220e11i		−	0.517713i	−0.965916	−	0.258856i	$-0.916654\pi$
$971$		−	4.60220e11i		−	0.517713i	0.965916	−	0.258856i	$-0.0833456\pi$
$972$		0			0
$973$		−	1.22546e12i		−	1.36725i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.41831e12			−1.55665			−0.778327	−	0.627860i	$-0.783931\pi$
$977$	−1.41831e12			−1.55665			−0.778327	+	0.627860i	$0.783931\pi$
$978$		0			0
$979$	−4.21360e11			−0.458694
$980$		0			0
$981$	9.72638e9	+	2.94494e11i	0.0105021	+	0.317981i
$982$		0			0
$983$	1.48643e12			1.59196			0.795978	−	0.605326i	$-0.206957\pi$
$983$	1.48643e12			1.59196			0.795978	+	0.605326i	$0.206957\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	7.79496e11	+	7.54177e11i	0.821382	+	0.794702i
$988$		0			0
$989$		−	4.00157e11i		−	0.418259i
$990$		0			0
$991$	5.45711e11			0.565807			0.282903	−	0.959148i	$-0.408702\pi$
$991$	5.45711e11			0.565807			0.282903	+	0.959148i	$0.408702\pi$
$992$		0			0
$993$	−1.22686e12	+	1.26804e12i	−1.26182	+	1.30418i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	1.51985e12i		−	1.53823i	−0.639112	−	0.769114i	$-0.720698\pi$
$997$		−	1.51985e12i		−	1.53823i	0.639112	−	0.769114i	$-0.279302\pi$
$998$		0			0
$999$	1.46372e8	−	1.61638e8i	0.000146959	−	0.000162286i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.9.b.e.149.8		20
3.2	odd	2	inner	300.9.b.e.149.14		20
5.2	odd	4		300.9.g.g.101.8	yes	10
5.3	odd	4		300.9.g.e.101.3	✓	10
5.4	even	2	inner	300.9.b.e.149.13		20
15.2	even	4		300.9.g.g.101.7	yes	10
15.8	even	4		300.9.g.e.101.4	yes	10
15.14	odd	2	inner	300.9.b.e.149.7		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.9.b.e.149.7		20	15.14	odd	2	inner
300.9.b.e.149.8		20	1.1	even	1	trivial
300.9.b.e.149.13		20	5.4	even	2	inner
300.9.b.e.149.14		20	3.2	odd	2	inner
300.9.g.e.101.3	✓	10	5.3	odd	4
300.9.g.e.101.4	yes	10	15.8	even	4
300.9.g.g.101.7	yes	10	15.2	even	4
300.9.g.g.101.8	yes	10	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	300.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-56.3224 + 58.2133i) q^{3} -2295.51i q^{7} +(-216.575 - 6557.42i) q^{9} +17396.4i q^{11} -16463.9i q^{13} -13497.4 q^{17} -195706. q^{19} +(133629. + 129288. i) q^{21} -206229. q^{23} +(393927. + 356722. i) q^{27} -367170. i q^{29} -933775. q^{31} +(-1.01270e6 - 979805. i) q^{33} -410.325i q^{37} +(958418. + 927287. i) q^{39} -3.34234e6i q^{41} +1.94035e6i q^{43} +5.83328e6 q^{47} +495447. q^{49} +(760207. - 785730. i) q^{51} +7.57320e6 q^{53} +(1.10226e7 - 1.13927e7i) q^{57} -2.32367e7i q^{59} +8.94563e6 q^{61} +(-1.50526e7 + 497149. i) q^{63} +2.59768e7i q^{67} +(1.16153e7 - 1.20053e7i) q^{69} -2.66873e7i q^{71} +1.77737e7i q^{73} +3.99335e7 q^{77} -3.48684e7 q^{79} +(-4.29529e7 + 2.84034e6i) q^{81} -4.67421e7 q^{83} +(2.13742e7 + 2.06799e7i) q^{87} +2.42212e7i q^{89} -3.77930e7 q^{91} +(5.25925e7 - 5.43581e7i) q^{93} -2.59451e7i q^{97} +(1.14075e8 - 3.76761e6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 24910 q^{9} + 752060 q^{19} - 285080 q^{21} - 821720 q^{31} - 7055740 q^{39} - 27783060 q^{49} - 1289490 q^{51} - 102747080 q^{61} + 25125180 q^{69} - 72422800 q^{79} - 151395550 q^{81} + 129038960 q^{91}+ \cdots - 104585010 q^{99}+O(q^{100}) \) 20 * q + 24910 * q^9 + 752060 * q^19 - 285080 * q^21 - 821720 * q^31 - 7055740 * q^39 - 27783060 * q^49 - 1289490 * q^51 - 102747080 * q^61 + 25125180 * q^69 - 72422800 * q^79 - 151395550 * q^81 + 129038960 * q^91 - 104585010 * q^99

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