LMFDB - Embedded newform 300.9.g.h.101.14

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 7378 x^{14} + 23156928 x^{12} - 101588726286 x^{10} + \cdots + 34\!\cdots\!81 $ x^16 - 7378x^14 + 23156928x^12 - 101588726286x^10 + 1484695155668670x^8 - 4373061557178808206x^6 + 42910255095788484704448x^4 - 588516817021165377771877458*x^2 + 3433683820292512484657849089281
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{38}\cdot 3^{20}\cdot 5^{18} $
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			101.14
Root			$-74.2942 - 32.2702i$ of defining polynomial
Character	$\chi$	$=$	300.101
Dual form			300.9.g.h.101.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(74.2942 + 32.2702i) q^{3} -4017.33 q^{7} +(4478.26 + 4794.99i) q^{9} +O(q^{10})$	$q+(74.2942 + 32.2702i) q^{3} -4017.33 q^{7} +(4478.26 + 4794.99i) q^{9} +11332.6i q^{11} +14669.4 q^{13} -121646. i q^{17} -22809.0 q^{19} +(-298464. - 129640. i) q^{21} -452115. i q^{23} +(177974. + 500754. i) q^{27} +925775. i q^{29} -1.24456e6 q^{31} +(-365706. + 841948. i) q^{33} +9962.64 q^{37} +(1.08985e6 + 473386. i) q^{39} -3.24930e6i q^{41} +986757. q^{43} +749904. i q^{47} +1.03741e7 q^{49} +(3.92556e6 - 9.03762e6i) q^{51} +3.98864e6i q^{53} +(-1.69458e6 - 736052. i) q^{57} -1.17788e7i q^{59} +2.04000e7 q^{61} +(-1.79906e7 - 1.92630e7i) q^{63} +1.35287e7 q^{67} +(1.45898e7 - 3.35895e7i) q^{69} +1.35989e7i q^{71} +5.39793e7 q^{73} -4.55268e7i q^{77} -1.08987e7 q^{79} +(-2.93705e6 + 4.29464e7i) q^{81} +5.54029e7i q^{83} +(-2.98750e7 + 6.87797e7i) q^{87} -8.17147e7i q^{89} -5.89319e7 q^{91} +(-9.24638e7 - 4.01623e7i) q^{93} +1.22851e8 q^{97} +(-5.43397e7 + 5.07504e7i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 14756 q^{9}+O(q^{10}) $ 16 * q + 14756 * q^9	$ 16 q + 14756 q^{9} + 48024 q^{19} + 4724 q^{21} + 470104 q^{31} + 2849664 q^{39} + 19816920 q^{49} + 5026040 q^{51} + 18849944 q^{61} + 38669180 q^{69} + 90778632 q^{79} + 16242056 q^{81} + 22375296 q^{91} - 16968560 q^{99}+O(q^{100}) $ 16 * q + 14756 * q^9 + 48024 * q^19 + 4724 * q^21 + 470104 * q^31 + 2849664 * q^39 + 19816920 * q^49 + 5026040 * q^51 + 18849944 * q^61 + 38669180 * q^69 + 90778632 * q^79 + 16242056 * q^81 + 22375296 * q^91 - 16968560 * q^99

Display 100 coefficients 10 coefficients

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
$3$	74.2942	+	32.2702i	0.917213	+	0.398398i
$3$	74.2942	+	32.2702i	0.917213	+	0.398398i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−4017.33			−1.67319			−0.836594	−	0.547823i	$-0.815457\pi$
$7$	−4017.33			−1.67319			−0.836594	+	0.547823i	$0.815457\pi$
$8$		0			0
$9$	4478.26	+	4794.99i	0.682558	+	0.730831i
$10$		0			0
$11$			11332.6i			0.774033i	0.922073	+	0.387016i	$0.126494\pi$
$11$			11332.6i			0.774033i	−0.922073	+	0.387016i	$0.873506\pi$
$12$		0			0
$13$	14669.4			0.513618			0.256809	−	0.966462i	$-0.417329\pi$
$13$	14669.4			0.513618			0.256809	+	0.966462i	$0.417329\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	121646.i		−	1.45648i	−0.685324	−	0.728238i	$-0.740339\pi$
$17$		−	121646.i		−	1.45648i	0.685324	−	0.728238i	$-0.259661\pi$
$18$		0			0
$19$	−22809.0			−0.175022			−0.0875109	−	0.996164i	$-0.527891\pi$
$19$	−22809.0			−0.175022			−0.0875109	+	0.996164i	$0.527891\pi$
$20$		0			0
$21$	−298464.	−	129640.i	−1.53467	−	0.666595i
$22$		0			0
$23$		−	452115.i		−	1.61561i	−0.589448	−	0.807806i	$-0.700655\pi$
$23$		−	452115.i		−	1.61561i	0.589448	−	0.807806i	$-0.299345\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	177974.	+	500754.i	0.334889	+	0.942258i
$28$		0			0
$29$			925775.i			1.30892i	0.756096	+	0.654460i	$0.227104\pi$
$29$			925775.i			1.30892i	−0.756096	+	0.654460i	$0.772896\pi$
$30$		0			0
$31$	−1.24456e6			−1.34763			−0.673814	−	0.738901i	$-0.735345\pi$
$31$	−1.24456e6			−1.34763			−0.673814	+	0.738901i	$0.735345\pi$
$32$		0			0
$33$	−365706.	+	841948.i	−0.308373	+	0.709953i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	9962.64			0.00531579			0.00265789	−	0.999996i	$-0.499154\pi$
$37$	9962.64			0.00531579			0.00265789	+	0.999996i	$0.499154\pi$
$38$		0			0
$39$	1.08985e6	+	473386.i	0.471097	+	0.204624i
$40$		0			0
$41$		−	3.24930e6i		−	1.14988i	−0.818194	−	0.574942i	$-0.805024\pi$
$41$		−	3.24930e6i		−	1.14988i	0.818194	−	0.574942i	$-0.194976\pi$
$42$		0			0
$43$	986757.			0.288627			0.144313	−	0.989532i	$-0.453903\pi$
$43$	986757.			0.288627			0.144313	+	0.989532i	$0.453903\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			749904.i			0.153679i	0.997043	+	0.0768395i	$0.0244829\pi$
$47$			749904.i			0.153679i	−0.997043	+	0.0768395i	$0.975517\pi$
$48$		0			0
$49$	1.03741e7			1.79956
$50$		0			0
$51$	3.92556e6	−	9.03762e6i	0.580257	−	1.33590i
$52$		0			0
$53$			3.98864e6i			0.505500i	0.967532	+	0.252750i	$0.0813350\pi$
$53$			3.98864e6i			0.505500i	−0.967532	+	0.252750i	$0.918665\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−1.69458e6	−	736052.i	−0.160532	−	0.0697283i
$58$		0			0
$59$		−	1.17788e7i		−	0.972061i	−0.873942	−	0.486030i	$-0.838444\pi$
$59$		−	1.17788e7i		−	0.972061i	0.873942	−	0.486030i	$-0.161556\pi$
$60$		0			0
$61$	2.04000e7			1.47337			0.736684	−	0.676238i	$-0.236391\pi$
$61$	2.04000e7			1.47337			0.736684	+	0.676238i	$0.236391\pi$
$62$		0			0
$63$	−1.79906e7	−	1.92630e7i	−1.14205	−	1.22282i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	1.35287e7			0.671363			0.335682	−	0.941976i	$-0.391033\pi$
$67$	1.35287e7			0.671363			0.335682	+	0.941976i	$0.391033\pi$
$68$		0			0
$69$	1.45898e7	−	3.35895e7i	0.643657	−	1.48186i
$70$		0			0
$71$			1.35989e7i			0.535145i	0.963538	+	0.267573i	$0.0862215\pi$
$71$			1.35989e7i			0.535145i	−0.963538	+	0.267573i	$0.913778\pi$
$72$		0			0
$73$	5.39793e7			1.90080			0.950399	−	0.311035i	$-0.100676\pi$
$73$	5.39793e7			1.90080			0.950399	+	0.311035i	$0.100676\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	4.55268e7i		−	1.29510i
$78$		0			0
$79$	−1.08987e7			−0.279811			−0.139906	−	0.990165i	$-0.544680\pi$
$79$	−1.08987e7			−0.279811			−0.139906	+	0.990165i	$0.544680\pi$
$80$		0			0
$81$	−2.93705e6	+	4.29464e7i	−0.0682292	+	0.997670i
$82$		0			0
$83$			5.54029e7i			1.16740i	0.811969	+	0.583701i	$0.198396\pi$
$83$			5.54029e7i			1.16740i	−0.811969	+	0.583701i	$0.801604\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−2.98750e7	+	6.87797e7i	−0.521471	+	1.20056i
$88$		0			0
$89$		−	8.17147e7i		−	1.30239i	−0.758912	−	0.651194i	$-0.774269\pi$
$89$		−	8.17147e7i		−	1.30239i	0.758912	−	0.651194i	$-0.225731\pi$
$90$		0			0
$91$	−5.89319e7			−0.859380
$92$		0			0
$93$	−9.24638e7	−	4.01623e7i	−1.23606	−	0.536892i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.22851e8			1.38768			0.693841	−	0.720128i	$-0.255917\pi$
$97$	1.22851e8			1.38768			0.693841	+	0.720128i	$0.255917\pi$
$98$		0			0
$99$	−5.43397e7	+	5.07504e7i	−0.565687	+	0.528322i
$100$		0			0
$101$		−	7.32849e6i		−	0.0704254i	−0.999380	−	0.0352127i	$-0.988789\pi$
$101$		−	7.32849e6i		−	0.0704254i	0.999380	−	0.0352127i	$-0.0112109\pi$
$102$		0			0
$103$	9.35738e7			0.831391			0.415696	−	0.909504i	$-0.363538\pi$
$103$	9.35738e7			0.831391			0.415696	+	0.909504i	$0.363538\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	1.65967e8i		−	1.26615i	−0.774090	−	0.633076i	$-0.781792\pi$
$107$		−	1.65967e8i		−	1.26615i	0.774090	−	0.633076i	$-0.218208\pi$
$108$		0			0
$109$	2.76976e7			0.196217			0.0981085	−	0.995176i	$-0.468721\pi$
$109$	2.76976e7			0.196217			0.0981085	+	0.995176i	$0.468721\pi$
$110$		0			0
$111$	740167.	+	321497.i	0.00487571	+	0.00211780i
$112$		0			0
$113$			4.57224e7i			0.280424i	0.990122	+	0.140212i	$0.0447784\pi$
$113$			4.57224e7i			0.280424i	−0.990122	+	0.140212i	$0.955222\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	6.56936e7	+	7.03397e7i	0.350574	+	0.375368i
$118$		0			0
$119$			4.88693e8i			2.43696i
$120$		0			0
$121$	8.59308e7			0.400873
$122$		0			0
$123$	1.04856e8	−	2.41404e8i	0.458112	−	1.05469i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	2.26014e8			0.868802			0.434401	−	0.900720i	$-0.356960\pi$
$127$	2.26014e8			0.868802			0.434401	+	0.900720i	$0.356960\pi$
$128$		0			0
$129$	7.33104e7	+	3.18429e7i	0.264732	+	0.114988i
$130$		0			0
$131$		−	5.76919e8i		−	1.95898i	−0.201494	−	0.979490i	$-0.564580\pi$
$131$		−	5.76919e8i		−	1.95898i	0.201494	−	0.979490i	$-0.435420\pi$
$132$		0			0
$133$	9.16313e7			0.292844
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	3.85337e8i		−	1.09385i	−0.837181	−	0.546926i	$-0.815798\pi$
$137$		−	3.85337e8i		−	1.09385i	0.837181	−	0.546926i	$-0.184202\pi$
$138$		0			0
$139$	−2.39587e8			−0.641807			−0.320903	−	0.947112i	$-0.603987\pi$
$139$	−2.39587e8			−0.641807			−0.320903	+	0.947112i	$0.603987\pi$
$140$		0			0
$141$	−2.41996e7	+	5.57135e7i	−0.0612254	+	0.140956i
$142$		0			0
$143$			1.66243e8i			0.397557i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	7.70736e8	+	3.34775e8i	1.65058	+	0.716941i
$148$		0			0
$149$		−	4.07729e6i		−	0.00827230i	−0.999991	−	0.00413615i	$-0.998683\pi$
$149$		−	4.07729e6i		−	0.00827230i	0.999991	−	0.00413615i	$-0.00131658\pi$
$150$		0			0
$151$	−2.75381e8			−0.529696			−0.264848	−	0.964290i	$-0.585322\pi$
$151$	−2.75381e8			−0.529696			−0.264848	+	0.964290i	$0.585322\pi$
$152$		0			0
$153$	5.83292e8	−	5.44764e8i	1.06444	−	0.994130i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	6.32870e8			1.04164			0.520818	−	0.853668i	$-0.325627\pi$
$157$	6.32870e8			1.04164			0.520818	+	0.853668i	$0.325627\pi$
$158$		0			0
$159$	−1.28714e8	+	2.96333e8i	−0.201390	+	0.463651i
$160$		0			0
$161$			1.81629e9i			2.70322i
$162$		0			0
$163$	−6.06865e8			−0.859690			−0.429845	−	0.902903i	$-0.641432\pi$
$163$	−6.06865e8			−0.859690			−0.429845	+	0.902903i	$0.641432\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	9.43132e8i		−	1.21257i	−0.795248	−	0.606285i	$-0.792659\pi$
$167$		−	9.43132e8i		−	1.21257i	0.795248	−	0.606285i	$-0.207341\pi$
$168$		0			0
$169$	−6.00538e8			−0.736197
$170$		0			0
$171$	−1.02145e8	−	1.09369e8i	−0.119463	−	0.127911i
$172$		0			0
$173$			7.01076e8i			0.782673i	0.920248	+	0.391337i	$0.127987\pi$
$173$			7.01076e8i			0.782673i	−0.920248	+	0.391337i	$0.872013\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	3.80105e8	−	8.75098e8i	0.387267	−	0.891586i
$178$		0			0
$179$		−	1.09833e9i		−	1.06984i	−0.844901	−	0.534922i	$-0.820341\pi$
$179$		−	1.09833e9i		−	1.06984i	0.844901	−	0.534922i	$-0.179659\pi$
$180$		0			0
$181$	−5.20407e8			−0.484874			−0.242437	−	0.970167i	$-0.577947\pi$
$181$	−5.20407e8			−0.484874			−0.242437	+	0.970167i	$0.577947\pi$
$182$		0			0
$183$	1.51560e9	+	6.58313e8i	1.35139	+	0.586987i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	1.37857e9			1.12736
$188$		0			0
$189$	−7.14978e8	−	2.01169e9i	−0.560332	−	1.57657i
$190$		0			0
$191$		−	9.77875e8i		−	0.734767i	−0.930069	−	0.367384i	$-0.880254\pi$
$191$		−	9.77875e8i		−	0.734767i	0.930069	−	0.367384i	$-0.119746\pi$
$192$		0			0
$193$	7.30466e8			0.526467			0.263233	−	0.964732i	$-0.415211\pi$
$193$	7.30466e8			0.526467			0.263233	+	0.964732i	$0.415211\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			4.83486e8i			0.321010i	0.987035	+	0.160505i	$0.0513123\pi$
$197$			4.83486e8i			0.321010i	−0.987035	+	0.160505i	$0.948688\pi$
$198$		0			0
$199$	2.43526e9			1.55286			0.776431	−	0.630203i	$-0.217028\pi$
$199$	2.43526e9			1.55286			0.776431	+	0.630203i	$0.217028\pi$
$200$		0			0
$201$	1.00511e9	+	4.36575e8i	0.615783	+	0.267470i
$202$		0			0
$203$		−	3.71914e9i		−	2.19007i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	2.16788e9	−	2.02469e9i	1.18074	−	1.10275i
$208$		0			0
$209$		−	2.58486e8i		−	0.135473i
$210$		0			0
$211$	3.19886e9			1.61386			0.806929	−	0.590649i	$-0.201128\pi$
$211$	3.19886e9			1.61386			0.806929	+	0.590649i	$0.201128\pi$
$212$		0			0
$213$	−4.38841e8	+	1.01032e9i	−0.213201	+	0.490842i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	4.99981e9			2.25483
$218$		0			0
$219$	4.01035e9	+	1.74192e9i	1.74344	+	0.757274i
$220$		0			0
$221$		−	1.78448e9i		−	0.748072i
$222$		0			0
$223$	1.61131e9			0.651569			0.325785	−	0.945444i	$-0.394372\pi$
$223$	1.61131e9			0.651569			0.325785	+	0.945444i	$0.394372\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			5.21746e8i			0.196497i	0.995162	+	0.0982485i	$0.0313240\pi$
$227$			5.21746e8i			0.196497i	−0.995162	+	0.0982485i	$0.968676\pi$
$228$		0			0
$229$	−5.87443e8			−0.213611			−0.106805	−	0.994280i	$-0.534062\pi$
$229$	−5.87443e8			−0.213611			−0.106805	+	0.994280i	$0.534062\pi$
$230$		0			0
$231$	1.46916e9	−	3.38238e9i	0.515966	−	1.18788i
$232$		0			0
$233$		−	2.28192e9i		−	0.774241i	−0.922029	−	0.387120i	$-0.873470\pi$
$233$		−	2.28192e9i		−	0.774241i	0.922029	−	0.387120i	$-0.126530\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−8.09708e8	−	3.51703e8i	−0.256646	−	0.111476i
$238$		0			0
$239$		−	2.24611e9i		−	0.688398i	−0.938897	−	0.344199i	$-0.888151\pi$
$239$		−	2.24611e9i		−	0.688398i	0.938897	−	0.344199i	$-0.111849\pi$
$240$		0			0
$241$	−4.70089e9			−1.39352			−0.696758	−	0.717306i	$-0.745375\pi$
$241$	−4.70089e9			−1.39352			−0.696758	+	0.717306i	$0.745375\pi$
$242$		0			0
$243$	−1.60410e9	+	3.09589e9i	−0.460050	+	0.887893i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−3.34595e8			−0.0898943
$248$		0			0
$249$	−1.78787e9	+	4.11612e9i	−0.465091	+	1.07076i
$250$		0			0
$251$		−	3.14963e9i		−	0.793533i	−0.917920	−	0.396767i	$-0.870132\pi$
$251$		−	3.14963e9i		−	0.793533i	0.917920	−	0.396767i	$-0.129868\pi$
$252$		0			0
$253$	5.12364e9			1.25054
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	2.30731e9i		−	0.528899i	−0.964400	−	0.264449i	$-0.914810\pi$
$257$		−	2.30731e9i		−	0.528899i	0.964400	−	0.264449i	$-0.0851902\pi$
$258$		0			0
$259$	−4.00232e7			−0.00889432
$260$		0			0
$261$	−4.43908e9	+	4.14586e9i	−0.956600	+	0.893414i
$262$		0			0
$263$			1.60101e9i			0.334634i	0.985903	+	0.167317i	$0.0535104\pi$
$263$			1.60101e9i			0.334634i	−0.985903	+	0.167317i	$0.946490\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	2.63695e9	−	6.07093e9i	0.518869	−	1.19457i
$268$		0			0
$269$			9.07288e9i			1.73275i	0.499394	+	0.866375i	$0.333556\pi$
$269$			9.07288e9i			1.73275i	−0.499394	+	0.866375i	$0.666444\pi$
$270$		0			0
$271$	−4.37176e9			−0.810549			−0.405274	−	0.914195i	$-0.632824\pi$
$271$	−4.37176e9			−0.810549			−0.405274	+	0.914195i	$0.632824\pi$
$272$		0			0
$273$	−4.37830e9	−	1.90175e9i	−0.788234	−	0.342375i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	7.47715e9			1.27004			0.635020	−	0.772496i	$-0.280992\pi$
$277$	7.47715e9			1.27004			0.635020	+	0.772496i	$0.280992\pi$
$278$		0			0
$279$	−5.57348e9	−	5.96766e9i	−0.919834	−	0.984888i
$280$		0			0
$281$			8.81988e9i			1.41461i	0.706907	+	0.707306i	$0.250090\pi$
$281$			8.81988e9i			1.41461i	−0.706907	+	0.707306i	$0.749910\pi$
$282$		0			0
$283$	9.19993e7			0.0143430			0.00717148	−	0.999974i	$-0.497717\pi$
$283$	9.19993e7			0.0143430			0.00717148	+	0.999974i	$0.497717\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			1.30535e10i			1.92397i
$288$		0			0
$289$	−7.82208e9			−1.12132
$290$		0			0
$291$	9.12709e9	+	3.96442e9i	1.27280	+	0.552850i
$292$		0			0
$293$		−	2.23825e9i		−	0.303695i	−0.988404	−	0.151848i	$-0.951478\pi$
$293$		−	2.23825e9i		−	0.303695i	0.988404	−	0.151848i	$-0.0485223\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−5.67486e9	+	2.01691e9i	−0.729338	+	0.259215i
$298$		0			0
$299$		−	6.63227e9i		−	0.829807i
$300$		0			0
$301$	−3.96413e9			−0.482927
$302$		0			0
$303$	2.36492e8	−	5.44465e8i	0.0280573	−	0.0645950i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−8.85192e9			−0.996515			−0.498258	−	0.867029i	$-0.666027\pi$
$307$	−8.85192e9			−0.996515			−0.498258	+	0.867029i	$0.666027\pi$
$308$		0			0
$309$	6.95199e9	+	3.01965e9i	0.762562	+	0.331225i
$310$		0			0
$311$			1.13950e10i			1.21807i	0.793144	+	0.609034i	$0.208443\pi$
$311$			1.13950e10i			1.21807i	−0.793144	+	0.609034i	$0.791557\pi$
$312$		0			0
$313$	6.46179e9			0.673248			0.336624	−	0.941639i	$-0.390715\pi$
$313$	6.46179e9			0.673248			0.336624	+	0.941639i	$0.390715\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	1.61882e10i		−	1.60310i	−0.597928	−	0.801550i	$-0.704009\pi$
$317$		−	1.61882e10i		−	1.60310i	0.597928	−	0.801550i	$-0.295991\pi$
$318$		0			0
$319$	−1.04914e10			−1.01315
$320$		0			0
$321$	5.35578e9	−	1.23304e10i	0.504432	−	1.16133i
$322$		0			0
$323$			2.77463e9i			0.254915i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	2.05777e9	+	8.93809e8i	0.179973	+	0.0781725i
$328$		0			0
$329$		−	3.01261e9i		−	0.257134i
$330$		0			0
$331$	1.24611e10			1.03811			0.519055	−	0.854741i	$-0.326284\pi$
$331$	1.24611e10			1.03811			0.519055	+	0.854741i	$0.326284\pi$
$332$		0			0
$333$	4.46153e7	+	4.77707e7i	0.00362833	+	0.00388495i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.07049e10			−0.829971			−0.414985	−	0.909828i	$-0.636213\pi$
$337$	−1.07049e10			−0.829971			−0.414985	+	0.909828i	$0.636213\pi$
$338$		0			0
$339$	−1.47547e9	+	3.39691e9i	−0.111720	+	0.257208i
$340$		0			0
$341$		−	1.41041e10i		−	1.04311i
$342$		0			0
$343$	−1.85171e10			−1.33782
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	3.65014e9i		−	0.251763i	−0.992045	−	0.125881i	$-0.959824\pi$
$347$		−	3.65014e9i		−	0.251763i	0.992045	−	0.125881i	$-0.0401759\pi$
$348$		0			0
$349$	1.48706e10			1.00237			0.501184	−	0.865341i	$-0.332898\pi$
$349$	1.48706e10			1.00237			0.501184	+	0.865341i	$0.332898\pi$
$350$		0			0
$351$	2.61077e9	+	7.34578e9i	0.172005	+	0.483960i
$352$		0			0
$353$			1.09523e10i			0.705356i	0.935745	+	0.352678i	$0.114729\pi$
$353$			1.09523e10i			0.705356i	−0.935745	+	0.352678i	$0.885271\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−1.57702e10	+	3.63071e10i	−0.970880	+	2.23521i
$358$		0			0
$359$			1.21546e10i			0.731748i	0.930664	+	0.365874i	$0.119230\pi$
$359$			1.21546e10i			0.731748i	−0.930664	+	0.365874i	$0.880770\pi$
$360$		0			0
$361$	−1.64633e10			−0.969367
$362$		0			0
$363$	6.38416e9	+	2.77301e9i	0.367686	+	0.159707i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−2.71742e10			−1.49793			−0.748966	−	0.662609i	$-0.769449\pi$
$367$	−2.71742e10			−1.49793			−0.748966	+	0.662609i	$0.769449\pi$
$368$		0			0
$369$	1.55803e10	−	1.45512e10i	0.840372	−	0.784863i
$370$		0			0
$371$		−	1.60237e10i		−	0.845797i
$372$		0			0
$373$	−3.00563e10			−1.55274			−0.776372	−	0.630275i	$-0.782942\pi$
$373$	−3.00563e10			−1.55274			−0.776372	+	0.630275i	$0.782942\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			1.35806e10i			0.672285i
$378$		0			0
$379$	2.41230e7			0.00116916			0.000584581	−	1.00000i	$-0.499814\pi$
$379$	2.41230e7			0.00116916			0.000584581		1.00000i	$0.499814\pi$
$380$		0			0
$381$	1.67915e10	+	7.29353e9i	0.796876	+	0.346129i
$382$		0			0
$383$			1.28273e10i			0.596129i	0.954546	+	0.298065i	$0.0963411\pi$
$383$			1.28273e10i			0.596129i	−0.954546	+	0.298065i	$0.903659\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	4.41896e9	+	4.73149e9i	0.197004	+	0.210937i
$388$		0			0
$389$			1.34652e10i			0.588051i	0.955798	+	0.294025i	$0.0949950\pi$
$389$			1.34652e10i			0.588051i	−0.955798	+	0.294025i	$0.905005\pi$
$390$		0			0
$391$	−5.49981e10			−2.35310
$392$		0			0
$393$	1.86173e10	−	4.28618e10i	0.780454	−	1.79680i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−7.09168e9			−0.285488			−0.142744	−	0.989760i	$-0.545592\pi$
$397$	−7.09168e9			−0.285488			−0.142744	+	0.989760i	$0.545592\pi$
$398$		0			0
$399$	6.80767e9	+	2.95696e9i	0.268601	+	0.116669i
$400$		0			0
$401$		−	1.60936e10i		−	0.622409i	−0.950343	−	0.311205i	$-0.899268\pi$
$401$		−	1.60936e10i		−	0.622409i	0.950343	−	0.311205i	$-0.100732\pi$
$402$		0			0
$403$	−1.82570e10			−0.692165
$404$		0			0
$405$		0			0
$406$		0			0
$407$			1.12903e8i			0.00411459i
$408$		0			0
$409$	1.45211e10			0.518928			0.259464	−	0.965753i	$-0.416454\pi$
$409$	1.45211e10			0.518928			0.259464	+	0.965753i	$0.416454\pi$
$410$		0			0
$411$	1.24349e10	−	2.86283e10i	0.435789	−	1.00330i
$412$		0			0
$413$			4.73193e10i			1.62644i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−1.77999e10	−	7.73154e9i	−0.588673	−	0.255695i
$418$		0			0
$419$		−	3.60535e10i		−	1.16975i	−0.811125	−	0.584873i	$-0.801144\pi$
$419$		−	3.60535e10i		−	1.16975i	0.811125	−	0.584873i	$-0.198856\pi$
$420$		0			0
$421$	−8.78217e9			−0.279559			−0.139780	−	0.990183i	$-0.544639\pi$
$421$	−8.78217e9			−0.279559			−0.139780	+	0.990183i	$0.544639\pi$
$422$		0			0
$423$	−3.59578e9	+	3.35827e9i	−0.112313	+	0.104895i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−8.19535e10			−2.46522
$428$		0			0
$429$	−5.36470e9	+	1.23509e10i	−0.158386	+	0.364644i
$430$		0			0
$431$			2.58709e10i			0.749726i	0.927080	+	0.374863i	$0.122310\pi$
$431$			2.58709e10i			0.749726i	−0.927080	+	0.374863i	$0.877690\pi$
$432$		0			0
$433$	1.71429e10			0.487677			0.243838	−	0.969816i	$-0.421593\pi$
$433$	1.71429e10			0.487677			0.243838	+	0.969816i	$0.421593\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			1.03123e10i			0.282767i
$438$		0			0
$439$	−3.21261e10			−0.864967			−0.432483	−	0.901642i	$-0.642363\pi$
$439$	−3.21261e10			−0.864967			−0.432483	+	0.901642i	$0.642363\pi$
$440$		0			0
$441$	4.64580e10	+	4.97437e10i	1.22830	+	1.31518i
$442$		0			0
$443$		−	4.31206e10i		−	1.11962i	−0.828621	−	0.559810i	$-0.810874\pi$
$443$		−	4.31206e10i		−	1.11962i	0.828621	−	0.559810i	$-0.189126\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	1.31575e8	−	3.02919e8i	0.00329567	−	0.00758746i
$448$		0			0
$449$		−	5.97995e10i		−	1.47134i	−0.677341	−	0.735669i	$-0.736868\pi$
$449$		−	5.97995e10i		−	1.47134i	0.677341	−	0.735669i	$-0.263132\pi$
$450$		0			0
$451$	3.68231e10			0.890048
$452$		0			0
$453$	−2.04592e10	−	8.88662e9i	−0.485844	−	0.211030i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	9.83318e9			0.225439			0.112720	−	0.993627i	$-0.464044\pi$
$457$	9.83318e9			0.225439			0.112720	+	0.993627i	$0.464044\pi$
$458$		0			0
$459$	6.09149e10	−	2.16499e10i	1.37238	−	0.487758i
$460$		0			0
$461$		−	1.36056e9i		−	0.0301242i	−0.999887	−	0.0150621i	$-0.995205\pi$
$461$		−	1.36056e9i		−	0.0301242i	0.999887	−	0.0150621i	$-0.00479460\pi$
$462$		0			0
$463$	−4.49710e10			−0.978608			−0.489304	−	0.872113i	$-0.662749\pi$
$463$	−4.49710e10			−0.978608			−0.489304	+	0.872113i	$0.662749\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			2.97127e10i			0.624705i	0.949966	+	0.312353i	$0.101117\pi$
$467$			2.97127e10i			0.624705i	−0.949966	+	0.312353i	$0.898883\pi$
$468$		0			0
$469$	−5.43493e10			−1.12332
$470$		0			0
$471$	4.70186e10	+	2.04229e10i	0.955402	+	0.414986i
$472$		0			0
$473$			1.11825e10i			0.223407i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1.91255e10	+	1.78622e10i	−0.369435	+	0.345033i
$478$		0			0
$479$			7.97953e10i			1.51578i	0.652385	+	0.757888i	$0.273769\pi$
$479$			7.97953e10i			1.51578i	−0.652385	+	0.757888i	$0.726231\pi$
$480$		0			0
$481$	1.46146e8			0.00273028
$482$		0			0
$483$	−5.86122e10	+	1.34940e11i	−1.07696	+	2.47943i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	4.05153e10			0.720284			0.360142	−	0.932898i	$-0.382728\pi$
$487$	4.05153e10			0.720284			0.360142	+	0.932898i	$0.382728\pi$
$488$		0			0
$489$	−4.50866e10	−	1.95837e10i	−0.788519	−	0.342499i
$490$		0			0
$491$			1.34994e10i			0.232268i	0.993234	+	0.116134i	$0.0370502\pi$
$491$			1.34994e10i			0.232268i	−0.993234	+	0.116134i	$0.962950\pi$
$492$		0			0
$493$	1.12617e11			1.90641
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	5.46314e10i		−	0.895399i
$498$		0			0
$499$	1.82581e10			0.294478			0.147239	−	0.989101i	$-0.452961\pi$
$499$	1.82581e10			0.294478			0.147239	+	0.989101i	$0.452961\pi$
$500$		0			0
$501$	3.04351e10	−	7.00692e10i	0.483085	−	1.11218i
$502$		0			0
$503$		−	7.00331e10i		−	1.09403i	−0.837121	−	0.547017i	$-0.815763\pi$
$503$		−	7.00331e10i		−	1.09403i	0.837121	−	0.547017i	$-0.184237\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−4.46165e10	−	1.93795e10i	−0.675249	−	0.293299i
$508$		0			0
$509$			1.18160e10i			0.176034i	0.996119	+	0.0880172i	$0.0280530\pi$
$509$			1.18160e10i			0.176034i	−0.996119	+	0.0880172i	$0.971947\pi$
$510$		0			0
$511$	−2.16852e11			−3.18039
$512$		0			0
$513$	−4.05941e9	−	1.14217e10i	−0.0586129	−	0.164916i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−8.49837e9			−0.118953
$518$		0			0
$519$	−2.26239e10	+	5.20859e10i	−0.311815	+	0.717878i
$520$		0			0
$521$			1.02514e11i			1.39134i	0.718360	+	0.695671i	$0.244893\pi$
$521$			1.02514e11i			1.39134i	−0.718360	+	0.695671i	$0.755107\pi$
$522$		0			0
$523$	2.79035e10			0.372952			0.186476	−	0.982460i	$-0.440293\pi$
$523$	2.79035e10			0.372952			0.186476	+	0.982460i	$0.440293\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			1.51396e11i			1.96279i
$528$		0			0
$529$	−1.26097e11			−1.61020
$530$		0			0
$531$	5.64792e10	−	5.27486e10i	0.710413	−	0.663488i
$532$		0			0
$533$		−	4.76654e10i		−	0.590601i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	3.54434e10	−	8.15996e10i	0.426224	−	0.981275i
$538$		0			0
$539$			1.17566e11i			1.39292i
$540$		0			0
$541$	−1.60899e11			−1.87830			−0.939149	−	0.343509i	$-0.888384\pi$
$541$	−1.60899e11			−1.87830			−0.939149	+	0.343509i	$0.888384\pi$
$542$		0			0
$543$	−3.86632e10	−	1.67937e10i	−0.444732	−	0.193173i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.48569e11			1.65951			0.829754	−	0.558129i	$-0.188481\pi$
$547$	1.48569e11			1.65951			0.829754	+	0.558129i	$0.188481\pi$
$548$		0			0
$549$	9.13566e10	+	9.78177e10i	1.00566	+	1.07678i
$550$		0			0
$551$		−	2.11160e10i		−	0.229090i
$552$		0			0
$553$	4.37835e10			0.468177
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	7.20464e10i		−	0.748499i	−0.927328	−	0.374250i	$-0.877900\pi$
$557$		−	7.20464e10i		−	0.748499i	0.927328	−	0.374250i	$-0.122100\pi$
$558$		0			0
$559$	1.44752e10			0.148244
$560$		0			0
$561$	1.02420e11	+	4.44868e10i	1.03403	+	0.449138i
$562$		0			0
$563$		−	3.59816e10i		−	0.358135i	−0.983837	−	0.179067i	$-0.942692\pi$
$563$		−	3.59816e10i		−	0.358135i	0.983837	−	0.179067i	$-0.0573080\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	1.17991e10	−	1.72530e11i	0.114160	−	1.66929i
$568$		0			0
$569$		−	1.80528e10i		−	0.172225i	−0.996285	−	0.0861124i	$-0.972556\pi$
$569$		−	1.80528e10i		−	0.172225i	0.996285	−	0.0861124i	$-0.0274444\pi$
$570$		0			0
$571$	−5.51665e10			−0.518956			−0.259478	−	0.965749i	$-0.583551\pi$
$571$	−5.51665e10			−0.518956			−0.259478	+	0.965749i	$0.583551\pi$
$572$		0			0
$573$	3.15563e10	−	7.26505e10i	0.292730	−	0.673938i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	1.58383e11			1.42891			0.714455	−	0.699681i	$-0.246675\pi$
$577$	1.58383e11			1.42891			0.714455	+	0.699681i	$0.246675\pi$
$578$		0			0
$579$	5.42694e10	+	2.35723e10i	0.482882	+	0.209743i
$580$		0			0
$581$		−	2.22572e11i		−	1.95328i
$582$		0			0
$583$	−4.52017e10			−0.391274
$584$		0			0
$585$		0			0
$586$		0			0
$587$			7.05361e10i			0.594100i	0.954862	+	0.297050i	$0.0960027\pi$
$587$			7.05361e10i			0.594100i	−0.954862	+	0.297050i	$0.903997\pi$
$588$		0			0
$589$	2.83872e10			0.235864
$590$		0			0
$591$	−1.56022e10	+	3.59202e10i	−0.127890	+	0.294435i
$592$		0			0
$593$		−	7.16827e10i		−	0.579689i	−0.957074	−	0.289845i	$-0.906396\pi$
$593$		−	7.16827e10i		−	0.579689i	0.957074	−	0.289845i	$-0.0936036\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	1.80926e11	+	7.85864e10i	1.42430	+	0.618657i
$598$		0			0
$599$		−	1.98040e11i		−	1.53832i	−0.639057	−	0.769159i	$-0.720675\pi$
$599$		−	1.98040e11i		−	1.53832i	0.639057	−	0.769159i	$-0.279325\pi$
$600$		0			0
$601$	1.76093e11			1.34972			0.674861	−	0.737945i	$-0.264204\pi$
$601$	1.76093e11			1.34972			0.674861	+	0.737945i	$0.264204\pi$
$602$		0			0
$603$	6.05852e10	+	6.48700e10i	0.458244	+	0.490653i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−1.53412e11			−1.13007			−0.565035	−	0.825067i	$-0.691137\pi$
$607$	−1.53412e11			−1.13007			−0.565035	+	0.825067i	$0.691137\pi$
$608$		0			0
$609$	1.20017e11	−	2.76310e11i	0.872520	−	2.00876i
$610$		0			0
$611$			1.10007e10i			0.0789322i
$612$		0			0
$613$	1.12547e11			0.797062			0.398531	−	0.917155i	$-0.369520\pi$
$613$	1.12547e11			0.797062			0.398531	+	0.917155i	$0.369520\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	9.70972e10i		−	0.669986i	−0.942221	−	0.334993i	$-0.891266\pi$
$617$		−	9.70972e10i		−	0.669986i	0.942221	−	0.334993i	$-0.108734\pi$
$618$		0			0
$619$	2.43396e10			0.165787			0.0828936	−	0.996558i	$-0.473584\pi$
$619$	2.43396e10			0.165787			0.0828936	+	0.996558i	$0.473584\pi$
$620$		0			0
$621$	2.26398e11	−	8.04645e10i	1.52232	−	0.541051i
$622$		0			0
$623$			3.28275e11i			2.17914i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	8.34140e9	−	1.92040e10i	0.0539720	−	0.124257i
$628$		0			0
$629$		−	1.21192e9i		−	0.00774232i
$630$		0			0
$631$	−4.38956e10			−0.276888			−0.138444	−	0.990370i	$-0.544210\pi$
$631$	−4.38956e10			−0.276888			−0.138444	+	0.990370i	$0.544210\pi$
$632$		0			0
$633$	2.37657e11	+	1.03228e11i	1.48025	+	0.642958i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1.52182e11			0.924286
$638$		0			0
$639$	−6.52067e10	+	6.08996e10i	−0.391101	+	0.365268i
$640$		0			0
$641$			1.27326e11i			0.754196i	0.926173	+	0.377098i	$0.123078\pi$
$641$			1.27326e11i			0.754196i	−0.926173	+	0.377098i	$0.876922\pi$
$642$		0			0
$643$	−2.50491e11			−1.46537			−0.732686	−	0.680567i	$-0.761734\pi$
$643$	−2.50491e11			−1.46537			−0.732686	+	0.680567i	$0.761734\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	2.85872e11i		−	1.63138i	−0.578491	−	0.815689i	$-0.696358\pi$
$647$		−	2.85872e11i		−	1.63138i	0.578491	−	0.815689i	$-0.303642\pi$
$648$		0			0
$649$	1.33485e11			0.752407
$650$		0			0
$651$	3.71457e11	+	1.61345e11i	2.06816	+	0.898322i
$652$		0			0
$653$		−	2.78895e11i		−	1.53387i	−0.641726	−	0.766934i	$-0.721781\pi$
$653$		−	2.78895e11i		−	1.53387i	0.641726	−	0.766934i	$-0.278219\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	2.41733e11	+	2.58830e11i	1.29740	+	1.38916i
$658$		0			0
$659$		−	3.09719e11i		−	1.64220i	−0.570784	−	0.821100i	$-0.693361\pi$
$659$		−	3.09719e11i		−	1.64220i	0.570784	−	0.821100i	$-0.306639\pi$
$660$		0			0
$661$	−2.59385e11			−1.35875			−0.679374	−	0.733792i	$-0.737748\pi$
$661$	−2.59385e11			−1.35875			−0.679374	+	0.733792i	$0.737748\pi$
$662$		0			0
$663$	5.75857e10	−	1.32577e11i	0.298031	−	0.686141i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	4.18556e11			2.11471
$668$		0			0
$669$	1.19711e11	+	5.19975e10i	0.597627	+	0.259584i
$670$		0			0
$671$			2.31185e11i			1.14043i
$672$		0			0
$673$	1.68948e10			0.0823557			0.0411778	−	0.999152i	$-0.486889\pi$
$673$	1.68948e10			0.0823557			0.0411778	+	0.999152i	$0.486889\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	2.66225e11i		−	1.26734i	−0.773602	−	0.633672i	$-0.781547\pi$
$677$		−	2.66225e11i		−	1.26734i	0.773602	−	0.633672i	$-0.218453\pi$
$678$		0			0
$679$	−4.93531e11			−2.32186
$680$		0			0
$681$	−1.68369e10	+	3.87627e10i	−0.0782841	+	0.180230i
$682$		0			0
$683$		−	2.27073e11i		−	1.04348i	−0.853106	−	0.521738i	$-0.825284\pi$
$683$		−	2.27073e11i		−	1.04348i	0.853106	−	0.521738i	$-0.174716\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−4.36436e10	−	1.89569e10i	−0.195927	−	0.0851022i
$688$		0			0
$689$			5.85111e10i			0.259634i
$690$		0			0
$691$	−3.55909e11			−1.56109			−0.780543	−	0.625103i	$-0.785057\pi$
$691$	−3.55909e11			−1.56109			−0.780543	+	0.625103i	$0.785057\pi$
$692$		0			0
$693$	2.18300e11	−	2.03881e11i	0.946502	−	0.883983i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−3.95265e11			−1.67478
$698$		0			0
$699$	7.36380e10	−	1.69533e11i	0.308456	−	0.710143i
$700$		0			0
$701$		−	3.36766e11i		−	1.39462i	−0.716769	−	0.697311i	$-0.754380\pi$
$701$		−	3.36766e11i		−	1.39462i	0.716769	−	0.697311i	$-0.245620\pi$
$702$		0			0
$703$	−2.27238e8			−0.000930379
$704$		0			0
$705$		0			0
$706$		0			0
$707$			2.94409e10i			0.117835i
$708$		0			0
$709$	2.37614e11			0.940345			0.470172	−	0.882575i	$-0.344192\pi$
$709$	2.37614e11			0.940345			0.470172	+	0.882575i	$0.344192\pi$
$710$		0			0
$711$	−4.88071e10	−	5.22590e10i	−0.190987	−	0.204495i
$712$		0			0
$713$			5.62685e11i			2.17724i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	7.24825e10	−	1.66873e11i	0.274256	−	0.631407i
$718$		0			0
$719$			3.53231e11i			1.32173i	0.750504	+	0.660866i	$0.229811\pi$
$719$			3.53231e11i			1.32173i	−0.750504	+	0.660866i	$0.770189\pi$
$720$		0			0
$721$	−3.75916e11			−1.39107
$722$		0			0
$723$	−3.49249e11	−	1.51699e11i	−1.27815	−	0.555174i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	1.44118e11			0.515919			0.257960	−	0.966156i	$-0.416950\pi$
$727$	1.44118e11			0.515919			0.257960	+	0.966156i	$0.416950\pi$
$728$		0			0
$729$	−2.19080e11	+	1.78242e11i	−0.775699	+	0.631103i
$730$		0			0
$731$		−	1.20035e11i		−	0.420378i
$732$		0			0
$733$	2.62582e11			0.909596			0.454798	−	0.890595i	$-0.349712\pi$
$733$	2.62582e11			0.909596			0.454798	+	0.890595i	$0.349712\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			1.53316e11i			0.519657i
$738$		0			0
$739$	−1.83611e11			−0.615633			−0.307816	−	0.951446i	$-0.599598\pi$
$739$	−1.83611e11			−0.615633			−0.307816	+	0.951446i	$0.599598\pi$
$740$		0			0
$741$	−2.48585e10	−	1.07975e10i	−0.0824522	−	0.0358137i
$742$		0			0
$743$			2.01511e11i			0.661216i	0.943768	+	0.330608i	$0.107254\pi$
$743$			2.01511e11i			0.661216i	−0.943768	+	0.330608i	$0.892746\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−2.65656e11	+	2.48109e11i	−0.853174	+	0.796819i
$748$		0			0
$749$			6.66742e11i			2.11851i
$750$		0			0
$751$	3.41725e11			1.07428			0.537139	−	0.843494i	$-0.319505\pi$
$751$	3.41725e11			1.07428			0.537139	+	0.843494i	$0.319505\pi$
$752$		0			0
$753$	1.01639e11	−	2.34000e11i	0.316142	−	0.727839i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	4.06291e11			1.23724			0.618620	−	0.785690i	$-0.287692\pi$
$757$	4.06291e11			1.23724			0.618620	+	0.785690i	$0.287692\pi$
$758$		0			0
$759$	3.80657e11	+	1.65341e11i	1.14701	+	0.498211i
$760$		0			0
$761$			3.14906e11i			0.938951i	0.882946	+	0.469475i	$0.155557\pi$
$761$			3.14906e11i			0.938951i	−0.882946	+	0.469475i	$0.844443\pi$
$762$		0			0
$763$	−1.11270e11			−0.328308
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	1.72789e11i		−	0.499268i
$768$		0			0
$769$	−2.34345e11			−0.670118			−0.335059	−	0.942197i	$-0.608756\pi$
$769$	−2.34345e11			−0.670118			−0.335059	+	0.942197i	$0.608756\pi$
$770$		0			0
$771$	7.44573e10	−	1.71419e11i	0.210712	−	0.485113i
$772$		0			0
$773$		−	5.06053e10i		−	0.141735i	−0.997486	−	0.0708677i	$-0.977423\pi$
$773$		−	5.06053e10i		−	0.141735i	0.997486	−	0.0708677i	$-0.0225768\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−2.97349e9	−	1.29156e9i	−0.00815798	−	0.00354348i
$778$		0			0
$779$			7.41133e10i			0.201255i
$780$		0			0
$781$	−1.54112e11			−0.414220
$782$		0			0
$783$	−4.63586e11	+	1.64764e11i	−1.23334	+	0.438343i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−6.23463e11			−1.62522			−0.812610	−	0.582808i	$-0.801954\pi$
$787$	−6.23463e11			−1.62522			−0.812610	+	0.582808i	$0.801954\pi$
$788$		0			0
$789$	−5.16649e10	+	1.18946e11i	−0.133318	+	0.306931i
$790$		0			0
$791$		−	1.83682e11i		−	0.469202i
$792$		0			0
$793$	2.99257e11			0.756748
$794$		0			0
$795$		0			0
$796$		0			0
$797$			4.31198e11i			1.06867i	0.845273	+	0.534335i	$0.179438\pi$
$797$			4.31198e11i			1.06867i	−0.845273	+	0.534335i	$0.820562\pi$
$798$		0			0
$799$	9.12231e10			0.223830
$800$		0			0
$801$	3.91821e11	−	3.65940e11i	0.951826	−	0.888955i
$802$		0			0
$803$			6.11726e11i			1.47128i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−2.92784e11	+	6.74062e11i	−0.690324	+	1.58930i
$808$		0			0
$809$		−	5.07535e11i		−	1.18487i	−0.805617	−	0.592437i	$-0.798166\pi$
$809$		−	5.07535e11i		−	1.18487i	0.805617	−	0.592437i	$-0.201834\pi$
$810$		0			0
$811$	−1.82026e10			−0.0420774			−0.0210387	−	0.999779i	$-0.506697\pi$
$811$	−1.82026e10			−0.0420774			−0.0210387	+	0.999779i	$0.506697\pi$
$812$		0			0
$813$	−3.24796e11	−	1.41078e11i	−0.743445	−	0.322921i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−2.25070e10			−0.0505160
$818$		0			0
$819$	−2.63913e11	−	2.82578e11i	−0.586576	−	0.628062i
$820$		0			0
$821$			9.48913e9i			0.0208859i	0.999945	+	0.0104430i	$0.00332416\pi$
$821$			9.48913e9i			0.0208859i	−0.999945	+	0.0104430i	$0.996676\pi$
$822$		0			0
$823$	−2.82548e11			−0.615875			−0.307937	−	0.951407i	$-0.599639\pi$
$823$	−2.82548e11			−0.615875			−0.307937	+	0.951407i	$0.599639\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			2.73392e10i			0.0584472i	0.999573	+	0.0292236i	$0.00930348\pi$
$827$			2.73392e10i			0.0584472i	−0.999573	+	0.0292236i	$0.990697\pi$
$828$		0			0
$829$	−3.56575e11			−0.754976			−0.377488	−	0.926015i	$-0.623212\pi$
$829$	−3.56575e11			−0.754976			−0.377488	+	0.926015i	$0.623212\pi$
$830$		0			0
$831$	5.55509e11	+	2.41289e11i	1.16490	+	0.505981i
$832$		0			0
$833$		−	1.26197e12i		−	2.62102i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−2.21499e11	−	6.23220e11i	−0.451305	−	1.26981i
$838$		0			0
$839$		−	1.47820e11i		−	0.298323i	−0.988813	−	0.149161i	$-0.952343\pi$
$839$		−	1.47820e11i		−	0.298323i	0.988813	−	0.149161i	$-0.0476574\pi$
$840$		0			0
$841$	−3.56812e11			−0.713273
$842$		0			0
$843$	−2.84620e11	+	6.55266e11i	−0.563579	+	1.29750i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−3.45212e11			−0.670737
$848$		0			0
$849$	6.83502e9	+	2.96884e9i	0.0131555	+	0.00571421i
$850$		0			0
$851$		−	4.50426e9i		−	0.00858825i
$852$		0			0
$853$	4.87703e11			0.921212			0.460606	−	0.887605i	$-0.347632\pi$
$853$	4.87703e11			0.921212			0.460606	+	0.887605i	$0.347632\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	5.65321e11i		−	1.04802i	−0.851711	−	0.524012i	$-0.824435\pi$
$857$		−	5.65321e11i		−	1.04802i	0.851711	−	0.524012i	$-0.175565\pi$
$858$		0			0
$859$	9.50163e11			1.74512			0.872561	−	0.488506i	$-0.162458\pi$
$859$	9.50163e11			1.74512			0.872561	+	0.488506i	$0.162458\pi$
$860$		0			0
$861$	−4.21239e11	+	9.69799e11i	−0.766508	+	1.76469i
$862$		0			0
$863$		−	8.75521e10i		−	0.157842i	−0.996881	−	0.0789211i	$-0.974852\pi$
$863$		−	8.75521e10i		−	0.157842i	0.996881	−	0.0789211i	$-0.0251475\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−5.81135e11	−	2.52420e11i	−1.02849	−	0.446733i
$868$		0			0
$869$		−	1.23510e11i		−	0.216583i
$870$		0			0
$871$	1.98459e11			0.344824
$872$		0			0
$873$	5.50157e11	+	5.89067e11i	0.947174	+	1.01416i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−9.80932e11			−1.65821			−0.829107	−	0.559090i	$-0.811151\pi$
$877$	−9.80932e11			−1.65821			−0.829107	+	0.559090i	$0.811151\pi$
$878$		0			0
$879$	7.22289e10	−	1.66289e11i	0.120992	−	0.278553i
$880$		0			0
$881$			1.03446e12i			1.71716i	0.512677	+	0.858582i	$0.328654\pi$
$881$			1.03446e12i			1.71716i	−0.512677	+	0.858582i	$0.671346\pi$
$882$		0			0
$883$	−2.29618e11			−0.377714			−0.188857	−	0.982005i	$-0.560478\pi$
$883$	−2.29618e11			−0.377714			−0.188857	+	0.982005i	$0.560478\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			9.73588e11i			1.57283i	0.617701	+	0.786413i	$0.288064\pi$
$887$			9.73588e11i			1.57283i	−0.617701	+	0.786413i	$0.711936\pi$
$888$		0			0
$889$	−9.07972e11			−1.45367
$890$		0			0
$891$	−4.86695e11	−	3.32844e10i	−0.772229	−	0.0528117i
$892$		0			0
$893$		−	1.71046e10i		−	0.0268972i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	2.14025e11	−	4.92739e11i	0.330594	−	0.761110i
$898$		0			0
$899$		−	1.15218e12i		−	1.76394i
$900$		0			0
$901$	4.85203e11			0.736249
$902$		0			0
$903$	−2.94512e11	−	1.27923e11i	−0.442947	−	0.192397i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−6.17806e11			−0.912900			−0.456450	−	0.889749i	$-0.650879\pi$
$907$	−6.17806e11			−0.912900			−0.456450	+	0.889749i	$0.650879\pi$
$908$		0			0
$909$	3.51400e10	−	3.28189e10i	0.0514691	−	0.0480694i
$910$		0			0
$911$		−	9.19187e11i		−	1.33454i	−0.744817	−	0.667268i	$-0.767463\pi$
$911$		−	9.19187e11i		−	1.33454i	0.744817	−	0.667268i	$-0.232537\pi$
$912$		0			0
$913$	−6.27860e11			−0.903607
$914$		0			0
$915$		0			0
$916$		0			0
$917$			2.31767e12i			3.27774i
$918$		0			0
$919$	9.42778e10			0.132174			0.0660872	−	0.997814i	$-0.478948\pi$
$919$	9.42778e10			0.132174			0.0660872	+	0.997814i	$0.478948\pi$
$920$		0			0
$921$	−6.57646e11	−	2.85654e11i	−0.914016	−	0.397010i
$922$		0			0
$923$			1.99489e11i			0.274860i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	4.19048e11	+	4.48685e11i	0.567473	+	0.607607i
$928$		0			0
$929$		−	3.95905e11i		−	0.531531i	−0.964038	−	0.265765i	$-0.914375\pi$
$929$		−	3.95905e11i		−	0.531531i	0.964038	−	0.265765i	$-0.0856246\pi$
$930$		0			0
$931$	−2.36623e11			−0.314962
$932$		0			0
$933$	−3.67718e11	+	8.46580e11i	−0.485276	+	1.11723i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1.96530e11			0.254959			0.127480	−	0.991841i	$-0.459311\pi$
$937$	1.96530e11			0.254959			0.127480	+	0.991841i	$0.459311\pi$
$938$		0			0
$939$	4.80073e11	+	2.08523e11i	0.617512	+	0.268221i
$940$		0			0
$941$		−	5.42960e9i		−	0.00692483i	−0.999994	−	0.00346242i	$-0.998898\pi$
$941$		−	5.42960e9i		−	0.00692483i	0.999994	−	0.00346242i	$-0.00110212\pi$
$942$		0			0
$943$	−1.46906e12			−1.85777
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	7.49964e11i		−	0.932482i	−0.884658	−	0.466241i	$-0.845608\pi$
$947$		−	7.49964e11i		−	0.932482i	0.884658	−	0.466241i	$-0.154392\pi$
$948$		0			0
$949$	7.91846e11			0.976283
$950$		0			0
$951$	5.22396e11	−	1.20269e12i	0.638672	−	1.47038i
$952$		0			0
$953$		−	1.34634e11i		−	0.163224i	−0.996664	−	0.0816119i	$-0.973993\pi$
$953$		−	1.34634e11i		−	0.163224i	0.996664	−	0.0816119i	$-0.0260068\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−7.79454e11	−	3.38561e11i	−0.929271	−	0.403636i
$958$		0			0
$959$			1.54803e12i			1.83022i
$960$		0			0
$961$	6.96044e11			0.816100
$962$		0			0
$963$	7.95808e11	−	7.43242e11i	0.925343	−	0.864222i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1.09741e12			−1.25506			−0.627530	−	0.778592i	$-0.715934\pi$
$967$	−1.09741e12			−1.25506			−0.627530	+	0.778592i	$0.715934\pi$
$968$		0			0
$969$	−8.95381e10	+	2.06139e11i	−0.101558	+	0.233811i
$970$		0			0
$971$		−	5.97472e11i		−	0.672111i	−0.941842	−	0.336055i	$-0.890907\pi$
$971$		−	5.97472e11i		−	0.672111i	0.941842	−	0.336055i	$-0.109093\pi$
$972$		0			0
$973$	9.62500e11			1.07386
$974$		0			0
$975$		0			0
$976$		0			0
$977$			1.84630e11i			0.202639i	0.994854	+	0.101320i	$0.0323064\pi$
$977$			1.84630e11i			0.202639i	−0.994854	+	0.101320i	$0.967694\pi$
$978$		0			0
$979$	9.26041e11			1.00809
$980$		0			0
$981$	1.24037e11	+	1.32810e11i	0.133929	+	0.143402i
$982$		0			0
$983$			6.79919e11i			0.728187i	0.931362	+	0.364093i	$0.118621\pi$
$983$			6.79919e11i			0.728187i	−0.931362	+	0.364093i	$0.881379\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	9.72176e10	−	2.23819e11i	0.102442	−	0.235846i
$988$		0			0
$989$		−	4.46127e11i		−	0.466309i
$990$		0			0
$991$	7.10490e11			0.736653			0.368327	−	0.929696i	$-0.379931\pi$
$991$	7.10490e11			0.736653			0.368327	+	0.929696i	$0.379931\pi$
$992$		0			0
$993$	9.25786e11	+	4.02122e11i	0.952168	+	0.413581i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	3.85789e11			0.390453			0.195227	−	0.980758i	$-0.437456\pi$
$997$	3.85789e11			0.390453			0.195227	+	0.980758i	$0.437456\pi$
$998$		0			0
$999$	1.77309e9	+	4.98884e9i	0.00178020	+	0.00500884i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.9.g.h.101.14		16
3.2	odd	2	inner	300.9.g.h.101.13		16
5.2	odd	4		60.9.b.a.29.11	yes	16
5.3	odd	4		60.9.b.a.29.6	yes	16
5.4	even	2	inner	300.9.g.h.101.3		16
15.2	even	4		60.9.b.a.29.5	✓	16
15.8	even	4		60.9.b.a.29.12	yes	16
15.14	odd	2	inner	300.9.g.h.101.4		16
20.3	even	4		240.9.c.d.209.11		16
20.7	even	4		240.9.c.d.209.6		16
60.23	odd	4		240.9.c.d.209.5		16
60.47	odd	4		240.9.c.d.209.12		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.9.b.a.29.5	✓	16	15.2	even	4
60.9.b.a.29.6	yes	16	5.3	odd	4
60.9.b.a.29.11	yes	16	5.2	odd	4
60.9.b.a.29.12	yes	16	15.8	even	4
240.9.c.d.209.5		16	60.23	odd	4
240.9.c.d.209.6		16	20.7	even	4
240.9.c.d.209.11		16	20.3	even	4
240.9.c.d.209.12		16	60.47	odd	4
300.9.g.h.101.3		16	5.4	even	2	inner
300.9.g.h.101.4		16	15.14	odd	2	inner
300.9.g.h.101.13		16	3.2	odd	2	inner
300.9.g.h.101.14		16	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+(74.2942 + 32.2702i) q^{3} -4017.33 q^{7} +(4478.26 + 4794.99i) q^{9} +O(q^{10})\)	\(q+(74.2942 + 32.2702i) q^{3} -4017.33 q^{7} +(4478.26 + 4794.99i) q^{9} +11332.6i q^{11} +14669.4 q^{13} -121646. i q^{17} -22809.0 q^{19} +(-298464. - 129640. i) q^{21} -452115. i q^{23} +(177974. + 500754. i) q^{27} +925775. i q^{29} -1.24456e6 q^{31} +(-365706. + 841948. i) q^{33} +9962.64 q^{37} +(1.08985e6 + 473386. i) q^{39} -3.24930e6i q^{41} +986757. q^{43} +749904. i q^{47} +1.03741e7 q^{49} +(3.92556e6 - 9.03762e6i) q^{51} +3.98864e6i q^{53} +(-1.69458e6 - 736052. i) q^{57} -1.17788e7i q^{59} +2.04000e7 q^{61} +(-1.79906e7 - 1.92630e7i) q^{63} +1.35287e7 q^{67} +(1.45898e7 - 3.35895e7i) q^{69} +1.35989e7i q^{71} +5.39793e7 q^{73} -4.55268e7i q^{77} -1.08987e7 q^{79} +(-2.93705e6 + 4.29464e7i) q^{81} +5.54029e7i q^{83} +(-2.98750e7 + 6.87797e7i) q^{87} -8.17147e7i q^{89} -5.89319e7 q^{91} +(-9.24638e7 - 4.01623e7i) q^{93} +1.22851e8 q^{97} +(-5.43397e7 + 5.07504e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 14756 q^{9}+O(q^{10}) \) 16 * q + 14756 * q^9	\( 16 q + 14756 q^{9} + 48024 q^{19} + 4724 q^{21} + 470104 q^{31} + 2849664 q^{39} + 19816920 q^{49} + 5026040 q^{51} + 18849944 q^{61} + 38669180 q^{69} + 90778632 q^{79} + 16242056 q^{81} + 22375296 q^{91} - 16968560 q^{99}+O(q^{100}) \) 16 * q + 14756 * q^9 + 48024 * q^19 + 4724 * q^21 + 470104 * q^31 + 2849664 * q^39 + 19816920 * q^49 + 5026040 * q^51 + 18849944 * q^61 + 38669180 * q^69 + 90778632 * q^79 + 16242056 * q^81 + 22375296 * q^91 - 16968560 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more