LMFDB - Embedded newform 300.9.g.h.101.9

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			101.9
Root			$-23.4026 + 77.5456i$ of defining polynomial
Character	$\chi$	$=$	300.101
Dual form			300.9.g.h.101.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(23.4026 - 77.5456i) q^{3} +256.799 q^{7} +(-5465.63 - 3629.54i) q^{9} +O(q^{10})$	$q+(23.4026 - 77.5456i) q^{3} +256.799 q^{7} +(-5465.63 - 3629.54i) q^{9} +12464.5i q^{11} +3381.78 q^{13} -50943.0i q^{17} +46365.2 q^{19} +(6009.77 - 19913.6i) q^{21} -319641. i q^{23} +(-409365. + 338895. i) q^{27} +310614. i q^{29} -834932. q^{31} +(966570. + 291703. i) q^{33} -2.88249e6 q^{37} +(79142.5 - 262242. i) q^{39} +2.24041e6i q^{41} +5.12207e6 q^{43} +3.54045e6i q^{47} -5.69886e6 q^{49} +(-3.95040e6 - 1.19220e6i) q^{51} -557473. i q^{53} +(1.08507e6 - 3.59542e6i) q^{57} +1.90288e7i q^{59} -4.81127e6 q^{61} +(-1.40357e6 - 932062. i) q^{63} +1.45363e7 q^{67} +(-2.47867e7 - 7.48043e6i) q^{69} -3.50379e6i q^{71} -4.64813e7 q^{73} +3.20088e6i q^{77} -4.67477e7 q^{79} +(1.66996e7 + 3.96755e7i) q^{81} +1.10689e7i q^{83} +(2.40867e7 + 7.26917e6i) q^{87} +7.11764e7i q^{89} +868438. q^{91} +(-1.95396e7 + 6.47453e7i) q^{93} +6.59215e7 q^{97} +(4.52405e7 - 6.81266e7i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	23.4026	−	77.5456i	0.288921	−	0.957353i
$3$	23.4026	−	77.5456i	0.288921	−	0.957353i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	256.799			0.106955			0.0534775	−	0.998569i	$-0.482969\pi$
$7$	256.799			0.106955			0.0534775	+	0.998569i	$0.482969\pi$
$8$		0			0
$9$	−5465.63	−	3629.54i	−0.833049	−	0.553199i
$10$		0			0
$11$			12464.5i			0.851345i	0.904877	+	0.425672i	$0.139962\pi$
$11$			12464.5i			0.851345i	−0.904877	+	0.425672i	$0.860038\pi$
$12$		0			0
$13$	3381.78			0.118406			0.0592028	−	0.998246i	$-0.481144\pi$
$13$	3381.78			0.118406			0.0592028	+	0.998246i	$0.481144\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	50943.0i		−	0.609942i	−0.952362	−	0.304971i	$-0.901353\pi$
$17$		−	50943.0i		−	0.609942i	0.952362	−	0.304971i	$-0.0986468\pi$
$18$		0			0
$19$	46365.2			0.355777			0.177888	−	0.984051i	$-0.443073\pi$
$19$	46365.2			0.355777			0.177888	+	0.984051i	$0.443073\pi$
$20$		0			0
$21$	6009.77	−	19913.6i	0.0309016	−	0.102394i
$22$		0			0
$23$		−	319641.i		−	1.14222i	−0.820873	−	0.571111i	$-0.806512\pi$
$23$		−	319641.i		−	1.14222i	0.820873	−	0.571111i	$-0.193488\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−409365.	+	338895.i	−0.770292	+	0.637691i
$28$		0			0
$29$			310614.i			0.439166i	0.975594	+	0.219583i	$0.0704696\pi$
$29$			310614.i			0.439166i	−0.975594	+	0.219583i	$0.929530\pi$
$30$		0			0
$31$	−834932.			−0.904075			−0.452038	−	0.891999i	$-0.649303\pi$
$31$	−834932.			−0.904075			−0.452038	+	0.891999i	$0.649303\pi$
$32$		0			0
$33$	966570.	+	291703.i	0.815038	+	0.245972i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−2.88249e6			−1.53801			−0.769007	−	0.639240i	$-0.779249\pi$
$37$	−2.88249e6			−1.53801			−0.769007	+	0.639240i	$0.779249\pi$
$38$		0			0
$39$	79142.5	−	262242.i	0.0342099	−	0.113356i
$40$		0			0
$41$			2.24041e6i			0.792852i	0.918067	+	0.396426i	$0.129750\pi$
$41$			2.24041e6i			0.792852i	−0.918067	+	0.396426i	$0.870250\pi$
$42$		0			0
$43$	5.12207e6			1.49821			0.749103	−	0.662453i	$-0.230485\pi$
$43$	5.12207e6			1.49821			0.749103	+	0.662453i	$0.230485\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			3.54045e6i			0.725549i	0.931877	+	0.362775i	$0.118170\pi$
$47$			3.54045e6i			0.725549i	−0.931877	+	0.362775i	$0.881830\pi$
$48$		0			0
$49$	−5.69886e6			−0.988561
$50$		0			0
$51$	−3.95040e6	−	1.19220e6i	−0.583930	−	0.176225i
$52$		0			0
$53$		−	557473.i		−	0.0706514i	−0.999376	−	0.0353257i	$-0.988753\pi$
$53$		−	557473.i		−	0.0706514i	0.999376	−	0.0353257i	$-0.0112469\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	1.08507e6	−	3.59542e6i	0.102791	−	0.340604i
$58$		0			0
$59$			1.90288e7i			1.57037i	0.619259	+	0.785186i	$0.287433\pi$
$59$			1.90288e7i			1.57037i	−0.619259	+	0.785186i	$0.712567\pi$
$60$		0			0
$61$	−4.81127e6			−0.347488			−0.173744	−	0.984791i	$-0.555587\pi$
$61$	−4.81127e6			−0.347488			−0.173744	+	0.984791i	$0.555587\pi$
$62$		0			0
$63$	−1.40357e6	−	932062.i	−0.0890987	−	0.0591674i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	1.45363e7			0.721364			0.360682	−	0.932689i	$-0.382544\pi$
$67$	1.45363e7			0.721364			0.360682	+	0.932689i	$0.382544\pi$
$68$		0			0
$69$	−2.47867e7	−	7.48043e6i	−1.09351	−	0.330012i
$70$		0			0
$71$		−	3.50379e6i		−	0.137881i	−0.997621	−	0.0689405i	$-0.978038\pi$
$71$		−	3.50379e6i		−	0.137881i	0.997621	−	0.0689405i	$-0.0219619\pi$
$72$		0			0
$73$	−4.64813e7			−1.63677			−0.818384	−	0.574672i	$-0.805130\pi$
$73$	−4.64813e7			−1.63677			−0.818384	+	0.574672i	$0.805130\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			3.20088e6i			0.0910556i
$78$		0			0
$79$	−4.67477e7			−1.20020			−0.600098	−	0.799926i	$-0.704872\pi$
$79$	−4.67477e7			−1.20020			−0.600098	+	0.799926i	$0.704872\pi$
$80$		0			0
$81$	1.66996e7	+	3.96755e7i	0.387942	+	0.921684i
$82$		0			0
$83$			1.10689e7i			0.233233i	0.993177	+	0.116617i	$0.0372049\pi$
$83$			1.10689e7i			0.233233i	−0.993177	+	0.116617i	$0.962795\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	2.40867e7	+	7.26917e6i	0.420437	+	0.126884i
$88$		0			0
$89$			7.11764e7i			1.13443i	0.823571	+	0.567213i	$0.191978\pi$
$89$			7.11764e7i			1.13443i	−0.823571	+	0.567213i	$0.808022\pi$
$90$		0			0
$91$	868438.			0.0126641
$92$		0			0
$93$	−1.95396e7	+	6.47453e7i	−0.261207	+	0.865519i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	6.59215e7			0.744629			0.372314	−	0.928107i	$-0.378564\pi$
$97$	6.59215e7			0.744629			0.372314	+	0.928107i	$0.378564\pi$
$98$		0			0
$99$	4.52405e7	−	6.81266e7i	0.470963	−	0.709212i
$100$		0			0
$101$		−	1.71876e8i		−	1.65169i	−0.563896	−	0.825846i	$-0.690698\pi$
$101$		−	1.71876e8i		−	1.65169i	0.563896	−	0.825846i	$-0.309302\pi$
$102$		0			0
$103$	1.23564e8			1.09785			0.548927	−	0.835870i	$-0.315037\pi$
$103$	1.23564e8			1.09785			0.548927	+	0.835870i	$0.315037\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			2.15424e8i			1.64346i	0.569878	+	0.821729i	$0.306990\pi$
$107$			2.15424e8i			1.64346i	−0.569878	+	0.821729i	$0.693010\pi$
$108$		0			0
$109$	9.20961e7			0.652432			0.326216	−	0.945295i	$-0.394226\pi$
$109$	9.20961e7			0.652432			0.326216	+	0.945295i	$0.394226\pi$
$110$		0			0
$111$	−6.74577e7	+	2.23524e8i	−0.444365	+	1.47242i
$112$		0			0
$113$		−	5.81056e7i		−	0.356372i	−0.983997	−	0.178186i	$-0.942977\pi$
$113$		−	5.81056e7i		−	0.356372i	0.983997	−	0.178186i	$-0.0570229\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−1.84836e7	−	1.22743e7i	−0.0986376	−	0.0655018i
$118$		0			0
$119$		−	1.30821e7i		−	0.0652363i
$120$		0			0
$121$	5.89941e7			0.275212
$122$		0			0
$123$	1.73734e8	+	5.24315e7i	0.759039	+	0.229072i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−3.40183e8			−1.30767			−0.653833	−	0.756639i	$-0.726840\pi$
$127$	−3.40183e8			−1.30767			−0.653833	+	0.756639i	$0.726840\pi$
$128$		0			0
$129$	1.19870e8	−	3.97194e8i	0.432864	−	1.43431i
$130$		0			0
$131$		−	2.57060e8i		−	0.872871i	−0.899736	−	0.436435i	$-0.856241\pi$
$131$		−	2.57060e8i		−	0.872871i	0.899736	−	0.436435i	$-0.143759\pi$
$132$		0			0
$133$	1.19065e7			0.0380521
$134$		0			0
$135$		0			0
$136$		0			0
$137$			5.91829e8i			1.68002i	0.542572	+	0.840010i	$0.317451\pi$
$137$			5.91829e8i			1.68002i	−0.542572	+	0.840010i	$0.682549\pi$
$138$		0			0
$139$	5.53864e8			1.48369			0.741846	−	0.670571i	$-0.233951\pi$
$139$	5.53864e8			1.48369			0.741846	+	0.670571i	$0.233951\pi$
$140$		0			0
$141$	2.74546e8	+	8.28557e7i	0.694606	+	0.209626i
$142$		0			0
$143$			4.21523e7i			0.100804i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−1.33368e8	+	4.41921e8i	−0.285616	+	0.946401i
$148$		0			0
$149$			2.07326e8i			0.420638i	0.977633	+	0.210319i	$0.0674503\pi$
$149$			2.07326e8i			0.420638i	−0.977633	+	0.210319i	$0.932550\pi$
$150$		0			0
$151$	−2.47750e8			−0.476548			−0.238274	−	0.971198i	$-0.576582\pi$
$151$	−2.47750e8			−0.476548			−0.238274	+	0.971198i	$0.576582\pi$
$152$		0			0
$153$	−1.84899e8	+	2.78436e8i	−0.337419	+	0.508112i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	1.15674e9			1.90388			0.951938	−	0.306290i	$-0.0990879\pi$
$157$	1.15674e9			1.90388			0.951938	+	0.306290i	$0.0990879\pi$
$158$		0			0
$159$	−4.32296e7	−	1.30463e7i	−0.0676383	−	0.0204127i
$160$		0			0
$161$		−	8.20834e7i		−	0.122166i
$162$		0			0
$163$	2.04950e8			0.290334			0.145167	−	0.989407i	$-0.453628\pi$
$163$	2.04950e8			0.290334			0.145167	+	0.989407i	$0.453628\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			9.71371e8i			1.24888i	0.781074	+	0.624438i	$0.214672\pi$
$167$			9.71371e8i			1.24888i	−0.781074	+	0.624438i	$0.785328\pi$
$168$		0			0
$169$	−8.04294e8			−0.985980
$170$		0			0
$171$	−2.53415e8	−	1.68284e8i	−0.296380	−	0.196815i
$172$		0			0
$173$			1.39561e9i			1.55804i	0.626998	+	0.779020i	$0.284283\pi$
$173$			1.39561e9i			1.55804i	−0.626998	+	0.779020i	$0.715717\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	1.47560e9	+	4.45323e8i	1.50340	+	0.453714i
$178$		0			0
$179$			1.35408e8i			0.131896i	0.997823	+	0.0659481i	$0.0210072\pi$
$179$			1.35408e8i			0.131896i	−0.997823	+	0.0659481i	$0.978993\pi$
$180$		0			0
$181$	1.18630e9			1.10530			0.552651	−	0.833413i	$-0.313616\pi$
$181$	1.18630e9			1.10530			0.552651	+	0.833413i	$0.313616\pi$
$182$		0			0
$183$	−1.12596e8	+	3.73093e8i	−0.100397	+	0.332669i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	6.34981e8			0.519271
$188$		0			0
$189$	−1.05124e8	+	8.70279e7i	−0.0823866	+	0.0682042i
$190$		0			0
$191$			1.52674e9i			1.14718i	0.819143	+	0.573589i	$0.194449\pi$
$191$			1.52674e9i			1.14718i	−0.819143	+	0.573589i	$0.805551\pi$
$192$		0			0
$193$	−1.02295e9			−0.737267			−0.368634	−	0.929575i	$-0.620174\pi$
$193$	−1.02295e9			−0.737267			−0.368634	+	0.929575i	$0.620174\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			2.45145e9i			1.62764i	0.581117	+	0.813820i	$0.302616\pi$
$197$			2.45145e9i			1.62764i	−0.581117	+	0.813820i	$0.697384\pi$
$198$		0			0
$199$	−1.75331e9			−1.11801			−0.559006	−	0.829164i	$-0.688817\pi$
$199$	−1.75331e9			−1.11801			−0.559006	+	0.829164i	$0.688817\pi$
$200$		0			0
$201$	3.40187e8	−	1.12722e9i	0.208417	−	0.690600i
$202$		0			0
$203$			7.97652e7i			0.0469710i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−1.16015e9	+	1.74704e9i	−0.631876	+	0.951527i
$208$		0			0
$209$			5.77921e8i			0.302889i
$210$		0			0
$211$	−3.63722e9			−1.83501			−0.917507	−	0.397719i	$-0.869802\pi$
$211$	−3.63722e9			−1.83501			−0.917507	+	0.397719i	$0.869802\pi$
$212$		0			0
$213$	−2.71703e8	−	8.19979e7i	−0.132001	−	0.0398368i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−2.14410e8			−0.0966954
$218$		0			0
$219$	−1.08778e9	+	3.60442e9i	−0.472897	+	1.56696i
$220$		0			0
$221$		−	1.72278e8i		−	0.0722205i
$222$		0			0
$223$	2.29784e9			0.929181			0.464590	−	0.885526i	$-0.346202\pi$
$223$	2.29784e9			0.929181			0.464590	+	0.885526i	$0.346202\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			2.44052e9i			0.919134i	0.888143	+	0.459567i	$0.151995\pi$
$227$			2.44052e9i			0.919134i	−0.888143	+	0.459567i	$0.848005\pi$
$228$		0			0
$229$	1.38104e9			0.502186			0.251093	−	0.967963i	$-0.419210\pi$
$229$	1.38104e9			0.502186			0.251093	+	0.967963i	$0.419210\pi$
$230$		0			0
$231$	2.48214e8	+	7.49090e7i	0.0871723	+	0.0263079i
$232$		0			0
$233$			1.38282e9i			0.469184i	0.972094	+	0.234592i	$0.0753754\pi$
$233$			1.38282e9i			0.469184i	−0.972094	+	0.234592i	$0.924625\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−1.09402e9	+	3.62508e9i	−0.346762	+	1.14901i
$238$		0			0
$239$		−	4.95875e9i		−	1.51978i	−0.650053	−	0.759889i	$-0.725253\pi$
$239$		−	4.95875e9i		−	1.51978i	0.650053	−	0.759889i	$-0.274747\pi$
$240$		0			0
$241$	5.26001e8			0.155926			0.0779629	−	0.996956i	$-0.475158\pi$
$241$	5.26001e8			0.155926			0.0779629	+	0.996956i	$0.475158\pi$
$242$		0			0
$243$	3.46747e9	−	3.66471e8i	0.994461	−	0.105103i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	1.56797e8			0.0421260
$248$		0			0
$249$	8.58341e8	+	2.59040e8i	0.223286	+	0.0673860i
$250$		0			0
$251$			3.97738e9i			1.00208i	0.865424	+	0.501040i	$0.167049\pi$
$251$			3.97738e9i			1.00208i	−0.865424	+	0.501040i	$0.832951\pi$
$252$		0			0
$253$	3.98417e9			0.972425
$254$		0			0
$255$		0			0
$256$		0			0
$257$			1.36791e9i			0.313562i	0.987633	+	0.156781i	$0.0501117\pi$
$257$			1.36791e9i			0.313562i	−0.987633	+	0.156781i	$0.949888\pi$
$258$		0			0
$259$	−7.40219e8			−0.164498
$260$		0			0
$261$	1.12738e9	−	1.69770e9i	0.242946	−	0.365847i
$262$		0			0
$263$			4.59660e9i			0.960758i	0.877061	+	0.480379i	$0.159501\pi$
$263$			4.59660e9i			0.960758i	−0.877061	+	0.480379i	$0.840499\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	5.51942e9	+	1.66571e9i	1.08605	+	0.327760i
$268$		0			0
$269$		−	1.55008e9i		−	0.296037i	−0.988985	−	0.148019i	$-0.952710\pi$
$269$		−	1.55008e9i		−	0.296037i	0.988985	−	0.148019i	$-0.0472895\pi$
$270$		0			0
$271$	3.32135e7			0.00615797			0.00307898	−	0.999995i	$-0.499020\pi$
$271$	3.32135e7			0.00615797			0.00307898	+	0.999995i	$0.499020\pi$
$272$		0			0
$273$	2.03237e7	−	6.73435e7i	0.00365892	−	0.0121240i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.73846e9			−0.295287			−0.147644	−	0.989041i	$-0.547169\pi$
$277$	−1.73846e9			−0.295287			−0.147644	+	0.989041i	$0.547169\pi$
$278$		0			0
$279$	4.56344e9	+	3.03042e9i	0.753139	+	0.500134i
$280$		0			0
$281$		−	3.54161e9i		−	0.568036i	−0.958819	−	0.284018i	$-0.908332\pi$
$281$		−	3.54161e9i		−	0.568036i	0.958819	−	0.284018i	$-0.0916675\pi$
$282$		0			0
$283$	−8.01341e9			−1.24931			−0.624657	−	0.780899i	$-0.714761\pi$
$283$	−8.01341e9			−1.24931			−0.624657	+	0.780899i	$0.714761\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			5.75335e8i			0.0847995i
$288$		0			0
$289$	4.38057e9			0.627971
$290$		0			0
$291$	1.54273e9	−	5.11192e9i	0.215139	−	0.712873i
$292$		0			0
$293$		−	2.36667e9i		−	0.321120i	−0.987026	−	0.160560i	$-0.948670\pi$
$293$		−	2.36667e9i		−	0.321120i	0.987026	−	0.160560i	$-0.0513300\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−4.22417e9	−	5.10255e9i	−0.542895	−	0.655784i
$298$		0			0
$299$		−	1.08095e9i		−	0.135245i
$300$		0			0
$301$	1.31534e9			0.160241
$302$		0			0
$303$	−1.33282e10	−	4.02234e9i	−1.58125	−	0.477209i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	1.04832e10			1.18016			0.590078	−	0.807346i	$-0.299097\pi$
$307$	1.04832e10			1.18016			0.590078	+	0.807346i	$0.299097\pi$
$308$		0			0
$309$	2.89173e9	−	9.58188e9i	0.317193	−	1.05103i
$310$		0			0
$311$			3.67560e9i			0.392904i	0.980513	+	0.196452i	$0.0629420\pi$
$311$			3.67560e9i			0.392904i	−0.980513	+	0.196452i	$0.937058\pi$
$312$		0			0
$313$	−4.54244e9			−0.473273			−0.236637	−	0.971598i	$-0.576045\pi$
$313$	−4.54244e9			−0.473273			−0.236637	+	0.971598i	$0.576045\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	6.11764e9i		−	0.605824i	−0.953018	−	0.302912i	$-0.902041\pi$
$317$		−	6.11764e9i		−	0.605824i	0.953018	−	0.302912i	$-0.0979589\pi$
$318$		0			0
$319$	−3.87166e9			−0.373881
$320$		0			0
$321$	1.67052e10	+	5.04148e9i	1.57337	+	0.474830i
$322$		0			0
$323$		−	2.36198e9i		−	0.217003i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	2.15529e9	−	7.14164e9i	0.188501	−	0.624608i
$328$		0			0
$329$			9.09183e8i			0.0776011i
$330$		0			0
$331$	−9.30830e9			−0.775458			−0.387729	−	0.921773i	$-0.626740\pi$
$331$	−9.30830e9			−0.775458			−0.387729	+	0.921773i	$0.626740\pi$
$332$		0			0
$333$	1.57546e10	+	1.04621e10i	1.28124	+	0.850828i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−6.19237e9			−0.480106			−0.240053	−	0.970760i	$-0.577165\pi$
$337$	−6.19237e9			−0.480106			−0.240053	+	0.970760i	$0.577165\pi$
$338$		0			0
$339$	−4.50583e9	−	1.35982e9i	−0.341174	−	0.102964i
$340$		0			0
$341$		−	1.04070e10i		−	0.769680i
$342$		0			0
$343$	−2.94385e9			−0.212686
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	2.13615e10i		−	1.47338i	−0.676231	−	0.736690i	$-0.736388\pi$
$347$		−	2.13615e10i		−	1.47338i	0.676231	−	0.736690i	$-0.263612\pi$
$348$		0			0
$349$	2.38609e9			0.160837			0.0804185	−	0.996761i	$-0.474374\pi$
$349$	2.38609e9			0.160837			0.0804185	+	0.996761i	$0.474374\pi$
$350$		0			0
$351$	−1.38438e9	+	1.14607e9i	−0.0912069	+	0.0755062i
$352$		0			0
$353$		−	1.60900e10i		−	1.03623i	−0.855310	−	0.518117i	$-0.826633\pi$
$353$		−	1.60900e10i		−	1.03623i	0.855310	−	0.518117i	$-0.173367\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−1.01446e9	−	3.06155e8i	−0.0624542	−	0.0188482i
$358$		0			0
$359$			1.72981e10i			1.04141i	0.853738	+	0.520703i	$0.174330\pi$
$359$			1.72981e10i			1.04141i	−0.853738	+	0.520703i	$0.825670\pi$
$360$		0			0
$361$	−1.48338e10			−0.873423
$362$		0			0
$363$	1.38062e9	−	4.57473e9i	0.0795145	−	0.263475i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−1.66116e10			−0.915687			−0.457844	−	0.889033i	$-0.651378\pi$
$367$	−1.66116e10			−0.915687			−0.457844	+	0.889033i	$0.651378\pi$
$368$		0			0
$369$	8.13166e9	−	1.22453e10i	0.438605	−	0.660485i
$370$		0			0
$371$		−	1.43159e8i		−	0.00755652i
$372$		0			0
$373$	3.18030e10			1.64298			0.821492	−	0.570220i	$-0.193142\pi$
$373$	3.18030e10			1.64298			0.821492	+	0.570220i	$0.193142\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			1.05043e9i			0.0519997i
$378$		0			0
$379$	−1.39442e10			−0.675830			−0.337915	−	0.941177i	$-0.609722\pi$
$379$	−1.39442e10			−0.675830			−0.337915	+	0.941177i	$0.609722\pi$
$380$		0			0
$381$	−7.96116e9	+	2.63797e10i	−0.377813	+	1.25190i
$382$		0			0
$383$		−	1.58220e10i		−	0.735304i	−0.929963	−	0.367652i	$-0.880162\pi$
$383$		−	1.58220e10i		−	0.735304i	0.929963	−	0.367652i	$-0.119838\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−2.79954e10	−	1.85908e10i	−1.24808	−	0.828807i
$388$		0			0
$389$			3.32728e10i			1.45308i	0.687122	+	0.726542i	$0.258874\pi$
$389$			3.32728e10i			1.45308i	−0.687122	+	0.726542i	$0.741126\pi$
$390$		0			0
$391$	−1.62834e10			−0.696689
$392$		0			0
$393$	−1.99339e10	−	6.01589e9i	−0.835645	−	0.252191i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−2.94761e10			−1.18661			−0.593305	−	0.804978i	$-0.702177\pi$
$397$	−2.94761e10			−1.18661			−0.593305	+	0.804978i	$0.702177\pi$
$398$		0			0
$399$	2.78644e8	−	9.23299e8i	0.0109941	−	0.0364293i
$400$		0			0
$401$		−	2.57220e10i		−	0.994780i	−0.867527	−	0.497390i	$-0.834292\pi$
$401$		−	2.57220e10i		−	0.994780i	0.867527	−	0.497390i	$-0.165708\pi$
$402$		0			0
$403$	−2.82356e9			−0.107048
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	3.59289e10i		−	1.30938i
$408$		0			0
$409$	−4.13595e10			−1.47803			−0.739014	−	0.673690i	$-0.764708\pi$
$409$	−4.13595e10			−1.47803			−0.739014	+	0.673690i	$0.764708\pi$
$410$		0			0
$411$	4.58938e10	+	1.38504e10i	1.60837	+	0.485393i
$412$		0			0
$413$			4.88657e9i			0.167959i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	1.29619e10	−	4.29497e10i	0.428670	−	1.42042i
$418$		0			0
$419$		−	2.60034e10i		−	0.843674i	−0.906672	−	0.421837i	$-0.861385\pi$
$419$		−	2.60034e10i		−	0.843674i	0.906672	−	0.421837i	$-0.138615\pi$
$420$		0			0
$421$	−3.62239e10			−1.15310			−0.576550	−	0.817062i	$-0.695601\pi$
$421$	−3.62239e10			−1.15310			−0.576550	+	0.817062i	$0.695601\pi$
$422$		0			0
$423$	1.28502e10	−	1.93508e10i	0.401373	−	0.604418i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−1.23553e9			−0.0371656
$428$		0			0
$429$	3.26873e9	+	9.86475e8i	0.0965050	+	0.0291244i
$430$		0			0
$431$			1.08452e10i			0.314289i	0.987576	+	0.157144i	$0.0502288\pi$
$431$			1.08452e10i			0.314289i	−0.987576	+	0.157144i	$0.949771\pi$
$432$		0			0
$433$	−5.70222e10			−1.62216			−0.811078	−	0.584939i	$-0.801119\pi$
$433$	−5.70222e10			−1.62216			−0.811078	+	0.584939i	$0.801119\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	1.48202e10i		−	0.406376i
$438$		0			0
$439$	7.30245e8			0.0196612			0.00983060	−	0.999952i	$-0.496871\pi$
$439$	7.30245e8			0.0196612			0.00983060	+	0.999952i	$0.496871\pi$
$440$		0			0
$441$	3.11479e10	+	2.06842e10i	0.823520	+	0.546871i
$442$		0			0
$443$		−	4.47298e10i		−	1.16140i	−0.814117	−	0.580701i	$-0.802779\pi$
$443$		−	4.47298e10i		−	1.16140i	0.814117	−	0.580701i	$-0.197221\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	1.60772e10	+	4.85197e9i	0.402699	+	0.121531i
$448$		0			0
$449$			2.24769e10i			0.553033i	0.961009	+	0.276517i	$0.0891801\pi$
$449$			2.24769e10i			0.553033i	−0.961009	+	0.276517i	$0.910820\pi$
$450$		0			0
$451$	−2.79257e10			−0.674991
$452$		0			0
$453$	−5.79801e9	+	1.92119e10i	−0.137685	+	0.456225i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	7.54064e10			1.72880			0.864398	−	0.502809i	$-0.167700\pi$
$457$	7.54064e10			1.72880			0.864398	+	0.502809i	$0.167700\pi$
$458$		0			0
$459$	1.72643e10	+	2.08543e10i	0.388955	+	0.469834i
$460$		0			0
$461$		−	2.50914e10i		−	0.555547i	−0.960647	−	0.277774i	$-0.910403\pi$
$461$		−	2.50914e10i		−	0.555547i	0.960647	−	0.277774i	$-0.0895965\pi$
$462$		0			0
$463$	−9.80089e9			−0.213276			−0.106638	−	0.994298i	$-0.534009\pi$
$463$	−9.80089e9			−0.213276			−0.106638	+	0.994298i	$0.534009\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	5.86194e10i		−	1.23246i	−0.787565	−	0.616232i	$-0.788659\pi$
$467$		−	5.86194e10i		−	1.23246i	0.787565	−	0.616232i	$-0.211341\pi$
$468$		0			0
$469$	3.73290e9			0.0771534
$470$		0			0
$471$	2.70708e10	−	8.97004e10i	0.550070	−	1.82268i
$472$		0			0
$473$			6.38443e10i			1.27549i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−2.02337e9	+	3.04695e9i	−0.0390843	+	0.0588561i
$478$		0			0
$479$		−	3.02045e10i		−	0.573759i	−0.957967	−	0.286879i	$-0.907382\pi$
$479$		−	3.02045e10i		−	0.573759i	0.957967	−	0.286879i	$-0.0926179\pi$
$480$		0			0
$481$	−9.74794e9			−0.182109
$482$		0			0
$483$	−6.36520e9	−	1.92097e9i	−0.116956	−	0.0352965i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−2.62697e10			−0.467023			−0.233512	−	0.972354i	$-0.575022\pi$
$487$	−2.62697e10			−0.467023			−0.233512	+	0.972354i	$0.575022\pi$
$488$		0			0
$489$	4.79638e9	−	1.58930e10i	0.0838837	−	0.277952i
$490$		0			0
$491$			5.10056e10i			0.877591i	0.898587	+	0.438795i	$0.144595\pi$
$491$			5.10056e10i			0.877591i	−0.898587	+	0.438795i	$0.855405\pi$
$492$		0			0
$493$	1.58236e10			0.267866
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	8.99769e8i		−	0.0147471i
$498$		0			0
$499$	2.74914e10			0.443398			0.221699	−	0.975115i	$-0.428840\pi$
$499$	2.74914e10			0.443398			0.221699	+	0.975115i	$0.428840\pi$
$500$		0			0
$501$	7.53255e10	+	2.27326e10i	1.19562	+	0.360827i
$502$		0			0
$503$		−	2.31491e10i		−	0.361628i	−0.983517	−	0.180814i	$-0.942127\pi$
$503$		−	2.31491e10i		−	0.361628i	0.983517	−	0.180814i	$-0.0578733\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1.88226e10	+	6.23695e10i	−0.284871	+	0.943931i
$508$		0			0
$509$		−	9.43836e10i		−	1.40613i	−0.711125	−	0.703065i	$-0.751814\pi$
$509$		−	9.43836e10i		−	1.40613i	0.711125	−	0.703065i	$-0.248186\pi$
$510$		0			0
$511$	−1.19363e10			−0.175060
$512$		0			0
$513$	−1.89803e10	+	1.57129e10i	−0.274052	+	0.226876i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−4.41301e10			−0.617692
$518$		0			0
$519$	1.08223e11	+	3.26609e10i	1.49159	+	0.450151i
$520$		0			0
$521$			8.47409e10i			1.15012i	0.818112	+	0.575059i	$0.195021\pi$
$521$			8.47409e10i			1.15012i	−0.818112	+	0.575059i	$0.804979\pi$
$522$		0			0
$523$	6.40087e10			0.855523			0.427762	−	0.903892i	$-0.359302\pi$
$523$	6.40087e10			0.855523			0.427762	+	0.903892i	$0.359302\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			4.25339e10i			0.551433i
$528$		0			0
$529$	−2.38592e10			−0.304672
$530$		0			0
$531$	6.90657e10	−	1.04004e11i	0.868729	−	1.30820i
$532$		0			0
$533$			7.57658e9i			0.0938781i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	1.05003e10	+	3.16890e9i	0.126271	+	0.0381076i
$538$		0			0
$539$		−	7.10336e10i		−	0.841606i
$540$		0			0
$541$	−1.54593e11			−1.80468			−0.902341	−	0.431022i	$-0.858153\pi$
$541$	−1.54593e11			−1.80468			−0.902341	+	0.431022i	$0.858153\pi$
$542$		0			0
$543$	2.77626e10	−	9.19925e10i	0.319345	−	1.05816i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−8.26663e10			−0.923378			−0.461689	−	0.887042i	$-0.652756\pi$
$547$	−8.26663e10			−0.923378			−0.461689	+	0.887042i	$0.652756\pi$
$548$		0			0
$549$	2.62966e10	+	1.74627e10i	0.289475	+	0.192230i
$550$		0			0
$551$			1.44017e10i			0.156245i
$552$		0			0
$553$	−1.20048e10			−0.128367
$554$		0			0
$555$		0			0
$556$		0			0
$557$			9.69894e10i			1.00764i	0.863810	+	0.503818i	$0.168072\pi$
$557$			9.69894e10i			1.00764i	−0.863810	+	0.503818i	$0.831928\pi$
$558$		0			0
$559$	1.73217e10			0.177396
$560$		0			0
$561$	1.48602e10	−	4.92399e10i	0.150028	−	0.497126i
$562$		0			0
$563$			2.23100e10i			0.222057i	0.993817	+	0.111029i	$0.0354145\pi$
$563$			2.23100e10i			0.222057i	−0.993817	+	0.111029i	$0.964585\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	4.28844e9	+	1.01886e10i	0.0414923	+	0.0985787i
$568$		0			0
$569$			1.30461e11i			1.24461i	0.782776	+	0.622303i	$0.213803\pi$
$569$			1.30461e11i			1.24461i	−0.782776	+	0.622303i	$0.786197\pi$
$570$		0			0
$571$	−9.73124e10			−0.915427			−0.457714	−	0.889100i	$-0.651332\pi$
$571$	−9.73124e10			−0.915427			−0.457714	+	0.889100i	$0.651332\pi$
$572$		0			0
$573$	1.18392e11	+	3.57297e10i	1.09825	+	0.331444i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	1.19953e11			1.08220			0.541101	−	0.840958i	$-0.318008\pi$
$577$	1.19953e11			1.08220			0.541101	+	0.840958i	$0.318008\pi$
$578$		0			0
$579$	−2.39397e10	+	7.93252e10i	−0.213012	+	0.705825i
$580$		0			0
$581$			2.84247e9i			0.0249454i
$582$		0			0
$583$	6.94865e9			0.0601487
$584$		0			0
$585$		0			0
$586$		0			0
$587$			1.54595e11i			1.30209i	0.759038	+	0.651046i	$0.225670\pi$
$587$			1.54595e11i			1.30209i	−0.759038	+	0.651046i	$0.774330\pi$
$588$		0			0
$589$	−3.87118e10			−0.321649
$590$		0			0
$591$	1.90099e11	+	5.73704e10i	1.55823	+	0.470260i
$592$		0			0
$593$			1.65495e11i			1.33834i	0.743110	+	0.669169i	$0.233350\pi$
$593$			1.65495e11i			1.33834i	−0.743110	+	0.669169i	$0.766650\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−4.10321e10	+	1.35961e11i	−0.323017	+	1.07033i
$598$		0			0
$599$		−	1.77549e11i		−	1.37914i	−0.724217	−	0.689572i	$-0.757798\pi$
$599$		−	1.77549e11i		−	1.37914i	0.724217	−	0.689572i	$-0.242202\pi$
$600$		0			0
$601$	−1.38738e11			−1.06340			−0.531700	−	0.846933i	$-0.678447\pi$
$601$	−1.38738e11			−1.06340			−0.531700	+	0.846933i	$0.678447\pi$
$602$		0			0
$603$	−7.94500e10	−	5.27600e10i	−0.600931	−	0.399058i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	1.41461e11			1.04204			0.521018	−	0.853546i	$-0.325552\pi$
$607$	1.41461e11			1.04204			0.521018	+	0.853546i	$0.325552\pi$
$608$		0			0
$609$	6.18544e9	+	1.86672e9i	0.0449678	+	0.0135709i
$610$		0			0
$611$			1.19730e10i			0.0859090i
$612$		0			0
$613$	1.18052e11			0.836047			0.418024	−	0.908436i	$-0.362723\pi$
$613$	1.18052e11			0.836047			0.418024	+	0.908436i	$0.362723\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			1.21381e11i			0.837550i	0.908090	+	0.418775i	$0.137540\pi$
$617$			1.21381e11i			0.837550i	−0.908090	+	0.418775i	$0.862460\pi$
$618$		0			0
$619$	−2.48460e11			−1.69237			−0.846183	−	0.532892i	$-0.821105\pi$
$619$	−2.48460e11			−1.69237			−0.846183	+	0.532892i	$0.821105\pi$
$620$		0			0
$621$	1.08325e11	+	1.30850e11i	0.728385	+	0.879845i
$622$		0			0
$623$			1.82780e10i			0.121333i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	4.48152e10	+	1.35249e10i	0.289971	+	0.0875110i
$628$		0			0
$629$			1.46842e11i			0.938099i
$630$		0			0
$631$	−2.51072e10			−0.158373			−0.0791866	−	0.996860i	$-0.525232\pi$
$631$	−2.51072e10			−0.158373			−0.0791866	+	0.996860i	$0.525232\pi$
$632$		0			0
$633$	−8.51204e10	+	2.82050e11i	−0.530175	+	1.75676i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−1.92723e10			−0.117051
$638$		0			0
$639$	−1.27171e10	+	1.91504e10i	−0.0762757	+	0.114862i
$640$		0			0
$641$		−	1.60061e11i		−	0.948097i	−0.880499	−	0.474049i	$-0.842792\pi$
$641$		−	1.60061e11i		−	0.948097i	0.880499	−	0.474049i	$-0.157208\pi$
$642$		0			0
$643$	−2.15414e11			−1.26017			−0.630086	−	0.776525i	$-0.716980\pi$
$643$	−2.15414e11			−1.26017			−0.630086	+	0.776525i	$0.716980\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	6.77038e10i		−	0.386363i	−0.981163	−	0.193182i	$-0.938119\pi$
$647$		−	6.77038e10i		−	0.386363i	0.981163	−	0.193182i	$-0.0618806\pi$
$648$		0			0
$649$	−2.37185e11			−1.33693
$650$		0			0
$651$	−5.01775e9	+	1.66265e10i	−0.0279373	+	0.0925716i
$652$		0			0
$653$			6.26731e10i			0.344690i	0.985037	+	0.172345i	$0.0551343\pi$
$653$			6.26731e10i			0.344690i	−0.985037	+	0.172345i	$0.944866\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	2.54050e11	+	1.68706e11i	1.36351	+	0.905458i
$658$		0			0
$659$			3.04415e11i			1.61408i	0.590499	+	0.807038i	$0.298931\pi$
$659$			3.04415e11i			1.61408i	−0.590499	+	0.807038i	$0.701069\pi$
$660$		0			0
$661$	−2.00579e11			−1.05070			−0.525351	−	0.850886i	$-0.676066\pi$
$661$	−2.00579e11			−1.05070			−0.525351	+	0.850886i	$0.676066\pi$
$662$		0			0
$663$	−1.33594e10	−	4.03176e9i	−0.0691405	−	0.0208660i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	9.92847e10			0.501625
$668$		0			0
$669$	5.37755e10	−	1.78187e11i	0.268460	−	0.889554i
$670$		0			0
$671$		−	5.99702e10i		−	0.295832i
$672$		0			0
$673$	8.29793e9			0.0404492			0.0202246	−	0.999795i	$-0.493562\pi$
$673$	8.29793e9			0.0404492			0.0202246	+	0.999795i	$0.493562\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			3.92697e11i			1.86940i	0.355433	+	0.934702i	$0.384333\pi$
$677$			3.92697e11i			1.86940i	−0.355433	+	0.934702i	$0.615667\pi$
$678$		0			0
$679$	1.69286e10			0.0796418
$680$		0			0
$681$	1.89252e11	+	5.71146e10i	0.879936	+	0.265557i
$682$		0			0
$683$		−	2.74429e11i		−	1.26109i	−0.776151	−	0.630547i	$-0.782830\pi$
$683$		−	2.74429e11i		−	1.26109i	0.776151	−	0.630547i	$-0.217170\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	3.23200e10	−	1.07094e11i	0.145092	−	0.480769i
$688$		0			0
$689$		−	1.88525e9i		−	0.00836551i
$690$		0			0
$691$	−1.37801e11			−0.604420			−0.302210	−	0.953241i	$-0.597724\pi$
$691$	−1.37801e11			−0.604420			−0.302210	+	0.953241i	$0.597724\pi$
$692$		0			0
$693$	1.16177e10	−	1.74948e10i	0.0503719	−	0.0758538i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	1.14133e11			0.483594
$698$		0			0
$699$	1.07232e11	+	3.23617e10i	0.449175	+	0.135557i
$700$		0			0
$701$		−	6.74100e9i		−	0.0279159i	−0.999903	−	0.0139580i	$-0.995557\pi$
$701$		−	6.74100e9i		−	0.0279159i	0.999903	−	0.0139580i	$-0.00444310\pi$
$702$		0			0
$703$	−1.33647e11			−0.547190
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	4.41375e10i		−	0.176657i
$708$		0			0
$709$	2.20795e11			0.873786			0.436893	−	0.899513i	$-0.356079\pi$
$709$	2.20795e11			0.873786			0.436893	+	0.899513i	$0.356079\pi$
$710$		0			0
$711$	2.55506e11	+	1.69673e11i	0.999822	+	0.663948i
$712$		0			0
$713$			2.66878e11i			1.03266i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−3.84529e11	−	1.16048e11i	−1.45496	−	0.439096i
$718$		0			0
$719$		−	9.64414e10i		−	0.360867i	−0.983587	−	0.180434i	$-0.942250\pi$
$719$		−	9.64414e10i		−	0.360867i	0.983587	−	0.180434i	$-0.0577501\pi$
$720$		0			0
$721$	3.17312e10			0.117421
$722$		0			0
$723$	1.23098e10	−	4.07890e10i	0.0450503	−	0.149276i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−4.98426e10			−0.178428			−0.0892141	−	0.996012i	$-0.528436\pi$
$727$	−4.98426e10			−0.178428			−0.0892141	+	0.996012i	$0.528436\pi$
$728$		0			0
$729$	5.27297e10	−	2.77464e11i	0.186700	−	0.982417i
$730$		0			0
$731$		−	2.60934e11i		−	0.913819i
$732$		0			0
$733$	5.26848e11			1.82503			0.912514	−	0.409046i	$-0.134138\pi$
$733$	5.26848e11			1.82503			0.912514	+	0.409046i	$0.134138\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			1.81188e11i			0.614129i
$738$		0			0
$739$	−2.03810e10			−0.0683357			−0.0341679	−	0.999416i	$-0.510878\pi$
$739$	−2.03810e10			−0.0683357			−0.0341679	+	0.999416i	$0.510878\pi$
$740$		0			0
$741$	3.66946e9	−	1.21589e10i	0.0121711	−	0.0403294i
$742$		0			0
$743$		−	2.59879e11i		−	0.852738i	−0.904549	−	0.426369i	$-0.859793\pi$
$743$		−	2.59879e11i		−	0.852738i	0.904549	−	0.426369i	$-0.140207\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	4.01748e10	−	6.04983e10i	0.129024	−	0.194295i
$748$		0			0
$749$			5.53206e10i			0.175776i
$750$		0			0
$751$	−3.65510e11			−1.14905			−0.574526	−	0.818486i	$-0.694814\pi$
$751$	−3.65510e11			−1.14905			−0.574526	+	0.818486i	$0.694814\pi$
$752$		0			0
$753$	3.08428e11	+	9.30812e10i	0.959344	+	0.289522i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−5.23181e11			−1.59319			−0.796597	−	0.604511i	$-0.793369\pi$
$757$	−5.23181e11			−1.59319			−0.796597	+	0.604511i	$0.793369\pi$
$758$		0			0
$759$	9.32401e10	−	3.08955e11i	0.280954	−	0.930954i
$760$		0			0
$761$			2.74492e11i			0.818447i	0.912434	+	0.409223i	$0.134200\pi$
$761$			2.74492e11i			0.818447i	−0.912434	+	0.409223i	$0.865800\pi$
$762$		0			0
$763$	2.36502e10			0.0697808
$764$		0			0
$765$		0			0
$766$		0			0
$767$			6.43512e10i			0.185941i
$768$		0			0
$769$	2.78182e11			0.795471			0.397735	−	0.917500i	$-0.369796\pi$
$769$	2.78182e11			0.795471			0.397735	+	0.917500i	$0.369796\pi$
$770$		0			0
$771$	1.06075e11	+	3.20126e10i	0.300190	+	0.0905948i
$772$		0			0
$773$			2.91471e11i			0.816352i	0.912903	+	0.408176i	$0.133835\pi$
$773$			2.91471e11i			0.816352i	−0.912903	+	0.408176i	$0.866165\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−1.73231e10	+	5.74007e10i	−0.0475270	+	0.157483i
$778$		0			0
$779$			1.03877e11i			0.282078i
$780$		0			0
$781$	4.36731e10			0.117384
$782$		0			0
$783$	−1.05265e11	−	1.27154e11i	−0.280052	−	0.338286i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−2.88982e10			−0.0753307			−0.0376654	−	0.999290i	$-0.511992\pi$
$787$	−2.88982e10			−0.0753307			−0.0376654	+	0.999290i	$0.511992\pi$
$788$		0			0
$789$	3.56446e11	+	1.07573e11i	0.919785	+	0.277583i
$790$		0			0
$791$		−	1.49214e10i		−	0.0381158i
$792$		0			0
$793$	−1.62707e10			−0.0411445
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	3.73379e11i		−	0.925372i	−0.886522	−	0.462686i	$-0.846886\pi$
$797$		−	3.73379e11i		−	0.925372i	0.886522	−	0.462686i	$-0.153114\pi$
$798$		0			0
$799$	1.80361e11			0.442543
$800$		0			0
$801$	2.58338e11	−	3.89024e11i	0.627564	−	0.945033i
$802$		0			0
$803$		−	5.79368e11i		−	1.39345i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−1.20202e11	−	3.62760e10i	−0.283412	−	0.0855314i
$808$		0			0
$809$		−	4.29442e11i		−	1.00256i	−0.865285	−	0.501280i	$-0.832863\pi$
$809$		−	4.29442e11i		−	1.00256i	0.865285	−	0.501280i	$-0.167137\pi$
$810$		0			0
$811$	−1.94016e11			−0.448491			−0.224246	−	0.974533i	$-0.571992\pi$
$811$	−1.94016e11			−0.448491			−0.224246	+	0.974533i	$0.571992\pi$
$812$		0			0
$813$	7.77283e8	−	2.57556e9i	0.00177917	−	0.00589535i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	2.37486e11			0.533027
$818$		0			0
$819$	−4.74656e9	−	3.15203e9i	−0.0105498	−	0.00700575i
$820$		0			0
$821$		−	6.65867e11i		−	1.46560i	−0.680445	−	0.732799i	$-0.738213\pi$
$821$		−	6.65867e11i		−	1.46560i	0.680445	−	0.732799i	$-0.261787\pi$
$822$		0			0
$823$	−3.53299e11			−0.770093			−0.385047	−	0.922897i	$-0.625815\pi$
$823$	−3.53299e11			−0.770093			−0.385047	+	0.922897i	$0.625815\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	8.61188e11i		−	1.84109i	−0.390631	−	0.920547i	$-0.627743\pi$
$827$		−	8.61188e11i		−	1.84109i	0.390631	−	0.920547i	$-0.372257\pi$
$828$		0			0
$829$	−1.44570e11			−0.306097			−0.153048	−	0.988219i	$-0.548909\pi$
$829$	−1.44570e11			−0.306097			−0.153048	+	0.988219i	$0.548909\pi$
$830$		0			0
$831$	−4.06844e10	+	1.34810e11i	−0.0853147	+	0.282694i
$832$		0			0
$833$			2.90317e11i			0.602965i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	3.41792e11	−	2.82955e11i	0.696402	−	0.576521i
$838$		0			0
$839$			3.35487e11i			0.677060i	0.940956	+	0.338530i	$0.109930\pi$
$839$			3.35487e11i			0.677060i	−0.940956	+	0.338530i	$0.890070\pi$
$840$		0			0
$841$	4.03766e11			0.807133
$842$		0			0
$843$	−2.74636e11	−	8.28830e10i	−0.543811	−	0.164118i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	1.51496e10			0.0294353
$848$		0			0
$849$	−1.87535e11	+	6.21405e11i	−0.360954	+	1.19604i
$850$		0			0
$851$			9.21360e11i			1.75675i
$852$		0			0
$853$	−5.14823e11			−0.972438			−0.486219	−	0.873837i	$-0.661624\pi$
$853$	−5.14823e11			−0.972438			−0.486219	+	0.873837i	$0.661624\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	6.92341e11i		−	1.28350i	−0.766913	−	0.641752i	$-0.778208\pi$
$857$		−	6.92341e11i		−	1.28350i	0.766913	−	0.641752i	$-0.221792\pi$
$858$		0			0
$859$	−9.80593e11			−1.80101			−0.900505	−	0.434846i	$-0.856803\pi$
$859$	−9.80593e11			−1.80101			−0.900505	+	0.434846i	$0.856803\pi$
$860$		0			0
$861$	4.46147e10	+	1.34643e10i	0.0811830	+	0.0245004i
$862$		0			0
$863$		−	5.66878e11i		−	1.02199i	−0.859584	−	0.510994i	$-0.829277\pi$
$863$		−	5.66878e11i		−	1.02199i	0.859584	−	0.510994i	$-0.170723\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	1.02517e11	−	3.39694e11i	0.181434	−	0.601190i
$868$		0			0
$869$		−	5.82689e11i		−	1.02178i
$870$		0			0
$871$	4.91585e10			0.0854135
$872$		0			0
$873$	−3.60303e11	−	2.39265e11i	−0.620312	−	0.411928i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.73028e11			0.292495			0.146248	−	0.989248i	$-0.453280\pi$
$877$	1.73028e11			0.292495			0.146248	+	0.989248i	$0.453280\pi$
$878$		0			0
$879$	−1.83525e11	−	5.53862e10i	−0.307425	−	0.0927783i
$880$		0			0
$881$			4.19964e11i			0.697121i	0.937286	+	0.348560i	$0.113329\pi$
$881$			4.19964e11i			0.697121i	−0.937286	+	0.348560i	$0.886671\pi$
$882$		0			0
$883$	−1.46098e11			−0.240326			−0.120163	−	0.992754i	$-0.538342\pi$
$883$	−1.46098e11			−0.240326			−0.120163	+	0.992754i	$0.538342\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	6.10763e11i		−	0.986685i	−0.869835	−	0.493342i	$-0.835775\pi$
$887$		−	6.10763e11i		−	0.986685i	0.869835	−	0.493342i	$-0.164225\pi$
$888$		0			0
$889$	−8.73585e10			−0.139861
$890$		0			0
$891$	−4.94537e11	+	2.08153e11i	−0.784671	+	0.330272i
$892$		0			0
$893$			1.64154e11i			0.258134i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−8.38233e10	−	2.52972e10i	−0.129478	−	0.0390753i
$898$		0			0
$899$		−	2.59341e11i		−	0.397039i
$900$		0			0
$901$	−2.83993e10			−0.0430932
$902$		0			0
$903$	3.07825e10	−	1.01999e11i	0.0462969	−	0.153407i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−4.54224e11			−0.671184			−0.335592	−	0.942008i	$-0.608936\pi$
$907$	−4.54224e11			−0.671184			−0.335592	+	0.942008i	$0.608936\pi$
$908$		0			0
$909$	−6.23830e11	+	9.39410e11i	−0.913715	+	1.37594i
$910$		0			0
$911$		−	9.69211e11i		−	1.40716i	−0.710614	−	0.703582i	$-0.751583\pi$
$911$		−	9.69211e11i		−	1.40716i	0.710614	−	0.703582i	$-0.248417\pi$
$912$		0			0
$913$	−1.37968e11			−0.198562
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	6.60128e10i		−	0.0933579i
$918$		0			0
$919$	7.66094e11			1.07404			0.537019	−	0.843570i	$-0.319550\pi$
$919$	7.66094e11			1.07404			0.537019	+	0.843570i	$0.319550\pi$
$920$		0			0
$921$	2.45334e11	−	8.12924e11i	0.340972	−	1.12983i
$922$		0			0
$923$		−	1.18490e10i		−	0.0163259i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−6.75358e11	−	4.48482e11i	−0.914566	−	0.607332i
$928$		0			0
$929$			5.97981e11i			0.802832i	0.915896	+	0.401416i	$0.131482\pi$
$929$			5.97981e11i			0.802832i	−0.915896	+	0.401416i	$0.868518\pi$
$930$		0			0
$931$	−2.64229e11			−0.351707
$932$		0			0
$933$	2.85027e11	+	8.60187e10i	0.376148	+	0.113518i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	2.06963e11			0.268494			0.134247	−	0.990948i	$-0.457138\pi$
$937$	2.06963e11			0.268494			0.134247	+	0.990948i	$0.457138\pi$
$938$		0			0
$939$	−1.06305e11	+	3.52246e11i	−0.136739	+	0.453089i
$940$		0			0
$941$			1.17181e12i			1.49452i	0.664534	+	0.747258i	$0.268630\pi$
$941$			1.17181e12i			1.49452i	−0.664534	+	0.747258i	$0.731370\pi$
$942$		0			0
$943$	7.16126e11			0.905614
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	1.15839e11i		−	0.144031i	−0.997404	−	0.0720156i	$-0.977057\pi$
$947$		−	1.15839e11i		−	0.144031i	0.997404	−	0.0720156i	$-0.0229431\pi$
$948$		0			0
$949$	−1.57190e11			−0.193802
$950$		0			0
$951$	−4.74396e11	−	1.43169e11i	−0.579988	−	0.175035i
$952$		0			0
$953$			9.28962e11i			1.12623i	0.826379	+	0.563114i	$0.190397\pi$
$953$			9.28962e11i			1.12623i	−0.826379	+	0.563114i	$0.809603\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−9.06069e10	+	3.00230e11i	−0.108022	+	0.357937i
$958$		0			0
$959$			1.51981e11i			0.179686i
$960$		0			0
$961$	−1.55779e11			−0.182648
$962$		0			0
$963$	7.81889e11	−	1.17743e12i	0.909160	−	1.36908i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1.26346e12			−1.44496			−0.722478	−	0.691394i	$-0.756997\pi$
$967$	−1.26346e12			−1.44496			−0.722478	+	0.691394i	$0.756997\pi$
$968$		0			0
$969$	−1.83161e11	−	5.52765e10i	−0.207749	−	0.0626968i
$970$		0			0
$971$			7.05422e11i			0.793546i	0.917917	+	0.396773i	$0.129870\pi$
$971$			7.05422e11i			0.793546i	−0.917917	+	0.396773i	$0.870130\pi$
$972$		0			0
$973$	1.42232e11			0.158688
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	7.66667e11i		−	0.841450i	−0.907188	−	0.420725i	$-0.861776\pi$
$977$		−	7.66667e11i		−	0.841450i	0.907188	−	0.420725i	$-0.138224\pi$
$978$		0			0
$979$	−8.87182e11			−0.965788
$980$		0			0
$981$	−5.03364e11	−	3.34266e11i	−0.543508	−	0.360925i
$982$		0			0
$983$		−	1.79128e12i		−	1.91845i	−0.282648	−	0.959224i	$-0.591213\pi$
$983$		−	1.79128e12i		−	1.91845i	0.282648	−	0.959224i	$-0.408787\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	7.05031e10	+	2.12773e10i	0.0742916	+	0.0224206i
$988$		0			0
$989$		−	1.63722e12i		−	1.71129i
$990$		0			0
$991$	2.71910e11			0.281923			0.140962	−	0.990015i	$-0.454981\pi$
$991$	2.71910e11			0.281923			0.140962	+	0.990015i	$0.454981\pi$
$992$		0			0
$993$	−2.17839e11	+	7.21818e11i	−0.224046	+	0.742387i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	2.04659e11			0.207133			0.103567	−	0.994623i	$-0.466974\pi$
$997$	2.04659e11			0.207133			0.103567	+	0.994623i	$0.466974\pi$
$998$		0			0
$999$	1.17999e12	−	9.76861e11i	1.18472	−	0.980778i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.9.g.h.101.9		16
3.2	odd	2	inner	300.9.g.h.101.10		16
5.2	odd	4		60.9.b.a.29.1	✓	16
5.3	odd	4		60.9.b.a.29.16	yes	16
5.4	even	2	inner	300.9.g.h.101.8		16
15.2	even	4		60.9.b.a.29.15	yes	16
15.8	even	4		60.9.b.a.29.2	yes	16
15.14	odd	2	inner	300.9.g.h.101.7		16
20.3	even	4		240.9.c.d.209.1		16
20.7	even	4		240.9.c.d.209.16		16
60.23	odd	4		240.9.c.d.209.15		16
60.47	odd	4		240.9.c.d.209.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.9.b.a.29.1	✓	16	5.2	odd	4
60.9.b.a.29.2	yes	16	15.8	even	4
60.9.b.a.29.15	yes	16	15.2	even	4
60.9.b.a.29.16	yes	16	5.3	odd	4
240.9.c.d.209.1		16	20.3	even	4
240.9.c.d.209.2		16	60.47	odd	4
240.9.c.d.209.15		16	60.23	odd	4
240.9.c.d.209.16		16	20.7	even	4
300.9.g.h.101.7		16	15.14	odd	2	inner
300.9.g.h.101.8		16	5.4	even	2	inner
300.9.g.h.101.9		16	1.1	even	1	trivial
300.9.g.h.101.10		16	3.2	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q+(23.4026 - 77.5456i) q^{3} +256.799 q^{7} +(-5465.63 - 3629.54i) q^{9} +O(q^{10})\)	\(q+(23.4026 - 77.5456i) q^{3} +256.799 q^{7} +(-5465.63 - 3629.54i) q^{9} +12464.5i q^{11} +3381.78 q^{13} -50943.0i q^{17} +46365.2 q^{19} +(6009.77 - 19913.6i) q^{21} -319641. i q^{23} +(-409365. + 338895. i) q^{27} +310614. i q^{29} -834932. q^{31} +(966570. + 291703. i) q^{33} -2.88249e6 q^{37} +(79142.5 - 262242. i) q^{39} +2.24041e6i q^{41} +5.12207e6 q^{43} +3.54045e6i q^{47} -5.69886e6 q^{49} +(-3.95040e6 - 1.19220e6i) q^{51} -557473. i q^{53} +(1.08507e6 - 3.59542e6i) q^{57} +1.90288e7i q^{59} -4.81127e6 q^{61} +(-1.40357e6 - 932062. i) q^{63} +1.45363e7 q^{67} +(-2.47867e7 - 7.48043e6i) q^{69} -3.50379e6i q^{71} -4.64813e7 q^{73} +3.20088e6i q^{77} -4.67477e7 q^{79} +(1.66996e7 + 3.96755e7i) q^{81} +1.10689e7i q^{83} +(2.40867e7 + 7.26917e6i) q^{87} +7.11764e7i q^{89} +868438. q^{91} +(-1.95396e7 + 6.47453e7i) q^{93} +6.59215e7 q^{97} +(4.52405e7 - 6.81266e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 14756 q^{9}+O(q^{10}) \) 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

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