LMFDB - Embedded newform 300.9.g.h.101.5

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			101.5
Root			$51.0255 + 62.9079i$ of defining polynomial
Character	$\chi$	$=$	300.101
Dual form			300.9.g.h.101.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-51.0255 - 62.9079i) q^{3} -459.019 q^{7} +(-1353.80 + 6419.81i) q^{9} +O(q^{10})$	$q+(-51.0255 - 62.9079i) q^{3} -459.019 q^{7} +(-1353.80 + 6419.81i) q^{9} +9139.47i q^{11} +48094.6 q^{13} +53336.1i q^{17} -122304. q^{19} +(23421.7 + 28875.9i) q^{21} +263928. i q^{23} +(472935. - 242410. i) q^{27} -1.04331e6i q^{29} +1.39629e6 q^{31} +(574944. - 466346. i) q^{33} -2.84955e6 q^{37} +(-2.45405e6 - 3.02553e6i) q^{39} +2.57085e6i q^{41} -1.51750e6 q^{43} -2.59155e6i q^{47} -5.55410e6 q^{49} +(3.35526e6 - 2.72150e6i) q^{51} +2.74970e6i q^{53} +(6.24065e6 + 7.69391e6i) q^{57} -1.78378e7i q^{59} -8.28811e6 q^{61} +(621418. - 2.94681e6i) q^{63} -462829. q^{67} +(1.66031e7 - 1.34671e7i) q^{69} -3.15348e7i q^{71} +1.02650e7 q^{73} -4.19519e6i q^{77} +2.02668e7 q^{79} +(-3.93812e7 - 1.73822e7i) q^{81} +3.52318e7i q^{83} +(-6.56326e7 + 5.32356e7i) q^{87} +4.31268e7i q^{89} -2.20763e7 q^{91} +(-7.12462e7 - 8.78374e7i) q^{93} +5.78240e7 q^{97} +(-5.86737e7 - 1.23730e7i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−51.0255	−	62.9079i	−0.629944	−	0.776640i
$3$	−51.0255	−	62.9079i	−0.629944	−	0.776640i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−459.019			−0.191178			−0.0955890	−	0.995421i	$-0.530473\pi$
$7$	−459.019			−0.191178			−0.0955890	+	0.995421i	$0.530473\pi$
$8$		0			0
$9$	−1353.80	+	6419.81i	−0.206340	+	0.978480i
$10$		0			0
$11$			9139.47i			0.624238i	0.950043	+	0.312119i	$0.101039\pi$
$11$			9139.47i			0.624238i	−0.950043	+	0.312119i	$0.898961\pi$
$12$		0			0
$13$	48094.6			1.68392			0.841962	−	0.539537i	$-0.181401\pi$
$13$	48094.6			1.68392			0.841962	+	0.539537i	$0.181401\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			53336.1i			0.638595i	0.947655	+	0.319297i	$0.103447\pi$
$17$			53336.1i			0.638595i	−0.947655	+	0.319297i	$0.896553\pi$
$18$		0			0
$19$	−122304.			−0.938486			−0.469243	−	0.883069i	$-0.655473\pi$
$19$	−122304.			−0.938486			−0.469243	+	0.883069i	$0.655473\pi$
$20$		0			0
$21$	23421.7	+	28875.9i	0.120432	+	0.148477i
$22$		0			0
$23$			263928.i			0.943135i	0.881830	+	0.471568i	$0.156312\pi$
$23$			263928.i			0.943135i	−0.881830	+	0.471568i	$0.843688\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	472935.	−	242410.i	0.889910	−	0.456136i
$28$		0			0
$29$		−	1.04331e6i		−	1.47511i	−0.675290	−	0.737553i	$-0.735981\pi$
$29$		−	1.04331e6i		−	1.47511i	0.675290	−	0.737553i	$-0.264019\pi$
$30$		0			0
$31$	1.39629e6			1.51192			0.755958	−	0.654620i	$-0.227171\pi$
$31$	1.39629e6			1.51192			0.755958	+	0.654620i	$0.227171\pi$
$32$		0			0
$33$	574944.	−	466346.i	0.484808	−	0.393235i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−2.84955e6			−1.52044			−0.760221	−	0.649665i	$-0.774909\pi$
$37$	−2.84955e6			−1.52044			−0.760221	+	0.649665i	$0.774909\pi$
$38$		0			0
$39$	−2.45405e6	−	3.02553e6i	−1.06078	−	1.30780i
$40$		0			0
$41$			2.57085e6i			0.909790i	0.890545	+	0.454895i	$0.150323\pi$
$41$			2.57085e6i			0.909790i	−0.890545	+	0.454895i	$0.849677\pi$
$42$		0			0
$43$	−1.51750e6			−0.443868			−0.221934	−	0.975062i	$-0.571237\pi$
$43$	−1.51750e6			−0.443868			−0.221934	+	0.975062i	$0.571237\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	2.59155e6i		−	0.531089i	−0.964099	−	0.265545i	$-0.914448\pi$
$47$		−	2.59155e6i		−	0.531089i	0.964099	−	0.265545i	$-0.0855518\pi$
$48$		0			0
$49$	−5.55410e6			−0.963451
$50$		0			0
$51$	3.35526e6	−	2.72150e6i	0.495958	−	0.402279i
$52$		0			0
$53$			2.74970e6i			0.348483i	0.984703	+	0.174241i	$0.0557473\pi$
$53$			2.74970e6i			0.348483i	−0.984703	+	0.174241i	$0.944253\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	6.24065e6	+	7.69391e6i	0.591194	+	0.728866i
$58$		0			0
$59$		−	1.78378e7i		−	1.47208i	−0.676936	−	0.736042i	$-0.736693\pi$
$59$		−	1.78378e7i		−	1.47208i	0.676936	−	0.736042i	$-0.263307\pi$
$60$		0			0
$61$	−8.28811e6			−0.598599			−0.299300	−	0.954159i	$-0.596753\pi$
$61$	−8.28811e6			−0.598599			−0.299300	+	0.954159i	$0.596753\pi$
$62$		0			0
$63$	621418.	−	2.94681e6i	0.0394477	−	0.187064i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−462829.			−0.0229679			−0.0114840	−	0.999934i	$-0.503656\pi$
$67$	−462829.			−0.0229679			−0.0114840	+	0.999934i	$0.503656\pi$
$68$		0			0
$69$	1.66031e7	−	1.34671e7i	0.732477	−	0.594123i
$70$		0			0
$71$		−	3.15348e7i		−	1.24096i	−0.784224	−	0.620478i	$-0.786939\pi$
$71$		−	3.15348e7i		−	1.24096i	0.784224	−	0.620478i	$-0.213061\pi$
$72$		0			0
$73$	1.02650e7			0.361468			0.180734	−	0.983532i	$-0.442153\pi$
$73$	1.02650e7			0.361468			0.180734	+	0.983532i	$0.442153\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	4.19519e6i		−	0.119341i
$78$		0			0
$79$	2.02668e7			0.520328			0.260164	−	0.965564i	$-0.416223\pi$
$79$	2.02668e7			0.520328			0.260164	+	0.965564i	$0.416223\pi$
$80$		0			0
$81$	−3.93812e7	−	1.73822e7i	−0.914848	−	0.403799i
$82$		0			0
$83$			3.52318e7i			0.742373i	0.928558	+	0.371187i	$0.121049\pi$
$83$			3.52318e7i			0.742373i	−0.928558	+	0.371187i	$0.878951\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−6.56326e7	+	5.32356e7i	−1.14563	+	0.929234i
$88$		0			0
$89$			4.31268e7i			0.687365i	0.939086	+	0.343682i	$0.111674\pi$
$89$			4.31268e7i			0.687365i	−0.939086	+	0.343682i	$0.888326\pi$
$90$		0			0
$91$	−2.20763e7			−0.321929
$92$		0			0
$93$	−7.12462e7	−	8.78374e7i	−0.952423	−	1.17421i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	5.78240e7			0.653162			0.326581	−	0.945169i	$-0.394103\pi$
$97$	5.78240e7			0.653162			0.326581	+	0.945169i	$0.394103\pi$
$98$		0			0
$99$	−5.86737e7	−	1.23730e7i	−0.610805	−	0.128805i
$100$		0			0
$101$		−	4.41147e7i		−	0.423934i	−0.977277	−	0.211967i	$-0.932013\pi$
$101$		−	4.41147e7i		−	0.423934i	0.977277	−	0.211967i	$-0.0679869\pi$
$102$		0			0
$103$	−3.82195e7			−0.339575			−0.169788	−	0.985481i	$-0.554308\pi$
$103$	−3.82195e7			−0.339575			−0.169788	+	0.985481i	$0.554308\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	6.27380e7i		−	0.478625i	−0.970943	−	0.239313i	$-0.923078\pi$
$107$		−	6.27380e7i		−	0.478625i	0.970943	−	0.239313i	$-0.0769221\pi$
$108$		0			0
$109$	3.48648e7			0.246991			0.123496	−	0.992345i	$-0.460590\pi$
$109$	3.48648e7			0.246991			0.123496	+	0.992345i	$0.460590\pi$
$110$		0			0
$111$	1.45400e8	+	1.79259e8i	0.957793	+	1.18084i
$112$		0			0
$113$			2.69695e8i			1.65409i	0.562135	+	0.827046i	$0.309980\pi$
$113$			2.69695e8i			1.65409i	−0.562135	+	0.827046i	$0.690020\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−6.51102e7	+	3.08758e8i	−0.347461	+	1.64769i
$118$		0			0
$119$		−	2.44823e7i		−	0.122085i
$120$		0			0
$121$	1.30829e8			0.610327
$122$		0			0
$123$	1.61727e8	−	1.31179e8i	0.706579	−	0.573117i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−4.94693e8			−1.90161			−0.950803	−	0.309796i	$-0.899739\pi$
$127$	−4.94693e8			−1.90161			−0.950803	+	0.309796i	$0.899739\pi$
$128$		0			0
$129$	7.74310e7	+	9.54625e7i	0.279612	+	0.344726i
$130$		0			0
$131$			2.56571e8i			0.871208i	0.900138	+	0.435604i	$0.143465\pi$
$131$			2.56571e8i			0.871208i	−0.900138	+	0.435604i	$0.856535\pi$
$132$		0			0
$133$	5.61400e7			0.179418
$134$		0			0
$135$		0			0
$136$		0			0
$137$			8.06580e7i			0.228963i	0.993425	+	0.114481i	$0.0365206\pi$
$137$			8.06580e7i			0.228963i	−0.993425	+	0.114481i	$0.963479\pi$
$138$		0			0
$139$	−5.03868e8			−1.34976			−0.674882	−	0.737926i	$-0.735805\pi$
$139$	−5.03868e8			−1.34976			−0.674882	+	0.737926i	$0.735805\pi$
$140$		0			0
$141$	−1.63029e8	+	1.32235e8i	−0.412465	+	0.334557i
$142$		0			0
$143$			4.39559e8i			1.05117i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	2.83401e8	+	3.49397e8i	0.606921	+	0.748255i
$148$		0			0
$149$			9.26379e8i			1.87951i	0.341854	+	0.939753i	$0.388945\pi$
$149$			9.26379e8i			1.87951i	−0.341854	+	0.939753i	$0.611055\pi$
$150$		0			0
$151$	1.66966e8			0.321158			0.160579	−	0.987023i	$-0.448664\pi$
$151$	1.66966e8			0.321158			0.160579	+	0.987023i	$0.448664\pi$
$152$		0			0
$153$	−3.42407e8	−	7.22062e7i	−0.624852	−	0.131768i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−7.60014e8			−1.25090			−0.625451	−	0.780264i	$-0.715085\pi$
$157$	−7.60014e8			−1.25090			−0.625451	+	0.780264i	$0.715085\pi$
$158$		0			0
$159$	1.72978e8	−	1.40305e8i	0.270646	−	0.219525i
$160$		0			0
$161$		−	1.21148e8i		−	0.180307i
$162$		0			0
$163$	−1.02534e9			−1.45250			−0.726252	−	0.687429i	$-0.758739\pi$
$163$	−1.02534e9			−1.45250			−0.726252	+	0.687429i	$0.758739\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			1.26101e9i			1.62126i	0.585560	+	0.810629i	$0.300875\pi$
$167$			1.26101e9i			1.62126i	−0.585560	+	0.810629i	$0.699125\pi$
$168$		0			0
$169$	1.49736e9			1.83560
$170$		0			0
$171$	1.65575e8	−	7.85171e8i	0.193647	−	0.918290i
$172$		0			0
$173$		−	5.01044e7i		−	0.0559360i	−0.999609	−	0.0279680i	$-0.991096\pi$
$173$		−	5.01044e7i		−	0.0559360i	0.999609	−	0.0279680i	$-0.00890365\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−1.12214e9	+	9.10181e8i	−1.14328	+	0.927331i
$178$		0			0
$179$		−	1.00928e9i		−	0.983108i	−0.870847	−	0.491554i	$-0.836429\pi$
$179$		−	1.00928e9i		−	0.983108i	0.870847	−	0.491554i	$-0.163571\pi$
$180$		0			0
$181$	−1.77364e9			−1.65254			−0.826268	−	0.563277i	$-0.809540\pi$
$181$	−1.77364e9			−1.65254			−0.826268	+	0.563277i	$0.809540\pi$
$182$		0			0
$183$	4.22905e8	+	5.21387e8i	0.377084	+	0.464896i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−4.87463e8			−0.398635
$188$		0			0
$189$	−2.17086e8	+	1.11271e8i	−0.170131	+	0.0872033i
$190$		0			0
$191$		−	1.24397e9i		−	0.934712i	−0.884069	−	0.467356i	$-0.845207\pi$
$191$		−	1.24397e9i		−	0.934712i	0.884069	−	0.467356i	$-0.154793\pi$
$192$		0			0
$193$	−2.76609e7			−0.0199359			−0.00996797	−	0.999950i	$-0.503173\pi$
$193$	−2.76609e7			−0.0199359			−0.00996797	+	0.999950i	$0.503173\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			2.12130e9i			1.40844i	0.709984	+	0.704218i	$0.248702\pi$
$197$			2.12130e9i			1.40844i	−0.709984	+	0.704218i	$0.751298\pi$
$198$		0			0
$199$	−7.80182e8			−0.497489			−0.248745	−	0.968569i	$-0.580018\pi$
$199$	−7.80182e8			−0.497489			−0.248745	+	0.968569i	$0.580018\pi$
$200$		0			0
$201$	2.36161e7	+	2.91156e7i	0.0144685	+	0.0178378i
$202$		0			0
$203$			4.78901e8i			0.282008i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−1.69437e9	−	3.57305e8i	−0.922840	−	0.194607i
$208$		0			0
$209$		−	1.11780e9i		−	0.585839i
$210$		0			0
$211$	1.47843e9			0.745884			0.372942	−	0.927855i	$-0.378349\pi$
$211$	1.47843e9			0.745884			0.372942	+	0.927855i	$0.378349\pi$
$212$		0			0
$213$	−1.98379e9	+	1.60908e9i	−0.963776	+	0.781734i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−6.40921e8			−0.289045
$218$		0			0
$219$	−5.23779e8	−	6.45752e8i	−0.227705	−	0.280730i
$220$		0			0
$221$			2.56517e9i			1.07535i
$222$		0			0
$223$	9.39456e8			0.379889			0.189945	−	0.981795i	$-0.439169\pi$
$223$	9.39456e8			0.379889			0.189945	+	0.981795i	$0.439169\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			4.93076e9i			1.85699i	0.371339	+	0.928497i	$0.378899\pi$
$227$			4.93076e9i			1.85699i	−0.371339	+	0.928497i	$0.621101\pi$
$228$		0			0
$229$	−3.90618e9			−1.42040			−0.710199	−	0.704001i	$-0.751395\pi$
$229$	−3.90618e9			−1.42040			−0.710199	+	0.704001i	$0.751395\pi$
$230$		0			0
$231$	−2.63910e8	+	2.14062e8i	−0.0926847	+	0.0751780i
$232$		0			0
$233$		−	4.02506e7i		−	0.0136568i	−0.999977	−	0.00682839i	$-0.997826\pi$
$233$		−	4.02506e7i		−	0.0136568i	0.999977	−	0.00682839i	$-0.00217356\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−1.03413e9	−	1.27494e9i	−0.327778	−	0.404108i
$238$		0			0
$239$		−	5.06846e9i		−	1.55340i	−0.629869	−	0.776702i	$-0.716891\pi$
$239$		−	5.06846e9i		−	1.55340i	0.629869	−	0.776702i	$-0.283109\pi$
$240$		0			0
$241$	2.04398e9			0.605910			0.302955	−	0.953005i	$-0.402027\pi$
$241$	2.04398e9			0.605910			0.302955	+	0.953005i	$0.402027\pi$
$242$		0			0
$243$	9.15966e8	+	3.36432e9i	0.262697	+	0.964878i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−5.88218e9			−1.58034
$248$		0			0
$249$	2.21636e9	−	1.79772e9i	0.576557	−	0.467654i
$250$		0			0
$251$		−	2.48750e9i		−	0.626711i	−0.949636	−	0.313356i	$-0.898547\pi$
$251$		−	2.48750e9i		−	0.626711i	0.949636	−	0.313356i	$-0.101453\pi$
$252$		0			0
$253$	−2.41216e9			−0.588741
$254$		0			0
$255$		0			0
$256$		0			0
$257$			4.66441e9i			1.06921i	0.845101	+	0.534606i	$0.179540\pi$
$257$			4.66441e9i			1.06921i	−0.845101	+	0.534606i	$0.820460\pi$
$258$		0			0
$259$	1.30800e9			0.290675
$260$		0			0
$261$	6.69788e9	+	1.41243e9i	1.44336	+	0.304373i
$262$		0			0
$263$		−	5.71598e9i		−	1.19473i	−0.801971	−	0.597363i	$-0.796215\pi$
$263$		−	5.71598e9i		−	1.19473i	0.801971	−	0.597363i	$-0.203785\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	2.71301e9	−	2.20057e9i	0.533835	−	0.433001i
$268$		0			0
$269$			5.53953e9i			1.05795i	0.848639	+	0.528973i	$0.177423\pi$
$269$			5.53953e9i			1.05795i	−0.848639	+	0.528973i	$0.822577\pi$
$270$		0			0
$271$	−8.41658e9			−1.56048			−0.780241	−	0.625479i	$-0.784904\pi$
$271$	−8.41658e9			−1.56048			−0.780241	+	0.625479i	$0.784904\pi$
$272$		0			0
$273$	1.12645e9	+	1.38877e9i	0.202798	+	0.250023i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.24754e9			−0.211901			−0.105951	−	0.994371i	$-0.533789\pi$
$277$	−1.24754e9			−0.211901			−0.105951	+	0.994371i	$0.533789\pi$
$278$		0			0
$279$	−1.89029e9	+	8.96389e9i	−0.311969	+	1.47938i
$280$		0			0
$281$			6.20826e9i			0.995737i	0.867253	+	0.497868i	$0.165884\pi$
$281$			6.20826e9i			0.995737i	−0.867253	+	0.497868i	$0.834116\pi$
$282$		0			0
$283$	−4.82589e9			−0.752370			−0.376185	−	0.926545i	$-0.622764\pi$
$283$	−4.82589e9			−0.752370			−0.376185	+	0.926545i	$0.622764\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	1.18007e9i		−	0.173932i
$288$		0			0
$289$	4.13102e9			0.592197
$290$		0			0
$291$	−2.95050e9	−	3.63758e9i	−0.411456	−	0.507272i
$292$		0			0
$293$			4.75107e9i			0.644645i	0.946630	+	0.322323i	$0.104464\pi$
$293$			4.75107e9i			0.644645i	−0.946630	+	0.322323i	$0.895536\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	2.21550e9	+	4.32237e9i	0.284738	+	0.555516i
$298$		0			0
$299$			1.26935e10i			1.58817i
$300$		0			0
$301$	6.96559e8			0.0848579
$302$		0			0
$303$	−2.77516e9	+	2.25097e9i	−0.329244	+	0.267055i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	6.53537e9			0.735727			0.367863	−	0.929880i	$-0.380089\pi$
$307$	6.53537e9			0.735727			0.367863	+	0.929880i	$0.380089\pi$
$308$		0			0
$309$	1.95017e9	+	2.40431e9i	0.213914	+	0.263728i
$310$		0			0
$311$		−	9.42520e9i		−	1.00751i	−0.863847	−	0.503755i	$-0.831952\pi$
$311$		−	9.42520e9i		−	1.00751i	0.863847	−	0.503755i	$-0.168048\pi$
$312$		0			0
$313$	1.16410e10			1.21287			0.606433	−	0.795134i	$-0.292600\pi$
$313$	1.16410e10			1.21287			0.606433	+	0.795134i	$0.292600\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	9.36294e9i		−	0.927203i	−0.886044	−	0.463602i	$-0.846557\pi$
$317$		−	9.36294e9i		−	0.927203i	0.886044	−	0.463602i	$-0.153443\pi$
$318$		0			0
$319$	9.53534e9			0.920817
$320$		0			0
$321$	−3.94671e9	+	3.20124e9i	−0.371720	+	0.301507i
$322$		0			0
$323$		−	6.52324e9i		−	0.599312i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−1.77899e9	−	2.19327e9i	−0.155591	−	0.191823i
$328$		0			0
$329$			1.18957e9i			0.101533i
$330$		0			0
$331$	−1.04898e10			−0.873886			−0.436943	−	0.899489i	$-0.643939\pi$
$331$	−1.04898e10			−0.873886			−0.436943	+	0.899489i	$0.643939\pi$
$332$		0			0
$333$	3.85771e9	−	1.82936e10i	0.313728	−	1.48772i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	7.07467e9			0.548512			0.274256	−	0.961657i	$-0.411568\pi$
$337$	7.07467e9			0.548512			0.274256	+	0.961657i	$0.411568\pi$
$338$		0			0
$339$	1.69659e10	−	1.37613e10i	1.28463	−	1.04199i
$340$		0			0
$341$			1.27613e10i			0.943796i
$342$		0			0
$343$	5.19559e9			0.375369
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	8.94083e9i		−	0.616680i	−0.951276	−	0.308340i	$-0.900227\pi$
$347$		−	8.94083e9i		−	0.616680i	0.951276	−	0.308340i	$-0.0997734\pi$
$348$		0			0
$349$	−2.42213e10			−1.63266			−0.816329	−	0.577588i	$-0.803994\pi$
$349$	−2.42213e10			−1.63266			−0.816329	+	0.577588i	$0.803994\pi$
$350$		0			0
$351$	2.27456e10	−	1.16586e10i	1.49854	−	0.768099i
$352$		0			0
$353$			2.34151e10i			1.50798i	0.656884	+	0.753991i	$0.271874\pi$
$353$			2.34151e10i			1.50798i	−0.656884	+	0.753991i	$0.728126\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−1.54013e9	+	1.24922e9i	−0.0948164	+	0.0769070i
$358$		0			0
$359$		−	7.48179e9i		−	0.450431i	−0.974309	−	0.225215i	$-0.927691\pi$
$359$		−	7.48179e9i		−	0.450431i	0.974309	−	0.225215i	$-0.0723086\pi$
$360$		0			0
$361$	−2.02518e9			−0.119244
$362$		0			0
$363$	−6.67561e9	−	8.23017e9i	−0.384472	−	0.474004i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	3.08337e10			1.69966			0.849830	−	0.527057i	$-0.176705\pi$
$367$	3.08337e10			1.69966			0.849830	+	0.527057i	$0.176705\pi$
$368$		0			0
$369$	−1.65044e10	−	3.48040e9i	−0.890211	−	0.187726i
$370$		0			0
$371$		−	1.26216e9i		−	0.0666223i
$372$		0			0
$373$	−2.68065e10			−1.38486			−0.692428	−	0.721487i	$-0.743459\pi$
$373$	−2.68065e10			−1.38486			−0.692428	+	0.721487i	$0.743459\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	5.01777e10i		−	2.48397i
$378$		0			0
$379$	−2.56615e10			−1.24373			−0.621863	−	0.783126i	$-0.713624\pi$
$379$	−2.56615e10			−1.24373			−0.621863	+	0.783126i	$0.713624\pi$
$380$		0			0
$381$	2.52419e10	+	3.11201e10i	1.19791	+	1.47686i
$382$		0			0
$383$		−	1.75285e10i		−	0.814612i	−0.913292	−	0.407306i	$-0.866468\pi$
$383$		−	1.75285e10i		−	0.814612i	0.913292	−	0.407306i	$-0.133532\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	2.05438e9	−	9.74204e9i	0.0915877	−	0.434316i
$388$		0			0
$389$		−	3.24519e10i		−	1.41723i	−0.705594	−	0.708617i	$-0.749320\pi$
$389$		−	3.24519e10i		−	1.41723i	0.705594	−	0.708617i	$-0.250680\pi$
$390$		0			0
$391$	−1.40769e10			−0.602281
$392$		0			0
$393$	1.61403e10	−	1.30916e10i	0.676615	−	0.548813i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−4.58427e9			−0.184548			−0.0922738	−	0.995734i	$-0.529414\pi$
$397$	−4.58427e9			−0.184548			−0.0922738	+	0.995734i	$0.529414\pi$
$398$		0			0
$399$	−2.86457e9	−	3.53165e9i	−0.113023	−	0.139343i
$400$		0			0
$401$		−	3.56669e10i		−	1.37939i	−0.724100	−	0.689695i	$-0.757744\pi$
$401$		−	3.56669e10i		−	1.37939i	0.724100	−	0.689695i	$-0.242256\pi$
$402$		0			0
$403$	6.71538e10			2.54595
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	2.60434e10i		−	0.949117i
$408$		0			0
$409$	1.87379e10			0.669620			0.334810	−	0.942286i	$-0.391328\pi$
$409$	1.87379e10			0.669620			0.334810	+	0.942286i	$0.391328\pi$
$410$		0			0
$411$	5.07402e9	−	4.11561e9i	0.177822	−	0.144234i
$412$		0			0
$413$			8.18787e9i			0.281430i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	2.57101e10	+	3.16973e10i	0.850276	+	1.04828i
$418$		0			0
$419$		−	2.82911e10i		−	0.917898i	−0.888463	−	0.458949i	$-0.848226\pi$
$419$		−	2.82911e10i		−	0.917898i	0.888463	−	0.458949i	$-0.151774\pi$
$420$		0			0
$421$	−2.06273e10			−0.656618			−0.328309	−	0.944570i	$-0.606479\pi$
$421$	−2.06273e10			−0.656618			−0.328309	+	0.944570i	$0.606479\pi$
$422$		0			0
$423$	1.66372e10	+	3.50843e9i	0.519661	+	0.109585i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	3.80440e9			0.114439
$428$		0			0
$429$	2.76517e10	−	2.24287e10i	0.816381	−	0.662178i
$430$		0			0
$431$			1.72749e10i			0.500618i	0.968166	+	0.250309i	$0.0805322\pi$
$431$			1.72749e10i			0.500618i	−0.968166	+	0.250309i	$0.919468\pi$
$432$		0			0
$433$	−3.35377e10			−0.954075			−0.477037	−	0.878883i	$-0.658289\pi$
$433$	−3.35377e10			−0.954075			−0.477037	+	0.878883i	$0.658289\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	3.22796e10i		−	0.885120i
$438$		0			0
$439$	−3.58022e6			−9.63945e−5			−4.81972e−5	−	1.00000i	$-0.500015\pi$
$439$	−3.58022e6			−9.63945e−5			−4.81972e−5		1.00000i	$0.500015\pi$
$440$		0			0
$441$	7.51912e9	−	3.56563e10i	0.198798	−	0.942718i
$442$		0			0
$443$		−	1.35038e10i		−	0.350623i	−0.984513	−	0.175312i	$-0.943907\pi$
$443$		−	1.35038e10i		−	0.350623i	0.984513	−	0.175312i	$-0.0560933\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	5.82765e10	−	4.72690e10i	1.45970	−	1.18398i
$448$		0			0
$449$			6.60570e10i			1.62530i	0.582752	+	0.812650i	$0.301976\pi$
$449$			6.60570e10i			1.62530i	−0.582752	+	0.812650i	$0.698024\pi$
$450$		0			0
$451$	−2.34962e10			−0.567925
$452$		0			0
$453$	−8.51950e9	−	1.05034e10i	−0.202312	−	0.249424i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	2.96351e10			0.679425			0.339712	−	0.940529i	$-0.389670\pi$
$457$	2.96351e10			0.679425			0.339712	+	0.940529i	$0.389670\pi$
$458$		0			0
$459$	1.29292e10	+	2.52245e10i	0.291286	+	0.568292i
$460$		0			0
$461$		−	4.06549e10i		−	0.900139i	−0.892994	−	0.450069i	$-0.851399\pi$
$461$		−	4.06549e10i		−	0.900139i	0.892994	−	0.450069i	$-0.148601\pi$
$462$		0			0
$463$	−4.18983e10			−0.911743			−0.455872	−	0.890046i	$-0.650672\pi$
$463$	−4.18983e10			−0.911743			−0.455872	+	0.890046i	$0.650672\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			4.13906e10i			0.870230i	0.900375	+	0.435115i	$0.143292\pi$
$467$			4.13906e10i			0.870230i	−0.900375	+	0.435115i	$0.856708\pi$
$468$		0			0
$469$	2.12447e8			0.00439096
$470$		0			0
$471$	3.87801e10	+	4.78109e10i	0.787999	+	0.971501i
$472$		0			0
$473$		−	1.38691e10i		−	0.277079i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1.76525e10	−	3.72253e9i	−0.340984	−	0.0719059i
$478$		0			0
$479$			7.07355e10i			1.34368i	0.740697	+	0.671839i	$0.234495\pi$
$479$			7.07355e10i			1.34368i	−0.740697	+	0.671839i	$0.765505\pi$
$480$		0			0
$481$	−1.37048e11			−2.56031
$482$		0			0
$483$	−7.62115e9	+	6.18163e9i	−0.140034	+	0.113583i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−1.97837e10			−0.351715			−0.175857	−	0.984416i	$-0.556270\pi$
$487$	−1.97837e10			−0.351715			−0.175857	+	0.984416i	$0.556270\pi$
$488$		0			0
$489$	5.23185e10	+	6.45019e10i	0.914997	+	1.12807i
$490$		0			0
$491$			8.33438e10i			1.43399i	0.697076	+	0.716997i	$0.254484\pi$
$491$			8.33438e10i			1.43399i	−0.697076	+	0.716997i	$0.745516\pi$
$492$		0			0
$493$	5.56463e10			0.941995
$494$		0			0
$495$		0			0
$496$		0			0
$497$			1.44751e10i			0.237244i
$498$		0			0
$499$	−4.62385e10			−0.745764			−0.372882	−	0.927879i	$-0.621630\pi$
$499$	−4.62385e10			−0.745764			−0.372882	+	0.927879i	$0.621630\pi$
$500$		0			0
$501$	7.93274e10	−	6.43436e10i	1.25913	−	1.02130i
$502$		0			0
$503$		−	8.85743e10i		−	1.38368i	−0.722051	−	0.691840i	$-0.756800\pi$
$503$		−	8.85743e10i		−	1.38368i	0.722051	−	0.691840i	$-0.243200\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−7.64033e10	−	9.41954e10i	−1.15633	−	1.42560i
$508$		0			0
$509$		−	4.40674e10i		−	0.656518i	−0.944588	−	0.328259i	$-0.893538\pi$
$509$		−	4.40674e10i		−	0.656518i	0.944588	−	0.328259i	$-0.106462\pi$
$510$		0			0
$511$	−4.71185e9			−0.0691047
$512$		0			0
$513$	−5.78420e10	+	2.96478e10i	−0.835168	+	0.428078i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	2.36854e10			0.331526
$518$		0			0
$519$	−3.15196e9	+	2.55660e9i	−0.0434422	+	0.0352366i
$520$		0			0
$521$		−	1.51676e10i		−	0.205857i	−0.994689	−	0.102929i	$-0.967179\pi$
$521$		−	1.51676e10i		−	0.205857i	0.994689	−	0.102929i	$-0.0328213\pi$
$522$		0			0
$523$	2.51178e10			0.335719			0.167859	−	0.985811i	$-0.446315\pi$
$523$	2.51178e10			0.335719			0.167859	+	0.985811i	$0.446315\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			7.44724e10i			0.965502i
$528$		0			0
$529$	8.65301e9			0.110496
$530$		0			0
$531$	1.14515e11	+	2.41487e10i	1.44040	+	0.303750i
$532$		0			0
$533$			1.23644e11i			1.53202i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−6.34919e10	+	5.14992e10i	−0.763521	+	0.619303i
$538$		0			0
$539$		−	5.07616e10i		−	0.601423i
$540$		0			0
$541$	2.12945e10			0.248587			0.124294	−	0.992245i	$-0.460334\pi$
$541$	2.12945e10			0.248587			0.124294	+	0.992245i	$0.460334\pi$
$542$		0			0
$543$	9.05008e10	+	1.11576e11i	1.04101	+	1.28343i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	3.20728e10			0.358251			0.179125	−	0.983826i	$-0.442673\pi$
$547$	3.20728e10			0.358251			0.179125	+	0.983826i	$0.442673\pi$
$548$		0			0
$549$	1.12204e10	−	5.32081e10i	0.123515	−	0.585718i
$550$		0			0
$551$			1.27602e11i			1.38437i
$552$		0			0
$553$	−9.30285e9			−0.0994754
$554$		0			0
$555$		0			0
$556$		0			0
$557$			1.22209e11i			1.26965i	0.772657	+	0.634824i	$0.218927\pi$
$557$			1.22209e11i			1.26965i	−0.772657	+	0.634824i	$0.781073\pi$
$558$		0			0
$559$	−7.29833e10			−0.747440
$560$		0			0
$561$	2.48731e10	+	3.06653e10i	0.251118	+	0.309596i
$562$		0			0
$563$		−	1.65935e11i		−	1.65160i	−0.563965	−	0.825798i	$-0.690725\pi$
$563$		−	1.65935e11i		−	1.65160i	0.563965	−	0.825798i	$-0.309275\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	1.80767e10	+	7.97877e9i	0.174899	+	0.0771976i
$568$		0			0
$569$			4.71831e10i			0.450130i	0.974344	+	0.225065i	$0.0722594\pi$
$569$			4.71831e10i			0.450130i	−0.974344	+	0.225065i	$0.927741\pi$
$570$		0			0
$571$	−2.58965e10			−0.243611			−0.121805	−	0.992554i	$-0.538868\pi$
$571$	−2.58965e10			−0.243611			−0.121805	+	0.992554i	$0.538868\pi$
$572$		0			0
$573$	−7.82557e10	+	6.34744e10i	−0.725935	+	0.588817i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−8.31317e10			−0.750004			−0.375002	−	0.927024i	$-0.622358\pi$
$577$	−8.31317e10			−0.750004			−0.375002	+	0.927024i	$0.622358\pi$
$578$		0			0
$579$	1.41141e9	+	1.74009e9i	0.0125585	+	0.0154831i
$580$		0			0
$581$		−	1.61720e10i		−	0.141925i
$582$		0			0
$583$	−2.51308e10			−0.217536
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	1.44597e10i		−	0.121789i	−0.998144	−	0.0608944i	$-0.980605\pi$
$587$		−	1.44597e10i		−	0.121789i	0.998144	−	0.0608944i	$-0.0193953\pi$
$588$		0			0
$589$	−1.70772e11			−1.41891
$590$		0			0
$591$	1.33446e11	−	1.08240e11i	1.09385	−	0.887237i
$592$		0			0
$593$			1.00123e11i			0.809687i	0.914386	+	0.404843i	$0.132674\pi$
$593$			1.00123e11i			0.809687i	−0.914386	+	0.404843i	$0.867326\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	3.98092e10	+	4.90796e10i	0.313391	+	0.386370i
$598$		0			0
$599$		−	5.36658e10i		−	0.416860i	−0.978037	−	0.208430i	$-0.933165\pi$
$599$		−	5.36658e10i		−	0.416860i	0.978037	−	0.208430i	$-0.0668354\pi$
$600$		0			0
$601$	−2.45026e11			−1.87808			−0.939042	−	0.343802i	$-0.888285\pi$
$601$	−2.45026e11			−1.87808			−0.939042	+	0.343802i	$0.888285\pi$
$602$		0			0
$603$	6.26576e8	−	2.97127e9i	0.00473920	−	0.0224736i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−2.54344e11			−1.87355			−0.936777	−	0.349926i	$-0.886207\pi$
$607$	−2.54344e11			−1.87355			−0.936777	+	0.349926i	$0.886207\pi$
$608$		0			0
$609$	3.01266e10	−	2.44361e10i	0.219019	−	0.177649i
$610$		0			0
$611$		−	1.24639e11i		−	0.894314i
$612$		0			0
$613$	−1.26140e11			−0.893330			−0.446665	−	0.894701i	$-0.647388\pi$
$613$	−1.26140e11			−0.893330			−0.446665	+	0.894701i	$0.647388\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			2.04776e11i			1.41299i	0.707718	+	0.706495i	$0.249725\pi$
$617$			2.04776e11i			1.41299i	−0.707718	+	0.706495i	$0.750275\pi$
$618$		0			0
$619$	6.98668e10			0.475892			0.237946	−	0.971278i	$-0.423526\pi$
$619$	6.98668e10			0.475892			0.237946	+	0.971278i	$0.423526\pi$
$620$		0			0
$621$	6.39787e10	+	1.24821e11i	0.430198	+	0.839306i
$622$		0			0
$623$		−	1.97960e10i		−	0.131409i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−7.03183e10	+	5.70362e10i	−0.454986	+	0.369046i
$628$		0			0
$629$		−	1.51984e11i		−	0.970946i
$630$		0			0
$631$	−2.51868e11			−1.58875			−0.794373	−	0.607430i	$-0.792201\pi$
$631$	−2.51868e11			−1.58875			−0.794373	+	0.607430i	$0.792201\pi$
$632$		0			0
$633$	−7.54377e10	−	9.30050e10i	−0.469866	−	0.579284i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−2.67122e11			−1.62238
$638$		0			0
$639$	2.02447e11	+	4.26917e10i	1.21425	+	0.256059i
$640$		0			0
$641$		−	2.41559e11i		−	1.43084i	−0.698695	−	0.715420i	$-0.746235\pi$
$641$		−	2.41559e11i		−	1.43084i	0.698695	−	0.715420i	$-0.253765\pi$
$642$		0			0
$643$	−6.40818e9			−0.0374879			−0.0187439	−	0.999824i	$-0.505967\pi$
$643$	−6.40818e9			−0.0374879			−0.0187439	+	0.999824i	$0.505967\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			1.87104e11i			1.06774i	0.845566	+	0.533871i	$0.179263\pi$
$647$			1.87104e11i			1.06774i	−0.845566	+	0.533871i	$0.820737\pi$
$648$		0			0
$649$	1.63028e11			0.918931
$650$		0			0
$651$	3.27033e10	+	4.03190e10i	0.182082	+	0.224484i
$652$		0			0
$653$			2.67029e11i			1.46861i	0.678821	+	0.734304i	$0.262491\pi$
$653$			2.67029e11i			1.46861i	−0.678821	+	0.734304i	$0.737509\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−1.38968e10	+	6.58997e10i	−0.0745852	+	0.353689i
$658$		0			0
$659$		−	7.73297e10i		−	0.410020i	−0.978760	−	0.205010i	$-0.934277\pi$
$659$		−	7.73297e10i		−	0.410020i	0.978760	−	0.205010i	$-0.0657226\pi$
$660$		0			0
$661$	−2.86028e9			−0.0149831			−0.00749157	−	0.999972i	$-0.502385\pi$
$661$	−2.86028e9			−0.0149831			−0.00749157	+	0.999972i	$0.502385\pi$
$662$		0			0
$663$	1.61370e11	−	1.30889e11i	0.835156	−	0.677408i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	2.75360e11			1.39122
$668$		0			0
$669$	−4.79362e10	−	5.90992e10i	−0.239309	−	0.295037i
$670$		0			0
$671$		−	7.57489e10i		−	0.373668i
$672$		0			0
$673$	−3.48736e11			−1.69995			−0.849976	−	0.526821i	$-0.823384\pi$
$673$	−3.48736e11			−1.69995			−0.849976	+	0.526821i	$0.823384\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	2.13457e11i		−	1.01615i	−0.861314	−	0.508073i	$-0.830358\pi$
$677$		−	2.13457e11i		−	1.01615i	0.861314	−	0.508073i	$-0.169642\pi$
$678$		0			0
$679$	−2.65423e10			−0.124870
$680$		0			0
$681$	3.10184e11	−	2.51595e11i	1.44222	−	1.16980i
$682$		0			0
$683$			5.28052e10i			0.242657i	0.992612	+	0.121329i	$0.0387155\pi$
$683$			5.28052e10i			0.242657i	−0.992612	+	0.121329i	$0.961284\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	1.99315e11	+	2.45729e11i	0.894772	+	1.10314i
$688$		0			0
$689$			1.32245e11i			0.586818i
$690$		0			0
$691$	−3.18883e11			−1.39868			−0.699341	−	0.714788i	$-0.746523\pi$
$691$	−3.18883e11			−1.39868			−0.699341	+	0.714788i	$0.746523\pi$
$692$		0			0
$693$	2.69323e10	+	5.67943e9i	0.116772	+	0.0246247i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−1.37119e11			−0.580987
$698$		0			0
$699$	−2.53208e9	+	2.05381e9i	−0.0106064	+	0.00860302i
$700$		0			0
$701$			3.03198e11i			1.25561i	0.778371	+	0.627804i	$0.216046\pi$
$701$			3.03198e11i			1.25561i	−0.778371	+	0.627804i	$0.783954\pi$
$702$		0			0
$703$	3.48513e11			1.42691
$704$		0			0
$705$		0			0
$706$		0			0
$707$			2.02495e10i			0.0810468i
$708$		0			0
$709$	−4.44301e11			−1.75830			−0.879149	−	0.476547i	$-0.841888\pi$
$709$	−4.44301e11			−1.75830			−0.879149	+	0.476547i	$0.841888\pi$
$710$		0			0
$711$	−2.74372e10	+	1.30109e11i	−0.107365	+	0.509131i
$712$		0			0
$713$			3.68519e11i			1.42594i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−3.18846e11	+	2.58621e11i	−1.20644	+	0.978558i
$718$		0			0
$719$			3.95363e11i			1.47938i	0.672946	+	0.739692i	$0.265029\pi$
$719$			3.95363e11i			1.47938i	−0.672946	+	0.739692i	$0.734971\pi$
$720$		0			0
$721$	1.75435e10			0.0649194
$722$		0			0
$723$	−1.04295e11	−	1.28582e11i	−0.381690	−	0.470574i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−1.95379e10			−0.0699423			−0.0349712	−	0.999388i	$-0.511134\pi$
$727$	−1.95379e10			−0.0699423			−0.0349712	+	0.999388i	$0.511134\pi$
$728$		0			0
$729$	1.64905e11	−	2.29288e11i	0.583879	−	0.811841i
$730$		0			0
$731$		−	8.09373e10i		−	0.283452i
$732$		0			0
$733$	2.26211e11			0.783607			0.391803	−	0.920049i	$-0.371851\pi$
$733$	2.26211e11			0.783607			0.391803	+	0.920049i	$0.371851\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	4.23001e9i		−	0.0143374i
$738$		0			0
$739$	−1.33510e11			−0.447649			−0.223824	−	0.974630i	$-0.571854\pi$
$739$	−1.33510e11			−0.447649			−0.223824	+	0.974630i	$0.571854\pi$
$740$		0			0
$741$	3.00141e11	+	3.70035e11i	0.995526	+	1.22736i
$742$		0			0
$743$		−	2.51159e11i		−	0.824125i	−0.911156	−	0.412062i	$-0.864809\pi$
$743$		−	2.51159e11i		−	0.824125i	0.911156	−	0.412062i	$-0.135191\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−2.26181e11	−	4.76967e10i	−0.726398	−	0.153181i
$748$		0			0
$749$			2.87979e10i			0.0915027i
$750$		0			0
$751$	1.05457e11			0.331525			0.165762	−	0.986166i	$-0.446992\pi$
$751$	1.05457e11			0.331525			0.165762	+	0.986166i	$0.446992\pi$
$752$		0			0
$753$	−1.56483e11	+	1.26926e11i	−0.486729	+	0.394793i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	7.46755e10			0.227402			0.113701	−	0.993515i	$-0.463729\pi$
$757$	7.46755e10			0.227402			0.113701	+	0.993515i	$0.463729\pi$
$758$		0			0
$759$	1.23082e11	+	1.51744e11i	0.370874	+	0.457240i
$760$		0			0
$761$			4.79443e11i			1.42955i	0.699356	+	0.714773i	$0.253470\pi$
$761$			4.79443e11i			1.42955i	−0.699356	+	0.714773i	$0.746530\pi$
$762$		0			0
$763$	−1.60036e10			−0.0472193
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	8.57899e11i		−	2.47888i
$768$		0			0
$769$	1.54805e11			0.442669			0.221334	−	0.975198i	$-0.428959\pi$
$769$	1.54805e11			0.442669			0.221334	+	0.975198i	$0.428959\pi$
$770$		0			0
$771$	2.93428e11	−	2.38004e11i	0.830393	−	0.673544i
$772$		0			0
$773$		−	5.73981e10i		−	0.160761i	−0.996764	−	0.0803803i	$-0.974387\pi$
$773$		−	5.73981e10i		−	0.160761i	0.996764	−	0.0803803i	$-0.0256135\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−6.67412e10	−	8.22833e10i	−0.183109	−	0.225750i
$778$		0			0
$779$		−	3.14426e11i		−	0.853825i
$780$		0			0
$781$	2.88211e11			0.774652
$782$		0			0
$783$	−2.52909e11	−	4.93419e11i	−0.672849	−	1.31271i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	3.98550e11			1.03893			0.519463	−	0.854493i	$-0.326132\pi$
$787$	3.98550e11			1.03893			0.519463	+	0.854493i	$0.326132\pi$
$788$		0			0
$789$	−3.59580e11	+	2.91661e11i	−0.927872	+	0.752611i
$790$		0			0
$791$		−	1.23795e11i		−	0.316226i
$792$		0			0
$793$	−3.98613e11			−1.00800
$794$		0			0
$795$		0			0
$796$		0			0
$797$			3.04742e11i			0.755264i	0.925956	+	0.377632i	$0.123262\pi$
$797$			3.04742e11i			0.755264i	−0.925956	+	0.377632i	$0.876738\pi$
$798$		0			0
$799$	1.38223e11			0.339151
$800$		0			0
$801$	−2.76866e11	−	5.83849e10i	−0.672573	−	0.141831i
$802$		0			0
$803$			9.38171e10i			0.225642i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	3.48480e11	−	2.82657e11i	0.821643	−	0.666447i
$808$		0			0
$809$			1.47553e11i			0.344471i	0.985056	+	0.172236i	$0.0550990\pi$
$809$			1.47553e11i			0.344471i	−0.985056	+	0.172236i	$0.944901\pi$
$810$		0			0
$811$	−4.18158e11			−0.966622			−0.483311	−	0.875449i	$-0.660566\pi$
$811$	−4.18158e11			−0.966622			−0.483311	+	0.875449i	$0.660566\pi$
$812$		0			0
$813$	4.29460e11	+	5.29469e11i	0.983017	+	1.21193i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	1.85597e11			0.416564
$818$		0			0
$819$	2.98868e10	−	1.41726e11i	0.0664269	−	0.315002i
$820$		0			0
$821$			1.74589e11i			0.384277i	0.981368	+	0.192138i	$0.0615422\pi$
$821$			1.74589e11i			0.384277i	−0.981368	+	0.192138i	$0.938458\pi$
$822$		0			0
$823$	5.90822e11			1.28783			0.643913	−	0.765099i	$-0.277310\pi$
$823$	5.90822e11			1.28783			0.643913	+	0.765099i	$0.277310\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			4.79472e10i			0.102504i	0.998686	+	0.0512521i	$0.0163212\pi$
$827$			4.79472e10i			0.102504i	−0.998686	+	0.0512521i	$0.983679\pi$
$828$		0			0
$829$	1.65827e11			0.351104			0.175552	−	0.984470i	$-0.443829\pi$
$829$	1.65827e11			0.351104			0.175552	+	0.984470i	$0.443829\pi$
$830$		0			0
$831$	6.36561e10	+	7.84798e10i	0.133486	+	0.164571i
$832$		0			0
$833$		−	2.96234e11i		−	0.615255i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	6.60352e11	−	3.38473e11i	1.34547	−	0.689640i
$838$		0			0
$839$			5.82814e11i			1.17620i	0.808787	+	0.588101i	$0.200124\pi$
$839$			5.82814e11i			1.17620i	−0.808787	+	0.588101i	$0.799876\pi$
$840$		0			0
$841$	−5.88258e11			−1.17594
$842$		0			0
$843$	3.90548e11	−	3.16779e11i	0.773329	−	0.627259i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−6.00529e10			−0.116681
$848$		0			0
$849$	2.46243e11	+	3.03586e11i	0.473952	+	0.584321i
$850$		0			0
$851$		−	7.52076e11i		−	1.43398i
$852$		0			0
$853$	−4.89760e11			−0.925097			−0.462549	−	0.886594i	$-0.653065\pi$
$853$	−4.89760e11			−0.925097			−0.462549	+	0.886594i	$0.653065\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	1.11714e11i		−	0.207103i	−0.994624	−	0.103551i	$-0.966979\pi$
$857$		−	1.11714e11i		−	0.207103i	0.994624	−	0.103551i	$-0.0330206\pi$
$858$		0			0
$859$	−9.17638e10			−0.168538			−0.0842692	−	0.996443i	$-0.526856\pi$
$859$	−9.17638e10			−0.168538			−0.0842692	+	0.996443i	$0.526856\pi$
$860$		0			0
$861$	−7.42355e10	+	6.02135e10i	−0.135082	+	0.109567i
$862$		0			0
$863$		−	5.85110e11i		−	1.05486i	−0.849599	−	0.527429i	$-0.823156\pi$
$863$		−	5.85110e11i		−	1.05486i	0.849599	−	0.527429i	$-0.176844\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−2.10787e11	−	2.59874e11i	−0.373051	−	0.459924i
$868$		0			0
$869$			1.85228e11i			0.324809i
$870$		0			0
$871$	−2.22596e10			−0.0386762
$872$		0			0
$873$	−7.82819e10	+	3.71219e11i	−0.134773	+	0.639106i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	3.71675e11			0.628297			0.314148	−	0.949374i	$-0.398281\pi$
$877$	3.71675e11			0.628297			0.314148	+	0.949374i	$0.398281\pi$
$878$		0			0
$879$	2.98880e11	−	2.42426e11i	0.500658	−	0.406091i
$880$		0			0
$881$		−	8.23900e10i		−	0.136764i	−0.997659	−	0.0683818i	$-0.978216\pi$
$881$		−	8.23900e10i		−	0.136764i	0.997659	−	0.0683818i	$-0.0217836\pi$
$882$		0			0
$883$	9.90170e11			1.62880			0.814399	−	0.580306i	$-0.197067\pi$
$883$	9.90170e11			1.62880			0.814399	+	0.580306i	$0.197067\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	2.88048e11i		−	0.465339i	−0.972556	−	0.232670i	$-0.925254\pi$
$887$		−	2.88048e11i		−	0.465339i	0.972556	−	0.232670i	$-0.0747461\pi$
$888$		0			0
$889$	2.27073e11			0.363545
$890$		0			0
$891$	1.58864e11	−	3.59923e11i	0.252067	−	0.571083i
$892$		0			0
$893$			3.16958e11i			0.498420i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	7.98521e11	−	6.47692e11i	1.23344	−	1.00046i
$898$		0			0
$899$		−	1.45677e12i		−	2.23024i
$900$		0			0
$901$	−1.46658e11			−0.222539
$902$		0			0
$903$	−3.55423e10	−	4.38190e10i	−0.0534557	−	0.0659040i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	3.59541e11			0.531274			0.265637	−	0.964073i	$-0.414418\pi$
$907$	3.59541e11			0.531274			0.265637	+	0.964073i	$0.414418\pi$
$908$		0			0
$909$	2.83208e11	+	5.97223e10i	0.414811	+	0.0874744i
$910$		0			0
$911$		−	1.59446e11i		−	0.231494i	−0.993279	−	0.115747i	$-0.963074\pi$
$911$		−	1.59446e11i		−	0.231494i	0.993279	−	0.115747i	$-0.0369263\pi$
$912$		0			0
$913$	−3.22000e11			−0.463418
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	1.17771e11i		−	0.166556i
$918$		0			0
$919$	−6.75378e11			−0.946858			−0.473429	−	0.880832i	$-0.656984\pi$
$919$	−6.75378e11			−0.946858			−0.473429	+	0.880832i	$0.656984\pi$
$920$		0			0
$921$	−3.33470e11	−	4.11126e11i	−0.463467	−	0.571395i
$922$		0			0
$923$		−	1.51665e12i		−	2.08968i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	5.17414e10	−	2.45362e11i	0.0700680	−	0.332268i
$928$		0			0
$929$		−	7.68419e11i		−	1.03166i	−0.856692	−	0.515829i	$-0.827484\pi$
$929$		−	7.68419e11i		−	1.03166i	0.856692	−	0.515829i	$-0.172516\pi$
$930$		0			0
$931$	6.79292e11			0.904185
$932$		0			0
$933$	−5.92919e11	+	4.80926e11i	−0.782472	+	0.634675i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−2.45957e11			−0.319080			−0.159540	−	0.987191i	$-0.551001\pi$
$937$	−2.45957e11			−0.319080			−0.159540	+	0.987191i	$0.551001\pi$
$938$		0			0
$939$	−5.93988e11	−	7.32311e11i	−0.764039	−	0.941961i
$940$		0			0
$941$		−	2.05589e11i		−	0.262205i	−0.991369	−	0.131102i	$-0.958148\pi$
$941$		−	2.05589e11i		−	0.262205i	0.991369	−	0.131102i	$-0.0418517\pi$
$942$		0			0
$943$	−6.78519e11			−0.858055
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	1.25097e12i		−	1.55542i	−0.628624	−	0.777709i	$-0.716382\pi$
$947$		−	1.25097e12i		−	1.55542i	0.628624	−	0.777709i	$-0.283618\pi$
$948$		0			0
$949$	4.93693e11			0.608684
$950$		0			0
$951$	−5.89002e11	+	4.77749e11i	−0.720103	+	0.584087i
$952$		0			0
$953$		−	9.19955e11i		−	1.11531i	−0.830074	−	0.557654i	$-0.811702\pi$
$953$		−	9.19955e11i		−	1.11531i	0.830074	−	0.557654i	$-0.188298\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−4.86545e11	−	5.99848e11i	−0.580064	−	0.715143i
$958$		0			0
$959$		−	3.70235e10i		−	0.0437727i
$960$		0			0
$961$	1.09672e12			1.28589
$962$		0			0
$963$	4.02766e11	+	8.49345e10i	0.468326	+	0.0987595i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	7.65204e11			0.875127			0.437563	−	0.899188i	$-0.355842\pi$
$967$	7.65204e11			0.875127			0.437563	+	0.899188i	$0.355842\pi$
$968$		0			0
$969$	−4.10363e11	+	3.32852e11i	−0.465450	+	0.377534i
$970$		0			0
$971$			1.23934e12i			1.39417i	0.716990	+	0.697084i	$0.245519\pi$
$971$			1.23934e12i			1.39417i	−0.716990	+	0.697084i	$0.754481\pi$
$972$		0			0
$973$	2.31285e11			0.258045
$974$		0			0
$975$		0			0
$976$		0			0
$977$			1.59613e12i			1.75182i	0.482473	+	0.875911i	$0.339739\pi$
$977$			1.59613e12i			1.75182i	−0.482473	+	0.875911i	$0.660261\pi$
$978$		0			0
$979$	−3.94156e11			−0.429079
$980$		0			0
$981$	−4.71998e10	+	2.23825e11i	−0.0509641	+	0.241676i
$982$		0			0
$983$			1.82288e12i			1.95229i	0.217130	+	0.976143i	$0.430331\pi$
$983$			1.82288e12i			1.95229i	−0.217130	+	0.976143i	$0.569669\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	7.48332e10	−	6.06983e10i	0.0788543	−	0.0639599i
$988$		0			0
$989$		−	4.00510e11i		−	0.418628i
$990$		0			0
$991$	9.32448e11			0.966786			0.483393	−	0.875404i	$-0.339404\pi$
$991$	9.32448e11			0.966786			0.483393	+	0.875404i	$0.339404\pi$
$992$		0			0
$993$	5.35247e11	+	6.59890e11i	0.550500	+	0.678695i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	8.66862e10			0.0877343			0.0438671	−	0.999037i	$-0.486032\pi$
$997$	8.66862e10			0.0877343			0.0438671	+	0.999037i	$0.486032\pi$
$998$		0			0
$999$	−1.34765e12	+	6.90759e11i	−1.35306	+	0.693529i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.9.g.h.101.5		16
3.2	odd	2	inner	300.9.g.h.101.6		16
5.2	odd	4		60.9.b.a.29.4	yes	16
5.3	odd	4		60.9.b.a.29.13	yes	16
5.4	even	2	inner	300.9.g.h.101.12		16
15.2	even	4		60.9.b.a.29.14	yes	16
15.8	even	4		60.9.b.a.29.3	✓	16
15.14	odd	2	inner	300.9.g.h.101.11		16
20.3	even	4		240.9.c.d.209.4		16
20.7	even	4		240.9.c.d.209.13		16
60.23	odd	4		240.9.c.d.209.14		16
60.47	odd	4		240.9.c.d.209.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.9.b.a.29.3	✓	16	15.8	even	4
60.9.b.a.29.4	yes	16	5.2	odd	4
60.9.b.a.29.13	yes	16	5.3	odd	4
60.9.b.a.29.14	yes	16	15.2	even	4
240.9.c.d.209.3		16	60.47	odd	4
240.9.c.d.209.4		16	20.3	even	4
240.9.c.d.209.13		16	20.7	even	4
240.9.c.d.209.14		16	60.23	odd	4
300.9.g.h.101.5		16	1.1	even	1	trivial
300.9.g.h.101.6		16	3.2	odd	2	inner
300.9.g.h.101.11		16	15.14	odd	2	inner
300.9.g.h.101.12		16	5.4	even	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)

\(f(q)\)	\(=\)	\(q+(-51.0255 - 62.9079i) q^{3} -459.019 q^{7} +(-1353.80 + 6419.81i) q^{9} +O(q^{10})\)	\(q+(-51.0255 - 62.9079i) q^{3} -459.019 q^{7} +(-1353.80 + 6419.81i) q^{9} +9139.47i q^{11} +48094.6 q^{13} +53336.1i q^{17} -122304. q^{19} +(23421.7 + 28875.9i) q^{21} +263928. i q^{23} +(472935. - 242410. i) q^{27} -1.04331e6i q^{29} +1.39629e6 q^{31} +(574944. - 466346. i) q^{33} -2.84955e6 q^{37} +(-2.45405e6 - 3.02553e6i) q^{39} +2.57085e6i q^{41} -1.51750e6 q^{43} -2.59155e6i q^{47} -5.55410e6 q^{49} +(3.35526e6 - 2.72150e6i) q^{51} +2.74970e6i q^{53} +(6.24065e6 + 7.69391e6i) q^{57} -1.78378e7i q^{59} -8.28811e6 q^{61} +(621418. - 2.94681e6i) q^{63} -462829. q^{67} +(1.66031e7 - 1.34671e7i) q^{69} -3.15348e7i q^{71} +1.02650e7 q^{73} -4.19519e6i q^{77} +2.02668e7 q^{79} +(-3.93812e7 - 1.73822e7i) q^{81} +3.52318e7i q^{83} +(-6.56326e7 + 5.32356e7i) q^{87} +4.31268e7i q^{89} -2.20763e7 q^{91} +(-7.12462e7 - 8.78374e7i) q^{93} +5.78240e7 q^{97} +(-5.86737e7 - 1.23730e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 14756 q^{9}+O(q^{10}) \) 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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