LMFDB - Embedded newform 300.7.b.e.149.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,7,Mod(149,300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("300.149");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 300 = 2^{2} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	300.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$69.0162250860$
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} + 406 x^{14} + 67561 x^{12} + 5921226 x^{10} + 291565644 x^{8} + 7924637994 x^{6} + \cdots + 276002078881 $ x^16 + 406x^14 + 67561x^12 + 5921226x^10 + 291565644x^8 + 7924637994x^6 + 107459649193x^4 + 554332708870*x^2 + 276002078881
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{30}\cdot 3^{18}\cdot 5^{26} $
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			149.2
Root			$4.99240i$ of defining polynomial
Character	$\chi$	$=$	300.149
Dual form			300.7.b.e.149.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-24.5453 + 11.2485i) q^{3} +414.697i q^{7} +(475.944 - 552.194i) q^{9} +O(q^{10})$	$q+(-24.5453 + 11.2485i) q^{3} +414.697i q^{7} +(475.944 - 552.194i) q^{9} +61.5283i q^{11} -1379.66i q^{13} -6808.58 q^{17} +7822.89 q^{19} +(-4664.71 - 10178.9i) q^{21} -7459.40 q^{23} +(-5470.84 + 18907.4i) q^{27} -424.303i q^{29} -49671.1 q^{31} +(-692.099 - 1510.23i) q^{33} -2272.15i q^{37} +(15519.1 + 33864.3i) q^{39} -56271.8i q^{41} +131325. i q^{43} +84752.1 q^{47} -54324.8 q^{49} +(167119. - 76586.2i) q^{51} -166720. q^{53} +(-192015. + 87995.5i) q^{57} -400288. i q^{59} -83666.7 q^{61} +(228993. + 197373. i) q^{63} -425271. i q^{67} +(183093. - 83906.8i) q^{69} +346163. i q^{71} -171712. i q^{73} -25515.6 q^{77} +266214. q^{79} +(-78396.2 - 525627. i) q^{81} +1.01555e6 q^{83} +(4772.76 + 10414.6i) q^{87} -955706. i q^{89} +572143. q^{91} +(1.21919e6 - 558724. i) q^{93} +1.19431e6i q^{97} +(33975.6 + 29284.0i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2984 q^{9}+O(q^{10}) $ 16 * q - 2984 * q^9	$ 16 q - 2984 q^{9} + 30544 q^{19} - 1736 q^{21} + 70064 q^{31} - 79216 q^{39} - 405120 q^{49} + 858240 q^{51} - 271216 q^{61} + 509880 q^{69} - 1415408 q^{79} - 2396224 q^{81} - 4008224 q^{91} - 5300160 q^{99}+O(q^{100}) $ 16 * q - 2984 * q^9 + 30544 * q^19 - 1736 * q^21 + 70064 * q^31 - 79216 * q^39 - 405120 * q^49 + 858240 * q^51 - 271216 * q^61 + 509880 * q^69 - 1415408 * q^79 - 2396224 * q^81 - 4008224 * q^91 - 5300160 * 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
$3$	−24.5453	+	11.2485i	−0.909085	+	0.416610i
$3$	−24.5453	+	11.2485i	−0.909085	+	0.416610i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$			414.697i			1.20903i	0.796594	+	0.604515i	$0.206633\pi$
$7$			414.697i			1.20903i	−0.796594	+	0.604515i	$0.793367\pi$
$8$		0			0
$9$	475.944	−	552.194i	0.652872	−	0.757468i
$10$		0			0
$11$			61.5283i			0.0462271i	0.999733	+	0.0231135i	$0.00735792\pi$
$11$			61.5283i			0.0462271i	−0.999733	+	0.0231135i	$0.992642\pi$
$12$		0			0
$13$		−	1379.66i		−	0.627976i	−0.949427	−	0.313988i	$-0.898335\pi$
$13$		−	1379.66i		−	0.627976i	0.949427	−	0.313988i	$-0.101665\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−6808.58			−1.38583			−0.692915	−	0.721019i	$-0.743674\pi$
$17$	−6808.58			−1.38583			−0.692915	+	0.721019i	$0.743674\pi$
$18$		0			0
$19$	7822.89			1.14053			0.570264	−	0.821461i	$-0.306841\pi$
$19$	7822.89			1.14053			0.570264	+	0.821461i	$0.306841\pi$
$20$		0			0
$21$	−4664.71	−	10178.9i	−0.503694	−	1.09911i
$22$		0			0
$23$	−7459.40			−0.613084			−0.306542	−	0.951857i	$-0.599172\pi$
$23$	−7459.40			−0.613084			−0.306542	+	0.951857i	$0.599172\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−5470.84	+	18907.4i	−0.277947	+	0.960596i
$28$		0			0
$29$		−	424.303i		−	0.0173973i	−0.999962	−	0.00869865i	$-0.997231\pi$
$29$		−	424.303i		−	0.0173973i	0.999962	−	0.00869865i	$-0.00276890\pi$
$30$		0			0
$31$	−49671.1			−1.66732			−0.833660	−	0.552279i	$-0.813758\pi$
$31$	−49671.1			−1.66732			−0.833660	+	0.552279i	$0.813758\pi$
$32$		0			0
$33$	−692.099	−	1510.23i	−0.0192587	−	0.0420244i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	2272.15i		−	0.0448572i	−0.999748	−	0.0224286i	$-0.992860\pi$
$37$		−	2272.15i		−	0.0448572i	0.999748	−	0.0224286i	$-0.00713984\pi$
$38$		0			0
$39$	15519.1	+	33864.3i	0.261621	+	0.570884i
$40$		0			0
$41$		−	56271.8i		−	0.816469i	−0.912877	−	0.408234i	$-0.866145\pi$
$41$		−	56271.8i		−	0.816469i	0.912877	−	0.408234i	$-0.133855\pi$
$42$		0			0
$43$			131325.i			1.65174i	0.563858	+	0.825872i	$0.309316\pi$
$43$			131325.i			1.65174i	−0.563858	+	0.825872i	$0.690684\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	84752.1			0.816313			0.408157	−	0.912912i	$-0.366172\pi$
$47$	84752.1			0.816313			0.408157	+	0.912912i	$0.366172\pi$
$48$		0			0
$49$	−54324.8			−0.461754
$50$		0			0
$51$	167119.	−	76586.2i	1.25984	−	0.577351i
$52$		0			0
$53$	−166720.			−1.11985			−0.559924	−	0.828544i	$-0.689170\pi$
$53$	−166720.			−1.11985			−0.559924	+	0.828544i	$0.689170\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−192015.	+	87995.5i	−1.03684	+	0.475156i
$58$		0			0
$59$		−	400288.i		−	1.94902i	−0.224342	−	0.974511i	$-0.572023\pi$
$59$		−	400288.i		−	1.94902i	0.224342	−	0.974511i	$-0.427977\pi$
$60$		0			0
$61$	−83666.7			−0.368607			−0.184303	−	0.982869i	$-0.559003\pi$
$61$	−83666.7			−0.368607			−0.184303	+	0.982869i	$0.559003\pi$
$62$		0			0
$63$	228993.	+	197373.i	0.915802	+	0.789342i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	425271.i		−	1.41397i	−0.707227	−	0.706986i	$-0.750054\pi$
$67$		−	425271.i		−	1.41397i	0.707227	−	0.706986i	$-0.249946\pi$
$68$		0			0
$69$	183093.	−	83906.8i	0.557346	−	0.255417i
$70$		0			0
$71$			346163.i			0.967175i	0.875296	+	0.483587i	$0.160666\pi$
$71$			346163.i			0.967175i	−0.875296	+	0.483587i	$0.839334\pi$
$72$		0			0
$73$		−	171712.i		−	0.441401i	−0.975342	−	0.220700i	$-0.929166\pi$
$73$		−	171712.i		−	0.441401i	0.975342	−	0.220700i	$-0.0708343\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−25515.6			−0.0558899
$78$		0			0
$79$	266214.			0.539944			0.269972	−	0.962868i	$-0.412985\pi$
$79$	266214.			0.539944			0.269972	+	0.962868i	$0.412985\pi$
$80$		0			0
$81$	−78396.2	−	525627.i	−0.147516	−	0.989060i
$82$		0			0
$83$	1.01555e6			1.77610			0.888048	−	0.459751i	$-0.152061\pi$
$83$	1.01555e6			1.77610			0.888048	+	0.459751i	$0.152061\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	4772.76	+	10414.6i	0.00724789	+	0.0158156i
$88$		0			0
$89$		−	955706.i		−	1.35567i	−0.735214	−	0.677835i	$-0.762918\pi$
$89$		−	955706.i		−	1.35567i	0.735214	−	0.677835i	$-0.237082\pi$
$90$		0			0
$91$	572143.			0.759242
$92$		0			0
$93$	1.21919e6	−	558724.i	1.51574	−	0.694622i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			1.19431e6i			1.30859i	0.756240	+	0.654294i	$0.227034\pi$
$97$			1.19431e6i			1.30859i	−0.756240	+	0.654294i	$0.772966\pi$
$98$		0			0
$99$	33975.6	+	29284.0i	0.0350155	+	0.0301804i
$100$		0			0
$101$		−	173920.i		−	0.168805i	−0.996432	−	0.0844025i	$-0.973102\pi$
$101$		−	173920.i		−	0.168805i	0.996432	−	0.0844025i	$-0.0268981\pi$
$102$		0			0
$103$		−	32602.0i		−	0.0298355i	−0.999889	−	0.0149177i	$-0.995251\pi$
$103$		−	32602.0i		−	0.0298355i	0.999889	−	0.0149177i	$-0.00474864\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.24930e6			1.01980			0.509902	−	0.860233i	$-0.329682\pi$
$107$	1.24930e6			1.01980			0.509902	+	0.860233i	$0.329682\pi$
$108$		0			0
$109$	2.29804e6			1.77451			0.887255	−	0.461279i	$-0.152609\pi$
$109$	2.29804e6			1.77451			0.887255	+	0.461279i	$0.152609\pi$
$110$		0			0
$111$	25558.2	+	55770.6i	0.0186879	+	0.0407790i
$112$		0			0
$113$	1.94978e6			1.35130			0.675648	−	0.737224i	$-0.263864\pi$
$113$	1.94978e6			1.35130			0.675648	+	0.737224i	$0.263864\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−761842.	−	656642.i	−0.475672	−	0.409988i
$118$		0			0
$119$		−	2.82350e6i		−	1.67551i
$120$		0			0
$121$	1.76778e6			0.997863
$122$		0			0
$123$	632972.	+	1.38121e6i	0.340149	+	0.742240i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	2.24490e6i		−	1.09594i	−0.836498	−	0.547969i	$-0.815401\pi$
$127$		−	2.24490e6i		−	1.09594i	0.836498	−	0.547969i	$-0.184599\pi$
$128$		0			0
$129$	−1.47721e6	−	3.22342e6i	−0.688133	−	1.50158i
$130$		0			0
$131$		−	2.73023e6i		−	1.21446i	−0.794524	−	0.607232i	$-0.792280\pi$
$131$		−	2.73023e6i		−	1.21446i	0.794524	−	0.607232i	$-0.207720\pi$
$132$		0			0
$133$			3.24413e6i			1.37893i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.34993e6			−1.69169			−0.845845	−	0.533429i	$-0.820903\pi$
$137$	−4.34993e6			−1.69169			−0.845845	+	0.533429i	$0.820903\pi$
$138$		0			0
$139$	−53660.9			−0.0199808			−0.00999041	−	0.999950i	$-0.503180\pi$
$139$	−53660.9			−0.0199808			−0.00999041	+	0.999950i	$0.503180\pi$
$140$		0			0
$141$	−2.08027e6	+	953332.i	−0.742098	+	0.340084i
$142$		0			0
$143$	84888.3			0.0290295
$144$		0			0
$145$		0			0
$146$		0			0
$147$	1.33342e6	−	611071.i	0.419773	−	0.192371i
$148$		0			0
$149$		−	2.96446e6i		−	0.896163i	−0.893993	−	0.448081i	$-0.852107\pi$
$149$		−	2.96446e6i		−	0.896163i	0.893993	−	0.448081i	$-0.147893\pi$
$150$		0			0
$151$	−129392.			−0.0375817			−0.0187909	−	0.999823i	$-0.505982\pi$
$151$	−129392.			−0.0375817			−0.0187909	+	0.999823i	$0.505982\pi$
$152$		0			0
$153$	−3.24050e6	+	3.75966e6i	−0.904770	+	1.04972i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			38136.7i			0.00985472i	0.999988	+	0.00492736i	$0.00156843\pi$
$157$			38136.7i			0.00985472i	−0.999988	+	0.00492736i	$0.998432\pi$
$158$		0			0
$159$	4.09218e6	−	1.87534e6i	1.01804	−	0.466540i
$160$		0			0
$161$		−	3.09339e6i		−	0.741238i
$162$		0			0
$163$		−	879856.i		−	0.203165i	−0.994827	−	0.101582i	$-0.967609\pi$
$163$		−	879856.i		−	0.203165i	0.994827	−	0.101582i	$-0.0323906\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	682306.			0.146497			0.0732487	−	0.997314i	$-0.476663\pi$
$167$	682306.			0.146497			0.0732487	+	0.997314i	$0.476663\pi$
$168$		0			0
$169$	2.92334e6			0.605646
$170$		0			0
$171$	3.72325e6	−	4.31975e6i	0.744619	−	0.863914i
$172$		0			0
$173$	6.54016e6			1.26314			0.631568	−	0.775321i	$-0.282412\pi$
$173$	6.54016e6			1.26314			0.631568	+	0.775321i	$0.282412\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	4.50263e6	+	9.82519e6i	0.811982	+	1.77183i
$178$		0			0
$179$			7.06827e6i			1.23241i	0.787587	+	0.616203i	$0.211330\pi$
$179$			7.06827e6i			1.23241i	−0.787587	+	0.616203i	$0.788670\pi$
$180$		0			0
$181$	−6.16556e6			−1.03977			−0.519885	−	0.854237i	$-0.674025\pi$
$181$	−6.16556e6			−1.03977			−0.519885	+	0.854237i	$0.674025\pi$
$182$		0			0
$183$	2.05363e6	−	941123.i	0.335095	−	0.153565i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	418920.i		−	0.0640629i
$188$		0			0
$189$	−7.84085e6	−	2.26874e6i	−1.16139	−	0.336047i
$190$		0			0
$191$		−	4.08743e6i		−	0.586611i	−0.956019	−	0.293306i	$-0.905245\pi$
$191$		−	4.08743e6i		−	0.586611i	0.956019	−	0.293306i	$-0.0947554\pi$
$192$		0			0
$193$			56701.9i			0.00788725i	0.999992	+	0.00394363i	$0.00125530\pi$
$193$			56701.9i			0.00788725i	−0.999992	+	0.00394363i	$0.998745\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−6.15117e6			−0.804561			−0.402281	−	0.915516i	$-0.631782\pi$
$197$	−6.15117e6			−0.804561			−0.402281	+	0.915516i	$0.631782\pi$
$198$		0			0
$199$	8.00172e6			1.01537			0.507685	−	0.861543i	$-0.330501\pi$
$199$	8.00172e6			1.01537			0.507685	+	0.861543i	$0.330501\pi$
$200$		0			0
$201$	4.78365e6	+	1.04384e7i	0.589075	+	1.28542i
$202$		0			0
$203$	175957.			0.0210339
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−3.55025e6	+	4.11904e6i	−0.400266	+	0.464392i
$208$		0			0
$209$			481328.i			0.0527233i
$210$		0			0
$211$	−8.73879e6			−0.930259			−0.465130	−	0.885243i	$-0.653992\pi$
$211$	−8.73879e6			−0.930259			−0.465130	+	0.885243i	$0.653992\pi$
$212$		0			0
$213$	−3.89380e6	−	8.49666e6i	−0.402935	−	0.879244i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	2.05985e7i		−	2.01584i
$218$		0			0
$219$	1.93150e6	+	4.21473e6i	0.183892	+	0.401271i
$220$		0			0
$221$			9.39355e6i			0.870268i
$222$		0			0
$223$		−	1.51154e7i		−	1.36302i	−0.731807	−	0.681512i	$-0.761323\pi$
$223$		−	1.51154e7i		−	1.36302i	0.731807	−	0.681512i	$-0.238677\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1.32527e7			1.13299			0.566497	−	0.824064i	$-0.308298\pi$
$227$	1.32527e7			1.13299			0.566497	+	0.824064i	$0.308298\pi$
$228$		0			0
$229$	−8.97752e6			−0.747567			−0.373783	−	0.927516i	$-0.621940\pi$
$229$	−8.97752e6			−0.747567			−0.373783	+	0.927516i	$0.621940\pi$
$230$		0			0
$231$	626288.	−	287012.i	0.0508087	−	0.0232843i
$232$		0			0
$233$	−1.05430e7			−0.833479			−0.416739	−	0.909026i	$-0.636827\pi$
$233$	−1.05430e7			−0.833479			−0.416739	+	0.909026i	$0.636827\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−6.53429e6	+	2.99450e6i	−0.490855	+	0.224946i
$238$		0			0
$239$		−	4.63239e6i		−	0.339322i	−0.985503	−	0.169661i	$-0.945733\pi$
$239$		−	4.63239e6i		−	0.339322i	0.985503	−	0.169661i	$-0.0542672\pi$
$240$		0			0
$241$	2.14836e7			1.53481			0.767407	−	0.641161i	$-0.221547\pi$
$241$	2.14836e7			1.53481			0.767407	+	0.641161i	$0.221547\pi$
$242$		0			0
$243$	7.83676e6	+	1.20198e7i	0.546157	+	0.837683i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	1.07929e7i		−	0.716225i
$248$		0			0
$249$	−2.49269e7	+	1.14234e7i	−1.61462	+	0.739939i
$250$		0			0
$251$			2.26878e7i			1.43473i	0.696697	+	0.717366i	$0.254652\pi$
$251$			2.26878e7i			1.43473i	−0.696697	+	0.717366i	$0.745348\pi$
$252$		0			0
$253$		−	458964.i		−	0.0283411i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3.52158e6			0.207462			0.103731	−	0.994605i	$-0.466922\pi$
$257$	3.52158e6			0.207462			0.103731	+	0.994605i	$0.466922\pi$
$258$		0			0
$259$	942254.			0.0542337
$260$		0			0
$261$	−234298.	−	201944.i	−0.0131779	−	0.0113582i
$262$		0			0
$263$	−1.73580e7			−0.954185			−0.477093	−	0.878853i	$-0.658309\pi$
$263$	−1.73580e7			−0.954185			−0.477093	+	0.878853i	$0.658309\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	1.07502e7	+	2.34581e7i	0.564786	+	1.23242i
$268$		0			0
$269$			3.42910e7i			1.76167i	0.473428	+	0.880833i	$0.343016\pi$
$269$			3.42910e7i			1.76167i	−0.473428	+	0.880833i	$0.656984\pi$
$270$		0			0
$271$	7.75571e6			0.389685			0.194843	−	0.980835i	$-0.437580\pi$
$271$	7.75571e6			0.389685			0.194843	+	0.980835i	$0.437580\pi$
$272$		0			0
$273$	−1.40434e7	+	6.43573e6i	−0.690216	+	0.316308i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	2.75252e6i		−	0.129507i	−0.997901	−	0.0647533i	$-0.979374\pi$
$277$		−	2.75252e6i		−	0.129507i	0.997901	−	0.0647533i	$-0.0206260\pi$
$278$		0			0
$279$	−2.36406e7	+	2.74281e7i	−1.08855	+	1.26294i
$280$		0			0
$281$			9.11525e6i			0.410818i	0.978676	+	0.205409i	$0.0658525\pi$
$281$			9.11525e6i			0.410818i	−0.978676	+	0.205409i	$0.934148\pi$
$282$		0			0
$283$		−	2.81125e7i		−	1.24034i	−0.784468	−	0.620169i	$-0.787064\pi$
$283$		−	2.81125e7i		−	1.24034i	0.784468	−	0.620169i	$-0.212936\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2.33358e7			0.987135
$288$		0			0
$289$	2.22192e7			0.920525
$290$		0			0
$291$	−1.34342e7	−	2.93148e7i	−0.545171	−	1.18962i
$292$		0			0
$293$	−3.34576e7			−1.33012			−0.665062	−	0.746788i	$-0.731595\pi$
$293$	−3.34576e7			−1.33012			−0.665062	+	0.746788i	$0.731595\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−1.16334e6	−	336611.i	−0.0444056	−	0.0128487i
$298$		0			0
$299$			1.02915e7i			0.385002i
$300$		0			0
$301$	−5.44602e7			−1.99701
$302$		0			0
$303$	1.95633e6	+	4.26892e6i	0.0703259	+	0.153458i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			1.57467e7i			0.544219i	0.962266	+	0.272110i	$0.0877213\pi$
$307$			1.57467e7i			0.544219i	−0.962266	+	0.272110i	$0.912279\pi$
$308$		0			0
$309$	366723.	+	800226.i	0.0124298	+	0.0271230i
$310$		0			0
$311$		−	7.00699e6i		−	0.232943i	−0.993194	−	0.116472i	$-0.962842\pi$
$311$		−	7.00699e6i		−	0.232943i	0.993194	−	0.116472i	$-0.0371584\pi$
$312$		0			0
$313$		−	5.79389e6i		−	0.188946i	−0.995527	−	0.0944728i	$-0.969883\pi$
$313$		−	5.79389e6i		−	0.188946i	0.995527	−	0.0944728i	$-0.0301166\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3.96429e7			−1.24448			−0.622239	−	0.782827i	$-0.713777\pi$
$317$	−3.96429e7			−1.24448			−0.622239	+	0.782827i	$0.713777\pi$
$318$		0			0
$319$	26106.6			0.000804227
$320$		0			0
$321$	−3.06645e7	+	1.40528e7i	−0.927089	+	0.424861i
$322$		0			0
$323$	−5.32628e7			−1.58058
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−5.64061e7	+	2.58495e7i	−1.61318	+	0.739279i
$328$		0			0
$329$			3.51465e7i			0.986947i
$330$		0			0
$331$	3.49643e6			0.0964142			0.0482071	−	0.998837i	$-0.484649\pi$
$331$	3.49643e6			0.0964142			0.0482071	+	0.998837i	$0.484649\pi$
$332$		0			0
$333$	−1.25467e6	−	1.08142e6i	−0.0339779	−	0.0292860i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	4.53770e7i		−	1.18562i	−0.805342	−	0.592811i	$-0.798018\pi$
$337$		−	4.53770e7i		−	1.18562i	0.805342	−	0.592811i	$-0.201982\pi$
$338$		0			0
$339$	−4.78580e7	+	2.19321e7i	−1.22844	+	0.562964i
$340$		0			0
$341$		−	3.05618e6i		−	0.0770753i
$342$		0			0
$343$			2.62604e7i			0.650756i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.83908e7			0.440162			0.220081	−	0.975482i	$-0.429368\pi$
$347$	1.83908e7			0.440162			0.220081	+	0.975482i	$0.429368\pi$
$348$		0			0
$349$	7.74052e7			1.82093			0.910467	−	0.413583i	$-0.135723\pi$
$349$	7.74052e7			1.82093			0.910467	+	0.413583i	$0.135723\pi$
$350$		0			0
$351$	2.60859e7	+	7.54792e6i	0.603232	+	0.174544i
$352$		0			0
$353$	4.76509e7			1.08330			0.541648	−	0.840605i	$-0.317801\pi$
$353$	4.76509e7			1.08330			0.541648	+	0.840605i	$0.317801\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	3.17601e7	+	6.93037e7i	0.698034	+	1.52318i
$358$		0			0
$359$		−	8.95438e7i		−	1.93532i	−0.252262	−	0.967659i	$-0.581174\pi$
$359$		−	8.95438e7i		−	1.93532i	0.252262	−	0.967659i	$-0.418826\pi$
$360$		0			0
$361$	1.41516e7			0.300805
$362$		0			0
$363$	−4.33906e7	+	1.98848e7i	−0.907143	+	0.415720i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	7.10716e7i		−	1.43780i	−0.695115	−	0.718899i	$-0.744647\pi$
$367$		−	7.10716e7i		−	1.43780i	0.695115	−	0.718899i	$-0.255353\pi$
$368$		0			0
$369$	−3.10730e7	−	2.67822e7i	−0.618449	−	0.533049i
$370$		0			0
$371$		−	6.91382e7i		−	1.35393i
$372$		0			0
$373$		−	9.06999e7i		−	1.74775i	−0.486146	−	0.873877i	$-0.661598\pi$
$373$		−	9.06999e7i		−	1.74775i	0.486146	−	0.873877i	$-0.338402\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−585395.			−0.0109251
$378$		0			0
$379$	1.66542e7			0.305918			0.152959	−	0.988233i	$-0.451120\pi$
$379$	1.66542e7			0.305918			0.152959	+	0.988233i	$0.451120\pi$
$380$		0			0
$381$	2.52517e7	+	5.51018e7i	0.456579	+	0.996302i
$382$		0			0
$383$	1.04612e7			0.186202			0.0931010	−	0.995657i	$-0.470322\pi$
$383$	1.04612e7			0.186202			0.0931010	+	0.995657i	$0.470322\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	7.25170e7	+	6.25034e7i	1.25114	+	1.07838i
$388$		0			0
$389$			1.01213e8i			1.71945i	0.510758	+	0.859724i	$0.329365\pi$
$389$			1.01213e8i			1.71945i	−0.510758	+	0.859724i	$0.670635\pi$
$390$		0			0
$391$	5.07879e7			0.849631
$392$		0			0
$393$	3.07109e7	+	6.70143e7i	0.505958	+	1.10405i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	1.01774e8i		−	1.62655i	−0.581883	−	0.813273i	$-0.697684\pi$
$397$		−	1.01774e8i		−	1.62655i	0.581883	−	0.813273i	$-0.302316\pi$
$398$		0			0
$399$	−3.64915e7	−	7.96281e7i	−0.574478	−	1.25357i
$400$		0			0
$401$			1.02772e8i			1.59382i	0.604095	+	0.796912i	$0.293535\pi$
$401$			1.02772e8i			1.59382i	−0.604095	+	0.796912i	$0.706465\pi$
$402$		0			0
$403$			6.85294e7i			1.04704i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	139801.			0.00207362
$408$		0			0
$409$	−6.56844e7			−0.960047			−0.480023	−	0.877256i	$-0.659372\pi$
$409$	−6.56844e7			−0.960047			−0.480023	+	0.877256i	$0.659372\pi$
$410$		0			0
$411$	1.06770e8	−	4.89301e7i	1.53789	−	0.704775i
$412$		0			0
$413$	1.65998e8			2.35642
$414$		0			0
$415$		0			0
$416$		0			0
$417$	1.31712e6	−	603603.i	0.0181643	−	0.00832421i
$418$		0			0
$419$		−	6.34334e7i		−	0.862334i	−0.902272	−	0.431167i	$-0.858102\pi$
$419$		−	6.34334e7i		−	0.862334i	0.902272	−	0.431167i	$-0.141898\pi$
$420$		0			0
$421$	−9.10207e7			−1.21981			−0.609907	−	0.792473i	$-0.708793\pi$
$421$	−9.10207e7			−1.21981			−0.609907	+	0.792473i	$0.708793\pi$
$422$		0			0
$423$	4.03372e7	−	4.67996e7i	0.532948	−	0.618331i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	3.46964e7i		−	0.445657i
$428$		0			0
$429$	−2.08361e6	+	954864.i	−0.0263903	+	0.0120940i
$430$		0			0
$431$		−	4.20911e7i		−	0.525725i	−0.964833	−	0.262862i	$-0.915333\pi$
$431$		−	4.20911e7i		−	0.525725i	0.964833	−	0.262862i	$-0.0846666\pi$
$432$		0			0
$433$		−	92285.8i		−	0.00113677i	−1.00000		0.000568383i	$-0.999819\pi$
$433$		−	92285.8i		−	0.00113677i	1.00000		0.000568383i	$-0.000180922\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−5.83540e7			−0.699240
$438$		0			0
$439$	1.42372e8			1.68280			0.841398	−	0.540416i	$-0.181733\pi$
$439$	1.42372e8			1.68280			0.841398	+	0.540416i	$0.181733\pi$
$440$		0			0
$441$	−2.58556e7	+	2.99979e7i	−0.301466	+	0.349764i
$442$		0			0
$443$	8.16658e7			0.939354			0.469677	−	0.882838i	$-0.344370\pi$
$443$	8.16658e7			0.939354			0.469677	+	0.882838i	$0.344370\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	3.33457e7	+	7.27636e7i	0.373350	+	0.814688i
$448$		0			0
$449$		−	1.31510e8i		−	1.45284i	−0.687250	−	0.726421i	$-0.741183\pi$
$449$		−	1.31510e8i		−	1.45284i	0.687250	−	0.726421i	$-0.258817\pi$
$450$		0			0
$451$	3.46231e6			0.0377430
$452$		0			0
$453$	3.17597e6	−	1.45546e6i	0.0341650	−	0.0156569i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	8.80615e7i		−	0.922651i	−0.887231	−	0.461326i	$-0.847374\pi$
$457$		−	8.80615e7i		−	0.922651i	0.887231	−	0.461326i	$-0.152626\pi$
$458$		0			0
$459$	3.72487e7	−	1.28733e8i	0.385188	−	1.33122i
$460$		0			0
$461$		−	7.89270e7i		−	0.805606i	−0.915287	−	0.402803i	$-0.868036\pi$
$461$		−	7.89270e7i		−	0.805606i	0.915287	−	0.402803i	$-0.131964\pi$
$462$		0			0
$463$		−	1.04058e8i		−	1.04841i	−0.851592	−	0.524205i	$-0.824363\pi$
$463$		−	1.04058e8i		−	1.04841i	0.851592	−	0.524205i	$-0.175637\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−2.93827e7			−0.288497			−0.144248	−	0.989542i	$-0.546076\pi$
$467$	−2.93827e7			−0.288497			−0.144248	+	0.989542i	$0.546076\pi$
$468$		0			0
$469$	1.76359e8			1.70954
$470$		0			0
$471$	−428980.	−	936077.i	−0.00410558	−	0.00895878i
$472$		0			0
$473$	−8.08021e6			−0.0763553
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−7.93492e7	+	9.20617e7i	−0.731118	+	0.848250i
$478$		0			0
$479$		−	1.71918e7i		−	0.156429i	−0.996937	−	0.0782143i	$-0.975078\pi$
$479$		−	1.71918e7i		−	0.156429i	0.996937	−	0.0782143i	$-0.0249218\pi$
$480$		0			0
$481$	−3.13480e6			−0.0281692
$482$		0			0
$483$	3.47959e7	+	7.59283e7i	0.308807	+	0.673848i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			8.72586e7i			0.755477i	0.925912	+	0.377739i	$0.123298\pi$
$487$			8.72586e7i			0.755477i	−0.925912	+	0.377739i	$0.876702\pi$
$488$		0			0
$489$	9.89704e6	+	2.15963e7i	0.0846406	+	0.184694i
$490$		0			0
$491$		−	1.79828e8i		−	1.51919i	−0.650396	−	0.759596i	$-0.725397\pi$
$491$		−	1.79828e8i		−	1.51919i	0.650396	−	0.759596i	$-0.274603\pi$
$492$		0			0
$493$			2.88890e6i			0.0241097i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.43553e8			−1.16934
$498$		0			0
$499$	8.70160e7			0.700321			0.350161	−	0.936690i	$-0.386127\pi$
$499$	8.70160e7			0.700321			0.350161	+	0.936690i	$0.386127\pi$
$500$		0			0
$501$	−1.67474e7	+	7.67490e6i	−0.133179	+	0.0610323i
$502$		0			0
$503$	4.25009e7			0.333960			0.166980	−	0.985960i	$-0.446598\pi$
$503$	4.25009e7			0.333960			0.166980	+	0.985960i	$0.446598\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−7.17542e7	+	3.28831e7i	−0.550584	+	0.252318i
$508$		0			0
$509$		−	1.86035e8i		−	1.41072i	−0.708850	−	0.705359i	$-0.750786\pi$
$509$		−	1.86035e8i		−	1.41072i	0.708850	−	0.705359i	$-0.249214\pi$
$510$		0			0
$511$	7.12087e7			0.533667
$512$		0			0
$513$	−4.27977e7	+	1.47911e8i	−0.317007	+	1.09559i
$514$		0			0
$515$		0			0
$516$		0			0
$517$			5.21465e6i			0.0377358i
$518$		0			0
$519$	−1.60530e8	+	7.35668e7i	−1.14830	+	0.526235i
$520$		0			0
$521$			5.95626e7i			0.421173i	0.977575	+	0.210587i	$0.0675374\pi$
$521$			5.95626e7i			0.421173i	−0.977575	+	0.210587i	$0.932463\pi$
$522$		0			0
$523$			1.77836e8i			1.24313i	0.783364	+	0.621563i	$0.213502\pi$
$523$			1.77836e8i			1.24313i	−0.783364	+	0.621563i	$0.786498\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	3.38190e8			2.31062
$528$		0			0
$529$	−9.23933e7			−0.624127
$530$		0			0
$531$	−2.21037e8	−	1.90515e8i	−1.47632	−	1.27246i
$532$		0			0
$533$	−7.76362e7			−0.512723
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−7.95072e7	−	1.73493e8i	−0.513433	−	1.12036i
$538$		0			0
$539$		−	3.34251e6i		−	0.0213455i
$540$		0			0
$541$	1.08545e8			0.685519			0.342760	−	0.939423i	$-0.388638\pi$
$541$	1.08545e8			0.685519			0.342760	+	0.939423i	$0.388638\pi$
$542$		0			0
$543$	1.51336e8	−	6.93532e7i	0.945239	−	0.433178i
$544$		0			0
$545$		0			0
$546$		0			0
$547$			1.25572e7i			0.0767241i	0.999264	+	0.0383620i	$0.0122140\pi$
$547$			1.25572e7i			0.0767241i	−0.999264	+	0.0383620i	$0.987786\pi$
$548$		0			0
$549$	−3.98207e7	+	4.62003e7i	−0.240653	+	0.279208i
$550$		0			0
$551$		−	3.31927e6i		−	0.0198421i
$552$		0			0
$553$			1.10398e8i			0.652809i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	2.86738e7			0.165928			0.0829641	−	0.996553i	$-0.473561\pi$
$557$	2.86738e7			0.165928			0.0829641	+	0.996553i	$0.473561\pi$
$558$		0			0
$559$	1.81185e8			1.03726
$560$		0			0
$561$	4.71221e6	+	1.02825e7i	0.0266892	+	0.0582386i
$562$		0			0
$563$	−1.61064e8			−0.902555			−0.451278	−	0.892384i	$-0.649032\pi$
$563$	−1.61064e8			−0.902555			−0.451278	+	0.892384i	$0.649032\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	2.17976e8	−	3.25107e7i	1.19580	−	0.178352i
$568$		0			0
$569$			1.14947e8i			0.623965i	0.950088	+	0.311982i	$0.100993\pi$
$569$			1.14947e8i			0.623965i	−0.950088	+	0.311982i	$0.899007\pi$
$570$		0			0
$571$	−8.55993e7			−0.459792			−0.229896	−	0.973215i	$-0.573839\pi$
$571$	−8.55993e7			−0.459792			−0.229896	+	0.973215i	$0.573839\pi$
$572$		0			0
$573$	4.59774e7	+	1.00327e8i	0.244388	+	0.533280i
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	8.75149e7i		−	0.455569i	−0.973712	−	0.227785i	$-0.926852\pi$
$577$		−	8.75149e7i		−	0.455569i	0.973712	−	0.227785i	$-0.0731482\pi$
$578$		0			0
$579$	−637810.	−	1.39177e6i	−0.00328591	−	0.00717019i
$580$		0			0
$581$			4.21145e8i			2.14735i
$582$		0			0
$583$		−	1.02580e7i		−	0.0517673i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.61940e8			−0.800646			−0.400323	−	0.916374i	$-0.631102\pi$
$587$	−1.61940e8			−0.800646			−0.400323	+	0.916374i	$0.631102\pi$
$588$		0			0
$589$	−3.88571e8			−1.90163
$590$		0			0
$591$	1.50982e8	−	6.91913e7i	0.731415	−	0.335188i
$592$		0			0
$593$	−2.85726e8			−1.37021			−0.685104	−	0.728446i	$-0.740243\pi$
$593$	−2.85726e8			−1.37021			−0.685104	+	0.728446i	$0.740243\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−1.96405e8	+	9.00071e7i	−0.923058	+	0.423013i
$598$		0			0
$599$		−	4.98255e7i		−	0.231831i	−0.993259	−	0.115916i	$-0.963020\pi$
$599$		−	4.98255e7i		−	0.231831i	0.993259	−	0.115916i	$-0.0369802\pi$
$600$		0			0
$601$	7.73983e7			0.356540			0.178270	−	0.983982i	$-0.442950\pi$
$601$	7.73983e7			0.356540			0.178270	+	0.983982i	$0.442950\pi$
$602$		0			0
$603$	−2.34832e8	−	2.02405e8i	−1.07104	−	0.923143i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			4.82366e7i			0.215680i	0.994168	+	0.107840i	$0.0343935\pi$
$607$			4.82366e7i			0.215680i	−0.994168	+	0.107840i	$0.965607\pi$
$608$		0			0
$609$	−4.31892e6	+	1.97925e6i	−0.0191216	+	0.00876292i
$610$		0			0
$611$		−	1.16929e8i		−	0.512625i
$612$		0			0
$613$		−	1.34171e8i		−	0.582474i	−0.956651	−	0.291237i	$-0.905933\pi$
$613$		−	1.34171e8i		−	0.582474i	0.956651	−	0.291237i	$-0.0940668\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−4.66923e7			−0.198788			−0.0993939	−	0.995048i	$-0.531690\pi$
$617$	−4.66923e7			−0.198788			−0.0993939	+	0.995048i	$0.531690\pi$
$618$		0			0
$619$	1.05294e8			0.443948			0.221974	−	0.975053i	$-0.428750\pi$
$619$	1.05294e8			0.443948			0.221974	+	0.975053i	$0.428750\pi$
$620$		0			0
$621$	4.08092e7	−	1.41038e8i	0.170405	−	0.588927i
$622$		0			0
$623$	3.96329e8			1.63905
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−5.41421e6	−	1.18144e7i	−0.0219651	−	0.0479300i
$628$		0			0
$629$			1.54701e7i			0.0621644i
$630$		0			0
$631$	2.38661e7			0.0949936			0.0474968	−	0.998871i	$-0.484876\pi$
$631$	2.38661e7			0.0949936			0.0474968	+	0.998871i	$0.484876\pi$
$632$		0			0
$633$	2.14496e8	−	9.82981e7i	0.845685	−	0.387555i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			7.49500e7i			0.289970i
$638$		0			0
$639$	1.91149e8	+	1.64754e8i	0.732604	+	0.631441i
$640$		0			0
$641$			8.49890e7i			0.322692i	0.986898	+	0.161346i	$0.0515836\pi$
$641$			8.49890e7i			0.322692i	−0.986898	+	0.161346i	$0.948416\pi$
$642$		0			0
$643$		−	3.40735e8i		−	1.28169i	−0.767670	−	0.640845i	$-0.778584\pi$
$643$		−	3.40735e8i		−	1.28169i	0.767670	−	0.640845i	$-0.221416\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	3.26674e8			1.20615			0.603076	−	0.797684i	$-0.293942\pi$
$647$	3.26674e8			1.20615			0.603076	+	0.797684i	$0.293942\pi$
$648$		0			0
$649$	2.46290e7			0.0900976
$650$		0			0
$651$	2.31701e8	+	5.05596e8i	0.839819	+	1.83257i
$652$		0			0
$653$	−2.15329e8			−0.773326			−0.386663	−	0.922221i	$-0.626372\pi$
$653$	−2.15329e8			−0.773326			−0.386663	+	0.922221i	$0.626372\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−9.48186e7	−	8.17254e7i	−0.334347	−	0.288178i
$658$		0			0
$659$			3.75646e8i			1.31257i	0.754513	+	0.656285i	$0.227873\pi$
$659$			3.75646e8i			1.31257i	−0.754513	+	0.656285i	$0.772127\pi$
$660$		0			0
$661$	−3.65705e8			−1.26627			−0.633135	−	0.774041i	$-0.718232\pi$
$661$	−3.65705e8			−1.26627			−0.633135	+	0.774041i	$0.718232\pi$
$662$		0			0
$663$	−1.05663e8	−	2.30568e8i	−0.362563	−	0.791148i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			3.16504e6i			0.0106660i
$668$		0			0
$669$	1.70025e8	+	3.71011e8i	0.567850	+	1.23911i
$670$		0			0
$671$		−	5.14787e6i		−	0.0170396i
$672$		0			0
$673$		−	2.47794e8i		−	0.812917i	−0.913669	−	0.406459i	$-0.866764\pi$
$673$		−	2.47794e8i		−	0.812917i	0.913669	−	0.406459i	$-0.133236\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.27016e8			−0.409347			−0.204674	−	0.978830i	$-0.565613\pi$
$677$	−1.27016e8			−0.409347			−0.204674	+	0.978830i	$0.565613\pi$
$678$		0			0
$679$	−4.95278e8			−1.58212
$680$		0			0
$681$	−3.25292e8	+	1.49073e8i	−1.02999	+	0.472017i
$682$		0			0
$683$	−3.06787e8			−0.962884			−0.481442	−	0.876478i	$-0.659887\pi$
$683$	−3.06787e8			−0.962884			−0.481442	+	0.876478i	$0.659887\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	2.20356e8	−	1.00983e8i	0.679602	−	0.311444i
$688$		0			0
$689$			2.30017e8i			0.703238i
$690$		0			0
$691$	−1.85102e8			−0.561017			−0.280509	−	0.959852i	$-0.590503\pi$
$691$	−1.85102e8			−0.561017			−0.280509	+	0.959852i	$0.590503\pi$
$692$		0			0
$693$	−1.21440e7	+	1.40896e7i	−0.0364890	+	0.0423348i
$694$		0			0
$695$		0			0
$696$		0			0
$697$			3.83131e8i			1.13149i
$698$		0			0
$699$	2.58780e8	−	1.18592e8i	0.757703	−	0.347236i
$700$		0			0
$701$			4.66462e8i			1.35414i	0.735920	+	0.677068i	$0.236750\pi$
$701$			4.66462e8i			1.35414i	−0.735920	+	0.677068i	$0.763250\pi$
$702$		0			0
$703$		−	1.77748e7i		−	0.0511609i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	7.21241e7			0.204090
$708$		0			0
$709$	−5.51189e8			−1.54654			−0.773271	−	0.634076i	$-0.781381\pi$
$709$	−5.51189e8			−1.54654			−0.773271	+	0.634076i	$0.781381\pi$
$710$		0			0
$711$	1.26703e8	−	1.47002e8i	0.352515	−	0.408991i
$712$		0			0
$713$	3.70517e8			1.02221
$714$		0			0
$715$		0			0
$716$		0			0
$717$	5.21073e7	+	1.13703e8i	0.141365	+	0.308472i
$718$		0			0
$719$			6.52168e8i			1.75458i	0.479963	+	0.877289i	$0.340650\pi$
$719$			6.52168e8i			1.75458i	−0.479963	+	0.877289i	$0.659350\pi$
$720$		0			0
$721$	1.35200e7			0.0360720
$722$		0			0
$723$	−5.27321e8	+	2.41657e8i	−1.39528	+	0.639419i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	2.30050e7i		−	0.0598713i	−0.999552	−	0.0299357i	$-0.990470\pi$
$727$		−	2.30050e7i		−	0.0598713i	0.999552	−	0.0299357i	$-0.00953024\pi$
$728$		0			0
$729$	−3.27560e8	−	2.06879e8i	−0.845490	−	0.533990i
$730$		0			0
$731$		−	8.94138e8i		−	2.28904i
$732$		0			0
$733$			2.39083e8i			0.607068i	0.952821	+	0.303534i	$0.0981665\pi$
$733$			2.39083e8i			0.607068i	−0.952821	+	0.303534i	$0.901833\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2.61662e7			0.0653638
$738$		0			0
$739$	−4.07576e8			−1.00989			−0.504946	−	0.863151i	$-0.668488\pi$
$739$	−4.07576e8			−1.00989			−0.504946	+	0.863151i	$0.668488\pi$
$740$		0			0
$741$	1.21404e8	+	2.64916e8i	0.298386	+	0.651109i
$742$		0			0
$743$	−2.87465e8			−0.700841			−0.350420	−	0.936593i	$-0.613961\pi$
$743$	−2.87465e8			−0.700841			−0.350420	+	0.936593i	$0.613961\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	4.83344e8	−	5.60780e8i	1.15956	−	1.34534i
$748$		0			0
$749$			5.18083e8i			1.23297i
$750$		0			0
$751$	4.86133e8			1.14772			0.573859	−	0.818954i	$-0.305446\pi$
$751$	4.86133e8			1.14772			0.573859	+	0.818954i	$0.305446\pi$
$752$		0			0
$753$	−2.55203e8	−	5.56878e8i	−0.597724	−	1.30429i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	2.22098e8i		−	0.511986i	−0.966679	−	0.255993i	$-0.917598\pi$
$757$		−	2.22098e8i		−	0.511986i	0.966679	−	0.255993i	$-0.0824024\pi$
$758$		0			0
$759$	5.16264e6	+	1.12654e7i	0.0118072	+	0.0257645i
$760$		0			0
$761$			1.36605e8i			0.309964i	0.987917	+	0.154982i	$0.0495320\pi$
$761$			1.36605e8i			0.309964i	−0.987917	+	0.154982i	$0.950468\pi$
$762$		0			0
$763$			9.52992e8i			2.14544i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−5.52263e8			−1.22394
$768$		0			0
$769$	6.76897e8			1.48848			0.744241	−	0.667911i	$-0.232811\pi$
$769$	6.76897e8			1.48848			0.744241	+	0.667911i	$0.232811\pi$
$770$		0			0
$771$	−8.64383e7	+	3.96124e7i	−0.188601	+	0.0864308i
$772$		0			0
$773$	−6.32394e8			−1.36914			−0.684572	−	0.728945i	$-0.740011\pi$
$773$	−6.32394e8			−1.36914			−0.684572	+	0.728945i	$0.740011\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−2.31279e7	+	1.05989e7i	−0.0493030	+	0.0225943i
$778$		0			0
$779$		−	4.40208e8i		−	0.931206i
$780$		0			0
$781$	−2.12988e7			−0.0447097
$782$		0			0
$783$	8.02247e6	+	2.32129e6i	0.0167118	+	0.00483553i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			7.43145e8i			1.52458i	0.647238	+	0.762288i	$0.275924\pi$
$787$			7.43145e8i			1.52458i	−0.647238	+	0.762288i	$0.724076\pi$
$788$		0			0
$789$	4.26058e8	−	1.95251e8i	0.867436	−	0.397523i
$790$		0			0
$791$			8.08569e8i			1.63376i
$792$		0			0
$793$			1.15432e8i			0.231476i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−4.32008e7			−0.0853329			−0.0426664	−	0.999089i	$-0.513585\pi$
$797$	−4.32008e7			−0.0853329			−0.0426664	+	0.999089i	$0.513585\pi$
$798$		0			0
$799$	−5.77042e8			−1.13127
$800$		0			0
$801$	−5.27735e8	−	4.54862e8i	−1.02688	−	0.885079i
$802$		0			0
$803$	1.05652e7			0.0204047
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−3.85721e8	−	8.41683e8i	−0.733927	−	1.60150i
$808$		0			0
$809$		−	4.44561e8i		−	0.839627i	−0.907610	−	0.419813i	$-0.862096\pi$
$809$		−	4.44561e8i		−	0.839627i	0.907610	−	0.419813i	$-0.137904\pi$
$810$		0			0
$811$	−3.11585e8			−0.584136			−0.292068	−	0.956398i	$-0.594343\pi$
$811$	−3.11585e8			−0.584136			−0.292068	+	0.956398i	$0.594343\pi$
$812$		0			0
$813$	−1.90366e8	+	8.72399e7i	−0.354257	+	0.162347i
$814$		0			0
$815$		0			0
$816$		0			0
$817$			1.02734e9i			1.88386i
$818$		0			0
$819$	2.72308e8	−	3.15934e8i	0.495688	−	0.575102i
$820$		0			0
$821$		−	1.06060e9i		−	1.91656i	−0.285833	−	0.958280i	$-0.592270\pi$
$821$		−	1.06060e9i		−	1.91656i	0.285833	−	0.958280i	$-0.407730\pi$
$822$		0			0
$823$			4.65269e7i			0.0834651i	0.999129	+	0.0417325i	$0.0132877\pi$
$823$			4.65269e7i			0.0834651i	−0.999129	+	0.0417325i	$0.986712\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−4.51328e8			−0.797951			−0.398975	−	0.916962i	$-0.630634\pi$
$827$	−4.51328e8			−0.797951			−0.398975	+	0.916962i	$0.630634\pi$
$828$		0			0
$829$	−3.04859e8			−0.535100			−0.267550	−	0.963544i	$-0.586214\pi$
$829$	−3.04859e8			−0.535100			−0.267550	+	0.963544i	$0.586214\pi$
$830$		0			0
$831$	3.09617e7	+	6.75616e7i	0.0539538	+	0.117733i
$832$		0			0
$833$	3.69875e8			0.639912
$834$		0			0
$835$		0			0
$836$		0			0
$837$	2.71743e8	−	9.39152e8i	0.463427	−	1.60162i
$838$		0			0
$839$		−	2.25607e8i		−	0.382003i	−0.981590	−	0.191001i	$-0.938827\pi$
$839$		−	2.25607e8i		−	0.382003i	0.981590	−	0.191001i	$-0.0611735\pi$
$840$		0			0
$841$	5.94643e8			0.999697
$842$		0			0
$843$	−1.02533e8	−	2.23737e8i	−0.171151	−	0.373469i
$844$		0			0
$845$		0			0
$846$		0			0
$847$			7.33092e8i			1.20645i
$848$		0			0
$849$	3.16222e8	+	6.90029e8i	0.516737	+	1.12757i
$850$		0			0
$851$			1.69489e7i			0.0275012i
$852$		0			0
$853$		−	2.50662e8i		−	0.403869i	−0.979399	−	0.201935i	$-0.935277\pi$
$853$		−	2.50662e8i		−	0.403869i	0.979399	−	0.201935i	$-0.0647229\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	2.24351e8			0.356440			0.178220	−	0.983991i	$-0.442966\pi$
$857$	2.24351e8			0.356440			0.178220	+	0.983991i	$0.442966\pi$
$858$		0			0
$859$	−1.58303e8			−0.249753			−0.124876	−	0.992172i	$-0.539853\pi$
$859$	−1.58303e8			−0.249753			−0.124876	+	0.992172i	$0.539853\pi$
$860$		0			0
$861$	−5.72784e8	+	2.62492e8i	−0.897390	+	0.411250i
$862$		0			0
$863$	4.43360e8			0.689801			0.344900	−	0.938639i	$-0.387913\pi$
$863$	4.43360e8			0.689801			0.344900	+	0.938639i	$0.387913\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−5.45378e8	+	2.49932e8i	−0.836836	+	0.383500i
$868$		0			0
$869$			1.63797e7i			0.0249601i
$870$		0			0
$871$	−5.86730e8			−0.887941
$872$		0			0
$873$	6.59493e8	+	5.68426e8i	0.991214	+	0.854340i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	7.04336e8i		−	1.04419i	−0.852886	−	0.522097i	$-0.825150\pi$
$877$		−	7.04336e8i		−	1.04419i	0.852886	−	0.522097i	$-0.174850\pi$
$878$		0			0
$879$	8.21227e8	−	3.76347e8i	1.20920	−	0.554143i
$880$		0			0
$881$		−	1.02359e9i		−	1.49692i	−0.663181	−	0.748459i	$-0.730794\pi$
$881$		−	1.02359e9i		−	1.49692i	0.663181	−	0.748459i	$-0.269206\pi$
$882$		0			0
$883$		−	3.14880e8i		−	0.457365i	−0.973501	−	0.228682i	$-0.926558\pi$
$883$		−	3.14880e8i		−	0.457365i	0.973501	−	0.228682i	$-0.0734418\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	6.28597e8			0.900744			0.450372	−	0.892841i	$-0.351291\pi$
$887$	6.28597e8			0.900744			0.450372	+	0.892841i	$0.351291\pi$
$888$		0			0
$889$	9.30955e8			1.32502
$890$		0			0
$891$	3.23409e7	−	4.82358e6i	0.0457213	−	0.00681925i
$892$		0			0
$893$	6.63006e8			0.931029
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−1.15763e8	−	2.52607e8i	−0.160396	−	0.350000i
$898$		0			0
$899$			2.10756e7i			0.0290069i
$900$		0			0
$901$	1.13512e9			1.55192
$902$		0			0
$903$	1.33674e9	−	6.12594e8i	1.81545	−	0.831973i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			6.20458e8i			0.831555i	0.909466	+	0.415777i	$0.136490\pi$
$907$			6.20458e8i			0.831555i	−0.909466	+	0.415777i	$0.863510\pi$
$908$		0			0
$909$	−9.60376e7	−	8.27761e7i	−0.127864	−	0.110208i
$910$		0			0
$911$			1.24142e9i			1.64197i	0.570953	+	0.820983i	$0.306574\pi$
$911$			1.24142e9i			1.64197i	−0.570953	+	0.820983i	$0.693426\pi$
$912$		0			0
$913$			6.24849e7i			0.0821037i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.13222e9			1.46832
$918$		0			0
$919$	−6.59663e8			−0.849916			−0.424958	−	0.905213i	$-0.639711\pi$
$919$	−6.59663e8			−0.849916			−0.424958	+	0.905213i	$0.639711\pi$
$920$		0			0
$921$	−1.77126e8	−	3.86507e8i	−0.226727	−	0.494741i
$922$		0			0
$923$	4.77588e8			0.607363
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−1.80027e7	−	1.55167e7i	−0.0225994	−	0.0194787i
$928$		0			0
$929$		−	1.38845e9i		−	1.73174i	−0.500270	−	0.865870i	$-0.666766\pi$
$929$		−	1.38845e9i		−	1.73174i	0.500270	−	0.865870i	$-0.333234\pi$
$930$		0			0
$931$	−4.24977e8			−0.526643
$932$		0			0
$933$	7.88179e7	+	1.71989e8i	0.0970466	+	0.211765i
$934$		0			0
$935$		0			0
$936$		0			0
$937$			9.75190e8i			1.18541i	0.805418	+	0.592707i	$0.201941\pi$
$937$			9.75190e8i			1.18541i	−0.805418	+	0.592707i	$0.798059\pi$
$938$		0			0
$939$	6.51724e7	+	1.42213e8i	0.0787167	+	0.171768i
$940$		0			0
$941$			7.89692e8i			0.947739i	0.880595	+	0.473870i	$0.157143\pi$
$941$			7.89692e8i			0.947739i	−0.880595	+	0.473870i	$0.842857\pi$
$942$		0			0
$943$			4.19754e8i			0.500564i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−5.10202e8			−0.600747			−0.300374	−	0.953822i	$-0.597111\pi$
$947$	−5.10202e8			−0.600747			−0.300374	+	0.953822i	$0.597111\pi$
$948$		0			0
$949$	−2.36905e8			−0.277189
$950$		0			0
$951$	9.73047e8	−	4.45922e8i	1.13134	−	0.518462i
$952$		0			0
$953$	6.26272e8			0.723576			0.361788	−	0.932260i	$-0.382166\pi$
$953$	6.26272e8			0.723576			0.361788	+	0.932260i	$0.382166\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−640795.	+	293659.i	−0.000731111	+	0.000335049i
$958$		0			0
$959$		−	1.80390e9i		−	2.04530i
$960$		0			0
$961$	1.57971e9			1.77995
$962$		0			0
$963$	5.94598e8	−	6.89858e8i	0.665801	−	0.772469i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	4.99324e8i		−	0.552209i	−0.961128	−	0.276104i	$-0.910956\pi$
$967$		−	4.99324e8i		−	0.552209i	0.961128	−	0.276104i	$-0.0890435\pi$
$968$		0			0
$969$	1.30735e9	−	5.99125e8i	1.43688	−	0.658485i
$970$		0			0
$971$		−	4.94342e7i		−	0.0539970i	−0.999635	−	0.0269985i	$-0.991405\pi$
$971$		−	4.94342e7i		−	0.0539970i	0.999635	−	0.0269985i	$-0.00859494\pi$
$972$		0			0
$973$		−	2.22530e7i		−	0.0241574i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	4.90224e7			0.0525667			0.0262834	−	0.999655i	$-0.491633\pi$
$977$	4.90224e7			0.0525667			0.0262834	+	0.999655i	$0.491633\pi$
$978$		0			0
$979$	5.88029e7			0.0626687
$980$		0			0
$981$	1.09374e9	−	1.26897e9i	1.15853	−	1.34414i
$982$		0			0
$983$	−2.97397e8			−0.313095			−0.156548	−	0.987670i	$-0.550036\pi$
$983$	−2.97397e8			−0.313095			−0.156548	+	0.987670i	$0.550036\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−3.95344e8	−	8.62681e8i	−0.411172	−	0.897219i
$988$		0			0
$989$		−	9.79607e8i		−	1.01266i
$990$		0			0
$991$	−8.08393e8			−0.830619			−0.415309	−	0.909680i	$-0.636327\pi$
$991$	−8.08393e8			−0.830619			−0.415309	+	0.909680i	$0.636327\pi$
$992$		0			0
$993$	−8.58210e7	+	3.93295e7i	−0.0876488	+	0.0401671i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	1.39077e9i		−	1.40336i	−0.712493	−	0.701680i	$-0.752434\pi$
$997$		−	1.39077e9i		−	1.40336i	0.712493	−	0.701680i	$-0.247566\pi$
$998$		0			0
$999$	4.29605e7	+	1.24306e7i	0.0430896	+	0.0124679i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.7.b.e.149.2		16
3.2	odd	2	inner	300.7.b.e.149.16		16
5.2	odd	4		60.7.g.a.41.6	yes	8
5.3	odd	4		300.7.g.h.101.3		8
5.4	even	2	inner	300.7.b.e.149.15		16
15.2	even	4		60.7.g.a.41.5	✓	8
15.8	even	4		300.7.g.h.101.4		8
15.14	odd	2	inner	300.7.b.e.149.1		16
20.7	even	4		240.7.l.c.161.3		8
60.47	odd	4		240.7.l.c.161.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.7.g.a.41.5	✓	8	15.2	even	4
60.7.g.a.41.6	yes	8	5.2	odd	4
240.7.l.c.161.3		8	20.7	even	4
240.7.l.c.161.4		8	60.47	odd	4
300.7.b.e.149.1		16	15.14	odd	2	inner
300.7.b.e.149.2		16	1.1	even	1	trivial
300.7.b.e.149.15		16	5.4	even	2	inner
300.7.b.e.149.16		16	3.2	odd	2	inner
300.7.g.h.101.3		8	5.3	odd	4
300.7.g.h.101.4		8	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	300.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-24.5453 + 11.2485i) q^{3} +414.697i q^{7} +(475.944 - 552.194i) q^{9} +O(q^{10})\)	\(q+(-24.5453 + 11.2485i) q^{3} +414.697i q^{7} +(475.944 - 552.194i) q^{9} +61.5283i q^{11} -1379.66i q^{13} -6808.58 q^{17} +7822.89 q^{19} +(-4664.71 - 10178.9i) q^{21} -7459.40 q^{23} +(-5470.84 + 18907.4i) q^{27} -424.303i q^{29} -49671.1 q^{31} +(-692.099 - 1510.23i) q^{33} -2272.15i q^{37} +(15519.1 + 33864.3i) q^{39} -56271.8i q^{41} +131325. i q^{43} +84752.1 q^{47} -54324.8 q^{49} +(167119. - 76586.2i) q^{51} -166720. q^{53} +(-192015. + 87995.5i) q^{57} -400288. i q^{59} -83666.7 q^{61} +(228993. + 197373. i) q^{63} -425271. i q^{67} +(183093. - 83906.8i) q^{69} +346163. i q^{71} -171712. i q^{73} -25515.6 q^{77} +266214. q^{79} +(-78396.2 - 525627. i) q^{81} +1.01555e6 q^{83} +(4772.76 + 10414.6i) q^{87} -955706. i q^{89} +572143. q^{91} +(1.21919e6 - 558724. i) q^{93} +1.19431e6i q^{97} +(33975.6 + 29284.0i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2984 q^{9}+O(q^{10}) \) 16 * q - 2984 * q^9	\( 16 q - 2984 q^{9} + 30544 q^{19} - 1736 q^{21} + 70064 q^{31} - 79216 q^{39} - 405120 q^{49} + 858240 q^{51} - 271216 q^{61} + 509880 q^{69} - 1415408 q^{79} - 2396224 q^{81} - 4008224 q^{91} - 5300160 q^{99}+O(q^{100}) \) 16 * q - 2984 * q^9 + 30544 * q^19 - 1736 * q^21 + 70064 * q^31 - 79216 * q^39 - 405120 * q^49 + 858240 * q^51 - 271216 * q^61 + 509880 * q^69 - 1415408 * q^79 - 2396224 * q^81 - 4008224 * q^91 - 5300160 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more