LMFDB - Embedded newform 45.6.b.d.19.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,6,Mod(19,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("45.19");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	45.b (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:	$7.21727189158$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-5}, \sqrt{-14})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 38x^{2} + 81 $ x^4 + 38*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2}\cdot 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			19.1
Root			$-1.50559i$ of defining polynomial
Character	$\chi$	$=$	45.19
Dual form			45.6.b.d.19.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.47214i q^{2} +12.0000 q^{4} +(-50.1996 + 24.5967i) q^{5} -224.499i q^{7} -196.774i q^{8} +O(q^{10})$	\(q-4.47214i q^{2} +12.0000 q^{4} +(-50.1996 + 24.5967i) q^{5} -224.499i q^{7} -196.774i q^{8} +(110.000 + 224.499i) q^{10} -501.996 q^{11} +224.499i q^{13} -1003.99 q^{14} -496.000 q^{16} -1668.11i q^{17} -484.000 q^{19} +(-602.395 + 295.161i) q^{20} +2244.99i q^{22} +2262.90i q^{23} +(1915.00 - 2469.49i) q^{25} +1003.99 q^{26} -2693.99i q^{28} +5521.96 q^{29} +3608.00 q^{31} -4078.59i q^{32} -7460.00 q^{34} +(5521.96 + 11269.8i) q^{35} -7408.48i q^{37} +2164.51i q^{38} +(4840.00 + 9877.98i) q^{40} +11043.9 q^{41} +12572.0i q^{43} -6023.95 q^{44} +10120.0 q^{46} -9543.54i q^{47} -33593.0 q^{49} +(-11043.9 - 8564.14i) q^{50} +2693.99i q^{52} +4771.77i q^{53} +(25200.0 - 12347.5i) q^{55} -44175.6 q^{56} -24694.9i q^{58} -5521.96 q^{59} +21362.0 q^{61} -16135.5i q^{62} -34112.0 q^{64} +(-5521.96 - 11269.8i) q^{65} +34572.9i q^{67} -20017.3i q^{68} +(50400.0 - 24694.9i) q^{70} +33131.7 q^{71} -2693.99i q^{73} -33131.7 q^{74} -5808.00 q^{76} +112698. i q^{77} -99616.0 q^{79} +(24899.0 - 12200.0i) q^{80} -49389.9i q^{82} -57118.1i q^{83} +(41030.0 + 83738.3i) q^{85} +56223.6 q^{86} +98779.8i q^{88} +120479. q^{89} +50400.0 q^{91} +27154.8i q^{92} -42680.0 q^{94} +(24296.6 - 11904.8i) q^{95} -64206.8i q^{97} +150232. i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 48 q^{4}+O(q^{10}) $ 4 * q + 48 * q^4	$ 4 q + 48 q^{4} + 440 q^{10} - 1984 q^{16} - 1936 q^{19} + 7660 q^{25} + 14432 q^{31} - 29840 q^{34} + 19360 q^{40} + 40480 q^{46} - 134372 q^{49} + 100800 q^{55} + 85448 q^{61} - 136448 q^{64} + 201600 q^{70} - 23232 q^{76} - 398464 q^{79} + 164120 q^{85} + 201600 q^{91} - 170720 q^{94}+O(q^{100}) $ 4 * q + 48 * q^4 + 440 * q^10 - 1984 * q^16 - 1936 * q^19 + 7660 * q^25 + 14432 * q^31 - 29840 * q^34 + 19360 * q^40 + 40480 * q^46 - 134372 * q^49 + 100800 * q^55 + 85448 * q^61 - 136448 * q^64 + 201600 * q^70 - 23232 * q^76 - 398464 * q^79 + 164120 * q^85 + 201600 * q^91 - 170720 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$1$	$-1$

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 $

$2$		−	4.47214i		−	0.790569i	−0.918559	−	0.395285i	$-0.870646\pi$
$2$		−	4.47214i		−	0.790569i	0.918559	−	0.395285i	$-0.129354\pi$
$3$		0			0
$3$		0			0
$4$	12.0000			0.375000
$5$	−50.1996	+	24.5967i	−0.897998	+	0.440000i
$5$	−50.1996	+	24.5967i	−0.897998	+	0.440000i
$6$		0			0
$7$		−	224.499i		−	1.73169i	−0.500312	−	0.865845i	$-0.666781\pi$
$7$		−	224.499i		−	1.73169i	0.500312	−	0.865845i	$-0.333219\pi$
$8$		−	196.774i		−	1.08703i
$9$		0			0
$10$	110.000	+	224.499i	0.347851	+	0.709930i
$11$	−501.996			−1.25089			−0.625444	−	0.780269i	$-0.715082\pi$
$11$	−501.996			−1.25089			−0.625444	+	0.780269i	$0.715082\pi$
$12$		0			0
$13$			224.499i			0.368432i	0.982886	+	0.184216i	$0.0589745\pi$
$13$			224.499i			0.368432i	−0.982886	+	0.184216i	$0.941025\pi$
$14$	−1003.99			−1.36902
$15$		0			0
$16$	−496.000			−0.484375
$17$		−	1668.11i		−	1.39991i	−0.714185	−	0.699957i	$-0.753202\pi$
$17$		−	1668.11i		−	1.39991i	0.714185	−	0.699957i	$-0.246798\pi$
$18$		0			0
$19$	−484.000			−0.307582			−0.153791	−	0.988103i	$-0.549148\pi$
$19$	−484.000			−0.307582			−0.153791	+	0.988103i	$0.549148\pi$
$20$	−602.395	+	295.161i	−0.336749	+	0.165000i
$21$		0			0
$22$			2244.99i			0.988914i
$23$			2262.90i			0.891961i	0.895043	+	0.445981i	$0.147145\pi$
$23$			2262.90i			0.891961i	−0.895043	+	0.445981i	$0.852855\pi$
$24$		0			0
$25$	1915.00	−	2469.49i	0.612800	−	0.790238i
$26$	1003.99			0.291271
$27$		0			0
$28$		−	2693.99i		−	0.649384i
$29$	5521.96			1.21926			0.609632	−	0.792684i	$-0.291317\pi$
$29$	5521.96			1.21926			0.609632	+	0.792684i	$0.291317\pi$
$30$		0			0
$31$	3608.00			0.674314			0.337157	−	0.941448i	$-0.390535\pi$
$31$	3608.00			0.674314			0.337157	+	0.941448i	$0.390535\pi$
$32$		−	4078.59i		−	0.704101i
$33$		0			0
$34$	−7460.00			−1.10673
$35$	5521.96	+	11269.8i	0.761944	+	1.55505i
$36$		0			0
$37$		−	7408.48i		−	0.889662i	−0.895615	−	0.444831i	$-0.853264\pi$
$37$		−	7408.48i		−	0.889662i	0.895615	−	0.444831i	$-0.146736\pi$
$38$			2164.51i			0.243165i
$39$		0			0
$40$	4840.00	+	9877.98i	0.478294	+	0.976153i
$41$	11043.9			1.02604			0.513019	−	0.858377i	$-0.328527\pi$
$41$	11043.9			1.02604			0.513019	+	0.858377i	$0.328527\pi$
$42$		0			0
$43$			12572.0i			1.03689i	0.855111	+	0.518444i	$0.173489\pi$
$43$			12572.0i			1.03689i	−0.855111	+	0.518444i	$0.826511\pi$
$44$	−6023.95			−0.469083
$45$		0			0
$46$	10120.0			0.705157
$47$		−	9543.54i		−	0.630180i	−0.949062	−	0.315090i	$-0.897965\pi$
$47$		−	9543.54i		−	0.630180i	0.949062	−	0.315090i	$-0.102035\pi$
$48$		0			0
$49$	−33593.0			−1.99875
$50$	−11043.9	−	8564.14i	−0.624738	−	0.484461i
$51$		0			0
$52$			2693.99i			0.138162i
$53$			4771.77i			0.233340i	0.993171	+	0.116670i	$0.0372220\pi$
$53$			4771.77i			0.233340i	−0.993171	+	0.116670i	$0.962778\pi$
$54$		0			0
$55$	25200.0	−	12347.5i	1.12329	−	0.550391i
$56$	−44175.6			−1.88240
$57$		0			0
$58$		−	24694.9i		−	0.963913i
$59$	−5521.96			−0.206520			−0.103260	−	0.994654i	$-0.532927\pi$
$59$	−5521.96			−0.206520			−0.103260	+	0.994654i	$0.532927\pi$
$60$		0			0
$61$	21362.0			0.735051			0.367525	−	0.930013i	$-0.380205\pi$
$61$	21362.0			0.735051			0.367525	+	0.930013i	$0.380205\pi$
$62$		−	16135.5i		−	0.533092i
$63$		0			0
$64$	−34112.0			−1.04102
$65$	−5521.96	−	11269.8i	−0.162110	−	0.330851i
$66$		0			0
$67$			34572.9i			0.940912i	0.882423	+	0.470456i	$0.155911\pi$
$67$			34572.9i			0.940912i	−0.882423	+	0.470456i	$0.844089\pi$
$68$		−	20017.3i		−	0.524968i
$69$		0			0
$70$	50400.0	−	24694.9i	1.22938	−	0.602369i
$71$	33131.7			0.780007			0.390003	−	0.920813i	$-0.372474\pi$
$71$	33131.7			0.780007			0.390003	+	0.920813i	$0.372474\pi$
$72$		0			0
$73$		−	2693.99i		−	0.0591683i	−0.999562	−	0.0295842i	$-0.990582\pi$
$73$		−	2693.99i		−	0.0591683i	0.999562	−	0.0295842i	$-0.00941831\pi$
$74$	−33131.7			−0.703339
$75$		0			0
$76$	−5808.00			−0.115343
$77$			112698.i			2.16615i
$78$		0			0
$79$	−99616.0			−1.79581			−0.897907	−	0.440185i	$-0.854913\pi$
$79$	−99616.0			−1.79581			−0.897907	+	0.440185i	$0.854913\pi$
$80$	24899.0	−	12200.0i	0.434968	−	0.213125i
$81$		0			0
$82$		−	49389.9i		−	0.811154i
$83$		−	57118.1i		−	0.910078i	−0.890471	−	0.455039i	$-0.849625\pi$
$83$		−	57118.1i		−	0.910078i	0.890471	−	0.455039i	$-0.150375\pi$
$84$		0			0
$85$	41030.0	+	83738.3i	0.615962	+	1.25712i
$86$	56223.6			0.819733
$87$		0			0
$88$			98779.8i			1.35976i
$89$	120479.			1.61227			0.806133	−	0.591735i	$-0.201557\pi$
$89$	120479.			1.61227			0.806133	+	0.591735i	$0.201557\pi$
$90$		0			0
$91$	50400.0			0.638009
$92$			27154.8i			0.334485i
$93$		0			0
$94$	−42680.0			−0.498201
$95$	24296.6	−	11904.8i	0.276208	−	0.135336i
$96$		0			0
$97$		−	64206.8i		−	0.692870i	−0.938074	−	0.346435i	$-0.887392\pi$
$97$		−	64206.8i		−	0.692870i	0.938074	−	0.346435i	$-0.112608\pi$
$98$			150232.i			1.58015i
$99$		0			0
$100$	22980.0	−	29633.9i	0.229800	−	0.296339i
$101$	−60741.5			−0.592492			−0.296246	−	0.955112i	$-0.595735\pi$
$101$	−60741.5			−0.592492			−0.296246	+	0.955112i	$0.595735\pi$
$102$		0			0
$103$		−	150639.i		−	1.39909i	−0.714590	−	0.699544i	$-0.753387\pi$
$103$		−	150639.i		−	1.39909i	0.714590	−	0.699544i	$-0.246613\pi$
$104$	44175.6			0.400497
$105$		0			0
$106$	21340.0			0.184472
$107$			1914.07i			0.0161622i	0.999967	+	0.00808108i	$0.00257232\pi$
$107$			1914.07i			0.0161622i	−0.999967	+	0.00808108i	$0.997428\pi$
$108$		0			0
$109$	1694.00			0.0136567			0.00682837	−	0.999977i	$-0.497826\pi$
$109$	1694.00			0.0136567			0.00682837	+	0.999977i	$0.497826\pi$
$110$	−55219.6	−	112698.i	−0.435122	−	0.888043i
$111$		0			0
$112$			111352.i			0.838787i
$113$		−	148712.i		−	1.09559i	−0.836611	−	0.547797i	$-0.815467\pi$
$113$		−	148712.i		−	1.09559i	0.836611	−	0.547797i	$-0.184533\pi$
$114$		0			0
$115$	−55660.0	−	113597.i	−0.392463	−	0.800979i
$116$	66263.5			0.457224
$117$		0			0
$118$			24694.9i			0.163269i
$119$	−374489.			−2.42422
$120$		0			0
$121$	90949.0			0.564722
$122$		−	95533.8i		−	0.581109i
$123$		0			0
$124$	43296.0			0.252868
$125$	−35390.7	+	171070.i	−0.202588	+	0.979264i
$126$		0			0
$127$			173089.i			0.952271i	0.879372	+	0.476135i	$0.157963\pi$
$127$			173089.i			0.952271i	−0.879372	+	0.476135i	$0.842037\pi$
$128$			22038.7i			0.118894i
$129$		0			0
$130$	−50400.0	+	24694.9i	−0.261560	+	0.128159i
$131$	−270576.			−1.37756			−0.688780	−	0.724970i	$-0.741854\pi$
$131$	−270576.			−1.37756			−0.688780	+	0.724970i	$0.741854\pi$
$132$		0			0
$133$			108658.i			0.532637i
$134$	154615.			0.743856
$135$		0			0
$136$	−328240.			−1.52175
$137$		−	319020.i		−	1.45217i	−0.687607	−	0.726083i	$-0.741339\pi$
$137$		−	319020.i		−	1.45217i	0.687607	−	0.726083i	$-0.258661\pi$
$138$		0			0
$139$	226556.			0.994577			0.497289	−	0.867585i	$-0.334329\pi$
$139$	226556.			0.994577			0.497289	+	0.867585i	$0.334329\pi$
$140$	66263.5	+	135237.i	0.285729	+	0.583145i
$141$		0			0
$142$		−	148170.i		−	0.616649i
$143$		−	112698.i		−	0.460867i
$144$		0			0
$145$	−277200.	+	135822.i	−1.09490	+	0.536476i
$146$	−12047.9			−0.0467767
$147$		0			0
$148$		−	88901.8i		−	0.333623i
$149$	160137.			0.590915			0.295458	−	0.955356i	$-0.404528\pi$
$149$	160137.			0.590915			0.295458	+	0.955356i	$0.404528\pi$
$150$		0			0
$151$	8888.00			0.0317221			0.0158610	−	0.999874i	$-0.494951\pi$
$151$	8888.00			0.0317221			0.0158610	+	0.999874i	$0.494951\pi$
$152$			95238.6i			0.334352i
$153$		0			0
$154$	504000.			1.71249
$155$	−181120.	+	88745.1i	−0.605533	+	0.296698i
$156$		0			0
$157$		−	506246.i		−	1.63913i	−0.572988	−	0.819564i	$-0.694216\pi$
$157$		−	506246.i		−	1.63913i	0.572988	−	0.819564i	$-0.305784\pi$
$158$			445496.i			1.41972i
$159$		0			0
$160$	100320.	+	204743.i	0.309804	+	0.632281i
$161$	508020.			1.54460
$162$		0			0
$163$			222254.i			0.655211i	0.944815	+	0.327606i	$0.106242\pi$
$163$			222254.i			0.655211i	−0.944815	+	0.327606i	$0.893758\pi$
$164$	132527.			0.384764
$165$		0			0
$166$	−255440.			−0.719480
$167$		−	97555.2i		−	0.270682i	−0.990799	−	0.135341i	$-0.956787\pi$
$167$		−	97555.2i		−	0.270682i	0.990799	−	0.135341i	$-0.0432130\pi$
$168$		0			0
$169$	320893.			0.864258
$170$	374489.	−	183492.i	0.993841	−	0.486961i
$171$		0			0
$172$			150864.i			0.388833i
$173$			276669.i			0.702821i	0.936221	+	0.351411i	$0.114298\pi$
$173$			276669.i			0.702821i	−0.936221	+	0.351411i	$0.885702\pi$
$174$		0			0
$175$	−554400.	−	429916.i	−1.36845	−	1.06118i
$176$	248990.			0.605899
$177$		0			0
$178$		−	538799.i		−	1.27461i
$179$	28613.8			0.0667487			0.0333743	−	0.999443i	$-0.489375\pi$
$179$	28613.8			0.0667487			0.0333743	+	0.999443i	$0.489375\pi$
$180$		0			0
$181$	−81862.0			−0.185732			−0.0928658	−	0.995679i	$-0.529603\pi$
$181$	−81862.0			−0.185732			−0.0928658	+	0.995679i	$0.529603\pi$
$182$		−	225396.i		−	0.504391i
$183$		0			0
$184$	445280.			0.969591
$185$	182225.	+	371903.i	0.391451	+	0.798914i
$186$		0			0
$187$			837383.i			1.75114i
$188$		−	114522.i		−	0.236318i
$189$		0			0
$190$	−53240.0	−	108658.i	−0.106993	−	0.218362i
$191$	−486936.			−0.965803			−0.482902	−	0.875675i	$-0.660417\pi$
$191$	−486936.			−0.965803			−0.482902	+	0.875675i	$0.660417\pi$
$192$		0			0
$193$			541044.i			1.04554i	0.852475	+	0.522768i	$0.175101\pi$
$193$			541044.i			1.04554i	−0.852475	+	0.522768i	$0.824899\pi$
$194$	−287142.			−0.547762
$195$		0			0
$196$	−403116.			−0.749531
$197$			812976.i			1.49249i	0.665670	+	0.746246i	$0.268146\pi$
$197$			812976.i			1.49249i	−0.665670	+	0.746246i	$0.731854\pi$
$198$		0			0
$199$	146144.			0.261606			0.130803	−	0.991408i	$-0.458244\pi$
$199$	146144.			0.261606			0.130803	+	0.991408i	$0.458244\pi$
$200$	−485932.	−	376822.i	−0.859015	−	0.666134i
$201$		0			0
$202$			271644.i			0.468406i
$203$		−	1.23968e6i		−	2.11139i
$204$		0			0
$205$	−554400.	+	271644.i	−0.921380	+	0.451457i
$206$	−673679.			−1.10608
$207$		0			0
$208$		−	111352.i		−	0.178459i
$209$	242966.			0.384751
$210$		0			0
$211$	846692.			1.30924			0.654620	−	0.755958i	$-0.272829\pi$
$211$	846692.			1.30924			0.654620	+	0.755958i	$0.272829\pi$
$212$			57261.2i			0.0875026i
$213$		0			0
$214$	8560.00			0.0127773
$215$	−309230.	−	631108.i	−0.456231	−	0.931124i
$216$		0			0
$217$		−	809994.i		−	1.16770i
$218$		−	7575.80i		−	0.0107966i
$219$		0			0
$220$	302400.	−	148170.i	0.421236	−	0.206397i
$221$	374489.			0.515773
$222$		0			0
$223$		−	812463.i		−	1.09406i	−0.837113	−	0.547031i	$-0.815758\pi$
$223$		−	812463.i		−	1.09406i	0.837113	−	0.547031i	$-0.184242\pi$
$224$	−915641.			−1.21928
$225$		0			0
$226$	−665060.			−0.866143
$227$			355034.i			0.457304i	0.973508	+	0.228652i	$0.0734318\pi$
$227$			355034.i			0.457304i	−0.973508	+	0.228652i	$0.926568\pi$
$228$		0			0
$229$	−411094.			−0.518027			−0.259014	−	0.965874i	$-0.583397\pi$
$229$	−411094.			−0.518027			−0.259014	+	0.965874i	$0.583397\pi$
$230$	−508020.	+	248919.i	−0.633230	+	0.310269i
$231$		0			0
$232$		−	1.08658e6i		−	1.32538i
$233$			738193.i			0.890800i	0.895332	+	0.445400i	$0.146939\pi$
$233$			738193.i			0.890800i	−0.895332	+	0.445400i	$0.853061\pi$
$234$		0			0
$235$	234740.	+	479082.i	0.277279	+	0.565900i
$236$	−66263.5			−0.0774451
$237$		0			0
$238$			1.67477e6i			1.91651i
$239$	−121483.			−0.137569			−0.0687845	−	0.997632i	$-0.521912\pi$
$239$	−121483.			−0.137569			−0.0687845	+	0.997632i	$0.521912\pi$
$240$		0			0
$241$	1.12224e6			1.24464			0.622320	−	0.782763i	$-0.286190\pi$
$241$	1.12224e6			1.24464			0.622320	+	0.782763i	$0.286190\pi$
$242$		−	406736.i		−	0.446452i
$243$		0			0
$244$	256344.			0.275644
$245$	1.68636e6	−	826279.i	1.79487	−	0.879450i
$246$		0			0
$247$		−	108658.i		−	0.113323i
$248$		−	709961.i		−	0.733002i
$249$		0			0
$250$	765050.	+	158272.i	0.774176	+	0.160160i
$251$	16565.9			0.0165970			0.00829851	−	0.999966i	$-0.497358\pi$
$251$	16565.9			0.0165970			0.00829851	+	0.999966i	$0.497358\pi$
$252$		0			0
$253$		−	1.13597e6i		−	1.11574i
$254$	774078.			0.752836
$255$		0			0
$256$	−993024.			−0.947021
$257$			1.10779e6i			1.04622i	0.852264	+	0.523111i	$0.175229\pi$
$257$			1.10779e6i			1.04622i	−0.852264	+	0.523111i	$0.824771\pi$
$258$		0			0
$259$	−1.66320e6			−1.54062
$260$	−66263.5	−	135237.i	−0.0607912	−	0.124069i
$261$		0			0
$262$			1.21005e6i			1.08906i
$263$			1.82582e6i			1.62768i	0.581090	+	0.813840i	$0.302627\pi$
$263$			1.82582e6i			1.62768i	−0.581090	+	0.813840i	$0.697373\pi$
$264$		0			0
$265$	−117370.	−	239541.i	−0.102670	−	0.209539i
$266$	485932.			0.421087
$267$		0			0
$268$			414875.i			0.352842i
$269$	1.34184e6			1.13062			0.565312	−	0.824877i	$-0.308756\pi$
$269$	1.34184e6			1.13062			0.565312	+	0.824877i	$0.308756\pi$
$270$		0			0
$271$	724592.			0.599336			0.299668	−	0.954044i	$-0.403124\pi$
$271$	724592.			0.599336			0.299668	+	0.954044i	$0.403124\pi$
$272$			827381.i			0.678084i
$273$		0			0
$274$	−1.42670e6			−1.14804
$275$	−961322.	+	1.23968e6i	−0.766544	+	0.988500i
$276$		0			0
$277$			2.00007e6i			1.56619i	0.621901	+	0.783096i	$0.286361\pi$
$277$			2.00007e6i			1.56619i	−0.621901	+	0.783096i	$0.713639\pi$
$278$		−	1.01319e6i		−	0.786282i
$279$		0			0
$280$	2.21760e6	−	1.08658e6i	1.69039	−	0.828258i
$281$	2.20878e6			1.66873			0.834367	−	0.551209i	$-0.185833\pi$
$281$	2.20878e6			1.66873			0.834367	+	0.551209i	$0.185833\pi$
$282$		0			0
$283$		−	442264.i		−	0.328258i	−0.986439	−	0.164129i	$-0.947519\pi$
$283$		−	442264.i		−	0.328258i	0.986439	−	0.164129i	$-0.0524813\pi$
$284$	397581.			0.292503
$285$		0			0
$286$	−504000.			−0.364347
$287$		−	2.47935e6i		−	1.77678i
$288$		0			0
$289$	−1.36272e6			−0.959761
$290$	607415.	+	1.23968e6i	0.424122	+	0.865592i
$291$		0			0
$292$		−	32327.9i		−	0.0221881i
$293$		−	1.01093e6i		−	0.687943i	−0.938980	−	0.343971i	$-0.888228\pi$
$293$		−	1.01093e6i		−	0.687943i	0.938980	−	0.343971i	$-0.111772\pi$
$294$		0			0
$295$	277200.	−	135822.i	0.185455	−	0.0908690i
$296$	−1.45780e6			−0.967092
$297$		0			0
$298$		−	716153.i		−	0.467160i
$299$	−508020.			−0.328627
$300$		0			0
$301$	2.82240e6			1.79557
$302$		−	39748.3i		−	0.0250785i
$303$		0			0
$304$	240064.			0.148985
$305$	−1.07236e6	+	525436.i	−0.660074	+	0.323422i
$306$		0			0
$307$		−	2.37341e6i		−	1.43723i	−0.695408	−	0.718615i	$-0.744776\pi$
$307$		−	2.37341e6i		−	1.43723i	0.695408	−	0.718615i	$-0.255224\pi$
$308$			1.35237e6i			0.812307i
$309$		0			0
$310$	396880.	+	809994.i	0.234561	+	0.478716i
$311$	−2.57323e6			−1.50861			−0.754307	−	0.656522i	$-0.772027\pi$
$311$	−2.57323e6			−1.50861			−0.754307	+	0.656522i	$0.772027\pi$
$312$		0			0
$313$			1.53603e6i			0.886212i	0.896469	+	0.443106i	$0.146123\pi$
$313$			1.53603e6i			0.886212i	−0.896469	+	0.443106i	$0.853877\pi$
$314$	−2.26400e6			−1.29584
$315$		0			0
$316$	−1.19539e6			−0.673430
$317$		−	2.26708e6i		−	1.26712i	−0.773692	−	0.633562i	$-0.781592\pi$
$317$		−	2.26708e6i		−	1.26712i	0.773692	−	0.633562i	$-0.218408\pi$
$318$		0			0
$319$	−2.77200e6			−1.52516
$320$	1.71241e6	−	839044.i	0.934830	−	0.458047i
$321$		0			0
$322$		−	2.27193e6i		−	1.22111i
$323$			807364.i			0.430589i
$324$		0			0
$325$	554400.	+	429916.i	0.291149	+	0.225775i
$326$	993952.			0.517990
$327$		0			0
$328$		−	2.17315e6i		−	1.11534i
$329$	−2.14252e6			−1.09128
$330$		0			0
$331$	1.65981e6			0.832701			0.416350	−	0.909204i	$-0.363309\pi$
$331$	1.65981e6			0.832701			0.416350	+	0.909204i	$0.363309\pi$
$332$		−	685417.i		−	0.341279i
$333$		0			0
$334$	−436280.			−0.213993
$335$	−850381.	−	1.73555e6i	−0.414001	−	0.844937i
$336$		0			0
$337$			822566.i			0.394544i	0.980349	+	0.197272i	$0.0632083\pi$
$337$			822566.i			0.394544i	−0.980349	+	0.197272i	$0.936792\pi$
$338$		−	1.43508e6i		−	0.683256i
$339$		0			0
$340$	492360.	+	1.00486e6i	0.230986	+	0.471420i
$341$	−1.81120e6			−0.843492
$342$		0			0
$343$			3.76845e6i			1.72953i
$344$	2.47384e6			1.12713
$345$		0			0
$346$	1.23730e6			0.555629
$347$		−	977323.i		−	0.435727i	−0.975979	−	0.217863i	$-0.930091\pi$
$347$		−	977323.i		−	0.435727i	0.975979	−	0.217863i	$-0.0699087\pi$
$348$		0			0
$349$	−774994.			−0.340592			−0.170296	−	0.985393i	$-0.554472\pi$
$349$	−774994.			−0.340592			−0.170296	+	0.985393i	$0.554472\pi$
$350$	−1.92264e6	+	2.47935e6i	−0.838936	+	1.08185i
$351$		0			0
$352$			2.04743e6i			0.880752i
$353$			2.76069e6i			1.17918i	0.807702	+	0.589591i	$0.200711\pi$
$353$			2.76069e6i			1.17918i	−0.807702	+	0.589591i	$0.799289\pi$
$354$		0			0
$355$	−1.66320e6	+	814933.i	−0.700444	+	0.343203i
$356$	1.44575e6			0.604599
$357$		0			0
$358$		−	127965.i		−	0.0527694i
$359$	−1.12648e6			−0.461304			−0.230652	−	0.973036i	$-0.574086\pi$
$359$	−1.12648e6			−0.461304			−0.230652	+	0.973036i	$0.574086\pi$
$360$		0			0
$361$	−2.24184e6			−0.905393
$362$			366098.i			0.146834i
$363$		0			0
$364$	604800.			0.239254
$365$	66263.5	+	135237.i	0.0260341	+	0.0531330i
$366$		0			0
$367$		−	264236.i		−	0.102406i	−0.998688	−	0.0512031i	$-0.983694\pi$
$367$		−	264236.i		−	0.102406i	0.998688	−	0.0512031i	$-0.0163056\pi$
$368$		−	1.12240e6i		−	0.432044i
$369$		0			0
$370$	1.66320e6	−	814933.i	0.631597	−	0.309469i
$371$	1.07126e6			0.404073
$372$		0			0
$373$		−	3.07227e6i		−	1.14337i	−0.820472	−	0.571687i	$-0.806289\pi$
$373$		−	3.07227e6i		−	1.14337i	0.820472	−	0.571687i	$-0.193711\pi$
$374$	3.74489e6			1.38440
$375$		0			0
$376$	−1.87792e6			−0.685027
$377$			1.23968e6i			0.449216i
$378$		0			0
$379$	1.52304e6			0.544643			0.272322	−	0.962206i	$-0.412209\pi$
$379$	1.52304e6			0.544643			0.272322	+	0.962206i	$0.412209\pi$
$380$	291559.	−	142858.i	0.103578	−	0.0507511i
$381$		0			0
$382$			2.17764e6i			0.763534i
$383$		−	2.40625e6i		−	0.838193i	−0.907942	−	0.419096i	$-0.862347\pi$
$383$		−	2.40625e6i		−	0.838193i	0.907942	−	0.419096i	$-0.137653\pi$
$384$		0			0
$385$	−2.77200e6	−	5.65739e6i	−0.953106	−	1.94520i
$386$	2.41962e6			0.826569
$387$		0			0
$388$		−	770482.i		−	0.259826i
$389$	1.26453e6			0.423696			0.211848	−	0.977303i	$-0.432052\pi$
$389$	1.26453e6			0.423696			0.211848	+	0.977303i	$0.432052\pi$
$390$		0			0
$391$	3.77476e6			1.24867
$392$			6.61023e6i			2.17271i
$393$		0			0
$394$	3.63574e6			1.17992
$395$	5.00068e6	−	2.45023e6i	1.61264	−	0.790158i
$396$		0			0
$397$			1.56319e6i			0.497778i	0.968532	+	0.248889i	$0.0800654\pi$
$397$			1.56319e6i			0.497778i	−0.968532	+	0.248889i	$0.919935\pi$
$398$		−	653576.i		−	0.206818i
$399$		0			0
$400$	−949840.	+	1.22487e6i	−0.296825	+	0.382772i
$401$	−1.36844e6			−0.424977			−0.212488	−	0.977164i	$-0.568157\pi$
$401$	−1.36844e6			−0.424977			−0.212488	+	0.977164i	$0.568157\pi$
$402$		0			0
$403$			809994.i			0.248439i
$404$	−728898.			−0.222184
$405$		0			0
$406$	−5.54400e6			−1.66920
$407$			3.71903e6i			1.11287i
$408$		0			0
$409$	4.31427e6			1.27526			0.637630	−	0.770343i	$-0.279915\pi$
$409$	4.31427e6			1.27526			0.637630	+	0.770343i	$0.279915\pi$
$410$	1.21483e6	+	2.47935e6i	0.356908	+	0.728414i
$411$		0			0
$412$		−	1.80767e6i		−	0.524658i
$413$			1.23968e6i			0.357629i
$414$		0			0
$415$	1.40492e6	+	2.86731e6i	0.400434	+	0.817248i
$416$	915641.			0.259413
$417$		0			0
$418$		−	1.08658e6i		−	0.304172i
$419$	−689241.			−0.191794			−0.0958972	−	0.995391i	$-0.530572\pi$
$419$	−689241.			−0.191794			−0.0958972	+	0.995391i	$0.530572\pi$
$420$		0			0
$421$	−1.03556e6			−0.284755			−0.142377	−	0.989812i	$-0.545475\pi$
$421$	−1.03556e6			−0.284755			−0.142377	+	0.989812i	$0.545475\pi$
$422$		−	3.78652e6i		−	1.03505i
$423$		0			0
$424$	938960.			0.253649
$425$	−4.11938e6	−	3.19442e6i	−1.10627	−	0.857868i
$426$		0			0
$427$		−	4.79576e6i		−	1.27288i
$428$			22968.9i			0.00606081i
$429$		0			0
$430$	−2.82240e6	+	1.38292e6i	−0.736118	+	0.360682i
$431$	2.78307e6			0.721656			0.360828	−	0.932632i	$-0.382494\pi$
$431$	2.78307e6			0.721656			0.360828	+	0.932632i	$0.382494\pi$
$432$		0			0
$433$			400058.i			0.102542i	0.998685	+	0.0512712i	$0.0163273\pi$
$433$			400058.i			0.102542i	−0.998685	+	0.0512712i	$0.983673\pi$
$434$	−3.62240e6			−0.923150
$435$		0			0
$436$	20328.0			0.00512128
$437$		−	1.09524e6i		−	0.274351i
$438$		0			0
$439$	3.10006e6			0.767731			0.383866	−	0.923389i	$-0.374593\pi$
$439$	3.10006e6			0.767731			0.383866	+	0.923389i	$0.374593\pi$
$440$	−2.42966e6	−	4.95870e6i	−0.598293	−	1.22106i
$441$		0			0
$442$		−	1.67477e6i		−	0.407754i
$443$			219797.i			0.0532122i	0.999646	+	0.0266061i	$0.00846999\pi$
$443$			219797.i			0.0532122i	−0.999646	+	0.0266061i	$0.991530\pi$
$444$		0			0
$445$	−6.04800e6	+	2.96339e6i	−1.44781	+	0.709397i
$446$	−3.63345e6			−0.864931
$447$		0			0
$448$			7.65813e6i			1.80272i
$449$	2.47283e6			0.578867			0.289434	−	0.957198i	$-0.406533\pi$
$449$	2.47283e6			0.578867			0.289434	+	0.957198i	$0.406533\pi$
$450$		0			0
$451$	−5.54400e6			−1.28346
$452$		−	1.78454e6i		−	0.410848i
$453$		0			0
$454$	1.58776e6			0.361531
$455$	−2.53006e6	+	1.23968e6i	−0.572931	+	0.280724i
$456$		0			0
$457$			2.89649e6i			0.648757i	0.945927	+	0.324378i	$0.105155\pi$
$457$			2.89649e6i			0.648757i	−0.945927	+	0.324378i	$0.894845\pi$
$458$			1.83847e6i			0.409536i
$459$		0			0
$460$	−667920.	−	1.36316e6i	−0.147174	−	0.300367i
$461$	−7.05154e6			−1.54537			−0.772683	−	0.634792i	$-0.781086\pi$
$461$	−7.05154e6			−1.54537			−0.772683	+	0.634792i	$0.781086\pi$
$462$		0			0
$463$		−	2.89178e6i		−	0.626920i	−0.949601	−	0.313460i	$-0.898512\pi$
$463$		−	2.89178e6i		−	0.626920i	0.949601	−	0.313460i	$-0.101488\pi$
$464$	−2.73889e6			−0.590581
$465$		0			0
$466$	3.30130e6			0.704239
$467$			7.71866e6i			1.63776i	0.573967	+	0.818879i	$0.305404\pi$
$467$			7.71866e6i			1.63776i	−0.573967	+	0.818879i	$0.694596\pi$
$468$		0			0
$469$	7.76160e6			1.62937
$470$	2.14252e6	−	1.04979e6i	0.447384	−	0.219209i
$471$		0			0
$472$			1.08658e6i			0.224494i
$473$		−	6.31108e6i		−	1.29703i
$474$		0			0
$475$	−926860.	+	1.19524e6i	−0.188486	+	0.243063i
$476$	−4.49387e6			−0.909082
$477$		0			0
$478$			543289.i			0.108758i
$479$	−209834.			−0.0417867			−0.0208933	−	0.999782i	$-0.506651\pi$
$479$	−209834.			−0.0417867			−0.0208933	+	0.999782i	$0.506651\pi$
$480$		0			0
$481$	1.66320e6			0.327779
$482$		−	5.01882e6i		−	0.983975i
$483$		0			0
$484$	1.09139e6			0.211771
$485$	1.57928e6	+	3.22316e6i	0.304863	+	0.622196i
$486$		0			0
$487$			777891.i			0.148626i	0.997235	+	0.0743132i	$0.0236765\pi$
$487$			777891.i			0.148626i	−0.997235	+	0.0743132i	$0.976324\pi$
$488$		−	4.20349e6i		−	0.799024i
$489$		0			0
$490$	−3.69523e6	−	7.54161e6i	−0.695266	−	1.41897i
$491$	9.93400e6			1.85960			0.929802	−	0.368061i	$-0.119978\pi$
$491$	9.93400e6			1.85960			0.929802	+	0.368061i	$0.119978\pi$
$492$		0			0
$493$		−	9.21121e6i		−	1.70687i
$494$	−485932.			−0.0895897
$495$		0			0
$496$	−1.78957e6			−0.326621
$497$		−	7.43806e6i		−	1.35073i
$498$		0			0
$499$	−5.10332e6			−0.917489			−0.458745	−	0.888568i	$-0.651701\pi$
$499$	−5.10332e6			−0.917489			−0.458745	+	0.888568i	$0.651701\pi$
$500$	−424689.	+	2.05284e6i	−0.0759706	+	0.367224i
$501$		0			0
$502$		−	74084.8i		−	0.0131211i
$503$			4.00824e6i			0.706373i	0.935553	+	0.353186i	$0.114902\pi$
$503$			4.00824e6i			0.706373i	−0.935553	+	0.353186i	$0.885098\pi$
$504$		0			0
$505$	3.04920e6	−	1.49404e6i	0.532056	−	0.260696i
$506$	−5.08020e6			−0.882073
$507$		0			0
$508$			2.07707e6i			0.357101i
$509$	−1.96029e6			−0.335372			−0.167686	−	0.985840i	$-0.553629\pi$
$509$	−1.96029e6			−0.335372			−0.167686	+	0.985840i	$0.553629\pi$
$510$		0			0
$511$	−604800.			−0.102461
$512$			5.14618e6i			0.867580i
$513$		0			0
$514$	4.95418e6			0.827111
$515$	3.70523e6	+	7.56202e6i	0.615598	+	1.25638i
$516$		0			0
$517$			4.79082e6i			0.788285i
$518$			7.43806e6i			1.21797i
$519$		0			0
$520$	−2.21760e6	+	1.08658e6i	−0.359646	+	0.176219i
$521$	9.65338e6			1.55806			0.779032	−	0.626984i	$-0.215711\pi$
$521$	9.65338e6			1.55806			0.779032	+	0.626984i	$0.215711\pi$
$522$		0			0
$523$		−	462020.i		−	0.0738595i	−0.999318	−	0.0369298i	$-0.988242\pi$
$523$		−	462020.i		−	0.0738595i	0.999318	−	0.0369298i	$-0.0117578\pi$
$524$	−3.24691e6			−0.516585
$525$		0			0
$526$	8.16532e6			1.28679
$527$		−	6.01853e6i		−	0.943982i
$528$		0			0
$529$	1.31562e6			0.204405
$530$	−1.07126e6	+	524895.i	−0.165655	+	0.0811675i
$531$		0			0
$532$			1.30389e6i			0.199739i
$533$			2.47935e6i			0.378025i
$534$		0			0
$535$	−47080.0	−	96085.8i	−0.00711135	−	0.0145136i
$536$	6.80305e6			1.02280
$537$		0			0
$538$		−	6.00087e6i		−	0.893837i
$539$	1.68636e7			2.50021
$540$		0			0
$541$	−1.08238e7			−1.58996			−0.794978	−	0.606639i	$-0.792517\pi$
$541$	−1.08238e7			−1.58996			−0.794978	+	0.606639i	$0.792517\pi$
$542$		−	3.24047e6i		−	0.473817i
$543$		0			0
$544$	−6.80352e6			−0.985681
$545$	−85038.1	+	41666.9i	−0.0122637	+	0.00600897i
$546$		0			0
$547$			2.02229e6i			0.288985i	0.989506	+	0.144493i	$0.0461550\pi$
$547$			2.02229e6i			0.288985i	−0.989506	+	0.144493i	$0.953845\pi$
$548$		−	3.82824e6i		−	0.544562i
$549$		0			0
$550$	5.54400e6	+	4.29916e6i	0.781478	+	0.606007i
$551$	−2.67263e6			−0.375024
$552$		0			0
$553$			2.23637e7i			3.10979i
$554$	8.94457e6			1.23818
$555$		0			0
$556$	2.71867e6			0.372967
$557$			3.92593e6i			0.536173i	0.963395	+	0.268086i	$0.0863913\pi$
$557$			3.92593e6i			0.536173i	−0.963395	+	0.268086i	$0.913609\pi$
$558$		0			0
$559$	−2.82240e6			−0.382023
$560$	−2.73889e6	−	5.58981e6i	−0.369066	−	0.753229i
$561$		0			0
$562$		−	9.87798e6i		−	1.31925i
$563$			1.12906e7i			1.50123i	0.660739	+	0.750616i	$0.270243\pi$
$563$			1.12906e7i			1.50123i	−0.660739	+	0.750616i	$0.729757\pi$
$564$		0			0
$565$	3.65783e6	+	7.46528e6i	0.482061	+	0.983841i
$566$	−1.97786e6			−0.259511
$567$		0			0
$568$		−	6.51946e6i		−	0.847893i
$569$	−1.10770e7			−1.43431			−0.717155	−	0.696913i	$-0.754556\pi$
$569$	−1.10770e7			−1.43431			−0.717155	+	0.696913i	$0.754556\pi$
$570$		0			0
$571$	3.28605e6			0.421778			0.210889	−	0.977510i	$-0.432364\pi$
$571$	3.28605e6			0.421778			0.210889	+	0.977510i	$0.432364\pi$
$572$		−	1.35237e6i		−	0.172825i
$573$		0			0
$574$	−1.10880e7			−1.40467
$575$	5.58822e6	+	4.33346e6i	0.704862	+	0.546594i
$576$		0			0
$577$		−	1.42539e7i		−	1.78236i	−0.453653	−	0.891179i	$-0.649879\pi$
$577$		−	1.42539e7i		−	1.78236i	0.453653	−	0.891179i	$-0.350121\pi$
$578$			6.09428e6i			0.758757i
$579$		0			0
$580$	−3.32640e6	+	1.62987e6i	−0.410586	+	0.201179i
$581$	−1.28230e7			−1.57597
$582$		0			0
$583$		−	2.39541e6i		−	0.291883i
$584$	−530108.			−0.0643179
$585$		0			0
$586$	−4.52102e6			−0.543867
$587$		−	6.12026e6i		−	0.733119i	−0.930395	−	0.366560i	$-0.880536\pi$
$587$		−	6.12026e6i		−	0.733119i	0.930395	−	0.366560i	$-0.119464\pi$
$588$		0			0
$589$	−1.74627e6			−0.207407
$590$	−607415.	−	1.23968e6i	−0.0718382	−	0.146615i
$591$		0			0
$592$			3.67461e6i			0.430930i
$593$		−	1.38377e7i		−	1.61595i	−0.589215	−	0.807976i	$-0.700563\pi$
$593$		−	1.38377e7i		−	1.61595i	0.589215	−	0.807976i	$-0.299437\pi$
$594$		0			0
$595$	1.87992e7	−	9.21121e6i	2.17694	−	1.06666i
$596$	1.92164e6			0.221593
$597$		0			0
$598$			2.27193e6i			0.259802i
$599$	−6.37133e6			−0.725543			−0.362772	−	0.931878i	$-0.618169\pi$
$599$	−6.37133e6			−0.725543			−0.362772	+	0.931878i	$0.618169\pi$
$600$		0			0
$601$	2.06116e6			0.232769			0.116384	−	0.993204i	$-0.462870\pi$
$601$	2.06116e6			0.232769			0.116384	+	0.993204i	$0.462870\pi$
$602$		−	1.26222e7i		−	1.41952i
$603$		0			0
$604$	106656.			0.0118958
$605$	−4.56560e6	+	2.23705e6i	−0.507119	+	0.248478i
$606$		0			0
$607$			8.89040e6i			0.979377i	0.871898	+	0.489688i	$0.162889\pi$
$607$			8.89040e6i			0.979377i	−0.871898	+	0.489688i	$0.837111\pi$
$608$			1.97404e6i			0.216569i
$609$		0			0
$610$	2.34982e6	+	4.79576e6i	0.255688	+	0.521834i
$611$	2.14252e6			0.232178
$612$		0			0
$613$		−	9.20605e6i		−	0.989514i	−0.869031	−	0.494757i	$-0.835257\pi$
$613$		−	9.20605e6i		−	0.989514i	0.869031	−	0.494757i	$-0.164743\pi$
$614$	−1.06142e7			−1.13623
$615$		0			0
$616$	2.21760e7			2.35468
$617$			1.07622e7i			1.13812i	0.822296	+	0.569061i	$0.192693\pi$
$617$			1.07622e7i			1.13812i	−0.822296	+	0.569061i	$0.807307\pi$
$618$		0			0
$619$	−1.40448e7			−1.47329			−0.736644	−	0.676281i	$-0.763591\pi$
$619$	−1.40448e7			−1.47329			−0.736644	+	0.676281i	$0.763591\pi$
$620$	−2.17344e6	+	1.06494e6i	−0.227075	+	0.111262i
$621$		0			0
$622$			1.15078e7i			1.19266i
$623$		−	2.70475e7i		−	2.79194i
$624$		0			0
$625$	−2.43118e6	−	9.45816e6i	−0.248952	−	0.968516i
$626$	6.86931e6			0.700612
$627$		0			0
$628$		−	6.07495e6i		−	0.614673i
$629$	−1.23581e7			−1.24545
$630$		0			0
$631$	−1.00120e7			−1.00104			−0.500518	−	0.865726i	$-0.666857\pi$
$631$	−1.00120e7			−1.00104			−0.500518	+	0.865726i	$0.666857\pi$
$632$			1.96018e7i			1.95211i
$633$		0			0
$634$	−1.01387e7			−1.00175
$635$	−4.25743e6	−	8.68900e6i	−0.418999	−	0.855137i
$636$		0			0
$637$		−	7.54161e6i		−	0.736403i
$638$			1.23968e7i			1.20575i
$639$		0			0
$640$	−542080.	−	1.10633e6i	−0.0523135	−	0.106767i
$641$	−9.60720e6			−0.923532			−0.461766	−	0.887002i	$-0.652784\pi$
$641$	−9.60720e6			−0.923532			−0.461766	+	0.887002i	$0.652784\pi$
$642$		0			0
$643$		−	2.21761e6i		−	0.211523i	−0.994392	−	0.105761i	$-0.966272\pi$
$643$		−	2.21761e6i		−	0.211523i	0.994392	−	0.105761i	$-0.0337280\pi$
$644$	6.09624e6			0.579225
$645$		0			0
$646$	3.61064e6			0.340410
$647$			5.71481e6i			0.536712i	0.963320	+	0.268356i	$0.0864803\pi$
$647$			5.71481e6i			0.536712i	−0.963320	+	0.268356i	$0.913520\pi$
$648$		0			0
$649$	2.77200e6			0.258334
$650$	1.92264e6	−	2.47935e6i	0.178491	−	0.230173i
$651$		0			0
$652$			2.66705e6i			0.245704i
$653$		−	4.71603e6i		−	0.432807i	−0.976304	−	0.216403i	$-0.930567\pi$
$653$		−	4.71603e6i		−	0.432807i	0.976304	−	0.216403i	$-0.0694326\pi$
$654$		0			0
$655$	1.35828e7	−	6.65529e6i	1.23705	−	0.606127i
$656$	−5.47778e6			−0.496987
$657$		0			0
$658$			9.58164e6i			0.862730i
$659$	2.01496e7			1.80740			0.903698	−	0.428171i	$-0.140842\pi$
$659$	2.01496e7			1.80740			0.903698	+	0.428171i	$0.140842\pi$
$660$		0			0
$661$	−564742.			−0.0502743			−0.0251372	−	0.999684i	$-0.508002\pi$
$661$	−564742.			−0.0502743			−0.0251372	+	0.999684i	$0.508002\pi$
$662$		−	7.42290e6i		−	0.658308i
$663$		0			0
$664$	−1.12394e7			−0.989285
$665$	−2.67263e6	−	5.45457e6i	−0.234360	−	0.478307i
$666$		0			0
$667$			1.24956e7i			1.08754i
$668$		−	1.17066e6i		−	0.101506i
$669$		0			0
$670$	−7.76160e6	+	3.80302e6i	−0.667981	+	0.327297i
$671$	−1.07236e7			−0.919466
$672$		0			0
$673$			1.07198e7i			0.912328i	0.889896	+	0.456164i	$0.150777\pi$
$673$			1.07198e7i			0.912328i	−0.889896	+	0.456164i	$0.849223\pi$
$674$	3.67863e6			0.311915
$675$		0			0
$676$	3.85072e6			0.324097
$677$			3.36926e6i			0.282529i	0.989972	+	0.141264i	$0.0451168\pi$
$677$			3.36926e6i			0.282529i	−0.989972	+	0.141264i	$0.954883\pi$
$678$		0			0
$679$	−1.44144e7			−1.19984
$680$	1.64775e7	−	8.07364e6i	1.36653	−	0.669571i
$681$		0			0
$682$			8.09994e6i			0.666839i
$683$			1.55817e7i			1.27810i	0.769166	+	0.639049i	$0.220672\pi$
$683$			1.55817e7i			1.27810i	−0.769166	+	0.639049i	$0.779328\pi$
$684$		0			0
$685$	7.84685e6	+	1.60147e7i	0.638953	+	1.30404i
$686$	1.68530e7			1.36731
$687$		0			0
$688$		−	6.23570e6i		−	0.502243i
$689$	−1.07126e6			−0.0859699
$690$		0			0
$691$	9.73771e6			0.775821			0.387911	−	0.921697i	$-0.373197\pi$
$691$	9.73771e6			0.775821			0.387911	+	0.921697i	$0.373197\pi$
$692$			3.32002e6i			0.263558i
$693$		0			0
$694$	−4.37072e6			−0.344472
$695$	−1.13730e7	+	5.57254e6i	−0.893128	+	0.437614i
$696$		0			0
$697$		−	1.84224e7i		−	1.43637i
$698$			3.46588e6i			0.269262i
$699$		0			0
$700$	−6.65280e6	−	5.15900e6i	−0.513168	−	0.397942i
$701$	1.72451e7			1.32547			0.662735	−	0.748854i	$-0.269396\pi$
$701$	1.72451e7			1.32547			0.662735	+	0.748854i	$0.269396\pi$
$702$		0			0
$703$			3.58571e6i			0.273644i
$704$	1.71241e7			1.30219
$705$		0			0
$706$	1.23462e7			0.932225
$707$			1.36364e7i			1.02601i
$708$		0			0
$709$	1.81083e7			1.35289			0.676443	−	0.736495i	$-0.263520\pi$
$709$	1.81083e7			1.35289			0.676443	+	0.736495i	$0.263520\pi$
$710$	3.64449e6	+	7.43806e6i	0.271326	+	0.553750i
$711$		0			0
$712$		−	2.37071e7i		−	1.75259i
$713$			8.16455e6i			0.601462i
$714$		0			0
$715$	2.77200e6	+	5.65739e6i	0.202781	+	0.413857i
$716$	343365.			0.0250307
$717$		0			0
$718$			5.03777e6i			0.364693i
$719$	−1.77044e7			−1.27720			−0.638600	−	0.769539i	$-0.720486\pi$
$719$	−1.77044e7			−1.27720			−0.638600	+	0.769539i	$0.720486\pi$
$720$		0			0
$721$	−3.38184e7			−2.42279
$722$			1.00258e7i			0.715776i
$723$		0			0
$724$	−982344.			−0.0696494
$725$	1.05745e7	−	1.36364e7i	0.747165	−	0.963509i
$726$		0			0
$727$		−	7.80607e6i		−	0.547768i	−0.961763	−	0.273884i	$-0.911692\pi$
$727$		−	7.80607e6i		−	0.547768i	0.961763	−	0.273884i	$-0.0883084\pi$
$728$		−	9.91741e6i		−	0.693537i
$729$		0			0
$730$	604800.	−	296339.i	0.0420054	−	0.0205817i
$731$	2.09714e7			1.45156
$732$		0			0
$733$		−	2.11490e7i		−	1.45388i	−0.686700	−	0.726941i	$-0.740941\pi$
$733$		−	2.11490e7i		−	1.45388i	0.686700	−	0.726941i	$-0.259059\pi$
$734$	−1.18170e6			−0.0809593
$735$		0			0
$736$	9.22944e6			0.628031
$737$		−	1.73555e7i		−	1.17698i
$738$		0			0
$739$	9.20256e6			0.619865			0.309933	−	0.950759i	$-0.399693\pi$
$739$	9.20256e6			0.619865			0.309933	+	0.950759i	$0.399693\pi$
$740$	2.18669e6	+	4.46283e6i	0.146794	+	0.299593i
$741$		0			0
$742$		−	4.79082e6i		−	0.319448i
$743$		−	4.78756e6i		−	0.318157i	−0.987266	−	0.159079i	$-0.949148\pi$
$743$		−	4.78756e6i		−	0.318157i	0.987266	−	0.159079i	$-0.0508523\pi$
$744$		0			0
$745$	−8.03880e6	+	3.93884e6i	−0.530641	+	0.260003i
$746$	−1.37396e7			−0.903916
$747$		0			0
$748$			1.00486e7i			0.656676i
$749$	429709.			0.0279878
$750$		0			0
$751$	7.03369e6			0.455075			0.227538	−	0.973769i	$-0.426933\pi$
$751$	7.03369e6			0.455075			0.227538	+	0.973769i	$0.426933\pi$
$752$			4.73359e6i			0.305244i
$753$		0			0
$754$	5.54400e6			0.355136
$755$	−446174.	+	218616.i	−0.0284864	+	0.0139577i
$756$		0			0
$757$			1.99807e7i			1.26727i	0.773631	+	0.633637i	$0.218439\pi$
$757$			1.99807e7i			1.26727i	−0.773631	+	0.633637i	$0.781561\pi$
$758$		−	6.81122e6i		−	0.430578i
$759$		0			0
$760$	−2.34256e6	−	4.78094e6i	−0.147115	−	0.300247i
$761$	−1.15851e7			−0.725165			−0.362582	−	0.931952i	$-0.618105\pi$
$761$	−1.15851e7			−0.725165			−0.362582	+	0.931952i	$0.618105\pi$
$762$		0			0
$763$		−	380302.i		−	0.0236492i
$764$	−5.84323e6			−0.362176
$765$		0			0
$766$	−1.07611e7			−0.662650
$767$		−	1.23968e6i		−	0.0760886i
$768$		0			0
$769$	1.88308e7			1.14830			0.574148	−	0.818752i	$-0.305333\pi$
$769$	1.88308e7			1.14830			0.574148	+	0.818752i	$0.305333\pi$
$770$	−2.53006e7	+	1.23968e7i	−1.53781	+	0.753497i
$771$		0			0
$772$			6.49252e6i			0.392076i
$773$		−	9.80638e6i		−	0.590283i	−0.955454	−	0.295141i	$-0.904633\pi$
$773$		−	9.80638e6i		−	0.590283i	0.955454	−	0.295141i	$-0.0953667\pi$
$774$		0			0
$775$	6.90932e6	−	8.90993e6i	0.413220	−	0.532869i
$776$	−1.26342e7			−0.753173
$777$		0			0
$778$		−	5.65514e6i		−	0.334961i
$779$	−5.34525e6			−0.315591
$780$		0			0
$781$	−1.66320e7			−0.975701
$782$		−	1.68812e7i		−	0.987160i
$783$		0			0
$784$	1.66621e7			0.968145
$785$	1.24520e7	+	2.54134e7i	0.721216	+	1.47193i
$786$		0			0
$787$			3.21012e7i			1.84750i	0.382998	+	0.923749i	$0.374892\pi$
$787$			3.21012e7i			1.84750i	−0.382998	+	0.923749i	$0.625108\pi$
$788$			9.75571e6i			0.559685i
$789$		0			0
$790$	−1.09578e7	−	2.23637e7i	−0.624675	−	1.27490i
$791$	−3.33857e7			−1.89723
$792$		0			0
$793$			4.79576e6i			0.270816i
$794$	6.99080e6			0.393528
$795$		0			0
$796$	1.75373e6			0.0981024
$797$		−	3.06076e7i		−	1.70680i	−0.521256	−	0.853400i	$-0.674536\pi$
$797$		−	3.06076e7i		−	1.70680i	0.521256	−	0.853400i	$-0.325464\pi$
$798$		0			0
$799$	−1.59196e7			−0.882198
$800$	−1.00720e7	−	7.81050e6i	−0.556407	−	0.431473i
$801$		0			0
$802$			6.11985e6i			0.335974i
$803$			1.35237e6i			0.0740130i
$804$		0			0
$805$	−2.55024e7	+	1.24956e7i	−1.38705	+	0.679624i
$806$	3.62240e6			0.196408
$807$		0			0
$808$			1.19524e7i			0.644058i
$809$	1.30208e7			0.699464			0.349732	−	0.936850i	$-0.386273\pi$
$809$	1.30208e7			0.699464			0.349732	+	0.936850i	$0.386273\pi$
$810$		0			0
$811$	−2.58466e7			−1.37991			−0.689956	−	0.723851i	$-0.742370\pi$
$811$	−2.58466e7			−1.37991			−0.689956	+	0.723851i	$0.742370\pi$
$812$		−	1.48761e7i		−	0.791771i
$813$		0			0
$814$	1.66320e7			0.879799
$815$	−5.46674e6	−	1.11571e7i	−0.288293	−	0.588378i
$816$		0			0
$817$		−	6.08483e6i		−	0.318929i
$818$		−	1.92940e7i		−	1.00818i
$819$		0			0
$820$	−6.65280e6	+	3.25973e6i	−0.345517	+	0.169296i
$821$	1.88575e7			0.976396			0.488198	−	0.872733i	$-0.337654\pi$
$821$	1.88575e7			0.976396			0.488198	+	0.872733i	$0.337654\pi$
$822$		0			0
$823$		−	1.90126e7i		−	0.978459i	−0.872155	−	0.489230i	$-0.837278\pi$
$823$		−	1.90126e7i		−	0.978459i	0.872155	−	0.489230i	$-0.162722\pi$
$824$	−2.96419e7			−1.52085
$825$		0			0
$826$	5.54400e6			0.282731
$827$		−	823839.i		−	0.0418869i	−0.999781	−	0.0209435i	$-0.993333\pi$
$827$		−	823839.i		−	0.0418869i	0.999781	−	0.0209435i	$-0.00666700\pi$
$828$		0			0
$829$	3.60900e7			1.82390			0.911948	−	0.410305i	$-0.134578\pi$
$829$	3.60900e7			1.82390			0.911948	+	0.410305i	$0.134578\pi$
$830$	1.28230e7	−	6.28299e6i	0.646091	−	0.316571i
$831$		0			0
$832$		−	7.65813e6i		−	0.383543i
$833$			5.60367e7i			2.79808i
$834$		0			0
$835$	2.39954e6	+	4.89723e6i	0.119100	+	0.243072i
$836$	2.91559e6			0.144282
$837$		0			0
$838$			3.08238e6i			0.151627i
$839$	−1.23692e7			−0.606647			−0.303324	−	0.952888i	$-0.598096\pi$
$839$	−1.23692e7			−0.606647			−0.303324	+	0.952888i	$0.598096\pi$
$840$		0			0
$841$	9.98085e6			0.486606
$842$			4.63117e6i			0.225118i
$843$		0			0
$844$	1.01603e7			0.490965
$845$	−1.61087e7	+	7.89292e6i	−0.776102	+	0.380274i
$846$		0			0
$847$		−	2.04180e7i		−	0.977923i
$848$		−	2.36680e6i		−	0.113024i
$849$		0			0
$850$	−1.42859e7	+	1.84224e7i	−0.678204	+	0.874580i
$851$	1.67647e7			0.793544
$852$		0			0
$853$			9.97159e6i			0.469237i	0.972088	+	0.234618i	$0.0753840\pi$
$853$			9.97159e6i			0.469237i	−0.972088	+	0.234618i	$0.924616\pi$
$854$	−2.14473e7			−1.00630
$855$		0			0
$856$	376640.			0.0175688
$857$			1.48619e7i			0.691231i	0.938376	+	0.345616i	$0.112330\pi$
$857$			1.48619e7i			0.691231i	−0.938376	+	0.345616i	$0.887670\pi$
$858$		0			0
$859$	1.41452e7			0.654071			0.327036	−	0.945012i	$-0.393950\pi$
$859$	1.41452e7			0.654071			0.327036	+	0.945012i	$0.393950\pi$
$860$	−3.71075e6	−	7.57329e6i	−0.171087	−	0.349171i
$861$		0			0
$862$		−	1.24462e7i		−	0.570519i
$863$		−	1.72325e6i		−	0.0787627i	−0.999224	−	0.0393814i	$-0.987461\pi$
$863$		−	1.72325e6i		−	0.0787627i	0.999224	−	0.0393814i	$-0.0125387\pi$
$864$		0			0
$865$	−6.80515e6	−	1.38887e7i	−0.309241	−	0.631132i
$866$	1.78911e6			0.0810668
$867$		0			0
$868$		−	9.71993e6i		−	0.437889i
$869$	5.00068e7			2.24636
$870$		0			0
$871$	−7.76160e6			−0.346662
$872$		−	333335.i		−	0.0148453i
$873$		0			0
$874$	−4.89808e6			−0.216894
$875$	3.84052e7	+	7.94520e6i	1.69578	+	0.350820i
$876$		0			0
$877$		−	8.53435e6i		−	0.374689i	−0.982294	−	0.187345i	$-0.940012\pi$
$877$		−	8.53435e6i		−	0.374689i	0.982294	−	0.187345i	$-0.0599881\pi$
$878$		−	1.38639e7i		−	0.606945i
$879$		0			0
$880$	−1.24992e7	+	6.12434e6i	−0.544096	+	0.266596i
$881$	7.09019e6			0.307764			0.153882	−	0.988089i	$-0.450822\pi$
$881$	7.09019e6			0.307764			0.153882	+	0.988089i	$0.450822\pi$
$882$		0			0
$883$			7.54183e6i			0.325518i	0.986666	+	0.162759i	$0.0520393\pi$
$883$			7.54183e6i			0.325518i	−0.986666	+	0.162759i	$0.947961\pi$
$884$	4.49387e6			0.193415
$885$		0			0
$886$	982960.			0.0420680
$887$		−	4.11968e7i		−	1.75815i	−0.476687	−	0.879073i	$-0.658163\pi$
$887$		−	4.11968e7i		−	1.75815i	0.476687	−	0.879073i	$-0.341837\pi$
$888$		0			0
$889$	3.88584e7			1.64904
$890$	1.32527e7	+	2.70475e7i	0.560827	+	1.14459i
$891$		0			0
$892$		−	9.74956e6i		−	0.410273i
$893$			4.61907e6i			0.193832i
$894$		0			0
$895$	−1.43640e6	+	703806.i	−0.0599401	+	0.0293694i
$896$	4.94767e6			0.205888
$897$		0			0
$898$		−	1.10588e7i		−	0.457635i
$899$	1.99232e7			0.822167
$900$		0			0
$901$	7.95982e6			0.326656
$902$			2.47935e7i			1.01466i
$903$		0			0
$904$	−2.92626e7			−1.19095
$905$	4.10944e6	−	2.01354e6i	0.166787	−	0.0817219i
$906$		0			0
$907$		−	3.99367e7i		−	1.61196i	−0.591945	−	0.805979i	$-0.701640\pi$
$907$		−	3.99367e7i		−	1.61196i	0.591945	−	0.805979i	$-0.298360\pi$
$908$			4.26041e6i			0.171489i
$909$		0			0
$910$	5.54400e6	+	1.13148e7i	0.221932	+	0.452942i
$911$	−9.30901e6			−0.371627			−0.185814	−	0.982585i	$-0.559492\pi$
$911$	−9.30901e6			−0.371627			−0.185814	+	0.982585i	$0.559492\pi$
$912$		0			0
$913$			2.86731e7i			1.13841i
$914$	1.29535e7			0.512887
$915$		0			0
$916$	−4.93313e6			−0.194260
$917$			6.07441e7i			2.38551i
$918$		0			0
$919$	−2.88035e7			−1.12501			−0.562506	−	0.826793i	$-0.690163\pi$
$919$	−2.88035e7			−1.12501			−0.562506	+	0.826793i	$0.690163\pi$
$920$	−2.23529e7	+	1.09524e7i	−0.870691	+	0.426620i
$921$		0			0
$922$			3.15354e7i			1.22172i
$923$			7.43806e6i			0.287379i
$924$		0			0
$925$	−1.82952e7	−	1.41872e7i	−0.703045	−	0.545185i
$926$	−1.29324e7			−0.495624
$927$		0			0
$928$		−	2.25218e7i		−	0.858485i
$929$	2.03981e7			0.775444			0.387722	−	0.921776i	$-0.373262\pi$
$929$	2.03981e7			0.775444			0.387722	+	0.921776i	$0.373262\pi$
$930$		0			0
$931$	1.62590e7			0.614780
$932$			8.85832e6i			0.334050i
$933$		0			0
$934$	3.45189e7			1.29476
$935$	−2.05969e7	−	4.20363e7i	−0.770500	−	1.57252i
$936$		0			0
$937$			4.91357e7i			1.82830i	0.405371	+	0.914152i	$0.367142\pi$
$937$			4.91357e7i			1.82830i	−0.405371	+	0.914152i	$0.632858\pi$
$938$		−	3.47109e7i		−	1.28813i
$939$		0			0
$940$	2.81688e6	+	5.74898e6i	0.103980	+	0.212213i
$941$	3.80960e7			1.40251			0.701254	−	0.712912i	$-0.252624\pi$
$941$	3.80960e7			1.40251			0.701254	+	0.712912i	$0.252624\pi$
$942$		0			0
$943$			2.49913e7i			0.915186i
$944$	2.73889e6			0.100033
$945$		0			0
$946$	−2.82240e7			−1.02539
$947$			1.45947e6i			0.0528836i	0.999650	+	0.0264418i	$0.00841767\pi$
$947$			1.45947e6i			0.0528836i	−0.999650	+	0.0264418i	$0.991582\pi$
$948$		0			0
$949$	604800.			0.0217995
$950$	5.34525e6	+	4.14504e6i	0.192158	+	0.149012i
$951$		0			0
$952$			7.36897e7i			2.63520i
$953$		−	1.94841e7i		−	0.694940i	−0.937691	−	0.347470i	$-0.887041\pi$
$953$		−	1.94841e7i		−	0.694940i	0.937691	−	0.347470i	$-0.112959\pi$
$954$		0			0
$955$	2.44440e7	−	1.19770e7i	0.867289	−	0.424953i
$956$	−1.45780e6			−0.0515884
$957$		0			0
$958$			938408.i			0.0330353i
$959$	−7.16198e7			−2.51470
$960$		0			0
$961$	−1.56115e7			−0.545300
$962$		−	7.43806e6i		−	0.259132i
$963$		0			0
$964$	1.34669e7			0.466740
$965$	−1.33079e7	−	2.71602e7i	−0.460036	−	0.938889i
$966$		0			0
$967$		−	1.65207e7i		−	0.568149i	−0.958802	−	0.284074i	$-0.908314\pi$
$967$		−	1.65207e7i		−	0.568149i	0.958802	−	0.284074i	$-0.0916862\pi$
$968$		−	1.78964e7i		−	0.613871i
$969$		0			0
$970$	1.44144e7	−	7.06275e6i	0.491889	−	0.241015i
$971$	−1.02422e7			−0.348615			−0.174308	−	0.984691i	$-0.555769\pi$
$971$	−1.02422e7			−0.348615			−0.174308	+	0.984691i	$0.555769\pi$
$972$		0			0
$973$		−	5.08617e7i		−	1.72230i
$974$	3.47883e6			0.117500
$975$		0			0
$976$	−1.05956e7			−0.356040
$977$			1.95090e7i			0.653881i	0.945045	+	0.326941i	$0.106018\pi$
$977$			1.95090e7i			0.653881i	−0.945045	+	0.326941i	$0.893982\pi$
$978$		0			0
$979$	−6.04800e7			−2.01676
$980$	2.02363e7	−	9.91534e6i	0.673078	−	0.329794i
$981$		0			0
$982$		−	4.44262e7i		−	1.47015i
$983$			1.91895e7i			0.633403i	0.948525	+	0.316701i	$0.102575\pi$
$983$			1.91895e7i			0.633403i	−0.948525	+	0.316701i	$0.897425\pi$
$984$		0			0
$985$	−1.99966e7	−	4.08111e7i	−0.656697	−	1.34026i
$986$	−4.11938e7			−1.34940
$987$		0			0
$988$		−	1.30389e6i		−	0.0424961i
$989$	−2.84491e7			−0.924864
$990$		0			0
$991$	5.38361e7			1.74136			0.870682	−	0.491846i	$-0.163678\pi$
$991$	5.38361e7			1.74136			0.870682	+	0.491846i	$0.163678\pi$
$992$		−	1.47155e7i		−	0.474785i
$993$		0			0
$994$	−3.32640e7			−1.06785
$995$	−7.33637e6	+	3.59467e6i	−0.234922	+	0.115107i
$996$		0			0
$997$			3.25008e6i			0.103551i	0.998659	+	0.0517757i	$0.0164881\pi$
$997$			3.25008e6i			0.103551i	−0.998659	+	0.0517757i	$0.983512\pi$
$998$			2.28227e7i			0.725339i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.6.b.d.19.1	✓	4
3.2	odd	2	inner	45.6.b.d.19.4	yes	4
4.3	odd	2		720.6.f.k.289.2		4
5.2	odd	4		225.6.a.v.1.4		4
5.3	odd	4		225.6.a.v.1.1		4
5.4	even	2	inner	45.6.b.d.19.3	yes	4
12.11	even	2		720.6.f.k.289.3		4
15.2	even	4		225.6.a.v.1.2		4
15.8	even	4		225.6.a.v.1.3		4
15.14	odd	2	inner	45.6.b.d.19.2	yes	4
20.19	odd	2		720.6.f.k.289.1		4
60.59	even	2		720.6.f.k.289.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.6.b.d.19.1	✓	4	1.1	even	1	trivial
45.6.b.d.19.2	yes	4	15.14	odd	2	inner
45.6.b.d.19.3	yes	4	5.4	even	2	inner
45.6.b.d.19.4	yes	4	3.2	odd	2	inner
225.6.a.v.1.1		4	5.3	odd	4
225.6.a.v.1.2		4	15.2	even	4
225.6.a.v.1.3		4	15.8	even	4
225.6.a.v.1.4		4	5.2	odd	4
720.6.f.k.289.1		4	20.19	odd	2
720.6.f.k.289.2		4	4.3	odd	2
720.6.f.k.289.3		4	12.11	even	2
720.6.f.k.289.4		4	60.59	even	2

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:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	45.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.47214i q^{2} +12.0000 q^{4} +(-50.1996 + 24.5967i) q^{5} -224.499i q^{7} -196.774i q^{8} +O(q^{10})\)	\(q-4.47214i q^{2} +12.0000 q^{4} +(-50.1996 + 24.5967i) q^{5} -224.499i q^{7} -196.774i q^{8} +(110.000 + 224.499i) q^{10} -501.996 q^{11} +224.499i q^{13} -1003.99 q^{14} -496.000 q^{16} -1668.11i q^{17} -484.000 q^{19} +(-602.395 + 295.161i) q^{20} +2244.99i q^{22} +2262.90i q^{23} +(1915.00 - 2469.49i) q^{25} +1003.99 q^{26} -2693.99i q^{28} +5521.96 q^{29} +3608.00 q^{31} -4078.59i q^{32} -7460.00 q^{34} +(5521.96 + 11269.8i) q^{35} -7408.48i q^{37} +2164.51i q^{38} +(4840.00 + 9877.98i) q^{40} +11043.9 q^{41} +12572.0i q^{43} -6023.95 q^{44} +10120.0 q^{46} -9543.54i q^{47} -33593.0 q^{49} +(-11043.9 - 8564.14i) q^{50} +2693.99i q^{52} +4771.77i q^{53} +(25200.0 - 12347.5i) q^{55} -44175.6 q^{56} -24694.9i q^{58} -5521.96 q^{59} +21362.0 q^{61} -16135.5i q^{62} -34112.0 q^{64} +(-5521.96 - 11269.8i) q^{65} +34572.9i q^{67} -20017.3i q^{68} +(50400.0 - 24694.9i) q^{70} +33131.7 q^{71} -2693.99i q^{73} -33131.7 q^{74} -5808.00 q^{76} +112698. i q^{77} -99616.0 q^{79} +(24899.0 - 12200.0i) q^{80} -49389.9i q^{82} -57118.1i q^{83} +(41030.0 + 83738.3i) q^{85} +56223.6 q^{86} +98779.8i q^{88} +120479. q^{89} +50400.0 q^{91} +27154.8i q^{92} -42680.0 q^{94} +(24296.6 - 11904.8i) q^{95} -64206.8i q^{97} +150232. i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 48 q^{4}+O(q^{10}) \) 4 * q + 48 * q^4	\( 4 q + 48 q^{4} + 440 q^{10} - 1984 q^{16} - 1936 q^{19} + 7660 q^{25} + 14432 q^{31} - 29840 q^{34} + 19360 q^{40} + 40480 q^{46} - 134372 q^{49} + 100800 q^{55} + 85448 q^{61} - 136448 q^{64} + 201600 q^{70} - 23232 q^{76} - 398464 q^{79} + 164120 q^{85} + 201600 q^{91} - 170720 q^{94}+O(q^{100}) \) 4 * q + 48 * q^4 + 440 * q^10 - 1984 * q^16 - 1936 * q^19 + 7660 * q^25 + 14432 * q^31 - 29840 * q^34 + 19360 * q^40 + 40480 * q^46 - 134372 * q^49 + 100800 * q^55 + 85448 * q^61 - 136448 * q^64 + 201600 * q^70 - 23232 * q^76 - 398464 * q^79 + 164120 * q^85 + 201600 * q^91 - 170720 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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