LMFDB - Embedded newform 342.9.d.a.37.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [342,9,Mod(37,342)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(342, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("342.37");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 342 = 2 \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	342.d (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:	$139.323484641$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 46118 x^{10} + 738386961 x^{8} + 5214446299656 x^{6} + \cdots + 92\!\cdots\!64 $ x^12 + 46118x^10 + 738386961x^8 + 5214446299656x^6 + 17370647163698184x^4 + 24830681474333400768*x^2 + 9218779084612644462864
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{21}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 38)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.11
Root			$-111.533i$ of defining polynomial
Character	$\chi$	$=$	342.37
Dual form			342.9.d.a.37.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+11.3137i q^{2} -128.000 q^{4} +419.091 q^{5} -2431.38 q^{7} -1448.15i q^{8} +O(q^{10})$	\(q+11.3137i q^{2} -128.000 q^{4} +419.091 q^{5} -2431.38 q^{7} -1448.15i q^{8} +4741.48i q^{10} -18851.2 q^{11} -52190.4i q^{13} -27507.9i q^{14} +16384.0 q^{16} -163673. q^{17} +(-127361. - 27617.3i) q^{19} -53643.7 q^{20} -213277. i q^{22} +294819. q^{23} -214988. q^{25} +590467. q^{26} +311217. q^{28} +1.01197e6i q^{29} +188192. i q^{31} +185364. i q^{32} -1.85175e6i q^{34} -1.01897e6 q^{35} -1.31226e6i q^{37} +(312455. - 1.44093e6i) q^{38} -606909. i q^{40} +662199. i q^{41} -1.24898e6 q^{43} +2.41295e6 q^{44} +3.33550e6i q^{46} +3.35309e6 q^{47} +146807. q^{49} -2.43231e6i q^{50} +6.68038e6i q^{52} +6.52766e6i q^{53} -7.90038e6 q^{55} +3.52101e6i q^{56} -1.14492e7 q^{58} +5.48176e6i q^{59} +8.15096e6 q^{61} -2.12914e6 q^{62} -2.09715e6 q^{64} -2.18726e7i q^{65} +1.27314e7i q^{67} +2.09502e7 q^{68} -1.15283e7i q^{70} -2.99568e7i q^{71} +2.59027e7 q^{73} +1.48465e7 q^{74} +(1.63022e7 + 3.53502e6i) q^{76} +4.58344e7 q^{77} -5.24410e7i q^{79} +6.86639e6 q^{80} -7.49193e6 q^{82} +3.87316e7 q^{83} -6.85940e7 q^{85} -1.41306e7i q^{86} +2.72995e7i q^{88} +4.25848e7i q^{89} +1.26895e8i q^{91} -3.77368e7 q^{92} +3.79359e7i q^{94} +(-5.33759e7 - 1.15742e7i) q^{95} -1.29599e8i q^{97} +1.66093e6i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 1536 q^{4} - 558 q^{5} - 5422 q^{7}+O(q^{10}) $ 12 * q - 1536 * q^4 - 558 * q^5 - 5422 * q^7	$ 12 q - 1536 q^{4} - 558 q^{5} - 5422 q^{7} + 12546 q^{11} + 196608 q^{16} - 270810 q^{17} + 41512 q^{19} + 71424 q^{20} + 823956 q^{23} + 865538 q^{25} + 431616 q^{26} + 694016 q^{28} + 1194378 q^{35} - 998784 q^{38} + 7586646 q^{43} - 1605888 q^{44} + 20260530 q^{47} - 19498842 q^{49} - 14858554 q^{55} - 5506560 q^{58} - 41363266 q^{61} - 32266752 q^{62} - 25165824 q^{64} + 34663680 q^{68} + 87906498 q^{73} + 2149632 q^{74} - 5313536 q^{76} + 78817962 q^{77} - 9142272 q^{80} - 49609728 q^{82} + 55944960 q^{83} + 25440254 q^{85} - 105466368 q^{92} - 81396774 q^{95}+O(q^{100}) $ 12 * q - 1536 * q^4 - 558 * q^5 - 5422 * q^7 + 12546 * q^11 + 196608 * q^16 - 270810 * q^17 + 41512 * q^19 + 71424 * q^20 + 823956 * q^23 + 865538 * q^25 + 431616 * q^26 + 694016 * q^28 + 1194378 * q^35 - 998784 * q^38 + 7586646 * q^43 - 1605888 * q^44 + 20260530 * q^47 - 19498842 * q^49 - 14858554 * q^55 - 5506560 * q^58 - 41363266 * q^61 - 32266752 * q^62 - 25165824 * q^64 + 34663680 * q^68 + 87906498 * q^73 + 2149632 * q^74 - 5313536 * q^76 + 78817962 * q^77 - 9142272 * q^80 - 49609728 * q^82 + 55944960 * q^83 + 25440254 * q^85 - 105466368 * q^92 - 81396774 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/342\mathbb{Z}\right)^\times$.

$n$	$191$	$325$
$\chi(n)$	$1$	$-1$

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 $

$2$			11.3137i			0.707107i
$2$			11.3137i			0.707107i
$3$		0			0
$3$		0			0
$4$	−128.000			−0.500000
$5$	419.091			0.670546			0.335273	−	0.942121i	$-0.391171\pi$
$5$	419.091			0.670546			0.335273	+	0.942121i	$0.391171\pi$
$6$		0			0
$7$	−2431.38			−1.01265			−0.506326	−	0.862342i	$-0.668997\pi$
$7$	−2431.38			−1.01265			−0.506326	+	0.862342i	$0.668997\pi$
$8$		−	1448.15i		−	0.353553i
$9$		0			0
$10$			4741.48i			0.474148i
$11$	−18851.2			−1.28756			−0.643781	−	0.765209i	$-0.722635\pi$
$11$	−18851.2			−1.28756			−0.643781	+	0.765209i	$0.722635\pi$
$12$		0			0
$13$		−	52190.4i		−	1.82733i	−0.406466	−	0.913666i	$-0.633239\pi$
$13$		−	52190.4i		−	1.82733i	0.406466	−	0.913666i	$-0.366761\pi$
$14$		−	27507.9i		−	0.716054i
$15$		0			0
$16$	16384.0			0.250000
$17$	−163673.			−1.95967			−0.979833	−	0.199819i	$-0.935964\pi$
$17$	−163673.			−1.95967			−0.979833	+	0.199819i	$0.935964\pi$
$18$		0			0
$19$	−127361.	−	27617.3i	−0.977287	−	0.211918i
$19$	−127361.	−	27617.3i	−0.977287	−	0.211918i
$20$	−53643.7			−0.335273
$21$		0			0
$22$		−	213277.i		−	0.910444i
$23$	294819.			1.05352			0.526762	−	0.850013i	$-0.323406\pi$
$23$	294819.			1.05352			0.526762	+	0.850013i	$0.323406\pi$
$24$		0			0
$25$	−214988.			−0.550368
$26$	590467.			1.29212
$27$		0			0
$28$	311217.			0.506326
$29$			1.01197e6i			1.43080i	0.698717	+	0.715398i	$0.253754\pi$
$29$			1.01197e6i			1.43080i	−0.698717	+	0.715398i	$0.746246\pi$
$30$		0			0
$31$			188192.i			0.203776i	0.994796	+	0.101888i	$0.0324884\pi$
$31$			188192.i			0.203776i	−0.994796	+	0.101888i	$0.967512\pi$
$32$			185364.i			0.176777i
$33$		0			0
$34$		−	1.85175e6i		−	1.38569i
$35$	−1.01897e6			−0.679030
$36$		0			0
$37$		−	1.31226e6i		−	0.700184i	−0.936715	−	0.350092i	$-0.886150\pi$
$37$		−	1.31226e6i		−	0.700184i	0.936715	−	0.350092i	$-0.113850\pi$
$38$	312455.	−	1.44093e6i	0.149849	−	0.691047i
$39$		0			0
$40$		−	606909.i		−	0.237074i
$41$			662199.i			0.234344i	0.993112	+	0.117172i	$0.0373828\pi$
$41$			662199.i			0.234344i	−0.993112	+	0.117172i	$0.962617\pi$
$42$		0			0
$43$	−1.24898e6			−0.365328			−0.182664	−	0.983175i	$-0.558472\pi$
$43$	−1.24898e6			−0.365328			−0.182664	+	0.983175i	$0.558472\pi$
$44$	2.41295e6			0.643781
$45$		0			0
$46$			3.33550e6i			0.744954i
$47$	3.35309e6			0.687154			0.343577	−	0.939124i	$-0.388361\pi$
$47$	3.35309e6			0.687154			0.343577	+	0.939124i	$0.388361\pi$
$48$		0			0
$49$	146807.			0.0254661
$50$		−	2.43231e6i		−	0.389169i
$51$		0			0
$52$			6.68038e6i			0.913666i
$53$			6.52766e6i			0.827283i	0.910440	+	0.413641i	$0.135743\pi$
$53$			6.52766e6i			0.827283i	−0.910440	+	0.413641i	$0.864257\pi$
$54$		0			0
$55$	−7.90038e6			−0.863370
$56$			3.52101e6i			0.358027i
$57$		0			0
$58$	−1.14492e7			−1.01173
$59$			5.48176e6i			0.452389i	0.974082	+	0.226194i	$0.0726284\pi$
$59$			5.48176e6i			0.452389i	−0.974082	+	0.226194i	$0.927372\pi$
$60$		0			0
$61$	8.15096e6			0.588694			0.294347	−	0.955699i	$-0.404898\pi$
$61$	8.15096e6			0.588694			0.294347	+	0.955699i	$0.404898\pi$
$62$	−2.12914e6			−0.144092
$63$		0			0
$64$	−2.09715e6			−0.125000
$65$		−	2.18726e7i		−	1.22531i
$66$		0			0
$67$			1.27314e7i			0.631796i	0.948793	+	0.315898i	$0.102306\pi$
$67$			1.27314e7i			0.631796i	−0.948793	+	0.315898i	$0.897694\pi$
$68$	2.09502e7			0.979833
$69$		0			0
$70$		−	1.15283e7i		−	0.480147i
$71$		−	2.99568e7i		−	1.17886i	−0.807819	−	0.589430i	$-0.799352\pi$
$71$		−	2.99568e7i		−	1.17886i	0.807819	−	0.589430i	$-0.200648\pi$
$72$		0			0
$73$	2.59027e7			0.912124			0.456062	−	0.889948i	$-0.349260\pi$
$73$	2.59027e7			0.912124			0.456062	+	0.889948i	$0.349260\pi$
$74$	1.48465e7			0.495105
$75$		0			0
$76$	1.63022e7	+	3.53502e6i	0.488644	+	0.105959i
$77$	4.58344e7			1.30385
$78$		0			0
$79$		−	5.24410e7i		−	1.34636i	−0.739477	−	0.673182i	$-0.764927\pi$
$79$		−	5.24410e7i		−	1.34636i	0.739477	−	0.673182i	$-0.235073\pi$
$80$	6.86639e6			0.167636
$81$		0			0
$82$	−7.49193e6			−0.165706
$83$	3.87316e7			0.816118			0.408059	−	0.912955i	$-0.366206\pi$
$83$	3.87316e7			0.816118			0.408059	+	0.912955i	$0.366206\pi$
$84$		0			0
$85$	−6.85940e7			−1.31405
$86$		−	1.41306e7i		−	0.258326i
$87$		0			0
$88$			2.72995e7i			0.455222i
$89$			4.25848e7i			0.678726i	0.940655	+	0.339363i	$0.110212\pi$
$89$			4.25848e7i			0.678726i	−0.940655	+	0.339363i	$0.889788\pi$
$90$		0			0
$91$			1.26895e8i			1.85045i
$92$	−3.77368e7			−0.526762
$93$		0			0
$94$			3.79359e7i			0.485892i
$95$	−5.33759e7	−	1.15742e7i	−0.655316	−	0.142101i
$96$		0			0
$97$		−	1.29599e8i		−	1.46391i	−0.681352	−	0.731956i	$-0.738608\pi$
$97$		−	1.29599e8i		−	1.46391i	0.681352	−	0.731956i	$-0.261392\pi$
$98$			1.66093e6i			0.0180072i
$99$		0			0
$100$	2.75184e7			0.275184
$101$	1.59646e8			1.53417			0.767083	−	0.641548i	$-0.221708\pi$
$101$	1.59646e8			1.53417			0.767083	+	0.641548i	$0.221708\pi$
$102$		0			0
$103$		−	300515.i		−	0.00267003i	−0.999999	−	0.00133502i	$-0.999575\pi$
$103$		−	300515.i		−	0.00267003i	0.999999	−	0.00133502i	$-0.000424949\pi$
$104$	−7.55798e7			−0.646060
$105$		0			0
$106$	−7.38520e7			−0.584977
$107$			1.64311e8i			1.25352i	0.779211	+	0.626761i	$0.215620\pi$
$107$			1.64311e8i			1.25352i	−0.779211	+	0.626761i	$0.784380\pi$
$108$		0			0
$109$			6.14943e7i			0.435641i	0.975989	+	0.217821i	$0.0698948\pi$
$109$			6.14943e7i			0.435641i	−0.975989	+	0.217821i	$0.930105\pi$
$110$		−	8.93826e7i		−	0.610495i
$111$		0			0
$112$	−3.98357e7			−0.253163
$113$		−	1.30584e8i		−	0.800894i	−0.916320	−	0.400447i	$-0.868855\pi$
$113$		−	1.30584e8i		−	0.800894i	0.916320	−	0.400447i	$-0.131145\pi$
$114$		0			0
$115$	1.23556e8			0.706436
$116$		−	1.29533e8i		−	0.715398i
$117$		0			0
$118$	−6.20190e7			−0.319887
$119$	3.97952e8			1.98446
$120$		0			0
$121$	1.41009e8			0.657818
$122$			9.22176e7i			0.416269i
$123$		0			0
$124$		−	2.40885e7i		−	0.101888i
$125$	−2.53807e8			−1.03959
$126$		0			0
$127$			769315.i			0.00295726i	0.999999	+	0.00147863i	$0.000470663\pi$
$127$			769315.i			0.00295726i	−0.999999	+	0.00147863i	$0.999529\pi$
$128$		−	2.37266e7i		−	0.0883883i
$129$		0			0
$130$	2.47460e8			0.866425
$131$	1.33442e8			0.453114			0.226557	−	0.973998i	$-0.427253\pi$
$131$	1.33442e8			0.453114			0.226557	+	0.973998i	$0.427253\pi$
$132$		0			0
$133$	3.09663e8	+	6.71482e7i	0.989653	+	0.214599i
$134$	−1.44039e8			−0.446747
$135$		0			0
$136$			2.37024e8i			0.692846i
$137$	4.62601e8			1.31318			0.656590	−	0.754247i	$-0.271998\pi$
$137$	4.62601e8			1.31318			0.656590	+	0.754247i	$0.271998\pi$
$138$		0			0
$139$	2.41952e8			0.648142			0.324071	−	0.946033i	$-0.394948\pi$
$139$	2.41952e8			0.648142			0.324071	+	0.946033i	$0.394948\pi$
$140$	1.30428e8			0.339515
$141$		0			0
$142$	3.38923e8			0.833580
$143$			9.83853e8i			2.35281i
$144$		0			0
$145$			4.24110e8i			0.959414i
$146$			2.93056e8i			0.644969i
$147$		0			0
$148$			1.67969e8i			0.350092i
$149$	−6.56868e8			−1.33270			−0.666351	−	0.745639i	$-0.732145\pi$
$149$	−6.56868e8			−1.33270			−0.666351	+	0.745639i	$0.732145\pi$
$150$		0			0
$151$		−	1.30987e8i		−	0.251953i	−0.992033	−	0.125977i	$-0.959794\pi$
$151$		−	1.30987e8i		−	0.251953i	0.992033	−	0.125977i	$-0.0402064\pi$
$152$	−3.99942e7	+	1.84439e8i	−0.0749243	+	0.345523i
$153$		0			0
$154$			5.18558e8i			0.921964i
$155$			7.88694e7i			0.136641i
$156$		0			0
$157$	−4.82803e8			−0.794641			−0.397321	−	0.917680i	$-0.630060\pi$
$157$	−4.82803e8			−0.794641			−0.397321	+	0.917680i	$0.630060\pi$
$158$	5.93302e8			0.952023
$159$		0			0
$160$			7.76843e7i			0.118537i
$161$	−7.16817e8			−1.06685
$162$		0			0
$163$	8.10364e8			1.14797			0.573984	−	0.818867i	$-0.305397\pi$
$163$	8.10364e8			1.14797			0.573984	+	0.818867i	$0.305397\pi$
$164$		−	8.47615e7i		−	0.117172i
$165$		0			0
$166$			4.38198e8i			0.577083i
$167$			1.43055e9i			1.83924i	0.392812	+	0.919619i	$0.371502\pi$
$167$			1.43055e9i			1.83924i	−0.392812	+	0.919619i	$0.628498\pi$
$168$		0			0
$169$	−1.90811e9			−2.33914
$170$		−	7.76053e8i		−	0.929171i
$171$		0			0
$172$	1.59870e8			0.182664
$173$		−	6.71000e8i		−	0.749097i	−0.927207	−	0.374548i	$-0.877798\pi$
$173$		−	6.71000e8i		−	0.749097i	0.927207	−	0.374548i	$-0.122202\pi$
$174$		0			0
$175$	5.22716e8			0.557332
$176$	−3.08858e8			−0.321891
$177$		0			0
$178$	−4.81792e8			−0.479932
$179$		−	6.56448e8i		−	0.639423i	−0.947515	−	0.319711i	$-0.896414\pi$
$179$		−	6.56448e8i		−	0.639423i	0.947515	−	0.319711i	$-0.103586\pi$
$180$		0			0
$181$			5.45978e8i			0.508699i	0.967112	+	0.254349i	$0.0818613\pi$
$181$			5.45978e8i			0.508699i	−0.967112	+	0.254349i	$0.918139\pi$
$182$	−1.43565e9			−1.30847
$183$		0			0
$184$		−	4.26944e8i		−	0.372477i
$185$		−	5.49956e8i		−	0.469506i
$186$		0			0
$187$	3.08544e9			2.52319
$188$	−4.29196e8			−0.343577
$189$		0			0
$190$	1.30947e8	−	6.03880e8i	0.100480	−	0.463379i
$191$	1.08420e7			0.00814658			0.00407329	−	0.999992i	$-0.498703\pi$
$191$	1.08420e7			0.00814658			0.00407329	+	0.999992i	$0.498703\pi$
$192$		0			0
$193$			6.56281e8i			0.472999i	0.971632	+	0.236500i	$0.0760002\pi$
$193$			6.56281e8i			0.472999i	−0.971632	+	0.236500i	$0.924000\pi$
$194$	1.46625e9			1.03514
$195$		0			0
$196$	−1.87913e7			−0.0127330
$197$	−4.53130e8			−0.300855			−0.150428	−	0.988621i	$-0.548065\pi$
$197$	−4.53130e8			−0.300855			−0.150428	+	0.988621i	$0.548065\pi$
$198$		0			0
$199$	−1.59794e9			−1.01894			−0.509470	−	0.860489i	$-0.670158\pi$
$199$	−1.59794e9			−1.01894			−0.509470	+	0.860489i	$0.670158\pi$
$200$			3.11335e8i			0.194585i
$201$		0			0
$202$			1.80619e9i			1.08482i
$203$		−	2.46049e9i		−	1.44890i
$204$		0			0
$205$			2.77522e8i			0.157138i
$206$	3.39994e6			0.00188800
$207$		0			0
$208$		−	8.55088e8i		−	0.456833i
$209$	2.40091e9	+	5.20620e8i	1.25832	+	0.272858i
$210$		0			0
$211$			2.51512e9i			1.26891i	0.772961	+	0.634453i	$0.218775\pi$
$211$			2.51512e9i			1.26891i	−0.772961	+	0.634453i	$0.781225\pi$
$212$		−	8.35540e8i		−	0.413641i
$213$		0			0
$214$	−1.85897e9			−0.886374
$215$	−5.23438e8			−0.244969
$216$		0			0
$217$		−	4.57565e8i		−	0.206355i
$218$	−6.95729e8			−0.308045
$219$		0			0
$220$	1.01125e9			0.431685
$221$			8.54218e9i			3.58096i
$222$		0			0
$223$		−	1.22087e9i		−	0.493687i	−0.969055	−	0.246843i	$-0.920607\pi$
$223$		−	1.22087e9i		−	0.493687i	0.969055	−	0.246843i	$-0.0793933\pi$
$224$		−	4.50690e8i		−	0.179013i
$225$		0			0
$226$	1.47739e9			0.566318
$227$			8.23442e8i			0.310120i	0.987905	+	0.155060i	$0.0495571\pi$
$227$			8.23442e8i			0.310120i	−0.987905	+	0.155060i	$0.950443\pi$
$228$		0			0
$229$	−5.17641e9			−1.88229			−0.941145	−	0.338002i	$-0.890249\pi$
$229$	−5.17641e9			−1.88229			−0.941145	+	0.338002i	$0.890249\pi$
$230$			1.39788e9i			0.499526i
$231$		0			0
$232$	1.46550e9			0.505863
$233$	−1.44033e9			−0.488697			−0.244348	−	0.969687i	$-0.578574\pi$
$233$	−1.44033e9			−0.488697			−0.244348	+	0.969687i	$0.578574\pi$
$234$		0			0
$235$	1.40525e9			0.460769
$236$		−	7.01665e8i		−	0.226194i
$237$		0			0
$238$			4.50231e9i			1.40323i
$239$	1.19815e9			0.367215			0.183608	−	0.983000i	$-0.441222\pi$
$239$	1.19815e9			0.367215			0.183608	+	0.983000i	$0.441222\pi$
$240$		0			0
$241$			4.95635e9i			1.46924i	0.678477	+	0.734621i	$0.262640\pi$
$241$			4.95635e9i			1.46924i	−0.678477	+	0.734621i	$0.737360\pi$
$242$			1.59534e9i			0.465148i
$243$		0			0
$244$	−1.04332e9			−0.294347
$245$	6.15254e7			0.0170762
$246$		0			0
$247$	−1.44136e9	+	6.64703e9i	−0.387244	+	1.78583i
$248$	2.72531e8			0.0720458
$249$		0			0
$250$		−	2.87150e9i		−	0.735103i
$251$	4.34101e8			0.109369			0.0546847	−	0.998504i	$-0.482585\pi$
$251$	4.34101e8			0.109369			0.0546847	+	0.998504i	$0.482585\pi$
$252$		0			0
$253$	−5.55770e9			−1.35648
$254$	−8.70381e6			−0.00209110
$255$		0			0
$256$	2.68435e8			0.0625000
$257$			5.24516e9i			1.20234i	0.799122	+	0.601169i	$0.205298\pi$
$257$			5.24516e9i			1.20234i	−0.799122	+	0.601169i	$0.794702\pi$
$258$		0			0
$259$			3.19060e9i			0.709044i
$260$			2.79969e9i			0.612655i
$261$		0			0
$262$			1.50973e9i			0.320400i
$263$	2.01646e9			0.421470			0.210735	−	0.977543i	$-0.432414\pi$
$263$	2.01646e9			0.421470			0.210735	+	0.977543i	$0.432414\pi$
$264$		0			0
$265$			2.73568e9i			0.554731i
$266$	−7.59696e8	+	3.50344e9i	−0.151745	+	0.699790i
$267$		0			0
$268$		−	1.62962e9i		−	0.315898i
$269$		−	6.18818e9i		−	1.18183i	−0.806735	−	0.590913i	$-0.798768\pi$
$269$		−	6.18818e9i		−	1.18183i	0.806735	−	0.590913i	$-0.201232\pi$
$270$		0			0
$271$	2.96901e8			0.0550471			0.0275235	−	0.999621i	$-0.491238\pi$
$271$	2.96901e8			0.0550471			0.0275235	+	0.999621i	$0.491238\pi$
$272$	−2.68162e9			−0.489916
$273$		0			0
$274$			5.23373e9i			0.928559i
$275$	4.05278e9			0.708634
$276$		0			0
$277$	3.28776e9			0.558446			0.279223	−	0.960226i	$-0.409923\pi$
$277$	3.28776e9			0.558446			0.279223	+	0.960226i	$0.409923\pi$
$278$			2.73738e9i			0.458306i
$279$		0			0
$280$			1.47563e9i			0.240073i
$281$		−	3.61700e9i		−	0.580127i	−0.957007	−	0.290063i	$-0.906324\pi$
$281$		−	3.61700e9i		−	0.580127i	0.957007	−	0.290063i	$-0.0936764\pi$
$282$		0			0
$283$	1.06697e10			1.66344			0.831720	−	0.555195i	$-0.187356\pi$
$283$	1.06697e10			1.66344			0.831720	+	0.555195i	$0.187356\pi$
$284$			3.83447e9i			0.589430i
$285$		0			0
$286$	−1.11310e10			−1.66368
$287$		−	1.61006e9i		−	0.237309i
$288$		0			0
$289$	1.98132e10			2.84029
$290$	−4.79825e9			−0.678408
$291$		0			0
$292$	−3.31555e9			−0.456062
$293$		−	1.30694e9i		−	0.177331i	−0.996061	−	0.0886653i	$-0.971740\pi$
$293$		−	1.30694e9i		−	0.177331i	0.996061	−	0.0886653i	$-0.0282602\pi$
$294$		0			0
$295$			2.29736e9i			0.303347i
$296$	−1.90035e9			−0.247553
$297$		0			0
$298$		−	7.43161e9i		−	0.942362i
$299$		−	1.53867e10i		−	1.92514i
$300$		0			0
$301$	3.03675e9			0.369951
$302$	1.48195e9			0.178158
$303$		0			0
$304$	−2.08668e9	−	4.52482e8i	−0.244322	−	0.0529795i
$305$	3.41600e9			0.394746
$306$		0			0
$307$			8.88173e9i			0.999871i	0.866063	+	0.499935i	$0.166643\pi$
$307$			8.88173e9i			0.999871i	−0.866063	+	0.499935i	$0.833357\pi$
$308$	−5.86681e9			−0.651927
$309$		0			0
$310$	−8.92306e8			−0.0966200
$311$	−1.33648e9			−0.142863			−0.0714317	−	0.997445i	$-0.522757\pi$
$311$	−1.33648e9			−0.142863			−0.0714317	+	0.997445i	$0.522757\pi$
$312$		0			0
$313$	−8.88761e9			−0.925993			−0.462996	−	0.886360i	$-0.653226\pi$
$313$	−8.88761e9			−0.925993			−0.462996	+	0.886360i	$0.653226\pi$
$314$		−	5.46229e9i		−	0.561896i
$315$		0			0
$316$			6.71245e9i			0.673182i
$317$			1.20507e10i			1.19337i	0.802475	+	0.596685i	$0.203516\pi$
$317$			1.20507e10i			1.19337i	−0.802475	+	0.596685i	$0.796484\pi$
$318$		0			0
$319$		−	1.90769e10i		−	1.84224i
$320$	−8.78898e8			−0.0838182
$321$		0			0
$322$		−	8.10986e9i		−	0.754380i
$323$	2.08456e10	+	4.52022e9i	1.91516	+	0.415288i
$324$		0			0
$325$			1.12203e10i			1.00571i
$326$			9.16822e9i			0.811736i
$327$		0			0
$328$	9.58967e8			0.0828530
$329$	−8.15265e9			−0.695849
$330$		0			0
$331$		−	7.18195e9i		−	0.598316i	−0.954204	−	0.299158i	$-0.903294\pi$
$331$		−	7.18195e9i		−	0.598316i	0.954204	−	0.299158i	$-0.0967058\pi$
$332$	−4.95765e9			−0.408059
$333$		0			0
$334$	−1.61848e10			−1.30054
$335$			5.33562e9i			0.423648i
$336$		0			0
$337$		−	7.28217e9i		−	0.564601i	−0.959326	−	0.282300i	$-0.908903\pi$
$337$		−	7.28217e9i		−	0.564601i	0.959326	−	0.282300i	$-0.0910975\pi$
$338$		−	2.15878e10i		−	1.65402i
$339$		0			0
$340$	8.78003e9			0.657023
$341$		−	3.54764e9i		−	0.262375i
$342$		0			0
$343$	1.36595e10			0.986865
$344$			1.80872e9i			0.129163i
$345$		0			0
$346$	7.59150e9			0.529691
$347$	−1.25319e8			−0.00864367			−0.00432183	−	0.999991i	$-0.501376\pi$
$347$	−1.25319e8			−0.00864367			−0.00432183	+	0.999991i	$0.501376\pi$
$348$		0			0
$349$	1.83163e9			0.123463			0.0617313	−	0.998093i	$-0.480338\pi$
$349$	1.83163e9			0.123463			0.0617313	+	0.998093i	$0.480338\pi$
$350$			5.91386e9i			0.394093i
$351$		0			0
$352$		−	3.49433e9i		−	0.227611i
$353$	−6.31655e9			−0.406800			−0.203400	−	0.979096i	$-0.565199\pi$
$353$	−6.31655e9			−0.406800			−0.203400	+	0.979096i	$0.565199\pi$
$354$		0			0
$355$		−	1.25546e10i		−	0.790480i
$356$		−	5.45085e9i		−	0.339363i
$357$		0			0
$358$	7.42686e9			0.452140
$359$	−7.50786e9			−0.452000			−0.226000	−	0.974127i	$-0.572565\pi$
$359$	−7.50786e9			−0.452000			−0.226000	+	0.974127i	$0.572565\pi$
$360$		0			0
$361$	1.54581e10	+	7.03475e9i	0.910182	+	0.414209i
$362$	−6.17704e9			−0.359704
$363$		0			0
$364$		−	1.62425e10i		−	0.925227i
$365$	1.08556e10			0.611621
$366$		0			0
$367$	−3.53863e9			−0.195061			−0.0975306	−	0.995233i	$-0.531094\pi$
$367$	−3.53863e9			−0.195061			−0.0975306	+	0.995233i	$0.531094\pi$
$368$	4.83032e9			0.263381
$369$		0			0
$370$	6.22204e9			0.331991
$371$		−	1.58712e10i		−	0.837750i
$372$		0			0
$373$		−	3.19012e10i		−	1.64805i	−0.566551	−	0.824026i	$-0.691723\pi$
$373$		−	3.19012e10i		−	1.64805i	0.566551	−	0.824026i	$-0.308277\pi$
$374$			3.49077e10i			1.78417i
$375$		0			0
$376$		−	4.85580e9i		−	0.242946i
$377$	5.28154e10			2.61454
$378$		0			0
$379$			2.71860e10i			1.31762i	0.752311	+	0.658808i	$0.228939\pi$
$379$			2.71860e10i			1.31762i	−0.752311	+	0.658808i	$0.771061\pi$
$380$	6.83212e9	+	1.48150e9i	0.327658	+	0.0710503i
$381$		0			0
$382$			1.22663e8i			0.00576050i
$383$			3.88214e9i			0.180416i	0.995923	+	0.0902082i	$0.0287532\pi$
$383$			3.88214e9i			0.180416i	−0.995923	+	0.0902082i	$0.971247\pi$
$384$		0			0
$385$	1.92088e10			0.874294
$386$	−7.42497e9			−0.334461
$387$		0			0
$388$			1.65887e10i			0.731956i
$389$	−4.06187e10			−1.77390			−0.886948	−	0.461870i	$-0.847179\pi$
$389$	−4.06187e10			−1.77390			−0.886948	+	0.461870i	$0.847179\pi$
$390$		0			0
$391$	−4.82540e10			−2.06455
$392$		−	2.12599e8i		−	0.00900361i
$393$		0			0
$394$		−	5.12658e9i		−	0.212737i
$395$		−	2.19776e10i		−	0.902799i
$396$		0			0
$397$	−3.37520e10			−1.35874			−0.679372	−	0.733794i	$-0.737748\pi$
$397$	−3.37520e10			−1.35874			−0.679372	+	0.733794i	$0.737748\pi$
$398$		−	1.80786e10i		−	0.720499i
$399$		0			0
$400$	−3.52236e9			−0.137592
$401$		−	2.89932e10i		−	1.12129i	−0.828055	−	0.560646i	$-0.810553\pi$
$401$		−	2.89932e10i		−	1.12129i	0.828055	−	0.560646i	$-0.189447\pi$
$402$		0			0
$403$	9.82180e9			0.372367
$404$	−2.04347e10			−0.767083
$405$		0			0
$406$	2.78373e10			1.02453
$407$			2.47377e10i			0.901531i
$408$		0			0
$409$		−	2.62952e10i		−	0.939686i	−0.882750	−	0.469843i	$-0.844310\pi$
$409$		−	2.62952e10i		−	0.939686i	0.882750	−	0.469843i	$-0.155690\pi$
$410$	−3.13980e9			−0.111114
$411$		0			0
$412$			3.84659e7i			0.00133502i
$413$		−	1.33282e10i		−	0.458113i
$414$		0			0
$415$	1.62321e10			0.547245
$416$	9.67422e9			0.323030
$417$		0			0
$418$	−5.89015e9	+	2.71632e10i	−0.192939	+	0.889766i
$419$	3.04930e10			0.989335			0.494667	−	0.869082i	$-0.335290\pi$
$419$	3.04930e10			0.989335			0.494667	+	0.869082i	$0.335290\pi$
$420$		0			0
$421$			1.37997e9i			0.0439280i	0.999759	+	0.0219640i	$0.00699192\pi$
$421$			1.37997e9i			0.0439280i	−0.999759	+	0.0219640i	$0.993008\pi$
$422$	−2.84554e10			−0.897252
$423$		0			0
$424$	9.45306e9			0.292489
$425$	3.51877e10			1.07854
$426$		0			0
$427$	−1.98181e10			−0.596143
$428$		−	2.10318e10i		−	0.626761i
$429$		0			0
$430$		−	5.92203e9i		−	0.173219i
$431$		−	5.60782e10i		−	1.62512i	−0.582879	−	0.812559i	$-0.698074\pi$
$431$		−	5.60782e10i		−	1.62512i	0.582879	−	0.812559i	$-0.301926\pi$
$432$		0			0
$433$		−	4.24360e10i		−	1.20721i	−0.797284	−	0.603605i	$-0.793731\pi$
$433$		−	4.24360e10i		−	1.20721i	0.797284	−	0.603605i	$-0.206269\pi$
$434$	5.17676e9			0.145915
$435$		0			0
$436$		−	7.87127e9i		−	0.217821i
$437$	−3.75485e10	−	8.14212e9i	−1.02960	−	0.223260i
$438$		0			0
$439$			4.34205e9i			0.116906i	0.998290	+	0.0584531i	$0.0186168\pi$
$439$			4.34205e9i			0.116906i	−0.998290	+	0.0584531i	$0.981383\pi$
$440$			1.14410e10i			0.305247i
$441$		0			0
$442$	−9.66437e10			−2.53212
$443$	2.62509e10			0.681599			0.340799	−	0.940136i	$-0.389302\pi$
$443$	2.62509e10			0.681599			0.340799	+	0.940136i	$0.389302\pi$
$444$		0			0
$445$			1.78469e10i			0.455117i
$446$	1.38126e10			0.349089
$447$		0			0
$448$	5.09897e9			0.126582
$449$			6.19394e10i			1.52399i	0.647584	+	0.761994i	$0.275780\pi$
$449$			6.19394e10i			1.52399i	−0.647584	+	0.761994i	$0.724220\pi$
$450$		0			0
$451$		−	1.24833e10i		−	0.301732i
$452$			1.67147e10i			0.400447i
$453$		0			0
$454$	−9.31619e9			−0.219288
$455$			5.31805e10i			1.24081i
$456$		0			0
$457$	−1.13543e10			−0.260312			−0.130156	−	0.991494i	$-0.541548\pi$
$457$	−1.13543e10			−0.260312			−0.130156	+	0.991494i	$0.541548\pi$
$458$		−	5.85644e10i		−	1.33098i
$459$		0			0
$460$	−1.58152e10			−0.353218
$461$	2.30267e10			0.509833			0.254917	−	0.966963i	$-0.417952\pi$
$461$	2.30267e10			0.509833			0.254917	+	0.966963i	$0.417952\pi$
$462$		0			0
$463$	4.80160e10			1.04487			0.522435	−	0.852679i	$-0.325024\pi$
$463$	4.80160e10			1.04487			0.522435	+	0.852679i	$0.325024\pi$
$464$			1.65802e10i			0.357699i
$465$		0			0
$466$		−	1.62955e10i		−	0.345561i
$467$	4.22787e10			0.888903			0.444452	−	0.895803i	$-0.353399\pi$
$467$	4.22787e10			0.888903			0.444452	+	0.895803i	$0.353399\pi$
$468$		0			0
$469$		−	3.09549e10i		−	0.639790i
$470$			1.58986e10i			0.325813i
$471$		0			0
$472$	7.93843e9			0.159943
$473$	2.35449e10			0.470383
$474$		0			0
$475$	2.73810e10	+	5.93738e9i	0.537868	+	0.116633i
$476$	−5.09378e10			−0.992231
$477$		0			0
$478$			1.35556e10i			0.259661i
$479$	5.20093e10			0.987959			0.493980	−	0.869474i	$-0.335542\pi$
$479$	5.20093e10			0.987959			0.493980	+	0.869474i	$0.335542\pi$
$480$		0			0
$481$	−6.84873e10			−1.27947
$482$	−5.60747e10			−1.03891
$483$		0			0
$484$	−1.80492e10			−0.328909
$485$		−	5.43138e10i		−	0.981620i
$486$		0			0
$487$		−	3.08222e10i		−	0.547959i	−0.961735	−	0.273980i	$-0.911660\pi$
$487$		−	3.08222e10i		−	0.547959i	0.961735	−	0.273980i	$-0.0883401\pi$
$488$		−	1.18039e10i		−	0.208135i
$489$		0			0
$490$			6.96081e8i			0.0120747i
$491$	−8.16859e10			−1.40547			−0.702735	−	0.711452i	$-0.748038\pi$
$491$	−8.16859e10			−1.40547			−0.702735	+	0.711452i	$0.748038\pi$
$492$		0			0
$493$		−	1.65633e11i		−	2.80388i
$494$	−7.52026e10	−	1.63071e10i	−1.26277	−	0.273823i
$495$		0			0
$496$			3.08333e9i			0.0509440i
$497$			7.28364e10i			1.19378i
$498$		0			0
$499$	7.92783e10			1.27865			0.639326	−	0.768936i	$-0.279214\pi$
$499$	7.92783e10			1.27865			0.639326	+	0.768936i	$0.279214\pi$
$500$	3.24873e10			0.519797
$501$		0			0
$502$			4.91129e9i			0.0773359i
$503$	−2.38054e10			−0.371881			−0.185940	−	0.982561i	$-0.559533\pi$
$503$	−2.38054e10			−0.371881			−0.185940	+	0.982561i	$0.559533\pi$
$504$		0			0
$505$	6.69062e10			1.02873
$506$		−	6.28782e10i		−	0.959175i
$507$		0			0
$508$		−	9.84724e7i		−	0.00147863i
$509$		−	1.02167e11i		−	1.52208i	−0.648702	−	0.761042i	$-0.724688\pi$
$509$		−	1.02167e11i		−	1.52208i	0.648702	−	0.761042i	$-0.275312\pi$
$510$		0			0
$511$	−6.29793e10			−0.923665
$512$			3.03700e9i			0.0441942i
$513$		0			0
$514$	−5.93422e10			−0.850181
$515$		−	1.25943e8i		−	0.00179038i
$516$		0			0
$517$	−6.32099e10			−0.884755
$518$	−3.60975e10			−0.501370
$519$		0			0
$520$	−3.16748e10			−0.433213
$521$		−	1.32024e10i		−	0.179186i	−0.995978	−	0.0895929i	$-0.971443\pi$
$521$		−	1.32024e10i		−	0.179186i	0.995978	−	0.0895929i	$-0.0285566\pi$
$522$		0			0
$523$		−	8.84421e10i		−	1.18210i	−0.806637	−	0.591048i	$-0.798715\pi$
$523$		−	8.84421e10i		−	1.18210i	0.806637	−	0.591048i	$-0.201285\pi$
$524$	−1.70806e10			−0.226557
$525$		0			0
$526$			2.28136e10i			0.298024i
$527$		−	3.08019e10i		−	0.399333i
$528$		0			0
$529$	8.60731e9			0.109912
$530$	−3.09507e10			−0.392254
$531$		0			0
$532$	−3.96369e10	−	8.59498e9i	−0.494827	−	0.107300i
$533$	3.45605e10			0.428224
$534$		0			0
$535$			6.88614e10i			0.840544i
$536$	1.84370e10			0.223374
$537$		0			0
$538$	7.00112e10			0.835677
$539$	−2.76749e9			−0.0327892
$540$		0			0
$541$	−7.15785e10			−0.835591			−0.417796	−	0.908541i	$-0.637197\pi$
$541$	−7.15785e10			−0.835591			−0.417796	+	0.908541i	$0.637197\pi$
$542$			3.35905e9i			0.0389242i
$543$		0			0
$544$		−	3.03391e10i		−	0.346423i
$545$			2.57717e10i			0.292117i
$546$		0			0
$547$		−	4.33427e10i		−	0.484135i	−0.970259	−	0.242068i	$-0.922174\pi$
$547$		−	4.33427e10i		−	0.484135i	0.970259	−	0.242068i	$-0.0778256\pi$
$548$	−5.92129e10			−0.656590
$549$		0			0
$550$			4.58519e10i			0.501080i
$551$	2.79480e10	−	1.28886e11i	0.303211	−	1.39830i
$552$		0			0
$553$			1.27504e11i			1.36340i
$554$			3.71968e10i			0.394881i
$555$		0			0
$556$	−3.09699e10			−0.324071
$557$	2.63069e9			0.0273306			0.0136653	−	0.999907i	$-0.495650\pi$
$557$	2.63069e9			0.0273306			0.0136653	+	0.999907i	$0.495650\pi$
$558$		0			0
$559$			6.51850e10i			0.667576i
$560$	−1.66948e10			−0.169758
$561$		0			0
$562$	4.09217e10			0.410212
$563$			4.17573e10i			0.415622i	0.978169	+	0.207811i	$0.0666340\pi$
$563$			4.17573e10i			0.415622i	−0.978169	+	0.207811i	$0.933366\pi$
$564$		0			0
$565$		−	5.47265e10i		−	0.537036i
$566$			1.20714e11i			1.17623i
$567$		0			0
$568$	−4.33821e10			−0.416790
$569$			1.72406e11i			1.64476i	0.568938	+	0.822381i	$0.307355\pi$
$569$			1.72406e11i			1.64476i	−0.568938	+	0.822381i	$0.692645\pi$
$570$		0			0
$571$	7.55025e10			0.710259			0.355129	−	0.934817i	$-0.384437\pi$
$571$	7.55025e10			0.710259			0.355129	+	0.934817i	$0.384437\pi$
$572$		−	1.25933e11i		−	1.17640i
$573$		0			0
$574$	1.82157e10			0.167803
$575$	−6.33824e10			−0.579826
$576$		0			0
$577$	6.69917e10			0.604391			0.302195	−	0.953246i	$-0.402280\pi$
$577$	6.69917e10			0.604391			0.302195	+	0.953246i	$0.402280\pi$
$578$			2.24160e11i			2.00839i
$579$		0			0
$580$		−	5.42860e10i		−	0.479707i
$581$	−9.41713e10			−0.826445
$582$		0			0
$583$		−	1.23054e11i		−	1.06518i
$584$		−	3.75111e10i		−	0.322484i
$585$		0			0
$586$	1.47863e10			0.125392
$587$	−3.78872e10			−0.319110			−0.159555	−	0.987189i	$-0.551006\pi$
$587$	−3.78872e10			−0.319110			−0.159555	+	0.987189i	$0.551006\pi$
$588$		0			0
$589$	5.19735e9	−	2.39683e10i	0.0431838	−	0.199148i
$590$	−2.59916e10			−0.214499
$591$		0			0
$592$		−	2.15000e10i		−	0.175046i
$593$	−4.38042e10			−0.354239			−0.177120	−	0.984189i	$-0.556678\pi$
$593$	−4.38042e10			−0.354239			−0.177120	+	0.984189i	$0.556678\pi$
$594$		0			0
$595$	1.66778e11			1.33067
$596$	8.40790e10			0.666351
$597$		0			0
$598$	1.74081e11			1.36128
$599$		−	1.32046e11i		−	1.02569i	−0.858480	−	0.512847i	$-0.828591\pi$
$599$		−	1.32046e11i		−	1.02569i	0.858480	−	0.512847i	$-0.171409\pi$
$600$		0			0
$601$			2.59370e10i			0.198802i	0.995047	+	0.0994012i	$0.0316927\pi$
$601$			2.59370e10i			0.198802i	−0.995047	+	0.0994012i	$0.968307\pi$
$602$			3.43570e10i			0.261595i
$603$		0			0
$604$			1.67663e10i			0.125977i
$605$	5.90957e10			0.441097
$606$		0			0
$607$			1.84790e11i			1.36120i	0.732654	+	0.680601i	$0.238281\pi$
$607$			1.84790e11i			1.36120i	−0.732654	+	0.680601i	$0.761719\pi$
$608$	5.11926e9	−	2.36081e10i	0.0374621	−	0.172762i
$609$		0			0
$610$			3.86476e10i			0.279128i
$611$		−	1.74999e11i		−	1.25566i
$612$		0			0
$613$	1.13917e11			0.806767			0.403383	−	0.915031i	$-0.367834\pi$
$613$	1.13917e11			0.806767			0.403383	+	0.915031i	$0.367834\pi$
$614$	−1.00485e11			−0.707015
$615$		0			0
$616$		−	6.63754e10i		−	0.460982i
$617$	2.19273e11			1.51302			0.756511	−	0.653981i	$-0.226902\pi$
$617$	2.19273e11			1.51302			0.756511	+	0.653981i	$0.226902\pi$
$618$		0			0
$619$	1.11363e11			0.758540			0.379270	−	0.925286i	$-0.376175\pi$
$619$	1.11363e11			0.758540			0.379270	+	0.925286i	$0.376175\pi$
$620$		−	1.00953e10i		−	0.0683206i
$621$		0			0
$622$		−	1.51206e10i		−	0.101020i
$623$		−	1.03540e11i		−	0.687314i
$624$		0			0
$625$	−2.23887e10			−0.146727
$626$		−	1.00552e11i		−	0.654776i
$627$		0			0
$628$	6.17987e10			0.397321
$629$			2.14782e11i			1.37213i
$630$		0			0
$631$	−1.41858e11			−0.894819			−0.447409	−	0.894329i	$-0.647653\pi$
$631$	−1.41858e11			−0.894819			−0.447409	+	0.894329i	$0.647653\pi$
$632$	−7.59427e10			−0.476012
$633$		0			0
$634$	−1.36338e11			−0.843841
$635$			3.22413e8i			0.00198298i
$636$		0			0
$637$		−	7.66191e9i		−	0.0465350i
$638$	2.15831e11			1.30266
$639$		0			0
$640$		−	9.94360e9i		−	0.0592684i
$641$			1.83546e11i			1.08721i	0.839341	+	0.543606i	$0.182941\pi$
$641$			1.83546e11i			1.08721i	−0.839341	+	0.543606i	$0.817059\pi$
$642$		0			0
$643$	−2.98123e11			−1.74402			−0.872011	−	0.489487i	$-0.837184\pi$
$643$	−2.98123e11			−1.74402			−0.872011	+	0.489487i	$0.837184\pi$
$644$	9.17526e10			0.533427
$645$		0			0
$646$	−5.11404e10	+	2.35841e11i	−0.293653	+	1.35422i
$647$	4.95090e9			0.0282531			0.0141266	−	0.999900i	$-0.495503\pi$
$647$	4.95090e9			0.0282531			0.0141266	+	0.999900i	$0.495503\pi$
$648$		0			0
$649$		−	1.03338e11i		−	0.582479i
$650$	−1.26943e11			−0.711141
$651$		0			0
$652$	−1.03727e11			−0.573984
$653$	6.17884e10			0.339824			0.169912	−	0.985459i	$-0.445652\pi$
$653$	6.17884e10			0.339824			0.169912	+	0.985459i	$0.445652\pi$
$654$		0			0
$655$	5.59244e10			0.303834
$656$			1.08495e10i			0.0585859i
$657$		0			0
$658$		−	9.22367e10i		−	0.492040i
$659$			2.67718e11i			1.41950i	0.704453	+	0.709751i	$0.251193\pi$
$659$			2.67718e11i			1.41950i	−0.704453	+	0.709751i	$0.748807\pi$
$660$		0			0
$661$			8.84369e10i			0.463263i	0.972804	+	0.231631i	$0.0744063\pi$
$661$			8.84369e10i			0.463263i	−0.972804	+	0.231631i	$0.925594\pi$
$662$	8.12545e10			0.423073
$663$		0			0
$664$		−	5.60894e10i		−	0.288541i
$665$	1.29777e11	+	2.81412e10i	0.663608	+	0.143899i
$666$		0			0
$667$			2.98349e11i			1.50738i
$668$		−	1.83111e11i		−	0.919619i
$669$		0			0
$670$	−6.03656e10			−0.299565
$671$	−1.53655e11			−0.757980
$672$		0			0
$673$			3.47693e11i			1.69487i	0.530902	+	0.847433i	$0.321853\pi$
$673$			3.47693e11i			1.69487i	−0.530902	+	0.847433i	$0.678147\pi$
$674$	8.23884e10			0.399233
$675$		0			0
$676$	2.44238e11			1.16957
$677$			3.61885e10i			0.172273i	0.996283	+	0.0861364i	$0.0274521\pi$
$677$			3.61885e10i			0.172273i	−0.996283	+	0.0861364i	$0.972548\pi$
$678$		0			0
$679$			3.15104e11i			1.48243i
$680$			9.93347e10i			0.464585i
$681$		0			0
$682$	4.01370e10			0.185527
$683$		−	1.32775e11i		−	0.610144i	−0.952329	−	0.305072i	$-0.901320\pi$
$683$		−	1.32775e11i		−	0.610144i	0.952329	−	0.305072i	$-0.0986805\pi$
$684$		0			0
$685$	1.93872e11			0.880548
$686$			1.54539e11i			0.697819i
$687$		0			0
$688$	−2.04634e10			−0.0913320
$689$	3.40681e11			1.51172
$690$		0			0
$691$	−1.82168e11			−0.799025			−0.399512	−	0.916728i	$-0.630821\pi$
$691$	−1.82168e11			−0.799025			−0.399512	+	0.916728i	$0.630821\pi$
$692$			8.58880e10i			0.374548i
$693$		0			0
$694$		−	1.41782e9i		−	0.00611200i
$695$	1.01400e11			0.434609
$696$		0			0
$697$		−	1.08384e11i		−	0.459235i
$698$			2.07225e10i			0.0873012i
$699$		0			0
$700$	−6.69077e10			−0.278666
$701$	−1.01373e11			−0.419808			−0.209904	−	0.977722i	$-0.567315\pi$
$701$	−1.01373e11			−0.419808			−0.209904	+	0.977722i	$0.567315\pi$
$702$		0			0
$703$	−3.62411e10	+	1.67131e11i	−0.148382	+	0.684281i
$704$	3.95338e10			0.160945
$705$		0			0
$706$		−	7.14636e10i		−	0.287651i
$707$	−3.88160e11			−1.55358
$708$		0			0
$709$	4.52517e11			1.79081			0.895407	−	0.445249i	$-0.146885\pi$
$709$	4.52517e11			1.79081			0.895407	+	0.445249i	$0.146885\pi$
$710$	1.42040e11			0.558954
$711$		0			0
$712$	6.16694e10			0.239966
$713$			5.54825e10i			0.214683i
$714$		0			0
$715$			4.12324e11i			1.57766i
$716$			8.40253e10i			0.319711i
$717$		0			0
$718$		−	8.49417e10i		−	0.319612i
$719$	1.80755e11			0.676353			0.338177	−	0.941083i	$-0.390190\pi$
$719$	1.80755e11			0.676353			0.338177	+	0.941083i	$0.390190\pi$
$720$		0			0
$721$			7.30665e8i			0.00270382i
$722$	−7.95891e10	+	1.74889e11i	−0.292890	+	0.643596i
$723$		0			0
$724$		−	6.98852e10i		−	0.254349i
$725$		−	2.17562e11i		−	0.787464i
$726$		0			0
$727$	3.76607e11			1.34819			0.674094	−	0.738645i	$-0.264534\pi$
$727$	3.76607e11			1.34819			0.674094	+	0.738645i	$0.264534\pi$
$728$	1.83763e11			0.654234
$729$		0			0
$730$			1.22817e11i			0.432481i
$731$	2.04425e11			0.715921
$732$		0			0
$733$	−1.71833e11			−0.595238			−0.297619	−	0.954685i	$-0.596192\pi$
$733$	−1.71833e11			−0.595238			−0.297619	+	0.954685i	$0.596192\pi$
$734$		−	4.00350e10i		−	0.137929i
$735$		0			0
$736$			5.46488e10i			0.186238i
$737$		−	2.40002e11i		−	0.813477i
$738$		0			0
$739$	−1.46848e11			−0.492368			−0.246184	−	0.969223i	$-0.579177\pi$
$739$	−1.46848e11			−0.492368			−0.246184	+	0.969223i	$0.579177\pi$
$740$			7.03944e10i			0.234753i
$741$		0			0
$742$	1.79562e11			0.592379
$743$			4.41397e11i			1.44835i	0.689615	+	0.724177i	$0.257780\pi$
$743$			4.41397e11i			1.44835i	−0.689615	+	0.724177i	$0.742220\pi$
$744$		0			0
$745$	−2.75287e11			−0.893637
$746$	3.60920e11			1.16535
$747$		0			0
$748$	−3.94936e11			−1.26160
$749$		−	3.99503e11i		−	1.26938i
$750$		0			0
$751$		−	4.93724e11i		−	1.55212i	−0.630661	−	0.776059i	$-0.717216\pi$
$751$		−	4.93724e11i		−	1.55212i	0.630661	−	0.776059i	$-0.282784\pi$
$752$	5.49371e10			0.171789
$753$		0			0
$754$			5.97538e11i			1.84876i
$755$		−	5.48954e10i		−	0.168946i
$756$		0			0
$757$	1.16205e11			0.353868			0.176934	−	0.984223i	$-0.443382\pi$
$757$	1.16205e11			0.353868			0.176934	+	0.984223i	$0.443382\pi$
$758$	−3.07575e11			−0.931695
$759$		0			0
$760$	−1.67612e10	+	7.72966e10i	−0.0502402	+	0.231689i
$761$	2.30030e10			0.0685875			0.0342938	−	0.999412i	$-0.489082\pi$
$761$	2.30030e10			0.0685875			0.0342938	+	0.999412i	$0.489082\pi$
$762$		0			0
$763$		−	1.49516e11i		−	0.441153i
$764$	−1.38777e9			−0.00407329
$765$		0			0
$766$	−4.39214e10			−0.127574
$767$	2.86095e11			0.826664
$768$		0			0
$769$	−2.82954e11			−0.809115			−0.404557	−	0.914513i	$-0.632574\pi$
$769$	−2.82954e11			−0.809115			−0.404557	+	0.914513i	$0.632574\pi$
$770$			2.17323e11i			0.618219i
$771$		0			0
$772$		−	8.40040e10i		−	0.236500i
$773$			9.47008e10i			0.265238i	0.991167	+	0.132619i	$0.0423387\pi$
$773$			9.47008e10i			0.265238i	−0.991167	+	0.132619i	$0.957661\pi$
$774$		0			0
$775$		−	4.04588e10i		−	0.112152i
$776$	−1.87679e11			−0.517571
$777$		0			0
$778$		−	4.59549e11i		−	1.25433i
$779$	1.82882e10	−	8.43384e10i	0.0496616	−	0.229021i
$780$		0			0
$781$			5.64722e11i			1.51786i
$782$		−	5.45932e11i		−	1.45986i
$783$		0			0
$784$	2.40528e9			0.00636651
$785$	−2.02338e11			−0.532843
$786$		0			0
$787$		−	5.15172e11i		−	1.34293i	−0.741037	−	0.671464i	$-0.765666\pi$
$787$		−	5.15172e11i		−	1.34293i	0.741037	−	0.671464i	$-0.234334\pi$
$788$	5.80006e10			0.150428
$789$		0			0
$790$	2.48648e11			0.638375
$791$			3.17499e11i			0.811028i
$792$		0			0
$793$		−	4.25402e11i		−	1.07574i
$794$		−	3.81860e11i		−	0.960777i
$795$		0			0
$796$	2.04536e11			0.509470
$797$			1.73952e11i			0.431119i	0.976491	+	0.215559i	$0.0691575\pi$
$797$			1.73952e11i			0.431119i	−0.976491	+	0.215559i	$0.930843\pi$
$798$		0			0
$799$	−5.48812e11			−1.34659
$800$		−	3.98509e10i		−	0.0972923i
$801$		0			0
$802$	3.28021e11			0.792874
$803$	−4.88297e11			−1.17442
$804$		0			0
$805$	−3.00412e11			−0.715375
$806$			1.11121e11i			0.263303i
$807$		0			0
$808$		−	2.31192e11i		−	0.542409i
$809$	−5.80910e11			−1.35617			−0.678086	−	0.734983i	$-0.737190\pi$
$809$	−5.80910e11			−1.35617			−0.678086	+	0.734983i	$0.737190\pi$
$810$		0			0
$811$			6.95097e11i			1.60680i	0.595439	+	0.803401i	$0.296978\pi$
$811$			6.95097e11i			1.60680i	−0.595439	+	0.803401i	$0.703022\pi$
$812$			3.14943e11i			0.724450i
$813$		0			0
$814$	−2.79875e11			−0.637479
$815$	3.39616e11			0.769765
$816$		0			0
$817$	1.59072e11	+	3.44936e10i	0.357031	+	0.0774195i
$818$	2.97496e11			0.664459
$819$		0			0
$820$		−	3.55228e10i		−	0.0785691i
$821$	−1.93809e11			−0.426581			−0.213290	−	0.976989i	$-0.568418\pi$
$821$	−1.93809e11			−0.426581			−0.213290	+	0.976989i	$0.568418\pi$
$822$		0			0
$823$	3.76429e10			0.0820511			0.0410255	−	0.999158i	$-0.486938\pi$
$823$	3.76429e10			0.0820511			0.0410255	+	0.999158i	$0.486938\pi$
$824$	−4.35192e8			−0.000944000
$825$		0			0
$826$	1.50792e11			0.323935
$827$			1.92643e11i			0.411843i	0.978569	+	0.205921i	$0.0660191\pi$
$827$			1.92643e11i			0.411843i	−0.978569	+	0.205921i	$0.933981\pi$
$828$		0			0
$829$		−	1.61600e11i		−	0.342156i	−0.985258	−	0.171078i	$-0.945275\pi$
$829$		−	1.61600e11i		−	0.342156i	0.985258	−	0.171078i	$-0.0547250\pi$
$830$			1.83645e11i			0.386961i
$831$		0			0
$832$			1.09451e11i			0.228417i
$833$	−2.40283e10			−0.0499050
$834$		0			0
$835$			5.99532e11i			1.23329i
$836$	−3.07317e11	−	6.66394e10i	−0.629160	−	0.136429i
$837$		0			0
$838$			3.44988e11i			0.699565i
$839$		−	4.81679e11i		−	0.972097i	−0.873932	−	0.486049i	$-0.838438\pi$
$839$		−	4.81679e11i		−	0.972097i	0.873932	−	0.486049i	$-0.161562\pi$
$840$		0			0
$841$	−5.23846e11			−1.04718
$842$	−1.56126e10			−0.0310618
$843$		0			0
$844$		−	3.21936e11i		−	0.634453i
$845$	−7.99673e11			−1.56850
$846$		0			0
$847$	−3.42847e11			−0.666142
$848$			1.06949e11i			0.206821i
$849$		0			0
$850$			3.98103e11i			0.762641i
$851$		−	3.86879e11i		−	0.737661i
$852$		0			0
$853$	6.06370e11			1.14536			0.572679	−	0.819780i	$-0.305904\pi$
$853$	6.06370e11			1.14536			0.572679	+	0.819780i	$0.305904\pi$
$854$		−	2.24216e11i		−	0.421536i
$855$		0			0
$856$	2.37948e11			0.443187
$857$			5.32627e11i			0.987416i	0.869628	+	0.493708i	$0.164359\pi$
$857$			5.32627e11i			0.987416i	−0.869628	+	0.493708i	$0.835641\pi$
$858$		0			0
$859$	1.13776e11			0.208966			0.104483	−	0.994527i	$-0.466681\pi$
$859$	1.13776e11			0.208966			0.104483	+	0.994527i	$0.466681\pi$
$860$	6.70001e10			0.122485
$861$		0			0
$862$	6.34452e11			1.14913
$863$			5.64549e10i			0.101779i	0.998704	+	0.0508896i	$0.0162057\pi$
$863$			5.64549e10i			0.101779i	−0.998704	+	0.0508896i	$0.983794\pi$
$864$		0			0
$865$		−	2.81210e11i		−	0.502304i
$866$	4.80108e11			0.853626
$867$		0			0
$868$			5.85683e10i			0.103177i
$869$			9.88576e11i			1.73353i
$870$		0			0
$871$	6.64457e11			1.15450
$872$	8.90533e10			0.154022
$873$		0			0
$874$	9.21176e10	−	4.24813e11i	0.157869	−	0.728034i
$875$	6.17101e11			1.05275
$876$		0			0
$877$			5.39767e11i			0.912447i	0.889865	+	0.456223i	$0.150798\pi$
$877$			5.39767e11i			0.912447i	−0.889865	+	0.456223i	$0.849202\pi$
$878$	−4.91247e10			−0.0826651
$879$		0			0
$880$	−1.29440e11			−0.215843
$881$	−1.13071e12			−1.87692			−0.938461	−	0.345385i	$-0.887748\pi$
$881$	−1.13071e12			−1.87692			−0.938461	+	0.345385i	$0.887748\pi$
$882$		0			0
$883$	7.01399e11			1.15378			0.576889	−	0.816822i	$-0.304266\pi$
$883$	7.01399e11			1.15378			0.576889	+	0.816822i	$0.304266\pi$
$884$		−	1.09340e12i		−	1.79048i
$885$		0			0
$886$			2.96995e11i			0.481963i
$887$			9.54811e11i			1.54249i	0.636537	+	0.771246i	$0.280366\pi$
$887$			9.54811e11i			1.54249i	−0.636537	+	0.771246i	$0.719634\pi$
$888$		0			0
$889$		−	1.87050e9i		−	0.00299468i
$890$	−2.01915e11			−0.321816
$891$		0			0
$892$			1.56272e11i			0.246843i
$893$	−4.27054e11	−	9.26036e10i	−0.671547	−	0.145620i
$894$		0			0
$895$		−	2.75111e11i		−	0.428762i
$896$			5.76883e10i			0.0895067i
$897$		0			0
$898$	−7.00764e11			−1.07762
$899$	−1.90445e11			−0.291562
$900$		0			0
$901$		−	1.06840e12i		−	1.62120i
$902$	1.41232e11			0.213357
$903$		0			0
$904$	−1.89105e11			−0.283159
$905$			2.28815e11i			0.341106i
$906$		0			0
$907$		−	6.62886e10i		−	0.0979511i	−0.998800	−	0.0489756i	$-0.984404\pi$
$907$		−	6.62886e10i		−	0.0979511i	0.998800	−	0.0489756i	$-0.0155956\pi$
$908$		−	1.05401e11i		−	0.155060i
$909$		0			0
$910$	−6.01669e11			−0.877388
$911$		−	4.30099e11i		−	0.624446i	−0.950009	−	0.312223i	$-0.898926\pi$
$911$		−	4.30099e11i		−	0.624446i	0.950009	−	0.312223i	$-0.101074\pi$
$912$		0			0
$913$	−7.30138e11			−1.05080
$914$		−	1.28459e11i		−	0.184069i
$915$		0			0
$916$	6.62580e11			0.941145
$917$	−3.24449e11			−0.458848
$918$		0			0
$919$	6.09038e11			0.853852			0.426926	−	0.904287i	$-0.359597\pi$
$919$	6.09038e11			0.853852			0.426926	+	0.904287i	$0.359597\pi$
$920$		−	1.78928e11i		−	0.249763i
$921$		0			0
$922$			2.60517e11i			0.360506i
$923$	−1.56346e12			−2.15417
$924$		0			0
$925$			2.82119e11i			0.385359i
$926$			5.43240e11i			0.738835i
$927$		0			0
$928$	−1.87583e11			−0.252931
$929$	−3.42134e11			−0.459338			−0.229669	−	0.973269i	$-0.573764\pi$
$929$	−3.42134e11			−0.459338			−0.229669	+	0.973269i	$0.573764\pi$
$930$		0			0
$931$	−1.86975e10	−	4.05441e9i	−0.0248877	−	0.00539671i
$932$	1.84363e11			0.244348
$933$		0			0
$934$			4.78329e11i			0.628550i
$935$	1.29308e12			1.69192
$936$		0			0
$937$	−6.89868e11			−0.894968			−0.447484	−	0.894292i	$-0.647680\pi$
$937$	−6.89868e11			−0.894968			−0.447484	+	0.894292i	$0.647680\pi$
$938$	3.50214e11			0.452400
$939$		0			0
$940$	−1.79872e11			−0.230384
$941$			4.74893e11i			0.605672i	0.953043	+	0.302836i	$0.0979334\pi$
$941$			4.74893e11i			0.605672i	−0.953043	+	0.302836i	$0.902067\pi$
$942$		0			0
$943$			1.95229e11i			0.246887i
$944$			8.98131e10i			0.113097i
$945$		0			0
$946$			2.66380e11i			0.332611i
$947$	1.27186e12			1.58140			0.790699	−	0.612205i	$-0.209717\pi$
$947$	1.27186e12			1.58140			0.790699	+	0.612205i	$0.209717\pi$
$948$		0			0
$949$		−	1.35187e12i		−	1.66675i
$950$	−6.71738e10	+	3.09781e11i	−0.0824718	+	0.380330i
$951$		0			0
$952$		−	5.76296e11i		−	0.701613i
$953$		−	4.65803e10i		−	0.0564716i	−0.999601	−	0.0282358i	$-0.991011\pi$
$953$		−	4.65803e10i		−	0.0564716i	0.999601	−	0.0282358i	$-0.00898893\pi$
$954$		0			0
$955$	4.54378e9			0.00546266
$956$	−1.53364e11			−0.183608
$957$		0			0
$958$			5.88418e11i			0.698593i
$959$	−1.12476e12			−1.32980
$960$		0			0
$961$	8.17475e11			0.958475
$962$		−	7.74846e11i		−	0.904722i
$963$		0			0
$964$		−	6.34413e11i		−	0.734621i
$965$			2.75042e11i			0.317168i
$966$		0			0
$967$	−1.15462e12			−1.32049			−0.660245	−	0.751050i	$-0.729547\pi$
$967$	−1.15462e12			−1.32049			−0.660245	+	0.751050i	$0.729547\pi$
$968$		−	2.04203e11i		−	0.232574i
$969$		0			0
$970$	6.14491e11			0.694110
$971$		−	6.13870e10i		−	0.0690557i	−0.999404	−	0.0345279i	$-0.989007\pi$
$971$		−	6.13870e10i		−	0.0690557i	0.999404	−	0.0345279i	$-0.0109927\pi$
$972$		0			0
$973$	−5.88277e11			−0.656343
$974$	3.48714e11			0.387466
$975$		0			0
$976$	1.33545e11			0.147173
$977$		−	6.15166e11i		−	0.675171i	−0.941295	−	0.337586i	$-0.890390\pi$
$977$		−	6.15166e11i		−	0.675171i	0.941295	−	0.337586i	$-0.109610\pi$
$978$		0			0
$979$		−	8.02775e11i		−	0.873903i
$980$	−7.87525e9			−0.00853808
$981$		0			0
$982$		−	9.24171e11i		−	0.993817i
$983$		−	3.06094e11i		−	0.327824i	−0.986475	−	0.163912i	$-0.947589\pi$
$983$		−	3.06094e11i		−	0.327824i	0.986475	−	0.163912i	$-0.0524114\pi$
$984$		0			0
$985$	−1.89903e11			−0.201737
$986$	1.87392e12			1.98264
$987$		0			0
$988$	1.84494e11	−	8.50820e11i	0.193622	−	0.892914i
$989$	−3.68224e11			−0.384882
$990$		0			0
$991$		−	1.16697e12i		−	1.20994i	−0.796247	−	0.604972i	$-0.793184\pi$
$991$		−	1.16697e12i		−	1.20994i	0.796247	−	0.604972i	$-0.206816\pi$
$992$	−3.48839e10			−0.0360229
$993$		0			0
$994$	−8.24050e11			−0.844127
$995$	−6.69683e11			−0.683246
$996$		0			0
$997$	−6.19793e11			−0.627287			−0.313643	−	0.949541i	$-0.601550\pi$
$997$	−6.19793e11			−0.627287			−0.313643	+	0.949541i	$0.601550\pi$
$998$			8.96931e11i			0.904143i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.9.d.a.37.11		12
3.2	odd	2		38.9.b.a.37.6	✓	12
12.11	even	2		304.9.e.e.113.2		12
19.18	odd	2	inner	342.9.d.a.37.5		12
57.56	even	2		38.9.b.a.37.7	yes	12
228.227	odd	2		304.9.e.e.113.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.9.b.a.37.6	✓	12	3.2	odd	2
38.9.b.a.37.7	yes	12	57.56	even	2
304.9.e.e.113.2		12	12.11	even	2
304.9.e.e.113.11		12	228.227	odd	2
342.9.d.a.37.5		12	19.18	odd	2	inner
342.9.d.a.37.11		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	342.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+11.3137i q^{2} -128.000 q^{4} +419.091 q^{5} -2431.38 q^{7} -1448.15i q^{8} +O(q^{10})\)	\(q+11.3137i q^{2} -128.000 q^{4} +419.091 q^{5} -2431.38 q^{7} -1448.15i q^{8} +4741.48i q^{10} -18851.2 q^{11} -52190.4i q^{13} -27507.9i q^{14} +16384.0 q^{16} -163673. q^{17} +(-127361. - 27617.3i) q^{19} -53643.7 q^{20} -213277. i q^{22} +294819. q^{23} -214988. q^{25} +590467. q^{26} +311217. q^{28} +1.01197e6i q^{29} +188192. i q^{31} +185364. i q^{32} -1.85175e6i q^{34} -1.01897e6 q^{35} -1.31226e6i q^{37} +(312455. - 1.44093e6i) q^{38} -606909. i q^{40} +662199. i q^{41} -1.24898e6 q^{43} +2.41295e6 q^{44} +3.33550e6i q^{46} +3.35309e6 q^{47} +146807. q^{49} -2.43231e6i q^{50} +6.68038e6i q^{52} +6.52766e6i q^{53} -7.90038e6 q^{55} +3.52101e6i q^{56} -1.14492e7 q^{58} +5.48176e6i q^{59} +8.15096e6 q^{61} -2.12914e6 q^{62} -2.09715e6 q^{64} -2.18726e7i q^{65} +1.27314e7i q^{67} +2.09502e7 q^{68} -1.15283e7i q^{70} -2.99568e7i q^{71} +2.59027e7 q^{73} +1.48465e7 q^{74} +(1.63022e7 + 3.53502e6i) q^{76} +4.58344e7 q^{77} -5.24410e7i q^{79} +6.86639e6 q^{80} -7.49193e6 q^{82} +3.87316e7 q^{83} -6.85940e7 q^{85} -1.41306e7i q^{86} +2.72995e7i q^{88} +4.25848e7i q^{89} +1.26895e8i q^{91} -3.77368e7 q^{92} +3.79359e7i q^{94} +(-5.33759e7 - 1.15742e7i) q^{95} -1.29599e8i q^{97} +1.66093e6i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 1536 q^{4} - 558 q^{5} - 5422 q^{7}+O(q^{10}) \) 12 * q - 1536 * q^4 - 558 * q^5 - 5422 * q^7	\( 12 q - 1536 q^{4} - 558 q^{5} - 5422 q^{7} + 12546 q^{11} + 196608 q^{16} - 270810 q^{17} + 41512 q^{19} + 71424 q^{20} + 823956 q^{23} + 865538 q^{25} + 431616 q^{26} + 694016 q^{28} + 1194378 q^{35} - 998784 q^{38} + 7586646 q^{43} - 1605888 q^{44} + 20260530 q^{47} - 19498842 q^{49} - 14858554 q^{55} - 5506560 q^{58} - 41363266 q^{61} - 32266752 q^{62} - 25165824 q^{64} + 34663680 q^{68} + 87906498 q^{73} + 2149632 q^{74} - 5313536 q^{76} + 78817962 q^{77} - 9142272 q^{80} - 49609728 q^{82} + 55944960 q^{83} + 25440254 q^{85} - 105466368 q^{92} - 81396774 q^{95}+O(q^{100}) \) 12 * q - 1536 * q^4 - 558 * q^5 - 5422 * q^7 + 12546 * q^11 + 196608 * q^16 - 270810 * q^17 + 41512 * q^19 + 71424 * q^20 + 823956 * q^23 + 865538 * q^25 + 431616 * q^26 + 694016 * q^28 + 1194378 * q^35 - 998784 * q^38 + 7586646 * q^43 - 1605888 * q^44 + 20260530 * q^47 - 19498842 * q^49 - 14858554 * q^55 - 5506560 * q^58 - 41363266 * q^61 - 32266752 * q^62 - 25165824 * q^64 + 34663680 * q^68 + 87906498 * q^73 + 2149632 * q^74 - 5313536 * q^76 + 78817962 * q^77 - 9142272 * q^80 - 49609728 * q^82 + 55944960 * q^83 + 25440254 * q^85 - 105466368 * q^92 - 81396774 * q^95

Introduction

L-functions

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