LMFDB - Embedded newform 304.9.e.e.113.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,9,Mod(113,304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(304, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("304.113");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 304 = 2^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	304.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$123.843097459$
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:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{25}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 38)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			113.2
Root			$-111.533i$ of defining polynomial
Character	$\chi$	$=$	304.113
Dual form			304.9.e.e.113.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-124.261i q^{3} -419.091 q^{5} +2431.38 q^{7} -8879.72 q^{9} +O(q^{10})$	$q-124.261i q^{3} -419.091 q^{5} +2431.38 q^{7} -8879.72 q^{9} -18851.2 q^{11} -52190.4i q^{13} +52076.6i q^{15} +163673. q^{17} +(127361. + 27617.3i) q^{19} -302125. i q^{21} +294819. q^{23} -214988. q^{25} +288125. i q^{27} -1.01197e6i q^{29} -188192. i q^{31} +2.34246e6i q^{33} -1.01897e6 q^{35} -1.31226e6i q^{37} -6.48522e6 q^{39} -662199. i q^{41} +1.24898e6 q^{43} +3.72141e6 q^{45} +3.35309e6 q^{47} +146807. q^{49} -2.03381e7i q^{51} -6.52766e6i q^{53} +7.90038e6 q^{55} +(3.43175e6 - 1.58260e7i) q^{57} +5.48176e6i q^{59} +8.15096e6 q^{61} -2.15900e7 q^{63} +2.18726e7i q^{65} -1.27314e7i q^{67} -3.66344e7i q^{69} -2.99568e7i q^{71} +2.59027e7 q^{73} +2.67145e7i q^{75} -4.58344e7 q^{77} +5.24410e7i q^{79} -2.24572e7 q^{81} +3.87316e7 q^{83} -6.85940e7 q^{85} -1.25749e8 q^{87} -4.25848e7i q^{89} -1.26895e8i q^{91} -2.33848e7 q^{93} +(-5.33759e7 - 1.15742e7i) q^{95} -1.29599e8i q^{97} +1.67393e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 558 q^{5} + 5422 q^{7} - 15592 q^{9}+O(q^{10}) $ 12 * q + 558 * q^5 + 5422 * q^7 - 15592 * q^9	$ 12 q + 558 q^{5} + 5422 q^{7} - 15592 q^{9} + 12546 q^{11} + 270810 q^{17} - 41512 q^{19} + 823956 q^{23} + 865538 q^{25} + 1194378 q^{35} - 5786100 q^{39} - 7586646 q^{43} + 2226046 q^{45} + 20260530 q^{47} - 19498842 q^{49} + 14858554 q^{55} + 14430564 q^{57} - 41363266 q^{61} - 84235798 q^{63} + 87906498 q^{73} - 78817962 q^{77} - 100904812 q^{81} + 55944960 q^{83} + 25440254 q^{85} - 119189604 q^{87} + 105500856 q^{93} - 81396774 q^{95} + 85554938 q^{99}+O(q^{100}) $ 12 * q + 558 * q^5 + 5422 * q^7 - 15592 * q^9 + 12546 * q^11 + 270810 * q^17 - 41512 * q^19 + 823956 * q^23 + 865538 * q^25 + 1194378 * q^35 - 5786100 * q^39 - 7586646 * q^43 + 2226046 * q^45 + 20260530 * q^47 - 19498842 * q^49 + 14858554 * q^55 + 14430564 * q^57 - 41363266 * q^61 - 84235798 * q^63 + 87906498 * q^73 - 78817962 * q^77 - 100904812 * q^81 + 55944960 * q^83 + 25440254 * q^85 - 119189604 * q^87 + 105500856 * q^93 - 81396774 * q^95 + 85554938 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/304\mathbb{Z}\right)^\times$.

$n$	$97$	$191$	$229$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		−	124.261i		−	1.53408i	−0.641598	−	0.767041i	$-0.721728\pi$
$3$		−	124.261i		−	1.53408i	0.641598	−	0.767041i	$-0.278272\pi$
$4$		0			0
$5$	−419.091			−0.670546			−0.335273	−	0.942121i	$-0.608829\pi$
$5$	−419.091			−0.670546			−0.335273	+	0.942121i	$0.608829\pi$
$6$		0			0
$7$	2431.38			1.01265			0.506326	−	0.862342i	$-0.331003\pi$
$7$	2431.38			1.01265			0.506326	+	0.862342i	$0.331003\pi$
$8$		0			0
$9$	−8879.72			−1.35341
$10$		0			0
$11$	−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	$-0.366761\pi$
$14$		0			0
$15$			52076.6i			1.02867i
$16$		0			0
$17$	163673.			1.95967			0.979833	−	0.199819i	$-0.0640356\pi$
$17$	163673.			1.95967			0.979833	+	0.199819i	$0.0640356\pi$
$18$		0			0
$19$	127361.	+	27617.3i	0.977287	+	0.211918i
$19$	127361.	+	27617.3i	0.977287	+	0.211918i
$20$		0			0
$21$		−	302125.i		−	1.55349i
$22$		0			0
$23$	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
$26$		0			0
$27$			288125.i			0.542159i
$28$		0			0
$29$		−	1.01197e6i		−	1.43080i	−0.698717	−	0.715398i	$-0.746246\pi$
$29$		−	1.01197e6i		−	1.43080i	0.698717	−	0.715398i	$-0.253754\pi$
$30$		0			0
$31$		−	188192.i		−	0.203776i	−0.994796	−	0.101888i	$-0.967512\pi$
$31$		−	188192.i		−	0.203776i	0.994796	−	0.101888i	$-0.0324884\pi$
$32$		0			0
$33$			2.34246e6i			1.97523i
$34$		0			0
$35$	−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	$-0.113850\pi$
$38$		0			0
$39$	−6.48522e6			−2.80328
$40$		0			0
$41$		−	662199.i		−	0.234344i	−0.993112	−	0.117172i	$-0.962617\pi$
$41$		−	662199.i		−	0.234344i	0.993112	−	0.117172i	$-0.0373828\pi$
$42$		0			0
$43$	1.24898e6			0.365328			0.182664	−	0.983175i	$-0.441528\pi$
$43$	1.24898e6			0.365328			0.182664	+	0.983175i	$0.441528\pi$
$44$		0			0
$45$	3.72141e6			0.907523
$46$		0			0
$47$	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
$50$		0			0
$51$		−	2.03381e7i		−	3.00629i
$52$		0			0
$53$		−	6.52766e6i		−	0.827283i	−0.910440	−	0.413641i	$-0.864257\pi$
$53$		−	6.52766e6i		−	0.827283i	0.910440	−	0.413641i	$-0.135743\pi$
$54$		0			0
$55$	7.90038e6			0.863370
$56$		0			0
$57$	3.43175e6	−	1.58260e7i	0.325099	−	1.49924i
$58$		0			0
$59$			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	$0.404898\pi$
$62$		0			0
$63$	−2.15900e7			−1.37053
$64$		0			0
$65$			2.18726e7i			1.22531i
$66$		0			0
$67$		−	1.27314e7i		−	0.631796i	−0.948793	−	0.315898i	$-0.897694\pi$
$67$		−	1.27314e7i		−	0.631796i	0.948793	−	0.315898i	$-0.102306\pi$
$68$		0			0
$69$		−	3.66344e7i		−	1.61619i
$70$		0			0
$71$		−	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	$0.349260\pi$
$74$		0			0
$75$			2.67145e7i			0.844310i
$76$		0			0
$77$	−4.58344e7			−1.30385
$78$		0			0
$79$			5.24410e7i			1.34636i	0.739477	+	0.673182i	$0.235073\pi$
$79$			5.24410e7i			1.34636i	−0.739477	+	0.673182i	$0.764927\pi$
$80$		0			0
$81$	−2.24572e7			−0.521693
$82$		0			0
$83$	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
$86$		0			0
$87$	−1.25749e8			−2.19496
$88$		0			0
$89$		−	4.25848e7i		−	0.678726i	−0.940655	−	0.339363i	$-0.889788\pi$
$89$		−	4.25848e7i		−	0.678726i	0.940655	−	0.339363i	$-0.110212\pi$
$90$		0			0
$91$		−	1.26895e8i		−	1.85045i
$92$		0			0
$93$	−2.33848e7			−0.312609
$94$		0			0
$95$	−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	$-0.261392\pi$
$98$		0			0
$99$	1.67393e8			1.74260
$100$		0			0
$101$	−1.59646e8			−1.53417			−0.767083	−	0.641548i	$-0.778292\pi$
$101$	−1.59646e8			−1.53417			−0.767083	+	0.641548i	$0.778292\pi$
$102$		0			0
$103$			300515.i			0.00267003i	0.999999	+	0.00133502i	$0.000424949\pi$
$103$			300515.i			0.00267003i	−0.999999	+	0.00133502i	$0.999575\pi$
$104$		0			0
$105$			1.26618e8i			1.04169i
$106$		0			0
$107$			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	$0.930105\pi$
$110$		0			0
$111$	−1.63062e8			−1.07414
$112$		0			0
$113$			1.30584e8i			0.800894i	0.916320	+	0.400447i	$0.131145\pi$
$113$			1.30584e8i			0.800894i	−0.916320	+	0.400447i	$0.868855\pi$
$114$		0			0
$115$	−1.23556e8			−0.706436
$116$		0			0
$117$			4.63436e8i			2.47313i
$118$		0			0
$119$	3.97952e8			1.98446
$120$		0			0
$121$	1.41009e8			0.657818
$122$		0			0
$123$	−8.22854e7			−0.359503
$124$		0			0
$125$	2.53807e8			1.03959
$126$		0			0
$127$		−	769315.i		−	0.00295726i	−0.999999	−	0.00147863i	$-0.999529\pi$
$127$		−	769315.i		−	0.00295726i	0.999999	−	0.00147863i	$-0.000470663\pi$
$128$		0			0
$129$		−	1.55200e8i		−	0.560443i
$130$		0			0
$131$	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
$134$		0			0
$135$		−	1.20751e8i		−	0.363542i
$136$		0			0
$137$	−4.62601e8			−1.31318			−0.656590	−	0.754247i	$-0.728002\pi$
$137$	−4.62601e8			−1.31318			−0.656590	+	0.754247i	$0.728002\pi$
$138$		0			0
$139$	−2.41952e8			−0.648142			−0.324071	−	0.946033i	$-0.605052\pi$
$139$	−2.41952e8			−0.648142			−0.324071	+	0.946033i	$0.605052\pi$
$140$		0			0
$141$		−	4.16658e8i		−	1.05415i
$142$		0			0
$143$			9.83853e8i			2.35281i
$144$		0			0
$145$			4.24110e8i			0.959414i
$146$		0			0
$147$		−	1.82423e7i		−	0.0390670i
$148$		0			0
$149$	6.56868e8			1.33270			0.666351	−	0.745639i	$-0.267855\pi$
$149$	6.56868e8			1.33270			0.666351	+	0.745639i	$0.267855\pi$
$150$		0			0
$151$			1.30987e8i			0.251953i	0.992033	+	0.125977i	$0.0402064\pi$
$151$			1.30987e8i			0.251953i	−0.992033	+	0.125977i	$0.959794\pi$
$152$		0			0
$153$	−1.45337e9			−2.65223
$154$		0			0
$155$			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	$0.630060\pi$
$158$		0			0
$159$	−8.11131e8			−1.26912
$160$		0			0
$161$	7.16817e8			1.06685
$162$		0			0
$163$	−8.10364e8			−1.14797			−0.573984	−	0.818867i	$-0.694603\pi$
$163$	−8.10364e8			−1.14797			−0.573984	+	0.818867i	$0.694603\pi$
$164$		0			0
$165$		−	9.81706e8i		−	1.32448i
$166$		0			0
$167$			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
$170$		0			0
$171$	−1.13093e9	−	2.45234e8i	−1.32267	−	0.286811i
$172$		0			0
$173$			6.71000e8i			0.749097i	0.927207	+	0.374548i	$0.122202\pi$
$173$			6.71000e8i			0.749097i	−0.927207	+	0.374548i	$0.877798\pi$
$174$		0			0
$175$	−5.22716e8			−0.557332
$176$		0			0
$177$	6.81167e8			0.694001
$178$		0			0
$179$		−	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	$0.918139\pi$
$182$		0			0
$183$		−	1.01284e9i		−	0.903105i
$184$		0			0
$185$			5.49956e8i			0.469506i
$186$		0			0
$187$	−3.08544e9			−2.52319
$188$		0			0
$189$			7.00542e8i			0.549019i
$190$		0			0
$191$	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	$0.924000\pi$
$194$		0			0
$195$	2.71790e9			1.87973
$196$		0			0
$197$	4.53130e8			0.300855			0.150428	−	0.988621i	$-0.451935\pi$
$197$	4.53130e8			0.300855			0.150428	+	0.988621i	$0.451935\pi$
$198$		0			0
$199$	1.59794e9			1.01894			0.509470	−	0.860489i	$-0.329842\pi$
$199$	1.59794e9			1.01894			0.509470	+	0.860489i	$0.329842\pi$
$200$		0			0
$201$	−1.58201e9			−0.969227
$202$		0			0
$203$		−	2.46049e9i		−	1.44890i
$204$		0			0
$205$			2.77522e8i			0.157138i
$206$		0			0
$207$	−2.61791e9			−1.42585
$208$		0			0
$209$	−2.40091e9	−	5.20620e8i	−1.25832	−	0.272858i
$210$		0			0
$211$		−	2.51512e9i		−	1.26891i	−0.772961	−	0.634453i	$-0.781225\pi$
$211$		−	2.51512e9i		−	1.26891i	0.772961	−	0.634453i	$-0.218775\pi$
$212$		0			0
$213$	−3.72245e9			−1.80847
$214$		0			0
$215$	−5.23438e8			−0.244969
$216$		0			0
$217$		−	4.57565e8i		−	0.206355i
$218$		0			0
$219$		−	3.21869e9i		−	1.39927i
$220$		0			0
$221$		−	8.54218e9i		−	3.58096i
$222$		0			0
$223$			1.22087e9i			0.493687i	0.969055	+	0.246843i	$0.0793933\pi$
$223$			1.22087e9i			0.493687i	−0.969055	+	0.246843i	$0.920607\pi$
$224$		0			0
$225$	1.90903e9			0.744873
$226$		0			0
$227$			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	$0.890249\pi$
$230$		0			0
$231$			5.69542e9i			2.00022i
$232$		0			0
$233$	1.44033e9			0.488697			0.244348	−	0.969687i	$-0.421426\pi$
$233$	1.44033e9			0.488697			0.244348	+	0.969687i	$0.421426\pi$
$234$		0			0
$235$	−1.40525e9			−0.460769
$236$		0			0
$237$	6.51636e9			2.06543
$238$		0			0
$239$	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	$0.737360\pi$
$242$		0			0
$243$			4.68093e9i			1.34248i
$244$		0			0
$245$	−6.15254e7			−0.0170762
$246$		0			0
$247$	1.44136e9	−	6.64703e9i	0.387244	−	1.78583i
$248$		0			0
$249$		−	4.81282e9i		−	1.25199i
$250$		0			0
$251$	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
$254$		0			0
$255$			8.52354e9i			2.01585i
$256$		0			0
$257$		−	5.24516e9i		−	1.20234i	−0.799122	−	0.601169i	$-0.794702\pi$
$257$		−	5.24516e9i		−	1.20234i	0.799122	−	0.601169i	$-0.205298\pi$
$258$		0			0
$259$		−	3.19060e9i		−	0.709044i
$260$		0			0
$261$			8.98605e9i			1.93645i
$262$		0			0
$263$	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
$266$		0			0
$267$	−5.29162e9			−1.04122
$268$		0			0
$269$			6.18818e9i			1.18183i	0.806735	+	0.590913i	$0.201232\pi$
$269$			6.18818e9i			1.18183i	−0.806735	+	0.590913i	$0.798768\pi$
$270$		0			0
$271$	−2.96901e8			−0.0550471			−0.0275235	−	0.999621i	$-0.508762\pi$
$271$	−2.96901e8			−0.0550471			−0.0275235	+	0.999621i	$0.508762\pi$
$272$		0			0
$273$	−1.57680e10			−2.83875
$274$		0			0
$275$	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	$0.409923\pi$
$278$		0			0
$279$			1.67109e9i			0.275793i
$280$		0			0
$281$			3.61700e9i			0.580127i	0.957007	+	0.290063i	$0.0936764\pi$
$281$			3.61700e9i			0.580127i	−0.957007	+	0.290063i	$0.906324\pi$
$282$		0			0
$283$	−1.06697e10			−1.66344			−0.831720	−	0.555195i	$-0.812644\pi$
$283$	−1.06697e10			−1.66344			−0.831720	+	0.555195i	$0.812644\pi$
$284$		0			0
$285$	−1.43822e9	+	6.63253e9i	−0.217994	+	1.00531i
$286$		0			0
$287$		−	1.61006e9i		−	0.237309i
$288$		0			0
$289$	1.98132e10			2.84029
$290$		0			0
$291$	−1.61041e10			−2.24576
$292$		0			0
$293$			1.30694e9i			0.177331i	0.996061	+	0.0886653i	$0.0282602\pi$
$293$			1.30694e9i			0.177331i	−0.996061	+	0.0886653i	$0.971740\pi$
$294$		0			0
$295$		−	2.29736e9i		−	0.303347i
$296$		0			0
$297$		−	5.43151e9i		−	0.698064i
$298$		0			0
$299$		−	1.53867e10i		−	1.92514i
$300$		0			0
$301$	3.03675e9			0.369951
$302$		0			0
$303$			1.98377e10i			2.35354i
$304$		0			0
$305$	−3.41600e9			−0.394746
$306$		0			0
$307$		−	8.88173e9i		−	0.999871i	−0.866063	−	0.499935i	$-0.833357\pi$
$307$		−	8.88173e9i		−	0.999871i	0.866063	−	0.499935i	$-0.166643\pi$
$308$		0			0
$309$	3.73422e7			0.00409605
$310$		0			0
$311$	−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	$0.653226\pi$
$314$		0			0
$315$	9.04817e9			0.919006
$316$		0			0
$317$		−	1.20507e10i		−	1.19337i	−0.802475	−	0.596685i	$-0.796484\pi$
$317$		−	1.20507e10i		−	1.19337i	0.802475	−	0.596685i	$-0.203516\pi$
$318$		0			0
$319$			1.90769e10i			1.84224i
$320$		0			0
$321$	2.04174e10			1.92301
$322$		0			0
$323$	2.08456e10	+	4.52022e9i	1.91516	+	0.415288i
$324$		0			0
$325$			1.12203e10i			1.00571i
$326$		0			0
$327$	7.64133e9			0.668310
$328$		0			0
$329$	8.15265e9			0.695849
$330$		0			0
$331$			7.18195e9i			0.598316i	0.954204	+	0.299158i	$0.0967058\pi$
$331$			7.18195e9i			0.598316i	−0.954204	+	0.299158i	$0.903294\pi$
$332$		0			0
$333$			1.16525e10i			0.947636i
$334$		0			0
$335$			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	$-0.0910975\pi$
$338$		0			0
$339$	1.62264e10			1.22864
$340$		0			0
$341$			3.54764e9i			0.262375i
$342$		0			0
$343$	−1.36595e10			−0.986865
$344$		0			0
$345$			1.53532e10i			1.08373i
$346$		0			0
$347$	−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	$0.480338\pi$
$350$		0			0
$351$	1.50374e10			0.990704
$352$		0			0
$353$	6.31655e9			0.406800			0.203400	−	0.979096i	$-0.434801\pi$
$353$	6.31655e9			0.406800			0.203400	+	0.979096i	$0.434801\pi$
$354$		0			0
$355$			1.25546e10i			0.790480i
$356$		0			0
$357$		−	4.94498e10i		−	3.04433i
$358$		0			0
$359$	−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
$362$		0			0
$363$		−	1.75219e10i		−	1.00915i
$364$		0			0
$365$	−1.08556e10			−0.611621
$366$		0			0
$367$	3.53863e9			0.195061			0.0975306	−	0.995233i	$-0.468906\pi$
$367$	3.53863e9			0.195061			0.0975306	+	0.995233i	$0.468906\pi$
$368$		0			0
$369$			5.88014e9i			0.317163i
$370$		0			0
$371$		−	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	$-0.308277\pi$
$374$		0			0
$375$		−	3.15382e10i		−	1.59482i
$376$		0			0
$377$	−5.28154e10			−2.61454
$378$		0			0
$379$		−	2.71860e10i		−	1.31762i	−0.752311	−	0.658808i	$-0.771061\pi$
$379$		−	2.71860e10i		−	1.31762i	0.752311	−	0.658808i	$-0.228939\pi$
$380$		0			0
$381$	−9.55956e7			−0.00453668
$382$		0			0
$383$			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
$386$		0			0
$387$	−1.10906e10			−0.494438
$388$		0			0
$389$	4.06187e10			1.77390			0.886948	−	0.461870i	$-0.152821\pi$
$389$	4.06187e10			1.77390			0.886948	+	0.461870i	$0.152821\pi$
$390$		0			0
$391$	4.82540e10			2.06455
$392$		0			0
$393$		−	1.65816e10i		−	0.695115i
$394$		0			0
$395$		−	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	$0.737748\pi$
$398$		0			0
$399$	8.34389e9	−	3.84790e10i	0.329213	−	1.51821i
$400$		0			0
$401$			2.89932e10i			1.12129i	0.828055	+	0.560646i	$0.189447\pi$
$401$			2.89932e10i			1.12129i	−0.828055	+	0.560646i	$0.810553\pi$
$402$		0			0
$403$	−9.82180e9			−0.372367
$404$		0			0
$405$	9.41160e9			0.349819
$406$		0			0
$407$			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	$-0.155690\pi$
$410$		0			0
$411$			5.74831e10i			2.01453i
$412$		0			0
$413$			1.33282e10i			0.458113i
$414$		0			0
$415$	−1.62321e10			−0.547245
$416$		0			0
$417$			3.00651e10i			0.994303i
$418$		0			0
$419$	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	$0.993008\pi$
$422$		0			0
$423$	−2.97745e10			−0.930001
$424$		0			0
$425$	−3.51877e10			−1.07854
$426$		0			0
$427$	1.98181e10			0.596143
$428$		0			0
$429$	1.22254e11			3.60940
$430$		0			0
$431$		−	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	$-0.206269\pi$
$434$		0			0
$435$	5.27002e10			1.47182
$436$		0			0
$437$	3.75485e10	+	8.14212e9i	1.02960	+	0.223260i
$438$		0			0
$439$		−	4.34205e9i		−	0.116906i	−0.998290	−	0.0584531i	$-0.981383\pi$
$439$		−	4.34205e9i		−	0.116906i	0.998290	−	0.0584531i	$-0.0186168\pi$
$440$		0			0
$441$	−1.30360e9			−0.0344660
$442$		0			0
$443$	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
$446$		0			0
$447$		−	8.16228e10i		−	2.04447i
$448$		0			0
$449$		−	6.19394e10i		−	1.52399i	−0.647584	−	0.761994i	$-0.724220\pi$
$449$		−	6.19394e10i		−	1.52399i	0.647584	−	0.761994i	$-0.275780\pi$
$450$		0			0
$451$			1.24833e10i			0.301732i
$452$		0			0
$453$	1.62765e10			0.386517
$454$		0			0
$455$			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	$0.541548\pi$
$458$		0			0
$459$			4.71584e10i			1.06245i
$460$		0			0
$461$	−2.30267e10			−0.509833			−0.254917	−	0.966963i	$-0.582048\pi$
$461$	−2.30267e10			−0.509833			−0.254917	+	0.966963i	$0.582048\pi$
$462$		0			0
$463$	−4.80160e10			−1.04487			−0.522435	−	0.852679i	$-0.674976\pi$
$463$	−4.80160e10			−1.04487			−0.522435	+	0.852679i	$0.674976\pi$
$464$		0			0
$465$	9.80037e9			0.209619
$466$		0			0
$467$	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
$470$		0			0
$471$			5.99934e10i			1.21905i
$472$		0			0
$473$	−2.35449e10			−0.470383
$474$		0			0
$475$	−2.73810e10	−	5.93738e9i	−0.537868	−	0.116633i
$476$		0			0
$477$			5.79638e10i			1.11965i
$478$		0			0
$479$	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
$482$		0			0
$483$		−	8.90722e10i		−	1.63664i
$484$		0			0
$485$			5.43138e10i			0.981620i
$486$		0			0
$487$			3.08222e10i			0.547959i	0.961735	+	0.273980i	$0.0883401\pi$
$487$			3.08222e10i			0.547959i	−0.961735	+	0.273980i	$0.911660\pi$
$488$		0			0
$489$			1.00696e11i			1.76108i
$490$		0			0
$491$	−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
$494$		0			0
$495$	−7.01531e10			−1.16849
$496$		0			0
$497$		−	7.28364e10i		−	1.19378i
$498$		0			0
$499$	−7.92783e10			−1.27865			−0.639326	−	0.768936i	$-0.720786\pi$
$499$	−7.92783e10			−1.27865			−0.639326	+	0.768936i	$0.720786\pi$
$500$		0			0
$501$	1.77761e11			2.82154
$502$		0			0
$503$	−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
$506$		0			0
$507$			2.37103e11i			3.58844i
$508$		0			0
$509$			1.02167e11i			1.52208i	0.648702	+	0.761042i	$0.275312\pi$
$509$			1.02167e11i			1.52208i	−0.648702	+	0.761042i	$0.724688\pi$
$510$		0			0
$511$	6.29793e10			0.923665
$512$		0			0
$513$	−7.95726e9	+	3.66960e10i	−0.114893	+	0.529845i
$514$		0			0
$515$		−	1.25943e8i		−	0.00179038i
$516$		0			0
$517$	−6.32099e10			−0.884755
$518$		0			0
$519$	8.33789e10			1.14918
$520$		0			0
$521$			1.32024e10i			0.179186i	0.995978	+	0.0895929i	$0.0285566\pi$
$521$			1.32024e10i			0.179186i	−0.995978	+	0.0895929i	$0.971443\pi$
$522$		0			0
$523$			8.84421e10i			1.18210i	0.806637	+	0.591048i	$0.201285\pi$
$523$			8.84421e10i			1.18210i	−0.806637	+	0.591048i	$0.798715\pi$
$524$		0			0
$525$			6.49531e10i			0.854993i
$526$		0			0
$527$		−	3.08019e10i		−	0.399333i
$528$		0			0
$529$	8.60731e9			0.109912
$530$		0			0
$531$		−	4.86764e10i		−	0.612267i
$532$		0			0
$533$	−3.45605e10			−0.428224
$534$		0			0
$535$		−	6.88614e10i		−	0.840544i
$536$		0			0
$537$	−8.15706e10			−0.980927
$538$		0			0
$539$	−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	$0.637197\pi$
$542$		0			0
$543$	6.78436e10			0.780386
$544$		0			0
$545$		−	2.57717e10i		−	0.292117i
$546$		0			0
$547$			4.33427e10i			0.484135i	0.970259	+	0.242068i	$0.0778256\pi$
$547$			4.33427e10i			0.484135i	−0.970259	+	0.242068i	$0.922174\pi$
$548$		0			0
$549$	−7.23782e10			−0.796744
$550$		0			0
$551$	2.79480e10	−	1.28886e11i	0.303211	−	1.39830i
$552$		0			0
$553$			1.27504e11i			1.36340i
$554$		0			0
$555$	6.83379e10			0.720261
$556$		0			0
$557$	−2.63069e9			−0.0273306			−0.0136653	−	0.999907i	$-0.504350\pi$
$557$	−2.63069e9			−0.0273306			−0.0136653	+	0.999907i	$0.504350\pi$
$558$		0			0
$559$		−	6.51850e10i		−	0.667576i
$560$		0			0
$561$			3.83399e11i			3.87079i
$562$		0			0
$563$			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
$566$		0			0
$567$	−5.46019e10			−0.528294
$568$		0			0
$569$		−	1.72406e11i		−	1.64476i	−0.568938	−	0.822381i	$-0.692645\pi$
$569$		−	1.72406e11i		−	1.64476i	0.568938	−	0.822381i	$-0.307355\pi$
$570$		0			0
$571$	−7.55025e10			−0.710259			−0.355129	−	0.934817i	$-0.615563\pi$
$571$	−7.55025e10			−0.710259			−0.355129	+	0.934817i	$0.615563\pi$
$572$		0			0
$573$		−	1.34723e9i		−	0.0124975i
$574$		0			0
$575$	−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	$0.402280\pi$
$578$		0			0
$579$	8.15499e10			0.725620
$580$		0			0
$581$	9.41713e10			0.826445
$582$		0			0
$583$			1.23054e11i			1.06518i
$584$		0			0
$585$		−	1.94222e11i		−	1.65835i
$586$		0			0
$587$	−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
$590$		0			0
$591$		−	5.63062e10i		−	0.461537i
$592$		0			0
$593$	4.38042e10			0.354239			0.177120	−	0.984189i	$-0.443322\pi$
$593$	4.38042e10			0.354239			0.177120	+	0.984189i	$0.443322\pi$
$594$		0			0
$595$	−1.66778e11			−1.33067
$596$		0			0
$597$		−	1.98561e11i		−	1.56314i
$598$		0			0
$599$		−	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	$0.968307\pi$
$602$		0			0
$603$			1.13051e11i			0.855079i
$604$		0			0
$605$	−5.90957e10			−0.441097
$606$		0			0
$607$		−	1.84790e11i		−	1.36120i	−0.732654	−	0.680601i	$-0.761719\pi$
$607$		−	1.84790e11i		−	1.36120i	0.732654	−	0.680601i	$-0.238281\pi$
$608$		0			0
$609$	−3.05743e11			−2.22273
$610$		0			0
$611$		−	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	$0.367834\pi$
$614$		0			0
$615$	3.44851e10			0.241063
$616$		0			0
$617$	−2.19273e11			−1.51302			−0.756511	−	0.653981i	$-0.773098\pi$
$617$	−2.19273e11			−1.51302			−0.756511	+	0.653981i	$0.773098\pi$
$618$		0			0
$619$	−1.11363e11			−0.758540			−0.379270	−	0.925286i	$-0.623825\pi$
$619$	−1.11363e11			−0.758540			−0.379270	+	0.925286i	$0.623825\pi$
$620$		0			0
$621$			8.49449e10i			0.571177i
$622$		0			0
$623$		−	1.03540e11i		−	0.687314i
$624$		0			0
$625$	−2.23887e10			−0.146727
$626$		0			0
$627$	−6.46926e10	+	2.98339e11i	−0.418586	+	1.93037i
$628$		0			0
$629$		−	2.14782e11i		−	1.37213i
$630$		0			0
$631$	1.41858e11			0.894819			0.447409	−	0.894329i	$-0.352347\pi$
$631$	1.41858e11			0.894819			0.447409	+	0.894329i	$0.352347\pi$
$632$		0			0
$633$	−3.12531e11			−1.94661
$634$		0			0
$635$			3.22413e8i			0.00198298i
$636$		0			0
$637$		−	7.66191e9i		−	0.0465350i
$638$		0			0
$639$			2.66008e11i			1.59548i
$640$		0			0
$641$		−	1.83546e11i		−	1.08721i	−0.839341	−	0.543606i	$-0.817059\pi$
$641$		−	1.83546e11i		−	1.08721i	0.839341	−	0.543606i	$-0.182941\pi$
$642$		0			0
$643$	2.98123e11			1.74402			0.872011	−	0.489487i	$-0.162816\pi$
$643$	2.98123e11			1.74402			0.872011	+	0.489487i	$0.162816\pi$
$644$		0			0
$645$			6.50428e10i			0.375803i
$646$		0			0
$647$	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
$650$		0			0
$651$	−5.68574e10			−0.316565
$652$		0			0
$653$	−6.17884e10			−0.339824			−0.169912	−	0.985459i	$-0.554348\pi$
$653$	−6.17884e10			−0.339824			−0.169912	+	0.985459i	$0.554348\pi$
$654$		0			0
$655$	−5.59244e10			−0.303834
$656$		0			0
$657$	−2.30009e11			−1.23448
$658$		0			0
$659$			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	$0.925594\pi$
$662$		0			0
$663$	−1.06146e12			−5.49349
$664$		0			0
$665$	−1.29777e11	−	2.81412e10i	−0.663608	−	0.143899i
$666$		0			0
$667$		−	2.98349e11i		−	1.50738i
$668$		0			0
$669$	1.51707e11			0.757356
$670$		0			0
$671$	−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	$0.678147\pi$
$674$		0			0
$675$		−	6.19434e10i		−	0.298387i
$676$		0			0
$677$		−	3.61885e10i		−	0.172273i	−0.996283	−	0.0861364i	$-0.972548\pi$
$677$		−	3.61885e10i		−	0.172273i	0.996283	−	0.0861364i	$-0.0274521\pi$
$678$		0			0
$679$		−	3.15104e11i		−	1.48243i
$680$		0			0
$681$	1.02322e11			0.475750
$682$		0			0
$683$		−	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
$686$		0			0
$687$			6.43224e11i			2.88759i
$688$		0			0
$689$	−3.40681e11			−1.51172
$690$		0			0
$691$	1.82168e11			0.799025			0.399512	−	0.916728i	$-0.369179\pi$
$691$	1.82168e11			0.799025			0.399512	+	0.916728i	$0.369179\pi$
$692$		0			0
$693$	4.06997e11			1.76465
$694$		0			0
$695$	1.01400e11			0.434609
$696$		0			0
$697$		−	1.08384e11i		−	0.459235i
$698$		0			0
$699$		−	1.78977e11i		−	0.749701i
$700$		0			0
$701$	1.01373e11			0.419808			0.209904	−	0.977722i	$-0.432685\pi$
$701$	1.01373e11			0.419808			0.209904	+	0.977722i	$0.432685\pi$
$702$		0			0
$703$	3.62411e10	−	1.67131e11i	0.148382	−	0.684281i
$704$		0			0
$705$			1.74618e11i			0.706857i
$706$		0			0
$707$	−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	$0.146885\pi$
$710$		0			0
$711$		−	4.65661e11i		−	1.82218i
$712$		0			0
$713$		−	5.54825e10i		−	0.214683i
$714$		0			0
$715$		−	4.12324e11i		−	1.57766i
$716$		0			0
$717$		−	1.48883e11i		−	0.563339i
$718$		0			0
$719$	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
$722$		0			0
$723$	6.15879e11			2.25394
$724$		0			0
$725$			2.17562e11i			0.787464i
$726$		0			0
$727$	−3.76607e11			−1.34819			−0.674094	−	0.738645i	$-0.735466\pi$
$727$	−3.76607e11			−1.34819			−0.674094	+	0.738645i	$0.735466\pi$
$728$		0			0
$729$	4.34315e11			1.53778
$730$		0			0
$731$	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	$0.596192\pi$
$734$		0			0
$735$			7.64519e9i			0.0261962i
$736$		0			0
$737$			2.40002e11i			0.813477i
$738$		0			0
$739$	1.46848e11			0.492368			0.246184	−	0.969223i	$-0.420823\pi$
$739$	1.46848e11			0.492368			0.246184	+	0.969223i	$0.420823\pi$
$740$		0			0
$741$	−8.25965e11	−	1.79105e11i	−2.73961	−	0.594065i
$742$		0			0
$743$			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
$746$		0			0
$747$	−3.43926e11			−1.10454
$748$		0			0
$749$			3.99503e11i			1.26938i
$750$		0			0
$751$			4.93724e11i			1.55212i	0.630661	+	0.776059i	$0.282784\pi$
$751$			4.93724e11i			1.55212i	−0.630661	+	0.776059i	$0.717216\pi$
$752$		0			0
$753$		−	5.39417e10i		−	0.167782i
$754$		0			0
$755$		−	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	$0.443382\pi$
$758$		0			0
$759$			6.90603e11i			2.08095i
$760$		0			0
$761$	−2.30030e10			−0.0685875			−0.0342938	−	0.999412i	$-0.510918\pi$
$761$	−2.30030e10			−0.0685875			−0.0342938	+	0.999412i	$0.510918\pi$
$762$		0			0
$763$			1.49516e11i			0.441153i
$764$		0			0
$765$	6.09095e11			1.77844
$766$		0			0
$767$	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	$0.632574\pi$
$770$		0			0
$771$	−6.51767e11			−1.84449
$772$		0			0
$773$		−	9.47008e10i		−	0.265238i	−0.991167	−	0.132619i	$-0.957661\pi$
$773$		−	9.47008e10i		−	0.265238i	0.991167	−	0.132619i	$-0.0423387\pi$
$774$		0			0
$775$			4.04588e10i			0.112152i
$776$		0			0
$777$	−3.96466e11			−1.08773
$778$		0			0
$779$	1.82882e10	−	8.43384e10i	0.0496616	−	0.229021i
$780$		0			0
$781$			5.64722e11i			1.51786i
$782$		0			0
$783$	2.91576e11			0.775718
$784$		0			0
$785$	2.02338e11			0.532843
$786$		0			0
$787$			5.15172e11i			1.34293i	0.741037	+	0.671464i	$0.234334\pi$
$787$			5.15172e11i			1.34293i	−0.741037	+	0.671464i	$0.765666\pi$
$788$		0			0
$789$		−	2.50567e11i		−	0.646570i
$790$		0			0
$791$			3.17499e11i			0.811028i
$792$		0			0
$793$		−	4.25402e11i		−	1.07574i
$794$		0			0
$795$	3.39938e11			0.851003
$796$		0			0
$797$		−	1.73952e11i		−	0.431119i	−0.976491	−	0.215559i	$-0.930843\pi$
$797$		−	1.73952e11i		−	0.431119i	0.976491	−	0.215559i	$-0.0691575\pi$
$798$		0			0
$799$	5.48812e11			1.34659
$800$		0			0
$801$			3.78141e11i			0.918594i
$802$		0			0
$803$	−4.88297e11			−1.17442
$804$		0			0
$805$	−3.00412e11			−0.715375
$806$		0			0
$807$	7.68947e11			1.81302
$808$		0			0
$809$	5.80910e11			1.35617			0.678086	−	0.734983i	$-0.262810\pi$
$809$	5.80910e11			1.35617			0.678086	+	0.734983i	$0.262810\pi$
$810$		0			0
$811$		−	6.95097e11i		−	1.60680i	−0.595439	−	0.803401i	$-0.703022\pi$
$811$		−	6.95097e11i		−	1.60680i	0.595439	−	0.803401i	$-0.296978\pi$
$812$		0			0
$813$			3.68931e10i			0.0844468i
$814$		0			0
$815$	3.39616e11			0.769765
$816$		0			0
$817$	1.59072e11	+	3.44936e10i	0.357031	+	0.0774195i
$818$		0			0
$819$			1.12679e12i			2.50442i
$820$		0			0
$821$	1.93809e11			0.426581			0.213290	−	0.976989i	$-0.431582\pi$
$821$	1.93809e11			0.426581			0.213290	+	0.976989i	$0.431582\pi$
$822$		0			0
$823$	−3.76429e10			−0.0820511			−0.0410255	−	0.999158i	$-0.513062\pi$
$823$	−3.76429e10			−0.0820511			−0.0410255	+	0.999158i	$0.513062\pi$
$824$		0			0
$825$		−	5.03601e11i		−	1.08710i
$826$		0			0
$827$			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	$-0.0547250\pi$
$830$		0			0
$831$		−	4.08540e11i		−	0.856702i
$832$		0			0
$833$	2.40283e10			0.0499050
$834$		0			0
$835$		−	5.99532e11i		−	1.23329i
$836$		0			0
$837$	5.42228e10			0.110479
$838$		0			0
$839$		−	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
$842$		0			0
$843$	4.49451e11			0.889962
$844$		0			0
$845$	7.99673e11			1.56850
$846$		0			0
$847$	3.42847e11			0.666142
$848$		0			0
$849$			1.32583e12i			2.55185i
$850$		0			0
$851$		−	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	$0.305904\pi$
$854$		0			0
$855$	4.73963e11	+	1.02775e11i	0.886911	+	0.192320i
$856$		0			0
$857$		−	5.32627e11i		−	0.987416i	−0.869628	−	0.493708i	$-0.835641\pi$
$857$		−	5.32627e11i		−	0.987416i	0.869628	−	0.493708i	$-0.164359\pi$
$858$		0			0
$859$	−1.13776e11			−0.208966			−0.104483	−	0.994527i	$-0.533319\pi$
$859$	−1.13776e11			−0.208966			−0.104483	+	0.994527i	$0.533319\pi$
$860$		0			0
$861$	−2.00067e11			−0.364051
$862$		0			0
$863$			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
$866$		0			0
$867$		−	2.46200e12i		−	4.35724i
$868$		0			0
$869$		−	9.88576e11i		−	1.73353i
$870$		0			0
$871$	−6.64457e11			−1.15450
$872$		0			0
$873$			1.15080e12i			1.98127i
$874$		0			0
$875$	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	$0.849202\pi$
$878$		0			0
$879$	1.62401e11			0.272040
$880$		0			0
$881$	1.13071e12			1.87692			0.938461	−	0.345385i	$-0.112252\pi$
$881$	1.13071e12			1.87692			0.938461	+	0.345385i	$0.112252\pi$
$882$		0			0
$883$	−7.01399e11			−1.15378			−0.576889	−	0.816822i	$-0.695734\pi$
$883$	−7.01399e11			−1.15378			−0.576889	+	0.816822i	$0.695734\pi$
$884$		0			0
$885$	−2.85471e11			−0.465360
$886$		0			0
$887$			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
$890$		0			0
$891$	4.23345e11			0.671712
$892$		0			0
$893$	4.27054e11	+	9.26036e10i	0.671547	+	0.145620i
$894$		0			0
$895$			2.75111e11i			0.428762i
$896$		0			0
$897$	−1.91197e12			−2.95332
$898$		0			0
$899$	−1.90445e11			−0.291562
$900$		0			0
$901$		−	1.06840e12i		−	1.62120i
$902$		0			0
$903$		−	3.77349e11i		−	0.567535i
$904$		0			0
$905$		−	2.28815e11i		−	0.341106i
$906$		0			0
$907$			6.62886e10i			0.0979511i	0.998800	+	0.0489756i	$0.0155956\pi$
$907$			6.62886e10i			0.0979511i	−0.998800	+	0.0489756i	$0.984404\pi$
$908$		0			0
$909$	1.41761e12			2.07635
$910$		0			0
$911$		−	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
$914$		0			0
$915$			4.24474e11i			0.605573i
$916$		0			0
$917$	3.24449e11			0.458848
$918$		0			0
$919$	−6.09038e11			−0.853852			−0.426926	−	0.904287i	$-0.640403\pi$
$919$	−6.09038e11			−0.853852			−0.426926	+	0.904287i	$0.640403\pi$
$920$		0			0
$921$	−1.10365e12			−1.53388
$922$		0			0
$923$	−1.56346e12			−2.15417
$924$		0			0
$925$			2.82119e11i			0.385359i
$926$		0			0
$927$		−	2.66849e9i		−	0.00361365i
$928$		0			0
$929$	3.42134e11			0.459338			0.229669	−	0.973269i	$-0.426236\pi$
$929$	3.42134e11			0.459338			0.229669	+	0.973269i	$0.426236\pi$
$930$		0			0
$931$	1.86975e10	+	4.05441e9i	0.0248877	+	0.00539671i
$932$		0			0
$933$			1.66072e11i			0.219164i
$934$		0			0
$935$	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	$0.647680\pi$
$938$		0			0
$939$			1.10438e12i			1.42055i
$940$		0			0
$941$		−	4.74893e11i		−	0.605672i	−0.953043	−	0.302836i	$-0.902067\pi$
$941$		−	4.74893e11i		−	0.605672i	0.953043	−	0.302836i	$-0.0979334\pi$
$942$		0			0
$943$		−	1.95229e11i		−	0.246887i
$944$		0			0
$945$		−	2.93591e11i		−	0.368142i
$946$		0			0
$947$	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
$950$		0			0
$951$	−1.49743e12			−1.83073
$952$		0			0
$953$			4.65803e10i			0.0564716i	0.999601	+	0.0282358i	$0.00898893\pi$
$953$			4.65803e10i			0.0564716i	−0.999601	+	0.0282358i	$0.991011\pi$
$954$		0			0
$955$	−4.54378e9			−0.00546266
$956$		0			0
$957$	2.37051e12			2.82615
$958$		0			0
$959$	−1.12476e12			−1.32980
$960$		0			0
$961$	8.17475e11			0.958475
$962$		0			0
$963$		−	1.45904e12i		−	1.69653i
$964$		0			0
$965$		−	2.75042e11i		−	0.317168i
$966$		0			0
$967$	1.15462e12			1.32049			0.660245	−	0.751050i	$-0.270453\pi$
$967$	1.15462e12			1.32049			0.660245	+	0.751050i	$0.270453\pi$
$968$		0			0
$969$	5.61685e11	−	2.59029e12i	0.637086	−	2.93801i
$970$		0			0
$971$		−	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
$974$		0			0
$975$	1.39424e12			1.54284
$976$		0			0
$977$			6.15166e11i			0.675171i	0.941295	+	0.337586i	$0.109610\pi$
$977$			6.15166e11i			0.675171i	−0.941295	+	0.337586i	$0.890390\pi$
$978$		0			0
$979$			8.02775e11i			0.873903i
$980$		0			0
$981$		−	5.46052e11i		−	0.589601i
$982$		0			0
$983$		−	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
$986$		0			0
$987$		−	1.01305e12i		−	1.06749i
$988$		0			0
$989$	3.68224e11			0.384882
$990$		0			0
$991$			1.16697e12i			1.20994i	0.796247	+	0.604972i	$0.206816\pi$
$991$			1.16697e12i			1.20994i	−0.796247	+	0.604972i	$0.793184\pi$
$992$		0			0
$993$	8.92435e11			0.917866
$994$		0			0
$995$	−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	$0.601550\pi$
$998$		0			0
$999$	3.78095e11			0.379611

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.9.b.a.37.6	✓	12	4.3	odd	2
38.9.b.a.37.7	yes	12	76.75	even	2
304.9.e.e.113.2		12	1.1	even	1	trivial
304.9.e.e.113.11		12	19.18	odd	2	inner
342.9.d.a.37.5		12	228.227	odd	2
342.9.d.a.37.11		12	12.11	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:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	304.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-124.261i q^{3} -419.091 q^{5} +2431.38 q^{7} -8879.72 q^{9} +O(q^{10})\)	\(q-124.261i q^{3} -419.091 q^{5} +2431.38 q^{7} -8879.72 q^{9} -18851.2 q^{11} -52190.4i q^{13} +52076.6i q^{15} +163673. q^{17} +(127361. + 27617.3i) q^{19} -302125. i q^{21} +294819. q^{23} -214988. q^{25} +288125. i q^{27} -1.01197e6i q^{29} -188192. i q^{31} +2.34246e6i q^{33} -1.01897e6 q^{35} -1.31226e6i q^{37} -6.48522e6 q^{39} -662199. i q^{41} +1.24898e6 q^{43} +3.72141e6 q^{45} +3.35309e6 q^{47} +146807. q^{49} -2.03381e7i q^{51} -6.52766e6i q^{53} +7.90038e6 q^{55} +(3.43175e6 - 1.58260e7i) q^{57} +5.48176e6i q^{59} +8.15096e6 q^{61} -2.15900e7 q^{63} +2.18726e7i q^{65} -1.27314e7i q^{67} -3.66344e7i q^{69} -2.99568e7i q^{71} +2.59027e7 q^{73} +2.67145e7i q^{75} -4.58344e7 q^{77} +5.24410e7i q^{79} -2.24572e7 q^{81} +3.87316e7 q^{83} -6.85940e7 q^{85} -1.25749e8 q^{87} -4.25848e7i q^{89} -1.26895e8i q^{91} -2.33848e7 q^{93} +(-5.33759e7 - 1.15742e7i) q^{95} -1.29599e8i q^{97} +1.67393e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 558 q^{5} + 5422 q^{7} - 15592 q^{9}+O(q^{10}) \) 12 * q + 558 * q^5 + 5422 * q^7 - 15592 * q^9	\( 12 q + 558 q^{5} + 5422 q^{7} - 15592 q^{9} + 12546 q^{11} + 270810 q^{17} - 41512 q^{19} + 823956 q^{23} + 865538 q^{25} + 1194378 q^{35} - 5786100 q^{39} - 7586646 q^{43} + 2226046 q^{45} + 20260530 q^{47} - 19498842 q^{49} + 14858554 q^{55} + 14430564 q^{57} - 41363266 q^{61} - 84235798 q^{63} + 87906498 q^{73} - 78817962 q^{77} - 100904812 q^{81} + 55944960 q^{83} + 25440254 q^{85} - 119189604 q^{87} + 105500856 q^{93} - 81396774 q^{95} + 85554938 q^{99}+O(q^{100}) \) 12 * q + 558 * q^5 + 5422 * q^7 - 15592 * q^9 + 12546 * q^11 + 270810 * q^17 - 41512 * q^19 + 823956 * q^23 + 865538 * q^25 + 1194378 * q^35 - 5786100 * q^39 - 7586646 * q^43 + 2226046 * q^45 + 20260530 * q^47 - 19498842 * q^49 + 14858554 * q^55 + 14430564 * q^57 - 41363266 * q^61 - 84235798 * q^63 + 87906498 * q^73 - 78817962 * q^77 - 100904812 * q^81 + 55944960 * q^83 + 25440254 * q^85 - 119189604 * q^87 + 105500856 * q^93 - 81396774 * q^95 + 85554938 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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