LMFDB - Embedded newform 304.9.e.e.113.4

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			113.4
Root			$-61.5968i$ of defining polynomial
Character	$\chi$	$=$	304.113
Dual form			304.9.e.e.113.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-74.3247i q^{3} -154.845 q^{5} -1585.07 q^{7} +1036.83 q^{9} +O(q^{10})$	$q-74.3247i q^{3} -154.845 q^{5} -1585.07 q^{7} +1036.83 q^{9} +25549.2 q^{11} -10369.3i q^{13} +11508.8i q^{15} -132967. q^{17} +(-13145.5 + 129656. i) q^{19} +117810. i q^{21} +334889. q^{23} -366648. q^{25} -564707. i q^{27} +605325. i q^{29} +141494. i q^{31} -1.89894e6i q^{33} +245440. q^{35} -2.45048e6i q^{37} -770699. q^{39} -2.38955e6i q^{41} -1.53889e6 q^{43} -160548. q^{45} -5.01339e6 q^{47} -3.25235e6 q^{49} +9.88272e6i q^{51} +1.18301e7i q^{53} -3.95616e6 q^{55} +(9.63667e6 + 977036. i) q^{57} +7.32369e6i q^{59} -1.12872e7 q^{61} -1.64346e6 q^{63} +1.60564e6i q^{65} +1.87052e7i q^{67} -2.48905e7i q^{69} -2.17638e7i q^{71} -8.89170e6 q^{73} +2.72510e7i q^{75} -4.04974e7 q^{77} -5.26495e7i q^{79} -3.51690e7 q^{81} -3.76587e7 q^{83} +2.05892e7 q^{85} +4.49906e7 q^{87} +1.06654e8i q^{89} +1.64362e7i q^{91} +1.05165e7 q^{93} +(2.03551e6 - 2.00766e7i) q^{95} -2.45621e7i q^{97} +2.64903e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	74.3247i		−	0.917589i	−0.888542	−	0.458795i	$-0.848281\pi$
$3$		−	74.3247i		−	0.917589i	0.888542	−	0.458795i	$-0.151719\pi$
$4$		0			0
$5$	−154.845			−0.247752			−0.123876	−	0.992298i	$-0.539532\pi$
$5$	−154.845			−0.247752			−0.123876	+	0.992298i	$0.539532\pi$
$6$		0			0
$7$	−1585.07			−0.660172			−0.330086	−	0.943951i	$-0.607078\pi$
$7$	−1585.07			−0.660172			−0.330086	+	0.943951i	$0.607078\pi$
$8$		0			0
$9$	1036.83			0.158030
$10$		0			0
$11$	25549.2			1.74505			0.872524	−	0.488572i	$-0.162482\pi$
$11$	25549.2			1.74505			0.872524	+	0.488572i	$0.162482\pi$
$12$		0			0
$13$		−	10369.3i		−	0.363060i	−0.983385	−	0.181530i	$-0.941895\pi$
$13$		−	10369.3i		−	0.363060i	0.983385	−	0.181530i	$-0.0581049\pi$
$14$		0			0
$15$			11508.8i			0.227334i
$16$		0			0
$17$	−132967.			−1.59202			−0.796008	−	0.605286i	$-0.793059\pi$
$17$	−132967.			−1.59202			−0.796008	+	0.605286i	$0.793059\pi$
$18$		0			0
$19$	−13145.5	+	129656.i	−0.100870	+	0.994900i
$19$	−13145.5	+	129656.i	−0.100870	+	0.994900i
$20$		0			0
$21$			117810.i			0.605767i
$22$		0			0
$23$	334889.			1.19671			0.598355	−	0.801231i	$-0.295821\pi$
$23$	334889.			1.19671			0.598355	+	0.801231i	$0.295821\pi$
$24$		0			0
$25$	−366648.			−0.938619
$26$		0			0
$27$		−	564707.i		−	1.06260i
$28$		0			0
$29$			605325.i			0.855848i	0.903815	+	0.427924i	$0.140755\pi$
$29$			605325.i			0.855848i	−0.903815	+	0.427924i	$0.859245\pi$
$30$		0			0
$31$			141494.i			0.153211i	0.997061	+	0.0766055i	$0.0244082\pi$
$31$			141494.i			0.153211i	−0.997061	+	0.0766055i	$0.975592\pi$
$32$		0			0
$33$		−	1.89894e6i		−	1.60124i
$34$		0			0
$35$	245440.			0.163559
$36$		0			0
$37$		−	2.45048e6i		−	1.30751i	−0.756706	−	0.653755i	$-0.773193\pi$
$37$		−	2.45048e6i		−	1.30751i	0.756706	−	0.653755i	$-0.226807\pi$
$38$		0			0
$39$	−770699.			−0.333140
$40$		0			0
$41$		−	2.38955e6i		−	0.845629i	−0.906216	−	0.422815i	$-0.861042\pi$
$41$		−	2.38955e6i		−	0.845629i	0.906216	−	0.422815i	$-0.138958\pi$
$42$		0			0
$43$	−1.53889e6			−0.450126			−0.225063	−	0.974344i	$-0.572259\pi$
$43$	−1.53889e6			−0.450126			−0.225063	+	0.974344i	$0.572259\pi$
$44$		0			0
$45$	−160548.			−0.0391521
$46$		0			0
$47$	−5.01339e6			−1.02740			−0.513701	−	0.857970i	$-0.671726\pi$
$47$	−5.01339e6			−1.02740			−0.513701	+	0.857970i	$0.671726\pi$
$48$		0			0
$49$	−3.25235e6			−0.564173
$50$		0			0
$51$			9.88272e6i			1.46082i
$52$		0			0
$53$			1.18301e7i			1.49929i	0.661840	+	0.749645i	$0.269776\pi$
$53$			1.18301e7i			1.49929i	−0.661840	+	0.749645i	$0.730224\pi$
$54$		0			0
$55$	−3.95616e6			−0.432338
$56$		0			0
$57$	9.63667e6	+	977036.i	0.912909	+	0.0925574i
$58$		0			0
$59$			7.32369e6i			0.604396i	0.953245	+	0.302198i	$0.0977204\pi$
$59$			7.32369e6i			0.604396i	−0.953245	+	0.302198i	$0.902280\pi$
$60$		0			0
$61$	−1.12872e7			−0.815203			−0.407602	−	0.913160i	$-0.633635\pi$
$61$	−1.12872e7			−0.815203			−0.407602	+	0.913160i	$0.633635\pi$
$62$		0			0
$63$	−1.64346e6			−0.104327
$64$		0			0
$65$			1.60564e6i			0.0899486i
$66$		0			0
$67$			1.87052e7i			0.928245i	0.885771	+	0.464122i	$0.153630\pi$
$67$			1.87052e7i			0.928245i	−0.885771	+	0.464122i	$0.846370\pi$
$68$		0			0
$69$		−	2.48905e7i		−	1.09809i
$70$		0			0
$71$		−	2.17638e7i		−	0.856450i	−0.903672	−	0.428225i	$-0.859139\pi$
$71$		−	2.17638e7i		−	0.856450i	0.903672	−	0.428225i	$-0.140861\pi$
$72$		0			0
$73$	−8.89170e6			−0.313107			−0.156554	−	0.987669i	$-0.550038\pi$
$73$	−8.89170e6			−0.313107			−0.156554	+	0.987669i	$0.550038\pi$
$74$		0			0
$75$			2.72510e7i			0.861267i
$76$		0			0
$77$	−4.04974e7			−1.15203
$78$		0			0
$79$		−	5.26495e7i		−	1.35172i	−0.737031	−	0.675859i	$-0.763773\pi$
$79$		−	5.26495e7i		−	1.35172i	0.737031	−	0.675859i	$-0.236227\pi$
$80$		0			0
$81$	−3.51690e7			−0.816997
$82$		0			0
$83$	−3.76587e7			−0.793511			−0.396756	−	0.917924i	$-0.629864\pi$
$83$	−3.76587e7			−0.793511			−0.396756	+	0.917924i	$0.629864\pi$
$84$		0			0
$85$	2.05892e7			0.394424
$86$		0			0
$87$	4.49906e7			0.785317
$88$		0			0
$89$			1.06654e8i			1.69988i	0.526883	+	0.849938i	$0.323360\pi$
$89$			1.06654e8i			1.69988i	−0.526883	+	0.849938i	$0.676640\pi$
$90$		0			0
$91$			1.64362e7i			0.239682i
$92$		0			0
$93$	1.05165e7			0.140585
$94$		0			0
$95$	2.03551e6	−	2.00766e7i	0.0249907	−	0.246488i
$96$		0			0
$97$		−	2.45621e7i		−	0.277446i	−0.990331	−	0.138723i	$-0.955700\pi$
$97$		−	2.45621e7i		−	0.277446i	0.990331	−	0.138723i	$-0.0442998\pi$
$98$		0			0
$99$	2.64903e7			0.275769
$100$		0			0
$101$	4.35309e7			0.418323			0.209162	−	0.977881i	$-0.432927\pi$
$101$	4.35309e7			0.418323			0.209162	+	0.977881i	$0.432927\pi$
$102$		0			0
$103$		−	2.50911e7i		−	0.222931i	−0.993768	−	0.111465i	$-0.964446\pi$
$103$		−	2.50911e7i		−	0.222931i	0.993768	−	0.111465i	$-0.0355544\pi$
$104$		0			0
$105$		−	1.82423e7i		−	0.150080i
$106$		0			0
$107$			2.68160e7i			0.204578i	0.994755	+	0.102289i	$0.0326167\pi$
$107$			2.68160e7i			0.204578i	−0.994755	+	0.102289i	$0.967383\pi$
$108$		0			0
$109$			1.10533e8i			0.783041i	0.920169	+	0.391521i	$0.128051\pi$
$109$			1.10533e8i			0.783041i	−0.920169	+	0.391521i	$0.871949\pi$
$110$		0			0
$111$	−1.82132e8			−1.19976
$112$		0			0
$113$			1.51426e8i			0.928723i	0.885646	+	0.464361i	$0.153716\pi$
$113$			1.51426e8i			0.928723i	−0.885646	+	0.464361i	$0.846284\pi$
$114$		0			0
$115$	−5.18557e7			−0.296487
$116$		0			0
$117$		−	1.07513e7i		−	0.0573743i
$118$		0			0
$119$	2.10762e8			1.05100
$120$		0			0
$121$	4.38405e8			2.04519
$122$		0			0
$123$	−1.77602e8			−0.775941
$124$		0			0
$125$	1.17260e8			0.480296
$126$		0			0
$127$			4.59385e8i			1.76588i	0.469482	+	0.882942i	$0.344441\pi$
$127$			4.59385e8i			1.76588i	−0.469482	+	0.882942i	$0.655559\pi$
$128$		0			0
$129$			1.14378e8i			0.413031i
$130$		0			0
$131$	3.08184e8			1.04646			0.523232	−	0.852190i	$-0.324726\pi$
$131$	3.08184e8			1.04646			0.523232	+	0.852190i	$0.324726\pi$
$132$		0			0
$133$	2.08366e7	−	2.05515e8i	0.0665916	−	0.656805i
$134$		0			0
$135$			8.74419e7i			0.263260i
$136$		0			0
$137$	−4.53360e8			−1.28695			−0.643473	−	0.765468i	$-0.722507\pi$
$137$	−4.53360e8			−1.28695			−0.643473	+	0.765468i	$0.722507\pi$
$138$		0			0
$139$	3.86640e8			1.03573			0.517866	−	0.855462i	$-0.326726\pi$
$139$	3.86640e8			1.03573			0.517866	+	0.855462i	$0.326726\pi$
$140$		0			0
$141$			3.72619e8i			0.942732i
$142$		0			0
$143$		−	2.64929e8i		−	0.633556i
$144$		0			0
$145$		−	9.37314e7i		−	0.212038i
$146$		0			0
$147$			2.41730e8i			0.517679i
$148$		0			0
$149$	5.42202e8			1.10006			0.550029	−	0.835145i	$-0.314617\pi$
$149$	5.42202e8			1.10006			0.550029	+	0.835145i	$0.314617\pi$
$150$		0			0
$151$		−	1.75171e7i		−	0.0336942i	−0.999858	−	0.0168471i	$-0.994637\pi$
$151$		−	1.75171e7i		−	0.0336942i	0.999858	−	0.0168471i	$-0.00536285\pi$
$152$		0			0
$153$	−1.37864e8			−0.251586
$154$		0			0
$155$		−	2.19095e7i		−	0.0379583i
$156$		0			0
$157$	−6.98783e8			−1.15012			−0.575061	−	0.818111i	$-0.695022\pi$
$157$	−6.98783e8			−1.15012			−0.575061	+	0.818111i	$0.695022\pi$
$158$		0			0
$159$	8.79271e8			1.37573
$160$		0			0
$161$	−5.30823e8			−0.790035
$162$		0			0
$163$	−3.03342e8			−0.429716			−0.214858	−	0.976645i	$-0.568929\pi$
$163$	−3.03342e8			−0.429716			−0.214858	+	0.976645i	$0.568929\pi$
$164$		0			0
$165$			2.94041e8i			0.396709i
$166$		0			0
$167$			1.38262e9i			1.77761i	0.458286	+	0.888805i	$0.348464\pi$
$167$			1.38262e9i			1.77761i	−0.458286	+	0.888805i	$0.651536\pi$
$168$		0			0
$169$	7.08207e8			0.868188
$170$		0			0
$171$	−1.36297e7	+	1.34432e8i	−0.0159405	+	0.157224i
$172$		0			0
$173$			6.31095e8i			0.704547i	0.935897	+	0.352274i	$0.114591\pi$
$173$			6.31095e8i			0.704547i	−0.935897	+	0.352274i	$0.885409\pi$
$174$		0			0
$175$	5.81164e8			0.619650
$176$		0			0
$177$	5.44331e8			0.554588
$178$		0			0
$179$			1.63061e9i			1.58832i	0.607711	+	0.794158i	$0.292088\pi$
$179$			1.63061e9i			1.58832i	−0.607711	+	0.794158i	$0.707912\pi$
$180$		0			0
$181$		−	8.11994e8i		−	0.756551i	−0.925693	−	0.378276i	$-0.876517\pi$
$181$		−	8.11994e8i		−	0.756551i	0.925693	−	0.378276i	$-0.123483\pi$
$182$		0			0
$183$			8.38916e8i			0.748022i
$184$		0			0
$185$			3.79444e8i			0.323938i
$186$		0			0
$187$	−3.39720e9			−2.77814
$188$		0			0
$189$			8.95101e8i			0.701496i
$190$		0			0
$191$	1.14327e9			0.859041			0.429521	−	0.903057i	$-0.358683\pi$
$191$	1.14327e9			0.859041			0.429521	+	0.903057i	$0.358683\pi$
$192$		0			0
$193$			1.21153e9i			0.873185i	0.899659	+	0.436592i	$0.143815\pi$
$193$			1.21153e9i			0.873185i	−0.899659	+	0.436592i	$0.856185\pi$
$194$		0			0
$195$	1.19339e8			0.0825359
$196$		0			0
$197$	4.99652e8			0.331744			0.165872	−	0.986147i	$-0.446956\pi$
$197$	4.99652e8			0.331744			0.165872	+	0.986147i	$0.446956\pi$
$198$		0			0
$199$	−2.04764e9			−1.30570			−0.652848	−	0.757489i	$-0.726426\pi$
$199$	−2.04764e9			−1.30570			−0.652848	+	0.757489i	$0.726426\pi$
$200$		0			0
$201$	1.39026e9			0.851747
$202$		0			0
$203$		−	9.59484e8i		−	0.565007i
$204$		0			0
$205$			3.70009e8i			0.209506i
$206$		0			0
$207$	3.47224e8			0.189116
$208$		0			0
$209$	−3.35857e8	+	3.31262e9i	−0.176023	+	1.73615i
$210$		0			0
$211$		−	6.24079e8i		−	0.314854i	−0.987531	−	0.157427i	$-0.949680\pi$
$211$		−	6.24079e8i		−	0.314854i	0.987531	−	0.157427i	$-0.0503199\pi$
$212$		0			0
$213$	−1.61759e9			−0.785870
$214$		0			0
$215$	2.38289e8			0.111520
$216$		0			0
$217$		−	2.24277e8i		−	0.101146i
$218$		0			0
$219$			6.60873e8i			0.287304i
$220$		0			0
$221$			1.37878e9i			0.577997i
$222$		0			0
$223$		−	2.29270e9i		−	0.927102i	−0.886070	−	0.463551i	$-0.846575\pi$
$223$		−	2.29270e9i		−	0.927102i	0.886070	−	0.463551i	$-0.153425\pi$
$224$		0			0
$225$	−3.80153e8			−0.148330
$226$		0			0
$227$			3.90434e9i			1.47043i	0.677835	+	0.735214i	$0.262918\pi$
$227$			3.90434e9i			1.47043i	−0.677835	+	0.735214i	$0.737082\pi$
$228$		0			0
$229$	−3.02177e8			−0.109880			−0.0549401	−	0.998490i	$-0.517497\pi$
$229$	−3.02177e8			−0.109880			−0.0549401	+	0.998490i	$0.517497\pi$
$230$		0			0
$231$			3.00996e9i			1.05709i
$232$		0			0
$233$	−4.59616e9			−1.55945			−0.779726	−	0.626121i	$-0.784641\pi$
$233$	−4.59616e9			−1.55945			−0.779726	+	0.626121i	$0.784641\pi$
$234$		0			0
$235$	7.76297e8			0.254540
$236$		0			0
$237$	−3.91316e9			−1.24032
$238$		0			0
$239$	1.85947e9			0.569897			0.284949	−	0.958543i	$-0.408023\pi$
$239$	1.85947e9			0.569897			0.284949	+	0.958543i	$0.408023\pi$
$240$		0			0
$241$		−	4.35931e8i		−	0.129226i	−0.997910	−	0.0646130i	$-0.979419\pi$
$241$		−	4.35931e8i		−	0.129226i	0.997910	−	0.0646130i	$-0.0205813\pi$
$242$		0			0
$243$		−	1.09111e9i		−	0.312928i
$244$		0			0
$245$	5.03609e8			0.139775
$246$		0			0
$247$	1.34445e9	+	1.36310e8i	0.361208	+	0.0366219i
$248$		0			0
$249$			2.79897e9i			0.728117i
$250$		0			0
$251$	−1.73172e9			−0.436297			−0.218148	−	0.975916i	$-0.570002\pi$
$251$	−1.73172e9			−0.436297			−0.218148	+	0.975916i	$0.570002\pi$
$252$		0			0
$253$	8.55615e9			2.08832
$254$		0			0
$255$		−	1.53029e9i		−	0.361920i
$256$		0			0
$257$			7.49270e9i			1.71754i	0.512364	+	0.858768i	$0.328770\pi$
$257$			7.49270e9i			1.71754i	−0.512364	+	0.858768i	$0.671230\pi$
$258$		0			0
$259$			3.88419e9i			0.863181i
$260$		0			0
$261$			6.27621e8i			0.135249i
$262$		0			0
$263$	4.66900e9			0.975890			0.487945	−	0.872874i	$-0.337747\pi$
$263$	4.66900e9			0.975890			0.487945	+	0.872874i	$0.337747\pi$
$264$		0			0
$265$		−	1.83183e9i		−	0.371452i
$266$		0			0
$267$	7.92703e9			1.55979
$268$		0			0
$269$			2.06571e9i			0.394512i	0.980352	+	0.197256i	$0.0632030\pi$
$269$			2.06571e9i			0.394512i	−0.980352	+	0.197256i	$0.936797\pi$
$270$		0			0
$271$	−1.24856e9			−0.231490			−0.115745	−	0.993279i	$-0.536926\pi$
$271$	−1.24856e9			−0.231490			−0.115745	+	0.993279i	$0.536926\pi$
$272$		0			0
$273$	1.22161e9			0.219929
$274$		0			0
$275$	−9.36758e9			−1.63793
$276$		0			0
$277$	−1.05391e8			−0.0179013			−0.00895067	−	0.999960i	$-0.502849\pi$
$277$	−1.05391e8			−0.0179013			−0.00895067	+	0.999960i	$0.502849\pi$
$278$		0			0
$279$			1.46705e8i			0.0242119i
$280$		0			0
$281$		−	9.25009e9i		−	1.48361i	−0.670614	−	0.741807i	$-0.733969\pi$
$281$		−	9.25009e9i		−	1.48361i	0.670614	−	0.741807i	$-0.266031\pi$
$282$		0			0
$283$	−7.95944e9			−1.24090			−0.620450	−	0.784246i	$-0.713050\pi$
$283$	−7.95944e9			−1.24090			−0.620450	+	0.784246i	$0.713050\pi$
$284$		0			0
$285$	−1.49219e9	−	1.51289e8i	−0.226175	−	0.0229312i
$286$		0			0
$287$			3.78760e9i			0.558261i
$288$		0			0
$289$	1.07044e10			1.53451
$290$		0			0
$291$	−1.82557e9			−0.254582
$292$		0			0
$293$		−	1.46284e9i		−	0.198484i	−0.995063	−	0.0992420i	$-0.968358\pi$
$293$		−	1.46284e9i		−	0.198484i	0.995063	−	0.0992420i	$-0.0316418\pi$
$294$		0			0
$295$		−	1.13403e9i		−	0.149740i
$296$		0			0
$297$		−	1.44278e10i		−	1.85428i
$298$		0			0
$299$		−	3.47258e9i		−	0.434477i
$300$		0			0
$301$	2.43926e9			0.297161
$302$		0			0
$303$		−	3.23542e9i		−	0.383849i
$304$		0			0
$305$	1.74776e9			0.201968
$306$		0			0
$307$			1.59981e9i			0.180101i	0.995937	+	0.0900504i	$0.0287028\pi$
$307$			1.59981e9i			0.180101i	−0.995937	+	0.0900504i	$0.971297\pi$
$308$		0			0
$309$	−1.86489e9			−0.204559
$310$		0			0
$311$	1.36371e10			1.45775			0.728873	−	0.684649i	$-0.240045\pi$
$311$	1.36371e10			1.45775			0.728873	+	0.684649i	$0.240045\pi$
$312$		0			0
$313$	−6.84472e9			−0.713146			−0.356573	−	0.934268i	$-0.616055\pi$
$313$	−6.84472e9			−0.713146			−0.356573	+	0.934268i	$0.616055\pi$
$314$		0			0
$315$	2.54481e8			0.0258471
$316$		0			0
$317$		−	4.91136e9i		−	0.486368i	−0.969980	−	0.243184i	$-0.921808\pi$
$317$		−	4.91136e9i		−	0.486368i	0.969980	−	0.243184i	$-0.0781919\pi$
$318$		0			0
$319$			1.54656e10i			1.49349i
$320$		0			0
$321$	1.99310e9			0.187719
$322$		0			0
$323$	1.74791e9	−	1.72400e10i	0.160587	−	1.58390i
$324$		0			0
$325$			3.80190e9i			0.340775i
$326$		0			0
$327$	8.21531e9			0.718510
$328$		0			0
$329$	7.94658e9			0.678261
$330$		0			0
$331$			1.69481e10i			1.41192i	0.708252	+	0.705960i	$0.249484\pi$
$331$			1.69481e10i			1.41192i	−0.708252	+	0.705960i	$0.750516\pi$
$332$		0			0
$333$		−	2.54074e9i		−	0.206626i
$334$		0			0
$335$		−	2.89640e9i		−	0.229974i
$336$		0			0
$337$			2.45355e10i			1.90229i	0.308750	+	0.951143i	$0.400089\pi$
$337$			2.45355e10i			1.90229i	−0.308750	+	0.951143i	$0.599911\pi$
$338$		0			0
$339$	1.12547e10			0.852186
$340$		0			0
$341$			3.61505e9i			0.267360i
$342$		0			0
$343$	1.42928e10			1.03262
$344$		0			0
$345$			3.85416e9i			0.272053i
$346$		0			0
$347$	−1.97537e10			−1.36248			−0.681239	−	0.732061i	$-0.738559\pi$
$347$	−1.97537e10			−1.36248			−0.681239	+	0.732061i	$0.738559\pi$
$348$		0			0
$349$	−7.25657e9			−0.489136			−0.244568	−	0.969632i	$-0.578646\pi$
$349$	−7.25657e9			−0.489136			−0.244568	+	0.969632i	$0.578646\pi$
$350$		0			0
$351$	−5.85564e9			−0.385786
$352$		0			0
$353$	−1.86135e10			−1.19875			−0.599375	−	0.800468i	$-0.704584\pi$
$353$	−1.86135e10			−1.19875			−0.599375	+	0.800468i	$0.704584\pi$
$354$		0			0
$355$			3.37002e9i			0.212187i
$356$		0			0
$357$		−	1.56648e10i		−	0.964390i
$358$		0			0
$359$	−1.90002e10			−1.14388			−0.571940	−	0.820296i	$-0.693809\pi$
$359$	−1.90002e10			−1.14388			−0.571940	+	0.820296i	$0.693809\pi$
$360$		0			0
$361$	−1.66380e10	−	3.40879e9i	−0.979650	−	0.200711i
$362$		0			0
$363$		−	3.25843e10i		−	1.87664i
$364$		0			0
$365$	1.37683e9			0.0775729
$366$		0			0
$367$	1.69812e10			0.936061			0.468030	−	0.883712i	$-0.344964\pi$
$367$	1.69812e10			0.936061			0.468030	+	0.883712i	$0.344964\pi$
$368$		0			0
$369$		−	2.47756e9i		−	0.133635i
$370$		0			0
$371$		−	1.87516e10i		−	0.989789i
$372$		0			0
$373$		−	3.35760e10i		−	1.73458i	−0.497805	−	0.867289i	$-0.665860\pi$
$373$		−	3.35760e10i		−	1.73458i	0.497805	−	0.867289i	$-0.334140\pi$
$374$		0			0
$375$		−	8.71530e9i		−	0.440714i
$376$		0			0
$377$	6.27683e9			0.310724
$378$		0			0
$379$			3.39145e9i			0.164372i	0.996617	+	0.0821860i	$0.0261902\pi$
$379$			3.39145e9i			0.164372i	−0.996617	+	0.0821860i	$0.973810\pi$
$380$		0			0
$381$	3.41437e10			1.62036
$382$		0			0
$383$		−	4.19332e10i		−	1.94878i	−0.224864	−	0.974390i	$-0.572194\pi$
$383$		−	4.19332e10i		−	1.94878i	0.224864	−	0.974390i	$-0.427806\pi$
$384$		0			0
$385$	6.27081e9			0.285417
$386$		0			0
$387$	−1.59558e9			−0.0711334
$388$		0			0
$389$	−1.91799e10			−0.837620			−0.418810	−	0.908074i	$-0.637553\pi$
$389$	−1.91799e10			−0.837620			−0.418810	+	0.908074i	$0.637553\pi$
$390$		0			0
$391$	−4.45291e10			−1.90518
$392$		0			0
$393$		−	2.29057e10i		−	0.960224i
$394$		0			0
$395$			8.15250e9i			0.334890i
$396$		0			0
$397$	−3.73055e10			−1.50180			−0.750899	−	0.660417i	$-0.770379\pi$
$397$	−3.73055e10			−1.50180			−0.750899	+	0.660417i	$0.770379\pi$
$398$		0			0
$399$	−1.52748e10	−	1.54867e9i	−0.602677	−	0.0611038i
$400$		0			0
$401$			1.19174e10i			0.460895i	0.973085	+	0.230448i	$0.0740191\pi$
$401$			1.19174e10i			0.460895i	−0.973085	+	0.230448i	$0.925981\pi$
$402$		0			0
$403$	1.46720e9			0.0556247
$404$		0			0
$405$	5.44574e9			0.202412
$406$		0			0
$407$		−	6.26080e10i		−	2.28167i
$408$		0			0
$409$			1.57301e10i			0.562133i	0.959688	+	0.281066i	$0.0906881\pi$
$409$			1.57301e10i			0.562133i	−0.959688	+	0.281066i	$0.909312\pi$
$410$		0			0
$411$			3.36958e10i			1.18089i
$412$		0			0
$413$		−	1.16086e10i		−	0.399005i
$414$		0			0
$415$	5.83125e9			0.196594
$416$		0			0
$417$		−	2.87369e10i		−	0.950377i
$418$		0			0
$419$	−3.53760e10			−1.14776			−0.573882	−	0.818938i	$-0.694563\pi$
$419$	−3.53760e10			−1.14776			−0.573882	+	0.818938i	$0.694563\pi$
$420$		0			0
$421$		−	2.22556e10i		−	0.708454i	−0.935159	−	0.354227i	$-0.884744\pi$
$421$		−	2.22556e10i		−	0.708454i	0.935159	−	0.354227i	$-0.115256\pi$
$422$		0			0
$423$	−5.19805e9			−0.162360
$424$		0			0
$425$	4.87520e10			1.49430
$426$		0			0
$427$	1.78910e10			0.538174
$428$		0			0
$429$	−1.96908e10			−0.581345
$430$		0			0
$431$			6.16218e10i			1.78577i	0.450285	+	0.892885i	$0.351322\pi$
$431$			6.16218e10i			1.78577i	−0.450285	+	0.892885i	$0.648678\pi$
$432$		0			0
$433$			4.72412e9i			0.134391i	0.997740	+	0.0671954i	$0.0214051\pi$
$433$			4.72412e9i			0.134391i	−0.997740	+	0.0671954i	$0.978595\pi$
$434$		0			0
$435$	−6.96656e9			−0.194563
$436$		0			0
$437$	−4.40228e9	+	4.34204e10i	−0.120712	+	1.19061i
$438$		0			0
$439$		−	4.36060e10i		−	1.17405i	−0.809568	−	0.587027i	$-0.800298\pi$
$439$		−	4.36060e10i		−	1.17405i	0.809568	−	0.587027i	$-0.199702\pi$
$440$		0			0
$441$	−3.37214e9			−0.0891562
$442$		0			0
$443$	4.41307e10			1.14585			0.572923	−	0.819610i	$-0.305810\pi$
$443$	4.41307e10			1.14585			0.572923	+	0.819610i	$0.305810\pi$
$444$		0			0
$445$		−	1.65148e10i		−	0.421147i
$446$		0			0
$447$		−	4.02990e10i		−	1.00940i
$448$		0			0
$449$		−	2.32868e10i		−	0.572961i	−0.958086	−	0.286481i	$-0.907515\pi$
$449$		−	2.32868e10i		−	0.572961i	0.958086	−	0.286481i	$-0.0924854\pi$
$450$		0			0
$451$		−	6.10511e10i		−	1.47566i
$452$		0			0
$453$	−1.30196e9			−0.0309174
$454$		0			0
$455$		−	2.54505e9i		−	0.0593815i
$456$		0			0
$457$	−4.60367e10			−1.05545			−0.527727	−	0.849414i	$-0.676956\pi$
$457$	−4.60367e10			−1.05545			−0.527727	+	0.849414i	$0.676956\pi$
$458$		0			0
$459$			7.50873e10i			1.69167i
$460$		0			0
$461$	3.61112e10			0.799537			0.399769	−	0.916616i	$-0.369091\pi$
$461$	3.61112e10			0.799537			0.399769	+	0.916616i	$0.369091\pi$
$462$		0			0
$463$	−3.50543e9			−0.0762812			−0.0381406	−	0.999272i	$-0.512143\pi$
$463$	−3.50543e9			−0.0762812			−0.0381406	+	0.999272i	$0.512143\pi$
$464$		0			0
$465$	−1.62842e9			−0.0348301
$466$		0			0
$467$	−4.73893e9			−0.0996352			−0.0498176	−	0.998758i	$-0.515864\pi$
$467$	−4.73893e9			−0.0996352			−0.0498176	+	0.998758i	$0.515864\pi$
$468$		0			0
$469$		−	2.96490e10i		−	0.612801i
$470$		0			0
$471$			5.19369e10i			1.05534i
$472$		0			0
$473$	−3.93175e10			−0.785492
$474$		0			0
$475$	4.81977e9	−	4.75382e10i	0.0946787	−	0.933832i
$476$		0			0
$477$			1.22659e10i			0.236933i
$478$		0			0
$479$	7.54648e9			0.143352			0.0716758	−	0.997428i	$-0.477165\pi$
$479$	7.54648e9			0.143352			0.0716758	+	0.997428i	$0.477165\pi$
$480$		0			0
$481$	−2.54099e10			−0.474704
$482$		0			0
$483$			3.94533e10i			0.724927i
$484$		0			0
$485$			3.80332e9i			0.0687378i
$486$		0			0
$487$			8.92286e10i			1.58631i	0.609020	+	0.793155i	$0.291563\pi$
$487$			8.92286e10i			1.58631i	−0.609020	+	0.793155i	$0.708437\pi$
$488$		0			0
$489$			2.25458e10i			0.394303i
$490$		0			0
$491$	3.26865e10			0.562397			0.281198	−	0.959650i	$-0.409268\pi$
$491$	3.26865e10			0.562397			0.281198	+	0.959650i	$0.409268\pi$
$492$		0			0
$493$		−	8.04881e10i		−	1.36252i
$494$		0			0
$495$	−4.10188e9			−0.0683223
$496$		0			0
$497$			3.44973e10i			0.565404i
$498$		0			0
$499$	9.51579e10			1.53477			0.767384	−	0.641188i	$-0.221558\pi$
$499$	9.51579e10			1.53477			0.767384	+	0.641188i	$0.221558\pi$
$500$		0			0
$501$	1.02763e11			1.63112
$502$		0			0
$503$	−1.90098e10			−0.296966			−0.148483	−	0.988915i	$-0.547439\pi$
$503$	−1.90098e10			−0.296966			−0.148483	+	0.988915i	$0.547439\pi$
$504$		0			0
$505$	−6.74053e9			−0.103640
$506$		0			0
$507$		−	5.26373e10i		−	0.796640i
$508$		0			0
$509$		−	1.06082e11i		−	1.58041i	−0.612845	−	0.790203i	$-0.709975\pi$
$509$		−	1.06082e11i		−	1.58041i	0.612845	−	0.790203i	$-0.290025\pi$
$510$		0			0
$511$	1.40940e10			0.206705
$512$		0			0
$513$	7.32178e10	+	7.42335e9i	1.05718	+	0.107184i
$514$		0			0
$515$			3.88522e9i			0.0552315i
$516$		0			0
$517$	−1.28088e11			−1.79286
$518$		0			0
$519$	4.69060e10			0.646485
$520$		0			0
$521$			5.55106e10i			0.753399i	0.926335	+	0.376700i	$0.122941\pi$
$521$			5.55106e10i			0.753399i	−0.926335	+	0.376700i	$0.877059\pi$
$522$		0			0
$523$		−	1.05842e11i		−	1.41465i	−0.706888	−	0.707326i	$-0.749902\pi$
$523$		−	1.05842e11i		−	1.41465i	0.706888	−	0.707326i	$-0.250098\pi$
$524$		0			0
$525$		−	4.31948e10i		−	0.568584i
$526$		0			0
$527$		−	1.88139e10i		−	0.243914i
$528$		0			0
$529$	3.38395e10			0.432116
$530$		0			0
$531$			7.59345e9i			0.0955126i
$532$		0			0
$533$	−2.47780e10			−0.307014
$534$		0			0
$535$		−	4.15232e9i		−	0.0506846i
$536$		0			0
$537$	1.21194e11			1.45742
$538$		0			0
$539$	−8.30950e10			−0.984509
$540$		0			0
$541$	6.69196e10			0.781204			0.390602	−	0.920560i	$-0.372267\pi$
$541$	6.69196e10			0.781204			0.390602	+	0.920560i	$0.372267\pi$
$542$		0			0
$543$	−6.03512e10			−0.694204
$544$		0			0
$545$		−	1.71154e10i		−	0.194000i
$546$		0			0
$547$		−	9.76775e9i		−	0.109105i	−0.998511	−	0.0545526i	$-0.982627\pi$
$547$		−	9.76775e9i		−	0.109105i	0.998511	−	0.0545526i	$-0.0173732\pi$
$548$		0			0
$549$	−1.17029e10			−0.128826
$550$		0			0
$551$	−7.84842e10	−	7.95730e9i	−0.851483	−	0.0863295i
$552$		0			0
$553$			8.34533e10i			0.892366i
$554$		0			0
$555$	2.82021e10			0.297242
$556$		0			0
$557$	1.59399e10			0.165602			0.0828010	−	0.996566i	$-0.473613\pi$
$557$	1.59399e10			0.165602			0.0828010	+	0.996566i	$0.473613\pi$
$558$		0			0
$559$			1.59573e10i			0.163423i
$560$		0			0
$561$			2.52496e11i			2.54919i
$562$		0			0
$563$			1.84760e10i			0.183897i	0.995764	+	0.0919483i	$0.0293094\pi$
$563$			1.84760e10i			0.183897i	−0.995764	+	0.0919483i	$0.970691\pi$
$564$		0			0
$565$		−	2.34475e10i		−	0.230093i
$566$		0			0
$567$	5.57455e10			0.539358
$568$		0			0
$569$			5.39118e10i			0.514322i	0.966369	+	0.257161i	$0.0827870\pi$
$569$			5.39118e10i			0.514322i	−0.966369	+	0.257161i	$0.917213\pi$
$570$		0			0
$571$	−5.13331e9			−0.0482896			−0.0241448	−	0.999708i	$-0.507686\pi$
$571$	−5.13331e9			−0.0482896			−0.0241448	+	0.999708i	$0.507686\pi$
$572$		0			0
$573$		−	8.49730e10i		−	0.788247i
$574$		0			0
$575$	−1.22786e11			−1.12326
$576$		0			0
$577$	−4.51019e9			−0.0406903			−0.0203452	−	0.999793i	$-0.506477\pi$
$577$	−4.51019e9			−0.0406903			−0.0203452	+	0.999793i	$0.506477\pi$
$578$		0			0
$579$	9.00469e10			0.801225
$580$		0			0
$581$	5.96918e10			0.523854
$582$		0			0
$583$			3.02251e11i			2.61633i
$584$		0			0
$585$			1.66478e9i			0.0142146i
$586$		0			0
$587$	5.32145e10			0.448206			0.224103	−	0.974565i	$-0.428055\pi$
$587$	5.32145e10			0.448206			0.224103	+	0.974565i	$0.428055\pi$
$588$		0			0
$589$	−1.83455e10	−	1.86000e9i	−0.152430	−	0.0154544i
$590$		0			0
$591$		−	3.71365e10i		−	0.304404i
$592$		0			0
$593$	−2.81590e10			−0.227719			−0.113859	−	0.993497i	$-0.536321\pi$
$593$	−2.81590e10			−0.227719			−0.113859	+	0.993497i	$0.536321\pi$
$594$		0			0
$595$	−3.26354e10			−0.260388
$596$		0			0
$597$			1.52191e11i			1.19809i
$598$		0			0
$599$		−	1.53599e10i		−	0.119311i	−0.998219	−	0.0596554i	$-0.981000\pi$
$599$		−	1.53599e10i		−	0.119311i	0.998219	−	0.0596554i	$-0.0190002\pi$
$600$		0			0
$601$			1.08743e11i			0.833494i	0.909023	+	0.416747i	$0.136830\pi$
$601$			1.08743e11i			0.833494i	−0.909023	+	0.416747i	$0.863170\pi$
$602$		0			0
$603$			1.93941e10i			0.146690i
$604$		0			0
$605$	−6.78846e10			−0.506699
$606$		0			0
$607$			1.12346e11i			0.827565i	0.910376	+	0.413783i	$0.135793\pi$
$607$			1.12346e11i			0.827565i	−0.910376	+	0.413783i	$0.864207\pi$
$608$		0			0
$609$	−7.13134e10			−0.518444
$610$		0			0
$611$			5.19856e10i			0.373008i
$612$		0			0
$613$	−2.12496e10			−0.150490			−0.0752451	−	0.997165i	$-0.523974\pi$
$613$	−2.12496e10			−0.150490			−0.0752451	+	0.997165i	$0.523974\pi$
$614$		0			0
$615$	2.75008e10			0.192240
$616$		0			0
$617$	−2.05896e11			−1.42072			−0.710359	−	0.703840i	$-0.751467\pi$
$617$	−2.05896e11			−1.42072			−0.710359	+	0.703840i	$0.751467\pi$
$618$		0			0
$619$	−1.59468e9			−0.0108620			−0.00543100	−	0.999985i	$-0.501729\pi$
$619$	−1.59468e9			−0.0108620			−0.00543100	+	0.999985i	$0.501729\pi$
$620$		0			0
$621$		−	1.89114e11i		−	1.27162i
$622$		0			0
$623$		−	1.69054e11i		−	1.12221i
$624$		0			0
$625$	1.25065e11			0.819625
$626$		0			0
$627$	2.46210e11	+	2.49625e10i	1.59307	+	0.161517i
$628$		0			0
$629$			3.25833e11i			2.08158i
$630$		0			0
$631$	−2.84477e11			−1.79444			−0.897222	−	0.441580i	$-0.854418\pi$
$631$	−2.84477e11			−1.79444			−0.897222	+	0.441580i	$0.854418\pi$
$632$		0			0
$633$	−4.63845e10			−0.288907
$634$		0			0
$635$		−	7.11334e10i		−	0.437501i
$636$		0			0
$637$			3.37247e10i			0.204829i
$638$		0			0
$639$		−	2.25655e10i		−	0.135345i
$640$		0			0
$641$			5.30174e10i			0.314041i	0.987595	+	0.157021i	$0.0501889\pi$
$641$			5.30174e10i			0.314041i	−0.987595	+	0.157021i	$0.949811\pi$
$642$		0			0
$643$	−5.79754e10			−0.339156			−0.169578	−	0.985517i	$-0.554241\pi$
$643$	−5.79754e10			−0.339156			−0.169578	+	0.985517i	$0.554241\pi$
$644$		0			0
$645$		−	1.77108e10i		−	0.102329i
$646$		0			0
$647$	2.75434e11			1.57181			0.785904	−	0.618348i	$-0.212198\pi$
$647$	2.75434e11			1.57181			0.785904	+	0.618348i	$0.212198\pi$
$648$		0			0
$649$			1.87115e11i			1.05470i
$650$		0			0
$651$	−1.66694e10			−0.0928101
$652$		0			0
$653$	−2.69428e11			−1.48180			−0.740900	−	0.671615i	$-0.765601\pi$
$653$	−2.69428e11			−1.48180			−0.740900	+	0.671615i	$0.765601\pi$
$654$		0			0
$655$	−4.77206e10			−0.259263
$656$		0			0
$657$	−9.21921e9			−0.0494803
$658$		0			0
$659$			8.80932e10i			0.467090i	0.972346	+	0.233545i	$0.0750326\pi$
$659$			8.80932e10i			0.467090i	−0.972346	+	0.233545i	$0.924967\pi$
$660$		0			0
$661$		−	2.53825e10i		−	0.132962i	−0.997788	−	0.0664811i	$-0.978823\pi$
$661$		−	2.53825e10i		−	0.132962i	0.997788	−	0.0664811i	$-0.0211772\pi$
$662$		0			0
$663$	1.02477e11			0.530364
$664$		0			0
$665$	−3.22643e9	+	3.18229e10i	−0.0164982	+	0.162724i
$666$		0			0
$667$			2.02716e11i			1.02420i
$668$		0			0
$669$	−1.70404e11			−0.850699
$670$		0			0
$671$	−2.88379e11			−1.42257
$672$		0			0
$673$		−	1.21341e11i		−	0.591490i	−0.955267	−	0.295745i	$-0.904432\pi$
$673$		−	1.21341e11i		−	0.591490i	0.955267	−	0.295745i	$-0.0955679\pi$
$674$		0			0
$675$			2.07049e11i			0.997373i
$676$		0			0
$677$		−	3.38640e11i		−	1.61207i	−0.591870	−	0.806034i	$-0.701610\pi$
$677$		−	3.38640e11i		−	1.61207i	0.591870	−	0.806034i	$-0.298390\pi$
$678$		0			0
$679$			3.89328e10i			0.183162i
$680$		0			0
$681$	2.90189e11			1.34925
$682$		0			0
$683$			1.36488e11i			0.627208i	0.949554	+	0.313604i	$0.101536\pi$
$683$			1.36488e11i			0.627208i	−0.949554	+	0.313604i	$0.898464\pi$
$684$		0			0
$685$	7.02004e10			0.318843
$686$		0			0
$687$			2.24592e10i			0.100825i
$688$		0			0
$689$	1.22671e11			0.544332
$690$		0			0
$691$	−3.85706e11			−1.69178			−0.845890	−	0.533358i	$-0.820930\pi$
$691$	−3.85706e11			−1.69178			−0.845890	+	0.533358i	$0.820930\pi$
$692$		0			0
$693$	−4.19891e10			−0.182055
$694$		0			0
$695$	−5.98692e10			−0.256604
$696$		0			0
$697$			3.17730e11i			1.34626i
$698$		0			0
$699$			3.41609e11i			1.43094i
$700$		0			0
$701$	4.23246e11			1.75275			0.876377	−	0.481626i	$-0.159954\pi$
$701$	4.23246e11			1.75275			0.876377	+	0.481626i	$0.159954\pi$
$702$		0			0
$703$	3.17721e11	+	3.22128e10i	1.30084	+	0.131889i
$704$		0			0
$705$		−	5.76981e10i		−	0.233563i
$706$		0			0
$707$	−6.89996e10			−0.276165
$708$		0			0
$709$	4.73862e10			0.187529			0.0937643	−	0.995594i	$-0.470110\pi$
$709$	4.73862e10			0.187529			0.0937643	+	0.995594i	$0.470110\pi$
$710$		0			0
$711$		−	5.45888e10i		−	0.213612i
$712$		0			0
$713$			4.73846e10i			0.183349i
$714$		0			0
$715$			4.10229e10i			0.156965i
$716$		0			0
$717$		−	1.38204e11i		−	0.522932i
$718$		0			0
$719$	1.26152e11			0.472039			0.236020	−	0.971748i	$-0.424157\pi$
$719$	1.26152e11			0.472039			0.236020	+	0.971748i	$0.424157\pi$
$720$		0			0
$721$			3.97711e10i			0.147173i
$722$		0			0
$723$	−3.24005e10			−0.118576
$724$		0			0
$725$		−	2.21941e11i		−	0.803315i
$726$		0			0
$727$	−8.49520e10			−0.304114			−0.152057	−	0.988372i	$-0.548590\pi$
$727$	−8.49520e10			−0.304114			−0.152057	+	0.988372i	$0.548590\pi$
$728$		0			0
$729$	−3.11841e11			−1.10414
$730$		0			0
$731$	2.04622e11			0.716608
$732$		0			0
$733$	5.23738e11			1.81426			0.907128	−	0.420855i	$-0.138270\pi$
$733$	5.23738e11			1.81426			0.907128	+	0.420855i	$0.138270\pi$
$734$		0			0
$735$		−	3.74306e10i		−	0.128256i
$736$		0			0
$737$			4.77903e11i			1.61983i
$738$		0			0
$739$	−3.44499e11			−1.15508			−0.577538	−	0.816364i	$-0.695986\pi$
$739$	−3.44499e11			−1.15508			−0.577538	+	0.816364i	$0.695986\pi$
$740$		0			0
$741$	1.01312e10	−	9.99260e10i	0.0336039	−	0.331441i
$742$		0			0
$743$			4.53922e10i			0.148945i	0.997223	+	0.0744726i	$0.0237273\pi$
$743$			4.53922e10i			0.148945i	−0.997223	+	0.0744726i	$0.976273\pi$
$744$		0			0
$745$	−8.39571e10			−0.272541
$746$		0			0
$747$	−3.90458e10			−0.125398
$748$		0			0
$749$		−	4.25054e10i		−	0.135057i
$750$		0			0
$751$			8.96270e10i			0.281760i	0.990027	+	0.140880i	$0.0449931\pi$
$751$			8.96270e10i			0.281760i	−0.990027	+	0.140880i	$0.955007\pi$
$752$		0			0
$753$			1.28709e11i			0.400341i
$754$		0			0
$755$			2.71243e9i			0.00834779i
$756$		0			0
$757$	−5.94573e11			−1.81060			−0.905298	−	0.424778i	$-0.860352\pi$
$757$	−5.94573e11			−1.81060			−0.905298	+	0.424778i	$0.860352\pi$
$758$		0			0
$759$		−	6.35934e11i		−	1.91622i
$760$		0			0
$761$	−4.29954e11			−1.28199			−0.640993	−	0.767547i	$-0.721477\pi$
$761$	−4.29954e11			−1.28199			−0.640993	+	0.767547i	$0.721477\pi$
$762$		0			0
$763$		−	1.75202e11i		−	0.516942i
$764$		0			0
$765$	2.13476e10			0.0623308
$766$		0			0
$767$	7.59419e10			0.219432
$768$		0			0
$769$	3.94606e11			1.12839			0.564194	−	0.825642i	$-0.309187\pi$
$769$	3.94606e11			1.12839			0.564194	+	0.825642i	$0.309187\pi$
$770$		0			0
$771$	5.56893e11			1.57599
$772$		0			0
$773$		−	3.34209e10i		−	0.0936053i	−0.998904	−	0.0468026i	$-0.985097\pi$
$773$		−	3.34209e10i		−	0.0936053i	0.998904	−	0.0468026i	$-0.0149032\pi$
$774$		0			0
$775$		−	5.18783e10i		−	0.143807i
$776$		0			0
$777$	2.88692e11			0.792046
$778$		0			0
$779$	3.09820e11	+	3.14118e10i	0.841316	+	0.0852988i
$780$		0			0
$781$		−	5.56049e11i		−	1.49455i
$782$		0			0
$783$	3.41831e11			0.909420
$784$		0			0
$785$	1.08203e11			0.284944
$786$		0			0
$787$		−	5.01627e11i		−	1.30762i	−0.756658	−	0.653811i	$-0.773169\pi$
$787$		−	5.01627e11i		−	1.30762i	0.756658	−	0.653811i	$-0.226831\pi$
$788$		0			0
$789$		−	3.47022e11i		−	0.895467i
$790$		0			0
$791$		−	2.40021e11i		−	0.613116i
$792$		0			0
$793$			1.17041e11i			0.295967i
$794$		0			0
$795$	−1.36150e11			−0.340840
$796$		0			0
$797$		−	3.80292e11i		−	0.942505i	−0.881998	−	0.471253i	$-0.843802\pi$
$797$		−	3.80292e11i		−	0.942505i	0.881998	−	0.471253i	$-0.156198\pi$
$798$		0			0
$799$	6.66614e11			1.63564
$800$		0			0
$801$			1.10582e11i			0.268631i
$802$		0			0
$803$	−2.27176e11			−0.546387
$804$		0			0
$805$	8.21951e10			0.195732
$806$		0			0
$807$	1.53533e11			0.362000
$808$		0			0
$809$	−2.33585e11			−0.545319			−0.272660	−	0.962111i	$-0.587903\pi$
$809$	−2.33585e11			−0.545319			−0.272660	+	0.962111i	$0.587903\pi$
$810$		0			0
$811$		−	2.37656e10i		−	0.0549371i	−0.999623	−	0.0274685i	$-0.991255\pi$
$811$		−	2.37656e10i		−	0.0549371i	0.999623	−	0.0274685i	$-0.00874461\pi$
$812$		0			0
$813$			9.27991e10i			0.212413i
$814$		0			0
$815$	4.69709e10			0.106463
$816$		0			0
$817$	2.02295e10	−	1.99527e11i	0.0454043	−	0.447831i
$818$		0			0
$819$			1.70416e10i			0.0378769i
$820$		0			0
$821$	−5.15004e11			−1.13354			−0.566771	−	0.823875i	$-0.691808\pi$
$821$	−5.15004e11			−1.13354			−0.566771	+	0.823875i	$0.691808\pi$
$822$		0			0
$823$	−2.28392e11			−0.497831			−0.248916	−	0.968525i	$-0.580074\pi$
$823$	−2.28392e11			−0.497831			−0.248916	+	0.968525i	$0.580074\pi$
$824$		0			0
$825$			6.96243e11i			1.50295i
$826$		0			0
$827$		−	6.97217e10i		−	0.149055i	−0.997219	−	0.0745274i	$-0.976255\pi$
$827$		−	6.97217e10i		−	0.149055i	0.997219	−	0.0745274i	$-0.0237448\pi$
$828$		0			0
$829$			4.08026e11i			0.863913i	0.901894	+	0.431956i	$0.142177\pi$
$829$			4.08026e11i			0.863913i	−0.901894	+	0.431956i	$0.857823\pi$
$830$		0			0
$831$			7.83318e9i			0.0164261i
$832$		0			0
$833$	4.32454e11			0.898173
$834$		0			0
$835$		−	2.14091e11i		−	0.440406i
$836$		0			0
$837$	7.99024e10			0.162801
$838$		0			0
$839$			9.01495e11i			1.81935i	0.415325	+	0.909673i	$0.363668\pi$
$839$			9.01495e11i			1.81935i	−0.415325	+	0.909673i	$0.636332\pi$
$840$		0			0
$841$	1.33828e11			0.267524
$842$		0			0
$843$	−6.87511e11			−1.36135
$844$		0			0
$845$	−1.09662e11			−0.215095
$846$		0			0
$847$	−6.94903e11			−1.35018
$848$		0			0
$849$			5.91583e11i			1.13864i
$850$		0			0
$851$		−	8.20639e11i		−	1.56471i
$852$		0			0
$853$	−5.99808e10			−0.113296			−0.0566482	−	0.998394i	$-0.518041\pi$
$853$	−5.99808e10			−0.113296			−0.0566482	+	0.998394i	$0.518041\pi$
$854$		0			0
$855$	2.11049e9	−	2.08161e10i	0.00394928	−	0.0389524i
$856$		0			0
$857$			2.70130e10i			0.0500783i	0.999686	+	0.0250391i	$0.00797104\pi$
$857$			2.70130e10i			0.0500783i	−0.999686	+	0.0250391i	$0.992029\pi$
$858$		0			0
$859$	7.98513e10			0.146659			0.0733296	−	0.997308i	$-0.476637\pi$
$859$	7.98513e10			0.146659			0.0733296	+	0.997308i	$0.476637\pi$
$860$		0			0
$861$	2.81513e11			0.512254
$862$		0			0
$863$			1.02525e12i			1.84836i	0.381954	+	0.924181i	$0.375251\pi$
$863$			1.02525e12i			1.84836i	−0.381954	+	0.924181i	$0.624749\pi$
$864$		0			0
$865$		−	9.77217e10i		−	0.174553i
$866$		0			0
$867$		−	7.95602e11i		−	1.40805i
$868$		0			0
$869$		−	1.34516e12i		−	2.35881i
$870$		0			0
$871$	1.93960e11			0.337008
$872$		0			0
$873$		−	2.54669e10i		−	0.0438448i
$874$		0			0
$875$	−1.85865e11			−0.317078
$876$		0			0
$877$			9.55040e11i			1.61445i	0.590247	+	0.807223i	$0.299030\pi$
$877$			9.55040e11i			1.61445i	−0.590247	+	0.807223i	$0.700970\pi$
$878$		0			0
$879$	−1.08725e11			−0.182127
$880$		0			0
$881$	1.13499e12			1.88403			0.942014	−	0.335573i	$-0.108930\pi$
$881$	1.13499e12			1.88403			0.942014	+	0.335573i	$0.108930\pi$
$882$		0			0
$883$	−2.78656e10			−0.0458379			−0.0229190	−	0.999737i	$-0.507296\pi$
$883$	−2.78656e10			−0.0458379			−0.0229190	+	0.999737i	$0.507296\pi$
$884$		0			0
$885$	−8.42868e10			−0.137400
$886$		0			0
$887$		−	6.71858e11i		−	1.08538i	−0.839932	−	0.542691i	$-0.817405\pi$
$887$		−	6.71858e11i		−	1.08538i	0.839932	−	0.542691i	$-0.182595\pi$
$888$		0			0
$889$		−	7.28159e11i		−	1.16579i
$890$		0			0
$891$	−8.98542e11			−1.42570
$892$		0			0
$893$	6.59035e10	−	6.50018e11i	0.103634	−	1.02216i
$894$		0			0
$895$		−	2.52491e11i		−	0.393508i
$896$		0			0
$897$	−2.58098e11			−0.398672
$898$		0			0
$899$	−8.56496e10			−0.131125
$900$		0			0
$901$		−	1.57301e12i		−	2.38689i
$902$		0			0
$903$		−	1.81297e11i		−	0.272672i
$904$		0			0
$905$			1.25733e11i			0.187437i
$906$		0			0
$907$		−	6.08911e11i		−	0.899757i	−0.893090	−	0.449878i	$-0.851467\pi$
$907$		−	6.08911e11i		−	0.899757i	0.893090	−	0.449878i	$-0.148533\pi$
$908$		0			0
$909$	4.51343e10			0.0661075
$910$		0			0
$911$		−	7.07571e10i		−	0.102730i	−0.998680	−	0.0513649i	$-0.983643\pi$
$911$		−	7.07571e10i		−	0.102730i	0.998680	−	0.0513649i	$-0.0163572\pi$
$912$		0			0
$913$	−9.62151e11			−1.38471
$914$		0			0
$915$		−	1.29902e11i		−	0.185324i
$916$		0			0
$917$	−4.88493e11			−0.690846
$918$		0			0
$919$	2.81891e11			0.395202			0.197601	−	0.980283i	$-0.436685\pi$
$919$	2.81891e11			0.395202			0.197601	+	0.980283i	$0.436685\pi$
$920$		0			0
$921$	1.18906e11			0.165259
$922$		0			0
$923$	−2.25677e11			−0.310943
$924$		0			0
$925$			8.98465e11i			1.22725i
$926$		0			0
$927$		−	2.60153e10i		−	0.0352297i
$928$		0			0
$929$	−1.08860e12			−1.46152			−0.730762	−	0.682632i	$-0.760835\pi$
$929$	−1.08860e12			−1.46152			−0.730762	+	0.682632i	$0.760835\pi$
$930$		0			0
$931$	4.27537e10	−	4.21687e11i	0.0569082	−	0.561296i
$932$		0			0
$933$		−	1.01358e12i		−	1.33761i
$934$		0			0
$935$	5.26038e11			0.688289
$936$		0			0
$937$	−8.13586e11			−1.05547			−0.527734	−	0.849410i	$-0.676958\pi$
$937$	−8.13586e11			−1.05547			−0.527734	+	0.849410i	$0.676958\pi$
$938$		0			0
$939$			5.08732e11i			0.654375i
$940$		0			0
$941$		−	1.38430e12i		−	1.76552i	−0.469827	−	0.882759i	$-0.655684\pi$
$941$		−	1.38430e12i		−	1.76552i	0.469827	−	0.882759i	$-0.344316\pi$
$942$		0			0
$943$		−	8.00232e11i		−	1.01197i
$944$		0			0
$945$		−	1.38602e11i		−	0.173797i
$946$		0			0
$947$	−5.32762e11			−0.662420			−0.331210	−	0.943557i	$-0.607457\pi$
$947$	−5.32762e11			−0.662420			−0.331210	+	0.943557i	$0.607457\pi$
$948$		0			0
$949$			9.22011e10i			0.113677i
$950$		0			0
$951$	−3.65036e11			−0.446286
$952$		0			0
$953$			5.73373e11i			0.695129i	0.937656	+	0.347564i	$0.112991\pi$
$953$			5.73373e11i			0.695129i	−0.937656	+	0.347564i	$0.887009\pi$
$954$		0			0
$955$	−1.77029e11			−0.212829
$956$		0			0
$957$	1.14948e12			1.37042
$958$		0			0
$959$	7.18608e11			0.849606
$960$		0			0
$961$	8.32871e11			0.976526
$962$		0			0
$963$			2.78038e10i			0.0323295i
$964$		0			0
$965$		−	1.87600e11i		−	0.216333i
$966$		0			0
$967$	1.36489e12			1.56096			0.780479	−	0.625182i	$-0.214975\pi$
$967$	1.36489e12			1.56096			0.780479	+	0.625182i	$0.214975\pi$
$968$		0			0
$969$	−1.28136e12	−	1.29913e11i	−1.45337	−	0.147353i
$970$		0			0
$971$			1.57922e12i			1.77650i	0.459361	+	0.888250i	$0.348078\pi$
$971$			1.57922e12i			1.77650i	−0.459361	+	0.888250i	$0.651922\pi$
$972$		0			0
$973$	−6.12853e11			−0.683762
$974$		0			0
$975$	2.82575e11			0.312691
$976$		0			0
$977$		−	5.51758e11i		−	0.605578i	−0.953058	−	0.302789i	$-0.902082\pi$
$977$		−	5.51758e11i		−	0.605578i	0.953058	−	0.302789i	$-0.0979178\pi$
$978$		0			0
$979$			2.72493e12i			2.96636i
$980$		0			0
$981$			1.14604e11i			0.123744i
$982$		0			0
$983$		−	8.60977e11i		−	0.922099i	−0.887374	−	0.461050i	$-0.847473\pi$
$983$		−	8.60977e11i		−	0.922099i	0.887374	−	0.461050i	$-0.152527\pi$
$984$		0			0
$985$	−7.73684e10			−0.0821900
$986$		0			0
$987$		−	5.90628e11i		−	0.622365i
$988$		0			0
$989$	−5.15358e11			−0.538671
$990$		0			0
$991$		−	1.27354e11i		−	0.132044i	−0.997818	−	0.0660220i	$-0.978969\pi$
$991$		−	1.27354e11i		−	0.132044i	0.997818	−	0.0660220i	$-0.0210308\pi$
$992$		0			0
$993$	1.25967e12			1.29556
$994$		0			0
$995$	3.17067e11			0.323488
$996$		0			0
$997$	−1.40891e12			−1.42595			−0.712975	−	0.701190i	$-0.752653\pi$
$997$	−1.40891e12			−1.42595			−0.712975	+	0.701190i	$0.752653\pi$
$998$		0			0
$999$	−1.38381e12			−1.38935

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.9.e.e.113.4		12
4.3	odd	2		38.9.b.a.37.4	✓	12
12.11	even	2		342.9.d.a.37.10		12
19.18	odd	2	inner	304.9.e.e.113.9		12
76.75	even	2		38.9.b.a.37.9	yes	12
228.227	odd	2		342.9.d.a.37.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.9.b.a.37.4	✓	12	4.3	odd	2
38.9.b.a.37.9	yes	12	76.75	even	2
304.9.e.e.113.4		12	1.1	even	1	trivial
304.9.e.e.113.9		12	19.18	odd	2	inner
342.9.d.a.37.4		12	228.227	odd	2
342.9.d.a.37.10		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)

\(f(q)\)	\(=\)	\(q-74.3247i q^{3} -154.845 q^{5} -1585.07 q^{7} +1036.83 q^{9} +O(q^{10})\)	\(q-74.3247i q^{3} -154.845 q^{5} -1585.07 q^{7} +1036.83 q^{9} +25549.2 q^{11} -10369.3i q^{13} +11508.8i q^{15} -132967. q^{17} +(-13145.5 + 129656. i) q^{19} +117810. i q^{21} +334889. q^{23} -366648. q^{25} -564707. i q^{27} +605325. i q^{29} +141494. i q^{31} -1.89894e6i q^{33} +245440. q^{35} -2.45048e6i q^{37} -770699. q^{39} -2.38955e6i q^{41} -1.53889e6 q^{43} -160548. q^{45} -5.01339e6 q^{47} -3.25235e6 q^{49} +9.88272e6i q^{51} +1.18301e7i q^{53} -3.95616e6 q^{55} +(9.63667e6 + 977036. i) q^{57} +7.32369e6i q^{59} -1.12872e7 q^{61} -1.64346e6 q^{63} +1.60564e6i q^{65} +1.87052e7i q^{67} -2.48905e7i q^{69} -2.17638e7i q^{71} -8.89170e6 q^{73} +2.72510e7i q^{75} -4.04974e7 q^{77} -5.26495e7i q^{79} -3.51690e7 q^{81} -3.76587e7 q^{83} +2.05892e7 q^{85} +4.49906e7 q^{87} +1.06654e8i q^{89} +1.64362e7i q^{91} +1.05165e7 q^{93} +(2.03551e6 - 2.00766e7i) q^{95} -2.45621e7i q^{97} +2.64903e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 558 q^{5} + 5422 q^{7} - 15592 q^{9}+O(q^{10}) \) 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

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