LMFDB - Embedded newform 76.9.c.b.37.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,9,Mod(37,76)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(76, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("76.37");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 76 = 2^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	76.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$30.9607743646$
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} + 62471 x^{10} + 1417247091 x^{8} + 14731726590693 x^{6} + \cdots + 18\!\cdots\!00 $ x^12 + 62471x^10 + 1417247091x^8 + 14731726590693x^6 + 76361585248208952x^4 + 190911088284097639680*x^2 + 180929129574476434636800
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{19}\cdot 3^{4}\cdot 7 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.2
Root			$-140.007i$ of defining polynomial
Character	$\chi$	$=$	76.37
Dual form			76.9.c.b.37.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-140.007i q^{3} -899.569 q^{5} -1723.79 q^{7} -13041.1 q^{9} +O(q^{10})$	$q-140.007i q^{3} -899.569 q^{5} -1723.79 q^{7} -13041.1 q^{9} +12145.8 q^{11} +15520.7i q^{13} +125946. i q^{15} -145834. q^{17} +(116191. - 59019.3i) q^{19} +241343. i q^{21} +113326. q^{23} +418599. q^{25} +907260. i q^{27} -1.29740e6i q^{29} -532079. i q^{31} -1.70050e6i q^{33} +1.55066e6 q^{35} +79715.0i q^{37} +2.17302e6 q^{39} +5.19911e6i q^{41} +579489. q^{43} +1.17314e7 q^{45} +2.29893e6 q^{47} -2.79336e6 q^{49} +2.04179e7i q^{51} +2.81266e6i q^{53} -1.09259e7 q^{55} +(-8.26314e6 - 1.62676e7i) q^{57} +1.69400e7i q^{59} +2.16451e7 q^{61} +2.24800e7 q^{63} -1.39620e7i q^{65} +3.62133e7i q^{67} -1.58664e7i q^{69} -2.75860e7i q^{71} +1.71116e6 q^{73} -5.86070e7i q^{75} -2.09367e7 q^{77} +1.73629e7i q^{79} +4.14606e7 q^{81} +1.65377e7 q^{83} +1.31188e8 q^{85} -1.81646e8 q^{87} -3.60825e6i q^{89} -2.67544e7i q^{91} -7.44951e7 q^{93} +(-1.04522e8 + 5.30919e7i) q^{95} +5.40415e7i q^{97} -1.58394e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 568 q^{5} + 1734 q^{7} - 46210 q^{9}+O(q^{10}) $ 12 * q - 568 * q^5 + 1734 * q^7 - 46210 * q^9	$ 12 q - 568 q^{5} + 1734 q^{7} - 46210 q^{9} + 1976 q^{11} - 47446 q^{17} - 113178 q^{19} + 378950 q^{23} - 502656 q^{25} - 1876832 q^{35} - 1463394 q^{39} + 6234396 q^{43} - 507200 q^{45} + 4902224 q^{47} - 16736298 q^{49} - 38366664 q^{55} - 19396170 q^{57} + 16602912 q^{61} - 11274020 q^{63} + 111268614 q^{73} - 134425928 q^{77} + 193590584 q^{81} + 34030388 q^{83} + 387605748 q^{85} - 217889946 q^{87} - 474880524 q^{93} - 256537036 q^{95} + 276136420 q^{99}+O(q^{100}) $ 12 * q - 568 * q^5 + 1734 * q^7 - 46210 * q^9 + 1976 * q^11 - 47446 * q^17 - 113178 * q^19 + 378950 * q^23 - 502656 * q^25 - 1876832 * q^35 - 1463394 * q^39 + 6234396 * q^43 - 507200 * q^45 + 4902224 * q^47 - 16736298 * q^49 - 38366664 * q^55 - 19396170 * q^57 + 16602912 * q^61 - 11274020 * q^63 + 111268614 * q^73 - 134425928 * q^77 + 193590584 * q^81 + 34030388 * q^83 + 387605748 * q^85 - 217889946 * q^87 - 474880524 * q^93 - 256537036 * q^95 + 276136420 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$.

$n$	$21$	$39$
$\chi(n)$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		−	140.007i		−	1.72849i	−0.503074	−	0.864243i	$-0.667798\pi$
$3$		−	140.007i		−	1.72849i	0.503074	−	0.864243i	$-0.332202\pi$
$4$		0			0
$5$	−899.569			−1.43931			−0.719655	−	0.694332i	$-0.755700\pi$
$5$	−899.569			−1.43931			−0.719655	+	0.694332i	$0.755700\pi$
$6$		0			0
$7$	−1723.79			−0.717945			−0.358973	−	0.933348i	$-0.616873\pi$
$7$	−1723.79			−0.717945			−0.358973	+	0.933348i	$0.616873\pi$
$8$		0			0
$9$	−13041.1			−1.98767
$10$		0			0
$11$	12145.8			0.829572			0.414786	−	0.909919i	$-0.363856\pi$
$11$	12145.8			0.829572			0.414786	+	0.909919i	$0.363856\pi$
$12$		0			0
$13$			15520.7i			0.543424i	0.962379	+	0.271712i	$0.0875897\pi$
$13$			15520.7i			0.543424i	−0.962379	+	0.271712i	$0.912410\pi$
$14$		0			0
$15$			125946.i			2.48783i
$16$		0			0
$17$	−145834.			−1.74608			−0.873039	−	0.487650i	$-0.837854\pi$
$17$	−145834.			−1.74608			−0.873039	+	0.487650i	$0.837854\pi$
$18$		0			0
$19$	116191.	−	59019.3i	0.891573	−	0.452876i
$19$	116191.	−	59019.3i	0.891573	−	0.452876i
$20$		0			0
$21$			241343.i			1.24096i
$22$		0			0
$23$	113326.			0.404965			0.202482	−	0.979286i	$-0.435099\pi$
$23$	113326.			0.404965			0.202482	+	0.979286i	$0.435099\pi$
$24$		0			0
$25$	418599.			1.07161
$26$		0			0
$27$			907260.i			1.70717i
$28$		0			0
$29$		−	1.29740e6i		−	1.83435i	−0.398482	−	0.917176i	$-0.630463\pi$
$29$		−	1.29740e6i		−	1.83435i	0.398482	−	0.917176i	$-0.369537\pi$
$30$		0			0
$31$		−	532079.i		−	0.576142i	−0.957609	−	0.288071i	$-0.906986\pi$
$31$		−	532079.i		−	0.576142i	0.957609	−	0.288071i	$-0.0930139\pi$
$32$		0			0
$33$		−	1.70050e6i		−	1.43390i
$34$		0			0
$35$	1.55066e6			1.03335
$36$		0			0
$37$			79715.0i			0.0425337i	0.999774	+	0.0212669i	$0.00676996\pi$
$37$			79715.0i			0.0425337i	−0.999774	+	0.0212669i	$0.993230\pi$
$38$		0			0
$39$	2.17302e6			0.939300
$40$		0			0
$41$			5.19911e6i			1.83990i	0.392037	+	0.919949i	$0.371771\pi$
$41$			5.19911e6i			1.83990i	−0.392037	+	0.919949i	$0.628229\pi$
$42$		0			0
$43$	579489.			0.169501			0.0847504	−	0.996402i	$-0.472991\pi$
$43$	579489.			0.169501			0.0847504	+	0.996402i	$0.472991\pi$
$44$		0			0
$45$	1.17314e7			2.86087
$46$		0			0
$47$	2.29893e6			0.471122			0.235561	−	0.971860i	$-0.424307\pi$
$47$	2.29893e6			0.471122			0.235561	+	0.971860i	$0.424307\pi$
$48$		0			0
$49$	−2.79336e6			−0.484555
$50$		0			0
$51$			2.04179e7i			3.01807i
$52$		0			0
$53$			2.81266e6i			0.356463i	0.983989	+	0.178232i	$0.0570376\pi$
$53$			2.81266e6i			0.356463i	−0.983989	+	0.178232i	$0.942962\pi$
$54$		0			0
$55$	−1.09259e7			−1.19401
$56$		0			0
$57$	−8.26314e6	−	1.62676e7i	−0.782791	−	1.54107i
$58$		0			0
$59$			1.69400e7i			1.39799i	0.715125	+	0.698996i	$0.246370\pi$
$59$			1.69400e7i			1.39799i	−0.715125	+	0.698996i	$0.753630\pi$
$60$		0			0
$61$	2.16451e7			1.56329			0.781646	−	0.623723i	$-0.214381\pi$
$61$	2.16451e7			1.56329			0.781646	+	0.623723i	$0.214381\pi$
$62$		0			0
$63$	2.24800e7			1.42704
$64$		0			0
$65$		−	1.39620e7i		−	0.782155i
$66$		0			0
$67$			3.62133e7i			1.79708i	0.438887	+	0.898542i	$0.355373\pi$
$67$			3.62133e7i			1.79708i	−0.438887	+	0.898542i	$0.644627\pi$
$68$		0			0
$69$		−	1.58664e7i		−	0.699976i
$70$		0			0
$71$		−	2.75860e7i		−	1.08556i	−0.839874	−	0.542782i	$-0.817371\pi$
$71$		−	2.75860e7i		−	1.08556i	0.839874	−	0.542782i	$-0.182629\pi$
$72$		0			0
$73$	1.71116e6			0.0602558			0.0301279	−	0.999546i	$-0.490409\pi$
$73$	1.71116e6			0.0602558			0.0301279	+	0.999546i	$0.490409\pi$
$74$		0			0
$75$		−	5.86070e7i		−	1.85227i
$76$		0			0
$77$	−2.09367e7			−0.595587
$78$		0			0
$79$			1.73629e7i			0.445774i	0.974844	+	0.222887i	$0.0715480\pi$
$79$			1.73629e7i			0.445774i	−0.974844	+	0.222887i	$0.928452\pi$
$80$		0			0
$81$	4.14606e7			0.963153
$82$		0			0
$83$	1.65377e7			0.348469			0.174234	−	0.984704i	$-0.444255\pi$
$83$	1.65377e7			0.348469			0.174234	+	0.984704i	$0.444255\pi$
$84$		0			0
$85$	1.31188e8			2.51315
$86$		0			0
$87$	−1.81646e8			−3.17065
$88$		0			0
$89$		−	3.60825e6i		−	0.0575091i	−0.999587	−	0.0287545i	$-0.990846\pi$
$89$		−	3.60825e6i		−	0.0575091i	0.999587	−	0.0287545i	$-0.00915411\pi$
$90$		0			0
$91$		−	2.67544e7i		−	0.390148i
$92$		0			0
$93$	−7.44951e7			−0.995854
$94$		0			0
$95$	−1.04522e8	+	5.30919e7i	−1.28325	+	0.651829i
$96$		0			0
$97$			5.40415e7i			0.610436i	0.952282	+	0.305218i	$0.0987294\pi$
$97$			5.40415e7i			0.610436i	−0.952282	+	0.305218i	$0.901271\pi$
$98$		0			0
$99$	−1.58394e8			−1.64891
$100$		0			0
$101$	−1.59464e8			−1.53241			−0.766207	−	0.642594i	$-0.777858\pi$
$101$	−1.59464e8			−1.53241			−0.766207	+	0.642594i	$0.777858\pi$
$102$		0			0
$103$			2.59129e7i			0.230233i	0.993352	+	0.115117i	$0.0367241\pi$
$103$			2.59129e7i			0.230233i	−0.993352	+	0.115117i	$0.963276\pi$
$104$		0			0
$105$		−	2.17105e8i		−	1.78612i
$106$		0			0
$107$		−	1.33757e8i		−	1.02043i	−0.860047	−	0.510215i	$-0.829566\pi$
$107$		−	1.33757e8i		−	1.02043i	0.860047	−	0.510215i	$-0.170434\pi$
$108$		0			0
$109$			1.83790e8i			1.30202i	0.759071	+	0.651008i	$0.225653\pi$
$109$			1.83790e8i			1.30202i	−0.759071	+	0.651008i	$0.774347\pi$
$110$		0			0
$111$	1.11607e7			0.0735189
$112$		0			0
$113$			2.85263e8i			1.74957i	0.484508	+	0.874787i	$0.338999\pi$
$113$			2.85263e8i			1.74957i	−0.484508	+	0.874787i	$0.661001\pi$
$114$		0			0
$115$	−1.01944e8			−0.582870
$116$		0			0
$117$		−	2.02407e8i		−	1.08014i
$118$		0			0
$119$	2.51387e8			1.25359
$120$		0			0
$121$	−6.68394e7			−0.311811
$122$		0			0
$123$	7.27915e8			3.18024
$124$		0			0
$125$	−2.51647e7			−0.103074
$126$		0			0
$127$		−	3.38203e8i		−	1.30006i	−0.759910	−	0.650028i	$-0.774757\pi$
$127$		−	3.38203e8i		−	1.30006i	0.759910	−	0.650028i	$-0.225243\pi$
$128$		0			0
$129$		−	8.11328e7i		−	0.292980i
$130$		0			0
$131$	−3.26795e8			−1.10966			−0.554831	−	0.831963i	$-0.687217\pi$
$131$	−3.26795e8			−1.10966			−0.554831	+	0.831963i	$0.687217\pi$
$132$		0			0
$133$	−2.00288e8	+	1.01737e8i	−0.640101	+	0.325140i
$134$		0			0
$135$		−	8.16143e8i		−	2.45715i
$136$		0			0
$137$	−2.97450e8			−0.844367			−0.422183	−	0.906510i	$-0.638736\pi$
$137$	−2.97450e8			−0.844367			−0.422183	+	0.906510i	$0.638736\pi$
$138$		0			0
$139$	−1.63371e7			−0.0437638			−0.0218819	−	0.999761i	$-0.506966\pi$
$139$	−1.63371e7			−0.0437638			−0.0218819	+	0.999761i	$0.506966\pi$
$140$		0			0
$141$		−	3.21867e8i		−	0.814329i
$142$		0			0
$143$			1.88511e8i			0.450809i
$144$		0			0
$145$			1.16710e9i			2.64020i
$146$		0			0
$147$			3.91092e8i			0.837547i
$148$		0			0
$149$	1.09846e8			0.222864			0.111432	−	0.993772i	$-0.464456\pi$
$149$	1.09846e8			0.222864			0.111432	+	0.993772i	$0.464456\pi$
$150$		0			0
$151$			1.71132e8i			0.329172i	0.986363	+	0.164586i	$0.0526288\pi$
$151$			1.71132e8i			0.329172i	−0.986363	+	0.164586i	$0.947371\pi$
$152$		0			0
$153$	1.90184e9			3.47062
$154$		0			0
$155$			4.78642e8i			0.829247i
$156$		0			0
$157$	1.56063e8			0.256864			0.128432	−	0.991718i	$-0.459006\pi$
$157$	1.56063e8			0.256864			0.128432	+	0.991718i	$0.459006\pi$
$158$		0			0
$159$	3.93794e8			0.616142
$160$		0			0
$161$	−1.95349e8			−0.290742
$162$		0			0
$163$	−1.02176e8			−0.144743			−0.0723714	−	0.997378i	$-0.523057\pi$
$163$	−1.02176e8			−0.144743			−0.0723714	+	0.997378i	$0.523057\pi$
$164$		0			0
$165$			1.52971e9i			2.06383i
$166$		0			0
$167$		−	1.19664e8i		−	0.153851i	−0.997037	−	0.0769253i	$-0.975490\pi$
$167$		−	1.19664e8i		−	0.153851i	0.997037	−	0.0769253i	$-0.0245103\pi$
$168$		0			0
$169$	5.74838e8			0.704691
$170$		0			0
$171$	−1.51525e9	+	7.69675e8i	−1.77215	+	0.900167i
$172$		0			0
$173$			1.44194e9i			1.60976i	0.593437	+	0.804881i	$0.297771\pi$
$173$			1.44194e9i			1.60976i	−0.593437	+	0.804881i	$0.702229\pi$
$174$		0			0
$175$	−7.21575e8			−0.769360
$176$		0			0
$177$	2.37172e9			2.41641
$178$		0			0
$179$		−	1.24827e9i		−	1.21589i	−0.793977	−	0.607947i	$-0.791993\pi$
$179$		−	1.24827e9i		−	1.21589i	0.793977	−	0.607947i	$-0.208007\pi$
$180$		0			0
$181$		−	1.14382e9i		−	1.06572i	−0.846202	−	0.532862i	$-0.821116\pi$
$181$		−	1.14382e9i		−	1.06572i	0.846202	−	0.532862i	$-0.178884\pi$
$182$		0			0
$183$		−	3.03047e9i		−	2.70213i
$184$		0			0
$185$		−	7.17091e7i		−	0.0612192i
$186$		0			0
$187$	−1.77127e9			−1.44850
$188$		0			0
$189$		−	1.56392e9i		−	1.22565i
$190$		0			0
$191$	2.22286e9			1.67024			0.835118	−	0.550070i	$-0.185399\pi$
$191$	2.22286e9			1.67024			0.835118	+	0.550070i	$0.185399\pi$
$192$		0			0
$193$			1.82148e9i			1.31279i	0.754416	+	0.656396i	$0.227920\pi$
$193$			1.82148e9i			1.31279i	−0.754416	+	0.656396i	$0.772080\pi$
$194$		0			0
$195$	−1.95478e9			−1.35194
$196$		0			0
$197$	−1.38616e9			−0.920337			−0.460169	−	0.887832i	$-0.652211\pi$
$197$	−1.38616e9			−0.920337			−0.460169	+	0.887832i	$0.652211\pi$
$198$		0			0
$199$	−1.29383e9			−0.825022			−0.412511	−	0.910953i	$-0.635348\pi$
$199$	−1.29383e9			−0.825022			−0.412511	+	0.910953i	$0.635348\pi$
$200$		0			0
$201$	5.07013e9			3.10624
$202$		0			0
$203$			2.23644e9i			1.31696i
$204$		0			0
$205$		−	4.67696e9i		−	2.64819i
$206$		0			0
$207$	−1.47789e9			−0.804935
$208$		0			0
$209$	1.41122e9	−	7.16834e8i	0.739624	−	0.375693i
$210$		0			0
$211$			8.39759e8i			0.423667i	0.977306	+	0.211834i	$0.0679435\pi$
$211$			8.39759e8i			0.423667i	−0.977306	+	0.211834i	$0.932057\pi$
$212$		0			0
$213$	−3.86225e9			−1.87638
$214$		0			0
$215$	−5.21290e8			−0.243964
$216$		0			0
$217$			9.17191e8i			0.413638i
$218$		0			0
$219$		−	2.39575e8i		−	0.104151i
$220$		0			0
$221$		−	2.26345e9i		−	0.948860i
$222$		0			0
$223$		−	1.95754e7i		−	0.00791572i	−0.999992	−	0.00395786i	$-0.998740\pi$
$223$		−	1.95754e7i		−	0.00791572i	0.999992	−	0.00395786i	$-0.00125983\pi$
$224$		0			0
$225$	−5.45899e9			−2.13001
$226$		0			0
$227$			3.20875e9i			1.20846i	0.796809	+	0.604231i	$0.206519\pi$
$227$			3.20875e9i			1.20846i	−0.796809	+	0.604231i	$0.793481\pi$
$228$		0			0
$229$	1.57011e9			0.570939			0.285469	−	0.958388i	$-0.407851\pi$
$229$	1.57011e9			0.570939			0.285469	+	0.958388i	$0.407851\pi$
$230$		0			0
$231$			2.93129e9i			1.02946i
$232$		0			0
$233$	−3.95641e9			−1.34239			−0.671193	−	0.741282i	$-0.734218\pi$
$233$	−3.95641e9			−1.34239			−0.671193	+	0.741282i	$0.734218\pi$
$234$		0			0
$235$	−2.06804e9			−0.678091
$236$		0			0
$237$	2.43094e9			0.770514
$238$		0			0
$239$	−3.09583e9			−0.948825			−0.474412	−	0.880303i	$-0.657339\pi$
$239$	−3.09583e9			−0.948825			−0.474412	+	0.880303i	$0.657339\pi$
$240$		0			0
$241$			4.53015e9i			1.34290i	0.741049	+	0.671451i	$0.234329\pi$
$241$			4.53015e9i			1.34290i	−0.741049	+	0.671451i	$0.765671\pi$
$242$		0			0
$243$			1.47742e8i			0.0423721i
$244$		0			0
$245$	2.51282e9			0.697425
$246$		0			0
$247$	9.16022e8	+	1.80336e9i	0.246104	+	0.484502i
$248$		0			0
$249$		−	2.31541e9i		−	0.602324i
$250$		0			0
$251$	−5.84455e9			−1.47250			−0.736251	−	0.676709i	$-0.763406\pi$
$251$	−5.84455e9			−1.47250			−0.736251	+	0.676709i	$0.763406\pi$
$252$		0			0
$253$	1.37643e9			0.335947
$254$		0			0
$255$		−	1.83673e10i		−	4.34394i
$256$		0			0
$257$		−	6.76945e9i		−	1.55175i	−0.630888	−	0.775874i	$-0.717309\pi$
$257$		−	6.76945e9i		−	1.55175i	0.630888	−	0.775874i	$-0.282691\pi$
$258$		0			0
$259$		−	1.37412e8i		−	0.0305369i
$260$		0			0
$261$			1.69195e10i			3.64608i
$262$		0			0
$263$	−8.99742e8			−0.188059			−0.0940297	−	0.995569i	$-0.529975\pi$
$263$	−8.99742e8			−0.188059			−0.0940297	+	0.995569i	$0.529975\pi$
$264$		0			0
$265$		−	2.53019e9i		−	0.513061i
$266$		0			0
$267$	−5.05181e8			−0.0994036
$268$		0			0
$269$		−	4.03192e9i		−	0.770022i	−0.922912	−	0.385011i	$-0.874198\pi$
$269$		−	4.03192e9i		−	0.770022i	0.922912	−	0.385011i	$-0.125802\pi$
$270$		0			0
$271$	8.54365e9			1.58404			0.792021	−	0.610494i	$-0.209029\pi$
$271$	8.54365e9			1.58404			0.792021	+	0.610494i	$0.209029\pi$
$272$		0			0
$273$	−3.74581e9			−0.674366
$274$		0			0
$275$	5.08420e9			0.888981
$276$		0			0
$277$	8.71433e9			1.48018			0.740091	−	0.672507i	$-0.234782\pi$
$277$	8.71433e9			1.48018			0.740091	+	0.672507i	$0.234782\pi$
$278$		0			0
$279$			6.93889e9i			1.14518i
$280$		0			0
$281$		−	7.80446e9i		−	1.25175i	−0.779923	−	0.625875i	$-0.784742\pi$
$281$		−	7.80446e9i		−	1.25175i	0.779923	−	0.625875i	$-0.215258\pi$
$282$		0			0
$283$	−1.86112e9			−0.290154			−0.145077	−	0.989420i	$-0.546343\pi$
$283$	−1.86112e9			−0.290154			−0.145077	+	0.989420i	$0.546343\pi$
$284$		0			0
$285$	7.43326e9	+	1.46338e10i	1.12668	+	2.21808i
$286$		0			0
$287$		−	8.96216e9i		−	1.32095i
$288$		0			0
$289$	1.42919e10			2.04879
$290$		0			0
$291$	7.56621e9			1.05513
$292$		0			0
$293$		−	2.62638e9i		−	0.356358i	−0.983998	−	0.178179i	$-0.942979\pi$
$293$		−	2.62638e9i		−	0.356358i	0.983998	−	0.178179i	$-0.0570206\pi$
$294$		0			0
$295$		−	1.52387e10i		−	2.01215i
$296$		0			0
$297$			1.10194e10i			1.41622i
$298$		0			0
$299$			1.75890e9i			0.220067i
$300$		0			0
$301$	−9.98915e8			−0.121692
$302$		0			0
$303$			2.23261e10i			2.64876i
$304$		0			0
$305$	−1.94712e10			−2.25006
$306$		0			0
$307$			1.13623e10i			1.27912i	0.768741	+	0.639561i	$0.220884\pi$
$307$			1.13623e10i			1.27912i	−0.768741	+	0.639561i	$0.779116\pi$
$308$		0			0
$309$	3.62800e9			0.397955
$310$		0			0
$311$	1.14839e10			1.22758			0.613788	−	0.789471i	$-0.289645\pi$
$311$	1.14839e10			1.22758			0.613788	+	0.789471i	$0.289645\pi$
$312$		0			0
$313$	6.19156e9			0.645094			0.322547	−	0.946554i	$-0.395461\pi$
$313$	6.19156e9			0.645094			0.322547	+	0.946554i	$0.395461\pi$
$314$		0			0
$315$	−2.02223e10			−2.05395
$316$		0			0
$317$			7.26753e9i			0.719697i	0.933011	+	0.359849i	$0.117172\pi$
$317$			7.26753e9i			0.719697i	−0.933011	+	0.359849i	$0.882828\pi$
$318$		0			0
$319$		−	1.57579e10i		−	1.52173i
$320$		0			0
$321$	−1.87270e10			−1.76380
$322$		0			0
$323$	−1.69446e10	+	8.60703e9i	−1.55676	+	0.790757i
$324$		0			0
$325$			6.49696e9i			0.582340i
$326$		0			0
$327$	2.57320e10			2.25052
$328$		0			0
$329$	−3.96286e9			−0.338240
$330$		0			0
$331$			5.55820e9i			0.463044i	0.972830	+	0.231522i	$0.0743705\pi$
$331$			5.55820e9i			0.463044i	−0.972830	+	0.231522i	$0.925630\pi$
$332$		0			0
$333$		−	1.03957e9i		−	0.0845428i
$334$		0			0
$335$		−	3.25763e10i		−	2.58656i
$336$		0			0
$337$			2.06254e10i			1.59913i	0.600582	+	0.799563i	$0.294936\pi$
$337$			2.06254e10i			1.59913i	−0.600582	+	0.799563i	$0.705064\pi$
$338$		0			0
$339$	3.99390e10			3.02411
$340$		0			0
$341$		−	6.46251e9i		−	0.477951i
$342$		0			0
$343$	1.47524e10			1.06583
$344$		0			0
$345$			1.42730e10i			1.00748i
$346$		0			0
$347$	−1.03538e10			−0.714135			−0.357068	−	0.934079i	$-0.616223\pi$
$347$	−1.03538e10			−0.714135			−0.357068	+	0.934079i	$0.616223\pi$
$348$		0			0
$349$	3.69749e9			0.249233			0.124617	−	0.992205i	$-0.460230\pi$
$349$	3.69749e9			0.249233			0.124617	+	0.992205i	$0.460230\pi$
$350$		0			0
$351$	−1.40813e10			−0.927716
$352$		0			0
$353$	−1.56735e10			−1.00941			−0.504706	−	0.863291i	$-0.668399\pi$
$353$	−1.56735e10			−1.00941			−0.504706	+	0.863291i	$0.668399\pi$
$354$		0			0
$355$			2.48155e10i			1.56246i
$356$		0			0
$357$		−	3.51960e10i		−	2.16681i
$358$		0			0
$359$	5.74277e9			0.345735			0.172867	−	0.984945i	$-0.444697\pi$
$359$	5.74277e9			0.345735			0.172867	+	0.984945i	$0.444697\pi$
$360$		0			0
$361$	1.00170e10	−	1.37150e10i	0.589806	−	0.807545i
$362$		0			0
$363$			9.35801e9i			0.538961i
$364$		0			0
$365$	−1.53931e9			−0.0867268
$366$		0			0
$367$	−3.15256e10			−1.73780			−0.868900	−	0.494989i	$-0.835172\pi$
$367$	−3.15256e10			−1.73780			−0.868900	+	0.494989i	$0.835172\pi$
$368$		0			0
$369$		−	6.78021e10i		−	3.65711i
$370$		0			0
$371$		−	4.84843e9i		−	0.255921i
$372$		0			0
$373$		−	2.61709e10i		−	1.35202i	−0.736893	−	0.676009i	$-0.763708\pi$
$373$		−	2.61709e10i		−	1.35202i	0.736893	−	0.676009i	$-0.236292\pi$
$374$		0			0
$375$			3.52324e9i			0.178163i
$376$		0			0
$377$	2.01366e10			0.996830
$378$		0			0
$379$		−	2.94000e10i		−	1.42492i	−0.701712	−	0.712461i	$-0.747581\pi$
$379$		−	2.94000e10i		−	1.42492i	0.701712	−	0.712461i	$-0.252419\pi$
$380$		0			0
$381$	−4.73509e10			−2.24713
$382$		0			0
$383$			4.12124e10i			1.91528i	0.287961	+	0.957642i	$0.407023\pi$
$383$			4.12124e10i			1.91528i	−0.287961	+	0.957642i	$0.592977\pi$
$384$		0			0
$385$	1.88340e10			0.857234
$386$		0			0
$387$	−7.55717e9			−0.336911
$388$		0			0
$389$	−1.98270e10			−0.865881			−0.432940	−	0.901423i	$-0.642524\pi$
$389$	−1.98270e10			−0.865881			−0.432940	+	0.901423i	$0.642524\pi$
$390$		0			0
$391$	−1.65268e10			−0.707100
$392$		0			0
$393$			4.57538e10i			1.91804i
$394$		0			0
$395$		−	1.56191e10i		−	0.641606i
$396$		0			0
$397$	−3.11874e10			−1.25550			−0.627750	−	0.778415i	$-0.716024\pi$
$397$	−3.11874e10			−1.25550			−0.627750	+	0.778415i	$0.716024\pi$
$398$		0			0
$399$	1.42439e10	+	2.80418e10i	0.562001	+	1.10641i
$400$		0			0
$401$		−	1.47850e10i		−	0.571798i	−0.958260	−	0.285899i	$-0.907708\pi$
$401$		−	1.47850e10i		−	0.571798i	0.958260	−	0.285899i	$-0.0922922\pi$
$402$		0			0
$403$	8.25825e9			0.313089
$404$		0			0
$405$	−3.72966e10			−1.38628
$406$		0			0
$407$			9.68199e8i			0.0352848i
$408$		0			0
$409$		−	1.60573e10i		−	0.573825i	−0.957957	−	0.286913i	$-0.907371\pi$
$409$		−	1.60573e10i		−	0.573825i	0.957957	−	0.286913i	$-0.0926289\pi$
$410$		0			0
$411$			4.16452e10i			1.45948i
$412$		0			0
$413$		−	2.92009e10i		−	1.00368i
$414$		0			0
$415$	−1.48768e10			−0.501555
$416$		0			0
$417$			2.28731e9i			0.0756452i
$418$		0			0
$419$	2.68823e10			0.872189			0.436095	−	0.899901i	$-0.356361\pi$
$419$	2.68823e10			0.872189			0.436095	+	0.899901i	$0.356361\pi$
$420$		0			0
$421$			8.28140e9i			0.263618i	0.991275	+	0.131809i	$0.0420786\pi$
$421$			8.28140e9i			0.263618i	−0.991275	+	0.131809i	$0.957921\pi$
$422$		0			0
$423$	−2.99805e10			−0.936435
$424$		0			0
$425$	−6.10461e10			−1.87112
$426$		0			0
$427$	−3.73115e10			−1.12236
$428$		0			0
$429$	2.63929e10			0.779217
$430$		0			0
$431$			2.46588e10i			0.714599i	0.933990	+	0.357300i	$0.116302\pi$
$431$			2.46588e10i			0.714599i	−0.933990	+	0.357300i	$0.883698\pi$
$432$		0			0
$433$			4.25433e10i			1.21026i	0.796126	+	0.605131i	$0.206879\pi$
$433$			4.25433e10i			1.21026i	−0.796126	+	0.605131i	$0.793121\pi$
$434$		0			0
$435$	1.63403e11			4.56355
$436$		0			0
$437$	1.31674e10	−	6.68840e9i	0.361056	−	0.183399i
$438$		0			0
$439$			2.67244e10i			0.719531i	0.933043	+	0.359765i	$0.117143\pi$
$439$			2.67244e10i			0.719531i	−0.933043	+	0.359765i	$0.882857\pi$
$440$		0			0
$441$	3.64285e10			0.963134
$442$		0			0
$443$	−1.68035e10			−0.436300			−0.218150	−	0.975915i	$-0.570002\pi$
$443$	−1.68035e10			−0.436300			−0.218150	+	0.975915i	$0.570002\pi$
$444$		0			0
$445$			3.24587e9i			0.0827734i
$446$		0			0
$447$		−	1.53793e10i		−	0.385218i
$448$		0			0
$449$		−	2.62692e10i		−	0.646340i	−0.946341	−	0.323170i	$-0.895251\pi$
$449$		−	2.62692e10i		−	0.646340i	0.946341	−	0.323170i	$-0.104749\pi$
$450$		0			0
$451$			6.31472e10i			1.52633i
$452$		0			0
$453$	2.39597e10			0.568969
$454$		0			0
$455$			2.40674e10i			0.561544i
$456$		0			0
$457$	3.60465e10			0.826415			0.413207	−	0.910637i	$-0.364408\pi$
$457$	3.60465e10			0.826415			0.413207	+	0.910637i	$0.364408\pi$
$458$		0			0
$459$		−	1.32310e11i		−	2.98085i
$460$		0			0
$461$	−7.49465e10			−1.65939			−0.829693	−	0.558219i	$-0.811485\pi$
$461$	−7.49465e10			−1.65939			−0.829693	+	0.558219i	$0.811485\pi$
$462$		0			0
$463$	6.14046e10			1.33622			0.668108	−	0.744064i	$-0.267104\pi$
$463$	6.14046e10			1.33622			0.668108	+	0.744064i	$0.267104\pi$
$464$		0			0
$465$	6.70134e10			1.43334
$466$		0			0
$467$	−6.56341e10			−1.37995			−0.689973	−	0.723835i	$-0.742378\pi$
$467$	−6.56341e10			−1.37995			−0.689973	+	0.723835i	$0.742378\pi$
$468$		0			0
$469$		−	6.24239e10i		−	1.29021i
$470$		0			0
$471$		−	2.18501e10i		−	0.443986i
$472$		0			0
$473$	7.03834e9			0.140613
$474$		0			0
$475$	4.86373e10	−	2.47054e10i	0.955422	−	0.485308i
$476$		0			0
$477$		−	3.66802e10i		−	0.708530i
$478$		0			0
$479$	−4.18821e10			−0.795585			−0.397792	−	0.917475i	$-0.630224\pi$
$479$	−4.18821e10			−0.795585			−0.397792	+	0.917475i	$0.630224\pi$
$480$		0			0
$481$	−1.23723e9			−0.0231138
$482$		0			0
$483$			2.73503e10i			0.502544i
$484$		0			0
$485$		−	4.86140e10i		−	0.878607i
$486$		0			0
$487$		−	1.54680e9i		−	0.0274992i	−0.999905	−	0.0137496i	$-0.995623\pi$
$487$		−	1.54680e9i		−	0.0274992i	0.999905	−	0.0137496i	$-0.00437677\pi$
$488$		0			0
$489$			1.43053e10i			0.250186i
$490$		0			0
$491$	−4.40272e10			−0.757522			−0.378761	−	0.925495i	$-0.623650\pi$
$491$	−4.40272e10			−0.757522			−0.378761	+	0.925495i	$0.623650\pi$
$492$		0			0
$493$			1.89206e11i			3.20292i
$494$		0			0
$495$	1.42486e11			2.37330
$496$		0			0
$497$			4.75524e10i			0.779375i
$498$		0			0
$499$	−1.45043e10			−0.233935			−0.116968	−	0.993136i	$-0.537317\pi$
$499$	−1.45043e10			−0.233935			−0.116968	+	0.993136i	$0.537317\pi$
$500$		0			0
$501$	−1.67539e10			−0.265929
$502$		0			0
$503$	−9.31407e10			−1.45501			−0.727507	−	0.686100i	$-0.759321\pi$
$503$	−9.31407e10			−1.45501			−0.727507	+	0.686100i	$0.759321\pi$
$504$		0			0
$505$	1.43448e11			2.20562
$506$		0			0
$507$		−	8.04816e10i		−	1.21805i
$508$		0			0
$509$			4.21005e10i			0.627215i	0.949553	+	0.313608i	$0.101538\pi$
$509$			4.21005e10i			0.627215i	−0.949553	+	0.313608i	$0.898462\pi$
$510$		0			0
$511$	−2.94967e9			−0.0432604
$512$		0			0
$513$	5.35458e10	+	1.05415e11i	0.773136	+	1.52207i
$514$		0			0
$515$		−	2.33105e10i		−	0.331377i
$516$		0			0
$517$	2.79222e10			0.390830
$518$		0			0
$519$	2.01882e11			2.78245
$520$		0			0
$521$			6.94132e10i			0.942088i	0.882110	+	0.471044i	$0.156123\pi$
$521$			6.94132e10i			0.942088i	−0.882110	+	0.471044i	$0.843877\pi$
$522$		0			0
$523$			8.35450e10i			1.11664i	0.829625	+	0.558320i	$0.188554\pi$
$523$			8.35450e10i			1.11664i	−0.829625	+	0.558320i	$0.811446\pi$
$524$		0			0
$525$			1.01026e11i			1.32983i
$526$		0			0
$527$			7.75954e10i			1.00599i
$528$		0			0
$529$	−6.54683e10			−0.836004
$530$		0			0
$531$		−	2.20916e11i		−	2.77874i
$532$		0			0
$533$	−8.06940e10			−0.999844
$534$		0			0
$535$			1.20324e11i			1.46871i
$536$		0			0
$537$	−1.74767e11			−2.10166
$538$		0			0
$539$	−3.39275e10			−0.401973
$540$		0			0
$541$	4.22537e10			0.493259			0.246630	−	0.969110i	$-0.420677\pi$
$541$	4.22537e10			0.493259			0.246630	+	0.969110i	$0.420677\pi$
$542$		0			0
$543$	−1.60144e11			−1.84209
$544$		0			0
$545$		−	1.65332e11i		−	1.87400i
$546$		0			0
$547$			3.32086e10i			0.370938i	0.982650	+	0.185469i	$0.0593804\pi$
$547$			3.32086e10i			0.370938i	−0.982650	+	0.185469i	$0.940620\pi$
$548$		0			0
$549$	−2.82275e11			−3.10730
$550$		0			0
$551$	−7.65718e10	−	1.50746e11i	−0.830735	−	1.63546i
$552$		0			0
$553$		−	2.99300e10i		−	0.320041i
$554$		0			0
$555$	−1.00398e10			−0.105817
$556$		0			0
$557$	5.09046e10			0.528855			0.264427	−	0.964406i	$-0.414817\pi$
$557$	5.09046e10			0.528855			0.264427	+	0.964406i	$0.414817\pi$
$558$		0			0
$559$			8.99409e9i			0.0921107i
$560$		0			0
$561$			2.47991e11i			2.50371i
$562$		0			0
$563$			5.64065e9i			0.0561430i	0.999606	+	0.0280715i	$0.00893662\pi$
$563$			5.64065e9i			0.0561430i	−0.999606	+	0.0280715i	$0.991063\pi$
$564$		0			0
$565$		−	2.56614e11i		−	2.51818i
$566$		0			0
$567$	−7.14692e10			−0.691491
$568$		0			0
$569$		−	1.70474e11i		−	1.62634i	−0.582029	−	0.813168i	$-0.697741\pi$
$569$		−	1.70474e11i		−	1.62634i	0.582029	−	0.813168i	$-0.302259\pi$
$570$		0			0
$571$	1.29925e11			1.22222			0.611109	−	0.791546i	$-0.290724\pi$
$571$	1.29925e11			1.22222			0.611109	+	0.791546i	$0.290724\pi$
$572$		0			0
$573$		−	3.11216e11i		−	2.88698i
$574$		0			0
$575$	4.74380e10			0.433966
$576$		0			0
$577$	1.02668e11			0.926255			0.463128	−	0.886292i	$-0.346727\pi$
$577$	1.02668e11			0.926255			0.463128	+	0.886292i	$0.346727\pi$
$578$		0			0
$579$	2.55021e11			2.26914
$580$		0			0
$581$	−2.85075e10			−0.250181
$582$		0			0
$583$			3.41620e10i			0.295712i
$584$		0			0
$585$			1.82079e11i			1.55466i
$586$		0			0
$587$	−1.05311e11			−0.886993			−0.443496	−	0.896276i	$-0.646262\pi$
$587$	−1.05311e11			−0.886993			−0.443496	+	0.896276i	$0.646262\pi$
$588$		0			0
$589$	−3.14029e10	−	6.18227e10i	−0.260921	−	0.513673i
$590$		0			0
$591$			1.94072e11i			1.59079i
$592$		0			0
$593$	−5.55119e9			−0.0448918			−0.0224459	−	0.999748i	$-0.507145\pi$
$593$	−5.55119e9			−0.0448918			−0.0224459	+	0.999748i	$0.507145\pi$
$594$		0			0
$595$	−2.26140e11			−1.80430
$596$		0			0
$597$			1.81146e11i			1.42604i
$598$		0			0
$599$		−	4.38065e10i		−	0.340276i	−0.985420	−	0.170138i	$-0.945579\pi$
$599$		−	4.38065e10i		−	0.340276i	0.985420	−	0.170138i	$-0.0544214\pi$
$600$		0			0
$601$		−	1.21257e11i		−	0.929415i	−0.885464	−	0.464707i	$-0.846160\pi$
$601$		−	1.21257e11i		−	0.929415i	0.885464	−	0.464707i	$-0.153840\pi$
$602$		0			0
$603$		−	4.72260e11i		−	3.57201i
$604$		0			0
$605$	6.01267e10			0.448792
$606$		0			0
$607$			1.63331e10i			0.120314i	0.998189	+	0.0601568i	$0.0191601\pi$
$607$			1.63331e10i			0.120314i	−0.998189	+	0.0601568i	$0.980840\pi$
$608$		0			0
$609$	3.13119e11			2.27636
$610$		0			0
$611$			3.56810e10i			0.256019i
$612$		0			0
$613$	6.00607e10			0.425352			0.212676	−	0.977123i	$-0.431782\pi$
$613$	6.00607e10			0.425352			0.212676	+	0.977123i	$0.431782\pi$
$614$		0			0
$615$	−6.54809e11			−4.57735
$616$		0			0
$617$	2.47002e11			1.70436			0.852178	−	0.523253i	$-0.175282\pi$
$617$	2.47002e11			1.70436			0.852178	+	0.523253i	$0.175282\pi$
$618$		0			0
$619$	−6.39869e10			−0.435841			−0.217921	−	0.975967i	$-0.569927\pi$
$619$	−6.39869e10			−0.435841			−0.217921	+	0.975967i	$0.569927\pi$
$620$		0			0
$621$			1.02816e11i			0.691343i
$622$		0			0
$623$			6.21985e9i			0.0412883i
$624$		0			0
$625$	−1.40878e11			−0.923258
$626$		0			0
$627$	−1.00362e11	−	1.97582e11i	−0.649381	−	1.27843i
$628$		0			0
$629$		−	1.16252e10i		−	0.0742672i
$630$		0			0
$631$	1.42026e11			0.895881			0.447941	−	0.894063i	$-0.352158\pi$
$631$	1.42026e11			0.895881			0.447941	+	0.894063i	$0.352158\pi$
$632$		0			0
$633$	1.17572e11			0.732303
$634$		0			0
$635$			3.04237e11i			1.87119i
$636$		0			0
$637$		−	4.33550e10i		−	0.263319i
$638$		0			0
$639$			3.59751e11i			2.15774i
$640$		0			0
$641$			4.73933e10i			0.280728i	0.990100	+	0.140364i	$0.0448272\pi$
$641$			4.73933e10i			0.280728i	−0.990100	+	0.140364i	$0.955173\pi$
$642$		0			0
$643$	−1.49643e11			−0.875411			−0.437705	−	0.899118i	$-0.644209\pi$
$643$	−1.49643e11			−0.875411			−0.437705	+	0.899118i	$0.644209\pi$
$644$		0			0
$645$			7.29845e10i			0.421689i
$646$		0			0
$647$	−2.81495e11			−1.60640			−0.803201	−	0.595708i	$-0.796872\pi$
$647$	−2.81495e11			−1.60640			−0.803201	+	0.595708i	$0.796872\pi$
$648$		0			0
$649$			2.05749e11i			1.15974i
$650$		0			0
$651$	1.28414e11			0.714969
$652$		0			0
$653$	−1.12609e11			−0.619329			−0.309665	−	0.950846i	$-0.600217\pi$
$653$	−1.12609e11			−0.619329			−0.309665	+	0.950846i	$0.600217\pi$
$654$		0			0
$655$	2.93975e11			1.59715
$656$		0			0
$657$	−2.23154e10			−0.119768
$658$		0			0
$659$		−	9.41903e10i		−	0.499418i	−0.968321	−	0.249709i	$-0.919665\pi$
$659$		−	9.41903e10i		−	0.499418i	0.968321	−	0.249709i	$-0.0803350\pi$
$660$		0			0
$661$			7.59207e10i			0.397699i	0.980030	+	0.198850i	$0.0637205\pi$
$661$			7.59207e10i			0.397699i	−0.980030	+	0.198850i	$0.936279\pi$
$662$		0			0
$663$	−3.16900e11			−1.64009
$664$		0			0
$665$	1.80173e11	−	9.15191e10i	0.921303	−	0.467978i
$666$		0			0
$667$		−	1.47029e11i		−	0.742848i
$668$		0			0
$669$	−2.74070e9			−0.0136822
$670$		0			0
$671$	2.62896e11			1.29686
$672$		0			0
$673$			2.50245e11i			1.21985i	0.792460	+	0.609924i	$0.208800\pi$
$673$			2.50245e11i			1.21985i	−0.792460	+	0.609924i	$0.791200\pi$
$674$		0			0
$675$			3.79778e11i			1.82943i
$676$		0			0
$677$		−	1.25548e11i		−	0.597663i	−0.954306	−	0.298832i	$-0.903403\pi$
$677$		−	1.25548e11i		−	0.597663i	0.954306	−	0.298832i	$-0.0965969\pi$
$678$		0			0
$679$		−	9.31560e10i		−	0.438260i
$680$		0			0
$681$	4.49249e11			2.08881
$682$		0			0
$683$			2.77435e10i			0.127491i	0.997966	+	0.0637453i	$0.0203045\pi$
$683$			2.77435e10i			0.127491i	−0.997966	+	0.0637453i	$0.979695\pi$
$684$		0			0
$685$	2.67576e11			1.21531
$686$		0			0
$687$		−	2.19828e11i		−	0.986860i
$688$		0			0
$689$	−4.36546e10			−0.193710
$690$		0			0
$691$	−2.05317e11			−0.900559			−0.450279	−	0.892888i	$-0.648676\pi$
$691$	−2.05317e11			−0.900559			−0.450279	+	0.892888i	$0.648676\pi$
$692$		0			0
$693$	2.73037e11			1.18383
$694$		0			0
$695$	1.46963e10			0.0629898
$696$		0			0
$697$		−	7.58209e11i		−	3.21261i
$698$		0			0
$699$			5.53927e11i			2.32030i
$700$		0			0
$701$	−1.53321e9			−0.00634935			−0.00317468	−	0.999995i	$-0.501011\pi$
$701$	−1.53321e9			−0.00634935			−0.00317468	+	0.999995i	$0.501011\pi$
$702$		0			0
$703$	4.70472e9	+	9.26215e9i	0.0192625	+	0.0379219i
$704$		0			0
$705$			2.89541e11i			1.17207i
$706$		0			0
$707$	2.74881e11			1.10019
$708$		0			0
$709$	−2.68252e11			−1.06159			−0.530796	−	0.847500i	$-0.678107\pi$
$709$	−2.68252e11			−1.06159			−0.530796	+	0.847500i	$0.678107\pi$
$710$		0			0
$711$		−	2.26431e11i		−	0.886049i
$712$		0			0
$713$		−	6.02983e10i		−	0.233317i
$714$		0			0
$715$		−	1.69579e11i		−	0.648854i
$716$		0			0
$717$			4.33440e11i			1.64003i
$718$		0			0
$719$	3.42763e11			1.28256			0.641282	−	0.767306i	$-0.278403\pi$
$719$	3.42763e11			1.28256			0.641282	+	0.767306i	$0.278403\pi$
$720$		0			0
$721$		−	4.46683e10i		−	0.165295i
$722$		0			0
$723$	6.34255e11			2.32119
$724$		0			0
$725$		−	5.43092e11i		−	1.96572i
$726$		0			0
$727$	−1.30708e11			−0.467914			−0.233957	−	0.972247i	$-0.575168\pi$
$727$	−1.30708e11			−0.467914			−0.233957	+	0.972247i	$0.575168\pi$
$728$		0			0
$729$	2.92708e11			1.03639
$730$		0			0
$731$	−8.45093e10			−0.295962
$732$		0			0
$733$	1.18718e10			0.0411246			0.0205623	−	0.999789i	$-0.493454\pi$
$733$	1.18718e10			0.0411246			0.0205623	+	0.999789i	$0.493454\pi$
$734$		0			0
$735$		−	3.51814e11i		−	1.20549i
$736$		0			0
$737$			4.39838e11i			1.49081i
$738$		0			0
$739$	3.02471e11			1.01416			0.507079	−	0.861899i	$-0.330725\pi$
$739$	3.02471e11			1.01416			0.507079	+	0.861899i	$0.330725\pi$
$740$		0			0
$741$	2.52484e11	−	1.28250e11i	0.837455	−	0.425387i
$742$		0			0
$743$		−	2.15054e11i		−	0.705655i	−0.935688	−	0.352827i	$-0.885220\pi$
$743$		−	2.15054e11i		−	0.705655i	0.935688	−	0.352827i	$-0.114780\pi$
$744$		0			0
$745$	−9.88143e10			−0.320771
$746$		0			0
$747$	−2.15670e11			−0.692640
$748$		0			0
$749$			2.30569e11i			0.732612i
$750$		0			0
$751$			1.90681e11i			0.599442i	0.954027	+	0.299721i	$0.0968936\pi$
$751$			1.90681e11i			0.599442i	−0.954027	+	0.299721i	$0.903106\pi$
$752$		0			0
$753$			8.18280e11i			2.54520i
$754$		0			0
$755$		−	1.53945e11i		−	0.473781i
$756$		0			0
$757$	−3.02110e11			−0.919986			−0.459993	−	0.887923i	$-0.652148\pi$
$757$	−3.02110e11			−0.919986			−0.459993	+	0.887923i	$0.652148\pi$
$758$		0			0
$759$		−	1.92710e11i		−	0.580680i
$760$		0			0
$761$	4.71298e11			1.40526			0.702631	−	0.711554i	$-0.252008\pi$
$761$	4.71298e11			1.40526			0.702631	+	0.711554i	$0.252008\pi$
$762$		0			0
$763$		−	3.16815e11i		−	0.934776i
$764$		0			0
$765$	−1.71083e12			−4.99530
$766$		0			0
$767$	−2.62921e11			−0.759702
$768$		0			0
$769$	−5.12876e11			−1.46658			−0.733292	−	0.679914i	$-0.762017\pi$
$769$	−5.12876e11			−1.46658			−0.733292	+	0.679914i	$0.762017\pi$
$770$		0			0
$771$	−9.47774e11			−2.68218
$772$		0			0
$773$		−	1.62223e10i		−	0.0454354i	−0.999742	−	0.0227177i	$-0.992768\pi$
$773$		−	1.62223e10i		−	0.0454354i	0.999742	−	0.0227177i	$-0.00723190\pi$
$774$		0			0
$775$		−	2.22728e11i		−	0.617402i
$776$		0			0
$777$	−1.92386e10			−0.0527826
$778$		0			0
$779$	3.06848e11	+	6.04089e11i	0.833247	+	1.64040i
$780$		0			0
$781$		−	3.35053e11i		−	0.900553i
$782$		0			0
$783$	1.17708e12			3.13155
$784$		0			0
$785$	−1.40390e11			−0.369707
$786$		0			0
$787$			3.30712e11i			0.862086i	0.902331	+	0.431043i	$0.141854\pi$
$787$			3.30712e11i			0.862086i	−0.902331	+	0.431043i	$0.858146\pi$
$788$		0			0
$789$			1.25971e11i			0.325058i
$790$		0			0
$791$		−	4.91733e11i		−	1.25610i
$792$		0			0
$793$			3.35947e11i			0.849529i
$794$		0			0
$795$	−3.54245e11			−0.886819
$796$		0			0
$797$			4.39885e11i			1.09020i	0.838371	+	0.545100i	$0.183508\pi$
$797$			4.39885e11i			1.09020i	−0.838371	+	0.545100i	$0.816492\pi$
$798$		0			0
$799$	−3.35262e11			−0.822617
$800$		0			0
$801$			4.70554e10i			0.114309i
$802$		0			0
$803$	2.07833e10			0.0499865
$804$		0			0
$805$	1.75730e11			0.418468
$806$		0			0
$807$	−5.64499e11			−1.33097
$808$		0			0
$809$	−1.38159e11			−0.322541			−0.161270	−	0.986910i	$-0.551559\pi$
$809$	−1.38159e11			−0.322541			−0.161270	+	0.986910i	$0.551559\pi$
$810$		0			0
$811$			6.25363e11i			1.44560i	0.691056	+	0.722801i	$0.257146\pi$
$811$			6.25363e11i			1.44560i	−0.691056	+	0.722801i	$0.742854\pi$
$812$		0			0
$813$		−	1.19618e12i		−	2.73799i
$814$		0			0
$815$	9.19140e10			0.208330
$816$		0			0
$817$	6.73313e10	−	3.42010e10i	0.151122	−	0.0767629i
$818$		0			0
$819$			3.48906e11i			0.775485i
$820$		0			0
$821$	−3.49028e10			−0.0768224			−0.0384112	−	0.999262i	$-0.512230\pi$
$821$	−3.49028e10			−0.0768224			−0.0384112	+	0.999262i	$0.512230\pi$
$822$		0			0
$823$	7.54263e11			1.64408			0.822041	−	0.569429i	$-0.192836\pi$
$823$	7.54263e11			1.64408			0.822041	+	0.569429i	$0.192836\pi$
$824$		0			0
$825$		−	7.11826e11i		−	1.53659i
$826$		0			0
$827$		−	1.90166e11i		−	0.406547i	−0.979122	−	0.203273i	$-0.934842\pi$
$827$		−	1.90166e11i		−	0.406547i	0.979122	−	0.203273i	$-0.0651580\pi$
$828$		0			0
$829$			5.67264e11i			1.20107i	0.799600	+	0.600533i	$0.205045\pi$
$829$			5.67264e11i			1.20107i	−0.799600	+	0.600533i	$0.794955\pi$
$830$		0			0
$831$		−	1.22007e12i		−	2.55847i
$832$		0			0
$833$	4.07368e11			0.846071
$834$		0			0
$835$			1.07646e11i			0.221439i
$836$		0			0
$837$	4.82734e11			0.983572
$838$		0			0
$839$			8.88701e11i			1.79353i	0.442511	+	0.896763i	$0.354088\pi$
$839$			8.88701e11i			1.79353i	−0.442511	+	0.896763i	$0.645912\pi$
$840$		0			0
$841$	−1.18301e12			−2.36485
$842$		0			0
$843$	−1.09268e12			−2.16363
$844$		0			0
$845$	−5.17106e11			−1.01427
$846$		0			0
$847$	1.15217e11			0.223863
$848$		0			0
$849$			2.60571e11i			0.501528i
$850$		0			0
$851$			9.03376e9i			0.0172246i
$852$		0			0
$853$	8.30200e11			1.56815			0.784073	−	0.620668i	$-0.213139\pi$
$853$	8.30200e11			1.56815			0.784073	+	0.620668i	$0.213139\pi$
$854$		0			0
$855$	1.36307e12	−	6.92376e11i	2.55067	−	1.29562i
$856$		0			0
$857$		−	2.70008e11i		−	0.500557i	−0.968174	−	0.250279i	$-0.919478\pi$
$857$		−	2.70008e11i		−	0.500557i	0.968174	−	0.250279i	$-0.0805222\pi$
$858$		0			0
$859$	4.38318e11			0.805039			0.402520	−	0.915411i	$-0.368135\pi$
$859$	4.38318e11			0.805039			0.402520	+	0.915411i	$0.368135\pi$
$860$		0			0
$861$	−1.25477e12			−2.28324
$862$		0			0
$863$			1.16257e11i			0.209593i	0.994494	+	0.104797i	$0.0334192\pi$
$863$			1.16257e11i			0.209593i	−0.994494	+	0.104797i	$0.966581\pi$
$864$		0			0
$865$		−	1.29712e12i		−	2.31695i
$866$		0			0
$867$		−	2.00097e12i		−	3.54130i
$868$		0			0
$869$			2.10886e11i			0.369801i
$870$		0			0
$871$	−5.62056e11			−0.976578
$872$		0			0
$873$		−	7.04760e11i		−	1.21334i
$874$		0			0
$875$	4.33785e10			0.0740018
$876$		0			0
$877$		−	1.20324e11i		−	0.203402i	−0.994815	−	0.101701i	$-0.967572\pi$
$877$		−	1.20324e11i		−	0.203402i	0.994815	−	0.101701i	$-0.0324285\pi$
$878$		0			0
$879$	−3.67713e11			−0.615961
$880$		0			0
$881$	4.18299e11			0.694358			0.347179	−	0.937799i	$-0.387140\pi$
$881$	4.18299e11			0.694358			0.347179	+	0.937799i	$0.387140\pi$
$882$		0			0
$883$	5.84880e11			0.962108			0.481054	−	0.876691i	$-0.340254\pi$
$883$	5.84880e11			0.962108			0.481054	+	0.876691i	$0.340254\pi$
$884$		0			0
$885$	−2.13353e12			−3.47797
$886$		0			0
$887$			1.08014e12i			1.74496i	0.488653	+	0.872478i	$0.337488\pi$
$887$			1.08014e12i			1.74496i	−0.488653	+	0.872478i	$0.662512\pi$
$888$		0			0
$889$			5.82989e11i			0.933369i
$890$		0			0
$891$	5.03570e11			0.799004
$892$		0			0
$893$	2.67114e11	−	1.35681e11i	0.420040	−	0.213360i
$894$		0			0
$895$			1.12290e12i			1.75005i
$896$		0			0
$897$	2.46259e11			0.380383
$898$		0			0
$899$	−6.90321e11			−1.05685
$900$		0			0
$901$		−	4.10183e11i		−	0.622412i
$902$		0			0
$903$			1.39856e11i			0.210343i
$904$		0			0
$905$			1.02895e12i			1.53391i
$906$		0			0
$907$			4.61051e11i			0.681271i	0.940195	+	0.340636i	$0.110642\pi$
$907$			4.61051e11i			0.681271i	−0.940195	+	0.340636i	$0.889358\pi$
$908$		0			0
$909$	2.07958e12			3.04593
$910$		0			0
$911$		−	8.54642e10i		−	0.124083i	−0.998074	−	0.0620413i	$-0.980239\pi$
$911$		−	8.54642e10i		−	0.124083i	0.998074	−	0.0620413i	$-0.0197610\pi$
$912$		0			0
$913$	2.00863e11			0.289080
$914$		0			0
$915$			2.72612e12i			3.88920i
$916$		0			0
$917$	5.63325e11			0.796676
$918$		0			0
$919$	−9.09978e11			−1.27576			−0.637880	−	0.770136i	$-0.720188\pi$
$919$	−9.09978e11			−1.27576			−0.637880	+	0.770136i	$0.720188\pi$
$920$		0			0
$921$	1.59080e12			2.21094
$922$		0			0
$923$	4.28155e11			0.589921
$924$		0			0
$925$			3.33686e10i			0.0455797i
$926$		0			0
$927$		−	3.37933e11i		−	0.457627i
$928$		0			0
$929$	−1.32475e11			−0.177856			−0.0889282	−	0.996038i	$-0.528344\pi$
$929$	−1.32475e11			−0.177856			−0.0889282	+	0.996038i	$0.528344\pi$
$930$		0			0
$931$	−3.24563e11	+	1.64862e11i	−0.432016	+	0.219443i
$932$		0			0
$933$		−	1.60783e12i		−	2.12185i
$934$		0			0
$935$	1.59338e12			2.08484
$936$		0			0
$937$	−5.04349e11			−0.654294			−0.327147	−	0.944973i	$-0.606087\pi$
$937$	−5.04349e11			−0.654294			−0.327147	+	0.944973i	$0.606087\pi$
$938$		0			0
$939$		−	8.66864e11i		−	1.11504i
$940$		0			0
$941$			1.20682e12i			1.53916i	0.638548	+	0.769582i	$0.279535\pi$
$941$			1.20682e12i			1.53916i	−0.638548	+	0.769582i	$0.720465\pi$
$942$		0			0
$943$			5.89193e11i			0.745094i
$944$		0			0
$945$			1.40686e12i			1.76410i
$946$		0			0
$947$	3.79553e11			0.471925			0.235962	−	0.971762i	$-0.424176\pi$
$947$	3.79553e11			0.471925			0.235962	+	0.971762i	$0.424176\pi$
$948$		0			0
$949$			2.65584e10i			0.0327444i
$950$		0			0
$951$	1.01751e12			1.24399
$952$		0			0
$953$		−	3.40784e11i		−	0.413150i	−0.978431	−	0.206575i	$-0.933768\pi$
$953$		−	3.40784e11i		−	0.413150i	0.978431	−	0.206575i	$-0.0662318\pi$
$954$		0			0
$955$	−1.99961e12			−2.40399
$956$		0			0
$957$	−2.20623e12			−2.63028
$958$		0			0
$959$	5.12739e11			0.606209
$960$		0			0
$961$	5.69783e11			0.668060
$962$		0			0
$963$			1.74434e12i			2.02827i
$964$		0			0
$965$		−	1.63855e12i		−	1.88952i
$966$		0			0
$967$	−9.85663e11			−1.12726			−0.563628	−	0.826029i	$-0.690595\pi$
$967$	−9.85663e11			−1.12726			−0.563628	+	0.826029i	$0.690595\pi$
$968$		0			0
$969$	1.20505e12	+	2.37237e12i	1.36681	+	2.69083i
$970$		0			0
$971$			7.27131e11i			0.817967i	0.912542	+	0.408984i	$0.134117\pi$
$971$			7.27131e11i			0.817967i	−0.912542	+	0.408984i	$0.865883\pi$
$972$		0			0
$973$	2.81616e10			0.0314200
$974$		0			0
$975$	9.09623e11			1.00657
$976$		0			0
$977$		−	1.03612e12i		−	1.13718i	−0.822620	−	0.568592i	$-0.807489\pi$
$977$		−	1.03612e12i		−	1.13718i	0.822620	−	0.568592i	$-0.192511\pi$
$978$		0			0
$979$		−	4.38249e10i		−	0.0477079i
$980$		0			0
$981$		−	2.39682e12i		−	2.58797i
$982$		0			0
$983$			5.55008e11i			0.594409i	0.954814	+	0.297204i	$0.0960543\pi$
$983$			5.55008e11i			0.594409i	−0.954814	+	0.297204i	$0.903946\pi$
$984$		0			0
$985$	1.24694e12			1.32465
$986$		0			0
$987$			5.54830e11i			0.584644i
$988$		0			0
$989$	6.56710e10			0.0686418
$990$		0			0
$991$		−	1.39169e12i		−	1.44294i	−0.692444	−	0.721472i	$-0.743466\pi$
$991$		−	1.39169e12i		−	1.44294i	0.692444	−	0.721472i	$-0.256534\pi$
$992$		0			0
$993$	7.78189e11			0.800365
$994$		0			0
$995$	1.16389e12			1.18746
$996$		0			0
$997$	8.16802e11			0.826678			0.413339	−	0.910577i	$-0.364362\pi$
$997$	8.16802e11			0.826678			0.413339	+	0.910577i	$0.364362\pi$
$998$		0			0
$999$	−7.23222e10			−0.0726122

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.9.c.b.37.2	✓	12
4.3	odd	2		304.9.e.c.113.11		12
19.18	odd	2	inner	76.9.c.b.37.11	yes	12
76.75	even	2		304.9.e.c.113.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.9.c.b.37.2	✓	12	1.1	even	1	trivial
76.9.c.b.37.11	yes	12	19.18	odd	2	inner
304.9.e.c.113.2		12	76.75	even	2
304.9.e.c.113.11		12	4.3	odd	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:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	76.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-140.007i q^{3} -899.569 q^{5} -1723.79 q^{7} -13041.1 q^{9} +O(q^{10})\)	\(q-140.007i q^{3} -899.569 q^{5} -1723.79 q^{7} -13041.1 q^{9} +12145.8 q^{11} +15520.7i q^{13} +125946. i q^{15} -145834. q^{17} +(116191. - 59019.3i) q^{19} +241343. i q^{21} +113326. q^{23} +418599. q^{25} +907260. i q^{27} -1.29740e6i q^{29} -532079. i q^{31} -1.70050e6i q^{33} +1.55066e6 q^{35} +79715.0i q^{37} +2.17302e6 q^{39} +5.19911e6i q^{41} +579489. q^{43} +1.17314e7 q^{45} +2.29893e6 q^{47} -2.79336e6 q^{49} +2.04179e7i q^{51} +2.81266e6i q^{53} -1.09259e7 q^{55} +(-8.26314e6 - 1.62676e7i) q^{57} +1.69400e7i q^{59} +2.16451e7 q^{61} +2.24800e7 q^{63} -1.39620e7i q^{65} +3.62133e7i q^{67} -1.58664e7i q^{69} -2.75860e7i q^{71} +1.71116e6 q^{73} -5.86070e7i q^{75} -2.09367e7 q^{77} +1.73629e7i q^{79} +4.14606e7 q^{81} +1.65377e7 q^{83} +1.31188e8 q^{85} -1.81646e8 q^{87} -3.60825e6i q^{89} -2.67544e7i q^{91} -7.44951e7 q^{93} +(-1.04522e8 + 5.30919e7i) q^{95} +5.40415e7i q^{97} -1.58394e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 568 q^{5} + 1734 q^{7} - 46210 q^{9}+O(q^{10}) \) 12 * q - 568 * q^5 + 1734 * q^7 - 46210 * q^9	\( 12 q - 568 q^{5} + 1734 q^{7} - 46210 q^{9} + 1976 q^{11} - 47446 q^{17} - 113178 q^{19} + 378950 q^{23} - 502656 q^{25} - 1876832 q^{35} - 1463394 q^{39} + 6234396 q^{43} - 507200 q^{45} + 4902224 q^{47} - 16736298 q^{49} - 38366664 q^{55} - 19396170 q^{57} + 16602912 q^{61} - 11274020 q^{63} + 111268614 q^{73} - 134425928 q^{77} + 193590584 q^{81} + 34030388 q^{83} + 387605748 q^{85} - 217889946 q^{87} - 474880524 q^{93} - 256537036 q^{95} + 276136420 q^{99}+O(q^{100}) \) 12 * q - 568 * q^5 + 1734 * q^7 - 46210 * q^9 + 1976 * q^11 - 47446 * q^17 - 113178 * q^19 + 378950 * q^23 - 502656 * q^25 - 1876832 * q^35 - 1463394 * q^39 + 6234396 * q^43 - 507200 * q^45 + 4902224 * q^47 - 16736298 * q^49 - 38366664 * q^55 - 19396170 * q^57 + 16602912 * q^61 - 11274020 * q^63 + 111268614 * q^73 - 134425928 * q^77 + 193590584 * q^81 + 34030388 * q^83 + 387605748 * q^85 - 217889946 * q^87 - 474880524 * q^93 - 256537036 * q^95 + 276136420 * q^99

Introduction

L-functions

Modular forms

Varieties

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