LMFDB - Embedded newform 256.11.c.m.255.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,11,Mod(255,256)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(256, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 11, names="a")

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

chi := DirichletCharacter("256.255");

S:= CuspForms(chi, 11);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 11 $
Character orbit:	$[\chi]$	$=$	256.c (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:	$162.651456684$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{14} + 61429 x^{12} - 23865589 x^{10} + 9433993075 x^{8} - 796642244899 x^{6} + \cdots + 11\!\cdots\!56 $ x^16 - x^14 + 61429x^12 - 23865589x^10 + 9433993075x^8 - 796642244899x^6 + 422044223147719x^4 - 3251830843526311x^2 + 11210521480874115856
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{178}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			255.15
Root			$11.1953 - 8.43092i$ of defining polynomial
Character	$\chi$	$=$	256.255
Dual form			256.11.c.m.255.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+352.099i q^{3} -3773.58 q^{5} -15618.2i q^{7} -64924.4 q^{9} +115158. i q^{11} +546374. q^{13} -1.32867e6i q^{15} -1.50100e6 q^{17} +3.45753e6i q^{19} +5.49914e6 q^{21} +3.90931e6i q^{23} +4.47430e6 q^{25} -2.06872e6i q^{27} -1.57435e7 q^{29} +1.36335e7i q^{31} -4.05471e7 q^{33} +5.89365e7i q^{35} +7.96225e7 q^{37} +1.92377e8i q^{39} +1.01409e7 q^{41} -4.09302e7i q^{43} +2.44998e8 q^{45} +2.36040e8i q^{47} +3.85474e7 q^{49} -5.28499e8i q^{51} -3.09768e8 q^{53} -4.34559e8i q^{55} -1.21739e9 q^{57} +6.26191e7i q^{59} -1.02129e9 q^{61} +1.01400e9i q^{63} -2.06179e9 q^{65} +6.25414e8i q^{67} -1.37646e9 q^{69} +2.05543e9i q^{71} -2.44117e9 q^{73} +1.57539e9i q^{75} +1.79856e9 q^{77} -3.04489e9i q^{79} -3.10533e9 q^{81} +2.75511e9i q^{83} +5.66413e9 q^{85} -5.54325e9i q^{87} -2.51922e8 q^{89} -8.53337e9i q^{91} -4.80035e9 q^{93} -1.30473e10i q^{95} -1.56384e10 q^{97} -7.47658e9i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 70992 q^{9} - 741600 q^{17} + 37585520 q^{25} - 148617408 q^{33} - 185338656 q^{41} - 160864496 q^{49} - 2801425728 q^{57} - 1678056960 q^{65} - 5600145440 q^{73} - 23818130352 q^{81} - 12325192224 q^{89}+ \cdots - 19395727072 q^{97}+O(q^{100}) $ 16 * q + 70992 * q^9 - 741600 * q^17 + 37585520 * q^25 - 148617408 * q^33 - 185338656 * q^41 - 160864496 * q^49 - 2801425728 * q^57 - 1678056960 * q^65 - 5600145440 * q^73 - 23818130352 * q^81 - 12325192224 * q^89 - 19395727072 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/256\mathbb{Z}\right)^\times$.

$n$	$5$	$255$
$\chi(n)$	$1$	$-1$

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
$3$	352.099i	1.44897i	0.689293	+	0.724483i	$0.257921\pi$
$3$	352.099i	1.44897i	−0.689293	+	0.724483i	$0.742079\pi$
$4$	0	0
$5$	−3773.58	−1.20755	−0.603773	−	0.797156i	$-0.706337\pi$
$5$	−3773.58	−1.20755	−0.603773	+	0.797156i	$0.706337\pi$
$6$	0	0
$7$	− 15618.2i	− 0.929267i	−0.885503	−	0.464633i	$-0.846186\pi$
$7$	− 15618.2i	− 0.929267i	0.885503	−	0.464633i	$-0.153814\pi$
$8$	0	0
$9$	−64924.4	−1.09950
$10$	0	0
$11$	115158.i	0.715042i	0.933905	+	0.357521i	$0.116378\pi$
$11$	115158.i	0.715042i	−0.933905	+	0.357521i	$0.883622\pi$
$12$	0	0
$13$	546374.	1.47154	0.735772	−	0.677230i	$-0.236820\pi$
$13$	546374.	1.47154	0.735772	+	0.677230i	$0.236820\pi$
$14$	0	0
$15$	− 1.32867e6i	− 1.74969i
$16$	0	0
$17$	−1.50100e6	−1.05715	−0.528573	−	0.848888i	$-0.677273\pi$
$17$	−1.50100e6	−1.05715	−0.528573	+	0.848888i	$0.677273\pi$
$18$	0	0
$19$	3.45753e6i	1.39636i	0.715922	+	0.698180i	$0.246007\pi$
$19$	3.45753e6i	1.39636i	−0.715922	+	0.698180i	$0.753993\pi$
$20$	0	0
$21$	5.49914e6	1.34648
$22$	0	0
$23$	3.90931e6i	0.607380i	0.952771	+	0.303690i	$0.0982188\pi$
$23$	3.90931e6i	0.607380i	−0.952771	+	0.303690i	$0.901781\pi$
$24$	0	0
$25$	4.47430e6	0.458168
$26$	0	0
$27$	− 2.06872e6i	− 0.144173i
$28$	0	0
$29$	−1.57435e7	−0.767557	−0.383778	−	0.923425i	$-0.625377\pi$
$29$	−1.57435e7	−0.767557	−0.383778	+	0.923425i	$0.625377\pi$
$30$	0	0
$31$	1.36335e7i	0.476212i	0.971239	+	0.238106i	$0.0765265\pi$
$31$	1.36335e7i	0.476212i	−0.971239	+	0.238106i	$0.923473\pi$
$32$	0	0
$33$	−4.05471e7	−1.03607
$34$	0	0
$35$	5.89365e7i	1.12213i
$36$	0	0
$37$	7.96225e7	1.14823	0.574113	−	0.818776i	$-0.305347\pi$
$37$	7.96225e7	1.14823	0.574113	+	0.818776i	$0.305347\pi$
$38$	0	0
$39$	1.92377e8i	2.13222i
$40$	0	0
$41$	1.01409e7	0.0875298	0.0437649	−	0.999042i	$-0.486065\pi$
$41$	1.01409e7	0.0875298	0.0437649	+	0.999042i	$0.486065\pi$
$42$	0	0
$43$	− 4.09302e7i	− 0.278421i	−0.990263	−	0.139210i	$-0.955544\pi$
$43$	− 4.09302e7i	− 0.278421i	0.990263	−	0.139210i	$-0.0444565\pi$
$44$	0	0
$45$	2.44998e8	1.32770
$46$	0	0
$47$	2.36040e8i	1.02919i	0.857433	+	0.514595i	$0.172058\pi$
$47$	2.36040e8i	1.02919i	−0.857433	+	0.514595i	$0.827942\pi$
$48$	0	0
$49$	3.85474e7	0.136463
$50$	0	0
$51$	− 5.28499e8i	− 1.53177i
$52$	0	0
$53$	−3.09768e8	−0.740725	−0.370362	−	0.928887i	$-0.620766\pi$
$53$	−3.09768e8	−0.740725	−0.370362	+	0.928887i	$0.620766\pi$
$54$	0	0
$55$	− 4.34559e8i	− 0.863447i
$56$	0	0
$57$	−1.21739e9	−2.02328
$58$	0	0
$59$	6.26191e7i	0.0875885i	0.999041	+	0.0437942i	$0.0139446\pi$
$59$	6.26191e7i	0.0875885i	−0.999041	+	0.0437942i	$0.986055\pi$
$60$	0	0
$61$	−1.02129e9	−1.20921	−0.604603	−	0.796527i	$-0.706668\pi$
$61$	−1.02129e9	−1.20921	−0.604603	+	0.796527i	$0.706668\pi$
$62$	0	0
$63$	1.01400e9i	1.02173i
$64$	0	0
$65$	−2.06179e9	−1.77696
$66$	0	0
$67$	6.25414e8i	0.463227i	0.972808	+	0.231613i	$0.0744004\pi$
$67$	6.25414e8i	0.463227i	−0.972808	+	0.231613i	$0.925600\pi$
$68$	0	0
$69$	−1.37646e9	−0.880073
$70$	0	0
$71$	2.05543e9i	1.13923i	0.821913	+	0.569614i	$0.192907\pi$
$71$	2.05543e9i	1.13923i	−0.821913	+	0.569614i	$0.807093\pi$
$72$	0	0
$73$	−2.44117e9	−1.17756	−0.588781	−	0.808292i	$-0.700392\pi$
$73$	−2.44117e9	−1.17756	−0.588781	+	0.808292i	$0.700392\pi$
$74$	0	0
$75$	1.57539e9i	0.663870i
$76$	0	0
$77$	1.79856e9	0.664465
$78$	0	0
$79$	− 3.04489e9i	− 0.989548i	−0.869022	−	0.494774i	$-0.835251\pi$
$79$	− 3.04489e9i	− 0.989548i	0.869022	−	0.494774i	$-0.164749\pi$
$80$	0	0
$81$	−3.10533e9	−0.890599
$82$	0	0
$83$	2.75511e9i	0.699438i	0.936855	+	0.349719i	$0.113723\pi$
$83$	2.75511e9i	0.699438i	−0.936855	+	0.349719i	$0.886277\pi$
$84$	0	0
$85$	5.66413e9	1.27655
$86$	0	0
$87$	− 5.54325e9i	− 1.11216i
$88$	0	0
$89$	−2.51922e8	−0.0451145	−0.0225573	−	0.999746i	$-0.507181\pi$
$89$	−2.51922e8	−0.0451145	−0.0225573	+	0.999746i	$0.507181\pi$
$90$	0	0
$91$	− 8.53337e9i	− 1.36746i
$92$	0	0
$93$	−4.80035e9	−0.690014
$94$	0	0
$95$	− 1.30473e10i	− 1.68617i
$96$	0	0
$97$	−1.56384e10	−1.82110	−0.910549	−	0.413401i	$-0.864341\pi$
$97$	−1.56384e10	−1.82110	−0.910549	+	0.413401i	$0.864341\pi$
$98$	0	0
$99$	− 7.47658e9i	− 0.786189i
$100$	0	0
$101$	3.85788e9	0.367064	0.183532	−	0.983014i	$-0.441247\pi$
$101$	3.85788e9	0.367064	0.183532	+	0.983014i	$0.441247\pi$
$102$	0	0
$103$	− 9.28145e9i	− 0.800626i	−0.916378	−	0.400313i	$-0.868901\pi$
$103$	− 9.28145e9i	− 0.800626i	0.916378	−	0.400313i	$-0.131099\pi$
$104$	0	0
$105$	−2.07515e10	−1.62593
$106$	0	0
$107$	3.91137e9i	0.278875i	0.990231	+	0.139438i	$0.0445294\pi$
$107$	3.91137e9i	0.278875i	−0.990231	+	0.139438i	$0.955471\pi$
$108$	0	0
$109$	1.09799e10	0.713619	0.356809	−	0.934177i	$-0.383865\pi$
$109$	1.09799e10	0.713619	0.356809	+	0.934177i	$0.383865\pi$
$110$	0	0
$111$	2.80350e10i	1.66374i
$112$	0	0
$113$	2.02898e10	1.10125	0.550624	−	0.834753i	$-0.314390\pi$
$113$	2.02898e10	1.10125	0.550624	+	0.834753i	$0.314390\pi$
$114$	0	0
$115$	− 1.47521e10i	− 0.733440i
$116$	0	0
$117$	−3.54730e10	−1.61796
$118$	0	0
$119$	2.34428e10i	0.982371i
$120$	0	0
$121$	1.26760e10	0.488714
$122$	0	0
$123$	3.57058e9i	0.126828i
$124$	0	0
$125$	1.99672e10	0.654287
$126$	0	0
$127$	− 6.26328e9i	− 0.189576i	−0.995497	−	0.0947880i	$-0.969783\pi$
$127$	− 6.26328e9i	− 0.189576i	0.995497	−	0.0947880i	$-0.0302173\pi$
$128$	0	0
$129$	1.44115e10	0.403422
$130$	0	0
$131$	− 4.99170e9i	− 0.129387i	−0.997905	−	0.0646937i	$-0.979393\pi$
$131$	− 4.99170e9i	− 0.129387i	0.997905	−	0.0646937i	$-0.0206070\pi$
$132$	0	0
$133$	5.40003e10	1.29759
$134$	0	0
$135$	7.80649e9i	0.174095i
$136$	0	0
$137$	5.31439e10	1.10116	0.550580	−	0.834782i	$-0.314406\pi$
$137$	5.31439e10	1.10116	0.550580	+	0.834782i	$0.314406\pi$
$138$	0	0
$139$	− 3.24537e10i	− 0.625446i	−0.949844	−	0.312723i	$-0.898759\pi$
$139$	− 3.24537e10i	− 0.625446i	0.949844	−	0.312723i	$-0.101241\pi$
$140$	0	0
$141$	−8.31093e10	−1.49126
$142$	0	0
$143$	6.29195e10i	1.05222i
$144$	0	0
$145$	5.94093e10	0.926860
$146$	0	0
$147$	1.35725e10i	0.197730i
$148$	0	0
$149$	−9.33485e10	−1.27109	−0.635544	−	0.772064i	$-0.719224\pi$
$149$	−9.33485e10	−1.27109	−0.635544	+	0.772064i	$0.719224\pi$
$150$	0	0
$151$	− 1.79317e8i	− 0.00228422i	−0.999999	−	0.00114211i	$-0.999636\pi$
$151$	− 1.79317e8i	− 0.00228422i	0.999999	−	0.00114211i	$-0.000363545\pi$
$152$	0	0
$153$	9.74513e10	1.16233
$154$	0	0
$155$	− 5.14473e10i	− 0.575048i
$156$	0	0
$157$	−4.41399e10	−0.462736	−0.231368	−	0.972866i	$-0.574320\pi$
$157$	−4.41399e10	−0.462736	−0.231368	+	0.972866i	$0.574320\pi$
$158$	0	0
$159$	− 1.09069e11i	− 1.07328i
$160$	0	0
$161$	6.10563e10	0.564419
$162$	0	0
$163$	− 1.06005e11i	− 0.921272i	−0.887589	−	0.460636i	$-0.847621\pi$
$163$	− 1.06005e11i	− 0.921272i	0.887589	−	0.460636i	$-0.152379\pi$
$164$	0	0
$165$	1.53008e11	1.25110
$166$	0	0
$167$	1.65219e11i	1.27198i	0.771699	+	0.635988i	$0.219407\pi$
$167$	1.65219e11i	1.27198i	−0.771699	+	0.635988i	$0.780593\pi$
$168$	0	0
$169$	1.60666e11	1.16544
$170$	0	0
$171$	− 2.24478e11i	− 1.53530i
$172$	0	0
$173$	2.73295e11	1.76361	0.881803	−	0.471617i	$-0.156330\pi$
$173$	2.73295e11	1.76361	0.881803	+	0.471617i	$0.156330\pi$
$174$	0	0
$175$	− 6.98805e10i	− 0.425761i
$176$	0	0
$177$	−2.20481e10	−0.126913
$178$	0	0
$179$	− 1.55618e11i	− 0.846824i	−0.905937	−	0.423412i	$-0.860832\pi$
$179$	− 1.55618e11i	− 0.846824i	0.905937	−	0.423412i	$-0.139168\pi$
$180$	0	0
$181$	1.55132e10	0.0798562	0.0399281	−	0.999203i	$-0.487287\pi$
$181$	1.55132e10	0.0798562	0.0399281	+	0.999203i	$0.487287\pi$
$182$	0	0
$183$	− 3.59595e11i	− 1.75210i
$184$	0	0
$185$	−3.00462e11	−1.38654
$186$	0	0
$187$	− 1.72852e11i	− 0.755904i
$188$	0	0
$189$	−3.23097e10	−0.133975
$190$	0	0
$191$	2.19148e11i	0.862127i	0.902322	+	0.431063i	$0.141861\pi$
$191$	2.19148e11i	0.862127i	−0.902322	+	0.431063i	$0.858139\pi$
$192$	0	0
$193$	1.83821e11	0.686451	0.343225	−	0.939253i	$-0.388481\pi$
$193$	1.83821e11	0.686451	0.343225	+	0.939253i	$0.388481\pi$
$194$	0	0
$195$	− 7.25952e11i	− 2.57475i
$196$	0	0
$197$	−2.89571e10	−0.0975943	−0.0487971	−	0.998809i	$-0.515539\pi$
$197$	−2.89571e10	−0.0975943	−0.0487971	+	0.998809i	$0.515539\pi$
$198$	0	0
$199$	− 3.25273e11i	− 1.04228i	−0.853473	−	0.521138i	$-0.825508\pi$
$199$	− 3.25273e11i	− 1.04228i	0.853473	−	0.521138i	$-0.174492\pi$
$200$	0	0
$201$	−2.20207e11	−0.671199
$202$	0	0
$203$	2.45884e11i	0.713265i
$204$	0	0
$205$	−3.82674e10	−0.105696
$206$	0	0
$207$	− 2.53810e11i	− 0.667815i
$208$	0	0
$209$	−3.98163e11	−0.998457
$210$	0	0
$211$	6.20459e11i	1.48354i	0.670652	+	0.741772i	$0.266014\pi$
$211$	6.20459e11i	1.48354i	−0.670652	+	0.741772i	$0.733986\pi$
$212$	0	0
$213$	−7.23713e11	−1.65070
$214$	0	0
$215$	1.54454e11i	0.336206i
$216$	0	0
$217$	2.12931e11	0.442528
$218$	0	0
$219$	− 8.59533e11i	− 1.70625i
$220$	0	0
$221$	−8.20105e11	−1.55564
$222$	0	0
$223$	− 5.25673e11i	− 0.953217i	−0.879116	−	0.476608i	$-0.841866\pi$
$223$	− 5.25673e11i	− 0.953217i	0.879116	−	0.476608i	$-0.158134\pi$
$224$	0	0
$225$	−2.90491e11	−0.503756
$226$	0	0
$227$	− 8.73761e11i	− 1.44965i	−0.688933	−	0.724825i	$-0.741921\pi$
$227$	− 8.73761e11i	− 1.44965i	0.688933	−	0.724825i	$-0.258079\pi$
$228$	0	0
$229$	8.30219e11	1.31830	0.659151	−	0.752010i	$-0.270916\pi$
$229$	8.30219e11	1.31830	0.659151	+	0.752010i	$0.270916\pi$
$230$	0	0
$231$	6.33272e11i	0.962787i
$232$	0	0
$233$	−8.33377e11	−1.21356	−0.606781	−	0.794869i	$-0.707540\pi$
$233$	−8.33377e11	−1.21356	−0.606781	+	0.794869i	$0.707540\pi$
$234$	0	0
$235$	− 8.90716e11i	− 1.24280i
$236$	0	0
$237$	1.07210e12	1.43382
$238$	0	0
$239$	9.19325e11i	1.17891i	0.807802	+	0.589454i	$0.200657\pi$
$239$	9.19325e11i	1.17891i	−0.807802	+	0.589454i	$0.799343\pi$
$240$	0	0
$241$	−1.56744e12	−1.92799	−0.963997	−	0.265912i	$-0.914327\pi$
$241$	−1.56744e12	−1.92799	−0.963997	+	0.265912i	$0.914327\pi$
$242$	0	0
$243$	− 1.21554e12i	− 1.43462i
$244$	0	0
$245$	−1.45462e11	−0.164785
$246$	0	0
$247$	1.88910e12i	2.05481i
$248$	0	0
$249$	−9.70072e11	−1.01346
$250$	0	0
$251$	− 1.83947e12i	− 1.84639i	−0.384331	−	0.923195i	$-0.625568\pi$
$251$	− 1.83947e12i	− 1.84639i	0.384331	−	0.923195i	$-0.374432\pi$
$252$	0	0
$253$	−4.50189e11	−0.434303
$254$	0	0
$255$	1.99433e12i	1.84968i
$256$	0	0
$257$	−4.69431e11	−0.418703	−0.209352	−	0.977840i	$-0.567135\pi$
$257$	−4.69431e11	−0.418703	−0.209352	+	0.977840i	$0.567135\pi$
$258$	0	0
$259$	− 1.24356e12i	− 1.06701i
$260$	0	0
$261$	1.02214e12	0.843929
$262$	0	0
$263$	− 2.02740e12i	− 1.61124i	−0.592433	−	0.805619i	$-0.701833\pi$
$263$	− 2.02740e12i	− 1.61124i	0.592433	−	0.805619i	$-0.298167\pi$
$264$	0	0
$265$	1.16893e12	0.894459
$266$	0	0
$267$	− 8.87015e10i	− 0.0653694i
$268$	0	0
$269$	−1.54905e12	−1.09977	−0.549886	−	0.835239i	$-0.685329\pi$
$269$	−1.54905e12	−1.09977	−0.549886	+	0.835239i	$0.685329\pi$
$270$	0	0
$271$	− 2.74756e12i	− 1.87975i	−0.341514	−	0.939877i	$-0.610940\pi$
$271$	− 2.74756e12i	− 1.87975i	0.341514	−	0.939877i	$-0.389060\pi$
$272$	0	0
$273$	3.00459e12	1.98140
$274$	0	0
$275$	5.15253e11i	0.327610i
$276$	0	0
$277$	−3.44260e11	−0.211100	−0.105550	−	0.994414i	$-0.533660\pi$
$277$	−3.44260e11	−0.211100	−0.105550	+	0.994414i	$0.533660\pi$
$278$	0	0
$279$	− 8.85149e11i	− 0.523595i
$280$	0	0
$281$	5.16881e11	0.295025	0.147513	−	0.989060i	$-0.452873\pi$
$281$	5.16881e11	0.295025	0.147513	+	0.989060i	$0.452873\pi$
$282$	0	0
$283$	− 9.80974e11i	− 0.540413i	−0.962802	−	0.270206i	$-0.912908\pi$
$283$	− 9.80974e11i	− 0.540413i	0.962802	−	0.270206i	$-0.0870919\pi$
$284$	0	0
$285$	4.59392e12	2.44320
$286$	0	0
$287$	− 1.58382e11i	− 0.0813385i
$288$	0	0
$289$	2.36996e11	0.117558
$290$	0	0
$291$	− 5.50625e12i	− 2.63871i
$292$	0	0
$293$	−3.78962e12	−1.75492	−0.877461	−	0.479648i	$-0.840764\pi$
$293$	−3.78962e12	−1.75492	−0.877461	+	0.479648i	$0.840764\pi$
$294$	0	0
$295$	− 2.36298e11i	− 0.105767i
$296$	0	0
$297$	2.38230e11	0.103090
$298$	0	0
$299$	2.13594e12i	0.893787i
$300$	0	0
$301$	−6.39256e11	−0.258727
$302$	0	0
$303$	1.35836e12i	0.531864i
$304$	0	0
$305$	3.85393e12	1.46017
$306$	0	0
$307$	4.74199e12i	1.73888i	0.494041	+	0.869438i	$0.335519\pi$
$307$	4.74199e12i	1.73888i	−0.494041	+	0.869438i	$0.664481\pi$
$308$	0	0
$309$	3.26798e12	1.16008
$310$	0	0
$311$	− 1.77863e12i	− 0.611340i	−0.952138	−	0.305670i	$-0.901120\pi$
$311$	− 1.77863e12i	− 0.611340i	0.952138	−	0.305670i	$-0.0988804\pi$
$312$	0	0
$313$	2.19482e12	0.730595	0.365297	−	0.930891i	$-0.380967\pi$
$313$	2.19482e12	0.730595	0.365297	+	0.930891i	$0.380967\pi$
$314$	0	0
$315$	− 3.82642e12i	− 1.23379i
$316$	0	0
$317$	−2.30472e12	−0.719981	−0.359991	−	0.932956i	$-0.617220\pi$
$317$	−2.30472e12	−0.719981	−0.359991	+	0.932956i	$0.617220\pi$
$318$	0	0
$319$	− 1.81299e12i	− 0.548836i
$320$	0	0
$321$	−1.37719e12	−0.404080
$322$	0	0
$323$	− 5.18974e12i	− 1.47616i
$324$	0	0
$325$	2.44464e12	0.674215
$326$	0	0
$327$	3.86601e12i	1.03401i
$328$	0	0
$329$	3.68651e12	0.956393
$330$	0	0
$331$	− 3.59393e12i	− 0.904544i	−0.891880	−	0.452272i	$-0.850614\pi$
$331$	− 3.59393e12i	− 0.904544i	0.891880	−	0.452272i	$-0.149386\pi$
$332$	0	0
$333$	−5.16944e12	−1.26247
$334$	0	0
$335$	− 2.36005e12i	− 0.559368i
$336$	0	0
$337$	−1.56362e12	−0.359734	−0.179867	−	0.983691i	$-0.557567\pi$
$337$	−1.56362e12	−0.359734	−0.179867	+	0.983691i	$0.557567\pi$
$338$	0	0
$339$	7.14401e12i	1.59567i
$340$	0	0
$341$	−1.57001e12	−0.340512
$342$	0	0
$343$	− 5.01379e12i	− 1.05608i
$344$	0	0
$345$	5.19419e12	1.06273
$346$	0	0
$347$	1.77379e10i	0.00352578i	0.999998	+	0.00176289i	$0.000561145\pi$
$347$	1.77379e10i	0.00352578i	−0.999998	+	0.00176289i	$0.999439\pi$
$348$	0	0
$349$	−1.94015e12	−0.374722	−0.187361	−	0.982291i	$-0.559993\pi$
$349$	−1.94015e12	−0.374722	−0.187361	+	0.982291i	$0.559993\pi$
$350$	0	0
$351$	− 1.13030e12i	− 0.212157i
$352$	0	0
$353$	4.41804e12	0.806038	0.403019	−	0.915192i	$-0.367961\pi$
$353$	4.41804e12	0.806038	0.403019	+	0.915192i	$0.367961\pi$
$354$	0	0
$355$	− 7.75633e12i	− 1.37567i
$356$	0	0
$357$	−8.25419e12	−1.42342
$358$	0	0
$359$	4.43076e12i	0.743029i	0.928427	+	0.371515i	$0.121161\pi$
$359$	4.43076e12i	0.743029i	−0.928427	+	0.371515i	$0.878839\pi$
$360$	0	0
$361$	−5.82343e12	−0.949824
$362$	0	0
$363$	4.46320e12i	0.708130i
$364$	0	0
$365$	9.21196e12	1.42196
$366$	0	0
$367$	2.67602e12i	0.401939i	0.979598	+	0.200969i	$0.0644092\pi$
$367$	2.67602e12i	0.401939i	−0.979598	+	0.200969i	$0.935591\pi$
$368$	0	0
$369$	−6.58390e11	−0.0962390
$370$	0	0
$371$	4.83801e12i	0.688331i
$372$	0	0
$373$	9.92009e12	1.37395	0.686976	−	0.726680i	$-0.258938\pi$
$373$	9.92009e12	1.37395	0.686976	+	0.726680i	$0.258938\pi$
$374$	0	0
$375$	7.03044e12i	0.948039i
$376$	0	0
$377$	−8.60182e12	−1.12949
$378$	0	0
$379$	− 6.20132e11i	− 0.0793028i	−0.999214	−	0.0396514i	$-0.987375\pi$
$379$	− 6.20132e11i	− 0.0793028i	0.999214	−	0.0396514i	$-0.0126247\pi$
$380$	0	0
$381$	2.20529e12	0.274689
$382$	0	0
$383$	5.18547e12i	0.629207i	0.949223	+	0.314604i	$0.101872\pi$
$383$	5.18547e12i	0.629207i	−0.949223	+	0.314604i	$0.898128\pi$
$384$	0	0
$385$	−6.78703e12	−0.802373
$386$	0	0
$387$	2.65737e12i	0.306124i
$388$	0	0
$389$	−1.84987e12	−0.207679	−0.103840	−	0.994594i	$-0.533113\pi$
$389$	−1.84987e12	−0.207679	−0.103840	+	0.994594i	$0.533113\pi$
$390$	0	0
$391$	− 5.86786e12i	− 0.642090i
$392$	0	0
$393$	1.75757e12	0.187478
$394$	0	0
$395$	1.14902e13i	1.19493i
$396$	0	0
$397$	6.84533e12	0.694132	0.347066	−	0.937841i	$-0.387178\pi$
$397$	6.84533e12	0.694132	0.347066	+	0.937841i	$0.387178\pi$
$398$	0	0
$399$	1.90134e13i	1.88017i
$400$	0	0
$401$	−1.07383e13	−1.03565	−0.517825	−	0.855486i	$-0.673258\pi$
$401$	−1.07383e13	−1.03565	−0.517825	+	0.855486i	$0.673258\pi$
$402$	0	0
$403$	7.44901e12i	0.700766i
$404$	0	0
$405$	1.17182e13	1.07544
$406$	0	0
$407$	9.16919e12i	0.821030i
$408$	0	0
$409$	−1.79443e13	−1.56787	−0.783936	−	0.620841i	$-0.786791\pi$
$409$	−1.79443e13	−1.56787	−0.783936	+	0.620841i	$0.786791\pi$
$410$	0	0
$411$	1.87119e13i	1.59554i
$412$	0	0
$413$	9.77997e11	0.0813931
$414$	0	0
$415$	− 1.03967e13i	− 0.844604i
$416$	0	0
$417$	1.14269e13	0.906250
$418$	0	0
$419$	1.01675e13i	0.787303i	0.919260	+	0.393652i	$0.128788\pi$
$419$	1.01675e13i	0.787303i	−0.919260	+	0.393652i	$0.871212\pi$
$420$	0	0
$421$	−1.38694e13	−1.04869	−0.524346	−	0.851505i	$-0.675690\pi$
$421$	−1.38694e13	−1.04869	−0.524346	+	0.851505i	$0.675690\pi$
$422$	0	0
$423$	− 1.53247e13i	− 1.13160i
$424$	0	0
$425$	−6.71591e12	−0.484351
$426$	0	0
$427$	1.59507e13i	1.12368i
$428$	0	0
$429$	−2.21539e13	−1.52462
$430$	0	0
$431$	3.51756e11i	0.0236513i	0.999930	+	0.0118257i	$0.00376431\pi$
$431$	3.51756e11i	0.0236513i	−0.999930	+	0.0118257i	$0.996236\pi$
$432$	0	0
$433$	−1.73769e13	−1.14165	−0.570823	−	0.821073i	$-0.693376\pi$
$433$	−1.73769e13	−1.14165	−0.570823	+	0.821073i	$0.693376\pi$
$434$	0	0
$435$	2.09179e13i	1.34299i
$436$	0	0
$437$	−1.35165e13	−0.848122
$438$	0	0
$439$	− 1.05808e13i	− 0.648928i	−0.945898	−	0.324464i	$-0.894816\pi$
$439$	− 1.05808e13i	− 0.648928i	0.945898	−	0.324464i	$-0.105184\pi$
$440$	0	0
$441$	−2.50267e12	−0.150041
$442$	0	0
$443$	− 2.18951e13i	− 1.28330i	−0.766996	−	0.641651i	$-0.778250\pi$
$443$	− 2.18951e13i	− 1.28330i	0.766996	−	0.641651i	$-0.221750\pi$
$444$	0	0
$445$	9.50649e11	0.0544779
$446$	0	0
$447$	− 3.28679e13i	− 1.84176i
$448$	0	0
$449$	−3.53942e13	−1.93954	−0.969772	−	0.244011i	$-0.921537\pi$
$449$	−3.53942e13	−1.93954	−0.969772	+	0.244011i	$0.921537\pi$
$450$	0	0
$451$	1.16780e12i	0.0625875i
$452$	0	0
$453$	6.31374e10	0.00330975
$454$	0	0
$455$	3.22014e13i	1.65127i
$456$	0	0
$457$	1.93883e13	0.972653	0.486326	−	0.873777i	$-0.338337\pi$
$457$	1.93883e13	0.972653	0.486326	+	0.873777i	$0.338337\pi$
$458$	0	0
$459$	3.10514e12i	0.152412i
$460$	0	0
$461$	−1.69421e13	−0.813696	−0.406848	−	0.913496i	$-0.633372\pi$
$461$	−1.69421e13	−0.813696	−0.406848	+	0.913496i	$0.633372\pi$
$462$	0	0
$463$	− 4.19900e13i	− 1.97352i	−0.162198	−	0.986758i	$-0.551858\pi$
$463$	− 4.19900e13i	− 1.97352i	0.162198	−	0.986758i	$-0.448142\pi$
$464$	0	0
$465$	1.81145e13	0.833224
$466$	0	0
$467$	9.55773e12i	0.430299i	0.976581	+	0.215150i	$0.0690239\pi$
$467$	9.55773e12i	0.430299i	−0.976581	+	0.215150i	$0.930976\pi$
$468$	0	0
$469$	9.76783e12	0.430461
$470$	0	0
$471$	− 1.55416e13i	− 0.670488i
$472$	0	0
$473$	4.71346e12	0.199083
$474$	0	0
$475$	1.54700e13i	0.639768i
$476$	0	0
$477$	2.01115e13	0.814427
$478$	0	0
$479$	4.05634e13i	1.60863i	0.594201	+	0.804317i	$0.297468\pi$
$479$	4.05634e13i	1.60863i	−0.594201	+	0.804317i	$0.702532\pi$
$480$	0	0
$481$	4.35036e13	1.68966
$482$	0	0
$483$	2.14978e13i	0.817823i
$484$	0	0
$485$	5.90127e13	2.19906
$486$	0	0
$487$	− 5.27934e13i	− 1.92723i	−0.267286	−	0.963617i	$-0.586127\pi$
$487$	− 5.27934e13i	− 1.92723i	0.267286	−	0.963617i	$-0.413873\pi$
$488$	0	0
$489$	3.73242e13	1.33489
$490$	0	0
$491$	2.41468e13i	0.846158i	0.906093	+	0.423079i	$0.139051\pi$
$491$	2.41468e13i	0.846158i	−0.906093	+	0.423079i	$0.860949\pi$
$492$	0	0
$493$	2.36309e13	0.811419
$494$	0	0
$495$	2.82135e13i	0.949360i
$496$	0	0
$497$	3.21021e13	1.05865
$498$	0	0
$499$	2.21286e11i	0.00715238i	0.999994	+	0.00357619i	$0.00113834\pi$
$499$	2.21286e11i	0.00715238i	−0.999994	+	0.00357619i	$0.998862\pi$
$500$	0	0
$501$	−5.81735e13	−1.84305
$502$	0	0
$503$	9.04543e12i	0.280924i	0.990086	+	0.140462i	$0.0448588\pi$
$503$	9.04543e12i	0.280924i	−0.990086	+	0.140462i	$0.955141\pi$
$504$	0	0
$505$	−1.45580e13	−0.443247
$506$	0	0
$507$	5.65702e13i	1.68868i
$508$	0	0
$509$	−2.24139e13	−0.656036	−0.328018	−	0.944671i	$-0.606381\pi$
$509$	−2.24139e13	−0.656036	−0.328018	+	0.944671i	$0.606381\pi$
$510$	0	0
$511$	3.81267e13i	1.09427i
$512$	0	0
$513$	7.15266e12	0.201317
$514$	0	0
$515$	3.50243e13i	0.966793i
$516$	0	0
$517$	−2.71819e13	−0.735915
$518$	0	0
$519$	9.62269e13i	2.55540i
$520$	0	0
$521$	−2.19502e12	−0.0571807	−0.0285904	−	0.999591i	$-0.509102\pi$
$521$	−2.19502e12	−0.0571807	−0.0285904	+	0.999591i	$0.509102\pi$
$522$	0	0
$523$	2.91933e13i	0.746062i	0.927819	+	0.373031i	$0.121681\pi$
$523$	2.91933e13i	0.746062i	−0.927819	+	0.373031i	$0.878319\pi$
$524$	0	0
$525$	2.46048e13	0.616913
$526$	0	0
$527$	− 2.04639e13i	− 0.503425i
$528$	0	0
$529$	2.61438e13	0.631089
$530$	0	0
$531$	− 4.06551e12i	− 0.0963036i
$532$	0	0
$533$	5.54070e12	0.128804
$534$	0	0
$535$	− 1.47599e13i	− 0.336755i
$536$	0	0
$537$	5.47927e13	1.22702
$538$	0	0
$539$	4.43905e12i	0.0975768i
$540$	0	0
$541$	4.86387e12	0.104953	0.0524766	−	0.998622i	$-0.483288\pi$
$541$	4.86387e12	0.104953	0.0524766	+	0.998622i	$0.483288\pi$
$542$	0	0
$543$	5.46218e12i	0.115709i
$544$	0	0
$545$	−4.14336e13	−0.861728
$546$	0	0
$547$	3.29297e13i	0.672436i	0.941784	+	0.336218i	$0.109148\pi$
$547$	3.29297e13i	0.672436i	−0.941784	+	0.336218i	$0.890852\pi$
$548$	0	0
$549$	6.63067e13	1.32952
$550$	0	0
$551$	− 5.44335e13i	− 1.07179i
$552$	0	0
$553$	−4.75557e13	−0.919554
$554$	0	0
$555$	− 1.05792e14i	− 2.00904i
$556$	0	0
$557$	4.29354e13	0.800828	0.400414	−	0.916334i	$-0.368866\pi$
$557$	4.29354e13	0.800828	0.400414	+	0.916334i	$0.368866\pi$
$558$	0	0
$559$	− 2.23632e13i	− 0.409709i
$560$	0	0
$561$	6.08610e13	1.09528
$562$	0	0
$563$	− 1.01247e14i	− 1.78994i	−0.446126	−	0.894970i	$-0.647197\pi$
$563$	− 1.01247e14i	− 1.78994i	0.446126	−	0.894970i	$-0.352803\pi$
$564$	0	0
$565$	−7.65652e13	−1.32981
$566$	0	0
$567$	4.84996e13i	0.827604i
$568$	0	0
$569$	3.75395e12	0.0629401	0.0314701	−	0.999505i	$-0.489981\pi$
$569$	3.75395e12	0.0629401	0.0314701	+	0.999505i	$0.489981\pi$
$570$	0	0
$571$	4.82773e13i	0.795358i	0.917525	+	0.397679i	$0.130184\pi$
$571$	4.82773e13i	0.795358i	−0.917525	+	0.397679i	$0.869816\pi$
$572$	0	0
$573$	−7.71618e13	−1.24919
$574$	0	0
$575$	1.74914e13i	0.278283i
$576$	0	0
$577$	−3.01929e13	−0.472091	−0.236046	−	0.971742i	$-0.575851\pi$
$577$	−3.01929e13	−0.472091	−0.236046	+	0.971742i	$0.575851\pi$
$578$	0	0
$579$	6.47232e13i	0.994644i
$580$	0	0
$581$	4.30299e13	0.649964
$582$	0	0
$583$	− 3.56723e13i	− 0.529649i
$584$	0	0
$585$	1.33860e14	1.95377
$586$	0	0
$587$	− 5.82710e13i	− 0.836107i	−0.908422	−	0.418053i	$-0.862712\pi$
$587$	− 5.82710e13i	− 0.836107i	0.908422	−	0.418053i	$-0.137288\pi$
$588$	0	0
$589$	−4.71383e13	−0.664963
$590$	0	0
$591$	− 1.01958e13i	− 0.141411i
$592$	0	0
$593$	−3.13152e13	−0.427052	−0.213526	−	0.976937i	$-0.568495\pi$
$593$	−3.13152e13	−0.427052	−0.213526	+	0.976937i	$0.568495\pi$
$594$	0	0
$595$	− 8.84635e13i	− 1.18626i
$596$	0	0
$597$	1.14528e14	1.51022
$598$	0	0
$599$	7.34955e13i	0.953074i	0.879154	+	0.476537i	$0.158108\pi$
$599$	7.34955e13i	0.953074i	−0.879154	+	0.476537i	$0.841892\pi$
$600$	0	0
$601$	−1.07112e14	−1.36605	−0.683024	−	0.730396i	$-0.739335\pi$
$601$	−1.07112e14	−1.36605	−0.683024	+	0.730396i	$0.739335\pi$
$602$	0	0
$603$	− 4.06046e13i	− 0.509318i
$604$	0	0
$605$	−4.78339e13	−0.590145
$606$	0	0
$607$	− 1.36295e13i	− 0.165401i	−0.996574	−	0.0827004i	$-0.973646\pi$
$607$	− 1.36295e13i	− 0.165401i	0.996574	−	0.0827004i	$-0.0263545\pi$
$608$	0	0
$609$	−8.65756e13	−1.03350
$610$	0	0
$611$	1.28966e14i	1.51450i
$612$	0	0
$613$	−1.16805e14	−1.34946	−0.674729	−	0.738066i	$-0.735739\pi$
$613$	−1.16805e14	−1.34946	−0.674729	+	0.738066i	$0.735739\pi$
$614$	0	0
$615$	− 1.34739e13i	− 0.153150i
$616$	0	0
$617$	−4.22466e13	−0.472461	−0.236230	−	0.971697i	$-0.575912\pi$
$617$	−4.22466e13	−0.472461	−0.236230	+	0.971697i	$0.575912\pi$
$618$	0	0
$619$	− 1.04957e14i	− 1.15493i	−0.816415	−	0.577466i	$-0.804042\pi$
$619$	− 1.04957e14i	− 1.15493i	0.816415	−	0.577466i	$-0.195958\pi$
$620$	0	0
$621$	8.08727e12	0.0875677
$622$	0	0
$623$	3.93457e12i	0.0419234i
$624$	0	0
$625$	−1.19042e14	−1.24825
$626$	0	0
$627$	− 1.40193e14i	− 1.44673i
$628$	0	0
$629$	−1.19513e14	−1.21384
$630$	0	0
$631$	− 2.87613e12i	− 0.0287516i	−0.999897	−	0.0143758i	$-0.995424\pi$
$631$	− 2.87613e12i	− 0.0287516i	0.999897	−	0.0143758i	$-0.00457612\pi$
$632$	0	0
$633$	−2.18463e14	−2.14960
$634$	0	0
$635$	2.36350e13i	0.228922i
$636$	0	0
$637$	2.10613e13	0.200811
$638$	0	0
$639$	− 1.33447e14i	− 1.25258i
$640$	0	0
$641$	−3.00941e13	−0.278094	−0.139047	−	0.990286i	$-0.544404\pi$
$641$	−3.00941e13	−0.278094	−0.139047	+	0.990286i	$0.544404\pi$
$642$	0	0
$643$	− 1.66359e14i	− 1.51353i	−0.653689	−	0.756764i	$-0.726779\pi$
$643$	− 1.66359e14i	− 1.51353i	0.653689	−	0.756764i	$-0.273221\pi$
$644$	0	0
$645$	−5.43829e13	−0.487151
$646$	0	0
$647$	− 1.73339e14i	− 1.52889i	−0.644690	−	0.764444i	$-0.723013\pi$
$647$	− 1.73339e14i	− 1.52889i	0.644690	−	0.764444i	$-0.276987\pi$
$648$	0	0
$649$	−7.21111e12	−0.0626295
$650$	0	0
$651$	7.49728e13i	0.641207i
$652$	0	0
$653$	1.89861e14	1.59908	0.799541	−	0.600612i	$-0.205076\pi$
$653$	1.89861e14	1.59908	0.799541	+	0.600612i	$0.205076\pi$
$654$	0	0
$655$	1.88366e13i	0.156241i
$656$	0	0
$657$	1.58492e14	1.29473
$658$	0	0
$659$	− 9.80342e13i	− 0.788771i	−0.918945	−	0.394385i	$-0.870958\pi$
$659$	− 9.80342e13i	− 0.788771i	0.918945	−	0.394385i	$-0.129042\pi$
$660$	0	0
$661$	1.54197e14	1.22199	0.610996	−	0.791634i	$-0.290769\pi$
$661$	1.54197e14	1.22199	0.610996	+	0.791634i	$0.290769\pi$
$662$	0	0
$663$	− 2.88758e14i	− 2.25406i
$664$	0	0
$665$	−2.03775e14	−1.56690
$666$	0	0
$667$	− 6.15461e13i	− 0.466199i
$668$	0	0
$669$	1.85089e14	1.38118
$670$	0	0
$671$	− 1.17610e14i	− 0.864634i
$672$	0	0
$673$	−6.70067e13	−0.485336	−0.242668	−	0.970109i	$-0.578023\pi$
$673$	−6.70067e13	−0.485336	−0.242668	+	0.970109i	$0.578023\pi$
$674$	0	0
$675$	− 9.25608e12i	− 0.0660554i
$676$	0	0
$677$	1.71430e14	1.20543	0.602715	−	0.797956i	$-0.294085\pi$
$677$	1.71430e14	1.20543	0.602715	+	0.797956i	$0.294085\pi$
$678$	0	0
$679$	2.44243e14i	1.69229i
$680$	0	0
$681$	3.07650e14	2.10049
$682$	0	0
$683$	− 5.05521e13i	− 0.340123i	−0.985433	−	0.170061i	$-0.945603\pi$
$683$	− 5.05521e13i	− 0.340123i	0.985433	−	0.170061i	$-0.0543966\pi$
$684$	0	0
$685$	−2.00543e14	−1.32970
$686$	0	0
$687$	2.92319e14i	1.91017i
$688$	0	0
$689$	−1.69249e14	−1.09001
$690$	0	0
$691$	5.33777e13i	0.338821i	0.985546	+	0.169410i	$0.0541863\pi$
$691$	5.33777e13i	0.338821i	−0.985546	+	0.169410i	$0.945814\pi$
$692$	0	0
$693$	−1.16771e14	−0.730580
$694$	0	0
$695$	1.22467e14i	0.755256i
$696$	0	0
$697$	−1.52214e13	−0.0925317
$698$	0	0
$699$	− 2.93431e14i	− 1.75841i
$700$	0	0
$701$	−3.55178e13	−0.209824	−0.104912	−	0.994481i	$-0.533456\pi$
$701$	−3.55178e13	−0.209824	−0.104912	+	0.994481i	$0.533456\pi$
$702$	0	0
$703$	2.75297e14i	1.60334i
$704$	0	0
$705$	3.13620e14	1.80077
$706$	0	0
$707$	− 6.02531e13i	− 0.341101i
$708$	0	0
$709$	2.13491e14	1.19165	0.595824	−	0.803115i	$-0.296826\pi$
$709$	2.13491e14	1.19165	0.595824	+	0.803115i	$0.296826\pi$
$710$	0	0
$711$	1.97688e14i	1.08801i
$712$	0	0
$713$	−5.32977e13	−0.289242
$714$	0	0
$715$	− 2.37432e14i	− 1.27060i
$716$	0	0
$717$	−3.23693e14	−1.70820
$718$	0	0
$719$	− 2.65916e14i	− 1.38389i	−0.721952	−	0.691943i	$-0.756755\pi$
$719$	− 2.65916e14i	− 1.38389i	0.721952	−	0.691943i	$-0.243245\pi$
$720$	0	0
$721$	−1.44959e14	−0.743995
$722$	0	0
$723$	− 5.51893e14i	− 2.79360i
$724$	0	0
$725$	−7.04410e13	−0.351670
$726$	0	0
$727$	1.56999e14i	0.773082i	0.922272	+	0.386541i	$0.126330\pi$
$727$	1.56999e14i	0.773082i	−0.922272	+	0.386541i	$0.873670\pi$
$728$	0	0
$729$	2.44622e14	1.18812
$730$	0	0
$731$	6.14361e13i	0.294332i
$732$	0	0
$733$	2.64144e14	1.24831	0.624153	−	0.781302i	$-0.285444\pi$
$733$	2.64144e14	1.24831	0.624153	+	0.781302i	$0.285444\pi$
$734$	0	0
$735$	− 5.12169e13i	− 0.238768i
$736$	0	0
$737$	−7.20216e13	−0.331227
$738$	0	0
$739$	2.68052e14i	1.21618i	0.793869	+	0.608089i	$0.208064\pi$
$739$	2.68052e14i	1.21618i	−0.793869	+	0.608089i	$0.791936\pi$
$740$	0	0
$741$	−6.65150e14	−2.97734
$742$	0	0
$743$	9.68880e13i	0.427884i	0.976846	+	0.213942i	$0.0686303\pi$
$743$	9.68880e13i	0.427884i	−0.976846	+	0.213942i	$0.931370\pi$
$744$	0	0
$745$	3.52258e14	1.53490
$746$	0	0
$747$	− 1.78874e14i	− 0.769032i
$748$	0	0
$749$	6.10885e13	0.259149
$750$	0	0
$751$	2.13814e13i	0.0895026i	0.998998	+	0.0447513i	$0.0142495\pi$
$751$	2.13814e13i	0.0895026i	−0.998998	+	0.0447513i	$0.985750\pi$
$752$	0	0
$753$	6.47674e14	2.67536
$754$	0	0
$755$	6.76669e11i	0.00275830i
$756$	0	0
$757$	3.45330e14	1.38917	0.694584	−	0.719412i	$-0.255589\pi$
$757$	3.45330e14	1.38917	0.694584	+	0.719412i	$0.255589\pi$
$758$	0	0
$759$	− 1.58511e14i	− 0.629290i
$760$	0	0
$761$	−1.01279e14	−0.396822	−0.198411	−	0.980119i	$-0.563578\pi$
$761$	−1.01279e14	−0.396822	−0.198411	+	0.980119i	$0.563578\pi$
$762$	0	0
$763$	− 1.71486e14i	− 0.663142i
$764$	0	0
$765$	−3.67740e14	−1.40357
$766$	0	0
$767$	3.42135e13i	0.128890i
$768$	0	0
$769$	−2.20211e14	−0.818854	−0.409427	−	0.912343i	$-0.634271\pi$
$769$	−2.20211e14	−0.818854	−0.409427	+	0.912343i	$0.634271\pi$
$770$	0	0
$771$	− 1.65286e14i	− 0.606687i
$772$	0	0
$773$	−1.15285e14	−0.417709	−0.208855	−	0.977947i	$-0.566974\pi$
$773$	−1.15285e14	−0.417709	−0.208855	+	0.977947i	$0.566974\pi$
$774$	0	0
$775$	6.10005e13i	0.218185i
$776$	0	0
$777$	4.37855e14	1.54606
$778$	0	0
$779$	3.50623e13i	0.122223i
$780$	0	0
$781$	−2.36700e14	−0.814596
$782$	0	0
$783$	3.25689e13i	0.110661i
$784$	0	0
$785$	1.66566e14	0.558775
$786$	0	0
$787$	2.91930e14i	0.966953i	0.875357	+	0.483476i	$0.160626\pi$
$787$	2.91930e14i	0.966953i	−0.875357	+	0.483476i	$0.839374\pi$
$788$	0	0
$789$	7.13843e14	2.33463
$790$	0	0
$791$	− 3.16890e14i	− 1.02335i
$792$	0	0
$793$	−5.58007e14	−1.77940
$794$	0	0
$795$	4.11580e14i	1.29604i
$796$	0	0
$797$	3.56212e14	1.10769	0.553843	−	0.832621i	$-0.313161\pi$
$797$	3.56212e14	1.10769	0.553843	+	0.832621i	$0.313161\pi$
$798$	0	0
$799$	− 3.54295e14i	− 1.08800i
$800$	0	0
$801$	1.63559e13	0.0496035
$802$	0	0
$803$	− 2.81121e14i	− 0.842007i
$804$	0	0
$805$	−2.30401e14	−0.681562
$806$	0	0
$807$	− 5.45417e14i	− 1.59353i
$808$	0	0
$809$	2.42201e14	0.698930	0.349465	−	0.936949i	$-0.386363\pi$
$809$	2.42201e14	0.698930	0.349465	+	0.936949i	$0.386363\pi$
$810$	0	0
$811$	4.06752e14i	1.15938i	0.814838	+	0.579689i	$0.196826\pi$
$811$	4.06752e14i	1.15938i	−0.814838	+	0.579689i	$0.803174\pi$
$812$	0	0
$813$	9.67412e14	2.72370
$814$	0	0
$815$	4.00018e14i	1.11248i
$816$	0	0
$817$	1.41517e14	0.388776
$818$	0	0
$819$	5.54024e14i	1.50352i
$820$	0	0
$821$	−4.04166e14	−1.08354	−0.541768	−	0.840528i	$-0.682245\pi$
$821$	−4.04166e14	−1.08354	−0.541768	+	0.840528i	$0.682245\pi$
$822$	0	0
$823$	1.42442e14i	0.377259i	0.982048	+	0.188630i	$0.0604045\pi$
$823$	1.42442e14i	0.377259i	−0.982048	+	0.188630i	$0.939595\pi$
$824$	0	0
$825$	−1.81420e14	−0.474695
$826$	0	0
$827$	3.85554e13i	0.0996684i	0.998758	+	0.0498342i	$0.0158693\pi$
$827$	3.85554e13i	0.0996684i	−0.998758	+	0.0498342i	$0.984131\pi$
$828$	0	0
$829$	−2.35400e14	−0.601221	−0.300610	−	0.953747i	$-0.597190\pi$
$829$	−2.35400e14	−0.601221	−0.300610	+	0.953747i	$0.597190\pi$
$830$	0	0
$831$	− 1.21214e14i	− 0.305877i
$832$	0	0
$833$	−5.78595e13	−0.144261
$834$	0	0
$835$	− 6.23469e14i	− 1.53597i
$836$	0	0
$837$	2.82040e13	0.0686568
$838$	0	0
$839$	− 5.54263e14i	− 1.33323i	−0.745401	−	0.666616i	$-0.767742\pi$
$839$	− 5.54263e14i	− 1.33323i	0.745401	−	0.666616i	$-0.232258\pi$
$840$	0	0
$841$	−1.72850e14	−0.410857
$842$	0	0
$843$	1.81993e14i	0.427481i
$844$	0	0
$845$	−6.06286e14	−1.40732
$846$	0	0
$847$	− 1.97976e14i	− 0.454146i
$848$	0	0
$849$	3.45400e14	0.783039
$850$	0	0
$851$	3.11269e14i	0.697410i
$852$	0	0
$853$	−2.17078e14	−0.480696	−0.240348	−	0.970687i	$-0.577262\pi$
$853$	−2.17078e14	−0.480696	−0.240348	+	0.970687i	$0.577262\pi$
$854$	0	0
$855$	8.47086e14i	1.85395i
$856$	0	0
$857$	−7.39171e14	−1.59897	−0.799485	−	0.600685i	$-0.794894\pi$
$857$	−7.39171e14	−1.59897	−0.799485	+	0.600685i	$0.794894\pi$
$858$	0	0
$859$	− 3.16235e14i	− 0.676152i	−0.941119	−	0.338076i	$-0.890224\pi$
$859$	− 3.16235e14i	− 0.676152i	0.941119	−	0.338076i	$-0.109776\pi$
$860$	0	0
$861$	5.57661e13	0.117857
$862$	0	0
$863$	3.17121e14i	0.662478i	0.943547	+	0.331239i	$0.107467\pi$
$863$	3.17121e14i	0.662478i	−0.943547	+	0.331239i	$0.892533\pi$
$864$	0	0
$865$	−1.03130e15	−2.12964
$866$	0	0
$867$	8.34458e13i	0.170337i
$868$	0	0
$869$	3.50645e14	0.707569
$870$	0	0
$871$	3.41710e14i	0.681658i
$872$	0	0
$873$	1.01531e15	2.00230
$874$	0	0
$875$	− 3.11852e14i	− 0.608007i
$876$	0	0
$877$	5.34325e14	1.02993	0.514965	−	0.857211i	$-0.327805\pi$
$877$	5.34325e14	1.02993	0.514965	+	0.857211i	$0.327805\pi$
$878$	0	0
$879$	− 1.33432e15i	− 2.54282i
$880$	0	0
$881$	−1.20104e13	−0.0226297	−0.0113148	−	0.999936i	$-0.503602\pi$
$881$	−1.20104e13	−0.0226297	−0.0113148	+	0.999936i	$0.503602\pi$
$882$	0	0
$883$	6.89886e14i	1.28521i	0.766198	+	0.642605i	$0.222146\pi$
$883$	6.89886e14i	1.28521i	−0.766198	+	0.642605i	$0.777854\pi$
$884$	0	0
$885$	8.32003e13	0.153253
$886$	0	0
$887$	7.40300e13i	0.134831i	0.997725	+	0.0674154i	$0.0214753\pi$
$887$	7.40300e13i	0.134831i	−0.997725	+	0.0674154i	$0.978525\pi$
$888$	0	0
$889$	−9.78211e13	−0.176167
$890$	0	0
$891$	− 3.57604e14i	− 0.636816i
$892$	0	0
$893$	−8.16114e14	−1.43712
$894$	0	0
$895$	5.87236e14i	1.02258i
$896$	0	0
$897$	−7.52063e14	−1.29507
$898$	0	0
$899$	− 2.14639e14i	− 0.365519i
$900$	0	0
$901$	4.64960e14	0.783054
$902$	0	0
$903$	− 2.25081e14i	− 0.374887i
$904$	0	0
$905$	−5.85404e13	−0.0964301
$906$	0	0
$907$	1.10129e15i	1.79418i	0.441852	+	0.897088i	$0.354322\pi$
$907$	1.10129e15i	1.79418i	−0.441852	+	0.897088i	$0.645678\pi$
$908$	0	0
$909$	−2.50471e14	−0.403587
$910$	0	0
$911$	5.75678e14i	0.917461i	0.888575	+	0.458730i	$0.151696\pi$
$911$	5.75678e14i	0.917461i	−0.888575	+	0.458730i	$0.848304\pi$
$912$	0	0
$913$	−3.17274e14	−0.500128
$914$	0	0
$915$	1.35696e15i	2.11574i
$916$	0	0
$917$	−7.79613e13	−0.120235
$918$	0	0
$919$	− 2.49831e14i	− 0.381126i	−0.981675	−	0.190563i	$-0.938969\pi$
$919$	− 2.49831e14i	− 0.381126i	0.981675	−	0.190563i	$-0.0610314\pi$
$920$	0	0
$921$	−1.66965e15	−2.51957
$922$	0	0
$923$	1.12303e15i	1.67642i
$924$	0	0
$925$	3.56255e14	0.526081
$926$	0	0
$927$	6.02593e14i	0.880289i
$928$	0	0
$929$	−5.75740e14	−0.832046	−0.416023	−	0.909354i	$-0.636576\pi$
$929$	−5.75740e14	−0.832046	−0.416023	+	0.909354i	$0.636576\pi$
$930$	0	0
$931$	1.33279e14i	0.190552i
$932$	0	0
$933$	6.26252e14	0.885811
$934$	0	0
$935$	6.52272e14i	0.912789i
$936$	0	0
$937$	−4.19947e14	−0.581428	−0.290714	−	0.956810i	$-0.593893\pi$
$937$	−4.19947e14	−0.581428	−0.290714	+	0.956810i	$0.593893\pi$
$938$	0	0
$939$	7.72792e14i	1.05861i
$940$	0	0
$941$	−4.76423e14	−0.645721	−0.322860	−	0.946447i	$-0.604644\pi$
$941$	−4.76423e14	−0.645721	−0.322860	+	0.946447i	$0.604644\pi$
$942$	0	0
$943$	3.96438e13i	0.0531639i
$944$	0	0
$945$	1.21923e14	0.161781
$946$	0	0
$947$	7.31112e14i	0.959918i	0.877291	+	0.479959i	$0.159348\pi$
$947$	7.31112e14i	0.959918i	−0.877291	+	0.479959i	$0.840652\pi$
$948$	0	0
$949$	−1.33379e15	−1.73283
$950$	0	0
$951$	− 8.11488e14i	− 1.04323i
$952$	0	0
$953$	−4.33318e14	−0.551242	−0.275621	−	0.961266i	$-0.588884\pi$
$953$	−4.33318e14	−0.551242	−0.275621	+	0.961266i	$0.588884\pi$
$954$	0	0
$955$	− 8.26974e14i	− 1.04106i
$956$	0	0
$957$	6.38352e14	0.795244
$958$	0	0
$959$	− 8.30012e14i	− 1.02327i
$960$	0	0
$961$	6.33755e14	0.773222
$962$	0	0
$963$	− 2.53943e14i	− 0.306623i
$964$	0	0
$965$	−6.93665e14	−0.828921
$966$	0	0
$967$	1.28203e15i	1.51623i	0.652120	+	0.758115i	$0.273880\pi$
$967$	1.28203e15i	1.51623i	−0.652120	+	0.758115i	$0.726120\pi$
$968$	0	0
$969$	1.82730e15	2.13890
$970$	0	0
$971$	1.03910e15i	1.20381i	0.798566	+	0.601907i	$0.205592\pi$
$971$	1.03910e15i	1.20381i	−0.798566	+	0.601907i	$0.794408\pi$
$972$	0	0
$973$	−5.06868e14	−0.581207
$974$	0	0
$975$	8.60755e14i	0.976914i
$976$	0	0
$977$	−8.98564e13	−0.100943	−0.0504715	−	0.998726i	$-0.516072\pi$
$977$	−8.98564e13	−0.100943	−0.0504715	+	0.998726i	$0.516072\pi$
$978$	0	0
$979$	− 2.90109e13i	− 0.0322588i
$980$	0	0
$981$	−7.12864e14	−0.784624
$982$	0	0
$983$	5.65525e14i	0.616147i	0.951363	+	0.308073i	$0.0996842\pi$
$983$	5.65525e14i	0.616147i	−0.951363	+	0.308073i	$0.900316\pi$
$984$	0	0
$985$	1.09272e14	0.117850
$986$	0	0
$987$	1.29802e15i	1.38578i
$988$	0	0
$989$	1.60009e14	0.169107
$990$	0	0
$991$	1.22093e15i	1.27739i	0.769461	+	0.638693i	$0.220525\pi$
$991$	1.22093e15i	1.27739i	−0.769461	+	0.638693i	$0.779475\pi$
$992$	0	0
$993$	1.26542e15	1.31065
$994$	0	0
$995$	1.22744e15i	1.25860i
$996$	0	0
$997$	4.49639e14	0.456445	0.228222	−	0.973609i	$-0.426709\pi$
$997$	4.49639e14	0.456445	0.228222	+	0.973609i	$0.426709\pi$
$998$	0	0
$999$	− 1.64717e14i	− 0.165543i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.11.c.m.255.15		16
4.3	odd	2	inner	256.11.c.m.255.1		16
8.3	odd	2	inner	256.11.c.m.255.16		16
8.5	even	2	inner	256.11.c.m.255.2		16
16.3	odd	4		8.11.d.b.3.5	✓	8
16.5	even	4		8.11.d.b.3.6	yes	8
16.11	odd	4		32.11.d.b.15.1		8
16.13	even	4		32.11.d.b.15.2		8
48.5	odd	4		72.11.b.b.19.3		8
48.11	even	4		288.11.b.b.271.7		8
48.29	odd	4		288.11.b.b.271.2		8
48.35	even	4		72.11.b.b.19.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.11.d.b.3.5	✓	8	16.3	odd	4
8.11.d.b.3.6	yes	8	16.5	even	4
32.11.d.b.15.1		8	16.11	odd	4
32.11.d.b.15.2		8	16.13	even	4
72.11.b.b.19.3		8	48.5	odd	4
72.11.b.b.19.4		8	48.35	even	4
256.11.c.m.255.1		16	4.3	odd	2	inner
256.11.c.m.255.2		16	8.5	even	2	inner
256.11.c.m.255.15		16	1.1	even	1	trivial
256.11.c.m.255.16		16	8.3	odd	2	inner
288.11.b.b.271.2		8	48.29	odd	4
288.11.b.b.271.7		8	48.11	even	4

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:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	256.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+352.099i q^{3} -3773.58 q^{5} -15618.2i q^{7} -64924.4 q^{9} +115158. i q^{11} +546374. q^{13} -1.32867e6i q^{15} -1.50100e6 q^{17} +3.45753e6i q^{19} +5.49914e6 q^{21} +3.90931e6i q^{23} +4.47430e6 q^{25} -2.06872e6i q^{27} -1.57435e7 q^{29} +1.36335e7i q^{31} -4.05471e7 q^{33} +5.89365e7i q^{35} +7.96225e7 q^{37} +1.92377e8i q^{39} +1.01409e7 q^{41} -4.09302e7i q^{43} +2.44998e8 q^{45} +2.36040e8i q^{47} +3.85474e7 q^{49} -5.28499e8i q^{51} -3.09768e8 q^{53} -4.34559e8i q^{55} -1.21739e9 q^{57} +6.26191e7i q^{59} -1.02129e9 q^{61} +1.01400e9i q^{63} -2.06179e9 q^{65} +6.25414e8i q^{67} -1.37646e9 q^{69} +2.05543e9i q^{71} -2.44117e9 q^{73} +1.57539e9i q^{75} +1.79856e9 q^{77} -3.04489e9i q^{79} -3.10533e9 q^{81} +2.75511e9i q^{83} +5.66413e9 q^{85} -5.54325e9i q^{87} -2.51922e8 q^{89} -8.53337e9i q^{91} -4.80035e9 q^{93} -1.30473e10i q^{95} -1.56384e10 q^{97} -7.47658e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 70992 q^{9} - 741600 q^{17} + 37585520 q^{25} - 148617408 q^{33} - 185338656 q^{41} - 160864496 q^{49} - 2801425728 q^{57} - 1678056960 q^{65} - 5600145440 q^{73} - 23818130352 q^{81} - 12325192224 q^{89}+ \cdots - 19395727072 q^{97}+O(q^{100}) \) 16 * q + 70992 * q^9 - 741600 * q^17 + 37585520 * q^25 - 148617408 * q^33 - 185338656 * q^41 - 160864496 * q^49 - 2801425728 * q^57 - 1678056960 * q^65 - 5600145440 * q^73 - 23818130352 * q^81 - 12325192224 * q^89 - 19395727072 * q^97

Introduction

L-functions

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