LMFDB - Embedded newform 256.11.c.k.255.6

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4 x^{9} - 9850 x^{8} - 22678 x^{7} + 31760900 x^{6} + 262382084 x^{5} - 36066825359 x^{4} + \cdots + 11\!\cdots\!28 $ x^10 - 4x^9 - 9850x^8 - 22678x^7 + 31760900x^6 + 262382084x^5 - 36066825359x^4 - 492294015528x^3 + 12219836291940x^2 + 252889346123568*x + 1185399433622628
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{85}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 128)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			255.6
Root			$-59.8851 + 1.41421i$ of defining polynomial
Character	$\chi$	$=$	256.255
Dual form			256.11.c.k.255.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+121.681i q^{3} +2688.94 q^{5} +30265.0i q^{7} +44242.7 q^{9} +O(q^{10})$	$q+121.681i q^{3} +2688.94 q^{5} +30265.0i q^{7} +44242.7 q^{9} -33112.3i q^{11} +510285. q^{13} +327194. i q^{15} -2.64540e6 q^{17} +3.77448e6i q^{19} -3.68269e6 q^{21} -6.21786e6i q^{23} -2.53522e6 q^{25} +1.25687e7i q^{27} -3.22858e7 q^{29} +1.24973e7i q^{31} +4.02915e6 q^{33} +8.13809e7i q^{35} +6.65303e7 q^{37} +6.20921e7i q^{39} -9.97485e6 q^{41} +2.31833e8i q^{43} +1.18966e8 q^{45} -4.86716e7i q^{47} -6.33498e8 q^{49} -3.21896e8i q^{51} -1.33989e8 q^{53} -8.90370e7i q^{55} -4.59284e8 q^{57} -8.52337e8i q^{59} -4.10213e8 q^{61} +1.33901e9i q^{63} +1.37213e9 q^{65} -1.27704e9i q^{67} +7.56598e8 q^{69} +1.06151e8i q^{71} -3.60337e8 q^{73} -3.08489e8i q^{75} +1.00214e9 q^{77} +2.41105e9i q^{79} +1.08311e9 q^{81} -2.05140e9i q^{83} -7.11332e9 q^{85} -3.92858e9i q^{87} +6.58898e9 q^{89} +1.54438e10i q^{91} -1.52068e9 q^{93} +1.01494e10i q^{95} -1.54022e9 q^{97} -1.46498e9i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 6232 q^{5} - 218038 q^{9}+O(q^{10}) $ 10 * q + 6232 * q^5 - 218038 * q^9	$ 10 q + 6232 q^{5} - 218038 q^{9} + 485560 q^{13} - 1468772 q^{17} + 15545024 q^{21} + 4383462 q^{25} + 25263800 q^{29} - 122737264 q^{33} + 178255288 q^{37} + 91656876 q^{41} - 718346408 q^{45} - 842454934 q^{49} - 798458664 q^{53} - 1612102544 q^{57} + 102636184 q^{61} + 376325920 q^{65} + 207224128 q^{69} + 2854265572 q^{73} + 9037740608 q^{77} + 11990017466 q^{81} - 5473132400 q^{85} - 11790814556 q^{89} - 24576098304 q^{93} - 9363277860 q^{97}+O(q^{100}) $ 10 * q + 6232 * q^5 - 218038 * q^9 + 485560 * q^13 - 1468772 * q^17 + 15545024 * q^21 + 4383462 * q^25 + 25263800 * q^29 - 122737264 * q^33 + 178255288 * q^37 + 91656876 * q^41 - 718346408 * q^45 - 842454934 * q^49 - 798458664 * q^53 - 1612102544 * q^57 + 102636184 * q^61 + 376325920 * q^65 + 207224128 * q^69 + 2854265572 * q^73 + 9037740608 * q^77 + 11990017466 * q^81 - 5473132400 * q^85 - 11790814556 * q^89 - 24576098304 * q^93 - 9363277860 * q^97

Display 100 coefficients 10 coefficients

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
$3$	121.681i	0.500746i	0.968149	+	0.250373i	$0.0805533\pi$
$3$	121.681i	0.500746i	−0.968149	+	0.250373i	$0.919447\pi$
$4$	0	0
$5$	2688.94	0.860461	0.430231	−	0.902719i	$-0.358432\pi$
$5$	2688.94	0.860461	0.430231	+	0.902719i	$0.358432\pi$
$6$	0	0
$7$	30265.0i	1.80074i	0.435125	+	0.900370i	$0.356704\pi$
$7$	30265.0i	1.80074i	−0.435125	+	0.900370i	$0.643296\pi$
$8$	0	0
$9$	44242.7	0.749253
$10$	0	0
$11$	− 33112.3i	− 0.205601i	−0.994702	−	0.102801i	$-0.967220\pi$
$11$	− 33112.3i	− 0.205601i	0.994702	−	0.102801i	$-0.0327804\pi$
$12$	0	0
$13$	510285.	1.37434	0.687172	−	0.726495i	$-0.258852\pi$
$13$	510285.	1.37434	0.687172	+	0.726495i	$0.258852\pi$
$14$	0	0
$15$	327194.i	0.430873i
$16$	0	0
$17$	−2.64540e6	−1.86314	−0.931572	−	0.363557i	$-0.881562\pi$
$17$	−2.64540e6	−1.86314	−0.931572	+	0.363557i	$0.881562\pi$
$18$	0	0
$19$	3.77448e6i	1.52436i	0.647362	+	0.762182i	$0.275872\pi$
$19$	3.77448e6i	1.52436i	−0.647362	+	0.762182i	$0.724128\pi$
$20$	0	0
$21$	−3.68269e6	−0.901714
$22$	0	0
$23$	− 6.21786e6i	− 0.966056i	−0.875605	−	0.483028i	$-0.839537\pi$
$23$	− 6.21786e6i	− 0.966056i	0.875605	−	0.483028i	$-0.160463\pi$
$24$	0	0
$25$	−2.53522e6	−0.259606
$26$	0	0
$27$	1.25687e7i	0.875932i
$28$	0	0
$29$	−3.22858e7	−1.57406	−0.787031	−	0.616914i	$-0.788383\pi$
$29$	−3.22858e7	−1.57406	−0.787031	+	0.616914i	$0.788383\pi$
$30$	0	0
$31$	1.24973e7i	0.436523i	0.975890	+	0.218261i	$0.0700385\pi$
$31$	1.24973e7i	0.436523i	−0.975890	+	0.218261i	$0.929961\pi$
$32$	0	0
$33$	4.02915e6	0.102954
$34$	0	0
$35$	8.13809e7i	1.54947i
$36$	0	0
$37$	6.65303e7	0.959425	0.479713	−	0.877426i	$-0.340741\pi$
$37$	6.65303e7	0.959425	0.479713	+	0.877426i	$0.340741\pi$
$38$	0	0
$39$	6.20921e7i	0.688198i
$40$	0	0
$41$	−9.97485e6	−0.0860969	−0.0430484	−	0.999073i	$-0.513707\pi$
$41$	−9.97485e6	−0.0860969	−0.0430484	+	0.999073i	$0.513707\pi$
$42$	0	0
$43$	2.31833e8i	1.57700i	0.615033	+	0.788501i	$0.289143\pi$
$43$	2.31833e8i	1.57700i	−0.615033	+	0.788501i	$0.710857\pi$
$44$	0	0
$45$	1.18966e8	0.644703
$46$	0	0
$47$	− 4.86716e7i	− 0.212220i	−0.994354	−	0.106110i	$-0.966160\pi$
$47$	− 4.86716e7i	− 0.212220i	0.994354	−	0.106110i	$-0.0338396\pi$
$48$	0	0
$49$	−6.33498e8	−2.24267
$50$	0	0
$51$	− 3.21896e8i	− 0.932962i
$52$	0	0
$53$	−1.33989e8	−0.320397	−0.160198	−	0.987085i	$-0.551213\pi$
$53$	−1.33989e8	−0.320397	−0.160198	+	0.987085i	$0.551213\pi$
$54$	0	0
$55$	− 8.90370e7i	− 0.176912i
$56$	0	0
$57$	−4.59284e8	−0.763320
$58$	0	0
$59$	− 8.52337e8i	− 1.19221i	−0.802908	−	0.596103i	$-0.796715\pi$
$59$	− 8.52337e8i	− 1.19221i	0.802908	−	0.596103i	$-0.203285\pi$
$60$	0	0
$61$	−4.10213e8	−0.485691	−0.242846	−	0.970065i	$-0.578081\pi$
$61$	−4.10213e8	−0.485691	−0.242846	+	0.970065i	$0.578081\pi$
$62$	0	0
$63$	1.33901e9i	1.34921i
$64$	0	0
$65$	1.37213e9	1.18257
$66$	0	0
$67$	− 1.27704e9i	− 0.945869i	−0.881098	−	0.472934i	$-0.843195\pi$
$67$	− 1.27704e9i	− 0.945869i	0.881098	−	0.472934i	$-0.156805\pi$
$68$	0	0
$69$	7.56598e8	0.483749
$70$	0	0
$71$	1.06151e8i	0.0588347i	0.999567	+	0.0294174i	$0.00936519\pi$
$71$	1.06151e8i	0.0588347i	−0.999567	+	0.0294174i	$0.990635\pi$
$72$	0	0
$73$	−3.60337e8	−0.173818	−0.0869089	−	0.996216i	$-0.527699\pi$
$73$	−3.60337e8	−0.173818	−0.0869089	+	0.996216i	$0.527699\pi$
$74$	0	0
$75$	− 3.08489e8i	− 0.129997i
$76$	0	0
$77$	1.00214e9	0.370234
$78$	0	0
$79$	2.41105e9i	0.783557i	0.920060	+	0.391778i	$0.128140\pi$
$79$	2.41105e9i	0.783557i	−0.920060	+	0.391778i	$0.871860\pi$
$80$	0	0
$81$	1.08311e9	0.310634
$82$	0	0
$83$	− 2.05140e9i	− 0.520788i	−0.965502	−	0.260394i	$-0.916148\pi$
$83$	− 2.05140e9i	− 0.520788i	0.965502	−	0.260394i	$-0.0838524\pi$
$84$	0	0
$85$	−7.11332e9	−1.60316
$86$	0	0
$87$	− 3.92858e9i	− 0.788205i
$88$	0	0
$89$	6.58898e9	1.17996	0.589981	−	0.807417i	$-0.299135\pi$
$89$	6.58898e9	1.17996	0.589981	+	0.807417i	$0.299135\pi$
$90$	0	0
$91$	1.54438e10i	2.47484i
$92$	0	0
$93$	−1.52068e9	−0.218587
$94$	0	0
$95$	1.01494e10i	1.31166i
$96$	0	0
$97$	−1.54022e9	−0.179360	−0.0896799	−	0.995971i	$-0.528584\pi$
$97$	−1.54022e9	−0.179360	−0.0896799	+	0.995971i	$0.528584\pi$
$98$	0	0
$99$	− 1.46498e9i	− 0.154047i
$100$	0	0
$101$	8.77466e9	0.834879	0.417440	−	0.908705i	$-0.362928\pi$
$101$	8.77466e9	0.834879	0.417440	+	0.908705i	$0.362928\pi$
$102$	0	0
$103$	9.25173e9i	0.798062i	0.916937	+	0.399031i	$0.130653\pi$
$103$	9.25173e9i	0.798062i	−0.916937	+	0.399031i	$0.869347\pi$
$104$	0	0
$105$	−9.90254e9	−0.775890
$106$	0	0
$107$	1.42627e10i	1.01691i	0.861088	+	0.508456i	$0.169783\pi$
$107$	1.42627e10i	1.01691i	−0.861088	+	0.508456i	$0.830217\pi$
$108$	0	0
$109$	−1.17215e9	−0.0761814	−0.0380907	−	0.999274i	$-0.512128\pi$
$109$	−1.17215e9	−0.0761814	−0.0380907	+	0.999274i	$0.512128\pi$
$110$	0	0
$111$	8.09550e9i	0.480428i
$112$	0	0
$113$	1.16559e10	0.632637	0.316318	−	0.948653i	$-0.397553\pi$
$113$	1.16559e10	0.632637	0.316318	+	0.948653i	$0.397553\pi$
$114$	0	0
$115$	− 1.67195e10i	− 0.831253i
$116$	0	0
$117$	2.25763e10	1.02973
$118$	0	0
$119$	− 8.00631e10i	− 3.35504i
$120$	0	0
$121$	2.48410e10	0.957728
$122$	0	0
$123$	− 1.21375e9i	− 0.0431127i
$124$	0	0
$125$	−3.30762e10	−1.08384
$126$	0	0
$127$	5.27548e10i	1.59677i	0.602144	+	0.798387i	$0.294313\pi$
$127$	5.27548e10i	1.59677i	−0.602144	+	0.798387i	$0.705687\pi$
$128$	0	0
$129$	−2.82097e10	−0.789678
$130$	0	0
$131$	− 6.15388e10i	− 1.59512i	−0.603241	−	0.797559i	$-0.706124\pi$
$131$	− 6.15388e10i	− 1.59512i	0.603241	−	0.797559i	$-0.293876\pi$
$132$	0	0
$133$	−1.14235e11	−2.74499
$134$	0	0
$135$	3.37964e10i	0.753706i
$136$	0	0
$137$	−3.06375e10	−0.634821	−0.317410	−	0.948288i	$-0.602813\pi$
$137$	−3.06375e10	−0.634821	−0.317410	+	0.948288i	$0.602813\pi$
$138$	0	0
$139$	1.64390e10i	0.316811i	0.987374	+	0.158406i	$0.0506354\pi$
$139$	1.64390e10i	0.316811i	−0.987374	+	0.158406i	$0.949365\pi$
$140$	0	0
$141$	5.92243e9	0.106268
$142$	0	0
$143$	− 1.68967e10i	− 0.282567i
$144$	0	0
$145$	−8.68146e10	−1.35442
$146$	0	0
$147$	− 7.70848e10i	− 1.12301i
$148$	0	0
$149$	2.16257e10	0.294469	0.147234	−	0.989102i	$-0.452963\pi$
$149$	2.16257e10	0.294469	0.147234	+	0.989102i	$0.452963\pi$
$150$	0	0
$151$	− 1.16957e11i	− 1.48985i	−0.667147	−	0.744926i	$-0.732485\pi$
$151$	− 1.16957e11i	− 1.48985i	0.667147	−	0.744926i	$-0.267515\pi$
$152$	0	0
$153$	−1.17039e11	−1.39597
$154$	0	0
$155$	3.36044e10i	0.375611i
$156$	0	0
$157$	−7.13296e10	−0.747776	−0.373888	−	0.927474i	$-0.621976\pi$
$157$	−7.13296e10	−0.747776	−0.373888	+	0.927474i	$0.621976\pi$
$158$	0	0
$159$	− 1.63039e10i	− 0.160438i
$160$	0	0
$161$	1.88184e11	1.73962
$162$	0	0
$163$	2.67639e10i	0.232601i	0.993214	+	0.116300i	$0.0371035\pi$
$163$	2.67639e10i	0.232601i	−0.993214	+	0.116300i	$0.962896\pi$
$164$	0	0
$165$	1.08341e10	0.0885879
$166$	0	0
$167$	9.70185e9i	0.0746917i	0.999302	+	0.0373459i	$0.0118903\pi$
$167$	9.70185e9i	0.0746917i	−0.999302	+	0.0373459i	$0.988110\pi$
$168$	0	0
$169$	1.22532e11	0.888823
$170$	0	0
$171$	1.66993e11i	1.14214i
$172$	0	0
$173$	−2.37621e10	−0.153340	−0.0766698	−	0.997057i	$-0.524429\pi$
$173$	−2.37621e10	−0.153340	−0.0766698	+	0.997057i	$0.524429\pi$
$174$	0	0
$175$	− 7.67285e10i	− 0.467484i
$176$	0	0
$177$	1.03714e11	0.596993
$178$	0	0
$179$	2.62781e11i	1.42997i	0.699138	+	0.714987i	$0.253567\pi$
$179$	2.62781e11i	1.42997i	−0.699138	+	0.714987i	$0.746433\pi$
$180$	0	0
$181$	−3.05356e11	−1.57186	−0.785931	−	0.618314i	$-0.787816\pi$
$181$	−3.05356e11	−1.57186	−0.785931	+	0.618314i	$0.787816\pi$
$182$	0	0
$183$	− 4.99152e10i	− 0.243208i
$184$	0	0
$185$	1.78896e11	0.825548
$186$	0	0
$187$	8.75952e10i	0.383065i
$188$	0	0
$189$	−3.80391e11	−1.57733
$190$	0	0
$191$	1.65071e11i	0.649389i	0.945819	+	0.324694i	$0.105261\pi$
$191$	1.65071e11i	0.649389i	−0.945819	+	0.324694i	$0.894739\pi$
$192$	0	0
$193$	−3.00796e11	−1.12327	−0.561636	−	0.827384i	$-0.689828\pi$
$193$	−3.00796e11	−1.12327	−0.561636	+	0.827384i	$0.689828\pi$
$194$	0	0
$195$	1.66962e11i	0.592168i
$196$	0	0
$197$	−3.68648e11	−1.24245	−0.621227	−	0.783631i	$-0.713366\pi$
$197$	−3.68648e11	−1.24245	−0.621227	+	0.783631i	$0.713366\pi$
$198$	0	0
$199$	2.81728e11i	0.902744i	0.892336	+	0.451372i	$0.149065\pi$
$199$	2.81728e11i	0.902744i	−0.892336	+	0.451372i	$0.850935\pi$
$200$	0	0
$201$	1.55392e11	0.473640
$202$	0	0
$203$	− 9.77131e11i	− 2.83448i
$204$	0	0
$205$	−2.68218e10	−0.0740830
$206$	0	0
$207$	− 2.75095e11i	− 0.723820i
$208$	0	0
$209$	1.24982e11	0.313411
$210$	0	0
$211$	− 4.97682e10i	− 0.118998i	−0.998228	−	0.0594990i	$-0.981050\pi$
$211$	− 4.97682e10i	− 0.118998i	0.998228	−	0.0594990i	$-0.0189503\pi$
$212$	0	0
$213$	−1.29166e10	−0.0294613
$214$	0	0
$215$	6.23384e11i	1.35695i
$216$	0	0
$217$	−3.78230e11	−0.786064
$218$	0	0
$219$	− 4.38463e10i	− 0.0870386i
$220$	0	0
$221$	−1.34991e12	−2.56060
$222$	0	0
$223$	− 7.53050e11i	− 1.36552i	−0.730641	−	0.682762i	$-0.760778\pi$
$223$	− 7.53050e11i	− 1.36552i	0.730641	−	0.682762i	$-0.239222\pi$
$224$	0	0
$225$	−1.12165e11	−0.194511
$226$	0	0
$227$	− 4.78118e11i	− 0.793241i	−0.917983	−	0.396621i	$-0.870183\pi$
$227$	− 4.78118e11i	− 0.793241i	0.917983	−	0.396621i	$-0.129817\pi$
$228$	0	0
$229$	−3.93341e11	−0.624586	−0.312293	−	0.949986i	$-0.601097\pi$
$229$	−3.93341e11	−0.624586	−0.312293	+	0.949986i	$0.601097\pi$
$230$	0	0
$231$	1.21942e11i	0.185393i
$232$	0	0
$233$	−2.62943e11	−0.382896	−0.191448	−	0.981503i	$-0.561318\pi$
$233$	−2.62943e11	−0.382896	−0.191448	+	0.981503i	$0.561318\pi$
$234$	0	0
$235$	− 1.30875e11i	− 0.182607i
$236$	0	0
$237$	−2.93380e11	−0.392363
$238$	0	0
$239$	5.35317e11i	0.686470i	0.939250	+	0.343235i	$0.111523\pi$
$239$	5.35317e11i	0.686470i	−0.939250	+	0.343235i	$0.888477\pi$
$240$	0	0
$241$	5.32117e11	0.654518	0.327259	−	0.944935i	$-0.393875\pi$
$241$	5.32117e11	0.654518	0.327259	+	0.944935i	$0.393875\pi$
$242$	0	0
$243$	8.73962e11i	1.03148i
$244$	0	0
$245$	−1.70344e12	−1.92973
$246$	0	0
$247$	1.92606e12i	2.09500i
$248$	0	0
$249$	2.49618e11	0.260782
$250$	0	0
$251$	− 1.33488e12i	− 1.33991i	−0.742402	−	0.669954i	$-0.766314\pi$
$251$	− 1.33488e12i	− 1.33991i	0.742402	−	0.669954i	$-0.233686\pi$
$252$	0	0
$253$	−2.05888e11	−0.198622
$254$	0	0
$255$	− 8.65558e11i	− 0.802778i
$256$	0	0
$257$	−9.86756e11	−0.880125	−0.440062	−	0.897967i	$-0.645044\pi$
$257$	−9.86756e11	−0.880125	−0.440062	+	0.897967i	$0.645044\pi$
$258$	0	0
$259$	2.01354e12i	1.72768i
$260$	0	0
$261$	−1.42841e12	−1.17937
$262$	0	0
$263$	− 1.68123e12i	− 1.33613i	−0.744103	−	0.668065i	$-0.767123\pi$
$263$	− 1.68123e12i	− 1.33613i	0.744103	−	0.668065i	$-0.232877\pi$
$264$	0	0
$265$	−3.60287e11	−0.275689
$266$	0	0
$267$	8.01756e11i	0.590862i
$268$	0	0
$269$	−4.11806e10	−0.0292369	−0.0146185	−	0.999893i	$-0.504653\pi$
$269$	−4.11806e10	−0.0292369	−0.0146185	+	0.999893i	$0.504653\pi$
$270$	0	0
$271$	2.68592e11i	0.183758i	0.995770	+	0.0918791i	$0.0292873\pi$
$271$	2.68592e11i	0.183758i	−0.995770	+	0.0918791i	$0.970713\pi$
$272$	0	0
$273$	−1.87922e12	−1.23927
$274$	0	0
$275$	8.39468e10i	0.0533754i
$276$	0	0
$277$	7.53091e11	0.461794	0.230897	−	0.972978i	$-0.425834\pi$
$277$	7.53091e11	0.461794	0.230897	+	0.972978i	$0.425834\pi$
$278$	0	0
$279$	5.52912e11i	0.327066i
$280$	0	0
$281$	−1.09457e12	−0.624760	−0.312380	−	0.949957i	$-0.601126\pi$
$281$	−1.09457e12	−0.624760	−0.312380	+	0.949957i	$0.601126\pi$
$282$	0	0
$283$	2.91466e12i	1.60567i	0.596202	+	0.802834i	$0.296676\pi$
$283$	2.91466e12i	1.60567i	−0.596202	+	0.802834i	$0.703324\pi$
$284$	0	0
$285$	−1.23499e12	−0.656807
$286$	0	0
$287$	− 3.01889e11i	− 0.155038i
$288$	0	0
$289$	4.98214e12	2.47131
$290$	0	0
$291$	− 1.87416e11i	− 0.0898137i
$292$	0	0
$293$	−2.36019e12	−1.09297	−0.546485	−	0.837469i	$-0.684035\pi$
$293$	−2.36019e12	−1.09297	−0.546485	+	0.837469i	$0.684035\pi$
$294$	0	0
$295$	− 2.29189e12i	− 1.02585i
$296$	0	0
$297$	4.16177e11	0.180093
$298$	0	0
$299$	− 3.17288e12i	− 1.32769i
$300$	0	0
$301$	−7.01643e12	−2.83977
$302$	0	0
$303$	1.06771e12i	0.418063i
$304$	0	0
$305$	−1.10304e12	−0.417918
$306$	0	0
$307$	− 1.55346e12i	− 0.569649i	−0.958580	−	0.284825i	$-0.908065\pi$
$307$	− 1.55346e12i	− 0.569649i	0.958580	−	0.284825i	$-0.0919354\pi$
$308$	0	0
$309$	−1.12576e12	−0.399627
$310$	0	0
$311$	− 5.98675e10i	− 0.0205773i	−0.999947	−	0.0102887i	$-0.996725\pi$
$311$	− 5.98675e10i	− 0.0205773i	0.999947	−	0.0102887i	$-0.00327504\pi$
$312$	0	0
$313$	4.16246e12	1.38557	0.692785	−	0.721144i	$-0.256383\pi$
$313$	4.16246e12	1.38557	0.692785	+	0.721144i	$0.256383\pi$
$314$	0	0
$315$	3.60051e12i	1.16094i
$316$	0	0
$317$	1.31147e12	0.409698	0.204849	−	0.978794i	$-0.434330\pi$
$317$	1.31147e12	0.409698	0.204849	+	0.978794i	$0.434330\pi$
$318$	0	0
$319$	1.06906e12i	0.323629i
$320$	0	0
$321$	−1.73550e12	−0.509214
$322$	0	0
$323$	− 9.98500e12i	− 2.84011i
$324$	0	0
$325$	−1.29368e12	−0.356789
$326$	0	0
$327$	− 1.42628e11i	− 0.0381475i
$328$	0	0
$329$	1.47305e12	0.382153
$330$	0	0
$331$	− 5.23994e11i	− 0.131882i	−0.997824	−	0.0659411i	$-0.978995\pi$
$331$	− 5.23994e11i	− 0.131882i	0.997824	−	0.0659411i	$-0.0210049\pi$
$332$	0	0
$333$	2.94348e12	0.718852
$334$	0	0
$335$	− 3.43389e12i	− 0.813884i
$336$	0	0
$337$	2.91746e12	0.671205	0.335602	−	0.942004i	$-0.391060\pi$
$337$	2.91746e12	0.671205	0.335602	+	0.942004i	$0.391060\pi$
$338$	0	0
$339$	1.41831e12i	0.316790i
$340$	0	0
$341$	4.13813e11	0.0897496
$342$	0	0
$343$	− 1.06237e13i	− 2.23772i
$344$	0	0
$345$	2.03445e12	0.416247
$346$	0	0
$347$	1.67790e11i	0.0333518i	0.999861	+	0.0166759i	$0.00530835\pi$
$347$	1.67790e11i	0.0333518i	−0.999861	+	0.0166759i	$0.994692\pi$
$348$	0	0
$349$	4.90701e12	0.947742	0.473871	−	0.880594i	$-0.342856\pi$
$349$	4.90701e12	0.947742	0.473871	+	0.880594i	$0.342856\pi$
$350$	0	0
$351$	6.41360e12i	1.20383i
$352$	0	0
$353$	−5.40830e12	−0.986705	−0.493353	−	0.869829i	$-0.664229\pi$
$353$	−5.40830e12	−0.986705	−0.493353	+	0.869829i	$0.664229\pi$
$354$	0	0
$355$	2.85435e11i	0.0506250i
$356$	0	0
$357$	9.74218e12	1.68002
$358$	0	0
$359$	− 4.03290e11i	− 0.0676309i	−0.999428	−	0.0338155i	$-0.989234\pi$
$359$	− 4.03290e11i	− 0.0676309i	0.999428	−	0.0338155i	$-0.0107658\pi$
$360$	0	0
$361$	−8.11562e12	−1.32369
$362$	0	0
$363$	3.02269e12i	0.479579i
$364$	0	0
$365$	−9.68925e11	−0.149564
$366$	0	0
$367$	1.04392e13i	1.56796i	0.620786	+	0.783980i	$0.286813\pi$
$367$	1.04392e13i	1.56796i	−0.620786	+	0.783980i	$0.713187\pi$
$368$	0	0
$369$	−4.41314e11	−0.0645083
$370$	0	0
$371$	− 4.05517e12i	− 0.576952i
$372$	0	0
$373$	−9.02668e12	−1.25021	−0.625106	−	0.780540i	$-0.714944\pi$
$373$	−9.02668e12	−1.25021	−0.625106	+	0.780540i	$0.714944\pi$
$374$	0	0
$375$	− 4.02476e12i	− 0.542730i
$376$	0	0
$377$	−1.64749e13	−2.16330
$378$	0	0
$379$	1.03303e13i	1.32105i	0.750806	+	0.660523i	$0.229666\pi$
$379$	1.03303e13i	1.32105i	−0.750806	+	0.660523i	$0.770334\pi$
$380$	0	0
$381$	−6.41928e12	−0.799579
$382$	0	0
$383$	5.20386e12i	0.631439i	0.948853	+	0.315720i	$0.102246\pi$
$383$	5.20386e12i	0.631439i	−0.948853	+	0.315720i	$0.897754\pi$
$384$	0	0
$385$	2.69471e12	0.318572
$386$	0	0
$387$	1.02569e13i	1.18157i
$388$	0	0
$389$	6.47926e12	0.727407	0.363704	−	0.931515i	$-0.381512\pi$
$389$	6.47926e12	0.727407	0.363704	+	0.931515i	$0.381512\pi$
$390$	0	0
$391$	1.64487e13i	1.79990i
$392$	0	0
$393$	7.48812e12	0.798749
$394$	0	0
$395$	6.48317e12i	0.674220i
$396$	0	0
$397$	6.67280e12	0.676636	0.338318	−	0.941032i	$-0.390142\pi$
$397$	6.67280e12	0.676636	0.338318	+	0.941032i	$0.390142\pi$
$398$	0	0
$399$	− 1.39002e13i	− 1.37454i
$400$	0	0
$401$	4.31383e12	0.416046	0.208023	−	0.978124i	$-0.433297\pi$
$401$	4.31383e12	0.416046	0.208023	+	0.978124i	$0.433297\pi$
$402$	0	0
$403$	6.37716e12i	0.599932i
$404$	0	0
$405$	2.91243e12	0.267288
$406$	0	0
$407$	− 2.20297e12i	− 0.197259i
$408$	0	0
$409$	1.82143e13	1.59146	0.795731	−	0.605650i	$-0.207087\pi$
$409$	1.82143e13	1.59146	0.795731	+	0.605650i	$0.207087\pi$
$410$	0	0
$411$	− 3.72802e12i	− 0.317884i
$412$	0	0
$413$	2.57960e13	2.14685
$414$	0	0
$415$	− 5.51611e12i	− 0.448118i
$416$	0	0
$417$	−2.00032e12	−0.158642
$418$	0	0
$419$	− 8.44732e12i	− 0.654107i	−0.945006	−	0.327054i	$-0.893944\pi$
$419$	− 8.44732e12i	− 0.654107i	0.945006	−	0.327054i	$-0.106056\pi$
$420$	0	0
$421$	4.27063e12	0.322910	0.161455	−	0.986880i	$-0.448381\pi$
$421$	4.27063e12	0.322910	0.161455	+	0.986880i	$0.448381\pi$
$422$	0	0
$423$	− 2.15336e12i	− 0.159007i
$424$	0	0
$425$	6.70666e12	0.483684
$426$	0	0
$427$	− 1.24151e13i	− 0.874603i
$428$	0	0
$429$	2.05601e12	0.141494
$430$	0	0
$431$	7.53083e12i	0.506356i	0.967420	+	0.253178i	$0.0814759\pi$
$431$	7.53083e12i	0.506356i	−0.967420	+	0.253178i	$0.918524\pi$
$432$	0	0
$433$	−1.89649e13	−1.24598	−0.622989	−	0.782231i	$-0.714082\pi$
$433$	−1.89649e13	−1.24598	−0.622989	+	0.782231i	$0.714082\pi$
$434$	0	0
$435$	− 1.05637e13i	− 0.678220i
$436$	0	0
$437$	2.34692e13	1.47262
$438$	0	0
$439$	7.59980e12i	0.466100i	0.972465	+	0.233050i	$0.0748707\pi$
$439$	7.59980e12i	0.466100i	−0.972465	+	0.233050i	$0.925129\pi$
$440$	0	0
$441$	−2.80276e13	−1.68032
$442$	0	0
$443$	− 1.74464e13i	− 1.02256i	−0.859414	−	0.511280i	$-0.829172\pi$
$443$	− 1.74464e13i	− 1.02256i	0.859414	−	0.511280i	$-0.170828\pi$
$444$	0	0
$445$	1.77174e13	1.01531
$446$	0	0
$447$	2.63145e12i	0.147454i
$448$	0	0
$449$	1.92870e13	1.05690	0.528448	−	0.848965i	$-0.322774\pi$
$449$	1.92870e13	1.05690	0.528448	+	0.848965i	$0.322774\pi$
$450$	0	0
$451$	3.30290e11i	0.0177016i
$452$	0	0
$453$	1.42315e13	0.746037
$454$	0	0
$455$	4.15274e13i	2.12950i
$456$	0	0
$457$	1.97035e13	0.988468	0.494234	−	0.869329i	$-0.335449\pi$
$457$	1.97035e13	0.988468	0.494234	+	0.869329i	$0.335449\pi$
$458$	0	0
$459$	− 3.32491e13i	− 1.63199i
$460$	0	0
$461$	−1.70800e13	−0.820321	−0.410161	−	0.912013i	$-0.634527\pi$
$461$	−1.70800e13	−0.820321	−0.410161	+	0.912013i	$0.634527\pi$
$462$	0	0
$463$	− 2.03199e13i	− 0.955029i	−0.878624	−	0.477514i	$-0.841538\pi$
$463$	− 2.03199e13i	− 0.955029i	0.878624	−	0.477514i	$-0.158462\pi$
$464$	0	0
$465$	−4.08903e12	−0.188086
$466$	0	0
$467$	3.06149e13i	1.37831i	0.724612	+	0.689157i	$0.242019\pi$
$467$	3.06149e13i	1.37831i	−0.724612	+	0.689157i	$0.757981\pi$
$468$	0	0
$469$	3.86497e13	1.70326
$470$	0	0
$471$	− 8.67948e12i	− 0.374446i
$472$	0	0
$473$	7.67651e12	0.324234
$474$	0	0
$475$	− 9.56913e12i	− 0.395735i
$476$	0	0
$477$	−5.92801e12	−0.240058
$478$	0	0
$479$	− 2.59599e13i	− 1.02950i	−0.857341	−	0.514749i	$-0.827885\pi$
$479$	− 2.59599e13i	− 1.02950i	0.857341	−	0.514749i	$-0.172115\pi$
$480$	0	0
$481$	3.39494e13	1.31858
$482$	0	0
$483$	2.28985e13i	0.871106i
$484$	0	0
$485$	−4.14157e12	−0.154332
$486$	0	0
$487$	− 4.57345e13i	− 1.66955i	−0.550592	−	0.834774i	$-0.685598\pi$
$487$	− 4.57345e13i	− 1.66955i	0.550592	−	0.834774i	$-0.314402\pi$
$488$	0	0
$489$	−3.25666e12	−0.116474
$490$	0	0
$491$	4.82141e12i	0.168953i	0.996425	+	0.0844766i	$0.0269218\pi$
$491$	4.82141e12i	0.168953i	−0.996425	+	0.0844766i	$0.973078\pi$
$492$	0	0
$493$	8.54088e13	2.93270
$494$	0	0
$495$	− 3.93923e12i	− 0.132552i
$496$	0	0
$497$	−3.21267e12	−0.105946
$498$	0	0
$499$	1.54166e13i	0.498295i	0.968466	+	0.249148i	$0.0801504\pi$
$499$	1.54166e13i	0.498295i	−0.968466	+	0.249148i	$0.919850\pi$
$500$	0	0
$501$	−1.18053e12	−0.0374016
$502$	0	0
$503$	5.50859e13i	1.71080i	0.517965	+	0.855402i	$0.326690\pi$
$503$	5.50859e13i	1.71080i	−0.517965	+	0.855402i	$0.673310\pi$
$504$	0	0
$505$	2.35946e13	0.718381
$506$	0	0
$507$	1.49098e13i	0.445075i
$508$	0	0
$509$	9.04337e12	0.264692	0.132346	−	0.991204i	$-0.457749\pi$
$509$	9.04337e12	0.264692	0.132346	+	0.991204i	$0.457749\pi$
$510$	0	0
$511$	− 1.09056e13i	− 0.313001i
$512$	0	0
$513$	−4.74402e13	−1.33524
$514$	0	0
$515$	2.48774e13i	0.686702i
$516$	0	0
$517$	−1.61163e12	−0.0436327
$518$	0	0
$519$	− 2.89140e12i	− 0.0767842i
$520$	0	0
$521$	−2.29993e13	−0.599136	−0.299568	−	0.954075i	$-0.596843\pi$
$521$	−2.29993e13	−0.599136	−0.299568	+	0.954075i	$0.596843\pi$
$522$	0	0
$523$	2.96086e13i	0.756676i	0.925667	+	0.378338i	$0.123504\pi$
$523$	2.96086e13i	0.756676i	−0.925667	+	0.378338i	$0.876496\pi$
$524$	0	0
$525$	9.33642e12	0.234091
$526$	0	0
$527$	− 3.30603e13i	− 0.813305i
$528$	0	0
$529$	2.76467e12	0.0667367
$530$	0	0
$531$	− 3.77097e13i	− 0.893265i
$532$	0	0
$533$	−5.09001e12	−0.118327
$534$	0	0
$535$	3.83516e13i	0.875013i
$536$	0	0
$537$	−3.19755e13	−0.716054
$538$	0	0
$539$	2.09765e13i	0.461095i
$540$	0	0
$541$	−4.97902e13	−1.07438	−0.537189	−	0.843462i	$-0.680514\pi$
$541$	−4.97902e13	−1.07438	−0.537189	+	0.843462i	$0.680514\pi$
$542$	0	0
$543$	− 3.71562e13i	− 0.787104i
$544$	0	0
$545$	−3.15183e12	−0.0655511
$546$	0	0
$547$	− 1.59283e13i	− 0.325261i	−0.986687	−	0.162631i	$-0.948002\pi$
$547$	− 1.59283e13i	− 0.325261i	0.986687	−	0.162631i	$-0.0519979\pi$
$548$	0	0
$549$	−1.81489e13	−0.363906
$550$	0	0
$551$	− 1.21862e14i	− 2.39944i
$552$	0	0
$553$	−7.29705e13	−1.41098
$554$	0	0
$555$	2.17683e13i	0.413390i
$556$	0	0
$557$	1.45122e13	0.270682	0.135341	−	0.990799i	$-0.456787\pi$
$557$	1.45122e13	0.270682	0.135341	+	0.990799i	$0.456787\pi$
$558$	0	0
$559$	1.18301e14i	2.16734i
$560$	0	0
$561$	−1.06587e13	−0.191818
$562$	0	0
$563$	5.60100e13i	0.990202i	0.868835	+	0.495101i	$0.164869\pi$
$563$	5.60100e13i	0.990202i	−0.868835	+	0.495101i	$0.835131\pi$
$564$	0	0
$565$	3.13421e13	0.544359
$566$	0	0
$567$	3.27805e13i	0.559371i
$568$	0	0
$569$	4.30086e12	0.0721097	0.0360549	−	0.999350i	$-0.488521\pi$
$569$	4.30086e12	0.0721097	0.0360549	+	0.999350i	$0.488521\pi$
$570$	0	0
$571$	− 1.05773e14i	− 1.74258i	−0.490770	−	0.871289i	$-0.663284\pi$
$571$	− 1.05773e14i	− 1.74258i	0.490770	−	0.871289i	$-0.336716\pi$
$572$	0	0
$573$	−2.00861e13	−0.325179
$574$	0	0
$575$	1.57636e13i	0.250794i
$576$	0	0
$577$	6.44373e13	1.00753	0.503766	−	0.863840i	$-0.331948\pi$
$577$	6.44373e13	1.00753	0.503766	+	0.863840i	$0.331948\pi$
$578$	0	0
$579$	− 3.66012e13i	− 0.562474i
$580$	0	0
$581$	6.20858e13	0.937804
$582$	0	0
$583$	4.43667e12i	0.0658740i
$584$	0	0
$585$	6.07065e13	0.886045
$586$	0	0
$587$	5.38824e13i	0.773136i	0.922261	+	0.386568i	$0.126340\pi$
$587$	5.38824e13i	0.773136i	−0.922261	+	0.386568i	$0.873660\pi$
$588$	0	0
$589$	−4.71707e13	−0.665420
$590$	0	0
$591$	− 4.48575e13i	− 0.622154i
$592$	0	0
$593$	−1.11219e14	−1.51672	−0.758361	−	0.651834i	$-0.774000\pi$
$593$	−1.11219e14	−1.51672	−0.758361	+	0.651834i	$0.774000\pi$
$594$	0	0
$595$	− 2.15285e14i	− 2.88688i
$596$	0	0
$597$	−3.42810e13	−0.452046
$598$	0	0
$599$	− 1.16949e14i	− 1.51657i	−0.651921	−	0.758287i	$-0.726037\pi$
$599$	− 1.16949e14i	− 1.51657i	0.651921	−	0.758287i	$-0.273963\pi$
$600$	0	0
$601$	3.83099e13	0.488583	0.244292	−	0.969702i	$-0.421445\pi$
$601$	3.83099e13	0.488583	0.244292	+	0.969702i	$0.421445\pi$
$602$	0	0
$603$	− 5.64997e13i	− 0.708695i
$604$	0	0
$605$	6.67960e13	0.824088
$606$	0	0
$607$	1.08595e14i	1.31785i	0.752208	+	0.658926i	$0.228989\pi$
$607$	1.08595e14i	1.31785i	−0.752208	+	0.658926i	$0.771011\pi$
$608$	0	0
$609$	1.18899e14	1.41935
$610$	0	0
$611$	− 2.48364e13i	− 0.291663i
$612$	0	0
$613$	−3.85493e13	−0.445363	−0.222681	−	0.974891i	$-0.571481\pi$
$613$	−3.85493e13	−0.445363	−0.222681	+	0.974891i	$0.571481\pi$
$614$	0	0
$615$	− 3.26371e12i	− 0.0370968i
$616$	0	0
$617$	1.55996e14	1.74457	0.872283	−	0.489002i	$-0.162639\pi$
$617$	1.55996e14	1.74457	0.872283	+	0.489002i	$0.162639\pi$
$618$	0	0
$619$	1.16264e14i	1.27936i	0.768643	+	0.639678i	$0.220932\pi$
$619$	1.16264e14i	1.27936i	−0.768643	+	0.639678i	$0.779068\pi$
$620$	0	0
$621$	7.81503e13	0.846199
$622$	0	0
$623$	1.99416e14i	2.12481i
$624$	0	0
$625$	−6.41821e13	−0.672998
$626$	0	0
$627$	1.52079e13i	0.156939i
$628$	0	0
$629$	−1.75999e14	−1.78755
$630$	0	0
$631$	5.58442e12i	0.0558253i	0.999610	+	0.0279127i	$0.00888603\pi$
$631$	5.58442e12i	0.0558253i	−0.999610	+	0.0279127i	$0.991114\pi$
$632$	0	0
$633$	6.05586e12	0.0595878
$634$	0	0
$635$	1.41855e14i	1.37396i
$636$	0	0
$637$	−3.23264e14	−3.08220
$638$	0	0
$639$	4.69642e12i	0.0440821i
$640$	0	0
$641$	3.89575e13	0.359999	0.179999	−	0.983667i	$-0.442390\pi$
$641$	3.89575e13	0.359999	0.179999	+	0.983667i	$0.442390\pi$
$642$	0	0
$643$	− 3.61241e13i	− 0.328656i	−0.986406	−	0.164328i	$-0.947454\pi$
$643$	− 3.61241e13i	− 0.328656i	0.986406	−	0.164328i	$-0.0525456\pi$
$644$	0	0
$645$	−7.58542e13	−0.679487
$646$	0	0
$647$	1.66446e14i	1.46809i	0.679100	+	0.734046i	$0.262370\pi$
$647$	1.66446e14i	1.46809i	−0.679100	+	0.734046i	$0.737630\pi$
$648$	0	0
$649$	−2.82228e13	−0.245119
$650$	0	0
$651$	− 4.60236e13i	− 0.393618i
$652$	0	0
$653$	−2.13416e14	−1.79747	−0.898734	−	0.438493i	$-0.855512\pi$
$653$	−2.13416e14	−1.79747	−0.898734	+	0.438493i	$0.855512\pi$
$654$	0	0
$655$	− 1.65474e14i	− 1.37254i
$656$	0	0
$657$	−1.59423e13	−0.130234
$658$	0	0
$659$	1.80608e14i	1.45315i	0.687088	+	0.726574i	$0.258889\pi$
$659$	1.80608e14i	1.45315i	−0.687088	+	0.726574i	$0.741111\pi$
$660$	0	0
$661$	−1.76768e13	−0.140086	−0.0700431	−	0.997544i	$-0.522314\pi$
$661$	−1.76768e13	−0.140086	−0.0700431	+	0.997544i	$0.522314\pi$
$662$	0	0
$663$	− 1.64258e14i	− 1.28221i
$664$	0	0
$665$	−3.07171e14	−2.36195
$666$	0	0
$667$	2.00749e14i	1.52063i
$668$	0	0
$669$	9.16321e13	0.683781
$670$	0	0
$671$	1.35831e13i	0.0998587i
$672$	0	0
$673$	1.60902e14	1.16543	0.582715	−	0.812676i	$-0.301990\pi$
$673$	1.60902e14	1.16543	0.582715	+	0.812676i	$0.301990\pi$
$674$	0	0
$675$	− 3.18643e13i	− 0.227397i
$676$	0	0
$677$	−1.09256e14	−0.768248	−0.384124	−	0.923282i	$-0.625496\pi$
$677$	−1.09256e14	−0.768248	−0.384124	+	0.923282i	$0.625496\pi$
$678$	0	0
$679$	− 4.66149e13i	− 0.322980i
$680$	0	0
$681$	5.81780e13	0.397213
$682$	0	0
$683$	1.57116e13i	0.105710i	0.998602	+	0.0528552i	$0.0168322\pi$
$683$	1.57116e13i	0.105710i	−0.998602	+	0.0528552i	$0.983168\pi$
$684$	0	0
$685$	−8.23826e13	−0.546239
$686$	0	0
$687$	− 4.78623e13i	− 0.312759i
$688$	0	0
$689$	−6.83723e13	−0.440336
$690$	0	0
$691$	− 1.42414e14i	− 0.903986i	−0.892022	−	0.451993i	$-0.850713\pi$
$691$	− 1.42414e14i	− 0.903986i	0.892022	−	0.451993i	$-0.149287\pi$
$692$	0	0
$693$	4.43375e13	0.277399
$694$	0	0
$695$	4.42035e13i	0.272604i
$696$	0	0
$697$	2.63875e13	0.160411
$698$	0	0
$699$	− 3.19952e13i	− 0.191734i
$700$	0	0
$701$	7.44711e13	0.439944	0.219972	−	0.975506i	$-0.429403\pi$
$701$	7.44711e13	0.439944	0.219972	+	0.975506i	$0.429403\pi$
$702$	0	0
$703$	2.51117e14i	1.46251i
$704$	0	0
$705$	1.59251e13	0.0914398
$706$	0	0
$707$	2.65566e14i	1.50340i
$708$	0	0
$709$	−2.39122e14	−1.33472	−0.667358	−	0.744737i	$-0.732575\pi$
$709$	−2.39122e14	−1.33472	−0.667358	+	0.744737i	$0.732575\pi$
$710$	0	0
$711$	1.06671e14i	0.587083i
$712$	0	0
$713$	7.77063e13	0.421705
$714$	0	0
$715$	− 4.54342e13i	− 0.243138i
$716$	0	0
$717$	−6.51381e13	−0.343747
$718$	0	0
$719$	− 5.65489e13i	− 0.294293i	−0.989115	−	0.147146i	$-0.952991\pi$
$719$	− 5.65489e13i	− 0.294293i	0.989115	−	0.147146i	$-0.0470089\pi$
$720$	0	0
$721$	−2.80004e14	−1.43710
$722$	0	0
$723$	6.47486e13i	0.327747i
$724$	0	0
$725$	8.18516e13	0.408636
$726$	0	0
$727$	− 1.80601e14i	− 0.889298i	−0.895705	−	0.444649i	$-0.853328\pi$
$727$	− 1.80601e14i	− 0.889298i	0.895705	−	0.444649i	$-0.146672\pi$
$728$	0	0
$729$	−4.23881e13	−0.205876
$730$	0	0
$731$	− 6.13290e14i	− 2.93818i
$732$	0	0
$733$	1.92425e14	0.909373	0.454686	−	0.890652i	$-0.349751\pi$
$733$	1.92425e14	0.909373	0.454686	+	0.890652i	$0.349751\pi$
$734$	0	0
$735$	− 2.07277e14i	− 0.966304i
$736$	0	0
$737$	−4.22857e13	−0.194472
$738$	0	0
$739$	4.62941e13i	0.210041i	0.994470	+	0.105020i	$0.0334908\pi$
$739$	4.62941e13i	0.210041i	−0.994470	+	0.105020i	$0.966509\pi$
$740$	0	0
$741$	−2.34365e14	−1.04906
$742$	0	0
$743$	3.11220e14i	1.37443i	0.726452	+	0.687217i	$0.241168\pi$
$743$	3.11220e14i	1.37443i	−0.726452	+	0.687217i	$0.758832\pi$
$744$	0	0
$745$	5.81503e13	0.253379
$746$	0	0
$747$	− 9.07596e13i	− 0.390202i
$748$	0	0
$749$	−4.31661e14	−1.83119
$750$	0	0
$751$	− 1.61606e14i	− 0.676484i	−0.941059	−	0.338242i	$-0.890168\pi$
$751$	− 1.61606e14i	− 0.676484i	0.941059	−	0.338242i	$-0.109832\pi$
$752$	0	0
$753$	1.62431e14	0.670954
$754$	0	0
$755$	− 3.14492e14i	− 1.28196i
$756$	0	0
$757$	−4.17141e14	−1.67804	−0.839022	−	0.544097i	$-0.816872\pi$
$757$	−4.17141e14	−1.67804	−0.839022	+	0.544097i	$0.816872\pi$
$758$	0	0
$759$	− 2.50527e13i	− 0.0994593i
$760$	0	0
$761$	4.50768e13	0.176616	0.0883080	−	0.996093i	$-0.471854\pi$
$761$	4.50768e13	0.176616	0.0883080	+	0.996093i	$0.471854\pi$
$762$	0	0
$763$	− 3.54750e13i	− 0.137183i
$764$	0	0
$765$	−3.14712e14	−1.20118
$766$	0	0
$767$	− 4.34935e14i	− 1.63850i
$768$	0	0
$769$	1.82550e13	0.0678812	0.0339406	−	0.999424i	$-0.489194\pi$
$769$	1.82550e13	0.0678812	0.0339406	+	0.999424i	$0.489194\pi$
$770$	0	0
$771$	− 1.20070e14i	− 0.440719i
$772$	0	0
$773$	4.36967e14	1.58326	0.791628	−	0.611003i	$-0.209234\pi$
$773$	4.36967e14	1.58326	0.791628	+	0.611003i	$0.209234\pi$
$774$	0	0
$775$	− 3.16833e13i	− 0.113324i
$776$	0	0
$777$	−2.45011e14	−0.865127
$778$	0	0
$779$	− 3.76499e13i	− 0.131243i
$780$	0	0
$781$	3.51491e12	0.0120965
$782$	0	0
$783$	− 4.05789e14i	− 1.37877i
$784$	0	0
$785$	−1.91801e14	−0.643432
$786$	0	0
$787$	4.98660e14i	1.65170i	0.563891	+	0.825849i	$0.309304\pi$
$787$	4.98660e14i	1.65170i	−0.563891	+	0.825849i	$0.690696\pi$
$788$	0	0
$789$	2.04574e14	0.669062
$790$	0	0
$791$	3.52767e14i	1.13921i
$792$	0	0
$793$	−2.09325e14	−0.667507
$794$	0	0
$795$	− 4.38402e13i	− 0.138050i
$796$	0	0
$797$	8.93513e12	0.0277849	0.0138925	−	0.999903i	$-0.495578\pi$
$797$	8.93513e12	0.0277849	0.0138925	+	0.999903i	$0.495578\pi$
$798$	0	0
$799$	1.28756e14i	0.395397i
$800$	0	0
$801$	2.91514e14	0.884091
$802$	0	0
$803$	1.19316e13i	0.0357372i
$804$	0	0
$805$	5.06016e14	1.49687
$806$	0	0
$807$	− 5.01091e12i	− 0.0146403i
$808$	0	0
$809$	3.61148e14	1.04218	0.521090	−	0.853502i	$-0.325526\pi$
$809$	3.61148e14	1.04218	0.521090	+	0.853502i	$0.325526\pi$
$810$	0	0
$811$	− 8.05369e12i	− 0.0229557i	−0.999934	−	0.0114779i	$-0.996346\pi$
$811$	− 8.05369e12i	− 0.0229557i	0.999934	−	0.0114779i	$-0.00365359\pi$
$812$	0	0
$813$	−3.26826e13	−0.0920162
$814$	0	0
$815$	7.19665e13i	0.200144i
$816$	0	0
$817$	−8.75047e14	−2.40393
$818$	0	0
$819$	6.83274e14i	1.85428i
$820$	0	0
$821$	−6.81853e14	−1.82799	−0.913997	−	0.405721i	$-0.867020\pi$
$821$	−6.81853e14	−1.82799	−0.913997	+	0.405721i	$0.867020\pi$
$822$	0	0
$823$	− 2.85674e12i	− 0.00756610i	−0.999993	−	0.00378305i	$-0.998796\pi$
$823$	− 2.85674e12i	− 0.00756610i	0.999993	−	0.00378305i	$-0.00120418\pi$
$824$	0	0
$825$	−1.02148e13	−0.0267275
$826$	0	0
$827$	4.38934e14i	1.13468i	0.823485	+	0.567338i	$0.192027\pi$
$827$	4.38934e14i	1.13468i	−0.823485	+	0.567338i	$0.807973\pi$
$828$	0	0
$829$	4.60079e14	1.17506	0.587530	−	0.809203i	$-0.300100\pi$
$829$	4.60079e14	1.17506	0.587530	+	0.809203i	$0.300100\pi$
$830$	0	0
$831$	9.16371e13i	0.231242i
$832$	0	0
$833$	1.67585e15	4.17841
$834$	0	0
$835$	2.60877e13i	0.0642693i
$836$	0	0
$837$	−1.57074e14	−0.382364
$838$	0	0
$839$	1.05802e13i	0.0254499i	0.999919	+	0.0127249i	$0.00405058\pi$
$839$	1.05802e13i	0.0254499i	−0.999919	+	0.0127249i	$0.995949\pi$
$840$	0	0
$841$	6.21666e14	1.47767
$842$	0	0
$843$	− 1.33189e14i	− 0.312846i
$844$	0	0
$845$	3.29481e14	0.764798
$846$	0	0
$847$	7.51814e14i	1.72462i
$848$	0	0
$849$	−3.54660e14	−0.804032
$850$	0	0
$851$	− 4.13677e14i	− 0.926858i
$852$	0	0
$853$	2.94235e14	0.651552	0.325776	−	0.945447i	$-0.394375\pi$
$853$	2.94235e14	0.651552	0.325776	+	0.945447i	$0.394375\pi$
$854$	0	0
$855$	4.49034e14i	0.982763i
$856$	0	0
$857$	6.02180e14	1.30263	0.651316	−	0.758806i	$-0.274217\pi$
$857$	6.02180e14	1.30263	0.651316	+	0.758806i	$0.274217\pi$
$858$	0	0
$859$	5.52016e14i	1.18028i	0.807300	+	0.590141i	$0.200928\pi$
$859$	5.52016e14i	1.18028i	−0.807300	+	0.590141i	$0.799072\pi$
$860$	0	0
$861$	3.67343e13	0.0776347
$862$	0	0
$863$	− 6.88007e14i	− 1.43727i	−0.695387	−	0.718636i	$-0.744767\pi$
$863$	− 6.88007e14i	− 1.43727i	0.695387	−	0.718636i	$-0.255233\pi$
$864$	0	0
$865$	−6.38949e13	−0.131943
$866$	0	0
$867$	6.06233e14i	1.23750i
$868$	0	0
$869$	7.98353e13	0.161100
$870$	0	0
$871$	− 6.51654e14i	− 1.29995i
$872$	0	0
$873$	−6.81436e13	−0.134386
$874$	0	0
$875$	− 1.00105e15i	− 1.95172i
$876$	0	0
$877$	5.78717e14	1.11550	0.557748	−	0.830010i	$-0.311665\pi$
$877$	5.78717e14	1.11550	0.557748	+	0.830010i	$0.311665\pi$
$878$	0	0
$879$	− 2.87191e14i	− 0.547301i
$880$	0	0
$881$	−6.04795e14	−1.13954	−0.569768	−	0.821805i	$-0.692967\pi$
$881$	−6.04795e14	−1.13954	−0.569768	+	0.821805i	$0.692967\pi$
$882$	0	0
$883$	− 2.86926e14i	− 0.534522i	−0.963624	−	0.267261i	$-0.913881\pi$
$883$	− 2.86926e14i	− 0.534522i	0.963624	−	0.267261i	$-0.0861187\pi$
$884$	0	0
$885$	2.78880e14	0.513689
$886$	0	0
$887$	1.21370e14i	0.221051i	0.993873	+	0.110525i	$0.0352534\pi$
$887$	1.21370e14i	0.221051i	−0.993873	+	0.110525i	$0.964747\pi$
$888$	0	0
$889$	−1.59663e15	−2.87538
$890$	0	0
$891$	− 3.58643e13i	− 0.0638667i
$892$	0	0
$893$	1.83710e14	0.323501
$894$	0	0
$895$	7.06601e14i	1.23044i
$896$	0	0
$897$	3.86080e14	0.664837
$898$	0	0
$899$	− 4.03484e14i	− 0.687113i
$900$	0	0
$901$	3.54453e14	0.596946
$902$	0	0
$903$	− 8.53768e14i	− 1.42200i
$904$	0	0
$905$	−8.21086e14	−1.35253
$906$	0	0
$907$	8.79272e14i	1.43247i	0.697857	+	0.716237i	$0.254137\pi$
$907$	8.79272e14i	1.43247i	−0.697857	+	0.716237i	$0.745863\pi$
$908$	0	0
$909$	3.88214e14	0.625536
$910$	0	0
$911$	1.15401e15i	1.83915i	0.392915	+	0.919575i	$0.371467\pi$
$911$	1.15401e15i	1.83915i	−0.392915	+	0.919575i	$0.628533\pi$
$912$	0	0
$913$	−6.79267e13	−0.107075
$914$	0	0
$915$	− 1.34219e14i	− 0.209271i
$916$	0	0
$917$	1.86247e15	2.87239
$918$	0	0
$919$	6.02115e14i	0.918548i	0.888295	+	0.459274i	$0.151890\pi$
$919$	6.02115e14i	0.918548i	−0.888295	+	0.459274i	$0.848110\pi$
$920$	0	0
$921$	1.89027e14	0.285250
$922$	0	0
$923$	5.41674e13i	0.0808592i
$924$	0	0
$925$	−1.68669e14	−0.249073
$926$	0	0
$927$	4.09321e14i	0.597951i
$928$	0	0
$929$	6.38405e13	0.0922609	0.0461305	−	0.998935i	$-0.485311\pi$
$929$	6.38405e13	0.0922609	0.0461305	+	0.998935i	$0.485311\pi$
$930$	0	0
$931$	− 2.39112e15i	− 3.41864i
$932$	0	0
$933$	7.28476e12	0.0103040
$934$	0	0
$935$	2.35538e14i	0.329612i
$936$	0	0
$937$	4.16317e14	0.576402	0.288201	−	0.957570i	$-0.406943\pi$
$937$	4.16317e14	0.576402	0.288201	+	0.957570i	$0.406943\pi$
$938$	0	0
$939$	5.06494e14i	0.693819i
$940$	0	0
$941$	9.14113e14	1.23894	0.619472	−	0.785018i	$-0.287347\pi$
$941$	9.14113e14	1.23894	0.619472	+	0.785018i	$0.287347\pi$
$942$	0	0
$943$	6.20223e13i	0.0831743i
$944$	0	0
$945$	−1.02285e15	−1.35723
$946$	0	0
$947$	− 4.75991e14i	− 0.624955i	−0.949925	−	0.312478i	$-0.898841\pi$
$947$	− 4.75991e14i	− 0.624955i	0.949925	−	0.312478i	$-0.101159\pi$
$948$	0	0
$949$	−1.83874e14	−0.238886
$950$	0	0
$951$	1.59582e14i	0.205155i
$952$	0	0
$953$	3.25881e14	0.414566	0.207283	−	0.978281i	$-0.433538\pi$
$953$	3.25881e14	0.414566	0.207283	+	0.978281i	$0.433538\pi$
$954$	0	0
$955$	4.43867e14i	0.558774i
$956$	0	0
$957$	−1.30084e14	−0.162056
$958$	0	0
$959$	− 9.27247e14i	− 1.14315i
$960$	0	0
$961$	6.63446e14	0.809448
$962$	0	0
$963$	6.31020e14i	0.761924i
$964$	0	0
$965$	−8.08822e14	−0.966532
$966$	0	0
$967$	− 1.32849e15i	− 1.57117i	−0.618751	−	0.785587i	$-0.712361\pi$
$967$	− 1.32849e15i	− 1.57117i	0.618751	−	0.785587i	$-0.287639\pi$
$968$	0	0
$969$	1.21499e15	1.42218
$970$	0	0
$971$	− 1.33037e15i	− 1.54126i	−0.637285	−	0.770628i	$-0.719943\pi$
$971$	− 1.33037e15i	− 1.54126i	0.637285	−	0.770628i	$-0.280057\pi$
$972$	0	0
$973$	−4.97526e14	−0.570495
$974$	0	0
$975$	− 1.57417e14i	− 0.178660i
$976$	0	0
$977$	−1.00671e15	−1.13092	−0.565462	−	0.824774i	$-0.691302\pi$
$977$	−1.00671e15	−1.13092	−0.565462	+	0.824774i	$0.691302\pi$
$978$	0	0
$979$	− 2.18176e14i	− 0.242602i
$980$	0	0
$981$	−5.18588e13	−0.0570792
$982$	0	0
$983$	− 6.92063e14i	− 0.754011i	−0.926211	−	0.377006i	$-0.876954\pi$
$983$	− 6.92063e14i	− 0.754011i	0.926211	−	0.377006i	$-0.123046\pi$
$984$	0	0
$985$	−9.91272e14	−1.06908
$986$	0	0
$987$	1.79242e14i	0.191362i
$988$	0	0
$989$	1.44150e15	1.52347
$990$	0	0
$991$	1.78144e15i	1.86381i	0.362700	+	0.931906i	$0.381855\pi$
$991$	1.78144e15i	1.86381i	−0.362700	+	0.931906i	$0.618145\pi$
$992$	0	0
$993$	6.37603e13	0.0660395
$994$	0	0
$995$	7.57550e14i	0.776776i
$996$	0	0
$997$	8.40996e14	0.853726	0.426863	−	0.904316i	$-0.359619\pi$
$997$	8.40996e14	0.853726	0.426863	+	0.904316i	$0.359619\pi$
$998$	0	0
$999$	8.36197e14i	0.840391i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.11.c.k.255.6		10
4.3	odd	2	inner	256.11.c.k.255.5		10
8.3	odd	2		256.11.c.j.255.6		10
8.5	even	2		256.11.c.j.255.5		10
16.3	odd	4		128.11.d.f.63.11	yes	20
16.5	even	4		128.11.d.f.63.12	yes	20
16.11	odd	4		128.11.d.f.63.10	yes	20
16.13	even	4		128.11.d.f.63.9	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.11.d.f.63.9	✓	20	16.13	even	4
128.11.d.f.63.10	yes	20	16.11	odd	4
128.11.d.f.63.11	yes	20	16.3	odd	4
128.11.d.f.63.12	yes	20	16.5	even	4
256.11.c.j.255.5		10	8.5	even	2
256.11.c.j.255.6		10	8.3	odd	2
256.11.c.k.255.5		10	4.3	odd	2	inner
256.11.c.k.255.6		10	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+121.681i q^{3} +2688.94 q^{5} +30265.0i q^{7} +44242.7 q^{9} +O(q^{10})\)	\(q+121.681i q^{3} +2688.94 q^{5} +30265.0i q^{7} +44242.7 q^{9} -33112.3i q^{11} +510285. q^{13} +327194. i q^{15} -2.64540e6 q^{17} +3.77448e6i q^{19} -3.68269e6 q^{21} -6.21786e6i q^{23} -2.53522e6 q^{25} +1.25687e7i q^{27} -3.22858e7 q^{29} +1.24973e7i q^{31} +4.02915e6 q^{33} +8.13809e7i q^{35} +6.65303e7 q^{37} +6.20921e7i q^{39} -9.97485e6 q^{41} +2.31833e8i q^{43} +1.18966e8 q^{45} -4.86716e7i q^{47} -6.33498e8 q^{49} -3.21896e8i q^{51} -1.33989e8 q^{53} -8.90370e7i q^{55} -4.59284e8 q^{57} -8.52337e8i q^{59} -4.10213e8 q^{61} +1.33901e9i q^{63} +1.37213e9 q^{65} -1.27704e9i q^{67} +7.56598e8 q^{69} +1.06151e8i q^{71} -3.60337e8 q^{73} -3.08489e8i q^{75} +1.00214e9 q^{77} +2.41105e9i q^{79} +1.08311e9 q^{81} -2.05140e9i q^{83} -7.11332e9 q^{85} -3.92858e9i q^{87} +6.58898e9 q^{89} +1.54438e10i q^{91} -1.52068e9 q^{93} +1.01494e10i q^{95} -1.54022e9 q^{97} -1.46498e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 6232 q^{5} - 218038 q^{9}+O(q^{10}) \) 10 * q + 6232 * q^5 - 218038 * q^9	\( 10 q + 6232 q^{5} - 218038 q^{9} + 485560 q^{13} - 1468772 q^{17} + 15545024 q^{21} + 4383462 q^{25} + 25263800 q^{29} - 122737264 q^{33} + 178255288 q^{37} + 91656876 q^{41} - 718346408 q^{45} - 842454934 q^{49} - 798458664 q^{53} - 1612102544 q^{57} + 102636184 q^{61} + 376325920 q^{65} + 207224128 q^{69} + 2854265572 q^{73} + 9037740608 q^{77} + 11990017466 q^{81} - 5473132400 q^{85} - 11790814556 q^{89} - 24576098304 q^{93} - 9363277860 q^{97}+O(q^{100}) \) 10 * q + 6232 * q^5 - 218038 * q^9 + 485560 * q^13 - 1468772 * q^17 + 15545024 * q^21 + 4383462 * q^25 + 25263800 * q^29 - 122737264 * q^33 + 178255288 * q^37 + 91656876 * q^41 - 718346408 * q^45 - 842454934 * q^49 - 798458664 * q^53 - 1612102544 * q^57 + 102636184 * q^61 + 376325920 * q^65 + 207224128 * q^69 + 2854265572 * q^73 + 9037740608 * q^77 + 11990017466 * q^81 - 5473132400 * q^85 - 11790814556 * q^89 - 24576098304 * q^93 - 9363277860 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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