LMFDB - Embedded newform 256.12.b.p.129.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [256,12,Mod(129,256)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("256.129"); S:= CuspForms(chi, 12); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(256, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 12, names="a")

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	256.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,-98736,0,-627628] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$196.695854223$
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} - 2 x^{11} - 2789 x^{10} - 107880 x^{9} + 4082152 x^{8} + 294993082 x^{7} - 2405623951 x^{6} + \cdots + 10\!\cdots\!25 $ x^12 - 2x^11 - 2789x^10 - 107880x^9 + 4082152x^8 + 294993082x^7 - 2405623951x^6 - 289286387330x^5 + 868103289142x^4 + 58558685920500x^3 - 1223688973325375x^2 + 34567751484157450*x + 1035378318383260825
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{110}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 128)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.2
Root			$6.00626 - 18.8120i$ of defining polynomial
Character	$\chi$	$=$	256.129
Dual form			256.12.b.p.129.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-597.717i q^{3} +9917.53i q^{5} -77908.4 q^{7} -180119. q^{9} -854628. i q^{11} +1.11758e6i q^{13} +5.92788e6 q^{15} +1.43133e6 q^{17} -9.95811e6i q^{19} +4.65672e7i q^{21} -3.13742e7 q^{23} -4.95293e7 q^{25} +1.77624e6i q^{27} -1.87059e8i q^{29} -6.40413e7 q^{31} -5.10826e8 q^{33} -7.72659e8i q^{35} -3.73906e8i q^{37} +6.67994e8 q^{39} -1.34692e9 q^{41} +5.87529e8i q^{43} -1.78633e9i q^{45} -8.78487e8 q^{47} +4.09239e9 q^{49} -8.55533e8i q^{51} +1.76776e9i q^{53} +8.47580e9 q^{55} -5.95213e9 q^{57} -7.95505e9i q^{59} +4.23552e9i q^{61} +1.40328e10 q^{63} -1.10836e10 q^{65} -1.38472e10i q^{67} +1.87529e10i q^{69} -3.21284e9 q^{71} -1.70921e9 q^{73} +2.96045e10i q^{75} +6.65827e10i q^{77} -8.12305e9 q^{79} -3.08458e10 q^{81} -4.35036e8i q^{83} +1.41953e10i q^{85} -1.11808e11 q^{87} +3.52435e10 q^{89} -8.70686e10i q^{91} +3.82786e10i q^{93} +9.87598e10 q^{95} +2.36079e10 q^{97} +1.53934e11i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 98736 q^{7} - 627628 q^{9} + 9656528 q^{15} + 8255272 q^{17} - 19842640 q^{23} - 103266004 q^{25} - 492266240 q^{31} - 675982576 q^{33} - 1296172112 q^{39} - 973831672 q^{41} + 1207950176 q^{47}+ \cdots + 437793250984 q^{97}+O(q^{100}) $ 12 * q - 98736 * q^7 - 627628 * q^9 + 9656528 * q^15 + 8255272 * q^17 - 19842640 * q^23 - 103266004 * q^25 - 492266240 * q^31 - 675982576 * q^33 - 1296172112 * q^39 - 973831672 * q^41 + 1207950176 * q^47 + 3693758124 * q^49 + 2710831952 * q^55 + 1566705296 * q^57 + 29053202992 * q^63 + 2753811376 * q^65 - 32362135024 * q^71 - 71927337384 * q^73 - 12207892384 * q^79 + 162481241980 * q^81 - 265373263632 * q^87 - 45442832232 * q^89 + 565435645136 * q^95 + 437793250984 * 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
$3$	− 597.717i	− 1.42013i	−0.704135	−	0.710066i	$-0.748665\pi$
$3$	− 597.717i	− 1.42013i	0.704135	−	0.710066i	$-0.251335\pi$
$4$	0	0
$5$	9917.53i	1.41928i	0.704564	+	0.709641i	$0.251143\pi$
$5$	9917.53i	1.41928i	−0.704564	+	0.709641i	$0.748857\pi$
$6$	0	0
$7$	−77908.4	−1.75204	−0.876022	−	0.482271i	$-0.839812\pi$
$7$	−77908.4	−1.75204	−0.876022	+	0.482271i	$0.839812\pi$
$8$	0	0
$9$	−180119.	−1.01678
$10$	0	0
$11$	− 854628.i	− 1.59999i	−0.600007	−	0.799995i	$-0.704835\pi$
$11$	− 854628.i	− 1.59999i	0.600007	−	0.799995i	$-0.295165\pi$
$12$	0	0
$13$	1.11758e6i	0.834812i	0.908720	+	0.417406i	$0.137061\pi$
$13$	1.11758e6i	0.834812i	−0.908720	+	0.417406i	$0.862939\pi$
$14$	0	0
$15$	5.92788e6	2.01557
$16$	0	0
$17$	1.43133e6	0.244496	0.122248	−	0.992500i	$-0.460990\pi$
$17$	1.43133e6	0.244496	0.122248	+	0.992500i	$0.460990\pi$
$18$	0	0
$19$	− 9.95811e6i	− 0.922639i	−0.887234	−	0.461320i	$-0.847376\pi$
$19$	− 9.95811e6i	− 0.922639i	0.887234	−	0.461320i	$-0.152624\pi$
$20$	0	0
$21$	4.65672e7i	2.48813i
$22$	0	0
$23$	−3.13742e7	−1.01641	−0.508206	−	0.861236i	$-0.669691\pi$
$23$	−3.13742e7	−1.01641	−0.508206	+	0.861236i	$0.669691\pi$
$24$	0	0
$25$	−4.95293e7	−1.01436
$26$	0	0
$27$	1.77624e6i	0.0238232i
$28$	0	0
$29$	− 1.87059e8i	− 1.69352i	−0.531978	−	0.846758i	$-0.678551\pi$
$29$	− 1.87059e8i	− 1.69352i	0.531978	−	0.846758i	$-0.321449\pi$
$30$	0	0
$31$	−6.40413e7	−0.401764	−0.200882	−	0.979615i	$-0.564381\pi$
$31$	−6.40413e7	−0.401764	−0.200882	+	0.979615i	$0.564381\pi$
$32$	0	0
$33$	−5.10826e8	−2.27220
$34$	0	0
$35$	− 7.72659e8i	− 2.48664i
$36$	0	0
$37$	− 3.73906e8i	− 0.886448i	−0.896411	−	0.443224i	$-0.853835\pi$
$37$	− 3.73906e8i	− 0.886448i	0.896411	−	0.443224i	$-0.146165\pi$
$38$	0	0
$39$	6.67994e8	1.18554
$40$	0	0
$41$	−1.34692e9	−1.81564	−0.907822	−	0.419356i	$-0.862256\pi$
$41$	−1.34692e9	−1.81564	−0.907822	+	0.419356i	$0.862256\pi$
$42$	0	0
$43$	5.87529e8i	0.609470i	0.952437	+	0.304735i	$0.0985679\pi$
$43$	5.87529e8i	0.609470i	−0.952437	+	0.304735i	$0.901432\pi$
$44$	0	0
$45$	− 1.78633e9i	− 1.44309i
$46$	0	0
$47$	−8.78487e8	−0.558724	−0.279362	−	0.960186i	$-0.590123\pi$
$47$	−8.78487e8	−0.558724	−0.279362	+	0.960186i	$0.590123\pi$
$48$	0	0
$49$	4.09239e9	2.06966
$50$	0	0
$51$	− 8.55533e8i	− 0.347217i
$52$	0	0
$53$	1.76776e9i	0.580638i	0.956930	+	0.290319i	$0.0937613\pi$
$53$	1.76776e9i	0.580638i	−0.956930	+	0.290319i	$0.906239\pi$
$54$	0	0
$55$	8.47580e9	2.27084
$56$	0	0
$57$	−5.95213e9	−1.31027
$58$	0	0
$59$	− 7.95505e9i	− 1.44863i	−0.689471	−	0.724314i	$-0.742157\pi$
$59$	− 7.95505e9i	− 1.44863i	0.689471	−	0.724314i	$-0.257843\pi$
$60$	0	0
$61$	4.23552e9i	0.642085i	0.947065	+	0.321043i	$0.104033\pi$
$61$	4.23552e9i	0.642085i	−0.947065	+	0.321043i	$0.895967\pi$
$62$	0	0
$63$	1.40328e10	1.78143
$64$	0	0
$65$	−1.10836e10	−1.18483
$66$	0	0
$67$	− 1.38472e10i	− 1.25300i	−0.779421	−	0.626500i	$-0.784487\pi$
$67$	− 1.38472e10i	− 1.25300i	0.779421	−	0.626500i	$-0.215513\pi$
$68$	0	0
$69$	1.87529e10i	1.44344i
$70$	0	0
$71$	−3.21284e9	−0.211334	−0.105667	−	0.994402i	$-0.533698\pi$
$71$	−3.21284e9	−0.211334	−0.105667	+	0.994402i	$0.533698\pi$
$72$	0	0
$73$	−1.70921e9	−0.0964980	−0.0482490	−	0.998835i	$-0.515364\pi$
$73$	−1.70921e9	−0.0964980	−0.0482490	+	0.998835i	$0.515364\pi$
$74$	0	0
$75$	2.96045e10i	1.44052i
$76$	0	0
$77$	6.65827e10i	2.80325i
$78$	0	0
$79$	−8.12305e9	−0.297010	−0.148505	−	0.988912i	$-0.547446\pi$
$79$	−8.12305e9	−0.297010	−0.148505	+	0.988912i	$0.547446\pi$
$80$	0	0
$81$	−3.08458e10	−0.982943
$82$	0	0
$83$	− 4.35036e8i	− 0.0121226i	−0.999982	−	0.00606130i	$-0.998071\pi$
$83$	− 4.35036e8i	− 0.0121226i	0.999982	−	0.00606130i	$-0.00192938\pi$
$84$	0	0
$85$	1.41953e10i	0.347009i
$86$	0	0
$87$	−1.11808e11	−2.40502
$88$	0	0
$89$	3.52435e10	0.669011	0.334506	−	0.942394i	$-0.391431\pi$
$89$	3.52435e10	0.669011	0.334506	+	0.942394i	$0.391431\pi$
$90$	0	0
$91$	− 8.70686e10i	− 1.46263i
$92$	0	0
$93$	3.82786e10i	0.570558i
$94$	0	0
$95$	9.87598e10	1.30948
$96$	0	0
$97$	2.36079e10	0.279134	0.139567	−	0.990213i	$-0.455429\pi$
$97$	2.36079e10	0.279134	0.139567	+	0.990213i	$0.455429\pi$
$98$	0	0
$99$	1.53934e11i	1.62683i
$100$	0	0
$101$	− 6.81665e10i	− 0.645362i	−0.946508	−	0.322681i	$-0.895416\pi$
$101$	− 6.81665e10i	− 0.645362i	0.946508	−	0.322681i	$-0.104584\pi$
$102$	0	0
$103$	1.73734e10	0.147666	0.0738331	−	0.997271i	$-0.476477\pi$
$103$	1.73734e10	0.147666	0.0738331	+	0.997271i	$0.476477\pi$
$104$	0	0
$105$	−4.61831e11	−3.53136
$106$	0	0
$107$	2.44123e11i	1.68267i	0.540516	+	0.841334i	$0.318229\pi$
$107$	2.44123e11i	1.68267i	−0.540516	+	0.841334i	$0.681771\pi$
$108$	0	0
$109$	2.06877e11i	1.28785i	0.765088	+	0.643925i	$0.222695\pi$
$109$	2.06877e11i	1.28785i	−0.765088	+	0.643925i	$0.777305\pi$
$110$	0	0
$111$	−2.23490e11	−1.25887
$112$	0	0
$113$	−2.53926e11	−1.29651	−0.648255	−	0.761423i	$-0.724501\pi$
$113$	−2.53926e11	−1.29651	−0.648255	+	0.761423i	$0.724501\pi$
$114$	0	0
$115$	− 3.11155e11i	− 1.44257i
$116$	0	0
$117$	− 2.01296e11i	− 0.848816i
$118$	0	0
$119$	−1.11513e11	−0.428368
$120$	0	0
$121$	−4.45077e11	−1.55997
$122$	0	0
$123$	8.05077e11i	2.57845i
$124$	0	0
$125$	− 6.95353e9i	− 0.0203798i
$126$	0	0
$127$	4.19487e11	1.12667	0.563337	−	0.826227i	$-0.309517\pi$
$127$	4.19487e11	1.12667	0.563337	+	0.826227i	$0.309517\pi$
$128$	0	0
$129$	3.51176e11	0.865528
$130$	0	0
$131$	− 6.01138e10i	− 0.136139i	−0.997681	−	0.0680694i	$-0.978316\pi$
$131$	− 6.01138e10i	− 0.136139i	0.997681	−	0.0680694i	$-0.0216839\pi$
$132$	0	0
$133$	7.75820e11i	1.61650i
$134$	0	0
$135$	−1.76159e10	−0.0338118
$136$	0	0
$137$	5.14968e11	0.911628	0.455814	−	0.890075i	$-0.349348\pi$
$137$	5.14968e11	0.911628	0.455814	+	0.890075i	$0.349348\pi$
$138$	0	0
$139$	− 1.35230e11i	− 0.221051i	−0.993873	−	0.110526i	$-0.964747\pi$
$139$	− 1.35230e11i	− 0.221051i	0.993873	−	0.110526i	$-0.0352534\pi$
$140$	0	0
$141$	5.25087e11i	0.793461i
$142$	0	0
$143$	9.55112e11	1.33569
$144$	0	0
$145$	1.85516e12	2.40358
$146$	0	0
$147$	− 2.44609e12i	− 2.93919i
$148$	0	0
$149$	− 6.02610e11i	− 0.672220i	−0.941823	−	0.336110i	$-0.890889\pi$
$149$	− 6.02610e11i	− 0.672220i	0.941823	−	0.336110i	$-0.109111\pi$
$150$	0	0
$151$	−4.48422e11	−0.464851	−0.232426	−	0.972614i	$-0.574666\pi$
$151$	−4.48422e11	−0.464851	−0.232426	+	0.972614i	$0.574666\pi$
$152$	0	0
$153$	−2.57810e11	−0.248598
$154$	0	0
$155$	− 6.35131e11i	− 0.570216i
$156$	0	0
$157$	− 4.70264e11i	− 0.393454i	−0.980458	−	0.196727i	$-0.936969\pi$
$157$	− 4.70264e11i	− 0.393454i	0.980458	−	0.196727i	$-0.0630312\pi$
$158$	0	0
$159$	1.05662e12	0.824582
$160$	0	0
$161$	2.44431e12	1.78080
$162$	0	0
$163$	2.69683e12i	1.83578i	0.396832	+	0.917891i	$0.370110\pi$
$163$	2.69683e12i	1.83578i	−0.396832	+	0.917891i	$0.629890\pi$
$164$	0	0
$165$	− 5.06613e12i	− 3.22489i
$166$	0	0
$167$	−1.76031e9	−0.00104869	−0.000524347	−	1.00000i	$-0.500167\pi$
$167$	−1.76031e9	−0.00104869	−0.000524347		1.00000i	$0.500167\pi$
$168$	0	0
$169$	5.43183e11	0.303089
$170$	0	0
$171$	1.79364e12i	0.938117i
$172$	0	0
$173$	2.43335e12i	1.19386i	0.802295	+	0.596928i	$0.203612\pi$
$173$	2.43335e12i	1.19386i	−0.802295	+	0.596928i	$0.796388\pi$
$174$	0	0
$175$	3.85874e12	1.77720
$176$	0	0
$177$	−4.75487e12	−2.05724
$178$	0	0
$179$	− 1.36397e12i	− 0.554771i	−0.960759	−	0.277385i	$-0.910532\pi$
$179$	− 1.36397e12i	− 0.554771i	0.960759	−	0.277385i	$-0.0894679\pi$
$180$	0	0
$181$	− 3.49761e12i	− 1.33825i	−0.743148	−	0.669127i	$-0.766668\pi$
$181$	− 3.49761e12i	− 1.33825i	0.743148	−	0.669127i	$-0.233332\pi$
$182$	0	0
$183$	2.53164e12	0.911846
$184$	0	0
$185$	3.70823e12	1.25812
$186$	0	0
$187$	− 1.22326e12i	− 0.391191i
$188$	0	0
$189$	− 1.38384e11i	− 0.0417393i
$190$	0	0
$191$	−2.11756e12	−0.602771	−0.301385	−	0.953502i	$-0.597449\pi$
$191$	−2.11756e12	−0.602771	−0.301385	+	0.953502i	$0.597449\pi$
$192$	0	0
$193$	7.06704e12	1.89964	0.949822	−	0.312791i	$-0.101264\pi$
$193$	7.06704e12	1.89964	0.949822	+	0.312791i	$0.101264\pi$
$194$	0	0
$195$	6.62485e12i	1.68262i
$196$	0	0
$197$	5.20225e12i	1.24919i	0.780951	+	0.624593i	$0.214735\pi$
$197$	5.20225e12i	1.24919i	−0.780951	+	0.624593i	$0.785265\pi$
$198$	0	0
$199$	−4.13737e12	−0.939794	−0.469897	−	0.882721i	$-0.655709\pi$
$199$	−4.13737e12	−0.939794	−0.469897	+	0.882721i	$0.655709\pi$
$200$	0	0
$201$	−8.27672e12	−1.77943
$202$	0	0
$203$	1.45735e13i	2.96711i
$204$	0	0
$205$	− 1.33581e13i	− 2.57691i
$206$	0	0
$207$	5.65108e12	1.03346
$208$	0	0
$209$	−8.51047e12	−1.47621
$210$	0	0
$211$	4.35380e12i	0.716663i	0.933594	+	0.358332i	$0.116654\pi$
$211$	4.35380e12i	0.716663i	−0.933594	+	0.358332i	$0.883346\pi$
$212$	0	0
$213$	1.92037e12i	0.300122i
$214$	0	0
$215$	−5.82683e12	−0.865010
$216$	0	0
$217$	4.98935e12	0.703908
$218$	0	0
$219$	1.02162e12i	0.137040i
$220$	0	0
$221$	1.59963e12i	0.204108i
$222$	0	0
$223$	−8.26654e12	−1.00380	−0.501900	−	0.864926i	$-0.667365\pi$
$223$	−8.26654e12	−1.00380	−0.501900	+	0.864926i	$0.667365\pi$
$224$	0	0
$225$	8.92115e12	1.03138
$226$	0	0
$227$	− 5.55818e12i	− 0.612055i	−0.952023	−	0.306028i	$-0.901000\pi$
$227$	− 5.55818e12i	− 0.612055i	0.952023	−	0.306028i	$-0.0990000\pi$
$228$	0	0
$229$	− 4.62718e12i	− 0.485535i	−0.970084	−	0.242768i	$-0.921945\pi$
$229$	− 4.62718e12i	− 0.485535i	0.970084	−	0.242768i	$-0.0780553\pi$
$230$	0	0
$231$	3.97976e13	3.98099
$232$	0	0
$233$	4.76375e12	0.454456	0.227228	−	0.973842i	$-0.427034\pi$
$233$	4.76375e12	0.454456	0.227228	+	0.973842i	$0.427034\pi$
$234$	0	0
$235$	− 8.71242e12i	− 0.792986i
$236$	0	0
$237$	4.85529e12i	0.421793i
$238$	0	0
$239$	−9.23338e12	−0.765901	−0.382950	−	0.923769i	$-0.625092\pi$
$239$	−9.23338e12	−0.765901	−0.382950	+	0.923769i	$0.625092\pi$
$240$	0	0
$241$	−5.37441e12	−0.425831	−0.212915	−	0.977071i	$-0.568296\pi$
$241$	−5.37441e12	−0.425831	−0.212915	+	0.977071i	$0.568296\pi$
$242$	0	0
$243$	1.87517e13i	1.41973i
$244$	0	0
$245$	4.05864e13i	2.93743i
$246$	0	0
$247$	1.11289e13	0.770230
$248$	0	0
$249$	−2.60028e11	−0.0172157
$250$	0	0
$251$	1.53576e13i	0.973012i	0.873677	+	0.486506i	$0.161729\pi$
$251$	1.53576e13i	0.973012i	−0.873677	+	0.486506i	$0.838271\pi$
$252$	0	0
$253$	2.68133e13i	1.62625i
$254$	0	0
$255$	8.48477e12	0.492798
$256$	0	0
$257$	−1.73232e13	−0.963820	−0.481910	−	0.876221i	$-0.660057\pi$
$257$	−1.73232e13	−0.963820	−0.481910	+	0.876221i	$0.660057\pi$
$258$	0	0
$259$	2.91304e13i	1.55310i
$260$	0	0
$261$	3.36928e13i	1.72193i
$262$	0	0
$263$	1.50898e13	0.739483	0.369742	−	0.929135i	$-0.379446\pi$
$263$	1.50898e13	0.739483	0.369742	+	0.929135i	$0.379446\pi$
$264$	0	0
$265$	−1.75318e13	−0.824088
$266$	0	0
$267$	− 2.10656e13i	− 0.950084i
$268$	0	0
$269$	9.94693e12i	0.430578i	0.976550	+	0.215289i	$0.0690693\pi$
$269$	9.94693e12i	0.430578i	−0.976550	+	0.215289i	$0.930931\pi$
$270$	0	0
$271$	3.89708e13	1.61960	0.809800	−	0.586705i	$-0.199575\pi$
$271$	3.89708e13	1.61960	0.809800	+	0.586705i	$0.199575\pi$
$272$	0	0
$273$	−5.20424e13	−2.07712
$274$	0	0
$275$	4.23291e13i	1.62296i
$276$	0	0
$277$	− 3.84613e13i	− 1.41705i	−0.705686	−	0.708524i	$-0.749361\pi$
$277$	− 3.84613e13i	− 1.41705i	0.705686	−	0.708524i	$-0.250639\pi$
$278$	0	0
$279$	1.15350e13	0.408504
$280$	0	0
$281$	2.65313e13	0.903386	0.451693	−	0.892173i	$-0.350820\pi$
$281$	2.65313e13	0.903386	0.451693	+	0.892173i	$0.350820\pi$
$282$	0	0
$283$	6.09172e12i	0.199487i	0.995013	+	0.0997435i	$0.0318022\pi$
$283$	6.09172e12i	0.199487i	−0.995013	+	0.0997435i	$0.968198\pi$
$284$	0	0
$285$	− 5.90304e13i	− 1.85964i
$286$	0	0
$287$	1.04936e14	3.18109
$288$	0	0
$289$	−3.22232e13	−0.940222
$290$	0	0
$291$	− 1.41108e13i	− 0.396407i
$292$	0	0
$293$	− 7.06519e12i	− 0.191140i	−0.995423	−	0.0955700i	$-0.969533\pi$
$293$	− 7.06519e12i	− 0.191140i	0.995423	−	0.0955700i	$-0.0304674\pi$
$294$	0	0
$295$	7.88944e13	2.05601
$296$	0	0
$297$	1.51802e12	0.0381169
$298$	0	0
$299$	− 3.50631e13i	− 0.848513i
$300$	0	0
$301$	− 4.57734e13i	− 1.06782i
$302$	0	0
$303$	−4.07443e13	−0.916499
$304$	0	0
$305$	−4.20059e13	−0.911300
$306$	0	0
$307$	5.06007e13i	1.05900i	0.848311	+	0.529499i	$0.177620\pi$
$307$	5.06007e13i	1.05900i	−0.848311	+	0.529499i	$0.822380\pi$
$308$	0	0
$309$	− 1.03844e13i	− 0.209705i
$310$	0	0
$311$	−5.61302e13	−1.09399	−0.546997	−	0.837135i	$-0.684229\pi$
$311$	−5.61302e13	−1.09399	−0.546997	+	0.837135i	$0.684229\pi$
$312$	0	0
$313$	2.71351e13	0.510549	0.255275	−	0.966869i	$-0.417834\pi$
$313$	2.71351e13	0.510549	0.255275	+	0.966869i	$0.417834\pi$
$314$	0	0
$315$	1.39170e14i	2.52836i
$316$	0	0
$317$	1.67821e13i	0.294456i	0.989103	+	0.147228i	$0.0470352\pi$
$317$	1.67821e13i	0.294456i	−0.989103	+	0.147228i	$0.952965\pi$
$318$	0	0
$319$	−1.59866e14	−2.70961
$320$	0	0
$321$	1.45917e14	2.38961
$322$	0	0
$323$	− 1.42534e13i	− 0.225582i
$324$	0	0
$325$	− 5.53527e13i	− 0.846799i
$326$	0	0
$327$	1.23654e14	1.82892
$328$	0	0
$329$	6.84415e13	0.978908
$330$	0	0
$331$	1.15836e14i	1.60247i	0.598349	+	0.801235i	$0.295823\pi$
$331$	1.15836e14i	1.60247i	−0.598349	+	0.801235i	$0.704177\pi$
$332$	0	0
$333$	6.73475e13i	0.901319i
$334$	0	0
$335$	1.37330e14	1.77836
$336$	0	0
$337$	9.66885e13	1.21174	0.605871	−	0.795563i	$-0.292825\pi$
$337$	9.66885e13	1.21174	0.605871	+	0.795563i	$0.292825\pi$
$338$	0	0
$339$	1.51776e14i	1.84122i
$340$	0	0
$341$	5.47315e13i	0.642818i
$342$	0	0
$343$	−1.64781e14	−1.87409
$344$	0	0
$345$	−1.85982e14	−2.04865
$346$	0	0
$347$	− 1.01211e14i	− 1.07998i	−0.841672	−	0.539989i	$-0.818428\pi$
$347$	− 1.01211e14i	− 1.07998i	0.841672	−	0.539989i	$-0.181572\pi$
$348$	0	0
$349$	− 2.62411e13i	− 0.271295i	−0.990757	−	0.135648i	$-0.956688\pi$
$349$	− 2.62411e13i	− 0.271295i	0.990757	−	0.135648i	$-0.0433115\pi$
$350$	0	0
$351$	−1.98508e12	−0.0198879
$352$	0	0
$353$	1.68507e14	1.63627	0.818137	−	0.575023i	$-0.195007\pi$
$353$	1.68507e14	1.63627	0.818137	+	0.575023i	$0.195007\pi$
$354$	0	0
$355$	− 3.18635e13i	− 0.299942i
$356$	0	0
$357$	6.66532e13i	0.608339i
$358$	0	0
$359$	1.37349e14	1.21565	0.607823	−	0.794072i	$-0.292043\pi$
$359$	1.37349e14	1.21565	0.607823	+	0.794072i	$0.292043\pi$
$360$	0	0
$361$	1.73264e13	0.148737
$362$	0	0
$363$	2.66030e14i	2.21536i
$364$	0	0
$365$	− 1.69511e13i	− 0.136958i
$366$	0	0
$367$	1.95237e14	1.53073	0.765367	−	0.643594i	$-0.222558\pi$
$367$	1.95237e14	1.53073	0.765367	+	0.643594i	$0.222558\pi$
$368$	0	0
$369$	2.42605e14	1.84610
$370$	0	0
$371$	− 1.37723e14i	− 1.01730i
$372$	0	0
$373$	− 2.09480e14i	− 1.50225i	−0.660158	−	0.751127i	$-0.729511\pi$
$373$	− 2.09480e14i	− 1.50225i	0.660158	−	0.751127i	$-0.270489\pi$
$374$	0	0
$375$	−4.15624e12	−0.0289420
$376$	0	0
$377$	2.09053e14	1.41377
$378$	0	0
$379$	1.65455e14i	1.08684i	0.839461	+	0.543419i	$0.182871\pi$
$379$	1.65455e14i	1.08684i	−0.839461	+	0.543419i	$0.817129\pi$
$380$	0	0
$381$	− 2.50735e14i	− 1.60002i
$382$	0	0
$383$	−8.85015e12	−0.0548728	−0.0274364	−	0.999624i	$-0.508734\pi$
$383$	−8.85015e12	−0.0548728	−0.0274364	+	0.999624i	$0.508734\pi$
$384$	0	0
$385$	−6.60336e14	−3.97860
$386$	0	0
$387$	− 1.05825e14i	− 0.619694i
$388$	0	0
$389$	− 1.65069e14i	− 0.939598i	−0.882773	−	0.469799i	$-0.844326\pi$
$389$	− 1.65069e14i	− 0.939598i	0.882773	−	0.469799i	$-0.155674\pi$
$390$	0	0
$391$	−4.49070e13	−0.248509
$392$	0	0
$393$	−3.59311e13	−0.193335
$394$	0	0
$395$	− 8.05606e13i	− 0.421540i
$396$	0	0
$397$	3.35880e14i	1.70937i	0.519146	+	0.854685i	$0.326250\pi$
$397$	3.35880e14i	1.70937i	−0.519146	+	0.854685i	$0.673750\pi$
$398$	0	0
$399$	4.63721e14	2.29565
$400$	0	0
$401$	4.11031e14	1.97961	0.989807	−	0.142416i	$-0.0454871\pi$
$401$	4.11031e14	1.97961	0.989807	+	0.142416i	$0.0454871\pi$
$402$	0	0
$403$	− 7.15710e13i	− 0.335397i
$404$	0	0
$405$	− 3.05914e14i	− 1.39507i
$406$	0	0
$407$	−3.19551e14	−1.41831
$408$	0	0
$409$	−1.67536e14	−0.723818	−0.361909	−	0.932213i	$-0.617875\pi$
$409$	−1.67536e14	−0.723818	−0.361909	+	0.932213i	$0.617875\pi$
$410$	0	0
$411$	− 3.07805e14i	− 1.29463i
$412$	0	0
$413$	6.19765e14i	2.53806i
$414$	0	0
$415$	4.31448e12	0.0172054
$416$	0	0
$417$	−8.08296e13	−0.313922
$418$	0	0
$419$	2.34573e14i	0.887362i	0.896185	+	0.443681i	$0.146328\pi$
$419$	2.34573e14i	0.887362i	−0.896185	+	0.443681i	$0.853672\pi$
$420$	0	0
$421$	− 1.38723e13i	− 0.0511208i	−0.999673	−	0.0255604i	$-0.991863\pi$
$421$	− 1.38723e13i	− 0.0511208i	0.999673	−	0.0255604i	$-0.00813702\pi$
$422$	0	0
$423$	1.58232e14	0.568096
$424$	0	0
$425$	−7.08929e13	−0.248007
$426$	0	0
$427$	− 3.29983e14i	− 1.12496i
$428$	0	0
$429$	− 5.70887e14i	− 1.89686i
$430$	0	0
$431$	−1.23793e14	−0.400932	−0.200466	−	0.979701i	$-0.564246\pi$
$431$	−1.23793e14	−0.400932	−0.200466	+	0.979701i	$0.564246\pi$
$432$	0	0
$433$	−1.88629e14	−0.595559	−0.297779	−	0.954635i	$-0.596246\pi$
$433$	−1.88629e14	−0.595559	−0.297779	+	0.954635i	$0.596246\pi$
$434$	0	0
$435$	− 1.10886e15i	− 3.41340i
$436$	0	0
$437$	3.12428e14i	0.937781i
$438$	0	0
$439$	−5.98893e14	−1.75305	−0.876525	−	0.481356i	$-0.840144\pi$
$439$	−5.98893e14	−1.75305	−0.876525	+	0.481356i	$0.840144\pi$
$440$	0	0
$441$	−7.37116e14	−2.10438
$442$	0	0
$443$	− 1.17021e14i	− 0.325869i	−0.986637	−	0.162934i	$-0.947904\pi$
$443$	− 1.17021e14i	− 0.325869i	0.986637	−	0.162934i	$-0.0520959\pi$
$444$	0	0
$445$	3.49528e14i	0.949515i
$446$	0	0
$447$	−3.60190e14	−0.954642
$448$	0	0
$449$	−2.15520e14	−0.557355	−0.278678	−	0.960385i	$-0.589896\pi$
$449$	−2.15520e14	−0.557355	−0.278678	+	0.960385i	$0.589896\pi$
$450$	0	0
$451$	1.15112e15i	2.90501i
$452$	0	0
$453$	2.68030e14i	0.660150i
$454$	0	0
$455$	8.63505e14	2.07588
$456$	0	0
$457$	−5.87938e14	−1.37972	−0.689862	−	0.723941i	$-0.742329\pi$
$457$	−5.87938e14	−1.37972	−0.689862	+	0.723941i	$0.742329\pi$
$458$	0	0
$459$	2.54239e12i	0.00582468i
$460$	0	0
$461$	4.77652e14i	1.06846i	0.845340	+	0.534228i	$0.179398\pi$
$461$	4.77652e14i	1.06846i	−0.845340	+	0.534228i	$0.820602\pi$
$462$	0	0
$463$	−4.59413e14	−1.00348	−0.501739	−	0.865019i	$-0.667306\pi$
$463$	−4.59413e14	−1.00348	−0.501739	+	0.865019i	$0.667306\pi$
$464$	0	0
$465$	−3.79629e14	−0.809782
$466$	0	0
$467$	8.02310e13i	0.167147i	0.996502	+	0.0835737i	$0.0266334\pi$
$467$	8.02310e13i	0.167147i	−0.996502	+	0.0835737i	$0.973367\pi$
$468$	0	0
$469$	1.07881e15i	2.19531i
$470$	0	0
$471$	−2.81085e14	−0.558756
$472$	0	0
$473$	5.02118e14	0.975146
$474$	0	0
$475$	4.93218e14i	0.935888i
$476$	0	0
$477$	− 3.18406e14i	− 0.590378i
$478$	0	0
$479$	−3.43868e14	−0.623083	−0.311542	−	0.950232i	$-0.600845\pi$
$479$	−3.43868e14	−0.623083	−0.311542	+	0.950232i	$0.600845\pi$
$480$	0	0
$481$	4.17869e14	0.740018
$482$	0	0
$483$	− 1.46101e15i	− 2.52897i
$484$	0	0
$485$	2.34132e14i	0.396170i
$486$	0	0
$487$	−9.23408e14	−1.52751	−0.763756	−	0.645505i	$-0.776647\pi$
$487$	−9.23408e14	−1.52751	−0.763756	+	0.645505i	$0.776647\pi$
$488$	0	0
$489$	1.61194e15	2.60705
$490$	0	0
$491$	2.73483e13i	0.0432496i	0.999766	+	0.0216248i	$0.00688393\pi$
$491$	2.73483e13i	0.0432496i	−0.999766	+	0.0216248i	$0.993116\pi$
$492$	0	0
$493$	− 2.67744e14i	− 0.414058i
$494$	0	0
$495$	−1.52665e15	−2.30893
$496$	0	0
$497$	2.50307e14	0.370266
$498$	0	0
$499$	8.37898e12i	0.0121238i	0.999982	+	0.00606189i	$0.00192957\pi$
$499$	8.37898e12i	0.0121238i	−0.999982	+	0.00606189i	$0.998070\pi$
$500$	0	0
$501$	1.05217e12i	0.00148928i
$502$	0	0
$503$	6.87360e13	0.0951832	0.0475916	−	0.998867i	$-0.484845\pi$
$503$	6.87360e13	0.0951832	0.0475916	+	0.998867i	$0.484845\pi$
$504$	0	0
$505$	6.76043e14	0.915950
$506$	0	0
$507$	− 3.24670e14i	− 0.430426i
$508$	0	0
$509$	8.81020e13i	0.114298i	0.998366	+	0.0571489i	$0.0182010\pi$
$509$	8.81020e13i	0.114298i	−0.998366	+	0.0571489i	$0.981799\pi$
$510$	0	0
$511$	1.33161e14	0.169069
$512$	0	0
$513$	1.76880e13	0.0219802
$514$	0	0
$515$	1.72302e14i	0.209580i
$516$	0	0
$517$	7.50779e14i	0.893952i
$518$	0	0
$519$	1.45446e15	1.69543
$520$	0	0
$521$	−6.38668e14	−0.728899	−0.364450	−	0.931223i	$-0.618743\pi$
$521$	−6.38668e14	−0.728899	−0.364450	+	0.931223i	$0.618743\pi$
$522$	0	0
$523$	− 2.87455e14i	− 0.321226i	−0.987017	−	0.160613i	$-0.948653\pi$
$523$	− 2.87455e14i	− 0.321226i	0.987017	−	0.160613i	$-0.0513471\pi$
$524$	0	0
$525$	− 2.30644e15i	− 2.52386i
$526$	0	0
$527$	−9.16645e13	−0.0982297
$528$	0	0
$529$	3.15311e13	0.0330928
$530$	0	0
$531$	1.43285e15i	1.47293i
$532$	0	0
$533$	− 1.50529e15i	− 1.51572i
$534$	0	0
$535$	−2.42110e15	−2.38818
$536$	0	0
$537$	−8.15269e14	−0.787848
$538$	0	0
$539$	− 3.49747e15i	− 3.31143i
$540$	0	0
$541$	− 1.61414e15i	− 1.49746i	−0.662875	−	0.748730i	$-0.730664\pi$
$541$	− 1.61414e15i	− 1.49746i	0.662875	−	0.748730i	$-0.269336\pi$
$542$	0	0
$543$	−2.09058e15	−1.90050
$544$	0	0
$545$	−2.05170e15	−1.82782
$546$	0	0
$547$	4.15221e14i	0.362534i	0.983434	+	0.181267i	$0.0580198\pi$
$547$	4.15221e14i	0.362534i	−0.983434	+	0.181267i	$0.941980\pi$
$548$	0	0
$549$	− 7.62896e14i	− 0.652856i
$550$	0	0
$551$	−1.86275e15	−1.56250
$552$	0	0
$553$	6.32854e14	0.520374
$554$	0	0
$555$	− 2.21647e15i	− 1.78670i
$556$	0	0
$557$	1.67408e15i	1.32304i	0.749929	+	0.661519i	$0.230088\pi$
$557$	1.67408e15i	1.32304i	−0.749929	+	0.661519i	$0.769912\pi$
$558$	0	0
$559$	−6.56608e14	−0.508793
$560$	0	0
$561$	−7.31162e14	−0.555543
$562$	0	0
$563$	− 3.43677e14i	− 0.256068i	−0.991770	−	0.128034i	$-0.959133\pi$
$563$	− 3.43677e14i	− 0.256068i	0.991770	−	0.128034i	$-0.0408666\pi$
$564$	0	0
$565$	− 2.51832e15i	− 1.84011i
$566$	0	0
$567$	2.40315e15	1.72216
$568$	0	0
$569$	1.15642e15	0.812824	0.406412	−	0.913690i	$-0.366780\pi$
$569$	1.15642e15	0.812824	0.406412	+	0.913690i	$0.366780\pi$
$570$	0	0
$571$	− 6.39657e14i	− 0.441010i	−0.975386	−	0.220505i	$-0.929229\pi$
$571$	− 6.39657e14i	− 0.441010i	0.975386	−	0.220505i	$-0.0707705\pi$
$572$	0	0
$573$	1.26570e15i	0.856014i
$574$	0	0
$575$	1.55394e15	1.03101
$576$	0	0
$577$	9.67432e14	0.629729	0.314864	−	0.949137i	$-0.398041\pi$
$577$	9.67432e14	0.629729	0.314864	+	0.949137i	$0.398041\pi$
$578$	0	0
$579$	− 4.22409e15i	− 2.69775i
$580$	0	0
$581$	3.38929e13i	0.0212393i
$582$	0	0
$583$	1.51077e15	0.929014
$584$	0	0
$585$	1.99636e15	1.20471
$586$	0	0
$587$	− 3.13688e14i	− 0.185776i	−0.995677	−	0.0928878i	$-0.970390\pi$
$587$	− 3.13688e14i	− 0.185776i	0.995677	−	0.0928878i	$-0.0296098\pi$
$588$	0	0
$589$	6.37730e14i	0.370683i
$590$	0	0
$591$	3.10947e15	1.77401
$592$	0	0
$593$	−1.06041e15	−0.593845	−0.296922	−	0.954902i	$-0.595960\pi$
$593$	−1.06041e15	−0.593845	−0.296922	+	0.954902i	$0.595960\pi$
$594$	0	0
$595$	− 1.10593e15i	− 0.607975i
$596$	0	0
$597$	2.47298e15i	1.33463i
$598$	0	0
$599$	2.45392e15	1.30021	0.650104	−	0.759845i	$-0.274725\pi$
$599$	2.45392e15	1.30021	0.650104	+	0.759845i	$0.274725\pi$
$600$	0	0
$601$	2.25580e15	1.17352	0.586761	−	0.809760i	$-0.300403\pi$
$601$	2.25580e15	1.17352	0.586761	+	0.809760i	$0.300403\pi$
$602$	0	0
$603$	2.49414e15i	1.27402i
$604$	0	0
$605$	− 4.41406e15i	− 2.21403i
$606$	0	0
$607$	−1.33628e15	−0.658202	−0.329101	−	0.944295i	$-0.606746\pi$
$607$	−1.33628e15	−0.658202	−0.329101	+	0.944295i	$0.606746\pi$
$608$	0	0
$609$	8.71080e15	4.21369
$610$	0	0
$611$	− 9.81776e14i	− 0.466429i
$612$	0	0
$613$	5.40273e14i	0.252104i	0.992024	+	0.126052i	$0.0402307\pi$
$613$	5.40273e14i	0.252104i	−0.992024	+	0.126052i	$0.959769\pi$
$614$	0	0
$615$	−7.98438e15	−3.65955
$616$	0	0
$617$	−2.96580e15	−1.33528	−0.667641	−	0.744483i	$-0.732696\pi$
$617$	−2.96580e15	−1.33528	−0.667641	+	0.744483i	$0.732696\pi$
$618$	0	0
$619$	1.61833e15i	0.715763i	0.933767	+	0.357882i	$0.116501\pi$
$619$	1.61833e15i	0.715763i	−0.933767	+	0.357882i	$0.883499\pi$
$620$	0	0
$621$	− 5.57280e13i	− 0.0242142i
$622$	0	0
$623$	−2.74576e15	−1.17214
$624$	0	0
$625$	−2.34946e15	−0.985435
$626$	0	0
$627$	5.08686e15i	2.09642i
$628$	0	0
$629$	− 5.35185e14i	− 0.216733i
$630$	0	0
$631$	2.43830e15	0.970345	0.485173	−	0.874418i	$-0.338757\pi$
$631$	2.43830e15	0.970345	0.485173	+	0.874418i	$0.338757\pi$
$632$	0	0
$633$	2.60234e15	1.01776
$634$	0	0
$635$	4.16027e15i	1.59907i
$636$	0	0
$637$	4.57356e15i	1.72777i
$638$	0	0
$639$	5.78693e14	0.214879
$640$	0	0
$641$	−1.65014e15	−0.602283	−0.301142	−	0.953579i	$-0.597368\pi$
$641$	−1.65014e15	−0.602283	−0.301142	+	0.953579i	$0.597368\pi$
$642$	0	0
$643$	− 3.86777e14i	− 0.138771i	−0.997590	−	0.0693857i	$-0.977896\pi$
$643$	− 3.86777e14i	− 0.138771i	0.997590	−	0.0693857i	$-0.0221039\pi$
$644$	0	0
$645$	3.48280e15i	1.22843i
$646$	0	0
$647$	−1.56835e15	−0.543838	−0.271919	−	0.962320i	$-0.587658\pi$
$647$	−1.56835e15	−0.543838	−0.271919	+	0.962320i	$0.587658\pi$
$648$	0	0
$649$	−6.79860e15	−2.31779
$650$	0	0
$651$	− 2.98222e15i	− 0.999642i
$652$	0	0
$653$	− 1.49632e15i	− 0.493177i	−0.969120	−	0.246589i	$-0.920690\pi$
$653$	− 1.49632e15i	− 0.493177i	0.969120	−	0.246589i	$-0.0793096\pi$
$654$	0	0
$655$	5.96181e14	0.193219
$656$	0	0
$657$	3.07860e14	0.0981168
$658$	0	0
$659$	− 5.11682e15i	− 1.60373i	−0.597508	−	0.801863i	$-0.703842\pi$
$659$	− 5.11682e15i	− 1.60373i	0.597508	−	0.801863i	$-0.296158\pi$
$660$	0	0
$661$	− 7.66517e14i	− 0.236273i	−0.992997	−	0.118136i	$-0.962308\pi$
$661$	− 7.66517e14i	− 0.236273i	0.992997	−	0.118136i	$-0.0376920\pi$
$662$	0	0
$663$	9.56124e14	0.289861
$664$	0	0
$665$	−7.69422e15	−2.29427
$666$	0	0
$667$	5.86882e15i	1.72131i
$668$	0	0
$669$	4.94105e15i	1.42553i
$670$	0	0
$671$	3.61979e15	1.02733
$672$	0	0
$673$	2.37983e15	0.664451	0.332225	−	0.943200i	$-0.392201\pi$
$673$	2.37983e15	0.664451	0.332225	+	0.943200i	$0.392201\pi$
$674$	0	0
$675$	− 8.79757e13i	− 0.0241653i
$676$	0	0
$677$	3.07123e15i	0.829992i	0.909823	+	0.414996i	$0.136217\pi$
$677$	3.07123e15i	0.829992i	−0.909823	+	0.414996i	$0.863783\pi$
$678$	0	0
$679$	−1.83925e15	−0.489055
$680$	0	0
$681$	−3.32222e15	−0.869199
$682$	0	0
$683$	1.04011e15i	0.267773i	0.990997	+	0.133887i	$0.0427458\pi$
$683$	1.04011e15i	0.267773i	−0.990997	+	0.133887i	$0.957254\pi$
$684$	0	0
$685$	5.10721e15i	1.29386i
$686$	0	0
$687$	−2.76574e15	−0.689524
$688$	0	0
$689$	−1.97560e15	−0.484723
$690$	0	0
$691$	4.84984e14i	0.117111i	0.998284	+	0.0585556i	$0.0186495\pi$
$691$	4.84984e14i	0.117111i	−0.998284	+	0.0585556i	$0.981351\pi$
$692$	0	0
$693$	− 1.19928e16i	− 2.85028i
$694$	0	0
$695$	1.34115e15	0.313734
$696$	0	0
$697$	−1.92789e15	−0.443918
$698$	0	0
$699$	− 2.84738e15i	− 0.645387i
$700$	0	0
$701$	2.21947e15i	0.495222i	0.968859	+	0.247611i	$0.0796456\pi$
$701$	2.21947e15i	0.495222i	−0.968859	+	0.247611i	$0.920354\pi$
$702$	0	0
$703$	−3.72340e15	−0.817872
$704$	0	0
$705$	−5.20756e15	−1.12614
$706$	0	0
$707$	5.31074e15i	1.13070i
$708$	0	0
$709$	− 2.31287e15i	− 0.484837i	−0.970172	−	0.242419i	$-0.922059\pi$
$709$	− 2.31287e15i	− 0.484837i	0.970172	−	0.242419i	$-0.0779408\pi$
$710$	0	0
$711$	1.46311e15	0.301992
$712$	0	0
$713$	2.00924e15	0.408357
$714$	0	0
$715$	9.47235e15i	1.89572i
$716$	0	0
$717$	5.51895e15i	1.08768i
$718$	0	0
$719$	5.74239e15	1.11451	0.557255	−	0.830341i	$-0.311855\pi$
$719$	5.74239e15	1.11451	0.557255	+	0.830341i	$0.311855\pi$
$720$	0	0
$721$	−1.35354e15	−0.258718
$722$	0	0
$723$	3.21238e15i	0.604736i
$724$	0	0
$725$	9.26489e15i	1.71783i
$726$	0	0
$727$	−3.63122e15	−0.663152	−0.331576	−	0.943429i	$-0.607580\pi$
$727$	−3.63122e15	−0.663152	−0.331576	+	0.943429i	$0.607580\pi$
$728$	0	0
$729$	5.74398e15	1.03326
$730$	0	0
$731$	8.40950e14i	0.149013i
$732$	0	0
$733$	5.06817e14i	0.0884666i	0.999021	+	0.0442333i	$0.0140845\pi$
$733$	5.06817e14i	0.0884666i	−0.999021	+	0.0442333i	$0.985916\pi$
$734$	0	0
$735$	2.42592e16	4.17153
$736$	0	0
$737$	−1.18342e16	−2.00479
$738$	0	0
$739$	5.93189e15i	0.990030i	0.868884	+	0.495015i	$0.164838\pi$
$739$	5.93189e15i	0.990030i	−0.868884	+	0.495015i	$0.835162\pi$
$740$	0	0
$741$	− 6.65196e15i	− 1.09383i
$742$	0	0
$743$	−5.46043e15	−0.884684	−0.442342	−	0.896846i	$-0.645852\pi$
$743$	−5.46043e15	−0.884684	−0.442342	+	0.896846i	$0.645852\pi$
$744$	0	0
$745$	5.97640e15	0.954070
$746$	0	0
$747$	7.83581e13i	0.0123260i
$748$	0	0
$749$	− 1.90192e16i	− 2.94811i
$750$	0	0
$751$	−2.24714e15	−0.343249	−0.171625	−	0.985162i	$-0.554902\pi$
$751$	−2.24714e15	−0.343249	−0.171625	+	0.985162i	$0.554902\pi$
$752$	0	0
$753$	9.17951e15	1.38181
$754$	0	0
$755$	− 4.44724e15i	− 0.659755i
$756$	0	0
$757$	− 1.15766e15i	− 0.169260i	−0.996412	−	0.0846298i	$-0.973029\pi$
$757$	− 1.15766e15i	− 0.169260i	0.996412	−	0.0846298i	$-0.0269708\pi$
$758$	0	0
$759$	1.60267e16	2.30949
$760$	0	0
$761$	−1.06661e16	−1.51492	−0.757460	−	0.652882i	$-0.773560\pi$
$761$	−1.06661e16	−1.51492	−0.757460	+	0.652882i	$0.773560\pi$
$762$	0	0
$763$	− 1.61174e16i	− 2.25637i
$764$	0	0
$765$	− 2.55684e15i	− 0.352830i
$766$	0	0
$767$	8.89037e15	1.20933
$768$	0	0
$769$	1.34918e16	1.80915	0.904573	−	0.426319i	$-0.140190\pi$
$769$	1.34918e16	1.80915	0.904573	+	0.426319i	$0.140190\pi$
$770$	0	0
$771$	1.03544e16i	1.36875i
$772$	0	0
$773$	8.35886e15i	1.08933i	0.838654	+	0.544665i	$0.183343\pi$
$773$	8.35886e15i	1.08933i	−0.838654	+	0.544665i	$0.816657\pi$
$774$	0	0
$775$	3.17192e15	0.407533
$776$	0	0
$777$	1.74118e16	2.20560
$778$	0	0
$779$	1.34128e16i	1.67518i
$780$	0	0
$781$	2.74578e15i	0.338132i
$782$	0	0
$783$	3.32261e14	0.0403450
$784$	0	0
$785$	4.66386e15	0.558421
$786$	0	0
$787$	− 1.19233e16i	− 1.40778i	−0.710309	−	0.703890i	$-0.751445\pi$
$787$	− 1.19233e16i	− 1.40778i	0.710309	−	0.703890i	$-0.248555\pi$
$788$	0	0
$789$	− 9.01946e15i	− 1.05016i
$790$	0	0
$791$	1.97830e16	2.27154
$792$	0	0
$793$	−4.73352e15	−0.536021
$794$	0	0
$795$	1.04790e16i	1.17031i
$796$	0	0
$797$	1.60230e16i	1.76491i	0.470397	+	0.882455i	$0.344111\pi$
$797$	1.60230e16i	1.76491i	−0.470397	+	0.882455i	$0.655889\pi$
$798$	0	0
$799$	−1.25741e15	−0.136606
$800$	0	0
$801$	−6.34800e15	−0.680234
$802$	0	0
$803$	1.46073e15i	0.154396i
$804$	0	0
$805$	2.42416e16i	2.52745i
$806$	0	0
$807$	5.94545e15	0.611477
$808$	0	0
$809$	6.04038e15	0.612840	0.306420	−	0.951896i	$-0.400869\pi$
$809$	6.04038e15	0.612840	0.306420	+	0.951896i	$0.400869\pi$
$810$	0	0
$811$	5.33307e15i	0.533780i	0.963727	+	0.266890i	$0.0859960\pi$
$811$	5.33307e15i	0.533780i	−0.963727	+	0.266890i	$0.914004\pi$
$812$	0	0
$813$	− 2.32935e16i	− 2.30005i
$814$	0	0
$815$	−2.67459e16	−2.60549
$816$	0	0
$817$	5.85067e15	0.562321
$818$	0	0
$819$	1.56827e16i	1.48716i
$820$	0	0
$821$	− 1.73438e16i	− 1.62277i	−0.584513	−	0.811384i	$-0.698714\pi$
$821$	− 1.73438e16i	− 1.62277i	0.584513	−	0.811384i	$-0.301286\pi$
$822$	0	0
$823$	1.08731e16	1.00382	0.501909	−	0.864920i	$-0.332631\pi$
$823$	1.08731e16	1.00382	0.501909	+	0.864920i	$0.332631\pi$
$824$	0	0
$825$	2.53008e16	2.30482
$826$	0	0
$827$	− 1.66173e16i	− 1.49376i	−0.664961	−	0.746879i	$-0.731552\pi$
$827$	− 1.66173e16i	− 1.49376i	0.664961	−	0.746879i	$-0.268448\pi$
$828$	0	0
$829$	8.40012e15i	0.745136i	0.928005	+	0.372568i	$0.121523\pi$
$829$	8.40012e15i	0.745136i	−0.928005	+	0.372568i	$0.878477\pi$
$830$	0	0
$831$	−2.29890e16	−2.01240
$832$	0	0
$833$	5.85758e15	0.506023
$834$	0	0
$835$	− 1.74579e13i	− 0.00148839i
$836$	0	0
$837$	− 1.13753e14i	− 0.00957130i
$838$	0	0
$839$	3.09433e15	0.256966	0.128483	−	0.991712i	$-0.458989\pi$
$839$	3.09433e15	0.256966	0.128483	+	0.991712i	$0.458989\pi$
$840$	0	0
$841$	−2.27905e16	−1.86800
$842$	0	0
$843$	− 1.58582e16i	− 1.28293i
$844$	0	0
$845$	5.38704e15i	0.430168i
$846$	0	0
$847$	3.46752e16	2.73313
$848$	0	0
$849$	3.64113e15	0.283298
$850$	0	0
$851$	1.17310e16i	0.900997i
$852$	0	0
$853$	1.75509e16i	1.33070i	0.746533	+	0.665348i	$0.231717\pi$
$853$	1.75509e16i	1.33070i	−0.746533	+	0.665348i	$0.768283\pi$
$854$	0	0
$855$	−1.77885e16	−1.33145
$856$	0	0
$857$	1.24432e16	0.919468	0.459734	−	0.888057i	$-0.347945\pi$
$857$	1.24432e16	0.919468	0.459734	+	0.888057i	$0.347945\pi$
$858$	0	0
$859$	2.36795e16i	1.72747i	0.503950	+	0.863733i	$0.331880\pi$
$859$	2.36795e16i	1.72747i	−0.503950	+	0.863733i	$0.668120\pi$
$860$	0	0
$861$	− 6.27222e16i	− 4.51756i
$862$	0	0
$863$	−1.97862e16	−1.40703	−0.703514	−	0.710681i	$-0.748387\pi$
$863$	−1.97862e16	−1.40703	−0.703514	+	0.710681i	$0.748387\pi$
$864$	0	0
$865$	−2.41329e16	−1.69442
$866$	0	0
$867$	1.92603e16i	1.33524i
$868$	0	0
$869$	6.94219e15i	0.475212i
$870$	0	0
$871$	1.54753e16	1.04602
$872$	0	0
$873$	−4.25222e15	−0.283816
$874$	0	0
$875$	5.41738e14i	0.0357063i
$876$	0	0
$877$	− 1.18249e16i	− 0.769660i	−0.922987	−	0.384830i	$-0.874260\pi$
$877$	− 1.18249e16i	− 0.769660i	0.922987	−	0.384830i	$-0.125740\pi$
$878$	0	0
$879$	−4.22298e15	−0.271444
$880$	0	0
$881$	−2.40809e15	−0.152864	−0.0764319	−	0.997075i	$-0.524353\pi$
$881$	−2.40809e15	−0.152864	−0.0764319	+	0.997075i	$0.524353\pi$
$882$	0	0
$883$	− 9.26051e15i	− 0.580565i	−0.956941	−	0.290283i	$-0.906251\pi$
$883$	− 9.26051e15i	− 0.580565i	0.956941	−	0.290283i	$-0.0937493\pi$
$884$	0	0
$885$	− 4.71565e16i	− 2.91981i
$886$	0	0
$887$	−8.10231e15	−0.495483	−0.247742	−	0.968826i	$-0.579688\pi$
$887$	−8.10231e15	−0.495483	−0.247742	+	0.968826i	$0.579688\pi$
$888$	0	0
$889$	−3.26815e16	−1.97398
$890$	0	0
$891$	2.63617e16i	1.57270i
$892$	0	0
$893$	8.74807e15i	0.515500i
$894$	0	0
$895$	1.35272e16	0.787376
$896$	0	0
$897$	−2.09578e16	−1.20500
$898$	0	0
$899$	1.19795e16i	0.680394i
$900$	0	0
$901$	2.53025e15i	0.141964i
$902$	0	0
$903$	−2.73595e16	−1.51644
$904$	0	0
$905$	3.46876e16	1.89936
$906$	0	0
$907$	− 2.68684e16i	− 1.45346i	−0.686925	−	0.726729i	$-0.741040\pi$
$907$	− 2.68684e16i	− 1.45346i	0.686925	−	0.726729i	$-0.258960\pi$
$908$	0	0
$909$	1.22781e16i	0.656188i
$910$	0	0
$911$	2.55702e15	0.135016	0.0675078	−	0.997719i	$-0.478495\pi$
$911$	2.55702e15	0.135016	0.0675078	+	0.997719i	$0.478495\pi$
$912$	0	0
$913$	−3.71794e14	−0.0193960
$914$	0	0
$915$	2.51076e16i	1.29417i
$916$	0	0
$917$	4.68337e15i	0.238521i
$918$	0	0
$919$	−2.29786e16	−1.15635	−0.578173	−	0.815914i	$-0.696234\pi$
$919$	−2.29786e16	−1.15635	−0.578173	+	0.815914i	$0.696234\pi$
$920$	0	0
$921$	3.02449e16	1.50392
$922$	0	0
$923$	− 3.59060e15i	− 0.176424i
$924$	0	0
$925$	1.85193e16i	0.899177i
$926$	0	0
$927$	−3.12928e15	−0.150143
$928$	0	0
$929$	−1.51573e16	−0.718678	−0.359339	−	0.933207i	$-0.616998\pi$
$929$	−1.51573e16	−0.718678	−0.359339	+	0.933207i	$0.616998\pi$
$930$	0	0
$931$	− 4.07524e16i	− 1.90955i
$932$	0	0
$933$	3.35500e16i	1.55362i
$934$	0	0
$935$	1.21317e16	0.555211
$936$	0	0
$937$	5.80870e15	0.262731	0.131365	−	0.991334i	$-0.458064\pi$
$937$	5.80870e15	0.262731	0.131365	+	0.991334i	$0.458064\pi$
$938$	0	0
$939$	− 1.62191e16i	− 0.725047i
$940$	0	0
$941$	3.66030e16i	1.61724i	0.588332	+	0.808619i	$0.299785\pi$
$941$	3.66030e16i	1.61724i	−0.588332	+	0.808619i	$0.700215\pi$
$942$	0	0
$943$	4.22585e16	1.84544
$944$	0	0
$945$	1.37243e15	0.0592398
$946$	0	0
$947$	1.94247e16i	0.828762i	0.910104	+	0.414381i	$0.136002\pi$
$947$	1.94247e16i	0.828762i	−0.910104	+	0.414381i	$0.863998\pi$
$948$	0	0
$949$	− 1.91017e15i	− 0.0805577i
$950$	0	0
$951$	1.00310e16	0.418167
$952$	0	0
$953$	−3.99878e16	−1.64784	−0.823922	−	0.566703i	$-0.808219\pi$
$953$	−3.99878e16	−1.64784	−0.823922	+	0.566703i	$0.808219\pi$
$954$	0	0
$955$	− 2.10010e16i	− 0.855501i
$956$	0	0
$957$	9.55545e16i	3.84800i
$958$	0	0
$959$	−4.01204e16	−1.59721
$960$	0	0
$961$	−2.13072e16	−0.838586
$962$	0	0
$963$	− 4.39712e16i	− 1.71090i
$964$	0	0
$965$	7.00876e16i	2.69613i
$966$	0	0
$967$	4.34033e16	1.65074	0.825368	−	0.564596i	$-0.190968\pi$
$967$	4.34033e16	1.65074	0.825368	+	0.564596i	$0.190968\pi$
$968$	0	0
$969$	−8.51949e15	−0.320356
$970$	0	0
$971$	1.05627e16i	0.392707i	0.980533	+	0.196353i	$0.0629100\pi$
$971$	1.05627e16i	0.392707i	−0.980533	+	0.196353i	$0.937090\pi$
$972$	0	0
$973$	1.05356e16i	0.387292i
$974$	0	0
$975$	−3.30853e16	−1.20257
$976$	0	0
$977$	2.02103e16	0.726360	0.363180	−	0.931719i	$-0.381691\pi$
$977$	2.02103e16	0.726360	0.363180	+	0.931719i	$0.381691\pi$
$978$	0	0
$979$	− 3.01200e16i	− 1.07041i
$980$	0	0
$981$	− 3.72623e16i	− 1.30945i
$982$	0	0
$983$	−1.28410e16	−0.446226	−0.223113	−	0.974793i	$-0.571622\pi$
$983$	−1.28410e16	−0.446226	−0.223113	+	0.974793i	$0.571622\pi$
$984$	0	0
$985$	−5.15935e16	−1.77295
$986$	0	0
$987$	− 4.09086e16i	− 1.39018i
$988$	0	0
$989$	− 1.84332e16i	− 0.619473i
$990$	0	0
$991$	1.07409e16	0.356972	0.178486	−	0.983942i	$-0.442880\pi$
$991$	1.07409e16	0.356972	0.178486	+	0.983942i	$0.442880\pi$
$992$	0	0
$993$	6.92372e16	2.27572
$994$	0	0
$995$	− 4.10325e16i	− 1.33383i
$996$	0	0
$997$	− 4.69914e16i	− 1.51076i	−0.655288	−	0.755379i	$-0.727453\pi$
$997$	− 4.69914e16i	− 1.51076i	0.655288	−	0.755379i	$-0.272547\pi$
$998$	0	0
$999$	6.64147e14	0.0211180

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.12.b.p.129.2		12
4.3	odd	2		256.12.b.q.129.11		12
8.3	odd	2		256.12.b.q.129.2		12
8.5	even	2	inner	256.12.b.p.129.11		12
16.3	odd	4		128.12.a.g.1.1	yes	6
16.5	even	4		128.12.a.h.1.1	yes	6
16.11	odd	4		128.12.a.f.1.6	yes	6
16.13	even	4		128.12.a.e.1.6	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.12.a.e.1.6	✓	6	16.13	even	4
128.12.a.f.1.6	yes	6	16.11	odd	4
128.12.a.g.1.1	yes	6	16.3	odd	4
128.12.a.h.1.1	yes	6	16.5	even	4
256.12.b.p.129.2		12	1.1	even	1	trivial
256.12.b.p.129.11		12	8.5	even	2	inner
256.12.b.q.129.2		12	8.3	odd	2
256.12.b.q.129.11		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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	256.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-597.717i q^{3} +9917.53i q^{5} -77908.4 q^{7} -180119. q^{9} -854628. i q^{11} +1.11758e6i q^{13} +5.92788e6 q^{15} +1.43133e6 q^{17} -9.95811e6i q^{19} +4.65672e7i q^{21} -3.13742e7 q^{23} -4.95293e7 q^{25} +1.77624e6i q^{27} -1.87059e8i q^{29} -6.40413e7 q^{31} -5.10826e8 q^{33} -7.72659e8i q^{35} -3.73906e8i q^{37} +6.67994e8 q^{39} -1.34692e9 q^{41} +5.87529e8i q^{43} -1.78633e9i q^{45} -8.78487e8 q^{47} +4.09239e9 q^{49} -8.55533e8i q^{51} +1.76776e9i q^{53} +8.47580e9 q^{55} -5.95213e9 q^{57} -7.95505e9i q^{59} +4.23552e9i q^{61} +1.40328e10 q^{63} -1.10836e10 q^{65} -1.38472e10i q^{67} +1.87529e10i q^{69} -3.21284e9 q^{71} -1.70921e9 q^{73} +2.96045e10i q^{75} +6.65827e10i q^{77} -8.12305e9 q^{79} -3.08458e10 q^{81} -4.35036e8i q^{83} +1.41953e10i q^{85} -1.11808e11 q^{87} +3.52435e10 q^{89} -8.70686e10i q^{91} +3.82786e10i q^{93} +9.87598e10 q^{95} +2.36079e10 q^{97} +1.53934e11i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 98736 q^{7} - 627628 q^{9} + 9656528 q^{15} + 8255272 q^{17} - 19842640 q^{23} - 103266004 q^{25} - 492266240 q^{31} - 675982576 q^{33} - 1296172112 q^{39} - 973831672 q^{41} + 1207950176 q^{47}+ \cdots + 437793250984 q^{97}+O(q^{100}) \) 12 * q - 98736 * q^7 - 627628 * q^9 + 9656528 * q^15 + 8255272 * q^17 - 19842640 * q^23 - 103266004 * q^25 - 492266240 * q^31 - 675982576 * q^33 - 1296172112 * q^39 - 973831672 * q^41 + 1207950176 * q^47 + 3693758124 * q^49 + 2710831952 * q^55 + 1566705296 * q^57 + 29053202992 * q^63 + 2753811376 * q^65 - 32362135024 * q^71 - 71927337384 * q^73 - 12207892384 * q^79 + 162481241980 * q^81 - 265373263632 * q^87 - 45442832232 * q^89 + 565435645136 * q^95 + 437793250984 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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