LMFDB - Embedded newform 256.12.b.c.129.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,12,Mod(129,256)]

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([0, 1]))

N = Newforms(chi, 12, names="a")

//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);

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

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

Self dual:	no
Analytic conductor:	$196.695854223$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	256.129
Dual form			256.12.b.c.129.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-252.000i q^{3} -4830.00i q^{5} -16744.0 q^{7} +113643. q^{9} +O(q^{10})$	$q-252.000i q^{3} -4830.00i q^{5} -16744.0 q^{7} +113643. q^{9} +534612. i q^{11} -577738. i q^{13} -1.21716e6 q^{15} -6.90593e6 q^{17} -1.06614e7i q^{19} +4.21949e6i q^{21} +1.86433e7 q^{23} +2.54992e7 q^{25} -7.32791e7i q^{27} +1.28407e8i q^{29} +5.28432e7 q^{31} +1.34722e8 q^{33} +8.08735e7i q^{35} +1.82213e8i q^{37} -1.45590e8 q^{39} -3.08120e8 q^{41} -1.71257e7i q^{43} -5.48896e8i q^{45} -2.68735e9 q^{47} -1.69697e9 q^{49} +1.74030e9i q^{51} +1.59606e9i q^{53} +2.58218e9 q^{55} -2.68668e9 q^{57} -5.18920e9i q^{59} +6.95648e9i q^{61} -1.90284e9 q^{63} -2.79047e9 q^{65} +1.54818e10i q^{67} -4.69810e9i q^{69} +9.79149e9 q^{71} -1.46379e9 q^{73} -6.42580e9i q^{75} -8.95154e9i q^{77} -3.81168e10 q^{79} +1.66519e9 q^{81} +2.93351e10i q^{83} +3.33557e10i q^{85} +3.23585e10 q^{87} +2.49929e10 q^{89} +9.67365e9i q^{91} -1.33165e10i q^{93} -5.14947e10 q^{95} +7.50136e10 q^{97} +6.07549e10i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 33488 q^{7} + 227286 q^{9}+O(q^{10}) $ 2 * q - 33488 * q^7 + 227286 * q^9	$ 2 q - 33488 q^{7} + 227286 q^{9} - 2434320 q^{15} - 13811868 q^{17} + 37286544 q^{23} + 50998450 q^{25} + 105686336 q^{31} + 269444448 q^{33} - 291179952 q^{39} - 616240884 q^{41} - 5374696992 q^{47} - 3393930414 q^{49} + 5164351920 q^{55} - 5373355680 q^{57} - 3805676784 q^{63} - 5580949080 q^{65} + 19582970544 q^{71} - 2927582644 q^{73} - 76233691360 q^{79} + 3330376722 q^{81} + 64716941520 q^{87} + 49985834220 q^{89} - 102989317200 q^{95} + 150027137092 q^{97}+O(q^{100}) $ 2 * q - 33488 * q^7 + 227286 * q^9 - 2434320 * q^15 - 13811868 * q^17 + 37286544 * q^23 + 50998450 * q^25 + 105686336 * q^31 + 269444448 * q^33 - 291179952 * q^39 - 616240884 * q^41 - 5374696992 * q^47 - 3393930414 * q^49 + 5164351920 * q^55 - 5373355680 * q^57 - 3805676784 * q^63 - 5580949080 * q^65 + 19582970544 * q^71 - 2927582644 * q^73 - 76233691360 * q^79 + 3330376722 * q^81 + 64716941520 * q^87 + 49985834220 * q^89 - 102989317200 * q^95 + 150027137092 * 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
$3$	− 252.000i	− 0.598734i	−0.954138	−	0.299367i	$-0.903225\pi$
$3$	− 252.000i	− 0.598734i	0.954138	−	0.299367i	$-0.0967754\pi$
$4$	0	0
$5$	− 4830.00i	− 0.691213i	−0.938379	−	0.345607i	$-0.887673\pi$
$5$	− 4830.00i	− 0.691213i	0.938379	−	0.345607i	$-0.112327\pi$
$6$	0	0
$7$	−16744.0	−0.376548	−0.188274	−	0.982117i	$-0.560289\pi$
$7$	−16744.0	−0.376548	−0.188274	+	0.982117i	$0.560289\pi$
$8$	0	0
$9$	113643.	0.641518
$10$	0	0
$11$	534612.i	1.00087i	0.865773	+	0.500436i	$0.166827\pi$
$11$	534612.i	1.00087i	−0.865773	+	0.500436i	$0.833173\pi$
$12$	0	0
$13$	− 577738.i	− 0.431561i	−0.976442	−	0.215781i	$-0.930770\pi$
$13$	− 577738.i	− 0.431561i	0.976442	−	0.215781i	$-0.0692296\pi$
$14$	0	0
$15$	−1.21716e6	−0.413853
$16$	0	0
$17$	−6.90593e6	−1.17965	−0.589825	−	0.807531i	$-0.700803\pi$
$17$	−6.90593e6	−1.17965	−0.589825	+	0.807531i	$0.700803\pi$
$18$	0	0
$19$	− 1.06614e7i	− 0.987803i	−0.869518	−	0.493901i	$-0.835570\pi$
$19$	− 1.06614e7i	− 0.987803i	0.869518	−	0.493901i	$-0.164430\pi$
$20$	0	0
$21$	4.21949e6i	0.225452i
$22$	0	0
$23$	1.86433e7	0.603975	0.301988	−	0.953312i	$-0.402350\pi$
$23$	1.86433e7	0.603975	0.301988	+	0.953312i	$0.402350\pi$
$24$	0	0
$25$	2.54992e7	0.522224
$26$	0	0
$27$	− 7.32791e7i	− 0.982832i
$28$	0	0
$29$	1.28407e8i	1.16251i	0.813720	+	0.581257i	$0.197439\pi$
$29$	1.28407e8i	1.16251i	−0.813720	+	0.581257i	$0.802561\pi$
$30$	0	0
$31$	5.28432e7	0.331512	0.165756	−	0.986167i	$-0.446994\pi$
$31$	5.28432e7	0.331512	0.165756	+	0.986167i	$0.446994\pi$
$32$	0	0
$33$	1.34722e8	0.599256
$34$	0	0
$35$	8.08735e7i	0.260275i
$36$	0	0
$37$	1.82213e8i	0.431987i	0.976395	+	0.215993i	$0.0692990\pi$
$37$	1.82213e8i	0.431987i	−0.976395	+	0.215993i	$0.930701\pi$
$38$	0	0
$39$	−1.45590e8	−0.258390
$40$	0	0
$41$	−3.08120e8	−0.415345	−0.207673	−	0.978198i	$-0.566589\pi$
$41$	−3.08120e8	−0.415345	−0.207673	+	0.978198i	$0.566589\pi$
$42$	0	0
$43$	− 1.71257e7i	− 0.0177653i	−0.999961	−	0.00888264i	$-0.997173\pi$
$43$	− 1.71257e7i	− 0.0177653i	0.999961	−	0.00888264i	$-0.00282747\pi$
$44$	0	0
$45$	− 5.48896e8i	− 0.443426i
$46$	0	0
$47$	−2.68735e9	−1.70917	−0.854586	−	0.519310i	$-0.826189\pi$
$47$	−2.68735e9	−1.70917	−0.854586	+	0.519310i	$0.826189\pi$
$48$	0	0
$49$	−1.69697e9	−0.858212
$50$	0	0
$51$	1.74030e9i	0.706296i
$52$	0	0
$53$	1.59606e9i	0.524241i	0.965035	+	0.262120i	$0.0844217\pi$
$53$	1.59606e9i	0.524241i	−0.965035	+	0.262120i	$0.915578\pi$
$54$	0	0
$55$	2.58218e9	0.691817
$56$	0	0
$57$	−2.68668e9	−0.591431
$58$	0	0
$59$	− 5.18920e9i	− 0.944963i	−0.881341	−	0.472481i	$-0.843358\pi$
$59$	− 5.18920e9i	− 0.944963i	0.881341	−	0.472481i	$-0.156642\pi$
$60$	0	0
$61$	6.95648e9i	1.05457i	0.849689	+	0.527285i	$0.176790\pi$
$61$	6.95648e9i	1.05457i	−0.849689	+	0.527285i	$0.823210\pi$
$62$	0	0
$63$	−1.90284e9	−0.241562
$64$	0	0
$65$	−2.79047e9	−0.298301
$66$	0	0
$67$	1.54818e10i	1.40091i	0.713696	+	0.700456i	$0.247020\pi$
$67$	1.54818e10i	1.40091i	−0.713696	+	0.700456i	$0.752980\pi$
$68$	0	0
$69$	− 4.69810e9i	− 0.361620i
$70$	0	0
$71$	9.79149e9	0.644062	0.322031	−	0.946729i	$-0.395634\pi$
$71$	9.79149e9	0.644062	0.322031	+	0.946729i	$0.395634\pi$
$72$	0	0
$73$	−1.46379e9	−0.0826425	−0.0413212	−	0.999146i	$-0.513157\pi$
$73$	−1.46379e9	−0.0826425	−0.0413212	+	0.999146i	$0.513157\pi$
$74$	0	0
$75$	− 6.42580e9i	− 0.312673i
$76$	0	0
$77$	− 8.95154e9i	− 0.376876i
$78$	0	0
$79$	−3.81168e10	−1.39370	−0.696848	−	0.717219i	$-0.745415\pi$
$79$	−3.81168e10	−1.39370	−0.696848	+	0.717219i	$0.745415\pi$
$80$	0	0
$81$	1.66519e9	0.0530635
$82$	0	0
$83$	2.93351e10i	0.817444i	0.912659	+	0.408722i	$0.134025\pi$
$83$	2.93351e10i	0.817444i	−0.912659	+	0.408722i	$0.865975\pi$
$84$	0	0
$85$	3.33557e10i	0.815390i
$86$	0	0
$87$	3.23585e10	0.696037
$88$	0	0
$89$	2.49929e10	0.474430	0.237215	−	0.971457i	$-0.423765\pi$
$89$	2.49929e10	0.474430	0.237215	+	0.971457i	$0.423765\pi$
$90$	0	0
$91$	9.67365e9i	0.162503i
$92$	0	0
$93$	− 1.33165e10i	− 0.198488i
$94$	0	0
$95$	−5.14947e10	−0.682782
$96$	0	0
$97$	7.50136e10	0.886942	0.443471	−	0.896289i	$-0.353747\pi$
$97$	7.50136e10	0.886942	0.443471	+	0.896289i	$0.353747\pi$
$98$	0	0
$99$	6.07549e10i	0.642078i
$100$	0	0
$101$	− 8.17430e10i	− 0.773896i	−0.922101	−	0.386948i	$-0.873529\pi$
$101$	− 8.17430e10i	− 0.773896i	0.922101	−	0.386948i	$-0.126471\pi$
$102$	0	0
$103$	−2.25755e11	−1.91881	−0.959407	−	0.282025i	$-0.908994\pi$
$103$	−2.25755e11	−1.91881	−0.959407	+	0.282025i	$0.908994\pi$
$104$	0	0
$105$	2.03801e10	0.155835
$106$	0	0
$107$	9.02413e10i	0.622006i	0.950409	+	0.311003i	$0.100665\pi$
$107$	9.02413e10i	0.622006i	−0.950409	+	0.311003i	$0.899335\pi$
$108$	0	0
$109$	7.34827e10i	0.457445i	0.973492	+	0.228723i	$0.0734549\pi$
$109$	7.34827e10i	0.457445i	−0.973492	+	0.228723i	$0.926545\pi$
$110$	0	0
$111$	4.59178e10	0.258645
$112$	0	0
$113$	−8.51469e10	−0.434748	−0.217374	−	0.976088i	$-0.569749\pi$
$113$	−8.51469e10	−0.434748	−0.217374	+	0.976088i	$0.569749\pi$
$114$	0	0
$115$	− 9.00470e10i	− 0.417476i
$116$	0	0
$117$	− 6.56559e10i	− 0.276854i
$118$	0	0
$119$	1.15633e11	0.444195
$120$	0	0
$121$	−4.98320e8	−0.00174658
$122$	0	0
$123$	7.76464e10i	0.248681i
$124$	0	0
$125$	− 3.59001e11i	− 1.05218i
$126$	0	0
$127$	2.62717e11	0.705615	0.352808	−	0.935696i	$-0.385227\pi$
$127$	2.62717e11	0.705615	0.352808	+	0.935696i	$0.385227\pi$
$128$	0	0
$129$	−4.31568e9	−0.0106367
$130$	0	0
$131$	− 6.31529e11i	− 1.43021i	−0.699015	−	0.715107i	$-0.746378\pi$
$131$	− 6.31529e11i	− 1.43021i	0.699015	−	0.715107i	$-0.253622\pi$
$132$	0	0
$133$	1.78515e11i	0.371955i
$134$	0	0
$135$	−3.53938e11	−0.679347
$136$	0	0
$137$	2.97199e11	0.526119	0.263059	−	0.964780i	$-0.415268\pi$
$137$	2.97199e11	0.526119	0.263059	+	0.964780i	$0.415268\pi$
$138$	0	0
$139$	5.96794e11i	0.975535i	0.872974	+	0.487767i	$0.162189\pi$
$139$	5.96794e11i	0.975535i	−0.872974	+	0.487767i	$0.837811\pi$
$140$	0	0
$141$	6.77212e11i	1.02334i
$142$	0	0
$143$	3.08866e11	0.431938
$144$	0	0
$145$	6.20204e11	0.803546
$146$	0	0
$147$	4.27635e11i	0.513840i
$148$	0	0
$149$	1.11543e12i	1.24428i	0.782905	+	0.622142i	$0.213737\pi$
$149$	1.11543e12i	1.24428i	−0.782905	+	0.622142i	$0.786263\pi$
$150$	0	0
$151$	−8.24447e11	−0.854653	−0.427326	−	0.904097i	$-0.640544\pi$
$151$	−8.24447e11	−0.854653	−0.427326	+	0.904097i	$0.640544\pi$
$152$	0	0
$153$	−7.84811e11	−0.756767
$154$	0	0
$155$	− 2.55233e11i	− 0.229146i
$156$	0	0
$157$	1.31512e12i	1.10031i	0.835062	+	0.550156i	$0.185432\pi$
$157$	1.31512e12i	1.10031i	−0.835062	+	0.550156i	$0.814568\pi$
$158$	0	0
$159$	4.02206e11	0.313881
$160$	0	0
$161$	−3.12163e11	−0.227425
$162$	0	0
$163$	3.57833e11i	0.243584i	0.992556	+	0.121792i	$0.0388640\pi$
$163$	3.57833e11i	0.243584i	−0.992556	+	0.121792i	$0.961136\pi$
$164$	0	0
$165$	− 6.50708e11i	− 0.414214i
$166$	0	0
$167$	2.75483e12	1.64117	0.820587	−	0.571521i	$-0.193646\pi$
$167$	2.75483e12	1.64117	0.820587	+	0.571521i	$0.193646\pi$
$168$	0	0
$169$	1.45838e12	0.813755
$170$	0	0
$171$	− 1.21160e12i	− 0.633693i
$172$	0	0
$173$	− 9.50387e11i	− 0.466280i	−0.972443	−	0.233140i	$-0.925100\pi$
$173$	− 9.50387e11i	− 0.466280i	0.972443	−	0.233140i	$-0.0749001\pi$
$174$	0	0
$175$	−4.26959e11	−0.196642
$176$	0	0
$177$	−1.30768e12	−0.565781
$178$	0	0
$179$	− 1.68138e12i	− 0.683873i	−0.939723	−	0.341936i	$-0.888917\pi$
$179$	− 1.68138e12i	− 0.683873i	0.939723	−	0.341936i	$-0.111083\pi$
$180$	0	0
$181$	9.96774e11i	0.381386i	0.981650	+	0.190693i	$0.0610735\pi$
$181$	9.96774e11i	0.381386i	−0.981650	+	0.190693i	$0.938927\pi$
$182$	0	0
$183$	1.75303e12	0.631406
$184$	0	0
$185$	8.80090e11	0.298595
$186$	0	0
$187$	− 3.69200e12i	− 1.18068i
$188$	0	0
$189$	1.22698e12i	0.370083i
$190$	0	0
$191$	−2.76240e12	−0.786328	−0.393164	−	0.919468i	$-0.628619\pi$
$191$	−2.76240e12	−0.786328	−0.393164	+	0.919468i	$0.628619\pi$
$192$	0	0
$193$	5.44239e12	1.46293	0.731466	−	0.681878i	$-0.238836\pi$
$193$	5.44239e12	1.46293	0.731466	+	0.681878i	$0.238836\pi$
$194$	0	0
$195$	7.03200e11i	0.178603i
$196$	0	0
$197$	2.87609e12i	0.690619i	0.938489	+	0.345309i	$0.112226\pi$
$197$	2.87609e12i	0.690619i	−0.938489	+	0.345309i	$0.887774\pi$
$198$	0	0
$199$	7.28391e11	0.165452	0.0827262	−	0.996572i	$-0.473637\pi$
$199$	7.28391e11	0.165452	0.0827262	+	0.996572i	$0.473637\pi$
$200$	0	0
$201$	3.90142e12	0.838773
$202$	0	0
$203$	− 2.15004e12i	− 0.437742i
$204$	0	0
$205$	1.48822e12i	0.287092i
$206$	0	0
$207$	2.11868e12	0.387461
$208$	0	0
$209$	5.69972e12	0.988665
$210$	0	0
$211$	6.79317e12i	1.11820i	0.829101	+	0.559099i	$0.188853\pi$
$211$	6.79317e12i	1.11820i	−0.829101	+	0.559099i	$0.811147\pi$
$212$	0	0
$213$	− 2.46745e12i	− 0.385622i
$214$	0	0
$215$	−8.27172e10	−0.0122796
$216$	0	0
$217$	−8.84806e11	−0.124830
$218$	0	0
$219$	3.68875e11i	0.0494808i
$220$	0	0
$221$	3.98982e12i	0.509092i
$222$	0	0
$223$	−7.33486e12	−0.890667	−0.445333	−	0.895365i	$-0.646915\pi$
$223$	−7.33486e12	−0.890667	−0.445333	+	0.895365i	$0.646915\pi$
$224$	0	0
$225$	2.89781e12	0.335016
$226$	0	0
$227$	1.35984e12i	0.149743i	0.997193	+	0.0748713i	$0.0238546\pi$
$227$	1.35984e12i	0.149743i	−0.997193	+	0.0748713i	$0.976145\pi$
$228$	0	0
$229$	1.18244e13i	1.24075i	0.784305	+	0.620375i	$0.213020\pi$
$229$	1.18244e13i	1.24075i	−0.784305	+	0.620375i	$0.786980\pi$
$230$	0	0
$231$	−2.25579e12	−0.225649
$232$	0	0
$233$	1.75634e13	1.67552	0.837761	−	0.546038i	$-0.183865\pi$
$233$	1.75634e13	1.67552	0.837761	+	0.546038i	$0.183865\pi$
$234$	0	0
$235$	1.29799e13i	1.18140i
$236$	0	0
$237$	9.60545e12i	0.834452i
$238$	0	0
$239$	7.13958e12	0.592221	0.296111	−	0.955154i	$-0.404310\pi$
$239$	7.13958e12	0.592221	0.296111	+	0.955154i	$0.404310\pi$
$240$	0	0
$241$	−2.31307e11	−0.0183271	−0.00916357	−	0.999958i	$-0.502917\pi$
$241$	−2.31307e11	−0.0183271	−0.00916357	+	0.999958i	$0.502917\pi$
$242$	0	0
$243$	− 1.34008e13i	− 1.01460i
$244$	0	0
$245$	8.19634e12i	0.593207i
$246$	0	0
$247$	−6.15951e12	−0.426297
$248$	0	0
$249$	7.39245e12	0.489431
$250$	0	0
$251$	1.29831e13i	0.822567i	0.911507	+	0.411284i	$0.134919\pi$
$251$	1.29831e13i	0.822567i	−0.911507	+	0.411284i	$0.865081\pi$
$252$	0	0
$253$	9.96692e12i	0.604502i
$254$	0	0
$255$	8.40563e12	0.488201
$256$	0	0
$257$	2.39612e13	1.33314	0.666571	−	0.745442i	$-0.267761\pi$
$257$	2.39612e13	1.33314	0.666571	+	0.745442i	$0.267761\pi$
$258$	0	0
$259$	− 3.05098e12i	− 0.162664i
$260$	0	0
$261$	1.45925e13i	0.745774i
$262$	0	0
$263$	−2.42737e13	−1.18954	−0.594771	−	0.803895i	$-0.702757\pi$
$263$	−2.42737e13	−1.18954	−0.594771	+	0.803895i	$0.702757\pi$
$264$	0	0
$265$	7.70895e12	0.362362
$266$	0	0
$267$	− 6.29822e12i	− 0.284057i
$268$	0	0
$269$	2.58377e13i	1.11845i	0.829016	+	0.559225i	$0.188901\pi$
$269$	2.58377e13i	1.11845i	−0.829016	+	0.559225i	$0.811099\pi$
$270$	0	0
$271$	3.76793e12	0.156593	0.0782964	−	0.996930i	$-0.475052\pi$
$271$	3.76793e12	0.156593	0.0782964	+	0.996930i	$0.475052\pi$
$272$	0	0
$273$	2.43776e12	0.0972963
$274$	0	0
$275$	1.36322e13i	0.522680i
$276$	0	0
$277$	1.64189e13i	0.604931i	0.953160	+	0.302466i	$0.0978098\pi$
$277$	1.64189e13i	0.604931i	−0.953160	+	0.302466i	$0.902190\pi$
$278$	0	0
$279$	6.00526e12	0.212671
$280$	0	0
$281$	−2.10357e13	−0.716263	−0.358132	−	0.933671i	$-0.616586\pi$
$281$	−2.10357e13	−0.716263	−0.358132	+	0.933671i	$0.616586\pi$
$282$	0	0
$283$	1.67132e13i	0.547310i	0.961828	+	0.273655i	$0.0882327\pi$
$283$	1.67132e13i	0.547310i	−0.961828	+	0.273655i	$0.911767\pi$
$284$	0	0
$285$	1.29767e13i	0.408805i
$286$	0	0
$287$	5.15917e12	0.156397
$288$	0	0
$289$	1.34200e13	0.391575
$290$	0	0
$291$	− 1.89034e13i	− 0.531042i
$292$	0	0
$293$	2.39269e13i	0.647312i	0.946175	+	0.323656i	$0.104912\pi$
$293$	2.39269e13i	0.647312i	−0.946175	+	0.323656i	$0.895088\pi$
$294$	0	0
$295$	−2.50639e13	−0.653171
$296$	0	0
$297$	3.91759e13	0.983690
$298$	0	0
$299$	− 1.07709e13i	− 0.260652i
$300$	0	0
$301$	2.86753e11i	0.00668947i
$302$	0	0
$303$	−2.05992e13	−0.463358
$304$	0	0
$305$	3.35998e13	0.728933
$306$	0	0
$307$	− 1.53111e13i	− 0.320439i	−0.987081	−	0.160219i	$-0.948780\pi$
$307$	− 1.53111e13i	− 0.320439i	0.987081	−	0.160219i	$-0.0512202\pi$
$308$	0	0
$309$	5.68903e13i	1.14886i
$310$	0	0
$311$	4.98752e13	0.972080	0.486040	−	0.873936i	$-0.338441\pi$
$311$	4.98752e13	0.972080	0.486040	+	0.873936i	$0.338441\pi$
$312$	0	0
$313$	9.94808e13	1.87174	0.935870	−	0.352345i	$-0.114616\pi$
$313$	9.94808e13	1.87174	0.935870	+	0.352345i	$0.114616\pi$
$314$	0	0
$315$	9.19071e12i	0.166971i
$316$	0	0
$317$	8.33692e13i	1.46278i	0.681958	+	0.731392i	$0.261129\pi$
$317$	8.33692e13i	1.46278i	−0.681958	+	0.731392i	$0.738871\pi$
$318$	0	0
$319$	−6.86477e13	−1.16353
$320$	0	0
$321$	2.27408e13	0.372416
$322$	0	0
$323$	7.36271e13i	1.16526i
$324$	0	0
$325$	− 1.47319e13i	− 0.225372i
$326$	0	0
$327$	1.85176e13	0.273888
$328$	0	0
$329$	4.49970e13	0.643585
$330$	0	0
$331$	− 6.35840e13i	− 0.879618i	−0.898091	−	0.439809i	$-0.855046\pi$
$331$	− 6.35840e13i	− 0.879618i	0.898091	−	0.439809i	$-0.144954\pi$
$332$	0	0
$333$	2.07073e13i	0.277127i
$334$	0	0
$335$	7.47772e13	0.968329
$336$	0	0
$337$	1.21001e14	1.51644	0.758221	−	0.651997i	$-0.226069\pi$
$337$	1.21001e14	1.51644	0.758221	+	0.651997i	$0.226069\pi$
$338$	0	0
$339$	2.14570e13i	0.260298i
$340$	0	0
$341$	2.82506e13i	0.331802i
$342$	0	0
$343$	6.15223e13	0.699705
$344$	0	0
$345$	−2.26918e13	−0.249957
$346$	0	0
$347$	− 1.55662e14i	− 1.66100i	−0.557020	−	0.830499i	$-0.688055\pi$
$347$	− 1.55662e14i	− 1.66100i	0.557020	−	0.830499i	$-0.311945\pi$
$348$	0	0
$349$	− 2.56430e13i	− 0.265112i	−0.991176	−	0.132556i	$-0.957682\pi$
$349$	− 2.56430e13i	− 0.265112i	0.991176	−	0.132556i	$-0.0423184\pi$
$350$	0	0
$351$	−4.23361e13	−0.424152
$352$	0	0
$353$	2.49098e13	0.241885	0.120943	−	0.992659i	$-0.461408\pi$
$353$	2.49098e13	0.241885	0.120943	+	0.992659i	$0.461408\pi$
$354$	0	0
$355$	− 4.72929e13i	− 0.445184i
$356$	0	0
$357$	− 2.91395e13i	− 0.265954i
$358$	0	0
$359$	1.57584e14	1.39474	0.697370	−	0.716712i	$-0.254354\pi$
$359$	1.57584e14	1.39474	0.697370	+	0.716712i	$0.254354\pi$
$360$	0	0
$361$	2.82438e12	0.0242457
$362$	0	0
$363$	1.25577e11i	0.00104574i
$364$	0	0
$365$	7.07011e12i	0.0571236i
$366$	0	0
$367$	1.77901e14	1.39481	0.697406	−	0.716676i	$-0.254338\pi$
$367$	1.77901e14	1.39481	0.697406	+	0.716676i	$0.254338\pi$
$368$	0	0
$369$	−3.50157e13	−0.266452
$370$	0	0
$371$	− 2.67244e13i	− 0.197402i
$372$	0	0
$373$	5.51617e13i	0.395585i	0.980244	+	0.197792i	$0.0633772\pi$
$373$	5.51617e13i	0.395585i	−0.980244	+	0.197792i	$0.936623\pi$
$374$	0	0
$375$	−9.04683e13	−0.629976
$376$	0	0
$377$	7.41854e13	0.501696
$378$	0	0
$379$	1.46463e14i	0.962083i	0.876698	+	0.481042i	$0.159741\pi$
$379$	1.46463e14i	0.962083i	−0.876698	+	0.481042i	$0.840259\pi$
$380$	0	0
$381$	− 6.62047e13i	− 0.422476i
$382$	0	0
$383$	−2.31450e14	−1.43504	−0.717519	−	0.696539i	$-0.754722\pi$
$383$	−2.31450e14	−1.43504	−0.717519	+	0.696539i	$0.754722\pi$
$384$	0	0
$385$	−4.32360e13	−0.260502
$386$	0	0
$387$	− 1.94622e12i	− 0.0113967i
$388$	0	0
$389$	1.49872e14i	0.853093i	0.904466	+	0.426547i	$0.140270\pi$
$389$	1.49872e14i	0.853093i	−0.904466	+	0.426547i	$0.859730\pi$
$390$	0	0
$391$	−1.28749e14	−0.712480
$392$	0	0
$393$	−1.59145e14	−0.856317
$394$	0	0
$395$	1.84104e14i	0.963341i
$396$	0	0
$397$	2.08111e14i	1.05912i	0.848271	+	0.529562i	$0.177644\pi$
$397$	2.08111e14i	1.05912i	−0.848271	+	0.529562i	$0.822356\pi$
$398$	0	0
$399$	4.49857e13	0.222702
$400$	0	0
$401$	−1.33408e14	−0.642521	−0.321261	−	0.946991i	$-0.604107\pi$
$401$	−1.33408e14	−0.642521	−0.321261	+	0.946991i	$0.604107\pi$
$402$	0	0
$403$	− 3.05295e13i	− 0.143068i
$404$	0	0
$405$	− 8.04286e12i	− 0.0366782i
$406$	0	0
$407$	−9.74134e13	−0.432364
$408$	0	0
$409$	2.06168e14	0.890722	0.445361	−	0.895351i	$-0.353075\pi$
$409$	2.06168e14	0.890722	0.445361	+	0.895351i	$0.353075\pi$
$410$	0	0
$411$	− 7.48941e13i	− 0.315005i
$412$	0	0
$413$	8.68880e13i	0.355824i
$414$	0	0
$415$	1.41689e14	0.565028
$416$	0	0
$417$	1.50392e14	0.584085
$418$	0	0
$419$	− 7.34035e13i	− 0.277677i	−0.990315	−	0.138838i	$-0.955663\pi$
$419$	− 7.34035e13i	− 0.277677i	0.990315	−	0.138838i	$-0.0443369\pi$
$420$	0	0
$421$	− 1.71112e14i	− 0.630563i	−0.948998	−	0.315282i	$-0.897901\pi$
$421$	− 1.71112e14i	− 0.630563i	0.948998	−	0.315282i	$-0.102099\pi$
$422$	0	0
$423$	−3.05398e14	−1.09646
$424$	0	0
$425$	−1.76096e14	−0.616042
$426$	0	0
$427$	− 1.16479e14i	− 0.397096i
$428$	0	0
$429$	− 7.78341e13i	− 0.258616i
$430$	0	0
$431$	7.17758e13	0.232463	0.116231	−	0.993222i	$-0.462919\pi$
$431$	7.17758e13	0.232463	0.116231	+	0.993222i	$0.462919\pi$
$432$	0	0
$433$	9.98812e13	0.315356	0.157678	−	0.987491i	$-0.449599\pi$
$433$	9.98812e13	0.315356	0.157678	+	0.987491i	$0.449599\pi$
$434$	0	0
$435$	− 1.56291e14i	− 0.481110i
$436$	0	0
$437$	− 1.98764e14i	− 0.596608i
$438$	0	0
$439$	−2.90312e13	−0.0849788	−0.0424894	−	0.999097i	$-0.513529\pi$
$439$	−2.90312e13	−0.0849788	−0.0424894	+	0.999097i	$0.513529\pi$
$440$	0	0
$441$	−1.92848e14	−0.550558
$442$	0	0
$443$	3.28370e14i	0.914414i	0.889360	+	0.457207i	$0.151150\pi$
$443$	3.28370e14i	0.914414i	−0.889360	+	0.457207i	$0.848850\pi$
$444$	0	0
$445$	− 1.20716e14i	− 0.327932i
$446$	0	0
$447$	2.81089e14	0.744994
$448$	0	0
$449$	−6.12368e14	−1.58364	−0.791822	−	0.610752i	$-0.790867\pi$
$449$	−6.12368e14	−1.58364	−0.791822	+	0.610752i	$0.790867\pi$
$450$	0	0
$451$	− 1.64725e14i	− 0.415708i
$452$	0	0
$453$	2.07761e14i	0.511709i
$454$	0	0
$455$	4.67237e13	0.112325
$456$	0	0
$457$	−3.03483e14	−0.712189	−0.356095	−	0.934450i	$-0.615892\pi$
$457$	−3.03483e14	−0.712189	−0.356095	+	0.934450i	$0.615892\pi$
$458$	0	0
$459$	5.06060e14i	1.15940i
$460$	0	0
$461$	− 7.29308e14i	− 1.63138i	−0.578487	−	0.815691i	$-0.696357\pi$
$461$	− 7.29308e14i	− 1.63138i	0.578487	−	0.815691i	$-0.303643\pi$
$462$	0	0
$463$	−1.22188e14	−0.266891	−0.133445	−	0.991056i	$-0.542604\pi$
$463$	−1.22188e14	−0.266891	−0.133445	+	0.991056i	$0.542604\pi$
$464$	0	0
$465$	−6.43186e13	−0.137197
$466$	0	0
$467$	6.17381e14i	1.28621i	0.765780	+	0.643103i	$0.222353\pi$
$467$	6.17381e14i	1.28621i	−0.765780	+	0.643103i	$0.777647\pi$
$468$	0	0
$469$	− 2.59228e14i	− 0.527510i
$470$	0	0
$471$	3.31409e14	0.658794
$472$	0	0
$473$	9.15561e12	0.0177808
$474$	0	0
$475$	− 2.71858e14i	− 0.515854i
$476$	0	0
$477$	1.81381e14i	0.336310i
$478$	0	0
$479$	−1.05084e15	−1.90410	−0.952052	−	0.305938i	$-0.901030\pi$
$479$	−1.05084e15	−1.90410	−0.952052	+	0.305938i	$0.901030\pi$
$480$	0	0
$481$	1.05272e14	0.186429
$482$	0	0
$483$	7.86651e13i	0.136167i
$484$	0	0
$485$	− 3.62316e14i	− 0.613066i
$486$	0	0
$487$	−2.19910e14	−0.363777	−0.181889	−	0.983319i	$-0.558221\pi$
$487$	−2.19910e14	−0.363777	−0.181889	+	0.983319i	$0.558221\pi$
$488$	0	0
$489$	9.01739e13	0.145842
$490$	0	0
$491$	− 4.83863e14i	− 0.765199i	−0.923914	−	0.382599i	$-0.875029\pi$
$491$	− 4.83863e14i	− 0.765199i	0.923914	−	0.382599i	$-0.124971\pi$
$492$	0	0
$493$	− 8.86768e14i	− 1.37136i
$494$	0	0
$495$	2.93446e14	0.443813
$496$	0	0
$497$	−1.63949e14	−0.242520
$498$	0	0
$499$	1.08878e14i	0.157538i	0.996893	+	0.0787691i	$0.0250990\pi$
$499$	1.08878e14i	0.157538i	−0.996893	+	0.0787691i	$0.974901\pi$
$500$	0	0
$501$	− 6.94218e14i	− 0.982626i
$502$	0	0
$503$	5.06588e14	0.701506	0.350753	−	0.936468i	$-0.385926\pi$
$503$	5.06588e14	0.701506	0.350753	+	0.936468i	$0.385926\pi$
$504$	0	0
$505$	−3.94818e14	−0.534927
$506$	0	0
$507$	− 3.67512e14i	− 0.487222i
$508$	0	0
$509$	8.57534e13i	0.111251i	0.998452	+	0.0556254i	$0.0177153\pi$
$509$	8.57534e13i	0.111251i	−0.998452	+	0.0556254i	$0.982285\pi$
$510$	0	0
$511$	2.45097e13	0.0311188
$512$	0	0
$513$	−7.81259e14	−0.970844
$514$	0	0
$515$	1.09040e15i	1.32631i
$516$	0	0
$517$	− 1.43669e15i	− 1.71066i
$518$	0	0
$519$	−2.39498e14	−0.279178
$520$	0	0
$521$	−9.27575e14	−1.05862	−0.529312	−	0.848428i	$-0.677550\pi$
$521$	−9.27575e14	−1.05862	−0.529312	+	0.848428i	$0.677550\pi$
$522$	0	0
$523$	− 2.18187e13i	− 0.0243820i	−0.999926	−	0.0121910i	$-0.996119\pi$
$523$	− 2.18187e13i	− 0.0243820i	0.999926	−	0.0121910i	$-0.00388061\pi$
$524$	0	0
$525$	1.07594e14i	0.117736i
$526$	0	0
$527$	−3.64931e14	−0.391069
$528$	0	0
$529$	−6.05238e14	−0.635214
$530$	0	0
$531$	− 5.89717e14i	− 0.606211i
$532$	0	0
$533$	1.78013e14i	0.179247i
$534$	0	0
$535$	4.35865e14	0.429939
$536$	0	0
$537$	−4.23709e14	−0.409458
$538$	0	0
$539$	− 9.07218e14i	− 0.858961i
$540$	0	0
$541$	− 1.69527e15i	− 1.57273i	−0.617765	−	0.786363i	$-0.711962\pi$
$541$	− 1.69527e15i	− 1.57273i	0.617765	−	0.786363i	$-0.288038\pi$
$542$	0	0
$543$	2.51187e14	0.228349
$544$	0	0
$545$	3.54921e14	0.316192
$546$	0	0
$547$	− 7.52145e14i	− 0.656706i	−0.944555	−	0.328353i	$-0.893506\pi$
$547$	− 7.52145e14i	− 0.656706i	0.944555	−	0.328353i	$-0.106494\pi$
$548$	0	0
$549$	7.90555e14i	0.676526i
$550$	0	0
$551$	1.36900e15	1.14834
$552$	0	0
$553$	6.38228e14	0.524793
$554$	0	0
$555$	− 2.21783e14i	− 0.178779i
$556$	0	0
$557$	1.87489e14i	0.148174i	0.997252	+	0.0740870i	$0.0236043\pi$
$557$	1.87489e14i	0.148174i	−0.997252	+	0.0740870i	$0.976396\pi$
$558$	0	0
$559$	−9.89417e12	−0.00766681
$560$	0	0
$561$	−9.30383e14	−0.706913
$562$	0	0
$563$	− 2.44971e14i	− 0.182524i	−0.995827	−	0.0912618i	$-0.970910\pi$
$563$	− 2.44971e14i	− 0.182524i	0.995827	−	0.0912618i	$-0.0290900\pi$
$564$	0	0
$565$	4.11259e14i	0.300503i
$566$	0	0
$567$	−2.78819e13	−0.0199809
$568$	0	0
$569$	−1.35243e15	−0.950596	−0.475298	−	0.879825i	$-0.657660\pi$
$569$	−1.35243e15	−0.950596	−0.475298	+	0.879825i	$0.657660\pi$
$570$	0	0
$571$	1.43223e15i	0.987447i	0.869619	+	0.493723i	$0.164364\pi$
$571$	1.43223e15i	0.987447i	−0.869619	+	0.493723i	$0.835636\pi$
$572$	0	0
$573$	6.96126e14i	0.470801i
$574$	0	0
$575$	4.75389e14	0.315410
$576$	0	0
$577$	−8.77659e14	−0.571293	−0.285647	−	0.958335i	$-0.592208\pi$
$577$	−8.77659e14	−0.571293	−0.285647	+	0.958335i	$0.592208\pi$
$578$	0	0
$579$	− 1.37148e15i	− 0.875907i
$580$	0	0
$581$	− 4.91187e14i	− 0.307807i
$582$	0	0
$583$	−8.53271e14	−0.524698
$584$	0	0
$585$	−3.17118e14	−0.191365
$586$	0	0
$587$	− 2.43425e15i	− 1.44164i	−0.693124	−	0.720818i	$-0.743766\pi$
$587$	− 2.43425e15i	− 1.44164i	0.693124	−	0.720818i	$-0.256234\pi$
$588$	0	0
$589$	− 5.63383e14i	− 0.327469i
$590$	0	0
$591$	7.24775e14	0.413497
$592$	0	0
$593$	−3.03318e14	−0.169863	−0.0849313	−	0.996387i	$-0.527067\pi$
$593$	−3.03318e14	−0.169863	−0.0849313	+	0.996387i	$0.527067\pi$
$594$	0	0
$595$	− 5.58507e14i	− 0.307033i
$596$	0	0
$597$	− 1.83555e14i	− 0.0990619i
$598$	0	0
$599$	−1.70198e15	−0.901795	−0.450898	−	0.892576i	$-0.648896\pi$
$599$	−1.70198e15	−0.901795	−0.450898	+	0.892576i	$0.648896\pi$
$600$	0	0
$601$	−2.33922e15	−1.21692	−0.608458	−	0.793586i	$-0.708212\pi$
$601$	−2.33922e15	−1.21692	−0.608458	+	0.793586i	$0.708212\pi$
$602$	0	0
$603$	1.75940e15i	0.898710i
$604$	0	0
$605$	2.40689e12i	0.00120726i
$606$	0	0
$607$	2.49607e15	1.22947	0.614737	−	0.788732i	$-0.289262\pi$
$607$	2.49607e15	1.22947	0.614737	+	0.788732i	$0.289262\pi$
$608$	0	0
$609$	−5.41810e14	−0.262091
$610$	0	0
$611$	1.55258e15i	0.737612i
$612$	0	0
$613$	− 2.47301e15i	− 1.15397i	−0.816756	−	0.576983i	$-0.804230\pi$
$613$	− 2.47301e15i	− 1.15397i	0.816756	−	0.576983i	$-0.195770\pi$
$614$	0	0
$615$	3.75032e14	0.171892
$616$	0	0
$617$	−2.43368e13	−0.0109571	−0.00547854	−	0.999985i	$-0.501744\pi$
$617$	−2.43368e13	−0.0109571	−0.00547854	+	0.999985i	$0.501744\pi$
$618$	0	0
$619$	4.22545e15i	1.86885i	0.356160	+	0.934425i	$0.384086\pi$
$619$	4.22545e15i	1.86885i	−0.356160	+	0.934425i	$0.615914\pi$
$620$	0	0
$621$	− 1.36616e15i	− 0.593606i
$622$	0	0
$623$	−4.18481e14	−0.178645
$624$	0	0
$625$	−4.88896e14	−0.205058
$626$	0	0
$627$	− 1.43633e15i	− 0.591947i
$628$	0	0
$629$	− 1.25835e15i	− 0.509594i
$630$	0	0
$631$	−4.26326e15	−1.69660	−0.848302	−	0.529513i	$-0.822375\pi$
$631$	−4.26326e15	−1.69660	−0.848302	+	0.529513i	$0.822375\pi$
$632$	0	0
$633$	1.71188e15	0.669503
$634$	0	0
$635$	− 1.26892e15i	− 0.487731i
$636$	0	0
$637$	9.80401e14i	0.370371i
$638$	0	0
$639$	1.11273e15	0.413177
$640$	0	0
$641$	1.00830e15	0.368018	0.184009	−	0.982925i	$-0.441092\pi$
$641$	1.00830e15	0.368018	0.184009	+	0.982925i	$0.441092\pi$
$642$	0	0
$643$	− 3.03982e14i	− 0.109066i	−0.998512	−	0.0545328i	$-0.982633\pi$
$643$	− 3.03982e14i	− 0.109066i	0.998512	−	0.0545328i	$-0.0173670\pi$
$644$	0	0
$645$	2.08447e13i	0.00735221i
$646$	0	0
$647$	3.43583e15	1.19140	0.595700	−	0.803207i	$-0.296875\pi$
$647$	3.43583e15	1.19140	0.595700	+	0.803207i	$0.296875\pi$
$648$	0	0
$649$	2.77421e15	0.945788
$650$	0	0
$651$	2.22971e14i	0.0747400i
$652$	0	0
$653$	− 1.18539e15i	− 0.390695i	−0.980734	−	0.195347i	$-0.937417\pi$
$653$	− 1.18539e15i	− 0.390695i	0.980734	−	0.195347i	$-0.0625834\pi$
$654$	0	0
$655$	−3.05028e15	−0.988583
$656$	0	0
$657$	−1.66350e14	−0.0530167
$658$	0	0
$659$	2.26510e15i	0.709934i	0.934879	+	0.354967i	$0.115508\pi$
$659$	2.26510e15i	0.709934i	−0.934879	+	0.354967i	$0.884492\pi$
$660$	0	0
$661$	5.33012e15i	1.64297i	0.570232	+	0.821484i	$0.306853\pi$
$661$	5.33012e15i	1.64297i	−0.570232	+	0.821484i	$0.693147\pi$
$662$	0	0
$663$	1.00543e15	0.304810
$664$	0	0
$665$	8.62227e14	0.257100
$666$	0	0
$667$	2.39392e15i	0.702130i
$668$	0	0
$669$	1.84839e15i	0.533272i
$670$	0	0
$671$	−3.71902e15	−1.05549
$672$	0	0
$673$	4.74120e15	1.32375	0.661874	−	0.749615i	$-0.269761\pi$
$673$	4.74120e15	1.32375	0.661874	+	0.749615i	$0.269761\pi$
$674$	0	0
$675$	− 1.86856e15i	− 0.513259i
$676$	0	0
$677$	1.41307e15i	0.381880i	0.981602	+	0.190940i	$0.0611535\pi$
$677$	1.41307e15i	0.381880i	−0.981602	+	0.190940i	$0.938846\pi$
$678$	0	0
$679$	−1.25603e15	−0.333976
$680$	0	0
$681$	3.42680e14	0.0896559
$682$	0	0
$683$	− 3.03116e15i	− 0.780359i	−0.920739	−	0.390180i	$-0.872413\pi$
$683$	− 3.03116e15i	− 0.780359i	0.920739	−	0.390180i	$-0.127587\pi$
$684$	0	0
$685$	− 1.43547e15i	− 0.363660i
$686$	0	0
$687$	2.97975e15	0.742879
$688$	0	0
$689$	9.22102e14	0.226242
$690$	0	0
$691$	2.74731e15i	0.663405i	0.943384	+	0.331703i	$0.107623\pi$
$691$	2.74731e15i	0.663405i	−0.943384	+	0.331703i	$0.892377\pi$
$692$	0	0
$693$	− 1.01728e15i	− 0.241773i
$694$	0	0
$695$	2.88251e15	0.674303
$696$	0	0
$697$	2.12786e15	0.489962
$698$	0	0
$699$	− 4.42597e15i	− 1.00319i
$700$	0	0
$701$	5.72747e15i	1.27795i	0.769228	+	0.638974i	$0.220641\pi$
$701$	5.72747e15i	1.27795i	−0.769228	+	0.638974i	$0.779359\pi$
$702$	0	0
$703$	1.94265e15	0.426718
$704$	0	0
$705$	3.27093e15	0.707345
$706$	0	0
$707$	1.36870e15i	0.291409i
$708$	0	0
$709$	− 6.98326e14i	− 0.146388i	−0.997318	−	0.0731938i	$-0.976681\pi$
$709$	− 6.98326e14i	− 0.146388i	0.997318	−	0.0731938i	$-0.0233192\pi$
$710$	0	0
$711$	−4.33171e15	−0.894081
$712$	0	0
$713$	9.85170e14	0.200225
$714$	0	0
$715$	− 1.49182e15i	− 0.298561i
$716$	0	0
$717$	− 1.79917e15i	− 0.354583i
$718$	0	0
$719$	−9.70979e15	−1.88452	−0.942260	−	0.334882i	$-0.891304\pi$
$719$	−9.70979e15	−1.88452	−0.942260	+	0.334882i	$0.891304\pi$
$720$	0	0
$721$	3.78004e15	0.722525
$722$	0	0
$723$	5.82893e13i	0.0109731i
$724$	0	0
$725$	3.27427e15i	0.607093i
$726$	0	0
$727$	2.46469e15	0.450114	0.225057	−	0.974346i	$-0.427743\pi$
$727$	2.46469e15	0.450114	0.225057	+	0.974346i	$0.427743\pi$
$728$	0	0
$729$	−3.08202e15	−0.554413
$730$	0	0
$731$	1.18269e14i	0.0209568i
$732$	0	0
$733$	7.91285e15i	1.38121i	0.723230	+	0.690607i	$0.242657\pi$
$733$	7.91285e15i	1.38121i	−0.723230	+	0.690607i	$0.757343\pi$
$734$	0	0
$735$	2.06548e15	0.355173
$736$	0	0
$737$	−8.27677e15	−1.40213
$738$	0	0
$739$	8.40694e15i	1.40312i	0.712613	+	0.701558i	$0.247512\pi$
$739$	8.40694e15i	1.40312i	−0.712613	+	0.701558i	$0.752488\pi$
$740$	0	0
$741$	1.55220e15i	0.255239i
$742$	0	0
$743$	1.36287e15	0.220809	0.110404	−	0.993887i	$-0.464785\pi$
$743$	1.36287e15	0.220809	0.110404	+	0.993887i	$0.464785\pi$
$744$	0	0
$745$	5.38754e15	0.860065
$746$	0	0
$747$	3.33373e15i	0.524405i
$748$	0	0
$749$	− 1.51100e15i	− 0.234215i
$750$	0	0
$751$	−6.81722e15	−1.04133	−0.520664	−	0.853762i	$-0.674316\pi$
$751$	−6.81722e15	−1.04133	−0.520664	+	0.853762i	$0.674316\pi$
$752$	0	0
$753$	3.27173e15	0.492499
$754$	0	0
$755$	3.98208e15i	0.590747i
$756$	0	0
$757$	6.67049e14i	0.0975282i	0.998810	+	0.0487641i	$0.0155283\pi$
$757$	6.67049e14i	0.0975282i	−0.998810	+	0.0487641i	$0.984472\pi$
$758$	0	0
$759$	2.51166e15	0.361936
$760$	0	0
$761$	7.74408e15	1.09990	0.549951	−	0.835197i	$-0.314646\pi$
$761$	7.74408e15	1.09990	0.549951	+	0.835197i	$0.314646\pi$
$762$	0	0
$763$	− 1.23039e15i	− 0.172250i
$764$	0	0
$765$	3.79064e15i	0.523088i
$766$	0	0
$767$	−2.99800e15	−0.407809
$768$	0	0
$769$	2.52411e15	0.338465	0.169232	−	0.985576i	$-0.445871\pi$
$769$	2.52411e15	0.338465	0.169232	+	0.985576i	$0.445871\pi$
$770$	0	0
$771$	− 6.03822e15i	− 0.798197i
$772$	0	0
$773$	1.11453e16i	1.45246i	0.687453	+	0.726229i	$0.258729\pi$
$773$	1.11453e16i	1.45246i	−0.687453	+	0.726229i	$0.741271\pi$
$774$	0	0
$775$	1.34746e15	0.173124
$776$	0	0
$777$	−7.68847e14	−0.0973922
$778$	0	0
$779$	3.28500e15i	0.410279i
$780$	0	0
$781$	5.23465e15i	0.644624i
$782$	0	0
$783$	9.40952e15	1.14256
$784$	0	0
$785$	6.35201e15	0.760551
$786$	0	0
$787$	− 1.32271e16i	− 1.56172i	−0.624705	−	0.780861i	$-0.714781\pi$
$787$	− 1.32271e16i	− 1.56172i	0.624705	−	0.780861i	$-0.285219\pi$
$788$	0	0
$789$	6.11698e15i	0.712219i
$790$	0	0
$791$	1.42570e15	0.163703
$792$	0	0
$793$	4.01902e15	0.455112
$794$	0	0
$795$	− 1.94266e15i	− 0.216958i
$796$	0	0
$797$	2.30248e15i	0.253615i	0.991927	+	0.126807i	$0.0404730\pi$
$797$	2.30248e15i	0.253615i	−0.991927	+	0.126807i	$0.959527\pi$
$798$	0	0
$799$	1.85587e16	2.01623
$800$	0	0
$801$	2.84027e15	0.304355
$802$	0	0
$803$	− 7.82560e14i	− 0.0827146i
$804$	0	0
$805$	1.50775e15i	0.157199i
$806$	0	0
$807$	6.51110e15	0.669653
$808$	0	0
$809$	−5.60472e15	−0.568639	−0.284320	−	0.958730i	$-0.591768\pi$
$809$	−5.60472e15	−0.568639	−0.284320	+	0.958730i	$0.591768\pi$
$810$	0	0
$811$	− 5.08516e15i	− 0.508968i	−0.967077	−	0.254484i	$-0.918094\pi$
$811$	− 5.08516e15i	− 0.508968i	0.967077	−	0.254484i	$-0.0819056\pi$
$812$	0	0
$813$	− 9.49519e14i	− 0.0937574i
$814$	0	0
$815$	1.72833e15	0.168368
$816$	0	0
$817$	−1.82584e14	−0.0175486
$818$	0	0
$819$	1.09934e15i	0.104249i
$820$	0	0
$821$	− 2.79111e14i	− 0.0261150i	−0.999915	−	0.0130575i	$-0.995844\pi$
$821$	− 2.79111e14i	− 0.0261150i	0.999915	−	0.0130575i	$-0.00415644\pi$
$822$	0	0
$823$	−1.35265e16	−1.24878	−0.624391	−	0.781112i	$-0.714653\pi$
$823$	−1.35265e16	−1.24878	−0.624391	+	0.781112i	$0.714653\pi$
$824$	0	0
$825$	3.43531e15	0.312946
$826$	0	0
$827$	2.72544e14i	0.0244994i	0.999925	+	0.0122497i	$0.00389930\pi$
$827$	2.72544e14i	0.0244994i	−0.999925	+	0.0122497i	$0.996101\pi$
$828$	0	0
$829$	1.80459e16i	1.60077i	0.599486	+	0.800385i	$0.295372\pi$
$829$	1.80459e16i	1.60077i	−0.599486	+	0.800385i	$0.704628\pi$
$830$	0	0
$831$	4.13757e15	0.362193
$832$	0	0
$833$	1.17191e16	1.01239
$834$	0	0
$835$	− 1.33058e16i	− 1.13440i
$836$	0	0
$837$	− 3.87230e15i	− 0.325821i
$838$	0	0
$839$	−7.96183e15	−0.661184	−0.330592	−	0.943774i	$-0.607248\pi$
$839$	−7.96183e15	−0.661184	−0.330592	+	0.943774i	$0.607248\pi$
$840$	0	0
$841$	−4.28775e15	−0.351440
$842$	0	0
$843$	5.30100e15i	0.428851i
$844$	0	0
$845$	− 7.04397e15i	− 0.562478i
$846$	0	0
$847$	8.34387e12	0.000657671 0
$848$	0	0
$849$	4.21172e15	0.327693
$850$	0	0
$851$	3.39705e15i	0.260909i
$852$	0	0
$853$	1.49826e16i	1.13598i	0.823037	+	0.567988i	$0.192278\pi$
$853$	1.49826e16i	1.13598i	−0.823037	+	0.567988i	$0.807722\pi$
$854$	0	0
$855$	−5.85201e15	−0.438017
$856$	0	0
$857$	2.22561e16	1.64458	0.822290	−	0.569068i	$-0.192696\pi$
$857$	2.22561e16	1.64458	0.822290	+	0.569068i	$0.192696\pi$
$858$	0	0
$859$	5.44237e15i	0.397032i	0.980098	+	0.198516i	$0.0636122\pi$
$859$	5.44237e15i	0.397032i	−0.980098	+	0.198516i	$0.936388\pi$
$860$	0	0
$861$	− 1.30011e15i	− 0.0936403i
$862$	0	0
$863$	−1.08110e16	−0.768787	−0.384393	−	0.923169i	$-0.625589\pi$
$863$	−1.08110e16	−0.768787	−0.384393	+	0.923169i	$0.625589\pi$
$864$	0	0
$865$	−4.59037e15	−0.322299
$866$	0	0
$867$	− 3.38185e15i	− 0.234449i
$868$	0	0
$869$	− 2.03777e16i	− 1.39491i
$870$	0	0
$871$	8.94444e15	0.604579
$872$	0	0
$873$	8.52477e15	0.568989
$874$	0	0
$875$	6.01111e15i	0.396197i
$876$	0	0
$877$	− 2.81024e16i	− 1.82914i	−0.404431	−	0.914568i	$-0.632530\pi$
$877$	− 2.81024e16i	− 1.82914i	0.404431	−	0.914568i	$-0.367470\pi$
$878$	0	0
$879$	6.02957e15	0.387568
$880$	0	0
$881$	4.22209e15	0.268016	0.134008	−	0.990980i	$-0.457215\pi$
$881$	4.22209e15	0.268016	0.134008	+	0.990980i	$0.457215\pi$
$882$	0	0
$883$	− 5.16092e14i	− 0.0323551i	−0.999869	−	0.0161776i	$-0.994850\pi$
$883$	− 5.16092e14i	− 0.0323551i	0.999869	−	0.0161776i	$-0.00514970\pi$
$884$	0	0
$885$	6.31609e15i	0.391075i
$886$	0	0
$887$	5.71906e15	0.349740	0.174870	−	0.984592i	$-0.444050\pi$
$887$	5.71906e15	0.349740	0.174870	+	0.984592i	$0.444050\pi$
$888$	0	0
$889$	−4.39894e15	−0.265698
$890$	0	0
$891$	8.90230e14i	0.0531098i
$892$	0	0
$893$	2.86510e16i	1.68832i
$894$	0	0
$895$	−8.12109e15	−0.472702
$896$	0	0
$897$	−2.71427e15	−0.156061
$898$	0	0
$899$	6.78541e15i	0.385388i
$900$	0	0
$901$	− 1.10223e16i	− 0.618421i
$902$	0	0
$903$	7.22617e13	0.00400521
$904$	0	0
$905$	4.81442e15	0.263619
$906$	0	0
$907$	− 8.43778e13i	− 0.00456445i	−0.999997	−	0.00228222i	$-0.999274\pi$
$907$	− 8.43778e13i	− 0.00456445i	0.999997	−	0.00228222i	$-0.000726455\pi$
$908$	0	0
$909$	− 9.28952e15i	− 0.496468i
$910$	0	0
$911$	1.10091e16	0.581298	0.290649	−	0.956830i	$-0.406129\pi$
$911$	1.10091e16	0.581298	0.290649	+	0.956830i	$0.406129\pi$
$912$	0	0
$913$	−1.56829e16	−0.818158
$914$	0	0
$915$	− 8.46715e15i	− 0.436437i
$916$	0	0
$917$	1.05743e16i	0.538544i
$918$	0	0
$919$	−4.86351e15	−0.244746	−0.122373	−	0.992484i	$-0.539050\pi$
$919$	−4.86351e15	−0.244746	−0.122373	+	0.992484i	$0.539050\pi$
$920$	0	0
$921$	−3.85840e15	−0.191857
$922$	0	0
$923$	− 5.65691e15i	− 0.277952i
$924$	0	0
$925$	4.64630e15i	0.225594i
$926$	0	0
$927$	−2.56555e16	−1.23095
$928$	0	0
$929$	3.57534e15	0.169524	0.0847620	−	0.996401i	$-0.472987\pi$
$929$	3.57534e15	0.169524	0.0847620	+	0.996401i	$0.472987\pi$
$930$	0	0
$931$	1.80921e16i	0.847744i
$932$	0	0
$933$	− 1.25685e16i	− 0.582017i
$934$	0	0
$935$	−1.78323e16	−0.816102
$936$	0	0
$937$	−3.86373e16	−1.74759	−0.873795	−	0.486295i	$-0.838348\pi$
$937$	−3.86373e16	−1.74759	−0.873795	+	0.486295i	$0.838348\pi$
$938$	0	0
$939$	− 2.50692e16i	− 1.12067i
$940$	0	0
$941$	− 3.48997e16i	− 1.54198i	−0.636846	−	0.770991i	$-0.719761\pi$
$941$	− 3.48997e16i	− 1.54198i	0.636846	−	0.770991i	$-0.280239\pi$
$942$	0	0
$943$	−5.74437e15	−0.250858
$944$	0	0
$945$	5.92634e15	0.255806
$946$	0	0
$947$	2.85123e16i	1.21649i	0.793751	+	0.608243i	$0.208125\pi$
$947$	2.85123e16i	1.21649i	−0.793751	+	0.608243i	$0.791875\pi$
$948$	0	0
$949$	8.45688e14i	0.0356653i
$950$	0	0
$951$	2.10091e16	0.875817
$952$	0	0
$953$	−4.00334e16	−1.64973	−0.824863	−	0.565332i	$-0.808748\pi$
$953$	−4.00334e16	−1.64973	−0.824863	+	0.565332i	$0.808748\pi$
$954$	0	0
$955$	1.33424e16i	0.543520i
$956$	0	0
$957$	1.72992e16i	0.696644i
$958$	0	0
$959$	−4.97630e15	−0.198109
$960$	0	0
$961$	−2.26161e16	−0.890100
$962$	0	0
$963$	1.02553e16i	0.399028i
$964$	0	0
$965$	− 2.62867e16i	− 1.01120i
$966$	0	0
$967$	1.84953e16	0.703422	0.351711	−	0.936109i	$-0.385600\pi$
$967$	1.84953e16	0.703422	0.351711	+	0.936109i	$0.385600\pi$
$968$	0	0
$969$	1.85540e16	0.697682
$970$	0	0
$971$	− 2.14877e16i	− 0.798884i	−0.916759	−	0.399442i	$-0.869204\pi$
$971$	− 2.14877e16i	− 0.798884i	0.916759	−	0.399442i	$-0.130796\pi$
$972$	0	0
$973$	− 9.99271e15i	− 0.367335i
$974$	0	0
$975$	−3.71243e15	−0.134938
$976$	0	0
$977$	−8.73880e15	−0.314074	−0.157037	−	0.987593i	$-0.550194\pi$
$977$	−8.73880e15	−0.314074	−0.157037	+	0.987593i	$0.550194\pi$
$978$	0	0
$979$	1.33615e16i	0.474844i
$980$	0	0
$981$	8.35079e15i	0.293459i
$982$	0	0
$983$	1.18924e16	0.413263	0.206631	−	0.978419i	$-0.433750\pi$
$983$	1.18924e16	0.413263	0.206631	+	0.978419i	$0.433750\pi$
$984$	0	0
$985$	1.38915e16	0.477365
$986$	0	0
$987$	− 1.13392e16i	− 0.385336i
$988$	0	0
$989$	− 3.19279e14i	− 0.0107298i
$990$	0	0
$991$	−2.34409e16	−0.779056	−0.389528	−	0.921015i	$-0.627362\pi$
$991$	−2.34409e16	−0.779056	−0.389528	+	0.921015i	$0.627362\pi$
$992$	0	0
$993$	−1.60232e16	−0.526657
$994$	0	0
$995$	− 3.51813e15i	− 0.114363i
$996$	0	0
$997$	2.14004e16i	0.688016i	0.938967	+	0.344008i	$0.111785\pi$
$997$	2.14004e16i	0.688016i	−0.938967	+	0.344008i	$0.888215\pi$
$998$	0	0
$999$	1.33524e16	0.424571

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.12.b.c.129.1		2
4.3	odd	2		256.12.b.e.129.2		2
8.3	odd	2		256.12.b.e.129.1		2
8.5	even	2	inner	256.12.b.c.129.2		2
16.3	odd	4		64.12.a.b.1.1		1
16.5	even	4		16.12.a.a.1.1		1
16.11	odd	4		1.12.a.a.1.1	✓	1
16.13	even	4		64.12.a.f.1.1		1
48.5	odd	4		144.12.a.d.1.1		1
48.11	even	4		9.12.a.b.1.1		1
80.27	even	4		25.12.b.b.24.1		2
80.43	even	4		25.12.b.b.24.2		2
80.59	odd	4		25.12.a.b.1.1		1
112.11	odd	12		49.12.c.b.30.1		2
112.27	even	4		49.12.a.a.1.1		1
112.59	even	12		49.12.c.c.30.1		2
112.75	even	12		49.12.c.c.18.1		2
112.107	odd	12		49.12.c.b.18.1		2
144.11	even	12		81.12.c.b.28.1		2
144.43	odd	12		81.12.c.d.28.1		2
144.59	even	12		81.12.c.b.55.1		2
144.139	odd	12		81.12.c.d.55.1		2
176.43	even	4		121.12.a.b.1.1		1
208.155	odd	4		169.12.a.a.1.1		1
240.59	even	4		225.12.a.b.1.1		1
240.107	odd	4		225.12.b.d.199.2		2
240.203	odd	4		225.12.b.d.199.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.12.a.a.1.1	✓	1	16.11	odd	4
9.12.a.b.1.1		1	48.11	even	4
16.12.a.a.1.1		1	16.5	even	4
25.12.a.b.1.1		1	80.59	odd	4
25.12.b.b.24.1		2	80.27	even	4
25.12.b.b.24.2		2	80.43	even	4
49.12.a.a.1.1		1	112.27	even	4
49.12.c.b.18.1		2	112.107	odd	12
49.12.c.b.30.1		2	112.11	odd	12
49.12.c.c.18.1		2	112.75	even	12
49.12.c.c.30.1		2	112.59	even	12
64.12.a.b.1.1		1	16.3	odd	4
64.12.a.f.1.1		1	16.13	even	4
81.12.c.b.28.1		2	144.11	even	12
81.12.c.b.55.1		2	144.59	even	12
81.12.c.d.28.1		2	144.43	odd	12
81.12.c.d.55.1		2	144.139	odd	12
121.12.a.b.1.1		1	176.43	even	4
144.12.a.d.1.1		1	48.5	odd	4
169.12.a.a.1.1		1	208.155	odd	4
225.12.a.b.1.1		1	240.59	even	4
225.12.b.d.199.1		2	240.203	odd	4
225.12.b.d.199.2		2	240.107	odd	4
256.12.b.c.129.1		2	1.1	even	1	trivial
256.12.b.c.129.2		2	8.5	even	2	inner
256.12.b.e.129.1		2	8.3	odd	2
256.12.b.e.129.2		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-252.000i q^{3} -4830.00i q^{5} -16744.0 q^{7} +113643. q^{9} +O(q^{10})\)	\(q-252.000i q^{3} -4830.00i q^{5} -16744.0 q^{7} +113643. q^{9} +534612. i q^{11} -577738. i q^{13} -1.21716e6 q^{15} -6.90593e6 q^{17} -1.06614e7i q^{19} +4.21949e6i q^{21} +1.86433e7 q^{23} +2.54992e7 q^{25} -7.32791e7i q^{27} +1.28407e8i q^{29} +5.28432e7 q^{31} +1.34722e8 q^{33} +8.08735e7i q^{35} +1.82213e8i q^{37} -1.45590e8 q^{39} -3.08120e8 q^{41} -1.71257e7i q^{43} -5.48896e8i q^{45} -2.68735e9 q^{47} -1.69697e9 q^{49} +1.74030e9i q^{51} +1.59606e9i q^{53} +2.58218e9 q^{55} -2.68668e9 q^{57} -5.18920e9i q^{59} +6.95648e9i q^{61} -1.90284e9 q^{63} -2.79047e9 q^{65} +1.54818e10i q^{67} -4.69810e9i q^{69} +9.79149e9 q^{71} -1.46379e9 q^{73} -6.42580e9i q^{75} -8.95154e9i q^{77} -3.81168e10 q^{79} +1.66519e9 q^{81} +2.93351e10i q^{83} +3.33557e10i q^{85} +3.23585e10 q^{87} +2.49929e10 q^{89} +9.67365e9i q^{91} -1.33165e10i q^{93} -5.14947e10 q^{95} +7.50136e10 q^{97} +6.07549e10i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 33488 q^{7} + 227286 q^{9}+O(q^{10}) \) 2 * q - 33488 * q^7 + 227286 * q^9	\( 2 q - 33488 q^{7} + 227286 q^{9} - 2434320 q^{15} - 13811868 q^{17} + 37286544 q^{23} + 50998450 q^{25} + 105686336 q^{31} + 269444448 q^{33} - 291179952 q^{39} - 616240884 q^{41} - 5374696992 q^{47} - 3393930414 q^{49} + 5164351920 q^{55} - 5373355680 q^{57} - 3805676784 q^{63} - 5580949080 q^{65} + 19582970544 q^{71} - 2927582644 q^{73} - 76233691360 q^{79} + 3330376722 q^{81} + 64716941520 q^{87} + 49985834220 q^{89} - 102989317200 q^{95} + 150027137092 q^{97}+O(q^{100}) \) 2 * q - 33488 * q^7 + 227286 * q^9 - 2434320 * q^15 - 13811868 * q^17 + 37286544 * q^23 + 50998450 * q^25 + 105686336 * q^31 + 269444448 * q^33 - 291179952 * q^39 - 616240884 * q^41 - 5374696992 * q^47 - 3393930414 * q^49 + 5164351920 * q^55 - 5373355680 * q^57 - 3805676784 * q^63 - 5580949080 * q^65 + 19582970544 * q^71 - 2927582644 * q^73 - 76233691360 * q^79 + 3330376722 * q^81 + 64716941520 * q^87 + 49985834220 * q^89 - 102989317200 * q^95 + 150027137092 * q^97

Introduction

L-functions

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