LMFDB - Embedded newform 256.7.d.f.127.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,7,Mod(127,256)]

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, 1]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("256.127");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	256.d (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:	$58.8938454067$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{15})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 16 $ x^4 - 7*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 4)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.1
Root			$1.93649 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	256.127
Dual form			256.7.d.f.127.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-30.9839 q^{3} -10.0000i q^{5} +309.839i q^{7} +231.000 q^{9} +O(q^{10})$	$q-30.9839 q^{3} -10.0000i q^{5} +309.839i q^{7} +231.000 q^{9} +960.500 q^{11} +1466.00i q^{13} +309.839i q^{15} -4766.00 q^{17} +7529.08 q^{19} -9600.00i q^{21} -10472.5i q^{23} +15525.0 q^{25} +15430.0 q^{27} +25498.0i q^{29} -41890.2i q^{31} -29760.0 q^{33} +3098.39 q^{35} -1994.00i q^{37} -45422.3i q^{39} -29362.0 q^{41} +21533.8 q^{43} -2310.00i q^{45} +7560.06i q^{47} +21649.0 q^{49} +147669. q^{51} +192854. i q^{53} -9605.00i q^{55} -233280. q^{57} +78420.2 q^{59} -10918.0i q^{61} +71572.7i q^{63} +14660.0 q^{65} -394146. q^{67} +324480. i q^{69} +532241. i q^{71} -288626. q^{73} -481025. q^{75} +297600. i q^{77} +310706. i q^{79} -646479. q^{81} -204153. q^{83} +47660.0i q^{85} -790027. i q^{87} -310738. q^{89} -454223. q^{91} +1.29792e6i q^{93} -75290.8i q^{95} -1.45709e6 q^{97} +221875. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 924 q^{9}+O(q^{10}) $ 4 * q + 924 * q^9	$ 4 q + 924 q^{9} - 19064 q^{17} + 62100 q^{25} - 119040 q^{33} - 117448 q^{41} + 86596 q^{49} - 933120 q^{57} + 58640 q^{65} - 1154504 q^{73} - 2585916 q^{81} - 1242952 q^{89} - 5828344 q^{97}+O(q^{100}) $ 4 * q + 924 * q^9 - 19064 * q^17 + 62100 * q^25 - 119040 * q^33 - 117448 * q^41 + 86596 * q^49 - 933120 * q^57 + 58640 * q^65 - 1154504 * q^73 - 2585916 * q^81 - 1242952 * q^89 - 5828344 * 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
$3$	−30.9839	−1.14755	−0.573775	−	0.819013i	$-0.694522\pi$
$3$	−30.9839	−1.14755	−0.573775	+	0.819013i	$0.694522\pi$
$4$	0	0
$5$	− 10.0000i	− 0.0800000i	−0.999200	−	0.0400000i	$-0.987264\pi$
$5$	− 10.0000i	− 0.0800000i	0.999200	−	0.0400000i	$-0.0127358\pi$
$6$	0	0
$7$	309.839i	0.903320i	0.892190	+	0.451660i	$0.149168\pi$
$7$	309.839i	0.903320i	−0.892190	+	0.451660i	$0.850832\pi$
$8$	0	0
$9$	231.000	0.316872
$10$	0	0
$11$	960.500	0.721638	0.360819	−	0.932636i	$-0.382497\pi$
$11$	960.500	0.721638	0.360819	+	0.932636i	$0.382497\pi$
$12$	0	0
$13$	1466.00i	0.667274i	0.942702	+	0.333637i	$0.108276\pi$
$13$	1466.00i	0.667274i	−0.942702	+	0.333637i	$0.891724\pi$
$14$	0	0
$15$	309.839i	0.0918040i
$16$	0	0
$17$	−4766.00	−0.970079	−0.485040	−	0.874492i	$-0.661195\pi$
$17$	−4766.00	−0.970079	−0.485040	+	0.874492i	$0.661195\pi$
$18$	0	0
$19$	7529.08	1.09769	0.548847	−	0.835923i	$-0.315067\pi$
$19$	7529.08	1.09769	0.548847	+	0.835923i	$0.315067\pi$
$20$	0	0
$21$	− 9600.00i	− 1.03661i
$22$	0	0
$23$	− 10472.5i	− 0.860734i	−0.902654	−	0.430367i	$-0.858384\pi$
$23$	− 10472.5i	− 0.860734i	0.902654	−	0.430367i	$-0.141616\pi$
$24$	0	0
$25$	15525.0	0.993600
$26$	0	0
$27$	15430.0	0.783923
$28$	0	0
$29$	25498.0i	1.04547i	0.852495	+	0.522736i	$0.175089\pi$
$29$	25498.0i	1.04547i	−0.852495	+	0.522736i	$0.824911\pi$
$30$	0	0
$31$	− 41890.2i	− 1.40614i	−0.711123	−	0.703068i	$-0.751813\pi$
$31$	− 41890.2i	− 1.40614i	0.711123	−	0.703068i	$-0.248187\pi$
$32$	0	0
$33$	−29760.0	−0.828116
$34$	0	0
$35$	3098.39	0.0722656
$36$	0	0
$37$	− 1994.00i	− 0.0393659i	−0.999806	−	0.0196829i	$-0.993734\pi$
$37$	− 1994.00i	− 0.0393659i	0.999806	−	0.0196829i	$-0.00626568\pi$
$38$	0	0
$39$	− 45422.3i	− 0.765730i
$40$	0	0
$41$	−29362.0	−0.426024	−0.213012	−	0.977050i	$-0.568327\pi$
$41$	−29362.0	−0.426024	−0.213012	+	0.977050i	$0.568327\pi$
$42$	0	0
$43$	21533.8	0.270841	0.135421	−	0.990788i	$-0.456761\pi$
$43$	21533.8	0.270841	0.135421	+	0.990788i	$0.456761\pi$
$44$	0	0
$45$	− 2310.00i	− 0.0253498i
$46$	0	0
$47$	7560.06i	0.0728168i	0.999337	+	0.0364084i	$0.0115917\pi$
$47$	7560.06i	0.0728168i	−0.999337	+	0.0364084i	$0.988408\pi$
$48$	0	0
$49$	21649.0	0.184013
$50$	0	0
$51$	147669.	1.11322
$52$	0	0
$53$	192854.i	1.29539i	0.761899	+	0.647696i	$0.224267\pi$
$53$	192854.i	1.29539i	−0.761899	+	0.647696i	$0.775733\pi$
$54$	0	0
$55$	− 9605.00i	− 0.0577310i
$56$	0	0
$57$	−233280.	−1.25966
$58$	0	0
$59$	78420.2	0.381831	0.190916	−	0.981606i	$-0.438854\pi$
$59$	78420.2	0.381831	0.190916	+	0.981606i	$0.438854\pi$
$60$	0	0
$61$	− 10918.0i	− 0.0481009i	−0.999711	−	0.0240505i	$-0.992344\pi$
$61$	− 10918.0i	− 0.0481009i	0.999711	−	0.0240505i	$-0.00765624\pi$
$62$	0	0
$63$	71572.7i	0.286237i
$64$	0	0
$65$	14660.0	0.0533819
$66$	0	0
$67$	−394146.	−1.31049	−0.655243	−	0.755418i	$-0.727434\pi$
$67$	−394146.	−1.31049	−0.655243	+	0.755418i	$0.727434\pi$
$68$	0	0
$69$	324480.i	0.987735i
$70$	0	0
$71$	532241.i	1.48708i	0.668694	+	0.743538i	$0.266854\pi$
$71$	532241.i	1.48708i	−0.668694	+	0.743538i	$0.733146\pi$
$72$	0	0
$73$	−288626.	−0.741937	−0.370968	−	0.928646i	$-0.620974\pi$
$73$	−288626.	−0.741937	−0.370968	+	0.928646i	$0.620974\pi$
$74$	0	0
$75$	−481025.	−1.14021
$76$	0	0
$77$	297600.i	0.651870i
$78$	0	0
$79$	310706.i	0.630186i	0.949061	+	0.315093i	$0.102036\pi$
$79$	310706.i	0.630186i	−0.949061	+	0.315093i	$0.897964\pi$
$80$	0	0
$81$	−646479.	−1.21646
$82$	0	0
$83$	−204153.	−0.357043	−0.178522	−	0.983936i	$-0.557131\pi$
$83$	−204153.	−0.357043	−0.178522	+	0.983936i	$0.557131\pi$
$84$	0	0
$85$	47660.0i	0.0776064i
$86$	0	0
$87$	− 790027.i	− 1.19973i
$88$	0	0
$89$	−310738.	−0.440783	−0.220391	−	0.975412i	$-0.570733\pi$
$89$	−310738.	−0.440783	−0.220391	+	0.975412i	$0.570733\pi$
$90$	0	0
$91$	−454223.	−0.602761
$92$	0	0
$93$	1.29792e6i	1.61361i
$94$	0	0
$95$	− 75290.8i	− 0.0878155i
$96$	0	0
$97$	−1.45709e6	−1.59650	−0.798252	−	0.602324i	$-0.794242\pi$
$97$	−1.45709e6	−1.59650	−0.798252	+	0.602324i	$0.794242\pi$
$98$	0	0
$99$	221875.	0.228667
$100$	0	0
$101$	639158.i	0.620360i	0.950678	+	0.310180i	$0.100389\pi$
$101$	639158.i	0.620360i	−0.950678	+	0.310180i	$0.899611\pi$
$102$	0	0
$103$	1.38913e6i	1.27125i	0.771997	+	0.635626i	$0.219258\pi$
$103$	1.38913e6i	1.27125i	−0.771997	+	0.635626i	$0.780742\pi$
$104$	0	0
$105$	−96000.0	−0.0829284
$106$	0	0
$107$	1.14935e6	0.938209	0.469105	−	0.883143i	$-0.344577\pi$
$107$	1.14935e6	0.938209	0.469105	+	0.883143i	$0.344577\pi$
$108$	0	0
$109$	1.53574e6i	1.18587i	0.805250	+	0.592936i	$0.202031\pi$
$109$	1.53574e6i	1.18587i	−0.805250	+	0.592936i	$0.797969\pi$
$110$	0	0
$111$	61781.8i	0.0451743i
$112$	0	0
$113$	−601694.	−0.417004	−0.208502	−	0.978022i	$-0.566859\pi$
$113$	−601694.	−0.417004	−0.208502	+	0.978022i	$0.566859\pi$
$114$	0	0
$115$	−104725.	−0.0688587
$116$	0	0
$117$	338646.i	0.211441i
$118$	0	0
$119$	− 1.47669e6i	− 0.876292i
$120$	0	0
$121$	−849001.	−0.479239
$122$	0	0
$123$	909748.	0.488884
$124$	0	0
$125$	− 311500.i	− 0.159488i
$126$	0	0
$127$	− 1.67462e6i	− 0.817531i	−0.912640	−	0.408765i	$-0.865959\pi$
$127$	− 1.67462e6i	− 0.817531i	0.912640	−	0.408765i	$-0.134041\pi$
$128$	0	0
$129$	−667200.	−0.310804
$130$	0	0
$131$	2.84454e6	1.26531	0.632656	−	0.774433i	$-0.281965\pi$
$131$	2.84454e6	1.26531	0.632656	+	0.774433i	$0.281965\pi$
$132$	0	0
$133$	2.33280e6i	0.991568i
$134$	0	0
$135$	− 154300.i	− 0.0627139i
$136$	0	0
$137$	−3.81003e6	−1.48172	−0.740862	−	0.671658i	$-0.765583\pi$
$137$	−3.81003e6	−1.48172	−0.740862	+	0.671658i	$0.765583\pi$
$138$	0	0
$139$	−138839.	−0.0516971	−0.0258485	−	0.999666i	$-0.508229\pi$
$139$	−138839.	−0.0516971	−0.0258485	+	0.999666i	$0.508229\pi$
$140$	0	0
$141$	− 234240.i	− 0.0835610i
$142$	0	0
$143$	1.40809e6i	0.481530i
$144$	0	0
$145$	254980.	0.0836377
$146$	0	0
$147$	−670770.	−0.211165
$148$	0	0
$149$	3.27426e6i	0.989816i	0.868945	+	0.494908i	$0.164798\pi$
$149$	3.27426e6i	0.989816i	−0.868945	+	0.494908i	$0.835202\pi$
$150$	0	0
$151$	− 5.59352e6i	− 1.62463i	−0.583220	−	0.812314i	$-0.698207\pi$
$151$	− 5.59352e6i	− 1.62463i	0.583220	−	0.812314i	$-0.301793\pi$
$152$	0	0
$153$	−1.10095e6	−0.307391
$154$	0	0
$155$	−418902.	−0.112491
$156$	0	0
$157$	816794.i	0.211064i	0.994416	+	0.105532i	$0.0336545\pi$
$157$	816794.i	0.211064i	−0.994416	+	0.105532i	$0.966345\pi$
$158$	0	0
$159$	− 5.97536e6i	− 1.48653i
$160$	0	0
$161$	3.24480e6	0.777518
$162$	0	0
$163$	−1.84593e6	−0.426237	−0.213119	−	0.977026i	$-0.568362\pi$
$163$	−1.84593e6	−0.426237	−0.213119	+	0.977026i	$0.568362\pi$
$164$	0	0
$165$	297600.i	0.0662493i
$166$	0	0
$167$	7.96515e6i	1.71019i	0.518471	+	0.855095i	$0.326501\pi$
$167$	7.96515e6i	1.71019i	−0.518471	+	0.855095i	$0.673499\pi$
$168$	0	0
$169$	2.67765e6	0.554746
$170$	0	0
$171$	1.73922e6	0.347829
$172$	0	0
$173$	− 5.12653e6i	− 0.990115i	−0.868860	−	0.495057i	$-0.835147\pi$
$173$	− 5.12653e6i	− 0.990115i	0.868860	−	0.495057i	$-0.164853\pi$
$174$	0	0
$175$	4.81025e6i	0.897538i
$176$	0	0
$177$	−2.42976e6	−0.438171
$178$	0	0
$179$	−2.33411e6	−0.406969	−0.203485	−	0.979078i	$-0.565227\pi$
$179$	−2.33411e6	−0.406969	−0.203485	+	0.979078i	$0.565227\pi$
$180$	0	0
$181$	− 9.69156e6i	− 1.63440i	−0.576355	−	0.817199i	$-0.695525\pi$
$181$	− 9.69156e6i	− 1.63440i	0.576355	−	0.817199i	$-0.304475\pi$
$182$	0	0
$183$	338282.i	0.0551983i
$184$	0	0
$185$	−19940.0	−0.00314927
$186$	0	0
$187$	−4.57774e6	−0.700046
$188$	0	0
$189$	4.78080e6i	0.708134i
$190$	0	0
$191$	1.14164e7i	1.63844i	0.573479	+	0.819220i	$0.305593\pi$
$191$	1.14164e7i	1.63844i	−0.573479	+	0.819220i	$0.694407\pi$
$192$	0	0
$193$	−2.43033e6	−0.338060	−0.169030	−	0.985611i	$-0.554064\pi$
$193$	−2.43033e6	−0.338060	−0.169030	+	0.985611i	$0.554064\pi$
$194$	0	0
$195$	−454223.	−0.0612584
$196$	0	0
$197$	2.23065e6i	0.291764i	0.989302	+	0.145882i	$0.0466020\pi$
$197$	2.23065e6i	0.291764i	−0.989302	+	0.145882i	$0.953398\pi$
$198$	0	0
$199$	4.89576e6i	0.621242i	0.950534	+	0.310621i	$0.100537\pi$
$199$	4.89576e6i	0.621242i	−0.950534	+	0.310621i	$0.899463\pi$
$200$	0	0
$201$	1.22122e7	1.50385
$202$	0	0
$203$	−7.90027e6	−0.944395
$204$	0	0
$205$	293620.i	0.0340819i
$206$	0	0
$207$	− 2.41916e6i	− 0.272743i
$208$	0	0
$209$	7.23168e6	0.792137
$210$	0	0
$211$	3.90951e6	0.416174	0.208087	−	0.978110i	$-0.433276\pi$
$211$	3.90951e6	0.416174	0.208087	+	0.978110i	$0.433276\pi$
$212$	0	0
$213$	− 1.64909e7i	− 1.70650i
$214$	0	0
$215$	− 215338.i	− 0.0216673i
$216$	0	0
$217$	1.29792e7	1.27019
$218$	0	0
$219$	8.94275e6	0.851410
$220$	0	0
$221$	− 6.98696e6i	− 0.647308i
$222$	0	0
$223$	− 3.33114e6i	− 0.300385i	−0.988657	−	0.150192i	$-0.952011\pi$
$223$	− 3.33114e6i	− 0.300385i	0.988657	−	0.150192i	$-0.0479893\pi$
$224$	0	0
$225$	3.58628e6	0.314844
$226$	0	0
$227$	−1.35033e7	−1.15442	−0.577208	−	0.816597i	$-0.695858\pi$
$227$	−1.35033e7	−1.15442	−0.577208	+	0.816597i	$0.695858\pi$
$228$	0	0
$229$	− 1.59598e6i	− 0.132899i	−0.997790	−	0.0664493i	$-0.978833\pi$
$229$	− 1.59598e6i	− 0.132899i	0.997790	−	0.0664493i	$-0.0211671\pi$
$230$	0	0
$231$	− 9.22080e6i	− 0.748053i
$232$	0	0
$233$	−8.04383e6	−0.635909	−0.317954	−	0.948106i	$-0.602996\pi$
$233$	−8.04383e6	−0.635909	−0.317954	+	0.948106i	$0.602996\pi$
$234$	0	0
$235$	75600.6	0.00582535
$236$	0	0
$237$	− 9.62688e6i	− 0.723170i
$238$	0	0
$239$	− 1.12532e7i	− 0.824296i	−0.911117	−	0.412148i	$-0.864779\pi$
$239$	− 1.12532e7i	− 0.824296i	0.911117	−	0.412148i	$-0.135221\pi$
$240$	0	0
$241$	5.05104e6	0.360853	0.180426	−	0.983589i	$-0.442252\pi$
$241$	5.05104e6	0.360853	0.180426	+	0.983589i	$0.442252\pi$
$242$	0	0
$243$	8.78197e6	0.612031
$244$	0	0
$245$	− 216490.i	− 0.0147211i
$246$	0	0
$247$	1.10376e7i	0.732462i
$248$	0	0
$249$	6.32544e6	0.409725
$250$	0	0
$251$	−4.71590e6	−0.298225	−0.149112	−	0.988820i	$-0.547642\pi$
$251$	−4.71590e6	−0.298225	−0.149112	+	0.988820i	$0.547642\pi$
$252$	0	0
$253$	− 1.00589e7i	− 0.621138i
$254$	0	0
$255$	− 1.47669e6i	− 0.0890572i
$256$	0	0
$257$	2.34552e7	1.38178	0.690892	−	0.722958i	$-0.257218\pi$
$257$	2.34552e7	1.38178	0.690892	+	0.722958i	$0.257218\pi$
$258$	0	0
$259$	617818.	0.0355600
$260$	0	0
$261$	5.89004e6i	0.331281i
$262$	0	0
$263$	− 2.16993e7i	− 1.19283i	−0.802676	−	0.596415i	$-0.796591\pi$
$263$	− 2.16993e7i	− 1.19283i	0.802676	−	0.596415i	$-0.203409\pi$
$264$	0	0
$265$	1.92854e6	0.103631
$266$	0	0
$267$	9.62786e6	0.505820
$268$	0	0
$269$	− 2.94278e7i	− 1.51182i	−0.654674	−	0.755911i	$-0.727194\pi$
$269$	− 2.94278e7i	− 1.51182i	0.654674	−	0.755911i	$-0.272806\pi$
$270$	0	0
$271$	8.51474e6i	0.427822i	0.976853	+	0.213911i	$0.0686203\pi$
$271$	8.51474e6i	0.427822i	−0.976853	+	0.213911i	$0.931380\pi$
$272$	0	0
$273$	1.40736e7	0.691699
$274$	0	0
$275$	1.49118e7	0.717019
$276$	0	0
$277$	− 2.76226e7i	− 1.29965i	−0.760085	−	0.649824i	$-0.774843\pi$
$277$	− 2.76226e7i	− 1.29965i	0.760085	−	0.649824i	$-0.225157\pi$
$278$	0	0
$279$	− 9.67663e6i	− 0.445566i
$280$	0	0
$281$	−8.64008e6	−0.389403	−0.194701	−	0.980863i	$-0.562374\pi$
$281$	−8.64008e6	−0.389403	−0.194701	+	0.980863i	$0.562374\pi$
$282$	0	0
$283$	1.27350e7	0.561873	0.280937	−	0.959726i	$-0.409355\pi$
$283$	1.27350e7	0.561873	0.280937	+	0.959726i	$0.409355\pi$
$284$	0	0
$285$	2.33280e6i	0.100773i
$286$	0	0
$287$	− 9.09748e6i	− 0.384836i
$288$	0	0
$289$	−1.42281e6	−0.0589460
$290$	0	0
$291$	4.51462e7	1.83207
$292$	0	0
$293$	4.45415e7i	1.77077i	0.464859	+	0.885385i	$0.346105\pi$
$293$	4.45415e7i	1.77077i	−0.464859	+	0.885385i	$0.653895\pi$
$294$	0	0
$295$	− 784202.i	− 0.0305465i
$296$	0	0
$297$	1.48205e7	0.565709
$298$	0	0
$299$	1.53528e7	0.574345
$300$	0	0
$301$	6.67200e6i	0.244656i
$302$	0	0
$303$	− 1.98036e7i	− 0.711895i
$304$	0	0
$305$	−109180.	−0.00384808
$306$	0	0
$307$	−4.89051e7	−1.69020	−0.845102	−	0.534606i	$-0.820460\pi$
$307$	−4.89051e7	−1.69020	−0.845102	+	0.534606i	$0.820460\pi$
$308$	0	0
$309$	− 4.30406e7i	− 1.45883i
$310$	0	0
$311$	5.00220e7i	1.66295i	0.555559	+	0.831477i	$0.312504\pi$
$311$	5.00220e7i	1.66295i	−0.555559	+	0.831477i	$0.687496\pi$
$312$	0	0
$313$	−1.12719e6	−0.0367589	−0.0183795	−	0.999831i	$-0.505851\pi$
$313$	−1.12719e6	−0.0367589	−0.0183795	+	0.999831i	$0.505851\pi$
$314$	0	0
$315$	715727.	0.0228990
$316$	0	0
$317$	3.44882e7i	1.08266i	0.840810	+	0.541330i	$0.182079\pi$
$317$	3.44882e7i	1.08266i	−0.840810	+	0.541330i	$0.817921\pi$
$318$	0	0
$319$	2.44908e7i	0.754452i
$320$	0	0
$321$	−3.56112e7	−1.07664
$322$	0	0
$323$	−3.58836e7	−1.06485
$324$	0	0
$325$	2.27596e7i	0.663003i
$326$	0	0
$327$	− 4.75831e7i	− 1.36085i
$328$	0	0
$329$	−2.34240e6	−0.0657769
$330$	0	0
$331$	−4.02696e7	−1.11044	−0.555218	−	0.831705i	$-0.687365\pi$
$331$	−4.02696e7	−1.11044	−0.555218	+	0.831705i	$0.687365\pi$
$332$	0	0
$333$	− 460614.i	− 0.0124740i
$334$	0	0
$335$	3.94146e6i	0.104839i
$336$	0	0
$337$	−3.42531e7	−0.894973	−0.447487	−	0.894291i	$-0.647681\pi$
$337$	−3.42531e7	−0.894973	−0.447487	+	0.894291i	$0.647681\pi$
$338$	0	0
$339$	1.86428e7	0.478533
$340$	0	0
$341$	− 4.02355e7i	− 1.01472i
$342$	0	0
$343$	4.31599e7i	1.06954i
$344$	0	0
$345$	3.24480e6	0.0790188
$346$	0	0
$347$	−5.45496e7	−1.30558	−0.652790	−	0.757539i	$-0.726401\pi$
$347$	−5.45496e7	−1.30558	−0.652790	+	0.757539i	$0.726401\pi$
$348$	0	0
$349$	4.70009e7i	1.10568i	0.833287	+	0.552840i	$0.186456\pi$
$349$	4.70009e7i	1.10568i	−0.833287	+	0.552840i	$0.813544\pi$
$350$	0	0
$351$	2.26203e7i	0.523091i
$352$	0	0
$353$	−1.27231e7	−0.289248	−0.144624	−	0.989487i	$-0.546197\pi$
$353$	−1.27231e7	−0.289248	−0.144624	+	0.989487i	$0.546197\pi$
$354$	0	0
$355$	5.32241e6	0.118966
$356$	0	0
$357$	4.57536e7i	1.00559i
$358$	0	0
$359$	− 2.02153e7i	− 0.436915i	−0.975846	−	0.218457i	$-0.929898\pi$
$359$	− 2.02153e7i	− 0.436915i	0.975846	−	0.218457i	$-0.0701025\pi$
$360$	0	0
$361$	9.64116e6	0.204931
$362$	0	0
$363$	2.63053e7	0.549951
$364$	0	0
$365$	2.88626e6i	0.0593549i
$366$	0	0
$367$	1.11057e7i	0.224672i	0.993670	+	0.112336i	$0.0358333\pi$
$367$	1.11057e7i	0.224672i	−0.993670	+	0.112336i	$0.964167\pi$
$368$	0	0
$369$	−6.78262e6	−0.134995
$370$	0	0
$371$	−5.97536e7	−1.17015
$372$	0	0
$373$	− 687146.i	− 0.0132411i	−0.999978	−	0.00662053i	$-0.997893\pi$
$373$	− 687146.i	− 0.0132411i	0.999978	−	0.00662053i	$-0.00210739\pi$
$374$	0	0
$375$	9.65147e6i	0.183021i
$376$	0	0
$377$	−3.73801e7	−0.697615
$378$	0	0
$379$	−1.48499e7	−0.272775	−0.136388	−	0.990656i	$-0.543549\pi$
$379$	−1.48499e7	−0.272775	−0.136388	+	0.990656i	$0.543549\pi$
$380$	0	0
$381$	5.18861e7i	0.938158i
$382$	0	0
$383$	3.35885e7i	0.597853i	0.954276	+	0.298926i	$0.0966285\pi$
$383$	3.35885e7i	0.597853i	−0.954276	+	0.298926i	$0.903372\pi$
$384$	0	0
$385$	2.97600e6	0.0521496
$386$	0	0
$387$	4.97430e6	0.0858222
$388$	0	0
$389$	1.01122e8i	1.71789i	0.512066	+	0.858946i	$0.328880\pi$
$389$	1.01122e8i	1.71789i	−0.512066	+	0.858946i	$0.671120\pi$
$390$	0	0
$391$	4.99122e7i	0.834980i
$392$	0	0
$393$	−8.81347e7	−1.45201
$394$	0	0
$395$	3.10706e6	0.0504149
$396$	0	0
$397$	− 3.48266e7i	− 0.556595i	−0.960495	−	0.278297i	$-0.910230\pi$
$397$	− 3.48266e7i	− 0.556595i	0.960495	−	0.278297i	$-0.0897701\pi$
$398$	0	0
$399$	− 7.22792e7i	− 1.13787i
$400$	0	0
$401$	−6.88398e7	−1.06760	−0.533798	−	0.845612i	$-0.679236\pi$
$401$	−6.88398e7	−1.06760	−0.533798	+	0.845612i	$0.679236\pi$
$402$	0	0
$403$	6.14110e7	0.938277
$404$	0	0
$405$	6.46479e6i	0.0973171i
$406$	0	0
$407$	− 1.91524e6i	− 0.0284079i
$408$	0	0
$409$	−4.59959e7	−0.672278	−0.336139	−	0.941812i	$-0.609121\pi$
$409$	−4.59959e7	−0.672278	−0.336139	+	0.941812i	$0.609121\pi$
$410$	0	0
$411$	1.18050e8	1.70035
$412$	0	0
$413$	2.42976e7i	0.344916i
$414$	0	0
$415$	2.04153e6i	0.0285635i
$416$	0	0
$417$	4.30176e6	0.0593250
$418$	0	0
$419$	2.71153e7	0.368615	0.184307	−	0.982869i	$-0.440996\pi$
$419$	2.71153e7	0.368615	0.184307	+	0.982869i	$0.440996\pi$
$420$	0	0
$421$	9.42078e7i	1.26253i	0.775569	+	0.631263i	$0.217463\pi$
$421$	9.42078e7i	1.26253i	−0.775569	+	0.631263i	$0.782537\pi$
$422$	0	0
$423$	1.74637e6i	0.0230737i
$424$	0	0
$425$	−7.39922e7	−0.963871
$426$	0	0
$427$	3.38282e6	0.0434505
$428$	0	0
$429$	− 4.36282e7i	− 0.552580i
$430$	0	0
$431$	5.19187e7i	0.648473i	0.945976	+	0.324236i	$0.105107\pi$
$431$	5.19187e7i	0.648473i	−0.945976	+	0.324236i	$0.894893\pi$
$432$	0	0
$433$	8.40210e7	1.03496	0.517481	−	0.855695i	$-0.326870\pi$
$433$	8.40210e7	1.03496	0.517481	+	0.855695i	$0.326870\pi$
$434$	0	0
$435$	−7.90027e6	−0.0959785
$436$	0	0
$437$	− 7.88486e7i	− 0.944822i
$438$	0	0
$439$	− 1.48115e8i	− 1.75068i	−0.483512	−	0.875338i	$-0.660639\pi$
$439$	− 1.48115e8i	− 1.75068i	0.483512	−	0.875338i	$-0.339361\pi$
$440$	0	0
$441$	5.00092e6	0.0583088
$442$	0	0
$443$	−8.03735e7	−0.924489	−0.462245	−	0.886752i	$-0.652956\pi$
$443$	−8.03735e7	−0.924489	−0.462245	+	0.886752i	$0.652956\pi$
$444$	0	0
$445$	3.10738e6i	0.0352626i
$446$	0	0
$447$	− 1.01449e8i	− 1.13586i
$448$	0	0
$449$	−8.80925e7	−0.973196	−0.486598	−	0.873626i	$-0.661762\pi$
$449$	−8.80925e7	−0.973196	−0.486598	+	0.873626i	$0.661762\pi$
$450$	0	0
$451$	−2.82022e7	−0.307435
$452$	0	0
$453$	1.73309e8i	1.86434i
$454$	0	0
$455$	4.54223e6i	0.0482209i
$456$	0	0
$457$	3.75423e7	0.393344	0.196672	−	0.980469i	$-0.436987\pi$
$457$	3.75423e7	0.393344	0.196672	+	0.980469i	$0.436987\pi$
$458$	0	0
$459$	−7.35392e7	−0.760468
$460$	0	0
$461$	1.15260e8i	1.17646i	0.808695	+	0.588228i	$0.200174\pi$
$461$	1.15260e8i	1.17646i	−0.808695	+	0.588228i	$0.799826\pi$
$462$	0	0
$463$	1.03415e7i	0.104194i	0.998642	+	0.0520970i	$0.0165905\pi$
$463$	1.03415e7i	0.104194i	−0.998642	+	0.0520970i	$0.983410\pi$
$464$	0	0
$465$	1.29792e7	0.129089
$466$	0	0
$467$	1.64223e8	1.61243	0.806217	−	0.591620i	$-0.201511\pi$
$467$	1.64223e8	1.61243	0.806217	+	0.591620i	$0.201511\pi$
$468$	0	0
$469$	− 1.22122e8i	− 1.18379i
$470$	0	0
$471$	− 2.53074e7i	− 0.242206i
$472$	0	0
$473$	2.06832e7	0.195449
$474$	0	0
$475$	1.16889e8	1.09067
$476$	0	0
$477$	4.45493e7i	0.410474i
$478$	0	0
$479$	− 7.76230e7i	− 0.706291i	−0.935568	−	0.353146i	$-0.885112\pi$
$479$	− 7.76230e7i	− 0.706291i	0.935568	−	0.353146i	$-0.114888\pi$
$480$	0	0
$481$	2.92320e6	0.0262678
$482$	0	0
$483$	−1.00536e8	−0.892241
$484$	0	0
$485$	1.45709e7i	0.127720i
$486$	0	0
$487$	− 1.08071e7i	− 0.0935670i	−0.998905	−	0.0467835i	$-0.985103\pi$
$487$	− 1.08071e7i	− 0.0935670i	0.998905	−	0.0467835i	$-0.0148971\pi$
$488$	0	0
$489$	5.71939e7	0.489129
$490$	0	0
$491$	1.85067e8	1.56345	0.781724	−	0.623624i	$-0.214340\pi$
$491$	1.85067e8	1.56345	0.781724	+	0.623624i	$0.214340\pi$
$492$	0	0
$493$	− 1.21523e8i	− 1.01419i
$494$	0	0
$495$	− 2.21875e6i	− 0.0182934i
$496$	0	0
$497$	−1.64909e8	−1.34331
$498$	0	0
$499$	−6.83704e7	−0.550258	−0.275129	−	0.961407i	$-0.588721\pi$
$499$	−6.83704e7	−0.550258	−0.275129	+	0.961407i	$0.588721\pi$
$500$	0	0
$501$	− 2.46791e8i	− 1.96253i
$502$	0	0
$503$	1.31562e7i	0.103377i	0.998663	+	0.0516887i	$0.0164604\pi$
$503$	1.31562e7i	0.103377i	−0.998663	+	0.0516887i	$0.983540\pi$
$504$	0	0
$505$	6.39158e6	0.0496288
$506$	0	0
$507$	−8.29640e7	−0.636599
$508$	0	0
$509$	1.34186e8i	1.01755i	0.860900	+	0.508775i	$0.169901\pi$
$509$	1.34186e8i	1.01755i	−0.860900	+	0.508775i	$0.830099\pi$
$510$	0	0
$511$	− 8.94275e7i	− 0.670206i
$512$	0	0
$513$	1.16173e8	0.860508
$514$	0	0
$515$	1.38913e7	0.101700
$516$	0	0
$517$	7.26144e6i	0.0525474i
$518$	0	0
$519$	1.58840e8i	1.13621i
$520$	0	0
$521$	1.98565e8	1.40407	0.702036	−	0.712142i	$-0.252275\pi$
$521$	1.98565e8	1.40407	0.702036	+	0.712142i	$0.252275\pi$
$522$	0	0
$523$	−2.15512e8	−1.50649	−0.753245	−	0.657740i	$-0.771513\pi$
$523$	−2.15512e8	−1.50649	−0.753245	+	0.657740i	$0.771513\pi$
$524$	0	0
$525$	− 1.49040e8i	− 1.02997i
$526$	0	0
$527$	1.99649e8i	1.36406i
$528$	0	0
$529$	3.83616e7	0.259137
$530$	0	0
$531$	1.81151e7	0.120992
$532$	0	0
$533$	− 4.30447e7i	− 0.284275i
$534$	0	0
$535$	− 1.14935e7i	− 0.0750567i
$536$	0	0
$537$	7.23197e7	0.467018
$538$	0	0
$539$	2.07939e7	0.132791
$540$	0	0
$541$	1.44188e7i	0.0910623i	0.998963	+	0.0455311i	$0.0144980\pi$
$541$	1.44188e7i	0.0910623i	−0.998963	+	0.0455311i	$0.985502\pi$
$542$	0	0
$543$	3.00282e8i	1.87556i
$544$	0	0
$545$	1.53574e7	0.0948697
$546$	0	0
$547$	4.24129e7	0.259141	0.129571	−	0.991570i	$-0.458640\pi$
$547$	4.24129e7	0.259141	0.129571	+	0.991570i	$0.458640\pi$
$548$	0	0
$549$	− 2.52206e6i	− 0.0152419i
$550$	0	0
$551$	1.91976e8i	1.14761i
$552$	0	0
$553$	−9.62688e7	−0.569259
$554$	0	0
$555$	617818.	0.00361395
$556$	0	0
$557$	8.90848e7i	0.515511i	0.966210	+	0.257756i	$0.0829829\pi$
$557$	8.90848e7i	0.515511i	−0.966210	+	0.257756i	$0.917017\pi$
$558$	0	0
$559$	3.15685e7i	0.180725i
$560$	0	0
$561$	1.41836e8	0.803338
$562$	0	0
$563$	−2.41576e8	−1.35372	−0.676860	−	0.736112i	$-0.736660\pi$
$563$	−2.41576e8	−1.35372	−0.676860	+	0.736112i	$0.736660\pi$
$564$	0	0
$565$	6.01694e6i	0.0333603i
$566$	0	0
$567$	− 2.00304e8i	− 1.09886i
$568$	0	0
$569$	2.56141e7	0.139041	0.0695203	−	0.997581i	$-0.477853\pi$
$569$	2.56141e7	0.139041	0.0695203	+	0.997581i	$0.477853\pi$
$570$	0	0
$571$	−1.10781e8	−0.595057	−0.297528	−	0.954713i	$-0.596162\pi$
$571$	−1.10781e8	−0.595057	−0.297528	+	0.954713i	$0.596162\pi$
$572$	0	0
$573$	− 3.53725e8i	− 1.88019i
$574$	0	0
$575$	− 1.62586e8i	− 0.855225i
$576$	0	0
$577$	1.07272e8	0.558415	0.279208	−	0.960231i	$-0.409928\pi$
$577$	1.07272e8	0.558415	0.279208	+	0.960231i	$0.409928\pi$
$578$	0	0
$579$	7.53011e7	0.387941
$580$	0	0
$581$	− 6.32544e7i	− 0.322524i
$582$	0	0
$583$	1.85236e8i	0.934803i
$584$	0	0
$585$	3.38646e6	0.0169152
$586$	0	0
$587$	−2.16397e7	−0.106988	−0.0534941	−	0.998568i	$-0.517036\pi$
$587$	−2.16397e7	−0.106988	−0.0534941	+	0.998568i	$0.517036\pi$
$588$	0	0
$589$	− 3.15395e8i	− 1.54351i
$590$	0	0
$591$	− 6.91140e7i	− 0.334814i
$592$	0	0
$593$	2.00341e8	0.960738	0.480369	−	0.877066i	$-0.340503\pi$
$593$	2.00341e8	0.960738	0.480369	+	0.877066i	$0.340503\pi$
$594$	0	0
$595$	−1.47669e7	−0.0701033
$596$	0	0
$597$	− 1.51690e8i	− 0.712907i
$598$	0	0
$599$	− 1.37592e8i	− 0.640197i	−0.947384	−	0.320098i	$-0.896284\pi$
$599$	− 1.37592e8i	− 0.640197i	0.947384	−	0.320098i	$-0.103716\pi$
$600$	0	0
$601$	1.90306e8	0.876655	0.438327	−	0.898815i	$-0.355571\pi$
$601$	1.90306e8	0.876655	0.438327	+	0.898815i	$0.355571\pi$
$602$	0	0
$603$	−9.10477e7	−0.415257
$604$	0	0
$605$	8.49001e6i	0.0383391i
$606$	0	0
$607$	− 1.25461e8i	− 0.560974i	−0.959858	−	0.280487i	$-0.909504\pi$
$607$	− 1.25461e8i	− 0.560974i	0.959858	−	0.280487i	$-0.0904960\pi$
$608$	0	0
$609$	2.44781e8	1.08374
$610$	0	0
$611$	−1.10831e7	−0.0485888
$612$	0	0
$613$	9.91111e7i	0.430270i	0.976584	+	0.215135i	$0.0690191\pi$
$613$	9.91111e7i	0.430270i	−0.976584	+	0.215135i	$0.930981\pi$
$614$	0	0
$615$	− 9.09748e6i	− 0.0391107i
$616$	0	0
$617$	−3.70827e7	−0.157876	−0.0789379	−	0.996880i	$-0.525153\pi$
$617$	−3.70827e7	−0.157876	−0.0789379	+	0.996880i	$0.525153\pi$
$618$	0	0
$619$	4.05274e8	1.70874	0.854372	−	0.519662i	$-0.173942\pi$
$619$	4.05274e8	1.70874	0.854372	+	0.519662i	$0.173942\pi$
$620$	0	0
$621$	− 1.61591e8i	− 0.674749i
$622$	0	0
$623$	− 9.62786e7i	− 0.398168i
$624$	0	0
$625$	2.39463e8	0.980841
$626$	0	0
$627$	−2.24065e8	−0.909017
$628$	0	0
$629$	9.50340e6i	0.0381880i
$630$	0	0
$631$	− 2.52648e7i	− 0.100561i	−0.998735	−	0.0502803i	$-0.983989\pi$
$631$	− 2.52648e7i	− 0.100561i	0.998735	−	0.0502803i	$-0.0160115\pi$
$632$	0	0
$633$	−1.21132e8	−0.477581
$634$	0	0
$635$	−1.67462e7	−0.0654025
$636$	0	0
$637$	3.17374e7i	0.122787i
$638$	0	0
$639$	1.22948e8i	0.471213i
$640$	0	0
$641$	−4.21293e8	−1.59959	−0.799797	−	0.600270i	$-0.795060\pi$
$641$	−4.21293e8	−1.59959	−0.799797	+	0.600270i	$0.795060\pi$
$642$	0	0
$643$	8.17706e7	0.307584	0.153792	−	0.988103i	$-0.450851\pi$
$643$	8.17706e7	0.307584	0.153792	+	0.988103i	$0.450851\pi$
$644$	0	0
$645$	6.67200e6i	0.0248643i
$646$	0	0
$647$	− 1.84284e8i	− 0.680416i	−0.940350	−	0.340208i	$-0.889503\pi$
$647$	− 1.84284e8i	− 0.680416i	0.940350	−	0.340208i	$-0.110497\pi$
$648$	0	0
$649$	7.53226e7	0.275544
$650$	0	0
$651$	−4.02146e8	−1.45761
$652$	0	0
$653$	6.43842e7i	0.231228i	0.993294	+	0.115614i	$0.0368835\pi$
$653$	6.43842e7i	0.231228i	−0.993294	+	0.115614i	$0.963117\pi$
$654$	0	0
$655$	− 2.84454e7i	− 0.101225i
$656$	0	0
$657$	−6.66726e7	−0.235099
$658$	0	0
$659$	5.38099e8	1.88021	0.940103	−	0.340889i	$-0.110728\pi$
$659$	5.38099e8	1.88021	0.940103	+	0.340889i	$0.110728\pi$
$660$	0	0
$661$	− 2.83897e8i	− 0.983008i	−0.870875	−	0.491504i	$-0.836447\pi$
$661$	− 2.83897e8i	− 0.983008i	0.870875	−	0.491504i	$-0.163553\pi$
$662$	0	0
$663$	2.16483e8i	0.742819i
$664$	0	0
$665$	2.33280e7	0.0793255
$666$	0	0
$667$	2.67029e8	0.899872
$668$	0	0
$669$	1.03212e8i	0.344707i
$670$	0	0
$671$	− 1.04867e7i	− 0.0347115i
$672$	0	0
$673$	2.77693e8	0.911002	0.455501	−	0.890235i	$-0.349460\pi$
$673$	2.77693e8	0.911002	0.455501	+	0.890235i	$0.349460\pi$
$674$	0	0
$675$	2.39550e8	0.778906
$676$	0	0
$677$	9.23026e7i	0.297473i	0.988877	+	0.148737i	$0.0475207\pi$
$677$	9.23026e7i	0.297473i	−0.988877	+	0.148737i	$0.952479\pi$
$678$	0	0
$679$	− 4.51462e8i	− 1.44215i
$680$	0	0
$681$	4.18384e8	1.32475
$682$	0	0
$683$	2.70862e8	0.850132	0.425066	−	0.905162i	$-0.360251\pi$
$683$	2.70862e8	0.850132	0.425066	+	0.905162i	$0.360251\pi$
$684$	0	0
$685$	3.81003e7i	0.118538i
$686$	0	0
$687$	4.94496e7i	0.152508i
$688$	0	0
$689$	−2.82724e8	−0.864380
$690$	0	0
$691$	−1.92568e7	−0.0583645	−0.0291823	−	0.999574i	$-0.509290\pi$
$691$	−1.92568e7	−0.0583645	−0.0291823	+	0.999574i	$0.509290\pi$
$692$	0	0
$693$	6.87456e7i	0.206560i
$694$	0	0
$695$	1.38839e6i	0.00413577i
$696$	0	0
$697$	1.39939e8	0.413277
$698$	0	0
$699$	2.49229e8	0.729738
$700$	0	0
$701$	3.52234e8i	1.02253i	0.859423	+	0.511266i	$0.170823\pi$
$701$	3.52234e8i	1.02253i	−0.859423	+	0.511266i	$0.829177\pi$
$702$	0	0
$703$	− 1.50130e7i	− 0.0432117i
$704$	0	0
$705$	−2.34240e6	−0.00668488
$706$	0	0
$707$	−1.98036e8	−0.560384
$708$	0	0
$709$	− 4.62733e8i	− 1.29835i	−0.760639	−	0.649175i	$-0.775114\pi$
$709$	− 4.62733e8i	− 1.29835i	0.760639	−	0.649175i	$-0.224886\pi$
$710$	0	0
$711$	7.17731e7i	0.199689i
$712$	0	0
$713$	−4.38697e8	−1.21031
$714$	0	0
$715$	1.40809e7	0.0385224
$716$	0	0
$717$	3.48668e8i	0.945921i
$718$	0	0
$719$	− 4.60385e8i	− 1.23861i	−0.785150	−	0.619305i	$-0.787414\pi$
$719$	− 4.60385e8i	− 1.23861i	0.785150	−	0.619305i	$-0.212586\pi$
$720$	0	0
$721$	−4.30406e8	−1.14835
$722$	0	0
$723$	−1.56501e8	−0.414097
$724$	0	0
$725$	3.95856e8i	1.03878i
$726$	0	0
$727$	4.82173e8i	1.25487i	0.778668	+	0.627437i	$0.215896\pi$
$727$	4.82173e8i	1.25487i	−0.778668	+	0.627437i	$0.784104\pi$
$728$	0	0
$729$	1.99184e8	0.514128
$730$	0	0
$731$	−1.02630e8	−0.262738
$732$	0	0
$733$	− 5.08270e8i	− 1.29057i	−0.763941	−	0.645287i	$-0.776738\pi$
$733$	− 5.08270e8i	− 1.29057i	0.763941	−	0.645287i	$-0.223262\pi$
$734$	0	0
$735$	6.70770e6i	0.0168932i
$736$	0	0
$737$	−3.78577e8	−0.945696
$738$	0	0
$739$	1.27767e8	0.316582	0.158291	−	0.987392i	$-0.449402\pi$
$739$	1.27767e8	0.316582	0.158291	+	0.987392i	$0.449402\pi$
$740$	0	0
$741$	− 3.41988e8i	− 0.840537i
$742$	0	0
$743$	− 2.83312e8i	− 0.690716i	−0.938471	−	0.345358i	$-0.887758\pi$
$743$	− 2.83312e8i	− 0.690716i	0.938471	−	0.345358i	$-0.112242\pi$
$744$	0	0
$745$	3.27426e7	0.0791853
$746$	0	0
$747$	−4.71593e7	−0.113137
$748$	0	0
$749$	3.56112e8i	0.847503i
$750$	0	0
$751$	2.15309e8i	0.508325i	0.967161	+	0.254163i	$0.0817999\pi$
$751$	2.15309e8i	0.508325i	−0.967161	+	0.254163i	$0.918200\pi$
$752$	0	0
$753$	1.46117e8	0.342228
$754$	0	0
$755$	−5.59352e7	−0.129970
$756$	0	0
$757$	3.03985e8i	0.700753i	0.936609	+	0.350377i	$0.113946\pi$
$757$	3.03985e8i	0.700753i	−0.936609	+	0.350377i	$0.886054\pi$
$758$	0	0
$759$	3.11663e8i	0.712787i
$760$	0	0
$761$	−8.63611e8	−1.95959	−0.979793	−	0.200014i	$-0.935901\pi$
$761$	−8.63611e8	−1.95959	−0.979793	+	0.200014i	$0.935901\pi$
$762$	0	0
$763$	−4.75831e8	−1.07122
$764$	0	0
$765$	1.10095e7i	0.0245913i
$766$	0	0
$767$	1.14964e8i	0.254786i
$768$	0	0
$769$	−1.97898e7	−0.0435174	−0.0217587	−	0.999763i	$-0.506927\pi$
$769$	−1.97898e7	−0.0435174	−0.0217587	+	0.999763i	$0.506927\pi$
$770$	0	0
$771$	−7.26734e8	−1.58567
$772$	0	0
$773$	5.64048e8i	1.22117i	0.791949	+	0.610587i	$0.209067\pi$
$773$	5.64048e8i	1.22117i	−0.791949	+	0.610587i	$0.790933\pi$
$774$	0	0
$775$	− 6.50345e8i	− 1.39714i
$776$	0	0
$777$	−1.91424e7	−0.0408069
$778$	0	0
$779$	−2.21069e8	−0.467644
$780$	0	0
$781$	5.11217e8i	1.07313i
$782$	0	0
$783$	3.93433e8i	0.819570i
$784$	0	0
$785$	8.16794e6	0.0168851
$786$	0	0
$787$	4.04198e8	0.829220	0.414610	−	0.909999i	$-0.363918\pi$
$787$	4.04198e8	0.829220	0.414610	+	0.909999i	$0.363918\pi$
$788$	0	0
$789$	6.72328e8i	1.36883i
$790$	0	0
$791$	− 1.86428e8i	− 0.376688i
$792$	0	0
$793$	1.60058e7	0.0320965
$794$	0	0
$795$	−5.97536e7	−0.118922
$796$	0	0
$797$	− 2.15998e7i	− 0.0426654i	−0.999772	−	0.0213327i	$-0.993209\pi$
$797$	− 2.15998e7i	− 0.0426654i	0.999772	−	0.0213327i	$-0.00679092\pi$
$798$	0	0
$799$	− 3.60313e7i	− 0.0706381i
$800$	0	0
$801$	−7.17805e7	−0.139672
$802$	0	0
$803$	−2.77225e8	−0.535410
$804$	0	0
$805$	− 3.24480e7i	− 0.0622014i
$806$	0	0
$807$	9.11786e8i	1.73489i
$808$	0	0
$809$	−6.51310e8	−1.23011	−0.615053	−	0.788486i	$-0.710865\pi$
$809$	−6.51310e8	−1.23011	−0.615053	+	0.788486i	$0.710865\pi$
$810$	0	0
$811$	−5.52891e8	−1.03652	−0.518259	−	0.855223i	$-0.673420\pi$
$811$	−5.52891e8	−1.03652	−0.518259	+	0.855223i	$0.673420\pi$
$812$	0	0
$813$	− 2.63820e8i	− 0.490948i
$814$	0	0
$815$	1.84593e7i	0.0340990i
$816$	0	0
$817$	1.62130e8	0.297301
$818$	0	0
$819$	−1.04926e8	−0.190998
$820$	0	0
$821$	8.94865e8i	1.61707i	0.588450	+	0.808534i	$0.299738\pi$
$821$	8.94865e8i	1.61707i	−0.588450	+	0.808534i	$0.700262\pi$
$822$	0	0
$823$	8.44482e8i	1.51492i	0.652879	+	0.757462i	$0.273561\pi$
$823$	8.44482e8i	1.51492i	−0.652879	+	0.757462i	$0.726439\pi$
$824$	0	0
$825$	−4.62024e8	−0.822816
$826$	0	0
$827$	3.11099e7	0.0550024	0.0275012	−	0.999622i	$-0.491245\pi$
$827$	3.11099e7	0.0550024	0.0275012	+	0.999622i	$0.491245\pi$
$828$	0	0
$829$	− 4.05444e7i	− 0.0711652i	−0.999367	−	0.0355826i	$-0.988671\pi$
$829$	− 4.05444e7i	− 0.0711652i	0.999367	−	0.0355826i	$-0.0113287\pi$
$830$	0	0
$831$	8.55856e8i	1.49141i
$832$	0	0
$833$	−1.03179e8	−0.178508
$834$	0	0
$835$	7.96515e7	0.136815
$836$	0	0
$837$	− 6.46364e8i	− 1.10230i
$838$	0	0
$839$	− 4.17820e8i	− 0.707463i	−0.935347	−	0.353731i	$-0.884913\pi$
$839$	− 4.17820e8i	− 0.707463i	0.935347	−	0.353731i	$-0.115087\pi$
$840$	0	0
$841$	−5.53247e7	−0.0930103
$842$	0	0
$843$	2.67703e8	0.446859
$844$	0	0
$845$	− 2.67765e7i	− 0.0443797i
$846$	0	0
$847$	− 2.63053e8i	− 0.432906i
$848$	0	0
$849$	−3.94578e8	−0.644778
$850$	0	0
$851$	−2.08823e7	−0.0338835
$852$	0	0
$853$	4.28994e8i	0.691201i	0.938382	+	0.345601i	$0.112325\pi$
$853$	4.28994e8i	0.691201i	−0.938382	+	0.345601i	$0.887675\pi$
$854$	0	0
$855$	− 1.73922e7i	− 0.0278263i
$856$	0	0
$857$	−2.20598e8	−0.350476	−0.175238	−	0.984526i	$-0.556070\pi$
$857$	−2.20598e8	−0.350476	−0.175238	+	0.984526i	$0.556070\pi$
$858$	0	0
$859$	1.14768e9	1.81068	0.905339	−	0.424689i	$-0.139617\pi$
$859$	1.14768e9	1.81068	0.905339	+	0.424689i	$0.139617\pi$
$860$	0	0
$861$	2.81875e8i	0.441619i
$862$	0	0
$863$	− 6.44987e7i	− 0.100350i	−0.998740	−	0.0501752i	$-0.984022\pi$
$863$	− 6.44987e7i	− 0.100350i	0.998740	−	0.0501752i	$-0.0159780\pi$
$864$	0	0
$865$	−5.12653e7	−0.0792092
$866$	0	0
$867$	4.40842e7	0.0676435
$868$	0	0
$869$	2.98433e8i	0.454766i
$870$	0	0
$871$	− 5.77818e8i	− 0.874453i
$872$	0	0
$873$	−3.36587e8	−0.505888
$874$	0	0
$875$	9.65147e7	0.144069
$876$	0	0
$877$	5.15252e8i	0.763873i	0.924189	+	0.381937i	$0.124743\pi$
$877$	5.15252e8i	0.763873i	−0.924189	+	0.381937i	$0.875257\pi$
$878$	0	0
$879$	− 1.38007e9i	− 2.03205i
$880$	0	0
$881$	−1.18519e8	−0.173324	−0.0866622	−	0.996238i	$-0.527620\pi$
$881$	−1.18519e8	−0.173324	−0.0866622	+	0.996238i	$0.527620\pi$
$882$	0	0
$883$	−1.28669e8	−0.186893	−0.0934466	−	0.995624i	$-0.529788\pi$
$883$	−1.28669e8	−0.186893	−0.0934466	+	0.995624i	$0.529788\pi$
$884$	0	0
$885$	2.42976e7i	0.0350537i
$886$	0	0
$887$	5.09948e8i	0.730727i	0.930865	+	0.365363i	$0.119055\pi$
$887$	5.09948e8i	0.730727i	−0.930865	+	0.365363i	$0.880945\pi$
$888$	0	0
$889$	5.18861e8	0.738492
$890$	0	0
$891$	−6.20943e8	−0.877847
$892$	0	0
$893$	5.69203e7i	0.0799306i
$894$	0	0
$895$	2.33411e7i	0.0325576i
$896$	0	0
$897$	−4.75688e8	−0.659090
$898$	0	0
$899$	1.06812e9	1.47007
$900$	0	0
$901$	− 9.19142e8i	− 1.25663i
$902$	0	0
$903$	− 2.06724e8i	− 0.280756i
$904$	0	0
$905$	−9.69156e7	−0.130752
$906$	0	0
$907$	5.43212e8	0.728027	0.364014	−	0.931394i	$-0.381406\pi$
$907$	5.43212e8	0.728027	0.364014	+	0.931394i	$0.381406\pi$
$908$	0	0
$909$	1.47645e8i	0.196575i
$910$	0	0
$911$	5.99585e8i	0.793041i	0.918026	+	0.396520i	$0.129782\pi$
$911$	5.99585e8i	0.793041i	−0.918026	+	0.396520i	$0.870218\pi$
$912$	0	0
$913$	−1.96089e8	−0.257656
$914$	0	0
$915$	3.38282e6	0.00441586
$916$	0	0
$917$	8.81347e8i	1.14298i
$918$	0	0
$919$	− 9.66486e8i	− 1.24523i	−0.782529	−	0.622614i	$-0.786071\pi$
$919$	− 9.66486e8i	− 1.24523i	0.782529	−	0.622614i	$-0.213929\pi$
$920$	0	0
$921$	1.51527e9	1.93959
$922$	0	0
$923$	−7.80265e8	−0.992286
$924$	0	0
$925$	− 3.09568e7i	− 0.0391139i
$926$	0	0
$927$	3.20889e8i	0.402825i
$928$	0	0
$929$	7.22626e8	0.901295	0.450647	−	0.892702i	$-0.351193\pi$
$929$	7.22626e8	0.901295	0.450647	+	0.892702i	$0.351193\pi$
$930$	0	0
$931$	1.62997e8	0.201990
$932$	0	0
$933$	− 1.54988e9i	− 1.90832i
$934$	0	0
$935$	4.57774e7i	0.0560037i
$936$	0	0
$937$	4.12016e8	0.500836	0.250418	−	0.968138i	$-0.419432\pi$
$937$	4.12016e8	0.500836	0.250418	+	0.968138i	$0.419432\pi$
$938$	0	0
$939$	3.49246e7	0.0421827
$940$	0	0
$941$	7.87666e8i	0.945308i	0.881248	+	0.472654i	$0.156704\pi$
$941$	7.87666e8i	0.945308i	−0.881248	+	0.472654i	$0.843296\pi$
$942$	0	0
$943$	3.07495e8i	0.366693i
$944$	0	0
$945$	4.78080e7	0.0566507
$946$	0	0
$947$	−5.29504e8	−0.623475	−0.311738	−	0.950168i	$-0.600911\pi$
$947$	−5.29504e8	−0.623475	−0.311738	+	0.950168i	$0.600911\pi$
$948$	0	0
$949$	− 4.23126e8i	− 0.495075i
$950$	0	0
$951$	− 1.06858e9i	− 1.24241i
$952$	0	0
$953$	−1.18323e9	−1.36706	−0.683532	−	0.729920i	$-0.739557\pi$
$953$	−1.18323e9	−1.36706	−0.683532	+	0.729920i	$0.739557\pi$
$954$	0	0
$955$	1.14164e8	0.131075
$956$	0	0
$957$	− 7.58820e8i	− 0.865771i
$958$	0	0
$959$	− 1.18050e9i	− 1.33847i
$960$	0	0
$961$	−8.67284e8	−0.977218
$962$	0	0
$963$	2.65499e8	0.297293
$964$	0	0
$965$	2.43033e7i	0.0270448i
$966$	0	0
$967$	4.78688e8i	0.529387i	0.964333	+	0.264693i	$0.0852708\pi$
$967$	4.78688e8i	0.529387i	−0.964333	+	0.264693i	$0.914729\pi$
$968$	0	0
$969$	1.11181e9	1.22197
$970$	0	0
$971$	7.02914e8	0.767793	0.383897	−	0.923376i	$-0.374582\pi$
$971$	7.02914e8	0.767793	0.383897	+	0.923376i	$0.374582\pi$
$972$	0	0
$973$	− 4.30176e7i	− 0.0466990i
$974$	0	0
$975$	− 7.05182e8i	− 0.760830i
$976$	0	0
$977$	−9.21323e8	−0.987934	−0.493967	−	0.869481i	$-0.664454\pi$
$977$	−9.21323e8	−0.987934	−0.493967	+	0.869481i	$0.664454\pi$
$978$	0	0
$979$	−2.98464e8	−0.318085
$980$	0	0
$981$	3.54755e8i	0.375770i
$982$	0	0
$983$	2.44410e8i	0.257311i	0.991689	+	0.128655i	$0.0410661\pi$
$983$	2.44410e8i	0.257311i	−0.991689	+	0.128655i	$0.958934\pi$
$984$	0	0
$985$	2.23065e7	0.0233411
$986$	0	0
$987$	7.25766e7	0.0754823
$988$	0	0
$989$	− 2.25514e8i	− 0.233122i
$990$	0	0
$991$	− 1.78184e9i	− 1.83083i	−0.402516	−	0.915413i	$-0.631864\pi$
$991$	− 1.78184e9i	− 1.83083i	0.402516	−	0.915413i	$-0.368136\pi$
$992$	0	0
$993$	1.24771e9	1.27428
$994$	0	0
$995$	4.89576e7	0.0496994
$996$	0	0
$997$	− 1.36790e9i	− 1.38029i	−0.723673	−	0.690143i	$-0.757548\pi$
$997$	− 1.36790e9i	− 1.38029i	0.723673	−	0.690143i	$-0.242452\pi$
$998$	0	0
$999$	− 3.07674e7i	− 0.0308598i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.7.d.f.127.1		4
4.3	odd	2	inner	256.7.d.f.127.3		4
8.3	odd	2	inner	256.7.d.f.127.2		4
8.5	even	2	inner	256.7.d.f.127.4		4
16.3	odd	4		64.7.c.c.63.2		2
16.5	even	4		4.7.b.a.3.1	✓	2
16.11	odd	4		4.7.b.a.3.2	yes	2
16.13	even	4		64.7.c.c.63.1		2
48.5	odd	4		36.7.d.c.19.2		2
48.11	even	4		36.7.d.c.19.1		2
48.29	odd	4		576.7.g.h.127.1		2
48.35	even	4		576.7.g.h.127.2		2
80.27	even	4		100.7.d.a.99.2		4
80.37	odd	4		100.7.d.a.99.4		4
80.43	even	4		100.7.d.a.99.3		4
80.53	odd	4		100.7.d.a.99.1		4
80.59	odd	4		100.7.b.c.51.1		2
80.69	even	4		100.7.b.c.51.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.7.b.a.3.1	✓	2	16.5	even	4
4.7.b.a.3.2	yes	2	16.11	odd	4
36.7.d.c.19.1		2	48.11	even	4
36.7.d.c.19.2		2	48.5	odd	4
64.7.c.c.63.1		2	16.13	even	4
64.7.c.c.63.2		2	16.3	odd	4
100.7.b.c.51.1		2	80.59	odd	4
100.7.b.c.51.2		2	80.69	even	4
100.7.d.a.99.1		4	80.53	odd	4
100.7.d.a.99.2		4	80.27	even	4
100.7.d.a.99.3		4	80.43	even	4
100.7.d.a.99.4		4	80.37	odd	4
256.7.d.f.127.1		4	1.1	even	1	trivial
256.7.d.f.127.2		4	8.3	odd	2	inner
256.7.d.f.127.3		4	4.3	odd	2	inner
256.7.d.f.127.4		4	8.5	even	2	inner
576.7.g.h.127.1		2	48.29	odd	4
576.7.g.h.127.2		2	48.35	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	256.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-30.9839 q^{3} -10.0000i q^{5} +309.839i q^{7} +231.000 q^{9} +O(q^{10})\)	\(q-30.9839 q^{3} -10.0000i q^{5} +309.839i q^{7} +231.000 q^{9} +960.500 q^{11} +1466.00i q^{13} +309.839i q^{15} -4766.00 q^{17} +7529.08 q^{19} -9600.00i q^{21} -10472.5i q^{23} +15525.0 q^{25} +15430.0 q^{27} +25498.0i q^{29} -41890.2i q^{31} -29760.0 q^{33} +3098.39 q^{35} -1994.00i q^{37} -45422.3i q^{39} -29362.0 q^{41} +21533.8 q^{43} -2310.00i q^{45} +7560.06i q^{47} +21649.0 q^{49} +147669. q^{51} +192854. i q^{53} -9605.00i q^{55} -233280. q^{57} +78420.2 q^{59} -10918.0i q^{61} +71572.7i q^{63} +14660.0 q^{65} -394146. q^{67} +324480. i q^{69} +532241. i q^{71} -288626. q^{73} -481025. q^{75} +297600. i q^{77} +310706. i q^{79} -646479. q^{81} -204153. q^{83} +47660.0i q^{85} -790027. i q^{87} -310738. q^{89} -454223. q^{91} +1.29792e6i q^{93} -75290.8i q^{95} -1.45709e6 q^{97} +221875. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 924 q^{9}+O(q^{10}) \) 4 * q + 924 * q^9	\( 4 q + 924 q^{9} - 19064 q^{17} + 62100 q^{25} - 119040 q^{33} - 117448 q^{41} + 86596 q^{49} - 933120 q^{57} + 58640 q^{65} - 1154504 q^{73} - 2585916 q^{81} - 1242952 q^{89} - 5828344 q^{97}+O(q^{100}) \) 4 * q + 924 * q^9 - 19064 * q^17 + 62100 * q^25 - 119040 * q^33 - 117448 * q^41 + 86596 * q^49 - 933120 * q^57 + 58640 * q^65 - 1154504 * q^73 - 2585916 * q^81 - 1242952 * q^89 - 5828344 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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