LMFDB - Embedded newform 256.6.b.b.129.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,6,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, 6, names="a")

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

chi := DirichletCharacter("256.129");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 6 $
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:	$41.0582578721$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 32)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	256.129
Dual form			256.6.b.b.129.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.00000i q^{3} -14.0000i q^{5} -208.000 q^{7} +179.000 q^{9} +O(q^{10})$	$q+8.00000i q^{3} -14.0000i q^{5} -208.000 q^{7} +179.000 q^{9} -536.000i q^{11} +694.000i q^{13} +112.000 q^{15} -1278.00 q^{17} -1112.00i q^{19} -1664.00i q^{21} +3216.00 q^{23} +2929.00 q^{25} +3376.00i q^{27} +2918.00i q^{29} +2624.00 q^{31} +4288.00 q^{33} +2912.00i q^{35} +9458.00i q^{37} -5552.00 q^{39} -170.000 q^{41} -19928.0i q^{43} -2506.00i q^{45} -32.0000 q^{47} +26457.0 q^{49} -10224.0i q^{51} +22178.0i q^{53} -7504.00 q^{55} +8896.00 q^{57} +41480.0i q^{59} +15462.0i q^{61} -37232.0 q^{63} +9716.00 q^{65} +20744.0i q^{67} +25728.0i q^{69} +28592.0 q^{71} +53670.0 q^{73} +23432.0i q^{75} +111488. i q^{77} +69152.0 q^{79} +16489.0 q^{81} +37800.0i q^{83} +17892.0i q^{85} -23344.0 q^{87} +126806. q^{89} -144352. i q^{91} +20992.0i q^{93} -15568.0 q^{95} +62290.0 q^{97} -95944.0i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 416 q^{7} + 358 q^{9}+O(q^{10}) $ 2 * q - 416 * q^7 + 358 * q^9	$ 2 q - 416 q^{7} + 358 q^{9} + 224 q^{15} - 2556 q^{17} + 6432 q^{23} + 5858 q^{25} + 5248 q^{31} + 8576 q^{33} - 11104 q^{39} - 340 q^{41} - 64 q^{47} + 52914 q^{49} - 15008 q^{55} + 17792 q^{57} - 74464 q^{63} + 19432 q^{65} + 57184 q^{71} + 107340 q^{73} + 138304 q^{79} + 32978 q^{81} - 46688 q^{87} + 253612 q^{89} - 31136 q^{95} + 124580 q^{97}+O(q^{100}) $ 2 * q - 416 * q^7 + 358 * q^9 + 224 * q^15 - 2556 * q^17 + 6432 * q^23 + 5858 * q^25 + 5248 * q^31 + 8576 * q^33 - 11104 * q^39 - 340 * q^41 - 64 * q^47 + 52914 * q^49 - 15008 * q^55 + 17792 * q^57 - 74464 * q^63 + 19432 * q^65 + 57184 * q^71 + 107340 * q^73 + 138304 * q^79 + 32978 * q^81 - 46688 * q^87 + 253612 * q^89 - 31136 * q^95 + 124580 * 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
$3$	8.00000i	0.513200i	0.966518	+	0.256600i	$0.0826023\pi$
$3$	8.00000i	0.513200i	−0.966518	+	0.256600i	$0.917398\pi$
$4$	0	0
$5$	− 14.0000i	− 0.250440i	−0.992129	−	0.125220i	$-0.960036\pi$
$5$	− 14.0000i	− 0.250440i	0.992129	−	0.125220i	$-0.0399636\pi$
$6$	0	0
$7$	−208.000	−1.60442	−0.802210	−	0.597042i	$-0.796343\pi$
$7$	−208.000	−1.60442	−0.802210	+	0.597042i	$0.796343\pi$
$8$	0	0
$9$	179.000	0.736626
$10$	0	0
$11$	− 536.000i	− 1.33562i	−0.744332	−	0.667810i	$-0.767232\pi$
$11$	− 536.000i	− 1.33562i	0.744332	−	0.667810i	$-0.232768\pi$
$12$	0	0
$13$	694.000i	1.13894i	0.822012	+	0.569470i	$0.192852\pi$
$13$	694.000i	1.13894i	−0.822012	+	0.569470i	$0.807148\pi$
$14$	0	0
$15$	112.000	0.128526
$16$	0	0
$17$	−1278.00	−1.07253	−0.536264	−	0.844050i	$-0.680165\pi$
$17$	−1278.00	−1.07253	−0.536264	+	0.844050i	$0.680165\pi$
$18$	0	0
$19$	− 1112.00i	− 0.706677i	−0.935496	−	0.353338i	$-0.885046\pi$
$19$	− 1112.00i	− 0.706677i	0.935496	−	0.353338i	$-0.114954\pi$
$20$	0	0
$21$	− 1664.00i	− 0.823389i
$22$	0	0
$23$	3216.00	1.26764	0.633821	−	0.773480i	$-0.281486\pi$
$23$	3216.00	1.26764	0.633821	+	0.773480i	$0.281486\pi$
$24$	0	0
$25$	2929.00	0.937280
$26$	0	0
$27$	3376.00i	0.891237i
$28$	0	0
$29$	2918.00i	0.644303i	0.946688	+	0.322152i	$0.104406\pi$
$29$	2918.00i	0.644303i	−0.946688	+	0.322152i	$0.895594\pi$
$30$	0	0
$31$	2624.00	0.490410	0.245205	−	0.969471i	$-0.421145\pi$
$31$	2624.00	0.490410	0.245205	+	0.969471i	$0.421145\pi$
$32$	0	0
$33$	4288.00	0.685441
$34$	0	0
$35$	2912.00i	0.401810i
$36$	0	0
$37$	9458.00i	1.13578i	0.823104	+	0.567891i	$0.192241\pi$
$37$	9458.00i	1.13578i	−0.823104	+	0.567891i	$0.807759\pi$
$38$	0	0
$39$	−5552.00	−0.584505
$40$	0	0
$41$	−170.000	−0.0157939	−0.00789695	−	0.999969i	$-0.502514\pi$
$41$	−170.000	−0.0157939	−0.00789695	+	0.999969i	$0.502514\pi$
$42$	0	0
$43$	− 19928.0i	− 1.64359i	−0.569786	−	0.821793i	$-0.692974\pi$
$43$	− 19928.0i	− 1.64359i	0.569786	−	0.821793i	$-0.307026\pi$
$44$	0	0
$45$	− 2506.00i	− 0.184480i
$46$	0	0
$47$	−32.0000	−0.00211303	−0.00105651	−	0.999999i	$-0.500336\pi$
$47$	−32.0000	−0.00211303	−0.00105651	+	0.999999i	$0.500336\pi$
$48$	0	0
$49$	26457.0	1.57417
$50$	0	0
$51$	− 10224.0i	− 0.550422i
$52$	0	0
$53$	22178.0i	1.08451i	0.840215	+	0.542254i	$0.182429\pi$
$53$	22178.0i	1.08451i	−0.840215	+	0.542254i	$0.817571\pi$
$54$	0	0
$55$	−7504.00	−0.334492
$56$	0	0
$57$	8896.00	0.362667
$58$	0	0
$59$	41480.0i	1.55135i	0.631135	+	0.775673i	$0.282589\pi$
$59$	41480.0i	1.55135i	−0.631135	+	0.775673i	$0.717411\pi$
$60$	0	0
$61$	15462.0i	0.532036i	0.963968	+	0.266018i	$0.0857081\pi$
$61$	15462.0i	0.532036i	−0.963968	+	0.266018i	$0.914292\pi$
$62$	0	0
$63$	−37232.0	−1.18186
$64$	0	0
$65$	9716.00	0.285236
$66$	0	0
$67$	20744.0i	0.564554i	0.959333	+	0.282277i	$0.0910897\pi$
$67$	20744.0i	0.564554i	−0.959333	+	0.282277i	$0.908910\pi$
$68$	0	0
$69$	25728.0i	0.650554i
$70$	0	0
$71$	28592.0	0.673130	0.336565	−	0.941660i	$-0.390735\pi$
$71$	28592.0	0.673130	0.336565	+	0.941660i	$0.390735\pi$
$72$	0	0
$73$	53670.0	1.17876	0.589379	−	0.807857i	$-0.299373\pi$
$73$	53670.0	1.17876	0.589379	+	0.807857i	$0.299373\pi$
$74$	0	0
$75$	23432.0i	0.481012i
$76$	0	0
$77$	111488.i	2.14290i
$78$	0	0
$79$	69152.0	1.24663	0.623314	−	0.781971i	$-0.285786\pi$
$79$	69152.0	1.24663	0.623314	+	0.781971i	$0.285786\pi$
$80$	0	0
$81$	16489.0	0.279243
$82$	0	0
$83$	37800.0i	0.602277i	0.953580	+	0.301139i	$0.0973667\pi$
$83$	37800.0i	0.602277i	−0.953580	+	0.301139i	$0.902633\pi$
$84$	0	0
$85$	17892.0i	0.268603i
$86$	0	0
$87$	−23344.0	−0.330657
$88$	0	0
$89$	126806.	1.69693	0.848467	−	0.529249i	$-0.177526\pi$
$89$	126806.	1.69693	0.848467	+	0.529249i	$0.177526\pi$
$90$	0	0
$91$	− 144352.i	− 1.82734i
$92$	0	0
$93$	20992.0i	0.251679i
$94$	0	0
$95$	−15568.0	−0.176980
$96$	0	0
$97$	62290.0	0.672185	0.336093	−	0.941829i	$-0.390894\pi$
$97$	62290.0	0.672185	0.336093	+	0.941829i	$0.390894\pi$
$98$	0	0
$99$	− 95944.0i	− 0.983852i
$100$	0	0
$101$	− 6414.00i	− 0.0625641i	−0.999511	−	0.0312821i	$-0.990041\pi$
$101$	− 6414.00i	− 0.0625641i	0.999511	−	0.0312821i	$-0.00995902\pi$
$102$	0	0
$103$	−108432.	−1.00708	−0.503541	−	0.863972i	$-0.667970\pi$
$103$	−108432.	−1.00708	−0.503541	+	0.863972i	$0.667970\pi$
$104$	0	0
$105$	−23296.0	−0.206209
$106$	0	0
$107$	103976.i	0.877958i	0.898497	+	0.438979i	$0.144660\pi$
$107$	103976.i	0.877958i	−0.898497	+	0.438979i	$0.855340\pi$
$108$	0	0
$109$	2486.00i	0.0200417i	0.999950	+	0.0100209i	$0.00318979\pi$
$109$	2486.00i	0.0200417i	−0.999950	+	0.0100209i	$0.996810\pi$
$110$	0	0
$111$	−75664.0	−0.582884
$112$	0	0
$113$	15794.0	0.116358	0.0581790	−	0.998306i	$-0.481471\pi$
$113$	15794.0	0.116358	0.0581790	+	0.998306i	$0.481471\pi$
$114$	0	0
$115$	− 45024.0i	− 0.317468i
$116$	0	0
$117$	124226.i	0.838973i
$118$	0	0
$119$	265824.	1.72079
$120$	0	0
$121$	−126245.	−0.783882
$122$	0	0
$123$	− 1360.00i	− 0.00810543i
$124$	0	0
$125$	− 84756.0i	− 0.485172i
$126$	0	0
$127$	−1024.00	−0.00563366	−0.00281683	−	0.999996i	$-0.500897\pi$
$127$	−1024.00	−0.00563366	−0.00281683	+	0.999996i	$0.500897\pi$
$128$	0	0
$129$	159424.	0.843489
$130$	0	0
$131$	22664.0i	0.115387i	0.998334	+	0.0576937i	$0.0183747\pi$
$131$	22664.0i	0.115387i	−0.998334	+	0.0576937i	$0.981625\pi$
$132$	0	0
$133$	231296.i	1.13381i
$134$	0	0
$135$	47264.0	0.223201
$136$	0	0
$137$	53238.0	0.242337	0.121169	−	0.992632i	$-0.461336\pi$
$137$	53238.0	0.242337	0.121169	+	0.992632i	$0.461336\pi$
$138$	0	0
$139$	19816.0i	0.0869919i	0.999054	+	0.0434960i	$0.0138496\pi$
$139$	19816.0i	0.0869919i	−0.999054	+	0.0434960i	$0.986150\pi$
$140$	0	0
$141$	− 256.000i	− 0.00108441i
$142$	0	0
$143$	371984.	1.52119
$144$	0	0
$145$	40852.0	0.161359
$146$	0	0
$147$	211656.i	0.807862i
$148$	0	0
$149$	− 452190.i	− 1.66861i	−0.551302	−	0.834306i	$-0.685869\pi$
$149$	− 452190.i	− 1.66861i	0.551302	−	0.834306i	$-0.314131\pi$
$150$	0	0
$151$	−263280.	−0.939670	−0.469835	−	0.882754i	$-0.655687\pi$
$151$	−263280.	−0.939670	−0.469835	+	0.882754i	$0.655687\pi$
$152$	0	0
$153$	−228762.	−0.790051
$154$	0	0
$155$	− 36736.0i	− 0.122818i
$156$	0	0
$157$	− 353530.i	− 1.14466i	−0.820023	−	0.572331i	$-0.806039\pi$
$157$	− 353530.i	− 1.14466i	0.820023	−	0.572331i	$-0.193961\pi$
$158$	0	0
$159$	−177424.	−0.556570
$160$	0	0
$161$	−668928.	−2.03383
$162$	0	0
$163$	100936.i	0.297562i	0.988870	+	0.148781i	$0.0475349\pi$
$163$	100936.i	0.297562i	−0.988870	+	0.148781i	$0.952465\pi$
$164$	0	0
$165$	− 60032.0i	− 0.171662i
$166$	0	0
$167$	−284944.	−0.790621	−0.395310	−	0.918548i	$-0.629363\pi$
$167$	−284944.	−0.790621	−0.395310	+	0.918548i	$0.629363\pi$
$168$	0	0
$169$	−110343.	−0.297186
$170$	0	0
$171$	− 199048.i	− 0.520556i
$172$	0	0
$173$	484374.i	1.23045i	0.788350	+	0.615227i	$0.210936\pi$
$173$	484374.i	1.23045i	−0.788350	+	0.615227i	$0.789064\pi$
$174$	0	0
$175$	−609232.	−1.50379
$176$	0	0
$177$	−331840.	−0.796151
$178$	0	0
$179$	− 406680.i	− 0.948681i	−0.880341	−	0.474341i	$-0.842687\pi$
$179$	− 406680.i	− 0.948681i	0.880341	−	0.474341i	$-0.157313\pi$
$180$	0	0
$181$	− 570302.i	− 1.29392i	−0.762523	−	0.646962i	$-0.776039\pi$
$181$	− 570302.i	− 1.29392i	0.762523	−	0.646962i	$-0.223961\pi$
$182$	0	0
$183$	−123696.	−0.273041
$184$	0	0
$185$	132412.	0.284445
$186$	0	0
$187$	685008.i	1.43249i
$188$	0	0
$189$	− 702208.i	− 1.42992i
$190$	0	0
$191$	−138624.	−0.274951	−0.137475	−	0.990505i	$-0.543899\pi$
$191$	−138624.	−0.274951	−0.137475	+	0.990505i	$0.543899\pi$
$192$	0	0
$193$	34482.0	0.0666345	0.0333173	−	0.999445i	$-0.489393\pi$
$193$	34482.0	0.0666345	0.0333173	+	0.999445i	$0.489393\pi$
$194$	0	0
$195$	77728.0i	0.146383i
$196$	0	0
$197$	− 643598.i	− 1.18154i	−0.806839	−	0.590771i	$-0.798824\pi$
$197$	− 643598.i	− 1.18154i	0.806839	−	0.590771i	$-0.201176\pi$
$198$	0	0
$199$	1.10738e6	1.98227	0.991134	−	0.132865i	$-0.0424178\pi$
$199$	1.10738e6	1.98227	0.991134	+	0.132865i	$0.0424178\pi$
$200$	0	0
$201$	−165952.	−0.289729
$202$	0	0
$203$	− 606944.i	− 1.03373i
$204$	0	0
$205$	2380.00i	0.00395542i
$206$	0	0
$207$	575664.	0.933777
$208$	0	0
$209$	−596032.	−0.943852
$210$	0	0
$211$	− 229976.i	− 0.355612i	−0.984066	−	0.177806i	$-0.943100\pi$
$211$	− 229976.i	− 0.355612i	0.984066	−	0.177806i	$-0.0568999\pi$
$212$	0	0
$213$	228736.i	0.345450i
$214$	0	0
$215$	−278992.	−0.411619
$216$	0	0
$217$	−545792.	−0.786824
$218$	0	0
$219$	429360.i	0.604939i
$220$	0	0
$221$	− 886932.i	− 1.22155i
$222$	0	0
$223$	1.08947e6	1.46708	0.733540	−	0.679646i	$-0.237867\pi$
$223$	1.08947e6	1.46708	0.733540	+	0.679646i	$0.237867\pi$
$224$	0	0
$225$	524291.	0.690424
$226$	0	0
$227$	687048.i	0.884958i	0.896779	+	0.442479i	$0.145901\pi$
$227$	687048.i	0.884958i	−0.896779	+	0.442479i	$0.854099\pi$
$228$	0	0
$229$	699730.i	0.881743i	0.897570	+	0.440871i	$0.145330\pi$
$229$	699730.i	0.881743i	−0.897570	+	0.440871i	$0.854670\pi$
$230$	0	0
$231$	−891904.	−1.09974
$232$	0	0
$233$	−937722.	−1.13158	−0.565789	−	0.824550i	$-0.691428\pi$
$233$	−937722.	−1.13158	−0.565789	+	0.824550i	$0.691428\pi$
$234$	0	0
$235$	448.000i	0 0.000529186i
$236$	0	0
$237$	553216.i	0.639770i
$238$	0	0
$239$	−643488.	−0.728695	−0.364347	−	0.931263i	$-0.618708\pi$
$239$	−643488.	−0.728695	−0.364347	+	0.931263i	$0.618708\pi$
$240$	0	0
$241$	157282.	0.174436	0.0872181	−	0.996189i	$-0.472202\pi$
$241$	157282.	0.174436	0.0872181	+	0.996189i	$0.472202\pi$
$242$	0	0
$243$	952280.i	1.03454i
$244$	0	0
$245$	− 370398.i	− 0.394233i
$246$	0	0
$247$	771728.	0.804863
$248$	0	0
$249$	−302400.	−0.309089
$250$	0	0
$251$	− 1.58604e6i	− 1.58902i	−0.607250	−	0.794511i	$-0.707727\pi$
$251$	− 1.58604e6i	− 1.58902i	0.607250	−	0.794511i	$-0.292273\pi$
$252$	0	0
$253$	− 1.72378e6i	− 1.69309i
$254$	0	0
$255$	−143136.	−0.137847
$256$	0	0
$257$	−654334.	−0.617969	−0.308984	−	0.951067i	$-0.599989\pi$
$257$	−654334.	−0.617969	−0.308984	+	0.951067i	$0.599989\pi$
$258$	0	0
$259$	− 1.96726e6i	− 1.82227i
$260$	0	0
$261$	522322.i	0.474610i
$262$	0	0
$263$	−330192.	−0.294359	−0.147179	−	0.989110i	$-0.547019\pi$
$263$	−330192.	−0.294359	−0.147179	+	0.989110i	$0.547019\pi$
$264$	0	0
$265$	310492.	0.271604
$266$	0	0
$267$	1.01445e6i	0.870867i
$268$	0	0
$269$	1.56956e6i	1.32250i	0.750164	+	0.661252i	$0.229974\pi$
$269$	1.56956e6i	1.32250i	−0.750164	+	0.661252i	$0.770026\pi$
$270$	0	0
$271$	−957792.	−0.792224	−0.396112	−	0.918202i	$-0.629641\pi$
$271$	−957792.	−0.792224	−0.396112	+	0.918202i	$0.629641\pi$
$272$	0	0
$273$	1.15482e6	0.937791
$274$	0	0
$275$	− 1.56994e6i	− 1.25185i
$276$	0	0
$277$	− 565438.i	− 0.442778i	−0.975186	−	0.221389i	$-0.928941\pi$
$277$	− 565438.i	− 0.442778i	0.975186	−	0.221389i	$-0.0710590\pi$
$278$	0	0
$279$	469696.	0.361249
$280$	0	0
$281$	1.34127e6	1.01333	0.506664	−	0.862143i	$-0.330878\pi$
$281$	1.34127e6	1.01333	0.506664	+	0.862143i	$0.330878\pi$
$282$	0	0
$283$	− 734264.i	− 0.544987i	−0.962158	−	0.272494i	$-0.912152\pi$
$283$	− 734264.i	− 0.544987i	0.962158	−	0.272494i	$-0.0878483\pi$
$284$	0	0
$285$	− 124544.i	− 0.0908261i
$286$	0	0
$287$	35360.0	0.0253401
$288$	0	0
$289$	213427.	0.150316
$290$	0	0
$291$	498320.i	0.344966i
$292$	0	0
$293$	1.13320e6i	0.771149i	0.922677	+	0.385574i	$0.125997\pi$
$293$	1.13320e6i	0.771149i	−0.922677	+	0.385574i	$0.874003\pi$
$294$	0	0
$295$	580720.	0.388519
$296$	0	0
$297$	1.80954e6	1.19035
$298$	0	0
$299$	2.23190e6i	1.44377i
$300$	0	0
$301$	4.14502e6i	2.63700i
$302$	0	0
$303$	51312.0	0.0321079
$304$	0	0
$305$	216468.	0.133243
$306$	0	0
$307$	2.91377e6i	1.76445i	0.470829	+	0.882224i	$0.343955\pi$
$307$	2.91377e6i	1.76445i	−0.470829	+	0.882224i	$0.656045\pi$
$308$	0	0
$309$	− 867456.i	− 0.516834i
$310$	0	0
$311$	−1.43813e6	−0.843134	−0.421567	−	0.906797i	$-0.638520\pi$
$311$	−1.43813e6	−0.843134	−0.421567	+	0.906797i	$0.638520\pi$
$312$	0	0
$313$	1.37601e6	0.793888	0.396944	−	0.917843i	$-0.370071\pi$
$313$	1.37601e6	0.793888	0.396944	+	0.917843i	$0.370071\pi$
$314$	0	0
$315$	521248.i	0.295984i
$316$	0	0
$317$	− 1.23494e6i	− 0.690235i	−0.938559	−	0.345118i	$-0.887839\pi$
$317$	− 1.23494e6i	− 0.690235i	0.938559	−	0.345118i	$-0.112161\pi$
$318$	0	0
$319$	1.56405e6	0.860545
$320$	0	0
$321$	−831808.	−0.450568
$322$	0	0
$323$	1.42114e6i	0.757930i
$324$	0	0
$325$	2.03273e6i	1.06751i
$326$	0	0
$327$	−19888.0	−0.0102854
$328$	0	0
$329$	6656.00	0.00339019
$330$	0	0
$331$	− 1.48930e6i	− 0.747160i	−0.927598	−	0.373580i	$-0.878130\pi$
$331$	− 1.48930e6i	− 0.747160i	0.927598	−	0.373580i	$-0.121870\pi$
$332$	0	0
$333$	1.69298e6i	0.836646i
$334$	0	0
$335$	290416.	0.141387
$336$	0	0
$337$	838226.	0.402056	0.201028	−	0.979586i	$-0.435572\pi$
$337$	838226.	0.402056	0.201028	+	0.979586i	$0.435572\pi$
$338$	0	0
$339$	126352.i	0.0597149i
$340$	0	0
$341$	− 1.40646e6i	− 0.655002i
$342$	0	0
$343$	−2.00720e6	−0.921203
$344$	0	0
$345$	360192.	0.162924
$346$	0	0
$347$	− 350008.i	− 0.156047i	−0.996952	−	0.0780233i	$-0.975139\pi$
$347$	− 350008.i	− 0.156047i	0.996952	−	0.0780233i	$-0.0248609\pi$
$348$	0	0
$349$	− 383642.i	− 0.168602i	−0.996440	−	0.0843010i	$-0.973134\pi$
$349$	− 383642.i	− 0.168602i	0.996440	−	0.0843010i	$-0.0268657\pi$
$350$	0	0
$351$	−2.34294e6	−1.01507
$352$	0	0
$353$	4.09309e6	1.74829	0.874147	−	0.485661i	$-0.161421\pi$
$353$	4.09309e6	1.74829	0.874147	+	0.485661i	$0.161421\pi$
$354$	0	0
$355$	− 400288.i	− 0.168578i
$356$	0	0
$357$	2.12659e6i	0.883108i
$358$	0	0
$359$	3.14430e6	1.28762	0.643811	−	0.765185i	$-0.277352\pi$
$359$	3.14430e6	1.28762	0.643811	+	0.765185i	$0.277352\pi$
$360$	0	0
$361$	1.23955e6	0.500608
$362$	0	0
$363$	− 1.00996e6i	− 0.402288i
$364$	0	0
$365$	− 751380.i	− 0.295208i
$366$	0	0
$367$	1.47619e6	0.572108	0.286054	−	0.958214i	$-0.407656\pi$
$367$	1.47619e6	0.572108	0.286054	+	0.958214i	$0.407656\pi$
$368$	0	0
$369$	−30430.0	−0.0116342
$370$	0	0
$371$	− 4.61302e6i	− 1.74001i
$372$	0	0
$373$	3.73981e6i	1.39180i	0.718138	+	0.695901i	$0.244995\pi$
$373$	3.73981e6i	1.39180i	−0.718138	+	0.695901i	$0.755005\pi$
$374$	0	0
$375$	678048.	0.248990
$376$	0	0
$377$	−2.02509e6	−0.733823
$378$	0	0
$379$	1.89966e6i	0.679324i	0.940548	+	0.339662i	$0.110313\pi$
$379$	1.89966e6i	0.679324i	−0.940548	+	0.339662i	$0.889687\pi$
$380$	0	0
$381$	− 8192.00i	− 0.00289120i
$382$	0	0
$383$	−1.74310e6	−0.607192	−0.303596	−	0.952801i	$-0.598187\pi$
$383$	−1.74310e6	−0.607192	−0.303596	+	0.952801i	$0.598187\pi$
$384$	0	0
$385$	1.56083e6	0.536666
$386$	0	0
$387$	− 3.56711e6i	− 1.21071i
$388$	0	0
$389$	2.69147e6i	0.901812i	0.892571	+	0.450906i	$0.148899\pi$
$389$	2.69147e6i	0.901812i	−0.892571	+	0.450906i	$0.851101\pi$
$390$	0	0
$391$	−4.11005e6	−1.35958
$392$	0	0
$393$	−181312.	−0.0592168
$394$	0	0
$395$	− 968128.i	− 0.312205i
$396$	0	0
$397$	5.37353e6i	1.71113i	0.517695	+	0.855565i	$0.326790\pi$
$397$	5.37353e6i	1.71113i	−0.517695	+	0.855565i	$0.673210\pi$
$398$	0	0
$399$	−1.85037e6	−0.581870
$400$	0	0
$401$	156418.	0.0485765	0.0242882	−	0.999705i	$-0.492268\pi$
$401$	156418.	0.0485765	0.0242882	+	0.999705i	$0.492268\pi$
$402$	0	0
$403$	1.82106e6i	0.558548i
$404$	0	0
$405$	− 230846.i	− 0.0699334i
$406$	0	0
$407$	5.06949e6	1.51697
$408$	0	0
$409$	306086.	0.0904764	0.0452382	−	0.998976i	$-0.485595\pi$
$409$	306086.	0.0904764	0.0452382	+	0.998976i	$0.485595\pi$
$410$	0	0
$411$	425904.i	0.124368i
$412$	0	0
$413$	− 8.62784e6i	− 2.48901i
$414$	0	0
$415$	529200.	0.150834
$416$	0	0
$417$	−158528.	−0.0446443
$418$	0	0
$419$	6.70868e6i	1.86682i	0.358814	+	0.933409i	$0.383181\pi$
$419$	6.70868e6i	1.86682i	−0.358814	+	0.933409i	$0.616819\pi$
$420$	0	0
$421$	− 4.02347e6i	− 1.10636i	−0.833063	−	0.553179i	$-0.813415\pi$
$421$	− 4.02347e6i	− 1.10636i	0.833063	−	0.553179i	$-0.186585\pi$
$422$	0	0
$423$	−5728.00	−0.00155651
$424$	0	0
$425$	−3.74326e6	−1.00526
$426$	0	0
$427$	− 3.21610e6i	− 0.853610i
$428$	0	0
$429$	2.97587e6i	0.780676i
$430$	0	0
$431$	7.04304e6	1.82628	0.913139	−	0.407648i	$-0.133651\pi$
$431$	7.04304e6	1.82628	0.913139	+	0.407648i	$0.133651\pi$
$432$	0	0
$433$	−1.25142e6	−0.320763	−0.160381	−	0.987055i	$-0.551272\pi$
$433$	−1.25142e6	−0.320763	−0.160381	+	0.987055i	$0.551272\pi$
$434$	0	0
$435$	326816.i	0.0828095i
$436$	0	0
$437$	− 3.57619e6i	− 0.895813i
$438$	0	0
$439$	−1.25406e6	−0.310569	−0.155285	−	0.987870i	$-0.549629\pi$
$439$	−1.25406e6	−0.310569	−0.155285	+	0.987870i	$0.549629\pi$
$440$	0	0
$441$	4.73580e6	1.15957
$442$	0	0
$443$	5.18081e6i	1.25426i	0.778914	+	0.627131i	$0.215771\pi$
$443$	5.18081e6i	1.25426i	−0.778914	+	0.627131i	$0.784229\pi$
$444$	0	0
$445$	− 1.77528e6i	− 0.424979i
$446$	0	0
$447$	3.61752e6	0.856332
$448$	0	0
$449$	−5.73064e6	−1.34149	−0.670745	−	0.741688i	$-0.734025\pi$
$449$	−5.73064e6	−1.34149	−0.670745	+	0.741688i	$0.734025\pi$
$450$	0	0
$451$	91120.0i	0.0210947i
$452$	0	0
$453$	− 2.10624e6i	− 0.482239i
$454$	0	0
$455$	−2.02093e6	−0.457638
$456$	0	0
$457$	5.24153e6	1.17400	0.586999	−	0.809588i	$-0.300309\pi$
$457$	5.24153e6	1.17400	0.586999	+	0.809588i	$0.300309\pi$
$458$	0	0
$459$	− 4.31453e6i	− 0.955876i
$460$	0	0
$461$	− 173994.i	− 0.0381313i	−0.999818	−	0.0190657i	$-0.993931\pi$
$461$	− 173994.i	− 0.0381313i	0.999818	−	0.0190657i	$-0.00606916\pi$
$462$	0	0
$463$	−4.01277e6	−0.869945	−0.434972	−	0.900444i	$-0.643242\pi$
$463$	−4.01277e6	−0.869945	−0.434972	+	0.900444i	$0.643242\pi$
$464$	0	0
$465$	293888.	0.0630303
$466$	0	0
$467$	− 774616.i	− 0.164359i	−0.996618	−	0.0821796i	$-0.973812\pi$
$467$	− 774616.i	− 0.164359i	0.996618	−	0.0821796i	$-0.0261881\pi$
$468$	0	0
$469$	− 4.31475e6i	− 0.905782i
$470$	0	0
$471$	2.82824e6	0.587441
$472$	0	0
$473$	−1.06814e7	−2.19521
$474$	0	0
$475$	− 3.25705e6i	− 0.662354i
$476$	0	0
$477$	3.96986e6i	0.798876i
$478$	0	0
$479$	−2.33530e6	−0.465054	−0.232527	−	0.972590i	$-0.574699\pi$
$479$	−2.33530e6	−0.465054	−0.232527	+	0.972590i	$0.574699\pi$
$480$	0	0
$481$	−6.56385e6	−1.29359
$482$	0	0
$483$	− 5.35142e6i	− 1.04376i
$484$	0	0
$485$	− 872060.i	− 0.168342i
$486$	0	0
$487$	1.03947e6	0.198605	0.0993025	−	0.995057i	$-0.468339\pi$
$487$	1.03947e6	0.198605	0.0993025	+	0.995057i	$0.468339\pi$
$488$	0	0
$489$	−807488.	−0.152709
$490$	0	0
$491$	7.85092e6i	1.46966i	0.678251	+	0.734830i	$0.262738\pi$
$491$	7.85092e6i	1.46966i	−0.678251	+	0.734830i	$0.737262\pi$
$492$	0	0
$493$	− 3.72920e6i	− 0.691033i
$494$	0	0
$495$	−1.34322e6	−0.246396
$496$	0	0
$497$	−5.94714e6	−1.07998
$498$	0	0
$499$	− 2.71644e6i	− 0.488370i	−0.969729	−	0.244185i	$-0.921480\pi$
$499$	− 2.71644e6i	− 0.488370i	0.969729	−	0.244185i	$-0.0785204\pi$
$500$	0	0
$501$	− 2.27955e6i	− 0.405747i
$502$	0	0
$503$	−4.62034e6	−0.814242	−0.407121	−	0.913374i	$-0.633467\pi$
$503$	−4.62034e6	−0.814242	−0.407121	+	0.913374i	$0.633467\pi$
$504$	0	0
$505$	−89796.0	−0.0156685
$506$	0	0
$507$	− 882744.i	− 0.152516i
$508$	0	0
$509$	− 4.46198e6i	− 0.763366i	−0.924293	−	0.381683i	$-0.875345\pi$
$509$	− 4.46198e6i	− 0.763366i	0.924293	−	0.381683i	$-0.124655\pi$
$510$	0	0
$511$	−1.11634e7	−1.89122
$512$	0	0
$513$	3.75411e6	0.629816
$514$	0	0
$515$	1.51805e6i	0.252213i
$516$	0	0
$517$	17152.0i	0.00282220i
$518$	0	0
$519$	−3.87499e6	−0.631470
$520$	0	0
$521$	−3.74375e6	−0.604245	−0.302122	−	0.953269i	$-0.597695\pi$
$521$	−3.74375e6	−0.604245	−0.302122	+	0.953269i	$0.597695\pi$
$522$	0	0
$523$	9.28433e6i	1.48421i	0.670282	+	0.742107i	$0.266173\pi$
$523$	9.28433e6i	1.48421i	−0.670282	+	0.742107i	$0.733827\pi$
$524$	0	0
$525$	− 4.87386e6i	− 0.771746i
$526$	0	0
$527$	−3.35347e6	−0.525979
$528$	0	0
$529$	3.90631e6	0.606915
$530$	0	0
$531$	7.42492e6i	1.14276i
$532$	0	0
$533$	− 117980.i	− 0.0179883i
$534$	0	0
$535$	1.45566e6	0.219875
$536$	0	0
$537$	3.25344e6	0.486863
$538$	0	0
$539$	− 1.41810e7i	− 2.10249i
$540$	0	0
$541$	862150.i	0.126645i	0.997993	+	0.0633227i	$0.0201697\pi$
$541$	862150.i	0.126645i	−0.997993	+	0.0633227i	$0.979830\pi$
$542$	0	0
$543$	4.56242e6	0.664042
$544$	0	0
$545$	34804.0	0.00501924
$546$	0	0
$547$	− 1.75442e6i	− 0.250707i	−0.992112	−	0.125353i	$-0.959994\pi$
$547$	− 1.75442e6i	− 0.250707i	0.992112	−	0.125353i	$-0.0400065\pi$
$548$	0	0
$549$	2.76770e6i	0.391911i
$550$	0	0
$551$	3.24482e6	0.455314
$552$	0	0
$553$	−1.43836e7	−2.00012
$554$	0	0
$555$	1.05930e6i	0.145977i
$556$	0	0
$557$	− 1.00292e7i	− 1.36971i	−0.728678	−	0.684856i	$-0.759865\pi$
$557$	− 1.00292e7i	− 1.36971i	0.728678	−	0.684856i	$-0.240135\pi$
$558$	0	0
$559$	1.38300e7	1.87195
$560$	0	0
$561$	−5.48006e6	−0.735154
$562$	0	0
$563$	5.27460e6i	0.701324i	0.936502	+	0.350662i	$0.114043\pi$
$563$	5.27460e6i	0.701324i	−0.936502	+	0.350662i	$0.885957\pi$
$564$	0	0
$565$	− 221116.i	− 0.0291406i
$566$	0	0
$567$	−3.42971e6	−0.448023
$568$	0	0
$569$	−8.36940e6	−1.08371	−0.541856	−	0.840471i	$-0.682278\pi$
$569$	−8.36940e6	−1.08371	−0.541856	+	0.840471i	$0.682278\pi$
$570$	0	0
$571$	4.02702e6i	0.516884i	0.966027	+	0.258442i	$0.0832091\pi$
$571$	4.02702e6i	0.516884i	−0.966027	+	0.258442i	$0.916791\pi$
$572$	0	0
$573$	− 1.10899e6i	− 0.141105i
$574$	0	0
$575$	9.41966e6	1.18814
$576$	0	0
$577$	2.37568e6	0.297063	0.148532	−	0.988908i	$-0.452545\pi$
$577$	2.37568e6	0.297063	0.148532	+	0.988908i	$0.452545\pi$
$578$	0	0
$579$	275856.i	0.0341968i
$580$	0	0
$581$	− 7.86240e6i	− 0.966306i
$582$	0	0
$583$	1.18874e7	1.44849
$584$	0	0
$585$	1.73916e6	0.210112
$586$	0	0
$587$	− 3.44028e6i	− 0.412096i	−0.978542	−	0.206048i	$-0.933940\pi$
$587$	− 3.44028e6i	− 0.412096i	0.978542	−	0.206048i	$-0.0660603\pi$
$588$	0	0
$589$	− 2.91789e6i	− 0.346562i
$590$	0	0
$591$	5.14878e6	0.606368
$592$	0	0
$593$	−3.22942e6	−0.377127	−0.188564	−	0.982061i	$-0.560383\pi$
$593$	−3.22942e6	−0.377127	−0.188564	+	0.982061i	$0.560383\pi$
$594$	0	0
$595$	− 3.72154e6i	− 0.430953i
$596$	0	0
$597$	8.85901e6i	1.01730i
$598$	0	0
$599$	−9.29714e6	−1.05872	−0.529361	−	0.848397i	$-0.677568\pi$
$599$	−9.29714e6	−1.05872	−0.529361	+	0.848397i	$0.677568\pi$
$600$	0	0
$601$	1.12782e7	1.27366	0.636828	−	0.771006i	$-0.280246\pi$
$601$	1.12782e7	1.27366	0.636828	+	0.771006i	$0.280246\pi$
$602$	0	0
$603$	3.71318e6i	0.415865i
$604$	0	0
$605$	1.76743e6i	0.196315i
$606$	0	0
$607$	−7.00115e6	−0.771255	−0.385627	−	0.922655i	$-0.626015\pi$
$607$	−7.00115e6	−0.771255	−0.385627	+	0.922655i	$0.626015\pi$
$608$	0	0
$609$	4.85555e6	0.530512
$610$	0	0
$611$	− 22208.0i	− 0.00240661i
$612$	0	0
$613$	6.19432e6i	0.665798i	0.942962	+	0.332899i	$0.108027\pi$
$613$	6.19432e6i	0.665798i	−0.942962	+	0.332899i	$0.891973\pi$
$614$	0	0
$615$	−19040.0	−0.00202992
$616$	0	0
$617$	2.62407e6	0.277500	0.138750	−	0.990327i	$-0.455692\pi$
$617$	2.62407e6	0.277500	0.138750	+	0.990327i	$0.455692\pi$
$618$	0	0
$619$	− 2.83721e6i	− 0.297622i	−0.988866	−	0.148811i	$-0.952455\pi$
$619$	− 2.83721e6i	− 0.297622i	0.988866	−	0.148811i	$-0.0475446\pi$
$620$	0	0
$621$	1.08572e7i	1.12977i
$622$	0	0
$623$	−2.63756e7	−2.72259
$624$	0	0
$625$	7.96654e6	0.815774
$626$	0	0
$627$	− 4.76826e6i	− 0.484385i
$628$	0	0
$629$	− 1.20873e7i	− 1.21816i
$630$	0	0
$631$	1.29656e7	1.29634	0.648170	−	0.761496i	$-0.275535\pi$
$631$	1.29656e7	1.29634	0.648170	+	0.761496i	$0.275535\pi$
$632$	0	0
$633$	1.83981e6	0.182500
$634$	0	0
$635$	14336.0i	0.00141089i
$636$	0	0
$637$	1.83612e7i	1.79288i
$638$	0	0
$639$	5.11797e6	0.495844
$640$	0	0
$641$	−1.16798e7	−1.12276	−0.561382	−	0.827557i	$-0.689730\pi$
$641$	−1.16798e7	−1.12276	−0.561382	+	0.827557i	$0.689730\pi$
$642$	0	0
$643$	− 7.02732e6i	− 0.670289i	−0.942167	−	0.335145i	$-0.891215\pi$
$643$	− 7.02732e6i	− 0.670289i	0.942167	−	0.335145i	$-0.108785\pi$
$644$	0	0
$645$	− 2.23194e6i	− 0.211243i
$646$	0	0
$647$	−9.72821e6	−0.913634	−0.456817	−	0.889561i	$-0.651011\pi$
$647$	−9.72821e6	−0.913634	−0.456817	+	0.889561i	$0.651011\pi$
$648$	0	0
$649$	2.22333e7	2.07201
$650$	0	0
$651$	− 4.36634e6i	− 0.403798i
$652$	0	0
$653$	− 9.81425e6i	− 0.900688i	−0.892855	−	0.450344i	$-0.851301\pi$
$653$	− 9.81425e6i	− 0.900688i	0.892855	−	0.450344i	$-0.148699\pi$
$654$	0	0
$655$	317296.	0.0288976
$656$	0	0
$657$	9.60693e6	0.868303
$658$	0	0
$659$	− 1.46652e7i	− 1.31545i	−0.753259	−	0.657724i	$-0.771519\pi$
$659$	− 1.46652e7i	− 1.31545i	0.753259	−	0.657724i	$-0.228481\pi$
$660$	0	0
$661$	1.41836e7i	1.26265i	0.775517	+	0.631327i	$0.217489\pi$
$661$	1.41836e7i	1.26265i	−0.775517	+	0.631327i	$0.782511\pi$
$662$	0	0
$663$	7.09546e6	0.626897
$664$	0	0
$665$	3.23814e6	0.283950
$666$	0	0
$667$	9.38429e6i	0.816745i
$668$	0	0
$669$	8.71578e6i	0.752906i
$670$	0	0
$671$	8.28763e6	0.710598
$672$	0	0
$673$	5.49941e6	0.468035	0.234018	−	0.972232i	$-0.424813\pi$
$673$	5.49941e6	0.468035	0.234018	+	0.972232i	$0.424813\pi$
$674$	0	0
$675$	9.88830e6i	0.835338i
$676$	0	0
$677$	− 1.77375e7i	− 1.48737i	−0.668528	−	0.743687i	$-0.733075\pi$
$677$	− 1.77375e7i	− 1.48737i	0.668528	−	0.743687i	$-0.266925\pi$
$678$	0	0
$679$	−1.29563e7	−1.07847
$680$	0	0
$681$	−5.49638e6	−0.454160
$682$	0	0
$683$	− 9.39670e6i	− 0.770768i	−0.922756	−	0.385384i	$-0.874069\pi$
$683$	− 9.39670e6i	− 0.770768i	0.922756	−	0.385384i	$-0.125931\pi$
$684$	0	0
$685$	− 745332.i	− 0.0606909i
$686$	0	0
$687$	−5.59784e6	−0.452510
$688$	0	0
$689$	−1.53915e7	−1.23519
$690$	0	0
$691$	1.34767e7i	1.07371i	0.843673	+	0.536857i	$0.180389\pi$
$691$	1.34767e7i	1.07371i	−0.843673	+	0.536857i	$0.819611\pi$
$692$	0	0
$693$	1.99564e7i	1.57851i
$694$	0	0
$695$	277424.	0.0217862
$696$	0	0
$697$	217260.	0.0169394
$698$	0	0
$699$	− 7.50178e6i	− 0.580726i
$700$	0	0
$701$	2.15594e7i	1.65707i	0.559935	+	0.828536i	$0.310826\pi$
$701$	2.15594e7i	1.65707i	−0.559935	+	0.828536i	$0.689174\pi$
$702$	0	0
$703$	1.05173e7	0.802631
$704$	0	0
$705$	−3584.00	−0.000271578 0
$706$	0	0
$707$	1.33411e6i	0.100379i
$708$	0	0
$709$	6.38165e6i	0.476779i	0.971170	+	0.238390i	$0.0766195\pi$
$709$	6.38165e6i	0.476779i	−0.971170	+	0.238390i	$0.923380\pi$
$710$	0	0
$711$	1.23782e7	0.918298
$712$	0	0
$713$	8.43878e6	0.621664
$714$	0	0
$715$	− 5.20778e6i	− 0.380967i
$716$	0	0
$717$	− 5.14790e6i	− 0.373966i
$718$	0	0
$719$	1.63566e7	1.17997	0.589986	−	0.807413i	$-0.299133\pi$
$719$	1.63566e7	1.17997	0.589986	+	0.807413i	$0.299133\pi$
$720$	0	0
$721$	2.25539e7	1.61578
$722$	0	0
$723$	1.25826e6i	0.0895207i
$724$	0	0
$725$	8.54682e6i	0.603893i
$726$	0	0
$727$	−2.13130e7	−1.49558	−0.747788	−	0.663937i	$-0.768884\pi$
$727$	−2.13130e7	−1.49558	−0.747788	+	0.663937i	$0.768884\pi$
$728$	0	0
$729$	−3.61141e6	−0.251686
$730$	0	0
$731$	2.54680e7i	1.76279i
$732$	0	0
$733$	1.21571e7i	0.835737i	0.908507	+	0.417869i	$0.137223\pi$
$733$	1.21571e7i	0.835737i	−0.908507	+	0.417869i	$0.862777\pi$
$734$	0	0
$735$	2.96318e6	0.202321
$736$	0	0
$737$	1.11188e7	0.754030
$738$	0	0
$739$	1.92337e7i	1.29555i	0.761834	+	0.647773i	$0.224299\pi$
$739$	1.92337e7i	1.29555i	−0.761834	+	0.647773i	$0.775701\pi$
$740$	0	0
$741$	6.17382e6i	0.413056i
$742$	0	0
$743$	1.66565e6	0.110691	0.0553454	−	0.998467i	$-0.482374\pi$
$743$	1.66565e6	0.110691	0.0553454	+	0.998467i	$0.482374\pi$
$744$	0	0
$745$	−6.33066e6	−0.417886
$746$	0	0
$747$	6.76620e6i	0.443653i
$748$	0	0
$749$	− 2.16270e7i	− 1.40861i
$750$	0	0
$751$	−9.81290e6	−0.634888	−0.317444	−	0.948277i	$-0.602825\pi$
$751$	−9.81290e6	−0.634888	−0.317444	+	0.948277i	$0.602825\pi$
$752$	0	0
$753$	1.26883e7	0.815486
$754$	0	0
$755$	3.68592e6i	0.235331i
$756$	0	0
$757$	1.92753e7i	1.22254i	0.791423	+	0.611269i	$0.209341\pi$
$757$	1.92753e7i	1.22254i	−0.791423	+	0.611269i	$0.790659\pi$
$758$	0	0
$759$	1.37902e7	0.868893
$760$	0	0
$761$	1.17863e7	0.737762	0.368881	−	0.929477i	$-0.379741\pi$
$761$	1.17863e7	0.737762	0.368881	+	0.929477i	$0.379741\pi$
$762$	0	0
$763$	− 517088.i	− 0.0321553i
$764$	0	0
$765$	3.20267e6i	0.197860i
$766$	0	0
$767$	−2.87871e7	−1.76689
$768$	0	0
$769$	1.22941e7	0.749690	0.374845	−	0.927087i	$-0.377696\pi$
$769$	1.22941e7	0.749690	0.374845	+	0.927087i	$0.377696\pi$
$770$	0	0
$771$	− 5.23467e6i	− 0.317142i
$772$	0	0
$773$	− 2.57086e6i	− 0.154750i	−0.997002	−	0.0773749i	$-0.975346\pi$
$773$	− 2.57086e6i	− 0.154750i	0.997002	−	0.0773749i	$-0.0246538\pi$
$774$	0	0
$775$	7.68570e6	0.459652
$776$	0	0
$777$	1.57381e7	0.935190
$778$	0	0
$779$	189040.i	0.0111612i
$780$	0	0
$781$	− 1.53253e7i	− 0.899046i
$782$	0	0
$783$	−9.85117e6	−0.574227
$784$	0	0
$785$	−4.94942e6	−0.286669
$786$	0	0
$787$	− 5.54594e6i	− 0.319182i	−0.987183	−	0.159591i	$-0.948982\pi$
$787$	− 5.54594e6i	− 0.319182i	0.987183	−	0.159591i	$-0.0510176\pi$
$788$	0	0
$789$	− 2.64154e6i	− 0.151065i
$790$	0	0
$791$	−3.28515e6	−0.186687
$792$	0	0
$793$	−1.07306e7	−0.605958
$794$	0	0
$795$	2.48394e6i	0.139387i
$796$	0	0
$797$	2.09610e7i	1.16887i	0.811440	+	0.584436i	$0.198684\pi$
$797$	2.09610e7i	1.16887i	−0.811440	+	0.584436i	$0.801316\pi$
$798$	0	0
$799$	40896.0	0.00226628
$800$	0	0
$801$	2.26983e7	1.25000
$802$	0	0
$803$	− 2.87671e7i	− 1.57437i
$804$	0	0
$805$	9.36499e6i	0.509352i
$806$	0	0
$807$	−1.25565e7	−0.678709
$808$	0	0
$809$	−2.70297e7	−1.45201	−0.726005	−	0.687690i	$-0.758625\pi$
$809$	−2.70297e7	−1.45201	−0.726005	+	0.687690i	$0.758625\pi$
$810$	0	0
$811$	− 2.13052e6i	− 0.113745i	−0.998381	−	0.0568727i	$-0.981887\pi$
$811$	− 2.13052e6i	− 0.113745i	0.998381	−	0.0568727i	$-0.0181129\pi$
$812$	0	0
$813$	− 7.66234e6i	− 0.406570i
$814$	0	0
$815$	1.41310e6	0.0745212
$816$	0	0
$817$	−2.21599e7	−1.16148
$818$	0	0
$819$	− 2.58390e7i	− 1.34607i
$820$	0	0
$821$	− 1.58060e7i	− 0.818396i	−0.912446	−	0.409198i	$-0.865809\pi$
$821$	− 1.58060e7i	− 0.818396i	0.912446	−	0.409198i	$-0.134191\pi$
$822$	0	0
$823$	−2.28848e7	−1.17773	−0.588867	−	0.808230i	$-0.700426\pi$
$823$	−2.28848e7	−1.17773	−0.588867	+	0.808230i	$0.700426\pi$
$824$	0	0
$825$	1.25596e7	0.642450
$826$	0	0
$827$	− 2.55336e7i	− 1.29822i	−0.760695	−	0.649109i	$-0.775142\pi$
$827$	− 2.55336e7i	− 1.29822i	0.760695	−	0.649109i	$-0.224858\pi$
$828$	0	0
$829$	8.31786e6i	0.420364i	0.977662	+	0.210182i	$0.0674056\pi$
$829$	8.31786e6i	0.420364i	−0.977662	+	0.210182i	$0.932594\pi$
$830$	0	0
$831$	4.52350e6	0.227234
$832$	0	0
$833$	−3.38120e7	−1.68834
$834$	0	0
$835$	3.98922e6i	0.198003i
$836$	0	0
$837$	8.85862e6i	0.437072i
$838$	0	0
$839$	3.66261e7	1.79633	0.898164	−	0.439660i	$-0.144901\pi$
$839$	3.66261e7	1.79633	0.898164	+	0.439660i	$0.144901\pi$
$840$	0	0
$841$	1.19964e7	0.584873
$842$	0	0
$843$	1.07302e7i	0.520041i
$844$	0	0
$845$	1.54480e6i	0.0744271i
$846$	0	0
$847$	2.62590e7	1.25768
$848$	0	0
$849$	5.87411e6	0.279687
$850$	0	0
$851$	3.04169e7i	1.43976i
$852$	0	0
$853$	− 1.74802e7i	− 0.822571i	−0.911507	−	0.411286i	$-0.865080\pi$
$853$	− 1.74802e7i	− 0.822571i	0.911507	−	0.411286i	$-0.134920\pi$
$854$	0	0
$855$	−2.78667e6	−0.130368
$856$	0	0
$857$	−9.31062e6	−0.433038	−0.216519	−	0.976278i	$-0.569470\pi$
$857$	−9.31062e6	−0.433038	−0.216519	+	0.976278i	$0.569470\pi$
$858$	0	0
$859$	− 3.49525e7i	− 1.61620i	−0.589045	−	0.808101i	$-0.700496\pi$
$859$	− 3.49525e7i	− 1.61620i	0.589045	−	0.808101i	$-0.299504\pi$
$860$	0	0
$861$	282880.i	0.0130045i
$862$	0	0
$863$	2.02349e7	0.924858	0.462429	−	0.886656i	$-0.346978\pi$
$863$	2.02349e7	0.924858	0.462429	+	0.886656i	$0.346978\pi$
$864$	0	0
$865$	6.78124e6	0.308155
$866$	0	0
$867$	1.70742e6i	0.0771421i
$868$	0	0
$869$	− 3.70655e7i	− 1.66502i
$870$	0	0
$871$	−1.43963e7	−0.642994
$872$	0	0
$873$	1.11499e7	0.495149
$874$	0	0
$875$	1.76292e7i	0.778419i
$876$	0	0
$877$	− 1.98342e7i	− 0.870797i	−0.900238	−	0.435398i	$-0.856608\pi$
$877$	− 1.98342e7i	− 0.870797i	0.900238	−	0.435398i	$-0.143392\pi$
$878$	0	0
$879$	−9.06562e6	−0.395754
$880$	0	0
$881$	−3.81023e7	−1.65391	−0.826954	−	0.562270i	$-0.809928\pi$
$881$	−3.81023e7	−1.65391	−0.826954	+	0.562270i	$0.809928\pi$
$882$	0	0
$883$	2.41560e7i	1.04261i	0.853369	+	0.521307i	$0.174555\pi$
$883$	2.41560e7i	1.04261i	−0.853369	+	0.521307i	$0.825445\pi$
$884$	0	0
$885$	4.64576e6i	0.199388i
$886$	0	0
$887$	2.02368e7	0.863638	0.431819	−	0.901960i	$-0.357872\pi$
$887$	2.02368e7	0.863638	0.431819	+	0.901960i	$0.357872\pi$
$888$	0	0
$889$	212992.	0.00903876
$890$	0	0
$891$	− 8.83810e6i	− 0.372962i
$892$	0	0
$893$	35584.0i	0.00149323i
$894$	0	0
$895$	−5.69352e6	−0.237587
$896$	0	0
$897$	−1.78552e7	−0.740942
$898$	0	0
$899$	7.65683e6i	0.315973i
$900$	0	0
$901$	− 2.83435e7i	− 1.16316i
$902$	0	0
$903$	−3.31602e7	−1.35331
$904$	0	0
$905$	−7.98423e6	−0.324050
$906$	0	0
$907$	1.97976e7i	0.799088i	0.916714	+	0.399544i	$0.130831\pi$
$907$	1.97976e7i	0.799088i	−0.916714	+	0.399544i	$0.869169\pi$
$908$	0	0
$909$	− 1.14811e6i	− 0.0460863i
$910$	0	0
$911$	2.13242e7	0.851288	0.425644	−	0.904891i	$-0.360048\pi$
$911$	2.13242e7	0.851288	0.425644	+	0.904891i	$0.360048\pi$
$912$	0	0
$913$	2.02608e7	0.804414
$914$	0	0
$915$	1.73174e6i	0.0683803i
$916$	0	0
$917$	− 4.71411e6i	− 0.185130i
$918$	0	0
$919$	3.49941e7	1.36680	0.683401	−	0.730043i	$-0.260500\pi$
$919$	3.49941e7	1.36680	0.683401	+	0.730043i	$0.260500\pi$
$920$	0	0
$921$	−2.33101e7	−0.905515
$922$	0	0
$923$	1.98428e7i	0.766655i
$924$	0	0
$925$	2.77025e7i	1.06455i
$926$	0	0
$927$	−1.94093e7	−0.741842
$928$	0	0
$929$	−2.88107e7	−1.09525	−0.547627	−	0.836722i	$-0.684469\pi$
$929$	−2.88107e7	−1.09525	−0.547627	+	0.836722i	$0.684469\pi$
$930$	0	0
$931$	− 2.94202e7i	− 1.11243i
$932$	0	0
$933$	− 1.15050e7i	− 0.432697i
$934$	0	0
$935$	9.59011e6	0.358752
$936$	0	0
$937$	−1.28854e7	−0.479457	−0.239729	−	0.970840i	$-0.577059\pi$
$937$	−1.28854e7	−0.479457	−0.239729	+	0.970840i	$0.577059\pi$
$938$	0	0
$939$	1.10080e7i	0.407424i
$940$	0	0
$941$	− 5.27615e7i	− 1.94242i	−0.238227	−	0.971210i	$-0.576566\pi$
$941$	− 5.27615e7i	− 1.94242i	0.238227	−	0.971210i	$-0.423434\pi$
$942$	0	0
$943$	−546720.	−0.0200210
$944$	0	0
$945$	−9.83091e6	−0.358108
$946$	0	0
$947$	− 5.53961e6i	− 0.200726i	−0.994951	−	0.100363i	$-0.968000\pi$
$947$	− 5.53961e6i	− 0.200726i	0.994951	−	0.100363i	$-0.0320004\pi$
$948$	0	0
$949$	3.72470e7i	1.34253i
$950$	0	0
$951$	9.87950e6	0.354229
$952$	0	0
$953$	228102.	0.00813574	0.00406787	−	0.999992i	$-0.498705\pi$
$953$	228102.	0.00813574	0.00406787	+	0.999992i	$0.498705\pi$
$954$	0	0
$955$	1.94074e6i	0.0688586i
$956$	0	0
$957$	1.25124e7i	0.441632i
$958$	0	0
$959$	−1.10735e7	−0.388811
$960$	0	0
$961$	−2.17438e7	−0.759498
$962$	0	0
$963$	1.86117e7i	0.646726i
$964$	0	0
$965$	− 482748.i	− 0.0166879i
$966$	0	0
$967$	7.36709e6	0.253355	0.126678	−	0.991944i	$-0.459569\pi$
$967$	7.36709e6	0.253355	0.126678	+	0.991944i	$0.459569\pi$
$968$	0	0
$969$	−1.13691e7	−0.388970
$970$	0	0
$971$	− 1.91161e7i	− 0.650654i	−0.945602	−	0.325327i	$-0.894526\pi$
$971$	− 1.91161e7i	− 0.650654i	0.945602	−	0.325327i	$-0.105474\pi$
$972$	0	0
$973$	− 4.12173e6i	− 0.139572i
$974$	0	0
$975$	−1.62618e7	−0.547844
$976$	0	0
$977$	5.57040e7	1.86702	0.933512	−	0.358546i	$-0.116727\pi$
$977$	5.57040e7	1.86702	0.933512	+	0.358546i	$0.116727\pi$
$978$	0	0
$979$	− 6.79680e7i	− 2.26646i
$980$	0	0
$981$	444994.i	0.0147632i
$982$	0	0
$983$	−1.55469e7	−0.513167	−0.256584	−	0.966522i	$-0.582597\pi$
$983$	−1.55469e7	−0.513167	−0.256584	+	0.966522i	$0.582597\pi$
$984$	0	0
$985$	−9.01037e6	−0.295905
$986$	0	0
$987$	53248.0i	0.00173984i
$988$	0	0
$989$	− 6.40884e7i	− 2.08348i
$990$	0	0
$991$	−2.36890e7	−0.766237	−0.383118	−	0.923699i	$-0.625150\pi$
$991$	−2.36890e7	−0.766237	−0.383118	+	0.923699i	$0.625150\pi$
$992$	0	0
$993$	1.19144e7	0.383442
$994$	0	0
$995$	− 1.55033e7i	− 0.496438i
$996$	0	0
$997$	− 3.71720e6i	− 0.118434i	−0.998245	−	0.0592172i	$-0.981140\pi$
$997$	− 3.71720e6i	− 0.118434i	0.998245	−	0.0592172i	$-0.0188605\pi$
$998$	0	0
$999$	−3.19302e7	−1.01225

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.6.b.b.129.2		2
4.3	odd	2		256.6.b.h.129.1		2
8.3	odd	2		256.6.b.h.129.2		2
8.5	even	2	inner	256.6.b.b.129.1		2
16.3	odd	4		64.6.a.e.1.1		1
16.5	even	4		32.6.a.c.1.1	yes	1
16.11	odd	4		32.6.a.a.1.1	✓	1
16.13	even	4		64.6.a.c.1.1		1
48.5	odd	4		288.6.a.e.1.1		1
48.11	even	4		288.6.a.d.1.1		1
48.29	odd	4		576.6.a.v.1.1		1
48.35	even	4		576.6.a.u.1.1		1
80.27	even	4		800.6.c.a.449.2		2
80.37	odd	4		800.6.c.b.449.1		2
80.43	even	4		800.6.c.a.449.1		2
80.53	odd	4		800.6.c.b.449.2		2
80.59	odd	4		800.6.a.e.1.1		1
80.69	even	4		800.6.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.6.a.a.1.1	✓	1	16.11	odd	4
32.6.a.c.1.1	yes	1	16.5	even	4
64.6.a.c.1.1		1	16.13	even	4
64.6.a.e.1.1		1	16.3	odd	4
256.6.b.b.129.1		2	8.5	even	2	inner
256.6.b.b.129.2		2	1.1	even	1	trivial
256.6.b.h.129.1		2	4.3	odd	2
256.6.b.h.129.2		2	8.3	odd	2
288.6.a.d.1.1		1	48.11	even	4
288.6.a.e.1.1		1	48.5	odd	4
576.6.a.u.1.1		1	48.35	even	4
576.6.a.v.1.1		1	48.29	odd	4
800.6.a.a.1.1		1	80.69	even	4
800.6.a.e.1.1		1	80.59	odd	4
800.6.c.a.449.1		2	80.43	even	4
800.6.c.a.449.2		2	80.27	even	4
800.6.c.b.449.1		2	80.37	odd	4
800.6.c.b.449.2		2	80.53	odd	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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	256.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+8.00000i q^{3} -14.0000i q^{5} -208.000 q^{7} +179.000 q^{9} +O(q^{10})\)	\(q+8.00000i q^{3} -14.0000i q^{5} -208.000 q^{7} +179.000 q^{9} -536.000i q^{11} +694.000i q^{13} +112.000 q^{15} -1278.00 q^{17} -1112.00i q^{19} -1664.00i q^{21} +3216.00 q^{23} +2929.00 q^{25} +3376.00i q^{27} +2918.00i q^{29} +2624.00 q^{31} +4288.00 q^{33} +2912.00i q^{35} +9458.00i q^{37} -5552.00 q^{39} -170.000 q^{41} -19928.0i q^{43} -2506.00i q^{45} -32.0000 q^{47} +26457.0 q^{49} -10224.0i q^{51} +22178.0i q^{53} -7504.00 q^{55} +8896.00 q^{57} +41480.0i q^{59} +15462.0i q^{61} -37232.0 q^{63} +9716.00 q^{65} +20744.0i q^{67} +25728.0i q^{69} +28592.0 q^{71} +53670.0 q^{73} +23432.0i q^{75} +111488. i q^{77} +69152.0 q^{79} +16489.0 q^{81} +37800.0i q^{83} +17892.0i q^{85} -23344.0 q^{87} +126806. q^{89} -144352. i q^{91} +20992.0i q^{93} -15568.0 q^{95} +62290.0 q^{97} -95944.0i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 416 q^{7} + 358 q^{9}+O(q^{10}) \) 2 * q - 416 * q^7 + 358 * q^9	\( 2 q - 416 q^{7} + 358 q^{9} + 224 q^{15} - 2556 q^{17} + 6432 q^{23} + 5858 q^{25} + 5248 q^{31} + 8576 q^{33} - 11104 q^{39} - 340 q^{41} - 64 q^{47} + 52914 q^{49} - 15008 q^{55} + 17792 q^{57} - 74464 q^{63} + 19432 q^{65} + 57184 q^{71} + 107340 q^{73} + 138304 q^{79} + 32978 q^{81} - 46688 q^{87} + 253612 q^{89} - 31136 q^{95} + 124580 q^{97}+O(q^{100}) \) 2 * q - 416 * q^7 + 358 * q^9 + 224 * q^15 - 2556 * q^17 + 6432 * q^23 + 5858 * q^25 + 5248 * q^31 + 8576 * q^33 - 11104 * q^39 - 340 * q^41 - 64 * q^47 + 52914 * q^49 - 15008 * q^55 + 17792 * q^57 - 74464 * q^63 + 19432 * q^65 + 57184 * q^71 + 107340 * q^73 + 138304 * q^79 + 32978 * q^81 - 46688 * q^87 + 253612 * q^89 - 31136 * q^95 + 124580 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more