LMFDB - Embedded newform 784.6.f.c.783.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [784,6,Mod(783,784)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("784.783"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	784.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [14,0,-18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$125.740914733$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 3 x^{13} + 691 x^{12} - 8602 x^{11} + 416261 x^{10} - 3521447 x^{9} + 66162087 x^{8} + \cdots + 17213603549184 $ x^14 - 3x^13 + 691x^12 - 8602x^11 + 416261x^10 - 3521447x^9 + 66162087x^8 - 31847546x^7 + 3819865057x^6 + 14320412589x^5 + 280505627703x^4 + 1315235310354x^3 + 6574080917124x^2 + 11537944725888*x + 17213603549184
Coefficient ring:	$\Z[a_1, \ldots, a_{43}]$
Coefficient ring index:	$ 2^{27}\cdot 3^{6}\cdot 7^{4} $
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			783.5
Root			$-1.05545 + 1.82809i$ of defining polynomial
Character	$\chi$	$=$	784.783
Dual form			784.6.f.c.783.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-8.95216 q^{3} -71.7457i q^{5} -162.859 q^{9} +153.533i q^{11} -891.047i q^{13} +642.279i q^{15} +1521.01i q^{17} +1768.03 q^{19} -3837.69i q^{23} -2022.45 q^{25} +3633.31 q^{27} -2658.88 q^{29} +6179.19 q^{31} -1374.45i q^{33} +8528.02 q^{37} +7976.79i q^{39} -18788.9i q^{41} -19232.2i q^{43} +11684.4i q^{45} -7081.56 q^{47} -13616.3i q^{51} +25162.9 q^{53} +11015.3 q^{55} -15827.6 q^{57} -1291.54 q^{59} +23945.3i q^{61} -63928.8 q^{65} -16460.2i q^{67} +34355.6i q^{69} -25824.7i q^{71} -10443.1i q^{73} +18105.3 q^{75} -17196.2i q^{79} +7048.71 q^{81} +53986.4 q^{83} +109126. q^{85} +23802.7 q^{87} -30675.3i q^{89} -55317.1 q^{93} -126848. i q^{95} +70219.1i q^{97} -25004.2i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 18 q^{3} + 1076 q^{9} - 4270 q^{19} - 10868 q^{25} - 17910 q^{27} + 684 q^{29} + 6238 q^{31} - 6862 q^{37} + 16818 q^{47} - 3414 q^{53} + 56622 q^{55} + 68450 q^{57} - 98610 q^{59} - 34896 q^{65}+ \cdots - 585746 q^{93}+O(q^{100}) $ 14 * q - 18 * q^3 + 1076 * q^9 - 4270 * q^19 - 10868 * q^25 - 17910 * q^27 + 684 * q^29 + 6238 * q^31 - 6862 * q^37 + 16818 * q^47 - 3414 * q^53 + 56622 * q^55 + 68450 * q^57 - 98610 * q^59 - 34896 * q^65 - 295668 * q^75 + 291134 * q^81 + 305400 * q^83 - 176454 * q^85 - 190068 * q^87 - 585746 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$.

$n$	$197$	$687$	$689$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	−8.95216	−0.574281	−0.287141	−	0.957888i	$-0.592705\pi$
$3$	−8.95216	−0.574281	−0.287141	+	0.957888i	$0.592705\pi$
$4$	0	0
$5$	− 71.7457i	− 1.28343i	−0.766945	−	0.641713i	$-0.778224\pi$
$5$	− 71.7457i	− 1.28343i	0.766945	−	0.641713i	$-0.221776\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−162.859	−0.670201
$10$	0	0
$11$	153.533i	0.382578i	0.981534	+	0.191289i	$0.0612668\pi$
$11$	153.533i	0.382578i	−0.981534	+	0.191289i	$0.938733\pi$
$12$	0	0
$13$	− 891.047i	− 1.46232i	−0.682207	−	0.731159i	$-0.738980\pi$
$13$	− 891.047i	− 1.46232i	0.682207	−	0.731159i	$-0.261020\pi$
$14$	0	0
$15$	642.279i	0.737048i
$16$	0	0
$17$	1521.01i	1.27646i	0.769844	+	0.638232i	$0.220334\pi$
$17$	1521.01i	1.27646i	−0.769844	+	0.638232i	$0.779666\pi$
$18$	0	0
$19$	1768.03	1.12358	0.561791	−	0.827279i	$-0.310113\pi$
$19$	1768.03	1.12358	0.561791	+	0.827279i	$0.310113\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 3837.69i	− 1.51269i	−0.654172	−	0.756346i	$-0.726983\pi$
$23$	− 3837.69i	− 1.51269i	0.654172	−	0.756346i	$-0.273017\pi$
$24$	0	0
$25$	−2022.45	−0.647184
$26$	0	0
$27$	3633.31	0.959165
$28$	0	0
$29$	−2658.88	−0.587089	−0.293544	−	0.955945i	$-0.594835\pi$
$29$	−2658.88	−0.587089	−0.293544	+	0.955945i	$0.594835\pi$
$30$	0	0
$31$	6179.19	1.15485	0.577427	−	0.816442i	$-0.304057\pi$
$31$	6179.19	1.15485	0.577427	+	0.816442i	$0.304057\pi$
$32$	0	0
$33$	− 1374.45i	− 0.219707i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8528.02	1.02410	0.512052	−	0.858955i	$-0.328886\pi$
$37$	8528.02	1.02410	0.512052	+	0.858955i	$0.328886\pi$
$38$	0	0
$39$	7976.79i	0.839782i
$40$	0	0
$41$	− 18788.9i	− 1.74559i	−0.488090	−	0.872793i	$-0.662306\pi$
$41$	− 18788.9i	− 1.74559i	0.488090	−	0.872793i	$-0.337694\pi$
$42$	0	0
$43$	− 19232.2i	− 1.58620i	−0.609090	−	0.793101i	$-0.708465\pi$
$43$	− 19232.2i	− 1.58620i	0.609090	−	0.793101i	$-0.291535\pi$
$44$	0	0
$45$	11684.4i	0.860154i
$46$	0	0
$47$	−7081.56	−0.467611	−0.233805	−	0.972283i	$-0.575118\pi$
$47$	−7081.56	−0.467611	−0.233805	+	0.972283i	$0.575118\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	− 13616.3i	− 0.733049i
$52$	0	0
$53$	25162.9	1.23047	0.615236	−	0.788343i	$-0.289061\pi$
$53$	25162.9	1.23047	0.615236	+	0.788343i	$0.289061\pi$
$54$	0	0
$55$	11015.3	0.491011
$56$	0	0
$57$	−15827.6	−0.645252
$58$	0	0
$59$	−1291.54	−0.0483036	−0.0241518	−	0.999708i	$-0.507688\pi$
$59$	−1291.54	−0.0483036	−0.0241518	+	0.999708i	$0.507688\pi$
$60$	0	0
$61$	23945.3i	0.823939i	0.911198	+	0.411970i	$0.135159\pi$
$61$	23945.3i	0.823939i	−0.911198	+	0.411970i	$0.864841\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−63928.8	−1.87678
$66$	0	0
$67$	− 16460.2i	− 0.447968i	−0.974593	−	0.223984i	$-0.928094\pi$
$67$	− 16460.2i	− 0.447968i	0.974593	−	0.223984i	$-0.0719064\pi$
$68$	0	0
$69$	34355.6i	0.868710i
$70$	0	0
$71$	− 25824.7i	− 0.607980i	−0.952675	−	0.303990i	$-0.901681\pi$
$71$	− 25824.7i	− 0.607980i	0.952675	−	0.303990i	$-0.0983190\pi$
$72$	0	0
$73$	− 10443.1i	− 0.229363i	−0.993402	−	0.114681i	$-0.963415\pi$
$73$	− 10443.1i	− 0.229363i	0.993402	−	0.114681i	$-0.0365847\pi$
$74$	0	0
$75$	18105.3	0.371665
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 17196.2i	− 0.310003i	−0.987914	−	0.155001i	$-0.950462\pi$
$79$	− 17196.2i	− 0.310003i	0.987914	−	0.155001i	$-0.0495382\pi$
$80$	0	0
$81$	7048.71	0.119370
$82$	0	0
$83$	53986.4	0.860180	0.430090	−	0.902786i	$-0.358482\pi$
$83$	53986.4	0.860180	0.430090	+	0.902786i	$0.358482\pi$
$84$	0	0
$85$	109126.	1.63825
$86$	0	0
$87$	23802.7	0.337154
$88$	0	0
$89$	− 30675.3i	− 0.410501i	−0.978709	−	0.205250i	$-0.934199\pi$
$89$	− 30675.3i	− 0.410501i	0.978709	−	0.205250i	$-0.0658009\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−55317.1	−0.663211
$94$	0	0
$95$	− 126848.i	− 1.44203i
$96$	0	0
$97$	70219.1i	0.757750i	0.925448	+	0.378875i	$0.123689\pi$
$97$	70219.1i	0.757750i	−0.925448	+	0.378875i	$0.876311\pi$
$98$	0	0
$99$	− 25004.2i	− 0.256404i
$100$	0	0
$101$	63822.6i	0.622545i	0.950321	+	0.311273i	$0.100755\pi$
$101$	63822.6i	0.622545i	−0.950321	+	0.311273i	$0.899245\pi$
$102$	0	0
$103$	−107712.	−1.00039	−0.500197	−	0.865912i	$-0.666739\pi$
$103$	−107712.	−1.00039	−0.500197	+	0.865912i	$0.666739\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	131897.i	1.11372i	0.830607	+	0.556860i	$0.187994\pi$
$107$	131897.i	1.11372i	−0.830607	+	0.556860i	$0.812006\pi$
$108$	0	0
$109$	95557.8	0.770371	0.385186	−	0.922839i	$-0.374137\pi$
$109$	95557.8	0.770371	0.385186	+	0.922839i	$0.374137\pi$
$110$	0	0
$111$	−76344.2	−0.588124
$112$	0	0
$113$	−37554.2	−0.276670	−0.138335	−	0.990385i	$-0.544175\pi$
$113$	−37554.2	−0.276670	−0.138335	+	0.990385i	$0.544175\pi$
$114$	0	0
$115$	−275338.	−1.94143
$116$	0	0
$117$	145115.i	0.980047i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	137479.	0.853634
$122$	0	0
$123$	168201.i	1.00246i
$124$	0	0
$125$	− 79103.3i	− 0.452814i
$126$	0	0
$127$	358280.i	1.97112i	0.169318	+	0.985561i	$0.445844\pi$
$127$	358280.i	1.97112i	−0.169318	+	0.985561i	$0.554156\pi$
$128$	0	0
$129$	172170.i	0.910926i
$130$	0	0
$131$	−166872.	−0.849580	−0.424790	−	0.905292i	$-0.639652\pi$
$131$	−166872.	−0.849580	−0.424790	+	0.905292i	$0.639652\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 260675.i	− 1.23102i
$136$	0	0
$137$	−254808.	−1.15987	−0.579937	−	0.814661i	$-0.696923\pi$
$137$	−254808.	−1.15987	−0.579937	+	0.814661i	$0.696923\pi$
$138$	0	0
$139$	−190751.	−0.837393	−0.418696	−	0.908126i	$-0.637513\pi$
$139$	−190751.	−0.837393	−0.418696	+	0.908126i	$0.637513\pi$
$140$	0	0
$141$	63395.3	0.268540
$142$	0	0
$143$	136805.	0.559451
$144$	0	0
$145$	190763.i	0.753485i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−87373.5	−0.322414	−0.161207	−	0.986921i	$-0.551539\pi$
$149$	−87373.5	−0.322414	−0.161207	+	0.986921i	$0.551539\pi$
$150$	0	0
$151$	− 183085.i	− 0.653446i	−0.945120	−	0.326723i	$-0.894056\pi$
$151$	− 183085.i	− 0.653446i	0.945120	−	0.326723i	$-0.105944\pi$
$152$	0	0
$153$	− 247709.i	− 0.855488i
$154$	0	0
$155$	− 443330.i	− 1.48217i
$156$	0	0
$157$	300184.i	0.971939i	0.873976	+	0.485969i	$0.161533\pi$
$157$	300184.i	0.971939i	−0.873976	+	0.485969i	$0.838467\pi$
$158$	0	0
$159$	−225263.	−0.706637
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 594455.i	− 1.75247i	−0.481887	−	0.876233i	$-0.660048\pi$
$163$	− 594455.i	− 1.75247i	0.481887	−	0.876233i	$-0.339952\pi$
$164$	0	0
$165$	−98611.0	−0.281978
$166$	0	0
$167$	336959.	0.934946	0.467473	−	0.884007i	$-0.345165\pi$
$167$	336959.	0.934946	0.467473	+	0.884007i	$0.345165\pi$
$168$	0	0
$169$	−422671.	−1.13838
$170$	0	0
$171$	−287939.	−0.753025
$172$	0	0
$173$	− 75109.9i	− 0.190802i	−0.995439	−	0.0954008i	$-0.969587\pi$
$173$	− 75109.9i	− 0.190802i	0.995439	−	0.0954008i	$-0.0304133\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	11562.1	0.0277398
$178$	0	0
$179$	− 185323.i	− 0.432312i	−0.976359	−	0.216156i	$-0.930648\pi$
$179$	− 185323.i	− 0.432312i	0.976359	−	0.216156i	$-0.0693520\pi$
$180$	0	0
$181$	− 302350.i	− 0.685984i	−0.939338	−	0.342992i	$-0.888560\pi$
$181$	− 302350.i	− 0.685984i	0.939338	−	0.342992i	$-0.111440\pi$
$182$	0	0
$183$	− 214362.i	− 0.473173i
$184$	0	0
$185$	− 611849.i	− 1.31436i
$186$	0	0
$187$	−233525.	−0.488347
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 900500.i	− 1.78608i	−0.449979	−	0.893039i	$-0.648568\pi$
$191$	− 900500.i	− 1.78608i	0.449979	−	0.893039i	$-0.351432\pi$
$192$	0	0
$193$	−616078.	−1.19054	−0.595268	−	0.803528i	$-0.702954\pi$
$193$	−616078.	−1.19054	−0.595268	+	0.803528i	$0.702954\pi$
$194$	0	0
$195$	572301.	1.07780
$196$	0	0
$197$	−698830.	−1.28294	−0.641469	−	0.767149i	$-0.721675\pi$
$197$	−698830.	−1.28294	−0.641469	+	0.767149i	$0.721675\pi$
$198$	0	0
$199$	−728210.	−1.30354	−0.651769	−	0.758418i	$-0.725973\pi$
$199$	−728210.	−1.30354	−0.651769	+	0.758418i	$0.725973\pi$
$200$	0	0
$201$	147354.i	0.257260i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1.34802e6	−2.24033
$206$	0	0
$207$	625002.i	1.01381i
$208$	0	0
$209$	271450.i	0.429857i
$210$	0	0
$211$	− 504855.i	− 0.780657i	−0.920676	−	0.390329i	$-0.872361\pi$
$211$	− 504855.i	− 0.780657i	0.920676	−	0.390329i	$-0.127639\pi$
$212$	0	0
$213$	231187.i	0.349152i
$214$	0	0
$215$	−1.37983e6	−2.03577
$216$	0	0
$217$	0	0
$218$	0	0
$219$	93488.4i	0.131719i
$220$	0	0
$221$	1.35529e6	1.86660
$222$	0	0
$223$	1.06011e6	1.42755	0.713774	−	0.700377i	$-0.246985\pi$
$223$	1.06011e6	1.42755	0.713774	+	0.700377i	$0.246985\pi$
$224$	0	0
$225$	329374.	0.433743
$226$	0	0
$227$	−926821.	−1.19380	−0.596899	−	0.802316i	$-0.703601\pi$
$227$	−926821.	−1.19380	−0.596899	+	0.802316i	$0.703601\pi$
$228$	0	0
$229$	1.03321e6i	1.30196i	0.759093	+	0.650982i	$0.225643\pi$
$229$	1.03321e6i	1.30196i	−0.759093	+	0.650982i	$0.774357\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−677767.	−0.817882	−0.408941	−	0.912561i	$-0.634102\pi$
$233$	−677767.	−0.817882	−0.408941	+	0.912561i	$0.634102\pi$
$234$	0	0
$235$	508072.i	0.600144i
$236$	0	0
$237$	153943.i	0.178029i
$238$	0	0
$239$	− 447964.i	− 0.507281i	−0.967299	−	0.253640i	$-0.918372\pi$
$239$	− 447964.i	− 0.507281i	0.967299	−	0.253640i	$-0.0816280\pi$
$240$	0	0
$241$	− 335760.i	− 0.372380i	−0.982514	−	0.186190i	$-0.940386\pi$
$241$	− 335760.i	− 0.372380i	0.982514	−	0.186190i	$-0.0596140\pi$
$242$	0	0
$243$	−945996.	−1.02772
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1.57539e6i	− 1.64303i
$248$	0	0
$249$	−483295.	−0.493985
$250$	0	0
$251$	702313.	0.703633	0.351816	−	0.936069i	$-0.385564\pi$
$251$	702313.	0.703633	0.351816	+	0.936069i	$0.385564\pi$
$252$	0	0
$253$	589212.	0.578722
$254$	0	0
$255$	−976910.	−0.940815
$256$	0	0
$257$	− 304895.i	− 0.287951i	−0.989581	−	0.143975i	$-0.954011\pi$
$257$	− 304895.i	− 0.287951i	0.989581	−	0.143975i	$-0.0459886\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	433022.	0.393467
$262$	0	0
$263$	− 1.89594e6i	− 1.69019i	−0.534618	−	0.845094i	$-0.679545\pi$
$263$	− 1.89594e6i	− 1.69019i	0.534618	−	0.845094i	$-0.320455\pi$
$264$	0	0
$265$	− 1.80533e6i	− 1.57922i
$266$	0	0
$267$	274610.i	0.235743i
$268$	0	0
$269$	1.54749e6i	1.30391i	0.758258	+	0.651955i	$0.226051\pi$
$269$	1.54749e6i	1.30391i	−0.758258	+	0.651955i	$0.773949\pi$
$270$	0	0
$271$	1.56683e6	1.29598	0.647989	−	0.761650i	$-0.275610\pi$
$271$	1.56683e6	1.29598	0.647989	+	0.761650i	$0.275610\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 310513.i	− 0.247598i
$276$	0	0
$277$	−1.87694e6	−1.46978	−0.734888	−	0.678188i	$-0.762765\pi$
$277$	−1.87694e6	−1.46978	−0.734888	+	0.678188i	$0.762765\pi$
$278$	0	0
$279$	−1.00634e6	−0.773985
$280$	0	0
$281$	−1.48022e6	−1.11831	−0.559155	−	0.829063i	$-0.688874\pi$
$281$	−1.48022e6	−1.11831	−0.559155	+	0.829063i	$0.688874\pi$
$282$	0	0
$283$	373510.	0.277227	0.138614	−	0.990347i	$-0.455735\pi$
$283$	373510.	0.277227	0.138614	+	0.990347i	$0.455735\pi$
$284$	0	0
$285$	1.13557e6i	0.828133i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−893603.	−0.629361
$290$	0	0
$291$	− 628613.i	− 0.435162i
$292$	0	0
$293$	1.51194e6i	1.02888i	0.857525	+	0.514442i	$0.172001\pi$
$293$	1.51194e6i	1.02888i	−0.857525	+	0.514442i	$0.827999\pi$
$294$	0	0
$295$	92662.7i	0.0619941i
$296$	0	0
$297$	557833.i	0.366955i
$298$	0	0
$299$	−3.41956e6	−2.21204
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 571350.i	− 0.357516i
$304$	0	0
$305$	1.71797e6	1.05747
$306$	0	0
$307$	−1.02264e6	−0.619267	−0.309634	−	0.950856i	$-0.600206\pi$
$307$	−1.02264e6	−0.619267	−0.309634	+	0.950856i	$0.600206\pi$
$308$	0	0
$309$	964254.	0.574507
$310$	0	0
$311$	−1.06266e6	−0.623006	−0.311503	−	0.950245i	$-0.600832\pi$
$311$	−1.06266e6	−0.623006	−0.311503	+	0.950245i	$0.600832\pi$
$312$	0	0
$313$	357351.i	0.206174i	0.994672	+	0.103087i	$0.0328721\pi$
$313$	357351.i	0.206174i	−0.994672	+	0.103087i	$0.967128\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	433980.	0.242561	0.121281	−	0.992618i	$-0.461300\pi$
$317$	433980.	0.242561	0.121281	+	0.992618i	$0.461300\pi$
$318$	0	0
$319$	− 408226.i	− 0.224607i
$320$	0	0
$321$	− 1.18076e6i	− 0.639588i
$322$	0	0
$323$	2.68918e6i	1.43421i
$324$	0	0
$325$	1.80210e6i	0.946389i
$326$	0	0
$327$	−855449.	−0.442410
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 3.29209e6i	− 1.65159i	−0.563971	−	0.825795i	$-0.690727\pi$
$331$	− 3.29209e6i	− 1.65159i	0.563971	−	0.825795i	$-0.309273\pi$
$332$	0	0
$333$	−1.38886e6	−0.686355
$334$	0	0
$335$	−1.18095e6	−0.574934
$336$	0	0
$337$	1.52754e6	0.732684	0.366342	−	0.930480i	$-0.380610\pi$
$337$	1.52754e6	0.732684	0.366342	+	0.930480i	$0.380610\pi$
$338$	0	0
$339$	336191.	0.158887
$340$	0	0
$341$	948709.i	0.441822i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	2.46487e6	1.11493
$346$	0	0
$347$	1.32088e6i	0.588898i	0.955667	+	0.294449i	$0.0951362\pi$
$347$	1.32088e6i	0.588898i	−0.955667	+	0.294449i	$0.904864\pi$
$348$	0	0
$349$	196028.i	0.0861498i	0.999072	+	0.0430749i	$0.0137154\pi$
$349$	196028.i	0.0861498i	−0.999072	+	0.0430749i	$0.986285\pi$
$350$	0	0
$351$	− 3.23745e6i	− 1.40261i
$352$	0	0
$353$	4.46351e6i	1.90651i	0.302162	+	0.953257i	$0.402292\pi$
$353$	4.46351e6i	1.90651i	−0.302162	+	0.953257i	$0.597708\pi$
$354$	0	0
$355$	−1.85281e6	−0.780298
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.43460e6i	1.40650i	0.710943	+	0.703250i	$0.248269\pi$
$359$	3.43460e6i	1.40650i	−0.710943	+	0.703250i	$0.751731\pi$
$360$	0	0
$361$	649815.	0.262435
$362$	0	0
$363$	−1.23073e6	−0.490226
$364$	0	0
$365$	−749249.	−0.294370
$366$	0	0
$367$	142703.	0.0553053	0.0276527	−	0.999618i	$-0.491197\pi$
$367$	142703.	0.0553053	0.0276527	+	0.999618i	$0.491197\pi$
$368$	0	0
$369$	3.05994e6i	1.16989i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−1.98908e6	−0.740252	−0.370126	−	0.928981i	$-0.620686\pi$
$373$	−1.98908e6	−0.740252	−0.370126	+	0.928981i	$0.620686\pi$
$374$	0	0
$375$	708145.i	0.260042i
$376$	0	0
$377$	2.36919e6i	0.858511i
$378$	0	0
$379$	− 2.27226e6i	− 0.812568i	−0.913747	−	0.406284i	$-0.866824\pi$
$379$	− 2.27226e6i	− 0.812568i	0.913747	−	0.406284i	$-0.133176\pi$
$380$	0	0
$381$	− 3.20738e6i	− 1.13198i
$382$	0	0
$383$	1.50492e6	0.524223	0.262112	−	0.965038i	$-0.415581\pi$
$383$	1.50492e6	0.524223	0.262112	+	0.965038i	$0.415581\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.13214e6i	1.06307i
$388$	0	0
$389$	2.83982e6	0.951518	0.475759	−	0.879576i	$-0.342173\pi$
$389$	2.83982e6	0.951518	0.475759	+	0.879576i	$0.342173\pi$
$390$	0	0
$391$	5.83715e6	1.93090
$392$	0	0
$393$	1.49386e6	0.487898
$394$	0	0
$395$	−1.23376e6	−0.397866
$396$	0	0
$397$	− 3.22840e6i	− 1.02804i	−0.857777	−	0.514022i	$-0.828155\pi$
$397$	− 3.22840e6i	− 1.02804i	0.857777	−	0.514022i	$-0.171845\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.04298e6	−1.87668	−0.938340	−	0.345715i	$-0.887637\pi$
$401$	−6.04298e6	−1.87668	−0.938340	+	0.345715i	$0.887637\pi$
$402$	0	0
$403$	− 5.50594e6i	− 1.68876i
$404$	0	0
$405$	− 505715.i	− 0.153203i
$406$	0	0
$407$	1.30933e6i	0.391800i
$408$	0	0
$409$	− 2.89551e6i	− 0.855888i	−0.903805	−	0.427944i	$-0.859238\pi$
$409$	− 2.89551e6i	− 0.855888i	0.903805	−	0.427944i	$-0.140762\pi$
$410$	0	0
$411$	2.28108e6	0.666094
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 3.87330e6i	− 1.10398i
$416$	0	0
$417$	1.70763e6	0.480899
$418$	0	0
$419$	3.80628e6	1.05917	0.529585	−	0.848257i	$-0.322348\pi$
$419$	3.80628e6	1.05917	0.529585	+	0.848257i	$0.322348\pi$
$420$	0	0
$421$	−4.30950e6	−1.18501	−0.592504	−	0.805567i	$-0.701861\pi$
$421$	−4.30950e6	−1.18501	−0.592504	+	0.805567i	$0.701861\pi$
$422$	0	0
$423$	1.15330e6	0.313393
$424$	0	0
$425$	− 3.07616e6i	− 0.826107i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1.22470e6	−0.321282
$430$	0	0
$431$	2.69850e6i	0.699727i	0.936801	+	0.349863i	$0.113772\pi$
$431$	2.69850e6i	0.699727i	−0.936801	+	0.349863i	$0.886228\pi$
$432$	0	0
$433$	− 3.11101e6i	− 0.797410i	−0.917079	−	0.398705i	$-0.869460\pi$
$433$	− 3.11101e6i	− 0.797410i	0.917079	−	0.398705i	$-0.130540\pi$
$434$	0	0
$435$	− 1.70774e6i	− 0.432712i
$436$	0	0
$437$	− 6.78513e6i	− 1.69963i
$438$	0	0
$439$	6.68403e6	1.65530	0.827651	−	0.561243i	$-0.189677\pi$
$439$	6.68403e6	1.65530	0.827651	+	0.561243i	$0.189677\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 1.03501e6i	− 0.250573i	−0.992121	−	0.125287i	$-0.960015\pi$
$443$	− 1.03501e6i	− 0.250573i	0.992121	−	0.125287i	$-0.0399851\pi$
$444$	0	0
$445$	−2.20082e6	−0.526848
$446$	0	0
$447$	782181.	0.185156
$448$	0	0
$449$	5.42168e6	1.26917	0.634583	−	0.772855i	$-0.281172\pi$
$449$	5.42168e6	1.26917	0.634583	+	0.772855i	$0.281172\pi$
$450$	0	0
$451$	2.88471e6	0.667823
$452$	0	0
$453$	1.63900e6i	0.375262i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2.76733e6	0.619828	0.309914	−	0.950765i	$-0.399700\pi$
$457$	2.76733e6	0.619828	0.309914	+	0.950765i	$0.399700\pi$
$458$	0	0
$459$	5.52629e6i	1.22434i
$460$	0	0
$461$	− 414792.i	− 0.0909029i	−0.998967	−	0.0454515i	$-0.985527\pi$
$461$	− 414792.i	− 0.0909029i	0.998967	−	0.0454515i	$-0.0144726\pi$
$462$	0	0
$463$	3.28204e6i	0.711526i	0.934576	+	0.355763i	$0.115779\pi$
$463$	3.28204e6i	0.711526i	−0.934576	+	0.355763i	$0.884221\pi$
$464$	0	0
$465$	3.96876e6i	0.851183i
$466$	0	0
$467$	−3.30005e6	−0.700210	−0.350105	−	0.936710i	$-0.613854\pi$
$467$	−3.30005e6	−0.700210	−0.350105	+	0.936710i	$0.613854\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 2.68730e6i	− 0.558166i
$472$	0	0
$473$	2.95278e6	0.606846
$474$	0	0
$475$	−3.57574e6	−0.727164
$476$	0	0
$477$	−4.09801e6	−0.824663
$478$	0	0
$479$	−2.33933e6	−0.465857	−0.232928	−	0.972494i	$-0.574831\pi$
$479$	−2.33933e6	−0.465857	−0.232928	+	0.972494i	$0.574831\pi$
$480$	0	0
$481$	− 7.59886e6i	− 1.49757i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.03792e6	0.972517
$486$	0	0
$487$	− 3.06869e6i	− 0.586314i	−0.956064	−	0.293157i	$-0.905294\pi$
$487$	− 3.06869e6i	− 0.586314i	0.956064	−	0.293157i	$-0.0947058\pi$
$488$	0	0
$489$	5.32165e6i	1.00641i
$490$	0	0
$491$	− 4.39524e6i	− 0.822770i	−0.911462	−	0.411385i	$-0.865045\pi$
$491$	− 4.39524e6i	− 0.822770i	0.911462	−	0.411385i	$-0.134955\pi$
$492$	0	0
$493$	− 4.04417e6i	− 0.749398i
$494$	0	0
$495$	−1.79394e6	−0.329076
$496$	0	0
$497$	0	0
$498$	0	0
$499$	7.77470e6i	1.39776i	0.715240	+	0.698879i	$0.246318\pi$
$499$	7.77470e6i	1.39776i	−0.715240	+	0.698879i	$0.753682\pi$
$500$	0	0
$501$	−3.01651e6	−0.536922
$502$	0	0
$503$	−5.37075e6	−0.946488	−0.473244	−	0.880931i	$-0.656917\pi$
$503$	−5.37075e6	−0.946488	−0.473244	+	0.880931i	$0.656917\pi$
$504$	0	0
$505$	4.57900e6	0.798991
$506$	0	0
$507$	3.78382e6	0.653748
$508$	0	0
$509$	2.18473e6i	0.373769i	0.982382	+	0.186884i	$0.0598389\pi$
$509$	2.18473e6i	0.373769i	−0.982382	+	0.186884i	$0.940161\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	6.42379e6	1.07770
$514$	0	0
$515$	7.72787e6i	1.28393i
$516$	0	0
$517$	− 1.08725e6i	− 0.178898i
$518$	0	0
$519$	672396.i	0.109574i
$520$	0	0
$521$	4.97007e6i	0.802174i	0.916040	+	0.401087i	$0.131367\pi$
$521$	4.97007e6i	0.802174i	−0.916040	+	0.401087i	$0.868633\pi$
$522$	0	0
$523$	1.17726e7	1.88199	0.940997	−	0.338416i	$-0.109891\pi$
$523$	1.17726e7	1.88199	0.940997	+	0.338416i	$0.109891\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.39858e6i	1.47413i
$528$	0	0
$529$	−8.29153e6	−1.28824
$530$	0	0
$531$	210339.	0.0323731
$532$	0	0
$533$	−1.67418e7	−2.55260
$534$	0	0
$535$	9.46305e6	1.42938
$536$	0	0
$537$	1.65904e6i	0.248269i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	6.46645e6	0.949889	0.474945	−	0.880016i	$-0.342468\pi$
$541$	6.46645e6	0.949889	0.474945	+	0.880016i	$0.342468\pi$
$542$	0	0
$543$	2.70669e6i	0.393948i
$544$	0	0
$545$	− 6.85586e6i	− 0.988715i
$546$	0	0
$547$	1.14808e7i	1.64061i	0.571927	+	0.820305i	$0.306196\pi$
$547$	1.14808e7i	1.64061i	−0.571927	+	0.820305i	$0.693804\pi$
$548$	0	0
$549$	− 3.89970e6i	− 0.552205i
$550$	0	0
$551$	−4.70097e6	−0.659642
$552$	0	0
$553$	0	0
$554$	0	0
$555$	5.47737e6i	0.754813i
$556$	0	0
$557$	−9.06301e6	−1.23775	−0.618877	−	0.785488i	$-0.712412\pi$
$557$	−9.06301e6	−1.23775	−0.618877	+	0.785488i	$0.712412\pi$
$558$	0	0
$559$	−1.71368e7	−2.31953
$560$	0	0
$561$	2.09055e6	0.280449
$562$	0	0
$563$	2.64960e6	0.352297	0.176148	−	0.984364i	$-0.443636\pi$
$563$	2.64960e6	0.352297	0.176148	+	0.984364i	$0.443636\pi$
$564$	0	0
$565$	2.69435e6i	0.355086i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	9.08250e6	1.17605	0.588023	−	0.808844i	$-0.299906\pi$
$569$	9.08250e6	1.17605	0.588023	+	0.808844i	$0.299906\pi$
$570$	0	0
$571$	− 1.53684e6i	− 0.197260i	−0.995124	−	0.0986299i	$-0.968554\pi$
$571$	− 1.53684e6i	− 0.197260i	0.995124	−	0.0986299i	$-0.0314460\pi$
$572$	0	0
$573$	8.06142e6i	1.02571i
$574$	0	0
$575$	7.76153e6i	0.978989i
$576$	0	0
$577$	− 4.23757e6i	− 0.529880i	−0.964265	−	0.264940i	$-0.914648\pi$
$577$	− 4.23757e6i	− 0.529880i	0.964265	−	0.264940i	$-0.0853522\pi$
$578$	0	0
$579$	5.51523e6	0.683702
$580$	0	0
$581$	0	0
$582$	0	0
$583$	3.86334e6i	0.470751i
$584$	0	0
$585$	1.04114e7	1.25782
$586$	0	0
$587$	−1.20181e7	−1.43960	−0.719799	−	0.694182i	$-0.755766\pi$
$587$	−1.20181e7	−1.43960	−0.719799	+	0.694182i	$0.755766\pi$
$588$	0	0
$589$	1.09250e7	1.29757
$590$	0	0
$591$	6.25604e6	0.736768
$592$	0	0
$593$	− 1.43625e7i	− 1.67723i	−0.544722	−	0.838617i	$-0.683365\pi$
$593$	− 1.43625e7i	− 1.67723i	0.544722	−	0.838617i	$-0.316635\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6.51905e6	0.748597
$598$	0	0
$599$	− 1.05888e7i	− 1.20581i	−0.797813	−	0.602905i	$-0.794010\pi$
$599$	− 1.05888e7i	− 1.20581i	0.797813	−	0.602905i	$-0.205990\pi$
$600$	0	0
$601$	− 2.68606e6i	− 0.303340i	−0.988431	−	0.151670i	$-0.951535\pi$
$601$	− 2.68606e6i	− 0.303340i	0.988431	−	0.151670i	$-0.0484652\pi$
$602$	0	0
$603$	2.68068e6i	0.300229i
$604$	0	0
$605$	− 9.86350e6i	− 1.09558i
$606$	0	0
$607$	8.02313e6	0.883836	0.441918	−	0.897055i	$-0.354298\pi$
$607$	8.02313e6	0.883836	0.441918	+	0.897055i	$0.354298\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6.31000e6i	0.683796i
$612$	0	0
$613$	8.22402e6	0.883961	0.441980	−	0.897025i	$-0.354276\pi$
$613$	8.22402e6	0.883961	0.441980	+	0.897025i	$0.354276\pi$
$614$	0	0
$615$	1.20677e7	1.28658
$616$	0	0
$617$	1.44777e7	1.53104	0.765519	−	0.643414i	$-0.222482\pi$
$617$	1.44777e7	1.53104	0.765519	+	0.643414i	$0.222482\pi$
$618$	0	0
$619$	−1.18114e7	−1.23901	−0.619507	−	0.784991i	$-0.712668\pi$
$619$	−1.18114e7	−1.23901	−0.619507	+	0.784991i	$0.712668\pi$
$620$	0	0
$621$	− 1.39435e7i	− 1.45092i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−1.19955e7	−1.22834
$626$	0	0
$627$	− 2.43006e6i	− 0.246859i
$628$	0	0
$629$	1.29712e7i	1.30723i
$630$	0	0
$631$	1.17328e7i	1.17308i	0.809920	+	0.586540i	$0.199510\pi$
$631$	1.17328e7i	1.17308i	−0.809920	+	0.586540i	$0.800490\pi$
$632$	0	0
$633$	4.51954e6i	0.448317i
$634$	0	0
$635$	2.57051e7	2.52979
$636$	0	0
$637$	0	0
$638$	0	0
$639$	4.20578e6i	0.407469i
$640$	0	0
$641$	−760587.	−0.0731145	−0.0365573	−	0.999332i	$-0.511639\pi$
$641$	−760587.	−0.0731145	−0.0365573	+	0.999332i	$0.511639\pi$
$642$	0	0
$643$	6.37262e6	0.607842	0.303921	−	0.952697i	$-0.401704\pi$
$643$	6.37262e6	0.607842	0.303921	+	0.952697i	$0.401704\pi$
$644$	0	0
$645$	1.23525e7	1.16911
$646$	0	0
$647$	7.88333e6	0.740371	0.370185	−	0.928958i	$-0.379294\pi$
$647$	7.88333e6	0.740371	0.370185	+	0.928958i	$0.379294\pi$
$648$	0	0
$649$	− 198295.i	− 0.0184799i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.21936e6	−0.111905	−0.0559524	−	0.998433i	$-0.517820\pi$
$653$	−1.21936e6	−0.111905	−0.0559524	+	0.998433i	$0.517820\pi$
$654$	0	0
$655$	1.19723e7i	1.09037i
$656$	0	0
$657$	1.70075e6i	0.153719i
$658$	0	0
$659$	− 7.56715e6i	− 0.678764i	−0.940649	−	0.339382i	$-0.889782\pi$
$659$	− 7.56715e6i	− 0.678764i	0.940649	−	0.339382i	$-0.110218\pi$
$660$	0	0
$661$	4.26736e6i	0.379888i	0.981795	+	0.189944i	$0.0608307\pi$
$661$	4.26736e6i	0.379888i	−0.981795	+	0.189944i	$0.939169\pi$
$662$	0	0
$663$	−1.21327e7	−1.07195
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.02040e7i	0.888084i
$668$	0	0
$669$	−9.49031e6	−0.819814
$670$	0	0
$671$	−3.67639e6	−0.315221
$672$	0	0
$673$	−1.12574e7	−0.958075	−0.479038	−	0.877794i	$-0.659014\pi$
$673$	−1.12574e7	−0.958075	−0.479038	+	0.877794i	$0.659014\pi$
$674$	0	0
$675$	−7.34819e6	−0.620756
$676$	0	0
$677$	1.84659e7i	1.54846i	0.632907	+	0.774228i	$0.281861\pi$
$677$	1.84659e7i	1.54846i	−0.632907	+	0.774228i	$0.718139\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	8.29704e6	0.685576
$682$	0	0
$683$	− 1.58950e6i	− 0.130380i	−0.997873	−	0.0651898i	$-0.979235\pi$
$683$	− 1.58950e6i	− 0.130380i	0.997873	−	0.0651898i	$-0.0207653\pi$
$684$	0	0
$685$	1.82814e7i	1.48861i
$686$	0	0
$687$	− 9.24944e6i	− 0.747694i
$688$	0	0
$689$	− 2.24213e7i	− 1.79934i
$690$	0	0
$691$	−5.56539e6	−0.443405	−0.221703	−	0.975114i	$-0.571161\pi$
$691$	−5.56539e6	−0.443405	−0.221703	+	0.975114i	$0.571161\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.36855e7i	1.07473i
$696$	0	0
$697$	2.85780e7	2.22818
$698$	0	0
$699$	6.06748e6	0.469694
$700$	0	0
$701$	−1.20878e7	−0.929078	−0.464539	−	0.885553i	$-0.653780\pi$
$701$	−1.20878e7	−0.929078	−0.464539	+	0.885553i	$0.653780\pi$
$702$	0	0
$703$	1.50778e7	1.15066
$704$	0	0
$705$	− 4.54834e6i	− 0.344651i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−1.58055e7	−1.18085	−0.590423	−	0.807094i	$-0.701039\pi$
$709$	−1.58055e7	−1.18085	−0.590423	+	0.807094i	$0.701039\pi$
$710$	0	0
$711$	2.80056e6i	0.207764i
$712$	0	0
$713$	− 2.37138e7i	− 1.74694i
$714$	0	0
$715$	− 9.81517e6i	− 0.718014i
$716$	0	0
$717$	4.01024e6i	0.291322i
$718$	0	0
$719$	2.10607e7	1.51932	0.759661	−	0.650319i	$-0.225365\pi$
$719$	2.10607e7	1.51932	0.759661	+	0.650319i	$0.225365\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	3.00577e6i	0.213851i
$724$	0	0
$725$	5.37745e6	0.379954
$726$	0	0
$727$	1.07408e7	0.753701	0.376851	−	0.926274i	$-0.377007\pi$
$727$	1.07408e7	0.753701	0.376851	+	0.926274i	$0.377007\pi$
$728$	0	0
$729$	6.75587e6	0.470828
$730$	0	0
$731$	2.92523e7	2.02473
$732$	0	0
$733$	− 1.10904e7i	− 0.762408i	−0.924491	−	0.381204i	$-0.875509\pi$
$733$	− 1.10904e7i	− 0.762408i	0.924491	−	0.381204i	$-0.124491\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.52718e6	0.171383
$738$	0	0
$739$	− 1.01206e7i	− 0.681702i	−0.940117	−	0.340851i	$-0.889285\pi$
$739$	− 1.01206e7i	− 0.681702i	0.940117	−	0.340851i	$-0.110715\pi$
$740$	0	0
$741$	1.41032e7i	0.943563i
$742$	0	0
$743$	− 1.69668e7i	− 1.12753i	−0.825934	−	0.563766i	$-0.809352\pi$
$743$	− 1.69668e7i	− 1.12753i	0.825934	−	0.563766i	$-0.190648\pi$
$744$	0	0
$745$	6.26868e6i	0.413795i
$746$	0	0
$747$	−8.79217e6	−0.576494
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 764861.i	− 0.0494861i	−0.999694	−	0.0247430i	$-0.992123\pi$
$751$	− 764861.i	− 0.0494861i	0.999694	−	0.0247430i	$-0.00787676\pi$
$752$	0	0
$753$	−6.28721e6	−0.404083
$754$	0	0
$755$	−1.31356e7	−0.838650
$756$	0	0
$757$	−1.42566e7	−0.904223	−0.452112	−	0.891961i	$-0.649329\pi$
$757$	−1.42566e7	−0.904223	−0.452112	+	0.891961i	$0.649329\pi$
$758$	0	0
$759$	−5.27472e6	−0.332349
$760$	0	0
$761$	5.12974e6i	0.321095i	0.987028	+	0.160548i	$0.0513260\pi$
$761$	5.12974e6i	0.321095i	−0.987028	+	0.160548i	$0.948674\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−1.77721e7	−1.09796
$766$	0	0
$767$	1.15083e6i	0.0706352i
$768$	0	0
$769$	− 2.22364e7i	− 1.35596i	−0.735079	−	0.677982i	$-0.762855\pi$
$769$	− 2.22364e7i	− 1.35596i	0.735079	−	0.677982i	$-0.237145\pi$
$770$	0	0
$771$	2.72947e6i	0.165365i
$772$	0	0
$773$	2.91884e7i	1.75696i	0.477780	+	0.878480i	$0.341442\pi$
$773$	2.91884e7i	1.75696i	−0.477780	+	0.878480i	$0.658558\pi$
$774$	0	0
$775$	−1.24971e7	−0.747403
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 3.32192e7i	− 1.96131i
$780$	0	0
$781$	3.96494e6	0.232600
$782$	0	0
$783$	−9.66054e6	−0.563115
$784$	0	0
$785$	2.15369e7	1.24741
$786$	0	0
$787$	1.53766e7	0.884962	0.442481	−	0.896778i	$-0.354098\pi$
$787$	1.53766e7	0.884962	0.442481	+	0.896778i	$0.354098\pi$
$788$	0	0
$789$	1.69727e7i	0.970643i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	2.13364e7	1.20486
$794$	0	0
$795$	1.61616e7i	0.906916i
$796$	0	0
$797$	− 1.57429e7i	− 0.877887i	−0.898515	−	0.438943i	$-0.855353\pi$
$797$	− 1.57429e7i	− 0.877887i	0.898515	−	0.438943i	$-0.144647\pi$
$798$	0	0
$799$	− 1.07711e7i	− 0.596888i
$800$	0	0
$801$	4.99575e6i	0.275118i
$802$	0	0
$803$	1.60336e6	0.0877492
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 1.38534e7i	− 0.748811i
$808$	0	0
$809$	−3.42322e7	−1.83892	−0.919461	−	0.393180i	$-0.871375\pi$
$809$	−3.42322e7	−1.83892	−0.919461	+	0.393180i	$0.871375\pi$
$810$	0	0
$811$	2.08905e7	1.11531	0.557656	−	0.830072i	$-0.311701\pi$
$811$	2.08905e7	1.11531	0.557656	+	0.830072i	$0.311701\pi$
$812$	0	0
$813$	−1.40265e7	−0.744256
$814$	0	0
$815$	−4.26496e7	−2.24916
$816$	0	0
$817$	− 3.40031e7i	− 1.78223i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.50646e7	1.29779	0.648894	−	0.760879i	$-0.275232\pi$
$821$	2.50646e7	1.29779	0.648894	+	0.760879i	$0.275232\pi$
$822$	0	0
$823$	1.59750e7i	0.822132i	0.911606	+	0.411066i	$0.134843\pi$
$823$	1.59750e7i	0.822132i	−0.911606	+	0.411066i	$0.865157\pi$
$824$	0	0
$825$	2.77976e6i	0.142191i
$826$	0	0
$827$	8.71482e6i	0.443093i	0.975150	+	0.221546i	$0.0711104\pi$
$827$	8.71482e6i	0.443093i	−0.975150	+	0.221546i	$0.928890\pi$
$828$	0	0
$829$	− 1.98070e7i	− 1.00100i	−0.865738	−	0.500498i	$-0.833150\pi$
$829$	− 1.98070e7i	− 1.00100i	0.865738	−	0.500498i	$-0.166850\pi$
$830$	0	0
$831$	1.68027e7	0.844065
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 2.41754e7i	− 1.19993i
$836$	0	0
$837$	2.24509e7	1.10770
$838$	0	0
$839$	2.72849e6	0.133819	0.0669094	−	0.997759i	$-0.478686\pi$
$839$	2.72849e6	0.133819	0.0669094	+	0.997759i	$0.478686\pi$
$840$	0	0
$841$	−1.34415e7	−0.655327
$842$	0	0
$843$	1.32512e7	0.642224
$844$	0	0
$845$	3.03248e7i	1.46102i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−3.34372e6	−0.159206
$850$	0	0
$851$	− 3.27279e7i	− 1.54915i
$852$	0	0
$853$	1.50708e7i	0.709194i	0.935019	+	0.354597i	$0.115382\pi$
$853$	1.50708e7i	0.709194i	−0.935019	+	0.354597i	$0.884618\pi$
$854$	0	0
$855$	2.06584e7i	0.966453i
$856$	0	0
$857$	1.67499e7i	0.779041i	0.921018	+	0.389521i	$0.127359\pi$
$857$	1.67499e7i	0.779041i	−0.921018	+	0.389521i	$0.872641\pi$
$858$	0	0
$859$	−9.42985e6	−0.436036	−0.218018	−	0.975945i	$-0.569959\pi$
$859$	−9.42985e6	−0.436036	−0.218018	+	0.975945i	$0.569959\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	6.11236e6i	0.279371i	0.990196	+	0.139686i	$0.0446092\pi$
$863$	6.11236e6i	0.279371i	−0.990196	+	0.139686i	$0.955391\pi$
$864$	0	0
$865$	−5.38882e6	−0.244880
$866$	0	0
$867$	7.99967e6	0.361430
$868$	0	0
$869$	2.64019e6	0.118600
$870$	0	0
$871$	−1.46668e7	−0.655072
$872$	0	0
$873$	− 1.14358e7i	− 0.507845i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.90816e7	−0.837751	−0.418875	−	0.908044i	$-0.637576\pi$
$877$	−1.90816e7	−0.837751	−0.418875	+	0.908044i	$0.637576\pi$
$878$	0	0
$879$	− 1.35352e7i	− 0.590869i
$880$	0	0
$881$	− 1.21256e7i	− 0.526338i	−0.964750	−	0.263169i	$-0.915232\pi$
$881$	− 1.21256e7i	− 0.526338i	0.964750	−	0.263169i	$-0.0847676\pi$
$882$	0	0
$883$	2.06429e7i	0.890980i	0.895287	+	0.445490i	$0.146971\pi$
$883$	2.06429e7i	0.890980i	−0.895287	+	0.445490i	$0.853029\pi$
$884$	0	0
$885$	− 829532.i	− 0.0356020i
$886$	0	0
$887$	−3.74806e6	−0.159955	−0.0799774	−	0.996797i	$-0.525485\pi$
$887$	−3.74806e6	−0.159955	−0.0799774	+	0.996797i	$0.525485\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.08221e6i	0.0456685i
$892$	0	0
$893$	−1.25204e7	−0.525399
$894$	0	0
$895$	−1.32962e7	−0.554841
$896$	0	0
$897$	3.06125e7	1.27033
$898$	0	0
$899$	−1.64297e7	−0.678002
$900$	0	0
$901$	3.82730e7i	1.57065i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.16923e7	−0.880410
$906$	0	0
$907$	4.18102e6i	0.168758i	0.996434	+	0.0843790i	$0.0268906\pi$
$907$	4.18102e6i	0.168758i	−0.996434	+	0.0843790i	$0.973109\pi$
$908$	0	0
$909$	− 1.03941e7i	− 0.417231i
$910$	0	0
$911$	− 1.56891e6i	− 0.0626329i	−0.999510	−	0.0313165i	$-0.990030\pi$
$911$	− 1.56891e6i	− 0.0626329i	0.999510	−	0.0313165i	$-0.00996997\pi$
$912$	0	0
$913$	8.28870e6i	0.329086i
$914$	0	0
$915$	−1.53796e7	−0.607283
$916$	0	0
$917$	0	0
$918$	0	0
$919$	3.71605e7i	1.45142i	0.688000	+	0.725710i	$0.258489\pi$
$919$	3.71605e7i	1.45142i	−0.688000	+	0.725710i	$0.741511\pi$
$920$	0	0
$921$	9.15487e6	0.355634
$922$	0	0
$923$	−2.30110e7	−0.889061
$924$	0	0
$925$	−1.72475e7	−0.662783
$926$	0	0
$927$	1.75418e7	0.670464
$928$	0	0
$929$	− 2.34333e7i	− 0.890827i	−0.895325	−	0.445414i	$-0.853057\pi$
$929$	− 2.34333e7i	− 0.890827i	0.895325	−	0.445414i	$-0.146943\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	9.51308e6	0.357781
$934$	0	0
$935$	1.67544e7i	0.626758i
$936$	0	0
$937$	− 1.07872e7i	− 0.401383i	−0.979654	−	0.200692i	$-0.935681\pi$
$937$	− 1.07872e7i	− 0.401383i	0.979654	−	0.200692i	$-0.0643189\pi$
$938$	0	0
$939$	− 3.19907e6i	− 0.118402i
$940$	0	0
$941$	− 5.01857e6i	− 0.184759i	−0.995724	−	0.0923795i	$-0.970553\pi$
$941$	− 5.01857e6i	− 0.184759i	0.995724	−	0.0923795i	$-0.0294473\pi$
$942$	0	0
$943$	−7.21059e7	−2.64053
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 7.28354e6i	− 0.263917i	−0.991255	−	0.131959i	$-0.957873\pi$
$947$	− 7.28354e6i	− 0.263917i	0.991255	−	0.131959i	$-0.0421266\pi$
$948$	0	0
$949$	−9.30530e6	−0.335402
$950$	0	0
$951$	−3.88506e6	−0.139298
$952$	0	0
$953$	−3.16792e6	−0.112991	−0.0564953	−	0.998403i	$-0.517993\pi$
$953$	−3.16792e6	−0.112991	−0.0564953	+	0.998403i	$0.517993\pi$
$954$	0	0
$955$	−6.46071e7	−2.29230
$956$	0	0
$957$	3.65450e6i	0.128988i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	9.55321e6	0.333688
$962$	0	0
$963$	− 2.14806e7i	− 0.746416i
$964$	0	0
$965$	4.42009e7i	1.52796i
$966$	0	0
$967$	− 4.48807e7i	− 1.54345i	−0.635954	−	0.771727i	$-0.719393\pi$
$967$	− 4.48807e7i	− 1.54345i	0.635954	−	0.771727i	$-0.280607\pi$
$968$	0	0
$969$	− 2.40739e7i	− 0.823641i
$970$	0	0
$971$	−3.40577e7	−1.15922	−0.579611	−	0.814893i	$-0.696796\pi$
$971$	−3.40577e7	−1.15922	−0.579611	+	0.814893i	$0.696796\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	− 1.61327e7i	− 0.543493i
$976$	0	0
$977$	−3.68984e7	−1.23672	−0.618359	−	0.785895i	$-0.712202\pi$
$977$	−3.68984e7	−1.23672	−0.618359	+	0.785895i	$0.712202\pi$
$978$	0	0
$979$	4.70967e6	0.157049
$980$	0	0
$981$	−1.55624e7	−0.516303
$982$	0	0
$983$	−4.60333e7	−1.51946	−0.759729	−	0.650240i	$-0.774668\pi$
$983$	−4.60333e7	−1.51946	−0.759729	+	0.650240i	$0.774668\pi$
$984$	0	0
$985$	5.01381e7i	1.64656i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−7.38074e7	−2.39943
$990$	0	0
$991$	− 190022.i	− 0.00614638i	−0.999995	−	0.00307319i	$-0.999022\pi$
$991$	− 190022.i	− 0.00614638i	0.999995	−	0.00307319i	$-0.000978228\pi$
$992$	0	0
$993$	2.94713e7i	0.948477i
$994$	0	0
$995$	5.22459e7i	1.67300i
$996$	0	0
$997$	− 1.93385e7i	− 0.616148i	−0.951362	−	0.308074i	$-0.900316\pi$
$997$	− 1.93385e7i	− 0.616148i	0.951362	−	0.308074i	$-0.0996844\pi$
$998$	0	0
$999$	3.09850e7	0.982285

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.f.c.783.5		14
4.3	odd	2		784.6.f.d.783.9		14
7.2	even	3		112.6.p.c.31.5	yes	14
7.3	odd	6		112.6.p.b.47.3	yes	14
7.6	odd	2		784.6.f.d.783.10		14
28.3	even	6		112.6.p.c.47.5	yes	14
28.23	odd	6		112.6.p.b.31.3	✓	14
28.27	even	2	inner	784.6.f.c.783.6		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.6.p.b.31.3	✓	14	28.23	odd	6
112.6.p.b.47.3	yes	14	7.3	odd	6
112.6.p.c.31.5	yes	14	7.2	even	3
112.6.p.c.47.5	yes	14	28.3	even	6
784.6.f.c.783.5		14	1.1	even	1	trivial
784.6.f.c.783.6		14	28.27	even	2	inner
784.6.f.d.783.9		14	4.3	odd	2
784.6.f.d.783.10		14	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	784.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-8.95216 q^{3} -71.7457i q^{5} -162.859 q^{9} +153.533i q^{11} -891.047i q^{13} +642.279i q^{15} +1521.01i q^{17} +1768.03 q^{19} -3837.69i q^{23} -2022.45 q^{25} +3633.31 q^{27} -2658.88 q^{29} +6179.19 q^{31} -1374.45i q^{33} +8528.02 q^{37} +7976.79i q^{39} -18788.9i q^{41} -19232.2i q^{43} +11684.4i q^{45} -7081.56 q^{47} -13616.3i q^{51} +25162.9 q^{53} +11015.3 q^{55} -15827.6 q^{57} -1291.54 q^{59} +23945.3i q^{61} -63928.8 q^{65} -16460.2i q^{67} +34355.6i q^{69} -25824.7i q^{71} -10443.1i q^{73} +18105.3 q^{75} -17196.2i q^{79} +7048.71 q^{81} +53986.4 q^{83} +109126. q^{85} +23802.7 q^{87} -30675.3i q^{89} -55317.1 q^{93} -126848. i q^{95} +70219.1i q^{97} -25004.2i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 18 q^{3} + 1076 q^{9} - 4270 q^{19} - 10868 q^{25} - 17910 q^{27} + 684 q^{29} + 6238 q^{31} - 6862 q^{37} + 16818 q^{47} - 3414 q^{53} + 56622 q^{55} + 68450 q^{57} - 98610 q^{59} - 34896 q^{65}+ \cdots - 585746 q^{93}+O(q^{100}) \) 14 * q - 18 * q^3 + 1076 * q^9 - 4270 * q^19 - 10868 * q^25 - 17910 * q^27 + 684 * q^29 + 6238 * q^31 - 6862 * q^37 + 16818 * q^47 - 3414 * q^53 + 56622 * q^55 + 68450 * q^57 - 98610 * q^59 - 34896 * q^65 - 295668 * q^75 + 291134 * q^81 + 305400 * q^83 - 176454 * q^85 - 190068 * q^87 - 585746 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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