LMFDB - Embedded newform 784.4.f.b.783.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,4,Mod(783,784)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("784.783");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	784.f (of order $2$, degree $1$, 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:	$46.2574974445$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			783.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	784.783
Dual form			784.4.f.b.783.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -5.19615i q^{5} -26.0000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -5.19615i q^{5} -26.0000 q^{9} +25.9808i q^{11} +69.2820i q^{13} +5.19615i q^{15} -36.3731i q^{17} +17.0000 q^{19} -140.296i q^{23} +98.0000 q^{25} +53.0000 q^{27} +90.0000 q^{29} -17.0000 q^{31} -25.9808i q^{33} +199.000 q^{37} -69.2820i q^{39} -187.061i q^{41} -252.879i q^{43} +135.100i q^{45} -567.000 q^{47} +36.3731i q^{51} -333.000 q^{53} +135.000 q^{55} -17.0000 q^{57} -801.000 q^{59} -358.535i q^{61} +360.000 q^{65} -216.506i q^{67} +140.296i q^{69} +488.438i q^{71} -403.568i q^{73} -98.0000 q^{75} -1193.38i q^{79} +649.000 q^{81} +468.000 q^{83} -189.000 q^{85} -90.0000 q^{87} -192.258i q^{89} +17.0000 q^{93} -88.3346i q^{95} +1392.57i q^{97} -675.500i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 52 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 52 * q^9	$ 2 q - 2 q^{3} - 52 q^{9} + 34 q^{19} + 196 q^{25} + 106 q^{27} + 180 q^{29} - 34 q^{31} + 398 q^{37} - 1134 q^{47} - 666 q^{53} + 270 q^{55} - 34 q^{57} - 1602 q^{59} + 720 q^{65} - 196 q^{75} + 1298 q^{81} + 936 q^{83} - 378 q^{85} - 180 q^{87} + 34 q^{93}+O(q^{100}) $ 2 * q - 2 * q^3 - 52 * q^9 + 34 * q^19 + 196 * q^25 + 106 * q^27 + 180 * q^29 - 34 * q^31 + 398 * q^37 - 1134 * q^47 - 666 * q^53 + 270 * q^55 - 34 * q^57 - 1602 * q^59 + 720 * q^65 - 196 * q^75 + 1298 * q^81 + 936 * q^83 - 378 * q^85 - 180 * q^87 + 34 * q^93

Display 100 coefficients 10 coefficients

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
$3$	−1.00000	−0.192450	−0.0962250	−	0.995360i	$-0.530677\pi$
$3$	−1.00000	−0.192450	−0.0962250	+	0.995360i	$0.530677\pi$
$4$	0	0
$5$	− 5.19615i	− 0.464758i	−0.972625	−	0.232379i	$-0.925349\pi$
$5$	− 5.19615i	− 0.464758i	0.972625	−	0.232379i	$-0.0746510\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−26.0000	−0.962963
$10$	0	0
$11$	25.9808i	0.712136i	0.934460	+	0.356068i	$0.115883\pi$
$11$	25.9808i	0.712136i	−0.934460	+	0.356068i	$0.884117\pi$
$12$	0	0
$13$	69.2820i	1.47811i	0.673647	+	0.739053i	$0.264727\pi$
$13$	69.2820i	1.47811i	−0.673647	+	0.739053i	$0.735273\pi$
$14$	0	0
$15$	5.19615i	0.0894427i
$16$	0	0
$17$	− 36.3731i	− 0.518927i	−0.965753	−	0.259464i	$-0.916454\pi$
$17$	− 36.3731i	− 0.518927i	0.965753	−	0.259464i	$-0.0835458\pi$
$18$	0	0
$19$	17.0000	0.205267	0.102633	−	0.994719i	$-0.467273\pi$
$19$	17.0000	0.205267	0.102633	+	0.994719i	$0.467273\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 140.296i	− 1.27190i	−0.771729	−	0.635951i	$-0.780608\pi$
$23$	− 140.296i	− 1.27190i	0.771729	−	0.635951i	$-0.219392\pi$
$24$	0	0
$25$	98.0000	0.784000
$26$	0	0
$27$	53.0000	0.377772
$28$	0	0
$29$	90.0000	0.576296	0.288148	−	0.957586i	$-0.406961\pi$
$29$	90.0000	0.576296	0.288148	+	0.957586i	$0.406961\pi$
$30$	0	0
$31$	−17.0000	−0.0984932	−0.0492466	−	0.998787i	$-0.515682\pi$
$31$	−17.0000	−0.0984932	−0.0492466	+	0.998787i	$0.515682\pi$
$32$	0	0
$33$	− 25.9808i	− 0.137051i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	199.000	0.884200	0.442100	−	0.896966i	$-0.354234\pi$
$37$	199.000	0.884200	0.442100	+	0.896966i	$0.354234\pi$
$38$	0	0
$39$	− 69.2820i	− 0.284462i
$40$	0	0
$41$	− 187.061i	− 0.712539i	−0.934383	−	0.356269i	$-0.884049\pi$
$41$	− 187.061i	− 0.712539i	0.934383	−	0.356269i	$-0.115951\pi$
$42$	0	0
$43$	− 252.879i	− 0.896831i	−0.893825	−	0.448416i	$-0.851988\pi$
$43$	− 252.879i	− 0.896831i	0.893825	−	0.448416i	$-0.148012\pi$
$44$	0	0
$45$	135.100i	0.447545i
$46$	0	0
$47$	−567.000	−1.75969	−0.879845	−	0.475260i	$-0.842354\pi$
$47$	−567.000	−1.75969	−0.879845	+	0.475260i	$0.842354\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	36.3731i	0.0998676i
$52$	0	0
$53$	−333.000	−0.863039	−0.431520	−	0.902104i	$-0.642022\pi$
$53$	−333.000	−0.863039	−0.431520	+	0.902104i	$0.642022\pi$
$54$	0	0
$55$	135.000	0.330971
$56$	0	0
$57$	−17.0000	−0.0395036
$58$	0	0
$59$	−801.000	−1.76748	−0.883740	−	0.467978i	$-0.844983\pi$
$59$	−801.000	−1.76748	−0.883740	+	0.467978i	$0.844983\pi$
$60$	0	0
$61$	− 358.535i	− 0.752551i	−0.926508	−	0.376276i	$-0.877205\pi$
$61$	− 358.535i	− 0.752551i	0.926508	−	0.376276i	$-0.122795\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	360.000	0.686962
$66$	0	0
$67$	− 216.506i	− 0.394783i	−0.980325	−	0.197391i	$-0.936753\pi$
$67$	− 216.506i	− 0.394783i	0.980325	−	0.197391i	$-0.0632470\pi$
$68$	0	0
$69$	140.296i	0.244778i
$70$	0	0
$71$	488.438i	0.816436i	0.912884	+	0.408218i	$0.133850\pi$
$71$	488.438i	0.816436i	−0.912884	+	0.408218i	$0.866150\pi$
$72$	0	0
$73$	− 403.568i	− 0.647042i	−0.946221	−	0.323521i	$-0.895133\pi$
$73$	− 403.568i	− 0.647042i	0.946221	−	0.323521i	$-0.104867\pi$
$74$	0	0
$75$	−98.0000	−0.150881
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 1193.38i	− 1.69957i	−0.527129	−	0.849785i	$-0.676732\pi$
$79$	− 1193.38i	− 1.69957i	0.527129	−	0.849785i	$-0.323268\pi$
$80$	0	0
$81$	649.000	0.890261
$82$	0	0
$83$	468.000	0.618912	0.309456	−	0.950914i	$-0.399853\pi$
$83$	468.000	0.618912	0.309456	+	0.950914i	$0.399853\pi$
$84$	0	0
$85$	−189.000	−0.241176
$86$	0	0
$87$	−90.0000	−0.110908
$88$	0	0
$89$	− 192.258i	− 0.228981i	−0.993424	−	0.114490i	$-0.963477\pi$
$89$	− 192.258i	− 0.228981i	0.993424	−	0.114490i	$-0.0365235\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	17.0000	0.0189550
$94$	0	0
$95$	− 88.3346i	− 0.0953993i
$96$	0	0
$97$	1392.57i	1.45767i	0.684690	+	0.728835i	$0.259938\pi$
$97$	1392.57i	1.45767i	−0.684690	+	0.728835i	$0.740062\pi$
$98$	0	0
$99$	− 675.500i	− 0.685760i
$100$	0	0
$101$	− 1678.36i	− 1.65349i	−0.562575	−	0.826746i	$-0.690189\pi$
$101$	− 1678.36i	− 1.65349i	0.562575	−	0.826746i	$-0.309811\pi$
$102$	0	0
$103$	−1331.00	−1.27328	−0.636638	−	0.771163i	$-0.719675\pi$
$103$	−1331.00	−1.27328	−0.636638	+	0.771163i	$0.719675\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1657.57i	− 1.49760i	−0.662794	−	0.748802i	$-0.730629\pi$
$107$	− 1657.57i	− 1.49760i	0.662794	−	0.748802i	$-0.269371\pi$
$108$	0	0
$109$	791.000	0.695083	0.347542	−	0.937665i	$-0.387017\pi$
$109$	791.000	0.695083	0.347542	+	0.937665i	$0.387017\pi$
$110$	0	0
$111$	−199.000	−0.170164
$112$	0	0
$113$	990.000	0.824171	0.412086	−	0.911145i	$-0.364800\pi$
$113$	990.000	0.824171	0.412086	+	0.911145i	$0.364800\pi$
$114$	0	0
$115$	−729.000	−0.591127
$116$	0	0
$117$	− 1801.33i	− 1.42336i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	656.000	0.492863
$122$	0	0
$123$	187.061i	0.137128i
$124$	0	0
$125$	− 1158.74i	− 0.829128i
$126$	0	0
$127$	− 1784.01i	− 1.24650i	−0.782023	−	0.623250i	$-0.785812\pi$
$127$	− 1784.01i	− 1.24650i	0.782023	−	0.623250i	$-0.214188\pi$
$128$	0	0
$129$	252.879i	0.172595i
$130$	0	0
$131$	−2619.00	−1.74674	−0.873371	−	0.487056i	$-0.838071\pi$
$131$	−2619.00	−1.74674	−0.873371	+	0.487056i	$0.838071\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 275.396i	− 0.175573i
$136$	0	0
$137$	−2061.00	−1.28528	−0.642639	−	0.766169i	$-0.722161\pi$
$137$	−2061.00	−1.28528	−0.642639	+	0.766169i	$0.722161\pi$
$138$	0	0
$139$	1108.00	0.676110	0.338055	−	0.941126i	$-0.390231\pi$
$139$	1108.00	0.676110	0.338055	+	0.941126i	$0.390231\pi$
$140$	0	0
$141$	567.000	0.338653
$142$	0	0
$143$	−1800.00	−1.05261
$144$	0	0
$145$	− 467.654i	− 0.267838i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2763.00	1.51915	0.759576	−	0.650418i	$-0.225406\pi$
$149$	2763.00	1.51915	0.759576	+	0.650418i	$0.225406\pi$
$150$	0	0
$151$	− 826.188i	− 0.445260i	−0.974903	−	0.222630i	$-0.928536\pi$
$151$	− 826.188i	− 0.445260i	0.974903	−	0.222630i	$-0.0714642\pi$
$152$	0	0
$153$	945.700i	0.499708i
$154$	0	0
$155$	88.3346i	0.0457755i
$156$	0	0
$157$	878.150i	0.446395i	0.974773	+	0.223197i	$0.0716494\pi$
$157$	878.150i	0.446395i	−0.974773	+	0.223197i	$0.928351\pi$
$158$	0	0
$159$	333.000	0.166092
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 2426.60i	− 1.16605i	−0.812454	−	0.583025i	$-0.801869\pi$
$163$	− 2426.60i	− 1.16605i	0.812454	−	0.583025i	$-0.198131\pi$
$164$	0	0
$165$	−135.000	−0.0636954
$166$	0	0
$167$	2700.00	1.25109	0.625546	−	0.780188i	$-0.284876\pi$
$167$	2700.00	1.25109	0.625546	+	0.780188i	$0.284876\pi$
$168$	0	0
$169$	−2603.00	−1.18480
$170$	0	0
$171$	−442.000	−0.197664
$172$	0	0
$173$	1387.37i	0.609711i	0.952399	+	0.304855i	$0.0986082\pi$
$173$	1387.37i	0.609711i	−0.952399	+	0.304855i	$0.901392\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	801.000	0.340152
$178$	0	0
$179$	− 2551.31i	− 1.06533i	−0.846326	−	0.532665i	$-0.821191\pi$
$179$	− 2551.31i	− 1.06533i	0.846326	−	0.532665i	$-0.178809\pi$
$180$	0	0
$181$	− 879.882i	− 0.361332i	−0.983545	−	0.180666i	$-0.942175\pi$
$181$	− 879.882i	− 0.361332i	0.983545	−	0.180666i	$-0.0578253\pi$
$182$	0	0
$183$	358.535i	0.144829i
$184$	0	0
$185$	− 1034.03i	− 0.410939i
$186$	0	0
$187$	945.000	0.369547
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1127.57i	0.427161i	0.976925	+	0.213580i	$0.0685126\pi$
$191$	1127.57i	0.427161i	−0.976925	+	0.213580i	$0.931487\pi$
$192$	0	0
$193$	1315.00	0.490444	0.245222	−	0.969467i	$-0.421139\pi$
$193$	1315.00	0.490444	0.245222	+	0.969467i	$0.421139\pi$
$194$	0	0
$195$	−360.000	−0.132206
$196$	0	0
$197$	3258.00	1.17829	0.589144	−	0.808028i	$-0.299465\pi$
$197$	3258.00	1.17829	0.589144	+	0.808028i	$0.299465\pi$
$198$	0	0
$199$	−1753.00	−0.624457	−0.312228	−	0.950007i	$-0.601075\pi$
$199$	−1753.00	−0.624457	−0.312228	+	0.950007i	$0.601075\pi$
$200$	0	0
$201$	216.506i	0.0759760i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−972.000	−0.331158
$206$	0	0
$207$	3647.70i	1.22480i
$208$	0	0
$209$	441.673i	0.146178i
$210$	0	0
$211$	3543.78i	1.15623i	0.815957	+	0.578113i	$0.196211\pi$
$211$	3543.78i	1.15623i	−0.815957	+	0.578113i	$0.803789\pi$
$212$	0	0
$213$	− 488.438i	− 0.157123i
$214$	0	0
$215$	−1314.00	−0.416810
$216$	0	0
$217$	0	0
$218$	0	0
$219$	403.568i	0.124523i
$220$	0	0
$221$	2520.00	0.767030
$222$	0	0
$223$	−2896.00	−0.869644	−0.434822	−	0.900517i	$-0.643189\pi$
$223$	−2896.00	−0.869644	−0.434822	+	0.900517i	$0.643189\pi$
$224$	0	0
$225$	−2548.00	−0.754963
$226$	0	0
$227$	3699.00	1.08155	0.540774	−	0.841168i	$-0.318132\pi$
$227$	3699.00	1.08155	0.540774	+	0.841168i	$0.318132\pi$
$228$	0	0
$229$	− 3815.71i	− 1.10109i	−0.834806	−	0.550544i	$-0.814420\pi$
$229$	− 3815.71i	− 1.10109i	0.834806	−	0.550544i	$-0.185580\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5661.00	−1.59169	−0.795846	−	0.605499i	$-0.792974\pi$
$233$	−5661.00	−1.59169	−0.795846	+	0.605499i	$0.792974\pi$
$234$	0	0
$235$	2946.22i	0.817830i
$236$	0	0
$237$	1193.38i	0.327083i
$238$	0	0
$239$	− 3481.42i	− 0.942236i	−0.882070	−	0.471118i	$-0.843851\pi$
$239$	− 3481.42i	− 0.942236i	0.882070	−	0.471118i	$-0.156149\pi$
$240$	0	0
$241$	− 2426.60i	− 0.648594i	−0.945955	−	0.324297i	$-0.894872\pi$
$241$	− 2426.60i	− 0.648594i	0.945955	−	0.324297i	$-0.105128\pi$
$242$	0	0
$243$	−2080.00	−0.549103
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1177.79i	0.303406i
$248$	0	0
$249$	−468.000	−0.119110
$250$	0	0
$251$	−6120.00	−1.53901	−0.769504	−	0.638642i	$-0.779496\pi$
$251$	−6120.00	−1.53901	−0.769504	+	0.638642i	$0.779496\pi$
$252$	0	0
$253$	3645.00	0.905768
$254$	0	0
$255$	189.000	0.0464143
$256$	0	0
$257$	3569.76i	0.866441i	0.901288	+	0.433220i	$0.142623\pi$
$257$	3569.76i	0.866441i	−0.901288	+	0.433220i	$0.857377\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−2340.00	−0.554952
$262$	0	0
$263$	− 3964.66i	− 0.929550i	−0.885429	−	0.464775i	$-0.846135\pi$
$263$	− 3964.66i	− 0.929550i	0.885429	−	0.464775i	$-0.153865\pi$
$264$	0	0
$265$	1730.32i	0.401104i
$266$	0	0
$267$	192.258i	0.0440673i
$268$	0	0
$269$	6157.44i	1.39563i	0.716276	+	0.697817i	$0.245845\pi$
$269$	6157.44i	1.39563i	−0.716276	+	0.697817i	$0.754155\pi$
$270$	0	0
$271$	5749.00	1.28866	0.644330	−	0.764748i	$-0.277136\pi$
$271$	5749.00	1.28866	0.644330	+	0.764748i	$0.277136\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2546.11i	0.558315i
$276$	0	0
$277$	−197.000	−0.0427313	−0.0213657	−	0.999772i	$-0.506801\pi$
$277$	−197.000	−0.0427313	−0.0213657	+	0.999772i	$0.506801\pi$
$278$	0	0
$279$	442.000	0.0948453
$280$	0	0
$281$	−3618.00	−0.768085	−0.384042	−	0.923315i	$-0.625468\pi$
$281$	−3618.00	−0.768085	−0.384042	+	0.923315i	$0.625468\pi$
$282$	0	0
$283$	3563.00	0.748404	0.374202	−	0.927347i	$-0.377917\pi$
$283$	3563.00	0.748404	0.374202	+	0.927347i	$0.377917\pi$
$284$	0	0
$285$	88.3346i	0.0183596i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	3590.00	0.730714
$290$	0	0
$291$	− 1392.57i	− 0.280529i
$292$	0	0
$293$	− 6048.32i	− 1.20596i	−0.797756	−	0.602981i	$-0.793980\pi$
$293$	− 6048.32i	− 1.20596i	0.797756	−	0.602981i	$-0.206020\pi$
$294$	0	0
$295$	4162.12i	0.821450i
$296$	0	0
$297$	1376.98i	0.269025i
$298$	0	0
$299$	9720.00	1.88001
$300$	0	0
$301$	0	0
$302$	0	0
$303$	1678.36i	0.318215i
$304$	0	0
$305$	−1863.00	−0.349754
$306$	0	0
$307$	7316.00	1.36009	0.680043	−	0.733173i	$-0.261961\pi$
$307$	7316.00	1.36009	0.680043	+	0.733173i	$0.261961\pi$
$308$	0	0
$309$	1331.00	0.245042
$310$	0	0
$311$	2547.00	0.464396	0.232198	−	0.972669i	$-0.425408\pi$
$311$	2547.00	0.464396	0.232198	+	0.972669i	$0.425408\pi$
$312$	0	0
$313$	9188.53i	1.65932i	0.558271	+	0.829659i	$0.311465\pi$
$313$	9188.53i	1.65932i	−0.558271	+	0.829659i	$0.688535\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	495.000	0.0877034	0.0438517	−	0.999038i	$-0.486037\pi$
$317$	495.000	0.0877034	0.0438517	+	0.999038i	$0.486037\pi$
$318$	0	0
$319$	2338.27i	0.410401i
$320$	0	0
$321$	1657.57i	0.288214i
$322$	0	0
$323$	− 618.342i	− 0.106519i
$324$	0	0
$325$	6789.64i	1.15884i
$326$	0	0
$327$	−791.000	−0.133769
$328$	0	0
$329$	0	0
$330$	0	0
$331$	9185.07i	1.52525i	0.646842	+	0.762624i	$0.276089\pi$
$331$	9185.07i	1.52525i	−0.646842	+	0.762624i	$0.723911\pi$
$332$	0	0
$333$	−5174.00	−0.851452
$334$	0	0
$335$	−1125.00	−0.183479
$336$	0	0
$337$	−5194.00	−0.839570	−0.419785	−	0.907623i	$-0.637895\pi$
$337$	−5194.00	−0.839570	−0.419785	+	0.907623i	$0.637895\pi$
$338$	0	0
$339$	−990.000	−0.158612
$340$	0	0
$341$	− 441.673i	− 0.0701406i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	729.000	0.113762
$346$	0	0
$347$	4494.67i	0.695351i	0.937615	+	0.347675i	$0.113029\pi$
$347$	4494.67i	0.695351i	−0.937615	+	0.347675i	$0.886971\pi$
$348$	0	0
$349$	− 9436.21i	− 1.44730i	−0.690165	−	0.723652i	$-0.742462\pi$
$349$	− 9436.21i	− 1.44730i	0.690165	−	0.723652i	$-0.257538\pi$
$350$	0	0
$351$	3671.95i	0.558388i
$352$	0	0
$353$	− 6230.19i	− 0.939375i	−0.882833	−	0.469688i	$-0.844367\pi$
$353$	− 6230.19i	− 0.939375i	0.882833	−	0.469688i	$-0.155633\pi$
$354$	0	0
$355$	2538.00	0.379445
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6739.41i	0.990787i	0.868669	+	0.495393i	$0.164976\pi$
$359$	6739.41i	0.990787i	−0.868669	+	0.495393i	$0.835024\pi$
$360$	0	0
$361$	−6570.00	−0.957866
$362$	0	0
$363$	−656.000	−0.0948514
$364$	0	0
$365$	−2097.00	−0.300718
$366$	0	0
$367$	12491.0	1.77663	0.888317	−	0.459230i	$-0.151875\pi$
$367$	12491.0	1.77663	0.888317	+	0.459230i	$0.151875\pi$
$368$	0	0
$369$	4863.60i	0.686149i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−7777.00	−1.07957	−0.539783	−	0.841804i	$-0.681494\pi$
$373$	−7777.00	−1.07957	−0.539783	+	0.841804i	$0.681494\pi$
$374$	0	0
$375$	1158.74i	0.159566i
$376$	0	0
$377$	6235.38i	0.851826i
$378$	0	0
$379$	− 6231.92i	− 0.844623i	−0.906451	−	0.422312i	$-0.861219\pi$
$379$	− 6231.92i	− 0.844623i	0.906451	−	0.422312i	$-0.138781\pi$
$380$	0	0
$381$	1784.01i	0.239889i
$382$	0	0
$383$	7281.00	0.971388	0.485694	−	0.874129i	$-0.338567\pi$
$383$	7281.00	0.971388	0.485694	+	0.874129i	$0.338567\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6574.86i	0.863615i
$388$	0	0
$389$	11583.0	1.50972	0.754860	−	0.655885i	$-0.227705\pi$
$389$	11583.0	1.50972	0.754860	+	0.655885i	$0.227705\pi$
$390$	0	0
$391$	−5103.00	−0.660025
$392$	0	0
$393$	2619.00	0.336160
$394$	0	0
$395$	−6201.00	−0.789889
$396$	0	0
$397$	− 2610.20i	− 0.329980i	−0.986295	−	0.164990i	$-0.947241\pi$
$397$	− 2610.20i	− 0.329980i	0.986295	−	0.164990i	$-0.0527592\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−7605.00	−0.947071	−0.473536	−	0.880775i	$-0.657022\pi$
$401$	−7605.00	−0.947071	−0.473536	+	0.880775i	$0.657022\pi$
$402$	0	0
$403$	− 1177.79i	− 0.145583i
$404$	0	0
$405$	− 3372.30i	− 0.413756i
$406$	0	0
$407$	5170.17i	0.629671i
$408$	0	0
$409$	4882.65i	0.590297i	0.955451	+	0.295149i	$0.0953692\pi$
$409$	4882.65i	0.590297i	−0.955451	+	0.295149i	$0.904631\pi$
$410$	0	0
$411$	2061.00	0.247352
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 2431.80i	− 0.287644i
$416$	0	0
$417$	−1108.00	−0.130117
$418$	0	0
$419$	−12780.0	−1.49008	−0.745040	−	0.667019i	$-0.767570\pi$
$419$	−12780.0	−1.49008	−0.745040	+	0.667019i	$0.767570\pi$
$420$	0	0
$421$	9802.00	1.13473	0.567364	−	0.823467i	$-0.307963\pi$
$421$	9802.00	1.13473	0.567364	+	0.823467i	$0.307963\pi$
$422$	0	0
$423$	14742.0	1.69452
$424$	0	0
$425$	− 3564.56i	− 0.406839i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	1800.00	0.202575
$430$	0	0
$431$	− 2405.82i	− 0.268873i	−0.990922	−	0.134436i	$-0.957078\pi$
$431$	− 2405.82i	− 0.268873i	0.990922	−	0.134436i	$-0.0429224\pi$
$432$	0	0
$433$	8390.05i	0.931178i	0.885001	+	0.465589i	$0.154158\pi$
$433$	8390.05i	0.931178i	−0.885001	+	0.465589i	$0.845842\pi$
$434$	0	0
$435$	467.654i	0.0515455i
$436$	0	0
$437$	− 2385.03i	− 0.261079i
$438$	0	0
$439$	1505.00	0.163621	0.0818106	−	0.996648i	$-0.473930\pi$
$439$	1505.00	0.163621	0.0818106	+	0.996648i	$0.473930\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 6479.60i	− 0.694933i	−0.937692	−	0.347466i	$-0.887042\pi$
$443$	− 6479.60i	− 0.694933i	0.937692	−	0.347466i	$-0.112958\pi$
$444$	0	0
$445$	−999.000	−0.106421
$446$	0	0
$447$	−2763.00	−0.292361
$448$	0	0
$449$	−18090.0	−1.90138	−0.950690	−	0.310142i	$-0.899623\pi$
$449$	−18090.0	−1.90138	−0.950690	+	0.310142i	$0.899623\pi$
$450$	0	0
$451$	4860.00	0.507425
$452$	0	0
$453$	826.188i	0.0856903i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−11737.0	−1.20139	−0.600693	−	0.799480i	$-0.705109\pi$
$457$	−11737.0	−1.20139	−0.600693	+	0.799480i	$0.705109\pi$
$458$	0	0
$459$	− 1927.77i	− 0.196036i
$460$	0	0
$461$	− 956.092i	− 0.0965936i	−0.998833	−	0.0482968i	$-0.984621\pi$
$461$	− 956.092i	− 0.0965936i	0.998833	−	0.0482968i	$-0.0153793\pi$
$462$	0	0
$463$	4922.49i	0.494098i	0.969003	+	0.247049i	$0.0794609\pi$
$463$	4922.49i	0.494098i	−0.969003	+	0.247049i	$0.920539\pi$
$464$	0	0
$465$	− 88.3346i	− 0.00880950i
$466$	0	0
$467$	8397.00	0.832049	0.416024	−	0.909353i	$-0.363423\pi$
$467$	8397.00	0.832049	0.416024	+	0.909353i	$0.363423\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 878.150i	− 0.0859087i
$472$	0	0
$473$	6570.00	0.638666
$474$	0	0
$475$	1666.00	0.160929
$476$	0	0
$477$	8658.00	0.831075
$478$	0	0
$479$	−15741.0	−1.50151	−0.750756	−	0.660579i	$-0.770311\pi$
$479$	−15741.0	−1.50151	−0.750756	+	0.660579i	$0.770311\pi$
$480$	0	0
$481$	13787.1i	1.30694i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7236.00	0.677464
$486$	0	0
$487$	1079.07i	0.100405i	0.998739	+	0.0502025i	$0.0159867\pi$
$487$	1079.07i	0.100405i	−0.998739	+	0.0502025i	$0.984013\pi$
$488$	0	0
$489$	2426.60i	0.224407i
$490$	0	0
$491$	1278.25i	0.117488i	0.998273	+	0.0587442i	$0.0187096\pi$
$491$	1278.25i	0.117488i	−0.998273	+	0.0587442i	$0.981290\pi$
$492$	0	0
$493$	− 3273.58i	− 0.299056i
$494$	0	0
$495$	−3510.00	−0.318713
$496$	0	0
$497$	0	0
$498$	0	0
$499$	12067.2i	1.08257i	0.840839	+	0.541285i	$0.182062\pi$
$499$	12067.2i	1.08257i	−0.840839	+	0.541285i	$0.817938\pi$
$500$	0	0
$501$	−2700.00	−0.240773
$502$	0	0
$503$	−2808.00	−0.248912	−0.124456	−	0.992225i	$-0.539718\pi$
$503$	−2808.00	−0.248912	−0.124456	+	0.992225i	$0.539718\pi$
$504$	0	0
$505$	−8721.00	−0.768474
$506$	0	0
$507$	2603.00	0.228014
$508$	0	0
$509$	− 12974.8i	− 1.12986i	−0.825140	−	0.564929i	$-0.808904\pi$
$509$	− 12974.8i	− 1.12986i	0.825140	−	0.564929i	$-0.191096\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	901.000	0.0775441
$514$	0	0
$515$	6916.08i	0.591765i
$516$	0	0
$517$	− 14731.1i	− 1.25314i
$518$	0	0
$519$	− 1387.37i	− 0.117339i
$520$	0	0
$521$	− 431.281i	− 0.0362663i	−0.999836	−	0.0181332i	$-0.994228\pi$
$521$	− 431.281i	− 0.0362663i	0.999836	−	0.0181332i	$-0.00577228\pi$
$522$	0	0
$523$	−18791.0	−1.57108	−0.785538	−	0.618813i	$-0.787614\pi$
$523$	−18791.0	−1.57108	−0.785538	+	0.618813i	$0.787614\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	618.342i	0.0511108i
$528$	0	0
$529$	−7516.00	−0.617737
$530$	0	0
$531$	20826.0	1.70202
$532$	0	0
$533$	12960.0	1.05321
$534$	0	0
$535$	−8613.00	−0.696023
$536$	0	0
$537$	2551.31i	0.205023i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	12547.0	0.997112	0.498556	−	0.866857i	$-0.333864\pi$
$541$	12547.0	0.997112	0.498556	+	0.866857i	$0.333864\pi$
$542$	0	0
$543$	879.882i	0.0695384i
$544$	0	0
$545$	− 4110.16i	− 0.323045i
$546$	0	0
$547$	2151.21i	0.168152i	0.996459	+	0.0840758i	$0.0267938\pi$
$547$	2151.21i	0.168152i	−0.996459	+	0.0840758i	$0.973206\pi$
$548$	0	0
$549$	9321.90i	0.724679i
$550$	0	0
$551$	1530.00	0.118294
$552$	0	0
$553$	0	0
$554$	0	0
$555$	1034.03i	0.0790852i
$556$	0	0
$557$	3519.00	0.267693	0.133846	−	0.991002i	$-0.457267\pi$
$557$	3519.00	0.267693	0.133846	+	0.991002i	$0.457267\pi$
$558$	0	0
$559$	17520.0	1.32561
$560$	0	0
$561$	−945.000	−0.0711193
$562$	0	0
$563$	−8577.00	−0.642056	−0.321028	−	0.947070i	$-0.604028\pi$
$563$	−8577.00	−0.642056	−0.321028	+	0.947070i	$0.604028\pi$
$564$	0	0
$565$	− 5144.19i	− 0.383040i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	10863.0	0.800353	0.400176	−	0.916438i	$-0.368949\pi$
$569$	10863.0	0.800353	0.400176	+	0.916438i	$0.368949\pi$
$570$	0	0
$571$	− 1456.65i	− 0.106758i	−0.998574	−	0.0533792i	$-0.983001\pi$
$571$	− 1456.65i	− 0.106758i	0.998574	−	0.0533792i	$-0.0169992\pi$
$572$	0	0
$573$	− 1127.57i	− 0.0822072i
$574$	0	0
$575$	− 13749.0i	− 0.997172i
$576$	0	0
$577$	− 652.983i	− 0.0471127i	−0.999723	−	0.0235564i	$-0.992501\pi$
$577$	− 652.983i	− 0.0471127i	0.999723	−	0.0235564i	$-0.00749892\pi$
$578$	0	0
$579$	−1315.00	−0.0943861
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 8651.59i	− 0.614601i
$584$	0	0
$585$	−9360.00	−0.661519
$586$	0	0
$587$	−9684.00	−0.680922	−0.340461	−	0.940259i	$-0.610583\pi$
$587$	−9684.00	−0.680922	−0.340461	+	0.940259i	$0.610583\pi$
$588$	0	0
$589$	−289.000	−0.0202174
$590$	0	0
$591$	−3258.00	−0.226762
$592$	0	0
$593$	7144.71i	0.494769i	0.968917	+	0.247385i	$0.0795711\pi$
$593$	7144.71i	0.494769i	−0.968917	+	0.247385i	$0.920429\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1753.00	0.120177
$598$	0	0
$599$	− 17059.0i	− 1.16362i	−0.813323	−	0.581812i	$-0.802344\pi$
$599$	− 17059.0i	− 1.16362i	0.813323	−	0.581812i	$-0.197656\pi$
$600$	0	0
$601$	26070.8i	1.76947i	0.466095	+	0.884735i	$0.345661\pi$
$601$	26070.8i	1.76947i	−0.466095	+	0.884735i	$0.654339\pi$
$602$	0	0
$603$	5629.17i	0.380161i
$604$	0	0
$605$	− 3408.68i	− 0.229062i
$606$	0	0
$607$	−12295.0	−0.822139	−0.411070	−	0.911604i	$-0.634845\pi$
$607$	−12295.0	−0.822139	−0.411070	+	0.911604i	$0.634845\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 39282.9i	− 2.60101i
$612$	0	0
$613$	1055.00	0.0695123	0.0347562	−	0.999396i	$-0.488935\pi$
$613$	1055.00	0.0695123	0.0347562	+	0.999396i	$0.488935\pi$
$614$	0	0
$615$	972.000	0.0637314
$616$	0	0
$617$	1494.00	0.0974816	0.0487408	−	0.998811i	$-0.484479\pi$
$617$	1494.00	0.0974816	0.0487408	+	0.998811i	$0.484479\pi$
$618$	0	0
$619$	5599.00	0.363559	0.181779	−	0.983339i	$-0.441814\pi$
$619$	5599.00	0.363559	0.181779	+	0.983339i	$0.441814\pi$
$620$	0	0
$621$	− 7435.69i	− 0.480490i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	6229.00	0.398656
$626$	0	0
$627$	− 441.673i	− 0.0281319i
$628$	0	0
$629$	− 7238.24i	− 0.458836i
$630$	0	0
$631$	− 19007.5i	− 1.19917i	−0.800310	−	0.599586i	$-0.795332\pi$
$631$	− 19007.5i	− 1.19917i	0.800310	−	0.599586i	$-0.204668\pi$
$632$	0	0
$633$	− 3543.78i	− 0.222516i
$634$	0	0
$635$	−9270.00	−0.579321
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 12699.4i	− 0.786198i
$640$	0	0
$641$	22023.0	1.35703	0.678515	−	0.734587i	$-0.262624\pi$
$641$	22023.0	1.35703	0.678515	+	0.734587i	$0.262624\pi$
$642$	0	0
$643$	−10036.0	−0.615523	−0.307761	−	0.951464i	$-0.599580\pi$
$643$	−10036.0	−0.615523	−0.307761	+	0.951464i	$0.599580\pi$
$644$	0	0
$645$	1314.00	0.0802150
$646$	0	0
$647$	−27693.0	−1.68273	−0.841363	−	0.540470i	$-0.818246\pi$
$647$	−27693.0	−1.68273	−0.841363	+	0.540470i	$0.818246\pi$
$648$	0	0
$649$	− 20810.6i	− 1.25869i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1287.00	0.0771274	0.0385637	−	0.999256i	$-0.487722\pi$
$653$	1287.00	0.0771274	0.0385637	+	0.999256i	$0.487722\pi$
$654$	0	0
$655$	13608.7i	0.811812i
$656$	0	0
$657$	10492.8i	0.623077i
$658$	0	0
$659$	− 5767.73i	− 0.340939i	−0.985363	−	0.170470i	$-0.945472\pi$
$659$	− 5767.73i	− 0.340939i	0.985363	−	0.170470i	$-0.0545284\pi$
$660$	0	0
$661$	− 10691.9i	− 0.629151i	−0.949232	−	0.314575i	$-0.898138\pi$
$661$	− 10691.9i	− 0.629151i	0.949232	−	0.314575i	$-0.101862\pi$
$662$	0	0
$663$	−2520.00	−0.147615
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 12626.7i	− 0.732992i
$668$	0	0
$669$	2896.00	0.167363
$670$	0	0
$671$	9315.00	0.535919
$672$	0	0
$673$	14518.0	0.831542	0.415771	−	0.909469i	$-0.363512\pi$
$673$	14518.0	0.831542	0.415771	+	0.909469i	$0.363512\pi$
$674$	0	0
$675$	5194.00	0.296174
$676$	0	0
$677$	− 8443.75i	− 0.479349i	−0.970853	−	0.239675i	$-0.922959\pi$
$677$	− 8443.75i	− 0.479349i	0.970853	−	0.239675i	$-0.0770408\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−3699.00	−0.208144
$682$	0	0
$683$	− 27201.9i	− 1.52394i	−0.647613	−	0.761969i	$-0.724233\pi$
$683$	− 27201.9i	− 1.52394i	0.647613	−	0.761969i	$-0.275767\pi$
$684$	0	0
$685$	10709.3i	0.597343i
$686$	0	0
$687$	3815.71i	0.211904i
$688$	0	0
$689$	− 23070.9i	− 1.27566i
$690$	0	0
$691$	−19907.0	−1.09594	−0.547972	−	0.836496i	$-0.684600\pi$
$691$	−19907.0	−1.09594	−0.547972	+	0.836496i	$0.684600\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 5757.34i	− 0.314228i
$696$	0	0
$697$	−6804.00	−0.369756
$698$	0	0
$699$	5661.00	0.306321
$700$	0	0
$701$	12294.0	0.662394	0.331197	−	0.943562i	$-0.392548\pi$
$701$	12294.0	0.662394	0.331197	+	0.943562i	$0.392548\pi$
$702$	0	0
$703$	3383.00	0.181497
$704$	0	0
$705$	− 2946.22i	− 0.157391i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−8161.00	−0.432289	−0.216144	−	0.976361i	$-0.569348\pi$
$709$	−8161.00	−0.432289	−0.216144	+	0.976361i	$0.569348\pi$
$710$	0	0
$711$	31028.0i	1.63662i
$712$	0	0
$713$	2385.03i	0.125274i
$714$	0	0
$715$	9353.07i	0.489210i
$716$	0	0
$717$	3481.42i	0.181333i
$718$	0	0
$719$	−13995.0	−0.725905	−0.362952	−	0.931808i	$-0.618231\pi$
$719$	−13995.0	−0.725905	−0.362952	+	0.931808i	$0.618231\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	2426.60i	0.124822i
$724$	0	0
$725$	8820.00	0.451816
$726$	0	0
$727$	−1232.00	−0.0628506	−0.0314253	−	0.999506i	$-0.510005\pi$
$727$	−1232.00	−0.0628506	−0.0314253	+	0.999506i	$0.510005\pi$
$728$	0	0
$729$	−15443.0	−0.784586
$730$	0	0
$731$	−9198.00	−0.465390
$732$	0	0
$733$	− 27600.2i	− 1.39077i	−0.718635	−	0.695387i	$-0.755233\pi$
$733$	− 27600.2i	− 1.39077i	0.718635	−	0.695387i	$-0.244767\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5625.00	0.281139
$738$	0	0
$739$	23734.3i	1.18143i	0.806879	+	0.590717i	$0.201155\pi$
$739$	23734.3i	1.18143i	−0.806879	+	0.590717i	$0.798845\pi$
$740$	0	0
$741$	− 1177.79i	− 0.0583905i
$742$	0	0
$743$	− 11275.7i	− 0.556748i	−0.960473	−	0.278374i	$-0.910205\pi$
$743$	− 11275.7i	− 0.556748i	0.960473	−	0.278374i	$-0.0897954\pi$
$744$	0	0
$745$	− 14357.0i	− 0.706038i
$746$	0	0
$747$	−12168.0	−0.595989
$748$	0	0
$749$	0	0
$750$	0	0
$751$	6226.72i	0.302552i	0.988492	+	0.151276i	$0.0483382\pi$
$751$	6226.72i	0.302552i	−0.988492	+	0.151276i	$0.951662\pi$
$752$	0	0
$753$	6120.00	0.296182
$754$	0	0
$755$	−4293.00	−0.206938
$756$	0	0
$757$	−17062.0	−0.819193	−0.409596	−	0.912267i	$-0.634330\pi$
$757$	−17062.0	−0.819193	−0.409596	+	0.912267i	$0.634330\pi$
$758$	0	0
$759$	−3645.00	−0.174315
$760$	0	0
$761$	26848.5i	1.27892i	0.768824	+	0.639460i	$0.220842\pi$
$761$	26848.5i	1.27892i	−0.768824	+	0.639460i	$0.779158\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	4914.00	0.232243
$766$	0	0
$767$	− 55494.9i	− 2.61252i
$768$	0	0
$769$	− 4330.13i	− 0.203054i	−0.994833	−	0.101527i	$-0.967627\pi$
$769$	− 4330.13i	− 0.203054i	0.994833	−	0.101527i	$-0.0323728\pi$
$770$	0	0
$771$	− 3569.76i	− 0.166747i
$772$	0	0
$773$	− 16851.1i	− 0.784079i	−0.919948	−	0.392039i	$-0.871770\pi$
$773$	− 16851.1i	− 0.784079i	0.919948	−	0.392039i	$-0.128230\pi$
$774$	0	0
$775$	−1666.00	−0.0772187
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 3180.05i	− 0.146261i
$780$	0	0
$781$	−12690.0	−0.581413
$782$	0	0
$783$	4770.00	0.217709
$784$	0	0
$785$	4563.00	0.207466
$786$	0	0
$787$	−27065.0	−1.22587	−0.612937	−	0.790132i	$-0.710012\pi$
$787$	−27065.0	−1.22587	−0.612937	+	0.790132i	$0.710012\pi$
$788$	0	0
$789$	3964.66i	0.178892i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	24840.0	1.11235
$794$	0	0
$795$	− 1730.32i	− 0.0771926i
$796$	0	0
$797$	− 13717.8i	− 0.609675i	−0.952404	−	0.304837i	$-0.901398\pi$
$797$	− 13717.8i	− 0.609675i	0.952404	−	0.304837i	$-0.0986021\pi$
$798$	0	0
$799$	20623.5i	0.913151i
$800$	0	0
$801$	4998.70i	0.220500i
$802$	0	0
$803$	10485.0	0.460782
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 6157.44i	− 0.268590i
$808$	0	0
$809$	−22329.0	−0.970390	−0.485195	−	0.874406i	$-0.661251\pi$
$809$	−22329.0	−0.970390	−0.485195	+	0.874406i	$0.661251\pi$
$810$	0	0
$811$	43252.0	1.87273	0.936364	−	0.351029i	$-0.114168\pi$
$811$	43252.0	1.87273	0.936364	+	0.351029i	$0.114168\pi$
$812$	0	0
$813$	−5749.00	−0.248003
$814$	0	0
$815$	−12609.0	−0.541931
$816$	0	0
$817$	− 4298.95i	− 0.184090i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	29691.0	1.26215	0.631074	−	0.775723i	$-0.282615\pi$
$821$	29691.0	1.26215	0.631074	+	0.775723i	$0.282615\pi$
$822$	0	0
$823$	12538.3i	0.531055i	0.964103	+	0.265527i	$0.0855461\pi$
$823$	12538.3i	0.531055i	−0.964103	+	0.265527i	$0.914454\pi$
$824$	0	0
$825$	− 2546.11i	− 0.107448i
$826$	0	0
$827$	6910.88i	0.290586i	0.989389	+	0.145293i	$0.0464125\pi$
$827$	6910.88i	0.290586i	−0.989389	+	0.145293i	$0.953587\pi$
$828$	0	0
$829$	43355.0i	1.81638i	0.418557	+	0.908191i	$0.362536\pi$
$829$	43355.0i	1.81638i	−0.418557	+	0.908191i	$0.637464\pi$
$830$	0	0
$831$	197.000	0.00822365
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 14029.6i	− 0.581455i
$836$	0	0
$837$	−901.000	−0.0372080
$838$	0	0
$839$	−9648.00	−0.397004	−0.198502	−	0.980101i	$-0.563608\pi$
$839$	−9648.00	−0.397004	−0.198502	+	0.980101i	$0.563608\pi$
$840$	0	0
$841$	−16289.0	−0.667883
$842$	0	0
$843$	3618.00	0.147818
$844$	0	0
$845$	13525.6i	0.550644i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−3563.00	−0.144030
$850$	0	0
$851$	− 27918.9i	− 1.12462i
$852$	0	0
$853$	− 8119.85i	− 0.325930i	−0.986632	−	0.162965i	$-0.947894\pi$
$853$	− 8119.85i	− 0.325930i	0.986632	−	0.162965i	$-0.0521058\pi$
$854$	0	0
$855$	2296.70i	0.0918660i
$856$	0	0
$857$	− 1387.37i	− 0.0552996i	−0.999618	−	0.0276498i	$-0.991198\pi$
$857$	− 1387.37i	− 0.0552996i	0.999618	−	0.0276498i	$-0.00880232\pi$
$858$	0	0
$859$	−1015.00	−0.0403159	−0.0201579	−	0.999797i	$-0.506417\pi$
$859$	−1015.00	−0.0403159	−0.0201579	+	0.999797i	$0.506417\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	3226.81i	0.127279i	0.997973	+	0.0636396i	$0.0202708\pi$
$863$	3226.81i	0.127279i	−0.997973	+	0.0636396i	$0.979729\pi$
$864$	0	0
$865$	7209.00	0.283368
$866$	0	0
$867$	−3590.00	−0.140626
$868$	0	0
$869$	31005.0	1.21033
$870$	0	0
$871$	15000.0	0.583531
$872$	0	0
$873$	− 36206.8i	− 1.40368i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−15541.0	−0.598383	−0.299192	−	0.954193i	$-0.596717\pi$
$877$	−15541.0	−0.598383	−0.299192	+	0.954193i	$0.596717\pi$
$878$	0	0
$879$	6048.32i	0.232087i
$880$	0	0
$881$	− 28724.3i	− 1.09846i	−0.835670	−	0.549232i	$-0.814920\pi$
$881$	− 28724.3i	− 1.09846i	0.835670	−	0.549232i	$-0.185080\pi$
$882$	0	0
$883$	− 45826.6i	− 1.74653i	−0.487244	−	0.873266i	$-0.661998\pi$
$883$	− 45826.6i	− 1.74653i	0.487244	−	0.873266i	$-0.338002\pi$
$884$	0	0
$885$	− 4162.12i	− 0.158088i
$886$	0	0
$887$	6129.00	0.232009	0.116004	−	0.993249i	$-0.462991\pi$
$887$	6129.00	0.232009	0.116004	+	0.993249i	$0.462991\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	16861.5i	0.633987i
$892$	0	0
$893$	−9639.00	−0.361206
$894$	0	0
$895$	−13257.0	−0.495120
$896$	0	0
$897$	−9720.00	−0.361808
$898$	0	0
$899$	−1530.00	−0.0567612
$900$	0	0
$901$	12112.2i	0.447855i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4572.00	−0.167932
$906$	0	0
$907$	7335.24i	0.268536i	0.990945	+	0.134268i	$0.0428684\pi$
$907$	7335.24i	0.268536i	−0.990945	+	0.134268i	$0.957132\pi$
$908$	0	0
$909$	43637.3i	1.59225i
$910$	0	0
$911$	2774.75i	0.100913i	0.998726	+	0.0504563i	$0.0160676\pi$
$911$	2774.75i	0.100913i	−0.998726	+	0.0504563i	$0.983932\pi$
$912$	0	0
$913$	12159.0i	0.440749i
$914$	0	0
$915$	1863.00	0.0673103
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 1186.45i	− 0.0425871i	−0.999773	−	0.0212935i	$-0.993222\pi$
$919$	− 1186.45i	− 0.0425871i	0.999773	−	0.0212935i	$-0.00677846\pi$
$920$	0	0
$921$	−7316.00	−0.261749
$922$	0	0
$923$	−33840.0	−1.20678
$924$	0	0
$925$	19502.0	0.693213
$926$	0	0
$927$	34606.0	1.22612
$928$	0	0
$929$	25414.4i	0.897544i	0.893646	+	0.448772i	$0.148139\pi$
$929$	25414.4i	0.897544i	−0.893646	+	0.448772i	$0.851861\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−2547.00	−0.0893730
$934$	0	0
$935$	− 4910.36i	− 0.171750i
$936$	0	0
$937$	13870.3i	0.483588i	0.970328	+	0.241794i	$0.0777358\pi$
$937$	13870.3i	0.483588i	−0.970328	+	0.241794i	$0.922264\pi$
$938$	0	0
$939$	− 9188.53i	− 0.319336i
$940$	0	0
$941$	− 10220.8i	− 0.354080i	−0.984204	−	0.177040i	$-0.943348\pi$
$941$	− 10220.8i	− 0.354080i	0.984204	−	0.177040i	$-0.0566523\pi$
$942$	0	0
$943$	−26244.0	−0.906280
$944$	0	0
$945$	0	0
$946$	0	0
$947$	13494.4i	0.463051i	0.972829	+	0.231526i	$0.0743717\pi$
$947$	13494.4i	0.463051i	−0.972829	+	0.231526i	$0.925628\pi$
$948$	0	0
$949$	27960.0	0.956396
$950$	0	0
$951$	−495.000	−0.0168785
$952$	0	0
$953$	6966.00	0.236780	0.118390	−	0.992967i	$-0.462227\pi$
$953$	6966.00	0.236780	0.118390	+	0.992967i	$0.462227\pi$
$954$	0	0
$955$	5859.00	0.198526
$956$	0	0
$957$	− 2338.27i	− 0.0789817i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29502.0	−0.990299
$962$	0	0
$963$	43096.9i	1.44214i
$964$	0	0
$965$	− 6832.94i	− 0.227938i
$966$	0	0
$967$	15979.9i	0.531416i	0.964054	+	0.265708i	$0.0856057\pi$
$967$	15979.9i	0.531416i	−0.964054	+	0.265708i	$0.914394\pi$
$968$	0	0
$969$	618.342i	0.0204995i
$970$	0	0
$971$	22149.0	0.732024	0.366012	−	0.930610i	$-0.380723\pi$
$971$	22149.0	0.732024	0.366012	+	0.930610i	$0.380723\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	− 6789.64i	− 0.223018i
$976$	0	0
$977$	26739.0	0.875595	0.437798	−	0.899074i	$-0.355759\pi$
$977$	26739.0	0.875595	0.437798	+	0.899074i	$0.355759\pi$
$978$	0	0
$979$	4995.00	0.163065
$980$	0	0
$981$	−20566.0	−0.669339
$982$	0	0
$983$	5247.00	0.170248	0.0851238	−	0.996370i	$-0.472871\pi$
$983$	5247.00	0.170248	0.0851238	+	0.996370i	$0.472871\pi$
$984$	0	0
$985$	− 16929.1i	− 0.547619i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−35478.0	−1.14068
$990$	0	0
$991$	35138.1i	1.12634i	0.826342	+	0.563168i	$0.190418\pi$
$991$	35138.1i	1.12634i	−0.826342	+	0.563168i	$0.809582\pi$
$992$	0	0
$993$	− 9185.07i	− 0.293534i
$994$	0	0
$995$	9108.86i	0.290221i
$996$	0	0
$997$	− 22262.0i	− 0.707168i	−0.935403	−	0.353584i	$-0.884963\pi$
$997$	− 22262.0i	− 0.707168i	0.935403	−	0.353584i	$-0.115037\pi$
$998$	0	0
$999$	10547.0	0.334026

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.4.f.b.783.1		2
4.3	odd	2		784.4.f.c.783.1		2
7.2	even	3		112.4.p.c.31.1	yes	2
7.3	odd	6		112.4.p.b.47.1	yes	2
7.6	odd	2		784.4.f.c.783.2		2
28.3	even	6		112.4.p.c.47.1	yes	2
28.23	odd	6		112.4.p.b.31.1	✓	2
28.27	even	2	inner	784.4.f.b.783.2		2
56.3	even	6		448.4.p.b.383.1		2
56.37	even	6		448.4.p.b.255.1		2
56.45	odd	6		448.4.p.c.383.1		2
56.51	odd	6		448.4.p.c.255.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.4.p.b.31.1	✓	2	28.23	odd	6
112.4.p.b.47.1	yes	2	7.3	odd	6
112.4.p.c.31.1	yes	2	7.2	even	3
112.4.p.c.47.1	yes	2	28.3	even	6
448.4.p.b.255.1		2	56.37	even	6
448.4.p.b.383.1		2	56.3	even	6
448.4.p.c.255.1		2	56.51	odd	6
448.4.p.c.383.1		2	56.45	odd	6
784.4.f.b.783.1		2	1.1	even	1	trivial
784.4.f.b.783.2		2	28.27	even	2	inner
784.4.f.c.783.1		2	4.3	odd	2
784.4.f.c.783.2		2	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -5.19615i q^{5} -26.0000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -5.19615i q^{5} -26.0000 q^{9} +25.9808i q^{11} +69.2820i q^{13} +5.19615i q^{15} -36.3731i q^{17} +17.0000 q^{19} -140.296i q^{23} +98.0000 q^{25} +53.0000 q^{27} +90.0000 q^{29} -17.0000 q^{31} -25.9808i q^{33} +199.000 q^{37} -69.2820i q^{39} -187.061i q^{41} -252.879i q^{43} +135.100i q^{45} -567.000 q^{47} +36.3731i q^{51} -333.000 q^{53} +135.000 q^{55} -17.0000 q^{57} -801.000 q^{59} -358.535i q^{61} +360.000 q^{65} -216.506i q^{67} +140.296i q^{69} +488.438i q^{71} -403.568i q^{73} -98.0000 q^{75} -1193.38i q^{79} +649.000 q^{81} +468.000 q^{83} -189.000 q^{85} -90.0000 q^{87} -192.258i q^{89} +17.0000 q^{93} -88.3346i q^{95} +1392.57i q^{97} -675.500i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 52 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 52 * q^9	\( 2 q - 2 q^{3} - 52 q^{9} + 34 q^{19} + 196 q^{25} + 106 q^{27} + 180 q^{29} - 34 q^{31} + 398 q^{37} - 1134 q^{47} - 666 q^{53} + 270 q^{55} - 34 q^{57} - 1602 q^{59} + 720 q^{65} - 196 q^{75} + 1298 q^{81} + 936 q^{83} - 378 q^{85} - 180 q^{87} + 34 q^{93}+O(q^{100}) \) 2 * q - 2 * q^3 - 52 * q^9 + 34 * q^19 + 196 * q^25 + 106 * q^27 + 180 * q^29 - 34 * q^31 + 398 * q^37 - 1134 * q^47 - 666 * q^53 + 270 * q^55 - 34 * q^57 - 1602 * q^59 + 720 * q^65 - 196 * q^75 + 1298 * q^81 + 936 * q^83 - 378 * q^85 - 180 * q^87 + 34 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more