LMFDB - Embedded newform 280.6.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [280,6,Mod(1,280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(280, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("280.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 280 = 2^{3} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	280.a (trivial)

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:	yes
Analytic conductor:	$44.9074695476$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 463x - 1890 $ x^3 - 463*x - 1890
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$23.3245$ of defining polynomial
Character	$\chi$	$=$	280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-21.3245 q^{3} -25.0000 q^{5} +49.0000 q^{7} +211.733 q^{9} +O(q^{10})$	$q-21.3245 q^{3} -25.0000 q^{5} +49.0000 q^{7} +211.733 q^{9} -381.382 q^{11} +372.295 q^{13} +533.112 q^{15} -1099.55 q^{17} +2859.15 q^{19} -1044.90 q^{21} +2943.73 q^{23} +625.000 q^{25} +666.754 q^{27} +2410.16 q^{29} -463.669 q^{31} +8132.76 q^{33} -1225.00 q^{35} +1806.73 q^{37} -7938.99 q^{39} -9019.08 q^{41} -33.7132 q^{43} -5293.32 q^{45} -15132.0 q^{47} +2401.00 q^{49} +23447.4 q^{51} -31344.7 q^{53} +9534.55 q^{55} -60969.9 q^{57} -30247.3 q^{59} -10021.8 q^{61} +10374.9 q^{63} -9307.37 q^{65} +52832.9 q^{67} -62773.5 q^{69} -11610.0 q^{71} -4765.45 q^{73} -13327.8 q^{75} -18687.7 q^{77} -28256.2 q^{79} -65669.3 q^{81} +36453.8 q^{83} +27488.9 q^{85} -51395.4 q^{87} -132386. q^{89} +18242.4 q^{91} +9887.49 q^{93} -71478.8 q^{95} -98024.6 q^{97} -80751.1 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{3} - 75 q^{5} + 147 q^{7} + 209 q^{9}+O(q^{10}) $ 3 * q + 6 * q^3 - 75 * q^5 + 147 * q^7 + 209 * q^9	$ 3 q + 6 q^{3} - 75 q^{5} + 147 q^{7} + 209 q^{9} - 578 q^{11} + 80 q^{13} - 150 q^{15} - 1244 q^{17} + 944 q^{19} + 294 q^{21} + 1096 q^{23} + 1875 q^{25} - 3006 q^{27} + 1868 q^{29} - 6620 q^{31} + 2662 q^{33} - 3675 q^{35} - 4058 q^{37} - 20910 q^{39} - 9602 q^{41} - 12340 q^{43} - 5225 q^{45} - 41026 q^{47} + 7203 q^{49} + 10006 q^{51} - 37610 q^{53} + 14450 q^{55} - 95456 q^{57} - 37664 q^{59} - 8386 q^{61} + 10241 q^{63} - 2000 q^{65} + 69340 q^{67} - 37712 q^{69} - 34016 q^{71} - 45314 q^{73} + 3750 q^{75} - 28322 q^{77} + 1382 q^{79} - 101005 q^{81} - 10128 q^{83} + 31100 q^{85} - 41510 q^{87} - 222810 q^{89} + 3920 q^{91} - 174292 q^{93} - 23600 q^{95} - 159476 q^{97} - 156568 q^{99}+O(q^{100}) $ 3 * q + 6 * q^3 - 75 * q^5 + 147 * q^7 + 209 * q^9 - 578 * q^11 + 80 * q^13 - 150 * q^15 - 1244 * q^17 + 944 * q^19 + 294 * q^21 + 1096 * q^23 + 1875 * q^25 - 3006 * q^27 + 1868 * q^29 - 6620 * q^31 + 2662 * q^33 - 3675 * q^35 - 4058 * q^37 - 20910 * q^39 - 9602 * q^41 - 12340 * q^43 - 5225 * q^45 - 41026 * q^47 + 7203 * q^49 + 10006 * q^51 - 37610 * q^53 + 14450 * q^55 - 95456 * q^57 - 37664 * q^59 - 8386 * q^61 + 10241 * q^63 - 2000 * q^65 + 69340 * q^67 - 37712 * q^69 - 34016 * q^71 - 45314 * q^73 + 3750 * q^75 - 28322 * q^77 + 1382 * q^79 - 101005 * q^81 - 10128 * q^83 + 31100 * q^85 - 41510 * q^87 - 222810 * q^89 + 3920 * q^91 - 174292 * q^93 - 23600 * q^95 - 159476 * q^97 - 156568 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−21.3245	−1.36797	−0.683983	−	0.729498i	$-0.739754\pi$
$3$	−21.3245	−1.36797	−0.683983	+	0.729498i	$0.739754\pi$
$4$	0	0
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	211.733	0.871329
$10$	0	0
$11$	−381.382	−0.950338	−0.475169	−	0.879894i	$-0.657613\pi$
$11$	−381.382	−0.950338	−0.475169	+	0.879894i	$0.657613\pi$
$12$	0	0
$13$	372.295	0.610982	0.305491	−	0.952195i	$-0.401179\pi$
$13$	372.295	0.610982	0.305491	+	0.952195i	$0.401179\pi$
$14$	0	0
$15$	533.112	0.611773
$16$	0	0
$17$	−1099.55	−0.922772	−0.461386	−	0.887200i	$-0.652648\pi$
$17$	−1099.55	−0.922772	−0.461386	+	0.887200i	$0.652648\pi$
$18$	0	0
$19$	2859.15	1.81699	0.908497	−	0.417892i	$-0.137231\pi$
$19$	2859.15	1.81699	0.908497	+	0.417892i	$0.137231\pi$
$20$	0	0
$21$	−1044.90	−0.517042
$22$	0	0
$23$	2943.73	1.16032	0.580161	−	0.814502i	$-0.302990\pi$
$23$	2943.73	1.16032	0.580161	+	0.814502i	$0.302990\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	666.754	0.176018
$28$	0	0
$29$	2410.16	0.532171	0.266086	−	0.963949i	$-0.414270\pi$
$29$	2410.16	0.532171	0.266086	+	0.963949i	$0.414270\pi$
$30$	0	0
$31$	−463.669	−0.0866570	−0.0433285	−	0.999061i	$-0.513796\pi$
$31$	−463.669	−0.0866570	−0.0433285	+	0.999061i	$0.513796\pi$
$32$	0	0
$33$	8132.76	1.30003
$34$	0	0
$35$	−1225.00	−0.169031
$36$	0	0
$37$	1806.73	0.216965	0.108483	−	0.994098i	$-0.465401\pi$
$37$	1806.73	0.216965	0.108483	+	0.994098i	$0.465401\pi$
$38$	0	0
$39$	−7938.99	−0.835802
$40$	0	0
$41$	−9019.08	−0.837920	−0.418960	−	0.908005i	$-0.637605\pi$
$41$	−9019.08	−0.837920	−0.418960	+	0.908005i	$0.637605\pi$
$42$	0	0
$43$	−33.7132	−0.00278054	−0.00139027	−	0.999999i	$-0.500443\pi$
$43$	−33.7132	−0.00278054	−0.00139027	+	0.999999i	$0.500443\pi$
$44$	0	0
$45$	−5293.32	−0.389670
$46$	0	0
$47$	−15132.0	−0.999199	−0.499600	−	0.866256i	$-0.666520\pi$
$47$	−15132.0	−0.999199	−0.499600	+	0.866256i	$0.666520\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	23447.4	1.26232
$52$	0	0
$53$	−31344.7	−1.53276	−0.766381	−	0.642386i	$-0.777945\pi$
$53$	−31344.7	−1.53276	−0.766381	+	0.642386i	$0.777945\pi$
$54$	0	0
$55$	9534.55	0.425004
$56$	0	0
$57$	−60969.9	−2.48558
$58$	0	0
$59$	−30247.3	−1.13125	−0.565623	−	0.824664i	$-0.691364\pi$
$59$	−30247.3	−1.13125	−0.565623	+	0.824664i	$0.691364\pi$
$60$	0	0
$61$	−10021.8	−0.344841	−0.172421	−	0.985023i	$-0.555159\pi$
$61$	−10021.8	−0.344841	−0.172421	+	0.985023i	$0.555159\pi$
$62$	0	0
$63$	10374.9	0.329331
$64$	0	0
$65$	−9307.37	−0.273240
$66$	0	0
$67$	52832.9	1.43786	0.718931	−	0.695081i	$-0.244632\pi$
$67$	52832.9	1.43786	0.718931	+	0.695081i	$0.244632\pi$
$68$	0	0
$69$	−62773.5	−1.58728
$70$	0	0
$71$	−11610.0	−0.273330	−0.136665	−	0.990617i	$-0.543638\pi$
$71$	−11610.0	−0.273330	−0.136665	+	0.990617i	$0.543638\pi$
$72$	0	0
$73$	−4765.45	−0.104664	−0.0523320	−	0.998630i	$-0.516665\pi$
$73$	−4765.45	−0.104664	−0.0523320	+	0.998630i	$0.516665\pi$
$74$	0	0
$75$	−13327.8	−0.273593
$76$	0	0
$77$	−18687.7	−0.359194
$78$	0	0
$79$	−28256.2	−0.509385	−0.254693	−	0.967022i	$-0.581974\pi$
$79$	−28256.2	−0.509385	−0.254693	+	0.967022i	$0.581974\pi$
$80$	0	0
$81$	−65669.3	−1.11211
$82$	0	0
$83$	36453.8	0.580828	0.290414	−	0.956901i	$-0.406207\pi$
$83$	36453.8	0.580828	0.290414	+	0.956901i	$0.406207\pi$
$84$	0	0
$85$	27488.9	0.412676
$86$	0	0
$87$	−51395.4	−0.727992
$88$	0	0
$89$	−132386.	−1.77160	−0.885802	−	0.464063i	$-0.846391\pi$
$89$	−132386.	−1.77160	−0.885802	+	0.464063i	$0.846391\pi$
$90$	0	0
$91$	18242.4	0.230930
$92$	0	0
$93$	9887.49	0.118544
$94$	0	0
$95$	−71478.8	−0.812584
$96$	0	0
$97$	−98024.6	−1.05780	−0.528902	−	0.848683i	$-0.677396\pi$
$97$	−98024.6	−1.05780	−0.528902	+	0.848683i	$0.677396\pi$
$98$	0	0
$99$	−80751.1	−0.828057
$100$	0	0
$101$	34873.5	0.340167	0.170083	−	0.985430i	$-0.445596\pi$
$101$	34873.5	0.340167	0.170083	+	0.985430i	$0.445596\pi$
$102$	0	0
$103$	−93279.1	−0.866346	−0.433173	−	0.901311i	$-0.642606\pi$
$103$	−93279.1	−0.866346	−0.433173	+	0.901311i	$0.642606\pi$
$104$	0	0
$105$	26122.5	0.231228
$106$	0	0
$107$	193063.	1.63020	0.815098	−	0.579324i	$-0.196683\pi$
$107$	193063.	1.63020	0.815098	+	0.579324i	$0.196683\pi$
$108$	0	0
$109$	−34782.7	−0.280412	−0.140206	−	0.990122i	$-0.544776\pi$
$109$	−34782.7	−0.280412	−0.140206	+	0.990122i	$0.544776\pi$
$110$	0	0
$111$	−38527.6	−0.296801
$112$	0	0
$113$	54953.9	0.404858	0.202429	−	0.979297i	$-0.435116\pi$
$113$	54953.9	0.404858	0.202429	+	0.979297i	$0.435116\pi$
$114$	0	0
$115$	−73593.3	−0.518912
$116$	0	0
$117$	78827.1	0.532366
$118$	0	0
$119$	−53878.1	−0.348775
$120$	0	0
$121$	−15598.9	−0.0968568
$122$	0	0
$123$	192327.	1.14625
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	312973.	1.72186	0.860930	−	0.508723i	$-0.169882\pi$
$127$	312973.	1.72186	0.860930	+	0.508723i	$0.169882\pi$
$128$	0	0
$129$	718.916	0.00380368
$130$	0	0
$131$	112963.	0.575118	0.287559	−	0.957763i	$-0.407156\pi$
$131$	112963.	0.575118	0.287559	+	0.957763i	$0.407156\pi$
$132$	0	0
$133$	140099.	0.686759
$134$	0	0
$135$	−16668.8	−0.0787175
$136$	0	0
$137$	−41357.4	−0.188257	−0.0941287	−	0.995560i	$-0.530007\pi$
$137$	−41357.4	−0.188257	−0.0941287	+	0.995560i	$0.530007\pi$
$138$	0	0
$139$	−391228.	−1.71749	−0.858743	−	0.512407i	$-0.828754\pi$
$139$	−391228.	−1.71749	−0.858743	+	0.512407i	$0.828754\pi$
$140$	0	0
$141$	322682.	1.36687
$142$	0	0
$143$	−141986.	−0.580640
$144$	0	0
$145$	−60254.1	−0.237994
$146$	0	0
$147$	−51200.0	−0.195424
$148$	0	0
$149$	−303054.	−1.11829	−0.559144	−	0.829071i	$-0.688870\pi$
$149$	−303054.	−1.11829	−0.559144	+	0.829071i	$0.688870\pi$
$150$	0	0
$151$	−112954.	−0.403144	−0.201572	−	0.979474i	$-0.564605\pi$
$151$	−112954.	−0.403144	−0.201572	+	0.979474i	$0.564605\pi$
$152$	0	0
$153$	−232812.	−0.804038
$154$	0	0
$155$	11591.7	0.0387542
$156$	0	0
$157$	−23023.9	−0.0745470	−0.0372735	−	0.999305i	$-0.511867\pi$
$157$	−23023.9	−0.0745470	−0.0372735	+	0.999305i	$0.511867\pi$
$158$	0	0
$159$	668410.	2.09677
$160$	0	0
$161$	144243.	0.438560
$162$	0	0
$163$	212947.	0.627774	0.313887	−	0.949460i	$-0.398369\pi$
$163$	212947.	0.627774	0.313887	+	0.949460i	$0.398369\pi$
$164$	0	0
$165$	−203319.	−0.581391
$166$	0	0
$167$	−509533.	−1.41378	−0.706889	−	0.707325i	$-0.749902\pi$
$167$	−509533.	−1.41378	−0.706889	+	0.707325i	$0.749902\pi$
$168$	0	0
$169$	−232690.	−0.626701
$170$	0	0
$171$	605377.	1.58320
$172$	0	0
$173$	251973.	0.640088	0.320044	−	0.947403i	$-0.396302\pi$
$173$	251973.	0.640088	0.320044	+	0.947403i	$0.396302\pi$
$174$	0	0
$175$	30625.0	0.0755929
$176$	0	0
$177$	645008.	1.54750
$178$	0	0
$179$	−187246.	−0.436797	−0.218398	−	0.975860i	$-0.570083\pi$
$179$	−187246.	−0.436797	−0.218398	+	0.975860i	$0.570083\pi$
$180$	0	0
$181$	−547053.	−1.24117	−0.620587	−	0.784138i	$-0.713106\pi$
$181$	−547053.	−1.24117	−0.620587	+	0.784138i	$0.713106\pi$
$182$	0	0
$183$	213709.	0.471731
$184$	0	0
$185$	−45168.3	−0.0970297
$186$	0	0
$187$	419350.	0.876945
$188$	0	0
$189$	32670.9	0.0665284
$190$	0	0
$191$	82048.2	0.162737	0.0813684	−	0.996684i	$-0.474071\pi$
$191$	82048.2	0.162737	0.0813684	+	0.996684i	$0.474071\pi$
$192$	0	0
$193$	834554.	1.61273	0.806364	−	0.591419i	$-0.201432\pi$
$193$	834554.	1.61273	0.806364	+	0.591419i	$0.201432\pi$
$194$	0	0
$195$	198475.	0.373782
$196$	0	0
$197$	−801077.	−1.47065	−0.735324	−	0.677716i	$-0.762970\pi$
$197$	−801077.	−1.47065	−0.735324	+	0.677716i	$0.762970\pi$
$198$	0	0
$199$	−728144.	−1.30342	−0.651710	−	0.758468i	$-0.725948\pi$
$199$	−728144.	−1.30342	−0.651710	+	0.758468i	$0.725948\pi$
$200$	0	0
$201$	−1.12663e6	−1.96695
$202$	0	0
$203$	118098.	0.201142
$204$	0	0
$205$	225477.	0.374729
$206$	0	0
$207$	623285.	1.01102
$208$	0	0
$209$	−1.09043e6	−1.72676
$210$	0	0
$211$	737969.	1.14112	0.570561	−	0.821255i	$-0.306726\pi$
$211$	737969.	1.14112	0.570561	+	0.821255i	$0.306726\pi$
$212$	0	0
$213$	247578.	0.373906
$214$	0	0
$215$	842.830	0.00124349
$216$	0	0
$217$	−22719.8	−0.0327533
$218$	0	0
$219$	101621.	0.143177
$220$	0	0
$221$	−409358.	−0.563797
$222$	0	0
$223$	−504580.	−0.679466	−0.339733	−	0.940522i	$-0.610337\pi$
$223$	−504580.	−0.679466	−0.339733	+	0.940522i	$0.610337\pi$
$224$	0	0
$225$	132333.	0.174266
$226$	0	0
$227$	793643.	1.02226	0.511129	−	0.859504i	$-0.329228\pi$
$227$	793643.	1.02226	0.511129	+	0.859504i	$0.329228\pi$
$228$	0	0
$229$	533635.	0.672444	0.336222	−	0.941783i	$-0.390851\pi$
$229$	533635.	0.672444	0.336222	+	0.941783i	$0.390851\pi$
$230$	0	0
$231$	398505.	0.491365
$232$	0	0
$233$	1.56974e6	1.89425	0.947125	−	0.320863i	$-0.103973\pi$
$233$	1.56974e6	1.89425	0.947125	+	0.320863i	$0.103973\pi$
$234$	0	0
$235$	378300.	0.446856
$236$	0	0
$237$	602549.	0.696821
$238$	0	0
$239$	318353.	0.360507	0.180253	−	0.983620i	$-0.442308\pi$
$239$	318353.	0.360507	0.180253	+	0.983620i	$0.442308\pi$
$240$	0	0
$241$	−719047.	−0.797471	−0.398736	−	0.917066i	$-0.630551\pi$
$241$	−719047.	−0.797471	−0.398736	+	0.917066i	$0.630551\pi$
$242$	0	0
$243$	1.23834e6	1.34532
$244$	0	0
$245$	−60025.0	−0.0638877
$246$	0	0
$247$	1.06445e6	1.11015
$248$	0	0
$249$	−777358.	−0.794553
$250$	0	0
$251$	1.63688e6	1.63995	0.819977	−	0.572397i	$-0.193986\pi$
$251$	1.63688e6	1.63995	0.819977	+	0.572397i	$0.193986\pi$
$252$	0	0
$253$	−1.12269e6	−1.10270
$254$	0	0
$255$	−586185.	−0.564526
$256$	0	0
$257$	713835.	0.674163	0.337082	−	0.941475i	$-0.390560\pi$
$257$	713835.	0.674163	0.337082	+	0.941475i	$0.390560\pi$
$258$	0	0
$259$	88530.0	0.0820051
$260$	0	0
$261$	510311.	0.463696
$262$	0	0
$263$	−1.93481e6	−1.72484	−0.862419	−	0.506195i	$-0.831052\pi$
$263$	−1.93481e6	−1.72484	−0.862419	+	0.506195i	$0.831052\pi$
$264$	0	0
$265$	783618.	0.685472
$266$	0	0
$267$	2.82306e6	2.42349
$268$	0	0
$269$	104742.	0.0882554	0.0441277	−	0.999026i	$-0.485949\pi$
$269$	104742.	0.0882554	0.0441277	+	0.999026i	$0.485949\pi$
$270$	0	0
$271$	1.09108e6	0.902471	0.451235	−	0.892405i	$-0.350983\pi$
$271$	1.09108e6	0.902471	0.451235	+	0.892405i	$0.350983\pi$
$272$	0	0
$273$	−389010.	−0.315904
$274$	0	0
$275$	−238364.	−0.190068
$276$	0	0
$277$	328777.	0.257455	0.128728	−	0.991680i	$-0.458911\pi$
$277$	328777.	0.257455	0.128728	+	0.991680i	$0.458911\pi$
$278$	0	0
$279$	−98174.0	−0.0755067
$280$	0	0
$281$	−37359.3	−0.0282249	−0.0141125	−	0.999900i	$-0.504492\pi$
$281$	−37359.3	−0.0282249	−0.0141125	+	0.999900i	$0.504492\pi$
$282$	0	0
$283$	−1.80297e6	−1.33820	−0.669101	−	0.743172i	$-0.733321\pi$
$283$	−1.80297e6	−1.33820	−0.669101	+	0.743172i	$0.733321\pi$
$284$	0	0
$285$	1.52425e6	1.11159
$286$	0	0
$287$	−441935.	−0.316704
$288$	0	0
$289$	−210838.	−0.148492
$290$	0	0
$291$	2.09032e6	1.44704
$292$	0	0
$293$	−1.56997e6	−1.06837	−0.534185	−	0.845368i	$-0.679381\pi$
$293$	−1.56997e6	−1.06837	−0.534185	+	0.845368i	$0.679381\pi$
$294$	0	0
$295$	756183.	0.505908
$296$	0	0
$297$	−254288.	−0.167276
$298$	0	0
$299$	1.09594e6	0.708936
$300$	0	0
$301$	−1651.95	−0.00105094
$302$	0	0
$303$	−743658.	−0.465336
$304$	0	0
$305$	250544.	0.154218
$306$	0	0
$307$	−2.27153e6	−1.37554	−0.687769	−	0.725930i	$-0.741410\pi$
$307$	−2.27153e6	−1.37554	−0.687769	+	0.725930i	$0.741410\pi$
$308$	0	0
$309$	1.98913e6	1.18513
$310$	0	0
$311$	−2.71344e6	−1.59082	−0.795408	−	0.606074i	$-0.792743\pi$
$311$	−2.71344e6	−1.59082	−0.795408	+	0.606074i	$0.792743\pi$
$312$	0	0
$313$	−3.03548e6	−1.75132	−0.875661	−	0.482926i	$-0.839574\pi$
$313$	−3.03548e6	−1.75132	−0.875661	+	0.482926i	$0.839574\pi$
$314$	0	0
$315$	−259373.	−0.147281
$316$	0	0
$317$	655683.	0.366476	0.183238	−	0.983069i	$-0.441342\pi$
$317$	655683.	0.366476	0.183238	+	0.983069i	$0.441342\pi$
$318$	0	0
$319$	−919192.	−0.505743
$320$	0	0
$321$	−4.11697e6	−2.23005
$322$	0	0
$323$	−3.14379e6	−1.67667
$324$	0	0
$325$	232684.	0.122196
$326$	0	0
$327$	741722.	0.383594
$328$	0	0
$329$	−741469.	−0.377662
$330$	0	0
$331$	−3.77092e6	−1.89181	−0.945904	−	0.324447i	$-0.894822\pi$
$331$	−3.77092e6	−1.89181	−0.945904	+	0.324447i	$0.894822\pi$
$332$	0	0
$333$	382545.	0.189048
$334$	0	0
$335$	−1.32082e6	−0.643032
$336$	0	0
$337$	−1.44388e6	−0.692557	−0.346278	−	0.938132i	$-0.612555\pi$
$337$	−1.44388e6	−0.692557	−0.346278	+	0.938132i	$0.612555\pi$
$338$	0	0
$339$	−1.17186e6	−0.553831
$340$	0	0
$341$	176835.	0.0823535
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	1.56934e6	0.709853
$346$	0	0
$347$	−2.00100e6	−0.892120	−0.446060	−	0.895003i	$-0.647173\pi$
$347$	−2.00100e6	−0.892120	−0.446060	+	0.895003i	$0.647173\pi$
$348$	0	0
$349$	688692.	0.302664	0.151332	−	0.988483i	$-0.451644\pi$
$349$	688692.	0.302664	0.151332	+	0.988483i	$0.451644\pi$
$350$	0	0
$351$	248229.	0.107544
$352$	0	0
$353$	805367.	0.343999	0.172000	−	0.985097i	$-0.444977\pi$
$353$	805367.	0.343999	0.172000	+	0.985097i	$0.444977\pi$
$354$	0	0
$355$	290251.	0.122237
$356$	0	0
$357$	1.14892e6	0.477112
$358$	0	0
$359$	−1.80310e6	−0.738386	−0.369193	−	0.929353i	$-0.620366\pi$
$359$	−1.80310e6	−0.738386	−0.369193	+	0.929353i	$0.620366\pi$
$360$	0	0
$361$	5.69866e6	2.30147
$362$	0	0
$363$	332638.	0.132497
$364$	0	0
$365$	119136.	0.0468071
$366$	0	0
$367$	2.17144e6	0.841554	0.420777	−	0.907164i	$-0.361757\pi$
$367$	2.17144e6	0.841554	0.420777	+	0.907164i	$0.361757\pi$
$368$	0	0
$369$	−1.90964e6	−0.730104
$370$	0	0
$371$	−1.53589e6	−0.579330
$372$	0	0
$373$	2.31643e6	0.862080	0.431040	−	0.902333i	$-0.358147\pi$
$373$	2.31643e6	0.862080	0.431040	+	0.902333i	$0.358147\pi$
$374$	0	0
$375$	333195.	0.122355
$376$	0	0
$377$	897291.	0.325147
$378$	0	0
$379$	−4.28068e6	−1.53079	−0.765393	−	0.643563i	$-0.777455\pi$
$379$	−4.28068e6	−1.53079	−0.765393	+	0.643563i	$0.777455\pi$
$380$	0	0
$381$	−6.67399e6	−2.35544
$382$	0	0
$383$	−5.33887e6	−1.85974	−0.929871	−	0.367886i	$-0.880082\pi$
$383$	−5.33887e6	−1.85974	−0.929871	+	0.367886i	$0.880082\pi$
$384$	0	0
$385$	467193.	0.160637
$386$	0	0
$387$	−7138.19	−0.00242276
$388$	0	0
$389$	1.66287e6	0.557167	0.278584	−	0.960412i	$-0.410135\pi$
$389$	1.66287e6	0.557167	0.278584	+	0.960412i	$0.410135\pi$
$390$	0	0
$391$	−3.23679e6	−1.07071
$392$	0	0
$393$	−2.40887e6	−0.786741
$394$	0	0
$395$	706405.	0.227804
$396$	0	0
$397$	274923.	0.0875458	0.0437729	−	0.999042i	$-0.486062\pi$
$397$	274923.	0.0875458	0.0437729	+	0.999042i	$0.486062\pi$
$398$	0	0
$399$	−2.98753e6	−0.939462
$400$	0	0
$401$	−1.99036e6	−0.618118	−0.309059	−	0.951043i	$-0.600014\pi$
$401$	−1.99036e6	−0.618118	−0.309059	+	0.951043i	$0.600014\pi$
$402$	0	0
$403$	−172621.	−0.0529459
$404$	0	0
$405$	1.64173e6	0.497353
$406$	0	0
$407$	−689055.	−0.206190
$408$	0	0
$409$	2.80249e6	0.828391	0.414196	−	0.910188i	$-0.364063\pi$
$409$	2.80249e6	0.828391	0.414196	+	0.910188i	$0.364063\pi$
$410$	0	0
$411$	881925.	0.257530
$412$	0	0
$413$	−1.48212e6	−0.427570
$414$	0	0
$415$	−911345.	−0.259754
$416$	0	0
$417$	8.34273e6	2.34946
$418$	0	0
$419$	−574142.	−0.159766	−0.0798830	−	0.996804i	$-0.525455\pi$
$419$	−574142.	−0.159766	−0.0798830	+	0.996804i	$0.525455\pi$
$420$	0	0
$421$	−3.80048e6	−1.04504	−0.522520	−	0.852627i	$-0.675008\pi$
$421$	−3.80048e6	−1.04504	−0.522520	+	0.852627i	$0.675008\pi$
$422$	0	0
$423$	−3.20395e6	−0.870631
$424$	0	0
$425$	−687221.	−0.184554
$426$	0	0
$427$	−491066.	−0.130338
$428$	0	0
$429$	3.02779e6	0.794295
$430$	0	0
$431$	2.21830e6	0.575211	0.287606	−	0.957749i	$-0.407141\pi$
$431$	2.21830e6	0.575211	0.287606	+	0.957749i	$0.407141\pi$
$432$	0	0
$433$	−1.16997e6	−0.299886	−0.149943	−	0.988695i	$-0.547909\pi$
$433$	−1.16997e6	−0.299886	−0.149943	+	0.988695i	$0.547909\pi$
$434$	0	0
$435$	1.28489e6	0.325568
$436$	0	0
$437$	8.41658e6	2.10830
$438$	0	0
$439$	4.00014e6	0.990636	0.495318	−	0.868712i	$-0.335051\pi$
$439$	4.00014e6	0.990636	0.495318	+	0.868712i	$0.335051\pi$
$440$	0	0
$441$	508371.	0.124476
$442$	0	0
$443$	−2.94996e6	−0.714179	−0.357089	−	0.934070i	$-0.616231\pi$
$443$	−2.94996e6	−0.714179	−0.357089	+	0.934070i	$0.616231\pi$
$444$	0	0
$445$	3.30965e6	0.792286
$446$	0	0
$447$	6.46246e6	1.52978
$448$	0	0
$449$	−3.44945e6	−0.807483	−0.403742	−	0.914873i	$-0.632291\pi$
$449$	−3.44945e6	−0.807483	−0.403742	+	0.914873i	$0.632291\pi$
$450$	0	0
$451$	3.43971e6	0.796307
$452$	0	0
$453$	2.40869e6	0.551486
$454$	0	0
$455$	−456061.	−0.103275
$456$	0	0
$457$	−6.94632e6	−1.55584	−0.777919	−	0.628365i	$-0.783724\pi$
$457$	−6.94632e6	−1.55584	−0.777919	+	0.628365i	$0.783724\pi$
$458$	0	0
$459$	−733132.	−0.162424
$460$	0	0
$461$	3.63913e6	0.797527	0.398764	−	0.917054i	$-0.369439\pi$
$461$	3.63913e6	0.797527	0.398764	+	0.917054i	$0.369439\pi$
$462$	0	0
$463$	5.29061e6	1.14697	0.573486	−	0.819215i	$-0.305591\pi$
$463$	5.29061e6	1.14697	0.573486	+	0.819215i	$0.305591\pi$
$464$	0	0
$465$	−247187.	−0.0530144
$466$	0	0
$467$	7.01082e6	1.48757	0.743783	−	0.668421i	$-0.233029\pi$
$467$	7.01082e6	1.48757	0.743783	+	0.668421i	$0.233029\pi$
$468$	0	0
$469$	2.58881e6	0.543461
$470$	0	0
$471$	490973.	0.101978
$472$	0	0
$473$	12857.6	0.00264245
$474$	0	0
$475$	1.78697e6	0.363399
$476$	0	0
$477$	−6.63671e6	−1.33554
$478$	0	0
$479$	−9.33895e6	−1.85977	−0.929885	−	0.367850i	$-0.880094\pi$
$479$	−9.33895e6	−1.85977	−0.929885	+	0.367850i	$0.880094\pi$
$480$	0	0
$481$	672638.	0.132562
$482$	0	0
$483$	−3.07590e6	−0.599935
$484$	0	0
$485$	2.45061e6	0.473065
$486$	0	0
$487$	3.02941e6	0.578809	0.289404	−	0.957207i	$-0.406543\pi$
$487$	3.02941e6	0.578809	0.289404	+	0.957207i	$0.406543\pi$
$488$	0	0
$489$	−4.54099e6	−0.858773
$490$	0	0
$491$	−5.58970e6	−1.04637	−0.523185	−	0.852219i	$-0.675256\pi$
$491$	−5.58970e6	−1.04637	−0.523185	+	0.852219i	$0.675256\pi$
$492$	0	0
$493$	−2.65010e6	−0.491072
$494$	0	0
$495$	2.01878e6	0.370319
$496$	0	0
$497$	−568892.	−0.103309
$498$	0	0
$499$	−3.82995e6	−0.688561	−0.344280	−	0.938867i	$-0.611877\pi$
$499$	−3.82995e6	−0.688561	−0.344280	+	0.938867i	$0.611877\pi$
$500$	0	0
$501$	1.08655e7	1.93400
$502$	0	0
$503$	6.93999e6	1.22304	0.611518	−	0.791231i	$-0.290559\pi$
$503$	6.93999e6	1.22304	0.611518	+	0.791231i	$0.290559\pi$
$504$	0	0
$505$	−871837.	−0.152127
$506$	0	0
$507$	4.96198e6	0.857305
$508$	0	0
$509$	−4.13814e6	−0.707963	−0.353981	−	0.935252i	$-0.615172\pi$
$509$	−4.13814e6	−0.707963	−0.353981	+	0.935252i	$0.615172\pi$
$510$	0	0
$511$	−233507.	−0.0395592
$512$	0	0
$513$	1.90635e6	0.319823
$514$	0	0
$515$	2.33198e6	0.387442
$516$	0	0
$517$	5.77108e6	0.949578
$518$	0	0
$519$	−5.37320e6	−0.875618
$520$	0	0
$521$	−1.43012e6	−0.230823	−0.115411	−	0.993318i	$-0.536819\pi$
$521$	−1.43012e6	−0.230823	−0.115411	+	0.993318i	$0.536819\pi$
$522$	0	0
$523$	1.67112e6	0.267149	0.133574	−	0.991039i	$-0.457354\pi$
$523$	1.67112e6	0.267149	0.133574	+	0.991039i	$0.457354\pi$
$524$	0	0
$525$	−653062.	−0.103408
$526$	0	0
$527$	509829.	0.0799646
$528$	0	0
$529$	2.22921e6	0.346347
$530$	0	0
$531$	−6.40435e6	−0.985687
$532$	0	0
$533$	−3.35776e6	−0.511954
$534$	0	0
$535$	−4.82658e6	−0.729046
$536$	0	0
$537$	3.99292e6	0.597523
$538$	0	0
$539$	−915698.	−0.135763
$540$	0	0
$541$	2.14416e6	0.314966	0.157483	−	0.987522i	$-0.449662\pi$
$541$	2.14416e6	0.314966	0.157483	+	0.987522i	$0.449662\pi$
$542$	0	0
$543$	1.16656e7	1.69788
$544$	0	0
$545$	869566.	0.125404
$546$	0	0
$547$	3.69979e6	0.528700	0.264350	−	0.964427i	$-0.414843\pi$
$547$	3.69979e6	0.528700	0.264350	+	0.964427i	$0.414843\pi$
$548$	0	0
$549$	−2.12194e6	−0.300470
$550$	0	0
$551$	6.89102e6	0.966952
$552$	0	0
$553$	−1.38455e6	−0.192529
$554$	0	0
$555$	963191.	0.132733
$556$	0	0
$557$	5.12724e6	0.700238	0.350119	−	0.936705i	$-0.386141\pi$
$557$	5.12724e6	0.700238	0.350119	+	0.936705i	$0.386141\pi$
$558$	0	0
$559$	−12551.2	−0.00169886
$560$	0	0
$561$	−8.94241e6	−1.19963
$562$	0	0
$563$	−7.94546e6	−1.05645	−0.528224	−	0.849105i	$-0.677142\pi$
$563$	−7.94546e6	−1.05645	−0.528224	+	0.849105i	$0.677142\pi$
$564$	0	0
$565$	−1.37385e6	−0.181058
$566$	0	0
$567$	−3.21779e6	−0.420340
$568$	0	0
$569$	−1.76137e6	−0.228071	−0.114035	−	0.993477i	$-0.536378\pi$
$569$	−1.76137e6	−0.228071	−0.114035	+	0.993477i	$0.536378\pi$
$570$	0	0
$571$	−286582.	−0.0367840	−0.0183920	−	0.999831i	$-0.505855\pi$
$571$	−286582.	−0.0367840	−0.0183920	+	0.999831i	$0.505855\pi$
$572$	0	0
$573$	−1.74963e6	−0.222618
$574$	0	0
$575$	1.83983e6	0.232064
$576$	0	0
$577$	−4.12433e6	−0.515720	−0.257860	−	0.966182i	$-0.583017\pi$
$577$	−4.12433e6	−0.515720	−0.257860	+	0.966182i	$0.583017\pi$
$578$	0	0
$579$	−1.77964e7	−2.20616
$580$	0	0
$581$	1.78624e6	0.219532
$582$	0	0
$583$	1.19543e7	1.45664
$584$	0	0
$585$	−1.97068e6	−0.238082
$586$	0	0
$587$	−1.43423e7	−1.71800	−0.859000	−	0.511975i	$-0.828914\pi$
$587$	−1.43423e7	−1.71800	−0.859000	+	0.511975i	$0.828914\pi$
$588$	0	0
$589$	−1.32570e6	−0.157455
$590$	0	0
$591$	1.70825e7	2.01180
$592$	0	0
$593$	−8.67127e6	−1.01262	−0.506309	−	0.862352i	$-0.668991\pi$
$593$	−8.67127e6	−1.01262	−0.506309	+	0.862352i	$0.668991\pi$
$594$	0	0
$595$	1.34695e6	0.155977
$596$	0	0
$597$	1.55273e7	1.78303
$598$	0	0
$599$	1.14786e7	1.30714	0.653571	−	0.756866i	$-0.273270\pi$
$599$	1.14786e7	1.30714	0.653571	+	0.756866i	$0.273270\pi$
$600$	0	0
$601$	1.21619e7	1.37346	0.686728	−	0.726914i	$-0.259046\pi$
$601$	1.21619e7	1.37346	0.686728	+	0.726914i	$0.259046\pi$
$602$	0	0
$603$	1.11865e7	1.25285
$604$	0	0
$605$	389972.	0.0433157
$606$	0	0
$607$	9.05671e6	0.997697	0.498849	−	0.866689i	$-0.333756\pi$
$607$	9.05671e6	0.997697	0.498849	+	0.866689i	$0.333756\pi$
$608$	0	0
$609$	−2.51838e6	−0.275155
$610$	0	0
$611$	−5.63357e6	−0.610493
$612$	0	0
$613$	5.57609e6	0.599347	0.299673	−	0.954042i	$-0.403122\pi$
$613$	5.57609e6	0.599347	0.299673	+	0.954042i	$0.403122\pi$
$614$	0	0
$615$	−4.80818e6	−0.512616
$616$	0	0
$617$	1.88756e6	0.199612	0.0998062	−	0.995007i	$-0.468178\pi$
$617$	1.88756e6	0.199612	0.0998062	+	0.995007i	$0.468178\pi$
$618$	0	0
$619$	−353063.	−0.0370361	−0.0185181	−	0.999829i	$-0.505895\pi$
$619$	−353063.	−0.0370361	−0.0185181	+	0.999829i	$0.505895\pi$
$620$	0	0
$621$	1.96274e6	0.204237
$622$	0	0
$623$	−6.48691e6	−0.669604
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	2.32528e7	2.36215
$628$	0	0
$629$	−1.98660e6	−0.200209
$630$	0	0
$631$	9.14879e6	0.914724	0.457362	−	0.889281i	$-0.348794\pi$
$631$	9.14879e6	0.914724	0.457362	+	0.889281i	$0.348794\pi$
$632$	0	0
$633$	−1.57368e7	−1.56102
$634$	0	0
$635$	−7.82433e6	−0.770039
$636$	0	0
$637$	893880.	0.0872832
$638$	0	0
$639$	−2.45823e6	−0.238161
$640$	0	0
$641$	−8.33065e6	−0.800818	−0.400409	−	0.916336i	$-0.631132\pi$
$641$	−8.33065e6	−0.800818	−0.400409	+	0.916336i	$0.631132\pi$
$642$	0	0
$643$	8.59733e6	0.820042	0.410021	−	0.912076i	$-0.365521\pi$
$643$	8.59733e6	0.820042	0.410021	+	0.912076i	$0.365521\pi$
$644$	0	0
$645$	−17972.9	−0.00170106
$646$	0	0
$647$	−1.65094e7	−1.55050	−0.775248	−	0.631657i	$-0.782375\pi$
$647$	−1.65094e7	−1.55050	−0.775248	+	0.631657i	$0.782375\pi$
$648$	0	0
$649$	1.15358e7	1.07507
$650$	0	0
$651$	484487.	0.0448053
$652$	0	0
$653$	7.65233e6	0.702280	0.351140	−	0.936323i	$-0.385794\pi$
$653$	7.65233e6	0.702280	0.351140	+	0.936323i	$0.385794\pi$
$654$	0	0
$655$	−2.82407e6	−0.257200
$656$	0	0
$657$	−1.00900e6	−0.0911967
$658$	0	0
$659$	−1.28219e7	−1.15011	−0.575056	−	0.818114i	$-0.695020\pi$
$659$	−1.28219e7	−1.15011	−0.575056	+	0.818114i	$0.695020\pi$
$660$	0	0
$661$	1.20104e7	1.06918	0.534592	−	0.845110i	$-0.320465\pi$
$661$	1.20104e7	1.06918	0.534592	+	0.845110i	$0.320465\pi$
$662$	0	0
$663$	8.72935e6	0.771255
$664$	0	0
$665$	−3.50246e6	−0.307128
$666$	0	0
$667$	7.09487e6	0.617490
$668$	0	0
$669$	1.07599e7	0.929486
$670$	0	0
$671$	3.82211e6	0.327716
$672$	0	0
$673$	−6.31342e6	−0.537312	−0.268656	−	0.963236i	$-0.586580\pi$
$673$	−6.31342e6	−0.537312	−0.268656	+	0.963236i	$0.586580\pi$
$674$	0	0
$675$	416721.	0.0352035
$676$	0	0
$677$	1.43615e6	0.120428	0.0602142	−	0.998185i	$-0.480822\pi$
$677$	1.43615e6	0.120428	0.0602142	+	0.998185i	$0.480822\pi$
$678$	0	0
$679$	−4.80320e6	−0.399813
$680$	0	0
$681$	−1.69240e7	−1.39841
$682$	0	0
$683$	1.90641e7	1.56374	0.781871	−	0.623441i	$-0.214266\pi$
$683$	1.90641e7	1.56374	0.781871	+	0.623441i	$0.214266\pi$
$684$	0	0
$685$	1.03394e6	0.0841913
$686$	0	0
$687$	−1.13795e7	−0.919880
$688$	0	0
$689$	−1.16695e7	−0.936491
$690$	0	0
$691$	1.58516e7	1.26292	0.631462	−	0.775407i	$-0.282455\pi$
$691$	1.58516e7	1.26292	0.631462	+	0.775407i	$0.282455\pi$
$692$	0	0
$693$	−3.95680e6	−0.312976
$694$	0	0
$695$	9.78070e6	0.768083
$696$	0	0
$697$	9.91696e6	0.773209
$698$	0	0
$699$	−3.34738e7	−2.59127
$700$	0	0
$701$	870113.	0.0668776	0.0334388	−	0.999441i	$-0.489354\pi$
$701$	870113.	0.0668776	0.0334388	+	0.999441i	$0.489354\pi$
$702$	0	0
$703$	5.16573e6	0.394224
$704$	0	0
$705$	−8.06705e6	−0.611283
$706$	0	0
$707$	1.70880e6	0.128571
$708$	0	0
$709$	1.14645e7	0.856524	0.428262	−	0.903655i	$-0.359126\pi$
$709$	1.14645e7	0.856524	0.428262	+	0.903655i	$0.359126\pi$
$710$	0	0
$711$	−5.98277e6	−0.443842
$712$	0	0
$713$	−1.36492e6	−0.100550
$714$	0	0
$715$	3.54966e6	0.259670
$716$	0	0
$717$	−6.78870e6	−0.493161
$718$	0	0
$719$	−1.00675e7	−0.726273	−0.363137	−	0.931736i	$-0.618294\pi$
$719$	−1.00675e7	−0.726273	−0.363137	+	0.931736i	$0.618294\pi$
$720$	0	0
$721$	−4.57068e6	−0.327448
$722$	0	0
$723$	1.53333e7	1.09091
$724$	0	0
$725$	1.50635e6	0.106434
$726$	0	0
$727$	−1.42656e7	−1.00104	−0.500522	−	0.865724i	$-0.666858\pi$
$727$	−1.42656e7	−1.00104	−0.500522	+	0.865724i	$0.666858\pi$
$728$	0	0
$729$	−1.04493e7	−0.728232
$730$	0	0
$731$	37069.5	0.00256580
$732$	0	0
$733$	7.44090e6	0.511524	0.255762	−	0.966740i	$-0.417674\pi$
$733$	7.44090e6	0.511524	0.255762	+	0.966740i	$0.417674\pi$
$734$	0	0
$735$	1.28000e6	0.0873961
$736$	0	0
$737$	−2.01495e7	−1.36646
$738$	0	0
$739$	−2.73511e6	−0.184231	−0.0921156	−	0.995748i	$-0.529363\pi$
$739$	−2.73511e6	−0.184231	−0.0921156	+	0.995748i	$0.529363\pi$
$740$	0	0
$741$	−2.26988e7	−1.51865
$742$	0	0
$743$	2.09318e7	1.39102	0.695511	−	0.718515i	$-0.255178\pi$
$743$	2.09318e7	1.39102	0.695511	+	0.718515i	$0.255178\pi$
$744$	0	0
$745$	7.57634e6	0.500114
$746$	0	0
$747$	7.71847e6	0.506093
$748$	0	0
$749$	9.46009e6	0.616156
$750$	0	0
$751$	−2.73024e7	−1.76645	−0.883225	−	0.468950i	$-0.844632\pi$
$751$	−2.73024e7	−1.76645	−0.883225	+	0.468950i	$0.844632\pi$
$752$	0	0
$753$	−3.49055e7	−2.24340
$754$	0	0
$755$	2.82385e6	0.180291
$756$	0	0
$757$	1.28294e7	0.813703	0.406851	−	0.913494i	$-0.366627\pi$
$757$	1.28294e7	0.813703	0.406851	+	0.913494i	$0.366627\pi$
$758$	0	0
$759$	2.39407e7	1.50845
$760$	0	0
$761$	2.04401e7	1.27945	0.639723	−	0.768606i	$-0.279049\pi$
$761$	2.04401e7	1.27945	0.639723	+	0.768606i	$0.279049\pi$
$762$	0	0
$763$	−1.70435e6	−0.105986
$764$	0	0
$765$	5.82029e6	0.359577
$766$	0	0
$767$	−1.12609e7	−0.691171
$768$	0	0
$769$	−1.17270e7	−0.715109	−0.357555	−	0.933892i	$-0.616389\pi$
$769$	−1.17270e7	−0.715109	−0.357555	+	0.933892i	$0.616389\pi$
$770$	0	0
$771$	−1.52221e7	−0.922232
$772$	0	0
$773$	−2.63342e7	−1.58515	−0.792576	−	0.609773i	$-0.791260\pi$
$773$	−2.63342e7	−1.58515	−0.792576	+	0.609773i	$0.791260\pi$
$774$	0	0
$775$	−289793.	−0.0173314
$776$	0	0
$777$	−1.88785e6	−0.112180
$778$	0	0
$779$	−2.57869e7	−1.52250
$780$	0	0
$781$	4.42786e6	0.259756
$782$	0	0
$783$	1.60699e6	0.0936715
$784$	0	0
$785$	575598.	0.0333384
$786$	0	0
$787$	−32107.6	−0.00184787	−0.000923933	−	1.00000i	$-0.500294\pi$
$787$	−32107.6	−0.00184787	−0.000923933		1.00000i	$0.500294\pi$
$788$	0	0
$789$	4.12587e7	2.35952
$790$	0	0
$791$	2.69274e6	0.153022
$792$	0	0
$793$	−3.73105e6	−0.210692
$794$	0	0
$795$	−1.67102e7	−0.937702
$796$	0	0
$797$	−1.60617e7	−0.895665	−0.447832	−	0.894118i	$-0.647804\pi$
$797$	−1.60617e7	−0.895665	−0.447832	+	0.894118i	$0.647804\pi$
$798$	0	0
$799$	1.66385e7	0.922033
$800$	0	0
$801$	−2.80305e7	−1.54365
$802$	0	0
$803$	1.81746e6	0.0994662
$804$	0	0
$805$	−3.60607e6	−0.196130
$806$	0	0
$807$	−2.23357e6	−0.120730
$808$	0	0
$809$	9.71020e6	0.521623	0.260812	−	0.965390i	$-0.416010\pi$
$809$	9.71020e6	0.521623	0.260812	+	0.965390i	$0.416010\pi$
$810$	0	0
$811$	−2.95547e7	−1.57788	−0.788942	−	0.614468i	$-0.789371\pi$
$811$	−2.95547e7	−1.57788	−0.788942	+	0.614468i	$0.789371\pi$
$812$	0	0
$813$	−2.32667e7	−1.23455
$814$	0	0
$815$	−5.32369e6	−0.280749
$816$	0	0
$817$	−96391.2	−0.00505222
$818$	0	0
$819$	3.86253e6	0.201216
$820$	0	0
$821$	3.69239e6	0.191183	0.0955915	−	0.995421i	$-0.469526\pi$
$821$	3.69239e6	0.191183	0.0955915	+	0.995421i	$0.469526\pi$
$822$	0	0
$823$	−1.37549e7	−0.707878	−0.353939	−	0.935268i	$-0.615158\pi$
$823$	−1.37549e7	−0.707878	−0.353939	+	0.935268i	$0.615158\pi$
$824$	0	0
$825$	5.08298e6	0.260006
$826$	0	0
$827$	4.94189e6	0.251263	0.125632	−	0.992077i	$-0.459904\pi$
$827$	4.94189e6	0.251263	0.125632	+	0.992077i	$0.459904\pi$
$828$	0	0
$829$	−3.72295e7	−1.88149	−0.940743	−	0.339119i	$-0.889871\pi$
$829$	−3.72295e7	−1.88149	−0.940743	+	0.339119i	$0.889871\pi$
$830$	0	0
$831$	−7.01099e6	−0.352190
$832$	0	0
$833$	−2.64003e6	−0.131825
$834$	0	0
$835$	1.27383e7	0.632260
$836$	0	0
$837$	−309153.	−0.0152532
$838$	0	0
$839$	−1.90655e7	−0.935066	−0.467533	−	0.883976i	$-0.654857\pi$
$839$	−1.90655e7	−0.935066	−0.467533	+	0.883976i	$0.654857\pi$
$840$	0	0
$841$	−1.47023e7	−0.716794
$842$	0	0
$843$	796667.	0.0386107
$844$	0	0
$845$	5.81724e6	0.280269
$846$	0	0
$847$	−764345.	−0.0366084
$848$	0	0
$849$	3.84473e7	1.83061
$850$	0	0
$851$	5.31854e6	0.251749
$852$	0	0
$853$	−2.09785e7	−0.987193	−0.493596	−	0.869691i	$-0.664318\pi$
$853$	−2.09785e7	−0.987193	−0.493596	+	0.869691i	$0.664318\pi$
$854$	0	0
$855$	−1.51344e7	−0.708028
$856$	0	0
$857$	−2.97938e7	−1.38571	−0.692857	−	0.721075i	$-0.743648\pi$
$857$	−2.97938e7	−1.38571	−0.692857	+	0.721075i	$0.743648\pi$
$858$	0	0
$859$	1.86748e6	0.0863520	0.0431760	−	0.999067i	$-0.486252\pi$
$859$	1.86748e6	0.0863520	0.0431760	+	0.999067i	$0.486252\pi$
$860$	0	0
$861$	9.42402e6	0.433240
$862$	0	0
$863$	2.28328e7	1.04359	0.521797	−	0.853070i	$-0.325262\pi$
$863$	2.28328e7	1.04359	0.521797	+	0.853070i	$0.325262\pi$
$864$	0	0
$865$	−6.29934e6	−0.286256
$866$	0	0
$867$	4.49601e6	0.203132
$868$	0	0
$869$	1.07764e7	0.484088
$870$	0	0
$871$	1.96694e7	0.878508
$872$	0	0
$873$	−2.07550e7	−0.921696
$874$	0	0
$875$	−765625.	−0.0338062
$876$	0	0
$877$	−4.10654e7	−1.80292	−0.901462	−	0.432858i	$-0.857505\pi$
$877$	−4.10654e7	−1.80292	−0.901462	+	0.432858i	$0.857505\pi$
$878$	0	0
$879$	3.34787e7	1.46149
$880$	0	0
$881$	−3.94268e7	−1.71140	−0.855700	−	0.517472i	$-0.826873\pi$
$881$	−3.94268e7	−1.71140	−0.855700	+	0.517472i	$0.826873\pi$
$882$	0	0
$883$	1.49358e7	0.644654	0.322327	−	0.946628i	$-0.395535\pi$
$883$	1.49358e7	0.644654	0.322327	+	0.946628i	$0.395535\pi$
$884$	0	0
$885$	−1.61252e7	−0.692065
$886$	0	0
$887$	−2.44515e7	−1.04351	−0.521754	−	0.853096i	$-0.674722\pi$
$887$	−2.44515e7	−1.04351	−0.521754	+	0.853096i	$0.674722\pi$
$888$	0	0
$889$	1.53357e7	0.650802
$890$	0	0
$891$	2.50451e7	1.05689
$892$	0	0
$893$	−4.32647e7	−1.81554
$894$	0	0
$895$	4.68115e6	0.195342
$896$	0	0
$897$	−2.33702e7	−0.969800
$898$	0	0
$899$	−1.11752e6	−0.0461164
$900$	0	0
$901$	3.44652e7	1.41439
$902$	0	0
$903$	35226.9	0.00143766
$904$	0	0
$905$	1.36763e7	0.555070
$906$	0	0
$907$	2.77487e7	1.12002	0.560009	−	0.828487i	$-0.310798\pi$
$907$	2.77487e7	1.12002	0.560009	+	0.828487i	$0.310798\pi$
$908$	0	0
$909$	7.38386e6	0.296397
$910$	0	0
$911$	−1.04000e6	−0.0415181	−0.0207591	−	0.999785i	$-0.506608\pi$
$911$	−1.04000e6	−0.0415181	−0.0207591	+	0.999785i	$0.506608\pi$
$912$	0	0
$913$	−1.39028e7	−0.551984
$914$	0	0
$915$	−5.34271e6	−0.210964
$916$	0	0
$917$	5.53517e6	0.217374
$918$	0	0
$919$	2.58322e7	1.00896	0.504479	−	0.863424i	$-0.331685\pi$
$919$	2.58322e7	1.00896	0.504479	+	0.863424i	$0.331685\pi$
$920$	0	0
$921$	4.84392e7	1.88169
$922$	0	0
$923$	−4.32236e6	−0.167000
$924$	0	0
$925$	1.12921e6	0.0433930
$926$	0	0
$927$	−1.97503e7	−0.754872
$928$	0	0
$929$	1.92846e7	0.733114	0.366557	−	0.930396i	$-0.380536\pi$
$929$	1.92846e7	0.733114	0.366557	+	0.930396i	$0.380536\pi$
$930$	0	0
$931$	6.86483e6	0.259571
$932$	0	0
$933$	5.78628e7	2.17618
$934$	0	0
$935$	−1.04837e7	−0.392182
$936$	0	0
$937$	8.79237e6	0.327158	0.163579	−	0.986530i	$-0.447696\pi$
$937$	8.79237e6	0.327158	0.163579	+	0.986530i	$0.447696\pi$
$938$	0	0
$939$	6.47299e7	2.39575
$940$	0	0
$941$	5.04584e7	1.85763	0.928816	−	0.370542i	$-0.120828\pi$
$941$	5.04584e7	1.85763	0.928816	+	0.370542i	$0.120828\pi$
$942$	0	0
$943$	−2.65497e7	−0.972257
$944$	0	0
$945$	−816773.	−0.0297524
$946$	0	0
$947$	−6.48456e6	−0.234966	−0.117483	−	0.993075i	$-0.537483\pi$
$947$	−6.48456e6	−0.234966	−0.117483	+	0.993075i	$0.537483\pi$
$948$	0	0
$949$	−1.77415e6	−0.0639478
$950$	0	0
$951$	−1.39821e7	−0.501327
$952$	0	0
$953$	−1.67568e7	−0.597666	−0.298833	−	0.954305i	$-0.596597\pi$
$953$	−1.67568e7	−0.597666	−0.298833	+	0.954305i	$0.596597\pi$
$954$	0	0
$955$	−2.05121e6	−0.0727781
$956$	0	0
$957$	1.96013e7	0.691838
$958$	0	0
$959$	−2.02651e6	−0.0711546
$960$	0	0
$961$	−2.84142e7	−0.992491
$962$	0	0
$963$	4.08778e7	1.42044
$964$	0	0
$965$	−2.08639e7	−0.721234
$966$	0	0
$967$	1.12777e7	0.387843	0.193921	−	0.981017i	$-0.437879\pi$
$967$	1.12777e7	0.387843	0.193921	+	0.981017i	$0.437879\pi$
$968$	0	0
$969$	6.70397e7	2.29363
$970$	0	0
$971$	−2.36109e7	−0.803644	−0.401822	−	0.915718i	$-0.631623\pi$
$971$	−2.36109e7	−0.803644	−0.401822	+	0.915718i	$0.631623\pi$
$972$	0	0
$973$	−1.91702e7	−0.649149
$974$	0	0
$975$	−4.96187e6	−0.167160
$976$	0	0
$977$	−4.36268e7	−1.46223	−0.731117	−	0.682252i	$-0.761001\pi$
$977$	−4.36268e7	−1.46223	−0.731117	+	0.682252i	$0.761001\pi$
$978$	0	0
$979$	5.04896e7	1.68362
$980$	0	0
$981$	−7.36463e6	−0.244331
$982$	0	0
$983$	−1.23492e7	−0.407618	−0.203809	−	0.979011i	$-0.565332\pi$
$983$	−1.23492e7	−0.407618	−0.203809	+	0.979011i	$0.565332\pi$
$984$	0	0
$985$	2.00269e7	0.657694
$986$	0	0
$987$	1.58114e7	0.516628
$988$	0	0
$989$	−99242.6	−0.00322632
$990$	0	0
$991$	5.32915e7	1.72375	0.861874	−	0.507122i	$-0.169291\pi$
$991$	5.32915e7	1.72375	0.861874	+	0.507122i	$0.169291\pi$
$992$	0	0
$993$	8.04128e7	2.58793
$994$	0	0
$995$	1.82036e7	0.582907
$996$	0	0
$997$	6.43068e6	0.204889	0.102445	−	0.994739i	$-0.467334\pi$
$997$	6.43068e6	0.204889	0.102445	+	0.994739i	$0.467334\pi$
$998$	0	0
$999$	1.20465e6	0.0381897

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.6.a.e.1.1	✓	3
4.3	odd	2		560.6.a.s.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.6.a.e.1.1	✓	3	1.1	even	1	trivial
560.6.a.s.1.3		3	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	280.a (trivial)

\(f(q)\)	\(=\)	\(q-21.3245 q^{3} -25.0000 q^{5} +49.0000 q^{7} +211.733 q^{9} +O(q^{10})\)	\(q-21.3245 q^{3} -25.0000 q^{5} +49.0000 q^{7} +211.733 q^{9} -381.382 q^{11} +372.295 q^{13} +533.112 q^{15} -1099.55 q^{17} +2859.15 q^{19} -1044.90 q^{21} +2943.73 q^{23} +625.000 q^{25} +666.754 q^{27} +2410.16 q^{29} -463.669 q^{31} +8132.76 q^{33} -1225.00 q^{35} +1806.73 q^{37} -7938.99 q^{39} -9019.08 q^{41} -33.7132 q^{43} -5293.32 q^{45} -15132.0 q^{47} +2401.00 q^{49} +23447.4 q^{51} -31344.7 q^{53} +9534.55 q^{55} -60969.9 q^{57} -30247.3 q^{59} -10021.8 q^{61} +10374.9 q^{63} -9307.37 q^{65} +52832.9 q^{67} -62773.5 q^{69} -11610.0 q^{71} -4765.45 q^{73} -13327.8 q^{75} -18687.7 q^{77} -28256.2 q^{79} -65669.3 q^{81} +36453.8 q^{83} +27488.9 q^{85} -51395.4 q^{87} -132386. q^{89} +18242.4 q^{91} +9887.49 q^{93} -71478.8 q^{95} -98024.6 q^{97} -80751.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{3} - 75 q^{5} + 147 q^{7} + 209 q^{9}+O(q^{10}) \) 3 * q + 6 * q^3 - 75 * q^5 + 147 * q^7 + 209 * q^9	\( 3 q + 6 q^{3} - 75 q^{5} + 147 q^{7} + 209 q^{9} - 578 q^{11} + 80 q^{13} - 150 q^{15} - 1244 q^{17} + 944 q^{19} + 294 q^{21} + 1096 q^{23} + 1875 q^{25} - 3006 q^{27} + 1868 q^{29} - 6620 q^{31} + 2662 q^{33} - 3675 q^{35} - 4058 q^{37} - 20910 q^{39} - 9602 q^{41} - 12340 q^{43} - 5225 q^{45} - 41026 q^{47} + 7203 q^{49} + 10006 q^{51} - 37610 q^{53} + 14450 q^{55} - 95456 q^{57} - 37664 q^{59} - 8386 q^{61} + 10241 q^{63} - 2000 q^{65} + 69340 q^{67} - 37712 q^{69} - 34016 q^{71} - 45314 q^{73} + 3750 q^{75} - 28322 q^{77} + 1382 q^{79} - 101005 q^{81} - 10128 q^{83} + 31100 q^{85} - 41510 q^{87} - 222810 q^{89} + 3920 q^{91} - 174292 q^{93} - 23600 q^{95} - 159476 q^{97} - 156568 q^{99}+O(q^{100}) \) 3 * q + 6 * q^3 - 75 * q^5 + 147 * q^7 + 209 * q^9 - 578 * q^11 + 80 * q^13 - 150 * q^15 - 1244 * q^17 + 944 * q^19 + 294 * q^21 + 1096 * q^23 + 1875 * q^25 - 3006 * q^27 + 1868 * q^29 - 6620 * q^31 + 2662 * q^33 - 3675 * q^35 - 4058 * q^37 - 20910 * q^39 - 9602 * q^41 - 12340 * q^43 - 5225 * q^45 - 41026 * q^47 + 7203 * q^49 + 10006 * q^51 - 37610 * q^53 + 14450 * q^55 - 95456 * q^57 - 37664 * q^59 - 8386 * q^61 + 10241 * q^63 - 2000 * q^65 + 69340 * q^67 - 37712 * q^69 - 34016 * q^71 - 45314 * q^73 + 3750 * q^75 - 28322 * q^77 + 1382 * q^79 - 101005 * q^81 - 10128 * q^83 + 31100 * q^85 - 41510 * q^87 - 222810 * q^89 + 3920 * q^91 - 174292 * q^93 - 23600 * q^95 - 159476 * q^97 - 156568 * q^99

Introduction

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Modular forms

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