LMFDB - Embedded newform 280.6.a.i.1.3

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:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 791x^{3} + 280x^{2} + 24832x + 39040 $ x^5 - 2x^4 - 791x^3 + 280x^2 + 24832x + 39040
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.78912$ 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)

$f(q)$	$=$	$q-2.78912 q^{3} -25.0000 q^{5} -49.0000 q^{7} -235.221 q^{9} +O(q^{10})$	$q-2.78912 q^{3} -25.0000 q^{5} -49.0000 q^{7} -235.221 q^{9} -635.141 q^{11} +647.764 q^{13} +69.7280 q^{15} +95.5465 q^{17} -1933.02 q^{19} +136.667 q^{21} +4479.59 q^{23} +625.000 q^{25} +1333.81 q^{27} -5318.93 q^{29} +3175.17 q^{31} +1771.48 q^{33} +1225.00 q^{35} +4933.07 q^{37} -1806.69 q^{39} -7380.78 q^{41} -17712.9 q^{43} +5880.52 q^{45} +15906.3 q^{47} +2401.00 q^{49} -266.491 q^{51} +33119.4 q^{53} +15878.5 q^{55} +5391.42 q^{57} +30741.6 q^{59} +52779.6 q^{61} +11525.8 q^{63} -16194.1 q^{65} -43991.6 q^{67} -12494.1 q^{69} -38984.3 q^{71} -7067.09 q^{73} -1743.20 q^{75} +31121.9 q^{77} +7257.36 q^{79} +53438.5 q^{81} +79420.8 q^{83} -2388.66 q^{85} +14835.1 q^{87} +103190. q^{89} -31740.4 q^{91} -8855.92 q^{93} +48325.5 q^{95} -43853.7 q^{97} +149398. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{3} - 125 q^{5} - 245 q^{7} + 372 q^{9}+O(q^{10}) $ 5 * q - 3 * q^3 - 125 * q^5 - 245 * q^7 + 372 * q^9	$ 5 q - 3 q^{3} - 125 q^{5} - 245 q^{7} + 372 q^{9} - 263 q^{11} - 729 q^{13} + 75 q^{15} - 1003 q^{17} - 2506 q^{19} + 147 q^{21} - 1066 q^{23} + 3125 q^{25} + 615 q^{27} + 3489 q^{29} + 7880 q^{31} + 4863 q^{33} + 6125 q^{35} + 13118 q^{37} + 27189 q^{39} + 23972 q^{41} + 3978 q^{43} - 9300 q^{45} + 9057 q^{47} + 12005 q^{49} + 96639 q^{51} + 2128 q^{53} + 6575 q^{55} + 61674 q^{57} - 15512 q^{59} + 4560 q^{61} - 18228 q^{63} + 18225 q^{65} + 7780 q^{67} + 126474 q^{69} + 32752 q^{71} + 189498 q^{73} - 1875 q^{75} + 12887 q^{77} + 42055 q^{79} + 294645 q^{81} + 58420 q^{83} + 25075 q^{85} - 765 q^{87} + 231324 q^{89} + 35721 q^{91} + 395736 q^{93} + 62650 q^{95} + 247569 q^{97} + 30606 q^{99}+O(q^{100}) $ 5 * q - 3 * q^3 - 125 * q^5 - 245 * q^7 + 372 * q^9 - 263 * q^11 - 729 * q^13 + 75 * q^15 - 1003 * q^17 - 2506 * q^19 + 147 * q^21 - 1066 * q^23 + 3125 * q^25 + 615 * q^27 + 3489 * q^29 + 7880 * q^31 + 4863 * q^33 + 6125 * q^35 + 13118 * q^37 + 27189 * q^39 + 23972 * q^41 + 3978 * q^43 - 9300 * q^45 + 9057 * q^47 + 12005 * q^49 + 96639 * q^51 + 2128 * q^53 + 6575 * q^55 + 61674 * q^57 - 15512 * q^59 + 4560 * q^61 - 18228 * q^63 + 18225 * q^65 + 7780 * q^67 + 126474 * q^69 + 32752 * q^71 + 189498 * q^73 - 1875 * q^75 + 12887 * q^77 + 42055 * q^79 + 294645 * q^81 + 58420 * q^83 + 25075 * q^85 - 765 * q^87 + 231324 * q^89 + 35721 * q^91 + 395736 * q^93 + 62650 * q^95 + 247569 * q^97 + 30606 * 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
$3$	−2.78912	−0.178922	−0.0894610	−	0.995990i	$-0.528514\pi$
$3$	−2.78912	−0.178922	−0.0894610	+	0.995990i	$0.528514\pi$
$4$	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
$9$	−235.221	−0.967987
$10$	0	0
$11$	−635.141	−1.58266	−0.791331	−	0.611388i	$-0.790612\pi$
$11$	−635.141	−1.58266	−0.791331	+	0.611388i	$0.790612\pi$
$12$	0	0
$13$	647.764	1.06306	0.531531	−	0.847039i	$-0.321617\pi$
$13$	647.764	1.06306	0.531531	+	0.847039i	$0.321617\pi$
$14$	0	0
$15$	69.7280	0.0800164
$16$	0	0
$17$	95.5465	0.0801849	0.0400925	−	0.999196i	$-0.487235\pi$
$17$	95.5465	0.0801849	0.0400925	+	0.999196i	$0.487235\pi$
$18$	0	0
$19$	−1933.02	−1.22844	−0.614218	−	0.789137i	$-0.710528\pi$
$19$	−1933.02	−1.22844	−0.614218	+	0.789137i	$0.710528\pi$
$20$	0	0
$21$	136.667	0.0676262
$22$	0	0
$23$	4479.59	1.76571	0.882853	−	0.469650i	$-0.155620\pi$
$23$	4479.59	1.76571	0.882853	+	0.469650i	$0.155620\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	1333.81	0.352116
$28$	0	0
$29$	−5318.93	−1.17444	−0.587218	−	0.809429i	$-0.699777\pi$
$29$	−5318.93	−1.17444	−0.587218	+	0.809429i	$0.699777\pi$
$30$	0	0
$31$	3175.17	0.593420	0.296710	−	0.954968i	$-0.404110\pi$
$31$	3175.17	0.593420	0.296710	+	0.954968i	$0.404110\pi$
$32$	0	0
$33$	1771.48	0.283173
$34$	0	0
$35$	1225.00	0.169031
$36$	0	0
$37$	4933.07	0.592397	0.296199	−	0.955126i	$-0.404281\pi$
$37$	4933.07	0.592397	0.296199	+	0.955126i	$0.404281\pi$
$38$	0	0
$39$	−1806.69	−0.190205
$40$	0	0
$41$	−7380.78	−0.685713	−0.342857	−	0.939388i	$-0.611395\pi$
$41$	−7380.78	−0.685713	−0.342857	+	0.939388i	$0.611395\pi$
$42$	0	0
$43$	−17712.9	−1.46090	−0.730449	−	0.682968i	$-0.760689\pi$
$43$	−17712.9	−1.46090	−0.730449	+	0.682968i	$0.760689\pi$
$44$	0	0
$45$	5880.52	0.432897
$46$	0	0
$47$	15906.3	1.05033	0.525165	−	0.851001i	$-0.324004\pi$
$47$	15906.3	1.05033	0.525165	+	0.851001i	$0.324004\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−266.491	−0.0143468
$52$	0	0
$53$	33119.4	1.61954	0.809771	−	0.586745i	$-0.199591\pi$
$53$	33119.4	1.61954	0.809771	+	0.586745i	$0.199591\pi$
$54$	0	0
$55$	15878.5	0.707788
$56$	0	0
$57$	5391.42	0.219794
$58$	0	0
$59$	30741.6	1.14973	0.574865	−	0.818248i	$-0.305055\pi$
$59$	30741.6	1.14973	0.574865	+	0.818248i	$0.305055\pi$
$60$	0	0
$61$	52779.6	1.81611	0.908054	−	0.418854i	$-0.137568\pi$
$61$	52779.6	1.81611	0.908054	+	0.418854i	$0.137568\pi$
$62$	0	0
$63$	11525.8	0.365865
$64$	0	0
$65$	−16194.1	−0.475416
$66$	0	0
$67$	−43991.6	−1.19724	−0.598622	−	0.801032i	$-0.704285\pi$
$67$	−43991.6	−1.19724	−0.598622	+	0.801032i	$0.704285\pi$
$68$	0	0
$69$	−12494.1	−0.315924
$70$	0	0
$71$	−38984.3	−0.917792	−0.458896	−	0.888490i	$-0.651755\pi$
$71$	−38984.3	−0.917792	−0.458896	+	0.888490i	$0.651755\pi$
$72$	0	0
$73$	−7067.09	−0.155215	−0.0776075	−	0.996984i	$-0.524728\pi$
$73$	−7067.09	−0.155215	−0.0776075	+	0.996984i	$0.524728\pi$
$74$	0	0
$75$	−1743.20	−0.0357844
$76$	0	0
$77$	31121.9	0.598190
$78$	0	0
$79$	7257.36	0.130831	0.0654155	−	0.997858i	$-0.479163\pi$
$79$	7257.36	0.130831	0.0654155	+	0.997858i	$0.479163\pi$
$80$	0	0
$81$	53438.5	0.904986
$82$	0	0
$83$	79420.8	1.26543	0.632716	−	0.774384i	$-0.281940\pi$
$83$	79420.8	1.26543	0.632716	+	0.774384i	$0.281940\pi$
$84$	0	0
$85$	−2388.66	−0.0358598
$86$	0	0
$87$	14835.1	0.210132
$88$	0	0
$89$	103190.	1.38090	0.690452	−	0.723378i	$-0.257412\pi$
$89$	103190.	1.38090	0.690452	+	0.723378i	$0.257412\pi$
$90$	0	0
$91$	−31740.4	−0.401800
$92$	0	0
$93$	−8855.92	−0.106176
$94$	0	0
$95$	48325.5	0.549373
$96$	0	0
$97$	−43853.7	−0.473235	−0.236617	−	0.971603i	$-0.576039\pi$
$97$	−43853.7	−0.473235	−0.236617	+	0.971603i	$0.576039\pi$
$98$	0	0
$99$	149398.	1.53200
$100$	0	0
$101$	116989.	1.14114	0.570572	−	0.821247i	$-0.306721\pi$
$101$	116989.	1.14114	0.570572	+	0.821247i	$0.306721\pi$
$102$	0	0
$103$	−104323.	−0.968918	−0.484459	−	0.874814i	$-0.660983\pi$
$103$	−104323.	−0.968918	−0.484459	+	0.874814i	$0.660983\pi$
$104$	0	0
$105$	−3416.67	−0.0302433
$106$	0	0
$107$	39521.9	0.333717	0.166858	−	0.985981i	$-0.446638\pi$
$107$	39521.9	0.333717	0.166858	+	0.985981i	$0.446638\pi$
$108$	0	0
$109$	63707.6	0.513600	0.256800	−	0.966465i	$-0.417332\pi$
$109$	63707.6	0.513600	0.256800	+	0.966465i	$0.417332\pi$
$110$	0	0
$111$	−13758.9	−0.105993
$112$	0	0
$113$	115550.	0.851282	0.425641	−	0.904892i	$-0.360049\pi$
$113$	115550.	0.851282	0.425641	+	0.904892i	$0.360049\pi$
$114$	0	0
$115$	−111990.	−0.789648
$116$	0	0
$117$	−152368.	−1.02903
$118$	0	0
$119$	−4681.78	−0.0303070
$120$	0	0
$121$	242353.	1.50482
$122$	0	0
$123$	20585.9	0.122689
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	96072.1	0.528552	0.264276	−	0.964447i	$-0.414867\pi$
$127$	96072.1	0.528552	0.264276	+	0.964447i	$0.414867\pi$
$128$	0	0
$129$	49403.5	0.261387
$130$	0	0
$131$	−309706.	−1.57678	−0.788390	−	0.615176i	$-0.789085\pi$
$131$	−309706.	−1.57678	−0.788390	+	0.615176i	$0.789085\pi$
$132$	0	0
$133$	94718.0	0.464305
$134$	0	0
$135$	−33345.4	−0.157471
$136$	0	0
$137$	137707.	0.626837	0.313418	−	0.949615i	$-0.398526\pi$
$137$	137707.	0.626837	0.313418	+	0.949615i	$0.398526\pi$
$138$	0	0
$139$	−76812.1	−0.337204	−0.168602	−	0.985684i	$-0.553925\pi$
$139$	−76812.1	−0.337204	−0.168602	+	0.985684i	$0.553925\pi$
$140$	0	0
$141$	−44364.7	−0.187927
$142$	0	0
$143$	−411421.	−1.68247
$144$	0	0
$145$	132973.	0.525223
$146$	0	0
$147$	−6696.67	−0.0255603
$148$	0	0
$149$	−76766.2	−0.283273	−0.141636	−	0.989919i	$-0.545236\pi$
$149$	−76766.2	−0.283273	−0.141636	+	0.989919i	$0.545236\pi$
$150$	0	0
$151$	260127.	0.928417	0.464209	−	0.885726i	$-0.346339\pi$
$151$	260127.	0.928417	0.464209	+	0.885726i	$0.346339\pi$
$152$	0	0
$153$	−22474.5	−0.0776179
$154$	0	0
$155$	−79379.2	−0.265386
$156$	0	0
$157$	227312.	0.735992	0.367996	−	0.929827i	$-0.380044\pi$
$157$	227312.	0.735992	0.367996	+	0.929827i	$0.380044\pi$
$158$	0	0
$159$	−92373.9	−0.289772
$160$	0	0
$161$	−219500.	−0.667374
$162$	0	0
$163$	176734.	0.521017	0.260508	−	0.965472i	$-0.416110\pi$
$163$	176734.	0.521017	0.260508	+	0.965472i	$0.416110\pi$
$164$	0	0
$165$	−44287.1	−0.126639
$166$	0	0
$167$	−118090.	−0.327660	−0.163830	−	0.986489i	$-0.552385\pi$
$167$	−118090.	−0.327660	−0.163830	+	0.986489i	$0.552385\pi$
$168$	0	0
$169$	48305.2	0.130100
$170$	0	0
$171$	454686.	1.18911
$172$	0	0
$173$	−60854.7	−0.154589	−0.0772946	−	0.997008i	$-0.524628\pi$
$173$	−60854.7	−0.154589	−0.0772946	+	0.997008i	$0.524628\pi$
$174$	0	0
$175$	−30625.0	−0.0755929
$176$	0	0
$177$	−85741.9	−0.205712
$178$	0	0
$179$	−544298.	−1.26971	−0.634855	−	0.772632i	$-0.718940\pi$
$179$	−544298.	−1.26971	−0.634855	+	0.772632i	$0.718940\pi$
$180$	0	0
$181$	−828218.	−1.87909	−0.939547	−	0.342421i	$-0.888753\pi$
$181$	−828218.	−1.87909	−0.939547	+	0.342421i	$0.888753\pi$
$182$	0	0
$183$	−147209.	−0.324942
$184$	0	0
$185$	−123327.	−0.264928
$186$	0	0
$187$	−60685.5	−0.126906
$188$	0	0
$189$	−65356.9	−0.133087
$190$	0	0
$191$	−252674.	−0.501160	−0.250580	−	0.968096i	$-0.580621\pi$
$191$	−252674.	−0.501160	−0.250580	+	0.968096i	$0.580621\pi$
$192$	0	0
$193$	206385.	0.398827	0.199414	−	0.979915i	$-0.436096\pi$
$193$	206385.	0.398827	0.199414	+	0.979915i	$0.436096\pi$
$194$	0	0
$195$	45167.3	0.0850623
$196$	0	0
$197$	−205916.	−0.378029	−0.189014	−	0.981974i	$-0.560529\pi$
$197$	−205916.	−0.378029	−0.189014	+	0.981974i	$0.560529\pi$
$198$	0	0
$199$	−434899.	−0.778494	−0.389247	−	0.921133i	$-0.627265\pi$
$199$	−434899.	−0.778494	−0.389247	+	0.921133i	$0.627265\pi$
$200$	0	0
$201$	122698.	0.214213
$202$	0	0
$203$	260627.	0.443895
$204$	0	0
$205$	184520.	0.306660
$206$	0	0
$207$	−1.05369e6	−1.70918
$208$	0	0
$209$	1.22774e6	1.94420
$210$	0	0
$211$	−866187.	−1.33939	−0.669693	−	0.742638i	$-0.733574\pi$
$211$	−866187.	−1.33939	−0.669693	+	0.742638i	$0.733574\pi$
$212$	0	0
$213$	108732.	0.164213
$214$	0	0
$215$	442824.	0.653333
$216$	0	0
$217$	−155583.	−0.224292
$218$	0	0
$219$	19711.0	0.0277714
$220$	0	0
$221$	61891.6	0.0852415
$222$	0	0
$223$	−573118.	−0.771759	−0.385880	−	0.922549i	$-0.626102\pi$
$223$	−573118.	−0.771759	−0.385880	+	0.922549i	$0.626102\pi$
$224$	0	0
$225$	−147013.	−0.193597
$226$	0	0
$227$	−1.30853e6	−1.68547	−0.842733	−	0.538332i	$-0.819055\pi$
$227$	−1.30853e6	−1.68547	−0.842733	+	0.538332i	$0.819055\pi$
$228$	0	0
$229$	1.46883e6	1.85090	0.925451	−	0.378867i	$-0.123686\pi$
$229$	1.46883e6	1.85090	0.925451	+	0.378867i	$0.123686\pi$
$230$	0	0
$231$	−86802.7	−0.107029
$232$	0	0
$233$	1.16716e6	1.40845	0.704223	−	0.709978i	$-0.251295\pi$
$233$	1.16716e6	1.40845	0.704223	+	0.709978i	$0.251295\pi$
$234$	0	0
$235$	−397658.	−0.469722
$236$	0	0
$237$	−20241.6	−0.0234086
$238$	0	0
$239$	1.09408e6	1.23895	0.619473	−	0.785018i	$-0.287346\pi$
$239$	1.09408e6	1.23895	0.619473	+	0.785018i	$0.287346\pi$
$240$	0	0
$241$	−473011.	−0.524600	−0.262300	−	0.964986i	$-0.584481\pi$
$241$	−473011.	−0.524600	−0.262300	+	0.964986i	$0.584481\pi$
$242$	0	0
$243$	−473163.	−0.514038
$244$	0	0
$245$	−60025.0	−0.0638877
$246$	0	0
$247$	−1.25214e6	−1.30590
$248$	0	0
$249$	−221514.	−0.226414
$250$	0	0
$251$	−1.04562e6	−1.04759	−0.523793	−	0.851846i	$-0.675483\pi$
$251$	−1.04562e6	−1.04759	−0.523793	+	0.851846i	$0.675483\pi$
$252$	0	0
$253$	−2.84517e6	−2.79452
$254$	0	0
$255$	6662.27	0.00641611
$256$	0	0
$257$	−1.55132e6	−1.46510	−0.732552	−	0.680711i	$-0.761671\pi$
$257$	−1.55132e6	−1.46510	−0.732552	+	0.680711i	$0.761671\pi$
$258$	0	0
$259$	−241720.	−0.223905
$260$	0	0
$261$	1.25112e6	1.13684
$262$	0	0
$263$	2.02912e6	1.80891	0.904457	−	0.426564i	$-0.140276\pi$
$263$	2.02912e6	1.80891	0.904457	+	0.426564i	$0.140276\pi$
$264$	0	0
$265$	−827984.	−0.724282
$266$	0	0
$267$	−287810.	−0.247074
$268$	0	0
$269$	−1.16292e6	−0.979876	−0.489938	−	0.871757i	$-0.662981\pi$
$269$	−1.16292e6	−0.979876	−0.489938	+	0.871757i	$0.662981\pi$
$270$	0	0
$271$	851689.	0.704462	0.352231	−	0.935913i	$-0.385423\pi$
$271$	851689.	0.704462	0.352231	+	0.935913i	$0.385423\pi$
$272$	0	0
$273$	88527.8	0.0718908
$274$	0	0
$275$	−396963.	−0.316532
$276$	0	0
$277$	1.58364e6	1.24010	0.620051	−	0.784561i	$-0.287112\pi$
$277$	1.58364e6	1.24010	0.620051	+	0.784561i	$0.287112\pi$
$278$	0	0
$279$	−746866.	−0.574423
$280$	0	0
$281$	2.19916e6	1.66146	0.830731	−	0.556673i	$-0.187923\pi$
$281$	2.19916e6	1.66146	0.830731	+	0.556673i	$0.187923\pi$
$282$	0	0
$283$	483943.	0.359193	0.179597	−	0.983740i	$-0.442521\pi$
$283$	483943.	0.359193	0.179597	+	0.983740i	$0.442521\pi$
$284$	0	0
$285$	−134786.	−0.0982949
$286$	0	0
$287$	361658.	0.259175
$288$	0	0
$289$	−1.41073e6	−0.993570
$290$	0	0
$291$	122313.	0.0846721
$292$	0	0
$293$	1.59170e6	1.08316	0.541580	−	0.840649i	$-0.317826\pi$
$293$	1.59170e6	1.08316	0.541580	+	0.840649i	$0.317826\pi$
$294$	0	0
$295$	−768539.	−0.514175
$296$	0	0
$297$	−847160.	−0.557281
$298$	0	0
$299$	2.90172e6	1.87705
$300$	0	0
$301$	867934.	0.552167
$302$	0	0
$303$	−326296.	−0.204176
$304$	0	0
$305$	−1.31949e6	−0.812188
$306$	0	0
$307$	−1.04407e6	−0.632244	−0.316122	−	0.948719i	$-0.602381\pi$
$307$	−1.04407e6	−0.632244	−0.316122	+	0.948719i	$0.602381\pi$
$308$	0	0
$309$	290969.	0.173361
$310$	0	0
$311$	2.15519e6	1.26353	0.631763	−	0.775162i	$-0.282332\pi$
$311$	2.15519e6	1.26353	0.631763	+	0.775162i	$0.282332\pi$
$312$	0	0
$313$	1.21322e6	0.699968	0.349984	−	0.936756i	$-0.386187\pi$
$313$	1.21322e6	0.699968	0.349984	+	0.936756i	$0.386187\pi$
$314$	0	0
$315$	−288146.	−0.163620
$316$	0	0
$317$	−2.72187e6	−1.52131	−0.760657	−	0.649153i	$-0.775123\pi$
$317$	−2.72187e6	−1.52131	−0.760657	+	0.649153i	$0.775123\pi$
$318$	0	0
$319$	3.37827e6	1.85873
$320$	0	0
$321$	−110231.	−0.0597093
$322$	0	0
$323$	−184693.	−0.0985020
$324$	0	0
$325$	404853.	0.212612
$326$	0	0
$327$	−177688.	−0.0918944
$328$	0	0
$329$	−779411.	−0.396987
$330$	0	0
$331$	−740074.	−0.371283	−0.185641	−	0.982618i	$-0.559436\pi$
$331$	−740074.	−0.371283	−0.185641	+	0.982618i	$0.559436\pi$
$332$	0	0
$333$	−1.16036e6	−0.573433
$334$	0	0
$335$	1.09979e6	0.535424
$336$	0	0
$337$	2.32874e6	1.11698	0.558491	−	0.829511i	$-0.311381\pi$
$337$	2.32874e6	1.11698	0.558491	+	0.829511i	$0.311381\pi$
$338$	0	0
$339$	−322282.	−0.152313
$340$	0	0
$341$	−2.01668e6	−0.939184
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	312352.	0.141285
$346$	0	0
$347$	3.98267e6	1.77562	0.887812	−	0.460207i	$-0.152225\pi$
$347$	3.98267e6	1.77562	0.887812	+	0.460207i	$0.152225\pi$
$348$	0	0
$349$	−740202.	−0.325302	−0.162651	−	0.986684i	$-0.552004\pi$
$349$	−740202.	−0.325302	−0.162651	+	0.986684i	$0.552004\pi$
$350$	0	0
$351$	863997.	0.374321
$352$	0	0
$353$	3.54099e6	1.51247	0.756237	−	0.654297i	$-0.227036\pi$
$353$	3.54099e6	1.51247	0.756237	+	0.654297i	$0.227036\pi$
$354$	0	0
$355$	974609.	0.410449
$356$	0	0
$357$	13058.0	0.00542260
$358$	0	0
$359$	1.36395e6	0.558552	0.279276	−	0.960211i	$-0.409906\pi$
$359$	1.36395e6	0.558552	0.279276	+	0.960211i	$0.409906\pi$
$360$	0	0
$361$	1.26047e6	0.509053
$362$	0	0
$363$	−675950.	−0.269245
$364$	0	0
$365$	176677.	0.0694142
$366$	0	0
$367$	−3.96959e6	−1.53844	−0.769221	−	0.638983i	$-0.779355\pi$
$367$	−3.96959e6	−1.53844	−0.769221	+	0.638983i	$0.779355\pi$
$368$	0	0
$369$	1.73611e6	0.663762
$370$	0	0
$371$	−1.62285e6	−0.612130
$372$	0	0
$373$	3.51347e6	1.30757	0.653785	−	0.756681i	$-0.273180\pi$
$373$	3.51347e6	1.30757	0.653785	+	0.756681i	$0.273180\pi$
$374$	0	0
$375$	43580.0	0.0160033
$376$	0	0
$377$	−3.44541e6	−1.24850
$378$	0	0
$379$	5.25780e6	1.88021	0.940104	−	0.340887i	$-0.110727\pi$
$379$	5.25780e6	1.88021	0.940104	+	0.340887i	$0.110727\pi$
$380$	0	0
$381$	−267957.	−0.0945697
$382$	0	0
$383$	864388.	0.301101	0.150550	−	0.988602i	$-0.451895\pi$
$383$	864388.	0.301101	0.150550	+	0.988602i	$0.451895\pi$
$384$	0	0
$385$	−778047.	−0.267519
$386$	0	0
$387$	4.16645e6	1.41413
$388$	0	0
$389$	1.38956e6	0.465590	0.232795	−	0.972526i	$-0.425213\pi$
$389$	1.38956e6	0.465590	0.232795	+	0.972526i	$0.425213\pi$
$390$	0	0
$391$	428009.	0.141583
$392$	0	0
$393$	863806.	0.282121
$394$	0	0
$395$	−181434.	−0.0585094
$396$	0	0
$397$	2.16507e6	0.689438	0.344719	−	0.938706i	$-0.387974\pi$
$397$	2.16507e6	0.689438	0.344719	+	0.938706i	$0.387974\pi$
$398$	0	0
$399$	−264180.	−0.0830744
$400$	0	0
$401$	2.93321e6	0.910925	0.455462	−	0.890255i	$-0.349474\pi$
$401$	2.93321e6	0.910925	0.455462	+	0.890255i	$0.349474\pi$
$402$	0	0
$403$	2.05676e6	0.630842
$404$	0	0
$405$	−1.33596e6	−0.404722
$406$	0	0
$407$	−3.13319e6	−0.937565
$408$	0	0
$409$	1.07468e6	0.317667	0.158834	−	0.987305i	$-0.449227\pi$
$409$	1.07468e6	0.317667	0.158834	+	0.987305i	$0.449227\pi$
$410$	0	0
$411$	−384081.	−0.112155
$412$	0	0
$413$	−1.50634e6	−0.434557
$414$	0	0
$415$	−1.98552e6	−0.565918
$416$	0	0
$417$	214238.	0.0603332
$418$	0	0
$419$	3.95564e6	1.10073	0.550366	−	0.834923i	$-0.314488\pi$
$419$	3.95564e6	1.10073	0.550366	+	0.834923i	$0.314488\pi$
$420$	0	0
$421$	−6.45761e6	−1.77569	−0.887844	−	0.460144i	$-0.847798\pi$
$421$	−6.45761e6	−1.77569	−0.887844	+	0.460144i	$0.847798\pi$
$422$	0	0
$423$	−3.74150e6	−1.01671
$424$	0	0
$425$	59716.6	0.0160370
$426$	0	0
$427$	−2.58620e6	−0.686424
$428$	0	0
$429$	1.14750e6	0.301030
$430$	0	0
$431$	1.58103e6	0.409966	0.204983	−	0.978765i	$-0.434286\pi$
$431$	1.58103e6	0.409966	0.204983	+	0.978765i	$0.434286\pi$
$432$	0	0
$433$	−4.27793e6	−1.09651	−0.548256	−	0.836310i	$-0.684708\pi$
$433$	−4.27793e6	−1.09651	−0.548256	+	0.836310i	$0.684708\pi$
$434$	0	0
$435$	−370878.	−0.0939741
$436$	0	0
$437$	−8.65913e6	−2.16906
$438$	0	0
$439$	2.58849e6	0.641039	0.320520	−	0.947242i	$-0.396142\pi$
$439$	2.58849e6	0.641039	0.320520	+	0.947242i	$0.396142\pi$
$440$	0	0
$441$	−564765.	−0.138284
$442$	0	0
$443$	5.55728e6	1.34540	0.672702	−	0.739913i	$-0.265133\pi$
$443$	5.55728e6	1.34540	0.672702	+	0.739913i	$0.265133\pi$
$444$	0	0
$445$	−2.57976e6	−0.617559
$446$	0	0
$447$	214110.	0.0506837
$448$	0	0
$449$	−4.89010e6	−1.14473	−0.572363	−	0.820000i	$-0.693973\pi$
$449$	−4.89010e6	−1.14473	−0.572363	+	0.820000i	$0.693973\pi$
$450$	0	0
$451$	4.68783e6	1.08525
$452$	0	0
$453$	−725525.	−0.166114
$454$	0	0
$455$	793511.	0.179690
$456$	0	0
$457$	−3.58347e6	−0.802625	−0.401313	−	0.915941i	$-0.631446\pi$
$457$	−3.58347e6	−0.802625	−0.401313	+	0.915941i	$0.631446\pi$
$458$	0	0
$459$	127441.	0.0282344
$460$	0	0
$461$	4.82310e6	1.05700	0.528499	−	0.848934i	$-0.322755\pi$
$461$	4.82310e6	1.05700	0.528499	+	0.848934i	$0.322755\pi$
$462$	0	0
$463$	6.55295e6	1.42064	0.710320	−	0.703879i	$-0.248550\pi$
$463$	6.55295e6	1.42064	0.710320	+	0.703879i	$0.248550\pi$
$464$	0	0
$465$	221398.	0.0474833
$466$	0	0
$467$	−3.76303e6	−0.798446	−0.399223	−	0.916854i	$-0.630720\pi$
$467$	−3.76303e6	−0.798446	−0.399223	+	0.916854i	$0.630720\pi$
$468$	0	0
$469$	2.15559e6	0.452516
$470$	0	0
$471$	−634000.	−0.131685
$472$	0	0
$473$	1.12502e7	2.31211
$474$	0	0
$475$	−1.20814e6	−0.245687
$476$	0	0
$477$	−7.79037e6	−1.56770
$478$	0	0
$479$	2.41704e6	0.481332	0.240666	−	0.970608i	$-0.422634\pi$
$479$	2.41704e6	0.481332	0.240666	+	0.970608i	$0.422634\pi$
$480$	0	0
$481$	3.19547e6	0.629755
$482$	0	0
$483$	612211.	0.119408
$484$	0	0
$485$	1.09634e6	0.211637
$486$	0	0
$487$	−1.29910e6	−0.248211	−0.124105	−	0.992269i	$-0.539606\pi$
$487$	−1.29910e6	−0.248211	−0.124105	+	0.992269i	$0.539606\pi$
$488$	0	0
$489$	−492933.	−0.0932214
$490$	0	0
$491$	−7.54721e6	−1.41281	−0.706403	−	0.707809i	$-0.749684\pi$
$491$	−7.54721e6	−1.41281	−0.706403	+	0.707809i	$0.749684\pi$
$492$	0	0
$493$	−508205.	−0.0941720
$494$	0	0
$495$	−3.73496e6	−0.685129
$496$	0	0
$497$	1.91023e6	0.346893
$498$	0	0
$499$	6.30866e6	1.13419	0.567095	−	0.823653i	$-0.308067\pi$
$499$	6.30866e6	1.13419	0.567095	+	0.823653i	$0.308067\pi$
$500$	0	0
$501$	329368.	0.0586256
$502$	0	0
$503$	2.16913e6	0.382266	0.191133	−	0.981564i	$-0.438784\pi$
$503$	2.16913e6	0.382266	0.191133	+	0.981564i	$0.438784\pi$
$504$	0	0
$505$	−2.92472e6	−0.510335
$506$	0	0
$507$	−134729.	−0.0232778
$508$	0	0
$509$	1.70072e6	0.290963	0.145482	−	0.989361i	$-0.453527\pi$
$509$	1.70072e6	0.290963	0.145482	+	0.989361i	$0.453527\pi$
$510$	0	0
$511$	346287.	0.0586657
$512$	0	0
$513$	−2.57829e6	−0.432552
$514$	0	0
$515$	2.60807e6	0.433313
$516$	0	0
$517$	−1.01028e7	−1.66232
$518$	0	0
$519$	169731.	0.0276594
$520$	0	0
$521$	−3.95171e6	−0.637809	−0.318905	−	0.947787i	$-0.603315\pi$
$521$	−3.95171e6	−0.637809	−0.318905	+	0.947787i	$0.603315\pi$
$522$	0	0
$523$	−3.98112e6	−0.636430	−0.318215	−	0.948019i	$-0.603083\pi$
$523$	−3.98112e6	−0.636430	−0.318215	+	0.948019i	$0.603083\pi$
$524$	0	0
$525$	85416.8	0.0135252
$526$	0	0
$527$	303376.	0.0475834
$528$	0	0
$529$	1.36304e7	2.11772
$530$	0	0
$531$	−7.23105e6	−1.11292
$532$	0	0
$533$	−4.78100e6	−0.728956
$534$	0	0
$535$	−988047.	−0.149243
$536$	0	0
$537$	1.51811e6	0.227179
$538$	0	0
$539$	−1.52497e6	−0.226095
$540$	0	0
$541$	−3.76236e6	−0.552672	−0.276336	−	0.961061i	$-0.589120\pi$
$541$	−3.76236e6	−0.552672	−0.276336	+	0.961061i	$0.589120\pi$
$542$	0	0
$543$	2.31000e6	0.336211
$544$	0	0
$545$	−1.59269e6	−0.229689
$546$	0	0
$547$	−9.97755e6	−1.42579	−0.712895	−	0.701271i	$-0.752616\pi$
$547$	−9.97755e6	−1.42579	−0.712895	+	0.701271i	$0.752616\pi$
$548$	0	0
$549$	−1.24149e7	−1.75797
$550$	0	0
$551$	1.02816e7	1.44272
$552$	0	0
$553$	−355610.	−0.0494495
$554$	0	0
$555$	343973.	0.0474015
$556$	0	0
$557$	−22305.0	−0.00304624	−0.00152312	−	0.999999i	$-0.500485\pi$
$557$	−22305.0	−0.00304624	−0.00152312	+	0.999999i	$0.500485\pi$
$558$	0	0
$559$	−1.14738e7	−1.55302
$560$	0	0
$561$	169259.	0.0227062
$562$	0	0
$563$	−1.36608e7	−1.81637	−0.908187	−	0.418566i	$-0.862533\pi$
$563$	−1.36608e7	−1.81637	−0.908187	+	0.418566i	$0.862533\pi$
$564$	0	0
$565$	−2.88875e6	−0.380705
$566$	0	0
$567$	−2.61849e6	−0.342052
$568$	0	0
$569$	59148.2	0.00765880	0.00382940	−	0.999993i	$-0.498781\pi$
$569$	59148.2	0.00765880	0.00382940	+	0.999993i	$0.498781\pi$
$570$	0	0
$571$	−1.28725e6	−0.165224	−0.0826120	−	0.996582i	$-0.526326\pi$
$571$	−1.28725e6	−0.165224	−0.0826120	+	0.996582i	$0.526326\pi$
$572$	0	0
$573$	704736.	0.0896686
$574$	0	0
$575$	2.79974e6	0.353141
$576$	0	0
$577$	−1.63635e6	−0.204615	−0.102307	−	0.994753i	$-0.532623\pi$
$577$	−1.63635e6	−0.204615	−0.102307	+	0.994753i	$0.532623\pi$
$578$	0	0
$579$	−575632.	−0.0713590
$580$	0	0
$581$	−3.89162e6	−0.478288
$582$	0	0
$583$	−2.10355e7	−2.56319
$584$	0	0
$585$	3.80919e6	0.460196
$586$	0	0
$587$	765306.	0.0916726	0.0458363	−	0.998949i	$-0.485405\pi$
$587$	765306.	0.0916726	0.0458363	+	0.998949i	$0.485405\pi$
$588$	0	0
$589$	−6.13766e6	−0.728979
$590$	0	0
$591$	574325.	0.0676377
$592$	0	0
$593$	8.76158e6	1.02317	0.511583	−	0.859234i	$-0.329059\pi$
$593$	8.76158e6	1.02317	0.511583	+	0.859234i	$0.329059\pi$
$594$	0	0
$595$	117044.	0.0135537
$596$	0	0
$597$	1.21298e6	0.139290
$598$	0	0
$599$	1.08109e7	1.23110	0.615551	−	0.788097i	$-0.288934\pi$
$599$	1.08109e7	1.23110	0.615551	+	0.788097i	$0.288934\pi$
$600$	0	0
$601$	4.15961e6	0.469750	0.234875	−	0.972026i	$-0.424532\pi$
$601$	4.15961e6	0.469750	0.234875	+	0.972026i	$0.424532\pi$
$602$	0	0
$603$	1.03477e7	1.15892
$604$	0	0
$605$	−6.05882e6	−0.672976
$606$	0	0
$607$	284415.	0.0313315	0.0156657	−	0.999877i	$-0.495013\pi$
$607$	284415.	0.0313315	0.0156657	+	0.999877i	$0.495013\pi$
$608$	0	0
$609$	−726921.	−0.0794226
$610$	0	0
$611$	1.03036e7	1.11656
$612$	0	0
$613$	−1.31198e6	−0.141019	−0.0705093	−	0.997511i	$-0.522462\pi$
$613$	−1.31198e6	−0.141019	−0.0705093	+	0.997511i	$0.522462\pi$
$614$	0	0
$615$	−514647.	−0.0548683
$616$	0	0
$617$	6.86830e6	0.726334	0.363167	−	0.931724i	$-0.381696\pi$
$617$	6.86830e6	0.726334	0.363167	+	0.931724i	$0.381696\pi$
$618$	0	0
$619$	6.31368e6	0.662302	0.331151	−	0.943578i	$-0.392563\pi$
$619$	6.31368e6	0.662302	0.331151	+	0.943578i	$0.392563\pi$
$620$	0	0
$621$	5.97494e6	0.621734
$622$	0	0
$623$	−5.05632e6	−0.521933
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−3.42431e6	−0.347860
$628$	0	0
$629$	471338.	0.0475013
$630$	0	0
$631$	−4.05827e6	−0.405758	−0.202879	−	0.979204i	$-0.565030\pi$
$631$	−4.05827e6	−0.405758	−0.202879	+	0.979204i	$0.565030\pi$
$632$	0	0
$633$	2.41590e6	0.239646
$634$	0	0
$635$	−2.40180e6	−0.236376
$636$	0	0
$637$	1.55528e6	0.151866
$638$	0	0
$639$	9.16993e6	0.888411
$640$	0	0
$641$	1.74409e7	1.67658	0.838289	−	0.545226i	$-0.183556\pi$
$641$	1.74409e7	1.67658	0.838289	+	0.545226i	$0.183556\pi$
$642$	0	0
$643$	−9.64375e6	−0.919854	−0.459927	−	0.887957i	$-0.652124\pi$
$643$	−9.64375e6	−0.919854	−0.459927	+	0.887957i	$0.652124\pi$
$644$	0	0
$645$	−1.23509e6	−0.116896
$646$	0	0
$647$	1.31048e7	1.23075	0.615373	−	0.788236i	$-0.289006\pi$
$647$	1.31048e7	1.23075	0.615373	+	0.788236i	$0.289006\pi$
$648$	0	0
$649$	−1.95252e7	−1.81963
$650$	0	0
$651$	433940.	0.0401308
$652$	0	0
$653$	898898.	0.0824950	0.0412475	−	0.999149i	$-0.486867\pi$
$653$	898898.	0.0824950	0.0412475	+	0.999149i	$0.486867\pi$
$654$	0	0
$655$	7.74265e6	0.705158
$656$	0	0
$657$	1.66233e6	0.150246
$658$	0	0
$659$	1.75567e6	0.157482	0.0787408	−	0.996895i	$-0.474910\pi$
$659$	1.75567e6	0.157482	0.0787408	+	0.996895i	$0.474910\pi$
$660$	0	0
$661$	−2.09766e7	−1.86737	−0.933686	−	0.358093i	$-0.883427\pi$
$661$	−2.09766e7	−1.86737	−0.933686	+	0.358093i	$0.883427\pi$
$662$	0	0
$663$	−172623.	−0.0152516
$664$	0	0
$665$	−2.36795e6	−0.207643
$666$	0	0
$667$	−2.38266e7	−2.07371
$668$	0	0
$669$	1.59849e6	0.138085
$670$	0	0
$671$	−3.35225e7	−2.87428
$672$	0	0
$673$	5.71791e6	0.486631	0.243315	−	0.969947i	$-0.421765\pi$
$673$	5.71791e6	0.486631	0.243315	+	0.969947i	$0.421765\pi$
$674$	0	0
$675$	833634.	0.0704233
$676$	0	0
$677$	1.05680e7	0.886176	0.443088	−	0.896478i	$-0.353883\pi$
$677$	1.05680e7	0.886176	0.443088	+	0.896478i	$0.353883\pi$
$678$	0	0
$679$	2.14883e6	0.178866
$680$	0	0
$681$	3.64965e6	0.301567
$682$	0	0
$683$	5.80775e6	0.476383	0.238191	−	0.971218i	$-0.423445\pi$
$683$	5.80775e6	0.476383	0.238191	+	0.971218i	$0.423445\pi$
$684$	0	0
$685$	−3.44267e6	−0.280330
$686$	0	0
$687$	−4.09675e6	−0.331167
$688$	0	0
$689$	2.14535e7	1.72167
$690$	0	0
$691$	8.18331e6	0.651979	0.325990	−	0.945373i	$-0.394303\pi$
$691$	8.18331e6	0.651979	0.325990	+	0.945373i	$0.394303\pi$
$692$	0	0
$693$	−7.32052e6	−0.579040
$694$	0	0
$695$	1.92030e6	0.150802
$696$	0	0
$697$	−705208.	−0.0549839
$698$	0	0
$699$	−3.25535e6	−0.252002
$700$	0	0
$701$	3.51754e6	0.270361	0.135181	−	0.990821i	$-0.456839\pi$
$701$	3.51754e6	0.270361	0.135181	+	0.990821i	$0.456839\pi$
$702$	0	0
$703$	−9.53572e6	−0.727722
$704$	0	0
$705$	1.10912e6	0.0840436
$706$	0	0
$707$	−5.73245e6	−0.431312
$708$	0	0
$709$	6.91036e6	0.516280	0.258140	−	0.966108i	$-0.416890\pi$
$709$	6.91036e6	0.516280	0.258140	+	0.966108i	$0.416890\pi$
$710$	0	0
$711$	−1.70708e6	−0.126643
$712$	0	0
$713$	1.42234e7	1.04781
$714$	0	0
$715$	1.02855e7	0.752422
$716$	0	0
$717$	−3.05151e6	−0.221675
$718$	0	0
$719$	1.18370e7	0.853923	0.426962	−	0.904270i	$-0.359584\pi$
$719$	1.18370e7	0.853923	0.426962	+	0.904270i	$0.359584\pi$
$720$	0	0
$721$	5.11183e6	0.366216
$722$	0	0
$723$	1.31928e6	0.0938626
$724$	0	0
$725$	−3.32433e6	−0.234887
$726$	0	0
$727$	1.87844e6	0.131814	0.0659071	−	0.997826i	$-0.479006\pi$
$727$	1.87844e6	0.131814	0.0659071	+	0.997826i	$0.479006\pi$
$728$	0	0
$729$	−1.16658e7	−0.813013
$730$	0	0
$731$	−1.69241e6	−0.117142
$732$	0	0
$733$	2.82256e7	1.94036	0.970180	−	0.242384i	$-0.0779293\pi$
$733$	2.82256e7	1.94036	0.970180	+	0.242384i	$0.0779293\pi$
$734$	0	0
$735$	167417.	0.0114309
$736$	0	0
$737$	2.79409e7	1.89483
$738$	0	0
$739$	1.08997e7	0.734180	0.367090	−	0.930185i	$-0.380354\pi$
$739$	1.08997e7	0.734180	0.367090	+	0.930185i	$0.380354\pi$
$740$	0	0
$741$	3.49237e6	0.233655
$742$	0	0
$743$	−6.95093e6	−0.461924	−0.230962	−	0.972963i	$-0.574187\pi$
$743$	−6.95093e6	−0.461924	−0.230962	+	0.972963i	$0.574187\pi$
$744$	0	0
$745$	1.91916e6	0.126683
$746$	0	0
$747$	−1.86814e7	−1.22492
$748$	0	0
$749$	−1.93657e6	−0.126133
$750$	0	0
$751$	−1.41025e7	−0.912423	−0.456212	−	0.889871i	$-0.650794\pi$
$751$	−1.41025e7	−0.912423	−0.456212	+	0.889871i	$0.650794\pi$
$752$	0	0
$753$	2.91636e6	0.187436
$754$	0	0
$755$	−6.50318e6	−0.415201
$756$	0	0
$757$	−1.60947e7	−1.02080	−0.510402	−	0.859936i	$-0.670503\pi$
$757$	−1.60947e7	−1.02080	−0.510402	+	0.859936i	$0.670503\pi$
$758$	0	0
$759$	7.93551e6	0.500000
$760$	0	0
$761$	9.56904e6	0.598972	0.299486	−	0.954101i	$-0.403185\pi$
$761$	9.56904e6	0.598972	0.299486	+	0.954101i	$0.403185\pi$
$762$	0	0
$763$	−3.12167e6	−0.194123
$764$	0	0
$765$	561863.	0.0347118
$766$	0	0
$767$	1.99133e7	1.22223
$768$	0	0
$769$	−7.23166e6	−0.440983	−0.220492	−	0.975389i	$-0.570766\pi$
$769$	−7.23166e6	−0.440983	−0.220492	+	0.975389i	$0.570766\pi$
$770$	0	0
$771$	4.32681e6	0.262139
$772$	0	0
$773$	2.80950e7	1.69114	0.845572	−	0.533862i	$-0.179260\pi$
$773$	2.80950e7	1.69114	0.845572	+	0.533862i	$0.179260\pi$
$774$	0	0
$775$	1.98448e6	0.118684
$776$	0	0
$777$	674187.	0.0400616
$778$	0	0
$779$	1.42672e7	0.842355
$780$	0	0
$781$	2.47605e7	1.45255
$782$	0	0
$783$	−7.09446e6	−0.413538
$784$	0	0
$785$	−5.68280e6	−0.329146
$786$	0	0
$787$	−2.91886e6	−0.167987	−0.0839937	−	0.996466i	$-0.526768\pi$
$787$	−2.91886e6	−0.167987	−0.0839937	+	0.996466i	$0.526768\pi$
$788$	0	0
$789$	−5.65945e6	−0.323655
$790$	0	0
$791$	−5.66195e6	−0.321754
$792$	0	0
$793$	3.41887e7	1.93063
$794$	0	0
$795$	2.30935e6	0.129590
$796$	0	0
$797$	−1.17657e7	−0.656102	−0.328051	−	0.944660i	$-0.606392\pi$
$797$	−1.17657e7	−0.656102	−0.328051	+	0.944660i	$0.606392\pi$
$798$	0	0
$799$	1.51980e6	0.0842206
$800$	0	0
$801$	−2.42725e7	−1.33670
$802$	0	0
$803$	4.48860e6	0.245653
$804$	0	0
$805$	5.48749e6	0.298459
$806$	0	0
$807$	3.24354e6	0.175321
$808$	0	0
$809$	−1.92033e7	−1.03158	−0.515792	−	0.856714i	$-0.672502\pi$
$809$	−1.92033e7	−1.03158	−0.515792	+	0.856714i	$0.672502\pi$
$810$	0	0
$811$	2.28490e7	1.21987	0.609937	−	0.792450i	$-0.291195\pi$
$811$	2.28490e7	1.21987	0.609937	+	0.792450i	$0.291195\pi$
$812$	0	0
$813$	−2.37546e6	−0.126044
$814$	0	0
$815$	−4.41836e6	−0.233006
$816$	0	0
$817$	3.42395e7	1.79462
$818$	0	0
$819$	7.46601e6	0.388937
$820$	0	0
$821$	−5.62532e6	−0.291266	−0.145633	−	0.989339i	$-0.546522\pi$
$821$	−5.62532e6	−0.291266	−0.145633	+	0.989339i	$0.546522\pi$
$822$	0	0
$823$	−9.83323e6	−0.506053	−0.253027	−	0.967459i	$-0.581426\pi$
$823$	−9.83323e6	−0.506053	−0.253027	+	0.967459i	$0.581426\pi$
$824$	0	0
$825$	1.10718e6	0.0566346
$826$	0	0
$827$	3.37080e7	1.71384	0.856918	−	0.515453i	$-0.172376\pi$
$827$	3.37080e7	1.71384	0.856918	+	0.515453i	$0.172376\pi$
$828$	0	0
$829$	−3.56551e7	−1.80192	−0.900960	−	0.433902i	$-0.857137\pi$
$829$	−3.56551e7	−1.80192	−0.900960	+	0.433902i	$0.857137\pi$
$830$	0	0
$831$	−4.41697e6	−0.221882
$832$	0	0
$833$	229407.	0.0114550
$834$	0	0
$835$	2.95226e6	0.146534
$836$	0	0
$837$	4.23509e6	0.208953
$838$	0	0
$839$	6.54004e6	0.320757	0.160378	−	0.987056i	$-0.448729\pi$
$839$	6.54004e6	0.320757	0.160378	+	0.987056i	$0.448729\pi$
$840$	0	0
$841$	7.77984e6	0.379298
$842$	0	0
$843$	−6.13371e6	−0.297272
$844$	0	0
$845$	−1.20763e6	−0.0581825
$846$	0	0
$847$	−1.18753e7	−0.568768
$848$	0	0
$849$	−1.34977e6	−0.0642676
$850$	0	0
$851$	2.20981e7	1.04600
$852$	0	0
$853$	1.10498e7	0.519974	0.259987	−	0.965612i	$-0.416282\pi$
$853$	1.10498e7	0.519974	0.259987	+	0.965612i	$0.416282\pi$
$854$	0	0
$855$	−1.13672e7	−0.531786
$856$	0	0
$857$	−8.97319e6	−0.417345	−0.208672	−	0.977986i	$-0.566914\pi$
$857$	−8.97319e6	−0.417345	−0.208672	+	0.977986i	$0.566914\pi$
$858$	0	0
$859$	−1.39741e7	−0.646162	−0.323081	−	0.946371i	$-0.604719\pi$
$859$	−1.39741e7	−0.646162	−0.323081	+	0.946371i	$0.604719\pi$
$860$	0	0
$861$	−1.00871e6	−0.0463722
$862$	0	0
$863$	6.83841e6	0.312556	0.156278	−	0.987713i	$-0.450050\pi$
$863$	6.83841e6	0.312556	0.156278	+	0.987713i	$0.450050\pi$
$864$	0	0
$865$	1.52137e6	0.0691343
$866$	0	0
$867$	3.93469e6	0.177772
$868$	0	0
$869$	−4.60944e6	−0.207061
$870$	0	0
$871$	−2.84962e7	−1.27274
$872$	0	0
$873$	1.03153e7	0.458085
$874$	0	0
$875$	765625.	0.0338062
$876$	0	0
$877$	1.82691e7	0.802081	0.401041	−	0.916060i	$-0.368649\pi$
$877$	1.82691e7	0.802081	0.401041	+	0.916060i	$0.368649\pi$
$878$	0	0
$879$	−4.43944e6	−0.193801
$880$	0	0
$881$	3.83363e7	1.66406	0.832032	−	0.554728i	$-0.187178\pi$
$881$	3.83363e7	1.66406	0.832032	+	0.554728i	$0.187178\pi$
$882$	0	0
$883$	−3.05501e7	−1.31859	−0.659296	−	0.751883i	$-0.729146\pi$
$883$	−3.05501e7	−1.31859	−0.659296	+	0.751883i	$0.729146\pi$
$884$	0	0
$885$	2.14355e6	0.0919972
$886$	0	0
$887$	2.46293e7	1.05110	0.525549	−	0.850763i	$-0.323860\pi$
$887$	2.46293e7	1.05110	0.525549	+	0.850763i	$0.323860\pi$
$888$	0	0
$889$	−4.70753e6	−0.199774
$890$	0	0
$891$	−3.39410e7	−1.43229
$892$	0	0
$893$	−3.07473e7	−1.29026
$894$	0	0
$895$	1.36075e7	0.567831
$896$	0	0
$897$	−8.09323e6	−0.335846
$898$	0	0
$899$	−1.68885e7	−0.696934
$900$	0	0
$901$	3.16444e6	0.129863
$902$	0	0
$903$	−2.42077e6	−0.0987949
$904$	0	0
$905$	2.07055e7	0.840356
$906$	0	0
$907$	2.34527e7	0.946617	0.473309	−	0.880897i	$-0.343060\pi$
$907$	2.34527e7	0.946617	0.473309	+	0.880897i	$0.343060\pi$
$908$	0	0
$909$	−2.75182e7	−1.10461
$910$	0	0
$911$	3.81648e7	1.52359	0.761793	−	0.647821i	$-0.224319\pi$
$911$	3.81648e7	1.52359	0.761793	+	0.647821i	$0.224319\pi$
$912$	0	0
$913$	−5.04434e7	−2.00275
$914$	0	0
$915$	3.68021e6	0.145318
$916$	0	0
$917$	1.51756e7	0.595967
$918$	0	0
$919$	1.13393e7	0.442890	0.221445	−	0.975173i	$-0.428923\pi$
$919$	1.13393e7	0.442890	0.221445	+	0.975173i	$0.428923\pi$
$920$	0	0
$921$	2.91204e6	0.113122
$922$	0	0
$923$	−2.52527e7	−0.975670
$924$	0	0
$925$	3.08317e6	0.118479
$926$	0	0
$927$	2.45389e7	0.937900
$928$	0	0
$929$	4.62543e7	1.75838	0.879189	−	0.476473i	$-0.158085\pi$
$929$	4.62543e7	1.75838	0.879189	+	0.476473i	$0.158085\pi$
$930$	0	0
$931$	−4.64118e6	−0.175491
$932$	0	0
$933$	−6.01107e6	−0.226072
$934$	0	0
$935$	1.51714e6	0.0567539
$936$	0	0
$937$	3.21365e7	1.19578	0.597888	−	0.801580i	$-0.296007\pi$
$937$	3.21365e7	1.19578	0.597888	+	0.801580i	$0.296007\pi$
$938$	0	0
$939$	−3.38381e6	−0.125240
$940$	0	0
$941$	2.41747e7	0.889992	0.444996	−	0.895532i	$-0.353205\pi$
$941$	2.41747e7	0.889992	0.444996	+	0.895532i	$0.353205\pi$
$942$	0	0
$943$	−3.30628e7	−1.21077
$944$	0	0
$945$	1.63392e6	0.0595185
$946$	0	0
$947$	1.31735e7	0.477337	0.238669	−	0.971101i	$-0.423289\pi$
$947$	1.31735e7	0.477337	0.238669	+	0.971101i	$0.423289\pi$
$948$	0	0
$949$	−4.57781e6	−0.165003
$950$	0	0
$951$	7.59162e6	0.272197
$952$	0	0
$953$	−2.51353e7	−0.896504	−0.448252	−	0.893907i	$-0.647953\pi$
$953$	−2.51353e7	−0.896504	−0.448252	+	0.893907i	$0.647953\pi$
$954$	0	0
$955$	6.31684e6	0.224126
$956$	0	0
$957$	−9.42239e6	−0.332569
$958$	0	0
$959$	−6.74764e6	−0.236922
$960$	0	0
$961$	−1.85475e7	−0.647852
$962$	0	0
$963$	−9.29637e6	−0.323033
$964$	0	0
$965$	−5.15962e6	−0.178361
$966$	0	0
$967$	9.98676e6	0.343446	0.171723	−	0.985145i	$-0.445067\pi$
$967$	9.98676e6	0.343446	0.171723	+	0.985145i	$0.445067\pi$
$968$	0	0
$969$	515132.	0.0176242
$970$	0	0
$971$	−5.43566e7	−1.85014	−0.925069	−	0.379800i	$-0.875993\pi$
$971$	−5.43566e7	−1.85014	−0.925069	+	0.379800i	$0.875993\pi$
$972$	0	0
$973$	3.76379e6	0.127451
$974$	0	0
$975$	−1.12918e6	−0.0380410
$976$	0	0
$977$	−2.28021e7	−0.764257	−0.382128	−	0.924109i	$-0.624809\pi$
$977$	−2.28021e7	−0.764257	−0.382128	+	0.924109i	$0.624809\pi$
$978$	0	0
$979$	−6.55403e7	−2.18550
$980$	0	0
$981$	−1.49854e7	−0.497158
$982$	0	0
$983$	2.12314e7	0.700801	0.350400	−	0.936600i	$-0.386046\pi$
$983$	2.12314e7	0.700801	0.350400	+	0.936600i	$0.386046\pi$
$984$	0	0
$985$	5.14790e6	0.169060
$986$	0	0
$987$	2.17387e6	0.0710298
$988$	0	0
$989$	−7.93467e7	−2.57951
$990$	0	0
$991$	−1.32994e7	−0.430179	−0.215089	−	0.976594i	$-0.569004\pi$
$991$	−1.32994e7	−0.430179	−0.215089	+	0.976594i	$0.569004\pi$
$992$	0	0
$993$	2.06415e6	0.0664307
$994$	0	0
$995$	1.08725e7	0.348153
$996$	0	0
$997$	3.48141e7	1.10922	0.554610	−	0.832111i	$-0.312868\pi$
$997$	3.48141e7	1.10922	0.554610	+	0.832111i	$0.312868\pi$
$998$	0	0
$999$	6.57980e6	0.208593

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.6.a.i.1.3	✓	5
4.3	odd	2		560.6.a.z.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.6.a.i.1.3	✓	5	1.1	even	1	trivial
560.6.a.z.1.3		5	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)

\(f(q)\)	\(=\)	\(q-2.78912 q^{3} -25.0000 q^{5} -49.0000 q^{7} -235.221 q^{9} +O(q^{10})\)	\(q-2.78912 q^{3} -25.0000 q^{5} -49.0000 q^{7} -235.221 q^{9} -635.141 q^{11} +647.764 q^{13} +69.7280 q^{15} +95.5465 q^{17} -1933.02 q^{19} +136.667 q^{21} +4479.59 q^{23} +625.000 q^{25} +1333.81 q^{27} -5318.93 q^{29} +3175.17 q^{31} +1771.48 q^{33} +1225.00 q^{35} +4933.07 q^{37} -1806.69 q^{39} -7380.78 q^{41} -17712.9 q^{43} +5880.52 q^{45} +15906.3 q^{47} +2401.00 q^{49} -266.491 q^{51} +33119.4 q^{53} +15878.5 q^{55} +5391.42 q^{57} +30741.6 q^{59} +52779.6 q^{61} +11525.8 q^{63} -16194.1 q^{65} -43991.6 q^{67} -12494.1 q^{69} -38984.3 q^{71} -7067.09 q^{73} -1743.20 q^{75} +31121.9 q^{77} +7257.36 q^{79} +53438.5 q^{81} +79420.8 q^{83} -2388.66 q^{85} +14835.1 q^{87} +103190. q^{89} -31740.4 q^{91} -8855.92 q^{93} +48325.5 q^{95} -43853.7 q^{97} +149398. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{3} - 125 q^{5} - 245 q^{7} + 372 q^{9}+O(q^{10}) \) 5 * q - 3 * q^3 - 125 * q^5 - 245 * q^7 + 372 * q^9	\( 5 q - 3 q^{3} - 125 q^{5} - 245 q^{7} + 372 q^{9} - 263 q^{11} - 729 q^{13} + 75 q^{15} - 1003 q^{17} - 2506 q^{19} + 147 q^{21} - 1066 q^{23} + 3125 q^{25} + 615 q^{27} + 3489 q^{29} + 7880 q^{31} + 4863 q^{33} + 6125 q^{35} + 13118 q^{37} + 27189 q^{39} + 23972 q^{41} + 3978 q^{43} - 9300 q^{45} + 9057 q^{47} + 12005 q^{49} + 96639 q^{51} + 2128 q^{53} + 6575 q^{55} + 61674 q^{57} - 15512 q^{59} + 4560 q^{61} - 18228 q^{63} + 18225 q^{65} + 7780 q^{67} + 126474 q^{69} + 32752 q^{71} + 189498 q^{73} - 1875 q^{75} + 12887 q^{77} + 42055 q^{79} + 294645 q^{81} + 58420 q^{83} + 25075 q^{85} - 765 q^{87} + 231324 q^{89} + 35721 q^{91} + 395736 q^{93} + 62650 q^{95} + 247569 q^{97} + 30606 q^{99}+O(q^{100}) \) 5 * q - 3 * q^3 - 125 * q^5 - 245 * q^7 + 372 * q^9 - 263 * q^11 - 729 * q^13 + 75 * q^15 - 1003 * q^17 - 2506 * q^19 + 147 * q^21 - 1066 * q^23 + 3125 * q^25 + 615 * q^27 + 3489 * q^29 + 7880 * q^31 + 4863 * q^33 + 6125 * q^35 + 13118 * q^37 + 27189 * q^39 + 23972 * q^41 + 3978 * q^43 - 9300 * q^45 + 9057 * q^47 + 12005 * q^49 + 96639 * q^51 + 2128 * q^53 + 6575 * q^55 + 61674 * q^57 - 15512 * q^59 + 4560 * q^61 - 18228 * q^63 + 18225 * q^65 + 7780 * q^67 + 126474 * q^69 + 32752 * q^71 + 189498 * q^73 - 1875 * q^75 + 12887 * q^77 + 42055 * q^79 + 294645 * q^81 + 58420 * q^83 + 25075 * q^85 - 765 * q^87 + 231324 * q^89 + 35721 * q^91 + 395736 * q^93 + 62650 * q^95 + 247569 * q^97 + 30606 * q^99

Introduction

L-functions

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