LMFDB - Embedded newform 280.6.a.h.1.4

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:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 766x^{2} + 2548x + 104520 $ x^4 - x^3 - 766x^2 + 2548x + 104520
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$19.3945$ 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)

$f(q)$	$=$	$q+20.3945 q^{3} -25.0000 q^{5} -49.0000 q^{7} +172.937 q^{9} +O(q^{10})$	$q+20.3945 q^{3} -25.0000 q^{5} -49.0000 q^{7} +172.937 q^{9} -632.610 q^{11} +1109.19 q^{13} -509.863 q^{15} -1155.89 q^{17} +779.318 q^{19} -999.332 q^{21} -2412.75 q^{23} +625.000 q^{25} -1428.91 q^{27} -358.492 q^{29} -5276.24 q^{31} -12901.8 q^{33} +1225.00 q^{35} -10589.8 q^{37} +22621.5 q^{39} -19412.9 q^{41} +16749.6 q^{43} -4323.42 q^{45} -25187.0 q^{47} +2401.00 q^{49} -23573.7 q^{51} -16325.6 q^{53} +15815.2 q^{55} +15893.8 q^{57} -17053.3 q^{59} +54300.2 q^{61} -8473.89 q^{63} -27729.9 q^{65} +5249.11 q^{67} -49206.9 q^{69} +18018.0 q^{71} +40283.4 q^{73} +12746.6 q^{75} +30997.9 q^{77} +27270.6 q^{79} -71165.5 q^{81} -31615.8 q^{83} +28897.1 q^{85} -7311.27 q^{87} +50459.6 q^{89} -54350.5 q^{91} -107606. q^{93} -19483.0 q^{95} -65919.5 q^{97} -109401. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 5 q^{3} - 100 q^{5} - 196 q^{7} + 567 q^{9}+O(q^{10}) $ 4 * q + 5 * q^3 - 100 * q^5 - 196 * q^7 + 567 * q^9	$ 4 q + 5 q^{3} - 100 q^{5} - 196 q^{7} + 567 q^{9} - 123 q^{11} + 789 q^{13} - 125 q^{15} + 1051 q^{17} + 958 q^{19} - 245 q^{21} - 530 q^{23} + 2500 q^{25} - 3169 q^{27} - 5541 q^{29} - 10440 q^{31} - 3737 q^{33} + 4900 q^{35} - 24048 q^{37} - 9131 q^{39} - 35414 q^{41} + 2174 q^{43} - 14175 q^{45} - 2287 q^{47} + 9604 q^{49} - 54017 q^{51} - 22238 q^{53} + 3075 q^{55} + 5890 q^{57} - 22656 q^{59} - 20250 q^{61} - 27783 q^{63} - 19725 q^{65} - 19456 q^{67} - 47006 q^{69} - 72288 q^{71} + 59464 q^{73} + 3125 q^{75} + 6027 q^{77} - 232001 q^{79} - 149604 q^{81} - 169620 q^{83} - 26275 q^{85} - 270309 q^{87} + 141434 q^{89} - 38661 q^{91} - 203808 q^{93} - 23950 q^{95} + 167159 q^{97} - 292198 q^{99}+O(q^{100}) $ 4 * q + 5 * q^3 - 100 * q^5 - 196 * q^7 + 567 * q^9 - 123 * q^11 + 789 * q^13 - 125 * q^15 + 1051 * q^17 + 958 * q^19 - 245 * q^21 - 530 * q^23 + 2500 * q^25 - 3169 * q^27 - 5541 * q^29 - 10440 * q^31 - 3737 * q^33 + 4900 * q^35 - 24048 * q^37 - 9131 * q^39 - 35414 * q^41 + 2174 * q^43 - 14175 * q^45 - 2287 * q^47 + 9604 * q^49 - 54017 * q^51 - 22238 * q^53 + 3075 * q^55 + 5890 * q^57 - 22656 * q^59 - 20250 * q^61 - 27783 * q^63 - 19725 * q^65 - 19456 * q^67 - 47006 * q^69 - 72288 * q^71 + 59464 * q^73 + 3125 * q^75 + 6027 * q^77 - 232001 * q^79 - 149604 * q^81 - 169620 * q^83 - 26275 * q^85 - 270309 * q^87 + 141434 * q^89 - 38661 * q^91 - 203808 * q^93 - 23950 * q^95 + 167159 * q^97 - 292198 * 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
$3$	20.3945	1.30831	0.654155	−	0.756361i	$-0.273025\pi$
$3$	20.3945	1.30831	0.654155	+	0.756361i	$0.273025\pi$
$4$	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
$9$	172.937	0.711673
$10$	0	0
$11$	−632.610	−1.57636	−0.788178	−	0.615447i	$-0.788975\pi$
$11$	−632.610	−1.57636	−0.788178	+	0.615447i	$0.788975\pi$
$12$	0	0
$13$	1109.19	1.82033	0.910163	−	0.414250i	$-0.135956\pi$
$13$	1109.19	1.82033	0.910163	+	0.414250i	$0.135956\pi$
$14$	0	0
$15$	−509.863	−0.585094
$16$	0	0
$17$	−1155.89	−0.970046	−0.485023	−	0.874501i	$-0.661189\pi$
$17$	−1155.89	−0.970046	−0.485023	+	0.874501i	$0.661189\pi$
$18$	0	0
$19$	779.318	0.495257	0.247629	−	0.968855i	$-0.420349\pi$
$19$	779.318	0.495257	0.247629	+	0.968855i	$0.420349\pi$
$20$	0	0
$21$	−999.332	−0.494494
$22$	0	0
$23$	−2412.75	−0.951026	−0.475513	−	0.879709i	$-0.657738\pi$
$23$	−2412.75	−0.951026	−0.475513	+	0.879709i	$0.657738\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	−1428.91	−0.377221
$28$	0	0
$29$	−358.492	−0.0791560	−0.0395780	−	0.999216i	$-0.512601\pi$
$29$	−358.492	−0.0791560	−0.0395780	+	0.999216i	$0.512601\pi$
$30$	0	0
$31$	−5276.24	−0.986098	−0.493049	−	0.870002i	$-0.664118\pi$
$31$	−5276.24	−0.986098	−0.493049	+	0.870002i	$0.664118\pi$
$32$	0	0
$33$	−12901.8	−2.06236
$34$	0	0
$35$	1225.00	0.169031
$36$	0	0
$37$	−10589.8	−1.27170	−0.635851	−	0.771812i	$-0.719351\pi$
$37$	−10589.8	−1.27170	−0.635851	+	0.771812i	$0.719351\pi$
$38$	0	0
$39$	22621.5	2.38155
$40$	0	0
$41$	−19412.9	−1.80356	−0.901778	−	0.432199i	$-0.857738\pi$
$41$	−19412.9	−1.80356	−0.901778	+	0.432199i	$0.857738\pi$
$42$	0	0
$43$	16749.6	1.38145	0.690724	−	0.723119i	$-0.257292\pi$
$43$	16749.6	1.38145	0.690724	+	0.723119i	$0.257292\pi$
$44$	0	0
$45$	−4323.42	−0.318270
$46$	0	0
$47$	−25187.0	−1.66315	−0.831577	−	0.555409i	$-0.812562\pi$
$47$	−25187.0	−1.66315	−0.831577	+	0.555409i	$0.812562\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−23573.7	−1.26912
$52$	0	0
$53$	−16325.6	−0.798323	−0.399162	−	0.916881i	$-0.630699\pi$
$53$	−16325.6	−0.798323	−0.399162	+	0.916881i	$0.630699\pi$
$54$	0	0
$55$	15815.2	0.704968
$56$	0	0
$57$	15893.8	0.647950
$58$	0	0
$59$	−17053.3	−0.637792	−0.318896	−	0.947790i	$-0.603312\pi$
$59$	−17053.3	−0.637792	−0.318896	+	0.947790i	$0.603312\pi$
$60$	0	0
$61$	54300.2	1.86843	0.934215	−	0.356710i	$-0.116101\pi$
$61$	54300.2	1.86843	0.934215	+	0.356710i	$0.116101\pi$
$62$	0	0
$63$	−8473.89	−0.268987
$64$	0	0
$65$	−27729.9	−0.814075
$66$	0	0
$67$	5249.11	0.142856	0.0714280	−	0.997446i	$-0.477244\pi$
$67$	5249.11	0.142856	0.0714280	+	0.997446i	$0.477244\pi$
$68$	0	0
$69$	−49206.9	−1.24424
$70$	0	0
$71$	18018.0	0.424191	0.212096	−	0.977249i	$-0.431971\pi$
$71$	18018.0	0.424191	0.212096	+	0.977249i	$0.431971\pi$
$72$	0	0
$73$	40283.4	0.884746	0.442373	−	0.896831i	$-0.354137\pi$
$73$	40283.4	0.884746	0.442373	+	0.896831i	$0.354137\pi$
$74$	0	0
$75$	12746.6	0.261662
$76$	0	0
$77$	30997.9	0.595807
$78$	0	0
$79$	27270.6	0.491618	0.245809	−	0.969318i	$-0.420946\pi$
$79$	27270.6	0.491618	0.245809	+	0.969318i	$0.420946\pi$
$80$	0	0
$81$	−71165.5	−1.20519
$82$	0	0
$83$	−31615.8	−0.503743	−0.251871	−	0.967761i	$-0.581046\pi$
$83$	−31615.8	−0.503743	−0.251871	+	0.967761i	$0.581046\pi$
$84$	0	0
$85$	28897.1	0.433818
$86$	0	0
$87$	−7311.27	−0.103561
$88$	0	0
$89$	50459.6	0.675256	0.337628	−	0.941280i	$-0.390375\pi$
$89$	50459.6	0.675256	0.337628	+	0.941280i	$0.390375\pi$
$90$	0	0
$91$	−54350.5	−0.688019
$92$	0	0
$93$	−107606.	−1.29012
$94$	0	0
$95$	−19483.0	−0.221486
$96$	0	0
$97$	−65919.5	−0.711352	−0.355676	−	0.934609i	$-0.615749\pi$
$97$	−65919.5	−0.711352	−0.355676	+	0.934609i	$0.615749\pi$
$98$	0	0
$99$	−109401.	−1.12185
$100$	0	0
$101$	−130616.	−1.27407	−0.637035	−	0.770835i	$-0.719840\pi$
$101$	−130616.	−1.27407	−0.637035	+	0.770835i	$0.719840\pi$
$102$	0	0
$103$	−208570.	−1.93713	−0.968564	−	0.248764i	$-0.919976\pi$
$103$	−208570.	−1.93713	−0.968564	+	0.248764i	$0.919976\pi$
$104$	0	0
$105$	24983.3	0.221145
$106$	0	0
$107$	54551.4	0.460624	0.230312	−	0.973117i	$-0.426025\pi$
$107$	54551.4	0.460624	0.230312	+	0.973117i	$0.426025\pi$
$108$	0	0
$109$	−146449.	−1.18064	−0.590322	−	0.807168i	$-0.700999\pi$
$109$	−146449.	−1.18064	−0.590322	+	0.807168i	$0.700999\pi$
$110$	0	0
$111$	−215975.	−1.66378
$112$	0	0
$113$	142407.	1.04914	0.524572	−	0.851366i	$-0.324225\pi$
$113$	142407.	1.04914	0.524572	+	0.851366i	$0.324225\pi$
$114$	0	0
$115$	60318.7	0.425312
$116$	0	0
$117$	191820.	1.29548
$118$	0	0
$119$	56638.4	0.366643
$120$	0	0
$121$	239144.	1.48490
$122$	0	0
$123$	−395916.	−2.35961
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	−262139.	−1.44219	−0.721096	−	0.692835i	$-0.756361\pi$
$127$	−262139.	−1.44219	−0.721096	+	0.692835i	$0.756361\pi$
$128$	0	0
$129$	341601.	1.80736
$130$	0	0
$131$	222968.	1.13518	0.567590	−	0.823311i	$-0.307876\pi$
$131$	222968.	1.13518	0.567590	+	0.823311i	$0.307876\pi$
$132$	0	0
$133$	−38186.6	−0.187190
$134$	0	0
$135$	35722.7	0.168698
$136$	0	0
$137$	−102303.	−0.465678	−0.232839	−	0.972515i	$-0.574802\pi$
$137$	−102303.	−0.465678	−0.232839	+	0.972515i	$0.574802\pi$
$138$	0	0
$139$	184351.	0.809299	0.404649	−	0.914472i	$-0.367394\pi$
$139$	184351.	0.809299	0.404649	+	0.914472i	$0.367394\pi$
$140$	0	0
$141$	−513678.	−2.17592
$142$	0	0
$143$	−701687.	−2.86948
$144$	0	0
$145$	8962.29	0.0353997
$146$	0	0
$147$	48967.3	0.186901
$148$	0	0
$149$	423023.	1.56098	0.780492	−	0.625166i	$-0.214969\pi$
$149$	423023.	1.56098	0.780492	+	0.625166i	$0.214969\pi$
$150$	0	0
$151$	−69906.6	−0.249503	−0.124751	−	0.992188i	$-0.539813\pi$
$151$	−69906.6	−0.249503	−0.124751	+	0.992188i	$0.539813\pi$
$152$	0	0
$153$	−199895.	−0.690356
$154$	0	0
$155$	131906.	0.440996
$156$	0	0
$157$	−144467.	−0.467757	−0.233878	−	0.972266i	$-0.575142\pi$
$157$	−144467.	−0.467757	−0.233878	+	0.972266i	$0.575142\pi$
$158$	0	0
$159$	−332952.	−1.04445
$160$	0	0
$161$	118225.	0.359454
$162$	0	0
$163$	−62627.8	−0.184628	−0.0923141	−	0.995730i	$-0.529426\pi$
$163$	−62627.8	−0.184628	−0.0923141	+	0.995730i	$0.529426\pi$
$164$	0	0
$165$	322544.	0.922316
$166$	0	0
$167$	−46791.8	−0.129831	−0.0649155	−	0.997891i	$-0.520678\pi$
$167$	−46791.8	−0.129831	−0.0649155	+	0.997891i	$0.520678\pi$
$168$	0	0
$169$	859018.	2.31359
$170$	0	0
$171$	134773.	0.352461
$172$	0	0
$173$	634898.	1.61283	0.806415	−	0.591349i	$-0.201405\pi$
$173$	634898.	1.61283	0.806415	+	0.591349i	$0.201405\pi$
$174$	0	0
$175$	−30625.0	−0.0755929
$176$	0	0
$177$	−347795.	−0.834430
$178$	0	0
$179$	399910.	0.932887	0.466444	−	0.884551i	$-0.345535\pi$
$179$	399910.	0.932887	0.466444	+	0.884551i	$0.345535\pi$
$180$	0	0
$181$	618783.	1.40392	0.701959	−	0.712217i	$-0.252309\pi$
$181$	618783.	1.40392	0.701959	+	0.712217i	$0.252309\pi$
$182$	0	0
$183$	1.10743e6	2.44449
$184$	0	0
$185$	264746.	0.568722
$186$	0	0
$187$	731225.	1.52914
$188$	0	0
$189$	70016.6	0.142576
$190$	0	0
$191$	237155.	0.470379	0.235190	−	0.971949i	$-0.424429\pi$
$191$	237155.	0.470379	0.235190	+	0.971949i	$0.424429\pi$
$192$	0	0
$193$	144602.	0.279435	0.139717	−	0.990191i	$-0.455381\pi$
$193$	144602.	0.279435	0.139717	+	0.990191i	$0.455381\pi$
$194$	0	0
$195$	−565537.	−1.06506
$196$	0	0
$197$	919283.	1.68766	0.843828	−	0.536614i	$-0.180297\pi$
$197$	919283.	1.68766	0.843828	+	0.536614i	$0.180297\pi$
$198$	0	0
$199$	−789944.	−1.41405	−0.707023	−	0.707190i	$-0.749962\pi$
$199$	−789944.	−1.41405	−0.707023	+	0.707190i	$0.749962\pi$
$200$	0	0
$201$	107053.	0.186900
$202$	0	0
$203$	17566.1	0.0299182
$204$	0	0
$205$	485321.	0.806575
$206$	0	0
$207$	−417252.	−0.676820
$208$	0	0
$209$	−493005.	−0.780702
$210$	0	0
$211$	−119677.	−0.185057	−0.0925283	−	0.995710i	$-0.529495\pi$
$211$	−119677.	−0.185057	−0.0925283	+	0.995710i	$0.529495\pi$
$212$	0	0
$213$	367469.	0.554974
$214$	0	0
$215$	−418741.	−0.617802
$216$	0	0
$217$	258536.	0.372710
$218$	0	0
$219$	821560.	1.15752
$220$	0	0
$221$	−1.28210e6	−1.76580
$222$	0	0
$223$	−1.14048e6	−1.53577	−0.767887	−	0.640585i	$-0.778692\pi$
$223$	−1.14048e6	−1.53577	−0.767887	+	0.640585i	$0.778692\pi$
$224$	0	0
$225$	108085.	0.142335
$226$	0	0
$227$	854276.	1.10036	0.550178	−	0.835047i	$-0.314560\pi$
$227$	854276.	1.10036	0.550178	+	0.835047i	$0.314560\pi$
$228$	0	0
$229$	−988016.	−1.24502	−0.622509	−	0.782613i	$-0.713886\pi$
$229$	−988016.	−1.24502	−0.622509	+	0.782613i	$0.713886\pi$
$230$	0	0
$231$	632187.	0.779499
$232$	0	0
$233$	95831.3	0.115642	0.0578212	−	0.998327i	$-0.481585\pi$
$233$	95831.3	0.115642	0.0578212	+	0.998327i	$0.481585\pi$
$234$	0	0
$235$	629676.	0.743785
$236$	0	0
$237$	556171.	0.643188
$238$	0	0
$239$	−188313.	−0.213248	−0.106624	−	0.994299i	$-0.534004\pi$
$239$	−188313.	−0.213248	−0.106624	+	0.994299i	$0.534004\pi$
$240$	0	0
$241$	−134729.	−0.149424	−0.0747120	−	0.997205i	$-0.523804\pi$
$241$	−134729.	−0.149424	−0.0747120	+	0.997205i	$0.523804\pi$
$242$	0	0
$243$	−1.10416e6	−1.19955
$244$	0	0
$245$	−60025.0	−0.0638877
$246$	0	0
$247$	864415.	0.901530
$248$	0	0
$249$	−644789.	−0.659051
$250$	0	0
$251$	−1.22709e6	−1.22939	−0.614696	−	0.788764i	$-0.710721\pi$
$251$	−1.22709e6	−1.22939	−0.614696	+	0.788764i	$0.710721\pi$
$252$	0	0
$253$	1.52633e6	1.49916
$254$	0	0
$255$	589343.	0.567568
$256$	0	0
$257$	904925.	0.854633	0.427317	−	0.904102i	$-0.359459\pi$
$257$	904925.	0.854633	0.427317	+	0.904102i	$0.359459\pi$
$258$	0	0
$259$	518902.	0.480658
$260$	0	0
$261$	−61996.3	−0.0563332
$262$	0	0
$263$	−416552.	−0.371347	−0.185673	−	0.982612i	$-0.559447\pi$
$263$	−416552.	−0.371347	−0.185673	+	0.982612i	$0.559447\pi$
$264$	0	0
$265$	408139.	0.357021
$266$	0	0
$267$	1.02910e6	0.883444
$268$	0	0
$269$	−1.04121e6	−0.877315	−0.438658	−	0.898654i	$-0.644546\pi$
$269$	−1.04121e6	−0.877315	−0.438658	+	0.898654i	$0.644546\pi$
$270$	0	0
$271$	1.43797e6	1.18939	0.594697	−	0.803950i	$-0.297272\pi$
$271$	1.43797e6	1.18939	0.594697	+	0.803950i	$0.297272\pi$
$272$	0	0
$273$	−1.10845e6	−0.900141
$274$	0	0
$275$	−395381.	−0.315271
$276$	0	0
$277$	−144054.	−0.112804	−0.0564022	−	0.998408i	$-0.517963\pi$
$277$	−144054.	−0.112804	−0.0564022	+	0.998408i	$0.517963\pi$
$278$	0	0
$279$	−912454.	−0.701779
$280$	0	0
$281$	208074.	0.157200	0.0786001	−	0.996906i	$-0.474955\pi$
$281$	208074.	0.157200	0.0786001	+	0.996906i	$0.474955\pi$
$282$	0	0
$283$	1.15156e6	0.854711	0.427355	−	0.904084i	$-0.359445\pi$
$283$	1.15156e6	0.854711	0.427355	+	0.904084i	$0.359445\pi$
$284$	0	0
$285$	−397346.	−0.289772
$286$	0	0
$287$	951230.	0.681680
$288$	0	0
$289$	−83786.2	−0.0590103
$290$	0	0
$291$	−1.34440e6	−0.930669
$292$	0	0
$293$	−1.19779e6	−0.815102	−0.407551	−	0.913183i	$-0.633617\pi$
$293$	−1.19779e6	−0.815102	−0.407551	+	0.913183i	$0.633617\pi$
$294$	0	0
$295$	426333.	0.285229
$296$	0	0
$297$	903942.	0.594634
$298$	0	0
$299$	−2.67621e6	−1.73118
$300$	0	0
$301$	−820733.	−0.522138
$302$	0	0
$303$	−2.66385e6	−1.66688
$304$	0	0
$305$	−1.35751e6	−0.835588
$306$	0	0
$307$	1.41795e6	0.858649	0.429325	−	0.903150i	$-0.358752\pi$
$307$	1.41795e6	0.858649	0.429325	+	0.903150i	$0.358752\pi$
$308$	0	0
$309$	−4.25368e6	−2.53436
$310$	0	0
$311$	2.39168e6	1.40217	0.701086	−	0.713076i	$-0.252699\pi$
$311$	2.39168e6	1.40217	0.701086	+	0.713076i	$0.252699\pi$
$312$	0	0
$313$	2.21255e6	1.27653	0.638266	−	0.769816i	$-0.279652\pi$
$313$	2.21255e6	1.27653	0.638266	+	0.769816i	$0.279652\pi$
$314$	0	0
$315$	211847.	0.120295
$316$	0	0
$317$	−568907.	−0.317975	−0.158988	−	0.987281i	$-0.550823\pi$
$317$	−568907.	−0.317975	−0.158988	+	0.987281i	$0.550823\pi$
$318$	0	0
$319$	226785.	0.124778
$320$	0	0
$321$	1.11255e6	0.602639
$322$	0	0
$323$	−900803.	−0.480423
$324$	0	0
$325$	693246.	0.364065
$326$	0	0
$327$	−2.98675e6	−1.54465
$328$	0	0
$329$	1.23417e6	0.628613
$330$	0	0
$331$	−244368.	−0.122596	−0.0612978	−	0.998120i	$-0.519524\pi$
$331$	−244368.	−0.122596	−0.0612978	+	0.998120i	$0.519524\pi$
$332$	0	0
$333$	−1.83137e6	−0.905036
$334$	0	0
$335$	−131228.	−0.0638871
$336$	0	0
$337$	−1.19959e6	−0.575387	−0.287693	−	0.957723i	$-0.592888\pi$
$337$	−1.19959e6	−0.575387	−0.287693	+	0.957723i	$0.592888\pi$
$338$	0	0
$339$	2.90432e6	1.37261
$340$	0	0
$341$	3.33780e6	1.55444
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	1.23017e6	0.556439
$346$	0	0
$347$	3.05183e6	1.36062	0.680311	−	0.732924i	$-0.261845\pi$
$347$	3.05183e6	1.36062	0.680311	+	0.732924i	$0.261845\pi$
$348$	0	0
$349$	−639914.	−0.281228	−0.140614	−	0.990065i	$-0.544908\pi$
$349$	−639914.	−0.281228	−0.140614	+	0.990065i	$0.544908\pi$
$350$	0	0
$351$	−1.58494e6	−0.686664
$352$	0	0
$353$	1.38572e6	0.591888	0.295944	−	0.955205i	$-0.404366\pi$
$353$	1.38572e6	0.591888	0.295944	+	0.955205i	$0.404366\pi$
$354$	0	0
$355$	−450451.	−0.189704
$356$	0	0
$357$	1.15511e6	0.479682
$358$	0	0
$359$	−130036.	−0.0532510	−0.0266255	−	0.999645i	$-0.508476\pi$
$359$	−130036.	−0.0532510	−0.0266255	+	0.999645i	$0.508476\pi$
$360$	0	0
$361$	−1.86876e6	−0.754720
$362$	0	0
$363$	4.87724e6	1.94271
$364$	0	0
$365$	−1.00708e6	−0.395671
$366$	0	0
$367$	−4.00901e6	−1.55372	−0.776859	−	0.629674i	$-0.783188\pi$
$367$	−4.00901e6	−1.55372	−0.776859	+	0.629674i	$0.783188\pi$
$368$	0	0
$369$	−3.35719e6	−1.28354
$370$	0	0
$371$	799953.	0.301738
$372$	0	0
$373$	3.25700e6	1.21212	0.606060	−	0.795419i	$-0.292749\pi$
$373$	3.25700e6	1.21212	0.606060	+	0.795419i	$0.292749\pi$
$374$	0	0
$375$	−318664.	−0.117019
$376$	0	0
$377$	−397637.	−0.144090
$378$	0	0
$379$	−5.35428e6	−1.91471	−0.957355	−	0.288913i	$-0.906706\pi$
$379$	−5.35428e6	−1.91471	−0.957355	+	0.288913i	$0.906706\pi$
$380$	0	0
$381$	−5.34621e6	−1.88683
$382$	0	0
$383$	−3.93926e6	−1.37220	−0.686100	−	0.727507i	$-0.740679\pi$
$383$	−3.93926e6	−1.37220	−0.686100	+	0.727507i	$0.740679\pi$
$384$	0	0
$385$	−774947.	−0.266453
$386$	0	0
$387$	2.89663e6	0.983139
$388$	0	0
$389$	−5.19770e6	−1.74156	−0.870778	−	0.491676i	$-0.836384\pi$
$389$	−5.19770e6	−1.74156	−0.870778	+	0.491676i	$0.836384\pi$
$390$	0	0
$391$	2.78886e6	0.922539
$392$	0	0
$393$	4.54733e6	1.48517
$394$	0	0
$395$	−681766.	−0.219858
$396$	0	0
$397$	4.20809e6	1.34001	0.670006	−	0.742356i	$-0.266292\pi$
$397$	4.20809e6	1.34001	0.670006	+	0.742356i	$0.266292\pi$
$398$	0	0
$399$	−778798.	−0.244902
$400$	0	0
$401$	−3.64863e6	−1.13310	−0.566551	−	0.824027i	$-0.691723\pi$
$401$	−3.64863e6	−1.13310	−0.566551	+	0.824027i	$0.691723\pi$
$402$	0	0
$403$	−5.85237e6	−1.79502
$404$	0	0
$405$	1.77914e6	0.538979
$406$	0	0
$407$	6.69924e6	2.00465
$408$	0	0
$409$	−3.31222e6	−0.979065	−0.489532	−	0.871985i	$-0.662833\pi$
$409$	−3.31222e6	−0.979065	−0.489532	+	0.871985i	$0.662833\pi$
$410$	0	0
$411$	−2.08642e6	−0.609251
$412$	0	0
$413$	835613.	0.241063
$414$	0	0
$415$	790395.	0.225281
$416$	0	0
$417$	3.75975e6	1.05881
$418$	0	0
$419$	−1.80415e6	−0.502039	−0.251019	−	0.967982i	$-0.580766\pi$
$419$	−1.80415e6	−0.502039	−0.251019	+	0.967982i	$0.580766\pi$
$420$	0	0
$421$	1.67987e6	0.461925	0.230963	−	0.972963i	$-0.425812\pi$
$421$	1.67987e6	0.461925	0.230963	+	0.972963i	$0.425812\pi$
$422$	0	0
$423$	−4.35576e6	−1.18362
$424$	0	0
$425$	−722428.	−0.194009
$426$	0	0
$427$	−2.66071e6	−0.706200
$428$	0	0
$429$	−1.43106e7	−3.75417
$430$	0	0
$431$	1.74106e6	0.451460	0.225730	−	0.974190i	$-0.427523\pi$
$431$	1.74106e6	0.451460	0.225730	+	0.974190i	$0.427523\pi$
$432$	0	0
$433$	5.78190e6	1.48201	0.741005	−	0.671500i	$-0.234350\pi$
$433$	5.78190e6	1.48201	0.741005	+	0.671500i	$0.234350\pi$
$434$	0	0
$435$	182782.	0.0463137
$436$	0	0
$437$	−1.88030e6	−0.471003
$438$	0	0
$439$	−4.72668e6	−1.17056	−0.585281	−	0.810830i	$-0.699016\pi$
$439$	−4.72668e6	−1.17056	−0.585281	+	0.810830i	$0.699016\pi$
$440$	0	0
$441$	415221.	0.101668
$442$	0	0
$443$	−392962.	−0.0951353	−0.0475676	−	0.998868i	$-0.515147\pi$
$443$	−392962.	−0.0951353	−0.0475676	+	0.998868i	$0.515147\pi$
$444$	0	0
$445$	−1.26149e6	−0.301984
$446$	0	0
$447$	8.62735e6	2.04225
$448$	0	0
$449$	4.80512e6	1.12483	0.562417	−	0.826853i	$-0.309871\pi$
$449$	4.80512e6	1.12483	0.562417	+	0.826853i	$0.309871\pi$
$450$	0	0
$451$	1.22808e7	2.84305
$452$	0	0
$453$	−1.42571e6	−0.326427
$454$	0	0
$455$	1.35876e6	0.307691
$456$	0	0
$457$	−4.04805e6	−0.906684	−0.453342	−	0.891337i	$-0.649768\pi$
$457$	−4.04805e6	−0.906684	−0.453342	+	0.891337i	$0.649768\pi$
$458$	0	0
$459$	1.65166e6	0.365921
$460$	0	0
$461$	−662439.	−0.145176	−0.0725878	−	0.997362i	$-0.523126\pi$
$461$	−662439.	−0.145176	−0.0725878	+	0.997362i	$0.523126\pi$
$462$	0	0
$463$	663797.	0.143907	0.0719537	−	0.997408i	$-0.477077\pi$
$463$	663797.	0.143907	0.0719537	+	0.997408i	$0.477077\pi$
$464$	0	0
$465$	2.69016e6	0.576960
$466$	0	0
$467$	−4.34200e6	−0.921293	−0.460646	−	0.887584i	$-0.652382\pi$
$467$	−4.34200e6	−0.921293	−0.460646	+	0.887584i	$0.652382\pi$
$468$	0	0
$469$	−257206.	−0.0539945
$470$	0	0
$471$	−2.94634e6	−0.611971
$472$	0	0
$473$	−1.05960e7	−2.17765
$474$	0	0
$475$	487074.	0.0990515
$476$	0	0
$477$	−2.82329e6	−0.568145
$478$	0	0
$479$	325224.	0.0647655	0.0323827	−	0.999476i	$-0.489690\pi$
$479$	325224.	0.0647655	0.0323827	+	0.999476i	$0.489690\pi$
$480$	0	0
$481$	−1.17462e7	−2.31491
$482$	0	0
$483$	2.41114e6	0.470277
$484$	0	0
$485$	1.64799e6	0.318126
$486$	0	0
$487$	−5.33595e6	−1.01951	−0.509753	−	0.860321i	$-0.670263\pi$
$487$	−5.33595e6	−1.01951	−0.509753	+	0.860321i	$0.670263\pi$
$488$	0	0
$489$	−1.27726e6	−0.241551
$490$	0	0
$491$	−5.72386e6	−1.07148	−0.535742	−	0.844382i	$-0.679968\pi$
$491$	−5.72386e6	−1.07148	−0.535742	+	0.844382i	$0.679968\pi$
$492$	0	0
$493$	414375.	0.0767850
$494$	0	0
$495$	2.73504e6	0.501707
$496$	0	0
$497$	−882884.	−0.160329
$498$	0	0
$499$	−2.21540e6	−0.398291	−0.199146	−	0.979970i	$-0.563817\pi$
$499$	−2.21540e6	−0.398291	−0.199146	+	0.979970i	$0.563817\pi$
$500$	0	0
$501$	−954296.	−0.169859
$502$	0	0
$503$	3.18589e6	0.561449	0.280725	−	0.959788i	$-0.409425\pi$
$503$	3.18589e6	0.561449	0.280725	+	0.959788i	$0.409425\pi$
$504$	0	0
$505$	3.26540e6	0.569782
$506$	0	0
$507$	1.75193e7	3.02689
$508$	0	0
$509$	4.01850e6	0.687495	0.343748	−	0.939062i	$-0.388303\pi$
$509$	4.01850e6	0.687495	0.343748	+	0.939062i	$0.388303\pi$
$510$	0	0
$511$	−1.97389e6	−0.334403
$512$	0	0
$513$	−1.11358e6	−0.186821
$514$	0	0
$515$	5.21425e6	0.866310
$516$	0	0
$517$	1.59336e7	2.62172
$518$	0	0
$519$	1.29484e7	2.11008
$520$	0	0
$521$	8.72499e6	1.40822	0.704110	−	0.710091i	$-0.251346\pi$
$521$	8.72499e6	1.40822	0.704110	+	0.710091i	$0.251346\pi$
$522$	0	0
$523$	−5.80567e6	−0.928107	−0.464053	−	0.885807i	$-0.653605\pi$
$523$	−5.80567e6	−0.928107	−0.464053	+	0.885807i	$0.653605\pi$
$524$	0	0
$525$	−624582.	−0.0988989
$526$	0	0
$527$	6.09872e6	0.956561
$528$	0	0
$529$	−614989.	−0.0955494
$530$	0	0
$531$	−2.94915e6	−0.453900
$532$	0	0
$533$	−2.15326e7	−3.28306
$534$	0	0
$535$	−1.36379e6	−0.205997
$536$	0	0
$537$	8.15596e6	1.22051
$538$	0	0
$539$	−1.51890e6	−0.225194
$540$	0	0
$541$	−1.59061e6	−0.233652	−0.116826	−	0.993152i	$-0.537272\pi$
$541$	−1.59061e6	−0.233652	−0.116826	+	0.993152i	$0.537272\pi$
$542$	0	0
$543$	1.26198e7	1.83676
$544$	0	0
$545$	3.66122e6	0.528000
$546$	0	0
$547$	8.05870e6	1.15159	0.575793	−	0.817595i	$-0.304693\pi$
$547$	8.05870e6	1.15159	0.575793	+	0.817595i	$0.304693\pi$
$548$	0	0
$549$	9.39050e6	1.32971
$550$	0	0
$551$	−279379.	−0.0392026
$552$	0	0
$553$	−1.33626e6	−0.185814
$554$	0	0
$555$	5.39937e6	0.744064
$556$	0	0
$557$	−6.70306e6	−0.915451	−0.457725	−	0.889094i	$-0.651336\pi$
$557$	−6.70306e6	−0.915451	−0.457725	+	0.889094i	$0.651336\pi$
$558$	0	0
$559$	1.85786e7	2.51468
$560$	0	0
$561$	1.49130e7	2.00059
$562$	0	0
$563$	−1.17463e7	−1.56182	−0.780911	−	0.624642i	$-0.785245\pi$
$563$	−1.17463e7	−1.56182	−0.780911	+	0.624642i	$0.785245\pi$
$564$	0	0
$565$	−3.56017e6	−0.469192
$566$	0	0
$567$	3.48711e6	0.455521
$568$	0	0
$569$	−2.88121e6	−0.373074	−0.186537	−	0.982448i	$-0.559726\pi$
$569$	−2.88121e6	−0.373074	−0.186537	+	0.982448i	$0.559726\pi$
$570$	0	0
$571$	−7.47204e6	−0.959067	−0.479534	−	0.877524i	$-0.659194\pi$
$571$	−7.47204e6	−0.959067	−0.479534	+	0.877524i	$0.659194\pi$
$572$	0	0
$573$	4.83666e6	0.615402
$574$	0	0
$575$	−1.50797e6	−0.190205
$576$	0	0
$577$	1.39214e7	1.74077	0.870386	−	0.492369i	$-0.163869\pi$
$577$	1.39214e7	1.74077	0.870386	+	0.492369i	$0.163869\pi$
$578$	0	0
$579$	2.94909e6	0.365587
$580$	0	0
$581$	1.54917e6	0.190397
$582$	0	0
$583$	1.03277e7	1.25844
$584$	0	0
$585$	−4.79551e6	−0.579355
$586$	0	0
$587$	−1.40665e7	−1.68496	−0.842481	−	0.538725i	$-0.818906\pi$
$587$	−1.40665e7	−1.68496	−0.842481	+	0.538725i	$0.818906\pi$
$588$	0	0
$589$	−4.11187e6	−0.488372
$590$	0	0
$591$	1.87483e7	2.20798
$592$	0	0
$593$	7.10729e6	0.829979	0.414990	−	0.909826i	$-0.363785\pi$
$593$	7.10729e6	0.829979	0.414990	+	0.909826i	$0.363785\pi$
$594$	0	0
$595$	−1.41596e6	−0.163968
$596$	0	0
$597$	−1.61105e7	−1.85001
$598$	0	0
$599$	−7.80502e6	−0.888806	−0.444403	−	0.895827i	$-0.646584\pi$
$599$	−7.80502e6	−0.888806	−0.444403	+	0.895827i	$0.646584\pi$
$600$	0	0
$601$	1.03763e7	1.17181	0.585906	−	0.810379i	$-0.300739\pi$
$601$	1.03763e7	1.17181	0.585906	+	0.810379i	$0.300739\pi$
$602$	0	0
$603$	907763.	0.101667
$604$	0	0
$605$	−5.97861e6	−0.664067
$606$	0	0
$607$	443646.	0.0488725	0.0244363	−	0.999701i	$-0.492221\pi$
$607$	443646.	0.0488725	0.0244363	+	0.999701i	$0.492221\pi$
$608$	0	0
$609$	358252.	0.0391422
$610$	0	0
$611$	−2.79373e7	−3.02748
$612$	0	0
$613$	1.64309e6	0.176608	0.0883041	−	0.996094i	$-0.471855\pi$
$613$	1.64309e6	0.176608	0.0883041	+	0.996094i	$0.471855\pi$
$614$	0	0
$615$	9.89790e6	1.05525
$616$	0	0
$617$	1.17811e7	1.24587	0.622934	−	0.782275i	$-0.285941\pi$
$617$	1.17811e7	1.24587	0.622934	+	0.782275i	$0.285941\pi$
$618$	0	0
$619$	659942.	0.0692275	0.0346138	−	0.999401i	$-0.488980\pi$
$619$	659942.	0.0692275	0.0346138	+	0.999401i	$0.488980\pi$
$620$	0	0
$621$	3.44760e6	0.358747
$622$	0	0
$623$	−2.47252e6	−0.255223
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−1.00546e7	−1.02140
$628$	0	0
$629$	1.22406e7	1.23361
$630$	0	0
$631$	−4.51716e6	−0.451639	−0.225820	−	0.974169i	$-0.572506\pi$
$631$	−4.51716e6	−0.451639	−0.225820	+	0.974169i	$0.572506\pi$
$632$	0	0
$633$	−2.44076e6	−0.242111
$634$	0	0
$635$	6.55348e6	0.644968
$636$	0	0
$637$	2.66317e6	0.260047
$638$	0	0
$639$	3.11598e6	0.301886
$640$	0	0
$641$	1.98530e6	0.190845	0.0954225	−	0.995437i	$-0.469580\pi$
$641$	1.98530e6	0.190845	0.0954225	+	0.995437i	$0.469580\pi$
$642$	0	0
$643$	974149.	0.0929176	0.0464588	−	0.998920i	$-0.485206\pi$
$643$	974149.	0.0929176	0.0464588	+	0.998920i	$0.485206\pi$
$644$	0	0
$645$	−8.54002e6	−0.808276
$646$	0	0
$647$	6.79603e6	0.638255	0.319128	−	0.947712i	$-0.396610\pi$
$647$	6.79603e6	0.638255	0.319128	+	0.947712i	$0.396610\pi$
$648$	0	0
$649$	1.07881e7	1.00539
$650$	0	0
$651$	5.27271e6	0.487620
$652$	0	0
$653$	9.76075e6	0.895778	0.447889	−	0.894089i	$-0.352176\pi$
$653$	9.76075e6	0.895778	0.447889	+	0.894089i	$0.352176\pi$
$654$	0	0
$655$	−5.57420e6	−0.507668
$656$	0	0
$657$	6.96647e6	0.629650
$658$	0	0
$659$	−4.71931e6	−0.423316	−0.211658	−	0.977344i	$-0.567886\pi$
$659$	−4.71931e6	−0.423316	−0.211658	+	0.977344i	$0.567886\pi$
$660$	0	0
$661$	−6.93075e6	−0.616988	−0.308494	−	0.951226i	$-0.599825\pi$
$661$	−6.93075e6	−0.616988	−0.308494	+	0.951226i	$0.599825\pi$
$662$	0	0
$663$	−2.61478e7	−2.31021
$664$	0	0
$665$	954665.	0.0837138
$666$	0	0
$667$	864950.	0.0752794
$668$	0	0
$669$	−2.32596e7	−2.00927
$670$	0	0
$671$	−3.43509e7	−2.94531
$672$	0	0
$673$	−5.56141e6	−0.473311	−0.236656	−	0.971594i	$-0.576051\pi$
$673$	−5.56141e6	−0.473311	−0.236656	+	0.971594i	$0.576051\pi$
$674$	0	0
$675$	−893069.	−0.0754441
$676$	0	0
$677$	−2.08907e7	−1.75179	−0.875895	−	0.482502i	$-0.839728\pi$
$677$	−2.08907e7	−1.75179	−0.875895	+	0.482502i	$0.839728\pi$
$678$	0	0
$679$	3.23006e6	0.268866
$680$	0	0
$681$	1.74225e7	1.43961
$682$	0	0
$683$	−1.75263e7	−1.43760	−0.718802	−	0.695214i	$-0.755309\pi$
$683$	−1.75263e7	−1.43760	−0.718802	+	0.695214i	$0.755309\pi$
$684$	0	0
$685$	2.55757e6	0.208258
$686$	0	0
$687$	−2.01501e7	−1.62887
$688$	0	0
$689$	−1.81082e7	−1.45321
$690$	0	0
$691$	−6.12700e6	−0.488150	−0.244075	−	0.969756i	$-0.578484\pi$
$691$	−6.12700e6	−0.488150	−0.244075	+	0.969756i	$0.578484\pi$
$692$	0	0
$693$	5.36067e6	0.424020
$694$	0	0
$695$	−4.60878e6	−0.361929
$696$	0	0
$697$	2.24390e7	1.74953
$698$	0	0
$699$	1.95443e6	0.151296
$700$	0	0
$701$	2.93306e6	0.225438	0.112719	−	0.993627i	$-0.464044\pi$
$701$	2.93306e6	0.225438	0.112719	+	0.993627i	$0.464044\pi$
$702$	0	0
$703$	−8.25286e6	−0.629819
$704$	0	0
$705$	1.28419e7	0.973101
$706$	0	0
$707$	6.40019e6	0.481554
$708$	0	0
$709$	8.31984e6	0.621583	0.310792	−	0.950478i	$-0.399406\pi$
$709$	8.31984e6	0.621583	0.310792	+	0.950478i	$0.399406\pi$
$710$	0	0
$711$	4.71609e6	0.349871
$712$	0	0
$713$	1.27302e7	0.937805
$714$	0	0
$715$	1.75422e7	1.28327
$716$	0	0
$717$	−3.84056e6	−0.278995
$718$	0	0
$719$	2.64463e6	0.190785	0.0953923	−	0.995440i	$-0.469589\pi$
$719$	2.64463e6	0.190785	0.0953923	+	0.995440i	$0.469589\pi$
$720$	0	0
$721$	1.02199e7	0.732166
$722$	0	0
$723$	−2.74774e6	−0.195493
$724$	0	0
$725$	−224057.	−0.0158312
$726$	0	0
$727$	−1.90915e7	−1.33969	−0.669845	−	0.742501i	$-0.733639\pi$
$727$	−1.90915e7	−1.33969	−0.669845	+	0.742501i	$0.733639\pi$
$728$	0	0
$729$	−5.22563e6	−0.364183
$730$	0	0
$731$	−1.93607e7	−1.34007
$732$	0	0
$733$	−2.30530e7	−1.58477	−0.792387	−	0.610019i	$-0.791162\pi$
$733$	−2.30530e7	−1.58477	−0.792387	+	0.610019i	$0.791162\pi$
$734$	0	0
$735$	−1.22418e6	−0.0835848
$736$	0	0
$737$	−3.32064e6	−0.225192
$738$	0	0
$739$	−3.85005e6	−0.259331	−0.129666	−	0.991558i	$-0.541390\pi$
$739$	−3.85005e6	−0.259331	−0.129666	+	0.991558i	$0.541390\pi$
$740$	0	0
$741$	1.76293e7	1.17948
$742$	0	0
$743$	1.18655e7	0.788521	0.394260	−	0.918999i	$-0.371001\pi$
$743$	1.18655e7	0.788521	0.394260	+	0.918999i	$0.371001\pi$
$744$	0	0
$745$	−1.05756e7	−0.698093
$746$	0	0
$747$	−5.46753e6	−0.358500
$748$	0	0
$749$	−2.67302e6	−0.174099
$750$	0	0
$751$	−2.75505e7	−1.78250	−0.891252	−	0.453509i	$-0.850172\pi$
$751$	−2.75505e7	−1.78250	−0.891252	+	0.453509i	$0.850172\pi$
$752$	0	0
$753$	−2.50258e7	−1.60843
$754$	0	0
$755$	1.74766e6	0.111581
$756$	0	0
$757$	8.49848e6	0.539016	0.269508	−	0.962998i	$-0.413139\pi$
$757$	8.49848e6	0.539016	0.269508	+	0.962998i	$0.413139\pi$
$758$	0	0
$759$	3.11287e7	1.96136
$760$	0	0
$761$	8.07105e6	0.505206	0.252603	−	0.967570i	$-0.418713\pi$
$761$	8.07105e6	0.505206	0.252603	+	0.967570i	$0.418713\pi$
$762$	0	0
$763$	7.17598e6	0.446242
$764$	0	0
$765$	4.99737e6	0.308737
$766$	0	0
$767$	−1.89155e7	−1.16099
$768$	0	0
$769$	−2.21247e7	−1.34916	−0.674578	−	0.738203i	$-0.735674\pi$
$769$	−2.21247e7	−1.34916	−0.674578	+	0.738203i	$0.735674\pi$
$770$	0	0
$771$	1.84555e7	1.11812
$772$	0	0
$773$	9.25522e6	0.557106	0.278553	−	0.960421i	$-0.410145\pi$
$773$	9.25522e6	0.557106	0.278553	+	0.960421i	$0.410145\pi$
$774$	0	0
$775$	−3.29765e6	−0.197220
$776$	0	0
$777$	1.05828e7	0.628849
$778$	0	0
$779$	−1.51288e7	−0.893225
$780$	0	0
$781$	−1.13984e7	−0.668677
$782$	0	0
$783$	512252.	0.0298593
$784$	0	0
$785$	3.61168e6	0.209187
$786$	0	0
$787$	9.03615e6	0.520052	0.260026	−	0.965602i	$-0.416269\pi$
$787$	9.03615e6	0.520052	0.260026	+	0.965602i	$0.416269\pi$
$788$	0	0
$789$	−8.49538e6	−0.485836
$790$	0	0
$791$	−6.97794e6	−0.396539
$792$	0	0
$793$	6.02295e7	3.40115
$794$	0	0
$795$	8.32381e6	0.467094
$796$	0	0
$797$	−1.26959e7	−0.707975	−0.353987	−	0.935250i	$-0.615174\pi$
$797$	−1.26959e7	−0.707975	−0.353987	+	0.935250i	$0.615174\pi$
$798$	0	0
$799$	2.91133e7	1.61334
$800$	0	0
$801$	8.72631e6	0.480562
$802$	0	0
$803$	−2.54837e7	−1.39467
$804$	0	0
$805$	−2.95562e6	−0.160753
$806$	0	0
$807$	−2.12349e7	−1.14780
$808$	0	0
$809$	−1.36780e7	−0.734769	−0.367384	−	0.930069i	$-0.619747\pi$
$809$	−1.36780e7	−0.734769	−0.367384	+	0.930069i	$0.619747\pi$
$810$	0	0
$811$	−1.96935e6	−0.105141	−0.0525703	−	0.998617i	$-0.516741\pi$
$811$	−1.96935e6	−0.105141	−0.0525703	+	0.998617i	$0.516741\pi$
$812$	0	0
$813$	2.93267e7	1.55610
$814$	0	0
$815$	1.56569e6	0.0825682
$816$	0	0
$817$	1.30533e7	0.684172
$818$	0	0
$819$	−9.39919e6	−0.489644
$820$	0	0
$821$	−3.48282e7	−1.80332	−0.901660	−	0.432446i	$-0.857651\pi$
$821$	−3.48282e7	−1.80332	−0.901660	+	0.432446i	$0.857651\pi$
$822$	0	0
$823$	−3.16152e7	−1.62703	−0.813516	−	0.581542i	$-0.802450\pi$
$823$	−3.16152e7	−1.62703	−0.813516	+	0.581542i	$0.802450\pi$
$824$	0	0
$825$	−8.06361e6	−0.412472
$826$	0	0
$827$	7.29573e6	0.370941	0.185471	−	0.982650i	$-0.440619\pi$
$827$	7.29573e6	0.370941	0.185471	+	0.982650i	$0.440619\pi$
$828$	0	0
$829$	−5.21633e6	−0.263620	−0.131810	−	0.991275i	$-0.542079\pi$
$829$	−5.21633e6	−0.263620	−0.131810	+	0.991275i	$0.542079\pi$
$830$	0	0
$831$	−2.93791e6	−0.147583
$832$	0	0
$833$	−2.77528e6	−0.138578
$834$	0	0
$835$	1.16979e6	0.0580622
$836$	0	0
$837$	7.53927e6	0.371976
$838$	0	0
$839$	2.50299e7	1.22759	0.613795	−	0.789465i	$-0.289642\pi$
$839$	2.50299e7	1.22759	0.613795	+	0.789465i	$0.289642\pi$
$840$	0	0
$841$	−2.03826e7	−0.993734
$842$	0	0
$843$	4.24358e6	0.205666
$844$	0	0
$845$	−2.14755e7	−1.03467
$846$	0	0
$847$	−1.17181e7	−0.561239
$848$	0	0
$849$	2.34854e7	1.11823
$850$	0	0
$851$	2.55506e7	1.20942
$852$	0	0
$853$	−1.69595e7	−0.798070	−0.399035	−	0.916936i	$-0.630655\pi$
$853$	−1.69595e7	−0.798070	−0.399035	+	0.916936i	$0.630655\pi$
$854$	0	0
$855$	−3.36932e6	−0.157626
$856$	0	0
$857$	7.04659e6	0.327738	0.163869	−	0.986482i	$-0.447603\pi$
$857$	7.04659e6	0.327738	0.163869	+	0.986482i	$0.447603\pi$
$858$	0	0
$859$	−191541.	−0.00885686	−0.00442843	−	0.999990i	$-0.501410\pi$
$859$	−191541.	−0.00885686	−0.00442843	+	0.999990i	$0.501410\pi$
$860$	0	0
$861$	1.93999e7	0.891849
$862$	0	0
$863$	2.97436e7	1.35946	0.679731	−	0.733462i	$-0.262097\pi$
$863$	2.97436e7	1.35946	0.679731	+	0.733462i	$0.262097\pi$
$864$	0	0
$865$	−1.58725e7	−0.721280
$866$	0	0
$867$	−1.70878e6	−0.0772037
$868$	0	0
$869$	−1.72517e7	−0.774964
$870$	0	0
$871$	5.82228e6	0.260044
$872$	0	0
$873$	−1.13999e7	−0.506250
$874$	0	0
$875$	765625.	0.0338062
$876$	0	0
$877$	1.07355e7	0.471327	0.235664	−	0.971835i	$-0.424274\pi$
$877$	1.07355e7	0.471327	0.235664	+	0.971835i	$0.424274\pi$
$878$	0	0
$879$	−2.44284e7	−1.06640
$880$	0	0
$881$	−1.90794e7	−0.828181	−0.414091	−	0.910236i	$-0.635900\pi$
$881$	−1.90794e7	−0.828181	−0.414091	+	0.910236i	$0.635900\pi$
$882$	0	0
$883$	1.69784e7	0.732817	0.366408	−	0.930454i	$-0.380587\pi$
$883$	1.69784e7	0.732817	0.366408	+	0.930454i	$0.380587\pi$
$884$	0	0
$885$	8.69487e6	0.373168
$886$	0	0
$887$	2.40321e7	1.02561	0.512804	−	0.858505i	$-0.328607\pi$
$887$	2.40321e7	1.02561	0.512804	+	0.858505i	$0.328607\pi$
$888$	0	0
$889$	1.28448e7	0.545097
$890$	0	0
$891$	4.50200e7	1.89982
$892$	0	0
$893$	−1.96287e7	−0.823689
$894$	0	0
$895$	−9.99774e6	−0.417200
$896$	0	0
$897$	−5.45799e7	−2.26492
$898$	0	0
$899$	1.89149e6	0.0780556
$900$	0	0
$901$	1.88705e7	0.774410
$902$	0	0
$903$	−1.67384e7	−0.683118
$904$	0	0
$905$	−1.54696e7	−0.627852
$906$	0	0
$907$	−2.79088e7	−1.12648	−0.563239	−	0.826294i	$-0.690445\pi$
$907$	−2.79088e7	−1.12648	−0.563239	+	0.826294i	$0.690445\pi$
$908$	0	0
$909$	−2.25883e7	−0.906722
$910$	0	0
$911$	566158.	0.0226017	0.0113009	−	0.999936i	$-0.496403\pi$
$911$	566158.	0.0226017	0.0113009	+	0.999936i	$0.496403\pi$
$912$	0	0
$913$	2.00005e7	0.794078
$914$	0	0
$915$	−2.76857e7	−1.09321
$916$	0	0
$917$	−1.09254e7	−0.429058
$918$	0	0
$919$	−3.03617e7	−1.18587	−0.592936	−	0.805249i	$-0.702031\pi$
$919$	−3.03617e7	−1.18587	−0.592936	+	0.805249i	$0.702031\pi$
$920$	0	0
$921$	2.89185e7	1.12338
$922$	0	0
$923$	1.99855e7	0.772167
$924$	0	0
$925$	−6.61865e6	−0.254340
$926$	0	0
$927$	−3.60694e7	−1.37860
$928$	0	0
$929$	−7.08323e6	−0.269272	−0.134636	−	0.990895i	$-0.542987\pi$
$929$	−7.08323e6	−0.269272	−0.134636	+	0.990895i	$0.542987\pi$
$930$	0	0
$931$	1.87114e6	0.0707510
$932$	0	0
$933$	4.87771e7	1.83448
$934$	0	0
$935$	−1.82806e7	−0.683851
$936$	0	0
$937$	−2.84659e7	−1.05920	−0.529598	−	0.848249i	$-0.677657\pi$
$937$	−2.84659e7	−1.05920	−0.529598	+	0.848249i	$0.677657\pi$
$938$	0	0
$939$	4.51239e7	1.67010
$940$	0	0
$941$	1.22125e7	0.449605	0.224803	−	0.974404i	$-0.427826\pi$
$941$	1.22125e7	0.449605	0.224803	+	0.974404i	$0.427826\pi$
$942$	0	0
$943$	4.68383e7	1.71523
$944$	0	0
$945$	−1.75041e6	−0.0637619
$946$	0	0
$947$	−3.64909e7	−1.32224	−0.661119	−	0.750281i	$-0.729918\pi$
$947$	−3.64909e7	−1.32224	−0.661119	+	0.750281i	$0.729918\pi$
$948$	0	0
$949$	4.46821e7	1.61053
$950$	0	0
$951$	−1.16026e7	−0.416010
$952$	0	0
$953$	3.31535e7	1.18249	0.591245	−	0.806492i	$-0.298637\pi$
$953$	3.31535e7	1.18249	0.591245	+	0.806492i	$0.298637\pi$
$954$	0	0
$955$	−5.92887e6	−0.210360
$956$	0	0
$957$	4.62518e6	0.163248
$958$	0	0
$959$	5.01284e6	0.176010
$960$	0	0
$961$	−790481.	−0.0276111
$962$	0	0
$963$	9.43393e6	0.327814
$964$	0	0
$965$	−3.61505e6	−0.124967
$966$	0	0
$967$	2.87941e7	0.990234	0.495117	−	0.868826i	$-0.335125\pi$
$967$	2.87941e7	0.990234	0.495117	+	0.868826i	$0.335125\pi$
$968$	0	0
$969$	−1.83714e7	−0.628541
$970$	0	0
$971$	1.25890e7	0.428493	0.214246	−	0.976780i	$-0.431270\pi$
$971$	1.25890e7	0.428493	0.214246	+	0.976780i	$0.431270\pi$
$972$	0	0
$973$	−9.03321e6	−0.305886
$974$	0	0
$975$	1.41384e7	0.476310
$976$	0	0
$977$	−5.05401e7	−1.69395	−0.846974	−	0.531634i	$-0.821578\pi$
$977$	−5.05401e7	−1.69395	−0.846974	+	0.531634i	$0.821578\pi$
$978$	0	0
$979$	−3.19212e7	−1.06444
$980$	0	0
$981$	−2.53263e7	−0.840233
$982$	0	0
$983$	368018.	0.0121475	0.00607374	−	0.999982i	$-0.498067\pi$
$983$	368018.	0.0121475	0.00607374	+	0.999982i	$0.498067\pi$
$984$	0	0
$985$	−2.29821e7	−0.754743
$986$	0	0
$987$	2.51702e7	0.822421
$988$	0	0
$989$	−4.04127e7	−1.31379
$990$	0	0
$991$	−1.12626e7	−0.364295	−0.182147	−	0.983271i	$-0.558305\pi$
$991$	−1.12626e7	−0.364295	−0.182147	+	0.983271i	$0.558305\pi$
$992$	0	0
$993$	−4.98378e6	−0.160393
$994$	0	0
$995$	1.97486e7	0.632381
$996$	0	0
$997$	3.40284e7	1.08418	0.542092	−	0.840319i	$-0.317632\pi$
$997$	3.40284e7	1.08418	0.542092	+	0.840319i	$0.317632\pi$
$998$	0	0
$999$	1.51319e7	0.479712

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.6.a.h.1.4	✓	4
4.3	odd	2		560.6.a.w.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.6.a.h.1.4	✓	4	1.1	even	1	trivial
560.6.a.w.1.1		4	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)

\(f(q)\)	\(=\)	\(q+20.3945 q^{3} -25.0000 q^{5} -49.0000 q^{7} +172.937 q^{9} +O(q^{10})\)	\(q+20.3945 q^{3} -25.0000 q^{5} -49.0000 q^{7} +172.937 q^{9} -632.610 q^{11} +1109.19 q^{13} -509.863 q^{15} -1155.89 q^{17} +779.318 q^{19} -999.332 q^{21} -2412.75 q^{23} +625.000 q^{25} -1428.91 q^{27} -358.492 q^{29} -5276.24 q^{31} -12901.8 q^{33} +1225.00 q^{35} -10589.8 q^{37} +22621.5 q^{39} -19412.9 q^{41} +16749.6 q^{43} -4323.42 q^{45} -25187.0 q^{47} +2401.00 q^{49} -23573.7 q^{51} -16325.6 q^{53} +15815.2 q^{55} +15893.8 q^{57} -17053.3 q^{59} +54300.2 q^{61} -8473.89 q^{63} -27729.9 q^{65} +5249.11 q^{67} -49206.9 q^{69} +18018.0 q^{71} +40283.4 q^{73} +12746.6 q^{75} +30997.9 q^{77} +27270.6 q^{79} -71165.5 q^{81} -31615.8 q^{83} +28897.1 q^{85} -7311.27 q^{87} +50459.6 q^{89} -54350.5 q^{91} -107606. q^{93} -19483.0 q^{95} -65919.5 q^{97} -109401. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 5 q^{3} - 100 q^{5} - 196 q^{7} + 567 q^{9}+O(q^{10}) \) 4 * q + 5 * q^3 - 100 * q^5 - 196 * q^7 + 567 * q^9	\( 4 q + 5 q^{3} - 100 q^{5} - 196 q^{7} + 567 q^{9} - 123 q^{11} + 789 q^{13} - 125 q^{15} + 1051 q^{17} + 958 q^{19} - 245 q^{21} - 530 q^{23} + 2500 q^{25} - 3169 q^{27} - 5541 q^{29} - 10440 q^{31} - 3737 q^{33} + 4900 q^{35} - 24048 q^{37} - 9131 q^{39} - 35414 q^{41} + 2174 q^{43} - 14175 q^{45} - 2287 q^{47} + 9604 q^{49} - 54017 q^{51} - 22238 q^{53} + 3075 q^{55} + 5890 q^{57} - 22656 q^{59} - 20250 q^{61} - 27783 q^{63} - 19725 q^{65} - 19456 q^{67} - 47006 q^{69} - 72288 q^{71} + 59464 q^{73} + 3125 q^{75} + 6027 q^{77} - 232001 q^{79} - 149604 q^{81} - 169620 q^{83} - 26275 q^{85} - 270309 q^{87} + 141434 q^{89} - 38661 q^{91} - 203808 q^{93} - 23950 q^{95} + 167159 q^{97} - 292198 q^{99}+O(q^{100}) \) 4 * q + 5 * q^3 - 100 * q^5 - 196 * q^7 + 567 * q^9 - 123 * q^11 + 789 * q^13 - 125 * q^15 + 1051 * q^17 + 958 * q^19 - 245 * q^21 - 530 * q^23 + 2500 * q^25 - 3169 * q^27 - 5541 * q^29 - 10440 * q^31 - 3737 * q^33 + 4900 * q^35 - 24048 * q^37 - 9131 * q^39 - 35414 * q^41 + 2174 * q^43 - 14175 * q^45 - 2287 * q^47 + 9604 * q^49 - 54017 * q^51 - 22238 * q^53 + 3075 * q^55 + 5890 * q^57 - 22656 * q^59 - 20250 * q^61 - 27783 * q^63 - 19725 * q^65 - 19456 * q^67 - 47006 * q^69 - 72288 * q^71 + 59464 * q^73 + 3125 * q^75 + 6027 * q^77 - 232001 * q^79 - 149604 * q^81 - 169620 * q^83 - 26275 * q^85 - 270309 * q^87 + 141434 * q^89 - 38661 * q^91 - 203808 * q^93 - 23950 * q^95 + 167159 * q^97 - 292198 * q^99

Introduction

L-functions

Modular forms

Varieties

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