LMFDB - Embedded newform 280.6.a.j.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:	$0$
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} - 1151x^{3} - 5642x^{2} + 193596x + 1258056 $ x^5 - 1151x^3 - 5642x^2 + 193596*x + 1258056
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-13.5022$ 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)

$f(q)$	$=$	$q+16.5022 q^{3} +25.0000 q^{5} +49.0000 q^{7} +29.3221 q^{9} +O(q^{10})$	$q+16.5022 q^{3} +25.0000 q^{5} +49.0000 q^{7} +29.3221 q^{9} -680.908 q^{11} +866.540 q^{13} +412.555 q^{15} +606.270 q^{17} +1620.36 q^{19} +808.607 q^{21} +2283.18 q^{23} +625.000 q^{25} -3526.15 q^{27} +8766.51 q^{29} +5335.92 q^{31} -11236.5 q^{33} +1225.00 q^{35} -11911.6 q^{37} +14299.8 q^{39} +2914.26 q^{41} +7428.74 q^{43} +733.052 q^{45} +2772.69 q^{47} +2401.00 q^{49} +10004.8 q^{51} +31234.5 q^{53} -17022.7 q^{55} +26739.5 q^{57} -12252.3 q^{59} -27291.1 q^{61} +1436.78 q^{63} +21663.5 q^{65} -15060.8 q^{67} +37677.4 q^{69} +66287.7 q^{71} -50596.1 q^{73} +10313.9 q^{75} -33364.5 q^{77} -49498.2 q^{79} -65314.5 q^{81} -46403.7 q^{83} +15156.7 q^{85} +144667. q^{87} +84779.2 q^{89} +42460.5 q^{91} +88054.3 q^{93} +40509.1 q^{95} +59800.5 q^{97} -19965.6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 15 q^{3} + 125 q^{5} + 245 q^{7} + 1132 q^{9}+O(q^{10}) $ 5 * q + 15 * q^3 + 125 * q^5 + 245 * q^7 + 1132 * q^9	$ 5 q + 15 q^{3} + 125 q^{5} + 245 q^{7} + 1132 q^{9} + 281 q^{11} + 909 q^{13} + 375 q^{15} + 1495 q^{17} - 422 q^{19} + 735 q^{21} - 62 q^{23} + 3125 q^{25} - 3363 q^{27} - 2047 q^{29} + 1636 q^{31} + 19181 q^{33} + 6125 q^{35} - 10358 q^{37} + 15685 q^{39} + 6424 q^{41} + 28306 q^{43} + 28300 q^{45} + 20955 q^{47} + 12005 q^{49} - 23577 q^{51} + 43748 q^{53} + 7025 q^{55} + 13690 q^{57} + 45788 q^{59} + 50432 q^{61} + 55468 q^{63} + 22725 q^{65} + 40712 q^{67} + 35050 q^{69} - 3096 q^{71} + 135438 q^{73} + 9375 q^{75} + 13769 q^{77} + 13191 q^{79} + 381101 q^{81} + 35108 q^{83} + 37375 q^{85} + 297289 q^{87} + 213772 q^{89} + 44541 q^{91} + 134244 q^{93} - 10550 q^{95} + 10659 q^{97} + 39462 q^{99}+O(q^{100}) $ 5 * q + 15 * q^3 + 125 * q^5 + 245 * q^7 + 1132 * q^9 + 281 * q^11 + 909 * q^13 + 375 * q^15 + 1495 * q^17 - 422 * q^19 + 735 * q^21 - 62 * q^23 + 3125 * q^25 - 3363 * q^27 - 2047 * q^29 + 1636 * q^31 + 19181 * q^33 + 6125 * q^35 - 10358 * q^37 + 15685 * q^39 + 6424 * q^41 + 28306 * q^43 + 28300 * q^45 + 20955 * q^47 + 12005 * q^49 - 23577 * q^51 + 43748 * q^53 + 7025 * q^55 + 13690 * q^57 + 45788 * q^59 + 50432 * q^61 + 55468 * q^63 + 22725 * q^65 + 40712 * q^67 + 35050 * q^69 - 3096 * q^71 + 135438 * q^73 + 9375 * q^75 + 13769 * q^77 + 13191 * q^79 + 381101 * q^81 + 35108 * q^83 + 37375 * q^85 + 297289 * q^87 + 213772 * q^89 + 44541 * q^91 + 134244 * q^93 - 10550 * q^95 + 10659 * q^97 + 39462 * 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
$3$	16.5022	1.05862	0.529308	−	0.848430i	$-0.322452\pi$
$3$	16.5022	1.05862	0.529308	+	0.848430i	$0.322452\pi$
$4$	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
$9$	29.3221	0.120667
$10$	0	0
$11$	−680.908	−1.69671	−0.848354	−	0.529430i	$-0.822406\pi$
$11$	−680.908	−1.69671	−0.848354	+	0.529430i	$0.822406\pi$
$12$	0	0
$13$	866.540	1.42210	0.711050	−	0.703141i	$-0.248220\pi$
$13$	866.540	1.42210	0.711050	+	0.703141i	$0.248220\pi$
$14$	0	0
$15$	412.555	0.473427
$16$	0	0
$17$	606.270	0.508796	0.254398	−	0.967100i	$-0.418123\pi$
$17$	606.270	0.508796	0.254398	+	0.967100i	$0.418123\pi$
$18$	0	0
$19$	1620.36	1.02974	0.514871	−	0.857268i	$-0.327840\pi$
$19$	1620.36	1.02974	0.514871	+	0.857268i	$0.327840\pi$
$20$	0	0
$21$	808.607	0.400119
$22$	0	0
$23$	2283.18	0.899954	0.449977	−	0.893040i	$-0.351432\pi$
$23$	2283.18	0.899954	0.449977	+	0.893040i	$0.351432\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	−3526.15	−0.930876
$28$	0	0
$29$	8766.51	1.93567	0.967836	−	0.251582i	$-0.0809507\pi$
$29$	8766.51	1.93567	0.967836	+	0.251582i	$0.0809507\pi$
$30$	0	0
$31$	5335.92	0.997251	0.498626	−	0.866817i	$-0.333838\pi$
$31$	5335.92	0.997251	0.498626	+	0.866817i	$0.333838\pi$
$32$	0	0
$33$	−11236.5	−1.79616
$34$	0	0
$35$	1225.00	0.169031
$36$	0	0
$37$	−11911.6	−1.43043	−0.715216	−	0.698903i	$-0.753672\pi$
$37$	−11911.6	−1.43043	−0.715216	+	0.698903i	$0.753672\pi$
$38$	0	0
$39$	14299.8	1.50546
$40$	0	0
$41$	2914.26	0.270750	0.135375	−	0.990794i	$-0.456776\pi$
$41$	2914.26	0.270750	0.135375	+	0.990794i	$0.456776\pi$
$42$	0	0
$43$	7428.74	0.612695	0.306347	−	0.951920i	$-0.400893\pi$
$43$	7428.74	0.612695	0.306347	+	0.951920i	$0.400893\pi$
$44$	0	0
$45$	733.052	0.0539639
$46$	0	0
$47$	2772.69	0.183087	0.0915434	−	0.995801i	$-0.470820\pi$
$47$	2772.69	0.183087	0.0915434	+	0.995801i	$0.470820\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	10004.8	0.538619
$52$	0	0
$53$	31234.5	1.52737	0.763685	−	0.645589i	$-0.223388\pi$
$53$	31234.5	1.52737	0.763685	+	0.645589i	$0.223388\pi$
$54$	0	0
$55$	−17022.7	−0.758791
$56$	0	0
$57$	26739.5	1.09010
$58$	0	0
$59$	−12252.3	−0.458236	−0.229118	−	0.973399i	$-0.573584\pi$
$59$	−12252.3	−0.458236	−0.229118	+	0.973399i	$0.573584\pi$
$60$	0	0
$61$	−27291.1	−0.939066	−0.469533	−	0.882915i	$-0.655578\pi$
$61$	−27291.1	−0.939066	−0.469533	+	0.882915i	$0.655578\pi$
$62$	0	0
$63$	1436.78	0.0456078
$64$	0	0
$65$	21663.5	0.635982
$66$	0	0
$67$	−15060.8	−0.409884	−0.204942	−	0.978774i	$-0.565701\pi$
$67$	−15060.8	−0.409884	−0.204942	+	0.978774i	$0.565701\pi$
$68$	0	0
$69$	37677.4	0.952705
$70$	0	0
$71$	66287.7	1.56058	0.780291	−	0.625416i	$-0.215071\pi$
$71$	66287.7	1.56058	0.780291	+	0.625416i	$0.215071\pi$
$72$	0	0
$73$	−50596.1	−1.11125	−0.555623	−	0.831435i	$-0.687520\pi$
$73$	−50596.1	−1.11125	−0.555623	+	0.831435i	$0.687520\pi$
$74$	0	0
$75$	10313.9	0.211723
$76$	0	0
$77$	−33364.5	−0.641295
$78$	0	0
$79$	−49498.2	−0.892322	−0.446161	−	0.894953i	$-0.647209\pi$
$79$	−49498.2	−0.892322	−0.446161	+	0.894953i	$0.647209\pi$
$80$	0	0
$81$	−65314.5	−1.10611
$82$	0	0
$83$	−46403.7	−0.739362	−0.369681	−	0.929159i	$-0.620533\pi$
$83$	−46403.7	−0.739362	−0.369681	+	0.929159i	$0.620533\pi$
$84$	0	0
$85$	15156.7	0.227540
$86$	0	0
$87$	144667.	2.04913
$88$	0	0
$89$	84779.2	1.13453	0.567263	−	0.823537i	$-0.308002\pi$
$89$	84779.2	1.13453	0.567263	+	0.823537i	$0.308002\pi$
$90$	0	0
$91$	42460.5	0.537503
$92$	0	0
$93$	88054.3	1.05571
$94$	0	0
$95$	40509.1	0.460514
$96$	0	0
$97$	59800.5	0.645321	0.322660	−	0.946515i	$-0.395423\pi$
$97$	59800.5	0.645321	0.322660	+	0.946515i	$0.395423\pi$
$98$	0	0
$99$	−19965.6	−0.204736
$100$	0	0
$101$	128944.	1.25776	0.628882	−	0.777501i	$-0.283513\pi$
$101$	128944.	1.25776	0.628882	+	0.777501i	$0.283513\pi$
$102$	0	0
$103$	74553.1	0.692425	0.346213	−	0.938156i	$-0.387468\pi$
$103$	74553.1	0.692425	0.346213	+	0.938156i	$0.387468\pi$
$104$	0	0
$105$	20215.2	0.178939
$106$	0	0
$107$	67856.2	0.572968	0.286484	−	0.958085i	$-0.407514\pi$
$107$	67856.2	0.572968	0.286484	+	0.958085i	$0.407514\pi$
$108$	0	0
$109$	35684.6	0.287683	0.143841	−	0.989601i	$-0.454054\pi$
$109$	35684.6	0.287683	0.143841	+	0.989601i	$0.454054\pi$
$110$	0	0
$111$	−196568.	−1.51428
$112$	0	0
$113$	−182267.	−1.34280	−0.671400	−	0.741096i	$-0.734307\pi$
$113$	−182267.	−1.34280	−0.671400	+	0.741096i	$0.734307\pi$
$114$	0	0
$115$	57079.5	0.402472
$116$	0	0
$117$	25408.7	0.171600
$118$	0	0
$119$	29707.2	0.192307
$120$	0	0
$121$	302585.	1.87882
$122$	0	0
$123$	48091.7	0.286621
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	−137290.	−0.755316	−0.377658	−	0.925945i	$-0.623271\pi$
$127$	−137290.	−0.755316	−0.377658	+	0.925945i	$0.623271\pi$
$128$	0	0
$129$	122590.	0.648608
$130$	0	0
$131$	−223764.	−1.13923	−0.569615	−	0.821912i	$-0.692908\pi$
$131$	−223764.	−1.13923	−0.569615	+	0.821912i	$0.692908\pi$
$132$	0	0
$133$	79397.8	0.389206
$134$	0	0
$135$	−88153.8	−0.416300
$136$	0	0
$137$	312697.	1.42338	0.711692	−	0.702492i	$-0.247929\pi$
$137$	312697.	1.42338	0.711692	+	0.702492i	$0.247929\pi$
$138$	0	0
$139$	126363.	0.554730	0.277365	−	0.960765i	$-0.410539\pi$
$139$	126363.	0.554730	0.277365	+	0.960765i	$0.410539\pi$
$140$	0	0
$141$	45755.5	0.193818
$142$	0	0
$143$	−590034.	−2.41289
$144$	0	0
$145$	219163.	0.865659
$146$	0	0
$147$	39621.7	0.151231
$148$	0	0
$149$	489964.	1.80800	0.903999	−	0.427534i	$-0.140618\pi$
$149$	489964.	1.80800	0.903999	+	0.427534i	$0.140618\pi$
$150$	0	0
$151$	−532572.	−1.90080	−0.950399	−	0.311035i	$-0.899324\pi$
$151$	−532572.	−1.90080	−0.950399	+	0.311035i	$0.899324\pi$
$152$	0	0
$153$	17777.1	0.0613948
$154$	0	0
$155$	133398.	0.445984
$156$	0	0
$157$	−21397.4	−0.0692805	−0.0346403	−	0.999400i	$-0.511029\pi$
$157$	−21397.4	−0.0692805	−0.0346403	+	0.999400i	$0.511029\pi$
$158$	0	0
$159$	515437.	1.61690
$160$	0	0
$161$	111876.	0.340151
$162$	0	0
$163$	177362.	0.522867	0.261433	−	0.965221i	$-0.415805\pi$
$163$	177362.	0.522867	0.261433	+	0.965221i	$0.415805\pi$
$164$	0	0
$165$	−280912.	−0.803267
$166$	0	0
$167$	−626136.	−1.73731	−0.868655	−	0.495417i	$-0.835015\pi$
$167$	−626136.	−1.73731	−0.868655	+	0.495417i	$0.835015\pi$
$168$	0	0
$169$	379598.	1.02237
$170$	0	0
$171$	47512.4	0.124256
$172$	0	0
$173$	−639004.	−1.62326	−0.811631	−	0.584170i	$-0.801420\pi$
$173$	−639004.	−1.62326	−0.811631	+	0.584170i	$0.801420\pi$
$174$	0	0
$175$	30625.0	0.0755929
$176$	0	0
$177$	−202190.	−0.485096
$178$	0	0
$179$	−649943.	−1.51615	−0.758076	−	0.652166i	$-0.773860\pi$
$179$	−649943.	−1.51615	−0.758076	+	0.652166i	$0.773860\pi$
$180$	0	0
$181$	−331779.	−0.752752	−0.376376	−	0.926467i	$-0.622830\pi$
$181$	−331779.	−0.752752	−0.376376	+	0.926467i	$0.622830\pi$
$182$	0	0
$183$	−450363.	−0.994110
$184$	0	0
$185$	−297791.	−0.639709
$186$	0	0
$187$	−412814.	−0.863277
$188$	0	0
$189$	−172781.	−0.351838
$190$	0	0
$191$	−795439.	−1.57770	−0.788848	−	0.614588i	$-0.789322\pi$
$191$	−795439.	−1.57770	−0.788848	+	0.614588i	$0.789322\pi$
$192$	0	0
$193$	−671732.	−1.29808	−0.649042	−	0.760753i	$-0.724830\pi$
$193$	−671732.	−1.29808	−0.649042	+	0.760753i	$0.724830\pi$
$194$	0	0
$195$	357495.	0.673261
$196$	0	0
$197$	820163.	1.50569	0.752843	−	0.658200i	$-0.228682\pi$
$197$	820163.	1.50569	0.752843	+	0.658200i	$0.228682\pi$
$198$	0	0
$199$	−749336.	−1.34136	−0.670678	−	0.741749i	$-0.733997\pi$
$199$	−749336.	−1.34136	−0.670678	+	0.741749i	$0.733997\pi$
$200$	0	0
$201$	−248536.	−0.433909
$202$	0	0
$203$	429559.	0.731615
$204$	0	0
$205$	72856.6	0.121083
$206$	0	0
$207$	66947.5	0.108595
$208$	0	0
$209$	−1.10332e6	−1.74717
$210$	0	0
$211$	659330.	1.01952	0.509761	−	0.860316i	$-0.329734\pi$
$211$	659330.	1.01952	0.509761	+	0.860316i	$0.329734\pi$
$212$	0	0
$213$	1.09389e6	1.65206
$214$	0	0
$215$	185718.	0.274005
$216$	0	0
$217$	261460.	0.376926
$218$	0	0
$219$	−834946.	−1.17638
$220$	0	0
$221$	525357.	0.723559
$222$	0	0
$223$	−34108.0	−0.0459298	−0.0229649	−	0.999736i	$-0.507311\pi$
$223$	−34108.0	−0.0459298	−0.0229649	+	0.999736i	$0.507311\pi$
$224$	0	0
$225$	18326.3	0.0241334
$226$	0	0
$227$	−516684.	−0.665519	−0.332759	−	0.943012i	$-0.607980\pi$
$227$	−516684.	−0.665519	−0.332759	+	0.943012i	$0.607980\pi$
$228$	0	0
$229$	−528510.	−0.665985	−0.332992	−	0.942929i	$-0.608058\pi$
$229$	−528510.	−0.665985	−0.332992	+	0.942929i	$0.608058\pi$
$230$	0	0
$231$	−550587.	−0.678885
$232$	0	0
$233$	−205854.	−0.248410	−0.124205	−	0.992257i	$-0.539638\pi$
$233$	−205854.	−0.248410	−0.124205	+	0.992257i	$0.539638\pi$
$234$	0	0
$235$	69317.3	0.0818789
$236$	0	0
$237$	−816828.	−0.944626
$238$	0	0
$239$	687315.	0.778326	0.389163	−	0.921169i	$-0.372764\pi$
$239$	687315.	0.778326	0.389163	+	0.921169i	$0.372764\pi$
$240$	0	0
$241$	−240336.	−0.266548	−0.133274	−	0.991079i	$-0.542549\pi$
$241$	−240336.	−0.266548	−0.133274	+	0.991079i	$0.542549\pi$
$242$	0	0
$243$	−220976.	−0.240066
$244$	0	0
$245$	60025.0	0.0638877
$246$	0	0
$247$	1.40411e6	1.46440
$248$	0	0
$249$	−765762.	−0.782700
$250$	0	0
$251$	−172718.	−0.173043	−0.0865216	−	0.996250i	$-0.527575\pi$
$251$	−172718.	−0.173043	−0.0865216	+	0.996250i	$0.527575\pi$
$252$	0	0
$253$	−1.55464e6	−1.52696
$254$	0	0
$255$	250119.	0.240878
$256$	0	0
$257$	1.72963e6	1.63351	0.816755	−	0.576985i	$-0.195771\pi$
$257$	1.72963e6	1.63351	0.816755	+	0.576985i	$0.195771\pi$
$258$	0	0
$259$	−583671.	−0.540653
$260$	0	0
$261$	257052.	0.233572
$262$	0	0
$263$	−371469.	−0.331156	−0.165578	−	0.986197i	$-0.552949\pi$
$263$	−371469.	−0.331156	−0.165578	+	0.986197i	$0.552949\pi$
$264$	0	0
$265$	780862.	0.683061
$266$	0	0
$267$	1.39904e6	1.20103
$268$	0	0
$269$	−321370.	−0.270785	−0.135392	−	0.990792i	$-0.543230\pi$
$269$	−321370.	−0.270785	−0.135392	+	0.990792i	$0.543230\pi$
$270$	0	0
$271$	−1.32420e6	−1.09529	−0.547645	−	0.836711i	$-0.684475\pi$
$271$	−1.32420e6	−1.09529	−0.547645	+	0.836711i	$0.684475\pi$
$272$	0	0
$273$	700690.	0.569009
$274$	0	0
$275$	−425568.	−0.339341
$276$	0	0
$277$	−288242.	−0.225714	−0.112857	−	0.993611i	$-0.536000\pi$
$277$	−288242.	−0.225714	−0.112857	+	0.993611i	$0.536000\pi$
$278$	0	0
$279$	156460.	0.120335
$280$	0	0
$281$	1.42553e6	1.07698	0.538492	−	0.842631i	$-0.318994\pi$
$281$	1.42553e6	1.07698	0.538492	+	0.842631i	$0.318994\pi$
$282$	0	0
$283$	361.445	0.000268272 0	0.000134136	−	1.00000i	$-0.499957\pi$
$283$	361.445	0.000268272 0	0.000134136		1.00000i	$0.499957\pi$
$284$	0	0
$285$	668488.	0.487508
$286$	0	0
$287$	142799.	0.102334
$288$	0	0
$289$	−1.05229e6	−0.741127
$290$	0	0
$291$	986839.	0.683146
$292$	0	0
$293$	−2.55592e6	−1.73932	−0.869659	−	0.493653i	$-0.835661\pi$
$293$	−2.55592e6	−1.73932	−0.869659	+	0.493653i	$0.835661\pi$
$294$	0	0
$295$	−306308.	−0.204929
$296$	0	0
$297$	2.40099e6	1.57942
$298$	0	0
$299$	1.97847e6	1.27982
$300$	0	0
$301$	364008.	0.231577
$302$	0	0
$303$	2.12787e6	1.33149
$304$	0	0
$305$	−682277.	−0.419963
$306$	0	0
$307$	1.23699e6	0.749066	0.374533	−	0.927214i	$-0.377803\pi$
$307$	1.23699e6	0.749066	0.374533	+	0.927214i	$0.377803\pi$
$308$	0	0
$309$	1.23029e6	0.733012
$310$	0	0
$311$	2.23564e6	1.31070	0.655348	−	0.755327i	$-0.272522\pi$
$311$	2.23564e6	1.31070	0.655348	+	0.755327i	$0.272522\pi$
$312$	0	0
$313$	−862839.	−0.497816	−0.248908	−	0.968527i	$-0.580072\pi$
$313$	−862839.	−0.497816	−0.248908	+	0.968527i	$0.580072\pi$
$314$	0	0
$315$	35919.5	0.0203964
$316$	0	0
$317$	529699.	0.296061	0.148030	−	0.988983i	$-0.452707\pi$
$317$	529699.	0.296061	0.148030	+	0.988983i	$0.452707\pi$
$318$	0	0
$319$	−5.96919e6	−3.28427
$320$	0	0
$321$	1.11978e6	0.606552
$322$	0	0
$323$	982377.	0.523928
$324$	0	0
$325$	541587.	0.284420
$326$	0	0
$327$	588873.	0.304546
$328$	0	0
$329$	135862.	0.0692003
$330$	0	0
$331$	−13918.8	−0.00698286	−0.00349143	−	0.999994i	$-0.501111\pi$
$331$	−13918.8	−0.00698286	−0.00349143	+	0.999994i	$0.501111\pi$
$332$	0	0
$333$	−349274.	−0.172606
$334$	0	0
$335$	−376520.	−0.183306
$336$	0	0
$337$	−1.91582e6	−0.918925	−0.459462	−	0.888197i	$-0.651958\pi$
$337$	−1.91582e6	−0.918925	−0.459462	+	0.888197i	$0.651958\pi$
$338$	0	0
$339$	−3.00780e6	−1.42151
$340$	0	0
$341$	−3.63327e6	−1.69204
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	941936.	0.426063
$346$	0	0
$347$	−2.90607e6	−1.29564	−0.647818	−	0.761796i	$-0.724318\pi$
$347$	−2.90607e6	−1.29564	−0.647818	+	0.761796i	$0.724318\pi$
$348$	0	0
$349$	1.24413e6	0.546766	0.273383	−	0.961905i	$-0.411857\pi$
$349$	1.24413e6	0.546766	0.273383	+	0.961905i	$0.411857\pi$
$350$	0	0
$351$	−3.05555e6	−1.32380
$352$	0	0
$353$	1.92704e6	0.823102	0.411551	−	0.911387i	$-0.364987\pi$
$353$	1.92704e6	0.823102	0.411551	+	0.911387i	$0.364987\pi$
$354$	0	0
$355$	1.65719e6	0.697914
$356$	0	0
$357$	490234.	0.203579
$358$	0	0
$359$	−2.48838e6	−1.01901	−0.509507	−	0.860466i	$-0.670172\pi$
$359$	−2.48838e6	−1.01901	−0.509507	+	0.860466i	$0.670172\pi$
$360$	0	0
$361$	149477.	0.0603680
$362$	0	0
$363$	4.99331e6	1.98894
$364$	0	0
$365$	−1.26490e6	−0.496964
$366$	0	0
$367$	2.28578e6	0.885867	0.442933	−	0.896555i	$-0.353938\pi$
$367$	2.28578e6	0.885867	0.442933	+	0.896555i	$0.353938\pi$
$368$	0	0
$369$	85452.2	0.0326706
$370$	0	0
$371$	1.53049e6	0.577292
$372$	0	0
$373$	−44622.0	−0.0166065	−0.00830323	−	0.999966i	$-0.502643\pi$
$373$	−44622.0	−0.0166065	−0.00830323	+	0.999966i	$0.502643\pi$
$374$	0	0
$375$	257847.	0.0946855
$376$	0	0
$377$	7.59653e6	2.75272
$378$	0	0
$379$	−648644.	−0.231957	−0.115979	−	0.993252i	$-0.537000\pi$
$379$	−648644.	−0.231957	−0.115979	+	0.993252i	$0.537000\pi$
$380$	0	0
$381$	−2.26558e6	−0.799590
$382$	0	0
$383$	4.15511e6	1.44739	0.723694	−	0.690121i	$-0.242443\pi$
$383$	4.15511e6	1.44739	0.723694	+	0.690121i	$0.242443\pi$
$384$	0	0
$385$	−834113.	−0.286796
$386$	0	0
$387$	217826.	0.0739320
$388$	0	0
$389$	98002.4	0.0328369	0.0164185	−	0.999865i	$-0.494774\pi$
$389$	98002.4	0.0328369	0.0164185	+	0.999865i	$0.494774\pi$
$390$	0	0
$391$	1.38422e6	0.457893
$392$	0	0
$393$	−3.69259e6	−1.20601
$394$	0	0
$395$	−1.23746e6	−0.399059
$396$	0	0
$397$	−3.35237e6	−1.06752	−0.533760	−	0.845636i	$-0.679221\pi$
$397$	−3.35237e6	−1.06752	−0.533760	+	0.845636i	$0.679221\pi$
$398$	0	0
$399$	1.31024e6	0.412019
$400$	0	0
$401$	3.80919e6	1.18297	0.591483	−	0.806318i	$-0.298543\pi$
$401$	3.80919e6	1.18297	0.591483	+	0.806318i	$0.298543\pi$
$402$	0	0
$403$	4.62378e6	1.41819
$404$	0	0
$405$	−1.63286e6	−0.494666
$406$	0	0
$407$	8.11074e6	2.42703
$408$	0	0
$409$	1.04873e6	0.309997	0.154998	−	0.987915i	$-0.450463\pi$
$409$	1.04873e6	0.309997	0.154998	+	0.987915i	$0.450463\pi$
$410$	0	0
$411$	5.16018e6	1.50682
$412$	0	0
$413$	−600365.	−0.173197
$414$	0	0
$415$	−1.16009e6	−0.330653
$416$	0	0
$417$	2.08526e6	0.587246
$418$	0	0
$419$	−698800.	−0.194454	−0.0972272	−	0.995262i	$-0.530997\pi$
$419$	−698800.	−0.194454	−0.0972272	+	0.995262i	$0.530997\pi$
$420$	0	0
$421$	6.80725e6	1.87183	0.935915	−	0.352225i	$-0.114575\pi$
$421$	6.80725e6	1.87183	0.935915	+	0.352225i	$0.114575\pi$
$422$	0	0
$423$	81301.0	0.0220925
$424$	0	0
$425$	378919.	0.101759
$426$	0	0
$427$	−1.33726e6	−0.354934
$428$	0	0
$429$	−9.73685e6	−2.55432
$430$	0	0
$431$	−2.31322e6	−0.599824	−0.299912	−	0.953967i	$-0.596957\pi$
$431$	−2.31322e6	−0.599824	−0.299912	+	0.953967i	$0.596957\pi$
$432$	0	0
$433$	1.32191e6	0.338831	0.169415	−	0.985545i	$-0.445812\pi$
$433$	1.32191e6	0.338831	0.169415	+	0.985545i	$0.445812\pi$
$434$	0	0
$435$	3.61666e6	0.916400
$436$	0	0
$437$	3.69958e6	0.926720
$438$	0	0
$439$	−5.15407e6	−1.27641	−0.638203	−	0.769868i	$-0.720322\pi$
$439$	−5.15407e6	−1.27641	−0.638203	+	0.769868i	$0.720322\pi$
$440$	0	0
$441$	70402.3	0.0172381
$442$	0	0
$443$	−7.67953e6	−1.85920	−0.929599	−	0.368574i	$-0.879846\pi$
$443$	−7.67953e6	−1.85920	−0.929599	+	0.368574i	$0.879846\pi$
$444$	0	0
$445$	2.11948e6	0.507375
$446$	0	0
$447$	8.08547e6	1.91398
$448$	0	0
$449$	−4.60651e6	−1.07834	−0.539171	−	0.842196i	$-0.681262\pi$
$449$	−4.60651e6	−1.07834	−0.539171	+	0.842196i	$0.681262\pi$
$450$	0	0
$451$	−1.98435e6	−0.459384
$452$	0	0
$453$	−8.78860e6	−2.01221
$454$	0	0
$455$	1.06151e6	0.240379
$456$	0	0
$457$	3.05616e6	0.684520	0.342260	−	0.939605i	$-0.388808\pi$
$457$	3.05616e6	0.684520	0.342260	+	0.939605i	$0.388808\pi$
$458$	0	0
$459$	−2.13780e6	−0.473626
$460$	0	0
$461$	2.72536e6	0.597272	0.298636	−	0.954367i	$-0.403468\pi$
$461$	2.72536e6	0.597272	0.298636	+	0.954367i	$0.403468\pi$
$462$	0	0
$463$	3.54716e6	0.769004	0.384502	−	0.923124i	$-0.374373\pi$
$463$	3.54716e6	0.769004	0.384502	+	0.923124i	$0.374373\pi$
$464$	0	0
$465$	2.20136e6	0.472126
$466$	0	0
$467$	−59303.2	−0.0125830	−0.00629152	−	0.999980i	$-0.502003\pi$
$467$	−59303.2	−0.0125830	−0.00629152	+	0.999980i	$0.502003\pi$
$468$	0	0
$469$	−737979.	−0.154922
$470$	0	0
$471$	−353103.	−0.0733414
$472$	0	0
$473$	−5.05829e6	−1.03956
$474$	0	0
$475$	1.01273e6	0.205948
$476$	0	0
$477$	915859.	0.184303
$478$	0	0
$479$	7.60686e6	1.51484	0.757420	−	0.652928i	$-0.226460\pi$
$479$	7.60686e6	1.51484	0.757420	+	0.652928i	$0.226460\pi$
$480$	0	0
$481$	−1.03219e7	−2.03422
$482$	0	0
$483$	1.84619e6	0.360089
$484$	0	0
$485$	1.49501e6	0.288596
$486$	0	0
$487$	524068.	0.100130	0.0500651	−	0.998746i	$-0.484057\pi$
$487$	524068.	0.100130	0.0500651	+	0.998746i	$0.484057\pi$
$488$	0	0
$489$	2.92686e6	0.553515
$490$	0	0
$491$	−3.55551e6	−0.665577	−0.332788	−	0.943002i	$-0.607989\pi$
$491$	−3.55551e6	−0.665577	−0.332788	+	0.943002i	$0.607989\pi$
$492$	0	0
$493$	5.31487e6	0.984862
$494$	0	0
$495$	−499141.	−0.0915609
$496$	0	0
$497$	3.24810e6	0.589845
$498$	0	0
$499$	2.66175e6	0.478537	0.239269	−	0.970953i	$-0.423092\pi$
$499$	2.66175e6	0.478537	0.239269	+	0.970953i	$0.423092\pi$
$500$	0	0
$501$	−1.03326e7	−1.83914
$502$	0	0
$503$	−685370.	−0.120783	−0.0603914	−	0.998175i	$-0.519235\pi$
$503$	−685370.	−0.120783	−0.0603914	+	0.998175i	$0.519235\pi$
$504$	0	0
$505$	3.22361e6	0.562489
$506$	0	0
$507$	6.26420e6	1.08230
$508$	0	0
$509$	−7.32031e6	−1.25238	−0.626188	−	0.779672i	$-0.715386\pi$
$509$	−7.32031e6	−1.25238	−0.626188	+	0.779672i	$0.715386\pi$
$510$	0	0
$511$	−2.47921e6	−0.420011
$512$	0	0
$513$	−5.71365e6	−0.958562
$514$	0	0
$515$	1.86383e6	0.309662
$516$	0	0
$517$	−1.88795e6	−0.310645
$518$	0	0
$519$	−1.05450e7	−1.71841
$520$	0	0
$521$	−3.30683e6	−0.533725	−0.266862	−	0.963735i	$-0.585987\pi$
$521$	−3.30683e6	−0.533725	−0.266862	+	0.963735i	$0.585987\pi$
$522$	0	0
$523$	−1.12452e7	−1.79768	−0.898841	−	0.438275i	$-0.855590\pi$
$523$	−1.12452e7	−1.79768	−0.898841	+	0.438275i	$0.855590\pi$
$524$	0	0
$525$	505379.	0.0800238
$526$	0	0
$527$	3.23500e6	0.507397
$528$	0	0
$529$	−1.22344e6	−0.190083
$530$	0	0
$531$	−359264.	−0.0552939
$532$	0	0
$533$	2.52533e6	0.385034
$534$	0	0
$535$	1.69640e6	0.256239
$536$	0	0
$537$	−1.07255e7	−1.60502
$538$	0	0
$539$	−1.63486e6	−0.242387
$540$	0	0
$541$	−1.23358e6	−0.181206	−0.0906032	−	0.995887i	$-0.528880\pi$
$541$	−1.23358e6	−0.181206	−0.0906032	+	0.995887i	$0.528880\pi$
$542$	0	0
$543$	−5.47507e6	−0.796875
$544$	0	0
$545$	892114.	0.128656
$546$	0	0
$547$	−7.56564e6	−1.08113	−0.540564	−	0.841303i	$-0.681789\pi$
$547$	−7.56564e6	−1.08113	−0.540564	+	0.841303i	$0.681789\pi$
$548$	0	0
$549$	−800231.	−0.113314
$550$	0	0
$551$	1.42049e7	1.99324
$552$	0	0
$553$	−2.42541e6	−0.337266
$554$	0	0
$555$	−4.91420e6	−0.677206
$556$	0	0
$557$	2.71053e6	0.370183	0.185091	−	0.982721i	$-0.440742\pi$
$557$	2.71053e6	0.370183	0.185091	+	0.982721i	$0.440742\pi$
$558$	0	0
$559$	6.43730e6	0.871313
$560$	0	0
$561$	−6.81233e6	−0.913879
$562$	0	0
$563$	4.74846e6	0.631366	0.315683	−	0.948865i	$-0.397766\pi$
$563$	4.74846e6	0.631366	0.315683	+	0.948865i	$0.397766\pi$
$564$	0	0
$565$	−4.55667e6	−0.600518
$566$	0	0
$567$	−3.20041e6	−0.418069
$568$	0	0
$569$	1.05692e7	1.36855	0.684275	−	0.729224i	$-0.260119\pi$
$569$	1.05692e7	1.36855	0.684275	+	0.729224i	$0.260119\pi$
$570$	0	0
$571$	−1.24557e7	−1.59874	−0.799372	−	0.600836i	$-0.794834\pi$
$571$	−1.24557e7	−1.59874	−0.799372	+	0.600836i	$0.794834\pi$
$572$	0	0
$573$	−1.31265e7	−1.67017
$574$	0	0
$575$	1.42699e6	0.179991
$576$	0	0
$577$	−7.83705e6	−0.979971	−0.489986	−	0.871731i	$-0.662998\pi$
$577$	−7.83705e6	−0.979971	−0.489986	+	0.871731i	$0.662998\pi$
$578$	0	0
$579$	−1.10850e7	−1.37417
$580$	0	0
$581$	−2.27378e6	−0.279452
$582$	0	0
$583$	−2.12678e7	−2.59150
$584$	0	0
$585$	635218.	0.0767421
$586$	0	0
$587$	7.73346e6	0.926358	0.463179	−	0.886265i	$-0.346709\pi$
$587$	7.73346e6	0.926358	0.463179	+	0.886265i	$0.346709\pi$
$588$	0	0
$589$	8.64612e6	1.02691
$590$	0	0
$591$	1.35345e7	1.59394
$592$	0	0
$593$	9.45660e6	1.10433	0.552165	−	0.833735i	$-0.313802\pi$
$593$	9.45660e6	1.10433	0.552165	+	0.833735i	$0.313802\pi$
$594$	0	0
$595$	742680.	0.0860022
$596$	0	0
$597$	−1.23657e7	−1.41998
$598$	0	0
$599$	−7.10373e6	−0.808946	−0.404473	−	0.914550i	$-0.632545\pi$
$599$	−7.10373e6	−0.808946	−0.404473	+	0.914550i	$0.632545\pi$
$600$	0	0
$601$	1.38522e7	1.56434	0.782172	−	0.623063i	$-0.214112\pi$
$601$	1.38522e7	1.56434	0.782172	+	0.623063i	$0.214112\pi$
$602$	0	0
$603$	−441614.	−0.0494594
$604$	0	0
$605$	7.56463e6	0.840232
$606$	0	0
$607$	1.01046e7	1.11313	0.556566	−	0.830804i	$-0.312119\pi$
$607$	1.01046e7	1.11313	0.556566	+	0.830804i	$0.312119\pi$
$608$	0	0
$609$	7.08866e6	0.774499
$610$	0	0
$611$	2.40265e6	0.260368
$612$	0	0
$613$	−1.01325e7	−1.08909	−0.544544	−	0.838732i	$-0.683297\pi$
$613$	−1.01325e7	−1.08909	−0.544544	+	0.838732i	$0.683297\pi$
$614$	0	0
$615$	1.20229e6	0.128181
$616$	0	0
$617$	6.55217e6	0.692903	0.346452	−	0.938068i	$-0.387387\pi$
$617$	6.55217e6	0.692903	0.346452	+	0.938068i	$0.387387\pi$
$618$	0	0
$619$	−7.46272e6	−0.782836	−0.391418	−	0.920213i	$-0.628015\pi$
$619$	−7.46272e6	−0.782836	−0.391418	+	0.920213i	$0.628015\pi$
$620$	0	0
$621$	−8.05084e6	−0.837745
$622$	0	0
$623$	4.15418e6	0.428810
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−1.82072e7	−1.84958
$628$	0	0
$629$	−7.22167e6	−0.727798
$630$	0	0
$631$	−1.16810e7	−1.16790	−0.583952	−	0.811788i	$-0.698494\pi$
$631$	−1.16810e7	−1.16790	−0.583952	+	0.811788i	$0.698494\pi$
$632$	0	0
$633$	1.08804e7	1.07928
$634$	0	0
$635$	−3.43224e6	−0.337788
$636$	0	0
$637$	2.08056e6	0.203157
$638$	0	0
$639$	1.94369e6	0.188311
$640$	0	0
$641$	5.88490e6	0.565710	0.282855	−	0.959163i	$-0.408718\pi$
$641$	5.88490e6	0.565710	0.282855	+	0.959163i	$0.408718\pi$
$642$	0	0
$643$	1.41490e7	1.34958	0.674789	−	0.738011i	$-0.264235\pi$
$643$	1.41490e7	1.34958	0.674789	+	0.738011i	$0.264235\pi$
$644$	0	0
$645$	3.06476e6	0.290066
$646$	0	0
$647$	−6.96657e6	−0.654272	−0.327136	−	0.944977i	$-0.606084\pi$
$647$	−6.96657e6	−0.654272	−0.327136	+	0.944977i	$0.606084\pi$
$648$	0	0
$649$	8.34272e6	0.777492
$650$	0	0
$651$	4.31466e6	0.399019
$652$	0	0
$653$	−1.87255e7	−1.71851	−0.859254	−	0.511549i	$-0.829072\pi$
$653$	−1.87255e7	−1.71851	−0.859254	+	0.511549i	$0.829072\pi$
$654$	0	0
$655$	−5.59409e6	−0.509479
$656$	0	0
$657$	−1.48358e6	−0.134091
$658$	0	0
$659$	349798.	0.0313765	0.0156882	−	0.999877i	$-0.495006\pi$
$659$	349798.	0.0313765	0.0156882	+	0.999877i	$0.495006\pi$
$660$	0	0
$661$	−1.66024e7	−1.47798	−0.738988	−	0.673718i	$-0.764696\pi$
$661$	−1.66024e7	−1.47798	−0.738988	+	0.673718i	$0.764696\pi$
$662$	0	0
$663$	8.66953e6	0.765970
$664$	0	0
$665$	1.98494e6	0.174058
$666$	0	0
$667$	2.00155e7	1.74202
$668$	0	0
$669$	−562857.	−0.0486220
$670$	0	0
$671$	1.85827e7	1.59332
$672$	0	0
$673$	1.35844e7	1.15612	0.578061	−	0.815994i	$-0.303810\pi$
$673$	1.35844e7	1.15612	0.578061	+	0.815994i	$0.303810\pi$
$674$	0	0
$675$	−2.20385e6	−0.186175
$676$	0	0
$677$	1.35375e7	1.13518	0.567591	−	0.823310i	$-0.307875\pi$
$677$	1.35375e7	1.13518	0.567591	+	0.823310i	$0.307875\pi$
$678$	0	0
$679$	2.93022e6	0.243908
$680$	0	0
$681$	−8.52641e6	−0.704529
$682$	0	0
$683$	−1.14434e7	−0.938646	−0.469323	−	0.883026i	$-0.655502\pi$
$683$	−1.14434e7	−0.938646	−0.469323	+	0.883026i	$0.655502\pi$
$684$	0	0
$685$	7.81742e6	0.636557
$686$	0	0
$687$	−8.72157e6	−0.705022
$688$	0	0
$689$	2.70659e7	2.17207
$690$	0	0
$691$	−1.79956e7	−1.43374	−0.716872	−	0.697205i	$-0.754427\pi$
$691$	−1.79956e7	−1.43374	−0.716872	+	0.697205i	$0.754427\pi$
$692$	0	0
$693$	−978316.	−0.0773831
$694$	0	0
$695$	3.15907e6	0.248083
$696$	0	0
$697$	1.76683e6	0.137757
$698$	0	0
$699$	−3.39704e6	−0.262971
$700$	0	0
$701$	−2.58623e6	−0.198779	−0.0993897	−	0.995049i	$-0.531689\pi$
$701$	−2.58623e6	−0.198779	−0.0993897	+	0.995049i	$0.531689\pi$
$702$	0	0
$703$	−1.93012e7	−1.47298
$704$	0	0
$705$	1.14389e6	0.0866782
$706$	0	0
$707$	6.31828e6	0.475390
$708$	0	0
$709$	1.92794e7	1.44038	0.720191	−	0.693776i	$-0.244054\pi$
$709$	1.92794e7	1.44038	0.720191	+	0.693776i	$0.244054\pi$
$710$	0	0
$711$	−1.45139e6	−0.107674
$712$	0	0
$713$	1.21828e7	0.897480
$714$	0	0
$715$	−1.47509e7	−1.07908
$716$	0	0
$717$	1.13422e7	0.823948
$718$	0	0
$719$	−2.70312e7	−1.95004	−0.975018	−	0.222124i	$-0.928701\pi$
$719$	−2.70312e7	−1.95004	−0.975018	+	0.222124i	$0.928701\pi$
$720$	0	0
$721$	3.65310e6	0.261712
$722$	0	0
$723$	−3.96606e6	−0.282172
$724$	0	0
$725$	5.47907e6	0.387134
$726$	0	0
$727$	1.17008e7	0.821069	0.410534	−	0.911845i	$-0.365342\pi$
$727$	1.17008e7	0.821069	0.410534	+	0.911845i	$0.365342\pi$
$728$	0	0
$729$	1.22248e7	0.851969
$730$	0	0
$731$	4.50382e6	0.311736
$732$	0	0
$733$	8.02273e6	0.551521	0.275761	−	0.961226i	$-0.411070\pi$
$733$	8.02273e6	0.551521	0.275761	+	0.961226i	$0.411070\pi$
$734$	0	0
$735$	990544.	0.0676325
$736$	0	0
$737$	1.02550e7	0.695453
$738$	0	0
$739$	2.82195e6	0.190081	0.0950405	−	0.995473i	$-0.469702\pi$
$739$	2.82195e6	0.190081	0.0950405	+	0.995473i	$0.469702\pi$
$740$	0	0
$741$	2.31709e7	1.55023
$742$	0	0
$743$	2.49290e7	1.65666	0.828328	−	0.560244i	$-0.189293\pi$
$743$	2.49290e7	1.65666	0.828328	+	0.560244i	$0.189293\pi$
$744$	0	0
$745$	1.22491e7	0.808562
$746$	0	0
$747$	−1.36065e6	−0.0892165
$748$	0	0
$749$	3.32495e6	0.216561
$750$	0	0
$751$	1.86631e7	1.20749	0.603747	−	0.797176i	$-0.293674\pi$
$751$	1.86631e7	1.20749	0.603747	+	0.797176i	$0.293674\pi$
$752$	0	0
$753$	−2.85023e6	−0.183186
$754$	0	0
$755$	−1.33143e7	−0.850062
$756$	0	0
$757$	2.26739e7	1.43809	0.719047	−	0.694961i	$-0.244579\pi$
$757$	2.26739e7	1.43809	0.719047	+	0.694961i	$0.244579\pi$
$758$	0	0
$759$	−2.56549e7	−1.61646
$760$	0	0
$761$	−1.11519e7	−0.698051	−0.349025	−	0.937113i	$-0.613487\pi$
$761$	−1.11519e7	−0.698051	−0.349025	+	0.937113i	$0.613487\pi$
$762$	0	0
$763$	1.74854e6	0.108734
$764$	0	0
$765$	444427.	0.0274566
$766$	0	0
$767$	−1.06171e7	−0.651657
$768$	0	0
$769$	872917.	0.0532301	0.0266151	−	0.999646i	$-0.491527\pi$
$769$	872917.	0.0532301	0.0266151	+	0.999646i	$0.491527\pi$
$770$	0	0
$771$	2.85427e7	1.72926
$772$	0	0
$773$	1.61242e7	0.970575	0.485287	−	0.874355i	$-0.338715\pi$
$773$	1.61242e7	0.970575	0.485287	+	0.874355i	$0.338715\pi$
$774$	0	0
$775$	3.33495e6	0.199450
$776$	0	0
$777$	−9.63184e6	−0.572343
$778$	0	0
$779$	4.72216e6	0.278803
$780$	0	0
$781$	−4.51358e7	−2.64785
$782$	0	0
$783$	−3.09121e7	−1.80187
$784$	0	0
$785$	−534934.	−0.0309832
$786$	0	0
$787$	1.16758e6	0.0671970	0.0335985	−	0.999435i	$-0.489303\pi$
$787$	1.16758e6	0.0671970	0.0335985	+	0.999435i	$0.489303\pi$
$788$	0	0
$789$	−6.13005e6	−0.350567
$790$	0	0
$791$	−8.93106e6	−0.507530
$792$	0	0
$793$	−2.36488e7	−1.33545
$794$	0	0
$795$	1.28859e7	0.723099
$796$	0	0
$797$	−1.67530e7	−0.934214	−0.467107	−	0.884201i	$-0.654704\pi$
$797$	−1.67530e7	−0.934214	−0.467107	+	0.884201i	$0.654704\pi$
$798$	0	0
$799$	1.68100e6	0.0931537
$800$	0	0
$801$	2.48590e6	0.136900
$802$	0	0
$803$	3.44513e7	1.88546
$804$	0	0
$805$	2.79689e6	0.152120
$806$	0	0
$807$	−5.30330e6	−0.286657
$808$	0	0
$809$	2.61543e7	1.40499	0.702493	−	0.711690i	$-0.252070\pi$
$809$	2.61543e7	1.40499	0.702493	+	0.711690i	$0.252070\pi$
$810$	0	0
$811$	−9.83522e6	−0.525088	−0.262544	−	0.964920i	$-0.584561\pi$
$811$	−9.83522e6	−0.525088	−0.262544	+	0.964920i	$0.584561\pi$
$812$	0	0
$813$	−2.18521e7	−1.15949
$814$	0	0
$815$	4.43405e6	0.233833
$816$	0	0
$817$	1.20373e7	0.630917
$818$	0	0
$819$	1.24503e6	0.0648589
$820$	0	0
$821$	3.96385e6	0.205239	0.102619	−	0.994721i	$-0.467278\pi$
$821$	3.96385e6	0.205239	0.102619	+	0.994721i	$0.467278\pi$
$822$	0	0
$823$	2.45198e7	1.26188	0.630939	−	0.775832i	$-0.282670\pi$
$823$	2.45198e7	1.26188	0.630939	+	0.775832i	$0.282670\pi$
$824$	0	0
$825$	−7.02280e6	−0.359232
$826$	0	0
$827$	−1.18524e7	−0.602620	−0.301310	−	0.953526i	$-0.597424\pi$
$827$	−1.18524e7	−0.602620	−0.301310	+	0.953526i	$0.597424\pi$
$828$	0	0
$829$	−8.98506e6	−0.454082	−0.227041	−	0.973885i	$-0.572905\pi$
$829$	−8.98506e6	−0.454082	−0.227041	+	0.973885i	$0.572905\pi$
$830$	0	0
$831$	−4.75663e6	−0.238944
$832$	0	0
$833$	1.45565e6	0.0726851
$834$	0	0
$835$	−1.56534e7	−0.776949
$836$	0	0
$837$	−1.88153e7	−0.928317
$838$	0	0
$839$	6.44598e6	0.316143	0.158072	−	0.987428i	$-0.449472\pi$
$839$	6.44598e6	0.316143	0.158072	+	0.987428i	$0.449472\pi$
$840$	0	0
$841$	5.63406e7	2.74683
$842$	0	0
$843$	2.35243e7	1.14011
$844$	0	0
$845$	9.48996e6	0.457217
$846$	0	0
$847$	1.48267e7	0.710125
$848$	0	0
$849$	5964.63	0.000283997 0
$850$	0	0
$851$	−2.71964e7	−1.28732
$852$	0	0
$853$	−1.80719e7	−0.850416	−0.425208	−	0.905096i	$-0.639799\pi$
$853$	−1.80719e7	−0.850416	−0.425208	+	0.905096i	$0.639799\pi$
$854$	0	0
$855$	1.18781e6	0.0555689
$856$	0	0
$857$	−1.49157e7	−0.693732	−0.346866	−	0.937915i	$-0.612754\pi$
$857$	−1.49157e7	−0.693732	−0.346866	+	0.937915i	$0.612754\pi$
$858$	0	0
$859$	2.47065e7	1.14243	0.571213	−	0.820802i	$-0.306473\pi$
$859$	2.47065e7	1.14243	0.571213	+	0.820802i	$0.306473\pi$
$860$	0	0
$861$	2.35649e6	0.108332
$862$	0	0
$863$	1.86455e7	0.852210	0.426105	−	0.904674i	$-0.359885\pi$
$863$	1.86455e7	0.852210	0.426105	+	0.904674i	$0.359885\pi$
$864$	0	0
$865$	−1.59751e7	−0.725945
$866$	0	0
$867$	−1.73652e7	−0.784568
$868$	0	0
$869$	3.37037e7	1.51401
$870$	0	0
$871$	−1.30508e7	−0.582896
$872$	0	0
$873$	1.75347e6	0.0778688
$874$	0	0
$875$	765625.	0.0338062
$876$	0	0
$877$	−2.50101e7	−1.09803	−0.549017	−	0.835811i	$-0.684998\pi$
$877$	−2.50101e7	−1.09803	−0.549017	+	0.835811i	$0.684998\pi$
$878$	0	0
$879$	−4.21783e7	−1.84127
$880$	0	0
$881$	−2.08928e7	−0.906893	−0.453446	−	0.891284i	$-0.649806\pi$
$881$	−2.08928e7	−0.906893	−0.453446	+	0.891284i	$0.649806\pi$
$882$	0	0
$883$	−2.81810e7	−1.21634	−0.608169	−	0.793808i	$-0.708096\pi$
$883$	−2.81810e7	−1.21634	−0.608169	+	0.793808i	$0.708096\pi$
$884$	0	0
$885$	−5.05476e6	−0.216941
$886$	0	0
$887$	−2.26733e7	−0.967620	−0.483810	−	0.875173i	$-0.660747\pi$
$887$	−2.26733e7	−0.967620	−0.483810	+	0.875173i	$0.660747\pi$
$888$	0	0
$889$	−6.72720e6	−0.285483
$890$	0	0
$891$	4.44732e7	1.87674
$892$	0	0
$893$	4.49277e6	0.188532
$894$	0	0
$895$	−1.62486e7	−0.678044
$896$	0	0
$897$	3.26490e7	1.35484
$898$	0	0
$899$	4.67774e7	1.93035
$900$	0	0
$901$	1.89365e7	0.777120
$902$	0	0
$903$	6.00693e6	0.245151
$904$	0	0
$905$	−8.29447e6	−0.336641
$906$	0	0
$907$	−1.17788e7	−0.475427	−0.237714	−	0.971335i	$-0.576398\pi$
$907$	−1.17788e7	−0.475427	−0.237714	+	0.971335i	$0.576398\pi$
$908$	0	0
$909$	3.78092e6	0.151771
$910$	0	0
$911$	2.08101e7	0.830765	0.415383	−	0.909647i	$-0.363648\pi$
$911$	2.08101e7	0.830765	0.415383	+	0.909647i	$0.363648\pi$
$912$	0	0
$913$	3.15966e7	1.25448
$914$	0	0
$915$	−1.12591e7	−0.444580
$916$	0	0
$917$	−1.09644e7	−0.430588
$918$	0	0
$919$	−2.51580e7	−0.982626	−0.491313	−	0.870983i	$-0.663483\pi$
$919$	−2.51580e7	−0.982626	−0.491313	+	0.870983i	$0.663483\pi$
$920$	0	0
$921$	2.04130e7	0.792973
$922$	0	0
$923$	5.74409e7	2.21930
$924$	0	0
$925$	−7.44478e6	−0.286087
$926$	0	0
$927$	2.18605e6	0.0835528
$928$	0	0
$929$	−2.56052e7	−0.973394	−0.486697	−	0.873571i	$-0.661798\pi$
$929$	−2.56052e7	−0.973394	−0.486697	+	0.873571i	$0.661798\pi$
$930$	0	0
$931$	3.89049e6	0.147106
$932$	0	0
$933$	3.68930e7	1.38752
$934$	0	0
$935$	−1.03204e7	−0.386069
$936$	0	0
$937$	1.37072e7	0.510033	0.255017	−	0.966937i	$-0.417919\pi$
$937$	1.37072e7	0.510033	0.255017	+	0.966937i	$0.417919\pi$
$938$	0	0
$939$	−1.42387e7	−0.526996
$940$	0	0
$941$	4.09560e6	0.150780	0.0753899	−	0.997154i	$-0.475980\pi$
$941$	4.09560e6	0.150780	0.0753899	+	0.997154i	$0.475980\pi$
$942$	0	0
$943$	6.65378e6	0.243663
$944$	0	0
$945$	−4.31954e6	−0.157347
$946$	0	0
$947$	7.04276e6	0.255193	0.127596	−	0.991826i	$-0.459274\pi$
$947$	7.04276e6	0.255193	0.127596	+	0.991826i	$0.459274\pi$
$948$	0	0
$949$	−4.38435e7	−1.58030
$950$	0	0
$951$	8.74119e6	0.313415
$952$	0	0
$953$	−4.98997e6	−0.177978	−0.0889889	−	0.996033i	$-0.528364\pi$
$953$	−4.98997e6	−0.177978	−0.0889889	+	0.996033i	$0.528364\pi$
$954$	0	0
$955$	−1.98860e7	−0.705567
$956$	0	0
$957$	−9.85047e7	−3.47678
$958$	0	0
$959$	1.53221e7	0.537988
$960$	0	0
$961$	−157159.	−0.00548949
$962$	0	0
$963$	1.98968e6	0.0691382
$964$	0	0
$965$	−1.67933e7	−0.580521
$966$	0	0
$967$	−1.12507e7	−0.386912	−0.193456	−	0.981109i	$-0.561970\pi$
$967$	−1.12507e7	−0.386912	−0.193456	+	0.981109i	$0.561970\pi$
$968$	0	0
$969$	1.62114e7	0.554639
$970$	0	0
$971$	3.06621e7	1.04365	0.521825	−	0.853053i	$-0.325252\pi$
$971$	3.06621e7	1.04365	0.521825	+	0.853053i	$0.325252\pi$
$972$	0	0
$973$	6.19177e6	0.209668
$974$	0	0
$975$	8.93737e6	0.301091
$976$	0	0
$977$	−1.26301e7	−0.423323	−0.211661	−	0.977343i	$-0.567887\pi$
$977$	−1.26301e7	−0.423323	−0.211661	+	0.977343i	$0.567887\pi$
$978$	0	0
$979$	−5.77269e7	−1.92496
$980$	0	0
$981$	1.04634e6	0.0347138
$982$	0	0
$983$	3.16382e7	1.04431	0.522153	−	0.852852i	$-0.325129\pi$
$983$	3.16382e7	1.04431	0.522153	+	0.852852i	$0.325129\pi$
$984$	0	0
$985$	2.05041e7	0.673364
$986$	0	0
$987$	2.24202e6	0.0732565
$988$	0	0
$989$	1.69611e7	0.551397
$990$	0	0
$991$	2.45397e7	0.793754	0.396877	−	0.917872i	$-0.370094\pi$
$991$	2.45397e7	0.793754	0.396877	+	0.917872i	$0.370094\pi$
$992$	0	0
$993$	−229691.	−0.00739216
$994$	0	0
$995$	−1.87334e7	−0.599872
$996$	0	0
$997$	−2.15206e7	−0.685674	−0.342837	−	0.939395i	$-0.611388\pi$
$997$	−2.15206e7	−0.685674	−0.342837	+	0.939395i	$0.611388\pi$
$998$	0	0
$999$	4.20023e7	1.33156

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.6.a.j.1.4	✓	5
4.3	odd	2		560.6.a.y.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.6.a.j.1.4	✓	5	1.1	even	1	trivial
560.6.a.y.1.2		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)

\(f(q)\)	\(=\)	\(q+16.5022 q^{3} +25.0000 q^{5} +49.0000 q^{7} +29.3221 q^{9} +O(q^{10})\)	\(q+16.5022 q^{3} +25.0000 q^{5} +49.0000 q^{7} +29.3221 q^{9} -680.908 q^{11} +866.540 q^{13} +412.555 q^{15} +606.270 q^{17} +1620.36 q^{19} +808.607 q^{21} +2283.18 q^{23} +625.000 q^{25} -3526.15 q^{27} +8766.51 q^{29} +5335.92 q^{31} -11236.5 q^{33} +1225.00 q^{35} -11911.6 q^{37} +14299.8 q^{39} +2914.26 q^{41} +7428.74 q^{43} +733.052 q^{45} +2772.69 q^{47} +2401.00 q^{49} +10004.8 q^{51} +31234.5 q^{53} -17022.7 q^{55} +26739.5 q^{57} -12252.3 q^{59} -27291.1 q^{61} +1436.78 q^{63} +21663.5 q^{65} -15060.8 q^{67} +37677.4 q^{69} +66287.7 q^{71} -50596.1 q^{73} +10313.9 q^{75} -33364.5 q^{77} -49498.2 q^{79} -65314.5 q^{81} -46403.7 q^{83} +15156.7 q^{85} +144667. q^{87} +84779.2 q^{89} +42460.5 q^{91} +88054.3 q^{93} +40509.1 q^{95} +59800.5 q^{97} -19965.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 15 q^{3} + 125 q^{5} + 245 q^{7} + 1132 q^{9}+O(q^{10}) \) 5 * q + 15 * q^3 + 125 * q^5 + 245 * q^7 + 1132 * q^9	\( 5 q + 15 q^{3} + 125 q^{5} + 245 q^{7} + 1132 q^{9} + 281 q^{11} + 909 q^{13} + 375 q^{15} + 1495 q^{17} - 422 q^{19} + 735 q^{21} - 62 q^{23} + 3125 q^{25} - 3363 q^{27} - 2047 q^{29} + 1636 q^{31} + 19181 q^{33} + 6125 q^{35} - 10358 q^{37} + 15685 q^{39} + 6424 q^{41} + 28306 q^{43} + 28300 q^{45} + 20955 q^{47} + 12005 q^{49} - 23577 q^{51} + 43748 q^{53} + 7025 q^{55} + 13690 q^{57} + 45788 q^{59} + 50432 q^{61} + 55468 q^{63} + 22725 q^{65} + 40712 q^{67} + 35050 q^{69} - 3096 q^{71} + 135438 q^{73} + 9375 q^{75} + 13769 q^{77} + 13191 q^{79} + 381101 q^{81} + 35108 q^{83} + 37375 q^{85} + 297289 q^{87} + 213772 q^{89} + 44541 q^{91} + 134244 q^{93} - 10550 q^{95} + 10659 q^{97} + 39462 q^{99}+O(q^{100}) \) 5 * q + 15 * q^3 + 125 * q^5 + 245 * q^7 + 1132 * q^9 + 281 * q^11 + 909 * q^13 + 375 * q^15 + 1495 * q^17 - 422 * q^19 + 735 * q^21 - 62 * q^23 + 3125 * q^25 - 3363 * q^27 - 2047 * q^29 + 1636 * q^31 + 19181 * q^33 + 6125 * q^35 - 10358 * q^37 + 15685 * q^39 + 6424 * q^41 + 28306 * q^43 + 28300 * q^45 + 20955 * q^47 + 12005 * q^49 - 23577 * q^51 + 43748 * q^53 + 7025 * q^55 + 13690 * q^57 + 45788 * q^59 + 50432 * q^61 + 55468 * q^63 + 22725 * q^65 + 40712 * q^67 + 35050 * q^69 - 3096 * q^71 + 135438 * q^73 + 9375 * q^75 + 13769 * q^77 + 13191 * q^79 + 381101 * q^81 + 35108 * q^83 + 37375 * q^85 + 297289 * q^87 + 213772 * q^89 + 44541 * q^91 + 134244 * q^93 - 10550 * q^95 + 10659 * q^97 + 39462 * q^99

Introduction

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