LMFDB - Embedded newform 280.6.a.j.1.2

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			1.2
Root			$14.5231$ 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)

$f(q)$	$=$	$q-11.5231 q^{3} +25.0000 q^{5} +49.0000 q^{7} -110.219 q^{9} +O(q^{10})$	$q-11.5231 q^{3} +25.0000 q^{5} +49.0000 q^{7} -110.219 q^{9} +311.119 q^{11} +474.375 q^{13} -288.077 q^{15} -1013.83 q^{17} -225.511 q^{19} -564.631 q^{21} -1771.33 q^{23} +625.000 q^{25} +4070.17 q^{27} +2812.20 q^{29} -4256.50 q^{31} -3585.05 q^{33} +1225.00 q^{35} +5266.46 q^{37} -5466.26 q^{39} +4860.86 q^{41} -16931.4 q^{43} -2755.47 q^{45} +800.206 q^{47} +2401.00 q^{49} +11682.5 q^{51} +36991.3 q^{53} +7777.98 q^{55} +2598.58 q^{57} -4245.05 q^{59} +40820.6 q^{61} -5400.72 q^{63} +11859.4 q^{65} +14788.0 q^{67} +20411.2 q^{69} +63953.5 q^{71} +68002.8 q^{73} -7201.92 q^{75} +15244.8 q^{77} +26146.8 q^{79} -20117.7 q^{81} +31651.6 q^{83} -25345.9 q^{85} -32405.2 q^{87} -78418.1 q^{89} +23244.4 q^{91} +49047.9 q^{93} -5637.78 q^{95} +123416. q^{97} -34291.2 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−11.5231	−0.739206	−0.369603	−	0.929190i	$-0.620506\pi$
$3$	−11.5231	−0.739206	−0.369603	+	0.929190i	$0.620506\pi$
$4$	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
$9$	−110.219	−0.453575
$10$	0	0
$11$	311.119	0.775256	0.387628	−	0.921816i	$-0.373294\pi$
$11$	311.119	0.775256	0.387628	+	0.921816i	$0.373294\pi$
$12$	0	0
$13$	474.375	0.778508	0.389254	−	0.921130i	$-0.372733\pi$
$13$	474.375	0.778508	0.389254	+	0.921130i	$0.372733\pi$
$14$	0	0
$15$	−288.077	−0.330583
$16$	0	0
$17$	−1013.83	−0.850834	−0.425417	−	0.904997i	$-0.639873\pi$
$17$	−1013.83	−0.850834	−0.425417	+	0.904997i	$0.639873\pi$
$18$	0	0
$19$	−225.511	−0.143312	−0.0716562	−	0.997429i	$-0.522828\pi$
$19$	−225.511	−0.143312	−0.0716562	+	0.997429i	$0.522828\pi$
$20$	0	0
$21$	−564.631	−0.279393
$22$	0	0
$23$	−1771.33	−0.698200	−0.349100	−	0.937085i	$-0.613513\pi$
$23$	−1771.33	−0.698200	−0.349100	+	0.937085i	$0.613513\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	4070.17	1.07449
$28$	0	0
$29$	2812.20	0.620942	0.310471	−	0.950583i	$-0.399513\pi$
$29$	2812.20	0.620942	0.310471	+	0.950583i	$0.399513\pi$
$30$	0	0
$31$	−4256.50	−0.795514	−0.397757	−	0.917491i	$-0.630211\pi$
$31$	−4256.50	−0.795514	−0.397757	+	0.917491i	$0.630211\pi$
$32$	0	0
$33$	−3585.05	−0.573074
$34$	0	0
$35$	1225.00	0.169031
$36$	0	0
$37$	5266.46	0.632433	0.316216	−	0.948687i	$-0.397587\pi$
$37$	5266.46	0.632433	0.316216	+	0.948687i	$0.397587\pi$
$38$	0	0
$39$	−5466.26	−0.575478
$40$	0	0
$41$	4860.86	0.451599	0.225800	−	0.974174i	$-0.427501\pi$
$41$	4860.86	0.451599	0.225800	+	0.974174i	$0.427501\pi$
$42$	0	0
$43$	−16931.4	−1.39644	−0.698219	−	0.715885i	$-0.746024\pi$
$43$	−16931.4	−1.39644	−0.698219	+	0.715885i	$0.746024\pi$
$44$	0	0
$45$	−2755.47	−0.202845
$46$	0	0
$47$	800.206	0.0528393	0.0264197	−	0.999651i	$-0.491589\pi$
$47$	800.206	0.0528393	0.0264197	+	0.999651i	$0.491589\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	11682.5	0.628941
$52$	0	0
$53$	36991.3	1.80888	0.904441	−	0.426599i	$-0.140289\pi$
$53$	36991.3	1.80888	0.904441	+	0.426599i	$0.140289\pi$
$54$	0	0
$55$	7777.98	0.346705
$56$	0	0
$57$	2598.58	0.105937
$58$	0	0
$59$	−4245.05	−0.158764	−0.0793822	−	0.996844i	$-0.525295\pi$
$59$	−4245.05	−0.158764	−0.0793822	+	0.996844i	$0.525295\pi$
$60$	0	0
$61$	40820.6	1.40461	0.702304	−	0.711878i	$-0.252155\pi$
$61$	40820.6	1.40461	0.702304	+	0.711878i	$0.252155\pi$
$62$	0	0
$63$	−5400.72	−0.171435
$64$	0	0
$65$	11859.4	0.348159
$66$	0	0
$67$	14788.0	0.402459	0.201230	−	0.979544i	$-0.435506\pi$
$67$	14788.0	0.402459	0.201230	+	0.979544i	$0.435506\pi$
$68$	0	0
$69$	20411.2	0.516113
$70$	0	0
$71$	63953.5	1.50563	0.752815	−	0.658232i	$-0.228695\pi$
$71$	63953.5	1.50563	0.752815	+	0.658232i	$0.228695\pi$
$72$	0	0
$73$	68002.8	1.49355	0.746775	−	0.665077i	$-0.231601\pi$
$73$	68002.8	1.49355	0.746775	+	0.665077i	$0.231601\pi$
$74$	0	0
$75$	−7201.92	−0.147841
$76$	0	0
$77$	15244.8	0.293019
$78$	0	0
$79$	26146.8	0.471358	0.235679	−	0.971831i	$-0.424269\pi$
$79$	26146.8	0.471358	0.235679	+	0.971831i	$0.424269\pi$
$80$	0	0
$81$	−20117.7	−0.340694
$82$	0	0
$83$	31651.6	0.504313	0.252157	−	0.967686i	$-0.418860\pi$
$83$	31651.6	0.504313	0.252157	+	0.967686i	$0.418860\pi$
$84$	0	0
$85$	−25345.9	−0.380504
$86$	0	0
$87$	−32405.2	−0.459003
$88$	0	0
$89$	−78418.1	−1.04940	−0.524700	−	0.851287i	$-0.675823\pi$
$89$	−78418.1	−1.04940	−0.524700	+	0.851287i	$0.675823\pi$
$90$	0	0
$91$	23244.4	0.294248
$92$	0	0
$93$	49047.9	0.588049
$94$	0	0
$95$	−5637.78	−0.0640913
$96$	0	0
$97$	123416.	1.33181	0.665907	−	0.746034i	$-0.268045\pi$
$97$	123416.	1.33181	0.665907	+	0.746034i	$0.268045\pi$
$98$	0	0
$99$	−34291.2	−0.351637
$100$	0	0
$101$	89194.4	0.870030	0.435015	−	0.900423i	$-0.356743\pi$
$101$	89194.4	0.870030	0.435015	+	0.900423i	$0.356743\pi$
$102$	0	0
$103$	68795.8	0.638953	0.319477	−	0.947594i	$-0.396493\pi$
$103$	68795.8	0.638953	0.319477	+	0.947594i	$0.396493\pi$
$104$	0	0
$105$	−14115.8	−0.124949
$106$	0	0
$107$	61266.4	0.517324	0.258662	−	0.965968i	$-0.416718\pi$
$107$	61266.4	0.517324	0.258662	+	0.965968i	$0.416718\pi$
$108$	0	0
$109$	93037.5	0.750053	0.375026	−	0.927014i	$-0.377634\pi$
$109$	93037.5	0.750053	0.375026	+	0.927014i	$0.377634\pi$
$110$	0	0
$111$	−60685.8	−0.467498
$112$	0	0
$113$	58230.0	0.428993	0.214497	−	0.976725i	$-0.431189\pi$
$113$	58230.0	0.428993	0.214497	+	0.976725i	$0.431189\pi$
$114$	0	0
$115$	−44283.3	−0.312245
$116$	0	0
$117$	−52285.0	−0.353112
$118$	0	0
$119$	−49677.9	−0.321585
$120$	0	0
$121$	−64255.7	−0.398978
$122$	0	0
$123$	−56012.0	−0.333825
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	−60699.3	−0.333944	−0.166972	−	0.985962i	$-0.553399\pi$
$127$	−60699.3	−0.333944	−0.166972	+	0.985962i	$0.553399\pi$
$128$	0	0
$129$	195102.	1.03225
$130$	0	0
$131$	−215671.	−1.09803	−0.549013	−	0.835814i	$-0.684996\pi$
$131$	−215671.	−1.09803	−0.549013	+	0.835814i	$0.684996\pi$
$132$	0	0
$133$	−11050.0	−0.0541670
$134$	0	0
$135$	101754.	0.480527
$136$	0	0
$137$	59809.7	0.272251	0.136126	−	0.990692i	$-0.456535\pi$
$137$	59809.7	0.272251	0.136126	+	0.990692i	$0.456535\pi$
$138$	0	0
$139$	−186421.	−0.818384	−0.409192	−	0.912448i	$-0.634189\pi$
$139$	−186421.	−0.818384	−0.409192	+	0.912448i	$0.634189\pi$
$140$	0	0
$141$	−9220.83	−0.0390591
$142$	0	0
$143$	147587.	0.603543
$144$	0	0
$145$	70304.9	0.277694
$146$	0	0
$147$	−27666.9	−0.105601
$148$	0	0
$149$	−397247.	−1.46587	−0.732934	−	0.680300i	$-0.761849\pi$
$149$	−397247.	−1.46587	−0.732934	+	0.680300i	$0.761849\pi$
$150$	0	0
$151$	350286.	1.25020	0.625102	−	0.780543i	$-0.285057\pi$
$151$	350286.	1.25020	0.625102	+	0.780543i	$0.285057\pi$
$152$	0	0
$153$	111744.	0.385917
$154$	0	0
$155$	−106412.	−0.355765
$156$	0	0
$157$	131696.	0.426407	0.213203	−	0.977008i	$-0.431610\pi$
$157$	131696.	0.426407	0.213203	+	0.977008i	$0.431610\pi$
$158$	0	0
$159$	−426254.	−1.33714
$160$	0	0
$161$	−86795.2	−0.263895
$162$	0	0
$163$	−415230.	−1.22411	−0.612054	−	0.790816i	$-0.709656\pi$
$163$	−415230.	−1.22411	−0.612054	+	0.790816i	$0.709656\pi$
$164$	0	0
$165$	−89626.3	−0.256286
$166$	0	0
$167$	507318.	1.40763	0.703815	−	0.710383i	$-0.251478\pi$
$167$	507318.	1.40763	0.703815	+	0.710383i	$0.251478\pi$
$168$	0	0
$169$	−146262.	−0.393925
$170$	0	0
$171$	24855.5	0.0650029
$172$	0	0
$173$	447178.	1.13597	0.567983	−	0.823040i	$-0.307724\pi$
$173$	447178.	1.13597	0.567983	+	0.823040i	$0.307724\pi$
$174$	0	0
$175$	30625.0	0.0755929
$176$	0	0
$177$	48916.1	0.117360
$178$	0	0
$179$	−86849.9	−0.202599	−0.101299	−	0.994856i	$-0.532300\pi$
$179$	−86849.9	−0.202599	−0.101299	+	0.994856i	$0.532300\pi$
$180$	0	0
$181$	420933.	0.955028	0.477514	−	0.878624i	$-0.341538\pi$
$181$	420933.	0.955028	0.477514	+	0.878624i	$0.341538\pi$
$182$	0	0
$183$	−470379.	−1.03829
$184$	0	0
$185$	131661.	0.282833
$186$	0	0
$187$	−315424.	−0.659614
$188$	0	0
$189$	199438.	0.406119
$190$	0	0
$191$	−416647.	−0.826389	−0.413194	−	0.910643i	$-0.635587\pi$
$191$	−416647.	−0.826389	−0.413194	+	0.910643i	$0.635587\pi$
$192$	0	0
$193$	−122306.	−0.236349	−0.118175	−	0.992993i	$-0.537704\pi$
$193$	−122306.	−0.236349	−0.118175	+	0.992993i	$0.537704\pi$
$194$	0	0
$195$	−136656.	−0.257361
$196$	0	0
$197$	479601.	0.880470	0.440235	−	0.897882i	$-0.354895\pi$
$197$	479601.	0.880470	0.440235	+	0.897882i	$0.354895\pi$
$198$	0	0
$199$	−515014.	−0.921906	−0.460953	−	0.887424i	$-0.652492\pi$
$199$	−515014.	−0.921906	−0.460953	+	0.887424i	$0.652492\pi$
$200$	0	0
$201$	−170403.	−0.297500
$202$	0	0
$203$	137798.	0.234694
$204$	0	0
$205$	121521.	0.201961
$206$	0	0
$207$	195234.	0.316686
$208$	0	0
$209$	−70160.9	−0.111104
$210$	0	0
$211$	−174819.	−0.270322	−0.135161	−	0.990824i	$-0.543155\pi$
$211$	−174819.	−0.270322	−0.135161	+	0.990824i	$0.543155\pi$
$212$	0	0
$213$	−736941.	−1.11297
$214$	0	0
$215$	−423285.	−0.624506
$216$	0	0
$217$	−208568.	−0.300676
$218$	0	0
$219$	−783601.	−1.10404
$220$	0	0
$221$	−480937.	−0.662381
$222$	0	0
$223$	464089.	0.624941	0.312471	−	0.949927i	$-0.398843\pi$
$223$	464089.	0.624941	0.312471	+	0.949927i	$0.398843\pi$
$224$	0	0
$225$	−68886.7	−0.0907150
$226$	0	0
$227$	−347266.	−0.447299	−0.223649	−	0.974670i	$-0.571797\pi$
$227$	−347266.	−0.447299	−0.223649	+	0.974670i	$0.571797\pi$
$228$	0	0
$229$	−402168.	−0.506779	−0.253390	−	0.967364i	$-0.581545\pi$
$229$	−402168.	−0.506779	−0.253390	+	0.967364i	$0.581545\pi$
$230$	0	0
$231$	−175668.	−0.216602
$232$	0	0
$233$	531421.	0.641282	0.320641	−	0.947201i	$-0.396102\pi$
$233$	531421.	0.641282	0.320641	+	0.947201i	$0.396102\pi$
$234$	0	0
$235$	20005.2	0.0236305
$236$	0	0
$237$	−301292.	−0.348430
$238$	0	0
$239$	−929567.	−1.05265	−0.526327	−	0.850282i	$-0.676431\pi$
$239$	−929567.	−1.05265	−0.526327	+	0.850282i	$0.676431\pi$
$240$	0	0
$241$	188014.	0.208520	0.104260	−	0.994550i	$-0.466753\pi$
$241$	188014.	0.208520	0.104260	+	0.994550i	$0.466753\pi$
$242$	0	0
$243$	−757233.	−0.822648
$244$	0	0
$245$	60025.0	0.0638877
$246$	0	0
$247$	−106977.	−0.111570
$248$	0	0
$249$	−364724.	−0.372791
$250$	0	0
$251$	−216396.	−0.216803	−0.108401	−	0.994107i	$-0.534573\pi$
$251$	−216396.	−0.216803	−0.108401	+	0.994107i	$0.534573\pi$
$252$	0	0
$253$	−551095.	−0.541284
$254$	0	0
$255$	292062.	0.281271
$256$	0	0
$257$	688263.	0.650013	0.325006	−	0.945712i	$-0.394634\pi$
$257$	688263.	0.650013	0.325006	+	0.945712i	$0.394634\pi$
$258$	0	0
$259$	258056.	0.239037
$260$	0	0
$261$	−309957.	−0.281644
$262$	0	0
$263$	−1.80250e6	−1.60689	−0.803443	−	0.595381i	$-0.797001\pi$
$263$	−1.80250e6	−1.60689	−0.803443	+	0.595381i	$0.797001\pi$
$264$	0	0
$265$	924783.	0.808956
$266$	0	0
$267$	903617.	0.775723
$268$	0	0
$269$	772788.	0.651148	0.325574	−	0.945517i	$-0.394443\pi$
$269$	772788.	0.651148	0.325574	+	0.945517i	$0.394443\pi$
$270$	0	0
$271$	1.01704e6	0.841226	0.420613	−	0.907240i	$-0.361815\pi$
$271$	1.01704e6	0.841226	0.420613	+	0.907240i	$0.361815\pi$
$272$	0	0
$273$	−267847.	−0.217510
$274$	0	0
$275$	194450.	0.155051
$276$	0	0
$277$	−1.19978e6	−0.939513	−0.469757	−	0.882796i	$-0.655658\pi$
$277$	−1.19978e6	−0.939513	−0.469757	+	0.882796i	$0.655658\pi$
$278$	0	0
$279$	469146.	0.360825
$280$	0	0
$281$	120955.	0.0913814	0.0456907	−	0.998956i	$-0.485451\pi$
$281$	120955.	0.0913814	0.0456907	+	0.998956i	$0.485451\pi$
$282$	0	0
$283$	−417432.	−0.309827	−0.154914	−	0.987928i	$-0.549510\pi$
$283$	−417432.	−0.309827	−0.154914	+	0.987928i	$0.549510\pi$
$284$	0	0
$285$	64964.5	0.0473766
$286$	0	0
$287$	238182.	0.170688
$288$	0	0
$289$	−391997.	−0.276082
$290$	0	0
$291$	−1.42214e6	−0.984485
$292$	0	0
$293$	1.96720e6	1.33869	0.669345	−	0.742952i	$-0.266575\pi$
$293$	1.96720e6	1.33869	0.669345	+	0.742952i	$0.266575\pi$
$294$	0	0
$295$	−106126.	−0.0710016
$296$	0	0
$297$	1.26631e6	0.833006
$298$	0	0
$299$	−840274.	−0.543554
$300$	0	0
$301$	−829638.	−0.527804
$302$	0	0
$303$	−1.02779e6	−0.643131
$304$	0	0
$305$	1.02052e6	0.628159
$306$	0	0
$307$	1.66012e6	1.00529	0.502646	−	0.864492i	$-0.332360\pi$
$307$	1.66012e6	1.00529	0.502646	+	0.864492i	$0.332360\pi$
$308$	0	0
$309$	−792739.	−0.472318
$310$	0	0
$311$	2.53484e6	1.48610	0.743051	−	0.669234i	$-0.233378\pi$
$311$	2.53484e6	1.48610	0.743051	+	0.669234i	$0.233378\pi$
$312$	0	0
$313$	232971.	0.134413	0.0672063	−	0.997739i	$-0.478591\pi$
$313$	232971.	0.134413	0.0672063	+	0.997739i	$0.478591\pi$
$314$	0	0
$315$	−135018.	−0.0766682
$316$	0	0
$317$	4009.73	0.00224113	0.00112057	−	0.999999i	$-0.499643\pi$
$317$	4009.73	0.00224113	0.00112057	+	0.999999i	$0.499643\pi$
$318$	0	0
$319$	874929.	0.481389
$320$	0	0
$321$	−705977.	−0.382409
$322$	0	0
$323$	228631.	0.121935
$324$	0	0
$325$	296484.	0.155702
$326$	0	0
$327$	−1.07208e6	−0.554443
$328$	0	0
$329$	39210.1	0.0199714
$330$	0	0
$331$	−593441.	−0.297720	−0.148860	−	0.988858i	$-0.547560\pi$
$331$	−593441.	−0.297720	−0.148860	+	0.988858i	$0.547560\pi$
$332$	0	0
$333$	−580462.	−0.286856
$334$	0	0
$335$	369699.	0.179985
$336$	0	0
$337$	3.45961e6	1.65940	0.829701	−	0.558207i	$-0.188511\pi$
$337$	3.45961e6	1.65940	0.829701	+	0.558207i	$0.188511\pi$
$338$	0	0
$339$	−670988.	−0.317114
$340$	0	0
$341$	−1.32428e6	−0.616727
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	510279.	0.230813
$346$	0	0
$347$	3.17320e6	1.41473	0.707366	−	0.706847i	$-0.249883\pi$
$347$	3.17320e6	1.41473	0.707366	+	0.706847i	$0.249883\pi$
$348$	0	0
$349$	80384.4	0.0353271	0.0176636	−	0.999844i	$-0.494377\pi$
$349$	80384.4	0.0353271	0.0176636	+	0.999844i	$0.494377\pi$
$350$	0	0
$351$	1.93078e6	0.836500
$352$	0	0
$353$	−1.40759e6	−0.601230	−0.300615	−	0.953746i	$-0.597192\pi$
$353$	−1.40759e6	−0.601230	−0.300615	+	0.953746i	$0.597192\pi$
$354$	0	0
$355$	1.59884e6	0.673338
$356$	0	0
$357$	572442.	0.237717
$358$	0	0
$359$	3.88927e6	1.59269	0.796347	−	0.604840i	$-0.206763\pi$
$359$	3.88927e6	1.59269	0.796347	+	0.604840i	$0.206763\pi$
$360$	0	0
$361$	−2.42524e6	−0.979462
$362$	0	0
$363$	740424.	0.294926
$364$	0	0
$365$	1.70007e6	0.667936
$366$	0	0
$367$	3.60112e6	1.39564	0.697819	−	0.716274i	$-0.254154\pi$
$367$	3.60112e6	1.39564	0.697819	+	0.716274i	$0.254154\pi$
$368$	0	0
$369$	−535758.	−0.204834
$370$	0	0
$371$	1.81257e6	0.683693
$372$	0	0
$373$	−4.36006e6	−1.62263	−0.811317	−	0.584607i	$-0.801249\pi$
$373$	−4.36006e6	−1.62263	−0.811317	+	0.584607i	$0.801249\pi$
$374$	0	0
$375$	−180048.	−0.0661166
$376$	0	0
$377$	1.33404e6	0.483408
$378$	0	0
$379$	947203.	0.338723	0.169362	−	0.985554i	$-0.445829\pi$
$379$	947203.	0.338723	0.169362	+	0.985554i	$0.445829\pi$
$380$	0	0
$381$	699442.	0.246854
$382$	0	0
$383$	1.38593e6	0.482775	0.241387	−	0.970429i	$-0.422398\pi$
$383$	1.38593e6	0.482775	0.241387	+	0.970429i	$0.422398\pi$
$384$	0	0
$385$	381121.	0.131042
$386$	0	0
$387$	1.86616e6	0.633389
$388$	0	0
$389$	−4.84462e6	−1.62325	−0.811626	−	0.584178i	$-0.801417\pi$
$389$	−4.84462e6	−1.62325	−0.811626	+	0.584178i	$0.801417\pi$
$390$	0	0
$391$	1.79584e6	0.594052
$392$	0	0
$393$	2.48519e6	0.811667
$394$	0	0
$395$	653670.	0.210798
$396$	0	0
$397$	3.21974e6	1.02529	0.512643	−	0.858602i	$-0.328667\pi$
$397$	3.21974e6	1.02529	0.512643	+	0.858602i	$0.328667\pi$
$398$	0	0
$399$	127330.	0.0400405
$400$	0	0
$401$	326263.	0.101323	0.0506613	−	0.998716i	$-0.483867\pi$
$401$	326263.	0.101323	0.0506613	+	0.998716i	$0.483867\pi$
$402$	0	0
$403$	−2.01917e6	−0.619314
$404$	0	0
$405$	−502942.	−0.152363
$406$	0	0
$407$	1.63850e6	0.490297
$408$	0	0
$409$	751750.	0.222211	0.111105	−	0.993809i	$-0.464561\pi$
$409$	751750.	0.222211	0.111105	+	0.993809i	$0.464561\pi$
$410$	0	0
$411$	−689191.	−0.201250
$412$	0	0
$413$	−208008.	−0.0600073
$414$	0	0
$415$	791290.	0.225536
$416$	0	0
$417$	2.14814e6	0.604954
$418$	0	0
$419$	−2.05259e6	−0.571174	−0.285587	−	0.958353i	$-0.592188\pi$
$419$	−2.05259e6	−0.571174	−0.285587	+	0.958353i	$0.592188\pi$
$420$	0	0
$421$	−2.90695e6	−0.799341	−0.399670	−	0.916659i	$-0.630875\pi$
$421$	−2.90695e6	−0.799341	−0.399670	+	0.916659i	$0.630875\pi$
$422$	0	0
$423$	−88197.7	−0.0239666
$424$	0	0
$425$	−633647.	−0.170167
$426$	0	0
$427$	2.00021e6	0.530892
$428$	0	0
$429$	−1.70066e6	−0.446143
$430$	0	0
$431$	−1.10147e6	−0.285614	−0.142807	−	0.989751i	$-0.545613\pi$
$431$	−1.10147e6	−0.285614	−0.142807	+	0.989751i	$0.545613\pi$
$432$	0	0
$433$	−4.73866e6	−1.21461	−0.607303	−	0.794470i	$-0.707749\pi$
$433$	−4.73866e6	−1.21461	−0.607303	+	0.794470i	$0.707749\pi$
$434$	0	0
$435$	−810129.	−0.205273
$436$	0	0
$437$	399454.	0.100061
$438$	0	0
$439$	−7.03778e6	−1.74291	−0.871454	−	0.490478i	$-0.836822\pi$
$439$	−7.03778e6	−1.74291	−0.871454	+	0.490478i	$0.836822\pi$
$440$	0	0
$441$	−264635.	−0.0647964
$442$	0	0
$443$	7.02169e6	1.69994	0.849968	−	0.526835i	$-0.176621\pi$
$443$	7.02169e6	1.69994	0.849968	+	0.526835i	$0.176621\pi$
$444$	0	0
$445$	−1.96045e6	−0.469306
$446$	0	0
$447$	4.57750e6	1.08358
$448$	0	0
$449$	5.87564e6	1.37543	0.687716	−	0.725980i	$-0.258613\pi$
$449$	5.87564e6	1.37543	0.687716	+	0.725980i	$0.258613\pi$
$450$	0	0
$451$	1.51231e6	0.350105
$452$	0	0
$453$	−4.03638e6	−0.924158
$454$	0	0
$455$	581109.	0.131592
$456$	0	0
$457$	−5.83788e6	−1.30757	−0.653785	−	0.756680i	$-0.726820\pi$
$457$	−5.83788e6	−1.30757	−0.653785	+	0.756680i	$0.726820\pi$
$458$	0	0
$459$	−4.12647e6	−0.914213
$460$	0	0
$461$	−1.84161e6	−0.403594	−0.201797	−	0.979427i	$-0.564678\pi$
$461$	−1.84161e6	−0.403594	−0.201797	+	0.979427i	$0.564678\pi$
$462$	0	0
$463$	1.05065e6	0.227775	0.113888	−	0.993494i	$-0.463670\pi$
$463$	1.05065e6	0.227775	0.113888	+	0.993494i	$0.463670\pi$
$464$	0	0
$465$	1.22620e6	0.262983
$466$	0	0
$467$	4.50459e6	0.955792	0.477896	−	0.878417i	$-0.341400\pi$
$467$	4.50459e6	0.955792	0.477896	+	0.878417i	$0.341400\pi$
$468$	0	0
$469$	724611.	0.152115
$470$	0	0
$471$	−1.51754e6	−0.315202
$472$	0	0
$473$	−5.26768e6	−1.08260
$474$	0	0
$475$	−140944.	−0.0286625
$476$	0	0
$477$	−4.07714e6	−0.820464
$478$	0	0
$479$	−1.43333e6	−0.285435	−0.142717	−	0.989763i	$-0.545584\pi$
$479$	−1.43333e6	−0.285435	−0.142717	+	0.989763i	$0.545584\pi$
$480$	0	0
$481$	2.49827e6	0.492354
$482$	0	0
$483$	1.00015e6	0.195073
$484$	0	0
$485$	3.08541e6	0.595606
$486$	0	0
$487$	−6.77756e6	−1.29494	−0.647472	−	0.762090i	$-0.724174\pi$
$487$	−6.77756e6	−1.29494	−0.647472	+	0.762090i	$0.724174\pi$
$488$	0	0
$489$	4.78473e6	0.904867
$490$	0	0
$491$	−557751.	−0.104409	−0.0522043	−	0.998636i	$-0.516625\pi$
$491$	−557751.	−0.104409	−0.0522043	+	0.998636i	$0.516625\pi$
$492$	0	0
$493$	−2.85110e6	−0.528318
$494$	0	0
$495$	−857280.	−0.157257
$496$	0	0
$497$	3.13372e6	0.569075
$498$	0	0
$499$	−6.90271e6	−1.24099	−0.620495	−	0.784210i	$-0.713068\pi$
$499$	−6.90271e6	−1.24099	−0.620495	+	0.784210i	$0.713068\pi$
$500$	0	0
$501$	−5.84586e6	−1.04053
$502$	0	0
$503$	−1.31502e6	−0.231746	−0.115873	−	0.993264i	$-0.536967\pi$
$503$	−1.31502e6	−0.231746	−0.115873	+	0.993264i	$0.536967\pi$
$504$	0	0
$505$	2.22986e6	0.389089
$506$	0	0
$507$	1.68538e6	0.291192
$508$	0	0
$509$	−2.76594e6	−0.473205	−0.236602	−	0.971607i	$-0.576034\pi$
$509$	−2.76594e6	−0.473205	−0.236602	+	0.971607i	$0.576034\pi$
$510$	0	0
$511$	3.33214e6	0.564509
$512$	0	0
$513$	−917867.	−0.153988
$514$	0	0
$515$	1.71990e6	0.285748
$516$	0	0
$517$	248960.	0.0409640
$518$	0	0
$519$	−5.15287e6	−0.839713
$520$	0	0
$521$	3.25666e6	0.525628	0.262814	−	0.964846i	$-0.415349\pi$
$521$	3.25666e6	0.525628	0.262814	+	0.964846i	$0.415349\pi$
$522$	0	0
$523$	−1.19952e7	−1.91758	−0.958788	−	0.284121i	$-0.908298\pi$
$523$	−1.19952e7	−1.91758	−0.958788	+	0.284121i	$0.908298\pi$
$524$	0	0
$525$	−352894.	−0.0558787
$526$	0	0
$527$	4.31538e6	0.676850
$528$	0	0
$529$	−3.29873e6	−0.512517
$530$	0	0
$531$	467884.	0.0720116
$532$	0	0
$533$	2.30587e6	0.351574
$534$	0	0
$535$	1.53166e6	0.231354
$536$	0	0
$537$	1.00078e6	0.149762
$538$	0	0
$539$	746998.	0.110751
$540$	0	0
$541$	−4.18583e6	−0.614878	−0.307439	−	0.951568i	$-0.599472\pi$
$541$	−4.18583e6	−0.614878	−0.307439	+	0.951568i	$0.599472\pi$
$542$	0	0
$543$	−4.85044e6	−0.705962
$544$	0	0
$545$	2.32594e6	0.335434
$546$	0	0
$547$	−318082.	−0.0454539	−0.0227270	−	0.999742i	$-0.507235\pi$
$547$	−318082.	−0.0454539	−0.0227270	+	0.999742i	$0.507235\pi$
$548$	0	0
$549$	−4.49920e6	−0.637095
$550$	0	0
$551$	−634181.	−0.0889886
$552$	0	0
$553$	1.28119e6	0.178157
$554$	0	0
$555$	−1.51714e6	−0.209071
$556$	0	0
$557$	−4.60496e6	−0.628909	−0.314454	−	0.949273i	$-0.601822\pi$
$557$	−4.60496e6	−0.628909	−0.314454	+	0.949273i	$0.601822\pi$
$558$	0	0
$559$	−8.03182e6	−1.08714
$560$	0	0
$561$	3.63465e6	0.487591
$562$	0	0
$563$	1.38243e7	1.83811	0.919055	−	0.394130i	$-0.128954\pi$
$563$	1.38243e7	1.83811	0.919055	+	0.394130i	$0.128954\pi$
$564$	0	0
$565$	1.45575e6	0.191852
$566$	0	0
$567$	−985766.	−0.128770
$568$	0	0
$569$	−2.09257e6	−0.270957	−0.135478	−	0.990780i	$-0.543257\pi$
$569$	−2.09257e6	−0.270957	−0.135478	+	0.990780i	$0.543257\pi$
$570$	0	0
$571$	157479.	0.0202131	0.0101065	−	0.999949i	$-0.496783\pi$
$571$	157479.	0.0202131	0.0101065	+	0.999949i	$0.496783\pi$
$572$	0	0
$573$	4.80105e6	0.610871
$574$	0	0
$575$	−1.10708e6	−0.139640
$576$	0	0
$577$	1.04875e7	1.31139	0.655693	−	0.755027i	$-0.272376\pi$
$577$	1.04875e7	1.31139	0.655693	+	0.755027i	$0.272376\pi$
$578$	0	0
$579$	1.40934e6	0.174711
$580$	0	0
$581$	1.55093e6	0.190613
$582$	0	0
$583$	1.15087e7	1.40235
$584$	0	0
$585$	−1.30712e6	−0.157916
$586$	0	0
$587$	6.02372e6	0.721555	0.360777	−	0.932652i	$-0.382511\pi$
$587$	6.02372e6	0.721555	0.360777	+	0.932652i	$0.382511\pi$
$588$	0	0
$589$	959887.	0.114007
$590$	0	0
$591$	−5.52648e6	−0.650849
$592$	0	0
$593$	−1.11080e7	−1.29717	−0.648586	−	0.761141i	$-0.724640\pi$
$593$	−1.11080e7	−1.29717	−0.648586	+	0.761141i	$0.724640\pi$
$594$	0	0
$595$	−1.24195e6	−0.143817
$596$	0	0
$597$	5.93455e6	0.681478
$598$	0	0
$599$	−1.63258e7	−1.85912	−0.929558	−	0.368676i	$-0.879811\pi$
$599$	−1.63258e7	−1.85912	−0.929558	+	0.368676i	$0.879811\pi$
$600$	0	0
$601$	−715562.	−0.0808093	−0.0404046	−	0.999183i	$-0.512865\pi$
$601$	−715562.	−0.0808093	−0.0404046	+	0.999183i	$0.512865\pi$
$602$	0	0
$603$	−1.62991e6	−0.182545
$604$	0	0
$605$	−1.60639e6	−0.178428
$606$	0	0
$607$	−1.35532e6	−0.149304	−0.0746520	−	0.997210i	$-0.523785\pi$
$607$	−1.35532e6	−0.149304	−0.0746520	+	0.997210i	$0.523785\pi$
$608$	0	0
$609$	−1.58785e6	−0.173487
$610$	0	0
$611$	379598.	0.0411358
$612$	0	0
$613$	−8.83851e6	−0.950009	−0.475005	−	0.879983i	$-0.657554\pi$
$613$	−8.83851e6	−0.950009	−0.475005	+	0.879983i	$0.657554\pi$
$614$	0	0
$615$	−1.40030e6	−0.149291
$616$	0	0
$617$	1.24032e7	1.31165	0.655827	−	0.754911i	$-0.272320\pi$
$617$	1.24032e7	1.31165	0.655827	+	0.754911i	$0.272320\pi$
$618$	0	0
$619$	−8.55619e6	−0.897540	−0.448770	−	0.893647i	$-0.648138\pi$
$619$	−8.55619e6	−0.897540	−0.448770	+	0.893647i	$0.648138\pi$
$620$	0	0
$621$	−7.20961e6	−0.750210
$622$	0	0
$623$	−3.84249e6	−0.396636
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	808469.	0.0821286
$628$	0	0
$629$	−5.33932e6	−0.538095
$630$	0	0
$631$	−9.61471e6	−0.961309	−0.480654	−	0.876910i	$-0.659601\pi$
$631$	−9.61471e6	−0.961309	−0.480654	+	0.876910i	$0.659601\pi$
$632$	0	0
$633$	2.01445e6	0.199824
$634$	0	0
$635$	−1.51748e6	−0.149344
$636$	0	0
$637$	1.13897e6	0.111215
$638$	0	0
$639$	−7.04887e6	−0.682917
$640$	0	0
$641$	−2.90811e6	−0.279554	−0.139777	−	0.990183i	$-0.544639\pi$
$641$	−2.90811e6	−0.279554	−0.139777	+	0.990183i	$0.544639\pi$
$642$	0	0
$643$	1.37160e7	1.30828	0.654139	−	0.756375i	$-0.273031\pi$
$643$	1.37160e7	1.30828	0.654139	+	0.756375i	$0.273031\pi$
$644$	0	0
$645$	4.87754e6	0.461638
$646$	0	0
$647$	−5.19080e6	−0.487499	−0.243749	−	0.969838i	$-0.578377\pi$
$647$	−5.19080e6	−0.487499	−0.243749	+	0.969838i	$0.578377\pi$
$648$	0	0
$649$	−1.32072e6	−0.123083
$650$	0	0
$651$	2.40335e6	0.222261
$652$	0	0
$653$	−9.56309e6	−0.877638	−0.438819	−	0.898576i	$-0.644603\pi$
$653$	−9.56309e6	−0.877638	−0.438819	+	0.898576i	$0.644603\pi$
$654$	0	0
$655$	−5.39177e6	−0.491052
$656$	0	0
$657$	−7.49518e6	−0.677437
$658$	0	0
$659$	−7.21741e6	−0.647393	−0.323697	−	0.946161i	$-0.604926\pi$
$659$	−7.21741e6	−0.647393	−0.323697	+	0.946161i	$0.604926\pi$
$660$	0	0
$661$	1.64664e6	0.146587	0.0732933	−	0.997310i	$-0.476649\pi$
$661$	1.64664e6	0.146587	0.0732933	+	0.997310i	$0.476649\pi$
$662$	0	0
$663$	5.54188e6	0.489636
$664$	0	0
$665$	−276251.	−0.0242242
$666$	0	0
$667$	−4.98133e6	−0.433541
$668$	0	0
$669$	−5.34773e6	−0.461960
$670$	0	0
$671$	1.27001e7	1.08893
$672$	0	0
$673$	1.15356e7	0.981757	0.490878	−	0.871228i	$-0.336676\pi$
$673$	1.15356e7	0.981757	0.490878	+	0.871228i	$0.336676\pi$
$674$	0	0
$675$	2.54385e6	0.214898
$676$	0	0
$677$	−1.38785e7	−1.16378	−0.581891	−	0.813267i	$-0.697687\pi$
$677$	−1.38785e7	−1.16378	−0.581891	+	0.813267i	$0.697687\pi$
$678$	0	0
$679$	6.04741e6	0.503379
$680$	0	0
$681$	4.00157e6	0.330646
$682$	0	0
$683$	−528511.	−0.0433513	−0.0216757	−	0.999765i	$-0.506900\pi$
$683$	−528511.	−0.0433513	−0.0216757	+	0.999765i	$0.506900\pi$
$684$	0	0
$685$	1.49524e6	0.121754
$686$	0	0
$687$	4.63421e6	0.374614
$688$	0	0
$689$	1.75477e7	1.40823
$690$	0	0
$691$	−5.02023e6	−0.399971	−0.199985	−	0.979799i	$-0.564089\pi$
$691$	−5.02023e6	−0.399971	−0.199985	+	0.979799i	$0.564089\pi$
$692$	0	0
$693$	−1.68027e6	−0.132906
$694$	0	0
$695$	−4.66052e6	−0.365993
$696$	0	0
$697$	−4.92810e6	−0.384236
$698$	0	0
$699$	−6.12361e6	−0.474039
$700$	0	0
$701$	1.46820e7	1.12847	0.564233	−	0.825615i	$-0.309172\pi$
$701$	1.46820e7	1.12847	0.564233	+	0.825615i	$0.309172\pi$
$702$	0	0
$703$	−1.18764e6	−0.0906355
$704$	0	0
$705$	−230521.	−0.0174678
$706$	0	0
$707$	4.37053e6	0.328841
$708$	0	0
$709$	−5.45077e6	−0.407233	−0.203616	−	0.979051i	$-0.565270\pi$
$709$	−5.45077e6	−0.407233	−0.203616	+	0.979051i	$0.565270\pi$
$710$	0	0
$711$	−2.88187e6	−0.213796
$712$	0	0
$713$	7.53966e6	0.555428
$714$	0	0
$715$	3.68968e6	0.269913
$716$	0	0
$717$	1.07115e7	0.778128
$718$	0	0
$719$	1.57039e7	1.13288	0.566441	−	0.824102i	$-0.308320\pi$
$719$	1.57039e7	1.13288	0.566441	+	0.824102i	$0.308320\pi$
$720$	0	0
$721$	3.37099e6	0.241502
$722$	0	0
$723$	−2.16650e6	−0.154139
$724$	0	0
$725$	1.75762e6	0.124188
$726$	0	0
$727$	2.45816e6	0.172494	0.0862470	−	0.996274i	$-0.472513\pi$
$727$	2.45816e6	0.172494	0.0862470	+	0.996274i	$0.472513\pi$
$728$	0	0
$729$	1.36142e7	0.948800
$730$	0	0
$731$	1.71656e7	1.18814
$732$	0	0
$733$	−2.32960e7	−1.60148	−0.800740	−	0.599012i	$-0.795560\pi$
$733$	−2.32960e7	−1.60148	−0.800740	+	0.599012i	$0.795560\pi$
$734$	0	0
$735$	−691673.	−0.0472261
$736$	0	0
$737$	4.60083e6	0.312009
$738$	0	0
$739$	−1.77474e7	−1.19543	−0.597713	−	0.801710i	$-0.703924\pi$
$739$	−1.77474e7	−1.19543	−0.597713	+	0.801710i	$0.703924\pi$
$740$	0	0
$741$	1.23270e6	0.0824731
$742$	0	0
$743$	1.02128e7	0.678689	0.339344	−	0.940662i	$-0.389795\pi$
$743$	1.02128e7	0.678689	0.339344	+	0.940662i	$0.389795\pi$
$744$	0	0
$745$	−9.93117e6	−0.655556
$746$	0	0
$747$	−3.48860e6	−0.228744
$748$	0	0
$749$	3.00205e6	0.195530
$750$	0	0
$751$	−2.60266e7	−1.68391	−0.841953	−	0.539551i	$-0.818594\pi$
$751$	−2.60266e7	−1.68391	−0.841953	+	0.539551i	$0.818594\pi$
$752$	0	0
$753$	2.49354e6	0.160262
$754$	0	0
$755$	8.75716e6	0.559108
$756$	0	0
$757$	7.76057e6	0.492214	0.246107	−	0.969243i	$-0.420849\pi$
$757$	7.76057e6	0.492214	0.246107	+	0.969243i	$0.420849\pi$
$758$	0	0
$759$	6.35031e6	0.400120
$760$	0	0
$761$	1.51975e7	0.951287	0.475643	−	0.879638i	$-0.342215\pi$
$761$	1.51975e7	0.951287	0.475643	+	0.879638i	$0.342215\pi$
$762$	0	0
$763$	4.55884e6	0.283493
$764$	0	0
$765$	2.79359e6	0.172587
$766$	0	0
$767$	−2.01375e6	−0.123599
$768$	0	0
$769$	3.21753e7	1.96203	0.981017	−	0.193924i	$-0.0621214\pi$
$769$	3.21753e7	1.96203	0.981017	+	0.193924i	$0.0621214\pi$
$770$	0	0
$771$	−7.93091e6	−0.480493
$772$	0	0
$773$	−1.34756e7	−0.811148	−0.405574	−	0.914062i	$-0.632928\pi$
$773$	−1.34756e7	−0.811148	−0.405574	+	0.914062i	$0.632928\pi$
$774$	0	0
$775$	−2.66031e6	−0.159103
$776$	0	0
$777$	−2.97360e6	−0.176698
$778$	0	0
$779$	−1.09618e6	−0.0647198
$780$	0	0
$781$	1.98972e7	1.16725
$782$	0	0
$783$	1.14461e7	0.667196
$784$	0	0
$785$	3.29240e6	0.190695
$786$	0	0
$787$	1.09897e6	0.0632483	0.0316242	−	0.999500i	$-0.489932\pi$
$787$	1.09897e6	0.0632483	0.0316242	+	0.999500i	$0.489932\pi$
$788$	0	0
$789$	2.07703e7	1.18782
$790$	0	0
$791$	2.85327e6	0.162144
$792$	0	0
$793$	1.93643e7	1.09350
$794$	0	0
$795$	−1.06563e7	−0.597985
$796$	0	0
$797$	2.35293e7	1.31209	0.656045	−	0.754721i	$-0.272228\pi$
$797$	2.35293e7	1.31209	0.656045	+	0.754721i	$0.272228\pi$
$798$	0	0
$799$	−811276.	−0.0449575
$800$	0	0
$801$	8.64314e6	0.475982
$802$	0	0
$803$	2.11570e7	1.15788
$804$	0	0
$805$	−2.16988e6	−0.118017
$806$	0	0
$807$	−8.90489e6	−0.481332
$808$	0	0
$809$	2.37197e7	1.27420	0.637099	−	0.770782i	$-0.280134\pi$
$809$	2.37197e7	1.27420	0.637099	+	0.770782i	$0.280134\pi$
$810$	0	0
$811$	6.51983e6	0.348084	0.174042	−	0.984738i	$-0.444317\pi$
$811$	6.51983e6	0.348084	0.174042	+	0.984738i	$0.444317\pi$
$812$	0	0
$813$	−1.17194e7	−0.621839
$814$	0	0
$815$	−1.03808e7	−0.547438
$816$	0	0
$817$	3.81821e6	0.200127
$818$	0	0
$819$	−2.56196e6	−0.133464
$820$	0	0
$821$	−2.72390e7	−1.41037	−0.705186	−	0.709023i	$-0.749136\pi$
$821$	−2.72390e7	−1.41037	−0.705186	+	0.709023i	$0.749136\pi$
$822$	0	0
$823$	−459380.	−0.0236414	−0.0118207	−	0.999930i	$-0.503763\pi$
$823$	−459380.	−0.0236414	−0.0118207	+	0.999930i	$0.503763\pi$
$824$	0	0
$825$	−2.24066e6	−0.114615
$826$	0	0
$827$	−2.41158e7	−1.22613	−0.613066	−	0.790032i	$-0.710064\pi$
$827$	−2.41158e7	−1.22613	−0.613066	+	0.790032i	$0.710064\pi$
$828$	0	0
$829$	3.32368e7	1.67971	0.839853	−	0.542814i	$-0.182641\pi$
$829$	3.32368e7	1.67971	0.839853	+	0.542814i	$0.182641\pi$
$830$	0	0
$831$	1.38252e7	0.694493
$832$	0	0
$833$	−2.43422e6	−0.121548
$834$	0	0
$835$	1.26829e7	0.629512
$836$	0	0
$837$	−1.73246e7	−0.854773
$838$	0	0
$839$	2.78277e7	1.36481	0.682406	−	0.730974i	$-0.260934\pi$
$839$	2.78277e7	1.36481	0.682406	+	0.730974i	$0.260934\pi$
$840$	0	0
$841$	−1.26027e7	−0.614432
$842$	0	0
$843$	−1.39377e6	−0.0675496
$844$	0	0
$845$	−3.65654e6	−0.176169
$846$	0	0
$847$	−3.14853e6	−0.150799
$848$	0	0
$849$	4.81010e6	0.229026
$850$	0	0
$851$	−9.32864e6	−0.441565
$852$	0	0
$853$	−201764.	−0.00949449	−0.00474724	−	0.999989i	$-0.501511\pi$
$853$	−201764.	−0.00949449	−0.00474724	+	0.999989i	$0.501511\pi$
$854$	0	0
$855$	621389.	0.0290702
$856$	0	0
$857$	2.84473e7	1.32309	0.661543	−	0.749907i	$-0.269902\pi$
$857$	2.84473e7	1.32309	0.661543	+	0.749907i	$0.269902\pi$
$858$	0	0
$859$	−1.50059e7	−0.693870	−0.346935	−	0.937889i	$-0.612778\pi$
$859$	−1.50059e7	−0.693870	−0.346935	+	0.937889i	$0.612778\pi$
$860$	0	0
$861$	−2.74459e6	−0.126174
$862$	0	0
$863$	3.09759e7	1.41579	0.707893	−	0.706320i	$-0.249646\pi$
$863$	3.09759e7	1.41579	0.707893	+	0.706320i	$0.249646\pi$
$864$	0	0
$865$	1.11795e7	0.508020
$866$	0	0
$867$	4.51701e6	0.204081
$868$	0	0
$869$	8.13478e6	0.365423
$870$	0	0
$871$	7.01504e6	0.313318
$872$	0	0
$873$	−1.36028e7	−0.604078
$874$	0	0
$875$	765625.	0.0338062
$876$	0	0
$877$	−3.21782e7	−1.41274	−0.706371	−	0.707842i	$-0.749669\pi$
$877$	−3.21782e7	−1.41274	−0.706371	+	0.707842i	$0.749669\pi$
$878$	0	0
$879$	−2.26682e7	−0.989567
$880$	0	0
$881$	1.10554e7	0.479883	0.239941	−	0.970787i	$-0.422872\pi$
$881$	1.10554e7	0.479883	0.239941	+	0.970787i	$0.422872\pi$
$882$	0	0
$883$	−2.58212e7	−1.11449	−0.557243	−	0.830349i	$-0.688141\pi$
$883$	−2.58212e7	−1.11449	−0.557243	+	0.830349i	$0.688141\pi$
$884$	0	0
$885$	1.22290e6	0.0524848
$886$	0	0
$887$	−1.09131e6	−0.0465734	−0.0232867	−	0.999729i	$-0.507413\pi$
$887$	−1.09131e6	−0.0465734	−0.0232867	+	0.999729i	$0.507413\pi$
$888$	0	0
$889$	−2.97426e6	−0.126219
$890$	0	0
$891$	−6.25900e6	−0.264126
$892$	0	0
$893$	−180455.	−0.00757253
$894$	0	0
$895$	−2.17125e6	−0.0906050
$896$	0	0
$897$	9.68254e6	0.401799
$898$	0	0
$899$	−1.19701e7	−0.493968
$900$	0	0
$901$	−3.75031e7	−1.53906
$902$	0	0
$903$	9.55998e6	0.390155
$904$	0	0
$905$	1.05233e7	0.427102
$906$	0	0
$907$	−1.01657e7	−0.410319	−0.205159	−	0.978729i	$-0.565771\pi$
$907$	−1.01657e7	−0.410319	−0.205159	+	0.978729i	$0.565771\pi$
$908$	0	0
$909$	−9.83090e6	−0.394624
$910$	0	0
$911$	2.31036e7	0.922323	0.461161	−	0.887316i	$-0.347433\pi$
$911$	2.31036e7	0.922323	0.461161	+	0.887316i	$0.347433\pi$
$912$	0	0
$913$	9.84743e6	0.390972
$914$	0	0
$915$	−1.17595e7	−0.464339
$916$	0	0
$917$	−1.05679e7	−0.415015
$918$	0	0
$919$	3.15416e6	0.123196	0.0615978	−	0.998101i	$-0.480380\pi$
$919$	3.15416e6	0.123196	0.0615978	+	0.998101i	$0.480380\pi$
$920$	0	0
$921$	−1.91296e7	−0.743118
$922$	0	0
$923$	3.03379e7	1.17215
$924$	0	0
$925$	3.29154e6	0.126487
$926$	0	0
$927$	−7.58259e6	−0.289813
$928$	0	0
$929$	677947.	0.0257725	0.0128862	−	0.999917i	$-0.495898\pi$
$929$	677947.	0.0257725	0.0128862	+	0.999917i	$0.495898\pi$
$930$	0	0
$931$	−541452.	−0.0204732
$932$	0	0
$933$	−2.92091e7	−1.09854
$934$	0	0
$935$	−7.88559e6	−0.294988
$936$	0	0
$937$	−2.68845e7	−1.00035	−0.500176	−	0.865924i	$-0.666731\pi$
$937$	−2.68845e7	−1.00035	−0.500176	+	0.865924i	$0.666731\pi$
$938$	0	0
$939$	−2.68454e6	−0.0993586
$940$	0	0
$941$	3.60897e7	1.32864	0.664322	−	0.747446i	$-0.268720\pi$
$941$	3.60897e7	1.32864	0.664322	+	0.747446i	$0.268720\pi$
$942$	0	0
$943$	−8.61018e6	−0.315307
$944$	0	0
$945$	4.98595e6	0.181622
$946$	0	0
$947$	−9.34860e6	−0.338744	−0.169372	−	0.985552i	$-0.554174\pi$
$947$	−9.34860e6	−0.338744	−0.169372	+	0.985552i	$0.554174\pi$
$948$	0	0
$949$	3.22588e7	1.16274
$950$	0	0
$951$	−46204.5	−0.00165666
$952$	0	0
$953$	−4.89023e6	−0.174420	−0.0872101	−	0.996190i	$-0.527795\pi$
$953$	−4.89023e6	−0.174420	−0.0872101	+	0.996190i	$0.527795\pi$
$954$	0	0
$955$	−1.04162e7	−0.369572
$956$	0	0
$957$	−1.00819e7	−0.355845
$958$	0	0
$959$	2.93067e6	0.102901
$960$	0	0
$961$	−1.05114e7	−0.367157
$962$	0	0
$963$	−6.75270e6	−0.234645
$964$	0	0
$965$	−3.05765e6	−0.105699
$966$	0	0
$967$	−5.55405e7	−1.91005	−0.955023	−	0.296532i	$-0.904170\pi$
$967$	−5.55405e7	−1.91005	−0.955023	+	0.296532i	$0.904170\pi$
$968$	0	0
$969$	−2.63453e6	−0.0901351
$970$	0	0
$971$	2.78179e7	0.946839	0.473419	−	0.880837i	$-0.343020\pi$
$971$	2.78179e7	0.946839	0.473419	+	0.880837i	$0.343020\pi$
$972$	0	0
$973$	−9.13462e6	−0.309320
$974$	0	0
$975$	−3.41641e6	−0.115096
$976$	0	0
$977$	−1.10788e7	−0.371328	−0.185664	−	0.982613i	$-0.559444\pi$
$977$	−1.10788e7	−0.371328	−0.185664	+	0.982613i	$0.559444\pi$
$978$	0	0
$979$	−2.43974e7	−0.813554
$980$	0	0
$981$	−1.02545e7	−0.340205
$982$	0	0
$983$	−1.74932e7	−0.577413	−0.288707	−	0.957418i	$-0.593225\pi$
$983$	−1.74932e7	−0.577413	−0.288707	+	0.957418i	$0.593225\pi$
$984$	0	0
$985$	1.19900e7	0.393758
$986$	0	0
$987$	−451821.	−0.0147630
$988$	0	0
$989$	2.99911e7	0.974992
$990$	0	0
$991$	1.61286e7	0.521690	0.260845	−	0.965381i	$-0.415999\pi$
$991$	1.61286e7	0.521690	0.260845	+	0.965381i	$0.415999\pi$
$992$	0	0
$993$	6.83826e6	0.220076
$994$	0	0
$995$	−1.28754e7	−0.412289
$996$	0	0
$997$	−5.32509e7	−1.69664	−0.848319	−	0.529485i	$-0.822385\pi$
$997$	−5.32509e7	−1.69664	−0.848319	+	0.529485i	$0.822385\pi$
$998$	0	0
$999$	2.14354e7	0.679543

Twists

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

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

\(f(q)\)	\(=\)	\(q-11.5231 q^{3} +25.0000 q^{5} +49.0000 q^{7} -110.219 q^{9} +O(q^{10})\)	\(q-11.5231 q^{3} +25.0000 q^{5} +49.0000 q^{7} -110.219 q^{9} +311.119 q^{11} +474.375 q^{13} -288.077 q^{15} -1013.83 q^{17} -225.511 q^{19} -564.631 q^{21} -1771.33 q^{23} +625.000 q^{25} +4070.17 q^{27} +2812.20 q^{29} -4256.50 q^{31} -3585.05 q^{33} +1225.00 q^{35} +5266.46 q^{37} -5466.26 q^{39} +4860.86 q^{41} -16931.4 q^{43} -2755.47 q^{45} +800.206 q^{47} +2401.00 q^{49} +11682.5 q^{51} +36991.3 q^{53} +7777.98 q^{55} +2598.58 q^{57} -4245.05 q^{59} +40820.6 q^{61} -5400.72 q^{63} +11859.4 q^{65} +14788.0 q^{67} +20411.2 q^{69} +63953.5 q^{71} +68002.8 q^{73} -7201.92 q^{75} +15244.8 q^{77} +26146.8 q^{79} -20117.7 q^{81} +31651.6 q^{83} -25345.9 q^{85} -32405.2 q^{87} -78418.1 q^{89} +23244.4 q^{91} +49047.9 q^{93} -5637.78 q^{95} +123416. q^{97} -34291.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 15 q^{3} + 125 q^{5} + 245 q^{7} + 1132 q^{9}+O(q^{10}) \) 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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