LMFDB - Embedded newform 280.6.a.h.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("280.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$44.9074695476$
Analytic rank:	$1$
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			1.3
Root			$18.7529$ 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)

$f(q)$	$=$	$q+19.7529 q^{3} -25.0000 q^{5} -49.0000 q^{7} +147.178 q^{9} +O(q^{10})$	$q+19.7529 q^{3} -25.0000 q^{5} -49.0000 q^{7} +147.178 q^{9} +292.876 q^{11} -1048.73 q^{13} -493.823 q^{15} +554.009 q^{17} -460.823 q^{19} -967.893 q^{21} +1428.67 q^{23} +625.000 q^{25} -1892.76 q^{27} -7927.45 q^{29} -1965.57 q^{31} +5785.15 q^{33} +1225.00 q^{35} -12282.9 q^{37} -20715.6 q^{39} -13523.7 q^{41} -8692.93 q^{43} -3679.45 q^{45} +26090.5 q^{47} +2401.00 q^{49} +10943.3 q^{51} -4436.24 q^{53} -7321.90 q^{55} -9102.60 q^{57} +46358.5 q^{59} -53221.8 q^{61} -7211.73 q^{63} +26218.3 q^{65} +18348.2 q^{67} +28220.5 q^{69} +463.444 q^{71} -11113.8 q^{73} +12345.6 q^{75} -14350.9 q^{77} -49849.7 q^{79} -73151.9 q^{81} -112538. q^{83} -13850.2 q^{85} -156590. q^{87} +132348. q^{89} +51387.9 q^{91} -38825.8 q^{93} +11520.6 q^{95} +105866. q^{97} +43104.9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	19.7529	1.26715	0.633575	−	0.773681i	$-0.281587\pi$
$3$	19.7529	1.26715	0.633575	+	0.773681i	$0.281587\pi$
$4$	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
$9$	147.178	0.605671
$10$	0	0
$11$	292.876	0.729797	0.364898	−	0.931047i	$-0.381104\pi$
$11$	292.876	0.729797	0.364898	+	0.931047i	$0.381104\pi$
$12$	0	0
$13$	−1048.73	−1.72110	−0.860551	−	0.509364i	$-0.829881\pi$
$13$	−1048.73	−1.72110	−0.860551	+	0.509364i	$0.829881\pi$
$14$	0	0
$15$	−493.823	−0.566687
$16$	0	0
$17$	554.009	0.464937	0.232469	−	0.972604i	$-0.425320\pi$
$17$	554.009	0.464937	0.232469	+	0.972604i	$0.425320\pi$
$18$	0	0
$19$	−460.823	−0.292853	−0.146427	−	0.989222i	$-0.546777\pi$
$19$	−460.823	−0.292853	−0.146427	+	0.989222i	$0.546777\pi$
$20$	0	0
$21$	−967.893	−0.478938
$22$	0	0
$23$	1428.67	0.563136	0.281568	−	0.959541i	$-0.409145\pi$
$23$	1428.67	0.563136	0.281568	+	0.959541i	$0.409145\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	−1892.76	−0.499674
$28$	0	0
$29$	−7927.45	−1.75040	−0.875202	−	0.483758i	$-0.839272\pi$
$29$	−7927.45	−1.75040	−0.875202	+	0.483758i	$0.839272\pi$
$30$	0	0
$31$	−1965.57	−0.367354	−0.183677	−	0.982987i	$-0.558800\pi$
$31$	−1965.57	−0.367354	−0.183677	+	0.982987i	$0.558800\pi$
$32$	0	0
$33$	5785.15	0.924762
$34$	0	0
$35$	1225.00	0.169031
$36$	0	0
$37$	−12282.9	−1.47501	−0.737506	−	0.675341i	$-0.763996\pi$
$37$	−12282.9	−1.47501	−0.737506	+	0.675341i	$0.763996\pi$
$38$	0	0
$39$	−20715.6	−2.18090
$40$	0	0
$41$	−13523.7	−1.25642	−0.628211	−	0.778043i	$-0.716213\pi$
$41$	−13523.7	−1.25642	−0.628211	+	0.778043i	$0.716213\pi$
$42$	0	0
$43$	−8692.93	−0.716960	−0.358480	−	0.933537i	$-0.616705\pi$
$43$	−8692.93	−0.716960	−0.358480	+	0.933537i	$0.616705\pi$
$44$	0	0
$45$	−3679.45	−0.270864
$46$	0	0
$47$	26090.5	1.72281	0.861404	−	0.507920i	$-0.169586\pi$
$47$	26090.5	1.72281	0.861404	+	0.507920i	$0.169586\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	10943.3	0.589146
$52$	0	0
$53$	−4436.24	−0.216933	−0.108466	−	0.994100i	$-0.534594\pi$
$53$	−4436.24	−0.216933	−0.108466	+	0.994100i	$0.534594\pi$
$54$	0	0
$55$	−7321.90	−0.326375
$56$	0	0
$57$	−9102.60	−0.371089
$58$	0	0
$59$	46358.5	1.73380	0.866901	−	0.498481i	$-0.166109\pi$
$59$	46358.5	1.73380	0.866901	+	0.498481i	$0.166109\pi$
$60$	0	0
$61$	−53221.8	−1.83132	−0.915661	−	0.401951i	$-0.868332\pi$
$61$	−53221.8	−1.83132	−0.915661	+	0.401951i	$0.868332\pi$
$62$	0	0
$63$	−7211.73	−0.228922
$64$	0	0
$65$	26218.3	0.769701
$66$	0	0
$67$	18348.2	0.499351	0.249676	−	0.968330i	$-0.419676\pi$
$67$	18348.2	0.499351	0.249676	+	0.968330i	$0.419676\pi$
$68$	0	0
$69$	28220.5	0.713579
$70$	0	0
$71$	463.444	0.0109107	0.00545534	−	0.999985i	$-0.498264\pi$
$71$	463.444	0.0109107	0.00545534	+	0.999985i	$0.498264\pi$
$72$	0	0
$73$	−11113.8	−0.244093	−0.122046	−	0.992524i	$-0.538946\pi$
$73$	−11113.8	−0.244093	−0.122046	+	0.992524i	$0.538946\pi$
$74$	0	0
$75$	12345.6	0.253430
$76$	0	0
$77$	−14350.9	−0.275837
$78$	0	0
$79$	−49849.7	−0.898658	−0.449329	−	0.893366i	$-0.648337\pi$
$79$	−49849.7	−0.898658	−0.449329	+	0.893366i	$0.648337\pi$
$80$	0	0
$81$	−73151.9	−1.23883
$82$	0	0
$83$	−112538.	−1.79310	−0.896548	−	0.442947i	$-0.853933\pi$
$83$	−112538.	−1.79310	−0.896548	+	0.442947i	$0.853933\pi$
$84$	0	0
$85$	−13850.2	−0.207926
$86$	0	0
$87$	−156590.	−2.21803
$88$	0	0
$89$	132348.	1.77110	0.885550	−	0.464543i	$-0.153781\pi$
$89$	132348.	1.77110	0.885550	+	0.464543i	$0.153781\pi$
$90$	0	0
$91$	51387.9	0.650516
$92$	0	0
$93$	−38825.8	−0.465493
$94$	0	0
$95$	11520.6	0.130968
$96$	0	0
$97$	105866.	1.14242	0.571212	−	0.820803i	$-0.306473\pi$
$97$	105866.	1.14242	0.571212	+	0.820803i	$0.306473\pi$
$98$	0	0
$99$	43104.9	0.442017
$100$	0	0
$101$	17941.9	0.175011	0.0875055	−	0.996164i	$-0.472110\pi$
$101$	17941.9	0.175011	0.0875055	+	0.996164i	$0.472110\pi$
$102$	0	0
$103$	−69432.1	−0.644863	−0.322431	−	0.946593i	$-0.604500\pi$
$103$	−69432.1	−0.644863	−0.322431	+	0.946593i	$0.604500\pi$
$104$	0	0
$105$	24197.3	0.214188
$106$	0	0
$107$	−129044.	−1.08963	−0.544816	−	0.838556i	$-0.683400\pi$
$107$	−129044.	−1.08963	−0.544816	+	0.838556i	$0.683400\pi$
$108$	0	0
$109$	20407.7	0.164524	0.0822619	−	0.996611i	$-0.473786\pi$
$109$	20407.7	0.164524	0.0822619	+	0.996611i	$0.473786\pi$
$110$	0	0
$111$	−242622.	−1.86906
$112$	0	0
$113$	−97622.3	−0.719206	−0.359603	−	0.933105i	$-0.617088\pi$
$113$	−97622.3	−0.719206	−0.359603	+	0.933105i	$0.617088\pi$
$114$	0	0
$115$	−35716.8	−0.251842
$116$	0	0
$117$	−154351.	−1.04242
$118$	0	0
$119$	−27146.4	−0.175730
$120$	0	0
$121$	−75274.8	−0.467397
$122$	0	0
$123$	−267133.	−1.59208
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	242725.	1.33538	0.667690	−	0.744440i	$-0.267283\pi$
$127$	242725.	1.33538	0.667690	+	0.744440i	$0.267283\pi$
$128$	0	0
$129$	−171711.	−0.908497
$130$	0	0
$131$	−27198.0	−0.138471	−0.0692354	−	0.997600i	$-0.522056\pi$
$131$	−27198.0	−0.138471	−0.0692354	+	0.997600i	$0.522056\pi$
$132$	0	0
$133$	22580.3	0.110688
$134$	0	0
$135$	47319.1	0.223461
$136$	0	0
$137$	−150445.	−0.684822	−0.342411	−	0.939550i	$-0.611243\pi$
$137$	−150445.	−0.684822	−0.342411	+	0.939550i	$0.611243\pi$
$138$	0	0
$139$	−213448.	−0.937033	−0.468516	−	0.883455i	$-0.655211\pi$
$139$	−213448.	−0.937033	−0.468516	+	0.883455i	$0.655211\pi$
$140$	0	0
$141$	515363.	2.18306
$142$	0	0
$143$	−307149.	−1.25605
$144$	0	0
$145$	198186.	0.782805
$146$	0	0
$147$	47426.8	0.181022
$148$	0	0
$149$	145572.	0.537169	0.268585	−	0.963256i	$-0.413444\pi$
$149$	145572.	0.537169	0.268585	+	0.963256i	$0.413444\pi$
$150$	0	0
$151$	−150775.	−0.538129	−0.269065	−	0.963122i	$-0.586715\pi$
$151$	−150775.	−0.538129	−0.269065	+	0.963122i	$0.586715\pi$
$152$	0	0
$153$	81538.0	0.281599
$154$	0	0
$155$	49139.3	0.164286
$156$	0	0
$157$	334088.	1.08171	0.540856	−	0.841115i	$-0.318100\pi$
$157$	334088.	1.08171	0.540856	+	0.841115i	$0.318100\pi$
$158$	0	0
$159$	−87628.7	−0.274887
$160$	0	0
$161$	−70005.0	−0.212845
$162$	0	0
$163$	373810.	1.10200	0.551001	−	0.834505i	$-0.314246\pi$
$163$	373810.	1.10200	0.551001	+	0.834505i	$0.314246\pi$
$164$	0	0
$165$	−144629.	−0.413566
$166$	0	0
$167$	−244046.	−0.677143	−0.338571	−	0.940941i	$-0.609944\pi$
$167$	−244046.	−0.677143	−0.338571	+	0.940941i	$0.609944\pi$
$168$	0	0
$169$	728549.	1.96219
$170$	0	0
$171$	−67823.0	−0.177373
$172$	0	0
$173$	523251.	1.32921	0.664607	−	0.747193i	$-0.268599\pi$
$173$	523251.	1.32921	0.664607	+	0.747193i	$0.268599\pi$
$174$	0	0
$175$	−30625.0	−0.0755929
$176$	0	0
$177$	915716.	2.19699
$178$	0	0
$179$	387379.	0.903656	0.451828	−	0.892105i	$-0.350772\pi$
$179$	387379.	0.903656	0.451828	+	0.892105i	$0.350772\pi$
$180$	0	0
$181$	−530318.	−1.20321	−0.601603	−	0.798795i	$-0.705471\pi$
$181$	−530318.	−1.20321	−0.601603	+	0.798795i	$0.705471\pi$
$182$	0	0
$183$	−1.05129e6	−2.32056
$184$	0	0
$185$	307072.	0.659645
$186$	0	0
$187$	162256.	0.339310
$188$	0	0
$189$	92745.4	0.188859
$190$	0	0
$191$	305356.	0.605652	0.302826	−	0.953046i	$-0.402070\pi$
$191$	305356.	0.605652	0.302826	+	0.953046i	$0.402070\pi$
$192$	0	0
$193$	−94494.4	−0.182605	−0.0913025	−	0.995823i	$-0.529103\pi$
$193$	−94494.4	−0.182605	−0.0913025	+	0.995823i	$0.529103\pi$
$194$	0	0
$195$	517889.	0.975327
$196$	0	0
$197$	106457.	0.195438	0.0977188	−	0.995214i	$-0.468845\pi$
$197$	106457.	0.195438	0.0977188	+	0.995214i	$0.468845\pi$
$198$	0	0
$199$	493792.	0.883916	0.441958	−	0.897036i	$-0.354284\pi$
$199$	493792.	0.883916	0.441958	+	0.897036i	$0.354284\pi$
$200$	0	0
$201$	362431.	0.632754
$202$	0	0
$203$	388445.	0.661591
$204$	0	0
$205$	338092.	0.561889
$206$	0	0
$207$	210270.	0.341075
$208$	0	0
$209$	−134964.	−0.213723
$210$	0	0
$211$	336286.	0.520000	0.260000	−	0.965609i	$-0.416277\pi$
$211$	336286.	0.520000	0.260000	+	0.965609i	$0.416277\pi$
$212$	0	0
$213$	9154.38	0.0138255
$214$	0	0
$215$	217323.	0.320634
$216$	0	0
$217$	96313.0	0.138847
$218$	0	0
$219$	−219530.	−0.309302
$220$	0	0
$221$	−581008.	−0.800205
$222$	0	0
$223$	−374333.	−0.504076	−0.252038	−	0.967717i	$-0.581101\pi$
$223$	−374333.	−0.504076	−0.252038	+	0.967717i	$0.581101\pi$
$224$	0	0
$225$	91986.3	0.121134
$226$	0	0
$227$	264483.	0.340669	0.170335	−	0.985386i	$-0.445515\pi$
$227$	264483.	0.340669	0.170335	+	0.985386i	$0.445515\pi$
$228$	0	0
$229$	1.04041e6	1.31104	0.655519	−	0.755179i	$-0.272450\pi$
$229$	1.04041e6	1.31104	0.655519	+	0.755179i	$0.272450\pi$
$230$	0	0
$231$	−283473.	−0.349527
$232$	0	0
$233$	582086.	0.702421	0.351210	−	0.936297i	$-0.385770\pi$
$233$	582086.	0.702421	0.351210	+	0.936297i	$0.385770\pi$
$234$	0	0
$235$	−652261.	−0.770463
$236$	0	0
$237$	−984677.	−1.13874
$238$	0	0
$239$	225666.	0.255547	0.127773	−	0.991803i	$-0.459217\pi$
$239$	225666.	0.255547	0.127773	+	0.991803i	$0.459217\pi$
$240$	0	0
$241$	1.27202e6	1.41075	0.705375	−	0.708834i	$-0.250778\pi$
$241$	1.27202e6	1.41075	0.705375	+	0.708834i	$0.250778\pi$
$242$	0	0
$243$	−985023.	−1.07012
$244$	0	0
$245$	−60025.0	−0.0638877
$246$	0	0
$247$	483280.	0.504030
$248$	0	0
$249$	−2.22295e6	−2.27212
$250$	0	0
$251$	573681.	0.574759	0.287380	−	0.957817i	$-0.407216\pi$
$251$	573681.	0.574759	0.287380	+	0.957817i	$0.407216\pi$
$252$	0	0
$253$	418424.	0.410975
$254$	0	0
$255$	−273582.	−0.263474
$256$	0	0
$257$	256444.	0.242192	0.121096	−	0.992641i	$-0.461359\pi$
$257$	256444.	0.242192	0.121096	+	0.992641i	$0.461359\pi$
$258$	0	0
$259$	601860.	0.557502
$260$	0	0
$261$	−1.16675e6	−1.06017
$262$	0	0
$263$	−808407.	−0.720677	−0.360339	−	0.932822i	$-0.617339\pi$
$263$	−808407.	−0.720677	−0.360339	+	0.932822i	$0.617339\pi$
$264$	0	0
$265$	110906.	0.0970153
$266$	0	0
$267$	2.61427e6	2.24425
$268$	0	0
$269$	−1.65026e6	−1.39050	−0.695252	−	0.718766i	$-0.744707\pi$
$269$	−1.65026e6	−1.39050	−0.695252	+	0.718766i	$0.744707\pi$
$270$	0	0
$271$	−2.10446e6	−1.74068	−0.870338	−	0.492456i	$-0.836099\pi$
$271$	−2.10446e6	−1.74068	−0.870338	+	0.492456i	$0.836099\pi$
$272$	0	0
$273$	1.01506e6	0.824302
$274$	0	0
$275$	183047.	0.145959
$276$	0	0
$277$	974062.	0.762759	0.381379	−	0.924419i	$-0.375449\pi$
$277$	974062.	0.762759	0.381379	+	0.924419i	$0.375449\pi$
$278$	0	0
$279$	−289289.	−0.222496
$280$	0	0
$281$	−847554.	−0.640327	−0.320163	−	0.947362i	$-0.603738\pi$
$281$	−847554.	−0.640327	−0.320163	+	0.947362i	$0.603738\pi$
$282$	0	0
$283$	−727633.	−0.540065	−0.270032	−	0.962851i	$-0.587034\pi$
$283$	−727633.	−0.540065	−0.270032	+	0.962851i	$0.587034\pi$
$284$	0	0
$285$	227565.	0.165956
$286$	0	0
$287$	662661.	0.474883
$288$	0	0
$289$	−1.11293e6	−0.783833
$290$	0	0
$291$	2.09116e6	1.44762
$292$	0	0
$293$	2.07252e6	1.41036	0.705179	−	0.709030i	$-0.250867\pi$
$293$	2.07252e6	1.41036	0.705179	+	0.709030i	$0.250867\pi$
$294$	0	0
$295$	−1.15896e6	−0.775380
$296$	0	0
$297$	−554344.	−0.364660
$298$	0	0
$299$	−1.49830e6	−0.969215
$300$	0	0
$301$	425954.	0.270986
$302$	0	0
$303$	354405.	0.221765
$304$	0	0
$305$	1.33054e6	0.818992
$306$	0	0
$307$	−98160.6	−0.0594417	−0.0297208	−	0.999558i	$-0.509462\pi$
$307$	−98160.6	−0.0594417	−0.0297208	+	0.999558i	$0.509462\pi$
$308$	0	0
$309$	−1.37149e6	−0.817138
$310$	0	0
$311$	2.51651e6	1.47536	0.737680	−	0.675150i	$-0.235921\pi$
$311$	2.51651e6	1.47536	0.737680	+	0.675150i	$0.235921\pi$
$312$	0	0
$313$	1.05770e6	0.610240	0.305120	−	0.952314i	$-0.401303\pi$
$313$	1.05770e6	0.610240	0.305120	+	0.952314i	$0.401303\pi$
$314$	0	0
$315$	180293.	0.102377
$316$	0	0
$317$	−1.31622e6	−0.735666	−0.367833	−	0.929892i	$-0.619900\pi$
$317$	−1.31622e6	−0.735666	−0.367833	+	0.929892i	$0.619900\pi$
$318$	0	0
$319$	−2.32176e6	−1.27744
$320$	0	0
$321$	−2.54900e6	−1.38073
$322$	0	0
$323$	−255300.	−0.136158
$324$	0	0
$325$	−655459.	−0.344221
$326$	0	0
$327$	403113.	0.208476
$328$	0	0
$329$	−1.27843e6	−0.651160
$330$	0	0
$331$	1.13446e6	0.569138	0.284569	−	0.958656i	$-0.408150\pi$
$331$	1.13446e6	0.569138	0.284569	+	0.958656i	$0.408150\pi$
$332$	0	0
$333$	−1.80777e6	−0.893372
$334$	0	0
$335$	−458705.	−0.223317
$336$	0	0
$337$	3.58788e6	1.72093	0.860466	−	0.509508i	$-0.170172\pi$
$337$	3.58788e6	1.72093	0.860466	+	0.509508i	$0.170172\pi$
$338$	0	0
$339$	−1.92833e6	−0.911342
$340$	0	0
$341$	−575668.	−0.268094
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	−705512.	−0.319122
$346$	0	0
$347$	305835.	0.136353	0.0681764	−	0.997673i	$-0.478282\pi$
$347$	305835.	0.136353	0.0681764	+	0.997673i	$0.478282\pi$
$348$	0	0
$349$	−2.10377e6	−0.924558	−0.462279	−	0.886734i	$-0.652968\pi$
$349$	−2.10377e6	−0.924558	−0.462279	+	0.886734i	$0.652968\pi$
$350$	0	0
$351$	1.98500e6	0.859990
$352$	0	0
$353$	−1.24649e6	−0.532417	−0.266209	−	0.963915i	$-0.585771\pi$
$353$	−1.24649e6	−0.532417	−0.266209	+	0.963915i	$0.585771\pi$
$354$	0	0
$355$	−11586.1	−0.00487940
$356$	0	0
$357$	−536222.	−0.222676
$358$	0	0
$359$	−2.13532e6	−0.874435	−0.437218	−	0.899356i	$-0.644036\pi$
$359$	−2.13532e6	−0.874435	−0.437218	+	0.899356i	$0.644036\pi$
$360$	0	0
$361$	−2.26374e6	−0.914237
$362$	0	0
$363$	−1.48690e6	−0.592263
$364$	0	0
$365$	277845.	0.109162
$366$	0	0
$367$	3.46581e6	1.34319	0.671597	−	0.740916i	$-0.265609\pi$
$367$	3.46581e6	1.34319	0.671597	+	0.740916i	$0.265609\pi$
$368$	0	0
$369$	−1.99039e6	−0.760979
$370$	0	0
$371$	217376.	0.0819929
$372$	0	0
$373$	2.46366e6	0.916873	0.458436	−	0.888727i	$-0.348410\pi$
$373$	2.46366e6	0.916873	0.458436	+	0.888727i	$0.348410\pi$
$374$	0	0
$375$	−308639.	−0.113337
$376$	0	0
$377$	8.31378e6	3.01263
$378$	0	0
$379$	−1.40816e6	−0.503565	−0.251782	−	0.967784i	$-0.581017\pi$
$379$	−1.40816e6	−0.503565	−0.251782	+	0.967784i	$0.581017\pi$
$380$	0	0
$381$	4.79452e6	1.69213
$382$	0	0
$383$	−4.90977e6	−1.71027	−0.855133	−	0.518408i	$-0.826525\pi$
$383$	−4.90977e6	−1.71027	−0.855133	+	0.518408i	$0.826525\pi$
$384$	0	0
$385$	358773.	0.123358
$386$	0	0
$387$	−1.27941e6	−0.434242
$388$	0	0
$389$	−2.93543e6	−0.983552	−0.491776	−	0.870722i	$-0.663652\pi$
$389$	−2.93543e6	−0.983552	−0.491776	+	0.870722i	$0.663652\pi$
$390$	0	0
$391$	791498.	0.261823
$392$	0	0
$393$	−537239.	−0.175463
$394$	0	0
$395$	1.24624e6	0.401892
$396$	0	0
$397$	−743292.	−0.236692	−0.118346	−	0.992972i	$-0.537759\pi$
$397$	−743292.	−0.236692	−0.118346	+	0.992972i	$0.537759\pi$
$398$	0	0
$399$	446027.	0.140258
$400$	0	0
$401$	−2.97161e6	−0.922849	−0.461424	−	0.887180i	$-0.652661\pi$
$401$	−2.97161e6	−0.922849	−0.461424	+	0.887180i	$0.652661\pi$
$402$	0	0
$403$	2.06136e6	0.632254
$404$	0	0
$405$	1.82880e6	0.554023
$406$	0	0
$407$	−3.59735e6	−1.07646
$408$	0	0
$409$	−303453.	−0.0896981	−0.0448491	−	0.998994i	$-0.514281\pi$
$409$	−303453.	−0.0896981	−0.0448491	+	0.998994i	$0.514281\pi$
$410$	0	0
$411$	−2.97174e6	−0.867772
$412$	0	0
$413$	−2.27157e6	−0.655315
$414$	0	0
$415$	2.81345e6	0.801897
$416$	0	0
$417$	−4.21622e6	−1.18736
$418$	0	0
$419$	1.68010e6	0.467519	0.233759	−	0.972294i	$-0.424897\pi$
$419$	1.68010e6	0.467519	0.233759	+	0.972294i	$0.424897\pi$
$420$	0	0
$421$	−4.44807e6	−1.22311	−0.611556	−	0.791201i	$-0.709456\pi$
$421$	−4.44807e6	−1.22311	−0.611556	+	0.791201i	$0.709456\pi$
$422$	0	0
$423$	3.83994e6	1.04346
$424$	0	0
$425$	346256.	0.0929875
$426$	0	0
$427$	2.60787e6	0.692175
$428$	0	0
$429$	−6.06709e6	−1.59161
$430$	0	0
$431$	−6.23706e6	−1.61728	−0.808642	−	0.588300i	$-0.799797\pi$
$431$	−6.23706e6	−1.61728	−0.808642	+	0.588300i	$0.799797\pi$
$432$	0	0
$433$	2.92016e6	0.748491	0.374246	−	0.927330i	$-0.377902\pi$
$433$	2.92016e6	0.748491	0.374246	+	0.927330i	$0.377902\pi$
$434$	0	0
$435$	3.91476e6	0.991931
$436$	0	0
$437$	−658365.	−0.164916
$438$	0	0
$439$	−2.82042e6	−0.698476	−0.349238	−	0.937034i	$-0.613560\pi$
$439$	−2.82042e6	−0.698476	−0.349238	+	0.937034i	$0.613560\pi$
$440$	0	0
$441$	353375.	0.0865245
$442$	0	0
$443$	−2.68145e6	−0.649172	−0.324586	−	0.945856i	$-0.605225\pi$
$443$	−2.68145e6	−0.649172	−0.324586	+	0.945856i	$0.605225\pi$
$444$	0	0
$445$	−3.30871e6	−0.792060
$446$	0	0
$447$	2.87547e6	0.680675
$448$	0	0
$449$	−4.99181e6	−1.16854	−0.584269	−	0.811560i	$-0.698619\pi$
$449$	−4.99181e6	−1.16854	−0.584269	+	0.811560i	$0.698619\pi$
$450$	0	0
$451$	−3.96076e6	−0.916933
$452$	0	0
$453$	−2.97825e6	−0.681891
$454$	0	0
$455$	−1.28470e6	−0.290919
$456$	0	0
$457$	−2.39421e6	−0.536255	−0.268127	−	0.963383i	$-0.586405\pi$
$457$	−2.39421e6	−0.536255	−0.268127	+	0.963383i	$0.586405\pi$
$458$	0	0
$459$	−1.04861e6	−0.232317
$460$	0	0
$461$	1.67008e6	0.366002	0.183001	−	0.983113i	$-0.441419\pi$
$461$	1.67008e6	0.366002	0.183001	+	0.983113i	$0.441419\pi$
$462$	0	0
$463$	−965618.	−0.209340	−0.104670	−	0.994507i	$-0.533379\pi$
$463$	−965618.	−0.209340	−0.104670	+	0.994507i	$0.533379\pi$
$464$	0	0
$465$	970645.	0.208175
$466$	0	0
$467$	−1.20320e6	−0.255297	−0.127648	−	0.991819i	$-0.540743\pi$
$467$	−1.20320e6	−0.255297	−0.127648	+	0.991819i	$0.540743\pi$
$468$	0	0
$469$	−899062.	−0.188737
$470$	0	0
$471$	6.59921e6	1.37069
$472$	0	0
$473$	−2.54595e6	−0.523235
$474$	0	0
$475$	−288014.	−0.0585706
$476$	0	0
$477$	−652917.	−0.131390
$478$	0	0
$479$	5.89186e6	1.17331	0.586656	−	0.809836i	$-0.300444\pi$
$479$	5.89186e6	1.17331	0.586656	+	0.809836i	$0.300444\pi$
$480$	0	0
$481$	1.28815e7	2.53865
$482$	0	0
$483$	−1.38280e6	−0.269707
$484$	0	0
$485$	−2.64665e6	−0.510907
$486$	0	0
$487$	7.54374e6	1.44133	0.720667	−	0.693282i	$-0.243836\pi$
$487$	7.54374e6	1.44133	0.720667	+	0.693282i	$0.243836\pi$
$488$	0	0
$489$	7.38385e6	1.39640
$490$	0	0
$491$	−2.06558e6	−0.386668	−0.193334	−	0.981133i	$-0.561930\pi$
$491$	−2.06558e6	−0.386668	−0.193334	+	0.981133i	$0.561930\pi$
$492$	0	0
$493$	−4.39188e6	−0.813828
$494$	0	0
$495$	−1.07762e6	−0.197676
$496$	0	0
$497$	−22708.8	−0.00412385
$498$	0	0
$499$	−5.39110e6	−0.969228	−0.484614	−	0.874728i	$-0.661040\pi$
$499$	−5.39110e6	−0.969228	−0.484614	+	0.874728i	$0.661040\pi$
$500$	0	0
$501$	−4.82062e6	−0.858042
$502$	0	0
$503$	9.10223e6	1.60409	0.802043	−	0.597266i	$-0.203746\pi$
$503$	9.10223e6	1.60409	0.802043	+	0.597266i	$0.203746\pi$
$504$	0	0
$505$	−448548.	−0.0782673
$506$	0	0
$507$	1.43910e7	2.48640
$508$	0	0
$509$	−1.01605e7	−1.73828	−0.869138	−	0.494569i	$-0.835326\pi$
$509$	−1.01605e7	−1.73828	−0.869138	+	0.494569i	$0.835326\pi$
$510$	0	0
$511$	544576.	0.0922584
$512$	0	0
$513$	872228.	0.146331
$514$	0	0
$515$	1.73580e6	0.288391
$516$	0	0
$517$	7.64126e6	1.25730
$518$	0	0
$519$	1.03357e7	1.68431
$520$	0	0
$521$	−2.52081e6	−0.406860	−0.203430	−	0.979089i	$-0.565209\pi$
$521$	−2.52081e6	−0.406860	−0.203430	+	0.979089i	$0.565209\pi$
$522$	0	0
$523$	−4.93799e6	−0.789398	−0.394699	−	0.918811i	$-0.629151\pi$
$523$	−4.93799e6	−0.789398	−0.394699	+	0.918811i	$0.629151\pi$
$524$	0	0
$525$	−604933.	−0.0957876
$526$	0	0
$527$	−1.08894e6	−0.170797
$528$	0	0
$529$	−4.39523e6	−0.682878
$530$	0	0
$531$	6.82296e6	1.05011
$532$	0	0
$533$	1.41828e7	2.16243
$534$	0	0
$535$	3.22611e6	0.487298
$536$	0	0
$537$	7.65186e6	1.14507
$538$	0	0
$539$	703195.	0.104257
$540$	0	0
$541$	−1.34361e7	−1.97369	−0.986846	−	0.161664i	$-0.948314\pi$
$541$	−1.34361e7	−1.97369	−0.986846	+	0.161664i	$0.948314\pi$
$542$	0	0
$543$	−1.04753e7	−1.52464
$544$	0	0
$545$	−510193.	−0.0735773
$546$	0	0
$547$	−680412.	−0.0972308	−0.0486154	−	0.998818i	$-0.515481\pi$
$547$	−680412.	−0.0972308	−0.0486154	+	0.998818i	$0.515481\pi$
$548$	0	0
$549$	−7.83308e6	−1.10918
$550$	0	0
$551$	3.65315e6	0.512611
$552$	0	0
$553$	2.44263e6	0.339661
$554$	0	0
$555$	6.06556e6	0.835870
$556$	0	0
$557$	516325.	0.0705156	0.0352578	−	0.999378i	$-0.488775\pi$
$557$	516325.	0.0705156	0.0352578	+	0.999378i	$0.488775\pi$
$558$	0	0
$559$	9.11657e6	1.23396
$560$	0	0
$561$	3.20503e6	0.429957
$562$	0	0
$563$	397195.	0.0528120	0.0264060	−	0.999651i	$-0.491594\pi$
$563$	397195.	0.0528120	0.0264060	+	0.999651i	$0.491594\pi$
$564$	0	0
$565$	2.44056e6	0.321638
$566$	0	0
$567$	3.58444e6	0.468235
$568$	0	0
$569$	8.21075e6	1.06317	0.531584	−	0.847005i	$-0.321597\pi$
$569$	8.21075e6	1.06317	0.531584	+	0.847005i	$0.321597\pi$
$570$	0	0
$571$	4.69460e6	0.602571	0.301286	−	0.953534i	$-0.402584\pi$
$571$	4.69460e6	0.602571	0.301286	+	0.953534i	$0.402584\pi$
$572$	0	0
$573$	6.03168e6	0.767452
$574$	0	0
$575$	892921.	0.112627
$576$	0	0
$577$	6.13497e6	0.767137	0.383568	−	0.923512i	$-0.374695\pi$
$577$	6.13497e6	0.767137	0.383568	+	0.923512i	$0.374695\pi$
$578$	0	0
$579$	−1.86654e6	−0.231388
$580$	0	0
$581$	5.51435e6	0.677726
$582$	0	0
$583$	−1.29927e6	−0.158317
$584$	0	0
$585$	3.85877e6	0.466186
$586$	0	0
$587$	4.81792e6	0.577118	0.288559	−	0.957462i	$-0.406824\pi$
$587$	4.81792e6	0.577118	0.288559	+	0.957462i	$0.406824\pi$
$588$	0	0
$589$	905780.	0.107581
$590$	0	0
$591$	2.10284e6	0.247649
$592$	0	0
$593$	−1.39237e7	−1.62599	−0.812994	−	0.582272i	$-0.802164\pi$
$593$	−1.39237e7	−1.62599	−0.812994	+	0.582272i	$0.802164\pi$
$594$	0	0
$595$	678661.	0.0785888
$596$	0	0
$597$	9.75383e6	1.12006
$598$	0	0
$599$	1.09921e7	1.25174	0.625869	−	0.779928i	$-0.284744\pi$
$599$	1.09921e7	1.25174	0.625869	+	0.779928i	$0.284744\pi$
$600$	0	0
$601$	1.54815e7	1.74835	0.874173	−	0.485615i	$-0.161404\pi$
$601$	1.54815e7	1.74835	0.874173	+	0.485615i	$0.161404\pi$
$602$	0	0
$603$	2.70045e6	0.302443
$604$	0	0
$605$	1.88187e6	0.209026
$606$	0	0
$607$	3.96214e6	0.436474	0.218237	−	0.975896i	$-0.429969\pi$
$607$	3.96214e6	0.436474	0.218237	+	0.975896i	$0.429969\pi$
$608$	0	0
$609$	7.67292e6	0.838335
$610$	0	0
$611$	−2.73619e7	−2.96513
$612$	0	0
$613$	−1.51043e7	−1.62349	−0.811747	−	0.584010i	$-0.801483\pi$
$613$	−1.51043e7	−1.62349	−0.811747	+	0.584010i	$0.801483\pi$
$614$	0	0
$615$	6.67831e6	0.711998
$616$	0	0
$617$	−1.23066e6	−0.130144	−0.0650722	−	0.997881i	$-0.520728\pi$
$617$	−1.23066e6	−0.130144	−0.0650722	+	0.997881i	$0.520728\pi$
$618$	0	0
$619$	1.60523e7	1.68388	0.841938	−	0.539574i	$-0.181415\pi$
$619$	1.60523e7	1.68388	0.841938	+	0.539574i	$0.181415\pi$
$620$	0	0
$621$	−2.70414e6	−0.281384
$622$	0	0
$623$	−6.48507e6	−0.669413
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−2.66593e6	−0.270820
$628$	0	0
$629$	−6.80482e6	−0.685788
$630$	0	0
$631$	−7.77324e6	−0.777192	−0.388596	−	0.921408i	$-0.627040\pi$
$631$	−7.77324e6	−0.777192	−0.388596	+	0.921408i	$0.627040\pi$
$632$	0	0
$633$	6.64264e6	0.658918
$634$	0	0
$635$	−6.06812e6	−0.597200
$636$	0	0
$637$	−2.51801e6	−0.245872
$638$	0	0
$639$	68208.8	0.00660828
$640$	0	0
$641$	1.12196e6	0.107853	0.0539264	−	0.998545i	$-0.482826\pi$
$641$	1.12196e6	0.107853	0.0539264	+	0.998545i	$0.482826\pi$
$642$	0	0
$643$	−1.84431e7	−1.75916	−0.879581	−	0.475749i	$-0.842177\pi$
$643$	−1.84431e7	−1.75916	−0.879581	+	0.475749i	$0.842177\pi$
$644$	0	0
$645$	4.29277e6	0.406292
$646$	0	0
$647$	−1.74466e7	−1.63851	−0.819257	−	0.573427i	$-0.805614\pi$
$647$	−1.74466e7	−1.63851	−0.819257	+	0.573427i	$0.805614\pi$
$648$	0	0
$649$	1.35773e7	1.26532
$650$	0	0
$651$	1.90246e6	0.175940
$652$	0	0
$653$	−1.59053e7	−1.45968	−0.729840	−	0.683618i	$-0.760406\pi$
$653$	−1.59053e7	−1.45968	−0.729840	+	0.683618i	$0.760406\pi$
$654$	0	0
$655$	679949.	0.0619260
$656$	0	0
$657$	−1.63571e6	−0.147840
$658$	0	0
$659$	−1.72417e7	−1.54656	−0.773278	−	0.634067i	$-0.781384\pi$
$659$	−1.72417e7	−1.54656	−0.773278	+	0.634067i	$0.781384\pi$
$660$	0	0
$661$	2.08188e7	1.85333	0.926665	−	0.375889i	$-0.122663\pi$
$661$	2.08188e7	1.85333	0.926665	+	0.375889i	$0.122663\pi$
$662$	0	0
$663$	−1.14766e7	−1.01398
$664$	0	0
$665$	−564508.	−0.0495012
$666$	0	0
$667$	−1.13257e7	−0.985716
$668$	0	0
$669$	−7.39417e6	−0.638740
$670$	0	0
$671$	−1.55874e7	−1.33649
$672$	0	0
$673$	1.60235e7	1.36370	0.681852	−	0.731490i	$-0.261175\pi$
$673$	1.60235e7	1.36370	0.681852	+	0.731490i	$0.261175\pi$
$674$	0	0
$675$	−1.18298e6	−0.0999348
$676$	0	0
$677$	7.03642e6	0.590038	0.295019	−	0.955491i	$-0.404674\pi$
$677$	7.03642e6	0.590038	0.295019	+	0.955491i	$0.404674\pi$
$678$	0	0
$679$	−5.18744e6	−0.431796
$680$	0	0
$681$	5.22431e6	0.431680
$682$	0	0
$683$	−2.16517e7	−1.77599	−0.887993	−	0.459857i	$-0.847901\pi$
$683$	−2.16517e7	−1.77599	−0.887993	+	0.459857i	$0.847901\pi$
$684$	0	0
$685$	3.76114e6	0.306262
$686$	0	0
$687$	2.05511e7	1.66128
$688$	0	0
$689$	4.65243e6	0.373364
$690$	0	0
$691$	−2.19450e7	−1.74840	−0.874198	−	0.485570i	$-0.838612\pi$
$691$	−2.19450e7	−1.74840	−0.874198	+	0.485570i	$0.838612\pi$
$692$	0	0
$693$	−2.11214e6	−0.167067
$694$	0	0
$695$	5.33619e6	0.419054
$696$	0	0
$697$	−7.49225e6	−0.584158
$698$	0	0
$699$	1.14979e7	0.890073
$700$	0	0
$701$	−2.02464e7	−1.55615	−0.778076	−	0.628170i	$-0.783804\pi$
$701$	−2.02464e7	−1.55615	−0.778076	+	0.628170i	$0.783804\pi$
$702$	0	0
$703$	5.66022e6	0.431962
$704$	0	0
$705$	−1.28841e7	−0.976293
$706$	0	0
$707$	−879154.	−0.0661479
$708$	0	0
$709$	−761476.	−0.0568906	−0.0284453	−	0.999595i	$-0.509056\pi$
$709$	−761476.	−0.0568906	−0.0284453	+	0.999595i	$0.509056\pi$
$710$	0	0
$711$	−7.33678e6	−0.544292
$712$	0	0
$713$	−2.80816e6	−0.206870
$714$	0	0
$715$	7.67872e6	0.561725
$716$	0	0
$717$	4.45756e6	0.323816
$718$	0	0
$719$	−2.85627e6	−0.206052	−0.103026	−	0.994679i	$-0.532853\pi$
$719$	−2.85627e6	−0.206052	−0.103026	+	0.994679i	$0.532853\pi$
$720$	0	0
$721$	3.40217e6	0.243735
$722$	0	0
$723$	2.51261e7	1.78763
$724$	0	0
$725$	−4.95465e6	−0.350081
$726$	0	0
$727$	−2.18741e7	−1.53495	−0.767474	−	0.641080i	$-0.778486\pi$
$727$	−2.18741e7	−1.53495	−0.767474	+	0.641080i	$0.778486\pi$
$728$	0	0
$729$	−1.68117e6	−0.117164
$730$	0	0
$731$	−4.81596e6	−0.333342
$732$	0	0
$733$	8.53911e6	0.587020	0.293510	−	0.955956i	$-0.405177\pi$
$733$	8.53911e6	0.587020	0.293510	+	0.955956i	$0.405177\pi$
$734$	0	0
$735$	−1.18567e6	−0.0809553
$736$	0	0
$737$	5.37374e6	0.364425
$738$	0	0
$739$	−2.21190e7	−1.48989	−0.744945	−	0.667126i	$-0.767524\pi$
$739$	−2.21190e7	−1.48989	−0.744945	+	0.667126i	$0.767524\pi$
$740$	0	0
$741$	9.54620e6	0.638682
$742$	0	0
$743$	1.47831e7	0.982411	0.491206	−	0.871044i	$-0.336556\pi$
$743$	1.47831e7	0.982411	0.491206	+	0.871044i	$0.336556\pi$
$744$	0	0
$745$	−3.63929e6	−0.240229
$746$	0	0
$747$	−1.65631e7	−1.08603
$748$	0	0
$749$	6.32318e6	0.411842
$750$	0	0
$751$	−1.23310e6	−0.0797811	−0.0398905	−	0.999204i	$-0.512701\pi$
$751$	−1.23310e6	−0.0797811	−0.0398905	+	0.999204i	$0.512701\pi$
$752$	0	0
$753$	1.13319e7	0.728307
$754$	0	0
$755$	3.76937e6	0.240659
$756$	0	0
$757$	−1.44186e7	−0.914499	−0.457249	−	0.889338i	$-0.651165\pi$
$757$	−1.44186e7	−0.914499	−0.457249	+	0.889338i	$0.651165\pi$
$758$	0	0
$759$	8.26510e6	0.520767
$760$	0	0
$761$	1.16309e7	0.728031	0.364016	−	0.931393i	$-0.381405\pi$
$761$	1.16309e7	0.728031	0.364016	+	0.931393i	$0.381405\pi$
$762$	0	0
$763$	−999979.	−0.0621841
$764$	0	0
$765$	−2.03845e6	−0.125935
$766$	0	0
$767$	−4.86177e7	−2.98405
$768$	0	0
$769$	−1.19892e7	−0.731098	−0.365549	−	0.930792i	$-0.619119\pi$
$769$	−1.19892e7	−0.731098	−0.365549	+	0.930792i	$0.619119\pi$
$770$	0	0
$771$	5.06552e6	0.306894
$772$	0	0
$773$	−3.02957e7	−1.82361	−0.911804	−	0.410625i	$-0.865311\pi$
$773$	−3.02957e7	−1.82361	−0.911804	+	0.410625i	$0.865311\pi$
$774$	0	0
$775$	−1.22848e6	−0.0734708
$776$	0	0
$777$	1.18885e7	0.706439
$778$	0	0
$779$	6.23202e6	0.367947
$780$	0	0
$781$	135732.	0.00796257
$782$	0	0
$783$	1.50048e7	0.874631
$784$	0	0
$785$	−8.35220e6	−0.483756
$786$	0	0
$787$	2.53512e7	1.45902	0.729511	−	0.683969i	$-0.239748\pi$
$787$	2.53512e7	1.45902	0.729511	+	0.683969i	$0.239748\pi$
$788$	0	0
$789$	−1.59684e7	−0.913207
$790$	0	0
$791$	4.78349e6	0.271834
$792$	0	0
$793$	5.58155e7	3.15189
$794$	0	0
$795$	2.19072e6	0.122933
$796$	0	0
$797$	7.99192e6	0.445661	0.222831	−	0.974857i	$-0.428470\pi$
$797$	7.99192e6	0.445661	0.222831	+	0.974857i	$0.428470\pi$
$798$	0	0
$799$	1.44543e7	0.800998
$800$	0	0
$801$	1.94788e7	1.07270
$802$	0	0
$803$	−3.25496e6	−0.178138
$804$	0	0
$805$	1.75013e6	0.0951874
$806$	0	0
$807$	−3.25975e7	−1.76198
$808$	0	0
$809$	−1.71087e7	−0.919066	−0.459533	−	0.888161i	$-0.651983\pi$
$809$	−1.71087e7	−0.919066	−0.459533	+	0.888161i	$0.651983\pi$
$810$	0	0
$811$	−1.71160e7	−0.913796	−0.456898	−	0.889519i	$-0.651040\pi$
$811$	−1.71160e7	−0.913796	−0.456898	+	0.889519i	$0.651040\pi$
$812$	0	0
$813$	−4.15693e7	−2.20570
$814$	0	0
$815$	−9.34526e6	−0.492830
$816$	0	0
$817$	4.00590e6	0.209964
$818$	0	0
$819$	7.56318e6	0.393999
$820$	0	0
$821$	−4.17632e6	−0.216240	−0.108120	−	0.994138i	$-0.534483\pi$
$821$	−4.17632e6	−0.216240	−0.108120	+	0.994138i	$0.534483\pi$
$822$	0	0
$823$	−1.13990e7	−0.586634	−0.293317	−	0.956015i	$-0.594759\pi$
$823$	−1.13990e7	−0.586634	−0.293317	+	0.956015i	$0.594759\pi$
$824$	0	0
$825$	3.61572e6	0.184952
$826$	0	0
$827$	2.04301e7	1.03874	0.519370	−	0.854549i	$-0.326167\pi$
$827$	2.04301e7	1.03874	0.519370	+	0.854549i	$0.326167\pi$
$828$	0	0
$829$	2.35087e6	0.118807	0.0594036	−	0.998234i	$-0.481080\pi$
$829$	2.35087e6	0.118807	0.0594036	+	0.998234i	$0.481080\pi$
$830$	0	0
$831$	1.92406e7	0.966530
$832$	0	0
$833$	1.33018e6	0.0664196
$834$	0	0
$835$	6.10115e6	0.302827
$836$	0	0
$837$	3.72036e6	0.183557
$838$	0	0
$839$	−2.11880e7	−1.03917	−0.519583	−	0.854420i	$-0.673913\pi$
$839$	−2.11880e7	−1.03917	−0.519583	+	0.854420i	$0.673913\pi$
$840$	0	0
$841$	4.23333e7	2.06391
$842$	0	0
$843$	−1.67417e7	−0.811391
$844$	0	0
$845$	−1.82137e7	−0.877520
$846$	0	0
$847$	3.68846e6	0.176659
$848$	0	0
$849$	−1.43729e7	−0.684344
$850$	0	0
$851$	−1.75482e7	−0.830632
$852$	0	0
$853$	3.39915e7	1.59955	0.799774	−	0.600301i	$-0.204952\pi$
$853$	3.39915e7	1.59955	0.799774	+	0.600301i	$0.204952\pi$
$854$	0	0
$855$	1.69558e6	0.0793235
$856$	0	0
$857$	5.41600e6	0.251899	0.125949	−	0.992037i	$-0.459802\pi$
$857$	5.41600e6	0.251899	0.125949	+	0.992037i	$0.459802\pi$
$858$	0	0
$859$	−4.89631e6	−0.226405	−0.113203	−	0.993572i	$-0.536111\pi$
$859$	−4.89631e6	−0.226405	−0.113203	+	0.993572i	$0.536111\pi$
$860$	0	0
$861$	1.30895e7	0.601748
$862$	0	0
$863$	−1.48827e6	−0.0680227	−0.0340113	−	0.999421i	$-0.510828\pi$
$863$	−1.48827e6	−0.0680227	−0.0340113	+	0.999421i	$0.510828\pi$
$864$	0	0
$865$	−1.30813e7	−0.594442
$866$	0	0
$867$	−2.19836e7	−0.993235
$868$	0	0
$869$	−1.45998e7	−0.655838
$870$	0	0
$871$	−1.92424e7	−0.859435
$872$	0	0
$873$	1.55812e7	0.691933
$874$	0	0
$875$	765625.	0.0338062
$876$	0	0
$877$	−4.40509e6	−0.193400	−0.0966999	−	0.995314i	$-0.530829\pi$
$877$	−4.40509e6	−0.193400	−0.0966999	+	0.995314i	$0.530829\pi$
$878$	0	0
$879$	4.09383e7	1.78714
$880$	0	0
$881$	−2.98766e7	−1.29686	−0.648428	−	0.761276i	$-0.724573\pi$
$881$	−2.98766e7	−1.29686	−0.648428	+	0.761276i	$0.724573\pi$
$882$	0	0
$883$	3.49305e7	1.50766	0.753829	−	0.657071i	$-0.228205\pi$
$883$	3.49305e7	1.50766	0.753829	+	0.657071i	$0.228205\pi$
$884$	0	0
$885$	−2.28929e7	−0.982523
$886$	0	0
$887$	−1.01042e7	−0.431213	−0.215606	−	0.976480i	$-0.569173\pi$
$887$	−1.01042e7	−0.431213	−0.215606	+	0.976480i	$0.569173\pi$
$888$	0	0
$889$	−1.18935e7	−0.504726
$890$	0	0
$891$	−2.14244e7	−0.904096
$892$	0	0
$893$	−1.20231e7	−0.504530
$894$	0	0
$895$	−9.68447e6	−0.404127
$896$	0	0
$897$	−2.95958e7	−1.22814
$898$	0	0
$899$	1.55820e7	0.643018
$900$	0	0
$901$	−2.45772e6	−0.100860
$902$	0	0
$903$	8.41383e6	0.343380
$904$	0	0
$905$	1.32579e7	0.538090
$906$	0	0
$907$	9.16678e6	0.369997	0.184999	−	0.982739i	$-0.440772\pi$
$907$	9.16678e6	0.369997	0.184999	+	0.982739i	$0.440772\pi$
$908$	0	0
$909$	2.64066e6	0.105999
$910$	0	0
$911$	4.50739e7	1.79940	0.899702	−	0.436504i	$-0.143784\pi$
$911$	4.50739e7	1.79940	0.899702	+	0.436504i	$0.143784\pi$
$912$	0	0
$913$	−3.29596e7	−1.30859
$914$	0	0
$915$	2.62821e7	1.03779
$916$	0	0
$917$	1.33270e6	0.0523370
$918$	0	0
$919$	−1.27358e6	−0.0497437	−0.0248719	−	0.999691i	$-0.507918\pi$
$919$	−1.27358e6	−0.0497437	−0.0248719	+	0.999691i	$0.507918\pi$
$920$	0	0
$921$	−1.93896e6	−0.0753216
$922$	0	0
$923$	−486029.	−0.0187784
$924$	0	0
$925$	−7.67679e6	−0.295002
$926$	0	0
$927$	−1.02189e7	−0.390575
$928$	0	0
$929$	−1.23381e7	−0.469037	−0.234519	−	0.972112i	$-0.575351\pi$
$929$	−1.23381e7	−0.469037	−0.234519	+	0.972112i	$0.575351\pi$
$930$	0	0
$931$	−1.10644e6	−0.0418362
$932$	0	0
$933$	4.97085e7	1.86950
$934$	0	0
$935$	−4.05640e6	−0.151744
$936$	0	0
$937$	−2.04380e7	−0.760483	−0.380241	−	0.924887i	$-0.624159\pi$
$937$	−2.04380e7	−0.760483	−0.380241	+	0.924887i	$0.624159\pi$
$938$	0	0
$939$	2.08926e7	0.773266
$940$	0	0
$941$	−1.12582e7	−0.414473	−0.207236	−	0.978291i	$-0.566447\pi$
$941$	−1.12582e7	−0.414473	−0.207236	+	0.978291i	$0.566447\pi$
$942$	0	0
$943$	−1.93209e7	−0.707537
$944$	0	0
$945$	−2.31863e6	−0.0844603
$946$	0	0
$947$	4.06658e7	1.47351	0.736757	−	0.676158i	$-0.236356\pi$
$947$	4.06658e7	1.47351	0.736757	+	0.676158i	$0.236356\pi$
$948$	0	0
$949$	1.16554e7	0.420109
$950$	0	0
$951$	−2.59992e7	−0.932200
$952$	0	0
$953$	1.13845e7	0.406053	0.203026	−	0.979173i	$-0.434922\pi$
$953$	1.13845e7	0.406053	0.203026	+	0.979173i	$0.434922\pi$
$954$	0	0
$955$	−7.63390e6	−0.270856
$956$	0	0
$957$	−4.58615e7	−1.61871
$958$	0	0
$959$	7.37183e6	0.258838
$960$	0	0
$961$	−2.47657e7	−0.865051
$962$	0	0
$963$	−1.89925e7	−0.659959
$964$	0	0
$965$	2.36236e6	0.0816635
$966$	0	0
$967$	−1.86570e7	−0.641616	−0.320808	−	0.947144i	$-0.603954\pi$
$967$	−1.86570e7	−0.641616	−0.320808	+	0.947144i	$0.603954\pi$
$968$	0	0
$969$	−5.04292e6	−0.172533
$970$	0	0
$971$	5.45410e7	1.85641	0.928207	−	0.372063i	$-0.121350\pi$
$971$	5.45410e7	1.85641	0.928207	+	0.372063i	$0.121350\pi$
$972$	0	0
$973$	1.04589e7	0.354165
$974$	0	0
$975$	−1.29472e7	−0.436179
$976$	0	0
$977$	4.01554e7	1.34588	0.672941	−	0.739696i	$-0.265031\pi$
$977$	4.01554e7	1.34588	0.672941	+	0.739696i	$0.265031\pi$
$978$	0	0
$979$	3.87616e7	1.29254
$980$	0	0
$981$	3.00357e6	0.0996473
$982$	0	0
$983$	−3.19971e7	−1.05615	−0.528076	−	0.849197i	$-0.677086\pi$
$983$	−3.19971e7	−1.05615	−0.528076	+	0.849197i	$0.677086\pi$
$984$	0	0
$985$	−2.66142e6	−0.0874024
$986$	0	0
$987$	−2.52528e7	−0.825118
$988$	0	0
$989$	−1.24194e7	−0.403746
$990$	0	0
$991$	2.45076e7	0.792713	0.396357	−	0.918097i	$-0.370274\pi$
$991$	2.45076e7	0.792713	0.396357	+	0.918097i	$0.370274\pi$
$992$	0	0
$993$	2.24088e7	0.721183
$994$	0	0
$995$	−1.23448e7	−0.395299
$996$	0	0
$997$	1.72201e7	0.548653	0.274326	−	0.961637i	$-0.411545\pi$
$997$	1.72201e7	0.548653	0.274326	+	0.961637i	$0.411545\pi$
$998$	0	0
$999$	2.32485e7	0.737025

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.6.a.h.1.3	✓	4
4.3	odd	2		560.6.a.w.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.6.a.h.1.3	✓	4	1.1	even	1	trivial
560.6.a.w.1.2		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)

\(f(q)\)	\(=\)	\(q+19.7529 q^{3} -25.0000 q^{5} -49.0000 q^{7} +147.178 q^{9} +O(q^{10})\)	\(q+19.7529 q^{3} -25.0000 q^{5} -49.0000 q^{7} +147.178 q^{9} +292.876 q^{11} -1048.73 q^{13} -493.823 q^{15} +554.009 q^{17} -460.823 q^{19} -967.893 q^{21} +1428.67 q^{23} +625.000 q^{25} -1892.76 q^{27} -7927.45 q^{29} -1965.57 q^{31} +5785.15 q^{33} +1225.00 q^{35} -12282.9 q^{37} -20715.6 q^{39} -13523.7 q^{41} -8692.93 q^{43} -3679.45 q^{45} +26090.5 q^{47} +2401.00 q^{49} +10943.3 q^{51} -4436.24 q^{53} -7321.90 q^{55} -9102.60 q^{57} +46358.5 q^{59} -53221.8 q^{61} -7211.73 q^{63} +26218.3 q^{65} +18348.2 q^{67} +28220.5 q^{69} +463.444 q^{71} -11113.8 q^{73} +12345.6 q^{75} -14350.9 q^{77} -49849.7 q^{79} -73151.9 q^{81} -112538. q^{83} -13850.2 q^{85} -156590. q^{87} +132348. q^{89} +51387.9 q^{91} -38825.8 q^{93} +11520.6 q^{95} +105866. q^{97} +43104.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 5 q^{3} - 100 q^{5} - 196 q^{7} + 567 q^{9}+O(q^{10}) \) 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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