LMFDB - Embedded newform 495.6.a.a.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,6,Mod(1,495)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("495.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	495.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:	$79.3899908074$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.307532.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 76x + 168 $ x^3 - x^2 - 76*x + 168
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-9.21967$ of defining polynomial
Character	$\chi$	$=$	495.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.21967 q^{2} +20.1236 q^{4} +25.0000 q^{5} +147.570 q^{7} -85.7438 q^{8} +O(q^{10})$	\(q+7.21967 q^{2} +20.1236 q^{4} +25.0000 q^{5} +147.570 q^{7} -85.7438 q^{8} +180.492 q^{10} -121.000 q^{11} -1124.39 q^{13} +1065.40 q^{14} -1263.00 q^{16} -1019.99 q^{17} -13.7131 q^{19} +503.090 q^{20} -873.580 q^{22} +2116.94 q^{23} +625.000 q^{25} -8117.69 q^{26} +2969.63 q^{28} -3120.98 q^{29} -9628.99 q^{31} -6374.61 q^{32} -7363.96 q^{34} +3689.24 q^{35} -3121.52 q^{37} -99.0039 q^{38} -2143.59 q^{40} +5884.38 q^{41} +20522.3 q^{43} -2434.95 q^{44} +15283.6 q^{46} -24718.5 q^{47} +4969.81 q^{49} +4512.29 q^{50} -22626.7 q^{52} -22761.1 q^{53} -3025.00 q^{55} -12653.2 q^{56} -22532.4 q^{58} +14533.5 q^{59} +43984.9 q^{61} -69518.1 q^{62} -5606.68 q^{64} -28109.6 q^{65} -31488.3 q^{67} -20525.8 q^{68} +26635.1 q^{70} +29173.2 q^{71} +49583.0 q^{73} -22536.3 q^{74} -275.956 q^{76} -17855.9 q^{77} -85075.4 q^{79} -31574.9 q^{80} +42483.3 q^{82} -77321.9 q^{83} -25499.6 q^{85} +148164. q^{86} +10375.0 q^{88} -70217.7 q^{89} -165925. q^{91} +42600.5 q^{92} -178459. q^{94} -342.827 q^{95} -21080.0 q^{97} +35880.4 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 7 q^{2} + 73 q^{4} + 75 q^{5} + 92 q^{7} - 231 q^{8}+O(q^{10}) $ 3 * q - 7 * q^2 + 73 * q^4 + 75 * q^5 + 92 * q^7 - 231 * q^8	$ 3 q - 7 q^{2} + 73 q^{4} + 75 q^{5} + 92 q^{7} - 231 q^{8} - 175 q^{10} - 363 q^{11} - 90 q^{13} + 784 q^{14} - 415 q^{16} - 1934 q^{17} + 2084 q^{19} + 1825 q^{20} + 847 q^{22} - 1220 q^{23} + 1875 q^{25} - 17062 q^{26} + 11120 q^{28} - 4402 q^{29} - 10688 q^{31} - 12439 q^{32} - 4094 q^{34} + 2300 q^{35} - 8190 q^{37} - 13792 q^{38} - 5775 q^{40} - 5974 q^{41} + 18868 q^{43} - 8833 q^{44} + 46220 q^{46} - 55500 q^{47} + 1907 q^{49} - 4375 q^{50} + 27330 q^{52} - 9206 q^{53} - 9075 q^{55} - 73248 q^{56} + 15366 q^{58} + 59196 q^{59} + 79902 q^{61} - 64616 q^{62} + 2129 q^{64} - 2250 q^{65} + 4468 q^{67} + 1218 q^{68} + 19600 q^{70} + 75164 q^{71} - 61290 q^{73} + 56766 q^{74} + 37816 q^{76} - 11132 q^{77} - 83564 q^{79} - 10375 q^{80} + 147410 q^{82} - 74764 q^{83} - 48350 q^{85} + 253432 q^{86} + 27951 q^{88} - 37342 q^{89} - 126488 q^{91} - 148164 q^{92} + 59252 q^{94} + 52100 q^{95} + 33486 q^{97} + 95249 q^{98}+O(q^{100}) $ 3 * q - 7 * q^2 + 73 * q^4 + 75 * q^5 + 92 * q^7 - 231 * q^8 - 175 * q^10 - 363 * q^11 - 90 * q^13 + 784 * q^14 - 415 * q^16 - 1934 * q^17 + 2084 * q^19 + 1825 * q^20 + 847 * q^22 - 1220 * q^23 + 1875 * q^25 - 17062 * q^26 + 11120 * q^28 - 4402 * q^29 - 10688 * q^31 - 12439 * q^32 - 4094 * q^34 + 2300 * q^35 - 8190 * q^37 - 13792 * q^38 - 5775 * q^40 - 5974 * q^41 + 18868 * q^43 - 8833 * q^44 + 46220 * q^46 - 55500 * q^47 + 1907 * q^49 - 4375 * q^50 + 27330 * q^52 - 9206 * q^53 - 9075 * q^55 - 73248 * q^56 + 15366 * q^58 + 59196 * q^59 + 79902 * q^61 - 64616 * q^62 + 2129 * q^64 - 2250 * q^65 + 4468 * q^67 + 1218 * q^68 + 19600 * q^70 + 75164 * q^71 - 61290 * q^73 + 56766 * q^74 + 37816 * q^76 - 11132 * q^77 - 83564 * q^79 - 10375 * q^80 + 147410 * q^82 - 74764 * q^83 - 48350 * q^85 + 253432 * q^86 + 27951 * q^88 - 37342 * q^89 - 126488 * q^91 - 148164 * q^92 + 59252 * q^94 + 52100 * q^95 + 33486 * q^97 + 95249 * q^98

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 $

$2$	7.21967	1.27627	0.638134	−	0.769925i	$-0.279706\pi$
$2$	7.21967	1.27627	0.638134	+	0.769925i	$0.279706\pi$
$3$	0	0
$3$	0	0
$4$	20.1236	0.628862
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	147.570	1.13829	0.569144	−	0.822238i	$-0.307275\pi$
$7$	147.570	1.13829	0.569144	+	0.822238i	$0.307275\pi$
$8$	−85.7438	−0.473672
$9$	0	0
$10$	180.492	0.570765
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	−1124.39	−1.84526	−0.922628	−	0.385690i	$-0.873963\pi$
$13$	−1124.39	−1.84526	−0.922628	+	0.385690i	$0.873963\pi$
$14$	1065.40	1.45276
$15$	0	0
$16$	−1263.00	−1.23339
$17$	−1019.99	−0.855996	−0.427998	−	0.903780i	$-0.640781\pi$
$17$	−1019.99	−0.855996	−0.427998	+	0.903780i	$0.640781\pi$
$18$	0	0
$19$	−13.7131	−0.00871467	−0.00435734	−	0.999991i	$-0.501387\pi$
$19$	−13.7131	−0.00871467	−0.00435734	+	0.999991i	$0.501387\pi$
$20$	503.090	0.281236
$21$	0	0
$22$	−873.580	−0.384810
$23$	2116.94	0.834430	0.417215	−	0.908808i	$-0.363006\pi$
$23$	2116.94	0.834430	0.417215	+	0.908808i	$0.363006\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	−8117.69	−2.35504
$27$	0	0
$28$	2969.63	0.715826
$29$	−3120.98	−0.689122	−0.344561	−	0.938764i	$-0.611972\pi$
$29$	−3120.98	−0.689122	−0.344561	+	0.938764i	$0.611972\pi$
$30$	0	0
$31$	−9628.99	−1.79960	−0.899801	−	0.436301i	$-0.856288\pi$
$31$	−9628.99	−1.79960	−0.899801	+	0.436301i	$0.856288\pi$
$32$	−6374.61	−1.10047
$33$	0	0
$34$	−7363.96	−1.09248
$35$	3689.24	0.509058
$36$	0	0
$37$	−3121.52	−0.374854	−0.187427	−	0.982279i	$-0.560015\pi$
$37$	−3121.52	−0.374854	−0.187427	+	0.982279i	$0.560015\pi$
$38$	−99.0039	−0.0111223
$39$	0	0
$40$	−2143.59	−0.211832
$41$	5884.38	0.546690	0.273345	−	0.961916i	$-0.411870\pi$
$41$	5884.38	0.546690	0.273345	+	0.961916i	$0.411870\pi$
$42$	0	0
$43$	20522.3	1.69260	0.846302	−	0.532704i	$-0.178824\pi$
$43$	20522.3	1.69260	0.846302	+	0.532704i	$0.178824\pi$
$44$	−2434.95	−0.189609
$45$	0	0
$46$	15283.6	1.06496
$47$	−24718.5	−1.63222	−0.816108	−	0.577899i	$-0.803873\pi$
$47$	−24718.5	−1.63222	−0.816108	+	0.577899i	$0.803873\pi$
$48$	0	0
$49$	4969.81	0.295699
$50$	4512.29	0.255254
$51$	0	0
$52$	−22626.7	−1.16041
$53$	−22761.1	−1.11302	−0.556511	−	0.830840i	$-0.687860\pi$
$53$	−22761.1	−1.11302	−0.556511	+	0.830840i	$0.687860\pi$
$54$	0	0
$55$	−3025.00	−0.134840
$56$	−12653.2	−0.539175
$57$	0	0
$58$	−22532.4	−0.879505
$59$	14533.5	0.543553	0.271776	−	0.962360i	$-0.412389\pi$
$59$	14533.5	0.543553	0.271776	+	0.962360i	$0.412389\pi$
$60$	0	0
$61$	43984.9	1.51349	0.756743	−	0.653712i	$-0.226789\pi$
$61$	43984.9	1.51349	0.756743	+	0.653712i	$0.226789\pi$
$62$	−69518.1	−2.29677
$63$	0	0
$64$	−5606.68	−0.171102
$65$	−28109.6	−0.825224
$66$	0	0
$67$	−31488.3	−0.856964	−0.428482	−	0.903550i	$-0.640951\pi$
$67$	−31488.3	−0.856964	−0.428482	+	0.903550i	$0.640951\pi$
$68$	−20525.8	−0.538303
$69$	0	0
$70$	26635.1	0.649694
$71$	29173.2	0.686812	0.343406	−	0.939187i	$-0.388419\pi$
$71$	29173.2	0.686812	0.343406	+	0.939187i	$0.388419\pi$
$72$	0	0
$73$	49583.0	1.08899	0.544497	−	0.838763i	$-0.316721\pi$
$73$	49583.0	1.08899	0.544497	+	0.838763i	$0.316721\pi$
$74$	−22536.3	−0.478414
$75$	0	0
$76$	−275.956	−0.00548033
$77$	−17855.9	−0.343207
$78$	0	0
$79$	−85075.4	−1.53369	−0.766843	−	0.641835i	$-0.778174\pi$
$79$	−85075.4	−1.53369	−0.766843	+	0.641835i	$0.778174\pi$
$80$	−31574.9	−0.551591
$81$	0	0
$82$	42483.3	0.697724
$83$	−77321.9	−1.23199	−0.615995	−	0.787750i	$-0.711246\pi$
$83$	−77321.9	−1.23199	−0.615995	+	0.787750i	$0.711246\pi$
$84$	0	0
$85$	−25499.6	−0.382813
$86$	148164.	2.16022
$87$	0	0
$88$	10375.0	0.142817
$89$	−70217.7	−0.939661	−0.469831	−	0.882757i	$-0.655685\pi$
$89$	−70217.7	−0.939661	−0.469831	+	0.882757i	$0.655685\pi$
$90$	0	0
$91$	−165925.	−2.10043
$92$	42600.5	0.524741
$93$	0	0
$94$	−178459.	−2.08315
$95$	−342.827	−0.00389732
$96$	0	0
$97$	−21080.0	−0.227479	−0.113739	−	0.993511i	$-0.536283\pi$
$97$	−21080.0	−0.227479	−0.113739	+	0.993511i	$0.536283\pi$
$98$	35880.4	0.377391
$99$	0	0
$100$	12577.2	0.125772
$101$	60279.4	0.587984	0.293992	−	0.955808i	$-0.405016\pi$
$101$	60279.4	0.587984	0.293992	+	0.955808i	$0.405016\pi$
$102$	0	0
$103$	83535.5	0.775851	0.387925	−	0.921691i	$-0.373192\pi$
$103$	83535.5	0.775851	0.387925	+	0.921691i	$0.373192\pi$
$104$	96409.0	0.874046
$105$	0	0
$106$	−164328.	−1.42052
$107$	−159288.	−1.34500	−0.672502	−	0.740095i	$-0.734781\pi$
$107$	−159288.	−1.34500	−0.672502	+	0.740095i	$0.734781\pi$
$108$	0	0
$109$	−31166.3	−0.251258	−0.125629	−	0.992077i	$-0.540095\pi$
$109$	−31166.3	−0.251258	−0.125629	+	0.992077i	$0.540095\pi$
$110$	−21839.5	−0.172092
$111$	0	0
$112$	−186380.	−1.40396
$113$	−141850.	−1.04504	−0.522521	−	0.852626i	$-0.675008\pi$
$113$	−141850.	−1.04504	−0.522521	+	0.852626i	$0.675008\pi$
$114$	0	0
$115$	52923.6	0.373168
$116$	−62805.3	−0.433363
$117$	0	0
$118$	104927.	0.693719
$119$	−150519.	−0.974370
$120$	0	0
$121$	14641.0	0.0909091
$122$	317556.	1.93162
$123$	0	0
$124$	−193770.	−1.13170
$125$	15625.0	0.0894427
$126$	0	0
$127$	−92350.0	−0.508075	−0.254037	−	0.967194i	$-0.581759\pi$
$127$	−92350.0	−0.508075	−0.254037	+	0.967194i	$0.581759\pi$
$128$	163509.	0.882098
$129$	0	0
$130$	−202942.	−1.05321
$131$	−210539.	−1.07190	−0.535949	−	0.844250i	$-0.680046\pi$
$131$	−210539.	−1.07190	−0.535949	+	0.844250i	$0.680046\pi$
$132$	0	0
$133$	−2023.64	−0.00991981
$134$	−227335.	−1.09372
$135$	0	0
$136$	87457.4	0.405461
$137$	−23859.0	−0.108605	−0.0543027	−	0.998525i	$-0.517294\pi$
$137$	−23859.0	−0.108605	−0.0543027	+	0.998525i	$0.517294\pi$
$138$	0	0
$139$	292430.	1.28377	0.641883	−	0.766803i	$-0.278154\pi$
$139$	292430.	1.28377	0.641883	+	0.766803i	$0.278154\pi$
$140$	74240.8	0.320127
$141$	0	0
$142$	210621.	0.876557
$143$	136051.	0.556366
$144$	0	0
$145$	−78024.5	−0.308185
$146$	357973.	1.38985
$147$	0	0
$148$	−62816.2	−0.235731
$149$	−350530.	−1.29348	−0.646740	−	0.762710i	$-0.723868\pi$
$149$	−350530.	−1.29348	−0.646740	+	0.762710i	$0.723868\pi$
$150$	0	0
$151$	−513284.	−1.83196	−0.915979	−	0.401225i	$-0.868584\pi$
$151$	−513284.	−1.83196	−0.915979	+	0.401225i	$0.868584\pi$
$152$	1175.81	0.00412790
$153$	0	0
$154$	−128914.	−0.438024
$155$	−240725.	−0.804806
$156$	0	0
$157$	154810.	0.501244	0.250622	−	0.968085i	$-0.419365\pi$
$157$	154810.	0.501244	0.250622	+	0.968085i	$0.419365\pi$
$158$	−614216.	−1.95740
$159$	0	0
$160$	−159365.	−0.492146
$161$	312397.	0.949821
$162$	0	0
$163$	184152.	0.542883	0.271442	−	0.962455i	$-0.412500\pi$
$163$	184152.	0.542883	0.271442	+	0.962455i	$0.412500\pi$
$164$	118415.	0.343793
$165$	0	0
$166$	−558238.	−1.57235
$167$	72082.3	0.200003	0.100002	−	0.994987i	$-0.468115\pi$
$167$	72082.3	0.200003	0.100002	+	0.994987i	$0.468115\pi$
$168$	0	0
$169$	892949.	2.40497
$170$	−184099.	−0.488572
$171$	0	0
$172$	412983.	1.06441
$173$	374895.	0.952345	0.476173	−	0.879352i	$-0.342024\pi$
$173$	374895.	0.952345	0.476173	+	0.879352i	$0.342024\pi$
$174$	0	0
$175$	92231.0	0.227658
$176$	152823.	0.371882
$177$	0	0
$178$	−506948.	−1.19926
$179$	−354545.	−0.827064	−0.413532	−	0.910490i	$-0.635705\pi$
$179$	−354545.	−0.827064	−0.413532	+	0.910490i	$0.635705\pi$
$180$	0	0
$181$	403999.	0.916609	0.458305	−	0.888795i	$-0.348457\pi$
$181$	403999.	0.916609	0.458305	+	0.888795i	$0.348457\pi$
$182$	−1.19792e6	−2.68072
$183$	0	0
$184$	−181515.	−0.395246
$185$	−78038.0	−0.167640
$186$	0	0
$187$	123418.	0.258093
$188$	−497425.	−1.02644
$189$	0	0
$190$	−2475.10	−0.00497403
$191$	−181282.	−0.359560	−0.179780	−	0.983707i	$-0.557539\pi$
$191$	−181282.	−0.359560	−0.179780	+	0.983707i	$0.557539\pi$
$192$	0	0
$193$	683762.	1.32133	0.660665	−	0.750681i	$-0.270274\pi$
$193$	683762.	1.32133	0.660665	+	0.750681i	$0.270274\pi$
$194$	−152190.	−0.290324
$195$	0	0
$196$	100010.	0.185954
$197$	442098.	0.811620	0.405810	−	0.913958i	$-0.366990\pi$
$197$	442098.	0.811620	0.405810	+	0.913958i	$0.366990\pi$
$198$	0	0
$199$	925615.	1.65691	0.828453	−	0.560059i	$-0.189222\pi$
$199$	925615.	1.65691	0.828453	+	0.560059i	$0.189222\pi$
$200$	−53589.8	−0.0947344
$201$	0	0
$202$	435197.	0.750425
$203$	−460562.	−0.784419
$204$	0	0
$205$	147110.	0.244487
$206$	603099.	0.990194
$207$	0	0
$208$	1.42009e6	2.27593
$209$	1659.28	0.00262757
$210$	0	0
$211$	605880.	0.936873	0.468436	−	0.883497i	$-0.344818\pi$
$211$	605880.	0.936873	0.468436	+	0.883497i	$0.344818\pi$
$212$	−458035.	−0.699938
$213$	0	0
$214$	−1.15001e6	−1.71659
$215$	513058.	0.756955
$216$	0	0
$217$	−1.42095e6	−2.04846
$218$	−225011.	−0.320673
$219$	0	0
$220$	−60873.8	−0.0847957
$221$	1.14686e6	1.57953
$222$	0	0
$223$	581147.	0.782571	0.391286	−	0.920269i	$-0.372030\pi$
$223$	581147.	0.782571	0.391286	+	0.920269i	$0.372030\pi$
$224$	−940699.	−1.25265
$225$	0	0
$226$	−1.02411e6	−1.33376
$227$	−1.51651e6	−1.95335	−0.976674	−	0.214726i	$-0.931114\pi$
$227$	−1.51651e6	−1.95335	−0.976674	+	0.214726i	$0.931114\pi$
$228$	0	0
$229$	916540.	1.15495	0.577474	−	0.816409i	$-0.304038\pi$
$229$	916540.	1.15495	0.577474	+	0.816409i	$0.304038\pi$
$230$	382091.	0.476263
$231$	0	0
$232$	267605.	0.326418
$233$	650940.	0.785509	0.392755	−	0.919643i	$-0.371522\pi$
$233$	650940.	0.785509	0.392755	+	0.919643i	$0.371522\pi$
$234$	0	0
$235$	−617963.	−0.729950
$236$	292467.	0.341820
$237$	0	0
$238$	−1.08670e6	−1.24356
$239$	1.06473e6	1.20572	0.602860	−	0.797847i	$-0.294028\pi$
$239$	1.06473e6	1.20572	0.602860	+	0.797847i	$0.294028\pi$
$240$	0	0
$241$	−1.21958e6	−1.35259	−0.676297	−	0.736629i	$-0.736417\pi$
$241$	−1.21958e6	−1.35259	−0.676297	+	0.736629i	$0.736417\pi$
$242$	105703.	0.116024
$243$	0	0
$244$	885133.	0.951774
$245$	124245.	0.132240
$246$	0	0
$247$	15418.8	0.0160808
$248$	825625.	0.852420
$249$	0	0
$250$	112807.	0.114153
$251$	−1.82454e6	−1.82797	−0.913987	−	0.405743i	$-0.867013\pi$
$251$	−1.82454e6	−1.82797	−0.913987	+	0.405743i	$0.867013\pi$
$252$	0	0
$253$	−256150.	−0.251590
$254$	−666736.	−0.648440
$255$	0	0
$256$	1.35990e6	1.29690
$257$	475038.	0.448637	0.224319	−	0.974516i	$-0.427984\pi$
$257$	475038.	0.448637	0.224319	+	0.974516i	$0.427984\pi$
$258$	0	0
$259$	−460642.	−0.426691
$260$	−565667.	−0.518952
$261$	0	0
$262$	−1.52002e6	−1.36803
$263$	1.07504e6	0.958376	0.479188	−	0.877712i	$-0.340931\pi$
$263$	1.07504e6	0.958376	0.479188	+	0.877712i	$0.340931\pi$
$264$	0	0
$265$	−569028.	−0.497759
$266$	−14610.0	−0.0126603
$267$	0	0
$268$	−633658.	−0.538912
$269$	−180662.	−0.152225	−0.0761126	−	0.997099i	$-0.524251\pi$
$269$	−180662.	−0.152225	−0.0761126	+	0.997099i	$0.524251\pi$
$270$	0	0
$271$	302884.	0.250526	0.125263	−	0.992124i	$-0.460023\pi$
$271$	302884.	0.250526	0.125263	+	0.992124i	$0.460023\pi$
$272$	1.28824e6	1.05578
$273$	0	0
$274$	−172254.	−0.138610
$275$	−75625.0	−0.0603023
$276$	0	0
$277$	1.08898e6	0.852745	0.426372	−	0.904548i	$-0.359791\pi$
$277$	1.08898e6	0.852745	0.426372	+	0.904548i	$0.359791\pi$
$278$	2.11125e6	1.63843
$279$	0	0
$280$	−316329.	−0.241126
$281$	942590.	0.712127	0.356063	−	0.934462i	$-0.384119\pi$
$281$	942590.	0.712127	0.356063	+	0.934462i	$0.384119\pi$
$282$	0	0
$283$	848952.	0.630111	0.315055	−	0.949073i	$-0.397977\pi$
$283$	848952.	0.630111	0.315055	+	0.949073i	$0.397977\pi$
$284$	587069.	0.431910
$285$	0	0
$286$	982240.	0.710072
$287$	868357.	0.622291
$288$	0	0
$289$	−379486.	−0.267271
$290$	−563311.	−0.393327
$291$	0	0
$292$	997788.	0.684827
$293$	−628367.	−0.427606	−0.213803	−	0.976877i	$-0.568585\pi$
$293$	−628367.	−0.427606	−0.213803	+	0.976877i	$0.568585\pi$
$294$	0	0
$295$	363339.	0.243084
$296$	267651.	0.177558
$297$	0	0
$298$	−2.53071e6	−1.65083
$299$	−2.38026e6	−1.53974
$300$	0	0
$301$	3.02847e6	1.92667
$302$	−3.70574e6	−2.33807
$303$	0	0
$304$	17319.6	0.0107486
$305$	1.09962e6	0.676852
$306$	0	0
$307$	1.88248e6	1.13994	0.569972	−	0.821664i	$-0.306954\pi$
$307$	1.88248e6	1.13994	0.569972	+	0.821664i	$0.306954\pi$
$308$	−359325.	−0.215830
$309$	0	0
$310$	−1.73795e6	−1.02715
$311$	1.53497e6	0.899908	0.449954	−	0.893052i	$-0.351440\pi$
$311$	1.53497e6	0.899908	0.449954	+	0.893052i	$0.351440\pi$
$312$	0	0
$313$	−2.22901e6	−1.28603	−0.643015	−	0.765854i	$-0.722317\pi$
$313$	−2.22901e6	−1.28603	−0.643015	+	0.765854i	$0.722317\pi$
$314$	1.11767e6	0.639722
$315$	0	0
$316$	−1.71202e6	−0.964477
$317$	−1.00659e6	−0.562605	−0.281303	−	0.959619i	$-0.590766\pi$
$317$	−1.00659e6	−0.562605	−0.281303	+	0.959619i	$0.590766\pi$
$318$	0	0
$319$	377639.	0.207778
$320$	−140167.	−0.0765193
$321$	0	0
$322$	2.25540e6	1.21223
$323$	13987.2	0.00745973
$324$	0	0
$325$	−702741.	−0.369051
$326$	1.32951e6	0.692865
$327$	0	0
$328$	−504549.	−0.258952
$329$	−3.64770e6	−1.85793
$330$	0	0
$331$	−2.96963e6	−1.48982	−0.744908	−	0.667167i	$-0.767507\pi$
$331$	−2.96963e6	−1.48982	−0.744908	+	0.667167i	$0.767507\pi$
$332$	−1.55599e6	−0.774752
$333$	0	0
$334$	520410.	0.255258
$335$	−787208.	−0.383246
$336$	0	0
$337$	69782.5	0.0334712	0.0167356	−	0.999860i	$-0.494673\pi$
$337$	69782.5	0.0334712	0.0167356	+	0.999860i	$0.494673\pi$
$338$	6.44680e6	3.06939
$339$	0	0
$340$	−513144.	−0.240737
$341$	1.16511e6	0.542600
$342$	0	0
$343$	−1.74681e6	−0.801697
$344$	−1.75966e6	−0.801739
$345$	0	0
$346$	2.70662e6	1.21545
$347$	1.04233e6	0.464708	0.232354	−	0.972631i	$-0.425357\pi$
$347$	1.04233e6	0.464708	0.232354	+	0.972631i	$0.425357\pi$
$348$	0	0
$349$	−3.75012e6	−1.64809	−0.824047	−	0.566522i	$-0.808289\pi$
$349$	−3.75012e6	−1.64809	−0.824047	+	0.566522i	$0.808289\pi$
$350$	665877.	0.290552
$351$	0	0
$352$	771328.	0.331805
$353$	−800792.	−0.342045	−0.171022	−	0.985267i	$-0.554707\pi$
$353$	−800792.	−0.342045	−0.171022	+	0.985267i	$0.554707\pi$
$354$	0	0
$355$	729330.	0.307152
$356$	−1.41303e6	−0.590917
$357$	0	0
$358$	−2.55970e6	−1.05556
$359$	−267422.	−0.109512	−0.0547559	−	0.998500i	$-0.517438\pi$
$359$	−267422.	−0.109512	−0.0547559	+	0.998500i	$0.517438\pi$
$360$	0	0
$361$	−2.47591e6	−0.999924
$362$	2.91674e6	1.16984
$363$	0	0
$364$	−3.33901e6	−1.32088
$365$	1.23957e6	0.487013
$366$	0	0
$367$	2.63334e6	1.02057	0.510283	−	0.860007i	$-0.329541\pi$
$367$	2.63334e6	1.02057	0.510283	+	0.860007i	$0.329541\pi$
$368$	−2.67369e6	−1.02918
$369$	0	0
$370$	−563408.	−0.213953
$371$	−3.35885e6	−1.26694
$372$	0	0
$373$	−3.93389e6	−1.46403	−0.732015	−	0.681289i	$-0.761420\pi$
$373$	−3.93389e6	−1.46403	−0.732015	+	0.681289i	$0.761420\pi$
$374$	891039.	0.329395
$375$	0	0
$376$	2.11946e6	0.773135
$377$	3.50918e6	1.27161
$378$	0	0
$379$	−1.80355e6	−0.644956	−0.322478	−	0.946577i	$-0.604516\pi$
$379$	−1.80355e6	−0.644956	−0.322478	+	0.946577i	$0.604516\pi$
$380$	−6898.91	−0.00245088
$381$	0	0
$382$	−1.30880e6	−0.458895
$383$	−2.19398e6	−0.764252	−0.382126	−	0.924110i	$-0.624808\pi$
$383$	−2.19398e6	−0.764252	−0.382126	+	0.924110i	$0.624808\pi$
$384$	0	0
$385$	−446398.	−0.153487
$386$	4.93653e6	1.68637
$387$	0	0
$388$	−424204.	−0.143053
$389$	−354221.	−0.118686	−0.0593430	−	0.998238i	$-0.518901\pi$
$389$	−354221.	−0.118686	−0.0593430	+	0.998238i	$0.518901\pi$
$390$	0	0
$391$	−2.15925e6	−0.714268
$392$	−426130.	−0.140064
$393$	0	0
$394$	3.19180e6	1.03584
$395$	−2.12689e6	−0.685885
$396$	0	0
$397$	168373.	0.0536162	0.0268081	−	0.999641i	$-0.491466\pi$
$397$	168373.	0.0536162	0.0268081	+	0.999641i	$0.491466\pi$
$398$	6.68263e6	2.11466
$399$	0	0
$400$	−789373.	−0.246679
$401$	4.59458e6	1.42687	0.713436	−	0.700721i	$-0.247138\pi$
$401$	4.59458e6	1.42687	0.713436	+	0.700721i	$0.247138\pi$
$402$	0	0
$403$	1.08267e7	3.32073
$404$	1.21304e6	0.369761
$405$	0	0
$406$	−3.32510e6	−1.00113
$407$	377704.	0.113023
$408$	0	0
$409$	3.59299e6	1.06206	0.531028	−	0.847354i	$-0.321806\pi$
$409$	3.59299e6	1.06206	0.531028	+	0.847354i	$0.321806\pi$
$410$	1.06208e6	0.312032
$411$	0	0
$412$	1.68103e6	0.487903
$413$	2.14471e6	0.618719
$414$	0	0
$415$	−1.93305e6	−0.550963
$416$	7.16752e6	2.03065
$417$	0	0
$418$	11979.5	0.00335349
$419$	4.10701e6	1.14285	0.571427	−	0.820653i	$-0.306390\pi$
$419$	4.10701e6	1.14285	0.571427	+	0.820653i	$0.306390\pi$
$420$	0	0
$421$	5.36278e6	1.47464	0.737318	−	0.675546i	$-0.236092\pi$
$421$	5.36278e6	1.47464	0.737318	+	0.675546i	$0.236092\pi$
$422$	4.37425e6	1.19570
$423$	0	0
$424$	1.95162e6	0.527208
$425$	−637491.	−0.171199
$426$	0	0
$427$	6.49083e6	1.72278
$428$	−3.20545e6	−0.845822
$429$	0	0
$430$	3.70411e6	0.966078
$431$	−2.52160e6	−0.653857	−0.326929	−	0.945049i	$-0.606014\pi$
$431$	−2.52160e6	−0.653857	−0.326929	+	0.945049i	$0.606014\pi$
$432$	0	0
$433$	2.99694e6	0.768171	0.384085	−	0.923298i	$-0.374517\pi$
$433$	2.99694e6	0.768171	0.384085	+	0.923298i	$0.374517\pi$
$434$	−1.02588e7	−2.61439
$435$	0	0
$436$	−627179.	−0.158007
$437$	−29029.8	−0.00727178
$438$	0	0
$439$	1.93618e6	0.479495	0.239748	−	0.970835i	$-0.422935\pi$
$439$	1.93618e6	0.479495	0.239748	+	0.970835i	$0.422935\pi$
$440$	259375.	0.0638699
$441$	0	0
$442$	8.27992e6	2.01591
$443$	7.28132e6	1.76279	0.881395	−	0.472380i	$-0.156605\pi$
$443$	7.28132e6	1.76279	0.881395	+	0.472380i	$0.156605\pi$
$444$	0	0
$445$	−1.75544e6	−0.420229
$446$	4.19569e6	0.998771
$447$	0	0
$448$	−827377.	−0.194764
$449$	−6.15007e6	−1.43967	−0.719837	−	0.694143i	$-0.755784\pi$
$449$	−6.15007e6	−1.43967	−0.719837	+	0.694143i	$0.755784\pi$
$450$	0	0
$451$	−712011.	−0.164833
$452$	−2.85454e6	−0.657188
$453$	0	0
$454$	−1.09487e7	−2.49300
$455$	−4.14813e6	−0.939342
$456$	0	0
$457$	−5.60231e6	−1.25481	−0.627403	−	0.778695i	$-0.715882\pi$
$457$	−5.60231e6	−1.25481	−0.627403	+	0.778695i	$0.715882\pi$
$458$	6.61711e6	1.47402
$459$	0	0
$460$	1.06501e6	0.234671
$461$	−2.85697e6	−0.626113	−0.313057	−	0.949734i	$-0.601353\pi$
$461$	−2.85697e6	−0.626113	−0.313057	+	0.949734i	$0.601353\pi$
$462$	0	0
$463$	2.42615e6	0.525976	0.262988	−	0.964799i	$-0.415292\pi$
$463$	2.42615e6	0.525976	0.262988	+	0.964799i	$0.415292\pi$
$464$	3.94179e6	0.849959
$465$	0	0
$466$	4.69957e6	1.00252
$467$	−167522.	−0.0355451	−0.0177726	−	0.999842i	$-0.505657\pi$
$467$	−167522.	−0.0355451	−0.0177726	+	0.999842i	$0.505657\pi$
$468$	0	0
$469$	−4.64672e6	−0.975471
$470$	−4.46149e6	−0.931612
$471$	0	0
$472$	−1.24616e6	−0.257466
$473$	−2.48320e6	−0.510339
$474$	0	0
$475$	−8570.68	−0.00174293
$476$	−3.02898e6	−0.612744
$477$	0	0
$478$	7.68702e6	1.53882
$479$	−5.39034e6	−1.07344	−0.536719	−	0.843761i	$-0.680337\pi$
$479$	−5.39034e6	−1.07344	−0.536719	+	0.843761i	$0.680337\pi$
$480$	0	0
$481$	3.50979e6	0.691701
$482$	−8.80496e6	−1.72627
$483$	0	0
$484$	294629.	0.0571693
$485$	−526999.	−0.101732
$486$	0	0
$487$	1.38574e6	0.264764	0.132382	−	0.991199i	$-0.457737\pi$
$487$	1.38574e6	0.264764	0.132382	+	0.991199i	$0.457737\pi$
$488$	−3.77143e6	−0.716896
$489$	0	0
$490$	897009.	0.168774
$491$	3.39592e6	0.635703	0.317851	−	0.948141i	$-0.397039\pi$
$491$	3.39592e6	0.635703	0.317851	+	0.948141i	$0.397039\pi$
$492$	0	0
$493$	3.18336e6	0.589886
$494$	111319.	0.0205234
$495$	0	0
$496$	1.21614e7	2.21962
$497$	4.30508e6	0.781790
$498$	0	0
$499$	2.83293e6	0.509313	0.254657	−	0.967032i	$-0.418038\pi$
$499$	2.83293e6	0.509313	0.254657	+	0.967032i	$0.418038\pi$
$500$	314431.	0.0562471
$501$	0	0
$502$	−1.31726e7	−2.33299
$503$	−9.23328e6	−1.62718	−0.813591	−	0.581437i	$-0.802491\pi$
$503$	−9.23328e6	−1.62718	−0.813591	+	0.581437i	$0.802491\pi$
$504$	0	0
$505$	1.50698e6	0.262954
$506$	−1.84932e6	−0.321096
$507$	0	0
$508$	−1.85841e6	−0.319509
$509$	5.70039e6	0.975238	0.487619	−	0.873057i	$-0.337866\pi$
$509$	5.70039e6	0.975238	0.487619	+	0.873057i	$0.337866\pi$
$510$	0	0
$511$	7.31695e6	1.23959
$512$	4.58570e6	0.773091
$513$	0	0
$514$	3.42961e6	0.572582
$515$	2.08839e6	0.346971
$516$	0	0
$517$	2.99094e6	0.492132
$518$	−3.32568e6	−0.544573
$519$	0	0
$520$	2.41023e6	0.390885
$521$	1.35063e6	0.217994	0.108997	−	0.994042i	$-0.465236\pi$
$521$	1.35063e6	0.217994	0.108997	+	0.994042i	$0.465236\pi$
$522$	0	0
$523$	440837.	0.0704731	0.0352366	−	0.999379i	$-0.488782\pi$
$523$	440837.	0.0704731	0.0352366	+	0.999379i	$0.488782\pi$
$524$	−4.23679e6	−0.674076
$525$	0	0
$526$	7.76144e6	1.22314
$527$	9.82143e6	1.54045
$528$	0	0
$529$	−1.95489e6	−0.303727
$530$	−4.10819e6	−0.635274
$531$	0	0
$532$	−40722.8	−0.00623819
$533$	−6.61632e6	−1.00878
$534$	0	0
$535$	−3.98220e6	−0.601504
$536$	2.69993e6	0.405919
$537$	0	0
$538$	−1.30432e6	−0.194280
$539$	−601347.	−0.0891565
$540$	0	0
$541$	7.01375e6	1.03028	0.515142	−	0.857105i	$-0.327739\pi$
$541$	7.01375e6	1.03028	0.515142	+	0.857105i	$0.327739\pi$
$542$	2.18672e6	0.319738
$543$	0	0
$544$	6.50201e6	0.941999
$545$	−779159.	−0.112366
$546$	0	0
$547$	−1.01822e7	−1.45503	−0.727517	−	0.686090i	$-0.759326\pi$
$547$	−1.01822e7	−1.45503	−0.727517	+	0.686090i	$0.759326\pi$
$548$	−480129.	−0.0682978
$549$	0	0
$550$	−545987.	−0.0769619
$551$	42798.3	0.00600547
$552$	0	0
$553$	−1.25546e7	−1.74578
$554$	7.86205e6	1.08833
$555$	0	0
$556$	5.88475e6	0.807311
$557$	1.09154e7	1.49074	0.745371	−	0.666650i	$-0.232272\pi$
$557$	1.09154e7	1.49074	0.745371	+	0.666650i	$0.232272\pi$
$558$	0	0
$559$	−2.30750e7	−3.12329
$560$	−4.65950e6	−0.627869
$561$	0	0
$562$	6.80519e6	0.908865
$563$	−1.25187e6	−0.166452	−0.0832261	−	0.996531i	$-0.526522\pi$
$563$	−1.25187e6	−0.166452	−0.0832261	+	0.996531i	$0.526522\pi$
$564$	0	0
$565$	−3.54626e6	−0.467357
$566$	6.12915e6	0.804191
$567$	0	0
$568$	−2.50142e6	−0.325324
$569$	−3.75381e6	−0.486061	−0.243031	−	0.970019i	$-0.578142\pi$
$569$	−3.75381e6	−0.486061	−0.243031	+	0.970019i	$0.578142\pi$
$570$	0	0
$571$	−7.39573e6	−0.949273	−0.474636	−	0.880182i	$-0.657420\pi$
$571$	−7.39573e6	−0.949273	−0.474636	+	0.880182i	$0.657420\pi$
$572$	2.73783e6	0.349877
$573$	0	0
$574$	6.26925e6	0.794210
$575$	1.32309e6	0.166886
$576$	0	0
$577$	−6.45181e6	−0.806755	−0.403378	−	0.915034i	$-0.632164\pi$
$577$	−6.45181e6	−0.806755	−0.403378	+	0.915034i	$0.632164\pi$
$578$	−2.73976e6	−0.341109
$579$	0	0
$580$	−1.57013e6	−0.193806
$581$	−1.14104e7	−1.40236
$582$	0	0
$583$	2.75410e6	0.335589
$584$	−4.25143e6	−0.515826
$585$	0	0
$586$	−4.53660e6	−0.545741
$587$	−5.91956e6	−0.709078	−0.354539	−	0.935041i	$-0.615362\pi$
$587$	−5.91956e6	−0.709078	−0.354539	+	0.935041i	$0.615362\pi$
$588$	0	0
$589$	132043.	0.0156829
$590$	2.62318e6	0.310241
$591$	0	0
$592$	3.94247e6	0.462343
$593$	2.54171e6	0.296817	0.148409	−	0.988926i	$-0.452585\pi$
$593$	2.54171e6	0.296817	0.148409	+	0.988926i	$0.452585\pi$
$594$	0	0
$595$	−3.76297e6	−0.435751
$596$	−7.05393e6	−0.813421
$597$	0	0
$598$	−1.71847e7	−1.96512
$599$	−1.17309e7	−1.33587	−0.667935	−	0.744220i	$-0.732821\pi$
$599$	−1.17309e7	−1.33587	−0.667935	+	0.744220i	$0.732821\pi$
$600$	0	0
$601$	−2.88018e6	−0.325262	−0.162631	−	0.986687i	$-0.551998\pi$
$601$	−2.88018e6	−0.325262	−0.162631	+	0.986687i	$0.551998\pi$
$602$	2.18646e7	2.45895
$603$	0	0
$604$	−1.03291e7	−1.15205
$605$	366025.	0.0406558
$606$	0	0
$607$	−1.33207e7	−1.46742	−0.733712	−	0.679460i	$-0.762214\pi$
$607$	−1.33207e7	−1.46742	−0.733712	+	0.679460i	$0.762214\pi$
$608$	87415.6	0.00959025
$609$	0	0
$610$	7.93890e6	0.863845
$611$	2.77931e7	3.01186
$612$	0	0
$613$	−8.85926e6	−0.952239	−0.476120	−	0.879381i	$-0.657957\pi$
$613$	−8.85926e6	−0.952239	−0.476120	+	0.879381i	$0.657957\pi$
$614$	1.35908e7	1.45487
$615$	0	0
$616$	1.53103e6	0.162567
$617$	−1.75148e7	−1.85222	−0.926110	−	0.377253i	$-0.876869\pi$
$617$	−1.75148e7	−1.85222	−0.926110	+	0.377253i	$0.876869\pi$
$618$	0	0
$619$	3.94768e6	0.414109	0.207055	−	0.978329i	$-0.433612\pi$
$619$	3.94768e6	0.414109	0.207055	+	0.978329i	$0.433612\pi$
$620$	−4.84424e6	−0.506112
$621$	0	0
$622$	1.10820e7	1.14852
$623$	−1.03620e7	−1.06960
$624$	0	0
$625$	390625.	0.0400000
$626$	−1.60927e7	−1.64132
$627$	0	0
$628$	3.11533e6	0.315213
$629$	3.18391e6	0.320873
$630$	0	0
$631$	−5.78181e6	−0.578083	−0.289042	−	0.957317i	$-0.593337\pi$
$631$	−5.78181e6	−0.578083	−0.289042	+	0.957317i	$0.593337\pi$
$632$	7.29469e6	0.726464
$633$	0	0
$634$	−7.26723e6	−0.718035
$635$	−2.30875e6	−0.227218
$636$	0	0
$637$	−5.58798e6	−0.545640
$638$	2.72643e6	0.265181
$639$	0	0
$640$	4.08773e6	0.394486
$641$	4.31966e6	0.415245	0.207623	−	0.978209i	$-0.433427\pi$
$641$	4.31966e6	0.415245	0.207623	+	0.978209i	$0.433427\pi$
$642$	0	0
$643$	1.95083e7	1.86076	0.930381	−	0.366593i	$-0.119476\pi$
$643$	1.95083e7	1.86076	0.930381	+	0.366593i	$0.119476\pi$
$644$	6.28654e6	0.597306
$645$	0	0
$646$	100983.	0.00952062
$647$	1.08322e7	1.01732	0.508660	−	0.860967i	$-0.330141\pi$
$647$	1.08322e7	1.01732	0.508660	+	0.860967i	$0.330141\pi$
$648$	0	0
$649$	−1.75856e6	−0.163887
$650$	−5.07355e6	−0.471009
$651$	0	0
$652$	3.70579e6	0.341399
$653$	−1.83320e6	−0.168239	−0.0841197	−	0.996456i	$-0.526808\pi$
$653$	−1.83320e6	−0.168239	−0.0841197	+	0.996456i	$0.526808\pi$
$654$	0	0
$655$	−5.26346e6	−0.479367
$656$	−7.43195e6	−0.674285
$657$	0	0
$658$	−2.63352e7	−2.37122
$659$	−1.77259e7	−1.58999	−0.794994	−	0.606618i	$-0.792526\pi$
$659$	−1.77259e7	−1.58999	−0.794994	+	0.606618i	$0.792526\pi$
$660$	0	0
$661$	−5.89810e6	−0.525060	−0.262530	−	0.964924i	$-0.584557\pi$
$661$	−5.89810e6	−0.525060	−0.262530	+	0.964924i	$0.584557\pi$
$662$	−2.14398e7	−1.90141
$663$	0	0
$664$	6.62987e6	0.583559
$665$	−50590.9	−0.00443627
$666$	0	0
$667$	−6.60694e6	−0.575024
$668$	1.45055e6	0.125775
$669$	0	0
$670$	−5.68338e6	−0.489125
$671$	−5.32217e6	−0.456333
$672$	0	0
$673$	−6.98177e6	−0.594193	−0.297097	−	0.954847i	$-0.596018\pi$
$673$	−6.98177e6	−0.594193	−0.297097	+	0.954847i	$0.596018\pi$
$674$	503806.	0.0427183
$675$	0	0
$676$	1.79693e7	1.51240
$677$	−5.68896e6	−0.477047	−0.238524	−	0.971137i	$-0.576663\pi$
$677$	−5.68896e6	−0.477047	−0.238524	+	0.971137i	$0.576663\pi$
$678$	0	0
$679$	−3.11076e6	−0.258936
$680$	2.18644e6	0.181328
$681$	0	0
$682$	8.41169e6	0.692504
$683$	−1.65539e6	−0.135784	−0.0678920	−	0.997693i	$-0.521627\pi$
$683$	−1.65539e6	−0.135784	−0.0678920	+	0.997693i	$0.521627\pi$
$684$	0	0
$685$	−596475.	−0.0485698
$686$	−1.26114e7	−1.02318
$687$	0	0
$688$	−2.59196e7	−2.08765
$689$	2.55923e7	2.05381
$690$	0	0
$691$	−8.36041e6	−0.666090	−0.333045	−	0.942911i	$-0.608076\pi$
$691$	−8.36041e6	−0.666090	−0.333045	+	0.942911i	$0.608076\pi$
$692$	7.54423e6	0.598894
$693$	0	0
$694$	7.52524e6	0.593092
$695$	7.31076e6	0.574117
$696$	0	0
$697$	−6.00199e6	−0.467965
$698$	−2.70746e7	−2.10341
$699$	0	0
$700$	1.85602e6	0.143165
$701$	−1.07851e7	−0.828948	−0.414474	−	0.910061i	$-0.636034\pi$
$701$	−1.07851e7	−0.828948	−0.414474	+	0.910061i	$0.636034\pi$
$702$	0	0
$703$	42805.7	0.00326673
$704$	678409.	0.0515893
$705$	0	0
$706$	−5.78145e6	−0.436541
$707$	8.89541e6	0.669295
$708$	0	0
$709$	−1.37457e7	−1.02695	−0.513476	−	0.858104i	$-0.671642\pi$
$709$	−1.37457e7	−1.02695	−0.513476	+	0.858104i	$0.671642\pi$
$710$	5.26552e6	0.392008
$711$	0	0
$712$	6.02073e6	0.445091
$713$	−2.03840e7	−1.50164
$714$	0	0
$715$	3.40127e6	0.248814
$716$	−7.13472e6	−0.520109
$717$	0	0
$718$	−1.93070e6	−0.139767
$719$	−2.23290e6	−0.161082	−0.0805409	−	0.996751i	$-0.525665\pi$
$719$	−2.23290e6	−0.161082	−0.0805409	+	0.996751i	$0.525665\pi$
$720$	0	0
$721$	1.23273e7	0.883141
$722$	−1.78753e7	−1.27617
$723$	0	0
$724$	8.12991e6	0.576421
$725$	−1.95061e6	−0.137824
$726$	0	0
$727$	7.70500e6	0.540675	0.270338	−	0.962766i	$-0.412865\pi$
$727$	7.70500e6	0.540675	0.270338	+	0.962766i	$0.412865\pi$
$728$	1.42270e7	0.994916
$729$	0	0
$730$	8.94932e6	0.621560
$731$	−2.09325e7	−1.44886
$732$	0	0
$733$	−5.23205e6	−0.359676	−0.179838	−	0.983696i	$-0.557557\pi$
$733$	−5.23205e6	−0.359676	−0.179838	+	0.983696i	$0.557557\pi$
$734$	1.90118e7	1.30252
$735$	0	0
$736$	−1.34947e7	−0.918266
$737$	3.81009e6	0.258384
$738$	0	0
$739$	1.90698e7	1.28450	0.642252	−	0.766494i	$-0.278000\pi$
$739$	1.90698e7	1.28450	0.642252	+	0.766494i	$0.278000\pi$
$740$	−1.57040e6	−0.105422
$741$	0	0
$742$	−2.42498e7	−1.61696
$743$	−2.31407e7	−1.53781	−0.768907	−	0.639361i	$-0.779199\pi$
$743$	−2.31407e7	−1.53781	−0.768907	+	0.639361i	$0.779199\pi$
$744$	0	0
$745$	−8.76326e6	−0.578462
$746$	−2.84013e7	−1.86849
$747$	0	0
$748$	2.48362e6	0.162305
$749$	−2.35061e7	−1.53100
$750$	0	0
$751$	1.96879e7	1.27380	0.636898	−	0.770948i	$-0.280217\pi$
$751$	1.96879e7	1.27380	0.636898	+	0.770948i	$0.280217\pi$
$752$	3.12194e7	2.01317
$753$	0	0
$754$	2.53351e7	1.62291
$755$	−1.28321e7	−0.819277
$756$	0	0
$757$	−8.22162e6	−0.521456	−0.260728	−	0.965412i	$-0.583963\pi$
$757$	−8.22162e6	−0.521456	−0.260728	+	0.965412i	$0.583963\pi$
$758$	−1.30210e7	−0.823137
$759$	0	0
$760$	29395.3	0.00184605
$761$	−2.18719e6	−0.136907	−0.0684534	−	0.997654i	$-0.521806\pi$
$761$	−2.18719e6	−0.136907	−0.0684534	+	0.997654i	$0.521806\pi$
$762$	0	0
$763$	−4.59921e6	−0.286004
$764$	−3.64805e6	−0.226114
$765$	0	0
$766$	−1.58398e7	−0.975391
$767$	−1.63413e7	−1.00299
$768$	0	0
$769$	3.97950e6	0.242668	0.121334	−	0.992612i	$-0.461283\pi$
$769$	3.97950e6	0.242668	0.121334	+	0.992612i	$0.461283\pi$
$770$	−3.22285e6	−0.195890
$771$	0	0
$772$	1.37597e7	0.830935
$773$	1.92979e7	1.16161	0.580807	−	0.814042i	$-0.302737\pi$
$773$	1.92979e7	1.16161	0.580807	+	0.814042i	$0.302737\pi$
$774$	0	0
$775$	−6.01812e6	−0.359920
$776$	1.80748e6	0.107750
$777$	0	0
$778$	−2.55735e6	−0.151475
$779$	−80693.1	−0.00476423
$780$	0	0
$781$	−3.52996e6	−0.207082
$782$	−1.55891e7	−0.911599
$783$	0	0
$784$	−6.27685e6	−0.364713
$785$	3.87024e6	0.224163
$786$	0	0
$787$	−1.54391e7	−0.888556	−0.444278	−	0.895889i	$-0.646540\pi$
$787$	−1.54391e7	−0.888556	−0.444278	+	0.895889i	$0.646540\pi$
$788$	8.89659e6	0.510397
$789$	0	0
$790$	−1.53554e7	−0.875374
$791$	−2.09328e7	−1.18956
$792$	0	0
$793$	−4.94559e7	−2.79277
$794$	1.21560e6	0.0684287
$795$	0	0
$796$	1.86267e7	1.04196
$797$	−2.66071e7	−1.48372	−0.741860	−	0.670555i	$-0.766056\pi$
$797$	−2.66071e7	−1.48372	−0.741860	+	0.670555i	$0.766056\pi$
$798$	0	0
$799$	2.52125e7	1.39717
$800$	−3.98413e6	−0.220094
$801$	0	0
$802$	3.31713e7	1.82107
$803$	−5.99954e6	−0.328344
$804$	0	0
$805$	7.80992e6	0.424773
$806$	7.81651e7	4.23814
$807$	0	0
$808$	−5.16858e6	−0.278511
$809$	3.15651e7	1.69565	0.847825	−	0.530276i	$-0.177912\pi$
$809$	3.15651e7	1.69565	0.847825	+	0.530276i	$0.177912\pi$
$810$	0	0
$811$	3.50230e7	1.86982	0.934912	−	0.354880i	$-0.115478\pi$
$811$	3.50230e7	1.86982	0.934912	+	0.354880i	$0.115478\pi$
$812$	−9.26816e6	−0.493291
$813$	0	0
$814$	2.72690e6	0.144247
$815$	4.60379e6	0.242785
$816$	0	0
$817$	−281424.	−0.0147505
$818$	2.59402e7	1.35547
$819$	0	0
$820$	2.96037e6	0.153749
$821$	6.00202e6	0.310770	0.155385	−	0.987854i	$-0.450338\pi$
$821$	6.00202e6	0.310770	0.155385	+	0.987854i	$0.450338\pi$
$822$	0	0
$823$	−2.98462e7	−1.53599	−0.767996	−	0.640455i	$-0.778746\pi$
$823$	−2.98462e7	−1.53599	−0.767996	+	0.640455i	$0.778746\pi$
$824$	−7.16265e6	−0.367499
$825$	0	0
$826$	1.54841e7	0.789652
$827$	3.81477e6	0.193956	0.0969782	−	0.995287i	$-0.469082\pi$
$827$	3.81477e6	0.193956	0.0969782	+	0.995287i	$0.469082\pi$
$828$	0	0
$829$	7.23404e6	0.365590	0.182795	−	0.983151i	$-0.441486\pi$
$829$	7.23404e6	0.365590	0.182795	+	0.983151i	$0.441486\pi$
$830$	−1.39560e7	−0.703177
$831$	0	0
$832$	6.30407e6	0.315728
$833$	−5.06913e6	−0.253117
$834$	0	0
$835$	1.80206e6	0.0894442
$836$	33390.7	0.00165238
$837$	0	0
$838$	2.96512e7	1.45859
$839$	2.18515e6	0.107171	0.0535853	−	0.998563i	$-0.482935\pi$
$839$	2.18515e6	0.107171	0.0535853	+	0.998563i	$0.482935\pi$
$840$	0	0
$841$	−1.07706e7	−0.525111
$842$	3.87175e7	1.88203
$843$	0	0
$844$	1.21925e7	0.589164
$845$	2.23237e7	1.07554
$846$	0	0
$847$	2.16057e6	0.103481
$848$	2.87472e7	1.37280
$849$	0	0
$850$	−4.60247e6	−0.218496
$851$	−6.60808e6	−0.312789
$852$	0	0
$853$	−1.41107e7	−0.664014	−0.332007	−	0.943277i	$-0.607726\pi$
$853$	−1.41107e7	−0.664014	−0.332007	+	0.943277i	$0.607726\pi$
$854$	4.68616e7	2.19873
$855$	0	0
$856$	1.36580e7	0.637091
$857$	−3.50068e7	−1.62817	−0.814087	−	0.580743i	$-0.802762\pi$
$857$	−3.50068e7	−1.62817	−0.814087	+	0.580743i	$0.802762\pi$
$858$	0	0
$859$	2.72242e6	0.125885	0.0629423	−	0.998017i	$-0.479952\pi$
$859$	2.72242e6	0.125885	0.0629423	+	0.998017i	$0.479952\pi$
$860$	1.03246e7	0.476020
$861$	0	0
$862$	−1.82051e7	−0.834498
$863$	−976336.	−0.0446244	−0.0223122	−	0.999751i	$-0.507103\pi$
$863$	−976336.	−0.0446244	−0.0223122	+	0.999751i	$0.507103\pi$
$864$	0	0
$865$	9.37238e6	0.425902
$866$	2.16369e7	0.980392
$867$	0	0
$868$	−2.85945e7	−1.28820
$869$	1.02941e7	0.462424
$870$	0	0
$871$	3.54050e7	1.58132
$872$	2.67232e6	0.119014
$873$	0	0
$874$	−209586.	−0.00928075
$875$	2.30578e6	0.101812
$876$	0	0
$877$	3.04885e7	1.33856	0.669279	−	0.743011i	$-0.266603\pi$
$877$	3.04885e7	1.33856	0.669279	+	0.743011i	$0.266603\pi$
$878$	1.39786e7	0.611965
$879$	0	0
$880$	3.82056e6	0.166311
$881$	−3.54112e7	−1.53710	−0.768548	−	0.639792i	$-0.779020\pi$
$881$	−3.54112e7	−1.53710	−0.768548	+	0.639792i	$0.779020\pi$
$882$	0	0
$883$	7.34069e6	0.316836	0.158418	−	0.987372i	$-0.449361\pi$
$883$	7.34069e6	0.316836	0.158418	+	0.987372i	$0.449361\pi$
$884$	2.30789e7	0.993308
$885$	0	0
$886$	5.25687e7	2.24979
$887$	−8.51565e6	−0.363420	−0.181710	−	0.983352i	$-0.558163\pi$
$887$	−8.51565e6	−0.363420	−0.181710	+	0.983352i	$0.558163\pi$
$888$	0	0
$889$	−1.36281e7	−0.578335
$890$	−1.26737e7	−0.536326
$891$	0	0
$892$	1.16948e7	0.492129
$893$	338967.	0.0142242
$894$	0	0
$895$	−8.86363e6	−0.369874
$896$	2.41290e7	1.00408
$897$	0	0
$898$	−4.44015e7	−1.83741
$899$	3.00519e7	1.24014
$900$	0	0
$901$	2.32160e7	0.952743
$902$	−5.14048e6	−0.210372
$903$	0	0
$904$	1.21628e7	0.495007
$905$	1.01000e7	0.409920
$906$	0	0
$907$	1.84383e7	0.744224	0.372112	−	0.928188i	$-0.378634\pi$
$907$	1.84383e7	0.744224	0.372112	+	0.928188i	$0.378634\pi$
$908$	−3.05176e7	−1.22839
$909$	0	0
$910$	−2.99481e7	−1.19885
$911$	−3.20540e7	−1.27964	−0.639819	−	0.768526i	$-0.720991\pi$
$911$	−3.20540e7	−1.27964	−0.639819	+	0.768526i	$0.720991\pi$
$912$	0	0
$913$	9.35595e6	0.371459
$914$	−4.04468e7	−1.60147
$915$	0	0
$916$	1.84441e7	0.726303
$917$	−3.10691e7	−1.22013
$918$	0	0
$919$	2.41323e7	0.942563	0.471282	−	0.881983i	$-0.343792\pi$
$919$	2.41323e7	0.942563	0.471282	+	0.881983i	$0.343792\pi$
$920$	−4.53787e6	−0.176759
$921$	0	0
$922$	−2.06264e7	−0.799089
$923$	−3.28019e7	−1.26735
$924$	0	0
$925$	−1.95095e6	−0.0749708
$926$	1.75160e7	0.671287
$927$	0	0
$928$	1.98950e7	0.758359
$929$	2.12500e7	0.807828	0.403914	−	0.914797i	$-0.367650\pi$
$929$	2.12500e7	0.807828	0.403914	+	0.914797i	$0.367650\pi$
$930$	0	0
$931$	−68151.4	−0.00257692
$932$	1.30993e7	0.493977
$933$	0	0
$934$	−1.20945e6	−0.0453652
$935$	3.08546e6	0.115422
$936$	0	0
$937$	−2.96484e7	−1.10320	−0.551598	−	0.834110i	$-0.685982\pi$
$937$	−2.96484e7	−1.10320	−0.551598	+	0.834110i	$0.685982\pi$
$938$	−3.35478e7	−1.24496
$939$	0	0
$940$	−1.24356e7	−0.459038
$941$	4.32235e7	1.59128	0.795638	−	0.605772i	$-0.207136\pi$
$941$	4.32235e7	1.59128	0.795638	+	0.605772i	$0.207136\pi$
$942$	0	0
$943$	1.24569e7	0.456175
$944$	−1.83558e7	−0.670415
$945$	0	0
$946$	−1.79279e7	−0.651330
$947$	−1.03051e7	−0.373402	−0.186701	−	0.982417i	$-0.559780\pi$
$947$	−1.03051e7	−0.373402	−0.186701	+	0.982417i	$0.559780\pi$
$948$	0	0
$949$	−5.57504e7	−2.00947
$950$	−61877.4	−0.00222445
$951$	0	0
$952$	1.29061e7	0.461531
$953$	−2.76302e7	−0.985488	−0.492744	−	0.870174i	$-0.664006\pi$
$953$	−2.76302e7	−0.985488	−0.492744	+	0.870174i	$0.664006\pi$
$954$	0	0
$955$	−4.53205e6	−0.160800
$956$	2.14263e7	0.758231
$957$	0	0
$958$	−3.89164e7	−1.37000
$959$	−3.52087e6	−0.123624
$960$	0	0
$961$	6.40882e7	2.23856
$962$	2.53395e7	0.882797
$963$	0	0
$964$	−2.45423e7	−0.850596
$965$	1.70940e7	0.590917
$966$	0	0
$967$	−1.92779e7	−0.662968	−0.331484	−	0.943461i	$-0.607549\pi$
$967$	−1.92779e7	−0.662968	−0.331484	+	0.943461i	$0.607549\pi$
$968$	−1.25537e6	−0.0430611
$969$	0	0
$970$	−3.80476e6	−0.129837
$971$	2.79869e7	0.952592	0.476296	−	0.879285i	$-0.341979\pi$
$971$	2.79869e7	0.952592	0.476296	+	0.879285i	$0.341979\pi$
$972$	0	0
$973$	4.31539e7	1.46129
$974$	1.00046e7	0.337910
$975$	0	0
$976$	−5.55527e7	−1.86673
$977$	−3.40783e7	−1.14220	−0.571099	−	0.820881i	$-0.693483\pi$
$977$	−3.40783e7	−1.14220	−0.571099	+	0.820881i	$0.693483\pi$
$978$	0	0
$979$	8.49634e6	0.283319
$980$	2.50026e6	0.0831610
$981$	0	0
$982$	2.45174e7	0.811328
$983$	5.50760e7	1.81794	0.908968	−	0.416866i	$-0.136872\pi$
$983$	5.50760e7	1.81794	0.908968	+	0.416866i	$0.136872\pi$
$984$	0	0
$985$	1.10524e7	0.362967
$986$	2.29828e7	0.752853
$987$	0	0
$988$	310281.	0.0101126
$989$	4.34446e7	1.41236
$990$	0	0
$991$	−6.15387e6	−0.199051	−0.0995255	−	0.995035i	$-0.531732\pi$
$991$	−6.15387e6	−0.199051	−0.0995255	+	0.995035i	$0.531732\pi$
$992$	6.13810e7	1.98041
$993$	0	0
$994$	3.10812e7	0.997774
$995$	2.31404e7	0.740990
$996$	0	0
$997$	1.41497e7	0.450828	0.225414	−	0.974263i	$-0.427627\pi$
$997$	1.41497e7	0.450828	0.225414	+	0.974263i	$0.427627\pi$
$998$	2.04528e7	0.650021
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.6.a.a.1.3		3
3.2	odd	2		165.6.a.e.1.1	✓	3
15.14	odd	2		825.6.a.f.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.a.e.1.1	✓	3	3.2	odd	2
495.6.a.a.1.3		3	1.1	even	1	trivial
825.6.a.f.1.3		3	15.14	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:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	495.a (trivial)

\(f(q)\)	\(=\)	\(q+7.21967 q^{2} +20.1236 q^{4} +25.0000 q^{5} +147.570 q^{7} -85.7438 q^{8} +O(q^{10})\)	\(q+7.21967 q^{2} +20.1236 q^{4} +25.0000 q^{5} +147.570 q^{7} -85.7438 q^{8} +180.492 q^{10} -121.000 q^{11} -1124.39 q^{13} +1065.40 q^{14} -1263.00 q^{16} -1019.99 q^{17} -13.7131 q^{19} +503.090 q^{20} -873.580 q^{22} +2116.94 q^{23} +625.000 q^{25} -8117.69 q^{26} +2969.63 q^{28} -3120.98 q^{29} -9628.99 q^{31} -6374.61 q^{32} -7363.96 q^{34} +3689.24 q^{35} -3121.52 q^{37} -99.0039 q^{38} -2143.59 q^{40} +5884.38 q^{41} +20522.3 q^{43} -2434.95 q^{44} +15283.6 q^{46} -24718.5 q^{47} +4969.81 q^{49} +4512.29 q^{50} -22626.7 q^{52} -22761.1 q^{53} -3025.00 q^{55} -12653.2 q^{56} -22532.4 q^{58} +14533.5 q^{59} +43984.9 q^{61} -69518.1 q^{62} -5606.68 q^{64} -28109.6 q^{65} -31488.3 q^{67} -20525.8 q^{68} +26635.1 q^{70} +29173.2 q^{71} +49583.0 q^{73} -22536.3 q^{74} -275.956 q^{76} -17855.9 q^{77} -85075.4 q^{79} -31574.9 q^{80} +42483.3 q^{82} -77321.9 q^{83} -25499.6 q^{85} +148164. q^{86} +10375.0 q^{88} -70217.7 q^{89} -165925. q^{91} +42600.5 q^{92} -178459. q^{94} -342.827 q^{95} -21080.0 q^{97} +35880.4 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 7 q^{2} + 73 q^{4} + 75 q^{5} + 92 q^{7} - 231 q^{8}+O(q^{10}) \) 3 * q - 7 * q^2 + 73 * q^4 + 75 * q^5 + 92 * q^7 - 231 * q^8	\( 3 q - 7 q^{2} + 73 q^{4} + 75 q^{5} + 92 q^{7} - 231 q^{8} - 175 q^{10} - 363 q^{11} - 90 q^{13} + 784 q^{14} - 415 q^{16} - 1934 q^{17} + 2084 q^{19} + 1825 q^{20} + 847 q^{22} - 1220 q^{23} + 1875 q^{25} - 17062 q^{26} + 11120 q^{28} - 4402 q^{29} - 10688 q^{31} - 12439 q^{32} - 4094 q^{34} + 2300 q^{35} - 8190 q^{37} - 13792 q^{38} - 5775 q^{40} - 5974 q^{41} + 18868 q^{43} - 8833 q^{44} + 46220 q^{46} - 55500 q^{47} + 1907 q^{49} - 4375 q^{50} + 27330 q^{52} - 9206 q^{53} - 9075 q^{55} - 73248 q^{56} + 15366 q^{58} + 59196 q^{59} + 79902 q^{61} - 64616 q^{62} + 2129 q^{64} - 2250 q^{65} + 4468 q^{67} + 1218 q^{68} + 19600 q^{70} + 75164 q^{71} - 61290 q^{73} + 56766 q^{74} + 37816 q^{76} - 11132 q^{77} - 83564 q^{79} - 10375 q^{80} + 147410 q^{82} - 74764 q^{83} - 48350 q^{85} + 253432 q^{86} + 27951 q^{88} - 37342 q^{89} - 126488 q^{91} - 148164 q^{92} + 59252 q^{94} + 52100 q^{95} + 33486 q^{97} + 95249 q^{98}+O(q^{100}) \) 3 * q - 7 * q^2 + 73 * q^4 + 75 * q^5 + 92 * q^7 - 231 * q^8 - 175 * q^10 - 363 * q^11 - 90 * q^13 + 784 * q^14 - 415 * q^16 - 1934 * q^17 + 2084 * q^19 + 1825 * q^20 + 847 * q^22 - 1220 * q^23 + 1875 * q^25 - 17062 * q^26 + 11120 * q^28 - 4402 * q^29 - 10688 * q^31 - 12439 * q^32 - 4094 * q^34 + 2300 * q^35 - 8190 * q^37 - 13792 * q^38 - 5775 * q^40 - 5974 * q^41 + 18868 * q^43 - 8833 * q^44 + 46220 * q^46 - 55500 * q^47 + 1907 * q^49 - 4375 * q^50 + 27330 * q^52 - 9206 * q^53 - 9075 * q^55 - 73248 * q^56 + 15366 * q^58 + 59196 * q^59 + 79902 * q^61 - 64616 * q^62 + 2129 * q^64 - 2250 * q^65 + 4468 * q^67 + 1218 * q^68 + 19600 * q^70 + 75164 * q^71 - 61290 * q^73 + 56766 * q^74 + 37816 * q^76 - 11132 * q^77 - 83564 * q^79 - 10375 * q^80 + 147410 * q^82 - 74764 * q^83 - 48350 * q^85 + 253432 * q^86 + 27951 * q^88 - 37342 * q^89 - 126488 * q^91 - 148164 * q^92 + 59252 * q^94 + 52100 * q^95 + 33486 * q^97 + 95249 * q^98

Introduction

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