Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [495,6,Mod(1,495)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// 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);
 
Copy content sage: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")
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 495.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,5,0,61] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(79.3899908074\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 33x^{2} - 8x + 116 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.74666\) of defining polynomial
Character \(\chi\) \(=\) 495.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.64192 q^{2} -25.0202 q^{4} +25.0000 q^{5} -12.8802 q^{7} +150.643 q^{8} -66.0481 q^{10} +121.000 q^{11} -485.167 q^{13} +34.0284 q^{14} +402.660 q^{16} -266.661 q^{17} -149.702 q^{19} -625.506 q^{20} -319.673 q^{22} +3213.11 q^{23} +625.000 q^{25} +1281.77 q^{26} +322.265 q^{28} -2948.81 q^{29} +2145.87 q^{31} -5884.38 q^{32} +704.498 q^{34} -322.004 q^{35} -808.357 q^{37} +395.500 q^{38} +3766.08 q^{40} -10105.2 q^{41} +2763.15 q^{43} -3027.45 q^{44} -8488.79 q^{46} -9973.36 q^{47} -16641.1 q^{49} -1651.20 q^{50} +12139.0 q^{52} -7126.92 q^{53} +3025.00 q^{55} -1940.31 q^{56} +7790.52 q^{58} +33337.2 q^{59} -11871.1 q^{61} -5669.22 q^{62} +2660.94 q^{64} -12129.2 q^{65} +4500.58 q^{67} +6671.93 q^{68} +850.710 q^{70} +45977.8 q^{71} -62039.1 q^{73} +2135.62 q^{74} +3745.57 q^{76} -1558.50 q^{77} -57486.6 q^{79} +10066.5 q^{80} +26697.2 q^{82} +90511.7 q^{83} -6666.53 q^{85} -7300.03 q^{86} +18227.8 q^{88} +127861. q^{89} +6249.03 q^{91} -80392.8 q^{92} +26348.9 q^{94} -3742.54 q^{95} +132338. q^{97} +43964.5 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 5 q^{2} + 61 q^{4} + 100 q^{5} - 90 q^{7} + 135 q^{8} + 125 q^{10} + 484 q^{11} + 820 q^{13} + 1687 q^{14} - 2671 q^{16} + 3800 q^{17} - 3394 q^{19} + 1525 q^{20} + 605 q^{22} + 3020 q^{23} + 2500 q^{25}+ \cdots + 393590 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.64192 −0.467030 −0.233515 − 0.972353i \(-0.575023\pi\)
−0.233515 + 0.972353i \(0.575023\pi\)
\(3\) 0 0
\(4\) −25.0202 −0.781883
\(5\) 25.0000 0.447214
\(6\) 0 0
\(7\) −12.8802 −0.0993520 −0.0496760 − 0.998765i \(-0.515819\pi\)
−0.0496760 + 0.998765i \(0.515819\pi\)
\(8\) 150.643 0.832193
\(9\) 0 0
\(10\) −66.0481 −0.208862
\(11\) 121.000 0.301511
\(12\) 0 0
\(13\) −485.167 −0.796219 −0.398110 − 0.917338i \(-0.630334\pi\)
−0.398110 + 0.917338i \(0.630334\pi\)
\(14\) 34.0284 0.0464004
\(15\) 0 0
\(16\) 402.660 0.393223
\(17\) −266.661 −0.223788 −0.111894 − 0.993720i \(-0.535692\pi\)
−0.111894 + 0.993720i \(0.535692\pi\)
\(18\) 0 0
\(19\) −149.702 −0.0951356 −0.0475678 − 0.998868i \(-0.515147\pi\)
−0.0475678 + 0.998868i \(0.515147\pi\)
\(20\) −625.506 −0.349669
\(21\) 0 0
\(22\) −319.673 −0.140815
\(23\) 3213.11 1.26650 0.633251 − 0.773946i \(-0.281720\pi\)
0.633251 + 0.773946i \(0.281720\pi\)
\(24\) 0 0
\(25\) 625.000 0.200000
\(26\) 1281.77 0.371859
\(27\) 0 0
\(28\) 322.265 0.0776816
\(29\) −2948.81 −0.651106 −0.325553 − 0.945524i \(-0.605550\pi\)
−0.325553 + 0.945524i \(0.605550\pi\)
\(30\) 0 0
\(31\) 2145.87 0.401051 0.200525 − 0.979689i \(-0.435735\pi\)
0.200525 + 0.979689i \(0.435735\pi\)
\(32\) −5884.38 −1.01584
\(33\) 0 0
\(34\) 704.498 0.104516
\(35\) −322.004 −0.0444315
\(36\) 0 0
\(37\) −808.357 −0.0970731 −0.0485366 − 0.998821i \(-0.515456\pi\)
−0.0485366 + 0.998821i \(0.515456\pi\)
\(38\) 395.500 0.0444312
\(39\) 0 0
\(40\) 3766.08 0.372168
\(41\) −10105.2 −0.938829 −0.469414 − 0.882978i \(-0.655535\pi\)
−0.469414 + 0.882978i \(0.655535\pi\)
\(42\) 0 0
\(43\) 2763.15 0.227894 0.113947 − 0.993487i \(-0.463651\pi\)
0.113947 + 0.993487i \(0.463651\pi\)
\(44\) −3027.45 −0.235746
\(45\) 0 0
\(46\) −8488.79 −0.591495
\(47\) −9973.36 −0.658562 −0.329281 − 0.944232i \(-0.606806\pi\)
−0.329281 + 0.944232i \(0.606806\pi\)
\(48\) 0 0
\(49\) −16641.1 −0.990129
\(50\) −1651.20 −0.0934061
\(51\) 0 0
\(52\) 12139.0 0.622550
\(53\) −7126.92 −0.348508 −0.174254 − 0.984701i \(-0.555751\pi\)
−0.174254 + 0.984701i \(0.555751\pi\)
\(54\) 0 0
\(55\) 3025.00 0.134840
\(56\) −1940.31 −0.0826800
\(57\) 0 0
\(58\) 7790.52 0.304086
\(59\) 33337.2 1.24681 0.623403 − 0.781901i \(-0.285750\pi\)
0.623403 + 0.781901i \(0.285750\pi\)
\(60\) 0 0
\(61\) −11871.1 −0.408476 −0.204238 − 0.978921i \(-0.565472\pi\)
−0.204238 + 0.978921i \(0.565472\pi\)
\(62\) −5669.22 −0.187303
\(63\) 0 0
\(64\) 2660.94 0.0812053
\(65\) −12129.2 −0.356080
\(66\) 0 0
\(67\) 4500.58 0.122485 0.0612423 − 0.998123i \(-0.480494\pi\)
0.0612423 + 0.998123i \(0.480494\pi\)
\(68\) 6671.93 0.174976
\(69\) 0 0
\(70\) 850.710 0.0207509
\(71\) 45977.8 1.08244 0.541218 − 0.840882i \(-0.317963\pi\)
0.541218 + 0.840882i \(0.317963\pi\)
\(72\) 0 0
\(73\) −62039.1 −1.36257 −0.681284 − 0.732019i \(-0.738578\pi\)
−0.681284 + 0.732019i \(0.738578\pi\)
\(74\) 2135.62 0.0453361
\(75\) 0 0
\(76\) 3745.57 0.0743848
\(77\) −1558.50 −0.0299557
\(78\) 0 0
\(79\) −57486.6 −1.03633 −0.518166 − 0.855280i \(-0.673385\pi\)
−0.518166 + 0.855280i \(0.673385\pi\)
\(80\) 10066.5 0.175855
\(81\) 0 0
\(82\) 26697.2 0.438461
\(83\) 90511.7 1.44215 0.721074 − 0.692858i \(-0.243649\pi\)
0.721074 + 0.692858i \(0.243649\pi\)
\(84\) 0 0
\(85\) −6666.53 −0.100081
\(86\) −7300.03 −0.106434
\(87\) 0 0
\(88\) 18227.8 0.250916
\(89\) 127861. 1.71105 0.855524 − 0.517764i \(-0.173235\pi\)
0.855524 + 0.517764i \(0.173235\pi\)
\(90\) 0 0
\(91\) 6249.03 0.0791059
\(92\) −80392.8 −0.990256
\(93\) 0 0
\(94\) 26348.9 0.307569
\(95\) −3742.54 −0.0425459
\(96\) 0 0
\(97\) 132338. 1.42809 0.714046 − 0.700099i \(-0.246861\pi\)
0.714046 + 0.700099i \(0.246861\pi\)
\(98\) 43964.5 0.462420
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 495.6.a.g.1.2 4
3.2 odd 2 55.6.a.b.1.3 ✓ 4
12.11 even 2 880.6.a.n.1.2 4
15.2 even 4 275.6.b.d.199.5 8
15.8 even 4 275.6.b.d.199.4 8
15.14 odd 2 275.6.a.d.1.2 4
33.32 even 2 605.6.a.c.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.6.a.b.1.3 ✓ 4 3.2 odd 2
275.6.a.d.1.2 4 15.14 odd 2
275.6.b.d.199.4 8 15.8 even 4
275.6.b.d.199.5 8 15.2 even 4
495.6.a.g.1.2 4 1.1 even 1 trivial
605.6.a.c.1.2 4 33.32 even 2
880.6.a.n.1.2 4 12.11 even 2