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Properties

Label 245.4.j.e.79.6
Level $245$
Weight $4$
Character 245.79
Analytic conductor $14.455$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [245,4,Mod(79,245)] 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("245.79"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(245, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,0,36,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.4554679514\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 55 x^{18} + 2042 x^{16} - 41247 x^{14} + 600234 x^{12} - 4812047 x^{10} + 27547801 x^{8} + \cdots + 12960000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2}\cdot 7^{8} \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 79.6
Root \(0.480883 - 0.277638i\) of defining polynomial
Character \(\chi\) \(=\) 245.79
Dual form 245.4.j.e.214.6

$q$-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+(1.34691 + 0.777638i) q^{2} +(-4.29677 + 2.48074i) q^{3} +(-2.79056 - 4.83339i) q^{4} +(6.75908 - 8.90588i) q^{5} -7.71648 q^{6} -21.1224i q^{8} +(-1.19182 + 2.06430i) q^{9} +(16.0294 - 6.73929i) q^{10} +(-14.9116 - 25.8276i) q^{11} +(23.9808 + 13.8453i) q^{12} +90.7316i q^{13} +(-6.94904 + 55.0341i) q^{15} +(-5.89890 + 10.2172i) q^{16} +(-25.6118 + 14.7870i) q^{17} +(-3.21055 + 1.85361i) q^{18} +(-31.1639 + 53.9774i) q^{19} +(-61.9072 - 7.81689i) q^{20} -46.3833i q^{22} +(-78.4790 - 45.3099i) q^{23} +(52.3992 + 90.7581i) q^{24} +(-33.6296 - 120.391i) q^{25} +(-70.5563 + 122.207i) q^{26} -145.787i q^{27} -193.070 q^{29} +(-52.1563 + 68.7221i) q^{30} +(76.0614 + 131.742i) q^{31} +(-162.231 + 93.6640i) q^{32} +(128.143 + 73.9837i) q^{33} -45.9957 q^{34} +13.3034 q^{36} +(88.7272 + 51.2267i) q^{37} +(-83.9498 + 48.4684i) q^{38} +(-225.082 - 389.853i) q^{39} +(-188.114 - 142.768i) q^{40} -266.744 q^{41} +387.125i q^{43} +(-83.2233 + 144.147i) q^{44} +(10.3288 + 24.5670i) q^{45} +(-70.4694 - 122.057i) q^{46} +(-131.894 - 76.1492i) q^{47} -58.5346i q^{48} +(48.3248 - 188.308i) q^{50} +(73.3655 - 127.073i) q^{51} +(438.541 - 253.192i) q^{52} +(-70.6661 + 40.7991i) q^{53} +(113.369 - 196.361i) q^{54} +(-330.807 - 41.7702i) q^{55} -309.238i q^{57} +(-260.047 - 150.138i) q^{58} +(-117.942 - 204.281i) q^{59} +(285.393 - 119.989i) q^{60} +(-255.227 + 442.065i) q^{61} +236.593i q^{62} -196.964 q^{64} +(808.045 + 613.263i) q^{65} +(115.065 + 199.298i) q^{66} +(300.835 - 173.687i) q^{67} +(142.943 + 82.5280i) q^{68} +449.609 q^{69} +317.014 q^{71} +(43.6029 + 25.1741i) q^{72} +(-614.792 + 354.950i) q^{73} +(79.6716 + 137.995i) q^{74} +(443.158 + 433.868i) q^{75} +347.858 q^{76} -700.129i q^{78} +(531.476 - 920.543i) q^{79} +(51.1220 + 121.594i) q^{80} +(329.480 + 570.676i) q^{81} +(-359.280 - 207.430i) q^{82} +503.810i q^{83} +(-41.4212 + 328.043i) q^{85} +(-301.043 + 521.422i) q^{86} +(829.578 - 478.957i) q^{87} +(-545.541 + 314.968i) q^{88} +(241.171 - 417.721i) q^{89} +(-5.19233 + 41.1215i) q^{90} +505.760i q^{92} +(-653.638 - 377.378i) q^{93} +(-118.433 - 205.132i) q^{94} +(270.077 + 642.380i) q^{95} +(464.712 - 804.906i) q^{96} -481.167i q^{97} +71.0879 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 36 q^{4} - 6 q^{5} + 24 q^{6} + 46 q^{9} + 16 q^{10} - 84 q^{11} + 16 q^{15} - 148 q^{16} - 72 q^{19} - 136 q^{20} - 72 q^{24} + 362 q^{25} + 620 q^{26} + 176 q^{29} - 52 q^{30} - 120 q^{31} + 1928 q^{34}+ \cdots - 10608 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.34691 + 0.777638i 0.476204 + 0.274937i 0.718833 − 0.695183i \(-0.244676\pi\)
−0.242629 + 0.970119i \(0.578010\pi\)
\(3\) −4.29677 + 2.48074i −0.826915 + 0.477419i −0.852795 − 0.522246i \(-0.825094\pi\)
0.0258805 + 0.999665i \(0.491761\pi\)
\(4\) −2.79056 − 4.83339i −0.348820 − 0.604174i
\(5\) 6.75908 − 8.90588i 0.604551 − 0.796567i
\(6\) −7.71648 −0.525040
\(7\) 0 0
\(8\) − 21.1224i − 0.933486i
\(9\) −1.19182 + 2.06430i −0.0441416 + 0.0764555i
\(10\) 16.0294 − 6.73929i 0.506895 − 0.213115i
\(11\) −14.9116 − 25.8276i −0.408728 − 0.707938i 0.586019 − 0.810297i \(-0.300694\pi\)
−0.994748 + 0.102359i \(0.967361\pi\)
\(12\) 23.9808 + 13.8453i 0.576888 + 0.333067i
\(13\) 90.7316i 1.93572i 0.251481 + 0.967862i \(0.419083\pi\)
−0.251481 + 0.967862i \(0.580917\pi\)
\(14\) 0 0
\(15\) −6.94904 + 55.0341i −0.119616 + 0.947317i
\(16\) −5.89890 + 10.2172i −0.0921703 + 0.159644i
\(17\) −25.6118 + 14.7870i −0.365399 + 0.210963i −0.671447 − 0.741053i \(-0.734327\pi\)
0.306047 + 0.952016i \(0.400993\pi\)
\(18\) −3.21055 + 1.85361i −0.0420408 + 0.0242723i
\(19\) −31.1639 + 53.9774i −0.376289 + 0.651751i −0.990519 − 0.137376i \(-0.956133\pi\)
0.614230 + 0.789127i \(0.289467\pi\)
\(20\) −61.9072 − 7.81689i −0.692144 − 0.0873955i
\(21\) 0 0
\(22\) − 46.3833i − 0.449498i
\(23\) −78.4790 − 45.3099i −0.711479 − 0.410772i 0.100130 − 0.994974i \(-0.468074\pi\)
−0.811608 + 0.584202i \(0.801408\pi\)
\(24\) 52.3992 + 90.7581i 0.445664 + 0.771913i
\(25\) −33.6296 − 120.391i −0.269037 − 0.963130i
\(26\) −70.5563 + 122.207i −0.532201 + 0.921800i
\(27\) − 145.787i − 1.03913i
\(28\) 0 0
\(29\) −193.070 −1.23628 −0.618141 − 0.786067i \(-0.712114\pi\)
−0.618141 + 0.786067i \(0.712114\pi\)
\(30\) −52.1563 + 68.7221i −0.317413 + 0.418229i
\(31\) 76.0614 + 131.742i 0.440679 + 0.763278i 0.997740 − 0.0671935i \(-0.0214045\pi\)
−0.557061 + 0.830471i \(0.688071\pi\)
\(32\) −162.231 + 93.6640i −0.896207 + 0.517425i
\(33\) 128.143 + 73.9837i 0.675967 + 0.390270i
\(34\) −45.9957 −0.232006
\(35\) 0 0
\(36\) 13.3034 0.0615899
\(37\) 88.7272 + 51.2267i 0.394234 + 0.227611i 0.683993 − 0.729488i \(-0.260242\pi\)
−0.289759 + 0.957100i \(0.593575\pi\)
\(38\) −83.9498 + 48.4684i −0.358380 + 0.206911i
\(39\) −225.082 − 389.853i −0.924152 − 1.60068i
\(40\) −188.114 − 142.768i −0.743584 − 0.564340i
\(41\) −266.744 −1.01606 −0.508030 − 0.861340i \(-0.669626\pi\)
−0.508030 + 0.861340i \(0.669626\pi\)
\(42\) 0 0
\(43\) 387.125i 1.37293i 0.727163 + 0.686465i \(0.240838\pi\)
−0.727163 + 0.686465i \(0.759162\pi\)
\(44\) −83.2233 + 144.147i −0.285145 + 0.493886i
\(45\) 10.3288 + 24.5670i 0.0342160 + 0.0813830i
\(46\) −70.4694 − 122.057i −0.225873 − 0.391223i
\(47\) −131.894 − 76.1492i −0.409335 − 0.236330i 0.281169 − 0.959658i \(-0.409278\pi\)
−0.690504 + 0.723329i \(0.742611\pi\)
\(48\) − 58.5346i − 0.176016i
\(49\) 0 0
\(50\) 48.3248 − 188.308i 0.136683 − 0.532614i
\(51\) 73.3655 − 127.073i 0.201436 − 0.348897i
\(52\) 438.541 − 253.192i 1.16951 − 0.675219i
\(53\) −70.6661 + 40.7991i −0.183146 + 0.105739i −0.588770 − 0.808301i \(-0.700388\pi\)
0.405624 + 0.914040i \(0.367054\pi\)
\(54\) 113.369 − 196.361i 0.285696 − 0.494840i
\(55\) −330.807 − 41.7702i −0.811017 − 0.102405i
\(56\) 0 0
\(57\) − 309.238i − 0.718590i
\(58\) −260.047 − 150.138i −0.588722 − 0.339899i
\(59\) −117.942 − 204.281i −0.260250 − 0.450765i 0.706059 − 0.708153i \(-0.250471\pi\)
−0.966308 + 0.257388i \(0.917138\pi\)
\(60\) 285.393 − 119.989i 0.614068 − 0.258174i
\(61\) −255.227 + 442.065i −0.535712 + 0.927880i 0.463417 + 0.886140i \(0.346623\pi\)
−0.999129 + 0.0417395i \(0.986710\pi\)
\(62\) 236.593i 0.484635i
\(63\) 0 0
\(64\) −196.964 −0.384696
\(65\) 808.045 + 613.263i 1.54193 + 1.17024i
\(66\) 115.065 + 199.298i 0.214599 + 0.371696i
\(67\) 300.835 − 173.687i 0.548550 − 0.316705i −0.199987 − 0.979799i \(-0.564090\pi\)
0.748537 + 0.663093i \(0.230757\pi\)
\(68\) 142.943 + 82.5280i 0.254917 + 0.147176i
\(69\) 449.609 0.784443
\(70\) 0 0
\(71\) 317.014 0.529896 0.264948 − 0.964263i \(-0.414645\pi\)
0.264948 + 0.964263i \(0.414645\pi\)
\(72\) 43.6029 + 25.1741i 0.0713702 + 0.0412056i
\(73\) −614.792 + 354.950i −0.985698 + 0.569093i −0.903986 − 0.427563i \(-0.859372\pi\)
−0.0817126 + 0.996656i \(0.526039\pi\)
\(74\) 79.6716 + 137.995i 0.125157 + 0.216779i
\(75\) 443.158 + 433.868i 0.682287 + 0.667983i
\(76\) 347.858 0.525028
\(77\) 0 0
\(78\) − 700.129i − 1.01633i
\(79\) 531.476 − 920.543i 0.756908 − 1.31100i −0.187513 − 0.982262i \(-0.560043\pi\)
0.944420 − 0.328740i \(-0.106624\pi\)
\(80\) 51.1220 + 121.594i 0.0714452 + 0.169933i
\(81\) 329.480 + 570.676i 0.451961 + 0.782820i
\(82\) −359.280 − 207.430i −0.483851 − 0.279352i
\(83\) 503.810i 0.666270i 0.942879 + 0.333135i \(0.108106\pi\)
−0.942879 + 0.333135i \(0.891894\pi\)
\(84\) 0 0
\(85\) −41.4212 + 328.043i −0.0528560 + 0.418603i
\(86\) −301.043 + 521.422i −0.377469 + 0.653795i
\(87\) 829.578 − 478.957i 1.02230 − 0.590225i
\(88\) −545.541 + 314.968i −0.660851 + 0.381542i
\(89\) 241.171 − 417.721i 0.287237 − 0.497509i −0.685912 − 0.727684i \(-0.740597\pi\)
0.973149 + 0.230175i \(0.0739299\pi\)
\(90\) −5.19233 + 41.1215i −0.00608133 + 0.0481621i
\(91\) 0 0
\(92\) 505.760i 0.573142i
\(93\) −653.638 − 377.378i −0.728807 − 0.420777i
\(94\) −118.433 − 205.132i −0.129951 − 0.225082i
\(95\) 270.077 + 642.380i 0.291677 + 0.693755i
\(96\) 464.712 − 804.906i 0.494057 − 0.855733i
\(97\) − 481.167i − 0.503661i −0.967771 − 0.251831i \(-0.918967\pi\)
0.967771 − 0.251831i \(-0.0810326\pi\)
\(98\) 0 0
\(99\) 71.0879 0.0721677
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 245.4.j.e.79.6 20
5.4 even 2 inner 245.4.j.e.79.5 20
7.2 even 3 35.4.b.a.29.5 ✓ 10
7.3 odd 6 245.4.j.f.214.5 20
7.4 even 3 inner 245.4.j.e.214.5 20
7.5 odd 6 245.4.b.d.99.5 10
7.6 odd 2 245.4.j.f.79.6 20
21.2 odd 6 315.4.d.c.64.6 10
28.23 odd 6 560.4.g.f.449.8 10
35.2 odd 12 175.4.a.j.1.3 5
35.4 even 6 inner 245.4.j.e.214.6 20
35.9 even 6 35.4.b.a.29.6 yes 10
35.12 even 12 1225.4.a.bh.1.3 5
35.19 odd 6 245.4.b.d.99.6 10
35.23 odd 12 175.4.a.i.1.3 5
35.24 odd 6 245.4.j.f.214.6 20
35.33 even 12 1225.4.a.be.1.3 5
35.34 odd 2 245.4.j.f.79.5 20
105.2 even 12 1575.4.a.bn.1.3 5
105.23 even 12 1575.4.a.bq.1.3 5
105.44 odd 6 315.4.d.c.64.5 10
140.79 odd 6 560.4.g.f.449.3 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.b.a.29.5 ✓ 10 7.2 even 3
35.4.b.a.29.6 yes 10 35.9 even 6
175.4.a.i.1.3 5 35.23 odd 12
175.4.a.j.1.3 5 35.2 odd 12
245.4.b.d.99.5 10 7.5 odd 6
245.4.b.d.99.6 10 35.19 odd 6
245.4.j.e.79.5 20 5.4 even 2 inner
245.4.j.e.79.6 20 1.1 even 1 trivial
245.4.j.e.214.5 20 7.4 even 3 inner
245.4.j.e.214.6 20 35.4 even 6 inner
245.4.j.f.79.5 20 35.34 odd 2
245.4.j.f.79.6 20 7.6 odd 2
245.4.j.f.214.5 20 7.3 odd 6
245.4.j.f.214.6 20 35.24 odd 6
315.4.d.c.64.5 10 105.44 odd 6
315.4.d.c.64.6 10 21.2 odd 6
560.4.g.f.449.3 10 140.79 odd 6
560.4.g.f.449.8 10 28.23 odd 6
1225.4.a.be.1.3 5 35.33 even 12
1225.4.a.bh.1.3 5 35.12 even 12
1575.4.a.bn.1.3 5 105.2 even 12
1575.4.a.bq.1.3 5 105.23 even 12