Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5415,2,Mod(1,5415)] 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("5415.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5415, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5415 = 3 \cdot 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5415.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,1,5,7,5,1,2,6,5,1,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(43.2389926945\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.8797896.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 8x^{3} + 5x^{2} + 13x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 285)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.290697\) of defining polynomial
Character \(\chi\) \(=\) 5415.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.290697 q^{2} +1.00000 q^{3} -1.91550 q^{4} +1.00000 q^{5} +0.290697 q^{6} -0.486575 q^{7} -1.13822 q^{8} +1.00000 q^{9} +0.290697 q^{10} +5.34764 q^{11} -1.91550 q^{12} +2.48657 q^{13} -0.141446 q^{14} +1.00000 q^{15} +3.50011 q^{16} -3.41561 q^{17} +0.290697 q^{18} -1.91550 q^{20} -0.486575 q^{21} +1.55454 q^{22} +6.41238 q^{23} -1.13822 q^{24} +1.00000 q^{25} +0.722840 q^{26} +1.00000 q^{27} +0.932031 q^{28} -2.76624 q^{29} +0.290697 q^{30} -4.83099 q^{31} +3.29392 q^{32} +5.34764 q^{33} -0.992907 q^{34} -0.486575 q^{35} -1.91550 q^{36} +6.31756 q^{37} +2.48657 q^{39} -1.13822 q^{40} -2.58139 q^{41} -0.141446 q^{42} -2.48657 q^{43} -10.2434 q^{44} +1.00000 q^{45} +1.86406 q^{46} +11.1107 q^{47} +3.50011 q^{48} -6.76325 q^{49} +0.290697 q^{50} -3.41561 q^{51} -4.76302 q^{52} -2.99678 q^{53} +0.290697 q^{54} +5.34764 q^{55} +0.553831 q^{56} -0.804139 q^{58} -12.7332 q^{59} -1.91550 q^{60} +11.8548 q^{61} -1.40436 q^{62} -0.486575 q^{63} -6.04269 q^{64} +2.48657 q^{65} +1.55454 q^{66} +15.1754 q^{67} +6.54258 q^{68} +6.41238 q^{69} -0.141446 q^{70} +9.92903 q^{71} -1.13822 q^{72} -11.0128 q^{73} +1.83650 q^{74} +1.00000 q^{75} -2.60202 q^{77} +0.722840 q^{78} -8.13272 q^{79} +3.50011 q^{80} +1.00000 q^{81} -0.750404 q^{82} -5.86106 q^{83} +0.932031 q^{84} -3.41561 q^{85} -0.722840 q^{86} -2.76624 q^{87} -6.08681 q^{88} +6.51043 q^{89} +0.290697 q^{90} -1.20990 q^{91} -12.2829 q^{92} -4.83099 q^{93} +3.22984 q^{94} +3.29392 q^{96} -2.00000 q^{97} -1.96606 q^{98} +5.34764 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + q^{2} + 5 q^{3} + 7 q^{4} + 5 q^{5} + q^{6} + 2 q^{7} + 6 q^{8} + 5 q^{9} + q^{10} + 5 q^{11} + 7 q^{12} + 8 q^{13} + 4 q^{14} + 5 q^{15} + 7 q^{16} + 10 q^{17} + q^{18} + 7 q^{20} + 2 q^{21}+ \cdots + 5 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.290697 0.205554 0.102777 − 0.994704i \(-0.467227\pi\)
0.102777 + 0.994704i \(0.467227\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.91550 −0.957748
\(5\) 1.00000 0.447214
\(6\) 0.290697 0.118677
\(7\) −0.486575 −0.183908 −0.0919539 − 0.995763i \(-0.529311\pi\)
−0.0919539 + 0.995763i \(0.529311\pi\)
\(8\) −1.13822 −0.402423
\(9\) 1.00000 0.333333
\(10\) 0.290697 0.0919265
\(11\) 5.34764 1.61237 0.806187 − 0.591661i \(-0.201528\pi\)
0.806187 + 0.591661i \(0.201528\pi\)
\(12\) −1.91550 −0.552956
\(13\) 2.48657 0.689652 0.344826 − 0.938667i \(-0.387938\pi\)
0.344826 + 0.938667i \(0.387938\pi\)
\(14\) −0.141446 −0.0378030
\(15\) 1.00000 0.258199
\(16\) 3.50011 0.875028
\(17\) −3.41561 −0.828406 −0.414203 − 0.910184i \(-0.635940\pi\)
−0.414203 + 0.910184i \(0.635940\pi\)
\(18\) 0.290697 0.0685180
\(19\) 0 0
\(20\) −1.91550 −0.428318
\(21\) −0.486575 −0.106179
\(22\) 1.55454 0.331430
\(23\) 6.41238 1.33707 0.668537 − 0.743679i \(-0.266921\pi\)
0.668537 + 0.743679i \(0.266921\pi\)
\(24\) −1.13822 −0.232339
\(25\) 1.00000 0.200000
\(26\) 0.722840 0.141761
\(27\) 1.00000 0.192450
\(28\) 0.932031 0.176137
\(29\) −2.76624 −0.513679 −0.256839 − 0.966454i \(-0.582681\pi\)
−0.256839 + 0.966454i \(0.582681\pi\)
\(30\) 0.290697 0.0530738
\(31\) −4.83099 −0.867671 −0.433836 − 0.900992i \(-0.642840\pi\)
−0.433836 + 0.900992i \(0.642840\pi\)
\(32\) 3.29392 0.582288
\(33\) 5.34764 0.930904
\(34\) −0.992907 −0.170282
\(35\) −0.486575 −0.0822461
\(36\) −1.91550 −0.319249
\(37\) 6.31756 1.03860 0.519301 − 0.854592i \(-0.326192\pi\)
0.519301 + 0.854592i \(0.326192\pi\)
\(38\) 0 0
\(39\) 2.48657 0.398171
\(40\) −1.13822 −0.179969
\(41\) −2.58139 −0.403146 −0.201573 − 0.979473i \(-0.564605\pi\)
−0.201573 + 0.979473i \(0.564605\pi\)
\(42\) −0.141446 −0.0218256
\(43\) −2.48657 −0.379199 −0.189600 − 0.981862i \(-0.560719\pi\)
−0.189600 + 0.981862i \(0.560719\pi\)
\(44\) −10.2434 −1.54425
\(45\) 1.00000 0.149071
\(46\) 1.86406 0.274841
\(47\) 11.1107 1.62066 0.810328 − 0.585976i \(-0.199289\pi\)
0.810328 + 0.585976i \(0.199289\pi\)
\(48\) 3.50011 0.505198
\(49\) −6.76325 −0.966178
\(50\) 0.290697 0.0411108
\(51\) −3.41561 −0.478281
\(52\) −4.76302 −0.660512
\(53\) −2.99678 −0.411639 −0.205820 − 0.978590i \(-0.565986\pi\)
−0.205820 + 0.978590i \(0.565986\pi\)
\(54\) 0.290697 0.0395589
\(55\) 5.34764 0.721075
\(56\) 0.553831 0.0740087
\(57\) 0 0
\(58\) −0.804139 −0.105589
\(59\) −12.7332 −1.65772 −0.828859 − 0.559458i \(-0.811009\pi\)
−0.828859 + 0.559458i \(0.811009\pi\)
\(60\) −1.91550 −0.247289
\(61\) 11.8548 1.51786 0.758929 − 0.651174i \(-0.225723\pi\)
0.758929 + 0.651174i \(0.225723\pi\)
\(62\) −1.40436 −0.178353
\(63\) −0.486575 −0.0613026
\(64\) −6.04269 −0.755336
\(65\) 2.48657 0.308422
\(66\) 1.55454 0.191351
\(67\) 15.1754 1.85397 0.926985 − 0.375097i \(-0.122391\pi\)
0.926985 + 0.375097i \(0.122391\pi\)
\(68\) 6.54258 0.793404
\(69\) 6.41238 0.771960
\(70\) −0.141446 −0.0169060
\(71\) 9.92903 1.17836 0.589180 − 0.808002i \(-0.299451\pi\)
0.589180 + 0.808002i \(0.299451\pi\)
\(72\) −1.13822 −0.134141
\(73\) −11.0128 −1.28896 −0.644478 − 0.764623i \(-0.722925\pi\)
−0.644478 + 0.764623i \(0.722925\pi\)
\(74\) 1.83650 0.213489
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −2.60202 −0.296528
\(78\) 0.722840 0.0818455
\(79\) −8.13272 −0.915002 −0.457501 − 0.889209i \(-0.651255\pi\)
−0.457501 + 0.889209i \(0.651255\pi\)
\(80\) 3.50011 0.391324
\(81\) 1.00000 0.111111
\(82\) −0.750404 −0.0828683
\(83\) −5.86106 −0.643335 −0.321668 − 0.946853i \(-0.604243\pi\)
−0.321668 + 0.946853i \(0.604243\pi\)
\(84\) 0.932031 0.101693
\(85\) −3.41561 −0.370475
\(86\) −0.722840 −0.0779459
\(87\) −2.76624 −0.296572
\(88\) −6.08681 −0.648856
\(89\) 6.51043 0.690104 0.345052 − 0.938584i \(-0.387861\pi\)
0.345052 + 0.938584i \(0.387861\pi\)
\(90\) 0.290697 0.0306422
\(91\) −1.20990 −0.126832
\(92\) −12.2829 −1.28058
\(93\) −4.83099 −0.500950
\(94\) 3.22984 0.333132
\(95\) 0 0
\(96\) 3.29392 0.336184
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −1.96606 −0.198602
\(99\) 5.34764 0.537458
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 5415.2.a.z.1.3 5
19.8 odd 6 285.2.i.f.121.3 yes 10
19.12 odd 6 285.2.i.f.106.3 ✓ 10
19.18 odd 2 5415.2.a.y.1.3 5
57.8 even 6 855.2.k.i.406.3 10
57.50 even 6 855.2.k.i.676.3 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.i.f.106.3 ✓ 10 19.12 odd 6
285.2.i.f.121.3 yes 10 19.8 odd 6
855.2.k.i.406.3 10 57.8 even 6
855.2.k.i.676.3 10 57.50 even 6
5415.2.a.y.1.3 5 19.18 odd 2
5415.2.a.z.1.3 5 1.1 even 1 trivial