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 \(-0.532773\pi\)
−0.102777 + 0.994704i \(0.532773\pi\)
\(3\) −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 \(-0.612062\pi\)
−0.344826 + 0.938667i \(0.612062\pi\)
\(14\) 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 \(-0.417319\pi\)
0.256839 + 0.966454i \(0.417319\pi\)
\(30\) 0.290697 0.0530738
\(31\) 4.83099 0.867671 0.433836 − 0.900992i \(-0.357160\pi\)
0.433836 + 0.900992i \(0.357160\pi\)
\(32\) −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 \(-0.673808\pi\)
−0.519301 + 0.854592i \(0.673808\pi\)
\(38\) 0 0
\(39\) 2.48657 0.398171
\(40\) 1.13822 0.179969
\(41\) 2.58139 0.403146 0.201573 − 0.979473i \(-0.435395\pi\)
0.201573 + 0.979473i \(0.435395\pi\)
\(42\) −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 \(-0.434014\pi\)
0.205820 + 0.978590i \(0.434014\pi\)
\(54\) 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 \(-0.188991\pi\)
0.828859 + 0.559458i \(0.188991\pi\)
\(60\) 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 \(-0.877609\pi\)
−0.926985 + 0.375097i \(0.877609\pi\)
\(68\) 6.54258 0.793404
\(69\) −6.41238 −0.771960
\(70\) 0.141446 0.0169060
\(71\) −9.92903 −1.17836 −0.589180 − 0.808002i \(-0.700549\pi\)
−0.589180 + 0.808002i \(0.700549\pi\)
\(72\) 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 \(-0.348745\pi\)
0.457501 + 0.889209i \(0.348745\pi\)
\(80\) 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 \(-0.612139\pi\)
−0.345052 + 0.938584i \(0.612139\pi\)
\(90\) −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 \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 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.y.1.3 5
19.7 even 3 285.2.i.f.106.3 ✓ 10
19.11 even 3 285.2.i.f.121.3 yes 10
19.18 odd 2 5415.2.a.z.1.3 5
57.11 odd 6 855.2.k.i.406.3 10
57.26 odd 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 19.7 even 3
285.2.i.f.121.3 yes 10 19.11 even 3
855.2.k.i.406.3 10 57.11 odd 6
855.2.k.i.676.3 10 57.26 odd 6
5415.2.a.y.1.3 5 1.1 even 1 trivial
5415.2.a.z.1.3 5 19.18 odd 2