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 1.2
Root \(1.64661\) 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)
 
\(f(q)\) \(=\) \(q-1.64661 q^{2} -1.00000 q^{3} +0.711327 q^{4} +1.00000 q^{5} +1.64661 q^{6} +4.47988 q^{7} +2.12194 q^{8} +1.00000 q^{9} -1.64661 q^{10} -3.44134 q^{11} -0.711327 q^{12} +2.47988 q^{13} -7.37662 q^{14} -1.00000 q^{15} -4.91667 q^{16} +7.62799 q^{17} -1.64661 q^{18} +0.711327 q^{20} -4.47988 q^{21} +5.66654 q^{22} +3.87057 q^{23} -2.12194 q^{24} +1.00000 q^{25} -4.08340 q^{26} -1.00000 q^{27} +3.18666 q^{28} -8.73456 q^{29} +1.64661 q^{30} -0.422654 q^{31} +3.85195 q^{32} +3.44134 q^{33} -12.5603 q^{34} +4.47988 q^{35} +0.711327 q^{36} +3.90253 q^{37} -2.47988 q^{39} +2.12194 q^{40} +5.29322 q^{41} +7.37662 q^{42} +2.47988 q^{43} -2.44791 q^{44} +1.00000 q^{45} -6.37332 q^{46} -0.677330 q^{47} +4.91667 q^{48} +13.0693 q^{49} -1.64661 q^{50} -7.62799 q^{51} +1.76401 q^{52} +11.4986 q^{53} +1.64661 q^{54} -3.44134 q^{55} +9.50605 q^{56} +14.3824 q^{58} -8.53053 q^{59} -0.711327 q^{60} +8.20233 q^{61} +0.695946 q^{62} +4.47988 q^{63} +3.49067 q^{64} +2.47988 q^{65} -5.66654 q^{66} -9.63457 q^{67} +5.42600 q^{68} -3.87057 q^{69} -7.37662 q^{70} -3.85189 q^{71} +2.12194 q^{72} +16.7852 q^{73} -6.42595 q^{74} -1.00000 q^{75} -15.4168 q^{77} +4.08340 q^{78} +12.1252 q^{79} -4.91667 q^{80} +1.00000 q^{81} -8.71588 q^{82} -2.03855 q^{83} -3.18666 q^{84} +7.62799 q^{85} -4.08340 q^{86} +8.73456 q^{87} -7.30232 q^{88} -3.14511 q^{89} -1.64661 q^{90} +11.1096 q^{91} +2.75324 q^{92} +0.422654 q^{93} +1.11530 q^{94} -3.85195 q^{96} +2.00000 q^{97} -21.5201 q^{98} -3.44134 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - q^{2} - 5 q^{3} + 7 q^{4} + 5 q^{5} + q^{6} + 2 q^{7} - 6 q^{8} + 5 q^{9} - q^{10} + 5 q^{11} - 7 q^{12} - 8 q^{13} - 4 q^{14} - 5 q^{15} + 7 q^{16} + 10 q^{17} - q^{18} + 7 q^{20} - 2 q^{21}+ \cdots + 5 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.64661 −1.16433 −0.582165 − 0.813071i \(-0.697794\pi\)
−0.582165 + 0.813071i \(0.697794\pi\)
\(3\) −1.00000 −0.577350
\(4\) 0.711327 0.355663
\(5\) 1.00000 0.447214
\(6\) 1.64661 0.672226
\(7\) 4.47988 1.69324 0.846618 − 0.532201i \(-0.178635\pi\)
0.846618 + 0.532201i \(0.178635\pi\)
\(8\) 2.12194 0.750220
\(9\) 1.00000 0.333333
\(10\) −1.64661 −0.520704
\(11\) −3.44134 −1.03760 −0.518801 − 0.854895i \(-0.673621\pi\)
−0.518801 + 0.854895i \(0.673621\pi\)
\(12\) −0.711327 −0.205342
\(13\) 2.47988 0.687795 0.343898 − 0.939007i \(-0.388253\pi\)
0.343898 + 0.939007i \(0.388253\pi\)
\(14\) −7.37662 −1.97148
\(15\) −1.00000 −0.258199
\(16\) −4.91667 −1.22917
\(17\) 7.62799 1.85006 0.925030 − 0.379894i \(-0.124039\pi\)
0.925030 + 0.379894i \(0.124039\pi\)
\(18\) −1.64661 −0.388110
\(19\) 0 0
\(20\) 0.711327 0.159058
\(21\) −4.47988 −0.977590
\(22\) 5.66654 1.20811
\(23\) 3.87057 0.807069 0.403535 − 0.914964i \(-0.367782\pi\)
0.403535 + 0.914964i \(0.367782\pi\)
\(24\) −2.12194 −0.433140
\(25\) 1.00000 0.200000
\(26\) −4.08340 −0.800820
\(27\) −1.00000 −0.192450
\(28\) 3.18666 0.602222
\(29\) −8.73456 −1.62197 −0.810983 − 0.585069i \(-0.801067\pi\)
−0.810983 + 0.585069i \(0.801067\pi\)
\(30\) 1.64661 0.300629
\(31\) −0.422654 −0.0759108 −0.0379554 − 0.999279i \(-0.512084\pi\)
−0.0379554 + 0.999279i \(0.512084\pi\)
\(32\) 3.85195 0.680935
\(33\) 3.44134 0.599060
\(34\) −12.5603 −2.15408
\(35\) 4.47988 0.757238
\(36\) 0.711327 0.118554
\(37\) 3.90253 0.641573 0.320786 − 0.947152i \(-0.396053\pi\)
0.320786 + 0.947152i \(0.396053\pi\)
\(38\) 0 0
\(39\) −2.47988 −0.397099
\(40\) 2.12194 0.335509
\(41\) 5.29322 0.826662 0.413331 − 0.910581i \(-0.364365\pi\)
0.413331 + 0.910581i \(0.364365\pi\)
\(42\) 7.37662 1.13824
\(43\) 2.47988 0.378178 0.189089 − 0.981960i \(-0.439446\pi\)
0.189089 + 0.981960i \(0.439446\pi\)
\(44\) −2.44791 −0.369037
\(45\) 1.00000 0.149071
\(46\) −6.37332 −0.939695
\(47\) −0.677330 −0.0987987 −0.0493994 − 0.998779i \(-0.515731\pi\)
−0.0493994 + 0.998779i \(0.515731\pi\)
\(48\) 4.91667 0.709660
\(49\) 13.0693 1.86705
\(50\) −1.64661 −0.232866
\(51\) −7.62799 −1.06813
\(52\) 1.76401 0.244624
\(53\) 11.4986 1.57945 0.789725 − 0.613462i \(-0.210223\pi\)
0.789725 + 0.613462i \(0.210223\pi\)
\(54\) 1.64661 0.224075
\(55\) −3.44134 −0.464030
\(56\) 9.50605 1.27030
\(57\) 0 0
\(58\) 14.3824 1.88850
\(59\) −8.53053 −1.11058 −0.555290 − 0.831657i \(-0.687393\pi\)
−0.555290 + 0.831657i \(0.687393\pi\)
\(60\) −0.711327 −0.0918319
\(61\) 8.20233 1.05020 0.525101 − 0.851040i \(-0.324028\pi\)
0.525101 + 0.851040i \(0.324028\pi\)
\(62\) 0.695946 0.0883852
\(63\) 4.47988 0.564412
\(64\) 3.49067 0.436334
\(65\) 2.47988 0.307591
\(66\) −5.66654 −0.697503
\(67\) −9.63457 −1.17705 −0.588525 − 0.808479i \(-0.700291\pi\)
−0.588525 + 0.808479i \(0.700291\pi\)
\(68\) 5.42600 0.657999
\(69\) −3.87057 −0.465962
\(70\) −7.37662 −0.881675
\(71\) −3.85189 −0.457135 −0.228567 − 0.973528i \(-0.573404\pi\)
−0.228567 + 0.973528i \(0.573404\pi\)
\(72\) 2.12194 0.250073
\(73\) 16.7852 1.96456 0.982280 − 0.187420i \(-0.0600126\pi\)
0.982280 + 0.187420i \(0.0600126\pi\)
\(74\) −6.42595 −0.747002
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −15.4168 −1.75690
\(78\) 4.08340 0.462354
\(79\) 12.1252 1.36420 0.682098 − 0.731261i \(-0.261068\pi\)
0.682098 + 0.731261i \(0.261068\pi\)
\(80\) −4.91667 −0.549700
\(81\) 1.00000 0.111111
\(82\) −8.71588 −0.962507
\(83\) −2.03855 −0.223759 −0.111880 − 0.993722i \(-0.535687\pi\)
−0.111880 + 0.993722i \(0.535687\pi\)
\(84\) −3.18666 −0.347693
\(85\) 7.62799 0.827372
\(86\) −4.08340 −0.440324
\(87\) 8.73456 0.936443
\(88\) −7.30232 −0.778430
\(89\) −3.14511 −0.333381 −0.166690 − 0.986009i \(-0.553308\pi\)
−0.166690 + 0.986009i \(0.553308\pi\)
\(90\) −1.64661 −0.173568
\(91\) 11.1096 1.16460
\(92\) 2.75324 0.287045
\(93\) 0.422654 0.0438271
\(94\) 1.11530 0.115034
\(95\) 0 0
\(96\) −3.85195 −0.393138
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −21.5201 −2.17386
\(99\) −3.44134 −0.345867
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.1.2 5
19.7 even 3 285.2.i.f.106.4 ✓ 10
19.11 even 3 285.2.i.f.121.4 yes 10
19.18 odd 2 5415.2.a.z.1.4 5
57.11 odd 6 855.2.k.i.406.2 10
57.26 odd 6 855.2.k.i.676.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.i.f.106.4 ✓ 10 19.7 even 3
285.2.i.f.121.4 yes 10 19.11 even 3
855.2.k.i.406.2 10 57.11 odd 6
855.2.k.i.676.2 10 57.26 odd 6
5415.2.a.y.1.2 5 1.1 even 1 trivial
5415.2.a.z.1.4 5 19.18 odd 2