Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [145,2,Mod(59,145)] 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("145.59"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(145, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 145 = 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 145.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-4] 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: no
Analytic conductor: \(1.15783082931\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 59.4
Root \(-1.18614 - 1.26217i\) of defining polynomial
Character \(\chi\) \(=\) 145.59
Dual form 145.2.b.a.59.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.73205i q^{2} +2.52434i q^{3} -1.00000 q^{4} +(-2.18614 - 0.469882i) q^{5} -4.37228 q^{6} -1.58457i q^{7} +1.73205i q^{8} -3.37228 q^{9} +(0.813859 - 3.78651i) q^{10} +6.37228 q^{11} -2.52434i q^{12} -0.939764i q^{13} +2.74456 q^{14} +(1.18614 - 5.51856i) q^{15} -5.00000 q^{16} +5.04868i q^{17} -5.84096i q^{18} -4.00000 q^{19} +(2.18614 + 0.469882i) q^{20} +4.00000 q^{21} +11.0371i q^{22} -3.46410i q^{23} -4.37228 q^{24} +(4.55842 + 2.05446i) q^{25} +1.62772 q^{26} -0.939764i q^{27} +1.58457i q^{28} +1.00000 q^{29} +(9.55842 + 2.05446i) q^{30} +2.37228 q^{31} -5.19615i q^{32} +16.0858i q^{33} -8.74456 q^{34} +(-0.744563 + 3.46410i) q^{35} +3.37228 q^{36} -10.0974i q^{37} -6.92820i q^{38} +2.37228 q^{39} +(0.813859 - 3.78651i) q^{40} +6.74456 q^{41} +6.92820i q^{42} -5.69349i q^{43} -6.37228 q^{44} +(7.37228 + 1.58457i) q^{45} +6.00000 q^{46} +5.69349i q^{47} -12.6217i q^{48} +4.48913 q^{49} +(-3.55842 + 7.89542i) q^{50} -12.7446 q^{51} +0.939764i q^{52} -0.939764i q^{53} +1.62772 q^{54} +(-13.9307 - 2.99422i) q^{55} +2.74456 q^{56} -10.0974i q^{57} +1.73205i q^{58} -0.744563 q^{59} +(-1.18614 + 5.51856i) q^{60} +6.00000 q^{61} +4.10891i q^{62} +5.34363i q^{63} -1.00000 q^{64} +(-0.441578 + 2.05446i) q^{65} -27.8614 q^{66} -8.51278i q^{67} -5.04868i q^{68} +8.74456 q^{69} +(-6.00000 - 1.28962i) q^{70} -4.74456 q^{71} -5.84096i q^{72} +6.92820i q^{73} +17.4891 q^{74} +(-5.18614 + 11.5070i) q^{75} +4.00000 q^{76} -10.0974i q^{77} +4.10891i q^{78} -5.62772 q^{79} +(10.9307 + 2.34941i) q^{80} -7.74456 q^{81} +11.6819i q^{82} -16.7306i q^{83} -4.00000 q^{84} +(2.37228 - 11.0371i) q^{85} +9.86141 q^{86} +2.52434i q^{87} +11.0371i q^{88} -10.7446 q^{89} +(-2.74456 + 12.7692i) q^{90} -1.48913 q^{91} +3.46410i q^{92} +5.98844i q^{93} -9.86141 q^{94} +(8.74456 + 1.87953i) q^{95} +13.1168 q^{96} -6.92820i q^{97} +7.77539i q^{98} -21.4891 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 3 q^{5} - 6 q^{6} - 2 q^{9} + 9 q^{10} + 14 q^{11} - 12 q^{14} - q^{15} - 20 q^{16} - 16 q^{19} + 3 q^{20} + 16 q^{21} - 6 q^{24} + q^{25} + 18 q^{26} + 4 q^{29} + 21 q^{30} - 2 q^{31}+ \cdots - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/145\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(117\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73205i 1.22474i 0.790569 + 0.612372i \(0.209785\pi\)
−0.790569 + 0.612372i \(0.790215\pi\)
\(3\) 2.52434i 1.45743i 0.684819 + 0.728714i \(0.259881\pi\)
−0.684819 + 0.728714i \(0.740119\pi\)
\(4\) −1.00000 −0.500000
\(5\) −2.18614 − 0.469882i −0.977672 − 0.210138i
\(6\) −4.37228 −1.78498
\(7\) − 1.58457i − 0.598913i −0.954110 − 0.299456i \(-0.903195\pi\)
0.954110 − 0.299456i \(-0.0968053\pi\)
\(8\) 1.73205i 0.612372i
\(9\) −3.37228 −1.12409
\(10\) 0.813859 − 3.78651i 0.257365 − 1.19740i
\(11\) 6.37228 1.92132 0.960658 − 0.277736i \(-0.0895839\pi\)
0.960658 + 0.277736i \(0.0895839\pi\)
\(12\) − 2.52434i − 0.728714i
\(13\) − 0.939764i − 0.260644i −0.991472 − 0.130322i \(-0.958399\pi\)
0.991472 − 0.130322i \(-0.0416010\pi\)
\(14\) 2.74456 0.733515
\(15\) 1.18614 − 5.51856i 0.306260 − 1.42489i
\(16\) −5.00000 −1.25000
\(17\) 5.04868i 1.22448i 0.790671 + 0.612242i \(0.209732\pi\)
−0.790671 + 0.612242i \(0.790268\pi\)
\(18\) − 5.84096i − 1.37673i
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 2.18614 + 0.469882i 0.488836 + 0.105069i
\(21\) 4.00000 0.872872
\(22\) 11.0371i 2.35312i
\(23\) − 3.46410i − 0.722315i −0.932505 − 0.361158i \(-0.882382\pi\)
0.932505 − 0.361158i \(-0.117618\pi\)
\(24\) −4.37228 −0.892488
\(25\) 4.55842 + 2.05446i 0.911684 + 0.410891i
\(26\) 1.62772 0.319222
\(27\) − 0.939764i − 0.180858i
\(28\) 1.58457i 0.299456i
\(29\) 1.00000 0.185695
\(30\) 9.55842 + 2.05446i 1.74512 + 0.375091i
\(31\) 2.37228 0.426074 0.213037 − 0.977044i \(-0.431664\pi\)
0.213037 + 0.977044i \(0.431664\pi\)
\(32\) − 5.19615i − 0.918559i
\(33\) 16.0858i 2.80018i
\(34\) −8.74456 −1.49968
\(35\) −0.744563 + 3.46410i −0.125854 + 0.585540i
\(36\) 3.37228 0.562047
\(37\) − 10.0974i − 1.65999i −0.557768 − 0.829997i \(-0.688342\pi\)
0.557768 − 0.829997i \(-0.311658\pi\)
\(38\) − 6.92820i − 1.12390i
\(39\) 2.37228 0.379869
\(40\) 0.813859 − 3.78651i 0.128682 − 0.598699i
\(41\) 6.74456 1.05332 0.526662 − 0.850075i \(-0.323443\pi\)
0.526662 + 0.850075i \(0.323443\pi\)
\(42\) 6.92820i 1.06904i
\(43\) − 5.69349i − 0.868248i −0.900853 − 0.434124i \(-0.857058\pi\)
0.900853 − 0.434124i \(-0.142942\pi\)
\(44\) −6.37228 −0.960658
\(45\) 7.37228 + 1.58457i 1.09899 + 0.236214i
\(46\) 6.00000 0.884652
\(47\) 5.69349i 0.830480i 0.909712 + 0.415240i \(0.136302\pi\)
−0.909712 + 0.415240i \(0.863698\pi\)
\(48\) − 12.6217i − 1.82178i
\(49\) 4.48913 0.641304
\(50\) −3.55842 + 7.89542i −0.503237 + 1.11658i
\(51\) −12.7446 −1.78460
\(52\) 0.939764i 0.130322i
\(53\) − 0.939764i − 0.129086i −0.997915 − 0.0645432i \(-0.979441\pi\)
0.997915 − 0.0645432i \(-0.0205590\pi\)
\(54\) 1.62772 0.221504
\(55\) −13.9307 − 2.99422i −1.87842 − 0.403741i
\(56\) 2.74456 0.366758
\(57\) − 10.0974i − 1.33743i
\(58\) 1.73205i 0.227429i
\(59\) −0.744563 −0.0969338 −0.0484669 − 0.998825i \(-0.515434\pi\)
−0.0484669 + 0.998825i \(0.515434\pi\)
\(60\) −1.18614 + 5.51856i −0.153130 + 0.712443i
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 4.10891i 0.521832i
\(63\) 5.34363i 0.673234i
\(64\) −1.00000 −0.125000
\(65\) −0.441578 + 2.05446i −0.0547710 + 0.254824i
\(66\) −27.8614 −3.42950
\(67\) − 8.51278i − 1.04000i −0.854166 − 0.520001i \(-0.825932\pi\)
0.854166 − 0.520001i \(-0.174068\pi\)
\(68\) − 5.04868i − 0.612242i
\(69\) 8.74456 1.05272
\(70\) −6.00000 − 1.28962i −0.717137 − 0.154139i
\(71\) −4.74456 −0.563076 −0.281538 − 0.959550i \(-0.590845\pi\)
−0.281538 + 0.959550i \(0.590845\pi\)
\(72\) − 5.84096i − 0.688364i
\(73\) 6.92820i 0.810885i 0.914121 + 0.405442i \(0.132883\pi\)
−0.914121 + 0.405442i \(0.867117\pi\)
\(74\) 17.4891 2.03307
\(75\) −5.18614 + 11.5070i −0.598844 + 1.32871i
\(76\) 4.00000 0.458831
\(77\) − 10.0974i − 1.15070i
\(78\) 4.10891i 0.465243i
\(79\) −5.62772 −0.633168 −0.316584 − 0.948565i \(-0.602536\pi\)
−0.316584 + 0.948565i \(0.602536\pi\)
\(80\) 10.9307 + 2.34941i 1.22209 + 0.262672i
\(81\) −7.74456 −0.860507
\(82\) 11.6819i 1.29005i
\(83\) − 16.7306i − 1.83642i −0.396092 − 0.918211i \(-0.629634\pi\)
0.396092 − 0.918211i \(-0.370366\pi\)
\(84\) −4.00000 −0.436436
\(85\) 2.37228 − 11.0371i 0.257310 − 1.19714i
\(86\) 9.86141 1.06338
\(87\) 2.52434i 0.270637i
\(88\) 11.0371i 1.17656i
\(89\) −10.7446 −1.13892 −0.569461 − 0.822019i \(-0.692848\pi\)
−0.569461 + 0.822019i \(0.692848\pi\)
\(90\) −2.74456 + 12.7692i −0.289302 + 1.34599i
\(91\) −1.48913 −0.156103
\(92\) 3.46410i 0.361158i
\(93\) 5.98844i 0.620972i
\(94\) −9.86141 −1.01713
\(95\) 8.74456 + 1.87953i 0.897173 + 0.192835i
\(96\) 13.1168 1.33873
\(97\) − 6.92820i − 0.703452i −0.936103 − 0.351726i \(-0.885595\pi\)
0.936103 − 0.351726i \(-0.114405\pi\)
\(98\) 7.77539i 0.785433i
\(99\) −21.4891 −2.15974
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 145.2.b.a.59.4 yes 4
3.2 odd 2 1305.2.c.e.784.2 4
4.3 odd 2 2320.2.d.c.929.1 4
5.2 odd 4 725.2.a.g.1.2 4
5.3 odd 4 725.2.a.g.1.3 4
5.4 even 2 inner 145.2.b.a.59.1 ✓ 4
15.2 even 4 6525.2.a.bk.1.4 4
15.8 even 4 6525.2.a.bk.1.1 4
15.14 odd 2 1305.2.c.e.784.4 4
20.19 odd 2 2320.2.d.c.929.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.b.a.59.1 ✓ 4 5.4 even 2 inner
145.2.b.a.59.4 yes 4 1.1 even 1 trivial
725.2.a.g.1.2 4 5.2 odd 4
725.2.a.g.1.3 4 5.3 odd 4
1305.2.c.e.784.2 4 3.2 odd 2
1305.2.c.e.784.4 4 15.14 odd 2
2320.2.d.c.929.1 4 4.3 odd 2
2320.2.d.c.929.4 4 20.19 odd 2
6525.2.a.bk.1.1 4 15.8 even 4
6525.2.a.bk.1.4 4 15.2 even 4