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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [230,2,Mod(139,230)] 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("230.139"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 230.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.83655924649\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 139.1
Root \(-1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 230.139
Dual form 230.2.b.a.139.4

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.61803i q^{3} -1.00000 q^{4} -2.23607 q^{5} -1.61803 q^{6} -1.85410i q^{7} +1.00000i q^{8} +0.381966 q^{9} +2.23607i q^{10} -5.61803 q^{11} +1.61803i q^{12} -2.61803i q^{13} -1.85410 q^{14} +3.61803i q^{15} +1.00000 q^{16} +0.854102i q^{17} -0.381966i q^{18} +0.145898 q^{19} +2.23607 q^{20} -3.00000 q^{21} +5.61803i q^{22} -1.00000i q^{23} +1.61803 q^{24} +5.00000 q^{25} -2.61803 q^{26} -5.47214i q^{27} +1.85410i q^{28} +9.70820 q^{29} +3.61803 q^{30} -2.14590 q^{31} -1.00000i q^{32} +9.09017i q^{33} +0.854102 q^{34} +4.14590i q^{35} -0.381966 q^{36} -9.70820i q^{37} -0.145898i q^{38} -4.23607 q^{39} -2.23607i q^{40} -5.61803 q^{41} +3.00000i q^{42} -11.2361i q^{43} +5.61803 q^{44} -0.854102 q^{45} -1.00000 q^{46} +1.70820i q^{47} -1.61803i q^{48} +3.56231 q^{49} -5.00000i q^{50} +1.38197 q^{51} +2.61803i q^{52} +2.00000i q^{53} -5.47214 q^{54} +12.5623 q^{55} +1.85410 q^{56} -0.236068i q^{57} -9.70820i q^{58} +6.00000 q^{59} -3.61803i q^{60} +2.85410 q^{61} +2.14590i q^{62} -0.708204i q^{63} -1.00000 q^{64} +5.85410i q^{65} +9.09017 q^{66} +5.23607i q^{67} -0.854102i q^{68} -1.61803 q^{69} +4.14590 q^{70} +0.381966 q^{71} +0.381966i q^{72} +16.4721i q^{73} -9.70820 q^{74} -8.09017i q^{75} -0.145898 q^{76} +10.4164i q^{77} +4.23607i q^{78} +7.70820 q^{79} -2.23607 q^{80} -7.70820 q^{81} +5.61803i q^{82} -7.70820i q^{83} +3.00000 q^{84} -1.90983i q^{85} -11.2361 q^{86} -15.7082i q^{87} -5.61803i q^{88} -3.70820 q^{89} +0.854102i q^{90} -4.85410 q^{91} +1.00000i q^{92} +3.47214i q^{93} +1.70820 q^{94} -0.326238 q^{95} -1.61803 q^{96} +13.0344i q^{97} -3.56231i q^{98} -2.14590 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 2 q^{6} + 6 q^{9} - 18 q^{11} + 6 q^{14} + 4 q^{16} + 14 q^{19} - 12 q^{21} + 2 q^{24} + 20 q^{25} - 6 q^{26} + 12 q^{29} + 10 q^{30} - 22 q^{31} - 10 q^{34} - 6 q^{36} - 8 q^{39} - 18 q^{41}+ \cdots - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) − 1.61803i − 0.934172i −0.884212 − 0.467086i \(-0.845304\pi\)
0.884212 − 0.467086i \(-0.154696\pi\)
\(4\) −1.00000 −0.500000
\(5\) −2.23607 −1.00000
\(6\) −1.61803 −0.660560
\(7\) − 1.85410i − 0.700785i −0.936603 − 0.350392i \(-0.886048\pi\)
0.936603 − 0.350392i \(-0.113952\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0.381966 0.127322
\(10\) 2.23607i 0.707107i
\(11\) −5.61803 −1.69390 −0.846950 − 0.531672i \(-0.821564\pi\)
−0.846950 + 0.531672i \(0.821564\pi\)
\(12\) 1.61803i 0.467086i
\(13\) − 2.61803i − 0.726112i −0.931767 − 0.363056i \(-0.881733\pi\)
0.931767 − 0.363056i \(-0.118267\pi\)
\(14\) −1.85410 −0.495530
\(15\) 3.61803i 0.934172i
\(16\) 1.00000 0.250000
\(17\) 0.854102i 0.207150i 0.994622 + 0.103575i \(0.0330282\pi\)
−0.994622 + 0.103575i \(0.966972\pi\)
\(18\) − 0.381966i − 0.0900303i
\(19\) 0.145898 0.0334713 0.0167357 − 0.999860i \(-0.494673\pi\)
0.0167357 + 0.999860i \(0.494673\pi\)
\(20\) 2.23607 0.500000
\(21\) −3.00000 −0.654654
\(22\) 5.61803i 1.19777i
\(23\) − 1.00000i − 0.208514i
\(24\) 1.61803 0.330280
\(25\) 5.00000 1.00000
\(26\) −2.61803 −0.513439
\(27\) − 5.47214i − 1.05311i
\(28\) 1.85410i 0.350392i
\(29\) 9.70820 1.80277 0.901384 − 0.433020i \(-0.142552\pi\)
0.901384 + 0.433020i \(0.142552\pi\)
\(30\) 3.61803 0.660560
\(31\) −2.14590 −0.385415 −0.192707 − 0.981256i \(-0.561727\pi\)
−0.192707 + 0.981256i \(0.561727\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 9.09017i 1.58240i
\(34\) 0.854102 0.146477
\(35\) 4.14590i 0.700785i
\(36\) −0.381966 −0.0636610
\(37\) − 9.70820i − 1.59602i −0.602645 − 0.798009i \(-0.705886\pi\)
0.602645 − 0.798009i \(-0.294114\pi\)
\(38\) − 0.145898i − 0.0236678i
\(39\) −4.23607 −0.678314
\(40\) − 2.23607i − 0.353553i
\(41\) −5.61803 −0.877390 −0.438695 − 0.898636i \(-0.644559\pi\)
−0.438695 + 0.898636i \(0.644559\pi\)
\(42\) 3.00000i 0.462910i
\(43\) − 11.2361i − 1.71348i −0.515745 − 0.856742i \(-0.672485\pi\)
0.515745 − 0.856742i \(-0.327515\pi\)
\(44\) 5.61803 0.846950
\(45\) −0.854102 −0.127322
\(46\) −1.00000 −0.147442
\(47\) 1.70820i 0.249167i 0.992209 + 0.124584i \(0.0397595\pi\)
−0.992209 + 0.124584i \(0.960241\pi\)
\(48\) − 1.61803i − 0.233543i
\(49\) 3.56231 0.508901
\(50\) − 5.00000i − 0.707107i
\(51\) 1.38197 0.193514
\(52\) 2.61803i 0.363056i
\(53\) 2.00000i 0.274721i 0.990521 + 0.137361i \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) −5.47214 −0.744663
\(55\) 12.5623 1.69390
\(56\) 1.85410 0.247765
\(57\) − 0.236068i − 0.0312680i
\(58\) − 9.70820i − 1.27475i
\(59\) 6.00000 0.781133 0.390567 − 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) − 3.61803i − 0.467086i
\(61\) 2.85410 0.365430 0.182715 − 0.983166i \(-0.441511\pi\)
0.182715 + 0.983166i \(0.441511\pi\)
\(62\) 2.14590i 0.272529i
\(63\) − 0.708204i − 0.0892253i
\(64\) −1.00000 −0.125000
\(65\) 5.85410i 0.726112i
\(66\) 9.09017 1.11892
\(67\) 5.23607i 0.639688i 0.947470 + 0.319844i \(0.103630\pi\)
−0.947470 + 0.319844i \(0.896370\pi\)
\(68\) − 0.854102i − 0.103575i
\(69\) −1.61803 −0.194788
\(70\) 4.14590 0.495530
\(71\) 0.381966 0.0453310 0.0226655 − 0.999743i \(-0.492785\pi\)
0.0226655 + 0.999743i \(0.492785\pi\)
\(72\) 0.381966i 0.0450151i
\(73\) 16.4721i 1.92792i 0.266051 + 0.963959i \(0.414281\pi\)
−0.266051 + 0.963959i \(0.585719\pi\)
\(74\) −9.70820 −1.12856
\(75\) − 8.09017i − 0.934172i
\(76\) −0.145898 −0.0167357
\(77\) 10.4164i 1.18706i
\(78\) 4.23607i 0.479640i
\(79\) 7.70820 0.867241 0.433620 − 0.901096i \(-0.357236\pi\)
0.433620 + 0.901096i \(0.357236\pi\)
\(80\) −2.23607 −0.250000
\(81\) −7.70820 −0.856467
\(82\) 5.61803i 0.620408i
\(83\) − 7.70820i − 0.846085i −0.906110 − 0.423043i \(-0.860962\pi\)
0.906110 − 0.423043i \(-0.139038\pi\)
\(84\) 3.00000 0.327327
\(85\) − 1.90983i − 0.207150i
\(86\) −11.2361 −1.21162
\(87\) − 15.7082i − 1.68410i
\(88\) − 5.61803i − 0.598884i
\(89\) −3.70820 −0.393069 −0.196534 − 0.980497i \(-0.562969\pi\)
−0.196534 + 0.980497i \(0.562969\pi\)
\(90\) 0.854102i 0.0900303i
\(91\) −4.85410 −0.508848
\(92\) 1.00000i 0.104257i
\(93\) 3.47214i 0.360044i
\(94\) 1.70820 0.176188
\(95\) −0.326238 −0.0334713
\(96\) −1.61803 −0.165140
\(97\) 13.0344i 1.32345i 0.749748 + 0.661724i \(0.230175\pi\)
−0.749748 + 0.661724i \(0.769825\pi\)
\(98\) − 3.56231i − 0.359847i
\(99\) −2.14590 −0.215671
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 230.2.b.a.139.1 ✓ 4
3.2 odd 2 2070.2.d.c.829.4 4
4.3 odd 2 1840.2.e.c.369.4 4
5.2 odd 4 1150.2.a.n.1.1 2
5.3 odd 4 1150.2.a.l.1.2 2
5.4 even 2 inner 230.2.b.a.139.4 yes 4
15.14 odd 2 2070.2.d.c.829.2 4
20.3 even 4 9200.2.a.bo.1.1 2
20.7 even 4 9200.2.a.by.1.2 2
20.19 odd 2 1840.2.e.c.369.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.b.a.139.1 ✓ 4 1.1 even 1 trivial
230.2.b.a.139.4 yes 4 5.4 even 2 inner
1150.2.a.l.1.2 2 5.3 odd 4
1150.2.a.n.1.1 2 5.2 odd 4
1840.2.e.c.369.1 4 20.19 odd 2
1840.2.e.c.369.4 4 4.3 odd 2
2070.2.d.c.829.2 4 15.14 odd 2
2070.2.d.c.829.4 4 3.2 odd 2
9200.2.a.bo.1.1 2 20.3 even 4
9200.2.a.by.1.2 2 20.7 even 4