Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2368,2,Mod(1,2368)] 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("2368.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2368, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2368 = 2^{6} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2368.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-2,0,1,0,7,0,3,0,0] 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: \(18.9085751986\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 296)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.254102\) of defining polynomial
Character \(\chi\) \(=\) 2368.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.93543 q^{3} +2.93543 q^{5} +4.68133 q^{7} +0.745898 q^{9} -0.762305 q^{11} -1.76231 q^{13} +5.68133 q^{15} +3.36266 q^{17} -7.36266 q^{19} +9.06040 q^{21} +3.25410 q^{23} +3.61676 q^{25} -4.36266 q^{27} +3.25410 q^{29} +3.06457 q^{31} -1.47539 q^{33} +13.7417 q^{35} +1.00000 q^{37} -3.41082 q^{39} -7.42723 q^{41} +12.2499 q^{43} +2.18953 q^{45} +0.302263 q^{47} +14.9149 q^{49} +6.50820 q^{51} -5.53579 q^{53} -2.23769 q^{55} -14.2499 q^{57} -10.2499 q^{59} -12.2981 q^{61} +3.49180 q^{63} -5.17313 q^{65} +13.1526 q^{67} +6.29809 q^{69} +0.173127 q^{71} -1.23769 q^{73} +7.00000 q^{75} -3.56860 q^{77} +4.61676 q^{79} -10.6813 q^{81} -3.53579 q^{83} +9.87086 q^{85} +6.29809 q^{87} +15.7417 q^{89} -8.24993 q^{91} +5.93126 q^{93} -21.6126 q^{95} -16.1208 q^{97} -0.568602 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} + q^{5} + 7 q^{7} + 3 q^{9} - 3 q^{13} + 10 q^{15} - 4 q^{17} - 8 q^{19} + 3 q^{21} + 9 q^{23} - 4 q^{25} + q^{27} + 9 q^{29} + 17 q^{31} - 9 q^{33} + 10 q^{35} + 3 q^{37} - 7 q^{39} - 16 q^{41}+ \cdots + 24 q^{99}+O(q^{100}) \) 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 0
\(3\) 1.93543 1.11742 0.558711 − 0.829362i \(-0.311296\pi\)
0.558711 + 0.829362i \(0.311296\pi\)
\(4\) 0 0
\(5\) 2.93543 1.31277 0.656383 − 0.754428i \(-0.272086\pi\)
0.656383 + 0.754428i \(0.272086\pi\)
\(6\) 0 0
\(7\) 4.68133 1.76938 0.884688 − 0.466183i \(-0.154371\pi\)
0.884688 + 0.466183i \(0.154371\pi\)
\(8\) 0 0
\(9\) 0.745898 0.248633
\(10\) 0 0
\(11\) −0.762305 −0.229844 −0.114922 − 0.993375i \(-0.536662\pi\)
−0.114922 + 0.993375i \(0.536662\pi\)
\(12\) 0 0
\(13\) −1.76231 −0.488775 −0.244388 − 0.969678i \(-0.578587\pi\)
−0.244388 + 0.969678i \(0.578587\pi\)
\(14\) 0 0
\(15\) 5.68133 1.46691
\(16\) 0 0
\(17\) 3.36266 0.815565 0.407783 − 0.913079i \(-0.366302\pi\)
0.407783 + 0.913079i \(0.366302\pi\)
\(18\) 0 0
\(19\) −7.36266 −1.68911 −0.844555 − 0.535469i \(-0.820135\pi\)
−0.844555 + 0.535469i \(0.820135\pi\)
\(20\) 0 0
\(21\) 9.06040 1.97714
\(22\) 0 0
\(23\) 3.25410 0.678527 0.339264 − 0.940691i \(-0.389822\pi\)
0.339264 + 0.940691i \(0.389822\pi\)
\(24\) 0 0
\(25\) 3.61676 0.723353
\(26\) 0 0
\(27\) −4.36266 −0.839595
\(28\) 0 0
\(29\) 3.25410 0.604272 0.302136 − 0.953265i \(-0.402300\pi\)
0.302136 + 0.953265i \(0.402300\pi\)
\(30\) 0 0
\(31\) 3.06457 0.550413 0.275206 − 0.961385i \(-0.411254\pi\)
0.275206 + 0.961385i \(0.411254\pi\)
\(32\) 0 0
\(33\) −1.47539 −0.256832
\(34\) 0 0
\(35\) 13.7417 2.32278
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 0 0
\(39\) −3.41082 −0.546169
\(40\) 0 0
\(41\) −7.42723 −1.15994 −0.579969 − 0.814638i \(-0.696935\pi\)
−0.579969 + 0.814638i \(0.696935\pi\)
\(42\) 0 0
\(43\) 12.2499 1.86810 0.934049 − 0.357146i \(-0.116250\pi\)
0.934049 + 0.357146i \(0.116250\pi\)
\(44\) 0 0
\(45\) 2.18953 0.326396
\(46\) 0 0
\(47\) 0.302263 0.0440895 0.0220448 − 0.999757i \(-0.492982\pi\)
0.0220448 + 0.999757i \(0.492982\pi\)
\(48\) 0 0
\(49\) 14.9149 2.13069
\(50\) 0 0
\(51\) 6.50820 0.911331
\(52\) 0 0
\(53\) −5.53579 −0.760399 −0.380200 − 0.924904i \(-0.624145\pi\)
−0.380200 + 0.924904i \(0.624145\pi\)
\(54\) 0 0
\(55\) −2.23769 −0.301731
\(56\) 0 0
\(57\) −14.2499 −1.88745
\(58\) 0 0
\(59\) −10.2499 −1.33443 −0.667214 − 0.744866i \(-0.732513\pi\)
−0.667214 + 0.744866i \(0.732513\pi\)
\(60\) 0 0
\(61\) −12.2981 −1.57461 −0.787305 − 0.616564i \(-0.788524\pi\)
−0.787305 + 0.616564i \(0.788524\pi\)
\(62\) 0 0
\(63\) 3.49180 0.439925
\(64\) 0 0
\(65\) −5.17313 −0.641647
\(66\) 0 0
\(67\) 13.1526 1.60684 0.803420 − 0.595413i \(-0.203011\pi\)
0.803420 + 0.595413i \(0.203011\pi\)
\(68\) 0 0
\(69\) 6.29809 0.758201
\(70\) 0 0
\(71\) 0.173127 0.0205464 0.0102732 − 0.999947i \(-0.496730\pi\)
0.0102732 + 0.999947i \(0.496730\pi\)
\(72\) 0 0
\(73\) −1.23769 −0.144861 −0.0724306 − 0.997373i \(-0.523076\pi\)
−0.0724306 + 0.997373i \(0.523076\pi\)
\(74\) 0 0
\(75\) 7.00000 0.808290
\(76\) 0 0
\(77\) −3.56860 −0.406680
\(78\) 0 0
\(79\) 4.61676 0.519426 0.259713 − 0.965686i \(-0.416372\pi\)
0.259713 + 0.965686i \(0.416372\pi\)
\(80\) 0 0
\(81\) −10.6813 −1.18681
\(82\) 0 0
\(83\) −3.53579 −0.388103 −0.194052 − 0.980991i \(-0.562163\pi\)
−0.194052 + 0.980991i \(0.562163\pi\)
\(84\) 0 0
\(85\) 9.87086 1.07065
\(86\) 0 0
\(87\) 6.29809 0.675227
\(88\) 0 0
\(89\) 15.7417 1.66862 0.834310 − 0.551296i \(-0.185866\pi\)
0.834310 + 0.551296i \(0.185866\pi\)
\(90\) 0 0
\(91\) −8.24993 −0.864828
\(92\) 0 0
\(93\) 5.93126 0.615043
\(94\) 0 0
\(95\) −21.6126 −2.21741
\(96\) 0 0
\(97\) −16.1208 −1.63682 −0.818409 − 0.574635i \(-0.805144\pi\)
−0.818409 + 0.574635i \(0.805144\pi\)
\(98\) 0 0
\(99\) −0.568602 −0.0571467
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 2368.2.a.bb.1.3 3
4.3 odd 2 2368.2.a.be.1.1 3
8.3 odd 2 592.2.a.i.1.3 3
8.5 even 2 296.2.a.c.1.1 ✓ 3
24.5 odd 2 2664.2.a.p.1.3 3
24.11 even 2 5328.2.a.bn.1.3 3
40.29 even 2 7400.2.a.k.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
296.2.a.c.1.1 ✓ 3 8.5 even 2
592.2.a.i.1.3 3 8.3 odd 2
2368.2.a.bb.1.3 3 1.1 even 1 trivial
2368.2.a.be.1.1 3 4.3 odd 2
2664.2.a.p.1.3 3 24.5 odd 2
5328.2.a.bn.1.3 3 24.11 even 2
7400.2.a.k.1.3 3 40.29 even 2