Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,10,Mod(1,112)] 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("112.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 112.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,70,0,1554] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(57.6840136504\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{11209}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 2802 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 28)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-52.4363\) of defining polynomial
Character \(\chi\) \(=\) 112.1

$q$-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)
 
\(f(q)\) \(=\) \(q+140.873 q^{3} -1234.58 q^{5} +2401.00 q^{7} +162.080 q^{9} -11925.2 q^{11} +70593.9 q^{13} -173918. q^{15} +352927. q^{17} -371842. q^{19} +338235. q^{21} -342653. q^{23} -428940. q^{25} -2.74996e6 q^{27} +1.67222e6 q^{29} -6.38470e6 q^{31} -1.67993e6 q^{33} -2.96422e6 q^{35} -2.11208e7 q^{37} +9.94475e6 q^{39} -1.97637e7 q^{41} +1.65541e7 q^{43} -200100. q^{45} -1.70957e7 q^{47} +5.76480e6 q^{49} +4.97178e7 q^{51} +6.53986e7 q^{53} +1.47226e7 q^{55} -5.23824e7 q^{57} -7.92905e7 q^{59} +9.62918e7 q^{61} +389153. q^{63} -8.71537e7 q^{65} -2.48459e8 q^{67} -4.82704e7 q^{69} -1.12455e8 q^{71} +4.24344e7 q^{73} -6.04259e7 q^{75} -2.86324e7 q^{77} -2.09568e8 q^{79} -3.90584e8 q^{81} -7.24865e8 q^{83} -4.35717e8 q^{85} +2.35570e8 q^{87} -1.00901e9 q^{89} +1.69496e8 q^{91} -8.99429e8 q^{93} +4.59069e8 q^{95} +3.96497e8 q^{97} -1.93283e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 70 q^{3} + 1554 q^{5} + 4802 q^{7} - 14498 q^{9} - 62388 q^{11} + 122766 q^{13} - 371552 q^{15} + 73584 q^{17} - 1171198 q^{19} + 168070 q^{21} - 2262384 q^{23} + 5394106 q^{25} - 315980 q^{27} - 1923360 q^{29}+ \cdots + 737855932 q^{99}+O(q^{100}) \) 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
\(3\) 140.873 1.00411 0.502054 − 0.864836i \(-0.332578\pi\)
0.502054 + 0.864836i \(0.332578\pi\)
\(4\) 0 0
\(5\) −1234.58 −0.883393 −0.441696 − 0.897165i \(-0.645623\pi\)
−0.441696 + 0.897165i \(0.645623\pi\)
\(6\) 0 0
\(7\) 2401.00 0.377964
\(8\) 0 0
\(9\) 162.080 0.00823449
\(10\) 0 0
\(11\) −11925.2 −0.245583 −0.122792 − 0.992432i \(-0.539185\pi\)
−0.122792 + 0.992432i \(0.539185\pi\)
\(12\) 0 0
\(13\) 70593.9 0.685523 0.342762 − 0.939422i \(-0.388638\pi\)
0.342762 + 0.939422i \(0.388638\pi\)
\(14\) 0 0
\(15\) −173918. −0.887022
\(16\) 0 0
\(17\) 352927. 1.02486 0.512431 − 0.858728i \(-0.328745\pi\)
0.512431 + 0.858728i \(0.328745\pi\)
\(18\) 0 0
\(19\) −371842. −0.654587 −0.327294 − 0.944923i \(-0.606137\pi\)
−0.327294 + 0.944923i \(0.606137\pi\)
\(20\) 0 0
\(21\) 338235. 0.379517
\(22\) 0 0
\(23\) −342653. −0.255317 −0.127658 − 0.991818i \(-0.540746\pi\)
−0.127658 + 0.991818i \(0.540746\pi\)
\(24\) 0 0
\(25\) −428940. −0.219617
\(26\) 0 0
\(27\) −2.74996e6 −0.995840
\(28\) 0 0
\(29\) 1.67222e6 0.439038 0.219519 − 0.975608i \(-0.429551\pi\)
0.219519 + 0.975608i \(0.429551\pi\)
\(30\) 0 0
\(31\) −6.38470e6 −1.24169 −0.620845 − 0.783934i \(-0.713210\pi\)
−0.620845 + 0.783934i \(0.713210\pi\)
\(32\) 0 0
\(33\) −1.67993e6 −0.246592
\(34\) 0 0
\(35\) −2.96422e6 −0.333891
\(36\) 0 0
\(37\) −2.11208e7 −1.85269 −0.926347 − 0.376672i \(-0.877069\pi\)
−0.926347 + 0.376672i \(0.877069\pi\)
\(38\) 0 0
\(39\) 9.94475e6 0.688340
\(40\) 0 0
\(41\) −1.97637e7 −1.09230 −0.546150 − 0.837688i \(-0.683907\pi\)
−0.546150 + 0.837688i \(0.683907\pi\)
\(42\) 0 0
\(43\) 1.65541e7 0.738410 0.369205 − 0.929348i \(-0.379630\pi\)
0.369205 + 0.929348i \(0.379630\pi\)
\(44\) 0 0
\(45\) −200100. −0.00727429
\(46\) 0 0
\(47\) −1.70957e7 −0.511030 −0.255515 − 0.966805i \(-0.582245\pi\)
−0.255515 + 0.966805i \(0.582245\pi\)
\(48\) 0 0
\(49\) 5.76480e6 0.142857
\(50\) 0 0
\(51\) 4.97178e7 1.02907
\(52\) 0 0
\(53\) 6.53986e7 1.13849 0.569243 − 0.822170i \(-0.307237\pi\)
0.569243 + 0.822170i \(0.307237\pi\)
\(54\) 0 0
\(55\) 1.47226e7 0.216946
\(56\) 0 0
\(57\) −5.23824e7 −0.657277
\(58\) 0 0
\(59\) −7.92905e7 −0.851897 −0.425949 − 0.904747i \(-0.640060\pi\)
−0.425949 + 0.904747i \(0.640060\pi\)
\(60\) 0 0
\(61\) 9.62918e7 0.890441 0.445220 − 0.895421i \(-0.353125\pi\)
0.445220 + 0.895421i \(0.353125\pi\)
\(62\) 0 0
\(63\) 389153. 0.00311235
\(64\) 0 0
\(65\) −8.71537e7 −0.605586
\(66\) 0 0
\(67\) −2.48459e8 −1.50632 −0.753160 − 0.657837i \(-0.771472\pi\)
−0.753160 + 0.657837i \(0.771472\pi\)
\(68\) 0 0
\(69\) −4.82704e7 −0.256366
\(70\) 0 0
\(71\) −1.12455e8 −0.525189 −0.262595 − 0.964906i \(-0.584578\pi\)
−0.262595 + 0.964906i \(0.584578\pi\)
\(72\) 0 0
\(73\) 4.24344e7 0.174890 0.0874450 − 0.996169i \(-0.472130\pi\)
0.0874450 + 0.996169i \(0.472130\pi\)
\(74\) 0 0
\(75\) −6.04259e7 −0.220520
\(76\) 0 0
\(77\) −2.86324e7 −0.0928217
\(78\) 0 0
\(79\) −2.09568e8 −0.605345 −0.302673 − 0.953095i \(-0.597879\pi\)
−0.302673 + 0.953095i \(0.597879\pi\)
\(80\) 0 0
\(81\) −3.90584e8 −1.00817
\(82\) 0 0
\(83\) −7.24865e8 −1.67651 −0.838255 − 0.545279i \(-0.816424\pi\)
−0.838255 + 0.545279i \(0.816424\pi\)
\(84\) 0 0
\(85\) −4.35717e8 −0.905355
\(86\) 0 0
\(87\) 2.35570e8 0.440842
\(88\) 0 0
\(89\) −1.00901e9 −1.70467 −0.852333 − 0.523000i \(-0.824813\pi\)
−0.852333 + 0.523000i \(0.824813\pi\)
\(90\) 0 0
\(91\) 1.69496e8 0.259103
\(92\) 0 0
\(93\) −8.99429e8 −1.24679
\(94\) 0 0
\(95\) 4.59069e8 0.578258
\(96\) 0 0
\(97\) 3.96497e8 0.454744 0.227372 − 0.973808i \(-0.426987\pi\)
0.227372 + 0.973808i \(0.426987\pi\)
\(98\) 0 0
\(99\) −1.93283e6 −0.00202225
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 112.10.a.d.1.2 2
4.3 odd 2 28.10.a.b.1.1 ✓ 2
12.11 even 2 252.10.a.b.1.2 2
28.3 even 6 196.10.e.d.177.1 4
28.11 odd 6 196.10.e.e.177.2 4
28.19 even 6 196.10.e.d.165.1 4
28.23 odd 6 196.10.e.e.165.2 4
28.27 even 2 196.10.a.b.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.10.a.b.1.1 ✓ 2 4.3 odd 2
112.10.a.d.1.2 2 1.1 even 1 trivial
196.10.a.b.1.2 2 28.27 even 2
196.10.e.d.165.1 4 28.19 even 6
196.10.e.d.177.1 4 28.3 even 6
196.10.e.e.165.2 4 28.23 odd 6
196.10.e.e.177.2 4 28.11 odd 6
252.10.a.b.1.2 2 12.11 even 2