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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [32,7,Mod(31,32)] 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("32.31"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(32, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 32.c (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: \(7.36173067584\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{17} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 31.2
Root \(1.22474 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 32.31
Dual form 32.7.c.b.31.3

$q$-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)
 
\(f(q)\) \(=\) \(q-27.1918i q^{3} -170.767 q^{5} +374.384i q^{7} -10.3959 q^{9} +1358.26i q^{11} -2957.84 q^{13} +4643.48i q^{15} -6656.09 q^{17} -9926.98i q^{19} +10180.2 q^{21} +15585.2i q^{23} +13536.5 q^{25} -19540.2i q^{27} -2083.13 q^{29} -28446.6i q^{31} +36933.6 q^{33} -63932.5i q^{35} -3331.58 q^{37} +80429.0i q^{39} -63122.2 q^{41} +88501.2i q^{43} +1775.29 q^{45} -10825.8i q^{47} -22514.1 q^{49} +180991. i q^{51} -48049.2 q^{53} -231947. i q^{55} -269933. q^{57} +147032. i q^{59} -124334. q^{61} -3892.07i q^{63} +505102. q^{65} -87535.6i q^{67} +423790. q^{69} -225599. i q^{71} +475959. q^{73} -368082. i q^{75} -508511. q^{77} -276262. i q^{79} -538912. q^{81} +485661. i q^{83} +1.13664e6 q^{85} +56644.0i q^{87} -1.06400e6 q^{89} -1.10737e6i q^{91} -773516. q^{93} +1.69520e6i q^{95} +1.46796e6 q^{97} -14120.4i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 56 q^{5} - 3804 q^{9} - 8696 q^{13} - 5304 q^{17} - 1920 q^{21} + 36588 q^{25} + 77576 q^{29} + 71232 q^{33} + 125256 q^{37} - 209848 q^{41} - 536568 q^{45} + 95556 q^{49} - 68664 q^{53} - 526656 q^{57}+ \cdots + 2918344 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/32\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(31\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 27.1918i − 1.00711i −0.863965 − 0.503553i \(-0.832026\pi\)
0.863965 − 0.503553i \(-0.167974\pi\)
\(4\) 0 0
\(5\) −170.767 −1.36614 −0.683069 − 0.730353i \(-0.739355\pi\)
−0.683069 + 0.730353i \(0.739355\pi\)
\(6\) 0 0
\(7\) 374.384i 1.09150i 0.837949 + 0.545749i \(0.183755\pi\)
−0.837949 + 0.545749i \(0.816245\pi\)
\(8\) 0 0
\(9\) −10.3959 −0.0142605
\(10\) 0 0
\(11\) 1358.26i 1.02048i 0.860032 + 0.510241i \(0.170444\pi\)
−0.860032 + 0.510241i \(0.829556\pi\)
\(12\) 0 0
\(13\) −2957.84 −1.34631 −0.673154 − 0.739503i \(-0.735061\pi\)
−0.673154 + 0.739503i \(0.735061\pi\)
\(14\) 0 0
\(15\) 4643.48i 1.37585i
\(16\) 0 0
\(17\) −6656.09 −1.35479 −0.677396 − 0.735619i \(-0.736891\pi\)
−0.677396 + 0.735619i \(0.736891\pi\)
\(18\) 0 0
\(19\) − 9926.98i − 1.44729i −0.690171 − 0.723646i \(-0.742465\pi\)
0.690171 − 0.723646i \(-0.257535\pi\)
\(20\) 0 0
\(21\) 10180.2 1.09925
\(22\) 0 0
\(23\) 15585.2i 1.28094i 0.767983 + 0.640470i \(0.221260\pi\)
−0.767983 + 0.640470i \(0.778740\pi\)
\(24\) 0 0
\(25\) 13536.5 0.866335
\(26\) 0 0
\(27\) − 19540.2i − 0.992743i
\(28\) 0 0
\(29\) −2083.13 −0.0854125 −0.0427063 − 0.999088i \(-0.513598\pi\)
−0.0427063 + 0.999088i \(0.513598\pi\)
\(30\) 0 0
\(31\) − 28446.6i − 0.954873i −0.878666 − 0.477437i \(-0.841566\pi\)
0.878666 − 0.477437i \(-0.158434\pi\)
\(32\) 0 0
\(33\) 36933.6 1.02773
\(34\) 0 0
\(35\) − 63932.5i − 1.49114i
\(36\) 0 0
\(37\) −3331.58 −0.0657727 −0.0328863 − 0.999459i \(-0.510470\pi\)
−0.0328863 + 0.999459i \(0.510470\pi\)
\(38\) 0 0
\(39\) 80429.0i 1.35587i
\(40\) 0 0
\(41\) −63122.2 −0.915863 −0.457931 − 0.888988i \(-0.651409\pi\)
−0.457931 + 0.888988i \(0.651409\pi\)
\(42\) 0 0
\(43\) 88501.2i 1.11313i 0.830806 + 0.556563i \(0.187880\pi\)
−0.830806 + 0.556563i \(0.812120\pi\)
\(44\) 0 0
\(45\) 1775.29 0.0194819
\(46\) 0 0
\(47\) − 10825.8i − 0.104271i −0.998640 − 0.0521357i \(-0.983397\pi\)
0.998640 − 0.0521357i \(-0.0166028\pi\)
\(48\) 0 0
\(49\) −22514.1 −0.191367
\(50\) 0 0
\(51\) 180991.i 1.36442i
\(52\) 0 0
\(53\) −48049.2 −0.322744 −0.161372 − 0.986894i \(-0.551592\pi\)
−0.161372 + 0.986894i \(0.551592\pi\)
\(54\) 0 0
\(55\) − 231947.i − 1.39412i
\(56\) 0 0
\(57\) −269933. −1.45758
\(58\) 0 0
\(59\) 147032.i 0.715906i 0.933740 + 0.357953i \(0.116525\pi\)
−0.933740 + 0.357953i \(0.883475\pi\)
\(60\) 0 0
\(61\) −124334. −0.547773 −0.273887 − 0.961762i \(-0.588309\pi\)
−0.273887 + 0.961762i \(0.588309\pi\)
\(62\) 0 0
\(63\) − 3892.07i − 0.0155654i
\(64\) 0 0
\(65\) 505102. 1.83924
\(66\) 0 0
\(67\) − 87535.6i − 0.291045i −0.989355 − 0.145523i \(-0.953514\pi\)
0.989355 − 0.145523i \(-0.0464863\pi\)
\(68\) 0 0
\(69\) 423790. 1.29004
\(70\) 0 0
\(71\) − 225599.i − 0.630322i −0.949038 − 0.315161i \(-0.897941\pi\)
0.949038 − 0.315161i \(-0.102059\pi\)
\(72\) 0 0
\(73\) 475959. 1.22349 0.611746 − 0.791054i \(-0.290468\pi\)
0.611746 + 0.791054i \(0.290468\pi\)
\(74\) 0 0
\(75\) − 368082.i − 0.872490i
\(76\) 0 0
\(77\) −508511. −1.11385
\(78\) 0 0
\(79\) − 276262.i − 0.560324i −0.959953 − 0.280162i \(-0.909612\pi\)
0.959953 − 0.280162i \(-0.0903882\pi\)
\(80\) 0 0
\(81\) −538912. −1.01406
\(82\) 0 0
\(83\) 485661.i 0.849374i 0.905340 + 0.424687i \(0.139616\pi\)
−0.905340 + 0.424687i \(0.860384\pi\)
\(84\) 0 0
\(85\) 1.13664e6 1.85083
\(86\) 0 0
\(87\) 56644.0i 0.0860194i
\(88\) 0 0
\(89\) −1.06400e6 −1.50929 −0.754645 − 0.656134i \(-0.772191\pi\)
−0.754645 + 0.656134i \(0.772191\pi\)
\(90\) 0 0
\(91\) − 1.10737e6i − 1.46949i
\(92\) 0 0
\(93\) −773516. −0.961658
\(94\) 0 0
\(95\) 1.69520e6i 1.97720i
\(96\) 0 0
\(97\) 1.46796e6 1.60842 0.804209 − 0.594346i \(-0.202589\pi\)
0.804209 + 0.594346i \(0.202589\pi\)
\(98\) 0 0
\(99\) − 14120.4i − 0.0145526i
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 32.7.c.b.31.2 ✓ 4
3.2 odd 2 288.7.g.b.127.4 4
4.3 odd 2 inner 32.7.c.b.31.3 yes 4
8.3 odd 2 64.7.c.e.63.2 4
8.5 even 2 64.7.c.e.63.3 4
12.11 even 2 288.7.g.b.127.3 4
16.3 odd 4 256.7.d.g.127.2 4
16.5 even 4 256.7.d.g.127.1 4
16.11 odd 4 256.7.d.d.127.3 4
16.13 even 4 256.7.d.d.127.4 4
24.5 odd 2 576.7.g.l.127.2 4
24.11 even 2 576.7.g.l.127.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.7.c.b.31.2 ✓ 4 1.1 even 1 trivial
32.7.c.b.31.3 yes 4 4.3 odd 2 inner
64.7.c.e.63.2 4 8.3 odd 2
64.7.c.e.63.3 4 8.5 even 2
256.7.d.d.127.3 4 16.11 odd 4
256.7.d.d.127.4 4 16.13 even 4
256.7.d.g.127.1 4 16.5 even 4
256.7.d.g.127.2 4 16.3 odd 4
288.7.g.b.127.3 4 12.11 even 2
288.7.g.b.127.4 4 3.2 odd 2
576.7.g.l.127.1 4 24.11 even 2
576.7.g.l.127.2 4 24.5 odd 2