Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1840,4,Mod(1,1840)] 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("1840.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1840.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,1,0,-50,0,-28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(108.563514411\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} - 204 x^{8} + 42 x^{7} + 12958 x^{6} + 5872 x^{5} - 259871 x^{4} - 149461 x^{3} + \cdots - 43712 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{7}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 920)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-7.77468\) of defining polynomial
Character \(\chi\) \(=\) 1840.1

$q$-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)
 
\(f(q)\) \(=\) \(q-7.77468 q^{3} -5.00000 q^{5} +16.1209 q^{7} +33.4457 q^{9} -36.5622 q^{11} -0.303192 q^{13} +38.8734 q^{15} +27.3995 q^{17} -97.9768 q^{19} -125.335 q^{21} -23.0000 q^{23} +25.0000 q^{25} -50.1132 q^{27} +234.227 q^{29} +180.736 q^{31} +284.259 q^{33} -80.6043 q^{35} -410.588 q^{37} +2.35722 q^{39} +226.455 q^{41} -269.818 q^{43} -167.228 q^{45} +66.2788 q^{47} -83.1177 q^{49} -213.022 q^{51} -47.3176 q^{53} +182.811 q^{55} +761.739 q^{57} -861.711 q^{59} +925.069 q^{61} +539.173 q^{63} +1.51596 q^{65} -180.150 q^{67} +178.818 q^{69} -911.066 q^{71} +36.9445 q^{73} -194.367 q^{75} -589.414 q^{77} -1193.78 q^{79} -513.420 q^{81} -757.034 q^{83} -136.997 q^{85} -1821.04 q^{87} -418.694 q^{89} -4.88772 q^{91} -1405.17 q^{93} +489.884 q^{95} +621.708 q^{97} -1222.85 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{3} - 50 q^{5} - 28 q^{7} + 139 q^{9} + 14 q^{11} + 11 q^{13} - 5 q^{15} + 68 q^{17} - 114 q^{19} - 232 q^{21} - 230 q^{23} + 250 q^{25} + 433 q^{27} - 273 q^{29} + 129 q^{31} + 98 q^{33} + 140 q^{35}+ \cdots - 4356 q^{99}+O(q^{100}) \) 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
\(3\) −7.77468 −1.49624 −0.748119 − 0.663564i \(-0.769043\pi\)
−0.748119 + 0.663564i \(0.769043\pi\)
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 16.1209 0.870445 0.435222 − 0.900323i \(-0.356670\pi\)
0.435222 + 0.900323i \(0.356670\pi\)
\(8\) 0 0
\(9\) 33.4457 1.23873
\(10\) 0 0
\(11\) −36.5622 −1.00217 −0.501087 − 0.865397i \(-0.667066\pi\)
−0.501087 + 0.865397i \(0.667066\pi\)
\(12\) 0 0
\(13\) −0.303192 −0.00646849 −0.00323424 − 0.999995i \(-0.501029\pi\)
−0.00323424 + 0.999995i \(0.501029\pi\)
\(14\) 0 0
\(15\) 38.8734 0.669138
\(16\) 0 0
\(17\) 27.3995 0.390903 0.195452 − 0.980713i \(-0.437383\pi\)
0.195452 + 0.980713i \(0.437383\pi\)
\(18\) 0 0
\(19\) −97.9768 −1.18302 −0.591511 − 0.806297i \(-0.701468\pi\)
−0.591511 + 0.806297i \(0.701468\pi\)
\(20\) 0 0
\(21\) −125.335 −1.30239
\(22\) 0 0
\(23\) −23.0000 −0.208514
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −50.1132 −0.357196
\(28\) 0 0
\(29\) 234.227 1.49982 0.749910 − 0.661540i \(-0.230097\pi\)
0.749910 + 0.661540i \(0.230097\pi\)
\(30\) 0 0
\(31\) 180.736 1.04713 0.523567 − 0.851984i \(-0.324601\pi\)
0.523567 + 0.851984i \(0.324601\pi\)
\(32\) 0 0
\(33\) 284.259 1.49949
\(34\) 0 0
\(35\) −80.6043 −0.389275
\(36\) 0 0
\(37\) −410.588 −1.82433 −0.912165 − 0.409824i \(-0.865590\pi\)
−0.912165 + 0.409824i \(0.865590\pi\)
\(38\) 0 0
\(39\) 2.35722 0.00967840
\(40\) 0 0
\(41\) 226.455 0.862594 0.431297 − 0.902210i \(-0.358056\pi\)
0.431297 + 0.902210i \(0.358056\pi\)
\(42\) 0 0
\(43\) −269.818 −0.956905 −0.478452 − 0.878114i \(-0.658802\pi\)
−0.478452 + 0.878114i \(0.658802\pi\)
\(44\) 0 0
\(45\) −167.228 −0.553976
\(46\) 0 0
\(47\) 66.2788 0.205697 0.102849 − 0.994697i \(-0.467204\pi\)
0.102849 + 0.994697i \(0.467204\pi\)
\(48\) 0 0
\(49\) −83.1177 −0.242326
\(50\) 0 0
\(51\) −213.022 −0.584884
\(52\) 0 0
\(53\) −47.3176 −0.122634 −0.0613168 − 0.998118i \(-0.519530\pi\)
−0.0613168 + 0.998118i \(0.519530\pi\)
\(54\) 0 0
\(55\) 182.811 0.448185
\(56\) 0 0
\(57\) 761.739 1.77008
\(58\) 0 0
\(59\) −861.711 −1.90144 −0.950722 − 0.310045i \(-0.899656\pi\)
−0.950722 + 0.310045i \(0.899656\pi\)
\(60\) 0 0
\(61\) 925.069 1.94169 0.970844 − 0.239712i \(-0.0770529\pi\)
0.970844 + 0.239712i \(0.0770529\pi\)
\(62\) 0 0
\(63\) 539.173 1.07825
\(64\) 0 0
\(65\) 1.51596 0.00289279
\(66\) 0 0
\(67\) −180.150 −0.328490 −0.164245 − 0.986420i \(-0.552519\pi\)
−0.164245 + 0.986420i \(0.552519\pi\)
\(68\) 0 0
\(69\) 178.818 0.311987
\(70\) 0 0
\(71\) −911.066 −1.52287 −0.761434 − 0.648242i \(-0.775504\pi\)
−0.761434 + 0.648242i \(0.775504\pi\)
\(72\) 0 0
\(73\) 36.9445 0.0592333 0.0296166 − 0.999561i \(-0.490571\pi\)
0.0296166 + 0.999561i \(0.490571\pi\)
\(74\) 0 0
\(75\) −194.367 −0.299248
\(76\) 0 0
\(77\) −589.414 −0.872336
\(78\) 0 0
\(79\) −1193.78 −1.70013 −0.850066 − 0.526677i \(-0.823438\pi\)
−0.850066 + 0.526677i \(0.823438\pi\)
\(80\) 0 0
\(81\) −513.420 −0.704279
\(82\) 0 0
\(83\) −757.034 −1.00115 −0.500574 − 0.865694i \(-0.666878\pi\)
−0.500574 + 0.865694i \(0.666878\pi\)
\(84\) 0 0
\(85\) −136.997 −0.174817
\(86\) 0 0
\(87\) −1821.04 −2.24409
\(88\) 0 0
\(89\) −418.694 −0.498668 −0.249334 − 0.968418i \(-0.580212\pi\)
−0.249334 + 0.968418i \(0.580212\pi\)
\(90\) 0 0
\(91\) −4.88772 −0.00563046
\(92\) 0 0
\(93\) −1405.17 −1.56676
\(94\) 0 0
\(95\) 489.884 0.529064
\(96\) 0 0
\(97\) 621.708 0.650772 0.325386 − 0.945581i \(-0.394506\pi\)
0.325386 + 0.945581i \(0.394506\pi\)
\(98\) 0 0
\(99\) −1222.85 −1.24142
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 1840.4.a.bb.1.2 10
4.3 odd 2 920.4.a.g.1.9 ✓ 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.4.a.g.1.9 ✓ 10 4.3 odd 2
1840.4.a.bb.1.2 10 1.1 even 1 trivial