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 1.5
Root \(-0.575045\) 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)
 
\(f(q)\) \(=\) \(q-0.575045 q^{3} -5.00000 q^{5} +24.8747 q^{7} -26.6693 q^{9} +5.05844 q^{11} -93.1604 q^{13} +2.87522 q^{15} -95.9572 q^{17} -58.3273 q^{19} -14.3041 q^{21} -23.0000 q^{23} +25.0000 q^{25} +30.8623 q^{27} +85.0540 q^{29} -172.280 q^{31} -2.90883 q^{33} -124.373 q^{35} +222.779 q^{37} +53.5714 q^{39} +171.389 q^{41} +406.942 q^{43} +133.347 q^{45} +166.455 q^{47} +275.750 q^{49} +55.1797 q^{51} +69.4544 q^{53} -25.2922 q^{55} +33.5408 q^{57} -662.361 q^{59} +190.331 q^{61} -663.391 q^{63} +465.802 q^{65} -106.999 q^{67} +13.2260 q^{69} +920.046 q^{71} -649.441 q^{73} -14.3761 q^{75} +125.827 q^{77} +543.783 q^{79} +702.325 q^{81} +131.724 q^{83} +479.786 q^{85} -48.9099 q^{87} +788.269 q^{89} -2317.34 q^{91} +99.0689 q^{93} +291.637 q^{95} +900.688 q^{97} -134.905 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{3} - 50 q^{5} - 28 q^{7} + 139 q^{9} + 14 q^{11} + 11 q^{13} - 5 q^{15} + 68 q^{17} - 114 q^{19} - 232 q^{21} - 230 q^{23} + 250 q^{25} + 433 q^{27} - 273 q^{29} + 129 q^{31} + 98 q^{33} + 140 q^{35}+ \cdots - 4356 q^{99}+O(q^{100}) \) 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
\(3\) −0.575045 −0.110667 −0.0553337 − 0.998468i \(-0.517622\pi\)
−0.0553337 + 0.998468i \(0.517622\pi\)
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 24.8747 1.34311 0.671554 − 0.740956i \(-0.265627\pi\)
0.671554 + 0.740956i \(0.265627\pi\)
\(8\) 0 0
\(9\) −26.6693 −0.987753
\(10\) 0 0
\(11\) 5.05844 0.138653 0.0693263 − 0.997594i \(-0.477915\pi\)
0.0693263 + 0.997594i \(0.477915\pi\)
\(12\) 0 0
\(13\) −93.1604 −1.98754 −0.993771 − 0.111437i \(-0.964455\pi\)
−0.993771 + 0.111437i \(0.964455\pi\)
\(14\) 0 0
\(15\) 2.87522 0.0494920
\(16\) 0 0
\(17\) −95.9572 −1.36900 −0.684501 − 0.729012i \(-0.739980\pi\)
−0.684501 + 0.729012i \(0.739980\pi\)
\(18\) 0 0
\(19\) −58.3273 −0.704274 −0.352137 − 0.935949i \(-0.614545\pi\)
−0.352137 + 0.935949i \(0.614545\pi\)
\(20\) 0 0
\(21\) −14.3041 −0.148638
\(22\) 0 0
\(23\) −23.0000 −0.208514
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 30.8623 0.219979
\(28\) 0 0
\(29\) 85.0540 0.544625 0.272313 − 0.962209i \(-0.412211\pi\)
0.272313 + 0.962209i \(0.412211\pi\)
\(30\) 0 0
\(31\) −172.280 −0.998143 −0.499072 − 0.866561i \(-0.666326\pi\)
−0.499072 + 0.866561i \(0.666326\pi\)
\(32\) 0 0
\(33\) −2.90883 −0.0153443
\(34\) 0 0
\(35\) −124.373 −0.600656
\(36\) 0 0
\(37\) 222.779 0.989857 0.494929 − 0.868934i \(-0.335194\pi\)
0.494929 + 0.868934i \(0.335194\pi\)
\(38\) 0 0
\(39\) 53.5714 0.219956
\(40\) 0 0
\(41\) 171.389 0.652842 0.326421 − 0.945225i \(-0.394157\pi\)
0.326421 + 0.945225i \(0.394157\pi\)
\(42\) 0 0
\(43\) 406.942 1.44321 0.721606 − 0.692304i \(-0.243404\pi\)
0.721606 + 0.692304i \(0.243404\pi\)
\(44\) 0 0
\(45\) 133.347 0.441736
\(46\) 0 0
\(47\) 166.455 0.516595 0.258298 − 0.966065i \(-0.416838\pi\)
0.258298 + 0.966065i \(0.416838\pi\)
\(48\) 0 0
\(49\) 275.750 0.803937
\(50\) 0 0
\(51\) 55.1797 0.151504
\(52\) 0 0
\(53\) 69.4544 0.180006 0.0900028 − 0.995942i \(-0.471312\pi\)
0.0900028 + 0.995942i \(0.471312\pi\)
\(54\) 0 0
\(55\) −25.2922 −0.0620073
\(56\) 0 0
\(57\) 33.5408 0.0779402
\(58\) 0 0
\(59\) −662.361 −1.46156 −0.730780 − 0.682613i \(-0.760843\pi\)
−0.730780 + 0.682613i \(0.760843\pi\)
\(60\) 0 0
\(61\) 190.331 0.399498 0.199749 − 0.979847i \(-0.435987\pi\)
0.199749 + 0.979847i \(0.435987\pi\)
\(62\) 0 0
\(63\) −663.391 −1.32666
\(64\) 0 0
\(65\) 465.802 0.888856
\(66\) 0 0
\(67\) −106.999 −0.195105 −0.0975527 − 0.995230i \(-0.531101\pi\)
−0.0975527 + 0.995230i \(0.531101\pi\)
\(68\) 0 0
\(69\) 13.2260 0.0230758
\(70\) 0 0
\(71\) 920.046 1.53788 0.768939 − 0.639322i \(-0.220785\pi\)
0.768939 + 0.639322i \(0.220785\pi\)
\(72\) 0 0
\(73\) −649.441 −1.04125 −0.520625 − 0.853785i \(-0.674301\pi\)
−0.520625 + 0.853785i \(0.674301\pi\)
\(74\) 0 0
\(75\) −14.3761 −0.0221335
\(76\) 0 0
\(77\) 125.827 0.186225
\(78\) 0 0
\(79\) 543.783 0.774435 0.387218 − 0.921988i \(-0.373436\pi\)
0.387218 + 0.921988i \(0.373436\pi\)
\(80\) 0 0
\(81\) 702.325 0.963408
\(82\) 0 0
\(83\) 131.724 0.174200 0.0870998 − 0.996200i \(-0.472240\pi\)
0.0870998 + 0.996200i \(0.472240\pi\)
\(84\) 0 0
\(85\) 479.786 0.612236
\(86\) 0 0
\(87\) −48.9099 −0.0602723
\(88\) 0 0
\(89\) 788.269 0.938835 0.469417 − 0.882976i \(-0.344464\pi\)
0.469417 + 0.882976i \(0.344464\pi\)
\(90\) 0 0
\(91\) −2317.34 −2.66948
\(92\) 0 0
\(93\) 99.0689 0.110462
\(94\) 0 0
\(95\) 291.637 0.314961
\(96\) 0 0
\(97\) 900.688 0.942794 0.471397 − 0.881921i \(-0.343750\pi\)
0.471397 + 0.881921i \(0.343750\pi\)
\(98\) 0 0
\(99\) −134.905 −0.136954
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.1.5 10
4.3 odd 2 920.4.a.g.1.6 ✓ 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.4.a.g.1.6 ✓ 10 4.3 odd 2
1840.4.a.bb.1.5 10 1.1 even 1 trivial