Properties

Label 37.6.a.a
Level $37$
Weight $6$
Character orbit 37.a
Self dual yes
Analytic conductor $5.934$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [37,6,Mod(1,37)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(37, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("37.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 37.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(5.93420133308\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 160x^{5} + 156x^{4} + 6495x^{3} - 2943x^{2} - 64880x + 53844 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{2} + (\beta_{2} + \beta_1 - 7) q^{3} + ( - \beta_{5} + \beta_{3} - 2 \beta_{2} + \beta_1 + 15) q^{4} + (\beta_{6} + 3 \beta_{5} - 2 \beta_{3} - 2 \beta_{2} + 6 \beta_1 - 9) q^{5} + ( - 3 \beta_{6} + \beta_{5} + \beta_{4} - 5 \beta_{3} - 2 \beta_{2} + 17 \beta_1 - 21) q^{6} + ( - 4 \beta_{6} - 3 \beta_{5} - 5 \beta_{4} + 3 \beta_{3} + 2 \beta_1 - 45) q^{7} + (11 \beta_{6} + 4 \beta_{4} + 13 \beta_{3} + 9 \beta_{2} - 14 \beta_1 - 68) q^{8} + (10 \beta_{6} - 13 \beta_{5} + 11 \beta_{4} + 5 \beta_{3} - 5 \beta_{2} - 17 \beta_1 + 48) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{2} + (\beta_{2} + \beta_1 - 7) q^{3} + ( - \beta_{5} + \beta_{3} - 2 \beta_{2} + \beta_1 + 15) q^{4} + (\beta_{6} + 3 \beta_{5} - 2 \beta_{3} - 2 \beta_{2} + 6 \beta_1 - 9) q^{5} + ( - 3 \beta_{6} + \beta_{5} + \beta_{4} - 5 \beta_{3} - 2 \beta_{2} + 17 \beta_1 - 21) q^{6} + ( - 4 \beta_{6} - 3 \beta_{5} - 5 \beta_{4} + 3 \beta_{3} + 2 \beta_1 - 45) q^{7} + (11 \beta_{6} + 4 \beta_{4} + 13 \beta_{3} + 9 \beta_{2} - 14 \beta_1 - 68) q^{8} + (10 \beta_{6} - 13 \beta_{5} + 11 \beta_{4} + 5 \beta_{3} - 5 \beta_{2} - 17 \beta_1 + 48) q^{9} + ( - 5 \beta_{6} + 5 \beta_{5} - 15 \beta_{4} - 15 \beta_{3} + 20 \beta_{2} + \cdots - 294) q^{10}+ \cdots + (2435 \beta_{6} + 2666 \beta_{5} - 4247 \beta_{4} - 2337 \beta_{3} + \cdots - 5281) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 8 q^{2} - 47 q^{3} + 106 q^{4} - 64 q^{5} - 141 q^{6} - 293 q^{7} - 474 q^{8} + 292 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 8 q^{2} - 47 q^{3} + 106 q^{4} - 64 q^{5} - 141 q^{6} - 293 q^{7} - 474 q^{8} + 292 q^{9} - 2017 q^{10} - 1457 q^{11} - 3917 q^{12} - 536 q^{13} - 488 q^{14} - 254 q^{15} + 2714 q^{16} - 3068 q^{17} + 4107 q^{18} - 1900 q^{19} + 1453 q^{20} + 2425 q^{21} + 4467 q^{22} - 3986 q^{23} + 11523 q^{24} + 12231 q^{25} + 911 q^{26} - 10697 q^{27} + 6486 q^{28} - 7436 q^{29} + 50276 q^{30} + 5776 q^{31} - 13366 q^{32} + 2973 q^{33} + 24128 q^{34} - 17714 q^{35} + 57889 q^{36} - 9583 q^{37} + 1248 q^{38} - 34826 q^{39} - 46751 q^{40} - 25089 q^{41} - 6232 q^{42} - 22538 q^{43} - 22817 q^{44} - 68648 q^{45} + 25485 q^{46} - 60861 q^{47} - 70825 q^{48} - 29182 q^{49} + 26797 q^{50} + 21508 q^{51} + 74493 q^{52} - 15681 q^{53} - 58620 q^{54} + 2930 q^{55} - 5542 q^{56} + 27032 q^{57} + 4979 q^{58} - 54536 q^{59} + 78104 q^{60} + 48694 q^{61} - 5601 q^{62} - 21062 q^{63} + 67074 q^{64} + 22480 q^{65} + 77598 q^{66} - 39724 q^{67} - 183104 q^{68} + 245960 q^{69} + 162468 q^{70} - 92187 q^{71} + 17685 q^{72} + 73251 q^{73} + 10952 q^{74} - 162813 q^{75} + 13504 q^{76} - 4605 q^{77} + 235693 q^{78} + 78604 q^{79} + 112473 q^{80} + 236431 q^{81} + 200777 q^{82} - 82223 q^{83} + 201198 q^{84} + 86716 q^{85} - 55686 q^{86} + 107506 q^{87} - 633 q^{88} + 181680 q^{89} - 732742 q^{90} - 14802 q^{91} - 684469 q^{92} - 37328 q^{93} + 34724 q^{94} - 222304 q^{95} + 397743 q^{96} + 39092 q^{97} - 318498 q^{98} - 29766 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 160x^{5} + 156x^{4} + 6495x^{3} - 2943x^{2} - 64880x + 53844 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -589\nu^{6} - 7152\nu^{5} + 107452\nu^{4} + 892264\nu^{3} - 7034699\nu^{2} - 21137864\nu + 118170684 ) / 3673920 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 201\nu^{6} - 964\nu^{5} - 28196\nu^{4} + 117064\nu^{3} + 739263\nu^{2} - 1398164\nu - 1062180 ) / 244928 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1051\nu^{6} - 2832\nu^{5} - 133648\nu^{4} + 417524\nu^{3} + 2519081\nu^{2} - 7485964\nu + 10836084 ) / 918480 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4193\nu^{6} - 156\nu^{5} - 637844\nu^{4} - 28568\nu^{3} + 21484423\nu^{2} + 17629348\nu - 83273748 ) / 3673920 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -461\nu^{6} + 2706\nu^{5} + 60632\nu^{4} - 374656\nu^{3} - 1201555\nu^{2} + 7801838\nu - 4986408 ) / 367392 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{3} - 2\beta_{2} - \beta _1 + 46 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -11\beta_{6} + 3\beta_{5} - 4\beta_{4} - 16\beta_{3} - 3\beta_{2} + 78\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 45\beta_{6} - 138\beta_{5} + 64\beta_{4} + 133\beta_{3} - 197\beta_{2} - 146\beta _1 + 3641 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -1250\beta_{6} + 516\beta_{5} - 336\beta_{4} - 2210\beta_{3} - 254\beta_{2} + 7253\beta _1 - 2720 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 6724\beta_{6} - 14953\beta_{5} + 9696\beta_{4} + 14917\beta_{3} - 19750\beta_{2} - 20489\beta _1 + 337886 \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
9.76658
7.47374
3.30071
0.860211
−4.74171
−5.27747
−10.3821
−10.7666 −23.6672 83.9191 70.6580 254.814 −64.6233 −558.991 317.134 −760.745
1.2 −8.47374 1.99035 39.8042 −10.3767 −16.8657 175.264 −66.1308 −239.039 87.9294
1.3 −4.30071 −0.151013 −13.5039 95.6487 0.649463 −203.420 195.699 −242.977 −411.358
1.4 −1.86021 19.8285 −28.5396 −83.4407 −36.8852 −62.6698 112.616 150.169 155.217
1.5 3.74171 −3.60982 −17.9996 −45.0957 −13.5069 −32.6141 −187.084 −229.969 −168.735
1.6 4.27747 −11.5828 −13.7033 12.1163 −49.5451 −81.5934 −195.494 −108.839 51.8273
1.7 9.38206 −29.8081 56.0230 −103.510 −279.661 −23.3438 225.385 645.520 −971.136
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(37\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.6.a.a 7
3.b odd 2 1 333.6.a.c 7
4.b odd 2 1 592.6.a.g 7
5.b even 2 1 925.6.a.a 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.6.a.a 7 1.a even 1 1 trivial
333.6.a.c 7 3.b odd 2 1
592.6.a.g 7 4.b odd 2 1
925.6.a.a 7 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} + 8T_{2}^{6} - 133T_{2}^{5} - 906T_{2}^{4} + 4326T_{2}^{3} + 19928T_{2}^{2} - 40920T_{2} - 109600 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(37))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} + 8 T^{6} - 133 T^{5} + \cdots - 109600 \) Copy content Toggle raw display
$3$ \( T^{7} + 47 T^{6} + 108 T^{5} + \cdots + 175797 \) Copy content Toggle raw display
$5$ \( T^{7} + 64 T^{6} + \cdots - 330953386760 \) Copy content Toggle raw display
$7$ \( T^{7} + 293 T^{6} + \cdots - 8969445254224 \) Copy content Toggle raw display
$11$ \( T^{7} + 1457 T^{6} + \cdots - 74\!\cdots\!65 \) Copy content Toggle raw display
$13$ \( T^{7} + 536 T^{6} + \cdots - 60\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( T^{7} + 3068 T^{6} + \cdots - 26\!\cdots\!68 \) Copy content Toggle raw display
$19$ \( T^{7} + 1900 T^{6} + \cdots + 15\!\cdots\!88 \) Copy content Toggle raw display
$23$ \( T^{7} + 3986 T^{6} + \cdots - 21\!\cdots\!12 \) Copy content Toggle raw display
$29$ \( T^{7} + 7436 T^{6} + \cdots + 54\!\cdots\!08 \) Copy content Toggle raw display
$31$ \( T^{7} - 5776 T^{6} + \cdots - 15\!\cdots\!52 \) Copy content Toggle raw display
$37$ \( (T + 1369)^{7} \) Copy content Toggle raw display
$41$ \( T^{7} + 25089 T^{6} + \cdots + 31\!\cdots\!45 \) Copy content Toggle raw display
$43$ \( T^{7} + 22538 T^{6} + \cdots - 26\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{7} + 60861 T^{6} + \cdots - 26\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{7} + 15681 T^{6} + \cdots - 80\!\cdots\!12 \) Copy content Toggle raw display
$59$ \( T^{7} + 54536 T^{6} + \cdots - 27\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{7} - 48694 T^{6} + \cdots + 41\!\cdots\!40 \) Copy content Toggle raw display
$67$ \( T^{7} + 39724 T^{6} + \cdots + 34\!\cdots\!52 \) Copy content Toggle raw display
$71$ \( T^{7} + 92187 T^{6} + \cdots - 58\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{7} - 73251 T^{6} + \cdots + 25\!\cdots\!45 \) Copy content Toggle raw display
$79$ \( T^{7} - 78604 T^{6} + \cdots + 54\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T^{7} + 82223 T^{6} + \cdots - 19\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{7} - 181680 T^{6} + \cdots - 14\!\cdots\!48 \) Copy content Toggle raw display
$97$ \( T^{7} - 39092 T^{6} + \cdots - 49\!\cdots\!20 \) Copy content Toggle raw display
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