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Properties

Label 37.10.a.a
Level $37$
Weight $10$
Character orbit 37.a
Self dual yes
Analytic conductor $19.056$
Analytic rank $1$
Dimension $13$
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] = [37,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("37.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(19.0563259381\)
Analytic rank: \(1\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 6 x^{12} - 4637 x^{11} + 28852 x^{10} + 8006690 x^{9} - 52024972 x^{8} - 6415977160 x^{7} + 44300013488 x^{6} + 2423840779488 x^{5} + \cdots - 99\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 2) q^{2} + ( - \beta_{5} + \beta_1 - 20) q^{3} + ( - \beta_{5} + \beta_{4} + 4 \beta_1 + 208) q^{4} + (\beta_{9} + \beta_{5} - \beta_{4} + 11 \beta_1 - 171) q^{5} + ( - 2 \beta_{9} + \beta_{7} + 12 \beta_{5} - 3 \beta_{4} - 15 \beta_1 - 329) q^{6} + ( - \beta_{11} - 2 \beta_{9} - 3 \beta_{7} + 15 \beta_{5} - 3 \beta_{4} - 35 \beta_1 - 948) q^{7} + (4 \beta_{11} + \beta_{10} - 2 \beta_{9} - \beta_{8} - \beta_{7} + 45 \beta_{5} - 3 \beta_{4} + \cdots - 1476) q^{8}+ \cdots + ( - \beta_{11} - 5 \beta_{10} + 2 \beta_{9} + 4 \beta_{8} + 3 \beta_{7} - \beta_{6} + \cdots + 5196) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 2) q^{2} + ( - \beta_{5} + \beta_1 - 20) q^{3} + ( - \beta_{5} + \beta_{4} + 4 \beta_1 + 208) q^{4} + (\beta_{9} + \beta_{5} - \beta_{4} + 11 \beta_1 - 171) q^{5} + ( - 2 \beta_{9} + \beta_{7} + 12 \beta_{5} - 3 \beta_{4} - 15 \beta_1 - 329) q^{6} + ( - \beta_{11} - 2 \beta_{9} - 3 \beta_{7} + 15 \beta_{5} - 3 \beta_{4} - 35 \beta_1 - 948) q^{7} + (4 \beta_{11} + \beta_{10} - 2 \beta_{9} - \beta_{8} - \beta_{7} + 45 \beta_{5} - 3 \beta_{4} + \cdots - 1476) q^{8}+ \cdots + (277427 \beta_{12} + 242684 \beta_{11} + 5375 \beta_{10} + \cdots - 245381541) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 32 q^{2} - 251 q^{3} + 2730 q^{4} - 2159 q^{5} - 4401 q^{6} - 12576 q^{7} - 20394 q^{8} + 69112 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q - 32 q^{2} - 251 q^{3} + 2730 q^{4} - 2159 q^{5} - 4401 q^{6} - 12576 q^{7} - 20394 q^{8} + 69112 q^{9} - 106605 q^{10} - 112451 q^{11} + 220963 q^{12} + 7129 q^{13} + 294824 q^{14} - 63644 q^{15} + 145178 q^{16} - 890862 q^{17} - 3066225 q^{18} - 1435874 q^{19} - 3193339 q^{20} - 4745036 q^{21} - 4350913 q^{22} - 2565799 q^{23} - 15286101 q^{24} - 1828304 q^{25} - 7543133 q^{26} - 10134680 q^{27} - 24344602 q^{28} - 2992323 q^{29} - 25604140 q^{30} - 8242245 q^{31} - 22320310 q^{32} - 18079398 q^{33} + 5045920 q^{34} - 26953204 q^{35} - 10407455 q^{36} - 24364093 q^{37} - 42175680 q^{38} - 79430765 q^{39} - 61032223 q^{40} - 50975109 q^{41} + 54850616 q^{42} - 18142836 q^{43} - 55265137 q^{44} + 136868596 q^{45} + 157343401 q^{46} - 14353596 q^{47} + 213610631 q^{48} + 213271999 q^{49} + 175451561 q^{50} + 151710418 q^{51} + 151573285 q^{52} + 74438872 q^{53} + 174228132 q^{54} + 118316889 q^{55} + 362406090 q^{56} + 248282906 q^{57} + 206405719 q^{58} - 251964328 q^{59} + 877937048 q^{60} + 202847323 q^{61} - 34418509 q^{62} - 227178410 q^{63} + 187231490 q^{64} - 341466470 q^{65} + 989905110 q^{66} - 12257509 q^{67} - 332496576 q^{68} - 768097033 q^{69} + 1098443332 q^{70} - 310979094 q^{71} - 588411507 q^{72} - 249752015 q^{73} + 59973152 q^{74} - 634307724 q^{75} + 365061440 q^{76} - 1143945802 q^{77} + 1471186393 q^{78} + 30429049 q^{79} + 885873041 q^{80} - 350972903 q^{81} - 1192633571 q^{82} - 2559788658 q^{83} - 2510580834 q^{84} - 3291393166 q^{85} - 1373534302 q^{86} - 1215098129 q^{87} + 107941215 q^{88} - 3063565514 q^{89} - 552233182 q^{90} - 1743876566 q^{91} - 2937303341 q^{92} - 1077794354 q^{93} + 49542148 q^{94} - 2168155374 q^{95} - 1504910121 q^{96} - 429307758 q^{97} - 1351241634 q^{98} - 3266142174 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 6 x^{12} - 4637 x^{11} + 28852 x^{10} + 8006690 x^{9} - 52024972 x^{8} - 6415977160 x^{7} + 44300013488 x^{6} + 2423840779488 x^{5} + \cdots - 99\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 29\!\cdots\!07 \nu^{12} + \cdots - 45\!\cdots\!60 ) / 17\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 48\!\cdots\!29 \nu^{12} + \cdots + 43\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 95\!\cdots\!01 \nu^{12} + \cdots - 34\!\cdots\!00 ) / 38\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 95\!\cdots\!01 \nu^{12} + \cdots - 66\!\cdots\!00 ) / 38\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11\!\cdots\!31 \nu^{12} + \cdots - 13\!\cdots\!00 ) / 38\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 89\!\cdots\!97 \nu^{12} + \cdots - 30\!\cdots\!00 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 15\!\cdots\!29 \nu^{12} + \cdots + 62\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 13\!\cdots\!73 \nu^{12} + \cdots + 86\!\cdots\!00 ) / 89\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 29\!\cdots\!65 \nu^{12} + \cdots + 70\!\cdots\!60 ) / 17\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 24\!\cdots\!83 \nu^{12} + \cdots + 22\!\cdots\!00 ) / 89\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 14\!\cdots\!17 \nu^{12} + \cdots + 47\!\cdots\!00 ) / 26\!\cdots\!00 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} + 716 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{11} - \beta_{10} + 2\beta_{9} + \beta_{8} + \beta_{7} - 39\beta_{5} - 3\beta_{4} + 1193\beta _1 - 780 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 33 \beta_{12} + 4 \beta_{11} + 38 \beta_{10} + 51 \beta_{9} + 2 \beta_{8} - 53 \beta_{7} - 65 \beta_{6} - 1811 \beta_{5} + 1664 \beta_{4} + 42 \beta_{3} + 3 \beta_{2} + 338 \beta _1 + 842060 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 95 \beta_{12} - 8248 \beta_{11} - 2286 \beta_{10} + 4897 \beta_{9} + 1818 \beta_{8} + 857 \beta_{7} + 841 \beta_{6} - 78847 \beta_{5} - 7426 \beta_{4} - 550 \beta_{3} - 435 \beta_{2} + 1656682 \beta _1 - 1099204 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 87801 \beta_{12} + 23888 \beta_{11} + 82984 \beta_{10} + 90707 \beta_{9} - 10708 \beta_{8} - 102779 \beta_{7} - 174513 \beta_{6} - 3095339 \beta_{5} + 2604136 \beta_{4} + \cdots + 1161780604 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 77019 \beta_{12} - 14041560 \beta_{11} - 4533532 \beta_{10} + 9403063 \beta_{9} + 3218392 \beta_{8} + 120845 \beta_{7} + 3114323 \beta_{6} - 129074879 \beta_{5} + \cdots - 1834152692 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 179166405 \beta_{12} + 65427560 \beta_{11} + 146968080 \beta_{10} + 136600335 \beta_{9} - 33066436 \beta_{8} - 150469967 \beta_{7} - 365414189 \beta_{6} + \cdots + 1720483979356 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 780822343 \beta_{12} - 22736645448 \beta_{11} - 8586838112 \beta_{10} + 16487608923 \beta_{9} + 5728929068 \beta_{8} - 1384310411 \beta_{7} + \cdots - 3661488436788 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 333503309421 \beta_{12} + 145590909016 \beta_{11} + 246156622536 \beta_{10} + 200220734599 \beta_{9} - 69488801708 \beta_{8} - 197439955631 \beta_{7} + \cdots + 26\!\cdots\!76 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2448672829367 \beta_{12} - 36293262572104 \beta_{11} - 15790210549896 \beta_{10} + 27823504604155 \beta_{9} + 10122145630260 \beta_{8} + \cdots - 77\!\cdots\!64 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 594920210706789 \beta_{12} + 296669226450296 \beta_{11} + 405858436830440 \beta_{10} + 291871207155279 \beta_{9} - 130249531899148 \beta_{8} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
39.5553
38.3784
25.8517
20.6648
20.0873
10.3415
1.19145
−2.67443
−20.4888
−22.7923
−26.7693
−36.3196
−41.0259
−41.5553 233.991 1214.84 1451.63 −9723.56 −11035.4 −29206.7 35068.8 −60322.9
1.2 −40.3784 50.3948 1118.42 −1136.25 −2034.86 800.534 −24486.1 −17143.4 45879.8
1.3 −27.8517 −234.213 263.715 −396.481 6523.22 −10779.8 6915.15 35172.6 11042.7
1.4 −22.6648 −201.289 1.69319 −1574.64 4562.17 8238.87 11566.0 20834.2 35688.8
1.5 −22.0873 104.113 −24.1499 861.677 −2299.57 −5593.71 11842.1 −8843.54 −19032.1
1.6 −12.3415 −116.116 −359.688 44.3181 1433.04 8404.17 10757.9 −6200.11 −546.952
1.7 −3.19145 133.736 −501.815 16.5967 −426.813 4316.78 3235.54 −1797.60 −52.9675
1.8 0.674428 −256.621 −511.545 2353.95 −173.072 2771.95 −690.308 46171.4 1587.57
1.9 18.4888 161.312 −170.163 −1979.14 2982.47 3622.73 −12612.4 6338.52 −36592.1
1.10 20.7923 116.492 −79.6812 −29.0102 2422.13 −10914.4 −12302.4 −6112.64 −603.188
1.11 24.7693 −84.2227 101.518 1373.33 −2086.14 −3156.66 −10167.4 −12589.5 34016.4
1.12 34.3196 −129.318 665.837 −1076.40 −4438.14 9336.44 5279.62 −2959.93 −36941.6
1.13 39.0259 −29.2595 1011.02 −2068.58 −1141.88 −8587.50 19474.9 −18826.9 −80728.4
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
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 37.10.a.a ✓ 13
3.b odd 2 1 333.10.a.c 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.10.a.a ✓ 13 1.a even 1 1 trivial
333.10.a.c 13 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{13} + 32 T_{2}^{12} - 4181 T_{2}^{11} - 126994 T_{2}^{10} + 6431510 T_{2}^{9} + 184919896 T_{2}^{8} - 4482533640 T_{2}^{7} - 123081322848 T_{2}^{6} + 1392378379072 T_{2}^{5} + \cdots + 79\!\cdots\!76 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(37))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} + 32 T^{12} + \cdots + 79\!\cdots\!76 \) Copy content Toggle raw display
$3$ \( T^{13} + 251 T^{12} + \cdots + 13\!\cdots\!12 \) Copy content Toggle raw display
$5$ \( T^{13} + 2159 T^{12} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{13} + 12576 T^{12} + \cdots - 44\!\cdots\!20 \) Copy content Toggle raw display
$11$ \( T^{13} + 112451 T^{12} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{13} - 7129 T^{12} + \cdots - 21\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{13} + 890862 T^{12} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{13} + 1435874 T^{12} + \cdots - 84\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{13} + 2565799 T^{12} + \cdots - 30\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{13} + 2992323 T^{12} + \cdots - 38\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( T^{13} + 8242245 T^{12} + \cdots + 25\!\cdots\!80 \) Copy content Toggle raw display
$37$ \( (T + 1874161)^{13} \) Copy content Toggle raw display
$41$ \( T^{13} + 50975109 T^{12} + \cdots + 18\!\cdots\!10 \) Copy content Toggle raw display
$43$ \( T^{13} + 18142836 T^{12} + \cdots + 85\!\cdots\!20 \) Copy content Toggle raw display
$47$ \( T^{13} + 14353596 T^{12} + \cdots + 66\!\cdots\!80 \) Copy content Toggle raw display
$53$ \( T^{13} - 74438872 T^{12} + \cdots + 15\!\cdots\!52 \) Copy content Toggle raw display
$59$ \( T^{13} + 251964328 T^{12} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{13} - 202847323 T^{12} + \cdots + 92\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{13} + 12257509 T^{12} + \cdots - 20\!\cdots\!28 \) Copy content Toggle raw display
$71$ \( T^{13} + 310979094 T^{12} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{13} + 249752015 T^{12} + \cdots + 13\!\cdots\!02 \) Copy content Toggle raw display
$79$ \( T^{13} - 30429049 T^{12} + \cdots - 37\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{13} + 2559788658 T^{12} + \cdots - 50\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{13} + 3063565514 T^{12} + \cdots + 15\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{13} + 429307758 T^{12} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
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