Properties

Label 37.8.b.a
Level $37$
Weight $8$
Character orbit 37.b
Analytic conductor $11.558$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [37,8,Mod(36,37)]
 
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([1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("37.36");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 37.b (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(11.5582459429\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 1702 x^{18} + 1194509 x^{16} + 450999516 x^{14} + 100204783492 x^{12} + 13461378480848 x^{10} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{19}\cdot 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{4} + 4) q^{3} + (\beta_{2} - 42) q^{4} + \beta_{12} q^{5} + ( - \beta_{11} + 12 \beta_1) q^{6} + (\beta_{8} + 2 \beta_{4} - 87) q^{7} + (\beta_{3} - 42 \beta_1) q^{8} + ( - \beta_{8} + \beta_{6} - 2 \beta_{4} + 2 \beta_{2} + 618) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{4} + 4) q^{3} + (\beta_{2} - 42) q^{4} + \beta_{12} q^{5} + ( - \beta_{11} + 12 \beta_1) q^{6} + (\beta_{8} + 2 \beta_{4} - 87) q^{7} + (\beta_{3} - 42 \beta_1) q^{8} + ( - \beta_{8} + \beta_{6} - 2 \beta_{4} + 2 \beta_{2} + 618) q^{9} + (\beta_{8} - \beta_{7} + 8 \beta_{4} - 43) q^{10} + (\beta_{10} + 6 \beta_{4} + 2 \beta_{2} + 176) q^{11} + (\beta_{8} - \beta_{5} - 17 \beta_{4} + 11 \beta_{2} - 1518) q^{12} + (\beta_{17} - 2 \beta_{12} - \beta_{11} - 14 \beta_1) q^{13} + (\beta_{15} - 13 \beta_{12} - \beta_{11} + \beta_{3} - 50 \beta_1) q^{14} + (\beta_{17} - \beta_{16} - 4 \beta_{12} - \beta_{11} - 2 \beta_{3} - 126 \beta_1) q^{15} + ( - \beta_{10} + \beta_{9} - 2 \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - 6 \beta_{4} + \cdots + 1792) q^{16}+ \cdots + (28 \beta_{14} + 219 \beta_{10} - 312 \beta_{9} - 6874 \beta_{8} + \cdots + 2156834) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 78 q^{3} - 844 q^{4} - 1746 q^{7} + 12362 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 78 q^{3} - 844 q^{4} - 1746 q^{7} + 12362 q^{9} - 882 q^{10} + 3498 q^{11} - 30374 q^{12} + 36116 q^{16} + 113482 q^{21} - 108112 q^{25} + 49278 q^{26} - 304110 q^{27} - 41192 q^{28} + 429776 q^{30} + 305646 q^{33} - 960356 q^{34} + 484758 q^{36} + 108732 q^{37} + 1049916 q^{38} - 496346 q^{40} - 1577742 q^{41} + 685266 q^{44} - 2906298 q^{46} - 1512786 q^{47} + 1522958 q^{48} + 3269246 q^{49} + 2999358 q^{53} + 405946 q^{58} + 3728310 q^{62} - 11995292 q^{63} - 11109700 q^{64} + 4251792 q^{65} + 3562224 q^{67} + 21605644 q^{70} - 15259086 q^{71} + 11088018 q^{73} - 2036544 q^{74} + 14882062 q^{75} - 2419122 q^{77} - 12178734 q^{78} - 17764972 q^{81} - 12873822 q^{83} + 9944396 q^{84} - 2698920 q^{85} + 15345336 q^{86} - 13219100 q^{90} + 48981192 q^{95} + 43111380 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 1702 x^{18} + 1194509 x^{16} + 450999516 x^{14} + 100204783492 x^{12} + 13461378480848 x^{10} + \cdots + 13\!\cdots\!96 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 170 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 298\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 24\!\cdots\!11 \nu^{18} + \cdots - 21\!\cdots\!80 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 72\!\cdots\!59 \nu^{18} + \cdots + 28\!\cdots\!76 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10\!\cdots\!63 \nu^{18} + \cdots - 55\!\cdots\!48 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 16\!\cdots\!05 \nu^{18} + \cdots - 31\!\cdots\!32 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 14\!\cdots\!37 \nu^{18} + \cdots + 72\!\cdots\!16 ) / 91\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 11\!\cdots\!43 \nu^{18} + \cdots + 15\!\cdots\!04 ) / 61\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 16\!\cdots\!83 \nu^{18} + \cdots + 31\!\cdots\!56 ) / 61\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 24\!\cdots\!11 \nu^{19} + \cdots + 35\!\cdots\!80 \nu ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 14\!\cdots\!31 \nu^{19} + \cdots - 10\!\cdots\!64 \nu ) / 54\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 72\!\cdots\!43 \nu^{19} + \cdots + 30\!\cdots\!36 \nu ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 43\!\cdots\!81 \nu^{18} + \cdots + 17\!\cdots\!36 ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 62\!\cdots\!83 \nu^{19} + \cdots - 12\!\cdots\!56 \nu ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 92\!\cdots\!41 \nu^{19} + \cdots - 43\!\cdots\!64 \nu ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 93\!\cdots\!19 \nu^{19} + \cdots - 40\!\cdots\!12 \nu ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 28\!\cdots\!65 \nu^{19} + \cdots - 52\!\cdots\!32 \nu ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 84\!\cdots\!35 \nu^{19} + \cdots - 51\!\cdots\!64 \nu ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 170 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 298\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{10} + \beta_{9} - 2\beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - 6\beta_{4} - 448\beta_{2} + 50688 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10 \beta_{19} - 5 \beta_{18} - 2 \beta_{17} - 8 \beta_{15} + 2 \beta_{13} + 250 \beta_{12} - 41 \beta_{11} - 581 \beta_{3} + 107960 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 11 \beta_{14} + 970 \beta_{10} - 651 \beta_{9} + 2768 \beta_{8} - 703 \beta_{7} + 465 \beta_{6} - 871 \beta_{5} - 1914 \beta_{4} + 193797 \beta_{2} - 18380527 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 8050 \beta_{19} + 4363 \beta_{18} + 538 \beta_{17} - 660 \beta_{16} + 6402 \beta_{15} - 2224 \beta_{13} - 208888 \beta_{12} + 61423 \beta_{11} + 286937 \beta_{3} - 43171486 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 9567 \beta_{14} - 604362 \beta_{10} + 339039 \beta_{9} - 1930120 \beta_{8} + 417323 \beta_{7} - 230277 \beta_{6} + 556419 \beta_{5} + 6774826 \beta_{4} - 85047949 \beta_{2} + \cdots + 7356193323 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4809690 \beta_{19} - 2757911 \beta_{18} + 52510 \beta_{17} + 678180 \beta_{16} - 3642650 \beta_{15} + 1593872 \beta_{13} + 130255496 \beta_{12} - 51030763 \beta_{11} + \cdots + 18274909902 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 5774891 \beta_{14} + 327064038 \beta_{10} - 165205439 \beta_{9} + 1091155264 \beta_{8} - 230683451 \beta_{7} + 119017373 \beta_{6} - 315170019 \beta_{5} + \cdots - 3116047703047 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2587179690 \beta_{19} + 1543681783 \beta_{18} - 147038606 \beta_{17} - 484360500 \beta_{16} + 1832478546 \beta_{15} - 961797400 \beta_{13} + \cdots - 8013645796550 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 3038838435 \beta_{14} - 166734036758 \beta_{10} + 78569642423 \beta_{9} - 564579581136 \beta_{8} + 121539360243 \beta_{7} - 61461655013 \beta_{6} + \cdots + 13\!\cdots\!47 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1325256976250 \beta_{19} - 813119758175 \beta_{18} + 111130456894 \beta_{17} + 296377317780 \beta_{16} - 875227443314 \beta_{15} + 531829170296 \beta_{13} + \cdots + 35\!\cdots\!58 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 1503100340603 \beta_{14} + 82533398918870 \beta_{10} - 37034306669503 \beta_{9} + 280047778113504 \beta_{8} - 61935527232219 \beta_{7} + \cdots - 61\!\cdots\!31 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 660983474302250 \beta_{19} + 413570565364279 \beta_{18} - 65919848222606 \beta_{17} - 166574015615220 \beta_{16} + 409096393166226 \beta_{15} + \cdots - 16\!\cdots\!62 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 722012791137891 \beta_{14} + \cdots + 27\!\cdots\!43 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 32\!\cdots\!10 \beta_{19} + \cdots + 75\!\cdots\!46 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 34\!\cdots\!31 \beta_{14} + \cdots - 12\!\cdots\!47 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 15\!\cdots\!70 \beta_{19} + \cdots - 34\!\cdots\!98 \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/37\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
36.1
21.7545i
18.8806i
16.0010i
14.8861i
12.3896i
10.3219i
7.77325i
7.52533i
4.18617i
0.118026i
0.118026i
4.18617i
7.52533i
7.77325i
10.3219i
12.3896i
14.8861i
16.0010i
18.8806i
21.7545i
21.7545i 44.4956 −345.259 104.884i 967.981i −454.964 4726.36i −207.137 −2281.70
36.2 18.8806i −54.3989 −228.476 103.526i 1027.08i −202.536 1897.05i 772.237 1954.63
36.3 16.0010i 52.6587 −128.032 434.946i 842.592i 1479.04 0.510187i 585.941 6959.56
36.4 14.8861i −0.807131 −93.5969 387.405i 12.0151i 696.938 512.130i −2186.35 −5766.96
36.5 12.3896i 76.2286 −25.5018 93.6237i 944.440i −1237.82 1269.91i 3623.79 −1159.96
36.6 10.3219i 6.68770 21.4577 296.142i 69.0300i −1262.47 1542.69i −2142.27 3056.76
36.7 7.77325i −62.8536 67.5766 184.143i 488.577i 1071.60 1520.27i 1763.58 1431.39
36.8 7.52533i −79.9440 71.3694 468.997i 601.605i −1534.47 1500.32i 4204.05 −3529.36
36.9 4.18617i 63.9664 110.476 270.412i 267.774i 565.484 998.302i 1904.70 −1131.99
36.10 0.118026i −7.03331 127.986 225.549i 0.830113i 6.20125 30.2130i −2137.53 26.6207
36.11 0.118026i −7.03331 127.986 225.549i 0.830113i 6.20125 30.2130i −2137.53 26.6207
36.12 4.18617i 63.9664 110.476 270.412i 267.774i 565.484 998.302i 1904.70 −1131.99
36.13 7.52533i −79.9440 71.3694 468.997i 601.605i −1534.47 1500.32i 4204.05 −3529.36
36.14 7.77325i −62.8536 67.5766 184.143i 488.577i 1071.60 1520.27i 1763.58 1431.39
36.15 10.3219i 6.68770 21.4577 296.142i 69.0300i −1262.47 1542.69i −2142.27 3056.76
36.16 12.3896i 76.2286 −25.5018 93.6237i 944.440i −1237.82 1269.91i 3623.79 −1159.96
36.17 14.8861i −0.807131 −93.5969 387.405i 12.0151i 696.938 512.130i −2186.35 −5766.96
36.18 16.0010i 52.6587 −128.032 434.946i 842.592i 1479.04 0.510187i 585.941 6959.56
36.19 18.8806i −54.3989 −228.476 103.526i 1027.08i −202.536 1897.05i 772.237 1954.63
36.20 21.7545i 44.4956 −345.259 104.884i 967.981i −454.964 4726.36i −207.137 −2281.70
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 36.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.8.b.a 20
37.b even 2 1 inner 37.8.b.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.8.b.a 20 1.a even 1 1 trivial
37.8.b.a 20 37.b even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(37, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 1702 T^{18} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$3$ \( (T^{10} - 39 T^{9} + \cdots - 118561742732844)^{2} \) Copy content Toggle raw display
$5$ \( T^{20} + 835306 T^{18} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{10} + 873 T^{9} + \cdots - 85\!\cdots\!36)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} - 1749 T^{9} + \cdots - 11\!\cdots\!04)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + 632138730 T^{18} + \cdots + 38\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{20} + 3321595492 T^{18} + \cdots + 26\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{20} + 6260069400 T^{18} + \cdots + 50\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( T^{20} + 28047457810 T^{18} + \cdots + 33\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{20} + 224854622446 T^{18} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$31$ \( T^{20} + 263739369210 T^{18} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{20} - 108732 T^{19} + \cdots + 59\!\cdots\!49 \) Copy content Toggle raw display
$41$ \( (T^{10} + 788871 T^{9} + \cdots + 23\!\cdots\!26)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + 2779292121648 T^{18} + \cdots + 79\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( (T^{10} + 756393 T^{9} + \cdots + 74\!\cdots\!84)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} - 1499679 T^{9} + \cdots + 14\!\cdots\!64)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + 23780290980220 T^{18} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{20} + 19651838390478 T^{18} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{10} - 1781112 T^{9} + \cdots + 25\!\cdots\!76)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + 7629543 T^{9} + \cdots + 12\!\cdots\!76)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} - 5544009 T^{9} + \cdots + 35\!\cdots\!06)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + 170634215807934 T^{18} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( (T^{10} + 6436911 T^{9} + \cdots + 50\!\cdots\!16)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + 261084499656316 T^{18} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{20} + 788101919779560 T^{18} + \cdots + 27\!\cdots\!96 \) Copy content Toggle raw display
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