Properties

Label 37.2.e.a
Level $37$
Weight $2$
Character orbit 37.e
Analytic conductor $0.295$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 37.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.295446487479\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12}) q^{3} - \zeta_{12}^{2} q^{4} + (2 \zeta_{12}^{3} + \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{5} + (\zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{6} + (2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{7} - 3 \zeta_{12}^{3} q^{8} + ( - 4 \zeta_{12}^{3} + \zeta_{12}^{2} + 2 \zeta_{12} - 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12} q^{2} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12}) q^{3} - \zeta_{12}^{2} q^{4} + (2 \zeta_{12}^{3} + \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{5} + (\zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{6} + (2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{7} - 3 \zeta_{12}^{3} q^{8} + ( - 4 \zeta_{12}^{3} + \zeta_{12}^{2} + 2 \zeta_{12} - 1) q^{9} + (\zeta_{12}^{3} - 2 \zeta_{12} - 2) q^{10} + ( - \zeta_{12}^{3} + 2 \zeta_{12} - 3) q^{11} + (2 \zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12} + 1) q^{12} + ( - 2 \zeta_{12}^{2} + 4) q^{13} + (4 \zeta_{12}^{2} - 2) q^{14} + (\zeta_{12}^{2} + \zeta_{12} + 1) q^{15} + ( - \zeta_{12}^{2} + 1) q^{16} + (\zeta_{12}^{2} - 2 \zeta_{12} + 1) q^{17} + (\zeta_{12}^{3} - 2 \zeta_{12}^{2} - \zeta_{12} + 4) q^{18} + (3 \zeta_{12}^{3} + \zeta_{12}^{2} - 3 \zeta_{12} - 2) q^{19} + (\zeta_{12}^{2} + 2 \zeta_{12} + 1) q^{20} + (4 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 2 \zeta_{12} + 6) q^{21} + (\zeta_{12}^{2} - 3 \zeta_{12} + 1) q^{22} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{23} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 3 \zeta_{12} - 6) q^{24} + ( - 8 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 4 \zeta_{12} + 2) q^{25} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}) q^{26} - 4 q^{27} + ( - 4 \zeta_{12}^{3} + 2 \zeta_{12}) q^{28} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{29} + (\zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{30} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 3) q^{31} + (5 \zeta_{12}^{3} - 5 \zeta_{12}) q^{32} + (4 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 4 \zeta_{12}) q^{33} + (\zeta_{12}^{3} - 2 \zeta_{12}^{2} + \zeta_{12}) q^{34} + ( - 4 \zeta_{12}^{2} - 6 \zeta_{12} - 4) q^{35} + (2 \zeta_{12}^{3} - 4 \zeta_{12} + 1) q^{36} + ( - 7 \zeta_{12}^{2} + 4) q^{37} + (\zeta_{12}^{3} - 2 \zeta_{12} - 3) q^{38} + (2 \zeta_{12}^{2} - 6 \zeta_{12} + 2) q^{39} + (3 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 3 \zeta_{12}) q^{40} - 3 \zeta_{12}^{2} q^{41} + ( - 6 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 6 \zeta_{12} - 4) q^{42} + ( - 4 \zeta_{12}^{2} + 2) q^{43} + ( - \zeta_{12}^{3} + 3 \zeta_{12}^{2} - \zeta_{12}) q^{44} + (4 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 3) q^{45} + (2 \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12} - 1) q^{46} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12} + 3) q^{47} + (\zeta_{12}^{3} - 2 \zeta_{12} + 1) q^{48} + (5 \zeta_{12}^{2} - 5) q^{49} + ( - 2 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2 \zeta_{12} + 8) q^{50} + ( - 5 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 3) q^{51} + ( - 2 \zeta_{12}^{2} - 2) q^{52} + ( - 4 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 2 \zeta_{12} + 6) q^{53} - 4 \zeta_{12} q^{54} + ( - 3 \zeta_{12}^{3} - \zeta_{12}^{2} + 3 \zeta_{12} + 2) q^{55} + ( - 6 \zeta_{12}^{2} + 12) q^{56} + (2 \zeta_{12}^{2} + 2) q^{57} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - \zeta_{12} - 2) q^{58} + ( - 2 \zeta_{12}^{2} + 10 \zeta_{12} - 2) q^{59} + ( - \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{60} + (6 \zeta_{12}^{3} + \zeta_{12}^{2} - 6 \zeta_{12} - 2) q^{61} + (6 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 3 \zeta_{12} + 3) q^{62} + (2 \zeta_{12}^{3} - 4 \zeta_{12} + 12) q^{63} - 7 q^{64} + (8 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 4 \zeta_{12} - 6) q^{65} + ( - 6 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 4) q^{66} + (\zeta_{12}^{3} + 9 \zeta_{12}^{2} + \zeta_{12}) q^{67} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{68} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{69} + ( - 4 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 4 \zeta_{12}) q^{70} - 6 \zeta_{12}^{2} q^{71} + ( - 6 \zeta_{12}^{2} + 3 \zeta_{12} - 6) q^{72} + ( - 7 \zeta_{12}^{3} + 4 \zeta_{12}) q^{74} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12} - 10) q^{75} + (\zeta_{12}^{2} + 3 \zeta_{12} + 1) q^{76} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 6 \zeta_{12}) q^{77} + (2 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 2 \zeta_{12}) q^{78} + (9 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 9 \zeta_{12} - 6) q^{79} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{80} + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} - 2 \zeta_{12}) q^{81} - 3 \zeta_{12}^{3} q^{82} + (6 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 3 \zeta_{12} + 3) q^{83} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12} - 6) q^{84} + q^{85} + ( - 4 \zeta_{12}^{3} + 2 \zeta_{12}) q^{86} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} + 2) q^{87} + (9 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 3) q^{88} + (3 \zeta_{12}^{2} - 4 \zeta_{12} + 3) q^{89} + (6 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 3 \zeta_{12} - 4) q^{90} + 12 \zeta_{12} q^{91} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} + 2) q^{92} + ( - 12 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 12 \zeta_{12} - 12) q^{93} + (3 \zeta_{12}^{2} + 3 \zeta_{12} + 3) q^{94} + ( - 10 \zeta_{12}^{3} - 9 \zeta_{12}^{2} + 5 \zeta_{12} + 9) q^{95} + (5 \zeta_{12}^{2} - 5 \zeta_{12} + 5) q^{96} + ( - 6 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{97} + (5 \zeta_{12}^{3} - 5 \zeta_{12}) q^{98} + (14 \zeta_{12}^{3} - 9 \zeta_{12}^{2} - 7 \zeta_{12} + 9) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 2 q^{4} - 6 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 2 q^{4} - 6 q^{5} - 2 q^{9} - 8 q^{10} - 12 q^{11} + 2 q^{12} + 12 q^{13} + 6 q^{15} + 2 q^{16} + 6 q^{17} + 12 q^{18} - 6 q^{19} + 6 q^{20} + 12 q^{21} + 6 q^{22} - 18 q^{24} + 4 q^{25} - 16 q^{27} + 2 q^{30} - 12 q^{33} - 4 q^{34} - 24 q^{35} + 4 q^{36} + 2 q^{37} - 12 q^{38} + 12 q^{39} + 12 q^{40} - 6 q^{41} - 12 q^{42} + 6 q^{44} - 2 q^{46} + 12 q^{47} + 4 q^{48} - 10 q^{49} + 24 q^{50} - 12 q^{52} + 12 q^{53} + 6 q^{55} + 36 q^{56} + 12 q^{57} - 4 q^{58} - 12 q^{59} - 6 q^{61} + 6 q^{62} + 48 q^{63} - 28 q^{64} - 12 q^{65} + 18 q^{67} - 12 q^{70} - 12 q^{71} - 36 q^{72} - 40 q^{75} + 6 q^{76} + 12 q^{77} - 12 q^{78} - 18 q^{79} - 2 q^{81} + 6 q^{83} - 24 q^{84} + 4 q^{85} + 6 q^{87} + 18 q^{89} - 8 q^{90} + 6 q^{92} - 36 q^{93} + 18 q^{94} + 18 q^{95} + 30 q^{96} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(\zeta_{12}^{2}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i 1.36603 2.36603i −0.500000 + 0.866025i 0.232051 + 0.133975i 2.73205i −1.73205 + 3.00000i 3.00000i −2.23205 3.86603i −0.267949
11.2 0.866025 0.500000i −0.366025 + 0.633975i −0.500000 + 0.866025i −3.23205 1.86603i 0.732051i 1.73205 3.00000i 3.00000i 1.23205 + 2.13397i −3.73205
27.1 −0.866025 0.500000i 1.36603 + 2.36603i −0.500000 0.866025i 0.232051 0.133975i 2.73205i −1.73205 3.00000i 3.00000i −2.23205 + 3.86603i −0.267949
27.2 0.866025 + 0.500000i −0.366025 0.633975i −0.500000 0.866025i −3.23205 + 1.86603i 0.732051i 1.73205 + 3.00000i 3.00000i 1.23205 2.13397i −3.73205
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.2.e.a 4
3.b odd 2 1 333.2.s.b 4
4.b odd 2 1 592.2.w.c 4
5.b even 2 1 925.2.n.a 4
5.c odd 4 1 925.2.m.a 4
5.c odd 4 1 925.2.m.b 4
37.c even 3 1 1369.2.b.d 4
37.e even 6 1 inner 37.2.e.a 4
37.e even 6 1 1369.2.b.d 4
37.g odd 12 1 1369.2.a.g 2
37.g odd 12 1 1369.2.a.h 2
111.h odd 6 1 333.2.s.b 4
148.j odd 6 1 592.2.w.c 4
185.l even 6 1 925.2.n.a 4
185.r odd 12 1 925.2.m.a 4
185.r odd 12 1 925.2.m.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.e.a 4 1.a even 1 1 trivial
37.2.e.a 4 37.e even 6 1 inner
333.2.s.b 4 3.b odd 2 1
333.2.s.b 4 111.h odd 6 1
592.2.w.c 4 4.b odd 2 1
592.2.w.c 4 148.j odd 6 1
925.2.m.a 4 5.c odd 4 1
925.2.m.a 4 185.r odd 12 1
925.2.m.b 4 5.c odd 4 1
925.2.m.b 4 185.r odd 12 1
925.2.n.a 4 5.b even 2 1
925.2.n.a 4 185.l even 6 1
1369.2.a.g 2 37.g odd 12 1
1369.2.a.h 2 37.g odd 12 1
1369.2.b.d 4 37.c even 3 1
1369.2.b.d 4 37.e even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(37, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} + 6 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$5$ \( T^{4} + 6 T^{3} + 11 T^{2} - 6 T + 1 \) Copy content Toggle raw display
$7$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$11$ \( (T^{2} + 6 T + 6)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 6 T + 12)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 6 T^{3} + 11 T^{2} + 6 T + 1 \) Copy content Toggle raw display
$19$ \( T^{4} + 6 T^{3} + 6 T^{2} - 36 T + 36 \) Copy content Toggle raw display
$23$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$29$ \( T^{4} + 14T^{2} + 1 \) Copy content Toggle raw display
$31$ \( T^{4} + 72T^{2} + 324 \) Copy content Toggle raw display
$37$ \( (T^{2} - T + 37)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 6 T - 18)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 12 T^{3} + 120 T^{2} + \cdots + 576 \) Copy content Toggle raw display
$59$ \( T^{4} + 12 T^{3} - 40 T^{2} + \cdots + 7744 \) Copy content Toggle raw display
$61$ \( T^{4} + 6 T^{3} - 21 T^{2} + \cdots + 1089 \) Copy content Toggle raw display
$67$ \( T^{4} - 18 T^{3} + 246 T^{2} + \cdots + 6084 \) Copy content Toggle raw display
$71$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} + 18 T^{3} + 54 T^{2} + \cdots + 2916 \) Copy content Toggle raw display
$83$ \( T^{4} - 6 T^{3} + 54 T^{2} + 108 T + 324 \) Copy content Toggle raw display
$89$ \( T^{4} - 18 T^{3} + 119 T^{2} + \cdots + 121 \) Copy content Toggle raw display
$97$ \( T^{4} + 78T^{2} + 1089 \) Copy content Toggle raw display
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