Properties

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

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 37.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.295446487479\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 i q^{2} - q^{3} -2 q^{4} -2 i q^{5} -2 i q^{6} + 3 q^{7} -2 q^{9} +O(q^{10})\) \( q + 2 i q^{2} - q^{3} -2 q^{4} -2 i q^{5} -2 i q^{6} + 3 q^{7} -2 q^{9} + 4 q^{10} -3 q^{11} + 2 q^{12} -6 i q^{13} + 6 i q^{14} + 2 i q^{15} -4 q^{16} + 2 i q^{17} -4 i q^{18} + 6 i q^{19} + 4 i q^{20} -3 q^{21} -6 i q^{22} + 4 i q^{23} + q^{25} + 12 q^{26} + 5 q^{27} -6 q^{28} -4 i q^{29} -4 q^{30} -8 i q^{32} + 3 q^{33} -4 q^{34} -6 i q^{35} + 4 q^{36} + ( -1 + 6 i ) q^{37} -12 q^{38} + 6 i q^{39} -3 q^{41} -6 i q^{42} -6 i q^{43} + 6 q^{44} + 4 i q^{45} -8 q^{46} + 3 q^{47} + 4 q^{48} + 2 q^{49} + 2 i q^{50} -2 i q^{51} + 12 i q^{52} + 9 q^{53} + 10 i q^{54} + 6 i q^{55} -6 i q^{57} + 8 q^{58} -4 i q^{59} -4 i q^{60} -6 q^{63} + 8 q^{64} -12 q^{65} + 6 i q^{66} -12 q^{67} -4 i q^{68} -4 i q^{69} + 12 q^{70} -3 q^{71} + 9 q^{73} + ( -12 - 2 i ) q^{74} - q^{75} -12 i q^{76} -9 q^{77} -12 q^{78} + 6 i q^{79} + 8 i q^{80} + q^{81} -6 i q^{82} + 9 q^{83} + 6 q^{84} + 4 q^{85} + 12 q^{86} + 4 i q^{87} -14 i q^{89} -8 q^{90} -18 i q^{91} -8 i q^{92} + 6 i q^{94} + 12 q^{95} + 8 i q^{96} + 12 i q^{97} + 4 i q^{98} + 6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 4q^{4} + 6q^{7} - 4q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 4q^{4} + 6q^{7} - 4q^{9} + 8q^{10} - 6q^{11} + 4q^{12} - 8q^{16} - 6q^{21} + 2q^{25} + 24q^{26} + 10q^{27} - 12q^{28} - 8q^{30} + 6q^{33} - 8q^{34} + 8q^{36} - 2q^{37} - 24q^{38} - 6q^{41} + 12q^{44} - 16q^{46} + 6q^{47} + 8q^{48} + 4q^{49} + 18q^{53} + 16q^{58} - 12q^{63} + 16q^{64} - 24q^{65} - 24q^{67} + 24q^{70} - 6q^{71} + 18q^{73} - 24q^{74} - 2q^{75} - 18q^{77} - 24q^{78} + 2q^{81} + 18q^{83} + 12q^{84} + 8q^{85} + 24q^{86} - 16q^{90} + 24q^{95} + 12q^{99} + O(q^{100}) \)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
36.1
1.00000i
1.00000i
2.00000i −1.00000 −2.00000 2.00000i 2.00000i 3.00000 0 −2.00000 4.00000
36.2 2.00000i −1.00000 −2.00000 2.00000i 2.00000i 3.00000 0 −2.00000 4.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.2.b.a 2
3.b odd 2 1 333.2.c.a 2
4.b odd 2 1 592.2.g.b 2
5.b even 2 1 925.2.c.b 2
5.c odd 4 1 925.2.d.a 2
5.c odd 4 1 925.2.d.d 2
8.b even 2 1 2368.2.g.f 2
8.d odd 2 1 2368.2.g.b 2
12.b even 2 1 5328.2.h.c 2
37.b even 2 1 inner 37.2.b.a 2
37.d odd 4 1 1369.2.a.a 1
37.d odd 4 1 1369.2.a.f 1
111.d odd 2 1 333.2.c.a 2
148.b odd 2 1 592.2.g.b 2
185.d even 2 1 925.2.c.b 2
185.h odd 4 1 925.2.d.a 2
185.h odd 4 1 925.2.d.d 2
296.e even 2 1 2368.2.g.f 2
296.h odd 2 1 2368.2.g.b 2
444.g even 2 1 5328.2.h.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.b.a 2 1.a even 1 1 trivial
37.2.b.a 2 37.b even 2 1 inner
333.2.c.a 2 3.b odd 2 1
333.2.c.a 2 111.d odd 2 1
592.2.g.b 2 4.b odd 2 1
592.2.g.b 2 148.b odd 2 1
925.2.c.b 2 5.b even 2 1
925.2.c.b 2 185.d even 2 1
925.2.d.a 2 5.c odd 4 1
925.2.d.a 2 185.h odd 4 1
925.2.d.d 2 5.c odd 4 1
925.2.d.d 2 185.h odd 4 1
1369.2.a.a 1 37.d odd 4 1
1369.2.a.f 1 37.d odd 4 1
2368.2.g.b 2 8.d odd 2 1
2368.2.g.b 2 296.h odd 2 1
2368.2.g.f 2 8.b even 2 1
2368.2.g.f 2 296.e even 2 1
5328.2.h.c 2 12.b even 2 1
5328.2.h.c 2 444.g even 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( 4 + T^{2} \)
$7$ \( ( -3 + T )^{2} \)
$11$ \( ( 3 + T )^{2} \)
$13$ \( 36 + T^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( 36 + T^{2} \)
$23$ \( 16 + T^{2} \)
$29$ \( 16 + T^{2} \)
$31$ \( T^{2} \)
$37$ \( 37 + 2 T + T^{2} \)
$41$ \( ( 3 + T )^{2} \)
$43$ \( 36 + T^{2} \)
$47$ \( ( -3 + T )^{2} \)
$53$ \( ( -9 + T )^{2} \)
$59$ \( 16 + T^{2} \)
$61$ \( T^{2} \)
$67$ \( ( 12 + T )^{2} \)
$71$ \( ( 3 + T )^{2} \)
$73$ \( ( -9 + T )^{2} \)
$79$ \( 36 + T^{2} \)
$83$ \( ( -9 + T )^{2} \)
$89$ \( 196 + T^{2} \)
$97$ \( 144 + T^{2} \)
show more
show less