Properties

Label 370.2.b.c
Level $370$
Weight $2$
Character orbit 370.b
Analytic conductor $2.954$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.95446487479\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + \beta_{2} q^{3} - q^{4} + ( -2 + \beta_{1} ) q^{5} -\beta_{3} q^{6} + ( 2 \beta_{1} + \beta_{2} ) q^{7} -\beta_{1} q^{8} -3 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + \beta_{2} q^{3} - q^{4} + ( -2 + \beta_{1} ) q^{5} -\beta_{3} q^{6} + ( 2 \beta_{1} + \beta_{2} ) q^{7} -\beta_{1} q^{8} -3 q^{9} + ( -1 - 2 \beta_{1} ) q^{10} + 2 \beta_{3} q^{11} -\beta_{2} q^{12} -4 \beta_{1} q^{13} + ( -2 - \beta_{3} ) q^{14} + ( -2 \beta_{2} - \beta_{3} ) q^{15} + q^{16} -2 \beta_{2} q^{17} -3 \beta_{1} q^{18} + ( -6 + \beta_{3} ) q^{19} + ( 2 - \beta_{1} ) q^{20} + ( -6 - 2 \beta_{3} ) q^{21} + 2 \beta_{2} q^{22} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{23} + \beta_{3} q^{24} + ( 3 - 4 \beta_{1} ) q^{25} + 4 q^{26} + ( -2 \beta_{1} - \beta_{2} ) q^{28} + ( -\beta_{2} + 2 \beta_{3} ) q^{30} + ( -4 + \beta_{3} ) q^{31} + \beta_{1} q^{32} + 12 \beta_{1} q^{33} + 2 \beta_{3} q^{34} + ( -2 - 4 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{35} + 3 q^{36} + \beta_{1} q^{37} + ( -6 \beta_{1} + \beta_{2} ) q^{38} + 4 \beta_{3} q^{39} + ( 1 + 2 \beta_{1} ) q^{40} + 2 q^{41} + ( -6 \beta_{1} - 2 \beta_{2} ) q^{42} -4 \beta_{1} q^{43} -2 \beta_{3} q^{44} + ( 6 - 3 \beta_{1} ) q^{45} + ( -4 - 2 \beta_{3} ) q^{46} + ( 2 \beta_{1} + \beta_{2} ) q^{47} + \beta_{2} q^{48} + ( -3 - 4 \beta_{3} ) q^{49} + ( 4 + 3 \beta_{1} ) q^{50} + 12 q^{51} + 4 \beta_{1} q^{52} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{53} + ( 2 \beta_{2} - 4 \beta_{3} ) q^{55} + ( 2 + \beta_{3} ) q^{56} + ( 6 \beta_{1} - 6 \beta_{2} ) q^{57} + ( 6 - \beta_{3} ) q^{59} + ( 2 \beta_{2} + \beta_{3} ) q^{60} + 12 q^{61} + ( -4 \beta_{1} + \beta_{2} ) q^{62} + ( -6 \beta_{1} - 3 \beta_{2} ) q^{63} - q^{64} + ( 4 + 8 \beta_{1} ) q^{65} -12 q^{66} + ( -8 \beta_{1} + \beta_{2} ) q^{67} + 2 \beta_{2} q^{68} + ( -12 - 4 \beta_{3} ) q^{69} + ( 4 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{70} + 2 \beta_{3} q^{71} + 3 \beta_{1} q^{72} -4 \beta_{1} q^{73} - q^{74} + ( 3 \beta_{2} + 4 \beta_{3} ) q^{75} + ( 6 - \beta_{3} ) q^{76} + ( 12 \beta_{1} + 4 \beta_{2} ) q^{77} + 4 \beta_{2} q^{78} + ( -4 - \beta_{3} ) q^{79} + ( -2 + \beta_{1} ) q^{80} -9 q^{81} + 2 \beta_{1} q^{82} + ( 12 \beta_{1} - \beta_{2} ) q^{83} + ( 6 + 2 \beta_{3} ) q^{84} + ( 4 \beta_{2} + 2 \beta_{3} ) q^{85} + 4 q^{86} -2 \beta_{2} q^{88} + ( -6 - 4 \beta_{3} ) q^{89} + ( 3 + 6 \beta_{1} ) q^{90} + ( 8 + 4 \beta_{3} ) q^{91} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{92} + ( 6 \beta_{1} - 4 \beta_{2} ) q^{93} + ( -2 - \beta_{3} ) q^{94} + ( 12 - 6 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{95} -\beta_{3} q^{96} -2 \beta_{1} q^{97} + ( -3 \beta_{1} - 4 \beta_{2} ) q^{98} -6 \beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} - 8q^{5} - 12q^{9} + O(q^{10}) \) \( 4q - 4q^{4} - 8q^{5} - 12q^{9} - 4q^{10} - 8q^{14} + 4q^{16} - 24q^{19} + 8q^{20} - 24q^{21} + 12q^{25} + 16q^{26} - 16q^{31} - 8q^{35} + 12q^{36} + 4q^{40} + 8q^{41} + 24q^{45} - 16q^{46} - 12q^{49} + 16q^{50} + 48q^{51} + 8q^{56} + 24q^{59} + 48q^{61} - 4q^{64} + 16q^{65} - 48q^{66} - 48q^{69} + 16q^{70} - 4q^{74} + 24q^{76} - 16q^{79} - 8q^{80} - 36q^{81} + 24q^{84} + 16q^{86} - 24q^{89} + 12q^{90} + 32q^{91} - 8q^{94} + 48q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} \)\(/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 3 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 3 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\(3 \beta_{1}\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{3} + 3 \beta_{2}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/370\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(297\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
1.22474 1.22474i
−1.22474 + 1.22474i
−1.22474 1.22474i
1.22474 + 1.22474i
1.00000i 2.44949i −1.00000 −2.00000 1.00000i −2.44949 4.44949i 1.00000i −3.00000 −1.00000 + 2.00000i
149.2 1.00000i 2.44949i −1.00000 −2.00000 1.00000i 2.44949 0.449490i 1.00000i −3.00000 −1.00000 + 2.00000i
149.3 1.00000i 2.44949i −1.00000 −2.00000 + 1.00000i 2.44949 0.449490i 1.00000i −3.00000 −1.00000 2.00000i
149.4 1.00000i 2.44949i −1.00000 −2.00000 + 1.00000i −2.44949 4.44949i 1.00000i −3.00000 −1.00000 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.b.c 4
3.b odd 2 1 3330.2.d.m 4
5.b even 2 1 inner 370.2.b.c 4
5.c odd 4 1 1850.2.a.s 2
5.c odd 4 1 1850.2.a.v 2
15.d odd 2 1 3330.2.d.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.b.c 4 1.a even 1 1 trivial
370.2.b.c 4 5.b even 2 1 inner
1850.2.a.s 2 5.c odd 4 1
1850.2.a.v 2 5.c odd 4 1
3330.2.d.m 4 3.b odd 2 1
3330.2.d.m 4 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\):

\( T_{3}^{2} + 6 \)
\( T_{7}^{4} + 20 T_{7}^{2} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( ( 6 + T^{2} )^{2} \)
$5$ \( ( 5 + 4 T + T^{2} )^{2} \)
$7$ \( 4 + 20 T^{2} + T^{4} \)
$11$ \( ( -24 + T^{2} )^{2} \)
$13$ \( ( 16 + T^{2} )^{2} \)
$17$ \( ( 24 + T^{2} )^{2} \)
$19$ \( ( 30 + 12 T + T^{2} )^{2} \)
$23$ \( 64 + 80 T^{2} + T^{4} \)
$29$ \( T^{4} \)
$31$ \( ( 10 + 8 T + T^{2} )^{2} \)
$37$ \( ( 1 + T^{2} )^{2} \)
$41$ \( ( -2 + T )^{4} \)
$43$ \( ( 16 + T^{2} )^{2} \)
$47$ \( 4 + 20 T^{2} + T^{4} \)
$53$ \( 8464 + 200 T^{2} + T^{4} \)
$59$ \( ( 30 - 12 T + T^{2} )^{2} \)
$61$ \( ( -12 + T )^{4} \)
$67$ \( 3364 + 140 T^{2} + T^{4} \)
$71$ \( ( -24 + T^{2} )^{2} \)
$73$ \( ( 16 + T^{2} )^{2} \)
$79$ \( ( 10 + 8 T + T^{2} )^{2} \)
$83$ \( 19044 + 300 T^{2} + T^{4} \)
$89$ \( ( -60 + 12 T + T^{2} )^{2} \)
$97$ \( ( 4 + T^{2} )^{2} \)
show more
show less