Properties

Label 370.2.q.a
Level $370$
Weight $2$
Character orbit 370.q
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.q (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.95446487479\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

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

Character values

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

\(n\) \(261\) \(297\)
\(\chi(n)\) \(\zeta_{12}\) \(\zeta_{12}^{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.500000 + 0.866025i 0.133975 0.500000i −0.500000 0.866025i −1.86603 1.23205i 0.366025 + 0.366025i −0.732051 + 2.73205i 1.00000 2.36603 + 1.36603i 2.00000 1.00000i
103.1 −0.500000 0.866025i 0.133975 + 0.500000i −0.500000 + 0.866025i −1.86603 + 1.23205i 0.366025 0.366025i −0.732051 2.73205i 1.00000 2.36603 1.36603i 2.00000 + 1.00000i
267.1 −0.500000 0.866025i 1.86603 0.500000i −0.500000 + 0.866025i −0.133975 + 2.23205i −1.36603 1.36603i 2.73205 0.732051i 1.00000 0.633975 0.366025i 2.00000 1.00000i
273.1 −0.500000 + 0.866025i 1.86603 + 0.500000i −0.500000 0.866025i −0.133975 2.23205i −1.36603 + 1.36603i 2.73205 + 0.732051i 1.00000 0.633975 + 0.366025i 2.00000 + 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.p even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.q.a 4
5.c odd 4 1 370.2.r.a yes 4
37.g odd 12 1 370.2.r.a yes 4
185.p even 12 1 inner 370.2.q.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.q.a 4 1.a even 1 1 trivial
370.2.q.a 4 185.p even 12 1 inner
370.2.r.a yes 4 5.c odd 4 1
370.2.r.a yes 4 37.g odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 4 T_{3}^{3} + 5 T_{3}^{2} - 2 T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( 1 - 2 T + 5 T^{2} - 4 T^{3} + T^{4} \)
$5$ \( 25 + 20 T + 11 T^{2} + 4 T^{3} + T^{4} \)
$7$ \( 64 - 32 T + 8 T^{2} - 4 T^{3} + T^{4} \)
$11$ \( ( 4 + T^{2} )^{2} \)
$13$ \( 121 + 22 T + 15 T^{2} - 2 T^{3} + T^{4} \)
$17$ \( 676 - 468 T + 134 T^{2} - 18 T^{3} + T^{4} \)
$19$ \( 484 - 308 T + 74 T^{2} - 10 T^{3} + T^{4} \)
$23$ \( ( -6 + T )^{4} \)
$29$ \( 144 + 144 T + 72 T^{2} + 12 T^{3} + T^{4} \)
$31$ \( 625 + 250 T + 50 T^{2} - 10 T^{3} + T^{4} \)
$37$ \( ( 37 + 11 T + T^{2} )^{2} \)
$41$ \( 9 + 36 T + 51 T^{2} + 12 T^{3} + T^{4} \)
$43$ \( ( 11 + T )^{4} \)
$47$ \( 2704 - 208 T + 8 T^{2} + 4 T^{3} + T^{4} \)
$53$ \( 3721 - 1586 T + 233 T^{2} - 16 T^{3} + T^{4} \)
$59$ \( 676 - 364 T + 50 T^{2} - 2 T^{3} + T^{4} \)
$61$ \( 144 + 36 T^{2} + 12 T^{3} + T^{4} \)
$67$ \( 6084 - 1404 T + 306 T^{2} - 30 T^{3} + T^{4} \)
$71$ \( 1024 - 256 T + 96 T^{2} + 8 T^{3} + T^{4} \)
$73$ \( T^{4} \)
$79$ \( 2500 + 500 T + 50 T^{2} + 10 T^{3} + T^{4} \)
$83$ \( 45796 + 7276 T + 650 T^{2} + 38 T^{3} + T^{4} \)
$89$ \( 484 - 220 T + 194 T^{2} - 26 T^{3} + T^{4} \)
$97$ \( 64 + 32 T^{2} + T^{4} \)
show more
show less