Properties

Label 3850.2.c.x
Level $3850$
Weight $2$
Character orbit 3850.c
Analytic conductor $30.742$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3850.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.7424047782\)
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: \( 2^{2} \)
Twist minimal: no (minimal twist has level 770)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -\zeta_{12}^{3} q^{2} + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{3} - q^{4} + ( 1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{6} -\zeta_{12}^{3} q^{7} + \zeta_{12}^{3} q^{8} + ( -1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{9} +O(q^{10})\) \( q -\zeta_{12}^{3} q^{2} + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{3} - q^{4} + ( 1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{6} -\zeta_{12}^{3} q^{7} + \zeta_{12}^{3} q^{8} + ( -1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{9} + q^{11} + ( -1 + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{12} + ( -2 + 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{13} - q^{14} + q^{16} + ( 2 - 4 \zeta_{12}^{2} ) q^{17} + ( 2 - 4 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{18} + ( -5 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{19} + ( 1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{21} -\zeta_{12}^{3} q^{22} + ( -3 + 6 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{23} + ( -1 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{24} + ( 2 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{26} -4 \zeta_{12}^{3} q^{27} + \zeta_{12}^{3} q^{28} + ( 3 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{29} + 2 q^{31} -\zeta_{12}^{3} q^{32} + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{33} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{34} + ( 1 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{36} + ( -1 + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{37} + ( 1 - 2 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{38} + 4 q^{39} + ( 3 + 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{41} + ( -1 + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{42} + 2 \zeta_{12}^{3} q^{43} - q^{44} + ( -3 + 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{46} + ( -4 + 8 \zeta_{12}^{2} ) q^{47} + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{48} - q^{49} + ( -6 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{51} + ( 2 - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{52} + ( 1 - 2 \zeta_{12}^{2} - 9 \zeta_{12}^{3} ) q^{53} -4 q^{54} + q^{56} + ( -6 + 12 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{57} + ( -1 + 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{58} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{59} + ( 2 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{61} -2 \zeta_{12}^{3} q^{62} + ( 2 - 4 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{63} - q^{64} + ( 1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{66} + 4 \zeta_{12}^{3} q^{67} + ( -2 + 4 \zeta_{12}^{2} ) q^{68} + ( 12 - 12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{69} + ( 6 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{71} + ( -2 + 4 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{72} + ( -6 + 12 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{73} + ( 1 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{74} + ( 5 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{76} -\zeta_{12}^{3} q^{77} -4 \zeta_{12}^{3} q^{78} + ( 7 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{79} + ( 1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{81} + ( 3 - 6 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{82} + ( -6 + 12 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{83} + ( -1 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{84} + 2 q^{86} + ( 4 - 8 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{87} + \zeta_{12}^{3} q^{88} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{89} + ( 2 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{91} + ( 3 - 6 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{92} + ( 2 - 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{93} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{94} + ( 1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{96} + ( -9 + 18 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{97} + \zeta_{12}^{3} q^{98} + ( -1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + 4q^{6} - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{4} + 4q^{6} - 4q^{9} + 4q^{11} - 4q^{14} + 4q^{16} - 20q^{19} + 4q^{21} - 4q^{24} + 8q^{26} + 12q^{29} + 8q^{31} + 4q^{36} + 16q^{39} + 12q^{41} - 4q^{44} - 12q^{46} - 4q^{49} - 24q^{51} - 16q^{54} + 4q^{56} + 8q^{61} - 4q^{64} + 4q^{66} + 48q^{69} + 24q^{71} + 4q^{74} + 20q^{76} + 28q^{79} + 4q^{81} - 4q^{84} + 8q^{86} + 8q^{91} + 4q^{96} - 4q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3850\mathbb{Z}\right)^\times\).

\(n\) \(1751\) \(2201\) \(2927\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1849.1
0.866025 + 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
1.00000i 0.732051i −1.00000 0 −0.732051 1.00000i 1.00000i 2.46410 0
1849.2 1.00000i 2.73205i −1.00000 0 2.73205 1.00000i 1.00000i −4.46410 0
1849.3 1.00000i 2.73205i −1.00000 0 2.73205 1.00000i 1.00000i −4.46410 0
1849.4 1.00000i 0.732051i −1.00000 0 −0.732051 1.00000i 1.00000i 2.46410 0
\(n\): 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 3850.2.c.x 4
5.b even 2 1 inner 3850.2.c.x 4
5.c odd 4 1 770.2.a.j 2
5.c odd 4 1 3850.2.a.bd 2
15.e even 4 1 6930.2.a.bv 2
20.e even 4 1 6160.2.a.t 2
35.f even 4 1 5390.2.a.bs 2
55.e even 4 1 8470.2.a.br 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.a.j 2 5.c odd 4 1
3850.2.a.bd 2 5.c odd 4 1
3850.2.c.x 4 1.a even 1 1 trivial
3850.2.c.x 4 5.b even 2 1 inner
5390.2.a.bs 2 35.f even 4 1
6160.2.a.t 2 20.e even 4 1
6930.2.a.bv 2 15.e even 4 1
8470.2.a.br 2 55.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3850, [\chi])\):

\( T_{3}^{4} + 8 T_{3}^{2} + 4 \)
\( T_{13}^{4} + 32 T_{13}^{2} + 64 \)
\( T_{17}^{2} + 12 \)
\( T_{19}^{2} + 10 T_{19} + 22 \)
\( T_{37}^{4} + 8 T_{37}^{2} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( 4 + 8 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 1 + T^{2} )^{2} \)
$11$ \( ( -1 + T )^{4} \)
$13$ \( 64 + 32 T^{2} + T^{4} \)
$17$ \( ( 12 + T^{2} )^{2} \)
$19$ \( ( 22 + 10 T + T^{2} )^{2} \)
$23$ \( 324 + 72 T^{2} + T^{4} \)
$29$ \( ( 6 - 6 T + T^{2} )^{2} \)
$31$ \( ( -2 + T )^{4} \)
$37$ \( 4 + 8 T^{2} + T^{4} \)
$41$ \( ( -18 - 6 T + T^{2} )^{2} \)
$43$ \( ( 4 + T^{2} )^{2} \)
$47$ \( ( 48 + T^{2} )^{2} \)
$53$ \( 6084 + 168 T^{2} + T^{4} \)
$59$ \( ( -48 + T^{2} )^{2} \)
$61$ \( ( -44 - 4 T + T^{2} )^{2} \)
$67$ \( ( 16 + T^{2} )^{2} \)
$71$ \( ( 24 - 12 T + T^{2} )^{2} \)
$73$ \( 8464 + 248 T^{2} + T^{4} \)
$79$ \( ( 22 - 14 T + T^{2} )^{2} \)
$83$ \( 5184 + 288 T^{2} + T^{4} \)
$89$ \( ( -12 + T^{2} )^{2} \)
$97$ \( 58564 + 488 T^{2} + T^{4} \)
show more
show less