Properties

Label 3850.2.c.y
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: \(\Q(i, \sqrt{33})\)
Defining polynomial: \(x^{4} + 17 x^{2} + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 17 x^{2} + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 9 \nu \)\()/8\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 25 \nu \)\()/8\)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} + 17 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 17\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-9 \beta_{2} + 25 \beta_{1}\)\()/2\)

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
3.37228i
2.37228i
2.37228i
3.37228i
1.00000i 2.00000i −1.00000 0 2.00000 1.00000i 1.00000i −1.00000 0
1849.2 1.00000i 2.00000i −1.00000 0 2.00000 1.00000i 1.00000i −1.00000 0
1849.3 1.00000i 2.00000i −1.00000 0 2.00000 1.00000i 1.00000i −1.00000 0
1849.4 1.00000i 2.00000i −1.00000 0 2.00000 1.00000i 1.00000i −1.00000 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.y 4
5.b even 2 1 inner 3850.2.c.y 4
5.c odd 4 1 770.2.a.k 2
5.c odd 4 1 3850.2.a.bc 2
15.e even 4 1 6930.2.a.bo 2
20.e even 4 1 6160.2.a.r 2
35.f even 4 1 5390.2.a.bq 2
55.e even 4 1 8470.2.a.bu 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.a.k 2 5.c odd 4 1
3850.2.a.bc 2 5.c odd 4 1
3850.2.c.y 4 1.a even 1 1 trivial
3850.2.c.y 4 5.b even 2 1 inner
5390.2.a.bq 2 35.f even 4 1
6160.2.a.r 2 20.e even 4 1
6930.2.a.bo 2 15.e even 4 1
8470.2.a.bu 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 \(S_{2}^{\mathrm{new}}(3850, [\chi])\):

\( T_{3}^{2} + 4 \)
\( T_{13}^{4} + 68 T_{13}^{2} + 1024 \)
\( T_{17}^{4} + 68 T_{17}^{2} + 1024 \)
\( T_{19}^{2} - 2 T_{19} - 32 \)
\( T_{37}^{4} + 116 T_{37}^{2} + 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( ( 4 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( ( 1 + T^{2} )^{2} \)
$11$ \( ( 1 + T )^{4} \)
$13$ \( 1024 + 68 T^{2} + T^{4} \)
$17$ \( 1024 + 68 T^{2} + T^{4} \)
$19$ \( ( -32 - 2 T + T^{2} )^{2} \)
$23$ \( 1024 + 68 T^{2} + T^{4} \)
$29$ \( ( -24 + 6 T + T^{2} )^{2} \)
$31$ \( ( -32 + 2 T + T^{2} )^{2} \)
$37$ \( 64 + 116 T^{2} + T^{4} \)
$41$ \( ( 4 + T )^{4} \)
$43$ \( ( 16 + T^{2} )^{2} \)
$47$ \( 1024 + 68 T^{2} + T^{4} \)
$53$ \( 256 + 164 T^{2} + T^{4} \)
$59$ \( ( -24 - 6 T + T^{2} )^{2} \)
$61$ \( ( 16 - 14 T + T^{2} )^{2} \)
$67$ \( ( 16 + T^{2} )^{2} \)
$71$ \( ( 4 + T )^{4} \)
$73$ \( 64 + 116 T^{2} + T^{4} \)
$79$ \( ( -32 + 2 T + T^{2} )^{2} \)
$83$ \( ( 64 + T^{2} )^{2} \)
$89$ \( ( -116 + 8 T + T^{2} )^{2} \)
$97$ \( 7744 + 308 T^{2} + T^{4} \)
show more
show less