Properties

Label 3850.2.c.c
Level $3850$
Weight $2$
Character orbit 3850.c
Analytic conductor $30.742$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

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
1.00000i
1.00000i
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
\(n\): 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.c 2
5.b even 2 1 inner 3850.2.c.c 2
5.c odd 4 1 770.2.a.e 1
5.c odd 4 1 3850.2.a.m 1
15.e even 4 1 6930.2.a.bk 1
20.e even 4 1 6160.2.a.a 1
35.f even 4 1 5390.2.a.c 1
55.e even 4 1 8470.2.a.bg 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.a.e 1 5.c odd 4 1
3850.2.a.m 1 5.c odd 4 1
3850.2.c.c 2 1.a even 1 1 trivial
3850.2.c.c 2 5.b even 2 1 inner
5390.2.a.c 1 35.f even 4 1
6160.2.a.a 1 20.e even 4 1
6930.2.a.bk 1 15.e even 4 1
8470.2.a.bg 1 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 \(S_{2}^{\mathrm{new}}(3850, [\chi])\):

\( T_{3}^{2} + 4 \)
\( T_{13}^{2} + 4 \)
\( T_{17}^{2} + 4 \)
\( T_{19} + 6 \)
\( T_{37}^{2} + 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 4 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( ( 6 + T )^{2} \)
$23$ \( 36 + T^{2} \)
$29$ \( ( 4 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( 64 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( 16 + T^{2} \)
$53$ \( 144 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( -2 + T )^{2} \)
$67$ \( 64 + T^{2} \)
$71$ \( ( 12 + T )^{2} \)
$73$ \( 36 + T^{2} \)
$79$ \( ( 10 + T )^{2} \)
$83$ \( 144 + T^{2} \)
$89$ \( ( 14 + T )^{2} \)
$97$ \( 16 + T^{2} \)
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