Properties

Label 3850.2.a.bj
Level $3850$
Weight $2$
Character orbit 3850.a
Self dual yes
Analytic conductor $30.742$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.7424047782\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 154)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.618034
1.61803
−1.00000 −1.23607 1.00000 0 1.23607 −1.00000 −1.00000 −1.47214 0
1.2 −1.00000 3.23607 1.00000 0 −3.23607 −1.00000 −1.00000 7.47214 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(7\) \(1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3850.2.a.bj 2
5.b even 2 1 154.2.a.d 2
5.c odd 4 2 3850.2.c.q 4
15.d odd 2 1 1386.2.a.m 2
20.d odd 2 1 1232.2.a.p 2
35.c odd 2 1 1078.2.a.w 2
35.i odd 6 2 1078.2.e.n 4
35.j even 6 2 1078.2.e.q 4
40.e odd 2 1 4928.2.a.bk 2
40.f even 2 1 4928.2.a.bt 2
55.d odd 2 1 1694.2.a.l 2
105.g even 2 1 9702.2.a.cu 2
140.c even 2 1 8624.2.a.bf 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.d 2 5.b even 2 1
1078.2.a.w 2 35.c odd 2 1
1078.2.e.n 4 35.i odd 6 2
1078.2.e.q 4 35.j even 6 2
1232.2.a.p 2 20.d odd 2 1
1386.2.a.m 2 15.d odd 2 1
1694.2.a.l 2 55.d odd 2 1
3850.2.a.bj 2 1.a even 1 1 trivial
3850.2.c.q 4 5.c odd 4 2
4928.2.a.bk 2 40.e odd 2 1
4928.2.a.bt 2 40.f even 2 1
8624.2.a.bf 2 140.c even 2 1
9702.2.a.cu 2 105.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3850))\):

\( T_{3}^{2} - 2T_{3} - 4 \) Copy content Toggle raw display
\( T_{13}^{2} - 2T_{13} - 4 \) Copy content Toggle raw display
\( T_{17}^{2} - 4T_{17} - 16 \) Copy content Toggle raw display
\( T_{19}^{2} + 10T_{19} + 20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$17$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$19$ \( T^{2} + 10T + 20 \) Copy content Toggle raw display
$23$ \( (T + 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 20 \) Copy content Toggle raw display
$31$ \( (T - 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 4T - 76 \) Copy content Toggle raw display
$41$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$43$ \( T^{2} - 12T + 16 \) Copy content Toggle raw display
$47$ \( (T - 2)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
$59$ \( T^{2} - 10T + 20 \) Copy content Toggle raw display
$61$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T - 176 \) Copy content Toggle raw display
$71$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$73$ \( T^{2} + 8T - 64 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 2T - 124 \) Copy content Toggle raw display
$89$ \( (T - 10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 16T + 44 \) Copy content Toggle raw display
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