Properties

Label 3850.2.c.q
Level $3850$
Weight $2$
Character orbit 3850.c
Analytic conductor $30.742$
Analytic rank $0$
Dimension $4$
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: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 154)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} - \beta_1) q^{3} - q^{4} + (\beta_{3} - 1) q^{6} - \beta_1 q^{7} + \beta_1 q^{8} + (2 \beta_{3} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} - \beta_1) q^{3} - q^{4} + (\beta_{3} - 1) q^{6} - \beta_1 q^{7} + \beta_1 q^{8} + (2 \beta_{3} - 3) q^{9} + q^{11} + ( - \beta_{2} + \beta_1) q^{12} + ( - \beta_{2} - \beta_1) q^{13} - q^{14} + q^{16} + ( - 2 \beta_{2} + 2 \beta_1) q^{17} + ( - 2 \beta_{2} + 3 \beta_1) q^{18} + (\beta_{3} + 5) q^{19} + (\beta_{3} - 1) q^{21} - \beta_1 q^{22} + 4 \beta_1 q^{23} + ( - \beta_{3} + 1) q^{24} + ( - \beta_{3} - 1) q^{26} + ( - 2 \beta_{2} + 10 \beta_1) q^{27} + \beta_1 q^{28} - 2 \beta_{3} q^{29} + 2 q^{31} - \beta_1 q^{32} + (\beta_{2} - \beta_1) q^{33} + ( - 2 \beta_{3} + 2) q^{34} + ( - 2 \beta_{3} + 3) q^{36} + ( - 4 \beta_{2} + 2 \beta_1) q^{37} + ( - \beta_{2} - 5 \beta_1) q^{38} + 4 q^{39} + ( - 2 \beta_{3} + 2) q^{41} + ( - \beta_{2} + \beta_1) q^{42} + ( - 2 \beta_{2} - 6 \beta_1) q^{43} - q^{44} + 4 q^{46} + 2 \beta_1 q^{47} + (\beta_{2} - \beta_1) q^{48} - q^{49} + ( - 4 \beta_{3} + 12) q^{51} + (\beta_{2} + \beta_1) q^{52} + (2 \beta_{2} + 4 \beta_1) q^{53} + ( - 2 \beta_{3} + 10) q^{54} + q^{56} + 4 \beta_{2} q^{57} + 2 \beta_{2} q^{58} + (\beta_{3} - 5) q^{59} + (\beta_{3} - 3) q^{61} - 2 \beta_1 q^{62} + ( - 2 \beta_{2} + 3 \beta_1) q^{63} - q^{64} + (\beta_{3} - 1) q^{66} + ( - 6 \beta_{2} + 2 \beta_1) q^{67} + (2 \beta_{2} - 2 \beta_1) q^{68} + ( - 4 \beta_{3} + 4) q^{69} + (2 \beta_{3} + 2) q^{71} + (2 \beta_{2} - 3 \beta_1) q^{72} + (4 \beta_{2} + 4 \beta_1) q^{73} + ( - 4 \beta_{3} + 2) q^{74} + ( - \beta_{3} - 5) q^{76} - \beta_1 q^{77} - 4 \beta_1 q^{78} + ( - 6 \beta_{3} + 11) q^{81} + (2 \beta_{2} - 2 \beta_1) q^{82} + ( - 5 \beta_{2} - \beta_1) q^{83} + ( - \beta_{3} + 1) q^{84} + ( - 2 \beta_{3} - 6) q^{86} + (2 \beta_{2} - 10 \beta_1) q^{87} + \beta_1 q^{88} - 10 q^{89} + ( - \beta_{3} - 1) q^{91} - 4 \beta_1 q^{92} + (2 \beta_{2} - 2 \beta_1) q^{93} + 2 q^{94} + (\beta_{3} - 1) q^{96} + ( - 2 \beta_{2} - 8 \beta_1) q^{97} + \beta_1 q^{98} + (2 \beta_{3} - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{6} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 4 q^{6} - 12 q^{9} + 4 q^{11} - 4 q^{14} + 4 q^{16} + 20 q^{19} - 4 q^{21} + 4 q^{24} - 4 q^{26} + 8 q^{31} + 8 q^{34} + 12 q^{36} + 16 q^{39} + 8 q^{41} - 4 q^{44} + 16 q^{46} - 4 q^{49} + 48 q^{51} + 40 q^{54} + 4 q^{56} - 20 q^{59} - 12 q^{61} - 4 q^{64} - 4 q^{66} + 16 q^{69} + 8 q^{71} + 8 q^{74} - 20 q^{76} + 44 q^{81} + 4 q^{84} - 24 q^{86} - 40 q^{89} - 4 q^{91} + 8 q^{94} - 4 q^{96} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{2} + 2\beta_1 \) Copy content Toggle raw display

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
1.61803i
0.618034i
0.618034i
1.61803i
1.00000i 3.23607i −1.00000 0 −3.23607 1.00000i 1.00000i −7.47214 0
1849.2 1.00000i 1.23607i −1.00000 0 1.23607 1.00000i 1.00000i 1.47214 0
1849.3 1.00000i 1.23607i −1.00000 0 1.23607 1.00000i 1.00000i 1.47214 0
1849.4 1.00000i 3.23607i −1.00000 0 −3.23607 1.00000i 1.00000i −7.47214 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.q 4
5.b even 2 1 inner 3850.2.c.q 4
5.c odd 4 1 154.2.a.d 2
5.c odd 4 1 3850.2.a.bj 2
15.e even 4 1 1386.2.a.m 2
20.e even 4 1 1232.2.a.p 2
35.f even 4 1 1078.2.a.w 2
35.k even 12 2 1078.2.e.n 4
35.l odd 12 2 1078.2.e.q 4
40.i odd 4 1 4928.2.a.bt 2
40.k even 4 1 4928.2.a.bk 2
55.e even 4 1 1694.2.a.l 2
105.k odd 4 1 9702.2.a.cu 2
140.j odd 4 1 8624.2.a.bf 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.d 2 5.c odd 4 1
1078.2.a.w 2 35.f even 4 1
1078.2.e.n 4 35.k even 12 2
1078.2.e.q 4 35.l odd 12 2
1232.2.a.p 2 20.e even 4 1
1386.2.a.m 2 15.e even 4 1
1694.2.a.l 2 55.e even 4 1
3850.2.a.bj 2 5.c odd 4 1
3850.2.c.q 4 1.a even 1 1 trivial
3850.2.c.q 4 5.b even 2 1 inner
4928.2.a.bk 2 40.k even 4 1
4928.2.a.bt 2 40.i odd 4 1
8624.2.a.bf 2 140.j odd 4 1
9702.2.a.cu 2 105.k odd 4 1

Hecke kernels

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

\( T_{3}^{4} + 12T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{13}^{4} + 12T_{13}^{2} + 16 \) Copy content Toggle raw display
\( T_{17}^{4} + 48T_{17}^{2} + 256 \) Copy content Toggle raw display
\( T_{19}^{2} - 10T_{19} + 20 \) Copy content Toggle raw display
\( T_{37}^{4} + 168T_{37}^{2} + 5776 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T - 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{4} + 48T^{2} + 256 \) Copy content Toggle raw display
$19$ \( (T^{2} - 10 T + 20)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$31$ \( (T - 2)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 168T^{2} + 5776 \) Copy content Toggle raw display
$41$ \( (T^{2} - 4 T - 16)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 112T^{2} + 256 \) Copy content Toggle raw display
$47$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 72T^{2} + 16 \) Copy content Toggle raw display
$59$ \( (T^{2} + 10 T + 20)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 6 T + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 368 T^{2} + 30976 \) Copy content Toggle raw display
$71$ \( (T^{2} - 4 T - 16)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 192T^{2} + 4096 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 252 T^{2} + 15376 \) Copy content Toggle raw display
$89$ \( (T + 10)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 168T^{2} + 1936 \) Copy content Toggle raw display
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