Properties

Label 2205.4.a.ba
Level $2205$
Weight $4$
Character orbit 2205.a
Self dual yes
Analytic conductor $130.099$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2205.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(130.099211563\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 315)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( -4 + \beta ) q^{4} -5 q^{5} + ( 4 - 11 \beta ) q^{8} +O(q^{10})\) \( q + \beta q^{2} + ( -4 + \beta ) q^{4} -5 q^{5} + ( 4 - 11 \beta ) q^{8} -5 \beta q^{10} + ( -4 + 4 \beta ) q^{11} + 22 \beta q^{13} + ( -12 - 15 \beta ) q^{16} + ( 42 - 26 \beta ) q^{17} + ( -22 + 44 \beta ) q^{19} + ( 20 - 5 \beta ) q^{20} + 16 q^{22} + ( -34 - 14 \beta ) q^{23} + 25 q^{25} + ( 88 + 22 \beta ) q^{26} + ( -152 - 30 \beta ) q^{29} + ( 114 - 18 \beta ) q^{31} + ( -92 + 61 \beta ) q^{32} + ( -104 + 16 \beta ) q^{34} + ( 18 - 30 \beta ) q^{37} + ( 176 + 22 \beta ) q^{38} + ( -20 + 55 \beta ) q^{40} + ( 114 - 52 \beta ) q^{41} + ( -60 + 166 \beta ) q^{43} + ( 32 - 16 \beta ) q^{44} + ( -56 - 48 \beta ) q^{46} + ( 272 - 30 \beta ) q^{47} + 25 \beta q^{50} + ( 88 - 66 \beta ) q^{52} + ( -378 - 52 \beta ) q^{53} + ( 20 - 20 \beta ) q^{55} + ( -120 - 182 \beta ) q^{58} + 284 q^{59} + ( 354 - 90 \beta ) q^{61} + ( -72 + 96 \beta ) q^{62} + ( 340 + 89 \beta ) q^{64} -110 \beta q^{65} + ( 396 - 98 \beta ) q^{67} + ( -272 + 120 \beta ) q^{68} + ( -488 + 162 \beta ) q^{71} + ( 104 - 290 \beta ) q^{73} + ( -120 - 12 \beta ) q^{74} + ( 264 - 154 \beta ) q^{76} + ( 136 + 328 \beta ) q^{79} + ( 60 + 75 \beta ) q^{80} + ( -208 + 62 \beta ) q^{82} + ( -284 + 300 \beta ) q^{83} + ( -210 + 130 \beta ) q^{85} + ( 664 + 106 \beta ) q^{86} + ( -192 + 16 \beta ) q^{88} + ( 202 - 476 \beta ) q^{89} + ( 80 + 8 \beta ) q^{92} + ( -120 + 242 \beta ) q^{94} + ( 110 - 220 \beta ) q^{95} + ( -572 - 482 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 7 q^{4} - 10 q^{5} - 3 q^{8} + O(q^{10}) \) \( 2 q + q^{2} - 7 q^{4} - 10 q^{5} - 3 q^{8} - 5 q^{10} - 4 q^{11} + 22 q^{13} - 39 q^{16} + 58 q^{17} + 35 q^{20} + 32 q^{22} - 82 q^{23} + 50 q^{25} + 198 q^{26} - 334 q^{29} + 210 q^{31} - 123 q^{32} - 192 q^{34} + 6 q^{37} + 374 q^{38} + 15 q^{40} + 176 q^{41} + 46 q^{43} + 48 q^{44} - 160 q^{46} + 514 q^{47} + 25 q^{50} + 110 q^{52} - 808 q^{53} + 20 q^{55} - 422 q^{58} + 568 q^{59} + 618 q^{61} - 48 q^{62} + 769 q^{64} - 110 q^{65} + 694 q^{67} - 424 q^{68} - 814 q^{71} - 82 q^{73} - 252 q^{74} + 374 q^{76} + 600 q^{79} + 195 q^{80} - 354 q^{82} - 268 q^{83} - 290 q^{85} + 1434 q^{86} - 368 q^{88} - 72 q^{89} + 168 q^{92} + 2 q^{94} - 1626 q^{97} + O(q^{100}) \)

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
−1.56155
2.56155
−1.56155 0 −5.56155 −5.00000 0 0 21.1771 0 7.80776
1.2 2.56155 0 −1.43845 −5.00000 0 0 −24.1771 0 −12.8078
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(7\) \(-1\)

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 2205.4.a.ba 2
3.b odd 2 1 2205.4.a.y 2
7.b odd 2 1 315.4.a.j yes 2
21.c even 2 1 315.4.a.h 2
35.c odd 2 1 1575.4.a.r 2
105.g even 2 1 1575.4.a.u 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.4.a.h 2 21.c even 2 1
315.4.a.j yes 2 7.b odd 2 1
1575.4.a.r 2 35.c odd 2 1
1575.4.a.u 2 105.g even 2 1
2205.4.a.y 2 3.b odd 2 1
2205.4.a.ba 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2205))\):

\( T_{2}^{2} - T_{2} - 4 \)
\( T_{11}^{2} + 4 T_{11} - 64 \)
\( T_{13}^{2} - 22 T_{13} - 1936 \)
\( T_{17}^{2} - 58 T_{17} - 2032 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4 - T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 5 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( -64 + 4 T + T^{2} \)
$13$ \( -1936 - 22 T + T^{2} \)
$17$ \( -2032 - 58 T + T^{2} \)
$19$ \( -8228 + T^{2} \)
$23$ \( 848 + 82 T + T^{2} \)
$29$ \( 24064 + 334 T + T^{2} \)
$31$ \( 9648 - 210 T + T^{2} \)
$37$ \( -3816 - 6 T + T^{2} \)
$41$ \( -3748 - 176 T + T^{2} \)
$43$ \( -116584 - 46 T + T^{2} \)
$47$ \( 62224 - 514 T + T^{2} \)
$53$ \( 151724 + 808 T + T^{2} \)
$59$ \( ( -284 + T )^{2} \)
$61$ \( 61056 - 618 T + T^{2} \)
$67$ \( 79592 - 694 T + T^{2} \)
$71$ \( 54112 + 814 T + T^{2} \)
$73$ \( -355744 + 82 T + T^{2} \)
$79$ \( -367232 - 600 T + T^{2} \)
$83$ \( -364544 + 268 T + T^{2} \)
$89$ \( -961652 + 72 T + T^{2} \)
$97$ \( -326408 + 1626 T + T^{2} \)
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