Properties

Label 2205.4.a.y
Level $2205$
Weight $4$
Character orbit 2205.a
Self dual yes
Analytic conductor $130.099$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - \beta q^{2} + (\beta - 4) q^{4} + 5 q^{5} + (11 \beta - 4) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + (\beta - 4) q^{4} + 5 q^{5} + (11 \beta - 4) q^{8} - 5 \beta q^{10} + ( - 4 \beta + 4) q^{11} + 22 \beta q^{13} + ( - 15 \beta - 12) q^{16} + (26 \beta - 42) q^{17} + (44 \beta - 22) q^{19} + (5 \beta - 20) q^{20} + 16 q^{22} + (14 \beta + 34) q^{23} + 25 q^{25} + ( - 22 \beta - 88) q^{26} + (30 \beta + 152) q^{29} + ( - 18 \beta + 114) q^{31} + ( - 61 \beta + 92) q^{32} + (16 \beta - 104) q^{34} + ( - 30 \beta + 18) q^{37} + ( - 22 \beta - 176) q^{38} + (55 \beta - 20) q^{40} + (52 \beta - 114) q^{41} + (166 \beta - 60) q^{43} + (16 \beta - 32) q^{44} + ( - 48 \beta - 56) q^{46} + (30 \beta - 272) q^{47} - 25 \beta q^{50} + ( - 66 \beta + 88) q^{52} + (52 \beta + 378) q^{53} + ( - 20 \beta + 20) q^{55} + ( - 182 \beta - 120) q^{58} - 284 q^{59} + ( - 90 \beta + 354) q^{61} + ( - 96 \beta + 72) q^{62} + (89 \beta + 340) q^{64} + 110 \beta q^{65} + ( - 98 \beta + 396) q^{67} + ( - 120 \beta + 272) q^{68} + ( - 162 \beta + 488) q^{71} + ( - 290 \beta + 104) q^{73} + (12 \beta + 120) q^{74} + ( - 154 \beta + 264) q^{76} + (328 \beta + 136) q^{79} + ( - 75 \beta - 60) q^{80} + (62 \beta - 208) q^{82} + ( - 300 \beta + 284) q^{83} + (130 \beta - 210) q^{85} + ( - 106 \beta - 664) q^{86} + (16 \beta - 192) q^{88} + (476 \beta - 202) q^{89} + ( - 8 \beta - 80) q^{92} + (242 \beta - 120) q^{94} + (220 \beta - 110) q^{95} + ( - 482 \beta - 572) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 7 q^{4} + 10 q^{5} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - 7 q^{4} + 10 q^{5} + 3 q^{8} - 5 q^{10} + 4 q^{11} + 22 q^{13} - 39 q^{16} - 58 q^{17} - 35 q^{20} + 32 q^{22} + 82 q^{23} + 50 q^{25} - 198 q^{26} + 334 q^{29} + 210 q^{31} + 123 q^{32} - 192 q^{34} + 6 q^{37} - 374 q^{38} + 15 q^{40} - 176 q^{41} + 46 q^{43} - 48 q^{44} - 160 q^{46} - 514 q^{47} - 25 q^{50} + 110 q^{52} + 808 q^{53} + 20 q^{55} - 422 q^{58} - 568 q^{59} + 618 q^{61} + 48 q^{62} + 769 q^{64} + 110 q^{65} + 694 q^{67} + 424 q^{68} + 814 q^{71} - 82 q^{73} + 252 q^{74} + 374 q^{76} + 600 q^{79} - 195 q^{80} - 354 q^{82} + 268 q^{83} - 290 q^{85} - 1434 q^{86} - 368 q^{88} + 72 q^{89} - 168 q^{92} + 2 q^{94} - 1626 q^{97}+O(q^{100}) \) 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
2.56155
−1.56155
−2.56155 0 −1.43845 5.00000 0 0 24.1771 0 −12.8078
1.2 1.56155 0 −5.56155 5.00000 0 0 −21.1771 0 7.80776
\(n\): 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.y 2
3.b odd 2 1 2205.4.a.ba 2
7.b odd 2 1 315.4.a.h 2
21.c even 2 1 315.4.a.j yes 2
35.c odd 2 1 1575.4.a.u 2
105.g even 2 1 1575.4.a.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.4.a.h 2 7.b odd 2 1
315.4.a.j yes 2 21.c even 2 1
1575.4.a.r 2 105.g even 2 1
1575.4.a.u 2 35.c odd 2 1
2205.4.a.y 2 1.a even 1 1 trivial
2205.4.a.ba 2 3.b odd 2 1

Hecke kernels

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

\( T_{2}^{2} + T_{2} - 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 4T_{11} - 64 \) Copy content Toggle raw display
\( T_{13}^{2} - 22T_{13} - 1936 \) Copy content Toggle raw display
\( T_{17}^{2} + 58T_{17} - 2032 \) Copy content Toggle raw display

Hecke characteristic polynomials

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