Properties

Label 1575.4.a.r
Level $1575$
Weight $4$
Character orbit 1575.a
Self dual yes
Analytic conductor $92.928$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1575.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(92.9280082590\)
Analytic rank: \(1\)
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.

\(f(q)\) \(=\) \( q - \beta q^{2} + (\beta - 4) q^{4} + 7 q^{7} + (11 \beta - 4) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + (\beta - 4) q^{4} + 7 q^{7} + (11 \beta - 4) q^{8} + (4 \beta - 4) q^{11} + 22 \beta q^{13} - 7 \beta q^{14} + ( - 15 \beta - 12) q^{16} + ( - 26 \beta + 42) q^{17} + ( - 44 \beta + 22) q^{19} - 16 q^{22} + (14 \beta + 34) q^{23} + ( - 22 \beta - 88) q^{26} + (7 \beta - 28) q^{28} + ( - 30 \beta - 152) q^{29} + (18 \beta - 114) q^{31} + ( - 61 \beta + 92) q^{32} + ( - 16 \beta + 104) q^{34} + (30 \beta - 18) q^{37} + (22 \beta + 176) q^{38} + (52 \beta - 114) q^{41} + ( - 166 \beta + 60) q^{43} + ( - 16 \beta + 32) q^{44} + ( - 48 \beta - 56) q^{46} + ( - 30 \beta + 272) q^{47} + 49 q^{49} + ( - 66 \beta + 88) q^{52} + (52 \beta + 378) q^{53} + (77 \beta - 28) q^{56} + (182 \beta + 120) q^{58} - 284 q^{59} + (90 \beta - 354) q^{61} + (96 \beta - 72) q^{62} + (89 \beta + 340) q^{64} + (98 \beta - 396) q^{67} + (120 \beta - 272) q^{68} + (162 \beta - 488) q^{71} + ( - 290 \beta + 104) q^{73} + ( - 12 \beta - 120) q^{74} + (154 \beta - 264) q^{76} + (28 \beta - 28) q^{77} + (328 \beta + 136) q^{79} + (62 \beta - 208) q^{82} + (300 \beta - 284) q^{83} + (106 \beta + 664) q^{86} + ( - 16 \beta + 192) q^{88} + (476 \beta - 202) q^{89} + 154 \beta q^{91} + ( - 8 \beta - 80) q^{92} + ( - 242 \beta + 120) q^{94} + ( - 482 \beta - 572) q^{97} - 49 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 7 q^{4} + 14 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - 7 q^{4} + 14 q^{7} + 3 q^{8} - 4 q^{11} + 22 q^{13} - 7 q^{14} - 39 q^{16} + 58 q^{17} - 32 q^{22} + 82 q^{23} - 198 q^{26} - 49 q^{28} - 334 q^{29} - 210 q^{31} + 123 q^{32} + 192 q^{34} - 6 q^{37} + 374 q^{38} - 176 q^{41} - 46 q^{43} + 48 q^{44} - 160 q^{46} + 514 q^{47} + 98 q^{49} + 110 q^{52} + 808 q^{53} + 21 q^{56} + 422 q^{58} - 568 q^{59} - 618 q^{61} - 48 q^{62} + 769 q^{64} - 694 q^{67} - 424 q^{68} - 814 q^{71} - 82 q^{73} - 252 q^{74} - 374 q^{76} - 28 q^{77} + 600 q^{79} - 354 q^{82} - 268 q^{83} + 1434 q^{86} + 368 q^{88} + 72 q^{89} + 154 q^{91} - 168 q^{92} - 2 q^{94} - 1626 q^{97} - 49 q^{98}+O(q^{100}) \) 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 0 0 7.00000 24.1771 0 0
1.2 1.56155 0 −5.56155 0 0 7.00000 −21.1771 0 0
\(n\): 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 1575.4.a.r 2
3.b odd 2 1 1575.4.a.u 2
5.b even 2 1 315.4.a.j yes 2
15.d odd 2 1 315.4.a.h 2
35.c odd 2 1 2205.4.a.ba 2
105.g even 2 1 2205.4.a.y 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.4.a.h 2 15.d odd 2 1
315.4.a.j yes 2 5.b even 2 1
1575.4.a.r 2 1.a even 1 1 trivial
1575.4.a.u 2 3.b odd 2 1
2205.4.a.y 2 105.g even 2 1
2205.4.a.ba 2 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1575))\):

\( T_{2}^{2} + T_{2} - 4 \) 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

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
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 7)^{2} \) 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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