Properties

Label 1575.4.a.z
Level $1575$
Weight $4$
Character orbit 1575.a
Self dual yes
Analytic conductor $92.928$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( 4 + \beta ) q^{2} + ( 10 + 8 \beta ) q^{4} + 7 q^{7} + ( 24 + 34 \beta ) q^{8} +O(q^{10})\) \( q + ( 4 + \beta ) q^{2} + ( 10 + 8 \beta ) q^{4} + 7 q^{7} + ( 24 + 34 \beta ) q^{8} + ( 7 + 32 \beta ) q^{11} + ( -25 - 4 \beta ) q^{13} + ( 28 + 7 \beta ) q^{14} + ( 84 + 96 \beta ) q^{16} + ( -25 + 44 \beta ) q^{17} + ( 18 + 44 \beta ) q^{19} + ( 92 + 135 \beta ) q^{22} + ( 122 - 68 \beta ) q^{23} + ( -108 - 41 \beta ) q^{26} + ( 70 + 56 \beta ) q^{28} + ( 13 - 24 \beta ) q^{29} + ( -60 - 180 \beta ) q^{31} + ( 336 + 196 \beta ) q^{32} + ( -12 + 151 \beta ) q^{34} + ( -282 + 60 \beta ) q^{37} + ( 160 + 194 \beta ) q^{38} + ( 164 - 124 \beta ) q^{41} + ( 130 - 68 \beta ) q^{43} + ( 582 + 376 \beta ) q^{44} + ( 352 - 150 \beta ) q^{46} + ( -175 - 132 \beta ) q^{47} + 49 q^{49} + ( -314 - 240 \beta ) q^{52} + ( -28 + 128 \beta ) q^{53} + ( 168 + 238 \beta ) q^{56} + ( 4 - 83 \beta ) q^{58} + 616 q^{59} + ( 168 - 108 \beta ) q^{61} + ( -600 - 780 \beta ) q^{62} + ( 1064 + 352 \beta ) q^{64} + ( 76 + 64 \beta ) q^{67} + ( 454 + 240 \beta ) q^{68} + 952 q^{71} + ( -338 + 344 \beta ) q^{73} + ( -1008 - 42 \beta ) q^{74} + ( 884 + 584 \beta ) q^{76} + ( 49 + 224 \beta ) q^{77} + ( 507 + 248 \beta ) q^{79} + ( 408 - 332 \beta ) q^{82} + ( -188 + 600 \beta ) q^{83} + ( 384 - 142 \beta ) q^{86} + ( 2344 + 1006 \beta ) q^{88} + ( 108 - 44 \beta ) q^{89} + ( -175 - 28 \beta ) q^{91} + ( 132 + 296 \beta ) q^{92} + ( -964 - 703 \beta ) q^{94} + ( -1371 - 220 \beta ) q^{97} + ( 196 + 49 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{2} + 20 q^{4} + 14 q^{7} + 48 q^{8} + O(q^{10}) \) \( 2 q + 8 q^{2} + 20 q^{4} + 14 q^{7} + 48 q^{8} + 14 q^{11} - 50 q^{13} + 56 q^{14} + 168 q^{16} - 50 q^{17} + 36 q^{19} + 184 q^{22} + 244 q^{23} - 216 q^{26} + 140 q^{28} + 26 q^{29} - 120 q^{31} + 672 q^{32} - 24 q^{34} - 564 q^{37} + 320 q^{38} + 328 q^{41} + 260 q^{43} + 1164 q^{44} + 704 q^{46} - 350 q^{47} + 98 q^{49} - 628 q^{52} - 56 q^{53} + 336 q^{56} + 8 q^{58} + 1232 q^{59} + 336 q^{61} - 1200 q^{62} + 2128 q^{64} + 152 q^{67} + 908 q^{68} + 1904 q^{71} - 676 q^{73} - 2016 q^{74} + 1768 q^{76} + 98 q^{77} + 1014 q^{79} + 816 q^{82} - 376 q^{83} + 768 q^{86} + 4688 q^{88} + 216 q^{89} - 350 q^{91} + 264 q^{92} - 1928 q^{94} - 2742 q^{97} + 392 q^{98} + O(q^{100}) \)

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
−1.41421
1.41421
2.58579 0 −1.31371 0 0 7.00000 −24.0833 0 0
1.2 5.41421 0 21.3137 0 0 7.00000 72.0833 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.z 2
3.b odd 2 1 175.4.a.c 2
5.b even 2 1 315.4.a.f 2
15.d odd 2 1 35.4.a.b 2
15.e even 4 2 175.4.b.c 4
21.c even 2 1 1225.4.a.m 2
35.c odd 2 1 2205.4.a.u 2
60.h even 2 1 560.4.a.r 2
105.g even 2 1 245.4.a.k 2
105.o odd 6 2 245.4.e.h 4
105.p even 6 2 245.4.e.i 4
120.i odd 2 1 2240.4.a.bn 2
120.m even 2 1 2240.4.a.bo 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.a.b 2 15.d odd 2 1
175.4.a.c 2 3.b odd 2 1
175.4.b.c 4 15.e even 4 2
245.4.a.k 2 105.g even 2 1
245.4.e.h 4 105.o odd 6 2
245.4.e.i 4 105.p even 6 2
315.4.a.f 2 5.b even 2 1
560.4.a.r 2 60.h even 2 1
1225.4.a.m 2 21.c even 2 1
1575.4.a.z 2 1.a even 1 1 trivial
2205.4.a.u 2 35.c odd 2 1
2240.4.a.bn 2 120.i odd 2 1
2240.4.a.bo 2 120.m even 2 1

Hecke kernels

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

\( T_{2}^{2} - 8 T_{2} + 14 \)
\( T_{11}^{2} - 14 T_{11} - 1999 \)
\( T_{13}^{2} + 50 T_{13} + 593 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 14 - 8 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -7 + T )^{2} \)
$11$ \( -1999 - 14 T + T^{2} \)
$13$ \( 593 + 50 T + T^{2} \)
$17$ \( -3247 + 50 T + T^{2} \)
$19$ \( -3548 - 36 T + T^{2} \)
$23$ \( 5636 - 244 T + T^{2} \)
$29$ \( -983 - 26 T + T^{2} \)
$31$ \( -61200 + 120 T + T^{2} \)
$37$ \( 72324 + 564 T + T^{2} \)
$41$ \( -3856 - 328 T + T^{2} \)
$43$ \( 7652 - 260 T + T^{2} \)
$47$ \( -4223 + 350 T + T^{2} \)
$53$ \( -31984 + 56 T + T^{2} \)
$59$ \( ( -616 + T )^{2} \)
$61$ \( 4896 - 336 T + T^{2} \)
$67$ \( -2416 - 152 T + T^{2} \)
$71$ \( ( -952 + T )^{2} \)
$73$ \( -122428 + 676 T + T^{2} \)
$79$ \( 134041 - 1014 T + T^{2} \)
$83$ \( -684656 + 376 T + T^{2} \)
$89$ \( 7792 - 216 T + T^{2} \)
$97$ \( 1782841 + 2742 T + T^{2} \)
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