Properties

Label 1225.4.a.m
Level $1225$
Weight $4$
Character orbit 1225.a
Self dual yes
Analytic conductor $72.277$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.2773397570\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + (\beta - 4) q^{2} + (4 \beta + 1) q^{3} + ( - 8 \beta + 10) q^{4} + ( - 15 \beta + 4) q^{6} + (34 \beta - 24) q^{8} + (8 \beta + 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 4) q^{2} + (4 \beta + 1) q^{3} + ( - 8 \beta + 10) q^{4} + ( - 15 \beta + 4) q^{6} + (34 \beta - 24) q^{8} + (8 \beta + 6) q^{9} + (32 \beta - 7) q^{11} + (32 \beta - 54) q^{12} + ( - 4 \beta + 25) q^{13} + ( - 96 \beta + 84) q^{16} + ( - 44 \beta - 25) q^{17} + ( - 26 \beta - 8) q^{18} + (44 \beta - 18) q^{19} + ( - 135 \beta + 92) q^{22} + ( - 68 \beta - 122) q^{23} + ( - 62 \beta + 248) q^{24} + (41 \beta - 108) q^{26} + ( - 76 \beta + 43) q^{27} + ( - 24 \beta - 13) q^{29} + ( - 180 \beta + 60) q^{31} + (196 \beta - 336) q^{32} + (4 \beta + 249) q^{33} + (151 \beta + 12) q^{34} + (32 \beta - 68) q^{36} + ( - 60 \beta - 282) q^{37} + ( - 194 \beta + 160) q^{38} + (96 \beta - 7) q^{39} + (124 \beta + 164) q^{41} + (68 \beta + 130) q^{43} + (376 \beta - 582) q^{44} + (150 \beta + 352) q^{46} + (132 \beta - 175) q^{47} + (240 \beta - 684) q^{48} + ( - 144 \beta - 377) q^{51} + ( - 240 \beta + 314) q^{52} + (128 \beta + 28) q^{53} + (347 \beta - 324) q^{54} + ( - 28 \beta + 334) q^{57} + (83 \beta + 4) q^{58} + 616 q^{59} + ( - 108 \beta - 168) q^{61} + (780 \beta - 600) q^{62} + ( - 352 \beta + 1064) q^{64} + (233 \beta - 988) q^{66} + ( - 64 \beta + 76) q^{67} + ( - 240 \beta + 454) q^{68} + ( - 556 \beta - 666) q^{69} - 952 q^{71} + (12 \beta + 400) q^{72} + (344 \beta + 338) q^{73} + ( - 42 \beta + 1008) q^{74} + (584 \beta - 884) q^{76} + ( - 391 \beta + 220) q^{78} + ( - 248 \beta + 507) q^{79} + ( - 120 \beta - 727) q^{81} + ( - 332 \beta - 408) q^{82} + ( - 600 \beta - 188) q^{83} + ( - 142 \beta - 384) q^{86} + ( - 76 \beta - 205) q^{87} + ( - 1006 \beta + 2344) q^{88} + (44 \beta + 108) q^{89} + (296 \beta - 132) q^{92} + (60 \beta - 1380) q^{93} + ( - 703 \beta + 964) q^{94} + ( - 1148 \beta + 1232) q^{96} + ( - 220 \beta + 1371) q^{97} + (136 \beta + 470) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{2} + 2 q^{3} + 20 q^{4} + 8 q^{6} - 48 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{2} + 2 q^{3} + 20 q^{4} + 8 q^{6} - 48 q^{8} + 12 q^{9} - 14 q^{11} - 108 q^{12} + 50 q^{13} + 168 q^{16} - 50 q^{17} - 16 q^{18} - 36 q^{19} + 184 q^{22} - 244 q^{23} + 496 q^{24} - 216 q^{26} + 86 q^{27} - 26 q^{29} + 120 q^{31} - 672 q^{32} + 498 q^{33} + 24 q^{34} - 136 q^{36} - 564 q^{37} + 320 q^{38} - 14 q^{39} + 328 q^{41} + 260 q^{43} - 1164 q^{44} + 704 q^{46} - 350 q^{47} - 1368 q^{48} - 754 q^{51} + 628 q^{52} + 56 q^{53} - 648 q^{54} + 668 q^{57} + 8 q^{58} + 1232 q^{59} - 336 q^{61} - 1200 q^{62} + 2128 q^{64} - 1976 q^{66} + 152 q^{67} + 908 q^{68} - 1332 q^{69} - 1904 q^{71} + 800 q^{72} + 676 q^{73} + 2016 q^{74} - 1768 q^{76} + 440 q^{78} + 1014 q^{79} - 1454 q^{81} - 816 q^{82} - 376 q^{83} - 768 q^{86} - 410 q^{87} + 4688 q^{88} + 216 q^{89} - 264 q^{92} - 2760 q^{93} + 1928 q^{94} + 2464 q^{96} + 2742 q^{97} + 940 q^{99}+O(q^{100}) \) 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
−1.41421
1.41421
−5.41421 −4.65685 21.3137 0 25.2132 0 −72.0833 −5.31371 0
1.2 −2.58579 6.65685 −1.31371 0 −17.2132 0 24.0833 17.3137 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 1225.4.a.m 2
5.b even 2 1 245.4.a.k 2
7.b odd 2 1 175.4.a.c 2
15.d odd 2 1 2205.4.a.u 2
21.c even 2 1 1575.4.a.z 2
35.c odd 2 1 35.4.a.b 2
35.f even 4 2 175.4.b.c 4
35.i odd 6 2 245.4.e.h 4
35.j even 6 2 245.4.e.i 4
105.g even 2 1 315.4.a.f 2
140.c even 2 1 560.4.a.r 2
280.c odd 2 1 2240.4.a.bn 2
280.n 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 35.c odd 2 1
175.4.a.c 2 7.b odd 2 1
175.4.b.c 4 35.f even 4 2
245.4.a.k 2 5.b even 2 1
245.4.e.h 4 35.i odd 6 2
245.4.e.i 4 35.j even 6 2
315.4.a.f 2 105.g even 2 1
560.4.a.r 2 140.c even 2 1
1225.4.a.m 2 1.a even 1 1 trivial
1575.4.a.z 2 21.c even 2 1
2205.4.a.u 2 15.d odd 2 1
2240.4.a.bn 2 280.c odd 2 1
2240.4.a.bo 2 280.n even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1225))\):

\( T_{2}^{2} + 8T_{2} + 14 \) Copy content Toggle raw display
\( T_{3}^{2} - 2T_{3} - 31 \) Copy content Toggle raw display
\( T_{19}^{2} + 36T_{19} - 3548 \) Copy content Toggle raw display

Hecke characteristic polynomials

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