Properties

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

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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: \(\Q(\sqrt{11}) \)
Defining polynomial: \( x^{2} - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 245)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{11}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{2} - 5 q^{3} + ( - 2 \beta + 4) q^{4} + ( - 5 \beta + 5) q^{6} + ( - 2 \beta - 18) q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 1) q^{2} - 5 q^{3} + ( - 2 \beta + 4) q^{4} + ( - 5 \beta + 5) q^{6} + ( - 2 \beta - 18) q^{8} - 2 q^{9} + (4 \beta + 33) q^{11} + (10 \beta - 20) q^{12} + (20 \beta - 5) q^{13} - 36 q^{16} + ( - 20 \beta - 35) q^{17} + ( - 2 \beta + 2) q^{18} + ( - 20 \beta + 70) q^{19} + (29 \beta + 11) q^{22} + ( - 28 \beta + 8) q^{23} + (10 \beta + 90) q^{24} + ( - 25 \beta + 225) q^{26} + 145 q^{27} + (48 \beta - 129) q^{29} + (60 \beta - 10) q^{31} + ( - 20 \beta + 180) q^{32} + ( - 20 \beta - 165) q^{33} + ( - 15 \beta - 185) q^{34} + (4 \beta - 8) q^{36} + (44 \beta - 164) q^{37} + (90 \beta - 290) q^{38} + ( - 100 \beta + 25) q^{39} + (100 \beta + 150) q^{41} + (12 \beta + 58) q^{43} + ( - 50 \beta + 44) q^{44} + (36 \beta - 316) q^{46} + ( - 40 \beta + 15) q^{47} + 180 q^{48} + (100 \beta + 175) q^{51} + (90 \beta - 460) q^{52} + ( - 120 \beta - 270) q^{53} + (145 \beta - 145) q^{54} + (100 \beta - 350) q^{57} + ( - 177 \beta + 657) q^{58} + ( - 40 \beta + 190) q^{59} + (60 \beta + 540) q^{61} + ( - 70 \beta + 670) q^{62} + (200 \beta - 112) q^{64} + ( - 145 \beta - 55) q^{66} + ( - 96 \beta - 234) q^{67} + ( - 10 \beta + 300) q^{68} + (140 \beta - 40) q^{69} + ( - 64 \beta - 528) q^{71} + (4 \beta + 36) q^{72} + ( - 200 \beta + 430) q^{73} + ( - 208 \beta + 648) q^{74} + ( - 220 \beta + 720) q^{76} + (125 \beta - 1125) q^{78} + ( - 348 \beta + 79) q^{79} - 671 q^{81} + (50 \beta + 950) q^{82} + ( - 200 \beta - 20) q^{83} + (46 \beta + 74) q^{86} + ( - 240 \beta + 645) q^{87} + ( - 138 \beta - 682) q^{88} + ( - 380 \beta - 120) q^{89} + ( - 128 \beta + 648) q^{92} + ( - 300 \beta + 50) q^{93} + (55 \beta - 455) q^{94} + (100 \beta - 900) q^{96} + (180 \beta - 815) q^{97} + ( - 8 \beta - 66) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 10 q^{3} + 8 q^{4} + 10 q^{6} - 36 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 10 q^{3} + 8 q^{4} + 10 q^{6} - 36 q^{8} - 4 q^{9} + 66 q^{11} - 40 q^{12} - 10 q^{13} - 72 q^{16} - 70 q^{17} + 4 q^{18} + 140 q^{19} + 22 q^{22} + 16 q^{23} + 180 q^{24} + 450 q^{26} + 290 q^{27} - 258 q^{29} - 20 q^{31} + 360 q^{32} - 330 q^{33} - 370 q^{34} - 16 q^{36} - 328 q^{37} - 580 q^{38} + 50 q^{39} + 300 q^{41} + 116 q^{43} + 88 q^{44} - 632 q^{46} + 30 q^{47} + 360 q^{48} + 350 q^{51} - 920 q^{52} - 540 q^{53} - 290 q^{54} - 700 q^{57} + 1314 q^{58} + 380 q^{59} + 1080 q^{61} + 1340 q^{62} - 224 q^{64} - 110 q^{66} - 468 q^{67} + 600 q^{68} - 80 q^{69} - 1056 q^{71} + 72 q^{72} + 860 q^{73} + 1296 q^{74} + 1440 q^{76} - 2250 q^{78} + 158 q^{79} - 1342 q^{81} + 1900 q^{82} - 40 q^{83} + 148 q^{86} + 1290 q^{87} - 1364 q^{88} - 240 q^{89} + 1296 q^{92} + 100 q^{93} - 910 q^{94} - 1800 q^{96} - 1630 q^{97} - 132 q^{99}+O(q^{100}) \) 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
−3.31662
3.31662
−4.31662 −5.00000 10.6332 0 21.5831 0 −11.3668 −2.00000 0
1.2 2.31662 −5.00000 −2.63325 0 −11.5831 0 −24.6332 −2.00000 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.p 2
5.b even 2 1 245.4.a.j yes 2
7.b odd 2 1 1225.4.a.q 2
15.d odd 2 1 2205.4.a.w 2
35.c odd 2 1 245.4.a.i 2
35.i odd 6 2 245.4.e.k 4
35.j even 6 2 245.4.e.j 4
105.g even 2 1 2205.4.a.x 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.4.a.i 2 35.c odd 2 1
245.4.a.j yes 2 5.b even 2 1
245.4.e.j 4 35.j even 6 2
245.4.e.k 4 35.i odd 6 2
1225.4.a.p 2 1.a even 1 1 trivial
1225.4.a.q 2 7.b odd 2 1
2205.4.a.w 2 15.d odd 2 1
2205.4.a.x 2 105.g even 2 1

Hecke kernels

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

\( T_{2}^{2} + 2T_{2} - 10 \) Copy content Toggle raw display
\( T_{3} + 5 \) Copy content Toggle raw display
\( T_{19}^{2} - 140T_{19} + 500 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T - 10 \) Copy content Toggle raw display
$3$ \( (T + 5)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 66T + 913 \) Copy content Toggle raw display
$13$ \( T^{2} + 10T - 4375 \) Copy content Toggle raw display
$17$ \( T^{2} + 70T - 3175 \) Copy content Toggle raw display
$19$ \( T^{2} - 140T + 500 \) Copy content Toggle raw display
$23$ \( T^{2} - 16T - 8560 \) Copy content Toggle raw display
$29$ \( T^{2} + 258T - 8703 \) Copy content Toggle raw display
$31$ \( T^{2} + 20T - 39500 \) Copy content Toggle raw display
$37$ \( T^{2} + 328T + 5600 \) Copy content Toggle raw display
$41$ \( T^{2} - 300T - 87500 \) Copy content Toggle raw display
$43$ \( T^{2} - 116T + 1780 \) Copy content Toggle raw display
$47$ \( T^{2} - 30T - 17375 \) Copy content Toggle raw display
$53$ \( T^{2} + 540T - 85500 \) Copy content Toggle raw display
$59$ \( T^{2} - 380T + 18500 \) Copy content Toggle raw display
$61$ \( T^{2} - 1080 T + 252000 \) Copy content Toggle raw display
$67$ \( T^{2} + 468T - 46620 \) Copy content Toggle raw display
$71$ \( T^{2} + 1056 T + 233728 \) Copy content Toggle raw display
$73$ \( T^{2} - 860T - 255100 \) Copy content Toggle raw display
$79$ \( T^{2} - 158 T - 1325903 \) Copy content Toggle raw display
$83$ \( T^{2} + 40T - 439600 \) Copy content Toggle raw display
$89$ \( T^{2} + 240 T - 1574000 \) Copy content Toggle raw display
$97$ \( T^{2} + 1630 T + 307825 \) Copy content Toggle raw display
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