Properties

Label 245.4.a.i
Level $245$
Weight $4$
Character orbit 245.a
Self dual yes
Analytic conductor $14.455$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4554679514\)
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: yes
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.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{2} - 5 q^{3} + (2 \beta + 4) q^{4} - 5 q^{5} + ( - 5 \beta - 5) q^{6} + ( - 2 \beta + 18) q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{2} - 5 q^{3} + (2 \beta + 4) q^{4} - 5 q^{5} + ( - 5 \beta - 5) q^{6} + ( - 2 \beta + 18) q^{8} - 2 q^{9} + ( - 5 \beta - 5) q^{10} + ( - 4 \beta + 33) q^{11} + ( - 10 \beta - 20) q^{12} + ( - 20 \beta - 5) q^{13} + 25 q^{15} - 36 q^{16} + (20 \beta - 35) q^{17} + ( - 2 \beta - 2) q^{18} + ( - 20 \beta - 70) q^{19} + ( - 10 \beta - 20) q^{20} + (29 \beta - 11) q^{22} + ( - 28 \beta - 8) q^{23} + (10 \beta - 90) q^{24} + 25 q^{25} + ( - 25 \beta - 225) q^{26} + 145 q^{27} + ( - 48 \beta - 129) q^{29} + (25 \beta + 25) q^{30} + (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} + (10 \beta - 90) q^{40} + (100 \beta - 150) q^{41} + (12 \beta - 58) q^{43} + (50 \beta + 44) q^{44} + 10 q^{45} + ( - 36 \beta - 316) q^{46} + (40 \beta + 15) q^{47} + 180 q^{48} + (25 \beta + 25) q^{50} + ( - 100 \beta + 175) q^{51} + ( - 90 \beta - 460) q^{52} + ( - 120 \beta + 270) q^{53} + (145 \beta + 145) q^{54} + (20 \beta - 165) q^{55} + (100 \beta + 350) q^{57} + ( - 177 \beta - 657) q^{58} + ( - 40 \beta - 190) q^{59} + (50 \beta + 100) q^{60} + (60 \beta - 540) q^{61} + (70 \beta + 670) q^{62} + ( - 200 \beta - 112) q^{64} + (100 \beta + 25) q^{65} + ( - 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} - 125 q^{75} + ( - 220 \beta - 720) q^{76} + (125 \beta + 1125) q^{78} + (348 \beta + 79) q^{79} + 180 q^{80} - 671 q^{81} + ( - 50 \beta + 950) q^{82} + (200 \beta - 20) q^{83} + ( - 100 \beta + 175) q^{85} + ( - 46 \beta + 74) q^{86} + (240 \beta + 645) q^{87} + ( - 138 \beta + 682) q^{88} + ( - 380 \beta + 120) q^{89} + (10 \beta + 10) q^{90} + ( - 128 \beta - 648) q^{92} + ( - 300 \beta - 50) q^{93} + (55 \beta + 455) q^{94} + (100 \beta + 350) q^{95} + (100 \beta + 900) q^{96} + ( - 180 \beta - 815) q^{97} + (8 \beta - 66) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 10 q^{3} + 8 q^{4} - 10 q^{5} - 10 q^{6} + 36 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 10 q^{3} + 8 q^{4} - 10 q^{5} - 10 q^{6} + 36 q^{8} - 4 q^{9} - 10 q^{10} + 66 q^{11} - 40 q^{12} - 10 q^{13} + 50 q^{15} - 72 q^{16} - 70 q^{17} - 4 q^{18} - 140 q^{19} - 40 q^{20} - 22 q^{22} - 16 q^{23} - 180 q^{24} + 50 q^{25} - 450 q^{26} + 290 q^{27} - 258 q^{29} + 50 q^{30} + 20 q^{31} - 360 q^{32} - 330 q^{33} + 370 q^{34} - 16 q^{36} + 328 q^{37} - 580 q^{38} + 50 q^{39} - 180 q^{40} - 300 q^{41} - 116 q^{43} + 88 q^{44} + 20 q^{45} - 632 q^{46} + 30 q^{47} + 360 q^{48} + 50 q^{50} + 350 q^{51} - 920 q^{52} + 540 q^{53} + 290 q^{54} - 330 q^{55} + 700 q^{57} - 1314 q^{58} - 380 q^{59} + 200 q^{60} - 1080 q^{61} + 1340 q^{62} - 224 q^{64} + 50 q^{65} + 110 q^{66} + 468 q^{67} + 600 q^{68} + 80 q^{69} - 1056 q^{71} - 72 q^{72} + 860 q^{73} + 1296 q^{74} - 250 q^{75} - 1440 q^{76} + 2250 q^{78} + 158 q^{79} + 360 q^{80} - 1342 q^{81} + 1900 q^{82} - 40 q^{83} + 350 q^{85} + 148 q^{86} + 1290 q^{87} + 1364 q^{88} + 240 q^{89} + 20 q^{90} - 1296 q^{92} - 100 q^{93} + 910 q^{94} + 700 q^{95} + 1800 q^{96} - 1630 q^{97} - 132 q^{99}+O(q^{100}) \) 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
−2.31662 −5.00000 −2.63325 −5.00000 11.5831 0 24.6332 −2.00000 11.5831
1.2 4.31662 −5.00000 10.6332 −5.00000 −21.5831 0 11.3668 −2.00000 −21.5831
\(n\): 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 245.4.a.i 2
3.b odd 2 1 2205.4.a.x 2
5.b even 2 1 1225.4.a.q 2
7.b odd 2 1 245.4.a.j yes 2
7.c even 3 2 245.4.e.k 4
7.d odd 6 2 245.4.e.j 4
21.c even 2 1 2205.4.a.w 2
35.c odd 2 1 1225.4.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.4.a.i 2 1.a even 1 1 trivial
245.4.a.j yes 2 7.b odd 2 1
245.4.e.j 4 7.d odd 6 2
245.4.e.k 4 7.c even 3 2
1225.4.a.p 2 35.c odd 2 1
1225.4.a.q 2 5.b even 2 1
2205.4.a.w 2 21.c even 2 1
2205.4.a.x 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(245))\):

\( T_{2}^{2} - 2T_{2} - 10 \) Copy content Toggle raw display
\( T_{3} + 5 \) 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
$5$ \( (T + 5)^{2} \) 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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