Properties

Label 414.4.a.j
Level $414$
Weight $4$
Character orbit 414.a
Self dual yes
Analytic conductor $24.427$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 414.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(24.4267907424\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
Defining polynomial: \( x^{2} - x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 46)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + ( - 2 \beta - 4) q^{5} + (2 \beta + 2) q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 4 q^{4} + ( - 2 \beta - 4) q^{5} + (2 \beta + 2) q^{7} + 8 q^{8} + ( - 4 \beta - 8) q^{10} + (8 \beta - 40) q^{11} + (7 \beta - 59) q^{13} + (4 \beta + 4) q^{14} + 16 q^{16} + ( - 24 \beta - 50) q^{17} + (18 \beta + 2) q^{19} + ( - 8 \beta - 16) q^{20} + (16 \beta - 80) q^{22} + 23 q^{23} + (20 \beta - 69) q^{25} + (14 \beta - 118) q^{26} + (8 \beta + 8) q^{28} + ( - 65 \beta + 25) q^{29} + (17 \beta + 25) q^{31} + 32 q^{32} + ( - 48 \beta - 100) q^{34} + ( - 16 \beta - 48) q^{35} + ( - 70 \beta + 44) q^{37} + (36 \beta + 4) q^{38} + ( - 16 \beta - 32) q^{40} + (21 \beta - 253) q^{41} + (56 \beta - 248) q^{43} + (32 \beta - 160) q^{44} + 46 q^{46} + ( - 93 \beta - 61) q^{47} + (12 \beta - 299) q^{49} + (40 \beta - 138) q^{50} + (28 \beta - 236) q^{52} + (124 \beta - 182) q^{53} + 32 \beta q^{55} + (16 \beta + 16) q^{56} + ( - 130 \beta + 50) q^{58} + (32 \beta - 412) q^{59} + ( - 212 \beta + 334) q^{61} + (34 \beta + 50) q^{62} + 64 q^{64} + (76 \beta + 96) q^{65} + (48 \beta + 96) q^{67} + ( - 96 \beta - 200) q^{68} + ( - 32 \beta - 96) q^{70} + (91 \beta + 307) q^{71} + (29 \beta - 1) q^{73} + ( - 140 \beta + 88) q^{74} + (72 \beta + 8) q^{76} + ( - 48 \beta + 80) q^{77} + ( - 74 \beta + 334) q^{79} + ( - 32 \beta - 64) q^{80} + (42 \beta - 506) q^{82} + (178 \beta - 286) q^{83} + (244 \beta + 680) q^{85} + (112 \beta - 496) q^{86} + (64 \beta - 320) q^{88} + (94 \beta + 196) q^{89} + ( - 90 \beta + 22) q^{91} + 92 q^{92} + ( - 186 \beta - 122) q^{94} + ( - 112 \beta - 368) q^{95} + ( - 164 \beta + 1158) q^{97} + (24 \beta - 598) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 8 q^{4} - 10 q^{5} + 6 q^{7} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} + 8 q^{4} - 10 q^{5} + 6 q^{7} + 16 q^{8} - 20 q^{10} - 72 q^{11} - 111 q^{13} + 12 q^{14} + 32 q^{16} - 124 q^{17} + 22 q^{19} - 40 q^{20} - 144 q^{22} + 46 q^{23} - 118 q^{25} - 222 q^{26} + 24 q^{28} - 15 q^{29} + 67 q^{31} + 64 q^{32} - 248 q^{34} - 112 q^{35} + 18 q^{37} + 44 q^{38} - 80 q^{40} - 485 q^{41} - 440 q^{43} - 288 q^{44} + 92 q^{46} - 215 q^{47} - 586 q^{49} - 236 q^{50} - 444 q^{52} - 240 q^{53} + 32 q^{55} + 48 q^{56} - 30 q^{58} - 792 q^{59} + 456 q^{61} + 134 q^{62} + 128 q^{64} + 268 q^{65} + 240 q^{67} - 496 q^{68} - 224 q^{70} + 705 q^{71} + 27 q^{73} + 36 q^{74} + 88 q^{76} + 112 q^{77} + 594 q^{79} - 160 q^{80} - 970 q^{82} - 394 q^{83} + 1604 q^{85} - 880 q^{86} - 576 q^{88} + 486 q^{89} - 46 q^{91} + 184 q^{92} - 430 q^{94} - 848 q^{95} + 2152 q^{97} - 1172 q^{98}+O(q^{100}) \) 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
3.70156
−2.70156
2.00000 0 4.00000 −11.4031 0 9.40312 8.00000 0 −22.8062
1.2 2.00000 0 4.00000 1.40312 0 −3.40312 8.00000 0 2.80625
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(23\) \(-1\)

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 414.4.a.j 2
3.b odd 2 1 46.4.a.c 2
12.b even 2 1 368.4.a.g 2
15.d odd 2 1 1150.4.a.k 2
15.e even 4 2 1150.4.b.i 4
21.c even 2 1 2254.4.a.d 2
24.f even 2 1 1472.4.a.l 2
24.h odd 2 1 1472.4.a.m 2
69.c even 2 1 1058.4.a.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
46.4.a.c 2 3.b odd 2 1
368.4.a.g 2 12.b even 2 1
414.4.a.j 2 1.a even 1 1 trivial
1058.4.a.f 2 69.c even 2 1
1150.4.a.k 2 15.d odd 2 1
1150.4.b.i 4 15.e even 4 2
1472.4.a.l 2 24.f even 2 1
1472.4.a.m 2 24.h odd 2 1
2254.4.a.d 2 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(414))\):

\( T_{5}^{2} + 10T_{5} - 16 \) Copy content Toggle raw display
\( T_{7}^{2} - 6T_{7} - 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 10T - 16 \) Copy content Toggle raw display
$7$ \( T^{2} - 6T - 32 \) Copy content Toggle raw display
$11$ \( T^{2} + 72T + 640 \) Copy content Toggle raw display
$13$ \( T^{2} + 111T + 2578 \) Copy content Toggle raw display
$17$ \( T^{2} + 124T - 2060 \) Copy content Toggle raw display
$19$ \( T^{2} - 22T - 3200 \) Copy content Toggle raw display
$23$ \( (T - 23)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 15T - 43250 \) Copy content Toggle raw display
$31$ \( T^{2} - 67T - 1840 \) Copy content Toggle raw display
$37$ \( T^{2} - 18T - 50144 \) Copy content Toggle raw display
$41$ \( T^{2} + 485T + 54286 \) Copy content Toggle raw display
$43$ \( T^{2} + 440T + 16256 \) Copy content Toggle raw display
$47$ \( T^{2} + 215T - 77096 \) Copy content Toggle raw display
$53$ \( T^{2} + 240T - 143204 \) Copy content Toggle raw display
$59$ \( T^{2} + 792T + 146320 \) Copy content Toggle raw display
$61$ \( T^{2} - 456T - 408692 \) Copy content Toggle raw display
$67$ \( T^{2} - 240T - 9216 \) Copy content Toggle raw display
$71$ \( T^{2} - 705T + 39376 \) Copy content Toggle raw display
$73$ \( T^{2} - 27T - 8438 \) Copy content Toggle raw display
$79$ \( T^{2} - 594T + 32080 \) Copy content Toggle raw display
$83$ \( T^{2} + 394T - 285952 \) Copy content Toggle raw display
$89$ \( T^{2} - 486T - 31520 \) Copy content Toggle raw display
$97$ \( T^{2} - 2152 T + 882092 \) Copy content Toggle raw display
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