Properties

Label 400.6.a.u
Level $400$
Weight $6$
Character orbit 400.a
Self dual yes
Analytic conductor $64.154$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1535279252\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{241}) \)
Defining polynomial: \( x^{2} - x - 60 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 200)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{241}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 4) q^{3} + ( - 2 \beta - 4) q^{7} + (8 \beta + 14) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 4) q^{3} + ( - 2 \beta - 4) q^{7} + (8 \beta + 14) q^{9} + (21 \beta + 100) q^{11} + (52 \beta + 296) q^{13} + (16 \beta - 139) q^{17} + ( - 59 \beta + 420) q^{19} + ( - 12 \beta - 498) q^{21} + ( - 98 \beta - 976) q^{23} + ( - 197 \beta + 1012) q^{27} + (148 \beta - 2340) q^{29} + (354 \beta + 2504) q^{31} + (184 \beta + 5461) q^{33} + ( - 484 \beta - 6250) q^{37} + (504 \beta + 13716) q^{39} + (936 \beta - 2667) q^{41} + (1076 \beta + 112) q^{43} + 13036 q^{47} + (16 \beta - 15827) q^{49} + ( - 75 \beta + 3300) q^{51} + ( - 360 \beta + 23406) q^{53} + (184 \beta - 12539) q^{57} + (416 \beta + 40888) q^{59} + (44 \beta - 23466) q^{61} + ( - 60 \beta - 3912) q^{63} + (323 \beta + 34404) q^{67} + ( - 1368 \beta - 27522) q^{69} + (344 \beta - 3724) q^{71} + ( - 72 \beta + 54411) q^{73} + ( - 284 \beta - 10522) q^{77} + ( - 3062 \beta + 54052) q^{79} + ( - 1720 \beta - 46831) q^{81} + ( - 3841 \beta + 13612) q^{83} + ( - 1748 \beta + 26308) q^{87} + (5024 \beta + 35495) q^{89} + ( - 800 \beta - 26248) q^{91} + (3920 \beta + 95330) q^{93} + (3800 \beta + 48426) q^{97} + (1094 \beta + 41888) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{3} - 8 q^{7} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{3} - 8 q^{7} + 28 q^{9} + 200 q^{11} + 592 q^{13} - 278 q^{17} + 840 q^{19} - 996 q^{21} - 1952 q^{23} + 2024 q^{27} - 4680 q^{29} + 5008 q^{31} + 10922 q^{33} - 12500 q^{37} + 27432 q^{39} - 5334 q^{41} + 224 q^{43} + 26072 q^{47} - 31654 q^{49} + 6600 q^{51} + 46812 q^{53} - 25078 q^{57} + 81776 q^{59} - 46932 q^{61} - 7824 q^{63} + 68808 q^{67} - 55044 q^{69} - 7448 q^{71} + 108822 q^{73} - 21044 q^{77} + 108104 q^{79} - 93662 q^{81} + 27224 q^{83} + 52616 q^{87} + 70990 q^{89} - 52496 q^{91} + 190660 q^{93} + 96852 q^{97} + 83776 q^{99}+O(q^{100}) \) 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
−7.26209
8.26209
0 −11.5242 0 0 0 27.0483 0 −110.193 0
1.2 0 19.5242 0 0 0 −35.0483 0 138.193 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)

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 400.6.a.u 2
4.b odd 2 1 200.6.a.e 2
5.b even 2 1 400.6.a.r 2
5.c odd 4 2 400.6.c.o 4
20.d odd 2 1 200.6.a.f yes 2
20.e even 4 2 200.6.c.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.6.a.e 2 4.b odd 2 1
200.6.a.f yes 2 20.d odd 2 1
200.6.c.f 4 20.e even 4 2
400.6.a.r 2 5.b even 2 1
400.6.a.u 2 1.a even 1 1 trivial
400.6.c.o 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 8T_{3} - 225 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(400))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 8T - 225 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 8T - 948 \) Copy content Toggle raw display
$11$ \( T^{2} - 200T - 96281 \) Copy content Toggle raw display
$13$ \( T^{2} - 592T - 564048 \) Copy content Toggle raw display
$17$ \( T^{2} + 278T - 42375 \) Copy content Toggle raw display
$19$ \( T^{2} - 840T - 662521 \) Copy content Toggle raw display
$23$ \( T^{2} + 1952 T - 1361988 \) Copy content Toggle raw display
$29$ \( T^{2} + 4680 T + 196736 \) Copy content Toggle raw display
$31$ \( T^{2} - 5008 T - 23931140 \) Copy content Toggle raw display
$37$ \( T^{2} + 12500 T - 17393196 \) Copy content Toggle raw display
$41$ \( T^{2} + 5334 T - 204026247 \) Copy content Toggle raw display
$43$ \( T^{2} - 224 T - 279011472 \) Copy content Toggle raw display
$47$ \( (T - 13036)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 46812 T + 516607236 \) Copy content Toggle raw display
$59$ \( T^{2} - 81776 T + 1630122048 \) Copy content Toggle raw display
$61$ \( T^{2} + 46932 T + 550186580 \) Copy content Toggle raw display
$67$ \( T^{2} - 68808 T + 1158491927 \) Copy content Toggle raw display
$71$ \( T^{2} + 7448 T - 14650800 \) Copy content Toggle raw display
$73$ \( T^{2} - 108822 T + 2959307577 \) Copy content Toggle raw display
$79$ \( T^{2} - 108104 T + 662040300 \) Copy content Toggle raw display
$83$ \( T^{2} - 27224 T - 3370254177 \) Copy content Toggle raw display
$89$ \( T^{2} - 70990 T - 4823083791 \) Copy content Toggle raw display
$97$ \( T^{2} - 96852 T - 1134962524 \) Copy content Toggle raw display
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