Properties

Label 804.2.a.f
Level 804
Weight 2
Character orbit 804.a
Self dual yes
Analytic conductor 6.420
Analytic rank 0
Dimension 5
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 804.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.41997232251\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.24571284.1
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
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 a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( 1 - \beta_{1} ) q^{5} + ( 1 - \beta_{2} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( 1 - \beta_{1} ) q^{5} + ( 1 - \beta_{2} ) q^{7} + q^{9} + ( 1 + \beta_{4} ) q^{11} + ( 1 - \beta_{4} ) q^{13} + ( 1 - \beta_{1} ) q^{15} + ( \beta_{2} - \beta_{3} ) q^{17} + ( 2 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{19} + ( 1 - \beta_{2} ) q^{21} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{23} + ( 2 + \beta_{2} ) q^{25} + q^{27} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{29} + ( -\beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{31} + ( 1 + \beta_{4} ) q^{33} + ( -2 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{35} + ( 2 - \beta_{3} + \beta_{4} ) q^{37} + ( 1 - \beta_{4} ) q^{39} + ( 2 + \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{41} + ( 2 \beta_{1} + \beta_{2} - \beta_{4} ) q^{43} + ( 1 - \beta_{1} ) q^{45} + ( -2 - \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{47} + ( 5 - \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{49} + ( \beta_{2} - \beta_{3} ) q^{51} + ( -2 + \beta_{1} - \beta_{4} ) q^{53} + ( 3 + 2 \beta_{3} + \beta_{4} ) q^{55} + ( 2 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{57} + ( -3 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{59} + ( 1 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{61} + ( 1 - \beta_{2} ) q^{63} + ( -1 - 2 \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{65} + q^{67} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{69} + ( -5 + \beta_{4} ) q^{71} + ( 3 - 2 \beta_{1} - \beta_{3} ) q^{73} + ( 2 + \beta_{2} ) q^{75} + ( -1 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} ) q^{77} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{79} + q^{81} + ( -4 - 3 \beta_{1} - \beta_{4} ) q^{83} + ( 1 - 2 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} ) q^{85} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{87} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{89} + ( 3 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} ) q^{91} + ( -\beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{93} + ( -5 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{95} + ( 3 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{4} ) q^{97} + ( 1 + \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 5q^{3} + 3q^{5} + 5q^{7} + 5q^{9} + O(q^{10}) \) \( 5q + 5q^{3} + 3q^{5} + 5q^{7} + 5q^{9} + 6q^{11} + 4q^{13} + 3q^{15} + q^{17} + 13q^{19} + 5q^{21} + 10q^{25} + 5q^{27} + 3q^{29} + 3q^{31} + 6q^{33} - 9q^{35} + 12q^{37} + 4q^{39} + 11q^{41} + 3q^{43} + 3q^{45} - 13q^{47} + 24q^{49} + q^{51} - 9q^{53} + 14q^{55} + 13q^{57} - 12q^{59} + 4q^{61} + 5q^{63} - 8q^{65} + 5q^{67} - 24q^{71} + 12q^{73} + 10q^{75} - 4q^{77} - 4q^{79} + 5q^{81} - 27q^{83} - 4q^{85} + 3q^{87} - 5q^{89} + 14q^{91} + 3q^{93} - 30q^{95} + 8q^{97} + 6q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 2 x^{4} - 15 x^{3} + 10 x^{2} + 64 x + 38\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 6 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{4} - 5 \nu^{3} - 8 \nu^{2} + 34 \nu + 34 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{4} - 3 \nu^{3} - 10 \nu^{2} + 18 \nu + 28 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2 \beta_{1} + 6\)
\(\nu^{3}\)\(=\)\(\beta_{4} - 2 \beta_{3} + \beta_{2} + 10 \beta_{1} + 9\)
\(\nu^{4}\)\(=\)\(5 \beta_{4} - 6 \beta_{3} + 13 \beta_{2} + 32 \beta_{1} + 59\)

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
3.96721
2.95721
−0.789611
−1.65253
−2.48228
0 1.00000 0 −2.96721 0 −0.804360 0 1.00000 0
1.2 0 1.00000 0 −1.95721 0 4.16931 0 1.00000 0
1.3 0 1.00000 0 1.78961 0 4.79729 0 1.00000 0
1.4 0 1.00000 0 2.65253 0 0.964067 0 1.00000 0
1.5 0 1.00000 0 3.48228 0 −4.12631 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

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 804.2.a.f 5
3.b odd 2 1 2412.2.a.j 5
4.b odd 2 1 3216.2.a.ba 5
12.b even 2 1 9648.2.a.ca 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
804.2.a.f 5 1.a even 1 1 trivial
2412.2.a.j 5 3.b odd 2 1
3216.2.a.ba 5 4.b odd 2 1
9648.2.a.ca 5 12.b even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(67\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} - 3 T_{5}^{4} - 13 T_{5}^{3} + 37 T_{5}^{2} + 36 T_{5} - 96 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(804))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - T )^{5} \)
$5$ \( 1 - 3 T + 12 T^{2} - 23 T^{3} + 91 T^{4} - 176 T^{5} + 455 T^{6} - 575 T^{7} + 1500 T^{8} - 1875 T^{9} + 3125 T^{10} \)
$7$ \( 1 - 5 T + 18 T^{2} - 51 T^{3} + 133 T^{4} - 288 T^{5} + 931 T^{6} - 2499 T^{7} + 6174 T^{8} - 12005 T^{9} + 16807 T^{10} \)
$11$ \( 1 - 6 T + 39 T^{2} - 164 T^{3} + 754 T^{4} - 2540 T^{5} + 8294 T^{6} - 19844 T^{7} + 51909 T^{8} - 87846 T^{9} + 161051 T^{10} \)
$13$ \( 1 - 4 T + 41 T^{2} - 148 T^{3} + 922 T^{4} - 2464 T^{5} + 11986 T^{6} - 25012 T^{7} + 90077 T^{8} - 114244 T^{9} + 371293 T^{10} \)
$17$ \( 1 - T + 21 T^{2} - 24 T^{3} + 478 T^{4} - 1366 T^{5} + 8126 T^{6} - 6936 T^{7} + 103173 T^{8} - 83521 T^{9} + 1419857 T^{10} \)
$19$ \( 1 - 13 T + 99 T^{2} - 572 T^{3} + 3194 T^{4} - 15246 T^{5} + 60686 T^{6} - 206492 T^{7} + 679041 T^{8} - 1694173 T^{9} + 2476099 T^{10} \)
$23$ \( 1 + 55 T^{2} + 72 T^{3} + 1447 T^{4} + 3474 T^{5} + 33281 T^{6} + 38088 T^{7} + 669185 T^{8} + 6436343 T^{10} \)
$29$ \( 1 - 3 T + 87 T^{2} - 212 T^{3} + 4036 T^{4} - 8402 T^{5} + 117044 T^{6} - 178292 T^{7} + 2121843 T^{8} - 2121843 T^{9} + 20511149 T^{10} \)
$31$ \( 1 - 3 T + 18 T^{2} + 11 T^{3} + 705 T^{4} - 4656 T^{5} + 21855 T^{6} + 10571 T^{7} + 536238 T^{8} - 2770563 T^{9} + 28629151 T^{10} \)
$37$ \( 1 - 12 T + 193 T^{2} - 1530 T^{3} + 13997 T^{4} - 80018 T^{5} + 517889 T^{6} - 2094570 T^{7} + 9776029 T^{8} - 22489932 T^{9} + 69343957 T^{10} \)
$41$ \( 1 - 11 T + 94 T^{2} - 341 T^{3} + 2905 T^{4} - 13348 T^{5} + 119105 T^{6} - 573221 T^{7} + 6478574 T^{8} - 31083371 T^{9} + 115856201 T^{10} \)
$43$ \( 1 - 3 T + 108 T^{2} - 133 T^{3} + 5901 T^{4} - 1440 T^{5} + 253743 T^{6} - 245917 T^{7} + 8586756 T^{8} - 10256403 T^{9} + 147008443 T^{10} \)
$47$ \( 1 + 13 T + 123 T^{2} + 600 T^{3} + 3814 T^{4} + 18094 T^{5} + 179258 T^{6} + 1325400 T^{7} + 12770229 T^{8} + 63435853 T^{9} + 229345007 T^{10} \)
$53$ \( 1 + 9 T + 240 T^{2} + 1675 T^{3} + 24397 T^{4} + 126916 T^{5} + 1293041 T^{6} + 4705075 T^{7} + 35730480 T^{8} + 71014329 T^{9} + 418195493 T^{10} \)
$59$ \( 1 + 12 T + 253 T^{2} + 2696 T^{3} + 27307 T^{4} + 234602 T^{5} + 1611113 T^{6} + 9384776 T^{7} + 51960887 T^{8} + 145408332 T^{9} + 714924299 T^{10} \)
$61$ \( 1 - 4 T + 137 T^{2} - 628 T^{3} + 12754 T^{4} - 55984 T^{5} + 777994 T^{6} - 2336788 T^{7} + 31096397 T^{8} - 55383364 T^{9} + 844596301 T^{10} \)
$67$ \( ( 1 - T )^{5} \)
$71$ \( 1 + 24 T + 555 T^{2} + 7492 T^{3} + 93850 T^{4} + 822088 T^{5} + 6663350 T^{6} + 37767172 T^{7} + 198640605 T^{8} + 609880344 T^{9} + 1804229351 T^{10} \)
$73$ \( 1 - 12 T + 357 T^{2} - 3118 T^{3} + 50553 T^{4} - 326970 T^{5} + 3690369 T^{6} - 16615822 T^{7} + 138879069 T^{8} - 340778892 T^{9} + 2073071593 T^{10} \)
$79$ \( 1 + 4 T + 179 T^{2} - 96 T^{3} + 13138 T^{4} - 48712 T^{5} + 1037902 T^{6} - 599136 T^{7} + 88253981 T^{8} + 155800324 T^{9} + 3077056399 T^{10} \)
$83$ \( 1 + 27 T + 558 T^{2} + 8237 T^{3} + 99931 T^{4} + 998672 T^{5} + 8294273 T^{6} + 56744693 T^{7} + 319057146 T^{8} + 1281374667 T^{9} + 3939040643 T^{10} \)
$89$ \( 1 + 5 T + 315 T^{2} + 612 T^{3} + 41488 T^{4} + 27302 T^{5} + 3692432 T^{6} + 4847652 T^{7} + 222065235 T^{8} + 313711205 T^{9} + 5584059449 T^{10} \)
$97$ \( 1 - 8 T + 137 T^{2} - 852 T^{3} + 5470 T^{4} - 41656 T^{5} + 530590 T^{6} - 8016468 T^{7} + 125036201 T^{8} - 708234248 T^{9} + 8587340257 T^{10} \)
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