Properties

Label 3.20.a.b
Level $3$
Weight $20$
Character orbit 3.a
Self dual yes
Analytic conductor $6.865$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.86450089669\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{87481}) \)
Defining polynomial: \( x^{2} - x - 21870 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{87481}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 351) q^{2} - 19683 q^{3} + ( - 702 \beta + 386242) q^{4} + ( - 4160 \beta + 3008070) q^{5} + (19683 \beta - 6908733) q^{6} + (63936 \beta + 56946032) q^{7} + ( - 108356 \beta + 504250812) q^{8} + 387420489 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 351) q^{2} - 19683 q^{3} + ( - 702 \beta + 386242) q^{4} + ( - 4160 \beta + 3008070) q^{5} + (19683 \beta - 6908733) q^{6} + (63936 \beta + 56946032) q^{7} + ( - 108356 \beta + 504250812) q^{8} + 387420489 q^{9} + ( - 4468230 \beta + 4331121210) q^{10} + (10995584 \beta - 3325035636) q^{11} + (13817466 \beta - 7602401286) q^{12} + ( - 39982464 \beta - 22036178074) q^{13} + ( - 34504496 \beta - 30350609712) q^{14} + (81881280 \beta - 59207841810) q^{15} + ( - 174233592 \beta + 59801810440) q^{16} + (582591360 \beta + 168140735874) q^{17} + ( - 387420489 \beta + 135984591639) q^{18} + (1038738816 \beta + 301059462548) q^{19} + ( - 3718431860 \beta + 3461095598220) q^{20} + ( - 1258452288 \beta - 1120868747856) q^{21} + (7184485620 \beta - 9824229663372) q^{22} + (4325606272 \beta + 1184126082984) q^{23} + (2132771148 \beta - 9925168732596) q^{24} + ( - 25027142400 \beta + 3600199539175) q^{25} + (8002333210 \beta + 23744654894682) q^{26} - 7625597484987 q^{27} + ( - 15281345952 \beta - 13342794902944) q^{28} + (14942987072 \beta + 140488625985246) q^{29} + (87948171090 \beta - 85249458776430) q^{30} + (20829313728 \beta + 20805074626856) q^{31} + ( - 64148050704 \beta - 106203054501648) q^{32} + ( - 216426079872 \beta + 65446676423388) q^{33} + (36348831486 \beta - 399673674585666) q^{34} + ( - 44571529600 \beta - 38111204008800) q^{35} + ( - 271969183278 \beta + 149638064512338) q^{36} + (1169925037056 \beta + 318997081994942) q^{37} + (63537861868 \beta - 712157321908116) q^{38} + (786974838912 \beta + 433738093030542) q^{39} + ( - 2423625810840 \beta + 18\!\cdots\!80) q^{40}+ \cdots + (42\!\cdots\!76 \beta - 12\!\cdots\!04) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 702 q^{2} - 39366 q^{3} + 772484 q^{4} + 6016140 q^{5} - 13817466 q^{6} + 113892064 q^{7} + 1008501624 q^{8} + 774840978 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 702 q^{2} - 39366 q^{3} + 772484 q^{4} + 6016140 q^{5} - 13817466 q^{6} + 113892064 q^{7} + 1008501624 q^{8} + 774840978 q^{9} + 8662242420 q^{10} - 6650071272 q^{11} - 15204802572 q^{12} - 44072356148 q^{13} - 60701219424 q^{14} - 118415683620 q^{15} + 119603620880 q^{16} + 336281471748 q^{17} + 271969183278 q^{18} + 602118925096 q^{19} + 6922191196440 q^{20} - 2241737495712 q^{21} - 19648459326744 q^{22} + 2368252165968 q^{23} - 19850337465192 q^{24} + 7200399078350 q^{25} + 47489309789364 q^{26} - 15251194969974 q^{27} - 26685589805888 q^{28} + 280977251970492 q^{29} - 170498917552860 q^{30} + 41610149253712 q^{31} - 212406109003296 q^{32} + 130893352846776 q^{33} - 799347349171332 q^{34} - 76222408017600 q^{35} + 299276129024676 q^{36} + 637994163989884 q^{37} - 14\!\cdots\!32 q^{38}+ \cdots - 25\!\cdots\!08 q^{99}+O(q^{100}) \) 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
148.386
−147.386
−536.316 −19683.0 −236654. −683163. 1.05563e7 1.13677e8 4.08105e8 3.87420e8 3.66391e8
1.2 1238.32 −19683.0 1.00914e6 6.69930e6 −2.43738e7 214621. 6.00397e8 3.87420e8 8.29585e9
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)

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 3.20.a.b 2
3.b odd 2 1 9.20.a.c 2
4.b odd 2 1 48.20.a.j 2
5.b even 2 1 75.20.a.b 2
5.c odd 4 2 75.20.b.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.20.a.b 2 1.a even 1 1 trivial
9.20.a.c 2 3.b odd 2 1
48.20.a.j 2 4.b odd 2 1
75.20.a.b 2 5.b even 2 1
75.20.b.b 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 702T_{2} - 664128 \) acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(3))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 702T - 664128 \) Copy content Toggle raw display
$3$ \( (T + 19683)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 6016140 T - 4576715617500 \) Copy content Toggle raw display
$7$ \( T^{2} - 113892064 T + 24397550813440 \) Copy content Toggle raw display
$11$ \( T^{2} + 6650071272 T - 84\!\cdots\!28 \) Copy content Toggle raw display
$13$ \( T^{2} + 44072356148 T - 77\!\cdots\!08 \) Copy content Toggle raw display
$17$ \( T^{2} - 336281471748 T - 23\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{2} - 602118925096 T - 75\!\cdots\!20 \) Copy content Toggle raw display
$23$ \( T^{2} - 2368252165968 T - 13\!\cdots\!80 \) Copy content Toggle raw display
$29$ \( T^{2} - 280977251970492 T + 19\!\cdots\!80 \) Copy content Toggle raw display
$31$ \( T^{2} - 41610149253712 T + 91\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{2} - 637994163989884 T - 97\!\cdots\!80 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 99\!\cdots\!60 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 83\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots - 97\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 17\!\cdots\!80 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 76\!\cdots\!40 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 18\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 60\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 75\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 23\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 53\!\cdots\!12 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 82\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
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