Properties

Label 3.15.b.a
Level $3$
Weight $15$
Character orbit 3.b
Analytic conductor $3.730$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 15 \)
Character orbit: \([\chi]\) \(=\) 3.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.72986904456\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.1929141760.2
Defining polynomial: \( x^{4} + 364x^{2} + 3640 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{10}\cdot 3^{7} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} + 3 \beta_1 + 549) q^{3} + (\beta_{3} + 8 \beta_{2} - 9824) q^{4} + (3 \beta_{3} - 30 \beta_{2} - 190 \beta_1) q^{5} + (9 \beta_{3} + 24 \beta_{2} + 1845 \beta_1 - 78624) q^{6} + ( - 11 \beta_{3} - 88 \beta_{2} + 206402) q^{7} + ( - 48 \beta_{3} + 480 \beta_{2} - 16768 \beta_1) q^{8} + ( - 189 \beta_{3} - 306 \beta_{2} + 19710 \beta_1 - 406215) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} + 3 \beta_1 + 549) q^{3} + (\beta_{3} + 8 \beta_{2} - 9824) q^{4} + (3 \beta_{3} - 30 \beta_{2} - 190 \beta_1) q^{5} + (9 \beta_{3} + 24 \beta_{2} + 1845 \beta_1 - 78624) q^{6} + ( - 11 \beta_{3} - 88 \beta_{2} + 206402) q^{7} + ( - 48 \beta_{3} + 480 \beta_{2} - 16768 \beta_1) q^{8} + ( - 189 \beta_{3} - 306 \beta_{2} + 19710 \beta_1 - 406215) q^{9} + ( - 10 \beta_{3} - 80 \beta_{2} + 4979520) q^{10} + (231 \beta_{3} - 2310 \beta_{2} + 64570 \beta_1) q^{11} + (1701 \beta_{3} + 2696 \beta_{2} - 177216 \beta_1 - 39358944) q^{12} + (676 \beta_{3} + 5408 \beta_{2} + 50424218) q^{13} + (528 \beta_{3} - 5280 \beta_{2} + 463010 \beta_1) q^{14} + ( - 8271 \beta_{3} - 16710 \beta_{2} - 117270 \beta_1 - 112432320) q^{15} + ( - 3264 \beta_{3} - 26112 \beta_{2} + 278499328) q^{16} + ( - 10716 \beta_{3} + 107160 \beta_{2} - 2659944 \beta_1) q^{17} + (21546 \beta_{3} + 66960 \beta_{2} + 2439801 \beta_1 - 516559680) q^{18} + ( - 2589 \beta_{3} - 20712 \beta_{2} + 328736810) q^{19} + (49632 \beta_{3} - 496320 \beta_{2} + 2099840 \beta_1) q^{20} + ( - 18711 \beta_{3} - 127994 \beta_{2} + \cdots + 486935946) q^{21}+ \cdots + (1364449779 \beta_{3} + 17632180170 \beta_{2} + \cdots - 21526343879040) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2196 q^{3} - 39296 q^{4} - 314496 q^{6} + 825608 q^{7} - 1624860 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2196 q^{3} - 39296 q^{4} - 314496 q^{6} + 825608 q^{7} - 1624860 q^{9} + 19918080 q^{10} - 157435776 q^{12} + 201696872 q^{13} - 449729280 q^{15} + 1113997312 q^{16} - 2066238720 q^{18} + 1314947240 q^{19} + 1947743784 q^{21} - 6769002240 q^{22} + 13425205248 q^{24} - 6882482780 q^{25} + 20862898164 q^{27} - 35011502848 q^{28} + 12293648640 q^{30} - 34970710072 q^{31} - 59537237760 q^{33} + 278847249408 q^{34} - 282390924672 q^{36} + 55576789928 q^{37} + 18888686856 q^{39} + 106207395840 q^{40} - 235283892480 q^{42} - 323678929048 q^{43} + 1007022481920 q^{45} - 273460142592 q^{46} + 1055038980096 q^{48} - 2246577120564 q^{49} + 2656416881664 q^{51} - 328297944832 q^{52} - 5027863591296 q^{54} - 832322050560 q^{55} + 1073653456968 q^{57} + 12943993870080 q^{58} - 9089286635520 q^{60} - 4171641626392 q^{61} + 2946514688712 q^{63} + 9874081841152 q^{64} - 14371860222720 q^{66} - 10964239937752 q^{67} + 15227331485184 q^{69} + 4380138846720 q^{70} + 24815696855040 q^{72} - 44644130922808 q^{73} + 36265563003060 q^{75} - 19249495285504 q^{76} - 5388124734720 q^{78} + 41215442578760 q^{79} - 17388818777916 q^{81} - 42500400284160 q^{82} - 61945380290304 q^{84} + 45292279818240 q^{85} - 41651639527680 q^{87} + 147397549178880 q^{88} - 5107487374080 q^{90} + 23445744391888 q^{91} - 145717264072728 q^{93} + 50952237493248 q^{94} + 81817830162432 q^{96} + 70529980615688 q^{97} - 86105375516160 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 364x^{2} + 3640 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 12\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{3} + 8\nu^{2} + 688\nu + 1456 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -16\nu^{3} + 80\nu^{2} - 5504\nu + 14560 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 8\beta_{2} - 26208 ) / 144 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{3} + 10\beta_{2} - 1032\beta_1 ) / 36 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
18.8072i
3.20795i
3.20795i
18.8072i
225.686i 1922.67 1042.26i −34550.1 23159.5i −235223. 433920.i 478389. 4.09983e6i 2.61037e6 4.00784e6i 5.22678e6
2.2 38.4954i −824.672 + 2025.56i 14902.1 122931.i 77974.7 + 31746.1i −65585.1 1.20437e6i −3.42280e6 3.34084e6i 4.73226e6
2.3 38.4954i −824.672 2025.56i 14902.1 122931.i 77974.7 31746.1i −65585.1 1.20437e6i −3.42280e6 + 3.34084e6i 4.73226e6
2.4 225.686i 1922.67 + 1042.26i −34550.1 23159.5i −235223. + 433920.i 478389. 4.09983e6i 2.61037e6 + 4.00784e6i 5.22678e6
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.15.b.a 4
3.b odd 2 1 inner 3.15.b.a 4
4.b odd 2 1 48.15.e.b 4
5.b even 2 1 75.15.c.d 4
5.c odd 4 2 75.15.d.b 8
12.b even 2 1 48.15.e.b 4
15.d odd 2 1 75.15.c.d 4
15.e even 4 2 75.15.d.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.15.b.a 4 1.a even 1 1 trivial
3.15.b.a 4 3.b odd 2 1 inner
48.15.e.b 4 4.b odd 2 1
48.15.e.b 4 12.b even 2 1
75.15.c.d 4 5.b even 2 1
75.15.c.d 4 15.d odd 2 1
75.15.d.b 8 5.c odd 4 2
75.15.d.b 8 15.e even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{15}^{\mathrm{new}}(3, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 52416 T^{2} + \cdots + 75479040 \) Copy content Toggle raw display
$3$ \( T^{4} - 2196 T^{3} + \cdots + 22876792454961 \) Copy content Toggle raw display
$5$ \( T^{4} + 15648272640 T^{2} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{2} - 412804 T - 31375221500)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 300096887650560 T^{2} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{2} - 100848436 T + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 43\!\cdots\!60 \) Copy content Toggle raw display
$19$ \( (T^{2} - 657473620 T + 10\!\cdots\!96)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 14\!\cdots\!40 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{2} + 17485355036 T - 45\!\cdots\!76)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 27788394964 T - 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{2} + 161839464524 T - 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 16\!\cdots\!60 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 17\!\cdots\!60 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{2} + 2085820813196 T - 90\!\cdots\!96)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 5482119968876 T + 73\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{2} + 22322065461404 T + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 20607721289380 T - 37\!\cdots\!24)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 44\!\cdots\!60 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{2} - 35264990307844 T - 46\!\cdots\!00)^{2} \) Copy content Toggle raw display
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