Properties

Label 3.17.b.a
Level $3$
Weight $17$
Character orbit 3.b
Analytic conductor $4.870$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 17 \)
Character orbit: \([\chi]\) \(=\) 3.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.86973631570\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} + \cdots)\)
Defining polynomial: \( x^{4} + 3814x^{2} + 2981440 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6}\cdot 3^{8} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + \beta_1 - 513) q^{3} + (\beta_{3} + 3 \beta_{2} - \beta_1 - 3116) q^{4} + (15 \beta_{3} - 45 \beta_{2} + 95 \beta_1) q^{5} + (81 \beta_{3} - 21 \beta_{2} - 2532 \beta_1 - 100764) q^{6} + (217 \beta_{3} + 651 \beta_{2} - 217 \beta_1 - 785386) q^{7} + (264 \beta_{3} - 792 \beta_{2} + 50528 \beta_1) q^{8} + ( - 729 \beta_{3} + 135 \beta_{2} - 154413 \beta_1 + 4654665) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + \beta_1 - 513) q^{3} + (\beta_{3} + 3 \beta_{2} - \beta_1 - 3116) q^{4} + (15 \beta_{3} - 45 \beta_{2} + 95 \beta_1) q^{5} + (81 \beta_{3} - 21 \beta_{2} - 2532 \beta_1 - 100764) q^{6} + (217 \beta_{3} + 651 \beta_{2} - 217 \beta_1 - 785386) q^{7} + (264 \beta_{3} - 792 \beta_{2} + 50528 \beta_1) q^{8} + ( - 729 \beta_{3} + 135 \beta_{2} - 154413 \beta_1 + 4654665) q^{9} + ( - 3130 \beta_{3} - 9390 \beta_{2} + 3130 \beta_1 - 4661640) q^{10} + ( - 7341 \beta_{3} + 22023 \beta_{2} + 618715 \beta_1) q^{11} + ( - 2187 \beta_{3} + 367 \beta_{2} - 463277 \beta_1 + 143123652) q^{12} + (10972 \beta_{3} + 32916 \beta_{2} - 10972 \beta_1 - 395182606) q^{13} + (57288 \beta_{3} - 171864 \beta_{2} - 3365950 \beta_1) q^{14} + (49005 \beta_{3} - 10275 \beta_{2} + 6442185 \beta_1 + 1707489720) q^{15} + (59304 \beta_{3} + 177912 \beta_{2} - 59304 \beta_1 - 3640317152) q^{16} + ( - 189948 \beta_{3} + 569844 \beta_{2} - 10252764 \beta_1) q^{17} + ( - 161838 \beta_{3} + 56214 \beta_{2} + \cdots + 10576244520) q^{18}+ \cdots + ( - 374876825841 \beta_{3} + \cdots + 47\!\cdots\!40) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2052 q^{3} - 12464 q^{4} - 403056 q^{6} - 3141544 q^{7} + 18618660 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2052 q^{3} - 12464 q^{4} - 403056 q^{6} - 3141544 q^{7} + 18618660 q^{9} - 18646560 q^{10} + 572494608 q^{12} - 1580730424 q^{13} + 6829958880 q^{15} - 14561268608 q^{16} + 42304978080 q^{18} - 56117116360 q^{19} + 124455437064 q^{21} - 173545812000 q^{22} + 100515572352 q^{24} - 8074048700 q^{25} - 317983667652 q^{27} + 746852001056 q^{28} - 1762329117600 q^{30} + 2471781156248 q^{31} - 3610697951520 q^{33} + 2721261612672 q^{34} - 1219654126512 q^{36} + 370563213896 q^{37} + 7022170227384 q^{39} - 11795287092480 q^{40} + 27587883687840 q^{42} - 28065022062664 q^{43} + 18795326443200 q^{45} - 43994579504832 q^{46} + 41041959355008 q^{48} + 29478262537164 q^{49} - 82841575222656 q^{51} + 42193089120416 q^{52} - 107063660756304 q^{54} + 290253653236800 q^{55} - 335129108488344 q^{57} + 8796421982880 q^{58} + 56126440892160 q^{60} + 362269793083208 q^{61} - 266698363786344 q^{63} - 653949742779392 q^{64} + 13\!\cdots\!80 q^{66}+ \cdots + 18\!\cdots\!60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3814x^{2} + 2981440 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 6\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{3} + 132\nu^{2} - 6690\nu + 251724 ) / 22 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9\nu^{3} + 396\nu^{2} + 20202\nu + 755172 ) / 22 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 3\beta_{2} - \beta _1 - 68652 ) / 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 11\beta_{3} - 33\beta_{2} - 3356\beta_1 ) / 9 \) 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
52.1196i
33.1293i
33.1293i
52.1196i
312.717i −5369.70 3770.02i −32256.2 276758.i −1.17895e6 + 1.67920e6i −7.10881e6 1.04072e7i 1.46206e7 + 4.04878e7i 8.65472e7
2.2 198.776i 4343.70 + 4917.21i 26024.2 482304.i 977423. 863422.i 5.53804e6 1.81999e7i −5.31128e6 + 4.27178e7i −9.58704e7
2.3 198.776i 4343.70 4917.21i 26024.2 482304.i 977423. + 863422.i 5.53804e6 1.81999e7i −5.31128e6 4.27178e7i −9.58704e7
2.4 312.717i −5369.70 + 3770.02i −32256.2 276758.i −1.17895e6 1.67920e6i −7.10881e6 1.04072e7i 1.46206e7 4.04878e7i 8.65472e7
\(n\): 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 3.17.b.a 4
3.b odd 2 1 inner 3.17.b.a 4
4.b odd 2 1 48.17.e.b 4
5.b even 2 1 75.17.c.d 4
5.c odd 4 2 75.17.d.b 8
12.b even 2 1 48.17.e.b 4
15.d odd 2 1 75.17.c.d 4
15.e even 4 2 75.17.d.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.17.b.a 4 1.a even 1 1 trivial
3.17.b.a 4 3.b odd 2 1 inner
48.17.e.b 4 4.b odd 2 1
48.17.e.b 4 12.b even 2 1
75.17.c.d 4 5.b even 2 1
75.17.c.d 4 15.d odd 2 1
75.17.d.b 8 5.c odd 4 2
75.17.d.b 8 15.e even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{17}^{\mathrm{new}}(3, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 137304 T^{2} + \cdots + 3863946240 \) Copy content Toggle raw display
$3$ \( T^{4} + 2052 T^{3} + \cdots + 18\!\cdots\!41 \) Copy content Toggle raw display
$5$ \( T^{4} + 309212805600 T^{2} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1570772 T - 39368833865900)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{2} + 790365212 T + 53\!\cdots\!60)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 28\!\cdots\!60 \) Copy content Toggle raw display
$19$ \( (T^{2} + 28058558180 T - 15\!\cdots\!64)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 32\!\cdots\!40 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{2} - 1235890578124 T + 38\!\cdots\!44)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 185281606948 T - 79\!\cdots\!60)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{2} + 14032511031332 T - 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 26\!\cdots\!60 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 12\!\cdots\!60 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{2} - 181134896541604 T + 77\!\cdots\!04)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 13387202681732 T - 28\!\cdots\!20)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{2} - 158814443769988 T - 92\!\cdots\!60)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + \cdots + 10\!\cdots\!96)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 20\!\cdots\!60 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots + 81\!\cdots\!20)^{2} \) Copy content Toggle raw display
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