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Properties

Label 3150.3.c.d
Level $3150$
Weight $3$
Character orbit 3150.c
Analytic conductor $85.831$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 3150.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(85.8312832735\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: 16.0.9671731157401600000000.1
Defining polynomial: \( x^{16} + 7x^{12} + 753x^{8} + 112x^{4} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{28}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 630)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + 2 q^{4} - \beta_{2} q^{7} - 2 \beta_{5} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + 2 q^{4} - \beta_{2} q^{7} - 2 \beta_{5} q^{8} + (\beta_{11} - \beta_{4}) q^{11} + ( - \beta_{14} - 2 \beta_{3}) q^{13} - \beta_{11} q^{14} + 4 q^{16} + (2 \beta_{10} - \beta_{9} - 2 \beta_{5}) q^{17} + (\beta_{13} - 2 \beta_{6} - 10) q^{19} + ( - \beta_{3} + 2 \beta_{2}) q^{22} + (\beta_{10} + 2 \beta_{9} + 2 \beta_{5} + \beta_1) q^{23} + (\beta_{8} - \beta_{7} - 4 \beta_{4}) q^{26} - 2 \beta_{2} q^{28} + (2 \beta_{11} + 2 \beta_{7} + \beta_{4}) q^{29} + (\beta_{15} - 4 \beta_{6} - 14) q^{31} - 4 \beta_{5} q^{32} + (\beta_{15} - \beta_{13} - 2 \beta_{6} + 4) q^{34} + (\beta_{14} - \beta_{12} + 2 \beta_{3} + 6 \beta_{2}) q^{37} + (4 \beta_{10} + 10 \beta_{5} - \beta_1) q^{38} + (2 \beta_{11} + \beta_{8} - 3 \beta_{7} - 8 \beta_{4}) q^{41} + (2 \beta_{14} + 8 \beta_{3} + 10 \beta_{2}) q^{43} + (2 \beta_{11} - 2 \beta_{4}) q^{44} + ( - 2 \beta_{15} - \beta_{6} - 4) q^{46} + (10 \beta_{10} - 3 \beta_{9} - 4 \beta_{5} - 2 \beta_1) q^{47} - 7 q^{49} + ( - 2 \beta_{14} - 4 \beta_{3}) q^{52} + (3 \beta_{10} + 4 \beta_{9} - 9 \beta_{5} + 3 \beta_1) q^{53} - 2 \beta_{11} q^{56} + (2 \beta_{14} + 2 \beta_{12} + \beta_{3} + 4 \beta_{2}) q^{58} + ( - \beta_{8} + 4 \beta_{7} + 6 \beta_{4}) q^{59} + ( - 2 \beta_{15} - 6 \beta_{6} - 10) q^{61} + (8 \beta_{10} - 2 \beta_{9} + 14 \beta_{5} - \beta_1) q^{62} + 8 q^{64} + ( - 3 \beta_{14} + 3 \beta_{12} - 8 \beta_{3} + 2 \beta_{2}) q^{67} + (4 \beta_{10} - 2 \beta_{9} - 4 \beta_{5}) q^{68} + (7 \beta_{11} - 2 \beta_{8} - 4 \beta_{7} + 23 \beta_{4}) q^{71} + (5 \beta_{14} + 20 \beta_{2}) q^{73} + (6 \beta_{11} - 2 \beta_{8} + 4 \beta_{4}) q^{74} + (2 \beta_{13} - 4 \beta_{6} - 20) q^{76} + (\beta_{10} - 7 \beta_{5}) q^{77} + ( - 6 \beta_{13} - 4 \beta_{6}) q^{79} + ( - 4 \beta_{14} - 2 \beta_{12} - 8 \beta_{3} + 4 \beta_{2}) q^{82} + (6 \beta_{10} - \beta_{9} - 12 \beta_{5} - 2 \beta_1) q^{83} + (10 \beta_{11} - 2 \beta_{8} + 2 \beta_{7} + 16 \beta_{4}) q^{86} + ( - 2 \beta_{3} + 4 \beta_{2}) q^{88} + ( - 16 \beta_{11} - 3 \beta_{8} - 26 \beta_{4}) q^{89} + (\beta_{15} + 2 \beta_{13} - 2 \beta_{6}) q^{91} + (2 \beta_{10} + 4 \beta_{9} + 4 \beta_{5} + 2 \beta_1) q^{92} + (3 \beta_{15} + \beta_{13} - 10 \beta_{6} + 8) q^{94} + (3 \beta_{14} - 6 \beta_{12} + 14 \beta_{3} + 12 \beta_{2}) q^{97} + 7 \beta_{5} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 32 q^{4} + 64 q^{16} - 160 q^{19} - 224 q^{31} + 64 q^{34} - 64 q^{46} - 112 q^{49} - 160 q^{61} + 128 q^{64} - 320 q^{76} + 128 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 7x^{12} + 753x^{8} + 112x^{4} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{12} + 105\nu^{8} + 15\nu^{4} + 42056 ) / 3132 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 14\nu^{12} + 96\nu^{8} + 10752\nu^{4} + 815 ) / 2349 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -33\nu^{14} - 247\nu^{10} - 25025\nu^{6} - 18304\nu^{2} ) / 8352 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 245 \nu^{15} - 374 \nu^{13} + 1419 \nu^{11} - 2490 \nu^{9} + 182157 \nu^{7} - 278358 \nu^{5} - 185272 \nu^{3} + 39712 \nu ) / 150336 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 245 \nu^{15} + 374 \nu^{13} + 1419 \nu^{11} + 2490 \nu^{9} + 182157 \nu^{7} + 278358 \nu^{5} - 185272 \nu^{3} - 39712 \nu ) / 150336 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 31\nu^{14} + 201\nu^{10} + 23295\nu^{6} - 10112\nu^{2} ) / 2592 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 23\nu^{14} + 72\nu^{12} + 177\nu^{10} + 504\nu^{8} + 17367\nu^{6} + 53064\nu^{4} + 12704\nu^{2} + 4032 ) / 2592 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 23\nu^{14} + 177\nu^{10} + 17367\nu^{6} + 12704\nu^{2} ) / 864 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2025 \nu^{14} + 8 \nu^{12} + 12879 \nu^{10} - 840 \nu^{8} + 1510569 \nu^{6} - 120 \nu^{4} - 655776 \nu^{2} - 336448 ) / 75168 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 403 \nu^{15} + 122 \nu^{13} - 2925 \nu^{11} + 1110 \nu^{9} - 303675 \nu^{7} + 92826 \nu^{5} - 115960 \nu^{3} + 149024 \nu ) / 50112 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 403 \nu^{15} + 122 \nu^{13} + 2925 \nu^{11} + 1110 \nu^{9} + 303675 \nu^{7} + 92826 \nu^{5} + 115960 \nu^{3} + 149024 \nu ) / 50112 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 257 \nu^{15} + 2051 \nu^{13} + 3291 \nu^{11} + 14325 \nu^{9} + 203901 \nu^{7} + 1556115 \nu^{5} + 1217444 \nu^{3} + 322064 \nu ) / 75168 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 91\nu^{15} + 137\nu^{13} + 537\nu^{11} + 927\nu^{9} + 67887\nu^{7} + 103737\nu^{5} - 69044\nu^{3} - 14800\nu ) / 8352 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1895 \nu^{15} - 415 \nu^{13} + 12957 \nu^{11} - 2361 \nu^{9} + 1425867 \nu^{7} - 311151 \nu^{5} - 25348 \nu^{3} + 588464 \nu ) / 75168 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2425 \nu^{15} + 1225 \nu^{13} - 17859 \nu^{11} + 9183 \nu^{9} - 1833429 \nu^{7} + 933753 \nu^{5} - 984964 \nu^{3} + 954928 \nu ) / 75168 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( 3\beta_{15} + 5\beta_{14} - 3\beta_{13} + \beta_{12} - 6\beta_{11} - 6\beta_{10} + 6\beta_{5} - 6\beta_{4} ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{9} - 3\beta_{8} - 9\beta_{6} - 27\beta_{3} + \beta_1 ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{15} - \beta_{14} - 9\beta_{13} + 7\beta_{12} - 12\beta_{11} + 12\beta_{10} + 60\beta_{5} + 60\beta_{4} ) / 24 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{8} - 9\beta_{7} + 42\beta_{2} + \beta _1 - 14 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 15 \beta_{15} - 35 \beta_{14} + 105 \beta_{13} + 65 \beta_{12} - 66 \beta_{11} - 66 \beta_{10} - 726 \beta_{5} + 726 \beta_{4} ) / 48 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -60\beta_{9} - 36\beta_{8} + 135\beta_{6} - 243\beta_{3} - 20\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 273 \beta_{15} + 533 \beta_{14} + 429 \beta_{13} + 13 \beta_{12} - 1218 \beta_{11} + 1218 \beta_{10} - 2262 \beta_{5} - 2262 \beta_{4} ) / 48 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -21\beta_{8} + 63\beta_{7} - 282\beta_{2} + 217\beta _1 - 2914 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 867 \beta_{15} - 1241 \beta_{14} + 459 \beta_{13} - 493 \beta_{12} + 3876 \beta_{11} + 3876 \beta_{10} - 1140 \beta_{5} + 1140 \beta_{4} ) / 24 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -1815\beta_{9} + 3135\beta_{8} + 4059\beta_{6} + 21033\beta_{3} - 605\beta_1 ) / 24 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 6141 \beta_{15} - 2225 \beta_{14} + 9879 \beta_{13} - 10057 \beta_{12} + 27462 \beta_{11} - 27462 \beta_{10} - 79998 \beta_{5} - 79998 \beta_{4} ) / 48 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( -540\beta_{8} + 1620\beta_{7} - 7245\beta_{2} - 564\beta _1 + 7567 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 699 \beta_{15} + 43105 \beta_{14} - 84579 \beta_{13} - 41707 \beta_{12} - 3126 \beta_{11} - 3126 \beta_{10} + 565806 \beta_{5} - 565806 \beta_{4} ) / 48 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 102921\beta_{9} + 32799\beta_{8} - 230139\beta_{6} + 220023\beta_{3} + 34307\beta_1 ) / 24 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 81435 \beta_{15} - 192455 \beta_{14} - 194895 \beta_{13} + 29585 \beta_{12} + 364188 \beta_{11} - 364188 \beta_{10} + 1125300 \beta_{5} + 1125300 \beta_{4} ) / 24 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
2.08559 + 0.941471i
−0.796626 − 0.359610i
−0.941471 + 2.08559i
0.359610 − 0.796626i
0.359610 + 0.796626i
−0.941471 − 2.08559i
−0.796626 + 0.359610i
2.08559 − 0.941471i
−0.359610 + 0.796626i
0.941471 − 2.08559i
0.796626 + 0.359610i
−2.08559 − 0.941471i
−2.08559 + 0.941471i
0.796626 − 0.359610i
0.941471 + 2.08559i
−0.359610 − 0.796626i
−1.41421 0 2.00000 0 0 − 2.64575i −2.82843 0 0
449.2 −1.41421 0 2.00000 0 0 − 2.64575i −2.82843 0 0
449.3 −1.41421 0 2.00000 0 0 − 2.64575i −2.82843 0 0
449.4 −1.41421 0 2.00000 0 0 − 2.64575i −2.82843 0 0
449.5 −1.41421 0 2.00000 0 0 2.64575i −2.82843 0 0
449.6 −1.41421 0 2.00000 0 0 2.64575i −2.82843 0 0
449.7 −1.41421 0 2.00000 0 0 2.64575i −2.82843 0 0
449.8 −1.41421 0 2.00000 0 0 2.64575i −2.82843 0 0
449.9 1.41421 0 2.00000 0 0 − 2.64575i 2.82843 0 0
449.10 1.41421 0 2.00000 0 0 − 2.64575i 2.82843 0 0
449.11 1.41421 0 2.00000 0 0 − 2.64575i 2.82843 0 0
449.12 1.41421 0 2.00000 0 0 − 2.64575i 2.82843 0 0
449.13 1.41421 0 2.00000 0 0 2.64575i 2.82843 0 0
449.14 1.41421 0 2.00000 0 0 2.64575i 2.82843 0 0
449.15 1.41421 0 2.00000 0 0 2.64575i 2.82843 0 0
449.16 1.41421 0 2.00000 0 0 2.64575i 2.82843 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3150.3.c.d 16
3.b odd 2 1 inner 3150.3.c.d 16
5.b even 2 1 inner 3150.3.c.d 16
5.c odd 4 1 630.3.e.a ✓ 8
5.c odd 4 1 3150.3.e.i 8
15.d odd 2 1 inner 3150.3.c.d 16
15.e even 4 1 630.3.e.a ✓ 8
15.e even 4 1 3150.3.e.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.3.e.a ✓ 8 5.c odd 4 1
630.3.e.a ✓ 8 15.e even 4 1
3150.3.c.d 16 1.a even 1 1 trivial
3150.3.c.d 16 3.b odd 2 1 inner
3150.3.c.d 16 5.b even 2 1 inner
3150.3.c.d 16 15.d odd 2 1 inner
3150.3.e.i 8 5.c odd 4 1
3150.3.e.i 8 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{4} + 32T_{11}^{2} + 144 \) acting on \(S_{3}^{\mathrm{new}}(3150, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{2} + 7)^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 32 T^{2} + 144)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} + 504 T^{6} + 58776 T^{4} + \cdots + 5588496)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 896 T^{6} + 139392 T^{4} + \cdots + 26873856)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 40 T^{3} + 196 T^{2} + \cdots - 29916)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} - 2408 T^{6} + \cdots + 75666805776)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 2792 T^{6} + \cdots + 19925016336)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 56 T^{3} - 300 T^{2} + \cdots - 111676)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + 2512 T^{6} + 1562208 T^{4} + \cdots + 58736896)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 5776 T^{6} + \cdots + 1274875842816)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 5584 T^{6} + \cdots + 7578747136)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 11488 T^{6} + \cdots + 3582449565696)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 12112 T^{6} + \cdots + 58107995136)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 9328 T^{6} + \cdots + 21234991124736)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 40 T^{3} - 3736 T^{2} + \cdots - 42096)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + 14096 T^{6} + \cdots + 75851279421696)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 23936 T^{6} + \cdots + 842087039037696)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 22200 T^{6} + \cdots + 48965006250000)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 7376 T^{2} + 7795264)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} - 5728 T^{6} + \cdots + 2702420361216)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 26224 T^{6} + \cdots + 70331973325056)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 45688 T^{6} + \cdots + 348149175467536)^{2} \) Copy content Toggle raw display
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