Properties

Label 3150.3.c.f
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: \( 2^{28} \)
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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} + 2 q^{4} + \beta_{2} q^{7} - 2 \beta_{6} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} + 2 q^{4} + \beta_{2} q^{7} - 2 \beta_{6} q^{8} + (\beta_{12} + 2 \beta_{7} - \beta_{5}) q^{11} + (\beta_{14} + 2 \beta_{4} + 4 \beta_{2}) q^{13} - \beta_{12} q^{14} + 4 q^{16} + (\beta_{13} + 2 \beta_{11} + 6 \beta_{6} - \beta_1) q^{17} + ( - \beta_{15} + \beta_{9} + 10) q^{19} + ( - 2 \beta_{10} + \beta_{4} - 2 \beta_{2}) q^{22} + (3 \beta_{11} + 2 \beta_{6} + \beta_1) q^{23} + ( - 4 \beta_{12} - 4 \beta_{5} - \beta_{3}) q^{26} + 2 \beta_{2} q^{28} + ( - 2 \beta_{12} + 2 \beta_{7} + 13 \beta_{5} + 2 \beta_{3}) q^{29} + (2 \beta_{15} + 3 \beta_{9} - 2 \beta_{8} + 14) q^{31} - 4 \beta_{6} q^{32} + ( - \beta_{15} + 2 \beta_{9} + 2 \beta_{8} - 12) q^{34} + ( - 4 \beta_{14} + 3 \beta_{10} + 8 \beta_{4} + 10 \beta_{2}) q^{37} + (2 \beta_{13} - 10 \beta_{6} - \beta_1) q^{38} + ( - 6 \beta_{12} - 6 \beta_{7} + 20 \beta_{5} - \beta_{3}) q^{41} + (6 \beta_{14} + 2 \beta_{10} + 4 \beta_{4} - 2 \beta_{2}) q^{43} + (2 \beta_{12} + 4 \beta_{7} - 2 \beta_{5}) q^{44} + ( - 2 \beta_{9} + 3 \beta_{8} - 4) q^{46} + ( - 3 \beta_{13} - 2 \beta_{11} + 5 \beta_1) q^{47} - 7 q^{49} + (2 \beta_{14} + 4 \beta_{4} + 8 \beta_{2}) q^{52} + (2 \beta_{13} - 3 \beta_{11} + 15 \beta_{6} - 7 \beta_1) q^{53} - 2 \beta_{12} q^{56} + ( - 4 \beta_{14} - 2 \beta_{10} - 13 \beta_{4} + 4 \beta_{2}) q^{58} + ( - 12 \beta_{12} - 9 \beta_{7} + 2 \beta_{5} + 2 \beta_{3}) q^{59} + (2 \beta_{15} - 6 \beta_{9} + 6 \beta_{8} - 18) q^{61} + ( - 4 \beta_{13} - 4 \beta_{11} - 14 \beta_{6} - 3 \beta_1) q^{62} + 8 q^{64} + ( - 4 \beta_{14} - \beta_{10} - 4 \beta_{4} + 14 \beta_{2}) q^{67} + (2 \beta_{13} + 4 \beta_{11} + 12 \beta_{6} - 2 \beta_1) q^{68} + ( - 13 \beta_{12} + 4 \beta_{7} + 7 \beta_{5}) q^{71} + (3 \beta_{14} + 4 \beta_{10} + 24 \beta_{2}) q^{73} + ( - 10 \beta_{12} - 6 \beta_{7} - 16 \beta_{5} + 4 \beta_{3}) q^{74} + ( - 2 \beta_{15} + 2 \beta_{9} + 20) q^{76} + ( - 2 \beta_{13} - \beta_{11} - 7 \beta_{6}) q^{77} + ( - 2 \beta_{15} - 14 \beta_{9} - 12 \beta_{8} + 8) q^{79} + (2 \beta_{14} + 6 \beta_{10} - 20 \beta_{4} + 12 \beta_{2}) q^{82} + ( - \beta_{13} + 10 \beta_{11} - 16 \beta_{6} + 23 \beta_1) q^{83} + (2 \beta_{12} - 4 \beta_{7} - 8 \beta_{5} - 6 \beta_{3}) q^{86} + ( - 4 \beta_{10} + 2 \beta_{4} - 4 \beta_{2}) q^{88} + (\beta_{7} - 26 \beta_{5} - 4 \beta_{3}) q^{89} + ( - 7 \beta_{9} - 2 \beta_{8} - 28) q^{91} + (6 \beta_{11} + 4 \beta_{6} + 2 \beta_1) q^{92} + (3 \beta_{15} - 10 \beta_{9} - 2 \beta_{8}) q^{94} + ( - 5 \beta_{14} - 2 \beta_{10} - 2 \beta_{4} + 32 \beta_{2}) q^{97} + 7 \beta_{6} 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 \( 16 q + 32 q^{4} + 64 q^{16} + 160 q^{19} + 224 q^{31} - 192 q^{34} - 64 q^{46} - 112 q^{49} - 288 q^{61} + 128 q^{64} + 320 q^{76} + 128 q^{79} - 448 q^{91}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( -\nu^{12} + 105\nu^{8} + 15\nu^{4} + 42056 ) / 9396 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -14\nu^{12} - 96\nu^{8} - 10752\nu^{4} - 815 ) / 2349 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{12} - 7\nu^{8} - 737\nu^{4} - 56 ) / 36 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 33\nu^{14} + 247\nu^{10} + 25025\nu^{6} + 18304\nu^{2} ) / 8352 \) 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
\(\beta_{6}\)\(=\) \( ( 245 \nu^{15} + 374 \nu^{13} + 1419 \nu^{11} + 2490 \nu^{9} + 182157 \nu^{7} + 278358 \nu^{5} - 185272 \nu^{3} - 39712 \nu ) / 150336 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 23\nu^{14} + 177\nu^{10} + 17367\nu^{6} + 12704\nu^{2} ) / 2592 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 31\nu^{14} + 201\nu^{10} + 23295\nu^{6} - 10112\nu^{2} ) / 2592 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 91 \nu^{15} + 137 \nu^{13} + 537 \nu^{11} + 927 \nu^{9} + 67887 \nu^{7} + 103737 \nu^{5} - 69044 \nu^{3} - 14800 \nu ) / 25056 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 91 \nu^{15} - 137 \nu^{13} + 537 \nu^{11} - 927 \nu^{9} + 67887 \nu^{7} - 103737 \nu^{5} - 69044 \nu^{3} + 14800 \nu ) / 12528 \) 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
\(\beta_{12}\)\(=\) \( ( 403 \nu^{15} + 122 \nu^{13} + 2925 \nu^{11} + 1110 \nu^{9} + 303675 \nu^{7} + 92826 \nu^{5} + 115960 \nu^{3} + 149024 \nu ) / 50112 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 25\nu^{14} + 159\nu^{10} + 18649\nu^{6} - 8096\nu^{2} ) / 928 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1349 \nu^{15} + 407 \nu^{13} + 9735 \nu^{11} + 3201 \nu^{9} + 1018545 \nu^{7} + 311271 \nu^{5} + 388916 \nu^{3} + 499664 \nu ) / 75168 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1349 \nu^{15} + 407 \nu^{13} - 9735 \nu^{11} + 3201 \nu^{9} - 1018545 \nu^{7} + 311271 \nu^{5} - 388916 \nu^{3} + 499664 \nu ) / 37584 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{15} + 2\beta_{14} - 2\beta_{12} + 2\beta_{11} + \beta_{10} - 2\beta_{9} + 2\beta_{6} - 2\beta_{5} ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{13} - 3\beta_{8} - 3\beta_{7} + 9\beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{15} + 2\beta_{14} - 4\beta_{12} - 4\beta_{11} - 5\beta_{10} - 10\beta_{9} + 20\beta_{6} + 20\beta_{5} ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9\beta_{3} - 42\beta_{2} + 3\beta _1 - 14 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5 \beta_{15} + 10 \beta_{14} - 22 \beta_{12} + 22 \beta_{11} - 55 \beta_{10} + 110 \beta_{9} - 242 \beta_{6} + 242 \beta_{5} ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -20\beta_{13} + 45\beta_{8} - 36\beta_{7} + 81\beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 91 \beta_{15} + 182 \beta_{14} - 406 \beta_{12} - 406 \beta_{11} + 169 \beta_{10} + 338 \beta_{9} - 754 \beta_{6} - 754 \beta_{5} ) / 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -63\beta_{3} + 282\beta_{2} + 651\beta _1 - 2914 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 289 \beta_{15} - 578 \beta_{14} + 1292 \beta_{12} - 1292 \beta_{11} - 85 \beta_{10} + 170 \beta_{9} - 380 \beta_{6} + 380 \beta_{5} ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -605\beta_{13} + 1353\beta_{8} + 3135\beta_{7} - 7011\beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 2047 \beta_{15} - 4094 \beta_{14} + 9154 \beta_{12} + 9154 \beta_{11} + 5963 \beta_{10} + 11926 \beta_{9} - 26666 \beta_{6} - 26666 \beta_{5} ) / 16 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( -1620\beta_{3} + 7245\beta_{2} - 1692\beta _1 + 7567 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 233 \beta_{15} + 466 \beta_{14} - 1042 \beta_{12} + 1042 \beta_{11} + 42173 \beta_{10} - 84346 \beta_{9} + 188602 \beta_{6} - 188602 \beta_{5} ) / 16 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 34307\beta_{13} - 76713\beta_{8} + 32799\beta_{7} - 73341\beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 27145 \beta_{15} - 54290 \beta_{14} + 121396 \beta_{12} + 121396 \beta_{11} - 83875 \beta_{10} - 167750 \beta_{9} + 375100 \beta_{6} + 375100 \beta_{5} ) / 8 \) Copy content Toggle raw display

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
0.359610 0.796626i
−0.796626 0.359610i
−0.941471 + 2.08559i
−0.941471 2.08559i
−0.796626 + 0.359610i
0.359610 + 0.796626i
2.08559 0.941471i
−0.359610 + 0.796626i
−2.08559 0.941471i
0.941471 2.08559i
0.796626 + 0.359610i
0.796626 0.359610i
0.941471 + 2.08559i
−2.08559 + 0.941471i
−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.f 16
3.b odd 2 1 inner 3150.3.c.f 16
5.b even 2 1 inner 3150.3.c.f 16
5.c odd 4 1 630.3.e.b 8
5.c odd 4 1 3150.3.e.f 8
15.d odd 2 1 inner 3150.3.c.f 16
15.e even 4 1 630.3.e.b 8
15.e even 4 1 3150.3.e.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.3.e.b 8 5.c odd 4 1
630.3.e.b 8 15.e even 4 1
3150.3.c.f 16 1.a even 1 1 trivial
3150.3.c.f 16 3.b odd 2 1 inner
3150.3.c.f 16 5.b even 2 1 inner
3150.3.c.f 16 15.d odd 2 1 inner
3150.3.e.f 8 5.c odd 4 1
3150.3.e.f 8 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{8} + 384T_{11}^{6} + 44832T_{11}^{4} + 1673216T_{11}^{2} + 15872256 \) 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
$11$ \( (T^{8} + 384 T^{6} + 44832 T^{4} + \cdots + 15872256)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 792 T^{6} + 149208 T^{4} + \cdots + 14470416)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 1152 T^{6} + 117888 T^{4} + \cdots + 331776)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 40 T^{3} + 20 T^{2} + 7600 T + 24900)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} - 616 T^{6} + 112792 T^{4} + \cdots + 80353296)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 4136 T^{6} + \cdots + 182633150736)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 56 T^{3} - 1468 T^{2} + \cdots - 53724)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + 9744 T^{6} + \cdots + 17685591019776)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 8656 T^{6} + \cdots + 11776317808896)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 11088 T^{6} + \cdots + 23646201405696)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 7264 T^{6} + \cdots + 147161235456)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 8464 T^{6} + \cdots + 8432658441216)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 16816 T^{6} + \cdots + 7297000079616)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 72 T^{3} - 3032 T^{2} + \cdots + 5776)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + 10384 T^{6} + \cdots + 913966592256)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 11136 T^{6} + \cdots + 13463498301696)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 21208 T^{6} + \cdots + 36190379032336)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 32 T^{3} - 13840 T^{2} + \cdots - 14801856)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} - 50528 T^{6} + \cdots + 36\!\cdots\!96)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 14448 T^{6} + \cdots + 416365629696)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 36376 T^{6} + \cdots + 615175117820176)^{2} \) Copy content Toggle raw display
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