Properties

Label 3800.2.d.q
Level $3800$
Weight $2$
Character orbit 3800.d
Analytic conductor $30.343$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.3431527681\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \( x^{12} + 24x^{10} + 194x^{8} + 618x^{6} + 733x^{4} + 286x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{8} - \beta_1) q^{7} + (\beta_{4} - \beta_{3} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{8} - \beta_1) q^{7} + (\beta_{4} - \beta_{3} - 1) q^{9} + ( - \beta_{10} + \beta_{9} + \beta_{6} - \beta_{3}) q^{11} + ( - \beta_{11} - \beta_{8} - \beta_{2}) q^{13} + ( - \beta_{8} + 3 \beta_{7} + \beta_{5} - \beta_1) q^{17} + q^{19} + (\beta_{9} - \beta_{4} + \beta_{3} + 2) q^{21} + (\beta_{11} + \beta_{8} + 2 \beta_{7} - 2 \beta_{5} - \beta_1) q^{23} + (2 \beta_{8} - \beta_{7} - 2 \beta_{5} - 3 \beta_1) q^{27} + (\beta_{10} + \beta_{9} + \beta_{3} - 2) q^{29} + (2 \beta_{10} - \beta_{9} + \beta_{4} + 2 \beta_{3} + 1) q^{31} + ( - \beta_{11} - \beta_{8}) q^{33} + (2 \beta_{7} - \beta_{5} - 3 \beta_1) q^{37} + (\beta_{10} - 2 \beta_{9} + \beta_{4} + 3 \beta_{3} - 2) q^{39} + ( - \beta_{10} - \beta_{9} + \beta_{4} + \beta_{3}) q^{41} + ( - \beta_{8} + 2 \beta_{7} + 2 \beta_{5} + \beta_{2} + 2 \beta_1) q^{43} + ( - \beta_{11} - \beta_{8} - \beta_{7}) q^{47} + ( - 3 \beta_{10} + 2 \beta_{6} - \beta_{4} - \beta_{3} - 4) q^{49} + ( - \beta_{10} - \beta_{9} + \beta_{6} - \beta_{4} - 3 \beta_{3} + 7) q^{51} + (\beta_{7} + \beta_{5} + 3 \beta_{2}) q^{53} + \beta_1 q^{57} + (\beta_{10} + 4 \beta_{3} - 1) q^{59} + ( - \beta_{9} - \beta_{4} + 4 \beta_{3} + 1) q^{61} + ( - \beta_{8} + 3 \beta_{7} + 3 \beta_{5} - \beta_{2} + 6 \beta_1) q^{63} + (2 \beta_{7} + 2 \beta_{5} - \beta_{2} - \beta_1) q^{67} + (\beta_{9} - 2 \beta_{4} - 3 \beta_{3} + 3) q^{69} + (2 \beta_{10} - \beta_{9} - \beta_{6} + 2 \beta_{4} + 2 \beta_{3} + 3) q^{71} + (\beta_{11} + 2 \beta_{7} - 2 \beta_{5} - 3 \beta_{2} - \beta_1) q^{73} + ( - \beta_{11} - 3 \beta_{8} + 6 \beta_{7} + 4 \beta_{5} + 2 \beta_{2}) q^{77} + ( - 3 \beta_{10} + 2 \beta_{9} - 2 \beta_{4} + \beta_{3} + 1) q^{79} + (2 \beta_{10} + 2 \beta_{9} - 2 \beta_{6} + 3 \beta_{3} + 3) q^{81} + (\beta_{11} + 5 \beta_{7} + \beta_{5} + \beta_{2} + \beta_1) q^{83} + (\beta_{11} - 3 \beta_{8} + 5 \beta_{7} + 2 \beta_{5} - \beta_{2} + 2 \beta_1) q^{87} + (\beta_{10} - 2 \beta_{9} + \beta_{6} - \beta_{4} + \beta_{3} - 1) q^{89} + ( - 2 \beta_{10} - \beta_{4} - \beta_{3} + 7) q^{91} + (3 \beta_{11} + 2 \beta_{8} + 7 \beta_{7} - \beta_{5} + \beta_{2} - \beta_1) q^{93} + (\beta_{11} + 2 \beta_{8} + 2 \beta_{7} - 4 \beta_{5} - \beta_{2} - 2 \beta_1) q^{97} + ( - \beta_{10} + 2 \beta_{9} + \beta_{6} + \beta_{4} + \beta_{3} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{9} + 6 q^{11} + 12 q^{19} + 30 q^{21} - 18 q^{29} + 10 q^{31} - 24 q^{39} + 6 q^{41} - 44 q^{49} + 66 q^{51} + 18 q^{61} + 22 q^{69} + 38 q^{71} + 32 q^{79} + 52 q^{81} - 28 q^{89} + 84 q^{91} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 24x^{10} + 194x^{8} + 618x^{6} + 733x^{4} + 286x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} + 27\nu^{9} + 260\nu^{7} + 1068\nu^{5} + 1702\nu^{3} + 637\nu ) / 60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -8\nu^{10} - 181\nu^{8} - 1320\nu^{6} - 3399\nu^{4} - 2321\nu^{2} - 126 ) / 270 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -8\nu^{10} - 181\nu^{8} - 1320\nu^{6} - 3399\nu^{4} - 2051\nu^{2} + 954 ) / 270 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{9} - 23\nu^{7} - 171\nu^{5} - 450\nu^{3} - 313\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -19\nu^{10} - 413\nu^{8} - 2730\nu^{6} - 5052\nu^{4} + 1592\nu^{2} + 2367 ) / 540 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -14\nu^{11} - 328\nu^{9} - 2535\nu^{7} - 7332\nu^{5} - 6863\nu^{3} - 1683\nu ) / 270 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -14\nu^{11} - 373\nu^{9} - 3570\nu^{7} - 15027\nu^{5} - 26843\nu^{3} - 13338\nu ) / 540 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -37\nu^{10} - 854\nu^{8} - 6375\nu^{6} - 16581\nu^{4} - 9334\nu^{2} + 1206 ) / 540 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 14\nu^{10} + 328\nu^{8} + 2535\nu^{6} + 7332\nu^{4} + 6773\nu^{2} + 1053 ) / 180 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 16\nu^{11} + 377\nu^{9} + 2940\nu^{7} + 8643\nu^{5} + 8377\nu^{3} + 2112\nu ) / 90 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{8} - \beta_{7} - 2\beta_{5} - 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{10} + 2\beta_{9} - 2\beta_{6} - 9\beta_{4} + 12\beta_{3} + 30 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{11} - 24\beta_{8} + 21\beta_{7} + 22\beta_{5} - 4\beta_{2} + 84\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -32\beta_{10} - 28\beta_{9} + 36\beta_{6} + 81\beta_{4} - 143\beta_{3} - 261 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -62\beta_{11} + 250\beta_{8} - 317\beta_{7} - 222\beta_{5} + 64\beta_{2} - 816\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 410\beta_{10} + 314\beta_{9} - 474\beta_{6} - 754\beta_{4} + 1667\beta_{3} + 2420 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 913\beta_{11} - 2546\beta_{8} + 4150\beta_{7} + 2232\beta_{5} - 788\beta_{2} + 8141\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -4846\beta_{10} - 3334\beta_{9} + 5634\beta_{6} + 7228\beta_{4} - 18963\beta_{3} - 23289 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -11735\beta_{11} + 25970\beta_{8} - 50356\beta_{7} - 22636\beta_{5} + 8968\beta_{2} - 82678\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(951\) \(1901\) \(1977\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3649.1
3.26143i
2.77008i
1.93590i
1.08999i
0.848258i
0.185519i
0.185519i
0.848258i
1.08999i
1.93590i
2.77008i
3.26143i
0 3.26143i 0 0 0 4.07225i 0 −7.63693 0
3649.2 0 2.77008i 0 0 0 2.31077i 0 −4.67334 0
3649.3 0 1.93590i 0 0 0 1.24708i 0 −0.747704 0
3649.4 0 1.08999i 0 0 0 4.19727i 0 1.81192 0
3649.5 0 0.848258i 0 0 0 1.74484i 0 2.28046 0
3649.6 0 0.185519i 0 0 0 4.45651i 0 2.96558 0
3649.7 0 0.185519i 0 0 0 4.45651i 0 2.96558 0
3649.8 0 0.848258i 0 0 0 1.74484i 0 2.28046 0
3649.9 0 1.08999i 0 0 0 4.19727i 0 1.81192 0
3649.10 0 1.93590i 0 0 0 1.24708i 0 −0.747704 0
3649.11 0 2.77008i 0 0 0 2.31077i 0 −4.67334 0
3649.12 0 3.26143i 0 0 0 4.07225i 0 −7.63693 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3649.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3800.2.d.q 12
5.b even 2 1 inner 3800.2.d.q 12
5.c odd 4 1 3800.2.a.ba 6
5.c odd 4 1 3800.2.a.bc yes 6
20.e even 4 1 7600.2.a.ch 6
20.e even 4 1 7600.2.a.cl 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3800.2.a.ba 6 5.c odd 4 1
3800.2.a.bc yes 6 5.c odd 4 1
3800.2.d.q 12 1.a even 1 1 trivial
3800.2.d.q 12 5.b even 2 1 inner
7600.2.a.ch 6 20.e even 4 1
7600.2.a.cl 6 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3800, [\chi])\):

\( T_{3}^{12} + 24T_{3}^{10} + 194T_{3}^{8} + 618T_{3}^{6} + 733T_{3}^{4} + 286T_{3}^{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{12} + 64T_{7}^{10} + 1538T_{7}^{8} + 17066T_{7}^{6} + 87493T_{7}^{4} + 194534T_{7}^{2} + 146689 \) Copy content Toggle raw display
\( T_{11}^{6} - 3T_{11}^{5} - 40T_{11}^{4} + 139T_{11}^{3} + 10T_{11}^{2} - 208T_{11} - 88 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 24 T^{10} + 194 T^{8} + 618 T^{6} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 64 T^{10} + 1538 T^{8} + \cdots + 146689 \) Copy content Toggle raw display
$11$ \( (T^{6} - 3 T^{5} - 40 T^{4} + 139 T^{3} + \cdots - 88)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 131 T^{10} + 6021 T^{8} + \cdots + 680625 \) Copy content Toggle raw display
$17$ \( T^{12} + 152 T^{10} + 8082 T^{8} + \cdots + 9903609 \) Copy content Toggle raw display
$19$ \( (T - 1)^{12} \) Copy content Toggle raw display
$23$ \( T^{12} + 196 T^{10} + 13806 T^{8} + \cdots + 6185169 \) Copy content Toggle raw display
$29$ \( (T^{6} + 9 T^{5} - 67 T^{4} - 705 T^{3} + \cdots + 21951)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 5 T^{5} - 136 T^{4} + 599 T^{3} + \cdots + 11000)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 244 T^{10} + \cdots + 54243225 \) Copy content Toggle raw display
$41$ \( (T^{6} - 3 T^{5} - 180 T^{4} + \cdots - 113472)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 199 T^{10} + \cdots + 37454400 \) Copy content Toggle raw display
$47$ \( T^{12} + 120 T^{10} + 5136 T^{8} + \cdots + 3207681 \) Copy content Toggle raw display
$53$ \( T^{12} + 553 T^{10} + \cdots + 86211225 \) Copy content Toggle raw display
$59$ \( (T^{6} - 193 T^{4} - 255 T^{3} + \cdots - 32328)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 9 T^{5} - 132 T^{4} + 443 T^{3} + \cdots + 12424)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 407 T^{10} + \cdots + 18504977089 \) Copy content Toggle raw display
$71$ \( (T^{6} - 19 T^{5} - 75 T^{4} + 3491 T^{3} + \cdots - 27576)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 619 T^{10} + \cdots + 549129907089 \) Copy content Toggle raw display
$79$ \( (T^{6} - 16 T^{5} - 285 T^{4} + \cdots - 553536)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 309 T^{10} + \cdots + 5063745600 \) Copy content Toggle raw display
$89$ \( (T^{6} + 14 T^{5} - 67 T^{4} - 727 T^{3} + \cdots + 3240)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 537 T^{10} + \cdots + 83400819264 \) Copy content Toggle raw display
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