Properties

Label 1900.2.l.a
Level $1900$
Weight $2$
Character orbit 1900.l
Analytic conductor $15.172$
Analytic rank $0$
Dimension $8$
CM discriminant -19
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1900.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.1715763840\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.2702336256.1
Defining polynomial: \(x^{8} + 9 x^{6} + 56 x^{4} + 225 x^{2} + 625\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{3}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 380)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{1} + \beta_{5} ) q^{7} -3 \beta_{1} q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{1} + \beta_{5} ) q^{7} -3 \beta_{1} q^{9} + ( -1 + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{11} + ( -1 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} ) q^{17} -\beta_{3} q^{19} + ( -2 + 2 \beta_{1} - \beta_{3} - \beta_{7} ) q^{23} + ( \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{43} + ( 2 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 3 \beta_{5} - \beta_{7} ) q^{47} + ( 6 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} ) q^{49} + ( 3 - 3 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{61} + ( 3 - 3 \beta_{1} + 3 \beta_{6} ) q^{63} + ( -4 + 4 \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{73} + ( -5 - 5 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 3 \beta_{7} ) q^{77} -9 q^{81} + ( 8 - 8 \beta_{1} - \beta_{3} - \beta_{7} ) q^{83} + ( 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{5} - 3 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 6q^{7} + O(q^{10}) \) \( 8q + 6q^{7} - 14q^{17} - 16q^{23} - 2q^{43} + 26q^{47} + 18q^{63} - 22q^{73} - 26q^{77} - 72q^{81} + 64q^{83} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 9 x^{6} + 56 x^{4} + 225 x^{2} + 625\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 9 \nu^{7} + 56 \nu^{5} + 154 \nu^{3} + 625 \nu \)\()/1750\)
\(\beta_{2}\)\(=\)\((\)\( -2 \nu^{7} - 9 \nu^{6} - 28 \nu^{5} - 56 \nu^{4} - 252 \nu^{3} - 504 \nu^{2} - 1710 \nu - 1325 \)\()/700\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} - 27 \)\()/28\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} - 20 \nu^{6} - 16 \nu^{5} - 180 \nu^{4} - 44 \nu^{3} - 620 \nu^{2} - 175 \nu - 2000 \)\()/500\)
\(\beta_{5}\)\(=\)\((\)\( -7 \nu^{7} - \nu^{6} - 28 \nu^{5} - 84 \nu^{4} - 252 \nu^{3} - 56 \nu^{2} - 1015 \nu - 925 \)\()/700\)
\(\beta_{6}\)\(=\)\((\)\( -7 \nu^{7} - 135 \nu^{6} + 112 \nu^{5} - 840 \nu^{4} + 308 \nu^{3} - 4060 \nu^{2} + 4725 \nu - 12875 \)\()/3500\)
\(\beta_{7}\)\(=\)\((\)\( 9 \nu^{7} + 56 \nu^{5} + 404 \nu^{3} + 625 \nu \)\()/500\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-3 \beta_{7} + \beta_{6} - 3 \beta_{5} + \beta_{4} + \beta_{3} - 3 \beta_{2} + 3 \beta_{1} - 1\)\()/10\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} - 7 \beta_{6} + 11 \beta_{5} + 3 \beta_{4} - 7 \beta_{3} - 9 \beta_{2} + 9 \beta_{1} - 23\)\()/10\)
\(\nu^{3}\)\(=\)\(2 \beta_{7} - 7 \beta_{1}\)
\(\nu^{4}\)\(=\)\((\)\(-9 \beta_{7} + 13 \beta_{6} - 49 \beta_{5} - 27 \beta_{4} - 7 \beta_{3} + 31 \beta_{2} - 31 \beta_{1} - 73\)\()/10\)
\(\nu^{5}\)\(=\)\((\)\(23 \beta_{7} + 79 \beta_{6} + 123 \beta_{5} - 101 \beta_{4} - 11 \beta_{3} + 33 \beta_{2} + 247 \beta_{1} + 101\)\()/10\)
\(\nu^{6}\)\(=\)\(28 \beta_{3} + 27\)
\(\nu^{7}\)\(=\)\((\)\(-277 \beta_{7} - 561 \beta_{6} - 557 \beta_{5} + 559 \beta_{4} - \beta_{3} + 3 \beta_{2} + 1397 \beta_{1} - 559\)\()/10\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(\beta_{1}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
493.1
−0.656712 2.13746i
0.656712 2.13746i
1.52274 + 1.63746i
−1.52274 + 1.63746i
−0.656712 + 2.13746i
0.656712 + 2.13746i
1.52274 1.63746i
−1.52274 1.63746i
0 0 0 0 0 −3.52622 + 3.52622i 0 3.00000i 0
493.2 0 0 0 0 0 1.25130 1.25130i 0 3.00000i 0
493.3 0 0 0 0 0 2.42815 2.42815i 0 3.00000i 0
493.4 0 0 0 0 0 2.84677 2.84677i 0 3.00000i 0
1557.1 0 0 0 0 0 −3.52622 3.52622i 0 3.00000i 0
1557.2 0 0 0 0 0 1.25130 + 1.25130i 0 3.00000i 0
1557.3 0 0 0 0 0 2.42815 + 2.42815i 0 3.00000i 0
1557.4 0 0 0 0 0 2.84677 + 2.84677i 0 3.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1557.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
5.c odd 4 1 inner
95.g even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.2.l.a 8
5.b even 2 1 380.2.l.a 8
5.c odd 4 1 380.2.l.a 8
5.c odd 4 1 inner 1900.2.l.a 8
15.d odd 2 1 3420.2.bb.c 8
15.e even 4 1 3420.2.bb.c 8
19.b odd 2 1 CM 1900.2.l.a 8
95.d odd 2 1 380.2.l.a 8
95.g even 4 1 380.2.l.a 8
95.g even 4 1 inner 1900.2.l.a 8
285.b even 2 1 3420.2.bb.c 8
285.j odd 4 1 3420.2.bb.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.l.a 8 5.b even 2 1
380.2.l.a 8 5.c odd 4 1
380.2.l.a 8 95.d odd 2 1
380.2.l.a 8 95.g even 4 1
1900.2.l.a 8 1.a even 1 1 trivial
1900.2.l.a 8 5.c odd 4 1 inner
1900.2.l.a 8 19.b odd 2 1 CM
1900.2.l.a 8 95.g even 4 1 inner
3420.2.bb.c 8 15.d odd 2 1
3420.2.bb.c 8 15.e even 4 1
3420.2.bb.c 8 285.b even 2 1
3420.2.bb.c 8 285.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{2}^{\mathrm{new}}(1900, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( T^{8} \)
$7$ \( 14884 - 19032 T + 12168 T^{2} - 3696 T^{3} + 605 T^{4} - 42 T^{5} + 18 T^{6} - 6 T^{7} + T^{8} \)
$11$ \( ( 196 - 47 T^{2} + T^{4} )^{2} \)
$13$ \( T^{8} \)
$17$ \( 412164 + 323568 T + 127008 T^{2} + 18564 T^{3} + 1645 T^{4} + 238 T^{5} + 98 T^{6} + 14 T^{7} + T^{8} \)
$19$ \( ( 19 + T^{2} )^{4} \)
$23$ \( ( 900 - 240 T + 32 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$29$ \( T^{8} \)
$31$ \( T^{8} \)
$37$ \( T^{8} \)
$41$ \( T^{8} \)
$43$ \( 2815684 - 718184 T + 91592 T^{2} + 69832 T^{3} + 25885 T^{4} + 86 T^{5} + 2 T^{6} + 2 T^{7} + T^{8} \)
$47$ \( 1004004 - 1719432 T + 1472328 T^{2} - 58656 T^{3} + 2365 T^{4} - 1222 T^{5} + 338 T^{6} - 26 T^{7} + T^{8} \)
$53$ \( T^{8} \)
$59$ \( T^{8} \)
$61$ \( ( 26896 - 347 T^{2} + T^{4} )^{2} \)
$67$ \( T^{8} \)
$71$ \( T^{8} \)
$73$ \( 235991044 + 82463216 T + 14407712 T^{2} + 1255892 T^{3} + 59965 T^{4} + 1606 T^{5} + 242 T^{6} + 22 T^{7} + T^{8} \)
$79$ \( T^{8} \)
$83$ \( ( 8100 - 2880 T + 512 T^{2} - 32 T^{3} + T^{4} )^{2} \)
$89$ \( T^{8} \)
$97$ \( T^{8} \)
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