Properties

Label 2100.2.d.g
Level 2100
Weight 2
Character orbit 2100.d
Analytic conductor 16.769
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2100.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 420)
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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{3} ) q^{3} + ( 1 + \beta_{2} + \beta_{3} ) q^{7} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{3} ) q^{3} + ( 1 + \beta_{2} + \beta_{3} ) q^{7} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{9} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{11} + ( \beta_{1} + \beta_{3} ) q^{13} + ( -2 - 2 \beta_{1} + 2 \beta_{3} ) q^{17} + ( -\beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{19} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{21} + ( -3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{23} + ( -3 + \beta_{1} + 2 \beta_{2} ) q^{27} + 2 \beta_{2} q^{29} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{31} + ( -2 + 3 \beta_{1} - \beta_{3} ) q^{33} + ( -2 - 2 \beta_{1} + 2 \beta_{3} ) q^{37} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{39} + ( -6 + 2 \beta_{1} - 2 \beta_{3} ) q^{41} + ( 6 + \beta_{1} - \beta_{3} ) q^{43} + ( 2 + \beta_{1} - \beta_{3} ) q^{47} + ( -3 - 3 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{49} + ( 6 + 2 \beta_{1} - 2 \beta_{2} ) q^{51} + ( -\beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{53} + ( 2 + 5 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{57} + ( 4 + 4 \beta_{1} - 4 \beta_{3} ) q^{59} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{61} + ( 3 + 5 \beta_{1} + \beta_{2} ) q^{63} + ( -6 - 3 \beta_{1} + 3 \beta_{3} ) q^{67} + ( 6 + \beta_{1} + 5 \beta_{2} + 3 \beta_{3} ) q^{69} + ( -5 \beta_{1} - 3 \beta_{2} - 5 \beta_{3} ) q^{71} + ( \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{73} + ( 2 + 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{77} + ( 8 + 2 \beta_{1} - 2 \beta_{3} ) q^{79} + ( 1 - 4 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{81} + ( 6 + \beta_{1} - \beta_{3} ) q^{83} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{87} + ( -10 - 2 \beta_{1} + 2 \beta_{3} ) q^{89} + ( -2 + \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{91} + ( -2 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{93} + ( -5 \beta_{1} - 4 \beta_{2} - 5 \beta_{3} ) q^{97} + ( -4 - \beta_{1} + \beta_{2} - 5 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{3} + 6q^{7} + O(q^{10}) \) \( 4q - 2q^{3} + 6q^{7} - 14q^{27} - 16q^{33} - 12q^{39} - 32q^{41} + 20q^{43} + 4q^{47} + 20q^{51} + 2q^{63} - 12q^{67} + 28q^{69} + 8q^{77} + 24q^{79} + 4q^{81} + 20q^{83} + 8q^{87} - 32q^{89} - 4q^{91} - 8q^{93} - 24q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} + \nu + 1 \)
\(\beta_{2}\)\(=\)\( 2 \nu^{3} + 4 \nu \)
\(\beta_{3}\)\(=\)\( -\nu^{2} + \nu - 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + \beta_{1} - 2\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{3} + \beta_{2} - 2 \beta_{1}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1301.1
0.618034i
0.618034i
1.61803i
1.61803i
0 −1.61803 0.618034i 0 0 0 0.381966 2.61803i 0 2.23607 + 2.00000i 0
1301.2 0 −1.61803 + 0.618034i 0 0 0 0.381966 + 2.61803i 0 2.23607 2.00000i 0
1301.3 0 0.618034 1.61803i 0 0 0 2.61803 + 0.381966i 0 −2.23607 2.00000i 0
1301.4 0 0.618034 + 1.61803i 0 0 0 2.61803 0.381966i 0 −2.23607 + 2.00000i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.d.g 4
3.b odd 2 1 2100.2.d.h 4
5.b even 2 1 420.2.d.d yes 4
5.c odd 4 1 2100.2.f.b 4
5.c odd 4 1 2100.2.f.h 4
7.b odd 2 1 2100.2.d.h 4
15.d odd 2 1 420.2.d.c 4
15.e even 4 1 2100.2.f.a 4
15.e even 4 1 2100.2.f.g 4
20.d odd 2 1 1680.2.f.f 4
21.c even 2 1 inner 2100.2.d.g 4
35.c odd 2 1 420.2.d.c 4
35.f even 4 1 2100.2.f.a 4
35.f even 4 1 2100.2.f.g 4
60.h even 2 1 1680.2.f.j 4
105.g even 2 1 420.2.d.d yes 4
105.k odd 4 1 2100.2.f.b 4
105.k odd 4 1 2100.2.f.h 4
140.c even 2 1 1680.2.f.j 4
420.o odd 2 1 1680.2.f.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.d.c 4 15.d odd 2 1
420.2.d.c 4 35.c odd 2 1
420.2.d.d yes 4 5.b even 2 1
420.2.d.d yes 4 105.g even 2 1
1680.2.f.f 4 20.d odd 2 1
1680.2.f.f 4 420.o odd 2 1
1680.2.f.j 4 60.h even 2 1
1680.2.f.j 4 140.c even 2 1
2100.2.d.g 4 1.a even 1 1 trivial
2100.2.d.g 4 21.c even 2 1 inner
2100.2.d.h 4 3.b odd 2 1
2100.2.d.h 4 7.b odd 2 1
2100.2.f.a 4 15.e even 4 1
2100.2.f.a 4 35.f even 4 1
2100.2.f.b 4 5.c odd 4 1
2100.2.f.b 4 105.k odd 4 1
2100.2.f.g 4 15.e even 4 1
2100.2.f.g 4 35.f even 4 1
2100.2.f.h 4 5.c odd 4 1
2100.2.f.h 4 105.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2100, [\chi])\):

\( T_{11}^{4} + 28 T_{11}^{2} + 16 \)
\( T_{13}^{4} + 12 T_{13}^{2} + 16 \)
\( T_{17}^{2} - 20 \)
\( T_{37}^{2} - 20 \)
\( T_{41}^{2} + 16 T_{41} + 44 \)
\( T_{43}^{2} - 10 T_{43} + 20 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} \)
$5$ 1
$7$ \( 1 - 6 T + 18 T^{2} - 42 T^{3} + 49 T^{4} \)
$11$ \( 1 - 16 T^{2} + 126 T^{4} - 1936 T^{6} + 14641 T^{8} \)
$13$ \( 1 - 40 T^{2} + 718 T^{4} - 6760 T^{6} + 28561 T^{8} \)
$17$ \( ( 1 + 14 T^{2} + 289 T^{4} )^{2} \)
$19$ \( 1 - 16 T^{2} + 286 T^{4} - 5776 T^{6} + 130321 T^{8} \)
$23$ \( ( 1 - 2 T + 2 T^{2} - 46 T^{3} + 529 T^{4} )( 1 + 2 T + 2 T^{2} + 46 T^{3} + 529 T^{4} ) \)
$29$ \( ( 1 - 10 T + 29 T^{2} )^{2}( 1 + 10 T + 29 T^{2} )^{2} \)
$31$ \( 1 - 112 T^{2} + 5038 T^{4} - 107632 T^{6} + 923521 T^{8} \)
$37$ \( ( 1 + 54 T^{2} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 + 16 T + 126 T^{2} + 656 T^{3} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 10 T + 106 T^{2} - 430 T^{3} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 - 2 T + 90 T^{2} - 94 T^{3} + 2209 T^{4} )^{2} \)
$53$ \( 1 - 104 T^{2} + 7342 T^{4} - 292136 T^{6} + 7890481 T^{8} \)
$59$ \( ( 1 + 38 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( 1 - 52 T^{2} + 2998 T^{4} - 193492 T^{6} + 13845841 T^{8} \)
$67$ \( ( 1 + 6 T + 98 T^{2} + 402 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( 1 - 32 T^{2} + 9838 T^{4} - 161312 T^{6} + 25411681 T^{8} \)
$73$ \( 1 - 120 T^{2} + 12638 T^{4} - 639480 T^{6} + 28398241 T^{8} \)
$79$ \( ( 1 - 12 T + 174 T^{2} - 948 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 - 10 T + 186 T^{2} - 830 T^{3} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 + 16 T + 222 T^{2} + 1424 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( 1 - 120 T^{2} + 17918 T^{4} - 1129080 T^{6} + 88529281 T^{8} \)
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