Properties

Label 2100.2.q.g
Level 2100
Weight 2
Character orbit 2100.q
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.q (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{2} ) q^{3} -\beta_{1} q^{7} + \beta_{2} q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{2} ) q^{3} -\beta_{1} q^{7} + \beta_{2} q^{9} + ( 3 + \beta_{1} + 3 \beta_{2} ) q^{11} - q^{13} + ( 1 - \beta_{1} + \beta_{2} ) q^{17} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{19} + ( \beta_{1} + \beta_{3} ) q^{21} + 2 \beta_{2} q^{23} + q^{27} + ( 2 - 2 \beta_{3} ) q^{29} + ( -4 - 4 \beta_{2} ) q^{31} + ( -\beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{33} + ( -2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{37} + ( 1 + \beta_{2} ) q^{39} + ( -3 - \beta_{3} ) q^{41} + ( 2 - 2 \beta_{3} ) q^{43} + ( -3 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} ) q^{47} + 7 \beta_{2} q^{49} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{51} + ( 5 - \beta_{1} + 5 \beta_{2} ) q^{53} + ( -2 + \beta_{3} ) q^{57} + ( -7 - 3 \beta_{1} - 7 \beta_{2} ) q^{59} + ( 4 \beta_{1} - \beta_{2} + 4 \beta_{3} ) q^{61} -\beta_{3} q^{63} -3 \beta_{1} q^{67} + 2 q^{69} + ( -2 - 2 \beta_{3} ) q^{71} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{73} + ( -3 \beta_{1} - 7 \beta_{2} - 3 \beta_{3} ) q^{77} + ( -\beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{79} + ( -1 - \beta_{2} ) q^{81} + ( -1 + \beta_{3} ) q^{83} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{87} + ( -3 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{89} + \beta_{1} q^{91} + 4 \beta_{2} q^{93} + ( -9 + 2 \beta_{3} ) q^{97} + ( -3 + \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{3} - 2q^{9} + O(q^{10}) \) \( 4q - 2q^{3} - 2q^{9} + 6q^{11} - 4q^{13} + 2q^{17} + 4q^{19} - 4q^{23} + 4q^{27} + 8q^{29} - 8q^{31} + 6q^{33} + 6q^{37} + 2q^{39} - 12q^{41} + 8q^{43} - 10q^{47} - 14q^{49} + 2q^{51} + 10q^{53} - 8q^{57} - 14q^{59} + 2q^{61} + 8q^{69} - 8q^{71} - 2q^{73} + 14q^{77} + 8q^{79} - 2q^{81} - 4q^{83} - 4q^{87} + 2q^{89} - 8q^{93} - 36q^{97} - 12q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 7 x^{2} + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/7\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(7 \beta_{2}\)
\(\nu^{3}\)\(=\)\(7 \beta_{3}\)

Character values

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

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1201.1
1.32288 2.29129i
−1.32288 + 2.29129i
1.32288 + 2.29129i
−1.32288 2.29129i
0 −0.500000 + 0.866025i 0 0 0 −1.32288 + 2.29129i 0 −0.500000 0.866025i 0
1201.2 0 −0.500000 + 0.866025i 0 0 0 1.32288 2.29129i 0 −0.500000 0.866025i 0
1801.1 0 −0.500000 0.866025i 0 0 0 −1.32288 2.29129i 0 −0.500000 + 0.866025i 0
1801.2 0 −0.500000 0.866025i 0 0 0 1.32288 + 2.29129i 0 −0.500000 + 0.866025i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.q.g 4
5.b even 2 1 2100.2.q.i yes 4
5.c odd 4 2 2100.2.bc.h 8
7.c even 3 1 inner 2100.2.q.g 4
35.j even 6 1 2100.2.q.i yes 4
35.l odd 12 2 2100.2.bc.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.2.q.g 4 1.a even 1 1 trivial
2100.2.q.g 4 7.c even 3 1 inner
2100.2.q.i yes 4 5.b even 2 1
2100.2.q.i yes 4 35.j even 6 1
2100.2.bc.h 8 5.c odd 4 2
2100.2.bc.h 8 35.l odd 12 2

Hecke kernels

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

\( T_{11}^{4} - 6 T_{11}^{3} + 34 T_{11}^{2} - 12 T_{11} + 4 \)
\( T_{13} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + T + T^{2} )^{2} \)
$5$ 1
$7$ \( 1 + 7 T^{2} + 49 T^{4} \)
$11$ \( 1 - 6 T + 12 T^{2} - 12 T^{3} + 59 T^{4} - 132 T^{5} + 1452 T^{6} - 7986 T^{7} + 14641 T^{8} \)
$13$ \( ( 1 + T + 13 T^{2} )^{4} \)
$17$ \( 1 - 2 T - 24 T^{2} + 12 T^{3} + 427 T^{4} + 204 T^{5} - 6936 T^{6} - 9826 T^{7} + 83521 T^{8} \)
$19$ \( 1 - 4 T - 19 T^{2} + 12 T^{3} + 560 T^{4} + 228 T^{5} - 6859 T^{6} - 27436 T^{7} + 130321 T^{8} \)
$23$ \( ( 1 + 2 T - 19 T^{2} + 46 T^{3} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 4 T + 34 T^{2} - 116 T^{3} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 7 T + 31 T^{2} )^{2}( 1 + 11 T + 31 T^{2} )^{2} \)
$37$ \( 1 - 6 T - 19 T^{2} + 114 T^{3} + 324 T^{4} + 4218 T^{5} - 26011 T^{6} - 303918 T^{7} + 1874161 T^{8} \)
$41$ \( ( 1 + 6 T + 84 T^{2} + 246 T^{3} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 4 T + 62 T^{2} - 172 T^{3} + 1849 T^{4} )^{2} \)
$47$ \( 1 + 10 T + 44 T^{2} - 380 T^{3} - 3773 T^{4} - 17860 T^{5} + 97196 T^{6} + 1038230 T^{7} + 4879681 T^{8} \)
$53$ \( 1 - 10 T - 24 T^{2} - 180 T^{3} + 7267 T^{4} - 9540 T^{5} - 67416 T^{6} - 1488770 T^{7} + 7890481 T^{8} \)
$59$ \( 1 + 14 T + 92 T^{2} - 196 T^{3} - 4229 T^{4} - 11564 T^{5} + 320252 T^{6} + 2875306 T^{7} + 12117361 T^{8} \)
$61$ \( 1 - 2 T - 7 T^{2} + 222 T^{3} - 3844 T^{4} + 13542 T^{5} - 26047 T^{6} - 453962 T^{7} + 13845841 T^{8} \)
$67$ \( 1 - 71 T^{2} + 552 T^{4} - 318719 T^{6} + 20151121 T^{8} \)
$71$ \( ( 1 + 4 T + 118 T^{2} + 284 T^{3} + 5041 T^{4} )^{2} \)
$73$ \( 1 + 2 T - 115 T^{2} - 54 T^{3} + 8540 T^{4} - 3942 T^{5} - 612835 T^{6} + 778034 T^{7} + 28398241 T^{8} \)
$79$ \( 1 - 8 T - 103 T^{2} - 72 T^{3} + 16592 T^{4} - 5688 T^{5} - 642823 T^{6} - 3944312 T^{7} + 38950081 T^{8} \)
$83$ \( ( 1 + 2 T + 160 T^{2} + 166 T^{3} + 6889 T^{4} )^{2} \)
$89$ \( 1 - 2 T - 112 T^{2} + 124 T^{3} + 5179 T^{4} + 11036 T^{5} - 887152 T^{6} - 1409938 T^{7} + 62742241 T^{8} \)
$97$ \( ( 1 + 18 T + 247 T^{2} + 1746 T^{3} + 9409 T^{4} )^{2} \)
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