Properties

Label 4100.2.d.c
Level $4100$
Weight $2$
Character orbit 4100.d
Analytic conductor $32.739$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4100 = 2^{2} \cdot 5^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4100.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.7386648287\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} + 2 x^{6} + 4 x^{5} + 14 x^{4} - 14 x^{3} + 8 x^{2} + 4 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 164)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -\beta_{2} - \beta_{4} + \beta_{6} ) q^{3} + ( \beta_{2} - \beta_{4} - \beta_{7} ) q^{7} + ( -2 + \beta_{3} + \beta_{5} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{2} - \beta_{4} + \beta_{6} ) q^{3} + ( \beta_{2} - \beta_{4} - \beta_{7} ) q^{7} + ( -2 + \beta_{3} + \beta_{5} ) q^{9} + ( -\beta_{1} - \beta_{3} - \beta_{5} ) q^{11} + ( 2 \beta_{4} - \beta_{6} ) q^{13} + ( \beta_{6} + 2 \beta_{7} ) q^{17} + ( -1 - \beta_{1} + \beta_{3} ) q^{19} + ( -1 - 2 \beta_{1} + \beta_{3} - 3 \beta_{5} ) q^{21} + ( 2 \beta_{2} - 2 \beta_{6} ) q^{23} + ( 4 \beta_{2} + 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{27} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{29} + ( -2 - 2 \beta_{3} + 2 \beta_{5} ) q^{31} + ( -5 \beta_{2} - 2 \beta_{4} + \beta_{7} ) q^{33} + ( -\beta_{2} - 2 \beta_{6} - \beta_{7} ) q^{37} + ( 6 + 2 \beta_{1} ) q^{39} - q^{41} + ( 2 \beta_{4} + 2 \beta_{7} ) q^{43} + ( \beta_{2} + 3 \beta_{4} - \beta_{6} ) q^{47} + ( -6 - 4 \beta_{1} - \beta_{3} - 3 \beta_{5} ) q^{49} + ( -2 - 2 \beta_{1} - 6 \beta_{3} + 4 \beta_{5} ) q^{51} + ( 2 \beta_{2} + 2 \beta_{4} - 3 \beta_{6} + 2 \beta_{7} ) q^{53} + ( 3 \beta_{2} + 4 \beta_{4} - 3 \beta_{6} + \beta_{7} ) q^{57} + ( -4 - 2 \beta_{3} ) q^{59} + ( 6 - 2 \beta_{1} ) q^{61} + ( -3 \beta_{2} - \beta_{4} + \beta_{6} + 4 \beta_{7} ) q^{63} + ( \beta_{2} + \beta_{4} - 4 \beta_{6} + \beta_{7} ) q^{67} + ( 6 - 2 \beta_{5} ) q^{69} + ( -1 - \beta_{1} + 3 \beta_{3} - 4 \beta_{5} ) q^{71} + ( \beta_{2} - 3 \beta_{7} ) q^{73} + ( \beta_{2} + 4 \beta_{4} - 3 \beta_{6} + \beta_{7} ) q^{77} + ( 3 - \beta_{1} - 3 \beta_{3} ) q^{79} + ( 4 - 6 \beta_{1} - 3 \beta_{3} - 3 \beta_{5} ) q^{81} + ( -4 \beta_{4} - 2 \beta_{7} ) q^{83} + ( -4 \beta_{2} - 2 \beta_{4} - 4 \beta_{6} - 2 \beta_{7} ) q^{87} + ( -2 - 4 \beta_{1} - 2 \beta_{3} ) q^{89} + ( 12 + 2 \beta_{1} + 6 \beta_{5} ) q^{91} + ( 4 \beta_{2} - 6 \beta_{6} - 6 \beta_{7} ) q^{93} + ( -3 \beta_{6} - 4 \beta_{7} ) q^{97} + ( -9 + 7 \beta_{1} + 7 \beta_{3} + 4 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 24q^{9} + O(q^{10}) \) \( 8q - 24q^{9} + 8q^{11} - 12q^{19} + 8q^{29} - 16q^{31} + 48q^{39} - 8q^{41} - 32q^{49} - 8q^{51} - 24q^{59} + 48q^{61} + 56q^{69} - 4q^{71} + 36q^{79} + 56q^{81} - 8q^{89} + 72q^{91} - 116q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} + 2 x^{6} + 4 x^{5} + 14 x^{4} - 14 x^{3} + 8 x^{2} + 4 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -19 \nu^{7} + 33 \nu^{6} - 46 \nu^{5} + 12 \nu^{4} - 413 \nu^{3} + 174 \nu^{2} + 94 \nu + 516 \)\()/317\)
\(\beta_{2}\)\(=\)\((\)\( 100 \nu^{7} - 157 \nu^{6} + 142 \nu^{5} + 404 \nu^{4} + 1840 \nu^{3} - 799 \nu^{2} + 1474 \nu + 254 \)\()/951\)
\(\beta_{3}\)\(=\)\((\)\( 100 \nu^{7} - 157 \nu^{6} + 142 \nu^{5} + 404 \nu^{4} + 1840 \nu^{3} - 799 \nu^{2} - 428 \nu + 254 \)\()/951\)
\(\beta_{4}\)\(=\)\((\)\( 154 \nu^{7} - 451 \nu^{6} + 523 \nu^{5} + 470 \nu^{4} + 1312 \nu^{3} - 3646 \nu^{2} + 1357 \nu + 239 \)\()/951\)
\(\beta_{5}\)\(=\)\((\)\( -358 \nu^{7} + 505 \nu^{6} - 166 \nu^{5} - 2093 \nu^{4} - 5446 \nu^{3} + 2461 \nu^{2} + 1304 \nu - 3458 \)\()/951\)
\(\beta_{6}\)\(=\)\((\)\( -508 \nu^{7} + 1216 \nu^{6} - 1330 \nu^{5} - 1748 \nu^{4} - 6304 \nu^{3} + 10792 \nu^{2} - 5662 \nu - 986 \)\()/951\)
\(\beta_{7}\)\(=\)\((\)\( -836 \nu^{7} + 1769 \nu^{6} - 2024 \nu^{5} - 2959 \nu^{4} - 11198 \nu^{3} + 11777 \nu^{2} - 9812 \nu - 1705 \)\()/951\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{3} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} + 2 \beta_{4} + 2 \beta_{2}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} + \beta_{5} + 2 \beta_{4} + 5 \beta_{3} + 5 \beta_{2} + 2 \beta_{1} - 1\)\()/2\)
\(\nu^{4}\)\(=\)\(\beta_{5} + 7 \beta_{3} + 6 \beta_{1} - 8\)
\(\nu^{5}\)\(=\)\((\)\(-6 \beta_{7} - \beta_{6} + 6 \beta_{5} - 16 \beta_{4} + 29 \beta_{3} - 29 \beta_{2} + 16 \beta_{1} - 12\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-16 \beta_{7} - 13 \beta_{6} - 70 \beta_{4} - 92 \beta_{2}\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-35 \beta_{7} - 11 \beta_{6} - 35 \beta_{5} - 108 \beta_{4} - 175 \beta_{3} - 175 \beta_{2} - 108 \beta_{1} + 95\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4100\mathbb{Z}\right)^\times\).

\(n\) \(1477\) \(2051\) \(3901\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1149.1
1.76639 + 1.76639i
−0.187509 + 0.187509i
0.626295 0.626295i
−1.20518 + 1.20518i
−1.20518 1.20518i
0.626295 + 0.626295i
−0.187509 0.187509i
1.76639 1.76639i
0 3.24028i 0 0 0 0.858626i 0 −7.49944 0
1149.2 0 2.92968i 0 0 0 2.77840i 0 −5.58303 0
1149.3 0 2.21551i 0 0 0 5.06479i 0 −1.90849 0
1149.4 0 0.0950939i 0 0 0 3.14501i 0 2.99096 0
1149.5 0 0.0950939i 0 0 0 3.14501i 0 2.99096 0
1149.6 0 2.21551i 0 0 0 5.06479i 0 −1.90849 0
1149.7 0 2.92968i 0 0 0 2.77840i 0 −5.58303 0
1149.8 0 3.24028i 0 0 0 0.858626i 0 −7.49944 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1149.8
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 4100.2.d.c 8
5.b even 2 1 inner 4100.2.d.c 8
5.c odd 4 1 164.2.a.a 4
5.c odd 4 1 4100.2.a.c 4
15.e even 4 1 1476.2.a.g 4
20.e even 4 1 656.2.a.i 4
35.f even 4 1 8036.2.a.i 4
40.i odd 4 1 2624.2.a.v 4
40.k even 4 1 2624.2.a.y 4
60.l odd 4 1 5904.2.a.bp 4
205.g odd 4 1 6724.2.a.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.2.a.a 4 5.c odd 4 1
656.2.a.i 4 20.e even 4 1
1476.2.a.g 4 15.e even 4 1
2624.2.a.v 4 40.i odd 4 1
2624.2.a.y 4 40.k even 4 1
4100.2.a.c 4 5.c odd 4 1
4100.2.d.c 8 1.a even 1 1 trivial
4100.2.d.c 8 5.b even 2 1 inner
5904.2.a.bp 4 60.l odd 4 1
6724.2.a.c 4 205.g odd 4 1
8036.2.a.i 4 35.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4100, [\chi])\):

\( T_{3}^{8} + 24 T_{3}^{6} + 184 T_{3}^{4} + 444 T_{3}^{2} + 4 \)
\( T_{7}^{8} + 44 T_{7}^{6} + 560 T_{7}^{4} + 2348 T_{7}^{2} + 1444 \)