Properties

Label 4100.2.a.c
Level 4100
Weight 2
Character orbit 4100.a
Self dual yes
Analytic conductor 32.739
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 4100 = 2^{2} \cdot 5^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4100.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.7386648287\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.25808.1
Defining polynomial: \(x^{4} - 10 x^{2} - 6 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 164)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{2} ) q^{3} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{7} + ( 3 - \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{2} ) q^{3} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{7} + ( 3 - \beta_{2} - \beta_{3} ) q^{9} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{11} -2 \beta_{1} q^{13} -2 \beta_{3} q^{17} + ( 2 - \beta_{1} - \beta_{2} ) q^{19} + ( -2 + 2 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{21} + ( 2 + 2 \beta_{2} ) q^{23} + ( 5 + 2 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{27} + ( -2 - 2 \beta_{1} + 2 \beta_{2} ) q^{29} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{31} + ( 8 + 2 \beta_{1} - 5 \beta_{2} + \beta_{3} ) q^{33} + ( -4 + \beta_{2} + \beta_{3} ) q^{37} + ( -6 + 2 \beta_{1} ) q^{39} - q^{41} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{43} + ( 2 + 3 \beta_{1} - \beta_{2} ) q^{47} + ( 5 - 4 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{49} + ( 4 + 2 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{51} + ( 4 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{53} + ( 4 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{57} + ( 2 + 2 \beta_{2} ) q^{59} + ( 6 + 2 \beta_{1} ) q^{61} + ( 6 + \beta_{1} - 3 \beta_{2} + 4 \beta_{3} ) q^{63} + ( -7 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{67} + ( -6 + 2 \beta_{3} ) q^{69} + ( -4 + \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{71} + ( -4 + \beta_{2} - 3 \beta_{3} ) q^{73} + ( -2 + 4 \beta_{1} - \beta_{2} - \beta_{3} ) q^{77} + ( -6 - \beta_{1} + 3 \beta_{2} ) q^{79} + ( 7 + 6 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{81} + ( 2 + 4 \beta_{1} - 2 \beta_{3} ) q^{83} + ( -12 - 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{87} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{89} + ( 12 - 2 \beta_{1} + 6 \beta_{3} ) q^{91} + ( 2 + 4 \beta_{2} - 6 \beta_{3} ) q^{93} + ( -2 + 4 \beta_{3} ) q^{97} + ( 16 + 7 \beta_{1} - 7 \beta_{2} - 4 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{3} + 12q^{9} + O(q^{10}) \) \( 4q - 2q^{3} + 12q^{9} + 4q^{11} + 4q^{17} + 6q^{19} + 12q^{23} + 10q^{27} - 4q^{29} - 8q^{31} + 20q^{33} - 16q^{37} - 24q^{39} - 4q^{41} - 4q^{43} + 6q^{47} + 16q^{49} - 4q^{51} + 16q^{53} - 4q^{57} + 12q^{59} + 24q^{61} + 10q^{63} - 28q^{67} - 28q^{69} - 2q^{71} - 8q^{73} - 8q^{77} - 18q^{79} + 28q^{81} + 12q^{83} - 44q^{87} + 4q^{89} + 36q^{91} + 28q^{93} - 16q^{97} + 58q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 10 x^{2} - 6 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 7 \nu - 3 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 3 \nu^{2} + 7 \nu - 12 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + 5\)
\(\nu^{3}\)\(=\)\(3 \beta_{2} + 7 \beta_{1} + 3\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.55466
−2.46810
3.31526
0.707500
0 −2.92968 0 0 0 −2.77840 0 5.58303 0
1.2 0 −2.21551 0 0 0 5.06479 0 1.90849 0
1.3 0 −0.0950939 0 0 0 −3.14501 0 −2.99096 0
1.4 0 3.24028 0 0 0 0.858626 0 7.49944 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(41\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4100.2.a.c 4
5.b even 2 1 164.2.a.a 4
5.c odd 4 2 4100.2.d.c 8
15.d odd 2 1 1476.2.a.g 4
20.d odd 2 1 656.2.a.i 4
35.c odd 2 1 8036.2.a.i 4
40.e odd 2 1 2624.2.a.y 4
40.f even 2 1 2624.2.a.v 4
60.h even 2 1 5904.2.a.bp 4
205.c even 2 1 6724.2.a.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.2.a.a 4 5.b even 2 1
656.2.a.i 4 20.d odd 2 1
1476.2.a.g 4 15.d odd 2 1
2624.2.a.v 4 40.f even 2 1
2624.2.a.y 4 40.e odd 2 1
4100.2.a.c 4 1.a even 1 1 trivial
4100.2.d.c 8 5.c odd 4 2
5904.2.a.bp 4 60.h even 2 1
6724.2.a.c 4 205.c even 2 1
8036.2.a.i 4 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 2 T_{3}^{3} - 10 T_{3}^{2} - 22 T_{3} - 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4100))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 2 T + 2 T^{2} - 4 T^{3} - 8 T^{4} - 12 T^{5} + 18 T^{6} + 54 T^{7} + 81 T^{8} \)
$5$ 1
$7$ \( 1 + 6 T^{2} - 26 T^{3} + 24 T^{4} - 182 T^{5} + 294 T^{6} + 2401 T^{8} \)
$11$ \( 1 - 4 T + 26 T^{2} - 114 T^{3} + 384 T^{4} - 1254 T^{5} + 3146 T^{6} - 5324 T^{7} + 14641 T^{8} \)
$13$ \( 1 + 12 T^{2} + 48 T^{3} + 118 T^{4} + 624 T^{5} + 2028 T^{6} + 28561 T^{8} \)
$17$ \( 1 - 4 T + 20 T^{2} - 124 T^{3} + 534 T^{4} - 2108 T^{5} + 5780 T^{6} - 19652 T^{7} + 83521 T^{8} \)
$19$ \( 1 - 6 T + 62 T^{2} - 208 T^{3} + 1448 T^{4} - 3952 T^{5} + 22382 T^{6} - 41154 T^{7} + 130321 T^{8} \)
$23$ \( 1 - 12 T + 108 T^{2} - 700 T^{3} + 3718 T^{4} - 16100 T^{5} + 57132 T^{6} - 146004 T^{7} + 279841 T^{8} \)
$29$ \( 1 + 4 T + 76 T^{2} + 348 T^{3} + 2870 T^{4} + 10092 T^{5} + 63916 T^{6} + 97556 T^{7} + 707281 T^{8} \)
$31$ \( 1 + 8 T + 92 T^{2} + 712 T^{3} + 3846 T^{4} + 22072 T^{5} + 88412 T^{6} + 238328 T^{7} + 923521 T^{8} \)
$37$ \( 1 + 16 T + 212 T^{2} + 1740 T^{3} + 12626 T^{4} + 64380 T^{5} + 290228 T^{6} + 810448 T^{7} + 1874161 T^{8} \)
$41$ \( ( 1 + T )^{4} \)
$43$ \( 1 + 4 T + 124 T^{2} + 244 T^{3} + 6678 T^{4} + 10492 T^{5} + 229276 T^{6} + 318028 T^{7} + 3418801 T^{8} \)
$47$ \( 1 - 6 T + 126 T^{2} - 640 T^{3} + 8608 T^{4} - 30080 T^{5} + 278334 T^{6} - 622938 T^{7} + 4879681 T^{8} \)
$53$ \( 1 - 16 T + 212 T^{2} - 1824 T^{3} + 15558 T^{4} - 96672 T^{5} + 595508 T^{6} - 2382032 T^{7} + 7890481 T^{8} \)
$59$ \( 1 - 12 T + 252 T^{2} - 1996 T^{3} + 22582 T^{4} - 117764 T^{5} + 877212 T^{6} - 2464548 T^{7} + 12117361 T^{8} \)
$61$ \( 1 - 24 T + 420 T^{2} - 4824 T^{3} + 44086 T^{4} - 294264 T^{5} + 1562820 T^{6} - 5447544 T^{7} + 13845841 T^{8} \)
$67$ \( 1 + 28 T + 538 T^{2} + 6638 T^{3} + 64208 T^{4} + 444746 T^{5} + 2415082 T^{6} + 8421364 T^{7} + 20151121 T^{8} \)
$71$ \( 1 + 2 T + 98 T^{2} - 268 T^{3} + 3408 T^{4} - 19028 T^{5} + 494018 T^{6} + 715822 T^{7} + 25411681 T^{8} \)
$73$ \( 1 + 8 T + 212 T^{2} + 1060 T^{3} + 19890 T^{4} + 77380 T^{5} + 1129748 T^{6} + 3112136 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + 18 T + 366 T^{2} + 4224 T^{3} + 45328 T^{4} + 333696 T^{5} + 2284206 T^{6} + 8874702 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 12 T + 252 T^{2} - 1644 T^{3} + 24598 T^{4} - 136452 T^{5} + 1736028 T^{6} - 6861444 T^{7} + 47458321 T^{8} \)
$89$ \( 1 - 4 T + 228 T^{2} - 796 T^{3} + 29014 T^{4} - 70844 T^{5} + 1805988 T^{6} - 2819876 T^{7} + 62742241 T^{8} \)
$97$ \( 1 + 16 T + 268 T^{2} + 3376 T^{3} + 38118 T^{4} + 327472 T^{5} + 2521612 T^{6} + 14602768 T^{7} + 88529281 T^{8} \)
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