Properties

Label 100.4.e.e
Level $100$
Weight $4$
Character orbit 100.e
Analytic conductor $5.900$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 100.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.90019100057\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 7x^{10} + 44x^{8} - 156x^{6} + 704x^{4} - 1792x^{2} + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{15} \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_{3}) q^{2} + \beta_{9} q^{3} + ( - \beta_{11} + \beta_{5} + \beta_{4} - 2 \beta_{3}) q^{4} + ( - \beta_{10} + \beta_{9} + 2 \beta_{8} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_1) q^{6} + ( - 2 \beta_{11} - \beta_{10} - \beta_{8} - \beta_{7} - 2 \beta_{6} - \beta_{5} + 2 \beta_{4} + \cdots - 1) q^{7}+ \cdots + (3 \beta_{5} + 3 \beta_{4} + 10 \beta_{3} + 3 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + \beta_{3}) q^{2} + \beta_{9} q^{3} + ( - \beta_{11} + \beta_{5} + \beta_{4} - 2 \beta_{3}) q^{4} + ( - \beta_{10} + \beta_{9} + 2 \beta_{8} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_1) q^{6} + ( - 2 \beta_{11} - \beta_{10} - \beta_{8} - \beta_{7} - 2 \beta_{6} - \beta_{5} + 2 \beta_{4} + \cdots - 1) q^{7}+ \cdots + ( - 28 \beta_{11} + 29 \beta_{10} + 29 \beta_{9} + 45 \beta_{7} - 45 \beta_{6} + \cdots + 14 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{2} + 8 q^{6} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{2} + 8 q^{6} + 12 q^{8} + 80 q^{12} - 116 q^{13} + 312 q^{16} + 332 q^{17} - 198 q^{18} - 144 q^{21} - 360 q^{22} - 164 q^{26} + 880 q^{28} + 376 q^{32} - 80 q^{33} + 460 q^{36} - 508 q^{37} - 1600 q^{38} - 656 q^{41} - 1160 q^{42} - 1432 q^{46} + 2720 q^{48} + 932 q^{52} + 644 q^{53} + 2048 q^{56} + 960 q^{57} - 1576 q^{58} - 896 q^{61} - 2440 q^{62} - 1680 q^{66} + 844 q^{68} + 3036 q^{72} - 1436 q^{73} + 800 q^{76} - 3120 q^{77} - 3720 q^{78} + 5988 q^{81} + 1352 q^{82} - 2552 q^{86} + 2400 q^{88} + 1840 q^{92} + 3280 q^{93} + 1088 q^{96} + 4772 q^{97} - 1698 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 7x^{10} + 44x^{8} - 156x^{6} + 704x^{4} - 1792x^{2} + 4096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} - 23\nu^{9} + 92\nu^{7} - 668\nu^{5} + 1152\nu^{3} - 8704\nu ) / 1280 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} - 8\nu^{9} + 27\nu^{7} - 128\nu^{5} + 412\nu^{3} - 1504\nu ) / 640 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 5 \nu^{11} + 32 \nu^{10} + 35 \nu^{9} - 256 \nu^{8} - 220 \nu^{7} + 864 \nu^{6} + 780 \nu^{5} - 4096 \nu^{4} - 3520 \nu^{3} + 13184 \nu^{2} + 8960 \nu - 48128 ) / 5120 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 13 \nu^{11} - 32 \nu^{10} + 99 \nu^{9} + 256 \nu^{8} - 436 \nu^{7} - 864 \nu^{6} + 1804 \nu^{5} + 4096 \nu^{4} - 6816 \nu^{3} - 13184 \nu^{2} + 20992 \nu + 48128 ) / 5120 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 13 \nu^{11} + 48 \nu^{10} + 99 \nu^{9} + 16 \nu^{8} - 436 \nu^{7} + 416 \nu^{6} + 1804 \nu^{5} + 576 \nu^{4} - 6816 \nu^{3} + 8576 \nu^{2} + 10752 \nu + 22528 ) / 5120 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 21 \nu^{11} + 48 \nu^{10} - 163 \nu^{9} + 16 \nu^{8} + 652 \nu^{7} + 416 \nu^{6} - 2828 \nu^{5} + 576 \nu^{4} + 10112 \nu^{3} + 8576 \nu^{2} - 22784 \nu + 22528 ) / 5120 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -17\nu^{10} + 31\nu^{8} - 324\nu^{6} + 636\nu^{4} - 4384\nu^{2} - 512 ) / 640 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 45 \nu^{11} - 12 \nu^{10} + 115 \nu^{9} - 44 \nu^{8} - 900 \nu^{7} - 144 \nu^{6} + 1740 \nu^{5} - 2224 \nu^{4} - 10720 \nu^{3} + 256 \nu^{2} + 2560 \nu - 28672 ) / 5120 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 45 \nu^{11} + 12 \nu^{10} + 115 \nu^{9} + 44 \nu^{8} - 900 \nu^{7} + 144 \nu^{6} + 1740 \nu^{5} + 2224 \nu^{4} - 10720 \nu^{3} - 256 \nu^{2} + 2560 \nu + 28672 ) / 5120 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3\nu^{11} - 7\nu^{9} + 50\nu^{7} - 156\nu^{5} + 824\nu^{3} - 448\nu ) / 256 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - 2\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} + \beta_{10} + \beta_{9} - 3\beta_{5} - 3\beta_{4} - 2\beta_{3} - 2\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{10} - 3\beta_{9} + \beta_{8} - \beta_{5} + \beta_{4} - \beta_{3} + \beta _1 - 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 3 \beta_{11} - \beta_{10} - \beta_{9} - 5 \beta_{7} + 5 \beta_{6} - 10 \beta_{5} - 10 \beta_{4} + 14 \beta_{3} - 8 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 7 \beta_{10} - 7 \beta_{9} - 11 \beta_{8} - 14 \beta_{7} - 14 \beta_{6} + 5 \beta_{5} - 5 \beta_{4} + 19 \beta_{3} - \beta _1 - 60 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 25 \beta_{11} - 19 \beta_{10} - 19 \beta_{9} + 3 \beta_{7} - 3 \beta_{6} - 12 \beta_{5} - 12 \beta_{4} - 74 \beta_{3} + 16 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 35 \beta_{10} + 35 \beta_{9} - 41 \beta_{8} - 18 \beta_{7} - 18 \beta_{6} + 47 \beta_{5} - 47 \beta_{4} + 65 \beta_{3} + 9 \beta _1 - 348 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 49 \beta_{11} + 3 \beta_{10} + 3 \beta_{9} - 47 \beta_{7} + 47 \beta_{6} - 80 \beta_{5} - 80 \beta_{4} - 358 \beta_{3} + 88 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 85 \beta_{10} + 85 \beta_{9} + 97 \beta_{8} + 234 \beta_{7} + 234 \beta_{6} - 47 \beta_{5} + 47 \beta_{4} - 281 \beta_{3} - 185 \beta _1 - 516 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 271 \beta_{11} - 3 \beta_{10} - 3 \beta_{9} - 345 \beta_{7} + 345 \beta_{6} + 392 \beta_{5} + 392 \beta_{4} + 1526 \beta_{3} + 72 \beta_{2} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/100\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(77\)
\(\chi(n)\) \(-1\) \(\beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−1.76129 + 0.947553i
−1.83244 0.801352i
−1.13579 1.64620i
1.76129 + 0.947553i
1.83244 0.801352i
1.13579 1.64620i
−1.76129 0.947553i
−1.83244 + 0.801352i
−1.13579 + 1.64620i
1.76129 0.947553i
1.83244 + 0.801352i
1.13579 + 1.64620i
−2.70884 + 0.813737i 2.61822 + 2.61822i 6.67566 4.40857i 0 −9.22289 4.96181i 17.7783 17.7783i −14.4959 + 17.3744i 13.2899i 0
7.2 −1.03109 + 2.63379i −5.55970 5.55970i −5.87372 5.43134i 0 20.3756 8.91056i 1.14202 1.14202i 20.3613 9.86997i 34.8205i 0
7.3 0.510409 + 2.78199i 4.02923 + 4.02923i −7.47897 + 2.83991i 0 −9.15273 + 13.2658i −14.4440 + 14.4440i −11.7179 19.3569i 5.46937i 0
7.4 0.813737 2.70884i −2.61822 2.61822i −6.67566 4.40857i 0 −9.22289 + 4.96181i −17.7783 + 17.7783i −17.3744 + 14.4959i 13.2899i 0
7.5 2.63379 1.03109i 5.55970 + 5.55970i 5.87372 5.43134i 0 20.3756 + 8.91056i −1.14202 + 1.14202i 9.86997 20.3613i 34.8205i 0
7.6 2.78199 + 0.510409i −4.02923 4.02923i 7.47897 + 2.83991i 0 −9.15273 13.2658i 14.4440 14.4440i 19.3569 + 11.7179i 5.46937i 0
43.1 −2.70884 0.813737i 2.61822 2.61822i 6.67566 + 4.40857i 0 −9.22289 + 4.96181i 17.7783 + 17.7783i −14.4959 17.3744i 13.2899i 0
43.2 −1.03109 2.63379i −5.55970 + 5.55970i −5.87372 + 5.43134i 0 20.3756 + 8.91056i 1.14202 + 1.14202i 20.3613 + 9.86997i 34.8205i 0
43.3 0.510409 2.78199i 4.02923 4.02923i −7.47897 2.83991i 0 −9.15273 13.2658i −14.4440 14.4440i −11.7179 + 19.3569i 5.46937i 0
43.4 0.813737 + 2.70884i −2.61822 + 2.61822i −6.67566 + 4.40857i 0 −9.22289 4.96181i −17.7783 17.7783i −17.3744 14.4959i 13.2899i 0
43.5 2.63379 + 1.03109i 5.55970 5.55970i 5.87372 + 5.43134i 0 20.3756 8.91056i −1.14202 1.14202i 9.86997 + 20.3613i 34.8205i 0
43.6 2.78199 0.510409i −4.02923 + 4.02923i 7.47897 2.83991i 0 −9.15273 + 13.2658i 14.4440 + 14.4440i 19.3569 11.7179i 5.46937i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.4.e.e 12
4.b odd 2 1 inner 100.4.e.e 12
5.b even 2 1 20.4.e.b 12
5.c odd 4 1 20.4.e.b 12
5.c odd 4 1 inner 100.4.e.e 12
15.d odd 2 1 180.4.k.e 12
15.e even 4 1 180.4.k.e 12
20.d odd 2 1 20.4.e.b 12
20.e even 4 1 20.4.e.b 12
20.e even 4 1 inner 100.4.e.e 12
40.e odd 2 1 320.4.n.k 12
40.f even 2 1 320.4.n.k 12
40.i odd 4 1 320.4.n.k 12
40.k even 4 1 320.4.n.k 12
60.h even 2 1 180.4.k.e 12
60.l odd 4 1 180.4.k.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.4.e.b 12 5.b even 2 1
20.4.e.b 12 5.c odd 4 1
20.4.e.b 12 20.d odd 2 1
20.4.e.b 12 20.e even 4 1
100.4.e.e 12 1.a even 1 1 trivial
100.4.e.e 12 4.b odd 2 1 inner
100.4.e.e 12 5.c odd 4 1 inner
100.4.e.e 12 20.e even 4 1 inner
180.4.k.e 12 15.d odd 2 1
180.4.k.e 12 15.e even 4 1
180.4.k.e 12 60.h even 2 1
180.4.k.e 12 60.l odd 4 1
320.4.n.k 12 40.e odd 2 1
320.4.n.k 12 40.f even 2 1
320.4.n.k 12 40.i odd 4 1
320.4.n.k 12 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 5064T_{3}^{8} + 4945680T_{3}^{4} + 757350400 \) acting on \(S_{4}^{\mathrm{new}}(100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 6 T^{11} + 18 T^{10} + \cdots + 262144 \) Copy content Toggle raw display
$3$ \( T^{12} + 5064 T^{8} + \cdots + 757350400 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 573704 T^{8} + \cdots + 473344000000 \) Copy content Toggle raw display
$11$ \( (T^{6} + 3000 T^{4} + 1778960 T^{2} + \cdots + 88064000)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 58 T^{5} + 1682 T^{4} + \cdots + 7296200)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 166 T^{5} + 13778 T^{4} + \cdots + 3024864200)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 24160 T^{4} + \cdots - 148035584000)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 150061064 T^{8} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{6} + 65648 T^{4} + \cdots + 234782887936)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 75640 T^{4} + \cdots + 4998782336000)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 254 T^{5} + \cdots + 6862252857800)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} + 164 T^{2} - 18428 T - 1791008)^{4} \) Copy content Toggle raw display
$43$ \( T^{12} + 14590421064 T^{8} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{12} + 61550198664 T^{8} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{6} - 322 T^{5} + \cdots + 273645700473800)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 688480 T^{4} + \cdots - 742151346176000)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 224 T^{2} - 55468 T - 11698768)^{4} \) Copy content Toggle raw display
$67$ \( T^{12} + 442279393224 T^{8} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{6} + 715960 T^{4} + \cdots + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 718 T^{5} + \cdots + 17037736128200)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 2383360 T^{4} + \cdots - 42\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 2795286470344 T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{6} + 1431168 T^{4} + \cdots + 84\!\cdots\!16)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 2386 T^{5} + \cdots + 60\!\cdots\!00)^{2} \) Copy content Toggle raw display
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