Properties

Label 100.4.e.c
Level $100$
Weight $4$
Character orbit 100.e
Analytic conductor $5.900$
Analytic rank $0$
Dimension $2$
CM discriminant -20
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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 i + 2) q^{2} + ( - 7 i - 7) q^{3} - 8 i q^{4} - 28 q^{6} + ( - 9 i + 9) q^{7} + ( - 16 i - 16) q^{8} + 71 i q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 i + 2) q^{2} + ( - 7 i - 7) q^{3} - 8 i q^{4} - 28 q^{6} + ( - 9 i + 9) q^{7} + ( - 16 i - 16) q^{8} + 71 i q^{9} + (56 i - 56) q^{12} - 36 i q^{14} - 64 q^{16} + (142 i + 142) q^{18} - 126 q^{21} + ( - 67 i - 67) q^{23} + 224 i q^{24} + ( - 308 i + 308) q^{27} + ( - 72 i - 72) q^{28} - 306 i q^{29} + (128 i - 128) q^{32} + 568 q^{36} + 252 q^{41} + (252 i - 252) q^{42} + ( - 297 i - 297) q^{43} - 268 q^{46} + (301 i - 301) q^{47} + (448 i + 448) q^{48} + 181 i q^{49} - 1232 i q^{54} - 288 q^{56} + ( - 612 i - 612) q^{58} + 952 q^{61} + (639 i + 639) q^{63} + 512 i q^{64} + ( - 549 i + 549) q^{67} + 938 i q^{69} + ( - 1136 i + 1136) q^{72} - 2395 q^{81} + ( - 504 i + 504) q^{82} + ( - 77 i - 77) q^{83} + 1008 i q^{84} - 1188 q^{86} + (2142 i - 2142) q^{87} - 1386 i q^{89} + (536 i - 536) q^{92} + 1204 i q^{94} + 1792 q^{96} + (362 i + 362) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} - 14 q^{3} - 56 q^{6} + 18 q^{7} - 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} - 14 q^{3} - 56 q^{6} + 18 q^{7} - 32 q^{8} - 112 q^{12} - 128 q^{16} + 284 q^{18} - 252 q^{21} - 134 q^{23} + 616 q^{27} - 144 q^{28} - 256 q^{32} + 1136 q^{36} + 504 q^{41} - 504 q^{42} - 594 q^{43} - 536 q^{46} - 602 q^{47} + 896 q^{48} - 576 q^{56} - 1224 q^{58} + 1904 q^{61} + 1278 q^{63} + 1098 q^{67} + 2272 q^{72} - 4790 q^{81} + 1008 q^{82} - 154 q^{83} - 2376 q^{86} - 4284 q^{87} - 1072 q^{92} + 3584 q^{96} + 724 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(51\) \(77\)
\(\chi(n)\) \(-1\) \(i\)

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
1.00000i
1.00000i
2.00000 2.00000i −7.00000 7.00000i 8.00000i 0 −28.0000 9.00000 9.00000i −16.0000 16.0000i 71.0000i 0
43.1 2.00000 + 2.00000i −7.00000 + 7.00000i 8.00000i 0 −28.0000 9.00000 + 9.00000i −16.0000 + 16.0000i 71.0000i 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
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.c yes 2
4.b odd 2 1 100.4.e.b 2
5.b even 2 1 100.4.e.b 2
5.c odd 4 1 100.4.e.b 2
5.c odd 4 1 inner 100.4.e.c yes 2
20.d odd 2 1 CM 100.4.e.c yes 2
20.e even 4 1 100.4.e.b 2
20.e even 4 1 inner 100.4.e.c yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.4.e.b 2 4.b odd 2 1
100.4.e.b 2 5.b even 2 1
100.4.e.b 2 5.c odd 4 1
100.4.e.b 2 20.e even 4 1
100.4.e.c yes 2 1.a even 1 1 trivial
100.4.e.c yes 2 5.c odd 4 1 inner
100.4.e.c yes 2 20.d odd 2 1 CM
100.4.e.c yes 2 20.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 14T_{3} + 98 \) acting on \(S_{4}^{\mathrm{new}}(100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 4T + 8 \) Copy content Toggle raw display
$3$ \( T^{2} + 14T + 98 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 18T + 162 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 134T + 8978 \) Copy content Toggle raw display
$29$ \( T^{2} + 93636 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( (T - 252)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 594T + 176418 \) Copy content Toggle raw display
$47$ \( T^{2} + 602T + 181202 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 952)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 1098 T + 602802 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 154T + 11858 \) Copy content Toggle raw display
$89$ \( T^{2} + 1920996 \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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