Properties

Label 100.7.b.c
Level $100$
Weight $7$
Character orbit 100.b
Analytic conductor $23.005$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 100.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(23.0054083620\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-15}) \)
Defining polynomial: \(x^{2} - x + 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 4)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-15}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 + \beta ) q^{2} -4 \beta q^{3} + ( -56 - 4 \beta ) q^{4} + ( 240 + 8 \beta ) q^{6} + 40 \beta q^{7} + ( 352 - 48 \beta ) q^{8} -231 q^{9} +O(q^{10})\) \( q + ( -2 + \beta ) q^{2} -4 \beta q^{3} + ( -56 - 4 \beta ) q^{4} + ( 240 + 8 \beta ) q^{6} + 40 \beta q^{7} + ( 352 - 48 \beta ) q^{8} -231 q^{9} + 124 \beta q^{11} + ( -960 + 224 \beta ) q^{12} -1466 q^{13} + ( -2400 - 80 \beta ) q^{14} + ( 2176 + 448 \beta ) q^{16} + 4766 q^{17} + ( 462 - 231 \beta ) q^{18} -972 \beta q^{19} + 9600 q^{21} + ( -7440 - 248 \beta ) q^{22} -1352 \beta q^{23} + ( -11520 - 1408 \beta ) q^{24} + ( 2932 - 1466 \beta ) q^{26} -1992 \beta q^{27} + ( 9600 - 2240 \beta ) q^{28} + 25498 q^{29} -5408 \beta q^{31} + ( -31232 + 1280 \beta ) q^{32} + 29760 q^{33} + ( -9532 + 4766 \beta ) q^{34} + ( 12936 + 924 \beta ) q^{36} -1994 q^{37} + ( 58320 + 1944 \beta ) q^{38} + 5864 \beta q^{39} + 29362 q^{41} + ( -19200 + 9600 \beta ) q^{42} -2780 \beta q^{43} + ( 29760 - 6944 \beta ) q^{44} + ( 81120 + 2704 \beta ) q^{46} -976 \beta q^{47} + ( 107520 - 8704 \beta ) q^{48} + 21649 q^{49} -19064 \beta q^{51} + ( 82096 + 5864 \beta ) q^{52} + 192854 q^{53} + ( 119520 + 3984 \beta ) q^{54} + ( 115200 + 14080 \beta ) q^{56} -233280 q^{57} + ( -50996 + 25498 \beta ) q^{58} + 10124 \beta q^{59} -10918 q^{61} + ( 324480 + 10816 \beta ) q^{62} -9240 \beta q^{63} + ( -14336 - 33792 \beta ) q^{64} + ( -59520 + 29760 \beta ) q^{66} -50884 \beta q^{67} + ( -266896 - 19064 \beta ) q^{68} -324480 q^{69} -68712 \beta q^{71} + ( -81312 + 11088 \beta ) q^{72} -288626 q^{73} + ( 3988 - 1994 \beta ) q^{74} + ( -233280 + 54432 \beta ) q^{76} -297600 q^{77} + ( -351840 - 11728 \beta ) q^{78} + 40112 \beta q^{79} -646479 q^{81} + ( -58724 + 29362 \beta ) q^{82} -26356 \beta q^{83} + ( -537600 - 38400 \beta ) q^{84} + ( 166800 + 5560 \beta ) q^{86} -101992 \beta q^{87} + ( 357120 + 43648 \beta ) q^{88} + 310738 q^{89} -58640 \beta q^{91} + ( -324480 + 75712 \beta ) q^{92} -1297920 q^{93} + ( 58560 + 1952 \beta ) q^{94} + ( 307200 + 124928 \beta ) q^{96} + 1457086 q^{97} + ( -43298 + 21649 \beta ) q^{98} -28644 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{2} - 112q^{4} + 480q^{6} + 704q^{8} - 462q^{9} + O(q^{10}) \) \( 2q - 4q^{2} - 112q^{4} + 480q^{6} + 704q^{8} - 462q^{9} - 1920q^{12} - 2932q^{13} - 4800q^{14} + 4352q^{16} + 9532q^{17} + 924q^{18} + 19200q^{21} - 14880q^{22} - 23040q^{24} + 5864q^{26} + 19200q^{28} + 50996q^{29} - 62464q^{32} + 59520q^{33} - 19064q^{34} + 25872q^{36} - 3988q^{37} + 116640q^{38} + 58724q^{41} - 38400q^{42} + 59520q^{44} + 162240q^{46} + 215040q^{48} + 43298q^{49} + 164192q^{52} + 385708q^{53} + 239040q^{54} + 230400q^{56} - 466560q^{57} - 101992q^{58} - 21836q^{61} + 648960q^{62} - 28672q^{64} - 119040q^{66} - 533792q^{68} - 648960q^{69} - 162624q^{72} - 577252q^{73} + 7976q^{74} - 466560q^{76} - 595200q^{77} - 703680q^{78} - 1292958q^{81} - 117448q^{82} - 1075200q^{84} + 333600q^{86} + 714240q^{88} + 621476q^{89} - 648960q^{92} - 2595840q^{93} + 117120q^{94} + 614400q^{96} + 2914172q^{97} - 86596q^{98} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
0.500000 1.93649i
0.500000 + 1.93649i
−2.00000 7.74597i 30.9839i −56.0000 + 30.9839i 0 240.000 61.9677i 309.839i 352.000 + 371.806i −231.000 0
51.2 −2.00000 + 7.74597i 30.9839i −56.0000 30.9839i 0 240.000 + 61.9677i 309.839i 352.000 371.806i −231.000 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
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.7.b.c 2
4.b odd 2 1 inner 100.7.b.c 2
5.b even 2 1 4.7.b.a 2
5.c odd 4 2 100.7.d.a 4
15.d odd 2 1 36.7.d.c 2
20.d odd 2 1 4.7.b.a 2
20.e even 4 2 100.7.d.a 4
40.e odd 2 1 64.7.c.c 2
40.f even 2 1 64.7.c.c 2
60.h even 2 1 36.7.d.c 2
80.k odd 4 2 256.7.d.f 4
80.q even 4 2 256.7.d.f 4
120.i odd 2 1 576.7.g.h 2
120.m even 2 1 576.7.g.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.7.b.a 2 5.b even 2 1
4.7.b.a 2 20.d odd 2 1
36.7.d.c 2 15.d odd 2 1
36.7.d.c 2 60.h even 2 1
64.7.c.c 2 40.e odd 2 1
64.7.c.c 2 40.f even 2 1
100.7.b.c 2 1.a even 1 1 trivial
100.7.b.c 2 4.b odd 2 1 inner
100.7.d.a 4 5.c odd 4 2
100.7.d.a 4 20.e even 4 2
256.7.d.f 4 80.k odd 4 2
256.7.d.f 4 80.q even 4 2
576.7.g.h 2 120.i odd 2 1
576.7.g.h 2 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(100, [\chi])\):

\( T_{3}^{2} + 960 \)
\( T_{13} + 1466 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 64 + 4 T + T^{2} \)
$3$ \( 960 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 96000 + T^{2} \)
$11$ \( 922560 + T^{2} \)
$13$ \( ( 1466 + T )^{2} \)
$17$ \( ( -4766 + T )^{2} \)
$19$ \( 56687040 + T^{2} \)
$23$ \( 109674240 + T^{2} \)
$29$ \( ( -25498 + T )^{2} \)
$31$ \( 1754787840 + T^{2} \)
$37$ \( ( 1994 + T )^{2} \)
$41$ \( ( -29362 + T )^{2} \)
$43$ \( 463704000 + T^{2} \)
$47$ \( 57154560 + T^{2} \)
$53$ \( ( -192854 + T )^{2} \)
$59$ \( 6149722560 + T^{2} \)
$61$ \( ( 10918 + T )^{2} \)
$67$ \( 155350887360 + T^{2} \)
$71$ \( 283280336640 + T^{2} \)
$73$ \( ( 288626 + T )^{2} \)
$79$ \( 96538352640 + T^{2} \)
$83$ \( 41678324160 + T^{2} \)
$89$ \( ( -310738 + T )^{2} \)
$97$ \( ( -1457086 + T )^{2} \)
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