# 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 $2$

# Related objects

## 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
 0.5 − 1.93649i 0.5 + 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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