# Properties

 Label 4.7.b.a Level $4$ Weight $7$ Character orbit 4.b Analytic conductor $0.920$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4 = 2^{2}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 4.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.920216334479$$ 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: yes 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.

 $$f(q)$$ $$=$$ $$q + ( 2 + \beta ) q^{2} -4 \beta q^{3} + ( -56 + 4 \beta ) q^{4} + 10 q^{5} + ( 240 - 8 \beta ) q^{6} + 40 \beta q^{7} + ( -352 - 48 \beta ) q^{8} -231 q^{9} +O(q^{10})$$ $$q + ( 2 + \beta ) q^{2} -4 \beta q^{3} + ( -56 + 4 \beta ) q^{4} + 10 q^{5} + ( 240 - 8 \beta ) q^{6} + 40 \beta q^{7} + ( -352 - 48 \beta ) q^{8} -231 q^{9} + ( 20 + 10 \beta ) q^{10} -124 \beta q^{11} + ( 960 + 224 \beta ) q^{12} + 1466 q^{13} + ( -2400 + 80 \beta ) q^{14} -40 \beta q^{15} + ( 2176 - 448 \beta ) q^{16} -4766 q^{17} + ( -462 - 231 \beta ) q^{18} + 972 \beta q^{19} + ( -560 + 40 \beta ) q^{20} + 9600 q^{21} + ( 7440 - 248 \beta ) q^{22} -1352 \beta q^{23} + ( -11520 + 1408 \beta ) q^{24} -15525 q^{25} + ( 2932 + 1466 \beta ) q^{26} -1992 \beta q^{27} + ( -9600 - 2240 \beta ) q^{28} + 25498 q^{29} + ( 2400 - 80 \beta ) q^{30} + 5408 \beta q^{31} + ( 31232 + 1280 \beta ) q^{32} -29760 q^{33} + ( -9532 - 4766 \beta ) q^{34} + 400 \beta q^{35} + ( 12936 - 924 \beta ) q^{36} + 1994 q^{37} + ( -58320 + 1944 \beta ) q^{38} -5864 \beta q^{39} + ( -3520 - 480 \beta ) q^{40} + 29362 q^{41} + ( 19200 + 9600 \beta ) q^{42} -2780 \beta q^{43} + ( 29760 + 6944 \beta ) q^{44} -2310 q^{45} + ( 81120 - 2704 \beta ) q^{46} -976 \beta q^{47} + ( -107520 - 8704 \beta ) q^{48} + 21649 q^{49} + ( -31050 - 15525 \beta ) q^{50} + 19064 \beta q^{51} + ( -82096 + 5864 \beta ) q^{52} -192854 q^{53} + ( 119520 - 3984 \beta ) q^{54} -1240 \beta q^{55} + ( 115200 - 14080 \beta ) q^{56} + 233280 q^{57} + ( 50996 + 25498 \beta ) q^{58} -10124 \beta q^{59} + ( 9600 + 2240 \beta ) q^{60} -10918 q^{61} + ( -324480 + 10816 \beta ) q^{62} -9240 \beta q^{63} + ( -14336 + 33792 \beta ) q^{64} + 14660 q^{65} + ( -59520 - 29760 \beta ) q^{66} -50884 \beta q^{67} + ( 266896 - 19064 \beta ) q^{68} -324480 q^{69} + ( -24000 + 800 \beta ) q^{70} + 68712 \beta q^{71} + ( 81312 + 11088 \beta ) q^{72} + 288626 q^{73} + ( 3988 + 1994 \beta ) q^{74} + 62100 \beta q^{75} + ( -233280 - 54432 \beta ) q^{76} + 297600 q^{77} + ( 351840 - 11728 \beta ) q^{78} -40112 \beta q^{79} + ( 21760 - 4480 \beta ) q^{80} -646479 q^{81} + ( 58724 + 29362 \beta ) q^{82} -26356 \beta q^{83} + ( -537600 + 38400 \beta ) q^{84} -47660 q^{85} + ( 166800 - 5560 \beta ) q^{86} -101992 \beta q^{87} + ( -357120 + 43648 \beta ) q^{88} + 310738 q^{89} + ( -4620 - 2310 \beta ) q^{90} + 58640 \beta q^{91} + ( 324480 + 75712 \beta ) q^{92} + 1297920 q^{93} + ( 58560 - 1952 \beta ) q^{94} + 9720 \beta q^{95} + ( 307200 - 124928 \beta ) q^{96} -1457086 q^{97} + ( 43298 + 21649 \beta ) q^{98} + 28644 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{2} - 112q^{4} + 20q^{5} + 480q^{6} - 704q^{8} - 462q^{9} + O(q^{10})$$ $$2q + 4q^{2} - 112q^{4} + 20q^{5} + 480q^{6} - 704q^{8} - 462q^{9} + 40q^{10} + 1920q^{12} + 2932q^{13} - 4800q^{14} + 4352q^{16} - 9532q^{17} - 924q^{18} - 1120q^{20} + 19200q^{21} + 14880q^{22} - 23040q^{24} - 31050q^{25} + 5864q^{26} - 19200q^{28} + 50996q^{29} + 4800q^{30} + 62464q^{32} - 59520q^{33} - 19064q^{34} + 25872q^{36} + 3988q^{37} - 116640q^{38} - 7040q^{40} + 58724q^{41} + 38400q^{42} + 59520q^{44} - 4620q^{45} + 162240q^{46} - 215040q^{48} + 43298q^{49} - 62100q^{50} - 164192q^{52} - 385708q^{53} + 239040q^{54} + 230400q^{56} + 466560q^{57} + 101992q^{58} + 19200q^{60} - 21836q^{61} - 648960q^{62} - 28672q^{64} + 29320q^{65} - 119040q^{66} + 533792q^{68} - 648960q^{69} - 48000q^{70} + 162624q^{72} + 577252q^{73} + 7976q^{74} - 466560q^{76} + 595200q^{77} + 703680q^{78} + 43520q^{80} - 1292958q^{81} + 117448q^{82} - 1075200q^{84} - 95320q^{85} + 333600q^{86} - 714240q^{88} + 621476q^{89} - 9240q^{90} + 648960q^{92} + 2595840q^{93} + 117120q^{94} + 614400q^{96} - 2914172q^{97} + 86596q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 0.5 − 1.93649i 0.5 + 1.93649i
2.00000 7.74597i 30.9839i −56.0000 30.9839i 10.0000 240.000 + 61.9677i 309.839i −352.000 + 371.806i −231.000 20.0000 77.4597i
3.2 2.00000 + 7.74597i 30.9839i −56.0000 + 30.9839i 10.0000 240.000 61.9677i 309.839i −352.000 371.806i −231.000 20.0000 + 77.4597i
 $$n$$: 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 4.7.b.a 2
3.b odd 2 1 36.7.d.c 2
4.b odd 2 1 inner 4.7.b.a 2
5.b even 2 1 100.7.b.c 2
5.c odd 4 2 100.7.d.a 4
8.b even 2 1 64.7.c.c 2
8.d odd 2 1 64.7.c.c 2
12.b even 2 1 36.7.d.c 2
16.e even 4 2 256.7.d.f 4
16.f odd 4 2 256.7.d.f 4
20.d odd 2 1 100.7.b.c 2
20.e even 4 2 100.7.d.a 4
24.f even 2 1 576.7.g.h 2
24.h odd 2 1 576.7.g.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.7.b.a 2 1.a even 1 1 trivial
4.7.b.a 2 4.b odd 2 1 inner
36.7.d.c 2 3.b odd 2 1
36.7.d.c 2 12.b even 2 1
64.7.c.c 2 8.b even 2 1
64.7.c.c 2 8.d odd 2 1
100.7.b.c 2 5.b even 2 1
100.7.b.c 2 20.d odd 2 1
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 16.e even 4 2
256.7.d.f 4 16.f odd 4 2
576.7.g.h 2 24.f even 2 1
576.7.g.h 2 24.h odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{7}^{\mathrm{new}}(4, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$64 - 4 T + T^{2}$$
$3$ $$960 + T^{2}$$
$5$ $$( -10 + T )^{2}$$
$7$ $$96000 + T^{2}$$
$11$ $$922560 + T^{2}$$
$13$ $$( -1466 + T )^{2}$$
$17$ $$( 4766 + T )^{2}$$
$19$ $$56687040 + T^{2}$$
$23$ $$109674240 + T^{2}$$
$29$ $$( -25498 + T )^{2}$$
$31$ $$1754787840 + T^{2}$$
$37$ $$( -1994 + T )^{2}$$
$41$ $$( -29362 + T )^{2}$$
$43$ $$463704000 + T^{2}$$
$47$ $$57154560 + T^{2}$$
$53$ $$( 192854 + T )^{2}$$
$59$ $$6149722560 + T^{2}$$
$61$ $$( 10918 + T )^{2}$$
$67$ $$155350887360 + T^{2}$$
$71$ $$283280336640 + T^{2}$$
$73$ $$( -288626 + T )^{2}$$
$79$ $$96538352640 + T^{2}$$
$83$ $$41678324160 + T^{2}$$
$89$ $$( -310738 + T )^{2}$$
$97$ $$( 1457086 + T )^{2}$$