# Properties

 Label 12.7.c.a Level 12 Weight 7 Character orbit 12.c Analytic conductor 2.761 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$12 = 2^{2} \cdot 3$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 12.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.76064900344$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-5})$$ Defining polynomial: $$x^{2} + 5$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 12\sqrt{-5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -3 + \beta ) q^{3} + 6 \beta q^{5} + 242 q^{7} + ( -711 - 6 \beta ) q^{9} +O(q^{10})$$ $$q + ( -3 + \beta ) q^{3} + 6 \beta q^{5} + 242 q^{7} + ( -711 - 6 \beta ) q^{9} -66 \beta q^{11} + 2618 q^{13} + ( -4320 - 18 \beta ) q^{15} + 264 \beta q^{17} + 5786 q^{19} + ( -726 + 242 \beta ) q^{21} -348 \beta q^{23} -10295 q^{25} + ( 6453 - 693 \beta ) q^{27} -462 \beta q^{29} -20446 q^{31} + ( 47520 + 198 \beta ) q^{33} + 1452 \beta q^{35} -46774 q^{37} + ( -7854 + 2618 \beta ) q^{39} + 132 \beta q^{41} + 68618 q^{43} + ( 25920 - 4266 \beta ) q^{45} -792 \beta q^{47} -59085 q^{49} + ( -190080 - 792 \beta ) q^{51} -6402 \beta q^{53} + 285120 q^{55} + ( -17358 + 5786 \beta ) q^{57} + 5574 \beta q^{59} + 24794 q^{61} + ( -172062 - 1452 \beta ) q^{63} + 15708 \beta q^{65} -84358 q^{67} + ( 250560 + 1044 \beta ) q^{69} -12084 \beta q^{71} -113806 q^{73} + ( 30885 - 10295 \beta ) q^{75} -15972 \beta q^{77} -159742 q^{79} + ( 479601 + 8532 \beta ) q^{81} -19206 \beta q^{83} -1140480 q^{85} + ( 332640 + 1386 \beta ) q^{87} + 46812 \beta q^{89} + 633556 q^{91} + ( 61338 - 20446 \beta ) q^{93} + 34716 \beta q^{95} + 899522 q^{97} + ( -285120 + 46926 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 6q^{3} + 484q^{7} - 1422q^{9} + O(q^{10})$$ $$2q - 6q^{3} + 484q^{7} - 1422q^{9} + 5236q^{13} - 8640q^{15} + 11572q^{19} - 1452q^{21} - 20590q^{25} + 12906q^{27} - 40892q^{31} + 95040q^{33} - 93548q^{37} - 15708q^{39} + 137236q^{43} + 51840q^{45} - 118170q^{49} - 380160q^{51} + 570240q^{55} - 34716q^{57} + 49588q^{61} - 344124q^{63} - 168716q^{67} + 501120q^{69} - 227612q^{73} + 61770q^{75} - 319484q^{79} + 959202q^{81} - 2280960q^{85} + 665280q^{87} + 1267112q^{91} + 122676q^{93} + 1799044q^{97} - 570240q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/12\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$7$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 − 2.23607i 2.23607i
0 −3.00000 26.8328i 0 160.997i 0 242.000 0 −711.000 + 160.997i 0
5.2 0 −3.00000 + 26.8328i 0 160.997i 0 242.000 0 −711.000 160.997i 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
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 12.7.c.a 2
3.b odd 2 1 inner 12.7.c.a 2
4.b odd 2 1 48.7.e.c 2
5.b even 2 1 300.7.g.e 2
5.c odd 4 2 300.7.b.c 4
7.b odd 2 1 588.7.c.e 2
8.b even 2 1 192.7.e.e 2
8.d odd 2 1 192.7.e.d 2
9.c even 3 2 324.7.g.c 4
9.d odd 6 2 324.7.g.c 4
12.b even 2 1 48.7.e.c 2
15.d odd 2 1 300.7.g.e 2
15.e even 4 2 300.7.b.c 4
21.c even 2 1 588.7.c.e 2
24.f even 2 1 192.7.e.d 2
24.h odd 2 1 192.7.e.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.7.c.a 2 1.a even 1 1 trivial
12.7.c.a 2 3.b odd 2 1 inner
48.7.e.c 2 4.b odd 2 1
48.7.e.c 2 12.b even 2 1
192.7.e.d 2 8.d odd 2 1
192.7.e.d 2 24.f even 2 1
192.7.e.e 2 8.b even 2 1
192.7.e.e 2 24.h odd 2 1
300.7.b.c 4 5.c odd 4 2
300.7.b.c 4 15.e even 4 2
300.7.g.e 2 5.b even 2 1
300.7.g.e 2 15.d odd 2 1
324.7.g.c 4 9.c even 3 2
324.7.g.c 4 9.d odd 6 2
588.7.c.e 2 7.b odd 2 1
588.7.c.e 2 21.c even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{7}^{\mathrm{new}}(12, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 6 T + 729 T^{2}$$
$5$ $$1 - 5330 T^{2} + 244140625 T^{4}$$
$7$ $$( 1 - 242 T + 117649 T^{2} )^{2}$$
$11$ $$1 - 406802 T^{2} + 3138428376721 T^{4}$$
$13$ $$( 1 - 2618 T + 4826809 T^{2} )^{2}$$
$17$ $$1 + 1905982 T^{2} + 582622237229761 T^{4}$$
$19$ $$( 1 - 5786 T + 47045881 T^{2} )^{2}$$
$23$ $$1 - 208876898 T^{2} + 21914624432020321 T^{4}$$
$29$ $$1 - 1035966962 T^{2} + 353814783205469041 T^{4}$$
$31$ $$( 1 + 20446 T + 887503681 T^{2} )^{2}$$
$37$ $$( 1 + 46774 T + 2565726409 T^{2} )^{2}$$
$41$ $$1 - 9487663202 T^{2} + 22563490300366186081 T^{4}$$
$43$ $$( 1 - 68618 T + 6321363049 T^{2} )^{2}$$
$47$ $$1 - 21106800578 T^{2} +$$$$11\!\cdots\!41$$$$T^{4}$$
$53$ $$1 - 14819087378 T^{2} +$$$$49\!\cdots\!41$$$$T^{4}$$
$59$ $$1 - 61991044562 T^{2} +$$$$17\!\cdots\!81$$$$T^{4}$$
$61$ $$( 1 - 24794 T + 51520374361 T^{2} )^{2}$$
$67$ $$( 1 + 84358 T + 90458382169 T^{2} )^{2}$$
$71$ $$1 - 151063967522 T^{2} +$$$$16\!\cdots\!41$$$$T^{4}$$
$73$ $$( 1 + 113806 T + 151334226289 T^{2} )^{2}$$
$79$ $$( 1 + 159742 T + 243087455521 T^{2} )^{2}$$
$83$ $$1 - 388294032818 T^{2} +$$$$10\!\cdots\!61$$$$T^{4}$$
$89$ $$1 + 583819025758 T^{2} +$$$$24\!\cdots\!21$$$$T^{4}$$
$97$ $$( 1 - 899522 T + 832972004929 T^{2} )^{2}$$