# Properties

 Label 7.7.b.b Level 7 Weight 7 Character orbit 7.b Analytic conductor 1.610 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.61037858534$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-510})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{-510}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -8 q^{2} + \beta q^{3} -\beta q^{5} -8 \beta q^{6} + ( 133 + 7 \beta ) q^{7} + 512 q^{8} -1311 q^{9} +O(q^{10})$$ $$q -8 q^{2} + \beta q^{3} -\beta q^{5} -8 \beta q^{6} + ( 133 + 7 \beta ) q^{7} + 512 q^{8} -1311 q^{9} + 8 \beta q^{10} + 874 q^{11} + 49 \beta q^{13} + ( -1064 - 56 \beta ) q^{14} + 2040 q^{15} -4096 q^{16} -132 \beta q^{17} + 10488 q^{18} + 69 \beta q^{19} + ( -14280 + 133 \beta ) q^{21} -6992 q^{22} + 4738 q^{23} + 512 \beta q^{24} + 13585 q^{25} -392 \beta q^{26} -582 \beta q^{27} + 11146 q^{29} -16320 q^{30} -608 \beta q^{31} + 874 \beta q^{33} + 1056 \beta q^{34} + ( 14280 - 133 \beta ) q^{35} + 3002 q^{37} -552 \beta q^{38} -99960 q^{39} -512 \beta q^{40} + 1274 \beta q^{41} + ( 114240 - 1064 \beta ) q^{42} + 31418 q^{43} + 1311 \beta q^{45} -37904 q^{46} -1604 \beta q^{47} -4096 \beta q^{48} + ( -82271 + 1862 \beta ) q^{49} -108680 q^{50} + 269280 q^{51} -76406 q^{53} + 4656 \beta q^{54} -874 \beta q^{55} + ( 68096 + 3584 \beta ) q^{56} -140760 q^{57} -89168 q^{58} + 2507 \beta q^{59} -6091 \beta q^{61} + 4864 \beta q^{62} + ( -174363 - 9177 \beta ) q^{63} + 262144 q^{64} + 99960 q^{65} -6992 \beta q^{66} + 495242 q^{67} + 4738 \beta q^{69} + ( -114240 + 1064 \beta ) q^{70} -184406 q^{71} -671232 q^{72} -1350 \beta q^{73} -24016 q^{74} + 13585 \beta q^{75} + ( 116242 + 6118 \beta ) q^{77} + 799680 q^{78} -534934 q^{79} + 4096 \beta q^{80} + 231561 q^{81} -10192 \beta q^{82} -15827 \beta q^{83} -269280 q^{85} -251344 q^{86} + 11146 \beta q^{87} + 447488 q^{88} -13938 \beta q^{89} -10488 \beta q^{90} + ( -699720 + 6517 \beta ) q^{91} + 1240320 q^{93} + 12832 \beta q^{94} + 140760 q^{95} + 18032 \beta q^{97} + ( 658168 - 14896 \beta ) q^{98} -1145814 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 16q^{2} + 266q^{7} + 1024q^{8} - 2622q^{9} + O(q^{10})$$ $$2q - 16q^{2} + 266q^{7} + 1024q^{8} - 2622q^{9} + 1748q^{11} - 2128q^{14} + 4080q^{15} - 8192q^{16} + 20976q^{18} - 28560q^{21} - 13984q^{22} + 9476q^{23} + 27170q^{25} + 22292q^{29} - 32640q^{30} + 28560q^{35} + 6004q^{37} - 199920q^{39} + 228480q^{42} + 62836q^{43} - 75808q^{46} - 164542q^{49} - 217360q^{50} + 538560q^{51} - 152812q^{53} + 136192q^{56} - 281520q^{57} - 178336q^{58} - 348726q^{63} + 524288q^{64} + 199920q^{65} + 990484q^{67} - 228480q^{70} - 368812q^{71} - 1342464q^{72} - 48032q^{74} + 232484q^{77} + 1599360q^{78} - 1069868q^{79} + 463122q^{81} - 538560q^{85} - 502688q^{86} + 894976q^{88} - 1399440q^{91} + 2480640q^{93} + 281520q^{95} + 1316336q^{98} - 2291628q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/7\mathbb{Z}\right)^\times$$.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
6.1
 − 22.5832i 22.5832i
−8.00000 45.1664i 0 45.1664i 361.331i 133.000 316.165i 512.000 −1311.00 361.331i
6.2 −8.00000 45.1664i 0 45.1664i 361.331i 133.000 + 316.165i 512.000 −1311.00 361.331i
 $$n$$: 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
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7.7.b.b 2
3.b odd 2 1 63.7.d.d 2
4.b odd 2 1 112.7.c.b 2
5.b even 2 1 175.7.d.e 2
5.c odd 4 2 175.7.c.c 4
7.b odd 2 1 inner 7.7.b.b 2
7.c even 3 2 49.7.d.d 4
7.d odd 6 2 49.7.d.d 4
8.b even 2 1 448.7.c.d 2
8.d odd 2 1 448.7.c.c 2
21.c even 2 1 63.7.d.d 2
28.d even 2 1 112.7.c.b 2
35.c odd 2 1 175.7.d.e 2
35.f even 4 2 175.7.c.c 4
56.e even 2 1 448.7.c.c 2
56.h odd 2 1 448.7.c.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.7.b.b 2 1.a even 1 1 trivial
7.7.b.b 2 7.b odd 2 1 inner
49.7.d.d 4 7.c even 3 2
49.7.d.d 4 7.d odd 6 2
63.7.d.d 2 3.b odd 2 1
63.7.d.d 2 21.c even 2 1
112.7.c.b 2 4.b odd 2 1
112.7.c.b 2 28.d even 2 1
175.7.c.c 4 5.c odd 4 2
175.7.c.c 4 35.f even 4 2
175.7.d.e 2 5.b even 2 1
175.7.d.e 2 35.c odd 2 1
448.7.c.c 2 8.d odd 2 1
448.7.c.c 2 56.e even 2 1
448.7.c.d 2 8.b even 2 1
448.7.c.d 2 56.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 8$$ acting on $$S_{7}^{\mathrm{new}}(7, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 8 T + 64 T^{2} )^{2}$$
$3$ $$1 + 582 T^{2} + 531441 T^{4}$$
$5$ $$1 - 29210 T^{2} + 244140625 T^{4}$$
$7$ $$1 - 266 T + 117649 T^{2}$$
$11$ $$( 1 - 874 T + 1771561 T^{2} )^{2}$$
$13$ $$1 - 4755578 T^{2} + 23298085122481 T^{4}$$
$17$ $$1 - 12730178 T^{2} + 582622237229761 T^{4}$$
$19$ $$1 - 84379322 T^{2} + 2213314919066161 T^{4}$$
$23$ $$( 1 - 4738 T + 148035889 T^{2} )^{2}$$
$29$ $$( 1 - 11146 T + 594823321 T^{2} )^{2}$$
$31$ $$1 - 1020892802 T^{2} + 787662783788549761 T^{4}$$
$37$ $$( 1 - 3002 T + 2565726409 T^{2} )^{2}$$
$41$ $$1 - 6189133442 T^{2} + 22563490300366186081 T^{4}$$
$43$ $$( 1 - 31418 T + 6321363049 T^{2} )^{2}$$
$47$ $$1 - 16309886018 T^{2} +$$$$11\!\cdots\!41$$$$T^{4}$$
$53$ $$( 1 + 76406 T + 22164361129 T^{2} )^{2}$$
$59$ $$1 - 71539567322 T^{2} +$$$$17\!\cdots\!81$$$$T^{4}$$
$61$ $$1 - 27356175482 T^{2} +$$$$26\!\cdots\!21$$$$T^{4}$$
$67$ $$( 1 - 495242 T + 90458382169 T^{2} )^{2}$$
$71$ $$( 1 + 184406 T + 128100283921 T^{2} )^{2}$$
$73$ $$1 - 298950552578 T^{2} +$$$$22\!\cdots\!21$$$$T^{4}$$
$79$ $$( 1 + 534934 T + 243087455521 T^{2} )^{2}$$
$83$ $$1 - 142873131578 T^{2} +$$$$10\!\cdots\!61$$$$T^{4}$$
$89$ $$1 - 597656180162 T^{2} +$$$$24\!\cdots\!21$$$$T^{4}$$
$97$ $$1 - 1002631840898 T^{2} +$$$$69\!\cdots\!41$$$$T^{4}$$