# Properties

 Label 7.7.d.a Level 7 Weight 7 Character orbit 7.d 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.d (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.61037858534$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 12 \zeta_{6} q^{2} + ( -7 - 7 \zeta_{6} ) q^{3} + ( -80 + 80 \zeta_{6} ) q^{4} + ( 210 - 105 \zeta_{6} ) q^{5} + ( 84 - 168 \zeta_{6} ) q^{6} -343 q^{7} -192 q^{8} -582 \zeta_{6} q^{9} +O(q^{10})$$ $$q + 12 \zeta_{6} q^{2} + ( -7 - 7 \zeta_{6} ) q^{3} + ( -80 + 80 \zeta_{6} ) q^{4} + ( 210 - 105 \zeta_{6} ) q^{5} + ( 84 - 168 \zeta_{6} ) q^{6} -343 q^{7} -192 q^{8} -582 \zeta_{6} q^{9} + ( 1260 + 1260 \zeta_{6} ) q^{10} + ( -1479 + 1479 \zeta_{6} ) q^{11} + ( 1120 - 560 \zeta_{6} ) q^{12} + ( -280 + 560 \zeta_{6} ) q^{13} -4116 \zeta_{6} q^{14} -2205 q^{15} + 2816 \zeta_{6} q^{16} + ( -1743 - 1743 \zeta_{6} ) q^{17} + ( 6984 - 6984 \zeta_{6} ) q^{18} + ( 7938 - 3969 \zeta_{6} ) q^{19} + ( -8400 + 16800 \zeta_{6} ) q^{20} + ( 2401 + 2401 \zeta_{6} ) q^{21} -17748 q^{22} + 5913 \zeta_{6} q^{23} + ( 1344 + 1344 \zeta_{6} ) q^{24} + ( 17450 - 17450 \zeta_{6} ) q^{25} + ( -6720 + 3360 \zeta_{6} ) q^{26} + ( -9177 + 18354 \zeta_{6} ) q^{27} + ( 27440 - 27440 \zeta_{6} ) q^{28} + 3978 q^{29} -26460 \zeta_{6} q^{30} + ( -7399 - 7399 \zeta_{6} ) q^{31} + ( -46080 + 46080 \zeta_{6} ) q^{32} + ( 20706 - 10353 \zeta_{6} ) q^{33} + ( 20916 - 41832 \zeta_{6} ) q^{34} + ( -72030 + 36015 \zeta_{6} ) q^{35} + 46560 q^{36} + 61577 \zeta_{6} q^{37} + ( 47628 + 47628 \zeta_{6} ) q^{38} + ( 5880 - 5880 \zeta_{6} ) q^{39} + ( -40320 + 20160 \zeta_{6} ) q^{40} + ( 63840 - 127680 \zeta_{6} ) q^{41} + ( -28812 + 57624 \zeta_{6} ) q^{42} -17414 q^{43} -118320 \zeta_{6} q^{44} + ( -61110 - 61110 \zeta_{6} ) q^{45} + ( -70956 + 70956 \zeta_{6} ) q^{46} + ( -35406 + 17703 \zeta_{6} ) q^{47} + ( 19712 - 39424 \zeta_{6} ) q^{48} + 117649 q^{49} + 209400 q^{50} + 36603 \zeta_{6} q^{51} + ( -22400 - 22400 \zeta_{6} ) q^{52} + ( 60513 - 60513 \zeta_{6} ) q^{53} + ( -220248 + 110124 \zeta_{6} ) q^{54} + ( -155295 + 310590 \zeta_{6} ) q^{55} + 65856 q^{56} -83349 q^{57} + 47736 \zeta_{6} q^{58} + ( -124551 - 124551 \zeta_{6} ) q^{59} + ( 176400 - 176400 \zeta_{6} ) q^{60} + ( 187922 - 93961 \zeta_{6} ) q^{61} + ( 88788 - 177576 \zeta_{6} ) q^{62} + 199626 \zeta_{6} q^{63} -372736 q^{64} + 88200 \zeta_{6} q^{65} + ( 124236 + 124236 \zeta_{6} ) q^{66} + ( 268777 - 268777 \zeta_{6} ) q^{67} + ( 278880 - 139440 \zeta_{6} ) q^{68} + ( 41391 - 82782 \zeta_{6} ) q^{69} + ( -432180 - 432180 \zeta_{6} ) q^{70} + 101922 q^{71} + 111744 \zeta_{6} q^{72} + ( 183393 + 183393 \zeta_{6} ) q^{73} + ( -738924 + 738924 \zeta_{6} ) q^{74} + ( -244300 + 122150 \zeta_{6} ) q^{75} + ( -317520 + 635040 \zeta_{6} ) q^{76} + ( 507297 - 507297 \zeta_{6} ) q^{77} + 70560 q^{78} -362231 \zeta_{6} q^{79} + ( 295680 + 295680 \zeta_{6} ) q^{80} + ( -231561 + 231561 \zeta_{6} ) q^{81} + ( 1532160 - 766080 \zeta_{6} ) q^{82} + ( -125160 + 250320 \zeta_{6} ) q^{83} + ( -384160 + 192080 \zeta_{6} ) q^{84} -549045 q^{85} -208968 \zeta_{6} q^{86} + ( -27846 - 27846 \zeta_{6} ) q^{87} + ( 283968 - 283968 \zeta_{6} ) q^{88} + ( -1541022 + 770511 \zeta_{6} ) q^{89} + ( 733320 - 1466640 \zeta_{6} ) q^{90} + ( 96040 - 192080 \zeta_{6} ) q^{91} -473040 q^{92} + 155379 \zeta_{6} q^{93} + ( -212436 - 212436 \zeta_{6} ) q^{94} + ( 1250235 - 1250235 \zeta_{6} ) q^{95} + ( 645120 - 322560 \zeta_{6} ) q^{96} + ( -874160 + 1748320 \zeta_{6} ) q^{97} + 1411788 \zeta_{6} q^{98} + 860778 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 12q^{2} - 21q^{3} - 80q^{4} + 315q^{5} - 686q^{7} - 384q^{8} - 582q^{9} + O(q^{10})$$ $$2q + 12q^{2} - 21q^{3} - 80q^{4} + 315q^{5} - 686q^{7} - 384q^{8} - 582q^{9} + 3780q^{10} - 1479q^{11} + 1680q^{12} - 4116q^{14} - 4410q^{15} + 2816q^{16} - 5229q^{17} + 6984q^{18} + 11907q^{19} + 7203q^{21} - 35496q^{22} + 5913q^{23} + 4032q^{24} + 17450q^{25} - 10080q^{26} + 27440q^{28} + 7956q^{29} - 26460q^{30} - 22197q^{31} - 46080q^{32} + 31059q^{33} - 108045q^{35} + 93120q^{36} + 61577q^{37} + 142884q^{38} + 5880q^{39} - 60480q^{40} - 34828q^{43} - 118320q^{44} - 183330q^{45} - 70956q^{46} - 53109q^{47} + 235298q^{49} + 418800q^{50} + 36603q^{51} - 67200q^{52} + 60513q^{53} - 330372q^{54} + 131712q^{56} - 166698q^{57} + 47736q^{58} - 373653q^{59} + 176400q^{60} + 281883q^{61} + 199626q^{63} - 745472q^{64} + 88200q^{65} + 372708q^{66} + 268777q^{67} + 418320q^{68} - 1296540q^{70} + 203844q^{71} + 111744q^{72} + 550179q^{73} - 738924q^{74} - 366450q^{75} + 507297q^{77} + 141120q^{78} - 362231q^{79} + 887040q^{80} - 231561q^{81} + 2298240q^{82} - 576240q^{84} - 1098090q^{85} - 208968q^{86} - 83538q^{87} + 283968q^{88} - 2311533q^{89} - 946080q^{92} + 155379q^{93} - 637308q^{94} + 1250235q^{95} + 967680q^{96} + 1411788q^{98} + 1721556q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$3$$ $$\chi(n)$$ $$\zeta_{6}$$

## 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 + 0.866025i 0.5 − 0.866025i
6.00000 + 10.3923i −10.5000 6.06218i −40.0000 + 69.2820i 157.500 90.9327i 145.492i −343.000 −192.000 −291.000 504.027i 1890.00 + 1091.19i
5.1 6.00000 10.3923i −10.5000 + 6.06218i −40.0000 69.2820i 157.500 + 90.9327i 145.492i −343.000 −192.000 −291.000 + 504.027i 1890.00 1091.19i
 $$n$$: 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.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7.7.d.a 2
3.b odd 2 1 63.7.m.a 2
4.b odd 2 1 112.7.s.a 2
7.b odd 2 1 49.7.d.b 2
7.c even 3 1 49.7.b.a 2
7.c even 3 1 49.7.d.b 2
7.d odd 6 1 inner 7.7.d.a 2
7.d odd 6 1 49.7.b.a 2
21.g even 6 1 63.7.m.a 2
21.g even 6 1 441.7.d.a 2
21.h odd 6 1 441.7.d.a 2
28.f even 6 1 112.7.s.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.7.d.a 2 1.a even 1 1 trivial
7.7.d.a 2 7.d odd 6 1 inner
49.7.b.a 2 7.c even 3 1
49.7.b.a 2 7.d odd 6 1
49.7.d.b 2 7.b odd 2 1
49.7.d.b 2 7.c even 3 1
63.7.m.a 2 3.b odd 2 1
63.7.m.a 2 21.g even 6 1
112.7.s.a 2 4.b odd 2 1
112.7.s.a 2 28.f even 6 1
441.7.d.a 2 21.g even 6 1
441.7.d.a 2 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 12 T_{2} + 144$$ acting on $$S_{7}^{\mathrm{new}}(7, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 12 T + 80 T^{2} - 768 T^{3} + 4096 T^{4}$$
$3$ $$1 + 21 T + 876 T^{2} + 15309 T^{3} + 531441 T^{4}$$
$5$ $$1 - 315 T + 48700 T^{2} - 4921875 T^{3} + 244140625 T^{4}$$
$7$ $$( 1 + 343 T )^{2}$$
$11$ $$1 + 1479 T + 415880 T^{2} + 2620138719 T^{3} + 3138428376721 T^{4}$$
$13$ $$1 - 9418418 T^{2} + 23298085122481 T^{4}$$
$17$ $$1 + 5229 T + 33251716 T^{2} + 126215348301 T^{3} + 582622237229761 T^{4}$$
$19$ $$1 - 11907 T + 94304764 T^{2} - 560175305067 T^{3} + 2213314919066161 T^{4}$$
$23$ $$1 - 5913 T - 113072320 T^{2} - 875336211657 T^{3} + 21914624432020321 T^{4}$$
$29$ $$( 1 - 3978 T + 594823321 T^{2} )^{2}$$
$31$ $$1 + 22197 T + 1051739284 T^{2} + 19699919207157 T^{3} + 787662783788549761 T^{4}$$
$37$ $$1 - 61577 T + 1226000520 T^{2} - 157989735086993 T^{3} + 6582952005840035281 T^{4}$$
$41$ $$1 + 2726428318 T^{2} + 22563490300366186081 T^{4}$$
$43$ $$( 1 + 17414 T + 6321363049 T^{2} )^{2}$$
$47$ $$1 + 53109 T + 11719403956 T^{2} + 572473346907861 T^{3} +$$$$11\!\cdots\!41$$$$T^{4}$$
$53$ $$1 - 60513 T - 18502537960 T^{2} - 1341231984999177 T^{3} +$$$$49\!\cdots\!41$$$$T^{4}$$
$59$ $$1 + 373653 T + 88719388444 T^{2} + 15760882936560573 T^{3} +$$$$17\!\cdots\!81$$$$T^{4}$$
$61$ $$1 - 281883 T + 78006382924 T^{2} - 14522717686001763 T^{3} +$$$$26\!\cdots\!21$$$$T^{4}$$
$67$ $$1 - 268777 T - 18217306440 T^{2} - 24313132584237313 T^{3} +$$$$81\!\cdots\!61$$$$T^{4}$$
$71$ $$( 1 - 101922 T + 128100283921 T^{2} )^{2}$$
$73$ $$1 - 550179 T + 252233203636 T^{2} - 83260913285455731 T^{3} +$$$$22\!\cdots\!21$$$$T^{4}$$
$79$ $$1 + 362231 T - 111876158160 T^{2} + 88053812100827351 T^{3} +$$$$59\!\cdots\!41$$$$T^{4}$$
$83$ $$1 - 606885669938 T^{2} +$$$$10\!\cdots\!61$$$$T^{4}$$
$89$ $$1 + 2311533 T + 2278042894324 T^{2} + 1148788654438953213 T^{3} +$$$$24\!\cdots\!21$$$$T^{4}$$
$97$ $$1 + 626523106942 T^{2} +$$$$69\!\cdots\!41$$$$T^{4}$$