# 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 $$6$$ and degree $$2$$)

## 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: $$1$$ 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.

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

## 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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
7.d Odd 1 yes

## Hecke kernels

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