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

Downloads

Learn more about

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
0.500000 + 0.866025i
0.500000 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:

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 \(S_{7}^{\mathrm{new}}(7, [\chi])\).