Properties

Label 16.7.f.a
Level 16
Weight 7
Character orbit 16.f
Analytic conductor 3.681
Analytic rank 0
Dimension 22
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 16.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.68086533792\)
Analytic rank: \(0\)
Dimension: \(22\)
Relative dimension: \(11\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 22q - 2q^{2} - 2q^{3} + 88q^{4} - 2q^{5} - 512q^{6} - 4q^{7} + 964q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 22q - 2q^{2} - 2q^{3} + 88q^{4} - 2q^{5} - 512q^{6} - 4q^{7} + 964q^{8} - 1124q^{10} + 1358q^{11} - 2348q^{12} - 2q^{13} + 7564q^{14} - 7976q^{16} - 4q^{17} - 1874q^{18} + 3934q^{19} - 16564q^{20} - 1460q^{21} + 25252q^{22} - 13124q^{23} - 32592q^{24} + 58952q^{26} + 35776q^{27} + 45176q^{28} - 33202q^{29} + 8452q^{30} - 39672q^{32} - 4q^{33} - 52588q^{34} - 112420q^{35} - 138484q^{36} + 3598q^{37} - 187280q^{38} + 254396q^{39} + 165160q^{40} + 353496q^{42} - 267986q^{43} + 380724q^{44} + 32706q^{45} + 248156q^{46} - 460824q^{48} + 168066q^{49} - 719674q^{50} + 301788q^{51} - 784892q^{52} - 221842q^{53} - 547552q^{54} - 232708q^{55} + 366424q^{56} + 1441912q^{58} + 39150q^{59} + 2405600q^{60} + 326494q^{61} + 182832q^{62} - 318656q^{64} + 186412q^{65} - 3392732q^{66} - 122786q^{67} - 2504400q^{68} - 543188q^{69} - 1186816q^{70} - 267012q^{71} + 1748596q^{72} + 3610396q^{74} - 275278q^{75} + 4245988q^{76} + 231180q^{77} + 3179052q^{78} - 2898968q^{80} - 354298q^{81} - 5004624q^{82} - 288322q^{83} - 7602776q^{84} + 340748q^{85} - 3240476q^{86} + 2029884q^{87} + 3308152q^{88} + 10273400q^{90} + 302396q^{91} + 7253656q^{92} - 1173344q^{93} + 3859056q^{94} - 5689424q^{96} - 4q^{97} - 9176950q^{98} - 271522q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −7.92168 + 1.11666i 8.95029 8.95029i 61.5061 17.6917i −127.202 + 127.202i −60.9069 + 80.8958i −458.680 −467.476 + 208.829i 568.785i 865.616 1149.70i
3.2 −7.06494 3.75322i −9.43202 + 9.43202i 35.8267 + 53.0325i 46.6419 46.6419i 102.037 31.2362i 647.751 −54.0705 509.137i 551.074i −504.579 + 154.465i
3.3 −6.31611 + 4.90986i −31.9540 + 31.9540i 15.7866 62.0224i 95.0213 95.0213i 44.9356 358.715i −293.582 204.811 + 469.251i 1313.12i −133.624 + 1066.71i
3.4 −4.06674 + 6.88924i 23.4827 23.4827i −30.9232 56.0335i 55.3108 55.3108i 66.2800 + 257.276i 386.163 511.785 + 14.8363i 373.878i 156.115 + 605.984i
3.5 −3.99319 6.93213i 31.6770 31.6770i −32.1089 + 55.3626i 69.9376 69.9376i −346.081 93.0969i −445.243 511.998 + 1.51028i 1277.86i −764.090 205.543i
3.6 −0.392939 7.99034i −15.4507 + 15.4507i −63.6912 + 6.27943i −66.8872 + 66.8872i 129.527 + 117.385i −121.702 75.2016 + 506.447i 251.553i 560.735 + 508.169i
3.7 1.73714 + 7.80912i −8.66104 + 8.66104i −57.9647 + 27.1311i −38.1039 + 38.1039i −82.6805 52.5897i −108.944 −312.563 405.523i 578.973i −363.750 231.366i
3.8 5.82644 5.48203i −6.79913 + 6.79913i 3.89476 63.8814i 158.437 158.437i −2.34170 + 76.8878i −213.507 −327.507 393.552i 636.544i 54.5675 1791.68i
3.9 5.94882 5.34898i 25.6946 25.6946i 6.77693 63.6402i −141.826 + 141.826i 15.4128 290.292i 411.977 −300.095 414.834i 591.425i −85.0738 + 1602.32i
3.10 7.25359 + 3.37422i 14.8973 14.8973i 41.2292 + 48.9505i 41.0202 41.0202i 158.326 57.7921i −67.2599 133.890 + 494.184i 285.141i 435.955 159.133i
3.11 7.98961 + 0.407597i −33.4050 + 33.4050i 63.6677 + 6.51308i −93.3489 + 93.3489i −280.509 + 253.277i 261.028 506.026 + 77.9878i 1502.79i −783.870 + 707.773i
11.1 −7.92168 1.11666i 8.95029 + 8.95029i 61.5061 + 17.6917i −127.202 127.202i −60.9069 80.8958i −458.680 −467.476 208.829i 568.785i 865.616 + 1149.70i
11.2 −7.06494 + 3.75322i −9.43202 9.43202i 35.8267 53.0325i 46.6419 + 46.6419i 102.037 + 31.2362i 647.751 −54.0705 + 509.137i 551.074i −504.579 154.465i
11.3 −6.31611 4.90986i −31.9540 31.9540i 15.7866 + 62.0224i 95.0213 + 95.0213i 44.9356 + 358.715i −293.582 204.811 469.251i 1313.12i −133.624 1066.71i
11.4 −4.06674 6.88924i 23.4827 + 23.4827i −30.9232 + 56.0335i 55.3108 + 55.3108i 66.2800 257.276i 386.163 511.785 14.8363i 373.878i 156.115 605.984i
11.5 −3.99319 + 6.93213i 31.6770 + 31.6770i −32.1089 55.3626i 69.9376 + 69.9376i −346.081 + 93.0969i −445.243 511.998 1.51028i 1277.86i −764.090 + 205.543i
11.6 −0.392939 + 7.99034i −15.4507 15.4507i −63.6912 6.27943i −66.8872 66.8872i 129.527 117.385i −121.702 75.2016 506.447i 251.553i 560.735 508.169i
11.7 1.73714 7.80912i −8.66104 8.66104i −57.9647 27.1311i −38.1039 38.1039i −82.6805 + 52.5897i −108.944 −312.563 + 405.523i 578.973i −363.750 + 231.366i
11.8 5.82644 + 5.48203i −6.79913 6.79913i 3.89476 + 63.8814i 158.437 + 158.437i −2.34170 76.8878i −213.507 −327.507 + 393.552i 636.544i 54.5675 + 1791.68i
11.9 5.94882 + 5.34898i 25.6946 + 25.6946i 6.77693 + 63.6402i −141.826 141.826i 15.4128 + 290.292i 411.977 −300.095 + 414.834i 591.425i −85.0738 1602.32i
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 16.7.f.a 22
4.b odd 2 1 64.7.f.a 22
8.b even 2 1 128.7.f.b 22
8.d odd 2 1 128.7.f.a 22
16.e even 4 1 64.7.f.a 22
16.e even 4 1 128.7.f.a 22
16.f odd 4 1 inner 16.7.f.a 22
16.f odd 4 1 128.7.f.b 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.7.f.a 22 1.a even 1 1 trivial
16.7.f.a 22 16.f odd 4 1 inner
64.7.f.a 22 4.b odd 2 1
64.7.f.a 22 16.e even 4 1
128.7.f.a 22 8.d odd 2 1
128.7.f.a 22 16.e even 4 1
128.7.f.b 22 8.b even 2 1
128.7.f.b 22 16.f odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(16, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database