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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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [16,7,Mod(3,16)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(16, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("16.3");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 16.f (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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.

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

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 3.11
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 \(S_{7}^{\mathrm{new}}(16, [\chi])\).