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Properties

Label 68.7.c.a
Level $68$
Weight $7$
Character orbit 68.c
Analytic conductor $15.644$
Analytic rank $0$
Dimension $48$
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] = [68,7,Mod(35,68)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(68, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("68.35");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 68 = 2^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 68.c (of order \(2\), degree \(1\), 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: \(15.6436776861\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 10 q^{2} + 90 q^{4} - 510 q^{6} + 1406 q^{8} - 11664 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 10 q^{2} + 90 q^{4} - 510 q^{6} + 1406 q^{8} - 11664 q^{9} - 1710 q^{10} - 6234 q^{12} - 5040 q^{13} + 9040 q^{14} - 1230 q^{16} - 6974 q^{18} + 1046 q^{20} - 35488 q^{21} - 13254 q^{22} + 22838 q^{24} + 173856 q^{25} + 44744 q^{26} + 42864 q^{28} - 83536 q^{29} - 95104 q^{30} - 53050 q^{32} + 9072 q^{33} + 24174 q^{36} + 62640 q^{37} + 399840 q^{38} - 58218 q^{40} + 70448 q^{41} - 90520 q^{42} - 356746 q^{44} - 103808 q^{45} - 177084 q^{46} + 808886 q^{48} - 758880 q^{49} - 560506 q^{50} - 85800 q^{52} + 744784 q^{53} - 263596 q^{54} - 701896 q^{56} - 125008 q^{57} - 351534 q^{58} - 64960 q^{60} + 196560 q^{61} + 1428016 q^{62} - 60246 q^{64} - 708656 q^{65} + 1089052 q^{66} + 428496 q^{69} - 1132824 q^{70} - 1043230 q^{72} - 454272 q^{73} - 1135082 q^{74} + 850224 q^{76} - 549200 q^{77} + 703076 q^{78} + 1608910 q^{80} + 3871872 q^{81} - 328536 q^{82} + 2825208 q^{84} + 3243644 q^{86} + 691398 q^{88} - 1729776 q^{89} + 214114 q^{90} + 1469284 q^{92} - 6618704 q^{93} - 551136 q^{94} - 5148618 q^{96} + 4548720 q^{97} - 2942042 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1 −7.98183 − 0.538920i − 26.1199i 63.4191 + 8.60314i −43.2305 −14.0765 + 208.485i − 300.830i −501.564 − 102.847i 46.7509 345.058 + 23.2978i
35.2 −7.98183 + 0.538920i 26.1199i 63.4191 − 8.60314i −43.2305 −14.0765 − 208.485i 300.830i −501.564 + 102.847i 46.7509 345.058 − 23.2978i
35.3 −7.72967 − 2.06209i − 0.610865i 55.4956 + 31.8785i 204.094 −1.25966 + 4.72178i − 420.119i −363.226 − 360.847i 728.627 −1577.58 − 420.859i
35.4 −7.72967 + 2.06209i 0.610865i 55.4956 − 31.8785i 204.094 −1.25966 − 4.72178i 420.119i −363.226 + 360.847i 728.627 −1577.58 + 420.859i
35.5 −7.60885 − 2.47090i 43.5409i 51.7893 + 37.6014i −9.10522 107.585 − 331.296i − 262.967i −301.148 − 414.070i −1166.81 69.2803 + 22.4981i
35.6 −7.60885 + 2.47090i − 43.5409i 51.7893 − 37.6014i −9.10522 107.585 + 331.296i 262.967i −301.148 + 414.070i −1166.81 69.2803 − 22.4981i
35.7 −7.28784 − 3.29961i − 19.6026i 42.2252 + 48.0940i −213.849 −64.6808 + 142.860i 273.992i −149.039 − 489.828i 344.739 1558.50 + 705.619i
35.8 −7.28784 + 3.29961i 19.6026i 42.2252 − 48.0940i −213.849 −64.6808 − 142.860i − 273.992i −149.039 + 489.828i 344.739 1558.50 − 705.619i
35.9 −6.72092 − 4.33927i − 48.9403i 26.3415 + 58.3277i 141.226 −212.365 + 328.924i 79.9919i 76.0604 − 506.319i −1666.15 −949.169 − 612.817i
35.10 −6.72092 + 4.33927i 48.9403i 26.3415 − 58.3277i 141.226 −212.365 − 328.924i − 79.9919i 76.0604 + 506.319i −1666.15 −949.169 + 612.817i
35.11 −6.51785 − 4.63871i 15.6750i 20.9647 + 60.4688i 40.3553 72.7119 − 102.167i 429.939i 143.853 − 491.376i 483.294 −263.030 − 187.196i
35.12 −6.51785 + 4.63871i − 15.6750i 20.9647 − 60.4688i 40.3553 72.7119 + 102.167i − 429.939i 143.853 + 491.376i 483.294 −263.030 + 187.196i
35.13 −5.18048 − 6.09612i 13.6083i −10.3253 + 63.1616i −106.433 82.9576 − 70.4973i − 664.386i 438.531 − 264.263i 543.815 551.371 + 648.826i
35.14 −5.18048 + 6.09612i − 13.6083i −10.3253 − 63.1616i −106.433 82.9576 + 70.4973i 664.386i 438.531 + 264.263i 543.815 551.371 − 648.826i
35.15 −4.79612 − 6.40291i − 21.5550i −17.9945 + 61.4182i 55.5815 −138.015 + 103.380i 57.5070i 479.559 − 179.352i 264.382 −266.575 − 355.883i
35.16 −4.79612 + 6.40291i 21.5550i −17.9945 − 61.4182i 55.5815 −138.015 − 103.380i − 57.5070i 479.559 + 179.352i 264.382 −266.575 + 355.883i
35.17 −3.59873 − 7.14487i 48.4992i −38.0983 + 51.4249i −205.612 346.520 − 174.536i 394.427i 504.530 + 87.1424i −1623.17 739.944 + 1469.07i
35.18 −3.59873 + 7.14487i − 48.4992i −38.0983 − 51.4249i −205.612 346.520 + 174.536i − 394.427i 504.530 − 87.1424i −1623.17 739.944 − 1469.07i
35.19 −3.33420 − 7.27208i 40.7968i −41.7662 + 48.4931i 247.305 296.678 − 136.025i − 20.5596i 491.903 + 142.042i −935.380 −824.564 − 1798.42i
35.20 −3.33420 + 7.27208i − 40.7968i −41.7662 − 48.4931i 247.305 296.678 + 136.025i 20.5596i 491.903 − 142.042i −935.380 −824.564 + 1798.42i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 68.7.c.a ✓ 48
4.b odd 2 1 inner 68.7.c.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.7.c.a ✓ 48 1.a even 1 1 trivial
68.7.c.a ✓ 48 4.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(68, [\chi])\).