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Properties

Label 49.5.d.c
Level $49$
Weight $5$
Character orbit 49.d
Analytic conductor $5.065$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [49,5,Mod(19,49)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(49, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("49.19");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 49.d (of order \(6\), degree \(2\), not 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: \(5.06512819111\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} - 140 x^{14} + 1120 x^{13} + 8358 x^{12} - 65072 x^{11} - 277636 x^{10} + \cdots + 4890937156 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 7^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{12} + \beta_{2} + 1) q^{2} + \beta_{5} q^{3} + (3 \beta_{12} - 3 \beta_{4} + \cdots + 5 \beta_{2}) q^{4}+ \cdots + (\beta_{13} - \beta_{12} + \beta_{7} + \cdots + 44) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{12} + \beta_{2} + 1) q^{2} + \beta_{5} q^{3} + (3 \beta_{12} - 3 \beta_{4} + \cdots + 5 \beta_{2}) q^{4}+ \cdots + ( - 167 \beta_{13} + 145 \beta_{4} + \cdots + 3779) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 12 q^{2} - 52 q^{4} - 744 q^{8} + 352 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 12 q^{2} - 52 q^{4} - 744 q^{8} + 352 q^{9} - 120 q^{11} - 1264 q^{15} + 300 q^{16} - 340 q^{18} + 3904 q^{22} - 1752 q^{23} + 2192 q^{25} + 2496 q^{29} - 456 q^{30} - 3156 q^{32} + 9880 q^{36} - 2368 q^{37} + 7672 q^{39} - 17104 q^{43} - 264 q^{44} - 7208 q^{46} - 11112 q^{50} + 11976 q^{51} - 5496 q^{53} - 18400 q^{57} + 17496 q^{58} - 2856 q^{60} + 7960 q^{64} - 30240 q^{65} + 7440 q^{67} + 19968 q^{71} + 1508 q^{72} + 1080 q^{74} + 44912 q^{78} + 14096 q^{79} - 3432 q^{81} + 23824 q^{85} - 44496 q^{86} + 44464 q^{88} - 64464 q^{92} - 9584 q^{93} - 22488 q^{95} + 60288 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} - 140 x^{14} + 1120 x^{13} + 8358 x^{12} - 65072 x^{11} - 277636 x^{10} + \cdots + 4890937156 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2657812 \nu^{14} - 18604684 \nu^{13} - 613098815 \nu^{12} + 3920453782 \nu^{11} + \cdots - 19\!\cdots\!00 ) / 140406754788818 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 60906948 \nu^{14} + 426348636 \nu^{13} + 7618462488 \nu^{12} - 51253307196 \nu^{11} + \cdots + 13\!\cdots\!92 ) / 15\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 75995425305526 \nu^{14} - 531967977138682 \nu^{13} + \cdots - 10\!\cdots\!24 ) / 40\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 22\!\cdots\!80 \nu^{15} + \cdots - 12\!\cdots\!84 ) / 10\!\cdots\!58 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16\!\cdots\!45 \nu^{15} + \cdots + 82\!\cdots\!84 ) / 76\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 20\!\cdots\!15 \nu^{15} + \cdots + 10\!\cdots\!56 ) / 76\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 27\!\cdots\!60 \nu^{15} + \cdots - 88\!\cdots\!56 ) / 76\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 28\!\cdots\!39 \nu^{15} + \cdots - 21\!\cdots\!84 ) / 76\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 31\!\cdots\!01 \nu^{15} + \cdots + 89\!\cdots\!92 ) / 76\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 31\!\cdots\!92 \nu^{15} + \cdots - 11\!\cdots\!72 ) / 76\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 32\!\cdots\!05 \nu^{15} + \cdots - 10\!\cdots\!84 ) / 76\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 32\!\cdots\!88 \nu^{15} + \cdots - 52\!\cdots\!08 ) / 76\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 48\!\cdots\!02 \nu^{15} + \cdots - 14\!\cdots\!54 ) / 10\!\cdots\!58 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 49\!\cdots\!95 \nu^{15} + \cdots - 12\!\cdots\!64 ) / 76\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 71\!\cdots\!53 \nu^{15} + \cdots + 67\!\cdots\!88 ) / 76\!\cdots\!06 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -\beta_{14} - 3\beta_{11} - \beta_{6} + 3\beta_{5} + 49\beta_{4} ) / 49 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4 \beta_{14} - 16 \beta_{11} - 10 \beta_{6} - 26 \beta_{5} + 49 \beta_{4} + 7 \beta_{3} + 98 \beta_{2} + \cdots + 1078 ) / 49 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4 \beta_{15} - 16 \beta_{14} + 28 \beta_{13} + 189 \beta_{12} - 230 \beta_{11} + 5 \beta_{10} + \cdots + 896 ) / 49 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 4 \beta_{15} + 232 \beta_{14} + 56 \beta_{13} + 378 \beta_{12} - 788 \beta_{11} - 180 \beta_{10} + \cdots + 32858 ) / 49 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 384 \beta_{15} + 530 \beta_{14} + 1617 \beta_{13} + 12040 \beta_{12} - 11185 \beta_{11} + 284 \beta_{10} + \cdots + 57673 ) / 49 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 60 \beta_{15} + 12774 \beta_{14} + 4711 \beta_{13} + 35175 \beta_{12} - 39448 \beta_{11} + \cdots + 1078763 ) / 49 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 12775 \beta_{15} + 61795 \beta_{14} + 82614 \beta_{13} + 631169 \beta_{12} - 495960 \beta_{11} + \cdots + 2815148 ) / 49 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 20888 \beta_{15} + 663632 \beta_{14} + 308602 \beta_{13} + 2361408 \beta_{12} - 1947752 \beta_{11} + \cdots + 37815064 ) / 49 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 74848 \beta_{15} + 3889940 \beta_{14} + 3879015 \beta_{13} + 31527720 \beta_{12} - 21378951 \beta_{11} + \cdots + 127675443 ) / 49 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 3537776 \beta_{15} + 32967670 \beta_{14} + 17113257 \beta_{13} + 140172375 \beta_{12} + \cdots + 1407014693 ) / 49 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 23703415 \beta_{15} + 206119485 \beta_{14} + 172651206 \beta_{13} + 1544571567 \beta_{12} + \cdots + 5707603496 ) / 49 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 313866724 \beta_{15} + 1577162152 \beta_{14} + 852650624 \beta_{13} + 7763728896 \beta_{12} + \cdots + 54807658094 ) / 49 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 2313988586 \beta_{15} + 10091412694 \beta_{14} + 7428707573 \beta_{13} + 74767672434 \beta_{12} + \cdots + 254320773581 ) / 49 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 21623639856 \beta_{15} + 73217791014 \beta_{14} + 39618499837 \beta_{13} + 410135555375 \beta_{12} + \cdots + 2199567099961 ) / 49 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 158042676361 \beta_{15} + 472674139117 \beta_{14} + 312396917826 \beta_{13} + 3581606038605 \beta_{12} + \cdots + 11250815893884 ) / 49 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(1 + \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−5.65368 + 0.382683i
−4.32803 − 0.382683i
−4.31743 + 0.923880i
−1.11701 − 0.923880i
5.31743 − 0.923880i
2.11701 + 0.923880i
5.32803 + 0.382683i
6.65368 − 0.382683i
−5.65368 − 0.382683i
−4.32803 + 0.382683i
−4.31743 − 0.923880i
−1.11701 + 0.923880i
5.31743 + 0.923880i
2.11701 − 0.923880i
5.32803 − 0.382683i
6.65368 + 0.382683i
−1.99543 + 3.45618i −3.81864 + 2.20470i 0.0365359 + 0.0632820i 37.9221 + 21.8943i − 17.5972i 0 −64.1453 −30.7786 + 53.3102i −151.342 + 87.3772i
19.2 −1.99543 + 3.45618i 3.81864 − 2.20470i 0.0365359 + 0.0632820i −37.9221 − 21.8943i 17.5972i 0 −64.1453 −30.7786 + 53.3102i 151.342 − 87.3772i
19.3 −0.858610 + 1.48716i −13.9722 + 8.06687i 6.52558 + 11.3026i −6.78727 − 3.91863i − 27.7052i 0 −49.8872 89.6489 − 155.277i 11.6552 − 6.72915i
19.4 −0.858610 + 1.48716i 13.9722 − 8.06687i 6.52558 + 11.3026i 6.78727 + 3.91863i 27.7052i 0 −49.8872 89.6489 − 155.277i −11.6552 + 6.72915i
19.5 2.35861 − 4.08523i −10.8855 + 6.28477i −3.12608 − 5.41453i 29.1467 + 16.8279i 59.2933i 0 45.9827 38.4968 − 66.6784i 137.491 − 79.3807i
19.6 2.35861 − 4.08523i 10.8855 − 6.28477i −3.12608 − 5.41453i −29.1467 − 16.8279i − 59.2933i 0 45.9827 38.4968 − 66.6784i −137.491 + 79.3807i
19.7 3.49543 − 6.05426i −6.83370 + 3.94544i −16.4360 − 28.4680i −19.0608 − 11.0048i 55.1639i 0 −117.950 −9.36707 + 16.2242i −133.251 + 76.9327i
19.8 3.49543 − 6.05426i 6.83370 − 3.94544i −16.4360 − 28.4680i 19.0608 + 11.0048i − 55.1639i 0 −117.950 −9.36707 + 16.2242i 133.251 − 76.9327i
31.1 −1.99543 − 3.45618i −3.81864 − 2.20470i 0.0365359 − 0.0632820i 37.9221 − 21.8943i 17.5972i 0 −64.1453 −30.7786 − 53.3102i −151.342 − 87.3772i
31.2 −1.99543 − 3.45618i 3.81864 + 2.20470i 0.0365359 − 0.0632820i −37.9221 + 21.8943i − 17.5972i 0 −64.1453 −30.7786 − 53.3102i 151.342 + 87.3772i
31.3 −0.858610 − 1.48716i −13.9722 − 8.06687i 6.52558 − 11.3026i −6.78727 + 3.91863i 27.7052i 0 −49.8872 89.6489 + 155.277i 11.6552 + 6.72915i
31.4 −0.858610 − 1.48716i 13.9722 + 8.06687i 6.52558 − 11.3026i 6.78727 − 3.91863i − 27.7052i 0 −49.8872 89.6489 + 155.277i −11.6552 − 6.72915i
31.5 2.35861 + 4.08523i −10.8855 − 6.28477i −3.12608 + 5.41453i 29.1467 − 16.8279i − 59.2933i 0 45.9827 38.4968 + 66.6784i 137.491 + 79.3807i
31.6 2.35861 + 4.08523i 10.8855 + 6.28477i −3.12608 + 5.41453i −29.1467 + 16.8279i 59.2933i 0 45.9827 38.4968 + 66.6784i −137.491 − 79.3807i
31.7 3.49543 + 6.05426i −6.83370 − 3.94544i −16.4360 + 28.4680i −19.0608 + 11.0048i − 55.1639i 0 −117.950 −9.36707 − 16.2242i −133.251 − 76.9327i
31.8 3.49543 + 6.05426i 6.83370 + 3.94544i −16.4360 + 28.4680i 19.0608 − 11.0048i 55.1639i 0 −117.950 −9.36707 − 16.2242i 133.251 + 76.9327i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.5.d.c 16
7.b odd 2 1 inner 49.5.d.c 16
7.c even 3 1 49.5.b.b ✓ 8
7.c even 3 1 inner 49.5.d.c 16
7.d odd 6 1 49.5.b.b ✓ 8
7.d odd 6 1 inner 49.5.d.c 16
21.g even 6 1 441.5.d.g 8
21.h odd 6 1 441.5.d.g 8
28.f even 6 1 784.5.c.g 8
28.g odd 6 1 784.5.c.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.5.b.b ✓ 8 7.c even 3 1
49.5.b.b ✓ 8 7.d odd 6 1
49.5.d.c 16 1.a even 1 1 trivial
49.5.d.c 16 7.b odd 2 1 inner
49.5.d.c 16 7.c even 3 1 inner
49.5.d.c 16 7.d odd 6 1 inner
441.5.d.g 8 21.g even 6 1
441.5.d.g 8 21.h odd 6 1
784.5.c.g 8 28.f even 6 1
784.5.c.g 8 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 6T_{2}^{7} + 63T_{2}^{6} - 54T_{2}^{5} + 1151T_{2}^{4} - 204T_{2}^{3} + 17766T_{2}^{2} + 24408T_{2} + 51076 \) acting on \(S_{5}^{\mathrm{new}}(49, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 6 T^{7} + \cdots + 51076)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 24\!\cdots\!96 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 41\!\cdots\!36 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 740816911074304)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 11\!\cdots\!16)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 77\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 32\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 14\!\cdots\!44)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 624 T^{3} + \cdots - 124444104944)^{4} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 26\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 14\!\cdots\!36)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 51\!\cdots\!16)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 4296135337184)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 27\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 20\!\cdots\!56)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 73\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 59\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 29541809698816)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 36\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 27\!\cdots\!76)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 10\!\cdots\!96)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 18\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 57\!\cdots\!76)^{2} \) Copy content Toggle raw display
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