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Properties

Label 49.7.b.c
Level $49$
Weight $7$
Character orbit 49.b
Analytic conductor $11.273$
Analytic rank $0$
Dimension $12$
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] = [49,7,Mod(48,49)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(49, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("49.48");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 49.b (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: \(11.2726500974\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} - 402 x^{10} + 3108 x^{9} + 50331 x^{8} - 626752 x^{7} - 710832 x^{6} + \cdots + 11212481308 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 7^{15} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 2) q^{2} + \beta_{3} q^{3} + ( - \beta_{5} - \beta_1 + 47) q^{4} + ( - \beta_{9} - 2 \beta_{3}) q^{5} + ( - \beta_{9} + 2 \beta_{8} + \cdots + \beta_{2}) q^{6}+ \cdots + ( - \beta_{11} + 2 \beta_{6} + \cdots - 105) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 2) q^{2} + \beta_{3} q^{3} + ( - \beta_{5} - \beta_1 + 47) q^{4} + ( - \beta_{9} - 2 \beta_{3}) q^{5} + ( - \beta_{9} + 2 \beta_{8} + \cdots + \beta_{2}) q^{6}+ \cdots + (653 \beta_{11} + 3939 \beta_{6} + \cdots - 383991) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 20 q^{2} + 564 q^{4} + 940 q^{8} - 1252 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 20 q^{2} + 564 q^{4} + 940 q^{8} - 1252 q^{9} - 3872 q^{11} + 22432 q^{15} + 37908 q^{16} - 19436 q^{18} + 79872 q^{22} + 40032 q^{23} - 28860 q^{25} + 34016 q^{29} + 21192 q^{30} - 82060 q^{32} - 281620 q^{36} - 169632 q^{37} + 492128 q^{39} - 120768 q^{43} - 404312 q^{44} - 118872 q^{46} - 1045524 q^{50} - 539424 q^{51} - 15856 q^{53} + 926944 q^{57} + 971304 q^{58} - 429912 q^{60} - 5748 q^{64} - 267904 q^{65} + 1508352 q^{67} + 2518624 q^{71} + 2550364 q^{72} - 3876024 q^{74} - 67592 q^{78} + 232224 q^{79} + 528204 q^{81} + 754128 q^{85} - 1560848 q^{86} + 1812720 q^{88} + 3652888 q^{92} + 164384 q^{93} + 2305600 q^{95} - 4474992 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} - 402 x^{10} + 3108 x^{9} + 50331 x^{8} - 626752 x^{7} - 710832 x^{6} + \cdots + 11212481308 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 38\!\cdots\!75 \nu^{11} + \cdots - 27\!\cdots\!24 ) / 21\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 72\!\cdots\!45 \nu^{11} + \cdots - 11\!\cdots\!74 ) / 53\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 52\!\cdots\!85 \nu^{11} + \cdots + 63\!\cdots\!38 ) / 10\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11\!\cdots\!25 \nu^{11} + \cdots + 77\!\cdots\!00 ) / 21\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 98\!\cdots\!43 \nu^{11} + \cdots + 12\!\cdots\!76 ) / 10\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 29\!\cdots\!75 \nu^{11} + \cdots + 62\!\cdots\!88 ) / 26\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 84\!\cdots\!85 \nu^{11} + \cdots - 13\!\cdots\!64 ) / 53\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 24\!\cdots\!24 \nu^{11} + \cdots + 61\!\cdots\!50 ) / 10\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 30\!\cdots\!10 \nu^{11} + \cdots + 42\!\cdots\!46 ) / 10\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 51\!\cdots\!17 \nu^{11} + \cdots + 11\!\cdots\!06 ) / 10\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 27\!\cdots\!09 \nu^{11} + \cdots + 54\!\cdots\!88 ) / 21\!\cdots\!68 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 2 \beta_{10} + 4 \beta_{7} + 49 \beta_{6} - 49 \beta_{5} + 21 \beta_{4} - 6 \beta_{3} + 4 \beta_{2} + \cdots + 147 ) / 686 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 28 \beta_{11} - 24 \beta_{10} - 28 \beta_{9} + 28 \beta_{8} - 64 \beta_{7} - 231 \beta_{6} + \cdots + 46977 ) / 686 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 28 \beta_{11} - 6 \beta_{10} + 504 \beta_{9} + 84 \beta_{8} + 1888 \beta_{7} + 4865 \beta_{6} + \cdots - 262311 ) / 686 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2660 \beta_{11} - 5032 \beta_{10} - 12992 \beta_{9} + 12208 \beta_{8} - 31768 \beta_{7} + \cdots + 5648615 ) / 686 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 9436 \beta_{11} + 52334 \beta_{10} + 238840 \beta_{9} - 71260 \beta_{8} + 603438 \beta_{7} + \cdots - 58593843 ) / 686 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 278236 \beta_{11} - 1289808 \beta_{10} - 4122132 \beta_{9} + 2978668 \beta_{8} - 9921320 \beta_{7} + \cdots + 850401069 ) / 686 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2516444 \beta_{11} + 18217232 \beta_{10} + 69742190 \beta_{9} - 32639194 \beta_{8} + \cdots - 10215672341 ) / 686 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 33941348 \beta_{11} - 320914128 \beta_{10} - 1106239904 \beta_{9} + 687099392 \beta_{8} + \cdots + 125572924883 ) / 686 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 394684780 \beta_{11} + 4795968542 \beta_{10} + 17492242572 \beta_{9} - 9320516688 \beta_{8} + \cdots - 1450785331653 ) / 686 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 3709281492 \beta_{11} - 75470883816 \beta_{10} - 266795607148 \beta_{9} + 155598126292 \beta_{8} + \cdots + 14789104826447 ) / 686 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 37143238356 \beta_{11} + 1121303885928 \beta_{10} + 4029755221338 \beta_{9} + \cdots - 129893851512607 ) / 686 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
48.1
8.96617 + 1.84776i
8.96617 − 1.84776i
2.85211 − 0.765367i
2.85211 + 0.765367i
−14.4693 − 1.84776i
−14.4693 + 1.84776i
6.78526 + 0.765367i
6.78526 − 0.765367i
−7.22315 − 0.765367i
−7.22315 + 0.765367i
5.08897 + 1.84776i
5.08897 − 1.84776i
−14.7548 − 34.1070i 153.704 32.1894i 503.243i 0 −1323.57 −434.290 − 474.948i
48.2 −14.7548 34.1070i 153.704 − 32.1894i − 503.243i 0 −1323.57 −434.290 474.948i
48.3 −7.00447 − 12.3267i −14.9374 116.369i 86.3419i 0 552.915 577.053 − 815.102i
48.4 −7.00447 12.3267i −14.9374 − 116.369i − 86.3419i 0 552.915 577.053 815.102i
48.5 −0.342163 − 9.71999i −63.8829 80.0017i 3.32582i 0 43.7568 634.522 − 27.3736i
48.6 −0.342163 9.71999i −63.8829 − 80.0017i − 3.32582i 0 43.7568 634.522 27.3736i
48.7 4.66904 − 37.0663i −42.2001 218.499i − 173.064i 0 −495.852 −644.907 1020.18i
48.8 4.66904 37.0663i −42.2001 − 218.499i 173.064i 0 −495.852 −644.907 − 1020.18i
48.9 12.9923 − 46.6500i 104.799 − 30.7903i − 606.091i 0 530.077 −1447.23 − 400.037i
48.10 12.9923 46.6500i 104.799 30.7903i 606.091i 0 530.077 −1447.23 400.037i
48.11 14.4401 − 6.33640i 144.517 196.245i − 91.4984i 0 1162.67 688.850 2833.79i
48.12 14.4401 6.33640i 144.517 − 196.245i 91.4984i 0 1162.67 688.850 − 2833.79i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 48.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.7.b.c ✓ 12
3.b odd 2 1 441.7.d.e 12
7.b odd 2 1 inner 49.7.b.c ✓ 12
7.c even 3 2 49.7.d.e 24
7.d odd 6 2 49.7.d.e 24
21.c even 2 1 441.7.d.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.7.b.c ✓ 12 1.a even 1 1 trivial
49.7.b.c ✓ 12 7.b odd 2 1 inner
49.7.d.e 24 7.c even 3 2
49.7.d.e 24 7.d odd 6 2
441.7.d.e 12 3.b odd 2 1
441.7.d.e 12 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 10T_{2}^{5} - 283T_{2}^{4} + 2580T_{2}^{3} + 14482T_{2}^{2} - 85888T_{2} - 30976 \) acting on \(S_{7}^{\mathrm{new}}(49, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 10 T^{5} + \cdots - 30976)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 20\!\cdots\!68 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots + 10\!\cdots\!12)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 79\!\cdots\!28 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 15\!\cdots\!52 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 96\!\cdots\!12 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots - 13\!\cdots\!84)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 13\!\cdots\!44)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 11\!\cdots\!92 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 39\!\cdots\!64)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 10\!\cdots\!48 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 83\!\cdots\!08)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 22\!\cdots\!12 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 24\!\cdots\!04)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 11\!\cdots\!28 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 23\!\cdots\!72 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 14\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 68\!\cdots\!92 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 37\!\cdots\!36)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 40\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 22\!\cdots\!72 \) Copy content Toggle raw display
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