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Properties

Label 6275.2.a.d
Level $6275$
Weight $2$
Character orbit 6275.a
Self dual yes
Analytic conductor $50.106$
Analytic rank $1$
Dimension $14$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6275,2,Mod(1,6275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6275, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6275.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6275 = 5^{2} \cdot 251 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6275.a (trivial)

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: yes
Analytic conductor: \(50.1061272684\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} - 21 x^{12} + 17 x^{11} + 173 x^{10} - 109 x^{9} - 702 x^{8} + 334 x^{7} + 1431 x^{6} + \cdots + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1255)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{10} q^{3} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{4} + (\beta_{13} - 1) q^{6} + \beta_{5} q^{7} + (\beta_{8} + \beta_{7} + \beta_{2} + \beta_1) q^{8} + ( - \beta_{13} - \beta_{12} + \beta_{10} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{10} q^{3} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{4} + (\beta_{13} - 1) q^{6} + \beta_{5} q^{7} + (\beta_{8} + \beta_{7} + \beta_{2} + \beta_1) q^{8} + ( - \beta_{13} - \beta_{12} + \beta_{10} + \cdots + 1) q^{9}+ \cdots + (2 \beta_{12} - \beta_{11} - 2 \beta_{9} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + q^{2} + 3 q^{3} + 15 q^{4} - 12 q^{6} + 3 q^{7} + 9 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + q^{2} + 3 q^{3} + 15 q^{4} - 12 q^{6} + 3 q^{7} + 9 q^{8} + 7 q^{9} - 16 q^{11} + 4 q^{12} - 9 q^{13} - 21 q^{14} + 5 q^{16} + 20 q^{17} - 14 q^{18} - 2 q^{19} - 12 q^{21} - 13 q^{22} + 16 q^{23} - 20 q^{24} - 32 q^{26} + 9 q^{27} - 7 q^{28} - 30 q^{29} - 19 q^{31} + 29 q^{32} + 25 q^{33} - 4 q^{34} - 23 q^{36} - 15 q^{37} + 28 q^{38} - 29 q^{39} - 55 q^{41} + 19 q^{42} - 14 q^{43} - 41 q^{44} - 3 q^{46} + 2 q^{47} - 7 q^{48} - 25 q^{49} - 4 q^{51} - 12 q^{52} - 14 q^{53} - 2 q^{54} - 5 q^{56} + 8 q^{57} - 19 q^{58} - 13 q^{59} - 49 q^{61} + 9 q^{62} - 10 q^{63} - 21 q^{64} + 7 q^{66} - q^{67} + 45 q^{68} - 60 q^{69} - 22 q^{71} + 5 q^{72} + 27 q^{73} - 19 q^{74} - 26 q^{76} + 7 q^{77} - 8 q^{78} - 20 q^{79} - 2 q^{81} - 12 q^{82} - 25 q^{83} + 12 q^{84} + 38 q^{86} - 52 q^{87} - 15 q^{88} - 22 q^{89} - 11 q^{91} - 35 q^{92} - 37 q^{93} - 5 q^{94} - 4 q^{96} + 27 q^{97} + 21 q^{98} - 37 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} - 21 x^{12} + 17 x^{11} + 173 x^{10} - 109 x^{9} - 702 x^{8} + 334 x^{7} + 1431 x^{6} + \cdots + 11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 22 \nu^{13} + 46 \nu^{12} + 909 \nu^{11} - 537 \nu^{10} - 11838 \nu^{9} + 231 \nu^{8} + \cdots - 1741 ) / 1823 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 22 \nu^{13} + 46 \nu^{12} + 909 \nu^{11} - 537 \nu^{10} - 11838 \nu^{9} + 231 \nu^{8} + \cdots + 3728 ) / 1823 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 43 \nu^{13} + 573 \nu^{12} - 368 \nu^{11} - 10137 \nu^{10} - 3047 \nu^{9} + 62442 \nu^{8} + \cdots + 3320 ) / 1823 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 116 \nu^{13} - 574 \nu^{12} - 2307 \nu^{11} + 9792 \nu^{10} + 18998 \nu^{9} - 59554 \nu^{8} + \cdots + 562 ) / 1823 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 135 \nu^{13} - 448 \nu^{12} - 2512 \nu^{11} + 7687 \nu^{10} + 18118 \nu^{9} - 47904 \nu^{8} + \cdots - 9121 ) / 1823 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 159 \nu^{13} + \nu^{12} + 2675 \nu^{11} + 345 \nu^{10} - 15951 \nu^{9} - 2888 \nu^{8} + \cdots + 10702 ) / 1823 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 181 \nu^{13} - 47 \nu^{12} - 3584 \nu^{11} + 192 \nu^{10} + 27789 \nu^{9} + 2657 \nu^{8} + \cdots - 8961 ) / 1823 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 180 \nu^{13} - 618 \nu^{12} + 3957 \nu^{11} + 11019 \nu^{10} - 32057 \nu^{9} - 71030 \nu^{8} + \cdots + 3654 ) / 1823 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 284 \nu^{13} + 732 \nu^{12} - 5271 \nu^{11} - 13618 \nu^{10} + 34820 \nu^{9} + 89991 \nu^{8} + \cdots + 433 ) / 1823 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 517 \nu^{13} + 742 \nu^{12} - 9512 \nu^{11} - 13814 \nu^{10} + 63079 \nu^{9} + 88456 \nu^{8} + \cdots - 11042 ) / 1823 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 860 \nu^{13} - 522 \nu^{12} + 16475 \nu^{11} + 11325 \nu^{10} - 117714 \nu^{9} - 83943 \nu^{8} + \cdots + 11989 ) / 1823 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1016 \nu^{13} + 693 \nu^{12} - 18446 \nu^{11} - 14312 \nu^{10} + 120947 \nu^{9} + 98712 \nu^{8} + \cdots - 1301 ) / 1823 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + \beta_{7} + \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + \beta_{9} + \beta_{7} - \beta_{6} - \beta_{4} - 6\beta_{3} + 6\beta_{2} + 7\beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} - 2 \beta_{10} + 10 \beta_{8} + 8 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{12} + 10 \beta_{11} - 2 \beta_{10} + 10 \beta_{9} + 4 \beta_{8} + 12 \beta_{7} - 12 \beta_{6} + \cdots + 63 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 11 \beta_{13} - \beta_{12} + 3 \beta_{11} - 24 \beta_{10} + 3 \beta_{9} + 79 \beta_{8} + 59 \beta_{7} + \cdots + 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3 \beta_{13} - 12 \beta_{12} + 78 \beta_{11} - 29 \beta_{10} + 78 \beta_{9} + 63 \beta_{8} + 111 \beta_{7} + \cdots + 322 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 90 \beta_{13} - 15 \beta_{12} + 51 \beta_{11} - 211 \beta_{10} + 50 \beta_{9} + 584 \beta_{8} + \cdots + 52 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 52 \beta_{13} - 105 \beta_{12} + 569 \beta_{11} - 303 \beta_{10} + 565 \beta_{9} + 679 \beta_{8} + \cdots + 1708 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 669 \beta_{13} - 157 \beta_{12} + 574 \beta_{11} - 1666 \beta_{10} + 558 \beta_{9} + 4232 \beta_{8} + \cdots + 355 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 596 \beta_{13} - 826 \beta_{12} + 4075 \beta_{11} - 2763 \beta_{10} + 4002 \beta_{9} + 6270 \beta_{8} + \cdots + 9360 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 4806 \beta_{13} - 1422 \beta_{12} + 5444 \beta_{11} - 12585 \beta_{10} + 5266 \beta_{9} + 30557 \beta_{8} + \cdots + 2698 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.36098
−2.32137
−1.79629
−1.63978
−1.47264
−0.820025
0.0760467
0.362450
0.767233
0.950807
2.12220
2.19060
2.20848
2.73326
−2.36098 2.36799 3.57420 0 −5.59077 −0.917581 −3.71666 2.60738 0
1.2 −2.32137 0.890400 3.38877 0 −2.06695 2.91471 −3.22385 −2.20719 0
1.3 −1.79629 −0.569249 1.22667 0 1.02254 0.815419 1.38913 −2.67596 0
1.4 −1.63978 −0.707757 0.688871 0 1.16056 2.78843 2.14996 −2.49908 0
1.5 −1.47264 3.27631 0.168654 0 −4.82481 −1.24555 2.69690 7.73421 0
1.6 −0.820025 −2.87122 −1.32756 0 2.35447 3.08857 2.72868 5.24388 0
1.7 0.0760467 1.91134 −1.99422 0 0.145351 0.586583 −0.303747 0.653231 0
1.8 0.362450 2.05244 −1.86863 0 0.743905 1.15414 −1.40218 1.21250 0
1.9 0.767233 −0.0406429 −1.41135 0 −0.0311826 −3.65596 −2.61730 −2.99835 0
1.10 0.950807 −2.41406 −1.09597 0 −2.29530 0.744814 −2.94367 2.82767 0
1.11 2.12220 0.0735395 2.50372 0 0.156065 3.40768 1.06900 −2.99459 0
1.12 2.19060 2.05752 2.79875 0 4.50722 −2.77493 1.74975 1.23340 0
1.13 2.20848 −1.88926 2.87740 0 −4.17239 −2.84547 1.93772 0.569297 0
1.14 2.73326 −1.13736 5.47069 0 −3.10871 −1.06085 9.48627 −1.70640 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(251\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6275.2.a.d 14
5.b even 2 1 1255.2.a.c ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1255.2.a.c ✓ 14 5.b even 2 1
6275.2.a.d 14 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6275))\):

\( T_{2}^{14} - T_{2}^{13} - 21 T_{2}^{12} + 17 T_{2}^{11} + 173 T_{2}^{10} - 109 T_{2}^{9} - 702 T_{2}^{8} + \cdots + 11 \) Copy content Toggle raw display
\( T_{3}^{14} - 3 T_{3}^{13} - 20 T_{3}^{12} + 60 T_{3}^{11} + 142 T_{3}^{10} - 427 T_{3}^{9} - 452 T_{3}^{8} + \cdots - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} - T^{13} + \cdots + 11 \) Copy content Toggle raw display
$3$ \( T^{14} - 3 T^{13} + \cdots - 1 \) Copy content Toggle raw display
$5$ \( T^{14} \) Copy content Toggle raw display
$7$ \( T^{14} - 3 T^{13} + \cdots + 1231 \) Copy content Toggle raw display
$11$ \( T^{14} + 16 T^{13} + \cdots + 19871 \) Copy content Toggle raw display
$13$ \( T^{14} + 9 T^{13} + \cdots + 2997 \) Copy content Toggle raw display
$17$ \( T^{14} - 20 T^{13} + \cdots - 120931 \) Copy content Toggle raw display
$19$ \( T^{14} + 2 T^{13} + \cdots + 255937 \) Copy content Toggle raw display
$23$ \( T^{14} - 16 T^{13} + \cdots - 77432689 \) Copy content Toggle raw display
$29$ \( T^{14} + 30 T^{13} + \cdots + 9816277 \) Copy content Toggle raw display
$31$ \( T^{14} + 19 T^{13} + \cdots + 55985803 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots - 8215894667 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 305343208 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 221442641 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 2000576377 \) Copy content Toggle raw display
$53$ \( T^{14} + 14 T^{13} + \cdots + 63502189 \) Copy content Toggle raw display
$59$ \( T^{14} + 13 T^{13} + \cdots + 4666093 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 16271307221 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 3307587701 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 3393614111 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots - 601806246367 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 10184520303 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 1270305517 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 65730414413 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 130731193453 \) Copy content Toggle raw display
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