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Properties

Label 6422.2.a.bq
Level $6422$
Weight $2$
Character orbit 6422.a
Self dual yes
Analytic conductor $51.280$
Analytic rank $0$
Dimension $15$
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] = [6422,2,Mod(1,6422)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6422, 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("6422.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6422.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: \(51.2799281781\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 31 x^{13} - 4 x^{12} + 373 x^{11} + 85 x^{10} - 2208 x^{9} - 636 x^{8} + 6791 x^{7} + \cdots + 392 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta_1 q^{3} + q^{4} + \beta_{10} q^{5} + \beta_1 q^{6} + (\beta_{12} + \beta_{10} + \beta_{3} + 1) q^{7} + q^{8} + (\beta_{14} - \beta_{13} + \beta_{5} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta_1 q^{3} + q^{4} + \beta_{10} q^{5} + \beta_1 q^{6} + (\beta_{12} + \beta_{10} + \beta_{3} + 1) q^{7} + q^{8} + (\beta_{14} - \beta_{13} + \beta_{5} + \cdots + 1) q^{9}+ \cdots + (\beta_{13} - 5 \beta_{12} + 3 \beta_{11} + \cdots - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 15 q^{2} + 15 q^{4} + q^{5} + 18 q^{7} + 15 q^{8} + 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 15 q^{2} + 15 q^{4} + q^{5} + 18 q^{7} + 15 q^{8} + 17 q^{9} + q^{10} + 4 q^{11} + 18 q^{14} + 23 q^{15} + 15 q^{16} + 2 q^{17} + 17 q^{18} - 15 q^{19} + q^{20} + 2 q^{21} + 4 q^{22} + 17 q^{23} + 8 q^{25} + 12 q^{27} + 18 q^{28} - 20 q^{29} + 23 q^{30} + 30 q^{31} + 15 q^{32} + 36 q^{33} + 2 q^{34} + 32 q^{35} + 17 q^{36} + 35 q^{37} - 15 q^{38} + q^{40} + 15 q^{41} + 2 q^{42} + q^{43} + 4 q^{44} - 11 q^{45} + 17 q^{46} + 29 q^{49} + 8 q^{50} - q^{51} - q^{53} + 12 q^{54} - 6 q^{55} + 18 q^{56} - 20 q^{58} - 7 q^{59} + 23 q^{60} - 2 q^{61} + 30 q^{62} + 42 q^{63} + 15 q^{64} + 36 q^{66} + 34 q^{67} + 2 q^{68} - 12 q^{69} + 32 q^{70} + 4 q^{71} + 17 q^{72} + 12 q^{73} + 35 q^{74} + 31 q^{75} - 15 q^{76} - 20 q^{77} + 23 q^{79} + q^{80} + 7 q^{81} + 15 q^{82} - 3 q^{83} + 2 q^{84} + 46 q^{85} + q^{86} + 22 q^{87} + 4 q^{88} + 17 q^{89} - 11 q^{90} + 17 q^{92} + 60 q^{93} - q^{95} + 18 q^{97} + 29 q^{98} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 31 x^{13} - 4 x^{12} + 373 x^{11} + 85 x^{10} - 2208 x^{9} - 636 x^{8} + 6791 x^{7} + \cdots + 392 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 193884074943 \nu^{14} - 201950287980 \nu^{13} - 5788594784501 \nu^{12} + \cdots + 85125155787042 ) / 16434237893906 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 273993038595 \nu^{14} - 454917364489 \nu^{13} - 7951800462895 \nu^{12} + \cdots + 81940496280496 ) / 16434237893906 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 738118820283 \nu^{14} + 1081004001560 \nu^{13} + 21481071019325 \nu^{12} + \cdots - 264120839469396 ) / 32868475787812 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 64988194927 \nu^{14} + 77426247650 \nu^{13} + 1897449624715 \nu^{12} + \cdots - 15343610161320 ) / 2347748270558 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 38607285770 \nu^{14} + 50021871766 \nu^{13} + 1121665530434 \nu^{12} + \cdots - 10333663483962 ) / 1173874135279 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 708637533972 \nu^{14} + 761490392103 \nu^{13} + 20542837721082 \nu^{12} + \cdots - 47541675350932 ) / 16434237893906 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 452379850816 \nu^{14} + 617043986580 \nu^{13} + 13044910106152 \nu^{12} + \cdots - 78689644084590 ) / 8217118946953 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1203557977309 \nu^{14} + 1313396214410 \nu^{13} + 35288418753293 \nu^{12} + \cdots - 162345305353616 ) / 16434237893906 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2559294464187 \nu^{14} + 3047607863230 \nu^{13} + 74715943940201 \nu^{12} + \cdots - 412603877563136 ) / 32868475787812 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 405661145241 \nu^{14} - 586147754198 \nu^{13} - 11722490860231 \nu^{12} + \cdots + 147404924606680 ) / 4695496541116 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3319521574605 \nu^{14} - 4358410626934 \nu^{13} - 96553802732003 \nu^{12} + \cdots + 982897019051052 ) / 32868475787812 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 123406974291 \nu^{14} - 177046669743 \nu^{13} - 3570779556163 \nu^{12} + \cdots + 32503592648054 ) / 1173874135279 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 994365464810 \nu^{14} - 1409343410986 \nu^{13} - 28742233187393 \nu^{12} + \cdots + 205796730419665 ) / 8217118946953 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} - \beta_{13} + \beta_{5} + \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{12} + 2\beta_{11} - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} + 6\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8 \beta_{14} - 9 \beta_{13} + 2 \beta_{11} - 2 \beta_{9} - \beta_{8} + \beta_{7} + 3 \beta_{6} + \cdots + 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{14} - 24 \beta_{12} + 25 \beta_{11} - 3 \beta_{10} - 12 \beta_{9} + 12 \beta_{8} - 12 \beta_{7} + \cdots - 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 61 \beta_{14} - 76 \beta_{13} - 5 \beta_{12} + 34 \beta_{11} - 7 \beta_{10} - 28 \beta_{9} - 11 \beta_{8} + \cdots + 220 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 17 \beta_{14} - 246 \beta_{12} + 268 \beta_{11} - 58 \beta_{10} - 122 \beta_{9} + 126 \beta_{8} + \cdots - 111 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 459 \beta_{14} - 648 \beta_{13} - 104 \beta_{12} + 450 \beta_{11} - 138 \beta_{10} - 325 \beta_{9} + \cdots + 1794 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 231 \beta_{14} - 11 \beta_{13} - 2450 \beta_{12} + 2797 \beta_{11} - 798 \beta_{10} - 1235 \beta_{9} + \cdots - 1071 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3408 \beta_{14} - 5631 \beta_{13} - 1570 \beta_{12} + 5447 \beta_{11} - 1937 \beta_{10} - 3627 \beta_{9} + \cdots + 14936 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2874 \beta_{14} - 361 \beta_{13} - 24318 \beta_{12} + 29103 \beta_{11} - 9648 \beta_{10} + \cdots - 9967 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 24758 \beta_{14} - 49902 \beta_{13} - 20938 \beta_{12} + 63141 \beta_{11} - 23926 \beta_{10} + \cdots + 126497 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 33990 \beta_{14} - 7308 \beta_{13} - 242106 \beta_{12} + 303419 \beta_{11} - 109603 \beta_{10} + \cdots - 89840 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 173034 \beta_{14} - 450638 \beta_{13} - 261645 \beta_{12} + 713868 \beta_{11} - 278091 \beta_{10} + \cdots + 1088558 \) 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
−2.98660
−2.65291
−2.60210
−1.54553
−1.43594
−1.35072
−1.02081
0.206045
0.503505
1.01084
1.38264
1.75888
2.61859
2.85344
3.26068
1.00000 −2.98660 1.00000 −3.18940 −2.98660 3.65892 1.00000 5.91977 −3.18940
1.2 1.00000 −2.65291 1.00000 −0.987196 −2.65291 −3.23991 1.00000 4.03794 −0.987196
1.3 1.00000 −2.60210 1.00000 0.883013 −2.60210 3.97763 1.00000 3.77094 0.883013
1.4 1.00000 −1.54553 1.00000 −1.89914 −1.54553 0.915609 1.00000 −0.611330 −1.89914
1.5 1.00000 −1.43594 1.00000 −2.70512 −1.43594 0.857586 1.00000 −0.938065 −2.70512
1.6 1.00000 −1.35072 1.00000 2.02622 −1.35072 −1.66961 1.00000 −1.17555 2.02622
1.7 1.00000 −1.02081 1.00000 1.43823 −1.02081 5.08222 1.00000 −1.95796 1.43823
1.8 1.00000 0.206045 1.00000 3.15439 0.206045 3.30532 1.00000 −2.95755 3.15439
1.9 1.00000 0.503505 1.00000 −1.68542 0.503505 −4.13863 1.00000 −2.74648 −1.68542
1.10 1.00000 1.01084 1.00000 0.455213 1.01084 −0.356053 1.00000 −1.97821 0.455213
1.11 1.00000 1.38264 1.00000 −3.52682 1.38264 0.0942941 1.00000 −1.08830 −3.52682
1.12 1.00000 1.75888 1.00000 3.17158 1.75888 5.12706 1.00000 0.0936587 3.17158
1.13 1.00000 2.61859 1.00000 3.36169 2.61859 2.21822 1.00000 3.85700 3.36169
1.14 1.00000 2.85344 1.00000 2.12918 2.85344 −0.00224701 1.00000 5.14213 2.12918
1.15 1.00000 3.26068 1.00000 −1.62641 3.26068 2.16959 1.00000 7.63201 −1.62641
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(13\) \(1\)
\(19\) \(1\)

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 6422.2.a.bq yes 15
13.b even 2 1 6422.2.a.bo ✓ 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6422.2.a.bo ✓ 15 13.b even 2 1
6422.2.a.bq yes 15 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6422))\):

\( T_{3}^{15} - 31 T_{3}^{13} - 4 T_{3}^{12} + 373 T_{3}^{11} + 85 T_{3}^{10} - 2208 T_{3}^{9} - 636 T_{3}^{8} + \cdots + 392 \) Copy content Toggle raw display
\( T_{5}^{15} - T_{5}^{14} - 41 T_{5}^{13} + 38 T_{5}^{12} + 666 T_{5}^{11} - 555 T_{5}^{10} - 5492 T_{5}^{9} + \cdots + 13117 \) Copy content Toggle raw display
\( T_{7}^{15} - 18 T_{7}^{14} + 95 T_{7}^{13} + 125 T_{7}^{12} - 2865 T_{7}^{11} + 7769 T_{7}^{10} + \cdots + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{15} \) Copy content Toggle raw display
$3$ \( T^{15} - 31 T^{13} + \cdots + 392 \) Copy content Toggle raw display
$5$ \( T^{15} - T^{14} + \cdots + 13117 \) Copy content Toggle raw display
$7$ \( T^{15} - 18 T^{14} + \cdots + 8 \) Copy content Toggle raw display
$11$ \( T^{15} - 4 T^{14} + \cdots - 219128 \) Copy content Toggle raw display
$13$ \( T^{15} \) Copy content Toggle raw display
$17$ \( T^{15} - 2 T^{14} + \cdots - 3037649 \) Copy content Toggle raw display
$19$ \( (T + 1)^{15} \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots - 405660352 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots + 31065433664 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 29984427256 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots + 522209344 \) Copy content Toggle raw display
$41$ \( T^{15} - 15 T^{14} + \cdots + 18752 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots - 529101272 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 1176125837312 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots - 1601243116096 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots + 27405374296 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots + 4856975148389 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots - 17735387144 \) Copy content Toggle raw display
$71$ \( T^{15} - 4 T^{14} + \cdots - 72939544 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots + 5733317071 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots - 57872680424 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots - 215836257496 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 2289295683584 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 1664366526976 \) Copy content Toggle raw display
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