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Properties

Label 7.13.d.a
Level $7$
Weight $13$
Character orbit 7.d
Analytic conductor $6.398$
Analytic rank $0$
Dimension $14$
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] = [7,13,Mod(3,7)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7.3");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 7.d (of order \(6\), degree \(2\), 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: \(6.39795672093\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} + 21402 x^{12} - 299884 x^{11} + 348641804 x^{10} - 4374947304 x^{9} + \cdots + 49\!\cdots\!64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{5}\cdot 7^{11} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (6 \beta_{2} + \beta_1 + 6) q^{2} + ( - \beta_{4} - \beta_{3} + \beta_1 - 1) q^{3} + (\beta_{9} - 2 \beta_{7} + \beta_{5} + \cdots + 1) q^{4}+ \cdots + ( - 2 \beta_{13} + 12 \beta_{12} + \cdots + 38074) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (6 \beta_{2} + \beta_1 + 6) q^{2} + ( - \beta_{4} - \beta_{3} + \beta_1 - 1) q^{3} + (\beta_{9} - 2 \beta_{7} + \beta_{5} + \cdots + 1) q^{4}+ \cdots + (2600400 \beta_{13} + \cdots - 393452723411) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 44 q^{2} - 3 q^{3} - 14404 q^{4} - 9075 q^{5} + 334278 q^{7} - 52112 q^{8} + 265974 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 44 q^{2} - 3 q^{3} - 14404 q^{4} - 9075 q^{5} + 334278 q^{7} - 52112 q^{8} + 265974 q^{9} + 3406596 q^{10} - 2085193 q^{11} - 12871740 q^{12} + 21936320 q^{14} + 21308298 q^{15} - 64542600 q^{16} + 46675437 q^{17} + 12781272 q^{18} - 153598539 q^{19} + 46954005 q^{21} + 138292528 q^{22} + 230058719 q^{23} - 265960656 q^{24} + 19748318 q^{25} + 590880528 q^{26} + 229618900 q^{28} - 714663796 q^{29} + 999571848 q^{30} - 5040285435 q^{31} + 1011711600 q^{32} - 2159472843 q^{33} + 2425404597 q^{35} + 17119036896 q^{36} - 5044805001 q^{37} + 3238109592 q^{38} - 5776828680 q^{39} - 35225923512 q^{40} + 27354512640 q^{42} + 40132389276 q^{43} - 11796090372 q^{44} - 18717971166 q^{45} - 12430801472 q^{46} - 53247355563 q^{47} + 16893431198 q^{49} + 178352838608 q^{50} + 1366166853 q^{51} - 136241043960 q^{52} - 35499894841 q^{53} - 154061995896 q^{54} + 42830066608 q^{56} + 371092140234 q^{57} - 21841743752 q^{58} - 138067877811 q^{59} - 146425754916 q^{60} - 287167557435 q^{61} + 240766542198 q^{63} + 1077334893984 q^{64} - 121576543704 q^{65} - 358267134060 q^{66} - 104132716721 q^{67} - 1346622176340 q^{68} + 620078986800 q^{70} + 809521279196 q^{71} - 435854437392 q^{72} + 4606244997 q^{73} - 229538257100 q^{74} - 692089720566 q^{75} + 408395115359 q^{77} + 1860221591760 q^{78} + 53291758879 q^{79} - 1497721732104 q^{80} - 82850395887 q^{81} - 27489500640 q^{82} - 28369303620 q^{84} + 402017692458 q^{85} + 1190361662368 q^{86} - 603786809646 q^{87} + 342437177984 q^{88} + 77396974197 q^{89} - 1437510415152 q^{91} - 3683682769320 q^{92} - 509894759883 q^{93} + 2011339035528 q^{94} + 690022114413 q^{95} + 5403247899840 q^{96} + 153246523724 q^{98} - 5499432555540 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} + 21402 x^{12} - 299884 x^{11} + 348641804 x^{10} - 4374947304 x^{9} + \cdots + 49\!\cdots\!64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 22\!\cdots\!32 \nu^{13} + \cdots - 73\!\cdots\!36 ) / 77\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10\!\cdots\!09 \nu^{13} + \cdots - 96\!\cdots\!52 ) / 66\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 34\!\cdots\!05 \nu^{13} + \cdots - 15\!\cdots\!84 ) / 22\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 63\!\cdots\!65 \nu^{13} + \cdots + 13\!\cdots\!76 ) / 17\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 50\!\cdots\!82 \nu^{13} + \cdots + 46\!\cdots\!48 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 51\!\cdots\!66 \nu^{13} + \cdots + 14\!\cdots\!12 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 46\!\cdots\!53 \nu^{13} + \cdots + 48\!\cdots\!84 ) / 40\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 25\!\cdots\!03 \nu^{13} + \cdots - 49\!\cdots\!36 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 74\!\cdots\!73 \nu^{13} + \cdots + 35\!\cdots\!88 ) / 20\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 63\!\cdots\!29 \nu^{13} + \cdots + 74\!\cdots\!64 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 11\!\cdots\!83 \nu^{13} + \cdots + 64\!\cdots\!96 ) / 22\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 62\!\cdots\!37 \nu^{13} + \cdots - 27\!\cdots\!04 ) / 44\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - 2\beta_{7} + \beta_{5} - \beta_{4} - 9\beta_{3} + 6118\beta_{2} - 9\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 3 \beta_{13} - 3 \beta_{12} + 2 \beta_{11} + 4 \beta_{10} + 2 \beta_{9} + 3 \beta_{8} + 22 \beta_{7} + \cdots + 58249 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 92 \beta_{13} + 20 \beta_{12} + 46 \beta_{11} - 70 \beta_{10} - 16070 \beta_{9} + 94 \beta_{8} + \cdots - 68123884 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 17102 \beta_{13} - 90122 \beta_{12} - 51306 \beta_{11} - 141410 \beta_{10} + 287968 \beta_{9} + \cdots + 593630 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2911164 \beta_{13} + 244920 \beta_{12} + 1940776 \beta_{11} - 526224 \beta_{10} + 1940776 \beta_{9} + \cdots + 931223141640 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 524628360 \beta_{13} + 2248886608 \beta_{12} + 262314180 \beta_{11} + 992244876 \beta_{10} + \cdots - 27434922361200 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 16982612760 \beta_{13} - 37085124000 \beta_{12} - 50947838280 \beta_{11} + 21277392600 \beta_{10} + \cdots + 6383302370368 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 11934579672840 \beta_{13} - 12792516681048 \beta_{12} + 7956386448560 \beta_{11} + \cdots + 51\!\cdots\!20 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 568011372358688 \beta_{13} + 873069057318752 \beta_{12} + 284005686179344 \beta_{11} + \cdots - 20\!\cdots\!80 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 60\!\cdots\!36 \beta_{13} + \cdots + 42\!\cdots\!28 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 14\!\cdots\!84 \beta_{13} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 18\!\cdots\!32 \beta_{13} + \cdots - 16\!\cdots\!04 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
−62.5634 − 108.363i
−30.3050 − 52.4897i
−28.0544 − 48.5917i
7.24567 + 12.5499i
23.6166 + 40.9051i
32.8000 + 56.8112i
58.2606 + 100.910i
−62.5634 + 108.363i
−30.3050 + 52.4897i
−28.0544 + 48.5917i
7.24567 − 12.5499i
23.6166 − 40.9051i
32.8000 − 56.8112i
58.2606 − 100.910i
−59.5634 − 103.167i 252.939 + 146.035i −5047.60 + 8742.70i −6566.60 + 3791.23i − 34793.3i 79775.1 + 86470.9i 714666. −223068. − 386366.i 782258. + 451637.i
3.2 −27.3050 − 47.2936i −1101.82 − 636.136i 556.878 − 964.542i −6532.39 + 3771.47i 69478.7i 112749. − 33599.9i −284504. 543617. + 941573.i 356733. + 205960.i
3.3 −25.0544 − 43.3956i 227.742 + 131.487i 792.550 − 1372.74i 6619.03 − 3821.50i − 13177.3i −106216. − 50590.8i −284674. −231143. − 400351.i −331672. − 191491.i
3.4 10.2457 + 17.7460i 953.136 + 550.294i 1838.05 − 3183.60i −7383.58 + 4262.91i 22552.5i 106852. + 49233.1i 159261. 339926. + 588768.i −151299. − 87352.7i
3.5 26.6166 + 46.1012i −511.735 − 295.450i 631.117 − 1093.13i 18749.1 − 10824.8i − 31455.5i 21923.9 + 115588.i 285236. −91138.9 − 157857.i 998074. + 576238.i
3.6 35.8000 + 62.0074i −397.435 − 229.459i −515.278 + 892.488i −23639.1 + 13648.0i − 32858.6i −92024.7 − 73299.0i 219486. −160417. − 277851.i −1.69256e6 − 977198.i
3.7 61.2606 + 106.106i 575.672 + 332.364i −5457.72 + 9453.05i 14216.0 − 8207.61i 81443.4i 44079.5 − 109079.i −835526. −44788.3 − 77575.7i 1.74176e6 + 1.00561e6i
5.1 −59.5634 + 103.167i 252.939 − 146.035i −5047.60 − 8742.70i −6566.60 − 3791.23i 34793.3i 79775.1 − 86470.9i 714666. −223068. + 386366.i 782258. − 451637.i
5.2 −27.3050 + 47.2936i −1101.82 + 636.136i 556.878 + 964.542i −6532.39 − 3771.47i − 69478.7i 112749. + 33599.9i −284504. 543617. − 941573.i 356733. − 205960.i
5.3 −25.0544 + 43.3956i 227.742 − 131.487i 792.550 + 1372.74i 6619.03 + 3821.50i 13177.3i −106216. + 50590.8i −284674. −231143. + 400351.i −331672. + 191491.i
5.4 10.2457 − 17.7460i 953.136 − 550.294i 1838.05 + 3183.60i −7383.58 − 4262.91i − 22552.5i 106852. − 49233.1i 159261. 339926. − 588768.i −151299. + 87352.7i
5.5 26.6166 − 46.1012i −511.735 + 295.450i 631.117 + 1093.13i 18749.1 + 10824.8i 31455.5i 21923.9 − 115588.i 285236. −91138.9 + 157857.i 998074. − 576238.i
5.6 35.8000 − 62.0074i −397.435 + 229.459i −515.278 − 892.488i −23639.1 − 13648.0i 32858.6i −92024.7 + 73299.0i 219486. −160417. + 277851.i −1.69256e6 + 977198.i
5.7 61.2606 − 106.106i 575.672 − 332.364i −5457.72 − 9453.05i 14216.0 + 8207.61i − 81443.4i 44079.5 + 109079.i −835526. −44788.3 + 77575.7i 1.74176e6 − 1.00561e6i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 7.13.d.a ✓ 14
3.b odd 2 1 63.13.m.a 14
4.b odd 2 1 112.13.s.a 14
7.b odd 2 1 49.13.d.c 14
7.c even 3 1 49.13.b.a 14
7.c even 3 1 49.13.d.c 14
7.d odd 6 1 inner 7.13.d.a ✓ 14
7.d odd 6 1 49.13.b.a 14
21.g even 6 1 63.13.m.a 14
28.f even 6 1 112.13.s.a 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.13.d.a ✓ 14 1.a even 1 1 trivial
7.13.d.a ✓ 14 7.d odd 6 1 inner
49.13.b.a 14 7.c even 3 1
49.13.b.a 14 7.d odd 6 1
49.13.d.c 14 7.b odd 2 1
49.13.d.c 14 7.c even 3 1
63.13.m.a 14 3.b odd 2 1
63.13.m.a 14 21.g even 6 1
112.13.s.a 14 4.b odd 2 1
112.13.s.a 14 28.f even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{13}^{\mathrm{new}}(7, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + \cdots + 97\!\cdots\!84 \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 37\!\cdots\!83 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 13\!\cdots\!75 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 97\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 12\!\cdots\!41 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 66\!\cdots\!03 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 55\!\cdots\!67 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 10\!\cdots\!09 \) Copy content Toggle raw display
$29$ \( (T^{7} + \cdots + 96\!\cdots\!32)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 58\!\cdots\!47 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 41\!\cdots\!09 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{7} + \cdots - 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 30\!\cdots\!63 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 80\!\cdots\!25 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 10\!\cdots\!47 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 21\!\cdots\!27 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 20\!\cdots\!29 \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots - 45\!\cdots\!48)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 12\!\cdots\!43 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 72\!\cdots\!61 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 74\!\cdots\!87 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
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