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Properties

Label 425.4.d.e
Level $425$
Weight $4$
Character orbit 425.d
Analytic conductor $25.076$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [425,4,Mod(101,425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(425, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("425.101");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 425.d (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: \(25.0758117524\)
Analytic rank: \(0\)
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: \( x^{14} + 255 x^{12} + 24987 x^{10} + 1174725 x^{8} + 26685559 x^{6} + 257857029 x^{4} + \cdots + 1069039579 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_1 q^{3} + ( - \beta_{3} + 8) q^{4} + \beta_{10} q^{6} + \beta_{9} q^{7} + (\beta_{6} - \beta_{5} - \beta_{3} + \cdots - 8) q^{8}+ \cdots + ( - \beta_{5} + \beta_{4} - \beta_{3} + \cdots - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + \beta_1 q^{3} + ( - \beta_{3} + 8) q^{4} + \beta_{10} q^{6} + \beta_{9} q^{7} + (\beta_{6} - \beta_{5} - \beta_{3} + \cdots - 8) q^{8}+ \cdots + (3 \beta_{13} - 7 \beta_{12} + \cdots + 30 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 112 q^{4} - 108 q^{8} - 132 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 112 q^{4} - 108 q^{8} - 132 q^{9} + 58 q^{13} + 408 q^{16} - 94 q^{17} + 132 q^{18} - 108 q^{19} + 54 q^{21} - 40 q^{26} - 464 q^{32} + 448 q^{33} - 292 q^{34} - 64 q^{36} + 1328 q^{38} + 1384 q^{42} - 1908 q^{43} + 184 q^{47} - 1044 q^{49} + 740 q^{51} + 72 q^{52} - 1274 q^{53} + 172 q^{59} + 4344 q^{64} + 3120 q^{66} + 1976 q^{67} - 992 q^{68} + 2176 q^{69} + 4592 q^{72} - 3660 q^{76} - 3616 q^{77} - 1002 q^{81} + 2584 q^{83} - 8240 q^{84} + 2012 q^{86} - 1876 q^{87} - 1500 q^{89} - 5650 q^{93} + 6008 q^{94} - 132 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 255 x^{12} + 24987 x^{10} + 1174725 x^{8} + 26685559 x^{6} + 257857029 x^{4} + \cdots + 1069039579 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1808113213 \nu^{12} - 348583423331 \nu^{10} - 22671360349516 \nu^{8} + \cdots + 43\!\cdots\!24 ) / 35\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 468043202 \nu^{12} + 96946274953 \nu^{10} + 7161927663671 \nu^{8} + 228635514792700 \nu^{6} + \cdots + 34\!\cdots\!95 ) / 715365119049588 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2687818939 \nu^{12} - 587789647241 \nu^{10} - 47234658064966 \nu^{8} + \cdots - 20\!\cdots\!94 ) / 35\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 454961125 \nu^{12} - 93141466772 \nu^{10} - 6757211425387 \nu^{8} + \cdots - 22\!\cdots\!19 ) / 188253978697260 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10763589013 \nu^{12} + 2055785830061 \nu^{10} + 130103038441546 \nu^{8} + \cdots - 67\!\cdots\!94 ) / 35\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10640537123 \nu^{12} + 2156039290462 \nu^{10} + 153752448628547 \nu^{8} + \cdots + 28\!\cdots\!93 ) / 17\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 35253395222 \nu^{13} + 8638816778785 \nu^{11} + 800341969485425 \nu^{9} + \cdots + 10\!\cdots\!61 \nu ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 9065504023 \nu^{13} - 1824374318282 \nu^{11} - 128172318960583 \nu^{9} + \cdots - 31\!\cdots\!39 \nu ) / 21\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1808113213 \nu^{13} - 348583423331 \nu^{11} - 22671360349516 \nu^{9} + \cdots + 43\!\cdots\!24 \nu ) / 35\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 6609835466 \nu^{13} - 1322150693455 \nu^{11} - 91674050180555 \nu^{9} + \cdots + 59\!\cdots\!57 \nu ) / 12\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 162393042373 \nu^{13} - 32588459501036 \nu^{11} + \cdots - 42\!\cdots\!41 \nu ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 287511608311 \nu^{13} - 68060840729696 \nu^{11} + \cdots - 79\!\cdots\!35 \nu ) / 21\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} - \beta_{3} + 2\beta_{2} - 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{12} + \beta_{11} - 2\beta_{10} - 2\beta_{9} + \beta_{8} - 60\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{7} - 2\beta_{6} + 105\beta_{5} - 67\beta_{4} + 139\beta_{3} - 152\beta_{2} + 2121 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -9\beta_{13} - 211\beta_{12} - 79\beta_{11} + 290\beta_{10} + 68\beta_{9} - 215\beta_{8} + 3999\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -1143\beta_{7} + 237\beta_{6} - 9877\beta_{5} + 4505\beta_{4} - 13753\beta_{3} + 10681\beta_{2} - 139473 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1380 \beta_{13} + 19164 \beta_{12} + 6554 \beta_{11} - 30654 \beta_{10} + 539 \beta_{9} + \cdots - 283242 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 128634 \beta_{7} - 25782 \beta_{6} + 884894 \beta_{5} - 317828 \beta_{4} + 1243838 \beta_{3} + \cdots + 9754139 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 154416 \beta_{13} - 1664518 \beta_{12} - 549314 \beta_{11} + 2879374 \beta_{10} + \cdots + 21013541 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 12588164 \beta_{7} + 2624818 \beta_{6} - 77103581 \beta_{5} + 23434375 \beta_{4} - 108281965 \beta_{3} + \cdots - 716006106 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 15212982 \beta_{13} + 141679686 \beta_{12} + 45985885 \beta_{11} - 255941168 \beta_{10} + \cdots - 1615075272 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1151494205 \beta_{7} - 251823442 \beta_{6} + 6604702457 \beta_{5} - 1790368497 \beta_{4} + \cdots + 54584081145 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1403317647 \beta_{13} - 11932160567 \beta_{12} - 3841533465 \beta_{11} + 22104322504 \beta_{10} + \cdots + 127359613155 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(326\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
− 6.85393i
6.85393i
− 1.91890i
1.91890i
− 7.31812i
7.31812i
− 9.11445i
9.11445i
− 1.50450i
1.50450i
− 7.35578i
7.35578i
− 3.36786i
3.36786i
−5.56585 − 6.85393i 22.9787 0 38.1480i − 29.0376i −83.3694 −19.9763 0
101.2 −5.56585 6.85393i 22.9787 0 − 38.1480i 29.0376i −83.3694 −19.9763 0
101.3 −4.07862 − 1.91890i 8.63518 0 7.82648i 17.2809i −2.59066 23.3178 0
101.4 −4.07862 1.91890i 8.63518 0 − 7.82648i − 17.2809i −2.59066 23.3178 0
101.5 −3.49565 − 7.31812i 4.21960 0 25.5816i 14.7420i 13.2150 −26.5549 0
101.6 −3.49565 7.31812i 4.21960 0 − 25.5816i − 14.7420i 13.2150 −26.5549 0
101.7 1.26449 − 9.11445i −6.40106 0 − 11.5251i 11.9335i −18.2100 −56.0731 0
101.8 1.26449 9.11445i −6.40106 0 11.5251i − 11.9335i −18.2100 −56.0731 0
101.9 2.51126 − 1.50450i −1.69360 0 − 3.77819i 12.3795i −24.3431 24.7365 0
101.10 2.51126 1.50450i −1.69360 0 3.77819i − 12.3795i −24.3431 24.7365 0
101.11 4.22651 − 7.35578i 9.86335 0 − 31.0892i − 19.4302i 7.87544 −27.1074 0
101.12 4.22651 7.35578i 9.86335 0 31.0892i 19.4302i 7.87544 −27.1074 0
101.13 5.13788 − 3.36786i 18.3978 0 − 17.3036i 29.8441i 53.4227 15.6575 0
101.14 5.13788 3.36786i 18.3978 0 17.3036i − 29.8441i 53.4227 15.6575 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.14
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 425.4.d.e ✓ 14
5.b even 2 1 425.4.d.f yes 14
5.c odd 4 2 425.4.c.g 28
17.b even 2 1 inner 425.4.d.e ✓ 14
85.c even 2 1 425.4.d.f yes 14
85.g odd 4 2 425.4.c.g 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
425.4.c.g 28 5.c odd 4 2
425.4.c.g 28 85.g odd 4 2
425.4.d.e ✓ 14 1.a even 1 1 trivial
425.4.d.e ✓ 14 17.b even 2 1 inner
425.4.d.f yes 14 5.b even 2 1
425.4.d.f yes 14 85.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(425, [\chi])\):

\( T_{2}^{7} - 56T_{2}^{5} + 18T_{2}^{4} + 957T_{2}^{3} - 616T_{2}^{2} - 4976T_{2} + 5472 \) Copy content Toggle raw display
\( T_{3}^{14} + 255 T_{3}^{12} + 24987 T_{3}^{10} + 1174725 T_{3}^{8} + 26685559 T_{3}^{6} + \cdots + 1069039579 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{7} - 56 T^{5} + \cdots + 5472)^{2} \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 1069039579 \) Copy content Toggle raw display
$5$ \( T^{14} \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 40\!\cdots\!11 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{7} - 29 T^{6} + \cdots - 5344477)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 69\!\cdots\!17 \) Copy content Toggle raw display
$19$ \( (T^{7} + 54 T^{6} + \cdots + 51025562880)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 55\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 17\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 28\!\cdots\!75 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{7} + \cdots + 879866889858624)^{2} \) Copy content Toggle raw display
$47$ \( (T^{7} + \cdots - 29\!\cdots\!36)^{2} \) Copy content Toggle raw display
$53$ \( (T^{7} + \cdots + 13\!\cdots\!07)^{2} \) Copy content Toggle raw display
$59$ \( (T^{7} + \cdots - 78\!\cdots\!80)^{2} \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{7} + \cdots - 18\!\cdots\!12)^{2} \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 60\!\cdots\!75 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 15\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 17\!\cdots\!31 \) Copy content Toggle raw display
$83$ \( (T^{7} + \cdots + 55\!\cdots\!24)^{2} \) Copy content Toggle raw display
$89$ \( (T^{7} + \cdots + 83\!\cdots\!20)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 43\!\cdots\!84 \) Copy content Toggle raw display
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