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Properties

Label 605.6.a.l
Level $605$
Weight $6$
Character orbit 605.a
Self dual yes
Analytic conductor $97.032$
Analytic rank $1$
Dimension $14$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [605,6,Mod(1,605)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(605, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("605.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 605.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: \(97.0322109869\)
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: \( x^{14} - 245x^{12} + 22615x^{10} - 991491x^{8} + 21193344x^{6} - 191794080x^{4} + 358937856x^{2} - 24060672 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{4}\cdot 11^{2} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} - 1) q^{3} + (\beta_{5} + \beta_{4} + 3) q^{4} - 25 q^{5} + (\beta_{12} - \beta_{10} + \cdots - \beta_1) q^{6}+ \cdots + (\beta_{7} - 2 \beta_{5} - 5 \beta_{4} + \cdots + 74) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} - 1) q^{3} + (\beta_{5} + \beta_{4} + 3) q^{4} - 25 q^{5} + (\beta_{12} - \beta_{10} + \cdots - \beta_1) q^{6}+ \cdots + (103 \beta_{13} + 323 \beta_{12} + \cdots - 5620 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 18 q^{3} + 42 q^{4} - 350 q^{5} + 1076 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 18 q^{3} + 42 q^{4} - 350 q^{5} + 1076 q^{9} + 202 q^{12} - 574 q^{14} + 450 q^{15} - 3114 q^{16} - 1050 q^{20} + 8920 q^{23} + 8750 q^{25} + 812 q^{26} - 17550 q^{27} - 1248 q^{31} + 46344 q^{34} - 24012 q^{36} - 23744 q^{37} - 9948 q^{38} + 67634 q^{42} - 26900 q^{45} + 37322 q^{47} + 100270 q^{48} + 17096 q^{49} + 8988 q^{53} - 5826 q^{56} - 70560 q^{58} - 50952 q^{59} - 5050 q^{60} - 65446 q^{64} - 222650 q^{67} + 29536 q^{69} + 14350 q^{70} - 99592 q^{71} - 11250 q^{75} - 471148 q^{78} + 77850 q^{80} + 78302 q^{81} - 539322 q^{82} - 333906 q^{86} - 649766 q^{89} - 66140 q^{91} + 19520 q^{92} - 201656 q^{93} - 383140 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 245x^{12} + 22615x^{10} - 991491x^{8} + 21193344x^{6} - 191794080x^{4} + 358937856x^{2} - 24060672 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 173825 \nu^{12} + 37231833 \nu^{10} - 2788207515 \nu^{8} + 87354958223 \nu^{6} + \cdots - 1372668939072 ) / 48445178880 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 83589 \nu^{12} - 17875673 \nu^{10} + 1334729491 \nu^{8} - 41528156399 \nu^{6} + \cdots - 23453155200 ) / 6055647360 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1297 \nu^{12} + 275963 \nu^{10} - 20438029 \nu^{8} + 627388605 \nu^{6} - 7277529738 \nu^{4} + \cdots - 945180000 ) / 61792320 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1297 \nu^{12} - 275963 \nu^{10} + 20438029 \nu^{8} - 627388605 \nu^{6} + 7277529738 \nu^{4} + \cdots - 1217551200 ) / 61792320 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 55625 \nu^{13} + 11791573 \nu^{11} - 867195767 \nu^{9} + 26211437811 \nu^{7} + \cdots + 550538951040 \nu ) / 87497925120 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 8911305 \nu^{12} + 1892531873 \nu^{10} - 139668556147 \nu^{8} + 4254228696071 \nu^{6} + \cdots + 7107527459520 ) / 48445178880 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2168731 \nu^{13} - 481295531 \nu^{11} + 38270525569 \nu^{9} - 1338247134285 \nu^{7} + \cdots + 163419268405056 \nu ) / 1429132776960 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3024785 \nu^{12} - 643371325 \nu^{10} + 47625196607 \nu^{8} - 1460889093483 \nu^{6} + \cdots + 2778123888768 ) / 6055647360 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 49798921 \nu^{13} + 10675524049 \nu^{11} - 801966091235 \nu^{9} + \cdots - 10\!\cdots\!72 \nu ) / 5716531107840 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 171020071 \nu^{13} - 36327095759 \nu^{11} + 2682572418781 \nu^{9} + \cdots - 824951139165504 \nu ) / 17149593323520 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 92474995 \nu^{13} + 19763091419 \nu^{11} - 1475891558401 \nu^{9} + \cdots - 10\!\cdots\!00 \nu ) / 5716531107840 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1668642691 \nu^{13} + 350883452651 \nu^{11} - 25416537553201 \nu^{9} + \cdots + 31\!\cdots\!96 \nu ) / 17149593323520 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + 35 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{12} + 3\beta_{11} + \beta_{8} + 24\beta_{6} + 59\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{9} - \beta_{7} + 82\beta_{5} + 143\beta_{4} - \beta_{3} - 31\beta_{2} + 2150 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{13} + 72\beta_{12} + 330\beta_{11} + 33\beta_{10} + 118\beta_{8} + 3633\beta_{6} + 4171\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 276\beta_{9} - 124\beta_{7} + 6693\beta_{5} + 14989\beta_{4} + 84\beta_{3} - 3428\beta_{2} + 155919 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 400 \beta_{13} + 3437 \beta_{12} + 31327 \beta_{11} + 6600 \beta_{10} + 10949 \beta_{8} + \cdots + 324091 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 30290 \beta_{9} - 11749 \beta_{7} + 554318 \beta_{5} + 1446335 \beta_{4} + 24483 \beta_{3} + \cdots + 12327282 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 42039 \beta_{13} + 60564 \beta_{12} + 2863678 \beta_{11} + 866869 \beta_{10} + 954250 \beta_{8} + \cdots + 26471963 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3055348 \beta_{9} - 1038328 \beta_{7} + 46606537 \beta_{5} + 135052893 \beta_{4} + 3405984 \beta_{3} + \cdots + 1019010915 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 4093676 \beta_{13} - 12034127 \beta_{12} + 258468811 \beta_{11} + 96415796 \beta_{10} + \cdots + 2223130539 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 295064994 \beta_{9} - 90156025 \beta_{7} + 3970205034 \beta_{5} + 12403263775 \beta_{4} + \cdots + 86310908046 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 385221019 \beta_{13} - 2245135744 \beta_{12} + 23204254258 \beta_{11} + 9861714009 \beta_{10} + \cdots + 189897682891 \beta_1 \) 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
−9.43476
−8.04824
−5.68495
−5.67919
−4.91854
−1.54192
−0.263821
0.263821
1.54192
4.91854
5.67919
5.68495
8.04824
9.43476
−9.43476 1.66038 57.0147 −25.0000 −15.6652 −23.0971 −236.008 −240.243 235.869
1.2 −8.04824 18.7316 32.7742 −25.0000 −150.756 44.9005 −6.23109 107.873 201.206
1.3 −5.68495 −23.7255 0.318700 −25.0000 134.878 36.5214 180.107 319.899 142.124
1.4 −5.67919 −26.7478 0.253144 −25.0000 151.906 200.831 180.296 472.444 141.980
1.5 −4.91854 3.91675 −7.80792 −25.0000 −19.2647 −218.732 195.797 −227.659 122.964
1.6 −1.54192 −6.32369 −29.6225 −25.0000 9.75064 −53.2547 95.0171 −203.011 38.5481
1.7 −0.263821 23.4882 −31.9304 −25.0000 −6.19670 −176.920 16.8662 308.697 6.59553
1.8 0.263821 23.4882 −31.9304 −25.0000 6.19670 176.920 −16.8662 308.697 −6.59553
1.9 1.54192 −6.32369 −29.6225 −25.0000 −9.75064 53.2547 −95.0171 −203.011 −38.5481
1.10 4.91854 3.91675 −7.80792 −25.0000 19.2647 218.732 −195.797 −227.659 −122.964
1.11 5.67919 −26.7478 0.253144 −25.0000 −151.906 −200.831 −180.296 472.444 −141.980
1.12 5.68495 −23.7255 0.318700 −25.0000 −134.878 −36.5214 −180.107 319.899 −142.124
1.13 8.04824 18.7316 32.7742 −25.0000 150.756 −44.9005 6.23109 107.873 −201.206
1.14 9.43476 1.66038 57.0147 −25.0000 15.6652 23.0971 236.008 −240.243 −235.869
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(11\) \(1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.6.a.l ✓ 14
11.b odd 2 1 inner 605.6.a.l ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.6.a.l ✓ 14 1.a even 1 1 trivial
605.6.a.l ✓ 14 11.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} - 245 T_{2}^{12} + 22615 T_{2}^{10} - 991491 T_{2}^{8} + 21193344 T_{2}^{6} - 191794080 T_{2}^{4} + \cdots - 24060672 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(605))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} - 245 T^{12} + \cdots - 24060672 \) Copy content Toggle raw display
$3$ \( (T^{7} + 9 T^{6} + \cdots + 11482371)^{2} \) Copy content Toggle raw display
$5$ \( (T + 25)^{14} \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots - 24\!\cdots\!07 \) Copy content Toggle raw display
$11$ \( T^{14} \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots - 55\!\cdots\!48 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots - 76\!\cdots\!52 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots - 28\!\cdots\!08 \) Copy content Toggle raw display
$23$ \( (T^{7} + \cdots + 79\!\cdots\!20)^{2} \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots - 36\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{7} + \cdots - 35\!\cdots\!68)^{2} \) Copy content Toggle raw display
$37$ \( (T^{7} + \cdots - 19\!\cdots\!92)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots - 19\!\cdots\!23 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots - 36\!\cdots\!75 \) Copy content Toggle raw display
$47$ \( (T^{7} + \cdots + 21\!\cdots\!81)^{2} \) Copy content Toggle raw display
$53$ \( (T^{7} + \cdots - 13\!\cdots\!96)^{2} \) Copy content Toggle raw display
$59$ \( (T^{7} + \cdots - 85\!\cdots\!48)^{2} \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots - 16\!\cdots\!75 \) Copy content Toggle raw display
$67$ \( (T^{7} + \cdots - 15\!\cdots\!85)^{2} \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots - 79\!\cdots\!12)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots - 34\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 80\!\cdots\!12 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 52\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( (T^{7} + \cdots - 24\!\cdots\!23)^{2} \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots - 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
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