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Properties

Label 65.4.c.a
Level $65$
Weight $4$
Character orbit 65.c
Analytic conductor $3.835$
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] = [65,4,Mod(51,65)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(65, 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("65.51");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 65 = 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 65.c (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: \(3.83512415037\)
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: \( x^{14} + 84x^{12} + 2674x^{10} + 40048x^{8} + 278769x^{6} + 727552x^{4} + 339456x^{2} + 9216 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 5^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} - 4) q^{4} - \beta_{7} q^{5} + (\beta_{12} - \beta_{7} + \beta_1) q^{6} + (\beta_{11} + 2 \beta_{7} + \beta_1) q^{7} + ( - \beta_{9} + \beta_{8} - 4 \beta_1) q^{8} + (\beta_{5} + \beta_{2} + 11) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} - 4) q^{4} - \beta_{7} q^{5} + (\beta_{12} - \beta_{7} + \beta_1) q^{6} + (\beta_{11} + 2 \beta_{7} + \beta_1) q^{7} + ( - \beta_{9} + \beta_{8} - 4 \beta_1) q^{8} + (\beta_{5} + \beta_{2} + 11) q^{9} + (\beta_{3} - 1) q^{10} + (\beta_{13} + \beta_{11} - 4 \beta_1) q^{11} + ( - \beta_{10} + \beta_{6} - \beta_{5} + \cdots - 7) q^{12}+ \cdots + (35 \beta_{13} - 3 \beta_{12} + \cdots - 124 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 56 q^{4} + 158 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 56 q^{4} + 158 q^{9} - 20 q^{10} - 108 q^{12} - 4 q^{13} - 152 q^{14} + 280 q^{16} - 100 q^{17} + 648 q^{22} - 532 q^{23} - 350 q^{25} - 344 q^{26} - 48 q^{27} + 588 q^{29} - 200 q^{30} + 540 q^{35} + 496 q^{36} + 148 q^{38} - 260 q^{39} + 240 q^{40} - 620 q^{42} + 728 q^{43} - 2008 q^{48} - 1302 q^{49} + 176 q^{51} - 528 q^{52} + 1040 q^{53} - 40 q^{55} + 512 q^{56} - 1460 q^{61} + 2964 q^{62} + 1296 q^{64} - 30 q^{65} - 600 q^{66} + 1604 q^{68} + 4160 q^{69} - 5928 q^{74} - 3568 q^{77} + 2560 q^{78} - 1024 q^{79} - 2890 q^{81} - 6660 q^{82} + 5256 q^{87} + 1132 q^{88} + 1360 q^{90} + 916 q^{91} + 6044 q^{92} + 2656 q^{94} - 3120 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 84x^{12} + 2674x^{10} + 40048x^{8} + 278769x^{6} + 727552x^{4} + 339456x^{2} + 9216 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 331 \nu^{12} + 17100 \nu^{10} + 135334 \nu^{8} - 4575008 \nu^{6} - 70189989 \nu^{4} + \cdots - 2753376 ) / 16283040 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1069 \nu^{12} + 72444 \nu^{10} + 1716106 \nu^{8} + 17134688 \nu^{6} + 67868029 \nu^{4} + \cdots + 20876288 ) / 2171072 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1315 \nu^{12} + 90892 \nu^{10} + 2243030 \nu^{8} + 24552176 \nu^{6} + 121124275 \nu^{4} + \cdots + 52996480 ) / 2171072 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 37951 \nu^{12} + 2341860 \nu^{10} + 46164334 \nu^{8} + 290585632 \nu^{6} - 209579889 \nu^{4} + \cdots + 286093824 ) / 32566080 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 24787 \nu^{13} - 2086080 \nu^{11} - 66485638 \nu^{9} - 994293784 \nu^{7} + \cdots - 6236239488 \nu ) / 195396480 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 43091 \nu^{13} - 2914860 \nu^{11} - 68779574 \nu^{9} - 678098672 \nu^{7} + \cdots + 9496576896 \nu ) / 195396480 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 43091 \nu^{13} - 2914860 \nu^{11} - 68779574 \nu^{9} - 678098672 \nu^{7} + \cdots + 5588647296 \nu ) / 195396480 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 13507 \nu^{12} - 876120 \nu^{10} - 19157318 \nu^{8} - 161447384 \nu^{6} - 382305267 \nu^{4} + \cdots + 243292672 ) / 5427680 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 118777 \nu^{13} - 8952540 \nu^{11} - 250798018 \nu^{9} - 3279360064 \nu^{7} + \cdots - 27815782848 \nu ) / 195396480 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 120997 \nu^{13} - 8606040 \nu^{11} - 220935178 \nu^{9} - 2536415704 \nu^{7} + \cdots - 8310501888 \nu ) / 195396480 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 51665 \nu^{13} - 4464660 \nu^{11} - 146134418 \nu^{9} - 2239705328 \nu^{7} + \cdots - 16117175424 \nu ) / 78158592 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} + \beta_{8} - 20\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} + \beta_{6} - \beta_{5} + 4\beta_{4} - 2\beta_{3} - 26\beta_{2} + 244 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{13} - 6\beta_{12} + 2\beta_{11} + 44\beta_{9} - 32\beta_{8} - 22\beta_{7} + 449\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -40\beta_{10} - 40\beta_{6} + 52\beta_{5} - 176\beta_{4} + 110\beta_{3} + 679\beta_{2} - 5562 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -220\beta_{13} + 280\beta_{12} - 104\beta_{11} - 1477\beta_{9} + 919\beta_{8} + 1248\beta_{7} - 10852\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1301\beta_{10} + 1315\beta_{6} - 1861\beta_{5} + 5930\beta_{4} - 4304\beta_{3} - 18204\beta_{2} + 136258 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 8636 \beta_{13} - 9680 \beta_{12} + 3694 \beta_{11} + 45606 \beta_{9} - 26038 \beta_{8} + \cdots + 277269 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 39620 \beta_{10} - 40660 \beta_{6} + 58980 \beta_{5} - 182632 \beta_{4} + 147020 \beta_{3} + \cdots - 3518112 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 296120 \beta_{13} + 302672 \beta_{12} - 115880 \beta_{11} - 1360877 \beta_{9} + 739177 \beta_{8} + \cdots - 7378776 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1174085 \beta_{10} + 1222089 \beta_{6} - 1779429 \beta_{5} + 5426088 \beta_{4} - 4690062 \beta_{3} + \cdots + 94333640 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 9476132 \beta_{13} - 9080954 \beta_{12} + 3462850 \beta_{11} + 39976604 \beta_{9} + \cdots + 202045845 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/65\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(41\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
− 5.37688i
− 4.29153i
− 4.27643i
− 3.65500i
− 2.10835i
− 0.742335i
− 0.170066i
0.170066i
0.742335i
2.10835i
3.65500i
4.27643i
4.29153i
5.37688i
− 5.37688i −0.936724 −20.9108 − 5.00000i 5.03665i 0.201771i 69.4197i −26.1225 −26.8844
51.2 − 4.29153i 5.89541 −10.4172 5.00000i − 25.3003i − 34.1818i 10.3735i 7.75582 21.4576
51.3 − 4.27643i −7.17327 −10.2878 5.00000i 30.6760i 5.20875i 9.78383i 24.4558 21.3821
51.4 − 3.65500i 8.59394 −5.35901 − 5.00000i − 31.4108i 19.4590i − 9.65282i 46.8559 −18.2750
51.5 − 2.10835i −3.08338 3.55484 − 5.00000i 6.50086i − 17.4384i − 24.3617i −17.4928 −10.5418
51.6 − 0.742335i 5.13862 7.44894 5.00000i − 3.81458i 9.97127i − 11.4683i −0.594575 3.71168
51.7 − 0.170066i −8.43460 7.97108 − 5.00000i 1.43444i 32.7758i − 2.71614i 44.1424 −0.850329
51.8 0.170066i −8.43460 7.97108 5.00000i − 1.43444i − 32.7758i 2.71614i 44.1424 −0.850329
51.9 0.742335i 5.13862 7.44894 − 5.00000i 3.81458i − 9.97127i 11.4683i −0.594575 3.71168
51.10 2.10835i −3.08338 3.55484 5.00000i − 6.50086i 17.4384i 24.3617i −17.4928 −10.5418
51.11 3.65500i 8.59394 −5.35901 5.00000i 31.4108i − 19.4590i 9.65282i 46.8559 −18.2750
51.12 4.27643i −7.17327 −10.2878 − 5.00000i − 30.6760i − 5.20875i − 9.78383i 24.4558 21.3821
51.13 4.29153i 5.89541 −10.4172 − 5.00000i 25.3003i 34.1818i − 10.3735i 7.75582 21.4576
51.14 5.37688i −0.936724 −20.9108 5.00000i − 5.03665i − 0.201771i − 69.4197i −26.1225 −26.8844
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.14
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 65.4.c.a ✓ 14
3.b odd 2 1 585.4.b.e 14
4.b odd 2 1 1040.4.k.d 14
5.b even 2 1 325.4.c.e 14
5.c odd 4 1 325.4.d.c 14
5.c odd 4 1 325.4.d.d 14
13.b even 2 1 inner 65.4.c.a ✓ 14
13.d odd 4 1 845.4.a.i 7
13.d odd 4 1 845.4.a.l 7
39.d odd 2 1 585.4.b.e 14
52.b odd 2 1 1040.4.k.d 14
65.d even 2 1 325.4.c.e 14
65.h odd 4 1 325.4.d.c 14
65.h odd 4 1 325.4.d.d 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.4.c.a ✓ 14 1.a even 1 1 trivial
65.4.c.a ✓ 14 13.b even 2 1 inner
325.4.c.e 14 5.b even 2 1
325.4.c.e 14 65.d even 2 1
325.4.d.c 14 5.c odd 4 1
325.4.d.c 14 65.h odd 4 1
325.4.d.d 14 5.c odd 4 1
325.4.d.d 14 65.h odd 4 1
585.4.b.e 14 3.b odd 2 1
585.4.b.e 14 39.d odd 2 1
845.4.a.i 7 13.d odd 4 1
845.4.a.l 7 13.d odd 4 1
1040.4.k.d 14 4.b odd 2 1
1040.4.k.d 14 52.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(65, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 84 T^{12} + \cdots + 9216 \) Copy content Toggle raw display
$3$ \( (T^{7} - 134 T^{5} + \cdots - 45496)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 25)^{7} \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 15872256000000 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 24\!\cdots\!13 \) Copy content Toggle raw display
$17$ \( (T^{7} + 50 T^{6} + \cdots + 44413747200)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 23\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( (T^{7} + \cdots - 5699551522752)^{2} \) Copy content Toggle raw display
$29$ \( (T^{7} + \cdots + 440579141474400)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 76\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{7} + \cdots - 97\!\cdots\!24)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 48\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( (T^{7} + \cdots - 405893642720256)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 71\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( (T^{7} + \cdots - 35\!\cdots\!44)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 22\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 42\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( (T^{7} + \cdots - 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 22\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 18\!\cdots\!84 \) Copy content Toggle raw display
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