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Properties

Label 608.2.h.a
Level $608$
Weight $2$
Character orbit 608.h
Analytic conductor $4.855$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [608,2,Mod(607,608)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(608, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("608.607");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 608 = 2^{5} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 608.h (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: \(4.85490444289\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 8 x^{18} - 12 x^{17} + 82 x^{16} - 68 x^{15} - 438 x^{14} + 516 x^{13} + 3622 x^{12} + \cdots + 12832 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{25} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} - \beta_{2} q^{5} - \beta_{14} q^{7} + ( - \beta_{7} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{3} - \beta_{2} q^{5} - \beta_{14} q^{7} + ( - \beta_{7} + 1) q^{9} - \beta_{16} q^{11} + \beta_{11} q^{13} - \beta_{3} q^{15} - \beta_1 q^{17} + ( - \beta_{12} + \beta_{10}) q^{19} - \beta_{19} q^{21} + (\beta_{16} - \beta_{14} + \beta_{10}) q^{23} + (\beta_{9} + 1) q^{25} + ( - \beta_{15} + \beta_{14} + \cdots - \beta_{4}) q^{27}+ \cdots + ( - 3 \beta_{16} + 2 \beta_{15} + \cdots - 4 \beta_{10}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 28 q^{9} - 8 q^{17} + 20 q^{25} + 12 q^{49} - 8 q^{57} + 8 q^{73} - 40 q^{77} + 36 q^{81} - 56 q^{85} + 16 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 8 x^{18} - 12 x^{17} + 82 x^{16} - 68 x^{15} - 438 x^{14} + 516 x^{13} + 3622 x^{12} + \cdots + 12832 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 32\!\cdots\!11 \nu^{19} + \cdots + 12\!\cdots\!44 ) / 34\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 21\!\cdots\!25 \nu^{19} + \cdots - 29\!\cdots\!76 ) / 34\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 86\!\cdots\!77 \nu^{19} + \cdots - 49\!\cdots\!28 ) / 80\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 14\!\cdots\!56 \nu^{19} + \cdots + 38\!\cdots\!64 ) / 80\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 31\!\cdots\!01 \nu^{19} + \cdots + 68\!\cdots\!12 ) / 13\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 13\!\cdots\!69 \nu^{19} + \cdots + 31\!\cdots\!20 ) / 53\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 48\!\cdots\!05 \nu^{19} + \cdots - 79\!\cdots\!40 ) / 17\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 15\!\cdots\!59 \nu^{19} + \cdots - 23\!\cdots\!16 ) / 40\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 18\!\cdots\!48 \nu^{19} + \cdots - 21\!\cdots\!54 ) / 43\!\cdots\!03 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 63\!\cdots\!78 \nu^{19} + \cdots - 48\!\cdots\!28 ) / 47\!\cdots\!22 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 11\!\cdots\!97 \nu^{19} + \cdots - 14\!\cdots\!28 ) / 83\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 13\!\cdots\!29 \nu^{19} + \cdots + 19\!\cdots\!92 ) / 80\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 17\!\cdots\!21 \nu^{19} + \cdots - 13\!\cdots\!20 ) / 83\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 29\!\cdots\!17 \nu^{19} + \cdots - 34\!\cdots\!96 ) / 13\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 30\!\cdots\!16 \nu^{19} + \cdots + 30\!\cdots\!72 ) / 10\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 22\!\cdots\!47 \nu^{19} + \cdots - 20\!\cdots\!88 ) / 67\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 10\!\cdots\!43 \nu^{19} + \cdots - 10\!\cdots\!56 ) / 20\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 43\!\cdots\!53 \nu^{19} + \cdots + 41\!\cdots\!96 ) / 83\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 53\!\cdots\!77 \nu^{19} + \cdots + 44\!\cdots\!32 ) / 83\!\cdots\!48 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{17} - \beta_{15} + 4 \beta_{12} + 2 \beta_{11} - 2 \beta_{10} - \beta_{9} + \beta_{8} + \cdots - \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{19} - 2 \beta_{16} + \beta_{15} + 3 \beta_{14} + \beta_{13} - 2 \beta_{11} - 2 \beta_{10} + \cdots + 4 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4 \beta_{19} - \beta_{17} + 8 \beta_{16} + 3 \beta_{15} - 12 \beta_{14} - 4 \beta_{12} + 14 \beta_{11} + \cdots + 16 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 7 \beta_{19} - 2 \beta_{18} - 5 \beta_{17} + 10 \beta_{16} - 10 \beta_{15} + 16 \beta_{14} + 3 \beta_{13} + \cdots - 36 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 23 \beta_{19} + 8 \beta_{18} - 47 \beta_{17} - 17 \beta_{15} + 78 \beta_{14} + 15 \beta_{13} + \cdots + 320 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 55 \beta_{19} - 66 \beta_{18} + 42 \beta_{17} + 60 \beta_{16} + 7 \beta_{15} - 163 \beta_{14} + \cdots + 16 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 209 \beta_{19} - 30 \beta_{18} + 65 \beta_{17} - 200 \beta_{16} - 65 \beta_{15} + 504 \beta_{14} + \cdots - 312 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 119 \beta_{19} - 114 \beta_{18} - 307 \beta_{17} - 42 \beta_{16} + 286 \beta_{15} - 194 \beta_{14} + \cdots + 1524 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3794 \beta_{19} - 1188 \beta_{18} + 1057 \beta_{17} + 4780 \beta_{16} - 549 \beta_{15} - 2802 \beta_{14} + \cdots - 8864 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 2910 \beta_{19} + 1638 \beta_{18} - 4432 \beta_{17} - 1792 \beta_{16} - 2121 \beta_{15} + 10726 \beta_{14} + \cdots + 7276 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 10900 \beta_{19} - 11538 \beta_{18} + 3277 \beta_{17} + 12256 \beta_{16} + 13079 \beta_{15} + \cdots + 41984 ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 10871 \beta_{19} - 7830 \beta_{18} + 21955 \beta_{17} + 12316 \beta_{16} - 1616 \beta_{15} + \cdots - 73176 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 189448 \beta_{19} + 24496 \beta_{18} - 166113 \beta_{17} - 191876 \beta_{16} + 40813 \beta_{15} + \cdots + 244776 ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 302625 \beta_{19} - 112132 \beta_{18} + 169444 \beta_{17} + 349206 \beta_{16} + 103209 \beta_{15} + \cdots - 218100 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 355968 \beta_{19} + 290056 \beta_{18} - 723327 \beta_{17} - 124792 \beta_{16} - 799923 \beta_{15} + \cdots - 1694944 ) / 8 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 820807 \beta_{19} + 59910 \beta_{18} - 1102651 \beta_{17} - 773734 \beta_{16} + 515290 \beta_{15} + \cdots + 4076740 ) / 4 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 11562009 \beta_{19} - 4629800 \beta_{18} + 11945555 \beta_{17} + 11998592 \beta_{16} + 2967005 \beta_{15} + \cdots - 10706848 ) / 8 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 17308271 \beta_{19} + 4159366 \beta_{18} - 13022706 \beta_{17} - 18472108 \beta_{16} + \cdots - 12084088 ) / 4 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 37184433 \beta_{19} - 13383486 \beta_{18} + 25196371 \beta_{17} + 41780232 \beta_{16} + 46991841 \beta_{15} + \cdots + 101163208 ) / 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/608\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
607.1
1.31020 + 0.0138258i
1.31020 − 0.0138258i
−1.94154 − 2.03293i
−1.94154 + 2.03293i
0.938000 + 0.427934i
0.938000 − 0.427934i
2.36892 + 0.309736i
2.36892 − 0.309736i
1.34434 + 1.43741i
1.34434 − 1.43741i
−1.74969 − 1.03206i
−1.74969 + 1.03206i
−1.32297 + 0.736218i
−1.32297 − 0.736218i
−2.09988 − 1.58982i
−2.09988 + 1.58982i
0.170451 + 0.261845i
0.170451 − 0.261845i
0.982164 − 2.30619i
0.982164 + 2.30619i
0 −3.24190 0 −1.40854 0 − 2.71228i 0 7.50990 0
607.2 0 −3.24190 0 −1.40854 0 2.71228i 0 7.50990 0
607.3 0 −2.50470 0 3.11605 0 − 3.41292i 0 3.27350 0
607.4 0 −2.50470 0 3.11605 0 3.41292i 0 3.27350 0
607.5 0 −1.64315 0 −3.57487 0 − 1.60242i 0 −0.300055 0
607.6 0 −1.64315 0 −3.57487 0 1.60242i 0 −0.300055 0
607.7 0 −1.47920 0 2.30899 0 − 0.820225i 0 −0.811963 0
607.8 0 −1.47920 0 2.30899 0 0.820225i 0 −0.811963 0
607.9 0 −0.573253 0 −0.441633 0 − 3.12330i 0 −2.67138 0
607.10 0 −0.573253 0 −0.441633 0 3.12330i 0 −2.67138 0
607.11 0 0.573253 0 −0.441633 0 − 3.12330i 0 −2.67138 0
607.12 0 0.573253 0 −0.441633 0 3.12330i 0 −2.67138 0
607.13 0 1.47920 0 2.30899 0 − 0.820225i 0 −0.811963 0
607.14 0 1.47920 0 2.30899 0 0.820225i 0 −0.811963 0
607.15 0 1.64315 0 −3.57487 0 − 1.60242i 0 −0.300055 0
607.16 0 1.64315 0 −3.57487 0 1.60242i 0 −0.300055 0
607.17 0 2.50470 0 3.11605 0 − 3.41292i 0 3.27350 0
607.18 0 2.50470 0 3.11605 0 3.41292i 0 3.27350 0
607.19 0 3.24190 0 −1.40854 0 − 2.71228i 0 7.50990 0
607.20 0 3.24190 0 −1.40854 0 2.71228i 0 7.50990 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 607.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.b odd 2 1 inner
76.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 608.2.h.a ✓ 20
3.b odd 2 1 5472.2.k.c 20
4.b odd 2 1 inner 608.2.h.a ✓ 20
8.b even 2 1 1216.2.h.e 20
8.d odd 2 1 1216.2.h.e 20
12.b even 2 1 5472.2.k.c 20
19.b odd 2 1 inner 608.2.h.a ✓ 20
57.d even 2 1 5472.2.k.c 20
76.d even 2 1 inner 608.2.h.a ✓ 20
152.b even 2 1 1216.2.h.e 20
152.g odd 2 1 1216.2.h.e 20
228.b odd 2 1 5472.2.k.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.2.h.a ✓ 20 1.a even 1 1 trivial
608.2.h.a ✓ 20 4.b odd 2 1 inner
608.2.h.a ✓ 20 19.b odd 2 1 inner
608.2.h.a ✓ 20 76.d even 2 1 inner
1216.2.h.e 20 8.b even 2 1
1216.2.h.e 20 8.d odd 2 1
1216.2.h.e 20 152.b even 2 1
1216.2.h.e 20 152.g odd 2 1
5472.2.k.c 20 3.b odd 2 1
5472.2.k.c 20 12.b even 2 1
5472.2.k.c 20 57.d even 2 1
5472.2.k.c 20 228.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(608, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{10} - 22 T^{8} + \cdots - 128)^{2} \) Copy content Toggle raw display
$5$ \( (T^{5} - 15 T^{3} + \cdots + 16)^{4} \) Copy content Toggle raw display
$7$ \( (T^{10} + 32 T^{8} + \cdots + 1444)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + 50 T^{8} + \cdots + 64)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + 58 T^{8} + \cdots + 2048)^{2} \) Copy content Toggle raw display
$17$ \( (T^{5} + 2 T^{4} + \cdots + 526)^{4} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 6131066257801 \) Copy content Toggle raw display
$23$ \( (T^{10} + 90 T^{8} + \cdots + 23104)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + 170 T^{8} + \cdots + 307328)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} - 200 T^{8} + \cdots - 37879808)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + 240 T^{8} + \cdots + 20891648)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + 296 T^{8} + \cdots + 13770752)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + 138 T^{8} + \cdots + 92416)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + 298 T^{8} + \cdots + 3717184)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + 426 T^{8} + \cdots + 1753267328)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} - 342 T^{8} + \cdots - 24332288)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 111 T^{3} + \cdots - 256)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} - 246 T^{8} + \cdots - 46208)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} - 232 T^{8} + \cdots - 2097152)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} - 2 T^{4} + \cdots - 1922)^{4} \) Copy content Toggle raw display
$79$ \( (T^{10} - 216 T^{8} + \cdots - 32768)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + 620 T^{8} + \cdots + 3596161024)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + 320 T^{8} + \cdots + 3964928)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + 248 T^{8} + \cdots + 524288)^{2} \) Copy content Toggle raw display
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