↑

Properties

Label 385.2.i.c
Level $385$
Weight $2$
Character orbit 385.i
Analytic conductor $3.074$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [385,2,Mod(221,385)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(385, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("385.221");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 385 = 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 385.i (of order \(3\), 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: \(3.07424047782\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3 x^{15} + 17 x^{14} - 28 x^{13} + 127 x^{12} - 178 x^{11} + 612 x^{10} - 527 x^{9} + 1556 x^{8} + \cdots + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{7} - \beta_{4}) q^{3} + (\beta_{11} - \beta_{8}) q^{4} + (\beta_{8} - 1) q^{5} + (\beta_{15} + \beta_{12} + \beta_{6} + \cdots - 1) q^{6}+ \cdots + (\beta_{15} - \beta_{14} + \beta_{13} + \cdots - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{7} - \beta_{4}) q^{3} + (\beta_{11} - \beta_{8}) q^{4} + (\beta_{8} - 1) q^{5} + (\beta_{15} + \beta_{12} + \beta_{6} + \cdots - 1) q^{6}+ \cdots + (\beta_{13} - \beta_{12} + \beta_{9} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 3 q^{2} - q^{3} - 9 q^{4} - 8 q^{5} - 6 q^{6} - q^{7} + 18 q^{8} - 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 3 q^{2} - q^{3} - 9 q^{4} - 8 q^{5} - 6 q^{6} - q^{7} + 18 q^{8} - 19 q^{9} - 3 q^{10} + 8 q^{11} - 9 q^{12} + 28 q^{13} - 9 q^{14} + 2 q^{15} - 7 q^{16} - 5 q^{17} - 27 q^{18} - q^{19} + 18 q^{20} - 18 q^{21} - 6 q^{22} + 2 q^{23} + 24 q^{24} - 8 q^{25} - 21 q^{26} - 10 q^{27} + 32 q^{28} + 52 q^{29} + 3 q^{30} - 2 q^{31} - 16 q^{32} + q^{33} - 52 q^{34} + 5 q^{35} + 108 q^{36} + q^{37} + 31 q^{38} - 19 q^{39} - 9 q^{40} - 6 q^{41} + 44 q^{42} + 8 q^{43} + 9 q^{44} - 19 q^{45} - 10 q^{46} - q^{47} - 42 q^{48} + 17 q^{49} + 6 q^{50} - 3 q^{51} - 37 q^{52} - 26 q^{53} + 5 q^{54} - 16 q^{55} + 40 q^{57} + q^{58} + 19 q^{59} - 9 q^{60} - 52 q^{62} - 21 q^{63} + 2 q^{64} - 14 q^{65} - 3 q^{66} + 13 q^{67} - 15 q^{68} - 28 q^{69} + 15 q^{70} - 18 q^{71} - 32 q^{72} - 11 q^{73} - 24 q^{74} - q^{75} - 36 q^{76} + 4 q^{77} - 66 q^{78} + 8 q^{79} - 7 q^{80} - 52 q^{81} - 41 q^{82} + 64 q^{83} + 138 q^{84} + 10 q^{85} - 28 q^{86} + 16 q^{87} + 9 q^{88} - 5 q^{89} + 54 q^{90} + 13 q^{91} + 60 q^{92} + 14 q^{93} + 5 q^{94} - q^{95} - q^{96} + 18 q^{97} + 22 q^{98} - 38 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3 x^{15} + 17 x^{14} - 28 x^{13} + 127 x^{12} - 178 x^{11} + 612 x^{10} - 527 x^{9} + 1556 x^{8} + \cdots + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 367626137496199 \nu^{15} + \cdots + 12\!\cdots\!70 ) / 37\!\cdots\!22 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 178336902592334 \nu^{15} + 477682850946139 \nu^{14} - 247188530701317 \nu^{13} + \cdots + 38\!\cdots\!89 ) / 12\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 100841646682033 \nu^{15} + \cdots + 43\!\cdots\!61 ) / 62\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 468401988811307 \nu^{15} + \cdots - 32\!\cdots\!54 ) / 12\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 14\!\cdots\!28 \nu^{15} + \cdots + 21\!\cdots\!21 ) / 12\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 49\!\cdots\!15 \nu^{15} + \cdots + 27\!\cdots\!41 ) / 18\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 12\!\cdots\!02 \nu^{15} + \cdots + 40\!\cdots\!21 ) / 37\!\cdots\!22 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12\!\cdots\!85 \nu^{15} + \cdots - 11\!\cdots\!36 ) / 37\!\cdots\!22 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 29\!\cdots\!00 \nu^{15} + \cdots - 33\!\cdots\!09 ) / 62\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 12\!\cdots\!02 \nu^{15} + \cdots + 40\!\cdots\!21 ) / 12\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 85\!\cdots\!87 \nu^{15} + \cdots - 48\!\cdots\!57 ) / 62\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 17\!\cdots\!99 \nu^{15} + \cdots - 10\!\cdots\!80 ) / 12\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 19\!\cdots\!15 \nu^{15} + \cdots - 17\!\cdots\!90 ) / 12\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 19\!\cdots\!19 \nu^{15} + \cdots + 40\!\cdots\!66 ) / 12\!\cdots\!74 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - 3\beta_{8} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} + 4\beta_{5} + \beta_{4} + \beta_{3} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{14} - \beta_{13} - 7\beta_{11} + \beta_{9} + 14\beta_{8} + \beta_{7} + 7\beta_{3} + \beta_{2} - \beta _1 - 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{15} + \beta_{14} - 7 \beta_{13} - \beta_{12} - 8 \beta_{11} - 8 \beta_{10} + \beta_{9} + \cdots + 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{15} + 9 \beta_{13} - 8 \beta_{12} + \beta_{9} + 2 \beta_{6} - 10 \beta_{5} - 10 \beta_{4} + \cdots + 77 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 10 \beta_{15} - 10 \beta_{14} + 54 \beta_{13} + 57 \beta_{11} + 53 \beta_{10} - \beta_{9} - 77 \beta_{8} + \cdots + 24 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( \beta_{15} - 53 \beta_{14} + 15 \beta_{13} + 53 \beta_{12} + 272 \beta_{11} + 27 \beta_{10} - 64 \beta_{9} + \cdots - 27 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 15 \beta_{15} - 91 \beta_{13} + 76 \beta_{12} - 75 \beta_{9} - 335 \beta_{6} + 648 \beta_{5} + \cdots - 551 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 106 \beta_{15} + 334 \beta_{14} - 576 \beta_{13} - 1679 \beta_{11} - 256 \beta_{10} + 320 \beta_{9} + \cdots - 2474 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 363 \beta_{15} + 527 \beta_{14} - 1578 \beta_{13} - 527 \beta_{12} - 2660 \beta_{11} - 2096 \beta_{10} + \cdots + 2096 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 708 \beta_{15} + 2778 \beta_{13} - 2070 \beta_{12} + 837 \beta_{9} + 2109 \beta_{6} - 4412 \beta_{5} + \cdots + 16730 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 3486 \beta_{15} - 3515 \beta_{14} + 14539 \beta_{13} + 17776 \beta_{11} + 13138 \beta_{10} + \cdots + 12898 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 995 \beta_{15} - 12761 \beta_{14} + 9902 \beta_{13} + 12761 \beta_{12} + 64506 \beta_{11} + 16161 \beta_{10} + \cdots - 16161 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 11187 \beta_{15} - 34186 \beta_{13} + 22999 \beta_{12} - 23289 \beta_{9} - 82821 \beta_{6} + \cdots - 175151 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/385\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(232\) \(276\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
221.1
1.27511 + 2.20855i
1.25936 + 2.18128i
0.827576 + 1.43340i
0.420010 + 0.727479i
0.139605 + 0.241804i
−0.531161 − 0.919997i
−0.735245 − 1.27348i
−1.15525 − 2.00096i
1.27511 − 2.20855i
1.25936 − 2.18128i
0.827576 − 1.43340i
0.420010 − 0.727479i
0.139605 − 0.241804i
−0.531161 + 0.919997i
−0.735245 + 1.27348i
−1.15525 + 2.00096i
−1.27511 − 2.20855i −0.937301 + 1.62345i −2.25180 + 3.90023i −0.500000 − 0.866025i 4.78064 −2.44383 + 1.01376i 6.38473 −0.257068 − 0.445255i −1.27511 + 2.20855i
221.2 −1.25936 − 2.18128i 1.64549 − 2.85007i −2.17198 + 3.76198i −0.500000 − 0.866025i −8.28906 −0.160806 − 2.64086i 5.90377 −3.91527 − 6.78146i −1.25936 + 2.18128i
221.3 −0.827576 − 1.43340i −1.71605 + 2.97229i −0.369765 + 0.640452i −0.500000 − 0.866025i 5.68065 2.63215 + 0.267914i −2.08627 −4.38966 − 7.60311i −0.827576 + 1.43340i
221.4 −0.420010 − 0.727479i 0.864835 − 1.49794i 0.647183 − 1.12095i −0.500000 − 0.866025i −1.45296 −2.55075 − 0.702629i −2.76733 0.00412044 + 0.00713681i −0.420010 + 0.727479i
221.5 −0.139605 − 0.241804i −0.137980 + 0.238988i 0.961021 − 1.66454i −0.500000 − 0.866025i 0.0770509 1.16250 + 2.37668i −1.09508 1.46192 + 2.53213i −0.139605 + 0.241804i
221.6 0.531161 + 0.919997i 1.35728 − 2.35087i 0.435737 − 0.754718i −0.500000 − 0.866025i 2.88373 0.964079 + 2.46385i 3.05043 −2.18440 − 3.78348i 0.531161 − 0.919997i
221.7 0.735245 + 1.27348i −0.359468 + 0.622616i −0.0811705 + 0.140591i −0.500000 − 0.866025i −1.05719 2.24301 − 1.40318i 2.70226 1.24157 + 2.15046i 0.735245 − 1.27348i
221.8 1.15525 + 2.00096i −1.21680 + 2.10756i −1.66923 + 2.89118i −0.500000 − 0.866025i −5.62286 −2.34636 + 1.22254i −3.09251 −1.46121 − 2.53090i 1.15525 − 2.00096i
331.1 −1.27511 + 2.20855i −0.937301 − 1.62345i −2.25180 − 3.90023i −0.500000 + 0.866025i 4.78064 −2.44383 − 1.01376i 6.38473 −0.257068 + 0.445255i −1.27511 − 2.20855i
331.2 −1.25936 + 2.18128i 1.64549 + 2.85007i −2.17198 − 3.76198i −0.500000 + 0.866025i −8.28906 −0.160806 + 2.64086i 5.90377 −3.91527 + 6.78146i −1.25936 − 2.18128i
331.3 −0.827576 + 1.43340i −1.71605 − 2.97229i −0.369765 − 0.640452i −0.500000 + 0.866025i 5.68065 2.63215 − 0.267914i −2.08627 −4.38966 + 7.60311i −0.827576 − 1.43340i
331.4 −0.420010 + 0.727479i 0.864835 + 1.49794i 0.647183 + 1.12095i −0.500000 + 0.866025i −1.45296 −2.55075 + 0.702629i −2.76733 0.00412044 − 0.00713681i −0.420010 − 0.727479i
331.5 −0.139605 + 0.241804i −0.137980 − 0.238988i 0.961021 + 1.66454i −0.500000 + 0.866025i 0.0770509 1.16250 − 2.37668i −1.09508 1.46192 − 2.53213i −0.139605 − 0.241804i
331.6 0.531161 − 0.919997i 1.35728 + 2.35087i 0.435737 + 0.754718i −0.500000 + 0.866025i 2.88373 0.964079 − 2.46385i 3.05043 −2.18440 + 3.78348i 0.531161 + 0.919997i
331.7 0.735245 − 1.27348i −0.359468 − 0.622616i −0.0811705 − 0.140591i −0.500000 + 0.866025i −1.05719 2.24301 + 1.40318i 2.70226 1.24157 − 2.15046i 0.735245 + 1.27348i
331.8 1.15525 − 2.00096i −1.21680 − 2.10756i −1.66923 − 2.89118i −0.500000 + 0.866025i −5.62286 −2.34636 − 1.22254i −3.09251 −1.46121 + 2.53090i 1.15525 + 2.00096i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 221.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 385.2.i.c ✓ 16
7.c even 3 1 inner 385.2.i.c ✓ 16
7.c even 3 1 2695.2.a.t 8
7.d odd 6 1 2695.2.a.s 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.i.c ✓ 16 1.a even 1 1 trivial
385.2.i.c ✓ 16 7.c even 3 1 inner
2695.2.a.s 8 7.d odd 6 1
2695.2.a.t 8 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 3 T_{2}^{15} + 17 T_{2}^{14} + 28 T_{2}^{13} + 127 T_{2}^{12} + 178 T_{2}^{11} + 612 T_{2}^{10} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(385, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 3 T^{15} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{16} + T^{15} + \cdots + 2304 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} + T^{15} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{8} \) Copy content Toggle raw display
$13$ \( (T^{8} - 14 T^{7} + \cdots - 5119)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 110502144 \) Copy content Toggle raw display
$19$ \( T^{16} + T^{15} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( T^{16} - 2 T^{15} + \cdots + 20736 \) Copy content Toggle raw display
$29$ \( (T^{8} - 26 T^{7} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 2332600209 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 125451972864 \) Copy content Toggle raw display
$41$ \( (T^{8} + 3 T^{7} + \cdots - 82944)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 4 T^{7} + \cdots + 37189)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 1220194181376 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 427993344 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 1460861412921 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 416160000 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 1912137984 \) Copy content Toggle raw display
$71$ \( (T^{8} + 9 T^{7} + \cdots - 1485459)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 457139606641 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 747685337344 \) Copy content Toggle raw display
$83$ \( (T^{8} - 32 T^{7} + \cdots + 307773)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 10715415980481 \) Copy content Toggle raw display
$97$ \( (T^{8} - 9 T^{7} + \cdots + 554768)^{2} \) Copy content Toggle raw display
show more
show less