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Properties

Label 374.2.g.f
Level $374$
Weight $2$
Character orbit 374.g
Analytic conductor $2.986$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [374,2,Mod(69,374)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(374, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("374.69");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 374 = 2 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 374.g (of order \(5\), degree \(4\), 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: \(2.98640503560\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 10 x^{14} - 8 x^{13} + 68 x^{12} + 28 x^{11} + 306 x^{10} + 228 x^{9} + 1035 x^{8} + \cdots + 3481 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{8} - \beta_{6} + \beta_{3} - 1) q^{2} + (\beta_{12} + \beta_{11} - \beta_{6}) q^{3} + \beta_{8} q^{4} + ( - \beta_{14} - 2 \beta_{13} + \cdots - 1) q^{5}+ \cdots + ( - \beta_{12} + \beta_{11} + \beta_{10} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{8} - \beta_{6} + \beta_{3} - 1) q^{2} + (\beta_{12} + \beta_{11} - \beta_{6}) q^{3} + \beta_{8} q^{4} + ( - \beta_{14} - 2 \beta_{13} + \cdots - 1) q^{5}+ \cdots + (3 \beta_{15} - 3 \beta_{14} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{2} + 5 q^{3} - 4 q^{4} - q^{5} + 2 q^{7} - 4 q^{8} - 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{2} + 5 q^{3} - 4 q^{4} - q^{5} + 2 q^{7} - 4 q^{8} - 13 q^{9} + 4 q^{10} - 4 q^{11} - 10 q^{12} - 10 q^{13} + 2 q^{14} - 6 q^{15} - 4 q^{16} + 4 q^{17} + 2 q^{18} + 10 q^{19} - q^{20} + 6 q^{22} - 24 q^{23} - 37 q^{25} + 15 q^{26} - 4 q^{27} - 13 q^{28} - 16 q^{29} - 6 q^{30} + 11 q^{31} + 16 q^{32} + 36 q^{33} - 16 q^{34} + 2 q^{36} + 5 q^{37} - 25 q^{38} - 30 q^{39} - q^{40} + 4 q^{41} + 35 q^{42} + 16 q^{43} + 6 q^{44} + 6 q^{45} - 14 q^{46} - 14 q^{47} + 5 q^{48} + 2 q^{49} + 13 q^{50} - 5 q^{51} + 15 q^{52} + 5 q^{53} - 34 q^{54} + 81 q^{55} + 22 q^{56} - 12 q^{57} - 16 q^{58} + 17 q^{59} + 4 q^{60} - 30 q^{61} - 4 q^{62} + 72 q^{63} - 4 q^{64} + 6 q^{65} + 11 q^{66} + 26 q^{67} + 4 q^{68} - 73 q^{69} - 10 q^{70} - q^{71} - 13 q^{72} - 43 q^{73} + 5 q^{74} + 67 q^{75} + 30 q^{76} + 40 q^{77} + 70 q^{78} - 13 q^{79} - q^{80} - 39 q^{81} - 21 q^{82} - 24 q^{83} - 35 q^{84} + q^{85} + 26 q^{86} - 92 q^{87} - 4 q^{88} + 24 q^{89} - 29 q^{90} - 10 q^{91} + 26 q^{92} + 11 q^{93} - 4 q^{94} - 39 q^{95} + 5 q^{96} + 3 q^{97} + 2 q^{98} + 65 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 10 x^{14} - 8 x^{13} + 68 x^{12} + 28 x^{11} + 306 x^{10} + 228 x^{9} + 1035 x^{8} + \cdots + 3481 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 58\!\cdots\!82 \nu^{15} + \cdots + 34\!\cdots\!69 ) / 16\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 59\!\cdots\!97 \nu^{15} + \cdots + 56\!\cdots\!87 ) / 16\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 12\!\cdots\!55 \nu^{15} + \cdots + 20\!\cdots\!57 ) / 16\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 13\!\cdots\!94 \nu^{15} + \cdots - 49\!\cdots\!19 ) / 16\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 15\!\cdots\!56 \nu^{15} + \cdots - 21\!\cdots\!83 ) / 16\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 17\!\cdots\!63 \nu^{15} + \cdots + 16\!\cdots\!98 ) / 16\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 18\!\cdots\!58 \nu^{15} + \cdots - 13\!\cdots\!47 ) / 16\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 21\!\cdots\!28 \nu^{15} + \cdots - 14\!\cdots\!07 ) / 16\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 37\!\cdots\!16 \nu^{15} + \cdots - 17\!\cdots\!19 ) / 16\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 636223997932127 \nu^{15} + \cdots + 18\!\cdots\!14 ) / 25\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 45\!\cdots\!09 \nu^{15} + \cdots - 53\!\cdots\!36 ) / 16\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 46\!\cdots\!52 \nu^{15} + \cdots + 17\!\cdots\!58 ) / 16\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 94\!\cdots\!16 \nu^{15} + \cdots + 23\!\cdots\!63 ) / 16\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 10\!\cdots\!12 \nu^{15} + \cdots - 30\!\cdots\!35 ) / 16\!\cdots\!10 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} - \beta_{9} + 3\beta_{8} + \beta_{7} + \beta_{4} - \beta_{3} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{15} + \beta_{14} + \beta_{13} - 3 \beta_{12} - 8 \beta_{11} + \beta_{10} - 4 \beta_{9} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 13 \beta_{15} + \beta_{14} + 2 \beta_{13} - 3 \beta_{12} - 15 \beta_{11} + 6 \beta_{10} - 17 \beta_{9} + \cdots - 31 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 33 \beta_{15} - 4 \beta_{14} + 12 \beta_{13} - 21 \beta_{12} + 40 \beta_{10} - 40 \beta_{9} - 106 \beta_{8} + \cdots - 124 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 36 \beta_{15} - 57 \beta_{14} + 15 \beta_{13} - 101 \beta_{12} + 203 \beta_{11} + 152 \beta_{10} + \cdots - 372 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 245 \beta_{15} - 304 \beta_{14} - 74 \beta_{13} - 213 \beta_{12} + 1212 \beta_{11} + 245 \beta_{10} + \cdots - 256 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2046 \beta_{15} - 853 \beta_{14} - 530 \beta_{13} + 78 \beta_{12} + 4377 \beta_{11} - 530 \beta_{10} + \cdots + 3446 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 8676 \beta_{15} - 1193 \beta_{14} - 2386 \beta_{13} + 3517 \beta_{12} + 10113 \beta_{11} + \cdots + 24071 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 24594 \beta_{15} + 4612 \beta_{14} - 7483 \beta_{13} + 18732 \beta_{12} - 32812 \beta_{10} + \cdots + 97662 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 28200 \beta_{15} + 45311 \beta_{14} - 11089 \beta_{13} + 65339 \beta_{12} - 141794 \beta_{11} + \cdots + 251479 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 176016 \beta_{15} + 209256 \beta_{14} + 39983 \beta_{13} + 140179 \beta_{12} - 858535 \beta_{11} + \cdots + 162643 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1459469 \beta_{15} + 632339 \beta_{14} + 391678 \beta_{13} - 65409 \beta_{12} - 3210328 \beta_{11} + \cdots - 2512580 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 6355247 \beta_{15} + 902094 \beta_{14} + 1804188 \beta_{13} - 2355164 \beta_{12} - 7407665 \beta_{11} + \cdots - 17455496 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 18028972 \beta_{15} - 3363966 \beta_{14} + 5453153 \beta_{13} - 13355104 \beta_{12} + 23710125 \beta_{10} + \cdots - 70469904 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/374\mathbb{Z}\right)^\times\).

\(n\) \(35\) \(309\)
\(\chi(n)\) \(\beta_{8}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
69.1
−1.44028 + 1.04643i
−0.979351 + 0.711540i
1.51735 − 1.10242i
3.02032 − 2.19439i
−1.44028 − 1.04643i
−0.979351 − 0.711540i
1.51735 + 1.10242i
3.02032 + 2.19439i
−0.631507 − 1.94358i
−0.379230 − 1.16715i
0.221941 + 0.683065i
0.670762 + 2.06439i
−0.631507 + 1.94358i
−0.379230 + 1.16715i
0.221941 − 0.683065i
0.670762 − 2.06439i
−0.809017 + 0.587785i −0.859156 − 2.64421i 0.309017 − 0.951057i −2.75895 − 2.00449i 2.24930 + 1.63421i 1.14902 − 3.53632i 0.309017 + 0.951057i −3.82665 + 2.78022i 3.41024
69.2 −0.809017 + 0.587785i −0.683096 − 2.10235i 0.309017 − 0.951057i 2.39077 + 1.73699i 1.78837 + 1.29933i 1.04021 − 3.20144i 0.309017 + 0.951057i −1.52622 + 1.10886i −2.95515
69.3 −0.809017 + 0.587785i 0.270560 + 0.832698i 0.309017 − 0.951057i 0.0973285 + 0.0707133i −0.708335 − 0.514635i 0.450819 − 1.38748i 0.309017 + 0.951057i 1.80687 − 1.31277i −0.120305
69.4 −0.809017 + 0.587785i 0.844641 + 2.59954i 0.309017 − 0.951057i −0.538167 − 0.391001i −2.21130 − 1.60660i 0.0960172 − 0.295511i 0.309017 + 0.951057i −3.61712 + 2.62799i 0.665210
103.1 −0.809017 − 0.587785i −0.859156 + 2.64421i 0.309017 + 0.951057i −2.75895 + 2.00449i 2.24930 − 1.63421i 1.14902 + 3.53632i 0.309017 − 0.951057i −3.82665 − 2.78022i 3.41024
103.2 −0.809017 − 0.587785i −0.683096 + 2.10235i 0.309017 + 0.951057i 2.39077 − 1.73699i 1.78837 − 1.29933i 1.04021 + 3.20144i 0.309017 − 0.951057i −1.52622 − 1.10886i −2.95515
103.3 −0.809017 − 0.587785i 0.270560 − 0.832698i 0.309017 + 0.951057i 0.0973285 − 0.0707133i −0.708335 + 0.514635i 0.450819 + 1.38748i 0.309017 − 0.951057i 1.80687 + 1.31277i −0.120305
103.4 −0.809017 − 0.587785i 0.844641 − 2.59954i 0.309017 + 0.951057i −0.538167 + 0.391001i −2.21130 + 1.60660i 0.0960172 + 0.295511i 0.309017 − 0.951057i −3.61712 − 2.62799i 0.665210
137.1 0.309017 + 0.951057i −0.844290 − 0.613412i −0.809017 + 0.587785i 1.01829 − 3.13399i 0.322490 − 0.992522i −2.98412 + 2.16809i −0.809017 − 0.587785i −0.590501 − 1.81737i 3.29527
137.2 0.309017 + 0.951057i −0.183821 − 0.133553i −0.809017 + 0.587785i −1.16324 + 3.58010i 0.0702132 − 0.216094i −1.91546 + 1.39166i −0.809017 − 0.587785i −0.911098 − 2.80407i −3.76434
137.3 0.309017 + 0.951057i 1.39007 + 1.00994i −0.809017 + 0.587785i 1.12983 − 3.47725i −0.530958 + 1.63412i 0.631141 − 0.458551i −0.809017 − 0.587785i −0.0147498 − 0.0453953i 3.65620
137.4 0.309017 + 0.951057i 2.56509 + 1.86365i −0.809017 + 0.587785i −0.675860 + 2.08008i −0.979779 + 3.01545i 2.53238 − 1.83988i −0.809017 − 0.587785i 2.17947 + 6.70771i −2.18713
273.1 0.309017 − 0.951057i −0.844290 + 0.613412i −0.809017 − 0.587785i 1.01829 + 3.13399i 0.322490 + 0.992522i −2.98412 − 2.16809i −0.809017 + 0.587785i −0.590501 + 1.81737i 3.29527
273.2 0.309017 − 0.951057i −0.183821 + 0.133553i −0.809017 − 0.587785i −1.16324 − 3.58010i 0.0702132 + 0.216094i −1.91546 − 1.39166i −0.809017 + 0.587785i −0.911098 + 2.80407i −3.76434
273.3 0.309017 − 0.951057i 1.39007 − 1.00994i −0.809017 − 0.587785i 1.12983 + 3.47725i −0.530958 − 1.63412i 0.631141 + 0.458551i −0.809017 + 0.587785i −0.0147498 + 0.0453953i 3.65620
273.4 0.309017 − 0.951057i 2.56509 − 1.86365i −0.809017 − 0.587785i −0.675860 − 2.08008i −0.979779 − 3.01545i 2.53238 + 1.83988i −0.809017 + 0.587785i 2.17947 − 6.70771i −2.18713
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 69.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 374.2.g.f ✓ 16
11.c even 5 1 inner 374.2.g.f ✓ 16
11.c even 5 1 4114.2.a.bi 8
11.d odd 10 1 4114.2.a.bg 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
374.2.g.f ✓ 16 1.a even 1 1 trivial
374.2.g.f ✓ 16 11.c even 5 1 inner
4114.2.a.bg 8 11.d odd 10 1
4114.2.a.bi 8 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(374, [\chi])\):

\( T_{3}^{16} - 5 T_{3}^{15} + 25 T_{3}^{14} - 82 T_{3}^{13} + 277 T_{3}^{12} - 600 T_{3}^{11} + 1499 T_{3}^{10} + \cdots + 361 \) Copy content Toggle raw display
\( T_{5}^{16} + T_{5}^{15} + 29 T_{5}^{14} + 37 T_{5}^{13} + 434 T_{5}^{12} + 469 T_{5}^{11} + 4221 T_{5}^{10} + \cdots + 6400 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} - 5 T^{15} + \cdots + 361 \) Copy content Toggle raw display
$5$ \( T^{16} + T^{15} + \cdots + 6400 \) Copy content Toggle raw display
$7$ \( T^{16} - 2 T^{15} + \cdots + 14641 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( T^{16} + 10 T^{15} + \cdots + 4748041 \) Copy content Toggle raw display
$17$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 1353062656 \) Copy content Toggle raw display
$23$ \( (T^{8} + 12 T^{7} + \cdots - 30305)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + 16 T^{15} + \cdots + 774400 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 1664760545536 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 32480690176 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 446844119296 \) Copy content Toggle raw display
$43$ \( (T^{8} - 8 T^{7} + \cdots + 2163376)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 64882278400 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 1733411861281 \) Copy content Toggle raw display
$59$ \( T^{16} - 17 T^{15} + \cdots + 2310400 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 295570544896 \) Copy content Toggle raw display
$67$ \( (T^{8} - 13 T^{7} + \cdots - 671744)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 437608710400 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 1758106276096 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 140723267161 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 513834597126400 \) Copy content Toggle raw display
$89$ \( (T^{8} - 12 T^{7} + \cdots - 1074256)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 71547507947776 \) Copy content Toggle raw display
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