Properties

Label 927.2.c.a
Level $927$
Weight $2$
Character orbit 927.c
Analytic conductor $7.402$
Analytic rank $0$
Dimension $16$
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] = [927,2,Mod(926,927)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(927, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("927.926");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 927 = 3^{2} \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 927.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: \(7.40213226737\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 34x^{14} + 564x^{12} - 5204x^{10} + 29740x^{8} - 99088x^{6} + 214644x^{4} - 222312x^{2} + 84100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{13} q^{2} + \beta_{3} q^{4} - \beta_{8} q^{5} + ( - \beta_{2} + 1) q^{7} + ( - \beta_{15} - \beta_{14}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{13} q^{2} + \beta_{3} q^{4} - \beta_{8} q^{5} + ( - \beta_{2} + 1) q^{7} + ( - \beta_{15} - \beta_{14}) q^{8} - \beta_{4} q^{10} + \beta_{10} q^{11} + (\beta_{2} - \beta_1 - 1) q^{13} + ( - \beta_{14} - 2 \beta_{13} - \beta_{9}) q^{14} + (\beta_{3} + \beta_{2} - 1) q^{16} + ( - \beta_{15} + \beta_{13} - \beta_{9}) q^{17} + ( - 2 \beta_{3} + \beta_{2} - \beta_1) q^{19} + \beta_{12} q^{20} + ( - \beta_{6} + \beta_{5}) q^{22} + ( - 2 \beta_{15} - \beta_{14} + \beta_{13}) q^{23} + ( - \beta_{3} - 2 \beta_1 + 5) q^{25} + (\beta_{14} + \beta_{13}) q^{26} + (\beta_1 - 1) q^{28} + (\beta_{14} - 2 \beta_{13} - 3 \beta_{9}) q^{29} + ( - \beta_{7} + \beta_{5}) q^{31} + ( - 3 \beta_{15} - 2 \beta_{14} + 4 \beta_{13} + \beta_{9}) q^{32} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 + 2) q^{34} + ( - \beta_{12} + \beta_{11} - \beta_{10} - \beta_{8}) q^{35} + ( - \beta_{7} - \beta_{4}) q^{37} + (2 \beta_{15} + 3 \beta_{14} - 4 \beta_{13}) q^{38} + (\beta_{7} - \beta_{5}) q^{40} + ( - 2 \beta_{15} + 2 \beta_{13} + 3 \beta_{9}) q^{41} + ( - \beta_{7} - \beta_{6} + \beta_{5}) q^{43} + \beta_{11} q^{44} + ( - 2 \beta_{3} + \beta_{2}) q^{46} - \beta_{10} q^{47} + (\beta_{3} - 3 \beta_{2} - 2 \beta_1 - 3) q^{49} + (\beta_{15} + \beta_{14} - 9 \beta_{13} - 2 \beta_{9}) q^{50} + (\beta_{2} - 2 \beta_1) q^{52} + \beta_{11} q^{53} + ( - 3 \beta_{3} + 3 \beta_{2} + 7 \beta_1 + 3) q^{55} + ( - 2 \beta_{14} - 2 \beta_{13} - \beta_{9}) q^{56} + (2 \beta_{2} + 3 \beta_1 - 1) q^{58} + (4 \beta_{15} - \beta_{14} - \beta_{13} - 2 \beta_{9}) q^{59} + (3 \beta_{2} - 2) q^{61} + (\beta_{11} - \beta_{10} - \beta_{8}) q^{62} + ( - 3 \beta_{3} + 3 \beta_{2} - \beta_1 + 2) q^{64} + (\beta_{12} - \beta_{11}) q^{65} + (\beta_{6} + \beta_{4}) q^{67} + (3 \beta_{14} - 2 \beta_{13}) q^{68} + (\beta_{6} - 2 \beta_{5} - 2 \beta_{4}) q^{70} + ( - \beta_{12} + \beta_{11} + \beta_{8}) q^{71} + (2 \beta_{7} - \beta_{6}) q^{73} + (\beta_{12} + \beta_{8}) q^{74} + (3 \beta_{3} - \beta_{2} - 2 \beta_1 - 6) q^{76} + (\beta_{12} - \beta_{11} + 2 \beta_{10} + \beta_{8}) q^{77} + (4 \beta_{3} - 2 \beta_{2} - \beta_1 - 3) q^{79} + (2 \beta_{12} - \beta_{11} + \beta_{10} + \beta_{8}) q^{80} + (\beta_{3} - 3 \beta_{2} - 3 \beta_1 - 1) q^{82} + ( - 2 \beta_{15} - 4 \beta_{14} + 5 \beta_{13} + 3 \beta_{9}) q^{83} + (\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4}) q^{85} + (\beta_{11} - 2 \beta_{10} - \beta_{8}) q^{86} + (\beta_{7} - 2 \beta_{6} + \beta_{4}) q^{88} + ( - \beta_{12} - \beta_{11} + \beta_{10} + \beta_{8}) q^{89} + ( - \beta_{3} + \beta_{2} + \beta_1 - 3) q^{91} + ( - 2 \beta_{15} + \beta_{14} - \beta_{13} + \beta_{9}) q^{92} + (\beta_{6} - \beta_{5}) q^{94} + ( - \beta_{12} - \beta_{11} - \beta_{8}) q^{95} + (6 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 1) q^{97} + ( - \beta_{15} - 4 \beta_{14} - 5 \beta_{9}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{7} - 8 q^{13} - 16 q^{16} + 8 q^{19} + 96 q^{25} - 24 q^{28} + 24 q^{34} - 32 q^{49} + 16 q^{52} - 8 q^{55} - 40 q^{58} - 32 q^{61} + 40 q^{64} - 80 q^{76} - 40 q^{79} + 8 q^{82} - 56 q^{91} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 34x^{14} + 564x^{12} - 5204x^{10} + 29740x^{8} - 99088x^{6} + 214644x^{4} - 222312x^{2} + 84100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 20114813547 \nu^{14} + 657518183386 \nu^{12} - 10224082165378 \nu^{10} + 82617743997017 \nu^{8} - 369286265262418 \nu^{6} + \cdots - 29\!\cdots\!16 ) / 19\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 31852621799 \nu^{14} - 927485431885 \nu^{12} + 13336321445016 \nu^{10} - 98606458894395 \nu^{8} + 433829284161384 \nu^{6} + \cdots + 44\!\cdots\!68 ) / 19\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 67192712276 \nu^{14} - 2088492923645 \nu^{12} + 31391992019638 \nu^{10} - 245522293476599 \nu^{8} + \cdots + 32\!\cdots\!40 ) / 19\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 161774372946 \nu^{14} - 5091711741364 \nu^{12} + 78990248483794 \nu^{10} - 659778233769221 \nu^{8} + \cdots - 15\!\cdots\!12 ) / 19\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 73887296181 \nu^{14} + 2422756571858 \nu^{12} - 38744050645988 \nu^{10} + 338216353003786 \nu^{8} + \cdots + 63\!\cdots\!72 ) / 175709793597834 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 939520497824 \nu^{14} + 31142130627676 \nu^{12} - 503664966944388 \nu^{10} + \cdots + 82\!\cdots\!38 ) / 19\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1612023264355 \nu^{14} + 53278766175273 \nu^{12} - 858756502358368 \nu^{10} + \cdots + 14\!\cdots\!86 ) / 19\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3564940540838 \nu^{15} - 67038268505007 \nu^{13} + 186895041080732 \nu^{11} + \cdots + 34\!\cdots\!04 \nu ) / 56\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 6816403149681 \nu^{15} + 221102481148514 \nu^{13} + \cdots + 79\!\cdots\!12 \nu ) / 56\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 6816403149681 \nu^{15} + 221102481148514 \nu^{13} + \cdots + 19\!\cdots\!32 \nu ) / 56\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2096532504071 \nu^{15} - 50836045660766 \nu^{13} + 526979880771778 \nu^{11} - 732288998761202 \nu^{9} + \cdots + 77\!\cdots\!04 \nu ) / 11\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 4109972533687 \nu^{15} - 138855224047101 \nu^{13} + \cdots - 12\!\cdots\!88 \nu ) / 11\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 21754241947832 \nu^{15} + 729304952533983 \nu^{13} + \cdots + 17\!\cdots\!64 \nu ) / 56\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 11911187 \nu^{15} - 394390138 \nu^{13} + 6361271963 \nu^{11} - 56162895403 \nu^{9} + 301920423430 \nu^{7} - 893755136446 \nu^{5} + \cdots - 993728799594 \nu ) / 88294348010 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 144495421804111 \nu^{15} + \cdots + 12\!\cdots\!92 \nu ) / 56\!\cdots\!60 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{10} - \beta_{9} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{3} - 3\beta_{2} + 3\beta _1 + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8\beta_{15} + 11\beta_{14} - 8\beta_{13} + \beta_{12} + \beta_{11} + 4\beta_{10} - 22\beta_{9} - \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -10\beta_{7} + 10\beta_{6} + 9\beta_{5} + 3\beta_{4} + 4\beta_{3} - 2\beta_{2} + 5\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 133\beta_{15} + 205\beta_{14} - 93\beta_{13} - 7\beta_{12} - 27\beta_{10} - 241\beta_{9} + 3\beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -103\beta_{7} + 114\beta_{6} + 74\beta_{5} + 4\beta_{4} - 115\beta_{3} + 146\beta_{2} - 181\beta _1 - 417 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 526 \beta_{15} + 880 \beta_{14} - 199 \beta_{13} - 110 \beta_{12} - 130 \beta_{11} - 431 \beta_{10} - 757 \beta_{9} + 185 \beta_{8} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -418\beta_{7} + 544\beta_{6} + 180\beta_{5} - 92\beta_{4} - 2464\beta_{3} + 2454\beta_{2} - 4268\beta _1 - 7962 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 442 \beta_{15} - 315 \beta_{14} + 1244 \beta_{13} - 1317 \beta_{12} - 2457 \beta_{11} - 5687 \beta_{10} + 1671 \beta_{9} + 3495 \beta_{8} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 6952 \beta_{7} - 7640 \beta_{6} - 5099 \beta_{5} - 406 \beta_{4} - 27249 \beta_{3} + 23475 \beta_{2} - 51378 \beta _1 - 85946 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 84667 \beta_{15} - 140704 \beta_{14} + 33297 \beta_{13} - 9188 \beta_{12} - 23959 \beta_{11} - 44459 \beta_{10} + 125659 \beta_{9} + 34550 \beta_{8} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 183964 \beta_{7} - 222330 \beta_{6} - 105600 \beta_{5} + 13846 \beta_{4} - 147172 \beta_{3} + 106182 \beta_{2} - 303754 \beta _1 - 456678 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1470719 \beta_{15} - 2522011 \beta_{14} + 373873 \beta_{13} - 291 \beta_{12} - 54284 \beta_{11} - 39329 \beta_{10} + 2003829 \beta_{9} + 81133 \beta_{8} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 2337014 \beta_{7} - 2925858 \beta_{6} - 1195382 \beta_{5} + 293856 \beta_{4} + 1136828 \beta_{3} - 1090018 \beta_{2} + 2009156 \beta _1 + 3636584 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 14469820 \beta_{15} - 25228700 \beta_{14} + 2580414 \beta_{13} + 1142700 \beta_{12} + 2637840 \beta_{11} + 5298572 \beta_{10} + 18771308 \beta_{9} - 3792314 \beta_{8} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/927\mathbb{Z}\right)^\times\).

\(n\) \(722\) \(829\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
926.1
−1.54287 + 0.912316i
1.54287 + 0.912316i
2.58516 1.10064i
−2.58516 1.10064i
−3.23870 + 1.20231i
3.23870 + 1.20231i
0.973303 0.103539i
−0.973303 0.103539i
0.973303 + 0.103539i
−0.973303 + 0.103539i
−3.23870 1.20231i
3.23870 1.20231i
2.58516 + 1.10064i
−2.58516 + 1.10064i
−1.54287 0.912316i
1.54287 0.912316i
2.10850i 0 −2.44579 −3.56582 0 0.463913 0.939944i 0 7.51854i
926.2 2.10850i 0 −2.44579 3.56582 0 0.463913 0.939944i 0 7.51854i
926.3 1.67237i 0 −0.796815 −3.92802 0 4.16190 2.01217i 0 6.56910i
926.4 1.67237i 0 −0.796815 3.92802 0 4.16190 2.01217i 0 6.56910i
926.5 0.793520i 0 1.37033 −2.13753 0 0.751881 2.67442i 0 1.69617i
926.6 0.793520i 0 1.37033 2.13753 0 0.751881 2.67442i 0 1.69617i
926.7 0.357385i 0 1.87228 −3.35955 0 −1.37769 1.38389i 0 1.20065i
926.8 0.357385i 0 1.87228 3.35955 0 −1.37769 1.38389i 0 1.20065i
926.9 0.357385i 0 1.87228 −3.35955 0 −1.37769 1.38389i 0 1.20065i
926.10 0.357385i 0 1.87228 3.35955 0 −1.37769 1.38389i 0 1.20065i
926.11 0.793520i 0 1.37033 −2.13753 0 0.751881 2.67442i 0 1.69617i
926.12 0.793520i 0 1.37033 2.13753 0 0.751881 2.67442i 0 1.69617i
926.13 1.67237i 0 −0.796815 −3.92802 0 4.16190 2.01217i 0 6.56910i
926.14 1.67237i 0 −0.796815 3.92802 0 4.16190 2.01217i 0 6.56910i
926.15 2.10850i 0 −2.44579 −3.56582 0 0.463913 0.939944i 0 7.51854i
926.16 2.10850i 0 −2.44579 3.56582 0 0.463913 0.939944i 0 7.51854i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 926.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
103.b odd 2 1 inner
309.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 927.2.c.a 16
3.b odd 2 1 inner 927.2.c.a 16
103.b odd 2 1 inner 927.2.c.a 16
309.c even 2 1 inner 927.2.c.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
927.2.c.a 16 1.a even 1 1 trivial
927.2.c.a 16 3.b odd 2 1 inner
927.2.c.a 16 103.b odd 2 1 inner
927.2.c.a 16 309.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 8T_{2}^{6} + 18T_{2}^{4} + 10T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(927, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 8 T^{6} + 18 T^{4} + 10 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 44 T^{6} + 694 T^{4} + \cdots + 10117)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 4 T^{3} - 2 T^{2} + 6 T - 2)^{4} \) Copy content Toggle raw display
$11$ \( (T^{8} - 82 T^{6} + 2072 T^{4} + \cdots + 40468)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 2 T^{3} - 8 T^{2} - 18 T - 5)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 24 T^{6} + 184 T^{4} + 512 T^{2} + \cdots + 400)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 2 T^{3} - 24 T^{2} + 24 T - 4)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + 32 T^{6} + 334 T^{4} + 1226 T^{2} + \cdots + 841)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 130 T^{6} + 4540 T^{4} + \cdots + 676)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 128 T^{6} + 5410 T^{4} + \cdots + 495733)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 120 T^{6} + 4204 T^{4} + \cdots + 40468)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 166 T^{6} + 8464 T^{4} + \cdots + 448900)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 252 T^{6} + 18962 T^{4} + \cdots + 1709773)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 82 T^{6} + 2072 T^{4} + \cdots + 40468)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 166 T^{6} + 8820 T^{4} + \cdots + 1011700)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 248 T^{6} + 18622 T^{4} + \cdots + 5004169)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 8 T^{3} - 48 T^{2} - 94 T - 29)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + 208 T^{6} + 13418 T^{4} + \cdots + 1709773)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} - 208 T^{6} + 5832 T^{4} + \cdots + 161872)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 388 T^{6} + 42236 T^{4} + \cdots + 1982932)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 10 T^{3} - 48 T^{2} + 42 T + 2)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + 298 T^{6} + 29134 T^{4} + \cdots + 5004169)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 470 T^{6} + 70652 T^{4} + \cdots + 89393812)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 4 T^{3} - 242 T^{2} + 1548 T + 2617)^{4} \) Copy content Toggle raw display
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