↑

Properties

Label 2175.2.c.p
Level $2175$
Weight $2$
Character orbit 2175.c
Analytic conductor $17.367$
Analytic rank $0$
Dimension $16$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2175,2,Mod(349,2175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2175, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2175.349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2175.c (of order \(2\), degree \(1\), not 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: \(17.3674624396\)
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: \( x^{16} + 28x^{14} + 308x^{12} + 1671x^{10} + 4568x^{8} + 5616x^{6} + 2105x^{4} + 256x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{6} q^{3} + (\beta_{2} - 2) q^{4} - \beta_{3} q^{6} + ( - \beta_{11} - \beta_{8} + \beta_1) q^{7} + (\beta_{8} + \beta_{7} - 2 \beta_1) q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{6} q^{3} + (\beta_{2} - 2) q^{4} - \beta_{3} q^{6} + ( - \beta_{11} - \beta_{8} + \beta_1) q^{7} + (\beta_{8} + \beta_{7} - 2 \beta_1) q^{8} - q^{9} + (\beta_{12} - \beta_{10} + \beta_{3} + 1) q^{11} + (\beta_{14} + 2 \beta_{6}) q^{12} + ( - \beta_{13} - \beta_{8} + \beta_{6}) q^{13} + (\beta_{12} - 2 \beta_{10} + \beta_{9} + \cdots - 2) q^{14}+ \cdots + ( - \beta_{12} + \beta_{10} - \beta_{3} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 24 q^{4} + 4 q^{6} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 24 q^{4} + 4 q^{6} - 16 q^{9} + 12 q^{11} - 18 q^{14} + 64 q^{16} - 4 q^{21} - 6 q^{24} + 36 q^{26} - 16 q^{29} + 16 q^{31} + 26 q^{34} + 24 q^{36} + 12 q^{39} + 4 q^{41} + 30 q^{44} + 48 q^{46} - 76 q^{49} + 24 q^{51} - 4 q^{54} + 116 q^{56} - 36 q^{59} + 24 q^{61} - 42 q^{64} - 6 q^{66} - 28 q^{69} + 48 q^{71} + 44 q^{74} - 20 q^{79} + 16 q^{81} + 28 q^{84} + 16 q^{86} - 68 q^{89} + 52 q^{91} + 86 q^{94} - 4 q^{96} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 28x^{14} + 308x^{12} + 1671x^{10} + 4568x^{8} + 5616x^{6} + 2105x^{4} + 256x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{14} - 194\nu^{12} - 2106\nu^{10} - 11193\nu^{8} - 29386\nu^{6} - 32436\nu^{4} - 6803\nu^{2} - 98 ) / 114 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4\nu^{14} + 119\nu^{12} + 1369\nu^{10} + 7593\nu^{8} + 20402\nu^{6} + 22381\nu^{4} + 3920\nu^{2} - 96 ) / 57 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -9\nu^{14} - 244\nu^{12} - 2591\nu^{10} - 13498\nu^{8} - 34989\nu^{6} - 39095\nu^{4} - 10036\nu^{2} - 278 ) / 57 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 49 \nu^{15} + 1358 \nu^{13} + 14704 \nu^{11} + 77667 \nu^{9} + 201446 \nu^{7} + 216412 \nu^{5} + \cdots - 1062 \nu ) / 228 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12\nu^{15} + 338\nu^{13} + 3727\nu^{11} + 20138\nu^{9} + 54081\nu^{7} + 62887\nu^{5} + 18296\nu^{3} + 1840\nu ) / 57 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -12\nu^{15} - 338\nu^{13} - 3727\nu^{11} - 20138\nu^{9} - 54081\nu^{7} - 62887\nu^{5} - 18239\nu^{3} - 1498\nu ) / 57 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -24\nu^{14} - 676\nu^{12} - 7435\nu^{10} - 39877\nu^{8} - 105122\nu^{6} - 115723\nu^{4} - 23482\nu^{2} - 184 ) / 57 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 67 \nu^{14} - 1846 \nu^{12} - 19886 \nu^{10} - 104663 \nu^{8} - 271424 \nu^{6} - 294716 \nu^{4} + \cdots - 520 ) / 114 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 7\nu^{15} + 194\nu^{13} + 2104\nu^{11} + 11169\nu^{9} + 29354\nu^{7} + 32872\nu^{5} + 8075\nu^{3} + 498\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -33\nu^{14} - 920\nu^{12} - 10026\nu^{10} - 53375\nu^{8} - 140111\nu^{6} - 154875\nu^{4} - 33917\nu^{2} - 690 ) / 57 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 41 \nu^{15} + 1139 \nu^{13} + 12384 \nu^{11} + 65920 \nu^{9} + 173790 \nu^{7} + 195324 \nu^{5} + \cdots + 1714 \nu ) / 57 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 91 \nu^{15} - 2522 \nu^{13} - 27302 \nu^{11} - 144141 \nu^{9} - 373506 \nu^{7} - 400388 \nu^{5} + \cdots + 2222 \nu ) / 114 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 157 \nu^{15} - 4362 \nu^{13} - 47392 \nu^{11} - 251689 \nu^{9} - 659808 \nu^{7} - 730126 \nu^{5} + \cdots - 4326 \nu ) / 114 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + \beta_{7} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{12} + \beta_{9} + \beta_{5} - 7\beta_{2} + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} - \beta_{14} + \beta_{11} - 10\beta_{8} - 9\beta_{7} - \beta_{6} + 39\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10\beta_{12} - 9\beta_{9} - 11\beta_{5} + 2\beta_{4} - 2\beta_{3} + 49\beta_{2} - 156 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -13\beta_{15} + 14\beta_{14} - 2\beta_{13} - 11\beta_{11} + 80\beta_{8} + 68\beta_{7} + 17\beta_{6} - 263\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -80\beta_{12} + \beta_{10} + 63\beta_{9} + 93\beta_{5} - 32\beta_{4} + 37\beta_{3} - 346\beta_{2} + 1055 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 125 \beta_{15} - 147 \beta_{14} + 32 \beta_{13} + 94 \beta_{11} - 597 \beta_{8} - 488 \beta_{7} + \cdots + 1812 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 597\beta_{12} - 15\beta_{10} - 403\beta_{9} - 722\beta_{5} + 352\beta_{4} - 463\beta_{3} + 2457\beta_{2} - 7297 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1074 \beta_{15} + 1379 \beta_{14} - 352 \beta_{13} - 740 \beta_{11} + 4340 \beta_{8} + \cdots - 12650 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 4340 \beta_{12} + 158 \beta_{10} + 2451 \beta_{9} + 5414 \beta_{5} - 3333 \beta_{4} + 4870 \beta_{3} + \cdots + 51146 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 8747 \beta_{15} - 12173 \beta_{14} + 3333 \beta_{13} + 5626 \beta_{11} - 31242 \beta_{8} + \cdots + 89078 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 31242 \beta_{12} - 1465 \beta_{10} - 14271 \beta_{9} - 39989 \beta_{5} + 29242 \beta_{4} - 46455 \beta_{3} + \cdots - 361513 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 69231 \beta_{15} + 103415 \beta_{14} - 29242 \beta_{13} - 42048 \beta_{11} + 224258 \beta_{8} + \cdots - 631102 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1451\) \(2002\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
349.1
− 2.72810i
− 2.59587i
− 2.57789i
− 1.98486i
− 1.64893i
− 0.510732i
− 0.485464i
− 0.135002i
0.135002i
0.485464i
0.510732i
1.64893i
1.98486i
2.57789i
2.59587i
2.72810i
− 2.72810i − 1.00000i −5.44251 0 −2.72810 − 4.61695i 9.39150i −1.00000 0
349.2 − 2.59587i 1.00000i −4.73855 0 2.59587 3.93826i 7.10893i −1.00000 0
349.3 − 2.57789i 1.00000i −4.64553 0 2.57789 − 4.69867i 6.81989i −1.00000 0
349.4 − 1.98486i − 1.00000i −1.93969 0 −1.98486 2.16633i − 0.119715i −1.00000 0
349.5 − 1.64893i 1.00000i −0.718971 0 1.64893 − 1.44112i − 2.11233i −1.00000 0
349.6 − 0.510732i 1.00000i 1.73915 0 0.510732 4.82343i − 1.90970i −1.00000 0
349.7 − 0.485464i − 1.00000i 1.76432 0 −0.485464 1.94912i − 1.82745i −1.00000 0
349.8 − 0.135002i − 1.00000i 1.98177 0 −0.135002 1.12340i − 0.537546i −1.00000 0
349.9 0.135002i 1.00000i 1.98177 0 −0.135002 − 1.12340i 0.537546i −1.00000 0
349.10 0.485464i 1.00000i 1.76432 0 −0.485464 − 1.94912i 1.82745i −1.00000 0
349.11 0.510732i − 1.00000i 1.73915 0 0.510732 − 4.82343i 1.90970i −1.00000 0
349.12 1.64893i − 1.00000i −0.718971 0 1.64893 1.44112i 2.11233i −1.00000 0
349.13 1.98486i 1.00000i −1.93969 0 −1.98486 − 2.16633i 0.119715i −1.00000 0
349.14 2.57789i − 1.00000i −4.64553 0 2.57789 4.69867i − 6.81989i −1.00000 0
349.15 2.59587i − 1.00000i −4.73855 0 2.59587 − 3.93826i − 7.10893i −1.00000 0
349.16 2.72810i 1.00000i −5.44251 0 −2.72810 4.61695i − 9.39150i −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 349.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2175.2.c.p 16
5.b even 2 1 inner 2175.2.c.p 16
5.c odd 4 1 2175.2.a.bc ✓ 8
5.c odd 4 1 2175.2.a.bd yes 8
15.e even 4 1 6525.2.a.by 8
15.e even 4 1 6525.2.a.bz 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2175.2.a.bc ✓ 8 5.c odd 4 1
2175.2.a.bd yes 8 5.c odd 4 1
2175.2.c.p 16 1.a even 1 1 trivial
2175.2.c.p 16 5.b even 2 1 inner
6525.2.a.by 8 15.e even 4 1
6525.2.a.bz 8 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2175, [\chi])\):

\( T_{2}^{16} + 28T_{2}^{14} + 308T_{2}^{12} + 1671T_{2}^{10} + 4568T_{2}^{8} + 5616T_{2}^{6} + 2105T_{2}^{4} + 256T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{16} + 94 T_{7}^{14} + 3535 T_{7}^{12} + 67742 T_{7}^{10} + 700423 T_{7}^{8} + 3873298 T_{7}^{6} + \cdots + 7935489 \) Copy content Toggle raw display
\( T_{11}^{8} - 6T_{11}^{7} - 36T_{11}^{6} + 238T_{11}^{5} + 265T_{11}^{4} - 2382T_{11}^{3} - 21T_{11}^{2} + 6052T_{11} - 1140 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 28 T^{14} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 94 T^{14} + \cdots + 7935489 \) Copy content Toggle raw display
$11$ \( (T^{8} - 6 T^{7} + \cdots - 1140)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 110313009 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 359254116 \) Copy content Toggle raw display
$19$ \( (T^{8} - 99 T^{6} + \cdots + 205760)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 224280576 \) Copy content Toggle raw display
$29$ \( (T + 1)^{16} \) Copy content Toggle raw display
$31$ \( (T^{8} - 8 T^{7} + \cdots + 1225792)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 438045477175296 \) Copy content Toggle raw display
$41$ \( (T^{8} - 2 T^{7} + \cdots - 2160)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 236896358400 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 5715905513616 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 107495424 \) Copy content Toggle raw display
$59$ \( (T^{8} + 18 T^{7} + \cdots + 10368)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 12 T^{7} + \cdots + 386112)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 23\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( (T^{8} - 24 T^{7} + \cdots - 16256)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 386154569465856 \) Copy content Toggle raw display
$79$ \( (T^{8} + 10 T^{7} + \cdots + 2032128)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 20\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{8} + 34 T^{7} + \cdots - 12296640)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 98108232601600 \) Copy content Toggle raw display
show more
show less