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Properties

Label 325.6.b.h
Level $325$
Weight $6$
Character orbit 325.b
Analytic conductor $52.125$
Analytic rank $0$
Dimension $18$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [325,6,Mod(274,325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(325, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("325.274");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 325.b (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: \(52.1247414392\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 378 x^{16} + 59175 x^{14} + 4956036 x^{12} + 239061247 x^{10} + 6629044458 x^{8} + \cdots + 134659641600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7}\cdot 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{10} + \beta_1) q^{2} + ( - \beta_{12} + \beta_{10}) q^{3} + (\beta_{4} - \beta_{3} - 2 \beta_{2} - 11) q^{4} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \cdots - 11) q^{6}+ \cdots + (\beta_{9} + \beta_{8} - 2 \beta_{6} + \cdots - 61) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{10} + \beta_1) q^{2} + ( - \beta_{12} + \beta_{10}) q^{3} + (\beta_{4} - \beta_{3} - 2 \beta_{2} - 11) q^{4} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \cdots - 11) q^{6}+ \cdots + ( - 222 \beta_{9} - 467 \beta_{8} + \cdots + 83645) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9} - 2844 q^{11} + 684 q^{14} - 2122 q^{16} + 816 q^{19} - 7824 q^{21} + 16938 q^{24} - 1690 q^{26} + 17474 q^{29} + 1496 q^{31} + 35578 q^{34} + 1024 q^{36} + 3718 q^{39} - 57352 q^{41} + 61198 q^{44} - 24112 q^{46} + 81814 q^{49} - 62012 q^{51} + 205522 q^{54} - 46004 q^{56} + 176284 q^{59} + 56330 q^{61} + 201690 q^{64} + 85154 q^{66} + 363494 q^{69} - 141124 q^{71} + 271352 q^{74} + 92746 q^{76} + 328146 q^{79} - 139870 q^{81} + 691312 q^{84} - 589840 q^{86} + 505396 q^{89} + 4056 q^{91} + 1003212 q^{94} - 638362 q^{96} + 1515552 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 378 x^{16} + 59175 x^{14} + 4956036 x^{12} + 239061247 x^{10} + 6629044458 x^{8} + \cdots + 134659641600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 830612285 \nu^{16} - 245221272933 \nu^{14} - 29588801099862 \nu^{12} + \cdots - 73\!\cdots\!00 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 169447858870357 \nu^{16} + \cdots + 10\!\cdots\!00 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 169447858870357 \nu^{16} + \cdots + 22\!\cdots\!00 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 37677558134451 \nu^{16} + \cdots + 18\!\cdots\!00 ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 237656384232177 \nu^{16} + \cdots - 64\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 777494511741213 \nu^{16} + \cdots + 25\!\cdots\!00 ) / 83\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 558459987519809 \nu^{16} + \cdots + 14\!\cdots\!00 ) / 54\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 12\!\cdots\!89 \nu^{16} + \cdots - 19\!\cdots\!00 ) / 54\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2516334673159 \nu^{17} + 947364487902807 \nu^{15} + \cdots + 73\!\cdots\!40 \nu ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 34\!\cdots\!11 \nu^{17} + \cdots - 12\!\cdots\!00 \nu ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 19\!\cdots\!97 \nu^{17} + \cdots + 18\!\cdots\!00 \nu ) / 62\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 11\!\cdots\!41 \nu^{17} + \cdots + 66\!\cdots\!00 \nu ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 14\!\cdots\!31 \nu^{17} + \cdots - 36\!\cdots\!00 \nu ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 18\!\cdots\!53 \nu^{17} + \cdots - 60\!\cdots\!00 \nu ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 23\!\cdots\!77 \nu^{17} + \cdots + 10\!\cdots\!00 \nu ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 19\!\cdots\!61 \nu^{17} + \cdots + 50\!\cdots\!00 \nu ) / 12\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} - 42 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{17} - \beta_{15} - \beta_{14} + \beta_{12} + 10\beta_{10} - 64\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{9} - 5\beta_{8} - 2\beta_{7} - 6\beta_{6} + \beta_{5} - 80\beta_{4} + 104\beta_{3} - 20\beta_{2} + 2727 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 103 \beta_{17} + 3 \beta_{16} + 102 \beta_{15} + 124 \beta_{14} - 5 \beta_{13} - 160 \beta_{12} + \cdots + 4570 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 434 \beta_{9} + 742 \beta_{8} + 252 \beta_{7} + 666 \beta_{6} - 112 \beta_{5} + 6361 \beta_{4} + \cdots - 197044 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 9129 \beta_{17} - 268 \beta_{16} - 8449 \beta_{15} - 12301 \beta_{14} + 1148 \beta_{13} + \cdots - 347132 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 47447 \beta_{9} - 82105 \beta_{8} - 24930 \beta_{7} - 60362 \beta_{6} + 8033 \beta_{5} + \cdots + 15086387 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 783523 \beta_{17} + 22051 \beta_{16} + 656598 \beta_{15} + 1134456 \beta_{14} - 162957 \beta_{13} + \cdots + 27357034 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 4660094 \beta_{9} + 8141794 \beta_{8} + 2249788 \beta_{7} + 5201326 \beta_{6} - 408424 \beta_{5} + \cdots - 1194597960 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 66743885 \beta_{17} - 2119988 \beta_{16} - 50012777 \beta_{15} - 101291897 \beta_{14} + \cdots - 2205498700 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 433649475 \beta_{9} - 764956629 \beta_{8} - 193756210 \beta_{7} - 442139062 \beta_{6} + \cdots + 96562819319 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 5675773839 \beta_{17} + 228223171 \beta_{16} + 3805789334 \beta_{15} + 8892673700 \beta_{14} + \cdots + 180405141354 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 39122957722 \beta_{9} + 69695889742 \beta_{8} + 16262319516 \beta_{7} + 37489497026 \beta_{6} + \cdots - 7909310507916 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 482491639857 \beta_{17} - 25013279628 \beta_{16} - 292314779297 \beta_{15} + \cdots - 14899319538108 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 3462757503871 \beta_{9} - 6229168169953 \beta_{8} - 1345655022242 \beta_{7} - 3181314038914 \beta_{6} + \cdots + 653582352971563 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 41016052426939 \beta_{17} + 2667432728627 \beta_{16} + 22798159849030 \beta_{15} + \cdots + 12\!\cdots\!14 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/325\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
274.1
− 9.17271i
− 9.23858i
− 7.11691i
− 6.50844i
− 8.28147i
− 5.30592i
− 4.20685i
− 0.838151i
0.603392i
− 0.603392i
0.838151i
4.20685i
5.30592i
8.28147i
6.50844i
7.11691i
9.23858i
9.17271i
− 10.1727i − 23.0461i −71.4841 0 −234.441 − 51.0548i 401.660i −288.123 0
274.2 − 8.23858i 11.6375i −35.8743 0 95.8765 195.071i 31.9185i 107.569 0
274.3 − 8.11691i 3.22994i −33.8842 0 26.2172 − 20.7248i 15.2938i 232.567 0
274.4 − 7.50844i 17.3551i −24.3767 0 130.310 149.455i − 57.2388i −58.2000 0
274.5 − 7.28147i − 14.9935i −21.0198 0 −109.175 − 135.391i − 79.9521i 18.1955 0
274.6 − 4.30592i 19.9083i 13.4591 0 85.7235 − 125.783i − 195.743i −153.341 0
274.7 − 3.20685i − 29.5845i 21.7161 0 −94.8732 11.5734i − 172.260i −632.245 0
274.8 − 1.83815i 11.9262i 28.6212 0 21.9221 − 112.158i − 111.431i 100.766 0
274.9 − 0.396608i − 11.4974i 31.8427 0 −4.55994 − 8.04695i − 25.3205i 110.811 0
274.10 0.396608i 11.4974i 31.8427 0 −4.55994 8.04695i 25.3205i 110.811 0
274.11 1.83815i − 11.9262i 28.6212 0 21.9221 112.158i 111.431i 100.766 0
274.12 3.20685i 29.5845i 21.7161 0 −94.8732 − 11.5734i 172.260i −632.245 0
274.13 4.30592i − 19.9083i 13.4591 0 85.7235 125.783i 195.743i −153.341 0
274.14 7.28147i 14.9935i −21.0198 0 −109.175 135.391i 79.9521i 18.1955 0
274.15 7.50844i − 17.3551i −24.3767 0 130.310 − 149.455i 57.2388i −58.2000 0
274.16 8.11691i − 3.22994i −33.8842 0 26.2172 20.7248i − 15.2938i 232.567 0
274.17 8.23858i − 11.6375i −35.8743 0 95.8765 − 195.071i − 31.9185i 107.569 0
274.18 10.1727i 23.0461i −71.4841 0 −234.441 51.0548i − 401.660i −288.123 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 274.18
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 325.6.b.h 18
5.b even 2 1 inner 325.6.b.h 18
5.c odd 4 1 325.6.a.h ✓ 9
5.c odd 4 1 325.6.a.i yes 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.6.a.h ✓ 9 5.c odd 4 1
325.6.a.i yes 9 5.c odd 4 1
325.6.b.h 18 1.a even 1 1 trivial
325.6.b.h 18 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} + 379 T_{2}^{16} + 58767 T_{2}^{14} + 4806917 T_{2}^{12} + 222215028 T_{2}^{10} + \cdots + 140175360000 \) acting on \(S_{6}^{\mathrm{new}}(325, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + \cdots + 140175360000 \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots + 33\!\cdots\!76 \) Copy content Toggle raw display
$5$ \( T^{18} \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 30\!\cdots\!76 \) Copy content Toggle raw display
$11$ \( (T^{9} + \cdots - 97\!\cdots\!12)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 28561)^{9} \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 72\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( (T^{9} + \cdots + 55\!\cdots\!84)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{9} + \cdots - 70\!\cdots\!44)^{2} \) Copy content Toggle raw display
$31$ \( (T^{9} + \cdots - 65\!\cdots\!60)^{2} \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 21\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( (T^{9} + \cdots + 68\!\cdots\!12)^{2} \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( (T^{9} + \cdots - 25\!\cdots\!12)^{2} \) Copy content Toggle raw display
$61$ \( (T^{9} + \cdots - 15\!\cdots\!48)^{2} \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 15\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{9} + \cdots - 43\!\cdots\!76)^{2} \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 14\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( (T^{9} + \cdots - 58\!\cdots\!96)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 19\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T^{9} + \cdots + 30\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
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