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Properties

Label 4225.2.a.cb
Level $4225$
Weight $2$
Character orbit 4225.a
Self dual yes
Analytic conductor $33.737$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4225,2,Mod(1,4225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4225 = 5^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4225.a (trivial)

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: yes
Analytic conductor: \(33.7367948540\)
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: \( x^{18} - 26x^{16} + 281x^{14} - 1632x^{12} + 5482x^{10} - 10620x^{8} + 11052x^{6} - 5165x^{4} + 760x^{2} - 29 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 845)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{14} q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{15} + \beta_{9} + \beta_{2} + 1) q^{6} - \beta_{10} q^{7} + \beta_{3} q^{8} + (\beta_{17} - \beta_{15} + \beta_{9} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{14} q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{15} + \beta_{9} + \beta_{2} + 1) q^{6} - \beta_{10} q^{7} + \beta_{3} q^{8} + (\beta_{17} - \beta_{15} + \beta_{9} + \cdots + 2) q^{9}+ \cdots + (\beta_{17} - 2 \beta_{13} + \beta_{9} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 16 q^{4} + 16 q^{6} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 16 q^{4} + 16 q^{6} + 18 q^{9} + 22 q^{11} + 4 q^{14} - 12 q^{16} + 28 q^{19} + 26 q^{21} + 34 q^{24} - 20 q^{29} + 32 q^{31} + 18 q^{34} + 32 q^{36} + 52 q^{41} + 50 q^{44} + 30 q^{46} + 44 q^{49} - 40 q^{51} + 90 q^{54} - 20 q^{56} + 76 q^{59} + 8 q^{61} - 68 q^{64} + 8 q^{66} + 30 q^{69} + 72 q^{71} + 30 q^{74} + 4 q^{76} + 16 q^{79} - 30 q^{81} + 78 q^{84} - 30 q^{86} + 94 q^{89} - 128 q^{94} - 18 q^{96} + 76 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 26x^{16} + 281x^{14} - 1632x^{12} + 5482x^{10} - 10620x^{8} + 11052x^{6} - 5165x^{4} + 760x^{2} - 29 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 4\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 51 \nu^{16} - 1009 \nu^{14} + 7704 \nu^{12} - 28208 \nu^{10} + 47962 \nu^{8} - 21998 \nu^{6} + \cdots - 1435 ) / 788 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 49 \nu^{17} + 1066 \nu^{15} - 9256 \nu^{13} + 40830 \nu^{11} - 95532 \nu^{9} + 110442 \nu^{7} + \cdots + 780 \nu ) / 394 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 149 \nu^{16} - 3141 \nu^{14} + 26216 \nu^{12} - 109868 \nu^{10} + 239026 \nu^{8} - 242882 \nu^{6} + \cdots - 2207 ) / 788 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 149 \nu^{17} + 3141 \nu^{15} - 26216 \nu^{13} + 109868 \nu^{11} - 239026 \nu^{9} + 242882 \nu^{7} + \cdots + 2207 \nu ) / 788 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 33 \nu^{16} + 734 \nu^{14} - 6503 \nu^{12} + 29122 \nu^{10} - 68499 \nu^{8} + 78062 \nu^{6} + \cdots + 268 ) / 197 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 61 \nu^{16} - 1315 \nu^{14} + 11370 \nu^{12} - 50596 \nu^{10} + 123342 \nu^{8} - 162110 \nu^{6} + \cdots + 3367 ) / 394 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 74 \nu^{17} + 1437 \nu^{15} - 10344 \nu^{13} + 31396 \nu^{11} - 17244 \nu^{9} - 114124 \nu^{7} + \cdots + 14779 \nu ) / 394 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 22 \nu^{17} - 555 \nu^{15} + 5780 \nu^{13} - 32154 \nu^{11} + 103190 \nu^{9} - 192174 \nu^{7} + \cdots + 17223 \nu ) / 394 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 113 \nu^{17} - 2591 \nu^{15} + 24208 \nu^{13} - 118788 \nu^{11} + 327774 \nu^{9} - 502366 \nu^{7} + \cdots + 4067 \nu ) / 788 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 110 \nu^{16} - 2381 \nu^{14} + 20626 \nu^{12} - 91426 \nu^{10} + 218874 \nu^{8} - 272552 \nu^{6} + \cdots + 1011 ) / 394 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 137 \nu^{17} - 3089 \nu^{15} + 28436 \nu^{13} - 138320 \nu^{11} + 383394 \nu^{9} - 605702 \nu^{7} + \cdots + 13149 \nu ) / 788 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 351 \nu^{16} + 7431 \nu^{14} - 62524 \nu^{12} + 266448 \nu^{10} - 602554 \nu^{8} + 680934 \nu^{6} + \cdots + 1185 ) / 788 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 18 \nu^{17} - 472 \nu^{15} + 5141 \nu^{13} - 30015 \nu^{11} + 100815 \nu^{9} - 193338 \nu^{7} + \cdots + 6910 \nu ) / 197 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 143 \nu^{16} + 3115 \nu^{14} - 27326 \nu^{12} + 124094 \nu^{10} - 311210 \nu^{8} + 424292 \nu^{6} + \cdots - 5471 ) / 394 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{13} + \beta_{9} + \beta_{6} - \beta_{4} + 6\beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{16} + \beta_{14} - \beta_{12} + \beta_{10} - 2\beta_{5} + 7\beta_{3} + 18\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{17} - \beta_{15} - 9\beta_{13} + 10\beta_{9} + \beta_{8} + 10\beta_{6} - 13\beta_{4} + 35\beta_{2} + 61 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 10 \beta_{16} + 12 \beta_{14} - 12 \beta_{12} - \beta_{11} + 10 \beta_{10} + 2 \beta_{7} + \cdots + 89 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 14 \beta_{17} - 12 \beta_{15} - 62 \beta_{13} + 79 \beta_{9} + 13 \beta_{8} + 76 \beta_{6} - 114 \beta_{4} + \cdots + 305 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 74 \beta_{16} + 102 \beta_{14} - 105 \beta_{12} - 13 \beta_{11} + 75 \beta_{10} + 27 \beta_{7} + \cdots + 473 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 132 \beta_{17} - 101 \beta_{15} - 393 \beta_{13} + 569 \beta_{9} + 119 \beta_{8} + 532 \beta_{6} + \cdots + 1606 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 494 \beta_{16} + 758 \beta_{14} - 807 \beta_{12} - 119 \beta_{11} + 512 \beta_{10} + \cdots + 2652 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1054 \beta_{17} - 740 \beta_{15} - 2421 \beta_{13} + 3904 \beta_{9} + 944 \beta_{8} + 3601 \beta_{6} + \cdots + 8814 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 3161 \beta_{16} + 5274 \beta_{14} - 5792 \beta_{12} - 944 \beta_{11} + 3365 \beta_{10} + \cdots + 15445 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 7716 \beta_{17} - 5070 \beta_{15} - 14804 \beta_{13} + 26038 \beta_{9} + 6940 \beta_{8} + 23940 \beta_{6} + \cdots + 49995 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 19874 \beta_{16} + 35402 \beta_{14} - 39918 \beta_{12} - 6940 \beta_{11} + 21744 \beta_{10} + \cdots + 92341 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 53714 \beta_{17} - 33532 \beta_{15} - 90640 \beta_{13} + 170666 \beta_{9} + 48728 \beta_{8} + \cdots + 291097 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 124172 \beta_{16} + 232744 \beta_{14} - 268122 \beta_{12} - 48728 \beta_{11} + 139368 \beta_{10} + \cdots + 562048 \beta_1 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.51756
−2.15692
−2.14790
−1.94561
−1.76651
−1.57695
−0.887876
−0.395078
−0.242854
0.242854
0.395078
0.887876
1.57695
1.76651
1.94561
2.14790
2.15692
2.51756
−2.51756 −2.61964 4.33809 0 6.59510 1.23067 −5.88628 3.86253 0
1.2 −2.15692 1.75445 2.65230 0 −3.78422 3.86123 −1.40696 0.0781086 0
1.3 −2.14790 −2.37435 2.61347 0 5.09986 −3.54879 −1.31766 2.63754 0
1.4 −1.94561 0.752344 1.78540 0 −1.46377 1.49584 0.417520 −2.43398 0
1.5 −1.76651 0.691389 1.12056 0 −1.22135 −4.36221 1.55354 −2.52198 0
1.6 −1.57695 −2.97563 0.486782 0 4.69244 −2.50257 2.38627 5.85440 0
1.7 −0.887876 1.26907 −1.21168 0 −1.12678 2.28128 2.85157 −1.38947 0
1.8 −0.395078 2.73651 −1.84391 0 −1.08113 −0.629405 1.51865 4.48846 0
1.9 −0.242854 −1.19348 −1.94102 0 0.289840 4.78046 0.957093 −1.57562 0
1.10 0.242854 1.19348 −1.94102 0 0.289840 −4.78046 −0.957093 −1.57562 0
1.11 0.395078 −2.73651 −1.84391 0 −1.08113 0.629405 −1.51865 4.48846 0
1.12 0.887876 −1.26907 −1.21168 0 −1.12678 −2.28128 −2.85157 −1.38947 0
1.13 1.57695 2.97563 0.486782 0 4.69244 2.50257 −2.38627 5.85440 0
1.14 1.76651 −0.691389 1.12056 0 −1.22135 4.36221 −1.55354 −2.52198 0
1.15 1.94561 −0.752344 1.78540 0 −1.46377 −1.49584 −0.417520 −2.43398 0
1.16 2.14790 2.37435 2.61347 0 5.09986 3.54879 1.31766 2.63754 0
1.17 2.15692 −1.75445 2.65230 0 −3.78422 −3.86123 1.40696 0.0781086 0
1.18 2.51756 2.61964 4.33809 0 6.59510 −1.23067 5.88628 3.86253 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(13\) \(1\)

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 4225.2.a.cb 18
5.b even 2 1 inner 4225.2.a.cb 18
5.c odd 4 2 845.2.b.h yes 18
13.b even 2 1 4225.2.a.ca 18
65.d even 2 1 4225.2.a.ca 18
65.f even 4 2 845.2.d.e 36
65.h odd 4 2 845.2.b.g ✓ 18
65.k even 4 2 845.2.d.e 36
65.o even 12 4 845.2.l.g 72
65.q odd 12 4 845.2.n.h 36
65.r odd 12 4 845.2.n.i 36
65.t even 12 4 845.2.l.g 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
845.2.b.g ✓ 18 65.h odd 4 2
845.2.b.h yes 18 5.c odd 4 2
845.2.d.e 36 65.f even 4 2
845.2.d.e 36 65.k even 4 2
845.2.l.g 72 65.o even 12 4
845.2.l.g 72 65.t even 12 4
845.2.n.h 36 65.q odd 12 4
845.2.n.i 36 65.r odd 12 4
4225.2.a.ca 18 13.b even 2 1
4225.2.a.ca 18 65.d even 2 1
4225.2.a.cb 18 1.a even 1 1 trivial
4225.2.a.cb 18 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4225))\):

\( T_{2}^{18} - 26 T_{2}^{16} + 281 T_{2}^{14} - 1632 T_{2}^{12} + 5482 T_{2}^{10} - 10620 T_{2}^{8} + \cdots - 29 \) Copy content Toggle raw display
\( T_{3}^{18} - 36 T_{3}^{16} + 534 T_{3}^{14} - 4223 T_{3}^{12} + 19291 T_{3}^{10} - 51839 T_{3}^{8} + \cdots - 4901 \) Copy content Toggle raw display
\( T_{7}^{18} - 85 T_{7}^{16} + 2943 T_{7}^{14} - 53626 T_{7}^{12} + 555634 T_{7}^{10} - 3314427 T_{7}^{8} + \cdots - 3572829 \) Copy content Toggle raw display
\( T_{11}^{9} - 11 T_{11}^{8} - T_{11}^{7} + 335 T_{11}^{6} - 577 T_{11}^{5} - 3115 T_{11}^{4} + \cdots - 15336 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} - 26 T^{16} + \cdots - 29 \) Copy content Toggle raw display
$3$ \( T^{18} - 36 T^{16} + \cdots - 4901 \) Copy content Toggle raw display
$5$ \( T^{18} \) Copy content Toggle raw display
$7$ \( T^{18} - 85 T^{16} + \cdots - 3572829 \) Copy content Toggle raw display
$11$ \( (T^{9} - 11 T^{8} + \cdots - 15336)^{2} \) Copy content Toggle raw display
$13$ \( T^{18} \) Copy content Toggle raw display
$17$ \( T^{18} - 153 T^{16} + \cdots - 53009216 \) Copy content Toggle raw display
$19$ \( (T^{9} - 14 T^{8} + \cdots + 356504)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots - 1663159541 \) Copy content Toggle raw display
$29$ \( (T^{9} + 10 T^{8} + \cdots + 50679)^{2} \) Copy content Toggle raw display
$31$ \( (T^{9} - 16 T^{8} + \cdots - 4544)^{2} \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots - 228661056 \) Copy content Toggle raw display
$41$ \( (T^{9} - 26 T^{8} + \cdots + 2460807)^{2} \) Copy content Toggle raw display
$43$ \( T^{18} - 293 T^{16} + \cdots - 21141 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots - 33109633009349 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots - 8958557504 \) Copy content Toggle raw display
$59$ \( (T^{9} - 38 T^{8} + \cdots - 20952)^{2} \) Copy content Toggle raw display
$61$ \( (T^{9} - 4 T^{8} + \cdots + 3297433)^{2} \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots - 190023072011109 \) Copy content Toggle raw display
$71$ \( (T^{9} - 36 T^{8} + \cdots + 508248)^{2} \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots - 158284670828544 \) Copy content Toggle raw display
$79$ \( (T^{9} - 8 T^{8} + \cdots + 53432392)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots - 269507038437029 \) Copy content Toggle raw display
$89$ \( (T^{9} - 47 T^{8} + \cdots + 21405033)^{2} \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots - 294008165722944 \) Copy content Toggle raw display
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