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Properties

Label 425.4.b.k
Level $425$
Weight $4$
Character orbit 425.b
Analytic conductor $25.076$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [425,4,Mod(324,425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(425, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("425.324");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 425.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: \(25.0758117524\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 141 x^{18} + 8410 x^{16} + 276474 x^{14} + 5469637 x^{12} + 66721609 x^{10} + 493855792 x^{8} + \cdots + 851705856 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{13} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{13} - \beta_{11}) q^{3} + ( - \beta_{4} + \beta_{3} - 6) q^{4} + (\beta_{7} - \beta_{4}) q^{6} + (\beta_{13} + 2 \beta_{11} + \beta_{8}) q^{7} + (\beta_{19} + \beta_{17} + \cdots - 6 \beta_1) q^{8}+ \cdots + (2 \beta_{12} - \beta_{10} + \beta_{7} + \cdots - 13) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{13} - \beta_{11}) q^{3} + ( - \beta_{4} + \beta_{3} - 6) q^{4} + (\beta_{7} - \beta_{4}) q^{6} + (\beta_{13} + 2 \beta_{11} + \beta_{8}) q^{7} + (\beta_{19} + \beta_{17} + \cdots - 6 \beta_1) q^{8}+ \cdots + (19 \beta_{12} - 44 \beta_{10} + \cdots - 746) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 122 q^{4} + 2 q^{6} - 256 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 122 q^{4} + 2 q^{6} - 256 q^{9} + 136 q^{11} - 134 q^{14} + 634 q^{16} - 452 q^{19} + 900 q^{21} - 274 q^{24} + 1162 q^{26} + 176 q^{29} + 1308 q^{31} + 34 q^{34} + 2100 q^{36} - 1472 q^{39} + 728 q^{41} + 320 q^{44} + 4264 q^{46} - 1808 q^{49} + 204 q^{51} + 1698 q^{54} + 4726 q^{56} - 404 q^{59} + 2552 q^{61} - 3594 q^{64} + 3936 q^{66} + 616 q^{69} + 3448 q^{71} + 904 q^{74} + 8244 q^{76} - 6708 q^{79} + 6348 q^{81} - 7958 q^{84} + 4696 q^{86} - 528 q^{89} + 7064 q^{91} - 2508 q^{94} + 2834 q^{96} - 14212 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 141 x^{18} + 8410 x^{16} + 276474 x^{14} + 5469637 x^{12} + 66721609 x^{10} + 493855792 x^{8} + \cdots + 851705856 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 972437657 \nu^{18} + 133834501845 \nu^{16} + 7776864750554 \nu^{14} + 248255388798474 \nu^{12} + \cdots + 46\!\cdots\!40 ) / 21\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 27999 \nu^{18} + 3531305 \nu^{16} + 183505988 \nu^{14} + 5083628338 \nu^{12} + \cdots + 11645973903360 ) / 602984094720 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 27999 \nu^{18} + 3531305 \nu^{16} + 183505988 \nu^{14} + 5083628338 \nu^{12} + \cdots + 3204196577280 ) / 602984094720 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1164734527 \nu^{18} - 109732827059 \nu^{16} - 3177364527654 \nu^{14} + \cdots + 17\!\cdots\!52 ) / 21\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1005892135 \nu^{18} - 129722777643 \nu^{16} - 6913778342230 \nu^{14} + \cdots + 35\!\cdots\!64 ) / 10\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 663366463 \nu^{18} + 70266121627 \nu^{16} + 2831184857366 \nu^{14} + 51933109263350 \nu^{12} + \cdots - 45\!\cdots\!16 ) / 53\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 230542229 \nu^{19} + 4217547055 \nu^{17} + 2463977298142 \nu^{15} + \cdots - 28\!\cdots\!40 \nu ) / 76\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1852944493 \nu^{18} - 256232699545 \nu^{16} - 14821375893026 \nu^{14} + \cdots - 14\!\cdots\!80 ) / 10\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 910731415 \nu^{18} - 122177002539 \nu^{16} - 6803669364214 \nu^{14} + \cdots + 32\!\cdots\!12 ) / 42\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 4574705 \nu^{19} - 610986621 \nu^{17} - 34179202170 \nu^{15} + \cdots - 34\!\cdots\!12 \nu ) / 733228659179520 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1464134443 \nu^{18} - 183498816431 \nu^{16} - 9468914062686 \nu^{14} + \cdots - 21\!\cdots\!12 ) / 53\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 16292161919 \nu^{19} - 2498858235331 \nu^{17} - 158377982713398 \nu^{15} + \cdots - 22\!\cdots\!92 \nu ) / 16\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 27513396319 \nu^{19} + 3604197736771 \nu^{17} + 199784915465878 \nu^{15} + \cdots + 48\!\cdots\!92 \nu ) / 16\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 6617728531 \nu^{19} + 978512450663 \nu^{17} + 60661055694686 \nu^{15} + \cdots + 16\!\cdots\!88 \nu ) / 32\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 14999543 \nu^{19} + 2129872907 \nu^{17} + 127682774486 \nu^{15} + \cdots + 22\!\cdots\!04 \nu ) / 366614329589760 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 16640287719 \nu^{19} - 2112435250907 \nu^{17} - 110506255964838 \nu^{15} + \cdots - 69\!\cdots\!24 \nu ) / 20\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 72153232703 \nu^{19} + 8942215055987 \nu^{17} + 450638465989926 \nu^{15} + \cdots - 10\!\cdots\!16 \nu ) / 80\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 1005892135 \nu^{19} + 129722777643 \nu^{17} + 6913778342230 \nu^{15} + \cdots - 35\!\cdots\!64 \nu ) / 10\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{4} + \beta_{3} - 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} + \beta_{17} + \beta_{16} + 9\beta_{11} - \beta_{8} - 22\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{12} - \beta_{10} - 2\beta_{9} + \beta_{7} + 4\beta_{6} - \beta_{5} + 34\beta_{4} - 28\beta_{3} - 2\beta_{2} + 305 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 36 \beta_{19} - 2 \beta_{18} - 36 \beta_{17} - 40 \beta_{16} + 10 \beta_{15} - 6 \beta_{14} + \cdots + 547 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 92 \beta_{12} + 34 \beta_{10} + 92 \beta_{9} - 34 \beta_{7} - 168 \beta_{6} + 66 \beta_{5} + \cdots - 7500 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1055 \beta_{19} + 116 \beta_{18} + 1079 \beta_{17} + 1367 \beta_{16} - 588 \beta_{15} + \cdots - 14384 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3406 \beta_{12} - 787 \beta_{10} - 3334 \beta_{9} + 939 \beta_{7} + 5140 \beta_{6} - 2803 \beta_{5} + \cdots + 195751 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 29074 \beta_{19} - 5238 \beta_{18} - 31162 \beta_{17} - 44814 \beta_{16} + 24198 \beta_{15} + \cdots + 391389 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 118552 \beta_{12} + 13332 \beta_{10} + 110720 \beta_{9} - 25244 \beta_{7} - 136840 \beta_{6} + \cdots - 5296898 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 782909 \beta_{19} + 210440 \beta_{18} + 894509 \beta_{17} + 1446477 \beta_{16} - 868344 \beta_{15} + \cdots - 10905130 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 4008314 \beta_{12} - 73829 \beta_{10} - 3525706 \beta_{9} + 686325 \beta_{7} + 3320900 \beta_{6} + \cdots + 146905117 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 20892256 \beta_{19} - 7868970 \beta_{18} - 25728912 \beta_{17} - 46336660 \beta_{16} + \cdots + 309256599 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 132915508 \beta_{12} - 7289146 \beta_{10} + 109806820 \beta_{9} - 18867254 \beta_{7} + \cdots - 4148911160 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 555966651 \beta_{19} + 280409308 \beta_{18} + 743174243 \beta_{17} + 1477449699 \beta_{16} + \cdots - 8891323428 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 4342275078 \beta_{12} + 487874697 \beta_{10} - 3378912718 \beta_{9} + 519497567 \beta_{7} + \cdots + 118830768147 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 14801942158 \beta_{19} - 9660299550 \beta_{18} - 21568394150 \beta_{17} - 46944485050 \beta_{16} + \cdots + 258452257361 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 140198612144 \beta_{12} - 22185947080 \beta_{10} + 103302859144 \beta_{9} - 14195958512 \beta_{7} + \cdots - 3441896581358 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 395036938137 \beta_{19} + 324769118448 \beta_{18} + 628943671513 \beta_{17} + 1487202482201 \beta_{16} + \cdots - 7580282871838 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(326\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
324.1
− 5.55245i
− 5.17600i
− 4.90944i
− 4.25398i
− 4.15022i
− 3.16532i
− 2.86867i
− 2.06883i
− 1.02999i
− 0.605497i
0.605497i
1.02999i
2.06883i
2.86867i
3.16532i
4.15022i
4.25398i
4.90944i
5.17600i
5.55245i
− 5.55245i 0.854197i −22.8297 0 4.74288 − 22.1474i 82.3411i 26.2703 0
324.2 − 5.17600i 5.66210i −18.7909 0 29.3070 − 8.60812i 55.8539i −5.05942 0
324.3 − 4.90944i − 8.54295i −16.1026 0 −41.9411 29.3547i 39.7790i −45.9820 0
324.4 − 4.25398i 8.37256i −10.0964 0 35.6167 − 5.95662i 8.91795i −43.0997 0
324.5 − 4.15022i − 4.36306i −9.22436 0 −18.1077 − 26.8391i 5.08135i 7.96367 0
324.6 − 3.16532i − 9.73823i −2.01923 0 −30.8246 10.5373i − 18.9310i −67.8331 0
324.7 − 2.86867i 9.10998i −0.229276 0 26.1335 4.02694i − 22.2917i −55.9918 0
324.8 − 2.06883i − 0.643144i 3.71993 0 −1.33056 − 2.16624i − 24.2466i 26.5864 0
324.9 − 1.02999i 0.400571i 6.93913 0 0.412584 33.7017i − 15.3871i 26.8395 0
324.10 − 0.605497i − 4.96930i 7.63337 0 −3.00890 29.7343i − 9.46596i 2.30608 0
324.11 0.605497i 4.96930i 7.63337 0 −3.00890 − 29.7343i 9.46596i 2.30608 0
324.12 1.02999i − 0.400571i 6.93913 0 0.412584 − 33.7017i 15.3871i 26.8395 0
324.13 2.06883i 0.643144i 3.71993 0 −1.33056 2.16624i 24.2466i 26.5864 0
324.14 2.86867i − 9.10998i −0.229276 0 26.1335 − 4.02694i 22.2917i −55.9918 0
324.15 3.16532i 9.73823i −2.01923 0 −30.8246 − 10.5373i 18.9310i −67.8331 0
324.16 4.15022i 4.36306i −9.22436 0 −18.1077 26.8391i − 5.08135i 7.96367 0
324.17 4.25398i − 8.37256i −10.0964 0 35.6167 5.95662i − 8.91795i −43.0997 0
324.18 4.90944i 8.54295i −16.1026 0 −41.9411 − 29.3547i − 39.7790i −45.9820 0
324.19 5.17600i − 5.66210i −18.7909 0 29.3070 8.60812i − 55.8539i −5.05942 0
324.20 5.55245i − 0.854197i −22.8297 0 4.74288 22.1474i − 82.3411i 26.2703 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 324.20
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 425.4.b.k 20
5.b even 2 1 inner 425.4.b.k 20
5.c odd 4 1 425.4.a.l ✓ 10
5.c odd 4 1 425.4.a.m yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
425.4.a.l ✓ 10 5.c odd 4 1
425.4.a.m yes 10 5.c odd 4 1
425.4.b.k 20 1.a even 1 1 trivial
425.4.b.k 20 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 \(S_{4}^{\mathrm{new}}(425, [\chi])\):

\( T_{2}^{20} + 141 T_{2}^{18} + 8410 T_{2}^{16} + 276474 T_{2}^{14} + 5469637 T_{2}^{12} + \cdots + 851705856 \) Copy content Toggle raw display
\( T_{3}^{20} + 398 T_{3}^{18} + 65141 T_{3}^{16} + 5648716 T_{3}^{14} + 278626718 T_{3}^{12} + \cdots + 29386530625 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 851705856 \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 29386530625 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 67\!\cdots\!09 \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots - 17834351412480)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 30\!\cdots\!41 \) Copy content Toggle raw display
$17$ \( (T^{2} + 289)^{10} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 18\!\cdots\!40)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots - 60\!\cdots\!27)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 95\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 67\!\cdots\!36)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 15\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 24\!\cdots\!81 \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 20\!\cdots\!20)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 94\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 41\!\cdots\!03)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 34\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots - 13\!\cdots\!55)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 23\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 72\!\cdots\!00 \) Copy content Toggle raw display
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