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Properties

Label 425.4.d.g
Level $425$
Weight $4$
Character orbit 425.d
Analytic conductor $25.076$
Analytic rank $0$
Dimension $16$
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] = [425,4,Mod(101,425)]
 
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([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("425.101");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 425.d (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: \(25.0758117524\)
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: \( x^{16} + 113 x^{14} + 5162 x^{12} + 121550 x^{10} + 1552797 x^{8} + 10318089 x^{6} + 30998912 x^{4} + \cdots + 2663424 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 85)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_{11} q^{3} + ( - \beta_1 + 6) q^{4} + (\beta_{12} - \beta_{11} - \beta_{8}) q^{6} + (\beta_{10} - \beta_{8}) q^{7} + ( - \beta_{5} + 6 \beta_{2} - \beta_1 - 4) q^{8} + ( - \beta_{4} - \beta_{3} + 2 \beta_{2} - 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + \beta_{11} q^{3} + ( - \beta_1 + 6) q^{4} + (\beta_{12} - \beta_{11} - \beta_{8}) q^{6} + (\beta_{10} - \beta_{8}) q^{7} + ( - \beta_{5} + 6 \beta_{2} - \beta_1 - 4) q^{8} + ( - \beta_{4} - \beta_{3} + 2 \beta_{2} - 7) q^{9} + (\beta_{15} + 4 \beta_{11} + \cdots - \beta_{8}) q^{11}+ \cdots + (9 \beta_{15} + 2 \beta_{14} + \cdots + 31 \beta_{8}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{2} + 98 q^{4} - 78 q^{8} - 116 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{2} + 98 q^{4} - 78 q^{8} - 116 q^{9} - 48 q^{13} + 490 q^{16} + 132 q^{17} + 470 q^{18} - 44 q^{19} - 136 q^{21} + 720 q^{26} + 1614 q^{32} - 2304 q^{33} - 278 q^{34} - 1646 q^{36} + 1168 q^{38} - 2692 q^{42} + 92 q^{43} - 268 q^{47} - 1008 q^{49} - 432 q^{51} - 248 q^{52} + 328 q^{53} + 1060 q^{59} - 1750 q^{64} - 2504 q^{66} + 284 q^{67} - 2778 q^{68} - 960 q^{69} + 5546 q^{72} + 2012 q^{76} - 3288 q^{77} + 8480 q^{81} - 3668 q^{83} + 8840 q^{84} + 336 q^{86} - 2704 q^{87} - 2468 q^{89} + 2448 q^{93} - 9140 q^{94} + 4470 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 113 x^{14} + 5162 x^{12} + 121550 x^{10} + 1552797 x^{8} + 10318089 x^{6} + 30998912 x^{4} + \cdots + 2663424 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + 14 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 943 \nu^{14} + 91938 \nu^{12} + 3428718 \nu^{10} + 60471672 \nu^{8} + 498880455 \nu^{6} + \cdots - 53921280 ) / 538811200 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 178 \nu^{14} - 67707 \nu^{12} - 5023972 \nu^{10} - 154297406 \nu^{8} - 2233601606 \nu^{6} + \cdots - 1879009344 ) / 107762240 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1595 \nu^{14} - 400484 \nu^{12} - 27188974 \nu^{10} - 802792684 \nu^{8} - 11342296411 \nu^{6} + \cdots - 27078114624 ) / 538811200 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 6125 \nu^{14} + 583588 \nu^{12} + 21281818 \nu^{10} + 364969668 \nu^{8} + 2842979037 \nu^{6} + \cdots - 13396478592 ) / 538811200 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 17699 \nu^{14} - 1928765 \nu^{12} - 84117030 \nu^{10} - 1856218462 \nu^{8} + \cdots - 13051070496 ) / 269405600 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 57275 \nu^{14} - 5472263 \nu^{12} - 204361838 \nu^{10} - 3762854418 \nu^{8} + \cdots - 85488796928 ) / 538811200 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 55535 \nu^{15} + 3872029 \nu^{13} + 78939694 \nu^{11} + 20727454 \nu^{9} + \cdots + 2177097862784 \nu ) / 27479371200 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 22226 \nu^{15} - 2551539 \nu^{13} - 118399824 \nu^{11} - 2830208382 \nu^{9} + \cdots - 632856360128 \nu ) / 9159790400 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 153857 \nu^{15} + 15409217 \nu^{13} + 604801002 \nu^{11} + 11658524078 \nu^{9} + \cdots - 900871738240 \nu ) / 36639161600 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 502871 \nu^{15} + 52818271 \nu^{13} + 2215147126 \nu^{11} + 46990171954 \nu^{9} + \cdots + 1461472995200 \nu ) / 109917484800 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 761323 \nu^{15} + 82118615 \nu^{13} + 3555796190 \nu^{11} + 78549752594 \nu^{9} + \cdots + 10311363885952 \nu ) / 109917484800 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 243569 \nu^{15} + 25960555 \nu^{13} + 1109605750 \nu^{11} + 24046412722 \nu^{9} + \cdots - 1412035779424 \nu ) / 27479371200 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 291603 \nu^{15} + 32112087 \nu^{13} + 1412134942 \nu^{11} + 31280027506 \nu^{9} + \cdots - 1513133193536 \nu ) / 18319580800 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 763127 \nu^{15} + 86686979 \nu^{13} + 3941865974 \nu^{11} + 90893523450 \nu^{9} + \cdots + 9731003237632 \nu ) / 36639161600 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{14} - \beta_{13} - \beta_{12} - \beta_{11} - \beta_{9} + \beta_{8} ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 - 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -19\beta_{14} + 19\beta_{13} + 29\beta_{12} + 11\beta_{11} - 2\beta_{10} + 29\beta_{9} - 19\beta_{8} ) / 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{5} - 2\beta_{4} + 2\beta_{3} + 5\beta_{2} - 28\beta _1 + 302 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 40 \beta_{15} + 409 \beta_{14} - 389 \beta_{13} - 829 \beta_{12} - 333 \beta_{11} + \cdots + 469 \beta_{8} ) / 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -4\beta_{7} + 6\beta_{6} + 50\beta_{5} + 80\beta_{4} - 92\beta_{3} - 294\beta_{2} + 743\beta _1 - 7116 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1820 \beta_{15} - 9427 \beta_{14} + 8167 \beta_{13} + 23077 \beta_{12} + 12207 \beta_{11} + \cdots - 12207 \beta_{8} ) / 10 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 208 \beta_{7} - 322 \beta_{6} - 1987 \beta_{5} - 2494 \beta_{4} + 3146 \beta_{3} + 12203 \beta_{2} + \cdots + 174960 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 61420 \beta_{15} + 227733 \beta_{14} - 174173 \beta_{13} - 633153 \beta_{12} - 434197 \beta_{11} + \cdots + 323613 \beta_{8} ) / 10 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 7896 \beta_{7} + 12404 \beta_{6} + 70568 \beta_{5} + 71376 \beta_{4} - 96784 \beta_{3} + \cdots - 4428218 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1867400 \beta_{15} - 5693643 \beta_{14} + 3752243 \beta_{13} + 17269933 \beta_{12} + \cdots - 8665443 \beta_{8} ) / 10 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 266064 \beta_{7} - 419396 \beta_{6} - 2342961 \beta_{5} - 1972162 \beta_{4} + 2838266 \beta_{3} + \cdots + 114512018 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 54279760 \beta_{15} + 145981609 \beta_{14} - 81219229 \beta_{13} - 470601909 \beta_{12} + \cdots + 233615149 \beta_{8} ) / 10 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 8452620 \beta_{7} + 13263210 \beta_{6} + 74507034 \beta_{5} + 53753448 \beta_{4} - 81274092 \beta_{3} + \cdots - 3010102032 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 1545805860 \beta_{15} - 3813456811 \beta_{14} + 1756367431 \beta_{13} + 12845602501 \beta_{12} + \cdots - 6331355551 \beta_{8} ) / 10 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
− 4.91331i
4.91331i
4.83075i
− 4.83075i
3.79130i
− 3.79130i
− 0.289794i
0.289794i
1.63349i
− 1.63349i
1.74738i
− 1.74738i
− 4.11126i
4.11126i
5.33303i
− 5.33303i
−4.91331 − 10.2724i 16.1406 0 50.4717i 21.7147i −39.9972 −78.5231 0
101.2 −4.91331 10.2724i 16.1406 0 − 50.4717i − 21.7147i −39.9972 −78.5231 0
101.3 −4.83075 − 1.17308i 15.3362 0 5.66685i 17.9463i −35.4393 25.6239 0
101.4 −4.83075 1.17308i 15.3362 0 − 5.66685i − 17.9463i −35.4393 25.6239 0
101.5 −3.79130 − 5.21292i 6.37398 0 19.7638i − 18.4051i 6.16475 −0.174523 0
101.6 −3.79130 5.21292i 6.37398 0 − 19.7638i 18.4051i 6.16475 −0.174523 0
101.7 −0.289794 − 2.72824i −7.91602 0 0.790629i 29.7500i 4.61237 19.5567 0
101.8 −0.289794 2.72824i −7.91602 0 − 0.790629i − 29.7500i 4.61237 19.5567 0
101.9 1.63349 − 9.29997i −5.33171 0 − 15.1914i − 29.4708i −21.7772 −59.4894 0
101.10 1.63349 9.29997i −5.33171 0 15.1914i 29.4708i −21.7772 −59.4894 0
101.11 1.74738 − 0.322521i −4.94666 0 − 0.563566i − 7.55169i −22.6227 26.8960 0
101.12 1.74738 0.322521i −4.94666 0 0.563566i 7.55169i −22.6227 26.8960 0
101.13 4.11126 − 3.82687i 8.90247 0 − 15.7333i − 15.9229i 3.71030 12.3551 0
101.14 4.11126 3.82687i 8.90247 0 15.7333i 15.9229i 3.71030 12.3551 0
101.15 5.33303 − 5.58969i 20.4412 0 − 29.8100i 7.17613i 66.3491 −4.24466 0
101.16 5.33303 5.58969i 20.4412 0 29.8100i − 7.17613i 66.3491 −4.24466 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.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.d.g 16
5.b even 2 1 85.4.d.b ✓ 16
5.c odd 4 1 425.4.c.e 16
5.c odd 4 1 425.4.c.f 16
15.d odd 2 1 765.4.g.c 16
17.b even 2 1 inner 425.4.d.g 16
85.c even 2 1 85.4.d.b ✓ 16
85.g odd 4 1 425.4.c.e 16
85.g odd 4 1 425.4.c.f 16
85.j even 4 1 1445.4.a.o 8
85.j even 4 1 1445.4.a.p 8
255.h odd 2 1 765.4.g.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.4.d.b ✓ 16 5.b even 2 1
85.4.d.b ✓ 16 85.c even 2 1
425.4.c.e 16 5.c odd 4 1
425.4.c.e 16 85.g odd 4 1
425.4.c.f 16 5.c odd 4 1
425.4.c.f 16 85.g odd 4 1
425.4.d.g 16 1.a even 1 1 trivial
425.4.d.g 16 17.b even 2 1 inner
765.4.g.c 16 15.d odd 2 1
765.4.g.c 16 255.h odd 2 1
1445.4.a.o 8 85.j even 4 1
1445.4.a.p 8 85.j even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(425, [\chi])\):

\( T_{2}^{8} + T_{2}^{7} - 56T_{2}^{6} - 38T_{2}^{5} + 975T_{2}^{4} + 101T_{2}^{3} - 5352T_{2}^{2} + 4096T_{2} + 1632 \) Copy content Toggle raw display
\( T_{3}^{16} + 274 T_{3}^{14} + 27237 T_{3}^{12} + 1231368 T_{3}^{10} + 27204604 T_{3}^{8} + \cdots + 120912016 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + T^{7} - 56 T^{6} + \cdots + 1632)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 120912016 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots - 1309766957360)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 33\!\cdots\!21 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots - 747381242730240)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots - 12\!\cdots\!12)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots - 20\!\cdots\!76)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots - 79\!\cdots\!84)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 18\!\cdots\!40)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 25\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 39\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots - 97\!\cdots\!04)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 24\!\cdots\!80)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 33\!\cdots\!56 \) Copy content Toggle raw display
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