Properties

Label 245.8.a.b
Level $245$
Weight $8$
Character orbit 245.a
Self dual yes
Analytic conductor $76.534$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [245,8,Mod(1,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 245.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: \(76.5343312436\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{11}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 35)
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 \(\beta = 2\sqrt{11}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 8) q^{2} + (6 \beta + 15) q^{3} + (16 \beta - 20) q^{4} - 125 q^{5} + (63 \beta + 384) q^{6} + ( - 20 \beta - 480) q^{8} + (180 \beta - 378) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 8) q^{2} + (6 \beta + 15) q^{3} + (16 \beta - 20) q^{4} - 125 q^{5} + (63 \beta + 384) q^{6} + ( - 20 \beta - 480) q^{8} + (180 \beta - 378) q^{9} + ( - 125 \beta - 1000) q^{10} + ( - 380 \beta - 3953) q^{11} + (120 \beta + 3924) q^{12} + (418 \beta + 8909) q^{13} + ( - 750 \beta - 1875) q^{15} + ( - 2688 \beta - 2160) q^{16} + ( - 2210 \beta + 1199) q^{17} + (1062 \beta + 4896) q^{18} + ( - 5542 \beta + 1806) q^{19} + ( - 2000 \beta + 2500) q^{20} + ( - 6993 \beta - 48344) q^{22} + (10690 \beta + 6922) q^{23} + ( - 3180 \beta - 12480) q^{24} + 15625 q^{25} + (12253 \beta + 89664) q^{26} + ( - 12690 \beta + 9045) q^{27} + ( - 23772 \beta - 63449) q^{29} + ( - 7875 \beta - 48000) q^{30} + (22554 \beta - 126384) q^{31} + ( - 21104 \beta - 74112) q^{32} + ( - 29418 \beta - 159615) q^{33} + ( - 16481 \beta - 87648) q^{34} + ( - 9648 \beta + 134280) q^{36} + ( - 43638 \beta - 132930) q^{37} + ( - 42530 \beta - 229400) q^{38} + (59724 \beta + 243987) q^{39} + (2500 \beta + 60000) q^{40} + ( - 37354 \beta + 55960) q^{41} + ( - 24578 \beta + 473786) q^{43} + ( - 55648 \beta - 188460) q^{44} + ( - 22500 \beta + 47250) q^{45} + (92442 \beta + 525736) q^{46} + (56742 \beta - 135637) q^{47} + ( - 53280 \beta - 742032) q^{48} + (15625 \beta + 125000) q^{50} + ( - 25956 \beta - 565455) q^{51} + (134184 \beta + 116092) q^{52} + ( - 65224 \beta - 633896) q^{53} + ( - 92475 \beta - 486000) q^{54} + (47500 \beta + 494125) q^{55} + ( - 72294 \beta - 1435998) q^{57} + ( - 253625 \beta - 1553560) q^{58} + ( - 170640 \beta + 680060) q^{59} + ( - 15000 \beta - 490500) q^{60} + ( - 11334 \beta + 906840) q^{61} + (54048 \beta - 18696) q^{62} + (101120 \beta - 1244992) q^{64} + ( - 52250 \beta - 1113625) q^{65} + ( - 394959 \beta - 2571312) q^{66} + (506344 \beta - 1094656) q^{67} + (63384 \beta - 1579820) q^{68} + (201882 \beta + 2925990) q^{69} + ( - 222048 \beta - 747464) q^{71} + ( - 78840 \beta + 23040) q^{72} + ( - 212396 \beta - 3584894) q^{73} + ( - 482034 \beta - 2983512) q^{74} + (93750 \beta + 234375) q^{75} + (139736 \beta - 3937688) q^{76} + (721779 \beta + 4579752) q^{78} + (187504 \beta - 3971487) q^{79} + (336000 \beta + 270000) q^{80} + ( - 529740 \beta - 2387799) q^{81} + ( - 242872 \beta - 1195896) q^{82} + (939444 \beta + 152356) q^{83} + (276250 \beta - 149875) q^{85} + (277162 \beta + 2708856) q^{86} + ( - 737274 \beta - 7227543) q^{87} + (261460 \beta + 2231840) q^{88} + ( - 247538 \beta + 8971764) q^{89} + ( - 132750 \beta - 612000) q^{90} + ( - 103048 \beta + 7387320) q^{92} + ( - 419994 \beta + 4058496) q^{93} + (318299 \beta + 1411552) q^{94} + (692750 \beta - 225750) q^{95} + ( - 761232 \beta - 6683136) q^{96} + (680782 \beta - 2129037) q^{97} + ( - 567900 \beta - 1515366) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 16 q^{2} + 30 q^{3} - 40 q^{4} - 250 q^{5} + 768 q^{6} - 960 q^{8} - 756 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 16 q^{2} + 30 q^{3} - 40 q^{4} - 250 q^{5} + 768 q^{6} - 960 q^{8} - 756 q^{9} - 2000 q^{10} - 7906 q^{11} + 7848 q^{12} + 17818 q^{13} - 3750 q^{15} - 4320 q^{16} + 2398 q^{17} + 9792 q^{18} + 3612 q^{19} + 5000 q^{20} - 96688 q^{22} + 13844 q^{23} - 24960 q^{24} + 31250 q^{25} + 179328 q^{26} + 18090 q^{27} - 126898 q^{29} - 96000 q^{30} - 252768 q^{31} - 148224 q^{32} - 319230 q^{33} - 175296 q^{34} + 268560 q^{36} - 265860 q^{37} - 458800 q^{38} + 487974 q^{39} + 120000 q^{40} + 111920 q^{41} + 947572 q^{43} - 376920 q^{44} + 94500 q^{45} + 1051472 q^{46} - 271274 q^{47} - 1484064 q^{48} + 250000 q^{50} - 1130910 q^{51} + 232184 q^{52} - 1267792 q^{53} - 972000 q^{54} + 988250 q^{55} - 2871996 q^{57} - 3107120 q^{58} + 1360120 q^{59} - 981000 q^{60} + 1813680 q^{61} - 37392 q^{62} - 2489984 q^{64} - 2227250 q^{65} - 5142624 q^{66} - 2189312 q^{67} - 3159640 q^{68} + 5851980 q^{69} - 1494928 q^{71} + 46080 q^{72} - 7169788 q^{73} - 5967024 q^{74} + 468750 q^{75} - 7875376 q^{76} + 9159504 q^{78} - 7942974 q^{79} + 540000 q^{80} - 4775598 q^{81} - 2391792 q^{82} + 304712 q^{83} - 299750 q^{85} + 5417712 q^{86} - 14455086 q^{87} + 4463680 q^{88} + 17943528 q^{89} - 1224000 q^{90} + 14774640 q^{92} + 8116992 q^{93} + 2823104 q^{94} - 451500 q^{95} - 13366272 q^{96} - 4258074 q^{97} - 3030732 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
−3.31662
3.31662
1.36675 −24.7995 −126.132 −125.000 −33.8947 0 −347.335 −1571.98 −170.844
1.2 14.6332 54.7995 86.1320 −125.000 801.895 0 −612.665 815.985 −1829.16
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(7\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.8.a.b 2
7.b odd 2 1 35.8.a.a 2
21.c even 2 1 315.8.a.c 2
28.d even 2 1 560.8.a.i 2
35.c odd 2 1 175.8.a.b 2
35.f even 4 2 175.8.b.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.8.a.a 2 7.b odd 2 1
175.8.a.b 2 35.c odd 2 1
175.8.b.c 4 35.f even 4 2
245.8.a.b 2 1.a even 1 1 trivial
315.8.a.c 2 21.c even 2 1
560.8.a.i 2 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(245))\):

\( T_{2}^{2} - 16T_{2} + 20 \) Copy content Toggle raw display
\( T_{3}^{2} - 30T_{3} - 1359 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 16T + 20 \) Copy content Toggle raw display
$3$ \( T^{2} - 30T - 1359 \) Copy content Toggle raw display
$5$ \( (T + 125)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 7906 T + 9272609 \) Copy content Toggle raw display
$13$ \( T^{2} - 17818 T + 71682425 \) Copy content Toggle raw display
$17$ \( T^{2} - 2398 T - 213462799 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots - 1348143980 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 4980234316 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 20838975695 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 6409132848 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 66117717036 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 58262616304 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 197893738100 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots - 123267405047 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 214640651072 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 818710818800 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 816706565136 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 10082635080448 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 1610731398080 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 10866534315332 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 14225767990465 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 38809208931248 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 77796446568160 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 15859623239687 \) Copy content Toggle raw display
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