Properties

Label 49.10.a.b
Level $49$
Weight $10$
Character orbit 49.a
Self dual yes
Analytic conductor $25.237$
Analytic rank $0$
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] = [49,10,Mod(1,49)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(49, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("49.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 49.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: \(25.2367559720\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{193}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 7)
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 \(\beta = \sqrt{193}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 3) q^{2} + ( - 11 \beta + 43) q^{3} + (6 \beta - 310) q^{4} + (95 \beta + 1119) q^{5} + ( - 10 \beta + 1994) q^{6} + (804 \beta + 1308) q^{8} + ( - 946 \beta + 5519) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 3) q^{2} + ( - 11 \beta + 43) q^{3} + (6 \beta - 310) q^{4} + (95 \beta + 1119) q^{5} + ( - 10 \beta + 1994) q^{6} + (804 \beta + 1308) q^{8} + ( - 946 \beta + 5519) q^{9} + ( - 1404 \beta - 21692) q^{10} + ( - 3326 \beta + 17658) q^{11} + (3668 \beta - 26068) q^{12} + ( - 10899 \beta + 13265) q^{13} + ( - 8224 \beta - 153568) q^{15} + ( - 6792 \beta - 376) q^{16} + ( - 9426 \beta + 231960) q^{17} + ( - 2681 \beta + 166021) q^{18} + (1887 \beta + 462713) q^{19} + ( - 22736 \beta - 236880) q^{20} + ( - 7680 \beta + 588944) q^{22} + ( - 38088 \beta + 389064) q^{23} + (20184 \beta - 1650648) q^{24} + (212610 \beta + 1040861) q^{25} + (19432 \beta + 2063712) q^{26} + (115126 \beta + 1399306) q^{27} + ( - 94682 \beta - 5001792) q^{29} + (178240 \beta + 2047936) q^{30} + (161430 \beta - 1233630) q^{31} + ( - 390896 \beta + 642288) q^{32} + ( - 337256 \beta + 7820392) q^{33} + ( - 203682 \beta + 1123338) q^{34} + (326374 \beta - 2806358) q^{36} + ( - 248130 \beta + 15367776) q^{37} + ( - 468374 \beta - 1752330) q^{38} + ( - 614572 \beta + 23708972) q^{39} + (1023936 \beta + 16204992) q^{40} + (860818 \beta + 9551724) q^{41} + (1048278 \beta + 2032550) q^{43} + (1137008 \beta - 9325488) q^{44} + ( - 534269 \beta - 11169149) q^{45} + ( - 274800 \beta + 6183792) q^{46} + ( - 1033182 \beta + 41097510) q^{47} + ( - 287920 \beta + 14403248) q^{48} + ( - 1678691 \beta - 44156313) q^{50} + ( - 2956878 \beta + 29985678) q^{51} + (3458280 \beta - 16733192) q^{52} + (4685568 \beta - 27594906) q^{53} + ( - 1744684 \beta - 26417236) q^{54} + ( - 2044284 \beta - 41222908) q^{55} + ( - 5008702 \beta + 15890558) q^{57} + (5285838 \beta + 33279002) q^{58} + ( - 1563825 \beta + 3534609) q^{59} + (1628032 \beta + 38082688) q^{60} + (3395319 \beta - 22158193) q^{61} + (749340 \beta - 27455100) q^{62} + (4007904 \beta + 73708576) q^{64} + ( - 10935806 \beta - 184989630) q^{65} + ( - 6808624 \beta + 41629232) q^{66} + ( - 7026216 \beta - 120960668) q^{67} + (4313820 \beta - 82822908) q^{68} + ( - 5917488 \beta + 97590576) q^{69} + (15075900 \beta + 103246908) q^{71} + (3199908 \beta - 139573860) q^{72} + (2840484 \beta + 249576594) q^{73} + ( - 14623386 \beta + 1785762) q^{74} + ( - 2307241 \beta - 406614007) q^{75} + (2191308 \beta - 141255884) q^{76} + ( - 21865256 \beta + 47485480) q^{78} + (16873716 \beta + 234267548) q^{79} + ( - 7635968 \beta - 124952064) q^{80} + (8178170 \beta - 292872817) q^{81} + ( - 12134178 \beta - 194793046) q^{82} + (21562275 \beta - 222011979) q^{83} + (11488506 \beta + 86737530) q^{85} + ( - 5177384 \beta - 208415304) q^{86} + (50948386 \beta - 14067170) q^{87} + (9846624 \beta - 493005408) q^{88} + ( - 1406968 \beta - 318133698) q^{89} + (12771956 \beta + 136621364) q^{90} + (14141664 \beta - 164715744) q^{92} + (20511420 \beta - 395761980) q^{93} + ( - 37997964 \beta + 76111596) q^{94} + (46069288 \beta + 552373992) q^{95} + ( - 23873696 \beta + 857490592) q^{96} + ( - 5731530 \beta + 816358032) q^{97} + ( - 35060662 \beta + 704708930) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{2} + 86 q^{3} - 620 q^{4} + 2238 q^{5} + 3988 q^{6} + 2616 q^{8} + 11038 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{2} + 86 q^{3} - 620 q^{4} + 2238 q^{5} + 3988 q^{6} + 2616 q^{8} + 11038 q^{9} - 43384 q^{10} + 35316 q^{11} - 52136 q^{12} + 26530 q^{13} - 307136 q^{15} - 752 q^{16} + 463920 q^{17} + 332042 q^{18} + 925426 q^{19} - 473760 q^{20} + 1177888 q^{22} + 778128 q^{23} - 3301296 q^{24} + 2081722 q^{25} + 4127424 q^{26} + 2798612 q^{27} - 10003584 q^{29} + 4095872 q^{30} - 2467260 q^{31} + 1284576 q^{32} + 15640784 q^{33} + 2246676 q^{34} - 5612716 q^{36} + 30735552 q^{37} - 3504660 q^{38} + 47417944 q^{39} + 32409984 q^{40} + 19103448 q^{41} + 4065100 q^{43} - 18650976 q^{44} - 22338298 q^{45} + 12367584 q^{46} + 82195020 q^{47} + 28806496 q^{48} - 88312626 q^{50} + 59971356 q^{51} - 33466384 q^{52} - 55189812 q^{53} - 52834472 q^{54} - 82445816 q^{55} + 31781116 q^{57} + 66558004 q^{58} + 7069218 q^{59} + 76165376 q^{60} - 44316386 q^{61} - 54910200 q^{62} + 147417152 q^{64} - 369979260 q^{65} + 83258464 q^{66} - 241921336 q^{67} - 165645816 q^{68} + 195181152 q^{69} + 206493816 q^{71} - 279147720 q^{72} + 499153188 q^{73} + 3571524 q^{74} - 813228014 q^{75} - 282511768 q^{76} + 94970960 q^{78} + 468535096 q^{79} - 249904128 q^{80} - 585745634 q^{81} - 389586092 q^{82} - 444023958 q^{83} + 173475060 q^{85} - 416830608 q^{86} - 28134340 q^{87} - 986010816 q^{88} - 636267396 q^{89} + 273242728 q^{90} - 329431488 q^{92} - 791523960 q^{93} + 152223192 q^{94} + 1104747984 q^{95} + 1714981184 q^{96} + 1632716064 q^{97} + 1409417860 q^{99}+O(q^{100}) \) 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
7.44622
−6.44622
−16.8924 −109.817 −226.645 2438.78 1855.08 0 12477.5 −7623.25 −41197.0
1.2 10.8924 195.817 −393.355 −200.782 2132.92 0 −9861.52 18661.3 −2187.01
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 49.10.a.b 2
7.b odd 2 1 7.10.a.a 2
7.c even 3 2 49.10.c.b 4
7.d odd 6 2 49.10.c.c 4
21.c even 2 1 63.10.a.d 2
28.d even 2 1 112.10.a.e 2
35.c odd 2 1 175.10.a.b 2
35.f even 4 2 175.10.b.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.10.a.a 2 7.b odd 2 1
49.10.a.b 2 1.a even 1 1 trivial
49.10.c.b 4 7.c even 3 2
49.10.c.c 4 7.d odd 6 2
63.10.a.d 2 21.c even 2 1
112.10.a.e 2 28.d even 2 1
175.10.a.b 2 35.c odd 2 1
175.10.b.b 4 35.f even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(49))\):

\( T_{2}^{2} + 6T_{2} - 184 \) Copy content Toggle raw display
\( T_{3}^{2} - 86T_{3} - 21504 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 6T - 184 \) Copy content Toggle raw display
$3$ \( T^{2} - 86T - 21504 \) Copy content Toggle raw display
$5$ \( T^{2} - 2238 T - 489664 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 35316 T - 1823214304 \) Copy content Toggle raw display
$13$ \( T^{2} - 26530 T - 22750162568 \) Copy content Toggle raw display
$17$ \( T^{2} - 463920 T + 36657492732 \) Copy content Toggle raw display
$19$ \( T^{2} - 925426 T + 213416091952 \) Copy content Toggle raw display
$23$ \( T^{2} - 778128 T - 128613482496 \) Copy content Toggle raw display
$29$ \( T^{2} + 10003584 T + 23287739754332 \) Copy content Toggle raw display
$31$ \( T^{2} + 2467260 T - 3507668488800 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 224285819284476 \) Copy content Toggle raw display
$41$ \( T^{2} - 19103448 T - 51779041048756 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 207953886197312 \) Copy content Toggle raw display
$47$ \( T^{2} - 82195020 T + 14\!\cdots\!68 \) Copy content Toggle raw display
$53$ \( T^{2} + 55189812 T - 34\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 459497424927744 \) Copy content Toggle raw display
$61$ \( T^{2} + 44316386 T - 17\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{2} + 241921336 T + 51\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{2} - 206493816 T - 33\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{2} - 499153188 T + 60\!\cdots\!28 \) Copy content Toggle raw display
$79$ \( T^{2} - 468535096 T - 70118242258304 \) Copy content Toggle raw display
$83$ \( T^{2} + 444023958 T - 40\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{2} + 636267396 T + 10\!\cdots\!72 \) Copy content Toggle raw display
$97$ \( T^{2} - 1632716064 T + 66\!\cdots\!24 \) Copy content Toggle raw display
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