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Properties

Label 49.6.c.h
Level $49$
Weight $6$
Character orbit 49.c
Analytic conductor $7.859$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [49,6,Mod(18,49)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(49, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("49.18");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 49.c (of order \(3\), degree \(2\), 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: \(7.85880717084\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.54095201243136.19
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 102x^{6} + 320x^{5} + 4283x^{4} - 9104x^{3} - 85298x^{2} + 89904x + 714364 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{6} - \beta_{3} - 2 \beta_1 + 3) q^{2} + (\beta_{7} - \beta_{5} - 2 \beta_{4} - 2 \beta_{2}) q^{3} - 5 \beta_{6} q^{4} + ( - 2 \beta_{5} - 8 \beta_{2}) q^{5} + (11 \beta_{7} - 11 \beta_{4}) q^{6} + ( - 17 \beta_{3} - 59) q^{8} + (48 \beta_{6} + 48 \beta_{3} + 31 \beta_1 - 79) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} - \beta_{3} - 2 \beta_1 + 3) q^{2} + (\beta_{7} - \beta_{5} - 2 \beta_{4} - 2 \beta_{2}) q^{3} - 5 \beta_{6} q^{4} + ( - 2 \beta_{5} - 8 \beta_{2}) q^{5} + (11 \beta_{7} - 11 \beta_{4}) q^{6} + ( - 17 \beta_{3} - 59) q^{8} + (48 \beta_{6} + 48 \beta_{3} + 31 \beta_1 - 79) q^{9} + (38 \beta_{7} - 38 \beta_{5} - 30 \beta_{4} - 30 \beta_{2}) q^{10} + ( - 12 \beta_{6} + 494 \beta_1) q^{11} + (45 \beta_{5} + 35 \beta_{2}) q^{12} + (64 \beta_{7} + 56 \beta_{4}) q^{13} + (136 \beta_{3} - 1092) q^{15} + ( - 135 \beta_{6} - 135 \beta_{3} - 324 \beta_1 + 459) q^{16} + ( - 7 \beta_{7} + 7 \beta_{5} + 152 \beta_{4} + 152 \beta_{2}) q^{17} + (175 \beta_{6} + 1406 \beta_1) q^{18} + (53 \beta_{5} - 78 \beta_{2}) q^{19} + (170 \beta_{7} - 70 \beta_{4}) q^{20} + ( - 458 \beta_{3} + 1110) q^{22} + ( - 344 \beta_{6} - 344 \beta_{3} - 1612 \beta_1 + 1956) q^{23} + (77 \beta_{7} - 77 \beta_{5} + 33 \beta_{4} + 33 \beta_{2}) q^{24} + ( - 320 \beta_{6} - 531 \beta_1) q^{25} + ( - 32 \beta_{5} + 336 \beta_{2}) q^{26} + ( - 220 \beta_{7} - 88 \beta_{4}) q^{27} + (784 \beta_{3} - 1230) q^{29} + (1364 \beta_{6} + 1364 \beta_{3} + 5720 \beta_1 - 7084) q^{30} + ( - 246 \beta_{7} + 246 \beta_{5} - 772 \beta_{4} - 772 \beta_{2}) q^{31} + ( - 185 \beta_{6} - 6860 \beta_1) q^{32} + ( - 386 \beta_{5} - 904 \beta_{2}) q^{33} + ( - 629 \beta_{7} + 353 \beta_{4}) q^{34} + ( - 395 \beta_{3} + 7115) q^{36} + (48 \beta_{6} + 48 \beta_{3} - 2326 \beta_1 + 2278) q^{37} + (153 \beta_{7} - 153 \beta_{5} + 215 \beta_{4} + 215 \beta_{2}) q^{38} + (1232 \beta_{6} - 1232 \beta_1) q^{39} + ( - 426 \beta_{5} + 370 \beta_{2}) q^{40} + ( - 1103 \beta_{7} - 224 \beta_{4}) q^{41} + ( - 980 \beta_{3} + 5602) q^{43} + ( - 2410 \beta_{6} - 2410 \beta_{3} + 1680 \beta_1 + 730) q^{44} + ( - 1694 \beta_{7} + 1694 \beta_{5} + 920 \beta_{4} + 920 \beta_{2}) q^{45} + ( - 2644 \beta_{6} - 12856 \beta_1) q^{46} + (590 \beta_{5} + 1644 \beta_{2}) q^{47} + (1539 \beta_{7} - 1593 \beta_{4}) q^{48} + (1491 \beta_{3} - 11513) q^{50} + ( - 1716 \beta_{6} - 1716 \beta_{3} - 15994 \beta_1 + 17710) q^{51} + (800 \beta_{7} - 800 \beta_{5} + 2240 \beta_{4} + 2240 \beta_{2}) q^{52} + (2592 \beta_{6} + 24434 \beta_1) q^{53} + ( - 308 \beta_{5} - 1364 \beta_{2}) q^{54} + ( - 580 \beta_{7} + 3784 \beta_{4}) q^{55} + ( - 704 \beta_{3} - 3542) q^{57} + (2798 \beta_{6} + 2798 \beta_{3} + 22844 \beta_1 - 25642) q^{58} + (1425 \beta_{7} - 1425 \beta_{5} - 2914 \beta_{4} - 2914 \beta_{2}) q^{59} + (5460 \beta_{6} + 19040 \beta_1) q^{60} + (3326 \beta_{5} - 680 \beta_{2}) q^{61} + (2350 \beta_{7} + 178 \beta_{4}) q^{62} + (3095 \beta_{3} - 11627) q^{64} + (6496 \beta_{6} + 6496 \beta_{3} - 19040 \beta_1 + 12544) q^{65} + (4774 \beta_{7} - 4774 \beta_{5} - 4510 \beta_{4} - 4510 \beta_{2}) q^{66} + ( - 11632 \beta_{6} + 11540 \beta_1) q^{67} + ( - 3075 \beta_{5} - 245 \beta_{2}) q^{68} + (4708 \beta_{7} - 5632 \beta_{4}) q^{69} + ( - 4312 \beta_{3} - 36300) q^{71} + ( - 2305 \beta_{6} - 2305 \beta_{3} + 20492 \beta_1 - 18187) q^{72} + ( - 4407 \beta_{7} + 4407 \beta_{5} + 5080 \beta_{4} + \cdots + 5080 \beta_{2}) q^{73}+ \cdots + (22764 \beta_{3} - 21950) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 10 q^{2} - 10 q^{4} - 540 q^{8} - 220 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 10 q^{2} - 10 q^{4} - 540 q^{8} - 220 q^{9} + 1952 q^{11} - 8192 q^{15} + 1566 q^{16} + 5974 q^{18} + 7048 q^{22} + 7136 q^{23} - 2764 q^{25} - 6704 q^{29} - 25608 q^{30} - 27810 q^{32} + 55340 q^{36} + 9208 q^{37} - 2464 q^{39} + 40896 q^{43} - 1900 q^{44} - 56712 q^{46} - 86140 q^{50} + 67408 q^{51} + 102920 q^{53} - 31152 q^{57} - 96972 q^{58} + 87080 q^{60} - 80636 q^{64} + 63168 q^{65} + 22896 q^{67} - 307648 q^{71} - 77358 q^{72} - 17596 q^{74} + 266112 q^{78} + 90688 q^{79} + 17204 q^{81} + 545312 q^{85} + 161860 q^{86} - 154812 q^{88} - 424400 q^{92} - 247760 q^{93} - 108224 q^{95} - 84544 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 102x^{6} + 320x^{5} + 4283x^{4} - 9104x^{3} - 85298x^{2} + 89904x + 714364 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{6} - 6\nu^{5} - 269\nu^{4} + 548\nu^{3} + 10445\nu^{2} - 10720\nu - 126196 ) / 26418 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1138 \nu^{7} + 17720 \nu^{6} + 83725 \nu^{5} - 2169924 \nu^{4} - 2195078 \nu^{3} + 171730312 \nu^{2} - 72148073 \nu - 3454764460 ) / 181452033 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2276 \nu^{7} - 7966 \nu^{6} - 249872 \nu^{5} + 644595 \nu^{4} + 11918032 \nu^{3} - 18525626 \nu^{2} - 365868560 \nu + 276769577 ) / 181452033 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11543 \nu^{7} + 802804 \nu^{6} - 1498300 \nu^{5} - 59253540 \nu^{4} + 99982619 \nu^{3} + 1516065188 \nu^{2} - 1433696680 \nu - 12421280816 ) / 362904066 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2276 \nu^{7} - 7966 \nu^{6} - 249872 \nu^{5} + 644595 \nu^{4} + 11918032 \nu^{3} - 18525626 \nu^{2} - 184416527 \nu + 95317544 ) / 25921719 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 50560 \nu^{7} - 163223 \nu^{6} - 5273074 \nu^{5} + 11674431 \nu^{4} + 179862812 \nu^{3} - 206017366 \nu^{2} - 1958469688 \nu + 122390548 ) / 362904066 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1358 \nu^{7} + 4753 \nu^{6} + 92813 \nu^{5} - 243915 \nu^{4} - 2139991 \nu^{3} + 3456278 \nu^{2} + 13230308 \nu - 7199444 ) / 3049614 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - 7\beta_{3} + 7 ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} - 7\beta_{3} + 14\beta_{2} - 14\beta _1 + 203 ) / 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{7} - 42\beta_{6} + 87\beta_{5} - 203\beta_{3} + 21\beta_{2} + 399 ) / 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -8\beta_{7} - 84\beta_{6} + 228\beta_{5} + 56\beta_{4} - 399\beta_{3} + 812\beta_{2} - 2324\beta _1 + 6055 ) / 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 576 \beta_{7} - 3920 \beta_{6} + 4341 \beta_{5} + 140 \beta_{4} - 5943 \beta_{3} + 1995 \beta_{2} - 3920 \beta _1 + 17115 ) / 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 2256 \beta_{7} - 11550 \beta_{6} + 14766 \beta_{5} + 7952 \beta_{4} - 16835 \beta_{3} + 36330 \beta_{2} - 158760 \beta _1 + 175455 ) / 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 58394 \beta_{7} - 227066 \beta_{6} + 185151 \beta_{5} + 27342 \beta_{4} - 159551 \beta_{3} + 120197 \beta_{2} - 441784 \beta _1 + 615251 ) / 7 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1 + \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1
−4.10797 + 1.22474i
−5.52218 − 1.22474i
5.10797 − 1.22474i
6.52218 + 1.22474i
−4.10797 − 1.22474i
−5.52218 + 1.22474i
5.10797 + 1.22474i
6.52218 − 1.22474i
−1.40754 − 2.43792i −3.27401 + 5.67075i 12.0377 − 20.8499i 22.9955 + 39.8294i 18.4331 0 −157.856 100.062 + 173.312i 64.7341 − 112.123i
18.2 −1.40754 − 2.43792i 3.27401 − 5.67075i 12.0377 − 20.8499i −22.9955 − 39.8294i −18.4331 0 −157.856 100.062 + 173.312i −64.7341 + 112.123i
18.3 3.90754 + 6.76805i −11.7593 + 20.3677i −14.5377 + 25.1800i 37.1377 + 64.3243i −183.799 0 22.8562 −155.062 − 268.575i −290.234 + 502.699i
18.4 3.90754 + 6.76805i 11.7593 − 20.3677i −14.5377 + 25.1800i −37.1377 − 64.3243i 183.799 0 22.8562 −155.062 − 268.575i 290.234 − 502.699i
30.1 −1.40754 + 2.43792i −3.27401 − 5.67075i 12.0377 + 20.8499i 22.9955 − 39.8294i 18.4331 0 −157.856 100.062 − 173.312i 64.7341 + 112.123i
30.2 −1.40754 + 2.43792i 3.27401 + 5.67075i 12.0377 + 20.8499i −22.9955 + 39.8294i −18.4331 0 −157.856 100.062 − 173.312i −64.7341 − 112.123i
30.3 3.90754 − 6.76805i −11.7593 − 20.3677i −14.5377 − 25.1800i 37.1377 − 64.3243i −183.799 0 22.8562 −155.062 + 268.575i −290.234 − 502.699i
30.4 3.90754 − 6.76805i 11.7593 + 20.3677i −14.5377 − 25.1800i −37.1377 + 64.3243i 183.799 0 22.8562 −155.062 + 268.575i 290.234 + 502.699i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 18.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.6.c.h 8
7.b odd 2 1 inner 49.6.c.h 8
7.c even 3 1 49.6.a.g ✓ 4
7.c even 3 1 inner 49.6.c.h 8
7.d odd 6 1 49.6.a.g ✓ 4
7.d odd 6 1 inner 49.6.c.h 8
21.g even 6 1 441.6.a.z 4
21.h odd 6 1 441.6.a.z 4
28.f even 6 1 784.6.a.bf 4
28.g odd 6 1 784.6.a.bf 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.6.a.g ✓ 4 7.c even 3 1
49.6.a.g ✓ 4 7.d odd 6 1
49.6.c.h 8 1.a even 1 1 trivial
49.6.c.h 8 7.b odd 2 1 inner
49.6.c.h 8 7.c even 3 1 inner
49.6.c.h 8 7.d odd 6 1 inner
441.6.a.z 4 21.g even 6 1
441.6.a.z 4 21.h odd 6 1
784.6.a.bf 4 28.f even 6 1
784.6.a.bf 4 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(49, [\chi])\):

\( T_{2}^{4} - 5T_{2}^{3} + 47T_{2}^{2} + 110T_{2} + 484 \) Copy content Toggle raw display
\( T_{3}^{8} + 596T_{3}^{6} + 331500T_{3}^{4} + 14134736T_{3}^{2} + 562448656 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 5 T^{3} + 47 T^{2} + 110 T + 484)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + 596 T^{6} + \cdots + 562448656 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 136166867931136 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 976 T^{3} + \cdots + 54791573776)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 1260672 T^{2} + \cdots + 76158337024)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 2613668 T^{6} + \cdots + 28\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{8} + 1359892 T^{6} + \cdots + 32\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( (T^{4} - 3568 T^{3} + \cdots + 25707632896)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 1676 T - 16661788)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} + 85120848 T^{6} + \cdots + 37\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( (T^{4} - 4604 T^{3} + \cdots + 27395970301456)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 251093444 T^{2} + \cdots + 14\!\cdots\!56)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 10224 T - 998756)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + 349180624 T^{6} + \cdots + 29\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( (T^{4} - 51460 T^{3} + \cdots + 22\!\cdots\!64)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 1249753044 T^{6} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{8} + 2284246736 T^{6} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( (T^{4} - 11448 T^{3} + \cdots + 14\!\cdots\!04)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 76912 T + 953601968)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + 6121721124 T^{6} + \cdots + 43\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( (T^{4} - 45344 T^{3} + \cdots + 149515235584)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 9034370100 T^{2} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 7617937028 T^{6} + \cdots + 18\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( (T^{4} - 11859566628 T^{2} + \cdots + 11\!\cdots\!44)^{2} \) Copy content Toggle raw display
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