Properties

Label 35.6.a.c
Level $35$
Weight $6$
Character orbit 35.a
Self dual yes
Analytic conductor $5.613$
Analytic rank $0$
Dimension $3$
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] = [35,6,Mod(1,35)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(35, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("35.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 35.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: \(5.61343369345\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.577880.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 98x - 232 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 2) q^{2} + (\beta_{2} + 9) q^{3} + (2 \beta_{2} + 38) q^{4} - 25 q^{5} + ( - 6 \beta_{2} + 17 \beta_1 - 34) q^{6} + 49 q^{7} + ( - 12 \beta_{2} + 22 \beta_1 - 44) q^{8} + (9 \beta_{2} - 24 \beta_1 + 166) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 2) q^{2} + (\beta_{2} + 9) q^{3} + (2 \beta_{2} + 38) q^{4} - 25 q^{5} + ( - 6 \beta_{2} + 17 \beta_1 - 34) q^{6} + 49 q^{7} + ( - 12 \beta_{2} + 22 \beta_1 - 44) q^{8} + (9 \beta_{2} - 24 \beta_1 + 166) q^{9} + ( - 25 \beta_1 + 50) q^{10} + ( - 7 \beta_{2} - 8 \beta_1 - 67) q^{11} + (38 \beta_{2} - 48 \beta_1 + 998) q^{12} + ( - 5 \beta_{2} - 32 \beta_1 + 629) q^{13} + (49 \beta_1 - 98) q^{14} + ( - 25 \beta_{2} - 225) q^{15} + (52 \beta_{2} - 96 \beta_1 + 516) q^{16} + (\beta_{2} - 96 \beta_1 - 61) q^{17} + ( - 102 \beta_{2} + 190 \beta_1 - 2060) q^{18} + (42 \beta_{2} + 152 \beta_1 + 418) q^{19} + ( - 50 \beta_{2} - 950) q^{20} + (49 \beta_{2} + 441) q^{21} + (26 \beta_{2} - 139 \beta_1 - 282) q^{22} + (58 \beta_{2} - 168 \beta_1 + 1082) q^{23} + ( - 132 \beta_{2} + 662 \beta_1 - 4684) q^{24} + 625 q^{25} + ( - 34 \beta_{2} + 525 \beta_1 - 3290) q^{26} + (19 \beta_{2} - 624 \beta_1 + 2643) q^{27} + (98 \beta_{2} + 1862) q^{28} + ( - 11 \beta_{2} + 72 \beta_1 - 3781) q^{29} + (150 \beta_{2} - 425 \beta_1 + 850) q^{30} + ( - 92 \beta_{2} + 200 \beta_1 + 3036) q^{31} + ( - 120 \beta_{2} + 36 \beta_1 - 6792) q^{32} + ( - 35 \beta_{2} + 32 \beta_1 - 2771) q^{33} + ( - 198 \beta_{2} - 245 \beta_1 - 6230) q^{34} - 1225 q^{35} + (704 \beta_{2} - 1728 \beta_1 + 12980) q^{36} + ( - 372 \beta_{2} - 1256 \beta_1 + 1890) q^{37} + (52 \beta_{2} + 1058 \beta_1 + 8524) q^{38} + (757 \beta_{2} - 424 \beta_1 + 4533) q^{39} + (300 \beta_{2} - 550 \beta_1 + 1100) q^{40} + ( - 734 \beta_{2} + 1176 \beta_1 + 3236) q^{41} + ( - 294 \beta_{2} + 833 \beta_1 - 1666) q^{42} + ( - 446 \beta_{2} + 1544 \beta_1 + 9222) q^{43} + ( - 210 \beta_{2} - 96 \beta_1 - 6882) q^{44} + ( - 225 \beta_{2} + 600 \beta_1 - 4150) q^{45} + ( - 684 \beta_{2} + 1210 \beta_1 - 14180) q^{46} + ( - 23 \beta_{2} + 1856 \beta_1 + 769) q^{47} + (900 \beta_{2} - 2880 \beta_1 + 23236) q^{48} + 2401 q^{49} + (625 \beta_1 - 1250) q^{50} + (323 \beta_{2} - 1656 \beta_1 + 1315) q^{51} + (1414 \beta_{2} - 1488 \beta_1 + 21646) q^{52} + (622 \beta_{2} + 1552 \beta_1 + 2604) q^{53} + ( - 1362 \beta_{2} + 1547 \beta_1 - 46774) q^{54} + (175 \beta_{2} + 200 \beta_1 + 1675) q^{55} + ( - 588 \beta_{2} + 1078 \beta_1 - 2156) q^{56} + ( - 190 \beta_{2} + 1576 \beta_1 + 15106) q^{57} + (210 \beta_{2} - 3725 \beta_1 + 12490) q^{58} + (1056 \beta_{2} + 64 \beta_1 + 8272) q^{59} + ( - 950 \beta_{2} + 1200 \beta_1 - 24950) q^{60} + ( - 1018 \beta_{2} - 120 \beta_1 + 2568) q^{61} + (952 \beta_{2} + 2700 \beta_1 + 8600) q^{62} + (441 \beta_{2} - 1176 \beta_1 + 8134) q^{63} + ( - 872 \beta_{2} - 4608 \beta_1 + 1368) q^{64} + (125 \beta_{2} + 800 \beta_1 - 15725) q^{65} + (274 \beta_{2} - 2987 \beta_1 + 8214) q^{66} + (648 \beta_{2} - 384 \beta_1 - 32308) q^{67} + (666 \beta_{2} - 5232 \beta_1 + 1410) q^{68} + (1754 \beta_{2} - 4248 \beta_1 + 31450) q^{69} + ( - 1225 \beta_1 + 2450) q^{70} + (1600 \beta_{2} - 2944 \beta_1 - 17072) q^{71} + ( - 4416 \beta_{2} + 9076 \beta_1 - 85352) q^{72} + ( - 2896 \beta_{2} + 560 \beta_1 + 10658) q^{73} + ( - 280 \beta_{2} - 3598 \beta_1 - 80724) q^{74} + (625 \beta_{2} + 5625) q^{75} + (460 \beta_{2} + 6192 \beta_1 + 38572) q^{76} + ( - 343 \beta_{2} - 392 \beta_1 - 3283) q^{77} + ( - 5390 \beta_{2} + 9741 \beta_1 - 49162) q^{78} + ( - 313 \beta_{2} + 4376 \beta_1 - 23953) q^{79} + ( - 1300 \beta_{2} + 2400 \beta_1 - 12900) q^{80} + (2952 \beta_{2} - 5232 \beta_1 - 335) q^{81} + (6756 \beta_{2} - 284 \beta_1 + 82888) q^{82} + ( - 788 \beta_{2} + 4912 \beta_1 - 7236) q^{83} + (1862 \beta_{2} - 2352 \beta_1 + 48902) q^{84} + ( - 25 \beta_{2} + 2400 \beta_1 + 1525) q^{85} + (5764 \beta_{2} + 8742 \beta_1 + 90596) q^{86} + ( - 4069 \beta_{2} + 1488 \beta_1 - 38789) q^{87} + (236 \beta_{2} - 4306 \beta_1 + 19812) q^{88} + ( - 2294 \beta_{2} - 4680 \beta_1 - 64972) q^{89} + (2550 \beta_{2} - 4750 \beta_1 + 51500) q^{90} + ( - 245 \beta_{2} - 1568 \beta_1 + 30821) q^{91} + (4668 \beta_{2} - 11856 \beta_1 + 84540) q^{92} + (2236 \beta_{2} + 5608 \beta_1 - 6052) q^{93} + (3850 \beta_{2} + 4297 \beta_1 + 121326) q^{94} + ( - 1050 \beta_{2} - 3800 \beta_1 - 10450) q^{95} + ( - 6936 \beta_{2} + 3492 \beta_1 - 101064) q^{96} + ( - 4915 \beta_{2} - 32 \beta_1 + 37135) q^{97} + (2401 \beta_1 - 4802) q^{98} + ( - 1198 \beta_{2} + 3328 \beta_1 - 20650) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 6 q^{2} + 26 q^{3} + 112 q^{4} - 75 q^{5} - 96 q^{6} + 147 q^{7} - 120 q^{8} + 489 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 6 q^{2} + 26 q^{3} + 112 q^{4} - 75 q^{5} - 96 q^{6} + 147 q^{7} - 120 q^{8} + 489 q^{9} + 150 q^{10} - 194 q^{11} + 2956 q^{12} + 1892 q^{13} - 294 q^{14} - 650 q^{15} + 1496 q^{16} - 184 q^{17} - 6078 q^{18} + 1212 q^{19} - 2800 q^{20} + 1274 q^{21} - 872 q^{22} + 3188 q^{23} - 13920 q^{24} + 1875 q^{25} - 9836 q^{26} + 7910 q^{27} + 5488 q^{28} - 11332 q^{29} + 2400 q^{30} + 9200 q^{31} - 20256 q^{32} - 8278 q^{33} - 18492 q^{34} - 3675 q^{35} + 38236 q^{36} + 6042 q^{37} + 25520 q^{38} + 12842 q^{39} + 3000 q^{40} + 10442 q^{41} - 4704 q^{42} + 28112 q^{43} - 20436 q^{44} - 12225 q^{45} - 41856 q^{46} + 2330 q^{47} + 68808 q^{48} + 7203 q^{49} - 3750 q^{50} + 3622 q^{51} + 63524 q^{52} + 7190 q^{53} - 138960 q^{54} + 4850 q^{55} - 5880 q^{56} + 45508 q^{57} + 37260 q^{58} + 23760 q^{59} - 73900 q^{60} + 8722 q^{61} + 24848 q^{62} + 23961 q^{63} + 4976 q^{64} - 47300 q^{65} + 24368 q^{66} - 97572 q^{67} + 3564 q^{68} + 92596 q^{69} + 7350 q^{70} - 52816 q^{71} - 251640 q^{72} + 34870 q^{73} - 241892 q^{74} + 16250 q^{75} + 115256 q^{76} - 9506 q^{77} - 142096 q^{78} - 71546 q^{79} - 37400 q^{80} - 3957 q^{81} + 241908 q^{82} - 20920 q^{83} + 144844 q^{84} + 4600 q^{85} + 266024 q^{86} - 112298 q^{87} + 59200 q^{88} - 192622 q^{89} + 151950 q^{90} + 92708 q^{91} + 248952 q^{92} - 20392 q^{93} + 360128 q^{94} - 30300 q^{95} - 296256 q^{96} + 116320 q^{97} - 14406 q^{98} - 60752 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 98x - 232 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - 4\nu - 66 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} + 4\beta _1 + 66 \) 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
−8.38673
−2.53323
10.9200
−10.3867 27.9421 75.8842 −25.0000 −290.227 49.0000 −455.813 537.760 259.668
1.2 −4.53323 −15.7249 −11.4498 −25.0000 71.2847 49.0000 196.968 4.27317 113.331
1.3 8.91996 13.7828 47.5657 −25.0000 122.942 49.0000 138.845 −53.0335 −222.999
\(n\): 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 35.6.a.c 3
3.b odd 2 1 315.6.a.i 3
4.b odd 2 1 560.6.a.q 3
5.b even 2 1 175.6.a.e 3
5.c odd 4 2 175.6.b.e 6
7.b odd 2 1 245.6.a.d 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.6.a.c 3 1.a even 1 1 trivial
175.6.a.e 3 5.b even 2 1
175.6.b.e 6 5.c odd 4 2
245.6.a.d 3 7.b odd 2 1
315.6.a.i 3 3.b odd 2 1
560.6.a.q 3 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + 6T_{2}^{2} - 86T_{2} - 420 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(35))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 6 T^{2} + \cdots - 420 \) Copy content Toggle raw display
$3$ \( T^{3} - 26 T^{2} + \cdots + 6056 \) Copy content Toggle raw display
$5$ \( (T + 25)^{3} \) Copy content Toggle raw display
$7$ \( (T - 49)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 194 T^{2} + \cdots - 3144468 \) Copy content Toggle raw display
$13$ \( T^{3} - 1892 T^{2} + \cdots - 171071930 \) Copy content Toggle raw display
$17$ \( T^{3} + 184 T^{2} + \cdots + 132716862 \) Copy content Toggle raw display
$19$ \( T^{3} - 1212 T^{2} + \cdots - 140259040 \) Copy content Toggle raw display
$23$ \( T^{3} - 3188 T^{2} + \cdots + 125424384 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots + 51669601050 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots + 8818293376 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots + 1043894647208 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots + 4748673370848 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots + 4763987701360 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots - 1072310433384 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots + 515502653472 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots + 7000620748800 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots - 1590789613952 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots + 26595134017984 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots - 77522711777280 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots + 11550435576632 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots - 40598762145400 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots - 529374252288 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots - 31660963259040 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots + 117394067785166 \) Copy content Toggle raw display
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