Properties

Label 35.4.a.b
Level $35$
Weight $4$
Character orbit 35.a
Self dual yes
Analytic conductor $2.065$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [35,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("35.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(2.06506685020\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 4) q^{2} + ( - 4 \beta + 1) q^{3} + (8 \beta + 10) q^{4} - 5 q^{5} + ( - 15 \beta - 4) q^{6} - 7 q^{7} + (34 \beta + 24) q^{8} + ( - 8 \beta + 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 4) q^{2} + ( - 4 \beta + 1) q^{3} + (8 \beta + 10) q^{4} - 5 q^{5} + ( - 15 \beta - 4) q^{6} - 7 q^{7} + (34 \beta + 24) q^{8} + ( - 8 \beta + 6) q^{9} + ( - 5 \beta - 20) q^{10} + ( - 32 \beta - 7) q^{11} + ( - 32 \beta - 54) q^{12} + (4 \beta + 25) q^{13} + ( - 7 \beta - 28) q^{14} + (20 \beta - 5) q^{15} + (96 \beta + 84) q^{16} + (44 \beta - 25) q^{17} + ( - 26 \beta + 8) q^{18} + (44 \beta + 18) q^{19} + ( - 40 \beta - 50) q^{20} + (28 \beta - 7) q^{21} + ( - 135 \beta - 92) q^{22} + ( - 68 \beta + 122) q^{23} + ( - 62 \beta - 248) q^{24} + 25 q^{25} + (41 \beta + 108) q^{26} + (76 \beta + 43) q^{27} + ( - 56 \beta - 70) q^{28} + (24 \beta - 13) q^{29} + (75 \beta + 20) q^{30} + ( - 180 \beta - 60) q^{31} + (196 \beta + 336) q^{32} + ( - 4 \beta + 249) q^{33} + (151 \beta - 12) q^{34} + 35 q^{35} + ( - 32 \beta - 68) q^{36} + ( - 60 \beta + 282) q^{37} + (194 \beta + 160) q^{38} + ( - 96 \beta - 7) q^{39} + ( - 170 \beta - 120) q^{40} + (124 \beta - 164) q^{41} + (105 \beta + 28) q^{42} + (68 \beta - 130) q^{43} + ( - 376 \beta - 582) q^{44} + (40 \beta - 30) q^{45} + ( - 150 \beta + 352) q^{46} + ( - 132 \beta - 175) q^{47} + ( - 240 \beta - 684) q^{48} + 49 q^{49} + (25 \beta + 100) q^{50} + (144 \beta - 377) q^{51} + (240 \beta + 314) q^{52} + (128 \beta - 28) q^{53} + (347 \beta + 324) q^{54} + (160 \beta + 35) q^{55} + ( - 238 \beta - 168) q^{56} + ( - 28 \beta - 334) q^{57} + (83 \beta - 4) q^{58} - 616 q^{59} + (160 \beta + 270) q^{60} + ( - 108 \beta + 168) q^{61} + ( - 780 \beta - 600) q^{62} + (56 \beta - 42) q^{63} + (352 \beta + 1064) q^{64} + ( - 20 \beta - 125) q^{65} + (233 \beta + 988) q^{66} + ( - 64 \beta - 76) q^{67} + (240 \beta + 454) q^{68} + ( - 556 \beta + 666) q^{69} + (35 \beta + 140) q^{70} - 952 q^{71} + (12 \beta - 400) q^{72} + ( - 344 \beta + 338) q^{73} + (42 \beta + 1008) q^{74} + ( - 100 \beta + 25) q^{75} + (584 \beta + 884) q^{76} + (224 \beta + 49) q^{77} + ( - 391 \beta - 220) q^{78} + (248 \beta + 507) q^{79} + ( - 480 \beta - 420) q^{80} + (120 \beta - 727) q^{81} + (332 \beta - 408) q^{82} + (600 \beta - 188) q^{83} + (224 \beta + 378) q^{84} + ( - 220 \beta + 125) q^{85} + (142 \beta - 384) q^{86} + (76 \beta - 205) q^{87} + ( - 1006 \beta - 2344) q^{88} + (44 \beta - 108) q^{89} + (130 \beta - 40) q^{90} + ( - 28 \beta - 175) q^{91} + (296 \beta + 132) q^{92} + (60 \beta + 1380) q^{93} + ( - 703 \beta - 964) q^{94} + ( - 220 \beta - 90) q^{95} + ( - 1148 \beta - 1232) q^{96} + (220 \beta + 1371) q^{97} + (49 \beta + 196) q^{98} + ( - 136 \beta + 470) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{2} + 2 q^{3} + 20 q^{4} - 10 q^{5} - 8 q^{6} - 14 q^{7} + 48 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{2} + 2 q^{3} + 20 q^{4} - 10 q^{5} - 8 q^{6} - 14 q^{7} + 48 q^{8} + 12 q^{9} - 40 q^{10} - 14 q^{11} - 108 q^{12} + 50 q^{13} - 56 q^{14} - 10 q^{15} + 168 q^{16} - 50 q^{17} + 16 q^{18} + 36 q^{19} - 100 q^{20} - 14 q^{21} - 184 q^{22} + 244 q^{23} - 496 q^{24} + 50 q^{25} + 216 q^{26} + 86 q^{27} - 140 q^{28} - 26 q^{29} + 40 q^{30} - 120 q^{31} + 672 q^{32} + 498 q^{33} - 24 q^{34} + 70 q^{35} - 136 q^{36} + 564 q^{37} + 320 q^{38} - 14 q^{39} - 240 q^{40} - 328 q^{41} + 56 q^{42} - 260 q^{43} - 1164 q^{44} - 60 q^{45} + 704 q^{46} - 350 q^{47} - 1368 q^{48} + 98 q^{49} + 200 q^{50} - 754 q^{51} + 628 q^{52} - 56 q^{53} + 648 q^{54} + 70 q^{55} - 336 q^{56} - 668 q^{57} - 8 q^{58} - 1232 q^{59} + 540 q^{60} + 336 q^{61} - 1200 q^{62} - 84 q^{63} + 2128 q^{64} - 250 q^{65} + 1976 q^{66} - 152 q^{67} + 908 q^{68} + 1332 q^{69} + 280 q^{70} - 1904 q^{71} - 800 q^{72} + 676 q^{73} + 2016 q^{74} + 50 q^{75} + 1768 q^{76} + 98 q^{77} - 440 q^{78} + 1014 q^{79} - 840 q^{80} - 1454 q^{81} - 816 q^{82} - 376 q^{83} + 756 q^{84} + 250 q^{85} - 768 q^{86} - 410 q^{87} - 4688 q^{88} - 216 q^{89} - 80 q^{90} - 350 q^{91} + 264 q^{92} + 2760 q^{93} - 1928 q^{94} - 180 q^{95} - 2464 q^{96} + 2742 q^{97} + 392 q^{98} + 940 q^{99}+O(q^{100}) \) 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
−1.41421
1.41421
2.58579 6.65685 −1.31371 −5.00000 17.2132 −7.00000 −24.0833 17.3137 −12.9289
1.2 5.41421 −4.65685 21.3137 −5.00000 −25.2132 −7.00000 72.0833 −5.31371 −27.0711
\(n\): 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 35.4.a.b 2
3.b odd 2 1 315.4.a.f 2
4.b odd 2 1 560.4.a.r 2
5.b even 2 1 175.4.a.c 2
5.c odd 4 2 175.4.b.c 4
7.b odd 2 1 245.4.a.k 2
7.c even 3 2 245.4.e.h 4
7.d odd 6 2 245.4.e.i 4
8.b even 2 1 2240.4.a.bn 2
8.d odd 2 1 2240.4.a.bo 2
15.d odd 2 1 1575.4.a.z 2
21.c even 2 1 2205.4.a.u 2
35.c odd 2 1 1225.4.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.a.b 2 1.a even 1 1 trivial
175.4.a.c 2 5.b even 2 1
175.4.b.c 4 5.c odd 4 2
245.4.a.k 2 7.b odd 2 1
245.4.e.h 4 7.c even 3 2
245.4.e.i 4 7.d odd 6 2
315.4.a.f 2 3.b odd 2 1
560.4.a.r 2 4.b odd 2 1
1225.4.a.m 2 35.c odd 2 1
1575.4.a.z 2 15.d odd 2 1
2205.4.a.u 2 21.c even 2 1
2240.4.a.bn 2 8.b even 2 1
2240.4.a.bo 2 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 8T_{2} + 14 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(35))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 8T + 14 \) Copy content Toggle raw display
$3$ \( T^{2} - 2T - 31 \) Copy content Toggle raw display
$5$ \( (T + 5)^{2} \) Copy content Toggle raw display
$7$ \( (T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 14T - 1999 \) Copy content Toggle raw display
$13$ \( T^{2} - 50T + 593 \) Copy content Toggle raw display
$17$ \( T^{2} + 50T - 3247 \) Copy content Toggle raw display
$19$ \( T^{2} - 36T - 3548 \) Copy content Toggle raw display
$23$ \( T^{2} - 244T + 5636 \) Copy content Toggle raw display
$29$ \( T^{2} + 26T - 983 \) Copy content Toggle raw display
$31$ \( T^{2} + 120T - 61200 \) Copy content Toggle raw display
$37$ \( T^{2} - 564T + 72324 \) Copy content Toggle raw display
$41$ \( T^{2} + 328T - 3856 \) Copy content Toggle raw display
$43$ \( T^{2} + 260T + 7652 \) Copy content Toggle raw display
$47$ \( T^{2} + 350T - 4223 \) Copy content Toggle raw display
$53$ \( T^{2} + 56T - 31984 \) Copy content Toggle raw display
$59$ \( (T + 616)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 336T + 4896 \) Copy content Toggle raw display
$67$ \( T^{2} + 152T - 2416 \) Copy content Toggle raw display
$71$ \( (T + 952)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 676T - 122428 \) Copy content Toggle raw display
$79$ \( T^{2} - 1014 T + 134041 \) Copy content Toggle raw display
$83$ \( T^{2} + 376T - 684656 \) Copy content Toggle raw display
$89$ \( T^{2} + 216T + 7792 \) Copy content Toggle raw display
$97$ \( T^{2} - 2742 T + 1782841 \) Copy content Toggle raw display
show more
show less