Properties

Label 35.4.e.a
Level $35$
Weight $4$
Character orbit 35.e
Analytic conductor $2.065$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 35.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.06506685020\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 \zeta_{6} q^{2} + ( - 2 \zeta_{6} + 2) q^{3} + (\zeta_{6} - 1) q^{4} - 5 \zeta_{6} q^{5} - 6 q^{6} + ( - 14 \zeta_{6} - 7) q^{7} - 21 q^{8} + 23 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 3 \zeta_{6} q^{2} + ( - 2 \zeta_{6} + 2) q^{3} + (\zeta_{6} - 1) q^{4} - 5 \zeta_{6} q^{5} - 6 q^{6} + ( - 14 \zeta_{6} - 7) q^{7} - 21 q^{8} + 23 \zeta_{6} q^{9} + (15 \zeta_{6} - 15) q^{10} + ( - 45 \zeta_{6} + 45) q^{11} + 2 \zeta_{6} q^{12} + 59 q^{13} + (63 \zeta_{6} - 42) q^{14} - 10 q^{15} + 71 \zeta_{6} q^{16} + ( - 54 \zeta_{6} + 54) q^{17} + ( - 69 \zeta_{6} + 69) q^{18} + 121 \zeta_{6} q^{19} + 5 q^{20} + (14 \zeta_{6} - 42) q^{21} - 135 q^{22} - 69 \zeta_{6} q^{23} + (42 \zeta_{6} - 42) q^{24} + (25 \zeta_{6} - 25) q^{25} - 177 \zeta_{6} q^{26} + 100 q^{27} + ( - 7 \zeta_{6} + 21) q^{28} - 162 q^{29} + 30 \zeta_{6} q^{30} + ( - 88 \zeta_{6} + 88) q^{31} + ( - 45 \zeta_{6} + 45) q^{32} - 90 \zeta_{6} q^{33} - 162 q^{34} + (105 \zeta_{6} - 70) q^{35} - 23 q^{36} + 259 \zeta_{6} q^{37} + ( - 363 \zeta_{6} + 363) q^{38} + ( - 118 \zeta_{6} + 118) q^{39} + 105 \zeta_{6} q^{40} + 195 q^{41} + (84 \zeta_{6} + 42) q^{42} - 286 q^{43} + 45 \zeta_{6} q^{44} + ( - 115 \zeta_{6} + 115) q^{45} + (207 \zeta_{6} - 207) q^{46} - 45 \zeta_{6} q^{47} + 142 q^{48} + (392 \zeta_{6} - 147) q^{49} + 75 q^{50} - 108 \zeta_{6} q^{51} + (59 \zeta_{6} - 59) q^{52} + (597 \zeta_{6} - 597) q^{53} - 300 \zeta_{6} q^{54} - 225 q^{55} + (294 \zeta_{6} + 147) q^{56} + 242 q^{57} + 486 \zeta_{6} q^{58} + ( - 360 \zeta_{6} + 360) q^{59} + ( - 10 \zeta_{6} + 10) q^{60} - 392 \zeta_{6} q^{61} - 264 q^{62} + ( - 483 \zeta_{6} + 322) q^{63} + 433 q^{64} - 295 \zeta_{6} q^{65} + (270 \zeta_{6} - 270) q^{66} + ( - 280 \zeta_{6} + 280) q^{67} + 54 \zeta_{6} q^{68} - 138 q^{69} + ( - 105 \zeta_{6} + 315) q^{70} + 48 q^{71} - 483 \zeta_{6} q^{72} + (668 \zeta_{6} - 668) q^{73} + ( - 777 \zeta_{6} + 777) q^{74} + 50 \zeta_{6} q^{75} - 121 q^{76} + (315 \zeta_{6} - 945) q^{77} - 354 q^{78} - 782 \zeta_{6} q^{79} + ( - 355 \zeta_{6} + 355) q^{80} + (421 \zeta_{6} - 421) q^{81} - 585 \zeta_{6} q^{82} + 768 q^{83} + ( - 42 \zeta_{6} + 28) q^{84} - 270 q^{85} + 858 \zeta_{6} q^{86} + (324 \zeta_{6} - 324) q^{87} + (945 \zeta_{6} - 945) q^{88} + 1194 \zeta_{6} q^{89} - 345 q^{90} + ( - 826 \zeta_{6} - 413) q^{91} + 69 q^{92} - 176 \zeta_{6} q^{93} + (135 \zeta_{6} - 135) q^{94} + ( - 605 \zeta_{6} + 605) q^{95} - 90 \zeta_{6} q^{96} + 902 q^{97} + ( - 735 \zeta_{6} + 1176) q^{98} + 1035 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} + 2 q^{3} - q^{4} - 5 q^{5} - 12 q^{6} - 28 q^{7} - 42 q^{8} + 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} + 2 q^{3} - q^{4} - 5 q^{5} - 12 q^{6} - 28 q^{7} - 42 q^{8} + 23 q^{9} - 15 q^{10} + 45 q^{11} + 2 q^{12} + 118 q^{13} - 21 q^{14} - 20 q^{15} + 71 q^{16} + 54 q^{17} + 69 q^{18} + 121 q^{19} + 10 q^{20} - 70 q^{21} - 270 q^{22} - 69 q^{23} - 42 q^{24} - 25 q^{25} - 177 q^{26} + 200 q^{27} + 35 q^{28} - 324 q^{29} + 30 q^{30} + 88 q^{31} + 45 q^{32} - 90 q^{33} - 324 q^{34} - 35 q^{35} - 46 q^{36} + 259 q^{37} + 363 q^{38} + 118 q^{39} + 105 q^{40} + 390 q^{41} + 168 q^{42} - 572 q^{43} + 45 q^{44} + 115 q^{45} - 207 q^{46} - 45 q^{47} + 284 q^{48} + 98 q^{49} + 150 q^{50} - 108 q^{51} - 59 q^{52} - 597 q^{53} - 300 q^{54} - 450 q^{55} + 588 q^{56} + 484 q^{57} + 486 q^{58} + 360 q^{59} + 10 q^{60} - 392 q^{61} - 528 q^{62} + 161 q^{63} + 866 q^{64} - 295 q^{65} - 270 q^{66} + 280 q^{67} + 54 q^{68} - 276 q^{69} + 525 q^{70} + 96 q^{71} - 483 q^{72} - 668 q^{73} + 777 q^{74} + 50 q^{75} - 242 q^{76} - 1575 q^{77} - 708 q^{78} - 782 q^{79} + 355 q^{80} - 421 q^{81} - 585 q^{82} + 1536 q^{83} + 14 q^{84} - 540 q^{85} + 858 q^{86} - 324 q^{87} - 945 q^{88} + 1194 q^{89} - 690 q^{90} - 1652 q^{91} + 138 q^{92} - 176 q^{93} - 135 q^{94} + 605 q^{95} - 90 q^{96} + 1804 q^{97} + 1617 q^{98} + 2070 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
0.500000 + 0.866025i
0.500000 0.866025i
−1.50000 2.59808i 1.00000 1.73205i −0.500000 + 0.866025i −2.50000 4.33013i −6.00000 −14.0000 12.1244i −21.0000 11.5000 + 19.9186i −7.50000 + 12.9904i
16.1 −1.50000 + 2.59808i 1.00000 + 1.73205i −0.500000 0.866025i −2.50000 + 4.33013i −6.00000 −14.0000 + 12.1244i −21.0000 11.5000 19.9186i −7.50000 12.9904i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.4.e.a 2
3.b odd 2 1 315.4.j.b 2
4.b odd 2 1 560.4.q.b 2
5.b even 2 1 175.4.e.b 2
5.c odd 4 2 175.4.k.b 4
7.b odd 2 1 245.4.e.a 2
7.c even 3 1 inner 35.4.e.a 2
7.c even 3 1 245.4.a.e 1
7.d odd 6 1 245.4.a.f 1
7.d odd 6 1 245.4.e.a 2
21.g even 6 1 2205.4.a.g 1
21.h odd 6 1 315.4.j.b 2
21.h odd 6 1 2205.4.a.e 1
28.g odd 6 1 560.4.q.b 2
35.i odd 6 1 1225.4.a.a 1
35.j even 6 1 175.4.e.b 2
35.j even 6 1 1225.4.a.b 1
35.l odd 12 2 175.4.k.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.e.a 2 1.a even 1 1 trivial
35.4.e.a 2 7.c even 3 1 inner
175.4.e.b 2 5.b even 2 1
175.4.e.b 2 35.j even 6 1
175.4.k.b 4 5.c odd 4 2
175.4.k.b 4 35.l odd 12 2
245.4.a.e 1 7.c even 3 1
245.4.a.f 1 7.d odd 6 1
245.4.e.a 2 7.b odd 2 1
245.4.e.a 2 7.d odd 6 1
315.4.j.b 2 3.b odd 2 1
315.4.j.b 2 21.h odd 6 1
560.4.q.b 2 4.b odd 2 1
560.4.q.b 2 28.g odd 6 1
1225.4.a.a 1 35.i odd 6 1
1225.4.a.b 1 35.j even 6 1
2205.4.a.e 1 21.h odd 6 1
2205.4.a.g 1 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 3T_{2} + 9 \) acting on \(S_{4}^{\mathrm{new}}(35, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$3$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$5$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$7$ \( T^{2} + 28T + 343 \) Copy content Toggle raw display
$11$ \( T^{2} - 45T + 2025 \) Copy content Toggle raw display
$13$ \( (T - 59)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 54T + 2916 \) Copy content Toggle raw display
$19$ \( T^{2} - 121T + 14641 \) Copy content Toggle raw display
$23$ \( T^{2} + 69T + 4761 \) Copy content Toggle raw display
$29$ \( (T + 162)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 88T + 7744 \) Copy content Toggle raw display
$37$ \( T^{2} - 259T + 67081 \) Copy content Toggle raw display
$41$ \( (T - 195)^{2} \) Copy content Toggle raw display
$43$ \( (T + 286)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 45T + 2025 \) Copy content Toggle raw display
$53$ \( T^{2} + 597T + 356409 \) Copy content Toggle raw display
$59$ \( T^{2} - 360T + 129600 \) Copy content Toggle raw display
$61$ \( T^{2} + 392T + 153664 \) Copy content Toggle raw display
$67$ \( T^{2} - 280T + 78400 \) Copy content Toggle raw display
$71$ \( (T - 48)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 668T + 446224 \) Copy content Toggle raw display
$79$ \( T^{2} + 782T + 611524 \) Copy content Toggle raw display
$83$ \( (T - 768)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 1194 T + 1425636 \) Copy content Toggle raw display
$97$ \( (T - 902)^{2} \) Copy content Toggle raw display
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