Properties

Label 35.5.c.c
Level $35$
Weight $5$
Character orbit 35.c
Analytic conductor $3.618$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 35.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.61794870793\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-6}) \)
Defining polynomial: \( x^{2} + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} - 5 q^{3} + 10 q^{4} + (10 \beta - 5) q^{5} - 5 \beta q^{6} + (14 \beta - 35) q^{7} + 26 \beta q^{8} - 56 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} - 5 q^{3} + 10 q^{4} + (10 \beta - 5) q^{5} - 5 \beta q^{6} + (14 \beta - 35) q^{7} + 26 \beta q^{8} - 56 q^{9} + ( - 5 \beta - 60) q^{10} + 89 q^{11} - 50 q^{12} + 5 q^{13} + ( - 35 \beta - 84) q^{14} + ( - 50 \beta + 25) q^{15} + 4 q^{16} + 485 q^{17} - 56 \beta q^{18} + 90 \beta q^{19} + (100 \beta - 50) q^{20} + ( - 70 \beta + 175) q^{21} + 89 \beta q^{22} - 286 \beta q^{23} - 130 \beta q^{24} + ( - 100 \beta - 575) q^{25} + 5 \beta q^{26} + 685 q^{27} + (140 \beta - 350) q^{28} + 191 q^{29} + (25 \beta + 300) q^{30} - 430 \beta q^{31} + 420 \beta q^{32} - 445 q^{33} + 485 \beta q^{34} + ( - 420 \beta - 665) q^{35} - 560 q^{36} + 666 \beta q^{37} - 540 q^{38} - 25 q^{39} + ( - 130 \beta - 1560) q^{40} + 1190 \beta q^{41} + (175 \beta + 420) q^{42} + 154 \beta q^{43} + 890 q^{44} + ( - 560 \beta + 280) q^{45} + 1716 q^{46} + 2195 q^{47} - 20 q^{48} + ( - 980 \beta + 49) q^{49} + ( - 575 \beta + 600) q^{50} - 2425 q^{51} + 50 q^{52} - 648 \beta q^{53} + 685 \beta q^{54} + (890 \beta - 445) q^{55} + ( - 910 \beta - 2184) q^{56} - 450 \beta q^{57} + 191 \beta q^{58} - 1480 \beta q^{59} + ( - 500 \beta + 250) q^{60} + 790 \beta q^{61} + 2580 q^{62} + ( - 784 \beta + 1960) q^{63} - 2456 q^{64} + (50 \beta - 25) q^{65} - 445 \beta q^{66} + 836 \beta q^{67} + 4850 q^{68} + 1430 \beta q^{69} + ( - 665 \beta + 2520) q^{70} + 4454 q^{71} - 1456 \beta q^{72} - 8650 q^{73} - 3996 q^{74} + (500 \beta + 2875) q^{75} + 900 \beta q^{76} + (1246 \beta - 3115) q^{77} - 25 \beta q^{78} + 5561 q^{79} + (40 \beta - 20) q^{80} + 1111 q^{81} - 7140 q^{82} - 1990 q^{83} + ( - 700 \beta + 1750) q^{84} + (4850 \beta - 2425) q^{85} - 924 q^{86} - 955 q^{87} + 2314 \beta q^{88} - 330 \beta q^{89} + (280 \beta + 3360) q^{90} + (70 \beta - 175) q^{91} - 2860 \beta q^{92} + 2150 \beta q^{93} + 2195 \beta q^{94} + ( - 450 \beta - 5400) q^{95} - 2100 \beta q^{96} - 9235 q^{97} + (49 \beta + 5880) q^{98} - 4984 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 10 q^{3} + 20 q^{4} - 10 q^{5} - 70 q^{7} - 112 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 10 q^{3} + 20 q^{4} - 10 q^{5} - 70 q^{7} - 112 q^{9} - 120 q^{10} + 178 q^{11} - 100 q^{12} + 10 q^{13} - 168 q^{14} + 50 q^{15} + 8 q^{16} + 970 q^{17} - 100 q^{20} + 350 q^{21} - 1150 q^{25} + 1370 q^{27} - 700 q^{28} + 382 q^{29} + 600 q^{30} - 890 q^{33} - 1330 q^{35} - 1120 q^{36} - 1080 q^{38} - 50 q^{39} - 3120 q^{40} + 840 q^{42} + 1780 q^{44} + 560 q^{45} + 3432 q^{46} + 4390 q^{47} - 40 q^{48} + 98 q^{49} + 1200 q^{50} - 4850 q^{51} + 100 q^{52} - 890 q^{55} - 4368 q^{56} + 500 q^{60} + 5160 q^{62} + 3920 q^{63} - 4912 q^{64} - 50 q^{65} + 9700 q^{68} + 5040 q^{70} + 8908 q^{71} - 17300 q^{73} - 7992 q^{74} + 5750 q^{75} - 6230 q^{77} + 11122 q^{79} - 40 q^{80} + 2222 q^{81} - 14280 q^{82} - 3980 q^{83} + 3500 q^{84} - 4850 q^{85} - 1848 q^{86} - 1910 q^{87} + 6720 q^{90} - 350 q^{91} - 10800 q^{95} - 18470 q^{97} + 11760 q^{98} - 9968 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(22\) \(31\)
\(\chi(n)\) \(-1\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1
2.44949i
2.44949i
2.44949i −5.00000 10.0000 −5.00000 24.4949i 12.2474i −35.0000 34.2929i 63.6867i −56.0000 −60.0000 + 12.2474i
34.2 2.44949i −5.00000 10.0000 −5.00000 + 24.4949i 12.2474i −35.0000 + 34.2929i 63.6867i −56.0000 −60.0000 12.2474i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.5.c.c 2
3.b odd 2 1 315.5.e.d 2
4.b odd 2 1 560.5.p.e 2
5.b even 2 1 35.5.c.d yes 2
5.c odd 4 2 175.5.d.e 4
7.b odd 2 1 35.5.c.d yes 2
15.d odd 2 1 315.5.e.c 2
20.d odd 2 1 560.5.p.d 2
21.c even 2 1 315.5.e.c 2
28.d even 2 1 560.5.p.d 2
35.c odd 2 1 inner 35.5.c.c 2
35.f even 4 2 175.5.d.e 4
105.g even 2 1 315.5.e.d 2
140.c even 2 1 560.5.p.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.5.c.c 2 1.a even 1 1 trivial
35.5.c.c 2 35.c odd 2 1 inner
35.5.c.d yes 2 5.b even 2 1
35.5.c.d yes 2 7.b odd 2 1
175.5.d.e 4 5.c odd 4 2
175.5.d.e 4 35.f even 4 2
315.5.e.c 2 15.d odd 2 1
315.5.e.c 2 21.c even 2 1
315.5.e.d 2 3.b odd 2 1
315.5.e.d 2 105.g even 2 1
560.5.p.d 2 20.d odd 2 1
560.5.p.d 2 28.d even 2 1
560.5.p.e 2 4.b odd 2 1
560.5.p.e 2 140.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(35, [\chi])\):

\( T_{2}^{2} + 6 \) Copy content Toggle raw display
\( T_{3} + 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 6 \) Copy content Toggle raw display
$3$ \( (T + 5)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 10T + 625 \) Copy content Toggle raw display
$7$ \( T^{2} + 70T + 2401 \) Copy content Toggle raw display
$11$ \( (T - 89)^{2} \) Copy content Toggle raw display
$13$ \( (T - 5)^{2} \) Copy content Toggle raw display
$17$ \( (T - 485)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 48600 \) Copy content Toggle raw display
$23$ \( T^{2} + 490776 \) Copy content Toggle raw display
$29$ \( (T - 191)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 1109400 \) Copy content Toggle raw display
$37$ \( T^{2} + 2661336 \) Copy content Toggle raw display
$41$ \( T^{2} + 8496600 \) Copy content Toggle raw display
$43$ \( T^{2} + 142296 \) Copy content Toggle raw display
$47$ \( (T - 2195)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 2519424 \) Copy content Toggle raw display
$59$ \( T^{2} + 13142400 \) Copy content Toggle raw display
$61$ \( T^{2} + 3744600 \) Copy content Toggle raw display
$67$ \( T^{2} + 4193376 \) Copy content Toggle raw display
$71$ \( (T - 4454)^{2} \) Copy content Toggle raw display
$73$ \( (T + 8650)^{2} \) Copy content Toggle raw display
$79$ \( (T - 5561)^{2} \) Copy content Toggle raw display
$83$ \( (T + 1990)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 653400 \) Copy content Toggle raw display
$97$ \( (T + 9235)^{2} \) Copy content Toggle raw display
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