Properties

Label 35.6.e.a
Level $35$
Weight $6$
Character orbit 35.e
Analytic conductor $5.613$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 35.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.61343369345\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - x^{11} + 110 x^{10} + 27 x^{9} + 9714 x^{8} - 3257 x^{7} + 240269 x^{6} - 608504 x^{5} + 5067668 x^{4} - 7120352 x^{3} + 9029136 x^{2} - 1794240 x + 313600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} - \beta_1 + 1) q^{2} + (\beta_{6} - 3 \beta_{4} + \beta_{3} + \beta_1) q^{3} + (\beta_{9} - 5 \beta_{4} - \beta_{3} - \beta_1) q^{4} + ( - 25 \beta_{4} + 25) q^{5} + (\beta_{8} - \beta_{7} - \beta_{5} - 5 \beta_{3} + \beta_{2} + 16) q^{6} + ( - 2 \beta_{10} + 3 \beta_{8} - 3 \beta_{6} - \beta_{5} - 9 \beta_{4} + 7 \beta_{3} + \cdots + 3) q^{7}+ \cdots + ( - 2 \beta_{11} - 3 \beta_{10} + 4 \beta_{9} + 3 \beta_{8} + 2 \beta_{7} - 3 \beta_{6} + \cdots - 65) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} - \beta_1 + 1) q^{2} + (\beta_{6} - 3 \beta_{4} + \beta_{3} + \beta_1) q^{3} + (\beta_{9} - 5 \beta_{4} - \beta_{3} - \beta_1) q^{4} + ( - 25 \beta_{4} + 25) q^{5} + (\beta_{8} - \beta_{7} - \beta_{5} - 5 \beta_{3} + \beta_{2} + 16) q^{6} + ( - 2 \beta_{10} + 3 \beta_{8} - 3 \beta_{6} - \beta_{5} - 9 \beta_{4} + 7 \beta_{3} + \cdots + 3) q^{7}+ \cdots + (461 \beta_{8} + 524 \beta_{7} - 3201 \beta_{5} - 1922 \beta_{3} + \cdots - 32765) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 5 q^{2} - 20 q^{3} - 31 q^{4} + 150 q^{5} + 192 q^{6} - 20 q^{7} - 270 q^{8} - 378 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 5 q^{2} - 20 q^{3} - 31 q^{4} + 150 q^{5} + 192 q^{6} - 20 q^{7} - 270 q^{8} - 378 q^{9} - 125 q^{10} + 924 q^{11} - 370 q^{12} - 300 q^{13} + 3409 q^{14} - 1000 q^{15} - 435 q^{16} - 1540 q^{17} - 195 q^{18} - 92 q^{19} - 1550 q^{20} + 7288 q^{21} - 13710 q^{22} + 3920 q^{23} + 7200 q^{24} - 3750 q^{25} - 2635 q^{26} + 4120 q^{27} - 6015 q^{28} + 2528 q^{29} + 2400 q^{30} + 7160 q^{31} - 9105 q^{32} - 4460 q^{33} + 4332 q^{34} - 2500 q^{35} - 52750 q^{36} + 14170 q^{37} + 46215 q^{38} + 15376 q^{39} - 3375 q^{40} + 8196 q^{41} + 10500 q^{42} - 48920 q^{43} + 27873 q^{44} + 9450 q^{45} - 6815 q^{46} - 42940 q^{47} + 23220 q^{48} - 17856 q^{49} - 6250 q^{50} + 42008 q^{51} + 36115 q^{52} + 2450 q^{53} - 19566 q^{54} + 46200 q^{55} + 67998 q^{56} - 194200 q^{57} + 36110 q^{58} + 64600 q^{59} + 9250 q^{60} - 73620 q^{61} + 222880 q^{62} - 57780 q^{63} - 315994 q^{64} - 3750 q^{65} + 139138 q^{66} + 142620 q^{67} - 124330 q^{68} + 34688 q^{69} + 34475 q^{70} - 308512 q^{71} + 117495 q^{72} + 5120 q^{73} - 2785 q^{74} - 12500 q^{75} + 15550 q^{76} + 44230 q^{77} - 428180 q^{78} + 222504 q^{79} + 10875 q^{80} + 43986 q^{81} - 31665 q^{82} + 359160 q^{83} - 19966 q^{84} - 77000 q^{85} + 207160 q^{86} + 209300 q^{87} + 45145 q^{88} - 41648 q^{89} - 9750 q^{90} - 31376 q^{91} - 584370 q^{92} + 198520 q^{93} + 333699 q^{94} + 2300 q^{95} - 61824 q^{96} - 147960 q^{97} - 101815 q^{98} - 381544 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 110 x^{10} + 27 x^{9} + 9714 x^{8} - 3257 x^{7} + 240269 x^{6} - 608504 x^{5} + 5067668 x^{4} - 7120352 x^{3} + 9029136 x^{2} - 1794240 x + 313600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 28\!\cdots\!87 \nu^{11} + \cdots + 21\!\cdots\!60 ) / 59\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 66\!\cdots\!49 \nu^{11} + \cdots - 11\!\cdots\!00 ) / 59\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 60\!\cdots\!45 \nu^{11} + \cdots + 10\!\cdots\!60 ) / 95\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 52\!\cdots\!28 \nu^{11} + \cdots + 10\!\cdots\!20 ) / 14\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 15\!\cdots\!47 \nu^{11} + \cdots - 26\!\cdots\!40 ) / 28\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 70\!\cdots\!29 \nu^{11} + \cdots + 10\!\cdots\!40 ) / 10\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 16\!\cdots\!16 \nu^{11} + \cdots + 73\!\cdots\!76 ) / 14\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 54\!\cdots\!01 \nu^{11} + \cdots + 98\!\cdots\!40 ) / 23\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 13\!\cdots\!81 \nu^{11} + \cdots + 24\!\cdots\!60 ) / 40\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 96\!\cdots\!41 \nu^{11} + \cdots + 17\!\cdots\!60 ) / 14\!\cdots\!40 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - 36\beta_{4} + \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{8} + 4\beta_{5} + 69\beta_{3} + 3\beta_{2} - 22 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{11} - 91\beta_{9} - 8\beta_{7} + 8\beta_{6} + 8\beta_{5} + 2468\beta_{4} + 91\beta_{2} - 197\beta _1 - 2468 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 206\beta_{10} - 395\beta_{9} + 460\beta_{6} + 6066\beta_{4} - 5449\beta_{3} - 5449\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 124\beta_{8} + 872\beta_{7} - 1168\beta_{5} - 25089\beta_{3} - 7983\beta_{2} + 194128 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 544 \beta_{11} - 18162 \beta_{10} + 43887 \beta_{9} + 18162 \beta_{8} + 544 \beta_{7} - 42148 \beta_{6} - 42148 \beta_{5} - 820902 \beta_{4} - 43887 \beta_{2} + 461245 \beta _1 + 820902 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 77928 \beta_{11} - 29640 \beta_{10} + 705963 \beta_{9} - 145752 \beta_{6} - 16333132 \beta_{4} + 2758077 \beta_{3} + 2758077 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -1548246\beta_{8} - 127104\beta_{7} + 3699708\beta_{5} + 40566289\beta_{3} + 4580067\beta_{2} - 92353626 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 6669096 \beta_{11} + 4420596 \beta_{10} - 63318823 \beta_{9} - 4420596 \beta_{8} - 6669096 \beta_{7} + 16749024 \beta_{6} + 16749024 \beta_{5} + 1428426792 \beta_{4} + \cdots - 1428426792 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 18921120 \beta_{11} + 132144410 \beta_{10} - 462055479 \beta_{9} + 324463252 \beta_{6} + 9631351150 \beta_{4} - 3651786501 \beta_{3} - 3651786501 \beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(22\) \(31\)
\(\chi(n)\) \(1\) \(-1 + \beta_{4}\)

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
4.86278 + 8.42258i
2.06025 + 3.56846i
0.675239 + 1.16955i
0.101659 + 0.176079i
−3.11366 5.39303i
−4.08626 7.07762i
4.86278 8.42258i
2.06025 3.56846i
0.675239 1.16955i
0.101659 0.176079i
−3.11366 + 5.39303i
−4.08626 + 7.07762i
−4.36278 7.55656i −2.36982 + 4.10465i −22.0677 + 38.2224i 12.5000 + 21.6506i 41.3561 8.78435 + 129.344i 105.888 110.268 + 190.990i 109.070 188.914i
11.2 −1.56025 2.70243i −13.9359 + 24.1376i 11.1312 19.2799i 12.5000 + 21.6506i 86.9736 −89.4698 93.8198i −169.326 −266.916 462.312i 39.0062 67.5608i
11.3 −0.175239 0.303524i 10.7955 18.6984i 15.9386 27.6064i 12.5000 + 21.6506i −7.56720 −129.103 11.8023i −22.3876 −111.586 193.273i 4.38099 7.58809i
11.4 0.398341 + 0.689946i −5.28741 + 9.15807i 15.6826 27.1631i 12.5000 + 21.6506i −8.42477 103.426 + 78.1675i 50.4819 65.5865 + 113.599i −9.95851 + 17.2487i
11.5 3.61366 + 6.25905i 7.96046 13.7879i −10.1171 + 17.5234i 12.5000 + 21.6506i 115.066 102.346 + 79.5760i 85.0347 −5.23796 9.07242i −90.3416 + 156.476i
11.6 4.58626 + 7.94364i −7.16289 + 12.4065i −26.0676 + 45.1505i 12.5000 + 21.6506i −131.404 −5.98242 129.504i −184.691 18.8861 + 32.7118i −114.657 + 198.591i
16.1 −4.36278 + 7.55656i −2.36982 4.10465i −22.0677 38.2224i 12.5000 21.6506i 41.3561 8.78435 129.344i 105.888 110.268 190.990i 109.070 + 188.914i
16.2 −1.56025 + 2.70243i −13.9359 24.1376i 11.1312 + 19.2799i 12.5000 21.6506i 86.9736 −89.4698 + 93.8198i −169.326 −266.916 + 462.312i 39.0062 + 67.5608i
16.3 −0.175239 + 0.303524i 10.7955 + 18.6984i 15.9386 + 27.6064i 12.5000 21.6506i −7.56720 −129.103 + 11.8023i −22.3876 −111.586 + 193.273i 4.38099 + 7.58809i
16.4 0.398341 0.689946i −5.28741 9.15807i 15.6826 + 27.1631i 12.5000 21.6506i −8.42477 103.426 78.1675i 50.4819 65.5865 113.599i −9.95851 17.2487i
16.5 3.61366 6.25905i 7.96046 + 13.7879i −10.1171 17.5234i 12.5000 21.6506i 115.066 102.346 79.5760i 85.0347 −5.23796 + 9.07242i −90.3416 156.476i
16.6 4.58626 7.94364i −7.16289 12.4065i −26.0676 45.1505i 12.5000 21.6506i −131.404 −5.98242 + 129.504i −184.691 18.8861 32.7118i −114.657 198.591i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.6
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 35.6.e.a 12
7.c even 3 1 inner 35.6.e.a 12
7.c even 3 1 245.6.a.i 6
7.d odd 6 1 245.6.a.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.6.e.a 12 1.a even 1 1 trivial
35.6.e.a 12 7.c even 3 1 inner
245.6.a.h 6 7.d odd 6 1
245.6.a.i 6 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 5 T_{2}^{11} + 124 T_{2}^{10} - 275 T_{2}^{9} + 10044 T_{2}^{8} - 22195 T_{2}^{7} + 309235 T_{2}^{6} + 471890 T_{2}^{5} + 3125728 T_{2}^{4} - 1125720 T_{2}^{3} + 1657728 T_{2}^{2} + \cdots + 254016 \) acting on \(S_{6}^{\mathrm{new}}(35, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 5 T^{11} + 124 T^{10} + \cdots + 254016 \) Copy content Toggle raw display
$3$ \( T^{12} + 20 T^{11} + \cdots + 47324438284176 \) Copy content Toggle raw display
$5$ \( (T^{2} - 25 T + 625)^{6} \) Copy content Toggle raw display
$7$ \( T^{12} + 20 T^{11} + \cdots + 22\!\cdots\!49 \) Copy content Toggle raw display
$11$ \( T^{12} - 924 T^{11} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{6} + 150 T^{5} + \cdots + 37\!\cdots\!36)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 1540 T^{11} + \cdots + 35\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{12} + 92 T^{11} + \cdots + 23\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( T^{12} - 3920 T^{11} + \cdots + 30\!\cdots\!01 \) Copy content Toggle raw display
$29$ \( (T^{6} - 1264 T^{5} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} - 7160 T^{11} + \cdots + 36\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{12} - 14170 T^{11} + \cdots + 46\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( (T^{6} - 4098 T^{5} + \cdots + 10\!\cdots\!25)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 24460 T^{5} + \cdots + 19\!\cdots\!56)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 42940 T^{11} + \cdots + 97\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} - 2450 T^{11} + \cdots + 72\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{12} - 64600 T^{11} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{12} + 73620 T^{11} + \cdots + 45\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{12} - 142620 T^{11} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{6} + 154256 T^{5} + \cdots + 99\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} - 5120 T^{11} + \cdots + 26\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{12} - 222504 T^{11} + \cdots + 51\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( (T^{6} - 179580 T^{5} + \cdots + 35\!\cdots\!64)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + 41648 T^{11} + \cdots + 22\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( (T^{6} + 73980 T^{5} + \cdots + 99\!\cdots\!84)^{2} \) Copy content Toggle raw display
show more
show less