Properties

Label 35.8.a.a
Level $35$
Weight $8$
Character orbit 35.a
Self dual yes
Analytic conductor $10.933$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 35.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.9334758919\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{11}) \)
Defining polynomial: \( x^{2} - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{11}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 8) q^{2} + ( - 6 \beta - 15) q^{3} + (16 \beta - 20) q^{4} + 125 q^{5} + ( - 63 \beta - 384) q^{6} - 343 q^{7} + ( - 20 \beta - 480) q^{8} + (180 \beta - 378) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 8) q^{2} + ( - 6 \beta - 15) q^{3} + (16 \beta - 20) q^{4} + 125 q^{5} + ( - 63 \beta - 384) q^{6} - 343 q^{7} + ( - 20 \beta - 480) q^{8} + (180 \beta - 378) q^{9} + (125 \beta + 1000) q^{10} + ( - 380 \beta - 3953) q^{11} + ( - 120 \beta - 3924) q^{12} + ( - 418 \beta - 8909) q^{13} + ( - 343 \beta - 2744) q^{14} + ( - 750 \beta - 1875) q^{15} + ( - 2688 \beta - 2160) q^{16} + (2210 \beta - 1199) q^{17} + (1062 \beta + 4896) q^{18} + (5542 \beta - 1806) q^{19} + (2000 \beta - 2500) q^{20} + (2058 \beta + 5145) q^{21} + ( - 6993 \beta - 48344) q^{22} + (10690 \beta + 6922) q^{23} + (3180 \beta + 12480) q^{24} + 15625 q^{25} + ( - 12253 \beta - 89664) q^{26} + (12690 \beta - 9045) q^{27} + ( - 5488 \beta + 6860) q^{28} + ( - 23772 \beta - 63449) q^{29} + ( - 7875 \beta - 48000) q^{30} + ( - 22554 \beta + 126384) q^{31} + ( - 21104 \beta - 74112) q^{32} + (29418 \beta + 159615) q^{33} + (16481 \beta + 87648) q^{34} - 42875 q^{35} + ( - 9648 \beta + 134280) q^{36} + ( - 43638 \beta - 132930) q^{37} + (42530 \beta + 229400) q^{38} + (59724 \beta + 243987) q^{39} + ( - 2500 \beta - 60000) q^{40} + (37354 \beta - 55960) q^{41} + (21609 \beta + 131712) q^{42} + ( - 24578 \beta + 473786) q^{43} + ( - 55648 \beta - 188460) q^{44} + (22500 \beta - 47250) q^{45} + (92442 \beta + 525736) q^{46} + ( - 56742 \beta + 135637) q^{47} + (53280 \beta + 742032) q^{48} + 117649 q^{49} + (15625 \beta + 125000) q^{50} + ( - 25956 \beta - 565455) q^{51} + ( - 134184 \beta - 116092) q^{52} + ( - 65224 \beta - 633896) q^{53} + (92475 \beta + 486000) q^{54} + ( - 47500 \beta - 494125) q^{55} + (6860 \beta + 164640) q^{56} + ( - 72294 \beta - 1435998) q^{57} + ( - 253625 \beta - 1553560) q^{58} + (170640 \beta - 680060) q^{59} + ( - 15000 \beta - 490500) q^{60} + (11334 \beta - 906840) q^{61} + ( - 54048 \beta + 18696) q^{62} + ( - 61740 \beta + 129654) q^{63} + (101120 \beta - 1244992) q^{64} + ( - 52250 \beta - 1113625) q^{65} + (394959 \beta + 2571312) q^{66} + (506344 \beta - 1094656) q^{67} + ( - 63384 \beta + 1579820) q^{68} + ( - 201882 \beta - 2925990) q^{69} + ( - 42875 \beta - 343000) q^{70} + ( - 222048 \beta - 747464) q^{71} + ( - 78840 \beta + 23040) q^{72} + (212396 \beta + 3584894) q^{73} + ( - 482034 \beta - 2983512) q^{74} + ( - 93750 \beta - 234375) q^{75} + ( - 139736 \beta + 3937688) q^{76} + (130340 \beta + 1355879) q^{77} + (721779 \beta + 4579752) q^{78} + (187504 \beta - 3971487) q^{79} + ( - 336000 \beta - 270000) q^{80} + ( - 529740 \beta - 2387799) q^{81} + (242872 \beta + 1195896) q^{82} + ( - 939444 \beta - 152356) q^{83} + (41160 \beta + 1345932) q^{84} + (276250 \beta - 149875) q^{85} + (277162 \beta + 2708856) q^{86} + (737274 \beta + 7227543) q^{87} + (261460 \beta + 2231840) q^{88} + (247538 \beta - 8971764) q^{89} + (132750 \beta + 612000) q^{90} + (143374 \beta + 3055787) q^{91} + ( - 103048 \beta + 7387320) q^{92} + ( - 419994 \beta + 4058496) q^{93} + ( - 318299 \beta - 1411552) q^{94} + (692750 \beta - 225750) q^{95} + (761232 \beta + 6683136) q^{96} + ( - 680782 \beta + 2129037) q^{97} + (117649 \beta + 941192) q^{98} + ( - 567900 \beta - 1515366) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 16 q^{2} - 30 q^{3} - 40 q^{4} + 250 q^{5} - 768 q^{6} - 686 q^{7} - 960 q^{8} - 756 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 16 q^{2} - 30 q^{3} - 40 q^{4} + 250 q^{5} - 768 q^{6} - 686 q^{7} - 960 q^{8} - 756 q^{9} + 2000 q^{10} - 7906 q^{11} - 7848 q^{12} - 17818 q^{13} - 5488 q^{14} - 3750 q^{15} - 4320 q^{16} - 2398 q^{17} + 9792 q^{18} - 3612 q^{19} - 5000 q^{20} + 10290 q^{21} - 96688 q^{22} + 13844 q^{23} + 24960 q^{24} + 31250 q^{25} - 179328 q^{26} - 18090 q^{27} + 13720 q^{28} - 126898 q^{29} - 96000 q^{30} + 252768 q^{31} - 148224 q^{32} + 319230 q^{33} + 175296 q^{34} - 85750 q^{35} + 268560 q^{36} - 265860 q^{37} + 458800 q^{38} + 487974 q^{39} - 120000 q^{40} - 111920 q^{41} + 263424 q^{42} + 947572 q^{43} - 376920 q^{44} - 94500 q^{45} + 1051472 q^{46} + 271274 q^{47} + 1484064 q^{48} + 235298 q^{49} + 250000 q^{50} - 1130910 q^{51} - 232184 q^{52} - 1267792 q^{53} + 972000 q^{54} - 988250 q^{55} + 329280 q^{56} - 2871996 q^{57} - 3107120 q^{58} - 1360120 q^{59} - 981000 q^{60} - 1813680 q^{61} + 37392 q^{62} + 259308 q^{63} - 2489984 q^{64} - 2227250 q^{65} + 5142624 q^{66} - 2189312 q^{67} + 3159640 q^{68} - 5851980 q^{69} - 686000 q^{70} - 1494928 q^{71} + 46080 q^{72} + 7169788 q^{73} - 5967024 q^{74} - 468750 q^{75} + 7875376 q^{76} + 2711758 q^{77} + 9159504 q^{78} - 7942974 q^{79} - 540000 q^{80} - 4775598 q^{81} + 2391792 q^{82} - 304712 q^{83} + 2691864 q^{84} - 299750 q^{85} + 5417712 q^{86} + 14455086 q^{87} + 4463680 q^{88} - 17943528 q^{89} + 1224000 q^{90} + 6111574 q^{91} + 14774640 q^{92} + 8116992 q^{93} - 2823104 q^{94} - 451500 q^{95} + 13366272 q^{96} + 4258074 q^{97} + 1882384 q^{98} - 3030732 q^{99}+O(q^{100}) \) 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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.31662
3.31662
1.36675 24.7995 −126.132 125.000 33.8947 −343.000 −347.335 −1571.98 170.844
1.2 14.6332 −54.7995 86.1320 125.000 −801.895 −343.000 −612.665 815.985 1829.16
\(n\): 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 35.8.a.a 2
3.b odd 2 1 315.8.a.c 2
4.b odd 2 1 560.8.a.i 2
5.b even 2 1 175.8.a.b 2
5.c odd 4 2 175.8.b.c 4
7.b odd 2 1 245.8.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.8.a.a 2 1.a even 1 1 trivial
175.8.a.b 2 5.b even 2 1
175.8.b.c 4 5.c odd 4 2
245.8.a.b 2 7.b odd 2 1
315.8.a.c 2 3.b odd 2 1
560.8.a.i 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 16T_{2} + 20 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(35))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 16T + 20 \) Copy content Toggle raw display
$3$ \( T^{2} + 30T - 1359 \) Copy content Toggle raw display
$5$ \( (T - 125)^{2} \) Copy content Toggle raw display
$7$ \( (T + 343)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 7906 T + 9272609 \) Copy content Toggle raw display
$13$ \( T^{2} + 17818 T + 71682425 \) Copy content Toggle raw display
$17$ \( T^{2} + 2398 T - 213462799 \) Copy content Toggle raw display
$19$ \( T^{2} + 3612 T - 1348143980 \) Copy content Toggle raw display
$23$ \( T^{2} - 13844 T - 4980234316 \) Copy content Toggle raw display
$29$ \( T^{2} + 126898 T - 20838975695 \) Copy content Toggle raw display
$31$ \( T^{2} - 252768 T - 6409132848 \) Copy content Toggle raw display
$37$ \( T^{2} + 265860 T - 66117717036 \) Copy content Toggle raw display
$41$ \( T^{2} + 111920 T - 58262616304 \) Copy content Toggle raw display
$43$ \( T^{2} - 947572 T + 197893738100 \) Copy content Toggle raw display
$47$ \( T^{2} - 271274 T - 123267405047 \) Copy content Toggle raw display
$53$ \( T^{2} + 1267792 T + 214640651072 \) Copy content Toggle raw display
$59$ \( T^{2} + 1360120 T - 818710818800 \) Copy content Toggle raw display
$61$ \( T^{2} + 1813680 T + 816706565136 \) Copy content Toggle raw display
$67$ \( T^{2} + 2189312 T - 10082635080448 \) Copy content Toggle raw display
$71$ \( T^{2} + 1494928 T - 1610731398080 \) Copy content Toggle raw display
$73$ \( T^{2} - 7169788 T + 10866534315332 \) Copy content Toggle raw display
$79$ \( T^{2} + 7942974 T + 14225767990465 \) Copy content Toggle raw display
$83$ \( T^{2} + 304712 T - 38809208931248 \) Copy content Toggle raw display
$89$ \( T^{2} + 17943528 T + 77796446568160 \) Copy content Toggle raw display
$97$ \( T^{2} - 4258074 T - 15859623239687 \) Copy content Toggle raw display
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