Properties

Label 35.10.a.b
Level $35$
Weight $10$
Character orbit 35.a
Self dual yes
Analytic conductor $18.026$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 35.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.0262542657\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) 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 \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 12) q^{2} + (54 \beta - 87) q^{3} + ( - 24 \beta - 360) q^{4} + 625 q^{5} + ( - 735 \beta + 1476) q^{6} + 2401 q^{7} + ( - 584 \beta + 10272) q^{8} + ( - 9396 \beta + 11214) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 12) q^{2} + (54 \beta - 87) q^{3} + ( - 24 \beta - 360) q^{4} + 625 q^{5} + ( - 735 \beta + 1476) q^{6} + 2401 q^{7} + ( - 584 \beta + 10272) q^{8} + ( - 9396 \beta + 11214) q^{9} + (625 \beta - 7500) q^{10} + (9124 \beta + 9283) q^{11} + ( - 17352 \beta + 20952) q^{12} + ( - 18334 \beta - 25545) q^{13} + (2401 \beta - 28812) q^{14} + (33750 \beta - 54375) q^{15} + (29568 \beta + 56384) q^{16} + ( - 15186 \beta - 186955) q^{17} + (123966 \beta - 209736) q^{18} + (31138 \beta - 71638) q^{19} + ( - 15000 \beta - 225000) q^{20} + (129654 \beta - 208887) q^{21} + ( - 100205 \beta - 38404) q^{22} + ( - 821442 \beta - 249454) q^{23} + (605496 \beta - 1145952) q^{24} + 390625 q^{25} + (194463 \beta + 159868) q^{26} + (360126 \beta - 3322269) q^{27} + ( - 57624 \beta - 864360) q^{28} + ( - 296772 \beta - 5788777) q^{29} + ( - 459375 \beta + 922500) q^{30} + ( - 2191070 \beta - 1976880) q^{31} + (576 \beta - 5699328) q^{32} + ( - 292506 \beta + 3133947) q^{33} + ( - 4723 \beta + 2121972) q^{34} + 1500625 q^{35} + (3113424 \beta - 2233008) q^{36} + (3997070 \beta - 1602706) q^{37} + ( - 445294 \beta + 1108760) q^{38} + (215628 \beta - 5697873) q^{39} + ( - 365000 \beta + 6420000) q^{40} + (10352502 \beta + 529496) q^{41} + ( - 1764735 \beta + 3543876) q^{42} + ( - 9725678 \beta + 7974090) q^{43} + ( - 3507432 \beta - 5093688) q^{44} + ( - 5872500 \beta + 7008750) q^{45} + (9607850 \beta - 3578088) q^{46} + ( - 9479722 \beta + 32750645) q^{47} + (472320 \beta + 7867968) q^{48} + 5764801 q^{49} + (390625 \beta - 4687500) q^{50} + ( - 8774388 \beta + 9704733) q^{51} + (7213320 \beta + 12716328) q^{52} + ( - 5256968 \beta - 12557344) q^{53} + ( - 7643781 \beta + 42748236) q^{54} + (5702500 \beta + 5801875) q^{55} + ( - 1402184 \beta + 24663072) q^{56} + ( - 6577458 \beta + 19684122) q^{57} + ( - 2227513 \beta + 67091148) q^{58} + (39092800 \beta - 58079604) q^{59} + ( - 10845000 \beta + 13095000) q^{60} + (15794106 \beta - 22344272) q^{61} + (24315960 \beta + 6194000) q^{62} + ( - 22559796 \beta + 26924814) q^{63} + ( - 20845056 \beta + 39527936) q^{64} + ( - 11458750 \beta - 15965625) q^{65} + (6644019 \beta - 39947412) q^{66} + (76121736 \beta + 59046248) q^{67} + (9953880 \beta + 70219512) q^{68} + (57994938 \beta - 333160446) q^{69} + (1500625 \beta - 18007500) q^{70} + ( - 76576000 \beta - 147082912) q^{71} + ( - 103064688 \beta + 159088320) q^{72} + (17548324 \beta - 28709666) q^{73} + ( - 49567546 \beta + 51209032) q^{74} + (21093750 \beta - 33984375) q^{75} + ( - 9490368 \beta + 19811184) q^{76} + (21906724 \beta + 22288483) q^{77} + ( - 8285409 \beta + 70099500) q^{78} + (28471816 \beta - 346426427) q^{79} + (18480000 \beta + 35240000) q^{80} + ( - 25792020 \beta + 223886673) q^{81} + ( - 123700528 \beta + 76466064) q^{82} + (92242900 \beta - 270339964) q^{83} + ( - 41662152 \beta + 50305752) q^{84} + ( - 9491250 \beta - 116846875) q^{85} + (124682226 \beta - 173494504) q^{86} + ( - 286774794 \beta + 375418095) q^{87} + (88300456 \beta + 52727648) q^{88} + ( - 97352322 \beta - 389521852) q^{89} + (77478750 \beta - 131085000) q^{90} + ( - 44019934 \beta - 61333545) q^{91} + (301706016 \beta + 247520304) q^{92} + (83871570 \beta - 774553680) q^{93} + (146507309 \beta - 468845516) q^{94} + (19461250 \beta - 44773750) q^{95} + ( - 307813824 \beta + 496090368) q^{96} + (45681470 \beta - 1336519703) q^{97} + (5764801 \beta - 69177612) q^{98} + (15093468 \beta - 581733270) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 24 q^{2} - 174 q^{3} - 720 q^{4} + 1250 q^{5} + 2952 q^{6} + 4802 q^{7} + 20544 q^{8} + 22428 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 24 q^{2} - 174 q^{3} - 720 q^{4} + 1250 q^{5} + 2952 q^{6} + 4802 q^{7} + 20544 q^{8} + 22428 q^{9} - 15000 q^{10} + 18566 q^{11} + 41904 q^{12} - 51090 q^{13} - 57624 q^{14} - 108750 q^{15} + 112768 q^{16} - 373910 q^{17} - 419472 q^{18} - 143276 q^{19} - 450000 q^{20} - 417774 q^{21} - 76808 q^{22} - 498908 q^{23} - 2291904 q^{24} + 781250 q^{25} + 319736 q^{26} - 6644538 q^{27} - 1728720 q^{28} - 11577554 q^{29} + 1845000 q^{30} - 3953760 q^{31} - 11398656 q^{32} + 6267894 q^{33} + 4243944 q^{34} + 3001250 q^{35} - 4466016 q^{36} - 3205412 q^{37} + 2217520 q^{38} - 11395746 q^{39} + 12840000 q^{40} + 1058992 q^{41} + 7087752 q^{42} + 15948180 q^{43} - 10187376 q^{44} + 14017500 q^{45} - 7156176 q^{46} + 65501290 q^{47} + 15735936 q^{48} + 11529602 q^{49} - 9375000 q^{50} + 19409466 q^{51} + 25432656 q^{52} - 25114688 q^{53} + 85496472 q^{54} + 11603750 q^{55} + 49326144 q^{56} + 39368244 q^{57} + 134182296 q^{58} - 116159208 q^{59} + 26190000 q^{60} - 44688544 q^{61} + 12388000 q^{62} + 53849628 q^{63} + 79055872 q^{64} - 31931250 q^{65} - 79894824 q^{66} + 118092496 q^{67} + 140439024 q^{68} - 666320892 q^{69} - 36015000 q^{70} - 294165824 q^{71} + 318176640 q^{72} - 57419332 q^{73} + 102418064 q^{74} - 67968750 q^{75} + 39622368 q^{76} + 44576966 q^{77} + 140199000 q^{78} - 692852854 q^{79} + 70480000 q^{80} + 447773346 q^{81} + 152932128 q^{82} - 540679928 q^{83} + 100611504 q^{84} - 233693750 q^{85} - 346989008 q^{86} + 750836190 q^{87} + 105455296 q^{88} - 779043704 q^{89} - 262170000 q^{90} - 122667090 q^{91} + 495040608 q^{92} - 1549107360 q^{93} - 937691032 q^{94} - 89547500 q^{95} + 992180736 q^{96} - 2673039406 q^{97} - 138355224 q^{98} - 1163466540 q^{99}+O(q^{100}) \) 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
−1.41421
1.41421
−14.8284 −239.735 −292.118 625.000 3554.89 2401.00 11923.8 37789.9 −9267.77
1.2 −9.17157 65.7351 −427.882 625.000 −602.894 2401.00 8620.20 −15361.9 −5732.23
\(n\): 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 35.10.a.b 2
3.b odd 2 1 315.10.a.b 2
5.b even 2 1 175.10.a.c 2
5.c odd 4 2 175.10.b.c 4
7.b odd 2 1 245.10.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.10.a.b 2 1.a even 1 1 trivial
175.10.a.c 2 5.b even 2 1
175.10.b.c 4 5.c odd 4 2
245.10.a.c 2 7.b odd 2 1
315.10.a.b 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 24T_{2} + 136 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(35))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 24T + 136 \) Copy content Toggle raw display
$3$ \( T^{2} + 174T - 15759 \) Copy content Toggle raw display
$5$ \( (T - 625)^{2} \) Copy content Toggle raw display
$7$ \( (T - 2401)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 18566 T - 579804919 \) Copy content Toggle raw display
$13$ \( T^{2} + 51090 T - 2036537423 \) Copy content Toggle raw display
$17$ \( T^{2} + 373910 T + 33107255257 \) Copy content Toggle raw display
$19$ \( T^{2} + 143276 T - 2624597308 \) Copy content Toggle raw display
$23$ \( T^{2} + 498908 T - 5335908376796 \) Copy content Toggle raw display
$29$ \( T^{2} + 11577554 T + 32805350195857 \) Copy content Toggle raw display
$31$ \( T^{2} + 3953760 T - 34498247424800 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 125243882156764 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 857114015266016 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 693124389149372 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 353683714337753 \) Copy content Toggle raw display
$53$ \( T^{2} + 25114688 T - 63398812089856 \) Copy content Toggle raw display
$59$ \( T^{2} + 116159208 T - 88\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( T^{2} + 44688544 T - 14\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{2} - 118092496 T - 42\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{2} + 294165824 T - 25\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{2} + 57419332 T - 16\!\cdots\!52 \) Copy content Toggle raw display
$79$ \( T^{2} + 692852854 T + 11\!\cdots\!81 \) Copy content Toggle raw display
$83$ \( T^{2} + 540679928 T + 50\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{2} + 779043704 T + 75\!\cdots\!32 \) Copy content Toggle raw display
$97$ \( T^{2} + 2673039406 T + 17\!\cdots\!09 \) Copy content Toggle raw display
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