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Properties

Label 35.3.h.a.26.4
Level $35$
Weight $3$
Character 35.26
Analytic conductor $0.954$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [35,3,Mod(26,35)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("35.26"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(35, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 5])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 35.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.953680925261\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 19 x^{10} - 26 x^{9} + 244 x^{8} - 338 x^{7} + 1249 x^{6} - 986 x^{5} + 3532 x^{4} + \cdots + 441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 26.4
Root \(0.242987 + 0.420865i\) of defining polynomial
Character \(\chi\) \(=\) 35.26
Dual form 35.3.h.a.31.4

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.242987 + 0.420865i) q^{2} +(-4.74894 + 2.74180i) q^{3} +(1.88192 + 3.25957i) q^{4} +(-1.93649 - 1.11803i) q^{5} -2.66488i q^{6} +(5.87843 + 3.80053i) q^{7} -3.77301 q^{8} +(10.5350 - 18.2471i) q^{9} +(0.941083 - 0.543334i) q^{10} +(3.29893 + 5.71391i) q^{11} +(-17.8742 - 10.3197i) q^{12} +3.29523i q^{13} +(-3.02789 + 1.55055i) q^{14} +12.2617 q^{15} +(-6.61087 + 11.4504i) q^{16} +(9.44949 - 5.45566i) q^{17} +(5.11970 + 8.86759i) q^{18} +(8.27840 + 4.77954i) q^{19} -8.41618i q^{20} +(-38.3366 - 1.93099i) q^{21} -3.20638 q^{22} +(5.80641 - 10.0570i) q^{23} +(17.9178 - 10.3449i) q^{24} +(2.50000 + 4.33013i) q^{25} +(-1.38685 - 0.800696i) q^{26} +66.1866i q^{27} +(-1.32539 + 26.3135i) q^{28} -23.6149 q^{29} +(-2.97943 + 5.16052i) q^{30} +(9.22362 - 5.32526i) q^{31} +(-10.7587 - 18.6347i) q^{32} +(-31.3328 - 18.0900i) q^{33} +5.30261i q^{34} +(-7.13441 - 13.9320i) q^{35} +79.3035 q^{36} +(31.3288 - 54.2630i) q^{37} +(-4.02308 + 2.32273i) q^{38} +(-9.03486 - 15.6488i) q^{39} +(7.30641 + 4.21836i) q^{40} -49.7889i q^{41} +(10.1280 - 15.6653i) q^{42} -7.82740 q^{43} +(-12.4166 + 21.5062i) q^{44} +(-40.8017 + 23.5569i) q^{45} +(2.82176 + 4.88743i) q^{46} +(26.4161 + 15.2513i) q^{47} -72.5028i q^{48} +(20.1119 + 44.6823i) q^{49} -2.42987 q^{50} +(-29.9167 + 51.8172i) q^{51} +(-10.7410 + 6.20134i) q^{52} +(31.0976 + 53.8627i) q^{53} +(-27.8556 - 16.0824i) q^{54} -14.7532i q^{55} +(-22.1794 - 14.3394i) q^{56} -52.4182 q^{57} +(5.73811 - 9.93870i) q^{58} +(-39.0701 + 22.5571i) q^{59} +(23.0755 + 39.9679i) q^{60} +(35.0955 + 20.2624i) q^{61} +5.17587i q^{62} +(131.278 - 67.2258i) q^{63} -42.4300 q^{64} +(3.68418 - 6.38118i) q^{65} +(15.2269 - 8.79125i) q^{66} +(-26.3967 - 45.7204i) q^{67} +(35.5663 + 20.5342i) q^{68} +63.6802i q^{69} +(7.59705 + 0.382659i) q^{70} -48.2295 q^{71} +(-39.7485 + 68.8464i) q^{72} +(-25.2610 + 14.5844i) q^{73} +(15.2249 + 26.3704i) q^{74} +(-23.7447 - 13.7090i) q^{75} +35.9787i q^{76} +(-2.32337 + 46.1265i) q^{77} +8.78140 q^{78} +(5.47308 - 9.47965i) q^{79} +(25.6038 - 14.7824i) q^{80} +(-86.6559 - 150.092i) q^{81} +(20.9544 + 12.0980i) q^{82} -14.0223i q^{83} +(-65.8520 - 128.595i) q^{84} -24.3985 q^{85} +(1.90195 - 3.29428i) q^{86} +(112.146 - 64.7474i) q^{87} +(-12.4469 - 21.5586i) q^{88} +(91.7289 + 52.9597i) q^{89} -22.8960i q^{90} +(-12.5236 + 19.3708i) q^{91} +43.7087 q^{92} +(-29.2016 + 50.5787i) q^{93} +(-12.8375 + 7.41174i) q^{94} +(-10.6874 - 18.5111i) q^{95} +(102.185 + 58.9966i) q^{96} -118.638i q^{97} +(-23.6922 - 2.39279i) q^{98} +139.016 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{2} - 6 q^{3} - 10 q^{4} - 2 q^{7} - 4 q^{8} + 14 q^{9} - 14 q^{11} + 18 q^{12} - 2 q^{14} - 20 q^{15} - 22 q^{16} + 48 q^{17} + 64 q^{18} - 30 q^{19} - 84 q^{21} - 88 q^{22} - 14 q^{23} - 36 q^{24}+ \cdots - 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(22\) \(31\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.242987 + 0.420865i −0.121493 + 0.210433i −0.920357 − 0.391080i \(-0.872102\pi\)
0.798863 + 0.601512i \(0.205435\pi\)
\(3\) −4.74894 + 2.74180i −1.58298 + 0.913934i −0.588558 + 0.808455i \(0.700304\pi\)
−0.994422 + 0.105479i \(0.966362\pi\)
\(4\) 1.88192 + 3.25957i 0.470479 + 0.814893i
\(5\) −1.93649 − 1.11803i −0.387298 − 0.223607i
\(6\) − 2.66488i − 0.444147i
\(7\) 5.87843 + 3.80053i 0.839776 + 0.542933i
\(8\) −3.77301 −0.471627
\(9\) 10.5350 − 18.2471i 1.17055 − 2.02745i
\(10\) 0.941083 − 0.543334i 0.0941083 − 0.0543334i
\(11\) 3.29893 + 5.71391i 0.299902 + 0.519446i 0.976113 − 0.217262i \(-0.0697126\pi\)
−0.676211 + 0.736708i \(0.736379\pi\)
\(12\) −17.8742 − 10.3197i −1.48952 − 0.859973i
\(13\) 3.29523i 0.253479i 0.991936 + 0.126740i \(0.0404513\pi\)
−0.991936 + 0.126740i \(0.959549\pi\)
\(14\) −3.02789 + 1.55055i −0.216278 + 0.110753i
\(15\) 12.2617 0.817447
\(16\) −6.61087 + 11.4504i −0.413179 + 0.715648i
\(17\) 9.44949 − 5.45566i 0.555852 − 0.320921i −0.195627 − 0.980678i \(-0.562674\pi\)
0.751479 + 0.659757i \(0.229341\pi\)
\(18\) 5.11970 + 8.86759i 0.284428 + 0.492644i
\(19\) 8.27840 + 4.77954i 0.435705 + 0.251555i 0.701774 − 0.712399i \(-0.252392\pi\)
−0.266069 + 0.963954i \(0.585725\pi\)
\(20\) − 8.41618i − 0.420809i
\(21\) −38.3366 − 1.93099i −1.82555 − 0.0919521i
\(22\) −3.20638 −0.145744
\(23\) 5.80641 − 10.0570i 0.252453 − 0.437261i −0.711748 − 0.702435i \(-0.752096\pi\)
0.964201 + 0.265174i \(0.0854294\pi\)
\(24\) 17.9178 − 10.3449i 0.746575 − 0.431035i
\(25\) 2.50000 + 4.33013i 0.100000 + 0.173205i
\(26\) −1.38685 − 0.800696i −0.0533403 − 0.0307960i
\(27\) 66.1866i 2.45135i
\(28\) −1.32539 + 26.3135i −0.0473355 + 0.939766i
\(29\) −23.6149 −0.814308 −0.407154 − 0.913360i \(-0.633479\pi\)
−0.407154 + 0.913360i \(0.633479\pi\)
\(30\) −2.97943 + 5.16052i −0.0993143 + 0.172017i
\(31\) 9.22362 − 5.32526i 0.297536 − 0.171783i −0.343799 − 0.939043i \(-0.611714\pi\)
0.641336 + 0.767261i \(0.278381\pi\)
\(32\) −10.7587 − 18.6347i −0.336210 − 0.582333i
\(33\) −31.3328 − 18.0900i −0.949479 − 0.548182i
\(34\) 5.30261i 0.155959i
\(35\) −7.13441 − 13.9320i −0.203840 − 0.398057i
\(36\) 79.3035 2.20288
\(37\) 31.3288 − 54.2630i 0.846724 − 1.46657i −0.0373921 − 0.999301i \(-0.511905\pi\)
0.884116 − 0.467268i \(-0.154762\pi\)
\(38\) −4.02308 + 2.32273i −0.105871 + 0.0611244i
\(39\) −9.03486 − 15.6488i −0.231663 − 0.401252i
\(40\) 7.30641 + 4.21836i 0.182660 + 0.105459i
\(41\) − 49.7889i − 1.21436i −0.794563 − 0.607182i \(-0.792300\pi\)
0.794563 − 0.607182i \(-0.207700\pi\)
\(42\) 10.1280 − 15.6653i 0.241142 − 0.372984i
\(43\) −7.82740 −0.182033 −0.0910163 − 0.995849i \(-0.529012\pi\)
−0.0910163 + 0.995849i \(0.529012\pi\)
\(44\) −12.4166 + 21.5062i −0.282195 + 0.488777i
\(45\) −40.8017 + 23.5569i −0.906704 + 0.523486i
\(46\) 2.82176 + 4.88743i 0.0613426 + 0.106249i
\(47\) 26.4161 + 15.2513i 0.562044 + 0.324496i 0.753966 − 0.656914i \(-0.228138\pi\)
−0.191921 + 0.981410i \(0.561472\pi\)
\(48\) − 72.5028i − 1.51047i
\(49\) 20.1119 + 44.6823i 0.410448 + 0.911884i
\(50\) −2.42987 −0.0485973
\(51\) −29.9167 + 51.8172i −0.586602 + 1.01602i
\(52\) −10.7410 + 6.20134i −0.206558 + 0.119257i
\(53\) 31.0976 + 53.8627i 0.586748 + 1.01628i 0.994655 + 0.103254i \(0.0329254\pi\)
−0.407907 + 0.913023i \(0.633741\pi\)
\(54\) −27.8556 − 16.0824i −0.515845 − 0.297823i
\(55\) − 14.7532i − 0.268241i
\(56\) −22.1794 − 14.3394i −0.396061 − 0.256062i
\(57\) −52.4182 −0.919617
\(58\) 5.73811 − 9.93870i 0.0989329 − 0.171357i
\(59\) −39.0701 + 22.5571i −0.662205 + 0.382324i −0.793117 − 0.609070i \(-0.791543\pi\)
0.130911 + 0.991394i \(0.458210\pi\)
\(60\) 23.0755 + 39.9679i 0.384592 + 0.666132i
\(61\) 35.0955 + 20.2624i 0.575336 + 0.332170i 0.759278 − 0.650767i \(-0.225552\pi\)
−0.183942 + 0.982937i \(0.558886\pi\)
\(62\) 5.17587i 0.0834817i
\(63\) 131.278 − 67.2258i 2.08377 − 1.06708i
\(64\) −42.4300 −0.662969
\(65\) 3.68418 − 6.38118i 0.0566797 − 0.0981721i
\(66\) 15.2269 − 8.79125i 0.230711 − 0.133201i
\(67\) −26.3967 − 45.7204i −0.393981 − 0.682395i 0.598990 − 0.800757i \(-0.295569\pi\)
−0.992970 + 0.118362i \(0.962236\pi\)
\(68\) 35.5663 + 20.5342i 0.523033 + 0.301973i
\(69\) 63.6802i 0.922901i
\(70\) 7.59705 + 0.382659i 0.108529 + 0.00546656i
\(71\) −48.2295 −0.679289 −0.339644 − 0.940554i \(-0.610307\pi\)
−0.339644 + 0.940554i \(0.610307\pi\)
\(72\) −39.7485 + 68.8464i −0.552063 + 0.956200i
\(73\) −25.2610 + 14.5844i −0.346040 + 0.199787i −0.662940 − 0.748673i \(-0.730692\pi\)
0.316900 + 0.948459i \(0.397358\pi\)
\(74\) 15.2249 + 26.3704i 0.205742 + 0.356356i
\(75\) −23.7447 − 13.7090i −0.316596 − 0.182787i
\(76\) 35.9787i 0.473404i
\(77\) −2.32337 + 46.1265i −0.0301736 + 0.599045i
\(78\) 8.78140 0.112582
\(79\) 5.47308 − 9.47965i 0.0692795 − 0.119996i −0.829305 − 0.558796i \(-0.811263\pi\)
0.898584 + 0.438801i \(0.144597\pi\)
\(80\) 25.6038 − 14.7824i 0.320047 − 0.184779i
\(81\) −86.6559 − 150.092i −1.06983 − 1.85299i
\(82\) 20.9544 + 12.0980i 0.255542 + 0.147537i
\(83\) − 14.0223i − 0.168943i −0.996426 − 0.0844717i \(-0.973080\pi\)
0.996426 − 0.0844717i \(-0.0269203\pi\)
\(84\) −65.8520 − 128.595i −0.783953 − 1.53089i
\(85\) −24.3985 −0.287041
\(86\) 1.90195 − 3.29428i 0.0221157 − 0.0383056i
\(87\) 112.146 − 64.7474i 1.28903 − 0.744223i
\(88\) −12.4469 − 21.5586i −0.141442 − 0.244985i
\(89\) 91.7289 + 52.9597i 1.03066 + 0.595053i 0.917174 − 0.398487i \(-0.130465\pi\)
0.113487 + 0.993539i \(0.463798\pi\)
\(90\) − 22.8960i − 0.254400i
\(91\) −12.5236 + 19.3708i −0.137622 + 0.212866i
\(92\) 43.7087 0.475095
\(93\) −29.2016 + 50.5787i −0.313996 + 0.543857i
\(94\) −12.8375 + 7.41174i −0.136569 + 0.0788483i
\(95\) −10.6874 − 18.5111i −0.112499 − 0.194853i
\(96\) 102.185 + 58.9966i 1.06443 + 0.614548i
\(97\) − 118.638i − 1.22307i −0.791218 − 0.611534i \(-0.790553\pi\)
0.791218 − 0.611534i \(-0.209447\pi\)
\(98\) −23.6922 − 2.39279i −0.241757 − 0.0244163i
\(99\) 139.016 1.40420
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 35.3.h.a.26.4 ✓ 12
3.2 odd 2 315.3.w.c.271.3 12
4.3 odd 2 560.3.bx.c.481.6 12
5.2 odd 4 175.3.j.b.124.6 24
5.3 odd 4 175.3.j.b.124.7 24
5.4 even 2 175.3.i.d.26.3 12
7.2 even 3 245.3.d.a.146.5 12
7.3 odd 6 inner 35.3.h.a.31.4 yes 12
7.4 even 3 245.3.h.c.31.4 12
7.5 odd 6 245.3.d.a.146.6 12
7.6 odd 2 245.3.h.c.166.4 12
21.17 even 6 315.3.w.c.136.3 12
28.3 even 6 560.3.bx.c.241.6 12
35.3 even 12 175.3.j.b.24.6 24
35.17 even 12 175.3.j.b.24.7 24
35.24 odd 6 175.3.i.d.101.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.3.h.a.26.4 ✓ 12 1.1 even 1 trivial
35.3.h.a.31.4 yes 12 7.3 odd 6 inner
175.3.i.d.26.3 12 5.4 even 2
175.3.i.d.101.3 12 35.24 odd 6
175.3.j.b.24.6 24 35.3 even 12
175.3.j.b.24.7 24 35.17 even 12
175.3.j.b.124.6 24 5.2 odd 4
175.3.j.b.124.7 24 5.3 odd 4
245.3.d.a.146.5 12 7.2 even 3
245.3.d.a.146.6 12 7.5 odd 6
245.3.h.c.31.4 12 7.4 even 3
245.3.h.c.166.4 12 7.6 odd 2
315.3.w.c.136.3 12 21.17 even 6
315.3.w.c.271.3 12 3.2 odd 2
560.3.bx.c.241.6 12 28.3 even 6
560.3.bx.c.481.6 12 4.3 odd 2