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Properties

Label 35.3.h.a.31.6
Level $35$
Weight $3$
Character 35.31
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 31.6
Root \(-1.68940 + 2.92612i\) of defining polynomial
Character \(\chi\) \(=\) 35.31
Dual form 35.3.h.a.26.6

$q$-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+(1.68940 + 2.92612i) q^{2} +(-1.83681 - 1.06048i) q^{3} +(-3.70812 + 6.42265i) q^{4} +(1.93649 - 1.11803i) q^{5} -7.16630i q^{6} +(4.91879 - 4.98051i) q^{7} -11.5428 q^{8} +(-2.25076 - 3.89842i) q^{9} +(6.54300 + 3.77760i) q^{10} +(-7.37106 + 12.7670i) q^{11} +(13.6222 - 7.86478i) q^{12} -12.3492i q^{13} +(22.8834 + 5.97891i) q^{14} -4.74262 q^{15} +(-4.66781 - 8.08488i) q^{16} +(-2.62269 - 1.51421i) q^{17} +(7.60484 - 13.1720i) q^{18} +(-22.5227 + 13.0035i) q^{19} +16.5832i q^{20} +(-14.3166 + 3.93196i) q^{21} -49.8106 q^{22} +(-1.54543 - 2.67676i) q^{23} +(21.2018 + 12.2409i) q^{24} +(2.50000 - 4.33013i) q^{25} +(36.1351 - 20.8626i) q^{26} +28.6362i q^{27} +(13.7486 + 50.0600i) q^{28} +53.5925 q^{29} +(-8.01216 - 13.8775i) q^{30} +(-24.5204 - 14.1568i) q^{31} +(-7.31395 + 12.6681i) q^{32} +(27.0785 - 15.6338i) q^{33} -10.2324i q^{34} +(3.95681 - 15.1441i) q^{35} +33.3843 q^{36} +(-1.08686 - 1.88250i) q^{37} +(-76.0995 - 43.9361i) q^{38} +(-13.0961 + 22.6831i) q^{39} +(-22.3524 + 12.9052i) q^{40} +42.3289i q^{41} +(-35.6918 - 35.2495i) q^{42} +27.3860 q^{43} +(-54.6655 - 94.6835i) q^{44} +(-8.71714 - 5.03284i) q^{45} +(5.22168 - 9.04421i) q^{46} +(44.6800 - 25.7960i) q^{47} +19.8005i q^{48} +(-0.611033 - 48.9962i) q^{49} +16.8940 q^{50} +(3.21159 + 5.56264i) q^{51} +(79.3144 + 45.7922i) q^{52} +(8.99520 - 15.5801i) q^{53} +(-83.7930 + 48.3779i) q^{54} +32.9644i q^{55} +(-56.7764 + 57.4888i) q^{56} +55.1599 q^{57} +(90.5390 + 156.818i) q^{58} +(19.1472 + 11.0546i) q^{59} +(17.5862 - 30.4602i) q^{60} +(-12.6950 + 7.32944i) q^{61} -95.6660i q^{62} +(-30.4871 - 7.96560i) q^{63} -86.7671 q^{64} +(-13.8068 - 23.9141i) q^{65} +(91.4925 + 52.8232i) q^{66} +(-7.55988 + 13.0941i) q^{67} +(19.4505 - 11.2298i) q^{68} +6.55559i q^{69} +(50.9981 - 14.0063i) q^{70} -58.9099 q^{71} +(25.9799 + 44.9985i) q^{72} +(-38.6399 - 22.3087i) q^{73} +(3.67227 - 6.36056i) q^{74} +(-9.18404 + 5.30241i) q^{75} -192.874i q^{76} +(27.3298 + 99.5101i) q^{77} -88.4978 q^{78} +(-9.00142 - 15.5909i) q^{79} +(-18.0783 - 10.4375i) q^{80} +(10.1114 - 17.5134i) q^{81} +(-123.860 + 71.5103i) q^{82} +72.1159i q^{83} +(27.8341 - 106.531i) q^{84} -6.77177 q^{85} +(46.2659 + 80.1348i) q^{86} +(-98.4392 - 56.8339i) q^{87} +(85.0823 - 147.367i) q^{88} +(-8.38533 + 4.84127i) q^{89} -34.0099i q^{90} +(-61.5052 - 60.7429i) q^{91} +22.9225 q^{92} +(30.0261 + 52.0068i) q^{93} +(150.964 + 87.1593i) q^{94} +(-29.0767 + 50.3623i) q^{95} +(26.8687 - 15.5126i) q^{96} -145.969i q^{97} +(142.336 - 84.5619i) q^{98} +66.3618 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 \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.68940 + 2.92612i 0.844698 + 1.46306i 0.885883 + 0.463908i \(0.153553\pi\)
−0.0411851 + 0.999152i \(0.513113\pi\)
\(3\) −1.83681 − 1.06048i −0.612270 − 0.353494i 0.161584 − 0.986859i \(-0.448340\pi\)
−0.773853 + 0.633365i \(0.781673\pi\)
\(4\) −3.70812 + 6.42265i −0.927030 + 1.60566i
\(5\) 1.93649 − 1.11803i 0.387298 − 0.223607i
\(6\) − 7.16630i − 1.19438i
\(7\) 4.91879 − 4.98051i 0.702684 − 0.711502i
\(8\) −11.5428 −1.44284
\(9\) −2.25076 − 3.89842i −0.250084 − 0.433158i
\(10\) 6.54300 + 3.77760i 0.654300 + 0.377760i
\(11\) −7.37106 + 12.7670i −0.670096 + 1.16064i 0.307780 + 0.951458i \(0.400414\pi\)
−0.977876 + 0.209183i \(0.932919\pi\)
\(12\) 13.6222 − 7.86478i 1.13518 − 0.655399i
\(13\) − 12.3492i − 0.949936i −0.880003 − 0.474968i \(-0.842460\pi\)
0.880003 − 0.474968i \(-0.157540\pi\)
\(14\) 22.8834 + 5.97891i 1.63453 + 0.427065i
\(15\) −4.74262 −0.316175
\(16\) −4.66781 − 8.08488i −0.291738 − 0.505305i
\(17\) −2.62269 − 1.51421i −0.154276 − 0.0890714i 0.420875 − 0.907119i \(-0.361723\pi\)
−0.575151 + 0.818047i \(0.695057\pi\)
\(18\) 7.60484 − 13.1720i 0.422491 − 0.731776i
\(19\) −22.5227 + 13.0035i −1.18541 + 0.684394i −0.957259 − 0.289233i \(-0.906600\pi\)
−0.228146 + 0.973627i \(0.573267\pi\)
\(20\) 16.5832i 0.829160i
\(21\) −14.3166 + 3.93196i −0.681744 + 0.187236i
\(22\) −49.8106 −2.26412
\(23\) −1.54543 − 2.67676i −0.0671925 − 0.116381i 0.830472 − 0.557060i \(-0.188071\pi\)
−0.897664 + 0.440680i \(0.854737\pi\)
\(24\) 21.2018 + 12.2409i 0.883409 + 0.510037i
\(25\) 2.50000 − 4.33013i 0.100000 − 0.173205i
\(26\) 36.1351 − 20.8626i 1.38981 − 0.802409i
\(27\) 28.6362i 1.06060i
\(28\) 13.7486 + 50.0600i 0.491023 + 1.78786i
\(29\) 53.5925 1.84802 0.924009 − 0.382372i \(-0.124893\pi\)
0.924009 + 0.382372i \(0.124893\pi\)
\(30\) −8.01216 − 13.8775i −0.267072 − 0.462582i
\(31\) −24.5204 − 14.1568i −0.790979 − 0.456672i 0.0493280 − 0.998783i \(-0.484292\pi\)
−0.840307 + 0.542111i \(0.817625\pi\)
\(32\) −7.31395 + 12.6681i −0.228561 + 0.395879i
\(33\) 27.0785 − 15.6338i 0.820559 − 0.473750i
\(34\) − 10.2324i − 0.300954i
\(35\) 3.95681 − 15.1441i 0.113052 − 0.432688i
\(36\) 33.3843 0.927341
\(37\) −1.08686 − 1.88250i −0.0293746 − 0.0508782i 0.850964 − 0.525223i \(-0.176018\pi\)
−0.880339 + 0.474345i \(0.842685\pi\)
\(38\) −76.0995 − 43.9361i −2.00262 − 1.15621i
\(39\) −13.0961 + 22.6831i −0.335797 + 0.581617i
\(40\) −22.3524 + 12.9052i −0.558811 + 0.322630i
\(41\) 42.3289i 1.03241i 0.856464 + 0.516207i \(0.172656\pi\)
−0.856464 + 0.516207i \(0.827344\pi\)
\(42\) −35.6918 − 35.2495i −0.849806 − 0.839274i
\(43\) 27.3860 0.636885 0.318442 − 0.947942i \(-0.396840\pi\)
0.318442 + 0.947942i \(0.396840\pi\)
\(44\) −54.6655 − 94.6835i −1.24240 − 2.15190i
\(45\) −8.71714 − 5.03284i −0.193714 − 0.111841i
\(46\) 5.22168 − 9.04421i 0.113515 − 0.196613i
\(47\) 44.6800 − 25.7960i 0.950637 − 0.548851i 0.0573585 − 0.998354i \(-0.481732\pi\)
0.893279 + 0.449503i \(0.148399\pi\)
\(48\) 19.8005i 0.412511i
\(49\) −0.611033 − 48.9962i −0.0124701 − 0.999922i
\(50\) 16.8940 0.337879
\(51\) 3.21159 + 5.56264i 0.0629724 + 0.109071i
\(52\) 79.3144 + 45.7922i 1.52528 + 0.880619i
\(53\) 8.99520 − 15.5801i 0.169721 − 0.293965i −0.768601 − 0.639729i \(-0.779047\pi\)
0.938322 + 0.345764i \(0.112380\pi\)
\(54\) −83.7930 + 48.3779i −1.55172 + 0.895887i
\(55\) 32.9644i 0.599352i
\(56\) −56.7764 + 57.4888i −1.01386 + 1.02659i
\(57\) 55.1599 0.967717
\(58\) 90.5390 + 156.818i 1.56102 + 2.70376i
\(59\) 19.1472 + 11.0546i 0.324529 + 0.187367i 0.653409 − 0.757005i \(-0.273338\pi\)
−0.328881 + 0.944371i \(0.606671\pi\)
\(60\) 17.5862 − 30.4602i 0.293103 − 0.507670i
\(61\) −12.6950 + 7.32944i −0.208114 + 0.120155i −0.600435 − 0.799674i \(-0.705006\pi\)
0.392321 + 0.919829i \(0.371672\pi\)
\(62\) − 95.6660i − 1.54300i
\(63\) −30.4871 − 7.96560i −0.483923 − 0.126438i
\(64\) −86.7671 −1.35574
\(65\) −13.8068 − 23.9141i −0.212412 − 0.367909i
\(66\) 91.4925 + 52.8232i 1.38625 + 0.800351i
\(67\) −7.55988 + 13.0941i −0.112834 + 0.195434i −0.916912 − 0.399090i \(-0.869326\pi\)
0.804078 + 0.594524i \(0.202660\pi\)
\(68\) 19.4505 − 11.2298i 0.286037 − 0.165144i
\(69\) 6.55559i 0.0950086i
\(70\) 50.9981 − 14.0063i 0.728544 − 0.200090i
\(71\) −58.9099 −0.829717 −0.414859 − 0.909886i \(-0.636169\pi\)
−0.414859 + 0.909886i \(0.636169\pi\)
\(72\) 25.9799 + 44.9985i 0.360832 + 0.624980i
\(73\) −38.6399 − 22.3087i −0.529314 − 0.305599i 0.211423 − 0.977395i \(-0.432190\pi\)
−0.740737 + 0.671795i \(0.765523\pi\)
\(74\) 3.67227 − 6.36056i 0.0496253 − 0.0859535i
\(75\) −9.18404 + 5.30241i −0.122454 + 0.0706988i
\(76\) − 192.874i − 2.53781i
\(77\) 27.3298 + 99.5101i 0.354932 + 1.29234i
\(78\) −88.4978 −1.13459
\(79\) −9.00142 − 15.5909i −0.113942 − 0.197353i 0.803414 − 0.595420i \(-0.203014\pi\)
−0.917356 + 0.398067i \(0.869681\pi\)
\(80\) −18.0783 − 10.4375i −0.225979 − 0.130469i
\(81\) 10.1114 − 17.5134i 0.124832 − 0.216215i
\(82\) −123.860 + 71.5103i −1.51048 + 0.872077i
\(83\) 72.1159i 0.868866i 0.900704 + 0.434433i \(0.143051\pi\)
−0.900704 + 0.434433i \(0.856949\pi\)
\(84\) 27.8341 − 106.531i 0.331358 − 1.26822i
\(85\) −6.77177 −0.0796679
\(86\) 46.2659 + 80.1348i 0.537975 + 0.931800i
\(87\) −98.4392 − 56.8339i −1.13148 − 0.653263i
\(88\) 85.0823 − 147.367i 0.966845 − 1.67462i
\(89\) −8.38533 + 4.84127i −0.0942172 + 0.0543963i −0.546368 − 0.837545i \(-0.683990\pi\)
0.452151 + 0.891941i \(0.350657\pi\)
\(90\) − 34.0099i − 0.377887i
\(91\) −61.5052 − 60.7429i −0.675881 − 0.667505i
\(92\) 22.9225 0.249158
\(93\) 30.0261 + 52.0068i 0.322862 + 0.559213i
\(94\) 150.964 + 87.1593i 1.60600 + 0.927226i
\(95\) −29.0767 + 50.3623i −0.306070 + 0.530129i
\(96\) 26.8687 − 15.5126i 0.279882 − 0.161590i
\(97\) − 145.969i − 1.50483i −0.658688 − 0.752416i \(-0.728888\pi\)
0.658688 − 0.752416i \(-0.271112\pi\)
\(98\) 142.336 − 84.5619i 1.45241 − 0.862877i
\(99\) 66.3618 0.670322
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.31.6 yes 12
3.2 odd 2 315.3.w.c.136.1 12
4.3 odd 2 560.3.bx.c.241.4 12
5.2 odd 4 175.3.j.b.24.2 24
5.3 odd 4 175.3.j.b.24.11 24
5.4 even 2 175.3.i.d.101.1 12
7.2 even 3 245.3.h.c.166.6 12
7.3 odd 6 245.3.d.a.146.1 12
7.4 even 3 245.3.d.a.146.2 12
7.5 odd 6 inner 35.3.h.a.26.6 ✓ 12
7.6 odd 2 245.3.h.c.31.6 12
21.5 even 6 315.3.w.c.271.1 12
28.19 even 6 560.3.bx.c.481.4 12
35.12 even 12 175.3.j.b.124.11 24
35.19 odd 6 175.3.i.d.26.1 12
35.33 even 12 175.3.j.b.124.2 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.3.h.a.26.6 ✓ 12 7.5 odd 6 inner
35.3.h.a.31.6 yes 12 1.1 even 1 trivial
175.3.i.d.26.1 12 35.19 odd 6
175.3.i.d.101.1 12 5.4 even 2
175.3.j.b.24.2 24 5.2 odd 4
175.3.j.b.24.11 24 5.3 odd 4
175.3.j.b.124.2 24 35.33 even 12
175.3.j.b.124.11 24 35.12 even 12
245.3.d.a.146.1 12 7.3 odd 6
245.3.d.a.146.2 12 7.4 even 3
245.3.h.c.31.6 12 7.6 odd 2
245.3.h.c.166.6 12 7.2 even 3
315.3.w.c.136.1 12 3.2 odd 2
315.3.w.c.271.1 12 21.5 even 6
560.3.bx.c.241.4 12 4.3 odd 2
560.3.bx.c.481.4 12 28.19 even 6