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Properties

Label 35.3.l.a.32.2
Level $35$
Weight $3$
Character 35.32
Analytic conductor $0.954$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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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(2,35)] 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("35.2"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(35, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([3, 4])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 35.l (of order \(12\), degree \(4\), 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: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 32.2
Character \(\chi\) \(=\) 35.32
Dual form 35.3.l.a.23.2

$q$-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+(-0.471334 + 1.75904i) q^{2} +(0.0524492 + 0.195743i) q^{3} +(0.592022 + 0.341804i) q^{4} +(-1.13206 + 4.87016i) q^{5} -0.369042 q^{6} +(-0.563209 - 6.97731i) q^{7} +(-6.03113 + 6.03113i) q^{8} +(7.75866 - 4.47947i) q^{9} +(-8.03325 - 4.28681i) q^{10} +(6.67094 - 11.5544i) q^{11} +(-0.0358547 + 0.133812i) q^{12} +(4.91251 - 4.91251i) q^{13} +(12.5388 + 2.29793i) q^{14} +(-1.01268 + 0.0338433i) q^{15} +(-6.39912 - 11.0836i) q^{16} +(-19.4385 + 5.20854i) q^{17} +(4.22265 + 15.7592i) q^{18} +(-14.0516 + 8.11267i) q^{19} +(-2.33484 + 2.49630i) q^{20} +(1.33622 - 0.476199i) q^{21} +(17.1805 + 17.1805i) q^{22} +(18.7480 + 5.02351i) q^{23} +(-1.49688 - 0.864224i) q^{24} +(-22.4369 - 11.0266i) q^{25} +(6.32588 + 10.9567i) q^{26} +(2.57341 + 2.57341i) q^{27} +(2.05144 - 4.32322i) q^{28} +29.5639i q^{29} +(0.417777 - 1.79729i) q^{30} +(9.49369 - 16.4435i) q^{31} +(-10.4420 + 2.79793i) q^{32} +(2.61158 + 0.699772i) q^{33} -36.6482i q^{34} +(34.6182 + 5.15580i) q^{35} +6.12440 q^{36} +(1.92294 - 7.17651i) q^{37} +(-7.64756 - 28.5411i) q^{38} +(1.21925 + 0.703933i) q^{39} +(-22.5450 - 36.2001i) q^{40} -52.1667 q^{41} +(0.207848 + 2.57492i) q^{42} +(-11.3609 + 11.3609i) q^{43} +(7.89869 - 4.56031i) q^{44} +(13.0325 + 42.8569i) q^{45} +(-17.6732 + 30.6108i) q^{46} +(14.2885 - 53.3255i) q^{47} +(1.83391 - 1.83391i) q^{48} +(-48.3656 + 7.85937i) q^{49} +(29.9716 - 34.2703i) q^{50} +(-2.03907 - 3.53178i) q^{51} +(4.58742 - 1.22920i) q^{52} +(-3.27711 - 12.2303i) q^{53} +(-5.73967 + 3.31380i) q^{54} +(48.7199 + 45.5688i) q^{55} +(45.4778 + 38.6842i) q^{56} +(-2.32499 - 2.32499i) q^{57} +(-52.0041 - 13.9345i) q^{58} +(12.0029 + 6.92989i) q^{59} +(-0.611094 - 0.326101i) q^{60} +(57.3589 + 99.3486i) q^{61} +(24.4502 + 24.4502i) q^{62} +(-35.6244 - 51.6117i) q^{63} -70.8797i q^{64} +(18.3634 + 29.4859i) q^{65} +(-2.46186 + 4.26406i) q^{66} +(56.0993 - 15.0318i) q^{67} +(-13.2883 - 3.56060i) q^{68} +3.93327i q^{69} +(-25.3860 + 58.4648i) q^{70} -86.5580 q^{71} +(-19.7773 + 73.8097i) q^{72} +(6.50169 + 24.2646i) q^{73} +(11.7175 + 6.76507i) q^{74} +(0.981586 - 4.97021i) q^{75} -11.0918 q^{76} +(-84.3758 - 40.0376i) q^{77} +(-1.81292 + 1.81292i) q^{78} +(-23.0541 + 13.3103i) q^{79} +(61.2231 - 18.6175i) q^{80} +(39.9464 - 69.1893i) q^{81} +(24.5880 - 91.7635i) q^{82} +(32.6537 - 32.6537i) q^{83} +(0.953838 + 0.174805i) q^{84} +(-3.36086 - 100.565i) q^{85} +(-14.6295 - 25.3390i) q^{86} +(-5.78692 + 1.55060i) q^{87} +(29.4528 + 109.919i) q^{88} +(-62.6554 + 36.1741i) q^{89} +(-81.5299 + 2.72471i) q^{90} +(-37.0428 - 31.5093i) q^{91} +(9.38217 + 9.38217i) q^{92} +(3.71665 + 0.995873i) q^{93} +(87.0673 + 50.2683i) q^{94} +(-23.6028 - 77.6173i) q^{95} +(-1.09535 - 1.89720i) q^{96} +(47.8969 + 47.8969i) q^{97} +(8.97139 - 88.7816i) q^{98} -119.529i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 2 q^{2} - 2 q^{3} - 4 q^{5} - 6 q^{7} - 36 q^{8} + 14 q^{10} - 24 q^{11} - 46 q^{12} - 8 q^{13} + 52 q^{15} + 20 q^{16} - 48 q^{17} - 4 q^{18} - 72 q^{20} + 56 q^{21} + 104 q^{22} - 86 q^{23} - 16 q^{25}+ \cdots + 482 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(22\) \(31\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.471334 + 1.75904i −0.235667 + 0.879522i 0.742180 + 0.670201i \(0.233792\pi\)
−0.977847 + 0.209321i \(0.932875\pi\)
\(3\) 0.0524492 + 0.195743i 0.0174831 + 0.0652477i 0.974116 − 0.226048i \(-0.0725805\pi\)
−0.956633 + 0.291295i \(0.905914\pi\)
\(4\) 0.592022 + 0.341804i 0.148005 + 0.0854510i
\(5\) −1.13206 + 4.87016i −0.226412 + 0.974032i
\(6\) −0.369042 −0.0615070
\(7\) −0.563209 − 6.97731i −0.0804585 − 0.996758i
\(8\) −6.03113 + 6.03113i −0.753891 + 0.753891i
\(9\) 7.75866 − 4.47947i 0.862074 − 0.497719i
\(10\) −8.03325 − 4.28681i −0.803325 − 0.428681i
\(11\) 6.67094 − 11.5544i 0.606449 − 1.05040i −0.385371 − 0.922762i \(-0.625927\pi\)
0.991821 − 0.127639i \(-0.0407400\pi\)
\(12\) −0.0358547 + 0.133812i −0.00298789 + 0.0111510i
\(13\) 4.91251 − 4.91251i 0.377885 − 0.377885i −0.492454 − 0.870339i \(-0.663900\pi\)
0.870339 + 0.492454i \(0.163900\pi\)
\(14\) 12.5388 + 2.29793i 0.895632 + 0.164138i
\(15\) −1.01268 + 0.0338433i −0.0675117 + 0.00225622i
\(16\) −6.39912 − 11.0836i −0.399945 − 0.692726i
\(17\) −19.4385 + 5.20854i −1.14344 + 0.306385i −0.780334 − 0.625362i \(-0.784951\pi\)
−0.363108 + 0.931747i \(0.618284\pi\)
\(18\) 4.22265 + 15.7592i 0.234592 + 0.875509i
\(19\) −14.0516 + 8.11267i −0.739555 + 0.426983i −0.821908 − 0.569621i \(-0.807090\pi\)
0.0823522 + 0.996603i \(0.473757\pi\)
\(20\) −2.33484 + 2.49630i −0.116742 + 0.124815i
\(21\) 1.33622 − 0.476199i 0.0636296 − 0.0226761i
\(22\) 17.1805 + 17.1805i 0.780931 + 0.780931i
\(23\) 18.7480 + 5.02351i 0.815131 + 0.218414i 0.642217 − 0.766523i \(-0.278015\pi\)
0.172914 + 0.984937i \(0.444682\pi\)
\(24\) −1.49688 − 0.864224i −0.0623700 − 0.0360093i
\(25\) −22.4369 − 11.0266i −0.897475 − 0.441064i
\(26\) 6.32588 + 10.9567i 0.243303 + 0.421413i
\(27\) 2.57341 + 2.57341i 0.0953114 + 0.0953114i
\(28\) 2.05144 − 4.32322i 0.0732657 − 0.154401i
\(29\) 29.5639i 1.01944i 0.860339 + 0.509722i \(0.170252\pi\)
−0.860339 + 0.509722i \(0.829748\pi\)
\(30\) 0.417777 − 1.79729i 0.0139259 − 0.0599098i
\(31\) 9.49369 − 16.4435i 0.306248 − 0.530437i −0.671290 − 0.741194i \(-0.734260\pi\)
0.977538 + 0.210757i \(0.0675930\pi\)
\(32\) −10.4420 + 2.79793i −0.326313 + 0.0874352i
\(33\) 2.61158 + 0.699772i 0.0791389 + 0.0212052i
\(34\) − 36.6482i − 1.07789i
\(35\) 34.6182 + 5.15580i 0.989091 + 0.147309i
\(36\) 6.12440 0.170122
\(37\) 1.92294 − 7.17651i 0.0519714 − 0.193960i −0.935059 − 0.354491i \(-0.884654\pi\)
0.987031 + 0.160531i \(0.0513208\pi\)
\(38\) −7.64756 − 28.5411i −0.201252 − 0.751081i
\(39\) 1.21925 + 0.703933i 0.0312627 + 0.0180496i
\(40\) −22.5450 − 36.2001i −0.563624 − 0.905003i
\(41\) −52.1667 −1.27236 −0.636179 − 0.771541i \(-0.719486\pi\)
−0.636179 + 0.771541i \(0.719486\pi\)
\(42\) 0.207848 + 2.57492i 0.00494876 + 0.0613076i
\(43\) −11.3609 + 11.3609i −0.264206 + 0.264206i −0.826760 − 0.562554i \(-0.809819\pi\)
0.562554 + 0.826760i \(0.309819\pi\)
\(44\) 7.89869 − 4.56031i 0.179516 − 0.103643i
\(45\) 13.0325 + 42.8569i 0.289610 + 0.952376i
\(46\) −17.6732 + 30.6108i −0.384199 + 0.665452i
\(47\) 14.2885 − 53.3255i 0.304011 − 1.13459i −0.629782 − 0.776772i \(-0.716856\pi\)
0.933793 − 0.357814i \(-0.116478\pi\)
\(48\) 1.83391 − 1.83391i 0.0382065 − 0.0382065i
\(49\) −48.3656 + 7.85937i −0.987053 + 0.160395i
\(50\) 29.9716 − 34.2703i 0.599431 − 0.685405i
\(51\) −2.03907 − 3.53178i −0.0399818 − 0.0692505i
\(52\) 4.58742 − 1.22920i 0.0882197 − 0.0236384i
\(53\) −3.27711 − 12.2303i −0.0618323 − 0.230761i 0.928094 − 0.372347i \(-0.121447\pi\)
−0.989926 + 0.141585i \(0.954780\pi\)
\(54\) −5.73967 + 3.31380i −0.106290 + 0.0613667i
\(55\) 48.7199 + 45.5688i 0.885817 + 0.828524i
\(56\) 45.4778 + 38.6842i 0.812104 + 0.690790i
\(57\) −2.32499 − 2.32499i −0.0407894 − 0.0407894i
\(58\) −52.0041 − 13.9345i −0.896623 − 0.240249i
\(59\) 12.0029 + 6.92989i 0.203439 + 0.117456i 0.598259 − 0.801303i \(-0.295859\pi\)
−0.394819 + 0.918759i \(0.629193\pi\)
\(60\) −0.611094 − 0.326101i −0.0101849 − 0.00543501i
\(61\) 57.3589 + 99.3486i 0.940311 + 1.62867i 0.764879 + 0.644174i \(0.222799\pi\)
0.175432 + 0.984492i \(0.443868\pi\)
\(62\) 24.4502 + 24.4502i 0.394358 + 0.394358i
\(63\) −35.6244 − 51.6117i −0.565466 − 0.819233i
\(64\) − 70.8797i − 1.10750i
\(65\) 18.3634 + 29.4859i 0.282514 + 0.453630i
\(66\) −2.46186 + 4.26406i −0.0373009 + 0.0646070i
\(67\) 56.0993 − 15.0318i 0.837302 − 0.224355i 0.185406 − 0.982662i \(-0.440640\pi\)
0.651897 + 0.758308i \(0.273973\pi\)
\(68\) −13.2883 − 3.56060i −0.195417 − 0.0523617i
\(69\) 3.93327i 0.0570040i
\(70\) −25.3860 + 58.4648i −0.362657 + 0.835211i
\(71\) −86.5580 −1.21913 −0.609563 − 0.792737i \(-0.708655\pi\)
−0.609563 + 0.792737i \(0.708655\pi\)
\(72\) −19.7773 + 73.8097i −0.274684 + 1.02514i
\(73\) 6.50169 + 24.2646i 0.0890643 + 0.332392i 0.996053 − 0.0887630i \(-0.0282914\pi\)
−0.906989 + 0.421155i \(0.861625\pi\)
\(74\) 11.7175 + 6.76507i 0.158344 + 0.0914199i
\(75\) 0.981586 − 4.97021i 0.0130878 − 0.0662694i
\(76\) −11.0918 −0.145944
\(77\) −84.3758 − 40.0376i −1.09579 − 0.519969i
\(78\) −1.81292 + 1.81292i −0.0232426 + 0.0232426i
\(79\) −23.0541 + 13.3103i −0.291824 + 0.168485i −0.638764 − 0.769402i \(-0.720554\pi\)
0.346940 + 0.937887i \(0.387221\pi\)
\(80\) 61.2231 − 18.6175i 0.765289 − 0.232718i
\(81\) 39.9464 − 69.1893i 0.493166 − 0.854189i
\(82\) 24.5880 − 91.7635i 0.299853 − 1.11907i
\(83\) 32.6537 − 32.6537i 0.393418 − 0.393418i −0.482486 − 0.875904i \(-0.660266\pi\)
0.875904 + 0.482486i \(0.160266\pi\)
\(84\) 0.953838 + 0.174805i 0.0113552 + 0.00208102i
\(85\) −3.36086 − 100.565i −0.0395395 − 1.18312i
\(86\) −14.6295 − 25.3390i −0.170110 − 0.294640i
\(87\) −5.78692 + 1.55060i −0.0665164 + 0.0178230i
\(88\) 29.4528 + 109.919i 0.334691 + 1.24908i
\(89\) −62.6554 + 36.1741i −0.703993 + 0.406451i −0.808833 − 0.588038i \(-0.799900\pi\)
0.104840 + 0.994489i \(0.466567\pi\)
\(90\) −81.5299 + 2.72471i −0.905888 + 0.0302745i
\(91\) −37.0428 − 31.5093i −0.407064 − 0.346256i
\(92\) 9.38217 + 9.38217i 0.101980 + 0.101980i
\(93\) 3.71665 + 0.995873i 0.0399640 + 0.0107083i
\(94\) 87.0673 + 50.2683i 0.926248 + 0.534769i
\(95\) −23.6028 − 77.6173i −0.248451 − 0.817024i
\(96\) −1.09535 − 1.89720i −0.0114099 − 0.0197625i
\(97\) 47.8969 + 47.8969i 0.493783 + 0.493783i 0.909496 − 0.415713i \(-0.136468\pi\)
−0.415713 + 0.909496i \(0.636468\pi\)
\(98\) 8.97139 − 88.7816i 0.0915448 − 0.905935i
\(99\) − 119.529i − 1.20736i
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.l.a.32.2 yes 24
3.2 odd 2 315.3.ca.a.172.5 24
5.2 odd 4 175.3.p.c.18.2 24
5.3 odd 4 inner 35.3.l.a.18.5 yes 24
5.4 even 2 175.3.p.c.32.5 24
7.2 even 3 inner 35.3.l.a.2.5 ✓ 24
7.3 odd 6 245.3.g.b.197.2 12
7.4 even 3 245.3.g.c.197.2 12
7.5 odd 6 245.3.m.b.177.5 24
7.6 odd 2 245.3.m.b.67.2 24
15.8 even 4 315.3.ca.a.298.2 24
21.2 odd 6 315.3.ca.a.37.2 24
35.2 odd 12 175.3.p.c.93.5 24
35.3 even 12 245.3.g.b.148.2 12
35.9 even 6 175.3.p.c.107.2 24
35.13 even 4 245.3.m.b.18.5 24
35.18 odd 12 245.3.g.c.148.2 12
35.23 odd 12 inner 35.3.l.a.23.2 yes 24
35.33 even 12 245.3.m.b.128.2 24
105.23 even 12 315.3.ca.a.163.5 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.3.l.a.2.5 ✓ 24 7.2 even 3 inner
35.3.l.a.18.5 yes 24 5.3 odd 4 inner
35.3.l.a.23.2 yes 24 35.23 odd 12 inner
35.3.l.a.32.2 yes 24 1.1 even 1 trivial
175.3.p.c.18.2 24 5.2 odd 4
175.3.p.c.32.5 24 5.4 even 2
175.3.p.c.93.5 24 35.2 odd 12
175.3.p.c.107.2 24 35.9 even 6
245.3.g.b.148.2 12 35.3 even 12
245.3.g.b.197.2 12 7.3 odd 6
245.3.g.c.148.2 12 35.18 odd 12
245.3.g.c.197.2 12 7.4 even 3
245.3.m.b.18.5 24 35.13 even 4
245.3.m.b.67.2 24 7.6 odd 2
245.3.m.b.128.2 24 35.33 even 12
245.3.m.b.177.5 24 7.5 odd 6
315.3.ca.a.37.2 24 21.2 odd 6
315.3.ca.a.163.5 24 105.23 even 12
315.3.ca.a.172.5 24 3.2 odd 2
315.3.ca.a.298.2 24 15.8 even 4