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Properties

Label 35.5.l.a.2.10
Level $35$
Weight $5$
Character 35.2
Analytic conductor $3.618$
Analytic rank $0$
Dimension $56$
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] = [35,5,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, 5); 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, 5, names="a")
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
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: \(3.61794870793\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 2.10
Character \(\chi\) \(=\) 35.2
Dual form 35.5.l.a.18.10

$q$-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+(3.05908 - 0.819678i) q^{2} +(-10.5242 - 2.81996i) q^{3} +(-5.17031 + 2.98508i) q^{4} +(-19.9856 - 15.0191i) q^{5} -34.5059 q^{6} +(9.03266 - 48.1603i) q^{7} +(-49.2000 + 49.2000i) q^{8} +(32.6590 + 18.8557i) q^{9} +(-73.4485 - 29.5629i) q^{10} +(38.5942 + 66.8471i) q^{11} +(62.8313 - 16.8356i) q^{12} +(38.9625 - 38.9625i) q^{13} +(-11.8443 - 154.730i) q^{14} +(167.980 + 214.423i) q^{15} +(-62.4173 + 108.110i) q^{16} +(48.5837 - 181.317i) q^{17} +(115.362 + 30.9112i) q^{18} +(-401.442 - 231.773i) q^{19} +(148.165 + 17.9948i) q^{20} +(-230.872 + 481.377i) q^{21} +(172.856 + 172.856i) q^{22} +(-132.900 - 495.990i) q^{23} +(656.534 - 379.050i) q^{24} +(173.851 + 600.334i) q^{25} +(87.2526 - 151.126i) q^{26} +(333.507 + 333.507i) q^{27} +(97.0606 + 275.967i) q^{28} +279.595i q^{29} +(689.622 + 518.248i) q^{30} +(-334.905 - 580.072i) q^{31} +(185.811 - 693.457i) q^{32} +(-217.668 - 812.347i) q^{33} -594.486i q^{34} +(-903.849 + 826.851i) q^{35} -225.143 q^{36} +(-2070.08 + 554.675i) q^{37} +(-1418.02 - 379.958i) q^{38} +(-519.922 + 300.177i) q^{39} +(1722.24 - 244.352i) q^{40} -2730.30 q^{41} +(-311.680 + 1661.81i) q^{42} +(1396.25 - 1396.25i) q^{43} +(-399.088 - 230.413i) q^{44} +(-369.515 - 867.353i) q^{45} +(-813.105 - 1408.34i) q^{46} +(3133.99 - 839.751i) q^{47} +(961.759 - 961.759i) q^{48} +(-2237.82 - 870.031i) q^{49} +(1023.91 + 1693.97i) q^{50} +(-1022.61 + 1771.21i) q^{51} +(-85.1420 + 317.754i) q^{52} +(3850.71 + 1031.80i) q^{53} +(1293.59 + 746.856i) q^{54} +(232.656 - 1915.63i) q^{55} +(1925.08 + 2813.89i) q^{56} +(3571.28 + 3571.28i) q^{57} +(229.178 + 855.303i) q^{58} +(1412.81 - 815.684i) q^{59} +(-1508.58 - 607.201i) q^{60} +(-2092.58 + 3624.46i) q^{61} +(-1499.97 - 1499.97i) q^{62} +(1203.09 - 1402.55i) q^{63} -4271.00i q^{64} +(-1363.87 + 193.507i) q^{65} +(-1331.73 - 2306.62i) q^{66} +(1390.20 - 5188.30i) q^{67} +(290.052 + 1082.49i) q^{68} +5594.69i q^{69} +(-2087.19 + 3270.27i) q^{70} +7243.25 q^{71} +(-2534.52 + 679.123i) q^{72} +(-1118.73 - 299.762i) q^{73} +(-5877.87 + 3393.59i) q^{74} +(-136.735 - 6808.30i) q^{75} +2767.44 q^{76} +(3567.98 - 1254.90i) q^{77} +(-1344.43 + 1344.43i) q^{78} +(-310.736 - 179.404i) q^{79} +(2871.17 - 1223.19i) q^{80} +(-4096.74 - 7095.76i) q^{81} +(-8352.20 + 2237.97i) q^{82} +(-6247.65 + 6247.65i) q^{83} +(-243.273 - 3178.04i) q^{84} +(-3694.20 + 2894.05i) q^{85} +(3126.78 - 5415.73i) q^{86} +(788.446 - 2942.52i) q^{87} +(-5187.71 - 1390.04i) q^{88} +(5665.46 + 3270.95i) q^{89} +(-1841.33 - 2350.42i) q^{90} +(-1524.51 - 2228.38i) q^{91} +(2167.71 + 2167.71i) q^{92} +(1888.83 + 7049.22i) q^{93} +(8898.81 - 5137.73i) q^{94} +(4542.06 + 10661.4i) q^{95} +(-3911.03 + 6774.11i) q^{96} +(-8795.78 - 8795.78i) q^{97} +(-7558.82 - 827.201i) q^{98} +2910.88i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q - 2 q^{2} - 2 q^{3} + 16 q^{5} - 144 q^{6} + 46 q^{7} + 108 q^{8} - 66 q^{10} + 296 q^{11} - 358 q^{12} - 8 q^{13} - 68 q^{15} + 468 q^{16} + 28 q^{17} - 868 q^{18} - 1032 q^{20} + 1280 q^{21} + 56 q^{22}+ \cdots - 78606 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.05908 − 0.819678i 0.764770 − 0.204919i 0.144710 − 0.989474i \(-0.453775\pi\)
0.620060 + 0.784555i \(0.287108\pi\)
\(3\) −10.5242 − 2.81996i −1.16936 − 0.313328i −0.378660 − 0.925536i \(-0.623615\pi\)
−0.790697 + 0.612207i \(0.790282\pi\)
\(4\) −5.17031 + 2.98508i −0.323144 + 0.186568i
\(5\) −19.9856 − 15.0191i −0.799425 − 0.600765i
\(6\) −34.5059 −0.958497
\(7\) 9.03266 − 48.1603i 0.184340 − 0.982863i
\(8\) −49.2000 + 49.2000i −0.768750 + 0.768750i
\(9\) 32.6590 + 18.8557i 0.403198 + 0.232786i
\(10\) −73.4485 − 29.5629i −0.734485 − 0.295629i
\(11\) 38.5942 + 66.8471i 0.318960 + 0.552455i 0.980271 − 0.197657i \(-0.0633331\pi\)
−0.661311 + 0.750112i \(0.730000\pi\)
\(12\) 62.8313 − 16.8356i 0.436328 − 0.116914i
\(13\) 38.9625 − 38.9625i 0.230547 − 0.230547i −0.582374 − 0.812921i \(-0.697876\pi\)
0.812921 + 0.582374i \(0.197876\pi\)
\(14\) −11.8443 − 154.730i −0.0604299 − 0.789439i
\(15\) 167.980 + 214.423i 0.746578 + 0.952992i
\(16\) −62.4173 + 108.110i −0.243818 + 0.422304i
\(17\) 48.5837 − 181.317i 0.168110 − 0.627394i −0.829513 − 0.558487i \(-0.811382\pi\)
0.997623 − 0.0689071i \(-0.0219512\pi\)
\(18\) 115.362 + 30.9112i 0.356056 + 0.0954048i
\(19\) −401.442 − 231.773i −1.11203 − 0.642030i −0.172675 − 0.984979i \(-0.555241\pi\)
−0.939354 + 0.342949i \(0.888574\pi\)
\(20\) 148.165 + 17.9948i 0.370413 + 0.0449871i
\(21\) −230.872 + 481.377i −0.523518 + 1.09156i
\(22\) 172.856 + 172.856i 0.357140 + 0.357140i
\(23\) −132.900 − 495.990i −0.251229 − 0.937600i −0.970150 − 0.242508i \(-0.922030\pi\)
0.718920 − 0.695092i \(-0.244637\pi\)
\(24\) 656.534 − 379.050i 1.13982 − 0.658073i
\(25\) 173.851 + 600.334i 0.278162 + 0.960534i
\(26\) 87.2526 − 151.126i 0.129072 − 0.223559i
\(27\) 333.507 + 333.507i 0.457486 + 0.457486i
\(28\) 97.0606 + 275.967i 0.123802 + 0.351998i
\(29\) 279.595i 0.332455i 0.986087 + 0.166228i \(0.0531587\pi\)
−0.986087 + 0.166228i \(0.946841\pi\)
\(30\) 689.622 + 518.248i 0.766247 + 0.575832i
\(31\) −334.905 − 580.072i −0.348496 − 0.603613i 0.637486 − 0.770462i \(-0.279974\pi\)
−0.985983 + 0.166849i \(0.946641\pi\)
\(32\) 185.811 − 693.457i 0.181456 − 0.677204i
\(33\) −217.668 − 812.347i −0.199879 − 0.745957i
\(34\) − 594.486i − 0.514261i
\(35\) −903.849 + 826.851i −0.737836 + 0.674980i
\(36\) −225.143 −0.173721
\(37\) −2070.08 + 554.675i −1.51211 + 0.405168i −0.917135 − 0.398577i \(-0.869504\pi\)
−0.594974 + 0.803745i \(0.702838\pi\)
\(38\) −1418.02 − 379.958i −0.982011 − 0.263129i
\(39\) −519.922 + 300.177i −0.341829 + 0.197355i
\(40\) 1722.24 − 244.352i 1.07640 − 0.152720i
\(41\) −2730.30 −1.62421 −0.812106 − 0.583510i \(-0.801679\pi\)
−0.812106 + 0.583510i \(0.801679\pi\)
\(42\) −311.680 + 1661.81i −0.176689 + 0.942070i
\(43\) 1396.25 − 1396.25i 0.755140 − 0.755140i −0.220293 − 0.975434i \(-0.570701\pi\)
0.975434 + 0.220293i \(0.0707014\pi\)
\(44\) −399.088 − 230.413i −0.206140 − 0.119015i
\(45\) −369.515 − 867.353i −0.182477 − 0.428322i
\(46\) −813.105 − 1408.34i −0.384265 − 0.665567i
\(47\) 3133.99 − 839.751i 1.41874 − 0.380150i 0.533702 − 0.845673i \(-0.320800\pi\)
0.885036 + 0.465523i \(0.154134\pi\)
\(48\) 961.759 − 961.759i 0.417430 − 0.417430i
\(49\) −2237.82 − 870.031i −0.932037 − 0.362362i
\(50\) 1023.91 + 1693.97i 0.409562 + 0.677587i
\(51\) −1022.61 + 1771.21i −0.393161 + 0.680974i
\(52\) −85.1420 + 317.754i −0.0314874 + 0.117513i
\(53\) 3850.71 + 1031.80i 1.37085 + 0.367318i 0.867789 − 0.496934i \(-0.165541\pi\)
0.503060 + 0.864251i \(0.332207\pi\)
\(54\) 1293.59 + 746.856i 0.443619 + 0.256124i
\(55\) 232.656 − 1915.63i 0.0769110 − 0.633267i
\(56\) 1925.08 + 2813.89i 0.613864 + 0.897287i
\(57\) 3571.28 + 3571.28i 1.09919 + 1.09919i
\(58\) 229.178 + 855.303i 0.0681266 + 0.254252i
\(59\) 1412.81 − 815.684i 0.405862 − 0.234325i −0.283148 − 0.959076i \(-0.591379\pi\)
0.689010 + 0.724752i \(0.258045\pi\)
\(60\) −1508.58 − 607.201i −0.419050 − 0.168667i
\(61\) −2092.58 + 3624.46i −0.562371 + 0.974056i 0.434918 + 0.900470i \(0.356778\pi\)
−0.997289 + 0.0735855i \(0.976556\pi\)
\(62\) −1499.97 − 1499.97i −0.390211 − 0.390211i
\(63\) 1203.09 − 1402.55i 0.303122 − 0.353376i
\(64\) − 4271.00i − 1.04272i
\(65\) −1363.87 + 193.507i −0.322810 + 0.0458006i
\(66\) −1331.73 − 2306.62i −0.305722 − 0.529526i
\(67\) 1390.20 − 5188.30i 0.309691 − 1.15578i −0.619141 − 0.785280i \(-0.712519\pi\)
0.928832 − 0.370502i \(-0.120814\pi\)
\(68\) 290.052 + 1082.49i 0.0627276 + 0.234103i
\(69\) 5594.69i 1.17511i
\(70\) −2087.19 + 3270.27i −0.425958 + 0.667401i
\(71\) 7243.25 1.43687 0.718433 − 0.695596i \(-0.244860\pi\)
0.718433 + 0.695596i \(0.244860\pi\)
\(72\) −2534.52 + 679.123i −0.488913 + 0.131004i
\(73\) −1118.73 − 299.762i −0.209932 − 0.0562510i 0.152320 − 0.988331i \(-0.451325\pi\)
−0.362252 + 0.932080i \(0.617992\pi\)
\(74\) −5877.87 + 3393.59i −1.07339 + 0.619721i
\(75\) −136.735 − 6808.30i −0.0243085 − 1.21036i
\(76\) 2767.44 0.479128
\(77\) 3567.98 − 1254.90i 0.601784 − 0.211654i
\(78\) −1344.43 + 1344.43i −0.220979 + 0.220979i
\(79\) −310.736 − 179.404i −0.0497895 − 0.0287460i 0.474899 − 0.880041i \(-0.342485\pi\)
−0.524688 + 0.851295i \(0.675818\pi\)
\(80\) 2871.17 − 1223.19i 0.448620 − 0.191124i
\(81\) −4096.74 − 7095.76i −0.624407 − 1.08151i
\(82\) −8352.20 + 2237.97i −1.24215 + 0.332833i
\(83\) −6247.65 + 6247.65i −0.906903 + 0.906903i −0.996021 − 0.0891185i \(-0.971595\pi\)
0.0891185 + 0.996021i \(0.471595\pi\)
\(84\) −243.273 − 3178.04i −0.0344774 − 0.450403i
\(85\) −3694.20 + 2894.05i −0.511308 + 0.400560i
\(86\) 3126.78 − 5415.73i 0.422766 − 0.732252i
\(87\) 788.446 − 2942.52i 0.104168 − 0.388759i
\(88\) −5187.71 − 1390.04i −0.669901 − 0.179499i
\(89\) 5665.46 + 3270.95i 0.715245 + 0.412947i 0.813000 − 0.582264i \(-0.197833\pi\)
−0.0977550 + 0.995211i \(0.531166\pi\)
\(90\) −1841.33 − 2350.42i −0.227324 − 0.290175i
\(91\) −1524.51 − 2228.38i −0.184097 − 0.269095i
\(92\) 2167.71 + 2167.71i 0.256109 + 0.256109i
\(93\) 1888.83 + 7049.22i 0.218387 + 0.815033i
\(94\) 8898.81 − 5137.73i 1.00711 − 0.581454i
\(95\) 4542.06 + 10661.4i 0.503275 + 1.18132i
\(96\) −3911.03 + 6774.11i −0.424374 + 0.735038i
\(97\) −8795.78 − 8795.78i −0.934826 − 0.934826i 0.0631762 − 0.998002i \(-0.479877\pi\)
−0.998002 + 0.0631762i \(0.979877\pi\)
\(98\) −7558.82 − 827.201i −0.787049 − 0.0861309i
\(99\) 2910.88i 0.296998i
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 35.5.l.a.2.10 ✓ 56
5.3 odd 4 inner 35.5.l.a.23.5 yes 56
7.4 even 3 inner 35.5.l.a.32.5 yes 56
35.18 odd 12 inner 35.5.l.a.18.10 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.5.l.a.2.10 ✓ 56 1.1 even 1 trivial
35.5.l.a.18.10 yes 56 35.18 odd 12 inner
35.5.l.a.23.5 yes 56 5.3 odd 4 inner
35.5.l.a.32.5 yes 56 7.4 even 3 inner