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Properties

Label 35.5.l.a.2.4
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 2.4
Character \(\chi\) \(=\) 35.2
Dual form 35.5.l.a.18.4

$q$-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+(-5.20038 + 1.39344i) q^{2} +(2.56504 + 0.687301i) q^{3} +(11.2458 - 6.49278i) q^{4} +(-19.5926 + 15.5283i) q^{5} -14.2969 q^{6} +(20.1640 - 44.6589i) q^{7} +(11.4758 - 11.4758i) q^{8} +(-64.0410 - 36.9741i) q^{9} +(80.2514 - 108.054i) q^{10} +(-91.3743 - 158.265i) q^{11} +(33.3085 - 8.92500i) q^{12} +(141.894 - 141.894i) q^{13} +(-42.6310 + 260.340i) q^{14} +(-60.9286 + 26.3647i) q^{15} +(-147.572 + 255.602i) q^{16} +(-110.602 + 412.771i) q^{17} +(384.558 + 103.042i) q^{18} +(-213.281 - 123.138i) q^{19} +(-119.514 + 301.840i) q^{20} +(82.4155 - 100.693i) q^{21} +(695.713 + 695.713i) q^{22} +(-132.073 - 492.904i) q^{23} +(37.3231 - 21.5485i) q^{24} +(142.743 - 608.481i) q^{25} +(-540.180 + 935.619i) q^{26} +(-290.953 - 290.953i) q^{27} +(-63.1996 - 633.147i) q^{28} +1081.57i q^{29} +(280.114 - 222.007i) q^{30} +(-227.280 - 393.660i) q^{31} +(344.058 - 1284.04i) q^{32} +(-125.603 - 468.758i) q^{33} -2300.68i q^{34} +(298.411 + 1188.10i) q^{35} -960.259 q^{36} +(72.2086 - 19.3482i) q^{37} +(1280.72 + 343.169i) q^{38} +(461.486 - 266.439i) q^{39} +(-46.6412 + 403.039i) q^{40} -1376.70 q^{41} +(-288.282 + 638.483i) q^{42} +(-493.766 + 493.766i) q^{43} +(-2055.16 - 1186.55i) q^{44} +(1828.88 - 270.029i) q^{45} +(1373.66 + 2379.25i) q^{46} +(826.377 - 221.427i) q^{47} +(-554.204 + 554.204i) q^{48} +(-1587.83 - 1801.00i) q^{49} +(105.563 + 3363.23i) q^{50} +(-567.396 + 982.759i) q^{51} +(674.427 - 2517.00i) q^{52} +(-2893.14 - 775.214i) q^{53} +(1918.49 + 1107.64i) q^{54} +(4247.85 + 1681.94i) q^{55} +(-281.097 - 743.891i) q^{56} +(-462.441 - 462.441i) q^{57} +(-1507.10 - 5624.57i) q^{58} +(1513.47 - 873.800i) q^{59} +(-514.012 + 692.090i) q^{60} +(1639.85 - 2840.31i) q^{61} +(1730.48 + 1730.48i) q^{62} +(-2942.54 + 2114.45i) q^{63} +2434.61i q^{64} +(-576.701 + 4983.44i) q^{65} +(1306.37 + 2262.70i) q^{66} +(-700.989 + 2616.13i) q^{67} +(1436.23 + 5360.07i) q^{68} -1355.09i q^{69} +(-3207.39 - 5762.74i) q^{70} +242.826 q^{71} +(-1159.22 + 310.613i) q^{72} +(4723.25 + 1265.59i) q^{73} +(-348.551 + 201.236i) q^{74} +(784.352 - 1462.67i) q^{75} -3198.02 q^{76} +(-8910.40 + 889.421i) q^{77} +(-2028.64 + 2028.64i) q^{78} +(3161.63 + 1825.37i) q^{79} +(-1077.75 - 7299.47i) q^{80} +(2448.57 + 4241.04i) q^{81} +(7159.35 - 1918.34i) q^{82} +(7249.03 - 7249.03i) q^{83} +(273.052 - 1667.48i) q^{84} +(-4242.66 - 9804.74i) q^{85} +(1879.74 - 3255.80i) q^{86} +(-743.364 + 2774.27i) q^{87} +(-2864.80 - 767.620i) q^{88} +(2122.71 + 1225.55i) q^{89} +(-9134.58 + 3952.68i) q^{90} +(-3475.67 - 9197.94i) q^{91} +(-4685.60 - 4685.60i) q^{92} +(-312.419 - 1165.96i) q^{93} +(-3988.92 + 2303.01i) q^{94} +(6090.85 - 899.297i) q^{95} +(1765.05 - 3057.15i) q^{96} +(1798.31 + 1798.31i) q^{97} +(10766.9 + 7153.34i) q^{98} +13513.9i 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.20038 + 1.39344i −1.30009 + 0.348359i −0.841486 − 0.540279i \(-0.818319\pi\)
−0.458608 + 0.888639i \(0.651652\pi\)
\(3\) 2.56504 + 0.687301i 0.285005 + 0.0763668i 0.398489 − 0.917173i \(-0.369535\pi\)
−0.113485 + 0.993540i \(0.536201\pi\)
\(4\) 11.2458 − 6.49278i 0.702865 − 0.405799i
\(5\) −19.5926 + 15.5283i −0.783706 + 0.621133i
\(6\) −14.2969 −0.397136
\(7\) 20.1640 − 44.6589i 0.411510 − 0.911405i
\(8\) 11.4758 − 11.4758i 0.179309 − 0.179309i
\(9\) −64.0410 − 36.9741i −0.790630 − 0.456470i
\(10\) 80.2514 − 108.054i 0.802514 − 1.08054i
\(11\) −91.3743 − 158.265i −0.755159 − 1.30797i −0.945295 − 0.326216i \(-0.894226\pi\)
0.190136 − 0.981758i \(-0.439107\pi\)
\(12\) 33.3085 − 8.92500i 0.231309 − 0.0619791i
\(13\) 141.894 − 141.894i 0.839607 − 0.839607i −0.149200 − 0.988807i \(-0.547670\pi\)
0.988807 + 0.149200i \(0.0476699\pi\)
\(14\) −42.6310 + 260.340i −0.217505 + 1.32827i
\(15\) −60.9286 + 26.3647i −0.270794 + 0.117177i
\(16\) −147.572 + 255.602i −0.576453 + 0.998446i
\(17\) −110.602 + 412.771i −0.382705 + 1.42827i 0.459048 + 0.888412i \(0.348191\pi\)
−0.841753 + 0.539863i \(0.818476\pi\)
\(18\) 384.558 + 103.042i 1.18691 + 0.318031i
\(19\) −213.281 − 123.138i −0.590805 − 0.341101i 0.174611 − 0.984638i \(-0.444133\pi\)
−0.765416 + 0.643536i \(0.777467\pi\)
\(20\) −119.514 + 301.840i −0.298784 + 0.754599i
\(21\) 82.4155 − 100.693i 0.186883 − 0.228329i
\(22\) 695.713 + 695.713i 1.43742 + 1.43742i
\(23\) −132.073 − 492.904i −0.249666 − 0.931766i −0.970981 − 0.239158i \(-0.923128\pi\)
0.721315 − 0.692608i \(-0.243538\pi\)
\(24\) 37.3231 − 21.5485i 0.0647970 − 0.0374106i
\(25\) 142.743 − 608.481i 0.228389 − 0.973570i
\(26\) −540.180 + 935.619i −0.799083 + 1.38405i
\(27\) −290.953 − 290.953i −0.399112 − 0.399112i
\(28\) −63.1996 − 633.147i −0.0806118 − 0.807585i
\(29\) 1081.57i 1.28605i 0.765844 + 0.643026i \(0.222321\pi\)
−0.765844 + 0.643026i \(0.777679\pi\)
\(30\) 280.114 − 222.007i 0.311238 − 0.246674i
\(31\) −227.280 − 393.660i −0.236503 − 0.409636i 0.723205 − 0.690633i \(-0.242668\pi\)
−0.959709 + 0.280997i \(0.909335\pi\)
\(32\) 344.058 − 1284.04i 0.335994 − 1.25395i
\(33\) −125.603 − 468.758i −0.115338 − 0.430448i
\(34\) − 2300.68i − 1.99021i
\(35\) 298.411 + 1188.10i 0.243601 + 0.969876i
\(36\) −960.259 −0.740941
\(37\) 72.2086 − 19.3482i 0.0527455 − 0.0141331i −0.232350 − 0.972632i \(-0.574641\pi\)
0.285095 + 0.958499i \(0.407975\pi\)
\(38\) 1280.72 + 343.169i 0.886927 + 0.237651i
\(39\) 461.486 − 266.439i 0.303410 − 0.175174i
\(40\) −46.6412 + 403.039i −0.0291507 + 0.251900i
\(41\) −1376.70 −0.818976 −0.409488 − 0.912315i \(-0.634293\pi\)
−0.409488 + 0.912315i \(0.634293\pi\)
\(42\) −288.282 + 638.483i −0.163425 + 0.361952i
\(43\) −493.766 + 493.766i −0.267045 + 0.267045i −0.827908 − 0.560864i \(-0.810469\pi\)
0.560864 + 0.827908i \(0.310469\pi\)
\(44\) −2055.16 − 1186.55i −1.06155 − 0.612886i
\(45\) 1828.88 − 270.029i 0.903149 − 0.133348i
\(46\) 1373.66 + 2379.25i 0.649178 + 1.12441i
\(47\) 826.377 − 221.427i 0.374095 − 0.100239i −0.0668723 − 0.997762i \(-0.521302\pi\)
0.440968 + 0.897523i \(0.354635\pi\)
\(48\) −554.204 + 554.204i −0.240540 + 0.240540i
\(49\) −1587.83 − 1801.00i −0.661319 − 0.750104i
\(50\) 105.563 + 3363.23i 0.0422251 + 1.34529i
\(51\) −567.396 + 982.759i −0.218145 + 0.377839i
\(52\) 674.427 − 2517.00i 0.249418 − 0.930841i
\(53\) −2893.14 − 775.214i −1.02995 − 0.275975i −0.296009 − 0.955185i \(-0.595656\pi\)
−0.733944 + 0.679210i \(0.762323\pi\)
\(54\) 1918.49 + 1107.64i 0.657917 + 0.379849i
\(55\) 4247.85 + 1681.94i 1.40425 + 0.556013i
\(56\) −281.097 − 743.891i −0.0896356 − 0.237210i
\(57\) −462.441 − 462.441i −0.142333 − 0.142333i
\(58\) −1507.10 − 5624.57i −0.448008 − 1.67199i
\(59\) 1513.47 − 873.800i 0.434779 − 0.251020i −0.266601 − 0.963807i \(-0.585901\pi\)
0.701381 + 0.712787i \(0.252567\pi\)
\(60\) −514.012 + 692.090i −0.142781 + 0.192247i
\(61\) 1639.85 − 2840.31i 0.440702 − 0.763318i −0.557040 − 0.830486i \(-0.688063\pi\)
0.997742 + 0.0671678i \(0.0213963\pi\)
\(62\) 1730.48 + 1730.48i 0.450177 + 0.450177i
\(63\) −2942.54 + 2114.45i −0.741381 + 0.532742i
\(64\) 2434.61i 0.594388i
\(65\) −576.701 + 4983.44i −0.136497 + 1.17951i
\(66\) 1306.37 + 2262.70i 0.299901 + 0.519444i
\(67\) −700.989 + 2616.13i −0.156157 + 0.582786i 0.842846 + 0.538154i \(0.180878\pi\)
−0.999003 + 0.0446321i \(0.985788\pi\)
\(68\) 1436.23 + 5360.07i 0.310603 + 1.15919i
\(69\) − 1355.09i − 0.284624i
\(70\) −3207.39 − 5762.74i −0.654569 − 1.17607i
\(71\) 242.826 0.0481701 0.0240851 − 0.999710i \(-0.492333\pi\)
0.0240851 + 0.999710i \(0.492333\pi\)
\(72\) −1159.22 + 310.613i −0.223616 + 0.0599177i
\(73\) 4723.25 + 1265.59i 0.886330 + 0.237491i 0.673136 − 0.739519i \(-0.264947\pi\)
0.213194 + 0.977010i \(0.431613\pi\)
\(74\) −348.551 + 201.236i −0.0636507 + 0.0367488i
\(75\) 784.352 − 1462.67i 0.139440 − 0.260031i
\(76\) −3198.02 −0.553674
\(77\) −8910.40 + 889.421i −1.50285 + 0.150012i
\(78\) −2028.64 + 2028.64i −0.333438 + 0.333438i
\(79\) 3161.63 + 1825.37i 0.506591 + 0.292480i 0.731431 − 0.681915i \(-0.238853\pi\)
−0.224840 + 0.974396i \(0.572186\pi\)
\(80\) −1077.75 − 7299.47i −0.168398 − 1.14054i
\(81\) 2448.57 + 4241.04i 0.373200 + 0.646402i
\(82\) 7159.35 − 1918.34i 1.06475 − 0.285298i
\(83\) 7249.03 − 7249.03i 1.05226 − 1.05226i 0.0537046 − 0.998557i \(-0.482897\pi\)
0.998557 − 0.0537046i \(-0.0171029\pi\)
\(84\) 273.052 − 1667.48i 0.0386979 − 0.236322i
\(85\) −4242.66 − 9804.74i −0.587220 − 1.35706i
\(86\) 1879.74 − 3255.80i 0.254156 − 0.440211i
\(87\) −743.364 + 2774.27i −0.0982116 + 0.366531i
\(88\) −2864.80 − 767.620i −0.369938 − 0.0991245i
\(89\) 2122.71 + 1225.55i 0.267986 + 0.154722i 0.627972 − 0.778236i \(-0.283885\pi\)
−0.359986 + 0.932958i \(0.617219\pi\)
\(90\) −9134.58 + 3952.68i −1.12773 + 0.487985i
\(91\) −3475.67 − 9197.94i −0.419716 − 1.11073i
\(92\) −4685.60 − 4685.60i −0.553591 − 0.553591i
\(93\) −312.419 − 1165.96i −0.0361220 − 0.134809i
\(94\) −3988.92 + 2303.01i −0.451440 + 0.260639i
\(95\) 6090.85 − 899.297i 0.674886 − 0.0996451i
\(96\) 1765.05 − 3057.15i 0.191520 − 0.331722i
\(97\) 1798.31 + 1798.31i 0.191126 + 0.191126i 0.796183 − 0.605056i \(-0.206849\pi\)
−0.605056 + 0.796183i \(0.706849\pi\)
\(98\) 10766.9 + 7153.34i 1.12108 + 0.744830i
\(99\) 13513.9i 1.37883i
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.2.4 ✓ 56
5.3 odd 4 inner 35.5.l.a.23.11 yes 56
7.4 even 3 inner 35.5.l.a.32.11 yes 56
35.18 odd 12 inner 35.5.l.a.18.4 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.5.l.a.2.4 ✓ 56 1.1 even 1 trivial
35.5.l.a.18.4 yes 56 35.18 odd 12 inner
35.5.l.a.23.11 yes 56 5.3 odd 4 inner
35.5.l.a.32.11 yes 56 7.4 even 3 inner