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Properties

Label 35.5.l.a.23.11
Level $35$
Weight $5$
Character 35.23
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 23.11
Character \(\chi\) \(=\) 35.23
Dual form 35.5.l.a.32.11

$q$-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+(1.39344 + 5.20038i) q^{2} +(-0.687301 + 2.56504i) q^{3} +(-11.2458 + 6.49278i) q^{4} +(-3.65159 + 24.7319i) q^{5} -14.2969 q^{6} +(-44.6589 - 20.1640i) q^{7} +(11.4758 + 11.4758i) q^{8} +(64.0410 + 36.9741i) q^{9} +(-133.703 + 15.4726i) q^{10} +(-91.3743 - 158.265i) q^{11} +(-8.92500 - 33.3085i) 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} +(412.771 + 110.602i) q^{17} +(-103.042 + 384.558i) 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} +(492.904 - 132.073i) q^{23} +(-37.3231 + 21.5485i) q^{24} +(-598.332 - 180.622i) q^{25} +(-540.180 + 935.619i) q^{26} +(-290.953 + 290.953i) q^{27} +(633.147 - 63.1996i) q^{28} -1081.57i q^{29} +(52.2065 - 353.589i) q^{30} +(-227.280 - 393.660i) q^{31} +(-1284.04 - 344.058i) q^{32} +(468.758 - 125.603i) q^{33} +2300.68i q^{34} +(661.769 - 1030.87i) q^{35} -960.259 q^{36} +(-19.3482 - 72.2086i) q^{37} +(-343.169 + 1280.72i) q^{38} +(-461.486 + 266.439i) q^{39} +(-325.722 + 241.912i) q^{40} -1376.70 q^{41} +(638.483 + 288.282i) q^{42} +(-493.766 - 493.766i) q^{43} +(2055.16 + 1186.55i) q^{44} +(-1148.29 + 1448.84i) q^{45} +(1373.66 + 2379.25i) q^{46} +(-221.427 - 826.377i) 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} +(-2517.00 - 674.427i) q^{52} +(775.214 - 2893.14i) 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} +(5624.57 - 1507.10i) q^{58} +(-1513.47 + 873.800i) q^{59} +(856.373 - 99.1026i) q^{60} +(1639.85 - 2840.31i) q^{61} +(1730.48 - 1730.48i) q^{62} +(-2114.45 - 2942.54i) q^{63} -2434.61i q^{64} +(-4027.43 + 2991.16i) q^{65} +(1306.37 + 2262.70i) q^{66} +(2616.13 + 700.989i) q^{67} +(-5360.07 + 1436.23i) q^{68} +1355.09i q^{69} +(6283.03 + 2005.00i) q^{70} +242.826 q^{71} +(310.613 + 1159.22i) q^{72} +(-1265.59 + 4723.25i) q^{73} +(348.551 - 201.236i) q^{74} +(874.536 - 1410.60i) q^{75} -3198.02 q^{76} +(889.421 + 8910.40i) q^{77} +(-2028.64 - 2028.64i) q^{78} +(-3161.63 - 1825.37i) q^{79} +(-5782.65 - 4583.09i) q^{80} +(2448.57 + 4241.04i) q^{81} +(-1918.34 - 7159.35i) 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} +(2774.27 + 743.364i) q^{87} +(767.620 - 2864.80i) 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} +(1165.96 - 312.419i) q^{93} +(3988.92 - 2303.01i) q^{94} +(-3824.24 + 4825.18i) q^{95} +(1765.05 - 3057.15i) q^{96} +(1798.31 - 1798.31i) q^{97} +(-7153.34 + 10766.9i) 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 \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.39344 + 5.20038i 0.348359 + 1.30009i 0.888639 + 0.458608i \(0.151652\pi\)
−0.540279 + 0.841486i \(0.681681\pi\)
\(3\) −0.687301 + 2.56504i −0.0763668 + 0.285005i −0.993540 − 0.113485i \(-0.963799\pi\)
0.917173 + 0.398489i \(0.130465\pi\)
\(4\) −11.2458 + 6.49278i −0.702865 + 0.405799i
\(5\) −3.65159 + 24.7319i −0.146064 + 0.989275i
\(6\) −14.2969 −0.397136
\(7\) −44.6589 − 20.1640i −0.911405 − 0.411510i
\(8\) 11.4758 + 11.4758i 0.179309 + 0.179309i
\(9\) 64.0410 + 36.9741i 0.790630 + 0.456470i
\(10\) −133.703 + 15.4726i −1.33703 + 0.154726i
\(11\) −91.3743 − 158.265i −0.755159 − 1.30797i −0.945295 − 0.326216i \(-0.894226\pi\)
0.190136 − 0.981758i \(-0.439107\pi\)
\(12\) −8.92500 − 33.3085i −0.0619791 − 0.231309i
\(13\) 141.894 + 141.894i 0.839607 + 0.839607i 0.988807 − 0.149200i \(-0.0476699\pi\)
−0.149200 + 0.988807i \(0.547670\pi\)
\(14\) 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
\(17\) 412.771 + 110.602i 1.42827 + 0.382705i 0.888412 − 0.459048i \(-0.151809\pi\)
0.539863 + 0.841753i \(0.318476\pi\)
\(18\) −103.042 + 384.558i −0.318031 + 1.18691i
\(19\) 213.281 + 123.138i 0.590805 + 0.341101i 0.765416 − 0.643536i \(-0.222533\pi\)
−0.174611 + 0.984638i \(0.555867\pi\)
\(20\) −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
\(23\) 492.904 − 132.073i 0.931766 − 0.249666i 0.239158 − 0.970981i \(-0.423128\pi\)
0.692608 + 0.721315i \(0.256462\pi\)
\(24\) −37.3231 + 21.5485i −0.0647970 + 0.0374106i
\(25\) −598.332 − 180.622i −0.957331 − 0.288995i
\(26\) −540.180 + 935.619i −0.799083 + 1.38405i
\(27\) −290.953 + 290.953i −0.399112 + 0.399112i
\(28\) 633.147 − 63.1996i 0.807585 − 0.0806118i
\(29\) − 1081.57i − 1.28605i −0.765844 − 0.643026i \(-0.777679\pi\)
0.765844 − 0.643026i \(-0.222321\pi\)
\(30\) 52.2065 − 353.589i 0.0580072 − 0.392877i
\(31\) −227.280 − 393.660i −0.236503 − 0.409636i 0.723205 − 0.690633i \(-0.242668\pi\)
−0.959709 + 0.280997i \(0.909335\pi\)
\(32\) −1284.04 − 344.058i −1.25395 − 0.335994i
\(33\) 468.758 − 125.603i 0.430448 − 0.115338i
\(34\) 2300.68i 1.99021i
\(35\) 661.769 − 1030.87i 0.540220 − 0.841524i
\(36\) −960.259 −0.740941
\(37\) −19.3482 − 72.2086i −0.0141331 − 0.0527455i 0.958499 − 0.285095i \(-0.0920253\pi\)
−0.972632 + 0.232350i \(0.925359\pi\)
\(38\) −343.169 + 1280.72i −0.237651 + 0.886927i
\(39\) −461.486 + 266.439i −0.303410 + 0.175174i
\(40\) −325.722 + 241.912i −0.203576 + 0.151195i
\(41\) −1376.70 −0.818976 −0.409488 − 0.912315i \(-0.634293\pi\)
−0.409488 + 0.912315i \(0.634293\pi\)
\(42\) 638.483 + 288.282i 0.361952 + 0.163425i
\(43\) −493.766 − 493.766i −0.267045 − 0.267045i 0.560864 − 0.827908i \(-0.310469\pi\)
−0.827908 + 0.560864i \(0.810469\pi\)
\(44\) 2055.16 + 1186.55i 1.06155 + 0.612886i
\(45\) −1148.29 + 1448.84i −0.567057 + 0.715476i
\(46\) 1373.66 + 2379.25i 0.649178 + 1.12441i
\(47\) −221.427 − 826.377i −0.100239 − 0.374095i 0.897523 − 0.440968i \(-0.145365\pi\)
−0.997762 + 0.0668723i \(0.978698\pi\)
\(48\) −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
\(52\) −2517.00 − 674.427i −0.930841 − 0.249418i
\(53\) 775.214 − 2893.14i 0.275975 − 1.02995i −0.679210 − 0.733944i \(-0.737677\pi\)
0.955185 − 0.296009i \(-0.0956559\pi\)
\(54\) −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
\(58\) 5624.57 − 1507.10i 1.67199 − 0.448008i
\(59\) −1513.47 + 873.800i −0.434779 + 0.251020i −0.701381 − 0.712787i \(-0.747433\pi\)
0.266601 + 0.963807i \(0.414099\pi\)
\(60\) 856.373 − 99.1026i 0.237881 − 0.0275285i
\(61\) 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
\(63\) −2114.45 − 2942.54i −0.532742 − 0.741381i
\(64\) − 2434.61i − 0.594388i
\(65\) −4027.43 + 2991.16i −0.953238 + 0.707966i
\(66\) 1306.37 + 2262.70i 0.299901 + 0.519444i
\(67\) 2616.13 + 700.989i 0.582786 + 0.156157i 0.538154 − 0.842846i \(-0.319122\pi\)
0.0446321 + 0.999003i \(0.485788\pi\)
\(68\) −5360.07 + 1436.23i −1.15919 + 0.310603i
\(69\) 1355.09i 0.284624i
\(70\) 6283.03 + 2005.00i 1.28225 + 0.409184i
\(71\) 242.826 0.0481701 0.0240851 − 0.999710i \(-0.492333\pi\)
0.0240851 + 0.999710i \(0.492333\pi\)
\(72\) 310.613 + 1159.22i 0.0599177 + 0.223616i
\(73\) −1265.59 + 4723.25i −0.237491 + 0.886330i 0.739519 + 0.673136i \(0.235053\pi\)
−0.977010 + 0.213194i \(0.931613\pi\)
\(74\) 348.551 − 201.236i 0.0636507 − 0.0367488i
\(75\) 874.536 − 1410.60i 0.155473 − 0.250774i
\(76\) −3198.02 −0.553674
\(77\) 889.421 + 8910.40i 0.150012 + 1.50285i
\(78\) −2028.64 − 2028.64i −0.333438 − 0.333438i
\(79\) −3161.63 − 1825.37i −0.506591 − 0.292480i 0.224840 − 0.974396i \(-0.427814\pi\)
−0.731431 + 0.681915i \(0.761147\pi\)
\(80\) −5782.65 − 4583.09i −0.903539 − 0.716108i
\(81\) 2448.57 + 4241.04i 0.373200 + 0.646402i
\(82\) −1918.34 − 7159.35i −0.285298 − 1.06475i
\(83\) 7249.03 + 7249.03i 1.05226 + 1.05226i 0.998557 + 0.0537046i \(0.0171029\pi\)
0.0537046 + 0.998557i \(0.482897\pi\)
\(84\) −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
\(87\) 2774.27 + 743.364i 0.366531 + 0.0982116i
\(88\) 767.620 − 2864.80i 0.0991245 − 0.369938i
\(89\) −2122.71 − 1225.55i −0.267986 − 0.154722i 0.359986 − 0.932958i \(-0.382781\pi\)
−0.627972 + 0.778236i \(0.716115\pi\)
\(90\) −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
\(93\) 1165.96 − 312.419i 0.134809 − 0.0361220i
\(94\) 3988.92 − 2303.01i 0.451440 − 0.260639i
\(95\) −3824.24 + 4825.18i −0.423738 + 0.534646i
\(96\) 1765.05 − 3057.15i 0.191520 − 0.331722i
\(97\) 1798.31 − 1798.31i 0.191126 − 0.191126i −0.605056 − 0.796183i \(-0.706849\pi\)
0.796183 + 0.605056i \(0.206849\pi\)
\(98\) −7153.34 + 10766.9i −0.744830 + 1.12108i
\(99\) − 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.23.11 yes 56
5.2 odd 4 inner 35.5.l.a.2.4 ✓ 56
7.4 even 3 inner 35.5.l.a.18.4 yes 56
35.32 odd 12 inner 35.5.l.a.32.11 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.5.l.a.2.4 ✓ 56 5.2 odd 4 inner
35.5.l.a.18.4 yes 56 7.4 even 3 inner
35.5.l.a.23.11 yes 56 1.1 even 1 trivial
35.5.l.a.32.11 yes 56 35.32 odd 12 inner