↑

Properties

Label 35.5.l.a.2.8
Level $35$
Weight $5$
Character 35.2
Analytic conductor $3.618$
Analytic rank $0$
Dimension $56$
Inner twists $4$

Related objects

Downloads

Learn more

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 2.8
Character \(\chi\) \(=\) 35.2
Dual form 35.5.l.a.18.8

$q$-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+(-0.167922 + 0.0449946i) q^{2} +(2.09038 + 0.560116i) q^{3} +(-13.8302 + 7.98489i) q^{4} +(-20.4734 + 14.3471i) q^{5} -0.376224 q^{6} +(20.9813 + 44.2807i) q^{7} +(3.92997 - 3.92997i) q^{8} +(-66.0921 - 38.1583i) q^{9} +(2.79240 - 3.33039i) q^{10} +(58.3974 + 101.147i) q^{11} +(-33.3829 + 8.94493i) q^{12} +(-115.132 + 115.132i) q^{13} +(-5.51563 - 6.49167i) q^{14} +(-50.8333 + 18.5233i) q^{15} +(127.275 - 220.447i) q^{16} +(87.3197 - 325.882i) q^{17} +(12.8152 + 3.43384i) q^{18} +(254.594 + 146.990i) q^{19} +(168.593 - 361.901i) q^{20} +(19.0566 + 104.316i) q^{21} +(-14.3573 - 14.3573i) q^{22} +(33.6016 + 125.403i) q^{23} +(10.4164 - 6.01389i) q^{24} +(213.323 - 587.468i) q^{25} +(14.1528 - 24.5135i) q^{26} +(-240.736 - 240.736i) q^{27} +(-643.753 - 444.879i) q^{28} +1163.18i q^{29} +(7.70259 - 5.39771i) q^{30} +(489.808 + 848.372i) q^{31} +(-34.4689 + 128.640i) q^{32} +(65.4186 + 244.146i) q^{33} +58.6517i q^{34} +(-1064.86 - 605.558i) q^{35} +1218.76 q^{36} +(-1593.21 + 426.898i) q^{37} +(-49.3657 - 13.2275i) q^{38} +(-305.156 + 176.182i) q^{39} +(-24.0764 + 136.843i) q^{40} +1695.98 q^{41} +(-7.89367 - 16.6595i) q^{42} +(-2407.70 + 2407.70i) q^{43} +(-1615.30 - 932.593i) q^{44} +(1900.59 - 166.997i) q^{45} +(-11.2849 - 19.5460i) q^{46} +(1486.34 - 398.264i) q^{47} +(389.529 - 389.529i) q^{48} +(-1520.57 + 1858.14i) q^{49} +(-9.38882 + 108.247i) q^{50} +(365.063 - 632.307i) q^{51} +(672.984 - 2511.61i) q^{52} +(798.228 + 213.885i) q^{53} +(51.2568 + 29.5931i) q^{54} +(-2646.76 - 1233.00i) q^{55} +(256.478 + 91.5659i) q^{56} +(449.867 + 449.867i) q^{57} +(-52.3367 - 195.323i) q^{58} +(5363.95 - 3096.88i) q^{59} +(555.130 - 662.080i) q^{60} +(102.925 - 178.272i) q^{61} +(-120.422 - 120.422i) q^{62} +(302.977 - 3727.22i) q^{63} +4049.65i q^{64} +(705.339 - 4008.94i) q^{65} +(-21.9705 - 38.0540i) q^{66} +(306.442 - 1143.66i) q^{67} +(1394.48 + 5204.26i) q^{68} +280.961i q^{69} +(206.060 + 53.7738i) q^{70} -1526.20 q^{71} +(-409.701 + 109.779i) q^{72} +(-4008.87 - 1074.17i) q^{73} +(248.326 - 143.371i) q^{74} +(774.977 - 1108.55i) q^{75} -4694.79 q^{76} +(-3253.62 + 4708.08i) q^{77} +(43.3152 - 43.3152i) q^{78} +(-2657.12 - 1534.09i) q^{79} +(557.009 + 6339.33i) q^{80} +(2722.43 + 4715.39i) q^{81} +(-284.793 + 76.3102i) q^{82} +(6156.34 - 6156.34i) q^{83} +(-1096.51 - 1290.54i) q^{84} +(2887.71 + 7924.70i) q^{85} +(295.972 - 512.639i) q^{86} +(-651.514 + 2431.48i) q^{87} +(627.005 + 168.006i) q^{88} +(-1427.41 - 824.115i) q^{89} +(-311.638 + 113.559i) q^{90} +(-7513.73 - 2682.50i) q^{91} +(-1466.05 - 1466.05i) q^{92} +(548.698 + 2047.77i) q^{93} +(-231.670 + 133.755i) q^{94} +(-7321.29 + 643.289i) q^{95} +(-144.106 + 249.599i) q^{96} +(5266.59 + 5266.59i) q^{97} +(171.731 - 380.440i) q^{98} -8913.38i 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.167922 + 0.0449946i −0.0419805 + 0.0112487i −0.279748 − 0.960073i \(-0.590251\pi\)
0.237768 + 0.971322i \(0.423584\pi\)
\(3\) 2.09038 + 0.560116i 0.232265 + 0.0622351i 0.373074 − 0.927802i \(-0.378304\pi\)
−0.140809 + 0.990037i \(0.544970\pi\)
\(4\) −13.8302 + 7.98489i −0.864390 + 0.499056i
\(5\) −20.4734 + 14.3471i −0.818937 + 0.573883i
\(6\) −0.376224 −0.0104507
\(7\) 20.9813 + 44.2807i 0.428190 + 0.903689i
\(8\) 3.92997 − 3.92997i 0.0614057 − 0.0614057i
\(9\) −66.0921 − 38.1583i −0.815952 − 0.471090i
\(10\) 2.79240 − 3.33039i 0.0279240 − 0.0333039i
\(11\) 58.3974 + 101.147i 0.482623 + 0.835928i 0.999801 − 0.0199503i \(-0.00635080\pi\)
−0.517178 + 0.855878i \(0.673017\pi\)
\(12\) −33.3829 + 8.94493i −0.231826 + 0.0621175i
\(13\) −115.132 + 115.132i −0.681252 + 0.681252i −0.960282 − 0.279030i \(-0.909987\pi\)
0.279030 + 0.960282i \(0.409987\pi\)
\(14\) −5.51563 − 6.49167i −0.0281409 − 0.0331208i
\(15\) −50.8333 + 18.5233i −0.225926 + 0.0823260i
\(16\) 127.275 − 220.447i 0.497168 − 0.861121i
\(17\) 87.3197 − 325.882i 0.302144 − 1.12762i −0.633231 − 0.773962i \(-0.718272\pi\)
0.935376 − 0.353655i \(-0.115061\pi\)
\(18\) 12.8152 + 3.43384i 0.0395532 + 0.0105983i
\(19\) 254.594 + 146.990i 0.705246 + 0.407174i 0.809298 − 0.587398i \(-0.199848\pi\)
−0.104052 + 0.994572i \(0.533181\pi\)
\(20\) 168.593 − 361.901i 0.421482 − 0.904754i
\(21\) 19.0566 + 104.316i 0.0432123 + 0.236543i
\(22\) −14.3573 − 14.3573i −0.0296638 − 0.0296638i
\(23\) 33.6016 + 125.403i 0.0635191 + 0.237056i 0.990386 − 0.138335i \(-0.0441750\pi\)
−0.926866 + 0.375391i \(0.877508\pi\)
\(24\) 10.4164 − 6.01389i 0.0180840 − 0.0104408i
\(25\) 213.323 − 587.468i 0.341317 − 0.939948i
\(26\) 14.1528 − 24.5135i 0.0209362 − 0.0362625i
\(27\) −240.736 − 240.736i −0.330228 − 0.330228i
\(28\) −643.753 − 444.879i −0.821114 − 0.567448i
\(29\) 1163.18i 1.38309i 0.722334 + 0.691544i \(0.243069\pi\)
−0.722334 + 0.691544i \(0.756931\pi\)
\(30\) 7.70259 − 5.39771i 0.00855843 − 0.00599745i
\(31\) 489.808 + 848.372i 0.509685 + 0.882801i 0.999937 + 0.0112200i \(0.00357152\pi\)
−0.490252 + 0.871581i \(0.663095\pi\)
\(32\) −34.4689 + 128.640i −0.0336610 + 0.125625i
\(33\) 65.4186 + 244.146i 0.0600722 + 0.224192i
\(34\) 58.6517i 0.0507367i
\(35\) −1064.86 − 605.558i −0.869272 − 0.494333i
\(36\) 1218.76 0.940400
\(37\) −1593.21 + 426.898i −1.16377 + 0.311832i −0.788472 − 0.615071i \(-0.789127\pi\)
−0.375301 + 0.926903i \(0.622461\pi\)
\(38\) −49.3657 − 13.2275i −0.0341868 − 0.00916032i
\(39\) −305.156 + 176.182i −0.200628 + 0.115833i
\(40\) −24.0764 + 136.843i −0.0150478 + 0.0855272i
\(41\) 1695.98 1.00891 0.504457 − 0.863437i \(-0.331693\pi\)
0.504457 + 0.863437i \(0.331693\pi\)
\(42\) −7.89367 − 16.6595i −0.00447487 − 0.00944414i
\(43\) −2407.70 + 2407.70i −1.30216 + 1.30216i −0.375228 + 0.926932i \(0.622436\pi\)
−0.926932 + 0.375228i \(0.877564\pi\)
\(44\) −1615.30 − 932.593i −0.834349 − 0.481711i
\(45\) 1900.59 − 166.997i 0.938564 − 0.0824674i
\(46\) −11.2849 − 19.5460i −0.00533313 − 0.00923726i
\(47\) 1486.34 − 398.264i 0.672858 − 0.180292i 0.0938159 − 0.995590i \(-0.470093\pi\)
0.579042 + 0.815298i \(0.303427\pi\)
\(48\) 389.529 − 389.529i 0.169067 − 0.169067i
\(49\) −1520.57 + 1858.14i −0.633306 + 0.773901i
\(50\) −9.38882 + 108.247i −0.00375553 + 0.0432989i
\(51\) 365.063 − 632.307i 0.140355 − 0.243102i
\(52\) 672.984 − 2511.61i 0.248885 − 0.928850i
\(53\) 798.228 + 213.885i 0.284168 + 0.0761426i 0.398088 − 0.917347i \(-0.369674\pi\)
−0.113920 + 0.993490i \(0.536341\pi\)
\(54\) 51.2568 + 29.5931i 0.0175778 + 0.0101485i
\(55\) −2646.76 − 1233.00i −0.874962 − 0.407603i
\(56\) 256.478 + 91.5659i 0.0817850 + 0.0291983i
\(57\) 449.867 + 449.867i 0.138463 + 0.138463i
\(58\) −52.3367 − 195.323i −0.0155579 − 0.0580628i
\(59\) 5363.95 − 3096.88i 1.54092 − 0.889652i 0.542142 − 0.840287i \(-0.317614\pi\)
0.998781 − 0.0493647i \(-0.0157197\pi\)
\(60\) 555.130 − 662.080i 0.154203 − 0.183911i
\(61\) 102.925 − 178.272i 0.0276607 − 0.0479097i −0.851864 − 0.523764i \(-0.824528\pi\)
0.879524 + 0.475854i \(0.157861\pi\)
\(62\) −120.422 − 120.422i −0.0313272 − 0.0313272i
\(63\) 302.977 − 3727.22i 0.0763359 − 0.939082i
\(64\) 4049.65i 0.988684i
\(65\) 705.339 − 4008.94i 0.166944 − 0.948862i
\(66\) −21.9705 − 38.0540i −0.00504373 − 0.00873599i
\(67\) 306.442 − 1143.66i 0.0682650 − 0.254769i −0.923357 − 0.383943i \(-0.874566\pi\)
0.991622 + 0.129174i \(0.0412326\pi\)
\(68\) 1394.48 + 5204.26i 0.301574 + 1.12549i
\(69\) 280.961i 0.0590129i
\(70\) 206.060 + 53.7738i 0.0420531 + 0.0109742i
\(71\) −1526.20 −0.302757 −0.151378 − 0.988476i \(-0.548371\pi\)
−0.151378 + 0.988476i \(0.548371\pi\)
\(72\) −409.701 + 109.779i −0.0790318 + 0.0211765i
\(73\) −4008.87 − 1074.17i −0.752274 − 0.201571i −0.137748 − 0.990467i \(-0.543986\pi\)
−0.614527 + 0.788896i \(0.710653\pi\)
\(74\) 248.326 − 143.371i 0.0453481 − 0.0261818i
\(75\) 774.977 − 1108.55i 0.137774 − 0.197075i
\(76\) −4694.79 −0.812810
\(77\) −3253.62 + 4708.08i −0.548764 + 0.794077i
\(78\) 43.3152 − 43.3152i 0.00711953 − 0.00711953i
\(79\) −2657.12 − 1534.09i −0.425753 − 0.245808i 0.271783 − 0.962359i \(-0.412387\pi\)
−0.697536 + 0.716550i \(0.745720\pi\)
\(80\) 557.009 + 6339.33i 0.0870326 + 0.990521i
\(81\) 2722.43 + 4715.39i 0.414942 + 0.718700i
\(82\) −284.793 + 76.3102i −0.0423548 + 0.0113489i
\(83\) 6156.34 − 6156.34i 0.893648 − 0.893648i −0.101217 − 0.994864i \(-0.532274\pi\)
0.994864 + 0.101217i \(0.0322736\pi\)
\(84\) −1096.51 − 1290.54i −0.155400 − 0.182900i
\(85\) 2887.71 + 7924.70i 0.399683 + 1.09684i
\(86\) 295.972 − 512.639i 0.0400179 − 0.0693130i
\(87\) −651.514 + 2431.48i −0.0860767 + 0.321242i
\(88\) 627.005 + 168.006i 0.0809666 + 0.0216949i
\(89\) −1427.41 − 824.115i −0.180206 − 0.104042i 0.407184 − 0.913346i \(-0.366511\pi\)
−0.587389 + 0.809305i \(0.699844\pi\)
\(90\) −311.638 + 113.559i −0.0384738 + 0.0140196i
\(91\) −7513.73 − 2682.50i −0.907345 − 0.323934i
\(92\) −1466.05 − 1466.05i −0.173210 − 0.173210i
\(93\) 548.698 + 2047.77i 0.0634406 + 0.236764i
\(94\) −231.670 + 133.755i −0.0262189 + 0.0151375i
\(95\) −7321.29 + 643.289i −0.811223 + 0.0712786i
\(96\) −144.106 + 249.599i −0.0156365 + 0.0270832i
\(97\) 5266.59 + 5266.59i 0.559740 + 0.559740i 0.929233 − 0.369494i \(-0.120469\pi\)
−0.369494 + 0.929233i \(0.620469\pi\)
\(98\) 171.731 − 380.440i 0.0178812 − 0.0396126i
\(99\) − 8913.38i − 0.909435i
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.2.8 ✓ 56
5.3 odd 4 inner 35.5.l.a.23.7 yes 56
7.4 even 3 inner 35.5.l.a.32.7 yes 56
35.18 odd 12 inner 35.5.l.a.18.8 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.5.l.a.2.8 ✓ 56 1.1 even 1 trivial
35.5.l.a.18.8 yes 56 35.18 odd 12 inner
35.5.l.a.23.7 yes 56 5.3 odd 4 inner
35.5.l.a.32.7 yes 56 7.4 even 3 inner