↑

Properties

Label 35.5.l.a.2.13
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 2.13
Character \(\chi\) \(=\) 35.2
Dual form 35.5.l.a.18.13

$q$-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+(6.17220 - 1.65384i) q^{2} +(2.01200 + 0.539113i) q^{3} +(21.5045 - 12.4156i) q^{4} +(3.38702 + 24.7695i) q^{5} +13.3101 q^{6} +(2.59467 - 48.9313i) q^{7} +(39.9028 - 39.9028i) q^{8} +(-66.3906 - 38.3306i) q^{9} +(61.8701 + 147.281i) q^{10} +(-2.47446 - 4.28589i) q^{11} +(49.9604 - 13.3869i) q^{12} +(-145.178 + 145.178i) q^{13} +(-64.9094 - 306.305i) q^{14} +(-6.53889 + 51.6622i) q^{15} +(-18.3547 + 31.7912i) q^{16} +(34.4477 - 128.561i) q^{17} +(-473.168 - 126.785i) q^{18} +(516.304 + 298.088i) q^{19} +(380.365 + 490.603i) q^{20} +(31.6000 - 97.0508i) q^{21} +(-22.3610 - 22.3610i) q^{22} +(-70.2844 - 262.305i) q^{23} +(101.796 - 58.7722i) q^{24} +(-602.056 + 167.790i) q^{25} +(-655.968 + 1136.17i) q^{26} +(-232.217 - 232.217i) q^{27} +(-551.715 - 1084.46i) q^{28} -870.666i q^{29} +(45.0814 + 329.684i) q^{30} +(-16.7638 - 29.0357i) q^{31} +(-294.398 + 1098.71i) q^{32} +(-2.66803 - 9.95722i) q^{33} -850.472i q^{34} +(1220.79 - 101.462i) q^{35} -1903.59 q^{36} +(2173.69 - 582.437i) q^{37} +(3679.72 + 985.978i) q^{38} +(-370.365 + 213.831i) q^{39} +(1123.52 + 853.220i) q^{40} +441.857 q^{41} +(34.5353 - 651.278i) q^{42} +(631.163 - 631.163i) q^{43} +(-106.424 - 61.4439i) q^{44} +(724.564 - 1774.29i) q^{45} +(-867.619 - 1502.76i) q^{46} +(748.705 - 200.615i) q^{47} +(-54.0686 + 54.0686i) q^{48} +(-2387.54 - 253.921i) q^{49} +(-3438.52 + 2031.33i) q^{50} +(138.617 - 240.092i) q^{51} +(-1319.50 + 4924.46i) q^{52} +(3286.50 + 880.614i) q^{53} +(-1817.34 - 1049.24i) q^{54} +(97.7783 - 75.8075i) q^{55} +(-1848.96 - 2056.03i) q^{56} +(878.099 + 878.099i) q^{57} +(-1439.94 - 5373.93i) q^{58} +(-3243.05 + 1872.37i) q^{59} +(500.803 + 1192.15i) q^{60} +(-1323.13 + 2291.74i) q^{61} +(-151.490 - 151.490i) q^{62} +(-2047.83 + 3149.12i) q^{63} +6680.99i q^{64} +(-4087.71 - 3104.27i) q^{65} +(-32.9352 - 57.0455i) q^{66} +(565.405 - 2110.12i) q^{67} +(-855.379 - 3192.32i) q^{68} -565.649i q^{69} +(7367.16 - 2645.23i) q^{70} -9824.83 q^{71} +(-4178.66 + 1119.67i) q^{72} +(4022.41 + 1077.80i) q^{73} +(12453.2 - 7189.84i) q^{74} +(-1301.79 + 13.0158i) q^{75} +14803.8 q^{76} +(-216.134 + 109.958i) q^{77} +(-1932.33 + 1932.33i) q^{78} +(6344.57 + 3663.04i) q^{79} +(-849.620 - 346.958i) q^{80} +(2762.75 + 4785.22i) q^{81} +(2727.23 - 730.760i) q^{82} +(-5252.26 + 5252.26i) q^{83} +(-525.405 - 2479.36i) q^{84} +(3301.05 + 417.815i) q^{85} +(2851.82 - 4939.50i) q^{86} +(469.388 - 1751.78i) q^{87} +(-269.757 - 72.2811i) q^{88} +(-8280.92 - 4780.99i) q^{89} +(1537.77 - 12149.6i) q^{90} +(6727.06 + 7480.44i) q^{91} +(-4768.11 - 4768.11i) q^{92} +(-18.0752 - 67.4574i) q^{93} +(4289.37 - 2476.47i) q^{94} +(-5634.76 + 13798.2i) q^{95} +(-1184.66 + 2051.89i) q^{96} +(-11539.7 - 11539.7i) q^{97} +(-15156.3 + 2381.34i) q^{98} +379.390i 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 6.17220 − 1.65384i 1.54305 − 0.413459i 0.615800 − 0.787902i \(-0.288833\pi\)
0.927250 + 0.374443i \(0.122166\pi\)
\(3\) 2.01200 + 0.539113i 0.223555 + 0.0599015i 0.368858 − 0.929486i \(-0.379749\pi\)
−0.145303 + 0.989387i \(0.546416\pi\)
\(4\) 21.5045 − 12.4156i 1.34403 − 0.775976i
\(5\) 3.38702 + 24.7695i 0.135481 + 0.990780i
\(6\) 13.3101 0.369724
\(7\) 2.59467 − 48.9313i 0.0529525 − 0.998597i
\(8\) 39.9028 − 39.9028i 0.623481 − 0.623481i
\(9\) −66.3906 − 38.3306i −0.819637 − 0.473217i
\(10\) 61.8701 + 147.281i 0.618701 + 1.47281i
\(11\) −2.47446 − 4.28589i −0.0204501 − 0.0354206i 0.855619 − 0.517606i \(-0.173177\pi\)
−0.876069 + 0.482185i \(0.839843\pi\)
\(12\) 49.9604 − 13.3869i 0.346947 − 0.0929643i
\(13\) −145.178 + 145.178i −0.859042 + 0.859042i −0.991225 − 0.132183i \(-0.957801\pi\)
0.132183 + 0.991225i \(0.457801\pi\)
\(14\) −64.9094 − 306.305i −0.331171 − 1.56278i
\(15\) −6.53889 + 51.6622i −0.0290617 + 0.229610i
\(16\) −18.3547 + 31.7912i −0.0716979 + 0.124184i
\(17\) 34.4477 − 128.561i 0.119196 − 0.444846i −0.880370 − 0.474287i \(-0.842706\pi\)
0.999567 + 0.0294407i \(0.00937264\pi\)
\(18\) −473.168 − 126.785i −1.46040 − 0.391312i
\(19\) 516.304 + 298.088i 1.43020 + 0.825729i 0.997136 − 0.0756326i \(-0.0240976\pi\)
0.433068 + 0.901361i \(0.357431\pi\)
\(20\) 380.365 + 490.603i 0.950912 + 1.22651i
\(21\) 31.6000 − 97.0508i 0.0716553 − 0.220070i
\(22\) −22.3610 − 22.3610i −0.0462005 − 0.0462005i
\(23\) −70.2844 − 262.305i −0.132863 − 0.495851i 0.867135 − 0.498074i \(-0.165959\pi\)
−0.999998 + 0.00222290i \(0.999292\pi\)
\(24\) 101.796 − 58.7722i 0.176730 − 0.102035i
\(25\) −602.056 + 167.790i −0.963290 + 0.268463i
\(26\) −655.968 + 1136.17i −0.970366 + 1.68072i
\(27\) −232.217 − 232.217i −0.318542 − 0.318542i
\(28\) −551.715 − 1084.46i −0.703718 − 1.38323i
\(29\) − 870.666i − 1.03528i −0.855600 − 0.517638i \(-0.826812\pi\)
0.855600 − 0.517638i \(-0.173188\pi\)
\(30\) 45.0814 + 329.684i 0.0500905 + 0.366315i
\(31\) −16.7638 − 29.0357i −0.0174441 − 0.0302141i 0.857172 − 0.515031i \(-0.172220\pi\)
−0.874616 + 0.484817i \(0.838886\pi\)
\(32\) −294.398 + 1098.71i −0.287498 + 1.07296i
\(33\) −2.66803 − 9.95722i −0.00244998 − 0.00914345i
\(34\) − 850.472i − 0.735703i
\(35\) 1220.79 − 101.462i 0.996564 − 0.0828264i
\(36\) −1903.59 −1.46882
\(37\) 2173.69 − 582.437i 1.58779 − 0.425447i 0.646464 − 0.762944i \(-0.276247\pi\)
0.941327 + 0.337497i \(0.109580\pi\)
\(38\) 3679.72 + 985.978i 2.54828 + 0.682810i
\(39\) −370.365 + 213.831i −0.243501 + 0.140586i
\(40\) 1123.52 + 853.220i 0.702202 + 0.533263i
\(41\) 441.857 0.262854 0.131427 − 0.991326i \(-0.458044\pi\)
0.131427 + 0.991326i \(0.458044\pi\)
\(42\) 34.5353 − 651.278i 0.0195778 − 0.369205i
\(43\) 631.163 − 631.163i 0.341353 − 0.341353i −0.515523 − 0.856876i \(-0.672402\pi\)
0.856876 + 0.515523i \(0.172402\pi\)
\(44\) −106.424 − 61.4439i −0.0549711 − 0.0317376i
\(45\) 724.564 − 1774.29i 0.357809 − 0.876191i
\(46\) −867.619 − 1502.76i −0.410028 − 0.710189i
\(47\) 748.705 − 200.615i 0.338934 − 0.0908171i −0.0853376 − 0.996352i \(-0.527197\pi\)
0.424272 + 0.905535i \(0.360530\pi\)
\(48\) −54.0686 + 54.0686i −0.0234673 + 0.0234673i
\(49\) −2387.54 − 253.921i −0.994392 − 0.105756i
\(50\) −3438.52 + 2031.33i −1.37541 + 0.812533i
\(51\) 138.617 − 240.092i 0.0532939 − 0.0923077i
\(52\) −1319.50 + 4924.46i −0.487982 + 1.82117i
\(53\) 3286.50 + 880.614i 1.16999 + 0.313497i 0.790948 − 0.611883i \(-0.209588\pi\)
0.379040 + 0.925380i \(0.376254\pi\)
\(54\) −1817.34 − 1049.24i −0.623230 − 0.359822i
\(55\) 97.7783 − 75.8075i 0.0323234 − 0.0250603i
\(56\) −1848.96 − 2056.03i −0.589591 − 0.655621i
\(57\) 878.099 + 878.099i 0.270267 + 0.270267i
\(58\) −1439.94 − 5373.93i −0.428044 − 1.59748i
\(59\) −3243.05 + 1872.37i −0.931642 + 0.537884i −0.887330 − 0.461134i \(-0.847443\pi\)
−0.0443115 + 0.999018i \(0.514109\pi\)
\(60\) 500.803 + 1192.15i 0.139112 + 0.331154i
\(61\) −1323.13 + 2291.74i −0.355586 + 0.615892i −0.987218 − 0.159376i \(-0.949052\pi\)
0.631632 + 0.775268i \(0.282385\pi\)
\(62\) −151.490 − 151.490i −0.0394094 − 0.0394094i
\(63\) −2047.83 + 3149.12i −0.515955 + 0.793429i
\(64\) 6680.99i 1.63110i
\(65\) −4087.71 − 3104.27i −0.967505 − 0.734738i
\(66\) −32.9352 − 57.0455i −0.00756089 − 0.0130958i
\(67\) 565.405 − 2110.12i 0.125953 − 0.470065i −0.873918 − 0.486073i \(-0.838429\pi\)
0.999872 + 0.0160079i \(0.00509568\pi\)
\(68\) −855.379 − 3192.32i −0.184987 − 0.690380i
\(69\) − 565.649i − 0.118809i
\(70\) 7367.16 − 2645.23i 1.50350 − 0.539844i
\(71\) −9824.83 −1.94898 −0.974492 − 0.224420i \(-0.927951\pi\)
−0.974492 + 0.224420i \(0.927951\pi\)
\(72\) −4178.66 + 1119.67i −0.806070 + 0.215986i
\(73\) 4022.41 + 1077.80i 0.754815 + 0.202252i 0.615653 − 0.788018i \(-0.288892\pi\)
0.139162 + 0.990270i \(0.455559\pi\)
\(74\) 12453.2 − 7189.84i 2.27414 − 1.31297i
\(75\) −1301.79 + 13.0158i −0.231430 + 0.00231392i
\(76\) 14803.8 2.56298
\(77\) −216.134 + 109.958i −0.0364538 + 0.0185458i
\(78\) −1932.33 + 1932.33i −0.317608 + 0.317608i
\(79\) 6344.57 + 3663.04i 1.01659 + 0.586931i 0.913116 − 0.407700i \(-0.133669\pi\)
0.103479 + 0.994632i \(0.467002\pi\)
\(80\) −849.620 − 346.958i −0.132753 − 0.0542122i
\(81\) 2762.75 + 4785.22i 0.421087 + 0.729344i
\(82\) 2727.23 − 730.760i 0.405597 − 0.108679i
\(83\) −5252.26 + 5252.26i −0.762413 + 0.762413i −0.976758 − 0.214345i \(-0.931238\pi\)
0.214345 + 0.976758i \(0.431238\pi\)
\(84\) −525.405 − 2479.36i −0.0744621 − 0.351383i
\(85\) 3301.05 + 417.815i 0.456893 + 0.0578291i
\(86\) 2851.82 − 4939.50i 0.385590 − 0.667861i
\(87\) 469.388 − 1751.78i 0.0620145 − 0.231441i
\(88\) −269.757 − 72.2811i −0.0348343 − 0.00933382i
\(89\) −8280.92 − 4780.99i −1.04544 − 0.603584i −0.124070 − 0.992274i \(-0.539595\pi\)
−0.921369 + 0.388689i \(0.872928\pi\)
\(90\) 1537.77 − 12149.6i 0.189848 − 1.49995i
\(91\) 6727.06 + 7480.44i 0.812348 + 0.903325i
\(92\) −4768.11 − 4768.11i −0.563340 − 0.563340i
\(93\) −18.0752 − 67.4574i −0.00208986 − 0.00779945i
\(94\) 4289.37 − 2476.47i 0.485443 − 0.280271i
\(95\) −5634.76 + 13798.2i −0.624350 + 1.52889i
\(96\) −1184.66 + 2051.89i −0.128543 + 0.222644i
\(97\) −11539.7 − 11539.7i −1.22646 − 1.22646i −0.965295 − 0.261160i \(-0.915895\pi\)
−0.261160 − 0.965295i \(-0.584105\pi\)
\(98\) −15156.3 + 2381.34i −1.57812 + 0.247953i
\(99\) 379.390i 0.0387093i
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.2.13 ✓ 56
5.3 odd 4 inner 35.5.l.a.23.2 yes 56
7.4 even 3 inner 35.5.l.a.32.2 yes 56
35.18 odd 12 inner 35.5.l.a.18.13 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.5.l.a.2.13 ✓ 56 1.1 even 1 trivial
35.5.l.a.18.13 yes 56 35.18 odd 12 inner
35.5.l.a.23.2 yes 56 5.3 odd 4 inner
35.5.l.a.32.2 yes 56 7.4 even 3 inner