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Properties

Label 35.5.l.a.2.3
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 2.3
Character \(\chi\) \(=\) 35.2
Dual form 35.5.l.a.18.3

$q$-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+(-5.55079 + 1.48733i) q^{2} +(15.0503 + 4.03273i) q^{3} +(14.7427 - 8.51169i) q^{4} +(-2.87000 - 24.8347i) q^{5} -89.5393 q^{6} +(40.0414 + 28.2434i) q^{7} +(-4.15850 + 4.15850i) q^{8} +(140.102 + 80.8879i) q^{9} +(52.8682 + 133.584i) q^{10} +(48.2961 + 83.6513i) q^{11} +(256.208 - 68.6507i) q^{12} +(-48.6898 + 48.6898i) q^{13} +(-264.268 - 97.2182i) q^{14} +(56.9571 - 385.345i) q^{15} +(-119.289 + 206.615i) q^{16} +(25.3859 - 94.7415i) q^{17} +(-897.983 - 240.614i) q^{18} +(217.882 + 125.794i) q^{19} +(-253.697 - 341.702i) q^{20} +(488.739 + 586.548i) q^{21} +(-392.499 - 392.499i) q^{22} +(-247.264 - 922.803i) q^{23} +(-79.3569 + 45.8168i) q^{24} +(-608.526 + 142.551i) q^{25} +(197.849 - 342.685i) q^{26} +(889.957 + 889.957i) q^{27} +(830.716 + 75.5632i) q^{28} +7.20762i q^{29} +(256.978 + 2223.68i) q^{30} +(-157.749 - 273.229i) q^{31} +(379.199 - 1415.19i) q^{32} +(389.530 + 1453.75i) q^{33} +563.647i q^{34} +(586.497 - 1075.47i) q^{35} +2753.97 q^{36} +(-1460.18 + 391.255i) q^{37} +(-1396.52 - 374.195i) q^{38} +(-929.152 + 536.446i) q^{39} +(115.210 + 91.3402i) q^{40} -2207.36 q^{41} +(-3585.28 - 2528.89i) q^{42} +(1414.89 - 1414.89i) q^{43} +(1424.03 + 822.164i) q^{44} +(1606.73 - 3711.54i) q^{45} +(2745.02 + 4754.52i) q^{46} +(1154.98 - 309.475i) q^{47} +(-2628.57 + 2628.57i) q^{48} +(805.624 + 2261.81i) q^{49} +(3165.78 - 1696.35i) q^{50} +(764.133 - 1323.52i) q^{51} +(-303.386 + 1132.25i) q^{52} +(-1051.86 - 281.844i) q^{53} +(-6263.62 - 3616.30i) q^{54} +(1938.85 - 1439.50i) q^{55} +(-283.962 + 49.0620i) q^{56} +(2771.91 + 2771.91i) q^{57} +(-10.7201 - 40.0079i) q^{58} +(-5245.99 + 3028.77i) q^{59} +(-2440.24 - 6165.82i) q^{60} +(464.144 - 803.921i) q^{61} +(1282.01 + 1282.01i) q^{62} +(3325.33 + 7195.81i) q^{63} +4602.15i q^{64} +(1348.94 + 1069.46i) q^{65} +(-4324.40 - 7490.08i) q^{66} +(-171.487 + 639.998i) q^{67} +(-432.154 - 1612.82i) q^{68} -14885.7i q^{69} +(-1655.94 + 6842.05i) q^{70} -6612.04 q^{71} +(-918.986 + 246.242i) q^{72} +(-1460.37 - 391.305i) q^{73} +(7523.24 - 4343.55i) q^{74} +(-9733.40 - 308.572i) q^{75} +4282.89 q^{76} +(-428.752 + 4713.56i) q^{77} +(4359.65 - 4359.65i) q^{78} +(396.547 + 228.947i) q^{79} +(5473.59 + 2369.53i) q^{80} +(3253.28 + 5634.85i) q^{81} +(12252.6 - 3283.08i) q^{82} +(-406.242 + 406.242i) q^{83} +(12197.8 + 4487.31i) q^{84} +(-2425.74 - 358.543i) q^{85} +(-5749.33 + 9958.13i) q^{86} +(-29.0664 + 108.477i) q^{87} +(-548.703 - 147.025i) q^{88} +(6858.49 + 3959.75i) q^{89} +(-3398.36 + 22991.7i) q^{90} +(-3324.77 + 574.443i) q^{91} +(-11500.0 - 11500.0i) q^{92} +(-1272.32 - 4748.35i) q^{93} +(-5950.74 + 3435.66i) q^{94} +(2498.74 - 5772.07i) q^{95} +(11414.1 - 19769.9i) q^{96} +(-4926.26 - 4926.26i) q^{97} +(-7835.90 - 11356.6i) q^{98} +15626.3i 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.55079 + 1.48733i −1.38770 + 0.371832i −0.873911 − 0.486086i \(-0.838424\pi\)
−0.513786 + 0.857918i \(0.671757\pi\)
\(3\) 15.0503 + 4.03273i 1.67226 + 0.448081i 0.965719 − 0.259590i \(-0.0835876\pi\)
0.706542 + 0.707671i \(0.250254\pi\)
\(4\) 14.7427 − 8.51169i 0.921418 − 0.531981i
\(5\) −2.87000 − 24.8347i −0.114800 − 0.993389i
\(6\) −89.5393 −2.48720
\(7\) 40.0414 + 28.2434i 0.817171 + 0.576395i
\(8\) −4.15850 + 4.15850i −0.0649765 + 0.0649765i
\(9\) 140.102 + 80.8879i 1.72965 + 0.998616i
\(10\) 52.8682 + 133.584i 0.528682 + 1.33584i
\(11\) 48.2961 + 83.6513i 0.399142 + 0.691333i 0.993620 − 0.112778i \(-0.0359749\pi\)
−0.594479 + 0.804111i \(0.702642\pi\)
\(12\) 256.208 − 68.6507i 1.77922 − 0.476741i
\(13\) −48.6898 + 48.6898i −0.288106 + 0.288106i −0.836331 − 0.548225i \(-0.815304\pi\)
0.548225 + 0.836331i \(0.315304\pi\)
\(14\) −264.268 − 97.2182i −1.34831 − 0.496011i
\(15\) 56.9571 − 385.345i 0.253143 − 1.71264i
\(16\) −119.289 + 206.615i −0.465974 + 0.807090i
\(17\) 25.3859 − 94.7415i 0.0878405 − 0.327825i −0.907996 − 0.418978i \(-0.862388\pi\)
0.995837 + 0.0911527i \(0.0290551\pi\)
\(18\) −897.983 − 240.614i −2.77155 − 0.742635i
\(19\) 217.882 + 125.794i 0.603552 + 0.348461i 0.770438 − 0.637516i \(-0.220038\pi\)
−0.166886 + 0.985976i \(0.553371\pi\)
\(20\) −253.697 − 341.702i −0.634243 − 0.854255i
\(21\) 488.739 + 586.548i 1.10825 + 1.33004i
\(22\) −392.499 − 392.499i −0.810947 − 0.810947i
\(23\) −247.264 − 922.803i −0.467418 − 1.74443i −0.648745 − 0.761006i \(-0.724706\pi\)
0.181326 − 0.983423i \(-0.441961\pi\)
\(24\) −79.3569 + 45.8168i −0.137772 + 0.0795430i
\(25\) −608.526 + 142.551i −0.973642 + 0.228082i
\(26\) 197.849 − 342.685i 0.292676 − 0.506930i
\(27\) 889.957 + 889.957i 1.22079 + 1.22079i
\(28\) 830.716 + 75.5632i 1.05959 + 0.0963816i
\(29\) 7.20762i 0.00857029i 0.999991 + 0.00428515i \(0.00136401\pi\)
−0.999991 + 0.00428515i \(0.998636\pi\)
\(30\) 256.978 + 2223.68i 0.285531 + 2.47076i
\(31\) −157.749 − 273.229i −0.164151 − 0.284318i 0.772203 − 0.635376i \(-0.219155\pi\)
−0.936353 + 0.351059i \(0.885822\pi\)
\(32\) 379.199 − 1415.19i 0.370311 − 1.38202i
\(33\) 389.530 + 1453.75i 0.357695 + 1.33494i
\(34\) 563.647i 0.487584i
\(35\) 586.497 − 1075.47i 0.478773 − 0.877939i
\(36\) 2753.97 2.12498
\(37\) −1460.18 + 391.255i −1.06661 + 0.285796i −0.749098 − 0.662459i \(-0.769513\pi\)
−0.317508 + 0.948256i \(0.602846\pi\)
\(38\) −1396.52 − 374.195i −0.967116 − 0.259138i
\(39\) −929.152 + 536.446i −0.610882 + 0.352693i
\(40\) 115.210 + 91.3402i 0.0720063 + 0.0570876i
\(41\) −2207.36 −1.31313 −0.656563 − 0.754272i \(-0.727990\pi\)
−0.656563 + 0.754272i \(0.727990\pi\)
\(42\) −3585.28 − 2528.89i −2.03247 − 1.43361i
\(43\) 1414.89 − 1414.89i 0.765217 − 0.765217i −0.212043 − 0.977260i \(-0.568012\pi\)
0.977260 + 0.212043i \(0.0680118\pi\)
\(44\) 1424.03 + 822.164i 0.735552 + 0.424671i
\(45\) 1606.73 − 3711.54i 0.793449 − 1.83286i
\(46\) 2745.02 + 4754.52i 1.29727 + 2.24694i
\(47\) 1154.98 − 309.475i 0.522851 − 0.140097i 0.0122659 − 0.999925i \(-0.496096\pi\)
0.510585 + 0.859827i \(0.329429\pi\)
\(48\) −2628.57 + 2628.57i −1.14087 + 1.14087i
\(49\) 805.624 + 2261.81i 0.335537 + 0.942027i
\(50\) 3165.78 − 1696.35i 1.26631 − 0.678540i
\(51\) 764.133 − 1323.52i 0.293784 − 0.508849i
\(52\) −303.386 + 1132.25i −0.112199 + 0.418732i
\(53\) −1051.86 − 281.844i −0.374459 − 0.100336i 0.0666807 − 0.997774i \(-0.478759\pi\)
−0.441140 + 0.897438i \(0.645426\pi\)
\(54\) −6263.62 − 3616.30i −2.14802 − 1.24016i
\(55\) 1938.85 − 1439.50i 0.640941 − 0.475868i
\(56\) −283.962 + 49.0620i −0.0905491 + 0.0156448i
\(57\) 2771.91 + 2771.91i 0.853157 + 0.853157i
\(58\) −10.7201 − 40.0079i −0.00318671 − 0.0118930i
\(59\) −5245.99 + 3028.77i −1.50703 + 0.870086i −0.507067 + 0.861906i \(0.669271\pi\)
−0.999967 + 0.00818011i \(0.997396\pi\)
\(60\) −2440.24 − 6165.82i −0.677844 − 1.71273i
\(61\) 464.144 − 803.921i 0.124736 − 0.216050i −0.796893 − 0.604120i \(-0.793525\pi\)
0.921630 + 0.388070i \(0.126858\pi\)
\(62\) 1282.01 + 1282.01i 0.333510 + 0.333510i
\(63\) 3325.33 + 7195.81i 0.837825 + 1.81300i
\(64\) 4602.15i 1.12357i
\(65\) 1348.94 + 1069.46i 0.319275 + 0.253126i
\(66\) −4324.40 − 7490.08i −0.992745 − 1.71949i
\(67\) −171.487 + 639.998i −0.0382016 + 0.142570i −0.982393 − 0.186828i \(-0.940179\pi\)
0.944191 + 0.329398i \(0.106846\pi\)
\(68\) −432.154 − 1612.82i −0.0934589 − 0.348793i
\(69\) − 14885.7i − 3.12658i
\(70\) −1655.94 + 6842.05i −0.337946 + 1.39634i
\(71\) −6612.04 −1.31165 −0.655827 − 0.754912i \(-0.727680\pi\)
−0.655827 + 0.754912i \(0.727680\pi\)
\(72\) −918.986 + 246.242i −0.177274 + 0.0475003i
\(73\) −1460.37 − 391.305i −0.274042 − 0.0734293i 0.119181 − 0.992873i \(-0.461973\pi\)
−0.393223 + 0.919443i \(0.628640\pi\)
\(74\) 7523.24 − 4343.55i 1.37386 − 0.793197i
\(75\) −9733.40 − 308.572i −1.73038 − 0.0548573i
\(76\) 4282.89 0.741498
\(77\) −428.752 + 4713.56i −0.0723144 + 0.795001i
\(78\) 4359.65 − 4359.65i 0.716577 − 0.716577i
\(79\) 396.547 + 228.947i 0.0635390 + 0.0366843i 0.531433 − 0.847100i \(-0.321654\pi\)
−0.467894 + 0.883785i \(0.654987\pi\)
\(80\) 5473.59 + 2369.53i 0.855248 + 0.370239i
\(81\) 3253.28 + 5634.85i 0.495852 + 0.858841i
\(82\) 12252.6 − 3283.08i 1.82222 − 0.488262i
\(83\) −406.242 + 406.242i −0.0589697 + 0.0589697i −0.735977 − 0.677007i \(-0.763277\pi\)
0.677007 + 0.735977i \(0.263277\pi\)
\(84\) 12197.8 + 4487.31i 1.72872 + 0.635956i
\(85\) −2425.74 − 358.543i −0.335742 − 0.0496254i
\(86\) −5749.33 + 9958.13i −0.777357 + 1.34642i
\(87\) −29.0664 + 108.477i −0.00384018 + 0.0143318i
\(88\) −548.703 − 147.025i −0.0708553 − 0.0189856i
\(89\) 6858.49 + 3959.75i 0.865861 + 0.499905i 0.865971 − 0.500095i \(-0.166702\pi\)
−0.000109570 1.00000i \(0.500035\pi\)
\(90\) −3398.36 + 22991.7i −0.419551 + 2.83848i
\(91\) −3324.77 + 574.443i −0.401494 + 0.0693688i
\(92\) −11500.0 − 11500.0i −1.35869 − 1.35869i
\(93\) −1272.32 − 4748.35i −0.147106 − 0.549006i
\(94\) −5950.74 + 3435.66i −0.673466 + 0.388826i
\(95\) 2498.74 − 5772.07i 0.276869 − 0.639565i
\(96\) 11414.1 − 19769.9i 1.23851 − 2.14517i
\(97\) −4926.26 − 4926.26i −0.523569 − 0.523569i 0.395078 − 0.918648i \(-0.370718\pi\)
−0.918648 + 0.395078i \(0.870718\pi\)
\(98\) −7835.90 − 11356.6i −0.815900 − 1.18248i
\(99\) 15626.3i 1.59436i
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.2.3 ✓ 56
5.3 odd 4 inner 35.5.l.a.23.12 yes 56
7.4 even 3 inner 35.5.l.a.32.12 yes 56
35.18 odd 12 inner 35.5.l.a.18.3 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.5.l.a.2.3 ✓ 56 1.1 even 1 trivial
35.5.l.a.18.3 yes 56 35.18 odd 12 inner
35.5.l.a.23.12 yes 56 5.3 odd 4 inner
35.5.l.a.32.12 yes 56 7.4 even 3 inner