Properties

Label 35.4.a.b.1.2
Level $35$
Weight $4$
Character 35.1
Self dual yes
Analytic conductor $2.065$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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] = [35,4,Mod(1,35)] 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("35.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(35, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 35.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(2.06506685020\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 35.1

$q$-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)
 
\(f(q)\) \(=\) \(q+5.41421 q^{2} -4.65685 q^{3} +21.3137 q^{4} -5.00000 q^{5} -25.2132 q^{6} -7.00000 q^{7} +72.0833 q^{8} -5.31371 q^{9} -27.0711 q^{10} -52.2548 q^{11} -99.2548 q^{12} +30.6569 q^{13} -37.8995 q^{14} +23.2843 q^{15} +219.765 q^{16} +37.2254 q^{17} -28.7696 q^{18} +80.2254 q^{19} -106.569 q^{20} +32.5980 q^{21} -282.919 q^{22} +25.8335 q^{23} -335.681 q^{24} +25.0000 q^{25} +165.983 q^{26} +150.480 q^{27} -149.196 q^{28} +20.9411 q^{29} +126.066 q^{30} -314.558 q^{31} +613.186 q^{32} +243.343 q^{33} +201.546 q^{34} +35.0000 q^{35} -113.255 q^{36} +197.147 q^{37} +434.357 q^{38} -142.765 q^{39} -360.416 q^{40} +11.3625 q^{41} +176.492 q^{42} -33.8335 q^{43} -1113.74 q^{44} +26.5685 q^{45} +139.868 q^{46} -361.676 q^{47} -1023.41 q^{48} +49.0000 q^{49} +135.355 q^{50} -173.353 q^{51} +653.411 q^{52} +153.019 q^{53} +814.732 q^{54} +261.274 q^{55} -504.583 q^{56} -373.598 q^{57} +113.380 q^{58} -616.000 q^{59} +496.274 q^{60} +15.2649 q^{61} -1703.09 q^{62} +37.1960 q^{63} +1561.80 q^{64} -153.284 q^{65} +1317.51 q^{66} -166.510 q^{67} +793.411 q^{68} -120.303 q^{69} +189.497 q^{70} -952.000 q^{71} -383.029 q^{72} -148.489 q^{73} +1067.40 q^{74} -116.421 q^{75} +1709.90 q^{76} +365.784 q^{77} -772.958 q^{78} +857.725 q^{79} -1098.82 q^{80} -557.294 q^{81} +61.5189 q^{82} +660.528 q^{83} +694.784 q^{84} -186.127 q^{85} -183.182 q^{86} -97.5198 q^{87} -3766.70 q^{88} -45.7746 q^{89} +143.848 q^{90} -214.598 q^{91} +550.607 q^{92} +1464.85 q^{93} -1958.19 q^{94} -401.127 q^{95} -2855.52 q^{96} +1682.13 q^{97} +265.296 q^{98} +277.667 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{2} + 2 q^{3} + 20 q^{4} - 10 q^{5} - 8 q^{6} - 14 q^{7} + 48 q^{8} + 12 q^{9} - 40 q^{10} - 14 q^{11} - 108 q^{12} + 50 q^{13} - 56 q^{14} - 10 q^{15} + 168 q^{16} - 50 q^{17} + 16 q^{18} + 36 q^{19}+ \cdots + 940 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.41421 1.91421 0.957107 − 0.289735i \(-0.0935673\pi\)
0.957107 + 0.289735i \(0.0935673\pi\)
\(3\) −4.65685 −0.896212 −0.448106 − 0.893980i \(-0.647901\pi\)
−0.448106 + 0.893980i \(0.647901\pi\)
\(4\) 21.3137 2.66421
\(5\) −5.00000 −0.447214
\(6\) −25.2132 −1.71554
\(7\) −7.00000 −0.377964
\(8\) 72.0833 3.18566
\(9\) −5.31371 −0.196804
\(10\) −27.0711 −0.856062
\(11\) −52.2548 −1.43231 −0.716156 − 0.697941i \(-0.754100\pi\)
−0.716156 + 0.697941i \(0.754100\pi\)
\(12\) −99.2548 −2.38770
\(13\) 30.6569 0.654052 0.327026 − 0.945015i \(-0.393953\pi\)
0.327026 + 0.945015i \(0.393953\pi\)
\(14\) −37.8995 −0.723505
\(15\) 23.2843 0.400798
\(16\) 219.765 3.43382
\(17\) 37.2254 0.531087 0.265544 − 0.964099i \(-0.414449\pi\)
0.265544 + 0.964099i \(0.414449\pi\)
\(18\) −28.7696 −0.376725
\(19\) 80.2254 0.968683 0.484341 − 0.874879i \(-0.339059\pi\)
0.484341 + 0.874879i \(0.339059\pi\)
\(20\) −106.569 −1.19147
\(21\) 32.5980 0.338736
\(22\) −282.919 −2.74175
\(23\) 25.8335 0.234202 0.117101 − 0.993120i \(-0.462640\pi\)
0.117101 + 0.993120i \(0.462640\pi\)
\(24\) −335.681 −2.85503
\(25\) 25.0000 0.200000
\(26\) 165.983 1.25200
\(27\) 150.480 1.07259
\(28\) −149.196 −1.00698
\(29\) 20.9411 0.134092 0.0670460 − 0.997750i \(-0.478643\pi\)
0.0670460 + 0.997750i \(0.478643\pi\)
\(30\) 126.066 0.767213
\(31\) −314.558 −1.82246 −0.911232 − 0.411894i \(-0.864867\pi\)
−0.911232 + 0.411894i \(0.864867\pi\)
\(32\) 613.186 3.38741
\(33\) 243.343 1.28365
\(34\) 201.546 1.01661
\(35\) 35.0000 0.169031
\(36\) −113.255 −0.524328
\(37\) 197.147 0.875968 0.437984 − 0.898983i \(-0.355693\pi\)
0.437984 + 0.898983i \(0.355693\pi\)
\(38\) 434.357 1.85427
\(39\) −142.765 −0.586170
\(40\) −360.416 −1.42467
\(41\) 11.3625 0.0432810 0.0216405 − 0.999766i \(-0.493111\pi\)
0.0216405 + 0.999766i \(0.493111\pi\)
\(42\) 176.492 0.648414
\(43\) −33.8335 −0.119990 −0.0599948 − 0.998199i \(-0.519108\pi\)
−0.0599948 + 0.998199i \(0.519108\pi\)
\(44\) −1113.74 −3.81598
\(45\) 26.5685 0.0880134
\(46\) 139.868 0.448313
\(47\) −361.676 −1.12247 −0.561233 − 0.827658i \(-0.689673\pi\)
−0.561233 + 0.827658i \(0.689673\pi\)
\(48\) −1023.41 −3.07743
\(49\) 49.0000 0.142857
\(50\) 135.355 0.382843
\(51\) −173.353 −0.475967
\(52\) 653.411 1.74254
\(53\) 153.019 0.396582 0.198291 − 0.980143i \(-0.436461\pi\)
0.198291 + 0.980143i \(0.436461\pi\)
\(54\) 814.732 2.05317
\(55\) 261.274 0.640549
\(56\) −504.583 −1.20407
\(57\) −373.598 −0.868145
\(58\) 113.380 0.256681
\(59\) −616.000 −1.35926 −0.679630 − 0.733555i \(-0.737860\pi\)
−0.679630 + 0.733555i \(0.737860\pi\)
\(60\) 496.274 1.06781
\(61\) 15.2649 0.0320406 0.0160203 − 0.999872i \(-0.494900\pi\)
0.0160203 + 0.999872i \(0.494900\pi\)
\(62\) −1703.09 −3.48858
\(63\) 37.1960 0.0743849
\(64\) 1561.80 3.05040
\(65\) −153.284 −0.292501
\(66\) 1317.51 2.45719
\(67\) −166.510 −0.303618 −0.151809 − 0.988410i \(-0.548510\pi\)
−0.151809 + 0.988410i \(0.548510\pi\)
\(68\) 793.411 1.41493
\(69\) −120.303 −0.209895
\(70\) 189.497 0.323561
\(71\) −952.000 −1.59129 −0.795645 − 0.605763i \(-0.792868\pi\)
−0.795645 + 0.605763i \(0.792868\pi\)
\(72\) −383.029 −0.626951
\(73\) −148.489 −0.238074 −0.119037 − 0.992890i \(-0.537981\pi\)
−0.119037 + 0.992890i \(0.537981\pi\)
\(74\) 1067.40 1.67679
\(75\) −116.421 −0.179242
\(76\) 1709.90 2.58078
\(77\) 365.784 0.541363
\(78\) −772.958 −1.12205
\(79\) 857.725 1.22154 0.610770 − 0.791808i \(-0.290860\pi\)
0.610770 + 0.791808i \(0.290860\pi\)
\(80\) −1098.82 −1.53565
\(81\) −557.294 −0.764464
\(82\) 61.5189 0.0828491
\(83\) 660.528 0.873523 0.436761 − 0.899577i \(-0.356125\pi\)
0.436761 + 0.899577i \(0.356125\pi\)
\(84\) 694.784 0.902466
\(85\) −186.127 −0.237509
\(86\) −183.182 −0.229686
\(87\) −97.5198 −0.120175
\(88\) −3766.70 −4.56286
\(89\) −45.7746 −0.0545180 −0.0272590 − 0.999628i \(-0.508678\pi\)
−0.0272590 + 0.999628i \(0.508678\pi\)
\(90\) 143.848 0.168477
\(91\) −214.598 −0.247209
\(92\) 550.607 0.623965
\(93\) 1464.85 1.63331
\(94\) −1958.19 −2.14864
\(95\) −401.127 −0.433208
\(96\) −2855.52 −3.03583
\(97\) 1682.13 1.76076 0.880382 − 0.474265i \(-0.157286\pi\)
0.880382 + 0.474265i \(0.157286\pi\)
\(98\) 265.296 0.273459
\(99\) 277.667 0.281885
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 35.4.a.b.1.2 ✓ 2
3.2 odd 2 315.4.a.f.1.1 2
4.3 odd 2 560.4.a.r.1.2 2
5.2 odd 4 175.4.b.c.99.4 4
5.3 odd 4 175.4.b.c.99.1 4
5.4 even 2 175.4.a.c.1.1 2
7.2 even 3 245.4.e.h.116.1 4
7.3 odd 6 245.4.e.i.226.1 4
7.4 even 3 245.4.e.h.226.1 4
7.5 odd 6 245.4.e.i.116.1 4
7.6 odd 2 245.4.a.k.1.2 2
8.3 odd 2 2240.4.a.bo.1.1 2
8.5 even 2 2240.4.a.bn.1.2 2
15.14 odd 2 1575.4.a.z.1.2 2
21.20 even 2 2205.4.a.u.1.1 2
35.34 odd 2 1225.4.a.m.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.b.1.2 ✓ 2 1.1 even 1 trivial
175.4.a.c.1.1 2 5.4 even 2
175.4.b.c.99.1 4 5.3 odd 4
175.4.b.c.99.4 4 5.2 odd 4
245.4.a.k.1.2 2 7.6 odd 2
245.4.e.h.116.1 4 7.2 even 3
245.4.e.h.226.1 4 7.4 even 3
245.4.e.i.116.1 4 7.5 odd 6
245.4.e.i.226.1 4 7.3 odd 6
315.4.a.f.1.1 2 3.2 odd 2
560.4.a.r.1.2 2 4.3 odd 2
1225.4.a.m.1.1 2 35.34 odd 2
1575.4.a.z.1.2 2 15.14 odd 2
2205.4.a.u.1.1 2 21.20 even 2
2240.4.a.bn.1.2 2 8.5 even 2
2240.4.a.bo.1.1 2 8.3 odd 2