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Properties

Label 35.4.a.c.1.3
Level $35$
Weight $4$
Character 35.1
Self dual yes
Analytic conductor $2.065$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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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] = [35,4,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 = [3] 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: \(3\)
Coefficient field: 3.3.14360.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 17x - 14 \) 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 1.3
Root \(4.48565\) 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)
 
\(f(q)\) \(=\) \(q+3.48565 q^{2} +0.850238 q^{3} +4.14976 q^{4} +5.00000 q^{5} +2.96363 q^{6} +7.00000 q^{7} -13.4206 q^{8} -26.2771 q^{9} +17.4283 q^{10} -6.90764 q^{11} +3.52829 q^{12} -22.1364 q^{13} +24.3996 q^{14} +4.25119 q^{15} -79.9776 q^{16} +88.3030 q^{17} -91.5928 q^{18} +36.9560 q^{19} +20.7488 q^{20} +5.95167 q^{21} -24.0776 q^{22} -95.5283 q^{23} -11.4107 q^{24} +25.0000 q^{25} -77.1598 q^{26} -45.2982 q^{27} +29.0483 q^{28} +269.029 q^{29} +14.8182 q^{30} +197.114 q^{31} -171.409 q^{32} -5.87314 q^{33} +307.793 q^{34} +35.0000 q^{35} -109.044 q^{36} +2.14546 q^{37} +128.816 q^{38} -18.8212 q^{39} -67.1029 q^{40} +174.127 q^{41} +20.7454 q^{42} -17.0345 q^{43} -28.6650 q^{44} -131.385 q^{45} -332.978 q^{46} -528.029 q^{47} -68.0000 q^{48} +49.0000 q^{49} +87.1413 q^{50} +75.0786 q^{51} -91.8608 q^{52} -641.114 q^{53} -157.894 q^{54} -34.5382 q^{55} -93.9441 q^{56} +31.4214 q^{57} +937.742 q^{58} -642.975 q^{59} +17.6414 q^{60} +142.967 q^{61} +687.070 q^{62} -183.940 q^{63} +42.3480 q^{64} -110.682 q^{65} -20.4717 q^{66} +478.797 q^{67} +366.436 q^{68} -81.2218 q^{69} +121.998 q^{70} +105.550 q^{71} +352.654 q^{72} +986.512 q^{73} +7.47834 q^{74} +21.2560 q^{75} +153.358 q^{76} -48.3534 q^{77} -65.6042 q^{78} -1099.86 q^{79} -399.888 q^{80} +670.967 q^{81} +606.947 q^{82} -1236.62 q^{83} +24.6980 q^{84} +441.515 q^{85} -59.3763 q^{86} +228.739 q^{87} +92.7045 q^{88} -711.698 q^{89} -457.964 q^{90} -154.955 q^{91} -396.420 q^{92} +167.594 q^{93} -1840.52 q^{94} +184.780 q^{95} -145.739 q^{96} -636.553 q^{97} +170.797 q^{98} +181.513 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 2 q^{3} + 13 q^{4} + 15 q^{5} + 24 q^{6} + 21 q^{7} - 15 q^{8} + 81 q^{9} - 15 q^{10} - 74 q^{11} - 152 q^{12} + 44 q^{13} - 21 q^{14} + 10 q^{15} - 79 q^{16} - 52 q^{17} - 411 q^{18} + 168 q^{19}+ \cdots - 3488 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.48565 1.23236 0.616182 − 0.787604i \(-0.288679\pi\)
0.616182 + 0.787604i \(0.288679\pi\)
\(3\) 0.850238 0.163628 0.0818142 − 0.996648i \(-0.473929\pi\)
0.0818142 + 0.996648i \(0.473929\pi\)
\(4\) 4.14976 0.518720
\(5\) 5.00000 0.447214
\(6\) 2.96363 0.201650
\(7\) 7.00000 0.377964
\(8\) −13.4206 −0.593112
\(9\) −26.2771 −0.973226
\(10\) 17.4283 0.551130
\(11\) −6.90764 −0.189339 −0.0946696 − 0.995509i \(-0.530179\pi\)
−0.0946696 + 0.995509i \(0.530179\pi\)
\(12\) 3.52829 0.0848774
\(13\) −22.1364 −0.472272 −0.236136 − 0.971720i \(-0.575881\pi\)
−0.236136 + 0.971720i \(0.575881\pi\)
\(14\) 24.3996 0.465790
\(15\) 4.25119 0.0731769
\(16\) −79.9776 −1.24965
\(17\) 88.3030 1.25980 0.629901 − 0.776676i \(-0.283096\pi\)
0.629901 + 0.776676i \(0.283096\pi\)
\(18\) −91.5928 −1.19937
\(19\) 36.9560 0.446225 0.223113 − 0.974793i \(-0.428378\pi\)
0.223113 + 0.974793i \(0.428378\pi\)
\(20\) 20.7488 0.231979
\(21\) 5.95167 0.0618457
\(22\) −24.0776 −0.233335
\(23\) −95.5283 −0.866045 −0.433022 − 0.901383i \(-0.642553\pi\)
−0.433022 + 0.901383i \(0.642553\pi\)
\(24\) −11.4107 −0.0970500
\(25\) 25.0000 0.200000
\(26\) −77.1598 −0.582010
\(27\) −45.2982 −0.322876
\(28\) 29.0483 0.196058
\(29\) 269.029 1.72267 0.861336 − 0.508035i \(-0.169628\pi\)
0.861336 + 0.508035i \(0.169628\pi\)
\(30\) 14.8182 0.0901805
\(31\) 197.114 1.14202 0.571012 − 0.820942i \(-0.306551\pi\)
0.571012 + 0.820942i \(0.306551\pi\)
\(32\) −171.409 −0.946911
\(33\) −5.87314 −0.0309813
\(34\) 307.793 1.55253
\(35\) 35.0000 0.169031
\(36\) −109.044 −0.504832
\(37\) 2.14546 0.00953276 0.00476638 − 0.999989i \(-0.498483\pi\)
0.00476638 + 0.999989i \(0.498483\pi\)
\(38\) 128.816 0.549912
\(39\) −18.8212 −0.0772771
\(40\) −67.1029 −0.265248
\(41\) 174.127 0.663271 0.331636 − 0.943408i \(-0.392400\pi\)
0.331636 + 0.943408i \(0.392400\pi\)
\(42\) 20.7454 0.0762164
\(43\) −17.0345 −0.0604125 −0.0302062 − 0.999544i \(-0.509616\pi\)
−0.0302062 + 0.999544i \(0.509616\pi\)
\(44\) −28.6650 −0.0982140
\(45\) −131.385 −0.435240
\(46\) −332.978 −1.06728
\(47\) −528.029 −1.63874 −0.819371 − 0.573264i \(-0.805677\pi\)
−0.819371 + 0.573264i \(0.805677\pi\)
\(48\) −68.0000 −0.204478
\(49\) 49.0000 0.142857
\(50\) 87.1413 0.246473
\(51\) 75.0786 0.206139
\(52\) −91.8608 −0.244977
\(53\) −641.114 −1.66158 −0.830790 − 0.556586i \(-0.812111\pi\)
−0.830790 + 0.556586i \(0.812111\pi\)
\(54\) −157.894 −0.397900
\(55\) −34.5382 −0.0846750
\(56\) −93.9441 −0.224175
\(57\) 31.4214 0.0730151
\(58\) 937.742 2.12296
\(59\) −642.975 −1.41878 −0.709391 − 0.704815i \(-0.751030\pi\)
−0.709391 + 0.704815i \(0.751030\pi\)
\(60\) 17.6414 0.0379583
\(61\) 142.967 0.300083 0.150042 − 0.988680i \(-0.452059\pi\)
0.150042 + 0.988680i \(0.452059\pi\)
\(62\) 687.070 1.40739
\(63\) −183.940 −0.367845
\(64\) 42.3480 0.0827109
\(65\) −110.682 −0.211206
\(66\) −20.4717 −0.0381802
\(67\) 478.797 0.873050 0.436525 − 0.899692i \(-0.356209\pi\)
0.436525 + 0.899692i \(0.356209\pi\)
\(68\) 366.436 0.653484
\(69\) −81.2218 −0.141710
\(70\) 121.998 0.208307
\(71\) 105.550 0.176430 0.0882150 − 0.996101i \(-0.471884\pi\)
0.0882150 + 0.996101i \(0.471884\pi\)
\(72\) 352.654 0.577232
\(73\) 986.512 1.58168 0.790839 − 0.612024i \(-0.209644\pi\)
0.790839 + 0.612024i \(0.209644\pi\)
\(74\) 7.47834 0.0117478
\(75\) 21.2560 0.0327257
\(76\) 153.358 0.231466
\(77\) −48.3534 −0.0715635
\(78\) −65.6042 −0.0952335
\(79\) −1099.86 −1.56638 −0.783190 − 0.621783i \(-0.786409\pi\)
−0.783190 + 0.621783i \(0.786409\pi\)
\(80\) −399.888 −0.558860
\(81\) 670.967 0.920394
\(82\) 606.947 0.817391
\(83\) −1236.62 −1.63538 −0.817691 − 0.575657i \(-0.804746\pi\)
−0.817691 + 0.575657i \(0.804746\pi\)
\(84\) 24.6980 0.0320806
\(85\) 441.515 0.563400
\(86\) −59.3763 −0.0744501
\(87\) 228.739 0.281878
\(88\) 92.7045 0.112299
\(89\) −711.698 −0.847638 −0.423819 − 0.905747i \(-0.639311\pi\)
−0.423819 + 0.905747i \(0.639311\pi\)
\(90\) −457.964 −0.536374
\(91\) −154.955 −0.178502
\(92\) −396.420 −0.449235
\(93\) 167.594 0.186867
\(94\) −1840.52 −2.01953
\(95\) 184.780 0.199558
\(96\) −145.739 −0.154942
\(97\) −636.553 −0.666311 −0.333156 − 0.942872i \(-0.608113\pi\)
−0.333156 + 0.942872i \(0.608113\pi\)
\(98\) 170.797 0.176052
\(99\) 181.513 0.184270
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.c.1.3 ✓ 3
3.2 odd 2 315.4.a.p.1.1 3
4.3 odd 2 560.4.a.u.1.2 3
5.2 odd 4 175.4.b.e.99.5 6
5.3 odd 4 175.4.b.e.99.2 6
5.4 even 2 175.4.a.f.1.1 3
7.2 even 3 245.4.e.m.116.1 6
7.3 odd 6 245.4.e.n.226.1 6
7.4 even 3 245.4.e.m.226.1 6
7.5 odd 6 245.4.e.n.116.1 6
7.6 odd 2 245.4.a.l.1.3 3
8.3 odd 2 2240.4.a.bv.1.2 3
8.5 even 2 2240.4.a.bt.1.2 3
15.14 odd 2 1575.4.a.ba.1.3 3
21.20 even 2 2205.4.a.bm.1.1 3
35.34 odd 2 1225.4.a.y.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.c.1.3 ✓ 3 1.1 even 1 trivial
175.4.a.f.1.1 3 5.4 even 2
175.4.b.e.99.2 6 5.3 odd 4
175.4.b.e.99.5 6 5.2 odd 4
245.4.a.l.1.3 3 7.6 odd 2
245.4.e.m.116.1 6 7.2 even 3
245.4.e.m.226.1 6 7.4 even 3
245.4.e.n.116.1 6 7.5 odd 6
245.4.e.n.226.1 6 7.3 odd 6
315.4.a.p.1.1 3 3.2 odd 2
560.4.a.u.1.2 3 4.3 odd 2
1225.4.a.y.1.1 3 35.34 odd 2
1575.4.a.ba.1.3 3 15.14 odd 2
2205.4.a.bm.1.1 3 21.20 even 2
2240.4.a.bt.1.2 3 8.5 even 2
2240.4.a.bv.1.2 3 8.3 odd 2