Properties

Label 35.4.a.b.1.1
Level $35$
Weight $4$
Character 35.1
Self dual yes
Analytic conductor $2.065$
Analytic rank $0$
Dimension $2$
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 = [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 1.1
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)
 
\(f(q)\) \(=\) \(q+2.58579 q^{2} +6.65685 q^{3} -1.31371 q^{4} -5.00000 q^{5} +17.2132 q^{6} -7.00000 q^{7} -24.0833 q^{8} +17.3137 q^{9} -12.9289 q^{10} +38.2548 q^{11} -8.74517 q^{12} +19.3431 q^{13} -18.1005 q^{14} -33.2843 q^{15} -51.7645 q^{16} -87.2254 q^{17} +44.7696 q^{18} -44.2254 q^{19} +6.56854 q^{20} -46.5980 q^{21} +98.9188 q^{22} +218.167 q^{23} -160.319 q^{24} +25.0000 q^{25} +50.0172 q^{26} -64.4802 q^{27} +9.19596 q^{28} -46.9411 q^{29} -86.0660 q^{30} +194.558 q^{31} +58.8141 q^{32} +254.657 q^{33} -225.546 q^{34} +35.0000 q^{35} -22.7452 q^{36} +366.853 q^{37} -114.357 q^{38} +128.765 q^{39} +120.416 q^{40} -339.362 q^{41} -120.492 q^{42} -226.167 q^{43} -50.2557 q^{44} -86.5685 q^{45} +564.132 q^{46} +11.6762 q^{47} -344.589 q^{48} +49.0000 q^{49} +64.6447 q^{50} -580.647 q^{51} -25.4113 q^{52} -209.019 q^{53} -166.732 q^{54} -191.274 q^{55} +168.583 q^{56} -294.402 q^{57} -121.380 q^{58} -616.000 q^{59} +43.7258 q^{60} +320.735 q^{61} +503.087 q^{62} -121.196 q^{63} +566.197 q^{64} -96.7157 q^{65} +658.488 q^{66} +14.5097 q^{67} +114.589 q^{68} +1452.30 q^{69} +90.5025 q^{70} -952.000 q^{71} -416.971 q^{72} +824.489 q^{73} +948.603 q^{74} +166.421 q^{75} +58.0993 q^{76} -267.784 q^{77} +332.958 q^{78} +156.275 q^{79} +258.823 q^{80} -896.706 q^{81} -877.519 q^{82} -1036.53 q^{83} +61.2162 q^{84} +436.127 q^{85} -584.818 q^{86} -312.480 q^{87} -921.301 q^{88} -170.225 q^{89} -223.848 q^{90} -135.402 q^{91} -286.607 q^{92} +1295.15 q^{93} +30.1921 q^{94} +221.127 q^{95} +391.517 q^{96} +1059.87 q^{97} +126.704 q^{98} +662.333 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{2} + 2 q^{3} + 20 q^{4} - 10 q^{5} - 8 q^{6} - 14 q^{7} + 48 q^{8} + 12 q^{9} - 40 q^{10} - 14 q^{11} - 108 q^{12} + 50 q^{13} - 56 q^{14} - 10 q^{15} + 168 q^{16} - 50 q^{17} + 16 q^{18} + 36 q^{19}+ \cdots + 940 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.58579 0.914214 0.457107 − 0.889412i \(-0.348886\pi\)
0.457107 + 0.889412i \(0.348886\pi\)
\(3\) 6.65685 1.28111 0.640556 − 0.767911i \(-0.278704\pi\)
0.640556 + 0.767911i \(0.278704\pi\)
\(4\) −1.31371 −0.164214
\(5\) −5.00000 −0.447214
\(6\) 17.2132 1.17121
\(7\) −7.00000 −0.377964
\(8\) −24.0833 −1.06434
\(9\) 17.3137 0.641248
\(10\) −12.9289 −0.408849
\(11\) 38.2548 1.04857 0.524285 − 0.851543i \(-0.324333\pi\)
0.524285 + 0.851543i \(0.324333\pi\)
\(12\) −8.74517 −0.210376
\(13\) 19.3431 0.412679 0.206339 − 0.978480i \(-0.433845\pi\)
0.206339 + 0.978480i \(0.433845\pi\)
\(14\) −18.1005 −0.345540
\(15\) −33.2843 −0.572931
\(16\) −51.7645 −0.808820
\(17\) −87.2254 −1.24443 −0.622214 − 0.782847i \(-0.713767\pi\)
−0.622214 + 0.782847i \(0.713767\pi\)
\(18\) 44.7696 0.586238
\(19\) −44.2254 −0.534000 −0.267000 − 0.963697i \(-0.586032\pi\)
−0.267000 + 0.963697i \(0.586032\pi\)
\(20\) 6.56854 0.0734385
\(21\) −46.5980 −0.484215
\(22\) 98.9188 0.958617
\(23\) 218.167 1.97786 0.988932 − 0.148371i \(-0.0474028\pi\)
0.988932 + 0.148371i \(0.0474028\pi\)
\(24\) −160.319 −1.36354
\(25\) 25.0000 0.200000
\(26\) 50.0172 0.377276
\(27\) −64.4802 −0.459601
\(28\) 9.19596 0.0620669
\(29\) −46.9411 −0.300578 −0.150289 − 0.988642i \(-0.548020\pi\)
−0.150289 + 0.988642i \(0.548020\pi\)
\(30\) −86.0660 −0.523781
\(31\) 194.558 1.12722 0.563609 − 0.826042i \(-0.309413\pi\)
0.563609 + 0.826042i \(0.309413\pi\)
\(32\) 58.8141 0.324905
\(33\) 254.657 1.34334
\(34\) −225.546 −1.13767
\(35\) 35.0000 0.169031
\(36\) −22.7452 −0.105302
\(37\) 366.853 1.63001 0.815003 − 0.579457i \(-0.196735\pi\)
0.815003 + 0.579457i \(0.196735\pi\)
\(38\) −114.357 −0.488190
\(39\) 128.765 0.528688
\(40\) 120.416 0.475987
\(41\) −339.362 −1.29267 −0.646336 − 0.763053i \(-0.723699\pi\)
−0.646336 + 0.763053i \(0.723699\pi\)
\(42\) −120.492 −0.442676
\(43\) −226.167 −0.802095 −0.401047 − 0.916057i \(-0.631354\pi\)
−0.401047 + 0.916057i \(0.631354\pi\)
\(44\) −50.2557 −0.172189
\(45\) −86.5685 −0.286775
\(46\) 564.132 1.80819
\(47\) 11.6762 0.0362372 0.0181186 − 0.999836i \(-0.494232\pi\)
0.0181186 + 0.999836i \(0.494232\pi\)
\(48\) −344.589 −1.03619
\(49\) 49.0000 0.142857
\(50\) 64.6447 0.182843
\(51\) −580.647 −1.59425
\(52\) −25.4113 −0.0677674
\(53\) −209.019 −0.541717 −0.270859 − 0.962619i \(-0.587308\pi\)
−0.270859 + 0.962619i \(0.587308\pi\)
\(54\) −166.732 −0.420173
\(55\) −191.274 −0.468935
\(56\) 168.583 0.402283
\(57\) −294.402 −0.684114
\(58\) −121.380 −0.274792
\(59\) −616.000 −1.35926 −0.679630 − 0.733555i \(-0.737860\pi\)
−0.679630 + 0.733555i \(0.737860\pi\)
\(60\) 43.7258 0.0940830
\(61\) 320.735 0.673212 0.336606 − 0.941646i \(-0.390721\pi\)
0.336606 + 0.941646i \(0.390721\pi\)
\(62\) 503.087 1.03052
\(63\) −121.196 −0.242369
\(64\) 566.197 1.10585
\(65\) −96.7157 −0.184556
\(66\) 658.488 1.22810
\(67\) 14.5097 0.0264573 0.0132286 − 0.999912i \(-0.495789\pi\)
0.0132286 + 0.999912i \(0.495789\pi\)
\(68\) 114.589 0.204352
\(69\) 1452.30 2.53387
\(70\) 90.5025 0.154530
\(71\) −952.000 −1.59129 −0.795645 − 0.605763i \(-0.792868\pi\)
−0.795645 + 0.605763i \(0.792868\pi\)
\(72\) −416.971 −0.682506
\(73\) 824.489 1.32191 0.660953 − 0.750427i \(-0.270152\pi\)
0.660953 + 0.750427i \(0.270152\pi\)
\(74\) 948.603 1.49017
\(75\) 166.421 0.256222
\(76\) 58.0993 0.0876901
\(77\) −267.784 −0.396322
\(78\) 332.958 0.483334
\(79\) 156.275 0.222561 0.111280 − 0.993789i \(-0.464505\pi\)
0.111280 + 0.993789i \(0.464505\pi\)
\(80\) 258.823 0.361715
\(81\) −896.706 −1.23005
\(82\) −877.519 −1.18178
\(83\) −1036.53 −1.37077 −0.685384 − 0.728182i \(-0.740366\pi\)
−0.685384 + 0.728182i \(0.740366\pi\)
\(84\) 61.2162 0.0795147
\(85\) 436.127 0.556525
\(86\) −584.818 −0.733286
\(87\) −312.480 −0.385074
\(88\) −921.301 −1.11603
\(89\) −170.225 −0.202740 −0.101370 − 0.994849i \(-0.532323\pi\)
−0.101370 + 0.994849i \(0.532323\pi\)
\(90\) −223.848 −0.262174
\(91\) −135.402 −0.155978
\(92\) −286.607 −0.324792
\(93\) 1295.15 1.44409
\(94\) 30.1921 0.0331285
\(95\) 221.127 0.238812
\(96\) 391.517 0.416240
\(97\) 1059.87 1.10942 0.554710 − 0.832044i \(-0.312829\pi\)
0.554710 + 0.832044i \(0.312829\pi\)
\(98\) 126.704 0.130602
\(99\) 662.333 0.672394
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.1.1 ✓ 2
3.2 odd 2 315.4.a.f.1.2 2
4.3 odd 2 560.4.a.r.1.1 2
5.2 odd 4 175.4.b.c.99.3 4
5.3 odd 4 175.4.b.c.99.2 4
5.4 even 2 175.4.a.c.1.2 2
7.2 even 3 245.4.e.h.116.2 4
7.3 odd 6 245.4.e.i.226.2 4
7.4 even 3 245.4.e.h.226.2 4
7.5 odd 6 245.4.e.i.116.2 4
7.6 odd 2 245.4.a.k.1.1 2
8.3 odd 2 2240.4.a.bo.1.2 2
8.5 even 2 2240.4.a.bn.1.1 2
15.14 odd 2 1575.4.a.z.1.1 2
21.20 even 2 2205.4.a.u.1.2 2
35.34 odd 2 1225.4.a.m.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.b.1.1 ✓ 2 1.1 even 1 trivial
175.4.a.c.1.2 2 5.4 even 2
175.4.b.c.99.2 4 5.3 odd 4
175.4.b.c.99.3 4 5.2 odd 4
245.4.a.k.1.1 2 7.6 odd 2
245.4.e.h.116.2 4 7.2 even 3
245.4.e.h.226.2 4 7.4 even 3
245.4.e.i.116.2 4 7.5 odd 6
245.4.e.i.226.2 4 7.3 odd 6
315.4.a.f.1.2 2 3.2 odd 2
560.4.a.r.1.1 2 4.3 odd 2
1225.4.a.m.1.2 2 35.34 odd 2
1575.4.a.z.1.1 2 15.14 odd 2
2205.4.a.u.1.2 2 21.20 even 2
2240.4.a.bn.1.1 2 8.5 even 2
2240.4.a.bo.1.2 2 8.3 odd 2