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Properties

Label 105.2.g.a.104.3
Level $105$
Weight $2$
Character 105.104
Analytic conductor $0.838$
Analytic rank $0$
Dimension $4$
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] = [105,2,Mod(104,105)] 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("105.104"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(105, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 105.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.838429221223\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 104.3
Root \(-1.93185i\) of defining polynomial
Character \(\chi\) \(=\) 105.104
Dual form 105.2.g.a.104.4

$q$-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+1.73205 q^{2} +(-1.00000 - 1.41421i) q^{3} +1.00000 q^{4} +(1.73205 + 1.41421i) q^{5} +(-1.73205 - 2.44949i) q^{6} +(1.00000 - 2.44949i) q^{7} -1.73205 q^{8} +(-1.00000 + 2.82843i) q^{9} +(3.00000 + 2.44949i) q^{10} +2.82843i q^{11} +(-1.00000 - 1.41421i) q^{12} -4.00000 q^{13} +(1.73205 - 4.24264i) q^{14} +(0.267949 - 3.86370i) q^{15} -5.00000 q^{16} +2.82843i q^{17} +(-1.73205 + 4.89898i) q^{18} +(1.73205 + 1.41421i) q^{20} +(-4.46410 + 1.03528i) q^{21} +4.89898i q^{22} +3.46410 q^{23} +(1.73205 + 2.44949i) q^{24} +(1.00000 + 4.89898i) q^{25} -6.92820 q^{26} +(5.00000 - 1.41421i) q^{27} +(1.00000 - 2.44949i) q^{28} -5.65685i q^{29} +(0.464102 - 6.69213i) q^{30} -9.79796i q^{31} -5.19615 q^{32} +(4.00000 - 2.82843i) q^{33} +4.89898i q^{34} +(5.19615 - 2.82843i) q^{35} +(-1.00000 + 2.82843i) q^{36} +(4.00000 + 5.65685i) q^{39} +(-3.00000 - 2.44949i) q^{40} +3.46410 q^{41} +(-7.73205 + 1.79315i) q^{42} -4.89898i q^{43} +2.82843i q^{44} +(-5.73205 + 3.48477i) q^{45} +6.00000 q^{46} +2.82843i q^{47} +(5.00000 + 7.07107i) q^{48} +(-5.00000 - 4.89898i) q^{49} +(1.73205 + 8.48528i) q^{50} +(4.00000 - 2.82843i) q^{51} -4.00000 q^{52} +(8.66025 - 2.44949i) q^{54} +(-4.00000 + 4.89898i) q^{55} +(-1.73205 + 4.24264i) q^{56} -9.79796i q^{58} -6.92820 q^{59} +(0.267949 - 3.86370i) q^{60} +9.79796i q^{61} -16.9706i q^{62} +(5.92820 + 5.27792i) q^{63} +1.00000 q^{64} +(-6.92820 - 5.65685i) q^{65} +(6.92820 - 4.89898i) q^{66} +4.89898i q^{67} +2.82843i q^{68} +(-3.46410 - 4.89898i) q^{69} +(9.00000 - 4.89898i) q^{70} +2.82843i q^{71} +(1.73205 - 4.89898i) q^{72} +8.00000 q^{73} +(5.92820 - 6.31319i) q^{75} +(6.92820 + 2.82843i) q^{77} +(6.92820 + 9.79796i) q^{78} +8.00000 q^{79} +(-8.66025 - 7.07107i) q^{80} +(-7.00000 - 5.65685i) q^{81} +6.00000 q^{82} +2.82843i q^{83} +(-4.46410 + 1.03528i) q^{84} +(-4.00000 + 4.89898i) q^{85} -8.48528i q^{86} +(-8.00000 + 5.65685i) q^{87} -4.89898i q^{88} -10.3923 q^{89} +(-9.92820 + 6.03579i) q^{90} +(-4.00000 + 9.79796i) q^{91} +3.46410 q^{92} +(-13.8564 + 9.79796i) q^{93} +4.89898i q^{94} +(5.19615 + 7.34847i) q^{96} +8.00000 q^{97} +(-8.66025 - 8.48528i) q^{98} +(-8.00000 - 2.82843i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 4 q^{4} + 4 q^{7} - 4 q^{9} + 12 q^{10} - 4 q^{12} - 16 q^{13} + 8 q^{15} - 20 q^{16} - 4 q^{21} + 4 q^{25} + 20 q^{27} + 4 q^{28} - 12 q^{30} + 16 q^{33} - 4 q^{36} + 16 q^{39} - 12 q^{40}+ \cdots - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\) \(71\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73205 1.22474 0.612372 − 0.790569i \(-0.290215\pi\)
0.612372 + 0.790569i \(0.290215\pi\)
\(3\) −1.00000 − 1.41421i −0.577350 − 0.816497i
\(4\) 1.00000 0.500000
\(5\) 1.73205 + 1.41421i 0.774597 + 0.632456i
\(6\) −1.73205 − 2.44949i −0.707107 − 1.00000i
\(7\) 1.00000 − 2.44949i 0.377964 − 0.925820i
\(8\) −1.73205 −0.612372
\(9\) −1.00000 + 2.82843i −0.333333 + 0.942809i
\(10\) 3.00000 + 2.44949i 0.948683 + 0.774597i
\(11\) 2.82843i 0.852803i 0.904534 + 0.426401i \(0.140219\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) −1.00000 − 1.41421i −0.288675 − 0.408248i
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 1.73205 − 4.24264i 0.462910 − 1.13389i
\(15\) 0.267949 − 3.86370i 0.0691842 − 0.997604i
\(16\) −5.00000 −1.25000
\(17\) 2.82843i 0.685994i 0.939336 + 0.342997i \(0.111442\pi\)
−0.939336 + 0.342997i \(0.888558\pi\)
\(18\) −1.73205 + 4.89898i −0.408248 + 1.15470i
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 1.73205 + 1.41421i 0.387298 + 0.316228i
\(21\) −4.46410 + 1.03528i −0.974147 + 0.225916i
\(22\) 4.89898i 1.04447i
\(23\) 3.46410 0.722315 0.361158 − 0.932505i \(-0.382382\pi\)
0.361158 + 0.932505i \(0.382382\pi\)
\(24\) 1.73205 + 2.44949i 0.353553 + 0.500000i
\(25\) 1.00000 + 4.89898i 0.200000 + 0.979796i
\(26\) −6.92820 −1.35873
\(27\) 5.00000 − 1.41421i 0.962250 − 0.272166i
\(28\) 1.00000 − 2.44949i 0.188982 − 0.462910i
\(29\) − 5.65685i − 1.05045i −0.850963 − 0.525226i \(-0.823981\pi\)
0.850963 − 0.525226i \(-0.176019\pi\)
\(30\) 0.464102 − 6.69213i 0.0847330 − 1.22181i
\(31\) − 9.79796i − 1.75977i −0.475191 − 0.879883i \(-0.657621\pi\)
0.475191 − 0.879883i \(-0.342379\pi\)
\(32\) −5.19615 −0.918559
\(33\) 4.00000 − 2.82843i 0.696311 − 0.492366i
\(34\) 4.89898i 0.840168i
\(35\) 5.19615 − 2.82843i 0.878310 − 0.478091i
\(36\) −1.00000 + 2.82843i −0.166667 + 0.471405i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 4.00000 + 5.65685i 0.640513 + 0.905822i
\(40\) −3.00000 − 2.44949i −0.474342 − 0.387298i
\(41\) 3.46410 0.541002 0.270501 − 0.962720i \(-0.412811\pi\)
0.270501 + 0.962720i \(0.412811\pi\)
\(42\) −7.73205 + 1.79315i −1.19308 + 0.276689i
\(43\) − 4.89898i − 0.747087i −0.927613 − 0.373544i \(-0.878143\pi\)
0.927613 − 0.373544i \(-0.121857\pi\)
\(44\) 2.82843i 0.426401i
\(45\) −5.73205 + 3.48477i −0.854484 + 0.519478i
\(46\) 6.00000 0.884652
\(47\) 2.82843i 0.412568i 0.978492 + 0.206284i \(0.0661372\pi\)
−0.978492 + 0.206284i \(0.933863\pi\)
\(48\) 5.00000 + 7.07107i 0.721688 + 1.02062i
\(49\) −5.00000 − 4.89898i −0.714286 − 0.699854i
\(50\) 1.73205 + 8.48528i 0.244949 + 1.20000i
\(51\) 4.00000 − 2.82843i 0.560112 − 0.396059i
\(52\) −4.00000 −0.554700
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 8.66025 − 2.44949i 1.17851 − 0.333333i
\(55\) −4.00000 + 4.89898i −0.539360 + 0.660578i
\(56\) −1.73205 + 4.24264i −0.231455 + 0.566947i
\(57\) 0 0
\(58\) − 9.79796i − 1.28654i
\(59\) −6.92820 −0.901975 −0.450988 − 0.892530i \(-0.648928\pi\)
−0.450988 + 0.892530i \(0.648928\pi\)
\(60\) 0.267949 − 3.86370i 0.0345921 − 0.498802i
\(61\) 9.79796i 1.25450i 0.778818 + 0.627250i \(0.215820\pi\)
−0.778818 + 0.627250i \(0.784180\pi\)
\(62\) − 16.9706i − 2.15526i
\(63\) 5.92820 + 5.27792i 0.746883 + 0.664955i
\(64\) 1.00000 0.125000
\(65\) −6.92820 − 5.65685i −0.859338 − 0.701646i
\(66\) 6.92820 − 4.89898i 0.852803 − 0.603023i
\(67\) 4.89898i 0.598506i 0.954174 + 0.299253i \(0.0967374\pi\)
−0.954174 + 0.299253i \(0.903263\pi\)
\(68\) 2.82843i 0.342997i
\(69\) −3.46410 − 4.89898i −0.417029 − 0.589768i
\(70\) 9.00000 − 4.89898i 1.07571 − 0.585540i
\(71\) 2.82843i 0.335673i 0.985815 + 0.167836i \(0.0536780\pi\)
−0.985815 + 0.167836i \(0.946322\pi\)
\(72\) 1.73205 − 4.89898i 0.204124 − 0.577350i
\(73\) 8.00000 0.936329 0.468165 − 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 0 0
\(75\) 5.92820 − 6.31319i 0.684530 − 0.728985i
\(76\) 0 0
\(77\) 6.92820 + 2.82843i 0.789542 + 0.322329i
\(78\) 6.92820 + 9.79796i 0.784465 + 1.10940i
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −8.66025 − 7.07107i −0.968246 − 0.790569i
\(81\) −7.00000 − 5.65685i −0.777778 − 0.628539i
\(82\) 6.00000 0.662589
\(83\) 2.82843i 0.310460i 0.987878 + 0.155230i \(0.0496119\pi\)
−0.987878 + 0.155230i \(0.950388\pi\)
\(84\) −4.46410 + 1.03528i −0.487073 + 0.112958i
\(85\) −4.00000 + 4.89898i −0.433861 + 0.531369i
\(86\) − 8.48528i − 0.914991i
\(87\) −8.00000 + 5.65685i −0.857690 + 0.606478i
\(88\) − 4.89898i − 0.522233i
\(89\) −10.3923 −1.10158 −0.550791 − 0.834643i \(-0.685674\pi\)
−0.550791 + 0.834643i \(0.685674\pi\)
\(90\) −9.92820 + 6.03579i −1.04652 + 0.636228i
\(91\) −4.00000 + 9.79796i −0.419314 + 1.02711i
\(92\) 3.46410 0.361158
\(93\) −13.8564 + 9.79796i −1.43684 + 1.01600i
\(94\) 4.89898i 0.505291i
\(95\) 0 0
\(96\) 5.19615 + 7.34847i 0.530330 + 0.750000i
\(97\) 8.00000 0.812277 0.406138 − 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) −8.66025 − 8.48528i −0.874818 − 0.857143i
\(99\) −8.00000 − 2.82843i −0.804030 − 0.284268i
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 105.2.g.a.104.3 yes 4
3.2 odd 2 inner 105.2.g.a.104.2 yes 4
4.3 odd 2 1680.2.k.c.209.4 4
5.2 odd 4 525.2.b.j.251.6 8
5.3 odd 4 525.2.b.j.251.3 8
5.4 even 2 105.2.g.c.104.2 yes 4
7.2 even 3 735.2.p.c.374.2 8
7.3 odd 6 735.2.p.a.509.2 8
7.4 even 3 735.2.p.c.509.1 8
7.5 odd 6 735.2.p.a.374.1 8
7.6 odd 2 105.2.g.c.104.4 yes 4
12.11 even 2 1680.2.k.c.209.1 4
15.2 even 4 525.2.b.j.251.4 8
15.8 even 4 525.2.b.j.251.5 8
15.14 odd 2 105.2.g.c.104.3 yes 4
20.19 odd 2 1680.2.k.a.209.2 4
21.2 odd 6 735.2.p.c.374.3 8
21.5 even 6 735.2.p.a.374.4 8
21.11 odd 6 735.2.p.c.509.4 8
21.17 even 6 735.2.p.a.509.3 8
21.20 even 2 105.2.g.c.104.1 yes 4
28.27 even 2 1680.2.k.a.209.1 4
35.4 even 6 735.2.p.a.509.4 8
35.9 even 6 735.2.p.a.374.3 8
35.13 even 4 525.2.b.j.251.2 8
35.19 odd 6 735.2.p.c.374.4 8
35.24 odd 6 735.2.p.c.509.3 8
35.27 even 4 525.2.b.j.251.7 8
35.34 odd 2 inner 105.2.g.a.104.1 ✓ 4
60.59 even 2 1680.2.k.a.209.3 4
84.83 odd 2 1680.2.k.a.209.4 4
105.44 odd 6 735.2.p.a.374.2 8
105.59 even 6 735.2.p.c.509.2 8
105.62 odd 4 525.2.b.j.251.1 8
105.74 odd 6 735.2.p.a.509.1 8
105.83 odd 4 525.2.b.j.251.8 8
105.89 even 6 735.2.p.c.374.1 8
105.104 even 2 inner 105.2.g.a.104.4 yes 4
140.139 even 2 1680.2.k.c.209.3 4
420.419 odd 2 1680.2.k.c.209.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.g.a.104.1 ✓ 4 35.34 odd 2 inner
105.2.g.a.104.2 yes 4 3.2 odd 2 inner
105.2.g.a.104.3 yes 4 1.1 even 1 trivial
105.2.g.a.104.4 yes 4 105.104 even 2 inner
105.2.g.c.104.1 yes 4 21.20 even 2
105.2.g.c.104.2 yes 4 5.4 even 2
105.2.g.c.104.3 yes 4 15.14 odd 2
105.2.g.c.104.4 yes 4 7.6 odd 2
525.2.b.j.251.1 8 105.62 odd 4
525.2.b.j.251.2 8 35.13 even 4
525.2.b.j.251.3 8 5.3 odd 4
525.2.b.j.251.4 8 15.2 even 4
525.2.b.j.251.5 8 15.8 even 4
525.2.b.j.251.6 8 5.2 odd 4
525.2.b.j.251.7 8 35.27 even 4
525.2.b.j.251.8 8 105.83 odd 4
735.2.p.a.374.1 8 7.5 odd 6
735.2.p.a.374.2 8 105.44 odd 6
735.2.p.a.374.3 8 35.9 even 6
735.2.p.a.374.4 8 21.5 even 6
735.2.p.a.509.1 8 105.74 odd 6
735.2.p.a.509.2 8 7.3 odd 6
735.2.p.a.509.3 8 21.17 even 6
735.2.p.a.509.4 8 35.4 even 6
735.2.p.c.374.1 8 105.89 even 6
735.2.p.c.374.2 8 7.2 even 3
735.2.p.c.374.3 8 21.2 odd 6
735.2.p.c.374.4 8 35.19 odd 6
735.2.p.c.509.1 8 7.4 even 3
735.2.p.c.509.2 8 105.59 even 6
735.2.p.c.509.3 8 35.24 odd 6
735.2.p.c.509.4 8 21.11 odd 6
1680.2.k.a.209.1 4 28.27 even 2
1680.2.k.a.209.2 4 20.19 odd 2
1680.2.k.a.209.3 4 60.59 even 2
1680.2.k.a.209.4 4 84.83 odd 2
1680.2.k.c.209.1 4 12.11 even 2
1680.2.k.c.209.2 4 420.419 odd 2
1680.2.k.c.209.3 4 140.139 even 2
1680.2.k.c.209.4 4 4.3 odd 2