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Properties

Label 105.2.m.a.97.7
Level $105$
Weight $2$
Character 105.97
Analytic conductor $0.838$
Analytic rank $0$
Dimension $16$
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(13,105)] 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("105.13"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(105, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 105.m (of order \(4\), degree \(2\), 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: \(0.838429221223\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4x^{14} + 6x^{12} - 12x^{10} + 33x^{8} - 48x^{6} + 96x^{4} - 256x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 97.7
Root \(-0.944649 + 1.05244i\) of defining polynomial
Character \(\chi\) \(=\) 105.97
Dual form 105.2.m.a.13.7

$q$-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+(1.48838 + 1.48838i) q^{2} +(-0.707107 - 0.707107i) q^{3} +2.43055i q^{4} +(1.28999 + 1.82645i) q^{5} -2.10489i q^{6} +(-1.75993 - 1.97552i) q^{7} +(-0.640825 + 0.640825i) q^{8} +1.00000i q^{9} +(-0.798469 + 4.63845i) q^{10} -2.67187 q^{11} +(1.71866 - 1.71866i) q^{12} +(-1.22714 - 1.22714i) q^{13} +(0.320879 - 5.55976i) q^{14} +(0.379340 - 2.20366i) q^{15} +2.95352 q^{16} +(4.74624 - 4.74624i) q^{17} +(-1.48838 + 1.48838i) q^{18} -6.01729 q^{19} +(-4.43929 + 3.13538i) q^{20} +(-0.152445 + 2.64136i) q^{21} +(-3.97676 - 3.97676i) q^{22} +(-0.175684 + 0.175684i) q^{23} +0.906263 q^{24} +(-1.67187 + 4.71220i) q^{25} -3.65291i q^{26} +(0.707107 - 0.707107i) q^{27} +(4.80159 - 4.27759i) q^{28} -0.304889i q^{29} +(3.84448 - 2.71528i) q^{30} +7.25379i q^{31} +(5.67761 + 5.67761i) q^{32} +(1.88930 + 1.88930i) q^{33} +14.1284 q^{34} +(1.33791 - 5.76281i) q^{35} -2.43055 q^{36} +(-0.735441 - 0.735441i) q^{37} +(-8.95602 - 8.95602i) q^{38} +1.73544i q^{39} +(-1.99709 - 0.343782i) q^{40} +7.05736i q^{41} +(-4.15824 + 3.70445i) q^{42} +(0.304889 - 0.304889i) q^{43} -6.49412i q^{44} +(-1.82645 + 1.28999i) q^{45} -0.522969 q^{46} +(-0.556866 + 0.556866i) q^{47} +(-2.08845 - 2.08845i) q^{48} +(-0.805321 + 6.95352i) q^{49} +(-9.50193 + 4.52517i) q^{50} -6.71220 q^{51} +(2.98263 - 2.98263i) q^{52} +(-4.99031 + 4.99031i) q^{53} +2.10489 q^{54} +(-3.44668 - 4.88005i) q^{55} +(2.39376 + 0.138155i) q^{56} +(4.25487 + 4.25487i) q^{57} +(0.453791 - 0.453791i) q^{58} +7.98837 q^{59} +(5.35610 + 0.922006i) q^{60} -5.53409i q^{61} +(-10.7964 + 10.7964i) q^{62} +(1.97552 - 1.75993i) q^{63} +10.9939i q^{64} +(0.658323 - 3.82432i) q^{65} +5.62399i q^{66} +(-3.43055 - 3.43055i) q^{67} +(11.5360 + 11.5360i) q^{68} +0.248455 q^{69} +(10.5686 - 6.58594i) q^{70} +15.3087 q^{71} +(-0.640825 - 0.640825i) q^{72} +(-10.0208 - 10.0208i) q^{73} -2.18923i q^{74} +(4.51422 - 2.14984i) q^{75} -14.6253i q^{76} +(4.70230 + 5.27832i) q^{77} +(-2.58300 + 2.58300i) q^{78} -11.2973i q^{79} +(3.81000 + 5.39447i) q^{80} -1.00000 q^{81} +(-10.5040 + 10.5040i) q^{82} +(4.88941 + 4.88941i) q^{83} +(-6.41995 - 0.370525i) q^{84} +(14.7914 + 2.54621i) q^{85} +0.907583 q^{86} +(-0.215589 + 0.215589i) q^{87} +(1.71220 - 1.71220i) q^{88} -6.91251 q^{89} +(-4.63845 - 0.798469i) q^{90} +(-0.264559 + 4.58392i) q^{91} +(-0.427009 - 0.427009i) q^{92} +(5.12921 - 5.12921i) q^{93} -1.65766 q^{94} +(-7.76222 - 10.9903i) q^{95} -8.02936i q^{96} +(8.84137 - 8.84137i) q^{97} +(-11.5481 + 9.15086i) q^{98} -2.67187i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{7} + 24 q^{8} - 16 q^{11} + 8 q^{15} - 48 q^{16} + 8 q^{21} - 16 q^{22} - 40 q^{23} + 24 q^{28} - 8 q^{30} + 48 q^{32} - 8 q^{35} - 16 q^{36} + 32 q^{37} - 16 q^{42} - 16 q^{43} + 64 q^{46}+ \cdots - 96 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(22\) \(31\) \(71\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.48838 + 1.48838i 1.05244 + 1.05244i 0.998546 + 0.0538973i \(0.0171644\pi\)
0.0538973 + 0.998546i \(0.482836\pi\)
\(3\) −0.707107 − 0.707107i −0.408248 − 0.408248i
\(4\) 2.43055i 1.21528i
\(5\) 1.28999 + 1.82645i 0.576899 + 0.816815i
\(6\) − 2.10489i − 0.859317i
\(7\) −1.75993 − 1.97552i −0.665189 − 0.746675i
\(8\) −0.640825 + 0.640825i −0.226566 + 0.226566i
\(9\) 1.00000i 0.333333i
\(10\) −0.798469 + 4.63845i −0.252498 + 1.46681i
\(11\) −2.67187 −0.805600 −0.402800 − 0.915288i \(-0.631963\pi\)
−0.402800 + 0.915288i \(0.631963\pi\)
\(12\) 1.71866 − 1.71866i 0.496134 − 0.496134i
\(13\) −1.22714 − 1.22714i −0.340348 − 0.340348i 0.516150 − 0.856498i \(-0.327365\pi\)
−0.856498 + 0.516150i \(0.827365\pi\)
\(14\) 0.320879 − 5.55976i 0.0857585 − 1.48591i
\(15\) 0.379340 − 2.20366i 0.0979452 − 0.568982i
\(16\) 2.95352 0.738380
\(17\) 4.74624 − 4.74624i 1.15113 − 1.15113i 0.164807 − 0.986326i \(-0.447300\pi\)
0.986326 − 0.164807i \(-0.0527002\pi\)
\(18\) −1.48838 + 1.48838i −0.350815 + 0.350815i
\(19\) −6.01729 −1.38046 −0.690231 − 0.723589i \(-0.742491\pi\)
−0.690231 + 0.723589i \(0.742491\pi\)
\(20\) −4.43929 + 3.13538i −0.992656 + 0.701092i
\(21\) −0.152445 + 2.64136i −0.0332662 + 0.576391i
\(22\) −3.97676 − 3.97676i −0.847848 − 0.847848i
\(23\) −0.175684 + 0.175684i −0.0366327 + 0.0366327i −0.725186 − 0.688553i \(-0.758246\pi\)
0.688553 + 0.725186i \(0.258246\pi\)
\(24\) 0.906263 0.184990
\(25\) −1.67187 + 4.71220i −0.334374 + 0.942440i
\(26\) − 3.65291i − 0.716394i
\(27\) 0.707107 − 0.707107i 0.136083 − 0.136083i
\(28\) 4.80159 − 4.27759i 0.907416 − 0.808389i
\(29\) − 0.304889i − 0.0566165i −0.999599 − 0.0283083i \(-0.990988\pi\)
0.999599 − 0.0283083i \(-0.00901200\pi\)
\(30\) 3.84448 − 2.71528i 0.701903 − 0.495739i
\(31\) 7.25379i 1.30282i 0.758726 + 0.651410i \(0.225822\pi\)
−0.758726 + 0.651410i \(0.774178\pi\)
\(32\) 5.67761 + 5.67761i 1.00367 + 1.00367i
\(33\) 1.88930 + 1.88930i 0.328885 + 0.328885i
\(34\) 14.1284 2.42301
\(35\) 1.33791 − 5.76281i 0.226148 − 0.974093i
\(36\) −2.43055 −0.405092
\(37\) −0.735441 − 0.735441i −0.120906 − 0.120906i 0.644065 − 0.764971i \(-0.277247\pi\)
−0.764971 + 0.644065i \(0.777247\pi\)
\(38\) −8.95602 − 8.95602i −1.45286 − 1.45286i
\(39\) 1.73544i 0.277893i
\(40\) −1.99709 − 0.343782i −0.315768 − 0.0543567i
\(41\) 7.05736i 1.10217i 0.834447 + 0.551087i \(0.185787\pi\)
−0.834447 + 0.551087i \(0.814213\pi\)
\(42\) −4.15824 + 3.70445i −0.641630 + 0.571608i
\(43\) 0.304889 − 0.304889i 0.0464952 − 0.0464952i −0.683477 − 0.729972i \(-0.739533\pi\)
0.729972 + 0.683477i \(0.239533\pi\)
\(44\) − 6.49412i − 0.979026i
\(45\) −1.82645 + 1.28999i −0.272272 + 0.192300i
\(46\) −0.522969 −0.0771076
\(47\) −0.556866 + 0.556866i −0.0812273 + 0.0812273i −0.746553 − 0.665326i \(-0.768293\pi\)
0.665326 + 0.746553i \(0.268293\pi\)
\(48\) −2.08845 − 2.08845i −0.301442 − 0.301442i
\(49\) −0.805321 + 6.95352i −0.115046 + 0.993360i
\(50\) −9.50193 + 4.52517i −1.34378 + 0.639955i
\(51\) −6.71220 −0.939896
\(52\) 2.98263 − 2.98263i 0.413617 − 0.413617i
\(53\) −4.99031 + 4.99031i −0.685472 + 0.685472i −0.961228 − 0.275756i \(-0.911072\pi\)
0.275756 + 0.961228i \(0.411072\pi\)
\(54\) 2.10489 0.286439
\(55\) −3.44668 − 4.88005i −0.464750 − 0.658026i
\(56\) 2.39376 + 0.138155i 0.319880 + 0.0184617i
\(57\) 4.25487 + 4.25487i 0.563571 + 0.563571i
\(58\) 0.453791 − 0.453791i 0.0595857 − 0.0595857i
\(59\) 7.98837 1.04000 0.519999 − 0.854167i \(-0.325932\pi\)
0.519999 + 0.854167i \(0.325932\pi\)
\(60\) 5.35610 + 0.922006i 0.691470 + 0.119031i
\(61\) − 5.53409i − 0.708567i −0.935138 − 0.354284i \(-0.884725\pi\)
0.935138 − 0.354284i \(-0.115275\pi\)
\(62\) −10.7964 + 10.7964i −1.37114 + 1.37114i
\(63\) 1.97552 − 1.75993i 0.248892 − 0.221730i
\(64\) 10.9939i 1.37423i
\(65\) 0.658323 − 3.82432i 0.0816549 − 0.474348i
\(66\) 5.62399i 0.692265i
\(67\) −3.43055 − 3.43055i −0.419109 − 0.419109i 0.465788 − 0.884896i \(-0.345771\pi\)
−0.884896 + 0.465788i \(0.845771\pi\)
\(68\) 11.5360 + 11.5360i 1.39894 + 1.39894i
\(69\) 0.248455 0.0299104
\(70\) 10.5686 − 6.58594i 1.26319 − 0.787170i
\(71\) 15.3087 1.81681 0.908407 − 0.418087i \(-0.137299\pi\)
0.908407 + 0.418087i \(0.137299\pi\)
\(72\) −0.640825 − 0.640825i −0.0755219 − 0.0755219i
\(73\) −10.0208 − 10.0208i −1.17285 − 1.17285i −0.981527 − 0.191323i \(-0.938722\pi\)
−0.191323 − 0.981527i \(-0.561278\pi\)
\(74\) − 2.18923i − 0.254493i
\(75\) 4.51422 − 2.14984i 0.521257 − 0.248242i
\(76\) − 14.6253i − 1.67764i
\(77\) 4.70230 + 5.27832i 0.535876 + 0.601521i
\(78\) −2.58300 + 2.58300i −0.292467 + 0.292467i
\(79\) − 11.2973i − 1.27104i −0.772084 − 0.635521i \(-0.780785\pi\)
0.772084 − 0.635521i \(-0.219215\pi\)
\(80\) 3.81000 + 5.39447i 0.425971 + 0.603120i
\(81\) −1.00000 −0.111111
\(82\) −10.5040 + 10.5040i −1.15998 + 1.15998i
\(83\) 4.88941 + 4.88941i 0.536682 + 0.536682i 0.922553 − 0.385871i \(-0.126099\pi\)
−0.385871 + 0.922553i \(0.626099\pi\)
\(84\) −6.41995 − 0.370525i −0.700474 − 0.0404276i
\(85\) 14.7914 + 2.54621i 1.60435 + 0.276175i
\(86\) 0.907583 0.0978671
\(87\) −0.215589 + 0.215589i −0.0231136 + 0.0231136i
\(88\) 1.71220 − 1.71220i 0.182521 − 0.182521i
\(89\) −6.91251 −0.732725 −0.366363 − 0.930472i \(-0.619397\pi\)
−0.366363 + 0.930472i \(0.619397\pi\)
\(90\) −4.63845 − 0.798469i −0.488935 − 0.0841660i
\(91\) −0.264559 + 4.58392i −0.0277333 + 0.480525i
\(92\) −0.427009 − 0.427009i −0.0445188 − 0.0445188i
\(93\) 5.12921 − 5.12921i 0.531874 − 0.531874i
\(94\) −1.65766 −0.170974
\(95\) −7.76222 − 10.9903i −0.796387 − 1.12758i
\(96\) − 8.02936i − 0.819493i
\(97\) 8.84137 − 8.84137i 0.897705 − 0.897705i −0.0975276 − 0.995233i \(-0.531093\pi\)
0.995233 + 0.0975276i \(0.0310934\pi\)
\(98\) −11.5481 + 9.15086i −1.16654 + 0.924376i
\(99\) − 2.67187i − 0.268533i
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.m.a.97.7 yes 16
3.2 odd 2 315.2.p.e.307.1 16
4.3 odd 2 1680.2.cz.d.97.8 16
5.2 odd 4 525.2.m.b.118.1 16
5.3 odd 4 inner 105.2.m.a.13.8 yes 16
5.4 even 2 525.2.m.b.307.2 16
7.2 even 3 735.2.v.a.472.8 32
7.3 odd 6 735.2.v.a.607.2 32
7.4 even 3 735.2.v.a.607.1 32
7.5 odd 6 735.2.v.a.472.7 32
7.6 odd 2 inner 105.2.m.a.97.8 yes 16
15.8 even 4 315.2.p.e.118.2 16
20.3 even 4 1680.2.cz.d.433.1 16
21.20 even 2 315.2.p.e.307.2 16
28.27 even 2 1680.2.cz.d.97.1 16
35.3 even 12 735.2.v.a.313.8 32
35.13 even 4 inner 105.2.m.a.13.7 ✓ 16
35.18 odd 12 735.2.v.a.313.7 32
35.23 odd 12 735.2.v.a.178.2 32
35.27 even 4 525.2.m.b.118.2 16
35.33 even 12 735.2.v.a.178.1 32
35.34 odd 2 525.2.m.b.307.1 16
105.83 odd 4 315.2.p.e.118.1 16
140.83 odd 4 1680.2.cz.d.433.8 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.m.a.13.7 ✓ 16 35.13 even 4 inner
105.2.m.a.13.8 yes 16 5.3 odd 4 inner
105.2.m.a.97.7 yes 16 1.1 even 1 trivial
105.2.m.a.97.8 yes 16 7.6 odd 2 inner
315.2.p.e.118.1 16 105.83 odd 4
315.2.p.e.118.2 16 15.8 even 4
315.2.p.e.307.1 16 3.2 odd 2
315.2.p.e.307.2 16 21.20 even 2
525.2.m.b.118.1 16 5.2 odd 4
525.2.m.b.118.2 16 35.27 even 4
525.2.m.b.307.1 16 35.34 odd 2
525.2.m.b.307.2 16 5.4 even 2
735.2.v.a.178.1 32 35.33 even 12
735.2.v.a.178.2 32 35.23 odd 12
735.2.v.a.313.7 32 35.18 odd 12
735.2.v.a.313.8 32 35.3 even 12
735.2.v.a.472.7 32 7.5 odd 6
735.2.v.a.472.8 32 7.2 even 3
735.2.v.a.607.1 32 7.4 even 3
735.2.v.a.607.2 32 7.3 odd 6
1680.2.cz.d.97.1 16 28.27 even 2
1680.2.cz.d.97.8 16 4.3 odd 2
1680.2.cz.d.433.1 16 20.3 even 4
1680.2.cz.d.433.8 16 140.83 odd 4