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Properties

Label 105.2.m.a.97.3
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 97.3
Root \(1.40927 - 0.118126i\) of defining polynomial
Character \(\chi\) \(=\) 105.97
Dual form 105.2.m.a.13.3

$q$-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+(-0.167056 - 0.167056i) q^{2} +(-0.707107 - 0.707107i) q^{3} -1.94418i q^{4} +(-2.23450 - 0.0836010i) q^{5} +0.236253i q^{6} +(-0.0627175 - 2.64501i) q^{7} +(-0.658899 + 0.658899i) q^{8} +1.00000i q^{9} +(0.359321 + 0.387253i) q^{10} +3.98602 q^{11} +(-1.37475 + 1.37475i) q^{12} +(-0.500437 - 0.500437i) q^{13} +(-0.431387 + 0.452341i) q^{14} +(1.52092 + 1.63915i) q^{15} -3.66822 q^{16} +(1.67840 - 1.67840i) q^{17} +(0.167056 - 0.167056i) q^{18} +7.21850 q^{19} +(-0.162536 + 4.34429i) q^{20} +(-1.82596 + 1.91465i) q^{21} +(-0.665888 - 0.665888i) q^{22} +(-5.16007 + 5.16007i) q^{23} +0.931824 q^{24} +(4.98602 + 0.373614i) q^{25} +0.167202i q^{26} +(0.707107 - 0.707107i) q^{27} +(-5.14238 + 0.121934i) q^{28} -3.65191i q^{29} +(0.0197510 - 0.527908i) q^{30} +4.93821i q^{31} +(1.93060 + 1.93060i) q^{32} +(-2.81854 - 2.81854i) q^{33} -0.560773 q^{34} +(-0.0809828 + 5.91553i) q^{35} +1.94418 q^{36} +(0.292275 + 0.292275i) q^{37} +(-1.20589 - 1.20589i) q^{38} +0.707725i q^{39} +(1.52740 - 1.41723i) q^{40} -7.63184i q^{41} +(0.624890 - 0.0148172i) q^{42} +(3.65191 - 3.65191i) q^{43} -7.74956i q^{44} +(0.0836010 - 2.23450i) q^{45} +1.72404 q^{46} +(0.305303 - 0.305303i) q^{47} +(2.59383 + 2.59383i) q^{48} +(-6.99213 + 0.331777i) q^{49} +(-0.770530 - 0.895358i) q^{50} -2.37361 q^{51} +(-0.972943 + 0.972943i) q^{52} +(5.39653 - 5.39653i) q^{53} -0.236253 q^{54} +(-8.90678 - 0.333235i) q^{55} +(1.78412 + 1.70147i) q^{56} +(-5.10425 - 5.10425i) q^{57} +(-0.610073 + 0.610073i) q^{58} -6.10959 q^{59} +(3.18681 - 2.95695i) q^{60} +7.11047i q^{61} +(0.824957 - 0.824957i) q^{62} +(2.64501 - 0.0627175i) q^{63} +6.69141i q^{64} +(1.07639 + 1.16007i) q^{65} +0.941708i q^{66} +(0.944185 + 0.944185i) q^{67} +(-3.26312 - 3.26312i) q^{68} +7.29744 q^{69} +(1.00175 - 0.974695i) q^{70} +1.19297 q^{71} +(-0.658899 - 0.658899i) q^{72} +(-1.38298 - 1.38298i) q^{73} -0.0976524i q^{74} +(-3.26147 - 3.78983i) q^{75} -14.0341i q^{76} +(-0.249993 - 10.5431i) q^{77} +(0.118230 - 0.118230i) q^{78} +8.64027i q^{79} +(8.19666 + 0.306667i) q^{80} -1.00000 q^{81} +(-1.27494 + 1.27494i) q^{82} +(11.9895 + 11.9895i) q^{83} +(3.72244 + 3.54999i) q^{84} +(-3.89070 + 3.61007i) q^{85} -1.22015 q^{86} +(-2.58229 + 2.58229i) q^{87} +(-2.62639 + 2.62639i) q^{88} -7.82581 q^{89} +(-0.387253 + 0.359321i) q^{90} +(-1.29227 + 1.35505i) q^{91} +(10.0321 + 10.0321i) q^{92} +(3.49184 - 3.49184i) q^{93} -0.102005 q^{94} +(-16.1298 - 0.603474i) q^{95} -2.73028i q^{96} +(-7.43671 + 7.43671i) q^{97} +(1.22350 + 1.11265i) q^{98} +3.98602i 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.167056 − 0.167056i −0.118126 − 0.118126i 0.645573 − 0.763699i \(-0.276619\pi\)
−0.763699 + 0.645573i \(0.776619\pi\)
\(3\) −0.707107 − 0.707107i −0.408248 − 0.408248i
\(4\) − 1.94418i − 0.972092i
\(5\) −2.23450 − 0.0836010i −0.999301 − 0.0373875i
\(6\) 0.236253i 0.0964497i
\(7\) −0.0627175 − 2.64501i −0.0237050 − 0.999719i
\(8\) −0.658899 + 0.658899i −0.232956 + 0.232956i
\(9\) 1.00000i 0.333333i
\(10\) 0.359321 + 0.387253i 0.113627 + 0.122460i
\(11\) 3.98602 1.20183 0.600915 − 0.799313i \(-0.294803\pi\)
0.600915 + 0.799313i \(0.294803\pi\)
\(12\) −1.37475 + 1.37475i −0.396855 + 0.396855i
\(13\) −0.500437 − 0.500437i −0.138796 − 0.138796i 0.634295 − 0.773091i \(-0.281291\pi\)
−0.773091 + 0.634295i \(0.781291\pi\)
\(14\) −0.431387 + 0.452341i −0.115293 + 0.120893i
\(15\) 1.52092 + 1.63915i 0.392699 + 0.423226i
\(16\) −3.66822 −0.917056
\(17\) 1.67840 − 1.67840i 0.407071 − 0.407071i −0.473645 − 0.880716i \(-0.657062\pi\)
0.880716 + 0.473645i \(0.157062\pi\)
\(18\) 0.167056 − 0.167056i 0.0393754 − 0.0393754i
\(19\) 7.21850 1.65604 0.828019 − 0.560700i \(-0.189468\pi\)
0.828019 + 0.560700i \(0.189468\pi\)
\(20\) −0.162536 + 4.34429i −0.0363441 + 0.971413i
\(21\) −1.82596 + 1.91465i −0.398456 + 0.417811i
\(22\) −0.665888 − 0.665888i −0.141968 − 0.141968i
\(23\) −5.16007 + 5.16007i −1.07595 + 1.07595i −0.0790800 + 0.996868i \(0.525198\pi\)
−0.996868 + 0.0790800i \(0.974802\pi\)
\(24\) 0.931824 0.190208
\(25\) 4.98602 + 0.373614i 0.997204 + 0.0747227i
\(26\) 0.167202i 0.0327910i
\(27\) 0.707107 − 0.707107i 0.136083 − 0.136083i
\(28\) −5.14238 + 0.121934i −0.971819 + 0.0230434i
\(29\) − 3.65191i − 0.678143i −0.940761 − 0.339071i \(-0.889887\pi\)
0.940761 − 0.339071i \(-0.110113\pi\)
\(30\) 0.0197510 − 0.527908i 0.00360602 − 0.0963823i
\(31\) 4.93821i 0.886929i 0.896292 + 0.443465i \(0.146251\pi\)
−0.896292 + 0.443465i \(0.853749\pi\)
\(32\) 1.93060 + 1.93060i 0.341284 + 0.341284i
\(33\) −2.81854 − 2.81854i −0.490645 − 0.490645i
\(34\) −0.560773 −0.0961717
\(35\) −0.0809828 + 5.91553i −0.0136886 + 0.999906i
\(36\) 1.94418 0.324031
\(37\) 0.292275 + 0.292275i 0.0480497 + 0.0480497i 0.730723 − 0.682674i \(-0.239183\pi\)
−0.682674 + 0.730723i \(0.739183\pi\)
\(38\) −1.20589 − 1.20589i −0.195622 − 0.195622i
\(39\) 0.707725i 0.113327i
\(40\) 1.52740 − 1.41723i 0.241503 − 0.224084i
\(41\) − 7.63184i − 1.19189i −0.803024 − 0.595947i \(-0.796777\pi\)
0.803024 − 0.595947i \(-0.203223\pi\)
\(42\) 0.624890 − 0.0148172i 0.0964226 − 0.00228634i
\(43\) 3.65191 − 3.65191i 0.556911 − 0.556911i −0.371516 − 0.928427i \(-0.621162\pi\)
0.928427 + 0.371516i \(0.121162\pi\)
\(44\) − 7.74956i − 1.16829i
\(45\) 0.0836010 − 2.23450i 0.0124625 − 0.333100i
\(46\) 1.72404 0.254196
\(47\) 0.305303 − 0.305303i 0.0445331 − 0.0445331i −0.684490 − 0.729023i \(-0.739975\pi\)
0.729023 + 0.684490i \(0.239975\pi\)
\(48\) 2.59383 + 2.59383i 0.374386 + 0.374386i
\(49\) −6.99213 + 0.331777i −0.998876 + 0.0473967i
\(50\) −0.770530 − 0.895358i −0.108969 − 0.126623i
\(51\) −2.37361 −0.332372
\(52\) −0.972943 + 0.972943i −0.134923 + 0.134923i
\(53\) 5.39653 − 5.39653i 0.741270 − 0.741270i −0.231553 − 0.972822i \(-0.574381\pi\)
0.972822 + 0.231553i \(0.0743805\pi\)
\(54\) −0.236253 −0.0321499
\(55\) −8.90678 − 0.333235i −1.20099 − 0.0449335i
\(56\) 1.78412 + 1.70147i 0.238413 + 0.227368i
\(57\) −5.10425 − 5.10425i −0.676075 − 0.676075i
\(58\) −0.610073 + 0.610073i −0.0801065 + 0.0801065i
\(59\) −6.10959 −0.795401 −0.397701 − 0.917515i \(-0.630192\pi\)
−0.397701 + 0.917515i \(0.630192\pi\)
\(60\) 3.18681 − 2.95695i 0.411415 − 0.381740i
\(61\) 7.11047i 0.910402i 0.890389 + 0.455201i \(0.150433\pi\)
−0.890389 + 0.455201i \(0.849567\pi\)
\(62\) 0.824957 − 0.824957i 0.104770 − 0.104770i
\(63\) 2.64501 − 0.0627175i 0.333240 − 0.00790166i
\(64\) 6.69141i 0.836426i
\(65\) 1.07639 + 1.16007i 0.133510 + 0.143889i
\(66\) 0.941708i 0.115916i
\(67\) 0.944185 + 0.944185i 0.115351 + 0.115351i 0.762426 − 0.647075i \(-0.224008\pi\)
−0.647075 + 0.762426i \(0.724008\pi\)
\(68\) −3.26312 − 3.26312i −0.395711 − 0.395711i
\(69\) 7.29744 0.878508
\(70\) 1.00175 − 0.974695i 0.119732 − 0.116498i
\(71\) 1.19297 0.141579 0.0707897 − 0.997491i \(-0.477448\pi\)
0.0707897 + 0.997491i \(0.477448\pi\)
\(72\) −0.658899 − 0.658899i −0.0776520 − 0.0776520i
\(73\) −1.38298 − 1.38298i −0.161865 − 0.161865i 0.621527 − 0.783393i \(-0.286513\pi\)
−0.783393 + 0.621527i \(0.786513\pi\)
\(74\) − 0.0976524i − 0.0113519i
\(75\) −3.26147 − 3.78983i −0.376602 − 0.437612i
\(76\) − 14.0341i − 1.60982i
\(77\) −0.249993 − 10.5431i −0.0284894 − 1.20149i
\(78\) 0.118230 − 0.118230i 0.0133869 − 0.0133869i
\(79\) 8.64027i 0.972106i 0.873929 + 0.486053i \(0.161564\pi\)
−0.873929 + 0.486053i \(0.838436\pi\)
\(80\) 8.19666 + 0.306667i 0.916415 + 0.0342864i
\(81\) −1.00000 −0.111111
\(82\) −1.27494 + 1.27494i −0.140794 + 0.140794i
\(83\) 11.9895 + 11.9895i 1.31602 + 1.31602i 0.916898 + 0.399122i \(0.130685\pi\)
0.399122 + 0.916898i \(0.369315\pi\)
\(84\) 3.72244 + 3.54999i 0.406151 + 0.387336i
\(85\) −3.89070 + 3.61007i −0.422006 + 0.391567i
\(86\) −1.22015 −0.131572
\(87\) −2.58229 + 2.58229i −0.276851 + 0.276851i
\(88\) −2.62639 + 2.62639i −0.279974 + 0.279974i
\(89\) −7.82581 −0.829534 −0.414767 − 0.909928i \(-0.636137\pi\)
−0.414767 + 0.909928i \(0.636137\pi\)
\(90\) −0.387253 + 0.359321i −0.0408201 + 0.0378758i
\(91\) −1.29227 + 1.35505i −0.135467 + 0.142048i
\(92\) 10.0321 + 10.0321i 1.04592 + 1.04592i
\(93\) 3.49184 − 3.49184i 0.362087 − 0.362087i
\(94\) −0.102005 −0.0105211
\(95\) −16.1298 − 0.603474i −1.65488 − 0.0619151i
\(96\) − 2.73028i − 0.278658i
\(97\) −7.43671 + 7.43671i −0.755083 + 0.755083i −0.975423 − 0.220340i \(-0.929283\pi\)
0.220340 + 0.975423i \(0.429283\pi\)
\(98\) 1.22350 + 1.11265i 0.123592 + 0.112395i
\(99\) 3.98602i 0.400610i
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.97.3 yes 16
3.2 odd 2 315.2.p.e.307.6 16
4.3 odd 2 1680.2.cz.d.97.5 16
5.2 odd 4 525.2.m.b.118.5 16
5.3 odd 4 inner 105.2.m.a.13.4 yes 16
5.4 even 2 525.2.m.b.307.6 16
7.2 even 3 735.2.v.a.472.4 32
7.3 odd 6 735.2.v.a.607.6 32
7.4 even 3 735.2.v.a.607.5 32
7.5 odd 6 735.2.v.a.472.3 32
7.6 odd 2 inner 105.2.m.a.97.4 yes 16
15.8 even 4 315.2.p.e.118.5 16
20.3 even 4 1680.2.cz.d.433.4 16
21.20 even 2 315.2.p.e.307.5 16
28.27 even 2 1680.2.cz.d.97.4 16
35.3 even 12 735.2.v.a.313.4 32
35.13 even 4 inner 105.2.m.a.13.3 ✓ 16
35.18 odd 12 735.2.v.a.313.3 32
35.23 odd 12 735.2.v.a.178.6 32
35.27 even 4 525.2.m.b.118.6 16
35.33 even 12 735.2.v.a.178.5 32
35.34 odd 2 525.2.m.b.307.5 16
105.83 odd 4 315.2.p.e.118.6 16
140.83 odd 4 1680.2.cz.d.433.5 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.m.a.13.3 ✓ 16 35.13 even 4 inner
105.2.m.a.13.4 yes 16 5.3 odd 4 inner
105.2.m.a.97.3 yes 16 1.1 even 1 trivial
105.2.m.a.97.4 yes 16 7.6 odd 2 inner
315.2.p.e.118.5 16 15.8 even 4
315.2.p.e.118.6 16 105.83 odd 4
315.2.p.e.307.5 16 21.20 even 2
315.2.p.e.307.6 16 3.2 odd 2
525.2.m.b.118.5 16 5.2 odd 4
525.2.m.b.118.6 16 35.27 even 4
525.2.m.b.307.5 16 35.34 odd 2
525.2.m.b.307.6 16 5.4 even 2
735.2.v.a.178.5 32 35.33 even 12
735.2.v.a.178.6 32 35.23 odd 12
735.2.v.a.313.3 32 35.18 odd 12
735.2.v.a.313.4 32 35.3 even 12
735.2.v.a.472.3 32 7.5 odd 6
735.2.v.a.472.4 32 7.2 even 3
735.2.v.a.607.5 32 7.4 even 3
735.2.v.a.607.6 32 7.3 odd 6
1680.2.cz.d.97.4 16 28.27 even 2
1680.2.cz.d.97.5 16 4.3 odd 2
1680.2.cz.d.433.4 16 20.3 even 4
1680.2.cz.d.433.5 16 140.83 odd 4