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Properties

Label 105.3.e.a.34.13
Level $105$
Weight $3$
Character 105.34
Analytic conductor $2.861$
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] = [105,3,Mod(34,105)] 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("105.34"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(105, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 105.e (of order \(2\), degree \(1\), 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: \(2.86104277578\)
Analytic rank: \(0\)
Dimension: \(16\)
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: \( x^{16} - 8 x^{15} + 72 x^{14} - 292 x^{13} + 1148 x^{12} - 2304 x^{11} + 4996 x^{10} - 4490 x^{9} + \cdots + 1849 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 34.13
Root \(1.36603 + 3.14303i\) of defining polynomial
Character \(\chi\) \(=\) 105.34
Dual form 105.3.e.a.34.3

$q$-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+2.30086i q^{2} -1.73205 q^{3} -1.29396 q^{4} +(-3.65761 + 3.40909i) q^{5} -3.98521i q^{6} +(-6.39480 - 2.84720i) q^{7} +6.22623i q^{8} +3.00000 q^{9} +(-7.84383 - 8.41565i) q^{10} -13.9015 q^{11} +2.24120 q^{12} +3.78588 q^{13} +(6.55102 - 14.7135i) q^{14} +(6.33517 - 5.90471i) q^{15} -19.5015 q^{16} +14.8567 q^{17} +6.90258i q^{18} +6.94906i q^{19} +(4.73278 - 4.41121i) q^{20} +(11.0761 + 4.93150i) q^{21} -31.9853i q^{22} +40.2857i q^{23} -10.7841i q^{24} +(1.75623 - 24.9382i) q^{25} +8.71077i q^{26} -5.19615 q^{27} +(8.27458 + 3.68415i) q^{28} +9.88885 q^{29} +(13.5859 + 14.5763i) q^{30} +34.7764i q^{31} -19.9653i q^{32} +24.0781 q^{33} +34.1833i q^{34} +(33.0960 - 11.3865i) q^{35} -3.88187 q^{36} +30.4393i q^{37} -15.9888 q^{38} -6.55733 q^{39} +(-21.2258 - 22.7731i) q^{40} +44.6778i q^{41} +(-11.3467 + 25.4846i) q^{42} -26.0309i q^{43} +17.9879 q^{44} +(-10.9728 + 10.2273i) q^{45} -92.6918 q^{46} +23.7033 q^{47} +33.7776 q^{48} +(32.7869 + 36.4146i) q^{49} +(57.3794 + 4.04085i) q^{50} -25.7326 q^{51} -4.89875 q^{52} -59.5338i q^{53} -11.9556i q^{54} +(50.8462 - 47.3914i) q^{55} +(17.7273 - 39.8155i) q^{56} -12.0361i q^{57} +22.7529i q^{58} -81.0347i q^{59} +(-8.19742 + 7.64043i) q^{60} -78.8493i q^{61} -80.0155 q^{62} +(-19.1844 - 8.54161i) q^{63} -32.0687 q^{64} +(-13.8473 + 12.9064i) q^{65} +55.4002i q^{66} +29.9557i q^{67} -19.2240 q^{68} -69.7770i q^{69} +(26.1986 + 76.1494i) q^{70} -117.475 q^{71} +18.6787i q^{72} +22.2333 q^{73} -70.0365 q^{74} +(-3.04189 + 43.1943i) q^{75} -8.99177i q^{76} +(88.8971 + 39.5803i) q^{77} -15.0875i q^{78} +142.799 q^{79} +(71.3289 - 66.4823i) q^{80} +9.00000 q^{81} -102.797 q^{82} -160.872 q^{83} +(-14.3320 - 6.38114i) q^{84} +(-54.3402 + 50.6480i) q^{85} +59.8935 q^{86} -17.1280 q^{87} -86.5538i q^{88} +66.3844i q^{89} +(-23.5315 - 25.2469i) q^{90} +(-24.2099 - 10.7792i) q^{91} -52.1279i q^{92} -60.2344i q^{93} +54.5380i q^{94} +(-23.6899 - 25.4169i) q^{95} +34.5809i q^{96} +45.1538 q^{97} +(-83.7849 + 75.4379i) q^{98} -41.7044 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 32 q^{4} + 48 q^{9} - 56 q^{11} - 84 q^{14} + 24 q^{15} + 112 q^{16} - 12 q^{21} + 16 q^{25} - 32 q^{29} - 72 q^{30} - 4 q^{35} - 96 q^{36} + 72 q^{39} + 568 q^{44} - 96 q^{46} - 152 q^{49} - 96 q^{50}+ \cdots - 168 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.30086i 1.15043i 0.818002 + 0.575215i \(0.195082\pi\)
−0.818002 + 0.575215i \(0.804918\pi\)
\(3\) −1.73205 −0.577350
\(4\) −1.29396 −0.323489
\(5\) −3.65761 + 3.40909i −0.731522 + 0.681818i
\(6\) − 3.98521i − 0.664201i
\(7\) −6.39480 − 2.84720i −0.913542 − 0.406744i
\(8\) 6.22623i 0.778279i
\(9\) 3.00000 0.333333
\(10\) −7.84383 − 8.41565i −0.784383 − 0.841565i
\(11\) −13.9015 −1.26377 −0.631885 − 0.775062i \(-0.717719\pi\)
−0.631885 + 0.775062i \(0.717719\pi\)
\(12\) 2.24120 0.186766
\(13\) 3.78588 0.291221 0.145611 − 0.989342i \(-0.453485\pi\)
0.145611 + 0.989342i \(0.453485\pi\)
\(14\) 6.55102 − 14.7135i 0.467930 − 1.05097i
\(15\) 6.33517 − 5.90471i 0.422345 − 0.393648i
\(16\) −19.5015 −1.21884
\(17\) 14.8567 0.873926 0.436963 − 0.899479i \(-0.356054\pi\)
0.436963 + 0.899479i \(0.356054\pi\)
\(18\) 6.90258i 0.383477i
\(19\) 6.94906i 0.365740i 0.983137 + 0.182870i \(0.0585387\pi\)
−0.983137 + 0.182870i \(0.941461\pi\)
\(20\) 4.73278 − 4.41121i 0.236639 − 0.220560i
\(21\) 11.0761 + 4.93150i 0.527434 + 0.234833i
\(22\) − 31.9853i − 1.45388i
\(23\) 40.2857i 1.75155i 0.482716 + 0.875777i \(0.339650\pi\)
−0.482716 + 0.875777i \(0.660350\pi\)
\(24\) − 10.7841i − 0.449339i
\(25\) 1.75623 − 24.9382i 0.0702494 − 0.997529i
\(26\) 8.71077i 0.335030i
\(27\) −5.19615 −0.192450
\(28\) 8.27458 + 3.68415i 0.295521 + 0.131577i
\(29\) 9.88885 0.340995 0.170497 − 0.985358i \(-0.445463\pi\)
0.170497 + 0.985358i \(0.445463\pi\)
\(30\) 13.5859 + 14.5763i 0.452864 + 0.485878i
\(31\) 34.7764i 1.12182i 0.827877 + 0.560909i \(0.189548\pi\)
−0.827877 + 0.560909i \(0.810452\pi\)
\(32\) − 19.9653i − 0.623916i
\(33\) 24.0781 0.729638
\(34\) 34.1833i 1.00539i
\(35\) 33.0960 − 11.3865i 0.945601 − 0.325327i
\(36\) −3.88187 −0.107830
\(37\) 30.4393i 0.822683i 0.911481 + 0.411342i \(0.134940\pi\)
−0.911481 + 0.411342i \(0.865060\pi\)
\(38\) −15.9888 −0.420758
\(39\) −6.55733 −0.168137
\(40\) −21.2258 − 22.7731i −0.530644 − 0.569328i
\(41\) 44.6778i 1.08970i 0.838532 + 0.544852i \(0.183414\pi\)
−0.838532 + 0.544852i \(0.816586\pi\)
\(42\) −11.3467 + 25.4846i −0.270159 + 0.606776i
\(43\) − 26.0309i − 0.605370i −0.953091 − 0.302685i \(-0.902117\pi\)
0.953091 − 0.302685i \(-0.0978831\pi\)
\(44\) 17.9879 0.408816
\(45\) −10.9728 + 10.2273i −0.243841 + 0.227273i
\(46\) −92.6918 −2.01504
\(47\) 23.7033 0.504326 0.252163 − 0.967685i \(-0.418858\pi\)
0.252163 + 0.967685i \(0.418858\pi\)
\(48\) 33.7776 0.703700
\(49\) 32.7869 + 36.4146i 0.669119 + 0.743155i
\(50\) 57.3794 + 4.04085i 1.14759 + 0.0808170i
\(51\) −25.7326 −0.504562
\(52\) −4.89875 −0.0942068
\(53\) − 59.5338i − 1.12328i −0.827382 − 0.561640i \(-0.810171\pi\)
0.827382 − 0.561640i \(-0.189829\pi\)
\(54\) − 11.9556i − 0.221400i
\(55\) 50.8462 − 47.3914i 0.924476 − 0.861661i
\(56\) 17.7273 − 39.8155i 0.316560 − 0.710991i
\(57\) − 12.0361i − 0.211160i
\(58\) 22.7529i 0.392291i
\(59\) − 81.0347i − 1.37347i −0.726908 − 0.686735i \(-0.759043\pi\)
0.726908 − 0.686735i \(-0.240957\pi\)
\(60\) −8.19742 + 7.64043i −0.136624 + 0.127341i
\(61\) − 78.8493i − 1.29261i −0.763079 − 0.646306i \(-0.776313\pi\)
0.763079 − 0.646306i \(-0.223687\pi\)
\(62\) −80.0155 −1.29057
\(63\) −19.1844 − 8.54161i −0.304514 − 0.135581i
\(64\) −32.0687 −0.501073
\(65\) −13.8473 + 12.9064i −0.213035 + 0.198560i
\(66\) 55.4002i 0.839398i
\(67\) 29.9557i 0.447101i 0.974692 + 0.223550i \(0.0717647\pi\)
−0.974692 + 0.223550i \(0.928235\pi\)
\(68\) −19.2240 −0.282705
\(69\) − 69.7770i − 1.01126i
\(70\) 26.1986 + 76.1494i 0.374266 + 1.08785i
\(71\) −117.475 −1.65457 −0.827286 − 0.561780i \(-0.810117\pi\)
−0.827286 + 0.561780i \(0.810117\pi\)
\(72\) 18.6787i 0.259426i
\(73\) 22.2333 0.304566 0.152283 − 0.988337i \(-0.451338\pi\)
0.152283 + 0.988337i \(0.451338\pi\)
\(74\) −70.0365 −0.946439
\(75\) −3.04189 + 43.1943i −0.0405585 + 0.575924i
\(76\) − 8.99177i − 0.118313i
\(77\) 88.8971 + 39.5803i 1.15451 + 0.514030i
\(78\) − 15.0875i − 0.193429i
\(79\) 142.799 1.80758 0.903791 − 0.427974i \(-0.140772\pi\)
0.903791 + 0.427974i \(0.140772\pi\)
\(80\) 71.3289 − 66.4823i 0.891611 − 0.831029i
\(81\) 9.00000 0.111111
\(82\) −102.797 −1.25363
\(83\) −160.872 −1.93822 −0.969108 − 0.246636i \(-0.920675\pi\)
−0.969108 + 0.246636i \(0.920675\pi\)
\(84\) −14.3320 − 6.38114i −0.170619 − 0.0759660i
\(85\) −54.3402 + 50.6480i −0.639296 + 0.595858i
\(86\) 59.8935 0.696436
\(87\) −17.1280 −0.196873
\(88\) − 86.5538i − 0.983566i
\(89\) 66.3844i 0.745892i 0.927853 + 0.372946i \(0.121652\pi\)
−0.927853 + 0.372946i \(0.878348\pi\)
\(90\) −23.5315 − 25.2469i −0.261461 − 0.280522i
\(91\) −24.2099 − 10.7792i −0.266043 − 0.118452i
\(92\) − 52.1279i − 0.566608i
\(93\) − 60.2344i − 0.647682i
\(94\) 54.5380i 0.580192i
\(95\) −23.6899 − 25.4169i −0.249368 − 0.267547i
\(96\) 34.5809i 0.360218i
\(97\) 45.1538 0.465503 0.232751 − 0.972536i \(-0.425227\pi\)
0.232751 + 0.972536i \(0.425227\pi\)
\(98\) −83.7849 + 75.4379i −0.854948 + 0.769775i
\(99\) −41.7044 −0.421257
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 105.3.e.a.34.13 yes 16
3.2 odd 2 315.3.e.e.244.4 16
4.3 odd 2 1680.3.bd.c.769.12 16
5.2 odd 4 525.3.h.e.76.4 16
5.3 odd 4 525.3.h.e.76.13 16
5.4 even 2 inner 105.3.e.a.34.4 yes 16
7.6 odd 2 inner 105.3.e.a.34.14 yes 16
15.14 odd 2 315.3.e.e.244.13 16
20.19 odd 2 1680.3.bd.c.769.6 16
21.20 even 2 315.3.e.e.244.3 16
28.27 even 2 1680.3.bd.c.769.5 16
35.13 even 4 525.3.h.e.76.14 16
35.27 even 4 525.3.h.e.76.3 16
35.34 odd 2 inner 105.3.e.a.34.3 ✓ 16
105.104 even 2 315.3.e.e.244.14 16
140.139 even 2 1680.3.bd.c.769.11 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.3.e.a.34.3 ✓ 16 35.34 odd 2 inner
105.3.e.a.34.4 yes 16 5.4 even 2 inner
105.3.e.a.34.13 yes 16 1.1 even 1 trivial
105.3.e.a.34.14 yes 16 7.6 odd 2 inner
315.3.e.e.244.3 16 21.20 even 2
315.3.e.e.244.4 16 3.2 odd 2
315.3.e.e.244.13 16 15.14 odd 2
315.3.e.e.244.14 16 105.104 even 2
525.3.h.e.76.3 16 35.27 even 4
525.3.h.e.76.4 16 5.2 odd 4
525.3.h.e.76.13 16 5.3 odd 4
525.3.h.e.76.14 16 35.13 even 4
1680.3.bd.c.769.5 16 28.27 even 2
1680.3.bd.c.769.6 16 20.19 odd 2
1680.3.bd.c.769.11 16 140.139 even 2
1680.3.bd.c.769.12 16 4.3 odd 2