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Properties

Label 10.11.c.c.3.1
Level $10$
Weight $11$
Character 10.3
Analytic conductor $6.354$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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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] = [10,11,Mod(3,10)] 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("10.3"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(10, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([3])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 10.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-96,128] 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: \(6.35357252674\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 1148x^{3} + 68121x^{2} - 299628x + 658952 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{8}\cdot 5^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 3.1
Root \(2.29143 - 2.29143i\) of defining polynomial
Character \(\chi\) \(=\) 10.3
Dual form 10.11.c.c.7.1

$q$-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+(-16.0000 + 16.0000i) q^{2} +(-266.441 - 266.441i) q^{3} -512.000i q^{4} +(2583.39 + 1758.32i) q^{5} +8526.10 q^{6} +(-13021.6 + 13021.6i) q^{7} +(8192.00 + 8192.00i) q^{8} +82932.3i q^{9} +(-69467.5 + 13201.2i) q^{10} +239729. q^{11} +(-136418. + 136418. i) q^{12} +(287980. + 287980. i) q^{13} -416690. i q^{14} +(-219833. - 1.15681e6i) q^{15} -262144. q^{16} +(-884527. + 884527. i) q^{17} +(-1.32692e6 - 1.32692e6i) q^{18} +197744. i q^{19} +(900261. - 1.32270e6i) q^{20} +6.93894e6 q^{21} +(-3.83567e6 + 3.83567e6i) q^{22} +(-3.55115e6 - 3.55115e6i) q^{23} -4.36536e6i q^{24} +(3.58223e6 + 9.08488e6i) q^{25} -9.21535e6 q^{26} +(6.36348e6 - 6.36348e6i) q^{27} +(6.66703e6 + 6.66703e6i) q^{28} +3.57499e7i q^{29} +(2.20263e7 + 1.49916e7i) q^{30} +9.65516e6 q^{31} +(4.19430e6 - 4.19430e6i) q^{32} +(-6.38737e7 - 6.38737e7i) q^{33} -2.83049e7i q^{34} +(-5.65359e7 + 1.07437e7i) q^{35} +4.24613e7 q^{36} +(1.34842e7 - 1.34842e7i) q^{37} +(-3.16390e6 - 3.16390e6i) q^{38} -1.53459e8i q^{39} +(6.75899e6 + 3.55673e7i) q^{40} +1.08475e8 q^{41} +(-1.11023e8 + 1.11023e8i) q^{42} +(8.62678e7 + 8.62678e7i) q^{43} -1.22741e8i q^{44} +(-1.45822e8 + 2.14247e8i) q^{45} +1.13637e8 q^{46} +(-1.76759e8 + 1.76759e8i) q^{47} +(6.98458e7 + 6.98458e7i) q^{48} -5.66463e7i q^{49} +(-2.02674e8 - 8.80424e7i) q^{50} +4.71348e8 q^{51} +(1.47446e8 - 1.47446e8i) q^{52} +(-1.48530e8 - 1.48530e8i) q^{53} +2.03631e8i q^{54} +(6.19316e8 + 4.21522e8i) q^{55} -2.13345e8 q^{56} +(5.26870e7 - 5.26870e7i) q^{57} +(-5.71999e8 - 5.71999e8i) q^{58} -7.98638e8i q^{59} +(-5.92287e8 + 1.12554e8i) q^{60} -4.24449e8 q^{61} +(-1.54483e8 + 1.54483e8i) q^{62} +(-1.07991e9 - 1.07991e9i) q^{63} +1.34218e8i q^{64} +(2.37604e8 + 1.25033e9i) q^{65} +2.04396e9 q^{66} +(1.12768e9 - 1.12768e9i) q^{67} +(4.52878e8 + 4.52878e8i) q^{68} +1.89234e9i q^{69} +(7.32675e8 - 1.07647e9i) q^{70} -2.60778e9 q^{71} +(-6.79381e8 + 6.79381e8i) q^{72} +(8.51903e8 + 8.51903e8i) q^{73} +4.31493e8i q^{74} +(1.46613e9 - 3.37503e9i) q^{75} +1.01245e8 q^{76} +(-3.12165e9 + 3.12165e9i) q^{77} +(2.45534e9 + 2.45534e9i) q^{78} +4.56153e8i q^{79} +(-6.77221e8 - 4.60934e8i) q^{80} +1.50609e9 q^{81} +(-1.73559e9 + 1.73559e9i) q^{82} +(7.60916e8 + 7.60916e8i) q^{83} -3.55274e9i q^{84} +(-3.84037e9 + 7.29799e8i) q^{85} -2.76057e9 q^{86} +(9.52524e9 - 9.52524e9i) q^{87} +(1.96386e9 + 1.96386e9i) q^{88} +2.03858e9i q^{89} +(-1.09480e9 - 5.76110e9i) q^{90} -7.49988e9 q^{91} +(-1.81819e9 + 1.81819e9i) q^{92} +(-2.57253e9 - 2.57253e9i) q^{93} -5.65629e9i q^{94} +(-3.47698e8 + 5.10851e8i) q^{95} -2.23507e9 q^{96} +(8.73239e9 - 8.73239e9i) q^{97} +(9.06341e8 + 9.06341e8i) q^{98} +1.98813e10i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 96 q^{2} + 128 q^{3} + 5460 q^{5} - 4096 q^{6} + 13512 q^{7} + 49152 q^{8} - 173280 q^{10} + 647832 q^{11} + 65536 q^{12} - 742902 q^{13} + 1577720 q^{15} - 1572864 q^{16} - 755118 q^{17} - 5683744 q^{18}+ \cdots + 12874047264 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).

\(n\) \(7\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −16.0000 + 16.0000i −0.500000 + 0.500000i
\(3\) −266.441 − 266.441i −1.09646 − 1.09646i −0.994821 − 0.101643i \(-0.967590\pi\)
−0.101643 − 0.994821i \(-0.532410\pi\)
\(4\) − 512.000i − 0.500000i
\(5\) 2583.39 + 1758.32i 0.826686 + 0.562663i
\(6\) 8526.10 1.09646
\(7\) −13021.6 + 13021.6i −0.774769 + 0.774769i −0.978936 − 0.204167i \(-0.934552\pi\)
0.204167 + 0.978936i \(0.434552\pi\)
\(8\) 8192.00 + 8192.00i 0.250000 + 0.250000i
\(9\) 82932.3i 1.40447i
\(10\) −69467.5 + 13201.2i −0.694675 + 0.132012i
\(11\) 239729. 1.48853 0.744266 − 0.667884i \(-0.232800\pi\)
0.744266 + 0.667884i \(0.232800\pi\)
\(12\) −136418. + 136418.i −0.548232 + 0.548232i
\(13\) 287980. + 287980.i 0.775613 + 0.775613i 0.979082 − 0.203468i \(-0.0652214\pi\)
−0.203468 + 0.979082i \(0.565221\pi\)
\(14\) − 416690.i − 0.774769i
\(15\) −219833. − 1.15681e6i −0.289492 − 1.52337i
\(16\) −262144. −0.250000
\(17\) −884527. + 884527.i −0.622969 + 0.622969i −0.946290 − 0.323320i \(-0.895201\pi\)
0.323320 + 0.946290i \(0.395201\pi\)
\(18\) −1.32692e6 − 1.32692e6i −0.702233 − 0.702233i
\(19\) 197744.i 0.0798611i 0.999202 + 0.0399305i \(0.0127137\pi\)
−0.999202 + 0.0399305i \(0.987286\pi\)
\(20\) 900261. − 1.32270e6i 0.281332 − 0.413343i
\(21\) 6.93894e6 1.69901
\(22\) −3.83567e6 + 3.83567e6i −0.744266 + 0.744266i
\(23\) −3.55115e6 − 3.55115e6i −0.551734 − 0.551734i 0.375207 − 0.926941i \(-0.377572\pi\)
−0.926941 + 0.375207i \(0.877572\pi\)
\(24\) − 4.36536e6i − 0.548232i
\(25\) 3.58223e6 + 9.08488e6i 0.366821 + 0.930292i
\(26\) −9.21535e6 −0.775613
\(27\) 6.36348e6 − 6.36348e6i 0.443482 − 0.443482i
\(28\) 6.66703e6 + 6.66703e6i 0.387385 + 0.387385i
\(29\) 3.57499e7i 1.74295i 0.490439 + 0.871476i \(0.336837\pi\)
−0.490439 + 0.871476i \(0.663163\pi\)
\(30\) 2.20263e7 + 1.49916e7i 0.906432 + 0.616940i
\(31\) 9.65516e6 0.337249 0.168625 − 0.985680i \(-0.446067\pi\)
0.168625 + 0.985680i \(0.446067\pi\)
\(32\) 4.19430e6 − 4.19430e6i 0.125000 − 0.125000i
\(33\) −6.38737e7 − 6.38737e7i −1.63212 − 1.63212i
\(34\) − 2.83049e7i − 0.622969i
\(35\) −5.65359e7 + 1.07437e7i −1.07643 + 0.204557i
\(36\) 4.24613e7 0.702233
\(37\) 1.34842e7 − 1.34842e7i 0.194453 − 0.194453i −0.603164 − 0.797617i \(-0.706094\pi\)
0.797617 + 0.603164i \(0.206094\pi\)
\(38\) −3.16390e6 − 3.16390e6i −0.0399305 − 0.0399305i
\(39\) − 1.53459e8i − 1.70086i
\(40\) 6.75899e6 + 3.55673e7i 0.0660058 + 0.347337i
\(41\) 1.08475e8 0.936286 0.468143 − 0.883653i \(-0.344923\pi\)
0.468143 + 0.883653i \(0.344923\pi\)
\(42\) −1.11023e8 + 1.11023e8i −0.849507 + 0.849507i
\(43\) 8.62678e7 + 8.62678e7i 0.586822 + 0.586822i 0.936769 − 0.349947i \(-0.113801\pi\)
−0.349947 + 0.936769i \(0.613801\pi\)
\(44\) − 1.22741e8i − 0.744266i
\(45\) −1.45822e8 + 2.14247e8i −0.790241 + 1.16105i
\(46\) 1.13637e8 0.551734
\(47\) −1.76759e8 + 1.76759e8i −0.770713 + 0.770713i −0.978231 − 0.207518i \(-0.933461\pi\)
0.207518 + 0.978231i \(0.433461\pi\)
\(48\) 6.98458e7 + 6.98458e7i 0.274116 + 0.274116i
\(49\) − 5.66463e7i − 0.200535i
\(50\) −2.02674e8 − 8.80424e7i −0.648556 − 0.281736i
\(51\) 4.71348e8 1.36613
\(52\) 1.47446e8 − 1.47446e8i 0.387807 − 0.387807i
\(53\) −1.48530e8 − 1.48530e8i −0.355169 − 0.355169i 0.506860 − 0.862029i \(-0.330806\pi\)
−0.862029 + 0.506860i \(0.830806\pi\)
\(54\) 2.03631e8i 0.443482i
\(55\) 6.19316e8 + 4.21522e8i 1.23055 + 0.837542i
\(56\) −2.13345e8 −0.387385
\(57\) 5.26870e7 − 5.26870e7i 0.0875648 − 0.0875648i
\(58\) −5.71999e8 − 5.71999e8i −0.871476 − 0.871476i
\(59\) − 7.98638e8i − 1.11709i −0.829473 − 0.558547i \(-0.811359\pi\)
0.829473 − 0.558547i \(-0.188641\pi\)
\(60\) −5.92287e8 + 1.12554e8i −0.761686 + 0.144746i
\(61\) −4.24449e8 −0.502546 −0.251273 − 0.967916i \(-0.580849\pi\)
−0.251273 + 0.967916i \(0.580849\pi\)
\(62\) −1.54483e8 + 1.54483e8i −0.168625 + 0.168625i
\(63\) −1.07991e9 − 1.07991e9i −1.08814 − 1.08814i
\(64\) 1.34218e8i 0.125000i
\(65\) 2.37604e8 + 1.25033e9i 0.204780 + 1.07760i
\(66\) 2.04396e9 1.63212
\(67\) 1.12768e9 − 1.12768e9i 0.835239 − 0.835239i −0.152989 − 0.988228i \(-0.548890\pi\)
0.988228 + 0.152989i \(0.0488899\pi\)
\(68\) 4.52878e8 + 4.52878e8i 0.311485 + 0.311485i
\(69\) 1.89234e9i 1.20991i
\(70\) 7.32675e8 − 1.07647e9i 0.435934 − 0.640491i
\(71\) −2.60778e9 −1.44537 −0.722685 − 0.691178i \(-0.757092\pi\)
−0.722685 + 0.691178i \(0.757092\pi\)
\(72\) −6.79381e8 + 6.79381e8i −0.351116 + 0.351116i
\(73\) 8.51903e8 + 8.51903e8i 0.410938 + 0.410938i 0.882065 − 0.471128i \(-0.156153\pi\)
−0.471128 + 0.882065i \(0.656153\pi\)
\(74\) 4.31493e8i 0.194453i
\(75\) 1.46613e9 − 3.37503e9i 0.617826 − 1.42224i
\(76\) 1.01245e8 0.0399305
\(77\) −3.12165e9 + 3.12165e9i −1.15327 + 1.15327i
\(78\) 2.45534e9 + 2.45534e9i 0.850432 + 0.850432i
\(79\) 4.56153e8i 0.148243i 0.997249 + 0.0741217i \(0.0236153\pi\)
−0.997249 + 0.0741217i \(0.976385\pi\)
\(80\) −6.77221e8 − 4.60934e8i −0.206672 − 0.140666i
\(81\) 1.50609e9 0.431942
\(82\) −1.73559e9 + 1.73559e9i −0.468143 + 0.468143i
\(83\) 7.60916e8 + 7.60916e8i 0.193173 + 0.193173i 0.797066 − 0.603893i \(-0.206384\pi\)
−0.603893 + 0.797066i \(0.706384\pi\)
\(84\) − 3.55274e9i − 0.849507i
\(85\) −3.84037e9 + 7.29799e8i −0.865522 + 0.164478i
\(86\) −2.76057e9 −0.586822
\(87\) 9.52524e9 − 9.52524e9i 1.91108 − 1.91108i
\(88\) 1.96386e9 + 1.96386e9i 0.372133 + 0.372133i
\(89\) 2.03858e9i 0.365072i 0.983199 + 0.182536i \(0.0584306\pi\)
−0.983199 + 0.182536i \(0.941569\pi\)
\(90\) −1.09480e9 − 5.76110e9i −0.185406 − 0.975647i
\(91\) −7.49988e9 −1.20184
\(92\) −1.81819e9 + 1.81819e9i −0.275867 + 0.275867i
\(93\) −2.57253e9 − 2.57253e9i −0.369782 − 0.369782i
\(94\) − 5.65629e9i − 0.770713i
\(95\) −3.47698e8 + 5.10851e8i −0.0449349 + 0.0660201i
\(96\) −2.23507e9 −0.274116
\(97\) 8.73239e9 − 8.73239e9i 1.01689 − 1.01689i 0.0170366 − 0.999855i \(-0.494577\pi\)
0.999855 − 0.0170366i \(-0.00542316\pi\)
\(98\) 9.06341e8 + 9.06341e8i 0.100268 + 0.100268i
\(99\) 1.98813e10i 2.09059i
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 10.11.c.c.3.1 ✓ 6
3.2 odd 2 90.11.g.c.73.2 6
4.3 odd 2 80.11.p.c.33.3 6
5.2 odd 4 inner 10.11.c.c.7.1 yes 6
5.3 odd 4 50.11.c.e.7.3 6
5.4 even 2 50.11.c.e.43.3 6
15.2 even 4 90.11.g.c.37.2 6
20.7 even 4 80.11.p.c.17.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.11.c.c.3.1 ✓ 6 1.1 even 1 trivial
10.11.c.c.7.1 yes 6 5.2 odd 4 inner
50.11.c.e.7.3 6 5.3 odd 4
50.11.c.e.43.3 6 5.4 even 2
80.11.p.c.17.3 6 20.7 even 4
80.11.p.c.33.3 6 4.3 odd 2
90.11.g.c.37.2 6 15.2 even 4
90.11.g.c.73.2 6 3.2 odd 2