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Properties

Label 10.9.c.b.3.1
Level $10$
Weight $9$
Character 10.3
Analytic conductor $4.074$
Analytic rank $0$
Dimension $4$
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] = [10,9,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, 9); 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, 9, names="a")
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 10.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.07378610061\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{601})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 301x^{2} + 22500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 3.1
Root \(-11.7577i\) of defining polynomial
Character \(\chi\) \(=\) 10.3
Dual form 10.9.c.b.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+(8.00000 - 8.00000i) q^{2} +(-39.7883 - 39.7883i) q^{3} -128.000i q^{4} +(-401.365 - 479.094i) q^{5} -636.612 q^{6} +(144.447 - 144.447i) q^{7} +(-1024.00 - 1024.00i) q^{8} -3394.79i q^{9} +(-7043.67 - 621.836i) q^{10} +13600.3 q^{11} +(-5092.90 + 5092.90i) q^{12} +(30883.7 + 30883.7i) q^{13} -2311.15i q^{14} +(-3092.72 + 35031.9i) q^{15} -16384.0 q^{16} +(4949.02 - 4949.02i) q^{17} +(-27158.3 - 27158.3i) q^{18} -176579. i q^{19} +(-61324.1 + 51374.7i) q^{20} -11494.6 q^{21} +(108802. - 108802. i) q^{22} +(-229482. - 229482. i) q^{23} +81486.3i q^{24} +(-68437.7 + 384583. i) q^{25} +494139. q^{26} +(-396123. + 396123. i) q^{27} +(-18489.2 - 18489.2i) q^{28} -144167. i q^{29} +(255514. + 304997. i) q^{30} +1.50971e6 q^{31} +(-131072. + 131072. i) q^{32} +(-541132. - 541132. i) q^{33} -79184.4i q^{34} +(-127179. - 11227.8i) q^{35} -434533. q^{36} +(-1.60715e6 + 1.60715e6i) q^{37} +(-1.41263e6 - 1.41263e6i) q^{38} -2.45761e6i q^{39} +(-79595.0 + 901590. i) q^{40} +3.62276e6 q^{41} +(-91956.5 + 91956.5i) q^{42} +(-575576. - 575576. i) q^{43} -1.74084e6i q^{44} +(-1.62642e6 + 1.36255e6i) q^{45} -3.67171e6 q^{46} +(3.11186e6 - 3.11186e6i) q^{47} +(651891. + 651891. i) q^{48} +5.72307e6i q^{49} +(2.52916e6 + 3.62417e6i) q^{50} -393826. q^{51} +(3.95311e6 - 3.95311e6i) q^{52} +(8.16813e6 + 8.16813e6i) q^{53} +6.33798e6i q^{54} +(-5.45868e6 - 6.51582e6i) q^{55} -295827. q^{56} +(-7.02575e6 + 7.02575e6i) q^{57} +(-1.15334e6 - 1.15334e6i) q^{58} -1.54905e7i q^{59} +(4.48409e6 + 395868. i) q^{60} -1.92688e7 q^{61} +(1.20777e7 - 1.20777e7i) q^{62} +(-490366. - 490366. i) q^{63} +2.09715e6i q^{64} +(2.40057e6 - 2.71918e7i) q^{65} -8.65811e6 q^{66} +(-1.00075e7 + 1.00075e7i) q^{67} +(-633475. - 633475. i) q^{68} +1.82614e7i q^{69} +(-1.10726e6 + 927613. i) q^{70} +1.84257e7 q^{71} +(-3.47626e6 + 3.47626e6i) q^{72} +(2.45702e7 + 2.45702e7i) q^{73} +2.57144e7i q^{74} +(1.80249e7 - 1.25789e7i) q^{75} -2.26020e7 q^{76} +(1.96452e6 - 1.96452e6i) q^{77} +(-1.96609e7 - 1.96609e7i) q^{78} +2.17799e7i q^{79} +(6.57596e6 + 7.84948e6i) q^{80} +9.24891e6 q^{81} +(2.89820e7 - 2.89820e7i) q^{82} +(4.00189e6 + 4.00189e6i) q^{83} +1.47130e6i q^{84} +(-4.35741e6 - 384685. i) q^{85} -9.20921e6 q^{86} +(-5.73617e6 + 5.73617e6i) q^{87} +(-1.39267e7 - 1.39267e7i) q^{88} -6.14525e7i q^{89} +(-2.11100e6 + 2.39118e7i) q^{90} +8.92209e6 q^{91} +(-2.93737e7 + 2.93737e7i) q^{92} +(-6.00686e7 - 6.00686e7i) q^{93} -4.97898e7i q^{94} +(-8.45978e7 + 7.08724e7i) q^{95} +1.04303e7 q^{96} +(1.55814e7 - 1.55814e7i) q^{97} +(4.57846e7 + 4.57846e7i) q^{98} -4.61701e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 32 q^{2} + 86 q^{3} - 870 q^{5} + 1376 q^{6} + 5726 q^{7} - 4096 q^{8} - 4640 q^{10} - 14732 q^{11} + 11008 q^{12} + 45576 q^{13} - 115090 q^{15} - 65536 q^{16} + 3616 q^{17} + 60032 q^{18} + 37120 q^{20}+ \cdots - 52691072 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.00000 − 8.00000i 0.500000 − 0.500000i
\(3\) −39.7883 − 39.7883i −0.491213 − 0.491213i 0.417475 − 0.908688i \(-0.362915\pi\)
−0.908688 + 0.417475i \(0.862915\pi\)
\(4\) − 128.000i − 0.500000i
\(5\) −401.365 − 479.094i −0.642184 − 0.766551i
\(6\) −636.612 −0.491213
\(7\) 144.447 − 144.447i 0.0601610 − 0.0601610i −0.676386 − 0.736547i \(-0.736455\pi\)
0.736547 + 0.676386i \(0.236455\pi\)
\(8\) −1024.00 − 1024.00i −0.250000 − 0.250000i
\(9\) − 3394.79i − 0.517420i
\(10\) −7043.67 − 621.836i −0.704367 − 0.0621836i
\(11\) 13600.3 0.928918 0.464459 − 0.885595i \(-0.346249\pi\)
0.464459 + 0.885595i \(0.346249\pi\)
\(12\) −5092.90 + 5092.90i −0.245607 + 0.245607i
\(13\) 30883.7 + 30883.7i 1.08132 + 1.08132i 0.996386 + 0.0849366i \(0.0270688\pi\)
0.0849366 + 0.996386i \(0.472931\pi\)
\(14\) − 2311.15i − 0.0601610i
\(15\) −3092.72 + 35031.9i −0.0610908 + 0.691989i
\(16\) −16384.0 −0.250000
\(17\) 4949.02 − 4949.02i 0.0592549 − 0.0592549i −0.676858 − 0.736113i \(-0.736659\pi\)
0.736113 + 0.676858i \(0.236659\pi\)
\(18\) −27158.3 − 27158.3i −0.258710 − 0.258710i
\(19\) − 176579.i − 1.35495i −0.735546 − 0.677475i \(-0.763074\pi\)
0.735546 − 0.677475i \(-0.236926\pi\)
\(20\) −61324.1 + 51374.7i −0.383275 + 0.321092i
\(21\) −11494.6 −0.0591038
\(22\) 108802. − 108802.i 0.464459 − 0.464459i
\(23\) −229482. − 229482.i −0.820043 − 0.820043i 0.166071 − 0.986114i \(-0.446892\pi\)
−0.986114 + 0.166071i \(0.946892\pi\)
\(24\) 81486.3i 0.245607i
\(25\) −68437.7 + 384583.i −0.175200 + 0.984533i
\(26\) 494139. 1.08132
\(27\) −396123. + 396123.i −0.745376 + 0.745376i
\(28\) −18489.2 − 18489.2i −0.0300805 − 0.0300805i
\(29\) − 144167.i − 0.203833i −0.994793 − 0.101917i \(-0.967503\pi\)
0.994793 − 0.101917i \(-0.0324975\pi\)
\(30\) 255514. + 304997.i 0.315449 + 0.376540i
\(31\) 1.50971e6 1.63473 0.817365 − 0.576120i \(-0.195434\pi\)
0.817365 + 0.576120i \(0.195434\pi\)
\(32\) −131072. + 131072.i −0.125000 + 0.125000i
\(33\) −541132. − 541132.i −0.456297 − 0.456297i
\(34\) − 79184.4i − 0.0592549i
\(35\) −127179. − 11227.8i −0.0847509 − 0.00748206i
\(36\) −434533. −0.258710
\(37\) −1.60715e6 + 1.60715e6i −0.857532 + 0.857532i −0.991047 − 0.133515i \(-0.957374\pi\)
0.133515 + 0.991047i \(0.457374\pi\)
\(38\) −1.41263e6 − 1.41263e6i −0.677475 − 0.677475i
\(39\) − 2.45761e6i − 1.06232i
\(40\) −79595.0 + 901590.i −0.0310918 + 0.352184i
\(41\) 3.62276e6 1.28205 0.641023 − 0.767522i \(-0.278510\pi\)
0.641023 + 0.767522i \(0.278510\pi\)
\(42\) −91956.5 + 91956.5i −0.0295519 + 0.0295519i
\(43\) −575576. − 575576.i −0.168356 − 0.168356i 0.617900 − 0.786256i \(-0.287983\pi\)
−0.786256 + 0.617900i \(0.787983\pi\)
\(44\) − 1.74084e6i − 0.464459i
\(45\) −1.62642e6 + 1.36255e6i −0.396628 + 0.332278i
\(46\) −3.67171e6 −0.820043
\(47\) 3.11186e6 − 3.11186e6i 0.637718 − 0.637718i −0.312274 − 0.949992i \(-0.601091\pi\)
0.949992 + 0.312274i \(0.101091\pi\)
\(48\) 651891. + 651891.i 0.122803 + 0.122803i
\(49\) 5.72307e6i 0.992761i
\(50\) 2.52916e6 + 3.62417e6i 0.404666 + 0.579867i
\(51\) −393826. −0.0582135
\(52\) 3.95311e6 − 3.95311e6i 0.540661 − 0.540661i
\(53\) 8.16813e6 + 8.16813e6i 1.03519 + 1.03519i 0.999358 + 0.0358294i \(0.0114073\pi\)
0.0358294 + 0.999358i \(0.488593\pi\)
\(54\) 6.33798e6i 0.745376i
\(55\) −5.45868e6 − 6.51582e6i −0.596536 − 0.712063i
\(56\) −295827. −0.0300805
\(57\) −7.02575e6 + 7.02575e6i −0.665569 + 0.665569i
\(58\) −1.15334e6 − 1.15334e6i −0.101917 − 0.101917i
\(59\) − 1.54905e7i − 1.27837i −0.769051 − 0.639187i \(-0.779271\pi\)
0.769051 − 0.639187i \(-0.220729\pi\)
\(60\) 4.48409e6 + 395868.i 0.345994 + 0.0305454i
\(61\) −1.92688e7 −1.39167 −0.695833 − 0.718204i \(-0.744965\pi\)
−0.695833 + 0.718204i \(0.744965\pi\)
\(62\) 1.20777e7 − 1.20777e7i 0.817365 − 0.817365i
\(63\) −490366. − 490366.i −0.0311285 − 0.0311285i
\(64\) 2.09715e6i 0.125000i
\(65\) 2.40057e6 − 2.71918e7i 0.134481 − 1.52330i
\(66\) −8.65811e6 −0.456297
\(67\) −1.00075e7 + 1.00075e7i −0.496624 + 0.496624i −0.910385 − 0.413761i \(-0.864215\pi\)
0.413761 + 0.910385i \(0.364215\pi\)
\(68\) −633475. − 633475.i −0.0296274 − 0.0296274i
\(69\) 1.82614e7i 0.805632i
\(70\) −1.10726e6 + 927613.i −0.0461165 + 0.0386344i
\(71\) 1.84257e7 0.725087 0.362543 − 0.931967i \(-0.381908\pi\)
0.362543 + 0.931967i \(0.381908\pi\)
\(72\) −3.47626e6 + 3.47626e6i −0.129355 + 0.129355i
\(73\) 2.45702e7 + 2.45702e7i 0.865203 + 0.865203i 0.991937 − 0.126734i \(-0.0404494\pi\)
−0.126734 + 0.991937i \(0.540449\pi\)
\(74\) 2.57144e7i 0.857532i
\(75\) 1.80249e7 − 1.25789e7i 0.569676 − 0.397555i
\(76\) −2.26020e7 −0.677475
\(77\) 1.96452e6 − 1.96452e6i 0.0558847 − 0.0558847i
\(78\) −1.96609e7 − 1.96609e7i −0.531160 − 0.531160i
\(79\) 2.17799e7i 0.559176i 0.960120 + 0.279588i \(0.0901978\pi\)
−0.960120 + 0.279588i \(0.909802\pi\)
\(80\) 6.57596e6 + 7.84948e6i 0.160546 + 0.191638i
\(81\) 9.24891e6 0.214857
\(82\) 2.89820e7 − 2.89820e7i 0.641023 − 0.641023i
\(83\) 4.00189e6 + 4.00189e6i 0.0843242 + 0.0843242i 0.748011 − 0.663687i \(-0.231009\pi\)
−0.663687 + 0.748011i \(0.731009\pi\)
\(84\) 1.47130e6i 0.0295519i
\(85\) −4.35741e6 − 384685.i −0.0834744 − 0.00736936i
\(86\) −9.20921e6 −0.168356
\(87\) −5.73617e6 + 5.73617e6i −0.100126 + 0.100126i
\(88\) −1.39267e7 − 1.39267e7i −0.232229 − 0.232229i
\(89\) − 6.14525e7i − 0.979444i −0.871879 − 0.489722i \(-0.837098\pi\)
0.871879 − 0.489722i \(-0.162902\pi\)
\(90\) −2.11100e6 + 2.39118e7i −0.0321750 + 0.364453i
\(91\) 8.92209e6 0.130107
\(92\) −2.93737e7 + 2.93737e7i −0.410022 + 0.410022i
\(93\) −6.00686e7 − 6.00686e7i −0.803001 − 0.803001i
\(94\) − 4.97898e7i − 0.637718i
\(95\) −8.45978e7 + 7.08724e7i −1.03864 + 0.870127i
\(96\) 1.04303e7 0.122803
\(97\) 1.55814e7 − 1.55814e7i 0.176003 − 0.176003i −0.613608 − 0.789611i \(-0.710282\pi\)
0.789611 + 0.613608i \(0.210282\pi\)
\(98\) 4.57846e7 + 4.57846e7i 0.496381 + 0.496381i
\(99\) − 4.61701e7i − 0.480640i
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 10.9.c.b.3.1 ✓ 4
3.2 odd 2 90.9.g.a.73.2 4
4.3 odd 2 80.9.p.a.33.2 4
5.2 odd 4 inner 10.9.c.b.7.1 yes 4
5.3 odd 4 50.9.c.b.7.2 4
5.4 even 2 50.9.c.b.43.2 4
15.2 even 4 90.9.g.a.37.2 4
20.7 even 4 80.9.p.a.17.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.9.c.b.3.1 ✓ 4 1.1 even 1 trivial
10.9.c.b.7.1 yes 4 5.2 odd 4 inner
50.9.c.b.7.2 4 5.3 odd 4
50.9.c.b.43.2 4 5.4 even 2
80.9.p.a.17.2 4 20.7 even 4
80.9.p.a.33.2 4 4.3 odd 2
90.9.g.a.37.2 4 15.2 even 4
90.9.g.a.73.2 4 3.2 odd 2