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Properties

Label 7.10.c.a.4.3
Level $7$
Weight $10$
Character 7.4
Analytic conductor $3.605$
Analytic rank $0$
Dimension $10$
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] = [7,10,Mod(2,7)] 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("7.2"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 7.c (of order \(3\), 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: \(3.60525085315\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 430 x^{8} + 61 x^{7} + 146753 x^{6} + 23608 x^{5} + 16136944 x^{4} + 30575648 x^{3} + \cdots + 761760000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{3}\cdot 7^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 4.3
Root \(-0.371984 + 0.644295i\) of defining polynomial
Character \(\chi\) \(=\) 7.4
Dual form 7.10.c.a.2.3

$q$-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+(-2.74397 - 4.75269i) q^{2} +(-1.70307 + 2.94981i) q^{3} +(240.941 - 417.323i) q^{4} +(-828.924 - 1435.74i) q^{5} +18.6927 q^{6} +(2822.68 - 5690.88i) q^{7} -5454.36 q^{8} +(9835.70 + 17035.9i) q^{9} +(-4549.08 + 7879.24i) q^{10} +(-16835.7 + 29160.2i) q^{11} +(820.681 + 1421.46i) q^{12} +47905.5 q^{13} +(-34792.3 + 2200.27i) q^{14} +5646.87 q^{15} +(-108395. - 187746. i) q^{16} +(172909. - 299487. i) q^{17} +(53977.7 - 93492.1i) q^{18} +(202522. + 350778. i) q^{19} -798888. q^{20} +(11979.8 + 18018.3i) q^{21} +184786. q^{22} +(926736. + 1.60515e6i) q^{23} +(9289.18 - 16089.3i) q^{24} +(-397666. + 688778. i) q^{25} +(-131451. - 227680. i) q^{26} -134047. q^{27} +(-1.69483e6 - 2.54913e6i) q^{28} +682809. q^{29} +(-15494.8 - 26837.8i) q^{30} +(4.54839e6 - 7.87804e6i) q^{31} +(-1.99118e6 + 3.44883e6i) q^{32} +(-57344.7 - 99324.0i) q^{33} -1.89782e6 q^{34} +(-1.05104e7 + 664678. i) q^{35} +9.47930e6 q^{36} +(7.77984e6 + 1.34751e7i) q^{37} +(1.11143e6 - 1.92505e6i) q^{38} +(-81586.5 + 141312. i) q^{39} +(4.52125e6 + 7.83104e6i) q^{40} -2.98719e7 q^{41} +(52763.5 - 106378. i) q^{42} +6.28733e6 q^{43} +(8.11281e6 + 1.40518e7i) q^{44} +(1.63061e7 - 2.82430e7i) q^{45} +(5.08587e6 - 8.80898e6i) q^{46} +(-5.16042e6 - 8.93811e6i) q^{47} +738421. q^{48} +(-2.44186e7 - 3.21270e7i) q^{49} +4.36473e6 q^{50} +(588952. + 1.02010e6i) q^{51} +(1.15424e7 - 1.99920e7i) q^{52} +(-3.32218e7 + 5.75419e7i) q^{53} +(367820. + 637083. i) q^{54} +5.58219e7 q^{55} +(-1.53959e7 + 3.10401e7i) q^{56} -1.37964e6 q^{57} +(-1.87361e6 - 3.24518e6i) q^{58} +(-3.52902e7 + 6.11245e7i) q^{59} +(1.36056e6 - 2.35657e6i) q^{60} +(4.21696e7 + 7.30399e7i) q^{61} -4.99225e7 q^{62} +(1.24712e8 - 7.88682e6i) q^{63} -8.91419e7 q^{64} +(-3.97100e7 - 6.87797e7i) q^{65} +(-314704. + 545084. i) q^{66} +(1.05869e8 - 1.83370e8i) q^{67} +(-8.33217e7 - 1.44317e8i) q^{68} -6.31320e6 q^{69} +(3.19992e7 + 4.81288e7i) q^{70} +2.31588e7 q^{71} +(-5.36475e7 - 9.29202e7i) q^{72} +(-1.24284e8 + 2.15266e8i) q^{73} +(4.26952e7 - 7.39503e7i) q^{74} +(-1.35451e6 - 2.34608e6i) q^{75} +1.95184e8 q^{76} +(1.18426e8 + 1.78120e8i) q^{77} +895483. q^{78} +(-1.33038e8 - 2.30429e8i) q^{79} +(-1.79703e8 + 3.11255e8i) q^{80} +(-1.93368e8 + 3.34923e8i) q^{81} +(8.19675e7 + 1.41972e8i) q^{82} +6.33299e8 q^{83} +(1.04059e7 - 658069. i) q^{84} -5.73312e8 q^{85} +(-1.72522e7 - 2.98817e7i) q^{86} +(-1.16287e6 + 2.01416e6i) q^{87} +(9.18278e7 - 1.59050e8i) q^{88} +(-3.11190e8 - 5.38997e8i) q^{89} -1.78973e8 q^{90} +(1.35222e8 - 2.72624e8i) q^{91} +8.93156e8 q^{92} +(1.54925e7 + 2.68338e7i) q^{93} +(-2.83201e7 + 4.90518e7i) q^{94} +(3.35751e8 - 5.81537e8i) q^{95} +(-6.78226e6 - 1.17472e7i) q^{96} -9.94856e8 q^{97} +(-8.56861e7 + 2.04210e8i) q^{98} -6.62362e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 18 q^{2} + 161 q^{3} - 940 q^{4} + 1533 q^{5} - 8708 q^{6} - 1036 q^{7} + 34272 q^{8} - 35734 q^{9} + 4298 q^{10} + 42213 q^{11} + 135604 q^{12} - 319676 q^{13} - 39522 q^{14} + 151394 q^{15} + 322064 q^{16}+ \cdots - 1900777180 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.74397 − 4.75269i −0.121267 − 0.210041i 0.799000 − 0.601331i \(-0.205363\pi\)
−0.920268 + 0.391289i \(0.872029\pi\)
\(3\) −1.70307 + 2.94981i −0.0121391 + 0.0210256i −0.872031 − 0.489450i \(-0.837197\pi\)
0.859892 + 0.510476i \(0.170531\pi\)
\(4\) 240.941 − 417.323i 0.470588 − 0.815083i
\(5\) −828.924 − 1435.74i −0.593129 − 1.02733i −0.993808 − 0.111112i \(-0.964559\pi\)
0.400679 − 0.916219i \(-0.368774\pi\)
\(6\) 18.6927 0.00588833
\(7\) 2822.68 − 5690.88i 0.444345 − 0.895856i
\(8\) −5454.36 −0.470803
\(9\) 9835.70 + 17035.9i 0.499705 + 0.865515i
\(10\) −4549.08 + 7879.24i −0.143855 + 0.249163i
\(11\) −16835.7 + 29160.2i −0.346707 + 0.600515i −0.985662 − 0.168729i \(-0.946034\pi\)
0.638955 + 0.769244i \(0.279367\pi\)
\(12\) 820.681 + 1421.46i 0.0114251 + 0.0197888i
\(13\) 47905.5 0.465200 0.232600 − 0.972572i \(-0.425277\pi\)
0.232600 + 0.972572i \(0.425277\pi\)
\(14\) −34792.3 + 2200.27i −0.242051 + 0.0153073i
\(15\) 5646.87 0.0288003
\(16\) −108395. − 187746.i −0.413495 − 0.716195i
\(17\) 172909. − 299487.i 0.502107 − 0.869676i −0.497890 − 0.867240i \(-0.665892\pi\)
0.999997 − 0.00243516i \(-0.000775137\pi\)
\(18\) 53977.7 − 93492.1i 0.121196 − 0.209917i
\(19\) 202522. + 350778.i 0.356518 + 0.617507i 0.987376 − 0.158391i \(-0.0506306\pi\)
−0.630859 + 0.775898i \(0.717297\pi\)
\(20\) −798888. −1.11648
\(21\) 11979.8 + 18018.3i 0.0134419 + 0.0202175i
\(22\) 184786. 0.168177
\(23\) 926736. + 1.60515e6i 0.690527 + 1.19603i 0.971665 + 0.236361i \(0.0759548\pi\)
−0.281138 + 0.959667i \(0.590712\pi\)
\(24\) 9289.18 − 16089.3i 0.00571514 − 0.00989891i
\(25\) −397666. + 688778.i −0.203605 + 0.352654i
\(26\) −131451. − 227680.i −0.0564136 − 0.0977113i
\(27\) −134047. −0.0485422
\(28\) −1.69483e6 − 2.54913e6i −0.521093 − 0.783757i
\(29\) 682809. 0.179270 0.0896351 − 0.995975i \(-0.471430\pi\)
0.0896351 + 0.995975i \(0.471430\pi\)
\(30\) −15494.8 − 26837.8i −0.00349254 − 0.00604925i
\(31\) 4.54839e6 − 7.87804e6i 0.884565 − 1.53211i 0.0383536 − 0.999264i \(-0.487789\pi\)
0.846211 − 0.532847i \(-0.178878\pi\)
\(32\) −1.99118e6 + 3.44883e6i −0.335688 + 0.581429i
\(33\) −57344.7 − 99324.0i −0.00841746 − 0.0145795i
\(34\) −1.89782e6 −0.243557
\(35\) −1.05104e7 + 664678.i −1.18389 + 0.0748695i
\(36\) 9.47930e6 0.940622
\(37\) 7.77984e6 + 1.34751e7i 0.682437 + 1.18202i 0.974235 + 0.225536i \(0.0724133\pi\)
−0.291798 + 0.956480i \(0.594253\pi\)
\(38\) 1.11143e6 − 1.92505e6i 0.0864679 − 0.149767i
\(39\) −81586.5 + 141312.i −0.00564713 + 0.00978111i
\(40\) 4.52125e6 + 7.83104e6i 0.279247 + 0.483670i
\(41\) −2.98719e7 −1.65095 −0.825477 − 0.564435i \(-0.809094\pi\)
−0.825477 + 0.564435i \(0.809094\pi\)
\(42\) 52763.5 − 106378.i 0.00261645 − 0.00527509i
\(43\) 6.28733e6 0.280452 0.140226 − 0.990120i \(-0.455217\pi\)
0.140226 + 0.990120i \(0.455217\pi\)
\(44\) 8.11281e6 + 1.40518e7i 0.326313 + 0.565191i
\(45\) 1.63061e7 − 2.82430e7i 0.592780 − 1.02672i
\(46\) 5.08587e6 − 8.80898e6i 0.167477 − 0.290078i
\(47\) −5.16042e6 − 8.93811e6i −0.154257 − 0.267181i 0.778531 − 0.627606i \(-0.215965\pi\)
−0.932788 + 0.360425i \(0.882632\pi\)
\(48\) 738421. 0.0200779
\(49\) −2.44186e7 − 3.21270e7i −0.605115 − 0.796138i
\(50\) 4.36473e6 0.0987626
\(51\) 588952. + 1.02010e6i 0.0121903 + 0.0211142i
\(52\) 1.15424e7 − 1.99920e7i 0.218918 − 0.379177i
\(53\) −3.32218e7 + 5.75419e7i −0.578339 + 1.00171i 0.417331 + 0.908754i \(0.362965\pi\)
−0.995670 + 0.0929577i \(0.970368\pi\)
\(54\) 367820. + 637083.i 0.00588659 + 0.0101959i
\(55\) 5.58219e7 0.822570
\(56\) −1.53959e7 + 3.10401e7i −0.209199 + 0.421772i
\(57\) −1.37964e6 −0.0173113
\(58\) −1.87361e6 − 3.24518e6i −0.0217396 − 0.0376541i
\(59\) −3.52902e7 + 6.11245e7i −0.379158 + 0.656721i −0.990940 − 0.134305i \(-0.957120\pi\)
0.611782 + 0.791027i \(0.290453\pi\)
\(60\) 1.36056e6 − 2.35657e6i 0.0135531 − 0.0234746i
\(61\) 4.21696e7 + 7.30399e7i 0.389955 + 0.675423i 0.992443 − 0.122706i \(-0.0391571\pi\)
−0.602488 + 0.798128i \(0.705824\pi\)
\(62\) −4.99225e7 −0.429076
\(63\) 1.24712e8 − 7.88682e6i 0.997418 − 0.0630768i
\(64\) −8.91419e7 −0.664159
\(65\) −3.97100e7 − 6.87797e7i −0.275924 − 0.477914i
\(66\) −314704. + 545084.i −0.00204153 + 0.00353603i
\(67\) 1.05869e8 − 1.83370e8i 0.641846 − 1.11171i −0.343175 − 0.939272i \(-0.611502\pi\)
0.985020 − 0.172438i \(-0.0551644\pi\)
\(68\) −8.33217e7 − 1.44317e8i −0.472572 − 0.818519i
\(69\) −6.31320e6 −0.0335296
\(70\) 3.19992e7 + 4.81288e7i 0.159293 + 0.239587i
\(71\) 2.31588e7 0.108157 0.0540785 − 0.998537i \(-0.482778\pi\)
0.0540785 + 0.998537i \(0.482778\pi\)
\(72\) −5.36475e7 − 9.29202e7i −0.235263 − 0.407487i
\(73\) −1.24284e8 + 2.15266e8i −0.512226 + 0.887201i 0.487674 + 0.873026i \(0.337846\pi\)
−0.999900 + 0.0141754i \(0.995488\pi\)
\(74\) 4.26952e7 − 7.39503e7i 0.165515 − 0.286680i
\(75\) −1.35451e6 − 2.34608e6i −0.00494318 − 0.00856184i
\(76\) 1.95184e8 0.671092
\(77\) 1.18426e8 + 1.78120e8i 0.383917 + 0.577436i
\(78\) 895483. 0.00273925
\(79\) −1.33038e8 − 2.30429e8i −0.384287 − 0.665604i 0.607383 − 0.794409i \(-0.292219\pi\)
−0.991670 + 0.128805i \(0.958886\pi\)
\(80\) −1.79703e8 + 3.11255e8i −0.490513 + 0.849593i
\(81\) −1.93368e8 + 3.34923e8i −0.499116 + 0.864494i
\(82\) 8.19675e7 + 1.41972e8i 0.200207 + 0.346769i
\(83\) 6.33299e8 1.46473 0.732365 − 0.680912i \(-0.238417\pi\)
0.732365 + 0.680912i \(0.238417\pi\)
\(84\) 1.04059e7 − 658069.i 0.0228046 − 0.00144216i
\(85\) −5.73312e8 −1.19126
\(86\) −1.72522e7 − 2.98817e7i −0.0340097 − 0.0589065i
\(87\) −1.16287e6 + 2.01416e6i −0.00217619 + 0.00376926i
\(88\) 9.18278e7 − 1.59050e8i 0.163231 − 0.282724i
\(89\) −3.11190e8 − 5.38997e8i −0.525740 − 0.910608i −0.999550 − 0.0299813i \(-0.990455\pi\)
0.473811 − 0.880627i \(-0.342878\pi\)
\(90\) −1.78973e8 −0.287539
\(91\) 1.35222e8 − 2.72624e8i 0.206709 − 0.416752i
\(92\) 8.93156e8 1.29982
\(93\) 1.54925e7 + 2.68338e7i 0.0214757 + 0.0371970i
\(94\) −2.83201e7 + 4.90518e7i −0.0374127 + 0.0648007i
\(95\) 3.35751e8 − 5.81537e8i 0.422922 − 0.732523i
\(96\) −6.78226e6 − 1.17472e7i −0.00814994 − 0.0141161i
\(97\) −9.94856e8 −1.14100 −0.570502 − 0.821296i \(-0.693251\pi\)
−0.570502 + 0.821296i \(0.693251\pi\)
\(98\) −8.56861e7 + 2.04210e8i −0.0938411 + 0.223645i
\(99\) −6.62362e8 −0.693006
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 7.10.c.a.4.3 yes 10
3.2 odd 2 63.10.e.b.46.3 10
4.3 odd 2 112.10.i.c.81.3 10
7.2 even 3 inner 7.10.c.a.2.3 ✓ 10
7.3 odd 6 49.10.a.f.1.3 5
7.4 even 3 49.10.a.e.1.3 5
7.5 odd 6 49.10.c.g.30.3 10
7.6 odd 2 49.10.c.g.18.3 10
21.2 odd 6 63.10.e.b.37.3 10
28.23 odd 6 112.10.i.c.65.3 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.10.c.a.2.3 ✓ 10 7.2 even 3 inner
7.10.c.a.4.3 yes 10 1.1 even 1 trivial
49.10.a.e.1.3 5 7.4 even 3
49.10.a.f.1.3 5 7.3 odd 6
49.10.c.g.18.3 10 7.6 odd 2
49.10.c.g.30.3 10 7.5 odd 6
63.10.e.b.37.3 10 21.2 odd 6
63.10.e.b.46.3 10 3.2 odd 2
112.10.i.c.65.3 10 28.23 odd 6
112.10.i.c.81.3 10 4.3 odd 2