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Properties

Label 196.8.e.b.165.2
Level $196$
Weight $8$
Character 196.165
Analytic conductor $61.227$
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] = [196,8,Mod(165,196)] 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("196.165"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(196, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 196.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-14,0,294] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(61.2274649949\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{1009})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 253x^{2} + 252x + 63504 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 165.2
Root \(-7.69119 + 13.3215i\) of defining polynomial
Character \(\chi\) \(=\) 196.165
Dual form 196.8.e.b.177.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(12.3824 - 21.4469i) q^{3} +(-101.206 - 175.294i) q^{5} +(786.853 + 1362.87i) q^{9} +(3763.59 - 6518.73i) q^{11} -1700.28 q^{13} -5012.69 q^{15} +(1240.93 - 2149.36i) q^{17} +(19908.0 + 34481.7i) q^{19} +(-12780.0 - 22135.7i) q^{23} +(18577.1 - 32176.5i) q^{25} +93133.0 q^{27} +148830. q^{29} +(16769.2 - 29045.1i) q^{31} +(-93204.5 - 161435. i) q^{33} +(-200279. - 346893. i) q^{37} +(-21053.6 + 36465.8i) q^{39} -362750. q^{41} -324652. q^{43} +(159269. - 275862. i) q^{45} +(354450. + 613925. i) q^{47} +(-30731.4 - 53228.3i) q^{51} +(92962.0 - 161015. i) q^{53} -1.52360e6 q^{55} +986034. q^{57} +(596858. - 1.03379e6i) q^{59} +(-1.25602e6 - 2.17549e6i) q^{61} +(172079. + 298050. i) q^{65} +(1.42992e6 - 2.47669e6i) q^{67} -632989. q^{69} +3.22083e6 q^{71} +(-2.50643e6 + 4.34127e6i) q^{73} +(-460058. - 796844. i) q^{75} +(-2.97104e6 - 5.14599e6i) q^{79} +(-567640. + 983181. i) q^{81} -1.02006e7 q^{83} -502359. q^{85} +(1.84287e6 - 3.19194e6i) q^{87} +(-926867. - 1.60538e6i) q^{89} +(-415286. - 719296. i) q^{93} +(4.02963e6 - 6.97952e6i) q^{95} -1.52142e7 q^{97} +1.18456e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 14 q^{3} + 294 q^{5} + 2258 q^{9} + 3492 q^{11} - 32340 q^{13} - 48512 q^{15} + 29232 q^{17} + 3206 q^{19} + 9360 q^{23} - 131146 q^{25} - 36344 q^{27} + 369408 q^{29} - 165060 q^{31} - 342832 q^{33}+ \cdots + 18168664 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/196\mathbb{Z}\right)^\times\).

\(n\) \(99\) \(101\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 12.3824 − 21.4469i 0.264777 − 0.458607i −0.702728 − 0.711458i \(-0.748035\pi\)
0.967505 + 0.252852i \(0.0813684\pi\)
\(4\) 0 0
\(5\) −101.206 − 175.294i −0.362086 − 0.627152i 0.626218 − 0.779648i \(-0.284602\pi\)
−0.988304 + 0.152496i \(0.951269\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 786.853 + 1362.87i 0.359787 + 0.623169i
\(10\) 0 0
\(11\) 3763.59 − 6518.73i 0.852567 − 1.47669i −0.0263178 − 0.999654i \(-0.508378\pi\)
0.878884 − 0.477035i \(-0.158288\pi\)
\(12\) 0 0
\(13\) −1700.28 −0.214644 −0.107322 − 0.994224i \(-0.534228\pi\)
−0.107322 + 0.994224i \(0.534228\pi\)
\(14\) 0 0
\(15\) −5012.69 −0.383488
\(16\) 0 0
\(17\) 1240.93 − 2149.36i 0.0612599 − 0.106105i −0.833769 − 0.552114i \(-0.813821\pi\)
0.895029 + 0.446008i \(0.147155\pi\)
\(18\) 0 0
\(19\) 19908.0 + 34481.7i 0.665871 + 1.15332i 0.979048 + 0.203628i \(0.0652732\pi\)
−0.313177 + 0.949695i \(0.601393\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −12780.0 − 22135.7i −0.219020 − 0.379354i 0.735488 − 0.677537i \(-0.236953\pi\)
−0.954509 + 0.298183i \(0.903619\pi\)
\(24\) 0 0
\(25\) 18577.1 − 32176.5i 0.237787 − 0.411859i
\(26\) 0 0
\(27\) 93133.0 0.910606
\(28\) 0 0
\(29\) 148830. 1.13317 0.566587 − 0.824002i \(-0.308263\pi\)
0.566587 + 0.824002i \(0.308263\pi\)
\(30\) 0 0
\(31\) 16769.2 − 29045.1i 0.101099 − 0.175109i −0.811039 − 0.584992i \(-0.801097\pi\)
0.912138 + 0.409884i \(0.134431\pi\)
\(32\) 0 0
\(33\) −93204.5 − 161435.i −0.451479 − 0.781985i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −200279. − 346893.i −0.650023 − 1.12587i −0.983117 − 0.182979i \(-0.941426\pi\)
0.333094 − 0.942894i \(-0.391907\pi\)
\(38\) 0 0
\(39\) −21053.6 + 36465.8i −0.0568328 + 0.0984373i
\(40\) 0 0
\(41\) −362750. −0.821985 −0.410992 − 0.911639i \(-0.634818\pi\)
−0.410992 + 0.911639i \(0.634818\pi\)
\(42\) 0 0
\(43\) −324652. −0.622699 −0.311350 − 0.950295i \(-0.600781\pi\)
−0.311350 + 0.950295i \(0.600781\pi\)
\(44\) 0 0
\(45\) 159269. − 275862.i 0.260548 − 0.451282i
\(46\) 0 0
\(47\) 354450. + 613925.i 0.497980 + 0.862527i 0.999997 − 0.00233074i \(-0.000741899\pi\)
−0.502017 + 0.864858i \(0.667409\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −30731.4 − 53228.3i −0.0324404 − 0.0561884i
\(52\) 0 0
\(53\) 92962.0 − 161015.i 0.0857709 − 0.148559i −0.819949 − 0.572437i \(-0.805998\pi\)
0.905719 + 0.423878i \(0.139331\pi\)
\(54\) 0 0
\(55\) −1.52360e6 −1.23481
\(56\) 0 0
\(57\) 986034. 0.705228
\(58\) 0 0
\(59\) 596858. − 1.03379e6i 0.378346 − 0.655315i −0.612476 − 0.790489i \(-0.709826\pi\)
0.990822 + 0.135175i \(0.0431596\pi\)
\(60\) 0 0
\(61\) −1.25602e6 − 2.17549e6i −0.708504 − 1.22716i −0.965412 − 0.260729i \(-0.916037\pi\)
0.256908 − 0.966436i \(-0.417296\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 172079. + 298050.i 0.0777198 + 0.134615i
\(66\) 0 0
\(67\) 1.42992e6 − 2.47669e6i 0.580829 − 1.00603i −0.414552 − 0.910026i \(-0.636062\pi\)
0.995381 − 0.0960004i \(-0.0306050\pi\)
\(68\) 0 0
\(69\) −632989. −0.231966
\(70\) 0 0
\(71\) 3.22083e6 1.06798 0.533990 − 0.845491i \(-0.320692\pi\)
0.533990 + 0.845491i \(0.320692\pi\)
\(72\) 0 0
\(73\) −2.50643e6 + 4.34127e6i −0.754095 + 1.30613i 0.191727 + 0.981448i \(0.438591\pi\)
−0.945823 + 0.324683i \(0.894742\pi\)
\(74\) 0 0
\(75\) −460058. − 796844.i −0.125921 − 0.218102i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −2.97104e6 − 5.14599e6i −0.677975 − 1.17429i −0.975590 − 0.219601i \(-0.929524\pi\)
0.297615 − 0.954686i \(-0.403809\pi\)
\(80\) 0 0
\(81\) −567640. + 983181.i −0.118679 + 0.205559i
\(82\) 0 0
\(83\) −1.02006e7 −1.95819 −0.979094 − 0.203407i \(-0.934799\pi\)
−0.979094 + 0.203407i \(0.934799\pi\)
\(84\) 0 0
\(85\) −502359. −0.0887255
\(86\) 0 0
\(87\) 1.84287e6 − 3.19194e6i 0.300038 − 0.519681i
\(88\) 0 0
\(89\) −926867. − 1.60538e6i −0.139365 − 0.241387i 0.787892 − 0.615814i \(-0.211173\pi\)
−0.927256 + 0.374427i \(0.877839\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −415286. − 719296.i −0.0535373 − 0.0927294i
\(94\) 0 0
\(95\) 4.02963e6 − 6.97952e6i 0.482205 − 0.835204i
\(96\) 0 0
\(97\) −1.52142e7 −1.69257 −0.846287 − 0.532727i \(-0.821167\pi\)
−0.846287 + 0.532727i \(0.821167\pi\)
\(98\) 0 0
\(99\) 1.18456e7 1.22697
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 196.8.e.b.165.2 4
7.2 even 3 inner 196.8.e.b.177.2 4
7.3 odd 6 196.8.a.a.1.2 2
7.4 even 3 28.8.a.b.1.1 ✓ 2
7.5 odd 6 196.8.e.c.177.1 4
7.6 odd 2 196.8.e.c.165.1 4
21.11 odd 6 252.8.a.f.1.1 2
28.11 odd 6 112.8.a.h.1.2 2
56.11 odd 6 448.8.a.q.1.1 2
56.53 even 6 448.8.a.o.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.8.a.b.1.1 ✓ 2 7.4 even 3
112.8.a.h.1.2 2 28.11 odd 6
196.8.a.a.1.2 2 7.3 odd 6
196.8.e.b.165.2 4 1.1 even 1 trivial
196.8.e.b.177.2 4 7.2 even 3 inner
196.8.e.c.165.1 4 7.6 odd 2
196.8.e.c.177.1 4 7.5 odd 6
252.8.a.f.1.1 2 21.11 odd 6
448.8.a.o.1.2 2 56.53 even 6
448.8.a.q.1.1 2 56.11 odd 6