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Properties

Label 91.2.e.b.53.1
Level $91$
Weight $2$
Character 91.53
Analytic conductor $0.727$
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] = [91,2,Mod(53,91)] 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("91.53"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(91, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 91.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.726638658394\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 53.1
Root \(-0.309017 - 0.535233i\) of defining polynomial
Character \(\chi\) \(=\) 91.53
Dual form 91.2.e.b.79.1

$q$-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+(0.190983 + 0.330792i) q^{2} +(-1.11803 + 1.93649i) q^{3} +(0.927051 - 1.60570i) q^{4} +(1.11803 + 1.93649i) q^{5} -0.854102 q^{6} +(-2.00000 + 1.73205i) q^{7} +1.47214 q^{8} +(-1.00000 - 1.73205i) q^{9} +(-0.427051 + 0.739674i) q^{10} +(1.50000 - 2.59808i) q^{11} +(2.07295 + 3.59045i) q^{12} -1.00000 q^{13} +(-0.954915 - 0.330792i) q^{14} -5.00000 q^{15} +(-1.57295 - 2.72443i) q^{16} +(3.73607 - 6.47106i) q^{17} +(0.381966 - 0.661585i) q^{18} +(-1.50000 - 2.59808i) q^{19} +4.14590 q^{20} +(-1.11803 - 5.80948i) q^{21} +1.14590 q^{22} +(1.88197 + 3.25966i) q^{23} +(-1.64590 + 2.85078i) q^{24} +(-0.190983 - 0.330792i) q^{26} -2.23607 q^{27} +(0.927051 + 4.81710i) q^{28} -4.47214 q^{29} +(-0.954915 - 1.65396i) q^{30} +(-2.50000 + 4.33013i) q^{31} +(2.07295 - 3.59045i) q^{32} +(3.35410 + 5.80948i) q^{33} +2.85410 q^{34} +(-5.59017 - 1.93649i) q^{35} -3.70820 q^{36} +(4.35410 + 7.54153i) q^{37} +(0.572949 - 0.992377i) q^{38} +(1.11803 - 1.93649i) q^{39} +(1.64590 + 2.85078i) q^{40} +4.47214 q^{41} +(1.70820 - 1.47935i) q^{42} -8.00000 q^{43} +(-2.78115 - 4.81710i) q^{44} +(2.23607 - 3.87298i) q^{45} +(-0.718847 + 1.24508i) q^{46} +(-0.736068 - 1.27491i) q^{47} +7.03444 q^{48} +(1.00000 - 6.92820i) q^{49} +(8.35410 + 14.4697i) q^{51} +(-0.927051 + 1.60570i) q^{52} +(-0.736068 + 1.27491i) q^{53} +(-0.427051 - 0.739674i) q^{54} +6.70820 q^{55} +(-2.94427 + 2.54981i) q^{56} +6.70820 q^{57} +(-0.854102 - 1.47935i) q^{58} +(-3.73607 + 6.47106i) q^{59} +(-4.63525 + 8.02850i) q^{60} +(-1.50000 - 2.59808i) q^{61} -1.90983 q^{62} +(5.00000 + 1.73205i) q^{63} -4.70820 q^{64} +(-1.11803 - 1.93649i) q^{65} +(-1.28115 + 2.21902i) q^{66} +(1.50000 - 2.59808i) q^{67} +(-6.92705 - 11.9980i) q^{68} -8.41641 q^{69} +(-0.427051 - 2.21902i) q^{70} +8.94427 q^{71} +(-1.47214 - 2.54981i) q^{72} +(-5.35410 + 9.27358i) q^{73} +(-1.66312 + 2.88061i) q^{74} -5.56231 q^{76} +(1.50000 + 7.79423i) q^{77} +0.854102 q^{78} +(-5.35410 - 9.27358i) q^{79} +(3.51722 - 6.09201i) q^{80} +(5.50000 - 9.52628i) q^{81} +(0.854102 + 1.47935i) q^{82} +(-10.3647 - 3.59045i) q^{84} +16.7082 q^{85} +(-1.52786 - 2.64634i) q^{86} +(5.00000 - 8.66025i) q^{87} +(2.20820 - 3.82472i) q^{88} +(1.11803 + 1.93649i) q^{89} +1.70820 q^{90} +(2.00000 - 1.73205i) q^{91} +6.97871 q^{92} +(-5.59017 - 9.68246i) q^{93} +(0.281153 - 0.486971i) q^{94} +(3.35410 - 5.80948i) q^{95} +(4.63525 + 8.02850i) q^{96} -17.4164 q^{97} +(2.48278 - 0.992377i) q^{98} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} - 3 q^{4} + 10 q^{6} - 8 q^{7} - 12 q^{8} - 4 q^{9} + 5 q^{10} + 6 q^{11} + 15 q^{12} - 4 q^{13} - 15 q^{14} - 20 q^{15} - 13 q^{16} + 6 q^{17} + 6 q^{18} - 6 q^{19} + 30 q^{20} + 18 q^{22}+ \cdots - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/91\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(66\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.190983 + 0.330792i 0.135045 + 0.233905i 0.925615 − 0.378467i \(-0.123549\pi\)
−0.790569 + 0.612372i \(0.790215\pi\)
\(3\) −1.11803 + 1.93649i −0.645497 + 1.11803i 0.338689 + 0.940898i \(0.390016\pi\)
−0.984186 + 0.177136i \(0.943317\pi\)
\(4\) 0.927051 − 1.60570i 0.463525 − 0.802850i
\(5\) 1.11803 + 1.93649i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) −0.854102 −0.348686
\(7\) −2.00000 + 1.73205i −0.755929 + 0.654654i
\(8\) 1.47214 0.520479
\(9\) −1.00000 − 1.73205i −0.333333 − 0.577350i
\(10\) −0.427051 + 0.739674i −0.135045 + 0.233905i
\(11\) 1.50000 − 2.59808i 0.452267 − 0.783349i −0.546259 − 0.837616i \(-0.683949\pi\)
0.998526 + 0.0542666i \(0.0172821\pi\)
\(12\) 2.07295 + 3.59045i 0.598409 + 1.03647i
\(13\) −1.00000 −0.277350
\(14\) −0.954915 − 0.330792i −0.255212 − 0.0884080i
\(15\) −5.00000 −1.29099
\(16\) −1.57295 − 2.72443i −0.393237 − 0.681107i
\(17\) 3.73607 − 6.47106i 0.906130 − 1.56946i 0.0867359 − 0.996231i \(-0.472356\pi\)
0.819394 − 0.573231i \(-0.194310\pi\)
\(18\) 0.381966 − 0.661585i 0.0900303 − 0.155937i
\(19\) −1.50000 − 2.59808i −0.344124 − 0.596040i 0.641071 − 0.767482i \(-0.278491\pi\)
−0.985194 + 0.171442i \(0.945157\pi\)
\(20\) 4.14590 0.927051
\(21\) −1.11803 − 5.80948i −0.243975 − 1.26773i
\(22\) 1.14590 0.244306
\(23\) 1.88197 + 3.25966i 0.392417 + 0.679686i 0.992768 − 0.120051i \(-0.0383057\pi\)
−0.600351 + 0.799737i \(0.704972\pi\)
\(24\) −1.64590 + 2.85078i −0.335968 + 0.581913i
\(25\) 0 0
\(26\) −0.190983 − 0.330792i −0.0374548 − 0.0648737i
\(27\) −2.23607 −0.430331
\(28\) 0.927051 + 4.81710i 0.175196 + 0.910346i
\(29\) −4.47214 −0.830455 −0.415227 − 0.909718i \(-0.636298\pi\)
−0.415227 + 0.909718i \(0.636298\pi\)
\(30\) −0.954915 − 1.65396i −0.174343 − 0.301971i
\(31\) −2.50000 + 4.33013i −0.449013 + 0.777714i −0.998322 − 0.0579057i \(-0.981558\pi\)
0.549309 + 0.835619i \(0.314891\pi\)
\(32\) 2.07295 − 3.59045i 0.366449 − 0.634708i
\(33\) 3.35410 + 5.80948i 0.583874 + 1.01130i
\(34\) 2.85410 0.489474
\(35\) −5.59017 − 1.93649i −0.944911 − 0.327327i
\(36\) −3.70820 −0.618034
\(37\) 4.35410 + 7.54153i 0.715810 + 1.23982i 0.962646 + 0.270762i \(0.0872757\pi\)
−0.246836 + 0.969057i \(0.579391\pi\)
\(38\) 0.572949 − 0.992377i 0.0929446 − 0.160985i
\(39\) 1.11803 − 1.93649i 0.179029 − 0.310087i
\(40\) 1.64590 + 2.85078i 0.260239 + 0.450748i
\(41\) 4.47214 0.698430 0.349215 − 0.937043i \(-0.386448\pi\)
0.349215 + 0.937043i \(0.386448\pi\)
\(42\) 1.70820 − 1.47935i 0.263582 − 0.228268i
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −2.78115 − 4.81710i −0.419275 − 0.726205i
\(45\) 2.23607 − 3.87298i 0.333333 − 0.577350i
\(46\) −0.718847 + 1.24508i −0.105988 + 0.183577i
\(47\) −0.736068 − 1.27491i −0.107367 − 0.185964i 0.807336 − 0.590092i \(-0.200909\pi\)
−0.914703 + 0.404128i \(0.867575\pi\)
\(48\) 7.03444 1.01533
\(49\) 1.00000 − 6.92820i 0.142857 − 0.989743i
\(50\) 0 0
\(51\) 8.35410 + 14.4697i 1.16981 + 2.02617i
\(52\) −0.927051 + 1.60570i −0.128559 + 0.222670i
\(53\) −0.736068 + 1.27491i −0.101107 + 0.175122i −0.912141 − 0.409877i \(-0.865572\pi\)
0.811034 + 0.584999i \(0.198905\pi\)
\(54\) −0.427051 − 0.739674i −0.0581143 − 0.100657i
\(55\) 6.70820 0.904534
\(56\) −2.94427 + 2.54981i −0.393445 + 0.340733i
\(57\) 6.70820 0.888523
\(58\) −0.854102 − 1.47935i −0.112149 − 0.194248i
\(59\) −3.73607 + 6.47106i −0.486395 + 0.842460i −0.999878 − 0.0156395i \(-0.995022\pi\)
0.513483 + 0.858100i \(0.328355\pi\)
\(60\) −4.63525 + 8.02850i −0.598409 + 1.03647i
\(61\) −1.50000 − 2.59808i −0.192055 − 0.332650i 0.753876 − 0.657017i \(-0.228182\pi\)
−0.945931 + 0.324367i \(0.894849\pi\)
\(62\) −1.90983 −0.242549
\(63\) 5.00000 + 1.73205i 0.629941 + 0.218218i
\(64\) −4.70820 −0.588525
\(65\) −1.11803 − 1.93649i −0.138675 − 0.240192i
\(66\) −1.28115 + 2.21902i −0.157699 + 0.273143i
\(67\) 1.50000 − 2.59808i 0.183254 − 0.317406i −0.759733 − 0.650236i \(-0.774670\pi\)
0.942987 + 0.332830i \(0.108004\pi\)
\(68\) −6.92705 − 11.9980i −0.840028 − 1.45497i
\(69\) −8.41641 −1.01322
\(70\) −0.427051 − 2.21902i −0.0510424 − 0.265224i
\(71\) 8.94427 1.06149 0.530745 − 0.847532i \(-0.321912\pi\)
0.530745 + 0.847532i \(0.321912\pi\)
\(72\) −1.47214 − 2.54981i −0.173493 − 0.300498i
\(73\) −5.35410 + 9.27358i −0.626650 + 1.08539i 0.361569 + 0.932345i \(0.382241\pi\)
−0.988219 + 0.153045i \(0.951092\pi\)
\(74\) −1.66312 + 2.88061i −0.193334 + 0.334864i
\(75\) 0 0
\(76\) −5.56231 −0.638040
\(77\) 1.50000 + 7.79423i 0.170941 + 0.888235i
\(78\) 0.854102 0.0967080
\(79\) −5.35410 − 9.27358i −0.602384 − 1.04336i −0.992459 − 0.122576i \(-0.960884\pi\)
0.390076 − 0.920783i \(-0.372449\pi\)
\(80\) 3.51722 − 6.09201i 0.393237 − 0.681107i
\(81\) 5.50000 − 9.52628i 0.611111 − 1.05848i
\(82\) 0.854102 + 1.47935i 0.0943198 + 0.163367i
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −10.3647 − 3.59045i −1.13089 − 0.391751i
\(85\) 16.7082 1.81226
\(86\) −1.52786 − 2.64634i −0.164754 − 0.285362i
\(87\) 5.00000 − 8.66025i 0.536056 − 0.928477i
\(88\) 2.20820 − 3.82472i 0.235395 − 0.407717i
\(89\) 1.11803 + 1.93649i 0.118511 + 0.205268i 0.919178 − 0.393842i \(-0.128854\pi\)
−0.800667 + 0.599110i \(0.795521\pi\)
\(90\) 1.70820 0.180061
\(91\) 2.00000 − 1.73205i 0.209657 − 0.181568i
\(92\) 6.97871 0.727581
\(93\) −5.59017 − 9.68246i −0.579674 − 1.00402i
\(94\) 0.281153 − 0.486971i 0.0289987 − 0.0502272i
\(95\) 3.35410 − 5.80948i 0.344124 − 0.596040i
\(96\) 4.63525 + 8.02850i 0.473084 + 0.819405i
\(97\) −17.4164 −1.76837 −0.884184 − 0.467139i \(-0.845285\pi\)
−0.884184 + 0.467139i \(0.845285\pi\)
\(98\) 2.48278 − 0.992377i 0.250799 − 0.100245i
\(99\) −6.00000 −0.603023
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 91.2.e.b.53.1 ✓ 4
3.2 odd 2 819.2.j.c.235.2 4
4.3 odd 2 1456.2.r.j.417.2 4
7.2 even 3 inner 91.2.e.b.79.1 yes 4
7.3 odd 6 637.2.a.e.1.2 2
7.4 even 3 637.2.a.f.1.2 2
7.5 odd 6 637.2.e.h.79.1 4
7.6 odd 2 637.2.e.h.508.1 4
13.12 even 2 1183.2.e.d.508.2 4
21.2 odd 6 819.2.j.c.352.2 4
21.11 odd 6 5733.2.a.v.1.1 2
21.17 even 6 5733.2.a.w.1.1 2
28.23 odd 6 1456.2.r.j.625.2 4
91.25 even 6 8281.2.a.z.1.1 2
91.38 odd 6 8281.2.a.ba.1.1 2
91.51 even 6 1183.2.e.d.170.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.e.b.53.1 ✓ 4 1.1 even 1 trivial
91.2.e.b.79.1 yes 4 7.2 even 3 inner
637.2.a.e.1.2 2 7.3 odd 6
637.2.a.f.1.2 2 7.4 even 3
637.2.e.h.79.1 4 7.5 odd 6
637.2.e.h.508.1 4 7.6 odd 2
819.2.j.c.235.2 4 3.2 odd 2
819.2.j.c.352.2 4 21.2 odd 6
1183.2.e.d.170.2 4 91.51 even 6
1183.2.e.d.508.2 4 13.12 even 2
1456.2.r.j.417.2 4 4.3 odd 2
1456.2.r.j.625.2 4 28.23 odd 6
5733.2.a.v.1.1 2 21.11 odd 6
5733.2.a.w.1.1 2 21.17 even 6
8281.2.a.z.1.1 2 91.25 even 6
8281.2.a.ba.1.1 2 91.38 odd 6