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Properties

Label 224.6.a.d.1.1
Level $224$
Weight $6$
Character 224.1
Self dual yes
Analytic conductor $35.926$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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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] = [224,6,Mod(1,224)] 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("224.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(224, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 224.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,14] 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: yes
Analytic conductor: \(35.9259756381\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{61}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 15 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(4.40512\) of defining polynomial
Character \(\chi\) \(=\) 224.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.810250 q^{3} +71.6717 q^{5} +49.0000 q^{7} -242.343 q^{9} -569.271 q^{11} +137.702 q^{13} -58.0720 q^{15} -418.657 q^{17} -2552.13 q^{19} -39.7022 q^{21} +127.036 q^{23} +2011.84 q^{25} +393.249 q^{27} +2312.40 q^{29} +3992.18 q^{31} +461.252 q^{33} +3511.92 q^{35} -3853.34 q^{37} -111.573 q^{39} -4940.82 q^{41} -13679.2 q^{43} -17369.2 q^{45} -27624.7 q^{47} +2401.00 q^{49} +339.216 q^{51} +37400.0 q^{53} -40800.7 q^{55} +2067.86 q^{57} -37000.0 q^{59} -3803.32 q^{61} -11874.8 q^{63} +9869.36 q^{65} +22454.7 q^{67} -102.931 q^{69} -55088.9 q^{71} -69256.6 q^{73} -1630.09 q^{75} -27894.3 q^{77} -40937.8 q^{79} +58570.8 q^{81} +19789.0 q^{83} -30005.8 q^{85} -1873.62 q^{87} +104148. q^{89} +6747.41 q^{91} -3234.66 q^{93} -182916. q^{95} -96649.8 q^{97} +137959. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 14 q^{3} + 34 q^{5} + 98 q^{7} - 266 q^{9} - 420 q^{11} - 490 q^{13} - 616 q^{15} - 1056 q^{17} - 1246 q^{19} + 686 q^{21} + 504 q^{23} + 306 q^{25} - 3556 q^{27} - 3904 q^{29} - 2044 q^{31} + 2672 q^{33}+ \cdots + 134428 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −0.810250 −0.0519775 −0.0259888 − 0.999662i \(-0.508273\pi\)
−0.0259888 + 0.999662i \(0.508273\pi\)
\(4\) 0 0
\(5\) 71.6717 1.28210 0.641052 − 0.767498i \(-0.278498\pi\)
0.641052 + 0.767498i \(0.278498\pi\)
\(6\) 0 0
\(7\) 49.0000 0.377964
\(8\) 0 0
\(9\) −242.343 −0.997298
\(10\) 0 0
\(11\) −569.271 −1.41853 −0.709264 − 0.704943i \(-0.750972\pi\)
−0.709264 + 0.704943i \(0.750972\pi\)
\(12\) 0 0
\(13\) 137.702 0.225987 0.112993 − 0.993596i \(-0.463956\pi\)
0.112993 + 0.993596i \(0.463956\pi\)
\(14\) 0 0
\(15\) −58.0720 −0.0666406
\(16\) 0 0
\(17\) −418.657 −0.351346 −0.175673 − 0.984449i \(-0.556210\pi\)
−0.175673 + 0.984449i \(0.556210\pi\)
\(18\) 0 0
\(19\) −2552.13 −1.62188 −0.810941 − 0.585128i \(-0.801044\pi\)
−0.810941 + 0.585128i \(0.801044\pi\)
\(20\) 0 0
\(21\) −39.7022 −0.0196457
\(22\) 0 0
\(23\) 127.036 0.0500734 0.0250367 − 0.999687i \(-0.492030\pi\)
0.0250367 + 0.999687i \(0.492030\pi\)
\(24\) 0 0
\(25\) 2011.84 0.643789
\(26\) 0 0
\(27\) 393.249 0.103815
\(28\) 0 0
\(29\) 2312.40 0.510584 0.255292 − 0.966864i \(-0.417828\pi\)
0.255292 + 0.966864i \(0.417828\pi\)
\(30\) 0 0
\(31\) 3992.18 0.746115 0.373058 − 0.927808i \(-0.378309\pi\)
0.373058 + 0.927808i \(0.378309\pi\)
\(32\) 0 0
\(33\) 461.252 0.0737316
\(34\) 0 0
\(35\) 3511.92 0.484589
\(36\) 0 0
\(37\) −3853.34 −0.462736 −0.231368 − 0.972866i \(-0.574320\pi\)
−0.231368 + 0.972866i \(0.574320\pi\)
\(38\) 0 0
\(39\) −111.573 −0.0117462
\(40\) 0 0
\(41\) −4940.82 −0.459029 −0.229514 − 0.973305i \(-0.573714\pi\)
−0.229514 + 0.973305i \(0.573714\pi\)
\(42\) 0 0
\(43\) −13679.2 −1.12821 −0.564103 − 0.825704i \(-0.690778\pi\)
−0.564103 + 0.825704i \(0.690778\pi\)
\(44\) 0 0
\(45\) −17369.2 −1.27864
\(46\) 0 0
\(47\) −27624.7 −1.82412 −0.912059 − 0.410058i \(-0.865508\pi\)
−0.912059 + 0.410058i \(0.865508\pi\)
\(48\) 0 0
\(49\) 2401.00 0.142857
\(50\) 0 0
\(51\) 339.216 0.0182621
\(52\) 0 0
\(53\) 37400.0 1.82886 0.914432 − 0.404740i \(-0.132638\pi\)
0.914432 + 0.404740i \(0.132638\pi\)
\(54\) 0 0
\(55\) −40800.7 −1.81870
\(56\) 0 0
\(57\) 2067.86 0.0843014
\(58\) 0 0
\(59\) −37000.0 −1.38379 −0.691897 − 0.721996i \(-0.743225\pi\)
−0.691897 + 0.721996i \(0.743225\pi\)
\(60\) 0 0
\(61\) −3803.32 −0.130869 −0.0654347 − 0.997857i \(-0.520843\pi\)
−0.0654347 + 0.997857i \(0.520843\pi\)
\(62\) 0 0
\(63\) −11874.8 −0.376943
\(64\) 0 0
\(65\) 9869.36 0.289738
\(66\) 0 0
\(67\) 22454.7 0.611111 0.305555 − 0.952174i \(-0.401158\pi\)
0.305555 + 0.952174i \(0.401158\pi\)
\(68\) 0 0
\(69\) −102.931 −0.00260269
\(70\) 0 0
\(71\) −55088.9 −1.29693 −0.648467 − 0.761243i \(-0.724590\pi\)
−0.648467 + 0.761243i \(0.724590\pi\)
\(72\) 0 0
\(73\) −69256.6 −1.52109 −0.760543 − 0.649287i \(-0.775067\pi\)
−0.760543 + 0.649287i \(0.775067\pi\)
\(74\) 0 0
\(75\) −1630.09 −0.0334625
\(76\) 0 0
\(77\) −27894.3 −0.536153
\(78\) 0 0
\(79\) −40937.8 −0.738000 −0.369000 − 0.929429i \(-0.620300\pi\)
−0.369000 + 0.929429i \(0.620300\pi\)
\(80\) 0 0
\(81\) 58570.8 0.991902
\(82\) 0 0
\(83\) 19789.0 0.315303 0.157652 − 0.987495i \(-0.449608\pi\)
0.157652 + 0.987495i \(0.449608\pi\)
\(84\) 0 0
\(85\) −30005.8 −0.450462
\(86\) 0 0
\(87\) −1873.62 −0.0265389
\(88\) 0 0
\(89\) 104148. 1.39373 0.696864 − 0.717203i \(-0.254578\pi\)
0.696864 + 0.717203i \(0.254578\pi\)
\(90\) 0 0
\(91\) 6747.41 0.0854149
\(92\) 0 0
\(93\) −3234.66 −0.0387812
\(94\) 0 0
\(95\) −182916. −2.07942
\(96\) 0 0
\(97\) −96649.8 −1.04297 −0.521485 − 0.853261i \(-0.674622\pi\)
−0.521485 + 0.853261i \(0.674622\pi\)
\(98\) 0 0
\(99\) 137959. 1.41469
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 224.6.a.d.1.1 yes 2
4.3 odd 2 224.6.a.c.1.2 ✓ 2
8.3 odd 2 448.6.a.y.1.1 2
8.5 even 2 448.6.a.s.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.6.a.c.1.2 ✓ 2 4.3 odd 2
224.6.a.d.1.1 yes 2 1.1 even 1 trivial
448.6.a.s.1.2 2 8.5 even 2
448.6.a.y.1.1 2 8.3 odd 2