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Properties

Label 175.6.a.h.1.3
Level $175$
Weight $6$
Character 175.1
Self dual yes
Analytic conductor $28.067$
Analytic rank $1$
Dimension $4$
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] = [175,6,Mod(1,175)] 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("175.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 175.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,1] 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: yes
Analytic conductor: \(28.0671684673\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 56x^{2} + 128x + 120 \) 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 1.3
Root \(3.41666\) of defining polynomial
Character \(\chi\) \(=\) 175.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.41666 q^{2} +18.8187 q^{3} -20.3264 q^{4} +64.2972 q^{6} +49.0000 q^{7} -178.782 q^{8} +111.144 q^{9} -628.373 q^{11} -382.517 q^{12} -1053.10 q^{13} +167.416 q^{14} +39.6086 q^{16} -347.901 q^{17} +379.741 q^{18} +2602.26 q^{19} +922.117 q^{21} -2146.94 q^{22} -1802.42 q^{23} -3364.44 q^{24} -3598.07 q^{26} -2481.36 q^{27} -995.994 q^{28} -4708.03 q^{29} -519.218 q^{31} +5856.34 q^{32} -11825.2 q^{33} -1188.66 q^{34} -2259.16 q^{36} +15343.6 q^{37} +8891.04 q^{38} -19817.9 q^{39} -7175.12 q^{41} +3150.56 q^{42} -7193.63 q^{43} +12772.6 q^{44} -6158.28 q^{46} -10399.3 q^{47} +745.382 q^{48} +2401.00 q^{49} -6547.06 q^{51} +21405.7 q^{52} -5098.91 q^{53} -8477.97 q^{54} -8760.30 q^{56} +48971.2 q^{57} -16085.8 q^{58} -9516.66 q^{59} -21511.4 q^{61} -1773.99 q^{62} +5446.05 q^{63} +18741.7 q^{64} -40402.6 q^{66} +521.317 q^{67} +7071.59 q^{68} -33919.3 q^{69} +23115.5 q^{71} -19870.5 q^{72} +16554.4 q^{73} +52424.0 q^{74} -52894.6 q^{76} -30790.3 q^{77} -67711.1 q^{78} +93816.3 q^{79} -73704.0 q^{81} -24514.9 q^{82} -66094.7 q^{83} -18743.3 q^{84} -24578.2 q^{86} -88599.1 q^{87} +112342. q^{88} -140245. q^{89} -51601.7 q^{91} +36636.8 q^{92} -9771.01 q^{93} -35530.9 q^{94} +110209. q^{96} +97104.1 q^{97} +8203.41 q^{98} -69839.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - 6 q^{3} - 15 q^{4} + 79 q^{6} + 196 q^{7} - 279 q^{8} - 206 q^{9} - 260 q^{11} + 165 q^{12} - 214 q^{13} + 49 q^{14} - 1247 q^{16} - 146 q^{17} + 688 q^{18} + 856 q^{19} - 294 q^{21} + 4267 q^{22}+ \cdots - 100868 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.41666 0.603986 0.301993 − 0.953310i \(-0.402348\pi\)
0.301993 + 0.953310i \(0.402348\pi\)
\(3\) 18.8187 1.20722 0.603610 − 0.797279i \(-0.293728\pi\)
0.603610 + 0.797279i \(0.293728\pi\)
\(4\) −20.3264 −0.635201
\(5\) 0 0
\(6\) 64.2972 0.729145
\(7\) 49.0000 0.377964
\(8\) −178.782 −0.987639
\(9\) 111.144 0.457383
\(10\) 0 0
\(11\) −628.373 −1.56580 −0.782899 − 0.622149i \(-0.786260\pi\)
−0.782899 + 0.622149i \(0.786260\pi\)
\(12\) −382.517 −0.766827
\(13\) −1053.10 −1.72826 −0.864131 − 0.503268i \(-0.832131\pi\)
−0.864131 + 0.503268i \(0.832131\pi\)
\(14\) 167.416 0.228285
\(15\) 0 0
\(16\) 39.6086 0.0386802
\(17\) −347.901 −0.291967 −0.145984 − 0.989287i \(-0.546635\pi\)
−0.145984 + 0.989287i \(0.546635\pi\)
\(18\) 379.741 0.276253
\(19\) 2602.26 1.65374 0.826868 − 0.562395i \(-0.190120\pi\)
0.826868 + 0.562395i \(0.190120\pi\)
\(20\) 0 0
\(21\) 922.117 0.456287
\(22\) −2146.94 −0.945721
\(23\) −1802.42 −0.710457 −0.355228 − 0.934780i \(-0.615597\pi\)
−0.355228 + 0.934780i \(0.615597\pi\)
\(24\) −3364.44 −1.19230
\(25\) 0 0
\(26\) −3598.07 −1.04385
\(27\) −2481.36 −0.655059
\(28\) −995.994 −0.240083
\(29\) −4708.03 −1.03955 −0.519774 − 0.854304i \(-0.673984\pi\)
−0.519774 + 0.854304i \(0.673984\pi\)
\(30\) 0 0
\(31\) −519.218 −0.0970387 −0.0485194 − 0.998822i \(-0.515450\pi\)
−0.0485194 + 0.998822i \(0.515450\pi\)
\(32\) 5856.34 1.01100
\(33\) −11825.2 −1.89026
\(34\) −1188.66 −0.176344
\(35\) 0 0
\(36\) −2259.16 −0.290530
\(37\) 15343.6 1.84257 0.921285 − 0.388888i \(-0.127141\pi\)
0.921285 + 0.388888i \(0.127141\pi\)
\(38\) 8891.04 0.998834
\(39\) −19817.9 −2.08639
\(40\) 0 0
\(41\) −7175.12 −0.666606 −0.333303 − 0.942820i \(-0.608163\pi\)
−0.333303 + 0.942820i \(0.608163\pi\)
\(42\) 3150.56 0.275591
\(43\) −7193.63 −0.593304 −0.296652 − 0.954986i \(-0.595870\pi\)
−0.296652 + 0.954986i \(0.595870\pi\)
\(44\) 12772.6 0.994596
\(45\) 0 0
\(46\) −6158.28 −0.429106
\(47\) −10399.3 −0.686688 −0.343344 − 0.939210i \(-0.611560\pi\)
−0.343344 + 0.939210i \(0.611560\pi\)
\(48\) 745.382 0.0466956
\(49\) 2401.00 0.142857
\(50\) 0 0
\(51\) −6547.06 −0.352469
\(52\) 21405.7 1.09779
\(53\) −5098.91 −0.249338 −0.124669 − 0.992198i \(-0.539787\pi\)
−0.124669 + 0.992198i \(0.539787\pi\)
\(54\) −8477.97 −0.395647
\(55\) 0 0
\(56\) −8760.30 −0.373292
\(57\) 48971.2 1.99643
\(58\) −16085.8 −0.627872
\(59\) −9516.66 −0.355922 −0.177961 − 0.984038i \(-0.556950\pi\)
−0.177961 + 0.984038i \(0.556950\pi\)
\(60\) 0 0
\(61\) −21511.4 −0.740192 −0.370096 − 0.928993i \(-0.620675\pi\)
−0.370096 + 0.928993i \(0.620675\pi\)
\(62\) −1773.99 −0.0586101
\(63\) 5446.05 0.172874
\(64\) 18741.7 0.571951
\(65\) 0 0
\(66\) −40402.6 −1.14169
\(67\) 521.317 0.0141878 0.00709390 − 0.999975i \(-0.497742\pi\)
0.00709390 + 0.999975i \(0.497742\pi\)
\(68\) 7071.59 0.185458
\(69\) −33919.3 −0.857678
\(70\) 0 0
\(71\) 23115.5 0.544198 0.272099 − 0.962269i \(-0.412282\pi\)
0.272099 + 0.962269i \(0.412282\pi\)
\(72\) −19870.5 −0.451729
\(73\) 16554.4 0.363585 0.181792 − 0.983337i \(-0.441810\pi\)
0.181792 + 0.983337i \(0.441810\pi\)
\(74\) 52424.0 1.11289
\(75\) 0 0
\(76\) −52894.6 −1.05045
\(77\) −30790.3 −0.591816
\(78\) −67711.1 −1.26015
\(79\) 93816.3 1.69126 0.845630 − 0.533769i \(-0.179225\pi\)
0.845630 + 0.533769i \(0.179225\pi\)
\(80\) 0 0
\(81\) −73704.0 −1.24818
\(82\) −24514.9 −0.402621
\(83\) −66094.7 −1.05310 −0.526552 − 0.850143i \(-0.676516\pi\)
−0.526552 + 0.850143i \(0.676516\pi\)
\(84\) −18743.3 −0.289834
\(85\) 0 0
\(86\) −24578.2 −0.358347
\(87\) −88599.1 −1.25496
\(88\) 112342. 1.54644
\(89\) −140245. −1.87677 −0.938387 − 0.345586i \(-0.887680\pi\)
−0.938387 + 0.345586i \(0.887680\pi\)
\(90\) 0 0
\(91\) −51601.7 −0.653221
\(92\) 36636.8 0.451282
\(93\) −9771.01 −0.117147
\(94\) −35530.9 −0.414750
\(95\) 0 0
\(96\) 110209. 1.22050
\(97\) 97104.1 1.04787 0.523936 − 0.851758i \(-0.324463\pi\)
0.523936 + 0.851758i \(0.324463\pi\)
\(98\) 8203.41 0.0862838
\(99\) −69839.9 −0.716169
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 175.6.a.h.1.3 yes 4
5.2 odd 4 175.6.b.g.99.6 8
5.3 odd 4 175.6.b.g.99.3 8
5.4 even 2 175.6.a.g.1.2 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.6.a.g.1.2 ✓ 4 5.4 even 2
175.6.a.h.1.3 yes 4 1.1 even 1 trivial
175.6.b.g.99.3 8 5.3 odd 4
175.6.b.g.99.6 8 5.2 odd 4