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Properties

Label 315.10.a.b.1.2
Level $315$
Weight $10$
Character 315.1
Self dual yes
Analytic conductor $162.236$
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] = [315,10,Mod(1,315)] 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("315.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(315, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 315.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,24] 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: \(162.236288392\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 315.1

$q$-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)
 
\(f(q)\) \(=\) \(q+14.8284 q^{2} -292.118 q^{4} -625.000 q^{5} +2401.00 q^{7} -11923.8 q^{8} -9267.77 q^{10} +16523.6 q^{11} +26311.4 q^{13} +35603.1 q^{14} -27246.9 q^{16} +144003. q^{17} -159710. q^{19} +182574. q^{20} +245019. q^{22} -2.07393e6 q^{23} +390625. q^{25} +390156. q^{26} -701375. q^{28} +4.94938e6 q^{29} +4.22040e6 q^{31} +5.70096e6 q^{32} +2.13533e6 q^{34} -1.50062e6 q^{35} -1.29081e7 q^{37} -2.36824e6 q^{38} +7.45238e6 q^{40} +2.87518e7 q^{41} +3.54825e7 q^{43} -4.82683e6 q^{44} -3.07532e7 q^{46} -5.95633e7 q^{47} +5.76480e6 q^{49} +5.79235e6 q^{50} -7.68602e6 q^{52} -2.31161e6 q^{53} -1.03272e7 q^{55} -2.86290e7 q^{56} +7.33915e7 q^{58} +1.68651e8 q^{59} -6.70167e7 q^{61} +6.25819e7 q^{62} +9.84867e7 q^{64} -1.64446e7 q^{65} -1.56259e8 q^{67} -4.20657e7 q^{68} -2.22519e7 q^{70} -6.95067e7 q^{71} -7.83438e7 q^{73} -1.91407e8 q^{74} +4.66540e7 q^{76} +3.96731e7 q^{77} -4.26957e8 q^{79} +1.70293e7 q^{80} +4.26344e8 q^{82} +5.31242e8 q^{83} -9.00016e7 q^{85} +5.26149e8 q^{86} -1.97024e8 q^{88} +1.14168e8 q^{89} +6.31736e7 q^{91} +6.05833e8 q^{92} -8.83231e8 q^{94} +9.98185e7 q^{95} -1.46573e9 q^{97} +8.54829e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 24 q^{2} - 720 q^{4} - 1250 q^{5} + 4802 q^{7} - 20544 q^{8} - 15000 q^{10} - 18566 q^{11} - 51090 q^{13} + 57624 q^{14} + 112768 q^{16} + 373910 q^{17} - 143276 q^{19} + 450000 q^{20} - 76808 q^{22}+ \cdots + 138355224 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 14.8284 0.655330 0.327665 − 0.944794i \(-0.393738\pi\)
0.327665 + 0.944794i \(0.393738\pi\)
\(3\) 0 0
\(4\) −292.118 −0.570542
\(5\) −625.000 −0.447214
\(6\) 0 0
\(7\) 2401.00 0.377964
\(8\) −11923.8 −1.02922
\(9\) 0 0
\(10\) −9267.77 −0.293073
\(11\) 16523.6 0.340280 0.170140 − 0.985420i \(-0.445578\pi\)
0.170140 + 0.985420i \(0.445578\pi\)
\(12\) 0 0
\(13\) 26311.4 0.255505 0.127752 − 0.991806i \(-0.459224\pi\)
0.127752 + 0.991806i \(0.459224\pi\)
\(14\) 35603.1 0.247691
\(15\) 0 0
\(16\) −27246.9 −0.103939
\(17\) 144003. 0.418167 0.209084 − 0.977898i \(-0.432952\pi\)
0.209084 + 0.977898i \(0.432952\pi\)
\(18\) 0 0
\(19\) −159710. −0.281151 −0.140576 − 0.990070i \(-0.544895\pi\)
−0.140576 + 0.990070i \(0.544895\pi\)
\(20\) 182574. 0.255154
\(21\) 0 0
\(22\) 245019. 0.222996
\(23\) −2.07393e6 −1.54533 −0.772663 − 0.634817i \(-0.781075\pi\)
−0.772663 + 0.634817i \(0.781075\pi\)
\(24\) 0 0
\(25\) 390625. 0.200000
\(26\) 390156. 0.167440
\(27\) 0 0
\(28\) −701375. −0.215645
\(29\) 4.94938e6 1.29945 0.649725 − 0.760169i \(-0.274884\pi\)
0.649725 + 0.760169i \(0.274884\pi\)
\(30\) 0 0
\(31\) 4.22040e6 0.820779 0.410389 − 0.911910i \(-0.365393\pi\)
0.410389 + 0.911910i \(0.365393\pi\)
\(32\) 5.70096e6 0.961110
\(33\) 0 0
\(34\) 2.13533e6 0.274038
\(35\) −1.50062e6 −0.169031
\(36\) 0 0
\(37\) −1.29081e7 −1.13228 −0.566142 − 0.824308i \(-0.691565\pi\)
−0.566142 + 0.824308i \(0.691565\pi\)
\(38\) −2.36824e6 −0.184247
\(39\) 0 0
\(40\) 7.45238e6 0.460283
\(41\) 2.87518e7 1.58905 0.794525 − 0.607231i \(-0.207720\pi\)
0.794525 + 0.607231i \(0.207720\pi\)
\(42\) 0 0
\(43\) 3.54825e7 1.58273 0.791363 − 0.611347i \(-0.209372\pi\)
0.791363 + 0.611347i \(0.209372\pi\)
\(44\) −4.82683e6 −0.194144
\(45\) 0 0
\(46\) −3.07532e7 −1.01270
\(47\) −5.95633e7 −1.78049 −0.890243 − 0.455485i \(-0.849466\pi\)
−0.890243 + 0.455485i \(0.849466\pi\)
\(48\) 0 0
\(49\) 5.76480e6 0.142857
\(50\) 5.79235e6 0.131066
\(51\) 0 0
\(52\) −7.68602e6 −0.145776
\(53\) −2.31161e6 −0.0402414 −0.0201207 − 0.999798i \(-0.506405\pi\)
−0.0201207 + 0.999798i \(0.506405\pi\)
\(54\) 0 0
\(55\) −1.03272e7 −0.152178
\(56\) −2.86290e7 −0.389010
\(57\) 0 0
\(58\) 7.33915e7 0.851569
\(59\) 1.68651e8 1.81198 0.905992 − 0.423296i \(-0.139127\pi\)
0.905992 + 0.423296i \(0.139127\pi\)
\(60\) 0 0
\(61\) −6.70167e7 −0.619725 −0.309863 − 0.950781i \(-0.600283\pi\)
−0.309863 + 0.950781i \(0.600283\pi\)
\(62\) 6.25819e7 0.537881
\(63\) 0 0
\(64\) 9.84867e7 0.733783
\(65\) −1.64446e7 −0.114265
\(66\) 0 0
\(67\) −1.56259e8 −0.947343 −0.473671 − 0.880702i \(-0.657072\pi\)
−0.473671 + 0.880702i \(0.657072\pi\)
\(68\) −4.20657e7 −0.238582
\(69\) 0 0
\(70\) −2.22519e7 −0.110771
\(71\) −6.95067e7 −0.324612 −0.162306 − 0.986740i \(-0.551893\pi\)
−0.162306 + 0.986740i \(0.551893\pi\)
\(72\) 0 0
\(73\) −7.83438e7 −0.322888 −0.161444 − 0.986882i \(-0.551615\pi\)
−0.161444 + 0.986882i \(0.551615\pi\)
\(74\) −1.91407e8 −0.742020
\(75\) 0 0
\(76\) 4.66540e7 0.160409
\(77\) 3.96731e7 0.128614
\(78\) 0 0
\(79\) −4.26957e8 −1.23328 −0.616641 − 0.787245i \(-0.711507\pi\)
−0.616641 + 0.787245i \(0.711507\pi\)
\(80\) 1.70293e7 0.0464828
\(81\) 0 0
\(82\) 4.26344e8 1.04135
\(83\) 5.31242e8 1.22869 0.614343 − 0.789039i \(-0.289421\pi\)
0.614343 + 0.789039i \(0.289421\pi\)
\(84\) 0 0
\(85\) −9.00016e7 −0.187010
\(86\) 5.26149e8 1.03721
\(87\) 0 0
\(88\) −1.97024e8 −0.350225
\(89\) 1.14168e8 0.192881 0.0964404 − 0.995339i \(-0.469254\pi\)
0.0964404 + 0.995339i \(0.469254\pi\)
\(90\) 0 0
\(91\) 6.31736e7 0.0965716
\(92\) 6.05833e8 0.881674
\(93\) 0 0
\(94\) −8.83231e8 −1.16681
\(95\) 9.98185e7 0.125735
\(96\) 0 0
\(97\) −1.46573e9 −1.68105 −0.840524 − 0.541774i \(-0.817753\pi\)
−0.840524 + 0.541774i \(0.817753\pi\)
\(98\) 8.54829e7 0.0936186
\(99\) 0 0
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 315.10.a.b.1.2 2
3.2 odd 2 35.10.a.b.1.1 ✓ 2
15.2 even 4 175.10.b.c.99.1 4
15.8 even 4 175.10.b.c.99.4 4
15.14 odd 2 175.10.a.c.1.2 2
21.20 even 2 245.10.a.c.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.10.a.b.1.1 ✓ 2 3.2 odd 2
175.10.a.c.1.2 2 15.14 odd 2
175.10.b.c.99.1 4 15.2 even 4
175.10.b.c.99.4 4 15.8 even 4
245.10.a.c.1.1 2 21.20 even 2
315.10.a.b.1.2 2 1.1 even 1 trivial