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Properties

Label 175.10.b.c.99.3
Level $175$
Weight $10$
Character 175.99
Analytic conductor $90.131$
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] = [175,10,Mod(99,175)] 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("175.99"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 175.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,1440] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(90.1312713287\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.3
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 175.99
Dual form 175.10.b.c.99.2

$q$-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)
 
\(f(q)\) \(=\) \(q+9.17157i q^{2} +65.7351i q^{3} +427.882 q^{4} -602.894 q^{6} -2401.00i q^{7} +8620.20i q^{8} +15361.9 q^{9} +35089.6 q^{11} +28126.9i q^{12} -77401.4i q^{13} +22020.9 q^{14} +140015. q^{16} +229907. i q^{17} +140893. i q^{18} -16433.6 q^{19} +157830. q^{21} +321827. i q^{22} -2.57284e6i q^{23} -566649. q^{24} +709892. q^{26} +2.30368e6i q^{27} -1.02735e6i q^{28} +6.62817e6 q^{29} -8.17416e6 q^{31} +5.69770e6i q^{32} +2.30662e6i q^{33} -2.10861e6 q^{34} +6.57308e6 q^{36} -9.70272e6i q^{37} -150722. i q^{38} +5.08798e6 q^{39} +2.98108e7 q^{41} +1.44755e6i q^{42} -1.95343e7i q^{43} +1.50142e7 q^{44} +2.35970e7 q^{46} -5.93794e6i q^{47} +9.20389e6i q^{48} -5.76480e6 q^{49} -1.51130e7 q^{51} -3.31187e7i q^{52} -2.74263e7i q^{53} -2.11284e7 q^{54} +2.06971e7 q^{56} -1.08026e6i q^{57} +6.07908e7i q^{58} -5.24915e7 q^{59} +2.23282e7 q^{61} -7.49699e7i q^{62} -3.68839e7i q^{63} +1.94308e7 q^{64} -2.11553e7 q^{66} -2.74351e8i q^{67} +9.83733e7i q^{68} +1.69126e8 q^{69} -3.63673e8 q^{71} +1.32423e8i q^{72} +2.09245e7i q^{73} +8.89892e7 q^{74} -7.03163e6 q^{76} -8.42501e7i q^{77} +4.66648e7i q^{78} +2.65896e8 q^{79} +1.50936e8 q^{81} +2.73412e8i q^{82} -9.43764e6i q^{83} +6.75326e7 q^{84} +1.79160e8 q^{86} +4.35704e8i q^{87} +3.02479e8i q^{88} +6.64876e8 q^{89} -1.85841e8 q^{91} -1.10087e9i q^{92} -5.37329e8i q^{93} +5.44603e7 q^{94} -3.74539e8 q^{96} +1.20731e9i q^{97} -5.28723e7i q^{98} +5.39042e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 1440 q^{4} + 5904 q^{6} - 44856 q^{9} + 37132 q^{11} + 115248 q^{14} + 225536 q^{16} + 286552 q^{19} - 835548 q^{21} + 4583808 q^{24} + 639472 q^{26} + 23155108 q^{29} - 7907520 q^{31} - 8487888 q^{34}+ \cdots + 2326933080 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 9.17157i 0.405330i 0.979248 + 0.202665i \(0.0649603\pi\)
−0.979248 + 0.202665i \(0.935040\pi\)
\(3\) 65.7351i 0.468545i 0.972171 + 0.234273i \(0.0752708\pi\)
−0.972171 + 0.234273i \(0.924729\pi\)
\(4\) 427.882 0.835708
\(5\) 0 0
\(6\) −602.894 −0.189915
\(7\) − 2401.00i − 0.377964i
\(8\) 8620.20i 0.744067i
\(9\) 15361.9 0.780465
\(10\) 0 0
\(11\) 35089.6 0.722622 0.361311 − 0.932445i \(-0.382329\pi\)
0.361311 + 0.932445i \(0.382329\pi\)
\(12\) 28126.9i 0.391567i
\(13\) − 77401.4i − 0.751629i −0.926695 − 0.375815i \(-0.877363\pi\)
0.926695 − 0.375815i \(-0.122637\pi\)
\(14\) 22020.9 0.153200
\(15\) 0 0
\(16\) 140015. 0.534115
\(17\) 229907.i 0.667626i 0.942639 + 0.333813i \(0.108335\pi\)
−0.942639 + 0.333813i \(0.891665\pi\)
\(18\) 140893.i 0.316346i
\(19\) −16433.6 −0.0289295 −0.0144647 − 0.999895i \(-0.504604\pi\)
−0.0144647 + 0.999895i \(0.504604\pi\)
\(20\) 0 0
\(21\) 157830. 0.177093
\(22\) 321827.i 0.292900i
\(23\) − 2.57284e6i − 1.91707i −0.284975 − 0.958535i \(-0.591985\pi\)
0.284975 − 0.958535i \(-0.408015\pi\)
\(24\) −566649. −0.348629
\(25\) 0 0
\(26\) 709892. 0.304658
\(27\) 2.30368e6i 0.834228i
\(28\) − 1.02735e6i − 0.315868i
\(29\) 6.62817e6 1.74022 0.870108 − 0.492862i \(-0.164049\pi\)
0.870108 + 0.492862i \(0.164049\pi\)
\(30\) 0 0
\(31\) −8.17416e6 −1.58970 −0.794851 − 0.606805i \(-0.792451\pi\)
−0.794851 + 0.606805i \(0.792451\pi\)
\(32\) 5.69770e6i 0.960560i
\(33\) 2.30662e6i 0.338581i
\(34\) −2.10861e6 −0.270609
\(35\) 0 0
\(36\) 6.57308e6 0.652241
\(37\) − 9.70272e6i − 0.851110i −0.904933 − 0.425555i \(-0.860079\pi\)
0.904933 − 0.425555i \(-0.139921\pi\)
\(38\) − 150722.i − 0.0117260i
\(39\) 5.08798e6 0.352172
\(40\) 0 0
\(41\) 2.98108e7 1.64758 0.823789 − 0.566896i \(-0.191856\pi\)
0.823789 + 0.566896i \(0.191856\pi\)
\(42\) 1.44755e6i 0.0717813i
\(43\) − 1.95343e7i − 0.871343i −0.900106 − 0.435672i \(-0.856511\pi\)
0.900106 − 0.435672i \(-0.143489\pi\)
\(44\) 1.50142e7 0.603900
\(45\) 0 0
\(46\) 2.35970e7 0.777046
\(47\) − 5.93794e6i − 0.177499i −0.996054 − 0.0887494i \(-0.971713\pi\)
0.996054 − 0.0887494i \(-0.0282870\pi\)
\(48\) 9.20389e6i 0.250257i
\(49\) −5.76480e6 −0.142857
\(50\) 0 0
\(51\) −1.51130e7 −0.312813
\(52\) − 3.31187e7i − 0.628142i
\(53\) − 2.74263e7i − 0.477448i −0.971088 − 0.238724i \(-0.923271\pi\)
0.971088 − 0.238724i \(-0.0767291\pi\)
\(54\) −2.11284e7 −0.338138
\(55\) 0 0
\(56\) 2.06971e7 0.281231
\(57\) − 1.08026e6i − 0.0135548i
\(58\) 6.07908e7i 0.705362i
\(59\) −5.24915e7 −0.563969 −0.281984 − 0.959419i \(-0.590993\pi\)
−0.281984 + 0.959419i \(0.590993\pi\)
\(60\) 0 0
\(61\) 2.23282e7 0.206476 0.103238 − 0.994657i \(-0.467080\pi\)
0.103238 + 0.994657i \(0.467080\pi\)
\(62\) − 7.49699e7i − 0.644354i
\(63\) − 3.68839e7i − 0.294988i
\(64\) 1.94308e7 0.144771
\(65\) 0 0
\(66\) −2.11553e7 −0.137237
\(67\) − 2.74351e8i − 1.66330i −0.555302 − 0.831649i \(-0.687397\pi\)
0.555302 − 0.831649i \(-0.312603\pi\)
\(68\) 9.83733e7i 0.557940i
\(69\) 1.69126e8 0.898234
\(70\) 0 0
\(71\) −3.63673e8 −1.69843 −0.849216 − 0.528046i \(-0.822925\pi\)
−0.849216 + 0.528046i \(0.822925\pi\)
\(72\) 1.32423e8i 0.580719i
\(73\) 2.09245e7i 0.0862387i 0.999070 + 0.0431193i \(0.0137296\pi\)
−0.999070 + 0.0431193i \(0.986270\pi\)
\(74\) 8.89892e7 0.344980
\(75\) 0 0
\(76\) −7.03163e6 −0.0241766
\(77\) − 8.42501e7i − 0.273125i
\(78\) 4.66648e7i 0.142746i
\(79\) 2.65896e8 0.768051 0.384025 − 0.923323i \(-0.374538\pi\)
0.384025 + 0.923323i \(0.374538\pi\)
\(80\) 0 0
\(81\) 1.50936e8 0.389592
\(82\) 2.73412e8i 0.667813i
\(83\) − 9.43764e6i − 0.0218279i −0.999940 − 0.0109140i \(-0.996526\pi\)
0.999940 − 0.0109140i \(-0.00347409\pi\)
\(84\) 6.75326e7 0.147998
\(85\) 0 0
\(86\) 1.79160e8 0.353182
\(87\) 4.35704e8i 0.815369i
\(88\) 3.02479e8i 0.537679i
\(89\) 6.64876e8 1.12327 0.561637 − 0.827384i \(-0.310172\pi\)
0.561637 + 0.827384i \(0.310172\pi\)
\(90\) 0 0
\(91\) −1.85841e8 −0.284089
\(92\) − 1.10087e9i − 1.60211i
\(93\) − 5.37329e8i − 0.744847i
\(94\) 5.44603e7 0.0719456
\(95\) 0 0
\(96\) −3.74539e8 −0.450066
\(97\) 1.20731e9i 1.38467i 0.721575 + 0.692336i \(0.243418\pi\)
−0.721575 + 0.692336i \(0.756582\pi\)
\(98\) − 5.28723e7i − 0.0579043i
\(99\) 5.39042e8 0.563981
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 175.10.b.c.99.3 4
5.2 odd 4 35.10.a.b.1.2 ✓ 2
5.3 odd 4 175.10.a.c.1.1 2
5.4 even 2 inner 175.10.b.c.99.2 4
15.2 even 4 315.10.a.b.1.1 2
35.27 even 4 245.10.a.c.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.10.a.b.1.2 ✓ 2 5.2 odd 4
175.10.a.c.1.1 2 5.3 odd 4
175.10.b.c.99.2 4 5.4 even 2 inner
175.10.b.c.99.3 4 1.1 even 1 trivial
245.10.a.c.1.2 2 35.27 even 4
315.10.a.b.1.1 2 15.2 even 4