Properties

Label 5.35.c.a
Level 5
Weight 35
Character orbit 5.c
Analytic conductor 36.613
Analytic rank 0
Dimension 32
CM No

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Newspace parameters

Level: \( N \) = \( 5 \)
Weight: \( k \) = \( 35 \)
Character orbit: \([\chi]\) = 5.c (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(36.6128270213\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \(32q \) \(\mathstrut +\mathstrut 131070q^{2} \) \(\mathstrut -\mathstrut 78988260q^{3} \) \(\mathstrut -\mathstrut 730721770860q^{5} \) \(\mathstrut -\mathstrut 44043049959816q^{6} \) \(\mathstrut +\mathstrut 5337954235100q^{7} \) \(\mathstrut -\mathstrut 4536119823236220q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(32q \) \(\mathstrut +\mathstrut 131070q^{2} \) \(\mathstrut -\mathstrut 78988260q^{3} \) \(\mathstrut -\mathstrut 730721770860q^{5} \) \(\mathstrut -\mathstrut 44043049959816q^{6} \) \(\mathstrut +\mathstrut 5337954235100q^{7} \) \(\mathstrut -\mathstrut 4536119823236220q^{8} \) \(\mathstrut +\mathstrut 141196165961877590q^{10} \) \(\mathstrut +\mathstrut 365981602487621544q^{11} \) \(\mathstrut -\mathstrut 5313878568511975560q^{12} \) \(\mathstrut -\mathstrut 11000397968713344520q^{13} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!80\)\(q^{15} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!48\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(49\!\cdots\!60\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!90\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!20\)\(q^{20} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!16\)\(q^{21} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!40\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!80\)\(q^{23} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!00\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(57\!\cdots\!96\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(79\!\cdots\!60\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(88\!\cdots\!80\)\(q^{28} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!80\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(84\!\cdots\!04\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!80\)\(q^{32} \) \(\mathstrut +\mathstrut \)\(73\!\cdots\!80\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!60\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!32\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!20\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!80\)\(q^{38} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!84\)\(q^{41} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!40\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!00\)\(q^{43} \) \(\mathstrut -\mathstrut \)\(54\!\cdots\!20\)\(q^{45} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!24\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!60\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!20\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(49\!\cdots\!50\)\(q^{50} \) \(\mathstrut +\mathstrut \)\(87\!\cdots\!84\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!00\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(82\!\cdots\!40\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!20\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!20\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!80\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!80\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(49\!\cdots\!60\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!44\)\(q^{61} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!60\)\(q^{62} \) \(\mathstrut +\mathstrut \)\(86\!\cdots\!60\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!20\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!72\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!60\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!60\)\(q^{68} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!40\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!76\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!80\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!80\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!00\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(58\!\cdots\!60\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!00\)\(q^{77} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!00\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!60\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!32\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(48\!\cdots\!60\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(98\!\cdots\!40\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(54\!\cdots\!40\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!64\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!20\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!60\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!70\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(97\!\cdots\!04\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(58\!\cdots\!40\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!80\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!04\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!60\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!70\)\(q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −169510. 169510.i −198644. + 198644.i 4.02874e10i −3.95922e11 + 6.52168e11i 6.73444e10 1.10969e14 + 1.10969e14i 3.91696e15 3.91696e15i 1.66771e16i 1.77662e17 4.34363e16i
2.2 −151262. 151262.i −1.00065e8 + 1.00065e8i 2.85804e10i 1.61018e11 7.45755e11i 3.02721e13 −1.52632e14 1.52632e14i 1.72447e15 1.72447e15i 3.34887e15i −1.37160e17 + 8.84484e16i
2.3 −134834. 134834.i 1.19387e8 1.19387e8i 1.91805e10i 6.75906e11 3.53875e11i −3.21948e13 2.58783e14 + 2.58783e14i 2.69752e14 2.69752e14i 1.18293e16i −1.38849e17 4.34207e16i
2.4 −115430. 115430.i 1.39345e8 1.39345e8i 9.46824e9i −7.62275e11 3.18319e10i −3.21691e13 −2.97793e14 2.97793e14i −8.90153e14 + 8.90153e14i 2.21567e16i 8.43150e16 + 9.16637e16i
2.5 −85294.4 85294.4i −1.68126e8 + 1.68126e8i 2.62960e9i −5.92285e11 + 4.80911e11i 2.86804e13 2.04951e14 + 2.04951e14i −1.68964e15 + 1.68964e15i 3.98554e16i 9.15376e16 + 9.49962e15i
2.6 −83255.7 83255.7i −2.65055e7 + 2.65055e7i 3.31684e9i 5.91968e11 + 4.81301e11i 4.41347e12 −1.23465e14 1.23465e14i −1.70647e15 + 1.70647e15i 1.52721e16i −9.21369e15 8.93558e16i
2.7 −39637.5 39637.5i 937048. 937048.i 1.40376e10i −4.84629e11 5.89246e11i −7.42844e10 1.00289e14 + 1.00289e14i −1.23738e15 + 1.23738e15i 1.66754e16i −4.14676e15 + 4.25657e16i
2.8 8626.63 + 8626.63i 9.89110e7 9.89110e7i 1.70310e10i −8.06672e10 + 7.58663e11i 1.70654e12 1.10848e14 + 1.10848e14i 2.95125e14 2.95125e14i 2.88960e15i −7.24059e15 + 5.84882e15i
2.9 26483.0 + 26483.0i −1.33847e8 + 1.33847e8i 1.57772e10i 6.94260e11 3.16354e11i −7.08935e12 1.17241e13 + 1.17241e13i 8.72802e14 8.72802e14i 1.91529e16i 2.67641e16 + 1.00081e16i
2.10 36635.8 + 36635.8i 1.35857e8 1.35857e8i 1.44955e10i 6.50279e11 3.99016e11i 9.95447e12 −1.47937e14 1.47937e14i 1.16045e15 1.16045e15i 2.02371e16i 3.84418e16 + 9.20523e15i
2.11 67956.3 + 67956.3i −7.73822e7 + 7.73822e7i 7.94376e9i −6.92309e11 + 3.20600e11i −1.05172e13 −2.59708e14 2.59708e14i 1.70731e15 1.70731e15i 4.70118e15i −6.88336e16 2.52599e16i
2.12 109228. + 109228.i 1.53588e7 1.53588e7i 6.68164e9i 1.05031e11 7.55675e11i 3.35523e12 5.89338e13 + 5.89338e13i 1.14670e15 1.14670e15i 1.62054e16i 9.40132e16 7.10686e16i
2.13 117370. + 117370.i −6.15595e7 + 6.15595e7i 1.03717e10i 3.42977e11 + 6.81501e11i −1.44505e13 2.43299e14 + 2.43299e14i 7.99081e14 7.99081e14i 9.09803e15i −3.97325e16 + 1.20243e17i
2.14 136964. + 136964.i 1.30224e8 1.30224e8i 2.03383e10i −7.62514e11 + 2.54690e10i 3.56719e13 1.04083e14 + 1.04083e14i −4.32593e14 + 4.32593e14i 1.72393e16i −1.07925e17 1.00949e17i
2.15 167195. + 167195.i 4.67349e7 4.67349e7i 3.87285e10i 6.83351e11 + 3.39275e11i 1.56277e13 −2.57154e14 2.57154e14i −3.60283e15 + 3.60283e15i 1.23089e16i 5.75278e16 + 1.70978e17i
2.16 174299. + 174299.i −1.58564e8 + 1.58564e8i 4.35807e10i −4.99549e11 5.76652e11i −5.52754e13 3.74774e13 + 3.74774e13i −4.60166e15 + 4.60166e15i 3.36082e16i 1.34389e16 1.87581e17i
3.1 −169510. + 169510.i −198644. 198644.i 4.02874e10i −3.95922e11 6.52168e11i 6.73444e10 1.10969e14 1.10969e14i 3.91696e15 + 3.91696e15i 1.66771e16i 1.77662e17 + 4.34363e16i
3.2 −151262. + 151262.i −1.00065e8 1.00065e8i 2.85804e10i 1.61018e11 + 7.45755e11i 3.02721e13 −1.52632e14 + 1.52632e14i 1.72447e15 + 1.72447e15i 3.34887e15i −1.37160e17 8.84484e16i
3.3 −134834. + 134834.i 1.19387e8 + 1.19387e8i 1.91805e10i 6.75906e11 + 3.53875e11i −3.21948e13 2.58783e14 2.58783e14i 2.69752e14 + 2.69752e14i 1.18293e16i −1.38849e17 + 4.34207e16i
3.4 −115430. + 115430.i 1.39345e8 + 1.39345e8i 9.46824e9i −7.62275e11 + 3.18319e10i −3.21691e13 −2.97793e14 + 2.97793e14i −8.90153e14 8.90153e14i 2.21567e16i 8.43150e16 9.16637e16i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.16
Significant digits:
Format:

Inner twists

This newform does not have CM; other inner twists have not been computed.

Hecke kernels

There are no other newforms in \(S_{35}^{\mathrm{new}}(5, [\chi])\).