Properties

Label 5.7.c.a
Level 5
Weight 7
Character orbit 5.c
Analytic conductor 1.150
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 5 \)
Weight: \( k \) = \( 7 \)
Character orbit: \([\chi]\) = 5.c (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(1.1502704181\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{201})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -2 + 2 \beta_{1} + \beta_{3} ) q^{2} \) \( + ( 7 + 8 \beta_{1} + \beta_{2} ) q^{3} \) \( + ( -49 \beta_{1} - 5 \beta_{2} - 5 \beta_{3} ) q^{4} \) \( + ( -25 + 60 \beta_{1} + 10 \beta_{2} - 5 \beta_{3} ) q^{5} \) \( + ( -128 - 10 \beta_{1} - 10 \beta_{2} + 10 \beta_{3} ) q^{6} \) \( + ( 143 - 143 \beta_{1} + 11 \beta_{3} ) q^{7} \) \( + ( 460 + 470 \beta_{1} + 10 \beta_{2} ) q^{8} \) \( + ( -516 \beta_{1} + 15 \beta_{2} + 15 \beta_{3} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -2 + 2 \beta_{1} + \beta_{3} ) q^{2} \) \( + ( 7 + 8 \beta_{1} + \beta_{2} ) q^{3} \) \( + ( -49 \beta_{1} - 5 \beta_{2} - 5 \beta_{3} ) q^{4} \) \( + ( -25 + 60 \beta_{1} + 10 \beta_{2} - 5 \beta_{3} ) q^{5} \) \( + ( -128 - 10 \beta_{1} - 10 \beta_{2} + 10 \beta_{3} ) q^{6} \) \( + ( 143 - 143 \beta_{1} + 11 \beta_{3} ) q^{7} \) \( + ( 460 + 470 \beta_{1} + 10 \beta_{2} ) q^{8} \) \( + ( -516 \beta_{1} + 15 \beta_{2} + 15 \beta_{3} ) q^{9} \) \( + ( -1050 + 285 \beta_{1} - 65 \beta_{2} + 20 \beta_{3} ) q^{10} \) \( + ( -338 + 75 \beta_{1} + 75 \beta_{2} - 75 \beta_{3} ) q^{11} \) \( + ( 808 - 808 \beta_{1} - 124 \beta_{3} ) q^{12} \) \( + ( 557 + 423 \beta_{1} - 134 \beta_{2} ) q^{13} \) \( + ( -418 \beta_{1} + 110 \beta_{2} + 110 \beta_{3} ) q^{14} \) \( + ( -25 + 1270 \beta_{1} + 95 \beta_{2} + 90 \beta_{3} ) q^{15} \) \( + ( -24 - 170 \beta_{1} - 170 \beta_{2} + 170 \beta_{3} ) q^{16} \) \( + ( -667 + 667 \beta_{1} + 306 \beta_{3} ) q^{17} \) \( + ( -438 + 3 \beta_{1} + 441 \beta_{2} ) q^{18} \) \( + ( 1110 \beta_{1} - 570 \beta_{2} - 570 \beta_{3} ) q^{19} \) \( + ( 4700 - 6295 \beta_{1} + 105 \beta_{2} - 665 \beta_{3} ) q^{20} \) \( + ( 902 + 55 \beta_{1} + 55 \beta_{2} - 55 \beta_{3} ) q^{21} \) \( + ( -6824 + 6824 \beta_{1} + 112 \beta_{3} ) q^{22} \) \( + ( -9593 - 9912 \beta_{1} - 319 \beta_{2} ) q^{23} \) \( + ( 7980 \beta_{1} + 540 \beta_{2} + 540 \beta_{3} ) q^{24} \) \( + ( 8125 + 5175 \beta_{1} + 175 \beta_{2} + 1225 \beta_{3} ) q^{25} \) \( + ( 11172 - 155 \beta_{1} - 155 \beta_{2} + 155 \beta_{3} ) q^{26} \) \( + ( 7320 - 7320 \beta_{1} - 1020 \beta_{3} ) q^{27} \) \( + ( -792 - 1628 \beta_{1} - 836 \beta_{2} ) q^{28} \) \( + ( 10440 \beta_{1} + 1120 \beta_{2} + 1120 \beta_{3} ) q^{29} \) \( + ( -11800 - 13130 \beta_{1} - 1730 \beta_{2} - 10 \beta_{3} ) q^{30} \) \( + ( -10018 + 1725 \beta_{1} + 1725 \beta_{2} - 1725 \beta_{3} ) q^{31} \) \( + ( -12392 + 12392 \beta_{1} - 404 \beta_{3} ) q^{32} \) \( + ( 5134 + 5996 \beta_{1} + 862 \beta_{2} ) q^{33} \) \( + ( -34853 \beta_{1} - 1585 \beta_{2} - 1585 \beta_{3} ) q^{34} \) \( + ( -7425 + 16335 \beta_{1} + 110 \beta_{2} - 2255 \beta_{3} ) q^{35} \) \( + ( -8364 - 1845 \beta_{1} - 1845 \beta_{2} + 1845 \beta_{3} ) q^{36} \) \( + ( 38113 - 38113 \beta_{1} + 2796 \beta_{3} ) q^{37} \) \( + ( 53640 + 55380 \beta_{1} + 1740 \beta_{2} ) q^{38} \) \( + ( -6117 \beta_{1} - 515 \beta_{2} - 515 \beta_{3} ) q^{39} \) \( + ( -29500 + 28300 \beta_{1} + 6800 \beta_{2} + 2850 \beta_{3} ) q^{40} \) \( + ( -48458 - 5025 \beta_{1} - 5025 \beta_{2} + 5025 \beta_{3} ) q^{41} \) \( + ( -7304 + 7304 \beta_{1} + 1232 \beta_{3} ) q^{42} \) \( + ( -17193 - 18832 \beta_{1} - 1639 \beta_{2} ) q^{43} \) \( + ( -62488 \beta_{1} - 2360 \beta_{2} - 2360 \beta_{3} ) q^{44} \) \( + ( 19050 + 32145 \beta_{1} - 3630 \beta_{2} - 4635 \beta_{3} ) q^{45} \) \( + ( 70272 + 10550 \beta_{1} + 10550 \beta_{2} - 10550 \beta_{3} ) q^{46} \) \( + ( -2297 + 2297 \beta_{1} - 5009 \beta_{3} ) q^{47} \) \( + ( -17168 - 19912 \beta_{1} - 2744 \beta_{2} ) q^{48} \) \( + ( 67676 \beta_{1} + 3025 \beta_{2} + 3025 \beta_{3} ) q^{49} \) \( + ( -43750 - 125450 \beta_{1} - 9200 \beta_{2} + 4975 \beta_{3} ) q^{50} \) \( + ( -39938 - 3115 \beta_{1} - 3115 \beta_{2} + 3115 \beta_{3} ) q^{51} \) \( + ( -42492 + 42492 \beta_{1} + 1666 \beta_{3} ) q^{52} \) \( + ( -11143 - 3667 \beta_{1} + 7476 \beta_{2} ) q^{53} \) \( + ( 141660 \beta_{1} + 10380 \beta_{2} + 10380 \beta_{3} ) q^{54} \) \( + ( 120950 + 20595 \beta_{1} - 5 \beta_{2} + 6565 \beta_{3} ) q^{55} \) \( + ( 120560 - 3740 \beta_{1} - 3740 \beta_{2} + 3740 \beta_{3} ) q^{56} \) \( + ( 45240 - 45240 \beta_{1} - 7440 \beta_{3} ) q^{57} \) \( + ( -130640 - 146680 \beta_{1} - 16040 \beta_{2} ) q^{58} \) \( + ( 20430 \beta_{1} - 1810 \beta_{2} - 1810 \beta_{3} ) q^{59} \) \( + ( 144200 + 9460 \beta_{1} + 10860 \beta_{2} - 10880 \beta_{3} ) q^{60} \) \( + ( -31138 + 3375 \beta_{1} + 3375 \beta_{2} - 3375 \beta_{3} ) q^{61} \) \( + ( -152464 + 152464 \beta_{1} + 332 \beta_{3} ) q^{62} \) \( + ( -92433 - 82632 \beta_{1} + 9801 \beta_{2} ) q^{63} \) \( + ( -32764 \beta_{1} - 22060 \beta_{2} - 22060 \beta_{3} ) q^{64} \) \( + ( -108775 - 110380 \beta_{1} + 9695 \beta_{2} - 4585 \beta_{3} ) q^{65} \) \( + ( -106736 - 7720 \beta_{1} - 7720 \beta_{2} + 7720 \beta_{3} ) q^{66} \) \( + ( -23217 + 23217 \beta_{1} + 27031 \beta_{3} ) q^{67} \) \( + ( 182348 + 205542 \beta_{1} + 23194 \beta_{2} ) q^{68} \) \( + ( -178347 \beta_{1} - 12145 \beta_{2} - 12145 \beta_{3} ) q^{69} \) \( + ( -28600 + 168410 \beta_{1} - 9790 \beta_{2} - 330 \beta_{3} ) q^{70} \) \( + ( 317422 + 10125 \beta_{1} + 10125 \beta_{2} - 10125 \beta_{3} ) q^{71} \) \( + ( 229260 - 229260 \beta_{1} + 8790 \beta_{3} ) q^{72} \) \( + ( 149857 + 126913 \beta_{1} - 22944 \beta_{2} ) q^{73} \) \( + ( -97423 \beta_{1} + 29725 \beta_{2} + 29725 \beta_{3} ) q^{74} \) \( + ( -100625 + 109100 \beta_{1} - 275 \beta_{2} + 16200 \beta_{3} ) q^{75} \) \( + ( -496080 - 22380 \beta_{1} - 22380 \beta_{2} + 22380 \beta_{3} ) q^{76} \) \( + ( -130834 + 130834 \beta_{1} - 23518 \beta_{3} ) q^{77} \) \( + ( 62704 + 71396 \beta_{1} + 8692 \beta_{2} ) q^{78} \) \( + ( 149040 \beta_{1} + 23120 \beta_{2} + 23120 \beta_{3} ) q^{79} \) \( + ( -254400 - 94090 \beta_{1} - 7890 \beta_{2} - 10930 \beta_{3} ) q^{80} \) \( + ( -182619 + 26415 \beta_{1} + 26415 \beta_{2} - 26415 \beta_{3} ) q^{81} \) \( + ( 599416 - 599416 \beta_{1} - 78608 \beta_{3} ) q^{82} \) \( + ( 164607 + 103828 \beta_{1} - 60779 \beta_{2} ) q^{83} \) \( + ( -102168 \beta_{1} - 7480 \beta_{2} - 7480 \beta_{3} ) q^{84} \) \( + ( -322675 + 82810 \beta_{1} - 20165 \beta_{2} + 6945 \beta_{3} ) q^{85} \) \( + ( 232672 + 22110 \beta_{1} + 22110 \beta_{2} - 22110 \beta_{3} ) q^{86} \) \( + ( -177240 + 177240 \beta_{1} + 27240 \beta_{3} ) q^{87} \) \( + ( -80480 - 13360 \beta_{1} + 67120 \beta_{2} ) q^{88} \) \( + ( -541680 \beta_{1} - 20640 \beta_{2} - 20640 \beta_{3} ) q^{89} \) \( + ( 253350 + 419070 \beta_{1} - 10980 \beta_{2} + 22065 \beta_{3} ) q^{90} \) \( + ( 306702 - 23815 \beta_{1} - 23815 \beta_{2} + 23815 \beta_{3} ) q^{91} \) \( + ( -581592 + 581592 \beta_{1} + 113156 \beta_{3} ) q^{92} \) \( + ( 102374 + 119956 \beta_{1} + 17582 \beta_{2} ) q^{93} \) \( + ( 504442 \beta_{1} + 12730 \beta_{2} + 12730 \beta_{3} ) q^{94} \) \( + ( 201000 - 851550 \beta_{1} + 45450 \beta_{2} - 8850 \beta_{3} ) q^{95} \) \( + ( -133088 - 9160 \beta_{1} - 9160 \beta_{2} + 9160 \beta_{3} ) q^{96} \) \( + ( -11647 + 11647 \beta_{1} + 6416 \beta_{3} ) q^{97} \) \( + ( -431802 - 514603 \beta_{1} - 82801 \beta_{2} ) q^{98} \) \( + ( 361833 \beta_{1} - 42645 \beta_{2} - 42645 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 10q^{2} \) \(\mathstrut +\mathstrut 30q^{3} \) \(\mathstrut -\mathstrut 70q^{5} \) \(\mathstrut -\mathstrut 552q^{6} \) \(\mathstrut +\mathstrut 550q^{7} \) \(\mathstrut +\mathstrut 1860q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 10q^{2} \) \(\mathstrut +\mathstrut 30q^{3} \) \(\mathstrut -\mathstrut 70q^{5} \) \(\mathstrut -\mathstrut 552q^{6} \) \(\mathstrut +\mathstrut 550q^{7} \) \(\mathstrut +\mathstrut 1860q^{8} \) \(\mathstrut -\mathstrut 4370q^{10} \) \(\mathstrut -\mathstrut 1052q^{11} \) \(\mathstrut +\mathstrut 3480q^{12} \) \(\mathstrut +\mathstrut 1960q^{13} \) \(\mathstrut -\mathstrut 90q^{15} \) \(\mathstrut -\mathstrut 776q^{16} \) \(\mathstrut -\mathstrut 3280q^{17} \) \(\mathstrut -\mathstrut 870q^{18} \) \(\mathstrut +\mathstrut 20340q^{20} \) \(\mathstrut +\mathstrut 3828q^{21} \) \(\mathstrut -\mathstrut 27520q^{22} \) \(\mathstrut -\mathstrut 39010q^{23} \) \(\mathstrut +\mathstrut 30400q^{25} \) \(\mathstrut +\mathstrut 44068q^{26} \) \(\mathstrut +\mathstrut 31320q^{27} \) \(\mathstrut -\mathstrut 4840q^{28} \) \(\mathstrut -\mathstrut 50640q^{30} \) \(\mathstrut -\mathstrut 33172q^{31} \) \(\mathstrut -\mathstrut 48760q^{32} \) \(\mathstrut +\mathstrut 22260q^{33} \) \(\mathstrut -\mathstrut 24970q^{35} \) \(\mathstrut -\mathstrut 40836q^{36} \) \(\mathstrut +\mathstrut 146860q^{37} \) \(\mathstrut +\mathstrut 218040q^{38} \) \(\mathstrut -\mathstrut 110100q^{40} \) \(\mathstrut -\mathstrut 213932q^{41} \) \(\mathstrut -\mathstrut 31680q^{42} \) \(\mathstrut -\mathstrut 72050q^{43} \) \(\mathstrut +\mathstrut 78210q^{45} \) \(\mathstrut +\mathstrut 323288q^{46} \) \(\mathstrut +\mathstrut 830q^{47} \) \(\mathstrut -\mathstrut 74160q^{48} \) \(\mathstrut -\mathstrut 203350q^{50} \) \(\mathstrut -\mathstrut 172212q^{51} \) \(\mathstrut -\mathstrut 173300q^{52} \) \(\mathstrut -\mathstrut 29620q^{53} \) \(\mathstrut +\mathstrut 470660q^{55} \) \(\mathstrut +\mathstrut 467280q^{56} \) \(\mathstrut +\mathstrut 195840q^{57} \) \(\mathstrut -\mathstrut 554640q^{58} \) \(\mathstrut +\mathstrut 620280q^{60} \) \(\mathstrut -\mathstrut 111052q^{61} \) \(\mathstrut -\mathstrut 610520q^{62} \) \(\mathstrut -\mathstrut 350130q^{63} \) \(\mathstrut -\mathstrut 406540q^{65} \) \(\mathstrut -\mathstrut 457824q^{66} \) \(\mathstrut -\mathstrut 146930q^{67} \) \(\mathstrut +\mathstrut 775780q^{68} \) \(\mathstrut -\mathstrut 133320q^{70} \) \(\mathstrut +\mathstrut 1310188q^{71} \) \(\mathstrut +\mathstrut 899460q^{72} \) \(\mathstrut +\mathstrut 553540q^{73} \) \(\mathstrut -\mathstrut 435450q^{75} \) \(\mathstrut -\mathstrut 2073840q^{76} \) \(\mathstrut -\mathstrut 476300q^{77} \) \(\mathstrut +\mathstrut 268200q^{78} \) \(\mathstrut -\mathstrut 1011520q^{80} \) \(\mathstrut -\mathstrut 624816q^{81} \) \(\mathstrut +\mathstrut 2554880q^{82} \) \(\mathstrut +\mathstrut 536870q^{83} \) \(\mathstrut -\mathstrut 1344920q^{85} \) \(\mathstrut +\mathstrut 1019128q^{86} \) \(\mathstrut -\mathstrut 763440q^{87} \) \(\mathstrut -\mathstrut 187680q^{88} \) \(\mathstrut +\mathstrut 947310q^{90} \) \(\mathstrut +\mathstrut 1131548q^{91} \) \(\mathstrut -\mathstrut 2552680q^{92} \) \(\mathstrut +\mathstrut 444660q^{93} \) \(\mathstrut +\mathstrut 912600q^{95} \) \(\mathstrut -\mathstrut 568992q^{96} \) \(\mathstrut -\mathstrut 59420q^{97} \) \(\mathstrut -\mathstrut 1892810q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut +\mathstrut \) \(101\) \(x^{2}\mathstrut +\mathstrut \) \(2500\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 51 \nu \)\()/50\)
\(\beta_{2}\)\(=\)\( \nu^{2} + \nu + 51 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} - 50 \nu^{2} + 101 \nu - 2550 \)\()/50\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut -\mathstrut \) \(102\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(51\) \(\beta_{3}\mathstrut -\mathstrut \) \(51\) \(\beta_{2}\mathstrut +\mathstrut \) \(151\) \(\beta_{1}\)\()/2\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
6.58872i
7.58872i
6.58872i
7.58872i
−9.58872 9.58872i 14.5887 14.5887i 119.887i 88.8309 87.9436i −279.774 59.5240 + 59.5240i 535.887 535.887i 303.338i −1695.04 8.50745i
2.2 4.58872 + 4.58872i 0.411277 0.411277i 21.8872i −123.831 17.0564i 3.77447 215.476 + 215.476i 394.113 394.113i 728.662i −489.959 646.493i
3.1 −9.58872 + 9.58872i 14.5887 + 14.5887i 119.887i 88.8309 + 87.9436i −279.774 59.5240 59.5240i 535.887 + 535.887i 303.338i −1695.04 + 8.50745i
3.2 4.58872 4.58872i 0.411277 + 0.411277i 21.8872i −123.831 + 17.0564i 3.77447 215.476 215.476i 394.113 + 394.113i 728.662i −489.959 + 646.493i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
5.c Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{7}^{\mathrm{new}}(5, [\chi])\).