# 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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{7}^{\mathrm{new}}(5, [\chi])$$.