# Properties

 Label 25.7.c.c Level 25 Weight 7 Character orbit 25.c Analytic conductor 5.751 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 25.c (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.75135209050$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{201})$$ Defining polynomial: $$x^{4} + 101 x^{2} + 2500$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 5) 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.

 $$f(q)$$ $$=$$ $$q + ( 3 - 3 \beta_{1} + \beta_{3} ) q^{2} + ( -8 - 7 \beta_{1} + \beta_{2} ) q^{3} + ( -49 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} ) q^{4} + ( -148 + 10 \beta_{1} + 10 \beta_{2} - 10 \beta_{3} ) q^{6} + ( -132 + 132 \beta_{1} + 11 \beta_{3} ) q^{7} + ( -470 - 460 \beta_{1} + 10 \beta_{2} ) q^{8} + ( -516 \beta_{1} - 15 \beta_{2} - 15 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( 3 - 3 \beta_{1} + \beta_{3} ) q^{2} + ( -8 - 7 \beta_{1} + \beta_{2} ) q^{3} + ( -49 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} ) q^{4} + ( -148 + 10 \beta_{1} + 10 \beta_{2} - 10 \beta_{3} ) q^{6} + ( -132 + 132 \beta_{1} + 11 \beta_{3} ) q^{7} + ( -470 - 460 \beta_{1} + 10 \beta_{2} ) q^{8} + ( -516 \beta_{1} - 15 \beta_{2} - 15 \beta_{3} ) q^{9} + ( -188 - 75 \beta_{1} - 75 \beta_{2} + 75 \beta_{3} ) q^{11} + ( -932 + 932 \beta_{1} - 124 \beta_{3} ) q^{12} + ( -423 - 557 \beta_{1} - 134 \beta_{2} ) q^{13} + ( -418 \beta_{1} - 110 \beta_{2} - 110 \beta_{3} ) q^{14} + ( -364 + 170 \beta_{1} + 170 \beta_{2} - 170 \beta_{3} ) q^{16} + ( 973 - 973 \beta_{1} + 306 \beta_{3} ) q^{17} + ( -3 + 438 \beta_{1} + 441 \beta_{2} ) q^{18} + ( 1110 \beta_{1} + 570 \beta_{2} + 570 \beta_{3} ) q^{19} + ( 1012 - 55 \beta_{1} - 55 \beta_{2} + 55 \beta_{3} ) q^{21} + ( 6936 - 6936 \beta_{1} + 112 \beta_{3} ) q^{22} + ( 9912 + 9593 \beta_{1} - 319 \beta_{2} ) q^{23} + ( 7980 \beta_{1} - 540 \beta_{2} - 540 \beta_{3} ) q^{24} + ( 10862 + 155 \beta_{1} + 155 \beta_{2} - 155 \beta_{3} ) q^{26} + ( -8340 + 8340 \beta_{1} - 1020 \beta_{3} ) q^{27} + ( 1628 + 792 \beta_{1} - 836 \beta_{2} ) q^{28} + ( 10440 \beta_{1} - 1120 \beta_{2} - 1120 \beta_{3} ) q^{29} + ( -6568 - 1725 \beta_{1} - 1725 \beta_{2} + 1725 \beta_{3} ) q^{31} + ( 11988 - 11988 \beta_{1} - 404 \beta_{3} ) q^{32} + ( -5996 - 5134 \beta_{1} + 862 \beta_{2} ) q^{33} + ( -34853 \beta_{1} + 1585 \beta_{2} + 1585 \beta_{3} ) q^{34} + ( -12054 + 1845 \beta_{1} + 1845 \beta_{2} - 1845 \beta_{3} ) q^{36} + ( -35317 + 35317 \beta_{1} + 2796 \beta_{3} ) q^{37} + ( -55380 - 53640 \beta_{1} + 1740 \beta_{2} ) q^{38} + ( -6117 \beta_{1} + 515 \beta_{2} + 515 \beta_{3} ) q^{39} + ( -58508 + 5025 \beta_{1} + 5025 \beta_{2} - 5025 \beta_{3} ) q^{41} + ( 8536 - 8536 \beta_{1} + 1232 \beta_{3} ) q^{42} + ( 18832 + 17193 \beta_{1} - 1639 \beta_{2} ) q^{43} + ( -62488 \beta_{1} + 2360 \beta_{2} + 2360 \beta_{3} ) q^{44} + ( 91372 - 10550 \beta_{1} - 10550 \beta_{2} + 10550 \beta_{3} ) q^{46} + ( -2712 + 2712 \beta_{1} - 5009 \beta_{3} ) q^{47} + ( 19912 + 17168 \beta_{1} - 2744 \beta_{2} ) q^{48} + ( 67676 \beta_{1} - 3025 \beta_{2} - 3025 \beta_{3} ) q^{49} + ( -46168 + 3115 \beta_{1} + 3115 \beta_{2} - 3115 \beta_{3} ) q^{51} + ( 44158 - 44158 \beta_{1} + 1666 \beta_{3} ) q^{52} + ( 3667 + 11143 \beta_{1} + 7476 \beta_{2} ) q^{53} + ( 141660 \beta_{1} - 10380 \beta_{2} - 10380 \beta_{3} ) q^{54} + ( 113080 + 3740 \beta_{1} + 3740 \beta_{2} - 3740 \beta_{3} ) q^{56} + ( -52680 + 52680 \beta_{1} - 7440 \beta_{3} ) q^{57} + ( 146680 + 130640 \beta_{1} - 16040 \beta_{2} ) q^{58} + ( 20430 \beta_{1} + 1810 \beta_{2} + 1810 \beta_{3} ) q^{59} + ( -24388 - 3375 \beta_{1} - 3375 \beta_{2} + 3375 \beta_{3} ) q^{61} + ( 152796 - 152796 \beta_{1} + 332 \beta_{3} ) q^{62} + ( 82632 + 92433 \beta_{1} + 9801 \beta_{2} ) q^{63} + ( -32764 \beta_{1} + 22060 \beta_{2} + 22060 \beta_{3} ) q^{64} + ( -122176 + 7720 \beta_{1} + 7720 \beta_{2} - 7720 \beta_{3} ) q^{66} + ( 50248 - 50248 \beta_{1} + 27031 \beta_{3} ) q^{67} + ( -205542 - 182348 \beta_{1} + 23194 \beta_{2} ) q^{68} + ( -178347 \beta_{1} + 12145 \beta_{2} + 12145 \beta_{3} ) q^{69} + ( 337672 - 10125 \beta_{1} - 10125 \beta_{2} + 10125 \beta_{3} ) q^{71} + ( -220470 + 220470 \beta_{1} + 8790 \beta_{3} ) q^{72} + ( -126913 - 149857 \beta_{1} - 22944 \beta_{2} ) q^{73} + ( -97423 \beta_{1} - 29725 \beta_{2} - 29725 \beta_{3} ) q^{74} + ( -540840 + 22380 \beta_{1} + 22380 \beta_{2} - 22380 \beta_{3} ) q^{76} + ( 107316 - 107316 \beta_{1} - 23518 \beta_{3} ) q^{77} + ( -71396 - 62704 \beta_{1} + 8692 \beta_{2} ) q^{78} + ( 149040 \beta_{1} - 23120 \beta_{2} - 23120 \beta_{3} ) q^{79} + ( -129789 - 26415 \beta_{1} - 26415 \beta_{2} + 26415 \beta_{3} ) q^{81} + ( -678024 + 678024 \beta_{1} - 78608 \beta_{3} ) q^{82} + ( -103828 - 164607 \beta_{1} - 60779 \beta_{2} ) q^{83} + ( -102168 \beta_{1} + 7480 \beta_{2} + 7480 \beta_{3} ) q^{84} + ( 276892 - 22110 \beta_{1} - 22110 \beta_{2} + 22110 \beta_{3} ) q^{86} + ( 204480 - 204480 \beta_{1} + 27240 \beta_{3} ) q^{87} + ( 13360 + 80480 \beta_{1} + 67120 \beta_{2} ) q^{88} + ( -541680 \beta_{1} + 20640 \beta_{2} + 20640 \beta_{3} ) q^{89} + ( 259072 + 23815 \beta_{1} + 23815 \beta_{2} - 23815 \beta_{3} ) q^{91} + ( 694748 - 694748 \beta_{1} + 113156 \beta_{3} ) q^{92} + ( -119956 - 102374 \beta_{1} + 17582 \beta_{2} ) q^{93} + ( 504442 \beta_{1} - 12730 \beta_{2} - 12730 \beta_{3} ) q^{94} + ( -151408 + 9160 \beta_{1} + 9160 \beta_{2} - 9160 \beta_{3} ) q^{96} + ( 18063 - 18063 \beta_{1} + 6416 \beta_{3} ) q^{97} + ( 514603 + 431802 \beta_{1} - 82801 \beta_{2} ) q^{98} + ( 361833 \beta_{1} + 42645 \beta_{2} + 42645 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 10q^{2} - 30q^{3} - 552q^{6} - 550q^{7} - 1860q^{8} + O(q^{10})$$ $$4q + 10q^{2} - 30q^{3} - 552q^{6} - 550q^{7} - 1860q^{8} - 1052q^{11} - 3480q^{12} - 1960q^{13} - 776q^{16} + 3280q^{17} + 870q^{18} + 3828q^{21} + 27520q^{22} + 39010q^{23} + 44068q^{26} - 31320q^{27} + 4840q^{28} - 33172q^{31} + 48760q^{32} - 22260q^{33} - 40836q^{36} - 146860q^{37} - 218040q^{38} - 213932q^{41} + 31680q^{42} + 72050q^{43} + 323288q^{46} - 830q^{47} + 74160q^{48} - 172212q^{51} + 173300q^{52} + 29620q^{53} + 467280q^{56} - 195840q^{57} + 554640q^{58} - 111052q^{61} + 610520q^{62} + 350130q^{63} - 457824q^{66} + 146930q^{67} - 775780q^{68} + 1310188q^{71} - 899460q^{72} - 553540q^{73} - 2073840q^{76} + 476300q^{77} - 268200q^{78} - 624816q^{81} - 2554880q^{82} - 536870q^{83} + 1019128q^{86} + 763440q^{87} + 187680q^{88} + 1131548q^{91} + 2552680q^{92} - 444660q^{93} - 568992q^{96} + 59420q^{97} + 1892810q^{98} + O(q^{100})$$

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

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/25\mathbb{Z}\right)^\times$$.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 − 6.58872i 7.58872i 6.58872i − 7.58872i
−4.58872 4.58872i −0.411277 + 0.411277i 21.8872i 0 3.77447 −215.476 215.476i −394.113 + 394.113i 728.662i 0
7.2 9.58872 + 9.58872i −14.5887 + 14.5887i 119.887i 0 −279.774 −59.5240 59.5240i −535.887 + 535.887i 303.338i 0
18.1 −4.58872 + 4.58872i −0.411277 0.411277i 21.8872i 0 3.77447 −215.476 + 215.476i −394.113 394.113i 728.662i 0
18.2 9.58872 9.58872i −14.5887 14.5887i 119.887i 0 −279.774 −59.5240 + 59.5240i −535.887 535.887i 303.338i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.7.c.c 4
3.b odd 2 1 225.7.g.c 4
5.b even 2 1 5.7.c.a 4
5.c odd 4 1 5.7.c.a 4
5.c odd 4 1 inner 25.7.c.c 4
15.d odd 2 1 45.7.g.a 4
15.e even 4 1 45.7.g.a 4
15.e even 4 1 225.7.g.c 4
20.d odd 2 1 80.7.p.b 4
20.e even 4 1 80.7.p.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.7.c.a 4 5.b even 2 1
5.7.c.a 4 5.c odd 4 1
25.7.c.c 4 1.a even 1 1 trivial
25.7.c.c 4 5.c odd 4 1 inner
45.7.g.a 4 15.d odd 2 1
45.7.g.a 4 15.e even 4 1
80.7.p.b 4 20.d odd 2 1
80.7.p.b 4 20.e even 4 1
225.7.g.c 4 3.b odd 2 1
225.7.g.c 4 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 10 T_{2}^{3} + 50 T_{2}^{2} + 880 T_{2} + 7744$$ acting on $$S_{7}^{\mathrm{new}}(25, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 10 T + 50 T^{2} + 240 T^{3} - 6592 T^{4} + 15360 T^{5} + 204800 T^{6} - 2621440 T^{7} + 16777216 T^{8}$$
$3$ $$1 + 30 T + 450 T^{2} + 22230 T^{3} + 1098018 T^{4} + 16205670 T^{5} + 239148450 T^{6} + 11622614670 T^{7} + 282429536481 T^{8}$$
$5$ 1
$7$ $$1 + 550 T + 151250 T^{2} + 78815550 T^{3} + 40412328098 T^{4} + 9272570641950 T^{5} + 2093494689151250 T^{6} + 895627478850746950 T^{7} +$$$$19\!\cdots\!01$$$$T^{8}$$
$11$ $$( 1 + 526 T + 2481666 T^{2} + 931841086 T^{3} + 3138428376721 T^{4} )^{2}$$
$13$ $$1 + 1960 T + 1920800 T^{2} + 6864764760 T^{3} + 22780068732638 T^{4} + 33134908326450840 T^{5} + 44750961903261504800 T^{6} +$$$$22\!\cdots\!40$$$$T^{7} +$$$$54\!\cdots\!61$$$$T^{8}$$
$17$ $$1 - 3280 T + 5379200 T^{2} - 52715999280 T^{3} + 451561024170878 T^{4} - 1272436070024950320 T^{5} +$$$$31\!\cdots\!00$$$$T^{6} -$$$$46\!\cdots\!20$$$$T^{7} +$$$$33\!\cdots\!21$$$$T^{8}$$
$19$ $$1 - 55109524 T^{2} + 4864046077848966 T^{4} -$$$$12\!\cdots\!64$$$$T^{6} +$$$$48\!\cdots\!21$$$$T^{8}$$
$23$ $$1 - 39010 T + 760890050 T^{2} - 12796505733210 T^{3} + 182810834786595458 T^{4} -$$$$18\!\cdots\!90$$$$T^{5} +$$$$16\!\cdots\!50$$$$T^{6} -$$$$12\!\cdots\!90$$$$T^{7} +$$$$48\!\cdots\!41$$$$T^{8}$$
$29$ $$1 - 1657037284 T^{2} + 1284148562793102246 T^{4} -$$$$58\!\cdots\!44$$$$T^{6} +$$$$12\!\cdots\!81$$$$T^{8}$$
$31$ $$( 1 + 16586 T + 1245680586 T^{2} + 14720136053066 T^{3} + 787662783788549761 T^{4} )^{2}$$
$37$ $$1 + 146860 T + 10783929800 T^{2} + 657351006913860 T^{3} + 36420548331850749038 T^{4} +$$$$16\!\cdots\!40$$$$T^{5} +$$$$70\!\cdots\!00$$$$T^{6} +$$$$24\!\cdots\!40$$$$T^{7} +$$$$43\!\cdots\!61$$$$T^{8}$$
$41$ $$( 1 + 106966 T + 7285264146 T^{2} + 508099650242806 T^{3} + 22563490300366186081 T^{4} )^{2}$$
$43$ $$1 - 72050 T + 2595601250 T^{2} - 482755757677050 T^{3} + 89644137077771169698 T^{4} -$$$$30\!\cdots\!50$$$$T^{5} +$$$$10\!\cdots\!50$$$$T^{6} -$$$$18\!\cdots\!50$$$$T^{7} +$$$$15\!\cdots\!01$$$$T^{8}$$
$47$ $$1 + 830 T + 344450 T^{2} + 6853931089830 T^{3} +$$$$13\!\cdots\!18$$$$T^{4} +$$$$73\!\cdots\!70$$$$T^{5} +$$$$40\!\cdots\!50$$$$T^{6} +$$$$10\!\cdots\!70$$$$T^{7} +$$$$13\!\cdots\!81$$$$T^{8}$$
$53$ $$1 - 29620 T + 438672200 T^{2} - 493381118739420 T^{3} +$$$$52\!\cdots\!18$$$$T^{4} -$$$$10\!\cdots\!80$$$$T^{5} +$$$$21\!\cdots\!00$$$$T^{6} -$$$$32\!\cdots\!80$$$$T^{7} +$$$$24\!\cdots\!81$$$$T^{8}$$
$59$ $$1 - 166570372564 T^{2} +$$$$10\!\cdots\!86$$$$T^{4} -$$$$29\!\cdots\!84$$$$T^{6} +$$$$31\!\cdots\!61$$$$T^{8}$$
$61$ $$( 1 + 55526 T + 101522017266 T^{2} + 2860720306768886 T^{3} +$$$$26\!\cdots\!21$$$$T^{4} )^{2}$$
$67$ $$1 - 146930 T + 10794212450 T^{2} - 2898062262927930 T^{3} -$$$$42\!\cdots\!22$$$$T^{4} -$$$$26\!\cdots\!70$$$$T^{5} +$$$$88\!\cdots\!50$$$$T^{6} -$$$$10\!\cdots\!70$$$$T^{7} +$$$$66\!\cdots\!21$$$$T^{8}$$
$71$ $$( 1 - 655094 T + 342881964426 T^{2} - 83917727394943574 T^{3} +$$$$16\!\cdots\!41$$$$T^{4} )^{2}$$
$73$ $$1 + 553540 T + 153203265800 T^{2} + 75685034633171340 T^{3} +$$$$37\!\cdots\!58$$$$T^{4} +$$$$11\!\cdots\!60$$$$T^{5} +$$$$35\!\cdots\!00$$$$T^{6} +$$$$19\!\cdots\!60$$$$T^{7} +$$$$52\!\cdots\!41$$$$T^{8}$$
$79$ $$1 - 713041150084 T^{2} +$$$$23\!\cdots\!46$$$$T^{4} -$$$$42\!\cdots\!44$$$$T^{6} +$$$$34\!\cdots\!81$$$$T^{8}$$
$83$ $$1 + 536870 T + 144114698450 T^{2} - 4448869644102930 T^{3} -$$$$11\!\cdots\!22$$$$T^{4} -$$$$14\!\cdots\!70$$$$T^{5} +$$$$15\!\cdots\!50$$$$T^{6} +$$$$18\!\cdots\!30$$$$T^{7} +$$$$11\!\cdots\!21$$$$T^{8}$$
$89$ $$1 - 1229834859844 T^{2} +$$$$77\!\cdots\!26$$$$T^{4} -$$$$30\!\cdots\!24$$$$T^{6} +$$$$61\!\cdots\!41$$$$T^{8}$$
$97$ $$1 - 59420 T + 1765368200 T^{2} - 49275595300926420 T^{3} +$$$$13\!\cdots\!18$$$$T^{4} -$$$$41\!\cdots\!80$$$$T^{5} +$$$$12\!\cdots\!00$$$$T^{6} -$$$$34\!\cdots\!80$$$$T^{7} +$$$$48\!\cdots\!81$$$$T^{8}$$