# Properties

 Label 525.2.r.h Level 525 Weight 2 Character orbit 525.r Analytic conductor 4.192 Analytic rank 0 Dimension 16 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 525.r (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{12} q^{3} + ( 1 + \beta_{3} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{4} -\beta_{3} q^{6} + ( \beta_{10} + \beta_{11} + \beta_{15} ) q^{7} + ( 2 \beta_{1} - \beta_{8} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{8} -\beta_{5} q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{12} q^{3} + ( 1 + \beta_{3} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{4} -\beta_{3} q^{6} + ( \beta_{10} + \beta_{11} + \beta_{15} ) q^{7} + ( 2 \beta_{1} - \beta_{8} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{8} -\beta_{5} q^{9} + ( -3 + \beta_{2} - \beta_{4} - 3 \beta_{5} + \beta_{9} ) q^{11} + ( -\beta_{1} + 2 \beta_{8} - \beta_{11} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{12} + ( -\beta_{1} + \beta_{8} + \beta_{11} - \beta_{12} + \beta_{14} - \beta_{15} ) q^{13} + ( 2 - \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} ) q^{14} + ( 1 - 2 \beta_{2} - \beta_{3} + \beta_{4} + 5 \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{9} ) q^{16} + ( -\beta_{11} - \beta_{12} - \beta_{13} - \beta_{15} ) q^{17} + \beta_{11} q^{18} + ( -\beta_{2} + 2 \beta_{3} + 3 \beta_{4} + \beta_{6} + \beta_{9} ) q^{19} + ( 1 - \beta_{3} - \beta_{7} ) q^{21} + ( -3 \beta_{1} - 3 \beta_{8} - \beta_{10} + 2 \beta_{11} + 3 \beta_{12} + \beta_{13} - \beta_{14} ) q^{22} + ( 2 \beta_{1} + 2 \beta_{11} - 2 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{23} + ( -1 + \beta_{2} + 2 \beta_{4} - \beta_{5} + \beta_{9} ) q^{24} + ( 3 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} ) q^{26} + ( -\beta_{8} + \beta_{12} ) q^{27} + ( 4 \beta_{1} + \beta_{8} + \beta_{10} - 2 \beta_{11} + 4 \beta_{12} - \beta_{13} ) q^{28} -4 q^{29} + ( -3 + \beta_{2} + \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} ) q^{31} + ( -2 \beta_{1} + 2 \beta_{10} - 3 \beta_{11} + 4 \beta_{12} - 2 \beta_{14} ) q^{32} + ( -\beta_{1} - 3 \beta_{8} + \beta_{10} + \beta_{11} + \beta_{14} + \beta_{15} ) q^{33} + ( -2 - 2 \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{6} - 3 \beta_{7} + 3 \beta_{9} ) q^{34} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{7} - \beta_{9} ) q^{36} + ( -3 \beta_{8} + 2 \beta_{11} - 2 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{37} + ( 2 \beta_{1} - 2 \beta_{10} - 2 \beta_{11} - 7 \beta_{12} - \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{38} + ( \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} ) q^{39} + ( 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{9} ) q^{41} + ( -2 \beta_{1} + 2 \beta_{8} + \beta_{12} + \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{42} + ( \beta_{1} - 2 \beta_{10} - 3 \beta_{11} + 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{43} + ( 1 + 3 \beta_{2} - \beta_{3} - 4 \beta_{4} - 7 \beta_{5} - 3 \beta_{6} - \beta_{7} - 4 \beta_{9} ) q^{44} + ( 2 - 2 \beta_{2} - 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{9} ) q^{46} + ( 3 \beta_{1} - \beta_{8} - \beta_{10} + \beta_{11} - 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{47} + ( -\beta_{1} + 5 \beta_{8} - 2 \beta_{10} - \beta_{11} - 5 \beta_{12} + 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{48} + ( 1 + \beta_{2} - \beta_{3} - 3 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} ) q^{49} + ( -1 + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{51} + ( \beta_{1} - \beta_{10} + 2 \beta_{11} - 10 \beta_{12} + 2 \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{52} + ( 3 \beta_{11} + 3 \beta_{12} - \beta_{13} - \beta_{15} ) q^{53} + ( -\beta_{3} - \beta_{4} ) q^{54} + ( 4 - 2 \beta_{2} - 4 \beta_{3} + \beta_{4} + 12 \beta_{5} + 2 \beta_{6} + 2 \beta_{9} ) q^{56} + ( 2 \beta_{1} - \beta_{10} - 3 \beta_{11} + \beta_{13} - \beta_{15} ) q^{57} -4 \beta_{1} q^{58} + ( 5 - \beta_{2} - 2 \beta_{3} + \beta_{4} + 3 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{59} -5 \beta_{5} q^{61} + ( 12 \beta_{8} - 4 \beta_{10} - 4 \beta_{11} - 12 \beta_{12} + 4 \beta_{13} - 4 \beta_{15} ) q^{62} + ( \beta_{11} + \beta_{15} ) q^{63} + ( -3 - \beta_{2} - 7 \beta_{3} + \beta_{4} - \beta_{7} + \beta_{9} ) q^{64} + ( -3 - \beta_{2} - 3 \beta_{4} - 3 \beta_{5} - \beta_{9} ) q^{66} + ( \beta_{1} - \beta_{10} + \beta_{14} ) q^{67} + ( -7 \beta_{1} + \beta_{8} - \beta_{10} - \beta_{11} - \beta_{14} - \beta_{15} ) q^{68} + ( 2 - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{7} + 2 \beta_{9} ) q^{69} + ( 8 - 2 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} - 2 \beta_{7} + 2 \beta_{9} ) q^{71} + ( 2 \beta_{1} - \beta_{8} + \beta_{10} + \beta_{11} + \beta_{14} + \beta_{15} ) q^{72} + ( -2 \beta_{11} - 3 \beta_{12} + 2 \beta_{13} + 2 \beta_{15} ) q^{73} + ( -4 - 2 \beta_{2} - 2 \beta_{3} - 7 \beta_{4} - 6 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{9} ) q^{74} + ( -7 - \beta_{2} + 6 \beta_{3} - \beta_{6} ) q^{76} + ( -\beta_{1} + 3 \beta_{8} - \beta_{10} + 5 \beta_{11} - 2 \beta_{12} - 4 \beta_{13} + 3 \beta_{14} + \beta_{15} ) q^{77} + ( 3 \beta_{1} - 4 \beta_{8} - 3 \beta_{11} + 4 \beta_{12} ) q^{78} + ( -1 + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{79} + ( -1 - \beta_{5} ) q^{81} + ( 3 \beta_{1} + 3 \beta_{8} + \beta_{10} - \beta_{11} + 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{82} + ( -3 \beta_{1} - \beta_{8} + \beta_{10} + 4 \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{83} + ( 1 + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} - \beta_{6} ) q^{84} + ( -2 + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{86} -4 \beta_{12} q^{87} + ( -2 \beta_{1} + 2 \beta_{10} + 9 \beta_{11} - 5 \beta_{12} + \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{88} + ( 1 - \beta_{2} - \beta_{3} + 3 \beta_{5} + \beta_{6} - \beta_{7} ) q^{89} + ( 5 + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - 5 \beta_{6} + \beta_{7} - 2 \beta_{9} ) q^{91} + ( 2 \beta_{1} - 18 \beta_{8} + 4 \beta_{10} + 2 \beta_{11} + 18 \beta_{12} - 4 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{92} + ( 2 \beta_{1} - 2 \beta_{8} + \beta_{10} + \beta_{13} ) q^{93} + ( 5 - \beta_{2} - 5 \beta_{4} + 5 \beta_{5} - \beta_{9} ) q^{94} + ( 2 \beta_{2} + 3 \beta_{3} + \beta_{4} - 4 \beta_{5} - 2 \beta_{6} - 2 \beta_{9} ) q^{96} + ( 2 \beta_{1} - 3 \beta_{8} + 2 \beta_{10} + 3 \beta_{12} - 2 \beta_{13} + 2 \beta_{15} ) q^{97} + ( 2 \beta_{1} - 4 \beta_{8} + 2 \beta_{10} - \beta_{11} + 12 \beta_{12} - 2 \beta_{14} ) q^{98} + ( -2 + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} + \beta_{9} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 14q^{4} - 4q^{6} + 8q^{9} + O(q^{10})$$ $$16q + 14q^{4} - 4q^{6} + 8q^{9} - 16q^{11} + 24q^{14} - 34q^{16} + 6q^{19} + 4q^{21} - 6q^{24} + 38q^{26} - 64q^{29} - 18q^{31} - 56q^{34} + 28q^{36} + 14q^{39} + 16q^{41} + 52q^{44} + 12q^{46} + 42q^{49} - 12q^{51} - 2q^{54} - 42q^{56} + 20q^{59} + 40q^{61} - 84q^{64} - 24q^{66} + 8q^{69} + 88q^{71} - 42q^{74} - 92q^{76} + 16q^{79} - 8q^{81} + 36q^{84} - 24q^{86} - 20q^{89} + 42q^{91} + 44q^{94} + 34q^{96} - 32q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 15 x^{14} + 158 x^{12} - 843 x^{10} + 3258 x^{8} - 4947 x^{6} + 5489 x^{4} - 1296 x^{2} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-62937 \nu^{14} + 791239 \nu^{12} - 7673262 \nu^{10} + 29204099 \nu^{8} - 68243802 \nu^{6} - 255473269 \nu^{4} + 868504023 \nu^{2} - 741055184$$$$)/ 245379408$$ $$\beta_{3}$$ $$=$$ $$($$$$-288167 \nu^{14} + 4051357 \nu^{12} - 40373487 \nu^{10} + 188912823 \nu^{8} - 628102215 \nu^{6} + 348548739 \nu^{4} - 82510384 \nu^{2} - 306634312$$$$)/ 644120946$$ $$\beta_{4}$$ $$=$$ $$($$$$1037621 \nu^{14} - 15868765 \nu^{12} + 166205130 \nu^{10} - 885323493 \nu^{8} + 3314176350 \nu^{6} - 4613443065 \nu^{4} + 3721195993 \nu^{2} - 227006720$$$$)/ 1288241892$$ $$\beta_{5}$$ $$=$$ $$($$$$1708867 \nu^{14} - 25153821 \nu^{12} + 263454282 \nu^{10} - 1373439057 \nu^{8} + 5253351390 \nu^{6} - 7312838841 \nu^{4} + 8800380035 \nu^{2} - 2077487664$$$$)/ 1717655856$$ $$\beta_{6}$$ $$=$$ $$($$$$-2751857 \nu^{14} + 38259711 \nu^{12} - 393558942 \nu^{10} + 1888414411 \nu^{8} - 6858770730 \nu^{6} + 5649659155 \nu^{4} - 6993607393 \nu^{2} - 3459116192$$$$)/ 1717655856$$ $$\beta_{7}$$ $$=$$ $$($$$$-9945497 \nu^{14} + 154655863 \nu^{12} - 1657786662 \nu^{10} + 9304722867 \nu^{8} - 37439086770 \nu^{6} + 68016698715 \nu^{4} - 78810687601 \nu^{2} + 22301372336$$$$)/ 5152967568$$ $$\beta_{8}$$ $$=$$ $$($$$$886745 \nu^{15} - 12263554 \nu^{13} + 124236945 \nu^{11} - 581320905 \nu^{9} + 2003691717 \nu^{7} - 1072551165 \nu^{5} + 253900240 \nu^{3} + 2571974473 \nu$$$$)/ 1288241892$$ $$\beta_{9}$$ $$=$$ $$($$$$-8142721 \nu^{14} + 123758843 \nu^{12} - 1302663054 \nu^{10} + 7014096807 \nu^{8} - 26865475554 \nu^{6} + 41077574727 \nu^{4} - 37852679129 \nu^{2} + 9670329352$$$$)/ 2576483784$$ $$\beta_{10}$$ $$=$$ $$($$$$2706149 \nu^{15} - 45046027 \nu^{13} + 488786070 \nu^{11} - 2912604567 \nu^{9} + 11846264610 \nu^{7} - 24619823295 \nu^{5} + 25536879829 \nu^{3} - 18897225584 \nu$$$$)/ 2944552896$$ $$\beta_{11}$$ $$=$$ $$($$$$-1708867 \nu^{15} + 25153821 \nu^{13} - 263454282 \nu^{11} + 1373439057 \nu^{9} - 5253351390 \nu^{7} + 7312838841 \nu^{5} - 8800380035 \nu^{3} + 2077487664 \nu$$$$)/ 1717655856$$ $$\beta_{12}$$ $$=$$ $$($$$$-38329289 \nu^{15} + 584160679 \nu^{13} - 6185671086 \nu^{11} + 33603542211 \nu^{9} - 130922033898 \nu^{7} + 209714263563 \nu^{5} - 221543026969 \nu^{3} + 52315090832 \nu$$$$)/ 20611870272$$ $$\beta_{13}$$ $$=$$ $$($$$$-58035667 \nu^{15} + 852889805 \nu^{13} - 8898554154 \nu^{11} + 46016072625 \nu^{9} - 172692898302 \nu^{7} + 219622750473 \nu^{5} - 189951485651 \nu^{3} - 79171609568 \nu$$$$)/ 20611870272$$ $$\beta_{14}$$ $$=$$ $$($$$$-126993521 \nu^{15} + 1897063711 \nu^{13} - 19953851358 \nu^{11} + 105905971323 \nu^{9} - 407836732602 \nu^{7} + 603856761795 \nu^{5} - 650515145473 \nu^{3} + 42923508464 \nu$$$$)/ 20611870272$$ $$\beta_{15}$$ $$=$$ $$($$$$68497297 \nu^{15} - 1000621895 \nu^{13} + 10437980742 \nu^{11} - 53765247483 \nu^{9} + 203160524994 \nu^{7} - 264738896163 \nu^{5} + 289888751657 \nu^{3} - 6421385992 \nu$$$$)/ 10305935136$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} + \beta_{6} + 4 \beta_{5} + \beta_{3} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{14} - \beta_{13} + \beta_{12} - 5 \beta_{11} + \beta_{10} - \beta_{8} + 6 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$7 \beta_{9} - \beta_{7} + 8 \beta_{6} + 25 \beta_{5} + 7 \beta_{4} - \beta_{3} - 8 \beta_{2} + 1$$ $$\nu^{5}$$ $$=$$ $$-8 \beta_{15} - 2 \beta_{14} - 8 \beta_{13} + 12 \beta_{12} - 39 \beta_{11} + 2 \beta_{10} - 2 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$57 \beta_{9} - 57 \beta_{7} + 10 \beta_{6} + 57 \beta_{4} - 63 \beta_{3} - 47 \beta_{2} - 109$$ $$\nu^{7}$$ $$=$$ $$-83 \beta_{15} - 83 \beta_{14} + 26 \beta_{13} - 83 \beta_{11} - 57 \beta_{10} + 121 \beta_{8} - 286 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$83 \beta_{9} - 317 \beta_{7} - 317 \beta_{6} - 1123 \beta_{5} + 173 \beta_{4} - 317 \beta_{3} + 83 \beta_{2} - 806$$ $$\nu^{9}$$ $$=$$ $$-256 \beta_{15} - 400 \beta_{14} + 656 \beta_{13} - 1092 \beta_{12} + 1101 \beta_{11} - 656 \beta_{10} + 1092 \beta_{8} - 1757 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-2157 \beta_{9} + 656 \beta_{7} - 2813 \beta_{6} - 7684 \beta_{5} - 1209 \beta_{4} + 1604 \beta_{3} + 2813 \beta_{2} - 656$$ $$\nu^{11}$$ $$=$$ $$2813 \beta_{15} + 2260 \beta_{14} + 2813 \beta_{13} - 9229 \beta_{12} + 11998 \beta_{11} - 2260 \beta_{10} + 2260 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$-19884 \beta_{9} + 19884 \beta_{7} - 5073 \beta_{6} - 19884 \beta_{4} + 28560 \beta_{3} + 14811 \beta_{2} + 33181$$ $$\nu^{13}$$ $$=$$ $$38706 \beta_{15} + 38706 \beta_{14} - 18822 \beta_{13} + 38706 \beta_{11} + 19884 \beta_{10} - 74880 \beta_{8} + 101509 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$-38706 \beta_{9} + 102571 \beta_{7} + 102571 \beta_{6} + 369454 \beta_{5} - 112524 \beta_{4} + 102571 \beta_{3} - 38706 \beta_{2} + 266883$$ $$\nu^{15}$$ $$=$$ $$151230 \beta_{15} + 141277 \beta_{14} - 292507 \beta_{13} + 591373 \beta_{12} - 282089 \beta_{11} + 292507 \beta_{10} - 591373 \beta_{8} + 574596 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$-1$$ $$1$$ $$\beta_{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
424.1
 −2.34074 + 1.35143i −2.11082 + 1.21868i −1.06407 + 0.614340i −0.427967 + 0.247087i 0.427967 − 0.247087i 1.06407 − 0.614340i 2.11082 − 1.21868i 2.34074 − 1.35143i −2.34074 − 1.35143i −2.11082 − 1.21868i −1.06407 − 0.614340i −0.427967 − 0.247087i 0.427967 + 0.247087i 1.06407 + 0.614340i 2.11082 + 1.21868i 2.34074 + 1.35143i
−2.34074 + 1.35143i 0.866025 + 0.500000i 2.65271 4.59463i 0 −2.70285 −2.06605 + 1.65271i 8.93406i 0.500000 + 0.866025i 0
424.2 −2.11082 + 1.21868i −0.866025 0.500000i 1.97036 3.41277i 0 2.43736 −2.46138 0.970361i 4.73024i 0.500000 + 0.866025i 0
424.3 −1.06407 + 0.614340i 0.866025 + 0.500000i −0.245174 + 0.424653i 0 −1.22868 2.33443 1.24517i 3.05984i 0.500000 + 0.866025i 0
424.4 −0.427967 + 0.247087i −0.866025 0.500000i −0.877896 + 1.52056i 0 0.494173 1.86373 + 1.87790i 1.85601i 0.500000 + 0.866025i 0
424.5 0.427967 0.247087i 0.866025 + 0.500000i −0.877896 + 1.52056i 0 0.494173 −1.86373 1.87790i 1.85601i 0.500000 + 0.866025i 0
424.6 1.06407 0.614340i −0.866025 0.500000i −0.245174 + 0.424653i 0 −1.22868 −2.33443 + 1.24517i 3.05984i 0.500000 + 0.866025i 0
424.7 2.11082 1.21868i 0.866025 + 0.500000i 1.97036 3.41277i 0 2.43736 2.46138 + 0.970361i 4.73024i 0.500000 + 0.866025i 0
424.8 2.34074 1.35143i −0.866025 0.500000i 2.65271 4.59463i 0 −2.70285 2.06605 1.65271i 8.93406i 0.500000 + 0.866025i 0
499.1 −2.34074 1.35143i 0.866025 0.500000i 2.65271 + 4.59463i 0 −2.70285 −2.06605 1.65271i 8.93406i 0.500000 0.866025i 0
499.2 −2.11082 1.21868i −0.866025 + 0.500000i 1.97036 + 3.41277i 0 2.43736 −2.46138 + 0.970361i 4.73024i 0.500000 0.866025i 0
499.3 −1.06407 0.614340i 0.866025 0.500000i −0.245174 0.424653i 0 −1.22868 2.33443 + 1.24517i 3.05984i 0.500000 0.866025i 0
499.4 −0.427967 0.247087i −0.866025 + 0.500000i −0.877896 1.52056i 0 0.494173 1.86373 1.87790i 1.85601i 0.500000 0.866025i 0
499.5 0.427967 + 0.247087i 0.866025 0.500000i −0.877896 1.52056i 0 0.494173 −1.86373 + 1.87790i 1.85601i 0.500000 0.866025i 0
499.6 1.06407 + 0.614340i −0.866025 + 0.500000i −0.245174 0.424653i 0 −1.22868 −2.33443 1.24517i 3.05984i 0.500000 0.866025i 0
499.7 2.11082 + 1.21868i 0.866025 0.500000i 1.97036 + 3.41277i 0 2.43736 2.46138 0.970361i 4.73024i 0.500000 0.866025i 0
499.8 2.34074 + 1.35143i −0.866025 + 0.500000i 2.65271 + 4.59463i 0 −2.70285 2.06605 + 1.65271i 8.93406i 0.500000 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 499.8 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.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.r.h 16
5.b even 2 1 inner 525.2.r.h 16
5.c odd 4 1 525.2.i.i 8
5.c odd 4 1 525.2.i.j yes 8
7.c even 3 1 inner 525.2.r.h 16
35.j even 6 1 inner 525.2.r.h 16
35.k even 12 1 3675.2.a.bq 4
35.k even 12 1 3675.2.a.bx 4
35.l odd 12 1 525.2.i.i 8
35.l odd 12 1 525.2.i.j yes 8
35.l odd 12 1 3675.2.a.br 4
35.l odd 12 1 3675.2.a.bw 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.i.i 8 5.c odd 4 1
525.2.i.i 8 35.l odd 12 1
525.2.i.j yes 8 5.c odd 4 1
525.2.i.j yes 8 35.l odd 12 1
525.2.r.h 16 1.a even 1 1 trivial
525.2.r.h 16 5.b even 2 1 inner
525.2.r.h 16 7.c even 3 1 inner
525.2.r.h 16 35.j even 6 1 inner
3675.2.a.bq 4 35.k even 12 1
3675.2.a.br 4 35.l odd 12 1
3675.2.a.bw 4 35.l odd 12 1
3675.2.a.bx 4 35.k even 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(525, [\chi])$$:

 $$T_{2}^{16} - \cdots$$ $$T_{11}^{8} + \cdots$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2} + 2 T^{4} - 7 T^{6} - 22 T^{8} - 31 T^{10} - 15 T^{12} + 132 T^{14} + 220 T^{16} + 528 T^{18} - 240 T^{20} - 1984 T^{22} - 5632 T^{24} - 7168 T^{26} + 8192 T^{28} + 16384 T^{30} + 65536 T^{32}$$
$3$ $$( 1 - T^{2} + T^{4} )^{4}$$
$5$ 
$7$ $$1 - 21 T^{2} + 329 T^{4} - 3318 T^{6} + 27414 T^{8} - 162582 T^{10} + 789929 T^{12} - 2470629 T^{14} + 5764801 T^{16}$$
$11$ $$( 1 + 8 T + 28 T^{2} + 120 T^{3} + 402 T^{4} + 396 T^{5} + 384 T^{6} - 2384 T^{7} - 29421 T^{8} - 26224 T^{9} + 46464 T^{10} + 527076 T^{11} + 5885682 T^{12} + 19326120 T^{13} + 49603708 T^{14} + 155897368 T^{15} + 214358881 T^{16} )^{2}$$
$13$ $$( 1 - 17 T^{2} + 425 T^{4} - 6726 T^{6} + 86438 T^{8} - 1136694 T^{10} + 12138425 T^{12} - 82055753 T^{14} + 815730721 T^{16} )^{2}$$
$17$ $$1 + 60 T^{2} + 1484 T^{4} + 20616 T^{6} + 278218 T^{8} + 6808500 T^{10} + 123849488 T^{12} + 666359940 T^{14} - 6309930269 T^{16} + 192578022660 T^{18} + 10344033087248 T^{20} + 164340638536500 T^{22} + 1940781283720138 T^{24} + 41561730251656584 T^{26} + 864611400048965324 T^{28} + 10102669593564055740 T^{30} + 48661191875666868481 T^{32}$$
$19$ $$( 1 - 3 T - 30 T^{2} - 125 T^{3} + 969 T^{4} + 3616 T^{5} + 6406 T^{6} - 66486 T^{7} - 238148 T^{8} - 1263234 T^{9} + 2312566 T^{10} + 24802144 T^{11} + 126281049 T^{12} - 309512375 T^{13} - 1411376430 T^{14} - 2681615217 T^{15} + 16983563041 T^{16} )^{2}$$
$23$ $$1 + 44 T^{2} - 388 T^{4} - 24248 T^{6} + 734474 T^{8} + 14717524 T^{10} - 547791216 T^{12} - 1264808844 T^{14} + 414286345891 T^{16} - 669083878476 T^{18} - 153294441676656 T^{20} + 2178721749218836 T^{22} + 57517382603277194 T^{24} - 1004510043908560952 T^{26} - 8502874279623884548 T^{28} +$$$$51\!\cdots\!96$$$$T^{30} +$$$$61\!\cdots\!61$$$$T^{32}$$
$29$ $$( 1 + 4 T + 29 T^{2} )^{16}$$
$31$ $$( 1 + 9 T + 13 T^{2} + 90 T^{3} + 749 T^{4} - 1629 T^{5} + 22230 T^{6} + 154161 T^{7} - 324118 T^{8} + 4778991 T^{9} + 21363030 T^{10} - 48529539 T^{11} + 691717229 T^{12} + 2576623590 T^{13} + 11537547853 T^{14} + 247613526999 T^{15} + 852891037441 T^{16} )^{2}$$
$37$ $$1 + 108 T^{2} + 7374 T^{4} + 467096 T^{6} + 22703481 T^{8} + 873938840 T^{10} + 33932606830 T^{12} + 1175343473028 T^{14} + 38616591751924 T^{16} + 1609045214575332 T^{18} + 63595168349119630 T^{20} + 2242287961638825560 T^{22} + 79745510544985799001 T^{24} +$$$$22\!\cdots\!04$$$$T^{26} +$$$$48\!\cdots\!94$$$$T^{28} +$$$$97\!\cdots\!12$$$$T^{30} +$$$$12\!\cdots\!41$$$$T^{32}$$
$41$ $$( 1 - 4 T + 104 T^{2} - 256 T^{3} + 4966 T^{4} - 10496 T^{5} + 174824 T^{6} - 275684 T^{7} + 2825761 T^{8} )^{4}$$
$43$ $$( 1 - 215 T^{2} + 21194 T^{4} - 1322001 T^{6} + 62692682 T^{8} - 2444379849 T^{10} + 72458068394 T^{12} - 1359093055535 T^{14} + 11688200277601 T^{16} )^{2}$$
$47$ $$1 + 212 T^{2} + 20956 T^{4} + 1454040 T^{6} + 90802154 T^{8} + 5296739420 T^{10} + 289183381584 T^{12} + 15685077059564 T^{14} + 793076937068387 T^{16} + 34648335224576876 T^{18} + 1411122652631194704 T^{20} + 57094694749782569180 T^{22} +$$$$21\!\cdots\!94$$$$T^{24} +$$$$76\!\cdots\!60$$$$T^{26} +$$$$24\!\cdots\!96$$$$T^{28} +$$$$54\!\cdots\!28$$$$T^{30} +$$$$56\!\cdots\!21$$$$T^{32}$$
$53$ $$1 + 236 T^{2} + 25324 T^{4} + 1967208 T^{6} + 148062890 T^{8} + 10417342148 T^{10} + 635434140816 T^{12} + 36036607646708 T^{14} + 1967961208523651 T^{16} + 101226830879602772 T^{18} + 5013881014859972496 T^{20} +$$$$23\!\cdots\!92$$$$T^{22} +$$$$92\!\cdots\!90$$$$T^{24} +$$$$34\!\cdots\!92$$$$T^{26} +$$$$12\!\cdots\!84$$$$T^{28} +$$$$32\!\cdots\!84$$$$T^{30} +$$$$38\!\cdots\!21$$$$T^{32}$$
$59$ $$( 1 - 10 T - 96 T^{2} + 1492 T^{3} + 4998 T^{4} - 116970 T^{5} + 158320 T^{6} + 3153670 T^{7} - 19452177 T^{8} + 186066530 T^{9} + 551111920 T^{10} - 24023181630 T^{11} + 60562570278 T^{12} + 1066667054108 T^{13} - 4049331229536 T^{14} - 24886514848190 T^{15} + 146830437604321 T^{16} )^{2}$$
$61$ $$( 1 - 5 T - 36 T^{2} - 305 T^{3} + 3721 T^{4} )^{8}$$
$67$ $$1 + 501 T^{2} + 138952 T^{4} + 26939913 T^{6} + 4029956509 T^{8} + 487571679936 T^{10} + 49074273490354 T^{12} + 4173403064575434 T^{14} + 302514422246172016 T^{16} + 18734406356879123226 T^{18} +$$$$98\!\cdots\!34$$$$T^{20} +$$$$44\!\cdots\!84$$$$T^{22} +$$$$16\!\cdots\!69$$$$T^{24} +$$$$49\!\cdots\!37$$$$T^{26} +$$$$11\!\cdots\!72$$$$T^{28} +$$$$18\!\cdots\!29$$$$T^{30} +$$$$16\!\cdots\!81$$$$T^{32}$$
$71$ $$( 1 - 22 T + 252 T^{2} - 1806 T^{3} + 12966 T^{4} - 128226 T^{5} + 1270332 T^{6} - 7874042 T^{7} + 25411681 T^{8} )^{4}$$
$73$ $$1 + 324 T^{2} + 50502 T^{4} + 5409800 T^{6} + 473911185 T^{8} + 35078829272 T^{10} + 2162596802518 T^{12} + 122333701830108 T^{14} + 7885309757584996 T^{16} + 651916297052645532 T^{18} + 61413945183735570838 T^{20} +$$$$53\!\cdots\!08$$$$T^{22} +$$$$38\!\cdots\!85$$$$T^{24} +$$$$23\!\cdots\!00$$$$T^{26} +$$$$11\!\cdots\!42$$$$T^{28} +$$$$39\!\cdots\!16$$$$T^{30} +$$$$65\!\cdots\!61$$$$T^{32}$$
$79$ $$( 1 - 8 T - 182 T^{2} + 1560 T^{3} + 19865 T^{4} - 148676 T^{5} - 1461174 T^{6} + 5521756 T^{7} + 104973092 T^{8} + 436218724 T^{9} - 9119186934 T^{10} - 73303066364 T^{11} + 773743359065 T^{12} + 4800207982440 T^{13} - 44241916904822 T^{14} - 153631271889272 T^{15} + 1517108809906561 T^{16} )^{2}$$
$83$ $$( 1 - 436 T^{2} + 89684 T^{4} - 11804316 T^{6} + 1127629270 T^{8} - 81319932924 T^{10} + 4256252060564 T^{12} - 142546002788884 T^{14} + 2252292232139041 T^{16} )^{2}$$
$89$ $$( 1 + 10 T - 260 T^{2} - 1572 T^{3} + 55478 T^{4} + 195874 T^{5} - 7285344 T^{6} - 5340058 T^{7} + 785732359 T^{8} - 475265162 T^{9} - 57707209824 T^{10} + 138085097906 T^{11} + 3480814046198 T^{12} - 8778141453828 T^{13} - 129215135649860 T^{14} + 442313348955290 T^{15} + 3936588805702081 T^{16} )^{2}$$
$97$ $$( 1 - 516 T^{2} + 120042 T^{4} - 17454416 T^{6} + 1892032563 T^{8} - 164228600144 T^{10} + 10627231949802 T^{12} - 429813554543364 T^{14} + 7837433594376961 T^{16} )^{2}$$