# Properties

 Label 275.2.h.d Level $275$ Weight $2$ Character orbit 275.h Analytic conductor $2.196$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$275 = 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 275.h (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.19588605559$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 7 x^{14} + 25 x^{12} + 57 x^{10} + 194 x^{8} + 303 x^{6} + 235 x^{4} + 33 x^{2} + 121$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 55) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + \beta_{12} + \beta_{13} ) q^{2} + ( -\beta_{5} + \beta_{13} + \beta_{15} ) q^{3} + ( \beta_{4} + \beta_{6} + \beta_{8} ) q^{4} + ( -2 + \beta_{3} + \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{11} ) q^{6} + ( \beta_{1} - \beta_{2} - \beta_{12} + \beta_{15} ) q^{7} + ( -\beta_{1} - \beta_{2} - \beta_{10} ) q^{8} + ( -1 + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{9} + \beta_{11} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} + \beta_{12} + \beta_{13} ) q^{2} + ( -\beta_{5} + \beta_{13} + \beta_{15} ) q^{3} + ( \beta_{4} + \beta_{6} + \beta_{8} ) q^{4} + ( -2 + \beta_{3} + \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{11} ) q^{6} + ( \beta_{1} - \beta_{2} - \beta_{12} + \beta_{15} ) q^{7} + ( -\beta_{1} - \beta_{2} - \beta_{10} ) q^{8} + ( -1 + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{9} + \beta_{11} ) q^{9} + ( -2 + \beta_{3} + \beta_{4} + \beta_{6} + \beta_{8} - 2 \beta_{9} + 2 \beta_{11} ) q^{11} + ( \beta_{1} + \beta_{2} - 2 \beta_{5} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{12} + ( \beta_{1} + 2 \beta_{5} + \beta_{12} + \beta_{13} ) q^{13} + ( 2 - 2 \beta_{4} + 2 \beta_{7} - \beta_{8} ) q^{14} + ( 2 + 2 \beta_{3} - \beta_{4} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - \beta_{11} ) q^{16} + ( -\beta_{1} + 2 \beta_{5} + \beta_{12} - 3 \beta_{14} ) q^{17} + ( -2 \beta_{1} + 3 \beta_{2} - \beta_{5} + \beta_{10} - \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{18} + ( -1 - \beta_{3} + \beta_{4} - 2 \beta_{7} + \beta_{11} ) q^{19} + ( -\beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{21} + ( 2 \beta_{2} - \beta_{10} - \beta_{12} - 2 \beta_{13} + 2 \beta_{14} - 3 \beta_{15} ) q^{22} + ( 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{12} - 3 \beta_{13} - 2 \beta_{14} ) q^{23} + ( -1 + 3 \beta_{3} + \beta_{4} + 2 \beta_{6} - \beta_{9} - 2 \beta_{11} ) q^{24} + ( 2 - 2 \beta_{3} + 3 \beta_{4} - \beta_{6} + 2 \beta_{7} + 3 \beta_{8} ) q^{26} + ( \beta_{1} + \beta_{10} - \beta_{12} + 2 \beta_{14} ) q^{27} -\beta_{5} q^{28} + ( 1 - \beta_{3} - \beta_{6} + \beta_{7} + 3 \beta_{9} ) q^{29} + ( 2 - 2 \beta_{3} - 3 \beta_{4} - \beta_{6} + 2 \beta_{9} + \beta_{11} ) q^{31} + ( -3 \beta_{2} + 5 \beta_{5} - \beta_{10} + 2 \beta_{12} - \beta_{13} - 3 \beta_{14} - \beta_{15} ) q^{32} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{5} + 2 \beta_{10} - 3 \beta_{13} + \beta_{14} ) q^{33} + ( 4 \beta_{4} + 4 \beta_{6} - 4 \beta_{7} + 3 \beta_{8} + \beta_{9} - 2 \beta_{11} ) q^{34} + ( 1 + \beta_{3} - \beta_{4} - 2 \beta_{7} - 3 \beta_{8} - \beta_{11} ) q^{36} + ( 2 \beta_{1} - \beta_{2} - \beta_{5} + \beta_{10} + \beta_{12} + \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{37} + ( 2 \beta_{1} + 2 \beta_{5} + 2 \beta_{10} - 2 \beta_{12} + \beta_{14} ) q^{38} + ( -2 - 5 \beta_{3} - 2 \beta_{4} - 5 \beta_{6} + 5 \beta_{7} + 2 \beta_{8} - 3 \beta_{9} + 5 \beta_{11} ) q^{39} + ( -5 + 2 \beta_{3} + 5 \beta_{4} - 3 \beta_{7} - 2 \beta_{8} - 2 \beta_{11} ) q^{41} + ( -3 \beta_{1} + \beta_{2} - 3 \beta_{5} ) q^{42} + ( -4 \beta_{1} - \beta_{5} - 2 \beta_{10} - \beta_{12} + 2 \beta_{13} - 2 \beta_{15} ) q^{43} + ( -2 + 2 \beta_{3} + \beta_{4} - \beta_{7} - \beta_{8} - 4 \beta_{9} - \beta_{11} ) q^{44} + ( -4 - 3 \beta_{3} - \beta_{4} - \beta_{6} - 4 \beta_{9} + \beta_{11} ) q^{46} + ( -5 \beta_{1} + 3 \beta_{2} - 4 \beta_{5} + 3 \beta_{10} + 4 \beta_{13} + \beta_{15} ) q^{47} + ( \beta_{1} - 3 \beta_{2} + 2 \beta_{5} - 2 \beta_{10} + 4 \beta_{12} - 2 \beta_{14} + \beta_{15} ) q^{48} + ( 4 - \beta_{3} - 4 \beta_{4} - \beta_{6} - \beta_{7} - 4 \beta_{8} + 5 \beta_{9} - \beta_{11} ) q^{49} + ( 3 - 3 \beta_{3} - 4 \beta_{4} - 2 \beta_{6} + 3 \beta_{7} - 4 \beta_{8} + 9 \beta_{9} ) q^{51} + ( \beta_{1} - \beta_{2} - 2 \beta_{5} - \beta_{10} + 2 \beta_{13} + 2 \beta_{15} ) q^{52} + ( -8 \beta_{1} + 3 \beta_{2} - 3 \beta_{5} + 5 \beta_{12} + 5 \beta_{13} + \beta_{14} - \beta_{15} ) q^{53} + ( 5 - 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} - 2 \beta_{11} ) q^{54} + ( -1 + \beta_{8} - \beta_{9} - 3 \beta_{11} ) q^{56} + ( 5 \beta_{1} - 2 \beta_{2} + 5 \beta_{5} - \beta_{14} + \beta_{15} ) q^{57} + ( 2 \beta_{1} - 2 \beta_{5} + 2 \beta_{13} + \beta_{15} ) q^{58} + ( -3 + 3 \beta_{3} - 2 \beta_{6} - 3 \beta_{7} - 2 \beta_{9} ) q^{59} + ( -4 + 3 \beta_{4} - 4 \beta_{7} + 4 \beta_{8} + \beta_{9} - 4 \beta_{11} ) q^{61} + ( \beta_{1} - \beta_{5} + \beta_{10} + \beta_{14} + \beta_{15} ) q^{62} + ( -\beta_{1} - \beta_{2} + \beta_{5} - \beta_{10} - \beta_{13} + \beta_{15} ) q^{63} + ( 4 - 2 \beta_{3} - \beta_{4} + 3 \beta_{6} + 4 \beta_{9} - 3 \beta_{11} ) q^{64} + ( 5 - \beta_{3} + 4 \beta_{4} - \beta_{6} - 3 \beta_{7} - 7 \beta_{8} + \beta_{9} - 4 \beta_{11} ) q^{66} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{5} - 3 \beta_{10} - \beta_{12} - \beta_{13} + \beta_{14} - 3 \beta_{15} ) q^{67} + ( \beta_{1} + 3 \beta_{5} + 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{68} + ( -3 + \beta_{3} + 3 \beta_{4} - 4 \beta_{7} - 2 \beta_{8} - \beta_{11} ) q^{69} + ( 9 - 6 \beta_{3} - 9 \beta_{4} - 6 \beta_{6} + 3 \beta_{7} - 9 \beta_{8} + 6 \beta_{9} + 3 \beta_{11} ) q^{71} + ( -2 \beta_{1} + \beta_{5} - 2 \beta_{10} + 2 \beta_{12} + \beta_{14} ) q^{72} + ( 5 \beta_{1} - 4 \beta_{2} - \beta_{5} + \beta_{10} - 5 \beta_{12} - \beta_{13} + \beta_{14} + 5 \beta_{15} ) q^{73} + ( -3 - 2 \beta_{3} + 3 \beta_{4} + 4 \beta_{7} + 2 \beta_{8} + 2 \beta_{11} ) q^{74} + ( 4 + \beta_{4} + \beta_{6} - \beta_{7} + \beta_{9} - 2 \beta_{11} ) q^{76} + ( \beta_{1} - 3 \beta_{2} + \beta_{5} - \beta_{10} - 4 \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{77} + ( -\beta_{1} + 5 \beta_{2} - 12 \beta_{5} - 7 \beta_{12} + \beta_{13} + 5 \beta_{14} ) q^{78} + ( -1 - 5 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{9} + \beta_{11} ) q^{79} + ( -2 + 2 \beta_{3} + 5 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} + 5 \beta_{8} - 8 \beta_{9} ) q^{81} + ( 3 \beta_{1} + 5 \beta_{5} + 3 \beta_{10} - 3 \beta_{12} - 2 \beta_{14} ) q^{82} + ( 2 \beta_{1} - \beta_{5} - \beta_{10} - 2 \beta_{12} - \beta_{14} ) q^{83} + ( -2 + 2 \beta_{3} + \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{84} + ( 3 + 3 \beta_{3} - 8 \beta_{4} + 3 \beta_{9} ) q^{86} + ( -\beta_{1} + \beta_{2} - 3 \beta_{5} - 3 \beta_{10} - 2 \beta_{12} - 2 \beta_{13} + \beta_{14} - 3 \beta_{15} ) q^{87} + ( 2 \beta_{1} - 5 \beta_{2} + 6 \beta_{5} - 5 \beta_{10} - 5 \beta_{13} - 3 \beta_{14} - \beta_{15} ) q^{88} + ( -1 - 6 \beta_{4} - 6 \beta_{6} + 6 \beta_{7} - 2 \beta_{8} - 4 \beta_{9} + 2 \beta_{11} ) q^{89} + ( -4 \beta_{3} + 2 \beta_{7} - \beta_{8} + 4 \beta_{11} ) q^{91} + ( -2 \beta_{1} + 3 \beta_{2} - \beta_{5} + \beta_{10} - \beta_{12} + \beta_{14} - 2 \beta_{15} ) q^{92} + ( -\beta_{1} + 2 \beta_{5} - 2 \beta_{10} + \beta_{12} - 3 \beta_{14} ) q^{93} + ( -1 + 9 \beta_{3} + \beta_{4} + 9 \beta_{6} - 7 \beta_{7} + \beta_{8} + 6 \beta_{9} - 7 \beta_{11} ) q^{94} + ( -6 - 3 \beta_{3} + 6 \beta_{4} + 6 \beta_{7} + 4 \beta_{8} + 3 \beta_{11} ) q^{96} + ( -4 \beta_{1} + 4 \beta_{2} + 4 \beta_{12} + 4 \beta_{13} + \beta_{14} - \beta_{15} ) q^{97} + ( \beta_{1} - \beta_{2} + 4 \beta_{5} + 2 \beta_{10} + 3 \beta_{12} + \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{98} + ( 6 - 2 \beta_{3} + 3 \beta_{4} + \beta_{7} - \beta_{8} + 5 \beta_{9} + \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 4q^{4} - 18q^{6} - 2q^{9} + O(q^{10})$$ $$16q + 4q^{4} - 18q^{6} - 2q^{9} - 6q^{11} + 12q^{14} + 16q^{16} - 6q^{19} + 8q^{21} - 6q^{24} + 40q^{26} - 2q^{29} + 8q^{31} + 16q^{34} + 10q^{36} - 30q^{39} - 52q^{41} - 4q^{44} - 62q^{46} + 10q^{49} - 42q^{51} + 40q^{54} - 20q^{56} - 2q^{59} - 40q^{61} + 8q^{64} + 58q^{66} - 26q^{69} + 36q^{71} - 48q^{74} + 56q^{76} - 38q^{79} + 68q^{81} - 12q^{84} + 22q^{86} - 24q^{89} - 20q^{91} - 14q^{94} - 86q^{96} + 72q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 7 x^{14} + 25 x^{12} + 57 x^{10} + 194 x^{8} + 303 x^{6} + 235 x^{4} + 33 x^{2} + 121$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$1829973 \nu^{15} + 24476532 \nu^{13} + 152757677 \nu^{11} + 557238290 \nu^{9} + 1500352252 \nu^{7} + 3810530967 \nu^{5} + 7929326490 \nu^{3} + 7498263355 \nu$$$$)/ 2438578648$$ $$\beta_{3}$$ $$=$$ $$($$$$608673 \nu^{14} + 4779625 \nu^{12} + 9344761 \nu^{10} - 2560494 \nu^{8} + 9430964 \nu^{6} + 74558203 \nu^{4} - 726449607 \nu^{2} + 31775447$$$$)/ 609644662$$ $$\beta_{4}$$ $$=$$ $$($$$$-2744937 \nu^{14} - 22433610 \nu^{12} - 99837091 \nu^{10} - 276654262 \nu^{8} - 814713604 \nu^{6} - 1632321387 \nu^{4} - 3415114020 \nu^{2} - 771926881$$$$)/ 2438578648$$ $$\beta_{5}$$ $$=$$ $$($$$$2744937 \nu^{15} + 22433610 \nu^{13} + 99837091 \nu^{11} + 276654262 \nu^{9} + 814713604 \nu^{7} + 1632321387 \nu^{5} + 3415114020 \nu^{3} + 771926881 \nu$$$$)/ 2438578648$$ $$\beta_{6}$$ $$=$$ $$($$$$-2924455 \nu^{14} - 43736988 \nu^{12} - 193036323 \nu^{10} - 488836470 \nu^{8} - 1053314964 \nu^{6} - 3520413317 \nu^{4} - 719547006 \nu^{2} + 236512683$$$$)/ 2438578648$$ $$\beta_{7}$$ $$=$$ $$($$$$1351905 \nu^{14} + 5558819 \nu^{12} + 7678521 \nu^{10} - 13223046 \nu^{8} + 42401020 \nu^{6} - 304413167 \nu^{4} - 664868085 \nu^{2} - 629543387$$$$)/ 609644662$$ $$\beta_{8}$$ $$=$$ $$($$$$-6960689 \nu^{14} - 30587176 \nu^{12} - 61040185 \nu^{10} - 38446538 \nu^{8} - 665668084 \nu^{6} + 797037357 \nu^{4} + 1844303058 \nu^{2} + 1738523017$$$$)/ 2438578648$$ $$\beta_{9}$$ $$=$$ $$($$$$3962283 \nu^{14} + 31992860 \nu^{12} + 118526613 \nu^{10} + 271533274 \nu^{8} + 833575532 \nu^{6} + 1781437793 \nu^{4} + 742925482 \nu^{2} - 383811549$$$$)/ 1219289324$$ $$\beta_{10}$$ $$=$$ $$($$$$-8234811 \nu^{15} - 67300830 \nu^{13} - 299511273 \nu^{11} - 829962786 \nu^{9} - 2444140812 \nu^{7} - 4896964161 \nu^{5} - 7806763412 \nu^{3} - 2315780643 \nu$$$$)/ 2438578648$$ $$\beta_{11}$$ $$=$$ $$($$$$770947 \nu^{14} + 4704418 \nu^{12} + 13758165 \nu^{10} + 28071690 \nu^{8} + 126275508 \nu^{6} + 152673185 \nu^{4} - 21444456 \nu^{2} + 243303327$$$$)/ 221688968$$ $$\beta_{12}$$ $$=$$ $$($$$$-10669503 \nu^{15} - 86419330 \nu^{13} - 336890317 \nu^{11} - 819720810 \nu^{9} - 2481864668 \nu^{7} - 5195196973 \nu^{5} - 4900964984 \nu^{3} - 4303783 \nu$$$$)/ 2438578648$$ $$\beta_{13}$$ $$=$$ $$($$$$8815096 \nu^{15} + 58503253 \nu^{13} + 198965251 \nu^{11} + 429083674 \nu^{9} + 1573766376 \nu^{7} + 2199079808 \nu^{5} + 1528330963 \nu^{3} + 352179707 \nu$$$$)/ 1219289324$$ $$\beta_{14}$$ $$=$$ $$($$$$-19639683 \nu^{15} - 162786416 \nu^{13} - 643887411 \nu^{11} - 1627587846 \nu^{9} - 4886486364 \nu^{7} - 10096782513 \nu^{5} - 8290421402 \nu^{3} - 4755604557 \nu$$$$)/ 2438578648$$ $$\beta_{15}$$ $$=$$ $$($$$$-1496498 \nu^{15} - 9707543 \nu^{13} - 31032777 \nu^{11} - 60570826 \nu^{9} - 228914540 \nu^{7} - 268358314 \nu^{5} + 33071203 \nu^{3} + 172220447 \nu$$$$)/ 110844484$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{9} - 2 \beta_{4} + \beta_{3} - 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{10} + 3 \beta_{5}$$ $$\nu^{4}$$ $$=$$ $$6 \beta_{9} + 4 \beta_{8} + \beta_{7} + 4 \beta_{6} + 4 \beta_{4} - \beta_{3} + 1$$ $$\nu^{5}$$ $$=$$ $$-\beta_{15} + 5 \beta_{14} - \beta_{13} - 10 \beta_{12} - \beta_{10} - 15 \beta_{5} + 5 \beta_{2}$$ $$\nu^{6}$$ $$=$$ $$-7 \beta_{11} - 22 \beta_{8} - 21 \beta_{7} - \beta_{4} + 7 \beta_{3} + 1$$ $$\nu^{7}$$ $$=$$ $$21 \beta_{15} - 21 \beta_{14} + 36 \beta_{13} + 36 \beta_{12} + 29 \beta_{5} - 28 \beta_{2} - 7 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$85 \beta_{11} - 78 \beta_{9} + 87 \beta_{8} + 85 \beta_{7} - 49 \beta_{6} - 7 \beta_{4} - 49 \beta_{3} - 87$$ $$\nu^{9}$$ $$=$$ $$-85 \beta_{15} + 36 \beta_{14} - 136 \beta_{13} - 45 \beta_{12} + 36 \beta_{10} - 36 \beta_{5} + 121 \beta_{2} - 85 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-342 \beta_{11} + 396 \beta_{9} - 230 \beta_{8} - 166 \beta_{7} + 166 \beta_{6} + 166 \beta_{4} + 529$$ $$\nu^{11}$$ $$=$$ $$166 \beta_{15} + 220 \beta_{13} - 342 \beta_{10} - 220 \beta_{5} - 342 \beta_{2} + 475 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$728 \beta_{11} - 1653 \beta_{9} - 728 \beta_{6} - 1170 \beta_{4} + 651 \beta_{3} - 1653$$ $$\nu^{13}$$ $$=$$ $$-728 \beta_{14} + 1002 \beta_{12} + 1379 \beta_{10} + 2823 \beta_{5} - 1002 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$4644 \beta_{9} + 3748 \beta_{8} + 3109 \beta_{7} + 2472 \beta_{6} + 3748 \beta_{4} - 3109 \beta_{3} + 3109$$ $$\nu^{15}$$ $$=$$ $$-3109 \beta_{15} + 5581 \beta_{14} - 4385 \beta_{13} - 8392 \beta_{12} - 3109 \beta_{10} - 13973 \beta_{5} + 5581 \beta_{2} + 1276 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/275\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$\chi(n)$$ $$\beta_{9}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 0.625353 − 1.92464i 0.381325 − 1.17360i −0.381325 + 1.17360i −0.625353 + 1.92464i 1.33858 + 0.972539i 0.649397 + 0.471815i −0.649397 − 0.471815i −1.33858 − 0.972539i 0.625353 + 1.92464i 0.381325 + 1.17360i −0.381325 − 1.17360i −0.625353 − 1.92464i 1.33858 − 0.972539i 0.649397 − 0.471815i −0.649397 + 0.471815i −1.33858 + 0.972539i
−1.63719 1.18949i 0.809808 2.49233i 0.647481 + 1.99274i 0 −4.29042 + 3.11717i −0.298456 0.918552i 0.0595923 0.183406i −3.12889 2.27327i 0
26.2 −0.998322 0.725323i −0.112477 + 0.346168i −0.147481 0.453901i 0 0.363371 0.264005i 0.798988 + 2.45903i −0.944641 + 2.90731i 2.31987 + 1.68548i 0
26.3 0.998322 + 0.725323i 0.112477 0.346168i −0.147481 0.453901i 0 0.363371 0.264005i −0.798988 2.45903i 0.944641 2.90731i 2.31987 + 1.68548i 0
26.4 1.63719 + 1.18949i −0.809808 + 2.49233i 0.647481 + 1.99274i 0 −4.29042 + 3.11717i 0.298456 + 0.918552i −0.0595923 + 0.183406i −3.12889 2.27327i 0
126.1 −0.511294 1.57360i −1.59764 1.16075i −0.596764 + 0.433574i 0 −1.00970 + 3.10753i −1.81468 + 1.31845i −1.68978 1.22769i 0.278050 + 0.855749i 0
126.2 −0.248048 0.763412i 1.42447 + 1.03494i 1.09676 0.796845i 0 0.436748 1.34417i 0.479022 0.348029i −2.17917 1.58326i 0.0309674 + 0.0953077i 0
126.3 0.248048 + 0.763412i −1.42447 1.03494i 1.09676 0.796845i 0 0.436748 1.34417i −0.479022 + 0.348029i 2.17917 + 1.58326i 0.0309674 + 0.0953077i 0
126.4 0.511294 + 1.57360i 1.59764 + 1.16075i −0.596764 + 0.433574i 0 −1.00970 + 3.10753i 1.81468 1.31845i 1.68978 + 1.22769i 0.278050 + 0.855749i 0
201.1 −1.63719 + 1.18949i 0.809808 + 2.49233i 0.647481 1.99274i 0 −4.29042 3.11717i −0.298456 + 0.918552i 0.0595923 + 0.183406i −3.12889 + 2.27327i 0
201.2 −0.998322 + 0.725323i −0.112477 0.346168i −0.147481 + 0.453901i 0 0.363371 + 0.264005i 0.798988 2.45903i −0.944641 2.90731i 2.31987 1.68548i 0
201.3 0.998322 0.725323i 0.112477 + 0.346168i −0.147481 + 0.453901i 0 0.363371 + 0.264005i −0.798988 + 2.45903i 0.944641 + 2.90731i 2.31987 1.68548i 0
201.4 1.63719 1.18949i −0.809808 2.49233i 0.647481 1.99274i 0 −4.29042 3.11717i 0.298456 0.918552i −0.0595923 0.183406i −3.12889 + 2.27327i 0
251.1 −0.511294 + 1.57360i −1.59764 + 1.16075i −0.596764 0.433574i 0 −1.00970 3.10753i −1.81468 1.31845i −1.68978 + 1.22769i 0.278050 0.855749i 0
251.2 −0.248048 + 0.763412i 1.42447 1.03494i 1.09676 + 0.796845i 0 0.436748 + 1.34417i 0.479022 + 0.348029i −2.17917 + 1.58326i 0.0309674 0.0953077i 0
251.3 0.248048 0.763412i −1.42447 + 1.03494i 1.09676 + 0.796845i 0 0.436748 + 1.34417i −0.479022 0.348029i 2.17917 1.58326i 0.0309674 0.0953077i 0
251.4 0.511294 1.57360i 1.59764 1.16075i −0.596764 0.433574i 0 −1.00970 3.10753i 1.81468 + 1.31845i 1.68978 1.22769i 0.278050 0.855749i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 251.4 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
11.c even 5 1 inner
55.j even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 275.2.h.d 16
5.b even 2 1 inner 275.2.h.d 16
5.c odd 4 2 55.2.j.a 16
11.c even 5 1 inner 275.2.h.d 16
11.c even 5 1 3025.2.a.bl 8
11.d odd 10 1 3025.2.a.bk 8
15.e even 4 2 495.2.ba.a 16
20.e even 4 2 880.2.cd.c 16
55.e even 4 2 605.2.j.d 16
55.h odd 10 1 3025.2.a.bk 8
55.j even 10 1 inner 275.2.h.d 16
55.j even 10 1 3025.2.a.bl 8
55.k odd 20 2 55.2.j.a 16
55.k odd 20 2 605.2.b.g 8
55.k odd 20 4 605.2.j.h 16
55.l even 20 2 605.2.b.f 8
55.l even 20 2 605.2.j.d 16
55.l even 20 4 605.2.j.g 16
165.v even 20 2 495.2.ba.a 16
220.v even 20 2 880.2.cd.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.j.a 16 5.c odd 4 2
55.2.j.a 16 55.k odd 20 2
275.2.h.d 16 1.a even 1 1 trivial
275.2.h.d 16 5.b even 2 1 inner
275.2.h.d 16 11.c even 5 1 inner
275.2.h.d 16 55.j even 10 1 inner
495.2.ba.a 16 15.e even 4 2
495.2.ba.a 16 165.v even 20 2
605.2.b.f 8 55.l even 20 2
605.2.b.g 8 55.k odd 20 2
605.2.j.d 16 55.e even 4 2
605.2.j.d 16 55.l even 20 2
605.2.j.g 16 55.l even 20 4
605.2.j.h 16 55.k odd 20 4
880.2.cd.c 16 20.e even 4 2
880.2.cd.c 16 220.v even 20 2
3025.2.a.bk 8 11.d odd 10 1
3025.2.a.bk 8 55.h odd 10 1
3025.2.a.bl 8 11.c even 5 1
3025.2.a.bl 8 55.j even 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(275, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$121 + 308 T^{2} + 345 T^{4} + 138 T^{6} + 159 T^{8} + 62 T^{10} + 15 T^{12} + 2 T^{14} + T^{16}$$
$3$ $$121 + 1463 T^{2} + 6730 T^{4} - 637 T^{6} + 969 T^{8} + 77 T^{10} + 30 T^{12} + 7 T^{14} + T^{16}$$
$5$ $$T^{16}$$
$7$ $$121 + 11 T^{2} + 757 T^{4} + 1553 T^{6} + 1330 T^{8} + 187 T^{10} + 47 T^{12} + 9 T^{14} + T^{16}$$
$11$ $$( 14641 + 3993 T - 2662 T^{2} - 209 T^{3} + 335 T^{4} - 19 T^{5} - 22 T^{6} + 3 T^{7} + T^{8} )^{2}$$
$13$ $$47265625 + 24062500 T^{2} + 5390625 T^{4} + 431250 T^{6} + 99375 T^{8} + 7750 T^{10} + 375 T^{12} + 10 T^{14} + T^{16}$$
$17$ $$1675346761 + 439066837 T^{2} + 53285010 T^{4} + 1993827 T^{6} + 793659 T^{8} + 4603 T^{10} + 2160 T^{12} + 73 T^{14} + T^{16}$$
$19$ $$( 121 + 198 T + 1433 T^{2} - 879 T^{3} + 280 T^{4} - 9 T^{5} + 3 T^{6} + 3 T^{7} + T^{8} )^{2}$$
$23$ $$( 26411 - 16366 T^{2} + 2232 T^{4} - 99 T^{6} + T^{8} )^{2}$$
$29$ $$( 121 + 1056 T + 3565 T^{2} + 579 T^{3} + 344 T^{4} + 99 T^{5} + 25 T^{6} + T^{7} + T^{8} )^{2}$$
$31$ $$( 1 - 2 T + T^{2} + 4 T^{3} + 9 T^{4} - 8 T^{5} + 39 T^{6} - 4 T^{7} + T^{8} )^{2}$$
$37$ $$15768841 + 92432967 T^{2} + 205855505 T^{4} - 11467003 T^{6} + 4278334 T^{8} + 127363 T^{10} + 2275 T^{12} + 28 T^{14} + T^{16}$$
$41$ $$( 3876961 + 1592921 T + 504915 T^{2} + 110009 T^{3} + 19764 T^{4} + 2919 T^{5} + 345 T^{6} + 26 T^{7} + T^{8} )^{2}$$
$43$ $$( 212531 - 79707 T^{2} + 8319 T^{4} - 173 T^{6} + T^{8} )^{2}$$
$47$ $$59639012521 - 994427192 T^{2} + 3392111583 T^{4} + 559988696 T^{6} + 38916105 T^{8} + 1250516 T^{10} + 16813 T^{12} - 22 T^{14} + T^{16}$$
$53$ $$239836452361 + 47877571753 T^{2} + 6382430355 T^{4} + 733116213 T^{6} + 150715374 T^{8} + 395017 T^{10} + 20325 T^{12} + 217 T^{14} + T^{16}$$
$59$ $$( 1 - 47 T + 891 T^{2} - 3591 T^{3} + 5874 T^{4} - 203 T^{5} + 129 T^{6} + T^{7} + T^{8} )^{2}$$
$61$ $$( 43681 - 31350 T + 23717 T^{2} - 7100 T^{3} + 1894 T^{4} - 310 T^{5} + 148 T^{6} + 20 T^{7} + T^{8} )^{2}$$
$67$ $$( 18491 - 46104 T^{2} + 5736 T^{4} - 149 T^{6} + T^{8} )^{2}$$
$71$ $$( 6305121 - 3593241 T + 1203579 T^{2} - 254907 T^{3} + 36774 T^{4} - 3429 T^{5} + 261 T^{6} - 18 T^{7} + T^{8} )^{2}$$
$73$ $$1636073786281 + 1265544147219 T^{2} + 371225608252 T^{4} - 3758349303 T^{6} + 216983455 T^{8} + 217893 T^{10} + 18012 T^{12} + 201 T^{14} + T^{16}$$
$79$ $$( 101761 + 175769 T + 125860 T^{2} + 22301 T^{3} + 7779 T^{4} + 1361 T^{5} + 210 T^{6} + 19 T^{7} + T^{8} )^{2}$$
$83$ $$45169425961 + 6210793413 T^{2} + 601069813 T^{4} + 53440521 T^{6} + 5270830 T^{8} + 154881 T^{10} + 2643 T^{12} + 33 T^{14} + T^{16}$$
$89$ $$( 1871 - 486 T - 128 T^{2} + 6 T^{3} + T^{4} )^{4}$$
$97$ $$25937424601 + 7716919716 T^{2} + 1089510015 T^{4} + 75459714 T^{6} + 14115739 T^{8} + 628914 T^{10} + 11515 T^{12} - 24 T^{14} + T^{16}$$