# Properties

 Label 76.2.i.a Level $76$ Weight $2$ Character orbit 76.i Analytic conductor $0.607$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 76.i (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.606863055362$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{9})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 6 x^{11} - 3 x^{10} + 70 x^{9} - 15 x^{8} - 426 x^{7} + 64 x^{6} + 1659 x^{5} + 267 x^{4} - 3969 x^{3} - 2088 x^{2} + 4446 x + 4161$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{5} + \beta_{7} + \beta_{9} + \beta_{11} ) q^{3} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} - \beta_{9} ) q^{5} + ( \beta_{3} - \beta_{5} + \beta_{6} + \beta_{9} ) q^{7} + ( 1 - \beta_{1} - \beta_{2} + 3 \beta_{5} - \beta_{6} - \beta_{8} + \beta_{10} - \beta_{11} ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{5} + \beta_{7} + \beta_{9} + \beta_{11} ) q^{3} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} - \beta_{9} ) q^{5} + ( \beta_{3} - \beta_{5} + \beta_{6} + \beta_{9} ) q^{7} + ( 1 - \beta_{1} - \beta_{2} + 3 \beta_{5} - \beta_{6} - \beta_{8} + \beta_{10} - \beta_{11} ) q^{9} + ( -\beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{11} + ( -1 - \beta_{1} - \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{10} ) q^{13} + ( 1 - \beta_{2} + \beta_{3} - 4 \beta_{4} + 3 \beta_{6} + \beta_{7} - 3 \beta_{9} + \beta_{10} - \beta_{11} ) q^{15} + ( -3 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} + 3 \beta_{9} - \beta_{10} ) q^{17} + ( 1 - \beta_{1} + 2 \beta_{5} - \beta_{6} - 3 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{19} + ( -\beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} ) q^{21} + ( -2 + \beta_{1} + \beta_{2} + 3 \beta_{4} - \beta_{6} - \beta_{7} + 2 \beta_{9} - \beta_{11} ) q^{23} + ( \beta_{1} - 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - 3 \beta_{9} + \beta_{10} ) q^{25} + ( -2 + \beta_{2} - \beta_{5} - 2 \beta_{6} - 3 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{27} + ( 3 - \beta_{3} - 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} - \beta_{9} + \beta_{10} ) q^{29} + ( -2 - \beta_{2} - \beta_{3} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + 3 \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{31} + ( 2 - \beta_{2} + \beta_{3} - \beta_{4} - 5 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} - \beta_{8} + 3 \beta_{9} - \beta_{10} + \beta_{11} ) q^{33} + ( 2 + \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{8} - \beta_{9} + \beta_{11} ) q^{35} + ( -1 + \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{5} + 4 \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{37} + ( 6 - 2 \beta_{1} - \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{39} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} - 6 \beta_{6} - \beta_{11} ) q^{41} + ( 4 - \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - 3 \beta_{6} + 4 \beta_{7} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{43} + ( -1 + \beta_{2} - \beta_{3} + 7 \beta_{4} - 8 \beta_{7} - \beta_{8} + 4 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{45} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{3} + 4 \beta_{4} + 3 \beta_{5} + \beta_{6} + \beta_{8} - 3 \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{47} + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} + 2 \beta_{10} - \beta_{11} ) q^{49} + ( -5 + 3 \beta_{1} - \beta_{3} - 5 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} - \beta_{8} + 5 \beta_{9} + 3 \beta_{10} - \beta_{11} ) q^{51} + ( -1 + \beta_{2} - \beta_{3} - 2 \beta_{4} + 3 \beta_{6} - 4 \beta_{7} - 3 \beta_{9} - \beta_{10} + \beta_{11} ) q^{53} + ( -3 - \beta_{1} + \beta_{2} + \beta_{3} - 4 \beta_{4} + 3 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + 3 \beta_{9} ) q^{55} + ( -3 + 2 \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} - 4 \beta_{9} - \beta_{10} ) q^{57} + ( -3 - 2 \beta_{3} + 5 \beta_{4} - \beta_{5} - 2 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} - 3 \beta_{9} - \beta_{10} ) q^{59} + ( -4 - \beta_{1} - 3 \beta_{2} - \beta_{3} + 4 \beta_{4} + 4 \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{61} + ( 2 - \beta_{1} - \beta_{3} + 2 \beta_{4} - 5 \beta_{6} + \beta_{8} - 6 \beta_{9} - \beta_{10} - \beta_{11} ) q^{63} + ( -4 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 5 \beta_{4} + 5 \beta_{5} - 2 \beta_{7} - \beta_{8} + 7 \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{65} + ( 3 - \beta_{1} + \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} + \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{67} + ( 3 + \beta_{3} - 4 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} + 3 \beta_{8} - 8 \beta_{9} + 3 \beta_{10} - 3 \beta_{11} ) q^{69} + ( 3 - \beta_{1} + \beta_{5} - 3 \beta_{6} + \beta_{8} - \beta_{9} + 3 \beta_{10} - 3 \beta_{11} ) q^{71} + ( 3 - \beta_{1} + \beta_{3} - \beta_{4} + 3 \beta_{5} + 4 \beta_{6} - \beta_{8} + \beta_{9} ) q^{73} + ( 2 - 2 \beta_{1} - 3 \beta_{2} + 3 \beta_{4} + 9 \beta_{5} + 2 \beta_{6} - 7 \beta_{7} - 4 \beta_{8} + 3 \beta_{9} - 3 \beta_{10} - \beta_{11} ) q^{75} + ( 2 \beta_{1} - \beta_{2} - 3 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - 4 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} ) q^{77} + ( 1 - \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{5} - 4 \beta_{6} - \beta_{7} - \beta_{8} - 3 \beta_{11} ) q^{79} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + 4 \beta_{7} - 4 \beta_{9} - \beta_{10} + \beta_{11} ) q^{81} + ( 1 - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} + 4 \beta_{7} + \beta_{8} + 3 \beta_{10} - \beta_{11} ) q^{83} + ( -3 + 4 \beta_{2} + 2 \beta_{3} + \beta_{4} - 5 \beta_{5} - \beta_{6} + 4 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{85} + ( -5 + 4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + \beta_{4} + \beta_{5} - 5 \beta_{6} - 8 \beta_{7} + \beta_{9} + 2 \beta_{11} ) q^{87} + ( -2 - \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 9 \beta_{5} + \beta_{6} + 9 \beta_{7} + \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{89} + ( 1 + \beta_{1} + 3 \beta_{2} - \beta_{3} - 5 \beta_{4} + 4 \beta_{6} - 4 \beta_{7} - 7 \beta_{9} - \beta_{10} ) q^{91} + ( -5 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - 4 \beta_{4} + 4 \beta_{5} + \beta_{6} + 6 \beta_{7} + \beta_{8} + 3 \beta_{9} + 3 \beta_{10} ) q^{93} + ( -4 - \beta_{2} - \beta_{3} + 7 \beta_{4} - 6 \beta_{5} - \beta_{7} + \beta_{8} + 5 \beta_{9} - \beta_{10} + 4 \beta_{11} ) q^{95} + ( -3 + 4 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + 7 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} + 3 \beta_{9} + \beta_{10} ) q^{97} + ( -1 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{6} + 6 \beta_{7} + 5 \beta_{9} + \beta_{10} + \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 3q^{3} + 3q^{7} - 3q^{9} + O(q^{10})$$ $$12q - 3q^{3} + 3q^{7} - 3q^{9} + 3q^{11} - 9q^{13} - 15q^{15} - 3q^{17} - 12q^{19} - 15q^{21} - 12q^{23} - 18q^{25} - 9q^{27} + 27q^{29} + 6q^{31} + 48q^{33} + 33q^{35} - 12q^{37} + 60q^{39} + 3q^{41} + 27q^{43} + 24q^{45} - 15q^{47} + 9q^{49} - 33q^{51} - 21q^{53} - 27q^{55} - 42q^{57} - 48q^{59} - 6q^{61} - 9q^{63} - 33q^{65} + 24q^{67} - 33q^{69} + 30q^{73} + 42q^{75} + 24q^{77} + 3q^{79} + 3q^{81} + 3q^{83} - 42q^{85} - 18q^{87} - 18q^{89} - 24q^{91} - 78q^{93} + 9q^{95} + 12q^{97} - 6q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 6 x^{11} - 3 x^{10} + 70 x^{9} - 15 x^{8} - 426 x^{7} + 64 x^{6} + 1659 x^{5} + 267 x^{4} - 3969 x^{3} - 2088 x^{2} + 4446 x + 4161$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-1861 \nu^{11} + 59764 \nu^{10} - 261706 \nu^{9} - 459237 \nu^{8} + 3512934 \nu^{7} + 1223776 \nu^{6} - 19762014 \nu^{5} - 6600456 \nu^{4} + 62490886 \nu^{3} + 42461956 \nu^{2} - 98039407 \nu - 104999089$$$$)/8740601$$ $$\beta_{3}$$ $$=$$ $$($$$$-3014 \nu^{11} - 25876 \nu^{10} + 250822 \nu^{9} + 84243 \nu^{8} - 2839567 \nu^{7} - 652331 \nu^{6} + 18054126 \nu^{5} + 5399484 \nu^{4} - 59109957 \nu^{3} - 38613723 \nu^{2} + 94201422 \nu + 91337490$$$$)/8740601$$ $$\beta_{4}$$ $$=$$ $$($$$$-7410 \nu^{11} + 92642 \nu^{10} - 242771 \nu^{9} - 590557 \nu^{8} + 2422014 \nu^{7} + 2513886 \nu^{6} - 10863570 \nu^{5} - 10833438 \nu^{4} + 27062496 \nu^{3} + 32215080 \nu^{2} - 29279114 \nu - 39192874$$$$)/8740601$$ $$\beta_{5}$$ $$=$$ $$($$$$7410 \nu^{11} + 11132 \nu^{10} - 276099 \nu^{9} + 170744 \nu^{8} + 2370458 \nu^{7} - 1428976 \nu^{6} - 11344066 \nu^{5} + 2408876 \nu^{4} + 33428312 \nu^{3} + 6822812 \nu^{2} - 44659861 \nu - 26703616$$$$)/8740601$$ $$\beta_{6}$$ $$=$$ $$($$$$8405 \nu^{11} + 8018 \nu^{10} - 346679 \nu^{9} + 455105 \nu^{8} + 2627431 \nu^{7} - 3363517 \nu^{6} - 11636805 \nu^{5} + 9113704 \nu^{4} + 31536935 \nu^{3} - 4569699 \nu^{2} - 40761460 \nu - 11328251$$$$)/8740601$$ $$\beta_{7}$$ $$=$$ $$($$$$15815 \nu^{11} - 89341 \nu^{10} - 80323 \nu^{9} + 1088115 \nu^{8} - 105905 \nu^{7} - 5646270 \nu^{6} - 277950 \nu^{5} + 17338641 \nu^{4} + 8455587 \nu^{3} - 27341345 \nu^{2} - 22774844 \nu + 1553197$$$$)/8740601$$ $$\beta_{8}$$ $$=$$ $$($$$$16036 \nu^{11} - 88198 \nu^{10} + 124775 \nu^{9} + 357074 \nu^{8} - 2014841 \nu^{7} + 1083449 \nu^{6} + 10730801 \nu^{5} - 7009805 \nu^{4} - 37334292 \nu^{3} + 5511793 \nu^{2} + 68308502 \nu + 33629265$$$$)/8740601$$ $$\beta_{9}$$ $$=$$ $$($$$$-1506 \nu^{11} + 8283 \nu^{10} + 6301 \nu^{9} - 90477 \nu^{8} - 1422 \nu^{7} + 485184 \nu^{6} + 4698 \nu^{5} - 1477233 \nu^{4} - 388357 \nu^{3} + 2375103 \nu^{2} + 1084425 \nu - 745423$$$$)/514153$$ $$\beta_{10}$$ $$=$$ $$($$$$32367 \nu^{11} - 175660 \nu^{10} + 8466 \nu^{9} + 1222749 \nu^{8} - 536869 \nu^{7} - 4743060 \nu^{6} + 1737702 \nu^{5} + 11702325 \nu^{4} - 5650797 \nu^{3} - 17409849 \nu^{2} + 16526830 \nu + 20539212$$$$)/8740601$$ $$\beta_{11}$$ $$=$$ $$($$$$382 \nu^{11} - 2101 \nu^{10} + 162 \nu^{9} + 12670 \nu^{8} + 605 \nu^{7} - 42146 \nu^{6} - 39078 \nu^{5} + 48159 \nu^{4} + 203904 \nu^{3} + 142404 \nu^{2} - 273582 \nu - 476163$$$$)/80189$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} + \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{11} - 3 \beta_{10} - \beta_{9} + \beta_{8} - 2 \beta_{7} + \beta_{6} + 2 \beta_{5} - 3 \beta_{4} + 3 \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$\beta_{11} - 9 \beta_{10} - 7 \beta_{9} + 6 \beta_{8} - 12 \beta_{7} + 8 \beta_{6} - 13 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 6 \beta_{1} + 5$$ $$\nu^{5}$$ $$=$$ $$8 \beta_{11} - 28 \beta_{10} - 22 \beta_{9} + 10 \beta_{8} - 34 \beta_{7} + 11 \beta_{6} + 12 \beta_{5} - 25 \beta_{4} - 5 \beta_{3} - 3 \beta_{2} + 9 \beta_{1} + 4$$ $$\nu^{6}$$ $$=$$ $$11 \beta_{11} - 62 \beta_{10} - 71 \beta_{9} + 26 \beta_{8} - 108 \beta_{7} + 26 \beta_{6} + 25 \beta_{5} - 66 \beta_{4} - 33 \beta_{3} - 16 \beta_{2} + 24 \beta_{1} - 6$$ $$\nu^{7}$$ $$=$$ $$26 \beta_{11} - 152 \beta_{10} - 261 \beta_{9} + 42 \beta_{8} - 327 \beta_{7} - 12 \beta_{6} + 87 \beta_{5} - 91 \beta_{4} - 98 \beta_{3} - 75 \beta_{2} + 45 \beta_{1} - 5$$ $$\nu^{8}$$ $$=$$ $$-12 \beta_{11} - 288 \beta_{10} - 804 \beta_{9} + 76 \beta_{8} - 866 \beta_{7} - 236 \beta_{6} + 233 \beta_{5} - 51 \beta_{4} - 362 \beta_{3} - 270 \beta_{2} + 141 \beta_{1} + 18$$ $$\nu^{9}$$ $$=$$ $$-236 \beta_{11} - 480 \beta_{10} - 2610 \beta_{9} + 88 \beta_{8} - 2462 \beta_{7} - 1334 \beta_{6} + 761 \beta_{5} + 402 \beta_{4} - 1062 \beta_{3} - 762 \beta_{2} + 417 \beta_{1} + 384$$ $$\nu^{10}$$ $$=$$ $$-1334 \beta_{11} - 495 \beta_{10} - 7824 \beta_{9} + 204 \beta_{8} - 6291 \beta_{7} - 5298 \beta_{6} + 2331 \beta_{5} + 2500 \beta_{4} - 3136 \beta_{3} - 2046 \beta_{2} + 1327 \beta_{1} + 1889$$ $$\nu^{11}$$ $$=$$ $$-5298 \beta_{11} + 678 \beta_{10} - 23140 \beta_{9} + 726 \beta_{8} - 16756 \beta_{7} - 18648 \beta_{6} + 7923 \beta_{5} + 9189 \beta_{4} - 8536 \beta_{3} - 4679 \beta_{2} + 3928 \beta_{1} + 7815$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$39$$ $$\chi(n)$$ $$-\beta_{7}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 2.75227 − 0.342020i −1.75227 − 0.342020i 2.26253 − 0.984808i −1.26253 − 0.984808i 2.26253 + 0.984808i −1.26253 + 0.984808i −1.25236 + 0.642788i 2.25236 + 0.642788i 2.75227 + 0.342020i −1.75227 + 0.342020i −1.25236 − 0.642788i 2.25236 − 0.642788i
0 −0.467454 + 2.65106i 0 −0.982226 0.824185i 0 1.67233 + 2.89656i 0 −3.99055 1.45244i 0
5.2 0 0.314751 1.78504i 0 0.216181 + 0.181398i 0 −0.579936 1.00448i 0 −0.268219 0.0976237i 0
9.1 0 −0.834130 + 0.699919i 0 3.00735 + 1.09458i 0 0.278396 0.482195i 0 −0.315057 + 1.78678i 0
9.2 0 1.86622 1.56594i 0 −2.06765 0.752564i 0 −1.48413 + 2.57059i 0 0.509650 2.89037i 0
17.1 0 −0.834130 0.699919i 0 3.00735 1.09458i 0 0.278396 + 0.482195i 0 −0.315057 1.78678i 0
17.2 0 1.86622 + 1.56594i 0 −2.06765 + 0.752564i 0 −1.48413 2.57059i 0 0.509650 + 2.89037i 0
25.1 0 −2.83638 1.03236i 0 −0.658711 3.73574i 0 −0.0695116 + 0.120398i 0 4.68114 + 3.92794i 0
25.2 0 0.456991 + 0.166331i 0 0.485063 + 2.75093i 0 1.68285 2.91479i 0 −2.11696 1.77634i 0
61.1 0 −0.467454 2.65106i 0 −0.982226 + 0.824185i 0 1.67233 2.89656i 0 −3.99055 + 1.45244i 0
61.2 0 0.314751 + 1.78504i 0 0.216181 0.181398i 0 −0.579936 + 1.00448i 0 −0.268219 + 0.0976237i 0
73.1 0 −2.83638 + 1.03236i 0 −0.658711 + 3.73574i 0 −0.0695116 0.120398i 0 4.68114 3.92794i 0
73.2 0 0.456991 0.166331i 0 0.485063 2.75093i 0 1.68285 + 2.91479i 0 −2.11696 + 1.77634i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 73.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.2.i.a 12
3.b odd 2 1 684.2.bo.c 12
4.b odd 2 1 304.2.u.e 12
19.e even 9 1 inner 76.2.i.a 12
19.e even 9 1 1444.2.a.h 6
19.e even 9 2 1444.2.e.g 12
19.f odd 18 1 1444.2.a.g 6
19.f odd 18 2 1444.2.e.h 12
57.l odd 18 1 684.2.bo.c 12
76.k even 18 1 5776.2.a.by 6
76.l odd 18 1 304.2.u.e 12
76.l odd 18 1 5776.2.a.bw 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.i.a 12 1.a even 1 1 trivial
76.2.i.a 12 19.e even 9 1 inner
304.2.u.e 12 4.b odd 2 1
304.2.u.e 12 76.l odd 18 1
684.2.bo.c 12 3.b odd 2 1
684.2.bo.c 12 57.l odd 18 1
1444.2.a.g 6 19.f odd 18 1
1444.2.a.h 6 19.e even 9 1
1444.2.e.g 12 19.e even 9 2
1444.2.e.h 12 19.f odd 18 2
5776.2.a.bw 6 76.l odd 18 1
5776.2.a.by 6 76.k even 18 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(76, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$361 - 912 T + 39 T^{2} + 729 T^{3} + 1137 T^{4} + 195 T^{5} + 433 T^{6} + 75 T^{7} + 24 T^{8} + 18 T^{9} + 6 T^{10} + 3 T^{11} + T^{12}$$
$5$ $$729 - 2916 T + 4131 T^{2} + 9747 T^{3} + 7398 T^{4} + 1251 T^{5} + 64 T^{6} + 36 T^{7} - 72 T^{8} - 19 T^{9} + 9 T^{10} + T^{12}$$
$7$ $$9 + 54 T + 414 T^{2} - 366 T^{3} + 1386 T^{4} + 447 T^{5} + 1141 T^{6} - 186 T^{7} + 201 T^{8} - 22 T^{9} + 21 T^{10} - 3 T^{11} + T^{12}$$
$11$ $$729 + 2430 T^{2} + 1890 T^{3} + 7452 T^{4} + 3231 T^{5} + 3331 T^{6} - 300 T^{7} + 591 T^{8} + 2 T^{9} + 33 T^{10} - 3 T^{11} + T^{12}$$
$13$ $$130321 + 308655 T + 948708 T^{2} + 1178665 T^{3} + 639711 T^{4} + 152361 T^{5} + 20109 T^{6} + 4077 T^{7} + 1377 T^{8} + 268 T^{9} + 45 T^{10} + 9 T^{11} + T^{12}$$
$17$ $$8982009 + 10924065 T + 5143824 T^{2} - 2532789 T^{3} + 653913 T^{4} - 136809 T^{5} + 39861 T^{6} + 999 T^{7} - 1179 T^{8} - 30 T^{9} + 9 T^{10} + 3 T^{11} + T^{12}$$
$19$ $$47045881 + 29713188 T + 4300593 T^{2} - 1378659 T^{3} - 492765 T^{4} + 11913 T^{5} + 24358 T^{6} + 627 T^{7} - 1365 T^{8} - 201 T^{9} + 33 T^{10} + 12 T^{11} + T^{12}$$
$23$ $$151585344 + 75792672 T + 22161600 T^{2} + 4777056 T^{3} + 856980 T^{4} - 36522 T^{5} - 40985 T^{6} - 3729 T^{7} + 714 T^{8} + 251 T^{9} + 69 T^{10} + 12 T^{11} + T^{12}$$
$29$ $$263169 - 872613 T + 19768293 T^{2} - 6261165 T^{3} - 1363392 T^{4} + 832302 T^{5} - 54107 T^{6} - 41454 T^{7} + 15840 T^{8} - 3023 T^{9} + 369 T^{10} - 27 T^{11} + T^{12}$$
$31$ $$130321 + 1584429 T + 19798323 T^{2} - 6277068 T^{3} + 3546369 T^{4} - 321024 T^{5} + 205549 T^{6} - 14955 T^{7} + 8508 T^{8} - 90 T^{9} + 126 T^{10} - 6 T^{11} + T^{12}$$
$37$ $$( 11096 + 7836 T + 78 T^{2} - 639 T^{3} - 81 T^{4} + 6 T^{5} + T^{6} )^{2}$$
$41$ $$360278361 - 150671178 T + 23782896 T^{2} + 21232152 T^{3} + 3831138 T^{4} + 1806273 T^{5} + 492832 T^{6} + 44274 T^{7} + 6492 T^{8} + 910 T^{9} - 21 T^{10} - 3 T^{11} + T^{12}$$
$43$ $$1203409 - 7325766 T + 18911724 T^{2} + 1881450 T^{3} + 771306 T^{4} - 370035 T^{5} + 137242 T^{6} - 50418 T^{7} + 11874 T^{8} - 2142 T^{9} + 309 T^{10} - 27 T^{11} + T^{12}$$
$47$ $$729 + 10935 T + 1256796 T^{2} + 812565 T^{3} + 3161673 T^{4} + 3504195 T^{5} + 1894231 T^{6} + 417405 T^{7} + 32919 T^{8} - 460 T^{9} - 51 T^{10} + 15 T^{11} + T^{12}$$
$53$ $$95004009 - 80003376 T + 34913754 T^{2} - 1832436 T^{3} + 1156302 T^{4} + 666045 T^{5} + 42652 T^{6} + 4470 T^{7} + 3522 T^{8} + 890 T^{9} + 177 T^{10} + 21 T^{11} + T^{12}$$
$59$ $$12621848409 - 3266938413 T + 812641086 T^{2} + 282191040 T^{3} + 70948224 T^{4} + 40087386 T^{5} + 11066715 T^{6} + 1690956 T^{7} + 183636 T^{8} + 16278 T^{9} + 1107 T^{10} + 48 T^{11} + T^{12}$$
$61$ $$1418049649 - 727194327 T + 276693828 T^{2} - 39761552 T^{3} + 9943830 T^{4} + 1422378 T^{5} + 449985 T^{6} - 18414 T^{7} + 8838 T^{8} - 560 T^{9} - 141 T^{10} + 6 T^{11} + T^{12}$$
$67$ $$6080256576 - 2720114784 T + 19649952 T^{2} + 205544736 T^{3} - 32415444 T^{4} - 4719942 T^{5} + 2284291 T^{6} - 406245 T^{7} + 59610 T^{8} - 6139 T^{9} + 483 T^{10} - 24 T^{11} + T^{12}$$
$71$ $$16842816 + 43879968 T + 19478880 T^{2} - 17404416 T^{3} + 13286916 T^{4} - 5907222 T^{5} + 1016551 T^{6} - 34011 T^{7} + 8442 T^{8} - 487 T^{9} + 45 T^{10} + T^{12}$$
$73$ $$36864 - 82944 T + 78336 T^{2} + 89088 T^{3} + 211536 T^{4} - 66444 T^{5} + 259069 T^{6} - 109335 T^{7} + 21024 T^{8} - 2947 T^{9} + 375 T^{10} - 30 T^{11} + T^{12}$$
$79$ $$1548343801 - 2218221177 T + 1473881775 T^{2} - 476164415 T^{3} + 80469864 T^{4} - 5080752 T^{5} + 330321 T^{6} + 10548 T^{7} - 7938 T^{8} + 259 T^{9} + 39 T^{10} - 3 T^{11} + T^{12}$$
$83$ $$11939714361 + 672659964 T + 958378392 T^{2} + 19603782 T^{3} + 53308368 T^{4} + 866295 T^{5} + 1386243 T^{6} - 14472 T^{7} + 24957 T^{8} - 114 T^{9} + 189 T^{10} - 3 T^{11} + T^{12}$$
$89$ $$7695324729 - 2430102546 T + 1314529155 T^{2} - 321329349 T^{3} + 125240742 T^{4} + 37993293 T^{5} + 1486998 T^{6} - 24948 T^{7} + 26244 T^{8} + 2205 T^{9} + 117 T^{10} + 18 T^{11} + T^{12}$$
$97$ $$16983563041 + 31607403735 T + 24477528468 T^{2} - 3870585684 T^{3} + 595015248 T^{4} - 66305706 T^{5} + 4516399 T^{6} - 14412 T^{7} - 19416 T^{8} + 2124 T^{9} - 39 T^{10} - 12 T^{11} + T^{12}$$