# Properties

 Label 35.3.i.a Level 35 Weight 3 Character orbit 35.i Analytic conductor 0.954 Analytic rank 0 Dimension 12 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$35 = 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 35.i (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.953680925261$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 15 x^{10} + 180 x^{8} - 669 x^{6} + 1980 x^{4} - 135 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -\beta_{1} - \beta_{11} ) q^{3} + ( 1 + \beta_{3} + \beta_{7} ) q^{4} + ( \beta_{1} - 2 \beta_{4} - \beta_{6} + \beta_{8} - \beta_{9} ) q^{5} + ( -2 + \beta_{2} - 4 \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{6} + ( -\beta_{1} + \beta_{4} - \beta_{8} + 3 \beta_{9} ) q^{7} + ( \beta_{1} - \beta_{4} + \beta_{8} - 2 \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{8} + ( -\beta_{2} + 3 \beta_{3} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -\beta_{1} - \beta_{11} ) q^{3} + ( 1 + \beta_{3} + \beta_{7} ) q^{4} + ( \beta_{1} - 2 \beta_{4} - \beta_{6} + \beta_{8} - \beta_{9} ) q^{5} + ( -2 + \beta_{2} - 4 \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{6} + ( -\beta_{1} + \beta_{4} - \beta_{8} + 3 \beta_{9} ) q^{7} + ( \beta_{1} - \beta_{4} + \beta_{8} - 2 \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{8} + ( -\beta_{2} + 3 \beta_{3} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{9} + ( -3 - \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + \beta_{5} + \beta_{7} - 2 \beta_{9} + \beta_{11} ) q^{10} + ( 1 + \beta_{3} - \beta_{5} + 2 \beta_{6} ) q^{11} + ( -6 \beta_{1} + 12 \beta_{4} - 5 \beta_{8} + 5 \beta_{9} ) q^{12} + ( 2 \beta_{1} - \beta_{4} - \beta_{8} - 4 \beta_{10} ) q^{13} + ( -8 + \beta_{2} - 5 \beta_{3} - 3 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} ) q^{14} + ( 4 - \beta_{1} + 3 \beta_{2} - 2 \beta_{4} + 2 \beta_{8} - 4 \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{15} + ( \beta_{3} + 4 \beta_{5} - 2 \beta_{6} ) q^{16} + ( 8 \beta_{1} - 2 \beta_{4} + 3 \beta_{9} + \beta_{11} ) q^{17} + ( \beta_{1} - 7 \beta_{4} + 6 \beta_{8} - 3 \beta_{9} + 4 \beta_{10} + 2 \beta_{11} ) q^{18} + ( 2 - \beta_{2} + \beta_{3} - 3 \beta_{6} - \beta_{7} ) q^{19} + ( -2 - 6 \beta_{1} - \beta_{2} - 4 \beta_{3} - 3 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} + 4 \beta_{10} ) q^{20} + ( 4 - 4 \beta_{2} + 6 \beta_{3} - 2 \beta_{5} - \beta_{6} + 5 \beta_{7} ) q^{21} + ( 2 \beta_{1} - \beta_{4} - 2 \beta_{8} + 4 \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{22} + ( -\beta_{1} - 3 \beta_{8} - 3 \beta_{9} + 3 \beta_{10} - 3 \beta_{11} ) q^{23} + ( 12 + 4 \beta_{2} - 12 \beta_{3} - \beta_{5} - 2 \beta_{7} ) q^{24} + ( 3 + 4 \beta_{1} + 3 \beta_{3} - 8 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} + 4 \beta_{8} - 2 \beta_{9} - 4 \beta_{10} - 2 \beta_{11} ) q^{25} + ( 2 + 3 \beta_{2} + \beta_{3} - 5 \beta_{6} + 3 \beta_{7} ) q^{26} + ( \beta_{1} + 4 \beta_{4} + \beta_{8} + 4 \beta_{10} ) q^{27} + ( -5 \beta_{1} + 5 \beta_{4} + 2 \beta_{8} + \beta_{9} - 7 \beta_{10} - 7 \beta_{11} ) q^{28} + ( -\beta_{2} + 3 \beta_{5} + 3 \beta_{6} ) q^{29} + ( 16 \beta_{1} + \beta_{2} - 8 \beta_{3} + 2 \beta_{5} - \beta_{6} - \beta_{7} - 3 \beta_{8} - 3 \beta_{9} - 3 \beta_{10} + 3 \beta_{11} ) q^{30} + ( -6 + 2 \beta_{2} + 6 \beta_{3} + 2 \beta_{5} - \beta_{7} ) q^{31} + ( 2 \beta_{1} + 3 \beta_{4} - 4 \beta_{10} - 2 \beta_{11} ) q^{32} + ( -2 \beta_{1} + 4 \beta_{4} - 6 \beta_{8} + 6 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{33} + ( 18 - 3 \beta_{2} + 36 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} + 6 \beta_{7} ) q^{34} + ( 18 - 5 \beta_{1} + 3 \beta_{2} - \beta_{3} + 12 \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + \beta_{9} + 7 \beta_{10} + 7 \beta_{11} ) q^{35} + ( -17 - 9 \beta_{2} + \beta_{5} + \beta_{6} ) q^{36} + ( -19 \beta_{1} - 2 \beta_{8} - 2 \beta_{9} - 3 \beta_{10} + 3 \beta_{11} ) q^{37} + ( -9 \beta_{1} + 3 \beta_{4} - 3 \beta_{9} ) q^{38} + ( -40 - 40 \beta_{3} - 2 \beta_{5} + 4 \beta_{6} - \beta_{7} ) q^{39} + ( -18 + 2 \beta_{1} - 3 \beta_{2} - 9 \beta_{3} - 4 \beta_{4} + \beta_{6} - 3 \beta_{7} - 7 \beta_{8} + 7 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{40} + ( -3 + \beta_{2} - 6 \beta_{3} + 4 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} ) q^{41} + ( 5 \beta_{1} - 26 \beta_{4} + 5 \beta_{8} - 8 \beta_{9} + 7 \beta_{10} + 7 \beta_{11} ) q^{42} + ( 11 \beta_{1} - 2 \beta_{4} - 2 \beta_{8} + 4 \beta_{9} + 7 \beta_{10} + 14 \beta_{11} ) q^{43} + ( 2 \beta_{2} + 7 \beta_{3} + 2 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{44} + ( -12 - 13 \beta_{1} + 2 \beta_{2} + 12 \beta_{3} + 9 \beta_{4} - 2 \beta_{5} - \beta_{7} + 6 \beta_{9} - \beta_{11} ) q^{45} + ( 19 + 19 \beta_{3} + 2 \beta_{7} ) q^{46} + ( -5 \beta_{1} + 10 \beta_{4} + 13 \beta_{8} - 13 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{47} + ( 15 \beta_{1} - 4 \beta_{4} + 12 \beta_{8} - 7 \beta_{10} ) q^{48} + ( -17 + 3 \beta_{2} + 20 \beta_{3} + 5 \beta_{5} - 8 \beta_{6} + 5 \beta_{7} ) q^{49} + ( -16 - 11 \beta_{1} - 2 \beta_{2} + 13 \beta_{4} - 8 \beta_{8} + 16 \beta_{9} - 6 \beta_{10} - 12 \beta_{11} ) q^{50} + ( 7 \beta_{2} - 30 \beta_{3} - 12 \beta_{5} + 6 \beta_{6} - 7 \beta_{7} ) q^{51} + ( 17 \beta_{1} - 17 \beta_{4} - 15 \beta_{9} - 2 \beta_{11} ) q^{52} + ( -9 \beta_{1} + 15 \beta_{4} - 22 \beta_{8} + 11 \beta_{9} - 4 \beta_{10} - 2 \beta_{11} ) q^{53} + ( 28 + 14 \beta_{3} + 5 \beta_{6} ) q^{54} + ( -14 - 2 \beta_{1} - 2 \beta_{2} - 28 \beta_{3} + 3 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} + 3 \beta_{10} ) q^{55} + ( 21 + 7 \beta_{2} - 14 \beta_{3} + 7 \beta_{6} - 7 \beta_{7} ) q^{56} + ( 8 \beta_{1} - 24 \beta_{4} + 11 \beta_{8} - 22 \beta_{9} - 5 \beta_{10} - 10 \beta_{11} ) q^{57} + ( 2 \beta_{1} + 7 \beta_{8} + 7 \beta_{9} + 4 \beta_{10} - 4 \beta_{11} ) q^{58} + ( 22 - 18 \beta_{2} - 22 \beta_{3} + 2 \beta_{5} + 9 \beta_{7} ) q^{59} + ( 64 - 11 \beta_{1} + 64 \beta_{3} + 20 \beta_{4} + 6 \beta_{5} - 12 \beta_{6} + 13 \beta_{7} - 14 \beta_{8} + 7 \beta_{9} + 8 \beta_{10} + 4 \beta_{11} ) q^{60} + ( -12 - 4 \beta_{2} - 6 \beta_{3} + 11 \beta_{6} - 4 \beta_{7} ) q^{61} + ( -2 \beta_{4} + \beta_{8} - \beta_{10} ) q^{62} + ( 5 \beta_{1} + 16 \beta_{4} - 9 \beta_{8} + 6 \beta_{9} - 7 \beta_{11} ) q^{63} + ( 31 + 9 \beta_{2} - 10 \beta_{5} - 10 \beta_{6} ) q^{64} + ( 20 \beta_{1} - 8 \beta_{2} + 9 \beta_{3} - 2 \beta_{5} + \beta_{6} + 8 \beta_{7} + 11 \beta_{8} + 11 \beta_{9} - \beta_{10} + \beta_{11} ) q^{65} + ( -6 + 6 \beta_{3} - 4 \beta_{5} ) q^{66} + ( \beta_{1} - 14 \beta_{4} + 14 \beta_{8} - 7 \beta_{9} + 12 \beta_{10} + 6 \beta_{11} ) q^{67} + ( 22 \beta_{1} - 44 \beta_{4} + 17 \beta_{8} - 17 \beta_{9} + 3 \beta_{10} + 3 \beta_{11} ) q^{68} + ( 20 + 8 \beta_{2} + 40 \beta_{3} + 7 \beta_{5} - 7 \beta_{6} - 16 \beta_{7} ) q^{69} + ( 16 + 19 \beta_{1} - 2 \beta_{2} - 18 \beta_{3} + 9 \beta_{4} + 6 \beta_{5} + 3 \beta_{6} - \beta_{7} - 9 \beta_{8} - \beta_{9} - 7 \beta_{10} ) q^{70} + ( 16 + 13 \beta_{2} ) q^{71} + ( -31 \beta_{1} - \beta_{8} - \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{72} + ( -8 \beta_{1} + 6 \beta_{4} + 9 \beta_{9} - 5 \beta_{11} ) q^{73} + ( -99 - 99 \beta_{3} + 5 \beta_{5} - 10 \beta_{6} - 12 \beta_{7} ) q^{74} + ( -24 + 20 \beta_{1} + 6 \beta_{2} - 12 \beta_{3} - 40 \beta_{4} + 4 \beta_{6} + 6 \beta_{7} + 8 \beta_{8} - 8 \beta_{9} - \beta_{10} - \beta_{11} ) q^{75} + ( -25 - \beta_{2} - 50 \beta_{3} - 9 \beta_{5} + 9 \beta_{6} + 2 \beta_{7} ) q^{76} + ( 3 \beta_{1} - 17 \beta_{4} - 4 \beta_{8} - 2 \beta_{9} - 14 \beta_{10} - 7 \beta_{11} ) q^{77} + ( -42 \beta_{1} + 44 \beta_{4} - 5 \beta_{8} + 10 \beta_{9} - 3 \beta_{10} - 6 \beta_{11} ) q^{78} + ( 3 \beta_{2} + 32 \beta_{3} - 24 \beta_{5} + 12 \beta_{6} - 3 \beta_{7} ) q^{79} + ( -28 - 5 \beta_{1} + 8 \beta_{2} + 28 \beta_{3} + 9 \beta_{4} - 5 \beta_{5} - 4 \beta_{7} + 7 \beta_{9} + 6 \beta_{11} ) q^{80} + ( 49 + 49 \beta_{3} + 8 \beta_{5} - 16 \beta_{6} + \beta_{7} ) q^{81} + ( -7 \beta_{1} + 14 \beta_{4} + 5 \beta_{8} - 5 \beta_{9} + \beta_{10} + \beta_{11} ) q^{82} + ( -27 \beta_{1} + 12 \beta_{4} - 37 \beta_{8} + 2 \beta_{10} ) q^{83} + ( -92 - 20 \beta_{2} + 16 \beta_{3} + 11 \beta_{5} + 9 \beta_{6} + 11 \beta_{7} ) q^{84} + ( -16 - 10 \beta_{1} - 12 \beta_{2} + 14 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 5 \beta_{8} - 10 \beta_{9} + 9 \beta_{10} + 18 \beta_{11} ) q^{85} + ( -23 \beta_{2} + 64 \beta_{3} + 10 \beta_{5} - 5 \beta_{6} + 23 \beta_{7} ) q^{86} + ( 21 \beta_{1} + 13 \beta_{9} + 8 \beta_{11} ) q^{87} + ( 5 \beta_{1} + 5 \beta_{4} + 16 \beta_{8} - 8 \beta_{9} + 6 \beta_{10} + 3 \beta_{11} ) q^{88} + ( 76 + 8 \beta_{2} + 38 \beta_{3} + 9 \beta_{6} + 8 \beta_{7} ) q^{89} + ( -35 - 10 \beta_{1} + 10 \beta_{2} - 70 \beta_{3} - 8 \beta_{4} - 7 \beta_{5} + 7 \beta_{6} - 20 \beta_{7} - 7 \beta_{8} - 5 \beta_{10} ) q^{90} + ( -17 - 4 \beta_{2} - 22 \beta_{3} + 19 \beta_{5} - 15 \beta_{6} - 9 \beta_{7} ) q^{91} + ( 7 \beta_{1} + 17 \beta_{4} - 10 \beta_{8} + 20 \beta_{9} + 14 \beta_{10} + 28 \beta_{11} ) q^{92} + ( 14 \beta_{1} + 9 \beta_{8} + 9 \beta_{9} - 9 \beta_{10} + 9 \beta_{11} ) q^{93} + ( 55 + 14 \beta_{2} - 55 \beta_{3} + 11 \beta_{5} - 7 \beta_{7} ) q^{94} + ( 48 + 11 \beta_{1} + 48 \beta_{3} - 9 \beta_{4} - 9 \beta_{7} + 8 \beta_{8} - 4 \beta_{9} - 14 \beta_{10} - 7 \beta_{11} ) q^{95} + ( -60 - 5 \beta_{2} - 30 \beta_{3} + 9 \beta_{6} - 5 \beta_{7} ) q^{96} + ( 10 \beta_{1} + 22 \beta_{4} - 16 \beta_{8} - 4 \beta_{10} ) q^{97} + ( 12 \beta_{1} - 40 \beta_{4} + 12 \beta_{8} - 29 \beta_{9} + 7 \beta_{10} + 21 \beta_{11} ) q^{98} + ( 33 - 4 \beta_{2} + 3 \beta_{5} + 3 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 6q^{4} - 18q^{9} + O(q^{10})$$ $$12q + 6q^{4} - 18q^{9} - 54q^{10} + 6q^{11} - 66q^{14} + 48q^{15} - 6q^{16} + 18q^{19} + 12q^{21} + 216q^{24} + 18q^{25} + 18q^{26} + 48q^{30} - 108q^{31} + 222q^{35} - 204q^{36} - 240q^{39} - 162q^{40} - 42q^{44} - 216q^{45} + 114q^{46} - 324q^{49} - 192q^{50} + 180q^{51} + 252q^{54} + 336q^{56} + 396q^{59} + 384q^{60} - 108q^{61} + 372q^{64} - 54q^{65} - 108q^{66} + 300q^{70} + 192q^{71} - 594q^{74} - 216q^{75} - 192q^{79} - 504q^{80} + 294q^{81} - 1200q^{84} - 192q^{85} - 384q^{86} + 684q^{89} - 72q^{91} + 990q^{94} + 288q^{95} - 540q^{96} + 396q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 15 x^{10} + 180 x^{8} - 669 x^{6} + 1980 x^{4} - 135 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{10} + 12 \nu^{8} - 144 \nu^{6} + 132 \nu^{4} - 9 \nu^{2} - 7767$$$$)/1575$$ $$\beta_{3}$$ $$=$$ $$($$$$12 \nu^{10} - 179 \nu^{8} + 2148 \nu^{6} - 7884 \nu^{4} + 23628 \nu^{2} - 1611$$$$)/1575$$ $$\beta_{4}$$ $$=$$ $$($$$$-12 \nu^{11} + 179 \nu^{9} - 2148 \nu^{7} + 7884 \nu^{5} - 23628 \nu^{3} + 1611 \nu$$$$)/1575$$ $$\beta_{5}$$ $$=$$ $$($$$$53 \nu^{10} - 811 \nu^{8} + 9767 \nu^{6} - 37971 \nu^{4} + 111777 \nu^{2} - 15069$$$$)/3150$$ $$\beta_{6}$$ $$=$$ $$($$$$-62 \nu^{10} + 919 \nu^{8} - 10958 \nu^{6} + 39159 \nu^{4} - 111858 \nu^{2} - 7269$$$$)/3150$$ $$\beta_{7}$$ $$=$$ $$($$$$-12 \nu^{10} + 179 \nu^{8} - 2148 \nu^{6} + 7884 \nu^{4} - 23313 \nu^{2} + 36$$$$)/315$$ $$\beta_{8}$$ $$=$$ $$($$$$53 \nu^{11} - 801 \nu^{9} + 9627 \nu^{7} - 36471 \nu^{5} + 108207 \nu^{3} - 15939 \nu$$$$)/1350$$ $$\beta_{9}$$ $$=$$ $$($$$$-404 \nu^{11} + 6003 \nu^{9} - 71826 \nu^{7} + 259653 \nu^{5} - 757746 \nu^{3} - 58743 \nu$$$$)/9450$$ $$\beta_{10}$$ $$=$$ $$($$$$-583 \nu^{11} + 8781 \nu^{9} - 105477 \nu^{7} + 396681 \nu^{5} - 1179567 \nu^{3} + 168489 \nu$$$$)/9450$$ $$\beta_{11}$$ $$=$$ $$($$$$-622 \nu^{11} + 9249 \nu^{9} - 110778 \nu^{7} + 401829 \nu^{5} - 1179918 \nu^{3} - 86229 \nu$$$$)/9450$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} + 5 \beta_{3} + 5$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{11} + \beta_{10} - 2 \beta_{9} + \beta_{8} - 9 \beta_{4} + 9 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$12 \beta_{7} - 2 \beta_{6} + 4 \beta_{5} + 45 \beta_{3} - 12 \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$14 \beta_{11} + 28 \beta_{10} - 16 \beta_{9} + 32 \beta_{8} - 93 \beta_{4} + 2 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$30 \beta_{6} + 30 \beta_{5} - 135 \beta_{2} - 453$$ $$\nu^{7}$$ $$=$$ $$-165 \beta_{11} + 165 \beta_{10} + 195 \beta_{9} + 195 \beta_{8} - 933 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-1488 \beta_{7} + 720 \beta_{6} - 360 \beta_{5} - 4785 \beta_{3} - 4785$$ $$\nu^{9}$$ $$=$$ $$-3696 \beta_{11} - 1848 \beta_{10} + 4416 \beta_{9} - 2208 \beta_{8} + 10737 \beta_{4} - 10377 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-16281 \beta_{7} + 4056 \beta_{6} - 8112 \beta_{5} - 51525 \beta_{3} + 16281 \beta_{2}$$ $$\nu^{11}$$ $$=$$ $$-20337 \beta_{11} - 40674 \beta_{10} + 24393 \beta_{9} - 48786 \beta_{8} + 116649 \beta_{4} - 4056 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/35\mathbb{Z}\right)^\times$$.

 $$n$$ $$22$$ $$31$$ $$\chi(n)$$ $$-1$$ $$-\beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 −2.85853 − 1.65037i −1.74002 − 1.00460i −0.226181 − 0.130586i 0.226181 + 0.130586i 1.74002 + 1.00460i 2.85853 + 1.65037i −2.85853 + 1.65037i −1.74002 + 1.00460i −0.226181 + 0.130586i 0.226181 − 0.130586i 1.74002 − 1.00460i 2.85853 − 1.65037i
−2.85853 1.65037i 2.20085 + 3.81198i 3.44745 + 5.97116i 3.20324 3.83918i 14.5289i 5.32175 + 4.54742i 9.55533i −5.18747 + 8.98496i −15.4926 + 5.68786i
19.2 −1.74002 1.00460i −0.784824 1.35935i 0.0184439 + 0.0319457i −4.64034 1.86205i 3.15374i 1.38623 6.86137i 7.96269i 3.26810 5.66052i 6.20367 + 7.90169i
19.3 −0.226181 0.130586i −1.88157 3.25898i −1.96589 3.40503i 4.98089 0.436783i 0.982826i 1.66053 + 6.80019i 2.07156i −2.58064 + 4.46979i −1.18362 0.551640i
19.4 0.226181 + 0.130586i 1.88157 + 3.25898i −1.96589 3.40503i 2.11218 + 4.53196i 0.982826i −1.66053 6.80019i 2.07156i −2.58064 + 4.46979i −0.114075 + 1.30086i
19.5 1.74002 + 1.00460i 0.784824 + 1.35935i 0.0184439 + 0.0319457i −3.93275 3.08763i 3.15374i −1.38623 + 6.86137i 7.96269i 3.26810 5.66052i −3.74123 9.32338i
19.6 2.85853 + 1.65037i −2.20085 3.81198i 3.44745 + 5.97116i −1.72321 + 4.69367i 14.5289i −5.32175 4.54742i 9.55533i −5.18747 + 8.98496i −12.6721 + 10.5731i
24.1 −2.85853 + 1.65037i 2.20085 3.81198i 3.44745 5.97116i 3.20324 + 3.83918i 14.5289i 5.32175 4.54742i 9.55533i −5.18747 8.98496i −15.4926 5.68786i
24.2 −1.74002 + 1.00460i −0.784824 + 1.35935i 0.0184439 0.0319457i −4.64034 + 1.86205i 3.15374i 1.38623 + 6.86137i 7.96269i 3.26810 + 5.66052i 6.20367 7.90169i
24.3 −0.226181 + 0.130586i −1.88157 + 3.25898i −1.96589 + 3.40503i 4.98089 + 0.436783i 0.982826i 1.66053 6.80019i 2.07156i −2.58064 4.46979i −1.18362 + 0.551640i
24.4 0.226181 0.130586i 1.88157 3.25898i −1.96589 + 3.40503i 2.11218 4.53196i 0.982826i −1.66053 + 6.80019i 2.07156i −2.58064 4.46979i −0.114075 1.30086i
24.5 1.74002 1.00460i 0.784824 1.35935i 0.0184439 0.0319457i −3.93275 + 3.08763i 3.15374i −1.38623 6.86137i 7.96269i 3.26810 + 5.66052i −3.74123 + 9.32338i
24.6 2.85853 1.65037i −2.20085 + 3.81198i 3.44745 5.97116i −1.72321 4.69367i 14.5289i −5.32175 + 4.54742i 9.55533i −5.18747 8.98496i −12.6721 10.5731i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 24.6 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.d odd 6 1 inner
35.i odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.3.i.a 12
3.b odd 2 1 315.3.bi.c 12
4.b odd 2 1 560.3.br.a 12
5.b even 2 1 inner 35.3.i.a 12
5.c odd 4 2 175.3.i.c 12
7.b odd 2 1 245.3.i.d 12
7.c even 3 1 245.3.c.a 12
7.c even 3 1 245.3.i.d 12
7.d odd 6 1 inner 35.3.i.a 12
7.d odd 6 1 245.3.c.a 12
15.d odd 2 1 315.3.bi.c 12
20.d odd 2 1 560.3.br.a 12
21.g even 6 1 315.3.bi.c 12
28.f even 6 1 560.3.br.a 12
35.c odd 2 1 245.3.i.d 12
35.i odd 6 1 inner 35.3.i.a 12
35.i odd 6 1 245.3.c.a 12
35.j even 6 1 245.3.c.a 12
35.j even 6 1 245.3.i.d 12
35.k even 12 2 175.3.i.c 12
105.p even 6 1 315.3.bi.c 12
140.s even 6 1 560.3.br.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.i.a 12 1.a even 1 1 trivial
35.3.i.a 12 5.b even 2 1 inner
35.3.i.a 12 7.d odd 6 1 inner
35.3.i.a 12 35.i odd 6 1 inner
175.3.i.c 12 5.c odd 4 2
175.3.i.c 12 35.k even 12 2
245.3.c.a 12 7.c even 3 1
245.3.c.a 12 7.d odd 6 1
245.3.c.a 12 35.i odd 6 1
245.3.c.a 12 35.j even 6 1
245.3.i.d 12 7.b odd 2 1
245.3.i.d 12 7.c even 3 1
245.3.i.d 12 35.c odd 2 1
245.3.i.d 12 35.j even 6 1
315.3.bi.c 12 3.b odd 2 1
315.3.bi.c 12 15.d odd 2 1
315.3.bi.c 12 21.g even 6 1
315.3.bi.c 12 105.p even 6 1
560.3.br.a 12 4.b odd 2 1
560.3.br.a 12 20.d odd 2 1
560.3.br.a 12 28.f even 6 1
560.3.br.a 12 140.s even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(35, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 9 T^{2} + 36 T^{4} + 11 T^{6} - 468 T^{8} - 1791 T^{10} - 6135 T^{12} - 28656 T^{14} - 119808 T^{16} + 45056 T^{18} + 2359296 T^{20} + 9437184 T^{22} + 16777216 T^{24}$$
$3$ $$1 - 18 T^{2} + 48 T^{4} + 700 T^{6} + 2808 T^{8} - 95466 T^{10} + 798418 T^{12} - 7732746 T^{14} + 18423288 T^{16} + 372008700 T^{18} + 2066242608 T^{20} - 62762119218 T^{22} + 282429536481 T^{24}$$
$5$ $$1 - 9 T^{2} + 306 T^{4} - 2160 T^{5} - 10025 T^{6} - 54000 T^{7} + 191250 T^{8} - 3515625 T^{10} + 244140625 T^{12}$$
$7$ $$1 + 162 T^{2} + 12348 T^{4} + 657874 T^{6} + 29647548 T^{8} + 933897762 T^{10} + 13841287201 T^{12}$$
$11$ $$( 1 - 3 T - 294 T^{2} + 599 T^{3} + 52884 T^{4} - 52407 T^{5} - 7129020 T^{6} - 6341247 T^{7} + 774274644 T^{8} + 1061165039 T^{9} - 63021511014 T^{10} - 77812273803 T^{11} + 3138428376721 T^{12} )^{2}$$
$13$ $$( 1 + 525 T^{2} + 176235 T^{4} + 35127322 T^{6} + 5033447835 T^{8} + 428258628525 T^{10} + 23298085122481 T^{12} )^{2}$$
$17$ $$1 - 978 T^{2} + 403953 T^{4} - 153666190 T^{6} + 70041800718 T^{8} - 23391307509426 T^{10} + 6380863165523493 T^{12} - 1953665394494768946 T^{14} +$$$$48\!\cdots\!38$$$$T^{16} -$$$$89\!\cdots\!90$$$$T^{18} +$$$$19\!\cdots\!93$$$$T^{20} -$$$$39\!\cdots\!78$$$$T^{22} +$$$$33\!\cdots\!21$$$$T^{24}$$
$19$ $$( 1 - 9 T + 786 T^{2} - 6831 T^{3} + 329514 T^{4} - 5767407 T^{5} + 138225724 T^{6} - 2082033927 T^{7} + 42942593994 T^{8} - 321370413111 T^{9} + 13349080550226 T^{10} - 55179596320209 T^{11} + 2213314919066161 T^{12} )^{2}$$
$23$ $$1 + 1791 T^{2} + 1735353 T^{4} + 949509974 T^{6} + 244638905769 T^{8} - 93367738199529 T^{10} - 102138108157049694 T^{12} - 26128121225494394889 T^{14} +$$$$19\!\cdots\!89$$$$T^{16} +$$$$20\!\cdots\!54$$$$T^{18} +$$$$10\!\cdots\!33$$$$T^{20} +$$$$30\!\cdots\!91$$$$T^{22} +$$$$48\!\cdots\!41$$$$T^{24}$$
$29$ $$( 1 + 1980 T^{2} - 4508 T^{3} + 1665180 T^{4} + 594823321 T^{6} )^{4}$$
$31$ $$( 1 + 54 T + 3969 T^{2} + 161838 T^{3} + 7973532 T^{4} + 261757926 T^{5} + 9562987675 T^{6} + 251549366886 T^{7} + 7363724246172 T^{8} + 143631820725678 T^{9} + 3385124527603329 T^{10} + 44259927496963254 T^{11} + 787662783788549761 T^{12} )^{2}$$
$37$ $$1 + 1311 T^{2} - 608082 T^{4} - 4472473351 T^{6} - 3878060107536 T^{8} + 1923001154428131 T^{10} + 11130752847775425036 T^{12} +$$$$36\!\cdots\!91$$$$T^{14} -$$$$13\!\cdots\!56$$$$T^{16} -$$$$29\!\cdots\!31$$$$T^{18} -$$$$75\!\cdots\!62$$$$T^{20} +$$$$30\!\cdots\!11$$$$T^{22} +$$$$43\!\cdots\!61$$$$T^{24}$$
$41$ $$( 1 - 9009 T^{2} + 35490546 T^{4} - 77874429761 T^{6} + 100287800755506 T^{8} - 71936191389151089 T^{10} + 22563490300366186081 T^{12} )^{2}$$
$43$ $$( 1 - 4302 T^{2} + 12294324 T^{4} - 28688891246 T^{6} + 42031847185524 T^{8} - 50282637594239502 T^{10} + 39959630797262576401 T^{12} )^{2}$$
$47$ $$1 - 4917 T^{2} + 4518390 T^{4} - 4665949483 T^{6} + 64359941563332 T^{8} - 77972699319432741 T^{10} -$$$$11\!\cdots\!28$$$$T^{12} -$$$$38\!\cdots\!21$$$$T^{14} +$$$$15\!\cdots\!52$$$$T^{16} -$$$$54\!\cdots\!03$$$$T^{18} +$$$$25\!\cdots\!90$$$$T^{20} -$$$$13\!\cdots\!17$$$$T^{22} +$$$$13\!\cdots\!81$$$$T^{24}$$
$53$ $$1 + 9987 T^{2} + 43544766 T^{4} + 176791767125 T^{6} + 794854751690700 T^{8} + 2566788752469653379 T^{10} +$$$$68\!\cdots\!68$$$$T^{12} +$$$$20\!\cdots\!99$$$$T^{14} +$$$$49\!\cdots\!00$$$$T^{16} +$$$$86\!\cdots\!25$$$$T^{18} +$$$$16\!\cdots\!86$$$$T^{20} +$$$$30\!\cdots\!87$$$$T^{22} +$$$$24\!\cdots\!81$$$$T^{24}$$
$59$ $$( 1 - 198 T + 20817 T^{2} - 1534302 T^{3} + 81357012 T^{4} - 3269775942 T^{5} + 144715968451 T^{6} - 11382090054102 T^{7} + 985832284285332 T^{8} - 64717677126453582 T^{9} + 3056569219609150257 T^{10} -$$$$10\!\cdots\!98$$$$T^{11} +$$$$17\!\cdots\!81$$$$T^{12} )^{2}$$
$61$ $$( 1 + 54 T + 9270 T^{2} + 448092 T^{3} + 43173882 T^{4} + 1646729082 T^{5} + 167471347078 T^{6} + 6127478914122 T^{7} + 597778705524762 T^{8} + 23085867588169212 T^{9} + 1777126791484794870 T^{10} + 38520517229795660454 T^{11} +$$$$26\!\cdots\!21$$$$T^{12} )^{2}$$
$67$ $$1 + 18594 T^{2} + 182612664 T^{4} + 1241779502180 T^{6} + 6668167043256336 T^{8} + 31414801721204265066 T^{10} +$$$$14\!\cdots\!18$$$$T^{12} +$$$$63\!\cdots\!86$$$$T^{14} +$$$$27\!\cdots\!76$$$$T^{16} +$$$$10\!\cdots\!80$$$$T^{18} +$$$$30\!\cdots\!84$$$$T^{20} +$$$$61\!\cdots\!94$$$$T^{22} +$$$$66\!\cdots\!21$$$$T^{24}$$
$71$ $$( 1 - 48 T + 10821 T^{2} - 468428 T^{3} + 54548661 T^{4} - 1219760688 T^{5} + 128100283921 T^{6} )^{4}$$
$73$ $$1 - 28914 T^{2} + 473704881 T^{4} - 5432049531118 T^{6} + 48062871940123566 T^{8} -$$$$34\!\cdots\!54$$$$T^{10} +$$$$20\!\cdots\!57$$$$T^{12} -$$$$97\!\cdots\!14$$$$T^{14} +$$$$38\!\cdots\!46$$$$T^{16} -$$$$12\!\cdots\!78$$$$T^{18} +$$$$30\!\cdots\!41$$$$T^{20} -$$$$53\!\cdots\!14$$$$T^{22} +$$$$52\!\cdots\!41$$$$T^{24}$$
$79$ $$( 1 + 96 T - 3885 T^{2} - 1175224 T^{3} - 26907300 T^{4} + 3977488764 T^{5} + 535804365705 T^{6} + 24823507376124 T^{7} - 1048041514491300 T^{8} - 285682211827211704 T^{9} - 5893967726486989485 T^{10} +$$$$90\!\cdots\!96$$$$T^{11} +$$$$59\!\cdots\!41$$$$T^{12} )^{2}$$
$83$ $$( 1 + 17814 T^{2} + 211711020 T^{4} + 1761588062494 T^{6} + 10047449546397420 T^{8} + 40122333823324876374 T^{10} +$$$$10\!\cdots\!61$$$$T^{12} )^{2}$$
$89$ $$( 1 - 342 T + 66990 T^{2} - 9576684 T^{3} + 1074495570 T^{4} - 104351555322 T^{5} + 9499427045782 T^{6} - 826568669705562 T^{7} + 67416260006372370 T^{8} - 4759432777445553324 T^{9} +$$$$26\!\cdots\!90$$$$T^{10} -$$$$10\!\cdots\!42$$$$T^{11} +$$$$24\!\cdots\!21$$$$T^{12} )^{2}$$
$97$ $$( 1 + 38226 T^{2} + 661356303 T^{4} + 7342899199036 T^{6} + 58549397989408143 T^{8} +$$$$29\!\cdots\!86$$$$T^{10} +$$$$69\!\cdots\!41$$$$T^{12} )^{2}$$