# Properties

 Label 4012.2.a.g Level 4012 Weight 2 Character orbit 4012.a Self dual yes Analytic conductor 32.036 Analytic rank 1 Dimension 12 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4012 = 2^{2} \cdot 17 \cdot 59$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4012.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.0359812909$$ Analytic rank: $$1$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + \beta_{9} q^{5} + ( -\beta_{2} + \beta_{8} - \beta_{9} ) q^{7} + ( \beta_{1} + \beta_{4} + \beta_{5} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + \beta_{9} q^{5} + ( -\beta_{2} + \beta_{8} - \beta_{9} ) q^{7} + ( \beta_{1} + \beta_{4} + \beta_{5} ) q^{9} + ( -1 + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{9} + \beta_{11} ) q^{11} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{8} - \beta_{9} ) q^{13} + ( -1 - \beta_{1} + \beta_{2} - \beta_{5} + \beta_{7} ) q^{15} - q^{17} + ( -1 - \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{19} + ( 1 + \beta_{3} - \beta_{8} + \beta_{9} ) q^{21} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{9} - 2 \beta_{11} ) q^{23} + ( 2 \beta_{1} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{9} ) q^{25} + ( -2 - \beta_{1} + 2 \beta_{2} - 2 \beta_{5} - \beta_{6} + \beta_{8} - \beta_{10} - \beta_{11} ) q^{27} + ( -1 + \beta_{3} - \beta_{6} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{29} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{31} + ( -1 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{11} ) q^{33} + ( -\beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 3 \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{35} + ( 1 + 3 \beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{37} + ( -2 - 3 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{39} + ( -2 - 2 \beta_{1} + \beta_{2} - 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} + \beta_{11} ) q^{41} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} - 3 \beta_{9} ) q^{43} + ( 2 + 3 \beta_{1} - 2 \beta_{2} + \beta_{4} + 3 \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{45} + ( -5 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{47} + ( 2 \beta_{1} - \beta_{4} - \beta_{6} + 2 \beta_{8} - 3 \beta_{9} - \beta_{10} - \beta_{11} ) q^{49} + \beta_{1} q^{51} + ( -3 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{7} + 2 \beta_{9} - \beta_{10} ) q^{53} + ( -2 - \beta_{1} - 2 \beta_{4} - 2 \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{55} + ( 3 + 5 \beta_{1} - \beta_{2} + \beta_{4} + 3 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{9} - \beta_{11} ) q^{57} + q^{59} + ( -3 - \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{10} ) q^{61} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{10} ) q^{63} + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} - 2 \beta_{8} + 3 \beta_{9} - \beta_{10} ) q^{65} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{10} ) q^{67} + ( -2 + 3 \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{6} + \beta_{8} + 2 \beta_{9} + 3 \beta_{11} ) q^{69} + ( -1 - \beta_{3} + 2 \beta_{4} + \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{71} + ( 1 - 3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{11} ) q^{73} + ( -5 - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{7} + 3 \beta_{8} - 4 \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{75} + ( -1 - \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} - \beta_{11} ) q^{77} + ( 5 + 4 \beta_{1} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{79} + ( -1 - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{81} + ( -1 - 3 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{83} -\beta_{9} q^{85} + ( -2 + 3 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + 3 \beta_{8} - 2 \beta_{9} + 2 \beta_{11} ) q^{87} + ( -2 - 3 \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{7} - 5 \beta_{8} + 6 \beta_{9} + 3 \beta_{10} + 4 \beta_{11} ) q^{89} + ( -3 + \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - 5 \beta_{9} - \beta_{10} + \beta_{11} ) q^{91} + ( -2 + 4 \beta_{1} - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - \beta_{7} + 2 \beta_{8} - 4 \beta_{9} - 2 \beta_{11} ) q^{93} + ( -5 - 2 \beta_{1} + 2 \beta_{2} - \beta_{4} - 3 \beta_{5} - \beta_{6} + 2 \beta_{8} - \beta_{9} - 2 \beta_{10} ) q^{95} + ( -2 - 3 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} + 3 \beta_{7} - 2 \beta_{8} + 4 \beta_{9} + \beta_{10} + \beta_{11} ) q^{97} + ( -1 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{7} + \beta_{10} + 3 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 4q^{3} + 3q^{5} + 2q^{7} + 6q^{9} + O(q^{10})$$ $$12q - 4q^{3} + 3q^{5} + 2q^{7} + 6q^{9} - 5q^{11} + 9q^{13} - 16q^{15} - 12q^{17} - 15q^{19} + 5q^{21} - 22q^{23} + 5q^{25} - 16q^{27} - 17q^{29} + 5q^{31} - 33q^{33} - 9q^{35} - 2q^{37} - 24q^{39} - 18q^{41} - 16q^{43} + 30q^{45} - 60q^{47} + 10q^{49} + 4q^{51} - 24q^{53} - q^{55} + 41q^{57} + 12q^{59} - 17q^{61} - 23q^{63} + 2q^{65} - 22q^{67} + 6q^{69} - 10q^{71} + 4q^{73} - 62q^{75} - 6q^{77} + 48q^{79} + 4q^{81} - 20q^{83} - 3q^{85} + 13q^{87} - 29q^{89} - 28q^{91} - 31q^{93} - 48q^{95} - 26q^{97} - 3q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 4 x^{11} - 13 x^{10} + 60 x^{9} + 48 x^{8} - 289 x^{7} - 89 x^{6} + 602 x^{5} + 161 x^{4} - 555 x^{3} - 197 x^{2} + 181 x + 82$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$2338 \nu^{11} - 11665 \nu^{10} - 20131 \nu^{9} + 163345 \nu^{8} - 30162 \nu^{7} - 687740 \nu^{6} + 380492 \nu^{5} + 1189576 \nu^{4} - 606967 \nu^{3} - 905001 \nu^{2} + 253934 \nu + 253800$$$$)/2599$$ $$\beta_{3}$$ $$=$$ $$($$$$-2547 \nu^{11} + 12262 \nu^{10} + 23568 \nu^{9} - 172850 \nu^{8} + 11984 \nu^{7} + 736444 \nu^{6} - 344410 \nu^{5} - 1285735 \nu^{4} + 587872 \nu^{3} + 974824 \nu^{2} - 256909 \nu - 271839$$$$)/2599$$ $$\beta_{4}$$ $$=$$ $$($$$$-4271 \nu^{11} + 20370 \nu^{10} + 40491 \nu^{9} - 289158 \nu^{8} + 7917 \nu^{7} + 1252500 \nu^{6} - 537826 \nu^{5} - 2258499 \nu^{4} + 946617 \nu^{3} + 1803505 \nu^{2} - 421447 \nu - 531563$$$$)/2599$$ $$\beta_{5}$$ $$=$$ $$($$$$4271 \nu^{11} - 20370 \nu^{10} - 40491 \nu^{9} + 289158 \nu^{8} - 7917 \nu^{7} - 1252500 \nu^{6} + 537826 \nu^{5} + 2258499 \nu^{4} - 946617 \nu^{3} - 1800906 \nu^{2} + 418848 \nu + 523766$$$$)/2599$$ $$\beta_{6}$$ $$=$$ $$($$$$5174 \nu^{11} - 24852 \nu^{10} - 48663 \nu^{9} + 352472 \nu^{8} - 13528 \nu^{7} - 1523042 \nu^{6} + 656567 \nu^{5} + 2730194 \nu^{4} - 1130395 \nu^{3} - 2141268 \nu^{2} + 488631 \nu + 604080$$$$)/2599$$ $$\beta_{7}$$ $$=$$ $$($$$$7177 \nu^{11} - 33844 \nu^{10} - 70684 \nu^{9} + 485510 \nu^{8} + 20451 \nu^{7} - 2146631 \nu^{6} + 789903 \nu^{5} + 3978285 \nu^{4} - 1463177 \nu^{3} - 3242526 \nu^{2} + 666268 \nu + 949639$$$$)/2599$$ $$\beta_{8}$$ $$=$$ $$($$$$7808 \nu^{11} - 36691 \nu^{10} - 76783 \nu^{9} + 523720 \nu^{8} + 23527 \nu^{7} - 2291872 \nu^{6} + 831696 \nu^{5} + 4179055 \nu^{4} - 1509014 \nu^{3} - 3334225 \nu^{2} + 673024 \nu + 955056$$$$)/2599$$ $$\beta_{9}$$ $$=$$ $$($$$$359 \nu^{11} - 1664 \nu^{10} - 3574 \nu^{9} + 23728 \nu^{8} + 1593 \nu^{7} - 103673 \nu^{6} + 36826 \nu^{5} + 188615 \nu^{4} - 68695 \nu^{3} - 150396 \nu^{2} + 31304 \nu + 43294$$$$)/113$$ $$\beta_{10}$$ $$=$$ $$($$$$8258 \nu^{11} - 38536 \nu^{10} - 81149 \nu^{9} + 547518 \nu^{8} + 25074 \nu^{7} - 2374515 \nu^{6} + 868264 \nu^{5} + 4272685 \nu^{4} - 1552486 \nu^{3} - 3360792 \nu^{2} + 674617 \nu + 952399$$$$)/2599$$ $$\beta_{11}$$ $$=$$ $$($$$$-9490 \nu^{11} + 44107 \nu^{10} + 93749 \nu^{9} - 627896 \nu^{8} - 32509 \nu^{7} + 2735205 \nu^{6} - 1007803 \nu^{5} - 4961670 \nu^{4} + 1855766 \nu^{3} + 3959645 \nu^{2} - 838245 \nu - 1146553$$$$)/2599$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + \beta_{4} + \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{11} + \beta_{10} - \beta_{8} + \beta_{6} + 2 \beta_{5} - 2 \beta_{2} + 7 \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{6} + 11 \beta_{5} + 8 \beta_{4} + 9 \beta_{1} + 17$$ $$\nu^{5}$$ $$=$$ $$16 \beta_{11} + 10 \beta_{10} + 7 \beta_{9} - 10 \beta_{8} - \beta_{7} + 8 \beta_{6} + 22 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 18 \beta_{2} + 54 \beta_{1} + 17$$ $$\nu^{6}$$ $$=$$ $$39 \beta_{11} + 17 \beta_{10} + 26 \beta_{9} - 5 \beta_{8} + \beta_{7} - 28 \beta_{6} + 107 \beta_{5} + 62 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 78 \beta_{1} + 125$$ $$\nu^{7}$$ $$=$$ $$200 \beta_{11} + 102 \beta_{10} + 119 \beta_{9} - 98 \beta_{8} - 11 \beta_{7} + 49 \beta_{6} + 231 \beta_{5} - 24 \beta_{4} + 33 \beta_{3} - 146 \beta_{2} + 456 \beta_{1} + 150$$ $$\nu^{8}$$ $$=$$ $$533 \beta_{11} + 220 \beta_{10} + 390 \beta_{9} - 102 \beta_{8} + 17 \beta_{7} - 306 \beta_{6} + 1050 \beta_{5} + 502 \beta_{4} + 49 \beta_{3} + 27 \beta_{2} + 740 \beta_{1} + 1045$$ $$\nu^{9}$$ $$=$$ $$2293 \beta_{11} + 1067 \beta_{10} + 1531 \beta_{9} - 991 \beta_{8} - 87 \beta_{7} + 226 \beta_{6} + 2466 \beta_{5} - 186 \beta_{4} + 408 \beta_{3} - 1192 \beta_{2} + 4122 \beta_{1} + 1479$$ $$\nu^{10}$$ $$=$$ $$6431 \beta_{11} + 2602 \beta_{10} + 4889 \beta_{9} - 1482 \beta_{8} + 204 \beta_{7} - 3113 \beta_{6} + 10537 \beta_{5} + 4247 \beta_{4} + 765 \beta_{3} + 202 \beta_{2} + 7578 \beta_{1} + 9443$$ $$\nu^{11}$$ $$=$$ $$25282 \beta_{11} + 11239 \beta_{10} + 17863 \beta_{9} - 10168 \beta_{8} - 604 \beta_{7} + 164 \beta_{6} + 26631 \beta_{5} - 914 \beta_{4} + 4595 \beta_{3} - 10026 \beta_{2} + 39156 \beta_{1} + 15700$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 3.27119 2.43162 2.36915 1.85534 1.18762 0.859886 −0.660935 −0.692763 −0.864716 −1.18842 −1.70453 −2.86344
0 −3.27119 0 3.82141 0 −1.10063 0 7.70067 0
1.2 0 −2.43162 0 −0.659543 0 4.52406 0 2.91279 0
1.3 0 −2.36915 0 2.61284 0 −2.25519 0 2.61289 0
1.4 0 −1.85534 0 −3.75753 0 −3.33364 0 0.442279 0
1.5 0 −1.18762 0 1.03337 0 2.52681 0 −1.58957 0
1.6 0 −0.859886 0 2.49803 0 −3.53399 0 −2.26060 0
1.7 0 0.660935 0 2.45345 0 3.10518 0 −2.56317 0
1.8 0 0.692763 0 −3.58411 0 3.81064 0 −2.52008 0
1.9 0 0.864716 0 −1.40497 0 −2.45790 0 −2.25227 0
1.10 0 1.18842 0 0.677556 0 1.46203 0 −1.58766 0
1.11 0 1.70453 0 −0.637713 0 1.14740 0 −0.0945840 0
1.12 0 2.86344 0 −0.0527883 0 −1.89476 0 5.19930 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4012.2.a.g 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4012.2.a.g 12 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$17$$ $$1$$
$$59$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4012))$$:

 $$T_{3}^{12} + \cdots$$ $$T_{5}^{12} - \cdots$$