# Properties

 Label 225.4.k.c Level $225$ Weight $4$ Character orbit 225.k Analytic conductor $13.275$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 225.k (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.2754297513$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 23x^{10} + 198x^{8} - 719x^{6} + 886x^{4} + 585x^{2} + 81$$ x^12 - 23*x^10 + 198*x^8 - 719*x^6 + 886*x^4 + 585*x^2 + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 45) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{11} - \beta_{7}) q^{2} + (\beta_{11} + \beta_{8} + \beta_{5}) q^{3} + ( - \beta_{6} - 2 \beta_{4} - 4 \beta_{3} - \beta_1) q^{4} + (\beta_{4} + 5 \beta_{3} - 3 \beta_{2} - \beta_1 + 17) q^{6} + (\beta_{11} + 6 \beta_{10} - 5 \beta_{9} - \beta_{7} - 5 \beta_{5}) q^{7} + (3 \beta_{8} - 3 \beta_{7} - 3 \beta_{5}) q^{8} + (\beta_{6} + 4 \beta_{4} + 22 \beta_{3} - 5 \beta_{2} - \beta_1 + 2) q^{9}+O(q^{10})$$ q + (b11 - b7) * q^2 + (b11 + b8 + b5) * q^3 + (-b6 - 2*b4 - 4*b3 - b1) * q^4 + (b4 + 5*b3 - 3*b2 - b1 + 17) * q^6 + (b11 + 6*b10 - 5*b9 - b7 - 5*b5) * q^7 + (3*b8 - 3*b7 - 3*b5) * q^8 + (b6 + 4*b4 + 22*b3 - 5*b2 - b1 + 2) * q^9 $$q + (\beta_{11} - \beta_{7}) q^{2} + (\beta_{11} + \beta_{8} + \beta_{5}) q^{3} + ( - \beta_{6} - 2 \beta_{4} - 4 \beta_{3} - \beta_1) q^{4} + (\beta_{4} + 5 \beta_{3} - 3 \beta_{2} - \beta_1 + 17) q^{6} + (\beta_{11} + 6 \beta_{10} - 5 \beta_{9} - \beta_{7} - 5 \beta_{5}) q^{7} + (3 \beta_{8} - 3 \beta_{7} - 3 \beta_{5}) q^{8} + (\beta_{6} + 4 \beta_{4} + 22 \beta_{3} - 5 \beta_{2} - \beta_1 + 2) q^{9} + ( - 7 \beta_{6} - 2 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - 3) q^{11} + (17 \beta_{11} + 10 \beta_{10} + 7 \beta_{9} - 7 \beta_{8} - 11 \beta_{7} - 3 \beta_{5}) q^{12} + (14 \beta_{11} + 3 \beta_{10} + 10 \beta_{9} - 3 \beta_{8}) q^{13} + ( - 12 \beta_{6} + 15 \beta_{4} + 18 \beta_{3} - 12 \beta_1) q^{14} + ( - \beta_{6} + 19 \beta_{4} + 11 \beta_{3} - 19 \beta_{2} + 11) q^{16} + ( - 3 \beta_{8} - 2 \beta_{7} + 27 \beta_{5}) q^{17} + (17 \beta_{11} + 5 \beta_{10} - 4 \beta_{9} - \beta_{8} + 14 \beta_{7} - 5 \beta_{5}) q^{18} + (4 \beta_{2} + 5 \beta_1 + 55) q^{19} + (6 \beta_{6} - 3 \beta_{4} - 33 \beta_{3} + 18 \beta_{2} - 12 \beta_1 - 51) q^{21} + ( - 40 \beta_{11} - 33 \beta_{10} + 45 \beta_{9} + 33 \beta_{8}) q^{22} + (9 \beta_{11} + 36 \beta_{10} + 42 \beta_{9} - 36 \beta_{8}) q^{23} + ( - 9 \beta_{6} + 3 \beta_{4} - 21 \beta_{3} - 12 \beta_1 - 3) q^{24} + ( - 32 \beta_{2} - 7 \beta_1 + 153) q^{26} + ( - 24 \beta_{11} + 9 \beta_{10} - 9 \beta_{9} - 33 \beta_{8} + 45 \beta_{7} + \cdots - 33 \beta_{5}) q^{27}+ \cdots + ( - 80 \beta_{6} + 133 \beta_{4} - 575 \beta_{3} - 68 \beta_{2} + \cdots - 208) q^{99}+O(q^{100})$$ q + (b11 - b7) * q^2 + (b11 + b8 + b5) * q^3 + (-b6 - 2*b4 - 4*b3 - b1) * q^4 + (b4 + 5*b3 - 3*b2 - b1 + 17) * q^6 + (b11 + 6*b10 - 5*b9 - b7 - 5*b5) * q^7 + (3*b8 - 3*b7 - 3*b5) * q^8 + (b6 + 4*b4 + 22*b3 - 5*b2 - b1 + 2) * q^9 + (-7*b6 - 2*b4 - 3*b3 + 2*b2 - 3) * q^11 + (17*b11 + 10*b10 + 7*b9 - 7*b8 - 11*b7 - 3*b5) * q^12 + (14*b11 + 3*b10 + 10*b9 - 3*b8) * q^13 + (-12*b6 + 15*b4 + 18*b3 - 12*b1) * q^14 + (-b6 + 19*b4 + 11*b3 - 19*b2 + 11) * q^16 + (-3*b8 - 2*b7 + 27*b5) * q^17 + (17*b11 + 5*b10 - 4*b9 - b8 + 14*b7 - 5*b5) * q^18 + (4*b2 + 5*b1 + 55) * q^19 + (6*b6 - 3*b4 - 33*b3 + 18*b2 - 12*b1 - 51) * q^21 + (-40*b11 - 33*b10 + 45*b9 + 33*b8) * q^22 + (9*b11 + 36*b10 + 42*b9 - 36*b8) * q^23 + (-9*b6 + 3*b4 - 21*b3 - 12*b1 - 3) * q^24 + (-32*b2 - 7*b1 + 153) * q^26 + (-24*b11 + 9*b10 - 9*b9 - 33*b8 + 45*b7 - 33*b5) * q^27 + (27*b8 - 19*b7 - 74*b5) * q^28 + (-7*b6 - 23*b4 - 117*b3 + 23*b2 - 117) * q^29 + (31*b6 + 2*b4 - 127*b3 + 31*b1) * q^31 + (49*b11 + 21*b9) * q^32 + (18*b11 + 38*b10 + 23*b9 + 33*b8 - 58*b7 - 22*b5) * q^33 + (28*b6 - 37*b4 - 39*b3 + 37*b2 - 39) * q^34 + (13*b6 + 49*b4 + 28*b3 - 29*b2 - 10*b1 + 191) * q^36 + (51*b8 - 24*b7 - 46*b5) * q^37 + (26*b11 - 21*b10 + 27*b9 - 26*b7 + 27*b5) * q^38 + (10*b6 + 69*b4 + 179*b3 - 32*b2 + 3*b1 + 153) * q^39 + (20*b6 + 49*b4 + 72*b3 + 20*b1) * q^41 + (21*b11 + 108*b10 - 162*b9 - 33*b8 - 27*b7 - 114*b5) * q^42 + (-8*b11 + 87*b10 + 128*b9 + 8*b7 + 128*b5) * q^43 + (-47*b2 + 62*b1 - 291) * q^44 + (12*b2 - 3*b1 - 72) * q^46 + (-59*b11 - 15*b10 + 96*b9 + 59*b7 + 96*b5) * q^47 + (-36*b11 - 41*b10 - 101*b9 + 24*b8 + 61*b7 - 146*b5) * q^48 + (13*b6 - 124*b4 - 189*b3 + 13*b1) * q^49 + (28*b6 - 37*b4 - 39*b3 - 53*b2 + 31*b1 - 20) * q^51 + (108*b11 - 21*b10 - 65*b9 - 108*b7 - 65*b5) * q^52 + (24*b8 - 70*b7 - 102*b5) * q^53 + (36*b6 + 75*b4 + 420*b3 + 6*b1 + 87) * q^54 + (-24*b6 - 30*b4 - 237*b3 + 30*b2 - 237) * q^56 + (26*b11 - 21*b10 + 27*b9 + 113*b8 - 18*b7 + 89*b5) * q^57 + (-175*b11 - 12*b10 + 3*b9 + 12*b8) * q^58 + (41*b6 - 149*b4 + 36*b3 + 41*b1) * q^59 + (10*b6 - 85*b4 + 448*b3 + 85*b2 + 448) * q^61 + (-153*b8 + 30*b7 + 213*b5) * q^62 + (21*b11 + 21*b10 - 141*b9 - 141*b8 - 6*b7 + 69*b5) * q^63 + (33*b2 - 36*b1 + 500) * q^64 + (-151*b6 + 48*b4 - 341*b3 - 31*b2 - 33*b1 - 315) * q^66 + (41*b11 - 21*b10 + 308*b9 + 21*b8) * q^67 + (80*b11 + 153*b10 - 54*b9 - 153*b8) * q^68 + (39*b6 + 117*b4 - 399*b3 + 12*b2 + 36*b1 - 72) * q^69 + (-211*b2 - 38*b1 + 81) * q^71 + (87*b11 + 78*b10 - 111*b9 - 48*b8 - 57*b7 - 78*b5) * q^72 + (192*b8 - 44*b7 + 10*b5) * q^73 + (-121*b6 + 100*b4 - 33*b3 - 100*b2 - 33) * q^74 + (-18*b6 - 153*b4 + 23*b3 - 18*b1) * q^76 + (-240*b11 - 3*b10 + 117*b9 + 3*b8) * q^77 + (220*b11 + 3*b10 + 15*b9 + 19*b8 + 78*b7 - 53*b5) * q^78 + (-56*b6 - 136*b4 - 154*b3 + 136*b2 - 154) * q^79 + (39*b6 - 33*b4 - 87*b3 + 210*b2 + 69*b1 - 300) * q^81 + (-51*b8 + 221*b7 + 42*b5) * q^82 + (-105*b11 - 354*b10 + 3*b9 + 105*b7 + 3*b5) * q^83 + (-120*b6 + 90*b4 - 93*b3 + 102*b2 - 147*b1 + 27) * q^84 + (49*b6 + 62*b4 + 531*b3 + 49*b1) * q^86 + (60*b11 - 13*b10 + 35*b9 + 12*b8 - 235*b7 + 32*b5) * q^87 + (-234*b11 - 93*b10 + 168*b9 + 234*b7 + 168*b5) * q^88 + (-156*b2 - 84*b1 - 549) * q^89 + (278*b2 - 143*b1 - 377) * q^91 + (-141*b11 - 261*b10 - 381*b9 + 141*b7 - 381*b5) * q^92 + (-96*b11 - 441*b10 + 12*b9 + 288*b8 + 30*b7 + 213*b5) * q^93 + (170*b6 - 8*b4 + 633*b3 + 170*b1) * q^94 + (28*b6 + 210*b4 + 791*b3 - 119*b2 + 588) * q^96 + (-78*b11 - 60*b10 - 80*b9 + 78*b7 - 80*b5) * q^97 + (-189*b8 - 248*b7 + 339*b5) * q^98 + (-80*b6 + 133*b4 - 575*b3 - 68*b2 - 184*b1 - 208) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 22 q^{4} + 168 q^{6} - 114 q^{9}+O(q^{10})$$ 12 * q + 22 * q^4 + 168 * q^6 - 114 * q^9 $$12 q + 22 q^{4} + 168 q^{6} - 114 q^{9} - 28 q^{11} - 54 q^{14} + 26 q^{16} + 656 q^{19} - 288 q^{21} + 126 q^{24} + 1736 q^{26} - 670 q^{29} + 704 q^{31} - 104 q^{34} + 2172 q^{36} + 780 q^{39} - 374 q^{41} - 3928 q^{44} - 804 q^{46} + 860 q^{49} - 360 q^{51} - 1278 q^{54} - 1410 q^{56} - 596 q^{59} + 2878 q^{61} + 6276 q^{64} - 1932 q^{66} + 1746 q^{69} + 280 q^{71} - 640 q^{74} - 408 q^{76} - 764 q^{79} - 2502 q^{81} + 1818 q^{84} - 3160 q^{86} - 6876 q^{89} - 2840 q^{91} - 4154 q^{94} + 2310 q^{96} + 1524 q^{99}+O(q^{100})$$ 12 * q + 22 * q^4 + 168 * q^6 - 114 * q^9 - 28 * q^11 - 54 * q^14 + 26 * q^16 + 656 * q^19 - 288 * q^21 + 126 * q^24 + 1736 * q^26 - 670 * q^29 + 704 * q^31 - 104 * q^34 + 2172 * q^36 + 780 * q^39 - 374 * q^41 - 3928 * q^44 - 804 * q^46 + 860 * q^49 - 360 * q^51 - 1278 * q^54 - 1410 * q^56 - 596 * q^59 + 2878 * q^61 + 6276 * q^64 - 1932 * q^66 + 1746 * q^69 + 280 * q^71 - 640 * q^74 - 408 * q^76 - 764 * q^79 - 2502 * q^81 + 1818 * q^84 - 3160 * q^86 - 6876 * q^89 - 2840 * q^91 - 4154 * q^94 + 2310 * q^96 + 1524 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 23x^{10} + 198x^{8} - 719x^{6} + 886x^{4} + 585x^{2} + 81$$ :

 $$\beta_{1}$$ $$=$$ $$( 17\nu^{10} - 260\nu^{8} + 1311\nu^{6} - 2275\nu^{4} - 1388\nu^{2} + 33786 ) / 7875$$ (17*v^10 - 260*v^8 + 1311*v^6 - 2275*v^4 - 1388*v^2 + 33786) / 7875 $$\beta_{2}$$ $$=$$ $$( -19\nu^{10} + 445\nu^{8} - 3627\nu^{6} + 9800\nu^{4} + 5566\nu^{2} - 24327 ) / 7875$$ (-19*v^10 + 445*v^8 - 3627*v^6 + 9800*v^4 + 5566*v^2 - 24327) / 7875 $$\beta_{3}$$ $$=$$ $$( -18\nu^{10} + 415\nu^{8} - 3594\nu^{6} + 13350\nu^{4} - 18398\nu^{2} - 5994 ) / 1125$$ (-18*v^10 + 415*v^8 - 3594*v^6 + 13350*v^4 - 18398*v^2 - 5994) / 1125 $$\beta_{4}$$ $$=$$ $$( -368\nu^{10} + 8665\nu^{8} - 77019\nu^{6} + 295225\nu^{4} - 411623\nu^{2} - 133794 ) / 7875$$ (-368*v^10 + 8665*v^8 - 77019*v^6 + 295225*v^4 - 411623*v^2 - 133794) / 7875 $$\beta_{5}$$ $$=$$ $$( 181\nu^{11} - 4055\nu^{9} + 33723\nu^{7} - 115325\nu^{5} + 124141\nu^{3} + 132273\nu ) / 23625$$ (181*v^11 - 4055*v^9 + 33723*v^7 - 115325*v^5 + 124141*v^3 + 132273*v) / 23625 $$\beta_{6}$$ $$=$$ $$( 106\nu^{10} - 2455\nu^{8} + 21423\nu^{6} - 80150\nu^{4} + 111416\nu^{2} + 29448 ) / 1575$$ (106*v^10 - 2455*v^8 + 21423*v^6 - 80150*v^4 + 111416*v^2 + 29448) / 1575 $$\beta_{7}$$ $$=$$ $$( -88\nu^{11} + 2015\nu^{9} - 17154\nu^{7} + 60725\nu^{5} - 67168\nu^{3} - 70929\nu ) / 6750$$ (-88*v^11 + 2015*v^9 - 17154*v^7 + 60725*v^5 - 67168*v^3 - 70929*v) / 6750 $$\beta_{8}$$ $$=$$ $$( -316\nu^{11} + 7355\nu^{9} - 64053\nu^{7} + 233450\nu^{5} - 264751\nu^{3} - 277353\nu ) / 23625$$ (-316*v^11 + 7355*v^9 - 64053*v^7 + 233450*v^5 - 264751*v^3 - 277353*v) / 23625 $$\beta_{9}$$ $$=$$ $$( -1528\nu^{11} + 35465\nu^{9} - 309924\nu^{7} + 1163225\nu^{5} - 1604758\nu^{3} - 514449\nu ) / 23625$$ (-1528*v^11 + 35465*v^9 - 309924*v^7 + 1163225*v^5 - 1604758*v^3 - 514449*v) / 23625 $$\beta_{10}$$ $$=$$ $$( 2818\nu^{11} - 65540\nu^{9} + 575244\nu^{7} - 2181725\nu^{5} + 3102073\nu^{3} + 800019\nu ) / 23625$$ (2818*v^11 - 65540*v^9 + 575244*v^7 - 2181725*v^5 + 3102073*v^3 + 800019*v) / 23625 $$\beta_{11}$$ $$=$$ $$( -811\nu^{11} + 18830\nu^{9} - 164763\nu^{7} + 620450\nu^{5} - 860821\nu^{3} - 275688\nu ) / 6750$$ (-811*v^11 + 18830*v^9 - 164763*v^7 + 620450*v^5 - 860821*v^3 - 275688*v) / 6750
 $$\nu$$ $$=$$ $$( 4\beta_{11} + 4\beta_{10} + \beta_{9} - 2\beta_{8} - 2\beta_{7} + 2\beta_{5} ) / 6$$ (4*b11 + 4*b10 + b9 - 2*b8 - 2*b7 + 2*b5) / 6 $$\nu^{2}$$ $$=$$ $$( \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 11 ) / 3$$ (b6 + b4 + b3 + b2 - b1 + 11) / 3 $$\nu^{3}$$ $$=$$ $$( 26\beta_{11} + 20\beta_{10} - 7\beta_{9} - 10\beta_{8} - 4\beta_{7} + 13\beta_{5} ) / 6$$ (26*b11 + 20*b10 - 7*b9 - 10*b8 - 4*b7 + 13*b5) / 6 $$\nu^{4}$$ $$=$$ $$( 14\beta_{6} + 8\beta_{4} + 35\beta_{3} - \beta_{2} - 5\beta _1 + 79 ) / 3$$ (14*b6 + 8*b4 + 35*b3 - b2 - 5*b1 + 79) / 3 $$\nu^{5}$$ $$=$$ $$( 154\beta_{11} + 94\beta_{10} - 92\beta_{9} - 92\beta_{8} + 58\beta_{7} + 113\beta_{5} ) / 6$$ (154*b11 + 94*b10 - 92*b9 - 92*b8 + 58*b7 + 113*b5) / 6 $$\nu^{6}$$ $$=$$ $$( 151\beta_{6} + 46\beta_{4} + 505\beta_{3} - 26\beta_{2} + 2\beta _1 + 560 ) / 3$$ (151*b6 + 46*b4 + 505*b3 - 26*b2 + 2*b1 + 560) / 3 $$\nu^{7}$$ $$=$$ $$( 692\beta_{11} + 350\beta_{10} - 565\beta_{9} - 868\beta_{8} + 1166\beta_{7} + 1102\beta_{5} ) / 6$$ (692*b11 + 350*b10 - 565*b9 - 868*b8 + 1166*b7 + 1102*b5) / 6 $$\nu^{8}$$ $$=$$ $$( 1430\beta_{6} + 242\beta_{4} + 5399\beta_{3} - 190\beta_{2} + 460\beta _1 + 3580 ) / 3$$ (1430*b6 + 242*b4 + 5399*b3 - 190*b2 + 460*b1 + 3580) / 3 $$\nu^{9}$$ $$=$$ $$( 262\beta_{11} - 218\beta_{10} - 947\beta_{9} - 7154\beta_{8} + 14116\beta_{7} + 11039\beta_{5} ) / 6$$ (262*b11 - 218*b10 - 947*b9 - 7154*b8 + 14116*b7 + 11039*b5) / 6 $$\nu^{10}$$ $$=$$ $$( 12181\beta_{6} + 1306\beta_{4} + 48394\beta_{3} - 953\beta_{2} + 7520\beta _1 + 17075 ) / 3$$ (12181*b6 + 1306*b4 + 48394*b3 - 953*b2 + 7520*b1 + 17075) / 3 $$\nu^{11}$$ $$=$$ $$( -45694\beta_{11} - 26842\beta_{10} + 29504\beta_{9} - 48850\beta_{8} + 140162\beta_{7} + 104395\beta_{5} ) / 6$$ (-45694*b11 - 26842*b10 + 29504*b9 - 48850*b8 + 140162*b7 + 104395*b5) / 6

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$\beta_{3}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −2.88506 − 0.500000i −0.0378788 + 0.500000i 1.98116 − 0.500000i −1.98116 + 0.500000i 0.0378788 − 0.500000i 2.88506 + 0.500000i −2.88506 + 0.500000i −0.0378788 − 0.500000i 1.98116 + 0.500000i −1.98116 − 0.500000i 0.0378788 + 0.500000i 2.88506 − 0.500000i
−3.96084 + 2.28679i −3.96084 3.36330i 6.45882 11.1870i 0 23.3794 + 4.26387i −17.4197 + 10.0573i 22.4912i 4.37646 + 26.6429i 0
49.2 −3.24635 + 1.87428i −3.24635 4.05724i 3.02587 5.24096i 0 18.1432 + 7.08665i 27.1492 15.6746i 7.30318i −5.92239 + 26.3425i 0
49.3 −0.151541 + 0.0874923i −0.151541 5.19394i −3.98469 + 6.90169i 0 0.477395 + 0.773837i 7.32979 4.23186i 2.79440i −26.9541 + 1.57419i 0
49.4 0.151541 0.0874923i 0.151541 + 5.19394i −3.98469 + 6.90169i 0 0.477395 + 0.773837i −7.32979 + 4.23186i 2.79440i −26.9541 + 1.57419i 0
49.5 3.24635 1.87428i 3.24635 + 4.05724i 3.02587 5.24096i 0 18.1432 + 7.08665i −27.1492 + 15.6746i 7.30318i −5.92239 + 26.3425i 0
49.6 3.96084 2.28679i 3.96084 + 3.36330i 6.45882 11.1870i 0 23.3794 + 4.26387i 17.4197 10.0573i 22.4912i 4.37646 + 26.6429i 0
124.1 −3.96084 2.28679i −3.96084 + 3.36330i 6.45882 + 11.1870i 0 23.3794 4.26387i −17.4197 10.0573i 22.4912i 4.37646 26.6429i 0
124.2 −3.24635 1.87428i −3.24635 + 4.05724i 3.02587 + 5.24096i 0 18.1432 7.08665i 27.1492 + 15.6746i 7.30318i −5.92239 26.3425i 0
124.3 −0.151541 0.0874923i −0.151541 + 5.19394i −3.98469 6.90169i 0 0.477395 0.773837i 7.32979 + 4.23186i 2.79440i −26.9541 1.57419i 0
124.4 0.151541 + 0.0874923i 0.151541 5.19394i −3.98469 6.90169i 0 0.477395 0.773837i −7.32979 4.23186i 2.79440i −26.9541 1.57419i 0
124.5 3.24635 + 1.87428i 3.24635 4.05724i 3.02587 + 5.24096i 0 18.1432 7.08665i −27.1492 15.6746i 7.30318i −5.92239 26.3425i 0
124.6 3.96084 + 2.28679i 3.96084 3.36330i 6.45882 + 11.1870i 0 23.3794 4.26387i 17.4197 + 10.0573i 22.4912i 4.37646 26.6429i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 124.6 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
9.c even 3 1 inner
45.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.k.c 12
5.b even 2 1 inner 225.4.k.c 12
5.c odd 4 1 45.4.e.b 6
5.c odd 4 1 225.4.e.c 6
9.c even 3 1 inner 225.4.k.c 12
15.e even 4 1 135.4.e.b 6
45.j even 6 1 inner 225.4.k.c 12
45.k odd 12 1 45.4.e.b 6
45.k odd 12 1 225.4.e.c 6
45.k odd 12 1 405.4.a.h 3
45.k odd 12 1 2025.4.a.s 3
45.l even 12 1 135.4.e.b 6
45.l even 12 1 405.4.a.j 3
45.l even 12 1 2025.4.a.q 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.e.b 6 5.c odd 4 1
45.4.e.b 6 45.k odd 12 1
135.4.e.b 6 15.e even 4 1
135.4.e.b 6 45.l even 12 1
225.4.e.c 6 5.c odd 4 1
225.4.e.c 6 45.k odd 12 1
225.4.k.c 12 1.a even 1 1 trivial
225.4.k.c 12 5.b even 2 1 inner
225.4.k.c 12 9.c even 3 1 inner
225.4.k.c 12 45.j even 6 1 inner
405.4.a.h 3 45.k odd 12 1
405.4.a.j 3 45.l even 12 1
2025.4.a.q 3 45.l even 12 1
2025.4.a.s 3 45.k odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} - 35T_{2}^{10} + 930T_{2}^{8} - 10307T_{2}^{6} + 86710T_{2}^{4} - 2655T_{2}^{2} + 81$$ acting on $$S_{4}^{\mathrm{new}}(225, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 35 T^{10} + 930 T^{8} + \cdots + 81$$
$3$ $$T^{12} + 57 T^{10} + \cdots + 387420489$$
$5$ $$T^{12}$$
$7$ $$T^{12} + \cdots + 811313702977761$$
$11$ $$(T^{6} + 14 T^{5} + 3012 T^{4} + \cdots + 1896428304)^{2}$$
$13$ $$T^{12} - 6504 T^{10} + \cdots + 32\!\cdots\!16$$
$17$ $$(T^{6} + 9716 T^{4} + \cdots + 24437192976)^{2}$$
$19$ $$(T^{3} - 164 T^{2} + 7292 T - 57316)^{4}$$
$23$ $$T^{12} - 38907 T^{10} + \cdots + 14\!\cdots\!61$$
$29$ $$(T^{6} + 335 T^{5} + \cdots + 11463342489)^{2}$$
$31$ $$(T^{6} - 352 T^{5} + \cdots + 97238137683600)^{2}$$
$37$ $$(T^{6} + 112188 T^{4} + \cdots + 11124119360656)^{2}$$
$41$ $$(T^{6} + 187 T^{5} + \cdots + 52247959475625)^{2}$$
$43$ $$T^{12} - 342492 T^{10} + \cdots + 56\!\cdots\!76$$
$47$ $$T^{12} - 251663 T^{10} + \cdots + 38\!\cdots\!21$$
$53$ $$(T^{6} + 320300 T^{4} + \cdots + 10565984287296)^{2}$$
$59$ $$(T^{6} + 298 T^{5} + \cdots + 16\!\cdots\!16)^{2}$$
$61$ $$(T^{6} - 1439 T^{5} + \cdots + 30\!\cdots\!09)^{2}$$
$67$ $$T^{12} - 1231447 T^{10} + \cdots + 18\!\cdots\!61$$
$71$ $$(T^{3} - 70 T^{2} - 685460 T + 223775052)^{4}$$
$73$ $$(T^{6} + 881280 T^{4} + \cdots + 18\!\cdots\!36)^{2}$$
$79$ $$(T^{6} + 382 T^{5} + \cdots + 19\!\cdots\!56)^{2}$$
$83$ $$T^{12} - 3012507 T^{10} + \cdots + 83\!\cdots\!81$$
$89$ $$(T^{3} + 1719 T^{2} + 238491 T - 125506395)^{4}$$
$97$ $$T^{12} - 214860 T^{10} + \cdots + 81\!\cdots\!16$$