# Properties

 Label 315.2.g.a Level 315 Weight 2 Character orbit 315.g Analytic conductor 2.515 Analytic rank 0 Dimension 16 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ = $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 315.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{12}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + ( 1 + \beta_{14} ) q^{4} -\beta_{4} q^{5} -\beta_{12} q^{7} + ( \beta_{3} + \beta_{6} ) q^{8} +O(q^{10})$$ $$q + \beta_{3} q^{2} + ( 1 + \beta_{14} ) q^{4} -\beta_{4} q^{5} -\beta_{12} q^{7} + ( \beta_{3} + \beta_{6} ) q^{8} + \beta_{13} q^{10} + \beta_{7} q^{11} + ( \beta_{8} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{13} + ( \beta_{1} + \beta_{4} + \beta_{9} - \beta_{15} ) q^{14} + q^{16} + ( \beta_{1} + \beta_{4} + \beta_{5} - \beta_{9} ) q^{17} + ( -\beta_{8} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{19} + ( -\beta_{1} - 2 \beta_{4} - 2 \beta_{5} + \beta_{15} ) q^{20} + \beta_{2} q^{22} + ( -2 \beta_{3} + \beta_{6} ) q^{23} + ( 1 - \beta_{2} + \beta_{10} - \beta_{12} - \beta_{14} ) q^{25} + ( -\beta_{4} + \beta_{5} ) q^{26} + ( -\beta_{2} - \beta_{8} - 2 \beta_{10} - \beta_{11} - \beta_{13} ) q^{28} + ( -2 \beta_{1} + \beta_{7} - 2 \beta_{9} ) q^{29} + ( -\beta_{8} - \beta_{10} - \beta_{11} + \beta_{13} ) q^{31} + ( -\beta_{3} - 2 \beta_{6} ) q^{32} + ( 3 \beta_{8} + 2 \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} ) q^{34} + ( -\beta_{3} - \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{9} ) q^{35} + ( -2 \beta_{2} + \beta_{10} - \beta_{12} ) q^{37} + ( 3 \beta_{4} + 3 \beta_{5} ) q^{38} + ( -2 \beta_{8} - \beta_{11} + 2 \beta_{12} ) q^{40} + ( -\beta_{1} - \beta_{4} - \beta_{5} - \beta_{9} + 2 \beta_{15} ) q^{41} + ( -2 \beta_{2} - \beta_{10} + \beta_{12} ) q^{43} + ( -\beta_{1} + \beta_{7} - \beta_{9} ) q^{44} + ( -6 - \beta_{14} ) q^{46} + ( -\beta_{1} + \beta_{9} ) q^{47} + ( 1 + 2 \beta_{8} + \beta_{10} - \beta_{12} + 2 \beta_{14} ) q^{49} + ( 2 \beta_{1} - \beta_{3} - \beta_{6} - 3 \beta_{7} + 2 \beta_{9} ) q^{50} + ( -\beta_{8} + \beta_{10} - \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{52} + 3 \beta_{6} q^{53} + ( \beta_{8} + \beta_{10} ) q^{55} + ( 2 \beta_{4} - \beta_{5} - 3 \beta_{7} - \beta_{15} ) q^{56} + ( 5 \beta_{2} - 2 \beta_{10} + 2 \beta_{12} ) q^{58} + ( \beta_{1} + 2 \beta_{4} + \beta_{9} - 2 \beta_{15} ) q^{59} + ( -4 \beta_{8} - 2 \beta_{10} + 2 \beta_{12} ) q^{61} + ( -\beta_{1} - 5 \beta_{4} - 5 \beta_{5} + \beta_{9} ) q^{62} + ( -5 - 3 \beta_{14} ) q^{64} + ( \beta_{1} + 2 \beta_{3} - 3 \beta_{6} + \beta_{7} + \beta_{9} ) q^{65} + ( 2 \beta_{2} - 2 \beta_{10} + 2 \beta_{12} ) q^{67} + ( 5 \beta_{4} + 5 \beta_{5} ) q^{68} + ( -3 + 3 \beta_{2} - \beta_{10} - \beta_{11} - 2 \beta_{14} ) q^{70} + ( 2 \beta_{1} + \beta_{7} + 2 \beta_{9} ) q^{71} + ( -\beta_{8} - 2 \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{73} + ( 3 \beta_{1} - 6 \beta_{7} + 3 \beta_{9} ) q^{74} + ( 5 \beta_{8} + 3 \beta_{10} + \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{76} + ( -\beta_{4} - \beta_{5} + \beta_{6} ) q^{77} -4 q^{79} -\beta_{4} q^{80} + ( \beta_{8} + 4 \beta_{10} + \beta_{11} + 3 \beta_{12} + \beta_{13} ) q^{82} + ( -\beta_{1} + \beta_{9} ) q^{83} + ( 2 + 3 \beta_{2} + 2 \beta_{10} - 2 \beta_{12} + 3 \beta_{14} ) q^{85} + ( \beta_{1} - 6 \beta_{7} + \beta_{9} ) q^{86} + ( \beta_{2} - \beta_{10} + \beta_{12} ) q^{88} + ( -3 \beta_{4} + 3 \beta_{5} ) q^{89} + ( 2 + 3 \beta_{8} + \beta_{10} - \beta_{11} - 2 \beta_{12} + \beta_{13} - 2 \beta_{14} ) q^{91} + ( -4 \beta_{3} - 3 \beta_{6} ) q^{92} + ( -2 \beta_{8} - \beta_{10} + \beta_{12} ) q^{94} + ( -3 \beta_{1} + 4 \beta_{3} - \beta_{6} - 3 \beta_{7} - 3 \beta_{9} ) q^{95} + ( \beta_{8} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{97} + ( \beta_{1} + 5 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{9} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 16q^{4} + O(q^{10})$$ $$16q + 16q^{4} + 16q^{16} + 16q^{25} - 96q^{46} + 16q^{49} - 80q^{64} - 48q^{70} - 64q^{79} + 32q^{85} + 32q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 32 x^{14} + 324 x^{12} + 1328 x^{10} + 2314 x^{8} + 1920 x^{6} + 780 x^{4} + 144 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$421 \nu^{15} + 12551 \nu^{13} + 107523 \nu^{11} + 279263 \nu^{9} - 67895 \nu^{7} - 640401 \nu^{5} - 458781 \nu^{3} - 77157 \nu$$$$)/1872$$ $$\beta_{2}$$ $$=$$ $$($$$$-68 \nu^{15} - 2211 \nu^{13} - 23101 \nu^{11} - 100054 \nu^{9} - 188622 \nu^{7} - 156497 \nu^{5} - 49071 \nu^{3} - 3072 \nu$$$$)/312$$ $$\beta_{3}$$ $$=$$ $$($$$$103 \nu^{14} + 3280 \nu^{12} + 32851 \nu^{10} + 131324 \nu^{8} + 214559 \nu^{6} + 152528 \nu^{4} + 42663 \nu^{2} + 3084$$$$)/312$$ $$\beta_{4}$$ $$=$$ $$($$$$-1028 \nu^{15} + 57 \nu^{14} - 32587 \nu^{13} + 1839 \nu^{12} - 323232 \nu^{11} + 18993 \nu^{10} - 1266631 \nu^{9} + 81897 \nu^{8} - 1984820 \nu^{7} + 163401 \nu^{6} - 1330083 \nu^{5} + 176883 \nu^{4} - 344256 \nu^{3} + 89325 \nu^{2} - 20979 \nu + 12465$$$$)/1872$$ $$\beta_{5}$$ $$=$$ $$($$$$-1028 \nu^{15} - 57 \nu^{14} - 32587 \nu^{13} - 1839 \nu^{12} - 323232 \nu^{11} - 18993 \nu^{10} - 1266631 \nu^{9} - 81897 \nu^{8} - 1984820 \nu^{7} - 163401 \nu^{6} - 1330083 \nu^{5} - 176883 \nu^{4} - 344256 \nu^{3} - 89325 \nu^{2} - 20979 \nu - 12465$$$$)/1872$$ $$\beta_{6}$$ $$=$$ $$($$$$-87 \nu^{14} - 2720 \nu^{12} - 26182 \nu^{10} - 96114 \nu^{8} - 129001 \nu^{6} - 65828 \nu^{4} - 9972 \nu^{2} + 66$$$$)/156$$ $$\beta_{7}$$ $$=$$ $$($$$$824 \nu^{15} + 25954 \nu^{13} + 253929 \nu^{11} + 966469 \nu^{9} + 1418954 \nu^{7} + 860592 \nu^{5} + 197043 \nu^{3} + 9891 \nu$$$$)/936$$ $$\beta_{8}$$ $$=$$ $$($$$$207 \nu^{15} + 6517 \nu^{13} + 63713 \nu^{11} + 242383 \nu^{9} + 357611 \nu^{7} + 225809 \nu^{5} + 59329 \nu^{3} + 4839 \nu$$$$)/208$$ $$\beta_{9}$$ $$=$$ $$($$$$-2059 \nu^{15} - 64577 \nu^{13} - 625989 \nu^{11} - 2335421 \nu^{9} - 3273079 \nu^{7} - 1844289 \nu^{5} - 369813 \nu^{3} - 11529 \nu$$$$)/1872$$ $$\beta_{10}$$ $$=$$ $$($$$$719 \nu^{15} - 70 \nu^{14} + 22773 \nu^{13} - 2138 \nu^{12} + 225511 \nu^{11} - 19500 \nu^{10} + 881083 \nu^{9} - 62540 \nu^{8} + 1375671 \nu^{7} - 51874 \nu^{6} + 932585 \nu^{5} + 7938 \nu^{4} + 264627 \nu^{3} + 12816 \nu^{2} + 24987 \nu + 600$$$$)/624$$ $$\beta_{11}$$ $$=$$ $$($$$$-127 \nu^{15} + 155 \nu^{14} - 4016 \nu^{13} + 4905 \nu^{12} - 39637 \nu^{11} + 48477 \nu^{10} - 153888 \nu^{9} + 188563 \nu^{8} - 237739 \nu^{7} + 291103 \nu^{6} - 161288 \nu^{5} + 192477 \nu^{4} - 47909 \nu^{3} + 52101 \nu^{2} - 5976 \nu + 4579$$$$)/208$$ $$\beta_{12}$$ $$=$$ $$($$$$-719 \nu^{15} - 70 \nu^{14} - 22773 \nu^{13} - 2138 \nu^{12} - 225511 \nu^{11} - 19500 \nu^{10} - 881083 \nu^{9} - 62540 \nu^{8} - 1375671 \nu^{7} - 51874 \nu^{6} - 932585 \nu^{5} + 7938 \nu^{4} - 264627 \nu^{3} + 12816 \nu^{2} - 24987 \nu + 600$$$$)/624$$ $$\beta_{13}$$ $$=$$ $$($$$$-959 \nu^{15} + 395 \nu^{14} - 30276 \nu^{13} + 12577 \nu^{12} - 297739 \nu^{11} + 125931 \nu^{10} - 1146568 \nu^{9} + 503149 \nu^{8} - 1735287 \nu^{7} + 821435 \nu^{6} - 1126148 \nu^{5} + 585369 \nu^{4} - 298887 \nu^{3} + 169119 \nu^{2} - 21576 \nu + 14337$$$$)/624$$ $$\beta_{14}$$ $$=$$ $$($$$$-214 \nu^{14} - 6749 \nu^{12} - 66222 \nu^{10} - 253811 \nu^{8} - 380182 \nu^{6} - 243405 \nu^{4} - 64134 \nu^{2} - 5247$$$$)/156$$ $$\beta_{15}$$ $$=$$ $$($$$$-1847 \nu^{15} - 3339 \nu^{14} - 58600 \nu^{13} - 105243 \nu^{12} - 582465 \nu^{11} - 1031277 \nu^{10} - 2294710 \nu^{9} - 3939399 \nu^{8} - 3655307 \nu^{7} - 5841759 \nu^{6} - 2572428 \nu^{5} - 3635331 \nu^{4} - 758553 \nu^{3} - 895941 \nu^{2} - 65322 \nu - 59715$$$$)/1872$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{7} - \beta_{5} - \beta_{4} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{15} - 2 \beta_{14} - \beta_{12} - \beta_{10} - \beta_{9} - 2 \beta_{4} + 2 \beta_{3} - \beta_{1} - 8$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{13} - 2 \beta_{12} - 3 \beta_{11} - \beta_{10} - 5 \beta_{9} - 3 \beta_{8} + 11 \beta_{7} + 12 \beta_{5} + 12 \beta_{4} - 13 \beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$-14 \beta_{15} + 17 \beta_{14} + 4 \beta_{13} + 7 \beta_{12} + 4 \beta_{11} + 11 \beta_{10} + 7 \beta_{9} + 4 \beta_{8} - 4 \beta_{6} - 2 \beta_{5} + 16 \beta_{4} - 24 \beta_{3} + 7 \beta_{1} + 47$$ $$\nu^{5}$$ $$=$$ $$($$$$-60 \beta_{13} + 29 \beta_{12} + 60 \beta_{11} + 31 \beta_{10} + 112 \beta_{9} + 80 \beta_{8} - 199 \beta_{7} - 177 \beta_{5} - 177 \beta_{4} + 193 \beta_{2} + 38 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$430 \beta_{15} - 558 \beta_{14} - 182 \beta_{13} - 241 \beta_{12} - 182 \beta_{11} - 423 \beta_{10} - 215 \beta_{9} - 182 \beta_{8} + 194 \beta_{6} + 86 \beta_{5} - 516 \beta_{4} + 922 \beta_{3} - 215 \beta_{1} - 1412$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$1099 \beta_{13} - 446 \beta_{12} - 1099 \beta_{11} - 653 \beta_{10} - 2070 \beta_{9} - 1561 \beta_{8} + 3599 \beta_{7} + 2848 \beta_{5} + 2848 \beta_{4} - 3101 \beta_{2} - 814 \beta_{1}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$-3524 \beta_{15} + 4662 \beta_{14} + 1700 \beta_{13} + 2122 \beta_{12} + 1700 \beta_{11} + 3822 \beta_{10} + 1762 \beta_{9} + 1700 \beta_{8} - 1840 \beta_{6} - 768 \beta_{5} + 4292 \beta_{4} - 8320 \beta_{3} + 1762 \beta_{1} + 11519$$ $$\nu^{9}$$ $$=$$ $$($$$$-19488 \beta_{13} + 7254 \beta_{12} + 19488 \beta_{11} + 12234 \beta_{10} + 36520 \beta_{9} + 28104 \beta_{8} - 63655 \beta_{7} - 47643 \beta_{5} - 47643 \beta_{4} + 51843 \beta_{2} + 15212 \beta_{1}$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$118974 \beta_{15} - 158222 \beta_{14} - 60316 \beta_{13} - 74227 \beta_{12} - 60316 \beta_{11} - 134543 \beta_{10} - 59487 \beta_{9} - 60316 \beta_{8} + 65532 \beta_{6} + 26524 \beta_{5} - 145498 \beta_{4} + 292774 \beta_{3} - 59487 \beta_{1} - 388448$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$340417 \beta_{13} - 121702 \beta_{12} - 340417 \beta_{11} - 218715 \beta_{10} - 635141 \beta_{9} - 492789 \beta_{8} + 1111241 \beta_{7} + 810948 \beta_{5} + 810948 \beta_{4} - 882279 \beta_{2} - 271105 \beta_{1}$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$-1016822 \beta_{15} + 1354137 \beta_{14} + 525872 \beta_{13} + 644851 \beta_{12} + 525872 \beta_{11} + 1170723 \beta_{10} + 508411 \beta_{9} + 525872 \beta_{8} - 571900 \beta_{6} - 228046 \beta_{5} + 1244868 \beta_{4} - 2547320 \beta_{3} + 508411 \beta_{1} + 3318917$$ $$\nu^{13}$$ $$=$$ $$($$$$-5905588 \beta_{13} + 2073133 \beta_{12} + 5905588 \beta_{11} + 3832455 \beta_{10} + 10991424 \beta_{9} + 8557224 \beta_{8} - 19274917 \beta_{7} - 13908839 \beta_{5} - 13908839 \beta_{4} + 15131535 \beta_{2} + 4741670 \beta_{1}$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$34945962 \beta_{15} - 46555626 \beta_{14} - 18222830 \beta_{13} - 22325383 \beta_{12} - 18222830 \beta_{11} - 40548213 \beta_{10} - 17472981 \beta_{9} - 18222830 \beta_{8} + 19822722 \beta_{6} + 7849646 \beta_{5} - 42795608 \beta_{4} + 88224710 \beta_{3} - 17472981 \beta_{1} - 114055124$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$102133549 \beta_{13} - 35564202 \beta_{12} - 102133549 \beta_{11} - 66569347 \beta_{10} - 189859604 \beta_{9} - 148028747 \beta_{8} + 333334649 \beta_{7} + 239345952 \beta_{5} + 239345952 \beta_{4} - 260383075 \beta_{2} - 82285540 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/315\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$136$$ $$281$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
314.1
 − 4.15354i 4.15354i 0.562029i − 0.562029i 0.351269i − 0.351269i − 2.11224i 2.11224i − 1.06294i 1.06294i − 0.698027i 0.698027i − 2.73933i 2.73933i − 0.852184i 0.852184i
−2.33441 0 3.44949 −1.33239 1.79576i 0 −2.53958 + 0.741964i −3.38371 0 3.11034 + 4.19204i
314.2 −2.33441 0 3.44949 −1.33239 + 1.79576i 0 −2.53958 0.741964i −3.38371 0 3.11034 4.19204i
314.3 −2.33441 0 3.44949 1.33239 1.79576i 0 2.53958 0.741964i −3.38371 0 −3.11034 + 4.19204i
314.4 −2.33441 0 3.44949 1.33239 + 1.79576i 0 2.53958 + 0.741964i −3.38371 0 −3.11034 4.19204i
314.5 −0.741964 0 −1.44949 −2.05542 0.880486i 0 1.24519 + 2.33441i 2.55940 0 1.52505 + 0.653289i
314.6 −0.741964 0 −1.44949 −2.05542 + 0.880486i 0 1.24519 2.33441i 2.55940 0 1.52505 0.653289i
314.7 −0.741964 0 −1.44949 2.05542 0.880486i 0 −1.24519 2.33441i 2.55940 0 −1.52505 + 0.653289i
314.8 −0.741964 0 −1.44949 2.05542 + 0.880486i 0 −1.24519 + 2.33441i 2.55940 0 −1.52505 0.653289i
314.9 0.741964 0 −1.44949 −2.05542 0.880486i 0 −1.24519 + 2.33441i −2.55940 0 −1.52505 0.653289i
314.10 0.741964 0 −1.44949 −2.05542 + 0.880486i 0 −1.24519 2.33441i −2.55940 0 −1.52505 + 0.653289i
314.11 0.741964 0 −1.44949 2.05542 0.880486i 0 1.24519 2.33441i −2.55940 0 1.52505 0.653289i
314.12 0.741964 0 −1.44949 2.05542 + 0.880486i 0 1.24519 + 2.33441i −2.55940 0 1.52505 + 0.653289i
314.13 2.33441 0 3.44949 −1.33239 1.79576i 0 2.53958 + 0.741964i 3.38371 0 −3.11034 4.19204i
314.14 2.33441 0 3.44949 −1.33239 + 1.79576i 0 2.53958 0.741964i 3.38371 0 −3.11034 + 4.19204i
314.15 2.33441 0 3.44949 1.33239 1.79576i 0 −2.53958 0.741964i 3.38371 0 3.11034 4.19204i
314.16 2.33441 0 3.44949 1.33239 + 1.79576i 0 −2.53958 + 0.741964i 3.38371 0 3.11034 + 4.19204i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 314.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
15.d odd 2 1 inner
21.c even 2 1 inner
35.c odd 2 1 inner
105.g even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.g.a 16
3.b odd 2 1 inner 315.2.g.a 16
4.b odd 2 1 5040.2.k.g 16
5.b even 2 1 inner 315.2.g.a 16
5.c odd 4 2 1575.2.b.h 16
7.b odd 2 1 inner 315.2.g.a 16
12.b even 2 1 5040.2.k.g 16
15.d odd 2 1 inner 315.2.g.a 16
15.e even 4 2 1575.2.b.h 16
20.d odd 2 1 5040.2.k.g 16
21.c even 2 1 inner 315.2.g.a 16
28.d even 2 1 5040.2.k.g 16
35.c odd 2 1 inner 315.2.g.a 16
35.f even 4 2 1575.2.b.h 16
60.h even 2 1 5040.2.k.g 16
84.h odd 2 1 5040.2.k.g 16
105.g even 2 1 inner 315.2.g.a 16
105.k odd 4 2 1575.2.b.h 16
140.c even 2 1 5040.2.k.g 16
420.o odd 2 1 5040.2.k.g 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.g.a 16 1.a even 1 1 trivial
315.2.g.a 16 3.b odd 2 1 inner
315.2.g.a 16 5.b even 2 1 inner
315.2.g.a 16 7.b odd 2 1 inner
315.2.g.a 16 15.d odd 2 1 inner
315.2.g.a 16 21.c even 2 1 inner
315.2.g.a 16 35.c odd 2 1 inner
315.2.g.a 16 105.g even 2 1 inner
1575.2.b.h 16 5.c odd 4 2
1575.2.b.h 16 15.e even 4 2
1575.2.b.h 16 35.f even 4 2
1575.2.b.h 16 105.k odd 4 2
5040.2.k.g 16 4.b odd 2 1
5040.2.k.g 16 12.b even 2 1
5040.2.k.g 16 20.d odd 2 1
5040.2.k.g 16 28.d even 2 1
5040.2.k.g 16 60.h even 2 1
5040.2.k.g 16 84.h odd 2 1
5040.2.k.g 16 140.c even 2 1
5040.2.k.g 16 420.o odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(315, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T^{2} + 3 T^{4} + 8 T^{6} + 16 T^{8} )^{4}$$
$3$ 
$5$ $$( 1 - 4 T^{2} + 30 T^{4} - 100 T^{6} + 625 T^{8} )^{2}$$
$7$ $$( 1 - 4 T^{2} + 6 T^{4} - 196 T^{6} + 2401 T^{8} )^{2}$$
$11$ $$( 1 - 20 T^{2} + 121 T^{4} )^{8}$$
$13$ $$( 1 + 20 T^{2} + 222 T^{4} + 3380 T^{6} + 28561 T^{8} )^{4}$$
$17$ $$( 1 - 4 T^{2} + 558 T^{4} - 1156 T^{6} + 83521 T^{8} )^{4}$$
$19$ $$( 1 - 4 T^{2} + 510 T^{4} - 1444 T^{6} + 130321 T^{8} )^{4}$$
$23$ $$( 1 + 56 T^{2} + 1818 T^{4} + 29624 T^{6} + 279841 T^{8} )^{4}$$
$29$ $$( 1 - 16 T^{2} + 1362 T^{4} - 13456 T^{6} + 707281 T^{8} )^{4}$$
$31$ $$( 1 - 52 T^{2} + 1422 T^{4} - 49972 T^{6} + 923521 T^{8} )^{4}$$
$37$ $$( 1 - 76 T^{2} + 3318 T^{4} - 104044 T^{6} + 1874161 T^{8} )^{4}$$
$41$ $$( 1 + 44 T^{2} + 3246 T^{4} + 73964 T^{6} + 2825761 T^{8} )^{4}$$
$43$ $$( 1 - 100 T^{2} + 6102 T^{4} - 184900 T^{6} + 3418801 T^{8} )^{4}$$
$47$ $$( 1 - 124 T^{2} + 7398 T^{4} - 273916 T^{6} + 4879681 T^{8} )^{4}$$
$53$ $$( 1 + 104 T^{2} + 6378 T^{4} + 292136 T^{6} + 7890481 T^{8} )^{4}$$
$59$ $$( 1 + 92 T^{2} + 4374 T^{4} + 320252 T^{6} + 12117361 T^{8} )^{4}$$
$61$ $$( 1 - 52 T^{2} + 6582 T^{4} - 193492 T^{6} + 13845841 T^{8} )^{4}$$
$67$ $$( 1 - 124 T^{2} + 12438 T^{4} - 556636 T^{6} + 20151121 T^{8} )^{4}$$
$71$ $$( 1 - 184 T^{2} + 18162 T^{4} - 927544 T^{6} + 25411681 T^{8} )^{4}$$
$73$ $$( 1 + 164 T^{2} + 13326 T^{4} + 873956 T^{6} + 28398241 T^{8} )^{4}$$
$79$ $$( 1 + 4 T + 79 T^{2} )^{16}$$
$83$ $$( 1 - 268 T^{2} + 30870 T^{4} - 1846252 T^{6} + 47458321 T^{8} )^{4}$$
$89$ $$( 1 + 140 T^{2} + 18798 T^{4} + 1108940 T^{6} + 62742241 T^{8} )^{4}$$
$97$ $$( 1 + 356 T^{2} + 50286 T^{4} + 3349604 T^{6} + 88529281 T^{8} )^{4}$$