# Properties

 Label 20.11.f.a Level 20 Weight 11 Character orbit 20.f Analytic conductor 12.707 Analytic rank 0 Dimension 10 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$20 = 2^{2} \cdot 5$$ Weight: $$k$$ = $$11$$ Character orbit: $$[\chi]$$ = 20.f (of order $$4$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$12.7071450535$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{22}\cdot 3^{2}\cdot 5^{6}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( 6 + 6 \beta_{1} - \beta_{2} ) q^{3}$$ $$+ ( 90 - 174 \beta_{1} + 2 \beta_{2} + \beta_{5} ) q^{5}$$ $$+ ( 2227 - 2227 \beta_{1} - 7 \beta_{3} + \beta_{5} + \beta_{8} ) q^{7}$$ $$+ ( 3 + 2767 \beta_{1} - 14 \beta_{2} + 16 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( 6 + 6 \beta_{1} - \beta_{2} ) q^{3}$$ $$+ ( 90 - 174 \beta_{1} + 2 \beta_{2} + \beta_{5} ) q^{5}$$ $$+ ( 2227 - 2227 \beta_{1} - 7 \beta_{3} + \beta_{5} + \beta_{8} ) q^{7}$$ $$+ ( 3 + 2767 \beta_{1} - 14 \beta_{2} + 16 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{9}$$ $$+ ( -20199 + 3 \beta_{1} - 49 \beta_{2} - 61 \beta_{3} + 17 \beta_{4} - 17 \beta_{5} - \beta_{6} - 6 \beta_{7} - 6 \beta_{8} + 5 \beta_{9} ) q^{11}$$ $$+ ( 23905 + 23860 \beta_{1} - 170 \beta_{2} - 31 \beta_{4} - 60 \beta_{5} + 14 \beta_{7} + 15 \beta_{9} ) q^{13}$$ $$+ ( 21484 - 106744 \beta_{1} + 457 \beta_{2} + 88 \beta_{3} - 14 \beta_{4} + \beta_{5} - 5 \beta_{6} - 15 \beta_{7} + 20 \beta_{8} + 35 \beta_{9} ) q^{15}$$ $$+ ( 104076 - 103836 \beta_{1} + 150 \beta_{2} - 2546 \beta_{3} + 205 \beta_{4} + 315 \beta_{5} - 5 \beta_{6} + 75 \beta_{9} ) q^{17}$$ $$+ ( 387 - 116801 \beta_{1} - 4256 \beta_{2} + 4484 \beta_{3} - 547 \beta_{4} + 313 \beta_{5} - 15 \beta_{6} + 14 \beta_{7} - 14 \beta_{8} + 129 \beta_{9} ) q^{19}$$ $$+ ( 454494 + 9 \beta_{1} - 7442 \beta_{2} - 7808 \beta_{3} + 771 \beta_{4} - 486 \beta_{5} - 3 \beta_{6} + 42 \beta_{7} + 42 \beta_{8} + 180 \beta_{9} ) q^{21}$$ $$+ ( -408261 - 408711 \beta_{1} + 7361 \beta_{2} - 100 \beta_{3} - 577 \beta_{4} - 950 \beta_{5} - 50 \beta_{6} - 177 \beta_{7} + 200 \beta_{9} ) q^{23}$$ $$+ ( -415066 + 538352 \beta_{1} + 24574 \beta_{2} + 16508 \beta_{3} + 191 \beta_{4} - 98 \beta_{5} + 30 \beta_{6} + 215 \beta_{7} - 245 \beta_{8} + 165 \beta_{9} ) q^{25}$$ $$+ ( -489586 + 489556 \beta_{1} - 100 \beta_{2} - 39624 \beta_{3} - 310 \beta_{4} - 178 \beta_{5} - 40 \beta_{6} - 98 \beta_{8} - 50 \beta_{9} ) q^{27}$$ $$+ ( -1311 - 645195 \beta_{1} - 58082 \beta_{2} + 57398 \beta_{3} + 1691 \beta_{4} - 589 \beta_{5} + 95 \beta_{6} - 342 \beta_{7} + 342 \beta_{8} - 437 \beta_{9} ) q^{29}$$ $$+ ( 2307146 + 180 \beta_{1} - 42535 \beta_{2} - 40735 \beta_{3} - 3485 \beta_{4} + 3295 \beta_{5} - 60 \beta_{6} + 175 \beta_{7} + 175 \beta_{8} - 960 \beta_{9} ) q^{31}$$ $$+ ( 3198583 + 3201673 \beta_{1} + 85462 \beta_{2} + 950 \beta_{3} + 3427 \beta_{4} + 7445 \beta_{5} + 475 \beta_{6} + 812 \beta_{7} - 1505 \beta_{9} ) q^{33}$$ $$+ ( -5493057 - 7560567 \beta_{1} + 143256 \beta_{2} + 90671 \beta_{3} - 1003 \beta_{4} + 393 \beta_{5} - 295 \beta_{6} - 1260 \beta_{7} + 930 \beta_{8} - 2185 \beta_{9} ) q^{35}$$ $$+ ( -8809329 + 8800194 \beta_{1} - 4650 \beta_{2} - 127596 \beta_{3} - 4095 \beta_{4} - 10968 \beta_{5} + 720 \beta_{6} + 492 \beta_{8} - 2325 \beta_{9} ) q^{37}$$ $$+ ( -6120 + 8716998 \beta_{1} - 150615 \beta_{2} + 145815 \beta_{3} + 9405 \beta_{4} - 9885 \beta_{5} - 360 \beta_{6} + 2325 \beta_{7} - 2325 \beta_{8} - 2040 \beta_{9} ) q^{39}$$ $$+ ( 2971602 - 3297 \beta_{1} - 26384 \beta_{2} - 21886 \beta_{3} - 10648 \beta_{4} - 3697 \beta_{5} + 1099 \beta_{6} - 2751 \beta_{7} - 2751 \beta_{8} - 1150 \beta_{9} ) q^{41}$$ $$+ ( 15634784 + 15641624 \beta_{1} + 37931 \beta_{2} - 4200 \beta_{3} + 7666 \beta_{4} - 5580 \beta_{5} - 2100 \beta_{6} - 1274 \beta_{7} - 180 \beta_{9} ) q^{43}$$ $$+ ( -14438663 - 34317120 \beta_{1} - 41080 \beta_{2} - 88676 \beta_{3} + 2263 \beta_{4} + 1185 \beta_{5} + 2950 \beta_{6} + 3850 \beta_{7} + 700 \beta_{8} + 2475 \beta_{9} ) q^{45}$$ $$+ ( -45025797 + 45051447 \beta_{1} + 9300 \beta_{2} + 194517 \beta_{3} - 1650 \beta_{4} + 30993 \beta_{5} - 3900 \beta_{6} + 693 \beta_{8} + 4650 \beta_{9} ) q^{47}$$ $$+ ( 23661 - 96870911 \beta_{1} + 396982 \beta_{2} - 375568 \beta_{3} - 30031 \beta_{4} + 41884 \beta_{5} + 2820 \beta_{6} - 6943 \beta_{7} + 6943 \beta_{8} + 7887 \beta_{9} ) q^{49}$$ $$+ ( 163264826 + 16842 \beta_{1} + 93239 \beta_{2} + 61671 \beta_{3} + 67988 \beta_{4} + 2412 \beta_{5} - 5614 \beta_{6} + 10466 \beta_{7} + 10466 \beta_{8} + 10170 \beta_{9} ) q^{51}$$ $$+ ( 70025850 + 69959655 \beta_{1} - 721400 \beta_{2} + 14350 \beta_{3} - 74252 \beta_{4} - 38035 \beta_{5} + 7175 \beta_{6} - 882 \beta_{7} + 14890 \beta_{9} ) q^{53}$$ $$+ ( -130448785 - 166003205 \beta_{1} - 1281835 \beta_{2} - 413295 \beta_{3} - 1190 \beta_{4} - 12030 \beta_{5} - 14475 \beta_{6} - 7175 \beta_{7} - 15225 \beta_{8} + 15075 \beta_{9} ) q^{55}$$ $$+ ( -256283016 + 256289136 \beta_{1} + 32100 \beta_{2} + 859316 \beta_{3} + 104190 \beta_{4} + 9348 \beta_{5} + 14010 \beta_{6} - 12822 \beta_{8} + 16050 \beta_{9} ) q^{57}$$ $$+ ( 27279 - 182124177 \beta_{1} + 1425148 \beta_{2} - 1440552 \beta_{3} - 80709 \beta_{4} - 45949 \beta_{5} - 16795 \beta_{6} + 6048 \beta_{7} - 6048 \beta_{8} + 9093 \beta_{9} ) q^{59}$$ $$+ ( 299693575 - 48636 \beta_{1} + 1350008 \beta_{2} + 1371182 \beta_{3} - 36524 \beta_{4} - 92111 \beta_{5} + 16212 \beta_{6} - 10388 \beta_{7} - 10388 \beta_{8} + 5625 \beta_{9} ) q^{61}$$ $$+ ( 335027747 + 335122577 \beta_{1} - 734847 \beta_{2} - 44300 \beta_{3} + 116447 \beta_{4} - 28610 \beta_{5} - 22150 \beta_{6} - 533 \beta_{7} - 9460 \beta_{9} ) q^{63}$$ $$+ ( -287347779 - 661762176 \beta_{1} - 1010272 \beta_{2} - 1903078 \beta_{3} + 734 \beta_{4} + 28729 \beta_{5} + 36905 \beta_{6} + 13215 \beta_{7} + 39255 \beta_{8} - 20210 \beta_{9} ) q^{65}$$ $$+ ( -698436064 + 698442004 \beta_{1} - 79800 \beta_{2} + 2647539 \beta_{3} - 287220 \beta_{4} + 8082 \beta_{5} - 41880 \beta_{6} + 42042 \beta_{8} - 39900 \beta_{9} ) q^{67}$$ $$+ ( -117273 - 482566990 \beta_{1} + 918324 \beta_{2} - 887786 \beta_{3} + 333738 \beta_{4} + 85873 \beta_{5} + 54360 \beta_{6} + 14294 \beta_{7} - 14294 \beta_{8} - 39091 \beta_{9} ) q^{69}$$ $$+ ( 991579746 + 126804 \beta_{1} - 123487 \beta_{2} - 90023 \beta_{3} - 141989 \beta_{4} + 313279 \beta_{5} - 42268 \beta_{6} - 32793 \beta_{7} - 32793 \beta_{8} - 59000 \beta_{9} ) q^{71}$$ $$+ ( 840296525 + 840330905 \beta_{1} + 153460 \beta_{2} + 98100 \beta_{3} + 9284 \beta_{4} + 389190 \beta_{5} + 49050 \beta_{6} + 23954 \beta_{7} - 60510 \beta_{9} ) q^{73}$$ $$+ ( -1016260848 - 1517041254 \beta_{1} + 800227 \beta_{2} + 3385524 \beta_{3} - 22602 \beta_{4} - 1704 \beta_{5} - 50610 \beta_{6} - 25830 \beta_{7} - 2310 \beta_{8} - 66480 \beta_{9} ) q^{75}$$ $$+ ( -1983695121 + 1983304851 \beta_{1} - 85550 \beta_{2} - 4564974 \beta_{3} + 220935 \beta_{4} - 475999 \beta_{5} + 87315 \beta_{6} - 42954 \beta_{8} - 42775 \beta_{9} ) q^{77}$$ $$+ ( -164196 - 1169547184 \beta_{1} - 3722252 \beta_{2} + 3403988 \beta_{3} - 113404 \beta_{4} - 562664 \beta_{5} - 104400 \beta_{6} - 19132 \beta_{7} + 19132 \beta_{8} - 54732 \beta_{9} ) q^{79}$$ $$+ ( 2607070651 - 333219 \beta_{1} - 356428 \beta_{2} - 97922 \beta_{3} - 320786 \beta_{4} - 304599 \beta_{5} + 111073 \beta_{6} + 85153 \beta_{7} + 85153 \beta_{8} - 18180 \beta_{9} ) q^{81}$$ $$+ ( 1701197646 + 1701381756 \beta_{1} + 7538759 \beta_{2} - 141500 \beta_{3} + 231886 \beta_{4} - 249770 \beta_{5} - 70750 \beta_{6} - 22974 \beta_{7} + 9380 \beta_{9} ) q^{83}$$ $$+ ( -2425660479 - 3026744269 \beta_{1} + 10960292 \beta_{2} + 932912 \beta_{3} + 45184 \beta_{4} - 100874 \beta_{5} + 35685 \beta_{6} - 16820 \beta_{7} - 146990 \beta_{8} + 66705 \beta_{9} ) q^{85}$$ $$+ ( -3607508714 + 3607974854 \beta_{1} + 158800 \beta_{2} - 5819996 \beta_{3} - 65720 \beta_{4} + 474406 \beta_{5} - 75980 \beta_{6} - 71134 \beta_{8} + 79400 \beta_{9} ) q^{87}$$ $$+ ( 382212 - 2987348712 \beta_{1} - 11594956 \beta_{2} + 12091644 \beta_{3} - 211602 \beta_{4} + 930778 \beta_{5} + 120940 \beta_{6} - 64806 \beta_{7} + 64806 \beta_{8} + 127404 \beta_{9} ) q^{89}$$ $$+ ( 5227956088 + 596052 \beta_{1} - 15749161 \beta_{2} - 16456849 \beta_{3} + 1229698 \beta_{4} + 342262 \beta_{5} - 198684 \beta_{6} + 13006 \beta_{7} + 13006 \beta_{8} + 155160 \beta_{9} ) q^{91}$$ $$+ ( 2627506821 + 2626526631 \beta_{1} - 3911626 \beta_{2} + 230250 \beta_{3} - 1214715 \beta_{4} - 501045 \beta_{5} + 115125 \beta_{6} - 119400 \beta_{7} + 211605 \beta_{9} ) q^{93}$$ $$+ ( -2340129153 - 5378346015 \beta_{1} + 190180 \beta_{2} + 14475944 \beta_{3} + 73853 \beta_{4} + 142865 \beta_{5} - 20825 \beta_{6} + 258150 \beta_{7} + 183300 \beta_{8} + 182025 \beta_{9} ) q^{95}$$ $$+ ( -4897757035 + 4898523745 \beta_{1} + 361200 \beta_{2} - 16960660 \beta_{3} + 241920 \beta_{4} + 1202068 \beta_{5} - 74970 \beta_{6} + 254758 \beta_{8} + 180600 \beta_{9} ) q^{97}$$ $$+ ( 687621 - 3874453067 \beta_{1} + 340927 \beta_{2} + 101917 \beta_{3} - 863661 \beta_{4} + 579959 \beta_{5} - 7785 \beta_{6} + 76522 \beta_{7} - 76522 \beta_{8} + 229207 \beta_{9} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q$$ $$\mathstrut +\mathstrut 62q^{3}$$ $$\mathstrut +\mathstrut 894q^{5}$$ $$\mathstrut +\mathstrut 22286q^{7}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$10q$$ $$\mathstrut +\mathstrut 62q^{3}$$ $$\mathstrut +\mathstrut 894q^{5}$$ $$\mathstrut +\mathstrut 22286q^{7}$$ $$\mathstrut -\mathstrut 201700q^{11}$$ $$\mathstrut +\mathstrut 239298q^{13}$$ $$\mathstrut +\mathstrut 213662q^{15}$$ $$\mathstrut +\mathstrut 1045442q^{17}$$ $$\mathstrut +\mathstrut 4578860q^{21}$$ $$\mathstrut -\mathstrut 4097986q^{23}$$ $$\mathstrut -\mathstrut 4233934q^{25}$$ $$\mathstrut -\mathstrut 4817488q^{27}$$ $$\mathstrut +\mathstrut 23221660q^{31}$$ $$\mathstrut +\mathstrut 31816220q^{33}$$ $$\mathstrut -\mathstrut 55388242q^{35}$$ $$\mathstrut -\mathstrut 87811974q^{37}$$ $$\mathstrut +\mathstrut 29776460q^{41}$$ $$\mathstrut +\mathstrut 156325470q^{43}$$ $$\mathstrut -\mathstrut 144135236q^{45}$$ $$\mathstrut -\mathstrut 450750018q^{47}$$ $$\mathstrut +\mathstrut 1632585820q^{51}$$ $$\mathstrut +\mathstrut 701393866q^{53}$$ $$\mathstrut -\mathstrut 1301185140q^{55}$$ $$\mathstrut -\mathstrut 2564330416q^{57}$$ $$\mathstrut +\mathstrut 2991488220q^{61}$$ $$\mathstrut +\mathstrut 3352397678q^{63}$$ $$\mathstrut -\mathstrut 2867494182q^{65}$$ $$\mathstrut -\mathstrut 6990333394q^{67}$$ $$\mathstrut +\mathstrut 9915200380q^{71}$$ $$\mathstrut +\mathstrut 8401915018q^{73}$$ $$\mathstrut -\mathstrut 10170758642q^{75}$$ $$\mathstrut -\mathstrut 19825815140q^{77}$$ $$\mathstrut +\mathstrut 26071184290q^{81}$$ $$\mathstrut +\mathstrut 16998617454q^{83}$$ $$\mathstrut -\mathstrut 24280829854q^{85}$$ $$\mathstrut -\mathstrut 36065578576q^{87}$$ $$\mathstrut +\mathstrut 52347612540q^{91}$$ $$\mathstrut +\mathstrut 26277966572q^{93}$$ $$\mathstrut -\mathstrut 23431125296q^{95}$$ $$\mathstrut -\mathstrut 48945511254q^{97}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10}\mathstrut +\mathstrut$$ $$75402$$ $$x^{8}\mathstrut +\mathstrut$$ $$1918432665$$ $$x^{6}\mathstrut +\mathstrut$$ $$20025190470928$$ $$x^{4}\mathstrut +\mathstrut$$ $$87673968468747264$$ $$x^{2}\mathstrut +\mathstrut$$ $$130532036219814297600$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-$$$$4541161$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$296647947642$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$5722368475906305$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$31903112343361645168$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$25133664076757128794624$$ $$\nu$$$$)/$$$$15\!\cdots\!80$$ $$\beta_{2}$$ $$=$$ $$($$$$90309214939079$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$904047821228016$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$6227581417142813094$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$78201849021081876576$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$132428737167523808186031$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$2260069007478252952167024$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$930157843265327067522500816$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$22968475390724984057882264064$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$1805231517372463168818722956800$$ $$\nu\mathstrut +\mathstrut$$ $$52385259255902137290124920729600$$$$)/$$$$10\!\cdots\!00$$ $$\beta_{3}$$ $$=$$ $$($$$$-$$$$90309214939079$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$904047821228016$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$6227581417142813094$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$78201849021081876576$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$132428737167523808186031$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$2260069007478252952167024$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$930157843265327067522500816$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$22968475390724984057882264064$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$1805231517372463168818722956800$$ $$\nu\mathstrut +\mathstrut$$ $$52385259255902137290124920729600$$$$)/$$$$10\!\cdots\!00$$ $$\beta_{4}$$ $$=$$ $$($$$$564930968079603$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$152264363749954640$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$40402590323793675358$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$10492758055115205175840$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$926283859724013800015867$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$225312408840219445590929360$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$7649837258133219253989741712$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$1654632245798721308199079119360$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$20155810120283317413406538227200$$ $$\nu\mathstrut +\mathstrut$$ $$3512747831129400612332645823283200$$$$)/$$$$10\!\cdots\!00$$ $$\beta_{5}$$ $$=$$ $$($$$$-$$$$914897678021365$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$10264353724844484$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$65259502203617343810$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$868157502369340021224$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$1488456653296048110752205$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$23667049632778133149251876$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$12170753631420671839254010240$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$232907169776827212260280212736$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$28439828591831587136855374356480$$ $$\nu\mathstrut +\mathstrut$$ $$695360187952896174027342207206400$$$$)/$$$$78\!\cdots\!00$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$2253724267404781$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$385008305068617900$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$160836016495010223186$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$27198638407018194545400$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$3671916890550722016927429$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$605011221932823664604849100$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$30080505844669410461372804704$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$4689003945240519268575872265600$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$65704724596784247112617769697280$$ $$\nu\mathstrut -\mathstrut$$ $$10763701260736589279793009312307200$$$$)/$$$$78\!\cdots\!00$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$2481088965145559$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$135178905838851588$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$170549183336661624054$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$8837159656564436548968$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$3695320236335913156752031$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$161766785127981432763711332$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$29186967316632454181845341056$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$615638286722972091755119844352$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$86305060834141789374336390789120$$ $$\nu\mathstrut -\mathstrut$$ $$845423982250231788778821364915200$$$$)/$$$$78\!\cdots\!00$$ $$\beta_{8}$$ $$=$$ $$($$$$3963051222809629$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$318817099901964112$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$274008990396578281794$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$21118094260708667212832$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$5985418659785267889989781$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$409445389500490511743541968$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$47493790672604282107476060016$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$2164940401726914480858865294848$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$132837375781014517934849148633600$$ $$\nu\mathstrut +\mathstrut$$ $$1458368937525229995257761060454400$$$$)/$$$$10\!\cdots\!00$$ $$\beta_{9}$$ $$=$$ $$($$$$-$$$$3049966507141015$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$283095972228992296$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$217551766471964851430$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$19170999256470648432656$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$4961909763207647629722175$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$401030649407404371931187944$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$40571339622373109254010004720$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$2820481676653063207819715549184$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$98996508941602272269342548124160$$ $$\nu\mathstrut +\mathstrut$$ $$5582232700423711503460798705459200$$$$)/$$$$52\!\cdots\!00$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-$$$$\beta_{9}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$5$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$10$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$4$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$3$$$$)/400$$ $$\nu^{2}$$ $$=$$ $$($$$$-$$$$56$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$50$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$50$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$113$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$570$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$65$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$10591$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$10477$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$339$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$6036298$$$$)/400$$ $$\nu^{3}$$ $$=$$ $$($$$$31877$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$13300$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$13300$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$46694$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$77845$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$280890$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$595543$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$565909$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$272252392$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$95631$$$$)/400$$ $$\nu^{4}$$ $$=$$ $$($$$$603088$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$136210$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$136210$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$896413$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$5258706$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$140677$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$80000603$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$79413953$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$2689239$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$29608641818$$$$)/80$$ $$\nu^{5}$$ $$=$$ $$($$$$-$$$$1081368341$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$444738700$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$444738700$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$1326829982$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$1618476205$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$8750702010$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$34578791359$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$34087868077$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$12538459613536$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$3244105023$$$$)/400$$ $$\nu^{6}$$ $$=$$ $$($$$$-$$$$129853710776$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$1828519150$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$1828519150$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$165607670273$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$1053820332570$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$20763313135$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$13124135663311$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$13052627744317$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$496823010819$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$4389850019166058$$$$)/400$$ $$\nu^{7}$$ $$=$$ $$($$$$36636410491877$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$13260748574500$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$13260748574500$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$41196564484694$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$41616277888645$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$283396083996090$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$1486096992186343$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$1476976684200709$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$503918914433368792$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$109909231475631$$$$)/400$$ $$\nu^{8}$$ $$=$$ $$($$$$1023382239364000$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$118088843084030$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$118088843084030$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$1198274565434749$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$7981333822915026$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$380616418067723$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$85511686914799595$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$85161902262658097$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$3594823696304247$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$27962454017557512554$$$$)/80$$ $$\nu^{9}$$ $$=$$ $$($$$$-$$$$1249013132125513781$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$399264973449115900$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$399264973449115900$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$1336152328007984462$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$1198304942206280605$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$9403774485975124410$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$57694463160351373519$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$57520184768586432157$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$19167353126541581612176$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$3747039396376541343$$$$)/400$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/20\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$17$$ $$\chi(n)$$ $$1$$ $$-\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 149.896i − 186.802i 56.3206i 75.9602i − 95.3750i − 149.896i 186.802i − 56.3206i − 75.9602i 95.3750i
0 −190.862 190.862i 0 2756.18 + 1472.78i 0 14849.7 14849.7i 0 13807.6i 0
13.2 0 −189.544 189.544i 0 −3114.06 261.220i 0 −3939.73 + 3939.73i 0 12805.1i 0
13.3 0 7.29050 + 7.29050i 0 2223.57 2195.76i 0 −14037.3 + 14037.3i 0 58942.7i 0
13.4 0 186.378 + 186.378i 0 −473.688 + 3088.89i 0 −7250.16 + 7250.16i 0 10424.8i 0
13.5 0 217.737 + 217.737i 0 −944.998 2978.69i 0 21520.5 21520.5i 0 35770.2i 0
17.1 0 −190.862 + 190.862i 0 2756.18 1472.78i 0 14849.7 + 14849.7i 0 13807.6i 0
17.2 0 −189.544 + 189.544i 0 −3114.06 + 261.220i 0 −3939.73 3939.73i 0 12805.1i 0
17.3 0 7.29050 7.29050i 0 2223.57 + 2195.76i 0 −14037.3 14037.3i 0 58942.7i 0
17.4 0 186.378 186.378i 0 −473.688 3088.89i 0 −7250.16 7250.16i 0 10424.8i 0
17.5 0 217.737 217.737i 0 −944.998 + 2978.69i 0 21520.5 + 21520.5i 0 35770.2i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 17.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
5.c Odd 1 yes

## Hecke kernels

There are no other newforms in $$S_{11}^{\mathrm{new}}(20, [\chi])$$.