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

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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:

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 \(S_{11}^{\mathrm{new}}(20, [\chi])\).