LMFDB - Embedded newform 825.6.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,6,Mod(1,825)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	825.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$132.316651346$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.788.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x - 3 $ x^3 - x^2 - 7*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.35386$ of defining polynomial
Character	$\chi$	$=$	825.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.08127 q^{2} +9.00000 q^{3} -30.8308 q^{4} -9.73147 q^{6} +139.508 q^{7} +67.9374 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-1.08127 q^{2} +9.00000 q^{3} -30.8308 q^{4} -9.73147 q^{6} +139.508 q^{7} +67.9374 q^{8} +81.0000 q^{9} -121.000 q^{11} -277.478 q^{12} +646.331 q^{13} -150.846 q^{14} +913.128 q^{16} +1378.32 q^{17} -87.5832 q^{18} +1908.90 q^{19} +1255.57 q^{21} +130.834 q^{22} -343.726 q^{23} +611.436 q^{24} -698.861 q^{26} +729.000 q^{27} -4301.15 q^{28} +53.5092 q^{29} +634.133 q^{31} -3161.34 q^{32} -1089.00 q^{33} -1490.34 q^{34} -2497.30 q^{36} -11674.2 q^{37} -2064.04 q^{38} +5816.98 q^{39} +18866.5 q^{41} -1357.62 q^{42} -13379.3 q^{43} +3730.53 q^{44} +371.662 q^{46} +2252.95 q^{47} +8218.15 q^{48} +2655.48 q^{49} +12404.9 q^{51} -19926.9 q^{52} +8900.71 q^{53} -788.249 q^{54} +9477.80 q^{56} +17180.1 q^{57} -57.8581 q^{58} -11276.7 q^{59} +11852.4 q^{61} -685.672 q^{62} +11300.1 q^{63} -25801.8 q^{64} +1177.51 q^{66} +57333.6 q^{67} -42494.7 q^{68} -3093.53 q^{69} +31075.8 q^{71} +5502.93 q^{72} -56010.8 q^{73} +12623.0 q^{74} -58853.0 q^{76} -16880.5 q^{77} -6289.75 q^{78} -883.036 q^{79} +6561.00 q^{81} -20399.9 q^{82} -93986.3 q^{83} -38710.3 q^{84} +14466.7 q^{86} +481.583 q^{87} -8220.42 q^{88} +21739.9 q^{89} +90168.4 q^{91} +10597.4 q^{92} +5707.20 q^{93} -2436.06 q^{94} -28452.0 q^{96} +158059. q^{97} -2871.30 q^{98} -9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} + 27 q^{3} - 20 q^{4} - 18 q^{6} - 152 q^{7} + 24 q^{8} + 243 q^{9}+O(q^{10}) $ 3 * q - 2 * q^2 + 27 * q^3 - 20 * q^4 - 18 * q^6 - 152 * q^7 + 24 * q^8 + 243 * q^9	\( 3 q - 2 q^{2} + 27 q^{3} - 20 q^{4} - 18 q^{6} - 152 q^{7} + 24 q^{8} + 243 q^{9} - 363 q^{11} - 180 q^{12} + 546 q^{13} - 8 q^{14} - 1360 q^{16} + 314 q^{17} - 162 q^{18} + 1808 q^{19} - 1368 q^{21} + 242 q^{22} - 4288 q^{23} + 216 q^{24} + 812 q^{26} + 2187 q^{27} - 5888 q^{28} + 5582 q^{29} + 6328 q^{31} + 736 q^{32} - 3267 q^{33} + 11596 q^{34} - 1620 q^{36} - 16866 q^{37} - 9584 q^{38} + 4914 q^{39} + 23282 q^{41} - 72 q^{42} - 20572 q^{43} + 2420 q^{44} + 16592 q^{46} - 3432 q^{47} - 12240 q^{48} + 11531 q^{49} + 2826 q^{51} - 21816 q^{52} - 16138 q^{53} - 1458 q^{54} + 15648 q^{56} + 16272 q^{57} - 17460 q^{58} + 21972 q^{59} + 8322 q^{61} - 5056 q^{62} - 12312 q^{63} + 22208 q^{64} + 2178 q^{66} + 84332 q^{67} - 59832 q^{68} - 38592 q^{69} + 50528 q^{71} + 1944 q^{72} + 53838 q^{73} + 79212 q^{74} - 52448 q^{76} + 18392 q^{77} + 7308 q^{78} + 6364 q^{79} + 19683 q^{81} + 68020 q^{82} - 96272 q^{83} - 52992 q^{84} + 143152 q^{86} + 50238 q^{87} - 2904 q^{88} - 38938 q^{89} + 104968 q^{91} - 24000 q^{92} + 56952 q^{93} + 49088 q^{94} + 6624 q^{96} + 103242 q^{97} - 9554 q^{98} - 29403 q^{99}+O(q^{100}) \) 3 * q - 2 * q^2 + 27 * q^3 - 20 * q^4 - 18 * q^6 - 152 * q^7 + 24 * q^8 + 243 * q^9 - 363 * q^11 - 180 * q^12 + 546 * q^13 - 8 * q^14 - 1360 * q^16 + 314 * q^17 - 162 * q^18 + 1808 * q^19 - 1368 * q^21 + 242 * q^22 - 4288 * q^23 + 216 * q^24 + 812 * q^26 + 2187 * q^27 - 5888 * q^28 + 5582 * q^29 + 6328 * q^31 + 736 * q^32 - 3267 * q^33 + 11596 * q^34 - 1620 * q^36 - 16866 * q^37 - 9584 * q^38 + 4914 * q^39 + 23282 * q^41 - 72 * q^42 - 20572 * q^43 + 2420 * q^44 + 16592 * q^46 - 3432 * q^47 - 12240 * q^48 + 11531 * q^49 + 2826 * q^51 - 21816 * q^52 - 16138 * q^53 - 1458 * q^54 + 15648 * q^56 + 16272 * q^57 - 17460 * q^58 + 21972 * q^59 + 8322 * q^61 - 5056 * q^62 - 12312 * q^63 + 22208 * q^64 + 2178 * q^66 + 84332 * q^67 - 59832 * q^68 - 38592 * q^69 + 50528 * q^71 + 1944 * q^72 + 53838 * q^73 + 79212 * q^74 - 52448 * q^76 + 18392 * q^77 + 7308 * q^78 + 6364 * q^79 + 19683 * q^81 + 68020 * q^82 - 96272 * q^83 - 52992 * q^84 + 143152 * q^86 + 50238 * q^87 - 2904 * q^88 - 38938 * q^89 + 104968 * q^91 - 24000 * q^92 + 56952 * q^93 + 49088 * q^94 + 6624 * q^96 + 103242 * q^97 - 9554 * q^98 - 29403 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.08127	−0.191144	−0.0955720	−	0.995423i	$-0.530468\pi$
$2$	−1.08127	−0.191144	−0.0955720	+	0.995423i	$0.530468\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−30.8308	−0.963464
$5$	0	0
$5$	0	0
$6$	−9.73147	−0.110357
$7$	139.508	1.07610	0.538052	−	0.842912i	$-0.319161\pi$
$7$	139.508	1.07610	0.538052	+	0.842912i	$0.319161\pi$
$8$	67.9374	0.375304
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	−277.478	−0.556256
$13$	646.331	1.06071	0.530355	−	0.847776i	$-0.322059\pi$
$13$	646.331	1.06071	0.530355	+	0.847776i	$0.322059\pi$
$14$	−150.846	−0.205691
$15$	0	0
$16$	913.128	0.891727
$17$	1378.32	1.15672	0.578358	−	0.815783i	$-0.303694\pi$
$17$	1378.32	1.15672	0.578358	+	0.815783i	$0.303694\pi$
$18$	−87.5832	−0.0637147
$19$	1908.90	1.21311	0.606553	−	0.795043i	$-0.292552\pi$
$19$	1908.90	1.21311	0.606553	+	0.795043i	$0.292552\pi$
$20$	0	0
$21$	1255.57	0.621289
$22$	130.834	0.0576321
$23$	−343.726	−0.135485	−0.0677427	−	0.997703i	$-0.521580\pi$
$23$	−343.726	−0.135485	−0.0677427	+	0.997703i	$0.521580\pi$
$24$	611.436	0.216682
$25$	0	0
$26$	−698.861	−0.202748
$27$	729.000	0.192450
$28$	−4301.15	−1.03679
$29$	53.5092	0.0118150	0.00590750	−	0.999983i	$-0.498120\pi$
$29$	53.5092	0.0118150	0.00590750	+	0.999983i	$0.498120\pi$
$30$	0	0
$31$	634.133	0.118516	0.0592579	−	0.998243i	$-0.481127\pi$
$31$	634.133	0.118516	0.0592579	+	0.998243i	$0.481127\pi$
$32$	−3161.34	−0.545753
$33$	−1089.00	−0.174078
$34$	−1490.34	−0.221099
$35$	0	0
$36$	−2497.30	−0.321155
$37$	−11674.2	−1.40192	−0.700960	−	0.713201i	$-0.747245\pi$
$37$	−11674.2	−1.40192	−0.700960	+	0.713201i	$0.747245\pi$
$38$	−2064.04	−0.231878
$39$	5816.98	0.612401
$40$	0	0
$41$	18866.5	1.75280	0.876399	−	0.481585i	$-0.159939\pi$
$41$	18866.5	1.75280	0.876399	+	0.481585i	$0.159939\pi$
$42$	−1357.62	−0.118756
$43$	−13379.3	−1.10347	−0.551736	−	0.834018i	$-0.686035\pi$
$43$	−13379.3	−1.10347	−0.551736	+	0.834018i	$0.686035\pi$
$44$	3730.53	0.290495
$45$	0	0
$46$	371.662	0.0258972
$47$	2252.95	0.148767	0.0743835	−	0.997230i	$-0.476301\pi$
$47$	2252.95	0.148767	0.0743835	+	0.997230i	$0.476301\pi$
$48$	8218.15	0.514839
$49$	2655.48	0.157998
$50$	0	0
$51$	12404.9	0.667830
$52$	−19926.9	−1.02196
$53$	8900.71	0.435246	0.217623	−	0.976033i	$-0.430170\pi$
$53$	8900.71	0.435246	0.217623	+	0.976033i	$0.430170\pi$
$54$	−788.249	−0.0367857
$55$	0	0
$56$	9477.80	0.403866
$57$	17180.1	0.700387
$58$	−57.8581	−0.00225837
$59$	−11276.7	−0.421746	−0.210873	−	0.977514i	$-0.567631\pi$
$59$	−11276.7	−0.421746	−0.210873	+	0.977514i	$0.567631\pi$
$60$	0	0
$61$	11852.4	0.407834	0.203917	−	0.978988i	$-0.434633\pi$
$61$	11852.4	0.407834	0.203917	+	0.978988i	$0.434633\pi$
$62$	−685.672	−0.0226536
$63$	11300.1	0.358701
$64$	−25801.8	−0.787409
$65$	0	0
$66$	1177.51	0.0332739
$67$	57333.6	1.56035	0.780175	−	0.625561i	$-0.215130\pi$
$67$	57333.6	1.56035	0.780175	+	0.625561i	$0.215130\pi$
$68$	−42494.7	−1.11445
$69$	−3093.53	−0.0782226
$70$	0	0
$71$	31075.8	0.731605	0.365803	−	0.930692i	$-0.380794\pi$
$71$	31075.8	0.731605	0.365803	+	0.930692i	$0.380794\pi$
$72$	5502.93	0.125101
$73$	−56010.8	−1.23017	−0.615084	−	0.788462i	$-0.710878\pi$
$73$	−56010.8	−1.23017	−0.615084	+	0.788462i	$0.710878\pi$
$74$	12623.0	0.267969
$75$	0	0
$76$	−58853.0	−1.16878
$77$	−16880.5	−0.324457
$78$	−6289.75	−0.117057
$79$	−883.036	−0.0159188	−0.00795941	−	0.999968i	$-0.502534\pi$
$79$	−883.036	−0.0159188	−0.00795941	+	0.999968i	$0.502534\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−20399.9	−0.335037
$83$	−93986.3	−1.49751	−0.748754	−	0.662848i	$-0.769348\pi$
$83$	−93986.3	−1.49751	−0.748754	+	0.662848i	$0.769348\pi$
$84$	−38710.3	−0.598589
$85$	0	0
$86$	14466.7	0.210922
$87$	481.583	0.00682139
$88$	−8220.42	−0.113159
$89$	21739.9	0.290926	0.145463	−	0.989364i	$-0.453533\pi$
$89$	21739.9	0.290926	0.145463	+	0.989364i	$0.453533\pi$
$90$	0	0
$91$	90168.4	1.14143
$92$	10597.4	0.130535
$93$	5707.20	0.0684251
$94$	−2436.06	−0.0284359
$95$	0	0
$96$	−28452.0	−0.315090
$97$	158059.	1.70565	0.852825	−	0.522197i	$-0.174888\pi$
$97$	158059.	1.70565	0.852825	+	0.522197i	$0.174888\pi$
$98$	−2871.30	−0.0302004
$99$	−9801.00	−0.100504
$100$	0	0
$101$	−156968.	−1.53111	−0.765556	−	0.643369i	$-0.777536\pi$
$101$	−156968.	−1.53111	−0.765556	+	0.643369i	$0.777536\pi$
$102$	−13413.0	−0.127652
$103$	−169126.	−1.57078	−0.785391	−	0.618999i	$-0.787538\pi$
$103$	−169126.	−1.57078	−0.785391	+	0.618999i	$0.787538\pi$
$104$	43910.0	0.398089
$105$	0	0
$106$	−9624.11	−0.0831948
$107$	13538.5	0.114317	0.0571587	−	0.998365i	$-0.481796\pi$
$107$	13538.5	0.114317	0.0571587	+	0.998365i	$0.481796\pi$
$108$	−22475.7	−0.185419
$109$	−74736.4	−0.602513	−0.301256	−	0.953543i	$-0.597406\pi$
$109$	−74736.4	−0.602513	−0.301256	+	0.953543i	$0.597406\pi$
$110$	0	0
$111$	−105068.	−0.809399
$112$	127389.	0.959590
$113$	205914.	1.51701	0.758507	−	0.651665i	$-0.225929\pi$
$113$	205914.	1.51701	0.758507	+	0.651665i	$0.225929\pi$
$114$	−18576.4	−0.133875
$115$	0	0
$116$	−1649.73	−0.0113833
$117$	52352.8	0.353570
$118$	12193.2	0.0806142
$119$	192286.	1.24475
$120$	0	0
$121$	14641.0	0.0909091
$122$	−12815.7	−0.0779550
$123$	169799.	1.01198
$124$	−19550.9	−0.114186
$125$	0	0
$126$	−12218.6	−0.0685636
$127$	214682.	1.18110	0.590550	−	0.807001i	$-0.298911\pi$
$127$	214682.	1.18110	0.590550	+	0.807001i	$0.298911\pi$
$128$	129062.	0.696261
$129$	−120414.	−0.637090
$130$	0	0
$131$	253067.	1.28842	0.644209	−	0.764849i	$-0.277187\pi$
$131$	253067.	1.28842	0.644209	+	0.764849i	$0.277187\pi$
$132$	33574.8	0.167718
$133$	266307.	1.30543
$134$	−61993.3	−0.298252
$135$	0	0
$136$	93639.2	0.434121
$137$	−239398.	−1.08973	−0.544865	−	0.838524i	$-0.683419\pi$
$137$	−239398.	−1.08973	−0.544865	+	0.838524i	$0.683419\pi$
$138$	3344.96	0.0149518
$139$	−166398.	−0.730484	−0.365242	−	0.930913i	$-0.619014\pi$
$139$	−166398.	−0.730484	−0.365242	+	0.930913i	$0.619014\pi$
$140$	0	0
$141$	20276.5	0.0858907
$142$	−33601.5	−0.139842
$143$	−78206.1	−0.319816
$144$	73963.4	0.297242
$145$	0	0
$146$	60563.0	0.235139
$147$	23899.3	0.0912203
$148$	359926.	1.35070
$149$	295512.	1.09046	0.545229	−	0.838287i	$-0.316443\pi$
$149$	295512.	1.09046	0.545229	+	0.838287i	$0.316443\pi$
$150$	0	0
$151$	428373.	1.52890	0.764452	−	0.644681i	$-0.223010\pi$
$151$	428373.	1.52890	0.764452	+	0.644681i	$0.223010\pi$
$152$	129686.	0.455284
$153$	111644.	0.385572
$154$	18252.4	0.0620181
$155$	0	0
$156$	−179342.	−0.590027
$157$	289003.	0.935735	0.467867	−	0.883799i	$-0.345022\pi$
$157$	289003.	0.935735	0.467867	+	0.883799i	$0.345022\pi$
$158$	954.804	0.00304279
$159$	80106.4	0.251290
$160$	0	0
$161$	−47952.5	−0.145796
$162$	−7094.24	−0.0212382
$163$	−288223.	−0.849688	−0.424844	−	0.905267i	$-0.639671\pi$
$163$	−288223.	−0.849688	−0.424844	+	0.905267i	$0.639671\pi$
$164$	−581671.	−1.68876
$165$	0	0
$166$	101625.	0.286240
$167$	180912.	0.501969	0.250985	−	0.967991i	$-0.419246\pi$
$167$	180912.	0.501969	0.250985	+	0.967991i	$0.419246\pi$
$168$	85300.2	0.233172
$169$	46451.1	0.125106
$170$	0	0
$171$	154621.	0.404369
$172$	412495.	1.06316
$173$	−218579.	−0.555255	−0.277628	−	0.960689i	$-0.589548\pi$
$173$	−218579.	−0.555255	−0.277628	+	0.960689i	$0.589548\pi$
$174$	−520.723	−0.00130387
$175$	0	0
$176$	−110489.	−0.268866
$177$	−101490.	−0.243495
$178$	−23506.8	−0.0556089
$179$	310916.	0.725287	0.362643	−	0.931928i	$-0.381874\pi$
$179$	310916.	0.725287	0.362643	+	0.931928i	$0.381874\pi$
$180$	0	0
$181$	−423637.	−0.961163	−0.480582	−	0.876950i	$-0.659574\pi$
$181$	−423637.	−0.961163	−0.480582	+	0.876950i	$0.659574\pi$
$182$	−97496.7	−0.218178
$183$	106672.	0.235463
$184$	−23351.8	−0.0508483
$185$	0	0
$186$	−6171.04	−0.0130791
$187$	−166776.	−0.348763
$188$	−69460.3	−0.143332
$189$	101701.	0.207096
$190$	0	0
$191$	861089.	1.70791	0.853954	−	0.520348i	$-0.174198\pi$
$191$	861089.	1.70791	0.853954	+	0.520348i	$0.174198\pi$
$192$	−232216.	−0.454611
$193$	−430583.	−0.832077	−0.416038	−	0.909347i	$-0.636582\pi$
$193$	−430583.	−0.832077	−0.416038	+	0.909347i	$0.636582\pi$
$194$	−170905.	−0.326025
$195$	0	0
$196$	−81870.6	−0.152226
$197$	184755.	0.339181	0.169590	−	0.985515i	$-0.445756\pi$
$197$	184755.	0.339181	0.169590	+	0.985515i	$0.445756\pi$
$198$	10597.6	0.0192107
$199$	322993.	0.578177	0.289089	−	0.957302i	$-0.406648\pi$
$199$	322993.	0.578177	0.289089	+	0.957302i	$0.406648\pi$
$200$	0	0
$201$	516002.	0.900869
$202$	169725.	0.292663
$203$	7464.96	0.0127142
$204$	−382452.	−0.643431
$205$	0	0
$206$	182871.	0.300246
$207$	−27841.8	−0.0451618
$208$	590183.	0.945864
$209$	−230977.	−0.365765
$210$	0	0
$211$	−394106.	−0.609406	−0.304703	−	0.952447i	$-0.598557\pi$
$211$	−394106.	−0.609406	−0.304703	+	0.952447i	$0.598557\pi$
$212$	−274417.	−0.419344
$213$	279682.	0.422393
$214$	−14638.9	−0.0218511
$215$	0	0
$216$	49526.3	0.0722274
$217$	88466.6	0.127535
$218$	80810.6	0.115167
$219$	−504097.	−0.710238
$220$	0	0
$221$	890849.	1.22694
$222$	113607.	0.154712
$223$	−445038.	−0.599287	−0.299643	−	0.954051i	$-0.596868\pi$
$223$	−445038.	−0.599287	−0.299643	+	0.954051i	$0.596868\pi$
$224$	−441032.	−0.587286
$225$	0	0
$226$	−222649.	−0.289968
$227$	−1.15887e6	−1.49270	−0.746349	−	0.665555i	$-0.768195\pi$
$227$	−1.15887e6	−1.49270	−0.746349	+	0.665555i	$0.768195\pi$
$228$	−529677.	−0.674798
$229$	885853.	1.11628	0.558140	−	0.829747i	$-0.311515\pi$
$229$	885853.	1.11628	0.558140	+	0.829747i	$0.311515\pi$
$230$	0	0
$231$	−151924.	−0.187326
$232$	3635.27	0.00443422
$233$	−750392.	−0.905520	−0.452760	−	0.891632i	$-0.649561\pi$
$233$	−750392.	−0.905520	−0.452760	+	0.891632i	$0.649561\pi$
$234$	−56607.8	−0.0675828
$235$	0	0
$236$	347669.	0.406337
$237$	−7947.32	−0.00919073
$238$	−207914.	−0.237926
$239$	617415.	0.699169	0.349585	−	0.936905i	$-0.386323\pi$
$239$	617415.	0.699169	0.349585	+	0.936905i	$0.386323\pi$
$240$	0	0
$241$	1.20275e6	1.33393	0.666966	−	0.745088i	$-0.267593\pi$
$241$	1.20275e6	1.33393	0.666966	+	0.745088i	$0.267593\pi$
$242$	−15830.9	−0.0173767
$243$	59049.0	0.0641500
$244$	−365421.	−0.392933
$245$	0	0
$246$	−183599.	−0.193434
$247$	1.23378e6	1.28675
$248$	43081.3	0.0444795
$249$	−845877.	−0.864587
$250$	0	0
$251$	1.55775e6	1.56068	0.780338	−	0.625358i	$-0.215047\pi$
$251$	1.55775e6	1.56068	0.780338	+	0.625358i	$0.215047\pi$
$252$	−348393.	−0.345596
$253$	41590.8	0.0408504
$254$	−232130.	−0.225760
$255$	0	0
$256$	686107.	0.654323
$257$	−92156.0	−0.0870344	−0.0435172	−	0.999053i	$-0.513856\pi$
$257$	−92156.0	−0.0870344	−0.0435172	+	0.999053i	$0.513856\pi$
$258$	130200.	0.121776
$259$	−1.62865e6	−1.50861
$260$	0	0
$261$	4334.24	0.00393833
$262$	−273635.	−0.246274
$263$	862122.	0.768563	0.384281	−	0.923216i	$-0.374449\pi$
$263$	862122.	0.768563	0.384281	+	0.923216i	$0.374449\pi$
$264$	−73983.8	−0.0653321
$265$	0	0
$266$	−287950.	−0.249525
$267$	195659.	0.167966
$268$	−1.76764e6	−1.50334
$269$	−1.25052e6	−1.05368	−0.526841	−	0.849964i	$-0.676624\pi$
$269$	−1.25052e6	−1.05368	−0.526841	+	0.849964i	$0.676624\pi$
$270$	0	0
$271$	844654.	0.698644	0.349322	−	0.937003i	$-0.386412\pi$
$271$	844654.	0.698644	0.349322	+	0.937003i	$0.386412\pi$
$272$	1.25858e6	1.03147
$273$	811515.	0.659007
$274$	258855.	0.208295
$275$	0	0
$276$	95376.2	0.0753646
$277$	1.52990e6	1.19802	0.599008	−	0.800743i	$-0.295562\pi$
$277$	1.52990e6	1.19802	0.599008	+	0.800743i	$0.295562\pi$
$278$	179922.	0.139628
$279$	51364.8	0.0395053
$280$	0	0
$281$	1.13284e6	0.855858	0.427929	−	0.903812i	$-0.359243\pi$
$281$	1.13284e6	0.855858	0.427929	+	0.903812i	$0.359243\pi$
$282$	−21924.5	−0.0164175
$283$	411138.	0.305156	0.152578	−	0.988291i	$-0.451242\pi$
$283$	411138.	0.305156	0.152578	+	0.988291i	$0.451242\pi$
$284$	−958094.	−0.704875
$285$	0	0
$286$	84562.2	0.0611310
$287$	2.63203e6	1.88619
$288$	−256068.	−0.181918
$289$	479901.	0.337993
$290$	0	0
$291$	1.42253e6	0.984757
$292$	1.72686e6	1.18522
$293$	−314013.	−0.213687	−0.106843	−	0.994276i	$-0.534074\pi$
$293$	−314013.	−0.213687	−0.106843	+	0.994276i	$0.534074\pi$
$294$	−25841.7	−0.0174362
$295$	0	0
$296$	−793115.	−0.526147
$297$	−88209.0	−0.0580259
$298$	−319529.	−0.208435
$299$	−222161.	−0.143711
$300$	0	0
$301$	−1.86652e6	−1.18745
$302$	−463189.	−0.292241
$303$	−1.41271e6	−0.883988
$304$	1.74307e6	1.08176
$305$	0	0
$306$	−120717.	−0.0736998
$307$	−797600.	−0.482991	−0.241496	−	0.970402i	$-0.577638\pi$
$307$	−797600.	−0.482991	−0.241496	+	0.970402i	$0.577638\pi$
$308$	520439.	0.312603
$309$	−1.52213e6	−0.906892
$310$	0	0
$311$	−618368.	−0.362532	−0.181266	−	0.983434i	$-0.558020\pi$
$311$	−618368.	−0.362532	−0.181266	+	0.983434i	$0.558020\pi$
$312$	395190.	0.229837
$313$	2.60175e6	1.50108	0.750541	−	0.660824i	$-0.229793\pi$
$313$	2.60175e6	1.50108	0.750541	+	0.660824i	$0.229793\pi$
$314$	−312491.	−0.178860
$315$	0	0
$316$	27224.7	0.0153372
$317$	729701.	0.407846	0.203923	−	0.978987i	$-0.434631\pi$
$317$	729701.	0.407846	0.203923	+	0.978987i	$0.434631\pi$
$318$	−86617.0	−0.0480325
$319$	−6474.61	−0.00356235
$320$	0	0
$321$	121847.	0.0660011
$322$	51849.8	0.0278681
$323$	2.63107e6	1.40322
$324$	−202281.	−0.107052
$325$	0	0
$326$	311648.	0.162413
$327$	−672628.	−0.347861
$328$	1.28174e6	0.657833
$329$	314304.	0.160089
$330$	0	0
$331$	2.87129e6	1.44048	0.720239	−	0.693726i	$-0.244032\pi$
$331$	2.87129e6	1.44048	0.720239	+	0.693726i	$0.244032\pi$
$332$	2.89768e6	1.44280
$333$	−945611.	−0.467307
$334$	−195616.	−0.0959484
$335$	0	0
$336$	1.14650e6	0.554020
$337$	−2.16558e6	−1.03872	−0.519361	−	0.854555i	$-0.673830\pi$
$337$	−2.16558e6	−1.03872	−0.519361	+	0.854555i	$0.673830\pi$
$338$	−50226.3	−0.0239133
$339$	1.85323e6	0.875849
$340$	0	0
$341$	−76730.1	−0.0357338
$342$	−167187.	−0.0772927
$343$	−1.97425e6	−0.906081
$344$	−908953.	−0.414138
$345$	0	0
$346$	236343.	0.106134
$347$	−392930.	−0.175183	−0.0875913	−	0.996156i	$-0.527917\pi$
$347$	−392930.	−0.175183	−0.0875913	+	0.996156i	$0.527917\pi$
$348$	−14847.6	−0.00657216
$349$	247840.	0.108920	0.0544601	−	0.998516i	$-0.482656\pi$
$349$	247840.	0.108920	0.0544601	+	0.998516i	$0.482656\pi$
$350$	0	0
$351$	471175.	0.204134
$352$	382522.	0.164551
$353$	2.90847e6	1.24230	0.621151	−	0.783691i	$-0.286665\pi$
$353$	2.90847e6	1.24230	0.621151	+	0.783691i	$0.286665\pi$
$354$	109738.	0.0465426
$355$	0	0
$356$	−670261.	−0.280297
$357$	1.73058e6	0.718655
$358$	−336185.	−0.138634
$359$	758403.	0.310573	0.155287	−	0.987869i	$-0.450370\pi$
$359$	758403.	0.310573	0.155287	+	0.987869i	$0.450370\pi$
$360$	0	0
$361$	1.16779e6	0.471627
$362$	458067.	0.183721
$363$	131769.	0.0524864
$364$	−2.77997e6	−1.09973
$365$	0	0
$366$	−115342.	−0.0450073
$367$	2.29924e6	0.891085	0.445542	−	0.895261i	$-0.353011\pi$
$367$	2.29924e6	0.891085	0.445542	+	0.895261i	$0.353011\pi$
$368$	−313866.	−0.120816
$369$	1.52819e6	0.584266
$370$	0	0
$371$	1.24172e6	0.468370
$372$	−175958.	−0.0659251
$373$	19163.9	0.00713203	0.00356601	−	0.999994i	$-0.498865\pi$
$373$	19163.9	0.00713203	0.00356601	+	0.999994i	$0.498865\pi$
$374$	180331.	0.0666640
$375$	0	0
$376$	153059.	0.0558329
$377$	34584.7	0.0125323
$378$	−109967.	−0.0395852
$379$	346535.	0.123922	0.0619610	−	0.998079i	$-0.480265\pi$
$379$	346535.	0.123922	0.0619610	+	0.998079i	$0.480265\pi$
$380$	0	0
$381$	1.93214e6	0.681909
$382$	−931073.	−0.326457
$383$	5.25066e6	1.82901	0.914507	−	0.404571i	$-0.132579\pi$
$383$	5.25066e6	1.82901	0.914507	+	0.404571i	$0.132579\pi$
$384$	1.16155e6	0.401987
$385$	0	0
$386$	465578.	0.159047
$387$	−1.08372e6	−0.367824
$388$	−4.87309e6	−1.64333
$389$	−4.34474e6	−1.45576	−0.727880	−	0.685704i	$-0.759494\pi$
$389$	−4.34474e6	−1.45576	−0.727880	+	0.685704i	$0.759494\pi$
$390$	0	0
$391$	−473763.	−0.156718
$392$	180406.	0.0592974
$393$	2.27760e6	0.743869
$394$	−199771.	−0.0648324
$395$	0	0
$396$	302173.	0.0968318
$397$	−3.90748e6	−1.24429	−0.622144	−	0.782903i	$-0.713738\pi$
$397$	−3.90748e6	−1.24429	−0.622144	+	0.782903i	$0.713738\pi$
$398$	−349244.	−0.110515
$399$	2.39676e6	0.753689
$400$	0	0
$401$	−3.05315e6	−0.948171	−0.474085	−	0.880479i	$-0.657221\pi$
$401$	−3.05315e6	−0.948171	−0.474085	+	0.880479i	$0.657221\pi$
$402$	−557940.	−0.172196
$403$	409860.	0.125711
$404$	4.83945e6	1.47517
$405$	0	0
$406$	−8071.67	−0.00243023
$407$	1.41258e6	0.422695
$408$	842753.	0.250640
$409$	−5.46050e6	−1.61408	−0.807038	−	0.590499i	$-0.798931\pi$
$409$	−5.46050e6	−1.61408	−0.807038	+	0.590499i	$0.798931\pi$
$410$	0	0
$411$	−2.15458e6	−0.629156
$412$	5.21428e6	1.51339
$413$	−1.57318e6	−0.453842
$414$	30104.6	0.00863241
$415$	0	0
$416$	−2.04327e6	−0.578886
$417$	−1.49758e6	−0.421745
$418$	249749.	0.0699139
$419$	3.02804e6	0.842609	0.421305	−	0.906919i	$-0.361572\pi$
$419$	3.02804e6	0.842609	0.421305	+	0.906919i	$0.361572\pi$
$420$	0	0
$421$	−3.31510e6	−0.911572	−0.455786	−	0.890089i	$-0.650642\pi$
$421$	−3.31510e6	−0.911572	−0.455786	+	0.890089i	$0.650642\pi$
$422$	426136.	0.116484
$423$	182489.	0.0495890
$424$	604691.	0.163350
$425$	0	0
$426$	−302413.	−0.0807378
$427$	1.65351e6	0.438871
$428$	−417404.	−0.110141
$429$	−703855.	−0.184646
$430$	0	0
$431$	−3.02139e6	−0.783455	−0.391728	−	0.920081i	$-0.628122\pi$
$431$	−3.02139e6	−0.783455	−0.391728	+	0.920081i	$0.628122\pi$
$432$	665670.	0.171613
$433$	739663.	0.189589	0.0947947	−	0.995497i	$-0.469781\pi$
$433$	739663.	0.189589	0.0947947	+	0.995497i	$0.469781\pi$
$434$	−95656.7	−0.0243776
$435$	0	0
$436$	2.30419e6	0.580499
$437$	−656138.	−0.164358
$438$	545067.	0.135758
$439$	89371.4	0.0221329	0.0110664	−	0.999939i	$-0.496477\pi$
$439$	89371.4	0.0221329	0.0110664	+	0.999939i	$0.496477\pi$
$440$	0	0
$441$	215094.	0.0526661
$442$	−963252.	−0.234522
$443$	−5.06164e6	−1.22541	−0.612706	−	0.790311i	$-0.709919\pi$
$443$	−5.06164e6	−1.22541	−0.612706	+	0.790311i	$0.709919\pi$
$444$	3.23933e6	0.779826
$445$	0	0
$446$	481208.	0.114550
$447$	2.65961e6	0.629577
$448$	−3.59956e6	−0.847334
$449$	6.64598e6	1.55576	0.777881	−	0.628411i	$-0.216294\pi$
$449$	6.64598e6	1.55576	0.777881	+	0.628411i	$0.216294\pi$
$450$	0	0
$451$	−2.28285e6	−0.528489
$452$	−6.34850e6	−1.46159
$453$	3.85536e6	0.882713
$454$	1.25306e6	0.285320
$455$	0	0
$456$	1.16717e6	0.262858
$457$	176313.	0.0394905	0.0197453	−	0.999805i	$-0.493714\pi$
$457$	176313.	0.0394905	0.0197453	+	0.999805i	$0.493714\pi$
$458$	−957850.	−0.213370
$459$	1.00479e6	0.222610
$460$	0	0
$461$	8.43740e6	1.84908	0.924542	−	0.381081i	$-0.124448\pi$
$461$	8.43740e6	1.84908	0.924542	+	0.381081i	$0.124448\pi$
$462$	164272.	0.0358062
$463$	−5.65739e6	−1.22649	−0.613244	−	0.789893i	$-0.710136\pi$
$463$	−5.65739e6	−1.22649	−0.613244	+	0.789893i	$0.710136\pi$
$464$	48860.8	0.0105357
$465$	0	0
$466$	811379.	0.173085
$467$	2.54278e6	0.539532	0.269766	−	0.962926i	$-0.413054\pi$
$467$	2.54278e6	0.539532	0.269766	+	0.962926i	$0.413054\pi$
$468$	−1.61408e6	−0.340652
$469$	7.99850e6	1.67910
$470$	0	0
$471$	2.60102e6	0.540247
$472$	−766107.	−0.158283
$473$	1.61889e6	0.332710
$474$	8593.23	0.00175675
$475$	0	0
$476$	−5.92835e6	−1.19927
$477$	720958.	0.145082
$478$	−667594.	−0.133642
$479$	6.89136e6	1.37235	0.686177	−	0.727434i	$-0.259287\pi$
$479$	6.89136e6	1.37235	0.686177	+	0.727434i	$0.259287\pi$
$480$	0	0
$481$	−7.54540e6	−1.48703
$482$	−1.30051e6	−0.254973
$483$	−431573.	−0.0841756
$484$	−451394.	−0.0875876
$485$	0	0
$486$	−63848.2	−0.0122619
$487$	−7.89949e6	−1.50930	−0.754652	−	0.656125i	$-0.772194\pi$
$487$	−7.89949e6	−1.50930	−0.754652	+	0.656125i	$0.772194\pi$
$488$	805223.	0.153062
$489$	−2.59401e6	−0.490568
$490$	0	0
$491$	−2.29000e6	−0.428679	−0.214340	−	0.976759i	$-0.568760\pi$
$491$	−2.29000e6	−0.428679	−0.214340	+	0.976759i	$0.568760\pi$
$492$	−5.23504e6	−0.975005
$493$	73752.6	0.0136666
$494$	−1.33406e6	−0.245955
$495$	0	0
$496$	579045.	0.105684
$497$	4.33533e6	0.787283
$498$	914625.	0.165261
$499$	−5.96310e6	−1.07206	−0.536032	−	0.844198i	$-0.680077\pi$
$499$	−5.96310e6	−1.07206	−0.536032	+	0.844198i	$0.680077\pi$
$500$	0	0
$501$	1.62821e6	0.289812
$502$	−1.68435e6	−0.298314
$503$	−2.17453e6	−0.383218	−0.191609	−	0.981471i	$-0.561371\pi$
$503$	−2.17453e6	−0.383218	−0.191609	+	0.981471i	$0.561371\pi$
$504$	767702.	0.134622
$505$	0	0
$506$	−44971.1	−0.00780831
$507$	418060.	0.0722301
$508$	−6.61883e6	−1.13795
$509$	−3.33531e6	−0.570613	−0.285307	−	0.958436i	$-0.592095\pi$
$509$	−3.33531e6	−0.570613	−0.285307	+	0.958436i	$0.592095\pi$
$510$	0	0
$511$	−7.81395e6	−1.32379
$512$	−4.87184e6	−0.821331
$513$	1.39159e6	0.233462
$514$	99645.9	0.0166361
$515$	0	0
$516$	3.71245e6	0.613814
$517$	−272607.	−0.0448549
$518$	1.76101e6	0.288362
$519$	−1.96721e6	−0.320577
$520$	0	0
$521$	9.81327e6	1.58387	0.791935	−	0.610605i	$-0.209074\pi$
$521$	9.81327e6	1.58387	0.791935	+	0.610605i	$0.209074\pi$
$522$	−4686.51	−0.000752789 0
$523$	−7.96466e6	−1.27325	−0.636624	−	0.771175i	$-0.719670\pi$
$523$	−7.96466e6	−1.27325	−0.636624	+	0.771175i	$0.719670\pi$
$524$	−7.80227e6	−1.24134
$525$	0	0
$526$	−932190.	−0.146906
$527$	874036.	0.137089
$528$	−994397.	−0.155230
$529$	−6.31820e6	−0.981644
$530$	0	0
$531$	−913410.	−0.140582
$532$	−8.21046e6	−1.25773
$533$	1.21940e7	1.85921
$534$	−211561.	−0.0321058
$535$	0	0
$536$	3.89509e6	0.585607
$537$	2.79824e6	0.418745
$538$	1.35215e6	0.201405
$539$	−321313.	−0.0476383
$540$	0	0
$541$	−5.50616e6	−0.808827	−0.404413	−	0.914576i	$-0.632524\pi$
$541$	−5.50616e6	−0.808827	−0.404413	+	0.914576i	$0.632524\pi$
$542$	−913303.	−0.133542
$543$	−3.81273e6	−0.554928
$544$	−4.35733e6	−0.631281
$545$	0	0
$546$	−877470.	−0.125965
$547$	−2.92918e6	−0.418580	−0.209290	−	0.977854i	$-0.567115\pi$
$547$	−2.92918e6	−0.418580	−0.209290	+	0.977854i	$0.567115\pi$
$548$	7.38084e6	1.04992
$549$	960047.	0.135945
$550$	0	0
$551$	102144.	0.0143328
$552$	−210167.	−0.0293573
$553$	−123191.	−0.0171303
$554$	−1.65424e6	−0.228994
$555$	0	0
$556$	5.13018e6	0.703795
$557$	−4.05453e6	−0.553736	−0.276868	−	0.960908i	$-0.589296\pi$
$557$	−4.05453e6	−0.553736	−0.276868	+	0.960908i	$0.589296\pi$
$558$	−55539.4	−0.00755120
$559$	−8.64745e6	−1.17047
$560$	0	0
$561$	−1.50099e6	−0.201358
$562$	−1.22491e6	−0.163592
$563$	1.65845e6	0.220512	0.110256	−	0.993903i	$-0.464833\pi$
$563$	1.65845e6	0.220512	0.110256	+	0.993903i	$0.464833\pi$
$564$	−625143.	−0.0827526
$565$	0	0
$566$	−444553.	−0.0583288
$567$	915312.	0.119567
$568$	2.11121e6	0.274575
$569$	1.05475e7	1.36575	0.682873	−	0.730537i	$-0.260730\pi$
$569$	1.05475e7	1.36575	0.682873	+	0.730537i	$0.260730\pi$
$570$	0	0
$571$	−1.46769e7	−1.88384	−0.941918	−	0.335843i	$-0.890979\pi$
$571$	−1.46769e7	−1.88384	−0.941918	+	0.335843i	$0.890979\pi$
$572$	2.41116e6	0.308131
$573$	7.74980e6	0.986062
$574$	−2.84595e6	−0.360534
$575$	0	0
$576$	−2.08995e6	−0.262470
$577$	8.25271e6	1.03195	0.515973	−	0.856605i	$-0.327430\pi$
$577$	8.25271e6	1.03195	0.515973	+	0.856605i	$0.327430\pi$
$578$	−518905.	−0.0646053
$579$	−3.87525e6	−0.480400
$580$	0	0
$581$	−1.31118e7	−1.61147
$582$	−1.53815e6	−0.188230
$583$	−1.07699e6	−0.131232
$584$	−3.80522e6	−0.461687
$585$	0	0
$586$	339534.	0.0408450
$587$	−1.01063e7	−1.21059	−0.605296	−	0.796000i	$-0.706945\pi$
$587$	−1.01063e7	−1.21059	−0.605296	+	0.796000i	$0.706945\pi$
$588$	−736835.	−0.0878875
$589$	1.21050e6	0.143772
$590$	0	0
$591$	1.66280e6	0.195826
$592$	−1.06600e7	−1.25013
$593$	−1.48922e7	−1.73909	−0.869543	−	0.493857i	$-0.835587\pi$
$593$	−1.48922e7	−1.73909	−0.869543	+	0.493857i	$0.835587\pi$
$594$	95378.1	0.0110913
$595$	0	0
$596$	−9.11088e6	−1.05062
$597$	2.90694e6	0.333811
$598$	240217.	0.0274695
$599$	1.59013e7	1.81078	0.905390	−	0.424580i	$-0.139578\pi$
$599$	1.59013e7	1.81078	0.905390	+	0.424580i	$0.139578\pi$
$600$	0	0
$601$	−2.84828e6	−0.321660	−0.160830	−	0.986982i	$-0.551417\pi$
$601$	−2.84828e6	−0.321660	−0.160830	+	0.986982i	$0.551417\pi$
$602$	2.01822e6	0.226974
$603$	4.64402e6	0.520117
$604$	−1.32071e7	−1.47304
$605$	0	0
$606$	1.52753e6	0.168969
$607$	−1.32016e6	−0.145430	−0.0727149	−	0.997353i	$-0.523166\pi$
$607$	−1.32016e6	−0.145430	−0.0727149	+	0.997353i	$0.523166\pi$
$608$	−6.03467e6	−0.662056
$609$	67184.6	0.00734052
$610$	0	0
$611$	1.45615e6	0.157799
$612$	−3.44207e6	−0.371485
$613$	978527.	0.105177	0.0525886	−	0.998616i	$-0.483253\pi$
$613$	978527.	0.105177	0.0525886	+	0.998616i	$0.483253\pi$
$614$	862424.	0.0923209
$615$	0	0
$616$	−1.14681e6	−0.121770
$617$	−1.29643e7	−1.37100	−0.685500	−	0.728073i	$-0.740416\pi$
$617$	−1.29643e7	−1.37100	−0.685500	+	0.728073i	$0.740416\pi$
$618$	1.64584e6	0.173347
$619$	1.27406e7	1.33648	0.668240	−	0.743946i	$-0.267048\pi$
$619$	1.27406e7	1.33648	0.668240	+	0.743946i	$0.267048\pi$
$620$	0	0
$621$	−250576.	−0.0260742
$622$	668626.	0.0692958
$623$	3.03289e6	0.313067
$624$	5.31165e6	0.546095
$625$	0	0
$626$	−2.81320e6	−0.286923
$627$	−2.07879e6	−0.211175
$628$	−8.91020e6	−0.901547
$629$	−1.60908e7	−1.62162
$630$	0	0
$631$	−1.93988e7	−1.93955	−0.969775	−	0.243999i	$-0.921541\pi$
$631$	−1.93988e7	−1.93955	−0.969775	+	0.243999i	$0.921541\pi$
$632$	−59991.1	−0.00597440
$633$	−3.54695e6	−0.351840
$634$	−789006.	−0.0779574
$635$	0	0
$636$	−2.46975e6	−0.242108
$637$	1.71632e6	0.167590
$638$	7000.83	0.000680923 0
$639$	2.51714e6	0.243868
$640$	0	0
$641$	6.77844e6	0.651605	0.325803	−	0.945438i	$-0.394366\pi$
$641$	6.77844e6	0.651605	0.325803	+	0.945438i	$0.394366\pi$
$642$	−131750.	−0.0126157
$643$	1.74257e7	1.66212	0.831060	−	0.556182i	$-0.187734\pi$
$643$	1.74257e7	1.66212	0.831060	+	0.556182i	$0.187734\pi$
$644$	1.47842e6	0.140470
$645$	0	0
$646$	−2.84491e6	−0.268217
$647$	−792517.	−0.0744300	−0.0372150	−	0.999307i	$-0.511849\pi$
$647$	−792517.	−0.0744300	−0.0372150	+	0.999307i	$0.511849\pi$
$648$	445737.	0.0417005
$649$	1.36448e6	0.127161
$650$	0	0
$651$	796200.	0.0736325
$652$	8.88616e6	0.818644
$653$	9.09663e6	0.834829	0.417414	−	0.908716i	$-0.362936\pi$
$653$	9.09663e6	0.834829	0.417414	+	0.908716i	$0.362936\pi$
$654$	727295.	0.0664915
$655$	0	0
$656$	1.72275e7	1.56302
$657$	−4.53687e6	−0.410056
$658$	−339849.	−0.0306000
$659$	−8.15992e6	−0.731935	−0.365968	−	0.930628i	$-0.619262\pi$
$659$	−8.15992e6	−0.731935	−0.365968	+	0.930628i	$0.619262\pi$
$660$	0	0
$661$	−1.85172e6	−0.164843	−0.0824216	−	0.996598i	$-0.526265\pi$
$661$	−1.85172e6	−0.164843	−0.0824216	+	0.996598i	$0.526265\pi$
$662$	−3.10465e6	−0.275339
$663$	8.01765e6	0.708375
$664$	−6.38518e6	−0.562022
$665$	0	0
$666$	1.02246e6	0.0893229
$667$	−18392.5	−0.00160076
$668$	−5.57768e6	−0.483629
$669$	−4.00534e6	−0.345998
$670$	0	0
$671$	−1.43414e6	−0.122966
$672$	−3.96929e6	−0.339070
$673$	−1.42740e6	−0.121481	−0.0607405	−	0.998154i	$-0.519346\pi$
$673$	−1.42740e6	−0.121481	−0.0607405	+	0.998154i	$0.519346\pi$
$674$	2.34158e6	0.198545
$675$	0	0
$676$	−1.43213e6	−0.120535
$677$	−941868.	−0.0789802	−0.0394901	−	0.999220i	$-0.512573\pi$
$677$	−941868.	−0.0789802	−0.0394901	+	0.999220i	$0.512573\pi$
$678$	−2.00385e6	−0.167413
$679$	2.20505e7	1.83545
$680$	0	0
$681$	−1.04299e7	−0.861809
$682$	82966.3	0.00683031
$683$	−9.56089e6	−0.784236	−0.392118	−	0.919915i	$-0.628257\pi$
$683$	−9.56089e6	−0.784236	−0.392118	+	0.919915i	$0.628257\pi$
$684$	−4.76709e6	−0.389595
$685$	0	0
$686$	2.13471e6	0.173192
$687$	7.97268e6	0.644484
$688$	−1.22170e7	−0.983996
$689$	5.75281e6	0.461670
$690$	0	0
$691$	−1.53765e7	−1.22507	−0.612536	−	0.790443i	$-0.709851\pi$
$691$	−1.53765e7	−1.22507	−0.612536	+	0.790443i	$0.709851\pi$
$692$	6.73897e6	0.534968
$693$	−1.36732e6	−0.108152
$694$	424865.	0.0334851
$695$	0	0
$696$	32717.5	0.00256010
$697$	2.60040e7	2.02749
$698$	−267983.	−0.0208194
$699$	−6.75352e6	−0.522802
$700$	0	0
$701$	1.72261e7	1.32401	0.662006	−	0.749499i	$-0.269705\pi$
$701$	1.72261e7	1.32401	0.662006	+	0.749499i	$0.269705\pi$
$702$	−509470.	−0.0390190
$703$	−2.22849e7	−1.70068
$704$	3.12202e6	0.237413
$705$	0	0
$706$	−3.14485e6	−0.237459
$707$	−2.18983e7	−1.64764
$708$	3.12902e6	0.234599
$709$	2.47878e7	1.85192	0.925959	−	0.377625i	$-0.123259\pi$
$709$	2.47878e7	1.85192	0.925959	+	0.377625i	$0.123259\pi$
$710$	0	0
$711$	−71525.9	−0.00530627
$712$	1.47695e6	0.109186
$713$	−217968.	−0.0160572
$714$	−1.87123e6	−0.137367
$715$	0	0
$716$	−9.58579e6	−0.698788
$717$	5.55673e6	0.403665
$718$	−820042.	−0.0593642
$719$	7.44502e6	0.537086	0.268543	−	0.963268i	$-0.413458\pi$
$719$	7.44502e6	0.537086	0.268543	+	0.963268i	$0.413458\pi$
$720$	0	0
$721$	−2.35944e7	−1.69032
$722$	−1.26271e6	−0.0901486
$723$	1.08248e7	0.770146
$724$	1.30611e7	0.926046
$725$	0	0
$726$	−142478.	−0.0100325
$727$	−3.56658e6	−0.250274	−0.125137	−	0.992139i	$-0.539937\pi$
$727$	−3.56658e6	−0.250274	−0.125137	+	0.992139i	$0.539937\pi$
$728$	6.12580e6	0.428385
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.84409e7	−1.27641
$732$	−3.28879e6	−0.226860
$733$	−1.38091e7	−0.949305	−0.474652	−	0.880173i	$-0.657426\pi$
$733$	−1.38091e7	−0.949305	−0.474652	+	0.880173i	$0.657426\pi$
$734$	−2.48611e6	−0.170326
$735$	0	0
$736$	1.08663e6	0.0739416
$737$	−6.93737e6	−0.470463
$738$	−1.65239e6	−0.111679
$739$	−7.18607e6	−0.484039	−0.242020	−	0.970271i	$-0.577810\pi$
$739$	−7.18607e6	−0.484039	−0.242020	+	0.970271i	$0.577810\pi$
$740$	0	0
$741$	1.11040e7	0.742908
$742$	−1.34264e6	−0.0895261
$743$	−2.18843e6	−0.145432	−0.0727161	−	0.997353i	$-0.523167\pi$
$743$	−2.18843e6	−0.145432	−0.0727161	+	0.997353i	$0.523167\pi$
$744$	387732.	0.0256803
$745$	0	0
$746$	−20721.5	−0.00136324
$747$	−7.61289e6	−0.499170
$748$	5.14186e6	0.336021
$749$	1.88873e6	0.123017
$750$	0	0
$751$	−1.41771e6	−0.0917250	−0.0458625	−	0.998948i	$-0.514604\pi$
$751$	−1.41771e6	−0.0917250	−0.0458625	+	0.998948i	$0.514604\pi$
$752$	2.05723e6	0.132660
$753$	1.40197e7	0.901056
$754$	−37395.5	−0.00239547
$755$	0	0
$756$	−3.13554e6	−0.199530
$757$	2.05513e7	1.30347	0.651734	−	0.758448i	$-0.274042\pi$
$757$	2.05513e7	1.30347	0.651734	+	0.758448i	$0.274042\pi$
$758$	−374699.	−0.0236870
$759$	374318.	0.0235850
$760$	0	0
$761$	−1.30587e7	−0.817409	−0.408705	−	0.912667i	$-0.634019\pi$
$761$	−1.30587e7	−0.817409	−0.408705	+	0.912667i	$0.634019\pi$
$762$	−2.08917e6	−0.130343
$763$	−1.04263e7	−0.648366
$764$	−2.65481e7	−1.64551
$765$	0	0
$766$	−5.67740e6	−0.349605
$767$	−7.28846e6	−0.447350
$768$	6.17497e6	0.377774
$769$	1.82667e7	1.11389	0.556946	−	0.830549i	$-0.311973\pi$
$769$	1.82667e7	1.11389	0.556946	+	0.830549i	$0.311973\pi$
$770$	0	0
$771$	−829404.	−0.0502493
$772$	1.32752e7	0.801676
$773$	1.49182e7	0.897981	0.448990	−	0.893537i	$-0.351784\pi$
$773$	1.49182e7	0.897981	0.448990	+	0.893537i	$0.351784\pi$
$774$	1.17180e6	0.0703074
$775$	0	0
$776$	1.07381e7	0.640138
$777$	−1.46578e7	−0.870996
$778$	4.69786e6	0.278260
$779$	3.60143e7	2.12633
$780$	0	0
$781$	−3.76018e6	−0.220587
$782$	512268.	0.0299558
$783$	39008.2	0.00227380
$784$	2.42479e6	0.140891
$785$	0	0
$786$	−2.46271e6	−0.142186
$787$	1.60565e7	0.924087	0.462043	−	0.886857i	$-0.347116\pi$
$787$	1.60565e7	0.924087	0.462043	+	0.886857i	$0.347116\pi$
$788$	−5.69616e6	−0.326788
$789$	7.75910e6	0.443730
$790$	0	0
$791$	2.87266e7	1.63246
$792$	−665854.	−0.0377195
$793$	7.66060e6	0.432593
$794$	4.22506e6	0.237838
$795$	0	0
$796$	−9.95816e6	−0.557053
$797$	−2.10741e7	−1.17518	−0.587588	−	0.809160i	$-0.699922\pi$
$797$	−2.10741e7	−1.17518	−0.587588	+	0.809160i	$0.699922\pi$
$798$	−2.59155e6	−0.144063
$799$	3.10528e6	0.172081
$800$	0	0
$801$	1.76093e6	0.0969755
$802$	3.30129e6	0.181237
$803$	6.77730e6	0.370909
$804$	−1.59088e7	−0.867955
$805$	0	0
$806$	−443171.	−0.0240289
$807$	−1.12547e7	−0.608344
$808$	−1.06640e7	−0.574633
$809$	2.38181e7	1.27949	0.639744	−	0.768588i	$-0.279040\pi$
$809$	2.38181e7	1.27949	0.639744	+	0.768588i	$0.279040\pi$
$810$	0	0
$811$	2.93046e7	1.56453	0.782264	−	0.622947i	$-0.214065\pi$
$811$	2.93046e7	1.56453	0.782264	+	0.622947i	$0.214065\pi$
$812$	−230151.	−0.0122496
$813$	7.60189e6	0.403362
$814$	−1.52739e6	−0.0807956
$815$	0	0
$816$	1.13272e7	0.595522
$817$	−2.55397e7	−1.33863
$818$	5.90430e6	0.308521
$819$	7.30364e6	0.380478
$820$	0	0
$821$	−3.14594e6	−0.162889	−0.0814446	−	0.996678i	$-0.525953\pi$
$821$	−3.14594e6	−0.162889	−0.0814446	+	0.996678i	$0.525953\pi$
$822$	2.32969e6	0.120259
$823$	−2.44990e7	−1.26081	−0.630404	−	0.776267i	$-0.717111\pi$
$823$	−2.44990e7	−1.26081	−0.630404	+	0.776267i	$0.717111\pi$
$824$	−1.14899e7	−0.589522
$825$	0	0
$826$	1.70104e6	0.0867492
$827$	−1.34163e7	−0.682135	−0.341067	−	0.940039i	$-0.610788\pi$
$827$	−1.34163e7	−0.682135	−0.341067	+	0.940039i	$0.610788\pi$
$828$	858386.	0.0435118
$829$	1.42349e7	0.719394	0.359697	−	0.933069i	$-0.382880\pi$
$829$	1.42349e7	0.719394	0.359697	+	0.933069i	$0.382880\pi$
$830$	0	0
$831$	1.37691e7	0.691675
$832$	−1.66765e7	−0.835213
$833$	3.66009e6	0.182759
$834$	1.61929e6	0.0806140
$835$	0	0
$836$	7.12121e6	0.352402
$837$	462283.	0.0228084
$838$	−3.27414e6	−0.161060
$839$	1.15794e7	0.567913	0.283957	−	0.958837i	$-0.408353\pi$
$839$	1.15794e7	0.567913	0.283957	+	0.958837i	$0.408353\pi$
$840$	0	0
$841$	−2.05083e7	−0.999860
$842$	3.58453e6	0.174242
$843$	1.01955e7	0.494130
$844$	1.21506e7	0.587140
$845$	0	0
$846$	−197320.	−0.00947865
$847$	2.04254e6	0.0978276
$848$	8.12749e6	0.388121
$849$	3.70025e6	0.176182
$850$	0	0
$851$	4.01273e6	0.189940
$852$	−8.62285e6	−0.406960
$853$	−6.61595e6	−0.311329	−0.155664	−	0.987810i	$-0.549752\pi$
$853$	−6.61595e6	−0.311329	−0.155664	+	0.987810i	$0.549752\pi$
$854$	−1.78790e6	−0.0838876
$855$	0	0
$856$	919772.	0.0429038
$857$	9.97147e6	0.463775	0.231887	−	0.972743i	$-0.425510\pi$
$857$	9.97147e6	0.463775	0.231887	+	0.972743i	$0.425510\pi$
$858$	761060.	0.0352940
$859$	−3.79757e7	−1.75599	−0.877996	−	0.478667i	$-0.841120\pi$
$859$	−3.79757e7	−1.75599	−0.877996	+	0.478667i	$0.841120\pi$
$860$	0	0
$861$	2.36883e7	1.08899
$862$	3.26695e6	0.149753
$863$	−8.18531e6	−0.374118	−0.187059	−	0.982349i	$-0.559896\pi$
$863$	−8.18531e6	−0.374118	−0.187059	+	0.982349i	$0.559896\pi$
$864$	−2.30462e6	−0.105030
$865$	0	0
$866$	−799779.	−0.0362389
$867$	4.31911e6	0.195140
$868$	−2.72750e6	−0.122876
$869$	106847.	0.00479970
$870$	0	0
$871$	3.70565e7	1.65508
$872$	−5.07740e6	−0.226126
$873$	1.28028e7	0.568550
$874$	709465.	0.0314161
$875$	0	0
$876$	1.55417e7	0.684288
$877$	3.02517e7	1.32816	0.664081	−	0.747660i	$-0.268823\pi$
$877$	3.02517e7	1.32816	0.664081	+	0.747660i	$0.268823\pi$
$878$	−96635.0	−0.00423056
$879$	−2.82611e6	−0.123372
$880$	0	0
$881$	6.11414e6	0.265397	0.132698	−	0.991156i	$-0.457636\pi$
$881$	6.11414e6	0.265397	0.132698	+	0.991156i	$0.457636\pi$
$882$	−232575.	−0.0100668
$883$	7.72672e6	0.333498	0.166749	−	0.985999i	$-0.446673\pi$
$883$	7.72672e6	0.333498	0.166749	+	0.985999i	$0.446673\pi$
$884$	−2.74656e7	−1.18211
$885$	0	0
$886$	5.47302e6	0.234230
$887$	2.48274e7	1.05955	0.529775	−	0.848138i	$-0.322276\pi$
$887$	2.48274e7	1.05955	0.529775	+	0.848138i	$0.322276\pi$
$888$	−7.13803e6	−0.303771
$889$	2.99499e7	1.27099
$890$	0	0
$891$	−793881.	−0.0335013
$892$	1.37209e7	0.577391
$893$	4.30065e6	0.180470
$894$	−2.87576e6	−0.120340
$895$	0	0
$896$	1.80051e7	0.749249
$897$	−1.99945e6	−0.0829715
$898$	−7.18613e6	−0.297375
$899$	33931.9	0.00140026
$900$	0	0
$901$	1.22680e7	0.503457
$902$	2.46838e6	0.101017
$903$	−1.67987e7	−0.685575
$904$	1.39893e7	0.569342
$905$	0	0
$906$	−4.16870e6	−0.168725
$907$	3.64608e6	0.147166	0.0735831	−	0.997289i	$-0.476557\pi$
$907$	3.64608e6	0.147166	0.0735831	+	0.997289i	$0.476557\pi$
$908$	3.57291e7	1.43816
$909$	−1.27144e7	−0.510371
$910$	0	0
$911$	−1.15898e7	−0.462678	−0.231339	−	0.972873i	$-0.574311\pi$
$911$	−1.15898e7	−0.462678	−0.231339	+	0.972873i	$0.574311\pi$
$912$	1.56876e7	0.624554
$913$	1.13723e7	0.451516
$914$	−190642.	−0.00754838
$915$	0	0
$916$	−2.73116e7	−1.07550
$917$	3.53048e7	1.38647
$918$	−1.08646e6	−0.0425506
$919$	−3.48544e7	−1.36135	−0.680674	−	0.732586i	$-0.738313\pi$
$919$	−3.48544e7	−1.36135	−0.680674	+	0.732586i	$0.738313\pi$
$920$	0	0
$921$	−7.17840e6	−0.278855
$922$	−9.12314e6	−0.353441
$923$	2.00853e7	0.776021
$924$	4.68395e6	0.180481
$925$	0	0
$926$	6.11719e6	0.234436
$927$	−1.36992e7	−0.523594
$928$	−169161.	−0.00644806
$929$	−4.11213e7	−1.56325	−0.781624	−	0.623750i	$-0.785608\pi$
$929$	−4.11213e7	−1.56325	−0.781624	+	0.623750i	$0.785608\pi$
$930$	0	0
$931$	5.06903e6	0.191669
$932$	2.31352e7	0.872436
$933$	−5.56532e6	−0.209308
$934$	−2.74945e6	−0.103128
$935$	0	0
$936$	3.55671e6	0.132696
$937$	1.53222e6	0.0570126	0.0285063	−	0.999594i	$-0.490925\pi$
$937$	1.53222e6	0.0570126	0.0285063	+	0.999594i	$0.490925\pi$
$938$	−8.64857e6	−0.320950
$939$	2.34157e7	0.866650
$940$	0	0
$941$	−1.91551e7	−0.705196	−0.352598	−	0.935775i	$-0.614702\pi$
$941$	−1.91551e7	−0.705196	−0.352598	+	0.935775i	$0.614702\pi$
$942$	−2.81242e6	−0.103265
$943$	−6.48491e6	−0.237479
$944$	−1.02970e7	−0.376082
$945$	0	0
$946$	−1.75047e6	−0.0635955
$947$	−2.83184e7	−1.02611	−0.513054	−	0.858356i	$-0.671486\pi$
$947$	−2.83184e7	−1.02611	−0.513054	+	0.858356i	$0.671486\pi$
$948$	245023.	0.00885494
$949$	−3.62015e7	−1.30485
$950$	0	0
$951$	6.56731e6	0.235470
$952$	1.30634e7	0.467159
$953$	5.47682e7	1.95342	0.976711	−	0.214559i	$-0.0688313\pi$
$953$	5.47682e7	1.95342	0.976711	+	0.214559i	$0.0688313\pi$
$954$	−779553.	−0.0277316
$955$	0	0
$956$	−1.90354e7	−0.673624
$957$	−58271.5	−0.00205673
$958$	−7.45145e6	−0.262318
$959$	−3.33979e7	−1.17266
$960$	0	0
$961$	−2.82270e7	−0.985954
$962$	8.15865e6	0.284237
$963$	1.09662e6	0.0381058
$964$	−3.70819e7	−1.28520
$965$	0	0
$966$	466648.	0.0160897
$967$	−2.29671e7	−0.789843	−0.394922	−	0.918715i	$-0.629228\pi$
$967$	−2.29671e7	−0.789843	−0.394922	+	0.918715i	$0.629228\pi$
$968$	994671.	0.0341186
$969$	2.36796e7	0.810149
$970$	0	0
$971$	1.46281e7	0.497898	0.248949	−	0.968517i	$-0.419915\pi$
$971$	1.46281e7	0.497898	0.248949	+	0.968517i	$0.419915\pi$
$972$	−1.82053e6	−0.0618062
$973$	−2.32138e7	−0.786076
$974$	8.54152e6	0.288495
$975$	0	0
$976$	1.08228e7	0.363676
$977$	1.54714e7	0.518552	0.259276	−	0.965803i	$-0.416516\pi$
$977$	1.54714e7	0.518552	0.259276	+	0.965803i	$0.416516\pi$
$978$	2.80483e6	0.0937691
$979$	−2.63053e6	−0.0877176
$980$	0	0
$981$	−6.05365e6	−0.200838
$982$	2.47612e6	0.0819395
$983$	−2.02694e7	−0.669047	−0.334524	−	0.942387i	$-0.608575\pi$
$983$	−2.02694e7	−0.669047	−0.334524	+	0.942387i	$0.608575\pi$
$984$	1.15357e7	0.379800
$985$	0	0
$986$	−79746.8	−0.00261229
$987$	2.82874e6	0.0924272
$988$	−3.80385e7	−1.23974
$989$	4.59881e6	0.149505
$990$	0	0
$991$	−4.43452e7	−1.43437	−0.717187	−	0.696881i	$-0.754570\pi$
$991$	−4.43452e7	−1.43437	−0.717187	+	0.696881i	$0.754570\pi$
$992$	−2.00471e6	−0.0646803
$993$	2.58416e7	0.831661
$994$	−4.68768e6	−0.150484
$995$	0	0
$996$	2.60791e7	0.832998
$997$	−2.32293e7	−0.740112	−0.370056	−	0.929009i	$-0.620662\pi$
$997$	−2.32293e7	−0.740112	−0.370056	+	0.929009i	$0.620662\pi$
$998$	6.44775e6	0.204919
$999$	−8.51050e6	−0.269800

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.h.1.2		3
5.4	even	2		165.6.a.d.1.2	✓	3
15.14	odd	2		495.6.a.c.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.a.d.1.2	✓	3	5.4	even	2
495.6.a.c.1.2		3	15.14	odd	2
825.6.a.h.1.2		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

\(f(q)\)	\(=\)	\(q-1.08127 q^{2} +9.00000 q^{3} -30.8308 q^{4} -9.73147 q^{6} +139.508 q^{7} +67.9374 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-1.08127 q^{2} +9.00000 q^{3} -30.8308 q^{4} -9.73147 q^{6} +139.508 q^{7} +67.9374 q^{8} +81.0000 q^{9} -121.000 q^{11} -277.478 q^{12} +646.331 q^{13} -150.846 q^{14} +913.128 q^{16} +1378.32 q^{17} -87.5832 q^{18} +1908.90 q^{19} +1255.57 q^{21} +130.834 q^{22} -343.726 q^{23} +611.436 q^{24} -698.861 q^{26} +729.000 q^{27} -4301.15 q^{28} +53.5092 q^{29} +634.133 q^{31} -3161.34 q^{32} -1089.00 q^{33} -1490.34 q^{34} -2497.30 q^{36} -11674.2 q^{37} -2064.04 q^{38} +5816.98 q^{39} +18866.5 q^{41} -1357.62 q^{42} -13379.3 q^{43} +3730.53 q^{44} +371.662 q^{46} +2252.95 q^{47} +8218.15 q^{48} +2655.48 q^{49} +12404.9 q^{51} -19926.9 q^{52} +8900.71 q^{53} -788.249 q^{54} +9477.80 q^{56} +17180.1 q^{57} -57.8581 q^{58} -11276.7 q^{59} +11852.4 q^{61} -685.672 q^{62} +11300.1 q^{63} -25801.8 q^{64} +1177.51 q^{66} +57333.6 q^{67} -42494.7 q^{68} -3093.53 q^{69} +31075.8 q^{71} +5502.93 q^{72} -56010.8 q^{73} +12623.0 q^{74} -58853.0 q^{76} -16880.5 q^{77} -6289.75 q^{78} -883.036 q^{79} +6561.00 q^{81} -20399.9 q^{82} -93986.3 q^{83} -38710.3 q^{84} +14466.7 q^{86} +481.583 q^{87} -8220.42 q^{88} +21739.9 q^{89} +90168.4 q^{91} +10597.4 q^{92} +5707.20 q^{93} -2436.06 q^{94} -28452.0 q^{96} +158059. q^{97} -2871.30 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} + 27 q^{3} - 20 q^{4} - 18 q^{6} - 152 q^{7} + 24 q^{8} + 243 q^{9}+O(q^{10}) \) 3 * q - 2 * q^2 + 27 * q^3 - 20 * q^4 - 18 * q^6 - 152 * q^7 + 24 * q^8 + 243 * q^9	\( 3 q - 2 q^{2} + 27 q^{3} - 20 q^{4} - 18 q^{6} - 152 q^{7} + 24 q^{8} + 243 q^{9} - 363 q^{11} - 180 q^{12} + 546 q^{13} - 8 q^{14} - 1360 q^{16} + 314 q^{17} - 162 q^{18} + 1808 q^{19} - 1368 q^{21} + 242 q^{22} - 4288 q^{23} + 216 q^{24} + 812 q^{26} + 2187 q^{27} - 5888 q^{28} + 5582 q^{29} + 6328 q^{31} + 736 q^{32} - 3267 q^{33} + 11596 q^{34} - 1620 q^{36} - 16866 q^{37} - 9584 q^{38} + 4914 q^{39} + 23282 q^{41} - 72 q^{42} - 20572 q^{43} + 2420 q^{44} + 16592 q^{46} - 3432 q^{47} - 12240 q^{48} + 11531 q^{49} + 2826 q^{51} - 21816 q^{52} - 16138 q^{53} - 1458 q^{54} + 15648 q^{56} + 16272 q^{57} - 17460 q^{58} + 21972 q^{59} + 8322 q^{61} - 5056 q^{62} - 12312 q^{63} + 22208 q^{64} + 2178 q^{66} + 84332 q^{67} - 59832 q^{68} - 38592 q^{69} + 50528 q^{71} + 1944 q^{72} + 53838 q^{73} + 79212 q^{74} - 52448 q^{76} + 18392 q^{77} + 7308 q^{78} + 6364 q^{79} + 19683 q^{81} + 68020 q^{82} - 96272 q^{83} - 52992 q^{84} + 143152 q^{86} + 50238 q^{87} - 2904 q^{88} - 38938 q^{89} + 104968 q^{91} - 24000 q^{92} + 56952 q^{93} + 49088 q^{94} + 6624 q^{96} + 103242 q^{97} - 9554 q^{98} - 29403 q^{99}+O(q^{100}) \) 3 * q - 2 * q^2 + 27 * q^3 - 20 * q^4 - 18 * q^6 - 152 * q^7 + 24 * q^8 + 243 * q^9 - 363 * q^11 - 180 * q^12 + 546 * q^13 - 8 * q^14 - 1360 * q^16 + 314 * q^17 - 162 * q^18 + 1808 * q^19 - 1368 * q^21 + 242 * q^22 - 4288 * q^23 + 216 * q^24 + 812 * q^26 + 2187 * q^27 - 5888 * q^28 + 5582 * q^29 + 6328 * q^31 + 736 * q^32 - 3267 * q^33 + 11596 * q^34 - 1620 * q^36 - 16866 * q^37 - 9584 * q^38 + 4914 * q^39 + 23282 * q^41 - 72 * q^42 - 20572 * q^43 + 2420 * q^44 + 16592 * q^46 - 3432 * q^47 - 12240 * q^48 + 11531 * q^49 + 2826 * q^51 - 21816 * q^52 - 16138 * q^53 - 1458 * q^54 + 15648 * q^56 + 16272 * q^57 - 17460 * q^58 + 21972 * q^59 + 8322 * q^61 - 5056 * q^62 - 12312 * q^63 + 22208 * q^64 + 2178 * q^66 + 84332 * q^67 - 59832 * q^68 - 38592 * q^69 + 50528 * q^71 + 1944 * q^72 + 53838 * q^73 + 79212 * q^74 - 52448 * q^76 + 18392 * q^77 + 7308 * q^78 + 6364 * q^79 + 19683 * q^81 + 68020 * q^82 - 96272 * q^83 - 52992 * q^84 + 143152 * q^86 + 50238 * q^87 - 2904 * q^88 - 38938 * q^89 + 104968 * q^91 - 24000 * q^92 + 56952 * q^93 + 49088 * q^94 + 6624 * q^96 + 103242 * q^97 - 9554 * q^98 - 29403 * q^99

Introduction

L-functions

Modular forms

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Database

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