LMFDB - Embedded newform 825.6.a.v.1.3

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:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 306 x^{11} - 206 x^{10} + 34574 x^{9} + 39928 x^{8} - 1788312 x^{7} - 2591628 x^{6} + \cdots + 522579400 $ x^13 - 306x^11 - 206x^10 + 34574x^9 + 39928x^8 - 1788312x^7 - 2591628x^6 + 42852537x^5 + 63733360x^4 - 448113518x^3 - 549984598x^2 + 1518551280*x + 522579400
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{2}\cdot 5^{7} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-7.57523$ 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)

$f(q)$	$=$	$q-8.57523 q^{2} -9.00000 q^{3} +41.5346 q^{4} +77.1771 q^{6} -178.462 q^{7} -81.7612 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-8.57523 q^{2} -9.00000 q^{3} +41.5346 q^{4} +77.1771 q^{6} -178.462 q^{7} -81.7612 q^{8} +81.0000 q^{9} +121.000 q^{11} -373.811 q^{12} +361.980 q^{13} +1530.36 q^{14} -627.985 q^{16} -934.004 q^{17} -694.594 q^{18} +753.929 q^{19} +1606.16 q^{21} -1037.60 q^{22} -3231.01 q^{23} +735.851 q^{24} -3104.06 q^{26} -729.000 q^{27} -7412.36 q^{28} +2606.34 q^{29} +662.215 q^{31} +8001.48 q^{32} -1089.00 q^{33} +8009.30 q^{34} +3364.30 q^{36} +12924.2 q^{37} -6465.11 q^{38} -3257.82 q^{39} -2538.56 q^{41} -13773.2 q^{42} -22022.8 q^{43} +5025.68 q^{44} +27706.7 q^{46} -20835.4 q^{47} +5651.87 q^{48} +15041.8 q^{49} +8406.03 q^{51} +15034.7 q^{52} +27450.5 q^{53} +6251.34 q^{54} +14591.3 q^{56} -6785.36 q^{57} -22350.0 q^{58} +7759.71 q^{59} +38469.2 q^{61} -5678.65 q^{62} -14455.5 q^{63} -48519.0 q^{64} +9338.43 q^{66} -35521.3 q^{67} -38793.4 q^{68} +29079.1 q^{69} -62677.1 q^{71} -6622.66 q^{72} -68950.2 q^{73} -110828. q^{74} +31314.1 q^{76} -21594.0 q^{77} +27936.5 q^{78} -17313.8 q^{79} +6561.00 q^{81} +21768.7 q^{82} +89017.9 q^{83} +66711.2 q^{84} +188850. q^{86} -23457.1 q^{87} -9893.11 q^{88} +129627. q^{89} -64599.8 q^{91} -134199. q^{92} -5959.94 q^{93} +178668. q^{94} -72013.3 q^{96} +136732. q^{97} -128987. q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9}+O(q^{10}) $ 13 * q - 13 * q^2 - 117 * q^3 + 209 * q^4 + 117 * q^6 - 304 * q^7 - 399 * q^8 + 1053 * q^9	\( 13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9} + 1573 q^{11} - 1881 q^{12} - 986 q^{13} - 610 q^{14} + 3501 q^{16} - 1476 q^{17} - 1053 q^{18} + 270 q^{19} + 2736 q^{21} - 1573 q^{22} - 9084 q^{23} + 3591 q^{24} + 2652 q^{26} - 9477 q^{27} - 10920 q^{28} + 11952 q^{29} + 19096 q^{31} - 11661 q^{32} - 14157 q^{33} - 1302 q^{34} + 16929 q^{36} - 39964 q^{37} - 1574 q^{38} + 8874 q^{39} + 35184 q^{41} + 5490 q^{42} + 96 q^{43} + 25289 q^{44} - 4120 q^{46} - 34984 q^{47} - 31509 q^{48} + 14557 q^{49} + 13284 q^{51} - 39002 q^{52} - 22984 q^{53} + 9477 q^{54} + 59802 q^{56} - 2430 q^{57} - 18896 q^{58} - 9192 q^{59} + 5438 q^{61} - 272 q^{62} - 24624 q^{63} + 106557 q^{64} + 14157 q^{66} - 71508 q^{67} - 127948 q^{68} + 81756 q^{69} + 101700 q^{71} - 32319 q^{72} - 77390 q^{73} + 13676 q^{74} + 139966 q^{76} - 36784 q^{77} - 23868 q^{78} + 93954 q^{79} + 85293 q^{81} - 53284 q^{82} - 185918 q^{83} + 98280 q^{84} + 370930 q^{86} - 107568 q^{87} - 48279 q^{88} - 18418 q^{89} + 174536 q^{91} - 274264 q^{92} - 171864 q^{93} + 64520 q^{94} + 104949 q^{96} - 94312 q^{97} - 145677 q^{98} + 127413 q^{99}+O(q^{100}) \) 13 * q - 13 * q^2 - 117 * q^3 + 209 * q^4 + 117 * q^6 - 304 * q^7 - 399 * q^8 + 1053 * q^9 + 1573 * q^11 - 1881 * q^12 - 986 * q^13 - 610 * q^14 + 3501 * q^16 - 1476 * q^17 - 1053 * q^18 + 270 * q^19 + 2736 * q^21 - 1573 * q^22 - 9084 * q^23 + 3591 * q^24 + 2652 * q^26 - 9477 * q^27 - 10920 * q^28 + 11952 * q^29 + 19096 * q^31 - 11661 * q^32 - 14157 * q^33 - 1302 * q^34 + 16929 * q^36 - 39964 * q^37 - 1574 * q^38 + 8874 * q^39 + 35184 * q^41 + 5490 * q^42 + 96 * q^43 + 25289 * q^44 - 4120 * q^46 - 34984 * q^47 - 31509 * q^48 + 14557 * q^49 + 13284 * q^51 - 39002 * q^52 - 22984 * q^53 + 9477 * q^54 + 59802 * q^56 - 2430 * q^57 - 18896 * q^58 - 9192 * q^59 + 5438 * q^61 - 272 * q^62 - 24624 * q^63 + 106557 * q^64 + 14157 * q^66 - 71508 * q^67 - 127948 * q^68 + 81756 * q^69 + 101700 * q^71 - 32319 * q^72 - 77390 * q^73 + 13676 * q^74 + 139966 * q^76 - 36784 * q^77 - 23868 * q^78 + 93954 * q^79 + 85293 * q^81 - 53284 * q^82 - 185918 * q^83 + 98280 * q^84 + 370930 * q^86 - 107568 * q^87 - 48279 * q^88 - 18418 * q^89 + 174536 * q^91 - 274264 * q^92 - 171864 * q^93 + 64520 * q^94 + 104949 * q^96 - 94312 * q^97 - 145677 * q^98 + 127413 * 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 $

$2$	−8.57523	−1.51590	−0.757950	−	0.652312i	$-0.773799\pi$
$2$	−8.57523	−1.51590	−0.757950	+	0.652312i	$0.773799\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	41.5346	1.29796
$5$	0	0
$5$	0	0
$6$	77.1771	0.875206
$7$	−178.462	−1.37658	−0.688290	−	0.725435i	$-0.741638\pi$
$7$	−178.462	−1.37658	−0.688290	+	0.725435i	$0.741638\pi$
$8$	−81.7612	−0.451671
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	−373.811	−0.749375
$13$	361.980	0.594054	0.297027	−	0.954869i	$-0.404005\pi$
$13$	361.980	0.594054	0.297027	+	0.954869i	$0.404005\pi$
$14$	1530.36	2.08676
$15$	0	0
$16$	−627.985	−0.613267
$17$	−934.004	−0.783838	−0.391919	−	0.920000i	$-0.628189\pi$
$17$	−934.004	−0.783838	−0.391919	+	0.920000i	$0.628189\pi$
$18$	−694.594	−0.505300
$19$	753.929	0.479122	0.239561	−	0.970881i	$-0.422996\pi$
$19$	753.929	0.479122	0.239561	+	0.970881i	$0.422996\pi$
$20$	0	0
$21$	1606.16	0.794769
$22$	−1037.60	−0.457061
$23$	−3231.01	−1.27356	−0.636780	−	0.771046i	$-0.719734\pi$
$23$	−3231.01	−1.27356	−0.636780	+	0.771046i	$0.719734\pi$
$24$	735.851	0.260773
$25$	0	0
$26$	−3104.06	−0.900527
$27$	−729.000	−0.192450
$28$	−7412.36	−1.78674
$29$	2606.34	0.575488	0.287744	−	0.957707i	$-0.407095\pi$
$29$	2606.34	0.575488	0.287744	+	0.957707i	$0.407095\pi$
$30$	0	0
$31$	662.215	0.123764	0.0618821	−	0.998083i	$-0.480290\pi$
$31$	662.215	0.123764	0.0618821	+	0.998083i	$0.480290\pi$
$32$	8001.48	1.38132
$33$	−1089.00	−0.174078
$34$	8009.30	1.18822
$35$	0	0
$36$	3364.30	0.432652
$37$	12924.2	1.55203	0.776013	−	0.630717i	$-0.217239\pi$
$37$	12924.2	1.55203	0.776013	+	0.630717i	$0.217239\pi$
$38$	−6465.11	−0.726302
$39$	−3257.82	−0.342977
$40$	0	0
$41$	−2538.56	−0.235846	−0.117923	−	0.993023i	$-0.537624\pi$
$41$	−2538.56	−0.235846	−0.117923	+	0.993023i	$0.537624\pi$
$42$	−13773.2	−1.20479
$43$	−22022.8	−1.81636	−0.908178	−	0.418583i	$-0.862527\pi$
$43$	−22022.8	−1.81636	−0.908178	+	0.418583i	$0.862527\pi$
$44$	5025.68	0.391348
$45$	0	0
$46$	27706.7	1.93059
$47$	−20835.4	−1.37581	−0.687903	−	0.725802i	$-0.741469\pi$
$47$	−20835.4	−1.37581	−0.687903	+	0.725802i	$0.741469\pi$
$48$	5651.87	0.354070
$49$	15041.8	0.894974
$50$	0	0
$51$	8406.03	0.452549
$52$	15034.7	0.771056
$53$	27450.5	1.34233	0.671166	−	0.741307i	$-0.265794\pi$
$53$	27450.5	1.34233	0.671166	+	0.741307i	$0.265794\pi$
$54$	6251.34	0.291735
$55$	0	0
$56$	14591.3	0.621762
$57$	−6785.36	−0.276621
$58$	−22350.0	−0.872383
$59$	7759.71	0.290212	0.145106	−	0.989416i	$-0.453648\pi$
$59$	7759.71	0.290212	0.145106	+	0.989416i	$0.453648\pi$
$60$	0	0
$61$	38469.2	1.32370	0.661849	−	0.749638i	$-0.269772\pi$
$61$	38469.2	1.32370	0.661849	+	0.749638i	$0.269772\pi$
$62$	−5678.65	−0.187614
$63$	−14455.5	−0.458860
$64$	−48519.0	−1.48068
$65$	0	0
$66$	9338.43	0.263884
$67$	−35521.3	−0.966723	−0.483362	−	0.875421i	$-0.660584\pi$
$67$	−35521.3	−0.966723	−0.483362	+	0.875421i	$0.660584\pi$
$68$	−38793.4	−1.01739
$69$	29079.1	0.735290
$70$	0	0
$71$	−62677.1	−1.47558	−0.737790	−	0.675030i	$-0.764131\pi$
$71$	−62677.1	−1.47558	−0.737790	+	0.675030i	$0.764131\pi$
$72$	−6622.66	−0.150557
$73$	−68950.2	−1.51436	−0.757179	−	0.653208i	$-0.773423\pi$
$73$	−68950.2	−1.51436	−0.757179	+	0.653208i	$0.773423\pi$
$74$	−110828.	−2.35272
$75$	0	0
$76$	31314.1	0.621879
$77$	−21594.0	−0.415055
$78$	27936.5	0.519919
$79$	−17313.8	−0.312123	−0.156061	−	0.987747i	$-0.549880\pi$
$79$	−17313.8	−0.312123	−0.156061	+	0.987747i	$0.549880\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	21768.7	0.357518
$83$	89017.9	1.41835	0.709173	−	0.705035i	$-0.249069\pi$
$83$	89017.9	1.41835	0.709173	+	0.705035i	$0.249069\pi$
$84$	66711.2	1.03158
$85$	0	0
$86$	188850.	2.75342
$87$	−23457.1	−0.332258
$88$	−9893.11	−0.136184
$89$	129627.	1.73468	0.867341	−	0.497714i	$-0.165827\pi$
$89$	129627.	1.73468	0.867341	+	0.497714i	$0.165827\pi$
$90$	0	0
$91$	−64599.8	−0.817763
$92$	−134199.	−1.65302
$93$	−5959.94	−0.0714553
$94$	178668.	2.08559
$95$	0	0
$96$	−72013.3	−0.797507
$97$	136732.	1.47551	0.737755	−	0.675069i	$-0.235886\pi$
$97$	136732.	1.47551	0.737755	+	0.675069i	$0.235886\pi$
$98$	−128987.	−1.35669
$99$	9801.00	0.100504
$100$	0	0
$101$	51376.9	0.501146	0.250573	−	0.968098i	$-0.419381\pi$
$101$	51376.9	0.501146	0.250573	+	0.968098i	$0.419381\pi$
$102$	−72083.7	−0.686019
$103$	94899.7	0.881398	0.440699	−	0.897655i	$-0.354731\pi$
$103$	94899.7	0.881398	0.440699	+	0.897655i	$0.354731\pi$
$104$	−29595.9	−0.268317
$105$	0	0
$106$	−235394.	−2.03484
$107$	52232.6	0.441045	0.220522	−	0.975382i	$-0.429224\pi$
$107$	52232.6	0.441045	0.220522	+	0.975382i	$0.429224\pi$
$108$	−30278.7	−0.249792
$109$	138553.	1.11699	0.558496	−	0.829507i	$-0.311378\pi$
$109$	138553.	1.11699	0.558496	+	0.829507i	$0.311378\pi$
$110$	0	0
$111$	−116318.	−0.896063
$112$	112072.	0.844211
$113$	136642.	1.00667	0.503336	−	0.864091i	$-0.332106\pi$
$113$	136642.	1.00667	0.503336	+	0.864091i	$0.332106\pi$
$114$	58186.0	0.419330
$115$	0	0
$116$	108253.	0.746958
$117$	29320.4	0.198018
$118$	−66541.3	−0.439933
$119$	166685.	1.07902
$120$	0	0
$121$	14641.0	0.0909091
$122$	−329882.	−2.00659
$123$	22847.0	0.136165
$124$	27504.8	0.160640
$125$	0	0
$126$	123959.	0.695587
$127$	−23888.8	−0.131427	−0.0657136	−	0.997839i	$-0.520932\pi$
$127$	−23888.8	−0.131427	−0.0657136	+	0.997839i	$0.520932\pi$
$128$	160014.	0.863244
$129$	198205.	1.04867
$130$	0	0
$131$	67597.5	0.344154	0.172077	−	0.985084i	$-0.444952\pi$
$131$	67597.5	0.344154	0.172077	+	0.985084i	$0.444952\pi$
$132$	−45231.2	−0.225945
$133$	−134548.	−0.659550
$134$	304604.	1.46546
$135$	0	0
$136$	76365.3	0.354037
$137$	205843.	0.936991	0.468496	−	0.883466i	$-0.344796\pi$
$137$	205843.	0.936991	0.468496	+	0.883466i	$0.344796\pi$
$138$	−249360.	−1.11463
$139$	43495.6	0.190945	0.0954724	−	0.995432i	$-0.469564\pi$
$139$	43495.6	0.190945	0.0954724	+	0.995432i	$0.469564\pi$
$140$	0	0
$141$	187519.	0.794323
$142$	537470.	2.23683
$143$	43799.6	0.179114
$144$	−50866.8	−0.204422
$145$	0	0
$146$	591264.	2.29562
$147$	−135376.	−0.516713
$148$	536801.	2.01446
$149$	219355.	0.809436	0.404718	−	0.914442i	$-0.367370\pi$
$149$	219355.	0.809436	0.404718	+	0.914442i	$0.367370\pi$
$150$	0	0
$151$	−32928.1	−0.117523	−0.0587616	−	0.998272i	$-0.518715\pi$
$151$	−32928.1	−0.117523	−0.0587616	+	0.998272i	$0.518715\pi$
$152$	−61642.1	−0.216406
$153$	−75654.3	−0.261279
$154$	185173.	0.629182
$155$	0	0
$156$	−135312.	−0.445169
$157$	−457292.	−1.48062	−0.740311	−	0.672265i	$-0.765322\pi$
$157$	−457292.	−1.48062	−0.740311	+	0.672265i	$0.765322\pi$
$158$	148470.	0.473147
$159$	−247054.	−0.774996
$160$	0	0
$161$	576614.	1.75316
$162$	−56262.1	−0.168433
$163$	52822.0	0.155721	0.0778603	−	0.996964i	$-0.475191\pi$
$163$	52822.0	0.155721	0.0778603	+	0.996964i	$0.475191\pi$
$164$	−105438.	−0.306117
$165$	0	0
$166$	−763349.	−2.15007
$167$	327606.	0.908993	0.454497	−	0.890748i	$-0.349819\pi$
$167$	327606.	0.908993	0.454497	+	0.890748i	$0.349819\pi$
$168$	−131322.	−0.358974
$169$	−240264.	−0.647100
$170$	0	0
$171$	61068.2	0.159707
$172$	−914707.	−2.35755
$173$	−246387.	−0.625897	−0.312949	−	0.949770i	$-0.601317\pi$
$173$	−246387.	−0.625897	−0.312949	+	0.949770i	$0.601317\pi$
$174$	201150.	0.503670
$175$	0	0
$176$	−75986.2	−0.184907
$177$	−69837.4	−0.167554
$178$	−1.11158e6	−2.62961
$179$	302341.	0.705285	0.352643	−	0.935758i	$-0.385283\pi$
$179$	302341.	0.705285	0.352643	+	0.935758i	$0.385283\pi$
$180$	0	0
$181$	−465374.	−1.05586	−0.527929	−	0.849288i	$-0.677031\pi$
$181$	−465374.	−1.05586	−0.527929	+	0.849288i	$0.677031\pi$
$182$	553958.	1.23965
$183$	−346223.	−0.764237
$184$	264172.	0.575230
$185$	0	0
$186$	51107.8	0.108319
$187$	−113014.	−0.236336
$188$	−865390.	−1.78574
$189$	130099.	0.264923
$190$	0	0
$191$	294182.	0.583489	0.291745	−	0.956496i	$-0.405764\pi$
$191$	294182.	0.583489	0.291745	+	0.956496i	$0.405764\pi$
$192$	436671.	0.854872
$193$	280529.	0.542106	0.271053	−	0.962564i	$-0.412628\pi$
$193$	280529.	0.542106	0.271053	+	0.962564i	$0.412628\pi$
$194$	−1.17251e6	−2.23673
$195$	0	0
$196$	624756.	1.16164
$197$	241625.	0.443585	0.221792	−	0.975094i	$-0.428809\pi$
$197$	241625.	0.443585	0.221792	+	0.975094i	$0.428809\pi$
$198$	−84045.8	−0.152354
$199$	621763.	1.11299	0.556496	−	0.830850i	$-0.312145\pi$
$199$	621763.	1.11299	0.556496	+	0.830850i	$0.312145\pi$
$200$	0	0
$201$	319692.	0.558138
$202$	−440569.	−0.759688
$203$	−465134.	−0.792205
$204$	349141.	0.587389
$205$	0	0
$206$	−813787.	−1.33611
$207$	−261712.	−0.424520
$208$	−227318.	−0.364314
$209$	91225.4	0.144461
$210$	0	0
$211$	−446582.	−0.690550	−0.345275	−	0.938502i	$-0.612214\pi$
$211$	−446582.	−0.690550	−0.345275	+	0.938502i	$0.612214\pi$
$212$	1.14014e6	1.74229
$213$	564094.	0.851927
$214$	−447907.	−0.668580
$215$	0	0
$216$	59603.9	0.0869242
$217$	−118181.	−0.170371
$218$	−1.18813e6	−1.69325
$219$	620552.	0.874314
$220$	0	0
$221$	−338090.	−0.465642
$222$	997451.	1.35834
$223$	−29867.7	−0.0402198	−0.0201099	−	0.999798i	$-0.506402\pi$
$223$	−29867.7	−0.0402198	−0.0201099	+	0.999798i	$0.506402\pi$
$224$	−1.42796e6	−1.90150
$225$	0	0
$226$	−1.17174e6	−1.52602
$227$	−425464.	−0.548022	−0.274011	−	0.961726i	$-0.588351\pi$
$227$	−425464.	−0.548022	−0.274011	+	0.961726i	$0.588351\pi$
$228$	−281827.	−0.359042
$229$	−1.31835e6	−1.66127	−0.830637	−	0.556814i	$-0.812023\pi$
$229$	−1.31835e6	−1.66127	−0.830637	+	0.556814i	$0.812023\pi$
$230$	0	0
$231$	194346.	0.239632
$232$	−213098.	−0.259931
$233$	320062.	0.386228	0.193114	−	0.981176i	$-0.438141\pi$
$233$	320062.	0.386228	0.193114	+	0.981176i	$0.438141\pi$
$234$	−251429.	−0.300176
$235$	0	0
$236$	322296.	0.376683
$237$	155825.	0.180204
$238$	−1.42936e6	−1.63568
$239$	1.25534e6	1.42156	0.710781	−	0.703414i	$-0.248342\pi$
$239$	1.25534e6	1.42156	0.710781	+	0.703414i	$0.248342\pi$
$240$	0	0
$241$	−1.07813e6	−1.19572	−0.597859	−	0.801601i	$-0.703982\pi$
$241$	−1.07813e6	−1.19572	−0.597859	+	0.801601i	$0.703982\pi$
$242$	−125550.	−0.137809
$243$	−59049.0	−0.0641500
$244$	1.59780e6	1.71810
$245$	0	0
$246$	−195919.	−0.206413
$247$	272907.	0.284624
$248$	−54143.5	−0.0559007
$249$	−801161.	−0.818882
$250$	0	0
$251$	−918775.	−0.920502	−0.460251	−	0.887789i	$-0.652241\pi$
$251$	−918775.	−0.920502	−0.460251	+	0.887789i	$0.652241\pi$
$252$	−600401.	−0.595580
$253$	−390953.	−0.383993
$254$	204852.	0.199231
$255$	0	0
$256$	180449.	0.172089
$257$	−797355.	−0.753041	−0.376521	−	0.926408i	$-0.622880\pi$
$257$	−797355.	−0.753041	−0.376521	+	0.926408i	$0.622880\pi$
$258$	−1.69965e6	−1.58969
$259$	−2.30648e6	−2.13649
$260$	0	0
$261$	211114.	0.191829
$262$	−579664.	−0.521703
$263$	980563.	0.874150	0.437075	−	0.899425i	$-0.356014\pi$
$263$	980563.	0.874150	0.437075	+	0.899425i	$0.356014\pi$
$264$	89038.0	0.0786259
$265$	0	0
$266$	1.15378e6	0.999813
$267$	−1.16664e6	−1.00152
$268$	−1.47536e6	−1.25476
$269$	−1.32137e6	−1.11338	−0.556691	−	0.830719i	$-0.687929\pi$
$269$	−1.32137e6	−1.11338	−0.556691	+	0.830719i	$0.687929\pi$
$270$	0	0
$271$	−62864.5	−0.0519975	−0.0259987	−	0.999662i	$-0.508277\pi$
$271$	−62864.5	−0.0519975	−0.0259987	+	0.999662i	$0.508277\pi$
$272$	586540.	0.480702
$273$	581398.	0.472136
$274$	−1.76515e6	−1.42039
$275$	0	0
$276$	1.20779e6	0.954374
$277$	−1.44313e6	−1.13007	−0.565037	−	0.825066i	$-0.691138\pi$
$277$	−1.44313e6	−1.13007	−0.565037	+	0.825066i	$0.691138\pi$
$278$	−372984.	−0.289453
$279$	53639.4	0.0412547
$280$	0	0
$281$	−143939.	−0.108746	−0.0543731	−	0.998521i	$-0.517316\pi$
$281$	−143939.	−0.108746	−0.0543731	+	0.998521i	$0.517316\pi$
$282$	−1.60802e6	−1.20411
$283$	9060.12	0.00672462	0.00336231	−	0.999994i	$-0.498930\pi$
$283$	9060.12	0.00672462	0.00336231	+	0.999994i	$0.498930\pi$
$284$	−2.60327e6	−1.91524
$285$	0	0
$286$	−375591.	−0.271519
$287$	453037.	0.324660
$288$	648120.	0.460441
$289$	−547494.	−0.385598
$290$	0	0
$291$	−1.23059e6	−0.851886
$292$	−2.86382e6	−1.96557
$293$	1.85674e6	1.26352	0.631759	−	0.775165i	$-0.282333\pi$
$293$	1.85674e6	1.26352	0.631759	+	0.775165i	$0.282333\pi$
$294$	1.16088e6	0.783286
$295$	0	0
$296$	−1.05670e6	−0.701006
$297$	−88209.0	−0.0580259
$298$	−1.88102e6	−1.22703
$299$	−1.16956e6	−0.756563
$300$	0	0
$301$	3.93024e6	2.50036
$302$	282366.	0.178154
$303$	−462392.	−0.289337
$304$	−473456.	−0.293830
$305$	0	0
$306$	648753.	0.396074
$307$	504688.	0.305617	0.152808	−	0.988256i	$-0.451168\pi$
$307$	504688.	0.305617	0.152808	+	0.988256i	$0.451168\pi$
$308$	−896896.	−0.538723
$309$	−854097.	−0.508875
$310$	0	0
$311$	−448511.	−0.262950	−0.131475	−	0.991320i	$-0.541971\pi$
$311$	−448511.	−0.262950	−0.131475	+	0.991320i	$0.541971\pi$
$312$	266363.	0.154913
$313$	−825146.	−0.476069	−0.238035	−	0.971257i	$-0.576503\pi$
$313$	−825146.	−0.476069	−0.238035	+	0.971257i	$0.576503\pi$
$314$	3.92138e6	2.24448
$315$	0	0
$316$	−719123.	−0.405122
$317$	−3.26266e6	−1.82358	−0.911789	−	0.410659i	$-0.865299\pi$
$317$	−3.26266e6	−1.82358	−0.911789	+	0.410659i	$0.865299\pi$
$318$	2.11855e6	1.17482
$319$	315367.	0.173516
$320$	0	0
$321$	−470094.	−0.254637
$322$	−4.94460e6	−2.65761
$323$	−704172.	−0.375554
$324$	272508.	0.144217
$325$	0	0
$326$	−452961.	−0.236057
$327$	−1.24698e6	−0.644896
$328$	207556.	0.106525
$329$	3.71834e6	1.89391
$330$	0	0
$331$	−3.48175e6	−1.74674	−0.873369	−	0.487059i	$-0.838070\pi$
$331$	−3.48175e6	−1.74674	−0.873369	+	0.487059i	$0.838070\pi$
$332$	3.69732e6	1.84095
$333$	1.04686e6	0.517342
$334$	−2.80930e6	−1.37794
$335$	0	0
$336$	−1.00865e6	−0.487406
$337$	1.52759e6	0.732710	0.366355	−	0.930475i	$-0.380606\pi$
$337$	1.52759e6	0.732710	0.366355	+	0.930475i	$0.380606\pi$
$338$	2.06032e6	0.980939
$339$	−1.22978e6	−0.581203
$340$	0	0
$341$	80128.0	0.0373163
$342$	−523674.	−0.242101
$343$	315017.	0.144577
$344$	1.80061e6	0.820396
$345$	0	0
$346$	2.11283e6	0.948798
$347$	4.01479e6	1.78994	0.894972	−	0.446122i	$-0.147195\pi$
$347$	4.01479e6	1.78994	0.894972	+	0.446122i	$0.147195\pi$
$348$	−974279.	−0.431256
$349$	−511478.	−0.224783	−0.112391	−	0.993664i	$-0.535851\pi$
$349$	−511478.	−0.224783	−0.112391	+	0.993664i	$0.535851\pi$
$350$	0	0
$351$	−263883.	−0.114326
$352$	968179.	0.416485
$353$	722636.	0.308662	0.154331	−	0.988019i	$-0.450678\pi$
$353$	722636.	0.308662	0.154331	+	0.988019i	$0.450678\pi$
$354$	598872.	0.253995
$355$	0	0
$356$	5.38400e6	2.25154
$357$	−1.50016e6	−0.622970
$358$	−2.59264e6	−1.06914
$359$	393345.	0.161078	0.0805392	−	0.996751i	$-0.474336\pi$
$359$	393345.	0.161078	0.0805392	+	0.996751i	$0.474336\pi$
$360$	0	0
$361$	−1.90769e6	−0.770442
$362$	3.99069e6	1.60058
$363$	−131769.	−0.0524864
$364$	−2.68312e6	−1.06142
$365$	0	0
$366$	2.96894e6	1.15851
$367$	2.87578e6	1.11453	0.557264	−	0.830336i	$-0.311851\pi$
$367$	2.87578e6	1.11453	0.557264	+	0.830336i	$0.311851\pi$
$368$	2.02903e6	0.781032
$369$	−205623.	−0.0786152
$370$	0	0
$371$	−4.89888e6	−1.84783
$372$	−247543.	−0.0927458
$373$	−3.86030e6	−1.43665	−0.718323	−	0.695710i	$-0.755090\pi$
$373$	−3.86030e6	−1.43665	−0.718323	+	0.695710i	$0.755090\pi$
$374$	969125.	0.358262
$375$	0	0
$376$	1.70353e6	0.621412
$377$	943443.	0.341871
$378$	−1.11563e6	−0.401597
$379$	−1.63355e6	−0.584162	−0.292081	−	0.956394i	$-0.594348\pi$
$379$	−1.63355e6	−0.584162	−0.292081	+	0.956394i	$0.594348\pi$
$380$	0	0
$381$	214999.	0.0758795
$382$	−2.52268e6	−0.884512
$383$	−3.86858e6	−1.34758	−0.673790	−	0.738923i	$-0.735335\pi$
$383$	−3.86858e6	−1.34758	−0.673790	+	0.738923i	$0.735335\pi$
$384$	−1.44013e6	−0.498394
$385$	0	0
$386$	−2.40560e6	−0.821779
$387$	−1.78385e6	−0.605452
$388$	5.67912e6	1.91515
$389$	−10040.7	−0.00336428	−0.00168214	−	0.999999i	$-0.500535\pi$
$389$	−10040.7	−0.00336428	−0.00168214	+	0.999999i	$0.500535\pi$
$390$	0	0
$391$	3.01778e6	0.998264
$392$	−1.22984e6	−0.404234
$393$	−608378.	−0.198697
$394$	−2.07199e6	−0.672431
$395$	0	0
$396$	407080.	0.130449
$397$	3.77944e6	1.20352	0.601758	−	0.798679i	$-0.294467\pi$
$397$	3.77944e6	1.20352	0.601758	+	0.798679i	$0.294467\pi$
$398$	−5.33176e6	−1.68719
$399$	1.21093e6	0.380791
$400$	0	0
$401$	−5.77840e6	−1.79451	−0.897256	−	0.441510i	$-0.854443\pi$
$401$	−5.77840e6	−1.79451	−0.897256	+	0.441510i	$0.854443\pi$
$402$	−2.74143e6	−0.846082
$403$	239708.	0.0735226
$404$	2.13392e6	0.650465
$405$	0	0
$406$	3.98863e6	1.20090
$407$	1.56383e6	0.467954
$408$	−687287.	−0.204403
$409$	4.93158e6	1.45773	0.728866	−	0.684656i	$-0.240048\pi$
$409$	4.93158e6	1.45773	0.728866	+	0.684656i	$0.240048\pi$
$410$	0	0
$411$	−1.85259e6	−0.540972
$412$	3.94162e6	1.14401
$413$	−1.38482e6	−0.399501
$414$	2.24424e6	0.643530
$415$	0	0
$416$	2.89637e6	0.820580
$417$	−391460.	−0.110242
$418$	−782278.	−0.218988
$419$	−4.38879e6	−1.22126	−0.610632	−	0.791914i	$-0.709085\pi$
$419$	−4.38879e6	−1.22126	−0.610632	+	0.791914i	$0.709085\pi$
$420$	0	0
$421$	−6.32905e6	−1.74034	−0.870169	−	0.492754i	$-0.835990\pi$
$421$	−6.32905e6	−1.74034	−0.870169	+	0.492754i	$0.835990\pi$
$422$	3.82954e6	1.04681
$423$	−1.68767e6	−0.458602
$424$	−2.24438e6	−0.606293
$425$	0	0
$426$	−4.83723e6	−1.29144
$427$	−6.86531e6	−1.82218
$428$	2.16946e6	0.572456
$429$	−394196.	−0.103412
$430$	0	0
$431$	−4.39779e6	−1.14036	−0.570179	−	0.821520i	$-0.693126\pi$
$431$	−4.39779e6	−1.14036	−0.570179	+	0.821520i	$0.693126\pi$
$432$	457801.	0.118023
$433$	1.57360e6	0.403344	0.201672	−	0.979453i	$-0.435362\pi$
$433$	1.57360e6	0.403344	0.201672	+	0.979453i	$0.435362\pi$
$434$	1.01343e6	0.258266
$435$	0	0
$436$	5.75475e6	1.44981
$437$	−2.43595e6	−0.610190
$438$	−5.32137e6	−1.32537
$439$	−7.89299e6	−1.95470	−0.977351	−	0.211626i	$-0.932124\pi$
$439$	−7.89299e6	−1.95470	−0.977351	+	0.211626i	$0.932124\pi$
$440$	0	0
$441$	1.21839e6	0.298325
$442$	2.89920e6	0.705867
$443$	557226.	0.134903	0.0674516	−	0.997723i	$-0.478513\pi$
$443$	557226.	0.134903	0.0674516	+	0.997723i	$0.478513\pi$
$444$	−4.83121e6	−1.16305
$445$	0	0
$446$	256123.	0.0609693
$447$	−1.97420e6	−0.467328
$448$	8.65881e6	2.03828
$449$	2.27558e6	0.532691	0.266346	−	0.963878i	$-0.414184\pi$
$449$	2.27558e6	0.532691	0.266346	+	0.963878i	$0.414184\pi$
$450$	0	0
$451$	−307166.	−0.0711101
$452$	5.67537e6	1.30662
$453$	296352.	0.0678521
$454$	3.64845e6	0.830748
$455$	0	0
$456$	554779.	0.124942
$457$	−7.50427e6	−1.68081	−0.840404	−	0.541961i	$-0.817682\pi$
$457$	−7.50427e6	−1.68081	−0.840404	+	0.541961i	$0.817682\pi$
$458$	1.13051e7	2.51833
$459$	680889.	0.150850
$460$	0	0
$461$	436978.	0.0957652	0.0478826	−	0.998853i	$-0.484753\pi$
$461$	436978.	0.0957652	0.0478826	+	0.998853i	$0.484753\pi$
$462$	−1.66656e6	−0.363258
$463$	3.11651e6	0.675642	0.337821	−	0.941210i	$-0.390310\pi$
$463$	3.11651e6	0.675642	0.337821	+	0.941210i	$0.390310\pi$
$464$	−1.63674e6	−0.352928
$465$	0	0
$466$	−2.74460e6	−0.585484
$467$	1.89749e6	0.402613	0.201307	−	0.979528i	$-0.435481\pi$
$467$	1.89749e6	0.402613	0.201307	+	0.979528i	$0.435481\pi$
$468$	1.21781e6	0.257019
$469$	6.33922e6	1.33077
$470$	0	0
$471$	4.11562e6	0.854837
$472$	−634444.	−0.131081
$473$	−2.66476e6	−0.547652
$474$	−1.33623e6	−0.273172
$475$	0	0
$476$	6.92317e6	1.40051
$477$	2.22349e6	0.447444
$478$	−1.07648e7	−2.15495
$479$	−6.10287e6	−1.21533	−0.607666	−	0.794193i	$-0.707894\pi$
$479$	−6.10287e6	−1.21533	−0.607666	+	0.794193i	$0.707894\pi$
$480$	0	0
$481$	4.67830e6	0.921987
$482$	9.24522e6	1.81259
$483$	−5.18953e6	−1.01219
$484$	608108.	0.117996
$485$	0	0
$486$	506359.	0.0972451
$487$	7.30170e6	1.39509	0.697544	−	0.716542i	$-0.254276\pi$
$487$	7.30170e6	1.39509	0.697544	+	0.716542i	$0.254276\pi$
$488$	−3.14529e6	−0.597876
$489$	−475398.	−0.0899053
$490$	0	0
$491$	4.40884e6	0.825317	0.412658	−	0.910886i	$-0.364600\pi$
$491$	4.40884e6	0.825317	0.412658	+	0.910886i	$0.364600\pi$
$492$	948942.	0.176737
$493$	−2.43433e6	−0.451089
$494$	−2.34024e6	−0.431462
$495$	0	0
$496$	−415861.	−0.0759004
$497$	1.11855e7	2.03125
$498$	6.87014e6	1.24134
$499$	3.45620e6	0.621367	0.310683	−	0.950513i	$-0.399442\pi$
$499$	3.45620e6	0.621367	0.310683	+	0.950513i	$0.399442\pi$
$500$	0	0
$501$	−2.94845e6	−0.524808
$502$	7.87871e6	1.39539
$503$	−5.07651e6	−0.894634	−0.447317	−	0.894375i	$-0.647620\pi$
$503$	−5.07651e6	−0.894634	−0.447317	+	0.894375i	$0.647620\pi$
$504$	1.18190e6	0.207254
$505$	0	0
$506$	3.35251e6	0.582095
$507$	2.16237e6	0.373603
$508$	−992212.	−0.170587
$509$	7.63894e6	1.30689	0.653445	−	0.756974i	$-0.273323\pi$
$509$	7.63894e6	1.30689	0.653445	+	0.756974i	$0.273323\pi$
$510$	0	0
$511$	1.23050e7	2.08463
$512$	−6.66784e6	−1.12411
$513$	−549614.	−0.0922071
$514$	6.83750e6	1.14154
$515$	0	0
$516$	8.23237e6	1.36113
$517$	−2.52109e6	−0.414821
$518$	1.97786e7	3.23871
$519$	2.21748e6	0.361362
$520$	0	0
$521$	−9.46786e6	−1.52812	−0.764060	−	0.645145i	$-0.776797\pi$
$521$	−9.46786e6	−1.52812	−0.764060	+	0.645145i	$0.776797\pi$
$522$	−1.81035e6	−0.290794
$523$	1.12533e6	0.179898	0.0899490	−	0.995946i	$-0.471330\pi$
$523$	1.12533e6	0.179898	0.0899490	+	0.995946i	$0.471330\pi$
$524$	2.80764e6	0.446696
$525$	0	0
$526$	−8.40855e6	−1.32512
$527$	−618511.	−0.0970110
$528$	683876.	0.106756
$529$	4.00311e6	0.621953
$530$	0	0
$531$	628537.	0.0967374
$532$	−5.58839e6	−0.856067
$533$	−918907.	−0.140105
$534$	1.00042e7	1.51820
$535$	0	0
$536$	2.90427e6	0.436641
$537$	−2.72107e6	−0.407196
$538$	1.13311e7	1.68778
$539$	1.82006e6	0.269845
$540$	0	0
$541$	1.05433e7	1.54876	0.774379	−	0.632722i	$-0.218062\pi$
$541$	1.05433e7	1.54876	0.774379	+	0.632722i	$0.218062\pi$
$542$	539078.	0.0788230
$543$	4.18837e6	0.609600
$544$	−7.47341e6	−1.08273
$545$	0	0
$546$	−4.98562e6	−0.715711
$547$	−9.65329e6	−1.37945	−0.689727	−	0.724070i	$-0.742269\pi$
$547$	−9.65329e6	−1.37945	−0.689727	+	0.724070i	$0.742269\pi$
$548$	8.54962e6	1.21617
$549$	3.11601e6	0.441232
$550$	0	0
$551$	1.96499e6	0.275729
$552$	−2.37754e6	−0.332109
$553$	3.08987e6	0.429662
$554$	1.23752e7	1.71308
$555$	0	0
$556$	1.80657e6	0.247838
$557$	−4.21289e6	−0.575363	−0.287682	−	0.957726i	$-0.592885\pi$
$557$	−4.21289e6	−0.575363	−0.287682	+	0.957726i	$0.592885\pi$
$558$	−459970.	−0.0625381
$559$	−7.97180e6	−1.07901
$560$	0	0
$561$	1.01713e6	0.136449
$562$	1.23431e6	0.164848
$563$	−9.59902e6	−1.27631	−0.638155	−	0.769908i	$-0.720302\pi$
$563$	−9.59902e6	−1.27631	−0.638155	+	0.769908i	$0.720302\pi$
$564$	7.78851e6	1.03100
$565$	0	0
$566$	−77692.6	−0.0101939
$567$	−1.17089e6	−0.152953
$568$	5.12455e6	0.666477
$569$	4.20682e6	0.544720	0.272360	−	0.962195i	$-0.412196\pi$
$569$	4.20682e6	0.544720	0.272360	+	0.962195i	$0.412196\pi$
$570$	0	0
$571$	−1.22447e7	−1.57165	−0.785826	−	0.618447i	$-0.787762\pi$
$571$	−1.22447e7	−1.57165	−0.785826	+	0.618447i	$0.787762\pi$
$572$	1.81920e6	0.232482
$573$	−2.64764e6	−0.336878
$574$	−3.88490e6	−0.492153
$575$	0	0
$576$	−3.93004e6	−0.493561
$577$	−5.32012e6	−0.665245	−0.332623	−	0.943060i	$-0.607934\pi$
$577$	−5.32012e6	−0.665245	−0.332623	+	0.943060i	$0.607934\pi$
$578$	4.69489e6	0.584529
$579$	−2.52476e6	−0.312985
$580$	0	0
$581$	−1.58863e7	−1.95247
$582$	1.05526e7	1.29137
$583$	3.32151e6	0.404728
$584$	5.63745e6	0.683992
$585$	0	0
$586$	−1.59220e7	−1.91537
$587$	5.70534e6	0.683418	0.341709	−	0.939806i	$-0.388994\pi$
$587$	5.70534e6	0.683418	0.341709	+	0.939806i	$0.388994\pi$
$588$	−5.62280e6	−0.670671
$589$	499263.	0.0592981
$590$	0	0
$591$	−2.17463e6	−0.256104
$592$	−8.11620e6	−0.951806
$593$	−2.15575e6	−0.251746	−0.125873	−	0.992046i	$-0.540173\pi$
$593$	−2.15575e6	−0.251746	−0.125873	+	0.992046i	$0.540173\pi$
$594$	756413.	0.0879615
$595$	0	0
$596$	9.11083e6	1.05061
$597$	−5.59587e6	−0.642587
$598$	1.00293e7	1.14687
$599$	5.13968e6	0.585287	0.292643	−	0.956222i	$-0.405465\pi$
$599$	5.13968e6	0.585287	0.292643	+	0.956222i	$0.405465\pi$
$600$	0	0
$601$	−7.48666e6	−0.845478	−0.422739	−	0.906252i	$-0.638931\pi$
$601$	−7.48666e6	−0.845478	−0.422739	+	0.906252i	$0.638931\pi$
$602$	−3.37027e7	−3.79030
$603$	−2.87723e6	−0.322241
$604$	−1.36765e6	−0.152540
$605$	0	0
$606$	3.96512e6	0.438606
$607$	8.62794e6	0.950464	0.475232	−	0.879861i	$-0.342364\pi$
$607$	8.62794e6	0.950464	0.475232	+	0.879861i	$0.342364\pi$
$608$	6.03254e6	0.661822
$609$	4.18620e6	0.457380
$610$	0	0
$611$	−7.54200e6	−0.817304
$612$	−3.14227e6	−0.339129
$613$	1.44646e7	1.55473	0.777367	−	0.629047i	$-0.216555\pi$
$613$	1.44646e7	1.55473	0.777367	+	0.629047i	$0.216555\pi$
$614$	−4.32782e6	−0.463285
$615$	0	0
$616$	1.76555e6	0.187468
$617$	−5.82746e6	−0.616264	−0.308132	−	0.951344i	$-0.599704\pi$
$617$	−5.82746e6	−0.616264	−0.308132	+	0.951344i	$0.599704\pi$
$618$	7.32408e6	0.771404
$619$	−2.32647e6	−0.244045	−0.122023	−	0.992527i	$-0.538938\pi$
$619$	−2.32647e6	−0.244045	−0.122023	+	0.992527i	$0.538938\pi$
$620$	0	0
$621$	2.35541e6	0.245097
$622$	3.84609e6	0.398605
$623$	−2.31335e7	−2.38793
$624$	2.04586e6	0.210337
$625$	0	0
$626$	7.07582e6	0.721674
$627$	−821028.	−0.0834044
$628$	−1.89934e7	−1.92178
$629$	−1.20712e7	−1.21654
$630$	0	0
$631$	−398154.	−0.0398087	−0.0199043	−	0.999802i	$-0.506336\pi$
$631$	−398154.	−0.0398087	−0.0199043	+	0.999802i	$0.506336\pi$
$632$	1.41560e6	0.140977
$633$	4.01924e6	0.398689
$634$	2.79781e7	2.76436
$635$	0	0
$636$	−1.02613e7	−1.00591
$637$	5.44484e6	0.531663
$638$	−2.70435e6	−0.263033
$639$	−5.07684e6	−0.491860
$640$	0	0
$641$	1.81586e7	1.74557	0.872784	−	0.488107i	$-0.162312\pi$
$641$	1.81586e7	1.74557	0.872784	+	0.488107i	$0.162312\pi$
$642$	4.03116e6	0.386005
$643$	4.48264e6	0.427569	0.213784	−	0.976881i	$-0.431421\pi$
$643$	4.48264e6	0.427569	0.213784	+	0.976881i	$0.431421\pi$
$644$	2.39494e7	2.27552
$645$	0	0
$646$	6.03844e6	0.569303
$647$	−1.75122e7	−1.64467	−0.822336	−	0.569003i	$-0.807329\pi$
$647$	−1.75122e7	−1.64467	−0.822336	+	0.569003i	$0.807329\pi$
$648$	−536435.	−0.0501857
$649$	938925.	0.0875023
$650$	0	0
$651$	1.06362e6	0.0983639
$652$	2.19394e6	0.202118
$653$	1.89639e7	1.74039	0.870193	−	0.492711i	$-0.163994\pi$
$653$	1.89639e7	1.74039	0.870193	+	0.492711i	$0.163994\pi$
$654$	1.06931e7	0.977599
$655$	0	0
$656$	1.59418e6	0.144636
$657$	−5.58496e6	−0.504786
$658$	−3.18856e7	−2.87098
$659$	23732.7	0.00212879	0.00106440	−	0.999999i	$-0.499661\pi$
$659$	23732.7	0.00212879	0.00106440	+	0.999999i	$0.499661\pi$
$660$	0	0
$661$	1.09938e7	0.978686	0.489343	−	0.872091i	$-0.337237\pi$
$661$	1.09938e7	0.978686	0.489343	+	0.872091i	$0.337237\pi$
$662$	2.98568e7	2.64788
$663$	3.04281e6	0.268839
$664$	−7.27821e6	−0.640626
$665$	0	0
$666$	−8.97706e6	−0.784239
$667$	−8.42112e6	−0.732918
$668$	1.36070e7	1.17983
$669$	268810.	0.0232209
$670$	0	0
$671$	4.65477e6	0.399110
$672$	1.28517e7	1.09783
$673$	−1.19040e6	−0.101310	−0.0506552	−	0.998716i	$-0.516131\pi$
$673$	−1.19040e6	−0.101310	−0.0506552	+	0.998716i	$0.516131\pi$
$674$	−1.30994e7	−1.11072
$675$	0	0
$676$	−9.97925e6	−0.839907
$677$	2.03890e7	1.70972	0.854860	−	0.518859i	$-0.173643\pi$
$677$	2.03890e7	1.70972	0.854860	+	0.518859i	$0.173643\pi$
$678$	1.05456e7	0.881046
$679$	−2.44016e7	−2.03116
$680$	0	0
$681$	3.82918e6	0.316401
$682$	−687116.	−0.0565678
$683$	2.37020e7	1.94416	0.972082	−	0.234640i	$-0.0753911\pi$
$683$	2.37020e7	1.94416	0.972082	+	0.234640i	$0.0753911\pi$
$684$	2.53644e6	0.207293
$685$	0	0
$686$	−2.70134e6	−0.219164
$687$	1.18651e7	0.959137
$688$	1.38300e7	1.11391
$689$	9.93651e6	0.797418
$690$	0	0
$691$	−1.32012e7	−1.05176	−0.525881	−	0.850558i	$-0.676264\pi$
$691$	−1.32012e7	−1.05176	−0.525881	+	0.850558i	$0.676264\pi$
$692$	−1.02336e7	−0.812387
$693$	−1.74911e6	−0.138352
$694$	−3.44278e7	−2.71338
$695$	0	0
$696$	1.91788e6	0.150071
$697$	2.37102e6	0.184865
$698$	4.38604e6	0.340749
$699$	−2.88056e6	−0.222989
$700$	0	0
$701$	9.76550e6	0.750584	0.375292	−	0.926907i	$-0.377542\pi$
$701$	9.76550e6	0.750584	0.375292	+	0.926907i	$0.377542\pi$
$702$	2.26286e6	0.173306
$703$	9.74392e6	0.743610
$704$	−5.87080e6	−0.446442
$705$	0	0
$706$	−6.19677e6	−0.467900
$707$	−9.16884e6	−0.689868
$708$	−2.90067e6	−0.217478
$709$	−6.77780e6	−0.506376	−0.253188	−	0.967417i	$-0.581479\pi$
$709$	−6.77780e6	−0.506376	−0.253188	+	0.967417i	$0.581479\pi$
$710$	0	0
$711$	−1.40242e6	−0.104041
$712$	−1.05985e7	−0.783506
$713$	−2.13963e6	−0.157621
$714$	1.28642e7	0.944361
$715$	0	0
$716$	1.25576e7	0.915429
$717$	−1.12980e7	−0.820739
$718$	−3.37302e6	−0.244179
$719$	−9.77686e6	−0.705305	−0.352653	−	0.935754i	$-0.614720\pi$
$719$	−9.77686e6	−0.705305	−0.352653	+	0.935754i	$0.614720\pi$
$720$	0	0
$721$	−1.69360e7	−1.21331
$722$	1.63589e7	1.16791
$723$	9.70317e6	0.690348
$724$	−1.93291e7	−1.37046
$725$	0	0
$726$	1.12995e6	0.0795642
$727$	1.13969e7	0.799745	0.399873	−	0.916571i	$-0.369054\pi$
$727$	1.13969e7	0.799745	0.399873	+	0.916571i	$0.369054\pi$
$728$	5.28176e6	0.369360
$729$	531441.	0.0370370
$730$	0	0
$731$	2.05694e7	1.42373
$732$	−1.43802e7	−0.991946
$733$	6.58492e6	0.452679	0.226340	−	0.974048i	$-0.427324\pi$
$733$	6.58492e6	0.452679	0.226340	+	0.974048i	$0.427324\pi$
$734$	−2.46605e7	−1.68951
$735$	0	0
$736$	−2.58529e7	−1.75920
$737$	−4.29808e6	−0.291478
$738$	1.76327e6	0.119173
$739$	−1.51924e7	−1.02333	−0.511664	−	0.859186i	$-0.670971\pi$
$739$	−1.51924e7	−1.02333	−0.511664	+	0.859186i	$0.670971\pi$
$740$	0	0
$741$	−2.45616e6	−0.164328
$742$	4.20090e7	2.80112
$743$	6.92543e6	0.460230	0.230115	−	0.973163i	$-0.426090\pi$
$743$	6.92543e6	0.460230	0.230115	+	0.973163i	$0.426090\pi$
$744$	487292.	0.0322743
$745$	0	0
$746$	3.31030e7	2.17781
$747$	7.21045e6	0.472782
$748$	−4.69401e6	−0.306754
$749$	−9.32156e6	−0.607133
$750$	0	0
$751$	−1.53073e7	−0.990370	−0.495185	−	0.868788i	$-0.664900\pi$
$751$	−1.53073e7	−0.990370	−0.495185	+	0.868788i	$0.664900\pi$
$752$	1.30843e7	0.843737
$753$	8.26898e6	0.531452
$754$	−8.09024e6	−0.518242
$755$	0	0
$756$	5.40361e6	0.343858
$757$	−9.00842e6	−0.571359	−0.285679	−	0.958325i	$-0.592219\pi$
$757$	−9.00842e6	−0.571359	−0.285679	+	0.958325i	$0.592219\pi$
$758$	1.40080e7	0.885531
$759$	3.51857e6	0.221698
$760$	0	0
$761$	1.62103e7	1.01468	0.507339	−	0.861746i	$-0.330629\pi$
$761$	1.62103e7	1.01468	0.507339	+	0.861746i	$0.330629\pi$
$762$	−1.84367e6	−0.115026
$763$	−2.47265e7	−1.53763
$764$	1.22187e7	0.757343
$765$	0	0
$766$	3.31740e7	2.04280
$767$	2.80886e6	0.172402
$768$	−1.62404e6	−0.0993558
$769$	1.67909e7	1.02390	0.511950	−	0.859015i	$-0.328923\pi$
$769$	1.67909e7	1.02390	0.511950	+	0.859015i	$0.328923\pi$
$770$	0	0
$771$	7.17619e6	0.434768
$772$	1.16516e7	0.703629
$773$	−1.62605e7	−0.978780	−0.489390	−	0.872065i	$-0.662781\pi$
$773$	−1.62605e7	−0.978780	−0.489390	+	0.872065i	$0.662781\pi$
$774$	1.52969e7	0.917806
$775$	0	0
$776$	−1.11794e7	−0.666445
$777$	2.07583e7	1.23350
$778$	86101.7	0.00509991
$779$	−1.91389e6	−0.112999
$780$	0	0
$781$	−7.58393e6	−0.444904
$782$	−2.58781e7	−1.51327
$783$	−1.90002e6	−0.110753
$784$	−9.44604e6	−0.548858
$785$	0	0
$786$	5.21698e6	0.301205
$787$	5.01128e6	0.288411	0.144206	−	0.989548i	$-0.453937\pi$
$787$	5.01128e6	0.288411	0.144206	+	0.989548i	$0.453937\pi$
$788$	1.00358e7	0.575753
$789$	−8.82506e6	−0.504691
$790$	0	0
$791$	−2.43855e7	−1.38577
$792$	−801342.	−0.0453947
$793$	1.39251e7	0.786348
$794$	−3.24096e7	−1.82441
$795$	0	0
$796$	2.58247e7	1.44461
$797$	−3.07160e7	−1.71285	−0.856424	−	0.516273i	$-0.827319\pi$
$797$	−3.07160e7	−1.71285	−0.856424	+	0.516273i	$0.827319\pi$
$798$	−1.03840e7	−0.577242
$799$	1.94604e7	1.07841
$800$	0	0
$801$	1.04998e7	0.578227
$802$	4.95511e7	2.72030
$803$	−8.34297e6	−0.456596
$804$	1.32783e7	0.724438
$805$	0	0
$806$	−2.05556e6	−0.111453
$807$	1.18924e7	0.642812
$808$	−4.20064e6	−0.226353
$809$	6.14509e6	0.330108	0.165054	−	0.986284i	$-0.447220\pi$
$809$	6.14509e6	0.330108	0.165054	+	0.986284i	$0.447220\pi$
$810$	0	0
$811$	−1.46687e7	−0.783138	−0.391569	−	0.920149i	$-0.628068\pi$
$811$	−1.46687e7	−0.783138	−0.391569	+	0.920149i	$0.628068\pi$
$812$	−1.93191e7	−1.02825
$813$	565780.	0.0300208
$814$	−1.34102e7	−0.709371
$815$	0	0
$816$	−5.27886e6	−0.277533
$817$	−1.66036e7	−0.870257
$818$	−4.22894e7	−2.20978
$819$	−5.23258e6	−0.272588
$820$	0	0
$821$	−1.54951e7	−0.802300	−0.401150	−	0.916012i	$-0.631389\pi$
$821$	−1.54951e7	−0.802300	−0.401150	+	0.916012i	$0.631389\pi$
$822$	1.58864e7	0.820060
$823$	−2.53285e7	−1.30349	−0.651747	−	0.758436i	$-0.725964\pi$
$823$	−2.53285e7	−1.30349	−0.651747	+	0.758436i	$0.725964\pi$
$824$	−7.75912e6	−0.398102
$825$	0	0
$826$	1.18751e7	0.605603
$827$	6.01962e6	0.306059	0.153030	−	0.988222i	$-0.451097\pi$
$827$	6.01962e6	0.306059	0.153030	+	0.988222i	$0.451097\pi$
$828$	−1.08701e7	−0.551008
$829$	−2.70703e7	−1.36807	−0.684033	−	0.729451i	$-0.739775\pi$
$829$	−2.70703e7	−1.36807	−0.684033	+	0.729451i	$0.739775\pi$
$830$	0	0
$831$	1.29882e7	0.652449
$832$	−1.75629e7	−0.879605
$833$	−1.40491e7	−0.701514
$834$	3.35686e6	0.167116
$835$	0	0
$836$	3.78901e6	0.187504
$837$	−482755.	−0.0238184
$838$	3.76349e7	1.85132
$839$	2.29868e7	1.12739	0.563694	−	0.825984i	$-0.309380\pi$
$839$	2.29868e7	1.12739	0.563694	+	0.825984i	$0.309380\pi$
$840$	0	0
$841$	−1.37181e7	−0.668814
$842$	5.42731e7	2.63818
$843$	1.29546e6	0.0627847
$844$	−1.85486e7	−0.896303
$845$	0	0
$846$	1.44721e7	0.695196
$847$	−2.61287e6	−0.125144
$848$	−1.72385e7	−0.823208
$849$	−81541.1	−0.00388246
$850$	0	0
$851$	−4.17582e7	−1.97660
$852$	2.34294e7	1.10576
$853$	3.27587e7	1.54154	0.770768	−	0.637116i	$-0.219873\pi$
$853$	3.27587e7	1.54154	0.770768	+	0.637116i	$0.219873\pi$
$854$	5.88716e7	2.76224
$855$	0	0
$856$	−4.27060e6	−0.199207
$857$	5.91710e6	0.275205	0.137603	−	0.990488i	$-0.456060\pi$
$857$	5.91710e6	0.275205	0.137603	+	0.990488i	$0.456060\pi$
$858$	3.38032e6	0.156762
$859$	−3.88859e7	−1.79808	−0.899041	−	0.437865i	$-0.855735\pi$
$859$	−3.88859e7	−1.79808	−0.899041	+	0.437865i	$0.855735\pi$
$860$	0	0
$861$	−4.07734e6	−0.187443
$862$	3.77121e7	1.72867
$863$	−7.21944e6	−0.329972	−0.164986	−	0.986296i	$-0.552758\pi$
$863$	−7.21944e6	−0.329972	−0.164986	+	0.986296i	$0.552758\pi$
$864$	−5.83308e6	−0.265836
$865$	0	0
$866$	−1.34940e7	−0.611430
$867$	4.92745e6	0.222625
$868$	−4.90858e6	−0.221134
$869$	−2.09497e6	−0.0941086
$870$	0	0
$871$	−1.28580e7	−0.574286
$872$	−1.13283e7	−0.504514
$873$	1.10753e7	0.491837
$874$	2.08889e7	0.924988
$875$	0	0
$876$	2.57744e7	1.13482
$877$	−2.38159e7	−1.04561	−0.522803	−	0.852453i	$-0.675114\pi$
$877$	−2.38159e7	−1.04561	−0.522803	+	0.852453i	$0.675114\pi$
$878$	6.76842e7	2.96313
$879$	−1.67106e7	−0.729493
$880$	0	0
$881$	−9.82240e6	−0.426361	−0.213181	−	0.977013i	$-0.568382\pi$
$881$	−9.82240e6	−0.426361	−0.213181	+	0.977013i	$0.568382\pi$
$882$	−1.04480e7	−0.452231
$883$	175255.	0.00756430	0.00378215	−	0.999993i	$-0.498796\pi$
$883$	175255.	0.00756430	0.00378215	+	0.999993i	$0.498796\pi$
$884$	−1.40424e7	−0.604383
$885$	0	0
$886$	−4.77834e6	−0.204500
$887$	−4.60415e7	−1.96490	−0.982450	−	0.186524i	$-0.940278\pi$
$887$	−4.60415e7	−1.96490	−0.982450	+	0.186524i	$0.940278\pi$
$888$	9.51028e6	0.404726
$889$	4.26325e6	0.180920
$890$	0	0
$891$	793881.	0.0335013
$892$	−1.24054e6	−0.0522035
$893$	−1.57084e7	−0.659179
$894$	1.69292e7	0.708423
$895$	0	0
$896$	−2.85565e7	−1.18833
$897$	1.05261e7	0.436802
$898$	−1.95136e7	−0.807507
$899$	1.72596e6	0.0712248
$900$	0	0
$901$	−2.56388e7	−1.05217
$902$	2.63402e6	0.107796
$903$	−3.53722e7	−1.44358
$904$	−1.11720e7	−0.454685
$905$	0	0
$906$	−2.54129e6	−0.102857
$907$	−3.68866e7	−1.48885	−0.744425	−	0.667706i	$-0.767276\pi$
$907$	−3.68866e7	−1.48885	−0.744425	+	0.667706i	$0.767276\pi$
$908$	−1.76715e7	−0.711309
$909$	4.16153e6	0.167049
$910$	0	0
$911$	−3.44197e7	−1.37408	−0.687038	−	0.726622i	$-0.741089\pi$
$911$	−3.44197e7	−1.37408	−0.687038	+	0.726622i	$0.741089\pi$
$912$	4.26110e6	0.169643
$913$	1.07712e7	0.427647
$914$	6.43509e7	2.54794
$915$	0	0
$916$	−5.47570e7	−2.15626
$917$	−1.20636e7	−0.473755
$918$	−5.83878e6	−0.228673
$919$	−1.33270e7	−0.520527	−0.260263	−	0.965538i	$-0.583809\pi$
$919$	−1.33270e7	−0.520527	−0.260263	+	0.965538i	$0.583809\pi$
$920$	0	0
$921$	−4.54220e6	−0.176448
$922$	−3.74719e6	−0.145171
$923$	−2.26878e7	−0.876574
$924$	8.07206e6	0.311032
$925$	0	0
$926$	−2.67248e7	−1.02421
$927$	7.68688e6	0.293799
$928$	2.08546e7	0.794935
$929$	2.43196e7	0.924521	0.462260	−	0.886744i	$-0.347039\pi$
$929$	2.43196e7	0.924521	0.462260	+	0.886744i	$0.347039\pi$
$930$	0	0
$931$	1.13405e7	0.428802
$932$	1.32936e7	0.501307
$933$	4.03660e6	0.151814
$934$	−1.62714e7	−0.610322
$935$	0	0
$936$	−2.39727e6	−0.0894390
$937$	−3.18520e7	−1.18519	−0.592595	−	0.805501i	$-0.701896\pi$
$937$	−3.18520e7	−1.18519	−0.592595	+	0.805501i	$0.701896\pi$
$938$	−5.43603e7	−2.01732
$939$	7.42631e6	0.274859
$940$	0	0
$941$	5.35961e7	1.97315	0.986573	−	0.163318i	$-0.0522197\pi$
$941$	5.35961e7	1.97315	0.986573	+	0.163318i	$0.0522197\pi$
$942$	−3.52924e7	−1.29585
$943$	8.20212e6	0.300363
$944$	−4.87299e6	−0.177978
$945$	0	0
$946$	2.28509e7	0.830186
$947$	−1.82063e7	−0.659700	−0.329850	−	0.944033i	$-0.606998\pi$
$947$	−1.82063e7	−0.659700	−0.329850	+	0.944033i	$0.606998\pi$
$948$	6.47211e6	0.233897
$949$	−2.49586e7	−0.899610
$950$	0	0
$951$	2.93640e7	1.05284
$952$	−1.36283e7	−0.487360
$953$	1.33257e6	0.0475289	0.0237645	−	0.999718i	$-0.492435\pi$
$953$	1.33257e6	0.0475289	0.0237645	+	0.999718i	$0.492435\pi$
$954$	−1.90669e7	−0.678281
$955$	0	0
$956$	5.21399e7	1.84512
$957$	−2.83830e6	−0.100180
$958$	5.23335e7	1.84232
$959$	−3.67353e7	−1.28984
$960$	0	0
$961$	−2.81906e7	−0.984682
$962$	−4.01175e7	−1.39764
$963$	4.23084e6	0.147015
$964$	−4.47797e7	−1.55199
$965$	0	0
$966$	4.45014e7	1.53437
$967$	5.74029e7	1.97409	0.987047	−	0.160429i	$-0.0512877\pi$
$967$	5.74029e7	1.97409	0.987047	+	0.160429i	$0.0512877\pi$
$968$	−1.19707e6	−0.0410610
$969$	6.33755e6	0.216826
$970$	0	0
$971$	3.56758e7	1.21430	0.607149	−	0.794588i	$-0.292313\pi$
$971$	3.56758e7	1.21430	0.607149	+	0.794588i	$0.292313\pi$
$972$	−2.45258e6	−0.0832639
$973$	−7.76232e6	−0.262851
$974$	−6.26137e7	−2.11481
$975$	0	0
$976$	−2.41581e7	−0.811780
$977$	−5.30033e7	−1.77651	−0.888253	−	0.459354i	$-0.848081\pi$
$977$	−5.30033e7	−1.77651	−0.888253	+	0.459354i	$0.848081\pi$
$978$	4.07665e6	0.136288
$979$	1.56848e7	0.523026
$980$	0	0
$981$	1.12228e7	0.372331
$982$	−3.78068e7	−1.25110
$983$	−2.40146e7	−0.792667	−0.396333	−	0.918107i	$-0.629718\pi$
$983$	−2.40146e7	−0.792667	−0.396333	+	0.918107i	$0.629718\pi$
$984$	−1.86800e6	−0.0615020
$985$	0	0
$986$	2.08750e7	0.683806
$987$	−3.34650e7	−1.09345
$988$	1.13351e7	0.369430
$989$	7.11559e7	2.31324
$990$	0	0
$991$	5.40921e7	1.74964	0.874822	−	0.484444i	$-0.160978\pi$
$991$	5.40921e7	1.74964	0.874822	+	0.484444i	$0.160978\pi$
$992$	5.29870e6	0.170958
$993$	3.13358e7	1.00848
$994$	−9.59182e7	−3.07918
$995$	0	0
$996$	−3.32759e7	−1.06287
$997$	−4.73500e7	−1.50863	−0.754314	−	0.656513i	$-0.772031\pi$
$997$	−4.73500e7	−1.50863	−0.754314	+	0.656513i	$0.772031\pi$
$998$	−2.96378e7	−0.941931
$999$	−9.42174e6	−0.298688

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.v.1.3		13
5.2	odd	4		165.6.c.b.34.4	✓	26
5.3	odd	4		165.6.c.b.34.23	yes	26
5.4	even	2		825.6.a.y.1.11		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.c.b.34.4	✓	26	5.2	odd	4
165.6.c.b.34.23	yes	26	5.3	odd	4
825.6.a.v.1.3		13	1.1	even	1	trivial
825.6.a.y.1.11		13	5.4	even	2

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)

\(f(q)\)	\(=\)	\(q-8.57523 q^{2} -9.00000 q^{3} +41.5346 q^{4} +77.1771 q^{6} -178.462 q^{7} -81.7612 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-8.57523 q^{2} -9.00000 q^{3} +41.5346 q^{4} +77.1771 q^{6} -178.462 q^{7} -81.7612 q^{8} +81.0000 q^{9} +121.000 q^{11} -373.811 q^{12} +361.980 q^{13} +1530.36 q^{14} -627.985 q^{16} -934.004 q^{17} -694.594 q^{18} +753.929 q^{19} +1606.16 q^{21} -1037.60 q^{22} -3231.01 q^{23} +735.851 q^{24} -3104.06 q^{26} -729.000 q^{27} -7412.36 q^{28} +2606.34 q^{29} +662.215 q^{31} +8001.48 q^{32} -1089.00 q^{33} +8009.30 q^{34} +3364.30 q^{36} +12924.2 q^{37} -6465.11 q^{38} -3257.82 q^{39} -2538.56 q^{41} -13773.2 q^{42} -22022.8 q^{43} +5025.68 q^{44} +27706.7 q^{46} -20835.4 q^{47} +5651.87 q^{48} +15041.8 q^{49} +8406.03 q^{51} +15034.7 q^{52} +27450.5 q^{53} +6251.34 q^{54} +14591.3 q^{56} -6785.36 q^{57} -22350.0 q^{58} +7759.71 q^{59} +38469.2 q^{61} -5678.65 q^{62} -14455.5 q^{63} -48519.0 q^{64} +9338.43 q^{66} -35521.3 q^{67} -38793.4 q^{68} +29079.1 q^{69} -62677.1 q^{71} -6622.66 q^{72} -68950.2 q^{73} -110828. q^{74} +31314.1 q^{76} -21594.0 q^{77} +27936.5 q^{78} -17313.8 q^{79} +6561.00 q^{81} +21768.7 q^{82} +89017.9 q^{83} +66711.2 q^{84} +188850. q^{86} -23457.1 q^{87} -9893.11 q^{88} +129627. q^{89} -64599.8 q^{91} -134199. q^{92} -5959.94 q^{93} +178668. q^{94} -72013.3 q^{96} +136732. q^{97} -128987. q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9}+O(q^{10}) \) 13 * q - 13 * q^2 - 117 * q^3 + 209 * q^4 + 117 * q^6 - 304 * q^7 - 399 * q^8 + 1053 * q^9	\( 13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9} + 1573 q^{11} - 1881 q^{12} - 986 q^{13} - 610 q^{14} + 3501 q^{16} - 1476 q^{17} - 1053 q^{18} + 270 q^{19} + 2736 q^{21} - 1573 q^{22} - 9084 q^{23} + 3591 q^{24} + 2652 q^{26} - 9477 q^{27} - 10920 q^{28} + 11952 q^{29} + 19096 q^{31} - 11661 q^{32} - 14157 q^{33} - 1302 q^{34} + 16929 q^{36} - 39964 q^{37} - 1574 q^{38} + 8874 q^{39} + 35184 q^{41} + 5490 q^{42} + 96 q^{43} + 25289 q^{44} - 4120 q^{46} - 34984 q^{47} - 31509 q^{48} + 14557 q^{49} + 13284 q^{51} - 39002 q^{52} - 22984 q^{53} + 9477 q^{54} + 59802 q^{56} - 2430 q^{57} - 18896 q^{58} - 9192 q^{59} + 5438 q^{61} - 272 q^{62} - 24624 q^{63} + 106557 q^{64} + 14157 q^{66} - 71508 q^{67} - 127948 q^{68} + 81756 q^{69} + 101700 q^{71} - 32319 q^{72} - 77390 q^{73} + 13676 q^{74} + 139966 q^{76} - 36784 q^{77} - 23868 q^{78} + 93954 q^{79} + 85293 q^{81} - 53284 q^{82} - 185918 q^{83} + 98280 q^{84} + 370930 q^{86} - 107568 q^{87} - 48279 q^{88} - 18418 q^{89} + 174536 q^{91} - 274264 q^{92} - 171864 q^{93} + 64520 q^{94} + 104949 q^{96} - 94312 q^{97} - 145677 q^{98} + 127413 q^{99}+O(q^{100}) \) 13 * q - 13 * q^2 - 117 * q^3 + 209 * q^4 + 117 * q^6 - 304 * q^7 - 399 * q^8 + 1053 * q^9 + 1573 * q^11 - 1881 * q^12 - 986 * q^13 - 610 * q^14 + 3501 * q^16 - 1476 * q^17 - 1053 * q^18 + 270 * q^19 + 2736 * q^21 - 1573 * q^22 - 9084 * q^23 + 3591 * q^24 + 2652 * q^26 - 9477 * q^27 - 10920 * q^28 + 11952 * q^29 + 19096 * q^31 - 11661 * q^32 - 14157 * q^33 - 1302 * q^34 + 16929 * q^36 - 39964 * q^37 - 1574 * q^38 + 8874 * q^39 + 35184 * q^41 + 5490 * q^42 + 96 * q^43 + 25289 * q^44 - 4120 * q^46 - 34984 * q^47 - 31509 * q^48 + 14557 * q^49 + 13284 * q^51 - 39002 * q^52 - 22984 * q^53 + 9477 * q^54 + 59802 * q^56 - 2430 * q^57 - 18896 * q^58 - 9192 * q^59 + 5438 * q^61 - 272 * q^62 - 24624 * q^63 + 106557 * q^64 + 14157 * q^66 - 71508 * q^67 - 127948 * q^68 + 81756 * q^69 + 101700 * q^71 - 32319 * q^72 - 77390 * q^73 + 13676 * q^74 + 139966 * q^76 - 36784 * q^77 - 23868 * q^78 + 93954 * q^79 + 85293 * q^81 - 53284 * q^82 - 185918 * q^83 + 98280 * q^84 + 370930 * q^86 - 107568 * q^87 - 48279 * q^88 - 18418 * q^89 + 174536 * q^91 - 274264 * q^92 - 171864 * q^93 + 64520 * q^94 + 104949 * q^96 - 94312 * q^97 - 145677 * q^98 + 127413 * q^99

Introduction

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