LMFDB - Embedded newform 825.6.a.y.1.5

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:	$13$
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			1.5
Root			$4.33747$ 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)

$f(q)$	$=$	$q-3.33747 q^{2} +9.00000 q^{3} -20.8613 q^{4} -30.0373 q^{6} +103.473 q^{7} +176.423 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-3.33747 q^{2} +9.00000 q^{3} -20.8613 q^{4} -30.0373 q^{6} +103.473 q^{7} +176.423 q^{8} +81.0000 q^{9} +121.000 q^{11} -187.751 q^{12} -357.925 q^{13} -345.339 q^{14} +78.7535 q^{16} +2067.63 q^{17} -270.335 q^{18} +1320.17 q^{19} +931.258 q^{21} -403.834 q^{22} -2498.44 q^{23} +1587.81 q^{24} +1194.57 q^{26} +729.000 q^{27} -2158.58 q^{28} +1657.60 q^{29} +4690.57 q^{31} -5908.38 q^{32} +1089.00 q^{33} -6900.67 q^{34} -1689.76 q^{36} +5214.75 q^{37} -4406.02 q^{38} -3221.33 q^{39} +13577.0 q^{41} -3108.05 q^{42} +10764.0 q^{43} -2524.21 q^{44} +8338.46 q^{46} -2926.95 q^{47} +708.782 q^{48} -6100.32 q^{49} +18608.7 q^{51} +7466.78 q^{52} -21529.9 q^{53} -2433.02 q^{54} +18255.0 q^{56} +11881.5 q^{57} -5532.19 q^{58} -4541.09 q^{59} +3649.55 q^{61} -15654.6 q^{62} +8381.32 q^{63} +17198.9 q^{64} -3634.51 q^{66} -55473.7 q^{67} -43133.4 q^{68} -22485.9 q^{69} +4491.83 q^{71} +14290.3 q^{72} +33559.8 q^{73} -17404.1 q^{74} -27540.4 q^{76} +12520.2 q^{77} +10751.1 q^{78} -7912.05 q^{79} +6561.00 q^{81} -45312.9 q^{82} -20569.6 q^{83} -19427.2 q^{84} -35924.5 q^{86} +14918.4 q^{87} +21347.2 q^{88} -111119. q^{89} -37035.6 q^{91} +52120.6 q^{92} +42215.1 q^{93} +9768.61 q^{94} -53175.4 q^{96} -58271.1 q^{97} +20359.7 q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−3.33747	−0.589987	−0.294994	−	0.955499i	$-0.595318\pi$
$2$	−3.33747	−0.589987	−0.294994	+	0.955499i	$0.595318\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−20.8613	−0.651915
$5$	0	0
$5$	0	0
$6$	−30.0373	−0.340629
$7$	103.473	0.798146	0.399073	−	0.916919i	$-0.369332\pi$
$7$	103.473	0.798146	0.399073	+	0.916919i	$0.369332\pi$
$8$	176.423	0.974609
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	−187.751	−0.376383
$13$	−357.925	−0.587400	−0.293700	−	0.955898i	$-0.594887\pi$
$13$	−357.925	−0.587400	−0.293700	+	0.955898i	$0.594887\pi$
$14$	−345.339	−0.470896
$15$	0	0
$16$	78.7535	0.0769078
$17$	2067.63	1.73521	0.867603	−	0.497258i	$-0.165660\pi$
$17$	2067.63	1.73521	0.867603	+	0.497258i	$0.165660\pi$
$18$	−270.335	−0.196662
$19$	1320.17	0.838967	0.419484	−	0.907763i	$-0.362211\pi$
$19$	1320.17	0.838967	0.419484	+	0.907763i	$0.362211\pi$
$20$	0	0
$21$	931.258	0.460810
$22$	−403.834	−0.177888
$23$	−2498.44	−0.984802	−0.492401	−	0.870369i	$-0.663881\pi$
$23$	−2498.44	−0.984802	−0.492401	+	0.870369i	$0.663881\pi$
$24$	1587.81	0.562691
$25$	0	0
$26$	1194.57	0.346559
$27$	729.000	0.192450
$28$	−2158.58	−0.520323
$29$	1657.60	0.366003	0.183002	−	0.983113i	$-0.441419\pi$
$29$	1657.60	0.366003	0.183002	+	0.983113i	$0.441419\pi$
$30$	0	0
$31$	4690.57	0.876640	0.438320	−	0.898819i	$-0.355574\pi$
$31$	4690.57	0.876640	0.438320	+	0.898819i	$0.355574\pi$
$32$	−5908.38	−1.01998
$33$	1089.00	0.174078
$34$	−6900.67	−1.02375
$35$	0	0
$36$	−1689.76	−0.217305
$37$	5214.75	0.626223	0.313112	−	0.949716i	$-0.398629\pi$
$37$	5214.75	0.626223	0.313112	+	0.949716i	$0.398629\pi$
$38$	−4406.02	−0.494980
$39$	−3221.33	−0.339136
$40$	0	0
$41$	13577.0	1.26138	0.630688	−	0.776037i	$-0.282773\pi$
$41$	13577.0	1.26138	0.630688	+	0.776037i	$0.282773\pi$
$42$	−3108.05	−0.271872
$43$	10764.0	0.887774	0.443887	−	0.896083i	$-0.353599\pi$
$43$	10764.0	0.887774	0.443887	+	0.896083i	$0.353599\pi$
$44$	−2524.21	−0.196560
$45$	0	0
$46$	8338.46	0.581021
$47$	−2926.95	−0.193273	−0.0966363	−	0.995320i	$-0.530808\pi$
$47$	−2926.95	−0.193273	−0.0966363	+	0.995320i	$0.530808\pi$
$48$	708.782	0.0444027
$49$	−6100.32	−0.362963
$50$	0	0
$51$	18608.7	1.00182
$52$	7466.78	0.382935
$53$	−21529.9	−1.05282	−0.526408	−	0.850232i	$-0.676462\pi$
$53$	−21529.9	−1.05282	−0.526408	+	0.850232i	$0.676462\pi$
$54$	−2433.02	−0.113543
$55$	0	0
$56$	18255.0	0.777880
$57$	11881.5	0.484378
$58$	−5532.19	−0.215937
$59$	−4541.09	−0.169836	−0.0849181	−	0.996388i	$-0.527063\pi$
$59$	−4541.09	−0.169836	−0.0849181	+	0.996388i	$0.527063\pi$
$60$	0	0
$61$	3649.55	0.125578	0.0627892	−	0.998027i	$-0.480000\pi$
$61$	3649.55	0.125578	0.0627892	+	0.998027i	$0.480000\pi$
$62$	−15654.6	−0.517207
$63$	8381.32	0.266049
$64$	17198.9	0.524870
$65$	0	0
$66$	−3634.51	−0.102704
$67$	−55473.7	−1.50973	−0.754867	−	0.655878i	$-0.772299\pi$
$67$	−55473.7	−1.50973	−0.754867	+	0.655878i	$0.772299\pi$
$68$	−43133.4	−1.13121
$69$	−22485.9	−0.568575
$70$	0	0
$71$	4491.83	0.105749	0.0528746	−	0.998601i	$-0.483162\pi$
$71$	4491.83	0.105749	0.0528746	+	0.998601i	$0.483162\pi$
$72$	14290.3	0.324870
$73$	33559.8	0.737076	0.368538	−	0.929613i	$-0.379859\pi$
$73$	33559.8	0.737076	0.368538	+	0.929613i	$0.379859\pi$
$74$	−17404.1	−0.369464
$75$	0	0
$76$	−27540.4	−0.546935
$77$	12520.2	0.240650
$78$	10751.1	0.200086
$79$	−7912.05	−0.142634	−0.0713168	−	0.997454i	$-0.522720\pi$
$79$	−7912.05	−0.142634	−0.0713168	+	0.997454i	$0.522720\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−45312.9	−0.744196
$83$	−20569.6	−0.327741	−0.163870	−	0.986482i	$-0.552398\pi$
$83$	−20569.6	−0.327741	−0.163870	+	0.986482i	$0.552398\pi$
$84$	−19427.2	−0.300409
$85$	0	0
$86$	−35924.5	−0.523775
$87$	14918.4	0.211312
$88$	21347.2	0.293856
$89$	−111119.	−1.48701	−0.743504	−	0.668732i	$-0.766837\pi$
$89$	−111119.	−1.48701	−0.743504	+	0.668732i	$0.766837\pi$
$90$	0	0
$91$	−37035.6	−0.468831
$92$	52120.6	0.642007
$93$	42215.1	0.506128
$94$	9768.61	0.114028
$95$	0	0
$96$	−53175.4	−0.588888
$97$	−58271.1	−0.628816	−0.314408	−	0.949288i	$-0.601806\pi$
$97$	−58271.1	−0.628816	−0.314408	+	0.949288i	$0.601806\pi$
$98$	20359.7	0.214144
$99$	9801.00	0.100504
$100$	0	0
$101$	−56503.6	−0.551153	−0.275577	−	0.961279i	$-0.588869\pi$
$101$	−56503.6	−0.551153	−0.275577	+	0.961279i	$0.588869\pi$
$102$	−62106.0	−0.591062
$103$	95588.6	0.887796	0.443898	−	0.896077i	$-0.353595\pi$
$103$	95588.6	0.887796	0.443898	+	0.896077i	$0.353595\pi$
$104$	−63146.3	−0.572486
$105$	0	0
$106$	71855.5	0.621148
$107$	211645.	1.78710	0.893550	−	0.448964i	$-0.148207\pi$
$107$	211645.	1.78710	0.893550	+	0.448964i	$0.148207\pi$
$108$	−15207.9	−0.125461
$109$	370.739	0.00298884	0.00149442	−	0.999999i	$-0.499524\pi$
$109$	370.739	0.00298884	0.00149442	+	0.999999i	$0.499524\pi$
$110$	0	0
$111$	46932.8	0.361550
$112$	8148.87	0.0613836
$113$	−14124.9	−0.104061	−0.0520307	−	0.998645i	$-0.516569\pi$
$113$	−14124.9	−0.104061	−0.0520307	+	0.998645i	$0.516569\pi$
$114$	−39654.2	−0.285777
$115$	0	0
$116$	−34579.6	−0.238603
$117$	−28992.0	−0.195800
$118$	15155.8	0.100201
$119$	213944.	1.38495
$120$	0	0
$121$	14641.0	0.0909091
$122$	−12180.3	−0.0740897
$123$	122193.	0.728255
$124$	−97851.3	−0.571495
$125$	0	0
$126$	−27972.4	−0.156965
$127$	39049.1	0.214834	0.107417	−	0.994214i	$-0.465742\pi$
$127$	39049.1	0.214834	0.107417	+	0.994214i	$0.465742\pi$
$128$	131667.	0.710317
$129$	96876.0	0.512556
$130$	0	0
$131$	−152838.	−0.778131	−0.389066	−	0.921210i	$-0.627202\pi$
$131$	−152838.	−0.778131	−0.389066	+	0.921210i	$0.627202\pi$
$132$	−22717.9	−0.113484
$133$	136602.	0.669618
$134$	185142.	0.890724
$135$	0	0
$136$	364778.	1.69115
$137$	323866.	1.47422	0.737112	−	0.675771i	$-0.236189\pi$
$137$	323866.	1.47422	0.737112	+	0.675771i	$0.236189\pi$
$138$	75046.2	0.335452
$139$	195925.	0.860108	0.430054	−	0.902803i	$-0.358494\pi$
$139$	195925.	0.860108	0.430054	+	0.902803i	$0.358494\pi$
$140$	0	0
$141$	−26342.5	−0.111586
$142$	−14991.4	−0.0623907
$143$	−43309.0	−0.177108
$144$	6379.04	0.0256359
$145$	0	0
$146$	−112005.	−0.434865
$147$	−54902.9	−0.209557
$148$	−108786.	−0.408244
$149$	435048.	1.60536	0.802679	−	0.596411i	$-0.203407\pi$
$149$	435048.	1.60536	0.802679	+	0.596411i	$0.203407\pi$
$150$	0	0
$151$	−94062.6	−0.335718	−0.167859	−	0.985811i	$-0.553685\pi$
$151$	−94062.6	−0.335718	−0.167859	+	0.985811i	$0.553685\pi$
$152$	232908.	0.817665
$153$	167478.	0.578402
$154$	−41786.0	−0.141981
$155$	0	0
$156$	67201.0	0.221088
$157$	−194996.	−0.631360	−0.315680	−	0.948866i	$-0.602233\pi$
$157$	−194996.	−0.631360	−0.315680	+	0.948866i	$0.602233\pi$
$158$	26406.3	0.0841520
$159$	−193769.	−0.607844
$160$	0	0
$161$	−258521.	−0.786015
$162$	−21897.2	−0.0655542
$163$	154239.	0.454701	0.227350	−	0.973813i	$-0.426994\pi$
$163$	154239.	0.454701	0.227350	+	0.973813i	$0.426994\pi$
$164$	−283234.	−0.822309
$165$	0	0
$166$	68650.4	0.193363
$167$	−297517.	−0.825506	−0.412753	−	0.910843i	$-0.635433\pi$
$167$	−297517.	−0.825506	−0.412753	+	0.910843i	$0.635433\pi$
$168$	164295.	0.449109
$169$	−243182.	−0.654961
$170$	0	0
$171$	106934.	0.279656
$172$	−224551.	−0.578753
$173$	−49837.8	−0.126603	−0.0633014	−	0.997994i	$-0.520163\pi$
$173$	−49837.8	−0.126603	−0.0633014	+	0.997994i	$0.520163\pi$
$174$	−49789.8	−0.124671
$175$	0	0
$176$	9529.18	0.0231886
$177$	−40869.8	−0.0980550
$178$	370856.	0.877316
$179$	−825399.	−1.92545	−0.962724	−	0.270487i	$-0.912815\pi$
$179$	−825399.	−1.92545	−0.962724	+	0.270487i	$0.912815\pi$
$180$	0	0
$181$	−631320.	−1.43236	−0.716182	−	0.697914i	$-0.754112\pi$
$181$	−631320.	−1.43236	−0.716182	+	0.697914i	$0.754112\pi$
$182$	123605.	0.276604
$183$	32846.0	0.0725027
$184$	−440782.	−0.959797
$185$	0	0
$186$	−140892.	−0.298609
$187$	250184.	0.523184
$188$	61059.9	0.125997
$189$	75431.9	0.153603
$190$	0	0
$191$	367056.	0.728030	0.364015	−	0.931393i	$-0.381406\pi$
$191$	367056.	0.728030	0.364015	+	0.931393i	$0.381406\pi$
$192$	154790.	0.303034
$193$	239853.	0.463501	0.231751	−	0.972775i	$-0.425555\pi$
$193$	239853.	0.463501	0.231751	+	0.972775i	$0.425555\pi$
$194$	194478.	0.370994
$195$	0	0
$196$	127260.	0.236621
$197$	889966.	1.63383	0.816917	−	0.576755i	$-0.195681\pi$
$197$	889966.	1.63383	0.816917	+	0.576755i	$0.195681\pi$
$198$	−32710.6	−0.0592960
$199$	2447.34	0.00438089	0.00219045	−	0.999998i	$-0.499303\pi$
$199$	2447.34	0.00438089	0.00219045	+	0.999998i	$0.499303\pi$
$200$	0	0
$201$	−499264.	−0.871645
$202$	188579.	0.325174
$203$	171517.	0.292124
$204$	−388201.	−0.653102
$205$	0	0
$206$	−319024.	−0.523788
$207$	−202373.	−0.328267
$208$	−28187.9	−0.0451756
$209$	159740.	0.252958
$210$	0	0
$211$	779165.	1.20482	0.602411	−	0.798186i	$-0.294207\pi$
$211$	779165.	1.20482	0.602411	+	0.798186i	$0.294207\pi$
$212$	449141.	0.686347
$213$	40426.4	0.0610543
$214$	−706360.	−1.05437
$215$	0	0
$216$	128612.	0.187564
$217$	485348.	0.699687
$218$	−1237.33	−0.00176338
$219$	302038.	0.425551
$220$	0	0
$221$	−740058.	−1.01926
$222$	−156637.	−0.213310
$223$	1.27044e6	1.71077	0.855387	−	0.517990i	$-0.173320\pi$
$223$	1.27044e6	1.71077	0.855387	+	0.517990i	$0.173320\pi$
$224$	−611358.	−0.814096
$225$	0	0
$226$	47141.5	0.0613949
$227$	1755.97	0.00226179	0.00113089	−	0.999999i	$-0.499640\pi$
$227$	1755.97	0.00226179	0.00113089	+	0.999999i	$0.499640\pi$
$228$	−247863.	−0.315773
$229$	−185101.	−0.233249	−0.116624	−	0.993176i	$-0.537207\pi$
$229$	−185101.	−0.233249	−0.116624	+	0.993176i	$0.537207\pi$
$230$	0	0
$231$	112682.	0.138939
$232$	292439.	0.356710
$233$	172318.	0.207941	0.103971	−	0.994580i	$-0.466845\pi$
$233$	172318.	0.207941	0.103971	+	0.994580i	$0.466845\pi$
$234$	96759.9	0.115520
$235$	0	0
$236$	94733.0	0.110719
$237$	−71208.5	−0.0823495
$238$	−714033.	−0.817102
$239$	1.58068e6	1.78999	0.894994	−	0.446078i	$-0.147180\pi$
$239$	1.58068e6	1.78999	0.894994	+	0.446078i	$0.147180\pi$
$240$	0	0
$241$	1.44227e6	1.59957	0.799785	−	0.600287i	$-0.204947\pi$
$241$	1.44227e6	1.59957	0.799785	+	0.600287i	$0.204947\pi$
$242$	−48863.9	−0.0536352
$243$	59049.0	0.0641500
$244$	−76134.3	−0.0818664
$245$	0	0
$246$	−407816.	−0.429662
$247$	−472521.	−0.492810
$248$	827525.	0.854381
$249$	−185126.	−0.189221
$250$	0	0
$251$	1.18482e6	1.18705	0.593524	−	0.804816i	$-0.297736\pi$
$251$	1.18482e6	1.18705	0.593524	+	0.804816i	$0.297736\pi$
$252$	−174845.	−0.173441
$253$	−302311.	−0.296929
$254$	−130325.	−0.126749
$255$	0	0
$256$	−989801.	−0.943948
$257$	−307999.	−0.290882	−0.145441	−	0.989367i	$-0.546460\pi$
$257$	−307999.	−0.290882	−0.145441	+	0.989367i	$0.546460\pi$
$258$	−323321.	−0.302402
$259$	539586.	0.499818
$260$	0	0
$261$	134266.	0.122001
$262$	510093.	0.459088
$263$	1.92302e6	1.71433	0.857163	−	0.515045i	$-0.172225\pi$
$263$	1.92302e6	1.71433	0.857163	+	0.515045i	$0.172225\pi$
$264$	192125.	0.169658
$265$	0	0
$266$	−455905.	−0.395066
$267$	−1.00007e6	−0.858524
$268$	1.15725e6	0.984218
$269$	−1.14532e6	−0.965040	−0.482520	−	0.875885i	$-0.660278\pi$
$269$	−1.14532e6	−0.965040	−0.482520	+	0.875885i	$0.660278\pi$
$270$	0	0
$271$	−737503.	−0.610015	−0.305008	−	0.952350i	$-0.598659\pi$
$271$	−737503.	−0.610015	−0.305008	+	0.952350i	$0.598659\pi$
$272$	162833.	0.133451
$273$	−333321.	−0.270680
$274$	−1.08089e6	−0.869774
$275$	0	0
$276$	469085.	0.370663
$277$	1.91454e6	1.49922	0.749611	−	0.661878i	$-0.230240\pi$
$277$	1.91454e6	1.49922	0.749611	+	0.661878i	$0.230240\pi$
$278$	−653895.	−0.507453
$279$	379936.	0.292213
$280$	0	0
$281$	−1.09912e6	−0.830387	−0.415193	−	0.909733i	$-0.636286\pi$
$281$	−1.09912e6	−0.830387	−0.415193	+	0.909733i	$0.636286\pi$
$282$	87917.5	0.0658343
$283$	371541.	0.275766	0.137883	−	0.990449i	$-0.455970\pi$
$283$	371541.	0.275766	0.137883	+	0.990449i	$0.455970\pi$
$284$	−93705.2	−0.0689395
$285$	0	0
$286$	144543.	0.104491
$287$	1.40485e6	1.00676
$288$	−478578.	−0.339995
$289$	2.85525e6	2.01094
$290$	0	0
$291$	−524440.	−0.363047
$292$	−700100.	−0.480511
$293$	2.86231e6	1.94781	0.973907	−	0.226948i	$-0.0728748\pi$
$293$	2.86231e6	1.94781	0.973907	+	0.226948i	$0.0728748\pi$
$294$	183237.	0.123636
$295$	0	0
$296$	920003.	0.610323
$297$	88209.0	0.0580259
$298$	−1.45196e6	−0.947141
$299$	894254.	0.578473
$300$	0	0
$301$	1.11378e6	0.708573
$302$	313931.	0.198069
$303$	−508532.	−0.318209
$304$	103968.	0.0645231
$305$	0	0
$306$	−558954.	−0.341250
$307$	−2.67276e6	−1.61851	−0.809253	−	0.587460i	$-0.800128\pi$
$307$	−2.67276e6	−1.61851	−0.809253	+	0.587460i	$0.800128\pi$
$308$	−261188.	−0.156883
$309$	860298.	0.512569
$310$	0	0
$311$	1.84386e6	1.08100	0.540500	−	0.841344i	$-0.318235\pi$
$311$	1.84386e6	1.08100	0.540500	+	0.841344i	$0.318235\pi$
$312$	−568317.	−0.330525
$313$	1.87741e6	1.08318	0.541588	−	0.840644i	$-0.317823\pi$
$313$	1.87741e6	1.08318	0.541588	+	0.840644i	$0.317823\pi$
$314$	650794.	0.372494
$315$	0	0
$316$	165056.	0.0929849
$317$	−2.04246e6	−1.14158	−0.570789	−	0.821097i	$-0.693362\pi$
$317$	−2.04246e6	−1.14158	−0.570789	+	0.821097i	$0.693362\pi$
$318$	646699.	0.358620
$319$	200570.	0.110354
$320$	0	0
$321$	1.90481e6	1.03178
$322$	862807.	0.463739
$323$	2.72962e6	1.45578
$324$	−136871.	−0.0724350
$325$	0	0
$326$	−514769.	−0.268268
$327$	3336.65	0.00172561
$328$	2.39530e6	1.22935
$329$	−302860.	−0.154260
$330$	0	0
$331$	−2.48973e6	−1.24906	−0.624528	−	0.781002i	$-0.714709\pi$
$331$	−2.48973e6	−1.24906	−0.624528	+	0.781002i	$0.714709\pi$
$332$	429108.	0.213659
$333$	422395.	0.208741
$334$	992954.	0.487038
$335$	0	0
$336$	73339.8	0.0354398
$337$	−2.26543e6	−1.08662	−0.543308	−	0.839534i	$-0.682828\pi$
$337$	−2.26543e6	−1.08662	−0.543308	+	0.839534i	$0.682828\pi$
$338$	811615.	0.386419
$339$	−127124.	−0.0600799
$340$	0	0
$341$	567559.	0.264317
$342$	−356888.	−0.164993
$343$	−2.37029e6	−1.08784
$344$	1.89902e6	0.865232
$345$	0	0
$346$	166332.	0.0746941
$347$	887458.	0.395662	0.197831	−	0.980236i	$-0.436610\pi$
$347$	887458.	0.395662	0.197831	+	0.980236i	$0.436610\pi$
$348$	−311217.	−0.137757
$349$	3.37834e6	1.48470	0.742351	−	0.670011i	$-0.233711\pi$
$349$	3.37834e6	1.48470	0.742351	+	0.670011i	$0.233711\pi$
$350$	0	0
$351$	−260928.	−0.113045
$352$	−714914.	−0.307537
$353$	1.23199e6	0.526222	0.263111	−	0.964766i	$-0.415251\pi$
$353$	1.23199e6	0.526222	0.263111	+	0.964766i	$0.415251\pi$
$354$	136402.	0.0578512
$355$	0	0
$356$	2.31808e6	0.969402
$357$	1.92550e6	0.799600
$358$	2.75475e6	1.13599
$359$	1.03637e6	0.424405	0.212202	−	0.977226i	$-0.431936\pi$
$359$	1.03637e6	0.424405	0.212202	+	0.977226i	$0.431936\pi$
$360$	0	0
$361$	−733257.	−0.296134
$362$	2.10701e6	0.845076
$363$	131769.	0.0524864
$364$	772611.	0.305638
$365$	0	0
$366$	−109623.	−0.0427757
$367$	4.31603e6	1.67270	0.836352	−	0.548192i	$-0.184684\pi$
$367$	4.31603e6	1.67270	0.836352	+	0.548192i	$0.184684\pi$
$368$	−196761.	−0.0757389
$369$	1.09974e6	0.420458
$370$	0	0
$371$	−2.22777e6	−0.840301
$372$	−880661.	−0.329953
$373$	−2.03599e6	−0.757712	−0.378856	−	0.925456i	$-0.623682\pi$
$373$	−2.03599e6	−0.757712	−0.378856	+	0.925456i	$0.623682\pi$
$374$	−834981.	−0.308672
$375$	0	0
$376$	−516381.	−0.188365
$377$	−593297.	−0.214990
$378$	−251752.	−0.0906240
$379$	−3.67933e6	−1.31574	−0.657872	−	0.753130i	$-0.728543\pi$
$379$	−3.67933e6	−1.31574	−0.657872	+	0.753130i	$0.728543\pi$
$380$	0	0
$381$	351442.	0.124034
$382$	−1.22504e6	−0.429528
$383$	−3.35614e6	−1.16908	−0.584539	−	0.811366i	$-0.698725\pi$
$383$	−3.35614e6	−1.16908	−0.584539	+	0.811366i	$0.698725\pi$
$384$	1.18500e6	0.410102
$385$	0	0
$386$	−800501.	−0.273460
$387$	871884.	0.295925
$388$	1.21561e6	0.409935
$389$	5.94481e6	1.99188	0.995942	−	0.0900021i	$-0.0286874\pi$
$389$	5.94481e6	1.99188	0.995942	+	0.0900021i	$0.0286874\pi$
$390$	0	0
$391$	−5.16585e6	−1.70883
$392$	−1.07624e6	−0.353747
$393$	−1.37554e6	−0.449254
$394$	−2.97024e6	−0.963941
$395$	0	0
$396$	−204461.	−0.0655199
$397$	4.07424e6	1.29739	0.648695	−	0.761048i	$-0.275315\pi$
$397$	4.07424e6	1.29739	0.648695	+	0.761048i	$0.275315\pi$
$398$	−8167.95	−0.00258467
$399$	1.22942e6	0.386604
$400$	0	0
$401$	1.76526e6	0.548209	0.274105	−	0.961700i	$-0.411619\pi$
$401$	1.76526e6	0.548209	0.274105	+	0.961700i	$0.411619\pi$
$402$	1.66628e6	0.514260
$403$	−1.67887e6	−0.514939
$404$	1.17874e6	0.359305
$405$	0	0
$406$	−572433.	−0.172349
$407$	630985.	0.188813
$408$	3.28300e6	0.976384
$409$	−3.05109e6	−0.901877	−0.450938	−	0.892555i	$-0.648911\pi$
$409$	−3.05109e6	−0.901877	−0.450938	+	0.892555i	$0.648911\pi$
$410$	0	0
$411$	2.91479e6	0.851144
$412$	−1.99410e6	−0.578767
$413$	−469881.	−0.135554
$414$	675416.	0.193674
$415$	0	0
$416$	2.11476e6	0.599139
$417$	1.76333e6	0.496584
$418$	−533129.	−0.149242
$419$	−6.61264e6	−1.84009	−0.920046	−	0.391810i	$-0.871849\pi$
$419$	−6.61264e6	−1.84009	−0.920046	+	0.391810i	$0.871849\pi$
$420$	0	0
$421$	2.29303e6	0.630527	0.315263	−	0.949004i	$-0.397907\pi$
$421$	2.29303e6	0.630527	0.315263	+	0.949004i	$0.397907\pi$
$422$	−2.60044e6	−0.710830
$423$	−237083.	−0.0644242
$424$	−3.79837e6	−1.02608
$425$	0	0
$426$	−134922.	−0.0360213
$427$	377630.	0.100230
$428$	−4.41519e6	−1.16504
$429$	−389781.	−0.102253
$430$	0	0
$431$	5.53759e6	1.43591	0.717956	−	0.696089i	$-0.245078\pi$
$431$	5.53759e6	1.43591	0.717956	+	0.696089i	$0.245078\pi$
$432$	57411.3	0.0148009
$433$	−2.55617e6	−0.655195	−0.327598	−	0.944817i	$-0.606239\pi$
$433$	−2.55617e6	−0.655195	−0.327598	+	0.944817i	$0.606239\pi$
$434$	−1.61983e6	−0.412806
$435$	0	0
$436$	−7734.09	−0.00194847
$437$	−3.29835e6	−0.826216
$438$	−1.00804e6	−0.251070
$439$	−696199.	−0.172414	−0.0862070	−	0.996277i	$-0.527475\pi$
$439$	−696199.	−0.172414	−0.0862070	+	0.996277i	$0.527475\pi$
$440$	0	0
$441$	−494126.	−0.120988
$442$	2.46992e6	0.601351
$443$	6.82989e6	1.65350	0.826750	−	0.562570i	$-0.190187\pi$
$443$	6.82989e6	1.65350	0.826750	+	0.562570i	$0.190187\pi$
$444$	−979077.	−0.235700
$445$	0	0
$446$	−4.24006e6	−1.00933
$447$	3.91544e6	0.926854
$448$	1.77963e6	0.418923
$449$	−6.68298e6	−1.56442	−0.782212	−	0.623013i	$-0.785909\pi$
$449$	−6.68298e6	−1.56442	−0.782212	+	0.623013i	$0.785909\pi$
$450$	0	0
$451$	1.64282e6	0.380319
$452$	294664.	0.0678392
$453$	−846564.	−0.193827
$454$	−5860.49	−0.00133443
$455$	0	0
$456$	2.09617e6	0.472079
$457$	616838.	0.138159	0.0690797	−	0.997611i	$-0.477994\pi$
$457$	616838.	0.138159	0.0690797	+	0.997611i	$0.477994\pi$
$458$	617769.	0.137614
$459$	1.50730e6	0.333941
$460$	0	0
$461$	3.55459e6	0.779001	0.389500	−	0.921026i	$-0.372648\pi$
$461$	3.55459e6	0.779001	0.389500	+	0.921026i	$0.372648\pi$
$462$	−376074.	−0.0819725
$463$	−3.23948e6	−0.702299	−0.351150	−	0.936319i	$-0.614209\pi$
$463$	−3.23948e6	−0.702299	−0.351150	+	0.936319i	$0.614209\pi$
$464$	130542.	0.0281485
$465$	0	0
$466$	−575106.	−0.122683
$467$	7.20656e6	1.52910	0.764550	−	0.644564i	$-0.222961\pi$
$467$	7.20656e6	1.52910	0.764550	+	0.644564i	$0.222961\pi$
$468$	604809.	0.127645
$469$	−5.74004e6	−1.20499
$470$	0	0
$471$	−1.75497e6	−0.364516
$472$	−801154.	−0.165524
$473$	1.30244e6	0.267674
$474$	237656.	0.0485852
$475$	0	0
$476$	−4.46315e6	−0.902868
$477$	−1.74392e6	−0.350939
$478$	−5.27549e6	−1.05607
$479$	772579.	0.153852	0.0769261	−	0.997037i	$-0.475489\pi$
$479$	772579.	0.153852	0.0769261	+	0.997037i	$0.475489\pi$
$480$	0	0
$481$	−1.86649e6	−0.367844
$482$	−4.81353e6	−0.943726
$483$	−2.32669e6	−0.453806
$484$	−305430.	−0.0592650
$485$	0	0
$486$	−197074.	−0.0378477
$487$	−3.08873e6	−0.590143	−0.295072	−	0.955475i	$-0.595344\pi$
$487$	−3.08873e6	−0.590143	−0.295072	+	0.955475i	$0.595344\pi$
$488$	643865.	0.122390
$489$	1.38815e6	0.262522
$490$	0	0
$491$	4.82049e6	0.902375	0.451187	−	0.892429i	$-0.351001\pi$
$491$	4.82049e6	0.902375	0.451187	+	0.892429i	$0.351001\pi$
$492$	−2.54910e6	−0.474761
$493$	3.42731e6	0.635091
$494$	1.57703e6	0.290751
$495$	0	0
$496$	369399.	0.0674204
$497$	464783.	0.0844033
$498$	617854.	0.111638
$499$	−9.37761e6	−1.68594	−0.842968	−	0.537964i	$-0.819194\pi$
$499$	−9.37761e6	−1.68594	−0.842968	+	0.537964i	$0.819194\pi$
$500$	0	0
$501$	−2.67765e6	−0.476606
$502$	−3.95430e6	−0.700343
$503$	4.35545e6	0.767562	0.383781	−	0.923424i	$-0.374622\pi$
$503$	4.35545e6	0.767562	0.383781	+	0.923424i	$0.374622\pi$
$504$	1.47866e6	0.259293
$505$	0	0
$506$	1.00895e6	0.175184
$507$	−2.18864e6	−0.378142
$508$	−814615.	−0.140053
$509$	1.05979e7	1.81311	0.906556	−	0.422086i	$-0.138702\pi$
$509$	1.05979e7	1.81311	0.906556	+	0.422086i	$0.138702\pi$
$510$	0	0
$511$	3.47253e6	0.588294
$512$	−909912.	−0.153400
$513$	962402.	0.161459
$514$	1.02794e6	0.171617
$515$	0	0
$516$	−2.02096e6	−0.334143
$517$	−354161.	−0.0582739
$518$	−1.80086e6	−0.294886
$519$	−448540.	−0.0730942
$520$	0	0
$521$	6.52699e6	1.05346	0.526731	−	0.850032i	$-0.323418\pi$
$521$	6.52699e6	1.05346	0.526731	+	0.850032i	$0.323418\pi$
$522$	−448108.	−0.0719791
$523$	−9.35815e6	−1.49601	−0.748007	−	0.663690i	$-0.768989\pi$
$523$	−9.35815e6	−1.49601	−0.748007	+	0.663690i	$0.768989\pi$
$524$	3.18839e6	0.507275
$525$	0	0
$526$	−6.41801e6	−1.01143
$527$	9.69837e6	1.52115
$528$	85762.6	0.0133879
$529$	−194157.	−0.0301657
$530$	0	0
$531$	−367829.	−0.0566121
$532$	−2.84969e6	−0.436534
$533$	−4.85955e6	−0.740932
$534$	3.33771e6	0.506518
$535$	0	0
$536$	−9.78685e6	−1.47140
$537$	−7.42860e6	−1.11166
$538$	3.82247e6	0.569361
$539$	−738139.	−0.109438
$540$	0	0
$541$	9.93079e6	1.45878	0.729392	−	0.684096i	$-0.239803\pi$
$541$	9.93079e6	1.45878	0.729392	+	0.684096i	$0.239803\pi$
$542$	2.46140e6	0.359901
$543$	−5.68188e6	−0.826975
$544$	−1.22163e7	−1.76988
$545$	0	0
$546$	1.11245e6	0.159698
$547$	−8.79495e6	−1.25680	−0.628399	−	0.777891i	$-0.716289\pi$
$547$	−8.79495e6	−1.25680	−0.628399	+	0.777891i	$0.716289\pi$
$548$	−6.75625e6	−0.961068
$549$	295614.	0.0418595
$550$	0	0
$551$	2.18831e6	0.307065
$552$	−3.96704e6	−0.554139
$553$	−818685.	−0.113842
$554$	−6.38974e6	−0.884522
$555$	0	0
$556$	−4.08725e6	−0.560717
$557$	2.78134e6	0.379853	0.189927	−	0.981798i	$-0.439175\pi$
$557$	2.78134e6	0.379853	0.189927	+	0.981798i	$0.439175\pi$
$558$	−1.26803e6	−0.172402
$559$	−3.85271e6	−0.521479
$560$	0	0
$561$	2.25165e6	0.302061
$562$	3.66829e6	0.489918
$563$	−460410.	−0.0612172	−0.0306086	−	0.999531i	$-0.509745\pi$
$563$	−460410.	−0.0612172	−0.0306086	+	0.999531i	$0.509745\pi$
$564$	549539.	0.0727446
$565$	0	0
$566$	−1.24001e6	−0.162698
$567$	678887.	0.0886829
$568$	792462.	0.103064
$569$	8.74980e6	1.13297	0.566484	−	0.824073i	$-0.308303\pi$
$569$	8.74980e6	1.13297	0.566484	+	0.824073i	$0.308303\pi$
$570$	0	0
$571$	−8.95646e6	−1.14960	−0.574799	−	0.818295i	$-0.694920\pi$
$571$	−8.95646e6	−1.14960	−0.574799	+	0.818295i	$0.694920\pi$
$572$	903480.	0.115459
$573$	3.30351e6	0.420328
$574$	−4.68866e6	−0.593977
$575$	0	0
$576$	1.39311e6	0.174957
$577$	−1.14055e7	−1.42618	−0.713091	−	0.701071i	$-0.752706\pi$
$577$	−1.14055e7	−1.42618	−0.713091	+	0.701071i	$0.752706\pi$
$578$	−9.52931e6	−1.18643
$579$	2.15867e6	0.267603
$580$	0	0
$581$	−2.12840e6	−0.261585
$582$	1.75030e6	0.214193
$583$	−2.60512e6	−0.317436
$584$	5.92072e6	0.718361
$585$	0	0
$586$	−9.55288e6	−1.14919
$587$	1.05278e7	1.26108	0.630539	−	0.776157i	$-0.282834\pi$
$587$	1.05278e7	1.26108	0.630539	+	0.776157i	$0.282834\pi$
$588$	1.14534e6	0.136613
$589$	6.19234e6	0.735472
$590$	0	0
$591$	8.00969e6	0.943294
$592$	410680.	0.0481614
$593$	7.64531e6	0.892809	0.446404	−	0.894831i	$-0.352704\pi$
$593$	7.64531e6	0.892809	0.446404	+	0.894831i	$0.352704\pi$
$594$	−294395.	−0.0342345
$595$	0	0
$596$	−9.07566e6	−1.04656
$597$	22026.1	0.00252931
$598$	−2.98455e6	−0.341292
$599$	−1.28357e7	−1.46168	−0.730838	−	0.682550i	$-0.760871\pi$
$599$	−1.28357e7	−1.46168	−0.730838	+	0.682550i	$0.760871\pi$
$600$	0	0
$601$	−1.49774e7	−1.69142	−0.845710	−	0.533643i	$-0.820823\pi$
$601$	−1.49774e7	−1.69142	−0.845710	+	0.533643i	$0.820823\pi$
$602$	−3.71722e6	−0.418049
$603$	−4.49337e6	−0.503245
$604$	1.96227e6	0.218860
$605$	0	0
$606$	1.69721e6	0.187739
$607$	−273921.	−0.0301754	−0.0150877	−	0.999886i	$-0.504803\pi$
$607$	−273921.	−0.0301754	−0.0150877	+	0.999886i	$0.504803\pi$
$608$	−7.80005e6	−0.855733
$609$	1.54365e6	0.168658
$610$	0	0
$611$	1.04763e6	0.113528
$612$	−3.49381e6	−0.377069
$613$	−3.61183e6	−0.388219	−0.194109	−	0.980980i	$-0.562182\pi$
$613$	−3.61183e6	−0.388219	−0.194109	+	0.980980i	$0.562182\pi$
$614$	8.92027e6	0.954898
$615$	0	0
$616$	2.20886e6	0.234540
$617$	−1.11537e7	−1.17952	−0.589759	−	0.807579i	$-0.700777\pi$
$617$	−1.11537e7	−1.17952	−0.589759	+	0.807579i	$0.700777\pi$
$618$	−2.87122e6	−0.302409
$619$	−8.08693e6	−0.848314	−0.424157	−	0.905589i	$-0.639430\pi$
$619$	−8.08693e6	−0.848314	−0.424157	+	0.905589i	$0.639430\pi$
$620$	0	0
$621$	−1.82136e6	−0.189525
$622$	−6.15382e6	−0.637777
$623$	−1.14978e7	−1.18685
$624$	−253691.	−0.0260822
$625$	0	0
$626$	−6.26582e6	−0.639060
$627$	1.43766e6	0.146045
$628$	4.06787e6	0.411593
$629$	1.07822e7	1.08663
$630$	0	0
$631$	−1.09030e7	−1.09012	−0.545060	−	0.838397i	$-0.683493\pi$
$631$	−1.09030e7	−1.09012	−0.545060	+	0.838397i	$0.683493\pi$
$632$	−1.39587e6	−0.139012
$633$	7.01248e6	0.695605
$634$	6.81665e6	0.673516
$635$	0	0
$636$	4.04227e6	0.396262
$637$	2.18346e6	0.213205
$638$	−669396.	−0.0651075
$639$	363838.	0.0352497
$640$	0	0
$641$	−1.28118e7	−1.23159	−0.615794	−	0.787907i	$-0.711165\pi$
$641$	−1.28118e7	−1.23159	−0.615794	+	0.787907i	$0.711165\pi$
$642$	−6.35724e6	−0.608739
$643$	−4.42120e6	−0.421709	−0.210854	−	0.977517i	$-0.567625\pi$
$643$	−4.42120e6	−0.421709	−0.210854	+	0.977517i	$0.567625\pi$
$644$	5.39308e6	0.512415
$645$	0	0
$646$	−9.11003e6	−0.858892
$647$	−2.82953e6	−0.265738	−0.132869	−	0.991134i	$-0.542419\pi$
$647$	−2.82953e6	−0.265738	−0.132869	+	0.991134i	$0.542419\pi$
$648$	1.15751e6	0.108290
$649$	−549472.	−0.0512076
$650$	0	0
$651$	4.36813e6	0.403964
$652$	−3.21763e6	−0.296426
$653$	−1.51923e7	−1.39425	−0.697127	−	0.716948i	$-0.745539\pi$
$653$	−1.51923e7	−1.39425	−0.697127	+	0.716948i	$0.745539\pi$
$654$	−11136.0	−0.00101809
$655$	0	0
$656$	1.06924e6	0.0970096
$657$	2.71834e6	0.245692
$658$	1.01079e6	0.0910113
$659$	−1.45583e7	−1.30587	−0.652933	−	0.757416i	$-0.726462\pi$
$659$	−1.45583e7	−1.30587	−0.652933	+	0.757416i	$0.726462\pi$
$660$	0	0
$661$	5.60047e6	0.498564	0.249282	−	0.968431i	$-0.419805\pi$
$661$	5.60047e6	0.498564	0.249282	+	0.968431i	$0.419805\pi$
$662$	8.30941e6	0.736928
$663$	−6.66052e6	−0.588470
$664$	−3.62895e6	−0.319419
$665$	0	0
$666$	−1.40973e6	−0.123155
$667$	−4.14141e6	−0.360440
$668$	6.20658e6	0.538159
$669$	1.14340e7	0.987715
$670$	0	0
$671$	441596.	0.0378633
$672$	−5.50222e6	−0.470018
$673$	−2.22862e6	−0.189670	−0.0948351	−	0.995493i	$-0.530232\pi$
$673$	−2.22862e6	−0.189670	−0.0948351	+	0.995493i	$0.530232\pi$
$674$	7.56081e6	0.641089
$675$	0	0
$676$	5.07310e6	0.426979
$677$	−1.47850e7	−1.23980	−0.619898	−	0.784683i	$-0.712826\pi$
$677$	−1.47850e7	−1.23980	−0.619898	+	0.784683i	$0.712826\pi$
$678$	424274.	0.0354464
$679$	−6.02949e6	−0.501887
$680$	0	0
$681$	15803.7	0.00130584
$682$	−1.89421e6	−0.155944
$683$	415268.	0.0340625	0.0170313	−	0.999855i	$-0.494579\pi$
$683$	415268.	0.0340625	0.0170313	+	0.999855i	$0.494579\pi$
$684$	−2.23077e6	−0.182312
$685$	0	0
$686$	7.91078e6	0.641814
$687$	−1.66591e6	−0.134666
$688$	847703.	0.0682767
$689$	7.70610e6	0.618424
$690$	0	0
$691$	8.64233e6	0.688550	0.344275	−	0.938869i	$-0.388125\pi$
$691$	8.64233e6	0.688550	0.344275	+	0.938869i	$0.388125\pi$
$692$	1.03968e6	0.0825342
$693$	1.01414e6	0.0802167
$694$	−2.96187e6	−0.233436
$695$	0	0
$696$	2.63195e6	0.205947
$697$	2.80723e7	2.18875
$698$	−1.12751e7	−0.875956
$699$	1.55086e6	0.120055
$700$	0	0
$701$	−1.30512e7	−1.00313	−0.501564	−	0.865121i	$-0.667242\pi$
$701$	−1.30512e7	−1.00313	−0.501564	+	0.865121i	$0.667242\pi$
$702$	870839.	0.0666953
$703$	6.88434e6	0.525381
$704$	2.08107e6	0.158254
$705$	0	0
$706$	−4.11172e6	−0.310464
$707$	−5.84660e6	−0.439901
$708$	852597.	0.0639235
$709$	−1.56007e7	−1.16555	−0.582773	−	0.812635i	$-0.698032\pi$
$709$	−1.56007e7	−1.16555	−0.582773	+	0.812635i	$0.698032\pi$
$710$	0	0
$711$	−640876.	−0.0475445
$712$	−1.96039e7	−1.44925
$713$	−1.17191e7	−0.863317
$714$	−6.42630e6	−0.471754
$715$	0	0
$716$	1.72189e7	1.25523
$717$	1.42262e7	1.03345
$718$	−3.45887e6	−0.250394
$719$	−6.44874e6	−0.465214	−0.232607	−	0.972571i	$-0.574726\pi$
$719$	−6.44874e6	−0.465214	−0.232607	+	0.972571i	$0.574726\pi$
$720$	0	0
$721$	9.89085e6	0.708591
$722$	2.44723e6	0.174715
$723$	1.29804e7	0.923512
$724$	1.31701e7	0.933779
$725$	0	0
$726$	−439775.	−0.0309663
$727$	8.36468e6	0.586967	0.293483	−	0.955964i	$-0.405185\pi$
$727$	8.36468e6	0.586967	0.293483	+	0.955964i	$0.405185\pi$
$728$	−6.53394e6	−0.456927
$729$	531441.	0.0370370
$730$	0	0
$731$	2.22560e7	1.54047
$732$	−685209.	−0.0472656
$733$	3.08435e6	0.212033	0.106017	−	0.994364i	$-0.466190\pi$
$733$	3.08435e6	0.212033	0.106017	+	0.994364i	$0.466190\pi$
$734$	−1.44046e7	−0.986875
$735$	0	0
$736$	1.47617e7	1.00448
$737$	−6.71232e6	−0.455202
$738$	−3.67034e6	−0.248065
$739$	−2.28100e7	−1.53644	−0.768218	−	0.640189i	$-0.778856\pi$
$739$	−2.28100e7	−1.53644	−0.768218	+	0.640189i	$0.778856\pi$
$740$	0	0
$741$	−4.25269e6	−0.284524
$742$	7.43511e6	0.495767
$743$	1.21470e7	0.807232	0.403616	−	0.914929i	$-0.367753\pi$
$743$	1.21470e7	0.807232	0.403616	+	0.914929i	$0.367753\pi$
$744$	7.44772e6	0.493277
$745$	0	0
$746$	6.79507e6	0.447040
$747$	−1.66614e6	−0.109247
$748$	−5.21915e6	−0.341072
$749$	2.18996e7	1.42637
$750$	0	0
$751$	2.29619e7	1.48562	0.742809	−	0.669503i	$-0.233493\pi$
$751$	2.29619e7	1.48562	0.742809	+	0.669503i	$0.233493\pi$
$752$	−230508.	−0.0148642
$753$	1.06634e7	0.685342
$754$	1.98011e6	0.126842
$755$	0	0
$756$	−1.57361e6	−0.100136
$757$	8.97752e6	0.569399	0.284699	−	0.958617i	$-0.408106\pi$
$757$	8.97752e6	0.569399	0.284699	+	0.958617i	$0.408106\pi$
$758$	1.22797e7	0.776272
$759$	−2.72080e6	−0.171432
$760$	0	0
$761$	1.81766e7	1.13776	0.568880	−	0.822421i	$-0.307377\pi$
$761$	1.81766e7	1.13776	0.568880	+	0.822421i	$0.307377\pi$
$762$	−1.17293e6	−0.0731786
$763$	38361.5	0.00238553
$764$	−7.65726e6	−0.474613
$765$	0	0
$766$	1.12010e7	0.689741
$767$	1.62537e6	0.0997618
$768$	−8.90821e6	−0.544989
$769$	−1.53679e7	−0.937127	−0.468564	−	0.883430i	$-0.655228\pi$
$769$	−1.53679e7	−0.937127	−0.468564	+	0.883430i	$0.655228\pi$
$770$	0	0
$771$	−2.77199e6	−0.167941
$772$	−5.00363e6	−0.302163
$773$	732946.	0.0441188	0.0220594	−	0.999757i	$-0.492978\pi$
$773$	732946.	0.0441188	0.0220594	+	0.999757i	$0.492978\pi$
$774$	−2.90989e6	−0.174592
$775$	0	0
$776$	−1.02804e7	−0.612850
$777$	4.85628e6	0.288570
$778$	−1.98406e7	−1.17519
$779$	1.79239e7	1.05825
$780$	0	0
$781$	543511.	0.0318846
$782$	1.72409e7	1.00819
$783$	1.20839e6	0.0704373
$784$	−480422.	−0.0279147
$785$	0	0
$786$	4.59083e6	0.265054
$787$	2.25077e7	1.29537	0.647685	−	0.761908i	$-0.275737\pi$
$787$	2.25077e7	1.29537	0.647685	+	0.761908i	$0.275737\pi$
$788$	−1.85658e7	−1.06512
$789$	1.73071e7	0.989767
$790$	0	0
$791$	−1.46155e6	−0.0830562
$792$	1.72912e6	0.0979519
$793$	−1.30627e6	−0.0737648
$794$	−1.35977e7	−0.765444
$795$	0	0
$796$	−51054.7	−0.00285597
$797$	−5.07705e6	−0.283117	−0.141558	−	0.989930i	$-0.545211\pi$
$797$	−5.07705e6	−0.283117	−0.141558	+	0.989930i	$0.545211\pi$
$798$	−4.10314e6	−0.228092
$799$	−6.05185e6	−0.335368
$800$	0	0
$801$	−9.00063e6	−0.495669
$802$	−5.89149e6	−0.323437
$803$	4.06073e6	0.222237
$804$	1.04153e7	0.568238
$805$	0	0
$806$	5.60320e6	0.303807
$807$	−1.03079e7	−0.557166
$808$	−9.96853e6	−0.537159
$809$	1.15468e7	0.620282	0.310141	−	0.950691i	$-0.399624\pi$
$809$	1.15468e7	0.620282	0.310141	+	0.950691i	$0.399624\pi$
$810$	0	0
$811$	−2.62543e7	−1.40168	−0.700838	−	0.713320i	$-0.747190\pi$
$811$	−2.62543e7	−1.40168	−0.700838	+	0.713320i	$0.747190\pi$
$812$	−3.57806e6	−0.190440
$813$	−6.63753e6	−0.352192
$814$	−2.10590e6	−0.111398
$815$	0	0
$816$	1.46550e6	0.0770479
$817$	1.42103e7	0.744813
$818$	1.01829e7	0.532096
$819$	−2.99989e6	−0.156277
$820$	0	0
$821$	−3.96273e6	−0.205180	−0.102590	−	0.994724i	$-0.532713\pi$
$821$	−3.96273e6	−0.205180	−0.102590	+	0.994724i	$0.532713\pi$
$822$	−9.72803e6	−0.502164
$823$	−9.86765e6	−0.507825	−0.253913	−	0.967227i	$-0.581718\pi$
$823$	−9.86765e6	−0.507825	−0.253913	+	0.967227i	$0.581718\pi$
$824$	1.68640e7	0.865254
$825$	0	0
$826$	1.56821e6	0.0799752
$827$	5.90399e6	0.300180	0.150090	−	0.988672i	$-0.452044\pi$
$827$	5.90399e6	0.300180	0.150090	+	0.988672i	$0.452044\pi$
$828$	4.22177e6	0.214002
$829$	2.38090e7	1.20325	0.601624	−	0.798779i	$-0.294520\pi$
$829$	2.38090e7	1.20325	0.601624	+	0.798779i	$0.294520\pi$
$830$	0	0
$831$	1.72309e7	0.865576
$832$	−6.15593e6	−0.308309
$833$	−1.26132e7	−0.629816
$834$	−5.88505e6	−0.292978
$835$	0	0
$836$	−3.33239e6	−0.164907
$837$	3.41943e6	0.168709
$838$	2.20695e7	1.08563
$839$	1.88904e7	0.926479	0.463240	−	0.886233i	$-0.346687\pi$
$839$	1.88904e7	0.926479	0.463240	+	0.886233i	$0.346687\pi$
$840$	0	0
$841$	−1.77635e7	−0.866042
$842$	−7.65291e6	−0.372003
$843$	−9.89210e6	−0.479424
$844$	−1.62544e7	−0.785442
$845$	0	0
$846$	791257.	0.0380095
$847$	1.51495e6	0.0725587
$848$	−1.69556e6	−0.0809697
$849$	3.34387e6	0.159214
$850$	0	0
$851$	−1.30287e7	−0.616706
$852$	−843347.	−0.0398022
$853$	2.79626e7	1.31585	0.657924	−	0.753085i	$-0.271435\pi$
$853$	2.79626e7	1.31585	0.657924	+	0.753085i	$0.271435\pi$
$854$	−1.26033e6	−0.0591344
$855$	0	0
$856$	3.73391e7	1.74172
$857$	2.82523e6	0.131402	0.0657009	−	0.997839i	$-0.479072\pi$
$857$	2.82523e6	0.131402	0.0657009	+	0.997839i	$0.479072\pi$
$858$	1.30088e6	0.0603281
$859$	5.68353e6	0.262806	0.131403	−	0.991329i	$-0.458052\pi$
$859$	5.68353e6	0.262806	0.131403	+	0.991329i	$0.458052\pi$
$860$	0	0
$861$	1.26437e7	0.581254
$862$	−1.84816e7	−0.847170
$863$	−2.23836e6	−0.102306	−0.0511532	−	0.998691i	$-0.516290\pi$
$863$	−2.23836e6	−0.102306	−0.0511532	+	0.998691i	$0.516290\pi$
$864$	−4.30721e6	−0.196296
$865$	0	0
$866$	8.53116e6	0.386557
$867$	2.56972e7	1.16102
$868$	−1.01250e7	−0.456136
$869$	−957359.	−0.0430056
$870$	0	0
$871$	1.98555e7	0.886818
$872$	65406.9	0.00291295
$873$	−4.71996e6	−0.209605
$874$	1.10082e7	0.487457
$875$	0	0
$876$	−6.30090e6	−0.277423
$877$	6.04256e6	0.265291	0.132645	−	0.991164i	$-0.457653\pi$
$877$	6.04256e6	0.265291	0.132645	+	0.991164i	$0.457653\pi$
$878$	2.32355e6	0.101722
$879$	2.57608e7	1.12457
$880$	0	0
$881$	3.77877e7	1.64025	0.820126	−	0.572183i	$-0.193903\pi$
$881$	3.77877e7	1.64025	0.820126	+	0.572183i	$0.193903\pi$
$882$	1.64913e6	0.0713812
$883$	4.27448e7	1.84494	0.922469	−	0.386070i	$-0.126168\pi$
$883$	4.27448e7	1.84494	0.922469	+	0.386070i	$0.126168\pi$
$884$	1.54386e7	0.664471
$885$	0	0
$886$	−2.27946e7	−0.975544
$887$	−971018.	−0.0414398	−0.0207199	−	0.999785i	$-0.506596\pi$
$887$	−971018.	−0.0414398	−0.0207199	+	0.999785i	$0.506596\pi$
$888$	8.28002e6	0.352370
$889$	4.04053e6	0.171468
$890$	0	0
$891$	793881.	0.0335013
$892$	−2.65030e7	−1.11528
$893$	−3.86406e6	−0.162149
$894$	−1.30677e7	−0.546832
$895$	0	0
$896$	1.36240e7	0.566937
$897$	8.04828e6	0.333981
$898$	2.23043e7	0.922990
$899$	7.77509e6	0.320853
$900$	0	0
$901$	−4.45159e7	−1.82685
$902$	−5.48286e6	−0.224383
$903$	1.00241e7	0.409095
$904$	−2.49196e6	−0.101419
$905$	0	0
$906$	2.82538e6	0.114355
$907$	1.74008e7	0.702348	0.351174	−	0.936310i	$-0.385783\pi$
$907$	1.74008e7	0.702348	0.351174	+	0.936310i	$0.385783\pi$
$908$	−36631.7	−0.00147449
$909$	−4.57679e6	−0.183718
$910$	0	0
$911$	−1.56534e7	−0.624903	−0.312451	−	0.949934i	$-0.601150\pi$
$911$	−1.56534e7	−0.624903	−0.312451	+	0.949934i	$0.601150\pi$
$912$	935711.	0.0372524
$913$	−2.48892e6	−0.0988175
$914$	−2.05868e6	−0.0815123
$915$	0	0
$916$	3.86144e6	0.152058
$917$	−1.58146e7	−0.621062
$918$	−5.03059e6	−0.197021
$919$	−3.32140e7	−1.29728	−0.648639	−	0.761097i	$-0.724661\pi$
$919$	−3.32140e7	−1.29728	−0.648639	+	0.761097i	$0.724661\pi$
$920$	0	0
$921$	−2.40549e7	−0.934445
$922$	−1.18634e7	−0.459601
$923$	−1.60774e6	−0.0621171
$924$	−2.35069e6	−0.0905766
$925$	0	0
$926$	1.08117e7	0.414348
$927$	7.74268e6	0.295932
$928$	−9.79372e6	−0.373317
$929$	−7.05863e6	−0.268338	−0.134169	−	0.990958i	$-0.542836\pi$
$929$	−7.05863e6	−0.268338	−0.134169	+	0.990958i	$0.542836\pi$
$930$	0	0
$931$	−8.05345e6	−0.304514
$932$	−3.59477e6	−0.135560
$933$	1.65947e7	0.624116
$934$	−2.40517e7	−0.902150
$935$	0	0
$936$	−5.11485e6	−0.190829
$937$	−4.85427e6	−0.180624	−0.0903120	−	0.995914i	$-0.528786\pi$
$937$	−4.85427e6	−0.180624	−0.0903120	+	0.995914i	$0.528786\pi$
$938$	1.91572e7	0.710928
$939$	1.68967e7	0.625372
$940$	0	0
$941$	−3.79442e7	−1.39692	−0.698460	−	0.715649i	$-0.746131\pi$
$941$	−3.79442e7	−1.39692	−0.698460	+	0.715649i	$0.746131\pi$
$942$	5.85715e6	0.215060
$943$	−3.39213e7	−1.24220
$944$	−357627.	−0.0130617
$945$	0	0
$946$	−4.34687e6	−0.157924
$947$	−1.42196e7	−0.515244	−0.257622	−	0.966246i	$-0.582939\pi$
$947$	−1.42196e7	−0.515244	−0.257622	+	0.966246i	$0.582939\pi$
$948$	1.48550e6	0.0536849
$949$	−1.20119e7	−0.432958
$950$	0	0
$951$	−1.83821e7	−0.659090
$952$	3.77447e7	1.34978
$953$	−1.27436e7	−0.454529	−0.227264	−	0.973833i	$-0.572978\pi$
$953$	−1.27436e7	−0.454529	−0.227264	+	0.973833i	$0.572978\pi$
$954$	5.82030e6	0.207049
$955$	0	0
$956$	−3.29751e7	−1.16692
$957$	1.80513e6	0.0637130
$958$	−2.57846e6	−0.0907709
$959$	3.35114e7	1.17665
$960$	0	0
$961$	−6.62771e6	−0.231502
$962$	6.22937e6	0.217023
$963$	1.71433e7	0.595700
$964$	−3.00875e7	−1.04278
$965$	0	0
$966$	7.76526e6	0.267740
$967$	4.44841e7	1.52981	0.764906	−	0.644142i	$-0.222785\pi$
$967$	4.44841e7	1.52981	0.764906	+	0.644142i	$0.222785\pi$
$968$	2.58301e6	0.0886008
$969$	2.45666e7	0.840495
$970$	0	0
$971$	−1.58495e7	−0.539469	−0.269734	−	0.962935i	$-0.586936\pi$
$971$	−1.58495e7	−0.539469	−0.269734	+	0.962935i	$0.586936\pi$
$972$	−1.23184e6	−0.0418204
$973$	2.02730e7	0.686492
$974$	1.03086e7	0.348177
$975$	0	0
$976$	287415.	0.00965795
$977$	−3.72698e7	−1.24917	−0.624584	−	0.780958i	$-0.714731\pi$
$977$	−3.72698e7	−1.24917	−0.624584	+	0.780958i	$0.714731\pi$
$978$	−4.63292e6	−0.154885
$979$	−1.34454e7	−0.448350
$980$	0	0
$981$	30029.9	0.000996279 0
$982$	−1.60882e7	−0.532390
$983$	1.71461e7	0.565955	0.282978	−	0.959127i	$-0.408678\pi$
$983$	1.71461e7	0.565955	0.282978	+	0.959127i	$0.408678\pi$
$984$	2.15577e7	0.709764
$985$	0	0
$986$	−1.14385e7	−0.374695
$987$	−2.72574e6	−0.0890619
$988$	9.85740e6	0.321270
$989$	−2.68932e7	−0.874281
$990$	0	0
$991$	3.65029e7	1.18071	0.590354	−	0.807144i	$-0.298988\pi$
$991$	3.65029e7	1.18071	0.590354	+	0.807144i	$0.298988\pi$
$992$	−2.77137e7	−0.894159
$993$	−2.24076e7	−0.721143
$994$	−1.55120e6	−0.0497969
$995$	0	0
$996$	3.86197e6	0.123356
$997$	5.88370e6	0.187462	0.0937309	−	0.995598i	$-0.470121\pi$
$997$	5.88370e6	0.187462	0.0937309	+	0.995598i	$0.470121\pi$
$998$	3.12975e7	0.994681
$999$	3.80155e6	0.120517

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.y.1.5		13
5.2	odd	4		165.6.c.b.34.11	✓	26
5.3	odd	4		165.6.c.b.34.16	yes	26
5.4	even	2		825.6.a.v.1.9		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.c.b.34.11	✓	26	5.2	odd	4
165.6.c.b.34.16	yes	26	5.3	odd	4
825.6.a.v.1.9		13	5.4	even	2
825.6.a.y.1.5		13	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)

\(f(q)\)	\(=\)	\(q-3.33747 q^{2} +9.00000 q^{3} -20.8613 q^{4} -30.0373 q^{6} +103.473 q^{7} +176.423 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-3.33747 q^{2} +9.00000 q^{3} -20.8613 q^{4} -30.0373 q^{6} +103.473 q^{7} +176.423 q^{8} +81.0000 q^{9} +121.000 q^{11} -187.751 q^{12} -357.925 q^{13} -345.339 q^{14} +78.7535 q^{16} +2067.63 q^{17} -270.335 q^{18} +1320.17 q^{19} +931.258 q^{21} -403.834 q^{22} -2498.44 q^{23} +1587.81 q^{24} +1194.57 q^{26} +729.000 q^{27} -2158.58 q^{28} +1657.60 q^{29} +4690.57 q^{31} -5908.38 q^{32} +1089.00 q^{33} -6900.67 q^{34} -1689.76 q^{36} +5214.75 q^{37} -4406.02 q^{38} -3221.33 q^{39} +13577.0 q^{41} -3108.05 q^{42} +10764.0 q^{43} -2524.21 q^{44} +8338.46 q^{46} -2926.95 q^{47} +708.782 q^{48} -6100.32 q^{49} +18608.7 q^{51} +7466.78 q^{52} -21529.9 q^{53} -2433.02 q^{54} +18255.0 q^{56} +11881.5 q^{57} -5532.19 q^{58} -4541.09 q^{59} +3649.55 q^{61} -15654.6 q^{62} +8381.32 q^{63} +17198.9 q^{64} -3634.51 q^{66} -55473.7 q^{67} -43133.4 q^{68} -22485.9 q^{69} +4491.83 q^{71} +14290.3 q^{72} +33559.8 q^{73} -17404.1 q^{74} -27540.4 q^{76} +12520.2 q^{77} +10751.1 q^{78} -7912.05 q^{79} +6561.00 q^{81} -45312.9 q^{82} -20569.6 q^{83} -19427.2 q^{84} -35924.5 q^{86} +14918.4 q^{87} +21347.2 q^{88} -111119. q^{89} -37035.6 q^{91} +52120.6 q^{92} +42215.1 q^{93} +9768.61 q^{94} -53175.4 q^{96} -58271.1 q^{97} +20359.7 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 13 q^{2} + 117 q^{3} + 209 q^{4} + 117 q^{6} + 304 q^{7} + 399 q^{8} + 1053 q^{9}+O(q^{10}) \) 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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