LMFDB - Embedded newform 825.6.a.v.1.4

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			1.4
Root			$-4.72893$ 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)

$f(q)$	$=$	$q-5.72893 q^{2} -9.00000 q^{3} +0.820586 q^{4} +51.5603 q^{6} +158.171 q^{7} +178.625 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-5.72893 q^{2} -9.00000 q^{3} +0.820586 q^{4} +51.5603 q^{6} +158.171 q^{7} +178.625 q^{8} +81.0000 q^{9} +121.000 q^{11} -7.38528 q^{12} +78.2418 q^{13} -906.152 q^{14} -1049.59 q^{16} -88.2340 q^{17} -464.043 q^{18} +1865.42 q^{19} -1423.54 q^{21} -693.200 q^{22} +503.122 q^{23} -1607.62 q^{24} -448.241 q^{26} -729.000 q^{27} +129.793 q^{28} +1050.22 q^{29} -9057.51 q^{31} +297.011 q^{32} -1089.00 q^{33} +505.486 q^{34} +66.4675 q^{36} -3899.39 q^{37} -10686.9 q^{38} -704.176 q^{39} -9454.73 q^{41} +8155.37 q^{42} -8918.80 q^{43} +99.2910 q^{44} -2882.35 q^{46} -9431.40 q^{47} +9446.27 q^{48} +8211.19 q^{49} +794.106 q^{51} +64.2042 q^{52} -25507.5 q^{53} +4176.39 q^{54} +28253.3 q^{56} -16788.8 q^{57} -6016.65 q^{58} +9907.66 q^{59} -15614.3 q^{61} +51889.8 q^{62} +12811.9 q^{63} +31885.2 q^{64} +6238.80 q^{66} +16879.9 q^{67} -72.4036 q^{68} -4528.10 q^{69} +71467.7 q^{71} +14468.6 q^{72} -65733.0 q^{73} +22339.3 q^{74} +1530.74 q^{76} +19138.7 q^{77} +4034.17 q^{78} +104481. q^{79} +6561.00 q^{81} +54165.4 q^{82} +15457.9 q^{83} -1168.14 q^{84} +51095.2 q^{86} -9452.01 q^{87} +21613.6 q^{88} +2774.62 q^{89} +12375.6 q^{91} +412.855 q^{92} +81517.6 q^{93} +54031.8 q^{94} -2673.10 q^{96} -125401. q^{97} -47041.3 q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−5.72893	−1.01274	−0.506370	−	0.862316i	$-0.669013\pi$
$2$	−5.72893	−1.01274	−0.506370	+	0.862316i	$0.669013\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0.820586	0.0256433
$5$	0	0
$5$	0	0
$6$	51.5603	0.584706
$7$	158.171	1.22006	0.610032	−	0.792377i	$-0.291156\pi$
$7$	158.171	1.22006	0.610032	+	0.792377i	$0.291156\pi$
$8$	178.625	0.986770
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	−7.38528	−0.0148052
$13$	78.2418	0.128405	0.0642023	−	0.997937i	$-0.479550\pi$
$13$	78.2418	0.128405	0.0642023	+	0.997937i	$0.479550\pi$
$14$	−906.152	−1.23561
$15$	0	0
$16$	−1049.59	−1.02499
$17$	−88.2340	−0.0740480	−0.0370240	−	0.999314i	$-0.511788\pi$
$17$	−88.2340	−0.0740480	−0.0370240	+	0.999314i	$0.511788\pi$
$18$	−464.043	−0.337580
$19$	1865.42	1.18548	0.592738	−	0.805395i	$-0.298047\pi$
$19$	1865.42	1.18548	0.592738	+	0.805395i	$0.298047\pi$
$20$	0	0
$21$	−1423.54	−0.704405
$22$	−693.200	−0.305353
$23$	503.122	0.198314	0.0991571	−	0.995072i	$-0.468385\pi$
$23$	503.122	0.198314	0.0991571	+	0.995072i	$0.468385\pi$
$24$	−1607.62	−0.569712
$25$	0	0
$26$	−448.241	−0.130040
$27$	−729.000	−0.192450
$28$	129.793	0.0312865
$29$	1050.22	0.231892	0.115946	−	0.993255i	$-0.463010\pi$
$29$	1050.22	0.231892	0.115946	+	0.993255i	$0.463010\pi$
$30$	0	0
$31$	−9057.51	−1.69280	−0.846398	−	0.532550i	$-0.821234\pi$
$31$	−9057.51	−1.69280	−0.846398	+	0.532550i	$0.821234\pi$
$32$	297.011	0.0512741
$33$	−1089.00	−0.174078
$34$	505.486	0.0749914
$35$	0	0
$36$	66.4675	0.00854778
$37$	−3899.39	−0.468266	−0.234133	−	0.972205i	$-0.575225\pi$
$37$	−3899.39	−0.468266	−0.234133	+	0.972205i	$0.575225\pi$
$38$	−10686.9	−1.20058
$39$	−704.176	−0.0741344
$40$	0	0
$41$	−9454.73	−0.878394	−0.439197	−	0.898391i	$-0.644737\pi$
$41$	−9454.73	−0.878394	−0.439197	+	0.898391i	$0.644737\pi$
$42$	8155.37	0.713379
$43$	−8918.80	−0.735589	−0.367795	−	0.929907i	$-0.619887\pi$
$43$	−8918.80	−0.735589	−0.367795	+	0.929907i	$0.619887\pi$
$44$	99.2910	0.00773175
$45$	0	0
$46$	−2882.35	−0.200841
$47$	−9431.40	−0.622776	−0.311388	−	0.950283i	$-0.600794\pi$
$47$	−9431.40	−0.622776	−0.311388	+	0.950283i	$0.600794\pi$
$48$	9446.27	0.591776
$49$	8211.19	0.488558
$50$	0	0
$51$	794.106	0.0427516
$52$	64.2042	0.00329272
$53$	−25507.5	−1.24732	−0.623661	−	0.781695i	$-0.714355\pi$
$53$	−25507.5	−1.24732	−0.623661	+	0.781695i	$0.714355\pi$
$54$	4176.39	0.194902
$55$	0	0
$56$	28253.3	1.20392
$57$	−16788.8	−0.684435
$58$	−6016.65	−0.234847
$59$	9907.66	0.370545	0.185273	−	0.982687i	$-0.440683\pi$
$59$	9907.66	0.370545	0.185273	+	0.982687i	$0.440683\pi$
$60$	0	0
$61$	−15614.3	−0.537278	−0.268639	−	0.963241i	$-0.586574\pi$
$61$	−15614.3	−0.537278	−0.268639	+	0.963241i	$0.586574\pi$
$62$	51889.8	1.71436
$63$	12811.9	0.406688
$64$	31885.2	0.973058
$65$	0	0
$66$	6238.80	0.176295
$67$	16879.9	0.459392	0.229696	−	0.973262i	$-0.426227\pi$
$67$	16879.9	0.459392	0.229696	+	0.973262i	$0.426227\pi$
$68$	−72.4036	−0.00189884
$69$	−4528.10	−0.114497
$70$	0	0
$71$	71467.7	1.68253	0.841267	−	0.540620i	$-0.181810\pi$
$71$	71467.7	1.68253	0.841267	+	0.540620i	$0.181810\pi$
$72$	14468.6	0.328923
$73$	−65733.0	−1.44370	−0.721849	−	0.692050i	$-0.756708\pi$
$73$	−65733.0	−1.44370	−0.721849	+	0.692050i	$0.756708\pi$
$74$	22339.3	0.474232
$75$	0	0
$76$	1530.74	0.0303995
$77$	19138.7	0.367863
$78$	4034.17	0.0750789
$79$	104481.	1.88352	0.941762	−	0.336281i	$-0.109169\pi$
$79$	104481.	1.88352	0.941762	+	0.336281i	$0.109169\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	54165.4	0.889585
$83$	15457.9	0.246294	0.123147	−	0.992388i	$-0.460701\pi$
$83$	15457.9	0.246294	0.123147	+	0.992388i	$0.460701\pi$
$84$	−1168.14	−0.0180633
$85$	0	0
$86$	51095.2	0.744961
$87$	−9452.01	−0.133883
$88$	21613.6	0.297522
$89$	2774.62	0.0371303	0.0185651	−	0.999828i	$-0.494090\pi$
$89$	2774.62	0.0371303	0.0185651	+	0.999828i	$0.494090\pi$
$90$	0	0
$91$	12375.6	0.156662
$92$	412.855	0.00508544
$93$	81517.6	0.977336
$94$	54031.8	0.630710
$95$	0	0
$96$	−2673.10	−0.0296031
$97$	−125401.	−1.35323	−0.676614	−	0.736338i	$-0.736553\pi$
$97$	−125401.	−1.35323	−0.676614	+	0.736338i	$0.736553\pi$
$98$	−47041.3	−0.494782
$99$	9801.00	0.100504
$100$	0	0
$101$	50984.8	0.497322	0.248661	−	0.968591i	$-0.420010\pi$
$101$	50984.8	0.497322	0.248661	+	0.968591i	$0.420010\pi$
$102$	−4549.37	−0.0432963
$103$	51464.8	0.477989	0.238994	−	0.971021i	$-0.423182\pi$
$103$	51464.8	0.477989	0.238994	+	0.971021i	$0.423182\pi$
$104$	13975.9	0.126706
$105$	0	0
$106$	146131.	1.26321
$107$	−139655.	−1.17923	−0.589613	−	0.807686i	$-0.700720\pi$
$107$	−139655.	−1.17923	−0.589613	+	0.807686i	$0.700720\pi$
$108$	−598.208	−0.00493506
$109$	144034.	1.16118	0.580591	−	0.814195i	$-0.302822\pi$
$109$	144034.	1.16118	0.580591	+	0.814195i	$0.302822\pi$
$110$	0	0
$111$	35094.5	0.270353
$112$	−166014.	−1.25055
$113$	−62064.4	−0.457242	−0.228621	−	0.973515i	$-0.573422\pi$
$113$	−62064.4	−0.457242	−0.228621	+	0.973515i	$0.573422\pi$
$114$	96181.7	0.693155
$115$	0	0
$116$	861.799	0.00594649
$117$	6337.59	0.0428015
$118$	−56760.3	−0.375266
$119$	−13956.1	−0.0903434
$120$	0	0
$121$	14641.0	0.0909091
$122$	89453.4	0.544124
$123$	85092.5	0.507141
$124$	−7432.47	−0.0434089
$125$	0	0
$126$	−73398.3	−0.411870
$127$	−285166.	−1.56888	−0.784438	−	0.620207i	$-0.787049\pi$
$127$	−285166.	−1.56888	−0.784438	+	0.620207i	$0.787049\pi$
$128$	−192172.	−1.03673
$129$	80269.2	0.424693
$130$	0	0
$131$	−198363.	−1.00991	−0.504954	−	0.863146i	$-0.668491\pi$
$131$	−198363.	−1.00991	−0.504954	+	0.863146i	$0.668491\pi$
$132$	−893.619	−0.00446393
$133$	295056.	1.44636
$134$	−96703.8	−0.465245
$135$	0	0
$136$	−15760.7	−0.0730684
$137$	166328.	0.757120	0.378560	−	0.925577i	$-0.376419\pi$
$137$	166328.	0.757120	0.378560	+	0.925577i	$0.376419\pi$
$138$	25941.1	0.115955
$139$	−301037.	−1.32155	−0.660774	−	0.750585i	$-0.729772\pi$
$139$	−301037.	−1.32155	−0.660774	+	0.750585i	$0.729772\pi$
$140$	0	0
$141$	84882.6	0.359560
$142$	−409433.	−1.70397
$143$	9467.26	0.0387154
$144$	−85016.4	−0.341662
$145$	0	0
$146$	376580.	1.46209
$147$	−73900.7	−0.282069
$148$	−3199.79	−0.0120079
$149$	107457.	0.396525	0.198263	−	0.980149i	$-0.436470\pi$
$149$	107457.	0.396525	0.198263	+	0.980149i	$0.436470\pi$
$150$	0	0
$151$	24082.4	0.0859522	0.0429761	−	0.999076i	$-0.486316\pi$
$151$	24082.4	0.0859522	0.0429761	+	0.999076i	$0.486316\pi$
$152$	333210.	1.16979
$153$	−7146.95	−0.0246827
$154$	−109644.	−0.372550
$155$	0	0
$156$	−577.837	−0.00190105
$157$	479707.	1.55320	0.776600	−	0.629994i	$-0.216943\pi$
$157$	479707.	1.55320	0.776600	+	0.629994i	$0.216943\pi$
$158$	−598566.	−1.90752
$159$	229568.	0.720142
$160$	0	0
$161$	79579.5	0.241956
$162$	−37587.5	−0.112527
$163$	−437032.	−1.28838	−0.644190	−	0.764866i	$-0.722805\pi$
$163$	−437032.	−1.28838	−0.644190	+	0.764866i	$0.722805\pi$
$164$	−7758.42	−0.0225249
$165$	0	0
$166$	−88557.0	−0.249432
$167$	592592.	1.64424	0.822119	−	0.569316i	$-0.192792\pi$
$167$	592592.	1.64424	0.822119	+	0.569316i	$0.192792\pi$
$168$	−254280.	−0.695086
$169$	−365171.	−0.983512
$170$	0	0
$171$	151099.	0.395159
$172$	−7318.65	−0.0188630
$173$	−193454.	−0.491432	−0.245716	−	0.969342i	$-0.579023\pi$
$173$	−193454.	−0.491432	−0.245716	+	0.969342i	$0.579023\pi$
$174$	54149.9	0.135589
$175$	0	0
$176$	−127000.	−0.309045
$177$	−89169.0	−0.213934
$178$	−15895.6	−0.0376033
$179$	−373735.	−0.871828	−0.435914	−	0.899988i	$-0.643575\pi$
$179$	−373735.	−0.871828	−0.435914	+	0.899988i	$0.643575\pi$
$180$	0	0
$181$	−234887.	−0.532921	−0.266461	−	0.963846i	$-0.585854\pi$
$181$	−234887.	−0.532921	−0.266461	+	0.963846i	$0.585854\pi$
$182$	−70899.0	−0.158658
$183$	140529.	0.310198
$184$	89869.9	0.195691
$185$	0	0
$186$	−467008.	−0.989788
$187$	−10676.3	−0.0223263
$188$	−7739.28	−0.0159700
$189$	−115307.	−0.234802
$190$	0	0
$191$	261514.	0.518695	0.259348	−	0.965784i	$-0.416492\pi$
$191$	261514.	0.518695	0.259348	+	0.965784i	$0.416492\pi$
$192$	−286967.	−0.561796
$193$	−688977.	−1.33141	−0.665705	−	0.746215i	$-0.731869\pi$
$193$	−688977.	−1.33141	−0.665705	+	0.746215i	$0.731869\pi$
$194$	718412.	1.37047
$195$	0	0
$196$	6737.99	0.0125282
$197$	−625063.	−1.14752	−0.573758	−	0.819025i	$-0.694515\pi$
$197$	−625063.	−1.14752	−0.573758	+	0.819025i	$0.694515\pi$
$198$	−56149.2	−0.101784
$199$	1.02019e6	1.82619	0.913097	−	0.407742i	$-0.133684\pi$
$199$	1.02019e6	1.82619	0.913097	+	0.407742i	$0.133684\pi$
$200$	0	0
$201$	−151919.	−0.265230
$202$	−292088.	−0.503658
$203$	166115.	0.282924
$204$	651.632	0.00109629
$205$	0	0
$206$	−294838.	−0.484079
$207$	40752.9	0.0661047
$208$	−82121.4	−0.131613
$209$	225716.	0.357434
$210$	0	0
$211$	−544322.	−0.841685	−0.420843	−	0.907134i	$-0.638266\pi$
$211$	−544322.	−0.841685	−0.420843	+	0.907134i	$0.638266\pi$
$212$	−20931.1	−0.0319855
$213$	−643209.	−0.971412
$214$	800073.	1.19425
$215$	0	0
$216$	−130217.	−0.189904
$217$	−1.43264e6	−2.06532
$218$	−825163.	−1.17598
$219$	591597.	0.833520
$220$	0	0
$221$	−6903.58	−0.00950810
$222$	−201054.	−0.273798
$223$	−852497.	−1.14797	−0.573985	−	0.818866i	$-0.694603\pi$
$223$	−852497.	−1.14797	−0.573985	+	0.818866i	$0.694603\pi$
$224$	46978.7	0.0625577
$225$	0	0
$226$	355562.	0.463068
$227$	909729.	1.17178	0.585891	−	0.810390i	$-0.300744\pi$
$227$	909729.	1.17178	0.585891	+	0.810390i	$0.300744\pi$
$228$	−13776.6	−0.0175512
$229$	−561762.	−0.707886	−0.353943	−	0.935267i	$-0.615159\pi$
$229$	−561762.	−0.707886	−0.353943	+	0.935267i	$0.615159\pi$
$230$	0	0
$231$	−172249.	−0.212386
$232$	187596.	0.228825
$233$	604654.	0.729654	0.364827	−	0.931075i	$-0.381128\pi$
$233$	604654.	0.729654	0.364827	+	0.931075i	$0.381128\pi$
$234$	−36307.6	−0.0433468
$235$	0	0
$236$	8130.10	0.00950201
$237$	−940332.	−1.08745
$238$	79953.4	0.0914944
$239$	−1.13697e6	−1.28752	−0.643760	−	0.765227i	$-0.722627\pi$
$239$	−1.13697e6	−1.28752	−0.643760	+	0.765227i	$0.722627\pi$
$240$	0	0
$241$	261399.	0.289909	0.144954	−	0.989438i	$-0.453696\pi$
$241$	261399.	0.289909	0.144954	+	0.989438i	$0.453696\pi$
$242$	−83877.2	−0.0920673
$243$	−59049.0	−0.0641500
$244$	−12812.9	−0.0137776
$245$	0	0
$246$	−487489.	−0.513602
$247$	145954.	0.152220
$248$	−1.61789e6	−1.67040
$249$	−139121.	−0.142198
$250$	0	0
$251$	565375.	0.566438	0.283219	−	0.959055i	$-0.408598\pi$
$251$	565375.	0.566438	0.283219	+	0.959055i	$0.408598\pi$
$252$	10513.3	0.0104288
$253$	60877.8	0.0597940
$254$	1.63370e6	1.58886
$255$	0	0
$256$	80614.3	0.0768798
$257$	1.51946e6	1.43501	0.717507	−	0.696551i	$-0.245283\pi$
$257$	1.51946e6	1.43501	0.717507	+	0.696551i	$0.245283\pi$
$258$	−459856.	−0.430103
$259$	−616772.	−0.571315
$260$	0	0
$261$	85068.1	0.0772975
$262$	1.13641e6	1.02278
$263$	730370.	0.651109	0.325554	−	0.945523i	$-0.394449\pi$
$263$	730370.	0.651109	0.325554	+	0.945523i	$0.394449\pi$
$264$	−194522.	−0.171775
$265$	0	0
$266$	−1.69035e6	−1.46478
$267$	−24971.6	−0.0214372
$268$	13851.4	0.0117803
$269$	−475262.	−0.400453	−0.200227	−	0.979750i	$-0.564168\pi$
$269$	−475262.	−0.400453	−0.200227	+	0.979750i	$0.564168\pi$
$270$	0	0
$271$	−961313.	−0.795136	−0.397568	−	0.917573i	$-0.630146\pi$
$271$	−961313.	−0.795136	−0.397568	+	0.917573i	$0.630146\pi$
$272$	92609.1	0.0758982
$273$	−111381.	−0.0904488
$274$	−952883.	−0.766766
$275$	0	0
$276$	−3715.70	−0.00293608
$277$	−428260.	−0.335358	−0.167679	−	0.985842i	$-0.553627\pi$
$277$	−428260.	−0.335358	−0.167679	+	0.985842i	$0.553627\pi$
$278$	1.72462e6	1.33839
$279$	−733659.	−0.564265
$280$	0	0
$281$	−212752.	−0.160734	−0.0803670	−	0.996765i	$-0.525609\pi$
$281$	−212752.	−0.160734	−0.0803670	+	0.996765i	$0.525609\pi$
$282$	−486286.	−0.364141
$283$	563864.	0.418512	0.209256	−	0.977861i	$-0.432896\pi$
$283$	563864.	0.418512	0.209256	+	0.977861i	$0.432896\pi$
$284$	58645.4	0.0431458
$285$	0	0
$286$	−54237.2	−0.0392087
$287$	−1.49547e6	−1.07170
$288$	24057.9	0.0170914
$289$	−1.41207e6	−0.994517
$290$	0	0
$291$	1.12861e6	0.781287
$292$	−53939.6	−0.0370212
$293$	−1.80961e6	−1.23145	−0.615723	−	0.787963i	$-0.711136\pi$
$293$	−1.80961e6	−1.23145	−0.615723	+	0.787963i	$0.711136\pi$
$294$	423371.	0.285663
$295$	0	0
$296$	−696527.	−0.462071
$297$	−88209.0	−0.0580259
$298$	−615616.	−0.401577
$299$	39365.2	0.0254644
$300$	0	0
$301$	−1.41070e6	−0.897466
$302$	−137966.	−0.0870473
$303$	−458863.	−0.287129
$304$	−1.95792e6	−1.21510
$305$	0	0
$306$	40944.3	0.0249971
$307$	774049.	0.468730	0.234365	−	0.972149i	$-0.424699\pi$
$307$	774049.	0.468730	0.234365	+	0.972149i	$0.424699\pi$
$308$	15705.0	0.00943324
$309$	−463184.	−0.275967
$310$	0	0
$311$	1.35808e6	0.796205	0.398102	−	0.917341i	$-0.369669\pi$
$311$	1.35808e6	0.796205	0.398102	+	0.917341i	$0.369669\pi$
$312$	−125783.	−0.0731536
$313$	−3.38315e6	−1.95191	−0.975956	−	0.217968i	$-0.930057\pi$
$313$	−3.38315e6	−1.95191	−0.975956	+	0.217968i	$0.930057\pi$
$314$	−2.74821e6	−1.57299
$315$	0	0
$316$	85736.0	0.0482998
$317$	−668139.	−0.373438	−0.186719	−	0.982413i	$-0.559785\pi$
$317$	−668139.	−0.373438	−0.186719	+	0.982413i	$0.559785\pi$
$318$	−1.31518e6	−0.729317
$319$	127077.	0.0699182
$320$	0	0
$321$	1.25689e6	0.680826
$322$	−455905.	−0.245039
$323$	−164593.	−0.0877821
$324$	5383.87	0.00284926
$325$	0	0
$326$	2.50372e6	1.30479
$327$	−1.29631e6	−0.670409
$328$	−1.68885e6	−0.866773
$329$	−1.49178e6	−0.759827
$330$	0	0
$331$	94884.0	0.0476018	0.0238009	−	0.999717i	$-0.492423\pi$
$331$	94884.0	0.0476018	0.0238009	+	0.999717i	$0.492423\pi$
$332$	12684.5	0.00631581
$333$	−315851.	−0.156089
$334$	−3.39491e6	−1.66519
$335$	0	0
$336$	1.49413e6	0.722005
$337$	−507319.	−0.243336	−0.121668	−	0.992571i	$-0.538824\pi$
$337$	−507319.	−0.243336	−0.121668	+	0.992571i	$0.538824\pi$
$338$	2.09204e6	0.996043
$339$	558580.	0.263989
$340$	0	0
$341$	−1.09596e6	−0.510397
$342$	−865635.	−0.400193
$343$	−1.35961e6	−0.623993
$344$	−1.59312e6	−0.725858
$345$	0	0
$346$	1.10829e6	0.497693
$347$	−2.91869e6	−1.30126	−0.650631	−	0.759394i	$-0.725495\pi$
$347$	−2.91869e6	−1.30126	−0.650631	+	0.759394i	$0.725495\pi$
$348$	−7756.19	−0.00343321
$349$	−1.69598e6	−0.745346	−0.372673	−	0.927963i	$-0.621559\pi$
$349$	−1.69598e6	−0.745346	−0.372673	+	0.927963i	$0.621559\pi$
$350$	0	0
$351$	−57038.3	−0.0247115
$352$	35938.4	0.0154597
$353$	2.54669e6	1.08777	0.543887	−	0.839159i	$-0.316952\pi$
$353$	2.54669e6	1.08777	0.543887	+	0.839159i	$0.316952\pi$
$354$	510842.	0.216660
$355$	0	0
$356$	2276.81	0.000952144 0
$357$	125605.	0.0521598
$358$	2.14110e6	0.882936
$359$	−425376.	−0.174196	−0.0870978	−	0.996200i	$-0.527759\pi$
$359$	−425376.	−0.174196	−0.0870978	+	0.996200i	$0.527759\pi$
$360$	0	0
$361$	1.00369e6	0.405353
$362$	1.34565e6	0.539711
$363$	−131769.	−0.0524864
$364$	10155.3	0.00401733
$365$	0	0
$366$	−805081.	−0.314150
$367$	1.88623e6	0.731019	0.365509	−	0.930808i	$-0.380895\pi$
$367$	1.88623e6	0.731019	0.365509	+	0.930808i	$0.380895\pi$
$368$	−528070.	−0.203269
$369$	−765833.	−0.292798
$370$	0	0
$371$	−4.03456e6	−1.52181
$372$	66892.3	0.0250622
$373$	−903282.	−0.336164	−0.168082	−	0.985773i	$-0.553757\pi$
$373$	−903282.	−0.336164	−0.168082	+	0.985773i	$0.553757\pi$
$374$	61163.8	0.0226108
$375$	0	0
$376$	−1.68468e6	−0.614537
$377$	82171.3	0.0297760
$378$	660585.	0.237793
$379$	4.94830e6	1.76953	0.884766	−	0.466036i	$-0.154318\pi$
$379$	4.94830e6	1.76953	0.884766	+	0.466036i	$0.154318\pi$
$380$	0	0
$381$	2.56650e6	0.905791
$382$	−1.49820e6	−0.525304
$383$	1.62974e6	0.567703	0.283851	−	0.958868i	$-0.408388\pi$
$383$	1.62974e6	0.567703	0.283851	+	0.958868i	$0.408388\pi$
$384$	1.72955e6	0.598556
$385$	0	0
$386$	3.94710e6	1.34837
$387$	−722423.	−0.245196
$388$	−102902.	−0.0347013
$389$	5.33558e6	1.78775	0.893877	−	0.448311i	$-0.147974\pi$
$389$	5.33558e6	1.78775	0.893877	+	0.448311i	$0.147974\pi$
$390$	0	0
$391$	−44392.4	−0.0146848
$392$	1.46672e6	0.482094
$393$	1.78527e6	0.583071
$394$	3.58094e6	1.16214
$395$	0	0
$396$	8042.57	0.00257725
$397$	−3.89341e6	−1.23981	−0.619903	−	0.784678i	$-0.712828\pi$
$397$	−3.89341e6	−1.23981	−0.619903	+	0.784678i	$0.712828\pi$
$398$	−5.84457e6	−1.84946
$399$	−2.65550e6	−0.835055
$400$	0	0
$401$	401457.	0.124675	0.0623374	−	0.998055i	$-0.480145\pi$
$401$	401457.	0.124675	0.0623374	+	0.998055i	$0.480145\pi$
$402$	870335.	0.268609
$403$	−708676.	−0.217363
$404$	41837.5	0.0127530
$405$	0	0
$406$	−951662.	−0.286528
$407$	−471826.	−0.141187
$408$	141847.	0.0421861
$409$	−5.92484e6	−1.75133	−0.875666	−	0.482918i	$-0.839577\pi$
$409$	−5.92484e6	−1.75133	−0.875666	+	0.482918i	$0.839577\pi$
$410$	0	0
$411$	−1.49695e6	−0.437124
$412$	42231.4	0.0122572
$413$	1.56711e6	0.452089
$414$	−233470.	−0.0669469
$415$	0	0
$416$	23238.7	0.00658383
$417$	2.70933e6	0.762996
$418$	−1.29311e6	−0.361988
$419$	4.23888e6	1.17955	0.589774	−	0.807568i	$-0.299217\pi$
$419$	4.23888e6	1.17955	0.589774	+	0.807568i	$0.299217\pi$
$420$	0	0
$421$	5.04583e6	1.38748	0.693740	−	0.720225i	$-0.255962\pi$
$421$	5.04583e6	1.38748	0.693740	+	0.720225i	$0.255962\pi$
$422$	3.11838e6	0.852409
$423$	−763944.	−0.207592
$424$	−4.55627e6	−1.23082
$425$	0	0
$426$	3.68490e6	0.983788
$427$	−2.46974e6	−0.655514
$428$	−114599.	−0.0302393
$429$	−85205.3	−0.0223524
$430$	0	0
$431$	−7.38745e6	−1.91558	−0.957792	−	0.287461i	$-0.907189\pi$
$431$	−7.38745e6	−1.91558	−0.957792	+	0.287461i	$0.907189\pi$
$432$	765148.	0.197259
$433$	−1.66687e6	−0.427249	−0.213625	−	0.976916i	$-0.568527\pi$
$433$	−1.66687e6	−0.427249	−0.213625	+	0.976916i	$0.568527\pi$
$434$	8.20748e6	2.09163
$435$	0	0
$436$	118193.	0.0297766
$437$	938534.	0.235097
$438$	−3.38922e6	−0.844139
$439$	7.00345e6	1.73441	0.867203	−	0.497956i	$-0.165916\pi$
$439$	7.00345e6	1.73441	0.867203	+	0.497956i	$0.165916\pi$
$440$	0	0
$441$	665106.	0.162853
$442$	39550.1	0.00962924
$443$	1.09721e6	0.265633	0.132816	−	0.991141i	$-0.457598\pi$
$443$	1.09721e6	0.265633	0.132816	+	0.991141i	$0.457598\pi$
$444$	28798.1	0.00693276
$445$	0	0
$446$	4.88389e6	1.16260
$447$	−967117.	−0.228934
$448$	5.04332e6	1.18719
$449$	−1.62691e6	−0.380844	−0.190422	−	0.981702i	$-0.560986\pi$
$449$	−1.62691e6	−0.380844	−0.190422	+	0.981702i	$0.560986\pi$
$450$	0	0
$451$	−1.14402e6	−0.264846
$452$	−50929.2	−0.0117252
$453$	−216741.	−0.0496245
$454$	−5.21177e6	−1.18671
$455$	0	0
$456$	−2.99889e6	−0.675380
$457$	857428.	0.192047	0.0960235	−	0.995379i	$-0.469388\pi$
$457$	857428.	0.192047	0.0960235	+	0.995379i	$0.469388\pi$
$458$	3.21829e6	0.716905
$459$	64322.6	0.0142505
$460$	0	0
$461$	−5.70696e6	−1.25070	−0.625349	−	0.780345i	$-0.715043\pi$
$461$	−5.70696e6	−1.25070	−0.625349	+	0.780345i	$0.715043\pi$
$462$	986800.	0.215092
$463$	−5.95157e6	−1.29027	−0.645133	−	0.764070i	$-0.723198\pi$
$463$	−5.95157e6	−1.29027	−0.645133	+	0.764070i	$0.723198\pi$
$464$	−1.10230e6	−0.237686
$465$	0	0
$466$	−3.46402e6	−0.738950
$467$	5.25283e6	1.11455	0.557277	−	0.830326i	$-0.311846\pi$
$467$	5.25283e6	1.11455	0.557277	+	0.830326i	$0.311846\pi$
$468$	5200.54	0.00109757
$469$	2.66992e6	0.560488
$470$	0	0
$471$	−4.31737e6	−0.896740
$472$	1.76975e6	0.365643
$473$	−1.07918e6	−0.221789
$474$	5.38709e6	1.10131
$475$	0	0
$476$	−11452.2	−0.00231670
$477$	−2.06611e6	−0.415774
$478$	6.51361e6	1.30392
$479$	8.82420e6	1.75726	0.878631	−	0.477502i	$-0.158458\pi$
$479$	8.82420e6	1.75726	0.878631	+	0.477502i	$0.158458\pi$
$480$	0	0
$481$	−305095.	−0.0601275
$482$	−1.49754e6	−0.293602
$483$	−716216.	−0.139693
$484$	12014.2	0.00233121
$485$	0	0
$486$	338287.	0.0649673
$487$	−5.99283e6	−1.14501	−0.572506	−	0.819901i	$-0.694029\pi$
$487$	−5.99283e6	−1.14501	−0.572506	+	0.819901i	$0.694029\pi$
$488$	−2.78911e6	−0.530170
$489$	3.93328e6	0.743846
$490$	0	0
$491$	6.17263e6	1.15549	0.577745	−	0.816217i	$-0.303933\pi$
$491$	6.17263e6	1.15549	0.577745	+	0.816217i	$0.303933\pi$
$492$	69825.8	0.0130048
$493$	−92665.3	−0.0171712
$494$	−836159.	−0.154160
$495$	0	0
$496$	9.50663e6	1.73509
$497$	1.13041e7	2.05280
$498$	797013.	0.144010
$499$	−2.47210e6	−0.444441	−0.222221	−	0.974996i	$-0.571331\pi$
$499$	−2.47210e6	−0.444441	−0.222221	+	0.974996i	$0.571331\pi$
$500$	0	0
$501$	−5.33333e6	−0.949301
$502$	−3.23899e6	−0.573655
$503$	8.18126e6	1.44178	0.720892	−	0.693047i	$-0.243732\pi$
$503$	8.18126e6	1.44178	0.720892	+	0.693047i	$0.243732\pi$
$504$	2.28852e6	0.401308
$505$	0	0
$506$	−348764.	−0.0605558
$507$	3.28654e6	0.567831
$508$	−234003.	−0.0402312
$509$	1.19487e6	0.204421	0.102211	−	0.994763i	$-0.467408\pi$
$509$	1.19487e6	0.204421	0.102211	+	0.994763i	$0.467408\pi$
$510$	0	0
$511$	−1.03971e7	−1.76141
$512$	5.68768e6	0.958870
$513$	−1.35989e6	−0.228145
$514$	−8.70486e6	−1.45330
$515$	0	0
$516$	65867.8	0.0108905
$517$	−1.14120e6	−0.187774
$518$	3.53344e6	0.578593
$519$	1.74109e6	0.283728
$520$	0	0
$521$	1.09929e6	0.177426	0.0887128	−	0.996057i	$-0.471725\pi$
$521$	1.09929e6	0.177426	0.0887128	+	0.996057i	$0.471725\pi$
$522$	−487349.	−0.0782823
$523$	−7.07488e6	−1.13101	−0.565503	−	0.824746i	$-0.691318\pi$
$523$	−7.07488e6	−1.13101	−0.565503	+	0.824746i	$0.691318\pi$
$524$	−162774.	−0.0258974
$525$	0	0
$526$	−4.18424e6	−0.659404
$527$	799180.	0.125348
$528$	1.14300e6	0.178427
$529$	−6.18321e6	−0.960671
$530$	0	0
$531$	802521.	0.123515
$532$	242119.	0.0370894
$533$	−739755.	−0.112790
$534$	143060.	0.0217103
$535$	0	0
$536$	3.01517e6	0.453315
$537$	3.36361e6	0.503350
$538$	2.72274e6	0.405555
$539$	993554.	0.147306
$540$	0	0
$541$	−3.53440e6	−0.519186	−0.259593	−	0.965718i	$-0.583588\pi$
$541$	−3.53440e6	−0.519186	−0.259593	+	0.965718i	$0.583588\pi$
$542$	5.50729e6	0.805267
$543$	2.11398e6	0.307682
$544$	−26206.5	−0.00379675
$545$	0	0
$546$	638091.	0.0916011
$547$	−1.37763e6	−0.196863	−0.0984316	−	0.995144i	$-0.531383\pi$
$547$	−1.37763e6	−0.196863	−0.0984316	+	0.995144i	$0.531383\pi$
$548$	136487.	0.0194151
$549$	−1.26476e6	−0.179093
$550$	0	0
$551$	1.95911e6	0.274903
$552$	−808829.	−0.112982
$553$	1.65260e7	2.29802
$554$	2.45347e6	0.339630
$555$	0	0
$556$	−247027.	−0.0338889
$557$	−9.57607e6	−1.30782	−0.653912	−	0.756571i	$-0.726873\pi$
$557$	−9.57607e6	−1.30782	−0.653912	+	0.756571i	$0.726873\pi$
$558$	4.20308e6	0.571455
$559$	−697823.	−0.0944530
$560$	0	0
$561$	96086.8	0.0128901
$562$	1.21884e6	0.162782
$563$	−4.58027e6	−0.609003	−0.304502	−	0.952512i	$-0.598490\pi$
$563$	−4.58027e6	−0.609003	−0.304502	+	0.952512i	$0.598490\pi$
$564$	69653.5	0.00922031
$565$	0	0
$566$	−3.23033e6	−0.423844
$567$	1.03776e6	0.135563
$568$	1.27659e7	1.66027
$569$	−943189.	−0.122129	−0.0610644	−	0.998134i	$-0.519450\pi$
$569$	−943189.	−0.122129	−0.0610644	+	0.998134i	$0.519450\pi$
$570$	0	0
$571$	−581249.	−0.0746057	−0.0373029	−	0.999304i	$-0.511877\pi$
$571$	−581249.	−0.0746057	−0.0373029	+	0.999304i	$0.511877\pi$
$572$	7768.70	0.000992792 0
$573$	−2.35363e6	−0.299469
$574$	8.56742e6	1.08535
$575$	0	0
$576$	2.58270e6	0.324353
$577$	3.08163e6	0.385338	0.192669	−	0.981264i	$-0.438286\pi$
$577$	3.08163e6	0.385338	0.192669	+	0.981264i	$0.438286\pi$
$578$	8.08965e6	1.00719
$579$	6.20079e6	0.768689
$580$	0	0
$581$	2.44499e6	0.300495
$582$	−6.46571e6	−0.791241
$583$	−3.08641e6	−0.376082
$584$	−1.17415e7	−1.42460
$585$	0	0
$586$	1.03671e7	1.24713
$587$	−5.79374e6	−0.694007	−0.347004	−	0.937864i	$-0.612801\pi$
$587$	−5.79374e6	−0.694007	−0.347004	+	0.937864i	$0.612801\pi$
$588$	−60641.9	−0.00723318
$589$	−1.68961e7	−2.00677
$590$	0	0
$591$	5.62557e6	0.662518
$592$	4.09274e6	0.479966
$593$	1.14174e7	1.33331	0.666656	−	0.745366i	$-0.267725\pi$
$593$	1.14174e7	1.33331	0.666656	+	0.745366i	$0.267725\pi$
$594$	505343.	0.0587652
$595$	0	0
$596$	88178.1	0.0101682
$597$	−9.18168e6	−1.05435
$598$	−225520.	−0.0257889
$599$	1.33518e7	1.52045	0.760224	−	0.649661i	$-0.225089\pi$
$599$	1.33518e7	1.52045	0.760224	+	0.649661i	$0.225089\pi$
$600$	0	0
$601$	1774.23	0.000200366 0	0.000100183	−	1.00000i	$-0.499968\pi$
$601$	1774.23	0.000200366 0	0.000100183		1.00000i	$0.499968\pi$
$602$	8.08179e6	0.908901
$603$	1.36727e6	0.153131
$604$	19761.7	0.00220410
$605$	0	0
$606$	2.62879e6	0.290787
$607$	2.48398e6	0.273638	0.136819	−	0.990596i	$-0.456312\pi$
$607$	2.48398e6	0.273638	0.136819	+	0.990596i	$0.456312\pi$
$608$	554051.	0.0607842
$609$	−1.49504e6	−0.163346
$610$	0	0
$611$	−737930.	−0.0799672
$612$	−5864.69	−0.000632946 0
$613$	−3.19717e6	−0.343649	−0.171824	−	0.985128i	$-0.554966\pi$
$613$	−3.19717e6	−0.343649	−0.171824	+	0.985128i	$0.554966\pi$
$614$	−4.43447e6	−0.474702
$615$	0	0
$616$	3.41865e6	0.362997
$617$	3.43973e6	0.363757	0.181878	−	0.983321i	$-0.441782\pi$
$617$	3.43973e6	0.363757	0.181878	+	0.983321i	$0.441782\pi$
$618$	2.65354e6	0.279483
$619$	−580297.	−0.0608729	−0.0304364	−	0.999537i	$-0.509690\pi$
$619$	−580297.	−0.0608729	−0.0304364	+	0.999537i	$0.509690\pi$
$620$	0	0
$621$	−366776.	−0.0381656
$622$	−7.78035e6	−0.806349
$623$	438865.	0.0453013
$624$	739093.	0.0759867
$625$	0	0
$626$	1.93818e7	1.97678
$627$	−2.03144e6	−0.206365
$628$	393641.	0.0398292
$629$	344059.	0.0346742
$630$	0	0
$631$	−1.94969e7	−1.94936	−0.974680	−	0.223604i	$-0.928218\pi$
$631$	−1.94969e7	−1.94936	−0.974680	+	0.223604i	$0.928218\pi$
$632$	1.86629e7	1.85861
$633$	4.89890e6	0.485947
$634$	3.82772e6	0.378196
$635$	0	0
$636$	188380.	0.0184668
$637$	642458.	0.0627330
$638$	−728015.	−0.0708090
$639$	5.78888e6	0.560845
$640$	0	0
$641$	−1.50533e7	−1.44706	−0.723530	−	0.690293i	$-0.757482\pi$
$641$	−1.50533e7	−1.44706	−0.723530	+	0.690293i	$0.757482\pi$
$642$	−7.20065e6	−0.689500
$643$	−1.34527e7	−1.28316	−0.641580	−	0.767056i	$-0.721721\pi$
$643$	−1.34527e7	−1.28316	−0.641580	+	0.767056i	$0.721721\pi$
$644$	65301.9	0.00620456
$645$	0	0
$646$	942943.	0.0889005
$647$	−1.15666e7	−1.08629	−0.543146	−	0.839638i	$-0.682767\pi$
$647$	−1.15666e7	−1.08629	−0.543146	+	0.839638i	$0.682767\pi$
$648$	1.17196e6	0.109641
$649$	1.19883e6	0.111724
$650$	0	0
$651$	1.28938e7	1.19241
$652$	−358622.	−0.0330383
$653$	8.60996e6	0.790165	0.395083	−	0.918646i	$-0.370716\pi$
$653$	8.60996e6	0.790165	0.395083	+	0.918646i	$0.370716\pi$
$654$	7.42646e6	0.678950
$655$	0	0
$656$	9.92354e6	0.900341
$657$	−5.32438e6	−0.481233
$658$	8.54628e6	0.769507
$659$	−3.14337e6	−0.281957	−0.140978	−	0.990013i	$-0.545025\pi$
$659$	−3.14337e6	−0.281957	−0.140978	+	0.990013i	$0.545025\pi$
$660$	0	0
$661$	−2.75239e6	−0.245023	−0.122511	−	0.992467i	$-0.539095\pi$
$661$	−2.75239e6	−0.245023	−0.122511	+	0.992467i	$0.539095\pi$
$662$	−543584.	−0.0482083
$663$	62132.2	0.00548951
$664$	2.76116e6	0.243036
$665$	0	0
$666$	1.80948e6	0.158077
$667$	528390.	0.0459876
$668$	486273.	0.0421637
$669$	7.67247e6	0.662781
$670$	0	0
$671$	−1.88934e6	−0.161996
$672$	−422808.	−0.0361177
$673$	1.66398e7	1.41615	0.708076	−	0.706136i	$-0.249563\pi$
$673$	1.66398e7	1.41615	0.708076	+	0.706136i	$0.249563\pi$
$674$	2.90639e6	0.246436
$675$	0	0
$676$	−299655.	−0.0252205
$677$	−1.05991e7	−0.888789	−0.444395	−	0.895831i	$-0.646581\pi$
$677$	−1.05991e7	−0.888789	−0.444395	+	0.895831i	$0.646581\pi$
$678$	−3.20006e6	−0.267352
$679$	−1.98348e7	−1.65103
$680$	0	0
$681$	−8.18756e6	−0.676529
$682$	6.27867e6	0.516900
$683$	−544094.	−0.0446296	−0.0223148	−	0.999751i	$-0.507104\pi$
$683$	−544094.	−0.0446296	−0.0223148	+	0.999751i	$0.507104\pi$
$684$	123990.	0.0101332
$685$	0	0
$686$	7.78911e6	0.631943
$687$	5.05586e6	0.408698
$688$	9.36104e6	0.753969
$689$	−1.99575e6	−0.160162
$690$	0	0
$691$	−7.88699e6	−0.628371	−0.314185	−	0.949362i	$-0.601731\pi$
$691$	−7.88699e6	−0.628371	−0.314185	+	0.949362i	$0.601731\pi$
$692$	−158746.	−0.0126020
$693$	1.55024e6	0.122621
$694$	1.67210e7	1.31784
$695$	0	0
$696$	−1.68836e6	−0.132112
$697$	834228.	0.0650433
$698$	9.71616e6	0.754842
$699$	−5.44188e6	−0.421266
$700$	0	0
$701$	2.22952e7	1.71363	0.856814	−	0.515625i	$-0.172440\pi$
$701$	2.22952e7	1.71363	0.856814	+	0.515625i	$0.172440\pi$
$702$	326768.	0.0250263
$703$	−7.27400e6	−0.555118
$704$	3.85811e6	0.293388
$705$	0	0
$706$	−1.45898e7	−1.10163
$707$	8.06434e6	0.606765
$708$	−73170.9	−0.00548599
$709$	−8.40301e6	−0.627797	−0.313899	−	0.949457i	$-0.601635\pi$
$709$	−8.40301e6	−0.627797	−0.313899	+	0.949457i	$0.601635\pi$
$710$	0	0
$711$	8.46299e6	0.627841
$712$	495615.	0.0366390
$713$	−4.55703e6	−0.335706
$714$	−719580.	−0.0528243
$715$	0	0
$716$	−306682.	−0.0223566
$717$	1.02327e7	0.743350
$718$	2.43695e6	0.176415
$719$	1.39897e7	1.00922	0.504612	−	0.863346i	$-0.331636\pi$
$719$	1.39897e7	1.00922	0.504612	+	0.863346i	$0.331636\pi$
$720$	0	0
$721$	8.14027e6	0.583177
$722$	−5.75009e6	−0.410517
$723$	−2.35259e6	−0.167379
$724$	−192745.	−0.0136659
$725$	0	0
$726$	754895.	0.0531551
$727$	1.69990e7	1.19286	0.596428	−	0.802666i	$-0.296586\pi$
$727$	1.69990e7	1.19286	0.596428	+	0.802666i	$0.296586\pi$
$728$	2.21059e6	0.154589
$729$	531441.	0.0370370
$730$	0	0
$731$	786941.	0.0544689
$732$	115316.	0.00795450
$733$	9.14170e6	0.628445	0.314222	−	0.949349i	$-0.398256\pi$
$733$	9.14170e6	0.628445	0.314222	+	0.949349i	$0.398256\pi$
$734$	−1.08060e7	−0.740332
$735$	0	0
$736$	149433.	0.0101684
$737$	2.04247e6	0.138512
$738$	4.38740e6	0.296528
$739$	−1.54222e7	−1.03881	−0.519405	−	0.854528i	$-0.673846\pi$
$739$	−1.54222e7	−1.03881	−0.519405	+	0.854528i	$0.673846\pi$
$740$	0	0
$741$	−1.31358e6	−0.0878845
$742$	2.31137e7	1.54120
$743$	−2.34940e7	−1.56129	−0.780646	−	0.624973i	$-0.785110\pi$
$743$	−2.34940e7	−1.56129	−0.780646	+	0.624973i	$0.785110\pi$
$744$	1.45610e7	0.964407
$745$	0	0
$746$	5.17484e6	0.340447
$747$	1.25209e6	0.0820981
$748$	−8760.83	−0.000572521 0
$749$	−2.20894e7	−1.43873
$750$	0	0
$751$	9.67937e6	0.626250	0.313125	−	0.949712i	$-0.398624\pi$
$751$	9.67937e6	0.626250	0.313125	+	0.949712i	$0.398624\pi$
$752$	9.89906e6	0.638336
$753$	−5.08838e6	−0.327033
$754$	−470754.	−0.0301554
$755$	0	0
$756$	−94619.3	−0.00602109
$757$	−6.06490e6	−0.384666	−0.192333	−	0.981330i	$-0.561605\pi$
$757$	−6.06490e6	−0.384666	−0.192333	+	0.981330i	$0.561605\pi$
$758$	−2.83485e7	−1.79208
$759$	−547900.	−0.0345221
$760$	0	0
$761$	−2.42020e7	−1.51492	−0.757461	−	0.652880i	$-0.773561\pi$
$761$	−2.42020e7	−1.51492	−0.757461	+	0.652880i	$0.773561\pi$
$762$	−1.47033e7	−0.917331
$763$	2.27821e7	1.41672
$764$	214595.	0.0133011
$765$	0	0
$766$	−9.33665e6	−0.574936
$767$	775193.	0.0475797
$768$	−725529.	−0.0443866
$769$	3.47840e6	0.212111	0.106055	−	0.994360i	$-0.466178\pi$
$769$	3.47840e6	0.212111	0.106055	+	0.994360i	$0.466178\pi$
$770$	0	0
$771$	−1.36751e7	−0.828506
$772$	−565365.	−0.0341418
$773$	−1.02630e7	−0.617769	−0.308884	−	0.951100i	$-0.599956\pi$
$773$	−1.02630e7	−0.617769	−0.308884	+	0.951100i	$0.599956\pi$
$774$	4.13871e6	0.248320
$775$	0	0
$776$	−2.23997e7	−1.33533
$777$	5.55095e6	0.329849
$778$	−3.05672e7	−1.81053
$779$	−1.76370e7	−1.04131
$780$	0	0
$781$	8.64759e6	0.507303
$782$	254321.	0.0148719
$783$	−765613.	−0.0446277
$784$	−8.61834e6	−0.500764
$785$	0	0
$786$	−1.02277e7	−0.590500
$787$	2.49651e7	1.43680	0.718401	−	0.695629i	$-0.244874\pi$
$787$	2.49651e7	1.43680	0.718401	+	0.695629i	$0.244874\pi$
$788$	−512919.	−0.0294261
$789$	−6.57333e6	−0.375918
$790$	0	0
$791$	−9.81681e6	−0.557865
$792$	1.75070e6	0.0991742
$793$	−1.22169e6	−0.0689890
$794$	2.23050e7	1.25560
$795$	0	0
$796$	837151.	0.0468297
$797$	−2.85688e7	−1.59311	−0.796557	−	0.604564i	$-0.793347\pi$
$797$	−2.85688e7	−1.59311	−0.796557	+	0.604564i	$0.793347\pi$
$798$	1.52132e7	0.845694
$799$	832170.	0.0461153
$800$	0	0
$801$	224744.	0.0123768
$802$	−2.29992e6	−0.126263
$803$	−7.95370e6	−0.435292
$804$	−124663.	−0.00680138
$805$	0	0
$806$	4.05995e6	0.220132
$807$	4.27735e6	0.231202
$808$	9.10714e6	0.490743
$809$	1.00028e7	0.537340	0.268670	−	0.963232i	$-0.413416\pi$
$809$	1.00028e7	0.537340	0.268670	+	0.963232i	$0.413416\pi$
$810$	0	0
$811$	−7.88288e6	−0.420855	−0.210428	−	0.977609i	$-0.567486\pi$
$811$	−7.88288e6	−0.420855	−0.210428	+	0.977609i	$0.567486\pi$
$812$	136312.	0.00725511
$813$	8.65181e6	0.459072
$814$	2.70306e6	0.142986
$815$	0	0
$816$	−833482.	−0.0438198
$817$	−1.66373e7	−0.872023
$818$	3.39430e7	1.77364
$819$	1.00242e6	0.0522206
$820$	0	0
$821$	1.84331e7	0.954423	0.477211	−	0.878789i	$-0.341648\pi$
$821$	1.84331e7	0.954423	0.477211	+	0.878789i	$0.341648\pi$
$822$	8.57594e6	0.442693
$823$	1.93527e7	0.995961	0.497981	−	0.867188i	$-0.334075\pi$
$823$	1.93527e7	0.995961	0.497981	+	0.867188i	$0.334075\pi$
$824$	9.19288e6	0.471665
$825$	0	0
$826$	−8.97785e6	−0.457849
$827$	1.33049e7	0.676471	0.338235	−	0.941062i	$-0.390170\pi$
$827$	1.33049e7	0.676471	0.338235	+	0.941062i	$0.390170\pi$
$828$	33441.3	0.00169515
$829$	−1.61482e7	−0.816090	−0.408045	−	0.912962i	$-0.633789\pi$
$829$	−1.61482e7	−0.816090	−0.408045	+	0.912962i	$0.633789\pi$
$830$	0	0
$831$	3.85434e6	0.193619
$832$	2.49475e6	0.124945
$833$	−724505.	−0.0361767
$834$	−1.55216e7	−0.772717
$835$	0	0
$836$	185219.	0.00916581
$837$	6.60293e6	0.325779
$838$	−2.42842e7	−1.19458
$839$	2.17566e7	1.06705	0.533527	−	0.845783i	$-0.320866\pi$
$839$	2.17566e7	1.06705	0.533527	+	0.845783i	$0.320866\pi$
$840$	0	0
$841$	−1.94082e7	−0.946226
$842$	−2.89072e7	−1.40516
$843$	1.91477e6	0.0927998
$844$	−446663.	−0.0215836
$845$	0	0
$846$	4.37658e6	0.210237
$847$	2.31579e6	0.110915
$848$	2.67723e7	1.27849
$849$	−5.07478e6	−0.241628
$850$	0	0
$851$	−1.96187e6	−0.0928638
$852$	−527809.	−0.0249102
$853$	1.29447e7	0.609142	0.304571	−	0.952490i	$-0.401487\pi$
$853$	1.29447e7	0.609142	0.304571	+	0.952490i	$0.401487\pi$
$854$	1.41490e7	0.663866
$855$	0	0
$856$	−2.49458e7	−1.16362
$857$	−3.34203e7	−1.55438	−0.777192	−	0.629264i	$-0.783356\pi$
$857$	−3.34203e7	−1.55438	−0.777192	+	0.629264i	$0.783356\pi$
$858$	488135.	0.0226371
$859$	−4.22247e7	−1.95247	−0.976234	−	0.216719i	$-0.930464\pi$
$859$	−4.22247e7	−1.95247	−0.976234	+	0.216719i	$0.930464\pi$
$860$	0	0
$861$	1.34592e7	0.618745
$862$	4.23221e7	1.93999
$863$	9.62825e6	0.440069	0.220034	−	0.975492i	$-0.429383\pi$
$863$	9.62825e6	0.440069	0.220034	+	0.975492i	$0.429383\pi$
$864$	−216521.	−0.00986771
$865$	0	0
$866$	9.54936e6	0.432693
$867$	1.27086e7	0.574185
$868$	−1.17560e6	−0.0529617
$869$	1.26422e7	0.567904
$870$	0	0
$871$	1.32072e6	0.0589880
$872$	2.57281e7	1.14582
$873$	−1.01575e7	−0.451076
$874$	−5.37679e6	−0.238092
$875$	0	0
$876$	485457.	0.0213742
$877$	−1.37201e7	−0.602362	−0.301181	−	0.953567i	$-0.597381\pi$
$877$	−1.37201e7	−0.602362	−0.301181	+	0.953567i	$0.597381\pi$
$878$	−4.01222e7	−1.75650
$879$	1.62865e7	0.710976
$880$	0	0
$881$	2.99671e7	1.30078	0.650392	−	0.759599i	$-0.274605\pi$
$881$	2.99671e7	1.30078	0.650392	+	0.759599i	$0.274605\pi$
$882$	−3.81034e6	−0.164927
$883$	6.57264e6	0.283686	0.141843	−	0.989889i	$-0.454697\pi$
$883$	6.57264e6	0.283686	0.141843	+	0.989889i	$0.454697\pi$
$884$	−5664.99	−0.000243819 0
$885$	0	0
$886$	−6.28585e6	−0.269017
$887$	−1.99178e7	−0.850028	−0.425014	−	0.905187i	$-0.639731\pi$
$887$	−1.99178e7	−0.850028	−0.425014	+	0.905187i	$0.639731\pi$
$888$	6.26874e6	0.266777
$889$	−4.51051e7	−1.91413
$890$	0	0
$891$	793881.	0.0335013
$892$	−699548.	−0.0294378
$893$	−1.75935e7	−0.738285
$894$	5.54054e6	0.231851
$895$	0	0
$896$	−3.03961e7	−1.26488
$897$	−354287.	−0.0147019
$898$	9.32044e6	0.385696
$899$	−9.51241e6	−0.392547
$900$	0	0
$901$	2.25063e6	0.0923617
$902$	6.55402e6	0.268220
$903$	1.26963e7	0.518152
$904$	−1.10862e7	−0.451193
$905$	0	0
$906$	1.24170e6	0.0502568
$907$	−4.17948e7	−1.68696	−0.843478	−	0.537163i	$-0.819496\pi$
$907$	−4.17948e7	−1.68696	−0.843478	+	0.537163i	$0.819496\pi$
$908$	746511.	0.0300484
$909$	4.12977e6	0.165774
$910$	0	0
$911$	−2.46957e7	−0.985882	−0.492941	−	0.870063i	$-0.664078\pi$
$911$	−2.46957e7	−0.985882	−0.492941	+	0.870063i	$0.664078\pi$
$912$	1.76213e7	0.701536
$913$	1.87040e6	0.0742605
$914$	−4.91214e6	−0.194494
$915$	0	0
$916$	−460974.	−0.0181526
$917$	−3.13753e7	−1.23215
$918$	−368499.	−0.0144321
$919$	−2.76370e7	−1.07945	−0.539724	−	0.841842i	$-0.681471\pi$
$919$	−2.76370e7	−1.07945	−0.539724	+	0.841842i	$0.681471\pi$
$920$	0	0
$921$	−6.96644e6	−0.270621
$922$	3.26947e7	1.26663
$923$	5.59176e6	0.216045
$924$	−141345.	−0.00544628
$925$	0	0
$926$	3.40961e7	1.30670
$927$	4.16865e6	0.159330
$928$	311928.	0.0118901
$929$	−2.99531e7	−1.13868	−0.569340	−	0.822102i	$-0.692801\pi$
$929$	−2.99531e7	−1.13868	−0.569340	+	0.822102i	$0.692801\pi$
$930$	0	0
$931$	1.53173e7	0.579173
$932$	496171.	0.0187108
$933$	−1.22227e7	−0.459689
$934$	−3.00931e7	−1.12875
$935$	0	0
$936$	1.13205e6	0.0422353
$937$	−3.96082e7	−1.47379	−0.736896	−	0.676006i	$-0.763709\pi$
$937$	−3.96082e7	−1.47379	−0.736896	+	0.676006i	$0.763709\pi$
$938$	−1.52958e7	−0.567629
$939$	3.04483e7	1.12694
$940$	0	0
$941$	−3.50065e6	−0.128877	−0.0644383	−	0.997922i	$-0.520526\pi$
$941$	−3.50065e6	−0.128877	−0.0644383	+	0.997922i	$0.520526\pi$
$942$	2.47339e7	0.908165
$943$	−4.75688e6	−0.174198
$944$	−1.03989e7	−0.379804
$945$	0	0
$946$	6.18251e6	0.224614
$947$	−2.56533e7	−0.929540	−0.464770	−	0.885432i	$-0.653863\pi$
$947$	−2.56533e7	−0.929540	−0.464770	+	0.885432i	$0.653863\pi$
$948$	−771624.	−0.0278859
$949$	−5.14307e6	−0.185377
$950$	0	0
$951$	6.01325e6	0.215604
$952$	−2.49290e6	−0.0891482
$953$	1.43726e7	0.512629	0.256314	−	0.966593i	$-0.417492\pi$
$953$	1.43726e7	0.512629	0.256314	+	0.966593i	$0.417492\pi$
$954$	1.18366e7	0.421071
$955$	0	0
$956$	−932982.	−0.0330163
$957$	−1.14369e6	−0.0403673
$958$	−5.05532e7	−1.77965
$959$	2.63084e7	0.923735
$960$	0	0
$961$	5.34094e7	1.86556
$962$	1.74787e6	0.0608935
$963$	−1.13120e7	−0.393075
$964$	214501.	0.00743423
$965$	0	0
$966$	4.10315e6	0.141473
$967$	−3.41494e7	−1.17440	−0.587201	−	0.809441i	$-0.699770\pi$
$967$	−3.41494e7	−1.17440	−0.587201	+	0.809441i	$0.699770\pi$
$968$	2.61524e6	0.0897064
$969$	1.48134e6	0.0506810
$970$	0	0
$971$	8.76612e6	0.298373	0.149186	−	0.988809i	$-0.452335\pi$
$971$	8.76612e6	0.298373	0.149186	+	0.988809i	$0.452335\pi$
$972$	−48454.8	−0.00164502
$973$	−4.76154e7	−1.61237
$974$	3.43325e7	1.15960
$975$	0	0
$976$	1.63886e7	0.550703
$977$	−4.92659e7	−1.65124	−0.825620	−	0.564227i	$-0.809174\pi$
$977$	−4.92659e7	−1.65124	−0.825620	+	0.564227i	$0.809174\pi$
$978$	−2.25335e7	−0.753323
$979$	335729.	0.0111952
$980$	0	0
$981$	1.16668e7	0.387061
$982$	−3.53625e7	−1.17021
$983$	−5.54306e7	−1.82964	−0.914820	−	0.403861i	$-0.867668\pi$
$983$	−5.54306e7	−1.82964	−0.914820	+	0.403861i	$0.867668\pi$
$984$	1.51996e7	0.500432
$985$	0	0
$986$	530873.	0.0173899
$987$	1.34260e7	0.438686
$988$	119768.	0.00390344
$989$	−4.48725e6	−0.145878
$990$	0	0
$991$	−1.51099e7	−0.488739	−0.244370	−	0.969682i	$-0.578581\pi$
$991$	−1.51099e7	−0.488739	−0.244370	+	0.969682i	$0.578581\pi$
$992$	−2.69018e6	−0.0867966
$993$	−853956.	−0.0274829
$994$	−6.47606e7	−2.07895
$995$	0	0
$996$	−114161.	−0.00364643
$997$	3.98499e7	1.26966	0.634832	−	0.772650i	$-0.281069\pi$
$997$	3.98499e7	1.26966	0.634832	+	0.772650i	$0.281069\pi$
$998$	1.41625e7	0.450104
$999$	2.84266e6	0.0901178

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.v.1.4		13
5.2	odd	4		165.6.c.b.34.7	✓	26
5.3	odd	4		165.6.c.b.34.20	yes	26
5.4	even	2		825.6.a.y.1.10		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.c.b.34.7	✓	26	5.2	odd	4
165.6.c.b.34.20	yes	26	5.3	odd	4
825.6.a.v.1.4		13	1.1	even	1	trivial
825.6.a.y.1.10		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)

\(f(q)\)	\(=\)	\(q-5.72893 q^{2} -9.00000 q^{3} +0.820586 q^{4} +51.5603 q^{6} +158.171 q^{7} +178.625 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-5.72893 q^{2} -9.00000 q^{3} +0.820586 q^{4} +51.5603 q^{6} +158.171 q^{7} +178.625 q^{8} +81.0000 q^{9} +121.000 q^{11} -7.38528 q^{12} +78.2418 q^{13} -906.152 q^{14} -1049.59 q^{16} -88.2340 q^{17} -464.043 q^{18} +1865.42 q^{19} -1423.54 q^{21} -693.200 q^{22} +503.122 q^{23} -1607.62 q^{24} -448.241 q^{26} -729.000 q^{27} +129.793 q^{28} +1050.22 q^{29} -9057.51 q^{31} +297.011 q^{32} -1089.00 q^{33} +505.486 q^{34} +66.4675 q^{36} -3899.39 q^{37} -10686.9 q^{38} -704.176 q^{39} -9454.73 q^{41} +8155.37 q^{42} -8918.80 q^{43} +99.2910 q^{44} -2882.35 q^{46} -9431.40 q^{47} +9446.27 q^{48} +8211.19 q^{49} +794.106 q^{51} +64.2042 q^{52} -25507.5 q^{53} +4176.39 q^{54} +28253.3 q^{56} -16788.8 q^{57} -6016.65 q^{58} +9907.66 q^{59} -15614.3 q^{61} +51889.8 q^{62} +12811.9 q^{63} +31885.2 q^{64} +6238.80 q^{66} +16879.9 q^{67} -72.4036 q^{68} -4528.10 q^{69} +71467.7 q^{71} +14468.6 q^{72} -65733.0 q^{73} +22339.3 q^{74} +1530.74 q^{76} +19138.7 q^{77} +4034.17 q^{78} +104481. q^{79} +6561.00 q^{81} +54165.4 q^{82} +15457.9 q^{83} -1168.14 q^{84} +51095.2 q^{86} -9452.01 q^{87} +21613.6 q^{88} +2774.62 q^{89} +12375.6 q^{91} +412.855 q^{92} +81517.6 q^{93} +54031.8 q^{94} -2673.10 q^{96} -125401. q^{97} -47041.3 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9}+O(q^{10}) \) 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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