LMFDB - Embedded newform 825.6.a.w.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} - 3 x^{12} - 318 x^{11} + 776 x^{10} + 37929 x^{9} - 75673 x^{8} - 2114192 x^{7} + \cdots + 1037920000 $ x^13 - 3x^12 - 318x^11 + 776x^10 + 37929x^9 - 75673x^8 - 2114192x^7 + 3527024x^6 + 55736416x^5 - 75128864x^4 - 615221248x^3 + 450708480x^2 + 2256236800x + 1037920000
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{9}\cdot 5^{6} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$5.95684$ 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)

$f(q)$	$=$	$q-5.95684 q^{2} +9.00000 q^{3} +3.48390 q^{4} -53.6115 q^{6} -79.6742 q^{7} +169.866 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-5.95684 q^{2} +9.00000 q^{3} +3.48390 q^{4} -53.6115 q^{6} -79.6742 q^{7} +169.866 q^{8} +81.0000 q^{9} -121.000 q^{11} +31.3551 q^{12} +950.589 q^{13} +474.606 q^{14} -1123.35 q^{16} +383.081 q^{17} -482.504 q^{18} -523.437 q^{19} -717.068 q^{21} +720.777 q^{22} -4473.39 q^{23} +1528.79 q^{24} -5662.50 q^{26} +729.000 q^{27} -277.577 q^{28} +2763.50 q^{29} -8306.63 q^{31} +1255.89 q^{32} -1089.00 q^{33} -2281.95 q^{34} +282.196 q^{36} +2264.95 q^{37} +3118.03 q^{38} +8555.30 q^{39} +15121.4 q^{41} +4271.46 q^{42} +11610.9 q^{43} -421.552 q^{44} +26647.3 q^{46} +5277.54 q^{47} -10110.1 q^{48} -10459.0 q^{49} +3447.73 q^{51} +3311.76 q^{52} -32158.5 q^{53} -4342.53 q^{54} -13533.9 q^{56} -4710.94 q^{57} -16461.7 q^{58} +13413.7 q^{59} -22451.5 q^{61} +49481.2 q^{62} -6453.61 q^{63} +28466.0 q^{64} +6486.99 q^{66} -2853.23 q^{67} +1334.62 q^{68} -40260.5 q^{69} +38388.8 q^{71} +13759.1 q^{72} +65611.8 q^{73} -13492.0 q^{74} -1823.60 q^{76} +9640.58 q^{77} -50962.5 q^{78} +7286.04 q^{79} +6561.00 q^{81} -90076.0 q^{82} +9407.65 q^{83} -2498.19 q^{84} -69164.1 q^{86} +24871.5 q^{87} -20553.8 q^{88} +85598.0 q^{89} -75737.4 q^{91} -15584.8 q^{92} -74759.7 q^{93} -31437.5 q^{94} +11303.0 q^{96} -106380. q^{97} +62302.7 q^{98} -9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 3 q^{2} + 117 q^{3} + 229 q^{4} - 27 q^{6} - 284 q^{7} - 369 q^{8} + 1053 q^{9}+O(q^{10}) $ 13 * q - 3 * q^2 + 117 * q^3 + 229 * q^4 - 27 * q^6 - 284 * q^7 - 369 * q^8 + 1053 * q^9	\( 13 q - 3 q^{2} + 117 q^{3} + 229 q^{4} - 27 q^{6} - 284 q^{7} - 369 q^{8} + 1053 q^{9} - 1573 q^{11} + 2061 q^{12} - 366 q^{13} - 2758 q^{14} + 4141 q^{16} - 2056 q^{17} - 243 q^{18} - 310 q^{19} - 2556 q^{21} + 363 q^{22} - 3612 q^{23} - 3321 q^{24} + 2280 q^{26} + 9477 q^{27} - 7896 q^{28} - 4848 q^{29} - 24 q^{31} - 38111 q^{32} - 14157 q^{33} + 5518 q^{34} + 18549 q^{36} + 8420 q^{37} - 474 q^{38} - 3294 q^{39} - 15120 q^{41} - 24822 q^{42} - 35492 q^{43} - 27709 q^{44} - 20280 q^{46} - 46544 q^{47} + 37269 q^{48} + 81837 q^{49} - 18504 q^{51} - 107194 q^{52} - 42256 q^{53} - 2187 q^{54} - 196602 q^{56} - 2790 q^{57} - 114160 q^{58} - 65592 q^{59} - 52042 q^{61} - 94972 q^{62} - 23004 q^{63} + 185977 q^{64} + 3267 q^{66} - 80580 q^{67} - 61108 q^{68} - 32508 q^{69} - 77820 q^{71} - 29889 q^{72} - 103050 q^{73} - 240028 q^{74} - 271174 q^{76} + 34364 q^{77} + 20520 q^{78} - 112258 q^{79} + 85293 q^{81} - 64060 q^{82} - 292150 q^{83} - 71064 q^{84} - 319250 q^{86} - 43632 q^{87} + 44649 q^{88} - 295810 q^{89} - 24200 q^{91} - 121328 q^{92} - 216 q^{93} - 358144 q^{94} - 342999 q^{96} - 49072 q^{97} - 101815 q^{98} - 127413 q^{99}+O(q^{100}) \) 13 * q - 3 * q^2 + 117 * q^3 + 229 * q^4 - 27 * q^6 - 284 * q^7 - 369 * q^8 + 1053 * q^9 - 1573 * q^11 + 2061 * q^12 - 366 * q^13 - 2758 * q^14 + 4141 * q^16 - 2056 * q^17 - 243 * q^18 - 310 * q^19 - 2556 * q^21 + 363 * q^22 - 3612 * q^23 - 3321 * q^24 + 2280 * q^26 + 9477 * q^27 - 7896 * q^28 - 4848 * q^29 - 24 * q^31 - 38111 * q^32 - 14157 * q^33 + 5518 * q^34 + 18549 * q^36 + 8420 * q^37 - 474 * q^38 - 3294 * q^39 - 15120 * q^41 - 24822 * q^42 - 35492 * q^43 - 27709 * q^44 - 20280 * q^46 - 46544 * q^47 + 37269 * q^48 + 81837 * q^49 - 18504 * q^51 - 107194 * q^52 - 42256 * q^53 - 2187 * q^54 - 196602 * q^56 - 2790 * q^57 - 114160 * q^58 - 65592 * q^59 - 52042 * q^61 - 94972 * q^62 - 23004 * q^63 + 185977 * q^64 + 3267 * q^66 - 80580 * q^67 - 61108 * q^68 - 32508 * q^69 - 77820 * q^71 - 29889 * q^72 - 103050 * q^73 - 240028 * q^74 - 271174 * q^76 + 34364 * q^77 + 20520 * q^78 - 112258 * q^79 + 85293 * q^81 - 64060 * q^82 - 292150 * q^83 - 71064 * q^84 - 319250 * q^86 - 43632 * q^87 + 44649 * q^88 - 295810 * q^89 - 24200 * q^91 - 121328 * q^92 - 216 * q^93 - 358144 * q^94 - 342999 * q^96 - 49072 * q^97 - 101815 * 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 $

$2$	−5.95684	−1.05303	−0.526515	−	0.850166i	$-0.676502\pi$
$2$	−5.95684	−1.05303	−0.526515	+	0.850166i	$0.676502\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	3.48390	0.108872
$5$	0	0
$5$	0	0
$6$	−53.6115	−0.607967
$7$	−79.6742	−0.614572	−0.307286	−	0.951617i	$-0.599421\pi$
$7$	−79.6742	−0.614572	−0.307286	+	0.951617i	$0.599421\pi$
$8$	169.866	0.938384
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	31.3551	0.0628572
$13$	950.589	1.56003	0.780017	−	0.625758i	$-0.215210\pi$
$13$	950.589	1.56003	0.780017	+	0.625758i	$0.215210\pi$
$14$	474.606	0.647163
$15$	0	0
$16$	−1123.35	−1.09702
$17$	383.081	0.321490	0.160745	−	0.986996i	$-0.448610\pi$
$17$	383.081	0.321490	0.160745	+	0.986996i	$0.448610\pi$
$18$	−482.504	−0.351010
$19$	−523.437	−0.332645	−0.166322	−	0.986071i	$-0.553189\pi$
$19$	−523.437	−0.332645	−0.166322	+	0.986071i	$0.553189\pi$
$20$	0	0
$21$	−717.068	−0.354823
$22$	720.777	0.317500
$23$	−4473.39	−1.76326	−0.881632	−	0.471938i	$-0.843555\pi$
$23$	−4473.39	−1.76326	−0.881632	+	0.471938i	$0.843555\pi$
$24$	1528.79	0.541777
$25$	0	0
$26$	−5662.50	−1.64276
$27$	729.000	0.192450
$28$	−277.577	−0.0669096
$29$	2763.50	0.610188	0.305094	−	0.952322i	$-0.401312\pi$
$29$	2763.50	0.610188	0.305094	+	0.952322i	$0.401312\pi$
$30$	0	0
$31$	−8306.63	−1.55246	−0.776230	−	0.630449i	$-0.782871\pi$
$31$	−8306.63	−1.55246	−0.776230	+	0.630449i	$0.782871\pi$
$32$	1255.89	0.216809
$33$	−1089.00	−0.174078
$34$	−2281.95	−0.338539
$35$	0	0
$36$	282.196	0.0362906
$37$	2264.95	0.271991	0.135996	−	0.990709i	$-0.456577\pi$
$37$	2264.95	0.271991	0.135996	+	0.990709i	$0.456577\pi$
$38$	3118.03	0.350285
$39$	8555.30	0.900686
$40$	0	0
$41$	15121.4	1.40486	0.702431	−	0.711752i	$-0.252098\pi$
$41$	15121.4	1.40486	0.702431	+	0.711752i	$0.252098\pi$
$42$	4271.46	0.373639
$43$	11610.9	0.957622	0.478811	−	0.877918i	$-0.341068\pi$
$43$	11610.9	0.957622	0.478811	+	0.877918i	$0.341068\pi$
$44$	−421.552	−0.0328261
$45$	0	0
$46$	26647.3	1.85677
$47$	5277.54	0.348488	0.174244	−	0.984703i	$-0.444252\pi$
$47$	5277.54	0.348488	0.174244	+	0.984703i	$0.444252\pi$
$48$	−10110.1	−0.633364
$49$	−10459.0	−0.622301
$50$	0	0
$51$	3447.73	0.185613
$52$	3311.76	0.169844
$53$	−32158.5	−1.57255	−0.786277	−	0.617874i	$-0.787994\pi$
$53$	−32158.5	−1.57255	−0.786277	+	0.617874i	$0.787994\pi$
$54$	−4342.53	−0.202656
$55$	0	0
$56$	−13533.9	−0.576705
$57$	−4710.94	−0.192053
$58$	−16461.7	−0.642547
$59$	13413.7	0.501670	0.250835	−	0.968030i	$-0.419295\pi$
$59$	13413.7	0.501670	0.250835	+	0.968030i	$0.419295\pi$
$60$	0	0
$61$	−22451.5	−0.772541	−0.386271	−	0.922386i	$-0.626237\pi$
$61$	−22451.5	−0.772541	−0.386271	+	0.922386i	$0.626237\pi$
$62$	49481.2	1.63479
$63$	−6453.61	−0.204857
$64$	28466.0	0.868712
$65$	0	0
$66$	6486.99	0.183309
$67$	−2853.23	−0.0776515	−0.0388257	−	0.999246i	$-0.512362\pi$
$67$	−2853.23	−0.0776515	−0.0388257	+	0.999246i	$0.512362\pi$
$68$	1334.62	0.0350013
$69$	−40260.5	−1.01802
$70$	0	0
$71$	38388.8	0.903771	0.451885	−	0.892076i	$-0.350752\pi$
$71$	38388.8	0.903771	0.451885	+	0.892076i	$0.350752\pi$
$72$	13759.1	0.312795
$73$	65611.8	1.44104	0.720518	−	0.693436i	$-0.243904\pi$
$73$	65611.8	1.44104	0.720518	+	0.693436i	$0.243904\pi$
$74$	−13492.0	−0.286415
$75$	0	0
$76$	−1823.60	−0.0362157
$77$	9640.58	0.185300
$78$	−50962.5	−0.948450
$79$	7286.04	0.131348	0.0656740	−	0.997841i	$-0.479080\pi$
$79$	7286.04	0.131348	0.0656740	+	0.997841i	$0.479080\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−90076.0	−1.47936
$83$	9407.65	0.149895	0.0749473	−	0.997187i	$-0.476121\pi$
$83$	9407.65	0.149895	0.0749473	+	0.997187i	$0.476121\pi$
$84$	−2498.19	−0.0386303
$85$	0	0
$86$	−69164.1	−1.00840
$87$	24871.5	0.352292
$88$	−20553.8	−0.282934
$89$	85598.0	1.14548	0.572741	−	0.819736i	$-0.305880\pi$
$89$	85598.0	1.14548	0.572741	+	0.819736i	$0.305880\pi$
$90$	0	0
$91$	−75737.4	−0.958753
$92$	−15584.8	−0.191970
$93$	−74759.7	−0.896314
$94$	−31437.5	−0.366968
$95$	0	0
$96$	11303.0	0.125175
$97$	−106380.	−1.14797	−0.573984	−	0.818867i	$-0.694603\pi$
$97$	−106380.	−1.14797	−0.573984	+	0.818867i	$0.694603\pi$
$98$	62302.7	0.655302
$99$	−9801.00	−0.100504
$100$	0	0
$101$	−118603.	−1.15689	−0.578446	−	0.815720i	$-0.696341\pi$
$101$	−118603.	−1.15689	−0.578446	+	0.815720i	$0.696341\pi$
$102$	−20537.5	−0.195456
$103$	−127585.	−1.18497	−0.592484	−	0.805582i	$-0.701853\pi$
$103$	−127585.	−1.18497	−0.592484	+	0.805582i	$0.701853\pi$
$104$	161472.	1.46391
$105$	0	0
$106$	191563.	1.65595
$107$	−123820.	−1.04552	−0.522760	−	0.852480i	$-0.675098\pi$
$107$	−123820.	−1.04552	−0.522760	+	0.852480i	$0.675098\pi$
$108$	2539.76	0.0209524
$109$	115948.	0.934750	0.467375	−	0.884059i	$-0.345200\pi$
$109$	115948.	0.934750	0.467375	+	0.884059i	$0.345200\pi$
$110$	0	0
$111$	20384.6	0.157034
$112$	89501.8	0.674197
$113$	92650.3	0.682576	0.341288	−	0.939959i	$-0.389137\pi$
$113$	92650.3	0.682576	0.341288	+	0.939959i	$0.389137\pi$
$114$	28062.3	0.202237
$115$	0	0
$116$	9627.75	0.0664324
$117$	76997.7	0.520012
$118$	−79903.1	−0.528273
$119$	−30521.7	−0.197579
$120$	0	0
$121$	14641.0	0.0909091
$122$	133740.	0.813509
$123$	136093.	0.811097
$124$	−28939.5	−0.169019
$125$	0	0
$126$	38443.1	0.215721
$127$	−260060.	−1.43075	−0.715377	−	0.698739i	$-0.753745\pi$
$127$	−260060.	−1.43075	−0.715377	+	0.698739i	$0.753745\pi$
$128$	−209756.	−1.13159
$129$	104498.	0.552883
$130$	0	0
$131$	−176304.	−0.897604	−0.448802	−	0.893631i	$-0.648149\pi$
$131$	−176304.	−0.897604	−0.448802	+	0.893631i	$0.648149\pi$
$132$	−3793.97	−0.0189522
$133$	41704.5	0.204434
$134$	16996.2	0.0817693
$135$	0	0
$136$	65072.3	0.301682
$137$	−131304.	−0.597691	−0.298846	−	0.954301i	$-0.596602\pi$
$137$	−131304.	−0.597691	−0.298846	+	0.954301i	$0.596602\pi$
$138$	239825.	1.07201
$139$	269179.	1.18169	0.590845	−	0.806785i	$-0.298794\pi$
$139$	269179.	1.18169	0.590845	+	0.806785i	$0.298794\pi$
$140$	0	0
$141$	47497.9	0.201199
$142$	−228676.	−0.951697
$143$	−115021.	−0.470368
$144$	−90991.1	−0.365673
$145$	0	0
$146$	−390839.	−1.51745
$147$	−94131.2	−0.359286
$148$	7890.88	0.0296122
$149$	349328.	1.28905	0.644523	−	0.764585i	$-0.277056\pi$
$149$	349328.	1.28905	0.644523	+	0.764585i	$0.277056\pi$
$150$	0	0
$151$	359268.	1.28226	0.641130	−	0.767432i	$-0.278466\pi$
$151$	359268.	1.28226	0.641130	+	0.767432i	$0.278466\pi$
$152$	−88914.1	−0.312149
$153$	31029.5	0.107163
$154$	−57427.4	−0.195127
$155$	0	0
$156$	29805.8	0.0980595
$157$	405514.	1.31298	0.656488	−	0.754337i	$-0.272041\pi$
$157$	405514.	1.31298	0.656488	+	0.754337i	$0.272041\pi$
$158$	−43401.7	−0.138313
$159$	−289426.	−0.907915
$160$	0	0
$161$	356414.	1.08365
$162$	−39082.8	−0.117003
$163$	−103720.	−0.305768	−0.152884	−	0.988244i	$-0.548856\pi$
$163$	−103720.	−0.305768	−0.152884	+	0.988244i	$0.548856\pi$
$164$	52681.6	0.152950
$165$	0	0
$166$	−56039.9	−0.157844
$167$	−544901.	−1.51191	−0.755956	−	0.654622i	$-0.772828\pi$
$167$	−544901.	−1.51191	−0.755956	+	0.654622i	$0.772828\pi$
$168$	−121805.	−0.332961
$169$	532326.	1.43371
$170$	0	0
$171$	−42398.4	−0.110882
$172$	40451.2	0.104258
$173$	26306.8	0.0668270	0.0334135	−	0.999442i	$-0.489362\pi$
$173$	26306.8	0.0668270	0.0334135	+	0.999442i	$0.489362\pi$
$174$	−148155.	−0.370974
$175$	0	0
$176$	135925.	0.330764
$177$	120723.	0.289639
$178$	−509893.	−1.20623
$179$	491490.	1.14652	0.573261	−	0.819373i	$-0.305678\pi$
$179$	491490.	1.14652	0.573261	+	0.819373i	$0.305678\pi$
$180$	0	0
$181$	417291.	0.946767	0.473383	−	0.880856i	$-0.343033\pi$
$181$	417291.	0.946767	0.473383	+	0.880856i	$0.343033\pi$
$182$	451155.	1.00960
$183$	−202064.	−0.446027
$184$	−759876.	−1.65462
$185$	0	0
$186$	445331.	0.943845
$187$	−46352.8	−0.0969330
$188$	18386.4	0.0379405
$189$	−58082.5	−0.118274
$190$	0	0
$191$	−488382.	−0.968672	−0.484336	−	0.874882i	$-0.660939\pi$
$191$	−488382.	−0.968672	−0.484336	+	0.874882i	$0.660939\pi$
$192$	256194.	0.501551
$193$	−493974.	−0.954576	−0.477288	−	0.878747i	$-0.658380\pi$
$193$	−493974.	−0.954576	−0.477288	+	0.878747i	$0.658380\pi$
$194$	633687.	1.20884
$195$	0	0
$196$	−36438.2	−0.0677511
$197$	−382228.	−0.701710	−0.350855	−	0.936430i	$-0.614109\pi$
$197$	−382228.	−0.701710	−0.350855	+	0.936430i	$0.614109\pi$
$198$	58383.0	0.105833
$199$	−990145.	−1.77242	−0.886209	−	0.463287i	$-0.846670\pi$
$199$	−990145.	−1.77242	−0.886209	+	0.463287i	$0.846670\pi$
$200$	0	0
$201$	−25679.1	−0.0448321
$202$	706500.	1.21824
$203$	−220179.	−0.375005
$204$	12011.5	0.0202080
$205$	0	0
$206$	760003.	1.24781
$207$	−362345.	−0.587754
$208$	−1.06784e6	−1.71139
$209$	63335.9	0.100296
$210$	0	0
$211$	−438779.	−0.678483	−0.339242	−	0.940699i	$-0.610170\pi$
$211$	−438779.	−0.678483	−0.339242	+	0.940699i	$0.610170\pi$
$212$	−112037.	−0.171207
$213$	345499.	0.521792
$214$	737577.	1.10096
$215$	0	0
$216$	123832.	0.180592
$217$	661824.	0.954099
$218$	−690681.	−0.984320
$219$	590506.	0.831983
$220$	0	0
$221$	364152.	0.501536
$222$	−121428.	−0.165362
$223$	371756.	0.500606	0.250303	−	0.968168i	$-0.419470\pi$
$223$	371756.	0.500606	0.250303	+	0.968168i	$0.419470\pi$
$224$	−100062.	−0.133245
$225$	0	0
$226$	−551903.	−0.718773
$227$	−94720.0	−0.122005	−0.0610024	−	0.998138i	$-0.519430\pi$
$227$	−94720.0	−0.122005	−0.0610024	+	0.998138i	$0.519430\pi$
$228$	−16412.4	−0.0209091
$229$	398854.	0.502603	0.251302	−	0.967909i	$-0.419141\pi$
$229$	398854.	0.502603	0.251302	+	0.967909i	$0.419141\pi$
$230$	0	0
$231$	86765.2	0.106983
$232$	469423.	0.572591
$233$	−1.29802e6	−1.56636	−0.783182	−	0.621792i	$-0.786405\pi$
$233$	−1.29802e6	−1.56636	−0.783182	+	0.621792i	$0.786405\pi$
$234$	−458663.	−0.547588
$235$	0	0
$236$	46731.9	0.0546177
$237$	65574.3	0.0758338
$238$	181813.	0.208057
$239$	−1.37494e6	−1.55700	−0.778498	−	0.627647i	$-0.784018\pi$
$239$	−1.37494e6	−1.55700	−0.778498	+	0.627647i	$0.784018\pi$
$240$	0	0
$241$	1.54232e6	1.71054	0.855268	−	0.518185i	$-0.173392\pi$
$241$	1.54232e6	1.71054	0.855268	+	0.518185i	$0.173392\pi$
$242$	−87214.0	−0.0957300
$243$	59049.0	0.0641500
$244$	−78219.0	−0.0841080
$245$	0	0
$246$	−810684.	−0.854110
$247$	−497574.	−0.518937
$248$	−1.41101e6	−1.45680
$249$	84668.9	0.0865417
$250$	0	0
$251$	−1.02033e6	−1.02225	−0.511126	−	0.859506i	$-0.670771\pi$
$251$	−1.02033e6	−1.02225	−0.511126	+	0.859506i	$0.670771\pi$
$252$	−22483.7	−0.0223032
$253$	541280.	0.531644
$254$	1.54914e6	1.50663
$255$	0	0
$256$	338569.	0.322885
$257$	−1.85297e6	−1.74999	−0.874994	−	0.484134i	$-0.839135\pi$
$257$	−1.85297e6	−1.74999	−0.874994	+	0.484134i	$0.839135\pi$
$258$	−622477.	−0.582202
$259$	−180458.	−0.167158
$260$	0	0
$261$	223843.	0.203396
$262$	1.05022e6	0.945204
$263$	−759462.	−0.677044	−0.338522	−	0.940959i	$-0.609927\pi$
$263$	−759462.	−0.677044	−0.338522	+	0.940959i	$0.609927\pi$
$264$	−184984.	−0.163352
$265$	0	0
$266$	−248427.	−0.215275
$267$	770382.	0.661345
$268$	−9940.37	−0.00845406
$269$	−1.63257e6	−1.37560	−0.687798	−	0.725902i	$-0.741423\pi$
$269$	−1.63257e6	−1.37560	−0.687798	+	0.725902i	$0.741423\pi$
$270$	0	0
$271$	−1.05641e6	−0.873792	−0.436896	−	0.899512i	$-0.643922\pi$
$271$	−1.05641e6	−0.873792	−0.436896	+	0.899512i	$0.643922\pi$
$272$	−430333.	−0.352681
$273$	−681637.	−0.553537
$274$	782157.	0.629387
$275$	0	0
$276$	−140264.	−0.110834
$277$	−2.05929e6	−1.61257	−0.806283	−	0.591530i	$-0.798524\pi$
$277$	−2.05929e6	−1.61257	−0.806283	+	0.591530i	$0.798524\pi$
$278$	−1.60345e6	−1.24436
$279$	−672837.	−0.517487
$280$	0	0
$281$	−828418.	−0.625869	−0.312935	−	0.949775i	$-0.601312\pi$
$281$	−828418.	−0.625869	−0.312935	+	0.949775i	$0.601312\pi$
$282$	−282937.	−0.211869
$283$	892040.	0.662091	0.331046	−	0.943615i	$-0.392599\pi$
$283$	892040.	0.662091	0.331046	+	0.943615i	$0.392599\pi$
$284$	133743.	0.0983952
$285$	0	0
$286$	685163.	0.495312
$287$	−1.20479e6	−0.863388
$288$	101727.	0.0722697
$289$	−1.27311e6	−0.896644
$290$	0	0
$291$	−957418.	−0.662779
$292$	228585.	0.156888
$293$	−817443.	−0.556274	−0.278137	−	0.960541i	$-0.589717\pi$
$293$	−817443.	−0.556274	−0.278137	+	0.960541i	$0.589717\pi$
$294$	560724.	0.378339
$295$	0	0
$296$	384738.	0.255233
$297$	−88209.0	−0.0580259
$298$	−2.08089e6	−1.35740
$299$	−4.25235e6	−2.75075
$300$	0	0
$301$	−925088.	−0.588527
$302$	−2.14010e6	−1.35026
$303$	−1.06743e6	−0.667932
$304$	588002.	0.364918
$305$	0	0
$306$	−184838.	−0.112846
$307$	778608.	0.471490	0.235745	−	0.971815i	$-0.424247\pi$
$307$	778608.	0.471490	0.235745	+	0.971815i	$0.424247\pi$
$308$	33586.8	0.0201740
$309$	−1.14826e6	−0.684141
$310$	0	0
$311$	−1.29583e6	−0.759708	−0.379854	−	0.925047i	$-0.624026\pi$
$311$	−1.29583e6	−0.759708	−0.379854	+	0.925047i	$0.624026\pi$
$312$	1.45325e6	0.845190
$313$	918062.	0.529677	0.264839	−	0.964293i	$-0.414681\pi$
$313$	918062.	0.529677	0.264839	+	0.964293i	$0.414681\pi$
$314$	−2.41558e6	−1.38260
$315$	0	0
$316$	25383.8	0.0143001
$317$	−1.28990e6	−0.720955	−0.360478	−	0.932768i	$-0.617386\pi$
$317$	−1.28990e6	−0.720955	−0.360478	+	0.932768i	$0.617386\pi$
$318$	1.72406e6	0.956061
$319$	−334383.	−0.183979
$320$	0	0
$321$	−1.11438e6	−0.603631
$322$	−2.12310e6	−1.14112
$323$	−200519.	−0.106942
$324$	22857.9	0.0120969
$325$	0	0
$326$	617841.	0.321983
$327$	1.04353e6	0.539678
$328$	2.56861e6	1.31830
$329$	−420484.	−0.214171
$330$	0	0
$331$	−862434.	−0.432669	−0.216335	−	0.976319i	$-0.569410\pi$
$331$	−862434.	−0.432669	−0.216335	+	0.976319i	$0.569410\pi$
$332$	32775.3	0.0163193
$333$	183461.	0.0906638
$334$	3.24589e6	1.59209
$335$	0	0
$336$	805516.	0.389248
$337$	306776.	0.147145	0.0735726	−	0.997290i	$-0.476560\pi$
$337$	306776.	0.147145	0.0735726	+	0.997290i	$0.476560\pi$
$338$	−3.17098e6	−1.50974
$339$	833853.	0.394085
$340$	0	0
$341$	1.00510e6	0.468084
$342$	252561.	0.116762
$343$	2.17240e6	0.997021
$344$	1.97229e6	0.898617
$345$	0	0
$346$	−156705.	−0.0703708
$347$	−92651.8	−0.0413076	−0.0206538	−	0.999787i	$-0.506575\pi$
$347$	−92651.8	−0.0413076	−0.0206538	+	0.999787i	$0.506575\pi$
$348$	86649.7	0.0383547
$349$	−2.42638e6	−1.06634	−0.533170	−	0.846008i	$-0.678999\pi$
$349$	−2.42638e6	−1.06634	−0.533170	+	0.846008i	$0.678999\pi$
$350$	0	0
$351$	692979.	0.300229
$352$	−151963.	−0.0653704
$353$	2.84879e6	1.21681	0.608405	−	0.793626i	$-0.291809\pi$
$353$	2.84879e6	1.21681	0.608405	+	0.793626i	$0.291809\pi$
$354$	−719128.	−0.304999
$355$	0	0
$356$	298215.	0.124711
$357$	−274695.	−0.114072
$358$	−2.92773e6	−1.20732
$359$	−3.57819e6	−1.46530	−0.732651	−	0.680604i	$-0.761717\pi$
$359$	−3.57819e6	−1.46530	−0.732651	+	0.680604i	$0.761717\pi$
$360$	0	0
$361$	−2.20211e6	−0.889347
$362$	−2.48574e6	−0.996974
$363$	131769.	0.0524864
$364$	−263862.	−0.104381
$365$	0	0
$366$	1.20366e6	0.469680
$367$	−598899.	−0.232107	−0.116054	−	0.993243i	$-0.537024\pi$
$367$	−598899.	−0.232107	−0.116054	+	0.993243i	$0.537024\pi$
$368$	5.02517e6	1.93433
$369$	1.22484e6	0.468287
$370$	0	0
$371$	2.56220e6	0.966448
$372$	−260455.	−0.0975834
$373$	−1.18252e6	−0.440084	−0.220042	−	0.975490i	$-0.570619\pi$
$373$	−1.18252e6	−0.440084	−0.220042	+	0.975490i	$0.570619\pi$
$374$	276116.	0.102073
$375$	0	0
$376$	896474.	0.327015
$377$	2.62695e6	0.951915
$378$	345988.	0.124546
$379$	3.01681e6	1.07882	0.539411	−	0.842042i	$-0.318647\pi$
$379$	3.01681e6	1.07882	0.539411	+	0.842042i	$0.318647\pi$
$380$	0	0
$381$	−2.34054e6	−0.826046
$382$	2.90921e6	1.02004
$383$	−2.72763e6	−0.950140	−0.475070	−	0.879948i	$-0.657577\pi$
$383$	−2.72763e6	−0.950140	−0.475070	+	0.879948i	$0.657577\pi$
$384$	−1.88780e6	−0.653323
$385$	0	0
$386$	2.94252e6	1.00520
$387$	940481.	0.319207
$388$	−370617.	−0.124981
$389$	2.16455e6	0.725259	0.362629	−	0.931933i	$-0.381879\pi$
$389$	2.16455e6	0.725259	0.362629	+	0.931933i	$0.381879\pi$
$390$	0	0
$391$	−1.71367e6	−0.566872
$392$	−1.77663e6	−0.583958
$393$	−1.58674e6	−0.518232
$394$	2.27687e6	0.738921
$395$	0	0
$396$	−34145.7	−0.0109420
$397$	−2.00142e6	−0.637326	−0.318663	−	0.947868i	$-0.603234\pi$
$397$	−2.00142e6	−0.637326	−0.318663	+	0.947868i	$0.603234\pi$
$398$	5.89813e6	1.86641
$399$	375340.	0.118030
$400$	0	0
$401$	−4.97800e6	−1.54594	−0.772972	−	0.634440i	$-0.781231\pi$
$401$	−4.97800e6	−1.54594	−0.772972	+	0.634440i	$0.781231\pi$
$402$	152966.	0.0472095
$403$	−7.89619e6	−2.42189
$404$	−413202.	−0.125953
$405$	0	0
$406$	1.31157e6	0.394891
$407$	−274060.	−0.0820085
$408$	585651.	0.174176
$409$	5.60391e6	1.65647	0.828234	−	0.560383i	$-0.189346\pi$
$409$	5.60391e6	1.65647	0.828234	+	0.560383i	$0.189346\pi$
$410$	0	0
$411$	−1.18174e6	−0.345077
$412$	−444493.	−0.129010
$413$	−1.06872e6	−0.308312
$414$	2.15843e6	0.618923
$415$	0	0
$416$	1.19384e6	0.338230
$417$	2.42261e6	0.682249
$418$	−377282.	−0.105615
$419$	−3.23684e6	−0.900712	−0.450356	−	0.892849i	$-0.648703\pi$
$419$	−3.23684e6	−0.900712	−0.450356	+	0.892849i	$0.648703\pi$
$420$	0	0
$421$	−4.89299e6	−1.34546	−0.672728	−	0.739890i	$-0.734877\pi$
$421$	−4.89299e6	−1.34546	−0.672728	+	0.739890i	$0.734877\pi$
$422$	2.61373e6	0.714463
$423$	427481.	0.116163
$424$	−5.46262e6	−1.47566
$425$	0	0
$426$	−2.05808e6	−0.549463
$427$	1.78881e6	0.474782
$428$	−431378.	−0.113828
$429$	−1.03519e6	−0.271567
$430$	0	0
$431$	1.93157e6	0.500862	0.250431	−	0.968134i	$-0.419428\pi$
$431$	1.93157e6	0.500862	0.250431	+	0.968134i	$0.419428\pi$
$432$	−818920.	−0.211121
$433$	4.44543e6	1.13945	0.569723	−	0.821837i	$-0.307050\pi$
$433$	4.44543e6	1.13945	0.569723	+	0.821837i	$0.307050\pi$
$434$	−3.94238e6	−1.00469
$435$	0	0
$436$	403950.	0.101768
$437$	2.34154e6	0.586541
$438$	−3.51755e6	−0.876103
$439$	−75576.5	−0.0187165	−0.00935827	−	0.999956i	$-0.502979\pi$
$439$	−75576.5	−0.0187165	−0.00935827	+	0.999956i	$0.502979\pi$
$440$	0	0
$441$	−847181.	−0.207434
$442$	−2.16920e6	−0.528133
$443$	−678748.	−0.164323	−0.0821617	−	0.996619i	$-0.526182\pi$
$443$	−678748.	−0.164323	−0.0821617	+	0.996619i	$0.526182\pi$
$444$	71017.9	0.0170966
$445$	0	0
$446$	−2.21449e6	−0.527153
$447$	3.14396e6	0.744231
$448$	−2.26800e6	−0.533886
$449$	2.83771e6	0.664281	0.332141	−	0.943230i	$-0.392229\pi$
$449$	2.83771e6	0.664281	0.332141	+	0.943230i	$0.392229\pi$
$450$	0	0
$451$	−1.82969e6	−0.423582
$452$	322785.	0.0743133
$453$	3.23341e6	0.740313
$454$	564232.	0.128475
$455$	0	0
$456$	−800227.	−0.180219
$457$	230073.	0.0515318	0.0257659	−	0.999668i	$-0.491798\pi$
$457$	230073.	0.0515318	0.0257659	+	0.999668i	$0.491798\pi$
$458$	−2.37591e6	−0.529256
$459$	279266.	0.0618709
$460$	0	0
$461$	1.24653e6	0.273181	0.136590	−	0.990628i	$-0.456386\pi$
$461$	1.24653e6	0.273181	0.136590	+	0.990628i	$0.456386\pi$
$462$	−516846.	−0.112657
$463$	−5.91283e6	−1.28187	−0.640934	−	0.767596i	$-0.721453\pi$
$463$	−5.91283e6	−1.28187	−0.640934	+	0.767596i	$0.721453\pi$
$464$	−3.10437e6	−0.669388
$465$	0	0
$466$	7.73212e6	1.64943
$467$	−3.56624e6	−0.756690	−0.378345	−	0.925665i	$-0.623507\pi$
$467$	−3.56624e6	−0.756690	−0.378345	+	0.925665i	$0.623507\pi$
$468$	268252.	0.0566147
$469$	227329.	0.0477224
$470$	0	0
$471$	3.64963e6	0.758047
$472$	2.27852e6	0.470759
$473$	−1.40492e6	−0.288734
$474$	−390616.	−0.0798553
$475$	0	0
$476$	−106334.	−0.0215108
$477$	−2.60484e6	−0.524185
$478$	8.19027e6	1.63956
$479$	3.80121e6	0.756978	0.378489	−	0.925606i	$-0.376444\pi$
$479$	3.80121e6	0.756978	0.378489	+	0.925606i	$0.376444\pi$
$480$	0	0
$481$	2.15304e6	0.424316
$482$	−9.18736e6	−1.80125
$483$	3.20772e6	0.625647
$484$	51007.8	0.00989745
$485$	0	0
$486$	−351745.	−0.0675519
$487$	1.01729e7	1.94367	0.971834	−	0.235668i	$-0.0757278\pi$
$487$	1.01729e7	1.94367	0.971834	+	0.235668i	$0.0757278\pi$
$488$	−3.81375e6	−0.724941
$489$	−933477.	−0.176535
$490$	0	0
$491$	888060.	0.166241	0.0831206	−	0.996539i	$-0.473511\pi$
$491$	888060.	0.166241	0.0831206	+	0.996539i	$0.473511\pi$
$492$	474134.	0.0883057
$493$	1.05864e6	0.196170
$494$	2.96397e6	0.546457
$495$	0	0
$496$	9.33123e6	1.70308
$497$	−3.05859e6	−0.555432
$498$	−504359.	−0.0911310
$499$	−5.67063e6	−1.01948	−0.509742	−	0.860327i	$-0.670259\pi$
$499$	−5.67063e6	−1.01948	−0.509742	+	0.860327i	$0.670259\pi$
$500$	0	0
$501$	−4.90411e6	−0.872903
$502$	6.07796e6	1.07646
$503$	−7.03837e6	−1.24037	−0.620186	−	0.784455i	$-0.712943\pi$
$503$	−7.03837e6	−1.24037	−0.620186	+	0.784455i	$0.712943\pi$
$504$	−1.09625e6	−0.192235
$505$	0	0
$506$	−3.22432e6	−0.559837
$507$	4.79093e6	0.827752
$508$	−906025.	−0.155769
$509$	−9.24770e6	−1.58212	−0.791060	−	0.611739i	$-0.790470\pi$
$509$	−9.24770e6	−1.58212	−0.791060	+	0.611739i	$0.790470\pi$
$510$	0	0
$511$	−5.22757e6	−0.885620
$512$	4.69538e6	0.791582
$513$	−381586.	−0.0640175
$514$	1.10378e7	1.84279
$515$	0	0
$516$	364060.	0.0601934
$517$	−638583.	−0.105073
$518$	1.07496e6	0.176023
$519$	236761.	0.0385826
$520$	0	0
$521$	9.67418e6	1.56142	0.780710	−	0.624893i	$-0.214858\pi$
$521$	9.67418e6	1.56142	0.780710	+	0.624893i	$0.214858\pi$
$522$	−1.33340e6	−0.214182
$523$	−4.58700e6	−0.733287	−0.366644	−	0.930361i	$-0.619493\pi$
$523$	−4.58700e6	−0.733287	−0.366644	+	0.930361i	$0.619493\pi$
$524$	−614227.	−0.0977239
$525$	0	0
$526$	4.52399e6	0.712947
$527$	−3.18211e6	−0.499101
$528$	1.22333e6	0.190966
$529$	1.35749e7	2.10910
$530$	0	0
$531$	1.08651e6	0.167223
$532$	145294.	0.0222571
$533$	1.43743e7	2.19163
$534$	−4.58904e6	−0.696416
$535$	0	0
$536$	−484666.	−0.0728669
$537$	4.42341e6	0.661945
$538$	9.72495e6	1.44854
$539$	1.26554e6	0.187631
$540$	0	0
$541$	6.88879e6	1.01193	0.505964	−	0.862554i	$-0.331137\pi$
$541$	6.88879e6	1.01193	0.505964	+	0.862554i	$0.331137\pi$
$542$	6.29285e6	0.920129
$543$	3.75562e6	0.546616
$544$	481108.	0.0697021
$545$	0	0
$546$	4.06040e6	0.582890
$547$	3.92352e6	0.560670	0.280335	−	0.959902i	$-0.409554\pi$
$547$	3.92352e6	0.560670	0.280335	+	0.959902i	$0.409554\pi$
$548$	−457451.	−0.0650718
$549$	−1.81858e6	−0.257514
$550$	0	0
$551$	−1.44652e6	−0.202976
$552$	−6.83888e6	−0.955295
$553$	−580509.	−0.0807228
$554$	1.22668e7	1.69808
$555$	0	0
$556$	937792.	0.128653
$557$	546361.	0.0746176	0.0373088	−	0.999304i	$-0.488121\pi$
$557$	546361.	0.0746176	0.0373088	+	0.999304i	$0.488121\pi$
$558$	4.00798e6	0.544929
$559$	1.10372e7	1.49392
$560$	0	0
$561$	−417175.	−0.0559643
$562$	4.93475e6	0.659059
$563$	−802286.	−0.106674	−0.0533370	−	0.998577i	$-0.516986\pi$
$563$	−802286.	−0.106674	−0.0533370	+	0.998577i	$0.516986\pi$
$564$	165478.	0.0219050
$565$	0	0
$566$	−5.31373e6	−0.697202
$567$	−522742.	−0.0682858
$568$	6.52093e6	0.848084
$569$	3.79343e6	0.491192	0.245596	−	0.969372i	$-0.421016\pi$
$569$	3.79343e6	0.491192	0.245596	+	0.969372i	$0.421016\pi$
$570$	0	0
$571$	−1.24710e7	−1.60070	−0.800349	−	0.599535i	$-0.795352\pi$
$571$	−1.24710e7	−1.60070	−0.800349	+	0.599535i	$0.795352\pi$
$572$	−400723.	−0.0512099
$573$	−4.39544e6	−0.559263
$574$	7.17673e6	0.909174
$575$	0	0
$576$	2.30574e6	0.289571
$577$	−3.37909e6	−0.422532	−0.211266	−	0.977429i	$-0.567759\pi$
$577$	−3.37909e6	−0.422532	−0.211266	+	0.977429i	$0.567759\pi$
$578$	7.58368e6	0.944193
$579$	−4.44576e6	−0.551125
$580$	0	0
$581$	−749547.	−0.0921210
$582$	5.70318e6	0.697927
$583$	3.89118e6	0.474143
$584$	1.11452e7	1.35225
$585$	0	0
$586$	4.86938e6	0.585773
$587$	−1.63422e7	−1.95755	−0.978777	−	0.204926i	$-0.934305\pi$
$587$	−1.63422e7	−1.95755	−0.978777	+	0.204926i	$0.934305\pi$
$588$	−327944.	−0.0391161
$589$	4.34800e6	0.516418
$590$	0	0
$591$	−3.44006e6	−0.405132
$592$	−2.54433e6	−0.298380
$593$	48632.1	0.00567919	0.00283959	−	0.999996i	$-0.499096\pi$
$593$	48632.1	0.00567919	0.00283959	+	0.999996i	$0.499096\pi$
$594$	525447.	0.0611030
$595$	0	0
$596$	1.21703e6	0.140341
$597$	−8.91130e6	−1.02331
$598$	2.53306e7	2.89662
$599$	9.21613e6	1.04950	0.524749	−	0.851257i	$-0.324159\pi$
$599$	9.21613e6	1.04950	0.524749	+	0.851257i	$0.324159\pi$
$600$	0	0
$601$	7.14586e6	0.806990	0.403495	−	0.914982i	$-0.367795\pi$
$601$	7.14586e6	0.806990	0.403495	+	0.914982i	$0.367795\pi$
$602$	5.51060e6	0.619737
$603$	−231112.	−0.0258838
$604$	1.25165e6	0.139602
$605$	0	0
$606$	6.35850e6	0.703353
$607$	−9.23797e6	−1.01767	−0.508833	−	0.860865i	$-0.669923\pi$
$607$	−9.23797e6	−1.01767	−0.508833	+	0.860865i	$0.669923\pi$
$608$	−657381.	−0.0721204
$609$	−1.98161e6	−0.216509
$610$	0	0
$611$	5.01677e6	0.543653
$612$	108104.	0.0116671
$613$	251574.	0.0270405	0.0135203	−	0.999909i	$-0.495696\pi$
$613$	251574.	0.0270405	0.0135203	+	0.999909i	$0.495696\pi$
$614$	−4.63804e6	−0.496493
$615$	0	0
$616$	1.63760e6	0.173883
$617$	−1.01400e7	−1.07233	−0.536163	−	0.844114i	$-0.680127\pi$
$617$	−1.01400e7	−1.07233	−0.536163	+	0.844114i	$0.680127\pi$
$618$	6.84002e6	0.720421
$619$	−1.17316e6	−0.123064	−0.0615322	−	0.998105i	$-0.519599\pi$
$619$	−1.17316e6	−0.123064	−0.0615322	+	0.998105i	$0.519599\pi$
$620$	0	0
$621$	−3.26110e6	−0.339340
$622$	7.71904e6	0.799995
$623$	−6.81995e6	−0.703981
$624$	−9.61057e6	−0.988070
$625$	0	0
$626$	−5.46875e6	−0.557766
$627$	570023.	0.0579060
$628$	1.41277e6	0.142946
$629$	867661.	0.0874427
$630$	0	0
$631$	8.85619e6	0.885469	0.442735	−	0.896653i	$-0.354008\pi$
$631$	8.85619e6	0.885469	0.442735	+	0.896653i	$0.354008\pi$
$632$	1.23765e6	0.123255
$633$	−3.94901e6	−0.391723
$634$	7.68373e6	0.759188
$635$	0	0
$636$	−1.00833e6	−0.0988464
$637$	−9.94223e6	−0.970812
$638$	1.99187e6	0.193735
$639$	3.10949e6	0.301257
$640$	0	0
$641$	−925852.	−0.0890013	−0.0445006	−	0.999009i	$-0.514170\pi$
$641$	−925852.	−0.0890013	−0.0445006	+	0.999009i	$0.514170\pi$
$642$	6.63820e6	0.635642
$643$	−2.99028e6	−0.285223	−0.142611	−	0.989779i	$-0.545550\pi$
$643$	−2.99028e6	−0.285223	−0.142611	+	0.989779i	$0.545550\pi$
$644$	1.24171e6	0.117979
$645$	0	0
$646$	1.19446e6	0.112613
$647$	1.21048e7	1.13684	0.568418	−	0.822740i	$-0.307556\pi$
$647$	1.21048e7	1.13684	0.568418	+	0.822740i	$0.307556\pi$
$648$	1.11449e6	0.104265
$649$	−1.62306e6	−0.151259
$650$	0	0
$651$	5.95642e6	0.550849
$652$	−361349.	−0.0332896
$653$	−1.93820e6	−0.177876	−0.0889378	−	0.996037i	$-0.528347\pi$
$653$	−1.93820e6	−0.177876	−0.0889378	+	0.996037i	$0.528347\pi$
$654$	−6.21613e6	−0.568297
$655$	0	0
$656$	−1.69866e7	−1.54116
$657$	5.31456e6	0.480345
$658$	2.50476e6	0.225528
$659$	−1.03225e7	−0.925913	−0.462957	−	0.886381i	$-0.653211\pi$
$659$	−1.03225e7	−0.925913	−0.462957	+	0.886381i	$0.653211\pi$
$660$	0	0
$661$	−2.02254e6	−0.180050	−0.0900249	−	0.995940i	$-0.528695\pi$
$661$	−2.02254e6	−0.180050	−0.0900249	+	0.995940i	$0.528695\pi$
$662$	5.13738e6	0.455614
$663$	3.27737e6	0.289562
$664$	1.59804e6	0.140659
$665$	0	0
$666$	−1.09285e6	−0.0954717
$667$	−1.23622e7	−1.07592
$668$	−1.89838e6	−0.164605
$669$	3.34581e6	0.289025
$670$	0	0
$671$	2.71664e6	0.232930
$672$	−900560.	−0.0769289
$673$	7.54573e6	0.642190	0.321095	−	0.947047i	$-0.395949\pi$
$673$	7.54573e6	0.642190	0.321095	+	0.947047i	$0.395949\pi$
$674$	−1.82741e6	−0.154948
$675$	0	0
$676$	1.85457e6	0.156091
$677$	1.16590e7	0.977661	0.488831	−	0.872379i	$-0.337424\pi$
$677$	1.16590e7	0.977661	0.488831	+	0.872379i	$0.337424\pi$
$678$	−4.96713e6	−0.414984
$679$	8.47572e6	0.705509
$680$	0	0
$681$	−852480.	−0.0704395
$682$	−5.98723e6	−0.492907
$683$	2.12902e7	1.74633	0.873167	−	0.487421i	$-0.162062\pi$
$683$	2.12902e7	1.74633	0.873167	+	0.487421i	$0.162062\pi$
$684$	−147712.	−0.0120719
$685$	0	0
$686$	−1.29406e7	−1.04989
$687$	3.58969e6	0.290178
$688$	−1.30430e7	−1.05053
$689$	−3.05695e7	−2.45324
$690$	0	0
$691$	1.72139e7	1.37146	0.685732	−	0.727854i	$-0.259482\pi$
$691$	1.72139e7	1.37146	0.685732	+	0.727854i	$0.259482\pi$
$692$	91650.1	0.00727559
$693$	780887.	0.0617668
$694$	551911.	0.0434981
$695$	0	0
$696$	4.22481e6	0.330586
$697$	5.79273e6	0.451650
$698$	1.44536e7	1.12289
$699$	−1.16822e7	−0.904341
$700$	0	0
$701$	−2.17573e7	−1.67229	−0.836143	−	0.548512i	$-0.815195\pi$
$701$	−2.17573e7	−1.67229	−0.836143	+	0.548512i	$0.815195\pi$
$702$	−4.12796e6	−0.316150
$703$	−1.18556e6	−0.0904766
$704$	−3.44438e6	−0.261927
$705$	0	0
$706$	−1.69698e7	−1.28134
$707$	9.44962e6	0.710994
$708$	420587.	0.0315336
$709$	−2.30042e7	−1.71867	−0.859333	−	0.511416i	$-0.829121\pi$
$709$	−2.30042e7	−1.71867	−0.859333	+	0.511416i	$0.829121\pi$
$710$	0	0
$711$	590169.	0.0437827
$712$	1.45402e7	1.07490
$713$	3.71588e7	2.73740
$714$	1.63631e6	0.120122
$715$	0	0
$716$	1.71230e6	0.124824
$717$	−1.23744e7	−0.898933
$718$	2.13147e7	1.54301
$719$	−1.38917e7	−1.00215	−0.501074	−	0.865404i	$-0.667062\pi$
$719$	−1.38917e7	−1.00215	−0.501074	+	0.865404i	$0.667062\pi$
$720$	0	0
$721$	1.01652e7	0.728248
$722$	1.31176e7	0.936509
$723$	1.38809e7	0.987579
$724$	1.45380e6	0.103076
$725$	0	0
$726$	−784926.	−0.0552697
$727$	5.64670e6	0.396240	0.198120	−	0.980178i	$-0.436516\pi$
$727$	5.64670e6	0.396240	0.198120	+	0.980178i	$0.436516\pi$
$728$	−1.28652e7	−0.899679
$729$	531441.	0.0370370
$730$	0	0
$731$	4.44791e6	0.307866
$732$	−703971.	−0.0485598
$733$	1.57063e7	1.07973	0.539864	−	0.841752i	$-0.318476\pi$
$733$	1.57063e7	1.07973	0.539864	+	0.841752i	$0.318476\pi$
$734$	3.56754e6	0.244416
$735$	0	0
$736$	−5.61810e6	−0.382292
$737$	345241.	0.0234128
$738$	−7.29615e6	−0.493120
$739$	1.21584e7	0.818965	0.409482	−	0.912318i	$-0.365709\pi$
$739$	1.21584e7	0.818965	0.409482	+	0.912318i	$0.365709\pi$
$740$	0	0
$741$	−4.47816e6	−0.299609
$742$	−1.52626e7	−1.01770
$743$	−1.73437e7	−1.15258	−0.576289	−	0.817246i	$-0.695500\pi$
$743$	−1.73437e7	−1.15258	−0.576289	+	0.817246i	$0.695500\pi$
$744$	−1.26991e7	−0.841087
$745$	0	0
$746$	7.04407e6	0.463422
$747$	762020.	0.0499649
$748$	−161489.	−0.0105533
$749$	9.86529e6	0.642547
$750$	0	0
$751$	7.24104e6	0.468491	0.234245	−	0.972177i	$-0.424738\pi$
$751$	7.24104e6	0.468491	0.234245	+	0.972177i	$0.424738\pi$
$752$	−5.92852e6	−0.382297
$753$	−9.18300e6	−0.590197
$754$	−1.56483e7	−1.00239
$755$	0	0
$756$	−202354.	−0.0128768
$757$	1.58893e7	1.00778	0.503889	−	0.863769i	$-0.331902\pi$
$757$	1.58893e7	1.00778	0.503889	+	0.863769i	$0.331902\pi$
$758$	−1.79706e7	−1.13603
$759$	4.87152e6	0.306945
$760$	0	0
$761$	−2.63464e7	−1.64915	−0.824573	−	0.565756i	$-0.808585\pi$
$761$	−2.63464e7	−1.64915	−0.824573	+	0.565756i	$0.808585\pi$
$762$	1.39422e7	0.869851
$763$	−9.23803e6	−0.574471
$764$	−1.70148e6	−0.105461
$765$	0	0
$766$	1.62480e7	1.00053
$767$	1.27509e7	0.782622
$768$	3.04712e6	0.186418
$769$	3.59395e6	0.219157	0.109579	−	0.993978i	$-0.465050\pi$
$769$	3.59395e6	0.219157	0.109579	+	0.993978i	$0.465050\pi$
$770$	0	0
$771$	−1.66767e7	−1.01036
$772$	−1.72096e6	−0.103927
$773$	−5.97666e6	−0.359757	−0.179879	−	0.983689i	$-0.557571\pi$
$773$	−5.97666e6	−0.359757	−0.179879	+	0.983689i	$0.557571\pi$
$774$	−5.60229e6	−0.336135
$775$	0	0
$776$	−1.80703e7	−1.07724
$777$	−1.62413e6	−0.0965089
$778$	−1.28938e7	−0.763719
$779$	−7.91513e6	−0.467320
$780$	0	0
$781$	−4.64504e6	−0.272497
$782$	1.02081e7	0.596934
$783$	2.01459e6	0.117431
$784$	1.17491e7	0.682676
$785$	0	0
$786$	9.45195e6	0.545714
$787$	−2.67481e7	−1.53942	−0.769709	−	0.638395i	$-0.779599\pi$
$787$	−2.67481e7	−1.53942	−0.769709	+	0.638395i	$0.779599\pi$
$788$	−1.33165e6	−0.0763965
$789$	−6.83516e6	−0.390891
$790$	0	0
$791$	−7.38184e6	−0.419492
$792$	−1.66485e6	−0.0943112
$793$	−2.13422e7	−1.20519
$794$	1.19221e7	0.671123
$795$	0	0
$796$	−3.44957e6	−0.192966
$797$	−2.23935e7	−1.24875	−0.624377	−	0.781123i	$-0.714647\pi$
$797$	−2.23935e7	−1.24875	−0.624377	+	0.781123i	$0.714647\pi$
$798$	−2.23584e6	−0.124289
$799$	2.02173e6	0.112035
$800$	0	0
$801$	6.93344e6	0.381828
$802$	2.96531e7	1.62792
$803$	−7.93903e6	−0.434489
$804$	−89463.3	−0.00488096
$805$	0	0
$806$	4.70363e7	2.55032
$807$	−1.46931e7	−0.794201
$808$	−2.01466e7	−1.08561
$809$	2.12645e7	1.14231	0.571155	−	0.820842i	$-0.306495\pi$
$809$	2.12645e7	1.14231	0.571155	+	0.820842i	$0.306495\pi$
$810$	0	0
$811$	−2.02233e7	−1.07969	−0.539846	−	0.841764i	$-0.681517\pi$
$811$	−2.02233e7	−1.07969	−0.539846	+	0.841764i	$0.681517\pi$
$812$	−767083.	−0.0408275
$813$	−9.50767e6	−0.504484
$814$	1.63253e6	0.0863574
$815$	0	0
$816$	−3.87300e6	−0.203621
$817$	−6.07757e6	−0.318548
$818$	−3.33816e7	−1.74431
$819$	−6.13473e6	−0.319584
$820$	0	0
$821$	1.50533e7	0.779426	0.389713	−	0.920936i	$-0.372574\pi$
$821$	1.50533e7	0.779426	0.389713	+	0.920936i	$0.372574\pi$
$822$	7.03942e6	0.363377
$823$	−2.47959e7	−1.27609	−0.638044	−	0.770000i	$-0.720256\pi$
$823$	−2.47959e7	−1.27609	−0.638044	+	0.770000i	$0.720256\pi$
$824$	−2.16723e7	−1.11195
$825$	0	0
$826$	6.36622e6	0.324662
$827$	3.25347e7	1.65418	0.827091	−	0.562068i	$-0.189994\pi$
$827$	3.25347e7	1.65418	0.827091	+	0.562068i	$0.189994\pi$
$828$	−1.26237e6	−0.0639900
$829$	−1.34087e7	−0.677642	−0.338821	−	0.940851i	$-0.610028\pi$
$829$	−1.34087e7	−0.677642	−0.338821	+	0.940851i	$0.610028\pi$
$830$	0	0
$831$	−1.85336e7	−0.931016
$832$	2.70594e7	1.35522
$833$	−4.00665e6	−0.200064
$834$	−1.44311e7	−0.718429
$835$	0	0
$836$	220656.	0.0109194
$837$	−6.05553e6	−0.298771
$838$	1.92813e7	0.948477
$839$	−1.63668e7	−0.802711	−0.401355	−	0.915922i	$-0.631461\pi$
$839$	−1.63668e7	−0.802711	−0.401355	+	0.915922i	$0.631461\pi$
$840$	0	0
$841$	−1.28742e7	−0.627670
$842$	2.91468e7	1.41680
$843$	−7.45576e6	−0.361346
$844$	−1.52866e6	−0.0738678
$845$	0	0
$846$	−2.54644e6	−0.122323
$847$	−1.16651e6	−0.0558702
$848$	3.61251e7	1.72512
$849$	8.02836e6	0.382259
$850$	0	0
$851$	−1.01320e7	−0.479593
$852$	1.20368e6	0.0568085
$853$	2.49013e7	1.17179	0.585894	−	0.810388i	$-0.300744\pi$
$853$	2.49013e7	1.17179	0.585894	+	0.810388i	$0.300744\pi$
$854$	−1.06556e7	−0.499960
$855$	0	0
$856$	−2.10328e7	−0.981100
$857$	−3.99296e7	−1.85713	−0.928567	−	0.371164i	$-0.878959\pi$
$857$	−3.99296e7	−1.85713	−0.928567	+	0.371164i	$0.878959\pi$
$858$	6.16646e6	0.285968
$859$	1.99329e7	0.921697	0.460848	−	0.887479i	$-0.347545\pi$
$859$	1.99329e7	0.921697	0.460848	+	0.887479i	$0.347545\pi$
$860$	0	0
$861$	−1.08431e7	−0.498478
$862$	−1.15061e7	−0.527423
$863$	2.96515e7	1.35525	0.677625	−	0.735407i	$-0.263009\pi$
$863$	2.96515e7	1.35525	0.677625	+	0.735407i	$0.263009\pi$
$864$	915546.	0.0417249
$865$	0	0
$866$	−2.64807e7	−1.19987
$867$	−1.14580e7	−0.517678
$868$	2.30573e6	0.103875
$869$	−881610.	−0.0396029
$870$	0	0
$871$	−2.71225e6	−0.121139
$872$	1.96955e7	0.877155
$873$	−8.61676e6	−0.382656
$874$	−1.39482e7	−0.617645
$875$	0	0
$876$	2.05727e6	0.0905796
$877$	−2.12185e7	−0.931571	−0.465785	−	0.884898i	$-0.654228\pi$
$877$	−2.12185e7	−0.931571	−0.465785	+	0.884898i	$0.654228\pi$
$878$	450197.	0.0197091
$879$	−7.35699e6	−0.321165
$880$	0	0
$881$	1.51363e6	0.0657023	0.0328511	−	0.999460i	$-0.489541\pi$
$881$	1.51363e6	0.0657023	0.0328511	+	0.999460i	$0.489541\pi$
$882$	5.04652e6	0.218434
$883$	7.39920e6	0.319362	0.159681	−	0.987169i	$-0.448953\pi$
$883$	7.39920e6	0.319362	0.159681	+	0.987169i	$0.448953\pi$
$884$	1.26867e6	0.0546032
$885$	0	0
$886$	4.04319e6	0.173037
$887$	5.34001e6	0.227894	0.113947	−	0.993487i	$-0.463651\pi$
$887$	5.34001e6	0.227894	0.113947	+	0.993487i	$0.463651\pi$
$888$	3.46264e6	0.147359
$889$	2.07201e7	0.879301
$890$	0	0
$891$	−793881.	−0.0335013
$892$	1.29516e6	0.0545020
$893$	−2.76246e6	−0.115923
$894$	−1.87280e7	−0.783697
$895$	0	0
$896$	1.67121e7	0.695443
$897$	−3.82712e7	−1.58815
$898$	−1.69038e7	−0.699508
$899$	−2.29553e7	−0.947293
$900$	0	0
$901$	−1.23193e7	−0.505561
$902$	1.08992e7	0.446044
$903$	−8.32579e6	−0.339786
$904$	1.57381e7	0.640518
$905$	0	0
$906$	−1.92609e7	−0.779572
$907$	−1.11942e6	−0.0451830	−0.0225915	−	0.999745i	$-0.507192\pi$
$907$	−1.11942e6	−0.0451830	−0.0225915	+	0.999745i	$0.507192\pi$
$908$	−329995.	−0.0132829
$909$	−9.60686e6	−0.385631
$910$	0	0
$911$	9.45712e6	0.377540	0.188770	−	0.982021i	$-0.439550\pi$
$911$	9.45712e6	0.377540	0.188770	+	0.982021i	$0.439550\pi$
$912$	5.29202e6	0.210685
$913$	−1.13833e6	−0.0451949
$914$	−1.37051e6	−0.0542645
$915$	0	0
$916$	1.38957e6	0.0547194
$917$	1.40469e7	0.551642
$918$	−1.66354e6	−0.0651519
$919$	3.29076e7	1.28531	0.642655	−	0.766156i	$-0.277833\pi$
$919$	3.29076e7	1.28531	0.642655	+	0.766156i	$0.277833\pi$
$920$	0	0
$921$	7.00747e6	0.272215
$922$	−7.42537e6	−0.287667
$923$	3.64919e7	1.40991
$924$	302281.	0.0116475
$925$	0	0
$926$	3.52218e7	1.34985
$927$	−1.03344e7	−0.394989
$928$	3.47065e6	0.132294
$929$	−2.81978e7	−1.07195	−0.535976	−	0.844233i	$-0.680056\pi$
$929$	−2.81978e7	−1.07195	−0.535976	+	0.844233i	$0.680056\pi$
$930$	0	0
$931$	5.47464e6	0.207005
$932$	−4.52219e6	−0.170533
$933$	−1.16625e7	−0.438617
$934$	2.12435e7	0.796817
$935$	0	0
$936$	1.30793e7	0.487971
$937$	−2.78967e7	−1.03802	−0.519008	−	0.854769i	$-0.673699\pi$
$937$	−2.78967e7	−1.03802	−0.519008	+	0.854769i	$0.673699\pi$
$938$	−1.35416e6	−0.0502531
$939$	8.26256e6	0.305809
$940$	0	0
$941$	−394888.	−0.0145378	−0.00726891	−	0.999974i	$-0.502314\pi$
$941$	−394888.	−0.0145378	−0.00726891	+	0.999974i	$0.502314\pi$
$942$	−2.17402e7	−0.798246
$943$	−6.76441e7	−2.47714
$944$	−1.50682e7	−0.550341
$945$	0	0
$946$	8.36886e6	0.304045
$947$	−2.18179e7	−0.790567	−0.395283	−	0.918559i	$-0.629354\pi$
$947$	−2.18179e7	−0.790567	−0.395283	+	0.918559i	$0.629354\pi$
$948$	228454.	0.00825617
$949$	6.23699e7	2.24807
$950$	0	0
$951$	−1.16091e7	−0.416244
$952$	−5.18458e6	−0.185405
$953$	1.36384e7	0.486443	0.243221	−	0.969971i	$-0.421796\pi$
$953$	1.36384e7	0.486443	0.243221	+	0.969971i	$0.421796\pi$
$954$	1.55166e7	0.551982
$955$	0	0
$956$	−4.79014e6	−0.169513
$957$	−3.00945e6	−0.106220
$958$	−2.26432e7	−0.797120
$959$	1.04616e7	0.367324
$960$	0	0
$961$	4.03709e7	1.41013
$962$	−1.28253e7	−0.446818
$963$	−1.00294e7	−0.348507
$964$	5.37330e6	0.186229
$965$	0	0
$966$	−1.91079e7	−0.658825
$967$	2.00566e7	0.689749	0.344874	−	0.938649i	$-0.387921\pi$
$967$	2.00566e7	0.689749	0.344874	+	0.938649i	$0.387921\pi$
$968$	2.48700e6	0.0853077
$969$	−1.80467e6	−0.0617431
$970$	0	0
$971$	1.11918e7	0.380936	0.190468	−	0.981693i	$-0.438999\pi$
$971$	1.11918e7	0.380936	0.190468	+	0.981693i	$0.438999\pi$
$972$	205721.	0.00698414
$973$	−2.14466e7	−0.726234
$974$	−6.05982e7	−2.04674
$975$	0	0
$976$	2.52209e7	0.847492
$977$	−1.70098e7	−0.570115	−0.285058	−	0.958510i	$-0.592013\pi$
$977$	−1.70098e7	−0.570115	−0.285058	+	0.958510i	$0.592013\pi$
$978$	5.56057e6	0.185897
$979$	−1.03574e7	−0.345376
$980$	0	0
$981$	9.39175e6	0.311583
$982$	−5.29003e6	−0.175057
$983$	2.74333e6	0.0905512	0.0452756	−	0.998975i	$-0.485583\pi$
$983$	2.74333e6	0.0905512	0.0452756	+	0.998975i	$0.485583\pi$
$984$	2.31175e7	0.761121
$985$	0	0
$986$	−6.30616e6	−0.206573
$987$	−3.78436e6	−0.123651
$988$	−1.73350e6	−0.0564977
$989$	−5.19400e7	−1.68854
$990$	0	0
$991$	5.63471e7	1.82258	0.911292	−	0.411762i	$-0.135086\pi$
$991$	5.63471e7	1.82258	0.911292	+	0.411762i	$0.135086\pi$
$992$	−1.04322e7	−0.336588
$993$	−7.76191e6	−0.249802
$994$	1.82195e7	0.584886
$995$	0	0
$996$	294978.	0.00942196
$997$	−1.48994e7	−0.474714	−0.237357	−	0.971423i	$-0.576281\pi$
$997$	−1.48994e7	−0.474714	−0.237357	+	0.971423i	$0.576281\pi$
$998$	3.37790e7	1.07355
$999$	1.65115e6	0.0523448

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.w.1.4		13
5.2	odd	4		165.6.c.a.34.8	✓	26
5.3	odd	4		165.6.c.a.34.19	yes	26
5.4	even	2		825.6.a.x.1.10		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.c.a.34.8	✓	26	5.2	odd	4
165.6.c.a.34.19	yes	26	5.3	odd	4
825.6.a.w.1.4		13	1.1	even	1	trivial
825.6.a.x.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)

\(f(q)\)	\(=\)	\(q-5.95684 q^{2} +9.00000 q^{3} +3.48390 q^{4} -53.6115 q^{6} -79.6742 q^{7} +169.866 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-5.95684 q^{2} +9.00000 q^{3} +3.48390 q^{4} -53.6115 q^{6} -79.6742 q^{7} +169.866 q^{8} +81.0000 q^{9} -121.000 q^{11} +31.3551 q^{12} +950.589 q^{13} +474.606 q^{14} -1123.35 q^{16} +383.081 q^{17} -482.504 q^{18} -523.437 q^{19} -717.068 q^{21} +720.777 q^{22} -4473.39 q^{23} +1528.79 q^{24} -5662.50 q^{26} +729.000 q^{27} -277.577 q^{28} +2763.50 q^{29} -8306.63 q^{31} +1255.89 q^{32} -1089.00 q^{33} -2281.95 q^{34} +282.196 q^{36} +2264.95 q^{37} +3118.03 q^{38} +8555.30 q^{39} +15121.4 q^{41} +4271.46 q^{42} +11610.9 q^{43} -421.552 q^{44} +26647.3 q^{46} +5277.54 q^{47} -10110.1 q^{48} -10459.0 q^{49} +3447.73 q^{51} +3311.76 q^{52} -32158.5 q^{53} -4342.53 q^{54} -13533.9 q^{56} -4710.94 q^{57} -16461.7 q^{58} +13413.7 q^{59} -22451.5 q^{61} +49481.2 q^{62} -6453.61 q^{63} +28466.0 q^{64} +6486.99 q^{66} -2853.23 q^{67} +1334.62 q^{68} -40260.5 q^{69} +38388.8 q^{71} +13759.1 q^{72} +65611.8 q^{73} -13492.0 q^{74} -1823.60 q^{76} +9640.58 q^{77} -50962.5 q^{78} +7286.04 q^{79} +6561.00 q^{81} -90076.0 q^{82} +9407.65 q^{83} -2498.19 q^{84} -69164.1 q^{86} +24871.5 q^{87} -20553.8 q^{88} +85598.0 q^{89} -75737.4 q^{91} -15584.8 q^{92} -74759.7 q^{93} -31437.5 q^{94} +11303.0 q^{96} -106380. q^{97} +62302.7 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 3 q^{2} + 117 q^{3} + 229 q^{4} - 27 q^{6} - 284 q^{7} - 369 q^{8} + 1053 q^{9}+O(q^{10}) \) 13 * q - 3 * q^2 + 117 * q^3 + 229 * q^4 - 27 * q^6 - 284 * q^7 - 369 * q^8 + 1053 * q^9	\( 13 q - 3 q^{2} + 117 q^{3} + 229 q^{4} - 27 q^{6} - 284 q^{7} - 369 q^{8} + 1053 q^{9} - 1573 q^{11} + 2061 q^{12} - 366 q^{13} - 2758 q^{14} + 4141 q^{16} - 2056 q^{17} - 243 q^{18} - 310 q^{19} - 2556 q^{21} + 363 q^{22} - 3612 q^{23} - 3321 q^{24} + 2280 q^{26} + 9477 q^{27} - 7896 q^{28} - 4848 q^{29} - 24 q^{31} - 38111 q^{32} - 14157 q^{33} + 5518 q^{34} + 18549 q^{36} + 8420 q^{37} - 474 q^{38} - 3294 q^{39} - 15120 q^{41} - 24822 q^{42} - 35492 q^{43} - 27709 q^{44} - 20280 q^{46} - 46544 q^{47} + 37269 q^{48} + 81837 q^{49} - 18504 q^{51} - 107194 q^{52} - 42256 q^{53} - 2187 q^{54} - 196602 q^{56} - 2790 q^{57} - 114160 q^{58} - 65592 q^{59} - 52042 q^{61} - 94972 q^{62} - 23004 q^{63} + 185977 q^{64} + 3267 q^{66} - 80580 q^{67} - 61108 q^{68} - 32508 q^{69} - 77820 q^{71} - 29889 q^{72} - 103050 q^{73} - 240028 q^{74} - 271174 q^{76} + 34364 q^{77} + 20520 q^{78} - 112258 q^{79} + 85293 q^{81} - 64060 q^{82} - 292150 q^{83} - 71064 q^{84} - 319250 q^{86} - 43632 q^{87} + 44649 q^{88} - 295810 q^{89} - 24200 q^{91} - 121328 q^{92} - 216 q^{93} - 358144 q^{94} - 342999 q^{96} - 49072 q^{97} - 101815 q^{98} - 127413 q^{99}+O(q^{100}) \) 13 * q - 3 * q^2 + 117 * q^3 + 229 * q^4 - 27 * q^6 - 284 * q^7 - 369 * q^8 + 1053 * q^9 - 1573 * q^11 + 2061 * q^12 - 366 * q^13 - 2758 * q^14 + 4141 * q^16 - 2056 * q^17 - 243 * q^18 - 310 * q^19 - 2556 * q^21 + 363 * q^22 - 3612 * q^23 - 3321 * q^24 + 2280 * q^26 + 9477 * q^27 - 7896 * q^28 - 4848 * q^29 - 24 * q^31 - 38111 * q^32 - 14157 * q^33 + 5518 * q^34 + 18549 * q^36 + 8420 * q^37 - 474 * q^38 - 3294 * q^39 - 15120 * q^41 - 24822 * q^42 - 35492 * q^43 - 27709 * q^44 - 20280 * q^46 - 46544 * q^47 + 37269 * q^48 + 81837 * q^49 - 18504 * q^51 - 107194 * q^52 - 42256 * q^53 - 2187 * q^54 - 196602 * q^56 - 2790 * q^57 - 114160 * q^58 - 65592 * q^59 - 52042 * q^61 - 94972 * q^62 - 23004 * q^63 + 185977 * q^64 + 3267 * q^66 - 80580 * q^67 - 61108 * q^68 - 32508 * q^69 - 77820 * q^71 - 29889 * q^72 - 103050 * q^73 - 240028 * q^74 - 271174 * q^76 + 34364 * q^77 + 20520 * q^78 - 112258 * q^79 + 85293 * q^81 - 64060 * q^82 - 292150 * q^83 - 71064 * q^84 - 319250 * q^86 - 43632 * q^87 + 44649 * q^88 - 295810 * q^89 - 24200 * q^91 - 121328 * q^92 - 216 * q^93 - 358144 * q^94 - 342999 * q^96 - 49072 * q^97 - 101815 * q^98 - 127413 * q^99

Introduction

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