LMFDB - Embedded newform 825.4.a.u.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,4,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, 4, names="a")

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

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
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:	$48.6765757547$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.91035289.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 285x^{2} + 286x + 19616 $ x^4 - 2x^3 - 285x^2 + 286*x + 19616
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-12.6191$ of defining polynomial
Character	$\chi$	$=$	825.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155 q^{2} +3.00000 q^{3} -5.56155 q^{4} +4.68466 q^{6} -24.2670 q^{7} -21.1771 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+1.56155 q^{2} +3.00000 q^{3} -5.56155 q^{4} +4.68466 q^{6} -24.2670 q^{7} -21.1771 q^{8} +9.00000 q^{9} -11.0000 q^{11} -16.6847 q^{12} -84.2360 q^{13} -37.8942 q^{14} +11.4233 q^{16} +40.9641 q^{17} +14.0540 q^{18} +120.887 q^{19} -72.8010 q^{21} -17.1771 q^{22} +9.94182 q^{23} -63.5312 q^{24} -131.539 q^{26} +27.0000 q^{27} +134.962 q^{28} +196.860 q^{29} +151.123 q^{31} +187.255 q^{32} -33.0000 q^{33} +63.9675 q^{34} -50.0540 q^{36} +253.477 q^{37} +188.772 q^{38} -252.708 q^{39} -179.351 q^{41} -113.683 q^{42} +90.4851 q^{43} +61.1771 q^{44} +15.5247 q^{46} +483.434 q^{47} +34.2699 q^{48} +245.887 q^{49} +122.892 q^{51} +468.483 q^{52} -567.348 q^{53} +42.1619 q^{54} +513.904 q^{56} +362.662 q^{57} +307.407 q^{58} -491.696 q^{59} +127.008 q^{61} +235.986 q^{62} -218.403 q^{63} +201.022 q^{64} -51.5312 q^{66} -628.226 q^{67} -227.824 q^{68} +29.8255 q^{69} +309.976 q^{71} -190.594 q^{72} +1144.85 q^{73} +395.817 q^{74} -672.322 q^{76} +266.937 q^{77} -394.617 q^{78} -87.4620 q^{79} +81.0000 q^{81} -280.067 q^{82} -1040.13 q^{83} +404.887 q^{84} +141.297 q^{86} +590.579 q^{87} +232.948 q^{88} +390.723 q^{89} +2044.15 q^{91} -55.2920 q^{92} +453.368 q^{93} +754.908 q^{94} +561.764 q^{96} -165.548 q^{97} +383.966 q^{98} -99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 12 q^{3} - 14 q^{4} - 6 q^{6} - 11 q^{7} + 6 q^{8} + 36 q^{9}+O(q^{10}) $ 4 * q - 2 * q^2 + 12 * q^3 - 14 * q^4 - 6 * q^6 - 11 * q^7 + 6 * q^8 + 36 * q^9	$ 4 q - 2 q^{2} + 12 q^{3} - 14 q^{4} - 6 q^{6} - 11 q^{7} + 6 q^{8} + 36 q^{9} - 44 q^{11} - 42 q^{12} + 25 q^{13} - 3 q^{14} - 78 q^{16} + 85 q^{17} - 18 q^{18} + 118 q^{19} - 33 q^{21} + 22 q^{22} - 10 q^{23} + 18 q^{24} - 157 q^{26} + 108 q^{27} + 47 q^{28} + 251 q^{29} - 135 q^{31} + 246 q^{32} - 132 q^{33} + 289 q^{34} - 126 q^{36} + 419 q^{37} - 76 q^{38} + 75 q^{39} - 103 q^{41} - 9 q^{42} + 15 q^{43} + 154 q^{44} - 420 q^{46} + 665 q^{47} - 234 q^{48} + 1003 q^{49} + 255 q^{51} + 57 q^{52} - 116 q^{53} - 54 q^{54} + 77 q^{56} + 354 q^{57} + 189 q^{58} + 951 q^{59} - 350 q^{61} - 213 q^{62} - 99 q^{63} + 1538 q^{64} + 66 q^{66} + 266 q^{67} - 629 q^{68} - 30 q^{69} + 1526 q^{71} + 54 q^{72} - 566 q^{73} + 1091 q^{74} - 396 q^{76} + 121 q^{77} - 471 q^{78} + 2567 q^{79} + 324 q^{81} + 9 q^{82} - 961 q^{83} + 141 q^{84} + 1837 q^{86} + 753 q^{87} - 66 q^{88} + 1053 q^{89} + 2150 q^{91} + 460 q^{92} - 405 q^{93} + 2209 q^{94} + 738 q^{96} - 172 q^{97} - 1819 q^{98} - 396 q^{99}+O(q^{100}) $ 4 * q - 2 * q^2 + 12 * q^3 - 14 * q^4 - 6 * q^6 - 11 * q^7 + 6 * q^8 + 36 * q^9 - 44 * q^11 - 42 * q^12 + 25 * q^13 - 3 * q^14 - 78 * q^16 + 85 * q^17 - 18 * q^18 + 118 * q^19 - 33 * q^21 + 22 * q^22 - 10 * q^23 + 18 * q^24 - 157 * q^26 + 108 * q^27 + 47 * q^28 + 251 * q^29 - 135 * q^31 + 246 * q^32 - 132 * q^33 + 289 * q^34 - 126 * q^36 + 419 * q^37 - 76 * q^38 + 75 * q^39 - 103 * q^41 - 9 * q^42 + 15 * q^43 + 154 * q^44 - 420 * q^46 + 665 * q^47 - 234 * q^48 + 1003 * q^49 + 255 * q^51 + 57 * q^52 - 116 * q^53 - 54 * q^54 + 77 * q^56 + 354 * q^57 + 189 * q^58 + 951 * q^59 - 350 * q^61 - 213 * q^62 - 99 * q^63 + 1538 * q^64 + 66 * q^66 + 266 * q^67 - 629 * q^68 - 30 * q^69 + 1526 * q^71 + 54 * q^72 - 566 * q^73 + 1091 * q^74 - 396 * q^76 + 121 * q^77 - 471 * q^78 + 2567 * q^79 + 324 * q^81 + 9 * q^82 - 961 * q^83 + 141 * q^84 + 1837 * q^86 + 753 * q^87 - 66 * q^88 + 1053 * q^89 + 2150 * q^91 + 460 * q^92 - 405 * q^93 + 2209 * q^94 + 738 * q^96 - 172 * q^97 - 1819 * q^98 - 396 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.56155	0.552092	0.276046	−	0.961144i	$-0.410976\pi$
$2$	1.56155	0.552092	0.276046	+	0.961144i	$0.410976\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−5.56155	−0.695194
$5$	0	0
$5$	0	0
$6$	4.68466	0.318751
$7$	−24.2670	−1.31029	−0.655147	−	0.755501i	$-0.727394\pi$
$7$	−24.2670	−1.31029	−0.655147	+	0.755501i	$0.727394\pi$
$8$	−21.1771	−0.935904
$9$	9.00000	0.333333
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−16.6847	−0.401371
$13$	−84.2360	−1.79714	−0.898571	−	0.438828i	$-0.855394\pi$
$13$	−84.2360	−1.79714	−0.898571	+	0.438828i	$0.855394\pi$
$14$	−37.8942	−0.723404
$15$	0	0
$16$	11.4233	0.178489
$17$	40.9641	0.584426	0.292213	−	0.956353i	$-0.405608\pi$
$17$	40.9641	0.584426	0.292213	+	0.956353i	$0.405608\pi$
$18$	14.0540	0.184031
$19$	120.887	1.45966	0.729828	−	0.683630i	$-0.239600\pi$
$19$	120.887	1.45966	0.729828	+	0.683630i	$0.239600\pi$
$20$	0	0
$21$	−72.8010	−0.756499
$22$	−17.1771	−0.166462
$23$	9.94182	0.0901310	0.0450655	−	0.998984i	$-0.485650\pi$
$23$	9.94182	0.0901310	0.0450655	+	0.998984i	$0.485650\pi$
$24$	−63.5312	−0.540344
$25$	0	0
$26$	−131.539	−0.992189
$27$	27.0000	0.192450
$28$	134.962	0.910909
$29$	196.860	1.26055	0.630275	−	0.776372i	$-0.282942\pi$
$29$	196.860	1.26055	0.630275	+	0.776372i	$0.282942\pi$
$30$	0	0
$31$	151.123	0.875562	0.437781	−	0.899082i	$-0.355764\pi$
$31$	151.123	0.875562	0.437781	+	0.899082i	$0.355764\pi$
$32$	187.255	1.03445
$33$	−33.0000	−0.174078
$34$	63.9675	0.322657
$35$	0	0
$36$	−50.0540	−0.231731
$37$	253.477	1.12625	0.563126	−	0.826371i	$-0.309599\pi$
$37$	253.477	1.12625	0.563126	+	0.826371i	$0.309599\pi$
$38$	188.772	0.805865
$39$	−252.708	−1.03758
$40$	0	0
$41$	−179.351	−0.683170	−0.341585	−	0.939851i	$-0.610964\pi$
$41$	−179.351	−0.683170	−0.341585	+	0.939851i	$0.610964\pi$
$42$	−113.683	−0.417657
$43$	90.4851	0.320903	0.160452	−	0.987044i	$-0.448705\pi$
$43$	90.4851	0.320903	0.160452	+	0.987044i	$0.448705\pi$
$44$	61.1771	0.209609
$45$	0	0
$46$	15.5247	0.0497606
$47$	483.434	1.50034	0.750172	−	0.661243i	$-0.229971\pi$
$47$	483.434	1.50034	0.750172	+	0.661243i	$0.229971\pi$
$48$	34.2699	0.103051
$49$	245.887	0.716873
$50$	0	0
$51$	122.892	0.337419
$52$	468.483	1.24936
$53$	−567.348	−1.47040	−0.735200	−	0.677850i	$-0.762912\pi$
$53$	−567.348	−1.47040	−0.735200	+	0.677850i	$0.762912\pi$
$54$	42.1619	0.106250
$55$	0	0
$56$	513.904	1.22631
$57$	362.662	0.842733
$58$	307.407	0.695939
$59$	−491.696	−1.08497	−0.542487	−	0.840064i	$-0.682517\pi$
$59$	−491.696	−1.08497	−0.542487	+	0.840064i	$0.682517\pi$
$60$	0	0
$61$	127.008	0.266585	0.133292	−	0.991077i	$-0.457445\pi$
$61$	127.008	0.266585	0.133292	+	0.991077i	$0.457445\pi$
$62$	235.986	0.483391
$63$	−218.403	−0.436765
$64$	201.022	0.392621
$65$	0	0
$66$	−51.5312	−0.0961069
$67$	−628.226	−1.14552	−0.572762	−	0.819722i	$-0.694128\pi$
$67$	−628.226	−1.14552	−0.572762	+	0.819722i	$0.694128\pi$
$68$	−227.824	−0.406290
$69$	29.8255	0.0520372
$70$	0	0
$71$	309.976	0.518133	0.259066	−	0.965859i	$-0.416585\pi$
$71$	309.976	0.518133	0.259066	+	0.965859i	$0.416585\pi$
$72$	−190.594	−0.311968
$73$	1144.85	1.83554	0.917772	−	0.397109i	$-0.129986\pi$
$73$	1144.85	1.83554	0.917772	+	0.397109i	$0.129986\pi$
$74$	395.817	0.621795
$75$	0	0
$76$	−672.322	−1.01474
$77$	266.937	0.395069
$78$	−394.617	−0.572840
$79$	−87.4620	−0.124560	−0.0622800	−	0.998059i	$-0.519837\pi$
$79$	−87.4620	−0.124560	−0.0622800	+	0.998059i	$0.519837\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	−280.067	−0.377173
$83$	−1040.13	−1.37553	−0.687764	−	0.725934i	$-0.741408\pi$
$83$	−1040.13	−1.37553	−0.687764	+	0.725934i	$0.741408\pi$
$84$	404.887	0.525914
$85$	0	0
$86$	141.297	0.177168
$87$	590.579	0.727778
$88$	232.948	0.282186
$89$	390.723	0.465355	0.232677	−	0.972554i	$-0.425251\pi$
$89$	390.723	0.465355	0.232677	+	0.972554i	$0.425251\pi$
$90$	0	0
$91$	2044.15	2.35479
$92$	−55.2920	−0.0626585
$93$	453.368	0.505506
$94$	754.908	0.828328
$95$	0	0
$96$	561.764	0.597238
$97$	−165.548	−0.173287	−0.0866433	−	0.996239i	$-0.527614\pi$
$97$	−165.548	−0.173287	−0.0866433	+	0.996239i	$0.527614\pi$
$98$	383.966	0.395780
$99$	−99.0000	−0.100504
$100$	0	0
$101$	1510.35	1.48798	0.743990	−	0.668191i	$-0.232931\pi$
$101$	1510.35	1.48798	0.743990	+	0.668191i	$0.232931\pi$
$102$	191.903	0.186286
$103$	−1245.78	−1.19175	−0.595876	−	0.803076i	$-0.703195\pi$
$103$	−1245.78	−1.19175	−0.595876	+	0.803076i	$0.703195\pi$
$104$	1783.87	1.68195
$105$	0	0
$106$	−885.944	−0.811797
$107$	550.568	0.497434	0.248717	−	0.968576i	$-0.419991\pi$
$107$	550.568	0.497434	0.248717	+	0.968576i	$0.419991\pi$
$108$	−150.162	−0.133790
$109$	1380.30	1.21293	0.606463	−	0.795112i	$-0.292588\pi$
$109$	1380.30	1.21293	0.606463	+	0.795112i	$0.292588\pi$
$110$	0	0
$111$	760.430	0.650242
$112$	−277.209	−0.233873
$113$	−1017.96	−0.847445	−0.423723	−	0.905792i	$-0.639277\pi$
$113$	−1017.96	−0.847445	−0.423723	+	0.905792i	$0.639277\pi$
$114$	566.316	0.465267
$115$	0	0
$116$	−1094.85	−0.876326
$117$	−758.124	−0.599048
$118$	−767.810	−0.599005
$119$	−994.075	−0.765770
$120$	0	0
$121$	121.000	0.0909091
$122$	198.329	0.147179
$123$	−538.054	−0.394429
$124$	−840.477	−0.608686
$125$	0	0
$126$	−341.048	−0.241135
$127$	1181.86	0.825770	0.412885	−	0.910783i	$-0.364521\pi$
$127$	1181.86	0.825770	0.412885	+	0.910783i	$0.364521\pi$
$128$	−1184.13	−0.817683
$129$	271.455	0.185274
$130$	0	0
$131$	−2615.26	−1.74425	−0.872124	−	0.489284i	$-0.837258\pi$
$131$	−2615.26	−1.74425	−0.872124	+	0.489284i	$0.837258\pi$
$132$	183.531	0.121018
$133$	−2933.58	−1.91258
$134$	−981.009	−0.632435
$135$	0	0
$136$	−867.499	−0.546966
$137$	465.369	0.290213	0.145106	−	0.989416i	$-0.453648\pi$
$137$	465.369	0.290213	0.145106	+	0.989416i	$0.453648\pi$
$138$	46.5740	0.0287293
$139$	2809.52	1.71439	0.857196	−	0.514990i	$-0.172204\pi$
$139$	2809.52	1.71439	0.857196	+	0.514990i	$0.172204\pi$
$140$	0	0
$141$	1450.30	0.866224
$142$	484.045	0.286057
$143$	926.596	0.541859
$144$	102.810	0.0594963
$145$	0	0
$146$	1787.75	1.01339
$147$	737.662	0.413887
$148$	−1409.72	−0.782963
$149$	−583.496	−0.320818	−0.160409	−	0.987051i	$-0.551281\pi$
$149$	−583.496	−0.320818	−0.160409	+	0.987051i	$0.551281\pi$
$150$	0	0
$151$	1244.33	0.670610	0.335305	−	0.942110i	$-0.391161\pi$
$151$	1244.33	0.670610	0.335305	+	0.942110i	$0.391161\pi$
$152$	−2560.04	−1.36610
$153$	368.676	0.194809
$154$	416.836	0.218114
$155$	0	0
$156$	1405.45	0.721320
$157$	303.991	0.154529	0.0772647	−	0.997011i	$-0.475381\pi$
$157$	303.991	0.154529	0.0772647	+	0.997011i	$0.475381\pi$
$158$	−136.577	−0.0687687
$159$	−1702.04	−0.848936
$160$	0	0
$161$	−241.258	−0.118098
$162$	126.486	0.0613436
$163$	−259.982	−0.124929	−0.0624643	−	0.998047i	$-0.519896\pi$
$163$	−259.982	−0.124929	−0.0624643	+	0.998047i	$0.519896\pi$
$164$	997.472	0.474936
$165$	0	0
$166$	−1624.21	−0.759419
$167$	3468.36	1.60712	0.803562	−	0.595222i	$-0.202936\pi$
$167$	3468.36	1.60712	0.803562	+	0.595222i	$0.202936\pi$
$168$	1541.71	0.708010
$169$	4898.70	2.22972
$170$	0	0
$171$	1087.99	0.486552
$172$	−503.237	−0.223090
$173$	3864.34	1.69827	0.849134	−	0.528177i	$-0.177124\pi$
$173$	3864.34	1.69827	0.849134	+	0.528177i	$0.177124\pi$
$174$	922.220	0.401801
$175$	0	0
$176$	−125.656	−0.0538164
$177$	−1475.09	−0.626410
$178$	610.135	0.256919
$179$	163.870	0.0684258	0.0342129	−	0.999415i	$-0.489108\pi$
$179$	163.870	0.0684258	0.0342129	+	0.999415i	$0.489108\pi$
$180$	0	0
$181$	−3200.77	−1.31443	−0.657214	−	0.753704i	$-0.728265\pi$
$181$	−3200.77	−1.31443	−0.657214	+	0.753704i	$0.728265\pi$
$182$	3192.05	1.30006
$183$	381.023	0.153913
$184$	−210.539	−0.0843539
$185$	0	0
$186$	707.958	0.279086
$187$	−450.605	−0.176211
$188$	−2688.65	−1.04303
$189$	−655.209	−0.252166
$190$	0	0
$191$	2421.03	0.917171	0.458585	−	0.888650i	$-0.348356\pi$
$191$	2421.03	0.917171	0.458585	+	0.888650i	$0.348356\pi$
$192$	603.065	0.226680
$193$	2275.44	0.848651	0.424326	−	0.905510i	$-0.360511\pi$
$193$	2275.44	0.848651	0.424326	+	0.905510i	$0.360511\pi$
$194$	−258.511	−0.0956702
$195$	0	0
$196$	−1367.52	−0.498366
$197$	−1957.94	−0.708107	−0.354054	−	0.935225i	$-0.615197\pi$
$197$	−1957.94	−0.708107	−0.354054	+	0.935225i	$0.615197\pi$
$198$	−154.594	−0.0554874
$199$	−1975.36	−0.703665	−0.351833	−	0.936063i	$-0.614441\pi$
$199$	−1975.36	−0.703665	−0.351833	+	0.936063i	$0.614441\pi$
$200$	0	0
$201$	−1884.68	−0.661368
$202$	2358.50	0.821502
$203$	−4777.19	−1.65169
$204$	−683.471	−0.234571
$205$	0	0
$206$	−1945.35	−0.657958
$207$	89.4764	0.0300437
$208$	−962.252	−0.320770
$209$	−1329.76	−0.440103
$210$	0	0
$211$	1179.92	0.384973	0.192486	−	0.981300i	$-0.438345\pi$
$211$	1179.92	0.384973	0.192486	+	0.981300i	$0.438345\pi$
$212$	3155.34	1.02221
$213$	929.929	0.299144
$214$	859.741	0.274629
$215$	0	0
$216$	−571.781	−0.180115
$217$	−3667.29	−1.14724
$218$	2155.41	0.669647
$219$	3434.55	1.05975
$220$	0	0
$221$	−3450.65	−1.05030
$222$	1187.45	0.358993
$223$	−5742.73	−1.72449	−0.862247	−	0.506489i	$-0.830943\pi$
$223$	−5742.73	−1.72449	−0.862247	+	0.506489i	$0.830943\pi$
$224$	−4544.11	−1.35543
$225$	0	0
$226$	−1589.59	−0.467868
$227$	−713.160	−0.208520	−0.104260	−	0.994550i	$-0.533247\pi$
$227$	−713.160	−0.208520	−0.104260	+	0.994550i	$0.533247\pi$
$228$	−2016.97	−0.585863
$229$	4438.09	1.28069	0.640343	−	0.768089i	$-0.278792\pi$
$229$	4438.09	1.28069	0.640343	+	0.768089i	$0.278792\pi$
$230$	0	0
$231$	800.811	0.228093
$232$	−4168.91	−1.17975
$233$	614.179	0.172688	0.0863438	−	0.996265i	$-0.472482\pi$
$233$	614.179	0.172688	0.0863438	+	0.996265i	$0.472482\pi$
$234$	−1183.85	−0.330730
$235$	0	0
$236$	2734.60	0.754267
$237$	−262.386	−0.0719148
$238$	−1552.30	−0.422776
$239$	5191.00	1.40493	0.702464	−	0.711720i	$-0.252083\pi$
$239$	5191.00	1.40493	0.702464	+	0.711720i	$0.252083\pi$
$240$	0	0
$241$	−353.708	−0.0945408	−0.0472704	−	0.998882i	$-0.515052\pi$
$241$	−353.708	−0.0945408	−0.0472704	+	0.998882i	$0.515052\pi$
$242$	188.948	0.0501902
$243$	243.000	0.0641500
$244$	−706.360	−0.185328
$245$	0	0
$246$	−840.200	−0.217761
$247$	−10183.1	−2.62321
$248$	−3200.34	−0.819442
$249$	−3120.38	−0.794162
$250$	0	0
$251$	2612.91	0.657073	0.328537	−	0.944491i	$-0.393445\pi$
$251$	2612.91	0.657073	0.328537	+	0.944491i	$0.393445\pi$
$252$	1214.66	0.303636
$253$	−109.360	−0.0271755
$254$	1845.53	0.455901
$255$	0	0
$256$	−3457.26	−0.844057
$257$	−2434.38	−0.590866	−0.295433	−	0.955364i	$-0.595464\pi$
$257$	−2434.38	−0.590866	−0.295433	+	0.955364i	$0.595464\pi$
$258$	423.892	0.102288
$259$	−6151.12	−1.47572
$260$	0	0
$261$	1771.74	0.420183
$262$	−4083.87	−0.962986
$263$	7204.41	1.68914	0.844568	−	0.535449i	$-0.179857\pi$
$263$	7204.41	1.68914	0.844568	+	0.535449i	$0.179857\pi$
$264$	698.844	0.162920
$265$	0	0
$266$	−4580.93	−1.05592
$267$	1172.17	0.268673
$268$	3493.91	0.796361
$269$	5232.32	1.18595	0.592975	−	0.805221i	$-0.297953\pi$
$269$	5232.32	1.18595	0.592975	+	0.805221i	$0.297953\pi$
$270$	0	0
$271$	6231.13	1.39673	0.698366	−	0.715741i	$-0.253911\pi$
$271$	6231.13	1.39673	0.698366	+	0.715741i	$0.253911\pi$
$272$	467.944	0.104314
$273$	6132.46	1.35954
$274$	726.698	0.160224
$275$	0	0
$276$	−165.876	−0.0361759
$277$	4940.23	1.07159	0.535793	−	0.844349i	$-0.320013\pi$
$277$	4940.23	1.07159	0.535793	+	0.844349i	$0.320013\pi$
$278$	4387.22	0.946503
$279$	1360.10	0.291854
$280$	0	0
$281$	−1766.97	−0.375119	−0.187559	−	0.982253i	$-0.560058\pi$
$281$	−1766.97	−0.375119	−0.187559	+	0.982253i	$0.560058\pi$
$282$	2264.72	0.478235
$283$	5304.29	1.11416	0.557080	−	0.830459i	$-0.311922\pi$
$283$	5304.29	1.11416	0.557080	+	0.830459i	$0.311922\pi$
$284$	−1723.95	−0.360203
$285$	0	0
$286$	1446.93	0.299156
$287$	4352.32	0.895155
$288$	1685.29	0.344815
$289$	−3234.95	−0.658446
$290$	0	0
$291$	−496.643	−0.100047
$292$	−6367.15	−1.27606
$293$	−8356.39	−1.66616	−0.833081	−	0.553151i	$-0.813425\pi$
$293$	−8356.39	−1.66616	−0.833081	+	0.553151i	$0.813425\pi$
$294$	1151.90	0.228504
$295$	0	0
$296$	−5367.90	−1.05406
$297$	−297.000	−0.0580259
$298$	−911.160	−0.177121
$299$	−837.459	−0.161978
$300$	0	0
$301$	−2195.80	−0.420478
$302$	1943.09	0.370238
$303$	4531.06	0.859085
$304$	1380.93	0.260533
$305$	0	0
$306$	575.708	0.107552
$307$	−8558.29	−1.59103	−0.795517	−	0.605932i	$-0.792800\pi$
$307$	−8558.29	−1.59103	−0.795517	+	0.605932i	$0.792800\pi$
$308$	−1484.58	−0.274649
$309$	−3737.35	−0.688059
$310$	0	0
$311$	6135.19	1.11863	0.559316	−	0.828954i	$-0.311064\pi$
$311$	6135.19	1.11863	0.559316	+	0.828954i	$0.311064\pi$
$312$	5351.62	0.971076
$313$	4088.62	0.738346	0.369173	−	0.929361i	$-0.379641\pi$
$313$	4088.62	0.738346	0.369173	+	0.929361i	$0.379641\pi$
$314$	474.698	0.0853145
$315$	0	0
$316$	486.425	0.0865934
$317$	7957.42	1.40988	0.704942	−	0.709265i	$-0.250973\pi$
$317$	7957.42	1.40988	0.704942	+	0.709265i	$0.250973\pi$
$318$	−2657.83	−0.468691
$319$	−2165.46	−0.380070
$320$	0	0
$321$	1651.70	0.287194
$322$	−376.737	−0.0652011
$323$	4952.04	0.853062
$324$	−450.486	−0.0772438
$325$	0	0
$326$	−405.976	−0.0689721
$327$	4140.91	0.700283
$328$	3798.14	0.639382
$329$	−11731.5	−1.96589
$330$	0	0
$331$	−3113.97	−0.517097	−0.258549	−	0.965998i	$-0.583244\pi$
$331$	−3113.97	−0.517097	−0.258549	+	0.965998i	$0.583244\pi$
$332$	5784.72	0.956259
$333$	2281.29	0.375417
$334$	5416.02	0.887280
$335$	0	0
$336$	−831.627	−0.135027
$337$	−8267.16	−1.33632	−0.668161	−	0.744016i	$-0.732918\pi$
$337$	−8267.16	−1.33632	−0.668161	+	0.744016i	$0.732918\pi$
$338$	7649.58	1.23101
$339$	−3053.87	−0.489273
$340$	0	0
$341$	−1662.35	−0.263992
$342$	1698.95	0.268622
$343$	2356.63	0.370980
$344$	−1916.21	−0.300335
$345$	0	0
$346$	6034.37	0.937601
$347$	8673.03	1.34177	0.670883	−	0.741563i	$-0.265915\pi$
$347$	8673.03	1.34177	0.670883	+	0.741563i	$0.265915\pi$
$348$	−3284.54	−0.505947
$349$	3351.21	0.514000	0.257000	−	0.966411i	$-0.417266\pi$
$349$	3351.21	0.514000	0.257000	+	0.966411i	$0.417266\pi$
$350$	0	0
$351$	−2274.37	−0.345860
$352$	−2059.80	−0.311897
$353$	36.5920	0.00551727	0.00275864	−	0.999996i	$-0.499122\pi$
$353$	36.5920	0.00551727	0.00275864	+	0.999996i	$0.499122\pi$
$354$	−2303.43	−0.345836
$355$	0	0
$356$	−2173.03	−0.323512
$357$	−2982.22	−0.442118
$358$	255.892	0.0377773
$359$	−12789.2	−1.88019	−0.940095	−	0.340912i	$-0.889264\pi$
$359$	−12789.2	−1.88019	−0.940095	+	0.340912i	$0.889264\pi$
$360$	0	0
$361$	7754.77	1.13060
$362$	−4998.17	−0.725686
$363$	363.000	0.0524864
$364$	−11368.7	−1.63703
$365$	0	0
$366$	594.988	0.0849741
$367$	1293.66	0.184001	0.0920005	−	0.995759i	$-0.470674\pi$
$367$	1293.66	0.184001	0.0920005	+	0.995759i	$0.470674\pi$
$368$	113.568	0.0160874
$369$	−1614.16	−0.227723
$370$	0	0
$371$	13767.8	1.92666
$372$	−2521.43	−0.351425
$373$	−8638.61	−1.19917	−0.599585	−	0.800311i	$-0.704668\pi$
$373$	−8638.61	−1.19917	−0.599585	+	0.800311i	$0.704668\pi$
$374$	−703.643	−0.0972848
$375$	0	0
$376$	−10237.7	−1.40418
$377$	−16582.7	−2.26539
$378$	−1023.14	−0.139219
$379$	1853.53	0.251212	0.125606	−	0.992080i	$-0.459912\pi$
$379$	1853.53	0.251212	0.125606	+	0.992080i	$0.459912\pi$
$380$	0	0
$381$	3545.57	0.476758
$382$	3780.57	0.506363
$383$	3888.44	0.518773	0.259387	−	0.965774i	$-0.416480\pi$
$383$	3888.44	0.518773	0.259387	+	0.965774i	$0.416480\pi$
$384$	−3552.39	−0.472090
$385$	0	0
$386$	3553.22	0.468534
$387$	814.366	0.106968
$388$	920.701	0.120468
$389$	−10791.1	−1.40651	−0.703255	−	0.710938i	$-0.748271\pi$
$389$	−10791.1	−1.40651	−0.703255	+	0.710938i	$0.748271\pi$
$390$	0	0
$391$	407.257	0.0526749
$392$	−5207.18	−0.670924
$393$	−7845.79	−1.00704
$394$	−3057.42	−0.390941
$395$	0	0
$396$	550.594	0.0698696
$397$	−2875.92	−0.363573	−0.181787	−	0.983338i	$-0.558188\pi$
$397$	−2875.92	−0.363573	−0.181787	+	0.983338i	$0.558188\pi$
$398$	−3084.63	−0.388488
$399$	−8800.73	−1.10423
$400$	0	0
$401$	7615.88	0.948426	0.474213	−	0.880410i	$-0.342733\pi$
$401$	7615.88	0.948426	0.474213	+	0.880410i	$0.342733\pi$
$402$	−2943.03	−0.365136
$403$	−12730.0	−1.57351
$404$	−8399.92	−1.03443
$405$	0	0
$406$	−7459.84	−0.911886
$407$	−2788.24	−0.339578
$408$	−2602.50	−0.315791
$409$	−3791.51	−0.458382	−0.229191	−	0.973381i	$-0.573608\pi$
$409$	−3791.51	−0.458382	−0.229191	+	0.973381i	$0.573608\pi$
$410$	0	0
$411$	1396.11	0.167554
$412$	6928.48	0.828500
$413$	11932.0	1.42163
$414$	139.722	0.0165869
$415$	0	0
$416$	−15773.6	−1.85905
$417$	8428.57	0.989805
$418$	−2076.49	−0.242978
$419$	8806.66	1.02681	0.513405	−	0.858146i	$-0.328384\pi$
$419$	8806.66	1.02681	0.513405	+	0.858146i	$0.328384\pi$
$420$	0	0
$421$	−7730.71	−0.894945	−0.447473	−	0.894298i	$-0.647676\pi$
$421$	−7730.71	−0.894945	−0.447473	+	0.894298i	$0.647676\pi$
$422$	1842.51	0.212541
$423$	4350.91	0.500114
$424$	12014.8	1.37615
$425$	0	0
$426$	1452.13	0.165155
$427$	−3082.10	−0.349305
$428$	−3062.01	−0.345813
$429$	2779.79	0.312842
$430$	0	0
$431$	9017.97	1.00784	0.503922	−	0.863749i	$-0.331890\pi$
$431$	9017.97	1.00784	0.503922	+	0.863749i	$0.331890\pi$
$432$	308.429	0.0343502
$433$	−13799.1	−1.53150	−0.765751	−	0.643137i	$-0.777633\pi$
$433$	−13799.1	−1.53150	−0.765751	+	0.643137i	$0.777633\pi$
$434$	−5726.67	−0.633385
$435$	0	0
$436$	−7676.62	−0.843219
$437$	1201.84	0.131560
$438$	5363.24	0.585081
$439$	464.472	0.0504967	0.0252483	−	0.999681i	$-0.491962\pi$
$439$	464.472	0.0504967	0.0252483	+	0.999681i	$0.491962\pi$
$440$	0	0
$441$	2212.99	0.238958
$442$	−5388.37	−0.579861
$443$	5796.17	0.621635	0.310818	−	0.950470i	$-0.399397\pi$
$443$	5796.17	0.621635	0.310818	+	0.950470i	$0.399397\pi$
$444$	−4229.17	−0.452044
$445$	0	0
$446$	−8967.58	−0.952079
$447$	−1750.49	−0.185224
$448$	−4878.20	−0.514449
$449$	−8955.57	−0.941291	−0.470645	−	0.882323i	$-0.655979\pi$
$449$	−8955.57	−0.941291	−0.470645	+	0.882323i	$0.655979\pi$
$450$	0	0
$451$	1972.87	0.205984
$452$	5661.42	0.589139
$453$	3732.99	0.387177
$454$	−1113.64	−0.115122
$455$	0	0
$456$	−7680.13	−0.788717
$457$	−10067.8	−1.03053	−0.515267	−	0.857030i	$-0.672307\pi$
$457$	−10067.8	−1.03053	−0.515267	+	0.857030i	$0.672307\pi$
$458$	6930.31	0.707057
$459$	1106.03	0.112473
$460$	0	0
$461$	−8293.06	−0.837844	−0.418922	−	0.908022i	$-0.637592\pi$
$461$	−8293.06	−0.837844	−0.418922	+	0.908022i	$0.637592\pi$
$462$	1250.51	0.125928
$463$	9901.59	0.993879	0.496939	−	0.867785i	$-0.334457\pi$
$463$	9901.59	0.993879	0.496939	+	0.867785i	$0.334457\pi$
$464$	2248.79	0.224994
$465$	0	0
$466$	959.073	0.0953395
$467$	−10915.0	−1.08156	−0.540779	−	0.841165i	$-0.681870\pi$
$467$	−10915.0	−1.08156	−0.540779	+	0.841165i	$0.681870\pi$
$468$	4216.35	0.416454
$469$	15245.2	1.50097
$470$	0	0
$471$	911.973	0.0892176
$472$	10412.7	1.01543
$473$	−995.336	−0.0967560
$474$	−409.730	−0.0397036
$475$	0	0
$476$	5528.60	0.532359
$477$	−5106.13	−0.490134
$478$	8106.02	0.775650
$479$	5725.15	0.546114	0.273057	−	0.961998i	$-0.411965\pi$
$479$	5725.15	0.546114	0.273057	+	0.961998i	$0.411965\pi$
$480$	0	0
$481$	−21351.9	−2.02403
$482$	−552.334	−0.0521953
$483$	−723.775	−0.0681840
$484$	−672.948	−0.0631995
$485$	0	0
$486$	379.457	0.0354167
$487$	−16620.3	−1.54648	−0.773242	−	0.634111i	$-0.781366\pi$
$487$	−16620.3	−1.54648	−0.773242	+	0.634111i	$0.781366\pi$
$488$	−2689.65	−0.249498
$489$	−779.946	−0.0721276
$490$	0	0
$491$	−9769.75	−0.897969	−0.448985	−	0.893540i	$-0.648214\pi$
$491$	−9769.75	−0.897969	−0.448985	+	0.893540i	$0.648214\pi$
$492$	2992.42	0.274204
$493$	8064.17	0.736698
$494$	−15901.4	−1.44825
$495$	0	0
$496$	1726.32	0.156278
$497$	−7522.20	−0.678907
$498$	−4872.64	−0.438450
$499$	−1886.64	−0.169254	−0.0846270	−	0.996413i	$-0.526970\pi$
$499$	−1886.64	−0.169254	−0.0846270	+	0.996413i	$0.526970\pi$
$500$	0	0
$501$	10405.1	0.927873
$502$	4080.20	0.362765
$503$	2920.56	0.258889	0.129445	−	0.991587i	$-0.458681\pi$
$503$	2920.56	0.258889	0.129445	+	0.991587i	$0.458681\pi$
$504$	4625.14	0.408770
$505$	0	0
$506$	−170.771	−0.0150034
$507$	14696.1	1.28733
$508$	−6572.96	−0.574070
$509$	7956.30	0.692842	0.346421	−	0.938079i	$-0.387397\pi$
$509$	7956.30	0.692842	0.346421	+	0.938079i	$0.387397\pi$
$510$	0	0
$511$	−27782.1	−2.40510
$512$	4074.36	0.351686
$513$	3263.96	0.280911
$514$	−3801.41	−0.326212
$515$	0	0
$516$	−1509.71	−0.128801
$517$	−5317.78	−0.452371
$518$	−9605.30	−0.814735
$519$	11593.0	0.980496
$520$	0	0
$521$	20260.6	1.70371	0.851853	−	0.523780i	$-0.175479\pi$
$521$	20260.6	1.70371	0.851853	+	0.523780i	$0.175479\pi$
$522$	2766.66	0.231980
$523$	−3457.70	−0.289091	−0.144546	−	0.989498i	$-0.546172\pi$
$523$	−3457.70	−0.289091	−0.144546	+	0.989498i	$0.546172\pi$
$524$	14544.9	1.21259
$525$	0	0
$526$	11250.1	0.932559
$527$	6190.60	0.511702
$528$	−376.969	−0.0310709
$529$	−12068.2	−0.991876
$530$	0	0
$531$	−4425.27	−0.361658
$532$	16315.2	1.32961
$533$	15107.8	1.22775
$534$	1830.40	0.148332
$535$	0	0
$536$	13304.0	1.07210
$537$	491.610	0.0395056
$538$	8170.55	0.654753
$539$	−2704.76	−0.216145
$540$	0	0
$541$	−3020.10	−0.240008	−0.120004	−	0.992773i	$-0.538291\pi$
$541$	−3020.10	−0.240008	−0.120004	+	0.992773i	$0.538291\pi$
$542$	9730.24	0.771125
$543$	−9602.32	−0.758885
$544$	7670.71	0.604557
$545$	0	0
$546$	9576.16	0.750590
$547$	−17068.8	−1.33421	−0.667103	−	0.744965i	$-0.732466\pi$
$547$	−17068.8	−1.33421	−0.667103	+	0.744965i	$0.732466\pi$
$548$	−2588.17	−0.201754
$549$	1143.07	0.0888616
$550$	0	0
$551$	23797.9	1.83997
$552$	−631.616	−0.0487018
$553$	2122.44	0.163210
$554$	7714.42	0.591614
$555$	0	0
$556$	−15625.3	−1.19184
$557$	4965.05	0.377695	0.188847	−	0.982006i	$-0.439525\pi$
$557$	4965.05	0.377695	0.188847	+	0.982006i	$0.439525\pi$
$558$	2123.87	0.161130
$559$	−7622.10	−0.576709
$560$	0	0
$561$	−1351.81	−0.101736
$562$	−2759.21	−0.207100
$563$	−18839.1	−1.41025	−0.705126	−	0.709082i	$-0.749110\pi$
$563$	−18839.1	−1.41025	−0.705126	+	0.709082i	$0.749110\pi$
$564$	−8065.94	−0.602194
$565$	0	0
$566$	8282.92	0.615119
$567$	−1965.63	−0.145588
$568$	−6564.40	−0.484922
$569$	25785.9	1.89983	0.949914	−	0.312513i	$-0.101171\pi$
$569$	25785.9	1.89983	0.949914	+	0.312513i	$0.101171\pi$
$570$	0	0
$571$	1732.89	0.127004	0.0635018	−	0.997982i	$-0.479773\pi$
$571$	1732.89	0.127004	0.0635018	+	0.997982i	$0.479773\pi$
$572$	−5153.31	−0.376697
$573$	7263.09	0.529529
$574$	6796.38	0.494208
$575$	0	0
$576$	1809.20	0.130874
$577$	8872.77	0.640170	0.320085	−	0.947389i	$-0.396289\pi$
$577$	8872.77	0.640170	0.320085	+	0.947389i	$0.396289\pi$
$578$	−5051.54	−0.363523
$579$	6826.32	0.489969
$580$	0	0
$581$	25240.8	1.80235
$582$	−775.534	−0.0552352
$583$	6240.83	0.443343
$584$	−24244.6	−1.71789
$585$	0	0
$586$	−13048.9	−0.919875
$587$	−11144.0	−0.783582	−0.391791	−	0.920054i	$-0.628144\pi$
$587$	−11144.0	−0.783582	−0.391791	+	0.920054i	$0.628144\pi$
$588$	−4102.55	−0.287732
$589$	18268.8	1.27802
$590$	0	0
$591$	−5873.81	−0.408826
$592$	2895.54	0.201023
$593$	6643.23	0.460042	0.230021	−	0.973186i	$-0.426121\pi$
$593$	6643.23	0.460042	0.230021	+	0.973186i	$0.426121\pi$
$594$	−463.781	−0.0320356
$595$	0	0
$596$	3245.14	0.223031
$597$	−5926.07	−0.406261
$598$	−1307.74	−0.0894270
$599$	26138.0	1.78292	0.891462	−	0.453096i	$-0.149680\pi$
$599$	26138.0	1.78292	0.891462	+	0.453096i	$0.149680\pi$
$600$	0	0
$601$	8628.54	0.585633	0.292817	−	0.956169i	$-0.405407\pi$
$601$	8628.54	0.585633	0.292817	+	0.956169i	$0.405407\pi$
$602$	−3428.86	−0.232143
$603$	−5654.04	−0.381841
$604$	−6920.40	−0.466204
$605$	0	0
$606$	7075.50	0.474294
$607$	4043.74	0.270396	0.135198	−	0.990819i	$-0.456833\pi$
$607$	4043.74	0.270396	0.135198	+	0.990819i	$0.456833\pi$
$608$	22636.7	1.50994
$609$	−14331.6	−0.953604
$610$	0	0
$611$	−40722.6	−2.69633
$612$	−2050.41	−0.135430
$613$	5443.62	0.358671	0.179336	−	0.983788i	$-0.442605\pi$
$613$	5443.62	0.358671	0.179336	+	0.983788i	$0.442605\pi$
$614$	−13364.2	−0.878397
$615$	0	0
$616$	−5652.95	−0.369746
$617$	234.868	0.0153248	0.00766241	−	0.999971i	$-0.497561\pi$
$617$	234.868	0.0153248	0.00766241	+	0.999971i	$0.497561\pi$
$618$	−5836.06	−0.379872
$619$	−4517.44	−0.293330	−0.146665	−	0.989186i	$-0.546854\pi$
$619$	−4517.44	−0.293330	−0.146665	+	0.989186i	$0.546854\pi$
$620$	0	0
$621$	268.429	0.0173457
$622$	9580.42	0.617588
$623$	−9481.68	−0.609752
$624$	−2886.76	−0.185197
$625$	0	0
$626$	6384.59	0.407635
$627$	−3989.29	−0.254094
$628$	−1690.66	−0.107428
$629$	10383.4	0.658211
$630$	0	0
$631$	26533.3	1.67397	0.836983	−	0.547229i	$-0.184317\pi$
$631$	26533.3	1.67397	0.836983	+	0.547229i	$0.184317\pi$
$632$	1852.19	0.116576
$633$	3539.77	0.222264
$634$	12425.9	0.778386
$635$	0	0
$636$	9466.01	0.590175
$637$	−20712.6	−1.28832
$638$	−3381.47	−0.209834
$639$	2789.79	0.172711
$640$	0	0
$641$	−8233.77	−0.507355	−0.253677	−	0.967289i	$-0.581640\pi$
$641$	−8233.77	−0.507355	−0.253677	+	0.967289i	$0.581640\pi$
$642$	2579.22	0.158557
$643$	4844.30	0.297108	0.148554	−	0.988904i	$-0.452538\pi$
$643$	4844.30	0.297108	0.148554	+	0.988904i	$0.452538\pi$
$644$	1341.77	0.0821012
$645$	0	0
$646$	7732.87	0.470969
$647$	29856.8	1.81421	0.907104	−	0.420907i	$-0.138288\pi$
$647$	29856.8	1.81421	0.907104	+	0.420907i	$0.138288\pi$
$648$	−1715.34	−0.103989
$649$	5408.66	0.327132
$650$	0	0
$651$	−11001.9	−0.662362
$652$	1445.90	0.0868497
$653$	4212.65	0.252456	0.126228	−	0.992001i	$-0.459713\pi$
$653$	4212.65	0.252456	0.126228	+	0.992001i	$0.459713\pi$
$654$	6466.24	0.386621
$655$	0	0
$656$	−2048.78	−0.121938
$657$	10303.7	0.611848
$658$	−18319.4	−1.08535
$659$	14549.4	0.860039	0.430020	−	0.902820i	$-0.358507\pi$
$659$	14549.4	0.860039	0.430020	+	0.902820i	$0.358507\pi$
$660$	0	0
$661$	2494.86	0.146806	0.0734029	−	0.997302i	$-0.476614\pi$
$661$	2494.86	0.146806	0.0734029	+	0.997302i	$0.476614\pi$
$662$	−4862.63	−0.285485
$663$	−10351.9	−0.606389
$664$	22026.9	1.28736
$665$	0	0
$666$	3562.35	0.207265
$667$	1957.14	0.113615
$668$	−19289.5	−1.11726
$669$	−17228.2	−0.995637
$670$	0	0
$671$	−1397.09	−0.0803784
$672$	−13632.3	−0.782557
$673$	8223.98	0.471042	0.235521	−	0.971869i	$-0.424320\pi$
$673$	8223.98	0.471042	0.235521	+	0.971869i	$0.424320\pi$
$674$	−12909.6	−0.737773
$675$	0	0
$676$	−27244.4	−1.55009
$677$	9369.49	0.531904	0.265952	−	0.963986i	$-0.414314\pi$
$677$	9369.49	0.531904	0.265952	+	0.963986i	$0.414314\pi$
$678$	−4768.78	−0.270124
$679$	4017.34	0.227057
$680$	0	0
$681$	−2139.48	−0.120389
$682$	−2595.85	−0.145748
$683$	561.238	0.0314424	0.0157212	−	0.999876i	$-0.494996\pi$
$683$	561.238	0.0314424	0.0157212	+	0.999876i	$0.494996\pi$
$684$	−6050.90	−0.338248
$685$	0	0
$686$	3680.01	0.204815
$687$	13314.3	0.739405
$688$	1033.64	0.0572777
$689$	47791.1	2.64252
$690$	0	0
$691$	11456.5	0.630720	0.315360	−	0.948972i	$-0.397875\pi$
$691$	11456.5	0.630720	0.315360	+	0.948972i	$0.397875\pi$
$692$	−21491.7	−1.18063
$693$	2402.43	0.131690
$694$	13543.4	0.740779
$695$	0	0
$696$	−12506.7	−0.681130
$697$	−7346.96	−0.399263
$698$	5233.09	0.283775
$699$	1842.54	0.0997012
$700$	0	0
$701$	−2782.39	−0.149914	−0.0749569	−	0.997187i	$-0.523882\pi$
$701$	−2782.39	−0.149914	−0.0749569	+	0.997187i	$0.523882\pi$
$702$	−3551.55	−0.190947
$703$	30642.1	1.64394
$704$	−2211.24	−0.118380
$705$	0	0
$706$	57.1404	0.00304604
$707$	−36651.8	−1.94969
$708$	8203.79	0.435476
$709$	4824.70	0.255565	0.127783	−	0.991802i	$-0.459214\pi$
$709$	4824.70	0.255565	0.127783	+	0.991802i	$0.459214\pi$
$710$	0	0
$711$	−787.158	−0.0415200
$712$	−8274.37	−0.435527
$713$	1502.43	0.0789153
$714$	−4656.90	−0.244090
$715$	0	0
$716$	−911.371	−0.0475692
$717$	15573.0	0.811135
$718$	−19971.0	−1.03804
$719$	20717.6	1.07460	0.537298	−	0.843392i	$-0.319445\pi$
$719$	20717.6	1.07460	0.537298	+	0.843392i	$0.319445\pi$
$720$	0	0
$721$	30231.4	1.56155
$722$	12109.5	0.624195
$723$	−1061.12	−0.0545832
$724$	17801.3	0.913783
$725$	0	0
$726$	566.844	0.0289773
$727$	22731.2	1.15963	0.579816	−	0.814747i	$-0.303124\pi$
$727$	22731.2	1.15963	0.579816	+	0.814747i	$0.303124\pi$
$728$	−43289.2	−2.20385
$729$	729.000	0.0370370
$730$	0	0
$731$	3706.64	0.187544
$732$	−2119.08	−0.106999
$733$	−24311.8	−1.22507	−0.612536	−	0.790443i	$-0.709851\pi$
$733$	−24311.8	−1.22507	−0.612536	+	0.790443i	$0.709851\pi$
$734$	2020.11	0.101586
$735$	0	0
$736$	1861.65	0.0932357
$737$	6910.49	0.345388
$738$	−2520.60	−0.125724
$739$	16033.9	0.798126	0.399063	−	0.916924i	$-0.369335\pi$
$739$	16033.9	0.798126	0.399063	+	0.916924i	$0.369335\pi$
$740$	0	0
$741$	−30549.2	−1.51451
$742$	21499.2	1.06369
$743$	−19512.2	−0.963435	−0.481718	−	0.876326i	$-0.659987\pi$
$743$	−19512.2	−0.963435	−0.481718	+	0.876326i	$0.659987\pi$
$744$	−9601.01	−0.473105
$745$	0	0
$746$	−13489.7	−0.662053
$747$	−9361.15	−0.458509
$748$	2506.06	0.122501
$749$	−13360.6	−0.651785
$750$	0	0
$751$	8393.65	0.407841	0.203920	−	0.978987i	$-0.434632\pi$
$751$	8393.65	0.407841	0.203920	+	0.978987i	$0.434632\pi$
$752$	5522.41	0.267795
$753$	7838.73	0.379361
$754$	−25894.7	−1.25070
$755$	0	0
$756$	3643.98	0.175305
$757$	20208.6	0.970270	0.485135	−	0.874439i	$-0.338771\pi$
$757$	20208.6	0.970270	0.485135	+	0.874439i	$0.338771\pi$
$758$	2894.38	0.138692
$759$	−328.080	−0.0156898
$760$	0	0
$761$	3529.19	0.168112	0.0840558	−	0.996461i	$-0.473213\pi$
$761$	3529.19	0.168112	0.0840558	+	0.996461i	$0.473213\pi$
$762$	5536.59	0.263215
$763$	−33495.8	−1.58929
$764$	−13464.7	−0.637612
$765$	0	0
$766$	6072.01	0.286411
$767$	41418.5	1.94985
$768$	−10371.8	−0.487317
$769$	−26692.6	−1.25171	−0.625853	−	0.779941i	$-0.715249\pi$
$769$	−26692.6	−1.25171	−0.625853	+	0.779941i	$0.715249\pi$
$770$	0	0
$771$	−7303.14	−0.341136
$772$	−12655.0	−0.589977
$773$	32009.0	1.48937	0.744684	−	0.667417i	$-0.232600\pi$
$773$	32009.0	1.48937	0.744684	+	0.667417i	$0.232600\pi$
$774$	1271.67	0.0590561
$775$	0	0
$776$	3505.81	0.162180
$777$	−18453.4	−0.852008
$778$	−16850.9	−0.776523
$779$	−21681.3	−0.997194
$780$	0	0
$781$	−3409.74	−0.156223
$782$	635.954	0.0290814
$783$	5315.21	0.242593
$784$	2808.84	0.127954
$785$	0	0
$786$	−12251.6	−0.555980
$787$	8180.18	0.370511	0.185255	−	0.982690i	$-0.440689\pi$
$787$	8180.18	0.370511	0.185255	+	0.982690i	$0.440689\pi$
$788$	10889.2	0.492272
$789$	21613.2	0.975223
$790$	0	0
$791$	24702.8	1.11040
$792$	2096.53	0.0940619
$793$	−10698.6	−0.479091
$794$	−4490.91	−0.200726
$795$	0	0
$796$	10986.1	0.489184
$797$	−25471.0	−1.13203	−0.566015	−	0.824395i	$-0.691516\pi$
$797$	−25471.0	−1.13203	−0.566015	+	0.824395i	$0.691516\pi$
$798$	−13742.8	−0.609636
$799$	19803.4	0.876840
$800$	0	0
$801$	3516.51	0.155118
$802$	11892.6	0.523619
$803$	−12593.4	−0.553437
$804$	10481.7	0.459779
$805$	0	0
$806$	−19878.5	−0.868723
$807$	15697.0	0.684708
$808$	−31984.9	−1.39261
$809$	12381.1	0.538067	0.269034	−	0.963131i	$-0.413296\pi$
$809$	12381.1	0.538067	0.269034	+	0.963131i	$0.413296\pi$
$810$	0	0
$811$	29129.5	1.26125	0.630627	−	0.776086i	$-0.282798\pi$
$811$	29129.5	1.26125	0.630627	+	0.776086i	$0.282798\pi$
$812$	26568.6	1.14825
$813$	18693.4	0.806403
$814$	−4353.99	−0.187478
$815$	0	0
$816$	1403.83	0.0602255
$817$	10938.5	0.468409
$818$	−5920.64	−0.253069
$819$	18397.4	0.784929
$820$	0	0
$821$	−5279.00	−0.224407	−0.112204	−	0.993685i	$-0.535791\pi$
$821$	−5279.00	−0.224407	−0.112204	+	0.993685i	$0.535791\pi$
$822$	2180.09	0.0925055
$823$	19991.7	0.846738	0.423369	−	0.905957i	$-0.360847\pi$
$823$	19991.7	0.846738	0.423369	+	0.905957i	$0.360847\pi$
$824$	26382.0	1.11537
$825$	0	0
$826$	18632.4	0.784874
$827$	9918.83	0.417063	0.208532	−	0.978016i	$-0.433132\pi$
$827$	9918.83	0.417063	0.208532	+	0.978016i	$0.433132\pi$
$828$	−497.628	−0.0208862
$829$	−5671.76	−0.237622	−0.118811	−	0.992917i	$-0.537908\pi$
$829$	−5671.76	−0.237622	−0.118811	+	0.992917i	$0.537908\pi$
$830$	0	0
$831$	14820.7	0.618681
$832$	−16933.3	−0.705595
$833$	10072.5	0.418959
$834$	13161.7	0.546464
$835$	0	0
$836$	7395.54	0.305957
$837$	4080.31	0.168502
$838$	13752.1	0.566894
$839$	−29160.1	−1.19990	−0.599951	−	0.800036i	$-0.704813\pi$
$839$	−29160.1	−1.19990	−0.599951	+	0.800036i	$0.704813\pi$
$840$	0	0
$841$	14364.7	0.588984
$842$	−12071.9	−0.494092
$843$	−5300.90	−0.216575
$844$	−6562.21	−0.267631
$845$	0	0
$846$	6794.17	0.276109
$847$	−2936.31	−0.119118
$848$	−6480.98	−0.262450
$849$	15912.9	0.643260
$850$	0	0
$851$	2520.02	0.101510
$852$	−5171.85	−0.207963
$853$	18361.2	0.737017	0.368508	−	0.929624i	$-0.379869\pi$
$853$	18361.2	0.737017	0.368508	+	0.929624i	$0.379869\pi$
$854$	−4812.86	−0.192849
$855$	0	0
$856$	−11659.4	−0.465550
$857$	1755.28	0.0699642	0.0349821	−	0.999388i	$-0.488863\pi$
$857$	1755.28	0.0699642	0.0349821	+	0.999388i	$0.488863\pi$
$858$	4340.78	0.172718
$859$	−31317.3	−1.24392	−0.621962	−	0.783047i	$-0.713664\pi$
$859$	−31317.3	−1.24392	−0.621962	+	0.783047i	$0.713664\pi$
$860$	0	0
$861$	13057.0	0.516818
$862$	14082.0	0.556422
$863$	−42862.3	−1.69067	−0.845336	−	0.534235i	$-0.820600\pi$
$863$	−42862.3	−1.69067	−0.845336	+	0.534235i	$0.820600\pi$
$864$	5055.88	0.199079
$865$	0	0
$866$	−21548.0	−0.845530
$867$	−9704.84	−0.380154
$868$	20395.9	0.797558
$869$	962.082	0.0375563
$870$	0	0
$871$	52919.3	2.05867
$872$	−29230.8	−1.13518
$873$	−1489.93	−0.0577622
$874$	1876.74	0.0726335
$875$	0	0
$876$	−19101.4	−0.736733
$877$	38306.6	1.47494	0.737471	−	0.675379i	$-0.236020\pi$
$877$	38306.6	1.47494	0.737471	+	0.675379i	$0.236020\pi$
$878$	725.298	0.0278788
$879$	−25069.2	−0.961959
$880$	0	0
$881$	−12994.5	−0.496931	−0.248466	−	0.968641i	$-0.579926\pi$
$881$	−12994.5	−0.496931	−0.248466	+	0.968641i	$0.579926\pi$
$882$	3455.69	0.131927
$883$	−4256.48	−0.162222	−0.0811110	−	0.996705i	$-0.525847\pi$
$883$	−4256.48	−0.162222	−0.0811110	+	0.996705i	$0.525847\pi$
$884$	19191.0	0.730160
$885$	0	0
$886$	9051.03	0.343200
$887$	38188.8	1.44561	0.722804	−	0.691053i	$-0.242853\pi$
$887$	38188.8	1.44561	0.722804	+	0.691053i	$0.242853\pi$
$888$	−16103.7	−0.608563
$889$	−28680.1	−1.08200
$890$	0	0
$891$	−891.000	−0.0335013
$892$	31938.5	1.19886
$893$	58441.1	2.18999
$894$	−2733.48	−0.102261
$895$	0	0
$896$	28735.3	1.07141
$897$	−2512.38	−0.0935182
$898$	−13984.6	−0.519679
$899$	29750.0	1.10369
$900$	0	0
$901$	−23240.9	−0.859341
$902$	3080.73	0.113722
$903$	−6587.40	−0.242763
$904$	21557.3	0.793127
$905$	0	0
$906$	5829.26	0.213757
$907$	−31257.8	−1.14432	−0.572161	−	0.820142i	$-0.693895\pi$
$907$	−31257.8	−1.14432	−0.572161	+	0.820142i	$0.693895\pi$
$908$	3966.28	0.144962
$909$	13593.2	0.495993
$910$	0	0
$911$	−35781.4	−1.30131	−0.650653	−	0.759375i	$-0.725505\pi$
$911$	−35781.4	−1.30131	−0.650653	+	0.759375i	$0.725505\pi$
$912$	4142.80	0.150419
$913$	11441.4	0.414737
$914$	−15721.5	−0.568950
$915$	0	0
$916$	−24682.7	−0.890326
$917$	63464.6	2.28548
$918$	1727.12	0.0620954
$919$	31489.1	1.13028	0.565140	−	0.824995i	$-0.308822\pi$
$919$	31489.1	1.13028	0.565140	+	0.824995i	$0.308822\pi$
$920$	0	0
$921$	−25674.9	−0.918583
$922$	−12950.0	−0.462567
$923$	−26111.2	−0.931159
$924$	−4453.75	−0.158569
$925$	0	0
$926$	15461.9	0.548713
$927$	−11212.0	−0.397251
$928$	36862.9	1.30397
$929$	−40466.1	−1.42912	−0.714559	−	0.699575i	$-0.753373\pi$
$929$	−40466.1	−1.42912	−0.714559	+	0.699575i	$0.753373\pi$
$930$	0	0
$931$	29724.7	1.04639
$932$	−3415.79	−0.120051
$933$	18405.6	0.645843
$934$	−17044.4	−0.597120
$935$	0	0
$936$	16054.8	0.560651
$937$	39041.1	1.36117	0.680586	−	0.732668i	$-0.261725\pi$
$937$	39041.1	1.36117	0.680586	+	0.732668i	$0.261725\pi$
$938$	23806.1	0.828676
$939$	12265.9	0.426284
$940$	0	0
$941$	10842.0	0.375598	0.187799	−	0.982207i	$-0.439865\pi$
$941$	10842.0	0.375598	0.187799	+	0.982207i	$0.439865\pi$
$942$	1424.09	0.0492563
$943$	−1783.08	−0.0615748
$944$	−5616.79	−0.193656
$945$	0	0
$946$	−1554.27	−0.0534182
$947$	−17810.1	−0.611142	−0.305571	−	0.952169i	$-0.598847\pi$
$947$	−17810.1	−0.611142	−0.305571	+	0.952169i	$0.598847\pi$
$948$	1459.27	0.0499947
$949$	−96437.6	−3.29873
$950$	0	0
$951$	23872.3	0.813997
$952$	21051.6	0.716687
$953$	−45621.7	−1.55071	−0.775357	−	0.631523i	$-0.782430\pi$
$953$	−45621.7	−1.55071	−0.775357	+	0.631523i	$0.782430\pi$
$954$	−7973.49	−0.270599
$955$	0	0
$956$	−28870.0	−0.976697
$957$	−6496.37	−0.219433
$958$	8940.12	0.301506
$959$	−11293.1	−0.380264
$960$	0	0
$961$	−6952.93	−0.233390
$962$	−33342.0	−1.11745
$963$	4955.11	0.165811
$964$	1967.17	0.0657242
$965$	0	0
$966$	−1130.21	−0.0376439
$967$	−41111.6	−1.36718	−0.683588	−	0.729868i	$-0.739581\pi$
$967$	−41111.6	−1.36718	−0.683588	+	0.729868i	$0.739581\pi$
$968$	−2562.43	−0.0850821
$969$	14856.1	0.492515
$970$	0	0
$971$	22030.1	0.728093	0.364046	−	0.931381i	$-0.381395\pi$
$971$	22030.1	0.728093	0.364046	+	0.931381i	$0.381395\pi$
$972$	−1351.46	−0.0445967
$973$	−68178.7	−2.24636
$974$	−25953.5	−0.853802
$975$	0	0
$976$	1450.85	0.0475825
$977$	−34593.0	−1.13278	−0.566391	−	0.824137i	$-0.691661\pi$
$977$	−34593.0	−1.13278	−0.566391	+	0.824137i	$0.691661\pi$
$978$	−1217.93	−0.0398211
$979$	−4297.95	−0.140310
$980$	0	0
$981$	12422.7	0.404309
$982$	−15256.0	−0.495762
$983$	−20638.9	−0.669663	−0.334831	−	0.942278i	$-0.608679\pi$
$983$	−20638.9	−0.669663	−0.334831	+	0.942278i	$0.608679\pi$
$984$	11394.4	0.369147
$985$	0	0
$986$	12592.6	0.406725
$987$	−35194.5	−1.13501
$988$	56633.7	1.82364
$989$	899.586	0.0289233
$990$	0	0
$991$	−35229.7	−1.12927	−0.564636	−	0.825340i	$-0.690983\pi$
$991$	−35229.7	−1.12927	−0.564636	+	0.825340i	$0.690983\pi$
$992$	28298.4	0.905722
$993$	−9341.91	−0.298546
$994$	−11746.3	−0.374819
$995$	0	0
$996$	17354.2	0.552096
$997$	29366.2	0.932834	0.466417	−	0.884565i	$-0.345545\pi$
$997$	29366.2	0.932834	0.466417	+	0.884565i	$0.345545\pi$
$998$	−2946.09	−0.0934439
$999$	6843.87	0.216747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.4.a.u.1.3	✓	4
3.2	odd	2		2475.4.a.bd.1.1		4
5.2	odd	4		825.4.c.q.199.5		8
5.3	odd	4		825.4.c.q.199.4		8
5.4	even	2		825.4.a.v.1.2	yes	4
15.14	odd	2		2475.4.a.bb.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.4.a.u.1.3	✓	4	1.1	even	1	trivial
825.4.a.v.1.2	yes	4	5.4	even	2
825.4.c.q.199.4		8	5.3	odd	4
825.4.c.q.199.5		8	5.2	odd	4
2475.4.a.bb.1.4		4	15.14	odd	2
2475.4.a.bd.1.1		4	3.2	odd	2

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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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

\(f(q)\)	\(=\)	\(q+1.56155 q^{2} +3.00000 q^{3} -5.56155 q^{4} +4.68466 q^{6} -24.2670 q^{7} -21.1771 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+1.56155 q^{2} +3.00000 q^{3} -5.56155 q^{4} +4.68466 q^{6} -24.2670 q^{7} -21.1771 q^{8} +9.00000 q^{9} -11.0000 q^{11} -16.6847 q^{12} -84.2360 q^{13} -37.8942 q^{14} +11.4233 q^{16} +40.9641 q^{17} +14.0540 q^{18} +120.887 q^{19} -72.8010 q^{21} -17.1771 q^{22} +9.94182 q^{23} -63.5312 q^{24} -131.539 q^{26} +27.0000 q^{27} +134.962 q^{28} +196.860 q^{29} +151.123 q^{31} +187.255 q^{32} -33.0000 q^{33} +63.9675 q^{34} -50.0540 q^{36} +253.477 q^{37} +188.772 q^{38} -252.708 q^{39} -179.351 q^{41} -113.683 q^{42} +90.4851 q^{43} +61.1771 q^{44} +15.5247 q^{46} +483.434 q^{47} +34.2699 q^{48} +245.887 q^{49} +122.892 q^{51} +468.483 q^{52} -567.348 q^{53} +42.1619 q^{54} +513.904 q^{56} +362.662 q^{57} +307.407 q^{58} -491.696 q^{59} +127.008 q^{61} +235.986 q^{62} -218.403 q^{63} +201.022 q^{64} -51.5312 q^{66} -628.226 q^{67} -227.824 q^{68} +29.8255 q^{69} +309.976 q^{71} -190.594 q^{72} +1144.85 q^{73} +395.817 q^{74} -672.322 q^{76} +266.937 q^{77} -394.617 q^{78} -87.4620 q^{79} +81.0000 q^{81} -280.067 q^{82} -1040.13 q^{83} +404.887 q^{84} +141.297 q^{86} +590.579 q^{87} +232.948 q^{88} +390.723 q^{89} +2044.15 q^{91} -55.2920 q^{92} +453.368 q^{93} +754.908 q^{94} +561.764 q^{96} -165.548 q^{97} +383.966 q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 12 q^{3} - 14 q^{4} - 6 q^{6} - 11 q^{7} + 6 q^{8} + 36 q^{9}+O(q^{10}) \) 4 * q - 2 * q^2 + 12 * q^3 - 14 * q^4 - 6 * q^6 - 11 * q^7 + 6 * q^8 + 36 * q^9	\( 4 q - 2 q^{2} + 12 q^{3} - 14 q^{4} - 6 q^{6} - 11 q^{7} + 6 q^{8} + 36 q^{9} - 44 q^{11} - 42 q^{12} + 25 q^{13} - 3 q^{14} - 78 q^{16} + 85 q^{17} - 18 q^{18} + 118 q^{19} - 33 q^{21} + 22 q^{22} - 10 q^{23} + 18 q^{24} - 157 q^{26} + 108 q^{27} + 47 q^{28} + 251 q^{29} - 135 q^{31} + 246 q^{32} - 132 q^{33} + 289 q^{34} - 126 q^{36} + 419 q^{37} - 76 q^{38} + 75 q^{39} - 103 q^{41} - 9 q^{42} + 15 q^{43} + 154 q^{44} - 420 q^{46} + 665 q^{47} - 234 q^{48} + 1003 q^{49} + 255 q^{51} + 57 q^{52} - 116 q^{53} - 54 q^{54} + 77 q^{56} + 354 q^{57} + 189 q^{58} + 951 q^{59} - 350 q^{61} - 213 q^{62} - 99 q^{63} + 1538 q^{64} + 66 q^{66} + 266 q^{67} - 629 q^{68} - 30 q^{69} + 1526 q^{71} + 54 q^{72} - 566 q^{73} + 1091 q^{74} - 396 q^{76} + 121 q^{77} - 471 q^{78} + 2567 q^{79} + 324 q^{81} + 9 q^{82} - 961 q^{83} + 141 q^{84} + 1837 q^{86} + 753 q^{87} - 66 q^{88} + 1053 q^{89} + 2150 q^{91} + 460 q^{92} - 405 q^{93} + 2209 q^{94} + 738 q^{96} - 172 q^{97} - 1819 q^{98} - 396 q^{99}+O(q^{100}) \) 4 * q - 2 * q^2 + 12 * q^3 - 14 * q^4 - 6 * q^6 - 11 * q^7 + 6 * q^8 + 36 * q^9 - 44 * q^11 - 42 * q^12 + 25 * q^13 - 3 * q^14 - 78 * q^16 + 85 * q^17 - 18 * q^18 + 118 * q^19 - 33 * q^21 + 22 * q^22 - 10 * q^23 + 18 * q^24 - 157 * q^26 + 108 * q^27 + 47 * q^28 + 251 * q^29 - 135 * q^31 + 246 * q^32 - 132 * q^33 + 289 * q^34 - 126 * q^36 + 419 * q^37 - 76 * q^38 + 75 * q^39 - 103 * q^41 - 9 * q^42 + 15 * q^43 + 154 * q^44 - 420 * q^46 + 665 * q^47 - 234 * q^48 + 1003 * q^49 + 255 * q^51 + 57 * q^52 - 116 * q^53 - 54 * q^54 + 77 * q^56 + 354 * q^57 + 189 * q^58 + 951 * q^59 - 350 * q^61 - 213 * q^62 - 99 * q^63 + 1538 * q^64 + 66 * q^66 + 266 * q^67 - 629 * q^68 - 30 * q^69 + 1526 * q^71 + 54 * q^72 - 566 * q^73 + 1091 * q^74 - 396 * q^76 + 121 * q^77 - 471 * q^78 + 2567 * q^79 + 324 * q^81 + 9 * q^82 - 961 * q^83 + 141 * q^84 + 1837 * q^86 + 753 * q^87 - 66 * q^88 + 1053 * q^89 + 2150 * q^91 + 460 * q^92 - 405 * q^93 + 2209 * q^94 + 738 * q^96 - 172 * q^97 - 1819 * q^98 - 396 * q^99

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