LMFDB - Embedded newform 405.8.a.f.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,8,Mod(1,405)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("405.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	405.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:	$126.515935321$
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} - 5 x^{12} - 1170 x^{11} + 4622 x^{10} + 503384 x^{9} - 1392714 x^{8} - 97100172 x^{7} + \cdots + 4693741072256 $ x^13 - 5x^12 - 1170x^11 + 4622x^10 + 503384x^9 - 1392714x^8 - 97100172x^7 + 151526988x^6 + 8205832521x^5 - 4072720601x^4 - 282543812090x^3 - 118496553138x^2 + 3000776721014x + 4693741072256
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{24} $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$6.91113$ of defining polynomial
Character	$\chi$	$=$	405.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.91113 q^{2} -93.0585 q^{4} -125.000 q^{5} -1492.16 q^{7} +1306.71 q^{8} +O(q^{10})$	\(q-5.91113 q^{2} -93.0585 q^{4} -125.000 q^{5} -1492.16 q^{7} +1306.71 q^{8} +738.892 q^{10} -3576.16 q^{11} -109.642 q^{13} +8820.33 q^{14} +4187.38 q^{16} -24459.8 q^{17} +28498.8 q^{19} +11632.3 q^{20} +21139.1 q^{22} +54000.8 q^{23} +15625.0 q^{25} +648.110 q^{26} +138858. q^{28} -101327. q^{29} +10136.0 q^{31} -192011. q^{32} +144585. q^{34} +186519. q^{35} +572204. q^{37} -168460. q^{38} -163338. q^{40} +290903. q^{41} -437043. q^{43} +332792. q^{44} -319206. q^{46} +871931. q^{47} +1.40298e6 q^{49} -92361.4 q^{50} +10203.1 q^{52} +354451. q^{53} +447019. q^{55} -1.94981e6 q^{56} +598956. q^{58} +2.00531e6 q^{59} -2.54879e6 q^{61} -59915.5 q^{62} +599015. q^{64} +13705.3 q^{65} -4.04123e6 q^{67} +2.27619e6 q^{68} -1.10254e6 q^{70} +507433. q^{71} +3.39540e6 q^{73} -3.38237e6 q^{74} -2.65205e6 q^{76} +5.33618e6 q^{77} +842829. q^{79} -523422. q^{80} -1.71957e6 q^{82} +4.91734e6 q^{83} +3.05747e6 q^{85} +2.58342e6 q^{86} -4.67298e6 q^{88} -1.87078e6 q^{89} +163603. q^{91} -5.02523e6 q^{92} -5.15410e6 q^{94} -3.56235e6 q^{95} +3.31053e6 q^{97} -8.29322e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 8 q^{2} + 704 q^{4} - 1625 q^{5} - 1455 q^{7} + 1236 q^{8}+O(q^{10}) $ 13 * q + 8 * q^2 + 704 * q^4 - 1625 * q^5 - 1455 * q^7 + 1236 * q^8	\( 13 q + 8 q^{2} + 704 q^{4} - 1625 q^{5} - 1455 q^{7} + 1236 q^{8} - 1000 q^{10} + 1658 q^{11} - 13568 q^{13} + 12351 q^{14} + 52076 q^{16} + 6944 q^{17} - 45812 q^{19} - 88000 q^{20} - 17993 q^{22} + 96441 q^{23} + 203125 q^{25} + 126146 q^{26} - 216945 q^{28} + 39043 q^{29} - 158520 q^{31} + 725206 q^{32} - 441617 q^{34} + 181875 q^{35} - 505438 q^{37} + 615041 q^{38} - 154500 q^{40} + 1578883 q^{41} - 1082090 q^{43} + 498211 q^{44} - 2312547 q^{46} + 1690139 q^{47} - 1054816 q^{49} + 125000 q^{50} - 4100644 q^{52} - 102274 q^{53} - 207250 q^{55} + 389331 q^{56} - 5780521 q^{58} - 2908966 q^{59} - 3091451 q^{61} + 5212476 q^{62} - 5659352 q^{64} + 1696000 q^{65} - 1849533 q^{67} - 563369 q^{68} - 1543875 q^{70} - 2617958 q^{71} + 5310946 q^{73} - 11485004 q^{74} - 14421739 q^{76} + 1719660 q^{77} - 3632166 q^{79} - 6509500 q^{80} - 7347658 q^{82} + 1424115 q^{83} - 868000 q^{85} - 23193379 q^{86} - 20229447 q^{88} - 2816067 q^{89} - 16929702 q^{91} - 27930183 q^{92} - 11693189 q^{94} + 5726500 q^{95} - 12062476 q^{97} - 29617805 q^{98}+O(q^{100}) \) 13 * q + 8 * q^2 + 704 * q^4 - 1625 * q^5 - 1455 * q^7 + 1236 * q^8 - 1000 * q^10 + 1658 * q^11 - 13568 * q^13 + 12351 * q^14 + 52076 * q^16 + 6944 * q^17 - 45812 * q^19 - 88000 * q^20 - 17993 * q^22 + 96441 * q^23 + 203125 * q^25 + 126146 * q^26 - 216945 * q^28 + 39043 * q^29 - 158520 * q^31 + 725206 * q^32 - 441617 * q^34 + 181875 * q^35 - 505438 * q^37 + 615041 * q^38 - 154500 * q^40 + 1578883 * q^41 - 1082090 * q^43 + 498211 * q^44 - 2312547 * q^46 + 1690139 * q^47 - 1054816 * q^49 + 125000 * q^50 - 4100644 * q^52 - 102274 * q^53 - 207250 * q^55 + 389331 * q^56 - 5780521 * q^58 - 2908966 * q^59 - 3091451 * q^61 + 5212476 * q^62 - 5659352 * q^64 + 1696000 * q^65 - 1849533 * q^67 - 563369 * q^68 - 1543875 * q^70 - 2617958 * q^71 + 5310946 * q^73 - 11485004 * q^74 - 14421739 * q^76 + 1719660 * q^77 - 3632166 * q^79 - 6509500 * q^80 - 7347658 * q^82 + 1424115 * q^83 - 868000 * q^85 - 23193379 * q^86 - 20229447 * q^88 - 2816067 * q^89 - 16929702 * q^91 - 27930183 * q^92 - 11693189 * q^94 + 5726500 * q^95 - 12062476 * q^97 - 29617805 * q^98

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 $

$2$	−5.91113	−0.522475	−0.261238	−	0.965275i	$-0.584131\pi$
$2$	−5.91113	−0.522475	−0.261238	+	0.965275i	$0.584131\pi$
$3$	0	0
$3$	0	0
$4$	−93.0585	−0.727020
$5$	−125.000	−0.447214
$5$	−125.000	−0.447214
$6$	0	0
$7$	−1492.16	−1.64426	−0.822131	−	0.569299i	$-0.807215\pi$
$7$	−1492.16	−1.64426	−0.822131	+	0.569299i	$0.807215\pi$
$8$	1306.71	0.902325
$9$	0	0
$10$	738.892	0.233658
$11$	−3576.16	−0.810106	−0.405053	−	0.914293i	$-0.632747\pi$
$11$	−3576.16	−0.810106	−0.405053	+	0.914293i	$0.632747\pi$
$12$	0	0
$13$	−109.642	−0.0138413	−0.00692064	−	0.999976i	$-0.502203\pi$
$13$	−109.642	−0.0138413	−0.00692064	+	0.999976i	$0.502203\pi$
$14$	8820.33	0.859086
$15$	0	0
$16$	4187.38	0.255577
$17$	−24459.8	−1.20748	−0.603742	−	0.797180i	$-0.706324\pi$
$17$	−24459.8	−1.20748	−0.603742	+	0.797180i	$0.706324\pi$
$18$	0	0
$19$	28498.8	0.953210	0.476605	−	0.879117i	$-0.341867\pi$
$19$	28498.8	0.953210	0.476605	+	0.879117i	$0.341867\pi$
$20$	11632.3	0.325133
$21$	0	0
$22$	21139.1	0.423260
$23$	54000.8	0.925449	0.462725	−	0.886502i	$-0.346872\pi$
$23$	54000.8	0.925449	0.462725	+	0.886502i	$0.346872\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	648.110	0.00723172
$27$	0	0
$28$	138858.	1.19541
$29$	−101327.	−0.771491	−0.385746	−	0.922605i	$-0.626056\pi$
$29$	−101327.	−0.771491	−0.385746	+	0.922605i	$0.626056\pi$
$30$	0	0
$31$	10136.0	0.0611086	0.0305543	−	0.999533i	$-0.490273\pi$
$31$	10136.0	0.0611086	0.0305543	+	0.999533i	$0.490273\pi$
$32$	−192011.	−1.03586
$33$	0	0
$34$	144585.	0.630881
$35$	186519.	0.735336
$36$	0	0
$37$	572204.	1.85714	0.928570	−	0.371158i	$-0.121039\pi$
$37$	572204.	1.85714	0.928570	+	0.371158i	$0.121039\pi$
$38$	−168460.	−0.498029
$39$	0	0
$40$	−163338.	−0.403532
$41$	290903.	0.659182	0.329591	−	0.944124i	$-0.393089\pi$
$41$	290903.	0.659182	0.329591	+	0.944124i	$0.393089\pi$
$42$	0	0
$43$	−437043.	−0.838270	−0.419135	−	0.907924i	$-0.637667\pi$
$43$	−437043.	−0.838270	−0.419135	+	0.907924i	$0.637667\pi$
$44$	332792.	0.588963
$45$	0	0
$46$	−319206.	−0.483524
$47$	871931.	1.22501	0.612505	−	0.790467i	$-0.290162\pi$
$47$	871931.	1.22501	0.612505	+	0.790467i	$0.290162\pi$
$48$	0	0
$49$	1.40298e6	1.70359
$50$	−92361.4	−0.104495
$51$	0	0
$52$	10203.1	0.0100629
$53$	354451.	0.327032	0.163516	−	0.986541i	$-0.447716\pi$
$53$	354451.	0.327032	0.163516	+	0.986541i	$0.447716\pi$
$54$	0	0
$55$	447019.	0.362291
$56$	−1.94981e6	−1.48366
$57$	0	0
$58$	598956.	0.403085
$59$	2.00531e6	1.27116	0.635580	−	0.772035i	$-0.280761\pi$
$59$	2.00531e6	1.27116	0.635580	+	0.772035i	$0.280761\pi$
$60$	0	0
$61$	−2.54879e6	−1.43774	−0.718870	−	0.695145i	$-0.755340\pi$
$61$	−2.54879e6	−1.43774	−0.718870	+	0.695145i	$0.755340\pi$
$62$	−59915.5	−0.0319277
$63$	0	0
$64$	599015.	0.285633
$65$	13705.3	0.00619001
$66$	0	0
$67$	−4.04123e6	−1.64154	−0.820770	−	0.571258i	$-0.806455\pi$
$67$	−4.04123e6	−1.64154	−0.820770	+	0.571258i	$0.806455\pi$
$68$	2.27619e6	0.877865
$69$	0	0
$70$	−1.10254e6	−0.384195
$71$	507433.	0.168258	0.0841288	−	0.996455i	$-0.473189\pi$
$71$	507433.	0.168258	0.0841288	+	0.996455i	$0.473189\pi$
$72$	0	0
$73$	3.39540e6	1.02155	0.510776	−	0.859714i	$-0.329358\pi$
$73$	3.39540e6	1.02155	0.510776	+	0.859714i	$0.329358\pi$
$74$	−3.38237e6	−0.970309
$75$	0	0
$76$	−2.65205e6	−0.693003
$77$	5.33618e6	1.33203
$78$	0	0
$79$	842829.	0.192329	0.0961645	−	0.995365i	$-0.469343\pi$
$79$	842829.	0.192329	0.0961645	+	0.995365i	$0.469343\pi$
$80$	−523422.	−0.114298
$81$	0	0
$82$	−1.71957e6	−0.344406
$83$	4.91734e6	0.943968	0.471984	−	0.881607i	$-0.343538\pi$
$83$	4.91734e6	0.943968	0.471984	+	0.881607i	$0.343538\pi$
$84$	0	0
$85$	3.05747e6	0.540003
$86$	2.58342e6	0.437976
$87$	0	0
$88$	−4.67298e6	−0.730979
$89$	−1.87078e6	−0.281292	−0.140646	−	0.990060i	$-0.544918\pi$
$89$	−1.87078e6	−0.281292	−0.140646	+	0.990060i	$0.544918\pi$
$90$	0	0
$91$	163603.	0.0227587
$92$	−5.02523e6	−0.672820
$93$	0	0
$94$	−5.15410e6	−0.640037
$95$	−3.56235e6	−0.426289
$96$	0	0
$97$	3.31053e6	0.368295	0.184148	−	0.982899i	$-0.441048\pi$
$97$	3.31053e6	0.368295	0.184148	+	0.982899i	$0.441048\pi$
$98$	−8.29322e6	−0.890086
$99$	0	0
$100$	−1.45404e6	−0.145404
$101$	−1.58785e7	−1.53350	−0.766750	−	0.641946i	$-0.778127\pi$
$101$	−1.58785e7	−1.53350	−0.766750	+	0.641946i	$0.778127\pi$
$102$	0	0
$103$	9.69620e6	0.874321	0.437161	−	0.899383i	$-0.355984\pi$
$103$	9.69620e6	0.874321	0.437161	+	0.899383i	$0.355984\pi$
$104$	−143270.	−0.0124893
$105$	0	0
$106$	−2.09520e6	−0.170866
$107$	1.56533e7	1.23527	0.617635	−	0.786465i	$-0.288091\pi$
$107$	1.56533e7	1.23527	0.617635	+	0.786465i	$0.288091\pi$
$108$	0	0
$109$	1.72018e7	1.27227	0.636136	−	0.771577i	$-0.280532\pi$
$109$	1.72018e7	1.27227	0.636136	+	0.771577i	$0.280532\pi$
$110$	−2.64239e6	−0.189288
$111$	0	0
$112$	−6.24822e6	−0.420236
$113$	−1.51359e7	−0.986811	−0.493405	−	0.869799i	$-0.664248\pi$
$113$	−1.51359e7	−0.986811	−0.493405	+	0.869799i	$0.664248\pi$
$114$	0	0
$115$	−6.75010e6	−0.413874
$116$	9.42932e6	0.560889
$117$	0	0
$118$	−1.18537e7	−0.664150
$119$	3.64978e7	1.98542
$120$	0	0
$121$	−6.69828e6	−0.343728
$122$	1.50663e7	0.751183
$123$	0	0
$124$	−943245.	−0.0444272
$125$	−1.95312e6	−0.0894427
$126$	0	0
$127$	−4.00630e7	−1.73553	−0.867763	−	0.496979i	$-0.834443\pi$
$127$	−4.00630e7	−1.73553	−0.867763	+	0.496979i	$0.834443\pi$
$128$	2.10365e7	0.886622
$129$	0	0
$130$	−81013.7	−0.00323412
$131$	−6.41590e6	−0.249349	−0.124675	−	0.992198i	$-0.539789\pi$
$131$	−6.41590e6	−0.249349	−0.124675	+	0.992198i	$0.539789\pi$
$132$	0	0
$133$	−4.25246e7	−1.56733
$134$	2.38882e7	0.857664
$135$	0	0
$136$	−3.19618e7	−1.08954
$137$	5.35392e7	1.77889	0.889447	−	0.457038i	$-0.151090\pi$
$137$	5.35392e7	1.77889	0.889447	+	0.457038i	$0.151090\pi$
$138$	0	0
$139$	1.76934e7	0.558803	0.279401	−	0.960174i	$-0.409864\pi$
$139$	1.76934e7	0.558803	0.279401	+	0.960174i	$0.409864\pi$
$140$	−1.73572e7	−0.534604
$141$	0	0
$142$	−2.99950e6	−0.0879104
$143$	392098.	0.0112129
$144$	0	0
$145$	1.26658e7	0.345021
$146$	−2.00706e7	−0.533736
$147$	0	0
$148$	−5.32484e7	−1.35018
$149$	2.86103e7	0.708551	0.354275	−	0.935141i	$-0.384728\pi$
$149$	2.86103e7	0.708551	0.354275	+	0.935141i	$0.384728\pi$
$150$	0	0
$151$	6.86770e7	1.62327	0.811637	−	0.584162i	$-0.198577\pi$
$151$	6.86770e7	1.62327	0.811637	+	0.584162i	$0.198577\pi$
$152$	3.72395e7	0.860106
$153$	0	0
$154$	−3.15429e7	−0.695951
$155$	−1.26701e6	−0.0273286
$156$	0	0
$157$	−7.70685e7	−1.58938	−0.794690	−	0.607015i	$-0.792367\pi$
$157$	−7.70685e7	−1.58938	−0.794690	+	0.607015i	$0.792367\pi$
$158$	−4.98207e6	−0.100487
$159$	0	0
$160$	2.40013e7	0.463250
$161$	−8.05775e7	−1.52168
$162$	0	0
$163$	−9.79134e7	−1.77087	−0.885433	−	0.464767i	$-0.846138\pi$
$163$	−9.79134e7	−1.77087	−0.885433	+	0.464767i	$0.846138\pi$
$164$	−2.70710e7	−0.479238
$165$	0	0
$166$	−2.90671e7	−0.493200
$167$	−3.78656e7	−0.629125	−0.314562	−	0.949237i	$-0.601858\pi$
$167$	−3.78656e7	−0.629125	−0.314562	+	0.949237i	$0.601858\pi$
$168$	0	0
$169$	−6.27365e7	−0.999808
$170$	−1.80731e7	−0.282138
$171$	0	0
$172$	4.06705e7	0.609439
$173$	−2.11409e7	−0.310428	−0.155214	−	0.987881i	$-0.549607\pi$
$173$	−2.11409e7	−0.310428	−0.155214	+	0.987881i	$0.549607\pi$
$174$	0	0
$175$	−2.33149e7	−0.328852
$176$	−1.49747e7	−0.207045
$177$	0	0
$178$	1.10584e7	0.146968
$179$	−3.99517e7	−0.520655	−0.260328	−	0.965520i	$-0.583831\pi$
$179$	−3.99517e7	−0.520655	−0.260328	+	0.965520i	$0.583831\pi$
$180$	0	0
$181$	−7.24071e7	−0.907625	−0.453812	−	0.891097i	$-0.649936\pi$
$181$	−7.24071e7	−0.907625	−0.453812	+	0.891097i	$0.649936\pi$
$182$	−967080.	−0.0118908
$183$	0	0
$184$	7.05632e7	0.835056
$185$	−7.15254e7	−0.830538
$186$	0	0
$187$	8.74720e7	0.978191
$188$	−8.11406e7	−0.890606
$189$	0	0
$190$	2.10575e7	0.222725
$191$	9.44011e7	0.980303	0.490152	−	0.871637i	$-0.336941\pi$
$191$	9.44011e7	0.980303	0.490152	+	0.871637i	$0.336941\pi$
$192$	0	0
$193$	7.62748e7	0.763714	0.381857	−	0.924221i	$-0.375285\pi$
$193$	7.62748e7	0.763714	0.381857	+	0.924221i	$0.375285\pi$
$194$	−1.95690e7	−0.192425
$195$	0	0
$196$	−1.30560e8	−1.23855
$197$	9.58235e7	0.892977	0.446489	−	0.894789i	$-0.352674\pi$
$197$	9.58235e7	0.892977	0.446489	+	0.894789i	$0.352674\pi$
$198$	0	0
$199$	592156.	0.00532661	0.00266330	−	0.999996i	$-0.499152\pi$
$199$	592156.	0.00532661	0.00266330	+	0.999996i	$0.499152\pi$
$200$	2.04173e7	0.180465
$201$	0	0
$202$	9.38597e7	0.801215
$203$	1.51195e8	1.26853
$204$	0	0
$205$	−3.63629e7	−0.294795
$206$	−5.73155e7	−0.456811
$207$	0	0
$208$	−459113.	−0.00353751
$209$	−1.01916e8	−0.772202
$210$	0	0
$211$	−6.79066e7	−0.497649	−0.248825	−	0.968549i	$-0.580044\pi$
$211$	−6.79066e7	−0.497649	−0.248825	+	0.968549i	$0.580044\pi$
$212$	−3.29846e7	−0.237759
$213$	0	0
$214$	−9.25285e7	−0.645397
$215$	5.46303e7	0.374886
$216$	0	0
$217$	−1.51245e7	−0.100479
$218$	−1.01682e8	−0.664731
$219$	0	0
$220$	−4.15990e7	−0.263392
$221$	2.68183e6	0.0167131
$222$	0	0
$223$	9.11448e6	0.0550383	0.0275191	−	0.999621i	$-0.491239\pi$
$223$	9.11448e6	0.0550383	0.0275191	+	0.999621i	$0.491239\pi$
$224$	2.86509e8	1.70322
$225$	0	0
$226$	8.94704e7	0.515584
$227$	−3.07759e7	−0.174631	−0.0873154	−	0.996181i	$-0.527829\pi$
$227$	−3.07759e7	−0.174631	−0.0873154	+	0.996181i	$0.527829\pi$
$228$	0	0
$229$	−2.32505e8	−1.27941	−0.639704	−	0.768622i	$-0.720943\pi$
$229$	−2.32505e8	−1.27941	−0.639704	+	0.768622i	$0.720943\pi$
$230$	3.99007e7	0.216239
$231$	0	0
$232$	−1.32404e8	−0.696136
$233$	−2.43477e8	−1.26099	−0.630495	−	0.776193i	$-0.717148\pi$
$233$	−2.43477e8	−1.26099	−0.630495	+	0.776193i	$0.717148\pi$
$234$	0	0
$235$	−1.08991e8	−0.547841
$236$	−1.86612e8	−0.924159
$237$	0	0
$238$	−2.15743e8	−1.03733
$239$	−2.61739e8	−1.24015	−0.620076	−	0.784541i	$-0.712898\pi$
$239$	−2.61739e8	−1.24015	−0.620076	+	0.784541i	$0.712898\pi$
$240$	0	0
$241$	−2.36191e8	−1.08693	−0.543467	−	0.839431i	$-0.682889\pi$
$241$	−2.36191e8	−1.08693	−0.543467	+	0.839431i	$0.682889\pi$
$242$	3.95944e7	0.179589
$243$	0	0
$244$	2.37187e8	1.04526
$245$	−1.75373e8	−0.761871
$246$	0	0
$247$	−3.12467e6	−0.0131936
$248$	1.32448e7	0.0551398
$249$	0	0
$250$	1.15452e7	0.0467316
$251$	−4.89910e7	−0.195550	−0.0977751	−	0.995209i	$-0.531173\pi$
$251$	−4.89910e7	−0.195550	−0.0977751	+	0.995209i	$0.531173\pi$
$252$	0	0
$253$	−1.93115e8	−0.749712
$254$	2.36818e8	0.906769
$255$	0	0
$256$	−2.01023e8	−0.748871
$257$	3.72153e8	1.36759	0.683795	−	0.729674i	$-0.260328\pi$
$257$	3.72153e8	1.36759	0.683795	+	0.729674i	$0.260328\pi$
$258$	0	0
$259$	−8.53816e8	−3.05362
$260$	−1.27539e6	−0.00450026
$261$	0	0
$262$	3.79253e7	0.130279
$263$	−4.76949e8	−1.61669	−0.808346	−	0.588708i	$-0.799637\pi$
$263$	−4.76949e8	−1.61669	−0.808346	+	0.588708i	$0.799637\pi$
$264$	0	0
$265$	−4.43063e7	−0.146253
$266$	2.51369e8	0.818889
$267$	0	0
$268$	3.76071e8	1.19343
$269$	−2.27808e8	−0.713568	−0.356784	−	0.934187i	$-0.616127\pi$
$269$	−2.27808e8	−0.713568	−0.356784	+	0.934187i	$0.616127\pi$
$270$	0	0
$271$	−4.41893e8	−1.34873	−0.674365	−	0.738398i	$-0.735583\pi$
$271$	−4.41893e8	−1.34873	−0.674365	+	0.738398i	$0.735583\pi$
$272$	−1.02422e8	−0.308606
$273$	0	0
$274$	−3.16477e8	−0.929428
$275$	−5.58774e7	−0.162021
$276$	0	0
$277$	−3.27286e8	−0.925227	−0.462614	−	0.886560i	$-0.653088\pi$
$277$	−3.27286e8	−0.925227	−0.462614	+	0.886560i	$0.653088\pi$
$278$	−1.04588e8	−0.291961
$279$	0	0
$280$	2.43726e8	0.663512
$281$	5.98042e8	1.60790	0.803950	−	0.594696i	$-0.202728\pi$
$281$	5.98042e8	1.60790	0.803950	+	0.594696i	$0.202728\pi$
$282$	0	0
$283$	−1.05706e8	−0.277235	−0.138617	−	0.990346i	$-0.544266\pi$
$283$	−1.05706e8	−0.277235	−0.138617	+	0.990346i	$0.544266\pi$
$284$	−4.72210e7	−0.122327
$285$	0	0
$286$	−2.31774e6	−0.00585846
$287$	−4.34073e8	−1.08387
$288$	0	0
$289$	1.87943e8	0.458019
$290$	−7.48695e7	−0.180265
$291$	0	0
$292$	−3.15971e8	−0.742688
$293$	1.09165e8	0.253540	0.126770	−	0.991932i	$-0.459539\pi$
$293$	1.09165e8	0.253540	0.126770	+	0.991932i	$0.459539\pi$
$294$	0	0
$295$	−2.50664e8	−0.568480
$296$	7.47702e8	1.67574
$297$	0	0
$298$	−1.69119e8	−0.370200
$299$	−5.92077e6	−0.0128094
$300$	0	0
$301$	6.52135e8	1.37834
$302$	−4.05959e8	−0.848121
$303$	0	0
$304$	1.19335e8	0.243619
$305$	3.18599e8	0.642977
$306$	0	0
$307$	−7.01970e8	−1.38463	−0.692316	−	0.721595i	$-0.743410\pi$
$307$	−7.01970e8	−1.38463	−0.692316	+	0.721595i	$0.743410\pi$
$308$	−4.96577e8	−0.968409
$309$	0	0
$310$	7.48943e6	0.0142785
$311$	7.63124e8	1.43858	0.719289	−	0.694711i	$-0.244468\pi$
$311$	7.63124e8	1.43858	0.719289	+	0.694711i	$0.244468\pi$
$312$	0	0
$313$	8.70530e8	1.60464	0.802322	−	0.596891i	$-0.203598\pi$
$313$	8.70530e8	1.60464	0.802322	+	0.596891i	$0.203598\pi$
$314$	4.55562e8	0.830412
$315$	0	0
$316$	−7.84324e7	−0.139827
$317$	3.93809e8	0.694349	0.347175	−	0.937800i	$-0.387141\pi$
$317$	3.93809e8	0.694349	0.347175	+	0.937800i	$0.387141\pi$
$318$	0	0
$319$	3.62360e8	0.624990
$320$	−7.48769e7	−0.127739
$321$	0	0
$322$	4.76305e8	0.795040
$323$	−6.97074e8	−1.15099
$324$	0	0
$325$	−1.71316e6	−0.00276825
$326$	5.78779e8	0.925234
$327$	0	0
$328$	3.80125e8	0.594796
$329$	−1.30106e9	−2.01424
$330$	0	0
$331$	2.59009e8	0.392570	0.196285	−	0.980547i	$-0.437112\pi$
$331$	2.59009e8	0.392570	0.196285	+	0.980547i	$0.437112\pi$
$332$	−4.57601e8	−0.686283
$333$	0	0
$334$	2.23828e8	0.328702
$335$	5.05154e8	0.734119
$336$	0	0
$337$	3.93178e8	0.559608	0.279804	−	0.960057i	$-0.409730\pi$
$337$	3.93178e8	0.559608	0.279804	+	0.960057i	$0.409730\pi$
$338$	3.70844e8	0.522375
$339$	0	0
$340$	−2.84524e8	−0.392593
$341$	−3.62481e7	−0.0495045
$342$	0	0
$343$	−8.64615e8	−1.15689
$344$	−5.71086e8	−0.756392
$345$	0	0
$346$	1.24966e8	0.162191
$347$	−6.40520e8	−0.822962	−0.411481	−	0.911418i	$-0.634988\pi$
$347$	−6.40520e8	−0.822962	−0.411481	+	0.911418i	$0.634988\pi$
$348$	0	0
$349$	7.17808e8	0.903898	0.451949	−	0.892044i	$-0.350729\pi$
$349$	7.17808e8	0.903898	0.451949	+	0.892044i	$0.350729\pi$
$350$	1.37818e8	0.171817
$351$	0	0
$352$	6.86660e8	0.839155
$353$	1.48563e9	1.79763	0.898813	−	0.438333i	$-0.144431\pi$
$353$	1.48563e9	1.79763	0.898813	+	0.438333i	$0.144431\pi$
$354$	0	0
$355$	−6.34291e7	−0.0752471
$356$	1.74092e8	0.204505
$357$	0	0
$358$	2.36160e8	0.272029
$359$	9.55975e8	1.09048	0.545238	−	0.838281i	$-0.316439\pi$
$359$	9.55975e8	1.09048	0.545238	+	0.838281i	$0.316439\pi$
$360$	0	0
$361$	−8.16909e7	−0.0913900
$362$	4.28008e8	0.474212
$363$	0	0
$364$	−1.52247e7	−0.0165460
$365$	−4.24425e8	−0.456852
$366$	0	0
$367$	1.01414e9	1.07095	0.535474	−	0.844551i	$-0.320133\pi$
$367$	1.01414e9	1.07095	0.535474	+	0.844551i	$0.320133\pi$
$368$	2.26122e8	0.236524
$369$	0	0
$370$	4.22796e8	0.433936
$371$	−5.28895e8	−0.537726
$372$	0	0
$373$	6.38325e8	0.636885	0.318443	−	0.947942i	$-0.396840\pi$
$373$	6.38325e8	0.636885	0.318443	+	0.947942i	$0.396840\pi$
$374$	−5.17059e8	−0.511080
$375$	0	0
$376$	1.13936e9	1.10536
$377$	1.11097e7	0.0106784
$378$	0	0
$379$	1.42392e9	1.34353	0.671765	−	0.740764i	$-0.265537\pi$
$379$	1.42392e9	1.34353	0.671765	+	0.740764i	$0.265537\pi$
$380$	3.31507e8	0.309920
$381$	0	0
$382$	−5.58018e8	−0.512184
$383$	4.56913e8	0.415564	0.207782	−	0.978175i	$-0.433376\pi$
$383$	4.56913e8	0.415564	0.207782	+	0.978175i	$0.433376\pi$
$384$	0	0
$385$	−6.67022e8	−0.595700
$386$	−4.50871e8	−0.399022
$387$	0	0
$388$	−3.08073e8	−0.267758
$389$	6.64077e8	0.571998	0.285999	−	0.958230i	$-0.407675\pi$
$389$	6.64077e8	0.571998	0.285999	+	0.958230i	$0.407675\pi$
$390$	0	0
$391$	−1.32085e9	−1.11747
$392$	1.83329e9	1.53720
$393$	0	0
$394$	−5.66426e8	−0.466559
$395$	−1.05354e8	−0.0860121
$396$	0	0
$397$	9.63838e8	0.773103	0.386551	−	0.922268i	$-0.373666\pi$
$397$	9.63838e8	0.773103	0.386551	+	0.922268i	$0.373666\pi$
$398$	−3.50031e6	−0.00278302
$399$	0	0
$400$	6.54278e7	0.0511154
$401$	5.73962e8	0.444506	0.222253	−	0.974989i	$-0.428659\pi$
$401$	5.73962e8	0.444506	0.222253	+	0.974989i	$0.428659\pi$
$402$	0	0
$403$	−1.11134e6	−0.000845821 0
$404$	1.47763e9	1.11488
$405$	0	0
$406$	−8.93735e8	−0.662777
$407$	−2.04629e9	−1.50448
$408$	0	0
$409$	4.40151e8	0.318104	0.159052	−	0.987270i	$-0.449156\pi$
$409$	4.40151e8	0.318104	0.159052	+	0.987270i	$0.449156\pi$
$410$	2.14946e8	0.154023
$411$	0	0
$412$	−9.02314e8	−0.635649
$413$	−2.99224e9	−2.09012
$414$	0	0
$415$	−6.14668e8	−0.422155
$416$	2.10525e7	0.0143376
$417$	0	0
$418$	6.02440e8	0.403456
$419$	−2.09626e9	−1.39218	−0.696090	−	0.717955i	$-0.745078\pi$
$419$	−2.09626e9	−1.39218	−0.696090	+	0.717955i	$0.745078\pi$
$420$	0	0
$421$	6.70944e8	0.438227	0.219114	−	0.975699i	$-0.429683\pi$
$421$	6.70944e8	0.438227	0.219114	+	0.975699i	$0.429683\pi$
$422$	4.01405e8	0.260009
$423$	0	0
$424$	4.63163e8	0.295089
$425$	−3.82184e8	−0.241497
$426$	0	0
$427$	3.80319e9	2.36402
$428$	−1.45667e9	−0.898065
$429$	0	0
$430$	−3.22927e8	−0.195869
$431$	3.87384e8	0.233062	0.116531	−	0.993187i	$-0.462823\pi$
$431$	3.87384e8	0.233062	0.116531	+	0.993187i	$0.462823\pi$
$432$	0	0
$433$	−4.07848e7	−0.0241430	−0.0120715	−	0.999927i	$-0.503843\pi$
$433$	−4.07848e7	−0.0241430	−0.0120715	+	0.999927i	$0.503843\pi$
$434$	8.94032e7	0.0524975
$435$	0	0
$436$	−1.60077e9	−0.924967
$437$	1.53896e9	0.882148
$438$	0	0
$439$	−7.61384e8	−0.429514	−0.214757	−	0.976667i	$-0.568896\pi$
$439$	−7.61384e8	−0.429514	−0.214757	+	0.976667i	$0.568896\pi$
$440$	5.84123e8	0.326904
$441$	0	0
$442$	−1.58526e7	−0.00873219
$443$	1.26642e9	0.692094	0.346047	−	0.938217i	$-0.387524\pi$
$443$	1.26642e9	0.692094	0.346047	+	0.938217i	$0.387524\pi$
$444$	0	0
$445$	2.33847e8	0.125798
$446$	−5.38769e7	−0.0287561
$447$	0	0
$448$	−8.93824e8	−0.469655
$449$	4.51743e8	0.235521	0.117760	−	0.993042i	$-0.462429\pi$
$449$	4.51743e8	0.235521	0.117760	+	0.993042i	$0.462429\pi$
$450$	0	0
$451$	−1.04032e9	−0.534008
$452$	1.40853e9	0.717431
$453$	0	0
$454$	1.81921e8	0.0912403
$455$	−2.04504e7	−0.0101780
$456$	0	0
$457$	1.21974e9	0.597804	0.298902	−	0.954284i	$-0.403380\pi$
$457$	1.21974e9	0.597804	0.298902	+	0.954284i	$0.403380\pi$
$458$	1.37437e9	0.668459
$459$	0	0
$460$	6.28154e8	0.300894
$461$	−2.80664e9	−1.33424	−0.667119	−	0.744951i	$-0.732473\pi$
$461$	−2.80664e9	−1.33424	−0.667119	+	0.744951i	$0.732473\pi$
$462$	0	0
$463$	−2.36507e9	−1.10741	−0.553707	−	0.832712i	$-0.686787\pi$
$463$	−2.36507e9	−1.10741	−0.553707	+	0.832712i	$0.686787\pi$
$464$	−4.24293e8	−0.197176
$465$	0	0
$466$	1.43922e9	0.658837
$467$	2.03758e9	0.925773	0.462887	−	0.886417i	$-0.346814\pi$
$467$	2.03758e9	0.925773	0.462887	+	0.886417i	$0.346814\pi$
$468$	0	0
$469$	6.03014e9	2.69912
$470$	6.44262e8	0.286233
$471$	0	0
$472$	2.62036e9	1.14700
$473$	1.56293e9	0.679088
$474$	0	0
$475$	4.45294e8	0.190642
$476$	−3.39643e9	−1.44344
$477$	0	0
$478$	1.54717e9	0.647949
$479$	−1.67193e9	−0.695095	−0.347547	−	0.937662i	$-0.612985\pi$
$479$	−1.67193e9	−0.695095	−0.347547	+	0.937662i	$0.612985\pi$
$480$	0	0
$481$	−6.27377e7	−0.0257052
$482$	1.39615e9	0.567896
$483$	0	0
$484$	6.23332e8	0.249897
$485$	−4.13816e8	−0.164707
$486$	0	0
$487$	5.35542e7	0.0210108	0.0105054	−	0.999945i	$-0.496656\pi$
$487$	5.35542e7	0.0210108	0.0105054	+	0.999945i	$0.496656\pi$
$488$	−3.33052e9	−1.29731
$489$	0	0
$490$	1.03665e9	0.398059
$491$	1.47825e9	0.563589	0.281795	−	0.959475i	$-0.409070\pi$
$491$	1.47825e9	0.563589	0.281795	+	0.959475i	$0.409070\pi$
$492$	0	0
$493$	2.47843e9	0.931564
$494$	1.84703e7	0.00689335
$495$	0	0
$496$	4.24434e7	0.0156180
$497$	−7.57169e8	−0.276659
$498$	0	0
$499$	−1.97941e8	−0.0713156	−0.0356578	−	0.999364i	$-0.511353\pi$
$499$	−1.97941e8	−0.0713156	−0.0356578	+	0.999364i	$0.511353\pi$
$500$	1.81755e8	0.0650266
$501$	0	0
$502$	2.89592e8	0.102170
$503$	2.63594e9	0.923522	0.461761	−	0.887004i	$-0.347218\pi$
$503$	2.63594e9	0.923522	0.461761	+	0.887004i	$0.347218\pi$
$504$	0	0
$505$	1.98481e9	0.685802
$506$	1.14153e9	0.391706
$507$	0	0
$508$	3.72821e9	1.26176
$509$	1.12714e9	0.378849	0.189424	−	0.981895i	$-0.439338\pi$
$509$	1.12714e9	0.378849	0.189424	+	0.981895i	$0.439338\pi$
$510$	0	0
$511$	−5.06646e9	−1.67970
$512$	−1.50439e9	−0.495355
$513$	0	0
$514$	−2.19985e9	−0.714532
$515$	−1.21202e9	−0.391008
$516$	0	0
$517$	−3.11816e9	−0.992388
$518$	5.04702e9	1.59544
$519$	0	0
$520$	1.79088e7	0.00558540
$521$	−3.05729e9	−0.947120	−0.473560	−	0.880762i	$-0.657031\pi$
$521$	−3.05729e9	−0.947120	−0.473560	+	0.880762i	$0.657031\pi$
$522$	0	0
$523$	−1.25083e9	−0.382332	−0.191166	−	0.981558i	$-0.561227\pi$
$523$	−1.25083e9	−0.382332	−0.191166	+	0.981558i	$0.561227\pi$
$524$	5.97055e8	0.181282
$525$	0	0
$526$	2.81931e9	0.844681
$527$	−2.47925e8	−0.0737877
$528$	0	0
$529$	−4.88740e8	−0.143543
$530$	2.61900e8	0.0764136
$531$	0	0
$532$	3.95728e9	1.13948
$533$	−3.18953e7	−0.00912392
$534$	0	0
$535$	−1.95666e9	−0.552429
$536$	−5.28070e9	−1.48120
$537$	0	0
$538$	1.34660e9	0.372822
$539$	−5.01729e9	−1.38009
$540$	0	0
$541$	8.33089e8	0.226204	0.113102	−	0.993583i	$-0.463921\pi$
$541$	8.33089e8	0.226204	0.113102	+	0.993583i	$0.463921\pi$
$542$	2.61209e9	0.704678
$543$	0	0
$544$	4.69654e9	1.25078
$545$	−2.15022e9	−0.568977
$546$	0	0
$547$	2.71326e9	0.708820	0.354410	−	0.935090i	$-0.384682\pi$
$547$	2.71326e9	0.708820	0.354410	+	0.935090i	$0.384682\pi$
$548$	−4.98228e9	−1.29329
$549$	0	0
$550$	3.30299e8	0.0846521
$551$	−2.88769e9	−0.735394
$552$	0	0
$553$	−1.25763e9	−0.316239
$554$	1.93463e9	0.483408
$555$	0	0
$556$	−1.64652e9	−0.406260
$557$	6.65364e9	1.63142	0.815710	−	0.578460i	$-0.196346\pi$
$557$	6.65364e9	1.63142	0.815710	+	0.578460i	$0.196346\pi$
$558$	0	0
$559$	4.79183e7	0.0116027
$560$	7.81027e8	0.187935
$561$	0	0
$562$	−3.53510e9	−0.840088
$563$	−1.19753e9	−0.282818	−0.141409	−	0.989951i	$-0.545163\pi$
$563$	−1.19753e9	−0.282818	−0.141409	+	0.989951i	$0.545163\pi$
$564$	0	0
$565$	1.89199e9	0.441315
$566$	6.24843e8	0.144848
$567$	0	0
$568$	6.63066e8	0.151823
$569$	−4.56090e9	−1.03791	−0.518953	−	0.854803i	$-0.673678\pi$
$569$	−4.56090e9	−1.03791	−0.518953	+	0.854803i	$0.673678\pi$
$570$	0	0
$571$	−2.37885e9	−0.534739	−0.267369	−	0.963594i	$-0.586154\pi$
$571$	−2.37885e9	−0.534739	−0.267369	+	0.963594i	$0.586154\pi$
$572$	−3.64880e7	−0.00815200
$573$	0	0
$574$	2.56586e9	0.566294
$575$	8.43762e8	0.185090
$576$	0	0
$577$	2.36206e9	0.511889	0.255945	−	0.966691i	$-0.417613\pi$
$577$	2.36206e9	0.511889	0.255945	+	0.966691i	$0.417613\pi$
$578$	−1.11095e9	−0.239303
$579$	0	0
$580$	−1.17866e9	−0.250837
$581$	−7.33744e9	−1.55213
$582$	0	0
$583$	−1.26757e9	−0.264931
$584$	4.43679e9	0.921772
$585$	0	0
$586$	−6.45288e8	−0.132468
$587$	9.40803e9	1.91984	0.959921	−	0.280270i	$-0.0904241\pi$
$587$	9.40803e9	1.91984	0.959921	+	0.280270i	$0.0904241\pi$
$588$	0	0
$589$	2.88865e8	0.0582494
$590$	1.48171e9	0.297017
$591$	0	0
$592$	2.39603e9	0.474643
$593$	7.59035e9	1.49476	0.747379	−	0.664398i	$-0.231312\pi$
$593$	7.59035e9	1.49476	0.747379	+	0.664398i	$0.231312\pi$
$594$	0	0
$595$	−4.56223e9	−0.887907
$596$	−2.66243e9	−0.515130
$597$	0	0
$598$	3.49984e7	0.00669259
$599$	−4.75765e9	−0.904479	−0.452239	−	0.891897i	$-0.649375\pi$
$599$	−4.75765e9	−0.904479	−0.452239	+	0.891897i	$0.649375\pi$
$600$	0	0
$601$	−2.55008e9	−0.479175	−0.239587	−	0.970875i	$-0.577012\pi$
$601$	−2.55008e9	−0.479175	−0.239587	+	0.970875i	$0.577012\pi$
$602$	−3.85486e9	−0.720146
$603$	0	0
$604$	−6.39098e9	−1.18015
$605$	8.37285e8	0.153720
$606$	0	0
$607$	−1.51631e9	−0.275187	−0.137593	−	0.990489i	$-0.543937\pi$
$607$	−1.51631e9	−0.275187	−0.137593	+	0.990489i	$0.543937\pi$
$608$	−5.47207e9	−0.987390
$609$	0	0
$610$	−1.88328e9	−0.335939
$611$	−9.56004e7	−0.0169557
$612$	0	0
$613$	−5.33935e9	−0.936217	−0.468109	−	0.883671i	$-0.655064\pi$
$613$	−5.33935e9	−0.936217	−0.468109	+	0.883671i	$0.655064\pi$
$614$	4.14944e9	0.723436
$615$	0	0
$616$	6.97282e9	1.20192
$617$	−8.73155e9	−1.49656	−0.748279	−	0.663384i	$-0.769120\pi$
$617$	−8.73155e9	−1.49656	−0.748279	+	0.663384i	$0.769120\pi$
$618$	0	0
$619$	−4.46169e9	−0.756105	−0.378052	−	0.925784i	$-0.623406\pi$
$619$	−4.46169e9	−0.756105	−0.378052	+	0.925784i	$0.623406\pi$
$620$	1.17906e8	0.0198684
$621$	0	0
$622$	−4.51092e9	−0.751622
$623$	2.79149e9	0.462518
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	−5.14582e9	−0.838387
$627$	0	0
$628$	7.17188e9	1.15551
$629$	−1.39960e10	−2.24247
$630$	0	0
$631$	−6.60314e8	−0.104628	−0.0523139	−	0.998631i	$-0.516660\pi$
$631$	−6.60314e8	−0.104628	−0.0523139	+	0.998631i	$0.516660\pi$
$632$	1.10133e9	0.173543
$633$	0	0
$634$	−2.32786e9	−0.362780
$635$	5.00788e9	0.776151
$636$	0	0
$637$	−1.53826e8	−0.0235799
$638$	−2.14196e9	−0.326542
$639$	0	0
$640$	−2.62956e9	−0.396509
$641$	7.72820e8	0.115898	0.0579489	−	0.998320i	$-0.481544\pi$
$641$	7.72820e8	0.115898	0.0579489	+	0.998320i	$0.481544\pi$
$642$	0	0
$643$	−1.11126e9	−0.164845	−0.0824226	−	0.996597i	$-0.526266\pi$
$643$	−1.11126e9	−0.164845	−0.0824226	+	0.996597i	$0.526266\pi$
$644$	7.49843e9	1.10629
$645$	0	0
$646$	4.12050e9	0.601362
$647$	2.50450e9	0.363543	0.181772	−	0.983341i	$-0.441817\pi$
$647$	2.50450e9	0.363543	0.181772	+	0.983341i	$0.441817\pi$
$648$	0	0
$649$	−7.17132e9	−1.02978
$650$	1.01267e7	0.00144634
$651$	0	0
$652$	9.11168e9	1.28745
$653$	4.28585e9	0.602338	0.301169	−	0.953571i	$-0.402623\pi$
$653$	4.28585e9	0.602338	0.301169	+	0.953571i	$0.402623\pi$
$654$	0	0
$655$	8.01988e8	0.111512
$656$	1.21812e9	0.168472
$657$	0	0
$658$	7.69071e9	1.05239
$659$	2.19278e9	0.298467	0.149234	−	0.988802i	$-0.452319\pi$
$659$	2.19278e9	0.298467	0.149234	+	0.988802i	$0.452319\pi$
$660$	0	0
$661$	−4.27140e9	−0.575261	−0.287630	−	0.957741i	$-0.592867\pi$
$661$	−4.27140e9	−0.575261	−0.287630	+	0.957741i	$0.592867\pi$
$662$	−1.53104e9	−0.205108
$663$	0	0
$664$	6.42552e9	0.851766
$665$	5.31558e9	0.700930
$666$	0	0
$667$	−5.47172e9	−0.713976
$668$	3.52371e9	0.457386
$669$	0	0
$670$	−2.98603e9	−0.383559
$671$	9.11488e9	1.16472
$672$	0	0
$673$	−9.27633e9	−1.17307	−0.586534	−	0.809924i	$-0.699508\pi$
$673$	−9.27633e9	−1.17307	−0.586534	+	0.809924i	$0.699508\pi$
$674$	−2.32413e9	−0.292382
$675$	0	0
$676$	5.83817e9	0.726880
$677$	−4.38334e9	−0.542931	−0.271466	−	0.962448i	$-0.587508\pi$
$677$	−4.38334e9	−0.542931	−0.271466	+	0.962448i	$0.587508\pi$
$678$	0	0
$679$	−4.93982e9	−0.605574
$680$	3.99522e9	0.487259
$681$	0	0
$682$	2.14267e8	0.0258649
$683$	3.98108e9	0.478111	0.239055	−	0.971006i	$-0.423162\pi$
$683$	3.98108e9	0.478111	0.239055	+	0.971006i	$0.423162\pi$
$684$	0	0
$685$	−6.69240e9	−0.795546
$686$	5.11086e9	0.604448
$687$	0	0
$688$	−1.83006e9	−0.214243
$689$	−3.88627e7	−0.00452654
$690$	0	0
$691$	−9.79602e9	−1.12947	−0.564737	−	0.825271i	$-0.691022\pi$
$691$	−9.79602e9	−1.12947	−0.564737	+	0.825271i	$0.691022\pi$
$692$	1.96734e9	0.225688
$693$	0	0
$694$	3.78620e9	0.429977
$695$	−2.21167e9	−0.249904
$696$	0	0
$697$	−7.11544e9	−0.795952
$698$	−4.24306e9	−0.472264
$699$	0	0
$700$	2.16965e9	0.239082
$701$	6.13054e9	0.672181	0.336090	−	0.941830i	$-0.390895\pi$
$701$	6.13054e9	0.672181	0.336090	+	0.941830i	$0.390895\pi$
$702$	0	0
$703$	1.63071e10	1.77024
$704$	−2.14217e9	−0.231393
$705$	0	0
$706$	−8.78176e9	−0.939215
$707$	2.36931e10	2.52147
$708$	0	0
$709$	1.05803e10	1.11490	0.557452	−	0.830209i	$-0.311779\pi$
$709$	1.05803e10	1.11490	0.557452	+	0.830209i	$0.311779\pi$
$710$	3.74938e8	0.0393147
$711$	0	0
$712$	−2.44456e9	−0.253817
$713$	5.47354e8	0.0565529
$714$	0	0
$715$	−4.90122e7	−0.00501456
$716$	3.71785e9	0.378526
$717$	0	0
$718$	−5.65089e9	−0.569746
$719$	−9.86567e9	−0.989864	−0.494932	−	0.868932i	$-0.664807\pi$
$719$	−9.86567e9	−0.989864	−0.494932	+	0.868932i	$0.664807\pi$
$720$	0	0
$721$	−1.44682e10	−1.43761
$722$	4.82886e8	0.0477490
$723$	0	0
$724$	6.73810e9	0.659861
$725$	−1.58323e9	−0.154298
$726$	0	0
$727$	−2.53606e9	−0.244788	−0.122394	−	0.992482i	$-0.539057\pi$
$727$	−2.53606e9	−0.244788	−0.122394	+	0.992482i	$0.539057\pi$
$728$	2.13781e8	0.0205357
$729$	0	0
$730$	2.50883e9	0.238694
$731$	1.06900e10	1.01220
$732$	0	0
$733$	−3.86479e9	−0.362461	−0.181231	−	0.983441i	$-0.558008\pi$
$733$	−3.86479e9	−0.362461	−0.181231	+	0.983441i	$0.558008\pi$
$734$	−5.99474e9	−0.559544
$735$	0	0
$736$	−1.03687e10	−0.958634
$737$	1.44521e10	1.32982
$738$	0	0
$739$	−1.67871e10	−1.53010	−0.765052	−	0.643968i	$-0.777287\pi$
$739$	−1.67871e10	−1.53010	−0.765052	+	0.643968i	$0.777287\pi$
$740$	6.65605e9	0.603817
$741$	0	0
$742$	3.12637e9	0.280948
$743$	−1.43082e10	−1.27975	−0.639874	−	0.768480i	$-0.721014\pi$
$743$	−1.43082e10	−1.27975	−0.639874	+	0.768480i	$0.721014\pi$
$744$	0	0
$745$	−3.57629e9	−0.316873
$746$	−3.77323e9	−0.332757
$747$	0	0
$748$	−8.14002e9	−0.711164
$749$	−2.33571e10	−2.03110
$750$	0	0
$751$	−1.07967e10	−0.930149	−0.465075	−	0.885272i	$-0.653972\pi$
$751$	−1.07967e10	−0.930149	−0.465075	+	0.885272i	$0.653972\pi$
$752$	3.65110e9	0.313085
$753$	0	0
$754$	−6.56708e7	−0.00557921
$755$	−8.58462e9	−0.725950
$756$	0	0
$757$	1.23575e10	1.03536	0.517682	−	0.855573i	$-0.326795\pi$
$757$	1.23575e10	1.03536	0.517682	+	0.855573i	$0.326795\pi$
$758$	−8.41696e9	−0.701961
$759$	0	0
$760$	−4.65494e9	−0.384651
$761$	2.91011e9	0.239366	0.119683	−	0.992812i	$-0.461812\pi$
$761$	2.91011e9	0.239366	0.119683	+	0.992812i	$0.461812\pi$
$762$	0	0
$763$	−2.56677e10	−2.09195
$764$	−8.78483e9	−0.712700
$765$	0	0
$766$	−2.70087e9	−0.217122
$767$	−2.19867e8	−0.0175945
$768$	0	0
$769$	−1.64733e10	−1.30628	−0.653142	−	0.757236i	$-0.726549\pi$
$769$	−1.64733e10	−1.30628	−0.653142	+	0.757236i	$0.726549\pi$
$770$	3.94286e9	0.311239
$771$	0	0
$772$	−7.09802e9	−0.555235
$773$	1.06867e10	0.832177	0.416088	−	0.909324i	$-0.363401\pi$
$773$	1.06867e10	0.832177	0.416088	+	0.909324i	$0.363401\pi$
$774$	0	0
$775$	1.58376e8	0.0122217
$776$	4.32589e9	0.332322
$777$	0	0
$778$	−3.92545e9	−0.298855
$779$	8.29040e9	0.628339
$780$	0	0
$781$	−1.81466e9	−0.136306
$782$	7.80771e9	0.583848
$783$	0	0
$784$	5.87482e9	0.435400
$785$	9.63356e9	0.710793
$786$	0	0
$787$	−7.07872e9	−0.517658	−0.258829	−	0.965923i	$-0.583337\pi$
$787$	−7.07872e9	−0.517658	−0.258829	+	0.965923i	$0.583337\pi$
$788$	−8.91720e9	−0.649212
$789$	0	0
$790$	6.22759e8	0.0449392
$791$	2.25851e10	1.62258
$792$	0	0
$793$	2.79455e8	0.0199001
$794$	−5.69737e9	−0.403927
$795$	0	0
$796$	−5.51052e7	−0.00387255
$797$	−1.14235e10	−0.799274	−0.399637	−	0.916673i	$-0.630864\pi$
$797$	−1.14235e10	−0.799274	−0.399637	+	0.916673i	$0.630864\pi$
$798$	0	0
$799$	−2.13272e10	−1.47918
$800$	−3.00016e9	−0.207172
$801$	0	0
$802$	−3.39276e9	−0.232243
$803$	−1.21425e10	−0.827566
$804$	0	0
$805$	1.00722e10	0.680516
$806$	6.56926e6	0.000441920 0
$807$	0	0
$808$	−2.07485e10	−1.38371
$809$	−6.24087e9	−0.414405	−0.207203	−	0.978298i	$-0.566436\pi$
$809$	−6.24087e9	−0.414405	−0.207203	+	0.978298i	$0.566436\pi$
$810$	0	0
$811$	1.16407e10	0.766311	0.383156	−	0.923684i	$-0.374837\pi$
$811$	1.16407e10	0.766311	0.383156	+	0.923684i	$0.374837\pi$
$812$	−1.40700e10	−0.922249
$813$	0	0
$814$	1.20959e10	0.786054
$815$	1.22392e10	0.791955
$816$	0	0
$817$	−1.24552e10	−0.799048
$818$	−2.60179e9	−0.166202
$819$	0	0
$820$	3.38388e9	0.214322
$821$	−1.81563e10	−1.14505	−0.572527	−	0.819886i	$-0.694037\pi$
$821$	−1.81563e10	−1.14505	−0.572527	+	0.819886i	$0.694037\pi$
$822$	0	0
$823$	1.73491e10	1.08487	0.542434	−	0.840099i	$-0.317503\pi$
$823$	1.73491e10	1.08487	0.542434	+	0.840099i	$0.317503\pi$
$824$	1.26701e10	0.788922
$825$	0	0
$826$	1.76875e10	1.09204
$827$	−1.89554e10	−1.16537	−0.582684	−	0.812699i	$-0.697997\pi$
$827$	−1.89554e10	−1.16537	−0.582684	+	0.812699i	$0.697997\pi$
$828$	0	0
$829$	−2.42249e10	−1.47680	−0.738400	−	0.674363i	$-0.764418\pi$
$829$	−2.42249e10	−1.47680	−0.738400	+	0.674363i	$0.764418\pi$
$830$	3.63338e9	0.220566
$831$	0	0
$832$	−6.56774e7	−0.00395352
$833$	−3.43167e10	−2.05706
$834$	0	0
$835$	4.73319e9	0.281353
$836$	9.48416e9	0.561406
$837$	0	0
$838$	1.23912e10	0.727379
$839$	−1.27746e10	−0.746759	−0.373380	−	0.927679i	$-0.621801\pi$
$839$	−1.27746e10	−0.746759	−0.373380	+	0.927679i	$0.621801\pi$
$840$	0	0
$841$	−6.98277e9	−0.404801
$842$	−3.96604e9	−0.228963
$843$	0	0
$844$	6.31929e9	0.361801
$845$	7.84206e9	0.447128
$846$	0	0
$847$	9.99487e9	0.565178
$848$	1.48422e9	0.0835819
$849$	0	0
$850$	2.25914e9	0.126176
$851$	3.08994e10	1.71869
$852$	0	0
$853$	1.89284e10	1.04422	0.522111	−	0.852878i	$-0.325145\pi$
$853$	1.89284e10	1.04422	0.522111	+	0.852878i	$0.325145\pi$
$854$	−2.24812e10	−1.23514
$855$	0	0
$856$	2.04542e10	1.11461
$857$	3.22250e10	1.74888	0.874439	−	0.485135i	$-0.161230\pi$
$857$	3.22250e10	1.74888	0.874439	+	0.485135i	$0.161230\pi$
$858$	0	0
$859$	6.45420e9	0.347429	0.173715	−	0.984796i	$-0.444423\pi$
$859$	6.45420e9	0.347429	0.173715	+	0.984796i	$0.444423\pi$
$860$	−5.08382e9	−0.272549
$861$	0	0
$862$	−2.28988e9	−0.121769
$863$	−2.57288e10	−1.36264	−0.681320	−	0.731986i	$-0.738594\pi$
$863$	−2.57288e10	−1.36264	−0.681320	+	0.731986i	$0.738594\pi$
$864$	0	0
$865$	2.64261e9	0.138828
$866$	2.41084e8	0.0126141
$867$	0	0
$868$	1.40747e9	0.0730499
$869$	−3.01409e9	−0.155807
$870$	0	0
$871$	4.43089e8	0.0227210
$872$	2.24776e10	1.14800
$873$	0	0
$874$	−9.09698e9	−0.460900
$875$	2.91437e9	0.147067
$876$	0	0
$877$	−1.22061e10	−0.611054	−0.305527	−	0.952183i	$-0.598833\pi$
$877$	−1.22061e10	−0.611054	−0.305527	+	0.952183i	$0.598833\pi$
$878$	4.50064e9	0.224411
$879$	0	0
$880$	1.87184e9	0.0925932
$881$	−1.09967e10	−0.541808	−0.270904	−	0.962606i	$-0.587323\pi$
$881$	−1.09967e10	−0.541808	−0.270904	+	0.962606i	$0.587323\pi$
$882$	0	0
$883$	−6.12394e9	−0.299343	−0.149671	−	0.988736i	$-0.547822\pi$
$883$	−6.12394e9	−0.299343	−0.149671	+	0.988736i	$0.547822\pi$
$884$	−2.49567e8	−0.0121508
$885$	0	0
$886$	−7.48598e9	−0.361602
$887$	3.28161e10	1.57890	0.789448	−	0.613817i	$-0.210367\pi$
$887$	3.28161e10	1.57890	0.789448	+	0.613817i	$0.210367\pi$
$888$	0	0
$889$	5.97803e10	2.85366
$890$	−1.38230e9	−0.0657261
$891$	0	0
$892$	−8.48180e8	−0.0400139
$893$	2.48490e10	1.16769
$894$	0	0
$895$	4.99397e9	0.232844
$896$	−3.13897e10	−1.45784
$897$	0	0
$898$	−2.67031e9	−0.123054
$899$	−1.02705e9	−0.0471448
$900$	0	0
$901$	−8.66979e9	−0.394886
$902$	6.14945e9	0.279006
$903$	0	0
$904$	−1.97782e10	−0.890424
$905$	9.05089e9	0.405902
$906$	0	0
$907$	−6.74318e9	−0.300082	−0.150041	−	0.988680i	$-0.547940\pi$
$907$	−6.74318e9	−0.300082	−0.150041	+	0.988680i	$0.547940\pi$
$908$	2.86396e9	0.126960
$909$	0	0
$910$	1.20885e8	0.00531775
$911$	−1.49340e10	−0.654427	−0.327213	−	0.944950i	$-0.606110\pi$
$911$	−1.49340e10	−0.654427	−0.327213	+	0.944950i	$0.606110\pi$
$912$	0	0
$913$	−1.75852e10	−0.764714
$914$	−7.21002e9	−0.312338
$915$	0	0
$916$	2.16366e10	0.930154
$917$	9.57352e9	0.409996
$918$	0	0
$919$	1.32750e10	0.564198	0.282099	−	0.959385i	$-0.408969\pi$
$919$	1.32750e10	0.564198	0.282099	+	0.959385i	$0.408969\pi$
$920$	−8.82039e9	−0.373448
$921$	0	0
$922$	1.65904e10	0.697106
$923$	−5.56361e7	−0.00232890
$924$	0	0
$925$	8.94068e9	0.371428
$926$	1.39802e10	0.578596
$927$	0	0
$928$	1.94558e10	0.799155
$929$	1.00679e10	0.411988	0.205994	−	0.978553i	$-0.433957\pi$
$929$	1.00679e10	0.411988	0.205994	+	0.978553i	$0.433957\pi$
$930$	0	0
$931$	3.99833e10	1.62388
$932$	2.26576e10	0.916765
$933$	0	0
$934$	−1.20444e10	−0.483694
$935$	−1.09340e10	−0.437460
$936$	0	0
$937$	−2.03286e10	−0.807271	−0.403635	−	0.914920i	$-0.632254\pi$
$937$	−2.03286e10	−0.807271	−0.403635	+	0.914920i	$0.632254\pi$
$938$	−3.56450e10	−1.41022
$939$	0	0
$940$	1.01426e10	0.398291
$941$	−2.53707e10	−0.992589	−0.496294	−	0.868154i	$-0.665306\pi$
$941$	−2.53707e10	−0.992589	−0.496294	+	0.868154i	$0.665306\pi$
$942$	0	0
$943$	1.57090e10	0.610040
$944$	8.39701e9	0.324880
$945$	0	0
$946$	−9.23870e9	−0.354807
$947$	4.05421e10	1.55125	0.775625	−	0.631194i	$-0.217435\pi$
$947$	4.05421e10	1.55125	0.775625	+	0.631194i	$0.217435\pi$
$948$	0	0
$949$	−3.72279e8	−0.0141396
$950$	−2.63219e9	−0.0996058
$951$	0	0
$952$	4.76919e10	1.79149
$953$	9.94355e9	0.372149	0.186074	−	0.982536i	$-0.440423\pi$
$953$	9.94355e9	0.372149	0.186074	+	0.982536i	$0.440423\pi$
$954$	0	0
$955$	−1.18001e10	−0.438405
$956$	2.43570e10	0.901615
$957$	0	0
$958$	9.88300e9	0.363170
$959$	−7.98888e10	−2.92497
$960$	0	0
$961$	−2.74099e10	−0.996266
$962$	3.70851e8	0.0134303
$963$	0	0
$964$	2.19795e10	0.790222
$965$	−9.53436e9	−0.341543
$966$	0	0
$967$	4.29000e10	1.52568	0.762842	−	0.646585i	$-0.223803\pi$
$967$	4.29000e10	1.52568	0.762842	+	0.646585i	$0.223803\pi$
$968$	−8.75268e9	−0.310154
$969$	0	0
$970$	2.44612e9	0.0860552
$971$	−3.37004e10	−1.18132	−0.590661	−	0.806920i	$-0.701133\pi$
$971$	−3.37004e10	−1.18132	−0.590661	+	0.806920i	$0.701133\pi$
$972$	0	0
$973$	−2.64012e10	−0.918817
$974$	−3.16566e8	−0.0109776
$975$	0	0
$976$	−1.06728e10	−0.367453
$977$	3.12566e10	1.07229	0.536144	−	0.844127i	$-0.319880\pi$
$977$	3.12566e10	1.07229	0.536144	+	0.844127i	$0.319880\pi$
$978$	0	0
$979$	6.69020e9	0.227876
$980$	1.63199e10	0.553895
$981$	0	0
$982$	−8.73814e9	−0.294462
$983$	1.22350e10	0.410834	0.205417	−	0.978675i	$-0.434145\pi$
$983$	1.22350e10	0.410834	0.205417	+	0.978675i	$0.434145\pi$
$984$	0	0
$985$	−1.19779e10	−0.399352
$986$	−1.46503e10	−0.486719
$987$	0	0
$988$	2.90777e8	0.00959204
$989$	−2.36006e10	−0.775777
$990$	0	0
$991$	−4.59241e10	−1.49894	−0.749468	−	0.662040i	$-0.769691\pi$
$991$	−4.59241e10	−1.49894	−0.749468	+	0.662040i	$0.769691\pi$
$992$	−1.94623e9	−0.0632998
$993$	0	0
$994$	4.47572e9	0.144548
$995$	−7.40196e7	−0.00238213
$996$	0	0
$997$	−3.71958e10	−1.18867	−0.594334	−	0.804219i	$-0.702584\pi$
$997$	−3.71958e10	−1.18867	−0.594334	+	0.804219i	$0.702584\pi$
$998$	1.17006e9	0.0372606
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.8.a.f.1.5		13
3.2	odd	2		405.8.a.e.1.9		13
9.2	odd	6		45.8.e.a.31.5	yes	26
9.4	even	3		135.8.e.a.46.9		26
9.5	odd	6		45.8.e.a.16.5	✓	26
9.7	even	3		135.8.e.a.91.9		26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.8.e.a.16.5	✓	26	9.5	odd	6
45.8.e.a.31.5	yes	26	9.2	odd	6
135.8.e.a.46.9		26	9.4	even	3
135.8.e.a.91.9		26	9.7	even	3
405.8.a.e.1.9		13	3.2	odd	2
405.8.a.f.1.5		13	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	405.a (trivial)

\(f(q)\)	\(=\)	\(q-5.91113 q^{2} -93.0585 q^{4} -125.000 q^{5} -1492.16 q^{7} +1306.71 q^{8} +O(q^{10})\)	\(q-5.91113 q^{2} -93.0585 q^{4} -125.000 q^{5} -1492.16 q^{7} +1306.71 q^{8} +738.892 q^{10} -3576.16 q^{11} -109.642 q^{13} +8820.33 q^{14} +4187.38 q^{16} -24459.8 q^{17} +28498.8 q^{19} +11632.3 q^{20} +21139.1 q^{22} +54000.8 q^{23} +15625.0 q^{25} +648.110 q^{26} +138858. q^{28} -101327. q^{29} +10136.0 q^{31} -192011. q^{32} +144585. q^{34} +186519. q^{35} +572204. q^{37} -168460. q^{38} -163338. q^{40} +290903. q^{41} -437043. q^{43} +332792. q^{44} -319206. q^{46} +871931. q^{47} +1.40298e6 q^{49} -92361.4 q^{50} +10203.1 q^{52} +354451. q^{53} +447019. q^{55} -1.94981e6 q^{56} +598956. q^{58} +2.00531e6 q^{59} -2.54879e6 q^{61} -59915.5 q^{62} +599015. q^{64} +13705.3 q^{65} -4.04123e6 q^{67} +2.27619e6 q^{68} -1.10254e6 q^{70} +507433. q^{71} +3.39540e6 q^{73} -3.38237e6 q^{74} -2.65205e6 q^{76} +5.33618e6 q^{77} +842829. q^{79} -523422. q^{80} -1.71957e6 q^{82} +4.91734e6 q^{83} +3.05747e6 q^{85} +2.58342e6 q^{86} -4.67298e6 q^{88} -1.87078e6 q^{89} +163603. q^{91} -5.02523e6 q^{92} -5.15410e6 q^{94} -3.56235e6 q^{95} +3.31053e6 q^{97} -8.29322e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 8 q^{2} + 704 q^{4} - 1625 q^{5} - 1455 q^{7} + 1236 q^{8}+O(q^{10}) \) 13 * q + 8 * q^2 + 704 * q^4 - 1625 * q^5 - 1455 * q^7 + 1236 * q^8	\( 13 q + 8 q^{2} + 704 q^{4} - 1625 q^{5} - 1455 q^{7} + 1236 q^{8} - 1000 q^{10} + 1658 q^{11} - 13568 q^{13} + 12351 q^{14} + 52076 q^{16} + 6944 q^{17} - 45812 q^{19} - 88000 q^{20} - 17993 q^{22} + 96441 q^{23} + 203125 q^{25} + 126146 q^{26} - 216945 q^{28} + 39043 q^{29} - 158520 q^{31} + 725206 q^{32} - 441617 q^{34} + 181875 q^{35} - 505438 q^{37} + 615041 q^{38} - 154500 q^{40} + 1578883 q^{41} - 1082090 q^{43} + 498211 q^{44} - 2312547 q^{46} + 1690139 q^{47} - 1054816 q^{49} + 125000 q^{50} - 4100644 q^{52} - 102274 q^{53} - 207250 q^{55} + 389331 q^{56} - 5780521 q^{58} - 2908966 q^{59} - 3091451 q^{61} + 5212476 q^{62} - 5659352 q^{64} + 1696000 q^{65} - 1849533 q^{67} - 563369 q^{68} - 1543875 q^{70} - 2617958 q^{71} + 5310946 q^{73} - 11485004 q^{74} - 14421739 q^{76} + 1719660 q^{77} - 3632166 q^{79} - 6509500 q^{80} - 7347658 q^{82} + 1424115 q^{83} - 868000 q^{85} - 23193379 q^{86} - 20229447 q^{88} - 2816067 q^{89} - 16929702 q^{91} - 27930183 q^{92} - 11693189 q^{94} + 5726500 q^{95} - 12062476 q^{97} - 29617805 q^{98}+O(q^{100}) \) 13 * q + 8 * q^2 + 704 * q^4 - 1625 * q^5 - 1455 * q^7 + 1236 * q^8 - 1000 * q^10 + 1658 * q^11 - 13568 * q^13 + 12351 * q^14 + 52076 * q^16 + 6944 * q^17 - 45812 * q^19 - 88000 * q^20 - 17993 * q^22 + 96441 * q^23 + 203125 * q^25 + 126146 * q^26 - 216945 * q^28 + 39043 * q^29 - 158520 * q^31 + 725206 * q^32 - 441617 * q^34 + 181875 * q^35 - 505438 * q^37 + 615041 * q^38 - 154500 * q^40 + 1578883 * q^41 - 1082090 * q^43 + 498211 * q^44 - 2312547 * q^46 + 1690139 * q^47 - 1054816 * q^49 + 125000 * q^50 - 4100644 * q^52 - 102274 * q^53 - 207250 * q^55 + 389331 * q^56 - 5780521 * q^58 - 2908966 * q^59 - 3091451 * q^61 + 5212476 * q^62 - 5659352 * q^64 + 1696000 * q^65 - 1849533 * q^67 - 563369 * q^68 - 1543875 * q^70 - 2617958 * q^71 + 5310946 * q^73 - 11485004 * q^74 - 14421739 * q^76 + 1719660 * q^77 - 3632166 * q^79 - 6509500 * q^80 - 7347658 * q^82 + 1424115 * q^83 - 868000 * q^85 - 23193379 * q^86 - 20229447 * q^88 - 2816067 * q^89 - 16929702 * q^91 - 27930183 * q^92 - 11693189 * q^94 + 5726500 * q^95 - 12062476 * q^97 - 29617805 * q^98

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