LMFDB - Embedded newform 1075.6.a.a.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1075,6,Mod(1,1075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1075, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("1075.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1075 = 5^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1075.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:	$172.412606299$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 173x^{6} + 462x^{5} + 9118x^{4} - 14192x^{3} - 167688x^{2} + 106368x + 681984 $ x^8 - 4x^7 - 173x^6 + 462x^5 + 9118x^4 - 14192x^3 - 167688x^2 + 106368*x + 681984
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.08717$ of defining polynomial
Character	$\chi$	$=$	1075.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.08717 q^{2} -25.6605 q^{3} -15.2950 q^{4} -104.879 q^{6} +184.774 q^{7} -193.303 q^{8} +415.460 q^{9} +O(q^{10})$	\(q+4.08717 q^{2} -25.6605 q^{3} -15.2950 q^{4} -104.879 q^{6} +184.774 q^{7} -193.303 q^{8} +415.460 q^{9} -130.384 q^{11} +392.477 q^{12} +1178.72 q^{13} +755.202 q^{14} -300.622 q^{16} +493.240 q^{17} +1698.06 q^{18} +2427.14 q^{19} -4741.38 q^{21} -532.901 q^{22} +4184.13 q^{23} +4960.24 q^{24} +4817.63 q^{26} -4425.39 q^{27} -2826.12 q^{28} +1392.80 q^{29} +3167.29 q^{31} +4957.00 q^{32} +3345.71 q^{33} +2015.96 q^{34} -6354.46 q^{36} -12989.9 q^{37} +9920.13 q^{38} -30246.5 q^{39} +9526.42 q^{41} -19378.8 q^{42} +1849.00 q^{43} +1994.22 q^{44} +17101.3 q^{46} +9421.23 q^{47} +7714.10 q^{48} +17334.3 q^{49} -12656.8 q^{51} -18028.5 q^{52} +5074.60 q^{53} -18087.4 q^{54} -35717.3 q^{56} -62281.5 q^{57} +5692.63 q^{58} -12952.1 q^{59} -8998.97 q^{61} +12945.2 q^{62} +76766.0 q^{63} +29880.0 q^{64} +13674.5 q^{66} +41018.4 q^{67} -7544.11 q^{68} -107367. q^{69} -24420.1 q^{71} -80309.6 q^{72} -23475.4 q^{73} -53092.1 q^{74} -37123.1 q^{76} -24091.5 q^{77} -123623. q^{78} +21856.0 q^{79} +12601.0 q^{81} +38936.1 q^{82} +40429.1 q^{83} +72519.4 q^{84} +7557.18 q^{86} -35740.0 q^{87} +25203.6 q^{88} -114656. q^{89} +217796. q^{91} -63996.3 q^{92} -81274.1 q^{93} +38506.2 q^{94} -127199. q^{96} +26625.0 q^{97} +70848.2 q^{98} -54169.2 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{2} + 26 q^{3} + 122 q^{4} - 69 q^{6} + 136 q^{7} + 666 q^{8} + 546 q^{9}+O(q^{10}) $ 8 * q + 12 * q^2 + 26 * q^3 + 122 * q^4 - 69 * q^6 + 136 * q^7 + 666 * q^8 + 546 * q^9	\( 8 q + 12 q^{2} + 26 q^{3} + 122 q^{4} - 69 q^{6} + 136 q^{7} + 666 q^{8} + 546 q^{9} - 532 q^{11} + 4195 q^{12} + 2492 q^{13} - 4240 q^{14} + 1882 q^{16} + 2534 q^{17} + 3711 q^{18} - 1678 q^{19} - 2256 q^{21} - 11502 q^{22} + 2488 q^{23} + 19953 q^{24} + 4586 q^{26} + 8960 q^{27} - 18640 q^{28} - 4360 q^{29} + 5704 q^{31} + 18294 q^{32} + 12852 q^{33} + 30007 q^{34} + 67969 q^{36} + 3772 q^{37} + 6559 q^{38} + 11120 q^{39} - 10698 q^{41} - 78698 q^{42} + 14792 q^{43} - 356 q^{44} - 19389 q^{46} + 77864 q^{47} + 118727 q^{48} + 7188 q^{49} - 80246 q^{51} + 60736 q^{52} + 62352 q^{53} + 61026 q^{54} - 144528 q^{56} + 808 q^{57} - 52951 q^{58} - 26224 q^{59} - 82540 q^{61} + 9023 q^{62} + 61768 q^{63} + 153858 q^{64} - 48516 q^{66} - 27784 q^{67} - 40507 q^{68} - 93776 q^{69} - 9504 q^{71} + 186687 q^{72} - 14260 q^{73} - 15239 q^{74} + 1279 q^{76} + 218140 q^{77} - 264170 q^{78} + 160248 q^{79} + 161076 q^{81} + 47781 q^{82} + 77176 q^{83} + 16382 q^{84} + 22188 q^{86} - 268136 q^{87} - 129544 q^{88} - 265692 q^{89} + 401148 q^{91} - 190391 q^{92} + 123860 q^{93} + 248737 q^{94} + 950817 q^{96} - 144742 q^{97} + 292244 q^{98} + 239516 q^{99}+O(q^{100}) \) 8 * q + 12 * q^2 + 26 * q^3 + 122 * q^4 - 69 * q^6 + 136 * q^7 + 666 * q^8 + 546 * q^9 - 532 * q^11 + 4195 * q^12 + 2492 * q^13 - 4240 * q^14 + 1882 * q^16 + 2534 * q^17 + 3711 * q^18 - 1678 * q^19 - 2256 * q^21 - 11502 * q^22 + 2488 * q^23 + 19953 * q^24 + 4586 * q^26 + 8960 * q^27 - 18640 * q^28 - 4360 * q^29 + 5704 * q^31 + 18294 * q^32 + 12852 * q^33 + 30007 * q^34 + 67969 * q^36 + 3772 * q^37 + 6559 * q^38 + 11120 * q^39 - 10698 * q^41 - 78698 * q^42 + 14792 * q^43 - 356 * q^44 - 19389 * q^46 + 77864 * q^47 + 118727 * q^48 + 7188 * q^49 - 80246 * q^51 + 60736 * q^52 + 62352 * q^53 + 61026 * q^54 - 144528 * q^56 + 808 * q^57 - 52951 * q^58 - 26224 * q^59 - 82540 * q^61 + 9023 * q^62 + 61768 * q^63 + 153858 * q^64 - 48516 * q^66 - 27784 * q^67 - 40507 * q^68 - 93776 * q^69 - 9504 * q^71 + 186687 * q^72 - 14260 * q^73 - 15239 * q^74 + 1279 * q^76 + 218140 * q^77 - 264170 * q^78 + 160248 * q^79 + 161076 * q^81 + 47781 * q^82 + 77176 * q^83 + 16382 * q^84 + 22188 * q^86 - 268136 * q^87 - 129544 * q^88 - 265692 * q^89 + 401148 * q^91 - 190391 * q^92 + 123860 * q^93 + 248737 * q^94 + 950817 * q^96 - 144742 * q^97 + 292244 * q^98 + 239516 * 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 $

$2$	4.08717	0.722517	0.361258	−	0.932466i	$-0.382347\pi$
$2$	4.08717	0.722517	0.361258	+	0.932466i	$0.382347\pi$
$3$	−25.6605	−1.64612	−0.823060	−	0.567955i	$-0.807735\pi$
$3$	−25.6605	−1.64612	−0.823060	+	0.567955i	$0.807735\pi$
$4$	−15.2950	−0.477969
$5$	0	0
$5$	0	0
$6$	−104.879	−1.18935
$7$	184.774	1.42526	0.712631	−	0.701539i	$-0.247503\pi$
$7$	184.774	1.42526	0.712631	+	0.701539i	$0.247503\pi$
$8$	−193.303	−1.06786
$9$	415.460	1.70971
$10$	0	0
$11$	−130.384	−0.324894	−0.162447	−	0.986717i	$-0.551939\pi$
$11$	−130.384	−0.324894	−0.162447	+	0.986717i	$0.551939\pi$
$12$	392.477	0.786795
$13$	1178.72	1.93443	0.967213	−	0.253965i	$-0.0817349\pi$
$13$	1178.72	1.93443	0.967213	+	0.253965i	$0.0817349\pi$
$14$	755.202	1.02978
$15$	0	0
$16$	−300.622	−0.293576
$17$	493.240	0.413938	0.206969	−	0.978347i	$-0.433640\pi$
$17$	493.240	0.413938	0.206969	+	0.978347i	$0.433640\pi$
$18$	1698.06	1.23529
$19$	2427.14	1.54245	0.771224	−	0.636564i	$-0.219645\pi$
$19$	2427.14	1.54245	0.771224	+	0.636564i	$0.219645\pi$
$20$	0	0
$21$	−4741.38	−2.34615
$22$	−532.901	−0.234742
$23$	4184.13	1.64925	0.824623	−	0.565683i	$-0.191387\pi$
$23$	4184.13	1.64925	0.824623	+	0.565683i	$0.191387\pi$
$24$	4960.24	1.75782
$25$	0	0
$26$	4817.63	1.39766
$27$	−4425.39	−1.16827
$28$	−2826.12	−0.681232
$29$	1392.80	0.307535	0.153768	−	0.988107i	$-0.450859\pi$
$29$	1392.80	0.307535	0.153768	+	0.988107i	$0.450859\pi$
$30$	0	0
$31$	3167.29	0.591947	0.295974	−	0.955196i	$-0.404356\pi$
$31$	3167.29	0.591947	0.295974	+	0.955196i	$0.404356\pi$
$32$	4957.00	0.855744
$33$	3345.71	0.534815
$34$	2015.96	0.299078
$35$	0	0
$36$	−6354.46	−0.817189
$37$	−12989.9	−1.55992	−0.779961	−	0.625828i	$-0.784761\pi$
$37$	−12989.9	−1.55992	−0.779961	+	0.625828i	$0.784761\pi$
$38$	9920.13	1.11444
$39$	−30246.5	−3.18430
$40$	0	0
$41$	9526.42	0.885054	0.442527	−	0.896755i	$-0.354082\pi$
$41$	9526.42	0.885054	0.442527	+	0.896755i	$0.354082\pi$
$42$	−19378.8	−1.69514
$43$	1849.00	0.152499
$43$	1849.00	0.152499
$44$	1994.22	0.155289
$45$	0	0
$46$	17101.3	1.19161
$47$	9421.23	0.622104	0.311052	−	0.950393i	$-0.399319\pi$
$47$	9421.23	0.622104	0.311052	+	0.950393i	$0.399319\pi$
$48$	7714.10	0.483261
$49$	17334.3	1.03137
$50$	0	0
$51$	−12656.8	−0.681392
$52$	−18028.5	−0.924596
$53$	5074.60	0.248149	0.124074	−	0.992273i	$-0.460404\pi$
$53$	5074.60	0.248149	0.124074	+	0.992273i	$0.460404\pi$
$54$	−18087.4	−0.844093
$55$	0	0
$56$	−35717.3	−1.52198
$57$	−62281.5	−2.53905
$58$	5692.63	0.222199
$59$	−12952.1	−0.484408	−0.242204	−	0.970225i	$-0.577870\pi$
$59$	−12952.1	−0.484408	−0.242204	+	0.970225i	$0.577870\pi$
$60$	0	0
$61$	−8998.97	−0.309648	−0.154824	−	0.987942i	$-0.549481\pi$
$61$	−8998.97	−0.309648	−0.154824	+	0.987942i	$0.549481\pi$
$62$	12945.2	0.427692
$63$	76766.0	2.43679
$64$	29880.0	0.911866
$65$	0	0
$66$	13674.5	0.386413
$67$	41018.4	1.11633	0.558164	−	0.829731i	$-0.311506\pi$
$67$	41018.4	1.11633	0.558164	+	0.829731i	$0.311506\pi$
$68$	−7544.11	−0.197850
$69$	−107367.	−2.71486
$70$	0	0
$71$	−24420.1	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$71$	−24420.1	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$72$	−80309.6	−1.82573
$73$	−23475.4	−0.515593	−0.257796	−	0.966199i	$-0.582996\pi$
$73$	−23475.4	−0.515593	−0.257796	+	0.966199i	$0.582996\pi$
$74$	−53092.1	−1.12707
$75$	0	0
$76$	−37123.1	−0.737242
$77$	−24091.5	−0.463060
$78$	−123623.	−2.30071
$79$	21856.0	0.394006	0.197003	−	0.980403i	$-0.436879\pi$
$79$	21856.0	0.394006	0.197003	+	0.980403i	$0.436879\pi$
$80$	0	0
$81$	12601.0	0.213399
$82$	38936.1	0.639467
$83$	40429.1	0.644168	0.322084	−	0.946711i	$-0.395617\pi$
$83$	40429.1	0.644168	0.322084	+	0.946711i	$0.395617\pi$
$84$	72519.4	1.12139
$85$	0	0
$86$	7557.18	0.110183
$87$	−35740.0	−0.506240
$88$	25203.6	0.346941
$89$	−114656.	−1.53434	−0.767170	−	0.641444i	$-0.778336\pi$
$89$	−114656.	−1.53434	−0.767170	+	0.641444i	$0.778336\pi$
$90$	0	0
$91$	217796.	2.75707
$92$	−63996.3	−0.788289
$93$	−81274.1	−0.974416
$94$	38506.2	0.449481
$95$	0	0
$96$	−127199.	−1.40866
$97$	26625.0	0.287316	0.143658	−	0.989627i	$-0.454113\pi$
$97$	26625.0	0.287316	0.143658	+	0.989627i	$0.454113\pi$
$98$	70848.2	0.745184
$99$	−54169.2	−0.555475
$100$	0	0
$101$	−50646.9	−0.494025	−0.247013	−	0.969012i	$-0.579449\pi$
$101$	−50646.9	−0.494025	−0.247013	+	0.969012i	$0.579449\pi$
$102$	−51730.4	−0.492318
$103$	54043.0	0.501934	0.250967	−	0.967996i	$-0.419251\pi$
$103$	54043.0	0.501934	0.250967	+	0.967996i	$0.419251\pi$
$104$	−227850.	−2.06569
$105$	0	0
$106$	20740.8	0.179292
$107$	76911.7	0.649431	0.324715	−	0.945812i	$-0.394732\pi$
$107$	76911.7	0.649431	0.324715	+	0.945812i	$0.394732\pi$
$108$	67686.5	0.558396
$109$	53026.1	0.427487	0.213744	−	0.976890i	$-0.431434\pi$
$109$	53026.1	0.427487	0.213744	+	0.976890i	$0.431434\pi$
$110$	0	0
$111$	333328.	2.56782
$112$	−55547.0	−0.418423
$113$	151376.	1.11522	0.557610	−	0.830103i	$-0.311718\pi$
$113$	151376.	1.11522	0.557610	+	0.830103i	$0.311718\pi$
$114$	−254555.	−1.83451
$115$	0	0
$116$	−21302.9	−0.146992
$117$	489711.	3.30731
$118$	−52937.6	−0.349993
$119$	91137.7	0.589971
$120$	0	0
$121$	−144051.	−0.894444
$122$	−36780.4	−0.223726
$123$	−244452.	−1.45691
$124$	−48443.7	−0.282933
$125$	0	0
$126$	313756.	1.76062
$127$	−159994.	−0.880227	−0.440113	−	0.897942i	$-0.645062\pi$
$127$	−159994.	−0.880227	−0.440113	+	0.897942i	$0.645062\pi$
$128$	−36499.2	−0.196906
$129$	−47446.2	−0.251031
$130$	0	0
$131$	−74510.4	−0.379349	−0.189674	−	0.981847i	$-0.560743\pi$
$131$	−74510.4	−0.379349	−0.189674	+	0.981847i	$0.560743\pi$
$132$	−51172.7	−0.255625
$133$	448471.	2.19839
$134$	167649.	0.806566
$135$	0	0
$136$	−95344.7	−0.442027
$137$	27357.9	0.124532	0.0622661	−	0.998060i	$-0.480167\pi$
$137$	27357.9	0.124532	0.0622661	+	0.998060i	$0.480167\pi$
$138$	−438826.	−1.96153
$139$	1450.01	0.00636554	0.00318277	−	0.999995i	$-0.498987\pi$
$139$	1450.01	0.00636554	0.00318277	+	0.999995i	$0.498987\pi$
$140$	0	0
$141$	−241753.	−1.02406
$142$	−99809.0	−0.415383
$143$	−153686.	−0.628484
$144$	−124896.	−0.501930
$145$	0	0
$146$	−95948.2	−0.372524
$147$	−444806.	−1.69776
$148$	198681.	0.745595
$149$	361769.	1.33495	0.667476	−	0.744631i	$-0.267375\pi$
$149$	361769.	1.33495	0.667476	+	0.744631i	$0.267375\pi$
$150$	0	0
$151$	−203530.	−0.726419	−0.363209	−	0.931708i	$-0.618319\pi$
$151$	−203530.	−0.726419	−0.363209	+	0.931708i	$0.618319\pi$
$152$	−469173.	−1.64711
$153$	204921.	0.707715
$154$	−98466.1	−0.334568
$155$	0	0
$156$	462621.	1.52200
$157$	−385714.	−1.24887	−0.624434	−	0.781077i	$-0.714670\pi$
$157$	−385714.	−1.24887	−0.624434	+	0.781077i	$0.714670\pi$
$158$	89329.1	0.284676
$159$	−130217.	−0.408482
$160$	0	0
$161$	773116.	2.35061
$162$	51502.4	0.154184
$163$	237890.	0.701307	0.350653	−	0.936505i	$-0.385960\pi$
$163$	237890.	0.701307	0.350653	+	0.936505i	$0.385960\pi$
$164$	−145707.	−0.423029
$165$	0	0
$166$	165241.	0.465422
$167$	−204294.	−0.566845	−0.283423	−	0.958995i	$-0.591470\pi$
$167$	−204294.	−0.566845	−0.283423	+	0.958995i	$0.591470\pi$
$168$	916522.	2.50536
$169$	1.01809e6	2.74201
$170$	0	0
$171$	1.00838e6	2.63714
$172$	−28280.5	−0.0728896
$173$	767839.	1.95054	0.975270	−	0.221017i	$-0.0709375\pi$
$173$	767839.	1.95054	0.975270	+	0.221017i	$0.0709375\pi$
$174$	−146075.	−0.365767
$175$	0	0
$176$	39196.3	0.0953812
$177$	332358.	0.797393
$178$	−468619.	−1.10859
$179$	−21246.0	−0.0495616	−0.0247808	−	0.999693i	$-0.507889\pi$
$179$	−21246.0	−0.0495616	−0.0247808	+	0.999693i	$0.507889\pi$
$180$	0	0
$181$	−23603.2	−0.0535518	−0.0267759	−	0.999641i	$-0.508524\pi$
$181$	−23603.2	−0.0535518	−0.0267759	+	0.999641i	$0.508524\pi$
$182$	890171.	1.99203
$183$	230918.	0.509718
$184$	−808804.	−1.76116
$185$	0	0
$186$	−332181.	−0.704032
$187$	−64310.5	−0.134486
$188$	−144098.	−0.297347
$189$	−817696.	−1.66509
$190$	0	0
$191$	185710.	0.368343	0.184172	−	0.982894i	$-0.441040\pi$
$191$	185710.	0.368343	0.184172	+	0.982894i	$0.441040\pi$
$192$	−766735.	−1.50104
$193$	94410.2	0.182442	0.0912212	−	0.995831i	$-0.470923\pi$
$193$	94410.2	0.182442	0.0912212	+	0.995831i	$0.470923\pi$
$194$	108821.	0.207591
$195$	0	0
$196$	−265128.	−0.492965
$197$	132826.	0.243846	0.121923	−	0.992540i	$-0.461094\pi$
$197$	132826.	0.243846	0.121923	+	0.992540i	$0.461094\pi$
$198$	−221399.	−0.401340
$199$	−1.02908e6	−1.84211	−0.921055	−	0.389434i	$-0.872671\pi$
$199$	−1.02908e6	−1.84211	−0.921055	+	0.389434i	$0.872671\pi$
$200$	0	0
$201$	−1.05255e6	−1.83761
$202$	−207003.	−0.356942
$203$	257353.	0.438318
$204$	193585.	0.325685
$205$	0	0
$206$	220883.	0.362656
$207$	1.73834e6	2.81973
$208$	−354349.	−0.567902
$209$	−316460.	−0.501132
$210$	0	0
$211$	467519.	0.722925	0.361462	−	0.932387i	$-0.382278\pi$
$211$	467519.	0.722925	0.361462	+	0.932387i	$0.382278\pi$
$212$	−77616.0	−0.118607
$213$	626630.	0.946373
$214$	314351.	0.469225
$215$	0	0
$216$	855442.	1.24754
$217$	585231.	0.843680
$218$	216727.	0.308867
$219$	602391.	0.848727
$220$	0	0
$221$	581392.	0.800734
$222$	1.36237e6	1.85529
$223$	−1.34109e6	−1.80592	−0.902958	−	0.429730i	$-0.858609\pi$
$223$	−1.34109e6	−1.80592	−0.902958	+	0.429730i	$0.858609\pi$
$224$	915923.	1.21966
$225$	0	0
$226$	618700.	0.805766
$227$	−909685.	−1.17173	−0.585864	−	0.810410i	$-0.699245\pi$
$227$	−909685.	−1.17173	−0.585864	+	0.810410i	$0.699245\pi$
$228$	952596.	1.21359
$229$	1.28521e6	1.61952	0.809759	−	0.586763i	$-0.199598\pi$
$229$	1.28521e6	1.61952	0.809759	+	0.586763i	$0.199598\pi$
$230$	0	0
$231$	618199.	0.762252
$232$	−269233.	−0.328404
$233$	1.03177e6	1.24507	0.622536	−	0.782591i	$-0.286102\pi$
$233$	1.03177e6	1.24507	0.622536	+	0.782591i	$0.286102\pi$
$234$	2.00153e6	2.38959
$235$	0	0
$236$	198103.	0.231532
$237$	−560834.	−0.648580
$238$	372496.	0.426264
$239$	1.64071e6	1.85797	0.928984	−	0.370120i	$-0.120683\pi$
$239$	1.64071e6	1.85797	0.928984	+	0.370120i	$0.120683\pi$
$240$	0	0
$241$	−430616.	−0.477581	−0.238791	−	0.971071i	$-0.576751\pi$
$241$	−430616.	−0.477581	−0.238791	+	0.971071i	$0.576751\pi$
$242$	−588762.	−0.646251
$243$	752023.	0.816988
$244$	137639.	0.148002
$245$	0	0
$246$	−999119.	−1.05264
$247$	2.86092e6	2.98375
$248$	−612246.	−0.632116
$249$	−1.03743e6	−1.06038
$250$	0	0
$251$	374333.	0.375037	0.187518	−	0.982261i	$-0.439956\pi$
$251$	374333.	0.375037	0.187518	+	0.982261i	$0.439956\pi$
$252$	−1.17414e6	−1.16471
$253$	−545543.	−0.535830
$254$	−653923.	−0.635979
$255$	0	0
$256$	−1.10534e6	−1.05413
$257$	705602.	0.666387	0.333194	−	0.942858i	$-0.391874\pi$
$257$	705602.	0.666387	0.333194	+	0.942858i	$0.391874\pi$
$258$	−193921.	−0.181374
$259$	−2.40020e6	−2.22330
$260$	0	0
$261$	578653.	0.525796
$262$	−304537.	−0.274086
$263$	−611524.	−0.545160	−0.272580	−	0.962133i	$-0.587877\pi$
$263$	−611524.	−0.545160	−0.272580	+	0.962133i	$0.587877\pi$
$264$	−646736.	−0.571106
$265$	0	0
$266$	1.83298e6	1.58838
$267$	2.94212e6	2.52571
$268$	−627377.	−0.533570
$269$	320336.	0.269914	0.134957	−	0.990851i	$-0.456910\pi$
$269$	320336.	0.269914	0.134957	+	0.990851i	$0.456910\pi$
$270$	0	0
$271$	−431752.	−0.357117	−0.178559	−	0.983929i	$-0.557143\pi$
$271$	−431752.	−0.357117	−0.178559	+	0.983929i	$0.557143\pi$
$272$	−148279.	−0.121522
$273$	−5.58876e6	−4.53846
$274$	111817.	0.0899766
$275$	0	0
$276$	1.64217e6	1.29762
$277$	125219.	0.0980549	0.0490274	−	0.998797i	$-0.484388\pi$
$277$	125219.	0.0980549	0.0490274	+	0.998797i	$0.484388\pi$
$278$	5926.46	0.00459921
$279$	1.31588e6	1.01206
$280$	0	0
$281$	15143.4	0.0114408	0.00572042	−	0.999984i	$-0.498179\pi$
$281$	15143.4	0.0114408	0.00572042	+	0.999984i	$0.498179\pi$
$282$	−988087.	−0.739899
$283$	−1.90528e6	−1.41414	−0.707070	−	0.707144i	$-0.749983\pi$
$283$	−1.90528e6	−1.41414	−0.707070	+	0.707144i	$0.749983\pi$
$284$	373505.	0.274790
$285$	0	0
$286$	−628142.	−0.454091
$287$	1.76023e6	1.26143
$288$	2.05943e6	1.46307
$289$	−1.17657e6	−0.828655
$290$	0	0
$291$	−683210.	−0.472957
$292$	359057.	0.246437
$293$	477662.	0.325051	0.162526	−	0.986704i	$-0.448036\pi$
$293$	477662.	0.325051	0.162526	+	0.986704i	$0.448036\pi$
$294$	−1.81800e6	−1.22666
$295$	0	0
$296$	2.51099e6	1.66578
$297$	577000.	0.379564
$298$	1.47861e6	0.964526
$299$	4.93191e6	3.19034
$300$	0	0
$301$	341646.	0.217350
$302$	−831864.	−0.524850
$303$	1.29962e6	0.813225
$304$	−729651.	−0.452826
$305$	0	0
$306$	837548.	0.511336
$307$	1.12678e6	0.682328	0.341164	−	0.940004i	$-0.389179\pi$
$307$	1.12678e6	0.682328	0.341164	+	0.940004i	$0.389179\pi$
$308$	368480.	0.221328
$309$	−1.38677e6	−0.826243
$310$	0	0
$311$	−1.30351e6	−0.764209	−0.382105	−	0.924119i	$-0.624801\pi$
$311$	−1.30351e6	−0.764209	−0.382105	+	0.924119i	$0.624801\pi$
$312$	5.84674e6	3.40038
$313$	1.32275e6	0.763165	0.381583	−	0.924335i	$-0.375379\pi$
$313$	1.32275e6	0.763165	0.381583	+	0.924335i	$0.375379\pi$
$314$	−1.57648e6	−0.902329
$315$	0	0
$316$	−334287.	−0.188323
$317$	2.04785e6	1.14459	0.572295	−	0.820048i	$-0.306053\pi$
$317$	2.04785e6	1.14459	0.572295	+	0.820048i	$0.306053\pi$
$318$	−532217.	−0.295135
$319$	−181599.	−0.0999164
$320$	0	0
$321$	−1.97359e6	−1.06904
$322$	3.15986e6	1.69835
$323$	1.19716e6	0.638478
$324$	−192732.	−0.101998
$325$	0	0
$326$	972300.	0.506706
$327$	−1.36067e6	−0.703696
$328$	−1.84148e6	−0.945112
$329$	1.74079e6	0.886661
$330$	0	0
$331$	−816891.	−0.409821	−0.204910	−	0.978781i	$-0.565690\pi$
$331$	−816891.	−0.409821	−0.204910	+	0.978781i	$0.565690\pi$
$332$	−618364.	−0.307893
$333$	−5.39680e6	−2.66702
$334$	−834985.	−0.409555
$335$	0	0
$336$	1.42536e6	0.688774
$337$	−250202.	−0.120009	−0.0600047	−	0.998198i	$-0.519112\pi$
$337$	−250202.	−0.120009	−0.0600047	+	0.998198i	$0.519112\pi$
$338$	4.16110e6	1.98115
$339$	−3.88438e6	−1.83579
$340$	0	0
$341$	−412963.	−0.192320
$342$	4.12141e6	1.90538
$343$	97428.3	0.0447147
$344$	−357417.	−0.162847
$345$	0	0
$346$	3.13829e6	1.40930
$347$	−3.31702e6	−1.47885	−0.739426	−	0.673238i	$-0.764903\pi$
$347$	−3.31702e6	−1.47885	−0.739426	+	0.673238i	$0.764903\pi$
$348$	546643.	0.241967
$349$	1.28060e6	0.562796	0.281398	−	0.959591i	$-0.409202\pi$
$349$	1.28060e6	0.562796	0.281398	+	0.959591i	$0.409202\pi$
$350$	0	0
$351$	−5.21630e6	−2.25993
$352$	−646313.	−0.278026
$353$	−1.18634e6	−0.506727	−0.253363	−	0.967371i	$-0.581537\pi$
$353$	−1.18634e6	−0.506727	−0.253363	+	0.967371i	$0.581537\pi$
$354$	1.35840e6	0.576130
$355$	0	0
$356$	1.75366e6	0.733367
$357$	−2.33864e6	−0.971163
$358$	−86836.2	−0.0358091
$359$	−271890.	−0.111342	−0.0556708	−	0.998449i	$-0.517730\pi$
$359$	−271890.	−0.111342	−0.0556708	+	0.998449i	$0.517730\pi$
$360$	0	0
$361$	3.41490e6	1.37914
$362$	−96470.3	−0.0386921
$363$	3.69642e6	1.47236
$364$	−3.33120e6	−1.31779
$365$	0	0
$366$	943801.	0.368280
$367$	344456.	0.133496	0.0667481	−	0.997770i	$-0.478738\pi$
$367$	344456.	0.133496	0.0667481	+	0.997770i	$0.478738\pi$
$368$	−1.25784e6	−0.484179
$369$	3.95784e6	1.51319
$370$	0	0
$371$	937651.	0.353677
$372$	1.24309e6	0.465741
$373$	−1.46043e6	−0.543512	−0.271756	−	0.962366i	$-0.587604\pi$
$373$	−1.46043e6	−0.543512	−0.271756	+	0.962366i	$0.587604\pi$
$374$	−262848.	−0.0971686
$375$	0	0
$376$	−1.82115e6	−0.664319
$377$	1.64172e6	0.594904
$378$	−3.34206e6	−1.20305
$379$	2.18177e6	0.780207	0.390104	−	0.920771i	$-0.372439\pi$
$379$	2.18177e6	0.780207	0.390104	+	0.920771i	$0.372439\pi$
$380$	0	0
$381$	4.10552e6	1.44896
$382$	759031.	0.266134
$383$	−2.16635e6	−0.754627	−0.377313	−	0.926086i	$-0.623152\pi$
$383$	−2.16635e6	−0.754627	−0.377313	+	0.926086i	$0.623152\pi$
$384$	936586.	0.324130
$385$	0	0
$386$	385871.	0.131818
$387$	768185.	0.260728
$388$	−407230.	−0.137328
$389$	−4.47945e6	−1.50090	−0.750448	−	0.660929i	$-0.770162\pi$
$389$	−4.47945e6	−1.50090	−0.750448	+	0.660929i	$0.770162\pi$
$390$	0	0
$391$	2.06378e6	0.682686
$392$	−3.35077e6	−1.10136
$393$	1.91197e6	0.624454
$394$	542881.	0.176183
$395$	0	0
$396$	828519.	0.265500
$397$	807623.	0.257177	0.128589	−	0.991698i	$-0.458955\pi$
$397$	807623.	0.257177	0.128589	+	0.991698i	$0.458955\pi$
$398$	−4.20602e6	−1.33096
$399$	−1.15080e7	−3.61882
$400$	0	0
$401$	−4.66706e6	−1.44938	−0.724691	−	0.689075i	$-0.758017\pi$
$401$	−4.66706e6	−1.44938	−0.724691	+	0.689075i	$0.758017\pi$
$402$	−4.30196e6	−1.32770
$403$	3.73334e6	1.14508
$404$	774645.	0.236129
$405$	0	0
$406$	1.05185e6	0.316692
$407$	1.69368e6	0.506810
$408$	2.44659e6	0.727630
$409$	−4.17262e6	−1.23339	−0.616695	−	0.787202i	$-0.711529\pi$
$409$	−4.17262e6	−1.23339	−0.616695	+	0.787202i	$0.711529\pi$
$410$	0	0
$411$	−702017.	−0.204995
$412$	−826589.	−0.239909
$413$	−2.39321e6	−0.690408
$414$	7.10488e6	2.03730
$415$	0	0
$416$	5.84291e6	1.65537
$417$	−37208.0	−0.0104784
$418$	−1.29342e6	−0.362077
$419$	1.36605e6	0.380128	0.190064	−	0.981772i	$-0.439130\pi$
$419$	1.36605e6	0.380128	0.190064	+	0.981772i	$0.439130\pi$
$420$	0	0
$421$	−3.81386e6	−1.04872	−0.524360	−	0.851497i	$-0.675695\pi$
$421$	−3.81386e6	−1.04872	−0.524360	+	0.851497i	$0.675695\pi$
$422$	1.91083e6	0.522326
$423$	3.91414e6	1.06362
$424$	−980934.	−0.264987
$425$	0	0
$426$	2.56115e6	0.683770
$427$	−1.66277e6	−0.441330
$428$	−1.17637e6	−0.310408
$429$	3.94366e6	1.03456
$430$	0	0
$431$	5.01591e6	1.30064	0.650319	−	0.759661i	$-0.274635\pi$
$431$	5.01591e6	1.30064	0.650319	+	0.759661i	$0.274635\pi$
$432$	1.33037e6	0.342976
$433$	−1.08474e6	−0.278038	−0.139019	−	0.990290i	$-0.544395\pi$
$433$	−1.08474e6	−0.278038	−0.139019	+	0.990290i	$0.544395\pi$
$434$	2.39194e6	0.609573
$435$	0	0
$436$	−811035.	−0.204326
$437$	1.01555e7	2.54387
$438$	2.46208e6	0.613220
$439$	−6.43424e6	−1.59344	−0.796720	−	0.604348i	$-0.793434\pi$
$439$	−6.43424e6	−1.59344	−0.796720	+	0.604348i	$0.793434\pi$
$440$	0	0
$441$	7.20169e6	1.76335
$442$	2.37625e6	0.578544
$443$	940060.	0.227586	0.113793	−	0.993504i	$-0.463700\pi$
$443$	940060.	0.227586	0.113793	+	0.993504i	$0.463700\pi$
$444$	−5.09826e6	−1.22734
$445$	0	0
$446$	−5.48129e6	−1.30480
$447$	−9.28317e6	−2.19749
$448$	5.52104e6	1.29965
$449$	−4.71161e6	−1.10294	−0.551472	−	0.834194i	$-0.685934\pi$
$449$	−4.71161e6	−1.10294	−0.551472	+	0.834194i	$0.685934\pi$
$450$	0	0
$451$	−1.24209e6	−0.287549
$452$	−2.31530e6	−0.533041
$453$	5.22269e6	1.19577
$454$	−3.71804e6	−0.846593
$455$	0	0
$456$	1.20392e7	2.71135
$457$	−7.16888e6	−1.60569	−0.802843	−	0.596191i	$-0.796680\pi$
$457$	−7.16888e6	−1.60569	−0.802843	+	0.596191i	$0.796680\pi$
$458$	5.25288e6	1.17013
$459$	−2.18278e6	−0.483591
$460$	0	0
$461$	−7.38907e6	−1.61934	−0.809669	−	0.586887i	$-0.800353\pi$
$461$	−7.38907e6	−1.61934	−0.809669	+	0.586887i	$0.800353\pi$
$462$	2.52669e6	0.550740
$463$	−4.07277e6	−0.882954	−0.441477	−	0.897273i	$-0.645545\pi$
$463$	−4.07277e6	−0.882954	−0.441477	+	0.897273i	$0.645545\pi$
$464$	−418707.	−0.0902850
$465$	0	0
$466$	4.21704e6	0.899586
$467$	−1.64230e6	−0.348466	−0.174233	−	0.984704i	$-0.555745\pi$
$467$	−1.64230e6	−0.348466	−0.174233	+	0.984704i	$0.555745\pi$
$468$	−7.49013e6	−1.58079
$469$	7.57912e6	1.59106
$470$	0	0
$471$	9.89761e6	2.05579
$472$	2.50368e6	0.517278
$473$	−241080.	−0.0495459
$474$	−2.29223e6	−0.468610
$475$	0	0
$476$	−1.39395e6	−0.281988
$477$	2.10829e6	0.424262
$478$	6.70588e6	1.34241
$479$	772659.	0.153868	0.0769341	−	0.997036i	$-0.475487\pi$
$479$	772659.	0.153868	0.0769341	+	0.997036i	$0.475487\pi$
$480$	0	0
$481$	−1.53115e7	−3.01756
$482$	−1.76000e6	−0.345061
$483$	−1.98385e7	−3.86938
$484$	2.20326e6	0.427517
$485$	0	0
$486$	3.07365e6	0.590288
$487$	−3.82401e6	−0.730628	−0.365314	−	0.930884i	$-0.619038\pi$
$487$	−3.82401e6	−0.730628	−0.365314	+	0.930884i	$0.619038\pi$
$488$	1.73953e6	0.330660
$489$	−6.10438e6	−1.15444
$490$	0	0
$491$	−7.53498e6	−1.41052	−0.705259	−	0.708950i	$-0.749169\pi$
$491$	−7.53498e6	−1.41052	−0.705259	+	0.708950i	$0.749169\pi$
$492$	3.73890e6	0.696356
$493$	686986.	0.127301
$494$	1.16931e7	2.15581
$495$	0	0
$496$	−952156.	−0.173782
$497$	−4.51218e6	−0.819399
$498$	−4.24016e6	−0.766141
$499$	−7.94390e6	−1.42818	−0.714089	−	0.700055i	$-0.753159\pi$
$499$	−7.94390e6	−1.42818	−0.714089	+	0.700055i	$0.753159\pi$
$500$	0	0
$501$	5.24228e6	0.933095
$502$	1.52996e6	0.270970
$503$	1.07845e7	1.90056	0.950278	−	0.311402i	$-0.100799\pi$
$503$	1.07845e7	1.90056	0.950278	+	0.311402i	$0.100799\pi$
$504$	−1.48391e7	−2.60214
$505$	0	0
$506$	−2.22973e6	−0.387147
$507$	−2.61246e7	−4.51367
$508$	2.44711e6	0.420721
$509$	8.58816e6	1.46928	0.734642	−	0.678455i	$-0.237350\pi$
$509$	8.58816e6	1.46928	0.734642	+	0.678455i	$0.237350\pi$
$510$	0	0
$511$	−4.33764e6	−0.734855
$512$	−3.34974e6	−0.564724
$513$	−1.07410e7	−1.80199
$514$	2.88392e6	0.481476
$515$	0	0
$516$	725690.	0.119985
$517$	−1.22838e6	−0.202118
$518$	−9.81003e6	−1.60637
$519$	−1.97031e7	−3.21082
$520$	0	0
$521$	−8.60615e6	−1.38904	−0.694520	−	0.719473i	$-0.744383\pi$
$521$	−8.60615e6	−1.38904	−0.694520	+	0.719473i	$0.744383\pi$
$522$	2.36506e6	0.379896
$523$	1.45797e6	0.233075	0.116537	−	0.993186i	$-0.462821\pi$
$523$	1.45797e6	0.233075	0.116537	+	0.993186i	$0.462821\pi$
$524$	1.13964e6	0.181317
$525$	0	0
$526$	−2.49940e6	−0.393887
$527$	1.56223e6	0.245030
$528$	−1.00579e6	−0.157009
$529$	1.10706e7	1.72001
$530$	0	0
$531$	−5.38108e6	−0.828197
$532$	−6.85937e6	−1.05076
$533$	1.12290e7	1.71207
$534$	1.20250e7	1.82487
$535$	0	0
$536$	−7.92897e6	−1.19208
$537$	545183.	0.0815843
$538$	1.30927e6	0.195017
$539$	−2.26011e6	−0.335087
$540$	0	0
$541$	6.74347e6	0.990581	0.495291	−	0.868727i	$-0.335062\pi$
$541$	6.74347e6	0.990581	0.495291	+	0.868727i	$0.335062\pi$
$542$	−1.76464e6	−0.258023
$543$	605669.	0.0881527
$544$	2.44499e6	0.354225
$545$	0	0
$546$	−2.28422e7	−3.27911
$547$	−1.00578e7	−1.43725	−0.718626	−	0.695396i	$-0.755229\pi$
$547$	−1.00578e7	−1.43725	−0.718626	+	0.695396i	$0.755229\pi$
$548$	−418440.	−0.0595226
$549$	−3.73871e6	−0.529409
$550$	0	0
$551$	3.38052e6	0.474357
$552$	2.07543e7	2.89908
$553$	4.03841e6	0.561561
$554$	511790.	0.0708463
$555$	0	0
$556$	−22178.0	−0.00304253
$557$	1.05527e7	1.44120	0.720599	−	0.693352i	$-0.243867\pi$
$557$	1.05527e7	1.44120	0.720599	+	0.693352i	$0.243867\pi$
$558$	5.37823e6	0.731229
$559$	2.17945e6	0.294997
$560$	0	0
$561$	1.65024e6	0.221380
$562$	61893.7	0.00826620
$563$	−792356.	−0.105354	−0.0526768	−	0.998612i	$-0.516775\pi$
$563$	−792356.	−0.105354	−0.0526768	+	0.998612i	$0.516775\pi$
$564$	3.69762e6	0.489468
$565$	0	0
$566$	−7.78720e6	−1.02174
$567$	2.32833e6	0.304149
$568$	4.72047e6	0.613924
$569$	1.06869e7	1.38379	0.691894	−	0.721999i	$-0.256777\pi$
$569$	1.06869e7	1.38379	0.691894	+	0.721999i	$0.256777\pi$
$570$	0	0
$571$	−9.10904e6	−1.16918	−0.584591	−	0.811328i	$-0.698745\pi$
$571$	−9.10904e6	−1.16918	−0.584591	+	0.811328i	$0.698745\pi$
$572$	2.35063e6	0.300396
$573$	−4.76542e6	−0.606337
$574$	7.19437e6	0.911408
$575$	0	0
$576$	1.24139e7	1.55903
$577$	2.30196e6	0.287845	0.143923	−	0.989589i	$-0.454028\pi$
$577$	2.30196e6	0.287845	0.143923	+	0.989589i	$0.454028\pi$
$578$	−4.80885e6	−0.598717
$579$	−2.42261e6	−0.300322
$580$	0	0
$581$	7.47024e6	0.918109
$582$	−2.79240e6	−0.341720
$583$	−661645.	−0.0806221
$584$	4.53787e6	0.550579
$585$	0	0
$586$	1.95229e6	0.234855
$587$	7.44713e6	0.892060	0.446030	−	0.895018i	$-0.352837\pi$
$587$	7.44713e6	0.892060	0.446030	+	0.895018i	$0.352837\pi$
$588$	6.80331e6	0.811479
$589$	7.68744e6	0.913048
$590$	0	0
$591$	−3.40837e6	−0.401400
$592$	3.90506e6	0.457956
$593$	−7.82826e6	−0.914173	−0.457086	−	0.889422i	$-0.651107\pi$
$593$	−7.82826e6	−0.914173	−0.457086	+	0.889422i	$0.651107\pi$
$594$	2.35830e6	0.274241
$595$	0	0
$596$	−5.53327e6	−0.638066
$597$	2.64066e7	3.03233
$598$	2.01576e7	2.30508
$599$	−1.26163e7	−1.43669	−0.718347	−	0.695685i	$-0.755101\pi$
$599$	−1.26163e7	−1.43669	−0.718347	+	0.695685i	$0.755101\pi$
$600$	0	0
$601$	1.22325e7	1.38143	0.690717	−	0.723125i	$-0.257295\pi$
$601$	1.22325e7	1.38143	0.690717	+	0.723125i	$0.257295\pi$
$602$	1.39637e6	0.157039
$603$	1.70415e7	1.90860
$604$	3.11300e6	0.347206
$605$	0	0
$606$	5.31178e6	0.587569
$607$	−1.11500e7	−1.22830	−0.614151	−	0.789189i	$-0.710501\pi$
$607$	−1.11500e7	−1.22830	−0.614151	+	0.789189i	$0.710501\pi$
$608$	1.20313e7	1.31994
$609$	−6.60380e6	−0.721524
$610$	0	0
$611$	1.11050e7	1.20341
$612$	−3.13427e6	−0.338266
$613$	−4.57904e6	−0.492179	−0.246089	−	0.969247i	$-0.579146\pi$
$613$	−4.57904e6	−0.492179	−0.246089	+	0.969247i	$0.579146\pi$
$614$	4.60534e6	0.492994
$615$	0	0
$616$	4.65696e6	0.494482
$617$	7.65635e6	0.809671	0.404836	−	0.914389i	$-0.367329\pi$
$617$	7.65635e6	0.809671	0.404836	+	0.914389i	$0.367329\pi$
$618$	−5.66796e6	−0.596975
$619$	4.95647e6	0.519931	0.259965	−	0.965618i	$-0.416289\pi$
$619$	4.95647e6	0.519931	0.259965	+	0.965618i	$0.416289\pi$
$620$	0	0
$621$	−1.85164e7	−1.92676
$622$	−5.32766e6	−0.552154
$623$	−2.11854e7	−2.18684
$624$	9.09276e6	0.934834
$625$	0	0
$626$	5.40633e6	0.551400
$627$	8.12050e6	0.824924
$628$	5.89951e6	0.596921
$629$	−6.40716e6	−0.645712
$630$	0	0
$631$	1.90729e7	1.90697	0.953486	−	0.301438i	$-0.0974667\pi$
$631$	1.90729e7	1.90697	0.953486	+	0.301438i	$0.0974667\pi$
$632$	−4.22482e6	−0.420742
$633$	−1.19968e7	−1.19002
$634$	8.36992e6	0.826986
$635$	0	0
$636$	1.99166e6	0.195242
$637$	2.04323e7	1.99512
$638$	−742227.	−0.0721913
$639$	−1.01455e7	−0.982932
$640$	0	0
$641$	1.65287e7	1.58889	0.794445	−	0.607336i	$-0.207762\pi$
$641$	1.65287e7	1.58889	0.794445	+	0.607336i	$0.207762\pi$
$642$	−8.06640e6	−0.772400
$643$	1.19443e7	1.13928	0.569641	−	0.821893i	$-0.307082\pi$
$643$	1.19443e7	1.13928	0.569641	+	0.821893i	$0.307082\pi$
$644$	−1.18248e7	−1.12352
$645$	0	0
$646$	4.89300e6	0.461311
$647$	9.88438e6	0.928301	0.464150	−	0.885756i	$-0.346360\pi$
$647$	9.88438e6	0.928301	0.464150	+	0.885756i	$0.346360\pi$
$648$	−2.43581e6	−0.227880
$649$	1.68875e6	0.157381
$650$	0	0
$651$	−1.50173e7	−1.38880
$652$	−3.63854e6	−0.335203
$653$	−1.61632e6	−0.148336	−0.0741678	−	0.997246i	$-0.523630\pi$
$653$	−1.61632e6	−0.148336	−0.0741678	+	0.997246i	$0.523630\pi$
$654$	−5.56131e6	−0.508432
$655$	0	0
$656$	−2.86385e6	−0.259831
$657$	−9.75310e6	−0.881514
$658$	7.11493e6	0.640628
$659$	1.49221e7	1.33849	0.669246	−	0.743041i	$-0.266617\pi$
$659$	1.49221e7	1.33849	0.669246	+	0.743041i	$0.266617\pi$
$660$	0	0
$661$	−1.13725e7	−1.01240	−0.506202	−	0.862415i	$-0.668951\pi$
$661$	−1.13725e7	−1.01240	−0.506202	+	0.862415i	$0.668951\pi$
$662$	−3.33877e6	−0.296103
$663$	−1.49188e7	−1.31810
$664$	−7.81507e6	−0.687880
$665$	0	0
$666$	−2.20576e7	−1.92696
$667$	5.82766e6	0.507201
$668$	3.12468e6	0.270934
$669$	3.44131e7	2.97275
$670$	0	0
$671$	1.17332e6	0.100603
$672$	−2.35030e7	−2.00771
$673$	3.86161e6	0.328647	0.164324	−	0.986406i	$-0.447456\pi$
$673$	3.86161e6	0.328647	0.164324	+	0.986406i	$0.447456\pi$
$674$	−1.02262e6	−0.0867088
$675$	0	0
$676$	−1.55717e7	−1.31059
$677$	7.91159e6	0.663425	0.331713	−	0.943380i	$-0.392374\pi$
$677$	7.91159e6	0.663425	0.331713	+	0.943380i	$0.392374\pi$
$678$	−1.58761e7	−1.32639
$679$	4.91960e6	0.409501
$680$	0	0
$681$	2.33430e7	1.92880
$682$	−1.68785e6	−0.138955
$683$	1.70274e7	1.39668	0.698341	−	0.715765i	$-0.253922\pi$
$683$	1.70274e7	1.39668	0.698341	+	0.715765i	$0.253922\pi$
$684$	−1.54231e7	−1.26047
$685$	0	0
$686$	398206.	0.0323071
$687$	−3.29791e7	−2.66592
$688$	−555850.	−0.0447699
$689$	5.98153e6	0.480025
$690$	0	0
$691$	4.79185e6	0.381776	0.190888	−	0.981612i	$-0.438863\pi$
$691$	4.79185e6	0.381776	0.190888	+	0.981612i	$0.438863\pi$
$692$	−1.17441e7	−0.932298
$693$	−1.00090e7	−0.791698
$694$	−1.35572e7	−1.06850
$695$	0	0
$696$	6.90864e6	0.540592
$697$	4.69881e6	0.366358
$698$	5.23404e6	0.406629
$699$	−2.64758e7	−2.04954
$700$	0	0
$701$	1.01357e7	0.779038	0.389519	−	0.921018i	$-0.372641\pi$
$701$	1.01357e7	0.779038	0.389519	+	0.921018i	$0.372641\pi$
$702$	−2.13199e7	−1.63284
$703$	−3.15284e7	−2.40610
$704$	−3.89587e6	−0.296260
$705$	0	0
$706$	−4.84879e6	−0.366119
$707$	−9.35820e6	−0.704116
$708$	−5.08341e6	−0.381129
$709$	−1.09298e7	−0.816577	−0.408289	−	0.912853i	$-0.633874\pi$
$709$	−1.09298e7	−0.816577	−0.408289	+	0.912853i	$0.633874\pi$
$710$	0	0
$711$	9.08027e6	0.673635
$712$	2.21633e7	1.63846
$713$	1.32523e7	0.976267
$714$	−9.55841e6	−0.701682
$715$	0	0
$716$	324958.	0.0236889
$717$	−4.21015e7	−3.05844
$718$	−1.11126e6	−0.0804462
$719$	−6.65338e6	−0.479976	−0.239988	−	0.970776i	$-0.577144\pi$
$719$	−6.65338e6	−0.479976	−0.239988	+	0.970776i	$0.577144\pi$
$720$	0	0
$721$	9.98572e6	0.715387
$722$	1.39573e7	0.996455
$723$	1.10498e7	0.786156
$724$	361011.	0.0255961
$725$	0	0
$726$	1.51079e7	1.06381
$727$	1.56696e7	1.09957	0.549784	−	0.835307i	$-0.314710\pi$
$727$	1.56696e7	1.09957	0.549784	+	0.835307i	$0.314710\pi$
$728$	−4.21007e7	−2.94415
$729$	−2.23593e7	−1.55826
$730$	0	0
$731$	912000.	0.0631250
$732$	−3.53189e6	−0.243629
$733$	2.68601e7	1.84650	0.923248	−	0.384205i	$-0.125525\pi$
$733$	2.68601e7	1.84650	0.923248	+	0.384205i	$0.125525\pi$
$734$	1.40785e6	0.0964532
$735$	0	0
$736$	2.07407e7	1.41133
$737$	−5.34814e6	−0.362688
$738$	1.61764e7	1.09330
$739$	−532320.	−0.0358560	−0.0179280	−	0.999839i	$-0.505707\pi$
$739$	−532320.	−0.0358560	−0.0179280	+	0.999839i	$0.505707\pi$
$740$	0	0
$741$	−7.34124e7	−4.91161
$742$	3.83234e6	0.255538
$743$	2.24889e7	1.49450	0.747249	−	0.664544i	$-0.231374\pi$
$743$	2.24889e7	1.49450	0.747249	+	0.664544i	$0.231374\pi$
$744$	1.57105e7	1.04054
$745$	0	0
$746$	−5.96903e6	−0.392696
$747$	1.67967e7	1.10134
$748$	983630.	0.0642803
$749$	1.42112e7	0.925609
$750$	0	0
$751$	1.12267e7	0.726360	0.363180	−	0.931719i	$-0.381691\pi$
$751$	1.12267e7	0.726360	0.363180	+	0.931719i	$0.381691\pi$
$752$	−2.83223e6	−0.182635
$753$	−9.60556e6	−0.617356
$754$	6.71001e6	0.429828
$755$	0	0
$756$	1.25067e7	0.795861
$757$	6.60375e6	0.418842	0.209421	−	0.977826i	$-0.432842\pi$
$757$	6.60375e6	0.418842	0.209421	+	0.977826i	$0.432842\pi$
$758$	8.91725e6	0.563713
$759$	1.39989e7	0.882041
$760$	0	0
$761$	1.52837e7	0.956679	0.478339	−	0.878175i	$-0.341239\pi$
$761$	1.52837e7	0.956679	0.478339	+	0.878175i	$0.341239\pi$
$762$	1.67800e7	1.04690
$763$	9.79782e6	0.609282
$764$	−2.84044e6	−0.176057
$765$	0	0
$766$	−8.85426e6	−0.545231
$767$	−1.52669e7	−0.937051
$768$	2.83635e7	1.73523
$769$	1.93382e7	1.17923	0.589617	−	0.807683i	$-0.299279\pi$
$769$	1.93382e7	1.17923	0.589617	+	0.807683i	$0.299279\pi$
$770$	0	0
$771$	−1.81061e7	−1.09695
$772$	−1.44401e6	−0.0872019
$773$	2.28611e7	1.37610	0.688048	−	0.725665i	$-0.258468\pi$
$773$	2.28611e7	1.37610	0.688048	+	0.725665i	$0.258468\pi$
$774$	3.13970e6	0.188381
$775$	0	0
$776$	−5.14669e6	−0.306813
$777$	6.15902e7	3.65982
$778$	−1.83083e7	−1.08442
$779$	2.31219e7	1.36515
$780$	0	0
$781$	3.18398e6	0.186785
$782$	8.43502e6	0.493252
$783$	−6.16370e6	−0.359283
$784$	−5.21107e6	−0.302786
$785$	0	0
$786$	7.81456e6	0.451178
$787$	−1.74322e7	−1.00326	−0.501632	−	0.865081i	$-0.667267\pi$
$787$	−1.74322e7	−1.00326	−0.501632	+	0.865081i	$0.667267\pi$
$788$	−2.03157e6	−0.116551
$789$	1.56920e7	0.897398
$790$	0	0
$791$	2.79703e7	1.58948
$792$	1.04711e7	0.593169
$793$	−1.06073e7	−0.598992
$794$	3.30090e6	0.185815
$795$	0	0
$796$	1.57398e7	0.880471
$797$	−8.94753e6	−0.498951	−0.249475	−	0.968381i	$-0.580258\pi$
$797$	−8.94753e6	−0.498951	−0.249475	+	0.968381i	$0.580258\pi$
$798$	−4.70351e7	−2.61466
$799$	4.64693e6	0.257513
$800$	0	0
$801$	−4.76349e7	−2.62328
$802$	−1.90751e7	−1.04720
$803$	3.06082e6	0.167513
$804$	1.60988e7	0.878320
$805$	0	0
$806$	1.52588e7	0.827339
$807$	−8.21997e6	−0.444310
$808$	9.79019e6	0.527549
$809$	−9.71667e6	−0.521971	−0.260985	−	0.965343i	$-0.584047\pi$
$809$	−9.71667e6	−0.521971	−0.260985	+	0.965343i	$0.584047\pi$
$810$	0	0
$811$	−2.94873e6	−0.157428	−0.0787141	−	0.996897i	$-0.525081\pi$
$811$	−2.94873e6	−0.157428	−0.0787141	+	0.996897i	$0.525081\pi$
$812$	−3.93622e6	−0.209503
$813$	1.10790e7	0.587858
$814$	6.92236e6	0.366179
$815$	0	0
$816$	3.80490e6	0.200041
$817$	4.48778e6	0.235221
$818$	−1.70542e7	−0.891145
$819$	9.04856e7	4.71378
$820$	0	0
$821$	−2.62051e7	−1.35684	−0.678419	−	0.734675i	$-0.737335\pi$
$821$	−2.62051e7	−1.35684	−0.678419	+	0.734675i	$0.737335\pi$
$822$	−2.86926e6	−0.148112
$823$	−1.13095e7	−0.582030	−0.291015	−	0.956718i	$-0.593993\pi$
$823$	−1.13095e7	−0.582030	−0.291015	+	0.956718i	$0.593993\pi$
$824$	−1.04467e7	−0.535994
$825$	0	0
$826$	−9.78147e6	−0.498831
$827$	−1.42566e7	−0.724859	−0.362429	−	0.932011i	$-0.618053\pi$
$827$	−1.42566e7	−0.724859	−0.362429	+	0.932011i	$0.618053\pi$
$828$	−2.65879e7	−1.34775
$829$	1.53655e7	0.776536	0.388268	−	0.921546i	$-0.373073\pi$
$829$	1.53655e7	0.776536	0.388268	+	0.921546i	$0.373073\pi$
$830$	0	0
$831$	−3.21317e6	−0.161410
$832$	3.52202e7	1.76394
$833$	8.54996e6	0.426925
$834$	−152076.	−0.00757085
$835$	0	0
$836$	4.84025e6	0.239526
$837$	−1.40165e7	−0.691553
$838$	5.58327e6	0.274649
$839$	3.26736e7	1.60248	0.801240	−	0.598343i	$-0.204174\pi$
$839$	3.26736e7	1.60248	0.801240	+	0.598343i	$0.204174\pi$
$840$	0	0
$841$	−1.85712e7	−0.905422
$842$	−1.55879e7	−0.757718
$843$	−388587.	−0.0188330
$844$	−7.15071e6	−0.345536
$845$	0	0
$846$	1.59978e7	0.768482
$847$	−2.66168e7	−1.27482
$848$	−1.52554e6	−0.0728505
$849$	4.88903e7	2.32784
$850$	0	0
$851$	−5.43516e7	−2.57269
$852$	−9.58432e6	−0.452337
$853$	−3.34255e6	−0.157291	−0.0786457	−	0.996903i	$-0.525060\pi$
$853$	−3.34255e6	−0.157291	−0.0786457	+	0.996903i	$0.525060\pi$
$854$	−6.79604e6	−0.318868
$855$	0	0
$856$	−1.48672e7	−0.693500
$857$	−2.93627e7	−1.36566	−0.682831	−	0.730576i	$-0.739252\pi$
$857$	−2.93627e7	−1.36566	−0.682831	+	0.730576i	$0.739252\pi$
$858$	1.61184e7	0.747487
$859$	−8.99060e6	−0.415724	−0.207862	−	0.978158i	$-0.566651\pi$
$859$	−8.99060e6	−0.415724	−0.207862	+	0.978158i	$0.566651\pi$
$860$	0	0
$861$	−4.51683e7	−2.07647
$862$	2.05009e7	0.939733
$863$	1.75482e7	0.802059	0.401030	−	0.916065i	$-0.368652\pi$
$863$	1.75482e7	0.802059	0.401030	+	0.916065i	$0.368652\pi$
$864$	−2.19367e7	−0.999739
$865$	0	0
$866$	−4.43351e6	−0.200888
$867$	3.01914e7	1.36407
$868$	−8.95112e6	−0.403253
$869$	−2.84967e6	−0.128010
$870$	0	0
$871$	4.83492e7	2.15945
$872$	−1.02501e7	−0.456496
$873$	1.10616e7	0.491228
$874$	4.15071e7	1.83799
$875$	0	0
$876$	−9.21358e6	−0.405665
$877$	3.42578e7	1.50404	0.752022	−	0.659138i	$-0.229079\pi$
$877$	3.42578e7	1.50404	0.752022	+	0.659138i	$0.229079\pi$
$878$	−2.62978e7	−1.15129
$879$	−1.22570e7	−0.535073
$880$	0	0
$881$	1.29942e7	0.564038	0.282019	−	0.959409i	$-0.408996\pi$
$881$	1.29942e7	0.564038	0.282019	+	0.959409i	$0.408996\pi$
$882$	2.94346e7	1.27405
$883$	−640444.	−0.0276426	−0.0138213	−	0.999904i	$-0.504400\pi$
$883$	−640444.	−0.0276426	−0.0138213	+	0.999904i	$0.504400\pi$
$884$	−8.89239e6	−0.382726
$885$	0	0
$886$	3.84219e6	0.164435
$887$	−4.27483e7	−1.82436	−0.912179	−	0.409793i	$-0.865601\pi$
$887$	−4.27483e7	−1.82436	−0.912179	+	0.409793i	$0.865601\pi$
$888$	−6.44333e7	−2.74207
$889$	−2.95627e7	−1.25455
$890$	0	0
$891$	−1.64297e6	−0.0693321
$892$	2.05121e7	0.863172
$893$	2.28666e7	0.959563
$894$	−3.79419e7	−1.58773
$895$	0	0
$896$	−6.74408e6	−0.280642
$897$	−1.26555e8	−5.25169
$898$	−1.92572e7	−0.796895
$899$	4.41141e6	0.182045
$900$	0	0
$901$	2.50299e6	0.102718
$902$	−5.07664e6	−0.207759
$903$	−8.76681e6	−0.357785
$904$	−2.92614e7	−1.19090
$905$	0	0
$906$	2.13460e7	0.863966
$907$	1.02442e7	0.413485	0.206743	−	0.978395i	$-0.433714\pi$
$907$	1.02442e7	0.413485	0.206743	+	0.978395i	$0.433714\pi$
$908$	1.39137e7	0.560050
$909$	−2.10417e7	−0.844640
$910$	0	0
$911$	2.06052e7	0.822586	0.411293	−	0.911503i	$-0.365077\pi$
$911$	2.06052e7	0.822586	0.411293	+	0.911503i	$0.365077\pi$
$912$	1.87232e7	0.745405
$913$	−5.27131e6	−0.209287
$914$	−2.93004e7	−1.16014
$915$	0	0
$916$	−1.96573e7	−0.774080
$917$	−1.37676e7	−0.540672
$918$	−8.92140e6	−0.349403
$919$	−1.39914e7	−0.546479	−0.273240	−	0.961946i	$-0.588095\pi$
$919$	−1.39914e7	−0.546479	−0.273240	+	0.961946i	$0.588095\pi$
$920$	0	0
$921$	−2.89137e7	−1.12319
$922$	−3.02004e7	−1.17000
$923$	−2.87844e7	−1.11212
$924$	−9.45536e6	−0.364333
$925$	0	0
$926$	−1.66461e7	−0.637949
$927$	2.24527e7	0.858161
$928$	6.90412e6	0.263171
$929$	−2.20609e7	−0.838657	−0.419329	−	0.907835i	$-0.637734\pi$
$929$	−2.20609e7	−0.838657	−0.419329	+	0.907835i	$0.637734\pi$
$930$	0	0
$931$	4.20727e7	1.59084
$932$	−1.57810e7	−0.595106
$933$	3.34486e7	1.25798
$934$	−6.71236e6	−0.251773
$935$	0	0
$936$	−9.46625e7	−3.53174
$937$	−2.82986e7	−1.05297	−0.526485	−	0.850185i	$-0.676490\pi$
$937$	−2.82986e7	−1.05297	−0.526485	+	0.850185i	$0.676490\pi$
$938$	3.09772e7	1.14957
$939$	−3.39425e7	−1.25626
$940$	0	0
$941$	218548.	0.00804586	0.00402293	−	0.999992i	$-0.498719\pi$
$941$	218548.	0.00804586	0.00402293	+	0.999992i	$0.498719\pi$
$942$	4.04533e7	1.48534
$943$	3.98597e7	1.45967
$944$	3.89369e6	0.142211
$945$	0	0
$946$	−985335.	−0.0357978
$947$	−3.22539e7	−1.16871	−0.584356	−	0.811498i	$-0.698653\pi$
$947$	−3.22539e7	−1.16871	−0.584356	+	0.811498i	$0.698653\pi$
$948$	8.57797e6	0.310001
$949$	−2.76710e7	−0.997376
$950$	0	0
$951$	−5.25488e7	−1.88413
$952$	−1.76172e7	−0.630005
$953$	3.21935e6	0.114825	0.0574124	−	0.998351i	$-0.481715\pi$
$953$	3.21935e6	0.114825	0.0574124	+	0.998351i	$0.481715\pi$
$954$	8.61694e6	0.306537
$955$	0	0
$956$	−2.50948e7	−0.888052
$957$	4.65992e6	0.164474
$958$	3.15799e6	0.111172
$959$	5.05502e6	0.177491
$960$	0	0
$961$	−1.85974e7	−0.649598
$962$	−6.25808e7	−2.18023
$963$	3.19537e7	1.11034
$964$	6.58628e6	0.228269
$965$	0	0
$966$	−8.10835e7	−2.79569
$967$	−2.57522e7	−0.885620	−0.442810	−	0.896615i	$-0.646018\pi$
$967$	−2.57522e7	−0.885620	−0.442810	+	0.896615i	$0.646018\pi$
$968$	2.78455e7	0.955139
$969$	−3.07197e7	−1.05101
$970$	0	0
$971$	3.43740e7	1.16999	0.584994	−	0.811037i	$-0.301097\pi$
$971$	3.43740e7	1.16999	0.584994	+	0.811037i	$0.301097\pi$
$972$	−1.15022e7	−0.390495
$973$	267924.	0.00907256
$974$	−1.56294e7	−0.527891
$975$	0	0
$976$	2.70529e6	0.0909053
$977$	−500938.	−0.0167899	−0.00839494	−	0.999965i	$-0.502672\pi$
$977$	−500938.	−0.0167899	−0.00839494	+	0.999965i	$0.502672\pi$
$978$	−2.49497e7	−0.834099
$979$	1.49493e7	0.498498
$980$	0	0
$981$	2.20302e7	0.730880
$982$	−3.07968e7	−1.01912
$983$	2.62362e7	0.865998	0.432999	−	0.901394i	$-0.357455\pi$
$983$	2.62362e7	0.865998	0.432999	+	0.901394i	$0.357455\pi$
$984$	4.72534e7	1.55577
$985$	0	0
$986$	2.80783e6	0.0919769
$987$	−4.46696e7	−1.45955
$988$	−4.37577e7	−1.42614
$989$	7.73645e6	0.251508
$990$	0	0
$991$	−2.69649e7	−0.872196	−0.436098	−	0.899899i	$-0.643640\pi$
$991$	−2.69649e7	−0.872196	−0.436098	+	0.899899i	$0.643640\pi$
$992$	1.57002e7	0.506555
$993$	2.09618e7	0.674614
$994$	−1.84421e7	−0.592030
$995$	0	0
$996$	1.58675e7	0.506828
$997$	4.63300e7	1.47613	0.738064	−	0.674730i	$-0.235740\pi$
$997$	4.63300e7	1.47613	0.738064	+	0.674730i	$0.235740\pi$
$998$	−3.24681e7	−1.03188
$999$	5.74856e7	1.82241

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1075.6.a.a.1.5		8
5.4	even	2		43.6.a.a.1.4	✓	8
15.14	odd	2		387.6.a.c.1.5		8
20.19	odd	2		688.6.a.e.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.a.a.1.4	✓	8	5.4	even	2
387.6.a.c.1.5		8	15.14	odd	2
688.6.a.e.1.1		8	20.19	odd	2
1075.6.a.a.1.5		8	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:	\( N \)	\(=\)	\( 1075 = 5^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1075.a (trivial)

\(f(q)\)	\(=\)	\(q+4.08717 q^{2} -25.6605 q^{3} -15.2950 q^{4} -104.879 q^{6} +184.774 q^{7} -193.303 q^{8} +415.460 q^{9} +O(q^{10})\)	\(q+4.08717 q^{2} -25.6605 q^{3} -15.2950 q^{4} -104.879 q^{6} +184.774 q^{7} -193.303 q^{8} +415.460 q^{9} -130.384 q^{11} +392.477 q^{12} +1178.72 q^{13} +755.202 q^{14} -300.622 q^{16} +493.240 q^{17} +1698.06 q^{18} +2427.14 q^{19} -4741.38 q^{21} -532.901 q^{22} +4184.13 q^{23} +4960.24 q^{24} +4817.63 q^{26} -4425.39 q^{27} -2826.12 q^{28} +1392.80 q^{29} +3167.29 q^{31} +4957.00 q^{32} +3345.71 q^{33} +2015.96 q^{34} -6354.46 q^{36} -12989.9 q^{37} +9920.13 q^{38} -30246.5 q^{39} +9526.42 q^{41} -19378.8 q^{42} +1849.00 q^{43} +1994.22 q^{44} +17101.3 q^{46} +9421.23 q^{47} +7714.10 q^{48} +17334.3 q^{49} -12656.8 q^{51} -18028.5 q^{52} +5074.60 q^{53} -18087.4 q^{54} -35717.3 q^{56} -62281.5 q^{57} +5692.63 q^{58} -12952.1 q^{59} -8998.97 q^{61} +12945.2 q^{62} +76766.0 q^{63} +29880.0 q^{64} +13674.5 q^{66} +41018.4 q^{67} -7544.11 q^{68} -107367. q^{69} -24420.1 q^{71} -80309.6 q^{72} -23475.4 q^{73} -53092.1 q^{74} -37123.1 q^{76} -24091.5 q^{77} -123623. q^{78} +21856.0 q^{79} +12601.0 q^{81} +38936.1 q^{82} +40429.1 q^{83} +72519.4 q^{84} +7557.18 q^{86} -35740.0 q^{87} +25203.6 q^{88} -114656. q^{89} +217796. q^{91} -63996.3 q^{92} -81274.1 q^{93} +38506.2 q^{94} -127199. q^{96} +26625.0 q^{97} +70848.2 q^{98} -54169.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{2} + 26 q^{3} + 122 q^{4} - 69 q^{6} + 136 q^{7} + 666 q^{8} + 546 q^{9}+O(q^{10}) \) 8 * q + 12 * q^2 + 26 * q^3 + 122 * q^4 - 69 * q^6 + 136 * q^7 + 666 * q^8 + 546 * q^9	\( 8 q + 12 q^{2} + 26 q^{3} + 122 q^{4} - 69 q^{6} + 136 q^{7} + 666 q^{8} + 546 q^{9} - 532 q^{11} + 4195 q^{12} + 2492 q^{13} - 4240 q^{14} + 1882 q^{16} + 2534 q^{17} + 3711 q^{18} - 1678 q^{19} - 2256 q^{21} - 11502 q^{22} + 2488 q^{23} + 19953 q^{24} + 4586 q^{26} + 8960 q^{27} - 18640 q^{28} - 4360 q^{29} + 5704 q^{31} + 18294 q^{32} + 12852 q^{33} + 30007 q^{34} + 67969 q^{36} + 3772 q^{37} + 6559 q^{38} + 11120 q^{39} - 10698 q^{41} - 78698 q^{42} + 14792 q^{43} - 356 q^{44} - 19389 q^{46} + 77864 q^{47} + 118727 q^{48} + 7188 q^{49} - 80246 q^{51} + 60736 q^{52} + 62352 q^{53} + 61026 q^{54} - 144528 q^{56} + 808 q^{57} - 52951 q^{58} - 26224 q^{59} - 82540 q^{61} + 9023 q^{62} + 61768 q^{63} + 153858 q^{64} - 48516 q^{66} - 27784 q^{67} - 40507 q^{68} - 93776 q^{69} - 9504 q^{71} + 186687 q^{72} - 14260 q^{73} - 15239 q^{74} + 1279 q^{76} + 218140 q^{77} - 264170 q^{78} + 160248 q^{79} + 161076 q^{81} + 47781 q^{82} + 77176 q^{83} + 16382 q^{84} + 22188 q^{86} - 268136 q^{87} - 129544 q^{88} - 265692 q^{89} + 401148 q^{91} - 190391 q^{92} + 123860 q^{93} + 248737 q^{94} + 950817 q^{96} - 144742 q^{97} + 292244 q^{98} + 239516 q^{99}+O(q^{100}) \) 8 * q + 12 * q^2 + 26 * q^3 + 122 * q^4 - 69 * q^6 + 136 * q^7 + 666 * q^8 + 546 * q^9 - 532 * q^11 + 4195 * q^12 + 2492 * q^13 - 4240 * q^14 + 1882 * q^16 + 2534 * q^17 + 3711 * q^18 - 1678 * q^19 - 2256 * q^21 - 11502 * q^22 + 2488 * q^23 + 19953 * q^24 + 4586 * q^26 + 8960 * q^27 - 18640 * q^28 - 4360 * q^29 + 5704 * q^31 + 18294 * q^32 + 12852 * q^33 + 30007 * q^34 + 67969 * q^36 + 3772 * q^37 + 6559 * q^38 + 11120 * q^39 - 10698 * q^41 - 78698 * q^42 + 14792 * q^43 - 356 * q^44 - 19389 * q^46 + 77864 * q^47 + 118727 * q^48 + 7188 * q^49 - 80246 * q^51 + 60736 * q^52 + 62352 * q^53 + 61026 * q^54 - 144528 * q^56 + 808 * q^57 - 52951 * q^58 - 26224 * q^59 - 82540 * q^61 + 9023 * q^62 + 61768 * q^63 + 153858 * q^64 - 48516 * q^66 - 27784 * q^67 - 40507 * q^68 - 93776 * q^69 - 9504 * q^71 + 186687 * q^72 - 14260 * q^73 - 15239 * q^74 + 1279 * q^76 + 218140 * q^77 - 264170 * q^78 + 160248 * q^79 + 161076 * q^81 + 47781 * q^82 + 77176 * q^83 + 16382 * q^84 + 22188 * q^86 - 268136 * q^87 - 129544 * q^88 - 265692 * q^89 + 401148 * q^91 - 190391 * q^92 + 123860 * q^93 + 248737 * q^94 + 950817 * q^96 - 144742 * q^97 + 292244 * q^98 + 239516 * q^99

Introduction

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