LMFDB - Embedded newform 1075.6.a.l.1.25

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:	$52$
Twist minimal:	no (minimal twist has level 215)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.25
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)

$f(q)$	$=$	$q-0.00353772 q^{2} +4.42609 q^{3} -32.0000 q^{4} -0.0156583 q^{6} +78.1807 q^{7} +0.226414 q^{8} -223.410 q^{9} +O(q^{10})$	\(q-0.00353772 q^{2} +4.42609 q^{3} -32.0000 q^{4} -0.0156583 q^{6} +78.1807 q^{7} +0.226414 q^{8} -223.410 q^{9} +343.479 q^{11} -141.635 q^{12} +1056.05 q^{13} -0.276582 q^{14} +1024.00 q^{16} +1468.63 q^{17} +0.790361 q^{18} +1056.43 q^{19} +346.035 q^{21} -1.21513 q^{22} -1494.08 q^{23} +1.00213 q^{24} -3.73600 q^{26} -2064.37 q^{27} -2501.78 q^{28} +3072.92 q^{29} +9435.69 q^{31} -10.8679 q^{32} +1520.27 q^{33} -5.19561 q^{34} +7149.11 q^{36} -8632.32 q^{37} -3.73735 q^{38} +4674.16 q^{39} -18021.5 q^{41} -1.22418 q^{42} -1849.00 q^{43} -10991.3 q^{44} +5.28564 q^{46} +12673.8 q^{47} +4532.31 q^{48} -10694.8 q^{49} +6500.29 q^{51} -33793.5 q^{52} -8709.02 q^{53} +7.30318 q^{54} +17.7012 q^{56} +4675.86 q^{57} -10.8711 q^{58} -17359.9 q^{59} +41529.3 q^{61} -33.3808 q^{62} -17466.3 q^{63} -32767.9 q^{64} -5.37829 q^{66} -8888.44 q^{67} -46996.2 q^{68} -6612.93 q^{69} +11586.9 q^{71} -50.5831 q^{72} -75620.5 q^{73} +30.5387 q^{74} -33805.7 q^{76} +26853.4 q^{77} -16.5359 q^{78} +62221.3 q^{79} +45151.5 q^{81} +63.7550 q^{82} +104528. q^{83} -11073.1 q^{84} +6.54125 q^{86} +13601.0 q^{87} +77.7685 q^{88} +16370.8 q^{89} +82562.4 q^{91} +47810.5 q^{92} +41763.2 q^{93} -44.8363 q^{94} -48.1022 q^{96} -14099.6 q^{97} +37.8352 q^{98} -76736.5 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 52 q + 20 q^{2} + 54 q^{3} + 826 q^{4} - 162 q^{6} + 196 q^{7} + 960 q^{8} + 4098 q^{9}+O(q^{10}) $ 52 * q + 20 * q^2 + 54 * q^3 + 826 * q^4 - 162 * q^6 + 196 * q^7 + 960 * q^8 + 4098 * q^9	\( 52 q + 20 q^{2} + 54 q^{3} + 826 q^{4} - 162 q^{6} + 196 q^{7} + 960 q^{8} + 4098 q^{9} - 664 q^{11} - 523 q^{12} + 2704 q^{13} + 150 q^{14} + 13474 q^{16} + 7266 q^{17} + 4860 q^{18} - 1970 q^{19} + 800 q^{21} + 14477 q^{22} + 9522 q^{23} + 314 q^{24} + 5514 q^{26} + 22926 q^{27} + 9408 q^{28} - 7188 q^{29} - 11556 q^{31} + 48390 q^{32} + 26136 q^{33} + 16774 q^{34} + 51872 q^{36} + 42558 q^{37} + 46208 q^{38} + 4682 q^{39} - 7746 q^{41} + 174265 q^{42} - 96148 q^{43} - 48600 q^{44} + 16182 q^{46} + 87136 q^{47} - 2912 q^{48} + 142286 q^{49} - 3710 q^{51} + 146868 q^{52} + 127034 q^{53} - 49563 q^{54} - 2849 q^{56} + 101594 q^{57} + 9480 q^{58} - 55924 q^{59} + 73702 q^{61} + 186016 q^{62} + 50120 q^{63} + 157750 q^{64} + 58211 q^{66} + 131996 q^{67} + 298560 q^{68} + 128436 q^{69} - 56284 q^{71} + 343775 q^{72} + 128620 q^{73} - 17721 q^{74} - 170410 q^{76} + 448438 q^{77} + 237616 q^{78} + 106204 q^{79} + 478568 q^{81} + 249596 q^{82} + 348616 q^{83} - 131855 q^{84} - 36980 q^{86} + 267478 q^{87} + 525216 q^{88} + 80410 q^{89} + 226376 q^{91} + 581456 q^{92} + 902902 q^{93} + 180980 q^{94} + 38543 q^{96} + 316148 q^{97} + 295095 q^{98} + 68428 q^{99}+O(q^{100}) \) 52 * q + 20 * q^2 + 54 * q^3 + 826 * q^4 - 162 * q^6 + 196 * q^7 + 960 * q^8 + 4098 * q^9 - 664 * q^11 - 523 * q^12 + 2704 * q^13 + 150 * q^14 + 13474 * q^16 + 7266 * q^17 + 4860 * q^18 - 1970 * q^19 + 800 * q^21 + 14477 * q^22 + 9522 * q^23 + 314 * q^24 + 5514 * q^26 + 22926 * q^27 + 9408 * q^28 - 7188 * q^29 - 11556 * q^31 + 48390 * q^32 + 26136 * q^33 + 16774 * q^34 + 51872 * q^36 + 42558 * q^37 + 46208 * q^38 + 4682 * q^39 - 7746 * q^41 + 174265 * q^42 - 96148 * q^43 - 48600 * q^44 + 16182 * q^46 + 87136 * q^47 - 2912 * q^48 + 142286 * q^49 - 3710 * q^51 + 146868 * q^52 + 127034 * q^53 - 49563 * q^54 - 2849 * q^56 + 101594 * q^57 + 9480 * q^58 - 55924 * q^59 + 73702 * q^61 + 186016 * q^62 + 50120 * q^63 + 157750 * q^64 + 58211 * q^66 + 131996 * q^67 + 298560 * q^68 + 128436 * q^69 - 56284 * q^71 + 343775 * q^72 + 128620 * q^73 - 17721 * q^74 - 170410 * q^76 + 448438 * q^77 + 237616 * q^78 + 106204 * q^79 + 478568 * q^81 + 249596 * q^82 + 348616 * q^83 - 131855 * q^84 - 36980 * q^86 + 267478 * q^87 + 525216 * q^88 + 80410 * q^89 + 226376 * q^91 + 581456 * q^92 + 902902 * q^93 + 180980 * q^94 + 38543 * q^96 + 316148 * q^97 + 295095 * q^98 + 68428 * 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 $

$2$	−0.00353772	−0.000625387 0	−0.000312693	−	1.00000i	$-0.500100\pi$
$2$	−0.00353772	−0.000625387 0	−0.000312693		1.00000i	$0.500100\pi$
$3$	4.42609	0.283934	0.141967	−	0.989871i	$-0.454657\pi$
$3$	4.42609	0.283934	0.141967	+	0.989871i	$0.454657\pi$
$4$	−32.0000	−1.00000
$5$	0	0
$5$	0	0
$6$	−0.0156583	−0.000177569 0
$7$	78.1807	0.603051	0.301526	−	0.953458i	$-0.402504\pi$
$7$	78.1807	0.603051	0.301526	+	0.953458i	$0.402504\pi$
$8$	0.226414	0.00125077
$9$	−223.410	−0.919382
$10$	0	0
$11$	343.479	0.855890	0.427945	−	0.903805i	$-0.359238\pi$
$11$	343.479	0.855890	0.427945	+	0.903805i	$0.359238\pi$
$12$	−141.635	−0.283934
$13$	1056.05	1.73310	0.866552	−	0.499087i	$-0.166331\pi$
$13$	1056.05	1.73310	0.866552	+	0.499087i	$0.166331\pi$
$14$	−0.276582	−0.000377140 0
$15$	0	0
$16$	1024.00	0.999999
$17$	1468.63	1.23251	0.616255	−	0.787547i	$-0.288649\pi$
$17$	1468.63	1.23251	0.616255	+	0.787547i	$0.288649\pi$
$18$	0.790361	0.000574969 0
$19$	1056.43	0.671362	0.335681	−	0.941976i	$-0.391034\pi$
$19$	1056.43	0.671362	0.335681	+	0.941976i	$0.391034\pi$
$20$	0	0
$21$	346.035	0.171227
$22$	−1.21513	−0.000535263 0
$23$	−1494.08	−0.588917	−0.294458	−	0.955664i	$-0.595139\pi$
$23$	−1494.08	−0.588917	−0.294458	+	0.955664i	$0.595139\pi$
$24$	1.00213	0.000355137 0
$25$	0	0
$26$	−3.73600	−0.00108386
$27$	−2064.37	−0.544978
$28$	−2501.78	−0.603051
$29$	3072.92	0.678510	0.339255	−	0.940694i	$-0.389825\pi$
$29$	3072.92	0.678510	0.339255	+	0.940694i	$0.389825\pi$
$30$	0	0
$31$	9435.69	1.76347	0.881737	−	0.471741i	$-0.156374\pi$
$31$	9435.69	1.76347	0.881737	+	0.471741i	$0.156374\pi$
$32$	−10.8679	−0.00187616
$33$	1520.27	0.243016
$34$	−5.19561	−0.000770795 0
$35$	0	0
$36$	7149.11	0.919381
$37$	−8632.32	−1.03663	−0.518314	−	0.855190i	$-0.673440\pi$
$37$	−8632.32	−1.03663	−0.518314	+	0.855190i	$0.673440\pi$
$38$	−3.73735	−0.000419861 0
$39$	4674.16	0.492087
$40$	0	0
$41$	−18021.5	−1.67429	−0.837145	−	0.546981i	$-0.815777\pi$
$41$	−18021.5	−1.67429	−0.837145	+	0.546981i	$0.815777\pi$
$42$	−1.22418	−0.000107083 0
$43$	−1849.00	−0.152499
$43$	−1849.00	−0.152499
$44$	−10991.3	−0.855890
$45$	0	0
$46$	5.28564	0.000368301 0
$47$	12673.8	0.836876	0.418438	−	0.908245i	$-0.362578\pi$
$47$	12673.8	0.836876	0.418438	+	0.908245i	$0.362578\pi$
$48$	4532.31	0.283934
$49$	−10694.8	−0.636329
$50$	0	0
$51$	6500.29	0.349951
$52$	−33793.5	−1.73310
$53$	−8709.02	−0.425872	−0.212936	−	0.977066i	$-0.568303\pi$
$53$	−8709.02	−0.425872	−0.212936	+	0.977066i	$0.568303\pi$
$54$	7.30318	0.000340822 0
$55$	0	0
$56$	17.7012	0.000754281 0
$57$	4675.86	0.190622
$58$	−10.8711	−0.000424331 0
$59$	−17359.9	−0.649257	−0.324628	−	0.945842i	$-0.605239\pi$
$59$	−17359.9	−0.649257	−0.324628	+	0.945842i	$0.605239\pi$
$60$	0	0
$61$	41529.3	1.42899	0.714497	−	0.699638i	$-0.246656\pi$
$61$	41529.3	1.42899	0.714497	+	0.699638i	$0.246656\pi$
$62$	−33.3808	−0.00110285
$63$	−17466.3	−0.554434
$64$	−32767.9	−0.999998
$65$	0	0
$66$	−5.37829	−0.000151979 0
$67$	−8888.44	−0.241902	−0.120951	−	0.992659i	$-0.538594\pi$
$67$	−8888.44	−0.241902	−0.120951	+	0.992659i	$0.538594\pi$
$68$	−46996.2	−1.23251
$69$	−6612.93	−0.167213
$70$	0	0
$71$	11586.9	0.272786	0.136393	−	0.990655i	$-0.456449\pi$
$71$	11586.9	0.272786	0.136393	+	0.990655i	$0.456449\pi$
$72$	−50.5831	−0.00114994
$73$	−75620.5	−1.66086	−0.830428	−	0.557126i	$-0.811904\pi$
$73$	−75620.5	−1.66086	−0.830428	+	0.557126i	$0.811904\pi$
$74$	30.5387	0.000648294 0
$75$	0	0
$76$	−33805.7	−0.671361
$77$	26853.4	0.516146
$78$	−16.5359	−0.000307745 0
$79$	62221.3	1.12169	0.560843	−	0.827922i	$-0.310477\pi$
$79$	62221.3	1.12169	0.560843	+	0.827922i	$0.310477\pi$
$80$	0	0
$81$	45151.5	0.764644
$82$	63.7550	0.00104708
$83$	104528.	1.66547	0.832736	−	0.553671i	$-0.186773\pi$
$83$	104528.	1.66547	0.832736	+	0.553671i	$0.186773\pi$
$84$	−11073.1	−0.171227
$85$	0	0
$86$	6.54125	9.53706e−5 0
$87$	13601.0	0.192652
$88$	77.7685	0.00107053
$89$	16370.8	0.219076	0.109538	−	0.993983i	$-0.465063\pi$
$89$	16370.8	0.219076	0.109538	+	0.993983i	$0.465063\pi$
$90$	0	0
$91$	82562.4	1.04515
$92$	47810.5	0.588917
$93$	41763.2	0.500710
$94$	−44.8363	−0.000523371 0
$95$	0	0
$96$	−48.1022	−0.000532705 0
$97$	−14099.6	−0.152152	−0.0760759	−	0.997102i	$-0.524239\pi$
$97$	−14099.6	−0.152152	−0.0760759	+	0.997102i	$0.524239\pi$
$98$	37.8352	0.000397952 0
$99$	−76736.5	−0.786890
$100$	0	0
$101$	73684.5	0.718742	0.359371	−	0.933195i	$-0.382991\pi$
$101$	73684.5	0.718742	0.359371	+	0.933195i	$0.382991\pi$
$102$	−22.9962	−0.000218855 0
$103$	28407.3	0.263838	0.131919	−	0.991260i	$-0.457886\pi$
$103$	28407.3	0.263838	0.131919	+	0.991260i	$0.457886\pi$
$104$	239.104	0.00216772
$105$	0	0
$106$	30.8101	0.000266335 0
$107$	−38692.0	−0.326710	−0.163355	−	0.986567i	$-0.552232\pi$
$107$	−38692.0	−0.326710	−0.163355	+	0.986567i	$0.552232\pi$
$108$	66059.9	0.544977
$109$	75524.3	0.608864	0.304432	−	0.952534i	$-0.401533\pi$
$109$	75524.3	0.608864	0.304432	+	0.952534i	$0.401533\pi$
$110$	0	0
$111$	−38207.4	−0.294334
$112$	80056.9	0.603051
$113$	−105450.	−0.776875	−0.388438	−	0.921475i	$-0.626985\pi$
$113$	−105450.	−0.776875	−0.388438	+	0.921475i	$0.626985\pi$
$114$	−16.5419	−0.000119213 0
$115$	0	0
$116$	−98333.4	−0.678510
$117$	−235931.	−1.59338
$118$	61.4144	0.000406037 0
$119$	114819.	0.743267
$120$	0	0
$121$	−43073.3	−0.267451
$122$	−146.919	−0.000893674 0
$123$	−79764.7	−0.475388
$124$	−301942.	−1.76347
$125$	0	0
$126$	61.7910	0.000346736 0
$127$	−151292.	−0.832352	−0.416176	−	0.909284i	$-0.636630\pi$
$127$	−151292.	−0.832352	−0.416176	+	0.909284i	$0.636630\pi$
$128$	463.696	0.00250154
$129$	−8183.84	−0.0432995
$130$	0	0
$131$	160063.	0.814914	0.407457	−	0.913224i	$-0.366416\pi$
$131$	160063.	0.814914	0.407457	+	0.913224i	$0.366416\pi$
$132$	−48648.6	−0.243016
$133$	82592.4	0.404866
$134$	31.4448	0.000151282 0
$135$	0	0
$136$	332.519	0.00154159
$137$	243141.	1.10677	0.553384	−	0.832926i	$-0.313336\pi$
$137$	243141.	1.10677	0.553384	+	0.832926i	$0.313336\pi$
$138$	23.3947	0.000104573 0
$139$	213863.	0.938857	0.469429	−	0.882970i	$-0.344460\pi$
$139$	213863.	0.938857	0.469429	+	0.882970i	$0.344460\pi$
$140$	0	0
$141$	56095.3	0.237618
$142$	−40.9913	−0.000170597 0
$143$	362729.	1.48335
$144$	−228771.	−0.919380
$145$	0	0
$146$	267.524	0.00103868
$147$	−47336.1	−0.180675
$148$	276234.	1.03663
$149$	−121710.	−0.449117	−0.224558	−	0.974461i	$-0.572094\pi$
$149$	−121710.	−0.449117	−0.224558	+	0.974461i	$0.572094\pi$
$150$	0	0
$151$	−464982.	−1.65956	−0.829782	−	0.558088i	$-0.811535\pi$
$151$	−464982.	−1.65956	−0.829782	+	0.558088i	$0.811535\pi$
$152$	239.191	0.000839721 0
$153$	−328106.	−1.13315
$154$	−94.9999	−0.000322791 0
$155$	0	0
$156$	−149573.	−0.492087
$157$	−239887.	−0.776707	−0.388354	−	0.921510i	$-0.626956\pi$
$157$	−239887.	−0.776707	−0.388354	+	0.921510i	$0.626956\pi$
$158$	−220.122	−0.000701488 0
$159$	−38546.9	−0.120920
$160$	0	0
$161$	−116808.	−0.355147
$162$	−159.733	−0.000478198 0
$163$	509936.	1.50330	0.751651	−	0.659561i	$-0.229258\pi$
$163$	509936.	1.50330	0.751651	+	0.659561i	$0.229258\pi$
$164$	576687.	1.67429
$165$	0	0
$166$	−369.791	−0.00104156
$167$	567512.	1.57465	0.787325	−	0.616538i	$-0.211465\pi$
$167$	567512.	1.57465	0.787325	+	0.616538i	$0.211465\pi$
$168$	78.3472	0.000214166 0
$169$	743940.	2.00365
$170$	0	0
$171$	−236017.	−0.617238
$172$	59168.0	0.152499
$173$	192003.	0.487746	0.243873	−	0.969807i	$-0.421582\pi$
$173$	192003.	0.487746	0.243873	+	0.969807i	$0.421582\pi$
$174$	−48.1166	−0.000120482 0
$175$	0	0
$176$	351722.	0.855889
$177$	−76836.4	−0.184346
$178$	−57.9154	−0.000137007 0
$179$	181703.	0.423867	0.211933	−	0.977284i	$-0.432024\pi$
$179$	181703.	0.423867	0.211933	+	0.977284i	$0.432024\pi$
$180$	0	0
$181$	160168.	0.363395	0.181698	−	0.983354i	$-0.441841\pi$
$181$	160168.	0.363395	0.181698	+	0.983354i	$0.441841\pi$
$182$	−292.083	−0.000653623 0
$183$	183813.	0.405740
$184$	−338.281	−0.000736601 0
$185$	0	0
$186$	−147.747	−0.000313138 0
$187$	504443.	1.05489
$188$	−405561.	−0.836876
$189$	−161394.	−0.328650
$190$	0	0
$191$	−795626.	−1.57807	−0.789034	−	0.614349i	$-0.789419\pi$
$191$	−795626.	−1.57807	−0.789034	+	0.614349i	$0.789419\pi$
$192$	−145034.	−0.283933
$193$	142218.	0.274828	0.137414	−	0.990514i	$-0.456121\pi$
$193$	142218.	0.274828	0.137414	+	0.990514i	$0.456121\pi$
$194$	49.8804	9.51537e−5 0
$195$	0	0
$196$	342233.	0.636329
$197$	−613720.	−1.12669	−0.563345	−	0.826222i	$-0.690486\pi$
$197$	−613720.	−1.12669	−0.563345	+	0.826222i	$0.690486\pi$
$198$	271.472	0.000492111 0
$199$	95621.6	0.171168	0.0855841	−	0.996331i	$-0.472724\pi$
$199$	95621.6	0.171168	0.0855841	+	0.996331i	$0.472724\pi$
$200$	0	0
$201$	−39341.1	−0.0686841
$202$	−260.675	−0.000449492 0
$203$	240243.	0.409176
$204$	−208009.	−0.349951
$205$	0	0
$206$	−100.497	−0.000165001 0
$207$	333792.	0.541439
$208$	1.08139e6	1.73310
$209$	362861.	0.574612
$210$	0	0
$211$	−1.09309e6	−1.69024	−0.845120	−	0.534576i	$-0.820471\pi$
$211$	−1.09309e6	−1.69024	−0.845120	+	0.534576i	$0.820471\pi$
$212$	278688.	0.425872
$213$	51284.8	0.0774533
$214$	136.882	0.000204320 0
$215$	0	0
$216$	−467.403	−0.000681643 0
$217$	737688.	1.06347
$218$	−267.184	−0.000380776 0
$219$	−334703.	−0.471574
$220$	0	0
$221$	1.55094e6	2.13607
$222$	135.167	0.000184073 0
$223$	−69717.1	−0.0938808	−0.0469404	−	0.998898i	$-0.514947\pi$
$223$	−69717.1	−0.0938808	−0.0469404	+	0.998898i	$0.514947\pi$
$224$	−849.658	−0.00113142
$225$	0	0
$226$	373.054	0.000485848 0
$227$	−72397.0	−0.0932515	−0.0466258	−	0.998912i	$-0.514847\pi$
$227$	−72397.0	−0.0932515	−0.0466258	+	0.998912i	$0.514847\pi$
$228$	−149627.	−0.190622
$229$	−773691.	−0.974942	−0.487471	−	0.873139i	$-0.662081\pi$
$229$	−773691.	−0.974942	−0.487471	+	0.873139i	$0.662081\pi$
$230$	0	0
$231$	118856.	0.146551
$232$	695.752	0.000848662 0
$233$	1.17517e6	1.41812	0.709060	−	0.705149i	$-0.249120\pi$
$233$	1.17517e6	1.41812	0.709060	+	0.705149i	$0.249120\pi$
$234$	834.658	0.000996481 0
$235$	0	0
$236$	555516.	0.649257
$237$	275397.	0.318485
$238$	−406.196	−0.000464829 0
$239$	1.59060e6	1.80122	0.900608	−	0.434633i	$-0.143122\pi$
$239$	1.59060e6	1.80122	0.900608	+	0.434633i	$0.143122\pi$
$240$	0	0
$241$	863760.	0.957967	0.478984	−	0.877824i	$-0.341005\pi$
$241$	863760.	0.957967	0.478984	+	0.877824i	$0.341005\pi$
$242$	152.381	0.000167261 0
$243$	701487.	0.762086
$244$	−1.32894e6	−1.42899
$245$	0	0
$246$	282.185	0.000297301 0
$247$	1.11564e6	1.16354
$248$	2136.37	0.00220571
$249$	462650.	0.472884
$250$	0	0
$251$	−373603.	−0.374306	−0.187153	−	0.982331i	$-0.559926\pi$
$251$	−373603.	−0.374306	−0.187153	+	0.982331i	$0.559926\pi$
$252$	558922.	0.554434
$253$	−513184.	−0.504048
$254$	535.229	0.000520542 0
$255$	0	0
$256$	1.04857e6	0.999996
$257$	1.56964e6	1.48241	0.741204	−	0.671280i	$-0.234255\pi$
$257$	1.56964e6	1.48241	0.741204	+	0.671280i	$0.234255\pi$
$258$	28.9522	2.70789e−5 0
$259$	−674881.	−0.625140
$260$	0	0
$261$	−686520.	−0.623809
$262$	−566.257	−0.000509636 0
$263$	−121397.	−0.108222	−0.0541112	−	0.998535i	$-0.517233\pi$
$263$	−121397.	−0.108222	−0.0541112	+	0.998535i	$0.517233\pi$
$264$	344.210	0.000303958 0
$265$	0	0
$266$	−292.189	−0.000253198 0
$267$	72458.8	0.0622032
$268$	284430.	0.241901
$269$	−1.32256e6	−1.11438	−0.557192	−	0.830383i	$-0.688121\pi$
$269$	−1.32256e6	−1.11438	−0.557192	+	0.830383i	$0.688121\pi$
$270$	0	0
$271$	−681839.	−0.563974	−0.281987	−	0.959418i	$-0.590993\pi$
$271$	−681839.	−0.563974	−0.281987	+	0.959418i	$0.590993\pi$
$272$	1.50388e6	1.23251
$273$	365429.	0.296754
$274$	−860.166	−0.000692159 0
$275$	0	0
$276$	211614.	0.167213
$277$	−1.34688e6	−1.05470	−0.527351	−	0.849647i	$-0.676815\pi$
$277$	−1.34688e6	−1.05470	−0.527351	+	0.849647i	$0.676815\pi$
$278$	−756.589	−0.000587149 0
$279$	−2.10802e6	−1.62131
$280$	0	0
$281$	−426288.	−0.322060	−0.161030	−	0.986949i	$-0.551482\pi$
$281$	−426288.	−0.322060	−0.161030	+	0.986949i	$0.551482\pi$
$282$	−198.450	−0.000148603 0
$283$	891682.	0.661826	0.330913	−	0.943661i	$-0.392643\pi$
$283$	891682.	0.661826	0.330913	+	0.943661i	$0.392643\pi$
$284$	−370782.	−0.272786
$285$	0	0
$286$	−1283.24	−0.000927666 0
$287$	−1.40893e6	−1.00968
$288$	2427.99	0.00172491
$289$	737019.	0.519080
$290$	0	0
$291$	−62406.1	−0.0432011
$292$	2.41985e6	1.66086
$293$	1.65642e6	1.12720	0.563601	−	0.826047i	$-0.309416\pi$
$293$	1.65642e6	1.12720	0.563601	+	0.826047i	$0.309416\pi$
$294$	167.462	0.000112992 0
$295$	0	0
$296$	−1954.48	−0.00129659
$297$	−709068.	−0.466441
$298$	430.575	0.000280872 0
$299$	−1.57782e6	−1.02065
$300$	0	0
$301$	−144556.	−0.0919645
$302$	1644.98	0.00103787
$303$	326135.	0.204075
$304$	1.08178e6	0.671361
$305$	0	0
$306$	1160.75	0.000708655 0
$307$	413794.	0.250575	0.125288	−	0.992120i	$-0.460015\pi$
$307$	413794.	0.250575	0.125288	+	0.992120i	$0.460015\pi$
$308$	−859309.	−0.516146
$309$	125734.	0.0749126
$310$	0	0
$311$	1.24054e6	0.727296	0.363648	−	0.931536i	$-0.381531\pi$
$311$	1.24054e6	0.727296	0.363648	+	0.931536i	$0.381531\pi$
$312$	1058.30	0.000615489 0
$313$	1.63545e6	0.943575	0.471787	−	0.881712i	$-0.343609\pi$
$313$	1.63545e6	0.943575	0.471787	+	0.881712i	$0.343609\pi$
$314$	848.653	0.000485742 0
$315$	0	0
$316$	−1.99108e6	−1.12169
$317$	−2.54854e6	−1.42444	−0.712219	−	0.701957i	$-0.752310\pi$
$317$	−2.54854e6	−1.42444	−0.712219	+	0.701957i	$0.752310\pi$
$318$	136.368	7.56216e−5 0
$319$	1.05548e6	0.580730
$320$	0	0
$321$	−171254.	−0.0927639
$322$	413.235	0.000222104 0
$323$	1.55150e6	0.827460
$324$	−1.44485e6	−0.764644
$325$	0	0
$326$	−1804.01	−0.000940145 0
$327$	334278.	0.172877
$328$	−4080.32	−0.00209416
$329$	990844.	0.504680
$330$	0	0
$331$	−466025.	−0.233797	−0.116899	−	0.993144i	$-0.537295\pi$
$331$	−466025.	−0.233797	−0.116899	+	0.993144i	$0.537295\pi$
$332$	−3.34489e6	−1.66547
$333$	1.92854e6	0.953057
$334$	−2007.70	−0.000984766 0
$335$	0	0
$336$	354339.	0.171227
$337$	−1.93053e6	−0.925980	−0.462990	−	0.886364i	$-0.653223\pi$
$337$	−1.93053e6	−0.925980	−0.462990	+	0.886364i	$0.653223\pi$
$338$	−2631.85	−0.00125306
$339$	−466732.	−0.220581
$340$	0	0
$341$	3.24096e6	1.50934
$342$	834.961	0.000386012 0
$343$	−2.15011e6	−0.986791
$344$	−418.640	−0.000190741 0
$345$	0	0
$346$	−679.254	−0.000305030 0
$347$	1.86530e6	0.831622	0.415811	−	0.909451i	$-0.363498\pi$
$347$	1.86530e6	0.831622	0.415811	+	0.909451i	$0.363498\pi$
$348$	−435233.	−0.192652
$349$	2.46427e6	1.08299	0.541496	−	0.840703i	$-0.317858\pi$
$349$	2.46427e6	1.08299	0.541496	+	0.840703i	$0.317858\pi$
$350$	0	0
$351$	−2.18007e6	−0.944503
$352$	−3732.88	−0.00160579
$353$	162393.	0.0693635	0.0346818	−	0.999398i	$-0.488958\pi$
$353$	162393.	0.0693635	0.0346818	+	0.999398i	$0.488958\pi$
$354$	271.826	0.000115288 0
$355$	0	0
$356$	−523866.	−0.219076
$357$	508197.	0.211039
$358$	−642.814	−0.000265081 0
$359$	−4.55649e6	−1.86593	−0.932963	−	0.359972i	$-0.882786\pi$
$359$	−4.55649e6	−1.86593	−0.932963	+	0.359972i	$0.882786\pi$
$360$	0	0
$361$	−1.36006e6	−0.549273
$362$	−566.630	−0.000227263 0
$363$	−190647.	−0.0759386
$364$	−2.64200e6	−1.04515
$365$	0	0
$366$	−650.278	−0.000253744 0
$367$	809982.	0.313914	0.156957	−	0.987605i	$-0.449832\pi$
$367$	809982.	0.313914	0.156957	+	0.987605i	$0.449832\pi$
$368$	−1.52994e6	−0.588916
$369$	4.02617e6	1.53931
$370$	0	0
$371$	−680877.	−0.256823
$372$	−1.33642e6	−0.500710
$373$	1.09407e6	0.407169	0.203585	−	0.979057i	$-0.434741\pi$
$373$	1.09407e6	0.407169	0.203585	+	0.979057i	$0.434741\pi$
$374$	−1784.58	−0.000659716 0
$375$	0	0
$376$	2869.52	0.00104674
$377$	3.24514e6	1.17593
$378$	570.967	0.000205533 0
$379$	2.79365e6	0.999021	0.499510	−	0.866308i	$-0.333513\pi$
$379$	2.79365e6	0.999021	0.499510	+	0.866308i	$0.333513\pi$
$380$	0	0
$381$	−669633.	−0.236333
$382$	2814.71	0.000986903 0
$383$	−1.08096e6	−0.376540	−0.188270	−	0.982117i	$-0.560288\pi$
$383$	−1.08096e6	−0.376540	−0.188270	+	0.982117i	$0.560288\pi$
$384$	2052.36	0.000710274 0
$385$	0	0
$386$	−503.127	−0.000171874 0
$387$	413085.	0.140204
$388$	451187.	0.152152
$389$	5.14305e6	1.72324	0.861621	−	0.507552i	$-0.169449\pi$
$389$	5.14305e6	1.72324	0.861621	+	0.507552i	$0.169449\pi$
$390$	0	0
$391$	−2.19425e6	−0.725846
$392$	−2421.45	−0.000795903 0
$393$	708452.	0.231382
$394$	2171.17	0.000704617 0
$395$	0	0
$396$	2.45557e6	0.786890
$397$	−1.54920e6	−0.493322	−0.246661	−	0.969102i	$-0.579333\pi$
$397$	−1.54920e6	−0.493322	−0.246661	+	0.969102i	$0.579333\pi$
$398$	−338.283	−0.000107046 0
$399$	365562.	0.114955
$400$	0	0
$401$	3.99982e6	1.24217	0.621083	−	0.783744i	$-0.286693\pi$
$401$	3.99982e6	1.24217	0.621083	+	0.783744i	$0.286693\pi$
$402$	139.178	4.29541e−5 0
$403$	9.96452e6	3.05628
$404$	−2.35790e6	−0.718742
$405$	0	0
$406$	−849.913	−0.000255894 0
$407$	−2.96502e6	−0.887241
$408$	1471.76	0.000437710 0
$409$	−6.41702e6	−1.89682	−0.948408	−	0.317053i	$-0.897307\pi$
$409$	−6.41702e6	−1.89682	−0.948408	+	0.317053i	$0.897307\pi$
$410$	0	0
$411$	1.07616e6	0.314249
$412$	−909035.	−0.263838
$413$	−1.35721e6	−0.391535
$414$	−1180.86	−0.000338609 0
$415$	0	0
$416$	−11477.0	−0.00325158
$417$	946579.	0.266573
$418$	−1283.70	−0.000359355 0
$419$	3.66292e6	1.01928	0.509638	−	0.860389i	$-0.329779\pi$
$419$	3.66292e6	1.01928	0.509638	+	0.860389i	$0.329779\pi$
$420$	0	0
$421$	2.49763e6	0.686787	0.343394	−	0.939192i	$-0.388424\pi$
$421$	2.49763e6	0.686787	0.343394	+	0.939192i	$0.388424\pi$
$422$	3867.04	0.00105705
$423$	−2.83144e6	−0.769409
$424$	−1971.85	−0.000532670 0
$425$	0	0
$426$	−181.431	−4.84383e−5 0
$427$	3.24679e6	0.861757
$428$	1.23814e6	0.326709
$429$	1.60547e6	0.421173
$430$	0	0
$431$	2.03126e6	0.526711	0.263356	−	0.964699i	$-0.415171\pi$
$431$	2.03126e6	0.526711	0.263356	+	0.964699i	$0.415171\pi$
$432$	−2.11391e6	−0.544977
$433$	5.93651e6	1.52164	0.760819	−	0.648964i	$-0.224798\pi$
$433$	5.93651e6	1.52164	0.760819	+	0.648964i	$0.224798\pi$
$434$	−2609.74	−0.000665078 0
$435$	0	0
$436$	−2.41678e6	−0.608864
$437$	−1.57839e6	−0.395376
$438$	1184.09	0.000294916 0
$439$	6.21806e6	1.53990	0.769952	−	0.638102i	$-0.220280\pi$
$439$	6.21806e6	1.53990	0.769952	+	0.638102i	$0.220280\pi$
$440$	0	0
$441$	2.38932e6	0.585029
$442$	−5486.80	−0.00133587
$443$	2.82973e6	0.685071	0.342536	−	0.939505i	$-0.388714\pi$
$443$	2.82973e6	0.685071	0.342536	+	0.939505i	$0.388714\pi$
$444$	1.22264e6	0.294334
$445$	0	0
$446$	246.640	5.87118e−5 0
$447$	−538698.	−0.127519
$448$	−2.56182e6	−0.603050
$449$	2.43580e6	0.570197	0.285099	−	0.958498i	$-0.407974\pi$
$449$	2.43580e6	0.570197	0.285099	+	0.958498i	$0.407974\pi$
$450$	0	0
$451$	−6.18999e6	−1.43301
$452$	3.37441e6	0.776875
$453$	−2.05805e6	−0.471207
$454$	256.121	5.83183e−5 0
$455$	0	0
$456$	1058.68	0.000238425 0
$457$	3.89311e6	0.871979	0.435990	−	0.899952i	$-0.356398\pi$
$457$	3.89311e6	0.871979	0.435990	+	0.899952i	$0.356398\pi$
$458$	2737.10	0.000609716 0
$459$	−3.03180e6	−0.671690
$460$	0	0
$461$	−1.41045e6	−0.309105	−0.154553	−	0.987985i	$-0.549394\pi$
$461$	−1.41045e6	−0.309105	−0.154553	+	0.987985i	$0.549394\pi$
$462$	−420.478	−9.16513e−5 0
$463$	7.46201e6	1.61772	0.808860	−	0.588001i	$-0.200085\pi$
$463$	7.46201e6	1.61772	0.808860	+	0.588001i	$0.200085\pi$
$464$	3.14667e6	0.678509
$465$	0	0
$466$	−4157.44	−0.000886873 0
$467$	226020.	0.0479574	0.0239787	−	0.999712i	$-0.492367\pi$
$467$	226020.	0.0479574	0.0239787	+	0.999712i	$0.492367\pi$
$468$	7.54979e6	1.59338
$469$	−694905.	−0.145879
$470$	0	0
$471$	−1.06176e6	−0.220534
$472$	−3930.52	−0.000812073 0
$473$	−635092.	−0.130522
$474$	−974.279	−0.000199176 0
$475$	0	0
$476$	−3.67419e6	−0.743266
$477$	1.94568e6	0.391539
$478$	−5627.09	−0.00112646
$479$	3.60908e6	0.718717	0.359358	−	0.933200i	$-0.382996\pi$
$479$	3.60908e6	0.718717	0.359358	+	0.933200i	$0.382996\pi$
$480$	0	0
$481$	−9.11613e6	−1.79658
$482$	−3055.74	−0.000599100 0
$483$	−517004.	−0.100838
$484$	1.37835e6	0.267451
$485$	0	0
$486$	−2481.67	−0.000476598 0
$487$	−3.25236e6	−0.621407	−0.310704	−	0.950507i	$-0.600565\pi$
$487$	−3.25236e6	−0.621407	−0.310704	+	0.950507i	$0.600565\pi$
$488$	9402.83	0.00178735
$489$	2.25702e6	0.426839
$490$	0	0
$491$	−3.44360e6	−0.644628	−0.322314	−	0.946633i	$-0.604461\pi$
$491$	−3.44360e6	−0.644628	−0.322314	+	0.946633i	$0.604461\pi$
$492$	2.55247e6	0.475387
$493$	4.51298e6	0.836270
$494$	−3946.82	−0.000727662 0
$495$	0	0
$496$	9.66213e6	1.76347
$497$	905874.	0.164504
$498$	−1636.73	−0.000295735 0
$499$	5.93248e6	1.06656	0.533279	−	0.845939i	$-0.320959\pi$
$499$	5.93248e6	1.06656	0.533279	+	0.845939i	$0.320959\pi$
$500$	0	0
$501$	2.51186e6	0.447097
$502$	1321.70	0.000234086 0
$503$	8.79043e6	1.54914	0.774569	−	0.632489i	$-0.217967\pi$
$503$	8.79043e6	1.54914	0.774569	+	0.632489i	$0.217967\pi$
$504$	−3954.62	−0.000693472 0
$505$	0	0
$506$	1815.50	0.000315225 0
$507$	3.29275e6	0.568904
$508$	4.84134e6	0.832351
$509$	−9.23965e6	−1.58074	−0.790371	−	0.612629i	$-0.790112\pi$
$509$	−9.23965e6	−1.58074	−0.790371	+	0.612629i	$0.790112\pi$
$510$	0	0
$511$	−5.91206e6	−1.00158
$512$	−18547.8	−0.00312693
$513$	−2.18086e6	−0.365877
$514$	−5552.95	−0.000927078 0
$515$	0	0
$516$	261883.	0.0432995
$517$	4.35317e6	0.716275
$518$	2387.54	0.000390955 0
$519$	849824.	0.138488
$520$	0	0
$521$	−1.38904e6	−0.224192	−0.112096	−	0.993697i	$-0.535756\pi$
$521$	−1.38904e6	−0.224192	−0.112096	+	0.993697i	$0.535756\pi$
$522$	2428.72	0.000390122 0
$523$	−8.64626e6	−1.38221	−0.691105	−	0.722755i	$-0.742876\pi$
$523$	−8.64626e6	−1.38221	−0.691105	+	0.722755i	$0.742876\pi$
$524$	−5.12200e6	−0.814913
$525$	0	0
$526$	429.468	6.76809e−5 0
$527$	1.38575e7	2.17350
$528$	1.55675e6	0.243016
$529$	−4.20407e6	−0.653177
$530$	0	0
$531$	3.87836e6	0.596915
$532$	−2.64296e6	−0.404865
$533$	−1.90315e7	−2.90172
$534$	−256.339	−3.89011e−5 0
$535$	0	0
$536$	−2012.47	−0.000302564 0
$537$	804234.	0.120350
$538$	4678.85	0.000696921 0
$539$	−3.67343e6	−0.544628
$540$	0	0
$541$	4.59194e6	0.674533	0.337267	−	0.941409i	$-0.390498\pi$
$541$	4.59194e6	0.674533	0.337267	+	0.941409i	$0.390498\pi$
$542$	2412.16	0.000352702 0
$543$	708918.	0.103180
$544$	−15960.9	−0.00231238
$545$	0	0
$546$	−1292.79	−0.000185586 0
$547$	2.30374e6	0.329204	0.164602	−	0.986360i	$-0.447366\pi$
$547$	2.30374e6	0.329204	0.164602	+	0.986360i	$0.447366\pi$
$548$	−7.78051e6	−1.10677
$549$	−9.27806e6	−1.31379
$550$	0	0
$551$	3.24632e6	0.455526
$552$	−1497.26	−0.000209146 0
$553$	4.86450e6	0.676434
$554$	4764.89	0.000659597 0
$555$	0	0
$556$	−6.84363e6	−0.938857
$557$	−8.88996e6	−1.21412	−0.607061	−	0.794656i	$-0.707651\pi$
$557$	−8.88996e6	−1.21412	−0.607061	+	0.794656i	$0.707651\pi$
$558$	7457.60	0.00101394
$559$	−1.95263e6	−0.264296
$560$	0	0
$561$	2.23271e6	0.299520
$562$	1508.09	0.000201412 0
$563$	−67473.0	−0.00897138	−0.00448569	−	0.999990i	$-0.501428\pi$
$563$	−67473.0	−0.00897138	−0.00448569	+	0.999990i	$0.501428\pi$
$564$	−1.79505e6	−0.237618
$565$	0	0
$566$	−3154.52	−0.000413897 0
$567$	3.52997e6	0.461120
$568$	2623.44	0.000341194 0
$569$	−1.40102e7	−1.81411	−0.907057	−	0.421007i	$-0.861677\pi$
$569$	−1.40102e7	−1.81411	−0.907057	+	0.421007i	$0.861677\pi$
$570$	0	0
$571$	7.48697e6	0.960983	0.480492	−	0.876999i	$-0.340458\pi$
$571$	7.48697e6	0.960983	0.480492	+	0.876999i	$0.340458\pi$
$572$	−1.16073e7	−1.48335
$573$	−3.52152e6	−0.448067
$574$	4984.41	0.000631442 0
$575$	0	0
$576$	7.32067e6	0.919379
$577$	−8.22787e6	−1.02884	−0.514420	−	0.857538i	$-0.671993\pi$
$577$	−8.22787e6	−1.02884	−0.514420	+	0.857538i	$0.671993\pi$
$578$	−2607.37	−0.000324626 0
$579$	629469.	0.0780329
$580$	0	0
$581$	8.17207e6	1.00436
$582$	220.775	2.70174e−5 0
$583$	−2.99136e6	−0.364500
$584$	−17121.5	−0.00207736
$585$	0	0
$586$	−5859.96	−0.000704937 0
$587$	3.61821e6	0.433409	0.216705	−	0.976237i	$-0.430469\pi$
$587$	3.61821e6	0.433409	0.216705	+	0.976237i	$0.430469\pi$
$588$	1.51475e6	0.180675
$589$	9.96814e6	1.18393
$590$	0	0
$591$	−2.71638e6	−0.319906
$592$	−8.83949e6	−1.03663
$593$	−7.99199e6	−0.933293	−0.466646	−	0.884444i	$-0.654538\pi$
$593$	−7.99199e6	−0.933293	−0.466646	+	0.884444i	$0.654538\pi$
$594$	2508.49	0.000291706 0
$595$	0	0
$596$	3.89471e6	0.449116
$597$	423230.	0.0486005
$598$	5581.88	0.000638303 0
$599$	−6.36161e6	−0.724436	−0.362218	−	0.932093i	$-0.617980\pi$
$599$	−6.36161e6	−0.724436	−0.362218	+	0.932093i	$0.617980\pi$
$600$	0	0
$601$	2.73298e6	0.308638	0.154319	−	0.988021i	$-0.450682\pi$
$601$	2.73298e6	0.308638	0.154319	+	0.988021i	$0.450682\pi$
$602$	511.399	5.75134e−5 0
$603$	1.98576e6	0.222400
$604$	1.48794e7	1.65956
$605$	0	0
$606$	−1153.77	−0.000127626 0
$607$	1.59118e7	1.75286	0.876428	−	0.481532i	$-0.159920\pi$
$607$	1.59118e7	1.75286	0.876428	+	0.481532i	$0.159920\pi$
$608$	−11481.1	−0.00125958
$609$	1.06334e6	0.116179
$610$	0	0
$611$	1.33841e7	1.45039
$612$	1.04994e7	1.13315
$613$	−1.23223e7	−1.32447	−0.662234	−	0.749297i	$-0.730391\pi$
$613$	−1.23223e7	−1.32447	−0.662234	+	0.749297i	$0.730391\pi$
$614$	−1463.89	−0.000156706 0
$615$	0	0
$616$	6079.99	0.000645582 0
$617$	5.19439e6	0.549315	0.274658	−	0.961542i	$-0.411436\pi$
$617$	5.19439e6	0.549315	0.274658	+	0.961542i	$0.411436\pi$
$618$	−444.810	−4.68494e−5 0
$619$	1.15468e6	0.121125	0.0605624	−	0.998164i	$-0.480711\pi$
$619$	1.15468e6	0.121125	0.0605624	+	0.998164i	$0.480711\pi$
$620$	0	0
$621$	3.08434e6	0.320946
$622$	−4388.70	−0.000454841 0
$623$	1.27988e6	0.132114
$624$	4.78633e6	0.492086
$625$	0	0
$626$	−5785.76	−0.000590099 0
$627$	1.60606e6	0.163152
$628$	7.67638e6	0.776707
$629$	−1.26777e7	−1.27765
$630$	0	0
$631$	−7.68764e6	−0.768634	−0.384317	−	0.923201i	$-0.625563\pi$
$631$	−7.68764e6	−0.768634	−0.384317	+	0.923201i	$0.625563\pi$
$632$	14087.8	0.00140297
$633$	−4.83810e6	−0.479917
$634$	9016.03	0.000890825 0
$635$	0	0
$636$	1.23350e6	0.120920
$637$	−1.12942e7	−1.10282
$638$	−3734.00	−0.000363181 0
$639$	−2.58863e6	−0.250795
$640$	0	0
$641$	1.71146e6	0.164522	0.0822608	−	0.996611i	$-0.473786\pi$
$641$	1.71146e6	0.164522	0.0822608	+	0.996611i	$0.473786\pi$
$642$	605.850	5.80133e−5 0
$643$	356593.	0.0340131	0.0170065	−	0.999855i	$-0.494586\pi$
$643$	356593.	0.0340131	0.0170065	+	0.999855i	$0.494586\pi$
$644$	3.73786e6	0.355147
$645$	0	0
$646$	−5488.79	−0.000517482 0
$647$	−1.26524e7	−1.18826	−0.594132	−	0.804368i	$-0.702504\pi$
$647$	−1.26524e7	−1.18826	−0.594132	+	0.804368i	$0.702504\pi$
$648$	10222.9	0.000956396 0
$649$	−5.96275e6	−0.555693
$650$	0	0
$651$	3.26508e6	0.301954
$652$	−1.63179e7	−1.50330
$653$	−8.05675e6	−0.739396	−0.369698	−	0.929152i	$-0.620539\pi$
$653$	−8.05675e6	−0.739396	−0.369698	+	0.929152i	$0.620539\pi$
$654$	−1182.58	−0.000108115 0
$655$	0	0
$656$	−1.84540e7	−1.67429
$657$	1.68943e7	1.52696
$658$	−3505.33	−0.000315620 0
$659$	2.71544e6	0.243572	0.121786	−	0.992556i	$-0.461138\pi$
$659$	2.71544e6	0.243572	0.121786	+	0.992556i	$0.461138\pi$
$660$	0	0
$661$	−1.04028e7	−0.926076	−0.463038	−	0.886339i	$-0.653241\pi$
$661$	−1.04028e7	−0.926076	−0.463038	+	0.886339i	$0.653241\pi$
$662$	1648.67	0.000146214 0
$663$	6.86461e6	0.606502
$664$	23666.6	0.00208313
$665$	0	0
$666$	−6822.65	−0.000596029 0
$667$	−4.59118e6	−0.399586
$668$	−1.81604e7	−1.57465
$669$	−308574.	−0.0266560
$670$	0	0
$671$	1.42644e7	1.22306
$672$	−3760.66	−0.000321249 0
$673$	1.16759e7	0.993697	0.496849	−	0.867837i	$-0.334490\pi$
$673$	1.16759e7	0.993697	0.496849	+	0.867837i	$0.334490\pi$
$674$	6829.67	0.000579095 0
$675$	0	0
$676$	−2.38061e7	−2.00365
$677$	2.24509e7	1.88262	0.941309	−	0.337546i	$-0.109597\pi$
$677$	2.24509e7	1.88262	0.941309	+	0.337546i	$0.109597\pi$
$678$	1651.17	0.000137949 0
$679$	−1.10232e6	−0.0917553
$680$	0	0
$681$	−320436.	−0.0264773
$682$	−11465.6	−0.000943922 0
$683$	1.60650e7	1.31774	0.658871	−	0.752256i	$-0.271034\pi$
$683$	1.60650e7	1.31774	0.658871	+	0.752256i	$0.271034\pi$
$684$	7.55253e6	0.617237
$685$	0	0
$686$	7606.48	0.000617126 0
$687$	−3.42443e6	−0.276819
$688$	−1.89337e6	−0.152498
$689$	−9.19713e6	−0.738081
$690$	0	0
$691$	−901705.	−0.0718405	−0.0359202	−	0.999355i	$-0.511436\pi$
$691$	−901705.	−0.0718405	−0.0359202	+	0.999355i	$0.511436\pi$
$692$	−6.14410e6	−0.487745
$693$	−5.99931e6	−0.474535
$694$	−6598.92	−0.000520085 0
$695$	0	0
$696$	3079.46	0.000240964 0
$697$	−2.64669e7	−2.06358
$698$	−8717.92	−0.000677289 0
$699$	5.20143e6	0.402652
$700$	0	0
$701$	7.78394e6	0.598280	0.299140	−	0.954209i	$-0.403300\pi$
$701$	7.78394e6	0.598280	0.299140	+	0.954209i	$0.403300\pi$
$702$	7712.49	0.000590679 0
$703$	−9.11944e6	−0.695953
$704$	−1.12551e7	−0.855888
$705$	0	0
$706$	−574.502	−4.33790e−5 0
$707$	5.76071e6	0.433438
$708$	2.45876e6	0.184346
$709$	9.67167e6	0.722580	0.361290	−	0.932453i	$-0.382336\pi$
$709$	9.67167e6	0.722580	0.361290	+	0.932453i	$0.382336\pi$
$710$	0	0
$711$	−1.39008e7	−1.03126
$712$	3706.59	0.000274015 0
$713$	−1.40977e7	−1.03854
$714$	−1797.86	−0.000131981 0
$715$	0	0
$716$	−5.81449e6	−0.423866
$717$	7.04013e6	0.511426
$718$	16119.6	0.00116693
$719$	6.86720e6	0.495402	0.247701	−	0.968837i	$-0.420325\pi$
$719$	6.86720e6	0.495402	0.247701	+	0.968837i	$0.420325\pi$
$720$	0	0
$721$	2.22091e6	0.159108
$722$	4811.50	0.000343508 0
$723$	3.82308e6	0.271999
$724$	−5.12537e6	−0.363395
$725$	0	0
$726$	674.454	4.74910e−5 0
$727$	−6.13088e6	−0.430216	−0.215108	−	0.976590i	$-0.569010\pi$
$727$	−6.13088e6	−0.430216	−0.215108	+	0.976590i	$0.569010\pi$
$728$	18693.3	0.00130725
$729$	−7.86696e6	−0.548262
$730$	0	0
$731$	−2.71550e6	−0.187956
$732$	−5.88201e6	−0.405740
$733$	−2.82167e7	−1.93975	−0.969876	−	0.243597i	$-0.921672\pi$
$733$	−2.82167e7	−1.93975	−0.969876	+	0.243597i	$0.921672\pi$
$734$	−2865.49	−0.000196318 0
$735$	0	0
$736$	16237.5	0.00110490
$737$	−3.05299e6	−0.207041
$738$	−14243.5	−0.000962665 0
$739$	2.52195e7	1.69874	0.849368	−	0.527802i	$-0.176984\pi$
$739$	2.52195e7	1.69874	0.849368	+	0.527802i	$0.176984\pi$
$740$	0	0
$741$	4.93792e6	0.330368
$742$	2408.75	0.000160614 0
$743$	1.13603e6	0.0754949	0.0377475	−	0.999287i	$-0.487982\pi$
$743$	1.13603e6	0.0754949	0.0377475	+	0.999287i	$0.487982\pi$
$744$	9455.78	0.000626275 0
$745$	0	0
$746$	−3870.53	−0.000254638 0
$747$	−2.33526e7	−1.53120
$748$	−1.61422e7	−1.05489
$749$	−3.02497e6	−0.197023
$750$	0	0
$751$	1.35844e7	0.878905	0.439453	−	0.898266i	$-0.355172\pi$
$751$	1.35844e7	0.878905	0.439453	+	0.898266i	$0.355172\pi$
$752$	1.29779e7	0.836875
$753$	−1.65360e6	−0.106278
$754$	−11480.4	−0.000735410 0
$755$	0	0
$756$	5.16461e6	0.328649
$757$	2.47876e7	1.57215	0.786076	−	0.618130i	$-0.212109\pi$
$757$	2.47876e7	1.57215	0.786076	+	0.618130i	$0.212109\pi$
$758$	−9883.17	−0.000624774 0
$759$	−2.27140e6	−0.143116
$760$	0	0
$761$	−2.67533e7	−1.67462	−0.837308	−	0.546732i	$-0.815872\pi$
$761$	−2.67533e7	−1.67462	−0.837308	+	0.546732i	$0.815872\pi$
$762$	2368.97	0.000147799 0
$763$	5.90454e6	0.367177
$764$	2.54600e7	1.57807
$765$	0	0
$766$	3824.12	0.000235483 0
$767$	−1.83328e7	−1.12523
$768$	4.64108e6	0.283933
$769$	−2.06950e7	−1.26197	−0.630986	−	0.775794i	$-0.717350\pi$
$769$	−2.06950e7	−1.26197	−0.630986	+	0.775794i	$0.717350\pi$
$770$	0	0
$771$	6.94738e6	0.420906
$772$	−4.55096e6	−0.274828
$773$	−1.54618e7	−0.930701	−0.465350	−	0.885127i	$-0.654072\pi$
$773$	−1.54618e7	−0.930701	−0.465350	+	0.885127i	$0.654072\pi$
$774$	−1461.38	−8.76820e−5 0
$775$	0	0
$776$	−3192.35	−0.000190307 0
$777$	−2.98708e6	−0.177499
$778$	−18194.7	−0.00107769
$779$	−1.90384e7	−1.12405
$780$	0	0
$781$	3.97986e6	0.233475
$782$	7762.65	0.000453934 0
$783$	−6.34365e6	−0.369773
$784$	−1.09514e7	−0.636328
$785$	0	0
$786$	−2506.31	−0.000144703 0
$787$	−1.53082e7	−0.881021	−0.440511	−	0.897747i	$-0.645203\pi$
$787$	−1.53082e7	−0.881021	−0.440511	+	0.897747i	$0.645203\pi$
$788$	1.96390e7	1.12669
$789$	−537313.	−0.0307280
$790$	0	0
$791$	−8.24417e6	−0.468496
$792$	−17374.2	−0.000984221 0
$793$	4.38569e7	2.47660
$794$	5480.63	0.000308517 0
$795$	0	0
$796$	−3.05989e6	−0.171168
$797$	9.44001e6	0.526413	0.263207	−	0.964740i	$-0.415220\pi$
$797$	9.44001e6	0.526413	0.263207	+	0.964740i	$0.415220\pi$
$798$	−1293.26	−7.18914e−5 0
$799$	1.86131e7	1.03146
$800$	0	0
$801$	−3.65740e6	−0.201415
$802$	−14150.3	−0.000776835 0
$803$	−2.59740e7	−1.42151
$804$	1.25891e6	0.0686841
$805$	0	0
$806$	−35251.7	−0.00191136
$807$	−5.85378e6	−0.316412
$808$	16683.2	0.000898983 0
$809$	6.62446e6	0.355860	0.177930	−	0.984043i	$-0.443060\pi$
$809$	6.62446e6	0.355860	0.177930	+	0.984043i	$0.443060\pi$
$810$	0	0
$811$	−1.51429e7	−0.808455	−0.404228	−	0.914658i	$-0.632460\pi$
$811$	−1.51429e7	−0.808455	−0.404228	+	0.914658i	$0.632460\pi$
$812$	−7.68777e6	−0.409176
$813$	−3.01788e6	−0.160131
$814$	10489.4	0.000554869 0
$815$	0	0
$816$	6.65629e6	0.349951
$817$	−1.95334e6	−0.102382
$818$	22701.6	0.00118624
$819$	−1.84452e7	−0.960892
$820$	0	0
$821$	219211.	0.0113502	0.00567512	−	0.999984i	$-0.498194\pi$
$821$	219211.	0.0113502	0.00567512	+	0.999984i	$0.498194\pi$
$822$	−3807.17	−0.000196527 0
$823$	−3.47564e7	−1.78869	−0.894346	−	0.447375i	$-0.852359\pi$
$823$	−3.47564e7	−1.78869	−0.894346	+	0.447375i	$0.852359\pi$
$824$	6431.83	0.000330002 0
$825$	0	0
$826$	4801.42	0.000244861 0
$827$	−3.26993e7	−1.66255	−0.831275	−	0.555861i	$-0.812389\pi$
$827$	−3.26993e7	−1.66255	−0.831275	+	0.555861i	$0.812389\pi$
$828$	−1.06813e7	−0.541439
$829$	3.49311e7	1.76533	0.882664	−	0.470004i	$-0.155748\pi$
$829$	3.49311e7	1.76533	0.882664	+	0.470004i	$0.155748\pi$
$830$	0	0
$831$	−5.96142e6	−0.299466
$832$	−3.46044e7	−1.73310
$833$	−1.57067e7	−0.784281
$834$	−3348.73	−0.000166712 0
$835$	0	0
$836$	−1.16116e7	−0.574612
$837$	−1.94788e7	−0.961054
$838$	−12958.4	−0.000637442 0
$839$	2.60569e7	1.27796	0.638982	−	0.769222i	$-0.279356\pi$
$839$	2.60569e7	1.27796	0.638982	+	0.769222i	$0.279356\pi$
$840$	0	0
$841$	−1.10683e7	−0.539624
$842$	−8835.91	−0.000429508 0
$843$	−1.88679e6	−0.0914438
$844$	3.49788e7	1.69024
$845$	0	0
$846$	10016.9	0.000481178 0
$847$	−3.36750e6	−0.161287
$848$	−8.91802e6	−0.425872
$849$	3.94667e6	0.187915
$850$	0	0
$851$	1.28974e7	0.610488
$852$	−1.64111e6	−0.0774532
$853$	−1.36792e7	−0.643709	−0.321854	−	0.946789i	$-0.604306\pi$
$853$	−1.36792e7	−0.643709	−0.321854	+	0.946789i	$0.604306\pi$
$854$	−11486.2	−0.000538932 0
$855$	0	0
$856$	−8760.42	−0.000408640 0
$857$	7.35615e6	0.342136	0.171068	−	0.985259i	$-0.445278\pi$
$857$	7.35615e6	0.342136	0.171068	+	0.985259i	$0.445278\pi$
$858$	−5679.72	−0.000263396 0
$859$	7.11286e6	0.328898	0.164449	−	0.986386i	$-0.447415\pi$
$859$	7.11286e6	0.328898	0.164449	+	0.986386i	$0.447415\pi$
$860$	0	0
$861$	−6.23606e6	−0.286683
$862$	−7186.03	−0.000329398 0
$863$	−2.93200e7	−1.34010	−0.670050	−	0.742316i	$-0.733727\pi$
$863$	−2.93200e7	−1.34010	−0.670050	+	0.742316i	$0.733727\pi$
$864$	22435.3	0.00102246
$865$	0	0
$866$	−21001.7	−0.000951612 0
$867$	3.26211e6	0.147384
$868$	−2.36060e7	−1.06347
$869$	2.13717e7	0.960040
$870$	0	0
$871$	−9.38661e6	−0.419241
$872$	17099.8	0.000761551 0
$873$	3.14998e6	0.139886
$874$	5583.90	0.000247263 0
$875$	0	0
$876$	1.07105e7	0.471573
$877$	−3.15641e7	−1.38578	−0.692890	−	0.721044i	$-0.743663\pi$
$877$	−3.15641e7	−1.38578	−0.692890	+	0.721044i	$0.743663\pi$
$878$	−21997.8	−0.000963036 0
$879$	7.33147e6	0.320051
$880$	0	0
$881$	2.85426e7	1.23895	0.619475	−	0.785017i	$-0.287346\pi$
$881$	2.85426e7	1.23895	0.619475	+	0.785017i	$0.287346\pi$
$882$	−8452.74	−0.000365869 0
$883$	−1.51806e7	−0.655222	−0.327611	−	0.944813i	$-0.606244\pi$
$883$	−1.51806e7	−0.655222	−0.327611	+	0.944813i	$0.606244\pi$
$884$	−4.96301e7	−2.13607
$885$	0	0
$886$	−10010.8	−0.000428435 0
$887$	1.32033e7	0.563474	0.281737	−	0.959492i	$-0.409089\pi$
$887$	1.32033e7	0.563474	0.281737	+	0.959492i	$0.409089\pi$
$888$	−8650.71	−0.000368145 0
$889$	−1.18281e7	−0.501951
$890$	0	0
$891$	1.55086e7	0.654451
$892$	2.23094e6	0.0938808
$893$	1.33890e7	0.561847
$894$	1905.76	7.97490e−5 0
$895$	0	0
$896$	36252.1	0.00150856
$897$	−6.98356e6	−0.289798
$898$	−8617.17	−0.000356594 0
$899$	2.89951e7	1.19654
$900$	0	0
$901$	−1.27903e7	−0.524892
$902$	21898.5	0.000896185 0
$903$	−639819.	−0.0261118
$904$	−23875.4	−0.000971695 0
$905$	0	0
$906$	7280.82	0.000294686 0
$907$	3.96521e7	1.60047	0.800236	−	0.599685i	$-0.204707\pi$
$907$	3.96521e7	1.60047	0.800236	+	0.599685i	$0.204707\pi$
$908$	2.31670e6	0.0932515
$909$	−1.64618e7	−0.660798
$910$	0	0
$911$	1.22945e6	0.0490810	0.0245405	−	0.999699i	$-0.492188\pi$
$911$	1.22945e6	0.0490810	0.0245405	+	0.999699i	$0.492188\pi$
$912$	4.78807e6	0.190622
$913$	3.59031e7	1.42546
$914$	−13772.7	−0.000545324 0
$915$	0	0
$916$	2.47581e7	0.974942
$917$	1.25138e7	0.491435
$918$	10725.7	0.000420066 0
$919$	−9.08804e6	−0.354961	−0.177481	−	0.984124i	$-0.556795\pi$
$919$	−9.08804e6	−0.354961	−0.177481	+	0.984124i	$0.556795\pi$
$920$	0	0
$921$	1.83149e6	0.0711468
$922$	4989.79	0.000193310 0
$923$	1.22363e7	0.472767
$924$	−3.80338e6	−0.146551
$925$	0	0
$926$	−26398.5	−0.00101170
$927$	−6.34648e6	−0.242568
$928$	−33396.1	−0.00127299
$929$	−4.99370e7	−1.89838	−0.949190	−	0.314704i	$-0.898095\pi$
$929$	−4.99370e7	−1.89838	−0.949190	+	0.314704i	$0.898095\pi$
$930$	0	0
$931$	−1.12983e7	−0.427207
$932$	−3.76056e7	−1.41812
$933$	5.49076e6	0.206504
$934$	−799.597	−2.99919e−5 0
$935$	0	0
$936$	−53418.1	−0.00199296
$937$	3.54310e7	1.31836	0.659180	−	0.751985i	$-0.270903\pi$
$937$	3.54310e7	1.31836	0.659180	+	0.751985i	$0.270903\pi$
$938$	2458.38	9.12309e−5 0
$939$	7.23865e6	0.267913
$940$	0	0
$941$	1.82105e7	0.670421	0.335211	−	0.942143i	$-0.391193\pi$
$941$	1.82105e7	0.670421	0.335211	+	0.942143i	$0.391193\pi$
$942$	3756.22	0.000137919 0
$943$	2.69255e7	0.986017
$944$	−1.77765e7	−0.649256
$945$	0	0
$946$	2246.78	8.16268e−5 0
$947$	2.99743e7	1.08611	0.543056	−	0.839697i	$-0.317267\pi$
$947$	2.99743e7	1.08611	0.543056	+	0.839697i	$0.317267\pi$
$948$	−8.81271e6	−0.318485
$949$	−7.98587e7	−2.87844
$950$	0	0
$951$	−1.12801e7	−0.404446
$952$	25996.5	0.000929658 0
$953$	−4.88225e7	−1.74136	−0.870679	−	0.491851i	$-0.836320\pi$
$953$	−4.88225e7	−1.74136	−0.870679	+	0.491851i	$0.836320\pi$
$954$	−6883.27	−0.000244863 0
$955$	0	0
$956$	−5.08991e7	−1.80121
$957$	4.67166e6	0.164889
$958$	−12767.9	−0.000449476 0
$959$	1.90089e7	0.667439
$960$	0	0
$961$	6.04030e7	2.10984
$962$	32250.3	0.00112356
$963$	8.64417e6	0.300371
$964$	−2.76403e7	−0.957967
$965$	0	0
$966$	1829.01	6.30630e−5 0
$967$	5.21957e7	1.79502	0.897508	−	0.440999i	$-0.145376\pi$
$967$	5.21957e7	1.79502	0.897508	+	0.440999i	$0.145376\pi$
$968$	−9752.41	−0.000334521 0
$969$	6.86710e6	0.234944
$970$	0	0
$971$	−2.63198e7	−0.895849	−0.447924	−	0.894071i	$-0.647837\pi$
$971$	−2.63198e7	−0.895849	−0.447924	+	0.894071i	$0.647837\pi$
$972$	−2.24476e7	−0.762086
$973$	1.67200e7	0.566179
$974$	11505.9	0.000388620 0
$975$	0	0
$976$	4.25260e7	1.42899
$977$	−3.95342e6	−0.132506	−0.0662532	−	0.997803i	$-0.521105\pi$
$977$	−3.95342e6	−0.132506	−0.0662532	+	0.997803i	$0.521105\pi$
$978$	−7984.72	−0.000266939 0
$979$	5.62303e6	0.187505
$980$	0	0
$981$	−1.68729e7	−0.559779
$982$	12182.5	0.000403142 0
$983$	4.78744e7	1.58023	0.790114	−	0.612960i	$-0.210021\pi$
$983$	4.78744e7	1.58023	0.790114	+	0.612960i	$0.210021\pi$
$984$	−18059.9	−0.000594602 0
$985$	0	0
$986$	−15965.7	−0.000522992 0
$987$	4.38557e6	0.143296
$988$	−3.57004e7	−1.16354
$989$	2.76255e6	0.0898090
$990$	0	0
$991$	−8.72624e6	−0.282256	−0.141128	−	0.989991i	$-0.545073\pi$
$991$	−8.72624e6	−0.282256	−0.141128	+	0.989991i	$0.545073\pi$
$992$	−102546.	−0.00330856
$993$	−2.06267e6	−0.0663830
$994$	−3204.73	−0.000102879 0
$995$	0	0
$996$	−1.48048e7	−0.472884
$997$	4.45976e6	0.142093	0.0710467	−	0.997473i	$-0.477366\pi$
$997$	4.45976e6	0.142093	0.0710467	+	0.997473i	$0.477366\pi$
$998$	−20987.5	−0.000667012 0
$999$	1.78203e7	0.564939

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1075.6.a.l.1.25		52
5.2	odd	4		215.6.b.a.44.52	✓	104
5.3	odd	4		215.6.b.a.44.53	yes	104
5.4	even	2		1075.6.a.k.1.28		52

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
215.6.b.a.44.52	✓	104	5.2	odd	4
215.6.b.a.44.53	yes	104	5.3	odd	4
1075.6.a.k.1.28		52	5.4	even	2
1075.6.a.l.1.25		52	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)

\(f(q)\)	\(=\)	\(q-0.00353772 q^{2} +4.42609 q^{3} -32.0000 q^{4} -0.0156583 q^{6} +78.1807 q^{7} +0.226414 q^{8} -223.410 q^{9} +O(q^{10})\)	\(q-0.00353772 q^{2} +4.42609 q^{3} -32.0000 q^{4} -0.0156583 q^{6} +78.1807 q^{7} +0.226414 q^{8} -223.410 q^{9} +343.479 q^{11} -141.635 q^{12} +1056.05 q^{13} -0.276582 q^{14} +1024.00 q^{16} +1468.63 q^{17} +0.790361 q^{18} +1056.43 q^{19} +346.035 q^{21} -1.21513 q^{22} -1494.08 q^{23} +1.00213 q^{24} -3.73600 q^{26} -2064.37 q^{27} -2501.78 q^{28} +3072.92 q^{29} +9435.69 q^{31} -10.8679 q^{32} +1520.27 q^{33} -5.19561 q^{34} +7149.11 q^{36} -8632.32 q^{37} -3.73735 q^{38} +4674.16 q^{39} -18021.5 q^{41} -1.22418 q^{42} -1849.00 q^{43} -10991.3 q^{44} +5.28564 q^{46} +12673.8 q^{47} +4532.31 q^{48} -10694.8 q^{49} +6500.29 q^{51} -33793.5 q^{52} -8709.02 q^{53} +7.30318 q^{54} +17.7012 q^{56} +4675.86 q^{57} -10.8711 q^{58} -17359.9 q^{59} +41529.3 q^{61} -33.3808 q^{62} -17466.3 q^{63} -32767.9 q^{64} -5.37829 q^{66} -8888.44 q^{67} -46996.2 q^{68} -6612.93 q^{69} +11586.9 q^{71} -50.5831 q^{72} -75620.5 q^{73} +30.5387 q^{74} -33805.7 q^{76} +26853.4 q^{77} -16.5359 q^{78} +62221.3 q^{79} +45151.5 q^{81} +63.7550 q^{82} +104528. q^{83} -11073.1 q^{84} +6.54125 q^{86} +13601.0 q^{87} +77.7685 q^{88} +16370.8 q^{89} +82562.4 q^{91} +47810.5 q^{92} +41763.2 q^{93} -44.8363 q^{94} -48.1022 q^{96} -14099.6 q^{97} +37.8352 q^{98} -76736.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q + 20 q^{2} + 54 q^{3} + 826 q^{4} - 162 q^{6} + 196 q^{7} + 960 q^{8} + 4098 q^{9}+O(q^{10}) \) 52 * q + 20 * q^2 + 54 * q^3 + 826 * q^4 - 162 * q^6 + 196 * q^7 + 960 * q^8 + 4098 * q^9	\( 52 q + 20 q^{2} + 54 q^{3} + 826 q^{4} - 162 q^{6} + 196 q^{7} + 960 q^{8} + 4098 q^{9} - 664 q^{11} - 523 q^{12} + 2704 q^{13} + 150 q^{14} + 13474 q^{16} + 7266 q^{17} + 4860 q^{18} - 1970 q^{19} + 800 q^{21} + 14477 q^{22} + 9522 q^{23} + 314 q^{24} + 5514 q^{26} + 22926 q^{27} + 9408 q^{28} - 7188 q^{29} - 11556 q^{31} + 48390 q^{32} + 26136 q^{33} + 16774 q^{34} + 51872 q^{36} + 42558 q^{37} + 46208 q^{38} + 4682 q^{39} - 7746 q^{41} + 174265 q^{42} - 96148 q^{43} - 48600 q^{44} + 16182 q^{46} + 87136 q^{47} - 2912 q^{48} + 142286 q^{49} - 3710 q^{51} + 146868 q^{52} + 127034 q^{53} - 49563 q^{54} - 2849 q^{56} + 101594 q^{57} + 9480 q^{58} - 55924 q^{59} + 73702 q^{61} + 186016 q^{62} + 50120 q^{63} + 157750 q^{64} + 58211 q^{66} + 131996 q^{67} + 298560 q^{68} + 128436 q^{69} - 56284 q^{71} + 343775 q^{72} + 128620 q^{73} - 17721 q^{74} - 170410 q^{76} + 448438 q^{77} + 237616 q^{78} + 106204 q^{79} + 478568 q^{81} + 249596 q^{82} + 348616 q^{83} - 131855 q^{84} - 36980 q^{86} + 267478 q^{87} + 525216 q^{88} + 80410 q^{89} + 226376 q^{91} + 581456 q^{92} + 902902 q^{93} + 180980 q^{94} + 38543 q^{96} + 316148 q^{97} + 295095 q^{98} + 68428 q^{99}+O(q^{100}) \) 52 * q + 20 * q^2 + 54 * q^3 + 826 * q^4 - 162 * q^6 + 196 * q^7 + 960 * q^8 + 4098 * q^9 - 664 * q^11 - 523 * q^12 + 2704 * q^13 + 150 * q^14 + 13474 * q^16 + 7266 * q^17 + 4860 * q^18 - 1970 * q^19 + 800 * q^21 + 14477 * q^22 + 9522 * q^23 + 314 * q^24 + 5514 * q^26 + 22926 * q^27 + 9408 * q^28 - 7188 * q^29 - 11556 * q^31 + 48390 * q^32 + 26136 * q^33 + 16774 * q^34 + 51872 * q^36 + 42558 * q^37 + 46208 * q^38 + 4682 * q^39 - 7746 * q^41 + 174265 * q^42 - 96148 * q^43 - 48600 * q^44 + 16182 * q^46 + 87136 * q^47 - 2912 * q^48 + 142286 * q^49 - 3710 * q^51 + 146868 * q^52 + 127034 * q^53 - 49563 * q^54 - 2849 * q^56 + 101594 * q^57 + 9480 * q^58 - 55924 * q^59 + 73702 * q^61 + 186016 * q^62 + 50120 * q^63 + 157750 * q^64 + 58211 * q^66 + 131996 * q^67 + 298560 * q^68 + 128436 * q^69 - 56284 * q^71 + 343775 * q^72 + 128620 * q^73 - 17721 * q^74 - 170410 * q^76 + 448438 * q^77 + 237616 * q^78 + 106204 * q^79 + 478568 * q^81 + 249596 * q^82 + 348616 * q^83 - 131855 * q^84 - 36980 * q^86 + 267478 * q^87 + 525216 * q^88 + 80410 * q^89 + 226376 * q^91 + 581456 * q^92 + 902902 * q^93 + 180980 * q^94 + 38543 * q^96 + 316148 * q^97 + 295095 * q^98 + 68428 * q^99

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