LMFDB - Embedded newform 575.8.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [575,8,Mod(1,575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(575, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("575.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 575 = 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	575.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:	$179.621389653$
Analytic rank:	$1$
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} - 832x^{6} - 1059x^{5} + 203052x^{4} + 678328x^{3} - 13424272x^{2} - 73308944x - 37372224 $ x^8 - 832x^6 - 1059x^5 + 203052x^4 + 678328x^3 - 13424272x^2 - 73308944x - 37372224
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4}\cdot 5 $
Twist minimal:	no (minimal twist has level 23)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$19.9556$ of defining polynomial
Character	$\chi$	$=$	575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-19.9556 q^{2} +76.4323 q^{3} +270.224 q^{4} -1525.25 q^{6} -653.513 q^{7} -2838.16 q^{8} +3654.90 q^{9} +O(q^{10})$	\(q-19.9556 q^{2} +76.4323 q^{3} +270.224 q^{4} -1525.25 q^{6} -653.513 q^{7} -2838.16 q^{8} +3654.90 q^{9} -4337.65 q^{11} +20653.9 q^{12} +1039.41 q^{13} +13041.2 q^{14} +22048.4 q^{16} +4916.44 q^{17} -72935.5 q^{18} -16566.8 q^{19} -49949.5 q^{21} +86560.2 q^{22} +12167.0 q^{23} -216927. q^{24} -20741.9 q^{26} +112195. q^{27} -176595. q^{28} -105525. q^{29} +6798.84 q^{31} -76703.4 q^{32} -331537. q^{33} -98110.2 q^{34} +987642. q^{36} +407252. q^{37} +330599. q^{38} +79444.2 q^{39} +614719. q^{41} +996770. q^{42} +607028. q^{43} -1.17214e6 q^{44} -242799. q^{46} +907295. q^{47} +1.68521e6 q^{48} -396464. q^{49} +375775. q^{51} +280872. q^{52} -881135. q^{53} -2.23891e6 q^{54} +1.85478e6 q^{56} -1.26624e6 q^{57} +2.10581e6 q^{58} +1.60033e6 q^{59} +1.21442e6 q^{61} -135675. q^{62} -2.38852e6 q^{63} -1.29154e6 q^{64} +6.61600e6 q^{66} -783456. q^{67} +1.32854e6 q^{68} +929952. q^{69} +354699. q^{71} -1.03732e7 q^{72} -2.41783e6 q^{73} -8.12693e6 q^{74} -4.47674e6 q^{76} +2.83471e6 q^{77} -1.58535e6 q^{78} -4.43829e6 q^{79} +582055. q^{81} -1.22671e7 q^{82} -8.13014e6 q^{83} -1.34976e7 q^{84} -1.21136e7 q^{86} -8.06552e6 q^{87} +1.23110e7 q^{88} +4.11109e6 q^{89} -679265. q^{91} +3.28782e6 q^{92} +519651. q^{93} -1.81056e7 q^{94} -5.86262e6 q^{96} -3.08615e6 q^{97} +7.91165e6 q^{98} -1.58537e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 40 q^{3} + 640 q^{4} - 1745 q^{6} - 1446 q^{7} - 3177 q^{8} + 13878 q^{9}+O(q^{10}) $ 8 * q - 40 * q^3 + 640 * q^4 - 1745 * q^6 - 1446 * q^7 - 3177 * q^8 + 13878 * q^9	\( 8 q - 40 q^{3} + 640 q^{4} - 1745 q^{6} - 1446 q^{7} - 3177 q^{8} + 13878 q^{9} + 7588 q^{11} - 22733 q^{12} - 19862 q^{13} + 17544 q^{14} + 64336 q^{16} - 42070 q^{17} + 59129 q^{18} + 1050 q^{19} - 7698 q^{21} + 128220 q^{22} + 97336 q^{23} - 621188 q^{24} - 371761 q^{26} + 69500 q^{27} - 143050 q^{28} - 102578 q^{29} + 304172 q^{31} + 612824 q^{32} - 747242 q^{33} - 524530 q^{34} + 1868983 q^{36} - 286472 q^{37} + 762932 q^{38} + 1032828 q^{39} + 1324414 q^{41} + 1886168 q^{42} - 2052578 q^{43} - 867298 q^{44} - 675556 q^{47} - 1411151 q^{48} - 55404 q^{49} + 2775482 q^{51} + 1695409 q^{52} - 203654 q^{53} - 9897559 q^{54} - 5766846 q^{56} - 3908648 q^{57} + 5039991 q^{58} - 748892 q^{59} + 61822 q^{61} + 4939277 q^{62} - 1411632 q^{63} + 2702267 q^{64} + 3791866 q^{66} - 3235604 q^{67} - 4914980 q^{68} - 486680 q^{69} - 4951664 q^{71} + 7940241 q^{72} - 11019370 q^{73} + 356954 q^{74} + 21973240 q^{76} + 5284888 q^{77} + 1506779 q^{78} + 4202464 q^{79} + 10294096 q^{81} - 32636759 q^{82} - 518568 q^{83} + 7629190 q^{84} - 14681386 q^{86} - 4862532 q^{87} - 20589740 q^{88} + 4203864 q^{89} + 2488406 q^{91} + 7786880 q^{92} + 23367842 q^{93} + 12314327 q^{94} - 45317009 q^{96} - 18621134 q^{97} - 35756 q^{98} - 64729930 q^{99}+O(q^{100}) \) 8 * q - 40 * q^3 + 640 * q^4 - 1745 * q^6 - 1446 * q^7 - 3177 * q^8 + 13878 * q^9 + 7588 * q^11 - 22733 * q^12 - 19862 * q^13 + 17544 * q^14 + 64336 * q^16 - 42070 * q^17 + 59129 * q^18 + 1050 * q^19 - 7698 * q^21 + 128220 * q^22 + 97336 * q^23 - 621188 * q^24 - 371761 * q^26 + 69500 * q^27 - 143050 * q^28 - 102578 * q^29 + 304172 * q^31 + 612824 * q^32 - 747242 * q^33 - 524530 * q^34 + 1868983 * q^36 - 286472 * q^37 + 762932 * q^38 + 1032828 * q^39 + 1324414 * q^41 + 1886168 * q^42 - 2052578 * q^43 - 867298 * q^44 - 675556 * q^47 - 1411151 * q^48 - 55404 * q^49 + 2775482 * q^51 + 1695409 * q^52 - 203654 * q^53 - 9897559 * q^54 - 5766846 * q^56 - 3908648 * q^57 + 5039991 * q^58 - 748892 * q^59 + 61822 * q^61 + 4939277 * q^62 - 1411632 * q^63 + 2702267 * q^64 + 3791866 * q^66 - 3235604 * q^67 - 4914980 * q^68 - 486680 * q^69 - 4951664 * q^71 + 7940241 * q^72 - 11019370 * q^73 + 356954 * q^74 + 21973240 * q^76 + 5284888 * q^77 + 1506779 * q^78 + 4202464 * q^79 + 10294096 * q^81 - 32636759 * q^82 - 518568 * q^83 + 7629190 * q^84 - 14681386 * q^86 - 4862532 * q^87 - 20589740 * q^88 + 4203864 * q^89 + 2488406 * q^91 + 7786880 * q^92 + 23367842 * q^93 + 12314327 * q^94 - 45317009 * q^96 - 18621134 * q^97 - 35756 * q^98 - 64729930 * 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 $

$2$	−19.9556	−1.76384	−0.881919	−	0.471401i	$-0.843749\pi$
$2$	−19.9556	−1.76384	−0.881919	+	0.471401i	$0.843749\pi$
$3$	76.4323	1.63438	0.817189	−	0.576369i	$-0.195531\pi$
$3$	76.4323	1.63438	0.817189	+	0.576369i	$0.195531\pi$
$4$	270.224	2.11113
$5$	0	0
$5$	0	0
$6$	−1525.25	−2.88278
$7$	−653.513	−0.720130	−0.360065	−	0.932927i	$-0.617246\pi$
$7$	−653.513	−0.720130	−0.360065	+	0.932927i	$0.617246\pi$
$8$	−2838.16	−1.95985
$9$	3654.90	1.67119
$10$	0	0
$11$	−4337.65	−0.982608	−0.491304	−	0.870988i	$-0.663480\pi$
$11$	−4337.65	−0.982608	−0.491304	+	0.870988i	$0.663480\pi$
$12$	20653.9	3.45038
$13$	1039.41	0.131215	0.0656075	−	0.997846i	$-0.479101\pi$
$13$	1039.41	0.131215	0.0656075	+	0.997846i	$0.479101\pi$
$14$	13041.2	1.27019
$15$	0	0
$16$	22048.4	1.34573
$17$	4916.44	0.242705	0.121353	−	0.992609i	$-0.461277\pi$
$17$	4916.44	0.242705	0.121353	+	0.992609i	$0.461277\pi$
$18$	−72935.5	−2.94771
$19$	−16566.8	−0.554115	−0.277057	−	0.960853i	$-0.589359\pi$
$19$	−16566.8	−0.554115	−0.277057	+	0.960853i	$0.589359\pi$
$20$	0	0
$21$	−49949.5	−1.17697
$22$	86560.2	1.73316
$23$	12167.0	0.208514
$23$	12167.0	0.208514
$24$	−216927.	−3.20313
$25$	0	0
$26$	−20741.9	−0.231442
$27$	112195.	1.09698
$28$	−176595.	−1.52029
$29$	−105525.	−0.803457	−0.401728	−	0.915759i	$-0.631590\pi$
$29$	−105525.	−0.803457	−0.401728	+	0.915759i	$0.631590\pi$
$30$	0	0
$31$	6798.84	0.0409891	0.0204946	−	0.999790i	$-0.493476\pi$
$31$	6798.84	0.0409891	0.0204946	+	0.999790i	$0.493476\pi$
$32$	−76703.4	−0.413799
$33$	−331537.	−1.60595
$34$	−98110.2	−0.428093
$35$	0	0
$36$	987642.	3.52810
$37$	407252.	1.32177	0.660886	−	0.750486i	$-0.270180\pi$
$37$	407252.	1.32177	0.660886	+	0.750486i	$0.270180\pi$
$38$	330599.	0.977369
$39$	79444.2	0.214455
$40$	0	0
$41$	614719.	1.39294	0.696471	−	0.717585i	$-0.254752\pi$
$41$	614719.	1.39294	0.696471	+	0.717585i	$0.254752\pi$
$42$	996770.	2.07598
$43$	607028.	1.16431	0.582156	−	0.813077i	$-0.302209\pi$
$43$	607028.	1.16431	0.582156	+	0.813077i	$0.302209\pi$
$44$	−1.17214e6	−2.07441
$45$	0	0
$46$	−242799.	−0.367786
$47$	907295.	1.27469	0.637347	−	0.770577i	$-0.280032\pi$
$47$	907295.	1.27469	0.637347	+	0.770577i	$0.280032\pi$
$48$	1.68521e6	2.19943
$49$	−396464.	−0.481412
$50$	0	0
$51$	375775.	0.396672
$52$	280872.	0.277011
$53$	−881135.	−0.812974	−0.406487	−	0.913657i	$-0.633246\pi$
$53$	−881135.	−0.812974	−0.406487	+	0.913657i	$0.633246\pi$
$54$	−2.23891e6	−1.93490
$55$	0	0
$56$	1.85478e6	1.41135
$57$	−1.26624e6	−0.905633
$58$	2.10581e6	1.41717
$59$	1.60033e6	1.01444	0.507221	−	0.861816i	$-0.330673\pi$
$59$	1.60033e6	1.01444	0.507221	+	0.861816i	$0.330673\pi$
$60$	0	0
$61$	1.21442e6	0.685040	0.342520	−	0.939511i	$-0.388720\pi$
$61$	1.21442e6	0.685040	0.342520	+	0.939511i	$0.388720\pi$
$62$	−135675.	−0.0722982
$63$	−2.38852e6	−1.20348
$64$	−1.29154e6	−0.615853
$65$	0	0
$66$	6.61600e6	2.83264
$67$	−783456.	−0.318239	−0.159119	−	0.987259i	$-0.550865\pi$
$67$	−783456.	−0.318239	−0.159119	+	0.987259i	$0.550865\pi$
$68$	1.32854e6	0.512381
$69$	929952.	0.340791
$70$	0	0
$71$	354699.	0.117613	0.0588066	−	0.998269i	$-0.481270\pi$
$71$	354699.	0.117613	0.0588066	+	0.998269i	$0.481270\pi$
$72$	−1.03732e7	−3.27528
$73$	−2.41783e6	−0.727439	−0.363719	−	0.931509i	$-0.618493\pi$
$73$	−2.41783e6	−0.727439	−0.363719	+	0.931509i	$0.618493\pi$
$74$	−8.12693e6	−2.33139
$75$	0	0
$76$	−4.47674e6	−1.16981
$77$	2.83471e6	0.707606
$78$	−1.58535e6	−0.378264
$79$	−4.43829e6	−1.01279	−0.506397	−	0.862301i	$-0.669023\pi$
$79$	−4.43829e6	−1.01279	−0.506397	+	0.862301i	$0.669023\pi$
$80$	0	0
$81$	582055.	0.121693
$82$	−1.22671e7	−2.45692
$83$	−8.13014e6	−1.56072	−0.780360	−	0.625331i	$-0.784964\pi$
$83$	−8.13014e6	−1.56072	−0.780360	+	0.625331i	$0.784964\pi$
$84$	−1.34976e7	−2.48472
$85$	0	0
$86$	−1.21136e7	−2.05366
$87$	−8.06552e6	−1.31315
$88$	1.23110e7	1.92576
$89$	4.11109e6	0.618147	0.309074	−	0.951038i	$-0.399981\pi$
$89$	4.11109e6	0.618147	0.309074	+	0.951038i	$0.399981\pi$
$90$	0	0
$91$	−679265.	−0.0944918
$92$	3.28782e6	0.440200
$93$	519651.	0.0669917
$94$	−1.81056e7	−2.24836
$95$	0	0
$96$	−5.86262e6	−0.676304
$97$	−3.08615e6	−0.343333	−0.171667	−	0.985155i	$-0.554915\pi$
$97$	−3.08615e6	−0.343333	−0.171667	+	0.985155i	$0.554915\pi$
$98$	7.91165e6	0.849134
$99$	−1.58537e7	−1.64213
$100$	0	0
$101$	−1.64582e7	−1.58949	−0.794746	−	0.606942i	$-0.792396\pi$
$101$	−1.64582e7	−1.58949	−0.794746	+	0.606942i	$0.792396\pi$
$102$	−7.49879e6	−0.699666
$103$	−1.69239e7	−1.52605	−0.763026	−	0.646368i	$-0.776287\pi$
$103$	−1.69239e7	−1.52605	−0.763026	+	0.646368i	$0.776287\pi$
$104$	−2.95000e6	−0.257161
$105$	0	0
$106$	1.75835e7	1.43396
$107$	2.12029e6	0.167321	0.0836607	−	0.996494i	$-0.473339\pi$
$107$	2.12029e6	0.167321	0.0836607	+	0.996494i	$0.473339\pi$
$108$	3.03178e7	2.31587
$109$	6.43760e6	0.476136	0.238068	−	0.971248i	$-0.423486\pi$
$109$	6.43760e6	0.476136	0.238068	+	0.971248i	$0.423486\pi$
$110$	0	0
$111$	3.11272e7	2.16028
$112$	−1.44089e7	−0.969099
$113$	8.54511e6	0.557113	0.278557	−	0.960420i	$-0.410144\pi$
$113$	8.54511e6	0.557113	0.278557	+	0.960420i	$0.410144\pi$
$114$	2.52684e7	1.59739
$115$	0	0
$116$	−2.85154e7	−1.69620
$117$	3.79892e6	0.219285
$118$	−3.19354e7	−1.78931
$119$	−3.21295e6	−0.174779
$120$	0	0
$121$	−671952.	−0.0344818
$122$	−2.42345e7	−1.20830
$123$	4.69844e7	2.27659
$124$	1.83721e6	0.0865332
$125$	0	0
$126$	4.76643e7	2.12274
$127$	−3.53558e7	−1.53161	−0.765804	−	0.643074i	$-0.777659\pi$
$127$	−3.53558e7	−1.53161	−0.765804	+	0.643074i	$0.777659\pi$
$128$	3.55914e7	1.50006
$129$	4.63966e7	1.90293
$130$	0	0
$131$	−2.98530e7	−1.16022	−0.580108	−	0.814539i	$-0.696990\pi$
$131$	−2.98530e7	−1.16022	−0.580108	+	0.814539i	$0.696990\pi$
$132$	−8.95892e7	−3.39037
$133$	1.08266e7	0.399035
$134$	1.56343e7	0.561322
$135$	0	0
$136$	−1.39536e7	−0.475665
$137$	−1.62343e7	−0.539400	−0.269700	−	0.962944i	$-0.586925\pi$
$137$	−1.62343e7	−0.539400	−0.269700	+	0.962944i	$0.586925\pi$
$138$	−1.85577e7	−0.601101
$139$	2.38753e7	0.754046	0.377023	−	0.926204i	$-0.376948\pi$
$139$	2.38753e7	0.754046	0.377023	+	0.926204i	$0.376948\pi$
$140$	0	0
$141$	6.93467e7	2.08333
$142$	−7.07822e6	−0.207451
$143$	−4.50858e6	−0.128933
$144$	8.05847e7	2.24897
$145$	0	0
$146$	4.82492e7	1.28308
$147$	−3.03026e7	−0.786810
$148$	1.10049e8	2.79043
$149$	−1.49109e7	−0.369277	−0.184638	−	0.982807i	$-0.559111\pi$
$149$	−1.49109e7	−0.369277	−0.184638	+	0.982807i	$0.559111\pi$
$150$	0	0
$151$	−7.01442e6	−0.165795	−0.0828977	−	0.996558i	$-0.526417\pi$
$151$	−7.01442e6	−0.165795	−0.0828977	+	0.996558i	$0.526417\pi$
$152$	4.70191e7	1.08598
$153$	1.79691e7	0.405607
$154$	−5.65682e7	−1.24810
$155$	0	0
$156$	2.14677e7	0.452741
$157$	2.71207e7	0.559310	0.279655	−	0.960101i	$-0.409780\pi$
$157$	2.71207e7	0.559310	0.279655	+	0.960101i	$0.409780\pi$
$158$	8.85686e7	1.78640
$159$	−6.73472e7	−1.32871
$160$	0	0
$161$	−7.95129e6	−0.150158
$162$	−1.16152e7	−0.214647
$163$	−5.06776e7	−0.916557	−0.458278	−	0.888809i	$-0.651534\pi$
$163$	−5.06776e7	−0.916557	−0.458278	+	0.888809i	$0.651534\pi$
$164$	1.66112e8	2.94068
$165$	0	0
$166$	1.62242e8	2.75286
$167$	2.07036e7	0.343983	0.171992	−	0.985098i	$-0.444980\pi$
$167$	2.07036e7	0.343983	0.171992	+	0.985098i	$0.444980\pi$
$168$	1.41765e8	2.30667
$169$	−6.16682e7	−0.982783
$170$	0	0
$171$	−6.05498e7	−0.926033
$172$	1.64034e8	2.45801
$173$	−5.36427e7	−0.787679	−0.393840	−	0.919179i	$-0.628853\pi$
$173$	−5.36427e7	−0.787679	−0.393840	+	0.919179i	$0.628853\pi$
$174$	1.60952e8	2.31619
$175$	0	0
$176$	−9.56383e7	−1.32232
$177$	1.22317e8	1.65798
$178$	−8.20391e7	−1.09031
$179$	−3.38671e7	−0.441360	−0.220680	−	0.975346i	$-0.570828\pi$
$179$	−3.38671e7	−0.441360	−0.220680	+	0.975346i	$0.570828\pi$
$180$	0	0
$181$	1.04923e7	0.131521	0.0657607	−	0.997835i	$-0.479053\pi$
$181$	1.04923e7	0.131521	0.0657607	+	0.997835i	$0.479053\pi$
$182$	1.35551e7	0.166668
$183$	9.28212e7	1.11961
$184$	−3.45319e7	−0.408656
$185$	0	0
$186$	−1.03699e7	−0.118163
$187$	−2.13258e7	−0.238484
$188$	2.45173e8	2.69104
$189$	−7.33209e7	−0.789971
$190$	0	0
$191$	3.16922e7	0.329106	0.164553	−	0.986368i	$-0.447382\pi$
$191$	3.16922e7	0.329106	0.164553	+	0.986368i	$0.447382\pi$
$192$	−9.87152e7	−1.00654
$193$	−1.66918e8	−1.67130	−0.835649	−	0.549264i	$-0.814908\pi$
$193$	−1.66918e8	−1.67130	−0.835649	+	0.549264i	$0.814908\pi$
$194$	6.15858e7	0.605584
$195$	0	0
$196$	−1.07134e8	−1.01632
$197$	−1.19450e8	−1.11315	−0.556577	−	0.830796i	$-0.687885\pi$
$197$	−1.19450e8	−1.11315	−0.556577	+	0.830796i	$0.687885\pi$
$198$	3.16369e8	2.89645
$199$	2.02185e8	1.81871	0.909356	−	0.416019i	$-0.136575\pi$
$199$	2.02185e8	1.81871	0.909356	+	0.416019i	$0.136575\pi$
$200$	0	0
$201$	−5.98814e7	−0.520123
$202$	3.28433e8	2.80361
$203$	6.89620e7	0.578594
$204$	1.01543e8	0.837425
$205$	0	0
$206$	3.37725e8	2.69171
$207$	4.44692e7	0.348468
$208$	2.29172e7	0.176580
$209$	7.18608e7	0.544478
$210$	0	0
$211$	−1.01897e8	−0.746747	−0.373374	−	0.927681i	$-0.621799\pi$
$211$	−1.01897e8	−0.746747	−0.373374	+	0.927681i	$0.621799\pi$
$212$	−2.38104e8	−1.71629
$213$	2.71105e7	0.192224
$214$	−4.23115e7	−0.295128
$215$	0	0
$216$	−3.18427e8	−2.14992
$217$	−4.44313e6	−0.0295175
$218$	−1.28466e8	−0.839827
$219$	−1.84801e8	−1.18891
$220$	0	0
$221$	5.11017e6	0.0318466
$222$	−6.21160e8	−3.81038
$223$	−1.66667e8	−1.00643	−0.503214	−	0.864162i	$-0.667849\pi$
$223$	−1.66667e8	−1.00643	−0.503214	+	0.864162i	$0.667849\pi$
$224$	5.01266e7	0.297989
$225$	0	0
$226$	−1.70523e8	−0.982657
$227$	1.88738e8	1.07095	0.535474	−	0.844552i	$-0.320133\pi$
$227$	1.88738e8	1.07095	0.535474	+	0.844552i	$0.320133\pi$
$228$	−3.42167e8	−1.91191
$229$	−2.72204e8	−1.49785	−0.748927	−	0.662652i	$-0.769431\pi$
$229$	−2.72204e8	−1.49785	−0.748927	+	0.662652i	$0.769431\pi$
$230$	0	0
$231$	2.16664e8	1.15650
$232$	2.99497e8	1.57465
$233$	−2.31254e8	−1.19769	−0.598844	−	0.800866i	$-0.704373\pi$
$233$	−2.31254e8	−1.19769	−0.598844	+	0.800866i	$0.704373\pi$
$234$	−7.58096e7	−0.386784
$235$	0	0
$236$	4.32447e8	2.14161
$237$	−3.39229e8	−1.65529
$238$	6.41163e7	0.308283
$239$	−2.94824e8	−1.39692	−0.698458	−	0.715651i	$-0.746130\pi$
$239$	−2.94824e8	−1.39692	−0.698458	+	0.715651i	$0.746130\pi$
$240$	0	0
$241$	1.76524e8	0.812354	0.406177	−	0.913795i	$-0.366862\pi$
$241$	1.76524e8	0.812354	0.406177	+	0.913795i	$0.366862\pi$
$242$	1.34092e7	0.0608202
$243$	−2.00883e8	−0.898091
$244$	3.28167e8	1.44621
$245$	0	0
$246$	−9.37599e8	−4.01554
$247$	−1.72196e7	−0.0727082
$248$	−1.92962e7	−0.0803324
$249$	−6.21406e8	−2.55081
$250$	0	0
$251$	−3.25446e8	−1.29904	−0.649518	−	0.760346i	$-0.725029\pi$
$251$	−3.25446e8	−1.29904	−0.649518	+	0.760346i	$0.725029\pi$
$252$	−6.45437e8	−2.54069
$253$	−5.27762e7	−0.204888
$254$	7.05545e8	2.70151
$255$	0	0
$256$	−5.44929e8	−2.03002
$257$	4.52380e8	1.66241	0.831203	−	0.555969i	$-0.187653\pi$
$257$	4.52380e8	1.66241	0.831203	+	0.555969i	$0.187653\pi$
$258$	−9.25869e8	−3.35645
$259$	−2.66144e8	−0.951849
$260$	0	0
$261$	−3.85683e8	−1.34273
$262$	5.95734e8	2.04644
$263$	8.90957e7	0.302003	0.151002	−	0.988534i	$-0.451750\pi$
$263$	8.90957e7	0.302003	0.151002	+	0.988534i	$0.451750\pi$
$264$	9.40955e8	3.14742
$265$	0	0
$266$	−2.16051e8	−0.703833
$267$	3.14220e8	1.01029
$268$	−2.11709e8	−0.671842
$269$	2.72958e8	0.854993	0.427496	−	0.904017i	$-0.359396\pi$
$269$	2.72958e8	0.854993	0.427496	+	0.904017i	$0.359396\pi$
$270$	0	0
$271$	−1.53890e8	−0.469698	−0.234849	−	0.972032i	$-0.575460\pi$
$271$	−1.53890e8	−0.469698	−0.234849	+	0.972032i	$0.575460\pi$
$272$	1.08400e8	0.326615
$273$	−5.19178e7	−0.154435
$274$	3.23964e8	0.951414
$275$	0	0
$276$	2.51295e8	0.719454
$277$	5.75372e8	1.62656	0.813279	−	0.581873i	$-0.197680\pi$
$277$	5.75372e8	1.62656	0.813279	+	0.581873i	$0.197680\pi$
$278$	−4.76446e8	−1.33001
$279$	2.48491e7	0.0685007
$280$	0	0
$281$	5.26444e8	1.41540	0.707701	−	0.706512i	$-0.249732\pi$
$281$	5.26444e8	1.41540	0.707701	+	0.706512i	$0.249732\pi$
$282$	−1.38385e9	−3.67466
$283$	−8.78058e7	−0.230288	−0.115144	−	0.993349i	$-0.536733\pi$
$283$	−8.78058e7	−0.230288	−0.115144	+	0.993349i	$0.536733\pi$
$284$	9.58482e7	0.248296
$285$	0	0
$286$	8.99712e7	0.227417
$287$	−4.01727e8	−1.00310
$288$	−2.80343e8	−0.691538
$289$	−3.86167e8	−0.941094
$290$	0	0
$291$	−2.35882e8	−0.561137
$292$	−6.53357e8	−1.53572
$293$	−2.26978e8	−0.527167	−0.263583	−	0.964637i	$-0.584904\pi$
$293$	−2.26978e8	−0.527167	−0.263583	+	0.964637i	$0.584904\pi$
$294$	6.04706e8	1.38781
$295$	0	0
$296$	−1.15585e9	−2.59047
$297$	−4.86663e8	−1.07790
$298$	2.97555e8	0.651345
$299$	1.26464e7	0.0273602
$300$	0	0
$301$	−3.96701e8	−0.838456
$302$	1.39977e8	0.292436
$303$	−1.25794e9	−2.59783
$304$	−3.65270e8	−0.745688
$305$	0	0
$306$	−3.58583e8	−0.715426
$307$	−3.04912e8	−0.601437	−0.300719	−	0.953713i	$-0.597227\pi$
$307$	−3.04912e8	−0.601437	−0.300719	+	0.953713i	$0.597227\pi$
$308$	7.66007e8	1.49384
$309$	−1.29353e9	−2.49415
$310$	0	0
$311$	1.53838e8	0.290002	0.145001	−	0.989432i	$-0.453681\pi$
$311$	1.53838e8	0.290002	0.145001	+	0.989432i	$0.453681\pi$
$312$	−2.25475e8	−0.420299
$313$	−2.04764e8	−0.377440	−0.188720	−	0.982031i	$-0.560434\pi$
$313$	−2.04764e8	−0.377440	−0.188720	+	0.982031i	$0.560434\pi$
$314$	−5.41209e8	−0.986533
$315$	0	0
$316$	−1.19933e9	−2.13814
$317$	7.27132e8	1.28205	0.641026	−	0.767519i	$-0.278509\pi$
$317$	7.27132e8	1.28205	0.641026	+	0.767519i	$0.278509\pi$
$318$	1.34395e9	2.34363
$319$	4.57731e8	0.789483
$320$	0	0
$321$	1.62058e8	0.273466
$322$	1.58672e8	0.264854
$323$	−8.14494e7	−0.134487
$324$	1.57285e8	0.256910
$325$	0	0
$326$	1.01130e9	1.61666
$327$	4.92040e8	0.778186
$328$	−1.74467e9	−2.72995
$329$	−5.92929e8	−0.917946
$330$	0	0
$331$	8.28738e8	1.25609	0.628043	−	0.778179i	$-0.283856\pi$
$331$	8.28738e8	1.25609	0.628043	+	0.778179i	$0.283856\pi$
$332$	−2.19696e9	−3.29488
$333$	1.48846e9	2.20894
$334$	−4.13151e8	−0.606731
$335$	0	0
$336$	−1.10131e9	−1.58387
$337$	−5.44518e8	−0.775011	−0.387505	−	0.921867i	$-0.626663\pi$
$337$	−5.44518e8	−0.775011	−0.387505	+	0.921867i	$0.626663\pi$
$338$	1.23062e9	1.73347
$339$	6.53123e8	0.910534
$340$	0	0
$341$	−2.94910e7	−0.0402762
$342$	1.20831e9	1.63337
$343$	7.97290e8	1.06681
$344$	−1.72284e9	−2.28187
$345$	0	0
$346$	1.07047e9	1.38934
$347$	−5.55611e8	−0.713868	−0.356934	−	0.934130i	$-0.616178\pi$
$347$	−5.55611e8	−0.713868	−0.356934	+	0.934130i	$0.616178\pi$
$348$	−2.17950e9	−2.77223
$349$	−1.23854e9	−1.55963	−0.779816	−	0.626009i	$-0.784687\pi$
$349$	−1.23854e9	−1.55963	−0.779816	+	0.626009i	$0.784687\pi$
$350$	0	0
$351$	1.16616e8	0.143941
$352$	3.32712e8	0.406602
$353$	9.18376e8	1.11124	0.555622	−	0.831435i	$-0.312480\pi$
$353$	9.18376e8	1.11124	0.555622	+	0.831435i	$0.312480\pi$
$354$	−2.44090e9	−2.92441
$355$	0	0
$356$	1.11092e9	1.30499
$357$	−2.45574e8	−0.285656
$358$	6.75837e8	0.778487
$359$	1.34767e9	1.53728	0.768640	−	0.639682i	$-0.220934\pi$
$359$	1.34767e9	1.53728	0.768640	+	0.639682i	$0.220934\pi$
$360$	0	0
$361$	−6.19414e8	−0.692957
$362$	−2.09380e8	−0.231983
$363$	−5.13588e7	−0.0563562
$364$	−1.83554e8	−0.199484
$365$	0	0
$366$	−1.85230e9	−1.97482
$367$	−7.77642e8	−0.821199	−0.410600	−	0.911816i	$-0.634681\pi$
$367$	−7.77642e8	−0.821199	−0.410600	+	0.911816i	$0.634681\pi$
$368$	2.68263e8	0.280604
$369$	2.24674e9	2.32788
$370$	0	0
$371$	5.75833e8	0.585447
$372$	1.40422e8	0.141428
$373$	−6.44024e8	−0.642571	−0.321285	−	0.946982i	$-0.604115\pi$
$373$	−6.44024e8	−0.642571	−0.321285	+	0.946982i	$0.604115\pi$
$374$	4.25568e8	0.420647
$375$	0	0
$376$	−2.57505e9	−2.49821
$377$	−1.09683e8	−0.105426
$378$	1.46316e9	1.39338
$379$	1.46375e9	1.38112	0.690558	−	0.723277i	$-0.257365\pi$
$379$	1.46375e9	1.38112	0.690558	+	0.723277i	$0.257365\pi$
$380$	0	0
$381$	−2.70233e9	−2.50323
$382$	−6.32435e8	−0.580489
$383$	1.05720e9	0.961530	0.480765	−	0.876850i	$-0.340359\pi$
$383$	1.05720e9	0.961530	0.480765	+	0.876850i	$0.340359\pi$
$384$	2.72033e9	2.45167
$385$	0	0
$386$	3.33095e9	2.94790
$387$	2.21863e9	1.94579
$388$	−8.33952e8	−0.724820
$389$	2.67436e8	0.230354	0.115177	−	0.993345i	$-0.463256\pi$
$389$	2.67436e8	0.230354	0.115177	+	0.993345i	$0.463256\pi$
$390$	0	0
$391$	5.98183e7	0.0506075
$392$	1.12523e9	0.943495
$393$	−2.28174e9	−1.89623
$394$	2.38369e9	1.96342
$395$	0	0
$396$	−4.28405e9	−3.46674
$397$	−3.69295e8	−0.296215	−0.148107	−	0.988971i	$-0.547318\pi$
$397$	−3.69295e8	−0.296215	−0.148107	+	0.988971i	$0.547318\pi$
$398$	−4.03472e9	−3.20791
$399$	8.27501e8	0.652174
$400$	0	0
$401$	−1.98777e9	−1.53943	−0.769715	−	0.638387i	$-0.779602\pi$
$401$	−1.98777e9	−1.53943	−0.769715	+	0.638387i	$0.779602\pi$
$402$	1.19497e9	0.917412
$403$	7.06675e6	0.00537838
$404$	−4.44741e9	−3.35562
$405$	0	0
$406$	−1.37617e9	−1.02055
$407$	−1.76652e9	−1.29878
$408$	−1.06651e9	−0.777417
$409$	5.46117e8	0.394688	0.197344	−	0.980334i	$-0.436768\pi$
$409$	5.46117e8	0.394688	0.197344	+	0.980334i	$0.436768\pi$
$410$	0	0
$411$	−1.24082e9	−0.881583
$412$	−4.57324e9	−3.22169
$413$	−1.04583e9	−0.730530
$414$	−8.87407e8	−0.614641
$415$	0	0
$416$	−7.97259e7	−0.0542966
$417$	1.82485e9	1.23240
$418$	−1.43402e9	−0.960371
$419$	−1.16624e9	−0.774528	−0.387264	−	0.921969i	$-0.626580\pi$
$419$	−1.16624e9	−0.774528	−0.387264	+	0.921969i	$0.626580\pi$
$420$	0	0
$421$	−2.24993e9	−1.46954	−0.734769	−	0.678317i	$-0.762709\pi$
$421$	−2.24993e9	−1.46954	−0.734769	+	0.678317i	$0.762709\pi$
$422$	2.03341e9	1.31714
$423$	3.31607e9	2.13026
$424$	2.50080e9	1.59331
$425$	0	0
$426$	−5.41004e8	−0.339053
$427$	−7.93641e8	−0.493318
$428$	5.72953e8	0.353236
$429$	−3.44601e8	−0.210725
$430$	0	0
$431$	1.53220e9	0.921819	0.460909	−	0.887447i	$-0.347523\pi$
$431$	1.53220e9	0.921819	0.460909	+	0.887447i	$0.347523\pi$
$432$	2.47372e9	1.47624
$433$	−9.81704e8	−0.581129	−0.290565	−	0.956855i	$-0.593843\pi$
$433$	−9.81704e8	−0.581129	−0.290565	+	0.956855i	$0.593843\pi$
$434$	8.86651e7	0.0520641
$435$	0	0
$436$	1.73959e9	1.00518
$437$	−2.01568e8	−0.115541
$438$	3.68780e9	2.09705
$439$	−1.24756e9	−0.703777	−0.351889	−	0.936042i	$-0.614460\pi$
$439$	−1.24756e9	−0.703777	−0.351889	+	0.936042i	$0.614460\pi$
$440$	0	0
$441$	−1.44904e9	−0.804533
$442$	−1.01976e8	−0.0561722
$443$	2.07638e9	1.13473	0.567367	−	0.823465i	$-0.307962\pi$
$443$	2.07638e9	1.13473	0.567367	+	0.823465i	$0.307962\pi$
$444$	8.41132e9	4.56062
$445$	0	0
$446$	3.32594e9	1.77518
$447$	−1.13967e9	−0.603538
$448$	8.44036e8	0.443494
$449$	4.45747e8	0.232395	0.116197	−	0.993226i	$-0.462930\pi$
$449$	4.45747e8	0.232395	0.116197	+	0.993226i	$0.462930\pi$
$450$	0	0
$451$	−2.66644e9	−1.36872
$452$	2.30910e9	1.17614
$453$	−5.36128e8	−0.270972
$454$	−3.76637e9	−1.88898
$455$	0	0
$456$	3.59378e9	1.77490
$457$	−2.31240e9	−1.13333	−0.566665	−	0.823948i	$-0.691767\pi$
$457$	−2.31240e9	−1.13333	−0.566665	+	0.823948i	$0.691767\pi$
$458$	5.43197e9	2.64197
$459$	5.51599e8	0.266244
$460$	0	0
$461$	−1.38565e9	−0.658718	−0.329359	−	0.944205i	$-0.606833\pi$
$461$	−1.38565e9	−0.658718	−0.329359	+	0.944205i	$0.606833\pi$
$462$	−4.32364e9	−2.03987
$463$	−1.70685e9	−0.799212	−0.399606	−	0.916687i	$-0.630853\pi$
$463$	−1.70685e9	−0.799212	−0.399606	+	0.916687i	$0.630853\pi$
$464$	−2.32666e9	−1.08123
$465$	0	0
$466$	4.61480e9	2.11253
$467$	3.29906e7	0.0149893	0.00749465	−	0.999972i	$-0.497614\pi$
$467$	3.29906e7	0.0149893	0.00749465	+	0.999972i	$0.497614\pi$
$468$	1.02656e9	0.462939
$469$	5.11999e8	0.229173
$470$	0	0
$471$	2.07290e9	0.914125
$472$	−4.54199e9	−1.98815
$473$	−2.63308e9	−1.14406
$474$	6.76950e9	2.91966
$475$	0	0
$476$	−8.68218e8	−0.368981
$477$	−3.22046e9	−1.35864
$478$	5.88338e9	2.46393
$479$	1.56585e9	0.650993	0.325497	−	0.945543i	$-0.394469\pi$
$479$	1.56585e9	0.650993	0.325497	+	0.945543i	$0.394469\pi$
$480$	0	0
$481$	4.23300e8	0.173436
$482$	−3.52264e9	−1.43286
$483$	−6.07736e8	−0.245414
$484$	−1.81578e8	−0.0727953
$485$	0	0
$486$	4.00872e9	1.58409
$487$	5.67843e7	0.0222780	0.0111390	−	0.999938i	$-0.496454\pi$
$487$	5.67843e7	0.0222780	0.0111390	+	0.999938i	$0.496454\pi$
$488$	−3.44673e9	−1.34257
$489$	−3.87341e9	−1.49800
$490$	0	0
$491$	−2.55398e9	−0.973716	−0.486858	−	0.873481i	$-0.661857\pi$
$491$	−2.55398e9	−0.973716	−0.486858	+	0.873481i	$0.661857\pi$
$492$	1.26963e10	4.80618
$493$	−5.18807e8	−0.195003
$494$	3.43626e8	0.128245
$495$	0	0
$496$	1.49903e8	0.0551602
$497$	−2.31800e8	−0.0846968
$498$	1.24005e10	4.49921
$499$	−1.43002e8	−0.0515216	−0.0257608	−	0.999668i	$-0.508201\pi$
$499$	−1.43002e8	−0.0515216	−0.0257608	+	0.999668i	$0.508201\pi$
$500$	0	0
$501$	1.58242e9	0.562199
$502$	6.49446e9	2.29129
$503$	5.23549e8	0.183430	0.0917148	−	0.995785i	$-0.470765\pi$
$503$	5.23549e8	0.183430	0.0917148	+	0.995785i	$0.470765\pi$
$504$	6.77902e9	2.35863
$505$	0	0
$506$	1.05318e9	0.361389
$507$	−4.71344e9	−1.60624
$508$	−9.55400e9	−3.23342
$509$	−5.66266e9	−1.90330	−0.951652	−	0.307177i	$-0.900616\pi$
$509$	−5.66266e9	−1.90330	−0.951652	+	0.307177i	$0.900616\pi$
$510$	0	0
$511$	1.58009e9	0.523851
$512$	6.31866e9	2.08056
$513$	−1.85871e9	−0.607855
$514$	−9.02749e9	−2.93222
$515$	0	0
$516$	1.25375e10	4.01732
$517$	−3.93553e9	−1.25253
$518$	5.31106e9	1.67891
$519$	−4.10004e9	−1.28737
$520$	0	0
$521$	2.81823e9	0.873062	0.436531	−	0.899689i	$-0.356207\pi$
$521$	2.81823e9	0.873062	0.436531	+	0.899689i	$0.356207\pi$
$522$	7.69653e9	2.36836
$523$	−2.41696e9	−0.738776	−0.369388	−	0.929275i	$-0.620433\pi$
$523$	−2.41696e9	−0.738776	−0.369388	+	0.929275i	$0.620433\pi$
$524$	−8.06702e9	−2.44936
$525$	0	0
$526$	−1.77795e9	−0.532685
$527$	3.34260e7	0.00994827
$528$	−7.30986e9	−2.16118
$529$	1.48036e8	0.0434783
$530$	0	0
$531$	5.84904e9	1.69533
$532$	2.92561e9	0.842413
$533$	6.38942e8	0.182775
$534$	−6.27044e9	−1.78198
$535$	0	0
$536$	2.22358e9	0.623699
$537$	−2.58854e9	−0.721349
$538$	−5.44702e9	−1.50807
$539$	1.71972e9	0.473040
$540$	0	0
$541$	−3.24690e9	−0.881613	−0.440807	−	0.897602i	$-0.645308\pi$
$541$	−3.24690e9	−0.881613	−0.440807	+	0.897602i	$0.645308\pi$
$542$	3.07097e9	0.828472
$543$	8.01952e8	0.214956
$544$	−3.77107e8	−0.100431
$545$	0	0
$546$	1.03605e9	0.272399
$547$	−2.67521e9	−0.698878	−0.349439	−	0.936959i	$-0.613628\pi$
$547$	−2.67521e9	−0.698878	−0.349439	+	0.936959i	$0.613628\pi$
$548$	−4.38689e9	−1.13874
$549$	4.43860e9	1.14483
$550$	0	0
$551$	1.74821e9	0.445207
$552$	−2.63935e9	−0.667899
$553$	2.90048e9	0.729343
$554$	−1.14819e10	−2.86899
$555$	0	0
$556$	6.45169e9	1.59189
$557$	−1.29427e9	−0.317344	−0.158672	−	0.987331i	$-0.550721\pi$
$557$	−1.29427e9	−0.317344	−0.158672	+	0.987331i	$0.550721\pi$
$558$	−4.95877e8	−0.120824
$559$	6.30948e8	0.152775
$560$	0	0
$561$	−1.62998e9	−0.389773
$562$	−1.05055e10	−2.49654
$563$	1.28811e8	0.0304210	0.0152105	−	0.999884i	$-0.495158\pi$
$563$	1.28811e8	0.0304210	0.0152105	+	0.999884i	$0.495158\pi$
$564$	1.87392e10	4.39818
$565$	0	0
$566$	1.75221e9	0.406190
$567$	−3.80381e8	−0.0876350
$568$	−1.00669e9	−0.230504
$569$	−5.70575e9	−1.29843	−0.649217	−	0.760603i	$-0.724903\pi$
$569$	−5.70575e9	−1.29843	−0.649217	+	0.760603i	$0.724903\pi$
$570$	0	0
$571$	3.67874e9	0.826937	0.413469	−	0.910518i	$-0.364317\pi$
$571$	3.67874e9	0.826937	0.413469	+	0.910518i	$0.364317\pi$
$572$	−1.21833e9	−0.272193
$573$	2.42231e9	0.537883
$574$	8.01668e9	1.76931
$575$	0	0
$576$	−4.72044e9	−1.02921
$577$	−6.01407e9	−1.30333	−0.651664	−	0.758508i	$-0.725929\pi$
$577$	−6.01407e9	−1.30333	−0.651664	+	0.758508i	$0.725929\pi$
$578$	7.70618e9	1.65994
$579$	−1.27580e10	−2.73153
$580$	0	0
$581$	5.31315e9	1.12392
$582$	4.70715e9	0.989754
$583$	3.82206e9	0.798835
$584$	6.86220e9	1.42567
$585$	0	0
$586$	4.52948e9	0.929837
$587$	5.32427e9	1.08649	0.543246	−	0.839574i	$-0.317195\pi$
$587$	5.32427e9	1.08649	0.543246	+	0.839574i	$0.317195\pi$
$588$	−8.18851e9	−1.66106
$589$	−1.12635e8	−0.0227127
$590$	0	0
$591$	−9.12985e9	−1.81931
$592$	8.97925e9	1.77875
$593$	−5.72975e8	−0.112835	−0.0564176	−	0.998407i	$-0.517968\pi$
$593$	−5.72975e8	−0.112835	−0.0564176	+	0.998407i	$0.517968\pi$
$594$	9.71162e9	1.90125
$595$	0	0
$596$	−4.02929e9	−0.779590
$597$	1.54535e10	2.97246
$598$	−2.52367e8	−0.0482590
$599$	−1.42970e9	−0.271800	−0.135900	−	0.990723i	$-0.543393\pi$
$599$	−1.42970e9	−0.271800	−0.135900	+	0.990723i	$0.543393\pi$
$600$	0	0
$601$	4.44175e9	0.834629	0.417314	−	0.908762i	$-0.362971\pi$
$601$	4.44175e9	0.834629	0.417314	+	0.908762i	$0.362971\pi$
$602$	7.91638e9	1.47890
$603$	−2.86345e9	−0.531838
$604$	−1.89547e9	−0.350015
$605$	0	0
$606$	2.51029e10	4.58216
$607$	1.25069e9	0.226980	0.113490	−	0.993539i	$-0.463797\pi$
$607$	1.25069e9	0.226980	0.113490	+	0.993539i	$0.463797\pi$
$608$	1.27073e9	0.229292
$609$	5.27092e9	0.945641
$610$	0	0
$611$	9.43048e8	0.167259
$612$	4.85568e9	0.856288
$613$	−2.75470e9	−0.483018	−0.241509	−	0.970399i	$-0.577642\pi$
$613$	−2.75470e9	−0.483018	−0.241509	+	0.970399i	$0.577642\pi$
$614$	6.08469e9	1.06084
$615$	0	0
$616$	−8.04537e9	−1.38680
$617$	−9.76601e8	−0.167386	−0.0836930	−	0.996492i	$-0.526672\pi$
$617$	−9.76601e8	−0.167386	−0.0836930	+	0.996492i	$0.526672\pi$
$618$	2.58131e10	4.39927
$619$	9.92338e9	1.68168	0.840838	−	0.541286i	$-0.182062\pi$
$619$	9.92338e9	1.68168	0.840838	+	0.541286i	$0.182062\pi$
$620$	0	0
$621$	1.36508e9	0.228737
$622$	−3.06991e9	−0.511517
$623$	−2.68665e9	−0.445147
$624$	1.75162e9	0.288598
$625$	0	0
$626$	4.08618e9	0.665744
$627$	5.49249e9	0.889883
$628$	7.32868e9	1.18077
$629$	2.00223e9	0.320801
$630$	0	0
$631$	1.02281e10	1.62067	0.810333	−	0.585969i	$-0.199286\pi$
$631$	1.02281e10	1.62067	0.810333	+	0.585969i	$0.199286\pi$
$632$	1.25966e10	1.98492
$633$	−7.78824e9	−1.22047
$634$	−1.45103e10	−2.26133
$635$	0	0
$636$	−1.81988e10	−2.80507
$637$	−4.12087e8	−0.0631685
$638$	−9.13427e9	−1.39252
$639$	1.29639e9	0.196554
$640$	0	0
$641$	−1.79873e9	−0.269750	−0.134875	−	0.990863i	$-0.543063\pi$
$641$	−1.79873e9	−0.269750	−0.134875	+	0.990863i	$0.543063\pi$
$642$	−3.23397e9	−0.482351
$643$	1.13752e10	1.68741	0.843706	−	0.536806i	$-0.180369\pi$
$643$	1.13752e10	1.68741	0.843706	+	0.536806i	$0.180369\pi$
$644$	−2.14863e9	−0.317002
$645$	0	0
$646$	1.62537e9	0.237213
$647$	−1.94696e9	−0.282613	−0.141306	−	0.989966i	$-0.545130\pi$
$647$	−1.94696e9	−0.282613	−0.141306	+	0.989966i	$0.545130\pi$
$648$	−1.65197e9	−0.238500
$649$	−6.94166e9	−0.996798
$650$	0	0
$651$	−3.39598e8	−0.0482428
$652$	−1.36943e10	−1.93497
$653$	1.21092e8	0.0170184	0.00850919	−	0.999964i	$-0.497291\pi$
$653$	1.21092e8	0.0170184	0.00850919	+	0.999964i	$0.497291\pi$
$654$	−9.81894e9	−1.37260
$655$	0	0
$656$	1.35536e10	1.87452
$657$	−8.83694e9	−1.21569
$658$	1.18322e10	1.61911
$659$	−5.44558e9	−0.741216	−0.370608	−	0.928789i	$-0.620851\pi$
$659$	−5.44558e9	−0.741216	−0.370608	+	0.928789i	$0.620851\pi$
$660$	0	0
$661$	−1.04355e9	−0.140542	−0.0702712	−	0.997528i	$-0.522386\pi$
$661$	−1.04355e9	−0.140542	−0.0702712	+	0.997528i	$0.522386\pi$
$662$	−1.65379e10	−2.21553
$663$	3.90582e8	0.0520493
$664$	2.30747e10	3.05877
$665$	0	0
$666$	−2.97031e10	−3.89621
$667$	−1.28392e9	−0.167532
$668$	5.59460e9	0.726192
$669$	−1.27388e10	−1.64489
$670$	0	0
$671$	−5.26775e9	−0.673125
$672$	3.83130e9	0.487027
$673$	1.14044e9	0.144218	0.0721088	−	0.997397i	$-0.477027\pi$
$673$	1.14044e9	0.144218	0.0721088	+	0.997397i	$0.477027\pi$
$674$	1.08662e10	1.36699
$675$	0	0
$676$	−1.66642e10	−2.07478
$677$	1.61236e10	1.99711	0.998555	−	0.0537374i	$-0.0171134\pi$
$677$	1.61236e10	1.99711	0.998555	+	0.0537374i	$0.0171134\pi$
$678$	−1.30334e10	−1.60603
$679$	2.01684e9	0.247245
$680$	0	0
$681$	1.44257e10	1.75033
$682$	5.88509e8	0.0710408
$683$	1.72222e9	0.206832	0.103416	−	0.994638i	$-0.467023\pi$
$683$	1.72222e9	0.206832	0.103416	+	0.994638i	$0.467023\pi$
$684$	−1.63620e10	−1.95497
$685$	0	0
$686$	−1.59104e10	−1.88168
$687$	−2.08052e10	−2.44806
$688$	1.33840e10	1.56685
$689$	−9.15856e8	−0.106674
$690$	0	0
$691$	−1.54378e10	−1.77997	−0.889985	−	0.455990i	$-0.849285\pi$
$691$	−1.54378e10	−1.77997	−0.889985	+	0.455990i	$0.849285\pi$
$692$	−1.44956e10	−1.66289
$693$	1.03606e10	1.18255
$694$	1.10875e10	1.25915
$695$	0	0
$696$	2.28913e10	2.57358
$697$	3.02223e9	0.338074
$698$	2.47158e10	2.75094
$699$	−1.76753e10	−1.95748
$700$	0	0
$701$	−1.02580e10	−1.12473	−0.562367	−	0.826888i	$-0.690109\pi$
$701$	−1.02580e10	−1.12473	−0.562367	+	0.826888i	$0.690109\pi$
$702$	−2.32714e9	−0.253888
$703$	−6.74684e9	−0.732414
$704$	5.60224e9	0.605142
$705$	0	0
$706$	−1.83267e10	−1.96005
$707$	1.07557e10	1.14464
$708$	3.30529e10	3.50021
$709$	−8.51027e9	−0.896771	−0.448386	−	0.893840i	$-0.648001\pi$
$709$	−8.51027e9	−0.896771	−0.448386	+	0.893840i	$0.648001\pi$
$710$	0	0
$711$	−1.62215e10	−1.69257
$712$	−1.16679e10	−1.21147
$713$	8.27214e7	0.00854682
$714$	4.90056e9	0.503851
$715$	0	0
$716$	−9.15171e9	−0.931766
$717$	−2.25341e10	−2.28309
$718$	−2.68935e10	−2.71151
$719$	−4.11020e9	−0.412394	−0.206197	−	0.978511i	$-0.566109\pi$
$719$	−4.11020e9	−0.412394	−0.206197	+	0.978511i	$0.566109\pi$
$720$	0	0
$721$	1.10600e10	1.09896
$722$	1.23608e10	1.22226
$723$	1.34922e10	1.32769
$724$	2.83528e9	0.277658
$725$	0	0
$726$	1.02489e9	0.0994033
$727$	1.01166e9	0.0976485	0.0488242	−	0.998807i	$-0.484453\pi$
$727$	1.01166e9	0.0976485	0.0488242	+	0.998807i	$0.484453\pi$
$728$	1.92786e9	0.185190
$729$	−1.66269e10	−1.58951
$730$	0	0
$731$	2.98442e9	0.282585
$732$	2.50825e10	2.36365
$733$	1.31580e9	0.123403	0.0617017	−	0.998095i	$-0.480347\pi$
$733$	1.31580e9	0.123403	0.0617017	+	0.998095i	$0.480347\pi$
$734$	1.55183e10	1.44846
$735$	0	0
$736$	−9.33250e8	−0.0862831
$737$	3.39836e9	0.312704
$738$	−4.48349e10	−4.10600
$739$	−7.35134e9	−0.670056	−0.335028	−	0.942208i	$-0.608746\pi$
$739$	−7.35134e9	−0.670056	−0.335028	+	0.942208i	$0.608746\pi$
$740$	0	0
$741$	−1.31613e9	−0.118833
$742$	−1.14911e10	−1.03263
$743$	−1.06737e10	−0.954669	−0.477334	−	0.878722i	$-0.658397\pi$
$743$	−1.06737e10	−0.954669	−0.477334	+	0.878722i	$0.658397\pi$
$744$	−1.47485e9	−0.131294
$745$	0	0
$746$	1.28519e10	1.13339
$747$	−2.97149e10	−2.60826
$748$	−5.76274e9	−0.503470
$749$	−1.38563e9	−0.120493
$750$	0	0
$751$	−1.31746e10	−1.13501	−0.567504	−	0.823371i	$-0.692091\pi$
$751$	−1.31746e10	−1.13501	−0.567504	+	0.823371i	$0.692091\pi$
$752$	2.00044e10	1.71539
$753$	−2.48746e10	−2.12312
$754$	2.18879e9	0.185954
$755$	0	0
$756$	−1.98131e10	−1.66773
$757$	−3.55107e9	−0.297525	−0.148762	−	0.988873i	$-0.547529\pi$
$757$	−3.55107e9	−0.297525	−0.148762	+	0.988873i	$0.547529\pi$
$758$	−2.92100e10	−2.43607
$759$	−4.03381e9	−0.334864
$760$	0	0
$761$	−9.84137e9	−0.809486	−0.404743	−	0.914430i	$-0.632639\pi$
$761$	−9.84137e9	−0.809486	−0.404743	+	0.914430i	$0.632639\pi$
$762$	5.39264e10	4.41529
$763$	−4.20705e9	−0.342880
$764$	8.56399e9	0.694783
$765$	0	0
$766$	−2.10971e10	−1.69598
$767$	1.66339e9	0.133110
$768$	−4.16502e10	−3.31782
$769$	−2.42604e9	−0.192378	−0.0961892	−	0.995363i	$-0.530665\pi$
$769$	−2.42604e9	−0.192378	−0.0961892	+	0.995363i	$0.530665\pi$
$770$	0	0
$771$	3.45764e10	2.71700
$772$	−4.51054e10	−3.52832
$773$	−1.53603e10	−1.19611	−0.598055	−	0.801455i	$-0.704060\pi$
$773$	−1.53603e10	−1.19611	−0.598055	+	0.801455i	$0.704060\pi$
$774$	−4.42739e10	−3.43206
$775$	0	0
$776$	8.75899e9	0.672881
$777$	−2.03420e10	−1.55568
$778$	−5.33683e9	−0.406308
$779$	−1.01839e10	−0.771850
$780$	0	0
$781$	−1.53856e9	−0.115568
$782$	−1.19371e9	−0.0892635
$783$	−1.18394e10	−0.881379
$784$	−8.74139e9	−0.647850
$785$	0	0
$786$	4.55333e10	3.34465
$787$	−1.68198e10	−1.23001	−0.615005	−	0.788523i	$-0.710846\pi$
$787$	−1.68198e10	−1.23001	−0.615005	+	0.788523i	$0.710846\pi$
$788$	−3.22783e10	−2.35001
$789$	6.80979e9	0.493587
$790$	0	0
$791$	−5.58434e9	−0.401194
$792$	4.49953e10	3.21832
$793$	1.26228e9	0.0898874
$794$	7.36949e9	0.522475
$795$	0	0
$796$	5.46354e10	3.83953
$797$	2.52755e10	1.76846	0.884232	−	0.467049i	$-0.154683\pi$
$797$	2.52755e10	1.76846	0.884232	+	0.467049i	$0.154683\pi$
$798$	−1.65132e10	−1.15033
$799$	4.46066e9	0.309375
$800$	0	0
$801$	1.50256e10	1.03304
$802$	3.96670e10	2.71531
$803$	1.04877e10	0.714787
$804$	−1.61814e10	−1.09804
$805$	0	0
$806$	−1.41021e8	−0.00948660
$807$	2.08628e10	1.39738
$808$	4.67111e10	3.11516
$809$	−2.60561e10	−1.73017	−0.865085	−	0.501625i	$-0.832736\pi$
$809$	−2.60561e10	−1.73017	−0.865085	+	0.501625i	$0.832736\pi$
$810$	0	0
$811$	4.45254e8	0.0293113	0.0146557	−	0.999893i	$-0.495335\pi$
$811$	4.45254e8	0.0293113	0.0146557	+	0.999893i	$0.495335\pi$
$812$	1.86352e10	1.22148
$813$	−1.17622e10	−0.767665
$814$	3.52518e10	2.29085
$815$	0	0
$816$	8.28523e9	0.533813
$817$	−1.00565e10	−0.645162
$818$	−1.08981e10	−0.696166
$819$	−2.48265e9	−0.157914
$820$	0	0
$821$	1.81822e10	1.14669	0.573344	−	0.819314i	$-0.305646\pi$
$821$	1.81822e10	1.14669	0.573344	+	0.819314i	$0.305646\pi$
$822$	2.47613e10	1.55497
$823$	−1.52481e10	−0.953493	−0.476746	−	0.879041i	$-0.658184\pi$
$823$	−1.52481e10	−0.953493	−0.476746	+	0.879041i	$0.658184\pi$
$824$	4.80326e10	2.99083
$825$	0	0
$826$	2.08702e10	1.28854
$827$	−1.68693e10	−1.03712	−0.518558	−	0.855042i	$-0.673531\pi$
$827$	−1.68693e10	−1.03712	−0.518558	+	0.855042i	$0.673531\pi$
$828$	1.20166e10	0.735660
$829$	−2.98979e9	−0.182263	−0.0911317	−	0.995839i	$-0.529048\pi$
$829$	−2.98979e9	−0.182263	−0.0911317	+	0.995839i	$0.529048\pi$
$830$	0	0
$831$	4.39770e10	2.65841
$832$	−1.34243e9	−0.0808091
$833$	−1.94919e9	−0.116841
$834$	−3.64158e10	−2.17375
$835$	0	0
$836$	1.94185e10	1.14946
$837$	7.62795e8	0.0449644
$838$	2.32729e10	1.36614
$839$	3.09957e10	1.81190	0.905950	−	0.423384i	$-0.139158\pi$
$839$	3.09957e10	1.81190	0.905950	+	0.423384i	$0.139158\pi$
$840$	0	0
$841$	−6.11434e9	−0.354457
$842$	4.48985e10	2.59203
$843$	4.02373e10	2.31330
$844$	−2.75351e10	−1.57648
$845$	0	0
$846$	−6.61741e10	−3.75744
$847$	4.39129e8	0.0248314
$848$	−1.94276e10	−1.09404
$849$	−6.71120e9	−0.376377
$850$	0	0
$851$	4.95503e9	0.275609
$852$	7.32590e9	0.405810
$853$	9.53699e9	0.526126	0.263063	−	0.964779i	$-0.415267\pi$
$853$	9.53699e9	0.526126	0.263063	+	0.964779i	$0.415267\pi$
$854$	1.58376e10	0.870133
$855$	0	0
$856$	−6.01772e9	−0.327924
$857$	2.04118e10	1.10776	0.553882	−	0.832595i	$-0.313146\pi$
$857$	2.04118e10	1.10776	0.553882	+	0.832595i	$0.313146\pi$
$858$	6.87671e9	0.371685
$859$	1.51869e10	0.817510	0.408755	−	0.912644i	$-0.365963\pi$
$859$	1.51869e10	0.817510	0.408755	+	0.912644i	$0.365963\pi$
$860$	0	0
$861$	−3.07049e10	−1.63944
$862$	−3.05759e10	−1.62594
$863$	2.59491e10	1.37431	0.687154	−	0.726512i	$-0.258860\pi$
$863$	2.59491e10	1.37431	0.687154	+	0.726512i	$0.258860\pi$
$864$	−8.60573e9	−0.453931
$865$	0	0
$866$	1.95904e10	1.02502
$867$	−2.95157e10	−1.53810
$868$	−1.20064e9	−0.0623152
$869$	1.92518e10	0.995179
$870$	0	0
$871$	−8.14329e8	−0.0417577
$872$	−1.82709e10	−0.933154
$873$	−1.12796e10	−0.573776
$874$	4.02240e9	0.203796
$875$	0	0
$876$	−4.99376e10	−2.50994
$877$	1.17297e10	0.587202	0.293601	−	0.955928i	$-0.405146\pi$
$877$	1.17297e10	0.587202	0.293601	+	0.955928i	$0.405146\pi$
$878$	2.48957e10	1.24135
$879$	−1.73485e10	−0.861590
$880$	0	0
$881$	−2.57090e8	−0.0126669	−0.00633344	−	0.999980i	$-0.502016\pi$
$881$	−2.57090e8	−0.0126669	−0.00633344	+	0.999980i	$0.502016\pi$
$882$	2.89163e10	1.41907
$883$	1.93060e10	0.943692	0.471846	−	0.881681i	$-0.343588\pi$
$883$	1.93060e10	0.943692	0.471846	+	0.881681i	$0.343588\pi$
$884$	1.38089e9	0.0672321
$885$	0	0
$886$	−4.14353e10	−2.00149
$887$	1.15896e9	0.0557615	0.0278807	−	0.999611i	$-0.491124\pi$
$887$	1.15896e9	0.0557615	0.0278807	+	0.999611i	$0.491124\pi$
$888$	−8.83440e10	−4.23381
$889$	2.31055e10	1.10296
$890$	0	0
$891$	−2.52475e9	−0.119577
$892$	−4.50375e10	−2.12470
$893$	−1.50309e10	−0.706327
$894$	2.27428e10	1.06454
$895$	0	0
$896$	−2.32594e10	−1.08024
$897$	9.66597e8	0.0447169
$898$	−8.89512e9	−0.409907
$899$	−7.17447e8	−0.0329330
$900$	0	0
$901$	−4.33204e9	−0.197313
$902$	5.32102e10	2.41419
$903$	−3.03208e10	−1.37035
$904$	−2.42524e10	−1.09186
$905$	0	0
$906$	1.06987e10	0.477952
$907$	3.72007e9	0.165549	0.0827743	−	0.996568i	$-0.473622\pi$
$907$	3.72007e9	0.165549	0.0827743	+	0.996568i	$0.473622\pi$
$908$	5.10015e10	2.26091
$909$	−6.01532e10	−2.65635
$910$	0	0
$911$	−2.71597e9	−0.119017	−0.0595087	−	0.998228i	$-0.518953\pi$
$911$	−2.71597e9	−0.119017	−0.0595087	+	0.998228i	$0.518953\pi$
$912$	−2.79185e10	−1.21874
$913$	3.52657e10	1.53358
$914$	4.61452e10	1.99901
$915$	0	0
$916$	−7.35560e10	−3.16216
$917$	1.95094e10	0.835507
$918$	−1.10075e10	−0.469611
$919$	−2.47533e10	−1.05203	−0.526016	−	0.850475i	$-0.676315\pi$
$919$	−2.47533e10	−1.05203	−0.526016	+	0.850475i	$0.676315\pi$
$920$	0	0
$921$	−2.33051e10	−0.982977
$922$	2.76514e10	1.16187
$923$	3.68676e8	0.0154326
$924$	5.85477e10	2.44151
$925$	0	0
$926$	3.40612e10	1.40968
$927$	−6.18550e10	−2.55033
$928$	8.09413e9	0.332470
$929$	4.41128e10	1.80513	0.902567	−	0.430550i	$-0.141680\pi$
$929$	4.41128e10	1.80513	0.902567	+	0.430550i	$0.141680\pi$
$930$	0	0
$931$	6.56812e9	0.266758
$932$	−6.24904e10	−2.52847
$933$	1.17582e10	0.473973
$934$	−6.58346e8	−0.0264387
$935$	0	0
$936$	−1.07820e10	−0.429766
$937$	−1.57657e10	−0.626073	−0.313036	−	0.949741i	$-0.601346\pi$
$937$	−1.57657e10	−0.626073	−0.313036	+	0.949741i	$0.601346\pi$
$938$	−1.02172e10	−0.404225
$939$	−1.56506e10	−0.616880
$940$	0	0
$941$	4.32289e10	1.69126	0.845630	−	0.533769i	$-0.179225\pi$
$941$	4.32289e10	1.69126	0.845630	+	0.533769i	$0.179225\pi$
$942$	−4.13659e10	−1.61237
$943$	7.47928e9	0.290448
$944$	3.52847e10	1.36516
$945$	0	0
$946$	5.25445e10	2.01794
$947$	2.21647e9	0.0848081	0.0424040	−	0.999101i	$-0.486498\pi$
$947$	2.21647e9	0.0848081	0.0424040	+	0.999101i	$0.486498\pi$
$948$	−9.16678e10	−3.49452
$949$	−2.51311e9	−0.0954508
$950$	0	0
$951$	5.55764e10	2.09536
$952$	9.11889e9	0.342541
$953$	2.41997e10	0.905702	0.452851	−	0.891586i	$-0.350407\pi$
$953$	2.41997e10	0.905702	0.452851	+	0.891586i	$0.350407\pi$
$954$	6.42660e10	2.39642
$955$	0	0
$956$	−7.96686e10	−2.94907
$957$	3.49854e10	1.29031
$958$	−3.12474e10	−1.14825
$959$	1.06093e10	0.388438
$960$	0	0
$961$	−2.74664e10	−0.998320
$962$	−8.44718e9	−0.305914
$963$	7.74943e9	0.279626
$964$	4.77012e10	1.71498
$965$	0	0
$966$	1.21277e10	0.432871
$967$	−9.31181e9	−0.331163	−0.165581	−	0.986196i	$-0.552950\pi$
$967$	−9.31181e9	−0.331163	−0.165581	+	0.986196i	$0.552950\pi$
$968$	1.90711e9	0.0675790
$969$	−6.22537e9	−0.219802
$970$	0	0
$971$	−3.52264e10	−1.23481	−0.617407	−	0.786644i	$-0.711817\pi$
$971$	−3.52264e10	−1.23481	−0.617407	+	0.786644i	$0.711817\pi$
$972$	−5.42833e10	−1.89598
$973$	−1.56028e10	−0.543011
$974$	−1.13316e9	−0.0392948
$975$	0	0
$976$	2.67761e10	0.921877
$977$	2.58727e10	0.887588	0.443794	−	0.896129i	$-0.353632\pi$
$977$	2.58727e10	0.887588	0.443794	+	0.896129i	$0.353632\pi$
$978$	7.72960e10	2.64223
$979$	−1.78325e10	−0.607397
$980$	0	0
$981$	2.35288e10	0.795715
$982$	5.09661e10	1.71748
$983$	3.46133e10	1.16227	0.581133	−	0.813808i	$-0.302609\pi$
$983$	3.46133e10	1.16227	0.581133	+	0.813808i	$0.302609\pi$
$984$	−1.33349e11	−4.46178
$985$	0	0
$986$	1.03531e10	0.343954
$987$	−4.53190e10	−1.50027
$988$	−4.65315e9	−0.153496
$989$	7.38571e9	0.242776
$990$	0	0
$991$	−1.28569e10	−0.419641	−0.209820	−	0.977740i	$-0.567288\pi$
$991$	−1.28569e10	−0.419641	−0.209820	+	0.977740i	$0.567288\pi$
$992$	−5.21494e8	−0.0169613
$993$	6.33424e10	2.05292
$994$	4.62571e9	0.149391
$995$	0	0
$996$	−1.67919e11	−5.38507
$997$	−3.61960e10	−1.15672	−0.578360	−	0.815782i	$-0.696307\pi$
$997$	−3.61960e10	−1.15672	−0.578360	+	0.815782i	$0.696307\pi$
$998$	2.85368e9	0.0908758
$999$	4.56916e10	1.44996

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	575.8.a.b.1.1		8
5.4	even	2		23.8.a.b.1.8	✓	8
15.14	odd	2		207.8.a.f.1.1		8
20.19	odd	2		368.8.a.h.1.7		8
115.114	odd	2		529.8.a.c.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.8.a.b.1.8	✓	8	5.4	even	2
207.8.a.f.1.1		8	15.14	odd	2
368.8.a.h.1.7		8	20.19	odd	2
529.8.a.c.1.8		8	115.114	odd	2
575.8.a.b.1.1		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:	\( N \)	\(=\)	\( 575 = 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	575.a (trivial)

\(f(q)\)	\(=\)	\(q-19.9556 q^{2} +76.4323 q^{3} +270.224 q^{4} -1525.25 q^{6} -653.513 q^{7} -2838.16 q^{8} +3654.90 q^{9} +O(q^{10})\)	\(q-19.9556 q^{2} +76.4323 q^{3} +270.224 q^{4} -1525.25 q^{6} -653.513 q^{7} -2838.16 q^{8} +3654.90 q^{9} -4337.65 q^{11} +20653.9 q^{12} +1039.41 q^{13} +13041.2 q^{14} +22048.4 q^{16} +4916.44 q^{17} -72935.5 q^{18} -16566.8 q^{19} -49949.5 q^{21} +86560.2 q^{22} +12167.0 q^{23} -216927. q^{24} -20741.9 q^{26} +112195. q^{27} -176595. q^{28} -105525. q^{29} +6798.84 q^{31} -76703.4 q^{32} -331537. q^{33} -98110.2 q^{34} +987642. q^{36} +407252. q^{37} +330599. q^{38} +79444.2 q^{39} +614719. q^{41} +996770. q^{42} +607028. q^{43} -1.17214e6 q^{44} -242799. q^{46} +907295. q^{47} +1.68521e6 q^{48} -396464. q^{49} +375775. q^{51} +280872. q^{52} -881135. q^{53} -2.23891e6 q^{54} +1.85478e6 q^{56} -1.26624e6 q^{57} +2.10581e6 q^{58} +1.60033e6 q^{59} +1.21442e6 q^{61} -135675. q^{62} -2.38852e6 q^{63} -1.29154e6 q^{64} +6.61600e6 q^{66} -783456. q^{67} +1.32854e6 q^{68} +929952. q^{69} +354699. q^{71} -1.03732e7 q^{72} -2.41783e6 q^{73} -8.12693e6 q^{74} -4.47674e6 q^{76} +2.83471e6 q^{77} -1.58535e6 q^{78} -4.43829e6 q^{79} +582055. q^{81} -1.22671e7 q^{82} -8.13014e6 q^{83} -1.34976e7 q^{84} -1.21136e7 q^{86} -8.06552e6 q^{87} +1.23110e7 q^{88} +4.11109e6 q^{89} -679265. q^{91} +3.28782e6 q^{92} +519651. q^{93} -1.81056e7 q^{94} -5.86262e6 q^{96} -3.08615e6 q^{97} +7.91165e6 q^{98} -1.58537e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 40 q^{3} + 640 q^{4} - 1745 q^{6} - 1446 q^{7} - 3177 q^{8} + 13878 q^{9}+O(q^{10}) \) 8 * q - 40 * q^3 + 640 * q^4 - 1745 * q^6 - 1446 * q^7 - 3177 * q^8 + 13878 * q^9	\( 8 q - 40 q^{3} + 640 q^{4} - 1745 q^{6} - 1446 q^{7} - 3177 q^{8} + 13878 q^{9} + 7588 q^{11} - 22733 q^{12} - 19862 q^{13} + 17544 q^{14} + 64336 q^{16} - 42070 q^{17} + 59129 q^{18} + 1050 q^{19} - 7698 q^{21} + 128220 q^{22} + 97336 q^{23} - 621188 q^{24} - 371761 q^{26} + 69500 q^{27} - 143050 q^{28} - 102578 q^{29} + 304172 q^{31} + 612824 q^{32} - 747242 q^{33} - 524530 q^{34} + 1868983 q^{36} - 286472 q^{37} + 762932 q^{38} + 1032828 q^{39} + 1324414 q^{41} + 1886168 q^{42} - 2052578 q^{43} - 867298 q^{44} - 675556 q^{47} - 1411151 q^{48} - 55404 q^{49} + 2775482 q^{51} + 1695409 q^{52} - 203654 q^{53} - 9897559 q^{54} - 5766846 q^{56} - 3908648 q^{57} + 5039991 q^{58} - 748892 q^{59} + 61822 q^{61} + 4939277 q^{62} - 1411632 q^{63} + 2702267 q^{64} + 3791866 q^{66} - 3235604 q^{67} - 4914980 q^{68} - 486680 q^{69} - 4951664 q^{71} + 7940241 q^{72} - 11019370 q^{73} + 356954 q^{74} + 21973240 q^{76} + 5284888 q^{77} + 1506779 q^{78} + 4202464 q^{79} + 10294096 q^{81} - 32636759 q^{82} - 518568 q^{83} + 7629190 q^{84} - 14681386 q^{86} - 4862532 q^{87} - 20589740 q^{88} + 4203864 q^{89} + 2488406 q^{91} + 7786880 q^{92} + 23367842 q^{93} + 12314327 q^{94} - 45317009 q^{96} - 18621134 q^{97} - 35756 q^{98} - 64729930 q^{99}+O(q^{100}) \) 8 * q - 40 * q^3 + 640 * q^4 - 1745 * q^6 - 1446 * q^7 - 3177 * q^8 + 13878 * q^9 + 7588 * q^11 - 22733 * q^12 - 19862 * q^13 + 17544 * q^14 + 64336 * q^16 - 42070 * q^17 + 59129 * q^18 + 1050 * q^19 - 7698 * q^21 + 128220 * q^22 + 97336 * q^23 - 621188 * q^24 - 371761 * q^26 + 69500 * q^27 - 143050 * q^28 - 102578 * q^29 + 304172 * q^31 + 612824 * q^32 - 747242 * q^33 - 524530 * q^34 + 1868983 * q^36 - 286472 * q^37 + 762932 * q^38 + 1032828 * q^39 + 1324414 * q^41 + 1886168 * q^42 - 2052578 * q^43 - 867298 * q^44 - 675556 * q^47 - 1411151 * q^48 - 55404 * q^49 + 2775482 * q^51 + 1695409 * q^52 - 203654 * q^53 - 9897559 * q^54 - 5766846 * q^56 - 3908648 * q^57 + 5039991 * q^58 - 748892 * q^59 + 61822 * q^61 + 4939277 * q^62 - 1411632 * q^63 + 2702267 * q^64 + 3791866 * q^66 - 3235604 * q^67 - 4914980 * q^68 - 486680 * q^69 - 4951664 * q^71 + 7940241 * q^72 - 11019370 * q^73 + 356954 * q^74 + 21973240 * q^76 + 5284888 * q^77 + 1506779 * q^78 + 4202464 * q^79 + 10294096 * q^81 - 32636759 * q^82 - 518568 * q^83 + 7629190 * q^84 - 14681386 * q^86 - 4862532 * q^87 - 20589740 * q^88 + 4203864 * q^89 + 2488406 * q^91 + 7786880 * q^92 + 23367842 * q^93 + 12314327 * q^94 - 45317009 * q^96 - 18621134 * q^97 - 35756 * q^98 - 64729930 * q^99

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